LMFDB - Embedded newform 112.7.s.c.17.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,7,Mod(17,112)] 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("112.17"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	112.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$25.7660573654$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 $ x^8 - 2x^7 + 285x^6 + 282x^5 + 62091x^4 + 29260x^3 + 4838750x^2 + 2401000*x + 294122500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.4
Root			$-6.30576 - 10.9219i$ of defining polynomial
Character	$\chi$	$=$	112.17
Dual form			112.7.s.c.33.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(36.7384 + 21.2109i) q^{3} +(-162.347 + 93.7310i) q^{5} +(-141.244 + 312.569i) q^{7} +(535.305 + 927.175i) q^{9} +(555.924 - 962.889i) q^{11} +706.517i q^{13} -7952.48 q^{15} +(-7334.73 - 4234.71i) q^{17} +(-22.0061 + 12.7052i) q^{19} +(-11818.9 + 8487.36i) q^{21} +(-424.738 - 735.668i) q^{23} +(9758.49 - 16902.2i) q^{25} +14491.7i q^{27} -15109.4 q^{29} +(1205.81 + 696.173i) q^{31} +(40847.5 - 23583.3i) q^{33} +(-6366.92 - 63983.4i) q^{35} +(4646.47 + 8047.92i) q^{37} +(-14985.9 + 25956.3i) q^{39} +109829. i q^{41} +45569.8 q^{43} +(-173810. - 100349. i) q^{45} +(-119865. + 69204.2i) q^{47} +(-77749.5 - 88296.7i) q^{49} +(-179644. - 311152. i) q^{51} +(-30937.9 + 53586.1i) q^{53} +208429. i q^{55} -1077.96 q^{57} +(-68015.2 - 39268.6i) q^{59} +(-151296. + 87350.5i) q^{61} +(-365415. + 36362.0i) q^{63} +(-66222.6 - 114701. i) q^{65} +(-217974. + 377543. i) q^{67} -36036.3i q^{69} +561559. q^{71} +(-192170. - 110949. i) q^{73} +(717022. - 413973. i) q^{75} +(222448. + 309766. i) q^{77} +(333239. + 577187. i) q^{79} +(82854.9 - 143509. i) q^{81} -653635. i q^{83} +1.58769e6 q^{85} +(-555096. - 320485. i) q^{87} +(114603. - 66165.9i) q^{89} +(-220835. - 99791.1i) q^{91} +(29532.9 + 51152.5i) q^{93} +(2381.75 - 4125.31i) q^{95} +1.59480e6i q^{97} +1.19036e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37}+ \cdots + 4625928 q^{99}+O(q^{100}) $ 8 * q - 336 * q^5 - 652 * q^7 + 756 * q^9 + 1356 * q^11 - 27144 * q^15 - 17304 * q^17 + 32004 * q^19 + 9756 * q^21 + 4128 * q^23 + 4664 * q^25 - 30312 * q^29 + 3108 * q^31 + 3276 * q^33 - 98028 * q^35 - 6124 * q^37 - 100764 * q^39 + 297376 * q^43 - 172116 * q^45 - 313908 * q^47 + 32432 * q^49 - 253692 * q^51 + 278484 * q^53 - 81288 * q^57 + 835464 * q^59 - 995316 * q^61 - 1216188 * q^63 + 8316 * q^65 - 648808 * q^67 - 190128 * q^71 - 1617084 * q^73 + 2042208 * q^75 + 1456224 * q^77 - 70096 * q^79 + 1177920 * q^81 + 2190984 * q^85 + 2057076 * q^87 + 739116 * q^89 - 2233752 * q^91 - 23364 * q^93 - 725640 * q^95 + 4625928 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	36.7384	+	21.2109i	1.36068	+	0.785589i	0.989714	−	0.143057i	$-0.0456934\pi$
$3$	36.7384	+	21.2109i	1.36068	+	0.785589i	0.370966	+	0.928647i	$0.379027\pi$
$4$		0			0
$5$	−162.347	+	93.7310i	−1.29877	+	0.749848i	−0.980192	−	0.198047i	$-0.936540\pi$
$5$	−162.347	+	93.7310i	−1.29877	+	0.749848i	−0.318582	+	0.947895i	$0.603207\pi$
$6$		0			0
$7$	−141.244	+	312.569i	−0.411789	+	0.911279i
$7$	−141.244	+	312.569i	−0.411789	+	0.911279i
$8$		0			0
$9$	535.305	+	927.175i	0.734300	+	1.27185i
$10$		0			0
$11$	555.924	−	962.889i	0.417674	−	0.723432i	−0.578031	−	0.816015i	$-0.696179\pi$
$11$	555.924	−	962.889i	0.417674	−	0.723432i	0.995705	+	0.0925823i	$0.0295121\pi$
$12$		0			0
$13$			706.517i			0.321583i	0.986988	+	0.160791i	$0.0514046\pi$
$13$			706.517i			0.321583i	−0.986988	+	0.160791i	$0.948595\pi$
$14$		0			0
$15$	−7952.48			−2.35629
$16$		0			0
$17$	−7334.73	−	4234.71i	−1.49292	−	0.861939i	−0.492955	−	0.870055i	$-0.664083\pi$
$17$	−7334.73	−	4234.71i	−1.49292	−	0.861939i	−0.999967	+	0.00811595i	$0.997417\pi$
$18$		0			0
$19$	−22.0061	+	12.7052i	−0.00320835	+	0.00185234i	−0.501603	−	0.865098i	$-0.667256\pi$
$19$	−22.0061	+	12.7052i	−0.00320835	+	0.00185234i	0.498395	+	0.866950i	$0.333923\pi$
$20$		0			0
$21$	−11818.9	+	8487.36i	−1.27620	+	0.916463i
$22$		0			0
$23$	−424.738	−	735.668i	−0.0349090	−	0.0604642i	0.848043	−	0.529927i	$-0.177781\pi$
$23$	−424.738	−	735.668i	−0.0349090	−	0.0604642i	−0.882952	+	0.469463i	$0.844447\pi$
$24$		0			0
$25$	9758.49	−	16902.2i	0.624543	−	1.08174i
$26$		0			0
$27$			14491.7i			0.736255i
$28$		0			0
$29$	−15109.4			−0.619518			−0.309759	−	0.950815i	$-0.600248\pi$
$29$	−15109.4			−0.619518			−0.309759	+	0.950815i	$0.600248\pi$
$30$		0			0
$31$	1205.81	+	696.173i	0.0404755	+	0.0233686i	0.520101	−	0.854105i	$-0.325894\pi$
$31$	1205.81	+	696.173i	0.0404755	+	0.0233686i	−0.479626	+	0.877473i	$0.659228\pi$
$32$		0			0
$33$	40847.5	−	23583.3i	1.13664	−	0.656240i
$34$		0			0
$35$	−6366.92	−	63983.4i	−0.148500	−	1.49233i
$36$		0			0
$37$	4646.47	+	8047.92i	0.0917313	+	0.158883i	0.908240	−	0.418450i	$-0.137427\pi$
$37$	4646.47	+	8047.92i	0.0917313	+	0.158883i	−0.816508	+	0.577334i	$0.804093\pi$
$38$		0			0
$39$	−14985.9	+	25956.3i	−0.252632	+	0.437571i
$40$		0			0
$41$			109829.i			1.59356i	0.604272	+	0.796778i	$0.293464\pi$
$41$			109829.i			1.59356i	−0.604272	+	0.796778i	$0.706536\pi$
$42$		0			0
$43$	45569.8			0.573155			0.286577	−	0.958057i	$-0.407482\pi$
$43$	45569.8			0.573155			0.286577	+	0.958057i	$0.407482\pi$
$44$		0			0
$45$	−173810.	−	100349.i	−1.90738	−	1.10123i
$46$		0			0
$47$	−119865.	+	69204.2i	−1.15451	+	0.666559i	−0.949983	−	0.312301i	$-0.898900\pi$
$47$	−119865.	+	69204.2i	−1.15451	+	0.666559i	−0.204531	+	0.978860i	$0.565567\pi$
$48$		0			0
$49$	−77749.5	−	88296.7i	−0.660860	−	0.750510i
$50$		0			0
$51$	−179644.	−	311152.i	−1.35426	−	2.34565i
$52$		0			0
$53$	−30937.9	+	53586.1i	−0.207809	+	0.359935i	−0.951024	−	0.309117i	$-0.899966\pi$
$53$	−30937.9	+	53586.1i	−0.207809	+	0.359935i	0.743215	+	0.669052i	$0.233300\pi$
$54$		0			0
$55$			208429.i			1.25277i
$56$		0			0
$57$	−1077.96			−0.00582073
$58$		0			0
$59$	−68015.2	−	39268.6i	−0.331169	−	0.191201i	0.325191	−	0.945648i	$-0.394571\pi$
$59$	−68015.2	−	39268.6i	−0.331169	−	0.191201i	−0.656360	+	0.754448i	$0.727905\pi$
$60$		0			0
$61$	−151296.	+	87350.5i	−0.666556	+	0.384836i	−0.794770	−	0.606910i	$-0.792409\pi$
$61$	−151296.	+	87350.5i	−0.666556	+	0.384836i	0.128214	+	0.991746i	$0.459075\pi$
$62$		0			0
$63$	−365415.	+	36362.0i	−1.46138	+	0.145421i
$64$		0			0
$65$	−66222.6	−	114701.i	−0.241138	−	0.417664i
$66$		0			0
$67$	−217974.	+	377543.i	−0.724738	+	1.25528i	0.234344	+	0.972154i	$0.424706\pi$
$67$	−217974.	+	377543.i	−0.724738	+	1.25528i	−0.959082	+	0.283129i	$0.908627\pi$
$68$		0			0
$69$		−	36036.3i		−	0.109697i
$70$		0			0
$71$	561559.			1.56899			0.784495	−	0.620135i	$-0.212922\pi$
$71$	561559.			1.56899			0.784495	+	0.620135i	$0.212922\pi$
$72$		0			0
$73$	−192170.	−	110949.i	−0.493988	−	0.285204i	0.232239	−	0.972659i	$-0.425395\pi$
