LMFDB - Embedded newform 84.12.f.b.41.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(41,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.41");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.f (of order $2$, degree $1$, 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:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.12
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-138.135 + 397.575i) q^{3} +8832.29 q^{5} +(16667.3 + 41225.3i) q^{7} +(-138985. - 109838. i) q^{9} +O(q^{10})$	\(q+(-138.135 + 397.575i) q^{3} +8832.29 q^{5} +(16667.3 + 41225.3i) q^{7} +(-138985. - 109838. i) q^{9} -674199. i q^{11} +1.36330e6i q^{13} +(-1.22005e6 + 3.51150e6i) q^{15} -7.92109e6 q^{17} +8.91937e6i q^{19} +(-1.86925e7 + 931861. i) q^{21} -2.25397e7i q^{23} +2.91813e7 q^{25} +(6.28674e7 - 4.00844e7i) q^{27} +8.14998e7i q^{29} +2.35843e8i q^{31} +(2.68045e8 + 9.31303e7i) q^{33} +(1.47211e8 + 3.64114e8i) q^{35} -5.33115e8 q^{37} +(-5.42012e8 - 1.88319e8i) q^{39} +9.52106e8 q^{41} +1.43275e9 q^{43} +(-1.22755e9 - 9.70120e8i) q^{45} -2.52108e9 q^{47} +(-1.42173e9 + 1.37423e9i) q^{49} +(1.09418e9 - 3.14923e9i) q^{51} +2.77421e9i q^{53} -5.95472e9i q^{55} +(-3.54612e9 - 1.23207e9i) q^{57} -6.78699e9 q^{59} +4.33580e8i q^{61} +(2.21160e9 - 7.56039e9i) q^{63} +1.20410e10i q^{65} -1.56078e10 q^{67} +(8.96121e9 + 3.11351e9i) q^{69} +6.69911e9i q^{71} -2.11852e10i q^{73} +(-4.03095e9 + 1.16018e10i) q^{75} +(2.77941e10 - 1.12371e10i) q^{77} -2.95109e10 q^{79} +(7.25237e9 + 3.05315e10i) q^{81} -5.39016e10 q^{83} -6.99614e10 q^{85} +(-3.24023e10 - 1.12580e10i) q^{87} +3.45087e10 q^{89} +(-5.62023e10 + 2.27225e10i) q^{91} +(-9.37654e10 - 3.25782e10i) q^{93} +7.87785e10i q^{95} +2.62564e10i q^{97} +(-7.40525e10 + 9.37033e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) $ 28 * q + 9632 * q^7 + 267660 * q^9	$ 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) $ 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−138.135	+	397.575i	−0.328198	+	0.944609i
$3$	−138.135	+	397.575i	−0.328198	+	0.944609i
$4$		0			0
$5$	8832.29			1.26398			0.631988	−	0.774979i	$-0.282239\pi$
$5$	8832.29			1.26398			0.631988	+	0.774979i	$0.282239\pi$
$6$		0			0
$7$	16667.3	+	41225.3i	0.374823	+	0.927096i
$7$	16667.3	+	41225.3i	0.374823	+	0.927096i
$8$		0			0
$9$	−138985.	−	109838.i	−0.784572	−	0.620038i
$10$		0			0
$11$		−	674199.i		−	1.26220i	−0.775701	−	0.631100i	$-0.782604\pi$
$11$		−	674199.i		−	1.26220i	0.775701	−	0.631100i	$-0.217396\pi$
$12$		0			0
$13$			1.36330e6i			1.01836i	0.860660	+	0.509180i	$0.170051\pi$
$13$			1.36330e6i			1.01836i	−0.860660	+	0.509180i	$0.829949\pi$
$14$		0			0
$15$	−1.22005e6	+	3.51150e6i	−0.414834	+	1.19396i
$16$		0			0
$17$	−7.92109e6			−1.35306			−0.676529	−	0.736416i	$-0.736516\pi$
$17$	−7.92109e6			−1.35306			−0.676529	+	0.736416i	$0.736516\pi$
$18$		0			0
$19$			8.91937e6i			0.826398i	0.910641	+	0.413199i	$0.135589\pi$
$19$			8.91937e6i			0.826398i	−0.910641	+	0.413199i	$0.864411\pi$
$20$		0			0
$21$	−1.86925e7	+	931861.i	−0.998760	+	0.0497903i
$22$		0			0
$23$		−	2.25397e7i		−	0.730204i	−0.930967	−	0.365102i	$-0.881034\pi$
$23$		−	2.25397e7i		−	0.730204i	0.930967	−	0.365102i	$-0.118966\pi$
$24$		0			0
$25$	2.91813e7			0.597633
$26$		0			0
$27$	6.28674e7	−	4.00844e7i	0.843188	−	0.537619i
$28$		0			0
$29$			8.14998e7i			0.737849i	0.929459	+	0.368925i	$0.120274\pi$
$29$			8.14998e7i			0.737849i	−0.929459	+	0.368925i	$0.879726\pi$
$30$		0			0
$31$			2.35843e8i			1.47957i	0.672846	+	0.739783i	$0.265072\pi$
$31$			2.35843e8i			1.47957i	−0.672846	+	0.739783i	$0.734928\pi$
$32$		0			0
$33$	2.68045e8	+	9.31303e7i	1.19229	+	0.414252i
$34$		0			0
$35$	1.47211e8	+	3.64114e8i	0.473767	+	1.17183i
$36$		0			0
$37$	−5.33115e8			−1.26390			−0.631948	−	0.775011i	$-0.717744\pi$
$37$	−5.33115e8			−1.26390			−0.631948	+	0.775011i	$0.717744\pi$
$38$		0			0
$39$	−5.42012e8	−	1.88319e8i	−0.961953	−	0.334224i
$40$		0			0
$41$	9.52106e8			1.28344			0.641718	−	0.766941i	$-0.278222\pi$
$41$	9.52106e8			1.28344			0.641718	+	0.766941i	$0.278222\pi$
$42$		0			0
$43$	1.43275e9			1.48626			0.743130	−	0.669147i	$-0.233341\pi$
$43$	1.43275e9			1.48626			0.743130	+	0.669147i	$0.233341\pi$
$44$		0			0
$45$	−1.22755e9	−	9.70120e8i	−0.991680	−	0.783712i
$46$		0			0
$47$	−2.52108e9			−1.60342			−0.801711	−	0.597712i	$-0.796076\pi$
$47$	−2.52108e9			−1.60342			−0.801711	+	0.597712i	$0.796076\pi$
$48$		0			0
$49$	−1.42173e9	+	1.37423e9i	−0.719015	+	0.694995i
$50$		0			0
$51$	1.09418e9	−	3.14923e9i	0.444071	−	1.27811i
$52$		0			0
$53$			2.77421e9i			0.911217i	0.890180	+	0.455608i	$0.150578\pi$
$53$			2.77421e9i			0.911217i	−0.890180	+	0.455608i	$0.849422\pi$
$54$		0			0
$55$		−	5.95472e9i		−	1.59539i
$56$		0			0
$57$	−3.54612e9	−	1.23207e9i	−0.780623	−	0.271222i
$58$		0			0
$59$	−6.78699e9			−1.23592			−0.617961	−	0.786209i	$-0.712041\pi$
$59$	−6.78699e9			−1.23592			−0.617961	+	0.786209i	$0.712041\pi$
$60$		0			0
$61$			4.33580e8i			0.0657287i	0.999460	+	0.0328644i	$0.0104629\pi$
$61$			4.33580e8i			0.0657287i	−0.999460	+	0.0328644i	$0.989537\pi$
$62$		0			0
$63$	2.21160e9	−	7.56039e9i	0.280759	−	0.959778i
$64$		0			0
$65$			1.20410e10i			1.28718i
$66$		0			0
$67$	−1.56078e10			−1.41231			−0.706157	−	0.708055i	$-0.749573\pi$
$67$	−1.56078e10			−1.41231			−0.706157	+	0.708055i	$0.749573\pi$
$68$		0			0
$69$	8.96121e9	+	3.11351e9i	0.689758	+	0.239652i
$70$		0			0
$71$			6.69911e9i			0.440653i	0.975426	+	0.220326i	$0.0707122\pi$
$71$			6.69911e9i			0.440653i	−0.975426	+	0.220326i	$0.929288\pi$
$72$		0			0
$73$		−	2.11852e10i		−	1.19607i	−0.801471	−	0.598034i	$-0.795949\pi$
$73$		−	2.11852e10i		−	1.19607i	0.801471	−	0.598034i	$-0.204051\pi$
$74$		0			0
$75$	−4.03095e9	+	1.16018e10i	−0.196142	+	0.564530i
$76$		0			0
$77$	2.77941e10	−	1.12371e10i	1.17018	−	0.473102i
$78$		0			0
$79$	−2.95109e10			−1.07903			−0.539515	−	0.841976i	$-0.681392\pi$
$79$	−2.95109e10			−1.07903			−0.539515	+	0.841976i	$0.681392\pi$
$80$		0			0
$81$	7.25237e9	+	3.05315e10i	0.231107	+	0.972928i
