LMFDB - Embedded newform 288.8.f.a.143.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(143,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("288.143");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.f (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:	$89.9668873394$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.6
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-294.266 q^{5} +328.633i q^{7} +O(q^{10})$	$q-294.266 q^{5} +328.633i q^{7} +2680.12i q^{11} -1486.01i q^{13} -35589.5i q^{17} -16097.0 q^{19} -11593.2 q^{23} +8467.20 q^{25} -204036. q^{29} +149568. i q^{31} -96705.4i q^{35} +61595.2i q^{37} -232923. i q^{41} -619370. q^{43} +1.07968e6 q^{47} +715543. q^{49} -713736. q^{53} -788666. i q^{55} +1.30458e6i q^{59} +1.52680e6i q^{61} +437282. i q^{65} +4.81113e6 q^{67} +4.36754e6 q^{71} -1.23348e6 q^{73} -880774. q^{77} -5.34474e6i q^{79} -6.96837e6i q^{83} +1.04728e7i q^{85} -166492. i q^{89} +488352. q^{91} +4.73678e6 q^{95} +7.05299e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q+O(q^{10}) $ 28 * q	$ 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−294.266	−1.05280	−0.526398	−	0.850238i	$-0.676458\pi$
$5$	−294.266	−1.05280	−0.526398	+	0.850238i	$0.676458\pi$
$6$	0	0
$7$	328.633i	0.362133i	0.983471	+	0.181066i	$0.0579549\pi$
$7$	328.633i	0.362133i	−0.983471	+	0.181066i	$0.942045\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2680.12i	0.607126i	0.952811	+	0.303563i	$0.0981763\pi$
$11$	2680.12i	0.607126i	−0.952811	+	0.303563i	$0.901824\pi$
$12$	0	0
$13$	− 1486.01i	− 0.187595i	−0.995591	−	0.0937973i	$-0.970099\pi$
$13$	− 1486.01i	− 0.187595i	0.995591	−	0.0937973i	$-0.0299005\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 35589.5i	− 1.75692i	−0.477819	−	0.878458i	$-0.658573\pi$
$17$	− 35589.5i	− 1.75692i	0.477819	−	0.878458i	$-0.341427\pi$
$18$	0	0
$19$	−16097.0	−0.538402	−0.269201	−	0.963084i	$-0.586760\pi$
$19$	−16097.0	−0.538402	−0.269201	+	0.963084i	$0.586760\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−11593.2	−0.198680	−0.0993401	−	0.995054i	$-0.531673\pi$
$23$	−11593.2	−0.198680	−0.0993401	+	0.995054i	$0.531673\pi$
$24$	0	0
$25$	8467.20	0.108380
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−204036.	−1.55351	−0.776753	−	0.629806i	$-0.783135\pi$
$29$	−204036.	−1.55351	−0.776753	+	0.629806i	$0.783135\pi$
$30$	0	0
$31$	149568.i	0.901722i	0.892594	+	0.450861i	$0.148883\pi$
$31$	149568.i	0.901722i	−0.892594	+	0.450861i	$0.851117\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 96705.4i	− 0.381252i
$36$	0	0
$37$	61595.2i	0.199913i	0.994992	+	0.0999564i	$0.0318703\pi$
$37$	61595.2i	0.199913i	−0.994992	+	0.0999564i	$0.968130\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 232923.i	− 0.527799i	−0.964550	−	0.263900i	$-0.914991\pi$
$41$	− 232923.i	− 0.527799i	0.964550	−	0.263900i	$-0.0850088\pi$
$42$	0	0
$43$	−619370.	−1.18798	−0.593992	−	0.804471i	$-0.702449\pi$
$43$	−619370.	−1.18798	−0.593992	+	0.804471i	$0.702449\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.07968e6	1.51689	0.758445	−	0.651738i	$-0.225960\pi$
$47$	1.07968e6	1.51689	0.758445	+	0.651738i	$0.225960\pi$
$48$	0	0
$49$	715543.	0.868860
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−713736.	−0.658524	−0.329262	−	0.944239i	$-0.606800\pi$
$53$	−713736.	−0.658524	−0.329262	+	0.944239i	$0.606800\pi$
$54$	0	0
$55$	− 788666.i	− 0.639180i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.30458e6i	0.826965i	0.910512	+	0.413483i	$0.135688\pi$
$59$	1.30458e6i	0.826965i	−0.910512	+	0.413483i	$0.864312\pi$
$60$	0	0
$61$	1.52680e6i	0.861248i	0.902531	+	0.430624i	$0.141707\pi$
$61$	1.52680e6i	0.861248i	−0.902531	+	0.430624i	$0.858293\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	437282.i	0.197499i
$66$	0	0
$67$	4.81113e6	1.95427	0.977137	−	0.212610i	$-0.0681964\pi$
$67$	4.81113e6	1.95427	0.977137	+	0.212610i	$0.0681964\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.36754e6	1.44821	0.724107	−	0.689688i	$-0.242252\pi$
$71$	4.36754e6	1.44821	0.724107	+	0.689688i	$0.242252\pi$
$72$	0	0
$73$	−1.23348e6	−0.371108	−0.185554	−	0.982634i	$-0.559408\pi$
$73$	−1.23348e6	−0.371108	−0.185554	+	0.982634i	$0.559408\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−880774.	−0.219860
$78$	0	0
$79$	− 5.34474e6i	− 1.21964i	−0.792540	−	0.609820i	$-0.791242\pi$
$79$	− 5.34474e6i	− 1.21964i	0.792540	−	0.609820i	$-0.208758\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.96837e6i	− 1.33770i	−0.743398	−	0.668849i	$-0.766787\pi$
$83$	− 6.96837e6i	− 1.33770i	0.743398	−	0.668849i	$-0.233213\pi$
$84$	0	0
$85$	1.04728e7i	1.84968i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 166492.i	− 0.0250338i	−0.999922	−	0.0125169i	$-0.996016\pi$
