LMFDB - Embedded newform 112.10.i.f.65.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,91] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 58021 x^{16} + 299572 x^{15} + 1344281856 x^{14} - 13223849184 x^{13} + \cdots + 45\!\cdots\!49 $ x^18 - x^17 - 58021x^16 + 299572x^15 + 1344281856x^14 - 13223849184x^13 - 16002240964108x^12 + 235679191938382x^11 + 105022697526800314x^10 - 2027381674023318770x^9 - 376633377992039706452x^8 + 8787878212198864086436x^7 + 679531373791146081991682x^6 - 18701919138080616463625598x^5 - 488002400377889504476060425x^4 + 17409656239263783927309097669x^3 + 54946039381935966896110794812x^2 - 5699490442438655384248912605217x + 45543736045953959673944391022849
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{64}\cdot 3^{9}\cdot 7^{8} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.7
Root			$71.8084 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.10.i.f.81.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(76.8084 + 133.036i) q^{3} +(-1042.94 + 1806.43i) q^{5} +(489.665 - 6333.55i) q^{7} +(-1957.55 + 3390.58i) q^{9} +(-2801.38 - 4852.13i) q^{11} +115400. q^{13} -320427. q^{15} +(-272172. - 471416. i) q^{17} +(44741.2 - 77494.0i) q^{19} +(880200. - 421327. i) q^{21} +(967252. - 1.67533e6i) q^{23} +(-1.19890e6 - 2.07656e6i) q^{25} +2.42221e6 q^{27} -2.89342e6 q^{29} +(767039. + 1.32855e6i) q^{31} +(430339. - 745368. i) q^{33} +(1.09304e7 + 7.49008e6i) q^{35} +(7.31613e6 - 1.26719e7i) q^{37} +(8.86371e6 + 1.53524e7i) q^{39} +1.32906e7 q^{41} -6.41291e6 q^{43} +(-4.08323e6 - 7.07237e6i) q^{45} +(-2.47708e7 + 4.29043e7i) q^{47} +(-3.98741e7 - 6.20263e6i) q^{49} +(4.18102e7 - 7.24174e7i) q^{51} +(-3.02532e7 - 5.24001e7i) q^{53} +1.16867e7 q^{55} +1.37460e7 q^{57} +(6.25964e7 + 1.08420e8i) q^{59} +(5.29946e7 - 9.17893e7i) q^{61} +(2.05159e7 + 1.40585e7i) q^{63} +(-1.20356e8 + 2.08463e8i) q^{65} +(1.16276e8 + 2.01395e8i) q^{67} +2.97172e8 q^{69} -1.09836e8 q^{71} +(1.35469e8 + 2.34639e8i) q^{73} +(1.84171e8 - 3.18994e8i) q^{75} +(-3.21029e7 + 1.53667e7i) q^{77} +(1.39854e8 - 2.42235e8i) q^{79} +(2.24577e8 + 3.88978e8i) q^{81} +2.55171e8 q^{83} +1.13544e9 q^{85} +(-2.22239e8 - 3.84929e8i) q^{87} +(5.80101e8 - 1.00476e9i) q^{89} +(5.65075e7 - 7.30893e8i) q^{91} +(-1.17830e8 + 2.04088e8i) q^{93} +(9.33251e7 + 1.61644e8i) q^{95} +1.67001e8 q^{97} +2.19354e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 91 q^{3} + 801 q^{5} + 4820 q^{7} - 55886 q^{9} + 42719 q^{11} + 301620 q^{13} - 318842 q^{15} + 11357 q^{17} + 1237749 q^{19} - 352965 q^{21} - 95019 q^{23} - 848460 q^{25} - 983570 q^{27} + 12010116 q^{29}+ \cdots - 654760036 q^{99}+O(q^{100}) $ 18 * q + 91 * q^3 + 801 * q^5 + 4820 * q^7 - 55886 * q^9 + 42719 * q^11 + 301620 * q^13 - 318842 * q^15 + 11357 * q^17 + 1237749 * q^19 - 352965 * q^21 - 95019 * q^23 - 848460 * q^25 - 983570 * q^27 + 12010116 * q^29 + 10454399 * q^31 - 22110949 * q^33 + 14069675 * q^35 + 18487597 * q^37 - 26537570 * q^39 - 9446020 * q^41 - 15588088 * q^43 + 16445950 * q^45 - 6482527 * q^47 - 97119278 * q^49 + 91171769 * q^51 + 15314401 * q^53 + 27790478 * q^55 - 86196434 * q^57 + 84965083 * q^59 + 122850341 * q^61 - 682061514 * q^63 + 148004010 * q^65 + 27388339 * q^67 - 717062290 * q^69 + 473284416 * q^71 + 217201553 * q^73 - 86436700 * q^75 + 112464495 * q^77 + 326283565 * q^79 - 592248617 * q^81 - 1411349592 * q^83 + 1568978282 * q^85 + 787721718 * q^87 + 153096753 * q^89 + 878501576 * q^91 - 158673269 * q^93 - 1281844653 * q^95 + 919981372 * q^97 - 654760036 * q^99

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	76.8084	+	133.036i	0.547473	+	0.948251i	0.998447	+	0.0557142i	$0.0177436\pi$
$3$	76.8084	+	133.036i	0.547473	+	0.948251i	−0.450974	+	0.892537i	$0.648923\pi$
$4$		0			0
$5$	−1042.94	+	1806.43i	−0.746270	+	1.29258i	0.203329	+	0.979110i	$0.434824\pi$
$5$	−1042.94	+	1806.43i	−0.746270	+	1.29258i	−0.949599	+	0.313467i	$0.898510\pi$
$6$		0			0
$7$	489.665	−	6333.55i	0.0770828	−	0.997025i
$7$	489.665	−	6333.55i	0.0770828	−	0.997025i
$8$		0			0
$9$	−1957.55	+	3390.58i	−0.0994539	+	0.172259i
$10$		0			0
$11$	−2801.38	−	4852.13i	−0.0576905	−	0.0999230i	0.835738	−	0.549129i	$-0.185040\pi$
$11$	−2801.38	−	4852.13i	−0.0576905	−	0.0999230i	−0.893428	+	0.449206i	$0.851707\pi$
$12$		0			0
$13$	115400.			1.12063			0.560314	−	0.828280i	$-0.310680\pi$
$13$	115400.			1.12063			0.560314	+	0.828280i	$0.310680\pi$
$14$		0			0
$15$	−320427.			−1.63425
$16$		0			0
$17$	−272172.	−	471416.i	−0.790357	−	1.36894i	−0.925746	−	0.378146i	$-0.876562\pi$
$17$	−272172.	−	471416.i	−0.790357	−	1.36894i	0.135389	−	0.990793i	$-0.456772\pi$
$18$		0			0
$19$	44741.2	−	77494.0i	0.0787619	−	0.136420i	−0.823954	−	0.566657i	$-0.808237\pi$
$19$	44741.2	−	77494.0i	0.0787619	−	0.136420i	0.902716	+	0.430237i	$0.141570\pi$
$20$		0			0
$21$	880200.	−	421327.i	0.987631	−	0.472750i
$22$		0			0
$23$	967252.	−	1.67533e6i	0.720716	−	1.24832i	−0.239997	−	0.970774i	$-0.577146\pi$
$23$	967252.	−	1.67533e6i	0.720716	−	1.24832i	0.960713	−	0.277544i	$-0.0895203\pi$
$24$		0			0
$25$	−1.19890e6	−	2.07656e6i	−0.613837	−	1.06320i
$26$		0			0
$27$	2.42221e6			0.877153
$28$		0			0
$29$	−2.89342e6			−0.759663			−0.379831	−	0.925056i	$-0.624018\pi$
$29$	−2.89342e6			−0.759663			−0.379831	+	0.925056i	$0.624018\pi$
$30$		0			0
$31$	767039.	+	1.32855e6i	0.149173	+	0.258375i	0.930922	−	0.365218i	$-0.119006\pi$
$31$	767039.	+	1.32855e6i	0.149173	+	0.258375i	−0.781749	+	0.623593i	$0.785672\pi$
$32$		0			0
$33$	430339.	−	745368.i	0.0631681	−	0.109410i
$34$		0			0
$35$	1.09304e7	+	7.49008e6i	1.23121	+	0.843685i
$36$		0			0
$37$	7.31613e6	−	1.26719e7i	0.641762	−	1.11156i	−0.343277	−	0.939234i	$-0.611537\pi$
$37$	7.31613e6	−	1.26719e7i	0.641762	−	1.11156i	0.985039	−	0.172330i	$-0.0551296\pi$
$38$		0			0
$39$	8.86371e6	+	1.53524e7i	0.613514	+	1.06264i
$40$		0			0
$41$	1.32906e7			0.734543			0.367271	−	0.930114i	$-0.380292\pi$
$41$	1.32906e7			0.734543			0.367271	+	0.930114i	$0.380292\pi$
$42$		0			0
$43$	−6.41291e6			−0.286053			−0.143027	−	0.989719i	$-0.545683\pi$
$43$	−6.41291e6			−0.286053			−0.143027	+	0.989719i	$0.545683\pi$
$44$		0			0
$45$	−4.08323e6	−	7.07237e6i	−0.148439	−	0.257104i
$46$		0			0
$47$	−2.47708e7	+	4.29043e7i	−0.740457	+	1.28251i	0.211830	+	0.977307i	$0.432058\pi$
$47$	−2.47708e7	+	4.29043e7i	−0.740457	+	1.28251i	−0.952287	+	0.305203i	$0.901276\pi$
$48$		0			0
$49$	−3.98741e7	−	6.20263e6i	−0.988116	−	0.153707i
$50$		0			0
$51$	4.18102e7	−	7.24174e7i	0.865399	−	1.49891i
$52$		0			0
$53$	−3.02532e7	−	5.24001e7i	−0.526660	−	0.912202i	−0.999517	−	0.0310630i	$-0.990111\pi$
$53$	−3.02532e7	−	5.24001e7i	−0.526660	−	0.912202i	0.472857	−	0.881139i	$-0.343223\pi$
$54$		0			0
$55$	1.16867e7			0.172211
$56$		0			0
$57$	1.37460e7			0.172480
$58$		0			0
$59$	6.25964e7	+	1.08420e8i	0.672536	+	1.16487i	0.977183	+	0.212401i	$0.0681283\pi$
$59$	6.25964e7	+	1.08420e8i	0.672536	+	1.16487i	−0.304647	+	0.952465i	$0.598538\pi$
$60$		0			0
