LMFDB - Embedded newform 84.12.f.b.41.19

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			41.19
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(240.448 - 345.444i) q^{3} -113.306 q^{5} +(44008.9 + 6367.04i) q^{7} +(-61516.6 - 166123. i) q^{9} +O(q^{10})$	\(q+(240.448 - 345.444i) q^{3} -113.306 q^{5} +(44008.9 + 6367.04i) q^{7} +(-61516.6 - 166123. i) q^{9} +471963. i q^{11} +1.52458e6i q^{13} +(-27244.2 + 39141.0i) q^{15} +1.63168e6 q^{17} +2.97596e6i q^{19} +(1.27813e7 - 1.36717e7i) q^{21} +2.79258e7i q^{23} -4.88153e7 q^{25} +(-7.21777e7 - 1.86933e7i) q^{27} +1.72679e8i q^{29} +6.77394e7i q^{31} +(1.63037e8 + 1.13483e8i) q^{33} +(-4.98648e6 - 721424. i) q^{35} -2.11171e8 q^{37} +(5.26658e8 + 3.66582e8i) q^{39} +1.13330e9 q^{41} -1.22129e9 q^{43} +(6.97021e6 + 1.88227e7i) q^{45} -2.73123e9 q^{47} +(1.89625e9 + 5.60413e8i) q^{49} +(3.92333e8 - 5.63654e8i) q^{51} -4.10043e9i q^{53} -5.34763e7i q^{55} +(1.02803e9 + 7.15563e8i) q^{57} +9.33617e9 q^{59} +9.27679e9i q^{61} +(-1.64957e9 - 7.70257e9i) q^{63} -1.72744e8i q^{65} +9.31784e9 q^{67} +(9.64682e9 + 6.71471e9i) q^{69} -1.13341e9i q^{71} +1.18753e10i q^{73} +(-1.17375e10 + 1.68630e10i) q^{75} +(-3.00501e9 + 2.07706e10i) q^{77} +2.57625e9 q^{79} +(-2.38125e10 + 2.04386e10i) q^{81} +3.83358e10 q^{83} -1.84879e8 q^{85} +(5.96508e10 + 4.15202e10i) q^{87} +4.88289e10 q^{89} +(-9.70706e9 + 6.70952e10i) q^{91} +(2.34002e10 + 1.62878e10i) q^{93} -3.37194e8i q^{95} +1.23963e11i q^{97} +(7.84038e10 - 2.90336e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	240.448	−	345.444i	0.571287	−	0.820751i
$3$	240.448	−	345.444i	0.571287	−	0.820751i
$4$		0			0
$5$	−113.306			−0.0162150			−0.00810752	−	0.999967i	$-0.502581\pi$
$5$	−113.306			−0.0162150			−0.00810752	+	0.999967i	$0.502581\pi$
$6$		0			0
$7$	44008.9	+	6367.04i	0.989696	+	0.143185i
$7$	44008.9	+	6367.04i	0.989696	+	0.143185i
$8$		0			0
$9$	−61516.6	−	166123.i	−0.347263	−	0.937768i
$10$		0			0
$11$			471963.i			0.883585i	0.897117	+	0.441793i	$0.145657\pi$
$11$			471963.i			0.883585i	−0.897117	+	0.441793i	$0.854343\pi$
$12$		0			0
$13$			1.52458e6i			1.13884i	0.822048	+	0.569419i	$0.192832\pi$
$13$			1.52458e6i			1.13884i	−0.822048	+	0.569419i	$0.807168\pi$
$14$		0			0
$15$	−27244.2	+	39141.0i	−0.00926344	+	0.0133085i
$16$		0			0
$17$	1.63168e6			0.278718			0.139359	−	0.990242i	$-0.455496\pi$
$17$	1.63168e6			0.278718			0.139359	+	0.990242i	$0.455496\pi$
$18$		0			0
$19$			2.97596e6i			0.275729i	0.990451	+	0.137864i	$0.0440238\pi$
$19$			2.97596e6i			0.275729i	−0.990451	+	0.137864i	$0.955976\pi$
$20$		0			0
$21$	1.27813e7	−	1.36717e7i	0.682919	−	0.730494i
$22$		0			0
$23$			2.79258e7i			0.904697i	0.891841	+	0.452348i	$0.149414\pi$
$23$			2.79258e7i			0.904697i	−0.891841	+	0.452348i	$0.850586\pi$
$24$		0			0
$25$	−4.88153e7			−0.999737
$26$		0			0
$27$	−7.21777e7	−	1.86933e7i	−0.968060	−	0.250718i
$28$		0			0
$29$			1.72679e8i			1.56333i	0.623702	+	0.781663i	$0.285628\pi$
$29$			1.72679e8i			1.56333i	−0.623702	+	0.781663i	$0.714372\pi$
$30$		0			0
$31$			6.77394e7i			0.424964i	0.977165	+	0.212482i	$0.0681546\pi$
$31$			6.77394e7i			0.424964i	−0.977165	+	0.212482i	$0.931845\pi$
$32$		0			0
$33$	1.63037e8	+	1.13483e8i	0.725203	+	0.504780i
$34$		0			0
$35$	−4.98648e6	−	721424.i	−0.0160480	−	0.00232175i
$36$		0			0
$37$	−2.11171e8			−0.500640			−0.250320	−	0.968163i	$-0.580536\pi$
$37$	−2.11171e8			−0.500640			−0.250320	+	0.968163i	$0.580536\pi$
$38$		0			0
$39$	5.26658e8	+	3.66582e8i	0.934702	+	0.650603i
$40$		0			0
$41$	1.13330e9			1.52769			0.763845	−	0.645400i	$-0.223309\pi$
$41$	1.13330e9			1.52769			0.763845	+	0.645400i	$0.223309\pi$
$42$		0			0
$43$	−1.22129e9			−1.26690			−0.633451	−	0.773782i	$-0.718362\pi$
$43$	−1.22129e9			−1.26690			−0.633451	+	0.773782i	$0.718362\pi$
$44$		0			0
$45$	6.97021e6	+	1.88227e7i	0.00563089	+	0.0152059i
$46$		0			0
$47$	−2.73123e9			−1.73708			−0.868540	−	0.495618i	$-0.834942\pi$
$47$	−2.73123e9			−1.73708			−0.868540	+	0.495618i	$0.834942\pi$
$48$		0			0
$49$	1.89625e9	+	5.60413e8i	0.958996	+	0.283420i
$50$		0			0
$51$	3.92333e8	−	5.63654e8i	0.159228	−	0.228758i
$52$		0			0
$53$		−	4.10043e9i		−	1.34683i	−0.739266	−	0.673413i	$-0.764827\pi$
$53$		−	4.10043e9i		−	1.34683i	0.739266	−	0.673413i	$-0.235173\pi$
$54$		0			0
$55$		−	5.34763e7i		−	0.0143274i
$56$		0			0
$57$	1.02803e9	+	7.15563e8i	0.226305	+	0.157520i
$58$		0			0
$59$	9.33617e9			1.70013			0.850066	−	0.526676i	$-0.176562\pi$
$59$	9.33617e9			1.70013			0.850066	+	0.526676i	$0.176562\pi$
$60$		0			0
$61$			9.27679e9i			1.40632i	0.711033	+	0.703159i	$0.248228\pi$
$61$			9.27679e9i			1.40632i	−0.711033	+	0.703159i	$0.751772\pi$
$62$		0			0
$63$	−1.64957e9	−	7.70257e9i	−0.209410	−	0.977828i
$64$		0			0
$65$		−	1.72744e8i		−	0.0184663i
$66$		0			0
$67$	9.31784e9			0.843148			0.421574	−	0.906794i	$-0.361478\pi$
$67$	9.31784e9			0.843148			0.421574	+	0.906794i	$0.361478\pi$
$68$		0			0
$69$	9.64682e9	+	6.71471e9i	0.742531	+	0.516841i
$70$		0			0
$71$		−	1.13341e9i		−	0.0745535i	−0.999305	−	0.0372767i	$-0.988132\pi$
$71$		−	1.13341e9i		−	0.0745535i	0.999305	−	0.0372767i	$-0.0118683\pi$
$72$		0			0
$73$			1.18753e10i			0.670456i	0.942137	+	0.335228i	$0.108813\pi$
$73$			1.18753e10i			0.670456i	−0.942137	+	0.335228i	$0.891187\pi$
$74$		0			0
$75$	−1.17375e10	+	1.68630e10i	−0.571136	+	0.820535i
$76$		0			0
$77$	−3.00501e9	+	2.07706e10i	−0.126516	+	0.874481i
$78$		0			0
$79$	2.57625e9			0.0941974			0.0470987	−	0.998890i	$-0.485002\pi$
$79$	2.57625e9			0.0941974			0.0470987	+	0.998890i	$0.485002\pi$
$80$		0			0
$81$	−2.38125e10	+	2.04386e10i	−0.758817	+	0.651304i
$82$		0			0
$83$	3.83358e10			1.06826			0.534128	−	0.845404i	$-0.320640\pi$
$83$	3.83358e10			1.06826			0.534128	+	0.845404i	$0.320640\pi$
$84$		0			0
$85$	−1.84879e8			−0.00451943
$86$		0			0
$87$	5.96508e10	+	4.15202e10i	1.28310	+	0.893107i
$88$		0			0
$89$	4.88289e10			0.926898			0.463449	−	0.886124i	$-0.346612\pi$
$89$	4.88289e10			0.926898			0.463449	+	0.886124i	$0.346612\pi$