$89$	− 166492.i	− 0.0250338i	0.999922	−	0.0125169i	$-0.00398436\pi$
$90$	0	0
$91$	488352.	0.0679341
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.73678e6	0.566827
$96$	0	0
$97$	7.05299e6	0.784644	0.392322	−	0.919828i	$-0.371672\pi$
$97$	7.05299e6	0.784644	0.392322	+	0.919828i	$0.371672\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.23205e6	−0.505298	−0.252649	−	0.967558i	$-0.581302\pi$
$101$	−5.23205e6	−0.505298	−0.252649	+	0.967558i	$0.581302\pi$
$102$	0	0
$103$	1.46478e7i	1.32081i	0.750909	+	0.660406i	$0.229616\pi$
$103$	1.46478e7i	1.32081i	−0.750909	+	0.660406i	$0.770384\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.81950e6i	− 0.617072i	−0.951213	−	0.308536i	$-0.900161\pi$
$107$	− 7.81950e6i	− 0.617072i	0.951213	−	0.308536i	$-0.0998391\pi$
$108$	0	0
$109$	1.32338e7i	0.978794i	0.872061	+	0.489397i	$0.162783\pi$
$109$	1.32338e7i	0.978794i	−0.872061	+	0.489397i	$0.837217\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.35216e7i	− 1.53353i	−0.641927	−	0.766765i	$-0.721865\pi$
$113$	− 2.35216e7i	− 1.53353i	0.641927	−	0.766765i	$-0.278135\pi$
$114$	0	0
$115$	3.41147e6	0.209170
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.16959e7	0.636237
$120$	0	0
$121$	1.23042e7	0.631398
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.04979e7	0.938694
$126$	0	0
$127$	2.98987e7i	1.29521i	0.761978	+	0.647603i	$0.224228\pi$
$127$	2.98987e7i	1.29521i	−0.761978	+	0.647603i	$0.775772\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.48915e6i	0.368789i	0.982852	+	0.184394i	$0.0590324\pi$
$131$	9.48915e6i	0.368789i	−0.982852	+	0.184394i	$0.940968\pi$
$132$	0	0
$133$	− 5.28999e6i	− 0.194973i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.01971e7i	− 0.338809i	−0.985547	−	0.169404i	$-0.945816\pi$
$137$	− 1.01971e7i	− 0.338809i	0.985547	−	0.169404i	$-0.0541844\pi$
$138$	0	0
$139$	4.97951e7	1.57266	0.786330	−	0.617807i	$-0.211979\pi$
$139$	4.97951e7	1.57266	0.786330	+	0.617807i	$0.211979\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.98268e6	0.113894
$144$	0	0
$145$	6.00406e7	1.63553
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.45647e7	1.59898	0.799490	−	0.600680i	$-0.205103\pi$
$149$	6.45647e7	1.59898	0.799490	+	0.600680i	$0.205103\pi$
$150$	0	0
$151$	6.38149e7i	1.50835i	0.656672	+	0.754177i	$0.271964\pi$
$151$	6.38149e7i	1.50835i	−0.656672	+	0.754177i	$0.728036\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.40127e7i	− 0.949329i
$156$	0	0
$157$	− 1.74778e7i	− 0.360444i	−0.983626	−	0.180222i	$-0.942318\pi$
$157$	− 1.74778e7i	− 0.360444i	0.983626	−	0.180222i	$-0.0576816\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.80989e6i	− 0.0719486i
$162$	0	0
$163$	−52066.5	−0.000941677 0	−0.000470839	−	1.00000i	$-0.500150\pi$
$163$	−52066.5	−0.000941677 0	−0.000470839		1.00000i	$0.500150\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.62528e7	0.270036	0.135018	−	0.990843i	$-0.456891\pi$
$167$	1.62528e7	0.270036	0.135018	+	0.990843i	$0.456891\pi$
$168$	0	0
$169$	6.05403e7	0.964808
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.18628e8	1.74191	0.870954	−	0.491364i	$-0.163502\pi$
$173$	1.18628e8	1.74191	0.870954	+	0.491364i	$0.163502\pi$
$174$	0	0
$175$	2.78260e6i	0.0392480i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.22356e7i	− 0.550419i	−0.961384	−	0.275210i	$-0.911253\pi$
$179$	− 4.22356e7i	− 0.550419i	0.961384	−	0.275210i	$-0.0887472\pi$
$180$	0	0
$181$	1.16870e8i	1.46497i	0.680781	+	0.732487i	$0.261640\pi$
$181$	1.16870e8i	1.46497i	−0.680781	+	0.732487i	$0.738360\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.81253e7i	− 0.210468i
$186$	0	0
$187$	9.53841e7	1.06667
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.46598e8	−1.52234	−0.761171	−	0.648551i	$-0.775375\pi$
$191$	−1.46598e8	−1.52234	−0.761171	+	0.648551i	$0.775375\pi$
$192$	0	0
$193$	−9.92822e7	−0.994079	−0.497039	−	0.867728i	$-0.665580\pi$
$193$	−9.92822e7	−0.994079	−0.497039	+	0.867728i	$0.665580\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.90394e7	−0.270618	−0.135309	−	0.990803i	$-0.543203\pi$
$197$	−2.90394e7	−0.270618	−0.135309	+	0.990803i	$0.543203\pi$
$198$	0	0
$199$	1.90948e8i	1.71763i	0.512288	+	0.858814i	$0.328798\pi$
$199$	1.90948e8i	1.71763i	−0.512288	+	0.858814i	$0.671202\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.70528e7i	− 0.562576i
$204$	0	0
$205$	6.85412e7i	0.555665i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.31417e7i	− 0.326878i
$210$	0	0
$211$	1.21971e8	0.893854	0.446927	−	0.894570i	$-0.352518\pi$
$211$	1.21971e8	0.893854	0.446927	+	0.894570i	$0.352518\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.82259e8	1.25071
