LMFDB - Embedded newform 80.20.c.d.49.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(49,80)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 20, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("80.49");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$183.053357245$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.18
Character	$\chi$	$=$	80.49
Dual form			80.20.c.d.49.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15095.8i q^{3} +(913879. - 4.27063e6i) q^{5} -1.11343e8i q^{7} +9.34378e8 q^{9} +O(q^{10})$	\(q+15095.8i q^{3} +(913879. - 4.27063e6i) q^{5} -1.11343e8i q^{7} +9.34378e8 q^{9} +5.13079e9 q^{11} +2.56566e10i q^{13} +(6.44687e10 + 1.37957e10i) q^{15} +2.13846e11i q^{17} +1.45768e11 q^{19} +1.68081e12 q^{21} +1.33856e13i q^{23} +(-1.74031e13 - 7.80568e12i) q^{25} +3.16505e13i q^{27} -6.12640e13 q^{29} -5.73327e13 q^{31} +7.74534e13i q^{33} +(-4.75505e14 - 1.01754e14i) q^{35} +1.37048e14i q^{37} -3.87307e14 q^{39} -2.75660e15 q^{41} -1.86056e14i q^{43} +(8.53908e14 - 3.99039e15i) q^{45} +1.12347e16i q^{47} -9.98371e14 q^{49} -3.22817e15 q^{51} +1.94197e16i q^{53} +(4.68892e15 - 2.19117e16i) q^{55} +2.20049e15i q^{57} +2.98027e16 q^{59} +2.43149e16 q^{61} -1.04036e17i q^{63} +(1.09570e17 + 2.34470e16i) q^{65} +1.00771e17i q^{67} -2.02066e17 q^{69} +1.53588e16 q^{71} -4.79281e16i q^{73} +(1.17833e17 - 2.62715e17i) q^{75} -5.71277e17i q^{77} +2.02324e18 q^{79} +6.08202e17 q^{81} +8.87195e17i q^{83} +(9.13256e17 + 1.95429e17i) q^{85} -9.24830e17i q^{87} -3.09422e18 q^{89} +2.85668e18 q^{91} -8.65483e17i q^{93} +(1.33214e17 - 6.22523e17i) q^{95} +1.08500e19i q^{97} +4.79409e18 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 3935164 q^{5} - 10352560732 q^{9}+O(q^{10}) $ 28 * q - 3935164 * q^5 - 10352560732 * q^9	$ 28 q - 3935164 q^{5} - 10352560732 q^{9} - 18570995888 q^{11} - 79041415120 q^{15} + 1685869130672 q^{19} + 140419722832 q^{21} - 29558822439924 q^{25} + 160448197635496 q^{29} - 64251929934720 q^{31} + 88899634227664 q^{35} - 45\!\cdots\!04 q^{39}+ \cdots + 24\!\cdots\!92 q^{99}+O(q^{100}) $ 28 * q - 3935164 * q^5 - 10352560732 * q^9 - 18570995888 * q^11 - 79041415120 * q^15 + 1685869130672 * q^19 + 140419722832 * q^21 - 29558822439924 * q^25 + 160448197635496 * q^29 - 64251929934720 * q^31 + 88899634227664 * q^35 - 4563253596731104 * q^39 + 87868042694936 * q^41 + 3557698140217852 * q^45 - 28923781405647564 * q^49 - 14937379593456000 * q^51 - 28335039172010704 * q^55 + 83879503376835216 * q^59 + 2336306250785000 * q^61 + 23503427909569568 * q^65 - 46927873937137712 * q^69 - 104881848512783264 * q^71 - 1094024007292275040 * q^75 + 795721739900308672 * q^79 + 3593102866196377132 * q^81 + 834124526841485952 * q^85 + 1582238786752844568 * q^89 - 11017818924975052064 * q^91 - 30914057837649668144 * q^95 + 24571127463084047792 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			15095.8i			0.442797i	0.975183	+	0.221398i	$0.0710621\pi$
$3$			15095.8i			0.442797i	−0.975183	+	0.221398i	$0.928938\pi$
$4$		0			0
$5$	913879.	−	4.27063e6i	0.209254	−	0.977861i
$5$	913879.	−	4.27063e6i	0.209254	−	0.977861i
$6$		0			0
$7$		−	1.11343e8i		−	1.04287i	−0.853290	−	0.521437i	$-0.825396\pi$
$7$		−	1.11343e8i		−	1.04287i	0.853290	−	0.521437i	$-0.174604\pi$
$8$		0			0
$9$	9.34378e8			0.803931
$10$		0			0
$11$	5.13079e9			0.656075			0.328037	−	0.944665i	$-0.393613\pi$
$11$	5.13079e9			0.656075			0.328037	+	0.944665i	$0.393613\pi$
$12$		0			0
$13$			2.56566e10i			0.671023i	0.942036	+	0.335512i	$0.108909\pi$
$13$			2.56566e10i			0.671023i	−0.942036	+	0.335512i	$0.891091\pi$
$14$		0			0
$15$	6.44687e10	+	1.37957e10i	0.432994	+	0.0926570i
$16$		0			0
$17$			2.13846e11i			0.437356i	0.975797	+	0.218678i	$0.0701745\pi$
$17$			2.13846e11i			0.437356i	−0.975797	+	0.218678i	$0.929826\pi$
$18$		0			0
$19$	1.45768e11			0.103634			0.0518172	−	0.998657i	$-0.483499\pi$
$19$	1.45768e11			0.103634			0.0518172	+	0.998657i	$0.483499\pi$
$20$		0			0
$21$	1.68081e12			0.461781
$22$		0			0
$23$			1.33856e13i			1.54961i	0.632199	+	0.774806i	$0.282153\pi$
$23$			1.33856e13i			1.54961i	−0.632199	+	0.774806i	$0.717847\pi$
$24$		0			0
$25$	−1.74031e13	−	7.80568e12i	−0.912426	−	0.409242i
$26$		0			0
$27$			3.16505e13i			0.798775i
$28$		0			0
$29$	−6.12640e13			−0.784196			−0.392098	−	0.919923i	$-0.628251\pi$
$29$	−6.12640e13			−0.784196			−0.392098	+	0.919923i	$0.628251\pi$
$30$		0			0
$31$	−5.73327e13			−0.389463			−0.194731	−	0.980857i	$-0.562384\pi$
$31$	−5.73327e13			−0.389463			−0.194731	+	0.980857i	$0.562384\pi$
$32$		0			0
$33$			7.74534e13i			0.290508i
$34$		0			0
$35$	−4.75505e14	−	1.01754e14i	−1.01979	−	0.218225i
$36$		0			0
$37$			1.37048e14i			0.173363i	0.996236	+	0.0866815i	$0.0276262\pi$
$37$			1.37048e14i			0.173363i	−0.996236	+	0.0866815i	$0.972374\pi$
$38$		0			0
$39$	−3.87307e14			−0.297127
$40$		0			0
$41$	−2.75660e15			−1.31500			−0.657502	−	0.753453i	$-0.728387\pi$
$41$	−2.75660e15			−1.31500			−0.657502	+	0.753453i	$0.728387\pi$
$42$		0			0
$43$		−	1.86056e14i		−	0.0564538i	−0.999602	−	0.0282269i	$-0.991014\pi$
$43$		−	1.86056e14i		−	0.0564538i	0.999602	−	0.0282269i	$-0.00898609\pi$
$44$		0			0
$45$	8.53908e14	−	3.99039e15i	0.168226	−	0.786133i
$46$		0			0
$47$			1.12347e16i			1.46431i	0.681137	+	0.732156i	$0.261486\pi$
$47$			1.12347e16i			1.46431i	−0.681137	+	0.732156i	$0.738514\pi$
$48$		0			0
$49$	−9.98371e14			−0.0875849
$50$		0			0
$51$	−3.22817e15			−0.193660
$52$		0			0
$53$			1.94197e16i			0.808392i	0.914672	+	0.404196i	$0.132449\pi$
$53$			1.94197e16i			0.808392i	−0.914672	+	0.404196i	$0.867551\pi$
$54$		0			0
$55$	4.68892e15	−	2.19117e16i	0.137286	−	0.641550i
$56$		0			0
$57$			2.20049e15i			0.0458890i
$58$		0			0
$59$	2.98027e16			0.447880			0.223940	−	0.974603i	$-0.428108\pi$
$59$	2.98027e16			0.447880			0.223940	+	0.974603i	$0.428108\pi$
$60$		0			0
$61$	2.43149e16			0.266218			0.133109	−	0.991101i	$-0.457504\pi$
$61$	2.43149e16			0.266218			0.133109	+	0.991101i	$0.457504\pi$
