LMFDB - Embedded newform 256.11.c.m.255.6

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} + 61429 x^{12} - 23865589 x^{10} + 9433993075 x^{8} - 796642244899 x^{6} + \cdots + 11\!\cdots\!56 $ x^16 - x^14 + 61429x^12 - 23865589x^10 + 9433993075x^8 - 796642244899x^6 + 422044223147719x^4 - 3251830843526311x^2 + 11210521480874115856
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{178}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.6
Root			$6.91461 - 11.4287i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.m.255.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-196.684i q^{3} +5381.00 q^{5} +6613.83i q^{7} +20364.2 q^{9} -221096. i q^{11} +361689. q^{13} -1.05836e6i q^{15} +776217. q^{17} +1.46186e6i q^{19} +1.30084e6 q^{21} -2.34458e6i q^{23} +1.91895e7 q^{25} -1.56193e7i q^{27} -2.88750e7 q^{29} -3.00762e7i q^{31} -4.34861e7 q^{33} +3.55890e7i q^{35} -1.16919e7 q^{37} -7.11386e7i q^{39} -4.70330e7 q^{41} -3.52427e7i q^{43} +1.09580e8 q^{45} -4.23068e8i q^{47} +2.38733e8 q^{49} -1.52670e8i q^{51} +9.16056e6 q^{53} -1.18971e9i q^{55} +2.87526e8 q^{57} +1.03098e9i q^{59} -1.37140e8 q^{61} +1.34685e8i q^{63} +1.94625e9 q^{65} -7.94671e8i q^{67} -4.61143e8 q^{69} +2.04858e9i q^{71} +1.88707e9 q^{73} -3.77428e9i q^{75} +1.46229e9 q^{77} +1.04965e9i q^{79} -1.86960e9 q^{81} +3.77162e9i q^{83} +4.17682e9 q^{85} +5.67927e9i q^{87} +4.07338e9 q^{89} +2.39215e9i q^{91} -5.91551e9 q^{93} +7.86629e9i q^{95} +1.76770e9 q^{97} -4.50244e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 70992 q^{9} - 741600 q^{17} + 37585520 q^{25} - 148617408 q^{33} - 185338656 q^{41} - 160864496 q^{49} - 2801425728 q^{57} - 1678056960 q^{65} - 5600145440 q^{73} - 23818130352 q^{81} - 12325192224 q^{89}+ \cdots - 19395727072 q^{97}+O(q^{100}) $ 16 * q + 70992 * q^9 - 741600 * q^17 + 37585520 * q^25 - 148617408 * q^33 - 185338656 * q^41 - 160864496 * q^49 - 2801425728 * q^57 - 1678056960 * q^65 - 5600145440 * q^73 - 23818130352 * q^81 - 12325192224 * q^89 - 19395727072 * q^97

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
$3$	− 196.684i	− 0.809401i	−0.914449	−	0.404701i	$-0.867376\pi$
$3$	− 196.684i	− 0.809401i	0.914449	−	0.404701i	$-0.132624\pi$
$4$	0	0
$5$	5381.00	1.72192	0.860960	−	0.508673i	$-0.169864\pi$
$5$	5381.00	1.72192	0.860960	+	0.508673i	$0.169864\pi$
$6$	0	0
$7$	6613.83i	0.393516i	0.980452	+	0.196758i	$0.0630414\pi$
$7$	6613.83i	0.393516i	−0.980452	+	0.196758i	$0.936959\pi$
$8$	0	0
$9$	20364.2	0.344870
$10$	0	0
$11$	− 221096.i	− 1.37283i	−0.727210	−	0.686415i	$-0.759184\pi$
$11$	− 221096.i	− 1.37283i	0.727210	−	0.686415i	$-0.240816\pi$
$12$	0	0
$13$	361689.	0.974134	0.487067	−	0.873365i	$-0.338067\pi$
$13$	361689.	0.974134	0.487067	+	0.873365i	$0.338067\pi$
$14$	0	0
$15$	− 1.05836e6i	− 1.39372i
$16$	0	0
$17$	776217.	0.546687	0.273343	−	0.961917i	$-0.411870\pi$
$17$	776217.	0.546687	0.273343	+	0.961917i	$0.411870\pi$
$18$	0	0
$19$	1.46186e6i	0.590390i	0.955437	+	0.295195i	$0.0953846\pi$
$19$	1.46186e6i	0.590390i	−0.955437	+	0.295195i	$0.904615\pi$
$20$	0	0
$21$	1.30084e6	0.318513
$22$	0	0
$23$	− 2.34458e6i	− 0.364273i	−0.983273	−	0.182136i	$-0.941699\pi$
$23$	− 2.34458e6i	− 0.364273i	0.983273	−	0.182136i	$-0.0583012\pi$
$24$	0	0
$25$	1.91895e7	1.96501
$26$	0	0
$27$	− 1.56193e7i	− 1.08854i
$28$	0	0
$29$	−2.88750e7	−1.40777	−0.703886	−	0.710313i	$-0.748554\pi$
$29$	−2.88750e7	−1.40777	−0.703886	+	0.710313i	$0.748554\pi$
$30$	0	0
$31$	− 3.00762e7i	− 1.05054i	−0.850935	−	0.525272i	$-0.823964\pi$
$31$	− 3.00762e7i	− 1.05054i	0.850935	−	0.525272i	$-0.176036\pi$
$32$	0	0
$33$	−4.34861e7	−1.11117
$34$	0	0
$35$	3.55890e7i	0.677604i
$36$	0	0
$37$	−1.16919e7	−0.168607	−0.0843037	−	0.996440i	$-0.526867\pi$
$37$	−1.16919e7	−0.168607	−0.0843037	+	0.996440i	$0.526867\pi$
$38$	0	0
$39$	− 7.11386e7i	− 0.788465i
$40$	0	0
$41$	−4.70330e7	−0.405960	−0.202980	−	0.979183i	$-0.565063\pi$
$41$	−4.70330e7	−0.405960	−0.202980	+	0.979183i	$0.565063\pi$
$42$	0	0
$43$	− 3.52427e7i	− 0.239733i	−0.992790	−	0.119866i	$-0.961753\pi$
$43$	− 3.52427e7i	− 0.239733i	0.992790	−	0.119866i	$-0.0382466\pi$
$44$	0	0
$45$	1.09580e8	0.593838
$46$	0	0
$47$	− 4.23068e8i	− 1.84468i	−0.386380	−	0.922340i	$-0.626275\pi$
$47$	− 4.23068e8i	− 1.84468i	0.386380	−	0.922340i	$-0.373725\pi$
$48$	0	0
$49$	2.38733e8	0.845145
$50$	0	0
$51$	− 1.52670e8i	− 0.442489i
$52$	0	0
$53$	9.16056e6	0.0219050	0.0109525	−	0.999940i	$-0.496514\pi$
$53$	9.16056e6	0.0219050	0.0109525	+	0.999940i	$0.496514\pi$
$54$	0	0
$55$	− 1.18971e9i	− 2.36390i
$56$	0	0
$57$	2.87526e8	0.477862
$58$	0	0
$59$	1.03098e9i	1.44209i	0.692890	+	0.721044i	$0.256337\pi$
$59$	1.03098e9i	1.44209i	−0.692890	+	0.721044i	$0.743663\pi$
$60$	0	0
$61$	−1.37140e8	−0.162374	−0.0811869	−	0.996699i	$-0.525871\pi$
$61$	−1.37140e8	−0.162374	−0.0811869	+	0.996699i	$0.525871\pi$
$62$	0	0
$63$	1.34685e8i	0.135712i
$64$	0	0
$65$	1.94625e9	1.67738
$66$	0	0
$67$	− 7.94671e8i	− 0.588591i	−0.955715	−	0.294295i	$-0.904915\pi$
$67$	− 7.94671e8i	− 0.588591i	0.955715	−	0.294295i	$-0.0950849\pi$
$68$	0	0
$69$	−4.61143e8	−0.294843
$70$	0	0
