LMFDB - Embedded newform 256.11.c.k.255.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,11,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	256.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$162.651456684$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 9850 x^{8} - 22678 x^{7} + 31760900 x^{6} + 262382084 x^{5} - 36066825359 x^{4} + \cdots + 11\!\cdots\!28 $ x^10 - 4x^9 - 9850x^8 - 22678x^7 + 31760900x^6 + 262382084x^5 - 36066825359x^4 - 492294015528x^3 + 12219836291940x^2 + 252889346123568*x + 1185399433622628
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{85}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.10
Root			$32.0498 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.k.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+452.412i q^{3} +4038.75 q^{5} -26573.3i q^{7} -145628. q^{9} +O(q^{10})$	$q+452.412i q^{3} +4038.75 q^{5} -26573.3i q^{7} -145628. q^{9} +264936. i q^{11} -39524.4 q^{13} +1.82718e6i q^{15} +344589. q^{17} +2.00436e6i q^{19} +1.20221e7 q^{21} +2.18904e6i q^{23} +6.54589e6 q^{25} -3.91693e7i q^{27} +1.83209e7 q^{29} +2.39974e7i q^{31} -1.19860e8 q^{33} -1.07323e8i q^{35} +1.22621e8 q^{37} -1.78813e7i q^{39} +8.19724e7 q^{41} +1.09846e8i q^{43} -5.88154e8 q^{45} -1.39225e8i q^{47} -4.23666e8 q^{49} +1.55896e8i q^{51} -2.40522e8 q^{53} +1.07001e9i q^{55} -9.06799e8 q^{57} +1.70626e8i q^{59} +5.72042e8 q^{61} +3.86981e9i q^{63} -1.59629e8 q^{65} -5.78120e7i q^{67} -9.90348e8 q^{69} +2.82641e7i q^{71} +8.76382e8 q^{73} +2.96144e9i q^{75} +7.04022e9 q^{77} +4.19880e9i q^{79} +9.12149e9 q^{81} +1.91569e9i q^{83} +1.39171e9 q^{85} +8.28860e9i q^{87} -8.53411e9 q^{89} +1.05030e9i q^{91} -1.08567e10 q^{93} +8.09513e9i q^{95} -3.19621e9 q^{97} -3.85820e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6232 q^{5} - 218038 q^{9}+O(q^{10}) $ 10 * q + 6232 * q^5 - 218038 * q^9	$ 10 q + 6232 q^{5} - 218038 q^{9} + 485560 q^{13} - 1468772 q^{17} + 15545024 q^{21} + 4383462 q^{25} + 25263800 q^{29} - 122737264 q^{33} + 178255288 q^{37} + 91656876 q^{41} - 718346408 q^{45} - 842454934 q^{49} - 798458664 q^{53} - 1612102544 q^{57} + 102636184 q^{61} + 376325920 q^{65} + 207224128 q^{69} + 2854265572 q^{73} + 9037740608 q^{77} + 11990017466 q^{81} - 5473132400 q^{85} - 11790814556 q^{89} - 24576098304 q^{93} - 9363277860 q^{97}+O(q^{100}) $ 10 * q + 6232 * q^5 - 218038 * q^9 + 485560 * q^13 - 1468772 * q^17 + 15545024 * q^21 + 4383462 * q^25 + 25263800 * q^29 - 122737264 * q^33 + 178255288 * q^37 + 91656876 * q^41 - 718346408 * q^45 - 842454934 * q^49 - 798458664 * q^53 - 1612102544 * q^57 + 102636184 * q^61 + 376325920 * q^65 + 207224128 * q^69 + 2854265572 * q^73 + 9037740608 * q^77 + 11990017466 * q^81 - 5473132400 * q^85 - 11790814556 * q^89 - 24576098304 * q^93 - 9363277860 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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
$3$	452.412i	1.86178i	0.365302	+	0.930889i	$0.380966\pi$
$3$	452.412i	1.86178i	−0.365302	+	0.930889i	$0.619034\pi$
$4$	0	0
$5$	4038.75	1.29240	0.646200	−	0.763168i	$-0.276357\pi$
$5$	4038.75	1.29240	0.646200	+	0.763168i	$0.276357\pi$
$6$	0	0
$7$	− 26573.3i	− 1.58109i	−0.612406	−	0.790543i	$-0.709798\pi$
$7$	− 26573.3i	− 1.58109i	0.612406	−	0.790543i	$-0.290202\pi$
$8$	0	0
$9$	−145628.	−2.46622
$10$	0	0
$11$	264936.i	1.64504i	0.568735	+	0.822521i	$0.307433\pi$
$11$	264936.i	1.64504i	−0.568735	+	0.822521i	$0.692567\pi$
$12$	0	0
$13$	−39524.4	−0.106451	−0.0532254	−	0.998583i	$-0.516950\pi$
$13$	−39524.4	−0.106451	−0.0532254	+	0.998583i	$0.516950\pi$
$14$	0	0
$15$	1.82718e6i	2.40616i
$16$	0	0
$17$	344589.	0.242693	0.121346	−	0.992610i	$-0.461279\pi$
$17$	344589.	0.242693	0.121346	+	0.992610i	$0.461279\pi$
$18$	0	0
$19$	2.00436e6i	0.809485i	0.914431	+	0.404742i	$0.132639\pi$
$19$	2.00436e6i	0.809485i	−0.914431	+	0.404742i	$0.867361\pi$
$20$	0	0
$21$	1.20221e7	2.94363
$22$	0	0
$23$	2.18904e6i	0.340106i	0.985435	+	0.170053i	$0.0543939\pi$
$23$	2.18904e6i	0.340106i	−0.985435	+	0.170053i	$0.945606\pi$
$24$	0	0
$25$	6.54589e6	0.670299
$26$	0	0
$27$	− 3.91693e7i	− 2.72977i
$28$	0	0
$29$	1.83209e7	0.893216	0.446608	−	0.894730i	$-0.352632\pi$
$29$	1.83209e7	0.893216	0.446608	+	0.894730i	$0.352632\pi$
$30$	0	0
$31$	2.39974e7i	0.838215i	0.907937	+	0.419107i	$0.137657\pi$
$31$	2.39974e7i	0.838215i	−0.907937	+	0.419107i	$0.862343\pi$
$32$	0	0
$33$	−1.19860e8	−3.06270
$34$	0	0
$35$	− 1.07323e8i	− 2.04340i
$36$	0	0
$37$	1.22621e8	1.76831	0.884154	−	0.467196i	$-0.154736\pi$
$37$	1.22621e8	1.76831	0.884154	+	0.467196i	$0.154736\pi$
$38$	0	0
$39$	− 1.78813e7i	− 0.198188i
$40$	0	0
$41$	8.19724e7	0.707536	0.353768	−	0.935333i	$-0.384900\pi$
$41$	8.19724e7	0.707536	0.353768	+	0.935333i	$0.384900\pi$
$42$	0	0
$43$	1.09846e8i	0.747209i	0.927588	+	0.373604i	$0.121878\pi$
$43$	1.09846e8i	0.747209i	−0.927588	+	0.373604i	$0.878122\pi$
$44$	0	0
$45$	−5.88154e8	−3.18734
$46$	0	0
$47$	− 1.39225e8i	− 0.607055i	−0.952823	−	0.303527i	$-0.901836\pi$
$47$	− 1.39225e8i	− 0.607055i	0.952823	−	0.303527i	$-0.0981644\pi$
$48$	0	0
$49$	−4.23666e8	−1.49983
$50$	0	0
$51$	1.55896e8i	0.451840i
$52$	0	0
$53$	−2.40522e8	−0.575142	−0.287571	−	0.957759i	$-0.592848\pi$