$82$		0			0
$83$	−5.39016e10			−1.50201			−0.751004	−	0.660297i	$-0.770430\pi$
$83$	−5.39016e10			−1.50201			−0.751004	+	0.660297i	$0.770430\pi$
$84$		0			0
$85$	−6.99614e10			−1.71023
$86$		0			0
$87$	−3.24023e10	−	1.12580e10i	−0.696979	−	0.242161i
$88$		0			0
$89$	3.45087e10			0.655064			0.327532	−	0.944840i	$-0.393783\pi$
$89$	3.45087e10			0.655064			0.327532	+	0.944840i	$0.393783\pi$
$90$		0			0
$91$	−5.62023e10	+	2.27225e10i	−0.944118	+	0.381705i
$92$		0			0
$93$	−9.37654e10	−	3.25782e10i	−1.39761	−	0.485591i
$94$		0			0
$95$			7.87785e10i			1.04455i
$96$		0			0
$97$			2.62564e10i			0.310450i	0.987879	+	0.155225i	$0.0496102\pi$
$97$			2.62564e10i			0.310450i	−0.987879	+	0.155225i	$0.950390\pi$
$98$		0			0
$99$	−7.40525e10	+	9.37033e10i	−0.782612	+	0.990287i
$100$		0			0
$101$	5.61313e10			0.531419			0.265710	−	0.964053i	$-0.414394\pi$
$101$	5.61313e10			0.531419			0.265710	+	0.964053i	$0.414394\pi$
$102$		0			0
$103$		−	9.75174e10i		−	0.828853i	−0.910083	−	0.414426i	$-0.863982\pi$
$103$		−	9.75174e10i		−	0.828853i	0.910083	−	0.414426i	$-0.136018\pi$
$104$		0			0
$105$	−1.65098e11	+	8.23047e9i	−1.26241	+	0.0629337i
$106$		0			0
$107$		−	2.02351e11i		−	1.39474i	−0.716711	−	0.697370i	$-0.754353\pi$
$107$		−	2.02351e11i		−	1.39474i	0.716711	−	0.697370i	$-0.245647\pi$
$108$		0			0
$109$	−6.00804e8			−0.00374014			−0.00187007	−	0.999998i	$-0.500595\pi$
$109$	−6.00804e8			−0.00374014			−0.00187007	+	0.999998i	$0.500595\pi$
$110$		0			0
$111$	7.36417e10	−	2.11953e11i	0.414808	−	1.19389i
$112$		0			0
$113$			2.17953e11i			1.11284i	0.830902	+	0.556418i	$0.187825\pi$
$113$			2.17953e11i			1.11284i	−0.830902	+	0.556418i	$0.812175\pi$
$114$		0			0
$115$		−	1.99077e11i		−	0.922960i
$116$		0			0
$117$	1.49741e11	−	1.89477e11i	0.631422	−	0.798977i
$118$		0			0
$119$	−1.32023e11	−	3.26550e11i	−0.507157	−	1.25441i
$120$		0			0
$121$	−1.69233e11			−0.593150
$122$		0			0
$123$	−1.31519e11	+	3.78533e11i	−0.421221	+	1.21234i
$124$		0			0
$125$	−1.73527e11			−0.508582
$126$		0			0
$127$	−5.36248e10			−0.144027			−0.0720137	−	0.997404i	$-0.522943\pi$
$127$	−5.36248e10			−0.144027			−0.0720137	+	0.997404i	$0.522943\pi$
$128$		0			0
$129$	−1.97913e11	+	5.69626e11i	−0.487787	+	1.40393i
$130$		0			0
$131$	−3.16692e11			−0.717207			−0.358604	−	0.933490i	$-0.616747\pi$
$131$	−3.16692e11			−0.717207			−0.358604	+	0.933490i	$0.616747\pi$
$132$		0			0
$133$	−3.67704e11	+	1.48662e11i	−0.766151	+	0.309753i
$134$		0			0
$135$	5.55263e11	−	3.54037e11i	1.06577	−	0.679537i
$136$		0			0
$137$			8.47016e11i			1.49944i	0.661756	+	0.749719i	$0.269812\pi$
$137$			8.47016e11i			1.49944i	−0.661756	+	0.749719i	$0.730188\pi$
$138$		0			0
$139$			2.35441e11i			0.384858i	0.981311	+	0.192429i	$0.0616366\pi$
$139$			2.35441e11i			0.384858i	−0.981311	+	0.192429i	$0.938363\pi$
$140$		0			0
$141$	3.48248e11	−	1.00232e12i	0.526240	−	1.51461i
$142$		0			0
$143$	9.19133e11			1.28538
$144$		0			0
$145$			7.19830e11i			0.932623i
$146$		0			0
$147$	−3.49970e11	−	7.55072e11i	−0.420519	−	0.907284i
$148$		0			0
$149$		−	1.08981e12i		−	1.21570i	−0.794051	−	0.607851i	$-0.792032\pi$
$149$		−	1.08981e12i		−	1.21570i	0.794051	−	0.607851i	$-0.207968\pi$
$150$		0			0
$151$	9.84048e11			1.02010			0.510051	−	0.860144i	$-0.329627\pi$
$151$	9.84048e11			1.02010			0.510051	+	0.860144i	$0.329627\pi$
$152$		0			0
$153$	1.10091e12	+	8.70036e11i	1.06157	+	0.838946i
$154$		0			0
$155$			2.08304e12i			1.87013i
$156$		0			0
$157$			4.98975e11i			0.417475i	0.977972	+	0.208738i	$0.0669355\pi$
$157$			4.98975e11i			0.417475i	−0.977972	+	0.208738i	$0.933065\pi$
$158$		0			0
$159$	−1.10296e12	−	3.83214e11i	−0.860743	−	0.299060i
$160$		0			0
$161$	9.29205e11	−	3.75676e11i	0.676970	−	0.273698i
$162$		0			0
$163$	2.16163e12			1.47147			0.735733	−	0.677272i	$-0.236838\pi$
$163$	2.16163e12			1.47147			0.735733	+	0.677272i	$0.236838\pi$
$164$		0			0
$165$	2.36745e12	+	8.22554e11i	1.50702	+	0.523604i
$166$		0			0
$167$	−1.47075e11			−0.0876192			−0.0438096	−	0.999040i	$-0.513950\pi$
$167$	−1.47075e11			−0.0876192			−0.0438096	+	0.999040i	$0.513950\pi$
$168$		0			0
$169$	−6.64150e10			−0.0370586
$170$		0			0
$171$	9.79684e11	−	1.23965e12i	0.512398	−	0.648369i
$172$		0			0
$173$	6.88452e10			0.0337769			0.0168885	−	0.999857i	$-0.494624\pi$
$173$	6.88452e10			0.0337769			0.0168885	+	0.999857i	$0.494624\pi$
$174$		0			0
$175$	4.86374e11	+	1.20301e12i	0.224007	+	0.554063i
$176$		0			0
$177$	9.37519e11	−	2.69834e12i	0.405627	−	1.16746i
$178$		0			0
$179$		−	4.21115e12i		−	1.71281i	−0.516304	−	0.856405i	$-0.672693\pi$
$179$		−	4.21115e12i		−	1.71281i	0.516304	−	0.856405i	$-0.327307\pi$
$180$		0			0
$181$		−	5.67474e11i		−	0.217127i	−0.994090	−	0.108564i	$-0.965375\pi$
$181$		−	5.67474e11i		−	0.217127i	0.994090	−	0.108564i	$-0.0346251\pi$
$182$		0			0
$183$	−1.72381e11	−	5.98925e10i	−0.0620879	−	0.0215720i
$184$		0			0
$185$	−4.70863e12			−1.59753
$186$		0			0
$187$			5.34039e12i			1.70783i
$188$		0			0
$189$	2.70032e12	+	1.92363e12i	0.814471	+	0.580205i
$190$		0			0
$191$		−	4.11700e12i		−	1.17192i	−0.810341	−	0.585959i	$-0.800718\pi$
$191$		−	4.11700e12i		−	1.17192i	0.810341	−	0.585959i	$-0.199282\pi$
$192$		0			0
$193$	3.29033e12			0.884452			0.442226	−	0.896904i	$-0.354189\pi$
$193$	3.29033e12			0.884452			0.442226	+	0.896904i	$0.354189\pi$
$194$		0			0
$195$	−4.78721e12	−	1.66328e12i	−1.21588	−	0.422451i
$196$		0			0
$197$			3.20172e12i			0.768810i	0.923165	+	0.384405i	$0.125593\pi$
$197$			3.20172e12i			0.768810i	−0.923165	+	0.384405i	$0.874407\pi$
$198$		0			0
$199$			6.75865e11i			0.153521i	0.997050	+	0.0767605i	$0.0244577\pi$
$199$			6.75865e11i			0.153521i	−0.997050	+	0.0767605i	$0.975542\pi$
$200$		0			0
$201$	2.15598e12	−	6.20528e12i	0.463519	−	1.33408i
$202$		0			0
$203$	−3.35986e12	+	1.35838e12i	−0.684057	+	0.276563i
$204$		0			0
$205$	8.40928e12			1.62223
$206$		0			0
$207$	−2.47571e12	+	3.13267e12i	−0.452754	+	0.572898i
$208$		0			0
$209$	6.01343e12			1.04308
$210$		0			0
$211$	−8.46217e12			−1.39293			−0.696464	−	0.717592i	$-0.745244\pi$
