LMFDB - Embedded newform 256.11.c.l.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5593 x^{10} + 23492833 x^{8} - 43148127888 x^{6} + 59505890201856 x^{4} + \cdots + 42\!\cdots\!00 $ x^12 - 5593x^10 + 23492833x^8 - 43148127888x^6 + 59505890201856x^4 - 1615088885760000*x^2 + 42998169600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{96}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.10
Root			$48.2958 + 27.8836i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.l.255.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+355.190i q^{3} +2040.14 q^{5} +4614.42i q^{7} -67110.8 q^{9} +O(q^{10})$	$q+355.190i q^{3} +2040.14 q^{5} +4614.42i q^{7} -67110.8 q^{9} +17406.8i q^{11} +397731. q^{13} +724636. i q^{15} +2.37293e6 q^{17} +3.47998e6i q^{19} -1.63900e6 q^{21} -1.17512e7i q^{23} -5.60346e6 q^{25} -2.86346e6i q^{27} -3.22215e6 q^{29} -3.93679e7i q^{31} -6.18270e6 q^{33} +9.41406e6i q^{35} +1.05379e8 q^{37} +1.41270e8i q^{39} +7.77366e7 q^{41} -2.67293e6i q^{43} -1.36915e8 q^{45} +3.57121e7i q^{47} +2.61182e8 q^{49} +8.42841e8i q^{51} +6.33154e8 q^{53} +3.55122e7i q^{55} -1.23605e9 q^{57} -7.39465e8i q^{59} +7.02430e8 q^{61} -3.09677e8i q^{63} +8.11427e8 q^{65} +8.47285e8i q^{67} +4.17392e9 q^{69} +1.48372e9i q^{71} +1.11121e9 q^{73} -1.99029e9i q^{75} -8.03221e7 q^{77} -3.95504e9i q^{79} -2.94575e9 q^{81} -2.54010e9i q^{83} +4.84111e9 q^{85} -1.14448e9i q^{87} +6.76760e8 q^{89} +1.83530e9i q^{91} +1.39831e10 q^{93} +7.09964e9i q^{95} +1.01191e10 q^{97} -1.16818e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 375204 q^{9}+O(q^{10}) $ 12 * q - 375204 * q^9	$ 12 q - 375204 q^{9} - 1639752 q^{17} + 28703892 q^{25} + 358214544 q^{33} - 496702488 q^{41} + 1385670732 q^{49} - 5733526800 q^{57} + 8895346752 q^{65} - 12161297592 q^{73} - 7708157604 q^{81} + 5871785160 q^{89} + 30465806904 q^{97}+O(q^{100}) $ 12 * q - 375204 * q^9 - 1639752 * q^17 + 28703892 * q^25 + 358214544 * q^33 - 496702488 * q^41 + 1385670732 * q^49 - 5733526800 * q^57 + 8895346752 * q^65 - 12161297592 * q^73 - 7708157604 * q^81 + 5871785160 * q^89 + 30465806904 * 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
$3$	355.190i	1.46169i	0.682545	+	0.730843i	$0.260873\pi$
$3$	355.190i	1.46169i	−0.682545	+	0.730843i	$0.739127\pi$
$4$	0	0
$5$	2040.14	0.652845	0.326422	−	0.945224i	$-0.394157\pi$
$5$	2040.14	0.652845	0.326422	+	0.945224i	$0.394157\pi$
$6$	0	0
$7$	4614.42i	0.274554i	0.990533	+	0.137277i	$0.0438350\pi$
$7$	4614.42i	0.274554i	−0.990533	+	0.137277i	$0.956165\pi$
$8$	0	0
$9$	−67110.8	−1.13653
$10$	0	0
$11$	17406.8i	0.108082i	0.998539	+	0.0540411i	$0.0172102\pi$
$11$	17406.8i	0.108082i	−0.998539	+	0.0540411i	$0.982790\pi$
$12$	0	0
$13$	397731.	1.07121	0.535603	−	0.844470i	$-0.320084\pi$
$13$	397731.	1.07121	0.535603	+	0.844470i	$0.320084\pi$
$14$	0	0
$15$	724636.i	0.954254i
$16$	0	0
$17$	2.37293e6	1.67125	0.835623	−	0.549303i	$-0.185107\pi$
$17$	2.37293e6	1.67125	0.835623	+	0.549303i	$0.185107\pi$
$18$	0	0
$19$	3.47998e6i	1.40543i	0.711472	+	0.702714i	$0.248029\pi$
$19$	3.47998e6i	1.40543i	−0.711472	+	0.702714i	$0.751971\pi$
$20$	0	0
$21$	−1.63900e6	−0.401311
$22$	0	0
$23$	− 1.17512e7i	− 1.82576i	−0.408224	−	0.912882i	$-0.633852\pi$
$23$	− 1.17512e7i	− 1.82576i	0.408224	−	0.912882i	$-0.366148\pi$
$24$	0	0
$25$	−5.60346e6	−0.573794
$26$	0	0
$27$	− 2.86346e6i	− 0.199559i
$28$	0	0
$29$	−3.22215e6	−0.157093	−0.0785463	−	0.996910i	$-0.525028\pi$
$29$	−3.22215e6	−0.157093	−0.0785463	+	0.996910i	$0.525028\pi$
$30$	0	0
$31$	− 3.93679e7i	− 1.37510i	−0.726138	−	0.687549i	$-0.758687\pi$
$31$	− 3.93679e7i	− 1.37510i	0.726138	−	0.687549i	$-0.241313\pi$
$32$	0	0
$33$	−6.18270e6	−0.157982
$34$	0	0
$35$	9.41406e6i	0.179241i
$36$	0	0
$37$	1.05379e8	1.51966	0.759830	−	0.650122i	$-0.225282\pi$
$37$	1.05379e8	1.51966	0.759830	+	0.650122i	$0.225282\pi$
$38$	0	0
$39$	1.41270e8i	1.56577i
$40$	0	0
$41$	7.77366e7	0.670975	0.335488	−	0.942045i	$-0.391099\pi$
$41$	7.77366e7	0.670975	0.335488	+	0.942045i	$0.391099\pi$
$42$	0	0
$43$	− 2.67293e6i	− 0.0181822i	−0.999959	−	0.00909108i	$-0.997106\pi$
$43$	− 2.67293e6i	− 0.0181822i	0.999959	−	0.00909108i	$-0.00289382\pi$
$44$	0	0
$45$	−1.36915e8	−0.741975
$46$	0	0
$47$	3.57121e7i	0.155714i	0.996965	+	0.0778568i	$0.0248077\pi$
$47$	3.57121e7i	0.155714i	−0.996965	+	0.0778568i	$0.975192\pi$
$48$	0	0
$49$	2.61182e8	0.924620
$50$	0	0
$51$	8.42841e8i	2.44284i
$52$	0	0
$53$	6.33154e8	1.51401	0.757007	−	0.653407i	$-0.226661\pi$
$53$	6.33154e8	1.51401	0.757007	+	0.653407i	$0.226661\pi$
$54$	0	0
$55$	3.55122e7i	0.0705609i
$56$	0	0
$57$	−1.23605e9	−2.05429
$58$	0	0
$59$	− 7.39465e8i	− 1.03433i	−0.855887	−	0.517163i	$-0.826988\pi$
$59$	− 7.39465e8i	− 1.03433i	0.855887	−	0.517163i	$-0.173012\pi$
$60$	0	0
$61$	7.02430e8	0.831675	0.415838	−	0.909439i	$-0.363488\pi$
$61$	7.02430e8	0.831675	0.415838	+	0.909439i	$0.363488\pi$
$62$	0	0
$63$	− 3.09677e8i	− 0.312037i
$64$	0	0
$65$	8.11427e8	0.699331
$66$	0	0
$67$	8.47285e8i	0.627561i	0.949496	+	0.313780i	$0.101596\pi$
$67$	8.47285e8i	0.627561i	−0.949496	+	0.313780i	$0.898404\pi$