$62$		0			0
$63$		−	1.04036e17i		−	0.838398i
$64$		0			0
$65$	1.09570e17	+	2.34470e16i	0.656168	+	0.140414i
$66$		0			0
$67$			1.00771e17i			0.452506i	0.974069	+	0.226253i	$0.0726476\pi$
$67$			1.00771e17i			0.452506i	−0.974069	+	0.226253i	$0.927352\pi$
$68$		0			0
$69$	−2.02066e17			−0.686164
$70$		0			0
$71$	1.53588e16			0.0397561			0.0198780	−	0.999802i	$-0.493672\pi$
$71$	1.53588e16			0.0397561			0.0198780	+	0.999802i	$0.493672\pi$
$72$		0			0
$73$		−	4.79281e16i		−	0.0952848i	−0.998864	−	0.0476424i	$-0.984829\pi$
$73$		−	4.79281e16i		−	0.0952848i	0.998864	−	0.0476424i	$-0.0151708\pi$
$74$		0			0
$75$	1.17833e17	−	2.62715e17i	0.181211	−	0.404019i
$76$		0			0
$77$		−	5.71277e17i		−	0.684203i
$78$		0			0
$79$	2.02324e18			1.89929			0.949644	−	0.313330i	$-0.101445\pi$
$79$	2.02324e18			1.89929			0.949644	+	0.313330i	$0.101445\pi$
$80$		0			0
$81$	6.08202e17			0.450236
$82$		0			0
$83$			8.87195e17i			0.520927i	0.965484	+	0.260464i	$0.0838754\pi$
$83$			8.87195e17i			0.520927i	−0.965484	+	0.260464i	$0.916125\pi$
$84$		0			0
$85$	9.13256e17	+	1.95429e17i	0.427674	+	0.0915185i
$86$		0			0
$87$		−	9.24830e17i		−	0.347239i
$88$		0			0
$89$	−3.09422e18			−0.936151			−0.468075	−	0.883689i	$-0.655052\pi$
$89$	−3.09422e18			−0.936151			−0.468075	+	0.883689i	$0.655052\pi$
$90$		0			0
$91$	2.85668e18			0.699792
$92$		0			0
$93$		−	8.65483e17i		−	0.172453i
$94$		0			0
$95$	1.33214e17	−	6.22523e17i	0.0216859	−	0.101340i
$96$		0			0
$97$			1.08500e19i			1.44910i	0.689222	+	0.724550i	$0.257953\pi$
$97$			1.08500e19i			1.44910i	−0.689222	+	0.724550i	$0.742047\pi$
$98$		0			0
$99$	4.79409e18			0.527439
$100$		0			0
$101$	−1.81037e19			−1.64708			−0.823540	−	0.567258i	$-0.808004\pi$
$101$	−1.81037e19			−1.64708			−0.823540	+	0.567258i	$0.808004\pi$
$102$		0			0
$103$		−	1.13967e19i		−	0.860646i	−0.902675	−	0.430323i	$-0.858400\pi$
$103$		−	1.13967e19i		−	0.860646i	0.902675	−	0.430323i	$-0.141600\pi$
$104$		0			0
$105$	1.53606e18	−	7.17814e18i	0.0966295	−	0.451558i
$106$		0			0
$107$			8.79246e18i			0.462343i	0.972913	+	0.231172i	$0.0742559\pi$
$107$			8.79246e18i			0.462343i	−0.972913	+	0.231172i	$0.925744\pi$
$108$		0			0
$109$	−4.03734e19			−1.78051			−0.890254	−	0.455464i	$-0.849473\pi$
$109$	−4.03734e19			−1.78051			−0.890254	+	0.455464i	$0.849473\pi$
$110$		0			0
$111$	−2.06885e18			−0.0767646
$112$		0			0
$113$		−	2.36433e18i		−	0.0740395i	−0.999315	−	0.0370197i	$-0.988214\pi$
$113$		−	2.36433e18i		−	0.0740395i	0.999315	−	0.0370197i	$-0.0117864\pi$
$114$		0			0
$115$	5.71650e19	+	1.22328e19i	1.51531	+	0.324262i
$116$		0			0
$117$			2.39730e19i			0.539456i
$118$		0			0
$119$	2.38102e19			0.456107
$120$		0			0
$121$	−3.48341e19			−0.569566
$122$		0			0
$123$		−	4.16131e19i		−	0.582279i
$124$		0			0
$125$	−4.92396e19	+	6.71890e19i	−0.591111	+	0.806590i
$126$		0			0
$127$		−	2.00200e19i		−	0.206695i	−0.994645	−	0.103347i	$-0.967045\pi$
$127$		−	2.00200e19i		−	0.206695i	0.994645	−	0.103347i	$-0.0329553\pi$
$128$		0			0
$129$	2.80867e18			0.0249976
$130$		0			0
$131$	2.05795e20			1.58255			0.791274	−	0.611462i	$-0.209418\pi$
$131$	2.05795e20			1.58255			0.791274	+	0.611462i	$0.209418\pi$
$132$		0			0
$133$		−	1.62303e19i		−	0.108078i
$134$		0			0
$135$	1.35168e20	+	2.89247e19i	0.781091	+	0.167147i
$136$		0			0
$137$		−	2.09846e20i		−	1.05452i	−0.849703	−	0.527261i	$-0.823219\pi$
$137$		−	2.09846e20i		−	1.05452i	0.849703	−	0.527261i	$-0.176781\pi$
$138$		0			0
$139$	−1.10548e20			−0.484073			−0.242036	−	0.970267i	$-0.577815\pi$
$139$	−1.10548e20			−0.484073			−0.242036	+	0.970267i	$0.577815\pi$
$140$		0			0
$141$	−1.69598e20			−0.648393
$142$		0			0
$143$			1.31639e20i			0.440241i
$144$		0			0
$145$	−5.59879e19	+	2.61636e20i	−0.164096	+	0.766835i
$146$		0			0
$147$		−	1.50712e19i		−	0.0387823i
$148$		0			0
$149$	−4.66448e20			−1.05568			−0.527842	−	0.849343i	$-0.676999\pi$
$149$	−4.66448e20			−1.05568			−0.527842	+	0.849343i	$0.676999\pi$
$150$		0			0
$151$	−3.09887e20			−0.617907			−0.308953	−	0.951077i	$-0.599979\pi$
$151$	−3.09887e20			−0.617907			−0.308953	+	0.951077i	$0.599979\pi$
$152$		0			0
$153$			1.99813e20i			0.351604i
$154$		0			0
$155$	−5.23951e19	+	2.44847e20i	−0.0814966	+	0.380841i
$156$		0			0
$157$			1.11379e21i			1.53376i	0.641793	+	0.766878i	$0.278191\pi$
$157$			1.11379e21i			1.53376i	−0.641793	+	0.766878i	$0.721809\pi$
$158$		0			0
$159$	−2.93156e20			−0.357953
$160$		0			0
$161$	1.49039e21			1.61605
$162$		0			0
$163$		−	9.10162e20i		−	0.877680i	−0.898565	−	0.438840i	$-0.855389\pi$
$163$		−	9.10162e20i		−	0.877680i	0.898565	−	0.438840i	$-0.144611\pi$
$164$		0			0
$165$	3.30775e20	+	7.07830e19i	0.284076	+	0.0607899i
$166$		0			0
$167$			1.63033e21i			1.24873i	0.781133	+	0.624365i	$0.214642\pi$
$167$			1.63033e21i			1.24873i	−0.781133	+	0.624365i	$0.785358\pi$
$168$		0			0
$169$	8.03658e20			0.549728
$170$		0			0
$171$	1.36203e20			0.0833149
$172$		0			0
$173$		−	2.93415e21i		−	1.60710i	−0.595235	−	0.803552i	$-0.702941\pi$
$173$		−	2.93415e21i		−	1.60710i	0.595235	−	0.803552i	$-0.297059\pi$
$174$		0			0
$175$	−8.69108e20	+	1.93772e21i	−0.426788	+	0.951544i
$176$		0			0
$177$			4.49896e20i			0.198320i
$178$		0			0
$179$	2.60985e21			1.03398			0.516989	−	0.855992i	$-0.327053\pi$
$179$	2.60985e21			1.03398			0.516989	+	0.855992i	$0.327053\pi$
$180$		0			0
$181$	4.58307e21			1.63384			0.816922	−	0.576749i	$-0.195679\pi$
$181$	4.58307e21			1.63384			0.816922	+	0.576749i	$0.195679\pi$
$182$		0			0
$183$			3.67052e20i			0.117881i
$184$		0			0
$185$	5.85282e20	+	1.25245e20i	0.169525	+	0.0362769i
$186$		0			0
$187$			1.09720e21i			0.286938i
$188$		0			0
$189$	3.52406e21			0.833021
$190$		0			0
$191$	−8.88624e21			−1.90065			−0.950323	−	0.311267i	$-0.899247\pi$
$191$	−8.88624e21			−1.90065			−0.950323	+	0.311267i	$0.899247\pi$
$192$		0			0
$193$			5.59552e21i			1.08404i	0.840365	+	0.542021i	$0.182341\pi$
$193$			5.59552e21i			1.08404i	−0.840365	+	0.542021i	$0.817659\pi$