$53$	−2.40522e8	−0.575142	−0.287571	+	0.957759i	$0.592848\pi$
$54$	0	0
$55$	1.07001e9i	2.12605i
$56$	0	0
$57$	−9.06799e8	−1.50708
$58$	0	0
$59$	1.70626e8i	0.238664i	0.992854	+	0.119332i	$0.0380752\pi$
$59$	1.70626e8i	0.238664i	−0.992854	+	0.119332i	$0.961925\pi$
$60$	0	0
$61$	5.72042e8	0.677296	0.338648	−	0.940913i	$-0.390030\pi$
$61$	5.72042e8	0.677296	0.338648	+	0.940913i	$0.390030\pi$
$62$	0	0
$63$	3.86981e9i	3.89931i
$64$	0	0
$65$	−1.59629e8	−0.137577
$66$	0	0
$67$	− 5.78120e7i	− 0.0428197i	−0.999771	−	0.0214099i	$-0.993185\pi$
$67$	− 5.78120e7i	− 0.0428197i	0.999771	−	0.0214099i	$-0.00681549\pi$
$68$	0	0
$69$	−9.90348e8	−0.633202
$70$	0	0
$71$	2.82641e7i	0.0156654i	0.999969	+	0.00783272i	$0.00249326\pi$
$71$	2.82641e7i	0.0156654i	−0.999969	+	0.00783272i	$0.997507\pi$
$72$	0	0
$73$	8.76382e8	0.422746	0.211373	−	0.977406i	$-0.432207\pi$
$73$	8.76382e8	0.422746	0.211373	+	0.977406i	$0.432207\pi$
$74$	0	0
$75$	2.96144e9i	1.24795i
$76$	0	0
$77$	7.04022e9	2.60095
$78$	0	0
$79$	4.19880e9i	1.36455i	0.731096	+	0.682275i	$0.239009\pi$
$79$	4.19880e9i	1.36455i	−0.731096	+	0.682275i	$0.760991\pi$
$80$	0	0
$81$	9.12149e9	2.61602
$82$	0	0
$83$	1.91569e9i	0.486334i	0.969984	+	0.243167i	$0.0781863\pi$
$83$	1.91569e9i	0.486334i	−0.969984	+	0.243167i	$0.921814\pi$
$84$	0	0
$85$	1.39171e9	0.313656
$86$	0	0
$87$	8.28860e9i	1.66297i
$88$	0	0
$89$	−8.53411e9	−1.52830	−0.764149	−	0.645039i	$-0.776841\pi$
$89$	−8.53411e9	−1.52830	−0.764149	+	0.645039i	$0.776841\pi$
$90$	0	0
$91$	1.05030e9i	0.168308i
$92$	0	0
$93$	−1.08567e10	−1.56057
$94$	0	0
$95$	8.09513e9i	1.04618i
$96$	0	0
$97$	−3.19621e9	−0.372200	−0.186100	−	0.982531i	$-0.559585\pi$
$97$	−3.19621e9	−0.372200	−0.186100	+	0.982531i	$0.559585\pi$
$98$	0	0
$99$	− 3.85820e10i	− 4.05703i
$100$	0	0
$101$	−1.03559e10	−0.985330	−0.492665	−	0.870219i	$-0.663977\pi$
$101$	−1.03559e10	−0.985330	−0.492665	+	0.870219i	$0.663977\pi$
$102$	0	0
$103$	− 8.10306e9i	− 0.698977i	−0.936941	−	0.349489i	$-0.886355\pi$
$103$	− 8.10306e9i	− 0.698977i	0.936941	−	0.349489i	$-0.113645\pi$
$104$	0	0
$105$	4.85542e10	3.80435
$106$	0	0
$107$	5.63313e9i	0.401635i	0.979629	+	0.200817i	$0.0643598\pi$
$107$	5.63313e9i	0.401635i	−0.979629	+	0.200817i	$0.935640\pi$
$108$	0	0
$109$	−2.35404e10	−1.52996	−0.764982	−	0.644052i	$-0.777252\pi$
$109$	−2.35404e10	−1.52996	−0.764982	+	0.644052i	$0.777252\pi$
$110$	0	0
$111$	5.54754e10i	3.29220i
$112$	0	0
$113$	−1.72322e10	−0.935294	−0.467647	−	0.883915i	$-0.654898\pi$
$113$	−1.72322e10	−0.935294	−0.467647	+	0.883915i	$0.654898\pi$
$114$	0	0
$115$	8.84098e9i	0.439553i
$116$	0	0
$117$	5.75585e9	0.262531
$118$	0	0
$119$	− 9.15688e9i	− 0.383718i
$120$	0	0
$121$	−4.42534e10	−1.70616
$122$	0	0
$123$	3.70853e10i	1.31728i
$124$	0	0
$125$	−1.30037e10	−0.426106
$126$	0	0
$127$	− 4.93441e10i	− 1.49354i	−0.665083	−	0.746770i	$-0.731604\pi$
$127$	− 4.93441e10i	− 1.49354i	0.665083	−	0.746770i	$-0.268396\pi$
$128$	0	0
$129$	−4.96957e10	−1.39114
$130$	0	0
$131$	5.62471e10i	1.45795i	0.684539	+	0.728977i	$0.260004\pi$
$131$	5.62471e10i	1.45795i	−0.684539	+	0.728977i	$0.739996\pi$
$132$	0	0
$133$	5.32626e10	1.27987
$134$	0	0
$135$	− 1.58195e11i	− 3.52796i
$136$	0	0
$137$	−2.82609e10	−0.585576	−0.292788	−	0.956177i	$-0.594583\pi$
$137$	−2.82609e10	−0.585576	−0.292788	+	0.956177i	$0.594583\pi$
$138$	0	0
$139$	6.40824e10i	1.23499i	0.786573	+	0.617497i	$0.211853\pi$
$139$	6.40824e10i	1.23499i	−0.786573	+	0.617497i	$0.788147\pi$
$140$	0	0
$141$	6.29871e10	1.13020
$142$	0	0
$143$	− 1.04714e10i	− 0.175116i
$144$	0	0
$145$	7.39935e10	1.15439
$146$	0	0
$147$	− 1.91672e11i	− 2.79236i
$148$	0	0
$149$	−7.78696e10	−1.06032	−0.530159	−	0.847898i	$-0.677868\pi$
$149$	−7.78696e10	−1.06032	−0.530159	+	0.847898i	$0.677868\pi$
$150$	0	0
$151$	− 4.79460e10i	− 0.610756i	−0.952231	−	0.305378i	$-0.901217\pi$
$151$	− 4.79460e10i	− 0.610756i	0.952231	−	0.305378i	$-0.0987829\pi$
$152$	0	0
$153$	−5.01818e10	−0.598534
$154$	0	0
$155$	9.69194e10i	1.08331i
$156$	0	0
$157$	−7.52659e10	−0.789042	−0.394521	−	0.918887i	$-0.629089\pi$
$157$	−7.52659e10	−0.789042	−0.394521	+	0.918887i	$0.629089\pi$
$158$	0	0
$159$	− 1.08815e11i	− 1.07079i
$160$	0	0
$161$	5.81700e10	0.537737
$162$	0	0
$163$	− 5.87458e10i	− 0.510551i	−0.966868	−	0.255275i	$-0.917834\pi$
$163$	− 5.87458e10i	− 0.510551i	0.966868	−	0.255275i	$-0.0821661\pi$
$164$	0	0
$165$	−4.84085e11	−3.95824
$166$	0	0
$167$	− 1.31388e11i	− 1.01151i	−0.862676	−	0.505757i	$-0.831213\pi$
$167$	− 1.31388e11i	− 1.01151i	0.862676	−	0.505757i	$-0.168787\pi$
$168$	0	0
$169$	−1.36296e11	−0.988668
$170$	0	0
$171$	− 2.91891e11i	− 1.99637i
$172$	0	0
$173$	6.53110e10	0.421460	0.210730	−	0.977544i	$-0.432416\pi$
$173$	6.53110e10	0.421460	0.210730	+	0.977544i	$0.432416\pi$
$174$	0	0
$175$	− 1.73946e11i	− 1.05980i
$176$	0	0
$177$	−7.71935e10	−0.444339
$178$	0	0
$179$	2.59420e11i	1.41169i	0.708367	+	0.705844i	$0.249432\pi$
$179$	2.59420e11i	1.41169i	−0.708367	+	0.705844i	$0.750568\pi$
$180$	0	0
$181$	7.74303e10	0.398582	0.199291	−	0.979940i	$-0.436136\pi$
$181$	7.74303e10	0.398582	0.199291	+	0.979940i	$0.436136\pi$
$182$	0	0
$183$	2.58799e11i	1.26098i
$184$	0	0
$185$	4.95238e11	2.28536
$186$	0	0
$187$	9.12940e10i	0.399240i
$188$	0	0