$68$	0	0
$69$	4.17392e9	2.66869
$70$	0	0
$71$	1.48372e9i	0.822357i	0.911555	+	0.411178i	$0.134883\pi$
$71$	1.48372e9i	0.822357i	−0.911555	+	0.411178i	$0.865117\pi$
$72$	0	0
$73$	1.11121e9	0.536022	0.268011	−	0.963416i	$-0.413634\pi$
$73$	1.11121e9	0.536022	0.268011	+	0.963416i	$0.413634\pi$
$74$	0	0
$75$	− 1.99029e9i	− 0.838707i
$76$	0	0
$77$	−8.03221e7	−0.0296744
$78$	0	0
$79$	− 3.95504e9i	− 1.28533i	−0.766147	−	0.642666i	$-0.777828\pi$
$79$	− 3.95504e9i	− 1.28533i	0.766147	−	0.642666i	$-0.222172\pi$
$80$	0	0
$81$	−2.94575e9	−0.844834
$82$	0	0
$83$	− 2.54010e9i	− 0.644853i	−0.946594	−	0.322427i	$-0.895501\pi$
$83$	− 2.54010e9i	− 0.644853i	0.946594	−	0.322427i	$-0.104499\pi$
$84$	0	0
$85$	4.84111e9	1.09106
$86$	0	0
$87$	− 1.14448e9i	− 0.229620i
$88$	0	0
$89$	6.76760e8	0.121195	0.0605975	−	0.998162i	$-0.480699\pi$
$89$	6.76760e8	0.121195	0.0605975	+	0.998162i	$0.480699\pi$
$90$	0	0
$91$	1.83530e9i	0.294103i
$92$	0	0
$93$	1.39831e10	2.00996
$94$	0	0
$95$	7.09964e9i	0.917526i
$96$	0	0
$97$	1.01191e10	1.17837	0.589185	−	0.807998i	$-0.299449\pi$
$97$	1.01191e10	1.17837	0.589185	+	0.807998i	$0.299449\pi$
$98$	0	0
$99$	− 1.16818e9i	− 0.122838i
$100$	0	0
$101$	−8.45287e8	−0.0804262	−0.0402131	−	0.999191i	$-0.512804\pi$
$101$	−8.45287e8	−0.0804262	−0.0402131	+	0.999191i	$0.512804\pi$
$102$	0	0
$103$	− 3.10392e9i	− 0.267747i	−0.990998	−	0.133873i	$-0.957258\pi$
$103$	− 3.10392e9i	− 0.267747i	0.990998	−	0.133873i	$-0.0427416\pi$
$104$	0	0
$105$	−3.34378e9	−0.261994
$106$	0	0
$107$	2.88942e9i	0.206012i	0.994681	+	0.103006i	$0.0328460\pi$
$107$	2.88942e9i	0.206012i	−0.994681	+	0.103006i	$0.967154\pi$
$108$	0	0
$109$	2.12831e10	1.38325	0.691627	−	0.722255i	$-0.256894\pi$
$109$	2.12831e10	1.38325	0.691627	+	0.722255i	$0.256894\pi$
$110$	0	0
$111$	3.74296e10i	2.22127i
$112$	0	0
$113$	−2.73541e10	−1.48467	−0.742335	−	0.670029i	$-0.766282\pi$
$113$	−2.73541e10	−1.48467	−0.742335	+	0.670029i	$0.766282\pi$
$114$	0	0
$115$	− 2.39742e10i	− 1.19194i
$116$	0	0
$117$	−2.66920e10	−1.21745
$118$	0	0
$119$	1.09497e10i	0.458847i
$120$	0	0
$121$	2.56344e10	0.988318
$122$	0	0
$123$	2.76113e10i	0.980755i
$124$	0	0
$125$	−3.13551e10	−1.02744
$126$	0	0
$127$	− 4.53285e10i	− 1.37199i	−0.727604	−	0.685997i	$-0.759366\pi$
$127$	− 4.53285e10i	− 1.37199i	0.727604	−	0.685997i	$-0.240634\pi$
$128$	0	0
$129$	9.49397e8	0.0265766
$130$	0	0
$131$	− 1.45468e10i	− 0.377061i	−0.982067	−	0.188530i	$-0.939628\pi$
$131$	− 1.45468e10i	− 0.377061i	0.982067	−	0.188530i	$-0.0603724\pi$
$132$	0	0
$133$	−1.60581e10	−0.385865
$134$	0	0
$135$	− 5.84185e9i	− 0.130281i
$136$	0	0
$137$	2.95364e10	0.612004	0.306002	−	0.952031i	$-0.401009\pi$
$137$	2.95364e10	0.612004	0.306002	+	0.952031i	$0.401009\pi$
$138$	0	0
$139$	4.50816e10i	0.868811i	0.900717	+	0.434406i	$0.143042\pi$
$139$	4.50816e10i	0.868811i	−0.900717	+	0.434406i	$0.856958\pi$
$140$	0	0
$141$	−1.26846e10	−0.227604
$142$	0	0
$143$	6.92321e9i	0.115778i
$144$	0	0
$145$	−6.57364e9	−0.102557
$146$	0	0
$147$	9.27693e10i	1.35150i
$148$	0	0
$149$	−7.60264e10	−1.03522	−0.517610	−	0.855616i	$-0.673178\pi$
$149$	−7.60264e10	−1.03522	−0.517610	+	0.855616i	$0.673178\pi$
$150$	0	0
$151$	2.75147e10i	0.350494i	0.984524	+	0.175247i	$0.0560724\pi$
$151$	2.75147e10i	0.350494i	−0.984524	+	0.175247i	$0.943928\pi$
$152$	0	0
$153$	−1.59249e11	−1.89942
$154$	0	0
$155$	− 8.03159e10i	− 0.897725i
$156$	0	0
$157$	−1.81631e11	−1.90410	−0.952052	−	0.305936i	$-0.901031\pi$
$157$	−1.81631e11	−1.90410	−0.952052	+	0.305936i	$0.901031\pi$
$158$	0	0
$159$	2.24890e11i	2.21301i
$160$	0	0
$161$	5.42252e10	0.501270
$162$	0	0
$163$	1.18063e11i	1.02607i	0.858367	+	0.513035i	$0.171479\pi$
$163$	1.18063e11i	1.02607i	−0.858367	+	0.513035i	$0.828521\pi$
$164$	0	0
$165$	−1.26136e10	−0.103138
$166$	0	0
$167$	2.31920e11i	1.78549i	0.450565	+	0.892744i	$0.351223\pi$
$167$	2.31920e11i	1.78549i	−0.450565	+	0.892744i	$0.648777\pi$
$168$	0	0
$169$	2.03315e10	0.147481
$170$	0	0
$171$	− 2.33544e11i	− 1.59731i
$172$	0	0
$173$	−1.57528e11	−1.01655	−0.508274	−	0.861196i	$-0.669716\pi$
$173$	−1.57528e11	−1.01655	−0.508274	+	0.861196i	$0.669716\pi$
$174$	0	0
$175$	− 2.58567e10i	− 0.157537i
$176$	0	0
$177$	2.62650e11	1.51186
$178$	0	0
$179$	− 2.45948e11i	− 1.33838i	−0.743093	−	0.669188i	$-0.766642\pi$
$179$	− 2.45948e11i	− 1.33838i	0.743093	−	0.669188i	$-0.233358\pi$
$180$	0	0
$181$	6.24856e10	0.321652	0.160826	−	0.986983i	$-0.448584\pi$
$181$	6.24856e10	0.321652	0.160826	+	0.986983i	$0.448584\pi$
$182$	0	0
$183$	2.49496e11i	1.21565i
$184$	0	0
$185$	2.14988e11	0.992102
$186$	0	0
$187$	4.13050e10i	0.180632i
$188$	0	0
$189$	1.32132e10	0.0547897
$190$	0	0
$191$	− 1.69920e11i	− 0.668462i	−0.942491	−	0.334231i	$-0.891523\pi$
$191$	− 1.69920e11i	− 0.668462i	0.942491	−	0.334231i	$-0.108477\pi$
$192$	0	0
$193$	−4.08237e11	−1.52449	−0.762246	−	0.647287i	$-0.775903\pi$
$193$	−4.08237e11	−1.52449	−0.762246	+	0.647287i	$0.775903\pi$
$194$	0	0
$195$	2.88210e11i	1.02220i
$196$	0	0
$197$	5.34043e11	1.79989	0.899943	−	0.436009i	$-0.143608\pi$
$197$	5.34043e11	1.79989	0.899943	+	0.436009i	$0.143608\pi$