$61$	5.29946e7	−	9.17893e7i	0.490058	−	0.848805i	−0.509877	−	0.860247i	$-0.670309\pi$
$61$	5.29946e7	−	9.17893e7i	0.490058	−	0.848805i	0.999935	+	0.0114427i	$0.00364241\pi$
$62$		0			0
$63$	2.05159e7	+	1.40585e7i	0.164081	+	0.112436i
$64$		0			0
$65$	−1.20356e8	+	2.08463e8i	−0.836292	+	1.44850i
$66$		0			0
$67$	1.16276e8	+	2.01395e8i	0.704939	+	1.22099i	0.966713	+	0.255862i	$0.0823591\pi$
$67$	1.16276e8	+	2.01395e8i	0.704939	+	1.22099i	−0.261774	+	0.965129i	$0.584308\pi$
$68$		0			0
$69$	2.97172e8			1.57829
$70$		0			0
$71$	−1.09836e8			−0.512958			−0.256479	−	0.966550i	$-0.582562\pi$
$71$	−1.09836e8			−0.512958			−0.256479	+	0.966550i	$0.582562\pi$
$72$		0			0
$73$	1.35469e8	+	2.34639e8i	0.558323	+	0.967044i	0.997637	+	0.0687105i	$0.0218885\pi$
$73$	1.35469e8	+	2.34639e8i	0.558323	+	0.967044i	−0.439313	+	0.898334i	$0.644778\pi$
$74$		0			0
$75$	1.84171e8	−	3.18994e8i	0.672119	−	1.16414i
$76$		0			0
$77$	−3.21029e7	+	1.53667e7i	−0.104073	+	0.0498166i
$78$		0			0
$79$	1.39854e8	−	2.42235e8i	0.403975	−	0.699705i	−0.590227	−	0.807238i	$-0.700962\pi$
$79$	1.39854e8	−	2.42235e8i	0.403975	−	0.699705i	0.994202	+	0.107532i	$0.0342949\pi$
$80$		0			0
$81$	2.24577e8	+	3.88978e8i	0.579672	+	1.00402i
$82$		0			0
$83$	2.55171e8			0.590173			0.295086	−	0.955471i	$-0.404652\pi$
$83$	2.55171e8			0.590173			0.295086	+	0.955471i	$0.404652\pi$
$84$		0			0
$85$	1.13544e9			2.35928
$86$		0			0
$87$	−2.22239e8	−	3.84929e8i	−0.415895	−	0.720351i
$88$		0			0
$89$	5.80101e8	−	1.00476e9i	0.980050	−	1.69750i	0.317902	−	0.948124i	$-0.397022\pi$
$89$	5.80101e8	−	1.00476e9i	0.980050	−	1.69750i	0.662148	−	0.749373i	$-0.269645\pi$
$90$		0			0
$91$	5.65075e7	−	7.30893e8i	0.0863812	−	1.11729i
$92$		0			0
$93$	−1.17830e8	+	2.04088e8i	−0.163336	+	0.282907i
$94$		0			0
$95$	9.33251e7	+	1.61644e8i	0.117555	+	0.203612i
$96$		0			0
$97$	1.67001e8			0.191534			0.0957671	−	0.995404i	$-0.469470\pi$
$97$	1.67001e8			0.191534			0.0957671	+	0.995404i	$0.469470\pi$
$98$		0			0
$99$	2.19354e7			0.0229502
$100$		0			0
$101$	−9.45369e8	−	1.63743e9i	−0.903972	−	1.56573i	−0.822291	−	0.569067i	$-0.807304\pi$
$101$	−9.45369e8	−	1.63743e9i	−0.903972	−	1.56573i	−0.0816814	−	0.996658i	$-0.526029\pi$
$102$		0			0
$103$	3.89696e8	−	6.74973e8i	0.341160	−	0.590907i	−0.643488	−	0.765456i	$-0.722513\pi$
$103$	3.89696e8	−	6.74973e8i	0.341160	−	0.590907i	0.984648	+	0.174549i	$0.0558468\pi$
$104$		0			0
$105$	−1.56902e8	+	2.02944e9i	−0.125973	+	1.62939i
$106$		0			0
$107$	5.82472e8	−	1.00887e9i	0.429584	−	0.744061i	−0.567252	−	0.823544i	$-0.691994\pi$
$107$	5.82472e8	−	1.00887e9i	0.429584	−	0.744061i	0.996836	+	0.0794829i	$0.0253269\pi$
$108$		0			0
$109$	9.74921e8	+	1.68861e9i	0.661531	+	1.14581i	0.980213	+	0.197944i	$0.0634264\pi$
$109$	9.74921e8	+	1.68861e9i	0.661531	+	1.14581i	−0.318682	+	0.947862i	$0.603240\pi$
$110$		0			0
$111$	2.24776e9			1.40539
$112$		0			0
$113$	5.31881e7			0.0306875			0.0153437	−	0.999882i	$-0.495116\pi$
$113$	5.31881e7			0.0306875			0.0153437	+	0.999882i	$0.495116\pi$
$114$		0			0
$115$	2.01758e9	+	3.49455e9i	1.07570	+	1.86316i
$116$		0			0
$117$	−2.25902e8	+	3.91274e8i	−0.111451	+	0.193039i
$118$		0			0
$119$	−3.11901e9	+	1.49298e9i	−1.42579	+	0.682484i
$120$		0			0
$121$	1.16328e9	−	2.01486e9i	0.493344	−	0.854496i
$122$		0			0
$123$	1.02083e9	+	1.76813e9i	0.402143	+	0.696531i
$124$		0			0
$125$	9.27546e8			0.339813
$126$		0			0
$127$	−3.56359e9			−1.21555			−0.607773	−	0.794111i	$-0.707937\pi$
$127$	−3.56359e9			−1.21555			−0.607773	+	0.794111i	$0.707937\pi$
$128$		0			0
$129$	−4.92565e8	−	8.53148e8i	−0.156607	−	0.271250i
$130$		0			0
$131$	−2.49530e8	+	4.32198e8i	−0.0740290	+	0.128222i	−0.900664	−	0.434517i	$-0.856919\pi$
$131$	−2.49530e8	+	4.32198e8i	−0.0740290	+	0.128222i	0.826635	+	0.562739i	$0.190252\pi$
$132$		0			0
$133$	−4.68904e8	−	3.21317e8i	−0.129943	−	0.0890432i
$134$		0			0
$135$	−2.52623e9	+	4.37556e9i	−0.654593	+	1.13379i
$136$		0			0
$137$	−1.03847e9	−	1.79867e9i	−0.251854	−	0.436224i	0.712182	−	0.701995i	$-0.247707\pi$
$137$	−1.03847e9	−	1.79867e9i	−0.251854	−	0.436224i	−0.964036	+	0.265770i	$0.914374\pi$
$138$		0			0
$139$	8.36456e9			1.90054			0.950268	−	0.311432i	$-0.100809\pi$
$139$	8.36456e9			1.90054			0.950268	+	0.311432i	$0.100809\pi$
$140$		0			0
$141$	−7.61043e9			−1.62152
$142$		0			0
$143$	−3.23280e8	−	5.59937e8i	−0.0646497	−	0.111977i
$144$		0			0
$145$	3.01768e9	−	5.22677e9i	0.566913	−	0.981923i
$146$		0			0
$147$	−2.23749e9	−	5.78110e9i	−0.395214	−	1.02113i
$148$		0			0
$149$	−4.25680e9	+	7.37299e9i	−0.707530	+	1.22548i	0.258241	+	0.966081i	$0.416857\pi$
$149$	−4.25680e9	+	7.37299e9i	−0.707530	+	1.22548i	−0.965771	+	0.259397i	$0.916476\pi$
$150$		0			0
$151$	4.83281e8	+	8.37068e8i	0.0756491	+	0.131028i	0.901368	−	0.433053i	$-0.142564\pi$
$151$	4.83281e8	+	8.37068e8i	0.0756491	+	0.131028i	−0.825719	+	0.564081i	$0.809230\pi$
$152$		0			0
$153$	2.13116e9			0.314416
$154$		0			0
$155$	−3.19992e9			−0.445293
$156$		0			0
$157$	−4.03263e9	−	6.98472e9i	−0.529712	−	0.917488i	−0.999399	−	0.0346553i	$-0.988967\pi$
$157$	−4.03263e9	−	6.98472e9i	−0.529712	−	0.917488i	0.469687	−	0.882833i	$-0.344367\pi$
$158$		0			0
$159$	4.64740e9	−	8.04954e9i	0.576665	−	0.998812i
$160$		0			0
$161$	−1.01372e10	−	6.94649e9i	−1.18905	−	0.814796i
$162$		0			0
$163$	−2.44707e8	+	4.23845e8i	−0.0271520	+	0.0470287i	−0.879282	−	0.476301i	$-0.841977\pi$
$163$	−2.44707e8	+	4.23845e8i	−0.0271520	+	0.0470287i	0.852130	+	0.523330i	$0.175310\pi$
$164$		0			0
$165$	8.97638e8	+	1.55475e9i	0.0942808	+	0.163299i
$166$		0			0
$167$	−1.68953e10			−1.68090			−0.840450	−	0.541889i	$-0.817709\pi$
$167$	−1.68953e10			−1.68090			−0.840450	+	0.541889i	$0.817709\pi$
$168$		0			0
$169$	2.71273e9			0.255809
$170$		0			0
$171$	1.75166e8	+	3.03397e8i	0.0156664	+	0.0271349i
$172$		0			0
$173$	1.44389e9	−	2.50089e9i	0.122554	−	0.212269i	−0.798220	−	0.602366i	$-0.794225\pi$
$173$	1.44389e9	−	2.50089e9i	0.122554	−	0.212269i	0.920774	+	0.390096i	$0.127558\pi$
$174$		0			0
$175$	−1.37390e10	+	6.57648e9i	−1.10735	+	0.530057i
$176$		0			0
$177$	−9.61586e9	+	1.66552e10i	−0.736391	+	1.27547i
$178$		0			0
$179$	−1.04854e10	−	1.81612e10i	−0.763388	−	1.32223i	−0.941095	−	0.338143i	$-0.890201\pi$
$179$	−1.04854e10	−	1.81612e10i	−0.763388	−	1.32223i	0.177707	−	0.984083i	$-0.443132\pi$
$180$		0			0
$181$	1.86322e10			1.29036			0.645179	−	0.764031i	$-0.276783\pi$
$181$	1.86322e10			1.29036			0.645179	+	0.764031i	$0.276783\pi$
$182$		0			0
$183$	1.62817e10			1.07317
$184$		0			0
$185$	1.52606e10	+	2.64322e10i	0.957855	+	1.65905i
$186$		0			0
$187$	−1.52491e9	+	2.64123e9i	−0.0911923	+	0.157950i
$188$		0			0
$189$	1.18607e9	−	1.53412e10i	0.0676134	−	0.874543i
$190$		0			0
$191$	3.20581e9	−	5.55262e9i	0.174296	−	0.301890i	−0.765621	−	0.643291i	$-0.777568\pi$
$191$	3.20581e9	−	5.55262e9i	0.174296	−	0.301890i	0.939917	+	0.341402i	$0.110902\pi$
$192$		0			0
$193$	5.34416e8	+	9.25636e8i	0.0277250	+	0.0480211i	0.879555	−	0.475797i	$-0.157840\pi$
$193$	5.34416e8	+	9.25636e8i	0.0277250	+	0.0480211i	−0.851830	+	0.523818i	$0.824507\pi$