$216$	0	0
$217$	−4.91530e7	−0.326543
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.28864e7	−0.329588
$222$	0	0
$223$	− 2.53524e8i	− 1.53092i	−0.643483	−	0.765460i	$-0.722511\pi$
$223$	− 2.53524e8i	− 1.53092i	0.643483	−	0.765460i	$-0.277489\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.31259e7i	− 0.414936i	−0.978242	−	0.207468i	$-0.933478\pi$
$227$	− 7.31259e7i	− 0.414936i	0.978242	−	0.207468i	$-0.0665222\pi$
$228$	0	0
$229$	− 4.41665e7i	− 0.243035i	−0.992589	−	0.121518i	$-0.961224\pi$
$229$	− 4.41665e7i	− 0.243035i	0.992589	−	0.121518i	$-0.0387761\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.86368e8i	− 0.965219i	−0.875836	−	0.482609i	$-0.839689\pi$
$233$	− 1.86368e8i	− 0.965219i	0.875836	−	0.482609i	$-0.160311\pi$
$234$	0	0
$235$	−3.17714e8	−1.59698
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.04324e7	0.286337	0.143168	−	0.989698i	$-0.454271\pi$
$239$	6.04324e7	0.286337	0.143168	+	0.989698i	$0.454271\pi$
$240$	0	0
$241$	2.53357e8	1.16593	0.582965	−	0.812497i	$-0.301892\pi$
$241$	2.53357e8	1.16593	0.582965	+	0.812497i	$0.301892\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.10560e8	−0.914732
$246$	0	0
$247$	2.39203e7i	0.101001i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.32231e8i	1.72527i	0.505825	+	0.862636i	$0.331188\pi$
$251$	4.32231e8i	1.72527i	−0.505825	+	0.862636i	$0.668812\pi$
$252$	0	0
$253$	− 3.10710e7i	− 0.120624i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.53732e8i	1.29990i	0.759978	+	0.649948i	$0.225210\pi$
$257$	3.53732e8i	1.29990i	−0.759978	+	0.649948i	$0.774790\pi$
$258$	0	0
$259$	−2.02422e7	−0.0723950
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.41375e8	1.15714	0.578571	−	0.815632i	$-0.303611\pi$
$263$	3.41375e8	1.15714	0.578571	+	0.815632i	$0.303611\pi$
$264$	0	0
$265$	2.10028e8	0.693292
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.19128e8	−1.31284	−0.656422	−	0.754394i	$-0.727931\pi$
$269$	−4.19128e8	−1.31284	−0.656422	+	0.754394i	$0.727931\pi$
$270$	0	0
$271$	− 3.31720e8i	− 1.01246i	−0.862397	−	0.506232i	$-0.831038\pi$
$271$	− 3.31720e8i	− 1.01246i	0.862397	−	0.506232i	$-0.168962\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.26931e7i	0.0658005i
$276$	0	0
$277$	− 1.71149e8i	− 0.483834i	−0.970297	−	0.241917i	$-0.922224\pi$
$277$	− 1.71149e8i	− 0.483834i	0.970297	−	0.241917i	$-0.0777761\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.60548e8i	0.969374i	0.874688	+	0.484687i	$0.161066\pi$
$281$	3.60548e8i	0.969374i	−0.874688	+	0.484687i	$0.838934\pi$
$282$	0	0
$283$	−7.88163e7	−0.206711	−0.103356	−	0.994644i	$-0.532958\pi$
$283$	−7.88163e7	−0.206711	−0.103356	+	0.994644i	$0.532958\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.65462e7	0.191134
$288$	0	0
$289$	−8.56276e8	−2.08675
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.16444e7	−0.0270447	−0.0135223	−	0.999909i	$-0.504304\pi$
$293$	−1.16444e7	−0.0270447	−0.0135223	+	0.999909i	$0.504304\pi$
$294$	0	0
$295$	− 3.83892e8i	− 0.870626i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.72276e7i	0.0372713i
$300$	0	0
$301$	− 2.03545e8i	− 0.430208i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.49285e8i	− 0.906719i
$306$	0	0
$307$	−3.71307e8	−0.732401	−0.366201	−	0.930536i	$-0.619342\pi$
$307$	−3.71307e8	−0.732401	−0.366201	+	0.930536i	$0.619342\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.55339e8	−1.04688	−0.523440	−	0.852063i	$-0.675352\pi$
$311$	−5.55339e8	−1.04688	−0.523440	+	0.852063i	$0.675352\pi$
$312$	0	0
$313$	−6.90438e8	−1.27268	−0.636340	−	0.771408i	$-0.719553\pi$
$313$	−6.90438e8	−1.27268	−0.636340	+	0.771408i	$0.719553\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.87240e7	−0.121172	−0.0605858	−	0.998163i	$-0.519297\pi$
$317$	−6.87240e7	−0.121172	−0.0605858	+	0.998163i	$0.519297\pi$
$318$	0	0
$319$	− 5.46839e8i	− 0.943174i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.72883e8i	0.945927i
$324$	0	0
$325$	− 1.25824e7i	− 0.0203315i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.54820e8i	0.549315i
$330$	0	0
$331$	1.13679e8	0.172299	0.0861496	−	0.996282i	$-0.472544\pi$
$331$	1.13679e8	0.172299	0.0861496	+	0.996282i	$0.472544\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.41575e9	−2.05745
$336$	0	0
$337$	1.16559e8	0.165898	0.0829492	−	0.996554i	$-0.473566\pi$
$337$	1.16559e8	0.165898	0.0829492	+	0.996554i	$0.473566\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.00859e8	−0.547459
$342$	0	0
$343$	5.05795e8i	0.676776i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.74986e8i	− 0.995729i	−0.867255	−	0.497864i	$-0.834118\pi$
$347$	− 7.74986e8i	− 0.995729i	0.867255	−	0.497864i	$-0.165882\pi$