$198$	0	0
$199$	3.11923e11i	0.999499i	0.866170	+	0.499750i	$0.166575\pi$
$199$	3.11923e11i	0.999499i	−0.866170	+	0.499750i	$0.833425\pi$
$200$	0	0
$201$	−3.00947e11	−0.917297
$202$	0	0
$203$	− 1.48684e10i	− 0.0431304i
$204$	0	0
$205$	1.58594e11	0.438042
$206$	0	0
$207$	7.88635e11i	2.07503i
$208$	0	0
$209$	−6.05752e10	−0.151902
$210$	0	0
$211$	− 4.21246e11i	− 1.00722i	−0.863932	−	0.503609i	$-0.832005\pi$
$211$	− 4.21246e11i	− 1.00722i	0.863932	−	0.503609i	$-0.167995\pi$
$212$	0	0
$213$	−5.27002e11	−1.20203
$214$	0	0
$215$	− 5.45315e9i	− 0.0118701i
$216$	0	0
$217$	1.81660e11	0.377538
$218$	0	0
$219$	3.94691e11i	0.783495i
$220$	0	0
$221$	9.43789e11	1.79025
$222$	0	0
$223$	8.88553e11i	1.61124i	0.592436	+	0.805618i	$0.298166\pi$
$223$	8.88553e11i	1.61124i	−0.592436	+	0.805618i	$0.701834\pi$
$224$	0	0
$225$	3.76052e11	0.652132
$226$	0	0
$227$	3.64887e11i	0.605382i	0.953089	+	0.302691i	$0.0978849\pi$
$227$	3.64887e11i	0.605382i	−0.953089	+	0.302691i	$0.902115\pi$
$228$	0	0
$229$	2.01318e11	0.319672	0.159836	−	0.987144i	$-0.448903\pi$
$229$	2.01318e11	0.319672	0.159836	+	0.987144i	$0.448903\pi$
$230$	0	0
$231$	− 2.85296e10i	− 0.0433746i
$232$	0	0
$233$	−2.56863e11	−0.374043	−0.187021	−	0.982356i	$-0.559883\pi$
$233$	−2.56863e11	−0.374043	−0.187021	+	0.982356i	$0.559883\pi$
$234$	0	0
$235$	7.28577e10i	0.101657i
$236$	0	0
$237$	1.40479e12	1.87875
$238$	0	0
$239$	4.44780e11i	0.570368i	0.958473	+	0.285184i	$0.0920548\pi$
$239$	4.44780e11i	0.570368i	−0.958473	+	0.285184i	$0.907945\pi$
$240$	0	0
$241$	−1.72333e11	−0.211974	−0.105987	−	0.994368i	$-0.533800\pi$
$241$	−1.72333e11	−0.211974	−0.105987	+	0.994368i	$0.533800\pi$
$242$	0	0
$243$	− 1.21539e12i	− 1.43444i
$244$	0	0
$245$	5.32848e11	0.603633
$246$	0	0
$247$	1.38410e12i	1.50550i
$248$	0	0
$249$	9.02218e11	0.942573
$250$	0	0
$251$	1.18721e12i	1.19167i	0.803105	+	0.595837i	$0.203180\pi$
$251$	1.18721e12i	1.19167i	−0.803105	+	0.595837i	$0.796820\pi$
$252$	0	0
$253$	2.04551e11	0.197333
$254$	0	0
$255$	1.71951e12i	1.59479i
$256$	0	0
$257$	−1.21552e12	−1.08417	−0.542083	−	0.840325i	$-0.682364\pi$
$257$	−1.21552e12	−1.08417	−0.542083	+	0.840325i	$0.682364\pi$
$258$	0	0
$259$	4.86264e11i	0.417228i
$260$	0	0
$261$	2.16241e11	0.178540
$262$	0	0
$263$	1.68390e12i	1.33825i	0.743150	+	0.669125i	$0.233331\pi$
$263$	1.68390e12i	1.33825i	−0.743150	+	0.669125i	$0.766669\pi$
$264$	0	0
$265$	1.29172e12	0.988416
$266$	0	0
$267$	2.40378e11i	0.177149i
$268$	0	0
$269$	−1.13783e12	−0.807823	−0.403911	−	0.914798i	$-0.632350\pi$
$269$	−1.13783e12	−0.807823	−0.403911	+	0.914798i	$0.632350\pi$
$270$	0	0
$271$	8.11587e11i	0.555250i	0.960690	+	0.277625i	$0.0895473\pi$
$271$	8.11587e11i	0.555250i	−0.960690	+	0.277625i	$0.910453\pi$
$272$	0	0
$273$	−6.51879e11	−0.429887
$274$	0	0
$275$	− 9.75380e10i	− 0.0620170i
$276$	0	0
$277$	−5.74867e11	−0.352508	−0.176254	−	0.984345i	$-0.556398\pi$
$277$	−5.74867e11	−0.352508	−0.176254	+	0.984345i	$0.556398\pi$
$278$	0	0
$279$	2.64201e12i	1.56284i
$280$	0	0
$281$	−2.51448e12	−1.43521	−0.717606	−	0.696449i	$-0.754762\pi$
$281$	−2.51448e12	−1.43521	−0.717606	+	0.696449i	$0.754762\pi$
$282$	0	0
$283$	− 4.04579e11i	− 0.222880i	−0.993771	−	0.111440i	$-0.964454\pi$
$283$	− 4.04579e11i	− 0.222880i	0.993771	−	0.111440i	$-0.0355463\pi$
$284$	0	0
$285$	−2.52172e12	−1.34114
$286$	0	0
$287$	3.58710e11i	0.184219i
$288$	0	0
$289$	3.61481e12	1.79307
$290$	0	0
$291$	3.59419e12i	1.72241i
$292$	0	0
$293$	−1.49685e12	−0.693170	−0.346585	−	0.938019i	$-0.612659\pi$
$293$	−1.49685e12	−0.693170	−0.346585	+	0.938019i	$0.612659\pi$
$294$	0	0
$295$	− 1.50861e12i	− 0.675254i
$296$	0	0
$297$	4.98435e10	0.0215688
$298$	0	0
$299$	− 4.67383e12i	− 1.95577i
$300$	0	0
$301$	1.23340e10	0.00499198
$302$	0	0
$303$	− 3.00237e11i	− 0.117558i
$304$	0	0
$305$	1.43305e12	0.542955
$306$	0	0
$307$	− 4.66715e12i	− 1.71143i	−0.517444	−	0.855717i	$-0.673116\pi$
$307$	− 4.66715e12i	− 1.71143i	0.517444	−	0.855717i	$-0.326884\pi$
$308$	0	0
$309$	1.10248e12	0.391362
$310$	0	0
$311$	− 2.18146e12i	− 0.749799i	−0.927065	−	0.374900i	$-0.877677\pi$
$311$	− 2.18146e12i	− 0.749799i	0.927065	−	0.374900i	$-0.122323\pi$
$312$	0	0
$313$	−3.04540e12	−1.01373	−0.506865	−	0.862026i	$-0.669196\pi$
$313$	−3.04540e12	−1.01373	−0.506865	+	0.862026i	$0.669196\pi$
$314$	0	0
$315$	− 6.31785e11i	− 0.203712i
$316$	0	0
$317$	−4.57631e12	−1.42962	−0.714808	−	0.699321i	$-0.753486\pi$
$317$	−4.57631e12	−1.42962	−0.714808	+	0.699321i	$0.753486\pi$
$318$	0	0
$319$	− 5.60872e10i	− 0.0169789i
$320$	0	0
$321$	−1.02629e12	−0.301124
$322$	0	0
$323$	8.25775e12i	2.34882i
$324$	0	0
$325$	−2.22867e12	−0.614651
$326$	0	0
$327$	7.55953e12i	2.02188i
$328$	0	0
$329$	−1.64791e11	−0.0427517
$330$	0	0
$331$	− 4.31167e12i	− 1.08519i	−0.839995	−	0.542594i	$-0.817442\pi$
$331$	− 4.31167e12i	− 1.08519i	0.839995	−	0.542594i	$-0.182558\pi$
$332$	0	0
$333$	−7.07208e12	−1.72713
$334$	0	0
$335$	1.72858e12i	0.409700i
$336$	0	0
$337$	−3.87009e12	−0.890371	−0.445186	−	0.895438i	$-0.646862\pi$
$337$	−3.87009e12	−0.890371	−0.445186	+	0.895438i	$0.646862\pi$
$338$	0	0