$194$		0			0
$195$	−3.69774e10			−1.83139
$196$		0			0
$197$	−1.99832e10			−0.945293			−0.472647	−	0.881252i	$-0.656701\pi$
$197$	−1.99832e10			−0.945293			−0.472647	+	0.881252i	$0.656701\pi$
$198$		0			0
$199$	−7.90634e9	−	1.36942e10i	−0.357385	−	0.619009i	0.630138	−	0.776483i	$-0.282998\pi$
$199$	−7.90634e9	−	1.36942e10i	−0.357385	−	0.619009i	−0.987523	+	0.157474i	$0.949665\pi$
$200$		0			0
$201$	−1.78619e10	+	3.09377e10i	−0.771871	+	1.33692i
$202$		0			0
$203$	−1.41681e9	+	1.83256e10i	−0.0585569	+	0.757402i
$204$		0			0
$205$	−1.38613e10	+	2.40086e10i	−0.548167	+	0.949453i
$206$		0			0
$207$	3.78689e9	+	6.55909e9i	0.143356	+	0.248300i
$208$		0			0
$209$	−5.01348e8			−0.0181753
$210$		0			0
$211$	8.38594e9			0.291260			0.145630	−	0.989339i	$-0.453479\pi$
$211$	8.38594e9			0.291260			0.145630	+	0.989339i	$0.453479\pi$
$212$		0			0
$213$	−8.43631e9	−	1.46121e10i	−0.280831	−	0.486413i
$214$		0			0
$215$	6.68830e9	−	1.15845e10i	0.213473	−	0.369746i
$216$		0			0
$217$	8.79003e9	−	4.20754e9i	0.269105	−	0.128813i
$218$		0			0
$219$	−2.08102e10	+	3.60444e10i	−0.611334	+	1.05886i
$220$		0			0
$221$	−3.14087e10	−	5.44015e10i	−0.885697	−	1.53407i
$222$		0			0
$223$	3.61200e10			0.978084			0.489042	−	0.872260i	$-0.337347\pi$
$223$	3.61200e10			0.978084			0.489042	+	0.872260i	$0.337347\pi$
$224$		0			0
$225$	9.38764e9			0.244194
$226$		0			0
$227$	2.64920e10	+	4.58855e10i	0.662215	+	1.14699i	0.980032	+	0.198838i	$0.0637166\pi$
$227$	2.64920e10	+	4.58855e10i	0.662215	+	1.14699i	−0.317818	+	0.948152i	$0.602950\pi$
$228$		0			0
$229$	1.15511e10	−	2.00071e10i	0.277565	−	0.480756i	−0.693214	−	0.720732i	$-0.743806\pi$
$229$	1.15511e10	−	2.00071e10i	0.277565	−	0.480756i	0.970779	+	0.239975i	$0.0771393\pi$
$230$		0			0
$231$	−4.51010e9	−	3.09055e9i	−0.104216	−	0.0714138i
$232$		0			0
$233$	−7.82668e9	+	1.35562e10i	−0.173971	+	0.301326i	−0.939805	−	0.341712i	$-0.888993\pi$
$233$	−7.82668e9	+	1.35562e10i	−0.173971	+	0.301326i	0.765834	+	0.643038i	$0.222326\pi$
$234$		0			0
$235$	−5.16692e10	−	8.94936e10i	−1.10516	−	1.91420i
$236$		0			0
$237$	4.29680e10			0.884662
$238$		0			0
$239$	8.37136e10			1.65961			0.829804	−	0.558055i	$-0.188452\pi$
$239$	8.37136e10			1.65961			0.829804	+	0.558055i	$0.188452\pi$
$240$		0			0
$241$	−1.68849e10	−	2.92456e10i	−0.322420	−	0.558449i	0.658566	−	0.752523i	$-0.271163\pi$
$241$	−1.68849e10	−	2.92456e10i	−0.322420	−	0.558449i	−0.980987	+	0.194074i	$0.937830\pi$
$242$		0			0
$243$	−1.06605e10	+	1.84646e10i	−0.196133	+	0.339712i
$244$		0			0
$245$	5.27910e10	−	6.55608e10i	0.936080	−	1.16251i
$246$		0			0
$247$	5.16315e9	−	8.94283e9i	0.0882629	−	0.152876i
$248$		0			0
$249$	1.95992e10	+	3.39469e10i	0.323104	+	0.559632i
$250$		0			0
$251$	5.05222e10			0.803435			0.401717	−	0.915764i	$-0.368413\pi$
$251$	5.05222e10			0.803435			0.401717	+	0.915764i	$0.368413\pi$
$252$		0			0
$253$	−1.08386e10			−0.166314
$254$		0			0
$255$	8.72113e10	+	1.51054e11i	1.29164	+	2.23719i
$256$		0			0
$257$	−1.21395e10	+	2.10262e10i	−0.173581	+	0.300651i	−0.939669	−	0.342084i	$-0.888867\pi$
$257$	−1.21395e10	+	2.10262e10i	−0.173581	+	0.300651i	0.766088	+	0.642735i	$0.222200\pi$
$258$		0			0
$259$	−7.66757e10	−	5.25421e10i	−1.05879	−	0.725535i
$260$		0			0
$261$	5.66402e9	−	9.81038e9i	0.0755514	−	0.130859i
$262$		0			0
$263$	−7.90491e9	−	1.36917e10i	−0.101882	−	0.176464i	0.810578	−	0.585630i	$-0.199153\pi$
$263$	−7.90491e9	−	1.36917e10i	−0.101882	−	0.176464i	−0.912460	+	0.409166i	$0.865820\pi$
$264$		0			0
$265$	1.26210e11			1.57212
$266$		0			0
$267$	1.78226e11			2.14620
$268$		0			0
$269$	−2.72894e10	−	4.72666e10i	−0.317767	−	0.550389i	0.662255	−	0.749279i	$-0.269600\pi$
$269$	−2.72894e10	−	4.72666e10i	−0.317767	−	0.550389i	−0.980022	+	0.198890i	$0.936266\pi$
$270$		0			0
$271$	6.53339e9	−	1.13162e10i	0.0735829	−	0.127449i	−0.826886	−	0.562369i	$-0.809890\pi$
$271$	6.53339e9	−	1.13162e10i	0.0735829	−	0.127449i	0.900469	+	0.434920i	$0.143223\pi$
$272$		0			0
$273$	1.01575e11	−	4.86212e10i	1.10677	−	0.529778i
$274$		0			0
$275$	−6.71715e9	+	1.16344e10i	−0.0708252	+	0.122673i
$276$		0			0
$277$	−4.58397e9	−	7.93967e9i	−0.0467825	−	0.0810296i	0.841686	−	0.539967i	$-0.181563\pi$
$277$	−4.58397e9	−	7.93967e9i	−0.0467825	−	0.0810296i	−0.888468	+	0.458938i	$0.848230\pi$
$278$		0			0
$279$	−6.00608e9			−0.0593433
$280$		0			0
$281$	8.22953e10			0.787402			0.393701	−	0.919239i	$-0.371195\pi$
$281$	8.22953e10			0.787402			0.393701	+	0.919239i	$0.371195\pi$
$282$		0			0
$283$	6.51401e10	+	1.12826e11i	0.603684	+	1.04561i	0.992258	+	0.124193i	$0.0396343\pi$
$283$	6.51401e10	+	1.12826e11i	0.603684	+	1.04561i	−0.388574	+	0.921417i	$0.627032\pi$
$284$		0			0
$285$	−1.43363e10	+	2.48312e10i	−0.128717	+	0.222944i
$286$		0			0
$287$	6.50794e9	−	8.41766e10i	0.0566206	−	0.732357i
$288$		0			0
$289$	−8.88613e10	+	1.53912e11i	−0.749329	+	1.29788i
$290$		0			0
$291$	1.28271e10	+	2.22172e10i	0.104860	+	0.181623i
$292$		0			0
$293$	−2.34116e11			−1.85578			−0.927889	−	0.372855i	$-0.878379\pi$
$293$	−2.34116e11			−1.85578			−0.927889	+	0.372855i	$0.878379\pi$
$294$		0			0
$295$	−2.61138e11			−2.00757
$296$		0			0
$297$	−6.78553e9	−	1.17529e10i	−0.0506034	−	0.0876477i
$298$		0			0
$299$	1.11621e11	−	1.93334e11i	0.807656	−	1.39890i
$300$		0			0
$301$	−3.14017e9	+	4.06165e10i	−0.0220498	+	0.285202i
$302$		0			0
$303$	1.45224e11	−	2.51536e11i	0.989801	−	1.71439i
$304$		0			0
$305$	1.10541e11	+	1.91462e11i	0.731430	+	1.26687i
$306$		0			0
$307$	−2.41962e11			−1.55462			−0.777311	−	0.629117i	$-0.783417\pi$
$307$	−2.41962e11			−1.55462			−0.777311	+	0.629117i	$0.783417\pi$
$308$		0			0
$309$	1.19728e11			0.747104
$310$		0			0
$311$	1.43947e10	+	2.49324e10i	0.0872533	+	0.151127i	0.906349	−	0.422530i	$-0.138858\pi$
$311$	1.43947e10	+	2.49324e10i	0.0872533	+	0.151127i	−0.819096	+	0.573657i	$0.805524\pi$
$312$		0			0
$313$	7.28039e10	−	1.26100e11i	0.428751	−	0.742619i	−0.568012	−	0.823021i	$-0.692287\pi$
$313$	7.28039e10	−	1.26100e11i	0.428751	−	0.742619i	0.996763	+	0.0804021i	$0.0256204\pi$
$314$		0			0
$315$	−4.67926e10	+	2.23983e10i	−0.267781	+	0.128179i
$316$		0			0
$317$	−8.02286e10	+	1.38960e11i	−0.446234	+	0.772900i	−0.998137	−	0.0610084i	$-0.980568\pi$
$317$	−8.02286e10	+	1.38960e11i	−0.446234	+	0.772900i	0.551903	+	0.833908i	$0.313902\pi$
$318$		0			0
$319$	8.10557e9	+	1.40393e10i	0.0438254	+	0.0759077i
$320$		0			0
$321$	1.78955e11			0.940743
$322$		0			0
$323$	−4.87092e10			−0.249000
$324$		0			0
$325$	−1.38354e11	−	2.39635e11i	−0.687884	−	1.19145i
$326$		0			0
$327$	−1.49764e11	+	2.59399e11i	−0.724341	+	1.25460i
$328$		0			0
$329$	2.59607e11	+	1.77896e11i	1.22162	+	0.837114i
$330$		0			0
$331$	4.98105e10	−	8.62744e10i	0.228084	−	0.395054i	−0.729156	−	0.684347i	$-0.760087\pi$
$331$	4.98105e10	−	8.62744e10i	0.228084	−	0.395054i	0.957240	+	0.289294i	$0.0934205\pi$
$332$		0			0
$333$	2.86434e10	+	4.96119e10i	0.127651	+	0.221099i
$334$		0			0
$335$	−4.85075e11			−2.10430
$336$		0			0
$337$	−1.33308e11			−0.563018			−0.281509	−	0.959559i	$-0.590835\pi$
