LMFDB - Embedded newform 256.10.b.j.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,10,Mod(129,256)]

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([0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	256.b (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:	$131.849174058$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.10.b.j.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+228.000i q^{3} +666.000i q^{5} +6328.00 q^{7} -32301.0 q^{9} +O(q^{10})$	$q+228.000i q^{3} +666.000i q^{5} +6328.00 q^{7} -32301.0 q^{9} +30420.0i q^{11} -32338.0i q^{13} -151848. q^{15} +590994. q^{17} +34676.0i q^{19} +1.44278e6i q^{21} -1.04854e6 q^{23} +1.50957e6 q^{25} -2.87690e6i q^{27} +4.40941e6i q^{29} -7.40118e6 q^{31} -6.93576e6 q^{33} +4.21445e6i q^{35} -1.02345e7i q^{37} +7.37306e6 q^{39} -1.83527e7 q^{41} +252340. i q^{43} -2.15125e7i q^{45} -4.95171e7 q^{47} -310023. q^{49} +1.34747e8i q^{51} +6.63969e7i q^{53} -2.02597e7 q^{55} -7.90613e6 q^{57} +6.15237e7i q^{59} +3.56386e7i q^{61} -2.04401e8 q^{63} +2.15371e7 q^{65} +1.81742e8i q^{67} -2.39066e8i q^{69} -9.09050e7 q^{71} +2.62979e8 q^{73} +3.44182e8i q^{75} +1.92498e8i q^{77} -1.16503e8 q^{79} +2.01535e7 q^{81} -9.56372e6i q^{83} +3.93602e8i q^{85} -1.00534e9 q^{87} -6.11827e8 q^{89} -2.04635e8i q^{91} -1.68747e9i q^{93} -2.30942e7 q^{95} -2.59313e8 q^{97} -9.82596e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12656 q^{7} - 64602 q^{9}+O(q^{10}) $ 2 * q + 12656 * q^7 - 64602 * q^9	$ 2 q + 12656 q^{7} - 64602 q^{9} - 303696 q^{15} + 1181988 q^{17} - 2097072 q^{23} + 3019138 q^{25} - 14802368 q^{31} - 13871520 q^{33} + 14746128 q^{39} - 36705492 q^{41} - 99034272 q^{47} - 620046 q^{49} - 40519440 q^{55} - 15812256 q^{57} - 408801456 q^{63} + 43074216 q^{65} - 181809936 q^{71} + 525957356 q^{73} - 233005664 q^{79} + 40307058 q^{81} - 2010689136 q^{87} - 1223653428 q^{89} - 46188432 q^{95} - 518625596 q^{97}+O(q^{100}) $ 2 * q + 12656 * q^7 - 64602 * q^9 - 303696 * q^15 + 1181988 * q^17 - 2097072 * q^23 + 3019138 * q^25 - 14802368 * q^31 - 13871520 * q^33 + 14746128 * q^39 - 36705492 * q^41 - 99034272 * q^47 - 620046 * q^49 - 40519440 * q^55 - 15812256 * q^57 - 408801456 * q^63 + 43074216 * q^65 - 181809936 * q^71 + 525957356 * q^73 - 233005664 * q^79 + 40307058 * q^81 - 2010689136 * q^87 - 1223653428 * q^89 - 46188432 * q^95 - 518625596 * 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
$3$	228.000i	1.62513i	0.582868	+	0.812567i	$0.301931\pi$
$3$	228.000i	1.62513i	−0.582868	+	0.812567i	$0.698069\pi$
$4$	0	0
$5$	666.000i	0.476551i	0.971198	+	0.238275i	$0.0765821\pi$
$5$	666.000i	0.476551i	−0.971198	+	0.238275i	$0.923418\pi$
$6$	0	0
$7$	6328.00	0.996151	0.498076	−	0.867134i	$-0.334040\pi$
$7$	6328.00	0.996151	0.498076	+	0.867134i	$0.334040\pi$
$8$	0	0
$9$	−32301.0	−1.64106
$10$	0	0
$11$	30420.0i	0.626458i	0.949678	+	0.313229i	$0.101411\pi$
$11$	30420.0i	0.626458i	−0.949678	+	0.313229i	$0.898589\pi$
$12$	0	0
$13$	− 32338.0i	− 0.314028i	−0.987596	−	0.157014i	$-0.949813\pi$
$13$	− 32338.0i	− 0.314028i	0.987596	−	0.157014i	$-0.0501867\pi$
$14$	0	0
$15$	−151848.	−0.774459
$16$	0	0
$17$	590994.	1.71618	0.858090	−	0.513499i	$-0.171651\pi$
$17$	590994.	1.71618	0.858090	+	0.513499i	$0.171651\pi$
$18$	0	0
$19$	34676.0i	0.0610433i	0.999534	+	0.0305216i	$0.00971685\pi$
$19$	34676.0i	0.0610433i	−0.999534	+	0.0305216i	$0.990283\pi$
$20$	0	0
$21$	1.44278e6i	1.61888i
$22$	0	0
$23$	−1.04854e6	−0.781282	−0.390641	−	0.920543i	$-0.627747\pi$
$23$	−1.04854e6	−0.781282	−0.390641	+	0.920543i	$0.627747\pi$
$24$	0	0
$25$	1.50957e6	0.772899
$26$	0	0
$27$	− 2.87690e6i	− 1.04181i
$28$	0	0
$29$	4.40941e6i	1.15768i	0.815441	+	0.578841i	$0.196495\pi$
$29$	4.40941e6i	1.15768i	−0.815441	+	0.578841i	$0.803505\pi$
$30$	0	0
$31$	−7.40118e6	−1.43937	−0.719687	−	0.694299i	$-0.755715\pi$
$31$	−7.40118e6	−1.43937	−0.719687	+	0.694299i	$0.755715\pi$
$32$	0	0
$33$	−6.93576e6	−1.01808
$34$	0	0
$35$	4.21445e6i	0.474717i
$36$	0	0
$37$	− 1.02345e7i	− 0.897757i	−0.893593	−	0.448879i	$-0.851824\pi$
$37$	− 1.02345e7i	− 0.897757i	0.893593	−	0.448879i	$-0.148176\pi$
$38$	0	0
$39$	7.37306e6	0.510337
$40$	0	0
$41$	−1.83527e7	−1.01432	−0.507158	−	0.861853i	$-0.669304\pi$
$41$	−1.83527e7	−1.01432	−0.507158	+	0.861853i	$0.669304\pi$
$42$	0	0
$43$	252340.i	0.0112558i	0.999984	+	0.00562792i	$0.00179143\pi$
$43$	252340.i	0.0112558i	−0.999984	+	0.00562792i	$0.998209\pi$
$44$	0	0
$45$	− 2.15125e7i	− 0.782049i
$46$	0	0
$47$	−4.95171e7	−1.48018	−0.740091	−	0.672507i	$-0.765218\pi$
$47$	−4.95171e7	−1.48018	−0.740091	+	0.672507i	$0.765218\pi$
$48$	0	0
$49$	−310023.	−0.00768266
$50$	0	0
$51$	1.34747e8i	2.78902i
$52$	0	0
$53$	6.63969e7i	1.15586i	0.816085	+	0.577932i	$0.196140\pi$
$53$	6.63969e7i	1.15586i	−0.816085	+	0.577932i	$0.803860\pi$
$54$	0	0
$55$	−2.02597e7	−0.298539
$56$	0	0
$57$	−7.90613e6	−0.0992035
$58$	0	0
$59$	6.15237e7i	0.661011i	0.943804	+	0.330506i	$0.107219\pi$
$59$	6.15237e7i	0.661011i	−0.943804	+	0.330506i	$0.892781\pi$
$60$	0	0
$61$	3.56386e7i	0.329562i	0.986330	+	0.164781i	$0.0526917\pi$
$61$	3.56386e7i	0.329562i	−0.986330	+	0.164781i	$0.947308\pi$
$62$	0	0
$63$	−2.04401e8	−1.63474
$64$	0	0
$65$	2.15371e7	0.149650
$66$	0	0
$67$	1.81742e8i	1.10184i	0.834557	+	0.550921i	$0.185724\pi$