$71$	2.04858e9i	1.13543i	0.823225	+	0.567716i	$0.192173\pi$
$71$	2.04858e9i	1.13543i	−0.823225	+	0.567716i	$0.807827\pi$
$72$	0	0
$73$	1.88707e9	0.910277	0.455139	−	0.890421i	$-0.349590\pi$
$73$	1.88707e9	0.910277	0.455139	+	0.890421i	$0.349590\pi$
$74$	0	0
$75$	− 3.77428e9i	− 1.59048i
$76$	0	0
$77$	1.46229e9	0.540231
$78$	0	0
$79$	1.04965e9i	0.341120i	0.985347	+	0.170560i	$0.0545577\pi$
$79$	1.04965e9i	0.341120i	−0.985347	+	0.170560i	$0.945442\pi$
$80$	0	0
$81$	−1.86960e9	−0.536195
$82$	0	0
$83$	3.77162e9i	0.957496i	0.877952	+	0.478748i	$0.158909\pi$
$83$	3.77162e9i	0.957496i	−0.877952	+	0.478748i	$0.841091\pi$
$84$	0	0
$85$	4.17682e9	0.941351
$86$	0	0
$87$	5.67927e9i	1.13945i
$88$	0	0
$89$	4.07338e9	0.729465	0.364733	−	0.931112i	$-0.381160\pi$
$89$	4.07338e9	0.729465	0.364733	+	0.931112i	$0.381160\pi$
$90$	0	0
$91$	2.39215e9i	0.383337i
$92$	0	0
$93$	−5.91551e9	−0.850311
$94$	0	0
$95$	7.86629e9i	1.01660i
$96$	0	0
$97$	1.76770e9	0.205850	0.102925	−	0.994689i	$-0.467180\pi$
$97$	1.76770e9	0.205850	0.102925	+	0.994689i	$0.467180\pi$
$98$	0	0
$99$	− 4.50244e9i	− 0.473447i
$100$	0	0
$101$	−1.09549e10	−1.04232	−0.521160	−	0.853459i	$-0.674500\pi$
$101$	−1.09549e10	−1.04232	−0.521160	+	0.853459i	$0.674500\pi$
$102$	0	0
$103$	− 2.37007e8i	− 0.0204444i	−0.999948	−	0.0102222i	$-0.996746\pi$
$103$	− 2.37007e8i	− 0.0204444i	0.999948	−	0.0102222i	$-0.00325389\pi$
$104$	0	0
$105$	6.99981e9	0.548453
$106$	0	0
$107$	− 4.29761e9i	− 0.306414i	−0.988194	−	0.153207i	$-0.951040\pi$
$107$	− 4.29761e9i	− 0.306414i	0.988194	−	0.153207i	$-0.0489601\pi$
$108$	0	0
$109$	−2.92292e9	−0.189970	−0.0949849	−	0.995479i	$-0.530280\pi$
$109$	−2.92292e9	−0.189970	−0.0949849	+	0.995479i	$0.530280\pi$
$110$	0	0
$111$	2.29962e9i	0.136471i
$112$	0	0
$113$	2.48941e10	1.35115	0.675575	−	0.737291i	$-0.263895\pi$
$113$	2.48941e10	1.35115	0.675575	+	0.737291i	$0.263895\pi$
$114$	0	0
$115$	− 1.26162e10i	− 0.627248i
$116$	0	0
$117$	7.36551e9	0.335949
$118$	0	0
$119$	5.13377e9i	0.215130i
$120$	0	0
$121$	−2.29458e10	−0.884661
$122$	0	0
$123$	9.25066e9i	0.328585i
$124$	0	0
$125$	5.07100e10	1.66167
$126$	0	0
$127$	− 1.86927e10i	− 0.565788i	−0.959151	−	0.282894i	$-0.908706\pi$
$127$	− 1.86927e10i	− 0.565788i	0.959151	−	0.282894i	$-0.0912945\pi$
$128$	0	0
$129$	−6.93170e9	−0.194040
$130$	0	0
$131$	− 1.62805e10i	− 0.421998i	−0.977486	−	0.210999i	$-0.932328\pi$
$131$	− 1.62805e10i	− 0.421998i	0.977486	−	0.210999i	$-0.0676717\pi$
$132$	0	0
$133$	−9.66852e9	−0.232328
$134$	0	0
$135$	− 8.40477e10i	− 1.87438i
$136$	0	0
$137$	−3.24700e10	−0.672791	−0.336395	−	0.941721i	$-0.609208\pi$
$137$	−3.24700e10	−0.672791	−0.336395	+	0.941721i	$0.609208\pi$
$138$	0	0
$139$	− 3.25763e10i	− 0.627809i	−0.949455	−	0.313905i	$-0.898363\pi$
$139$	− 3.25763e10i	− 0.627809i	0.949455	−	0.313905i	$-0.101637\pi$
$140$	0	0
$141$	−8.32109e10	−1.49309
$142$	0	0
$143$	− 7.99678e10i	− 1.33732i
$144$	0	0
$145$	−1.55377e11	−2.42407
$146$	0	0
$147$	− 4.69550e10i	− 0.684061i
$148$	0	0
$149$	9.28139e10	1.26381	0.631905	−	0.775046i	$-0.282273\pi$
$149$	9.28139e10	1.26381	0.631905	+	0.775046i	$0.282273\pi$
$150$	0	0
$151$	− 1.62055e10i	− 0.206432i	−0.994659	−	0.103216i	$-0.967087\pi$
$151$	− 1.62055e10i	− 0.206432i	0.994659	−	0.103216i	$-0.0329133\pi$
$152$	0	0
$153$	1.58070e10	0.188536
$154$	0	0
$155$	− 1.61840e11i	− 1.80895i
$156$	0	0
$157$	2.39468e10	0.251043	0.125522	−	0.992091i	$-0.459940\pi$
$157$	2.39468e10	0.251043	0.125522	+	0.992091i	$0.459940\pi$
$158$	0	0
$159$	− 1.80174e9i	− 0.0177299i
$160$	0	0
$161$	1.55067e10	0.143347
$162$	0	0
$163$	7.89570e10i	0.686203i	0.939298	+	0.343101i	$0.111477\pi$
$163$	7.89570e10i	0.686203i	−0.939298	+	0.343101i	$0.888523\pi$
$164$	0	0
$165$	−2.33998e11	−1.91335
$166$	0	0
$167$	1.85927e11i	1.43140i	0.698408	+	0.715700i	$0.253892\pi$
$167$	1.85927e11i	1.43140i	−0.698408	+	0.715700i	$0.746108\pi$
$168$	0	0
$169$	−7.03958e9	−0.0510638
$170$	0	0
$171$	2.97697e10i	0.203608i
$172$	0	0
$173$	7.40712e9	0.0477990	0.0238995	−	0.999714i	$-0.492392\pi$
$173$	7.40712e9	0.0477990	0.0238995	+	0.999714i	$0.492392\pi$
$174$	0	0
$175$	1.26916e11i	0.773263i
$176$	0	0
$177$	2.02778e11	1.16723
$178$	0	0
$179$	− 3.09505e11i	− 1.68423i	−0.539295	−	0.842117i	$-0.681309\pi$
$179$	− 3.09505e11i	− 1.68423i	0.539295	−	0.842117i	$-0.318691\pi$
$180$	0	0
$181$	3.52000e11	1.81196	0.905982	−	0.423316i	$-0.139134\pi$
$181$	3.52000e11	1.81196	0.905982	+	0.423316i	$0.139134\pi$
$182$	0	0
$183$	2.69734e10i	0.131425i
$184$	0	0
$185$	−6.29141e10	−0.290328
$186$	0	0
$187$	− 1.71618e11i	− 0.750508i
$188$	0	0
$189$	1.03304e11	0.428358
$190$	0	0
$191$	4.23512e11i	1.66609i	0.553205	+	0.833045i	$0.313405\pi$
$191$	4.23512e11i	1.66609i	−0.553205	+	0.833045i	$0.686595\pi$
$192$	0	0
$193$	−2.50051e11	−0.933774	−0.466887	−	0.884317i	$-0.654625\pi$
$193$	−2.50051e11	−0.933774	−0.466887	+	0.884317i	$0.654625\pi$
$194$	0	0
$195$	− 3.82797e11i	− 1.35767i
$196$	0	0
$197$	−4.85950e11	−1.63780	−0.818899	−	0.573937i	$-0.805415\pi$
$197$	−4.85950e11	−1.63780	−0.818899	+	0.573937i	$0.805415\pi$
$198$	0	0
$199$	1.41021e11i	0.451876i	0.974142	+	0.225938i	$0.0725447\pi$