$189$	−1.04086e12	−4.31601
$190$	0	0
$191$	4.84284e11i	1.90517i	0.304274	+	0.952585i	$0.401586\pi$
$191$	4.84284e11i	1.90517i	−0.304274	+	0.952585i	$0.598414\pi$
$192$	0	0
$193$	4.44959e11	1.66163	0.830813	−	0.556552i	$-0.187876\pi$
$193$	4.44959e11	1.66163	0.830813	+	0.556552i	$0.187876\pi$
$194$	0	0
$195$	− 7.22183e10i	− 0.256138i
$196$	0	0
$197$	2.52721e11	0.851746	0.425873	−	0.904783i	$-0.359967\pi$
$197$	2.52721e11	0.851746	0.425873	+	0.904783i	$0.359967\pi$
$198$	0	0
$199$	− 1.61582e11i	− 0.517758i	−0.965910	−	0.258879i	$-0.916647\pi$
$199$	− 1.61582e11i	− 0.517758i	0.965910	−	0.258879i	$-0.0833531\pi$
$200$	0	0
$201$	2.61548e10	0.0797208
$202$	0	0
$203$	− 4.86847e11i	− 1.41225i
$204$	0	0
$205$	3.31066e11	0.914420
$206$	0	0
$207$	− 3.18785e11i	− 0.838776i
$208$	0	0
$209$	−5.31028e11	−1.33164
$210$	0	0
$211$	1.95577e11i	0.467633i	0.972281	+	0.233817i	$0.0751215\pi$
$211$	1.95577e11i	0.467633i	−0.972281	+	0.233817i	$0.924878\pi$
$212$	0	0
$213$	−1.27870e10	−0.0291656
$214$	0	0
$215$	4.43641e11i	0.965693i
$216$	0	0
$217$	6.37690e11	1.32529
$218$	0	0
$219$	3.96486e11i	0.787059i
$220$	0	0
$221$	−1.36197e10	−0.0258349
$222$	0	0
$223$	− 6.90001e11i	− 1.25120i	−0.780145	−	0.625598i	$-0.784855\pi$
$223$	− 6.90001e11i	− 1.25120i	0.780145	−	0.625598i	$-0.215145\pi$
$224$	0	0
$225$	−9.53263e11	−1.65310
$226$	0	0
$227$	8.41718e11i	1.39649i	0.715859	+	0.698244i	$0.246035\pi$
$227$	8.41718e11i	1.39649i	−0.715859	+	0.698244i	$0.753965\pi$
$228$	0	0
$229$	−9.76839e11	−1.55112	−0.775560	−	0.631274i	$-0.782532\pi$
$229$	−9.76839e11	−1.55112	−0.775560	+	0.631274i	$0.782532\pi$
$230$	0	0
$231$	3.18508e12i	4.84240i
$232$	0	0
$233$	−1.12910e12	−1.64420	−0.822099	−	0.569345i	$-0.807197\pi$
$233$	−1.12910e12	−1.64420	−0.822099	+	0.569345i	$0.807197\pi$
$234$	0	0
$235$	− 5.62295e11i	− 0.784558i
$236$	0	0
$237$	−1.89959e12	−2.54049
$238$	0	0
$239$	1.44608e12i	1.85440i	0.374572	+	0.927198i	$0.377789\pi$
$239$	1.44608e12i	1.85440i	−0.374572	+	0.927198i	$0.622211\pi$
$240$	0	0
$241$	5.37252e11	0.660835	0.330417	−	0.943835i	$-0.392811\pi$
$241$	5.37252e11	0.660835	0.330417	+	0.943835i	$0.392811\pi$
$242$	0	0
$243$	1.81376e12i	2.14067i
$244$	0	0
$245$	−1.71108e12	−1.93839
$246$	0	0
$247$	− 7.92214e10i	− 0.0861703i
$248$	0	0
$249$	−8.66681e11	−0.905446
$250$	0	0
$251$	9.49363e11i	0.952936i	0.879192	+	0.476468i	$0.158083\pi$
$251$	9.49363e11i	0.952936i	−0.879192	+	0.476468i	$0.841917\pi$
$252$	0	0
$253$	−5.79954e11	−0.559488
$254$	0	0
$255$	6.29627e11i	0.583959i
$256$	0	0
$257$	1.78544e12	1.59250	0.796252	−	0.604965i	$-0.206813\pi$
$257$	1.78544e12	1.59250	0.796252	+	0.604965i	$0.206813\pi$
$258$	0	0
$259$	− 3.25846e12i	− 2.79585i
$260$	0	0
$261$	−2.66803e12	−2.20287
$262$	0	0
$263$	− 3.13272e11i	− 0.248967i	−0.992222	−	0.124484i	$-0.960273\pi$
$263$	− 3.13272e11i	− 0.248967i	0.992222	−	0.124484i	$-0.0397274\pi$
$264$	0	0
$265$	−9.71408e11	−0.743314
$266$	0	0
$267$	− 3.86094e12i	− 2.84535i
$268$	0	0
$269$	7.62632e11	0.541444	0.270722	−	0.962658i	$-0.412738\pi$
$269$	7.62632e11	0.541444	0.270722	+	0.962658i	$0.412738\pi$
$270$	0	0
$271$	− 9.95106e11i	− 0.680805i	−0.940280	−	0.340403i	$-0.889437\pi$
$271$	− 9.95106e11i	− 0.680805i	0.940280	−	0.340403i	$-0.110563\pi$
$272$	0	0
$273$	−4.75166e11	−0.313352
$274$	0	0
$275$	1.73424e12i	1.10267i
$276$	0	0
$277$	−1.37163e12	−0.841079	−0.420540	−	0.907274i	$-0.638159\pi$
$277$	−1.37163e12	−0.841079	−0.420540	+	0.907274i	$0.638159\pi$
$278$	0	0
$279$	− 3.49468e12i	− 2.06722i
$280$	0	0
$281$	−3.29428e12	−1.88031	−0.940153	−	0.340752i	$-0.889318\pi$
$281$	−3.29428e12	−1.88031	−0.940153	+	0.340752i	$0.889318\pi$
$282$	0	0
$283$	− 8.03580e11i	− 0.442687i	−0.975196	−	0.221344i	$-0.928956\pi$
$283$	− 8.03580e11i	− 0.442687i	0.975196	−	0.221344i	$-0.0710443\pi$
$284$	0	0
$285$	−3.66234e12	−1.94775
$286$	0	0
$287$	− 2.17828e12i	− 1.11868i
$288$	0	0
$289$	−1.89725e12	−0.941100
$290$	0	0
$291$	− 1.44600e12i	− 0.692954i
$292$	0	0
$293$	1.69300e12	0.784006	0.392003	−	0.919964i	$-0.371782\pi$
$293$	1.69300e12	0.784006	0.392003	+	0.919964i	$0.371782\pi$
$294$	0	0
$295$	6.89118e11i	0.308449i
$296$	0	0
$297$	1.03773e13	4.49059
$298$	0	0
$299$	− 8.65205e10i	− 0.0362045i
$300$	0	0
$301$	2.91897e12	1.18140
$302$	0	0
$303$	− 4.68514e12i	− 1.83447i
$304$	0	0
$305$	2.31033e12	0.875338
$306$	0	0
$307$	3.19842e11i	0.117285i	0.998279	+	0.0586427i	$0.0186773\pi$
$307$	3.19842e11i	0.117285i	−0.998279	+	0.0586427i	$0.981323\pi$
$308$	0	0
$309$	3.66592e12	1.30134
$310$	0	0
$311$	− 2.38931e12i	− 0.821240i	−0.911807	−	0.410620i	$-0.865312\pi$
$311$	− 2.38931e12i	− 0.821240i	0.911807	−	0.410620i	$-0.134688\pi$
$312$	0	0
$313$	2.24282e12	0.746574	0.373287	−	0.927716i	$-0.378231\pi$
$313$	2.24282e12	0.746574	0.373287	+	0.927716i	$0.378231\pi$
$314$	0	0
$315$	1.56292e13i	5.03946i
$316$	0	0
$317$	6.40855e11	0.200200	0.100100	−	0.994977i	$-0.468084\pi$
$317$	6.40855e11	0.200200	0.100100	+	0.994977i	$0.468084\pi$
$318$	0	0
$319$	4.85386e12i	1.46938i
$320$	0	0
$321$	−2.54850e12	−0.747755
$322$	0	0
$323$	6.90682e11i	0.196456i
$324$	0	0
$325$	−2.58722e11	−0.0713538
$326$	0	0
$327$	− 1.06500e13i	− 2.84845i
$328$	0	0
$329$	−3.69967e12	−0.959806
$330$	0	0
