LMFDB - Embedded newform 126.7.s.b.53.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,7,Mod(53,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.53"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	126.s (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,256,0,0,896] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.9868145361$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 651 x^{14} + 38446 x^{13} - 2446277 x^{12} + 67189224 x^{11} + 2881043950 x^{10} + \cdots + 11\!\cdots\!24 $ x^16 - 2x^15 + 651x^14 + 38446x^13 - 2446277x^12 + 67189224x^11 + 2881043950x^10 - 16517569532x^9 + 8018840064606x^8 + 146392293066400x^7 - 712941546599108x^6 + 107460575619504864x^5 - 2054727486401987876x^4 - 100316168284549682000x^3 + 73199024581480812360x^2 - 21586912205181429993056*x + 1150535670231268848916624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.4
Root			$-2.96774 + 43.0220i$ of defining polynomial
Character	$\chi$	$=$	126.53
Dual form			126.7.s.b.107.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.89898 + 2.82843i) q^{2} +(16.0000 - 27.7128i) q^{4} +(201.597 - 116.392i) q^{5} +(342.188 + 23.5897i) q^{7} +181.019i q^{8} +(-658.414 + 1140.41i) q^{10} +(1601.69 + 924.739i) q^{11} +2878.87 q^{13} +(-1743.09 + 852.288i) q^{14} +(-512.000 - 886.810i) q^{16} +(-1754.82 - 1013.14i) q^{17} +(-1916.44 - 3319.37i) q^{19} -7449.10i q^{20} -10462.2 q^{22} +(-18166.2 + 10488.3i) q^{23} +(19281.8 - 33397.1i) q^{25} +(-14103.5 + 8142.68i) q^{26} +(6128.74 - 9105.55i) q^{28} +23707.9i q^{29} +(8305.58 - 14385.7i) q^{31} +(5016.55 + 2896.31i) q^{32} +11462.4 q^{34} +(71729.8 - 35072.4i) q^{35} +(13448.0 + 23292.6i) q^{37} +(18777.2 + 10841.0i) q^{38} +(21069.2 + 36493.0i) q^{40} +28806.3i q^{41} -50423.1 q^{43} +(51254.2 - 29591.6i) q^{44} +(59330.5 - 102763. i) q^{46} +(-29556.6 + 17064.5i) q^{47} +(116536. + 16144.2i) q^{49} +218149. i q^{50} +(46062.0 - 79781.7i) q^{52} +(179532. + 103653. i) q^{53} +430530. q^{55} +(-4270.19 + 61942.6i) q^{56} +(-67056.1 - 116145. i) q^{58} +(-246425. - 142274. i) q^{59} +(-115513. - 200075. i) q^{61} +93966.9i q^{62} -32768.0 q^{64} +(580373. - 335079. i) q^{65} +(-78894.9 + 136650. i) q^{67} +(-56154.1 + 32420.6i) q^{68} +(-252203. + 374701. i) q^{70} +402092. i q^{71} +(345730. - 598821. i) q^{73} +(-131763. - 76073.4i) q^{74} -122652. q^{76} +(526266. + 354218. i) q^{77} +(404402. + 700445. i) q^{79} +(-206436. - 119186. i) q^{80} +(-81476.6 - 141122. i) q^{82} -712652. i q^{83} -471688. q^{85} +(247022. - 142618. i) q^{86} +(-167396. + 289938. i) q^{88} +(-75600.8 + 43648.1i) q^{89} +(985116. + 67911.8i) q^{91} +671248. i q^{92} +(96531.3 - 167197. i) q^{94} +(-772699. - 446118. i) q^{95} -634602. q^{97} +(-616570. + 250523. i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 256 q^{4} + 896 q^{7} - 1568 q^{10} + 2184 q^{13} - 8192 q^{16} + 9548 q^{19} - 15616 q^{22} + 47036 q^{25} + 17024 q^{28} + 33096 q^{31} + 80192 q^{34} - 21796 q^{37} + 50176 q^{40} - 535096 q^{43}+ \cdots + 6864088 q^{97}+O(q^{100}) $ 16 * q + 256 * q^4 + 896 * q^7 - 1568 * q^10 + 2184 * q^13 - 8192 * q^16 + 9548 * q^19 - 15616 * q^22 + 47036 * q^25 + 17024 * q^28 + 33096 * q^31 + 80192 * q^34 - 21796 * q^37 + 50176 * q^40 - 535096 * q^43 + 47648 * q^46 + 826672 * q^49 + 34944 * q^52 + 1755656 * q^55 - 312128 * q^58 - 302904 * q^61 - 524288 * q^64 + 8356 * q^67 + 264992 * q^70 - 717668 * q^73 + 611072 * q^76 - 388104 * q^79 - 596064 * q^82 - 3439552 * q^85 - 249856 * q^88 + 5012364 * q^91 - 795648 * q^94 + 6864088 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−4.89898	+	2.82843i	−0.612372	+	0.353553i
$2$	−4.89898	+	2.82843i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	16.0000	−	27.7128i	0.250000	−	0.433013i
$5$	201.597	−	116.392i	1.61278	−	0.931138i	0.624056	−	0.781380i	$-0.285484\pi$
$5$	201.597	−	116.392i	1.61278	−	0.931138i	0.988723	−	0.149758i	$-0.0478494\pi$
$6$		0			0
$7$	342.188	+	23.5897i	0.997632	+	0.0687747i
$7$	342.188	+	23.5897i	0.997632	+	0.0687747i
$8$			181.019i			0.353553i
$9$		0			0
$10$	−658.414	+	1140.41i	−0.658414	+	1.14041i
$11$	1601.69	+	924.739i	1.20338	+	0.694770i	0.961304	−	0.275488i	$-0.0888395\pi$
$11$	1601.69	+	924.739i	1.20338	+	0.694770i	0.242072	+	0.970258i	$0.422173\pi$
$12$		0			0
$13$	2878.87			1.31037			0.655183	−	0.755470i	$-0.272592\pi$
$13$	2878.87			1.31037			0.655183	+	0.755470i	$0.272592\pi$
$14$	−1743.09	+	852.288i	−0.635238	+	0.310601i
$15$		0			0
$16$	−512.000	−	886.810i	−0.125000	−	0.216506i
$17$	−1754.82	−	1013.14i	−0.357178	−	0.206217i	0.310664	−	0.950520i	$-0.399449\pi$
$17$	−1754.82	−	1013.14i	−0.357178	−	0.206217i	−0.667842	+	0.744303i	$0.732782\pi$
$18$		0			0
$19$	−1916.44	−	3319.37i	−0.279405	−	0.483944i	0.691832	−	0.722059i	$-0.256804\pi$
$19$	−1916.44	−	3319.37i	−0.279405	−	0.483944i	−0.971237	+	0.238114i	$0.923471\pi$
$20$		−	7449.10i		−	0.931138i
$21$		0			0
$22$	−10462.2			−0.982553
$23$	−18166.2	+	10488.3i	−1.49307	+	0.862025i	−0.999968	−	0.00794652i	$-0.997471\pi$
$23$	−18166.2	+	10488.3i	−1.49307	+	0.862025i	−0.493102	+	0.869971i	$0.664137\pi$
$24$		0			0
$25$	19281.8	−	33397.1i	1.23404	−	2.13741i
$26$	−14103.5	+	8142.68i	−0.802432	+	0.463284i
$27$		0			0
$28$	6128.74	−	9105.55i	0.279188	−	0.414794i
$29$			23707.9i			0.972074i	0.873938	+	0.486037i	$0.161558\pi$
$29$			23707.9i			0.972074i	−0.873938	+	0.486037i	$0.838442\pi$
$30$		0			0
$31$	8305.58	−	14385.7i	0.278795	−	0.482887i	−0.692291	−	0.721619i	$-0.743398\pi$
$31$	8305.58	−	14385.7i	0.278795	−	0.482887i	0.971086	+	0.238732i	$0.0767317\pi$
$32$	5016.55	+	2896.31i	0.153093	+	0.0883883i
$33$		0			0
$34$	11462.4			0.291635
$35$	71729.8	−	35072.4i	1.67300	−	0.818015i
$36$		0			0
$37$	13448.0	+	23292.6i	0.265493	+	0.459847i	0.967693	−	0.252133i	$-0.0811321\pi$
$37$	13448.0	+	23292.6i	0.265493	+	0.459847i	−0.702200	+	0.711980i	$0.747799\pi$
$38$	18777.2	+	10841.0i	0.342200	+	0.197569i
$39$		0			0
$40$	21069.2	+	36493.0i	0.329207	+	0.570203i
$41$			28806.3i			0.417962i	0.977920	+	0.208981i	$0.0670146\pi$
$41$			28806.3i			0.417962i	−0.977920	+	0.208981i	$0.932985\pi$
$42$		0			0
$43$	−50423.1			−0.634197			−0.317098	−	0.948393i	$-0.602709\pi$
$43$	−50423.1			−0.634197			−0.317098	+	0.948393i	$0.602709\pi$
$44$	51254.2	−	29591.6i	0.601688	−	0.347385i
$45$		0			0
$46$	59330.5	−	102763.i	0.609544	−	1.05576i
$47$	−29556.6	+	17064.5i	−0.284682	+	0.164361i	−0.635541	−	0.772067i	$-0.719223\pi$
$47$	−29556.6	+	17064.5i	−0.284682	+	0.164361i	0.350859	+	0.936428i	$0.385890\pi$
$48$		0			0
$49$	116536.	+	16144.2i	0.990540	+	0.137224i
$50$			218149.i			1.74519i
$51$		0			0
$52$	46062.0	−	79781.7i	0.327591	−	0.567405i
$53$	179532.	+	103653.i	1.20591	+	0.696233i	0.961863	−	0.273530i	$-0.0881913\pi$
$53$	179532.	+	103653.i	1.20591	+	0.696233i	0.244048	+	0.969763i	$0.421525\pi$
$54$		0			0
$55$	430530.			2.58771
$56$	−4270.19	+	61942.6i	−0.0243155	+	0.352716i
$57$		0			0
$58$	−67056.1	−	116145.i	−0.343680	−	0.595271i
$59$	−246425.	−	142274.i	−1.19986	−	0.692737i	−0.239333	−	0.970938i	$-0.576929\pi$
$59$	−246425.	−	142274.i	−1.19986	−	0.692737i	−0.960523	+	0.278201i	$0.910262\pi$
$60$		0			0
$61$	−115513.	−	200075.i	−0.508912	−	0.881462i	−0.999947	−	0.0103219i	$-0.996714\pi$
$61$	−115513.	−	200075.i	−0.508912	−	0.881462i	0.491034	−	0.871140i	$-0.336619\pi$
$62$			93966.9i			0.394276i
$63$		0			0
$64$	−32768.0			−0.125000
$65$	580373.	−	335079.i	2.11333	−	1.22013i
$66$		0			0
$67$	−78894.9	+	136650.i	−0.262316	+	0.454345i	−0.966857	−	0.255319i	$-0.917820\pi$