$194$		0			0
$195$	−3.53952e20	+	1.65405e21i	−0.0621750	+	0.290549i
$196$		0			0
$197$			5.28706e21i			0.842918i	0.906847	+	0.421459i	$0.138482\pi$
$197$			5.28706e21i			0.842918i	−0.906847	+	0.421459i	$0.861518\pi$
$198$		0			0
$199$	−4.79409e21			−0.694387			−0.347194	−	0.937794i	$-0.612865\pi$
$199$	−4.79409e21			−0.694387			−0.347194	+	0.937794i	$0.612865\pi$
$200$		0			0
$201$	−1.52122e21			−0.200368
$202$		0			0
$203$			6.82132e21i			0.817817i
$204$		0			0
$205$	−2.51920e21	+	1.17724e22i	−0.275169	+	1.28589i
$206$		0			0
$207$			1.25072e22i			1.24578i
$208$		0			0
$209$	7.47906e20			0.0679919
$210$		0			0
$211$	2.00210e21			0.166265			0.0831326	−	0.996538i	$-0.473507\pi$
$211$	2.00210e21			0.166265			0.0831326	+	0.996538i	$0.473507\pi$
$212$		0			0
$213$			2.31854e20i			0.0176039i
$214$		0			0
$215$	−7.94577e20	−	1.70033e20i	−0.0552040	−	0.0118132i
$216$		0			0
$217$			6.38359e21i			0.406160i
$218$		0			0
$219$	7.23514e20			0.0421918
$220$		0			0
$221$	−5.48655e21			−0.293476
$222$		0			0
$223$			9.74077e21i			0.478296i	0.970983	+	0.239148i	$0.0768682\pi$
$223$			9.74077e21i			0.478296i	−0.970983	+	0.239148i	$0.923132\pi$
$224$		0			0
$225$	−1.62611e22	−	7.29346e21i	−0.733527	−	0.329003i
$226$		0			0
$227$			1.43612e22i			0.595586i	0.954630	+	0.297793i	$0.0962506\pi$
$227$			1.43612e22i			0.595586i	−0.954630	+	0.297793i	$0.903749\pi$
$228$		0			0
$229$	1.07150e22			0.408843			0.204421	−	0.978883i	$-0.434469\pi$
$229$	1.07150e22			0.408843			0.204421	+	0.978883i	$0.434469\pi$
$230$		0			0
$231$	8.62389e21			0.302963
$232$		0			0
$233$			4.52275e22i			1.46393i	0.681340	+	0.731967i	$0.261398\pi$
$233$			4.52275e22i			1.46393i	−0.681340	+	0.731967i	$0.738602\pi$
$234$		0			0
$235$	4.79795e22	+	1.02672e22i	1.43189	+	0.306413i
$236$		0			0
$237$			3.05425e22i			0.840999i
$238$		0			0
$239$	6.23485e22			1.58506			0.792530	−	0.609833i	$-0.208763\pi$
$239$	6.23485e22			1.58506			0.792530	+	0.609833i	$0.208763\pi$
$240$		0			0
$241$	4.65886e21			0.109425			0.0547127	−	0.998502i	$-0.482576\pi$
$241$	4.65886e21			0.109425			0.0547127	+	0.998502i	$0.482576\pi$
$242$		0			0
$243$			4.59674e22i			0.998138i
$244$		0			0
$245$	−9.12390e20	+	4.26368e21i	−0.0183275	+	0.0856459i
$246$		0			0
$247$			3.73992e21i			0.0695411i
$248$		0			0
$249$	−1.33929e22			−0.230665
$250$		0			0
$251$	3.37859e22			0.539306			0.269653	−	0.962958i	$-0.413091\pi$
$251$	3.37859e22			0.539306			0.269653	+	0.962958i	$0.413091\pi$
$252$		0			0
$253$			6.86786e22i			1.01666i
$254$		0			0
$255$	−2.95016e21	+	1.37863e22i	−0.0405241	+	0.189373i
$256$		0			0
$257$			3.92808e22i			0.500974i	0.968120	+	0.250487i	$0.0805907\pi$
$257$			3.92808e22i			0.500974i	−0.968120	+	0.250487i	$0.919409\pi$
$258$		0			0
$259$	1.52593e22			0.180796
$260$		0			0
$261$	−5.72437e22			−0.630439
$262$		0			0
$263$			7.17854e22i			0.735286i	0.929967	+	0.367643i	$0.119835\pi$
$263$			7.17854e22i			0.735286i	−0.929967	+	0.367643i	$0.880165\pi$
$264$		0			0
$265$	8.29345e22	+	1.77473e22i	0.790495	+	0.169159i
$266$		0			0
$267$		−	4.67097e22i		−	0.414525i
$268$		0			0
$269$	−1.95368e23			−1.61512			−0.807562	−	0.589783i	$-0.799213\pi$
$269$	−1.95368e23			−1.61512			−0.807562	+	0.589783i	$0.799213\pi$
$270$		0			0
$271$	1.44202e23			1.11112			0.555562	−	0.831475i	$-0.312503\pi$
$271$	1.44202e23			1.11112			0.555562	+	0.831475i	$0.312503\pi$
$272$		0			0
$273$			4.31240e22i			0.309866i
$274$		0			0
$275$	−8.92918e22	−	4.00493e22i	−0.598620	−	0.268494i
$276$		0			0
$277$			4.08455e22i			0.255615i	0.991799	+	0.127808i	$0.0407940\pi$
$277$			4.08455e22i			0.255615i	−0.991799	+	0.127808i	$0.959206\pi$
$278$		0			0
$279$	−5.35704e22			−0.313101
$280$		0			0
$281$	1.90463e23			1.04016			0.520080	−	0.854118i	$-0.325902\pi$
$281$	1.90463e23			1.04016			0.520080	+	0.854118i	$0.325902\pi$
$282$		0			0
$283$			1.94703e23i			0.994036i	0.867740	+	0.497018i	$0.165572\pi$
$283$			1.94703e23i			0.994036i	−0.867740	+	0.497018i	$0.834428\pi$
$284$		0			0
$285$	9.39749e21	+	2.01098e21i	0.0448731	+	0.00960244i
$286$		0			0
$287$			3.06928e23i			1.37138i
$288$		0			0
$289$	1.93343e23			0.808719
$290$		0			0
$291$	−1.63790e23			−0.641657
$292$		0			0
$293$			1.15214e23i			0.422926i	0.977386	+	0.211463i	$0.0678227\pi$
$293$			1.15214e23i			0.422926i	−0.977386	+	0.211463i	$0.932177\pi$
$294$		0			0
$295$	2.72360e22	−	1.27276e23i	0.0937205	−	0.437964i
$296$		0			0
$297$			1.62392e23i			0.524056i
$298$		0			0
$299$	−3.43429e23			−1.03983
$300$		0			0
$301$	−2.07160e22			−0.0588742
$302$		0			0
$303$		−	2.73290e23i		−	0.729322i
$304$		0			0
$305$	2.22208e22	−	1.03840e23i	0.0557072	−	0.260325i
$306$		0			0
$307$			3.17509e23i			0.748068i	0.927415	+	0.374034i	$0.122026\pi$
$307$			3.17509e23i			0.748068i	−0.927415	+	0.374034i	$0.877974\pi$
$308$		0			0
$309$	1.72042e23			0.381091
$310$		0			0
$311$	−9.54150e22			−0.198789			−0.0993946	−	0.995048i	$-0.531691\pi$
$311$	−9.54150e22			−0.198789			−0.0993946	+	0.995048i	$0.531691\pi$
$312$		0			0
$313$		−	6.83121e23i		−	1.33914i	−0.742748	−	0.669571i	$-0.766478\pi$
$313$		−	6.83121e23i		−	1.33914i	0.742748	−	0.669571i	$-0.233522\pi$
$314$		0			0
$315$	−4.44302e23	−	9.50767e22i	−0.819837	−	0.175438i
$316$		0			0
$317$		−	2.09244e23i		−	0.363572i	−0.983338	−	0.181786i	$-0.941812\pi$
$317$		−	2.09244e23i		−	0.363572i	0.983338	−	0.181786i	$-0.0581879\pi$
$318$		0			0
$319$	−3.14333e23			−0.514491
$320$		0			0
$321$	−1.32729e23			−0.204724
$322$		0			0
$323$			3.11719e22i			0.0453251i
$324$		0			0
$325$	2.00267e23	−	4.46506e23i	0.274611	−	0.612259i
$326$		0			0
$327$		−	6.09470e23i		−	0.788404i
$328$		0			0
$329$	1.25091e24			1.52709
$330$		0			0
$331$	3.13218e22			0.0360978			0.0180489	−	0.999837i	$-0.494255\pi$
$331$	3.13218e22			0.0360978			0.0180489	+	0.999837i	$0.494255\pi$
$332$		0			0
$333$			1.28055e23i			0.139372i
$334$		0			0
$335$	4.30355e23	+	9.20923e22i	0.442488	+	0.0946886i
$336$		0			0
$337$		−	2.60372e22i		−	0.0252994i	−0.999920	−	0.0126497i	$-0.995973\pi$