$339$	− 9.71589e12i	− 2.17012i
$340$	0	0
$341$	6.85267e11	0.148624
$342$	0	0
$343$	2.50867e12i	0.528411i
$344$	0	0
$345$	8.51538e12	1.74224
$346$	0	0
$347$	6.55294e12i	1.30253i	0.758849	+	0.651267i	$0.225762\pi$
$347$	6.55294e12i	1.30253i	−0.758849	+	0.651267i	$0.774238\pi$
$348$	0	0
$349$	4.28862e12	0.828306	0.414153	−	0.910207i	$-0.364078\pi$
$349$	4.28862e12	0.828306	0.414153	+	0.910207i	$0.364078\pi$
$350$	0	0
$351$	− 1.13889e12i	− 0.213769i
$352$	0	0
$353$	2.44634e12	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$353$	2.44634e12	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$354$	0	0
$355$	3.02700e12i	0.536871i
$356$	0	0
$357$	−3.88922e12	−0.670690
$358$	0	0
$359$	2.05242e12i	0.344187i	0.985081	+	0.172093i	$0.0550531\pi$
$359$	2.05242e12i	0.344187i	−0.985081	+	0.172093i	$0.944947\pi$
$360$	0	0
$361$	−5.97919e12	−0.975228
$362$	0	0
$363$	9.10509e12i	1.44461i
$364$	0	0
$365$	2.26703e12	0.349939
$366$	0	0
$367$	1.23281e12i	0.185168i	0.995705	+	0.0925841i	$0.0295127\pi$
$367$	1.23281e12i	0.185168i	−0.995705	+	0.0925841i	$0.970487\pi$
$368$	0	0
$369$	−5.21696e12	−0.762581
$370$	0	0
$371$	2.92164e12i	0.415678i
$372$	0	0
$373$	6.32417e11	0.0875910	0.0437955	−	0.999041i	$-0.486055\pi$
$373$	6.32417e11	0.0875910	0.0437955	+	0.999041i	$0.486055\pi$
$374$	0	0
$375$	− 1.11370e13i	− 1.50180i
$376$	0	0
$377$	−1.28155e12	−0.168279
$378$	0	0
$379$	2.30065e12i	0.294208i	0.989121	+	0.147104i	$0.0469952\pi$
$379$	2.30065e12i	0.294208i	−0.989121	+	0.147104i	$0.953005\pi$
$380$	0	0
$381$	1.61002e13	2.00543
$382$	0	0
$383$	− 3.84719e11i	− 0.0466821i	−0.999728	−	0.0233410i	$-0.992570\pi$
$383$	− 3.84719e11i	− 0.0466821i	0.999728	−	0.0233410i	$-0.00743036\pi$
$384$	0	0
$385$	−1.63868e11	−0.0193728
$386$	0	0
$387$	1.79382e11i	0.0206645i
$388$	0	0
$389$	4.53867e12	0.509542	0.254771	−	0.967001i	$-0.418000\pi$
$389$	4.53867e12	0.509542	0.254771	+	0.967001i	$0.418000\pi$
$390$	0	0
$391$	− 2.78849e13i	− 3.05130i
$392$	0	0
$393$	5.16688e12	0.551145
$394$	0	0
$395$	− 8.06882e12i	− 0.839121i
$396$	0	0
$397$	1.25602e13	1.27363	0.636815	−	0.771017i	$-0.280251\pi$
$397$	1.25602e13	1.27363	0.636815	+	0.771017i	$0.280251\pi$
$398$	0	0
$399$	− 5.70367e12i	− 0.564014i
$400$	0	0
$401$	1.69821e13	1.63783	0.818915	−	0.573914i	$-0.194576\pi$
$401$	1.69821e13	1.63783	0.818915	+	0.573914i	$0.194576\pi$
$402$	0	0
$403$	− 1.56578e13i	− 1.47301i
$404$	0	0
$405$	−6.00975e12	−0.551545
$406$	0	0
$407$	1.83431e12i	0.164248i
$408$	0	0
$409$	1.48399e13	1.29663	0.648314	−	0.761373i	$-0.275474\pi$
$409$	1.48399e13	1.29663	0.648314	+	0.761373i	$0.275474\pi$
$410$	0	0
$411$	1.04910e13i	0.894558i
$412$	0	0
$413$	3.41220e12	0.283978
$414$	0	0
$415$	− 5.18216e12i	− 0.420989i
$416$	0	0
$417$	−1.60125e13	−1.26993
$418$	0	0
$419$	4.51588e12i	0.349681i	0.984597	+	0.174841i	$0.0559410\pi$
$419$	4.51588e12i	0.349681i	−0.984597	+	0.174841i	$0.944059\pi$
$420$	0	0
$421$	3.78477e12	0.286173	0.143087	−	0.989710i	$-0.454297\pi$
$421$	3.78477e12	0.286173	0.143087	+	0.989710i	$0.454297\pi$
$422$	0	0
$423$	− 2.39667e12i	− 0.176973i
$424$	0	0
$425$	−1.32966e13	−0.958951
$426$	0	0
$427$	3.24131e12i	0.228339i
$428$	0	0
$429$	−2.45905e12	−0.169232
$430$	0	0
$431$	− 1.78977e13i	− 1.20340i	−0.798721	−	0.601701i	$-0.794490\pi$
$431$	− 1.78977e13i	− 1.20340i	0.798721	−	0.601701i	$-0.205510\pi$
$432$	0	0
$433$	−1.08277e13	−0.711374	−0.355687	−	0.934605i	$-0.615753\pi$
$433$	−1.08277e13	−0.711374	−0.355687	+	0.934605i	$0.615753\pi$
$434$	0	0
$435$	− 2.33489e12i	− 0.149906i
$436$	0	0
$437$	4.08941e13	2.56598
$438$	0	0
$439$	9.84406e12i	0.603742i	0.953349	+	0.301871i	$0.0976112\pi$
$439$	9.84406e12i	0.603742i	−0.953349	+	0.301871i	$0.902389\pi$
$440$	0	0
$441$	−1.75281e13	−1.05086
$442$	0	0
$443$	− 1.09267e13i	− 0.640429i	−0.947345	−	0.320214i	$-0.896245\pi$
$443$	− 1.09267e13i	− 0.640429i	0.947345	−	0.320214i	$-0.103755\pi$
$444$	0	0
$445$	1.38069e12	0.0791215
$446$	0	0
$447$	− 2.70038e13i	− 1.51317i
$448$	0	0
$449$	−1.28285e13	−0.702982	−0.351491	−	0.936191i	$-0.614325\pi$
$449$	−1.28285e13	−0.702982	−0.351491	+	0.936191i	$0.614325\pi$
$450$	0	0
$451$	1.35314e12i	0.0725205i
$452$	0	0
$453$	−9.77296e12	−0.512313
$454$	0	0
$455$	3.74427e12i	0.192004i
$456$	0	0
$457$	2.16658e13	1.08691	0.543456	−	0.839437i	$-0.317115\pi$
$457$	2.16658e13	1.08691	0.543456	+	0.839437i	$0.317115\pi$
$458$	0	0
$459$	− 6.79478e12i	− 0.333513i
$460$	0	0
$461$	−4.91074e12	−0.235853	−0.117927	−	0.993022i	$-0.537625\pi$
$461$	−4.91074e12	−0.235853	−0.117927	+	0.993022i	$0.537625\pi$
$462$	0	0
$463$	2.65357e12i	0.124717i	0.998054	+	0.0623584i	$0.0198622\pi$
$463$	2.65357e12i	0.124717i	−0.998054	+	0.0623584i	$0.980138\pi$
$464$	0	0
$465$	2.85274e13	1.31219
$466$	0	0
$467$	2.10084e13i	0.945819i	0.881111	+	0.472909i	$0.156796\pi$
$467$	2.10084e13i	0.945819i	−0.881111	+	0.472909i	$0.843204\pi$
$468$	0	0
$469$	−3.90973e12	−0.172299
$470$	0	0
$471$	− 6.45133e13i	− 2.78320i
$472$	0	0
$473$	4.65271e10	0.00196517
$474$	0	0
$475$	− 1.94999e13i	− 0.806426i
$476$	0	0
$477$	−4.24914e13	−1.72072
$478$	0	0