$67$	1.81742e8i	1.10184i	−0.834557	+	0.550921i	$0.814276\pi$
$68$	0	0
$69$	− 2.39066e8i	− 1.26969i
$70$	0	0
$71$	−9.09050e7	−0.424546	−0.212273	−	0.977210i	$-0.568087\pi$
$71$	−9.09050e7	−0.424546	−0.212273	+	0.977210i	$0.568087\pi$
$72$	0	0
$73$	2.62979e8	1.08385	0.541923	−	0.840428i	$-0.317696\pi$
$73$	2.62979e8	1.08385	0.541923	+	0.840428i	$0.317696\pi$
$74$	0	0
$75$	3.44182e8i	1.25607i
$76$	0	0
$77$	1.92498e8i	0.624047i
$78$	0	0
$79$	−1.16503e8	−0.336523	−0.168261	−	0.985742i	$-0.553815\pi$
$79$	−1.16503e8	−0.336523	−0.168261	+	0.985742i	$0.553815\pi$
$80$	0	0
$81$	2.01535e7	0.0520198
$82$	0	0
$83$	− 9.56372e6i	− 0.0221195i	−0.999939	−	0.0110598i	$-0.996479\pi$
$83$	− 9.56372e6i	− 0.0221195i	0.999939	−	0.0110598i	$-0.00352050\pi$
$84$	0	0
$85$	3.93602e8i	0.817847i
$86$	0	0
$87$	−1.00534e9	−1.88139
$88$	0	0
$89$	−6.11827e8	−1.03365	−0.516825	−	0.856091i	$-0.672886\pi$
$89$	−6.11827e8	−1.03365	−0.516825	+	0.856091i	$0.672886\pi$
$90$	0	0
$91$	− 2.04635e8i	− 0.312819i
$92$	0	0
$93$	− 1.68747e9i	− 2.33918i
$94$	0	0
$95$	−2.30942e7	−0.0290902
$96$	0	0
$97$	−2.59313e8	−0.297407	−0.148703	−	0.988882i	$-0.547510\pi$
$97$	−2.59313e8	−0.297407	−0.148703	+	0.988882i	$0.547510\pi$
$98$	0	0
$99$	− 9.82596e8i	− 1.02806i
$100$	0	0
$101$	− 1.56555e9i	− 1.49700i	−0.663137	−	0.748498i	$-0.730775\pi$
$101$	− 1.56555e9i	− 1.49700i	0.663137	−	0.748498i	$-0.269225\pi$
$102$	0	0
$103$	−3.77095e8	−0.330129	−0.165064	−	0.986283i	$-0.552783\pi$
$103$	−3.77095e8	−0.330129	−0.165064	+	0.986283i	$0.552783\pi$
$104$	0	0
$105$	−9.60894e8	−0.771478
$106$	0	0
$107$	2.17717e9i	1.60570i	0.596178	+	0.802852i	$0.296685\pi$
$107$	2.17717e9i	1.60570i	−0.596178	+	0.802852i	$0.703315\pi$
$108$	0	0
$109$	1.50811e9i	1.02333i	0.859185	+	0.511664i	$0.170971\pi$
$109$	1.50811e9i	1.02333i	−0.859185	+	0.511664i	$0.829029\pi$
$110$	0	0
$111$	2.33347e9	1.45898
$112$	0	0
$113$	−1.45355e9	−0.838640	−0.419320	−	0.907838i	$-0.637731\pi$
$113$	−1.45355e9	−0.838640	−0.419320	+	0.907838i	$0.637731\pi$
$114$	0	0
$115$	− 6.98325e8i	− 0.372321i
$116$	0	0
$117$	1.04455e9i	0.515339i
$118$	0	0
$119$	3.73981e9	1.70958
$120$	0	0
$121$	1.43257e9	0.607550
$122$	0	0
$123$	− 4.18443e9i	− 1.64840i
$124$	0	0
$125$	2.30615e9i	0.844877i
$126$	0	0
$127$	2.43679e9	0.831193	0.415597	−	0.909549i	$-0.363573\pi$
$127$	2.43679e9	0.831193	0.415597	+	0.909549i	$0.363573\pi$
$128$	0	0
$129$	−5.75335e7	−0.0182923
$130$	0	0
$131$	− 1.43358e9i	− 0.425305i	−0.977128	−	0.212653i	$-0.931790\pi$
$131$	− 1.43358e9i	− 0.425305i	0.977128	−	0.212653i	$-0.0682102\pi$
$132$	0	0
$133$	2.19430e8i	0.0608083i
$134$	0	0
$135$	1.91602e9	0.496475
$136$	0	0
$137$	−9.30903e8	−0.225768	−0.112884	−	0.993608i	$-0.536009\pi$
$137$	−9.30903e8	−0.225768	−0.112884	+	0.993608i	$0.536009\pi$
$138$	0	0
$139$	− 4.84316e9i	− 1.10043i	−0.835023	−	0.550215i	$-0.814546\pi$
$139$	− 4.84316e9i	− 1.10043i	0.835023	−	0.550215i	$-0.185454\pi$
$140$	0	0
$141$	− 1.12899e10i	− 2.40549i
$142$	0	0
$143$	9.83722e8	0.196725
$144$	0	0
$145$	−2.93666e9	−0.551694
$146$	0	0
$147$	− 7.06852e7i	− 0.0124854i
$148$	0	0
$149$	− 8.53269e9i	− 1.41823i	−0.705091	−	0.709117i	$-0.749094\pi$
$149$	− 8.53269e9i	− 1.41823i	0.705091	−	0.709117i	$-0.250906\pi$
$150$	0	0
$151$	7.14515e9	1.11845	0.559223	−	0.829017i	$-0.311099\pi$
$151$	7.14515e9	1.11845	0.559223	+	0.829017i	$0.311099\pi$
$152$	0	0
$153$	−1.90897e10	−2.81636
$154$	0	0
$155$	− 4.92919e9i	− 0.685935i
$156$	0	0
$157$	− 3.38239e9i	− 0.444299i	−0.975013	−	0.222149i	$-0.928693\pi$
$157$	− 3.38239e9i	− 0.444299i	0.975013	−	0.222149i	$-0.0713072\pi$
$158$	0	0
$159$	−1.51385e10	−1.87843
$160$	0	0
$161$	−6.63514e9	−0.778276
$162$	0	0
$163$	− 9.01515e8i	− 0.100030i	−0.998748	−	0.0500148i	$-0.984073\pi$
$163$	− 9.01515e8i	− 0.100030i	0.998748	−	0.0500148i	$-0.0159269\pi$
$164$	0	0
$165$	− 4.61922e9i	− 0.485166i
$166$	0	0
$167$	−4.05605e9	−0.403533	−0.201767	−	0.979434i	$-0.564668\pi$
$167$	−4.05605e9	−0.403533	−0.201767	+	0.979434i	$0.564668\pi$
$168$	0	0
$169$	9.55875e9	0.901387
$170$	0	0
$171$	− 1.12007e9i	− 0.100176i
$172$	0	0
$173$	− 1.02760e9i	− 0.0872202i	−0.999049	−	0.0436101i	$-0.986114\pi$
$173$	− 1.02760e9i	− 0.0872202i	0.999049	−	0.0436101i	$-0.0138859\pi$
$174$	0	0
$175$	9.55255e9	0.769925
$176$	0	0
$177$	−1.40274e10	−1.07423
$178$	0	0
$179$	1.48472e10i	1.08095i	0.841360	+	0.540476i	$0.181756\pi$
$179$	1.48472e10i	1.08095i	−0.841360	+	0.540476i	$0.818244\pi$
$180$	0	0
$181$	− 2.53270e10i	− 1.75400i	−0.480488	−	0.877001i	$-0.659541\pi$
$181$	− 2.53270e10i	− 1.75400i	0.480488	−	0.877001i	$-0.340459\pi$
$182$	0	0
$183$	−8.12561e9	−0.535582
$184$	0	0
$185$	6.81618e9	0.427827
$186$	0	0
$187$	1.79780e10i	1.07512i
$188$	0	0
$189$	− 1.82050e10i	− 1.03780i
$190$	0	0
$191$	−1.61656e10	−0.878904	−0.439452	−	0.898266i	$-0.644827\pi$
$191$	−1.61656e10	−0.878904	−0.439452	+	0.898266i	$0.644827\pi$
$192$	0	0
$193$	−1.80189e9	−0.0934802	−0.0467401	−	0.998907i	$-0.514883\pi$
$193$	−1.80189e9	−0.0934802	−0.0467401	+	0.998907i	$0.514883\pi$
$194$	0	0
$195$	4.91046e9i	0.243202i
$196$	0	0
$197$	1.86979e10i	0.884495i	0.896893	+	0.442247i	$0.145819\pi$
$197$	1.86979e10i	0.884495i	−0.896893	+	0.442247i	$0.854181\pi$
$198$	0	0
$199$	−2.89890e10	−1.31037	−0.655186	−	0.755468i	$-0.727410\pi$
$199$	−2.89890e10	−1.31037	−0.655186	+	0.755468i	$0.727410\pi$