$331$	− 5.41331e12i	− 1.36246i	−0.732070	−	0.681229i	$-0.761446\pi$
$331$	− 5.41331e12i	− 1.36246i	0.732070	−	0.681229i	$-0.238554\pi$
$332$	0	0
$333$	−1.78571e13	−4.36103
$334$	0	0
$335$	− 2.33488e11i	− 0.0553402i
$336$	0	0
$337$	3.90878e12	0.899273	0.449637	−	0.893212i	$-0.351553\pi$
$337$	3.90878e12	0.899273	0.449637	+	0.893212i	$0.351553\pi$
$338$	0	0
$339$	− 7.79605e12i	− 1.74131i
$340$	0	0
$341$	−6.35776e12	−1.37890
$342$	0	0
$343$	3.75191e12i	0.790281i
$344$	0	0
$345$	−3.99977e12	−0.818350
$346$	0	0
$347$	7.00065e12i	1.39152i	0.718272	+	0.695762i	$0.244933\pi$
$347$	7.00065e12i	1.39152i	−0.718272	+	0.695762i	$0.755067\pi$
$348$	0	0
$349$	7.03383e12	1.35852	0.679258	−	0.733900i	$-0.262302\pi$
$349$	7.03383e12	1.35852	0.679258	+	0.733900i	$0.262302\pi$
$350$	0	0
$351$	1.54814e12i	0.290587i
$352$	0	0
$353$	−3.00675e12	−0.548559	−0.274280	−	0.961650i	$-0.588439\pi$
$353$	−3.00675e12	−0.548559	−0.274280	+	0.961650i	$0.588439\pi$
$354$	0	0
$355$	1.14151e11i	0.0202460i
$356$	0	0
$357$	4.14268e12	0.714399
$358$	0	0
$359$	− 4.14903e12i	− 0.695784i	−0.937535	−	0.347892i	$-0.886898\pi$
$359$	− 4.14903e12i	− 0.695784i	0.937535	−	0.347892i	$-0.113102\pi$
$360$	0	0
$361$	2.11359e12	0.344734
$362$	0	0
$363$	− 2.00208e13i	− 3.17650i
$364$	0	0
$365$	3.53949e12	0.546357
$366$	0	0
$367$	7.53529e12i	1.13180i	0.824474	+	0.565900i	$0.191471\pi$
$367$	7.53529e12i	1.13180i	−0.824474	+	0.565900i	$0.808529\pi$
$368$	0	0
$369$	−1.19375e13	−1.74494
$370$	0	0
$371$	6.39147e12i	0.909350i
$372$	0	0
$373$	−1.38429e13	−1.91727	−0.958634	−	0.284642i	$-0.908125\pi$
$373$	−1.38429e13	−1.91727	−0.958634	+	0.284642i	$0.908125\pi$
$374$	0	0
$375$	− 5.88304e12i	− 0.793315i
$376$	0	0
$377$	−7.24123e11	−0.0950836
$378$	0	0
$379$	1.13137e13i	1.44680i	0.690428	+	0.723401i	$0.257422\pi$
$379$	1.13137e13i	1.44680i	−0.690428	+	0.723401i	$0.742578\pi$
$380$	0	0
$381$	2.23239e13	2.78064
$382$	0	0
$383$	− 5.03996e12i	− 0.611551i	−0.952104	−	0.305776i	$-0.901084\pi$
$383$	− 5.03996e12i	− 0.611551i	0.952104	−	0.305776i	$-0.0989158\pi$
$384$	0	0
$385$	2.84337e13	3.36147
$386$	0	0
$387$	− 1.59966e13i	− 1.84278i
$388$	0	0
$389$	−3.69609e11	−0.0414949	−0.0207475	−	0.999785i	$-0.506605\pi$
$389$	−3.69609e11	−0.0414949	−0.0207475	+	0.999785i	$0.506605\pi$
$390$	0	0
$391$	7.54319e11i	0.0825413i
$392$	0	0
$393$	−2.54469e13	−2.71439
$394$	0	0
$395$	1.69579e13i	1.76355i
$396$	0	0
$397$	9.38198e12	0.951354	0.475677	−	0.879620i	$-0.342203\pi$
$397$	9.38198e12	0.951354	0.475677	+	0.879620i	$0.342203\pi$
$398$	0	0
$399$	2.40967e13i	2.38283i
$400$	0	0
$401$	−1.01132e13	−0.975368	−0.487684	−	0.873020i	$-0.662158\pi$
$401$	−1.01132e13	−0.975368	−0.487684	+	0.873020i	$0.662158\pi$
$402$	0	0
$403$	− 9.48483e11i	− 0.0892286i
$404$	0	0
$405$	3.68394e13	3.38094
$406$	0	0
$407$	3.24868e13i	2.90894i
$408$	0	0
$409$	3.78872e11	0.0331037	0.0165518	−	0.999863i	$-0.494731\pi$
$409$	3.78872e11	0.0331037	0.0165518	+	0.999863i	$0.494731\pi$
$410$	0	0
$411$	− 1.27856e13i	− 1.09021i
$412$	0	0
$413$	4.53411e12	0.377348
$414$	0	0
$415$	7.73699e12i	0.628538i
$416$	0	0
$417$	−2.89917e13	−2.29928
$418$	0	0
$419$	− 9.86639e12i	− 0.763991i	−0.924164	−	0.381995i	$-0.875237\pi$
$419$	− 9.86639e12i	− 0.763991i	0.924164	−	0.381995i	$-0.124763\pi$
$420$	0	0
$421$	1.57732e13	1.19264	0.596319	−	0.802747i	$-0.296629\pi$
$421$	1.57732e13	1.19264	0.596319	+	0.802747i	$0.296629\pi$
$422$	0	0
$423$	2.02750e13i	1.49713i
$424$	0	0
$425$	2.25564e12	0.162677
$426$	0	0
$427$	− 1.52011e13i	− 1.07086i
$428$	0	0
$429$	4.73740e12	0.326027
$430$	0	0
$431$	− 5.70443e12i	− 0.383553i	−0.981439	−	0.191777i	$-0.938575\pi$
$431$	− 5.70443e12i	− 0.383553i	0.981439	−	0.191777i	$-0.0614249\pi$
$432$	0	0
$433$	2.47251e13	1.62442	0.812211	−	0.583364i	$-0.198264\pi$
$433$	2.47251e13	1.62442	0.812211	+	0.583364i	$0.198264\pi$
$434$	0	0
$435$	3.34756e13i	2.14922i
$436$	0	0
$437$	−4.38763e12	−0.275311
$438$	0	0
$439$	1.54091e13i	0.945051i	0.881317	+	0.472526i	$0.156658\pi$
$439$	1.54091e13i	0.945051i	−0.881317	+	0.472526i	$0.843342\pi$
$440$	0	0
$441$	6.16975e13	3.69892
$442$	0	0
$443$	3.34143e12i	0.195845i	0.995194	+	0.0979226i	$0.0312198\pi$
$443$	3.34143e12i	0.195845i	−0.995194	+	0.0979226i	$0.968780\pi$
$444$	0	0
$445$	−3.44671e13	−1.97517
$446$	0	0
$447$	− 3.52291e13i	− 1.97408i
$448$	0	0
$449$	2.41284e13	1.32220	0.661100	−	0.750298i	$-0.270090\pi$
$449$	2.41284e13	1.32220	0.661100	+	0.750298i	$0.270090\pi$
$450$	0	0
$451$	2.17174e13i	1.16393i
$452$	0	0
$453$	2.16914e13	1.13709
$454$	0	0
$455$	4.24188e12i	0.217521i
$456$	0	0
$457$	9.73250e12	0.488251	0.244126	−	0.969744i	$-0.421499\pi$
$457$	9.73250e12	0.488251	0.244126	+	0.969744i	$0.421499\pi$
$458$	0	0
$459$	− 1.34973e13i	− 0.662497i
$460$	0	0
$461$	1.53193e13	0.735755	0.367878	−	0.929874i	$-0.380085\pi$
$461$	1.53193e13	0.735755	0.367878	+	0.929874i	$0.380085\pi$
$462$	0	0
$463$	2.52007e13i	1.18442i	0.805783	+	0.592211i	$0.201745\pi$
$463$	2.52007e13i	1.18442i	−0.805783	+	0.592211i	$0.798255\pi$
$464$	0	0
$465$	−4.38475e13	−2.01688
$466$	0	0
$467$	1.77640e13i	0.799753i	0.916569	+	0.399876i	$0.130947\pi$
$467$	1.77640e13i	0.799753i	−0.916569	+	0.399876i	$0.869053\pi$
$468$	0	0
$469$	−1.53626e12	−0.0677017
$470$	0	0
$471$	− 3.40512e13i	− 1.46902i