$337$		−	2.60372e22i		−	0.0252994i	0.999920	−	0.0126497i	$-0.00402663\pi$
$338$		0			0
$339$	3.56915e22			0.0327845
$340$		0			0
$341$	−2.94162e23			−0.255517
$342$		0			0
$343$		−	1.15803e24i		−	0.951533i
$344$		0			0
$345$	−1.84664e23	+	8.62952e23i	−0.143582	+	0.670973i
$346$		0			0
$347$		−	1.38422e24i		−	1.01877i	−0.860540	−	0.509383i	$-0.829874\pi$
$347$		−	1.38422e24i		−	1.01877i	0.860540	−	0.509383i	$-0.170126\pi$
$348$		0			0
$349$	−3.12211e23			−0.217574			−0.108787	−	0.994065i	$-0.534697\pi$
$349$	−3.12211e23			−0.217574			−0.108787	+	0.994065i	$0.534697\pi$
$350$		0			0
$351$	−8.12044e23			−0.535997
$352$		0			0
$353$		−	2.44797e24i		−	1.53090i	−0.643496	−	0.765450i	$-0.722517\pi$
$353$		−	2.44797e24i		−	1.53090i	0.643496	−	0.765450i	$-0.277483\pi$
$354$		0			0
$355$	1.40361e22	−	6.55919e22i	0.00831911	−	0.0388759i
$356$		0			0
$357$			3.59434e23i			0.201963i
$358$		0			0
$359$	1.21046e24			0.644991			0.322495	−	0.946571i	$-0.395478\pi$
$359$	1.21046e24			0.644991			0.322495	+	0.946571i	$0.395478\pi$
$360$		0			0
$361$	−1.95717e24			−0.989260
$362$		0			0
$363$		−	5.25849e23i		−	0.252202i
$364$		0			0
$365$	−2.04684e23	−	4.38005e22i	−0.0931753	−	0.0199387i
$366$		0			0
$367$			8.61502e23i			0.372331i	0.982518	+	0.186165i	$0.0596060\pi$
$367$			8.61502e23i			0.372331i	−0.982518	+	0.186165i	$0.940394\pi$
$368$		0			0
$369$	−2.57570e24			−1.05717
$370$		0			0
$371$	2.16225e24			0.843050
$372$		0			0
$373$		−	5.22984e24i		−	1.93756i	−0.247925	−	0.968779i	$-0.579749\pi$
$373$		−	5.22984e24i		−	1.93756i	0.247925	−	0.968779i	$-0.420251\pi$
$374$		0			0
$375$	−1.01427e24	−	7.43311e23i	−0.357156	−	0.261742i
$376$		0			0
$377$		−	1.57183e24i		−	0.526214i
$378$		0			0
$379$	3.30936e24			1.05359			0.526795	−	0.849993i	$-0.323394\pi$
$379$	3.30936e24			1.05359			0.526795	+	0.849993i	$0.323394\pi$
$380$		0			0
$381$	3.02219e23			0.0915238
$382$		0			0
$383$		−	3.35295e24i		−	0.966138i	−0.875582	−	0.483069i	$-0.839522\pi$
$383$		−	3.35295e24i		−	0.966138i	0.875582	−	0.483069i	$-0.160478\pi$
$384$		0			0
$385$	−2.43972e24	−	5.22078e23i	−0.669056	−	0.143172i
$386$		0			0
$387$		−	1.73847e23i		−	0.0453850i
$388$		0			0
$389$	−5.25265e24			−1.30574			−0.652871	−	0.757469i	$-0.726436\pi$
$389$	−5.25265e24			−1.30574			−0.652871	+	0.757469i	$0.726436\pi$
$390$		0			0
$391$	−2.86245e24			−0.677733
$392$		0			0
$393$			3.10664e24i			0.700747i
$394$		0			0
$395$	1.84900e24	−	8.64053e24i	0.397433	−	1.85724i
$396$		0			0
$397$			4.47854e24i			0.917544i	0.888554	+	0.458772i	$0.151710\pi$
$397$			4.47854e24i			0.917544i	−0.888554	+	0.458772i	$0.848290\pi$
$398$		0			0
$399$	2.45009e23			0.0478564
$400$		0			0
$401$	4.13448e24			0.770104			0.385052	−	0.922895i	$-0.374184\pi$
$401$	4.13448e24			0.770104			0.385052	+	0.922895i	$0.374184\pi$
$402$		0			0
$403$		−	1.47096e24i		−	0.261339i
$404$		0			0
$405$	5.55823e23	−	2.59741e24i	0.0942136	−	0.440268i
$406$		0			0
$407$			7.03164e23i			0.113739i
$408$		0			0
$409$	2.40840e24			0.371841			0.185920	−	0.982565i	$-0.440473\pi$
$409$	2.40840e24			0.371841			0.185920	+	0.982565i	$0.440473\pi$
$410$		0			0
$411$	3.16780e24			0.466939
$412$		0			0
$413$		−	3.31832e24i		−	0.467082i
$414$		0			0
$415$	3.78888e24	+	8.10788e23i	0.509395	+	0.109006i
$416$		0			0
$417$		−	1.66881e24i		−	0.214346i
$418$		0			0
$419$	−5.23756e24			−0.642830			−0.321415	−	0.946938i	$-0.604158\pi$
$419$	−5.23756e24			−0.642830			−0.321415	+	0.946938i	$0.604158\pi$
$420$		0			0
$421$	−9.82692e23			−0.115276			−0.0576378	−	0.998338i	$-0.518357\pi$
$421$	−9.82692e23			−0.115276			−0.0576378	+	0.998338i	$0.518357\pi$
$422$		0			0
$423$			1.04975e25i			1.17721i
$424$		0			0
$425$	1.66921e24	−	3.72158e24i	0.178985	−	0.399055i
$426$		0			0
$427$		−	2.70729e24i		−	0.277632i
$428$		0			0
$429$	−1.98719e24			−0.194938
$430$		0			0
$431$	−6.32350e23			−0.0593504			−0.0296752	−	0.999560i	$-0.509447\pi$
$431$	−6.32350e23			−0.0593504			−0.0296752	+	0.999560i	$0.509447\pi$
$432$		0			0
$433$		−	4.44700e24i		−	0.399422i	−0.979855	−	0.199711i	$-0.936000\pi$
$433$		−	4.44700e24i		−	0.399422i	0.979855	−	0.199711i	$-0.0640004\pi$
$434$		0			0
$435$	−3.94961e24	−	8.45182e23i	−0.339552	−	0.0726612i
$436$		0			0
$437$			1.95119e24i			0.160593i
$438$		0			0
$439$	−1.90625e25			−1.50233			−0.751167	−	0.660112i	$-0.770509\pi$
$439$	−1.90625e25			−1.50233			−0.751167	+	0.660112i	$0.770509\pi$
$440$		0			0
$441$	−9.32856e23			−0.0704122
$442$		0			0
$443$			2.34035e25i			1.69218i	0.533041	+	0.846089i	$0.321049\pi$
$443$			2.34035e25i			1.69218i	−0.533041	+	0.846089i	$0.678951\pi$
$444$		0			0
$445$	−2.82774e24	+	1.32143e25i	−0.195893	+	0.915426i
$446$		0			0
$447$		−	7.04141e24i		−	0.467453i
$448$		0			0
$449$	1.52285e25			0.968983			0.484492	−	0.874796i	$-0.339005\pi$
$449$	1.52285e25			0.968983			0.484492	+	0.874796i	$0.339005\pi$
$450$		0			0
$451$	−1.41435e25			−0.862740
$452$		0			0
$453$		−	4.67800e24i		−	0.273607i
$454$		0			0
$455$	2.61066e24	−	1.21999e25i	0.146434	−	0.684300i
$456$		0			0
$457$			1.89866e25i			1.02151i	0.859725	+	0.510757i	$0.170635\pi$
$457$			1.89866e25i			1.02151i	−0.859725	+	0.510757i	$0.829365\pi$
$458$		0			0
$459$	−6.76831e24			−0.349349
$460$		0			0
$461$	2.73600e25			1.35506			0.677528	−	0.735497i	$-0.263051\pi$
$461$	2.73600e25			1.35506			0.677528	+	0.735497i	$0.263051\pi$
$462$		0			0
$463$			1.60950e25i			0.765019i	0.923952	+	0.382509i	$0.124940\pi$
$463$			1.60950e25i			0.765019i	−0.923952	+	0.382509i	$0.875060\pi$
$464$		0			0
$465$	−3.69616e24	−	7.90946e23i	−0.168635	−	0.0360864i
$466$		0			0
$467$			2.29748e25i			1.00633i	0.864190	+	0.503165i	$0.167831\pi$
$467$			2.29748e25i			1.00633i	−0.864190	+	0.503165i	$0.832169\pi$
$468$		0			0
$469$	1.12201e25			0.471906
$470$		0			0
$471$	−1.68135e25			−0.679142
$472$		0			0
$473$		−	9.54613e23i		−	0.0370379i
$474$		0			0
$475$	−2.53683e24	−	1.13782e24i	−0.0945586	−	0.0424116i
$476$		0			0