$90$		0			0
$91$	−9.70706e9	+	6.70952e10i	−0.163065	+	1.12710i
$92$		0			0
$93$	2.34002e10	+	1.62878e10i	0.348789	+	0.242776i
$94$		0			0
$95$		−	3.37194e8i		−	0.00447096i
$96$		0			0
$97$			1.23963e11i			1.46570i	0.680388	+	0.732852i	$0.261811\pi$
$97$			1.23963e11i			1.46570i	−0.680388	+	0.732852i	$0.738189\pi$
$98$		0			0
$99$	7.84038e10	−	2.90336e10i	0.828598	−	0.306836i
$100$		0			0
$101$	6.69904e10			0.634227			0.317114	−	0.948388i	$-0.397286\pi$
$101$	6.69904e10			0.634227			0.317114	+	0.948388i	$0.397286\pi$
$102$		0			0
$103$			1.17600e11i			0.999542i	0.866158	+	0.499771i	$0.166583\pi$
$103$			1.17600e11i			0.999542i	−0.866158	+	0.499771i	$0.833417\pi$
$104$		0			0
$105$	−1.44820e9	+	1.54909e9i	−0.0110736	+	0.0118450i
$106$		0			0
$107$		−	1.80768e11i		−	1.24598i	−0.782229	−	0.622991i	$-0.785917\pi$
$107$		−	1.80768e11i		−	1.24598i	0.782229	−	0.622991i	$-0.214083\pi$
$108$		0			0
$109$	−3.05519e10			−0.190192			−0.0950962	−	0.995468i	$-0.530316\pi$
$109$	−3.05519e10			−0.190192			−0.0950962	+	0.995468i	$0.530316\pi$
$110$		0			0
$111$	−5.07757e10	+	7.29480e10i	−0.286009	+	0.410901i
$112$		0			0
$113$			1.95662e10i			0.0999021i	0.998752	+	0.0499510i	$0.0159065\pi$
$113$			1.95662e10i			0.0999021i	−0.998752	+	0.0499510i	$0.984093\pi$
$114$		0			0
$115$		−	3.16417e9i		−	0.0146697i
$116$		0			0
$117$	2.53267e11	−	9.37870e10i	1.06797	−	0.395476i
$118$		0			0
$119$	7.18084e10	+	1.03889e10i	0.275846	+	0.0399083i
$120$		0			0
$121$	6.25624e10			0.219277
$122$		0			0
$123$	2.72501e11	−	3.91494e11i	0.872749	−	1.25385i
$124$		0			0
$125$	1.10636e10			0.0324258
$126$		0			0
$127$	−1.23978e11			−0.332985			−0.166492	−	0.986043i	$-0.553244\pi$
$127$	−1.23978e11			−0.332985			−0.166492	+	0.986043i	$0.553244\pi$
$128$		0			0
$129$	−2.93657e11	+	4.21889e11i	−0.723765	+	1.03981i
$130$		0			0
$131$	2.22104e11			0.502995			0.251497	−	0.967858i	$-0.419077\pi$
$131$	2.22104e11			0.502995			0.251497	+	0.967858i	$0.419077\pi$
$132$		0			0
$133$	−1.89480e10	+	1.30969e11i	−0.0394803	+	0.272888i
$134$		0			0
$135$	8.17817e9	+	2.11806e9i	0.0156971	+	0.00406540i
$136$		0			0
$137$			4.28893e11i			0.759252i	0.925140	+	0.379626i	$0.123947\pi$
$137$			4.28893e11i			0.759252i	−0.925140	+	0.379626i	$0.876053\pi$
$138$		0			0
$139$		−	2.91079e11i		−	0.475806i	−0.971289	−	0.237903i	$-0.923540\pi$
$139$		−	2.91079e11i		−	0.475806i	0.971289	−	0.237903i	$-0.0764600\pi$
$140$		0			0
$141$	−6.56719e11	+	9.43488e11i	−0.992371	+	1.42571i
$142$		0			0
$143$	−7.19546e11			−1.00626
$144$		0			0
$145$		−	1.95655e10i		−	0.0253494i
$146$		0			0
$147$	6.49541e11	−	5.20298e11i	0.780478	−	0.625183i
$148$		0			0
$149$		−	3.55911e11i		−	0.397024i	−0.980098	−	0.198512i	$-0.936389\pi$
$149$		−	3.55911e11i		−	0.397024i	0.980098	−	0.198512i	$-0.0636109\pi$
$150$		0			0
$151$	−1.84819e11			−0.191591			−0.0957954	−	0.995401i	$-0.530539\pi$
$151$	−1.84819e11			−0.191591			−0.0957954	+	0.995401i	$0.530539\pi$
$152$		0			0
$153$	−1.00375e11	−	2.71059e11i	−0.0967885	−	0.261373i
$154$		0			0
$155$		−	7.67528e9i		−	0.00689081i
$156$		0			0
$157$		−	1.75361e12i		−	1.46719i	−0.679589	−	0.733593i	$-0.737842\pi$
$157$		−	1.75361e12i		−	1.46719i	0.679589	−	0.733593i	$-0.262158\pi$
$158$		0			0
$159$	−1.41647e12	−	9.85939e11i	−1.10541	−	0.769424i
$160$		0			0
$161$	−1.77805e11	+	1.22899e12i	−0.129539	+	0.895375i
$162$		0			0
$163$	−2.54989e11			−0.173576			−0.0867880	−	0.996227i	$-0.527660\pi$
$163$	−2.54989e11			−0.173576			−0.0867880	+	0.996227i	$0.527660\pi$
$164$		0			0
$165$	−1.84731e10	−	1.28583e10i	−0.0117592	−	0.00818504i
$166$		0			0
$167$	−6.69349e11			−0.398760			−0.199380	−	0.979922i	$-0.563893\pi$
$167$	−6.69349e11			−0.398760			−0.199380	+	0.979922i	$0.563893\pi$
$168$		0			0
$169$	−5.32184e11			−0.296951
$170$		0			0
$171$	4.94375e11	−	1.83071e11i	0.258570	−	0.0957504i
$172$		0			0
$173$	−2.24751e12			−1.10268			−0.551338	−	0.834282i	$-0.685882\pi$
$173$	−2.24751e12			−1.10268			−0.551338	+	0.834282i	$0.685882\pi$
$174$		0			0
$175$	−2.14831e12	−	3.10809e11i	−0.989436	−	0.143148i
$176$		0			0
$177$	2.24486e12	−	3.22513e12i	0.971263	−	1.39538i
$178$		0			0
$179$		−	1.24133e12i		−	0.504890i	−0.967611	−	0.252445i	$-0.918765\pi$
$179$		−	1.24133e12i		−	0.504890i	0.967611	−	0.252445i	$-0.0812347\pi$
$180$		0			0
$181$		−	3.74804e12i		−	1.43407i	−0.697035	−	0.717037i	$-0.745498\pi$
$181$		−	3.74804e12i		−	1.43407i	0.697035	−	0.717037i	$-0.254502\pi$
$182$		0			0
$183$	3.20461e12	+	2.23058e12i	1.15424	+	0.803411i
$184$		0			0
$185$	2.39270e10			0.00811791
$186$		0			0
$187$			7.70091e11i			0.246271i
$188$		0			0
$189$	−3.05744e12	−	1.28223e12i	−0.922186	−	0.386746i
$190$		0			0
$191$		−	4.80965e12i		−	1.36908i	−0.728973	−	0.684542i	$-0.760002\pi$
$191$		−	4.80965e12i		−	1.36908i	0.728973	−	0.684542i	$-0.239998\pi$
$192$		0			0
$193$	−2.81562e12			−0.756849			−0.378425	−	0.925632i	$-0.623534\pi$
$193$	−2.81562e12			−0.756849			−0.378425	+	0.925632i	$0.623534\pi$
$194$		0			0
$195$	−5.96735e10	−	4.15360e10i	−0.0151562	−	0.0105496i
$196$		0			0
$197$		−	2.63844e10i		−	0.00633554i	−0.999995	−	0.00316777i	$-0.998992\pi$
$197$		−	2.63844e10i		−	0.00633554i	0.999995	−	0.00316777i	$-0.00100833\pi$
$198$		0			0
$199$		−	1.49140e12i		−	0.338767i	−0.985550	−	0.169383i	$-0.945822\pi$
$199$		−	1.49140e12i		−	0.338767i	0.985550	−	0.169383i	$-0.0541776\pi$
$200$		0			0
$201$	2.24045e12	−	3.21879e12i	0.481679	−	0.692014i
$202$		0			0
$203$	−1.09945e12	+	7.59940e12i	−0.223845	+	1.54722i
$204$		0			0
$205$	−1.28410e11			−0.0247716
$206$		0			0
$207$	4.63912e12	−	1.71790e12i	0.848396	−	0.314168i
$208$		0			0
$209$	−1.40454e12			−0.243630
$210$		0			0
$211$	−1.03449e13			−1.70283			−0.851416	−	0.524491i	$-0.824256\pi$
$211$	−1.03449e13			−1.70283			−0.851416	+	0.524491i	$0.824256\pi$
$212$		0			0
$213$	−3.91532e11	−	2.72527e11i	−0.0611898	−	0.0425914i
$214$		0			0
$215$	1.38380e11			0.0205429
$216$		0			0
$217$	−4.31299e11	+	2.98114e12i	−0.0608485	+	0.420585i
$218$		0			0