$200$	0	0
$201$	−4.14373e10	−1.79064
$202$	0	0
$203$	2.79027e10i	1.15323i
$204$	0	0
$205$	− 1.22229e10i	− 0.483374i
$206$	0	0
$207$	3.38688e10	1.28213
$208$	0	0
$209$	−1.05484e9	−0.0382411
$210$	0	0
$211$	− 1.97990e10i	− 0.687657i	−0.939033	−	0.343828i	$-0.888276\pi$
$211$	− 1.97990e10i	− 0.687657i	0.939033	−	0.343828i	$-0.111724\pi$
$212$	0	0
$213$	− 2.07263e10i	− 0.689945i
$214$	0	0
$215$	−1.68058e8	−0.00536398
$216$	0	0
$217$	−4.68347e10	−1.43383
$218$	0	0
$219$	5.99591e10i	1.76140i
$220$	0	0
$221$	− 1.91116e10i	− 0.538928i
$222$	0	0
$223$	−6.78768e10	−1.83802	−0.919009	−	0.394237i	$-0.871009\pi$
$223$	−6.78768e10	−1.83802	−0.919009	+	0.394237i	$0.871009\pi$
$224$	0	0
$225$	−4.87606e10	−1.26837
$226$	0	0
$227$	5.45606e10i	1.36384i	0.731428	+	0.681919i	$0.238854\pi$
$227$	5.45606e10i	1.36384i	−0.731428	+	0.681919i	$0.761146\pi$
$228$	0	0
$229$	− 4.63952e10i	− 1.11484i	−0.830230	−	0.557421i	$-0.811791\pi$
$229$	− 4.63952e10i	− 1.11484i	0.830230	−	0.557421i	$-0.188209\pi$
$230$	0	0
$231$	−4.38895e10	−1.01416
$232$	0	0
$233$	3.91389e8	0.00869975	0.00434988	−	0.999991i	$-0.498615\pi$
$233$	3.91389e8	0.00869975	0.00434988	+	0.999991i	$0.498615\pi$
$234$	0	0
$235$	− 3.29784e10i	− 0.705382i
$236$	0	0
$237$	− 2.65626e10i	− 0.546895i
$238$	0	0
$239$	9.06538e10	1.79720	0.898598	−	0.438772i	$-0.144586\pi$
$239$	9.06538e10	1.79720	0.898598	+	0.438772i	$0.144586\pi$
$240$	0	0
$241$	−6.77663e10	−1.29401	−0.647004	−	0.762486i	$-0.723978\pi$
$241$	−6.77663e10	−1.29401	−0.647004	+	0.762486i	$0.723978\pi$
$242$	0	0
$243$	− 5.20311e10i	− 0.957271i
$244$	0	0
$245$	− 2.06475e8i	− 0.00366118i
$246$	0	0
$247$	1.12135e9	0.0191693
$248$	0	0
$249$	2.18053e9	0.0359472
$250$	0	0
$251$	− 5.47163e10i	− 0.870131i	−0.900399	−	0.435066i	$-0.856725\pi$
$251$	− 5.47163e10i	− 0.870131i	0.900399	−	0.435066i	$-0.143275\pi$
$252$	0	0
$253$	− 3.18965e10i	− 0.489441i
$254$	0	0
$255$	−8.97413e10	−1.32911
$256$	0	0
$257$	−3.40900e10	−0.487447	−0.243724	−	0.969845i	$-0.578369\pi$
$257$	−3.40900e10	−0.487447	−0.243724	+	0.969845i	$0.578369\pi$
$258$	0	0
$259$	− 6.47639e10i	− 0.894302i
$260$	0	0
$261$	− 1.42428e11i	− 1.89983i
$262$	0	0
$263$	7.17361e10	0.924563	0.462282	−	0.886733i	$-0.347031\pi$
$263$	7.17361e10	0.924563	0.462282	+	0.886733i	$0.347031\pi$
$264$	0	0
$265$	−4.42203e10	−0.550828
$266$	0	0
$267$	− 1.39496e11i	− 1.67982i
$268$	0	0
$269$	2.31610e9i	0.0269695i	0.999909	+	0.0134847i	$0.00429246\pi$
$269$	2.31610e9i	0.0269695i	−0.999909	+	0.0134847i	$0.995708\pi$
$270$	0	0
$271$	8.04662e10	0.906258	0.453129	−	0.891445i	$-0.350308\pi$
$271$	8.04662e10	0.906258	0.453129	+	0.891445i	$0.350308\pi$
$272$	0	0
$273$	4.66567e10	0.508373
$274$	0	0
$275$	4.59211e10i	0.484189i
$276$	0	0
$277$	1.65644e11i	1.69051i	0.534367	+	0.845253i	$0.320550\pi$
$277$	1.65644e11i	1.69051i	−0.534367	+	0.845253i	$0.679450\pi$
$278$	0	0
$279$	2.39066e11	2.36210
$280$	0	0
$281$	−2.57177e10	−0.246067	−0.123034	−	0.992402i	$-0.539262\pi$
$281$	−2.57177e10	−0.246067	−0.123034	+	0.992402i	$0.539262\pi$
$282$	0	0
$283$	− 4.33126e10i	− 0.401398i	−0.979653	−	0.200699i	$-0.935679\pi$
$283$	− 4.33126e10i	− 0.401398i	0.979653	−	0.200699i	$-0.0643213\pi$
$284$	0	0
$285$	− 5.26548e9i	− 0.0472755i
$286$	0	0
$287$	−1.16136e11	−1.01041
$288$	0	0
$289$	2.30686e11	1.94528
$290$	0	0
$291$	− 5.91233e10i	− 0.483326i
$292$	0	0
$293$	4.83473e10i	0.383238i	0.981469	+	0.191619i	$0.0613737\pi$
$293$	4.83473e10i	0.383238i	−0.981469	+	0.191619i	$0.938626\pi$
$294$	0	0
$295$	−4.09748e10	−0.315005
$296$	0	0
$297$	8.75154e10	0.652650
$298$	0	0
$299$	3.39076e10i	0.245344i
$300$	0	0
$301$	1.59681e9i	0.0112125i
$302$	0	0
$303$	3.56945e11	2.43282
$304$	0	0
$305$	−2.37353e10	−0.157053
$306$	0	0
$307$	1.37971e11i	0.886470i	0.896406	+	0.443235i	$0.146169\pi$
$307$	1.37971e11i	0.886470i	−0.896406	+	0.443235i	$0.853831\pi$
$308$	0	0
$309$	− 8.59776e10i	− 0.536503i
$310$	0	0
$311$	2.04451e11	1.23928	0.619638	−	0.784887i	$-0.287279\pi$
$311$	2.04451e11	1.23928	0.619638	+	0.784887i	$0.287279\pi$
$312$	0	0
$313$	1.74184e11	1.02579	0.512897	−	0.858450i	$-0.328572\pi$
$313$	1.74184e11	1.02579	0.512897	+	0.858450i	$0.328572\pi$
$314$	0	0
$315$	− 1.36131e11i	− 0.779039i
$316$	0	0
$317$	− 8.42468e10i	− 0.468583i	−0.972166	−	0.234292i	$-0.924723\pi$
$317$	− 8.42468e10i	− 0.468583i	0.972166	−	0.234292i	$-0.0752770\pi$
$318$	0	0
$319$	−1.34134e11	−0.725239
$320$	0	0
$321$	−4.96395e11	−2.60949
$322$	0	0
$323$	2.04933e10i	0.104761i
$324$	0	0
$325$	− 4.88164e10i	− 0.242712i
$326$	0	0
$327$	−3.43850e11	−1.66305
$328$	0	0
$329$	−3.13344e11	−1.47449
$330$	0	0
$331$	2.88777e11i	1.32232i	0.750245	+	0.661160i	$0.229935\pi$
$331$	2.88777e11i	1.32232i	−0.750245	+	0.661160i	$0.770065\pi$
$332$	0	0
$333$	3.30585e11i	1.47327i
$334$	0	0
$335$	−1.21040e11	−0.525084
$336$	0	0
$337$	1.35030e11	0.570289	0.285144	−	0.958485i	$-0.407958\pi$
$337$	1.35030e11	0.570289	0.285144	+	0.958485i	$0.407958\pi$
$338$	0	0
$339$	− 3.31408e11i	− 1.36290i
$340$	0	0
$341$	− 2.25144e11i	− 0.901708i
$342$	0	0
$343$	−2.57319e11	−1.00380
$344$	0	0
$345$	1.59218e11	0.605071
$346$	0	0
$347$	3.91903e10i	0.145110i	0.997364	+	0.0725548i	$0.0231152\pi$
$347$	3.91903e10i	0.145110i	−0.997364	+	0.0725548i	$0.976885\pi$
$348$	0	0
$349$	− 4.58818e10i	− 0.165549i	−0.996568	−	0.0827744i	$-0.973622\pi$