$472$	0	0
$473$	−2.91021e13	−1.22919
$474$	0	0
$475$	1.31203e13i	0.542597i
$476$	0	0
$477$	3.50267e13	1.41843
$478$	0	0
$479$	2.44658e13i	0.970248i	0.874445	+	0.485124i	$0.161225\pi$
$479$	2.44658e13i	0.970248i	−0.874445	+	0.485124i	$0.838775\pi$
$480$	0	0
$481$	−4.84654e12	−0.188238
$482$	0	0
$483$	2.63168e13i	1.00115i
$484$	0	0
$485$	−1.29087e13	−0.481031
$486$	0	0
$487$	− 2.82407e13i	− 1.03094i	−0.856909	−	0.515468i	$-0.827618\pi$
$487$	− 2.82407e13i	− 1.03094i	0.856909	−	0.515468i	$-0.172382\pi$
$488$	0	0
$489$	2.65773e13	0.950532
$490$	0	0
$491$	− 3.59433e12i	− 0.125954i	−0.998015	−	0.0629768i	$-0.979941\pi$
$491$	− 3.59433e12i	− 0.125954i	0.998015	−	0.0629768i	$-0.0200594\pi$
$492$	0	0
$493$	6.31318e12	0.216777
$494$	0	0
$495$	− 1.55823e14i	− 5.24331i
$496$	0	0
$497$	7.51070e11	0.0247684
$498$	0	0
$499$	− 2.49464e13i	− 0.806315i	−0.915131	−	0.403158i	$-0.867913\pi$
$499$	− 2.49464e13i	− 0.806315i	0.915131	−	0.403158i	$-0.132087\pi$
$500$	0	0
$501$	5.94414e13	1.88322
$502$	0	0
$503$	− 7.87502e12i	− 0.244575i	−0.992495	−	0.122287i	$-0.960977\pi$
$503$	− 7.87502e12i	− 0.244575i	0.992495	−	0.122287i	$-0.0390230\pi$
$504$	0	0
$505$	−4.18250e13	−1.27344
$506$	0	0
$507$	− 6.16621e13i	− 1.84068i
$508$	0	0
$509$	2.35844e13	0.690296	0.345148	−	0.938548i	$-0.387829\pi$
$509$	2.35844e13	0.690296	0.345148	+	0.938548i	$0.387829\pi$
$510$	0	0
$511$	− 2.32884e13i	− 0.668397i
$512$	0	0
$513$	7.85095e13	2.20971
$514$	0	0
$515$	− 3.27262e13i	− 0.903358i
$516$	0	0
$517$	3.68856e13	0.998630
$518$	0	0
$519$	2.95475e13i	0.784664i
$520$	0	0
$521$	−3.87811e13	−1.01026	−0.505128	−	0.863044i	$-0.668555\pi$
$521$	−3.87811e13	−1.01026	−0.505128	+	0.863044i	$0.668555\pi$
$522$	0	0
$523$	− 8.37546e12i	− 0.214043i	−0.994257	−	0.107021i	$-0.965869\pi$
$523$	− 8.37546e12i	− 0.214043i	0.994257	−	0.107021i	$-0.0341313\pi$
$524$	0	0
$525$	7.86952e13	1.97311
$526$	0	0
$527$	8.26924e12i	0.203429i
$528$	0	0
$529$	3.66346e13	0.884328
$530$	0	0
$531$	− 2.48479e13i	− 0.588597i
$532$	0	0
$533$	−3.23991e12	−0.0753178
$534$	0	0
$535$	2.27508e13i	0.519073i
$536$	0	0
$537$	−1.17365e14	−2.62825
$538$	0	0
$539$	− 1.12244e14i	− 2.46729i
$540$	0	0
$541$	6.65654e13	1.43636	0.718178	−	0.695860i	$-0.244976\pi$
$541$	6.65654e13	1.43636	0.718178	+	0.695860i	$0.244976\pi$
$542$	0	0
$543$	3.50304e13i	0.742072i
$544$	0	0
$545$	−9.50738e13	−1.97733
$546$	0	0
$547$	1.96446e12i	0.0401150i	0.999799	+	0.0200575i	$0.00638493\pi$
$547$	1.96446e12i	0.0401150i	−0.999799	+	0.0200575i	$0.993615\pi$
$548$	0	0
$549$	−8.33052e13	−1.67036
$550$	0	0
$551$	3.67218e13i	0.723045i
$552$	0	0
$553$	1.11576e14	2.15747
$554$	0	0
$555$	2.24051e14i	4.25484i
$556$	0	0
$557$	1.89212e13	0.352917	0.176459	−	0.984308i	$-0.443536\pi$
$557$	1.89212e13	0.352917	0.176459	+	0.984308i	$0.443536\pi$
$558$	0	0
$559$	− 4.34160e12i	− 0.0795409i
$560$	0	0
$561$	−4.13025e13	−0.743296
$562$	0	0
$563$	− 7.54771e13i	− 1.33436i	−0.744896	−	0.667181i	$-0.767501\pi$
$563$	− 7.54771e13i	− 1.33436i	0.744896	−	0.667181i	$-0.232499\pi$
$564$	0	0
$565$	−6.95965e13	−1.20877
$566$	0	0
$567$	− 2.42388e14i	− 4.13615i
$568$	0	0
$569$	−2.28540e13	−0.383178	−0.191589	−	0.981475i	$-0.561364\pi$
$569$	−2.28540e13	−0.383178	−0.191589	+	0.981475i	$0.561364\pi$
$570$	0	0
$571$	6.68352e13i	1.10110i	0.834804	+	0.550548i	$0.185581\pi$
$571$	6.68352e13i	1.10110i	−0.834804	+	0.550548i	$0.814419\pi$
$572$	0	0
$573$	−2.19096e14	−3.54700
$574$	0	0
$575$	1.43292e13i	0.227973i
$576$	0	0
$577$	−7.20040e13	−1.12584	−0.562921	−	0.826511i	$-0.690323\pi$
$577$	−7.20040e13	−1.12584	−0.562921	+	0.826511i	$0.690323\pi$
$578$	0	0
$579$	2.01305e14i	3.09358i
$580$	0	0
$581$	5.09062e13	0.768936
$582$	0	0
$583$	− 6.37228e13i	− 0.946133i
$584$	0	0
$585$	2.32465e13	0.339295
$586$	0	0
$587$	− 4.61567e13i	− 0.662284i	−0.943581	−	0.331142i	$-0.892566\pi$
$587$	− 4.61567e13i	− 0.662284i	0.943581	−	0.331142i	$-0.107434\pi$
$588$	0	0
$589$	−4.80995e13	−0.678522
$590$	0	0
$591$	1.14334e14i	1.58576i
$592$	0	0
$593$	−4.20788e12	−0.0573839	−0.0286920	−	0.999588i	$-0.509134\pi$
$593$	−4.20788e12	−0.0573839	−0.0286920	+	0.999588i	$0.509134\pi$
$594$	0	0
$595$	− 3.69824e13i	− 0.495918i
$596$	0	0
$597$	7.31016e13	0.963951
$598$	0	0
$599$	9.97183e13i	1.29313i	0.762861	+	0.646563i	$0.223794\pi$
$599$	9.97183e13i	1.29313i	−0.762861	+	0.646563i	$0.776206\pi$
$600$	0	0
$601$	1.17039e14	1.49265	0.746323	−	0.665584i	$-0.231818\pi$
$601$	1.17039e14	1.49265	0.746323	+	0.665584i	$0.231818\pi$
$602$	0	0
$603$	8.41903e12i	0.105603i
$604$	0	0
$605$	−1.78729e14	−2.20504
$606$	0	0
$607$	5.53107e12i	0.0671221i	0.999437	+	0.0335610i	$0.0106848\pi$
$607$	5.53107e12i	0.0671221i	−0.999437	+	0.0335610i	$0.989315\pi$
$608$	0	0
$609$	2.20255e14	2.62930
$610$	0	0
$611$	5.50279e12i	0.0646214i
$612$	0	0
$613$	−5.31229e13	−0.613733	−0.306867	−	0.951753i	$-0.599281\pi$
$613$	−5.31229e13	−0.613733	−0.306867	+	0.951753i	$0.599281\pi$
$614$	0	0
$615$	1.49778e14i	1.70245i
$616$	0	0
$617$	1.15304e14	1.28950	0.644748	−	0.764395i	$-0.276962\pi$
$617$	1.15304e14	1.28950	0.644748	+	0.764395i	$0.276962\pi$
$618$	0	0
$619$	9.82062e13i	1.08065i	0.841456	+	0.540326i	$0.181699\pi$
$619$	9.82062e13i	1.08065i	−0.841456	+	0.540326i	$0.818301\pi$