$219$	4.10227e12	+	2.85540e12i	0.550277	+	0.383023i
$220$		0			0
$221$			2.48762e12i			0.317415i
$222$		0			0
$223$			1.16788e13i			1.41815i	0.705133	+	0.709075i	$0.250887\pi$
$223$			1.16788e13i			1.41815i	−0.705133	+	0.709075i	$0.749113\pi$
$224$		0			0
$225$	3.00295e12	+	8.10933e12i	0.347172	+	0.937521i
$226$		0			0
$227$	1.14378e13			1.25950			0.629751	−	0.776797i	$-0.283157\pi$
$227$	1.14378e13			1.25950			0.629751	+	0.776797i	$0.283157\pi$
$228$		0			0
$229$		−	1.18535e12i		−	0.124380i	−0.998064	−	0.0621901i	$-0.980192\pi$
$229$		−	1.18535e12i		−	0.124380i	0.998064	−	0.0621901i	$-0.0198085\pi$
$230$		0			0
$231$	6.45254e12	+	6.03231e12i	0.645453	+	0.603417i
$232$		0			0
$233$		−	1.79976e13i		−	1.71694i	−0.512860	−	0.858472i	$-0.671414\pi$
$233$		−	1.79976e13i		−	1.71694i	0.512860	−	0.858472i	$-0.328586\pi$
$234$		0			0
$235$	3.09465e11			0.0281669
$236$		0			0
$237$	6.19453e11	−	8.89950e11i	0.0538137	−	0.0773125i
$238$		0			0
$239$			1.99328e13i			1.65341i	0.562639	+	0.826703i	$0.309786\pi$
$239$			1.99328e13i			1.65341i	−0.562639	+	0.826703i	$0.690214\pi$
$240$		0			0
$241$			1.50138e13i			1.18959i	0.803879	+	0.594793i	$0.202766\pi$
$241$			1.50138e13i			1.18959i	−0.803879	+	0.594793i	$0.797234\pi$
$242$		0			0
$243$	1.33475e12	+	1.31403e13i	0.101056	+	0.994881i
$244$		0			0
$245$	−2.14857e11	−	6.34982e10i	−0.0155502	−	0.00459566i
$246$		0			0
$247$	−4.53709e12			−0.314010
$248$		0			0
$249$	9.21776e12	−	1.32429e13i	0.610280	−	0.876771i
$250$		0			0
$251$	7.21675e12			0.457231			0.228616	−	0.973517i	$-0.426580\pi$
$251$	7.21675e12			0.457231			0.228616	+	0.973517i	$0.426580\pi$
$252$		0			0
$253$	−1.31800e13			−0.799377
$254$		0			0
$255$	−4.44538e10	+	6.38654e10i	−0.00258189	+	0.00370932i
$256$		0			0
$257$	2.35142e13			1.30827			0.654136	−	0.756377i	$-0.273032\pi$
$257$	2.35142e13			1.30827			0.654136	+	0.756377i	$0.273032\pi$
$258$		0			0
$259$	−9.29344e12	−	1.34454e12i	−0.495482	−	0.0716843i
$260$		0			0
$261$	2.86858e13	−	1.06226e13i	1.46604	−	0.542885i
$262$		0			0
$263$		−	1.15132e13i		−	0.564209i	−0.959384	−	0.282105i	$-0.908967\pi$
$263$		−	1.15132e13i		−	0.564209i	0.959384	−	0.282105i	$-0.0910325\pi$
$264$		0			0
$265$			4.64603e11i			0.0218389i
$266$		0			0
$267$	1.17408e13	−	1.68677e13i	0.529525	−	0.760752i
$268$		0			0
$269$	−2.61870e13			−1.13357			−0.566784	−	0.823866i	$-0.691813\pi$
$269$	−2.61870e13			−1.13357			−0.566784	+	0.823866i	$0.691813\pi$
$270$		0			0
$271$			1.74929e13i			0.726993i	0.931596	+	0.363497i	$0.118417\pi$
$271$			1.74929e13i			0.726993i	−0.931596	+	0.363497i	$0.881583\pi$
$272$		0			0
$273$	2.08436e13	+	1.94861e13i	0.831914	+	0.777734i
$274$		0			0
$275$		−	2.30390e13i		−	0.883353i
$276$		0			0
$277$	−2.77920e13			−1.02395			−0.511977	−	0.858999i	$-0.671087\pi$
$277$	−2.77920e13			−1.02395			−0.511977	+	0.858999i	$0.671087\pi$
$278$		0			0
$279$	1.12530e13	−	4.16710e12i	0.398517	−	0.147574i
$280$		0			0
$281$		−	4.10085e12i		−	0.139633i	−0.997560	−	0.0698166i	$-0.977759\pi$
$281$		−	4.10085e12i		−	0.139633i	0.997560	−	0.0698166i	$-0.0222414\pi$
$282$		0			0
$283$			5.14769e13i			1.68573i	0.538129	+	0.842863i	$0.319131\pi$
$283$			5.14769e13i			1.68573i	−0.538129	+	0.842863i	$0.680869\pi$
$284$		0			0
$285$	−1.16482e11	−	8.10777e10i	−0.00366954	−	0.00255420i
$286$		0			0
$287$	4.98755e13	+	7.21579e12i	1.51195	+	0.218743i
$288$		0			0
$289$	−3.16095e13			−0.922316
$290$		0			0
$291$	4.28222e13	+	2.98065e13i	1.20298	+	0.837337i
$292$		0			0
$293$	−9.12690e12			−0.246917			−0.123459	−	0.992350i	$-0.539399\pi$
$293$	−9.12690e12			−0.246917			−0.123459	+	0.992350i	$0.539399\pi$
$294$		0			0
$295$	−1.05784e12			−0.0275677
$296$		0			0
$297$	8.82255e12	−	3.40652e13i	0.221531	−	0.855364i
$298$		0			0
$299$	−4.25752e13			−1.03030
$300$		0			0
$301$	−5.37478e13	−	7.77602e12i	−1.25385	−	0.181402i
$302$		0			0
$303$	1.61077e13	−	2.31414e13i	0.362325	−	0.520542i
$304$		0			0
$305$		−	1.05112e12i		−	0.0228035i
$306$		0			0
$307$			3.90280e11i			0.00816799i	0.999992	+	0.00408399i	$0.00129998\pi$
$307$			3.90280e11i			0.00816799i	−0.999992	+	0.00408399i	$0.998700\pi$
$308$		0			0
$309$	4.06241e13	+	2.82766e13i	0.820375	+	0.571025i
$310$		0			0
$311$	2.11599e13			0.412412			0.206206	−	0.978509i	$-0.433888\pi$
$311$	2.11599e13			0.412412			0.206206	+	0.978509i	$0.433888\pi$
$312$		0			0
$313$		−	6.94527e13i		−	1.30676i	−0.757031	−	0.653379i	$-0.773351\pi$
$313$		−	6.94527e13i		−	1.30676i	0.757031	−	0.653379i	$-0.226649\pi$
$314$		0			0
$315$	1.86907e11	+	8.72748e11i	0.00339560	+	0.0158555i
$316$		0			0
$317$			1.00483e14i			1.76306i	0.472126	+	0.881531i	$0.343487\pi$
$317$			1.00483e14i			1.76306i	−0.472126	+	0.881531i	$0.656513\pi$
$318$		0			0
$319$	−8.14979e13			−1.38133
$320$		0			0
$321$	−6.24454e13	−	4.34654e13i	−1.02264	−	0.711813i
$322$		0			0
$323$			4.85580e12i			0.0768506i
$324$		0			0
$325$		−	7.44228e13i		−	1.13854i
$326$		0			0
$327$	−7.34615e12	+	1.05540e13i	−0.108654	+	0.156100i
$328$		0			0
$329$	−1.20199e14	−	1.73898e13i	−1.71918	−	0.248724i
$330$		0			0
$331$	−2.85084e12			−0.0394384			−0.0197192	−	0.999806i	$-0.506277\pi$
$331$	−2.85084e12			−0.0394384			−0.0197192	+	0.999806i	$0.506277\pi$
$332$		0			0
$333$	1.29906e13	+	3.50804e13i	0.173854	+	0.469484i
$334$		0			0
$335$	−1.05577e12			−0.0136717
$336$		0			0
$337$	−8.58892e13			−1.07640			−0.538200	−	0.842817i	$-0.680896\pi$
$337$	−8.58892e13			−1.07640			−0.538200	+	0.842817i	$0.680896\pi$
$338$		0			0
$339$	6.75903e12	+	4.70465e12i	0.0819947	+	0.0570727i
$340$		0			0
$341$	−3.19705e13			−0.375492
$342$		0			0
$343$	7.98837e13	+	3.67367e13i	0.908533	+	0.417813i
$344$		0			0
$345$	−1.09304e12	−	7.60818e11i	−0.0120402	−	0.00838061i
$346$		0			0
$347$		−	4.52064e13i		−	0.482379i	−0.970478	−	0.241189i	$-0.922462\pi$
$347$		−	4.52064e13i		−	0.482379i	0.970478	−	0.241189i	$-0.0775375\pi$
$348$		0			0
$349$		−	1.57965e14i		−	1.63313i	−0.577255	−	0.816564i	$-0.695876\pi$
$349$		−	1.57965e14i		−	1.63313i	0.577255	−	0.816564i	$-0.304124\pi$