$67$	−78894.9	+	136650.i	−0.262316	+	0.454345i	0.704541	+	0.709663i	$0.251153\pi$
$68$	−56154.1	+	32420.6i	−0.178589	+	0.103108i
$69$		0			0
$70$	−252203.	+	374701.i	−0.735286	+	1.09242i
$71$			402092.i			1.12344i	0.827327	+	0.561720i	$0.189860\pi$
$71$			402092.i			1.12344i	−0.827327	+	0.561720i	$0.810140\pi$
$72$		0			0
$73$	345730.	−	598821.i	0.888726	−	1.53932i	0.0473444	−	0.998879i	$-0.484924\pi$
$73$	345730.	−	598821.i	0.888726	−	1.53932i	0.841382	−	0.540441i	$-0.181742\pi$
$74$	−131763.	−	76073.4i	−0.325161	−	0.187732i
$75$		0			0
$76$	−122652.			−0.279405
$77$	526266.	+	354218.i	1.15274	+	0.775887i
$78$		0			0
$79$	404402.	+	700445.i	0.820224	+	1.42067i	0.905515	+	0.424313i	$0.139485\pi$
$79$	404402.	+	700445.i	0.820224	+	1.42067i	−0.0852914	+	0.996356i	$0.527182\pi$
$80$	−206436.	−	119186.i	−0.403195	−	0.232784i
$81$		0			0
$82$	−81476.6	−	141122.i	−0.147772	−	0.255948i
$83$		−	712652.i		−	1.24636i	−0.782079	−	0.623180i	$-0.785841\pi$
$83$		−	712652.i		−	1.24636i	0.782079	−	0.623180i	$-0.214159\pi$
$84$		0			0
$85$	−471688.			−0.768066
$86$	247022.	−	142618.i	0.388365	−	0.224222i
$87$		0			0
$88$	−167396.	+	289938.i	−0.245638	+	0.425458i
$89$	−75600.8	+	43648.1i	−0.107240	+	0.0619150i	−0.552661	−	0.833406i	$-0.686387\pi$
$89$	−75600.8	+	43648.1i	−0.107240	+	0.0619150i	0.445421	+	0.895321i	$0.353054\pi$
$90$		0			0
$91$	985116.	+	67911.8i	1.30726	+	0.0901200i
$92$			671248.i			0.862025i
$93$		0			0
$94$	96531.3	−	167197.i	0.116221	−	0.201301i
$95$	−772699.	−	446118.i	−0.901238	−	0.520330i
$96$		0			0
$97$	−634602.			−0.695323			−0.347661	−	0.937620i	$-0.613024\pi$
$97$	−634602.			−0.695323			−0.347661	+	0.937620i	$0.613024\pi$
$98$	−616570.	+	250523.i	−0.655095	+	0.266177i
$99$		0			0
$100$	−617018.	−	1.06871e6i	−0.617018	−	1.06871i
$101$	−1.49639e6	−	863939.i	−1.45238	−	0.838530i	−0.453761	−	0.891123i	$-0.649918\pi$
$101$	−1.49639e6	−	863939.i	−1.45238	−	0.838530i	−0.998616	+	0.0525930i	$0.983251\pi$
$102$		0			0
$103$	−364317.	−	631016.i	−0.333402	−	0.577469i	0.649775	−	0.760127i	$-0.274863\pi$
$103$	−364317.	−	631016.i	−0.333402	−	0.577469i	−0.983177	+	0.182658i	$0.941530\pi$
$104$			521132.i			0.463284i
$105$		0			0
$106$	−1.17270e6			−0.984622
$107$	−154552.	+	89230.7i	−0.126161	+	0.0728389i	−0.561752	−	0.827306i	$-0.689873\pi$
$107$	−154552.	+	89230.7i	−0.126161	+	0.0728389i	0.435592	+	0.900144i	$0.356539\pi$
$108$		0			0
$109$	416837.	−	721983.i	0.321875	−	0.557504i	−0.659000	−	0.752143i	$-0.729020\pi$
$109$	416837.	−	721983.i	0.321875	−	0.557504i	0.980875	+	0.194639i	$0.0623536\pi$
$110$	−2.10916e6	+	1.21772e6i	−1.58464	+	0.914892i
$111$		0			0
$112$	−154281.	−	315534.i	−0.109814	−	0.224591i
$113$		−	1.71830e6i		−	1.19087i	−0.803403	−	0.595435i	$-0.796980\pi$
$113$		−	1.71830e6i		−	1.19087i	0.803403	−	0.595435i	$-0.203020\pi$
$114$		0			0
$115$	−2.44150e6	+	4.22881e6i	−1.60533	+	2.78051i
$116$	657013.	+	379327.i	0.420920	+	0.243019i
$117$		0			0
$118$	1.60964e6			0.979678
$119$	−576577.	−	388081.i	−0.342150	−	0.230293i
$120$		0			0
$121$	824503.	+	1.42808e6i	0.465411	+	0.806115i
$122$	1.13180e6	+	653443.i	0.623288	+	0.359855i
$123$		0			0
$124$	−265779.	−	460342.i	−0.139398	−	0.241444i
$125$		−	5.33975e6i		−	2.73395i
$126$		0			0
$127$	−1.92125e6			−0.937936			−0.468968	−	0.883215i	$-0.655374\pi$
$127$	−1.92125e6			−0.937936			−0.468968	+	0.883215i	$0.655374\pi$
$128$	160530.	−	92681.9i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−1.89549e6	+	3.28309e6i	−0.862763	+	1.49435i
$131$	834540.	−	481822.i	0.371222	−	0.214325i	−0.302770	−	0.953064i	$-0.597912\pi$
$131$	834540.	−	481822.i	0.371222	−	0.214325i	0.673992	+	0.738739i	$0.264578\pi$
$132$		0			0
$133$	−577480.	−	1.18106e6i	−0.245461	−	0.502015i
$134$		−	892594.i		−	0.370971i
$135$		0			0
$136$	183399.	−	317656.i	0.0729087	−	0.126282i
$137$	3.57459e6	+	2.06379e6i	1.39016	+	0.802608i	0.993332	−	0.115286i	$-0.0367784\pi$
$137$	3.57459e6	+	2.06379e6i	1.39016	+	0.802608i	0.396826	+	0.917894i	$0.370112\pi$
$138$		0			0
$139$	188837.			0.0703141			0.0351570	−	0.999382i	$-0.488807\pi$
$139$	188837.			0.0703141			0.0351570	+	0.999382i	$0.488807\pi$
$140$	175722.	−	2.54899e6i	0.0640387	−	0.928933i
$141$		0			0
$142$	−1.13729e6	−	1.96984e6i	−0.397196	−	0.687964i
$143$	4.61108e6	+	2.66221e6i	1.57686	+	0.910403i
$144$		0			0
$145$	2.75942e6	+	4.77945e6i	0.905135	+	1.56774i
$146$			3.91149e6i			1.25685i
$147$		0			0
$148$	860672.			0.265493
$149$	−2.37966e6	+	1.37390e6i	−0.719375	+	0.415332i	−0.814523	−	0.580132i	$-0.803001\pi$
$149$	−2.37966e6	+	1.37390e6i	−0.719375	+	0.415332i	0.0951474	+	0.995463i	$0.469668\pi$
$150$		0			0
$151$	111691.	−	193454.i	0.0324404	−	0.0561884i	−0.849349	−	0.527831i	$-0.823005\pi$
$151$	111691.	−	193454.i	0.0324404	−	0.0561884i	0.881790	+	0.471643i	$0.156339\pi$
$152$	600871.	−	346913.i	0.171100	−	0.0987847i
$153$		0			0
$154$	−3.58005e6	−	246801.i	−0.980227	−	0.0675748i
$155$		−	3.86682e6i		−	1.03839i
$156$		0			0
$157$	1.82443e6	−	3.16000e6i	0.471441	−	0.816560i	−0.528025	−	0.849229i	$-0.677067\pi$
$157$	1.82443e6	−	3.16000e6i	0.471441	−	0.816560i	0.999466	+	0.0326688i	$0.0104007\pi$
$158$	−3.96232e6	−	2.28765e6i	−1.00457	−	0.579986i
$159$		0			0
$160$	1.34843e6			0.329207
$161$	−6.46367e6	+	3.16042e6i	−1.54882	+	0.757298i
$162$		0			0
$163$	−656380.	−	1.13688e6i	−0.151563	−	0.262515i	0.780239	−	0.625481i	$-0.215097\pi$
$163$	−656380.	−	1.13688e6i	−0.151563	−	0.262515i	−0.931802	+	0.362967i	$0.881764\pi$
$164$	798305.	+	460901.i	0.180983	+	0.104490i
$165$		0			0
$166$	2.01568e6	+	3.49127e6i	0.440655	+	0.763236i
$167$			452350.i			0.0971237i	0.998820	+	0.0485618i	$0.0154638\pi$
$167$			452350.i			0.0971237i	−0.998820	+	0.0485618i	$0.984536\pi$
$168$		0			0
$169$	3.46111e6			0.717059
$170$	2.31079e6	−	1.33414e6i	0.470342	−	0.271552i
$171$		0			0
$172$	−806769.	+	1.39737e6i	−0.158549	+	0.274615i
$173$	−3.72957e6	+	2.15327e6i	−0.720311	+	0.415872i	−0.814867	−	0.579648i	$-0.803190\pi$
$173$	−3.72957e6	+	2.15327e6i	−0.720311	+	0.415872i	0.0945559	+	0.995520i	$0.469857\pi$
$174$		0			0
$175$	7.38583e6	−	1.09732e7i	1.37811	−	2.04748i
$176$		−	1.89387e6i		−	0.347385i
$177$		0			0
$178$	246911.	−	427663.i	0.0437805	−	0.0758300i
$179$	6.87588e6	+	3.96979e6i	1.19886	+	0.692163i	0.960302	−	0.278964i	$-0.0899911\pi$
$179$	6.87588e6	+	3.96979e6i	1.19886	+	0.692163i	0.238561	+	0.971128i	$0.423324\pi$
$180$		0			0
$181$	−3.78136e6			−0.637694			−0.318847	−	0.947806i	$-0.603296\pi$
$181$	−3.78136e6			−0.637694			−0.318847	+	0.947806i	$0.603296\pi$
$182$	−5.01814e6	+	2.45363e6i	−0.832394	+	0.407000i
$183$		0			0
$184$	−1.89858e6	−	3.28843e6i	−0.304772	−	0.527880i
$185$	5.42216e6	+	3.13049e6i	0.856362	+	0.494421i
$186$		0			0
$187$	−1.87379e6	−	3.24549e6i	−0.286547	−	0.496313i
$188$			1.09213e6i			0.164361i
$189$		0			0
$190$	5.04725e6			0.735858
$191$	2.10125e6	−	1.21316e6i	0.301563	−	0.174107i	−0.341582	−	0.939852i	$-0.610963\pi$
$191$	2.10125e6	−	1.21316e6i	0.301563	−	0.174107i	0.643145	+	0.765745i	$0.277629\pi$
$192$		0			0
$193$	−2.92704e6	+	5.06979e6i	−0.407153	+	0.705209i	−0.994569	−	0.104075i	$-0.966812\pi$
$193$	−2.92704e6	+	5.06979e6i	−0.407153	+	0.705209i	0.587417	+	0.809285i	$0.300145\pi$
$194$	3.10890e6	−	1.79493e6i	0.425797	−	0.245834i
$195$		0			0
$196$	2.31198e6	−	2.97123e6i	0.307055	−	0.394611i
$197$			167317.i			0.0218847i	0.999940	+	0.0109423i	$0.00348313\pi$
$197$			167317.i			0.0218847i	−0.999940	+	0.0109423i	$0.996517\pi$
$198$		0			0
$199$	3.05831e6	−	5.29714e6i	0.388080	−	0.672175i	−0.604111	−	0.796900i	$-0.706472\pi$