$199$	1.41021e11i	0.451876i	−0.974142	+	0.225938i	$0.927455\pi$
$200$	0	0
$201$	−1.56299e11	−0.476406
$202$	0	0
$203$	− 1.90975e11i	− 0.553982i
$204$	0	0
$205$	−2.53085e11	−0.699031
$206$	0	0
$207$	− 4.77456e10i	− 0.125627i
$208$	0	0
$209$	3.23212e11	0.810505
$210$	0	0
$211$	− 4.54045e11i	− 1.08564i	−0.839848	−	0.542821i	$-0.817356\pi$
$211$	− 4.54045e11i	− 1.08564i	0.839848	−	0.542821i	$-0.182644\pi$
$212$	0	0
$213$	4.02924e11	0.919019
$214$	0	0
$215$	− 1.89641e11i	− 0.412801i
$216$	0	0
$217$	1.98919e11	0.413406
$218$	0	0
$219$	− 3.71157e11i	− 0.736779i
$220$	0	0
$221$	2.80749e11	0.532546
$222$	0	0
$223$	− 7.10192e11i	− 1.28781i	−0.765106	−	0.643905i	$-0.777313\pi$
$223$	− 7.10192e11i	− 1.28781i	0.765106	−	0.643905i	$-0.222687\pi$
$224$	0	0
$225$	3.90780e11	0.677671
$226$	0	0
$227$	− 1.75996e11i	− 0.291994i	−0.989285	−	0.145997i	$-0.953361\pi$
$227$	− 1.75996e11i	− 0.291994i	0.989285	−	0.145997i	$-0.0466390\pi$
$228$	0	0
$229$	−3.96342e11	−0.629351	−0.314676	−	0.949199i	$-0.601896\pi$
$229$	−3.96342e11	−0.629351	−0.314676	+	0.949199i	$0.601896\pi$
$230$	0	0
$231$	− 2.87609e11i	− 0.437264i
$232$	0	0
$233$	9.14879e11	1.33225	0.666123	−	0.745842i	$-0.267953\pi$
$233$	9.14879e11	1.33225	0.666123	+	0.745842i	$0.267953\pi$
$234$	0	0
$235$	− 2.27653e12i	− 3.17639i
$236$	0	0
$237$	2.06449e11	0.276103
$238$	0	0
$239$	− 1.99248e11i	− 0.255508i	−0.991806	−	0.127754i	$-0.959223\pi$
$239$	− 1.99248e11i	− 0.255508i	0.991806	−	0.127754i	$-0.0407768\pi$
$240$	0	0
$241$	−5.21033e11	−0.640885	−0.320442	−	0.947268i	$-0.603832\pi$
$241$	−5.21033e11	−0.640885	−0.320442	+	0.947268i	$0.603832\pi$
$242$	0	0
$243$	− 5.54586e11i	− 0.654542i
$244$	0	0
$245$	1.28462e12	1.45527
$246$	0	0
$247$	5.28740e11i	0.575119i
$248$	0	0
$249$	7.41819e11	0.774999
$250$	0	0
$251$	8.76908e11i	0.880209i	0.897947	+	0.440104i	$0.145059\pi$
$251$	8.76908e11i	0.880209i	−0.897947	+	0.440104i	$0.854941\pi$
$252$	0	0
$253$	−5.18377e11	−0.500084
$254$	0	0
$255$	− 8.21516e11i	− 0.761930i
$256$	0	0
$257$	6.77292e11	0.604102	0.302051	−	0.953292i	$-0.402329\pi$
$257$	6.77292e11	0.604102	0.302051	+	0.953292i	$0.402329\pi$
$258$	0	0
$259$	− 7.73283e10i	− 0.0663498i
$260$	0	0
$261$	−5.88017e11	−0.485498
$262$	0	0
$263$	− 2.38906e12i	− 1.89867i	−0.314269	−	0.949334i	$-0.601759\pi$
$263$	− 2.38906e12i	− 1.89867i	0.314269	−	0.949334i	$-0.398241\pi$
$264$	0	0
$265$	4.92930e10	0.0377186
$266$	0	0
$267$	− 8.01170e11i	− 0.590430i
$268$	0	0
$269$	−9.15425e11	−0.649922	−0.324961	−	0.945727i	$-0.605351\pi$
$269$	−9.15425e11	−0.649922	−0.324961	+	0.945727i	$0.605351\pi$
$270$	0	0
$271$	7.01162e11i	0.479702i	0.970810	+	0.239851i	$0.0770986\pi$
$271$	7.01162e11i	0.479702i	−0.970810	+	0.239851i	$0.922901\pi$
$272$	0	0
$273$	4.70499e11	0.310274
$274$	0	0
$275$	− 4.24272e12i	− 2.69762i
$276$	0	0
$277$	−2.10823e12	−1.29276	−0.646382	−	0.763014i	$-0.723719\pi$
$277$	−2.10823e12	−1.29276	−0.646382	+	0.763014i	$0.723719\pi$
$278$	0	0
$279$	− 6.12477e11i	− 0.362301i
$280$	0	0
$281$	−2.87581e12	−1.64145	−0.820727	−	0.571320i	$-0.806431\pi$
$281$	−2.87581e12	−1.64145	−0.820727	+	0.571320i	$0.806431\pi$
$282$	0	0
$283$	− 2.12641e12i	− 1.17142i	−0.810519	−	0.585712i	$-0.800815\pi$
$283$	− 2.12641e12i	− 1.17142i	0.810519	−	0.585712i	$-0.199185\pi$
$284$	0	0
$285$	1.54718e12	0.822841
$286$	0	0
$287$	− 3.11068e11i	− 0.159752i
$288$	0	0
$289$	−1.41348e12	−0.701134
$290$	0	0
$291$	− 3.47680e11i	− 0.166615i
$292$	0	0
$293$	6.04754e11	0.280053	0.140027	−	0.990148i	$-0.455281\pi$
$293$	6.04754e11	0.280053	0.140027	+	0.990148i	$0.455281\pi$
$294$	0	0
$295$	5.54772e12i	2.48316i
$296$	0	0
$297$	−3.45337e12	−1.49438
$298$	0	0
$299$	− 8.48010e11i	− 0.354850i
$300$	0	0
$301$	2.33090e11	0.0943388
$302$	0	0
$303$	2.15466e12i	0.843655i
$304$	0	0
$305$	−7.37952e11	−0.279595
$306$	0	0
$307$	− 3.39388e11i	− 0.124453i	−0.998062	−	0.0622264i	$-0.980180\pi$
$307$	− 3.39388e11i	− 0.124453i	0.998062	−	0.0622264i	$-0.0198201\pi$
$308$	0	0
$309$	−4.66155e10	−0.0165477
$310$	0	0
$311$	− 4.79193e11i	− 0.164706i	−0.996603	−	0.0823528i	$-0.973757\pi$
$311$	− 4.79193e11i	− 0.164706i	0.996603	−	0.0823528i	$-0.0262434\pi$
$312$	0	0
$313$	3.15123e12	1.04896	0.524479	−	0.851423i	$-0.324260\pi$
$313$	3.15123e12	1.04896	0.524479	+	0.851423i	$0.324260\pi$
$314$	0	0
$315$	7.24742e11i	0.233685i
$316$	0	0
$317$	7.24188e11	0.226232	0.113116	−	0.993582i	$-0.463917\pi$
$317$	7.24188e11	0.226232	0.113116	+	0.993582i	$0.463917\pi$
$318$	0	0
$319$	6.38414e12i	1.93263i
$320$	0	0
$321$	−8.45274e11	−0.248012
$322$	0	0
$323$	1.13472e12i	0.322758i
$324$	0	0
$325$	6.94064e12	1.91418
$326$	0	0
$327$	5.74893e11i	0.153762i
$328$	0	0
$329$	2.79810e12	0.725911
$330$	0	0
$331$	− 1.89023e12i	− 0.475745i	−0.971296	−	0.237872i	$-0.923550\pi$
$331$	− 1.89023e12i	− 0.475745i	0.971296	−	0.237872i	$-0.0764500\pi$
$332$	0	0
$333$	−2.38096e11	−0.0581476
$334$	0	0
$335$	− 4.27612e12i	− 1.01351i
$336$	0	0
$337$	−2.44837e12	−0.563285	−0.281643	−	0.959519i	$-0.590879\pi$
$337$	−2.44837e12	−0.563285	−0.281643	+	0.959519i	$0.590879\pi$
$338$	0	0
$339$	− 4.89628e12i	− 1.09362i
$340$	0	0