$348$	0	0
$349$	− 2.75593e8i	− 0.347039i	−0.984830	−	0.173520i	$-0.944486\pi$
$349$	− 2.75593e8i	− 0.347039i	0.984830	−	0.173520i	$-0.0555140\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.91797e8i	0.958081i	0.877793	+	0.479041i	$0.159015\pi$
$353$	7.91797e8i	0.958081i	−0.877793	+	0.479041i	$0.840985\pi$
$354$	0	0
$355$	−1.28522e9	−1.52467
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.47158e8	−0.510071	−0.255035	−	0.966932i	$-0.582087\pi$
$359$	−4.47158e8	−0.510071	−0.255035	+	0.966932i	$0.582087\pi$
$360$	0	0
$361$	−6.34759e8	−0.710124
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.62970e8	0.390702
$366$	0	0
$367$	− 6.59899e8i	− 0.696861i	−0.937335	−	0.348431i	$-0.886715\pi$
$367$	− 6.59899e8i	− 0.696861i	0.937335	−	0.348431i	$-0.113285\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.34557e8i	− 0.238473i
$372$	0	0
$373$	− 1.64257e9i	− 1.63886i	−0.573177	−	0.819432i	$-0.694289\pi$
$373$	− 1.64257e9i	− 1.63886i	0.573177	−	0.819432i	$-0.305711\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.03199e8i	0.291429i
$378$	0	0
$379$	4.91034e8	0.463313	0.231656	−	0.972798i	$-0.425586\pi$
$379$	4.91034e8	0.463313	0.231656	+	0.972798i	$0.425586\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.15258e8	−0.195778	−0.0978889	−	0.995197i	$-0.531209\pi$
$383$	−2.15258e8	−0.195778	−0.0978889	+	0.995197i	$0.531209\pi$
$384$	0	0
$385$	2.59182e8	0.231468
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.68469e9	1.45109	0.725547	−	0.688172i	$-0.241587\pi$
$389$	1.68469e9	1.45109	0.725547	+	0.688172i	$0.241587\pi$
$390$	0	0
$391$	4.12595e8i	0.349064i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.57277e9i	1.28403i
$396$	0	0
$397$	− 8.46867e8i	− 0.679280i	−0.940556	−	0.339640i	$-0.889695\pi$
$397$	− 8.46867e8i	− 0.679280i	0.940556	−	0.339640i	$-0.110305\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 3.67705e8i	− 0.284770i	−0.989811	−	0.142385i	$-0.954523\pi$
$401$	− 3.67705e8i	− 0.284770i	0.989811	−	0.142385i	$-0.0454771\pi$
$402$	0	0
$403$	2.22260e8	0.169158
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.65082e8	−0.121372
$408$	0	0
$409$	2.91851e8	0.210925	0.105463	−	0.994423i	$-0.466368\pi$
$409$	2.91851e8	0.210925	0.105463	+	0.994423i	$0.466368\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.28726e8	−0.299471
$414$	0	0
$415$	2.05055e9i	1.40832i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.75874e8i	0.581691i	0.956770	+	0.290845i	$0.0939366\pi$
$419$	8.75874e8i	0.581691i	−0.956770	+	0.290845i	$0.906063\pi$
$420$	0	0
$421$	3.93556e8i	0.257051i	0.991706	+	0.128526i	$0.0410244\pi$
$421$	3.93556e8i	0.257051i	−0.991706	+	0.128526i	$0.958976\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.01344e8i	− 0.190415i
$426$	0	0
$427$	−5.01758e8	−0.311886
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.03454e9	1.22404	0.612020	−	0.790842i	$-0.290357\pi$
$431$	2.03454e9	1.22404	0.612020	+	0.790842i	$0.290357\pi$
$432$	0	0
$433$	1.34016e9	0.793318	0.396659	−	0.917966i	$-0.370169\pi$
$433$	1.34016e9	0.793318	0.396659	+	0.917966i	$0.370169\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.86615e8	0.106970
$438$	0	0
$439$	− 1.78225e9i	− 1.00541i	−0.864459	−	0.502704i	$-0.832339\pi$
$439$	− 1.78225e9i	− 1.00541i	0.864459	−	0.502704i	$-0.167661\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.08329e8i	0.387099i	0.981090	+	0.193550i	$0.0620000\pi$
$443$	7.08329e8i	0.387099i	−0.981090	+	0.193550i	$0.938000\pi$
$444$	0	0
$445$	4.89927e7i	0.0263555i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.08114e9i	− 0.563665i	−0.959464	−	0.281832i	$-0.909058\pi$
$449$	− 1.08114e9i	− 0.563665i	0.959464	−	0.281832i	$-0.0909422\pi$
$450$	0	0
$451$	6.24260e8	0.320441
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.43705e8	−0.0715208
$456$	0	0
$457$	7.45370e8	0.365313	0.182657	−	0.983177i	$-0.441530\pi$
$457$	7.45370e8	0.365313	0.182657	+	0.983177i	$0.441530\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.30861e8	−0.0622097	−0.0311048	−	0.999516i	$-0.509903\pi$
$461$	−1.30861e8	−0.0622097	−0.0311048	+	0.999516i	$0.509903\pi$
$462$	0	0
$463$	3.29457e9i	1.54264i	0.636447	+	0.771321i	$0.280403\pi$
$463$	3.29457e9i	1.54264i	−0.636447	+	0.771321i	$0.719597\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.84572e9i	− 1.74731i	−0.486549	−	0.873653i	$-0.661745\pi$
$467$	− 3.84572e9i	− 1.74731i	0.486549	−	0.873653i	$-0.338255\pi$
$468$	0	0
$469$	1.58110e9i	0.707707i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.65998e9i	− 0.721257i
$474$	0	0
$475$	−1.36296e8	−0.0583521
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.27149e9	0.528616	0.264308	−	0.964438i	$-0.414857\pi$