$199$	3.05831e6	−	5.29714e6i	0.388080	−	0.672175i	0.992191	+	0.124725i	$0.0398050\pi$
$200$	6.04552e6	+	3.49038e6i	0.755689	+	0.436297i
$201$		0			0
$202$	9.77435e6			1.18586
$203$	−559263.	+	8.11256e6i	−0.0668541	+	0.969772i
$204$		0			0
$205$	3.35283e6	+	5.80728e6i	0.389180	+	0.674079i
$206$	3.56956e6	+	2.06089e6i	0.408332	+	0.235751i
$207$		0			0
$208$	−1.47398e6	−	2.55301e6i	−0.163796	−	0.283703i
$209$		−	7.08883e6i		−	0.776490i
$210$		0			0
$211$	−1.40439e6			−0.149500			−0.0747500	−	0.997202i	$-0.523816\pi$
$211$	−1.40439e6			−0.149500			−0.0747500	+	0.997202i	$0.523816\pi$
$212$	5.74504e6	−	3.31690e6i	0.602956	−	0.348117i
$213$		0			0
$214$	504765.	−	874279.i	0.0515049	−	0.0892090i
$215$	−1.01652e7	+	5.86886e6i	−1.02282	+	0.590525i
$216$		0			0
$217$	3.18142e6	−	4.72668e6i	0.311345	−	0.462570i
$218$			4.71597e6i			0.455200i
$219$		0			0
$220$	6.88848e6	−	1.19312e7i	0.646927	−	1.12051i
$221$	−5.05189e6	−	2.91671e6i	−0.468034	−	0.270220i
$222$		0			0
$223$	−1.06098e7			−0.956734			−0.478367	−	0.878160i	$-0.658771\pi$
$223$	−1.06098e7			−0.956734			−0.478367	+	0.878160i	$0.658771\pi$
$224$	1.64828e6	+	1.10942e6i	0.146652	+	0.0987080i
$225$		0			0
$226$	4.86010e6	+	8.41793e6i	0.421036	+	0.729256i
$227$	1.89076e6	+	1.09163e6i	0.161643	+	0.0933249i	0.578640	−	0.815583i	$-0.303584\pi$
$227$	1.89076e6	+	1.09163e6i	0.161643	+	0.0933249i	−0.416996	+	0.908908i	$0.636917\pi$
$228$		0			0
$229$	−4.27871e6	−	7.41095e6i	−0.356292	−	0.617117i	0.631046	−	0.775746i	$-0.282626\pi$
$229$	−4.27871e6	−	7.41095e6i	−0.356292	−	0.617117i	−0.987338	+	0.158629i	$0.949293\pi$
$230$		−	2.76225e7i		−	2.27028i
$231$		0			0
$232$	−4.29159e6			−0.343680
$233$	−2.11307e7	+	1.21998e7i	−1.67050	+	0.964461i	−0.703135	+	0.711057i	$0.748217\pi$
$233$	−2.11307e7	+	1.21998e7i	−1.67050	+	0.964461i	−0.967360	+	0.253404i	$0.918450\pi$
$234$		0			0
$235$	−3.97235e6	+	6.88031e6i	−0.306086	+	0.530157i
$236$	−7.88561e6	+	4.55276e6i	−0.599928	+	0.346369i
$237$		0			0
$238$	3.92230e6	+	270395.i	0.290944	+	0.0200571i
$239$			1.57781e7i			1.15574i	0.816127	+	0.577872i	$0.196117\pi$
$239$			1.57781e7i			1.15574i	−0.816127	+	0.577872i	$0.803883\pi$
$240$		0			0
$241$	6.67779e6	−	1.15663e7i	0.477069	−	0.826309i	−0.522585	−	0.852587i	$-0.675032\pi$
$241$	6.67779e6	−	1.15663e7i	0.477069	−	0.826309i	0.999655	+	0.0262785i	$0.00836567\pi$
$242$	−8.07845e6	−	4.66410e6i	−0.570009	−	0.329095i
$243$		0			0
$244$	−7.39286e6			−0.508912
$245$	2.53724e7	−	1.03093e7i	1.72530	−	0.701018i
$246$		0			0
$247$	−5.51719e6	−	9.55606e6i	−0.366123	−	0.634144i
$248$	2.60409e6	+	1.50347e6i	0.170726	+	0.0985689i
$249$		0			0
$250$	1.51031e7	+	2.61593e7i	0.966599	+	1.67420i
$251$			1.33689e7i			0.845422i	0.906264	+	0.422711i	$0.138922\pi$
$251$			1.33689e7i			0.845422i	−0.906264	+	0.422711i	$0.861078\pi$
$252$		0			0
$253$	−3.87956e7			−2.39564
$254$	9.41217e6	−	5.43412e6i	0.574366	−	0.331610i
$255$		0			0
$256$	−524288.	+	908093.i	−0.0312500	+	0.0541266i
$257$	−1.86308e7	+	1.07565e7i	−1.09757	+	0.633681i	−0.935581	−	0.353112i	$-0.885123\pi$
$257$	−1.86308e7	+	1.07565e7i	−1.09757	+	0.633681i	−0.161987	+	0.986793i	$0.551790\pi$
$258$		0			0
$259$	4.05228e6	+	8.28769e6i	0.233238	+	0.477017i
$260$		−	2.14450e7i		−	1.22013i
$261$		0			0
$262$	−2.72560e6	+	4.72087e6i	−0.151551	+	0.262493i
$263$	−6.54426e6	−	3.77833e6i	−0.359744	−	0.207698i	0.309225	−	0.950989i	$-0.399930\pi$
$263$	−6.54426e6	−	3.77833e6i	−0.359744	−	0.207698i	−0.668968	+	0.743291i	$0.733264\pi$
$264$		0			0
$265$	4.82577e7			2.59316
$266$	6.16960e6	+	4.15262e6i	0.327802	+	0.220636i
$267$		0			0
$268$	2.52464e6	+	4.37280e6i	0.131158	+	0.227172i
$269$	1.19317e7	+	6.88879e6i	0.612980	+	0.353904i	0.774131	−	0.633025i	$-0.218187\pi$
$269$	1.19317e7	+	6.88879e6i	0.612980	+	0.353904i	−0.161151	+	0.986930i	$0.551521\pi$
$270$		0			0
$271$	1.80370e7	+	3.12409e7i	0.906266	+	1.56970i	0.819209	+	0.573495i	$0.194413\pi$
$271$	1.80370e7	+	3.12409e7i	0.906266	+	1.56970i	0.0870567	+	0.996203i	$0.472254\pi$
$272$			2.07492e6i			0.103108i
$273$		0			0
$274$	−2.33491e7			−1.13506
$275$	6.17671e7	−	3.56613e7i	2.97002	−	1.71474i
$276$		0			0
$277$	−8.28126e6	+	1.43436e7i	−0.389634	+	0.674867i	−0.992400	−	0.123051i	$-0.960732\pi$
$277$	−8.28126e6	+	1.43436e7i	−0.389634	+	0.674867i	0.602766	+	0.797918i	$0.294065\pi$
$278$	−925108.	+	534111.i	−0.0430584	+	0.0248598i
$279$		0			0
$280$	6.34878e6	+	1.29845e7i	0.289212	+	0.591494i
$281$		−	5.13711e6i		−	0.231526i	−0.993277	−	0.115763i	$-0.963069\pi$
$281$		−	5.13711e6i		−	0.231526i	0.993277	−	0.115763i	$-0.0369313\pi$
$282$		0			0
$283$	−1.28445e7	+	2.22473e7i	−0.566705	+	0.981562i	0.430183	+	0.902741i	$0.358449\pi$
$283$	−1.28445e7	+	2.22473e7i	−0.566705	+	0.981562i	−0.996889	+	0.0788209i	$0.974884\pi$
$284$	1.11431e7	+	6.43347e6i	0.486464	+	0.280860i
$285$		0			0
$286$	−3.01194e7			−1.28750
$287$	−679533.	+	9.85718e6i	−0.0287452	+	0.416972i
$288$		0			0
$289$	−1.00159e7	−	1.73480e7i	−0.414949	−	0.718713i
$290$	−2.70367e7	−	1.56096e7i	−1.10856	−	0.640027i
$291$		0			0
$292$	−1.10634e7	−	1.91623e7i	−0.444363	−	0.769660i
$293$		−	4.75324e7i		−	1.88968i	−0.327539	−	0.944838i	$-0.606219\pi$
$293$		−	4.75324e7i		−	1.88968i	0.327539	−	0.944838i	$-0.393781\pi$
$294$		0			0
$295$	−6.62382e7			−2.58013
$296$	−4.21641e6	+	2.43435e6i	−0.162580	+	0.0938658i
$297$		0			0
$298$	7.77193e6	−	1.34614e7i	0.293684	−	0.508675i
$299$	−5.22982e7	+	3.01944e7i	−1.95647	+	1.12957i
$300$		0			0
$301$	−1.72542e7	−	1.18947e6i	−0.632695	−	0.0436167i
$302$			1.26364e6i			0.0458777i
$303$		0			0
$304$	−1.96244e6	+	3.39904e6i	−0.0698514	+	0.120986i
$305$	−4.65744e7	−	2.68897e7i	−1.64153	−	0.947735i
$306$		0			0
$307$	−1.84851e7			−0.638863			−0.319431	−	0.947609i	$-0.603492\pi$
$307$	−1.84851e7			−0.638863			−0.319431	+	0.947609i	$0.603492\pi$
$308$	1.82366e7	−	8.91683e6i	0.624155	−	0.305182i
$309$		0			0
$310$	1.09370e7	+	1.89435e7i	0.367125	+	0.635879i
$311$	−2.64156e7	−	1.52510e7i	−0.878171	−	0.507012i	−0.00811606	−	0.999967i	$-0.502583\pi$
$311$	−2.64156e7	−	1.52510e7i	−0.878171	−	0.507012i	−0.870055	+	0.492955i	$0.835917\pi$
$312$		0			0
$313$	1.73123e7	+	2.99858e7i	0.564576	+	0.977874i	0.997089	+	0.0762462i	$0.0242935\pi$
$313$	1.73123e7	+	2.99858e7i	0.564576	+	0.977874i	−0.432513	+	0.901628i	$0.642373\pi$
$314$			2.06410e7i			0.666718i
$315$		0			0
$316$	2.58818e7			0.820224
$317$	−6.86236e6	+	3.96199e6i	−0.215425	+	0.124376i	−0.603830	−	0.797113i	$-0.706359\pi$
$317$	−6.86236e6	+	3.96199e6i	−0.215425	+	0.124376i	0.388405	+	0.921489i	$0.373026\pi$
$318$		0			0
$319$	−2.19236e7	+	3.79728e7i	−0.675368	+	1.16977i
$320$	−6.60594e6	+	3.81394e6i	−0.201597	+	0.116392i
$321$		0			0
$322$	2.27263e7	−	3.37648e7i	0.680710	−	1.01134i
$323$			7.76652e6i			0.230472i
$324$		0			0
$325$	5.55099e7	−	9.61459e7i	1.61704	−	2.80079i
$326$	6.43119e6	+	3.71305e6i	0.185626	+	0.107171i
$327$		0			0
$328$	−5.21450e6			−0.147772
$329$	−1.05164e7	+	5.14203e6i	−0.295312	+	0.144393i
$330$		0			0
$331$	−1.87990e7	−	3.25608e7i	−0.518383	−	0.897866i	−0.999772	−	0.0213589i	$-0.993201\pi$
$331$	−1.87990e7	−	3.25608e7i	−0.518383	−	0.897866i	0.481389	−	0.876507i	$-0.340133\pi$
$332$	−1.97496e7	−	1.14024e7i	−0.539689	−	0.311590i
$333$		0			0
$334$	−1.27944e6	−	2.21605e6i	−0.0343384	−	0.0594759i
$335$			3.67310e7i			0.977009i
$336$		0			0
$337$	−3.22770e6			−0.0843342			−0.0421671	−	0.999111i	$-0.513426\pi$
$337$	−3.22770e6			−0.0843342			−0.0421671	+	0.999111i	$0.513426\pi$
$338$	−1.69559e7	+	9.78948e6i	−0.439107	+	0.253519i
$339$		0			0
$340$	−7.54701e6	+	1.30718e7i	−0.192016	+	0.332582i