$341$	−6.64971e12	−1.44222
$342$	0	0
$343$	3.44718e12i	0.726095i
$344$	0	0
$345$	−2.48141e12	−0.507695
$346$	0	0
$347$	− 5.66030e12i	− 1.12510i	−0.826762	−	0.562551i	$-0.809820\pi$
$347$	− 5.66030e12i	− 1.12510i	0.826762	−	0.562551i	$-0.190180\pi$
$348$	0	0
$349$	−8.20588e12	−1.58489	−0.792443	−	0.609946i	$-0.791191\pi$
$349$	−8.20588e12	−1.58489	−0.792443	+	0.609946i	$0.791191\pi$
$350$	0	0
$351$	− 5.64935e12i	− 1.06038i
$352$	0	0
$353$	−7.02465e11	−0.128160	−0.0640798	−	0.997945i	$-0.520411\pi$
$353$	−7.02465e11	−0.128160	−0.0640798	+	0.997945i	$0.520411\pi$
$354$	0	0
$355$	1.10234e13i	1.95512i
$356$	0	0
$357$	1.00973e12	0.174127
$358$	0	0
$359$	− 5.97852e12i	− 1.00259i	−0.865278	−	0.501293i	$-0.832858\pi$
$359$	− 5.97852e12i	− 1.00259i	0.865278	−	0.501293i	$-0.167142\pi$
$360$	0	0
$361$	3.99402e12	0.651440
$362$	0	0
$363$	4.51309e12i	0.716045i
$364$	0	0
$365$	1.01543e13	1.56742
$366$	0	0
$367$	− 1.90765e12i	− 0.286530i	−0.989684	−	0.143265i	$-0.954240\pi$
$367$	− 1.90765e12i	− 0.286530i	0.989684	−	0.143265i	$-0.0457601\pi$
$368$	0	0
$369$	−9.57790e11	−0.140003
$370$	0	0
$371$	6.05864e10i	0.00861996i
$372$	0	0
$373$	−4.41936e12	−0.612090	−0.306045	−	0.952017i	$-0.599006\pi$
$373$	−4.41936e12	−0.612090	−0.306045	+	0.952017i	$0.599006\pi$
$374$	0	0
$375$	− 9.97387e12i	− 1.34495i
$376$	0	0
$377$	−1.04438e13	−1.37136
$378$	0	0
$379$	1.15746e13i	1.48017i	0.672516	+	0.740083i	$0.265214\pi$
$379$	1.15746e13i	1.48017i	−0.672516	+	0.740083i	$0.734786\pi$
$380$	0	0
$381$	−3.67657e12	−0.457949
$382$	0	0
$383$	− 7.38843e12i	− 0.896517i	−0.893904	−	0.448259i	$-0.852044\pi$
$383$	− 7.38843e12i	− 0.896517i	0.893904	−	0.448259i	$-0.147956\pi$
$384$	0	0
$385$	7.86857e12	0.930234
$386$	0	0
$387$	− 7.17691e11i	− 0.0826766i
$388$	0	0
$389$	8.41928e12	0.945207	0.472603	−	0.881275i	$-0.343314\pi$
$389$	8.41928e12	0.945207	0.472603	+	0.881275i	$0.343314\pi$
$390$	0	0
$391$	− 1.81991e12i	− 0.199143i
$392$	0	0
$393$	−3.20212e12	−0.341566
$394$	0	0
$395$	5.64815e12i	0.587382i
$396$	0	0
$397$	7.62252e12	0.772940	0.386470	−	0.922302i	$-0.373694\pi$
$397$	7.62252e12	0.772940	0.386470	+	0.922302i	$0.373694\pi$
$398$	0	0
$399$	1.90165e12i	0.188047i
$400$	0	0
$401$	1.08466e13	1.04610	0.523050	−	0.852302i	$-0.324794\pi$
$401$	1.08466e13	1.04610	0.523050	+	0.852302i	$0.324794\pi$
$402$	0	0
$403$	− 1.08782e13i	− 1.02337i
$404$	0	0
$405$	−1.00603e13	−0.923285
$406$	0	0
$407$	2.58503e12i	0.231469i
$408$	0	0
$409$	1.74748e13	1.52685	0.763425	−	0.645897i	$-0.223516\pi$
$409$	1.74748e13	1.52685	0.763425	+	0.645897i	$0.223516\pi$
$410$	0	0
$411$	6.38635e12i	0.544558i
$412$	0	0
$413$	−6.81875e12	−0.567485
$414$	0	0
$415$	2.02951e13i	1.64873i
$416$	0	0
$417$	−6.40725e12	−0.508149
$418$	0	0
$419$	1.67775e13i	1.29914i	0.760300	+	0.649572i	$0.225052\pi$
$419$	1.67775e13i	1.29914i	−0.760300	+	0.649572i	$0.774948\pi$
$420$	0	0
$421$	4.09777e12	0.309840	0.154920	−	0.987927i	$-0.450488\pi$
$421$	4.09777e12	0.309840	0.154920	+	0.987927i	$0.450488\pi$
$422$	0	0
$423$	− 8.61544e12i	− 0.636174i
$424$	0	0
$425$	1.48952e13	1.07424
$426$	0	0
$427$	− 9.07022e11i	− 0.0638967i
$428$	0	0
$429$	−1.57284e13	−1.08243
$430$	0	0
$431$	− 1.03214e13i	− 0.693985i	−0.937868	−	0.346993i	$-0.887203\pi$
$431$	− 1.03214e13i	− 0.693985i	0.937868	−	0.346993i	$-0.112797\pi$
$432$	0	0
$433$	2.17422e13	1.42844	0.714222	−	0.699919i	$-0.246781\pi$
$433$	2.17422e13	1.42844	0.714222	+	0.699919i	$0.246781\pi$
$434$	0	0
$435$	3.05602e13i	1.96205i
$436$	0	0
$437$	3.42746e12	0.215063
$438$	0	0
$439$	3.10509e13i	1.90437i	0.305524	+	0.952184i	$0.401168\pi$
$439$	3.10509e13i	1.90437i	−0.305524	+	0.952184i	$0.598832\pi$
$440$	0	0
$441$	4.86160e12	0.291465
$442$	0	0
$443$	2.32225e13i	1.36110i	0.732700	+	0.680552i	$0.238260\pi$
$443$	2.32225e13i	1.36110i	−0.732700	+	0.680552i	$0.761740\pi$
$444$	0	0
$445$	2.19188e13	1.25608
$446$	0	0
$447$	− 1.82550e13i	− 1.02293i
$448$	0	0
$449$	−1.05020e13	−0.575493	−0.287747	−	0.957707i	$-0.592906\pi$
$449$	−1.05020e13	−0.575493	−0.287747	+	0.957707i	$0.592906\pi$
$450$	0	0
$451$	1.03988e13i	0.557314i
$452$	0	0
$453$	−3.18736e12	−0.167086
$454$	0	0
$455$	1.28722e13i	0.660076i
$456$	0	0
$457$	−1.72386e13	−0.864813	−0.432407	−	0.901679i	$-0.642335\pi$
$457$	−1.72386e13	−0.864813	−0.432407	+	0.901679i	$0.642335\pi$
$458$	0	0
$459$	− 1.21240e13i	− 0.595090i
$460$	0	0
$461$	8.78179e11	0.0421773	0.0210886	−	0.999778i	$-0.493287\pi$
$461$	8.78179e11	0.0421773	0.0210886	+	0.999778i	$0.493287\pi$
$462$	0	0
$463$	3.73576e13i	1.75579i	0.478849	+	0.877897i	$0.341054\pi$
$463$	3.73576e13i	1.75579i	−0.478849	+	0.877897i	$0.658946\pi$
$464$	0	0
$465$	−3.18314e13	−1.46417
$466$	0	0
$467$	3.05749e13i	1.37652i	0.725466	+	0.688258i	$0.241624\pi$
$467$	3.05749e13i	1.37652i	−0.725466	+	0.688258i	$0.758376\pi$
$468$	0	0
$469$	5.25582e12	0.231620
$470$	0	0
$471$	− 4.70996e12i	− 0.203195i
$472$	0	0
$473$	−7.79201e12	−0.329112
$474$	0	0
$475$	2.80525e13i	1.16012i
$476$	0	0
$477$	1.86548e11	0.00755436
$478$	0	0
$479$	2.02856e13i	0.804470i	0.915536	+	0.402235i	$0.131767\pi$
$479$	2.02856e13i	0.804470i	−0.915536	+	0.402235i	$0.868233\pi$