$477$			1.81453e25i			0.649891i
$478$		0			0
$479$	5.56416e25			1.91519			0.957595	−	0.288117i	$-0.0930292\pi$
$479$	5.56416e25			1.91519			0.957595	+	0.288117i	$0.0930292\pi$
$480$		0			0
$481$	−3.51619e24			−0.116331
$482$		0			0
$483$			2.24987e25i			0.715582i
$484$		0			0
$485$	4.63364e25	+	9.91559e24i	1.41702	+	0.303230i
$486$		0			0
$487$			1.13213e24i			0.0332943i	0.999861	+	0.0166471i	$0.00529920\pi$
$487$			1.13213e24i			0.0332943i	−0.999861	+	0.0166471i	$0.994701\pi$
$488$		0			0
$489$	1.37396e25			0.388634
$490$		0			0
$491$	4.95007e25			1.34690			0.673452	−	0.739231i	$-0.264811\pi$
$491$	4.95007e25			1.34690			0.673452	+	0.739231i	$0.264811\pi$
$492$		0			0
$493$		−	1.31010e25i		−	0.342973i
$494$		0			0
$495$	4.38122e24	−	2.04738e25i	0.110369	−	0.515762i
$496$		0			0
$497$		−	1.71010e24i		−	0.0414606i
$498$		0			0
$499$	−1.16202e25			−0.271180			−0.135590	−	0.990765i	$-0.543293\pi$
$499$	−1.16202e25			−0.271180			−0.135590	+	0.990765i	$0.543293\pi$
$500$		0			0
$501$	−2.46111e25			−0.552934
$502$		0			0
$503$			3.58365e25i			0.775228i	0.921822	+	0.387614i	$0.126701\pi$
$503$			3.58365e25i			0.775228i	−0.921822	+	0.387614i	$0.873299\pi$
$504$		0			0
$505$	−1.65446e25	+	7.73143e25i	−0.344658	+	1.61062i
$506$		0			0
$507$			1.21319e25i			0.243418i
$508$		0			0
$509$	−5.60623e24			−0.108356			−0.0541779	−	0.998531i	$-0.517254\pi$
$509$	−5.60623e24			−0.108356			−0.0541779	+	0.998531i	$0.517254\pi$
$510$		0			0
$511$	−5.33646e24			−0.0993700
$512$		0			0
$513$			4.61363e24i			0.0827805i
$514$		0			0
$515$	−4.86710e25	−	1.04152e25i	−0.841592	−	0.180093i
$516$		0			0
$517$			5.76431e25i			0.960699i
$518$		0			0
$519$	4.42933e25			0.711620
$520$		0			0
$521$	−8.55203e25			−1.32468			−0.662339	−	0.749204i	$-0.730436\pi$
$521$	−8.55203e25			−1.32468			−0.662339	+	0.749204i	$0.730436\pi$
$522$		0			0
$523$			6.83826e25i			1.02136i	0.859770	+	0.510681i	$0.170607\pi$
$523$			6.83826e25i			1.02136i	−0.859770	+	0.510681i	$0.829393\pi$
$524$		0			0
$525$	−2.92514e25	−	1.31199e25i	−0.421341	−	0.188980i
$526$		0			0
$527$		−	1.22603e25i		−	0.170334i
$528$		0			0
$529$	−1.04559e26			−1.40130
$530$		0			0
$531$	2.78470e25			0.360064
$532$		0			0
$533$		−	7.07250e25i		−	0.882398i
$534$		0			0
$535$	3.75494e25	+	8.03524e24i	0.452107	+	0.0967471i
$536$		0			0
$537$			3.93978e25i			0.457843i
$538$		0			0
$539$	−5.12243e24			−0.0574622
$540$		0			0
$541$	−1.58765e25			−0.171942			−0.0859709	−	0.996298i	$-0.527399\pi$
$541$	−1.58765e25			−0.171942			−0.0859709	+	0.996298i	$0.527399\pi$
$542$		0			0
$543$			6.91852e25i			0.723461i
$544$		0			0
$545$	−3.68964e25	+	1.72420e26i	−0.372578	+	1.74109i
$546$		0			0
$547$		−	5.72050e25i		−	0.557897i	−0.960306	−	0.278949i	$-0.910014\pi$
$547$		−	5.72050e25i		−	0.557897i	0.960306	−	0.278949i	$-0.0899859\pi$
$548$		0			0
$549$	2.27193e25			0.214021
$550$		0			0
$551$	−8.93035e24			−0.0812696
$552$		0			0
$553$		−	2.25274e26i		−	1.98072i
$554$		0			0
$555$	−1.89068e24	+	8.83530e24i	−0.0160633	+	0.0750651i
$556$		0			0
$557$			1.53509e26i			1.26041i	0.776430	+	0.630203i	$0.217028\pi$
$557$			1.53509e26i			1.26041i	−0.776430	+	0.630203i	$0.782972\pi$
$558$		0			0
$559$	4.77357e24			0.0378818
$560$		0			0
$561$	−1.65631e25			−0.127055
$562$		0			0
$563$			1.54528e25i			0.114598i	0.998357	+	0.0572992i	$0.0182489\pi$
$563$			1.54528e25i			0.114598i	−0.998357	+	0.0572992i	$0.981751\pi$
$564$		0			0
$565$	−1.00972e25	−	2.16071e24i	−0.0724003	−	0.0154930i
$566$		0			0
$567$		−	6.77190e25i		−	0.469539i
$568$		0			0
$569$	1.31547e26			0.882090			0.441045	−	0.897485i	$-0.354608\pi$
$569$	1.31547e26			0.882090			0.441045	+	0.897485i	$0.354608\pi$
$570$		0			0
$571$	1.87201e26			1.21413			0.607064	−	0.794653i	$-0.292347\pi$
$571$	1.87201e26			1.21413			0.607064	+	0.794653i	$0.292347\pi$
$572$		0			0
$573$		−	1.34145e26i		−	0.841600i
$574$		0			0
$575$	1.04484e26	−	2.32951e26i	0.634167	−	1.41391i
$576$		0			0
$577$		−	1.15615e26i		−	0.678961i	−0.940613	−	0.339481i	$-0.889749\pi$
$577$		−	1.15615e26i		−	0.678961i	0.940613	−	0.339481i	$-0.110251\pi$
$578$		0			0
$579$	−8.44690e25			−0.480011
$580$		0			0
$581$	9.87829e25			0.543261
$582$		0			0
$583$			9.96384e25i			0.530366i
$584$		0			0
$585$	1.02380e26	+	2.19084e25i	0.527513	+	0.112883i
$586$		0			0
$587$		−	1.63752e26i		−	0.816817i	−0.912799	−	0.408408i	$-0.866084\pi$
$587$		−	1.63752e26i		−	0.816817i	0.912799	−	0.408408i	$-0.133916\pi$
$588$		0			0
$589$	−8.35728e24			−0.0403617
$590$		0			0
$591$	−7.98125e25			−0.373242
$592$		0			0
$593$		−	1.77679e26i		−	0.804665i	−0.915494	−	0.402333i	$-0.868200\pi$
$593$		−	1.77679e26i		−	0.804665i	0.915494	−	0.402333i	$-0.131800\pi$
$594$		0			0
$595$	2.17596e25	−	1.01685e26i	0.0954422	−	0.446010i
$596$		0			0
$597$		−	7.23706e25i		−	0.307472i
$598$		0			0
$599$	−4.45807e26			−1.83481			−0.917407	−	0.397950i	$-0.869722\pi$
$599$	−4.45807e26			−1.83481			−0.917407	+	0.397950i	$0.869722\pi$
$600$		0			0
$601$	4.47873e26			1.78586			0.892930	−	0.450195i	$-0.148646\pi$
$601$	4.47873e26			1.78586			0.892930	+	0.450195i	$0.148646\pi$
$602$		0			0
$603$			9.41580e25i			0.363783i
$604$		0			0
$605$	−3.18342e25	+	1.48764e26i	−0.119184	+	0.556956i
$606$		0			0
$607$		−	3.01201e26i		−	1.09286i	−0.837506	−	0.546429i	$-0.815987\pi$
$607$		−	3.01201e26i		−	1.09286i	0.837506	−	0.546429i	$-0.184013\pi$
$608$		0			0
$609$	−1.02973e26			−0.362127
$610$		0			0
$611$	−2.88246e26			−0.982588
$612$		0			0
$613$		−	3.17946e26i		−	1.05070i	−0.850886	−	0.525350i	$-0.823934\pi$
$613$		−	3.17946e26i		−	1.05070i	0.850886	−	0.525350i	$-0.176066\pi$
$614$		0			0
$615$	−1.77714e26	−	3.80293e25i	−0.569388	−	0.121844i
$616$		0			0
$617$		−	5.53205e26i		−	1.71861i	−0.511465	−	0.859304i	$-0.670897\pi$
$617$		−	5.53205e26i		−	1.71861i	0.511465	−	0.859304i	$-0.329103\pi$
$618$		0			0
$619$	4.27971e26			1.28930			0.644649	−	0.764478i	$-0.277003\pi$
$619$	4.27971e26			1.28930			0.644649	+	0.764478i	$0.277003\pi$
$620$		0			0