$73$	−192170.	−	110949.i	−0.493988	−	0.285204i	−0.726227	+	0.687454i	$0.758728\pi$
$74$		0			0
$75$	717022.	−	413973.i	1.69961	−	0.981269i
$76$		0			0
$77$	222448.	+	309766.i	0.487255	+	0.678519i
$78$		0			0
$79$	333239.	+	577187.i	0.675888	+	1.17067i	0.976208	+	0.216834i	$0.0695731\pi$
$79$	333239.	+	577187.i	0.675888	+	1.17067i	−0.300320	+	0.953838i	$0.597094\pi$
$80$		0			0
$81$	82854.9	−	143509.i	0.155906	−	0.270037i
$82$		0			0
$83$		−	653635.i		−	1.14314i	−0.820552	−	0.571572i	$-0.806334\pi$
$83$		−	653635.i		−	1.14314i	0.820552	−	0.571572i	$-0.193666\pi$
$84$		0			0
$85$	1.58769e6			2.58529
$86$		0			0
$87$	−555096.	−	320485.i	−0.842966	−	0.486687i
$88$		0			0
$89$	114603.	−	66165.9i	0.162564	−	0.0938565i	−0.416511	−	0.909131i	$-0.636747\pi$
$89$	114603.	−	66165.9i	0.162564	−	0.0938565i	0.579075	+	0.815274i	$0.303414\pi$
$90$		0			0
$91$	−220835.	−	99791.1i	−0.293052	−	0.132424i
$92$		0			0
$93$	29532.9	+	51152.5i	0.0367162	+	0.0635943i
$94$		0			0
$95$	2381.75	−	4125.31i	0.00277795	−	0.00481156i
$96$		0			0
$97$			1.59480e6i			1.74739i	0.486470	+	0.873697i	$0.338284\pi$
$97$			1.59480e6i			1.74739i	−0.486470	+	0.873697i	$0.661716\pi$
$98$		0			0
$99$	1.19036e6			1.22679
$100$		0			0
$101$	−1.12882e6	−	651726.i	−1.09562	−	0.632559i	−0.160556	−	0.987027i	$-0.551329\pi$
$101$	−1.12882e6	−	651726.i	−1.09562	−	0.632559i	−0.935068	+	0.354468i	$0.884662\pi$
$102$		0			0
$103$	5747.87	−	3318.53i	0.00526011	−	0.00303693i	−0.497368	−	0.867540i	$-0.665700\pi$
$103$	5747.87	−	3318.53i	0.00526011	−	0.00303693i	0.502628	+	0.864503i	$0.332367\pi$
$104$		0			0
$105$	1.12324e6	−	2.48570e6i	0.970294	−	2.14724i
$106$		0			0
$107$	655988.	+	1.13620e6i	0.535482	+	0.927481i	0.999140	+	0.0414673i	$0.0132032\pi$
$107$	655988.	+	1.13620e6i	0.535482	+	0.927481i	−0.463658	+	0.886014i	$0.653463\pi$
$108$		0			0
$109$	−771304.	+	1.33594e6i	−0.595589	+	1.03159i	0.397875	+	0.917440i	$0.369748\pi$
$109$	−771304.	+	1.33594e6i	−0.595589	+	1.03159i	−0.993464	+	0.114150i	$0.963586\pi$
$110$		0			0
$111$			394223.i			0.288253i
$112$		0			0
$113$	1.02212e6			0.708382			0.354191	−	0.935173i	$-0.384756\pi$
$113$	1.02212e6			0.708382			0.354191	+	0.935173i	$0.384756\pi$
$114$		0			0
$115$	137910.	+	79622.2i	0.0906779	+	0.0523529i
$116$		0			0
$117$	−655066.	+	378202.i	−0.409004	+	0.236138i
$118$		0			0
$119$	2.35962e6	−	1.69448e6i	1.40024	−	1.00553i
$120$		0			0
$121$	267678.	+	463631.i	0.151097	+	0.261708i
$122$		0			0
$123$	−2.32958e6	+	4.03495e6i	−1.25188	+	2.16832i
$124$		0			0
$125$			729599.i			0.373554i
$126$		0			0
$127$	−752429.			−0.367329			−0.183664	−	0.982989i	$-0.558796\pi$
$127$	−752429.			−0.367329			−0.183664	+	0.982989i	$0.558796\pi$
$128$		0			0
$129$	1.67416e6	+	966577.i	0.779880	+	0.450264i
$130$		0			0
$131$	−226468.	+	130752.i	−0.100738	+	0.0581611i	−0.549523	−	0.835479i	$-0.685190\pi$
$131$	−226468.	+	130752.i	−0.100738	+	0.0581611i	0.448785	+	0.893640i	$0.351857\pi$
$132$		0			0
$133$	−863.036	−	8672.95i	−0.000366838	−	0.00368648i
$134$		0			0
$135$	−1.35832e6	−	2.35268e6i	−0.552080	−	0.956230i
$136$		0			0
$137$	349472.	−	605303.i	0.135910	−	0.235402i	−0.790035	−	0.613062i	$-0.789938\pi$
$137$	349472.	−	605303.i	0.135910	−	0.235402i	0.925945	+	0.377659i	$0.123271\pi$
$138$		0			0
$139$			1.60298e6i			0.596875i	0.954429	+	0.298438i	$0.0964655\pi$
$139$			1.60298e6i			0.596875i	−0.954429	+	0.298438i	$0.903535\pi$
$140$		0			0
$141$	−5.87153e6			−2.09457
$142$		0			0
$143$	680297.	+	392770.i	0.232643	+	0.134317i
$144$		0			0
$145$	2.45297e6	−	1.41622e6i	0.804614	−	0.464544i
$146$		0			0
$147$	−983535.	−	4.89301e6i	−0.309626	−	1.54037i
$148$		0			0
$149$	2.08601e6	+	3.61308e6i	0.630607	+	1.09224i	0.987428	+	0.158070i	$0.0505272\pi$
$149$	2.08601e6	+	3.61308e6i	0.630607	+	1.09224i	−0.356821	+	0.934173i	$0.616139\pi$
$150$		0			0
$151$	3.10233e6	−	5.37339e6i	0.901067	−	1.56069i	0.0749560	−	0.997187i	$-0.476118\pi$
$151$	3.10233e6	−	5.37339e6i	0.901067	−	1.56069i	0.826111	−	0.563507i	$-0.190548\pi$
$152$		0			0
$153$		−	9.06744e6i		−	2.53169i
$154$		0			0
$155$	−261012.			−0.0700914
$156$		0			0
$157$	−1.70467e6	−	984193.i	−0.440496	−	0.254320i	0.263312	−	0.964711i	$-0.415185\pi$
$157$	−1.70467e6	−	984193.i	−0.440496	−	0.254320i	−0.703808	+	0.710390i	$0.748518\pi$
$158$		0			0
$159$	−2.27322e6	+	1.31244e6i	−0.565522	+	0.326504i
$160$		0			0
$161$	289938.	−	28851.4i	0.0694749	−	0.00691337i
$162$		0			0
$163$	609352.	+	1.05543e6i	0.140704	+	0.243706i	0.927762	−	0.373173i	$-0.121730\pi$
$163$	609352.	+	1.05543e6i	0.140704	+	0.243706i	−0.787058	+	0.616879i	$0.788397\pi$
$164$		0			0
$165$	−4.42097e6	+	7.65735e6i	−0.984160	+	1.70462i
$166$		0			0
$167$			638229.i			0.137034i	0.997650	+	0.0685168i	$0.0218267\pi$
$167$			638229.i			0.137034i	−0.997650	+	0.0685168i	$0.978173\pi$
$168$		0			0
$169$	4.32764e6			0.896585
$170$		0			0
$171$	−23560.0	−	13602.3i	−0.00471179	−	0.00272035i
$172$		0			0
$173$	538615.	−	310969.i	0.104026	−	0.0600592i	−0.447085	−	0.894492i	$-0.647538\pi$
$173$	538615.	−	310969.i	0.104026	−	0.0600592i	0.551110	+	0.834432i	$0.314204\pi$
$174$		0			0
$175$	3.90478e6	+	5.43753e6i	0.728588	+	1.01458i
$176$		0			0
$177$	−1.66585e6	−	2.88533e6i	−0.300410	−	0.520326i
$178$		0			0
$179$	−2.76061e6	+	4.78152e6i	−0.481334	+	0.833695i	−0.999771	−	0.0214210i	$-0.993181\pi$
$179$	−2.76061e6	+	4.78152e6i	−0.481334	+	0.833695i	0.518436	+	0.855116i	$0.326514\pi$
$180$		0			0
$181$			3.79950e6i			0.640753i	0.947290	+	0.320377i	$0.103809\pi$
$181$			3.79950e6i			0.640753i	−0.947290	+	0.320377i	$0.896191\pi$
$182$		0			0
$183$	−7.41114e6			−1.20929
$184$		0			0
$185$	−1.50868e6	−	871036.i	−0.238277	−	0.137569i
$186$		0			0
$187$	−8.15510e6	+	4.70835e6i	−1.24711	+	0.720019i
$188$		0			0
$189$	−4.52966e6	−	2.04686e6i	−0.670934	−	0.303182i
$190$		0			0
$191$	−2.60562e6	−	4.51307e6i	−0.373948	−	0.647697i	0.616221	−	0.787573i	$-0.288663\pi$
$191$	−2.60562e6	−	4.51307e6i	−0.373948	−	0.647697i	−0.990169	+	0.139876i	$0.955330\pi$
$192$		0			0
$193$	1.81264e6	−	3.13958e6i	0.252138	−	0.436716i	−0.711976	−	0.702204i	$-0.752200\pi$
$193$	1.81264e6	−	3.13958e6i	0.252138	−	0.436716i	0.964114	+	0.265487i	$0.0855329\pi$
$194$		0			0
$195$		−	5.61856e6i		−	0.757742i
$196$		0			0
$197$	−6.36761e6			−0.832871			−0.416435	−	0.909165i	$-0.636721\pi$
$197$	−6.36761e6			−0.832871			−0.416435	+	0.909165i	$0.636721\pi$
$198$		0			0
$199$	−3.64034e6	−	2.10175e6i	−0.461937	−	0.266699i	0.250921	−	0.968007i	$-0.419266\pi$
$199$	−3.64034e6	−	2.10175e6i	−0.461937	−	0.266699i	−0.712858	+	0.701308i	$0.752600\pi$
$200$		0			0
$201$	−1.60160e7	+	9.24687e6i	−1.97227	+	1.13869i
$202$		0			0
$203$	2.13411e6	−	4.72274e6i	0.255111	−	0.564554i
$204$		0			0