$479$	− 2.34898e13i	− 0.931542i	−0.884905	−	0.465771i	$-0.845777\pi$
$479$	− 2.34898e13i	− 0.931542i	0.884905	−	0.465771i	$-0.154223\pi$
$480$	0	0
$481$	4.19126e13	1.62787
$482$	0	0
$483$	1.92602e13i	0.732699i
$484$	0	0
$485$	2.06443e13	0.769293
$486$	0	0
$487$	1.37321e13i	0.501295i	0.968078	+	0.250648i	$0.0806436\pi$
$487$	1.37321e13i	0.501295i	−0.968078	+	0.250648i	$0.919356\pi$
$488$	0	0
$489$	−4.19349e13	−1.49979
$490$	0	0
$491$	− 4.88273e13i	− 1.71102i	−0.517786	−	0.855510i	$-0.673244\pi$
$491$	− 4.88273e13i	− 1.71102i	0.517786	−	0.855510i	$-0.326756\pi$
$492$	0	0
$493$	−7.64594e12	−0.262541
$494$	0	0
$495$	− 2.38325e12i	− 0.0801944i
$496$	0	0
$497$	−6.84651e12	−0.225781
$498$	0	0
$499$	− 1.69765e13i	− 0.548712i	−0.961628	−	0.274356i	$-0.911535\pi$
$499$	− 1.69765e13i	− 0.548712i	0.961628	−	0.274356i	$-0.0884646\pi$
$500$	0	0
$501$	−8.23758e13	−2.60982
$502$	0	0
$503$	1.44908e13i	0.450042i	0.974354	+	0.225021i	$0.0722450\pi$
$503$	1.44908e13i	0.450042i	−0.974354	+	0.225021i	$0.927755\pi$
$504$	0	0
$505$	−1.72450e12	−0.0525058
$506$	0	0
$507$	7.22156e12i	0.215571i
$508$	0	0
$509$	−4.56116e13	−1.33502	−0.667508	−	0.744602i	$-0.732639\pi$
$509$	−4.56116e13	−1.33502	−0.667508	+	0.744602i	$0.732639\pi$
$510$	0	0
$511$	5.12760e12i	0.147167i
$512$	0	0
$513$	9.96476e12	0.280466
$514$	0	0
$515$	− 6.33243e12i	− 0.174797i
$516$	0	0
$517$	−6.21632e11	−0.0168299
$518$	0	0
$519$	− 5.59524e13i	− 1.48587i
$520$	0	0
$521$	−1.65492e13	−0.431111	−0.215555	−	0.976492i	$-0.569156\pi$
$521$	−1.65492e13	−0.431111	−0.215555	+	0.976492i	$0.569156\pi$
$522$	0	0
$523$	7.10192e13i	1.81496i	0.420094	+	0.907481i	$0.361997\pi$
$523$	7.10192e13i	1.81496i	−0.420094	+	0.907481i	$0.638003\pi$
$524$	0	0
$525$	9.18404e12	0.230270
$526$	0	0
$527$	− 9.34173e13i	− 2.29813i
$528$	0	0
$529$	−9.66652e13	−2.33341
$530$	0	0
$531$	4.96261e13i	1.17554i
$532$	0	0
$533$	3.09183e13	0.718752
$534$	0	0
$535$	5.89482e12i	0.134494i
$536$	0	0
$537$	8.73582e13	1.95628
$538$	0	0
$539$	4.54634e12i	0.0999351i
$540$	0	0
$541$	2.64140e13	0.569965	0.284983	−	0.958533i	$-0.408012\pi$
$541$	2.64140e13	0.569965	0.284983	+	0.958533i	$0.408012\pi$
$542$	0	0
$543$	2.21942e13i	0.470155i
$544$	0	0
$545$	4.34204e13	0.903050
$546$	0	0
$547$	− 9.50792e13i	− 1.94155i	−0.239989	−	0.970776i	$-0.577144\pi$
$547$	− 9.50792e13i	− 1.94155i	0.239989	−	0.970776i	$-0.422856\pi$
$548$	0	0
$549$	−4.71406e13	−0.945221
$550$	0	0
$551$	− 1.12130e13i	− 0.220782i
$552$	0	0
$553$	1.82502e13	0.352892
$554$	0	0
$555$	7.63616e13i	1.45014i
$556$	0	0
$557$	−6.71585e13	−1.25264	−0.626318	−	0.779567i	$-0.715439\pi$
$557$	−6.71585e13	−1.25264	−0.626318	+	0.779567i	$0.715439\pi$
$558$	0	0
$559$	− 1.06311e12i	− 0.0194768i
$560$	0	0
$561$	−1.46711e13	−0.264028
$562$	0	0
$563$	9.49758e13i	1.67908i	0.543299	+	0.839539i	$0.317175\pi$
$563$	9.49758e13i	1.67908i	−0.543299	+	0.839539i	$0.682825\pi$
$564$	0	0
$565$	−5.58061e13	−0.969258
$566$	0	0
$567$	− 1.35929e13i	− 0.231952i
$568$	0	0
$569$	2.90418e13	0.486926	0.243463	−	0.969910i	$-0.421717\pi$
$569$	2.90418e13	0.486926	0.243463	+	0.969910i	$0.421717\pi$
$570$	0	0
$571$	− 5.48035e12i	− 0.0902876i	−0.998980	−	0.0451438i	$-0.985625\pi$
$571$	− 5.48035e12i	− 0.0902876i	0.998980	−	0.0451438i	$-0.0143746\pi$
$572$	0	0
$573$	6.03537e13	0.977082
$574$	0	0
$575$	6.58476e13i	1.04761i
$576$	0	0
$577$	−2.15948e13	−0.337653	−0.168827	−	0.985646i	$-0.553998\pi$
$577$	−2.15948e13	−0.337653	−0.168827	+	0.985646i	$0.553998\pi$
$578$	0	0
$579$	− 1.45001e14i	− 2.22833i
$580$	0	0
$581$	1.17211e13	0.177047
$582$	0	0
$583$	1.10212e13i	0.163638i
$584$	0	0
$585$	−5.44555e13	−0.794808
$586$	0	0
$587$	9.42256e12i	0.135201i	0.997712	+	0.0676003i	$0.0215343\pi$
$587$	9.42256e12i	0.135201i	−0.997712	+	0.0676003i	$0.978466\pi$
$588$	0	0
$589$	1.36999e14	1.93260
$590$	0	0
$591$	1.89686e14i	2.63087i
$592$	0	0
$593$	2.97508e13	0.405719	0.202860	−	0.979208i	$-0.434976\pi$
$593$	2.97508e13	0.405719	0.202860	+	0.979208i	$0.434976\pi$
$594$	0	0
$595$	2.23389e13i	0.299556i
$596$	0	0
$597$	−1.10792e14	−1.46095
$598$	0	0
$599$	8.17387e13i	1.05997i	0.848007	+	0.529985i	$0.177803\pi$
$599$	8.17387e13i	1.05997i	−0.848007	+	0.529985i	$0.822197\pi$
$600$	0	0
$601$	−8.49863e13	−1.08387	−0.541934	−	0.840421i	$-0.682308\pi$
$601$	−8.49863e13	−1.08387	−0.541934	+	0.840421i	$0.682308\pi$
$602$	0	0
$603$	− 5.68620e13i	− 0.713239i
$604$	0	0
$605$	5.22978e13	0.645218
$606$	0	0
$607$	3.30148e12i	0.0400650i	0.999799	+	0.0200325i	$0.00637697\pi$
$607$	3.30148e12i	0.0400650i	−0.999799	+	0.0200325i	$0.993623\pi$
$608$	0	0
$609$	5.28109e12	0.0630431
$610$	0	0
$611$	1.42038e13i	0.166801i
$612$	0	0
$613$	1.43588e14	1.65889	0.829443	−	0.558591i	$-0.188658\pi$
$613$	1.43588e14	1.65889	0.829443	+	0.558591i	$0.188658\pi$
$614$	0	0
$615$	5.63308e13i	0.640281i
$616$	0	0
$617$	−2.81944e13	−0.315309	−0.157655	−	0.987494i	$-0.550393\pi$
$617$	−2.81944e13	−0.315309	−0.157655	+	0.987494i	$0.550393\pi$
$618$	0	0
$619$	− 3.35388e13i	− 0.369057i	−0.982827	−	0.184529i	$-0.940924\pi$
$619$	− 3.35388e13i	− 0.369057i	0.982827	−	0.184529i	$-0.0590759\pi$
$620$	0	0