$620$	0	0
$621$	8.57431e13	0.928413
$622$	0	0
$623$	2.26780e14i	2.41637i
$624$	0	0
$625$	−1.16443e14	−1.22100
$626$	0	0
$627$	− 2.40243e14i	− 2.47921i
$628$	0	0
$629$	4.22540e13	0.429156
$630$	0	0
$631$	1.33565e14i	1.33520i	0.744522	+	0.667598i	$0.232677\pi$
$631$	1.33565e14i	1.33520i	−0.744522	+	0.667598i	$0.767323\pi$
$632$	0	0
$633$	−8.84814e13	−0.870629
$634$	0	0
$635$	− 1.99289e14i	− 1.93025i
$636$	0	0
$637$	1.67452e13	0.159659
$638$	0	0
$639$	− 4.11603e12i	− 0.0386344i
$640$	0	0
$641$	7.08012e13	0.654260	0.327130	−	0.944979i	$-0.393919\pi$
$641$	7.08012e13	0.654260	0.327130	+	0.944979i	$0.393919\pi$
$642$	0	0
$643$	− 1.39144e14i	− 1.26593i	−0.774182	−	0.632964i	$-0.781838\pi$
$643$	− 1.39144e14i	− 1.26593i	0.774182	−	0.632964i	$-0.218162\pi$
$644$	0	0
$645$	−2.00708e14	−1.79791
$646$	0	0
$647$	− 1.06845e14i	− 0.942392i	−0.882029	−	0.471196i	$-0.843823\pi$
$647$	− 1.06845e14i	− 0.942392i	0.882029	−	0.471196i	$-0.156177\pi$
$648$	0	0
$649$	−4.52050e13	−0.392612
$650$	0	0
$651$	2.88499e14i	2.46740i
$652$	0	0
$653$	−1.34851e14	−1.13577	−0.567884	−	0.823109i	$-0.692238\pi$
$653$	−1.34851e14	−1.13577	−0.567884	+	0.823109i	$0.692238\pi$
$654$	0	0
$655$	2.27168e14i	1.88426i
$656$	0	0
$657$	−1.27626e14	−1.04258
$658$	0	0
$659$	5.01002e13i	0.403100i	0.979478	+	0.201550i	$0.0645978\pi$
$659$	5.01002e13i	0.403100i	−0.979478	+	0.201550i	$0.935402\pi$
$660$	0	0
$661$	1.42059e14	1.12580	0.562901	−	0.826524i	$-0.309685\pi$
$661$	1.42059e14	1.12580	0.562901	+	0.826524i	$0.309685\pi$
$662$	0	0
$663$	− 6.16172e12i	− 0.0480988i
$664$	0	0
$665$	2.15114e14	1.65410
$666$	0	0
$667$	4.01051e13i	0.303788i
$668$	0	0
$669$	3.12165e14	2.32945
$670$	0	0
$671$	1.51554e14i	1.11418i
$672$	0	0
$673$	1.52130e14	1.10189	0.550946	−	0.834541i	$-0.314267\pi$
$673$	1.52130e14	1.10189	0.550946	+	0.834541i	$0.314267\pi$
$674$	0	0
$675$	− 2.56398e14i	− 1.82976i
$676$	0	0
$677$	1.31644e14	0.925674	0.462837	−	0.886443i	$-0.346832\pi$
$677$	1.31644e14	0.925674	0.462837	+	0.886443i	$0.346832\pi$
$678$	0	0
$679$	8.49338e13i	0.588480i
$680$	0	0
$681$	−3.80804e14	−2.59995
$682$	0	0
$683$	8.40230e13i	0.565321i	0.959220	+	0.282660i	$0.0912169\pi$
$683$	8.40230e13i	0.565321i	−0.959220	+	0.282660i	$0.908783\pi$
$684$	0	0
$685$	−1.14139e14	−0.756799
$686$	0	0
$687$	− 4.41934e14i	− 2.88784i
$688$	0	0
$689$	9.50649e12	0.0612244
$690$	0	0
$691$	2.60646e14i	1.65448i	0.561850	+	0.827239i	$0.310090\pi$
$691$	2.60646e14i	1.65448i	−0.561850	+	0.827239i	$0.689910\pi$
$692$	0	0
$693$	−1.02525e15	−6.41452
$694$	0	0
$695$	2.58813e14i	1.59611i
$696$	0	0
$697$	2.82468e13	0.171714
$698$	0	0
$699$	− 5.10820e14i	− 3.06113i
$700$	0	0
$701$	5.45247e13	0.322109	0.161055	−	0.986945i	$-0.448510\pi$
$701$	5.45247e13	0.322109	0.161055	+	0.986945i	$0.448510\pi$
$702$	0	0
$703$	2.45778e14i	1.43142i
$704$	0	0
$705$	2.54389e14	1.46067
$706$	0	0
$707$	2.75191e14i	1.55789i
$708$	0	0
$709$	−9.55149e13	−0.533138	−0.266569	−	0.963816i	$-0.585890\pi$
$709$	−9.55149e13	−0.533138	−0.266569	+	0.963816i	$0.585890\pi$
$710$	0	0
$711$	− 6.11462e14i	− 3.36528i
$712$	0	0
$713$	−5.25312e13	−0.285082
$714$	0	0
$715$	− 4.22915e13i	− 0.226320i
$716$	0	0
$717$	−6.54223e14	−3.45247
$718$	0	0
$719$	− 2.56842e14i	− 1.33666i	−0.743863	−	0.668332i	$-0.767009\pi$
$719$	− 2.56842e14i	− 1.33666i	0.743863	−	0.668332i	$-0.232991\pi$
$720$	0	0
$721$	−2.15325e14	−1.10514
$722$	0	0
$723$	2.43059e14i	1.23033i
$724$	0	0
$725$	1.19926e14	0.598722
$726$	0	0
$727$	3.15933e13i	0.155569i	0.996970	+	0.0777844i	$0.0247846\pi$
$727$	3.15933e13i	0.155569i	−0.996970	+	0.0777844i	$0.975215\pi$
$728$	0	0
$729$	−2.81954e14	−1.36943
$730$	0	0
$731$	3.78517e13i	0.181342i
$732$	0	0
$733$	−2.32577e14	−1.09912	−0.549562	−	0.835453i	$-0.685206\pi$
$733$	−2.32577e14	−1.09912	−0.549562	+	0.835453i	$0.685206\pi$
$734$	0	0
$735$	− 7.74114e14i	− 3.60885i
$736$	0	0
$737$	1.53164e13	0.0704402
$738$	0	0
$739$	− 2.47161e14i	− 1.12139i	−0.828021	−	0.560697i	$-0.810533\pi$
$739$	− 2.47161e14i	− 1.12139i	0.828021	−	0.560697i	$-0.189467\pi$
$740$	0	0
$741$	3.58407e13	0.160430
$742$	0	0
$743$	2.02988e12i	0.00896451i	0.999990	+	0.00448226i	$0.00142675\pi$
$743$	2.02988e12i	0.00896451i	−0.999990	+	0.00448226i	$0.998573\pi$
$744$	0	0
$745$	−3.14496e14	−1.37036
$746$	0	0
$747$	− 2.78977e14i	− 1.19941i
$748$	0	0
$749$	1.49691e14	0.635019
$750$	0	0
$751$	2.98896e14i	1.25118i	0.780151	+	0.625591i	$0.215142\pi$
$751$	2.98896e14i	1.25118i	−0.780151	+	0.625591i	$0.784858\pi$
$752$	0	0
$753$	−4.29503e14	−1.77416
$754$	0	0
$755$	− 1.93642e14i	− 0.789342i
$756$	0	0
$757$	4.64306e14	1.86778	0.933888	−	0.357566i	$-0.116393\pi$
$757$	4.64306e14	1.86778	0.933888	+	0.357566i	$0.116393\pi$
$758$	0	0
$759$	− 2.62378e14i	− 1.04164i
$760$	0	0
$761$	6.13990e13	0.240568	0.120284	−	0.992740i	$-0.461619\pi$
$761$	6.13990e13	0.240568	0.120284	+	0.992740i	$0.461619\pi$
$762$	0	0
$763$	6.25546e14i	2.41901i
$764$	0	0
$765$	−2.02672e14	−0.773545
$766$	0	0
$767$	− 6.74391e12i	− 0.0254059i
$768$	0	0
$769$	−3.64244e14	−1.35444	−0.677222	−	0.735779i	$-0.736816\pi$
$769$	−3.64244e14	−1.35444	−0.677222	+	0.735779i	$0.736816\pi$
$770$	0	0
$771$	8.07757e14i	2.96489i
$772$	0	0
$773$	−2.47696e14	−0.897474	−0.448737	−	0.893664i	$-0.648126\pi$