$337$	−1.33308e11			−0.563018			−0.281509	+	0.959559i	$0.590835\pi$
$338$		0			0
$339$	4.08529e9	+	7.07593e9i	0.0168006	+	0.0290995i
$340$		0			0
$341$	4.29753e9	−	7.44355e9i	0.0172117	−	0.0298116i
$342$		0			0
$343$	−5.88096e10	+	2.49507e11i	−0.229416	+	0.973328i
$344$		0			0
$345$	−3.09934e11	+	5.36821e11i	−1.17783	+	2.04006i
$346$		0			0
$347$	1.30972e11	+	2.26850e11i	0.484949	+	0.839956i	0.999850	−	0.0172934i	$-0.00550495\pi$
$347$	1.30972e11	+	2.26850e11i	0.484949	+	0.839956i	−0.514902	+	0.857249i	$0.672172\pi$
$348$		0			0
$349$	3.19105e11			1.15138			0.575690	−	0.817668i	$-0.304733\pi$
$349$	3.19105e11			1.15138			0.575690	+	0.817668i	$0.304733\pi$
$350$		0			0
$351$	2.79524e11			0.982963
$352$		0			0
$353$	−2.27064e11	−	3.93286e11i	−0.778326	−	1.34810i	−0.932906	−	0.360121i	$-0.882736\pi$
$353$	−2.27064e11	−	3.93286e11i	−0.778326	−	1.34810i	0.154579	−	0.987980i	$-0.450598\pi$
$354$		0			0
$355$	1.14553e11	−	1.98411e11i	0.382805	−	0.663037i
$356$		0			0
$357$	−4.38186e11	−	3.00267e11i	−1.42775	−	0.978364i
$358$		0			0
$359$	2.66513e11	−	4.61614e11i	0.846824	−	1.46674i	−0.0372037	−	0.999308i	$-0.511845\pi$
$359$	2.66513e11	−	4.61614e11i	0.846824	−	1.46674i	0.884028	−	0.467435i	$-0.154822\pi$
$360$		0			0
$361$	1.57340e11	+	2.72521e11i	0.487593	+	0.844536i
$362$		0			0
$363$	3.57398e11			1.08037
$364$		0			0
$365$	−5.65145e11			−1.66664
$366$		0			0
$367$	−4.82670e10	−	8.36008e10i	−0.138884	−	0.240554i	0.788190	−	0.615432i	$-0.211018\pi$
$367$	−4.82670e10	−	8.36008e10i	−0.138884	−	0.240554i	−0.927075	+	0.374877i	$0.877685\pi$
$368$		0			0
$369$	−2.60170e10	+	4.50628e10i	−0.0730532	+	0.126532i
$370$		0			0
$371$	−3.46693e11	+	1.65952e11i	−0.950084	+	0.454778i
$372$		0			0
$373$	3.26360e11	−	5.65272e11i	0.872985	−	1.51205i	0.0140911	−	0.999901i	$-0.495515\pi$
$373$	3.26360e11	−	5.65272e11i	0.872985	−	1.51205i	0.858894	−	0.512154i	$-0.171152\pi$
$374$		0			0
$375$	7.12433e10	+	1.23397e11i	0.186039	+	0.322229i
$376$		0			0
$377$	−3.33902e11			−0.851300
$378$		0			0
$379$	−4.03245e11			−1.00391			−0.501953	−	0.864895i	$-0.667385\pi$
$379$	−4.03245e11			−1.00391			−0.501953	+	0.864895i	$0.667385\pi$
$380$		0			0
$381$	−2.73714e11	−	4.74086e11i	−0.665479	−	1.15264i
$382$		0			0
$383$	4.67553e10	−	8.09826e10i	0.111029	−	0.192308i	−0.805156	−	0.593063i	$-0.797919\pi$
$383$	4.67553e10	−	8.09826e10i	0.111029	−	0.192308i	0.916185	+	0.400755i	$0.131252\pi$
$384$		0			0
$385$	5.72257e9	−	7.40184e10i	0.0132745	−	0.171698i
$386$		0			0
$387$	1.25536e10	−	2.17435e10i	0.0284491	−	0.0492753i
$388$		0			0
$389$	−2.59895e11	−	4.50151e11i	−0.575473	−	0.996748i	−0.995990	−	0.0894636i	$-0.971485\pi$
$389$	−2.59895e11	−	4.50151e11i	−0.575473	−	0.996748i	0.420517	−	0.907285i	$-0.361849\pi$
$390$		0			0
$391$	−1.05304e12			−2.27849
$392$		0			0
$393$	−7.66639e10			−0.162116
$394$		0			0
$395$	2.91721e11	+	5.05275e11i	0.602949	+	1.04434i
$396$		0			0
$397$	3.92781e11	−	6.80316e11i	0.793584	−	1.37453i	−0.130151	−	0.991494i	$-0.541546\pi$
$397$	3.92781e11	−	6.80316e11i	0.793584	−	1.37453i	0.923735	−	0.383033i	$-0.125120\pi$
$398$		0			0
$399$	6.73093e9	−	8.70609e10i	0.0132953	−	0.171967i
$400$		0			0
$401$	7.78409e10	−	1.34824e11i	0.150334	−	0.260387i	−0.781016	−	0.624511i	$-0.785298\pi$
$401$	7.78409e10	−	1.34824e11i	0.150334	−	0.260387i	0.931350	+	0.364124i	$0.118632\pi$
$402$		0			0
$403$	8.85166e10	+	1.53315e11i	0.167167	+	0.289543i
$404$		0			0
$405$	−9.36884e11			−1.73037
$406$		0			0
$407$	−8.19810e10			−0.148094
$408$		0			0
$409$	4.40712e11	+	7.63336e11i	0.778754	+	1.34884i	0.932660	+	0.360756i	$0.117481\pi$
$409$	4.40712e11	+	7.63336e11i	0.778754	+	1.34884i	−0.153906	+	0.988086i	$0.549185\pi$
$410$		0			0
$411$	1.59526e11	−	2.76306e11i	0.275767	−	0.477642i
$412$		0			0
$413$	7.17336e11	−	3.43368e11i	1.21324	−	0.580744i
$414$		0			0
$415$	−2.66128e11	+	4.60948e11i	−0.440428	+	0.762844i
$416$		0			0
$417$	6.42468e11	+	1.11279e12i	1.04049	+	1.80219i
$418$		0			0
$419$	8.11137e10			0.128567			0.0642837	−	0.997932i	$-0.479524\pi$
$419$	8.11137e10			0.128567			0.0642837	+	0.997932i	$0.479524\pi$
$420$		0			0
$421$	−8.85880e11			−1.37438			−0.687188	−	0.726480i	$-0.741155\pi$
$421$	−8.85880e11			−1.37438			−0.687188	+	0.726480i	$0.741155\pi$
$422$		0			0
$423$	−9.69803e10	−	1.67975e11i	−0.147283	−	0.255101i
$424$		0			0
$425$	−6.52615e11	+	1.13036e12i	−0.970301	+	1.68061i
$426$		0			0
$427$	−5.55402e11	−	3.80590e11i	−0.808504	−	0.554028i
$428$		0			0
$429$	4.96612e10	−	8.60157e10i	0.0707880	−	0.122608i
$430$		0			0
$431$	6.64077e11	+	1.15021e12i	0.926980	+	1.60558i	0.788345	+	0.615234i	$0.210938\pi$
$431$	6.64077e11	+	1.15021e12i	0.926980	+	1.60558i	0.138636	+	0.990343i	$0.455728\pi$
$432$		0			0
$433$	−1.00935e11			−0.137989			−0.0689945	−	0.997617i	$-0.521979\pi$
$433$	−1.00935e11			−0.137989			−0.0689945	+	0.997617i	$0.521979\pi$
$434$		0			0
$435$	9.27131e11			1.24148
$436$		0			0
$437$	−8.65520e10	−	1.49912e11i	−0.113530	−	0.196640i
$438$		0			0
$439$	3.29888e11	−	5.71382e11i	0.423912	−	0.734237i	−0.572406	−	0.819970i	$-0.693990\pi$
$439$	3.29888e11	−	5.71382e11i	0.423912	−	0.734237i	0.996318	+	0.0857333i	$0.0273233\pi$
$440$		0			0
$441$	9.90861e10	−	1.23054e11i	0.124750	−	0.154925i
$442$		0			0
$443$	−3.46961e11	+	6.00954e11i	−0.428019	+	0.741351i	−0.996697	−	0.0812091i	$-0.974122\pi$
$443$	−3.46961e11	+	6.00954e11i	−0.428019	+	0.741351i	0.568678	+	0.822560i	$0.307455\pi$
$444$		0			0
$445$	1.21002e12	+	2.09582e12i	1.46276	+	2.53358i
$446$		0			0
$447$	−1.30783e12			−1.54941
$448$		0			0
$449$	4.51714e11			0.524511			0.262256	−	0.964998i	$-0.415534\pi$
$449$	4.51714e11			0.524511			0.262256	+	0.964998i	$0.415534\pi$
$450$		0			0
$451$	−3.72320e10	−	6.44877e10i	−0.0423762	−	0.0733977i
$452$		0			0
$453$	−7.42401e10	+	1.28588e11i	−0.0828317	+	0.143469i
$454$		0			0
$455$	1.26137e12	+	8.64358e11i	1.37973	+	0.945458i
$456$		0			0
$457$	4.41262e11	−	7.64288e11i	0.473231	−	0.819661i	−0.526299	−	0.850300i	$-0.676421\pi$
$457$	4.41262e11	−	7.64288e11i	0.473231	−	0.819661i	0.999531	+	0.0306386i	$0.00975411\pi$
$458$		0			0
$459$	−6.59259e11	−	1.14187e12i	−0.693264	−	1.20077i
$460$		0			0
$461$	−6.35002e11			−0.654819			−0.327409	−	0.944883i	$-0.606176\pi$
$461$	−6.35002e11			−0.654819			−0.327409	+	0.944883i	$0.606176\pi$
$462$		0			0
$463$	1.39959e12			1.41543			0.707714	−	0.706499i	$-0.249726\pi$
$463$	1.39959e12			1.41543			0.707714	+	0.706499i	$0.249726\pi$
$464$		0			0
$465$	−2.45780e11	−	4.25704e11i	−0.243786	−	0.422250i
$466$		0			0
$467$	1.29327e11	−	2.24000e11i	0.125824	−	0.217933i	−0.796231	−	0.604993i	$-0.793176\pi$
$467$	1.29327e11	−	2.24000e11i	0.125824	−	0.217933i	0.922055	+	0.387060i	$0.126509\pi$
$468$		0			0
$469$	1.33248e12	−	6.37821e11i	1.27170	−	0.608724i
$470$		0			0
$471$	6.19479e11	−	1.07297e12i	0.580006	−	1.00460i
$472$		0			0
$473$	1.79650e10	+	3.11163e10i	0.0165026	+	0.0285833i
$474$		0			0
$475$	−2.14561e11			−0.193388
$476$		0			0
$477$	2.36889e11			0.209514