$349$	− 4.58818e10i	− 0.165549i	0.996568	−	0.0827744i	$-0.0263781\pi$
$350$	0	0
$351$	−9.30333e10	−0.327157
$352$	0	0
$353$	5.29590e11	1.81532	0.907660	−	0.419706i	$-0.137867\pi$
$353$	5.29590e11	1.81532	0.907660	+	0.419706i	$0.137867\pi$
$354$	0	0
$355$	− 6.05427e10i	− 0.202318i
$356$	0	0
$357$	8.52677e11i	2.77829i
$358$	0	0
$359$	4.54893e10	0.144539	0.0722693	−	0.997385i	$-0.476976\pi$
$359$	4.54893e10	0.144539	0.0722693	+	0.997385i	$0.476976\pi$
$360$	0	0
$361$	3.21485e11	0.996274
$362$	0	0
$363$	3.26626e11i	0.987350i
$364$	0	0
$365$	1.75144e11i	0.516508i
$366$	0	0
$367$	2.45167e11	0.705447	0.352723	−	0.935728i	$-0.385256\pi$
$367$	2.45167e11	0.705447	0.352723	+	0.935728i	$0.385256\pi$
$368$	0	0
$369$	5.92812e11	1.66456
$370$	0	0
$371$	4.20160e11i	1.15141i
$372$	0	0
$373$	1.60290e11i	0.428762i	0.976750	+	0.214381i	$0.0687734\pi$
$373$	1.60290e11i	0.428762i	−0.976750	+	0.214381i	$0.931227\pi$
$374$	0	0
$375$	−5.25803e11	−1.37304
$376$	0	0
$377$	1.42591e11	0.363544
$378$	0	0
$379$	3.55772e11i	0.885719i	0.896591	+	0.442859i	$0.146036\pi$
$379$	3.55772e11i	0.885719i	−0.896591	+	0.442859i	$0.853964\pi$
$380$	0	0
$381$	5.55589e11i	1.35080i
$382$	0	0
$383$	−4.97008e11	−1.18024	−0.590118	−	0.807317i	$-0.700919\pi$
$383$	−4.97008e11	−1.18024	−0.590118	+	0.807317i	$0.700919\pi$
$384$	0	0
$385$	−1.28204e11	−0.297390
$386$	0	0
$387$	− 8.15083e9i	− 0.0184715i
$388$	0	0
$389$	5.94268e11i	1.31586i	0.753080	+	0.657929i	$0.228568\pi$
$389$	5.94268e11i	1.31586i	−0.753080	+	0.657929i	$0.771432\pi$
$390$	0	0
$391$	−6.19678e11	−1.34082
$392$	0	0
$393$	3.26856e11	0.691178
$394$	0	0
$395$	− 7.75909e10i	− 0.160370i
$396$	0	0
$397$	− 1.18575e11i	− 0.239572i	−0.992800	−	0.119786i	$-0.961779\pi$
$397$	− 1.18575e11i	− 0.239572i	0.992800	−	0.119786i	$-0.0382209\pi$
$398$	0	0
$399$	−5.00300e10	−0.0988217
$400$	0	0
$401$	−5.27598e11	−1.01895	−0.509475	−	0.860485i	$-0.670161\pi$
$401$	−5.27598e11	−1.01895	−0.509475	+	0.860485i	$0.670161\pi$
$402$	0	0
$403$	2.39339e11i	0.452003i
$404$	0	0
$405$	1.34223e10i	0.0247901i
$406$	0	0
$407$	3.11334e11	0.562407
$408$	0	0
$409$	8.96872e10	0.158480	0.0792402	−	0.996856i	$-0.474751\pi$
$409$	8.96872e10	0.158480	0.0792402	+	0.996856i	$0.474751\pi$
$410$	0	0
$411$	− 2.12246e11i	− 0.366903i
$412$	0	0
$413$	3.89322e11i	0.658467i
$414$	0	0
$415$	6.36944e9	0.0105411
$416$	0	0
$417$	1.10424e12	1.78835
$418$	0	0
$419$	9.26538e11i	1.46859i	0.678831	+	0.734294i	$0.262487\pi$
$419$	9.26538e11i	1.46859i	−0.678831	+	0.734294i	$0.737513\pi$
$420$	0	0
$421$	− 1.22692e12i	− 1.90348i	−0.306905	−	0.951740i	$-0.599293\pi$
$421$	− 1.22692e12i	− 1.90348i	0.306905	−	0.951740i	$-0.400707\pi$
$422$	0	0
$423$	1.59945e12	2.42907
$424$	0	0
$425$	8.92146e11	1.32643
$426$	0	0
$427$	2.25521e11i	0.328293i
$428$	0	0
$429$	2.24289e11i	0.319705i
$430$	0	0
$431$	−9.56151e11	−1.33469	−0.667343	−	0.744751i	$-0.732568\pi$
$431$	−9.56151e11	−1.33469	−0.667343	+	0.744751i	$0.732568\pi$
$432$	0	0
$433$	7.42841e10	0.101555	0.0507774	−	0.998710i	$-0.483830\pi$
$433$	7.42841e10	0.101555	0.0507774	+	0.998710i	$0.483830\pi$
$434$	0	0
$435$	− 6.69559e11i	− 0.896577i
$436$	0	0
$437$	− 3.63590e10i	− 0.0476920i
$438$	0	0
$439$	1.66518e11	0.213979	0.106989	−	0.994260i	$-0.465879\pi$
$439$	1.66518e11	0.213979	0.106989	+	0.994260i	$0.465879\pi$
$440$	0	0
$441$	1.00141e10	0.0126077
$442$	0	0
$443$	− 6.41581e11i	− 0.791471i	−0.918365	−	0.395735i	$-0.870490\pi$
$443$	− 6.41581e11i	− 0.791471i	0.918365	−	0.395735i	$-0.129510\pi$
$444$	0	0
$445$	− 4.07477e11i	− 0.492587i
$446$	0	0
$447$	1.94545e12	2.30482
$448$	0	0
$449$	−2.77233e11	−0.321911	−0.160956	−	0.986962i	$-0.551458\pi$
$449$	−2.77233e11	−0.321911	−0.160956	+	0.986962i	$0.551458\pi$
$450$	0	0
$451$	− 5.58291e11i	− 0.635427i
$452$	0	0
$453$	1.62909e12i	1.81763i
$454$	0	0
$455$	1.36287e11	0.149074
$456$	0	0
$457$	−7.55228e11	−0.809944	−0.404972	−	0.914329i	$-0.632719\pi$
$457$	−7.55228e11	−0.809944	−0.404972	+	0.914329i	$0.632719\pi$
$458$	0	0
$459$	− 1.70023e12i	− 1.78793i
$460$	0	0
$461$	9.15740e11i	0.944318i	0.881514	+	0.472159i	$0.156525\pi$
$461$	9.15740e11i	0.944318i	−0.881514	+	0.472159i	$0.843475\pi$
$462$	0	0
$463$	−6.35894e11	−0.643088	−0.321544	−	0.946895i	$-0.604202\pi$
$463$	−6.35894e11	−0.643088	−0.321544	+	0.946895i	$0.604202\pi$
$464$	0	0
$465$	1.12385e12	1.11474
$466$	0	0
$467$	6.17286e11i	0.600566i	0.953850	+	0.300283i	$0.0970811\pi$
$467$	6.17286e11i	0.600566i	−0.953850	+	0.300283i	$0.902919\pi$
$468$	0	0
$469$	1.15007e12i	1.09760i
$470$	0	0
$471$	7.71184e11	0.722045
$472$	0	0
$473$	−7.67618e9	−0.00705132
$474$	0	0
$475$	5.23458e10i	0.0471803i
$476$	0	0
$477$	− 2.14469e12i	− 1.89684i
$478$	0	0
$479$	2.77942e11	0.241238	0.120619	−	0.992699i	$-0.461512\pi$
$479$	2.77942e11	0.241238	0.120619	+	0.992699i	$0.461512\pi$
$480$	0	0
$481$	−3.30963e11	−0.281921
$482$	0	0
$483$	− 1.51281e12i	− 1.26480i
$484$	0	0
$485$	− 1.72702e11i	− 0.141730i
$486$	0	0
$487$	4.99400e11	0.402317	0.201158	−	0.979559i	$-0.435529\pi$
$487$	4.99400e11	0.402317	0.201158	+	0.979559i	$0.435529\pi$
$488$	0	0
$489$	2.05545e11	0.162562
$490$	0	0
$491$	− 2.06241e12i	− 1.60143i	−0.599046	−	0.800715i	$-0.704453\pi$
$491$	− 2.06241e12i	− 1.60143i	0.599046	−	0.800715i	$-0.295547\pi$
$492$	0	0
$493$	2.60593e12i	1.98679i
$494$	0	0
$495$	6.54409e11	0.489921