$480$	0	0
$481$	−4.22883e12	−0.164246
$482$	0	0
$483$	− 3.04992e12i	− 0.116025i
$484$	0	0
$485$	9.51200e12	0.354457
$486$	0	0
$487$	2.31802e13i	0.846199i	0.906083	+	0.423100i	$0.139058\pi$
$487$	2.31802e13i	0.846199i	−0.906083	+	0.423100i	$0.860942\pi$
$488$	0	0
$489$	1.55296e13	0.555413
$490$	0	0
$491$	2.54719e13i	0.892593i	0.894885	+	0.446296i	$0.147257\pi$
$491$	2.54719e13i	0.892593i	−0.894885	+	0.446296i	$0.852743\pi$
$492$	0	0
$493$	−2.24133e13	−0.769611
$494$	0	0
$495$	− 2.42276e13i	− 0.815238i
$496$	0	0
$497$	−1.35489e13	−0.446811
$498$	0	0
$499$	− 6.30694e12i	− 0.203852i	−0.994792	−	0.101926i	$-0.967499\pi$
$499$	− 6.30694e12i	− 0.203852i	0.994792	−	0.101926i	$-0.0325005\pi$
$500$	0	0
$501$	3.65690e13	1.15858
$502$	0	0
$503$	− 5.04100e13i	− 1.56559i	−0.622283	−	0.782793i	$-0.713795\pi$
$503$	− 5.04100e13i	− 1.56559i	0.622283	−	0.782793i	$-0.286205\pi$
$504$	0	0
$505$	−5.89482e13	−1.79479
$506$	0	0
$507$	1.38458e12i	0.0413311i
$508$	0	0
$509$	6.42360e12	0.188014	0.0940068	−	0.995572i	$-0.470032\pi$
$509$	6.42360e12	0.188014	0.0940068	+	0.995572i	$0.470032\pi$
$510$	0	0
$511$	1.24808e13i	0.358209i
$512$	0	0
$513$	2.28334e13	0.642663
$514$	0	0
$515$	− 1.27533e12i	− 0.0352036i
$516$	0	0
$517$	−9.35384e13	−2.53243
$518$	0	0
$519$	− 1.45686e12i	− 0.0386886i
$520$	0	0
$521$	2.69887e13	0.703063	0.351531	−	0.936176i	$-0.385661\pi$
$521$	2.69887e13	0.703063	0.351531	+	0.936176i	$0.385661\pi$
$522$	0	0
$523$	6.68036e13i	1.70723i	0.520907	+	0.853613i	$0.325594\pi$
$523$	6.68036e13i	1.70723i	−0.520907	+	0.853613i	$0.674406\pi$
$524$	0	0
$525$	2.49625e13	0.625880
$526$	0	0
$527$	− 2.33456e13i	− 0.574318i
$528$	0	0
$529$	3.59294e13	0.867305
$530$	0	0
$531$	2.09952e13i	0.497332i
$532$	0	0
$533$	−1.70113e13	−0.395459
$534$	0	0
$535$	− 2.31255e13i	− 0.527620i
$536$	0	0
$537$	−6.08748e13	−1.36322
$538$	0	0
$539$	− 5.27827e13i	− 1.16024i
$540$	0	0
$541$	1.02428e11	0.00221021	0.00110511	−	0.999999i	$-0.499648\pi$
$541$	1.02428e11	0.00221021	0.00110511	+	0.999999i	$0.499648\pi$
$542$	0	0
$543$	− 6.92329e13i	− 1.46661i
$544$	0	0
$545$	−1.57282e13	−0.327113
$546$	0	0
$547$	− 1.84176e13i	− 0.376094i	−0.982160	−	0.188047i	$-0.939784\pi$
$547$	− 1.84176e13i	− 0.376094i	0.982160	−	0.188047i	$-0.0602158\pi$
$548$	0	0
$549$	−2.79275e12	−0.0559978
$550$	0	0
$551$	− 4.22114e13i	− 0.831135i
$552$	0	0
$553$	−6.94218e12	−0.134236
$554$	0	0
$555$	1.23742e13i	0.234992i
$556$	0	0
$557$	−7.26795e13	−1.35561	−0.677807	−	0.735240i	$-0.737069\pi$
$557$	−7.26795e13	−1.35561	−0.677807	+	0.735240i	$0.737069\pi$
$558$	0	0
$559$	− 1.27469e13i	− 0.233532i
$560$	0	0
$561$	−3.37546e13	−0.607462
$562$	0	0
$563$	1.23522e13i	0.218376i	0.994021	+	0.109188i	$0.0348250\pi$
$563$	1.23522e13i	0.218376i	−0.994021	+	0.109188i	$0.965175\pi$
$564$	0	0
$565$	1.33955e14	2.32657
$566$	0	0
$567$	− 1.23652e13i	− 0.211002i
$568$	0	0
$569$	9.79422e13	1.64213	0.821067	−	0.570832i	$-0.193379\pi$
$569$	9.79422e13	1.64213	0.821067	+	0.570832i	$0.193379\pi$
$570$	0	0
$571$	− 4.63676e13i	− 0.763895i	−0.924184	−	0.381948i	$-0.875254\pi$
$571$	− 4.63676e13i	− 0.763895i	0.924184	−	0.381948i	$-0.124746\pi$
$572$	0	0
$573$	8.32982e13	1.34854
$574$	0	0
$575$	− 4.49914e13i	− 0.715798i
$576$	0	0
$577$	−7.69976e13	−1.20392	−0.601961	−	0.798526i	$-0.705614\pi$
$577$	−7.69976e13	−1.20392	−0.601961	+	0.798526i	$0.705614\pi$
$578$	0	0
$579$	4.91811e13i	0.755798i
$580$	0	0
$581$	−2.49448e13	−0.376791
$582$	0	0
$583$	− 2.02536e12i	− 0.0300718i
$584$	0	0
$585$	3.96338e13	0.578477
$586$	0	0
$587$	− 2.22966e12i	− 0.0319926i	−0.999872	−	0.0159963i	$-0.994908\pi$
$587$	− 2.22966e12i	− 0.0319926i	0.999872	−	0.0159963i	$-0.00509199\pi$
$588$	0	0
$589$	4.39673e13	0.620230
$590$	0	0
$591$	9.55788e13i	1.32564i
$592$	0	0
$593$	4.14340e13	0.565045	0.282523	−	0.959261i	$-0.408829\pi$
$593$	4.14340e13	0.565045	0.282523	+	0.959261i	$0.408829\pi$
$594$	0	0
$595$	2.76248e13i	0.370437i
$596$	0	0
$597$	2.77367e13	0.365749
$598$	0	0
$599$	1.30467e14i	1.69187i	0.533290	+	0.845933i	$0.320955\pi$
$599$	1.30467e14i	1.69187i	−0.533290	+	0.845933i	$0.679045\pi$
$600$	0	0
$601$	−6.20993e13	−0.791980	−0.395990	−	0.918255i	$-0.629599\pi$
$601$	−6.20993e13	−0.791980	−0.395990	+	0.918255i	$0.629599\pi$
$602$	0	0
$603$	− 1.61828e13i	− 0.202987i
$604$	0	0
$605$	−1.23471e14	−1.52331
$606$	0	0
$607$	9.56403e13i	1.16064i	0.814389	+	0.580320i	$0.197072\pi$
$607$	9.56403e13i	1.16064i	−0.814389	+	0.580320i	$0.802928\pi$
$608$	0	0
$609$	−3.75617e13	−0.448393
$610$	0	0
$611$	− 1.53019e14i	− 1.79696i
$612$	0	0
$613$	−7.01720e13	−0.810702	−0.405351	−	0.914161i	$-0.632851\pi$
$613$	−7.01720e13	−0.810702	−0.405351	+	0.914161i	$0.632851\pi$
$614$	0	0
$615$	4.97778e13i	0.565796i
$616$	0	0
$617$	4.75331e13	0.531582	0.265791	−	0.964031i	$-0.414367\pi$
$617$	4.75331e13	0.531582	0.265791	+	0.964031i	$0.414367\pi$
$618$	0	0
$619$	− 7.55113e13i	− 0.830919i	−0.909612	−	0.415459i	$-0.863621\pi$
$619$	− 7.55113e13i	− 0.830919i	0.909612	−	0.415459i	$-0.136379\pi$
$620$	0	0
$621$	−3.66209e13	−0.396525
$622$	0	0
$623$	2.69406e13i	0.287057i
$624$	0	0