$211$	−8.46217e12			−1.39293			−0.696464	+	0.717592i	$0.745244\pi$
$212$		0			0
$213$	−2.66340e12	−	9.25380e11i	−0.416245	−	0.144621i
$214$		0			0
$215$	1.26545e13			1.87859
$216$		0			0
$217$	−9.72272e12	+	3.93088e12i	−1.37170	+	0.554576i
$218$		0			0
$219$	8.42269e12	+	2.92641e12i	1.12982	+	0.392547i
$220$		0			0
$221$		−	1.07988e13i		−	1.37790i
$222$		0			0
$223$			9.39037e12i			1.14027i	0.821552	+	0.570133i	$0.193108\pi$
$223$			9.39037e12i			1.14027i	−0.821552	+	0.570133i	$0.806892\pi$
$224$		0			0
$225$	−4.05575e12	−	3.20521e12i	−0.468886	−	0.370555i
$226$		0			0
$227$	5.62913e12			0.619867			0.309934	−	0.950758i	$-0.399693\pi$
$227$	5.62913e12			0.619867			0.309934	+	0.950758i	$0.399693\pi$
$228$		0			0
$229$			8.24055e10i			0.00864691i	0.999991	+	0.00432346i	$0.00137620\pi$
$229$			8.24055e10i			0.00864691i	−0.999991	+	0.00432346i	$0.998624\pi$
$230$		0			0
$231$	6.28260e11	+	1.26025e13i	0.0628454	+	1.26063i
$232$		0			0
$233$			1.34554e13i			1.28363i	0.766860	+	0.641814i	$0.221818\pi$
$233$			1.34554e13i			1.28363i	−0.766860	+	0.641814i	$0.778182\pi$
$234$		0			0
$235$	−2.22669e13			−2.02668
$236$		0			0
$237$	4.07648e12	−	1.17328e13i	0.354135	−	1.01926i
$238$		0			0
$239$			4.06540e12i			0.337221i	0.985683	+	0.168611i	$0.0539281\pi$
$239$			4.06540e12i			0.337221i	−0.985683	+	0.168611i	$0.946072\pi$
$240$		0			0
$241$			5.71196e12i			0.452576i	0.974060	+	0.226288i	$0.0726590\pi$
$241$			5.71196e12i			0.452576i	−0.974060	+	0.226288i	$0.927341\pi$
$242$		0			0
$243$	−1.31404e13	−	1.33410e12i	−0.994886	−	0.101008i
$244$		0			0
$245$	−1.25571e13	+	1.21376e13i	−0.908817	+	0.878456i
$246$		0			0
$247$	−1.21597e13			−0.841571
$248$		0			0
$249$	7.44569e12	−	2.14299e13i	0.492956	−	1.41881i
$250$		0			0
$251$	−2.51525e13			−1.59359			−0.796794	−	0.604251i	$-0.793472\pi$
$251$	−2.51525e13			−1.59359			−0.796794	+	0.604251i	$0.793472\pi$
$252$		0			0
$253$	−1.51962e13			−0.921664
$254$		0			0
$255$	9.66411e12	−	2.78149e13i	0.561294	−	1.61550i
$256$		0			0
$257$	−1.79007e13			−0.995949			−0.497974	−	0.867192i	$-0.665923\pi$
$257$	−1.79007e13			−0.995949			−0.497974	+	0.867192i	$0.665923\pi$
$258$		0			0
$259$	−8.88560e12	−	2.19778e13i	−0.473738	−	1.17175i
$260$		0			0
$261$	8.95176e12	−	1.13272e13i	0.457494	−	0.578896i
$262$		0			0
$263$			3.77086e12i			0.184792i	0.995722	+	0.0923960i	$0.0294526\pi$
$263$			3.77086e12i			0.184792i	−0.995722	+	0.0923960i	$0.970547\pi$
$264$		0			0
$265$			2.45026e13i			1.15175i
$266$		0			0
$267$	−4.76685e12	+	1.37198e13i	−0.214991	+	0.618779i
$268$		0			0
$269$	3.84201e12			0.166311			0.0831555	−	0.996537i	$-0.473500\pi$
$269$	3.84201e12			0.166311			0.0831555	+	0.996537i	$0.473500\pi$
$270$		0			0
$271$		−	8.19916e12i		−	0.340752i	−0.985379	−	0.170376i	$-0.945502\pi$
$271$		−	8.19916e12i		−	0.340752i	0.985379	−	0.170376i	$-0.0544982\pi$
$272$		0			0
$273$	−1.27040e12	−	2.54834e13i	−0.0507045	−	1.01710i
$274$		0			0
$275$		−	1.96740e13i		−	0.754333i
$276$		0			0
$277$	−5.31900e12			−0.195971			−0.0979854	−	0.995188i	$-0.531240\pi$
$277$	−5.31900e12			−0.195971			−0.0979854	+	0.995188i	$0.531240\pi$
$278$		0			0
$279$	2.59045e13	−	3.27786e13i	0.917387	−	1.16083i
$280$		0			0
$281$			1.34290e13i			0.457257i	0.973514	+	0.228628i	$0.0734241\pi$
$281$			1.34290e13i			0.457257i	−0.973514	+	0.228628i	$0.926576\pi$
$282$		0			0
$283$			4.84296e13i			1.58593i	0.609264	+	0.792967i	$0.291465\pi$
$283$			4.84296e13i			1.58593i	−0.609264	+	0.792967i	$0.708535\pi$
$284$		0			0
$285$	−3.13204e13	−	1.08820e13i	−0.986688	−	0.342818i
$286$		0			0
$287$	1.58691e13	+	3.92509e13i	0.481062	+	1.18987i
$288$		0			0
$289$	2.84718e13			0.830763
$290$		0			0
$291$	−1.04389e13	−	3.62693e12i	−0.293254	−	0.101889i
$292$		0			0
$293$	2.71164e13			0.733601			0.366801	−	0.930300i	$-0.380453\pi$
$293$	2.71164e13			0.733601			0.366801	+	0.930300i	$0.380453\pi$
$294$		0			0
$295$	−5.99447e13			−1.56217
$296$		0			0
$297$	−2.70248e13	−	4.23851e13i	−0.678582	−	1.06427i
$298$		0			0
$299$	3.07282e13			0.743612
$300$		0			0
$301$	2.38801e13	+	5.90657e13i	0.557085	+	1.37791i
$302$		0			0
$303$	−7.75368e12	+	2.23164e13i	−0.174411	+	0.501983i
$304$		0			0
$305$			3.82951e12i			0.0830795i
$306$		0			0
$307$			2.27973e13i			0.477115i	0.971128	+	0.238558i	$0.0766746\pi$
$307$			2.27973e13i			0.477115i	−0.971128	+	0.238558i	$0.923325\pi$
$308$		0			0
$309$	3.87705e13	+	1.34705e13i	0.782942	+	0.272028i
$310$		0			0
$311$	1.91597e12			0.0373429			0.0186714	−	0.999826i	$-0.494056\pi$
$311$	1.91597e12			0.0373429			0.0186714	+	0.999826i	$0.494056\pi$
$312$		0			0
$313$			6.23953e13i			1.17397i	0.809597	+	0.586987i	$0.199686\pi$
$313$			6.23953e13i			1.17397i	−0.809597	+	0.586987i	$0.800314\pi$
$314$		0			0
$315$	1.95335e13	−	6.67756e13i	0.354872	−	1.21314i
$316$		0			0
$317$		−	2.72235e13i		−	0.477659i	−0.971061	−	0.238830i	$-0.923236\pi$
$317$		−	2.72235e13i		−	0.477659i	0.971061	−	0.238830i	$-0.0767638\pi$
$318$		0			0
$319$	5.49471e13			0.931314
$320$		0			0
$321$	8.04495e13	+	2.79516e13i	1.31748	+	0.457751i
$322$		0			0
$323$		−	7.06512e13i		−	1.11816i
$324$		0			0
$325$			3.97827e13i			0.608606i
$326$		0			0
$327$	8.29919e10	−	2.38865e11i	0.00122751	−	0.00353296i
$328$		0			0
$329$	−4.20196e13	−	1.03932e14i	−0.601000	−	1.48653i
$330$		0			0
$331$	−6.74470e13			−0.933058			−0.466529	−	0.884506i	$-0.654496\pi$
$331$	−6.74470e13			−0.933058			−0.466529	+	0.884506i	$0.654496\pi$
$332$		0			0
$333$	7.40948e13	+	5.85562e13i	0.991618	+	0.783663i
$334$		0			0
$335$	−1.37853e14			−1.78513
$336$		0			0
$337$	1.46005e14			1.82980			0.914900	−	0.403681i	$-0.132270\pi$
$337$	1.46005e14			1.82980			0.914900	+	0.403681i	$0.132270\pi$
$338$		0			0
$339$	−8.66526e13	−	3.01069e13i	−1.05119	−	0.365231i
$340$		0			0
$341$	1.59005e14			1.86751
$342$		0			0
$343$	−8.03495e13	−	3.57064e13i	−0.913831	−	0.406096i
$344$		0			0
$345$	7.91480e13	+	2.74995e13i	0.871836	+	0.302914i
$346$		0			0
$347$			1.14206e14i			1.21864i	0.792924	+	0.609320i	$0.208558\pi$