$496$	0	0
$497$	−5.75247e11	−0.422912
$498$	0	0
$499$	− 1.21912e12i	− 0.880227i	−0.897942	−	0.440113i	$-0.854938\pi$
$499$	− 1.21912e12i	− 0.880227i	0.897942	−	0.440113i	$-0.145062\pi$
$500$	0	0
$501$	− 9.24780e11i	− 0.655796i
$502$	0	0
$503$	−1.80430e12	−1.25676	−0.628380	−	0.777906i	$-0.716282\pi$
$503$	−1.80430e12	−1.25676	−0.628380	+	0.777906i	$0.716282\pi$
$504$	0	0
$505$	1.04266e12	0.713395
$506$	0	0
$507$	2.17940e12i	1.46487i
$508$	0	0
$509$	− 2.03239e11i	− 0.134208i	−0.997746	−	0.0671039i	$-0.978624\pi$
$509$	− 2.03239e11i	− 0.134208i	0.997746	−	0.0671039i	$-0.0213759\pi$
$510$	0	0
$511$	1.66413e12	1.07967
$512$	0	0
$513$	9.97595e10	0.0635955
$514$	0	0
$515$	− 2.51145e11i	− 0.157323i
$516$	0	0
$517$	− 1.50631e12i	− 0.927272i
$518$	0	0
$519$	2.34293e11	0.141745
$520$	0	0
$521$	6.93093e11	0.412118	0.206059	−	0.978540i	$-0.433936\pi$
$521$	6.93093e11	0.412118	0.206059	+	0.978540i	$0.433936\pi$
$522$	0	0
$523$	1.97956e12i	1.15694i	0.815704	+	0.578470i	$0.196350\pi$
$523$	1.97956e12i	1.15694i	−0.815704	+	0.578470i	$0.803650\pi$
$524$	0	0
$525$	2.17798e12i	1.25123i
$526$	0	0
$527$	−4.37406e12	−2.47022
$528$	0	0
$529$	−7.01725e11	−0.389598
$530$	0	0
$531$	− 1.98728e12i	− 1.08476i
$532$	0	0
$533$	5.93491e11i	0.318524i
$534$	0	0
$535$	−1.45000e12	−0.765200
$536$	0	0
$537$	−3.38516e12	−1.75669
$538$	0	0
$539$	− 9.43090e9i	− 0.00481287i
$540$	0	0
$541$	− 2.95899e12i	− 1.48510i	−0.669790	−	0.742551i	$-0.733616\pi$
$541$	− 2.95899e12i	− 1.48510i	0.669790	−	0.742551i	$-0.266384\pi$
$542$	0	0
$543$	5.77456e12	2.85049
$544$	0	0
$545$	−1.00440e12	−0.487668
$546$	0	0
$547$	3.27526e12i	1.56424i	0.623130	+	0.782118i	$0.285861\pi$
$547$	3.27526e12i	1.56424i	−0.623130	+	0.782118i	$0.714139\pi$
$548$	0	0
$549$	− 1.15116e12i	− 0.540831i
$550$	0	0
$551$	−1.52901e11	−0.0706687
$552$	0	0
$553$	−7.37230e11	−0.335228
$554$	0	0
$555$	1.55409e12i	0.695276i
$556$	0	0
$557$	− 3.76405e12i	− 1.65694i	−0.560034	−	0.828470i	$-0.689212\pi$
$557$	− 3.76405e12i	− 1.65694i	0.560034	−	0.828470i	$-0.310788\pi$
$558$	0	0
$559$	8.16017e9	0.00353465
$560$	0	0
$561$	−4.09899e12	−1.74721
$562$	0	0
$563$	2.34987e12i	0.985725i	0.870107	+	0.492863i	$0.164049\pi$
$563$	2.34987e12i	0.985725i	−0.870107	+	0.492863i	$0.835951\pi$
$564$	0	0
$565$	− 9.68061e11i	− 0.399655i
$566$	0	0
$567$	1.27532e11	0.0518196
$568$	0	0
$569$	−2.66701e12	−1.06664	−0.533322	−	0.845912i	$-0.679057\pi$
$569$	−2.66701e12	−1.06664	−0.533322	+	0.845912i	$0.679057\pi$
$570$	0	0
$571$	1.72342e12i	0.678469i	0.940702	+	0.339234i	$0.110168\pi$
$571$	1.72342e12i	0.678469i	−0.940702	+	0.339234i	$0.889832\pi$
$572$	0	0
$573$	− 3.68576e12i	− 1.42834i
$574$	0	0
$575$	−1.58284e12	−0.603853
$576$	0	0
$577$	1.55856e12	0.585374	0.292687	−	0.956208i	$-0.405451\pi$
$577$	1.55856e12	0.585374	0.292687	+	0.956208i	$0.405451\pi$
$578$	0	0
$579$	− 4.10830e11i	− 0.151918i
$580$	0	0
$581$	− 6.05192e10i	− 0.0220344i
$582$	0	0
$583$	−2.01979e12	−0.724100
$584$	0	0
$585$	−6.95670e11	−0.245585
$586$	0	0
$587$	2.16623e12i	0.753065i	0.926403	+	0.376533i	$0.122884\pi$
$587$	2.16623e12i	0.753065i	−0.926403	+	0.376533i	$0.877116\pi$
$588$	0	0
$589$	− 2.56643e11i	− 0.0878641i
$590$	0	0
$591$	−4.26313e12	−1.43742
$592$	0	0
$593$	3.56244e12	1.18304	0.591522	−	0.806289i	$-0.298527\pi$
$593$	3.56244e12	1.18304	0.591522	+	0.806289i	$0.298527\pi$
$594$	0	0
$595$	2.49071e12i	0.814699i
$596$	0	0
$597$	− 6.60949e12i	− 2.12953i
$598$	0	0
$599$	−1.54407e12	−0.490056	−0.245028	−	0.969516i	$-0.578797\pi$
$599$	−1.54407e12	−0.490056	−0.245028	+	0.969516i	$0.578797\pi$
$600$	0	0
$601$	1.05277e12	0.329155	0.164577	−	0.986364i	$-0.447374\pi$
$601$	1.05277e12	0.329155	0.164577	+	0.986364i	$0.447374\pi$
$602$	0	0
$603$	− 5.87046e12i	− 1.80819i
$604$	0	0
$605$	9.54092e11i	0.289528i
$606$	0	0
$607$	−4.47471e12	−1.33787	−0.668937	−	0.743319i	$-0.733251\pi$
$607$	−4.47471e12	−1.33787	−0.668937	+	0.743319i	$0.733251\pi$
$608$	0	0
$609$	−6.36182e12	−1.87415
$610$	0	0
$611$	1.60129e12i	0.464818i
$612$	0	0
$613$	6.01862e12i	1.72157i	0.508969	+	0.860785i	$0.330027\pi$
$613$	6.01862e12i	1.72157i	−0.508969	+	0.860785i	$0.669973\pi$
$614$	0	0
$615$	2.78683e12	0.785547
$616$	0	0
$617$	−2.16191e12	−0.600557	−0.300278	−	0.953852i	$-0.597080\pi$
$617$	−2.16191e12	−0.600557	−0.300278	+	0.953852i	$0.597080\pi$
$618$	0	0
$619$	4.16924e12i	1.14143i	0.821149	+	0.570714i	$0.193334\pi$
$619$	4.16924e12i	1.14143i	−0.821149	+	0.570714i	$0.806666\pi$
$620$	0	0
$621$	3.01654e12i	0.813948i
$622$	0	0
$623$	−3.87164e12	−1.02967
$624$	0	0
$625$	1.41248e12	0.370273
$626$	0	0
$627$	− 2.40504e11i	− 0.0621468i
$628$	0	0
$629$	− 6.04853e12i	− 1.54071i
$630$	0	0
$631$	−4.10037e12	−1.02965	−0.514826	−	0.857295i	$-0.672143\pi$
$631$	−4.10037e12	−1.02965	−0.514826	+	0.857295i	$0.672143\pi$
$632$	0	0
$633$	4.51417e12	1.11753
$634$	0	0
$635$	1.62290e12i	0.396106i
$636$	0	0
$637$	1.00255e10i	0.00241257i
$638$	0	0
$639$	2.93632e12	0.696706
$640$	0	0
$641$	1.87188e12	0.437942	0.218971	−	0.975731i	$-0.429730\pi$
$641$	1.87188e12	0.437942	0.218971	+	0.975731i	$0.429730\pi$
$642$	0	0
$643$	− 1.34166e12i	− 0.309524i	−0.987952	−	0.154762i	$-0.950539\pi$
$643$	− 1.34166e12i	− 0.309524i	0.987952	−	0.154762i	$-0.0494610\pi$
$644$	0	0
$645$	− 3.83173e10i	− 0.00871719i
$646$	0	0