$341$	2.66060e7	−	1.53610e7i	0.670991	−	0.387397i
$342$		0			0
$343$	3.94964e7	+	8.27341e6i	0.978757	+	0.205023i
$344$		−	9.12755e6i		−	0.224222i
$345$		0			0
$346$	1.21807e7	−	2.10976e7i	0.294066	−	0.509337i
$347$	−1.14663e7	−	6.62007e6i	−0.274432	−	0.158443i	0.356468	−	0.934308i	$-0.383981\pi$
$347$	−1.14663e7	−	6.62007e6i	−0.274432	−	0.158443i	−0.630900	+	0.775864i	$0.717314\pi$
$348$		0			0
$349$	−4.24169e7			−0.997845			−0.498922	−	0.866647i	$-0.666271\pi$
$349$	−4.24169e7			−0.997845			−0.498922	+	0.866647i	$0.666271\pi$
$350$	−5.14607e6	+	7.46479e7i	−0.120025	+	1.74106i
$351$		0			0
$352$	5.35666e6	+	9.27801e6i	0.122819	+	0.212729i
$353$	3.81101e7	+	2.20029e7i	0.866395	+	0.500214i	0.866149	−	0.499786i	$-0.166588\pi$
$353$	3.81101e7	+	2.20029e7i	0.866395	+	0.500214i	0.000246744		1.00000i	$0.499921\pi$
$354$		0			0
$355$	4.68004e7	+	8.10606e7i	1.04608	+	1.81186i
$356$			2.79348e6i			0.0619150i
$357$		0			0
$358$	−4.49131e7			−0.978867
$359$	−1.36086e7	+	7.85691e6i	−0.294123	+	0.169812i	−0.639800	−	0.768542i	$-0.720983\pi$
$359$	−1.36086e7	+	7.85691e6i	−0.294123	+	0.169812i	0.345677	+	0.938354i	$0.387649\pi$
$360$		0			0
$361$	1.61774e7	−	2.80202e7i	0.343865	−	0.595592i
$362$	1.85248e7	−	1.06953e7i	0.390506	−	0.225459i
$363$		0			0
$364$	1.76439e7	−	2.62137e7i	0.365839	−	0.543532i
$365$		−	1.60961e8i		−	3.31011i
$366$		0			0
$367$	−2.82594e7	+	4.89467e7i	−0.571695	+	0.990205i	0.424697	+	0.905336i	$0.360381\pi$
$367$	−2.82594e7	+	4.89467e7i	−0.571695	+	0.990205i	−0.996392	+	0.0848697i	$0.972953\pi$
$368$	1.86022e7	+	1.07400e7i	0.373268	+	0.215506i
$369$		0			0
$370$	−3.54174e7			−0.699216
$371$	5.89887e7	+	3.97039e7i	1.15517	+	0.777521i
$372$		0			0
$373$	−3.66487e7	−	6.34774e7i	−0.706207	−	1.22319i	−0.966254	−	0.257590i	$-0.917071\pi$
$373$	−3.66487e7	−	6.34774e7i	−0.706207	−	1.22319i	0.260047	−	0.965596i	$-0.416262\pi$
$374$	1.83593e7	+	1.05997e7i	0.350947	+	0.202619i
$375$		0			0
$376$	−3.08900e6	−	5.35031e6i	−0.0581105	−	0.100650i
$377$			6.82521e7i			1.27377i
$378$		0			0
$379$	8.64522e7			1.58803			0.794014	−	0.607899i	$-0.207988\pi$
$379$	8.64522e7			1.58803			0.794014	+	0.607899i	$0.207988\pi$
$380$	−2.47264e7	+	1.42758e7i	−0.450619	+	0.260165i
$381$		0			0
$382$	−6.86265e6	+	1.18865e7i	−0.123112	+	0.213237i
$383$	6.16513e7	−	3.55944e7i	1.09735	−	0.633556i	0.161827	−	0.986819i	$-0.448261\pi$
$383$	6.16513e7	−	3.55944e7i	1.09735	−	0.633556i	0.935524	+	0.353263i	$0.114928\pi$
$384$		0			0
$385$	1.47322e8	+	1.01561e7i	2.58158	+	0.177969i
$386$		−	3.31157e7i		−	0.575801i
$387$		0			0
$388$	−1.01536e7	+	1.75866e7i	−0.173831	+	0.301084i
$389$	−4.23022e7	−	2.44232e7i	−0.718644	−	0.414910i	0.0956092	−	0.995419i	$-0.469520\pi$
$389$	−4.23022e7	−	2.44232e7i	−0.718644	−	0.414910i	−0.814254	+	0.580509i	$0.802853\pi$
$390$		0			0
$391$	4.25044e7			0.711056
$392$	−2.92242e6	+	2.10953e7i	−0.0485159	+	0.350209i
$393$		0			0
$394$	−473243.	−	819680.i	−0.00773740	−	0.0134016i
$395$	1.63053e8	+	9.41386e7i	2.64568	+	1.52748i
$396$		0			0
$397$	3.20245e7	+	5.54681e7i	0.511813	+	0.886486i	0.999906	+	0.0136946i	$0.00435927\pi$
$397$	3.20245e7	+	5.54681e7i	0.511813	+	0.886486i	−0.488093	+	0.872792i	$0.662307\pi$
$398$			3.46008e7i			0.548829i
$399$		0			0
$400$	−3.94891e7			−0.617018
$401$	3.95987e7	−	2.28623e7i	0.614112	−	0.354558i	−0.160461	−	0.987042i	$-0.551298\pi$
$401$	3.95987e7	−	2.28623e7i	0.614112	−	0.354558i	0.774573	+	0.632484i	$0.217965\pi$
$402$		0			0
$403$	2.39107e7	−	4.14146e7i	0.365324	−	0.632759i
$404$	−4.78843e7	+	2.76460e7i	−0.726189	+	0.419265i
$405$		0			0
$406$	−2.02060e7	−	4.13251e7i	−0.301927	−	0.617498i
$407$			4.97436e7i			0.737825i
$408$		0			0
$409$	2.63251e7	−	4.55964e7i	0.384769	−	0.666440i	−0.606968	−	0.794726i	$-0.707614\pi$
$409$	2.63251e7	−	4.55964e7i	0.384769	−	0.666440i	0.991737	+	0.128286i	$0.0409477\pi$
$410$	−3.28509e7	−	1.89665e7i	−0.476646	−	0.275192i
$411$		0			0
$412$	−2.33163e7			−0.333402
$413$	−8.09675e7	−	5.44974e7i	−1.14937	−	0.773616i
$414$		0			0
$415$	−8.29472e7	−	1.43669e8i	−1.16053	−	2.01010i
$416$	1.44420e7	+	8.33811e6i	0.200608	+	0.115821i
$417$		0			0
$418$	2.00502e7	+	3.47280e7i	0.274531	+	0.475501i
$419$		−	4.92688e7i		−	0.669777i	−0.942258	−	0.334889i	$-0.891301\pi$
$419$		−	4.92688e7i		−	0.669777i	0.942258	−	0.334889i	$-0.108699\pi$
$420$		0			0
$421$	2.07631e7			0.278257			0.139129	−	0.990274i	$-0.455570\pi$
$421$	2.07631e7			0.278257			0.139129	+	0.990274i	$0.455570\pi$
$422$	6.88010e6	−	3.97222e6i	0.0915497	−	0.0528563i
$423$		0			0
$424$	−1.87632e7	+	3.24988e7i	−0.246156	+	0.426354i
$425$	−6.76721e7	+	3.90705e7i	−0.881541	+	0.508958i
$426$		0			0
$427$	−3.48076e7	−	7.11882e7i	−0.447085	−	0.914375i
$428$			5.71077e6i			0.0728389i
$429$		0			0
$430$	3.31993e7	−	5.75028e7i	0.417564	−	0.723242i
$431$	5.29183e7	+	3.05524e7i	0.660959	+	0.381605i	0.792642	−	0.609687i	$-0.208705\pi$
$431$	5.29183e7	+	3.05524e7i	0.660959	+	0.381605i	−0.131683	+	0.991292i	$0.542038\pi$
$432$		0			0
$433$	5.48769e7			0.675968			0.337984	−	0.941152i	$-0.390255\pi$
$433$	5.48769e7			0.675968			0.337984	+	0.941152i	$0.390255\pi$
$434$	−2.21665e6	+	3.21543e7i	−0.0271162	+	0.393342i
$435$		0			0
$436$	−1.33388e7	−	2.31035e7i	−0.160937	−	0.278752i
$437$	6.96289e7	+	4.02003e7i	0.834344	+	0.481709i
$438$		0			0
$439$	3.56040e7	+	6.16680e7i	0.420829	+	0.728897i	0.996021	−	0.0891216i	$-0.0284060\pi$
$439$	3.56040e7	+	6.16680e7i	0.420829	+	0.728897i	−0.575192	+	0.818018i	$0.695073\pi$
$440$			7.79342e7i			0.914892i
$441$		0			0
$442$	3.29988e7			0.382148
$443$	−9.66802e7	+	5.58184e7i	−1.11206	+	0.642046i	−0.939361	−	0.342930i	$-0.888581\pi$
$443$	−9.66802e7	+	5.58184e7i	−1.11206	+	0.642046i	−0.172695	+	0.984975i	$0.555247\pi$
$444$		0			0
$445$	−1.01606e7	+	1.75987e7i	−0.115303	+	0.199710i
$446$	5.19770e7	−	3.00089e7i	0.585877	−	0.338256i
$447$		0			0
$448$	−1.12128e7	−	772988.i	−0.124704	−	0.00859683i
$449$		−	1.36487e8i		−	1.50783i	−0.656971	−	0.753916i	$-0.728163\pi$
$449$		−	1.36487e8i		−	1.50783i	0.656971	−	0.753916i	$-0.271837\pi$
$450$		0			0
$451$	−2.66383e7	+	4.61390e7i	−0.290387	+	0.502965i
$452$	−4.76190e7	−	2.74929e7i	−0.515662	−	0.297718i
$453$		0			0
$454$	−1.23504e7			−0.131981
$455$	2.06501e8	−	1.00969e8i	2.19224	−	1.07190i
$456$		0			0
$457$	9.21655e7	+	1.59635e8i	0.965650	+	1.67256i	0.707857	+	0.706356i	$0.249662\pi$
$457$	9.21655e7	+	1.59635e8i	0.965650	+	1.67256i	0.257793	+	0.966200i	$0.417005\pi$
$458$	4.19226e7	+	2.42041e7i	0.436367	+	0.251937i
$459$		0			0
$460$	7.81281e7	+	1.35322e8i	0.802664	+	1.39025i
$461$		−	1.37151e7i		−	0.139990i	−0.997547	−	0.0699950i	$-0.977702\pi$
$461$		−	1.37151e7i		−	0.139990i	0.997547	−	0.0699950i	$-0.0222983\pi$
$462$		0			0
$463$	−4.58979e7			−0.462435			−0.231217	−	0.972902i	$-0.574271\pi$
$463$	−4.58979e7			−0.462435			−0.231217	+	0.972902i	$0.574271\pi$
$464$	2.10244e7	−	1.21385e7i	0.210460	−	0.121509i
$465$		0			0
$466$	6.90124e7	−	1.19533e8i	0.681977	−	1.18122i
$467$	−1.07403e8	+	6.20094e7i	−1.05455	+	0.608845i	−0.923920	−	0.382586i	$-0.875033\pi$
$467$	−1.07403e8	+	6.20094e7i	−1.05455	+	0.608845i	−0.130630	+	0.991431i	$0.541700\pi$
$468$		0			0
$469$	−3.02204e7	+	4.48989e7i	−0.292942	+	0.435228i
$470$		−	4.49420e7i		−	0.432871i
$471$		0			0
$472$	2.57543e7	−	4.46077e7i	0.244920	−	0.424213i
$473$	−8.07624e7	−	4.66282e7i	−0.763178	−	0.440621i
$474$		0			0
$475$	−1.47810e8			−1.37918
$476$	−1.99801e7	+	9.76927e6i	−0.185257	+	0.0905819i
$477$		0			0
$478$	−4.46273e7	−	7.72968e7i	−0.408618	−	0.707746i
$479$	−2.14521e7	−	1.23854e7i	−0.195193	−	0.112695i	0.399218	−	0.916856i	$-0.369281\pi$