$479$	1.27149e9	0.528616	0.264308	+	0.964438i	$0.414857\pi$
$480$	0	0
$481$	9.15311e7	0.0375026
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.07545e9	−0.826070
$486$	0	0
$487$	− 2.64160e9i	− 1.03637i	−0.855268	−	0.518185i	$-0.826608\pi$
$487$	− 2.64160e9i	− 1.03637i	0.855268	−	0.518185i	$-0.173392\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 1.97319e9i	− 0.752285i	−0.926562	−	0.376143i	$-0.877250\pi$
$491$	− 1.97319e9i	− 0.752285i	0.926562	−	0.376143i	$-0.122750\pi$
$492$	0	0
$493$	7.26153e9i	2.72938i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.43532e9i	0.524446i
$498$	0	0
$499$	4.39385e9	1.58304	0.791522	−	0.611140i	$-0.209289\pi$
$499$	4.39385e9	1.58304	0.791522	+	0.611140i	$0.209289\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.42589e8	−0.0499574	−0.0249787	−	0.999688i	$-0.507952\pi$
$503$	−1.42589e8	−0.0499574	−0.0249787	+	0.999688i	$0.507952\pi$
$504$	0	0
$505$	1.53961e9	0.531975
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.09362e8	0.103981	0.0519906	−	0.998648i	$-0.483443\pi$
$509$	3.09362e8	0.103981	0.0519906	+	0.998648i	$0.483443\pi$
$510$	0	0
$511$	− 4.05361e8i	− 0.134391i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.31033e9i	− 1.39055i
$516$	0	0
$517$	2.89368e9i	0.920943i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 2.29488e8i	− 0.0710932i	−0.999368	−	0.0355466i	$-0.988683\pi$
$521$	− 2.29488e8i	− 0.0710932i	0.999368	−	0.0355466i	$-0.0113172\pi$
$522$	0	0
$523$	3.09139e9	0.944926	0.472463	−	0.881351i	$-0.343365\pi$
$523$	3.09139e9	0.944926	0.472463	+	0.881351i	$0.343365\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.32305e9	1.58425
$528$	0	0
$529$	−3.27042e9	−0.960526
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.46126e8	−0.0990123
$534$	0	0
$535$	2.30101e9i	0.649651i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.91774e9i	0.527508i
$540$	0	0
$541$	− 3.25383e9i	− 0.883497i	−0.897139	−	0.441748i	$-0.854358\pi$
$541$	− 3.25383e9i	− 0.883497i	0.897139	−	0.441748i	$-0.145642\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.89425e9i	− 1.03047i
$546$	0	0
$547$	−1.49458e9	−0.390449	−0.195224	−	0.980759i	$-0.562543\pi$
$547$	−1.49458e9	−0.390449	−0.195224	+	0.980759i	$0.562543\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.28435e9	0.836410
$552$	0	0
$553$	1.75646e9	0.441672
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.01514e9	0.984483	0.492241	−	0.870459i	$-0.336178\pi$
$557$	4.01514e9	0.984483	0.492241	+	0.870459i	$0.336178\pi$
$558$	0	0
$559$	9.20391e8i	0.222859i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.87925e9i	0.679987i	0.940428	+	0.339994i	$0.110425\pi$
$563$	2.87925e9i	0.679987i	−0.940428	+	0.339994i	$0.889575\pi$
$564$	0	0
$565$	6.92160e9i	1.61450i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.91695e9i	− 0.436233i	−0.975923	−	0.218116i	$-0.930009\pi$
$569$	− 1.91695e9i	− 0.436233i	0.975923	−	0.218116i	$-0.0699912\pi$
$570$	0	0
$571$	3.99834e9	0.898779	0.449389	−	0.893336i	$-0.351642\pi$
$571$	3.99834e9	0.898779	0.449389	+	0.893336i	$0.351642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.81616e7	−0.0215330
$576$	0	0
$577$	8.39033e9	1.81829	0.909146	−	0.416477i	$-0.136735\pi$
$577$	8.39033e9	1.81829	0.909146	+	0.416477i	$0.136735\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.29004e9	0.484424
$582$	0	0
$583$	− 1.91289e9i	− 0.399808i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.84669e9i	0.989036i	0.869167	+	0.494518i	$0.164655\pi$
$587$	4.84669e9i	0.989036i	−0.869167	+	0.494518i	$0.835345\pi$
$588$	0	0
$589$	− 2.40759e9i	− 0.485489i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.59521e9i	0.708000i	0.935245	+	0.354000i	$0.115179\pi$
$593$	3.59521e9i	0.708000i	−0.935245	+	0.354000i	$0.884821\pi$
$594$	0	0
$595$	−3.44170e9	−0.669828
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.60041e8	−0.0684476	−0.0342238	−	0.999414i	$-0.510896\pi$
$599$	−3.60041e8	−0.0684476	−0.0342238	+	0.999414i	$0.510896\pi$
$600$	0	0
$601$	5.21055e9	0.979091	0.489545	−	0.871978i	$-0.337163\pi$
$601$	5.21055e9	0.979091	0.489545	+	0.871978i	$0.337163\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.62069e9	−0.664733
$606$	0	0
$607$	7.88101e9i	1.43028i	0.698980	+	0.715141i	$0.253637\pi$
$607$	7.88101e9i	1.43028i	−0.698980	+	0.715141i	$0.746363\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.60442e9i	− 0.284560i
$612$	0	0
$613$	− 8.47669e9i	− 1.48633i	−0.669109	−	0.743164i	$-0.733324\pi$
$613$	− 8.47669e9i	− 1.48633i	0.669109	−	0.743164i	$-0.266676\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.30414e9i	0.223525i	0.993735	+	0.111762i	$0.0356495\pi$
$617$	1.30414e9i	0.223525i	−0.993735	+	0.111762i	$0.964350\pi$