$205$	−1.02944e7	−	1.78305e7i	−1.19492	−	2.06967i
$206$		0			0
$207$	454729.	−	787613.i	0.0512674	−	0.0887977i
$208$		0			0
$209$			28252.6i			0.00309470i
$210$		0			0
$211$	1.33548e7			1.42164			0.710822	−	0.703372i	$-0.248323\pi$
$211$	1.33548e7			1.42164			0.710822	+	0.703372i	$0.248323\pi$
$212$		0			0
$213$	2.06307e7	+	1.19112e7i	2.13489	+	1.23258i
$214$		0			0
$215$	−7.39811e6	+	4.27130e6i	−0.744398	+	0.429779i
$216$		0			0
$217$	−387914.	+	278567.i	−0.0379627	+	0.0272616i
$218$		0			0
$219$	−4.70667e6	−	8.15219e6i	−0.448107	−	0.776143i
$220$		0			0
$221$	2.99189e6	−	5.18211e6i	0.277185	−	0.480098i
$222$		0			0
$223$			6.98597e6i			0.629959i	0.949098	+	0.314980i	$0.101998\pi$
$223$			6.98597e6i			0.629959i	−0.949098	+	0.314980i	$0.898002\pi$
$224$		0			0
$225$	2.08951e7			1.83441
$226$		0			0
$227$	416145.	+	240261.i	0.0355768	+	0.0205403i	0.517683	−	0.855573i	$-0.326795\pi$
$227$	416145.	+	240261.i	0.0355768	+	0.0205403i	−0.482106	+	0.876113i	$0.660128\pi$
$228$		0			0
$229$	−9.92126e6	+	5.72804e6i	−0.826153	+	0.476980i	−0.852534	−	0.522672i	$-0.824935\pi$
$229$	−9.92126e6	+	5.72804e6i	−0.826153	+	0.476980i	0.0263807	+	0.999652i	$0.491602\pi$
$230$		0			0
$231$	1.60196e6	+	1.60986e7i	0.129962	+	1.30603i
$232$		0			0
$233$	−7.16013e6	−	1.24017e7i	−0.566048	−	0.980424i	−0.996951	−	0.0780261i	$-0.975138\pi$
$233$	−7.16013e6	−	1.24017e7i	−0.566048	−	0.980424i	0.430903	−	0.902398i	$-0.358195\pi$
$234$		0			0
$235$	1.29732e7	−	2.24702e7i	0.999636	−	1.73142i
$236$		0			0
$237$			2.82732e7i			2.12388i
$238$		0			0
$239$	−9.15640e6			−0.670704			−0.335352	−	0.942093i	$-0.608855\pi$
$239$	−9.15640e6			−0.670704			−0.335352	+	0.942093i	$0.608855\pi$
$240$		0			0
$241$	2.15763e7	+	1.24571e7i	1.54144	+	0.889951i	0.998748	+	0.0500197i	$0.0159284\pi$
$241$	2.15763e7	+	1.24571e7i	1.54144	+	0.889951i	0.542692	+	0.839932i	$0.317405\pi$
$242$		0			0
$243$	1.52370e7	−	8.79709e6i	1.06189	−	0.613084i
$244$		0			0
$245$	2.08985e7	+	7.04716e6i	1.42108	+	0.479199i
$246$		0			0
$247$	−8976.47	−	15547.7i	−0.000595682	−	0.00103175i
$248$		0			0
$249$	1.38642e7	−	2.40135e7i	0.898042	−	1.55545i
$250$		0			0
$251$			2.06023e7i			1.30285i	0.758714	+	0.651424i	$0.225828\pi$
$251$			2.06023e7i			1.30285i	−0.758714	+	0.651424i	$0.774172\pi$
$252$		0			0
$253$	−944488.			−0.0583223
$254$		0			0
$255$	5.83292e7	+	3.36764e7i	3.51776	+	2.03098i
$256$		0			0
$257$	1.07717e7	−	6.21904e6i	0.634577	−	0.366373i	−0.147945	−	0.988996i	$-0.547266\pi$
$257$	1.07717e7	−	6.21904e6i	0.634577	−	0.366373i	0.782523	+	0.622622i	$0.213933\pi$
$258$		0			0
$259$	−3.17181e6	+	315623.i	−0.182561	+	0.0181664i
$260$		0			0
$261$	−8.08815e6	−	1.40091e7i	−0.454912	−	0.787932i
$262$		0			0
$263$	1.42368e7	−	2.46588e7i	0.782609	−	1.35552i	−0.147808	−	0.989016i	$-0.547222\pi$
$263$	1.42368e7	−	2.46588e7i	0.782609	−	1.35552i	0.930417	−	0.366502i	$-0.119445\pi$
$264$		0			0
$265$		−	1.15994e7i		−	0.623299i
$266$		0			0
$267$	5.61375e6			0.294930
$268$		0			0
$269$	−1.26875e7	−	7.32511e6i	−0.651805	−	0.376320i	0.137342	−	0.990524i	$-0.456144\pi$
$269$	−1.26875e7	−	7.32511e6i	−0.651805	−	0.376320i	−0.789148	+	0.614204i	$0.789477\pi$
$270$		0			0
$271$	−1.28578e7	+	7.42345e6i	−0.646039	+	0.372991i	−0.786937	−	0.617033i	$-0.788334\pi$
$271$	−1.28578e7	+	7.42345e6i	−0.646039	+	0.372991i	0.140898	+	0.990024i	$0.455001\pi$
$272$		0			0
$273$	−5.99647e6	−	8.35028e6i	−0.294719	−	0.410405i
$274$		0			0
$275$	−1.08500e7	−	1.87927e7i	−0.521711	−	0.903630i
$276$		0			0
$277$	−1.13152e7	+	1.95985e7i	−0.532381	+	0.922111i	0.466904	+	0.884308i	$0.345369\pi$
$277$	−1.13152e7	+	1.95985e7i	−0.532381	+	0.922111i	−0.999285	+	0.0378029i	$0.987964\pi$
$278$		0			0
$279$			1.49066e6i			0.0686382i
$280$		0			0
$281$	−2.66513e7			−1.20116			−0.600579	−	0.799565i	$-0.705063\pi$
$281$	−2.66513e7			−1.20116			−0.600579	+	0.799565i	$0.705063\pi$
$282$		0			0
$283$	−2.25458e7	−	1.30169e7i	−0.994735	−	0.574310i	−0.0880485	−	0.996116i	$-0.528063\pi$
$283$	−2.25458e7	−	1.30169e7i	−0.994735	−	0.574310i	−0.906686	+	0.421806i	$0.861396\pi$
$284$		0			0
$285$	175003.	−	101038.i	0.00755981	−	0.00436466i
$286$		0			0
$287$	−3.43293e7	−	1.55127e7i	−1.45217	−	0.656209i
$288$		0			0
$289$	2.37967e7	+	4.12171e7i	0.985877	+	1.70759i
$290$		0			0
$291$	−3.38272e7	+	5.85903e7i	−1.37273	+	2.37765i
$292$		0			0
$293$		−	2.40799e6i		−	0.0957310i	−0.998854	−	0.0478655i	$-0.984758\pi$
$293$		−	2.40799e6i		−	0.0957310i	0.998854	−	0.0478655i	$-0.0152419\pi$
$294$		0			0
$295$	1.47227e7			0.573486
$296$		0			0
$297$	1.39539e7	+	8.05629e6i	0.532631	+	0.307515i
$298$		0			0
$299$	519762.	−	300085.i	0.0194442	−	0.0112261i
$300$		0			0
$301$	−6.43644e6	+	1.42437e7i	−0.236019	+	0.522304i
$302$		0			0
$303$	−2.76474e7	−	4.78867e7i	−0.993862	−	1.72142i
$304$		0			0
$305$	1.63749e7	−	2.83622e7i	0.577137	−	0.999631i
$306$		0			0
$307$			1.92678e7i			0.665911i	0.942942	+	0.332956i	$0.108046\pi$
$307$			1.92678e7i			0.665911i	−0.942942	+	0.332956i	$0.891954\pi$
$308$		0			0
$309$	281556.			0.00954311
$310$		0			0
$311$	3.26379e7	+	1.88435e7i	1.08503	+	0.626441i	0.932248	−	0.361819i	$-0.117844\pi$
$311$	3.26379e7	+	1.88435e7i	1.08503	+	0.626441i	0.152780	+	0.988260i	$0.451177\pi$
$312$		0			0
$313$	−3.47639e7	+	2.00710e7i	−1.13369	+	0.654539i	−0.944861	−	0.327470i	$-0.893804\pi$
$313$	−3.47639e7	+	2.00710e7i	−1.13369	+	0.654539i	−0.188833	+	0.982009i	$0.560470\pi$
$314$		0			0
$315$	5.59156e7	−	4.01539e7i	1.78896	−	1.28468i
$316$		0			0
$317$	−2.67147e7	−	4.62713e7i	−0.838636	−	1.45256i	−0.891036	−	0.453933i	$-0.850020\pi$
$317$	−2.67147e7	−	4.62713e7i	−0.838636	−	1.45256i	0.0524001	−	0.998626i	$-0.483313\pi$
$318$		0			0
$319$	−8.39969e6	+	1.45487e7i	−0.258757	+	0.448180i
$320$		0			0
$321$			5.56564e7i			1.68267i
$322$		0			0
$323$	215212.			0.00638643
$324$		0			0
$325$	1.19417e7	+	6.89454e6i	0.347869	+	0.200842i
$326$		0			0
$327$	−5.66729e7	+	3.27201e7i	−1.62081	+	0.935776i
$328$		0			0
$329$	−4.70087e6	−	4.72408e7i	−0.132005	−	1.32657i
$330$		0			0
$331$	1.04651e7	+	1.81261e7i	0.288575	+	0.499827i	0.973470	−	0.228815i	$-0.0734852\pi$
$331$	1.04651e7	+	1.81261e7i	0.288575	+	0.499827i	−0.684895	+	0.728642i	$0.740152\pi$
$332$		0			0
$333$	−4.97455e6	+	8.61618e6i	−0.134717	+	0.233336i
$334$		0			0
$335$		−	8.17238e7i		−	2.17377i
$336$		0			0
$337$	5.12346e6			0.133867			0.0669335	−	0.997757i	$-0.478678\pi$
$337$	5.12346e6			0.133867			0.0669335	+	0.997757i	$0.478678\pi$
$338$		0			0
$339$	3.75511e7	+	2.16802e7i	0.963882	+	0.556497i
$340$		0			0
$341$	1.34067e6	−	774038.i	0.0338111	−	0.0195209i
$342$		0			0
$343$	3.85804e7	−	1.18307e7i	0.956059	−	0.293176i
$344$		0			0
$345$	3.37772e6	+	5.85038e6i	0.0822557	+	0.142471i
$346$		0			0
$347$	667759.	−	1.15659e6i	0.0159820	−	0.0276817i	−0.857924	−	0.513777i	$-0.828246\pi$