$347$			1.14206e14i			1.21864i	−0.792924	+	0.609320i	$0.791442\pi$
$348$		0			0
$349$			1.28402e14i			1.32750i	0.747956	+	0.663748i	$0.231035\pi$
$349$			1.28402e14i			1.32750i	−0.747956	+	0.663748i	$0.768965\pi$
$350$		0			0
$351$	5.46468e13	+	8.57068e13i	0.547490	+	0.858670i
$352$		0			0
$353$	1.05142e14			1.02097			0.510486	−	0.859886i	$-0.329465\pi$
$353$	1.05142e14			1.02097			0.510486	+	0.859886i	$0.329465\pi$
$354$		0			0
$355$			5.91686e13i			0.556974i
$356$		0			0
$357$	1.48065e14	−	7.38136e12i	1.35138	−	0.0673691i
$358$		0			0
$359$		−	7.42230e13i		−	0.656930i	−0.944516	−	0.328465i	$-0.893469\pi$
$359$		−	7.42230e13i		−	0.656930i	0.944516	−	0.328465i	$-0.106531\pi$
$360$		0			0
$361$	3.69351e13			0.317066
$362$		0			0
$363$	2.33769e13	−	6.72826e13i	0.194671	−	0.560295i
$364$		0			0
$365$		−	1.87114e14i		−	1.51180i
$366$		0			0
$367$		−	1.96666e14i		−	1.54193i	−0.636876	−	0.770966i	$-0.719774\pi$
$367$		−	1.96666e14i		−	1.54193i	0.636876	−	0.770966i	$-0.280226\pi$
$368$		0			0
$369$	−1.32328e14	−	1.04577e14i	−1.00695	−	0.795779i
$370$		0			0
$371$	−1.14368e14	+	4.62386e13i	−0.844785	+	0.341545i
$372$		0			0
$373$	−1.45118e13			−0.104069			−0.0520347	−	0.998645i	$-0.516571\pi$
$373$	−1.45118e13			−0.104069			−0.0520347	+	0.998645i	$0.516571\pi$
$374$		0			0
$375$	2.39700e13	−	6.89898e13i	0.166916	−	0.480411i
$376$		0			0
$377$	−1.11108e14			−0.751397
$378$		0			0
$379$	1.98737e14			1.30546			0.652728	−	0.757592i	$-0.273624\pi$
$379$	1.98737e14			1.30546			0.652728	+	0.757592i	$0.273624\pi$
$380$		0			0
$381$	7.40745e12	−	2.13199e13i	0.0472695	−	0.136050i
$382$		0			0
$383$	2.14817e14			1.33191			0.665956	−	0.745991i	$-0.268024\pi$
$383$	2.14817e14			1.33191			0.665956	+	0.745991i	$0.268024\pi$
$384$		0			0
$385$	2.45485e14	−	9.92493e13i	1.47908	−	0.597989i
$386$		0			0
$387$	−1.99130e14	−	1.57370e14i	−1.16608	−	0.921537i
$388$		0			0
$389$			2.38756e14i			1.35904i	0.733658	+	0.679519i	$0.237811\pi$
$389$			2.38756e14i			1.35904i	−0.733658	+	0.679519i	$0.762189\pi$
$390$		0			0
$391$			1.78539e14i			0.988008i
$392$		0			0
$393$	4.37461e13	−	1.25909e14i	0.235386	−	0.677481i
$394$		0			0
$395$	−2.60649e14			−1.36387
$396$		0			0
$397$			2.22230e14i			1.13098i	0.824756	+	0.565489i	$0.191313\pi$
$397$			2.22230e14i			1.13098i	−0.824756	+	0.565489i	$0.808687\pi$
$398$		0			0
$399$	−8.31161e12	−	1.66725e14i	−0.0411466	−	0.825373i
$400$		0			0
$401$			1.83901e14i			0.885707i	0.896594	+	0.442853i	$0.146034\pi$
$401$			1.83901e14i			0.885707i	−0.896594	+	0.442853i	$0.853966\pi$
$402$		0			0
$403$	−3.21524e14			−1.50673
$404$		0			0
$405$	6.40551e13	+	2.69663e14i	0.292113	+	1.22976i
$406$		0			0
$407$			3.59426e14i			1.59529i
$408$		0			0
$409$		−	2.53752e13i		−	0.109631i	−0.998497	−	0.0548153i	$-0.982543\pi$
$409$		−	2.53752e13i		−	0.109631i	0.998497	−	0.0548153i	$-0.0174570\pi$
$410$		0			0
$411$	−3.36752e14	−	1.17002e14i	−1.41638	−	0.492113i
$412$		0			0
$413$	−1.13121e14	−	2.79796e14i	−0.463252	−	1.14582i
$414$		0			0
$415$	−4.76075e14			−1.89850
$416$		0			0
$417$	−9.36055e13	−	3.25226e13i	−0.363541	−	0.126310i
$418$		0			0
$419$	4.54359e14			1.71879			0.859394	−	0.511314i	$-0.170841\pi$
$419$	4.54359e14			1.71879			0.859394	+	0.511314i	$0.170841\pi$
$420$		0			0
$421$	−2.91306e14			−1.07349			−0.536745	−	0.843745i	$-0.680346\pi$
$421$	−2.91306e14			−1.07349			−0.536745	+	0.843745i	$0.680346\pi$
$422$		0			0
$423$	3.50391e14	+	2.76909e14i	1.25800	+	0.994182i
$424$		0			0
$425$	−2.31148e14			−0.808632
$426$		0			0
$427$	−1.78745e13	+	7.22662e12i	−0.0609368	+	0.0246367i
$428$		0			0
$429$	−1.26964e14	+	3.65424e14i	−0.421858	+	1.21418i
$430$		0			0
$431$		−	2.00039e14i		−	0.647872i	−0.946079	−	0.323936i	$-0.894994\pi$
$431$		−	2.00039e14i		−	0.647872i	0.946079	−	0.323936i	$-0.105006\pi$
$432$		0			0
$433$		−	2.71904e14i		−	0.858483i	−0.903190	−	0.429241i	$-0.858781\pi$
$433$		−	2.71904e14i		−	0.858483i	0.903190	−	0.429241i	$-0.141219\pi$
$434$		0			0
$435$	−2.86186e14	−	9.94336e13i	−0.880964	−	0.306085i
$436$		0			0
$437$	2.01040e14			0.603440
$438$		0			0
$439$		−	2.27715e14i		−	0.666556i	−0.942829	−	0.333278i	$-0.891845\pi$
$439$		−	2.27715e14i		−	0.666556i	0.942829	−	0.333278i	$-0.108155\pi$
$440$		0			0
$441$	3.48541e14	−	3.48376e13i	0.995042	−	0.0994571i
$442$		0			0
$443$			6.67383e14i			1.85846i	0.369496	+	0.929232i	$0.379530\pi$
$443$			6.67383e14i			1.85846i	−0.369496	+	0.929232i	$0.620470\pi$
$444$		0			0
$445$	3.04791e14			0.827985
$446$		0			0
$447$	4.33282e14	+	1.50541e14i	1.14836	+	0.398991i
$448$		0			0
$449$		−	3.91260e14i		−	1.01184i	−0.862581	−	0.505918i	$-0.831154\pi$
$449$		−	3.91260e14i		−	1.01184i	0.862581	−	0.505918i	$-0.168846\pi$
$450$		0			0
$451$		−	6.41909e14i		−	1.61995i
$452$		0			0
$453$	−1.35931e14	+	3.91233e14i	−0.334795	+	0.963597i
$454$		0			0
$455$	−4.96395e14	+	2.00692e14i	−1.19334	+	0.482466i
$456$		0			0
$457$	1.23255e14			0.289244			0.144622	−	0.989487i	$-0.453803\pi$
$457$	1.23255e14			0.289244			0.144622	+	0.989487i	$0.453803\pi$
$458$		0			0
$459$	−4.97978e14	+	3.17512e14i	−1.14088	+	0.727429i
$460$		0			0
$461$	−1.91725e14			−0.428868			−0.214434	−	0.976738i	$-0.568791\pi$
$461$	−1.91725e14			−0.428868			−0.214434	+	0.976738i	$0.568791\pi$
$462$		0			0
$463$	−1.46165e14			−0.319262			−0.159631	−	0.987177i	$-0.551030\pi$
$463$	−1.46165e14			−0.319262			−0.159631	+	0.987177i	$0.551030\pi$
$464$		0			0
$465$	−8.28164e14	−	2.87740e14i	−1.76655	−	0.613775i
$466$		0			0
$467$	5.34962e14			1.11450			0.557250	−	0.830345i	$-0.311857\pi$
$467$	5.34962e14			1.11450			0.557250	+	0.830345i	$0.311857\pi$
$468$		0			0
$469$	−2.60141e14	−	6.43438e14i	−0.529368	−	1.30935i
$470$		0			0
$471$	−1.98380e14	−	6.89258e13i	−0.394351	−	0.137015i
$472$		0			0
$473$		−	9.65960e14i		−	1.87596i
$474$		0			0
$475$			2.60279e14i			0.493883i
$476$		0			0
$477$	3.04713e14	−	3.85572e14i	0.564989	−	0.714915i
$478$		0			0
$479$	−2.74443e14			−0.497287			−0.248643	−	0.968595i	$-0.579985\pi$
$479$	−2.74443e14			−0.497287			−0.248643	+	0.968595i	$0.579985\pi$