$478$		0			0
$479$	−9.08428e11	−	1.57344e12i	−0.788462	−	1.36566i	−0.926909	−	0.375286i	$-0.877544\pi$
$479$	−9.08428e11	−	1.57344e12i	−0.788462	−	1.36566i	0.138447	−	0.990370i	$-0.455789\pi$
$480$		0			0
$481$	8.44284e11	−	1.46234e12i	0.719177	−	1.24565i
$482$		0			0
$483$	1.45515e11	−	1.88215e12i	0.121659	−	1.57360i
$484$		0			0
$485$	−1.74173e11	+	3.01676e11i	−0.142936	+	0.247573i
$486$		0			0
$487$	−1.17115e11	−	2.02850e11i	−0.0943481	−	0.163416i	0.814988	−	0.579477i	$-0.196743\pi$
$487$	−1.17115e11	−	2.02850e11i	−0.0943481	−	0.163416i	−0.909336	+	0.416062i	$0.863410\pi$
$488$		0			0
$489$	−7.51822e10			−0.0594601
$490$		0			0
$491$	−5.83805e11			−0.453316			−0.226658	−	0.973974i	$-0.572780\pi$
$491$	−5.83805e11			−0.453316			−0.226658	+	0.973974i	$0.572780\pi$
$492$		0			0
$493$	7.87509e11	+	1.36400e12i	0.600405	+	1.03993i
$494$		0			0
$495$	−2.28774e10	+	3.96247e10i	−0.0171270	+	0.0296649i
$496$		0			0
$497$	−5.37827e10	+	6.95651e11i	−0.0395402	+	0.511431i
$498$		0			0
$499$	−5.85659e11	+	1.01439e12i	−0.422856	+	0.732408i	−0.996218	−	0.0868946i	$-0.972306\pi$
$499$	−5.85659e11	+	1.01439e12i	−0.422856	+	0.732408i	0.573362	+	0.819302i	$0.305639\pi$
$500$		0			0
$501$	−1.29770e12	−	2.24768e12i	−0.920248	−	1.59392i
$502$		0			0
$503$	9.45139e11			0.658324			0.329162	−	0.944273i	$-0.393234\pi$
$503$	9.45139e11			0.658324			0.329162	+	0.944273i	$0.393234\pi$
$504$		0			0
$505$	3.94387e12			2.69843
$506$		0			0
$507$	2.08360e11	+	3.60891e11i	0.140049	+	0.242572i
$508$		0			0
$509$	−4.57200e11	+	7.91894e11i	−0.301909	+	0.522922i	−0.976568	−	0.215207i	$-0.930957\pi$
$509$	−4.57200e11	+	7.91894e11i	−0.301909	+	0.522922i	0.674659	+	0.738129i	$0.264291\pi$
$510$		0			0
$511$	1.55243e12	−	7.43103e11i	1.00720	−	0.482120i
$512$		0			0
$513$	1.08373e11	−	1.87707e11i	0.0690863	−	0.119661i
$514$		0			0
$515$	8.12862e11	+	1.40792e12i	0.509195	+	0.881952i
$516$		0			0
$517$	2.77570e11			0.170870
$518$		0			0
$519$	4.43611e11			0.268380
$520$		0			0
$521$	1.35247e12	+	2.34255e12i	0.804190	+	1.39290i	0.916837	+	0.399262i	$0.130734\pi$
$521$	1.35247e12	+	2.34255e12i	0.804190	+	1.39290i	−0.112647	+	0.993635i	$0.535933\pi$
$522$		0			0
$523$	−2.59251e11	+	4.49036e11i	−0.151518	+	0.262436i	−0.931786	−	0.363009i	$-0.881749\pi$
$523$	−2.59251e11	+	4.49036e11i	−0.151518	+	0.262436i	0.780268	+	0.625445i	$0.215083\pi$
$524$		0			0
$525$	−1.93018e12	−	1.32266e12i	−1.10887	−	0.759855i
$526$		0			0
$527$	4.17533e11	−	7.23189e11i	0.235800	−	0.408417i
$528$		0			0
$529$	−9.70576e11	−	1.68109e12i	−0.538864	−	0.933340i
$530$		0			0
$531$	−4.90143e11			−0.267545
$532$		0			0
$533$	1.53374e12			0.823150
$534$		0			0
$535$	1.21497e12	+	2.10439e12i	0.641171	+	1.11054i
$536$		0			0
$537$	1.61073e12	−	2.78986e12i	0.835869	−	1.44777i
$538$		0			0
$539$	8.16064e10	+	2.10850e11i	0.0416461	+	0.107603i
$540$		0			0
$541$	−1.09240e12	+	1.89208e12i	−0.548267	+	0.949626i	0.450126	+	0.892965i	$0.351379\pi$
$541$	−1.09240e12	+	1.89208e12i	−0.548267	+	0.949626i	−0.998393	+	0.0566615i	$0.981954\pi$
$542$		0			0
$543$	1.43111e12	+	2.47875e12i	0.706437	+	1.22358i
$544$		0			0
$545$	−4.06715e12			−1.97472
$546$		0			0
$547$	2.21998e12			1.06024			0.530122	−	0.847921i	$-0.322146\pi$
$547$	2.21998e12			1.06024			0.530122	+	0.847921i	$0.322146\pi$
$548$		0			0
$549$	2.07479e11	+	3.59365e11i	0.0974763	+	0.168834i
$550$		0			0
$551$	−1.29455e11	+	2.24223e11i	−0.0598325	+	0.103633i
$552$		0			0
$553$	−1.46573e12	−	1.00439e12i	−0.666484	−	0.456708i
$554$		0			0
$555$	−2.34429e12	+	4.06043e12i	−1.04880	+	1.81658i
$556$		0			0
$557$	1.99295e10	+	3.45190e10i	0.00877302	+	0.0151953i	0.870379	−	0.492383i	$-0.163874\pi$
$557$	1.99295e10	+	3.45190e10i	0.00877302	+	0.0151953i	−0.861606	+	0.507578i	$0.830541\pi$
$558$		0			0
$559$	−7.40051e11			−0.320560
$560$		0			0
$561$	−4.68504e11			−0.199701
$562$		0			0
$563$	5.36588e11	+	9.29397e11i	0.225088	+	0.389864i	0.956346	−	0.292237i	$-0.0943996\pi$
$563$	5.36588e11	+	9.29397e11i	0.225088	+	0.389864i	−0.731258	+	0.682101i	$0.761066\pi$
$564$		0			0
$565$	−5.54722e10	+	9.60806e10i	−0.0229011	+	0.0396659i
$566$		0			0
$567$	2.57358e12	−	1.23190e12i	1.04572	−	0.500554i
$568$		0			0
$569$	−2.35208e12	+	4.07393e12i	−0.940693	+	1.62933i	−0.176539	+	0.984294i	$0.556490\pi$
$569$	−2.35208e12	+	4.07393e12i	−0.940693	+	1.62933i	−0.764154	+	0.645034i	$0.776843\pi$
$570$		0			0
$571$	−1.25353e12	−	2.17118e12i	−0.493485	−	0.854740i	0.506487	−	0.862247i	$-0.330944\pi$
$571$	−1.25353e12	−	2.17118e12i	−0.493485	−	0.854740i	−0.999972	+	0.00750717i	$0.997610\pi$
$572$		0			0
$573$	9.84932e11			0.381690
$574$		0			0
$575$	−4.63856e12			−1.76961
$576$		0			0
$577$	−1.32891e12	−	2.30173e12i	−0.499118	−	0.864498i	0.500881	−	0.865516i	$-0.333009\pi$
$577$	−1.32891e12	−	2.30173e12i	−0.499118	−	0.864498i	−0.999999	+	0.00101805i	$0.999676\pi$
$578$		0			0
$579$	−8.20953e10	+	1.42193e11i	−0.0303574	+	0.0525806i
$580$		0			0
$581$	1.24948e11	−	1.61613e12i	0.0454922	−	0.588417i
$582$		0			0
$583$	−1.69501e11	+	2.93585e11i	−0.0607666	+	0.105251i
$584$		0			0
$585$	−4.71206e11	−	8.16153e11i	−0.166345	−	0.288118i
$586$		0			0
$587$	−5.14876e11			−0.178991			−0.0894955	−	0.995987i	$-0.528525\pi$
$587$	−5.14876e11			−0.178991			−0.0894955	+	0.995987i	$0.528525\pi$
$588$		0			0
$589$	1.37273e11			0.0469966
$590$		0			0
$591$	−1.53488e12	−	2.65848e12i	−0.517523	−	0.896376i
$592$		0			0
$593$	3.19333e11	−	5.53101e11i	0.106047	−	0.183678i	−0.808119	−	0.589020i	$-0.799514\pi$
$593$	3.19333e11	−	5.53101e11i	0.106047	−	0.183678i	0.914165	+	0.405341i	$0.132847\pi$
$594$		0			0
$595$	5.55985e11	−	7.19137e12i	0.181860	−	2.35226i
$596$		0			0
$597$	1.21455e12	−	2.10366e12i	0.391318	−	0.677782i
$598$		0			0
$599$	2.61951e12	+	4.53713e12i	0.831381	+	1.43999i	0.896943	+	0.442145i	$0.145783\pi$
$599$	2.61951e12	+	4.53713e12i	0.831381	+	1.43999i	−0.0655624	+	0.997848i	$0.520884\pi$
$600$		0			0
$601$	−5.90374e12			−1.84583			−0.922916	−	0.385002i	$-0.874201\pi$
$601$	−5.90374e12			−1.84583			−0.922916	+	0.385002i	$0.874201\pi$
$602$		0			0
$603$	−9.10462e11			−0.280436
$604$		0			0
$605$	2.42647e12	+	4.20277e12i	0.736335	+	1.27537i
$606$		0			0
$607$	−1.51269e12	+	2.62006e12i	−0.452274	+	0.783361i	−0.998527	−	0.0542592i	$-0.982720\pi$
$607$	−1.51269e12	+	2.62006e12i	−0.452274	+	0.783361i	0.546253	+	0.837620i	$0.316054\pi$
$608$		0			0
$609$	−2.54679e12	+	1.21908e12i	−0.750266	+	0.359131i
$610$		0			0
$611$	−2.85856e12	+	4.95117e12i	−0.829778	+	1.43722i
$612$		0			0
$613$	2.67847e12	+	4.63925e12i	0.766153	+	1.32702i	0.939635	+	0.342179i	$0.111165\pi$
$613$	2.67847e12	+	4.63925e12i	0.766153	+	1.32702i	−0.173482	+	0.984837i	$0.555502\pi$
$614$		0			0
$615$	−4.25867e12			−1.20043
$616$		0			0
$617$	3.53508e12			0.982011			0.491005	−	0.871157i	$-0.336630\pi$
$617$	3.53508e12			0.982011			0.491005	+	0.871157i	$0.336630\pi$
$618$		0			0
$619$	−3.56878e11	−	6.18131e11i	−0.0977039	−	0.169228i	0.813030	−	0.582222i	$-0.197816\pi$