$350$		0			0
$351$	2.84994e13	−	1.10041e14i	0.285527	−	1.10246i
$352$		0			0
$353$	5.92063e13			0.574920			0.287460	−	0.957793i	$-0.407189\pi$
$353$	5.92063e13			0.574920			0.287460	+	0.957793i	$0.407189\pi$
$354$		0			0
$355$			1.28423e11i			0.00120889i
$356$		0			0
$357$	2.08550e13	−	2.23078e13i	0.190342	−	0.203602i
$358$		0			0
$359$			3.18285e13i			0.281706i	0.990031	+	0.140853i	$0.0449845\pi$
$359$			3.18285e13i			0.281706i	−0.990031	+	0.140853i	$0.955015\pi$
$360$		0			0
$361$	1.07634e14			0.923974
$362$		0			0
$363$	1.50430e13	−	2.16118e13i	0.125270	−	0.179972i
$364$		0			0
$365$		−	1.34555e12i		−	0.0108715i
$366$		0			0
$367$			2.52552e12i			0.0198010i	0.999951	+	0.00990052i	$0.00315149\pi$
$367$			2.52552e12i			0.0198010i	−0.999951	+	0.00990052i	$0.996849\pi$
$368$		0			0
$369$	−6.97170e13	−	1.88268e14i	−0.530510	−	1.43262i
$370$		0			0
$371$	2.61076e13	−	1.80455e14i	0.192846	−	1.33295i
$372$		0			0
$373$	−1.93885e14			−1.39042			−0.695208	−	0.718809i	$-0.744688\pi$
$373$	−1.93885e14			−1.39042			−0.695208	+	0.718809i	$0.744688\pi$
$374$		0			0
$375$	2.66022e12	−	3.82186e12i	0.0185244	−	0.0266135i
$376$		0			0
$377$	−2.63262e14			−1.78037
$378$		0			0
$379$	−1.46192e13			−0.0960304			−0.0480152	−	0.998847i	$-0.515290\pi$
$379$	−1.46192e13			−0.0960304			−0.0480152	+	0.998847i	$0.515290\pi$
$380$		0			0
$381$	−2.98103e13	+	4.28275e13i	−0.190230	+	0.273297i
$382$		0			0
$383$	3.79797e13			0.235482			0.117741	−	0.993044i	$-0.462435\pi$
$383$	3.79797e13			0.235482			0.117741	+	0.993044i	$0.462435\pi$
$384$		0			0
$385$	3.40486e11	−	2.35344e12i	0.00205147	−	0.0141797i
$386$		0			0
$387$	7.51298e13	+	2.02885e14i	0.439949	+	1.18806i
$388$		0			0
$389$			5.11062e13i			0.290905i	0.989365	+	0.145452i	$0.0464637\pi$
$389$			5.11062e13i			0.290905i	−0.989365	+	0.145452i	$0.953536\pi$
$390$		0			0
$391$			4.55659e13i			0.252155i
$392$		0			0
$393$	5.34043e13	−	7.67244e13i	0.287354	−	0.412833i
$394$		0			0
$395$	−2.91905e11			−0.00152741
$396$		0			0
$397$			8.08783e13i			0.411608i	0.978593	+	0.205804i	$0.0659810\pi$
$397$			8.08783e13i			0.411608i	−0.978593	+	0.205804i	$0.934019\pi$
$398$		0			0
$399$	4.06864e13	+	3.80367e13i	0.201418	+	0.188301i
$400$		0			0
$401$			9.30846e13i			0.448316i	0.974553	+	0.224158i	$0.0719631\pi$
$401$			9.30846e13i			0.448316i	−0.974553	+	0.224158i	$0.928037\pi$
$402$		0			0
$403$	−1.03274e14			−0.483965
$404$		0			0
$405$	2.69810e12	−	2.31582e12i	0.0123043	−	0.0105609i
$406$		0			0
$407$		−	9.96652e13i		−	0.442358i
$408$		0			0
$409$			6.22974e13i			0.269148i	0.990904	+	0.134574i	$0.0429666\pi$
$409$			6.22974e13i			0.269148i	−0.990904	+	0.134574i	$0.957033\pi$
$410$		0			0
$411$	1.48159e14	+	1.03126e14i	0.623157	+	0.433751i
$412$		0			0
$413$	4.10875e14	+	5.94437e13i	1.68261	+	0.243434i
$414$		0			0
$415$	−4.34368e12			−0.0173218
$416$		0			0
$417$	−1.00552e14	−	6.99894e13i	−0.390518	−	0.271822i
$418$		0			0
$419$	3.19690e14			1.20935			0.604674	−	0.796473i	$-0.293303\pi$
$419$	3.19690e14			1.20935			0.604674	+	0.796473i	$0.293303\pi$
$420$		0			0
$421$	2.62841e14			0.968595			0.484298	−	0.874903i	$-0.339075\pi$
$421$	2.62841e14			0.968595			0.484298	+	0.874903i	$0.339075\pi$
$422$		0			0
$423$	1.68016e14	+	4.53719e14i	0.603224	+	1.62898i
$424$		0			0
$425$	−7.96508e13			−0.278645
$426$		0			0
$427$	−5.90656e13	+	4.08262e14i	−0.201364	+	1.39183i
$428$		0			0
$429$	−1.73013e14	+	2.48563e14i	−0.574863	+	0.825888i
$430$		0			0
$431$			4.43722e14i			1.43710i	0.695478	+	0.718548i	$0.255193\pi$
$431$			4.43722e14i			1.43710i	−0.695478	+	0.718548i	$0.744807\pi$
$432$		0			0
$433$		−	3.37507e14i		−	1.06561i	−0.846237	−	0.532807i	$-0.821137\pi$
$433$		−	3.37507e14i		−	1.06561i	0.846237	−	0.532807i	$-0.178863\pi$
$434$		0			0
$435$	−6.75880e12	−	4.70449e12i	−0.0208055	−	0.0144818i
$436$		0			0
$437$	−8.31062e13			−0.249451
$438$		0			0
$439$			2.82690e14i			0.827477i	0.910396	+	0.413738i	$0.135777\pi$
$439$			2.82690e14i			0.827477i	−0.910396	+	0.413738i	$0.864223\pi$
$440$		0			0
$441$	−2.35534e13	−	3.49485e14i	−0.0672421	−	0.997737i
$442$		0			0
$443$			2.15455e14i			0.599979i	0.953943	+	0.299989i	$0.0969831\pi$
$443$			2.15455e14i			0.599979i	−0.953943	+	0.299989i	$0.903017\pi$
$444$		0			0
$445$	−5.53261e12			−0.0150297
$446$		0			0
$447$	−1.22947e14	−	8.55780e13i	−0.325858	−	0.226815i
$448$		0			0
$449$		−	5.16411e14i		−	1.33549i	−0.744390	−	0.667746i	$-0.767259\pi$
$449$		−	5.16411e14i		−	1.33549i	0.744390	−	0.667746i	$-0.232741\pi$
$450$		0			0
$451$			5.34878e14i			1.34984i
$452$		0			0
$453$	−4.44394e13	+	6.38448e13i	−0.109453	+	0.157248i
$454$		0			0
$455$	1.09987e12	−	7.60229e12i	0.00264410	−	0.0182760i
$456$		0			0
$457$	5.55705e14			1.30408			0.652042	−	0.758183i	$-0.273913\pi$
$457$	5.55705e14			1.30408			0.652042	+	0.758183i	$0.273913\pi$
$458$		0			0
$459$	−1.17771e14	−	3.05014e13i	−0.269816	−	0.0698796i
$460$		0			0
$461$	−3.68184e14			−0.823588			−0.411794	−	0.911277i	$-0.635098\pi$
$461$	−3.68184e14			−0.823588			−0.411794	+	0.911277i	$0.635098\pi$
$462$		0			0
$463$	4.82936e14			1.05486			0.527429	−	0.849599i	$-0.323156\pi$
$463$	4.82936e14			1.05486			0.527429	+	0.849599i	$0.323156\pi$
$464$		0			0
$465$	−2.65138e12	−	1.84551e12i	−0.00565563	−	0.00393663i
$466$		0			0
$467$	4.21669e14			0.878473			0.439237	−	0.898371i	$-0.355249\pi$
$467$	4.21669e14			0.878473			0.439237	+	0.898371i	$0.355249\pi$
$468$		0			0
$469$	4.10068e14	+	5.93270e13i	0.834460	+	0.120726i
$470$		0			0
$471$	−6.05775e14	−	4.21652e14i	−1.20419	−	0.838184i
$472$		0			0
$473$		−	5.76405e14i		−	1.11942i
$474$		0			0
$475$		−	1.45272e14i		−	0.275656i
$476$		0			0
$477$	−6.81174e14	+	2.52244e14i	−1.26301	+	0.467703i
$478$		0			0
$479$	9.10387e14			1.64961			0.824804	−	0.565419i	$-0.191285\pi$
$479$	9.10387e14			1.64961			0.824804	+	0.565419i	$0.191285\pi$
$480$		0			0
$481$		−	3.21948e14i		−	0.570148i
$482$		0			0
$483$	3.81794e14	+	3.56929e14i	0.660875	+	0.617835i
$484$		0			0
$485$		−	1.40457e13i		−	0.0237665i