$618$	0	0
$619$	3.50616e9	0.594175	0.297087	−	0.954850i	$-0.403985\pi$
$619$	3.50616e9	0.594175	0.297087	+	0.954850i	$0.403985\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.47146e7	0.00906557
$624$	0	0
$625$	−6.69332e9	−1.09663
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.19214e9	0.351230
$630$	0	0
$631$	2.18649e9i	0.346453i	0.984882	+	0.173226i	$0.0554193\pi$
$631$	2.18649e9i	0.346453i	−0.984882	+	0.173226i	$0.944581\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 8.79814e9i	− 1.36359i
$636$	0	0
$637$	− 1.06330e9i	− 0.162993i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3.64284e9i	− 0.546308i	−0.961970	−	0.273154i	$-0.911933\pi$
$641$	− 3.64284e9i	− 0.546308i	0.961970	−	0.273154i	$-0.0880668\pi$
$642$	0	0
$643$	−1.02912e10	−1.52661	−0.763304	−	0.646040i	$-0.776424\pi$
$643$	−1.02912e10	−1.52661	−0.763304	+	0.646040i	$0.776424\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.42123e9	−1.22239	−0.611196	−	0.791479i	$-0.709311\pi$
$647$	−8.42123e9	−1.22239	−0.611196	+	0.791479i	$0.709311\pi$
$648$	0	0
$649$	−3.49641e9	−0.502072
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.12964e9	1.00201	0.501004	−	0.865445i	$-0.332964\pi$
$653$	7.12964e9	1.00201	0.501004	+	0.865445i	$0.332964\pi$
$654$	0	0
$655$	− 2.79233e9i	− 0.388260i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.28313e9i	− 0.310765i	−0.987854	−	0.155383i	$-0.950339\pi$
$659$	− 2.28313e9i	− 0.310765i	0.987854	−	0.155383i	$-0.0496610\pi$
$660$	0	0
$661$	− 1.04167e10i	− 1.40290i	−0.712719	−	0.701450i	$-0.752536\pi$
$661$	− 1.04167e10i	− 1.40290i	0.712719	−	0.701450i	$-0.247464\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.55666e9i	0.205267i
$666$	0	0
$667$	2.36542e9	0.308651
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.09201e9	−0.522887
$672$	0	0
$673$	5.55647e8	0.0702661	0.0351331	−	0.999383i	$-0.488814\pi$
$673$	5.55647e8	0.0702661	0.0351331	+	0.999383i	$0.488814\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.52244e9	−0.436298	−0.218149	−	0.975916i	$-0.570002\pi$
$677$	−3.52244e9	−0.436298	−0.218149	+	0.975916i	$0.570002\pi$
$678$	0	0
$679$	2.31785e9i	0.284145i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.97428e9i	1.19787i	0.800798	+	0.598934i	$0.204409\pi$
$683$	9.97428e9i	1.19787i	−0.800798	+	0.598934i	$0.795591\pi$
$684$	0	0
$685$	3.00065e9i	0.356697i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.06062e9i	0.123536i
$690$	0	0
$691$	−7.63799e9	−0.880655	−0.440328	−	0.897837i	$-0.645138\pi$
$691$	−7.63799e9	−0.880655	−0.440328	+	0.897837i	$0.645138\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.46530e10	−1.65569
$696$	0	0
$697$	−8.28962e9	−0.927299
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.19869e9	−0.350719	−0.175359	−	0.984504i	$-0.556109\pi$
$701$	−3.19869e9	−0.350719	−0.175359	+	0.984504i	$0.556109\pi$
$702$	0	0
$703$	− 9.91496e8i	− 0.107633i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.71942e9i	− 0.182985i
$708$	0	0
$709$	1.38637e10i	1.46088i	0.682975	+	0.730442i	$0.260686\pi$
$709$	1.38637e10i	1.46088i	−0.682975	+	0.730442i	$0.739314\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.73396e9i	− 0.179154i
$714$	0	0
$715$	−1.17197e9	−0.119907
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.52490e10	−1.53000	−0.765000	−	0.644030i	$-0.777261\pi$
$719$	−1.52490e10	−1.53000	−0.765000	+	0.644030i	$0.777261\pi$
$720$	0	0
$721$	−4.81374e9	−0.478309
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.72761e9	−0.168369
$726$	0	0
$727$	− 1.06784e10i	− 1.03071i	−0.856977	−	0.515355i	$-0.827660\pi$
$727$	− 1.06784e10i	− 1.03071i	0.856977	−	0.515355i	$-0.172340\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.20431e10i	2.08719i
$732$	0	0
$733$	− 2.09160e10i	− 1.96162i	−0.194974	−	0.980808i	$-0.562462\pi$
$733$	− 2.09160e10i	− 1.96162i	0.194974	−	0.980808i	$-0.437538\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.28944e10i	1.18649i
$738$	0	0
$739$	6.69063e9	0.609834	0.304917	−	0.952379i	$-0.401371\pi$
$739$	6.69063e9	0.609834	0.304917	+	0.952379i	$0.401371\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.80945e10	1.61840	0.809198	−	0.587536i	$-0.199902\pi$
$743$	1.80945e10	1.61840	0.809198	+	0.587536i	$0.199902\pi$
$744$	0	0
$745$	−1.89992e10	−1.68340
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.56975e9	0.223462
$750$	0	0
$751$	− 1.39247e10i	− 1.19963i	−0.800138	−	0.599815i	$-0.795241\pi$
$751$	− 1.39247e10i	− 1.19963i	0.800138	−	0.599815i	$-0.204759\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.87785e10i	− 1.58799i
$756$	0	0
$757$	− 1.41833e10i	− 1.18834i	−0.804338	−	0.594171i	$-0.797480\pi$