$621$	−3.36492e13	−0.364348
$622$	0	0
$623$	3.12286e12i	0.0332745i
$624$	0	0
$625$	−9.24743e12	−0.0969663
$626$	0	0
$627$	− 2.15157e13i	− 0.222033i
$628$	0	0
$629$	2.50058e14	2.53973
$630$	0	0
$631$	− 1.10629e14i	− 1.10591i	−0.833210	−	0.552957i	$-0.813500\pi$
$631$	− 1.10629e14i	− 1.10591i	0.833210	−	0.552957i	$-0.186500\pi$
$632$	0	0
$633$	1.49622e14	1.47224
$634$	0	0
$635$	− 9.24764e13i	− 0.895699i
$636$	0	0
$637$	1.03880e14	0.990458
$638$	0	0
$639$	− 9.95736e13i	− 0.934631i
$640$	0	0
$641$	2.73053e12	0.0252323	0.0126162	−	0.999920i	$-0.495984\pi$
$641$	2.73053e12	0.0252323	0.0126162	+	0.999920i	$0.495984\pi$
$642$	0	0
$643$	− 7.26452e13i	− 0.660925i	−0.943819	−	0.330462i	$-0.892795\pi$
$643$	− 7.26452e13i	− 0.660925i	0.943819	−	0.330462i	$-0.107205\pi$
$644$	0	0
$645$	1.93690e12	0.0173504
$646$	0	0
$647$	7.34487e13i	0.647832i	0.946086	+	0.323916i	$0.105000\pi$
$647$	7.34487e13i	0.647832i	−0.946086	+	0.323916i	$0.895000\pi$
$648$	0	0
$649$	1.28717e13	0.111792
$650$	0	0
$651$	6.45238e13i	0.551842i
$652$	0	0
$653$	9.19323e12	0.0774287	0.0387144	−	0.999250i	$-0.487674\pi$
$653$	9.19323e12	0.0774287	0.0387144	+	0.999250i	$0.487674\pi$
$654$	0	0
$655$	− 2.96775e13i	− 0.246162i
$656$	0	0
$657$	−7.45742e13	−0.609203
$658$	0	0
$659$	1.82524e14i	1.46856i	0.678844	+	0.734282i	$0.262481\pi$
$659$	1.82524e14i	1.46856i	−0.678844	+	0.734282i	$0.737519\pi$
$660$	0	0
$661$	6.12505e13	0.485403	0.242701	−	0.970101i	$-0.421967\pi$
$661$	6.12505e13	0.485403	0.242701	+	0.970101i	$0.421967\pi$
$662$	0	0
$663$	3.35224e14i	2.61678i
$664$	0	0
$665$	−3.27607e13	−0.251910
$666$	0	0
$667$	3.78643e13i	0.286814i
$668$	0	0
$669$	−3.15605e14	−2.35512
$670$	0	0
$671$	1.22270e13i	0.0898894i
$672$	0	0
$673$	5.26543e13	0.381381	0.190690	−	0.981650i	$-0.438927\pi$
$673$	5.26543e13	0.381381	0.190690	+	0.981650i	$0.438927\pi$
$674$	0	0
$675$	1.60453e13i	0.114506i
$676$	0	0
$677$	1.05166e14	0.739492	0.369746	−	0.929133i	$-0.379445\pi$
$677$	1.05166e14	0.739492	0.369746	+	0.929133i	$0.379445\pi$
$678$	0	0
$679$	4.66937e13i	0.323526i
$680$	0	0
$681$	−1.29604e14	−0.884878
$682$	0	0
$683$	− 1.27995e14i	− 0.861170i	−0.902550	−	0.430585i	$-0.858307\pi$
$683$	− 1.27995e14i	− 0.861170i	0.902550	−	0.430585i	$-0.141693\pi$
$684$	0	0
$685$	6.02583e13	0.399543
$686$	0	0
$687$	7.15061e13i	0.467261i
$688$	0	0
$689$	2.51825e14	1.62182
$690$	0	0
$691$	1.64774e14i	1.04592i	0.852358	+	0.522959i	$0.175172\pi$
$691$	1.64774e14i	1.04592i	−0.852358	+	0.522959i	$0.824828\pi$
$692$	0	0
$693$	5.39048e12	0.0337257
$694$	0	0
$695$	9.19728e13i	0.567199i
$696$	0	0
$697$	1.84464e14	1.12136
$698$	0	0
$699$	− 9.12350e13i	− 0.546733i
$700$	0	0
$701$	1.34675e14	0.795602	0.397801	−	0.917472i	$-0.369773\pi$
$701$	1.34675e14	0.795602	0.397801	+	0.917472i	$0.369773\pi$
$702$	0	0
$703$	3.66717e14i	2.13577i
$704$	0	0
$705$	−2.58783e13	−0.148590
$706$	0	0
$707$	− 3.90051e12i	− 0.0220813i
$708$	0	0
$709$	−1.24609e14	−0.695536	−0.347768	−	0.937581i	$-0.613060\pi$
$709$	−1.24609e14	−0.695536	−0.347768	+	0.937581i	$0.613060\pi$
$710$	0	0
$711$	2.65426e14i	1.46081i
$712$	0	0
$713$	−4.62621e14	−2.51060
$714$	0	0
$715$	1.41243e13i	0.0755853i
$716$	0	0
$717$	−1.57981e14	−0.833700
$718$	0	0
$719$	2.03322e14i	1.05813i	0.848581	+	0.529065i	$0.177457\pi$
$719$	2.03322e14i	1.05813i	−0.848581	+	0.529065i	$0.822543\pi$
$720$	0	0
$721$	1.43228e13	0.0735109
$722$	0	0
$723$	− 6.12108e13i	− 0.309839i
$724$	0	0
$725$	1.80552e13	0.0901389
$726$	0	0
$727$	− 2.12819e14i	− 1.04794i	−0.851736	−	0.523971i	$-0.824450\pi$
$727$	− 2.12819e14i	− 1.04794i	0.851736	−	0.523971i	$-0.175550\pi$
$728$	0	0
$729$	2.57749e14	1.25187
$730$	0	0
$731$	− 6.34268e12i	− 0.0303869i
$732$	0	0
$733$	−4.77128e13	−0.225484	−0.112742	−	0.993624i	$-0.535963\pi$
$733$	−4.77128e13	−0.225484	−0.112742	+	0.993624i	$0.535963\pi$
$734$	0	0
$735$	1.89262e14i	0.882322i
$736$	0	0
$737$	−1.47485e13	−0.0678282
$738$	0	0
$739$	8.34499e13i	0.378620i	0.981917	+	0.189310i	$0.0606251\pi$
$739$	8.34499e13i	0.378620i	−0.981917	+	0.189310i	$0.939375\pi$
$740$	0	0
$741$	−4.91617e14	−2.20057
$742$	0	0
$743$	− 2.85136e12i	− 0.0125924i	−0.999980	−	0.00629618i	$-0.997996\pi$
$743$	− 2.85136e12i	− 0.0125924i	0.999980	−	0.00629618i	$-0.00200415\pi$
$744$	0	0
$745$	−1.55104e14	−0.675838
$746$	0	0
$747$	1.70468e14i	0.732893i
$748$	0	0
$749$	−1.33330e13	−0.0565612
$750$	0	0
$751$	− 6.81300e13i	− 0.285193i	−0.989781	−	0.142596i	$-0.954455\pi$
$751$	− 6.81300e13i	− 0.285193i	0.989781	−	0.142596i	$-0.0455451\pi$
$752$	0	0
$753$	−4.21683e14	−1.74185
$754$	0	0
$755$	5.61339e13i	0.228818i
$756$	0	0
$757$	−7.24794e12	−0.0291565	−0.0145782	−	0.999894i	$-0.504641\pi$
$757$	−7.24794e12	−0.0291565	−0.0145782	+	0.999894i	$0.504641\pi$
$758$	0	0
$759$	7.26544e13i	0.288439i
$760$	0	0
$761$	4.15788e14	1.62910	0.814551	−	0.580092i	$-0.196983\pi$
$761$	4.15788e14	1.62910	0.814551	+	0.580092i	$0.196983\pi$
$762$	0	0
$763$	9.82091e13i	0.379777i
$764$	0	0
$765$	−3.24891e14	−1.24002
$766$	0	0
$767$	− 2.94108e14i	− 1.10798i
$768$	0	0
$769$	2.35876e14	0.877105	0.438552	−	0.898706i	$-0.355491\pi$