$773$	−2.47696e14	−0.897474	−0.448737	+	0.893664i	$0.648126\pi$
$774$	0	0
$775$	1.57084e14i	0.561854i
$776$	0	0
$777$	1.47417e15	5.20525
$778$	0	0
$779$	1.64303e14i	0.572740i
$780$	0	0
$781$	−7.48816e12	−0.0257703
$782$	0	0
$783$	− 7.17616e14i	− 2.43828i
$784$	0	0
$785$	−3.03980e14	−1.01976
$786$	0	0
$787$	− 7.60641e13i	− 0.251945i	−0.992034	−	0.125973i	$-0.959795\pi$
$787$	− 7.60641e13i	− 0.251945i	0.992034	−	0.125973i	$-0.0402051\pi$
$788$	0	0
$789$	1.41728e14	0.463522
$790$	0	0
$791$	4.57916e14i	1.47878i
$792$	0	0
$793$	−2.26096e13	−0.0720987
$794$	0	0
$795$	− 4.39477e14i	− 1.38389i
$796$	0	0
$797$	3.19667e14	0.994046	0.497023	−	0.867737i	$-0.334427\pi$
$797$	3.19667e14	0.994046	0.497023	+	0.867737i	$0.334427\pi$
$798$	0	0
$799$	− 4.79754e13i	− 0.147328i
$800$	0	0
$801$	1.24280e15	3.76912
$802$	0	0
$803$	2.32185e14i	0.695434i
$804$	0	0
$805$	2.34934e14	0.694971
$806$	0	0
$807$	3.45024e14i	1.00805i
$808$	0	0
$809$	6.68134e13	0.192806	0.0964031	−	0.995342i	$-0.469266\pi$
$809$	6.68134e13	0.192806	0.0964031	+	0.995342i	$0.469266\pi$
$810$	0	0
$811$	− 3.97609e14i	− 1.13332i	−0.823952	−	0.566660i	$-0.808236\pi$
$811$	− 3.97609e14i	− 1.13332i	0.823952	−	0.566660i	$-0.191764\pi$
$812$	0	0
$813$	4.50198e14	1.26751
$814$	0	0
$815$	− 2.37260e14i	− 0.659836i
$816$	0	0
$817$	−2.20171e14	−0.604854
$818$	0	0
$819$	− 1.52952e14i	− 0.415084i
$820$	0	0
$821$	5.66640e13	0.151912	0.0759559	−	0.997111i	$-0.475799\pi$
$821$	5.66640e13	0.151912	0.0759559	+	0.997111i	$0.475799\pi$
$822$	0	0
$823$	− 2.13809e14i	− 0.566275i	−0.959079	−	0.283138i	$-0.908625\pi$
$823$	− 2.13809e14i	− 0.566275i	0.959079	−	0.283138i	$-0.0913753\pi$
$824$	0	0
$825$	−7.84590e14	−2.05293
$826$	0	0
$827$	2.16053e14i	0.558513i	0.960216	+	0.279257i	$0.0900880\pi$
$827$	2.16053e14i	0.558513i	−0.960216	+	0.279257i	$0.909912\pi$
$828$	0	0
$829$	−2.02368e14	−0.516856	−0.258428	−	0.966031i	$-0.583204\pi$
$829$	−2.02368e14	−0.516856	−0.258428	+	0.966031i	$0.583204\pi$
$830$	0	0
$831$	− 6.20540e14i	− 1.56590i
$832$	0	0
$833$	−1.45991e14	−0.363999
$834$	0	0
$835$	− 5.30642e14i	− 1.30728i
$836$	0	0
$837$	9.39960e14	2.28814
$838$	0	0
$839$	3.25854e14i	0.783815i	0.920005	+	0.391908i	$0.128185\pi$
$839$	3.25854e14i	0.783815i	−0.920005	+	0.391908i	$0.871815\pi$
$840$	0	0
$841$	−8.50521e13	−0.202164
$842$	0	0
$843$	− 1.49037e15i	− 3.50071i
$844$	0	0
$845$	−5.50467e14	−1.27776
$846$	0	0
$847$	1.17596e15i	2.69759i
$848$	0	0
$849$	3.63550e14	0.824186
$850$	0	0
$851$	2.68423e14i	0.601412i
$852$	0	0
$853$	−6.43388e14	−1.42471	−0.712357	−	0.701817i	$-0.752372\pi$
$853$	−6.43388e14	−1.42471	−0.712357	+	0.701817i	$0.752372\pi$
$854$	0	0
$855$	− 1.17888e15i	− 2.58011i
$856$	0	0
$857$	6.72076e13	0.145383	0.0726916	−	0.997354i	$-0.476841\pi$
$857$	6.72076e13	0.145383	0.0726916	+	0.997354i	$0.476841\pi$
$858$	0	0
$859$	4.02839e13i	0.0861322i	0.999072	+	0.0430661i	$0.0137126\pi$
$859$	4.02839e13i	0.0861322i	−0.999072	+	0.0430661i	$0.986287\pi$
$860$	0	0
$861$	9.85480e14	2.08273
$862$	0	0
$863$	4.37664e14i	0.914296i	0.889391	+	0.457148i	$0.151129\pi$
$863$	4.37664e14i	0.914296i	−0.889391	+	0.457148i	$0.848871\pi$
$864$	0	0
$865$	2.63775e14	0.544694
$866$	0	0
$867$	− 8.58340e14i	− 1.75212i
$868$	0	0
$869$	−1.11241e15	−2.24474
$870$	0	0
$871$	2.28499e12i	0.00455819i
$872$	0	0
$873$	4.65456e14	0.917926
$874$	0	0
$875$	3.45552e14i	0.673711i
$876$	0	0
$877$	−3.51878e14	−0.678257	−0.339128	−	0.940740i	$-0.610132\pi$
$877$	−3.51878e14	−0.678257	−0.339128	+	0.940740i	$0.610132\pi$
$878$	0	0
$879$	7.65935e14i	1.45965i
$880$	0	0
$881$	5.05415e14	0.952288	0.476144	−	0.879367i	$-0.342034\pi$
$881$	5.05415e14	0.952288	0.476144	+	0.879367i	$0.342034\pi$
$882$	0	0
$883$	1.39419e14i	0.259727i	0.991532	+	0.129863i	$0.0414539\pi$
$883$	1.39419e14i	0.259727i	−0.991532	+	0.129863i	$0.958546\pi$
$884$	0	0
$885$	−3.11765e14	−0.574264
$886$	0	0
$887$	7.73698e14i	1.40914i	0.709635	+	0.704569i	$0.248860\pi$
$887$	7.73698e14i	1.40914i	−0.709635	+	0.704569i	$0.751140\pi$
$888$	0	0
$889$	−1.31124e15	−2.36141
$890$	0	0
$891$	2.41661e15i	4.30346i
$892$	0	0
$893$	2.79058e14	0.491402
$894$	0	0
$895$	1.04773e15i	1.82447i
$896$	0	0
$897$	3.91429e13	0.0674048
$898$	0	0
$899$	4.39653e14i	0.748707i
$900$	0	0
$901$	−8.28813e13	−0.139583
$902$	0	0
$903$	1.32058e15i	2.19951i
$904$	0	0
$905$	3.12722e14	0.515128
$906$	0	0
$907$	− 9.12730e14i	− 1.48698i	−0.668746	−	0.743491i	$-0.733169\pi$
$907$	− 9.12730e14i	− 1.48698i	0.668746	−	0.743491i	$-0.266831\pi$
$908$	0	0
$909$	1.50811e15	2.43004
$910$	0	0
$911$	− 8.96231e14i	− 1.42833i	−0.699978	−	0.714164i	$-0.746807\pi$
$911$	− 8.96231e14i	− 1.42833i	0.699978	−	0.714164i	$-0.253193\pi$
$912$	0	0
$913$	−5.07534e14	−0.800039
$914$	0	0
$915$	1.04522e15i	1.62969i
$916$	0	0
$917$	1.49467e15	2.30515
$918$	0	0
$919$	− 2.57273e14i	− 0.392480i	−0.980556	−	0.196240i	$-0.937127\pi$
$919$	− 2.57273e14i	− 0.392480i	0.980556	−	0.196240i	$-0.0628732\pi$
$920$	0	0
$921$	−1.44700e14	−0.218359
$922$	0	0
$923$	− 1.11712e12i	− 0.00166760i
$924$	0	0
$925$	8.02666e14	1.18529
$926$	0	0
$927$	1.18003e15i	1.72383i
$928$	0	0
$929$	1.07476e15	1.55322	0.776612	−	0.629980i	$-0.216937\pi$
$929$	1.07476e15	1.55322	0.776612	+	0.629980i	$0.216937\pi$