$480$		0			0
$481$		−	7.26794e14i		−	1.28710i
$482$		0			0
$483$	2.10038e13	+	4.21323e14i	0.0363571	+	0.729299i
$484$		0			0
$485$			2.31905e14i			0.392401i
$486$		0			0
$487$	−5.12082e14			−0.847092			−0.423546	−	0.905875i	$-0.639215\pi$
$487$	−5.12082e14			−0.847092			−0.423546	+	0.905875i	$0.639215\pi$
$488$		0			0
$489$	−2.98597e14	+	8.59411e14i	−0.482932	+	1.38996i
$490$		0			0
$491$			2.16539e14i			0.342443i	0.985233	+	0.171221i	$0.0547713\pi$
$491$			2.16539e14i			0.342443i	−0.985233	+	0.171221i	$0.945229\pi$
$492$		0			0
$493$		−	6.45568e14i		−	0.998352i
$494$		0			0
$495$	−6.54054e14	+	8.27615e14i	−0.989202	+	1.25170i
$496$		0			0
$497$	−2.76173e14	+	1.11656e14i	−0.408527	+	0.165167i
$498$		0			0
$499$	2.26752e14			0.328094			0.164047	−	0.986453i	$-0.447545\pi$
$499$	2.26752e14			0.328094			0.164047	+	0.986453i	$0.447545\pi$
$500$		0			0
$501$	2.03162e13	−	5.84735e13i	0.0287565	−	0.0827659i
$502$		0			0
$503$	−5.99152e14			−0.829685			−0.414843	−	0.909893i	$-0.636163\pi$
$503$	−5.99152e14			−0.829685			−0.414843	+	0.909893i	$0.636163\pi$
$504$		0			0
$505$	4.95768e14			0.671701
$506$		0			0
$507$	9.17422e12	−	2.64049e13i	0.0121626	−	0.0350059i
$508$		0			0
$509$	1.37350e15			1.78188			0.890942	−	0.454117i	$-0.150045\pi$
$509$	1.37350e15			1.78188			0.890942	+	0.454117i	$0.150045\pi$
$510$		0			0
$511$	8.73365e14	−	3.53100e14i	1.10887	−	0.448314i
$512$		0			0
$513$	3.57527e14	+	5.60737e14i	0.444287	+	0.696809i
$514$		0			0
$515$		−	8.61303e14i		−	1.04765i
$516$		0			0
$517$			1.69971e15i			2.02384i
$518$		0			0
$519$	−9.50991e12	+	2.73711e13i	−0.0110855	+	0.0319060i
$520$		0			0
$521$	4.11317e14			0.469428			0.234714	−	0.972064i	$-0.424585\pi$
$521$	4.11317e14			0.469428			0.234714	+	0.972064i	$0.424585\pi$
$522$		0			0
$523$			6.15252e13i			0.0687534i	0.999409	+	0.0343767i	$0.0109446\pi$
$523$			6.15252e13i			0.0687534i	−0.999409	+	0.0343767i	$0.989055\pi$
$524$		0			0
$525$	−5.45471e14	+	2.71929e13i	−0.596892	+	0.0297563i
$526$		0			0
$527$		−	1.86814e15i		−	2.00194i
$528$		0			0
$529$	4.44773e14			0.466802
$530$		0			0
$531$	9.43286e14	+	7.45468e14i	0.969669	+	0.766318i
$532$		0			0
$533$			1.29800e15i			1.30700i
$534$		0			0
$535$		−	1.78722e15i		−	1.76292i
$536$		0			0
$537$	1.67425e15	+	5.81707e14i	1.61794	+	0.562141i
$538$		0			0
$539$	9.26506e14	+	9.58527e14i	0.877223	+	0.907541i
$540$		0			0
$541$	1.90399e15			1.76637			0.883183	−	0.469029i	$-0.155396\pi$
$541$	1.90399e15			1.76637			0.883183	+	0.469029i	$0.155396\pi$
$542$		0			0
$543$	2.25614e14	+	7.83879e13i	0.205100	+	0.0712607i
$544$		0			0
$545$	−5.30648e12			−0.00472744
$546$		0			0
$547$	1.33498e15			1.16559			0.582794	−	0.812620i	$-0.301959\pi$
$547$	1.33498e15			1.16559			0.582794	+	0.812620i	$0.301959\pi$
$548$		0			0
$549$	4.76235e13	−	6.02609e13i	0.0407543	−	0.0515689i
$550$		0			0
$551$	−7.26927e14			−0.609757
$552$		0			0
$553$	−4.91867e14	−	1.21660e15i	−0.404445	−	1.00036i
$554$		0			0
$555$	6.50425e14	−	1.87203e15i	0.524307	−	1.50904i
$556$		0			0
$557$			1.32763e15i			1.04924i	0.851337	+	0.524619i	$0.175792\pi$
$557$			1.32763e15i			1.04924i	−0.851337	+	0.524619i	$0.824208\pi$
$558$		0			0
$559$			1.95327e15i			1.51355i
$560$		0			0
$561$	−2.12321e15	−	7.37694e14i	−1.61323	−	0.560506i
$562$		0			0
$563$	−1.06094e15			−0.790486			−0.395243	−	0.918577i	$-0.629340\pi$
$563$	−1.06094e15			−0.790486			−0.395243	+	0.918577i	$0.629340\pi$
$564$		0			0
$565$			1.92502e15i			1.40660i
$566$		0			0
$567$	−1.13779e15	+	8.07860e14i	−0.815374	+	0.578934i
$568$		0			0
$569$			1.33450e15i			0.937994i	0.883200	+	0.468997i	$0.155384\pi$
$569$			1.33450e15i			0.937994i	−0.883200	+	0.468997i	$0.844616\pi$
$570$		0			0
$571$	1.90913e15			1.31624			0.658122	−	0.752912i	$-0.271351\pi$
$571$	1.90913e15			1.31624			0.658122	+	0.752912i	$0.271351\pi$
$572$		0			0
$573$	1.63682e15	+	5.68700e14i	1.10700	+	0.384621i
$574$		0			0
$575$		−	6.57737e14i		−	0.436394i
$576$		0			0
$577$		−	9.11507e13i		−	0.0593325i	−0.999560	−	0.0296663i	$-0.990556\pi$
$577$		−	9.11507e13i		−	0.0593325i	0.999560	−	0.0296663i	$-0.00944445\pi$
$578$		0			0
$579$	−4.54509e14	+	1.30815e15i	−0.290276	+	0.835462i
$580$		0			0
$581$	−8.98396e14	−	2.22211e15i	−0.562988	−	1.39251i
$582$		0			0
$583$	1.87037e15			1.15014
$584$		0			0
$585$	1.32256e15	−	1.67352e15i	0.798102	−	1.00989i
$586$		0			0
$587$	2.05609e15			1.21768			0.608840	−	0.793293i	$-0.291635\pi$
$587$	2.05609e15			1.21768			0.608840	+	0.793293i	$0.291635\pi$
$588$		0			0
$589$	−2.10357e15			−1.22271
$590$		0			0
$591$	−1.27292e15	−	4.42268e14i	−0.726224	−	0.252322i
$592$		0			0
$593$	−2.10809e15			−1.18056			−0.590280	−	0.807198i	$-0.700983\pi$
$593$	−2.10809e15			−1.18056			−0.590280	+	0.807198i	$0.700983\pi$
$594$		0			0
$595$	−1.16607e15	−	2.88418e15i	−0.641034	−	1.58555i
$596$		0			0
$597$	−2.68707e14	−	9.33604e13i	−0.145017	−	0.0503853i
$598$		0			0
$599$			1.09829e15i			0.581930i	0.956734	+	0.290965i	$0.0939763\pi$
$599$			1.09829e15i			0.581930i	−0.956734	+	0.290965i	$0.906024\pi$
$600$		0			0
$601$		−	7.20536e14i		−	0.374840i	−0.982280	−	0.187420i	$-0.939987\pi$
$601$		−	7.20536e14i		−	0.374840i	0.982280	−	0.187420i	$-0.0600126\pi$
$602$		0			0
$603$	2.16925e15	+	1.71433e15i	1.10806	+	0.875688i
$604$		0			0
$605$	−1.49471e15			−0.749727
$606$		0			0
$607$		−	3.41473e15i		−	1.68197i	−0.541058	−	0.840985i	$-0.681976\pi$
$607$		−	3.41473e15i		−	1.68197i	0.541058	−	0.840985i	$-0.318024\pi$
$608$		0			0
$609$	−7.59465e13	−	1.52343e15i	−0.0367377	−	0.736934i
$610$		0			0
$611$		−	3.43697e15i		−	1.63286i
$612$		0			0
$613$	2.09343e15			0.976847			0.488424	−	0.872607i	$-0.337572\pi$
$613$	2.09343e15			0.976847			0.488424	+	0.872607i	$0.337572\pi$
$614$		0			0
$615$	−1.16161e15	+	3.34332e15i	−0.532413	+	1.53237i
$616$		0			0
$617$			2.31713e14i			0.104324i	0.998639	+	0.0521618i	$0.0166112\pi$
$617$			2.31713e14i			0.104324i	−0.998639	+	0.0521618i	$0.983389\pi$
$618$		0			0
$619$		−	7.77570e14i		−	0.343907i	−0.985105	−	0.171954i	$-0.944992\pi$