$621$	−4.23660e26			−1.23779
$622$		0			0
$623$			3.44519e26i			0.976287i
$624$		0			0
$625$	2.41941e26	+	2.71687e26i	0.665041	+	0.746807i
$626$		0			0
$627$			1.12902e25i			0.0301066i
$628$		0			0
$629$	−2.93071e25			−0.0758214
$630$		0			0
$631$	−7.21990e26			−1.81239			−0.906196	−	0.422858i	$-0.861027\pi$
$631$	−7.21990e26			−1.81239			−0.906196	+	0.422858i	$0.861027\pi$
$632$		0			0
$633$			3.02233e25i			0.0736218i
$634$		0			0
$635$	−8.54983e25	−	1.82959e25i	−0.202119	−	0.0432517i
$636$		0			0
$637$		−	2.56148e25i		−	0.0587715i
$638$		0			0
$639$	1.43509e25			0.0319611
$640$		0			0
$641$	−2.01056e26			−0.434677			−0.217338	−	0.976096i	$-0.569737\pi$
$641$	−2.01056e26			−0.434677			−0.217338	+	0.976096i	$0.569737\pi$
$642$		0			0
$643$			6.64148e26i			1.39399i	0.717075	+	0.696996i	$0.245481\pi$
$643$			6.64148e26i			1.39399i	−0.717075	+	0.696996i	$0.754519\pi$
$644$		0			0
$645$	2.56678e24	−	1.19948e25i	0.00523084	−	0.0244442i
$646$		0			0
$647$		−	7.47529e26i		−	1.47924i	−0.673027	−	0.739618i	$-0.735006\pi$
$647$		−	7.47529e26i		−	1.47924i	0.673027	−	0.739618i	$-0.264994\pi$
$648$		0			0
$649$	1.52911e26			0.293843
$650$		0			0
$651$	−9.63655e25			−0.179847
$652$		0			0
$653$		−	9.10300e26i		−	1.65010i	−0.565063	−	0.825048i	$-0.691148\pi$
$653$		−	9.10300e26i		−	1.65010i	0.565063	−	0.825048i	$-0.308852\pi$
$654$		0			0
$655$	1.88071e26	−	8.78873e26i	0.331154	−	1.54751i
$656$		0			0
$657$		−	4.47830e25i		−	0.0766024i
$658$		0			0
$659$	−1.58321e26			−0.263103			−0.131552	−	0.991309i	$-0.541996\pi$
$659$	−1.58321e26			−0.263103			−0.131552	+	0.991309i	$0.541996\pi$
$660$		0			0
$661$	5.01835e26			0.810302			0.405151	−	0.914250i	$-0.367219\pi$
$661$	5.01835e26			0.810302			0.405151	+	0.914250i	$0.367219\pi$
$662$		0			0
$663$		−	8.28240e25i		−	0.129950i
$664$		0			0
$665$	−6.93136e25	−	1.48325e25i	−0.105685	−	0.0226156i
$666$		0			0
$667$		−	8.20055e26i		−	1.21520i
$668$		0			0
$669$	−1.47045e26			−0.211788
$670$		0			0
$671$	1.24754e26			0.174659
$672$		0			0
$673$			3.71843e26i			0.506077i	0.967456	+	0.253038i	$0.0814299\pi$
$673$			3.71843e26i			0.506077i	−0.967456	+	0.253038i	$0.918570\pi$
$674$		0			0
$675$	2.47054e26	−	5.50818e26i	0.326893	−	0.728823i
$676$		0			0
$677$			1.31385e27i			1.69026i	0.534562	+	0.845129i	$0.320476\pi$
$677$			1.31385e27i			1.69026i	−0.534562	+	0.845129i	$0.679524\pi$
$678$		0			0
$679$	1.20807e27			1.51123
$680$		0			0
$681$	−2.16794e26			−0.263724
$682$		0			0
$683$			9.42374e26i			1.11488i	0.830218	+	0.557438i	$0.188216\pi$
$683$			9.42374e26i			1.11488i	−0.830218	+	0.557438i	$0.811784\pi$
$684$		0			0
$685$	−8.96177e26	−	1.91774e26i	−1.03118	−	0.220663i
$686$		0			0
$687$			1.61752e26i			0.181034i
$688$		0			0
$689$	−4.98244e26			−0.542450
$690$		0			0
$691$	3.80793e26			0.403318			0.201659	−	0.979456i	$-0.435367\pi$
$691$	3.80793e26			0.403318			0.201659	+	0.979456i	$0.435367\pi$
$692$		0			0
$693$		−	5.33789e26i		−	0.550052i
$694$		0			0
$695$	−1.01027e26	+	4.72110e26i	−0.101294	+	0.473356i
$696$		0			0
$697$		−	5.89486e26i		−	0.575125i
$698$		0			0
$699$	−6.82747e26			−0.648226
$700$		0			0
$701$	−1.20622e27			−1.11456			−0.557280	−	0.830324i	$-0.688155\pi$
$701$	−1.20622e27			−1.11456			−0.557280	+	0.830324i	$0.688155\pi$
$702$		0			0
$703$			1.99772e25i			0.0179664i
$704$		0			0
$705$	−1.54992e26	+	7.24290e26i	−0.135679	+	0.634038i
$706$		0			0
$707$			2.01572e27i			1.71770i
$708$		0			0
$709$	8.88683e26			0.737238			0.368619	−	0.929581i	$-0.379831\pi$
$709$	8.88683e26			0.737238			0.368619	+	0.929581i	$0.379831\pi$
$710$		0			0
$711$	1.89047e27			1.52690
$712$		0			0
$713$		−	7.67431e26i		−	0.603516i
$714$		0			0
$715$	5.62180e26	+	1.20302e26i	0.430495	+	0.0921222i
$716$		0			0
$717$			9.41201e26i			0.701860i
$718$		0			0
$719$	9.20987e25			0.0668850			0.0334425	−	0.999441i	$-0.489353\pi$
$719$	9.20987e25			0.0668850			0.0334425	+	0.999441i	$0.489353\pi$
$720$		0			0
$721$	−1.26894e27			−0.897545
$722$		0			0
$723$			7.03293e25i			0.0484532i
$724$		0			0
$725$	1.06619e27	+	4.78207e26i	0.715520	+	0.320926i
$726$		0			0
$727$		−	1.10807e27i		−	0.724419i	−0.932097	−	0.362209i	$-0.882023\pi$
$727$		−	1.10807e27i		−	0.724419i	0.932097	−	0.362209i	$-0.117977\pi$
$728$		0			0
$729$	1.29738e25			0.00826334
$730$		0			0
$731$	3.97872e25			0.0246904
$732$		0			0
$733$			1.16091e27i			0.701955i	0.936384	+	0.350977i	$0.114151\pi$
$733$			1.16091e27i			0.701955i	−0.936384	+	0.350977i	$0.885849\pi$
$734$		0			0
$735$	−6.43637e25	−	1.37733e25i	−0.0379237	−	0.00811535i
$736$		0			0
$737$			5.17034e26i			0.296878i
$738$		0			0
$739$	1.12945e27			0.632039			0.316019	−	0.948753i	$-0.397654\pi$
$739$	1.12945e27			0.632039			0.316019	+	0.948753i	$0.397654\pi$
$740$		0			0
$741$	−5.64571e25			−0.0307926
$742$		0			0
$743$		−	1.68679e27i		−	0.896744i	−0.893847	−	0.448372i	$-0.852004\pi$
$743$		−	1.68679e27i		−	0.896744i	0.893847	−	0.448372i	$-0.147996\pi$
$744$		0			0
$745$	−4.26277e26	+	1.99203e27i	−0.220906	+	1.03231i
$746$		0			0
$747$			8.28975e26i			0.418789i
$748$		0			0
$749$	9.78979e26			0.482165
$750$		0			0
$751$	−1.62477e27			−0.780211			−0.390105	−	0.920770i	$-0.627561\pi$
$751$	−1.62477e27			−0.780211			−0.390105	+	0.920770i	$0.627561\pi$
$752$		0			0
$753$			5.10025e26i			0.238803i
$754$		0			0
$755$	−2.83199e26	+	1.32342e27i	−0.129299	+	0.604227i
$756$		0			0
$757$			1.87866e26i			0.0836443i	0.999125	+	0.0418221i	$0.0133163\pi$
$757$			1.87866e26i			0.0836443i	−0.999125	+	0.0418221i	$0.986684\pi$
$758$		0			0
$759$	−1.03676e27			−0.450175
$760$		0			0
$761$	2.76754e27			1.17203			0.586015	−	0.810300i	$-0.300696\pi$
$761$	2.76754e27			1.17203			0.586015	+	0.810300i	$0.300696\pi$
$762$		0			0
$763$			4.49530e27i			1.85684i
$764$		0			0
$765$	8.53326e26	+	1.82604e26i	0.343820	+	0.0735745i
$766$		0			0
$767$			7.64636e26i			0.300538i
$768$		0			0
$769$	−8.04737e26			−0.308570			−0.154285	−	0.988026i	$-0.549307\pi$
$769$	−8.04737e26			−0.308570			−0.154285	+	0.988026i	$0.549307\pi$