$625$	8.54727e13	0.896247
$626$	0	0
$627$	− 6.35707e13i	− 0.656024i
$628$	0	0
$629$	−9.07546e12	−0.0921755
$630$	0	0
$631$	7.50751e13i	0.750498i	0.926924	+	0.375249i	$0.122443\pi$
$631$	7.50751e13i	0.750498i	−0.926924	+	0.375249i	$0.877557\pi$
$632$	0	0
$633$	−8.93037e13	−0.878720
$634$	0	0
$635$	− 1.00585e14i	− 0.974241i
$636$	0	0
$637$	8.63469e13	0.823284
$638$	0	0
$639$	4.17177e13i	0.391576i
$640$	0	0
$641$	1.92038e14	1.77458	0.887291	−	0.461210i	$-0.152584\pi$
$641$	1.92038e14	1.77458	0.887291	+	0.461210i	$0.152584\pi$
$642$	0	0
$643$	− 1.14745e14i	− 1.04395i	−0.852961	−	0.521975i	$-0.825195\pi$
$643$	− 1.14745e14i	− 1.04395i	0.852961	−	0.521975i	$-0.174805\pi$
$644$	0	0
$645$	−3.72995e13	−0.334121
$646$	0	0
$647$	− 5.82025e13i	− 0.513358i	−0.966497	−	0.256679i	$-0.917372\pi$
$647$	− 5.82025e13i	− 0.513358i	0.966497	−	0.256679i	$-0.0826284\pi$
$648$	0	0
$649$	2.27946e14	1.97974
$650$	0	0
$651$	− 3.91242e13i	− 0.334611i
$652$	0	0
$653$	1.62553e14	1.36908	0.684542	−	0.728974i	$-0.260002\pi$
$653$	1.62553e14	1.36908	0.684542	+	0.728974i	$0.260002\pi$
$654$	0	0
$655$	− 8.76052e13i	− 0.726647i
$656$	0	0
$657$	3.84287e13	0.313927
$658$	0	0
$659$	1.20273e14i	0.967704i	0.875150	+	0.483852i	$0.160763\pi$
$659$	1.20273e14i	0.967704i	−0.875150	+	0.483852i	$0.839237\pi$
$660$	0	0
$661$	−1.23883e14	−0.981758	−0.490879	−	0.871228i	$-0.663324\pi$
$661$	−1.23883e14	−0.981758	−0.490879	+	0.871228i	$0.663324\pi$
$662$	0	0
$663$	− 5.52190e13i	− 0.431043i
$664$	0	0
$665$	−5.20263e13	−0.400050
$666$	0	0
$667$	6.76999e13i	0.512813i
$668$	0	0
$669$	−1.39684e14	−1.04235
$670$	0	0
$671$	3.03211e13i	0.222911i
$672$	0	0
$673$	7.76604e13	0.562502	0.281251	−	0.959634i	$-0.409251\pi$
$673$	7.76604e13	0.562502	0.281251	+	0.959634i	$0.409251\pi$
$674$	0	0
$675$	− 2.99728e14i	− 2.13899i
$676$	0	0
$677$	1.79880e14	1.26485	0.632426	−	0.774621i	$-0.282059\pi$
$677$	1.79880e14	1.26485	0.632426	+	0.774621i	$0.282059\pi$
$678$	0	0
$679$	1.16913e13i	0.0810053i
$680$	0	0
$681$	−3.46157e13	−0.236340
$682$	0	0
$683$	1.46057e13i	0.0982698i	0.998792	+	0.0491349i	$0.0156464\pi$
$683$	1.46057e13i	0.0982698i	−0.998792	+	0.0491349i	$0.984354\pi$
$684$	0	0
$685$	−1.74721e14	−1.15849
$686$	0	0
$687$	7.79544e13i	0.509398i
$688$	0	0
$689$	3.31327e12	0.0213384
$690$	0	0
$691$	− 1.80286e14i	− 1.14438i	−0.820120	−	0.572191i	$-0.806094\pi$
$691$	− 1.80286e14i	− 1.14438i	0.820120	−	0.572191i	$-0.193906\pi$
$692$	0	0
$693$	2.97783e13	0.186309
$694$	0	0
$695$	− 1.75293e14i	− 1.08104i
$696$	0	0
$697$	−3.65078e13	−0.221933
$698$	0	0
$699$	− 1.79943e14i	− 1.07832i
$700$	0	0
$701$	2.61562e14	1.54520	0.772598	−	0.634895i	$-0.218957\pi$
$701$	2.61562e14	1.54520	0.772598	+	0.634895i	$0.218957\pi$
$702$	0	0
$703$	− 1.70920e13i	− 0.0995442i
$704$	0	0
$705$	−4.47758e14	−2.57097
$706$	0	0
$707$	− 7.24537e13i	− 0.410170i
$708$	0	0
$709$	2.37281e14	1.32444	0.662219	−	0.749310i	$-0.269615\pi$
$709$	2.37281e14	1.32444	0.662219	+	0.749310i	$0.269615\pi$
$710$	0	0
$711$	2.13752e13i	0.117642i
$712$	0	0
$713$	−7.05161e13	−0.382684
$714$	0	0
$715$	− 4.30307e14i	− 2.30276i
$716$	0	0
$717$	−3.91890e13	−0.206808
$718$	0	0
$719$	− 2.61945e14i	− 1.36322i	−0.731717	−	0.681609i	$-0.761281\pi$
$719$	− 2.61945e14i	− 1.36322i	0.731717	−	0.681609i	$-0.238719\pi$
$720$	0	0
$721$	1.56752e12	0.00804521
$722$	0	0
$723$	1.02479e14i	0.518733i
$724$	0	0
$725$	−5.54098e14	−2.76628
$726$	0	0
$727$	− 1.58213e14i	− 0.779058i	−0.921014	−	0.389529i	$-0.872638\pi$
$727$	− 1.58213e14i	− 0.779058i	0.921014	−	0.389529i	$-0.127362\pi$
$728$	0	0
$729$	−2.19476e14	−1.06598
$730$	0	0
$731$	− 2.73560e13i	− 0.131059i
$732$	0	0
$733$	−2.50230e14	−1.18255	−0.591276	−	0.806469i	$-0.701376\pi$
$733$	−2.50230e14	−1.18255	−0.591276	+	0.806469i	$0.701376\pi$
$734$	0	0
$735$	− 2.52665e14i	− 1.17790i
$736$	0	0
$737$	−1.75698e14	−0.808035
$738$	0	0
$739$	3.08452e14i	1.39948i	0.714400	+	0.699738i	$0.246700\pi$
$739$	3.08452e14i	1.39948i	−0.714400	+	0.699738i	$0.753300\pi$
$740$	0	0
$741$	1.03995e14	0.465502
$742$	0	0
$743$	2.42121e14i	1.06927i	0.845082	+	0.534637i	$0.179552\pi$
$743$	2.42121e14i	1.06927i	−0.845082	+	0.534637i	$0.820448\pi$
$744$	0	0
$745$	4.99431e14	2.17618
$746$	0	0
$747$	7.68060e13i	0.330211i
$748$	0	0
$749$	2.84237e13	0.120579
$750$	0	0
$751$	− 2.94510e14i	− 1.23282i	−0.787424	−	0.616412i	$-0.788586\pi$
$751$	− 2.94510e14i	− 1.23282i	0.787424	−	0.616412i	$-0.211414\pi$
$752$	0	0
$753$	1.72474e14	0.712442
$754$	0	0
$755$	− 8.72015e13i	− 0.355459i
$756$	0	0
$757$	2.90155e14	1.16721	0.583607	−	0.812036i	$-0.301641\pi$
$757$	2.90155e14	1.16721	0.583607	+	0.812036i	$0.301641\pi$
$758$	0	0
$759$	1.01957e14i	0.404769i
$760$	0	0
$761$	−4.11236e14	−1.61127	−0.805635	−	0.592412i	$-0.798176\pi$
$761$	−4.11236e14	−1.61127	−0.805635	+	0.592412i	$0.798176\pi$
$762$	0	0
$763$	− 1.93317e13i	− 0.0747562i
$764$	0	0
$765$	8.50577e13	0.324643
$766$	0	0
$767$	3.72895e14i	1.40479i
$768$	0	0
$769$	−1.42100e14	−0.528400	−0.264200	−	0.964468i	$-0.585108\pi$
$769$	−1.42100e14	−0.528400	−0.264200	+	0.964468i	$0.585108\pi$
$770$	0	0
$771$	− 1.33213e14i	− 0.488961i
$772$	0	0