$486$		0			0
$487$	7.22998e14			1.19599			0.597996	−	0.801499i	$-0.295964\pi$
$487$	7.22998e14			1.19599			0.597996	+	0.801499i	$0.295964\pi$
$488$		0			0
$489$	−6.13116e13	+	8.80846e13i	−0.0991617	+	0.142463i
$490$		0			0
$491$		−	3.25939e14i		−	0.515452i	−0.966218	−	0.257726i	$-0.917027\pi$
$491$		−	3.25939e14i		−	0.515452i	0.966218	−	0.257726i	$-0.0829731\pi$
$492$		0			0
$493$			2.81756e14i			0.435727i
$494$		0			0
$495$	−8.88363e12	+	3.28968e12i	−0.0134358	+	0.00497537i
$496$		0			0
$497$	7.21649e12	−	4.98804e13i	0.0106750	−	0.0737853i
$498$		0			0
$499$	2.89680e14			0.419145			0.209573	−	0.977793i	$-0.432793\pi$
$499$	2.89680e14			0.419145			0.209573	+	0.977793i	$0.432793\pi$
$500$		0			0
$501$	−1.60943e14	+	2.31223e14i	−0.227806	+	0.327283i
$502$		0			0
$503$	−1.11093e15			−1.53838			−0.769190	−	0.639020i	$-0.779340\pi$
$503$	−1.11093e15			−1.53838			−0.769190	+	0.639020i	$0.779340\pi$
$504$		0			0
$505$	−7.59042e12			−0.0102840
$506$		0			0
$507$	−1.27962e14	+	1.83840e14i	−0.169644	+	0.243723i
$508$		0			0
$509$	−8.51657e14			−1.10488			−0.552442	−	0.833551i	$-0.686304\pi$
$509$	−8.51657e14			−1.10488			−0.552442	+	0.833551i	$0.686304\pi$
$510$		0			0
$511$	−7.56107e13	+	5.22621e14i	−0.0959994	+	0.663548i
$512$		0			0
$513$	5.56305e13	−	2.14798e14i	0.0691301	−	0.266922i
$514$		0			0
$515$		−	1.33248e13i		−	0.0162076i
$516$		0			0
$517$		−	1.28904e15i		−	1.53486i
$518$		0			0
$519$	−5.40409e14	+	7.76389e14i	−0.629944	+	0.905022i
$520$		0			0
$521$	5.11623e14			0.583905			0.291953	−	0.956433i	$-0.405695\pi$
$521$	5.11623e14			0.583905			0.291953	+	0.956433i	$0.405695\pi$
$522$		0			0
$523$		−	1.57592e15i		−	1.76107i	−0.473985	−	0.880533i	$-0.657185\pi$
$523$		−	1.57592e15i		−	1.76107i	0.473985	−	0.880533i	$-0.342815\pi$
$524$		0			0
$525$	−6.23924e14	+	6.67388e14i	−0.682740	+	0.730302i
$526$		0			0
$527$			1.10529e14i			0.118445i
$528$		0			0
$529$	1.72957e14			0.181523
$530$		0			0
$531$	−5.74329e14	−	1.55095e15i	−0.590393	−	1.59433i
$532$		0			0
$533$			1.72781e15i			1.73979i
$534$		0			0
$535$			2.04822e13i			0.0202037i
$536$		0			0
$537$	−4.28812e14	−	2.98476e14i	−0.414389	−	0.288437i
$538$		0			0
$539$	−2.64494e14	+	8.94959e14i	−0.250425	+	0.847355i
$540$		0			0
$541$	−3.87278e14			−0.359284			−0.179642	−	0.983732i	$-0.557494\pi$
$541$	−3.87278e14			−0.359284			−0.179642	+	0.983732i	$0.557494\pi$
$542$		0			0
$543$	−1.29474e15	−	9.01207e14i	−1.17702	−	0.819267i
$544$		0			0
$545$	3.46172e12			0.00308398
$546$		0			0
$547$	−8.47364e13			−0.0739843			−0.0369922	−	0.999316i	$-0.511778\pi$
$547$	−8.47364e13			−0.0739843			−0.0369922	+	0.999316i	$0.511778\pi$
$548$		0			0
$549$	1.54109e15	−	5.70677e14i	1.31880	−	0.488362i
$550$		0			0
$551$	−5.13884e14			−0.431054
$552$		0			0
$553$	1.13378e14	+	1.64031e13i	0.0932267	+	0.0134877i
$554$		0			0
$555$	5.75320e12	−	8.26545e12i	0.00463765	−	0.00666278i
$556$		0			0
$557$			9.79014e14i			0.773722i	0.922138	+	0.386861i	$0.126441\pi$
$557$			9.79014e14i			0.773722i	−0.922138	+	0.386861i	$0.873559\pi$
$558$		0			0
$559$		−	1.86196e15i		−	1.44280i
$560$		0			0
$561$	2.66024e14	+	1.85167e14i	0.202127	+	0.140691i
$562$		0			0
$563$	−5.69442e14			−0.424281			−0.212140	−	0.977239i	$-0.568043\pi$
$563$	−5.69442e14			−0.424281			−0.212140	+	0.977239i	$0.568043\pi$
$564$		0			0
$565$		−	2.21697e12i		−	0.00161992i
$566$		0			0
$567$	−1.17810e15	+	7.47867e14i	−0.844255	+	0.535942i
$568$		0			0
$569$		−	1.99429e15i		−	1.40175i	−0.713282	−	0.700877i	$-0.752792\pi$
$569$		−	1.99429e15i		−	1.40175i	0.713282	−	0.700877i	$-0.247208\pi$
$570$		0			0
$571$	1.48496e15			1.02380			0.511900	−	0.859045i	$-0.328942\pi$
$571$	1.48496e15			1.02380			0.511900	+	0.859045i	$0.328942\pi$
$572$		0			0
$573$	−1.66147e15	−	1.15647e15i	−1.12368	−	0.782140i
$574$		0			0
$575$		−	1.36321e15i		−	0.904459i
$576$		0			0
$577$			2.20019e15i			1.43217i	0.698014	+	0.716084i	$0.254067\pi$
$577$			2.20019e15i			1.43217i	−0.698014	+	0.716084i	$0.745933\pi$
$578$		0			0
$579$	−6.77011e14	+	9.72642e14i	−0.432378	+	0.621185i
$580$		0			0
$581$	1.68712e15	+	2.44085e14i	1.05725	+	0.152958i
$582$		0			0
$583$	1.93525e15			1.19004
$584$		0			0
$585$	−2.86967e13	+	1.06266e13i	−0.0173171	+	0.00641267i
$586$		0			0
$587$	−2.65537e15			−1.57259			−0.786294	−	0.617852i	$-0.788003\pi$
$587$	−2.65537e15			−1.57259			−0.786294	+	0.617852i	$0.788003\pi$
$588$		0			0
$589$	−2.01590e14			−0.117175
$590$		0			0
$591$	−9.11436e12	−	6.34408e12i	−0.00519990	−	0.00361941i
$592$		0			0
$593$	−6.13671e14			−0.343664			−0.171832	−	0.985126i	$-0.554969\pi$
$593$	−6.13671e14			−0.343664			−0.171832	+	0.985126i	$0.554969\pi$
$594$		0			0
$595$	−8.13633e12	−	1.17713e12i	−0.00447286	−	0.000647115i
$596$		0			0
$597$	−5.15194e14	−	3.58603e14i	−0.278043	−	0.193533i
$598$		0			0
$599$		−	7.89153e12i		−	0.00418133i	−0.999998	−	0.00209066i	$-0.999335\pi$
$599$		−	7.89153e12i		−	0.00418133i	0.999998	−	0.00209066i	$-0.000665479\pi$
$600$		0			0
$601$		−	2.97556e15i		−	1.54796i	−0.633212	−	0.773979i	$-0.718264\pi$
$601$		−	2.97556e15i		−	1.54796i	0.633212	−	0.773979i	$-0.281736\pi$
$602$		0			0
$603$	−5.73202e14	−	1.54790e15i	−0.292794	−	0.790677i
$604$		0			0
$605$	−7.08870e12			−0.00355559
$606$		0			0
$607$		−	2.34064e15i		−	1.15292i	−0.817127	−	0.576458i	$-0.804435\pi$
$607$		−	2.34064e15i		−	1.15292i	0.817127	−	0.576458i	$-0.195565\pi$
$608$		0			0
$609$	2.36081e15	+	2.20706e15i	1.14200	+	1.06763i
$610$		0			0
$611$		−	4.16398e15i		−	1.97825i
$612$		0			0
$613$	−1.49622e15			−0.698174			−0.349087	−	0.937090i	$-0.613508\pi$
$613$	−1.49622e15			−0.698174			−0.349087	+	0.937090i	$0.613508\pi$
$614$		0			0
$615$	−3.08760e13	+	4.43586e13i	−0.0141517	+	0.0203313i
$616$		0			0
$617$			3.75120e15i			1.68889i	0.535640	+	0.844447i	$0.320071\pi$
$617$			3.75120e15i			1.68889i	−0.535640	+	0.844447i	$0.679929\pi$
$618$		0			0
$619$			1.64285e15i			0.726605i	0.931671	+	0.363302i	$0.118351\pi$
$619$			1.64285e15i			0.726605i	−0.931671	+	0.363302i	$0.881649\pi$