$770$		0			0
$771$	−5.92975e26			−0.221830
$772$		0			0
$773$		−	6.11228e26i		−	0.223099i	−0.993759	−	0.111550i	$-0.964419\pi$
$773$		−	6.11228e26i		−	0.223099i	0.993759	−	0.111550i	$-0.0355814\pi$
$774$		0			0
$775$	9.97768e26	+	4.47520e26i	0.355356	+	0.159385i
$776$		0			0
$777$			2.30352e26i			0.0800557i
$778$		0			0
$779$	−4.01824e26			−0.136279
$780$		0			0
$781$	7.88028e25			0.0260830
$782$		0			0
$783$		−	1.93904e27i		−	0.626396i
$784$		0			0
$785$	4.75658e27	+	1.01787e27i	1.49980	+	0.320944i
$786$		0			0
$787$			5.37488e27i			1.65428i	0.561996	+	0.827140i	$0.310033\pi$
$787$			5.37488e27i			1.65428i	−0.561996	+	0.827140i	$0.689967\pi$
$788$		0			0
$789$	−1.08366e27			−0.325582
$790$		0			0
$791$	−2.63252e26			−0.0772138
$792$		0			0
$793$			6.23837e26i			0.178639i
$794$		0			0
$795$	−2.67909e26	+	1.25196e27i	−0.0749031	+	0.350029i
$796$		0			0
$797$			4.51343e27i			1.23212i	0.787700	+	0.616059i	$0.211272\pi$
$797$			4.51343e27i			1.23212i	−0.787700	+	0.616059i	$0.788728\pi$
$798$		0			0
$799$	−2.40250e27			−0.640426
$800$		0			0
$801$	−2.89117e27			−0.752600
$802$		0			0
$803$		−	2.45909e26i		−	0.0625139i
$804$		0			0
$805$	1.36204e27	−	6.36492e27i	0.338165	−	1.58027i
$806$		0			0
$807$		−	2.94923e27i		−	0.715172i
$808$		0			0
$809$	5.36818e27			1.27150			0.635749	−	0.771895i	$-0.280691\pi$
$809$	5.36818e27			1.27150			0.635749	+	0.771895i	$0.280691\pi$
$810$		0			0
$811$	−5.66427e27			−1.31053			−0.655264	−	0.755400i	$-0.727443\pi$
$811$	−5.66427e27			−1.31053			−0.655264	+	0.755400i	$0.727443\pi$
$812$		0			0
$813$			2.17684e27i			0.492002i
$814$		0			0
$815$	−3.88697e27	−	8.31777e26i	−0.858250	−	0.183658i
$816$		0			0
$817$		−	2.71210e25i		−	0.00585055i
$818$		0			0
$819$	2.66922e27			0.562585
$820$		0			0
$821$	9.55870e26			0.196852			0.0984258	−	0.995144i	$-0.468619\pi$
$821$	9.55870e26			0.196852			0.0984258	+	0.995144i	$0.468619\pi$
$822$		0			0
$823$		−	5.19553e27i		−	1.04552i	−0.852480	−	0.522759i	$-0.824903\pi$
$823$		−	5.19553e27i		−	1.04552i	0.852480	−	0.522759i	$-0.175097\pi$
$824$		0			0
$825$	6.04576e26	−	1.34793e27i	0.118888	−	0.265067i
$826$		0			0
$827$			2.64812e27i			0.508903i	0.967086	+	0.254452i	$0.0818949\pi$
$827$			2.64812e27i			0.508903i	−0.967086	+	0.254452i	$0.918105\pi$
$828$		0			0
$829$	−1.37251e27			−0.257780			−0.128890	−	0.991659i	$-0.541141\pi$
$829$	−1.37251e27			−0.257780			−0.128890	+	0.991659i	$0.541141\pi$
$830$		0			0
$831$	−6.16596e26			−0.113186
$832$		0			0
$833$		−	2.13497e26i		−	0.0383058i
$834$		0			0
$835$	6.96254e27	+	1.48992e27i	1.22109	+	0.261302i
$836$		0			0
$837$		−	1.81461e27i		−	0.311093i
$838$		0			0
$839$	−2.96781e27			−0.497390			−0.248695	−	0.968582i	$-0.580002\pi$
$839$	−2.96781e27			−0.497390			−0.248695	+	0.968582i	$0.580002\pi$
$840$		0			0
$841$	−2.34998e27			−0.385037
$842$		0			0
$843$			2.87519e27i			0.460580i
$844$		0			0
$845$	7.34446e26	−	3.43213e27i	0.115033	−	0.537558i
$846$		0			0
$847$			3.87854e27i			0.593985i
$848$		0			0
$849$	−2.93920e27			−0.440156
$850$		0			0
$851$	−1.83447e27			−0.268645
$852$		0			0
$853$			5.57749e27i			0.798772i	0.916783	+	0.399386i	$0.130777\pi$
$853$			5.57749e27i			0.798772i	−0.916783	+	0.399386i	$0.869223\pi$
$854$		0			0
$855$	1.24473e26	−	5.81672e26i	0.0174340	−	0.0814704i
$856$		0			0
$857$		−	5.57196e27i		−	0.763291i	−0.924309	−	0.381645i	$-0.875358\pi$
$857$		−	5.57196e27i		−	0.763291i	0.924309	−	0.381645i	$-0.124642\pi$
$858$		0			0
$859$	8.12199e27			1.08825			0.544123	−	0.839005i	$-0.316862\pi$
$859$	8.12199e27			1.08825			0.544123	+	0.839005i	$0.316862\pi$
$860$		0			0
$861$	−4.63333e27			−0.607244
$862$		0			0
$863$		−	1.38696e28i		−	1.77813i	−0.457784	−	0.889063i	$-0.651357\pi$
$863$		−	1.38696e28i		−	1.77813i	0.457784	−	0.889063i	$-0.348643\pi$
$864$		0			0
$865$	−1.25307e28	−	2.68145e27i	−1.57152	−	0.336293i
$866$		0			0
$867$			2.91866e27i			0.358098i
$868$		0			0
$869$	1.03808e28			1.24608
$870$		0			0
$871$	−2.58544e27			−0.303642
$872$		0			0
$873$			1.01380e28i			1.16498i
$874$		0			0
$875$	7.48102e27	+	5.48248e27i	0.841171	+	0.616454i
$876$		0			0
$877$		−	3.23136e27i		−	0.355540i	−0.984072	−	0.177770i	$-0.943112\pi$
$877$		−	3.23136e27i		−	0.355540i	0.984072	−	0.177770i	$-0.0568883\pi$
$878$		0			0
$879$	−1.73925e27			−0.187270
$880$		0			0
$881$	−9.10113e27			−0.959012			−0.479506	−	0.877539i	$-0.659184\pi$
$881$	−9.10113e27			−0.959012			−0.479506	+	0.877539i	$0.659184\pi$
$882$		0			0
$883$		−	1.09948e28i		−	1.13386i	−0.823765	−	0.566932i	$-0.808130\pi$
$883$		−	1.09948e28i		−	1.13386i	0.823765	−	0.566932i	$-0.191870\pi$
$884$		0			0
$885$	1.92134e27	+	4.11150e26i	0.193929	+	0.0414992i
$886$		0			0
$887$		−	1.19162e28i		−	1.17723i	−0.808412	−	0.588617i	$-0.799673\pi$
$887$		−	1.19162e28i		−	1.17723i	0.808412	−	0.588617i	$-0.200327\pi$
$888$		0			0
$889$	−2.22909e27			−0.215557
$890$		0			0
$891$	3.12055e27			0.295388
$892$		0			0
$893$			1.63767e27i			0.151753i
$894$		0			0
$895$	2.38509e27	−	1.11457e28i	0.216364	−	1.01109i
$896$		0			0
$897$		−	5.18434e27i		−	0.460432i
$898$		0			0
$899$	3.51243e27			0.305415
$900$		0			0
$901$	−4.15282e27			−0.353555
$902$		0			0
$903$		−	3.12725e26i		−	0.0260693i
$904$		0			0
$905$	4.18837e27	−	1.95726e28i	0.341888	−	1.59767i
$906$		0			0
$907$		−	2.38969e27i		−	0.191017i	−0.995429	−	0.0955085i	$-0.969552\pi$
$907$		−	2.38969e27i		−	0.191017i	0.995429	−	0.0955085i	$-0.0304477\pi$
$908$		0			0
$909$	−1.69157e28			−1.32414
$910$		0			0
$911$	−1.06668e28			−0.817730			−0.408865	−	0.912595i	$-0.634075\pi$
$911$	−1.06668e28			−0.817730			−0.408865	+	0.912595i	$0.634075\pi$
$912$		0			0
$913$			4.55201e27i			0.341767i
$914$		0			0
$915$	1.56755e27	+	3.35441e26i	0.115271	+	0.0246670i
$916$		0			0
$917$		−	2.29138e28i		−	1.65040i
$918$		0			0
$919$	1.97334e28			1.39221			0.696105	−	0.717940i	$-0.254915\pi$
$919$	1.97334e28			1.39221			0.696105	+	0.717940i	$0.254915\pi$
$920$		0			0