$769$	2.35876e14	0.877105	0.438552	+	0.898706i	$0.355491\pi$
$770$	0	0
$771$	− 4.31740e14i	− 1.58471i
$772$	0	0
$773$	−1.51031e14	−0.547229	−0.273614	−	0.961839i	$-0.588219\pi$
$773$	−1.51031e14	−0.547229	−0.273614	+	0.961839i	$0.588219\pi$
$774$	0	0
$775$	2.20596e14i	0.789023i
$776$	0	0
$777$	−1.72716e14	−0.609857
$778$	0	0
$779$	2.70522e14i	0.943007i
$780$	0	0
$781$	−2.58268e13	−0.0888822
$782$	0	0
$783$	9.22649e12i	0.0313493i
$784$	0	0
$785$	−3.70552e14	−1.24308
$786$	0	0
$787$	− 2.53593e14i	− 0.839970i	−0.907531	−	0.419985i	$-0.862035\pi$
$787$	− 2.53593e14i	− 0.839970i	0.907531	−	0.419985i	$-0.137965\pi$
$788$	0	0
$789$	−5.98104e14	−1.95610
$790$	0	0
$791$	− 1.26223e14i	− 0.407621i
$792$	0	0
$793$	2.79378e14	0.890895
$794$	0	0
$795$	4.58806e14i	1.44475i
$796$	0	0
$797$	3.72131e14	1.15719	0.578595	−	0.815615i	$-0.303601\pi$
$797$	3.72131e14	1.15719	0.578595	+	0.815615i	$0.303601\pi$
$798$	0	0
$799$	8.47424e13i	0.260236i
$800$	0	0
$801$	−4.54179e13	−0.137741
$802$	0	0
$803$	1.93426e13i	0.0579344i
$804$	0	0
$805$	1.10627e14	0.327251
$806$	0	0
$807$	− 4.04146e14i	− 1.18078i
$808$	0	0
$809$	2.64826e14	0.764220	0.382110	−	0.924117i	$-0.375198\pi$
$809$	2.64826e14	0.764220	0.382110	+	0.924117i	$0.375198\pi$
$810$	0	0
$811$	− 3.72315e13i	− 0.106122i	−0.998591	−	0.0530611i	$-0.983102\pi$
$811$	− 3.72315e13i	− 0.106122i	0.998591	−	0.0530611i	$-0.0168978\pi$
$812$	0	0
$813$	−2.88267e14	−0.811601
$814$	0	0
$815$	2.40866e14i	0.669865i
$816$	0	0
$817$	9.30174e12	0.0255537
$818$	0	0
$819$	− 1.23168e14i	− 0.334256i
$820$	0	0
$821$	3.70079e14	0.992153	0.496076	−	0.868279i	$-0.334774\pi$
$821$	3.70079e14	0.992153	0.496076	+	0.868279i	$0.334774\pi$
$822$	0	0
$823$	3.32400e14i	0.880362i	0.897909	+	0.440181i	$0.145086\pi$
$823$	3.32400e14i	0.880362i	−0.897909	+	0.440181i	$0.854914\pi$
$824$	0	0
$825$	3.46445e13	0.0906494
$826$	0	0
$827$	− 5.93343e14i	− 1.53383i	−0.641746	−	0.766917i	$-0.721790\pi$
$827$	− 5.93343e14i	− 1.53383i	0.641746	−	0.766917i	$-0.278210\pi$
$828$	0	0
$829$	6.05560e14	1.54662	0.773312	−	0.634026i	$-0.218599\pi$
$829$	6.05560e14	1.54662	0.773312	+	0.634026i	$0.218599\pi$
$830$	0	0
$831$	− 2.04187e14i	− 0.515256i
$832$	0	0
$833$	6.19768e14	1.54527
$834$	0	0
$835$	4.73150e14i	1.16565i
$836$	0	0
$837$	−1.12728e14	−0.274413
$838$	0	0
$839$	− 7.06534e14i	− 1.69951i	−0.527178	−	0.849755i	$-0.676750\pi$
$839$	− 7.06534e14i	− 1.69951i	0.527178	−	0.849755i	$-0.323250\pi$
$840$	0	0
$841$	−4.10325e14	−0.975322
$842$	0	0
$843$	− 8.93116e14i	− 2.09783i
$844$	0	0
$845$	4.14792e13	0.0962823
$846$	0	0
$847$	1.18288e14i	0.271346i
$848$	0	0
$849$	1.43702e14	0.325781
$850$	0	0
$851$	− 1.23834e15i	− 2.77454i
$852$	0	0
$853$	1.44788e14	0.320618	0.160309	−	0.987067i	$-0.448751\pi$
$853$	1.44788e14	0.320618	0.160309	+	0.987067i	$0.448751\pi$
$854$	0	0
$855$	− 4.76462e14i	− 1.04279i
$856$	0	0
$857$	−2.49429e14	−0.539564	−0.269782	−	0.962921i	$-0.586952\pi$
$857$	−2.49429e14	−0.539564	−0.269782	+	0.962921i	$0.586952\pi$
$858$	0	0
$859$	5.01533e14i	1.07234i	0.844109	+	0.536171i	$0.180130\pi$
$859$	5.01533e14i	1.07234i	−0.844109	+	0.536171i	$0.819870\pi$
$860$	0	0
$861$	−1.27410e14	−0.269270
$862$	0	0
$863$	− 1.15087e14i	− 0.240420i	−0.992748	−	0.120210i	$-0.961643\pi$
$863$	− 1.15087e14i	− 0.240420i	0.992748	−	0.120210i	$-0.0383568\pi$
$864$	0	0
$865$	−3.21379e14	−0.663647
$866$	0	0
$867$	1.28394e15i	2.62090i
$868$	0	0
$869$	6.88444e13	0.138922
$870$	0	0
$871$	3.36992e14i	0.672246i
$872$	0	0
$873$	−6.79098e14	−1.33925
$874$	0	0
$875$	− 1.44686e14i	− 0.282088i
$876$	0	0
$877$	2.12863e14	0.410301	0.205150	−	0.978730i	$-0.434232\pi$
$877$	2.12863e14	0.410301	0.205150	+	0.978730i	$0.434232\pi$
$878$	0	0
$879$	− 5.31665e14i	− 1.01320i
$880$	0	0
$881$	−8.85797e14	−1.66899	−0.834497	−	0.551012i	$-0.814242\pi$
$881$	−8.85797e14	−1.66899	−0.834497	+	0.551012i	$0.814242\pi$
$882$	0	0
$883$	6.35751e13i	0.118436i	0.998245	+	0.0592180i	$0.0188607\pi$
$883$	6.35751e13i	0.118436i	−0.998245	+	0.0592180i	$0.981139\pi$
$884$	0	0
$885$	5.35843e14	0.987010
$886$	0	0
$887$	2.51644e14i	0.458319i	0.973389	+	0.229160i	$0.0735978\pi$
$887$	2.51644e14i	0.458319i	−0.973389	+	0.229160i	$0.926402\pi$
$888$	0	0
$889$	2.09165e14	0.376686
$890$	0	0
$891$	− 5.12760e13i	− 0.0913116i
$892$	0	0
$893$	−1.24277e14	−0.218844
$894$	0	0
$895$	− 5.01768e14i	− 0.873751i
$896$	0	0
$897$	1.66010e15	2.85872
$898$	0	0
$899$	1.26849e14i	0.216018i
$900$	0	0
$901$	1.50243e15	2.53029
$902$	0	0
$903$	4.38092e12i	0.00729670i
$904$	0	0
$905$	1.27479e14	0.209989
$906$	0	0
$907$	2.66687e14i	0.434476i	0.976119	+	0.217238i	$0.0697048\pi$
$907$	2.66687e14i	0.434476i	−0.976119	+	0.217238i	$0.930295\pi$
$908$	0	0
$909$	5.67279e13	0.0914065
$910$	0	0
$911$	− 1.18530e15i	− 1.88901i	−0.328495	−	0.944506i	$-0.606541\pi$
$911$	− 1.18530e15i	− 1.88901i	0.328495	−	0.944506i	$-0.393459\pi$
$912$	0	0
$913$	4.42150e13	0.0696972
$914$	0	0
$915$	5.09006e14i	0.793629i
$916$	0	0
$917$	6.71251e13	0.103523
$918$	0	0
$919$	2.34907e14i	0.358359i	0.983816	+	0.179179i	$0.0573443\pi$
$919$	2.34907e14i	0.358359i	−0.983816	+	0.179179i	$0.942656\pi$