$773$	2.58504e14	0.936635	0.468317	−	0.883560i	$-0.344860\pi$
$773$	2.58504e14	0.936635	0.468317	+	0.883560i	$0.344860\pi$
$774$	0	0
$775$	− 5.77147e14i	− 2.06433i
$776$	0	0
$777$	−1.52093e13	−0.0537036
$778$	0	0
$779$	− 6.87559e13i	− 0.239675i
$780$	0	0
$781$	4.52932e14	1.55875
$782$	0	0
$783$	4.51009e14i	1.53242i
$784$	0	0
$785$	1.28858e14	0.432277
$786$	0	0
$787$	− 4.24394e14i	− 1.40571i	−0.711333	−	0.702855i	$-0.751908\pi$
$787$	− 4.24394e14i	− 1.40571i	0.711333	−	0.702855i	$-0.248092\pi$
$788$	0	0
$789$	−4.69892e14	−1.53678
$790$	0	0
$791$	1.64645e14i	0.531700i
$792$	0	0
$793$	−4.96021e13	−0.158174
$794$	0	0
$795$	− 9.69516e12i	− 0.0305295i
$796$	0	0
$797$	−3.84509e14	−1.19568	−0.597839	−	0.801616i	$-0.703974\pi$
$797$	−3.84509e14	−1.19568	−0.597839	+	0.801616i	$0.703974\pi$
$798$	0	0
$799$	− 3.28393e14i	− 1.00846i
$800$	0	0
$801$	8.29511e13	0.251570
$802$	0	0
$803$	− 4.17223e14i	− 1.24966i
$804$	0	0
$805$	8.34414e13	0.246832
$806$	0	0
$807$	1.80050e14i	0.526048i
$808$	0	0
$809$	2.91643e14	0.841606	0.420803	−	0.907152i	$-0.361748\pi$
$809$	2.91643e14	0.841606	0.420803	+	0.907152i	$0.361748\pi$
$810$	0	0
$811$	1.84338e14i	0.525424i	0.964874	+	0.262712i	$0.0846169\pi$
$811$	1.84338e14i	0.525424i	−0.964874	+	0.262712i	$0.915383\pi$
$812$	0	0
$813$	1.37908e14	0.388271
$814$	0	0
$815$	4.24867e14i	1.18159i
$816$	0	0
$817$	5.15201e13	0.141536
$818$	0	0
$819$	4.87142e13i	0.132201i
$820$	0	0
$821$	3.88449e14	1.04140	0.520700	−	0.853740i	$-0.325671\pi$
$821$	3.88449e14	1.04140	0.520700	+	0.853740i	$0.325671\pi$
$822$	0	0
$823$	2.91752e14i	0.772707i	0.922351	+	0.386354i	$0.126266\pi$
$823$	2.91752e14i	0.772707i	−0.922351	+	0.386354i	$0.873734\pi$
$824$	0	0
$825$	−8.34477e14	−2.18346
$826$	0	0
$827$	2.48060e14i	0.641253i	0.947206	+	0.320626i	$0.103893\pi$
$827$	2.48060e14i	0.641253i	−0.947206	+	0.320626i	$0.896107\pi$
$828$	0	0
$829$	−1.64039e14	−0.418962	−0.209481	−	0.977813i	$-0.567177\pi$
$829$	−1.64039e14	−0.418962	−0.209481	+	0.977813i	$0.567177\pi$
$830$	0	0
$831$	4.14656e14i	1.04637i
$832$	0	0
$833$	1.85308e14	0.462030
$834$	0	0
$835$	1.00048e15i	2.46476i
$836$	0	0
$837$	−4.69770e14	−1.14356
$838$	0	0
$839$	5.19240e14i	1.24899i	0.781030	+	0.624494i	$0.214695\pi$
$839$	5.19240e14i	1.24899i	−0.781030	+	0.624494i	$0.785305\pi$
$840$	0	0
$841$	4.13060e14	0.981824
$842$	0	0
$843$	5.65628e14i	1.32860i
$844$	0	0
$845$	−3.78800e13	−0.0879278
$846$	0	0
$847$	− 1.51760e14i	− 0.348128i
$848$	0	0
$849$	−4.18231e14	−0.948152
$850$	0	0
$851$	2.74127e13i	0.0614191i
$852$	0	0
$853$	−8.40549e14	−1.86131	−0.930653	−	0.365903i	$-0.880760\pi$
$853$	−8.40549e14	−1.86131	−0.930653	+	0.365903i	$0.880760\pi$
$854$	0	0
$855$	1.60191e14i	0.350596i
$856$	0	0
$857$	−9.08205e13	−0.196463	−0.0982313	−	0.995164i	$-0.531319\pi$
$857$	−9.08205e13	−0.196463	−0.0982313	+	0.995164i	$0.531319\pi$
$858$	0	0
$859$	− 5.90738e14i	− 1.26307i	−0.775346	−	0.631537i	$-0.782424\pi$
$859$	− 5.90738e14i	− 1.26307i	0.775346	−	0.631537i	$-0.217576\pi$
$860$	0	0
$861$	−6.11823e13	−0.129303
$862$	0	0
$863$	5.07805e14i	1.06082i	0.847740	+	0.530411i	$0.177963\pi$
$863$	5.07805e14i	1.06082i	−0.847740	+	0.530411i	$0.822037\pi$
$864$	0	0
$865$	3.98577e13	0.0823060
$866$	0	0
$867$	2.78010e14i	0.567498i
$868$	0	0
$869$	2.32072e14	0.468300
$870$	0	0
$871$	− 2.87424e14i	− 0.573366i
$872$	0	0
$873$	3.59979e13	0.0709913
$874$	0	0
$875$	3.35387e14i	0.653892i
$876$	0	0
$877$	−2.07322e14	−0.399620	−0.199810	−	0.979835i	$-0.564032\pi$
$877$	−2.07322e14	−0.399620	−0.199810	+	0.979835i	$0.564032\pi$
$878$	0	0
$879$	− 1.18946e14i	− 0.226676i
$880$	0	0
$881$	3.62994e14	0.683943	0.341971	−	0.939710i	$-0.388905\pi$
$881$	3.62994e14	0.683943	0.341971	+	0.939710i	$0.388905\pi$
$882$	0	0
$883$	5.88143e12i	0.0109567i	0.999985	+	0.00547835i	$0.00174382\pi$
$883$	5.88143e12i	0.0109567i	−0.999985	+	0.00547835i	$0.998256\pi$
$884$	0	0
$885$	1.09115e15	2.00987
$886$	0	0
$887$	− 4.25024e13i	− 0.0774096i	−0.999251	−	0.0387048i	$-0.987677\pi$
$887$	− 4.25024e13i	− 0.0774096i	0.999251	−	0.0387048i	$-0.0123232\pi$
$888$	0	0
$889$	1.23630e14	0.222647
$890$	0	0
$891$	4.13360e14i	0.736105i
$892$	0	0
$893$	6.18468e14	1.08908
$894$	0	0
$895$	− 1.66545e15i	− 2.90012i
$896$	0	0
$897$	−1.66790e14	−0.287216
$898$	0	0
$899$	8.68450e14i	1.47893i
$900$	0	0
$901$	7.11058e12	0.0119752
$902$	0	0
$903$	− 4.58451e13i	− 0.0763579i
$904$	0	0
$905$	1.89411e15	3.12006
$906$	0	0
$907$	− 2.03986e14i	− 0.332326i	−0.986098	−	0.166163i	$-0.946862\pi$
$907$	− 2.03986e14i	− 0.332326i	0.986098	−	0.166163i	$-0.0531378\pi$
$908$	0	0
$909$	−2.23087e14	−0.359464
$910$	0	0
$911$	1.49719e14i	0.238608i	0.992858	+	0.119304i	$0.0380664\pi$
$911$	1.49719e14i	0.238608i	−0.992858	+	0.119304i	$0.961934\pi$
$912$	0	0
$913$	8.33888e14	1.31448
$914$	0	0
$915$	1.45144e14i	0.226304i
$916$	0	0
$917$	1.07676e14	0.166063
$918$	0	0
$919$	− 9.31781e14i	− 1.42146i	−0.703463	−	0.710732i	$-0.748364\pi$
$919$	− 9.31781e14i	− 1.42146i	0.703463	−	0.710732i	$-0.251636\pi$
$920$	0	0
$921$	−6.67524e13	−0.100732
$922$	0	0
$923$	7.40948e14i	1.10606i
$924$	0	0
$925$	−2.24362e14	−0.331315
$926$	0	0