$619$	−3.56878e11	−	6.18131e11i	−0.0977039	−	0.169228i	−0.910734	+	0.412994i	$0.864483\pi$
$620$		0			0
$621$	2.34289e12	−	4.05800e12i	0.632179	−	1.09497i
$622$		0			0
$623$	−6.07966e12	−	4.16609e12i	−1.61690	−	1.10798i
$624$		0			0
$625$	1.37422e12	−	2.38023e12i	0.360245	−	0.623962i
$626$		0			0
$627$	−3.85077e10	−	6.66973e10i	−0.00995048	−	0.0172347i
$628$		0			0
$629$	−7.96499e12			−2.02888
$630$		0			0
$631$	−6.27699e12			−1.57623			−0.788115	−	0.615528i	$-0.788943\pi$
$631$	−6.27699e12			−1.57623			−0.788115	+	0.615528i	$0.788943\pi$
$632$		0			0
$633$	6.44111e11	+	1.11563e12i	0.159457	+	0.276188i
$634$		0			0
$635$	3.71663e12	−	6.43739e12i	0.907125	−	1.57119i
$636$		0			0
$637$	−4.60148e12	−	7.15785e11i	−1.10731	−	0.172248i
$638$		0			0
$639$	2.15009e11	−	3.72407e11i	0.0510157	−	0.0883617i
$640$		0			0
$641$	8.96947e11	+	1.55356e12i	0.209848	+	0.363468i	0.951667	−	0.307133i	$-0.0993696\pi$
$641$	8.96947e11	+	1.55356e12i	0.209848	+	0.363468i	−0.741818	+	0.670601i	$0.766036\pi$
$642$		0			0
$643$	7.39512e12			1.70607			0.853033	−	0.521857i	$-0.174761\pi$
$643$	7.39512e12			1.70607			0.853033	+	0.521857i	$0.174761\pi$
$644$		0			0
$645$	2.05487e12			0.467483
$646$		0			0
$647$	−1.10642e12	−	1.91637e12i	−0.248227	−	0.429942i	0.714807	−	0.699322i	$-0.246515\pi$
$647$	−1.10642e12	−	1.91637e12i	−0.248227	−	0.429942i	−0.963034	+	0.269380i	$0.913181\pi$
$648$		0			0
$649$	3.50713e11	−	6.07452e11i	0.0775979	−	0.134404i
$650$		0			0
$651$	1.23490e12	+	8.46217e11i	0.269475	+	0.184658i
$652$		0			0
$653$	1.71344e12	−	2.96777e12i	0.368773	−	0.638734i	−0.620601	−	0.784127i	$-0.713111\pi$
$653$	1.71344e12	−	2.96777e12i	0.368773	−	0.638734i	0.989374	+	0.145393i	$0.0464445\pi$
$654$		0			0
$655$	−5.20491e11	−	9.01517e11i	−0.110491	−	0.191376i
$656$		0			0
$657$	−1.06075e12			−0.222110
$658$		0			0
$659$	5.15867e12			1.06550			0.532750	−	0.846273i	$-0.321159\pi$
$659$	5.15867e12			1.06550			0.532750	+	0.846273i	$0.321159\pi$
$660$		0			0
$661$	4.22535e12	+	7.31851e12i	0.860906	+	1.49113i	0.871056	+	0.491184i	$0.163436\pi$
$661$	4.22535e12	+	7.31851e12i	0.860906	+	1.49113i	−0.0101497	+	0.999948i	$0.503231\pi$
$662$		0			0
$663$	4.82491e12	−	8.35698e12i	0.969791	−	1.67973i
$664$		0			0
$665$	1.06948e12	−	5.11928e11i	0.212067	−	0.101511i
$666$		0			0
$667$	−2.79867e12	+	4.84744e12i	−0.547501	+	0.948300i
$668$		0			0
$669$	2.77432e12	+	4.80526e12i	0.535475	+	0.927469i
$670$		0			0
$671$	−5.93831e11			−0.113087
$672$		0			0
$673$	−4.58272e12			−0.861105			−0.430552	−	0.902566i	$-0.641681\pi$
$673$	−4.58272e12			−0.861105			−0.430552	+	0.902566i	$0.641681\pi$
$674$		0			0
$675$	−2.90399e12	−	5.02986e12i	−0.538429	−	0.932587i
$676$		0			0
$677$	−1.67750e12	+	2.90551e12i	−0.306911	+	0.531585i	−0.977685	−	0.210077i	$-0.932629\pi$
$677$	−1.67750e12	+	2.90551e12i	−0.306911	+	0.531585i	0.670774	+	0.741662i	$0.265962\pi$
$678$		0			0
$679$	8.17745e10	−	1.05771e12i	0.0147640	−	0.190964i
$680$		0			0
$681$	−4.06962e12	+	7.04879e12i	−0.725090	+	1.25589i
$682$		0			0
$683$	1.05762e12	+	1.83186e12i	0.185968	+	0.322106i	0.943902	−	0.330225i	$-0.107125\pi$
$683$	1.05762e12	+	1.83186e12i	0.185968	+	0.322106i	−0.757934	+	0.652331i	$0.773791\pi$
$684$		0			0
$685$	4.33224e12			0.751805
$686$		0			0
$687$	3.54889e12			0.607837
$688$		0			0
$689$	−3.49123e12	−	6.04699e12i	−0.590190	−	1.02224i
$690$		0			0
$691$	8.50655e11	−	1.47338e12i	0.141939	−	0.245846i	−0.786288	−	0.617861i	$-0.788000\pi$
$691$	8.50655e11	−	1.47338e12i	0.141939	−	0.245846i	0.928227	+	0.372015i	$0.121333\pi$
$692$		0			0
$693$	1.07410e10	−	1.38929e11i	0.00176907	−	0.0228819i
$694$		0			0
$695$	−8.72376e12	+	1.51100e13i	−1.41831	+	2.45659i
$696$		0			0
$697$	−3.61733e12	−	6.26540e12i	−0.580551	−	1.00554i
$698$		0			0
$699$	−2.40462e12			−0.380977
$700$		0			0
$701$	−4.34715e12			−0.679945			−0.339973	−	0.940435i	$-0.610418\pi$
$701$	−4.34715e12			−0.679945			−0.339973	+	0.940435i	$0.610418\pi$
$702$		0			0
$703$	−6.54665e11	−	1.13391e12i	−0.101093	−	0.175098i
$704$		0			0
$705$	7.93725e12	−	1.37477e13i	1.21009	−	2.09594i
$706$		0			0
$707$	−1.08336e13	+	5.18575e12i	−1.63075	+	0.780592i
$708$		0			0
$709$	1.65655e12	−	2.86922e12i	0.246204	−	0.426438i	−0.716265	−	0.697828i	$-0.754150\pi$
$709$	1.65655e12	−	2.86922e12i	0.246204	−	0.426438i	0.962469	+	0.271390i	$0.0874833\pi$
$710$		0			0
$711$	5.47545e11	+	9.48375e11i	0.0803538	+	0.139177i
$712$		0			0
$713$	2.96768e12			0.430045
$714$		0			0
$715$	1.34865e12			0.192984
$716$		0			0
$717$	6.42991e12	+	1.11369e13i	0.908591	+	1.57373i
$718$		0			0
$719$	−6.05718e11	+	1.04914e12i	−0.0845261	+	0.146403i	−0.905189	−	0.425009i	$-0.860271\pi$
$719$	−6.05718e11	+	1.04914e12i	−0.0845261	+	0.146403i	0.820663	+	0.571412i	$0.193604\pi$
$720$		0			0
$721$	−4.08415e12	−	2.79867e12i	−0.562851	−	0.385694i
$722$		0			0
$723$	2.59381e12	−	4.49261e12i	0.353033	−	0.611471i
$724$		0			0
$725$	3.46893e12	+	6.00836e12i	0.466309	+	0.807671i
$726$		0			0
$727$	7.42267e11			0.0985497			0.0492749	−	0.998785i	$-0.484309\pi$
$727$	7.42267e11			0.0985497			0.0492749	+	0.998785i	$0.484309\pi$
$728$		0			0
$729$	5.56541e12			0.729833
$730$		0			0
$731$	1.74541e12	+	3.02315e12i	0.226084	+	0.391589i
$732$		0			0
$733$	−9.56322e11	+	1.65640e12i	−0.122359	+	0.211932i	−0.920698	−	0.390277i	$-0.872379\pi$
$733$	−9.56322e11	+	1.65640e12i	−0.122359	+	0.211932i	0.798338	+	0.602209i	$0.205713\pi$
$734$		0			0
$735$	1.27767e13	+	1.98749e12i	1.61483	+	0.251196i
$736$		0			0
$737$	6.51463e11	−	1.12837e12i	0.0813367	−	0.140879i
$738$		0			0
$739$	4.89339e12	+	8.47560e12i	0.603545	+	1.04537i	0.992280	+	0.124021i	$0.0395790\pi$
$739$	4.89339e12	+	8.47560e12i	0.603545	+	1.04537i	−0.388734	+	0.921350i	$0.627088\pi$
$740$		0			0
$741$	1.58629e12			0.193286
$742$		0			0
$743$	5.65599e12			0.680862			0.340431	−	0.940269i	$-0.389427\pi$
$743$	5.65599e12			0.680862			0.340431	+	0.940269i	$0.389427\pi$
$744$		0			0
$745$	−8.87920e12	−	1.53792e13i	−1.05602	−	1.82907i
$746$		0			0
$747$	−4.99509e11	+	8.65176e11i	−0.0586950	+	0.101663i
$748$		0			0
$749$	−6.10452e12	−	4.18312e12i	−0.708734	−	0.485660i
$750$		0			0
$751$	4.13975e12	−	7.17027e12i	0.474892	−	0.822537i	−0.524695	−	0.851291i	$-0.675820\pi$
$751$	4.13975e12	−	7.17027e12i	0.474892	−	0.822537i	0.999587	+	0.0287535i	$0.00915378\pi$
$752$		0			0
$753$	3.88053e12	+	6.72128e12i	0.439859	+	0.761858i
$754$		0			0
$755$	−2.01614e12			−0.225819
$756$		0			0
$757$	−9.49224e12			−1.05060			−0.525300	−	0.850917i	$-0.676047\pi$
$757$	−9.49224e12			−1.05060			−0.525300	+	0.850917i	$0.676047\pi$
$758$		0			0
$759$	−8.32492e11	−	1.44192e12i	−0.0910525	−	0.157708i
$760$		0			0
$761$	−2.82368e11	+	4.89076e11i	−0.0305200	+	0.0528623i	−0.880882	−	0.473336i	$-0.843050\pi$
$761$	−2.82368e11	+	4.89076e11i	−0.0305200	+	0.0528623i	0.850362	+	0.526198i	$0.176383\pi$
$762$		0			0
$763$	1.11723e13	−	5.34786e12i	1.19339	−	0.571241i
$764$		0			0
$765$	−2.22268e12	+	3.84980e12i	−0.234640	+	0.406408i
$766$		0			0
$767$	7.22365e12	+	1.25117e13i	0.753663	+	1.30538i
$768$		0			0