$620$		0			0
$621$	5.22026e14	−	2.01562e15i	0.226824	−	0.875801i
$622$		0			0
$623$	2.14891e15	+	3.10895e14i	0.917347	+	0.132718i
$624$		0			0
$625$	2.38231e15			0.999211
$626$		0			0
$627$	−3.37720e14	+	4.85192e14i	−0.139183	+	0.199959i
$628$		0			0
$629$	−3.44564e14			−0.139537
$630$		0			0
$631$	2.10544e15			0.837879			0.418939	−	0.908014i	$-0.362402\pi$
$631$	2.10544e15			0.837879			0.418939	+	0.908014i	$0.362402\pi$
$632$		0			0
$633$	−2.48740e15	+	3.57358e15i	−0.972805	+	1.39760i
$634$		0			0
$635$	1.40475e13			0.00539936
$636$		0			0
$637$	−8.54395e14	+	2.89098e15i	−0.322769	+	1.09214i
$638$		0			0
$639$	−1.88286e14	+	6.97238e13i	−0.0699138	+	0.0258897i
$640$		0			0
$641$		−	5.36092e15i		−	1.95668i	−0.207004	−	0.978340i	$-0.566371\pi$
$641$		−	5.36092e15i		−	1.95668i	0.207004	−	0.978340i	$-0.433629\pi$
$642$		0			0
$643$		−	2.60720e15i		−	0.935437i	−0.883878	−	0.467718i	$-0.845076\pi$
$643$		−	2.60720e15i		−	0.935437i	0.883878	−	0.467718i	$-0.154924\pi$
$644$		0			0
$645$	3.32732e13	−	4.78026e13i	0.0117359	−	0.0168606i
$646$		0			0
$647$	3.92537e15			1.36115			0.680577	−	0.732677i	$-0.261729\pi$
$647$	3.92537e15			1.36115			0.680577	+	0.732677i	$0.261729\pi$
$648$		0			0
$649$			4.40633e15i			1.50221i
$650$		0			0
$651$	9.26112e14	+	8.65798e14i	0.310433	+	0.290216i
$652$		0			0
$653$			4.35551e15i			1.43554i	0.696279	+	0.717772i	$0.254838\pi$
$653$			4.35551e15i			1.43554i	−0.696279	+	0.717772i	$0.745162\pi$
$654$		0			0
$655$	−2.51657e13			−0.00815608
$656$		0			0
$657$	1.97276e15	−	7.30531e14i	0.628732	−	0.232825i
$658$		0			0
$659$			3.33227e15i			1.04441i	0.852821	+	0.522204i	$0.174890\pi$
$659$			3.33227e15i			1.04441i	−0.852821	+	0.522204i	$0.825110\pi$
$660$		0			0
$661$			5.98355e15i			1.84438i	0.386735	+	0.922191i	$0.373603\pi$
$661$			5.98355e15i			1.84438i	−0.386735	+	0.922191i	$0.626397\pi$
$662$		0			0
$663$	8.59335e14	+	5.98143e14i	0.260518	+	0.181335i
$664$		0			0
$665$	2.14693e12	−	1.48396e13i	0.000640175	−	0.00442489i
$666$		0			0
$667$	−4.82219e15			−1.41434
$668$		0			0
$669$	4.03438e15	+	2.80815e15i	1.16395	+	0.810170i
$670$		0			0
$671$	−4.37830e15			−1.24260
$672$		0			0
$673$	−5.22800e15			−1.45966			−0.729832	−	0.683627i	$-0.760402\pi$
$673$	−5.22800e15			−1.45966			−0.729832	+	0.683627i	$0.760402\pi$
$674$		0			0
$675$	3.52338e15	+	9.12519e14i	0.967806	+	0.250652i
$676$		0			0
$677$	4.82002e15			1.30260			0.651301	−	0.758820i	$-0.274224\pi$
$677$	4.82002e15			1.30260			0.651301	+	0.758820i	$0.274224\pi$
$678$		0			0
$679$	−7.89274e14	+	5.45546e15i	−0.209867	+	1.45060i
$680$		0			0
$681$	2.75019e15	−	3.95111e15i	0.719537	−	1.03374i
$682$		0			0
$683$			4.65834e15i			1.19927i	0.800273	+	0.599636i	$0.204688\pi$
$683$			4.65834e15i			1.19927i	−0.800273	+	0.599636i	$0.795312\pi$
$684$		0			0
$685$		−	4.85962e13i		−	0.0123113i
$686$		0			0
$687$	−4.09472e14	−	2.85015e14i	−0.102085	−	0.0710567i
$688$		0			0
$689$	6.25143e15			1.53382
$690$		0			0
$691$		−	1.84813e15i		−	0.446277i	−0.974787	−	0.223138i	$-0.928370\pi$
$691$		−	1.84813e15i		−	0.446277i	0.974787	−	0.223138i	$-0.0716302\pi$
$692$		0			0
$693$	3.63533e15	−	7.78537e14i	0.863994	−	0.185032i
$694$		0			0
$695$			3.29811e13i			0.00771522i
$696$		0			0
$697$	1.84919e15			0.425795
$698$		0			0
$699$	−6.21716e15	−	4.32748e15i	−1.40918	−	0.980868i
$700$		0			0
$701$		−	3.40698e15i		−	0.760188i	−0.924948	−	0.380094i	$-0.875892\pi$
$701$		−	3.40698e15i		−	0.760188i	0.924948	−	0.380094i	$-0.124108\pi$
$702$		0			0
$703$		−	6.28438e14i		−	0.138041i
$704$		0			0
$705$	7.44102e13	−	1.06903e14i	0.0160913	−	0.0231180i
$706$		0			0
$707$	2.94818e15	+	4.26530e14i	0.627692	+	0.0908119i
$708$		0			0
$709$	−4.55797e15			−0.955471			−0.477735	−	0.878504i	$-0.658542\pi$
$709$	−4.55797e15			−0.955471			−0.477735	+	0.878504i	$0.658542\pi$
$710$		0			0
$711$	−1.58482e14	−	4.27973e14i	−0.0327113	−	0.0883352i
$712$		0			0
$713$	−1.89168e15			−0.384463
$714$		0			0
$715$	8.15289e13			0.0163166
$716$		0			0
$717$	6.88567e15	+	4.79280e15i	1.35703	+	0.944569i
$718$		0			0
$719$	−7.27426e15			−1.41182			−0.705911	−	0.708301i	$-0.749462\pi$
$719$	−7.27426e15			−1.41182			−0.705911	+	0.708301i	$0.749462\pi$
$720$		0			0
$721$	−7.48761e14	+	5.17543e15i	−0.143120	+	0.989243i
$722$		0			0
$723$	5.18642e15	+	3.61003e15i	0.976354	+	0.679595i
$724$		0			0
$725$		−	8.42935e15i		−	1.56291i
$726$		0			0
$727$		−	5.59525e15i		−	1.02183i	−0.859630	−	0.510917i	$-0.829306\pi$
$727$		−	5.59525e15i		−	1.02183i	0.859630	−	0.510917i	$-0.170694\pi$
$728$		0			0
$729$	4.86018e15	+	2.69848e15i	0.874281	+	0.485420i
$730$		0			0
$731$	−1.99276e15			−0.353109
$732$		0			0
$733$			4.99585e14i			0.0872042i	0.999049	+	0.0436021i	$0.0138834\pi$
$733$			4.99585e14i			0.0872042i	−0.999049	+	0.0436021i	$0.986117\pi$
$734$		0			0
$735$	−7.35969e13	+	5.89530e13i	−0.0126555	+	0.0101374i
$736$		0			0
$737$			4.39768e15i			0.744993i
$738$		0			0
$739$	−9.66339e15			−1.61282			−0.806409	−	0.591358i	$-0.798592\pi$
$739$	−9.66339e15			−1.61282			−0.806409	+	0.591358i	$0.798592\pi$
$740$		0			0
$741$	−1.09093e15	+	1.56731e15i	−0.179390	+	0.257724i
$742$		0			0
$743$		−	8.79574e15i		−	1.42506i	−0.701640	−	0.712531i	$-0.747549\pi$
$743$		−	8.79574e15i		−	1.42506i	0.701640	−	0.712531i	$-0.252451\pi$
$744$		0			0
$745$			4.03269e13i			0.00643776i
$746$		0			0
$747$	−2.35829e15	−	6.36845e15i	−0.370966	−	1.00178i
$748$		0			0
$749$	1.15096e15	−	7.95543e15i	0.178406	−	1.23314i
$750$		0			0
$751$	2.90400e15			0.443586			0.221793	−	0.975094i	$-0.428809\pi$
$751$	2.90400e15			0.443586			0.221793	+	0.975094i	$0.428809\pi$
$752$		0			0
$753$	1.73525e15	−	2.49299e15i	0.261210	−	0.375273i
$754$		0			0
$755$	2.09412e13			0.00310665
$756$		0			0
$757$	2.79420e15			0.408536			0.204268	−	0.978915i	$-0.434519\pi$
$757$	2.79420e15			0.408536			0.204268	+	0.978915i	$0.434519\pi$
$758$		0			0
$759$	−3.16910e15	+	4.55295e15i	−0.456673	+	0.656089i
$760$		0			0
$761$	−1.29618e15			−0.184099			−0.0920494	−	0.995754i	$-0.529342\pi$