$757$	− 1.41833e10i	− 1.18834i	0.804338	−	0.594171i	$-0.202520\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.86744e9i	0.647123i	0.946207	+	0.323562i	$0.104880\pi$
$761$	7.86744e9i	0.647123i	−0.946207	+	0.323562i	$0.895120\pi$
$762$	0	0
$763$	−4.34906e9	−0.354454
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.93861e9	0.155134
$768$	0	0
$769$	1.63912e9	0.129978	0.0649890	−	0.997886i	$-0.479299\pi$
$769$	1.63912e9	0.129978	0.0649890	+	0.997886i	$0.479299\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.71642e9	0.211529	0.105764	−	0.994391i	$-0.466271\pi$
$773$	2.71642e9	0.211529	0.105764	+	0.994391i	$0.466271\pi$
$774$	0	0
$775$	1.26642e9i	0.0977288i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.74935e9i	0.284168i
$780$	0	0
$781$	1.17055e10i	0.879249i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.14311e9i	0.379474i
$786$	0	0
$787$	−8.31174e9	−0.607827	−0.303914	−	0.952700i	$-0.598293\pi$
$787$	−8.31174e9	−0.607827	−0.303914	+	0.952700i	$0.598293\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.72998e9	0.555342
$792$	0	0
$793$	2.26884e9	0.161565
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.72803e10	−1.20906	−0.604530	−	0.796582i	$-0.706639\pi$
$797$	−1.72803e10	−1.20906	−0.604530	+	0.796582i	$0.706639\pi$
$798$	0	0
$799$	− 3.84254e10i	− 2.66505i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 3.30586e9i	− 0.225310i
$804$	0	0
$805$	1.12112e9i	0.0757472i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.23227e10i	1.48227i	0.671356	+	0.741135i	$0.265712\pi$
$809$	2.23227e10i	1.48227i	−0.671356	+	0.741135i	$0.734288\pi$
$810$	0	0
$811$	−9.69423e9	−0.638176	−0.319088	−	0.947725i	$-0.603377\pi$
$811$	−9.69423e9	−0.638176	−0.319088	+	0.947725i	$0.603377\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.53214e7	0.000991394 0
$816$	0	0
$817$	9.96998e9	0.639613
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.46999e10	−1.55773	−0.778867	−	0.627189i	$-0.784205\pi$
$821$	−2.46999e10	−1.55773	−0.778867	+	0.627189i	$0.784205\pi$
$822$	0	0
$823$	2.00407e10i	1.25318i	0.779348	+	0.626591i	$0.215550\pi$
$823$	2.00407e10i	1.25318i	−0.779348	+	0.626591i	$0.784450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.11012e10i	1.91209i	0.293219	+	0.956045i	$0.405273\pi$
$827$	3.11012e10i	1.91209i	−0.293219	+	0.956045i	$0.594727\pi$
$828$	0	0
$829$	− 4.01877e9i	− 0.244992i	−0.992469	−	0.122496i	$-0.960910\pi$
$829$	− 4.01877e9i	− 0.244992i	0.992469	−	0.122496i	$-0.0390899\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.54659e10i	− 1.52651i
$834$	0	0
$835$	−4.78264e9	−0.284292
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.51383e10	0.884932	0.442466	−	0.896785i	$-0.354104\pi$
$839$	1.51383e10	0.884932	0.442466	+	0.896785i	$0.354104\pi$
$840$	0	0
$841$	2.43806e10	1.41338
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.78149e10	−1.01575
$846$	0	0
$847$	4.04355e9i	0.228650i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7.14083e8i	− 0.0397187i
$852$	0	0
$853$	− 1.53579e10i	− 0.847247i	−0.905838	−	0.423624i	$-0.860758\pi$
$853$	− 1.53579e10i	− 0.847247i	0.905838	−	0.423624i	$-0.139242\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.44454e10i	− 0.783968i	−0.919972	−	0.391984i	$-0.871789\pi$
$857$	− 1.44454e10i	− 0.783968i	0.919972	−	0.391984i	$-0.128211\pi$
$858$	0	0
$859$	7.57453e9	0.407736	0.203868	−	0.978998i	$-0.434649\pi$
$859$	7.57453e9	0.407736	0.203868	+	0.978998i	$0.434649\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.16885e10	−0.619045	−0.309522	−	0.950892i	$-0.600169\pi$
$863$	−1.16885e10	−0.619045	−0.309522	+	0.950892i	$0.600169\pi$
$864$	0	0
$865$	−3.49081e10	−1.83387
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.43245e10	0.740475
$870$	0	0
$871$	− 7.14939e9i	− 0.366611i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.73628e9i	0.339932i
$876$	0	0
$877$	1.94208e10i	0.972231i	0.873894	+	0.486116i	$0.161587\pi$
$877$	1.94208e10i	0.972231i	−0.873894	+	0.486116i	$0.838413\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.00292e10i	− 0.494139i	−0.968998	−	0.247069i	$-0.920532\pi$
$881$	− 1.00292e10i	− 0.494139i	0.968998	−	0.247069i	$-0.0794675\pi$
$882$	0	0
$883$	−3.58464e10	−1.75220	−0.876098	−	0.482133i	$-0.839862\pi$
$883$	−3.58464e10	−1.75220	−0.876098	+	0.482133i	$0.839862\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.96838e10	0.947059	0.473529	−	0.880778i	$-0.342980\pi$
$887$	1.96838e10	0.947059	0.473529	+	0.880778i	$0.342980\pi$
$888$	0	0
$889$	−9.82568e9	−0.469036
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.73796e10	−0.816696
$894$	0	0
$895$	1.24285e10i	0.579479i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.05172e10i	− 1.40083i