$347$	667759.	−	1.15659e6i	0.0159820	−	0.0276817i	0.873906	+	0.486095i	$0.161579\pi$
$348$		0			0
$349$		−	5.85989e7i		−	1.37852i	−0.724514	−	0.689260i	$-0.757936\pi$
$349$		−	5.85989e7i		−	1.37852i	0.724514	−	0.689260i	$-0.242064\pi$
$350$		0			0
$351$	−1.02386e7			−0.236767
$352$		0			0
$353$	−5.03777e7	−	2.90856e7i	−1.14529	−	0.661231i	−0.197552	−	0.980292i	$-0.563299\pi$
$353$	−5.03777e7	−	2.90856e7i	−1.14529	−	0.661231i	−0.947734	+	0.319061i	$0.896633\pi$
$354$		0			0
$355$	−9.11673e7	+	5.26354e7i	−2.03776	+	1.17650i
$356$		0			0
$357$	1.22630e8	−	1.22028e7i	2.69521	−	0.268197i
$358$		0			0
$359$	−2.79159e7	−	4.83518e7i	−0.603350	−	1.04503i	−0.992310	−	0.123778i	$-0.960499\pi$
$359$	−2.79159e7	−	4.83518e7i	−0.603350	−	1.04503i	0.388960	−	0.921254i	$-0.372834\pi$
$360$		0			0
$361$	−2.35226e7	+	4.07424e7i	−0.499993	+	0.866014i
$362$		0			0
$363$			2.27107e7i			0.474801i
$364$		0			0
$365$	4.15975e7			0.855439
$366$		0			0
$367$	−7.51777e6	−	4.34039e6i	−0.152087	−	0.0878072i	0.422025	−	0.906584i	$-0.361319\pi$
$367$	−7.51777e6	−	4.34039e6i	−0.152087	−	0.0878072i	−0.574112	+	0.818777i	$0.694653\pi$
$368$		0			0
$369$	−1.01831e8	+	5.87923e7i	−2.02676	+	1.17015i
$370$		0			0
$371$	−1.23795e7	−	1.72389e7i	−0.242428	−	0.337589i
$372$		0			0
$373$	−3.44474e7	−	5.96646e7i	−0.663789	−	1.14972i	−0.979612	−	0.200898i	$-0.935614\pi$
$373$	−3.44474e7	−	5.96646e7i	−0.663789	−	1.14972i	0.315823	−	0.948818i	$-0.397719\pi$
$374$		0			0
$375$	−1.54754e7	+	2.68043e7i	−0.293460	+	0.508288i
$376$		0			0
$377$		−	1.06751e7i		−	0.199226i
$378$		0			0
$379$	−2.82976e7			−0.519795			−0.259897	−	0.965636i	$-0.583689\pi$
$379$	−2.82976e7			−0.519795			−0.259897	+	0.965636i	$0.583689\pi$
$380$		0			0
$381$	−2.76430e7	−	1.59597e7i	−0.499817	−	0.288569i
$382$		0			0
$383$	9.34416e7	−	5.39485e7i	1.66320	−	0.960248i	0.692025	−	0.721874i	$-0.256719\pi$
$383$	9.34416e7	−	5.39485e7i	1.66320	−	0.960248i	0.971173	−	0.238374i	$-0.0766144\pi$
$384$		0			0
$385$	−6.51484e7	−	2.94393e7i	−1.14162	−	0.515876i
$386$		0			0
$387$	2.43937e7	+	4.22512e7i	0.420868	+	0.728964i
$388$		0			0
$389$	−3.23559e7	+	5.60420e7i	−0.549673	+	0.952061i	0.448624	+	0.893720i	$0.351914\pi$
$389$	−3.23559e7	+	5.60420e7i	−0.549673	+	0.952061i	−0.998297	+	0.0583403i	$0.981419\pi$
$390$		0			0
$391$			7.19456e6i			0.120358i
$392$		0			0
$393$	−1.10934e7			−0.182763
$394$		0			0
$395$	−1.08201e8	−	6.24697e7i	−1.75565	−	1.01363i
$396$		0			0
$397$	6.55387e7	−	3.78388e7i	1.04743	−	0.604736i	0.125504	−	0.992093i	$-0.459945\pi$
$397$	6.55387e7	−	3.78388e7i	1.04743	−	0.604736i	0.921930	+	0.387357i	$0.126612\pi$
$398$		0			0
$399$	152255.	−	336936.i	0.00239691	−	0.00530431i
$400$		0			0
$401$	−1.55716e6	−	2.69708e6i	−0.0241490	−	0.0418274i	0.853698	−	0.520768i	$-0.174354\pi$
$401$	−1.55716e6	−	2.69708e6i	−0.0241490	−	0.0418274i	−0.877847	+	0.478941i	$0.841021\pi$
$402$		0			0
$403$	−491858.	+	851923.i	−0.00751492	+	0.0130162i
$404$		0			0
$405$			3.10643e7i			0.467624i
$406$		0			0
$407$	1.03323e7			0.153255
$408$		0			0
$409$	1.05021e8	+	6.06337e7i	1.53499	+	0.886225i	0.999121	+	0.0419210i	$0.0133478\pi$
$409$	1.05021e8	+	6.06337e7i	1.53499	+	0.886225i	0.535865	+	0.844304i	$0.319986\pi$
$410$		0			0
$411$	2.56780e7	−	1.48252e7i	0.369859	−	0.213538i
$412$		0			0
$413$	2.18809e7	−	1.57130e7i	0.310609	−	0.223053i
$414$		0			0
$415$	6.12659e7	+	1.06116e8i	0.857184	+	1.48469i
$416$		0			0
$417$	−3.40006e7	+	5.88908e7i	−0.468899	+	0.812156i
$418$		0			0
$419$			1.07947e7i			0.146746i	0.997305	+	0.0733732i	$0.0233764\pi$
$419$			1.07947e7i			0.146746i	−0.997305	+	0.0733732i	$0.976624\pi$
$420$		0			0
$421$	−5.16567e6			−0.0692278			−0.0346139	−	0.999401i	$-0.511020\pi$
$421$	−5.16567e6			−0.0692278			−0.0346139	+	0.999401i	$0.511020\pi$
$422$		0			0
$423$	−1.28329e8	−	7.40907e7i	−1.69552	−	0.978909i
$424$		0			0
$425$	−1.43152e8	+	8.26487e7i	−1.86479	+	1.07664i
$426$		0			0
$427$	−5.93351e6	−	5.96280e7i	−0.0762128	−	0.765890i
$428$		0			0
$429$	1.66620e7	+	2.88595e7i	0.211036	+	0.365524i
$430$		0			0
$431$	−5.74292e7	+	9.94703e7i	−0.717301	+	1.24240i	0.244765	+	0.969582i	$0.421289\pi$
$431$	−5.74292e7	+	9.94703e7i	−0.717301	+	1.24240i	−0.962066	+	0.272818i	$0.912044\pi$
$432$		0			0
$433$		−	2.10551e7i		−	0.259355i	−0.991556	−	0.129677i	$-0.958606\pi$
$433$		−	2.10551e7i		−	0.259355i	0.991556	−	0.129677i	$-0.0413942\pi$
$434$		0			0
$435$	1.20157e8			1.45976
$436$		0			0
$437$	18693.7	+	10792.8i	0.000224001	+	0.000129327i
$438$		0			0
$439$	−5.69615e6	+	3.28867e6i	−0.0673268	+	0.0388711i	−0.533286	−	0.845935i	$-0.679043\pi$
$439$	−5.69615e6	+	3.28867e6i	−0.0673268	+	0.0388711i	0.465959	+	0.884806i	$0.345710\pi$
$440$		0			0
$441$	4.02469e7	−	1.19353e8i	0.469263	−	1.39161i
$442$		0			0
$443$	−3.89607e7	−	6.74819e7i	−0.448142	−	0.776205i	0.550123	−	0.835084i	$-0.314581\pi$
$443$	−3.89607e7	−	6.74819e7i	−0.448142	−	0.776205i	−0.998265	+	0.0588789i	$0.981247\pi$
$444$		0			0
$445$	−1.24036e7	+	2.14836e7i	−0.140756	+	0.243797i
$446$		0			0
$447$			1.76985e8i			1.98159i
$448$		0			0
$449$	1.49956e8			1.65663			0.828316	−	0.560261i	$-0.189299\pi$
$449$	1.49956e8			1.65663			0.828316	+	0.560261i	$0.189299\pi$
$450$		0			0
$451$	1.05754e8	+	6.10568e7i	1.15283	+	0.665587i
$452$		0			0
$453$	2.27949e8	−	1.31606e8i	2.45213	−	1.41574i
$454$		0			0
$455$	4.52054e7	−	4.49834e6i	0.479906	−	0.0477549i
$456$		0			0
$457$	6.34376e7	+	1.09877e8i	0.664658	+	1.15122i	0.979378	+	0.202037i	$0.0647562\pi$
$457$	6.34376e7	+	1.09877e8i	0.664658	+	1.15122i	−0.314719	+	0.949185i	$0.601910\pi$
$458$		0			0
$459$	6.13682e7	−	1.06293e8i	0.634607	−	1.09917i
$460$		0			0
$461$			8.34728e7i			0.852005i	0.904722	+	0.426002i	$0.140078\pi$
$461$			8.34728e7i			0.852005i	−0.904722	+	0.426002i	$0.859922\pi$
$462$		0			0
$463$	1.31072e8			1.32058			0.660292	−	0.751009i	$-0.270433\pi$
$463$	1.31072e8			1.32058			0.660292	+	0.751009i	$0.270433\pi$
$464$		0			0
$465$	−9.58915e6	−	5.53630e6i	−0.0953720	−	0.0550631i
$466$		0			0
$467$	7.53418e7	−	4.34986e7i	0.739751	−	0.427095i	−0.0822280	−	0.996614i	$-0.526204\pi$
$467$	7.53418e7	−	4.34986e7i	0.739751	−	0.427095i	0.821979	+	0.569518i	$0.192870\pi$
$468$		0			0
$469$	−8.72206e7	−	1.21457e8i	−0.845474	−	1.17735i
$470$		0			0
$471$	−4.17512e7	−	7.23153e7i	−0.399583	−	0.692098i
$472$		0			0
$473$	2.53333e7	−	4.38786e7i	0.239392	−	0.414639i
$474$		0			0
$475$			495936.i			0.00462748i
$476$		0			0
$477$	−6.62449e7			−0.610376
$478$		0			0
$479$	1.22097e8	+	7.04928e7i	1.11096	+	0.641413i	0.939078	−	0.343704i	$-0.111682\pi$
$479$	1.22097e8	+	7.04928e7i	1.11096	+	0.641413i	0.171883	+	0.985117i	$0.445015\pi$
$480$		0			0
$481$	−5.68599e6	+	3.28281e6i	−0.0510941	+	0.0294992i
$482$		0			0