$619$		−	7.77570e14i		−	0.343907i	0.985105	−	0.171954i	$-0.0550079\pi$
$620$		0			0
$621$	−9.03488e14	−	1.41701e15i	−0.392571	−	0.615700i
$622$		0			0
$623$	5.75168e14	+	1.42263e15i	0.245533	+	0.607307i
$624$		0			0
$625$	−2.95751e15			−1.24047
$626$		0			0
$627$	−8.30664e14	+	2.39079e15i	−0.342337	+	0.985303i
$628$		0			0
$629$	4.22285e15			1.71012
$630$		0			0
$631$	−1.49304e15			−0.594168			−0.297084	−	0.954851i	$-0.596014\pi$
$631$	−1.49304e15			−0.594168			−0.297084	+	0.954851i	$0.596014\pi$
$632$		0			0
$633$	1.16892e15	−	3.36435e15i	0.457156	−	1.31577i
$634$		0			0
$635$	−4.73630e14			−0.182047
$636$		0			0
$637$	−1.87348e15	−	1.93823e15i	−0.707755	−	0.732217i
$638$		0			0
$639$	7.35816e14	−	9.31074e14i	0.273221	−	0.345724i
$640$		0			0
$641$			4.52879e15i			1.65296i	0.562965	+	0.826481i	$0.309661\pi$
$641$			4.52879e15i			1.65296i	−0.562965	+	0.826481i	$0.690339\pi$
$642$		0			0
$643$			4.92986e15i			1.76878i	0.466748	+	0.884391i	$0.345426\pi$
$643$			4.92986e15i			1.76878i	−0.466748	+	0.884391i	$0.654574\pi$
$644$		0			0
$645$	−1.74802e15	+	5.03111e15i	−0.616551	+	1.77454i
$646$		0			0
$647$	2.40609e15			0.834332			0.417166	−	0.908830i	$-0.363023\pi$
$647$	2.40609e15			0.834332			0.417166	+	0.908830i	$0.363023\pi$
$648$		0			0
$649$			4.57578e15i			1.55998i
$650$		0			0
$651$	−2.19773e14	−	4.40850e15i	−0.0736681	−	1.47773i
$652$		0			0
$653$		−	3.48746e15i		−	1.14944i	−0.818350	−	0.574720i	$-0.805111\pi$
$653$		−	3.48746e15i		−	1.14944i	0.818350	−	0.574720i	$-0.194889\pi$
$654$		0			0
$655$	−2.79712e15			−0.906532
$656$		0			0
$657$	−2.32693e15	+	2.94441e15i	−0.741607	+	0.938402i
$658$		0			0
$659$		−	4.71622e14i		−	0.147817i	−0.997265	−	0.0739085i	$-0.976453\pi$
$659$		−	4.71622e14i		−	0.147817i	0.997265	−	0.0739085i	$-0.0235473\pi$
$660$		0			0
$661$		−	4.67965e15i		−	1.44246i	−0.692694	−	0.721232i	$-0.743576\pi$
$661$		−	4.67965e15i		−	1.44246i	0.692694	−	0.721232i	$-0.256424\pi$
$662$		0			0
$663$	4.29333e15	+	1.49169e15i	1.30158	+	0.452224i
$664$		0			0
$665$	−3.24767e15	+	1.31303e15i	−0.968395	+	0.391521i
$666$		0			0
$667$	1.83698e15			0.538781
$668$		0			0
$669$	−3.73338e15	−	1.29714e15i	−1.07711	−	0.374233i
$670$		0			0
$671$	2.92319e14			0.0829628
$672$		0			0
$673$	−3.26519e15			−0.911645			−0.455823	−	0.890071i	$-0.650655\pi$
$673$	−3.26519e15			−0.911645			−0.455823	+	0.890071i	$0.650655\pi$
$674$		0			0
$675$	1.83455e15	−	1.16971e15i	0.503917	−	0.321299i
$676$		0			0
$677$	−1.18223e15			−0.319495			−0.159747	−	0.987158i	$-0.551068\pi$
$677$	−1.18223e15			−0.319495			−0.159747	+	0.987158i	$0.551068\pi$
$678$		0			0
$679$	−1.08243e15	+	4.37625e14i	−0.287817	+	0.116364i
$680$		0			0
$681$	−7.77578e14	+	2.23800e15i	−0.203439	+	0.585532i
$682$		0			0
$683$		−	3.58812e15i		−	0.923746i	−0.886946	−	0.461873i	$-0.847178\pi$
$683$		−	3.58812e15i		−	0.923746i	0.886946	−	0.461873i	$-0.152822\pi$
$684$		0			0
$685$			7.48110e15i			1.89525i
$686$		0			0
$687$	−3.27624e13	−	1.13831e13i	−0.00816795	−	0.00283790i
$688$		0			0
$689$	−3.78206e15			−0.927947
$690$		0			0
$691$			4.35230e15i			1.05097i	0.850804	+	0.525484i	$0.176116\pi$
$691$			4.35230e15i			1.05097i	−0.850804	+	0.525484i	$0.823884\pi$
$692$		0			0
$693$	−5.09720e15	−	1.49106e15i	−1.21143	−	0.354374i
$694$		0			0
$695$			2.07949e15i			0.486451i
$696$		0			0
$697$	−7.54172e15			−1.73656
$698$		0			0
$699$	−5.34953e15	−	1.85866e15i	−1.21253	−	0.421284i
$700$		0			0
$701$		−	2.76253e15i		−	0.616393i	−0.951323	−	0.308197i	$-0.900275\pi$
$701$		−	2.76253e15i		−	0.616393i	0.951323	−	0.308197i	$-0.0997255\pi$
$702$		0			0
$703$		−	4.75505e15i		−	1.04448i
$704$		0			0
$705$	3.07583e15	−	8.85275e15i	0.665154	−	1.91442i
$706$		0			0
$707$	9.35558e14	+	2.31403e15i	0.199188	+	0.492677i
$708$		0			0
$709$	−6.99109e15			−1.46552			−0.732758	−	0.680489i	$-0.761767\pi$
$709$	−6.99109e15			−1.46552			−0.732758	+	0.680489i	$0.761767\pi$
$710$		0			0
$711$	4.10156e15	+	3.24141e15i	0.846576	+	0.669039i
$712$		0			0
$713$	5.31583e15			1.08039
$714$		0			0
$715$	8.11805e15			1.62468
$716$		0			0
$717$	−1.61630e15	−	5.61573e14i	−0.318542	−	0.110675i
$718$		0			0
$719$	1.40034e15			0.271784			0.135892	−	0.990724i	$-0.456610\pi$
$719$	1.40034e15			0.271784			0.135892	+	0.990724i	$0.456610\pi$
$720$		0			0
$721$	4.02019e15	−	1.62535e15i	0.768426	−	0.310673i
$722$		0			0
$723$	−2.27093e15	−	7.89019e14i	−0.427507	−	0.148534i
$724$		0			0
$725$			2.37827e15i			0.440963i
$726$		0			0
$727$			8.90831e15i			1.62688i	0.581647	+	0.813441i	$0.302408\pi$
$727$			8.90831e15i			1.62688i	−0.581647	+	0.813441i	$0.697592\pi$
$728$		0			0
$729$	2.34555e15	−	5.04000e15i	0.421932	−	0.906627i
$730$		0			0
$731$	−1.13490e16			−2.01099
$732$		0			0
$733$			1.12028e14i			0.0195549i	0.999952	+	0.00977746i	$0.00311231\pi$
$733$			1.12028e14i			0.0195549i	−0.999952	+	0.00977746i	$0.996888\pi$
$734$		0			0
$735$	−3.09104e15	−	6.66902e15i	−0.531525	−	1.14678i
$736$		0			0
$737$			1.05228e16i			1.78262i
$738$		0			0
$739$	5.21696e15			0.870709			0.435355	−	0.900259i	$-0.356623\pi$
$739$	5.21696e15			0.870709			0.435355	+	0.900259i	$0.356623\pi$
$740$		0			0
$741$	1.67968e15	−	4.83441e15i	0.276202	−	0.794956i
$742$		0			0
$743$		−	7.45447e14i		−	0.120775i	−0.998175	−	0.0603877i	$-0.980766\pi$
$743$		−	7.45447e14i		−	0.120775i	0.998175	−	0.0603877i	$-0.0192337\pi$
$744$		0			0
$745$		−	9.62554e15i		−	1.53662i
$746$		0			0
$747$	7.49150e15	+	5.92044e15i	1.17843	+	0.931302i
$748$		0			0
$749$	8.34196e15	−	3.37264e15i	1.29306	−	0.522781i
$750$		0			0
$751$	−1.48803e15			−0.227296			−0.113648	−	0.993521i	$-0.536254\pi$
$751$	−1.48803e15			−0.227296			−0.113648	+	0.993521i	$0.536254\pi$
$752$		0			0
$753$	3.47444e15	−	1.00000e16i	0.523013	−	1.50532i
$754$		0			0
$755$	8.69140e15			1.28938
$756$		0			0
$757$	−1.24719e16			−1.82350			−0.911751	−	0.410744i	$-0.865269\pi$
$757$	−1.24719e16			−1.82350			−0.911751	+	0.410744i	$0.865269\pi$
$758$		0			0
$759$	2.09913e15	−	6.04164e15i	0.302488	−	0.870612i
$760$		0			0
$761$	−3.09265e15			−0.439253			−0.219627	−	0.975584i	$-0.570484\pi$
