LMFDB - Embedded newform 288.8.f.a.143.2

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			143.2
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-527.660 q^{5} +1312.48i q^{7} +O(q^{10})$	$q-527.660 q^{5} +1312.48i q^{7} -961.214i q^{11} -10755.9i q^{13} -11544.1i q^{17} +46908.6 q^{19} +8879.25 q^{23} +200300. q^{25} -109471. q^{29} -239590. i q^{31} -692545. i q^{35} +366153. i q^{37} +40704.6i q^{41} -225674. q^{43} -713906. q^{47} -899069. q^{49} +786052. q^{53} +507194. i q^{55} +1.19024e6i q^{59} +1.03215e6i q^{61} +5.67543e6i q^{65} +1.72692e6 q^{67} +82775.2 q^{71} -2.92308e6 q^{73} +1.26158e6 q^{77} +1.21761e6i q^{79} -1.39552e6i q^{83} +6.09135e6i q^{85} -1.64007e6i q^{89} +1.41169e7 q^{91} -2.47518e7 q^{95} +175296. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−527.660	−1.88781	−0.943907	−	0.330212i	$-0.892880\pi$
$5$	−527.660	−1.88781	−0.943907	+	0.330212i	$0.892880\pi$
$6$	0	0
$7$	1312.48i	1.44627i	0.690705	+	0.723137i	$0.257301\pi$
$7$	1312.48i	1.44627i	−0.690705	+	0.723137i	$0.742699\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 961.214i	− 0.217744i	−0.994056	−	0.108872i	$-0.965276\pi$
$11$	− 961.214i	− 0.217744i	0.994056	−	0.108872i	$-0.0347238\pi$
$12$	0	0
$13$	− 10755.9i	− 1.35782i	−0.734220	−	0.678911i	$-0.762452\pi$
$13$	− 10755.9i	− 1.35782i	0.734220	−	0.678911i	$-0.237548\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 11544.1i	− 0.569886i	−0.958544	−	0.284943i	$-0.908025\pi$
$17$	− 11544.1i	− 0.569886i	0.958544	−	0.284943i	$-0.0919747\pi$
$18$	0	0
$19$	46908.6	1.56897	0.784485	−	0.620148i	$-0.212928\pi$
$19$	46908.6	1.56897	0.784485	+	0.620148i	$0.212928\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8879.25	0.152170	0.0760849	−	0.997101i	$-0.475758\pi$
$23$	8879.25	0.152170	0.0760849	+	0.997101i	$0.475758\pi$
$24$	0	0
$25$	200300.	2.56384
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−109471.	−0.833498	−0.416749	−	0.909022i	$-0.636831\pi$
$29$	−109471.	−0.833498	−0.416749	+	0.909022i	$0.636831\pi$
$30$	0	0
$31$	− 239590.i	− 1.44445i	−0.691659	−	0.722224i	$-0.743120\pi$
$31$	− 239590.i	− 1.44445i	0.691659	−	0.722224i	$-0.256880\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 692545.i	− 2.73030i
$36$	0	0
$37$	366153.i	1.18838i	0.804323	+	0.594192i	$0.202528\pi$
$37$	366153.i	1.18838i	−0.804323	+	0.594192i	$0.797472\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	40704.6i	0.0922359i	0.998936	+	0.0461180i	$0.0146850\pi$
$41$	40704.6i	0.0922359i	−0.998936	+	0.0461180i	$0.985315\pi$
$42$	0	0
$43$	−225674.	−0.432854	−0.216427	−	0.976299i	$-0.569440\pi$
$43$	−225674.	−0.432854	−0.216427	+	0.976299i	$0.569440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−713906.	−1.00300	−0.501498	−	0.865159i	$-0.667217\pi$
$47$	−713906.	−1.00300	−0.501498	+	0.865159i	$0.667217\pi$
$48$	0	0
$49$	−899069.	−1.09171
$50$	0	0
$51$	0	0
$52$	0	0
$53$	786052.	0.725246	0.362623	−	0.931936i	$-0.381881\pi$
$53$	786052.	0.725246	0.362623	+	0.931936i	$0.381881\pi$
$54$	0	0
$55$	507194.i	0.411060i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.19024e6i	0.754491i	0.926113	+	0.377246i	$0.123129\pi$
$59$	1.19024e6i	0.754491i	−0.926113	+	0.377246i	$0.876871\pi$
$60$	0	0