$900$	0	0
$901$	2.54015e10i	1.15697i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.43910e10i	− 1.54232i
$906$	0	0
$907$	4.16289e10	1.85255	0.926273	−	0.376853i	$-0.122994\pi$
$907$	4.16289e10	1.85255	0.926273	+	0.376853i	$0.122994\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.63565e9	0.334604	0.167302	−	0.985906i	$-0.446494\pi$
$911$	7.63565e9	0.334604	0.167302	+	0.985906i	$0.446494\pi$
$912$	0	0
$913$	1.86760e10	0.812151
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.11845e9	−0.133551
$918$	0	0
$919$	1.38559e10i	0.588886i	0.955669	+	0.294443i	$0.0951341\pi$
$919$	1.38559e10i	0.588886i	−0.955669	+	0.294443i	$0.904866\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 6.49021e9i	− 0.271677i
$924$	0	0
$925$	5.21539e8i	0.0216666i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.81860e10i	1.97181i	0.167294	+	0.985907i	$0.446497\pi$
$929$	4.81860e10i	1.97181i	−0.167294	+	0.985907i	$0.553503\pi$
$930$	0	0
$931$	−1.15181e10	−0.467796
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.80682e10	−1.12299
$936$	0	0
$937$	−5.58203e8	−0.0221668	−0.0110834	−	0.999939i	$-0.503528\pi$
$937$	−5.58203e8	−0.0221668	−0.0110834	+	0.999939i	$0.503528\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.60090e10	1.80003	0.900014	−	0.435862i	$-0.143556\pi$
$941$	4.60090e10	1.80003	0.900014	+	0.435862i	$0.143556\pi$
$942$	0	0
$943$	2.70031e9i	0.104863i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.84738e10i	− 0.706856i	−0.935462	−	0.353428i	$-0.885016\pi$
$947$	− 1.84738e10i	− 0.706856i	0.935462	−	0.353428i	$-0.114984\pi$
$948$	0	0
$949$	1.83296e9i	0.0696179i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.94712e10i	0.728732i	0.931256	+	0.364366i	$0.118714\pi$
$953$	1.94712e10i	0.728732i	−0.931256	+	0.364366i	$0.881286\pi$
$954$	0	0
$955$	4.31388e10	1.60272
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.35110e9	0.122694
$960$	0	0
$961$	5.14205e9	0.186898
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.92153e10	1.04656
$966$	0	0
$967$	1.72332e9i	0.0612876i	0.999530	+	0.0306438i	$0.00975574\pi$
$967$	1.72332e9i	0.0612876i	−0.999530	+	0.0306438i	$0.990244\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.13358e10i	0.397359i	0.980064	+	0.198680i	$0.0636653\pi$
$971$	1.13358e10i	0.397359i	−0.980064	+	0.198680i	$0.936335\pi$
$972$	0	0
$973$	1.63643e10i	0.569512i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.58110e10i	− 0.885470i	−0.896652	−	0.442735i	$-0.854008\pi$
$977$	− 2.58110e10i	− 0.885470i	0.896652	−	0.442735i	$-0.145992\pi$
$978$	0	0
$979$	4.46217e8	0.0151987
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.58165e10	−0.531097	−0.265549	−	0.964098i	$-0.585553\pi$
$983$	−1.58165e10	−0.531097	−0.265549	+	0.964098i	$0.585553\pi$
$984$	0	0
$985$	8.54530e9	0.284905
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.18046e9	0.236029
$990$	0	0
$991$	2.16501e10i	0.706646i	0.935501	+	0.353323i	$0.114948\pi$
$991$	2.16501e10i	0.706646i	−0.935501	+	0.353323i	$0.885052\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5.61894e10i	− 1.80831i
$996$	0	0
$997$	4.30355e9i	0.137529i	0.997633	+	0.0687644i	$0.0219057\pi$
$997$	4.30355e9i	0.137529i	−0.997633	+	0.0687644i	$0.978094\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.8.f.a.143.6		28
3.2	odd	2	inner	288.8.f.a.143.24		28
4.3	odd	2		72.8.f.a.35.22	yes	28
8.3	odd	2	inner	288.8.f.a.143.23		28
8.5	even	2		72.8.f.a.35.8	yes	28
12.11	even	2		72.8.f.a.35.7	✓	28
24.5	odd	2		72.8.f.a.35.21	yes	28
24.11	even	2	inner	288.8.f.a.143.5		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.7	✓	28	12.11	even	2
72.8.f.a.35.8	yes	28	8.5	even	2
72.8.f.a.35.21	yes	28	24.5	odd	2
72.8.f.a.35.22	yes	28	4.3	odd	2
288.8.f.a.143.5		28	24.11	even	2	inner
288.8.f.a.143.6		28	1.1	even	1	trivial
288.8.f.a.143.23		28	8.3	odd	2	inner
288.8.f.a.143.24		28	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-294.266 q^{5} +328.633i q^{7} +O(q^{10})\)	\(q-294.266 q^{5} +328.633i q^{7} +2680.12i q^{11} -1486.01i q^{13} -35589.5i q^{17} -16097.0 q^{19} -11593.2 q^{23} +8467.20 q^{25} -204036. q^{29} +149568. i q^{31} -96705.4i q^{35} +61595.2i q^{37} -232923. i q^{41} -619370. q^{43} +1.07968e6 q^{47} +715543. q^{49} -713736. q^{53} -788666. i q^{55} +1.30458e6i q^{59} +1.52680e6i q^{61} +437282. i q^{65} +4.81113e6 q^{67} +4.36754e6 q^{71} -1.23348e6 q^{73} -880774. q^{77} -5.34474e6i q^{79} -6.96837e6i q^{83} +1.04728e7i q^{85} -166492. i q^{89} +488352. q^{91} +4.73678e6 q^{95} +7.05299e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 28 * q	\( 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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