$761$	−3.09265e15			−0.439253			−0.219627	+	0.975584i	$0.570484\pi$
$762$		0			0
$763$	−1.00138e13	−	2.47683e13i	−0.00140189	−	0.00346747i
$764$		0			0
$765$	9.72356e15	+	7.68441e15i	1.34180	+	1.06041i
$766$		0			0
$767$		−	9.25267e15i		−	1.25861i
$768$		0			0
$769$			1.06465e16i			1.42762i	0.700338	+	0.713811i	$0.253033\pi$
$769$			1.06465e16i			1.42762i	−0.700338	+	0.713811i	$0.746967\pi$
$770$		0			0
$771$	2.47270e15	−	7.11686e15i	0.326868	−	0.940782i
$772$		0			0
$773$	−5.28652e15			−0.688942			−0.344471	−	0.938797i	$-0.611942\pi$
$773$	−5.28652e15			−0.688942			−0.344471	+	0.938797i	$0.611942\pi$
$774$		0			0
$775$			6.88222e15i			0.884238i
$776$		0			0
$777$	9.96525e15	−	4.96789e14i	1.26233	−	0.0629298i
$778$		0			0
$779$			8.49219e15i			1.06063i
$780$		0			0
$781$	4.51654e15			0.556192
$782$		0			0
$783$	3.26687e15	+	5.12368e15i	0.396681	+	0.622146i
$784$		0			0
$785$			4.40710e15i			0.527679i
$786$		0			0
$787$			8.57270e15i			1.01218i	0.862482	+	0.506088i	$0.168909\pi$
$787$			8.57270e15i			1.01218i	−0.862482	+	0.506088i	$0.831091\pi$
$788$		0			0
$789$	−1.49920e15	−	5.20886e14i	−0.174556	−	0.0606484i
$790$		0			0
$791$	−8.98518e15	+	3.63269e15i	−1.03171	+	0.417117i
$792$		0			0
$793$	−5.91098e14			−0.0669355
$794$		0			0
$795$	−9.74162e15	−	3.38466e15i	−1.08796	−	0.378004i
$796$		0			0
$797$	−3.31753e15			−0.365422			−0.182711	−	0.983167i	$-0.558487\pi$
$797$	−3.31753e15			−0.365422			−0.182711	+	0.983167i	$0.558487\pi$
$798$		0			0
$799$	1.99697e16			2.16952
$800$		0			0
$801$	−4.79618e15	−	3.79036e15i	−0.513945	−	0.406164i
$802$		0			0
$803$	−1.42830e16			−1.50968
$804$		0			0
$805$	8.20701e15	−	3.31808e15i	0.855673	−	0.345947i
$806$		0			0
$807$	−5.30715e14	+	1.52749e15i	−0.0545829	+	0.157099i
$808$		0			0
$809$		−	1.00470e16i		−	1.01934i	−0.860370	−	0.509670i	$-0.829768\pi$
$809$		−	1.00470e16i		−	1.01934i	0.860370	−	0.509670i	$-0.170232\pi$
$810$		0			0
$811$			6.54645e15i			0.655226i	0.944812	+	0.327613i	$0.106244\pi$
$811$			6.54645e15i			0.655226i	−0.944812	+	0.327613i	$0.893756\pi$
$812$		0			0
$813$	3.25978e15	+	1.13259e15i	0.321877	+	0.111834i
$814$		0			0
$815$	1.90922e16			1.85990
$816$		0			0
$817$			1.27792e16i			1.22824i
$818$		0			0
$819$	1.03070e16	+	3.01506e15i	0.977401	+	0.285914i
$820$		0			0
$821$		−	1.46809e16i		−	1.37361i	−0.726840	−	0.686806i	$-0.759012\pi$
$821$		−	1.46809e16i		−	1.37361i	0.726840	−	0.686806i	$-0.240988\pi$
$822$		0			0
$823$	−8.36776e15			−0.772521			−0.386260	−	0.922390i	$-0.626233\pi$
$823$	−8.36776e15			−0.772521			−0.386260	+	0.922390i	$0.626233\pi$
$824$		0			0
$825$	7.82189e15	+	2.71766e15i	0.712549	+	0.247571i
$826$		0			0
$827$		−	1.51787e16i		−	1.36444i	−0.731146	−	0.682221i	$-0.761014\pi$
$827$		−	1.51787e16i		−	1.36444i	0.731146	−	0.682221i	$-0.238986\pi$
$828$		0			0
$829$			1.18299e16i			1.04937i	0.851295	+	0.524687i	$0.175818\pi$
$829$			1.18299e16i			1.04937i	−0.851295	+	0.524687i	$0.824182\pi$
$830$		0			0
$831$	7.34739e14	−	2.11470e15i	0.0643172	−	0.185116i
$832$		0			0
$833$	1.12616e16	−	1.08854e16i	0.972868	−	0.940367i
$834$		0			0
$835$	−1.29901e15			−0.110749
$836$		0			0
$837$	9.45363e15	+	1.48268e16i	0.795442	+	1.24755i
$838$		0			0
$839$	1.20775e16			1.00296			0.501482	−	0.865168i	$-0.332788\pi$
$839$	1.20775e16			1.00296			0.501482	+	0.865168i	$0.332788\pi$
$840$		0			0
$841$	5.55829e15			0.455578
$842$		0			0
$843$	−5.33905e15	−	1.85502e15i	−0.431929	−	0.150071i
$844$		0			0
$845$	−5.86597e14			−0.0468412
$846$		0			0
$847$	−2.82065e15	−	6.97667e15i	−0.222326	−	0.549907i
$848$		0			0
$849$	−1.92544e16	−	6.68981e15i	−1.49809	−	0.520501i
$850$		0			0
$851$			1.20162e16i			0.922903i
$852$		0			0
$853$			2.32449e16i			1.76241i	0.472732	+	0.881206i	$0.343268\pi$
$853$			2.32449e16i			1.76241i	−0.472732	+	0.881206i	$0.656732\pi$
$854$		0			0
$855$	8.65286e15	−	1.09490e16i	0.647658	−	0.819522i
$856$		0			0
$857$	1.35819e16			1.00362			0.501808	−	0.864979i	$-0.332669\pi$
$857$	1.35819e16			1.00362			0.501808	+	0.864979i	$0.332669\pi$
$858$		0			0
$859$		−	1.98996e15i		−	0.145172i	−0.997362	−	0.0725858i	$-0.976875\pi$
$859$		−	1.98996e15i		−	0.145172i	0.997362	−	0.0725858i	$-0.0231251\pi$
$860$		0			0
$861$	−1.77972e16	+	8.87230e14i	−1.28184	+	0.0639027i
$862$		0			0
$863$			2.28322e16i			1.62364i	0.583909	+	0.811819i	$0.301522\pi$
$863$			2.28322e16i			1.62364i	−0.583909	+	0.811819i	$0.698478\pi$
$864$		0			0
$865$	6.08061e14			0.0426932
$866$		0			0
$867$	−3.93295e15	+	1.13197e16i	−0.272655	+	0.784746i
$868$		0			0
$869$			1.98962e16i			1.36195i
$870$		0			0
$871$		−	2.12781e16i		−	1.43825i
$872$		0			0
$873$	2.88395e15	−	3.64924e15i	0.192491	−	0.243570i
$874$		0			0
$875$	−2.89222e15	−	7.15369e15i	−0.190628	−	0.471504i
$876$		0			0
$877$	−1.18440e15			−0.0770907			−0.0385454	−	0.999257i	$-0.512272\pi$
$877$	−1.18440e15			−0.0770907			−0.0385454	+	0.999257i	$0.512272\pi$
$878$		0			0
$879$	−3.74572e15	+	1.07808e16i	−0.240767	+	0.692966i
$880$		0			0
$881$	−2.02832e16			−1.28756			−0.643781	−	0.765209i	$-0.722635\pi$
$881$	−2.02832e16			−1.28756			−0.643781	+	0.765209i	$0.722635\pi$
$882$		0			0
$883$	2.64816e15			0.166020			0.0830099	−	0.996549i	$-0.473547\pi$
$883$	2.64816e15			0.166020			0.0830099	+	0.996549i	$0.473547\pi$
$884$		0			0
$885$	8.28044e15	−	2.38325e16i	0.512702	−	1.47564i
$886$		0			0
$887$	1.18280e16			0.723322			0.361661	−	0.932310i	$-0.382210\pi$
$887$	1.18280e16			0.723322			0.361661	+	0.932310i	$0.382210\pi$
$888$		0			0
$889$	−8.93782e14	−	2.21070e15i	−0.0539849	−	0.133527i
$890$		0			0
$891$	2.05843e16	−	4.88954e15i	1.22803	−	0.291703i
$892$		0			0
$893$		−	2.24864e16i		−	1.32506i
$894$		0			0
$895$		−	3.71941e16i		−	2.16495i
$896$		0			0
$897$	−4.24464e15	+	1.22168e16i	−0.244052	+	0.702422i
$898$		0			0
$899$	−1.92212e16			−1.09170
$900$		0			0
$901$		−	2.19748e16i		−	1.23293i
$902$		0			0
$903$	−2.67817e16	+	1.33513e15i	−1.48442	+	0.0740013i
$904$		0			0
$905$		−	5.01210e15i		−	0.274443i
$906$		0			0