$61$	1.03215e6i	0.582220i	0.956690	+	0.291110i	$0.0940246\pi$
$61$	1.03215e6i	0.582220i	−0.956690	+	0.291110i	$0.905975\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.67543e6i	2.56332i
$66$	0	0
$67$	1.72692e6	0.701474	0.350737	−	0.936474i	$-0.385931\pi$
$67$	1.72692e6	0.701474	0.350737	+	0.936474i	$0.385931\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	82775.2	0.0274471	0.0137235	−	0.999906i	$-0.495632\pi$
$71$	82775.2	0.0274471	0.0137235	+	0.999906i	$0.495632\pi$
$72$	0	0
$73$	−2.92308e6	−0.879449	−0.439724	−	0.898133i	$-0.644924\pi$
$73$	−2.92308e6	−0.879449	−0.439724	+	0.898133i	$0.644924\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.26158e6	0.314917
$78$	0	0
$79$	1.21761e6i	0.277851i	0.990303	+	0.138926i	$0.0443649\pi$
$79$	1.21761e6i	0.277851i	−0.990303	+	0.138926i	$0.955635\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1.39552e6i	− 0.267894i	−0.990989	−	0.133947i	$-0.957235\pi$
$83$	− 1.39552e6i	− 0.267894i	0.990989	−	0.133947i	$-0.0427652\pi$
$84$	0	0
$85$	6.09135e6i	1.07584i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.64007e6i	− 0.246602i	−0.992369	−	0.123301i	$-0.960652\pi$
$89$	− 1.64007e6i	− 0.246602i	0.992369	−	0.123301i	$-0.0393480\pi$
$90$	0	0
$91$	1.41169e7	1.96378
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.47518e7	−2.96192
$96$	0	0
$97$	175296.	0.0195017	0.00975083	−	0.999952i	$-0.496896\pi$
$97$	175296.	0.0195017	0.00975083	+	0.999952i	$0.496896\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.77576e6	0.171498	0.0857489	−	0.996317i	$-0.472672\pi$
$101$	1.77576e6	0.171498	0.0857489	+	0.996317i	$0.472672\pi$
$102$	0	0
$103$	8.36138e6i	0.753959i	0.926222	+	0.376979i	$0.123037\pi$
$103$	8.36138e6i	0.753959i	−0.926222	+	0.376979i	$0.876963\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.16063e7i	− 1.70505i	−0.522684	−	0.852526i	$-0.675069\pi$
$107$	− 2.16063e7i	− 1.70505i	0.522684	−	0.852526i	$-0.324931\pi$
$108$	0	0
$109$	7.42811e6i	0.549396i	0.961531	+	0.274698i	$0.0885779\pi$
$109$	7.42811e6i	0.549396i	−0.961531	+	0.274698i	$0.911422\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.32692e7i	− 0.865106i	−0.901609	−	0.432553i	$-0.857613\pi$
$113$	− 1.32692e7i	− 0.865106i	0.901609	−	0.432553i	$-0.142387\pi$
$114$	0	0
$115$	−4.68522e6	−0.287268
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.51514e7	0.824212
$120$	0	0
$121$	1.85632e7	0.952588
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.44669e7	−2.95224
$126$	0	0
$127$	3.28675e7i	1.42381i	0.702273	+	0.711907i	$0.252168\pi$
$127$	3.28675e7i	1.42381i	−0.702273	+	0.711907i	$0.747832\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.89771e7i	1.12617i	0.826398	+	0.563086i	$0.190386\pi$
$131$	2.89771e7i	1.12617i	−0.826398	+	0.563086i	$0.809614\pi$
$132$	0	0
$133$	6.15667e7i	2.26916i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00356e7i	1.66248i	0.555911	+	0.831242i	$0.312369\pi$
$137$	5.00356e7i	1.66248i	−0.555911	+	0.831242i	$0.687631\pi$
$138$	0	0
$139$	−3.52978e7	−1.11480	−0.557399	−	0.830245i	$-0.688200\pi$
$139$	−3.52978e7	−1.11480	−0.557399	+	0.830245i	$0.688200\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.03387e7	−0.295658
$144$	0	0
$145$	5.77633e7	1.57349