$761$	−1.29618e15			−0.184099			−0.0920494	+	0.995754i	$0.529342\pi$
$762$		0			0
$763$	−1.34456e15	−	1.94525e14i	−0.188233	−	0.0272327i
$764$		0			0
$765$	1.13731e13	+	3.07126e13i	0.00156943	+	0.00423817i
$766$		0			0
$767$			1.42337e16i			1.93617i
$768$		0			0
$769$		−	1.10991e16i		−	1.48831i	−0.668008	−	0.744154i	$-0.732853\pi$
$769$		−	1.10991e16i		−	1.48831i	0.668008	−	0.744154i	$-0.267147\pi$
$770$		0			0
$771$	5.65394e15	−	8.12285e15i	0.747398	−	1.07377i
$772$		0			0
$773$	4.39635e15			0.572935			0.286467	−	0.958090i	$-0.407519\pi$
$773$	4.39635e15			0.572935			0.286467	+	0.958090i	$0.407519\pi$
$774$		0			0
$775$		−	3.30672e15i		−	0.424852i
$776$		0			0
$777$	−2.69905e15	+	2.88707e15i	−0.341897	+	0.365715i
$778$		0			0
$779$			3.37267e15i			0.421228i
$780$		0			0
$781$	5.34930e14			0.0658743
$782$		0			0
$783$	3.22793e15	−	1.24635e16i	0.391954	−	1.51339i
$784$		0			0
$785$			1.98695e14i			0.0237905i
$786$		0			0
$787$			3.03107e15i			0.357878i	0.983860	+	0.178939i	$0.0572664\pi$
$787$			3.03107e15i			0.357878i	−0.983860	+	0.178939i	$0.942734\pi$
$788$		0			0
$789$	−3.97718e15	−	2.76833e15i	−0.463075	−	0.322325i
$790$		0			0
$791$	−1.24579e14	+	8.61087e14i	−0.0143045	+	0.0988727i
$792$		0			0
$793$	−1.41432e16			−1.60157
$794$		0			0
$795$	1.60495e14	+	1.11713e14i	0.0179243	+	0.0124762i
$796$		0			0
$797$	−1.76272e16			−1.94162			−0.970808	−	0.239859i	$-0.922899\pi$
$797$	−1.76272e16			−1.94162			−0.970808	+	0.239859i	$0.922899\pi$
$798$		0			0
$799$	−4.45648e15			−0.484156
$800$		0			0
$801$	−3.00379e15	−	8.11159e15i	−0.321878	−	0.869215i
$802$		0			0
$803$	−5.60472e15			−0.592405
$804$		0			0
$805$	2.01464e13	−	1.39252e14i	0.00210048	−	0.0145185i
$806$		0			0
$807$	−6.29661e15	+	9.04615e15i	−0.647593	+	0.930377i
$808$		0			0
$809$			7.57791e15i			0.768834i	0.923159	+	0.384417i	$0.125598\pi$
$809$			7.57791e15i			0.768834i	−0.923159	+	0.384417i	$0.874402\pi$
$810$		0			0
$811$		−	1.26576e16i		−	1.26689i	−0.773789	−	0.633443i	$-0.781641\pi$
$811$		−	1.26576e16i		−	1.26689i	0.773789	−	0.633443i	$-0.218359\pi$
$812$		0			0
$813$	6.04282e15	+	4.20613e15i	0.596680	+	0.415322i
$814$		0			0
$815$	2.88918e13			0.00281454
$816$		0			0
$817$		−	3.63452e15i		−	0.349322i
$818$		0			0
$819$	1.17432e16	−	2.51490e15i	1.11359	−	0.238484i
$820$		0			0
$821$			1.09052e16i			1.02034i	0.860072	+	0.510172i	$0.170418\pi$
$821$			1.09052e16i			1.02034i	−0.860072	+	0.510172i	$0.829582\pi$
$822$		0			0
$823$	−2.29130e15			−0.211535			−0.105767	−	0.994391i	$-0.533730\pi$
$823$	−2.29130e15			−0.211535			−0.105767	+	0.994391i	$0.533730\pi$
$824$		0			0
$825$	−7.95870e15	−	5.53968e15i	−0.725012	−	0.504648i
$826$		0			0
$827$		−	4.99940e15i		−	0.449404i	−0.974427	−	0.224702i	$-0.927859\pi$
$827$		−	4.99940e15i		−	0.449404i	0.974427	−	0.224702i	$-0.0721409\pi$
$828$		0			0
$829$			2.13328e15i			0.189233i	0.995514	+	0.0946167i	$0.0301625\pi$
$829$			2.13328e15i			0.189233i	−0.995514	+	0.0946167i	$0.969837\pi$
$830$		0			0
$831$	−6.68252e15	+	9.60058e15i	−0.584972	+	0.840411i
$832$		0			0
$833$	3.09406e15	+	9.14413e14i	0.267289	+	0.0789941i
$834$		0			0
$835$	7.58413e13			0.00646592
$836$		0			0
$837$	1.26627e15	−	4.88927e15i	0.106546	−	0.411390i
$838$		0			0
$839$	2.56897e15			0.213338			0.106669	−	0.994295i	$-0.465982\pi$
$839$	2.56897e15			0.213338			0.106669	+	0.994295i	$0.465982\pi$
$840$		0			0
$841$	−1.76174e16			−1.44399
$842$		0			0
$843$	−1.41661e15	−	9.86040e14i	−0.114604	−	0.0797706i
$844$		0			0
$845$	6.02997e13			0.00481507
$846$		0			0
$847$	2.75331e15	+	3.98337e14i	0.217018	+	0.0313973i
$848$		0			0
$849$	1.77824e16	+	1.23775e16i	1.38356	+	0.963032i
$850$		0			0
$851$		−	5.89714e15i		−	0.452928i
$852$		0			0
$853$			2.63407e15i			0.199714i	0.995002	+	0.0998569i	$0.0318385\pi$
$853$			2.63407e15i			0.199714i	−0.995002	+	0.0998569i	$0.968162\pi$
$854$		0			0
$855$	−5.60157e13	+	2.07431e13i	−0.00419272	+	0.00155260i
$856$		0			0
$857$	−5.18512e15			−0.383146			−0.191573	−	0.981478i	$-0.561359\pi$
$857$	−5.18512e15			−0.383146			−0.191573	+	0.981478i	$0.561359\pi$
$858$		0			0
$859$		−	9.64925e15i		−	0.703933i	−0.936013	−	0.351966i	$-0.885513\pi$
$859$		−	9.64925e15i		−	0.703933i	0.936013	−	0.351966i	$-0.114487\pi$
$860$		0			0
$861$	1.44851e16	−	1.54942e16i	1.04329	−	1.11597i
$862$		0			0
$863$		−	1.04845e16i		−	0.745569i	−0.927918	−	0.372785i	$-0.878403\pi$
$863$		−	1.04845e16i		−	0.745569i	0.927918	−	0.372785i	$-0.121597\pi$
$864$		0			0
$865$	2.54656e14			0.0178799
$866$		0			0
$867$	−7.60044e15	+	1.09193e16i	−0.526907	+	0.756992i
$868$		0			0
$869$			1.21589e15i			0.0832314i
$870$		0			0
$871$			1.42058e16i			0.960208i
$872$		0			0
$873$	2.05930e16	−	7.62576e15i	1.37449	−	0.508985i
$874$		0			0
$875$	4.86897e14	+	7.04423e13i	0.0320917	+	0.00464290i
$876$		0			0
$877$	−1.66548e16			−1.08403			−0.542014	−	0.840369i	$-0.682338\pi$
$877$	−1.66548e16			−1.08403			−0.542014	+	0.840369i	$0.682338\pi$
$878$		0			0
$879$	−2.19455e15	+	3.15284e15i	−0.141061	+	0.202658i
$880$		0			0
$881$	4.96162e15			0.314960			0.157480	−	0.987522i	$-0.449663\pi$
$881$	4.96162e15			0.314960			0.157480	+	0.987522i	$0.449663\pi$
$882$		0			0
$883$	2.00877e16			1.25935			0.629674	−	0.776860i	$-0.283189\pi$
$883$	2.00877e16			1.25935			0.629674	+	0.776860i	$0.283189\pi$
$884$		0			0
$885$	−2.54357e14	+	3.65427e14i	−0.0157491	+	0.0226262i
$886$		0			0
$887$	1.27546e16			0.779989			0.389994	−	0.920817i	$-0.372477\pi$
$887$	1.27546e16			0.779989			0.389994	+	0.920817i	$0.372477\pi$
$888$		0			0
$889$	−5.45614e15	−	7.89373e14i	−0.329554	−	0.0476785i
$890$		0			0
$891$	−9.64627e15	−	1.12386e16i	−0.575483	−	0.670479i
$892$		0			0
$893$		−	8.12803e15i		−	0.478963i
$894$		0			0
$895$			1.40651e14i			0.00818681i
$896$		0			0
$897$	−1.02371e16	+	1.47074e16i	−0.588598	+	0.845622i
$898$		0			0
$899$	−1.16971e16			−0.664356
$900$		0			0
$901$		−	6.69057e15i		−	0.375385i
$902$		0			0
$903$	−1.56097e16	+	1.66972e16i	−0.865193	+	0.925465i
$904$		0			0