$479$	−2.14521e7	−	1.23854e7i	−0.195193	−	0.112695i	−0.594411	+	0.804161i	$0.702615\pi$
$480$		0			0
$481$	3.87151e7	+	6.70565e7i	0.347893	+	0.602568i
$482$			7.55506e7i			0.674678i
$483$		0			0
$484$	5.27682e7			0.465411
$485$	−1.27934e8	+	7.38628e7i	−1.12140	+	0.647441i
$486$		0			0
$487$	−8.96914e7	+	1.55350e8i	−0.776540	+	1.34501i	0.157384	+	0.987537i	$0.449694\pi$
$487$	−8.96914e7	+	1.55350e8i	−0.776540	+	1.34501i	−0.933925	+	0.357470i	$0.883640\pi$
$488$	3.62175e7	−	2.09102e7i	0.311644	−	0.179928i
$489$		0			0
$490$	−9.51399e7	+	1.22269e8i	−0.808676	+	1.03927i
$491$		−	8.70441e7i		−	0.735351i	−0.929954	−	0.367675i	$-0.880154\pi$
$491$		−	8.70441e7i		−	0.735351i	0.929954	−	0.367675i	$-0.119846\pi$
$492$		0			0
$493$	2.40195e7	−	4.16030e7i	0.200458	−	0.347204i
$494$	5.40572e7	+	3.12100e7i	0.448408	+	0.258888i
$495$		0			0
$496$	−1.70098e7			−0.139398
$497$	−9.48523e6	+	1.37591e8i	−0.0772643	+	1.12078i
$498$		0			0
$499$	−5.58407e7	−	9.67190e7i	−0.449417	−	0.778413i	0.548931	−	0.835868i	$-0.315035\pi$
$499$	−5.58407e7	−	9.67190e7i	−0.449417	−	0.778413i	−0.998348	+	0.0574544i	$0.981702\pi$
$500$	−1.47980e8	−	8.54361e7i	−1.18384	−	0.683488i
$501$		0			0
$502$	−3.78129e7	−	6.54938e7i	−0.298902	−	0.517713i
$503$		−	5.63868e7i		−	0.443071i	−0.975152	−	0.221536i	$-0.928893\pi$
$503$		−	5.63868e7i		−	0.443071i	0.975152	−	0.221536i	$-0.0711069\pi$
$504$		0			0
$505$	−4.02223e8			−3.12315
$506$	1.90059e8	−	1.09730e8i	1.46702	−	0.846985i
$507$		0			0
$508$	−3.07400e7	+	5.32433e7i	−0.234484	+	0.406138i
$509$	−3.14488e7	+	1.81570e7i	−0.238479	+	0.137686i	−0.614478	−	0.788934i	$-0.710633\pi$
$509$	−3.14488e7	+	1.81570e7i	−0.238479	+	0.137686i	0.375998	+	0.926620i	$0.377300\pi$
$510$		0			0
$511$	1.32431e8	−	1.96754e8i	0.992488	−	1.47455i
$512$		−	5.93164e6i		−	0.0441942i
$513$		0			0
$514$	6.08478e7	−	1.05392e8i	0.448080	−	0.776097i
$515$	−1.46891e8	−	8.48074e7i	−1.07541	−	0.620886i
$516$		0			0
$517$	−6.31208e7			−0.456773
$518$	−4.32931e7	−	2.91396e7i	−0.311480	−	0.209650i
$519$		0			0
$520$	6.06557e7	+	1.05059e8i	0.431382	+	0.747175i
$521$	−1.78363e8	−	1.02978e8i	−1.26122	−	0.728167i	−0.287912	−	0.957657i	$-0.592961\pi$
$521$	−1.78363e8	−	1.02978e8i	−1.26122	−	0.728167i	−0.973311	+	0.229490i	$0.926294\pi$
$522$		0			0
$523$	−2.31403e7	−	4.00802e7i	−0.161757	−	0.280172i	0.773742	−	0.633501i	$-0.218383\pi$
$523$	−2.31403e7	−	4.00802e7i	−0.161757	−	0.280172i	−0.935499	+	0.353329i	$0.885050\pi$
$524$		−	3.08366e7i		−	0.214325i
$525$		0			0
$526$	4.27469e7			0.293729
$527$	−2.91495e7	+	1.68295e7i	−0.199159	+	0.114985i
$528$		0			0
$529$	1.45989e8	−	2.52860e8i	0.986174	−	1.70810i
$530$	−2.36413e8	+	1.36493e8i	−1.58798	+	0.916819i
$531$		0			0
$532$	−4.19701e7	−	2.89333e6i	−0.278744	−	0.0192160i
$533$			8.29298e7i			0.547683i
$534$		0			0
$535$	−2.07715e7	+	3.59774e7i	−0.135646	+	0.234946i
$536$	−2.47363e7	−	1.42815e7i	−0.160635	−	0.0927427i
$537$		0			0
$538$	−7.79377e7			−0.500496
$539$	1.71726e8	+	1.33624e8i	1.09665	+	0.853329i
$540$		0			0
$541$	1.21144e8	+	2.09828e8i	0.765087	+	1.32517i	0.940200	+	0.340622i	$0.110638\pi$
$541$	1.21144e8	+	2.09828e8i	0.765087	+	1.32517i	−0.175113	+	0.984548i	$0.556029\pi$
$542$	−1.76725e8	−	1.02032e8i	−1.10994	−	0.640827i
$543$		0			0
$544$	−5.86876e6	−	1.01650e7i	−0.0364543	−	0.0631408i
$545$		−	1.94066e8i		−	1.19884i
$546$		0			0
$547$	−8.09693e7			−0.494718			−0.247359	−	0.968924i	$-0.579563\pi$
$547$	−8.09693e7			−0.494718			−0.247359	+	0.968924i	$0.579563\pi$
$548$	1.14387e8	−	6.60412e7i	0.695079	−	0.401304i
$549$		0			0
$550$	−2.01731e8	+	3.49408e8i	−1.21251	+	2.10012i
$551$	7.86954e7	−	4.54348e7i	0.470430	−	0.271603i
$552$		0			0
$553$	1.21858e8	+	2.49224e8i	0.720576	+	1.47372i
$554$		−	9.36918e7i		−	0.551026i
$555$		0			0
$556$	3.02139e6	−	5.23320e6i	0.0175785	−	0.0304469i
$557$	−6.33316e7	−	3.65645e7i	−0.366484	−	0.211590i	0.305437	−	0.952212i	$-0.401197\pi$
$557$	−6.33316e7	−	3.65645e7i	−0.366484	−	0.211590i	−0.671921	+	0.740623i	$0.734531\pi$
$558$		0			0
$559$	−1.45162e8			−0.831030
$560$	−6.78282e7	−	4.56536e7i	−0.386230	−	0.259963i
$561$		0			0
$562$	1.45299e7	+	2.51666e7i	0.0818568	+	0.141780i
$563$	2.05700e7	+	1.18761e7i	0.115268	+	0.0665501i	0.556526	−	0.830830i	$-0.312134\pi$
$563$	2.05700e7	+	1.18761e7i	0.115268	+	0.0665501i	−0.441257	+	0.897381i	$0.645467\pi$
$564$		0			0
$565$	−1.99997e8	−	3.46405e8i	−1.10886	−	1.92061i
$566$		−	1.45319e8i		−	0.801442i
$567$		0			0
$568$	−7.27864e7			−0.397196
$569$	−3.16370e6	+	1.82657e6i	−0.0171735	+	0.00991513i	−0.508562	−	0.861025i	$-0.669823\pi$
$569$	−3.16370e6	+	1.82657e6i	−0.0171735	+	0.00991513i	0.491389	+	0.870940i	$0.336489\pi$
$570$		0			0
$571$	−1.20273e8	+	2.08319e8i	−0.646041	+	1.11898i	0.338019	+	0.941139i	$0.390243\pi$
$571$	−1.20273e8	+	2.08319e8i	−0.646041	+	1.11898i	−0.984060	+	0.177837i	$0.943090\pi$
$572$	1.47554e8	−	8.51906e7i	0.788432	−	0.455201i
$573$		0			0
$574$	−2.45513e7	−	5.02121e7i	−0.129819	−	0.265505i
$575$			8.08930e8i			4.25508i
$576$		0			0
$577$	1.79305e7	−	3.10565e7i	0.0933392	−	0.161668i	−0.815575	−	0.578651i	$-0.803579\pi$
$577$	1.79305e7	−	3.10565e7i	0.0933392	−	0.161668i	0.908914	+	0.416983i	$0.136913\pi$
$578$	9.81350e7	+	5.66583e7i	0.508207	+	0.293413i
$579$		0			0
$580$	1.76603e8			0.905135
$581$	1.68113e7	−	2.43861e8i	0.0857180	−	1.24341i
$582$		0			0
$583$	1.91704e8	+	3.32041e8i	0.967444	+	1.67566i
$584$	1.08398e8	+	6.25838e7i	0.544232	+	0.314212i
$585$		0			0
$586$	1.34442e8	+	2.32860e8i	0.668101	+	1.15718i
$587$		−	1.23536e8i		−	0.610773i	−0.952228	−	0.305387i	$-0.901214\pi$
$587$		−	1.23536e8i		−	0.610773i	0.952228	−	0.305387i	$-0.0987857\pi$
$588$		0			0
$589$	−6.36687e7			−0.311587
$590$	3.24500e8	−	1.87350e8i	1.58000	−	0.912215i
$591$		0			0
$592$	1.37708e7	−	2.38516e7i	0.0663732	−	0.114962i
$593$	7.81814e7	−	4.51380e7i	0.374921	−	0.216461i	−0.300685	−	0.953723i	$-0.597215\pi$
$593$	7.81814e7	−	4.51380e7i	0.374921	−	0.216461i	0.675606	+	0.737263i	$0.263882\pi$
$594$		0			0
$595$	−1.61406e8	−	1.11270e7i	−0.766247	−	0.0528235i
$596$			8.79293e7i			0.415332i
$597$		0			0
$598$	1.70805e8	−	2.95843e8i	0.798725	−	1.38343i
$599$	1.71881e8	+	9.92358e7i	0.799739	+	0.461730i	0.843380	−	0.537318i	$-0.180562\pi$
$599$	1.71881e8	+	9.92358e7i	0.799739	+	0.461730i	−0.0436407	+	0.999047i	$0.513896\pi$
$600$		0			0
$601$	1.57998e8			0.727828			0.363914	−	0.931432i	$-0.381440\pi$
$601$	1.57998e8			0.727828			0.363914	+	0.931432i	$0.381440\pi$
$602$	8.78921e7	−	4.29750e7i	0.402866	−	0.196982i
$603$		0			0
$604$	−3.57410e6	−	6.19053e6i	−0.0162202	−	0.0280942i
$605$	3.32435e8	+	1.91932e8i	1.50121	+	0.866723i
$606$		0			0
$607$	7.84510e7	+	1.35881e8i	0.350778	+	0.607566i	0.986386	−	0.164446i	$-0.0525838\pi$
$607$	7.84510e7	+	1.35881e8i	0.350778	+	0.607566i	−0.635608	+	0.772012i	$0.719250\pi$
$608$		−	2.22024e7i		−	0.0987847i
$609$		0			0
$610$	3.04223e8			1.34030
$611$	−8.50896e7	+	4.91265e7i	−0.373038	+	0.215373i
$612$		0			0
$613$	1.40200e8	−	2.42834e8i	0.608649	−	1.05421i	−0.382814	−	0.923825i	$-0.625045\pi$
$613$	1.40200e8	−	2.42834e8i	0.608649	−	1.05421i	0.991463	−	0.130386i	$-0.0416216\pi$
$614$	9.05583e7	−	5.22839e7i	0.391222	−	0.225872i
$615$		0			0
$616$	−6.41203e7	+	9.52643e7i	−0.274317	+	0.407557i
$617$			663379.i			0.00282427i	0.999999	+	0.00141214i	$0.000449497\pi$
$617$			663379.i			0.00282427i	−0.999999	+	0.00141214i	$0.999551\pi$
$618$		0			0
$619$	−2.00330e8	+	3.46982e8i	−0.844645	+	1.46297i	0.0412848	+	0.999147i	$0.486855\pi$
$619$	−2.00330e8	+	3.46982e8i	−0.844645	+	1.46297i	−0.885929	+	0.463820i	$0.846478\pi$