$769$	−1.04208e13			−1.07456			−0.537281	−	0.843403i	$-0.680549\pi$
$769$	−1.04208e13			−1.07456			−0.537281	+	0.843403i	$0.680549\pi$
$770$		0			0
$771$	−3.72966e12			−0.380124
$772$		0			0
$773$	7.94519e12	+	1.37615e13i	0.800380	+	1.38630i	0.919366	+	0.393403i	$0.128702\pi$
$773$	7.94519e12	+	1.37615e13i	0.800380	+	1.38630i	−0.118986	+	0.992896i	$0.537964\pi$
$774$		0			0
$775$	1.83921e12	−	3.18560e12i	0.183136	−	0.317201i
$776$		0			0
$777$	1.10065e12	−	1.42363e13i	0.108331	−	1.40121i
$778$		0			0
$779$	5.94637e11	−	1.02994e12i	0.0578540	−	0.100206i
$780$		0			0
$781$	3.07692e11	+	5.32938e11i	0.0295928	+	0.0512562i
$782$		0			0
$783$	−7.00848e12			−0.666340
$784$		0			0
$785$	1.68232e13			1.58123
$786$		0			0
$787$	1.14447e12	+	1.98228e12i	0.106345	+	0.184195i	0.914287	−	0.405067i	$-0.132752\pi$
$787$	1.14447e12	+	1.98228e12i	0.106345	+	0.184195i	−0.807942	+	0.589262i	$0.799419\pi$
$788$		0			0
$789$	1.21433e12	−	2.10327e12i	0.111555	−	0.193219i
$790$		0			0
$791$	2.60443e10	−	3.36869e11i	0.00236548	−	0.0305962i
$792$		0			0
$793$	6.11559e12	−	1.05925e13i	0.549173	−	0.951195i
$794$		0			0
$795$	9.69396e12	+	1.67904e13i	0.860695	+	1.49077i
$796$		0			0
$797$	2.88305e12			0.253099			0.126549	−	0.991960i	$-0.459610\pi$
$797$	2.88305e12			0.253099			0.126549	+	0.991960i	$0.459610\pi$
$798$		0			0
$799$	2.69677e13			2.34090
$800$		0			0
$801$	2.27115e12	+	3.93375e12i	0.194940	+	0.337645i
$802$		0			0
$803$	7.58998e11	−	1.31462e12i	0.0644200	−	0.111579i
$804$		0			0
$805$	2.31208e13	−	1.10673e13i	1.94054	−	0.928879i
$806$		0			0
$807$	4.19211e12	−	7.26095e12i	0.347938	−	0.602646i
$808$		0			0
$809$	1.92244e12	+	3.32976e12i	0.157792	+	0.273303i	0.934072	−	0.357085i	$-0.116229\pi$
$809$	1.92244e12	+	3.32976e12i	0.157792	+	0.273303i	−0.776280	+	0.630388i	$0.782896\pi$
$810$		0			0
$811$	−1.88231e13			−1.52791			−0.763954	−	0.645271i	$-0.776745\pi$
$811$	−1.88231e13			−1.52791			−0.763954	+	0.645271i	$0.776745\pi$
$812$		0			0
$813$	2.00728e12			0.161139
$814$		0			0
$815$	−5.10432e11	−	8.84094e11i	−0.0405255	−	0.0701922i
$816$		0			0
$817$	−2.86921e11	+	4.96962e11i	−0.0225301	+	0.0390233i
$818$		0			0
$819$	2.36754e12	+	1.62235e12i	0.183873	+	0.125999i
$820$		0			0
$821$	−2.06087e12	+	3.56954e12i	−0.158310	+	0.274200i	−0.934259	−	0.356595i	$-0.883938\pi$
$821$	−2.06087e12	+	3.56954e12i	−0.158310	+	0.274200i	0.775950	+	0.630795i	$0.217271\pi$
$822$		0			0
$823$	−5.96737e12	−	1.03358e13i	−0.453402	−	0.785316i	0.545192	−	0.838311i	$-0.316457\pi$
$823$	−5.96737e12	−	1.03358e13i	−0.453402	−	0.785316i	−0.998595	+	0.0529951i	$0.983123\pi$
$824$		0			0
$825$	−2.06373e12			−0.155100
$826$		0			0
$827$	−1.91078e13			−1.42048			−0.710241	−	0.703959i	$-0.751414\pi$
$827$	−1.91078e13			−1.42048			−0.710241	+	0.703959i	$0.751414\pi$
$828$		0			0
$829$	7.02223e11	+	1.21629e12i	0.0516392	+	0.0894417i	0.890690	−	0.454612i	$-0.150222\pi$
$829$	7.02223e11	+	1.21629e12i	0.0516392	+	0.0894417i	−0.839050	+	0.544054i	$0.816889\pi$
$830$		0			0
$831$	7.04175e11	−	1.21967e12i	0.0512243	−	0.0887231i
$832$		0			0
$833$	7.92859e12	+	2.04854e13i	0.570549	+	1.47415i
$834$		0			0
$835$	1.76208e13	−	3.05202e13i	1.25440	−	2.17269i
$836$		0			0
$837$	1.85793e12	+	3.21803e12i	0.130847	+	0.226634i
$838$		0			0
$839$	−1.41615e13			−0.986692			−0.493346	−	0.869833i	$-0.664226\pi$
$839$	−1.41615e13			−0.986692			−0.493346	+	0.869833i	$0.664226\pi$
$840$		0			0
$841$	−6.13525e12			−0.422913
$842$		0			0
$843$	6.32096e12	+	1.09482e13i	0.431081	+	0.746655i
$844$		0			0
$845$	−2.82922e12	+	4.90036e12i	−0.190903	+	0.330653i
$846$		0			0
$847$	−1.21916e13	−	8.35428e12i	−0.813925	−	0.557743i
$848$		0			0
$849$	−1.00066e13	+	1.73320e13i	−0.661001	+	1.14489i
$850$		0			0
$851$	−1.41531e13	−	2.45139e13i	−0.925057	−	1.60224i
$852$		0			0
$853$	−4.16449e12			−0.269334			−0.134667	−	0.990891i	$-0.542996\pi$
$853$	−4.16449e12			−0.269334			−0.134667	+	0.990891i	$0.542996\pi$
$854$		0			0
$855$	−7.30755e11			−0.0467653
$856$		0			0
$857$	−8.79741e12	−	1.52376e13i	−0.557110	−	0.964943i	−0.997736	−	0.0672523i	$-0.978577\pi$
$857$	−8.79741e12	−	1.52376e13i	−0.557110	−	0.964943i	0.440626	−	0.897691i	$-0.354757\pi$
$858$		0			0
$859$	4.93082e12	−	8.54043e12i	0.308994	−	0.535193i	−0.669149	−	0.743128i	$-0.733341\pi$
$859$	4.93082e12	−	8.54043e12i	0.308994	−	0.535193i	0.978143	+	0.207936i	$0.0666745\pi$
$860$		0			0
$861$	1.16984e13	−	5.59968e12i	0.725457	−	0.347255i
$862$		0			0
$863$	−3.20684e12	+	5.55441e12i	−0.196802	+	0.340871i	−0.947490	−	0.319786i	$-0.896389\pi$
$863$	−3.20684e12	+	5.55441e12i	−0.196802	+	0.340871i	0.750688	+	0.660657i	$0.229722\pi$
$864$		0			0
$865$	3.01179e12	+	5.21658e12i	0.182916	+	0.316820i
$866$		0			0
$867$	−2.73012e13			−1.64095
$868$		0			0
$869$	−1.56714e12			−0.0932221
$870$		0			0
$871$	1.34182e13	+	2.32411e13i	0.789975	+	1.36828i
$872$		0			0
$873$	−3.26913e11	+	5.66230e11i	−0.0190488	+	0.0329935i
$874$		0			0
$875$	4.54187e11	−	5.87466e12i	0.0261938	−	0.338802i
$876$		0			0
$877$	9.75658e12	−	1.68989e13i	0.556928	−	0.964628i	−0.440822	−	0.897594i	$-0.645313\pi$
$877$	9.75658e12	−	1.68989e13i	0.556928	−	0.964628i	0.997751	−	0.0670339i	$-0.0213536\pi$
$878$		0			0
$879$	−1.79820e13	−	3.11458e13i	−1.01599	−	1.75975i
$880$		0			0
$881$	−2.42779e13			−1.35775			−0.678873	−	0.734255i	$-0.737532\pi$
$881$	−2.42779e13			−1.35775			−0.678873	+	0.734255i	$0.737532\pi$
$882$		0			0
$883$	−6.25628e12			−0.346332			−0.173166	−	0.984893i	$-0.555400\pi$
$883$	−6.25628e12			−0.346332			−0.173166	+	0.984893i	$0.555400\pi$
$884$		0			0
$885$	−2.00576e13	−	3.47408e13i	−1.09909	−	1.90368i
$886$		0			0
$887$	−1.64041e13	+	2.84127e13i	−0.889807	+	1.54119i	−0.0497050	+	0.998764i	$0.515828\pi$
$887$	−1.64041e13	+	2.84127e13i	−0.889807	+	1.54119i	−0.840102	+	0.542428i	$0.817505\pi$
$888$		0			0
$889$	−1.74497e12	+	2.25702e13i	−0.0936977	+	1.21193i
$890$		0			0
$891$	1.25825e12	−	2.17935e12i	0.0668832	−	0.115845i
$892$		0			0
$893$	2.21655e12	+	3.83918e12i	0.116640	+	0.202026i
$894$		0			0
$895$	4.37426e13			2.27877
$896$		0			0
$897$	3.42938e13			1.76868
$898$		0			0
$899$	−2.21937e12	−	3.84406e12i	−0.113321	−	0.196278i
$900$		0			0
$901$	−1.64682e13	+	2.85237e13i	−0.832499	+	1.44193i
$902$		0			0
$903$	−5.64464e12	+	2.70193e12i	−0.282515	+	0.135232i
$904$		0			0
$905$	−1.94323e13	+	3.36578e13i	−0.962956	+	1.66789i
$906$		0			0
$907$	1.03362e13	+	1.79029e13i	0.507143	+	0.878397i	0.999966	+	0.00826727i	$0.00263158\pi$
$907$	1.03362e13	+	1.79029e13i	0.507143	+	0.878397i	−0.492823	+	0.870129i	$0.664035\pi$
$908$		0			0
$909$	7.40243e12			0.359614
$910$		0			0
$911$	3.01133e12			0.144853			0.0724263	−	0.997374i	$-0.476926\pi$
$911$	3.01133e12			0.144853			0.0724263	+	0.997374i	$0.476926\pi$
$912$		0			0
$913$	−7.14829e11	−	1.23812e12i	−0.0340474	−	0.0589718i
$914$		0			0
$915$	−1.69809e13	+	2.94118e13i	−0.800877	+	1.38716i
$916$		0			0
$917$	2.61516e12	+	1.79204e12i	0.122134	+	0.0836924i
$918$		0			0
$919$	−1.19955e13	+	2.07768e13i	−0.554751	+	0.960856i	0.443172	+	0.896436i	$0.353853\pi$