$483$	1.12638e7	+	5.08990e6i	0.0999642	+	0.0451718i
$484$		0			0
$485$	−1.49482e8	−	2.58911e8i	−1.31028	−	2.26947i
$486$		0			0
$487$	7.15151e7	−	1.23868e8i	0.619171	−	1.07244i	−0.370466	−	0.928846i	$-0.620802\pi$
$487$	7.15151e7	−	1.23868e8i	0.619171	−	1.07244i	0.989637	−	0.143590i	$-0.0458646\pi$
$488$		0			0
$489$			5.16996e7i			0.442141i
$490$		0			0
$491$	−9.10418e7			−0.769124			−0.384562	−	0.923099i	$-0.625647\pi$
$491$	−9.10418e7			−0.769124			−0.384562	+	0.923099i	$0.625647\pi$
$492$		0			0
$493$	1.10824e8	+	6.39840e7i	0.924892	+	0.533987i
$494$		0			0
$495$	−1.93250e8	+	1.11573e8i	−1.59333	+	0.919908i
$496$		0			0
$497$	−7.93166e7	+	1.75526e8i	−0.646093	+	1.42979i
$498$		0			0
$499$	−2.83073e7	−	4.90296e7i	−0.227822	−	0.394600i	0.729340	−	0.684151i	$-0.239827\pi$
$499$	−2.83073e7	−	4.90296e7i	−0.227822	−	0.394600i	−0.957162	+	0.289551i	$0.906494\pi$
$500$		0			0
$501$	−1.35374e7	+	2.34475e7i	−0.107652	+	0.186459i
$502$		0			0
$503$			5.77667e7i			0.453914i	0.973905	+	0.226957i	$0.0728777\pi$
$503$			5.77667e7i			0.453914i	−0.973905	+	0.226957i	$0.927122\pi$
$504$		0			0
$505$	2.44348e8			1.89729
$506$		0			0
$507$	1.58991e8	+	9.17932e7i	1.21996	+	0.704347i
$508$		0			0
$509$	1.43165e8	−	8.26565e7i	1.08564	−	0.626793i	0.153225	−	0.988191i	$-0.451034\pi$
$509$	1.43165e8	−	8.26565e7i	1.08564	−	0.626793i	0.932411	+	0.361399i	$0.117701\pi$
$510$		0			0
$511$	6.18220e7	−	4.43954e7i	0.463320	−	0.332717i
$512$		0			0
$513$	−184121.	−	318906.i	−0.00136380	−	0.00236217i
$514$		0			0
$515$	−622099.	+	1.07751e6i	−0.00455447	+	0.00788857i
$516$		0			0
$517$			1.53889e8i			1.11362i
$518$		0			0
$519$	2.63838e7			0.188727
$520$		0			0
$521$	1.07568e6	+	621047.i	0.00760627	+	0.00439148i	0.503798	−	0.863821i	$-0.331935\pi$
$521$	1.07568e6	+	621047.i	0.00760627	+	0.00439148i	−0.496192	+	0.868213i	$0.665269\pi$
$522$		0			0
$523$	−3.51923e7	+	2.03183e7i	−0.246004	+	0.142030i	−0.617933	−	0.786231i	$-0.712030\pi$
$523$	−3.51923e7	+	2.03183e7i	−0.246004	+	0.142030i	0.371929	+	0.928261i	$0.378696\pi$
$524$		0			0
$525$	2.81202e7	+	2.82590e8i	0.194330	+	1.95289i
$526$		0			0
$527$	−5.89617e6	−	1.02125e7i	−0.0402845	−	0.0697749i
$528$		0			0
$529$	7.36571e7	−	1.27578e8i	0.497563	−	0.861804i
$530$		0			0
$531$		−	8.40828e7i		−	0.561595i
$532$		0			0
$533$	−7.75964e7			−0.512460
$534$		0			0
$535$	−2.12995e8	−	1.22973e8i	−1.39094	−	0.803060i
$536$		0			0
$537$	−2.02841e8	+	1.17110e8i	−1.30988	+	0.756262i
$538$		0			0
$539$	−1.28243e8	+	2.57778e7i	−0.818967	+	0.164619i
$540$		0			0
$541$	9.78002e7	+	1.69395e8i	0.617658	+	1.06981i	0.989912	+	0.141684i	$0.0452517\pi$
$541$	9.78002e7	+	1.69395e8i	0.617658	+	1.06981i	−0.372254	+	0.928131i	$0.621415\pi$
$542$		0			0
$543$	−8.05908e7	+	1.39587e8i	−0.503369	+	0.871860i
$544$		0			0
$545$		−	2.89180e8i		−	1.78640i
$546$		0			0
$547$	−9.54532e7			−0.583215			−0.291607	−	0.956538i	$-0.594190\pi$
$547$	−9.54532e7			−0.583215			−0.291607	+	0.956538i	$0.594190\pi$
$548$		0			0
$549$	−1.61979e8	−	9.35184e7i	−0.978905	−	0.565171i
$550$		0			0
$551$	332500.	−	191969.i	0.00198763	−	0.00114756i
$552$		0			0
$553$	−2.27479e8	+	2.26361e7i	−1.34513	+	0.133853i
$554$		0			0
$555$	−3.69509e7	−	6.40009e7i	−0.216146	−	0.374375i
$556$		0			0
$557$	6.01127e7	−	1.04118e8i	0.347857	−	0.602506i	−0.638012	−	0.770027i	$-0.720243\pi$
$557$	6.01127e7	−	1.04118e8i	0.347857	−	0.602506i	0.985869	+	0.167521i	$0.0535761\pi$
$558$		0			0
$559$			3.21959e7i			0.184317i
$560$		0			0
$561$	−3.99473e8			−2.26256
$562$		0			0
$563$	1.48035e7	+	8.54678e6i	0.0829542	+	0.0478936i	0.540903	−	0.841085i	$-0.318082\pi$
$563$	1.48035e7	+	8.54678e6i	0.0829542	+	0.0478936i	−0.457949	+	0.888978i	$0.651416\pi$
$564$		0			0
$565$	−1.65938e8	+	9.58046e7i	−0.920029	+	0.531179i
$566$		0			0
$567$	3.31537e7	+	4.61676e7i	0.181879	+	0.253273i
$568$		0			0
$569$	4.12844e7	+	7.15067e7i	0.224104	+	0.388159i	0.956050	−	0.293203i	$-0.0947212\pi$
$569$	4.12844e7	+	7.15067e7i	0.224104	+	0.388159i	−0.731946	+	0.681362i	$0.761388\pi$
$570$		0			0
$571$	1.49235e7	−	2.58483e7i	0.0801609	−	0.138843i	−0.823158	−	0.567812i	$-0.807790\pi$
$571$	1.49235e7	−	2.58483e7i	0.0801609	−	0.138843i	0.903319	+	0.428970i	$0.141123\pi$
$572$		0			0
$573$		−	2.21070e8i		−	1.17508i
$574$		0			0
$575$	−1.65792e7			−0.0872088
$576$		0			0
$577$	−2.57347e8	−	1.48579e8i	−1.33965	−	0.773448i	−0.352896	−	0.935663i	$-0.614803\pi$
$577$	−2.57347e8	−	1.48579e8i	−1.33965	−	0.773448i	−0.986756	+	0.162214i	$0.948136\pi$
$578$		0			0
$579$	1.33187e8	−	7.68954e7i	0.686159	−	0.396154i
$580$		0			0
$581$	2.04306e8	+	9.23218e7i	1.04172	+	0.470734i
$582$		0			0
$583$	3.43983e7	+	5.95795e7i	0.173592	+	0.300671i
$584$		0			0
$585$	7.08985e7	−	1.22800e8i	0.354136	−	0.613381i
$586$		0			0
$587$			3.09361e7i			0.152950i	0.997071	+	0.0764752i	$0.0243666\pi$
$587$			3.09361e7i			0.152950i	−0.997071	+	0.0764752i	$0.975633\pi$
$588$		0			0
$589$	−35380.1			−0.000173146
$590$		0			0
$591$	−2.33935e8	−	1.35063e8i	−1.13327	−	0.654294i
$592$		0			0
$593$	1.93620e8	−	1.11786e8i	0.928508	−	0.536075i	0.0421688	−	0.999110i	$-0.486573\pi$
$593$	1.93620e8	−	1.11786e8i	0.928508	−	0.536075i	0.886339	+	0.463036i	$0.153240\pi$
$594$		0			0
$595$	−2.24251e8	+	4.96263e8i	−1.06459	+	2.35592i
$596$		0			0
$597$	−8.91601e7	−	1.54430e8i	−0.419032	−	0.725785i
$598$		0			0
$599$	1.14788e8	−	1.98818e8i	0.534090	−	0.925071i	−0.465117	−	0.885249i	$-0.653988\pi$
$599$	1.14788e8	−	1.98818e8i	0.534090	−	0.925071i	0.999207	−	0.0398220i	$-0.0126791\pi$
$600$		0			0
$601$			2.45821e8i			1.13239i	0.824272	+	0.566194i	$0.191585\pi$
$601$			2.45821e8i			1.13239i	−0.824272	+	0.566194i	$0.808415\pi$
$602$		0			0
$603$	−4.66731e8			−2.12870
$604$		0			0
$605$	−8.69132e7	−	5.01794e7i	−0.392482	−	0.226600i
$606$		0			0
$607$	1.95506e7	−	1.12875e7i	0.0874164	−	0.0504699i	−0.455655	−	0.890157i	$-0.650595\pi$
$607$	1.95506e7	−	1.12875e7i	0.0874164	−	0.0504699i	0.543071	+	0.839687i	$0.317261\pi$
$608$		0			0
$609$	1.78577e8	−	1.28239e8i	0.790632	−	0.567765i
$610$		0			0
$611$	−4.88940e7	−	8.46868e7i	−0.214354	−	0.371272i
$612$		0			0
$613$	−4.89992e7	+	8.48691e7i	−0.212720	+	0.368441i	−0.952565	−	0.304336i	$-0.901565\pi$
$613$	−4.89992e7	+	8.48691e7i	−0.212720	+	0.368441i	0.739845	+	0.672777i	$0.234899\pi$
$614$		0			0
$615$		−	8.73416e8i		−	3.75488i
$616$		0			0
$617$	1.51949e7			0.0646907			0.0323454	−	0.999477i	$-0.489702\pi$
$617$	1.51949e7			0.0646907			0.0323454	+	0.999477i	$0.489702\pi$
$618$		0			0
$619$	3.56350e8	+	2.05739e8i	1.50247	+	0.867450i	0.999996	+	0.00285722i	$0.000909483\pi$
$619$	3.56350e8	+	2.05739e8i	1.50247	+	0.867450i	0.502472	+	0.864593i	$0.332424\pi$
$620$		0			0
$621$	1.06611e7	−	6.15518e6i	0.0445171	−	0.0257019i
$622$		0			0
$623$	4.49449e6	+	4.51667e7i	0.0185873	+	0.186790i