$907$	2.85504e15			0.154444			0.0772221	−	0.997014i	$-0.475395\pi$
$907$	2.85504e15			0.154444			0.0772221	+	0.997014i	$0.475395\pi$
$908$		0			0
$909$	−7.80138e15	−	6.16533e15i	−0.416937	−	0.329500i
$910$		0			0
$911$		−	2.07192e16i		−	1.09401i	−0.837130	−	0.547005i	$-0.815768\pi$
$911$		−	2.07192e16i		−	1.09401i	0.837130	−	0.547005i	$-0.184232\pi$
$912$		0			0
$913$			3.63404e16i			1.89584i
$914$		0			0
$915$	−1.52252e15	−	5.28988e14i	−0.0784776	−	0.0272665i
$916$		0			0
$917$	−5.27841e15	−	1.30557e16i	−0.268826	−	0.664920i
$918$		0			0
$919$	−1.86579e16			−0.938918			−0.469459	−	0.882954i	$-0.655551\pi$
$919$	−1.86579e16			−0.938918			−0.469459	+	0.882954i	$0.655551\pi$
$920$		0			0
$921$	−9.06365e15	−	3.14911e15i	−0.450687	−	0.156588i
$922$		0			0
$923$	−9.13287e15			−0.448743
$924$		0			0
$925$	−1.55570e16			−0.755346
$926$		0			0
$927$	−1.07111e16	+	1.35534e16i	−0.513920	+	0.650295i
$928$		0			0
$929$	2.49449e15			0.118276			0.0591379	−	0.998250i	$-0.481165\pi$
$929$	2.49449e15			0.118276			0.0591379	+	0.998250i	$0.481165\pi$
$930$		0			0
$931$	−1.22573e16	−	1.26809e16i	−0.574342	−	0.594193i
$932$		0			0
$933$	−2.64663e14	+	7.61743e14i	−0.0122559	+	0.0352744i
$934$		0			0
$935$			4.71679e16i			2.15865i
$936$		0			0
$937$			7.25594e15i			0.328190i	0.986445	+	0.164095i	$0.0524704\pi$
$937$			7.25594e15i			0.328190i	−0.986445	+	0.164095i	$0.947530\pi$
$938$		0			0
$939$	−2.48068e16	−	8.61896e15i	−1.10895	−	0.385296i
$940$		0			0
$941$	−2.08328e16			−0.920461			−0.460230	−	0.887800i	$-0.652233\pi$
$941$	−2.08328e16			−0.920461			−0.460230	+	0.887800i	$0.652233\pi$
$942$		0			0
$943$		−	2.14602e16i		−	0.937170i
$944$		0			0
$945$	2.38500e16	+	1.69900e16i	1.02947	+	0.733364i
$946$		0			0
$947$		−	1.74940e16i		−	0.746386i	−0.927754	−	0.373193i	$-0.878263\pi$
$947$		−	1.74940e16i		−	0.746386i	0.927754	−	0.373193i	$-0.121737\pi$
$948$		0			0
$949$	2.88816e16			1.21803
$950$		0			0
$951$	1.08234e16	+	3.76051e15i	0.451201	+	0.156767i
$952$		0			0
$953$			4.74959e16i			1.95725i	0.205661	+	0.978623i	$0.434065\pi$
$953$			4.74959e16i			1.95725i	−0.205661	+	0.978623i	$0.565935\pi$
$954$		0			0
$955$		−	3.63625e16i		−	1.48127i
$956$		0			0
$957$	−7.59010e15	+	2.18456e16i	−0.305655	+	0.879727i
$958$		0			0
$959$	−3.49185e16	+	1.41175e16i	−1.39012	+	0.562025i
$960$		0			0
$961$	−3.02136e16			−1.18912
$962$		0			0
$963$	−2.22257e16	+	2.81236e16i	−0.864792	+	1.09427i
$964$		0			0
$965$	2.90612e16			1.11793
$966$		0			0
$967$	−4.24033e16			−1.61270			−0.806352	−	0.591436i	$-0.798561\pi$
$967$	−4.24033e16			−1.61270			−0.806352	+	0.591436i	$0.798561\pi$
$968$		0			0
$969$	2.80891e16	+	9.75938e15i	1.05623	+	0.366979i
$970$		0			0
$971$	3.76749e16			1.40070			0.700352	−	0.713798i	$-0.253026\pi$
$971$	3.76749e16			1.40070			0.700352	+	0.713798i	$0.253026\pi$
$972$		0			0
$973$	−9.70613e15	+	3.92417e15i	−0.356801	+	0.144254i
$974$		0			0
$975$	−1.58166e16	−	5.49538e15i	−0.574895	−	0.199743i
$976$		0			0
$977$			3.16313e16i			1.13683i	0.822740	+	0.568417i	$0.192444\pi$
$977$			3.16313e16i			1.13683i	−0.822740	+	0.568417i	$0.807556\pi$
$978$		0			0
$979$		−	2.32657e16i		−	0.826822i
$980$		0			0
$981$	8.35025e13	+	6.59910e13i	0.00293441	+	0.00231902i
$982$		0			0
$983$	−2.31947e15			−0.0806018			−0.0403009	−	0.999188i	$-0.512832\pi$
$983$	−2.31947e15			−0.0806018			−0.0403009	+	0.999188i	$0.512832\pi$
$984$		0			0
$985$			2.82785e16i			0.971756i
$986$		0			0
$987$	4.71252e16	−	2.34929e15i	1.60143	−	0.0798348i
$988$		0			0
$989$		−	3.22938e16i		−	1.08527i
$990$		0			0
$991$	−2.66064e16			−0.884262			−0.442131	−	0.896951i	$-0.645777\pi$
$991$	−2.66064e16			−0.884262			−0.442131	+	0.896951i	$0.645777\pi$
$992$		0			0
$993$	9.31678e15	−	2.68152e16i	0.306228	−	0.881375i
$994$		0			0
$995$			5.96944e15i			0.194047i
$996$		0			0
$997$		−	3.50447e16i		−	1.12668i	−0.826226	−	0.563339i	$-0.809517\pi$
$997$		−	3.50447e16i		−	1.12668i	0.826226	−	0.563339i	$-0.190483\pi$
$998$		0			0
$999$	−3.35155e16	+	2.13696e16i	−1.06570	+	0.679494i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.12	yes	28
3.2	odd	2	inner	84.12.f.b.41.18	yes	28
7.6	odd	2	inner	84.12.f.b.41.17	yes	28
21.20	even	2	inner	84.12.f.b.41.11	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.11	✓	28	21.20	even	2	inner
84.12.f.b.41.12	yes	28	1.1	even	1	trivial
84.12.f.b.41.17	yes	28	7.6	odd	2	inner
84.12.f.b.41.18	yes	28	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-138.135 + 397.575i) q^{3} +8832.29 q^{5} +(16667.3 + 41225.3i) q^{7} +(-138985. - 109838. i) q^{9} +O(q^{10})\)	\(q+(-138.135 + 397.575i) q^{3} +8832.29 q^{5} +(16667.3 + 41225.3i) q^{7} +(-138985. - 109838. i) q^{9} -674199. i q^{11} +1.36330e6i q^{13} +(-1.22005e6 + 3.51150e6i) q^{15} -7.92109e6 q^{17} +8.91937e6i q^{19} +(-1.86925e7 + 931861. i) q^{21} -2.25397e7i q^{23} +2.91813e7 q^{25} +(6.28674e7 - 4.00844e7i) q^{27} +8.14998e7i q^{29} +2.35843e8i q^{31} +(2.68045e8 + 9.31303e7i) q^{33} +(1.47211e8 + 3.64114e8i) q^{35} -5.33115e8 q^{37} +(-5.42012e8 - 1.88319e8i) q^{39} +9.52106e8 q^{41} +1.43275e9 q^{43} +(-1.22755e9 - 9.70120e8i) q^{45} -2.52108e9 q^{47} +(-1.42173e9 + 1.37423e9i) q^{49} +(1.09418e9 - 3.14923e9i) q^{51} +2.77421e9i q^{53} -5.95472e9i q^{55} +(-3.54612e9 - 1.23207e9i) q^{57} -6.78699e9 q^{59} +4.33580e8i q^{61} +(2.21160e9 - 7.56039e9i) q^{63} +1.20410e10i q^{65} -1.56078e10 q^{67} +(8.96121e9 + 3.11351e9i) q^{69} +6.69911e9i q^{71} -2.11852e10i q^{73} +(-4.03095e9 + 1.16018e10i) q^{75} +(2.77941e10 - 1.12371e10i) q^{77} -2.95109e10 q^{79} +(7.25237e9 + 3.05315e10i) q^{81} -5.39016e10 q^{83} -6.99614e10 q^{85} +(-3.24023e10 - 1.12580e10i) q^{87} +3.45087e10 q^{89} +(-5.62023e10 + 2.27225e10i) q^{91} +(-9.37654e10 - 3.25782e10i) q^{93} +7.87785e10i q^{95} +2.62564e10i q^{97} +(-7.40525e10 + 9.37033e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 28 * q + 9632 * q^7 + 267660 * q^9	\( 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) \) 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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