$930$	0	0
$931$	− 8.49181e14i	− 1.21409i
$932$	0	0
$933$	1.08095e15	1.52897
$934$	0	0
$935$	3.68714e14i	0.515978i
$936$	0	0
$937$	1.65739e14	0.229470	0.114735	−	0.993396i	$-0.463398\pi$
$937$	1.65739e14	0.229470	0.114735	+	0.993396i	$0.463398\pi$
$938$	0	0
$939$	1.01468e15i	1.38995i
$940$	0	0
$941$	5.67712e14	0.769449	0.384724	−	0.923031i	$-0.374296\pi$
$941$	5.67712e14	0.769449	0.384724	+	0.923031i	$0.374296\pi$
$942$	0	0
$943$	1.79441e14i	0.240637i
$944$	0	0
$945$	−4.20377e15	−5.57801
$946$	0	0
$947$	− 3.20812e13i	− 0.0421211i	−0.999778	−	0.0210606i	$-0.993296\pi$
$947$	− 3.20812e13i	− 0.0421211i	0.999778	−	0.0210606i	$-0.00670428\pi$
$948$	0	0
$949$	−3.46385e13	−0.0450016
$950$	0	0
$951$	2.89931e14i	0.372728i
$952$	0	0
$953$	−1.37221e14	−0.174564	−0.0872821	−	0.996184i	$-0.527818\pi$
$953$	−1.37221e14	−0.174564	−0.0872821	+	0.996184i	$0.527818\pi$
$954$	0	0
$955$	1.95590e15i	2.46224i
$956$	0	0
$957$	−2.19594e15	−2.73566
$958$	0	0
$959$	7.50986e14i	0.925847i
$960$	0	0
$961$	2.43754e14	0.297396
$962$	0	0
$963$	− 8.20341e14i	− 0.990519i
$964$	0	0
$965$	1.79708e15	2.14749
$966$	0	0
$967$	− 4.29884e14i	− 0.508416i	−0.967150	−	0.254208i	$-0.918185\pi$
$967$	− 4.29884e14i	− 0.508416i	0.967150	−	0.254208i	$-0.0818148\pi$
$968$	0	0
$969$	−3.12473e14	−0.365758
$970$	0	0
$971$	− 5.57635e14i	− 0.646032i	−0.946393	−	0.323016i	$-0.895303\pi$
$971$	− 5.57635e14i	− 0.646032i	0.946393	−	0.323016i	$-0.104697\pi$
$972$	0	0
$973$	1.70288e15	1.95263
$974$	0	0
$975$	− 1.17049e14i	− 0.132845i
$976$	0	0
$977$	−5.95407e14	−0.668869	−0.334435	−	0.942419i	$-0.608545\pi$
$977$	−5.95407e14	−0.668869	−0.334435	+	0.942419i	$0.608545\pi$
$978$	0	0
$979$	− 2.26099e15i	− 2.51411i
$980$	0	0
$981$	3.42814e15	3.77323
$982$	0	0
$983$	− 5.94215e14i	− 0.647405i	−0.946159	−	0.323703i	$-0.895072\pi$
$983$	− 5.94215e14i	− 0.647405i	0.946159	−	0.323703i	$-0.104928\pi$
$984$	0	0
$985$	1.02068e15	1.10080
$986$	0	0
$987$	− 1.67378e15i	− 1.78695i
$988$	0	0
$989$	−2.40457e14	−0.254130
$990$	0	0
$991$	2.04216e14i	0.213659i	0.994277	+	0.106830i	$0.0340699\pi$
$991$	2.04216e14i	0.213659i	−0.994277	+	0.106830i	$0.965930\pi$
$992$	0	0
$993$	2.44905e15	2.53659
$994$	0	0
$995$	− 6.52589e14i	− 0.669151i
$996$	0	0
$997$	−7.24128e14	−0.735089	−0.367544	−	0.930006i	$-0.619801\pi$
$997$	−7.24128e14	−0.735089	−0.367544	+	0.930006i	$0.619801\pi$
$998$	0	0
$999$	− 4.80299e15i	− 4.82708i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.k.255.10		10
4.3	odd	2	inner	256.11.c.k.255.1		10
8.3	odd	2		256.11.c.j.255.10		10
8.5	even	2		256.11.c.j.255.1		10
16.3	odd	4		128.11.d.f.63.19	yes	20
16.5	even	4		128.11.d.f.63.20	yes	20
16.11	odd	4		128.11.d.f.63.2	yes	20
16.13	even	4		128.11.d.f.63.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.f.63.1	✓	20	16.13	even	4
128.11.d.f.63.2	yes	20	16.11	odd	4
128.11.d.f.63.19	yes	20	16.3	odd	4
128.11.d.f.63.20	yes	20	16.5	even	4
256.11.c.j.255.1		10	8.5	even	2
256.11.c.j.255.10		10	8.3	odd	2
256.11.c.k.255.1		10	4.3	odd	2	inner
256.11.c.k.255.10		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+452.412i q^{3} +4038.75 q^{5} -26573.3i q^{7} -145628. q^{9} +O(q^{10})\)	\(q+452.412i q^{3} +4038.75 q^{5} -26573.3i q^{7} -145628. q^{9} +264936. i q^{11} -39524.4 q^{13} +1.82718e6i q^{15} +344589. q^{17} +2.00436e6i q^{19} +1.20221e7 q^{21} +2.18904e6i q^{23} +6.54589e6 q^{25} -3.91693e7i q^{27} +1.83209e7 q^{29} +2.39974e7i q^{31} -1.19860e8 q^{33} -1.07323e8i q^{35} +1.22621e8 q^{37} -1.78813e7i q^{39} +8.19724e7 q^{41} +1.09846e8i q^{43} -5.88154e8 q^{45} -1.39225e8i q^{47} -4.23666e8 q^{49} +1.55896e8i q^{51} -2.40522e8 q^{53} +1.07001e9i q^{55} -9.06799e8 q^{57} +1.70626e8i q^{59} +5.72042e8 q^{61} +3.86981e9i q^{63} -1.59629e8 q^{65} -5.78120e7i q^{67} -9.90348e8 q^{69} +2.82641e7i q^{71} +8.76382e8 q^{73} +2.96144e9i q^{75} +7.04022e9 q^{77} +4.19880e9i q^{79} +9.12149e9 q^{81} +1.91569e9i q^{83} +1.39171e9 q^{85} +8.28860e9i q^{87} -8.53411e9 q^{89} +1.05030e9i q^{91} -1.08567e10 q^{93} +8.09513e9i q^{95} -3.19621e9 q^{97} -3.85820e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6232 q^{5} - 218038 q^{9}+O(q^{10}) \) 10 * q + 6232 * q^5 - 218038 * q^9	\( 10 q + 6232 q^{5} - 218038 q^{9} + 485560 q^{13} - 1468772 q^{17} + 15545024 q^{21} + 4383462 q^{25} + 25263800 q^{29} - 122737264 q^{33} + 178255288 q^{37} + 91656876 q^{41} - 718346408 q^{45} - 842454934 q^{49} - 798458664 q^{53} - 1612102544 q^{57} + 102636184 q^{61} + 376325920 q^{65} + 207224128 q^{69} + 2854265572 q^{73} + 9037740608 q^{77} + 11990017466 q^{81} - 5473132400 q^{85} - 11790814556 q^{89} - 24576098304 q^{93} - 9363277860 q^{97}+O(q^{100}) \) 10 * q + 6232 * q^5 - 218038 * q^9 + 485560 * q^13 - 1468772 * q^17 + 15545024 * q^21 + 4383462 * q^25 + 25263800 * q^29 - 122737264 * q^33 + 178255288 * q^37 + 91656876 * q^41 - 718346408 * q^45 - 842454934 * q^49 - 798458664 * q^53 - 1612102544 * q^57 + 102636184 * q^61 + 376325920 * q^65 + 207224128 * q^69 + 2854265572 * q^73 + 9037740608 * q^77 + 11990017466 * q^81 - 5473132400 * q^85 - 11790814556 * q^89 - 24576098304 * q^93 - 9363277860 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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