$624$		0			0
$625$	8.40904e7	+	1.45649e8i	0.344434	+	0.596578i
$626$		0			0
$627$	−599263.	+	1.03795e6i	−0.00243117	+	0.00421090i
$628$		0			0
$629$		−	7.87057e7i		−	0.316267i
$630$		0			0
$631$	3.87782e8			1.54348			0.771738	−	0.635941i	$-0.219388\pi$
$631$	3.87782e8			1.54348			0.771738	+	0.635941i	$0.219388\pi$
$632$		0			0
$633$	4.90635e8	+	2.83268e8i	1.93440	+	1.11683i
$634$		0			0
$635$	1.22155e8	−	7.05259e7i	0.477077	−	0.275440i
$636$		0			0
$637$	6.23832e7	−	5.49313e7i	0.241351	−	0.212521i
$638$		0			0
$639$	3.00605e8	+	5.20663e8i	1.15211	+	1.99551i
$640$		0			0
$641$	7.01490e7	−	1.21502e8i	0.266347	−	0.461326i	−0.701569	−	0.712602i	$-0.747517\pi$
$641$	7.01490e7	−	1.21502e8i	0.266347	−	0.461326i	0.967916	+	0.251276i	$0.0808500\pi$
$642$		0			0
$643$			2.07832e8i			0.781772i	0.920439	+	0.390886i	$0.127831\pi$
$643$			2.07832e8i			0.781772i	−0.920439	+	0.390886i	$0.872169\pi$
$644$		0			0
$645$	−3.62393e8			−1.35052
$646$		0			0
$647$	−6.02345e7	−	3.47764e7i	−0.222399	−	0.128402i	0.384662	−	0.923058i	$-0.374318\pi$
$647$	−6.02345e7	−	3.47764e7i	−0.222399	−	0.128402i	−0.607060	+	0.794656i	$0.707651\pi$
$648$		0			0
$649$	−7.56226e7	+	4.36607e7i	−0.276642	+	0.159719i
$650$		0			0
$651$	−2.01600e7	+	2.00610e6i	−0.0730714	+	0.00727125i
$652$		0			0
$653$	4.98072e7	+	8.62686e7i	0.178876	+	0.309823i	0.941496	−	0.337024i	$-0.109420\pi$
$653$	4.98072e7	+	8.62686e7i	0.178876	+	0.309823i	−0.762620	+	0.646847i	$0.776087\pi$
$654$		0			0
$655$	2.45109e7	−	4.24542e7i	0.0872240	−	0.151076i
$656$		0			0
$657$		−	2.37567e8i		−	0.837702i
$658$		0			0
$659$	−3.37346e8			−1.17874			−0.589372	−	0.807862i	$-0.700625\pi$
$659$	−3.37346e8			−1.17874			−0.589372	+	0.807862i	$0.700625\pi$
$660$		0			0
$661$	3.16534e8	+	1.82751e8i	1.09601	+	0.632784i	0.935171	−	0.354196i	$-0.115246\pi$
$661$	3.16534e8	+	1.82751e8i	1.09601	+	0.632784i	0.160842	+	0.986980i	$0.448579\pi$
$662$		0			0
$663$	2.19835e8	−	1.26922e8i	0.754319	−	0.435507i
$664$		0			0
$665$	953036.	+	1.32713e6i	0.00324074	+	0.00451284i
$666$		0			0
$667$	6.41755e6	+	1.11155e7i	0.0216268	+	0.0374587i
$668$		0			0
$669$	−1.48179e8	+	2.56653e8i	−0.494889	+	0.857173i
$670$		0			0
$671$			1.94241e8i			0.642944i
$672$		0			0
$673$	−4.77356e8			−1.56602			−0.783010	−	0.622008i	$-0.786317\pi$
$673$	−4.77356e8			−1.56602			−0.783010	+	0.622008i	$0.786317\pi$
$674$		0			0
$675$	2.44942e8	+	1.41417e8i	0.796438	+	0.459823i
$676$		0			0
$677$	−4.89050e8	+	2.82353e8i	−1.57611	+	0.909969i	−0.580718	+	0.814105i	$0.697228\pi$
$677$	−4.89050e8	+	2.82353e8i	−1.57611	+	0.909969i	−0.995394	+	0.0958642i	$0.969439\pi$
$678$		0			0
$679$	−4.98485e8	−	2.25255e8i	−1.59236	−	0.719558i
$680$		0			0
$681$	1.01923e7	+	1.76536e7i	0.0322724	+	0.0558975i
$682$		0			0
$683$	2.53875e7	−	4.39724e7i	0.0796814	−	0.138012i	−0.823431	−	0.567416i	$-0.807943\pi$
$683$	2.53875e7	−	4.39724e7i	0.0796814	−	0.138012i	0.903112	+	0.429404i	$0.141276\pi$
$684$		0			0
$685$			1.31025e8i			0.407646i
$686$		0			0
$687$	−4.85988e8			−1.49884
$688$		0			0
$689$	−3.78595e7	−	2.18582e7i	−0.115749	−	0.0668277i
$690$		0			0
$691$	−1.50527e8	+	8.69068e7i	−0.456226	+	0.263402i	−0.710456	−	0.703741i	$-0.751511\pi$
$691$	−1.50527e8	+	8.69068e7i	−0.456226	+	0.263402i	0.254230	+	0.967144i	$0.418178\pi$
$692$		0			0
$693$	−1.68130e8	+	3.72068e8i	−0.505180	+	1.11795i
$694$		0			0
$695$	−1.50249e8	−	2.60239e8i	−0.447566	−	0.775206i
$696$		0			0
$697$	4.65095e8	−	8.05569e8i	1.37355	−	2.37905i
$698$		0			0
$699$		−	6.07492e8i		−	1.77873i
$700$		0			0
$701$	1.13297e7			0.0328899			0.0164450	−	0.999865i	$-0.494765\pi$
$701$	1.13297e7			0.0328899			0.0164450	+	0.999865i	$0.494765\pi$
$702$		0			0
$703$	−204501.	−	118069.i	−0.000588613	−	0.000339836i
$704$		0			0
$705$	9.53225e8	−	5.50345e8i	2.72037	−	1.57061i
$706$		0			0
$707$	3.63148e8	−	2.60782e8i	1.02760	−	0.737938i
$708$		0			0
$709$	−9.56188e7	−	1.65617e8i	−0.268290	−	0.464692i	0.700130	−	0.714015i	$-0.253125\pi$
$709$	−9.56188e7	−	1.65617e8i	−0.268290	−	0.464692i	−0.968420	+	0.249323i	$0.919792\pi$
$710$		0			0
$711$	−3.56769e8	+	6.17942e8i	−0.992610	+	1.71925i
$712$		0			0
$713$		−	1.18276e6i		−	0.00326309i
$714$		0			0
$715$	−1.47259e8			−0.402868
$716$		0			0
$717$	−3.36391e8	−	1.94216e8i	−0.912614	−	0.526898i
$718$		0			0
$719$	2.80196e8	−	1.61771e8i	0.753832	−	0.435225i	−0.0732447	−	0.997314i	$-0.523335\pi$
$719$	2.80196e8	−	1.61771e8i	0.753832	−	0.435225i	0.827077	+	0.562089i	$0.190002\pi$
$720$		0			0
$721$	225420.	+	2.26533e6i	0.000601432	+	0.00604401i
$722$		0			0
$723$	5.28453e8	+	9.15308e8i	1.39827	+	2.42188i
$724$		0			0
$725$	−1.47445e8	+	2.55383e8i	−0.386916	+	0.670158i
$726$		0			0
$727$			1.14232e8i			0.297293i	0.988890	+	0.148646i	$0.0474916\pi$
$727$			1.14232e8i			0.297293i	−0.988890	+	0.148646i	$0.952508\pi$
$728$		0			0
$729$	6.25574e8			1.61472
$730$		0			0
$731$	−3.34242e8	−	1.92975e8i	−0.855675	−	0.494024i
$732$		0			0
$733$	−1.17409e8	+	6.77861e7i	−0.298119	+	0.172119i	−0.641597	−	0.767042i	$-0.721728\pi$
$733$	−1.17409e8	+	6.77861e7i	−0.298119	+	0.172119i	0.343479	+	0.939160i	$0.388395\pi$
$734$		0			0
$735$	6.18301e8	+	7.02177e8i	1.55718	+	1.76842i
$736$		0			0
$737$	2.42354e8	+	4.19770e8i	0.605408	+	1.04860i
$738$		0			0
$739$	3.16427e8	−	5.48068e8i	0.784044	−	1.35800i	−0.145524	−	0.989355i	$-0.546487\pi$
$739$	3.16427e8	−	5.48068e8i	0.784044	−	1.35800i	0.929568	−	0.368650i	$-0.120180\pi$
$740$		0			0
$741$		−	761596.i		−	0.00187185i
$742$		0			0
$743$	2.81712e8			0.686814			0.343407	−	0.939187i	$-0.388419\pi$
$743$	2.81712e8			0.686814			0.343407	+	0.939187i	$0.388419\pi$
$744$		0			0
$745$	−6.77316e8	−	3.91048e8i	−1.63803	−	0.945718i
$746$		0			0
$747$	6.06035e8	−	3.49894e8i	1.45390	−	0.839412i
$748$		0			0
$749$	−4.47796e8	+	4.45597e7i	−1.06570	+	0.106047i
$750$		0			0
$751$	3.04177e7	+	5.26850e7i	0.0718136	+	0.124385i	0.899696	−	0.436517i	$-0.143788\pi$
$751$	3.04177e7	+	5.26850e7i	0.0718136	+	0.124385i	−0.827883	+	0.560901i	$0.810455\pi$
$752$		0			0
$753$	−4.36993e8	+	7.56894e8i	−1.02350	+	1.77276i
$754$		0			0
$755$			1.16314e9i			2.70265i
$756$		0			0
$757$	6.73226e8			1.55193			0.775967	−	0.630774i	$-0.217262\pi$
$757$	6.73226e8			1.55193			0.775967	+	0.630774i	$0.217262\pi$
$758$		0			0
$759$	−3.46989e7	−	2.00334e7i	−0.0793580	−	0.0458174i
$760$		0			0
$761$	6.88035e8	−	3.97237e8i	1.56119	−	0.901355i	0.564055	−	0.825737i	$-0.309241\pi$
$761$	6.88035e8	−	3.97237e8i	1.56119	−	0.901355i	0.997137	−	0.0756172i	$-0.0240927\pi$
$762$		0			0
$763$	−3.08631e8	−	4.29778e8i	−0.694809	−	0.967545i
$764$		0			0
$765$	8.49900e8	+	1.47207e9i	1.89838	+	3.28809i
$766$		0			0
$767$	2.77440e7	−	4.80540e7i	0.0614869	−	0.106498i
$768$		0			0
$769$		−	8.61161e7i		−	0.189367i	−0.995507	−	0.0946837i	$-0.969816\pi$