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.26425e7	0.560754	0.280377	−	0.959890i	$-0.409541\pi$
$149$	2.26425e7	0.560754	0.280377	+	0.959890i	$0.409541\pi$
$150$	0	0
$151$	9.18770e6i	0.217164i	0.994088	+	0.108582i	$0.0346310\pi$
$151$	9.18770e6i	0.217164i	−0.994088	+	0.108582i	$0.965369\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.26422e8i	2.72685i
$156$	0	0
$157$	− 1.49030e7i	− 0.307345i	−0.988122	−	0.153672i	$-0.950890\pi$
$157$	− 1.49030e7i	− 0.307345i	0.988122	−	0.153672i	$-0.0491101\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.16539e7i	0.220079i
$162$	0	0
$163$	4.56940e7	0.826423	0.413212	−	0.910635i	$-0.364407\pi$
$163$	4.56940e7	0.826423	0.413212	+	0.910635i	$0.364407\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.75768e7	−0.458181	−0.229090	−	0.973405i	$-0.573575\pi$
$167$	−2.75768e7	−0.458181	−0.229090	+	0.973405i	$0.573575\pi$
$168$	0	0
$169$	−5.29398e7	−0.843683
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.78654e7	−1.43704	−0.718519	−	0.695508i	$-0.755179\pi$
$173$	−9.78654e7	−1.43704	−0.718519	+	0.695508i	$0.755179\pi$
$174$	0	0
$175$	2.62890e8i	3.70802i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2.60015e7i	− 0.338855i	−0.985543	−	0.169427i	$-0.945808\pi$
$179$	− 2.60015e7i	− 0.338855i	0.985543	−	0.169427i	$-0.0541918\pi$
$180$	0	0
$181$	8.07924e7i	1.01273i	0.862318	+	0.506367i	$0.169012\pi$
$181$	8.07924e7i	1.01273i	−0.862318	+	0.506367i	$0.830988\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.93205e8i	− 2.24345i
$186$	0	0
$187$	−1.10963e7	−0.124089
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.15025e8	−1.19447	−0.597236	−	0.802065i	$-0.703735\pi$
$191$	−1.15025e8	−1.19447	−0.597236	+	0.802065i	$0.703735\pi$
$192$	0	0
$193$	−1.06745e7	−0.106880	−0.0534400	−	0.998571i	$-0.517019\pi$
$193$	−1.06745e7	−0.106880	−0.0534400	+	0.998571i	$0.517019\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.17071e7	−0.481857	−0.240929	−	0.970543i	$-0.577452\pi$
$197$	−5.17071e7	−0.481857	−0.240929	+	0.970543i	$0.577452\pi$
$198$	0	0
$199$	− 2.11722e8i	− 1.90450i	−0.305321	−	0.952249i	$-0.598764\pi$
$199$	− 2.11722e8i	− 1.90450i	0.305321	−	0.952249i	$-0.401236\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.43678e8i	− 1.20547i
$204$	0	0
$205$	− 2.14782e7i	− 0.174124i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.50892e7i	− 0.341633i
$210$	0	0
$211$	3.28972e7	0.241085	0.120542	−	0.992708i	$-0.461537\pi$
$211$	3.28972e7	0.241085	0.120542	+	0.992708i	$0.461537\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.19079e8	0.817148
$216$	0	0
$217$	3.14457e8	2.08907
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.24166e8	−0.773805
$222$	0	0
$223$	2.58450e8i	1.56067i	0.625365	+	0.780333i	$0.284950\pi$
$223$	2.58450e8i	1.56067i	−0.625365	+	0.780333i	$0.715050\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.78780e7i	0.555386i	0.960670	+	0.277693i	$0.0895698\pi$
$227$	9.78780e7i	0.555386i	−0.960670	+	0.277693i	$0.910430\pi$
$228$	0	0
$229$	3.04204e7i	0.167394i	0.996491	+	0.0836972i	$0.0266729\pi$
$229$	3.04204e7i	0.167394i	−0.996491	+	0.0836972i	$0.973327\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.03447e8i	1.57158i	0.618493	+	0.785790i	$0.287743\pi$
$233$	3.03447e8i	1.57158i	−0.618493	+	0.785790i	$0.712257\pi$