$620$	−1.07161e8	−	6.18691e7i	−0.449635	−	0.259597i
$621$		0			0
$622$	1.72546e8			0.717024
$623$	−2.68993e7	+	1.31525e7i	−0.111244	+	0.0543930i
$624$		0			0
$625$	−3.20228e8	−	5.54651e8i	−1.31165	−	2.27185i
$626$	−1.69625e8	−	9.79332e7i	−0.691461	−	0.399215i
$627$		0			0
$628$	−5.83816e7	−	1.01120e8i	−0.235721	−	0.408280i
$629$		−	5.44990e7i		−	0.218996i
$630$		0			0
$631$	3.14143e8			1.25037			0.625187	−	0.780475i	$-0.285023\pi$
$631$	3.14143e8			1.25037			0.625187	+	0.780475i	$0.285023\pi$
$632$	−1.26794e8	+	7.32047e7i	−0.502283	+	0.289993i
$633$		0			0
$634$	2.24124e7	−	3.88194e7i	0.0879469	−	0.152328i
$635$	−3.87319e8	+	2.23619e8i	−1.51268	+	0.873347i
$636$		0			0
$637$	3.35493e8	+	4.64772e7i	1.29797	+	0.179813i
$638$		−	2.48038e8i		−	0.955114i
$639$		0			0
$640$	2.15749e7	−	3.73688e7i	0.0823017	−	0.142551i
$641$	4.34920e7	+	2.51101e7i	0.165133	+	0.0953398i	0.580289	−	0.814411i	$-0.302940\pi$
$641$	4.34920e7	+	2.51101e7i	0.165133	+	0.0953398i	−0.415156	+	0.909750i	$0.636273\pi$
$642$		0			0
$643$	1.71483e8			0.645043			0.322521	−	0.946562i	$-0.395470\pi$
$643$	1.71483e8			0.645043			0.322521	+	0.946562i	$0.395470\pi$
$644$	−1.58346e7	+	2.29693e8i	−0.0592855	+	0.859984i
$645$		0			0
$646$	−2.19670e7	−	3.80480e7i	−0.0814843	−	0.141135i
$647$	−1.05948e8	−	6.11689e7i	−0.391182	−	0.225849i	0.291490	−	0.956574i	$-0.405849\pi$
$647$	−1.05948e8	−	6.11689e7i	−0.391182	−	0.225849i	−0.682672	+	0.730725i	$0.739182\pi$
$648$		0			0
$649$	−2.63132e8	−	4.55758e8i	−0.962586	−	1.66725i
$650$			6.28023e8i			2.28684i
$651$		0			0
$652$	−4.20083e7			−0.151563
$653$	−2.27787e8	+	1.31513e8i	−0.818068	+	0.472312i	−0.849750	−	0.527186i	$-0.823247\pi$
$653$	−2.27787e8	+	1.31513e8i	−0.818068	+	0.472312i	0.0316820	+	0.999498i	$0.489914\pi$
$654$		0			0
$655$	1.12161e8	−	1.94268e8i	0.399132	−	0.691317i
$656$	2.55457e7	−	1.47488e7i	0.0904914	−	0.0522452i
$657$		0			0
$658$	3.69760e7	−	5.49357e7i	0.129790	−	0.192831i
$659$		−	8.49993e7i		−	0.297002i	−0.988912	−	0.148501i	$-0.952555\pi$
$659$		−	8.49993e7i		−	0.297002i	0.988912	−	0.148501i	$-0.0474448\pi$
$660$		0			0
$661$	−1.30383e8	+	2.25831e8i	−0.451459	+	0.781950i	−0.998477	−	0.0551712i	$-0.982430\pi$
$661$	−1.30383e8	+	2.25831e8i	−0.451459	+	0.781950i	0.547018	+	0.837121i	$0.315763\pi$
$662$	1.84192e8	+	1.06343e8i	0.634887	+	0.366552i
$663$		0			0
$664$	1.29004e8			0.440655
$665$	−2.53884e8	−	1.70884e8i	−0.863318	−	0.581080i
$666$		0			0
$667$	−2.48655e8	−	4.30683e8i	−0.837952	−	1.45138i
$668$	1.25359e7	+	7.23760e6i	0.0420558	+	0.0242809i
$669$		0			0
$670$	−1.03891e8	−	1.79945e8i	−0.345425	−	0.598294i
$671$		−	4.27279e8i		−	1.41431i
$672$		0			0
$673$	−4.91729e8			−1.61317			−0.806585	−	0.591118i	$-0.798687\pi$
$673$	−4.91729e8			−1.61317			−0.806585	+	0.591118i	$0.798687\pi$
$674$	1.58125e7	−	9.12932e6i	0.0516440	−	0.0298167i
$675$		0			0
$676$	5.53777e7	−	9.59170e7i	0.179265	−	0.310496i
$677$	2.55998e7	−	1.47800e7i	0.0825030	−	0.0476331i	−0.458181	−	0.888859i	$-0.651499\pi$
$677$	2.55998e7	−	1.47800e7i	0.0825030	−	0.0476331i	0.540684	+	0.841226i	$0.318165\pi$
$678$		0			0
$679$	−2.17153e8	−	1.49701e7i	−0.693676	−	0.0478206i
$680$		−	8.53847e7i		−	0.271552i
$681$		0			0
$682$	−8.68949e7	+	1.50506e8i	−0.273931	+	0.474462i
$683$	3.74678e8	+	2.16320e8i	1.17597	+	0.678945i	0.955078	−	0.296353i	$-0.0957706\pi$
$683$	3.74678e8	+	2.16320e8i	1.17597	+	0.678945i	0.220890	+	0.975299i	$0.429104\pi$
$684$		0			0
$685$	9.60836e8			2.98936
$686$	−2.16893e8	+	7.11814e7i	−0.671850	+	0.220493i
$687$		0			0
$688$	2.58166e7	+	4.47157e7i	0.0792746	+	0.137308i
$689$	5.16851e8	+	2.98404e8i	1.58018	+	0.912320i
$690$		0			0
$691$	8.43987e7	+	1.46183e8i	0.255801	+	0.443060i	0.965113	−	0.261835i	$-0.0843276\pi$
$691$	8.43987e7	+	1.46183e8i	0.255801	+	0.443060i	−0.709312	+	0.704895i	$0.750994\pi$
$692$			1.37809e8i			0.415872i
$693$		0			0
$694$	7.48976e7			0.224073
$695$	3.80690e7	−	2.19791e7i	0.113401	−	0.0654721i
$696$		0			0
$697$	2.91850e7	−	5.05498e7i	0.0861908	−	0.149287i
$698$	2.07800e8	−	1.19973e8i	0.611053	−	0.352791i
$699$		0			0
$700$	−1.85926e8	−	3.80254e8i	−0.542057	−	1.10861i
$701$			6.51460e7i			0.189118i	0.995519	+	0.0945591i	$0.0301441\pi$
$701$			6.51460e7i			0.189118i	−0.995519	+	0.0945591i	$0.969856\pi$
$702$		0			0
$703$	5.15446e7	−	8.92779e7i	0.148360	−	0.256967i
$704$	−5.24843e7	−	3.03018e7i	−0.150422	−	0.0868462i
$705$		0			0
$706$	−2.48934e8			−0.707409
$707$	−4.91665e8	−	3.30929e8i	−1.39127	−	0.936432i
$708$		0			0
$709$	3.19307e8	+	5.53055e8i	0.895920	+	1.55178i	0.832662	+	0.553781i	$0.186816\pi$
$709$	3.19307e8	+	5.53055e8i	0.895920	+	1.55178i	0.0632576	+	0.997997i	$0.479851\pi$
$710$	−4.58548e8	−	2.64743e8i	−1.28118	−	0.739689i
$711$		0			0
$712$	−7.90116e6	−	1.36852e7i	−0.0218902	−	0.0379150i
$713$			3.48444e8i			0.961313i
$714$		0			0
$715$	1.23944e9			3.39084
$716$	2.20028e8	−	1.27033e8i	0.599431	−	0.346082i
$717$		0			0
$718$	4.44454e7	−	7.69817e7i	0.120075	−	0.207976i
$719$	4.74064e7	−	2.73701e7i	0.127541	−	0.0736359i	−0.434872	−	0.900492i	$-0.643206\pi$
$719$	4.74064e7	−	2.73701e7i	0.127541	−	0.0736359i	0.562413	+	0.826856i	$0.309873\pi$
$720$		0			0
$721$	−1.09779e8	−	2.24520e8i	−0.292897	−	0.599031i
$722$			1.83027e8i			0.486299i
$723$		0			0
$724$	−6.05018e7	+	1.04792e8i	−0.159424	+	0.276130i
$725$	7.91775e8	+	4.57131e8i	2.07772	+	1.19957i
$726$		0			0
$727$	−5.25087e8			−1.36656			−0.683279	−	0.730157i	$-0.739447\pi$
$727$	−5.25087e8			−1.36656			−0.683279	+	0.730157i	$0.739447\pi$
$728$	−1.22934e7	+	1.78325e8i	−0.0318622	+	0.462187i
$729$		0			0
$730$	4.55267e8	+	7.88545e8i	1.17030	+	2.02702i
$731$	8.84832e7	+	5.10858e7i	0.226521	+	0.130782i
$732$		0			0
$733$	−4.95595e7	−	8.58395e7i	−0.125839	−	0.217959i	0.796222	−	0.605005i	$-0.206829\pi$
$733$	−4.95595e7	−	8.58395e7i	−0.125839	−	0.217959i	−0.922061	+	0.387046i	$0.873496\pi$
$734$		−	3.19719e8i		−	0.808499i
$735$		0			0
$736$	−1.21509e8			−0.304772
$737$	−2.52731e8	+	1.45914e8i	−0.631330	+	0.364499i
$738$		0			0
$739$	1.67177e8	−	2.89560e8i	0.414233	−	0.717472i	−0.581115	−	0.813821i	$-0.697383\pi$
$739$	1.67177e8	−	2.89560e8i	0.414233	−	0.717472i	0.995348	+	0.0963496i	$0.0307167\pi$
$740$	1.73509e8	−	1.00176e8i	0.428181	−	0.247210i
$741$		0			0
$742$	−4.01284e8	−	2.76637e7i	−0.982291	−	0.0677171i
$743$		−	1.95685e8i		−	0.477081i	−0.971133	−	0.238541i	$-0.923331\pi$
$743$		−	1.95685e8i		−	0.477081i	0.971133	−	0.238541i	$-0.0766690\pi$
$744$		0			0
$745$	−3.19822e8	+	5.53947e8i	−0.773462	+	1.33968i
$746$	3.59082e8	+	2.07316e8i	0.864923	+	0.499364i
$747$		0			0
$748$	−1.19922e8			−0.286547
$749$	−5.49908e7	+	2.68878e7i	−0.130871	+	0.0639897i
$750$		0			0
$751$	−1.30433e8	−	2.25916e8i	−0.307941	−	0.533369i	0.669971	−	0.742387i	$-0.266306\pi$
$751$	−1.30433e8	−	2.25916e8i	−0.307941	−	0.533369i	−0.977912	+	0.209018i	$0.932973\pi$
$752$	3.02659e7	+	1.74740e7i	0.0711705	+	0.0410903i
$753$		0			0
$754$	−1.93046e8	−	3.34366e8i	−0.450347	−	0.780023i
$755$		−	5.19997e7i		−	0.120826i
$756$		0			0
$757$	3.31458e8			0.764085			0.382042	−	0.924145i	$-0.375221\pi$
$757$	3.31458e8			0.764085			0.382042	+	0.924145i	$0.375221\pi$
$758$	−4.23527e8	+	2.44524e8i	−0.972465	+	0.561453i
$759$		0			0
$760$	8.07560e7	−	1.39873e8i	0.183964	−	0.318636i
$761$	1.86582e8	−	1.07723e8i	0.423365	−	0.244430i	−0.273151	−	0.961971i	$-0.588066\pi$
$761$	1.86582e8	−	1.07723e8i	0.423365	−	0.244430i	0.696516	+	0.717541i	$0.254733\pi$
$762$		0			0
$763$	1.59668e8	−	2.37221e8i	0.359455	−	0.534047i
$764$		−	7.76421e7i		−	0.174107i
$765$		0			0