$647$	4.94367e12	1.10912	0.554562	−	0.832142i	$-0.312886\pi$
$647$	4.94367e12	1.10912	0.554562	+	0.832142i	$0.312886\pi$
$648$	0	0
$649$	−1.87155e12	−0.414096
$650$	0	0
$651$	− 1.06783e13i	− 2.33017i
$652$	0	0
$653$	2.67139e12i	0.574947i	0.957789	+	0.287474i	$0.0928154\pi$
$653$	2.67139e12i	0.574947i	−0.957789	+	0.287474i	$0.907185\pi$
$654$	0	0
$655$	9.54763e11	0.202679
$656$	0	0
$657$	−8.49447e12	−1.77866
$658$	0	0
$659$	5.50089e12i	1.13618i	0.822965	+	0.568092i	$0.192318\pi$
$659$	5.50089e12i	1.13618i	−0.822965	+	0.568092i	$0.807682\pi$
$660$	0	0
$661$	− 1.06937e12i	− 0.217881i	−0.994048	−	0.108941i	$-0.965254\pi$
$661$	− 1.06937e12i	− 0.217881i	0.994048	−	0.108941i	$-0.0347459\pi$
$662$	0	0
$663$	4.35744e12	0.875831
$664$	0	0
$665$	−1.46140e11	−0.0289783
$666$	0	0
$667$	− 4.62342e12i	− 0.904476i
$668$	0	0
$669$	− 1.54759e13i	− 2.98702i
$670$	0	0
$671$	−1.08413e12	−0.206457
$672$	0	0
$673$	−4.96567e12	−0.933062	−0.466531	−	0.884505i	$-0.654496\pi$
$673$	−4.96567e12	−0.933062	−0.466531	+	0.884505i	$0.654496\pi$
$674$	0	0
$675$	− 4.34289e12i	− 0.805214i
$676$	0	0
$677$	2.75739e12i	0.504486i	0.967664	+	0.252243i	$0.0811682\pi$
$677$	2.75739e12i	0.504486i	−0.967664	+	0.252243i	$0.918832\pi$
$678$	0	0
$679$	−1.64093e12	−0.296262
$680$	0	0
$681$	−1.24398e13	−2.21642
$682$	0	0
$683$	5.05528e12i	0.888898i	0.895804	+	0.444449i	$0.146601\pi$
$683$	5.05528e12i	0.888898i	−0.895804	+	0.444449i	$0.853399\pi$
$684$	0	0
$685$	− 6.19982e11i	− 0.107590i
$686$	0	0
$687$	1.05781e13	1.81177
$688$	0	0
$689$	2.14714e12	0.362973
$690$	0	0
$691$	− 2.55414e12i	− 0.426181i	−0.977033	−	0.213090i	$-0.931647\pi$
$691$	− 2.55414e12i	− 0.426181i	0.977033	−	0.213090i	$-0.0683529\pi$
$692$	0	0
$693$	− 6.21787e12i	− 1.02410i
$694$	0	0
$695$	3.22554e12	0.524410
$696$	0	0
$697$	−1.08464e13	−1.74075
$698$	0	0
$699$	8.92367e10i	0.0141383i
$700$	0	0
$701$	− 8.11552e12i	− 1.26936i	−0.772774	−	0.634681i	$-0.781132\pi$
$701$	− 8.11552e12i	− 1.26936i	0.772774	−	0.634681i	$-0.218868\pi$
$702$	0	0
$703$	3.54892e11	0.0548020
$704$	0	0
$705$	7.51908e12	1.14634
$706$	0	0
$707$	− 9.90680e12i	− 1.49123i
$708$	0	0
$709$	2.04394e12i	0.303781i	0.988397	+	0.151890i	$0.0485360\pi$
$709$	2.04394e12i	0.303781i	−0.988397	+	0.151890i	$0.951464\pi$
$710$	0	0
$711$	3.76316e12	0.552254
$712$	0	0
$713$	7.76041e12	1.12456
$714$	0	0
$715$	6.55159e11i	0.0937496i
$716$	0	0
$717$	2.06691e13i	2.92069i
$718$	0	0
$719$	1.24231e13	1.73361	0.866804	−	0.498648i	$-0.166170\pi$
$719$	1.24231e13	1.73361	0.866804	+	0.498648i	$0.166170\pi$
$720$	0	0
$721$	−2.38626e12	−0.328858
$722$	0	0
$723$	− 1.54507e13i	− 2.10294i
$724$	0	0
$725$	6.65630e12i	0.894771i
$726$	0	0
$727$	5.37434e12	0.713543	0.356771	−	0.934192i	$-0.383878\pi$
$727$	5.37434e12	0.713543	0.356771	+	0.934192i	$0.383878\pi$
$728$	0	0
$729$	1.22598e13	1.60771
$730$	0	0
$731$	1.49131e11i	0.0193171i
$732$	0	0
$733$	1.28618e11i	0.0164563i	0.999966	+	0.00822815i	$0.00261913\pi$
$733$	1.28618e11i	0.0164563i	−0.999966	+	0.00822815i	$0.997381\pi$
$734$	0	0
$735$	4.70764e10	0.00594990
$736$	0	0
$737$	−5.52860e12	−0.690258
$738$	0	0
$739$	1.36726e13i	1.68636i	0.537630	+	0.843181i	$0.319320\pi$
$739$	1.36726e13i	1.68636i	−0.537630	+	0.843181i	$0.680680\pi$
$740$	0	0
$741$	2.55668e11i	0.0311527i
$742$	0	0
$743$	−1.31581e13	−1.58396	−0.791981	−	0.610546i	$-0.790950\pi$
$743$	−1.31581e13	−1.58396	−0.791981	+	0.610546i	$0.790950\pi$
$744$	0	0
$745$	5.68277e12	0.675861
$746$	0	0
$747$	3.08918e11i	0.0362995i
$748$	0	0
$749$	1.37771e13i	1.59952i
$750$	0	0
$751$	−2.08682e12	−0.239389	−0.119695	−	0.992811i	$-0.538192\pi$
$751$	−2.08682e12	−0.239389	−0.119695	+	0.992811i	$0.538192\pi$
$752$	0	0
$753$	1.24753e13	1.41408
$754$	0	0
$755$	4.75867e12i	0.532997i
$756$	0	0
$757$	− 5.54660e12i	− 0.613897i	−0.951726	−	0.306948i	$-0.900692\pi$
$757$	− 5.54660e12i	− 0.613897i	0.951726	−	0.306948i	$-0.0993079\pi$
$758$	0	0
$759$	7.27239e12	0.795407
$760$	0	0
$761$	1.13451e12	0.122625	0.0613123	−	0.998119i	$-0.480471\pi$
$761$	1.13451e12	0.122625	0.0613123	+	0.998119i	$0.480471\pi$
$762$	0	0
$763$	9.54335e12i	1.01939i
$764$	0	0
$765$	− 1.27137e13i	− 1.34214i
$766$	0	0
$767$	1.98955e12	0.207576
$768$	0	0
$769$	−2.61602e12	−0.269757	−0.134878	−	0.990862i	$-0.543064\pi$
$769$	−2.61602e12	−0.269757	−0.134878	+	0.990862i	$0.543064\pi$
$770$	0	0
$771$	− 7.77252e12i	− 0.792167i
$772$	0	0
$773$	5.33154e10i	0.00537088i	0.999996	+	0.00268544i	$0.000854803\pi$
$773$	5.33154e10i	0.00537088i	−0.999996	+	0.00268544i	$0.999145\pi$
$774$	0	0
$775$	−1.11726e13	−1.11249
$776$	0	0
$777$	1.47662e13	1.45336
$778$	0	0
$779$	− 6.36400e11i	− 0.0619172i
$780$	0	0
$781$	− 2.76533e12i	− 0.265961i
$782$	0	0
$783$	1.26854e13	1.20608
$784$	0	0
$785$	2.25267e12	0.211731
$786$	0	0
$787$	− 3.30783e12i	− 0.307367i	−0.988120	−	0.153683i	$-0.950886\pi$
$787$	− 3.30783e12i	− 0.307367i	0.988120	−	0.153683i	$-0.0491136\pi$
$788$	0	0
$789$	1.63558e13i	1.50254i
$790$	0	0
$791$	−9.19804e12	−0.835412
$792$	0	0
$793$	1.15248e12	0.103492
$794$	0	0
$795$	− 1.00822e13i	− 0.895169i
$796$	0	0
$797$	− 3.86873e12i	− 0.339630i	−0.985476	−	0.169815i	$-0.945683\pi$
$797$	− 3.86873e12i	− 0.339630i	0.985476	−	0.169815i	$-0.0543170\pi$
$798$	0	0
$799$	−2.92643e13	−2.54026
$800$	0	0
$801$	1.97626e13	1.69628
$802$	0	0
$803$	7.99981e12i	0.678984i
$804$	0	0