$234$	0	0
$235$	3.76700e8	1.89347
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.46584e8	−1.16835	−0.584173	−	0.811629i	$-0.698581\pi$
$239$	−2.46584e8	−1.16835	−0.584173	+	0.811629i	$0.698581\pi$
$240$	0	0
$241$	−3.24247e8	−1.49216	−0.746082	−	0.665854i	$-0.768067\pi$
$241$	−3.24247e8	−1.49216	−0.746082	+	0.665854i	$0.768067\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.74403e8	2.06094
$246$	0	0
$247$	− 5.04541e8i	− 2.13038i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.31952e8i	− 0.526692i	−0.964701	−	0.263346i	$-0.915174\pi$
$251$	− 1.31952e8i	− 0.526692i	0.964701	−	0.263346i	$-0.0848261\pi$
$252$	0	0
$253$	− 8.53486e6i	− 0.0331340i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.25505e8i	1.56365i	0.623500	+	0.781824i	$0.285710\pi$
$257$	4.25505e8i	1.56365i	−0.623500	+	0.781824i	$0.714290\pi$
$258$	0	0
$259$	−4.80570e8	−1.71873
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.98275e8	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$263$	2.98275e8	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$264$	0	0
$265$	−4.14768e8	−1.36913
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.01605e8	0.944726	0.472363	−	0.881404i	$-0.343401\pi$
$269$	3.01605e8	0.944726	0.472363	+	0.881404i	$0.343401\pi$
$270$	0	0
$271$	− 8.93027e7i	− 0.272566i	−0.990670	−	0.136283i	$-0.956484\pi$
$271$	− 8.93027e7i	− 0.272566i	0.990670	−	0.136283i	$-0.0435157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.92531e8i	− 0.558260i
$276$	0	0
$277$	2.27412e8i	0.642886i	0.946929	+	0.321443i	$0.104168\pi$
$277$	2.27412e8i	0.642886i	−0.946929	+	0.321443i	$0.895832\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.42040e8i	0.381889i	0.981601	+	0.190945i	$0.0611551\pi$
$281$	1.42040e8i	0.381889i	−0.981601	+	0.190945i	$0.938845\pi$
$282$	0	0
$283$	−6.43886e8	−1.68871	−0.844357	−	0.535781i	$-0.820017\pi$
$283$	−6.43886e8	−1.68871	−0.844357	+	0.535781i	$0.820017\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.34241e7	−0.133398
$288$	0	0
$289$	2.77073e8	0.675230
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.93386e8	−0.681402	−0.340701	−	0.940172i	$-0.610664\pi$
$293$	−2.93386e8	−0.681402	−0.340701	+	0.940172i	$0.610664\pi$
$294$	0	0
$295$	− 6.28045e8i	− 1.42434i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.55039e7i	− 0.206620i
$300$	0	0
$301$	− 2.96193e8i	− 0.626026i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.44623e8i	− 1.09912i
$306$	0	0
$307$	7.74955e8	1.52859	0.764297	−	0.644865i	$-0.223086\pi$
$307$	7.74955e8	1.52859	0.764297	+	0.644865i	$0.223086\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.49879e8	1.79063	0.895317	−	0.445429i	$-0.146949\pi$
$311$	9.49879e8	1.79063	0.895317	+	0.445429i	$0.146949\pi$
$312$	0	0
$313$	−7.46781e8	−1.37654	−0.688269	−	0.725456i	$-0.741629\pi$
$313$	−7.46781e8	−1.37654	−0.688269	+	0.725456i	$0.741629\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.87425e8	1.21204	0.606021	−	0.795448i	$-0.292765\pi$
$317$	6.87425e8	1.21204	0.606021	+	0.795448i	$0.292765\pi$
$318$	0	0
$319$	1.05225e8i	0.181489i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.41516e8i	− 0.894134i
$324$	0	0