$769$		−	8.61161e7i		−	0.189367i	0.995507	−	0.0946837i	$-0.0301840\pi$
$770$		0			0
$771$	5.27646e8			1.15128
$772$		0			0
$773$	−4.87520e8	−	2.81470e8i	−1.05549	−	0.609387i	−0.131308	−	0.991342i	$-0.541918\pi$
$773$	−4.87520e8	−	2.81470e8i	−1.05549	−	0.609387i	−0.924181	+	0.381954i	$0.875251\pi$
$774$		0			0
$775$	2.35337e7	−	1.35872e7i	0.0505574	−	0.0291894i
$776$		0			0
$777$	−1.23222e8	−	5.56815e7i	−0.262679	−	0.118699i
$778$		0			0
$779$	−1.39541e6	−	2.41692e6i	−0.00295181	−	0.00511269i
$780$		0			0
$781$	3.12184e8	−	5.40718e8i	0.655326	−	1.13506i
$782$		0			0
$783$		−	2.18962e8i		−	0.456124i
$784$		0			0
$785$	3.68997e8			0.762806
$786$		0			0
$787$	−7.72741e8	−	4.46142e8i	−1.58529	−	0.915269i	−0.994068	−	0.108763i	$-0.965311\pi$
$787$	−7.72741e8	−	4.46142e8i	−1.58529	−	0.915269i	−0.591225	−	0.806506i	$-0.701356\pi$
$788$		0			0
$789$	1.04607e9	−	6.03950e8i	2.12976	−	1.22962i
$790$		0			0
$791$	−1.44368e8	+	3.19484e8i	−0.291704	+	0.645534i
$792$		0			0
$793$	−6.17147e7	−	1.06893e8i	−0.123757	−	0.214353i
$794$		0			0
$795$	2.46033e8	−	4.26142e8i	0.489657	−	0.848111i
$796$		0			0
$797$		−	9.48842e7i		−	0.187421i	−0.995599	−	0.0937106i	$-0.970127\pi$
$797$		−	9.48842e7i		−	0.187421i	0.995599	−	0.0937106i	$-0.0298729\pi$
$798$		0			0
$799$	1.17224e9			2.29813
$800$		0			0
$801$	1.22695e8	+	7.08379e7i	0.238742	+	0.137838i
$802$		0			0
$803$	−2.13664e8	+	1.23359e8i	−0.412652	+	0.238245i
$804$		0			0
$805$	−4.43663e7	+	3.18601e7i	−0.0850482	+	0.0610745i
$806$		0			0
$807$	−3.10744e8	−	5.38225e8i	−0.591266	−	1.02410i
$808$		0			0
$809$	−4.13738e8	+	7.16615e8i	−0.781411	+	1.35344i	0.149709	+	0.988730i	$0.452166\pi$
$809$	−4.13738e8	+	7.16615e8i	−0.781411	+	1.35344i	−0.931120	+	0.364714i	$0.881167\pi$
$810$		0			0
$811$		−	9.28491e8i		−	1.74066i	−0.492465	−	0.870332i	$-0.663904\pi$
$811$		−	9.28491e8i		−	1.74066i	0.492465	−	0.870332i	$-0.336096\pi$
$812$		0			0
$813$	−6.29833e8			−1.17207
$814$		0			0
$815$	−1.97853e8	−	1.14230e8i	−0.365485	−	0.211013i
$816$		0			0
$817$	−1.00281e6	+	578975.i	−0.00183888	+	0.00106168i
$818$		0			0
$819$	−2.56904e7	−	2.58172e8i	−0.0467647	−	0.469956i
$820$		0			0
$821$	−2.77346e8	−	4.80377e8i	−0.501178	−	0.868065i	−0.999999	−	0.00136052i	$-0.999567\pi$
$821$	−2.77346e8	−	4.80377e8i	−0.501178	−	0.868065i	0.498821	−	0.866705i	$-0.333766\pi$
$822$		0			0
$823$	−4.45911e8	+	7.72340e8i	−0.799924	+	1.38551i	0.119741	+	0.992805i	$0.461793\pi$
$823$	−4.45911e8	+	7.72340e8i	−0.799924	+	1.38551i	−0.919665	+	0.392703i	$0.871540\pi$
$824$		0			0
$825$		−	9.20550e8i		−	1.63940i
$826$		0			0
$827$	1.38059e8			0.244089			0.122044	−	0.992525i	$-0.461055\pi$
$827$	1.38059e8			0.244089			0.122044	+	0.992525i	$0.461055\pi$
$828$		0			0
$829$	−1.20818e8	−	6.97541e7i	−0.212064	−	0.122435i	0.390206	−	0.920727i	$-0.372404\pi$
$829$	−1.20818e8	−	6.97541e7i	−0.212064	−	0.122435i	−0.602270	+	0.798292i	$0.705737\pi$
$830$		0			0
$831$	−8.31403e8	+	4.80011e8i	−1.44880	+	0.836465i
$832$		0			0
$833$	1.96360e8	+	9.76878e8i	0.339718	+	1.69007i
$834$		0			0
$835$	−5.98218e7	−	1.03614e8i	−0.102754	−	0.177976i
$836$		0			0
$837$	−1.00887e7	+	1.74742e7i	−0.0172052	+	0.0298003i
$838$		0			0
$839$			2.67389e8i			0.452750i	0.974040	+	0.226375i	$0.0726874\pi$
$839$			2.67389e8i			0.452750i	−0.974040	+	0.226375i	$0.927313\pi$
$840$		0			0
$841$	−3.66528e8			−0.616197
$842$		0			0
$843$	−9.79127e8	−	5.65299e8i	−1.63439	−	0.943617i
$844$		0			0
$845$	−7.02579e8	+	4.05634e8i	−1.16446	+	0.672302i
$846$		0			0
$847$	−1.82724e8	+	1.81827e7i	−0.300709	+	0.0299232i
$848$		0			0
$849$	−5.52198e8	−	9.56436e8i	−0.902344	−	1.56291i
$850$		0			0
$851$	3.94706e6	−	6.83651e6i	0.00640450	−	0.0110929i
$852$		0			0
$853$			6.69761e8i			1.07913i	0.841945	+	0.539564i	$0.181411\pi$
$853$			6.69761e8i			1.07913i	−0.841945	+	0.539564i	$0.818589\pi$
$854$		0			0
$855$	5.09985e6			0.00815941
$856$		0			0
$857$	−3.50523e8	−	2.02375e8i	−0.556897	−	0.321524i	0.195002	−	0.980803i	$-0.437529\pi$
$857$	−3.50523e8	−	2.02375e8i	−0.556897	−	0.321524i	−0.751899	+	0.659278i	$0.770862\pi$
$858$		0			0
$859$	2.97655e8	−	1.71851e8i	0.469607	−	0.271128i	−0.246468	−	0.969151i	$-0.579270\pi$
$859$	2.97655e8	−	1.71851e8i	0.469607	−	0.271128i	0.716075	+	0.698023i	$0.245937\pi$
$860$		0			0
$861$	−9.32162e8	−	1.29807e9i	−1.46043	−	2.03370i
$862$		0			0
$863$	3.74953e8	+	6.49438e8i	0.583371	+	1.01043i	0.995076	+	0.0991110i	$0.0315999\pi$
$863$	3.74953e8	+	6.49438e8i	0.583371	+	1.01043i	−0.411706	+	0.911317i	$0.635067\pi$
$864$		0			0
$865$	−5.82949e7	+	1.00970e8i	−0.0900705	+	0.156007i
$866$		0			0
$867$			2.01900e9i			3.09798i
$868$		0			0
$869$	7.41023e8			1.12920
$870$		0			0
$871$	−2.66740e8	−	1.54003e8i	−0.403677	−	0.233063i
$872$		0			0
$873$	−1.47866e9	+	8.53704e8i	−2.22242	+	1.28311i
$874$		0			0
$875$	−2.28050e8	−	1.03051e8i	−0.340412	−	0.153826i
$876$		0			0
$877$	7.44757e7	+	1.28996e8i	0.110412	+	0.191239i	0.915936	−	0.401323i	$-0.131450\pi$
$877$	7.44757e7	+	1.28996e8i	0.110412	+	0.191239i	−0.805524	+	0.592562i	$0.798116\pi$
$878$		0			0
$879$	5.10757e7	−	8.84658e7i	0.0752052	−	0.130259i
$880$		0			0
$881$			1.17850e9i			1.72346i	0.507368	+	0.861729i	$0.330618\pi$
$881$			1.17850e9i			1.72346i	−0.507368	+	0.861729i	$0.669382\pi$
$882$		0			0
$883$	−4.71777e8			−0.685259			−0.342630	−	0.939471i	$-0.611318\pi$
$883$	−4.71777e8			−0.685259			−0.342630	+	0.939471i	$0.611318\pi$
$884$		0			0
$885$	5.40890e8	+	3.12283e8i	0.780331	+	0.450524i
$886$		0			0
$887$	2.67515e7	−	1.54450e7i	0.0383334	−	0.0221318i	−0.480711	−	0.876879i	$-0.659621\pi$
$887$	2.67515e7	−	1.54450e7i	0.0383334	−	0.0221318i	0.519044	+	0.854747i	$0.326288\pi$
$888$		0			0
$889$	1.06276e8	−	2.35186e8i	0.151262	−	0.334739i
$890$		0			0
$891$	−9.21221e7	−	1.59560e8i	−0.130236	−	0.225575i
$892$		0			0
$893$	1.75851e6	−	3.04583e6i	0.00246939	−	0.00427712i
$894$		0			0
$895$		−	1.03502e9i		−	1.44371i
$896$		0			0
$897$	2.54603e7			0.0352765
$898$		0			0
$899$	−1.82190e7	−	1.05188e7i	−0.0250753	−	0.0144772i
$900$		0			0
$901$	4.53842e8	−	2.62026e8i	0.620484	−	0.358237i
$902$		0			0
$903$	−5.38586e8	+	3.86767e8i	−0.731462	+	0.525275i
$904$		0			0
$905$	−3.56131e8	−	6.16837e8i	−0.480467	−	0.832194i
$906$		0			0
$907$	−3.53032e8	+	6.11470e8i	−0.473143	+	0.819508i	−0.999527	−	0.0307389i	$-0.990214\pi$
$907$	−3.53032e8	+	6.11470e8i	−0.473143	+	0.819508i	0.526384	+	0.850247i	$0.323547\pi$
$908$		0			0
$909$		−	1.39549e9i		−	1.85795i
$910$		0			0
$911$	−4.46529e8			−0.590602			−0.295301	−	0.955404i	$-0.595420\pi$
$911$	−4.46529e8			−0.590602			−0.295301	+	0.955404i	$0.595420\pi$
$912$		0			0
$913$	−6.29378e8	−	3.63371e8i	−0.826988	−	0.477462i
$914$		0			0
$915$	1.20317e9	−	6.94653e8i	1.57060	−	0.906786i
$916$		0			0