$805$	− 4.41900e12i	− 0.370888i
$806$	0	0
$807$	−5.28071e11	−0.0438290
$808$	0	0
$809$	−7.39526e12	−0.606995	−0.303497	−	0.952832i	$-0.598154\pi$
$809$	−7.39526e12	−0.606995	−0.303497	+	0.952832i	$0.598154\pi$
$810$	0	0
$811$	− 8.92803e12i	− 0.724706i	−0.932041	−	0.362353i	$-0.881974\pi$
$811$	− 8.92803e12i	− 0.724706i	0.932041	−	0.362353i	$-0.118026\pi$
$812$	0	0
$813$	1.83463e13i	1.47279i
$814$	0	0
$815$	6.00409e11	0.0476692
$816$	0	0
$817$	−8.75014e9	−0.000687093 0
$818$	0	0
$819$	6.60991e12i	0.513355i
$820$	0	0
$821$	1.05534e13i	0.810674i	0.914167	+	0.405337i	$0.132846\pi$
$821$	1.05534e13i	0.810674i	−0.914167	+	0.405337i	$0.867154\pi$
$822$	0	0
$823$	9.16030e12	0.696002	0.348001	−	0.937494i	$-0.386861\pi$
$823$	9.16030e12	0.696002	0.348001	+	0.937494i	$0.386861\pi$
$824$	0	0
$825$	−1.04700e13	−0.786872
$826$	0	0
$827$	2.44096e13i	1.81462i	0.420462	+	0.907310i	$0.361868\pi$
$827$	2.44096e13i	1.81462i	−0.420462	+	0.907310i	$0.638132\pi$
$828$	0	0
$829$	9.12051e12i	0.670693i	0.942095	+	0.335346i	$0.108853\pi$
$829$	9.12051e12i	0.670693i	−0.942095	+	0.335346i	$0.891147\pi$
$830$	0	0
$831$	−3.77668e13	−2.74730
$832$	0	0
$833$	−1.83222e11	−0.0131848
$834$	0	0
$835$	− 2.70133e12i	− 0.192304i
$836$	0	0
$837$	2.12925e13i	1.49955i
$838$	0	0
$839$	−6.07575e12	−0.423322	−0.211661	−	0.977343i	$-0.567887\pi$
$839$	−6.07575e12	−0.423322	−0.211661	+	0.977343i	$0.567887\pi$
$840$	0	0
$841$	−4.93572e12	−0.340226
$842$	0	0
$843$	− 5.86364e12i	− 0.399893i
$844$	0	0
$845$	6.36613e12i	0.429556i
$846$	0	0
$847$	9.06531e12	0.605212
$848$	0	0
$849$	9.87527e12	0.652325
$850$	0	0
$851$	1.07312e13i	0.701402i
$852$	0	0
$853$	− 1.67917e13i	− 1.08599i	−0.839737	−	0.542993i	$-0.817291\pi$
$853$	− 1.67917e13i	− 1.08599i	0.839737	−	0.542993i	$-0.182709\pi$
$854$	0	0
$855$	7.45966e11	0.0477388
$856$	0	0
$857$	2.77707e13	1.75862	0.879312	−	0.476246i	$-0.158003\pi$
$857$	2.77707e13	1.75862	0.879312	+	0.476246i	$0.158003\pi$
$858$	0	0
$859$	− 1.85405e12i	− 0.116186i	−0.998311	−	0.0580928i	$-0.981498\pi$
$859$	− 1.85405e12i	− 0.116186i	0.998311	−	0.0580928i	$-0.0185019\pi$
$860$	0	0
$861$	− 2.64790e13i	− 1.64206i
$862$	0	0
$863$	8.72142e12	0.535228	0.267614	−	0.963526i	$-0.413765\pi$
$863$	8.72142e12	0.535228	0.267614	+	0.963526i	$0.413765\pi$
$864$	0	0
$865$	6.84383e11	0.0415649
$866$	0	0
$867$	5.25964e13i	3.16133i
$868$	0	0
$869$	− 3.54402e12i	− 0.210818i
$870$	0	0
$871$	5.87718e12	0.346009
$872$	0	0
$873$	8.37606e12	0.488063
$874$	0	0
$875$	1.45933e13i	0.841625i
$876$	0	0
$877$	2.76222e13i	1.57674i	0.615203	+	0.788369i	$0.289074\pi$
$877$	2.76222e13i	1.57674i	−0.615203	+	0.788369i	$0.710926\pi$
$878$	0	0
$879$	−1.10232e13	−0.622813
$880$	0	0
$881$	−1.00186e13	−0.560295	−0.280147	−	0.959957i	$-0.590383\pi$
$881$	−1.00186e13	−0.560295	−0.280147	+	0.959957i	$0.590383\pi$
$882$	0	0
$883$	9.43702e12i	0.522410i	0.965283	+	0.261205i	$0.0841199\pi$
$883$	9.43702e12i	0.522410i	−0.965283	+	0.261205i	$0.915880\pi$
$884$	0	0
$885$	− 9.34226e12i	− 0.511926i
$886$	0	0
$887$	3.75635e12	0.203756	0.101878	−	0.994797i	$-0.467515\pi$
$887$	3.75635e12	0.203756	0.101878	+	0.994797i	$0.467515\pi$
$888$	0	0
$889$	1.54200e13	0.827994
$890$	0	0
$891$	6.13070e11i	0.0325882i
$892$	0	0
$893$	− 1.71706e12i	− 0.0903552i
$894$	0	0
$895$	−9.88824e12	−0.515128
$896$	0	0
$897$	−7.73092e12	−0.398718
$898$	0	0
$899$	− 3.26348e13i	− 1.66634i
$900$	0	0
$901$	3.92402e13i	1.98367i
$902$	0	0
$903$	−3.64072e11	−0.0182219
$904$	0	0
$905$	1.68678e13	0.835871
$906$	0	0
$907$	− 2.36795e13i	− 1.16182i	−0.813967	−	0.580911i	$-0.802696\pi$
$907$	− 2.36795e13i	− 1.16182i	0.813967	−	0.580911i	$-0.197304\pi$
$908$	0	0
$909$	5.05688e13i	2.45666i
$910$	0	0
$911$	1.90030e13	0.914090	0.457045	−	0.889443i	$-0.348908\pi$
$911$	1.90030e13	0.914090	0.457045	+	0.889443i	$0.348908\pi$
$912$	0	0
$913$	2.90928e11	0.0138570
$914$	0	0
$915$	− 5.41165e12i	− 0.255232i
$916$	0	0
$917$	− 9.07168e12i	− 0.423668i
$918$	0	0
$919$	−5.56992e12	−0.257590	−0.128795	−	0.991671i	$-0.541111\pi$
$919$	−5.56992e12	−0.257590	−0.128795	+	0.991671i	$0.541111\pi$
$920$	0	0
$921$	−3.14573e13	−1.44063
$922$	0	0
$923$	2.93968e12i	0.133319i
$924$	0	0
$925$	− 1.54497e13i	− 0.693876i
$926$	0	0
$927$	1.21805e13	0.541761
$928$	0	0
$929$	3.54293e13	1.56060	0.780301	−	0.625404i	$-0.215066\pi$
$929$	3.54293e13	1.56060	0.780301	+	0.625404i	$0.215066\pi$
$930$	0	0
$931$	− 1.07504e10i	0 0.000468975i
$932$	0	0
$933$	4.66149e13i	2.01399i
$934$	0	0
$935$	−1.19734e13	−0.512347
$936$	0	0
$937$	2.45592e13	1.04084	0.520422	−	0.853909i	$-0.325775\pi$
$937$	2.45592e13	1.04084	0.520422	+	0.853909i	$0.325775\pi$
$938$	0	0
$939$	3.97140e13i	1.66705i
$940$	0	0
$941$	2.00516e13i	0.833675i	0.908981	+	0.416838i	$0.136862\pi$
$941$	2.00516e13i	0.833675i	−0.908981	+	0.416838i	$0.863138\pi$
$942$	0	0
$943$	1.92435e13	0.792468
$944$	0	0
$945$	1.21246e13	0.494564
$946$	0	0
$947$	− 1.51295e13i	− 0.611294i	−0.952145	−	0.305647i	$-0.901127\pi$
$947$	− 1.51295e13i	− 0.611294i	0.952145	−	0.305647i	$-0.0988727\pi$
$948$	0	0
$949$	− 8.50420e12i	− 0.340358i
$950$	0	0
$951$	1.92083e13	0.761510
$952$	0	0
$953$	−2.18751e13	−0.859075	−0.429538	−	0.903049i	$-0.641323\pi$
$953$	−2.18751e13	−0.859075	−0.429538	+	0.903049i	$0.641323\pi$
$954$	0	0
$955$	− 1.07663e13i	− 0.418843i
$956$	0	0