$905$			4.24675e14i			0.0232536i
$906$		0			0
$907$	2.31435e16			1.25195			0.625977	−	0.779842i	$-0.284700\pi$
$907$	2.31435e16			1.25195			0.625977	+	0.779842i	$0.284700\pi$
$908$		0			0
$909$	−4.12102e15	−	1.11286e16i	−0.220244	−	0.594758i
$910$		0			0
$911$		−	2.91442e16i		−	1.53887i	−0.638727	−	0.769433i	$-0.720539\pi$
$911$		−	2.91442e16i		−	1.53887i	0.638727	−	0.769433i	$-0.279461\pi$
$912$		0			0
$913$			1.80931e16i			0.943895i
$914$		0			0
$915$	−3.63102e14	−	2.52739e14i	−0.0187160	−	0.0130273i
$916$		0			0
$917$	9.77454e15	+	1.41414e15i	0.497812	+	0.0720214i
$918$		0			0
$919$	2.19996e16			1.10708			0.553540	−	0.832822i	$-0.313276\pi$
$919$	2.19996e16			1.10708			0.553540	+	0.832822i	$0.313276\pi$
$920$		0			0
$921$	1.34820e14	+	9.38419e13i	0.00670388	+	0.00466626i
$922$		0			0
$923$	1.72798e15			0.0849043
$924$		0			0
$925$	1.03084e16			0.500509
$926$		0			0
$927$	1.95360e16	−	7.23433e15i	0.937338	−	0.347104i
$928$		0			0
$929$	2.34636e14			0.0111252			0.00556261	−	0.999985i	$-0.498229\pi$
$929$	2.34636e14			0.0111252			0.00556261	+	0.999985i	$0.498229\pi$
$930$		0			0
$931$	−1.66777e15	+	5.64316e15i	−0.0781470	+	0.264423i
$932$		0			0
$933$	5.08785e15	−	7.30957e15i	0.235605	−	0.338487i
$934$		0			0
$935$		−	8.72561e13i		−	0.00399330i
$936$		0			0
$937$			2.41493e16i			1.09229i	0.837691	+	0.546144i	$0.183905\pi$
$937$			2.41493e16i			1.09229i	−0.837691	+	0.546144i	$0.816095\pi$
$938$		0			0
$939$	−2.39920e16	−	1.66997e16i	−1.07252	−	0.746533i
$940$		0			0
$941$	1.95434e16			0.863488			0.431744	−	0.901996i	$-0.357898\pi$
$941$	1.95434e16			0.863488			0.431744	+	0.901996i	$0.357898\pi$
$942$		0			0
$943$			3.16485e16i			1.38210i
$944$		0			0
$945$	3.46427e14	+	1.45285e14i	0.0149533	+	0.00627111i
$946$		0			0
$947$			1.04855e16i			0.447367i	0.974662	+	0.223684i	$0.0718082\pi$
$947$			1.04855e16i			0.447367i	−0.974662	+	0.223684i	$0.928192\pi$
$948$		0			0
$949$	−1.81049e16			−0.763541
$950$		0			0
$951$	3.47114e16	+	2.41610e16i	1.44703	+	1.00721i
$952$		0			0
$953$			4.69378e16i			1.93425i	0.254306	+	0.967124i	$0.418153\pi$
$953$			4.69378e16i			1.93425i	−0.254306	+	0.967124i	$0.581847\pi$
$954$		0			0
$955$			5.44963e14i			0.0221998i
$956$		0			0
$957$	−1.95960e16	+	2.81530e16i	−0.789136	+	1.13373i
$958$		0			0
$959$	−2.73078e15	+	1.88751e16i	−0.108714	+	0.751429i
$960$		0			0
$961$	2.08199e16			0.819406
$962$		0			0
$963$	−3.00297e16	+	1.11203e16i	−1.16844	+	0.432683i
$964$		0			0
$965$	3.19027e14			0.0122724
$966$		0			0
$967$	3.43829e16			1.30767			0.653834	−	0.756638i	$-0.273160\pi$
$967$	3.43829e16			1.30767			0.653834	+	0.756638i	$0.273160\pi$
$968$		0			0
$969$	1.67741e15	+	1.16757e15i	0.0630752	+	0.0439037i
$970$		0			0
$971$	2.39816e16			0.891603			0.445802	−	0.895132i	$-0.352919\pi$
$971$	2.39816e16			0.891603			0.445802	+	0.895132i	$0.352919\pi$
$972$		0			0
$973$	1.85331e15	−	1.28101e16i	0.0681284	−	0.470903i
$974$		0			0
$975$	−2.57089e16	−	1.78948e16i	−0.934456	−	0.650432i
$976$		0			0
$977$		−	4.52938e15i		−	0.162786i	−0.996682	−	0.0813932i	$-0.974063\pi$
$977$		−	4.52938e15i		−	0.162786i	0.996682	−	0.0813932i	$-0.0259370\pi$
$978$		0			0
$979$			2.30455e16i			0.818993i
$980$		0			0
$981$	1.87945e15	+	5.07537e15i	0.0660468	+	0.178356i
$982$		0			0
$983$	5.58087e16			1.93935			0.969677	−	0.244389i	$-0.0785875\pi$
$983$	5.58087e16			1.93935			0.969677	+	0.244389i	$0.0785875\pi$
$984$		0			0
$985$			2.98952e12i			0.000102731i
$986$		0			0
$987$	−3.49087e16	+	3.73406e16i	−1.18629	+	1.26893i
$988$		0			0
$989$		−	3.41056e16i		−	1.14616i
$990$		0			0
$991$	3.93480e16			1.30773			0.653864	−	0.756612i	$-0.273147\pi$
$991$	3.93480e16			1.30773			0.653864	+	0.756612i	$0.273147\pi$
$992$		0			0
$993$	−6.85478e14	+	9.84806e14i	−0.0225306	+	0.0323691i
$994$		0			0
$995$			1.68984e14i			0.00549312i
$996$		0			0
$997$		−	4.97785e16i		−	1.60036i	−0.599758	−	0.800181i	$-0.704737\pi$
$997$		−	4.97785e16i		−	1.60036i	0.599758	−	0.800181i	$-0.295263\pi$
$998$		0			0
$999$	1.52419e16	+	3.94749e15i	0.484650	+	0.125519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.19	yes	28
3.2	odd	2	inner	84.12.f.b.41.9	✓	28
7.6	odd	2	inner	84.12.f.b.41.10	yes	28
21.20	even	2	inner	84.12.f.b.41.20	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.9	✓	28	3.2	odd	2	inner
84.12.f.b.41.10	yes	28	7.6	odd	2	inner
84.12.f.b.41.19	yes	28	1.1	even	1	trivial
84.12.f.b.41.20	yes	28	21.20	even	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)

\(f(q)\)	\(=\)	\(q+(240.448 - 345.444i) q^{3} -113.306 q^{5} +(44008.9 + 6367.04i) q^{7} +(-61516.6 - 166123. i) q^{9} +O(q^{10})\)	\(q+(240.448 - 345.444i) q^{3} -113.306 q^{5} +(44008.9 + 6367.04i) q^{7} +(-61516.6 - 166123. i) q^{9} +471963. i q^{11} +1.52458e6i q^{13} +(-27244.2 + 39141.0i) q^{15} +1.63168e6 q^{17} +2.97596e6i q^{19} +(1.27813e7 - 1.36717e7i) q^{21} +2.79258e7i q^{23} -4.88153e7 q^{25} +(-7.21777e7 - 1.86933e7i) q^{27} +1.72679e8i q^{29} +6.77394e7i q^{31} +(1.63037e8 + 1.13483e8i) q^{33} +(-4.98648e6 - 721424. i) q^{35} -2.11171e8 q^{37} +(5.26658e8 + 3.66582e8i) q^{39} +1.13330e9 q^{41} -1.22129e9 q^{43} +(6.97021e6 + 1.88227e7i) q^{45} -2.73123e9 q^{47} +(1.89625e9 + 5.60413e8i) q^{49} +(3.92333e8 - 5.63654e8i) q^{51} -4.10043e9i q^{53} -5.34763e7i q^{55} +(1.02803e9 + 7.15563e8i) q^{57} +9.33617e9 q^{59} +9.27679e9i q^{61} +(-1.64957e9 - 7.70257e9i) q^{63} -1.72744e8i q^{65} +9.31784e9 q^{67} +(9.64682e9 + 6.71471e9i) q^{69} -1.13341e9i q^{71} +1.18753e10i q^{73} +(-1.17375e10 + 1.68630e10i) q^{75} +(-3.00501e9 + 2.07706e10i) q^{77} +2.57625e9 q^{79} +(-2.38125e10 + 2.04386e10i) q^{81} +3.83358e10 q^{83} -1.84879e8 q^{85} +(5.96508e10 + 4.15202e10i) q^{87} +4.88289e10 q^{89} +(-9.70706e9 + 6.70952e10i) q^{91} +(2.34002e10 + 1.62878e10i) q^{93} -3.37194e8i q^{95} +1.23963e11i q^{97} +(7.84038e10 - 2.90336e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 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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