$927$	− 4.82645e12i	− 0.00705066i
$928$	0	0
$929$	−9.29163e14	−1.34281	−0.671403	−	0.741093i	$-0.734308\pi$
$929$	−9.29163e14	−1.34281	−0.671403	+	0.741093i	$0.734308\pi$
$930$	0	0
$931$	3.48995e14i	0.498965i
$932$	0	0
$933$	−9.42498e13	−0.133313
$934$	0	0
$935$	− 9.23477e14i	− 1.29231i
$936$	0	0
$937$	3.08132e14	0.426618	0.213309	−	0.976985i	$-0.431576\pi$
$937$	3.08132e14	0.426618	0.213309	+	0.976985i	$0.431576\pi$
$938$	0	0
$939$	− 6.19798e14i	− 0.849029i
$940$	0	0
$941$	−6.11614e14	−0.828952	−0.414476	−	0.910060i	$-0.636035\pi$
$941$	−6.11614e14	−0.828952	−0.414476	+	0.910060i	$0.636035\pi$
$942$	0	0
$943$	1.10273e14i	0.147880i
$944$	0	0
$945$	5.55877e14	0.737598
$946$	0	0
$947$	7.23883e14i	0.950427i	0.879871	+	0.475213i	$0.157629\pi$
$947$	7.23883e14i	0.950427i	−0.879871	+	0.475213i	$0.842371\pi$
$948$	0	0
$949$	6.82532e14	0.886732
$950$	0	0
$951$	− 1.42437e14i	− 0.183113i
$952$	0	0
$953$	−1.17221e15	−1.49122	−0.745608	−	0.666385i	$-0.767841\pi$
$953$	−1.17221e15	−1.49122	−0.745608	+	0.666385i	$0.767841\pi$
$954$	0	0
$955$	2.27892e15i	2.86887i
$956$	0	0
$957$	1.25566e15	1.56427
$958$	0	0
$959$	− 2.14751e14i	− 0.264754i
$960$	0	0
$961$	−8.49472e13	−0.103641
$962$	0	0
$963$	− 8.75175e13i	− 0.105673i
$964$	0	0
$965$	−1.34552e15	−1.60788
$966$	0	0
$967$	1.04445e15i	1.23525i	0.786471	+	0.617627i	$0.211906\pi$
$967$	1.04445e15i	1.23525i	−0.786471	+	0.617627i	$0.788094\pi$
$968$	0	0
$969$	2.23183e14	0.261241
$970$	0	0
$971$	3.97256e13i	0.0460230i	0.999735	+	0.0230115i	$0.00732543\pi$
$971$	3.97256e13i	0.0460230i	−0.999735	+	0.0230115i	$0.992675\pi$
$972$	0	0
$973$	2.15454e14	0.247053
$974$	0	0
$975$	− 1.36512e15i	− 1.54934i
$976$	0	0
$977$	1.61456e15	1.81377	0.906884	−	0.421381i	$-0.138454\pi$
$977$	1.61456e15	1.81377	0.906884	+	0.421381i	$0.138454\pi$
$978$	0	0
$979$	− 9.00606e14i	− 1.00143i
$980$	0	0
$981$	−5.95230e13	−0.0655148
$982$	0	0
$983$	1.09516e14i	0.119319i	0.998219	+	0.0596597i	$0.0190016\pi$
$983$	1.09516e14i	0.119319i	−0.998219	+	0.0596597i	$0.980998\pi$
$984$	0	0
$985$	−2.61490e15	−2.82016
$986$	0	0
$987$	− 5.50343e14i	− 0.587554i
$988$	0	0
$989$	−8.26296e13	−0.0873281
$990$	0	0
$991$	1.73875e15i	1.81915i	0.415536	+	0.909577i	$0.363594\pi$
$991$	1.73875e15i	1.81915i	−0.415536	+	0.909577i	$0.636406\pi$
$992$	0	0
$993$	−3.71778e14	−0.385068
$994$	0	0
$995$	7.58836e14i	0.778094i
$996$	0	0
$997$	−1.61791e15	−1.64240	−0.821200	−	0.570641i	$-0.806695\pi$
$997$	−1.61791e15	−1.64240	−0.821200	+	0.570641i	$0.806695\pi$
$998$	0	0
$999$	1.82620e14i	0.183536i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.m.255.6		16
4.3	odd	2	inner	256.11.c.m.255.12		16
8.3	odd	2	inner	256.11.c.m.255.5		16
8.5	even	2	inner	256.11.c.m.255.11		16
16.3	odd	4		8.11.d.b.3.7	✓	8
16.5	even	4		8.11.d.b.3.8	yes	8
16.11	odd	4		32.11.d.b.15.8		8
16.13	even	4		32.11.d.b.15.7		8
48.5	odd	4		72.11.b.b.19.1		8
48.11	even	4		288.11.b.b.271.1		8
48.29	odd	4		288.11.b.b.271.8		8
48.35	even	4		72.11.b.b.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.11.d.b.3.7	✓	8	16.3	odd	4
8.11.d.b.3.8	yes	8	16.5	even	4
32.11.d.b.15.7		8	16.13	even	4
32.11.d.b.15.8		8	16.11	odd	4
72.11.b.b.19.1		8	48.5	odd	4
72.11.b.b.19.2		8	48.35	even	4
256.11.c.m.255.5		16	8.3	odd	2	inner
256.11.c.m.255.6		16	1.1	even	1	trivial
256.11.c.m.255.11		16	8.5	even	2	inner
256.11.c.m.255.12		16	4.3	odd	2	inner
288.11.b.b.271.1		8	48.11	even	4
288.11.b.b.271.8		8	48.29	odd	4

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)

\(f(q)\)	\(=\)	\(q-196.684i q^{3} +5381.00 q^{5} +6613.83i q^{7} +20364.2 q^{9} -221096. i q^{11} +361689. q^{13} -1.05836e6i q^{15} +776217. q^{17} +1.46186e6i q^{19} +1.30084e6 q^{21} -2.34458e6i q^{23} +1.91895e7 q^{25} -1.56193e7i q^{27} -2.88750e7 q^{29} -3.00762e7i q^{31} -4.34861e7 q^{33} +3.55890e7i q^{35} -1.16919e7 q^{37} -7.11386e7i q^{39} -4.70330e7 q^{41} -3.52427e7i q^{43} +1.09580e8 q^{45} -4.23068e8i q^{47} +2.38733e8 q^{49} -1.52670e8i q^{51} +9.16056e6 q^{53} -1.18971e9i q^{55} +2.87526e8 q^{57} +1.03098e9i q^{59} -1.37140e8 q^{61} +1.34685e8i q^{63} +1.94625e9 q^{65} -7.94671e8i q^{67} -4.61143e8 q^{69} +2.04858e9i q^{71} +1.88707e9 q^{73} -3.77428e9i q^{75} +1.46229e9 q^{77} +1.04965e9i q^{79} -1.86960e9 q^{81} +3.77162e9i q^{83} +4.17682e9 q^{85} +5.67927e9i q^{87} +4.07338e9 q^{89} +2.39215e9i q^{91} -5.91551e9 q^{93} +7.86629e9i q^{95} +1.76770e9 q^{97} -4.50244e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 70992 q^{9} - 741600 q^{17} + 37585520 q^{25} - 148617408 q^{33} - 185338656 q^{41} - 160864496 q^{49} - 2801425728 q^{57} - 1678056960 q^{65} - 5600145440 q^{73} - 23818130352 q^{81} - 12325192224 q^{89}+ \cdots - 19395727072 q^{97}+O(q^{100}) \) 16 * q + 70992 * q^9 - 741600 * q^17 + 37585520 * q^25 - 148617408 * q^33 - 185338656 * q^41 - 160864496 * q^49 - 2801425728 * q^57 - 1678056960 * q^65 - 5600145440 * q^73 - 23818130352 * q^81 - 12325192224 * q^89 - 19395727072 * q^97

Introduction

L-functions

Modular forms

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Database

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