$766$	−2.01352e8	+	3.48752e8i	−0.447992	+	0.775944i
$767$	−7.09427e8	−	4.09588e8i	−1.57225	−	0.907739i
$768$		0			0
$769$	1.95805e8			0.430571			0.215285	−	0.976551i	$-0.430932\pi$
$769$	1.95805e8			0.430571			0.215285	+	0.976551i	$0.430932\pi$
$770$	−7.50453e8	+	3.66935e8i	−1.64381	+	0.803743i
$771$		0			0
$772$	9.36654e7	+	1.62233e8i	0.203576	+	0.352605i
$773$	−6.34194e7	−	3.66152e7i	−0.137304	−	0.0792726i	0.429774	−	0.902936i	$-0.358593\pi$
$773$	−6.34194e7	−	3.66152e7i	−0.137304	−	0.0792726i	−0.567079	+	0.823664i	$0.691926\pi$
$774$		0			0
$775$	−3.20293e8	−	5.54764e8i	−0.688086	−	1.19180i
$776$		−	1.14875e8i		−	0.245834i
$777$		0			0
$778$	2.76317e8			0.586771
$779$	9.56190e7	−	5.52057e7i	0.202270	−	0.116781i
$780$		0			0
$781$	−3.71830e8	+	6.44028e8i	−0.780533	+	1.35192i
$782$	−2.08228e8	+	1.20221e8i	−0.435431	+	0.251396i
$783$		0			0
$784$	−4.53496e7	−	1.11611e8i	−0.0941077	−	0.231611i
$785$		−	8.49396e8i		−	1.75591i
$786$		0			0
$787$	−4.54960e8	+	7.88014e8i	−0.933360	+	1.61663i	−0.155826	+	0.987784i	$0.549804\pi$
$787$	−4.54960e8	+	7.88014e8i	−0.933360	+	1.61663i	−0.777533	+	0.628842i	$0.783529\pi$
$788$	4.63681e6	+	2.67707e6i	0.00947635	+	0.00547117i
$789$		0			0
$790$	−1.06506e9			−2.16019
$791$	4.05343e7	−	5.87982e8i	0.0819017	−	1.18805i
$792$		0			0
$793$	−3.32549e8	−	5.75991e8i	−0.666861	−	1.15504i
$794$	−3.13775e8	−	1.81158e8i	−0.626840	−	0.361906i
$795$		0			0
$796$	−9.78658e7	−	1.69509e8i	−0.194040	−	0.336087i
$797$			1.20237e8i			0.237500i	0.992924	+	0.118750i	$0.0378887\pi$
$797$			1.20237e8i			0.237500i	−0.992924	+	0.118750i	$0.962111\pi$
$798$		0			0
$799$	6.91551e7			0.135576
$800$	1.93456e8	−	1.11692e8i	0.377845	−	0.218149i
$801$		0			0
$802$	−1.29329e8	+	2.24004e8i	−0.250710	+	0.434243i
$803$	1.10751e9	−	6.39419e8i	2.13895	−	1.23492i
$804$		0			0
$805$	−9.35209e8	+	1.38945e9i	−1.79276	+	2.66352i
$806$			2.70519e8i			0.516645i
$807$		0			0
$808$	1.56390e8	−	2.70875e8i	0.296465	−	0.513493i
$809$	−6.21574e8	−	3.58866e8i	−1.17394	−	0.677777i	−0.219338	−	0.975649i	$-0.570390\pi$
$809$	−6.21574e8	−	3.58866e8i	−1.17394	−	0.677777i	−0.954606	+	0.297872i	$0.903723\pi$
$810$		0			0
$811$	7.64377e8			1.43300			0.716498	−	0.697589i	$-0.245744\pi$
$811$	7.64377e8			1.43300			0.716498	+	0.697589i	$0.245744\pi$
$812$	2.15874e8	+	1.45300e8i	0.403210	+	0.271392i
$813$		0			0
$814$	−1.40696e8	−	2.43693e8i	−0.260861	−	0.451824i
$815$	−2.64649e8	−	1.52795e8i	−0.488875	−	0.282252i
$816$		0			0
$817$	9.66329e7	+	1.67373e8i	0.177198	+	0.306916i
$818$			2.97835e8i			0.544146i
$819$		0			0
$820$	2.14581e8			0.389180
$821$	−2.20030e8	+	1.27034e8i	−0.397605	+	0.229557i	−0.685450	−	0.728120i	$-0.740394\pi$
$821$	−2.20030e8	+	1.27034e8i	−0.397605	+	0.229557i	0.287845	+	0.957677i	$0.407061\pi$
$822$		0			0
$823$	−4.44133e7	+	7.69261e7i	−0.0796734	+	0.137998i	−0.903109	−	0.429411i	$-0.858721\pi$
$823$	−4.44133e7	+	7.69261e7i	−0.0796734	+	0.137998i	0.823436	+	0.567410i	$0.192054\pi$
$824$	1.14226e8	−	6.59484e7i	0.204166	−	0.117875i
$825$		0			0
$826$	5.50800e8	+	3.79710e7i	0.977358	+	0.0673770i
$827$			8.42107e8i			1.48885i	0.667707	+	0.744424i	$0.267276\pi$
$827$			8.42107e8i			1.48885i	−0.667707	+	0.744424i	$0.732724\pi$
$828$		0			0
$829$	−9.28225e7	+	1.60773e8i	−0.162926	+	0.282196i	−0.935917	−	0.352221i	$-0.885426\pi$
$829$	−9.28225e7	+	1.60773e8i	−0.162926	+	0.282196i	0.772991	+	0.634417i	$0.218760\pi$
$830$	8.12713e8	+	4.69220e8i	1.42136	+	0.820620i
$831$		0			0
$832$	−9.43349e7			−0.163796
$833$	−1.88143e8	−	1.46398e8i	−0.325501	−	0.253279i
$834$		0			0
$835$	5.26500e7	+	9.11925e7i	0.0904356	+	0.156639i
$836$	−1.96451e8	−	1.13421e8i	−0.336230	−	0.194122i
$837$		0			0
$838$	1.39353e8	+	2.41367e8i	0.236802	+	0.410153i
$839$		−	4.80367e8i		−	0.813369i	−0.913569	−	0.406684i	$-0.866685\pi$
$839$		−	4.80367e8i		−	0.813369i	0.913569	−	0.406684i	$-0.133315\pi$
$840$		0			0
$841$	3.27581e7			0.0550719
$842$	−1.01718e8	+	5.87270e7i	−0.170397	+	0.0983787i
$843$		0			0
$844$	−2.24703e7	+	3.89197e7i	−0.0373750	+	0.0647354i
$845$	6.97749e8	−	4.02846e8i	1.15646	−	0.667680i
$846$		0			0
$847$	2.48447e8	+	5.08122e8i	0.408868	+	0.836215i
$848$		−	2.12282e8i		−	0.348117i
$849$		0			0
$850$	2.21016e8	−	3.82811e8i	0.359888	−	0.623344i
$851$	−4.88598e8	−	2.82092e8i	−0.792799	−	0.457723i
$852$		0			0
$853$	−5.70413e7			−0.0919057			−0.0459529	−	0.998944i	$-0.514632\pi$
$853$	−5.70413e7			−0.0919057			−0.0459529	+	0.998944i	$0.514632\pi$
$854$	3.71872e8	+	2.50299e8i	0.597063	+	0.401870i
$855$		0			0
$856$	−1.61525e7	−	2.79769e7i	−0.0257524	−	0.0446045i
$857$	2.69993e8	+	1.55880e8i	0.428953	+	0.247656i	0.698901	−	0.715219i	$-0.253673\pi$
$857$	2.69993e8	+	1.55880e8i	0.428953	+	0.247656i	−0.269947	+	0.962875i	$0.587006\pi$
$858$		0			0
$859$	−4.42571e8	−	7.66555e8i	−0.698238	−	1.20938i	−0.969077	−	0.246759i	$-0.920635\pi$
$859$	−4.42571e8	−	7.66555e8i	−0.698238	−	1.20938i	0.270839	−	0.962625i	$-0.412699\pi$
$860$			3.75607e8i			0.590525i
$861$		0			0
$862$	−3.45661e8			−0.539671
$863$	6.17709e8	−	3.56634e8i	0.961062	−	0.554869i	0.0645620	−	0.997914i	$-0.479435\pi$
$863$	6.17709e8	−	3.56634e8i	0.961062	−	0.554869i	0.896500	+	0.443045i	$0.146102\pi$
$864$		0			0
$865$	−5.01247e8	+	8.68186e8i	−0.774468	+	1.34142i
$866$	−2.68841e8	+	1.55215e8i	−0.413944	+	0.238991i
$867$		0			0
$868$	−8.00869e7	−	1.63793e8i	−0.122462	−	0.250459i
$869$			1.49587e9i			2.27947i
$870$		0			0
$871$	−2.27129e8	+	3.93398e8i	−0.343730	+	0.595358i
$872$	1.30693e8	+	7.54556e7i	0.197107	+	0.113800i
$873$		0			0
$874$	−4.54814e8			−0.681239
$875$	1.25963e8	−	1.82720e9i	0.188027	−	2.72748i
$876$		0			0
$877$	4.77839e8	+	8.27642e8i	0.708408	+	1.22700i	0.965448	+	0.260597i	$0.0839195\pi$
$877$	4.77839e8	+	8.27642e8i	0.708408	+	1.22700i	−0.257040	+	0.966401i	$0.582747\pi$
$878$	−3.48847e8	−	2.01407e8i	−0.515408	−	0.297571i
$879$		0			0
$880$	−2.20431e8	−	3.81798e8i	−0.323463	−	0.560255i
$881$		−	1.36744e9i		−	1.99977i	−0.0152446	−	0.999884i	$-0.504853\pi$
$881$		−	1.36744e9i		−	1.99977i	0.0152446	−	0.999884i	$-0.495147\pi$
$882$		0			0
$883$	−8.68635e8			−1.26170			−0.630849	−	0.775906i	$-0.717293\pi$
$883$	−8.68635e8			−1.26170			−0.630849	+	0.775906i	$0.717293\pi$
$884$	−1.61661e8	+	9.33348e7i	−0.234017	+	0.135110i
$885$		0			0
$886$	3.15756e8	−	5.46906e8i	0.453995	−	0.786342i
$887$	−1.04349e9	+	6.02460e8i	−1.49526	+	0.863291i	−0.999985	−	0.00544228i	$-0.998268\pi$
$887$	−1.04349e9	+	6.02460e8i	−1.49526	+	0.863291i	−0.495279	+	0.868734i	$0.664934\pi$
$888$		0			0
$889$	−6.57429e8	−	4.53218e7i	−0.935715	−	0.0645062i
$890$		−	1.14954e8i		−	0.163063i
$891$		0			0
$892$	−1.69756e8	+	2.94026e8i	−0.239183	+	0.414278i
$893$	1.13287e8	+	6.54062e7i	0.159083	+	0.0918469i
$894$		0			0
$895$	1.84821e9			2.57800
$896$	5.71177e7	−	2.79278e7i	0.0794047	−	0.0388251i
$897$		0			0
$898$	3.86044e8	+	6.68648e8i	0.533099	+	0.923355i
$899$	3.41055e8	+	1.96908e8i	0.469402	+	0.271009i
$900$		0			0
$901$	−2.10031e8	−	3.63784e8i	−0.287150	−	0.497359i
$902$		−	3.01378e8i		−	0.410670i
$903$		0			0
$904$	3.11046e8			0.421036
$905$	−7.62312e8	+	4.40121e8i	−1.02846	+	0.593781i
$906$		0			0
$907$	5.41540e8	−	9.37975e8i	0.725786	−	1.25710i	−0.232863	−	0.972510i	$-0.574809\pi$
$907$	5.41540e8	−	9.37975e8i	0.725786	−	1.25710i	0.958649	−	0.284590i	$-0.0918573\pi$
$908$	6.05042e7	−	3.49321e7i	0.0808217	−	0.0466624i
$909$		0			0
$910$	−7.26061e8	+	1.07872e9i	−0.963494	+	1.43148i
$911$			1.23656e9i			1.63554i	0.575544	+	0.817770i	$0.304790\pi$
$911$			1.23656e9i			1.63554i	−0.575544	+	0.817770i	$0.695210\pi$
$912$		0			0