$917$	−8.88163e6	−	8.92547e7i	−0.0115182	−	0.115751i
$918$		0			0
$919$	−3.00517e8	−	5.20511e8i	−0.387188	−	0.670630i	0.604882	−	0.796315i	$-0.293220\pi$
$919$	−3.00517e8	−	5.20511e8i	−0.387188	−	0.670630i	−0.992070	+	0.125685i	$0.959887\pi$
$920$		0			0
$921$	−4.08687e8	+	7.07867e8i	−0.523133	+	0.906092i
$922$		0			0
$923$			3.96751e8i			0.504560i
$924$		0			0
$925$	1.81370e8			0.229161
$926$		0			0
$927$	6.15373e6	+	3.55286e6i	0.00772501	+	0.00446004i
$928$		0			0
$929$	8.46917e8	−	4.88968e8i	1.05632	−	0.609864i	0.131904	−	0.991262i	$-0.457891\pi$
$929$	8.46917e8	−	4.88968e8i	1.05632	−	0.609864i	0.924411	+	0.381399i	$0.124557\pi$
$930$		0			0
$931$	2.83279e6	+	955242.i	0.00351047	+	0.00118376i
$932$		0			0
$933$	7.99375e8	+	1.38456e9i	0.984251	+	1.70477i
$934$		0			0
$935$	8.82636e8	−	1.52877e9i	1.07981	−	1.87028i
$936$		0			0
$937$			1.61274e8i			0.196041i	0.995184	+	0.0980205i	$0.0312511\pi$
$937$			1.61274e8i			0.196041i	−0.995184	+	0.0980205i	$0.968749\pi$
$938$		0			0
$939$	−1.70289e9			−2.05679
$940$		0			0
$941$	2.13148e8	+	1.23061e8i	0.255807	+	0.147690i	0.622420	−	0.782683i	$-0.286149\pi$
$941$	2.13148e8	+	1.23061e8i	0.255807	+	0.147690i	−0.366613	+	0.930373i	$0.619483\pi$
$942$		0			0
$943$	8.07980e7	−	4.66487e7i	0.0963530	−	0.0556294i
$944$		0			0
$945$	9.27230e8	−	9.22676e7i	1.09873	−	0.109334i
$946$		0			0
$947$	7.08264e8	+	1.22675e9i	0.833960	+	1.44446i	0.894874	+	0.446318i	$0.147265\pi$
$947$	7.08264e8	+	1.22675e9i	0.833960	+	1.44446i	−0.0609142	+	0.998143i	$0.519402\pi$
$948$		0			0
$949$	7.83876e7	−	1.35771e8i	0.0917168	−	0.158858i
$950$		0			0
$951$		−	2.26658e9i		−	2.63529i
$952$		0			0
$953$	7.96546e8			0.920306			0.460153	−	0.887840i	$-0.347795\pi$
$953$	7.96546e8			0.920306			0.460153	+	0.887840i	$0.347795\pi$
$954$		0			0
$955$	8.46029e8	+	4.88455e8i	0.971349	+	0.560808i
$956$		0			0
$957$	−6.17182e8	+	3.56330e8i	−0.704170	+	0.406553i
$958$		0			0
$959$	1.39838e8	+	1.94729e8i	0.158551	+	0.220788i
$960$		0			0
$961$	−4.42783e8	−	7.66922e8i	−0.498908	−	0.864134i
$962$		0			0
$963$	−7.02307e8	+	1.21643e9i	−0.786409	+	1.36210i
$964$		0			0
$965$			6.79601e8i			0.756262i
$966$		0			0
$967$	3.27333e6			0.00362002			0.00181001	−	0.999998i	$-0.499424\pi$
$967$	3.27333e6			0.00362002			0.00181001	+	0.999998i	$0.499424\pi$
$968$		0			0
$969$	7.90652e6	+	4.56483e6i	0.00868989	+	0.00501711i
$970$		0			0
$971$	6.30904e8	−	3.64252e8i	0.689137	−	0.397873i	−0.114152	−	0.993463i	$-0.536415\pi$
$971$	6.30904e8	−	3.64252e8i	0.689137	−	0.397873i	0.803289	+	0.595590i	$0.203082\pi$
$972$		0			0
$973$	−5.01041e8	−	2.26411e8i	−0.543920	−	0.245787i
$974$		0			0
$975$	2.92479e8	+	5.06589e8i	0.315559	+	0.546565i
$976$		0			0
$977$	−2.38256e8	+	4.12672e8i	−0.255482	+	0.442508i	−0.965026	−	0.262153i	$-0.915567\pi$
$977$	−2.38256e8	+	4.12672e8i	−0.255482	+	0.442508i	0.709544	+	0.704661i	$0.248901\pi$
$978$		0			0
$979$		−	1.47133e8i		−	0.156806i
$980$		0			0
$981$	−1.65153e9			−1.74936
$982$		0			0
$983$	1.44830e8	+	8.36176e7i	0.152475	+	0.0880313i	0.574296	−	0.818648i	$-0.305276\pi$
$983$	1.44830e8	+	8.36176e7i	0.152475	+	0.0880313i	−0.421822	+	0.906679i	$0.638609\pi$
$984$		0			0
$985$	1.03376e9	−	5.96842e8i	1.08171	−	0.624526i
$986$		0			0
$987$	8.29317e8	−	1.83526e9i	0.862520	−	1.90874i
$988$		0			0
$989$	−1.93552e7	−	3.35242e7i	−0.0200083	−	0.0346553i
$990$		0			0
$991$	−1.09069e8	+	1.88913e8i	−0.112068	+	0.194107i	−0.916604	−	0.399797i	$-0.869081\pi$
$991$	−1.09069e8	+	1.88913e8i	−0.112068	+	0.194107i	0.804536	+	0.593904i	$0.202414\pi$
$992$		0			0
$993$			8.87896e8i			0.906806i
$994$		0			0
$995$	7.87997e8			0.799936
$996$		0			0
$997$	−1.41152e8	−	8.14940e7i	−0.142430	−	0.0822319i	0.427092	−	0.904208i	$-0.359538\pi$
$997$	−1.41152e8	−	8.14940e7i	−0.142430	−	0.0822319i	−0.569521	+	0.821976i	$0.692871\pi$
$998$		0			0
$999$	−1.16628e8	+	6.73353e7i	−0.116979	+	0.0675377i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.7.s.c.17.4		8
4.3	odd	2		14.7.d.a.3.3	✓	8
7.5	odd	6	inner	112.7.s.c.33.4		8
12.11	even	2		126.7.n.c.73.2		8
28.3	even	6		98.7.b.c.97.1		8
28.11	odd	6		98.7.b.c.97.4		8
28.19	even	6		14.7.d.a.5.3	yes	8
28.23	odd	6		98.7.d.c.19.4		8
28.27	even	2		98.7.d.c.31.4		8
84.47	odd	6		126.7.n.c.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.7.d.a.3.3	✓	8	4.3	odd	2
14.7.d.a.5.3	yes	8	28.19	even	6
98.7.b.c.97.1		8	28.3	even	6
98.7.b.c.97.4		8	28.11	odd	6
98.7.d.c.19.4		8	28.23	odd	6
98.7.d.c.31.4		8	28.27	even	2
112.7.s.c.17.4		8	1.1	even	1	trivial
112.7.s.c.33.4		8	7.5	odd	6	inner
126.7.n.c.19.2		8	84.47	odd	6
126.7.n.c.73.2		8	12.11	even	2

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(36.7384 + 21.2109i) q^{3} +(-162.347 + 93.7310i) q^{5} +(-141.244 + 312.569i) q^{7} +(535.305 + 927.175i) q^{9} +(555.924 - 962.889i) q^{11} +706.517i q^{13} -7952.48 q^{15} +(-7334.73 - 4234.71i) q^{17} +(-22.0061 + 12.7052i) q^{19} +(-11818.9 + 8487.36i) q^{21} +(-424.738 - 735.668i) q^{23} +(9758.49 - 16902.2i) q^{25} +14491.7i q^{27} -15109.4 q^{29} +(1205.81 + 696.173i) q^{31} +(40847.5 - 23583.3i) q^{33} +(-6366.92 - 63983.4i) q^{35} +(4646.47 + 8047.92i) q^{37} +(-14985.9 + 25956.3i) q^{39} +109829. i q^{41} +45569.8 q^{43} +(-173810. - 100349. i) q^{45} +(-119865. + 69204.2i) q^{47} +(-77749.5 - 88296.7i) q^{49} +(-179644. - 311152. i) q^{51} +(-30937.9 + 53586.1i) q^{53} +208429. i q^{55} -1077.96 q^{57} +(-68015.2 - 39268.6i) q^{59} +(-151296. + 87350.5i) q^{61} +(-365415. + 36362.0i) q^{63} +(-66222.6 - 114701. i) q^{65} +(-217974. + 377543. i) q^{67} -36036.3i q^{69} +561559. q^{71} +(-192170. - 110949. i) q^{73} +(717022. - 413973. i) q^{75} +(222448. + 309766. i) q^{77} +(333239. + 577187. i) q^{79} +(82854.9 - 143509. i) q^{81} -653635. i q^{83} +1.58769e6 q^{85} +(-555096. - 320485. i) q^{87} +(114603. - 66165.9i) q^{89} +(-220835. - 99791.1i) q^{91} +(29532.9 + 51152.5i) q^{93} +(2381.75 - 4125.31i) q^{95} +1.59480e6i q^{97} +1.19036e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37}+ \cdots + 4625928 q^{99}+O(q^{100}) \) 8 * q - 336 * q^5 - 652 * q^7 + 756 * q^9 + 1356 * q^11 - 27144 * q^15 - 17304 * q^17 + 32004 * q^19 + 9756 * q^21 + 4128 * q^23 + 4664 * q^25 - 30312 * q^29 + 3108 * q^31 + 3276 * q^33 - 98028 * q^35 - 6124 * q^37 - 100764 * q^39 + 297376 * q^43 - 172116 * q^45 - 313908 * q^47 + 32432 * q^49 - 253692 * q^51 + 278484 * q^53 - 81288 * q^57 + 835464 * q^59 - 995316 * q^61 - 1216188 * q^63 + 8316 * q^65 - 648808 * q^67 - 190128 * q^71 - 1617084 * q^73 + 2042208 * q^75 + 1456224 * q^77 - 70096 * q^79 + 1177920 * q^81 + 2190984 * q^85 + 2057076 * q^87 + 739116 * q^89 - 2233752 * q^91 - 23364 * q^93 - 725640 * q^95 + 4625928 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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