$920$	0	0
$921$	1.65773e15	2.50158
$922$	0	0
$923$	5.90122e14i	0.880913i
$924$	0	0
$925$	−5.90488e14	−0.871972
$926$	0	0
$927$	2.08306e14i	0.304301i
$928$	0	0
$929$	−7.20857e14	−1.04177	−0.520883	−	0.853628i	$-0.674397\pi$
$929$	−7.20857e14	−1.04177	−0.520883	+	0.853628i	$0.674397\pi$
$930$	0	0
$931$	9.08909e14i	1.29949i
$932$	0	0
$933$	7.74832e14	1.09597
$934$	0	0
$935$	8.42680e13i	0.117925i
$936$	0	0
$937$	−1.04663e15	−1.44910	−0.724548	−	0.689225i	$-0.757951\pi$
$937$	−1.04663e15	−1.44910	−0.724548	+	0.689225i	$0.757951\pi$
$938$	0	0
$939$	− 1.08169e15i	− 1.48175i
$940$	0	0
$941$	9.99564e14	1.35476	0.677380	−	0.735633i	$-0.263115\pi$
$941$	9.99564e14	1.35476	0.677380	+	0.735633i	$0.263115\pi$
$942$	0	0
$943$	− 9.13502e14i	− 1.22504i
$944$	0	0
$945$	2.69567e13	0.0357691
$946$	0	0
$947$	− 7.94538e14i	− 1.04319i	−0.853192	−	0.521597i	$-0.825337\pi$
$947$	− 7.94538e14i	− 1.04319i	0.853192	−	0.521597i	$-0.174663\pi$
$948$	0	0
$949$	4.41963e14	0.574189
$950$	0	0
$951$	− 1.62546e15i	− 2.08965i
$952$	0	0
$953$	4.37313e14	0.556324	0.278162	−	0.960534i	$-0.410275\pi$
$953$	4.37313e14	0.556324	0.278162	+	0.960534i	$0.410275\pi$
$954$	0	0
$955$	− 3.46660e14i	− 0.436402i
$956$	0	0
$957$	1.99216e13	0.0248179
$958$	0	0
$959$	1.36293e14i	0.168028i
$960$	0	0
$961$	−7.30202e14	−0.890894
$962$	0	0
$963$	− 1.93911e14i	− 0.234138i
$964$	0	0
$965$	−8.32859e14	−0.995257
$966$	0	0
$967$	− 1.33282e15i	− 1.57630i	−0.615484	−	0.788149i	$-0.711040\pi$
$967$	− 1.33282e15i	− 1.57630i	0.615484	−	0.788149i	$-0.288960\pi$
$968$	0	0
$969$	−2.93307e15	−3.43323
$970$	0	0
$971$	1.40291e15i	1.62530i	0.582751	+	0.812651i	$0.301976\pi$
$971$	1.40291e15i	1.62530i	−0.582751	+	0.812651i	$0.698024\pi$
$972$	0	0
$973$	−2.08026e14	−0.238535
$974$	0	0
$975$	− 7.91601e14i	− 0.898427i
$976$	0	0
$977$	−2.14298e14	−0.240738	−0.120369	−	0.992729i	$-0.538408\pi$
$977$	−2.14298e14	−0.240738	−0.120369	+	0.992729i	$0.538408\pi$
$978$	0	0
$979$	1.17802e13i	0.0130990i
$980$	0	0
$981$	−1.42832e15	−1.57210
$982$	0	0
$983$	− 1.08912e15i	− 1.18661i	−0.804978	−	0.593305i	$-0.797823\pi$
$983$	− 1.08912e15i	− 1.18661i	0.804978	−	0.593305i	$-0.202177\pi$
$984$	0	0
$985$	1.08952e15	1.17505
$986$	0	0
$987$	− 5.85320e13i	− 0.0624896i
$988$	0	0
$989$	−3.14102e13	−0.0331963
$990$	0	0
$991$	− 1.51978e15i	− 1.59005i	−0.606575	−	0.795026i	$-0.707457\pi$
$991$	− 1.51978e15i	− 1.59005i	0.606575	−	0.795026i	$-0.292543\pi$
$992$	0	0
$993$	1.53146e15	1.58621
$994$	0	0
$995$	6.36367e14i	0.652517i
$996$	0	0
$997$	−1.08307e15	−1.09947	−0.549734	−	0.835340i	$-0.685271\pi$
$997$	−1.08307e15	−1.09947	−0.549734	+	0.835340i	$0.685271\pi$
$998$	0	0
$999$	− 3.01749e14i	− 0.303262i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.l.255.10		12
4.3	odd	2	inner	256.11.c.l.255.4		12
8.3	odd	2	inner	256.11.c.l.255.9		12
8.5	even	2	inner	256.11.c.l.255.3		12
16.3	odd	4		64.11.d.b.31.9	yes	12
16.5	even	4		64.11.d.b.31.10	yes	12
16.11	odd	4		64.11.d.b.31.4	yes	12
16.13	even	4		64.11.d.b.31.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.11.d.b.31.3	✓	12	16.13	even	4
64.11.d.b.31.4	yes	12	16.11	odd	4
64.11.d.b.31.9	yes	12	16.3	odd	4
64.11.d.b.31.10	yes	12	16.5	even	4
256.11.c.l.255.3		12	8.5	even	2	inner
256.11.c.l.255.4		12	4.3	odd	2	inner
256.11.c.l.255.9		12	8.3	odd	2	inner
256.11.c.l.255.10		12	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)

\(f(q)\)	\(=\)	\(q+355.190i q^{3} +2040.14 q^{5} +4614.42i q^{7} -67110.8 q^{9} +O(q^{10})\)	\(q+355.190i q^{3} +2040.14 q^{5} +4614.42i q^{7} -67110.8 q^{9} +17406.8i q^{11} +397731. q^{13} +724636. i q^{15} +2.37293e6 q^{17} +3.47998e6i q^{19} -1.63900e6 q^{21} -1.17512e7i q^{23} -5.60346e6 q^{25} -2.86346e6i q^{27} -3.22215e6 q^{29} -3.93679e7i q^{31} -6.18270e6 q^{33} +9.41406e6i q^{35} +1.05379e8 q^{37} +1.41270e8i q^{39} +7.77366e7 q^{41} -2.67293e6i q^{43} -1.36915e8 q^{45} +3.57121e7i q^{47} +2.61182e8 q^{49} +8.42841e8i q^{51} +6.33154e8 q^{53} +3.55122e7i q^{55} -1.23605e9 q^{57} -7.39465e8i q^{59} +7.02430e8 q^{61} -3.09677e8i q^{63} +8.11427e8 q^{65} +8.47285e8i q^{67} +4.17392e9 q^{69} +1.48372e9i q^{71} +1.11121e9 q^{73} -1.99029e9i q^{75} -8.03221e7 q^{77} -3.95504e9i q^{79} -2.94575e9 q^{81} -2.54010e9i q^{83} +4.84111e9 q^{85} -1.14448e9i q^{87} +6.76760e8 q^{89} +1.83530e9i q^{91} +1.39831e10 q^{93} +7.09964e9i q^{95} +1.01191e10 q^{97} -1.16818e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 375204 q^{9}+O(q^{10}) \) 12 * q - 375204 * q^9	\( 12 q - 375204 q^{9} - 1639752 q^{17} + 28703892 q^{25} + 358214544 q^{33} - 496702488 q^{41} + 1385670732 q^{49} - 5733526800 q^{57} + 8895346752 q^{65} - 12161297592 q^{73} - 7708157604 q^{81} + 5871785160 q^{89} + 30465806904 q^{97}+O(q^{100}) \) 12 * q - 375204 * q^9 - 1639752 * q^17 + 28703892 * q^25 + 358214544 * q^33 - 496702488 * q^41 + 1385670732 * q^49 - 5733526800 * q^57 + 8895346752 * q^65 - 12161297592 * q^73 - 7708157604 * q^81 + 5871785160 * q^89 + 30465806904 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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