$957$	− 3.05826e13i	− 1.17861i
$958$	0	0
$959$	−5.89076e12	−0.224899
$960$	0	0
$961$	2.83379e13	1.07180
$962$	0	0
$963$	− 7.03248e13i	− 2.63506i
$964$	0	0
$965$	− 1.20006e12i	− 0.0445480i
$966$	0	0
$967$	3.32239e13	1.22189	0.610945	−	0.791673i	$-0.290790\pi$
$967$	3.32239e13	1.22189	0.610945	+	0.791673i	$0.290790\pi$
$968$	0	0
$969$	−4.67247e12	−0.170251
$970$	0	0
$971$	− 3.71430e13i	− 1.34088i	−0.741963	−	0.670441i	$-0.766105\pi$
$971$	− 3.71430e13i	− 1.34088i	0.741963	−	0.670441i	$-0.233895\pi$
$972$	0	0
$973$	− 3.06475e13i	− 1.09619i
$974$	0	0
$975$	1.11301e13	0.394439
$976$	0	0
$977$	1.46872e13	0.515719	0.257860	−	0.966182i	$-0.416983\pi$
$977$	1.46872e13	0.515719	0.257860	+	0.966182i	$0.416983\pi$
$978$	0	0
$979$	− 1.86118e13i	− 0.647538i
$980$	0	0
$981$	− 4.87136e13i	− 1.67934i
$982$	0	0
$983$	−2.18746e13	−0.747220	−0.373610	−	0.927586i	$-0.621880\pi$
$983$	−2.18746e13	−0.747220	−0.373610	+	0.927586i	$0.621880\pi$
$984$	0	0
$985$	−1.24528e13	−0.421507
$986$	0	0
$987$	− 7.14425e13i	− 2.39624i
$988$	0	0
$989$	− 2.64588e11i	− 0.00879399i
$990$	0	0
$991$	4.20089e13	1.38360	0.691799	−	0.722091i	$-0.256819\pi$
$991$	4.20089e13	1.38360	0.691799	+	0.722091i	$0.256819\pi$
$992$	0	0
$993$	−6.58411e13	−2.14895
$994$	0	0
$995$	− 1.93067e13i	− 0.624458i
$996$	0	0
$997$	− 2.69704e12i	− 0.0864489i	−0.999065	−	0.0432245i	$-0.986237\pi$
$997$	− 2.69704e12i	− 0.0864489i	0.999065	−	0.0432245i	$-0.0137631\pi$
$998$	0	0
$999$	−2.94437e13	−0.935292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.10.b.j.129.2		2
4.3	odd	2		256.10.b.b.129.1		2
8.3	odd	2		256.10.b.b.129.2		2
8.5	even	2	inner	256.10.b.j.129.1		2
16.3	odd	4		64.10.a.i.1.1		1
16.5	even	4		4.10.a.a.1.1	✓	1
16.11	odd	4		16.10.a.a.1.1		1
16.13	even	4		64.10.a.a.1.1		1
48.5	odd	4		36.10.a.b.1.1		1
48.11	even	4		144.10.a.j.1.1		1
80.27	even	4		400.10.c.a.49.2		2
80.37	odd	4		100.10.c.a.49.1		2
80.43	even	4		400.10.c.a.49.1		2
80.53	odd	4		100.10.c.a.49.2		2
80.59	odd	4		400.10.a.k.1.1		1
80.69	even	4		100.10.a.a.1.1		1
112.5	odd	12		196.10.e.b.165.1		2
112.37	even	12		196.10.e.a.165.1		2
112.53	even	12		196.10.e.a.177.1		2
112.69	odd	4		196.10.a.a.1.1		1
112.101	odd	12		196.10.e.b.177.1		2
144.5	odd	12		324.10.e.b.217.1		2
144.85	even	12		324.10.e.e.217.1		2
144.101	odd	12		324.10.e.b.109.1		2
144.133	even	12		324.10.e.e.109.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.10.a.a.1.1	✓	1	16.5	even	4
16.10.a.a.1.1		1	16.11	odd	4
36.10.a.b.1.1		1	48.5	odd	4
64.10.a.a.1.1		1	16.13	even	4
64.10.a.i.1.1		1	16.3	odd	4
100.10.a.a.1.1		1	80.69	even	4
100.10.c.a.49.1		2	80.37	odd	4
100.10.c.a.49.2		2	80.53	odd	4
144.10.a.j.1.1		1	48.11	even	4
196.10.a.a.1.1		1	112.69	odd	4
196.10.e.a.165.1		2	112.37	even	12
196.10.e.a.177.1		2	112.53	even	12
196.10.e.b.165.1		2	112.5	odd	12
196.10.e.b.177.1		2	112.101	odd	12
256.10.b.b.129.1		2	4.3	odd	2
256.10.b.b.129.2		2	8.3	odd	2
256.10.b.j.129.1		2	8.5	even	2	inner
256.10.b.j.129.2		2	1.1	even	1	trivial
324.10.e.b.109.1		2	144.101	odd	12
324.10.e.b.217.1		2	144.5	odd	12
324.10.e.e.109.1		2	144.133	even	12
324.10.e.e.217.1		2	144.85	even	12
400.10.a.k.1.1		1	80.59	odd	4
400.10.c.a.49.1		2	80.43	even	4
400.10.c.a.49.2		2	80.27	even	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+228.000i q^{3} +666.000i q^{5} +6328.00 q^{7} -32301.0 q^{9} +O(q^{10})\)	\(q+228.000i q^{3} +666.000i q^{5} +6328.00 q^{7} -32301.0 q^{9} +30420.0i q^{11} -32338.0i q^{13} -151848. q^{15} +590994. q^{17} +34676.0i q^{19} +1.44278e6i q^{21} -1.04854e6 q^{23} +1.50957e6 q^{25} -2.87690e6i q^{27} +4.40941e6i q^{29} -7.40118e6 q^{31} -6.93576e6 q^{33} +4.21445e6i q^{35} -1.02345e7i q^{37} +7.37306e6 q^{39} -1.83527e7 q^{41} +252340. i q^{43} -2.15125e7i q^{45} -4.95171e7 q^{47} -310023. q^{49} +1.34747e8i q^{51} +6.63969e7i q^{53} -2.02597e7 q^{55} -7.90613e6 q^{57} +6.15237e7i q^{59} +3.56386e7i q^{61} -2.04401e8 q^{63} +2.15371e7 q^{65} +1.81742e8i q^{67} -2.39066e8i q^{69} -9.09050e7 q^{71} +2.62979e8 q^{73} +3.44182e8i q^{75} +1.92498e8i q^{77} -1.16503e8 q^{79} +2.01535e7 q^{81} -9.56372e6i q^{83} +3.93602e8i q^{85} -1.00534e9 q^{87} -6.11827e8 q^{89} -2.04635e8i q^{91} -1.68747e9i q^{93} -2.30942e7 q^{95} -2.59313e8 q^{97} -9.82596e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12656 q^{7} - 64602 q^{9}+O(q^{10}) \) 2 * q + 12656 * q^7 - 64602 * q^9	\( 2 q + 12656 q^{7} - 64602 q^{9} - 303696 q^{15} + 1181988 q^{17} - 2097072 q^{23} + 3019138 q^{25} - 14802368 q^{31} - 13871520 q^{33} + 14746128 q^{39} - 36705492 q^{41} - 99034272 q^{47} - 620046 q^{49} - 40519440 q^{55} - 15812256 q^{57} - 408801456 q^{63} + 43074216 q^{65} - 181809936 q^{71} + 525957356 q^{73} - 233005664 q^{79} + 40307058 q^{81} - 2010689136 q^{87} - 1223653428 q^{89} - 46188432 q^{95} - 518625596 q^{97}+O(q^{100}) \) 2 * q + 12656 * q^7 - 64602 * q^9 - 303696 * q^15 + 1181988 * q^17 - 2097072 * q^23 + 3019138 * q^25 - 14802368 * q^31 - 13871520 * q^33 + 14746128 * q^39 - 36705492 * q^41 - 99034272 * q^47 - 620046 * q^49 - 40519440 * q^55 - 15812256 * q^57 - 408801456 * q^63 + 43074216 * q^65 - 181809936 * q^71 + 525957356 * q^73 - 233005664 * q^79 + 40307058 * q^81 - 2010689136 * q^87 - 1223653428 * q^89 - 46188432 * q^95 - 518625596 * q^97

Introduction

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