$919$	−1.19955e13	+	2.07768e13i	−0.554751	+	0.960856i	−0.997923	+	0.0644199i	$0.979480\pi$
$920$		0			0
$921$	−1.85847e13	−	3.21897e13i	−0.851113	−	1.47417i
$922$		0			0
$923$	−1.26751e13			−0.574835
$924$		0			0
$925$	−3.50853e13			−1.57575
$926$		0			0
$927$	1.52570e12	+	2.64259e12i	0.0678594	+	0.117536i
$928$		0			0
$929$	−9.77237e12	+	1.69262e13i	−0.430457	+	0.745573i	−0.996913	−	0.0785193i	$-0.974981\pi$
$929$	−9.77237e12	+	1.69262e13i	−0.430457	+	0.745573i	0.566456	+	0.824092i	$0.308314\pi$
$930$		0			0
$931$	−2.26468e12	+	2.81249e12i	−0.0987946	+	0.122692i
$932$		0			0
$933$	−2.21127e12	+	3.83004e12i	−0.0955377	+	0.165476i
$934$		0			0
$935$	−3.18080e12	−	5.50930e12i	−0.136108	−	0.235746i
$936$		0			0
$937$	−8.02420e12			−0.340074			−0.170037	−	0.985438i	$-0.554389\pi$
$937$	−8.02420e12			−0.340074			−0.170037	+	0.985438i	$0.554389\pi$
$938$		0			0
$939$	2.23678e13			0.938919
$940$		0			0
$941$	−1.69527e13	−	2.93629e13i	−0.704831	−	1.22080i	−0.966752	−	0.255714i	$-0.917689\pi$
$941$	−1.69527e13	−	2.93629e13i	−0.704831	−	1.22080i	0.261921	−	0.965089i	$-0.415644\pi$
$942$		0			0
$943$	1.28554e13	−	2.22661e13i	0.529397	−	0.916942i
$944$		0			0
$945$	2.64758e13	+	1.81426e13i	1.07996	+	0.740041i
$946$		0			0
$947$	−4.11242e12	+	7.12291e12i	−0.166158	+	0.287795i	−0.937066	−	0.349152i	$-0.886470\pi$
$947$	−4.11242e12	+	7.12291e12i	−0.166158	+	0.287795i	0.770908	+	0.636947i	$0.219803\pi$
$948$		0			0
$949$	1.56331e13	+	2.70774e13i	0.625673	+	1.08370i
$950$		0			0
$951$	−2.46489e13			−0.977204
$952$		0			0
$953$	−1.97060e13			−0.773891			−0.386946	−	0.922102i	$-0.626470\pi$
$953$	−1.97060e13			−0.773891			−0.386946	+	0.922102i	$0.626470\pi$
$954$		0			0
$955$	6.68696e12	+	1.15822e13i	0.260144	+	0.450582i
$956$		0			0
$957$	−1.24515e12	+	2.15666e12i	−0.0479864	+	0.0831149i
$958$		0			0
$959$	−1.19005e13	+	5.69642e12i	−0.454340	+	0.217480i
$960$		0			0
$961$	1.20431e13	−	2.08593e13i	0.455495	−	0.788940i
$962$		0			0
$963$	2.28044e12	+	3.94983e12i	0.0854476	+	0.148000i
$964$		0			0
$965$	−2.22946e12			−0.0827614
$966$		0			0
$967$	−4.98171e13			−1.83214			−0.916072	−	0.401014i	$-0.868658\pi$
$967$	−4.98171e13			−1.83214			−0.916072	+	0.401014i	$0.868658\pi$
$968$		0			0
$969$	−3.74127e12	−	6.48008e12i	−0.136321	−	0.236115i
$970$		0			0
$971$	−7.04913e12	+	1.22095e13i	−0.254477	+	0.440768i	−0.964753	−	0.263156i	$-0.915237\pi$
$971$	−7.04913e12	+	1.22095e13i	−0.254477	+	0.440768i	0.710276	+	0.703923i	$0.248570\pi$
$972$		0			0
$973$	4.09583e12	−	5.29773e13i	0.146499	−	1.89488i
$974$		0			0
$975$	2.12534e13	−	3.68120e13i	0.753196	−	1.30457i
$976$		0			0
$977$	4.06319e12	+	7.03765e12i	0.142673	+	0.247117i	0.928502	−	0.371327i	$-0.121097\pi$
$977$	4.06319e12	+	7.03765e12i	0.142673	+	0.247117i	−0.785829	+	0.618443i	$0.787764\pi$
$978$		0			0
$979$	−6.50032e12			−0.226158
$980$		0			0
$981$	−7.63384e12			−0.263168
$982$		0			0
$983$	1.95336e13	+	3.38332e13i	0.667254	+	1.15572i	0.978669	+	0.205444i	$0.0658637\pi$
$983$	1.95336e13	+	3.38332e13i	0.667254	+	1.15572i	−0.311415	+	0.950274i	$0.600803\pi$
$984$		0			0
$985$	2.08413e13	−	3.60982e13i	0.705444	−	1.22186i
$986$		0			0
$987$	−3.72656e12	+	4.82010e13i	−0.124992	+	1.61670i
$988$		0			0
$989$	−6.20290e12	+	1.07437e13i	−0.206163	+	0.357085i
$990$		0			0
$991$	3.52693e12	+	6.10882e12i	0.116162	+	0.201199i	0.918244	−	0.396016i	$-0.129607\pi$
$991$	3.52693e12	+	6.10882e12i	0.116162	+	0.201199i	−0.802081	+	0.597215i	$0.796274\pi$
$992$		0			0
$993$	1.53035e13			0.499480
$994$		0			0
$995$	3.29835e13			1.06682
$996$		0			0
$997$	−8.54033e12	−	1.47923e13i	−0.273745	−	0.474140i	0.696073	−	0.717971i	$-0.254929\pi$
$997$	−8.54033e12	−	1.47923e13i	−0.273745	−	0.474140i	−0.969818	+	0.243831i	$0.921596\pi$
$998$		0			0
$999$	1.77212e13	−	3.06941e13i	0.562923	−	0.975012i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.f.65.7		18
4.3	odd	2		56.10.i.a.9.3	✓	18
7.4	even	3	inner	112.10.i.f.81.7		18
28.11	odd	6		56.10.i.a.25.3	yes	18
28.19	even	6		392.10.a.i.1.3		9
28.23	odd	6		392.10.a.l.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.10.i.a.9.3	✓	18	4.3	odd	2
56.10.i.a.25.3	yes	18	28.11	odd	6
112.10.i.f.65.7		18	1.1	even	1	trivial
112.10.i.f.81.7		18	7.4	even	3	inner
392.10.a.i.1.3		9	28.19	even	6
392.10.a.l.1.7		9	28.23	odd	6

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(76.8084 + 133.036i) q^{3} +(-1042.94 + 1806.43i) q^{5} +(489.665 - 6333.55i) q^{7} +(-1957.55 + 3390.58i) q^{9} +(-2801.38 - 4852.13i) q^{11} +115400. q^{13} -320427. q^{15} +(-272172. - 471416. i) q^{17} +(44741.2 - 77494.0i) q^{19} +(880200. - 421327. i) q^{21} +(967252. - 1.67533e6i) q^{23} +(-1.19890e6 - 2.07656e6i) q^{25} +2.42221e6 q^{27} -2.89342e6 q^{29} +(767039. + 1.32855e6i) q^{31} +(430339. - 745368. i) q^{33} +(1.09304e7 + 7.49008e6i) q^{35} +(7.31613e6 - 1.26719e7i) q^{37} +(8.86371e6 + 1.53524e7i) q^{39} +1.32906e7 q^{41} -6.41291e6 q^{43} +(-4.08323e6 - 7.07237e6i) q^{45} +(-2.47708e7 + 4.29043e7i) q^{47} +(-3.98741e7 - 6.20263e6i) q^{49} +(4.18102e7 - 7.24174e7i) q^{51} +(-3.02532e7 - 5.24001e7i) q^{53} +1.16867e7 q^{55} +1.37460e7 q^{57} +(6.25964e7 + 1.08420e8i) q^{59} +(5.29946e7 - 9.17893e7i) q^{61} +(2.05159e7 + 1.40585e7i) q^{63} +(-1.20356e8 + 2.08463e8i) q^{65} +(1.16276e8 + 2.01395e8i) q^{67} +2.97172e8 q^{69} -1.09836e8 q^{71} +(1.35469e8 + 2.34639e8i) q^{73} +(1.84171e8 - 3.18994e8i) q^{75} +(-3.21029e7 + 1.53667e7i) q^{77} +(1.39854e8 - 2.42235e8i) q^{79} +(2.24577e8 + 3.88978e8i) q^{81} +2.55171e8 q^{83} +1.13544e9 q^{85} +(-2.22239e8 - 3.84929e8i) q^{87} +(5.80101e8 - 1.00476e9i) q^{89} +(5.65075e7 - 7.30893e8i) q^{91} +(-1.17830e8 + 2.04088e8i) q^{93} +(9.33251e7 + 1.61644e8i) q^{95} +1.67001e8 q^{97} +2.19354e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 91 q^{3} + 801 q^{5} + 4820 q^{7} - 55886 q^{9} + 42719 q^{11} + 301620 q^{13} - 318842 q^{15} + 11357 q^{17} + 1237749 q^{19} - 352965 q^{21} - 95019 q^{23} - 848460 q^{25} - 983570 q^{27} + 12010116 q^{29}+ \cdots - 654760036 q^{99}+O(q^{100}) \) 18 * q + 91 * q^3 + 801 * q^5 + 4820 * q^7 - 55886 * q^9 + 42719 * q^11 + 301620 * q^13 - 318842 * q^15 + 11357 * q^17 + 1237749 * q^19 - 352965 * q^21 - 95019 * q^23 - 848460 * q^25 - 983570 * q^27 + 12010116 * q^29 + 10454399 * q^31 - 22110949 * q^33 + 14069675 * q^35 + 18487597 * q^37 - 26537570 * q^39 - 9446020 * q^41 - 15588088 * q^43 + 16445950 * q^45 - 6482527 * q^47 - 97119278 * q^49 + 91171769 * q^51 + 15314401 * q^53 + 27790478 * q^55 - 86196434 * q^57 + 84965083 * q^59 + 122850341 * q^61 - 682061514 * q^63 + 148004010 * q^65 + 27388339 * q^67 - 717062290 * q^69 + 473284416 * q^71 + 217201553 * q^73 - 86436700 * q^75 + 112464495 * q^77 + 326283565 * q^79 - 592248617 * q^81 - 1411349592 * q^83 + 1568978282 * q^85 + 787721718 * q^87 + 153096753 * q^89 + 878501576 * q^91 - 158673269 * q^93 - 1281844653 * q^95 + 919981372 * q^97 - 654760036 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more