$913$	6.59017e8	−	1.14145e9i	0.865933	−	1.49984i
$914$	−9.03034e8	−	5.21367e8i	−1.18268	−	0.682818i
$915$		0			0
$916$	−2.73838e8			−0.356292
$917$	2.96936e8	−	1.45187e8i	0.385083	−	0.188287i
$918$		0			0
$919$	−3.71841e8	−	6.44048e8i	−0.479083	−	0.829797i	0.520629	−	0.853783i	$-0.325698\pi$
$919$	−3.71841e8	−	6.44048e8i	−0.479083	−	0.829797i	−0.999712	+	0.0239862i	$0.992364\pi$
$920$	−7.65496e8	−	4.41959e8i	−0.983059	−	0.567569i
$921$		0			0
$922$	3.87923e7	+	6.71902e7i	0.0494940	+	0.0857261i
$923$			1.15757e9i			1.47212i
$924$		0			0
$925$	1.03721e9			1.31051
$926$	2.24853e8	−	1.29819e8i	0.283182	−	0.163495i
$927$		0			0
$928$	−6.86655e7	+	1.18932e8i	−0.0859200	+	0.148818i
$929$	−6.68483e8	+	3.85949e8i	−0.833764	+	0.481374i	−0.855140	−	0.518398i	$-0.826529\pi$
$929$	−6.68483e8	+	3.85949e8i	−0.833764	+	0.481374i	0.0213756	+	0.999772i	$0.493195\pi$
$930$		0			0
$931$	−1.69746e8	−	4.17766e8i	−0.210354	−	0.517707i
$932$			7.80787e8i			0.964461i
$933$		0			0
$934$	3.50778e8	−	6.07565e8i	0.430518	−	0.745679i
$935$	−7.55501e8	−	4.36188e8i	−0.924272	−	0.533629i
$936$		0			0
$937$	−1.09992e9			−1.33704			−0.668518	−	0.743696i	$-0.733071\pi$
$937$	−1.09992e9			−1.33704			−0.668518	+	0.743696i	$0.733071\pi$
$938$	2.10560e7	−	3.05435e8i	0.0255134	−	0.370092i
$939$		0			0
$940$	1.27115e8	+	2.20170e8i	0.153043	+	0.265078i
$941$	−2.28664e8	−	1.32019e8i	−0.274429	−	0.158441i	0.356470	−	0.934307i	$-0.383980\pi$
$941$	−2.28664e8	−	1.32019e8i	−0.274429	−	0.158441i	−0.630898	+	0.775865i	$0.717314\pi$
$942$		0			0
$943$	−3.02128e8	−	5.23301e8i	−0.360293	−	0.624046i
$944$			2.91376e8i			0.346369i
$945$		0			0
$946$	5.27538e8			0.623132
$947$	−8.98370e8	+	5.18674e8i	−1.05780	+	0.610724i	−0.924824	−	0.380395i	$-0.875788\pi$
$947$	−8.98370e8	+	5.18674e8i	−1.05780	+	0.610724i	−0.132980	+	0.991119i	$0.542455\pi$
$948$		0			0
$949$	9.95312e8	−	1.72393e9i	1.16456	−	2.01707i
$950$	7.24117e8	−	4.18069e8i	0.844575	−	0.487616i
$951$		0			0
$952$	7.02502e7	−	1.04372e8i	0.0814210	−	0.120968i
$953$			1.10083e9i			1.27186i	0.771746	+	0.635931i	$0.219384\pi$
$953$			1.10083e9i			1.27186i	−0.771746	+	0.635931i	$0.780616\pi$
$954$		0			0
$955$	2.82404e8	−	4.89138e8i	0.324236	−	0.561593i
$956$	4.37257e8	+	2.52450e8i	0.500452	+	0.288936i
$957$		0			0
$958$	1.40125e8			0.159374
$959$	1.17450e9	+	7.90527e8i	1.33167	+	0.896315i
$960$		0			0
$961$	3.05786e8	+	5.29638e8i	0.344547	+	0.596772i
$962$	−3.79329e8	−	2.19006e8i	−0.426080	−	0.245997i
$963$		0			0
$964$	−2.13689e8	−	3.70121e8i	−0.238535	−	0.413154i
$965$			1.36274e9i			1.51646i
$966$		0			0
$967$	−2.39410e8			−0.264766			−0.132383	−	0.991199i	$-0.542263\pi$
$967$	−2.39410e8			−0.264766			−0.132383	+	0.991199i	$0.542263\pi$
$968$	−2.58510e8	+	1.49251e8i	−0.285005	+	0.164548i
$969$		0			0
$970$	4.17831e8	−	7.23705e8i	0.457810	−	0.792951i
$971$	8.99011e8	−	5.19044e8i	0.981990	−	0.566952i	0.0791198	−	0.996865i	$-0.474789\pi$
$971$	8.99011e8	−	5.19044e8i	0.981990	−	0.566952i	0.902870	+	0.429913i	$0.141456\pi$
$972$		0			0
$973$	6.46177e7	+	4.45461e6i	0.0701476	+	0.00483583i
$974$		−	1.01474e9i		−	1.09819i
$975$		0			0
$976$	−1.18286e8	+	2.04877e8i	−0.127228	+	0.220366i
$977$	8.34797e8	+	4.81971e8i	0.895153	+	0.516817i	0.875625	−	0.482992i	$-0.160450\pi$
$977$	8.34797e8	+	4.81971e8i	0.895153	+	0.516817i	0.0195287	+	0.999809i	$0.493783\pi$
$978$		0			0
$979$	−1.61452e8			−0.172067
$980$	1.20260e8	−	8.68089e8i	0.127774	−	0.922329i
$981$		0			0
$982$	2.46198e8	+	4.26427e8i	0.259986	+	0.450309i
$983$	−1.25817e9	−	7.26406e8i	−1.32458	−	0.764749i	−0.340127	−	0.940379i	$-0.610470\pi$
$983$	−1.25817e9	−	7.26406e8i	−1.32458	−	0.764749i	−0.984456	+	0.175631i	$0.943804\pi$
$984$		0			0
$985$	1.94744e7	+	3.37306e7i	0.0203777	+	0.0352951i
$986$			2.71750e8i			0.283491i
$987$		0			0
$988$	−3.53100e8			−0.366123
$989$	9.15995e8	−	5.28850e8i	0.946901	−	0.546693i
$990$		0			0
$991$	8.66029e8	−	1.50001e9i	0.889839	−	1.54125i	0.0497742	−	0.998760i	$-0.484150\pi$
$991$	8.66029e8	−	1.50001e9i	0.889839	−	1.54125i	0.840065	−	0.542486i	$-0.182517\pi$
$992$	8.33308e7	−	4.81111e7i	0.0853632	−	0.0492845i
$993$		0			0
$994$	−3.42698e8	−	7.00883e8i	−0.348941	−	0.713652i
$995$		−	1.42385e9i		−	1.44543i
$996$		0			0
$997$	−7.31869e8	+	1.26763e9i	−0.738495	+	1.27911i	0.214678	+	0.976685i	$0.431130\pi$
$997$	−7.31869e8	+	1.26763e9i	−0.738495	+	1.27911i	−0.953173	+	0.302426i	$0.902203\pi$
$998$	5.47125e8	+	3.15883e8i	0.550421	+	0.317786i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.7.s.b.53.4	✓	16
3.2	odd	2	inner	126.7.s.b.53.5	yes	16
7.2	even	3	inner	126.7.s.b.107.5	yes	16
21.2	odd	6	inner	126.7.s.b.107.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.7.s.b.53.4	✓	16	1.1	even	1	trivial
126.7.s.b.53.5	yes	16	3.2	odd	2	inner
126.7.s.b.107.4	yes	16	21.2	odd	6	inner
126.7.s.b.107.5	yes	16	7.2	even	3	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	126.s (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.89898 + 2.82843i) q^{2} +(16.0000 - 27.7128i) q^{4} +(201.597 - 116.392i) q^{5} +(342.188 + 23.5897i) q^{7} +181.019i q^{8} +(-658.414 + 1140.41i) q^{10} +(1601.69 + 924.739i) q^{11} +2878.87 q^{13} +(-1743.09 + 852.288i) q^{14} +(-512.000 - 886.810i) q^{16} +(-1754.82 - 1013.14i) q^{17} +(-1916.44 - 3319.37i) q^{19} -7449.10i q^{20} -10462.2 q^{22} +(-18166.2 + 10488.3i) q^{23} +(19281.8 - 33397.1i) q^{25} +(-14103.5 + 8142.68i) q^{26} +(6128.74 - 9105.55i) q^{28} +23707.9i q^{29} +(8305.58 - 14385.7i) q^{31} +(5016.55 + 2896.31i) q^{32} +11462.4 q^{34} +(71729.8 - 35072.4i) q^{35} +(13448.0 + 23292.6i) q^{37} +(18777.2 + 10841.0i) q^{38} +(21069.2 + 36493.0i) q^{40} +28806.3i q^{41} -50423.1 q^{43} +(51254.2 - 29591.6i) q^{44} +(59330.5 - 102763. i) q^{46} +(-29556.6 + 17064.5i) q^{47} +(116536. + 16144.2i) q^{49} +218149. i q^{50} +(46062.0 - 79781.7i) q^{52} +(179532. + 103653. i) q^{53} +430530. q^{55} +(-4270.19 + 61942.6i) q^{56} +(-67056.1 - 116145. i) q^{58} +(-246425. - 142274. i) q^{59} +(-115513. - 200075. i) q^{61} +93966.9i q^{62} -32768.0 q^{64} +(580373. - 335079. i) q^{65} +(-78894.9 + 136650. i) q^{67} +(-56154.1 + 32420.6i) q^{68} +(-252203. + 374701. i) q^{70} +402092. i q^{71} +(345730. - 598821. i) q^{73} +(-131763. - 76073.4i) q^{74} -122652. q^{76} +(526266. + 354218. i) q^{77} +(404402. + 700445. i) q^{79} +(-206436. - 119186. i) q^{80} +(-81476.6 - 141122. i) q^{82} -712652. i q^{83} -471688. q^{85} +(247022. - 142618. i) q^{86} +(-167396. + 289938. i) q^{88} +(-75600.8 + 43648.1i) q^{89} +(985116. + 67911.8i) q^{91} +671248. i q^{92} +(96531.3 - 167197. i) q^{94} +(-772699. - 446118. i) q^{95} -634602. q^{97} +(-616570. + 250523. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 256 q^{4} + 896 q^{7} - 1568 q^{10} + 2184 q^{13} - 8192 q^{16} + 9548 q^{19} - 15616 q^{22} + 47036 q^{25} + 17024 q^{28} + 33096 q^{31} + 80192 q^{34} - 21796 q^{37} + 50176 q^{40} - 535096 q^{43}+ \cdots + 6864088 q^{97}+O(q^{100}) \) 16 * q + 256 * q^4 + 896 * q^7 - 1568 * q^10 + 2184 * q^13 - 8192 * q^16 + 9548 * q^19 - 15616 * q^22 + 47036 * q^25 + 17024 * q^28 + 33096 * q^31 + 80192 * q^34 - 21796 * q^37 + 50176 * q^40 - 535096 * q^43 + 47648 * q^46 + 826672 * q^49 + 34944 * q^52 + 1755656 * q^55 - 312128 * q^58 - 302904 * q^61 - 524288 * q^64 + 8356 * q^67 + 264992 * q^70 - 717668 * q^73 + 611072 * q^76 - 388104 * q^79 - 596064 * q^82 - 3439552 * q^85 - 249856 * q^88 + 5012364 * q^91 - 795648 * q^94 + 6864088 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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