$921$	−4.79305e27			−0.331242
$922$		0			0
$923$			3.94055e26i			0.0266773i
$924$		0			0
$925$	1.06975e27	−	2.38506e27i	0.0709475	−	0.158181i
$926$		0			0
$927$		−	1.06488e28i		−	0.691900i
$928$		0			0
$929$	9.50160e27			0.604849			0.302425	−	0.953173i	$-0.402204\pi$
$929$	9.50160e27			0.604849			0.302425	+	0.953173i	$0.402204\pi$
$930$		0			0
$931$	−1.45531e26			−0.00907680
$932$		0			0
$933$		−	1.44037e27i		−	0.0880233i
$934$		0			0
$935$	4.68572e27	+	1.00270e27i	0.280586	+	0.0600430i
$936$		0			0
$937$			1.05151e28i			0.617006i	0.951223	+	0.308503i	$0.0998279\pi$
$937$			1.05151e28i			0.617006i	−0.951223	+	0.308503i	$0.900172\pi$
$938$		0			0
$939$	1.03123e28			0.592968
$940$		0			0
$941$	1.33255e28			0.750901			0.375450	−	0.926842i	$-0.377488\pi$
$941$	1.33255e28			0.750901			0.375450	+	0.926842i	$0.377488\pi$
$942$		0			0
$943$		−	3.68987e28i		−	2.03775i
$944$		0			0
$945$	3.22056e27	−	1.50500e28i	0.174313	−	0.814579i
$946$		0			0
$947$			3.19308e28i			1.69389i	0.531682	+	0.846944i	$0.321560\pi$
$947$			3.19308e28i			1.69389i	−0.531682	+	0.846944i	$0.678440\pi$
$948$		0			0
$949$	1.22967e27			0.0639383
$950$		0			0
$951$	3.15872e27			0.160989
$952$		0			0
$953$			8.51393e27i			0.425351i	0.977123	+	0.212675i	$0.0682177\pi$
$953$			8.51393e27i			0.425351i	−0.977123	+	0.212675i	$0.931782\pi$
$954$		0			0
$955$	−8.12094e27	+	3.79499e28i	−0.397717	+	1.85857i
$956$		0			0
$957$		−	4.74511e27i		−	0.227815i
$958$		0			0
$959$	−2.33649e28			−1.09973
$960$		0			0
$961$	−1.83836e28			−0.848319
$962$		0			0
$963$			8.21548e27i			0.371692i
$964$		0			0
$965$	2.38964e28	+	5.11363e27i	1.06004	+	0.226840i
$966$		0			0
$967$		−	3.15222e28i		−	1.37109i	−0.728032	−	0.685543i	$-0.759565\pi$
$967$		−	3.15222e28i		−	1.37109i	0.728032	−	0.685543i	$-0.240435\pi$
$968$		0			0
$969$	−4.70565e26			−0.0200698
$970$		0			0
$971$	−4.45761e28			−1.86431			−0.932157	−	0.362054i	$-0.882076\pi$
$971$	−4.45761e28			−1.86431			−0.932157	+	0.362054i	$0.882076\pi$
$972$		0			0
$973$			1.23088e28i			0.504826i
$974$		0			0
$975$	6.74037e27	+	3.02320e27i	0.271106	+	0.121597i
$976$		0			0
$977$		−	1.64952e27i		−	0.0650668i	−0.999471	−	0.0325334i	$-0.989642\pi$
$977$		−	1.64952e27i		−	0.0650668i	0.999471	−	0.0325334i	$-0.0103575\pi$
$978$		0			0
$979$	−1.58758e28			−0.614185
$980$		0			0
$981$	−3.77240e28			−1.43141
$982$		0			0
$983$		−	3.97241e28i		−	1.47841i	−0.673478	−	0.739207i	$-0.735201\pi$
$983$		−	3.97241e28i		−	1.47841i	0.673478	−	0.739207i	$-0.264799\pi$
$984$		0			0
$985$	2.25791e28	+	4.83173e27i	0.824257	+	0.176384i
$986$		0			0
$987$			1.88835e28i			0.676192i
$988$		0			0
$989$	2.49047e27			0.0874815
$990$		0			0
$991$	−4.02833e28			−1.38812			−0.694058	−	0.719919i	$-0.744179\pi$
$991$	−4.02833e28			−1.38812			−0.694058	+	0.719919i	$0.744179\pi$
$992$		0			0
$993$			4.72828e26i			0.0159840i
$994$		0			0
$995$	−4.38121e27	+	2.04738e28i	−0.145303	+	0.679014i
$996$		0			0
$997$		−	1.65575e28i		−	0.538754i	−0.963035	−	0.269377i	$-0.913182\pi$
$997$		−	1.65575e28i		−	0.538754i	0.963035	−	0.269377i	$-0.0868178\pi$
$998$		0			0
$999$	−4.33763e27			−0.138478

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.c.d.49.18		28
4.3	odd	2		40.20.c.a.9.11	✓	28
5.4	even	2	inner	80.20.c.d.49.11		28
20.19	odd	2		40.20.c.a.9.18	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.20.c.a.9.11	✓	28	4.3	odd	2
40.20.c.a.9.18	yes	28	20.19	odd	2
80.20.c.d.49.11		28	5.4	even	2	inner
80.20.c.d.49.18		28	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15095.8i q^{3} +(913879. - 4.27063e6i) q^{5} -1.11343e8i q^{7} +9.34378e8 q^{9} +O(q^{10})\)	\(q+15095.8i q^{3} +(913879. - 4.27063e6i) q^{5} -1.11343e8i q^{7} +9.34378e8 q^{9} +5.13079e9 q^{11} +2.56566e10i q^{13} +(6.44687e10 + 1.37957e10i) q^{15} +2.13846e11i q^{17} +1.45768e11 q^{19} +1.68081e12 q^{21} +1.33856e13i q^{23} +(-1.74031e13 - 7.80568e12i) q^{25} +3.16505e13i q^{27} -6.12640e13 q^{29} -5.73327e13 q^{31} +7.74534e13i q^{33} +(-4.75505e14 - 1.01754e14i) q^{35} +1.37048e14i q^{37} -3.87307e14 q^{39} -2.75660e15 q^{41} -1.86056e14i q^{43} +(8.53908e14 - 3.99039e15i) q^{45} +1.12347e16i q^{47} -9.98371e14 q^{49} -3.22817e15 q^{51} +1.94197e16i q^{53} +(4.68892e15 - 2.19117e16i) q^{55} +2.20049e15i q^{57} +2.98027e16 q^{59} +2.43149e16 q^{61} -1.04036e17i q^{63} +(1.09570e17 + 2.34470e16i) q^{65} +1.00771e17i q^{67} -2.02066e17 q^{69} +1.53588e16 q^{71} -4.79281e16i q^{73} +(1.17833e17 - 2.62715e17i) q^{75} -5.71277e17i q^{77} +2.02324e18 q^{79} +6.08202e17 q^{81} +8.87195e17i q^{83} +(9.13256e17 + 1.95429e17i) q^{85} -9.24830e17i q^{87} -3.09422e18 q^{89} +2.85668e18 q^{91} -8.65483e17i q^{93} +(1.33214e17 - 6.22523e17i) q^{95} +1.08500e19i q^{97} +4.79409e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 3935164 q^{5} - 10352560732 q^{9}+O(q^{10}) \) 28 * q - 3935164 * q^5 - 10352560732 * q^9	\( 28 q - 3935164 q^{5} - 10352560732 q^{9} - 18570995888 q^{11} - 79041415120 q^{15} + 1685869130672 q^{19} + 140419722832 q^{21} - 29558822439924 q^{25} + 160448197635496 q^{29} - 64251929934720 q^{31} + 88899634227664 q^{35} - 45\!\cdots\!04 q^{39}+ \cdots + 24\!\cdots\!92 q^{99}+O(q^{100}) \) 28 * q - 3935164 * q^5 - 10352560732 * q^9 - 18570995888 * q^11 - 79041415120 * q^15 + 1685869130672 * q^19 + 140419722832 * q^21 - 29558822439924 * q^25 + 160448197635496 * q^29 - 64251929934720 * q^31 + 88899634227664 * q^35 - 4563253596731104 * q^39 + 87868042694936 * q^41 + 3557698140217852 * q^45 - 28923781405647564 * q^49 - 14937379593456000 * q^51 - 28335039172010704 * q^55 + 83879503376835216 * q^59 + 2336306250785000 * q^61 + 23503427909569568 * q^65 - 46927873937137712 * q^69 - 104881848512783264 * q^71 - 1094024007292275040 * q^75 + 795721739900308672 * q^79 + 3593102866196377132 * q^81 + 834124526841485952 * q^85 + 1582238786752844568 * q^89 - 11017818924975052064 * q^91 - 30914057837649668144 * q^95 + 24571127463084047792 * q^99

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