$325$	− 2.15440e9i	− 3.48124i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 9.36990e8i	− 1.45061i
$330$	0	0
$331$	4.77550e8	0.723803	0.361902	−	0.932216i	$-0.382128\pi$
$331$	4.77550e8	0.723803	0.361902	+	0.932216i	$0.382128\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.11229e8	−1.32425
$336$	0	0
$337$	3.20237e8	0.455792	0.227896	−	0.973685i	$-0.426815\pi$
$337$	3.20237e8	0.455792	0.227896	+	0.973685i	$0.426815\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.30297e8	−0.314520
$342$	0	0
$343$	− 9.91271e7i	− 0.132637i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.18689e8i	0.794913i	0.917621	+	0.397456i	$0.130107\pi$
$347$	6.18689e8i	0.794913i	−0.917621	+	0.397456i	$0.869893\pi$
$348$	0	0
$349$	− 5.94067e7i	− 0.0748078i	−0.999300	−	0.0374039i	$-0.988091\pi$
$349$	− 5.94067e7i	− 0.0748078i	0.999300	−	0.0374039i	$-0.0119088\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.32577e9i	− 1.60420i	−0.597192	−	0.802098i	$-0.703717\pi$
$353$	− 1.32577e9i	− 1.60420i	0.597192	−	0.802098i	$-0.296283\pi$
$354$	0	0
$355$	−4.36772e7	−0.0518150
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.17907e8	0.932982	0.466491	−	0.884526i	$-0.345518\pi$
$359$	8.17907e8	0.932982	0.466491	+	0.884526i	$0.345518\pi$
$360$	0	0
$361$	1.30654e9	1.46166
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.54239e9	1.66024
$366$	0	0
$367$	− 1.06854e9i	− 1.12839i	−0.825642	−	0.564195i	$-0.809187\pi$
$367$	− 1.06854e9i	− 1.12839i	0.825642	−	0.564195i	$-0.190813\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.03168e9i	1.04891i
$372$	0	0
$373$	2.96640e7i	0.0295970i	0.999890	+	0.0147985i	$0.00471069\pi$
$373$	2.96640e7i	0.0295970i	−0.999890	+	0.0147985i	$0.995289\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.17745e9i	1.13174i
$378$	0	0
$379$	−9.95464e8	−0.939265	−0.469633	−	0.882862i	$-0.655614\pi$
$379$	−9.95464e8	−0.939265	−0.469633	+	0.882862i	$0.655614\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.46129e9	1.32905	0.664523	−	0.747268i	$-0.268635\pi$
$383$	1.46129e9	1.32905	0.664523	+	0.747268i	$0.268635\pi$
$384$	0	0
$385$	−6.65684e8	−0.594505
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.67137e9	−1.43963	−0.719813	−	0.694169i	$-0.755772\pi$
$389$	−1.67137e9	−1.43963	−0.719813	+	0.694169i	$0.755772\pi$
$390$	0	0
$391$	− 1.02503e8i	− 0.0867195i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.42483e8i	− 0.524531i
$396$	0	0
$397$	4.73007e8i	0.379403i	0.981842	+	0.189701i	$0.0607520\pi$
$397$	4.73007e8i	0.379403i	−0.981842	+	0.189701i	$0.939248\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.63188e9i	1.26381i	0.775045	+	0.631906i	$0.217727\pi$
$401$	1.63188e9i	1.26381i	−0.775045	+	0.631906i	$0.782273\pi$
$402$	0	0
$403$	−2.57699e9	−1.96131
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.51952e8	0.258763
$408$	0	0
$409$	−3.20095e8	−0.231339	−0.115669	−	0.993288i	$-0.536901\pi$
$409$	−3.20095e8	−0.231339	−0.115669	+	0.993288i	$0.536901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.56218e9	−1.09120
$414$	0	0
$415$	7.36360e8i	0.505734i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.17937e9i	1.44738i	0.690128	+	0.723688i	$0.257554\pi$
$419$	2.17937e9i	1.44738i	−0.690128	+	0.723688i	$0.742446\pi$
$420$	0	0
$421$	− 1.38985e8i	− 0.0907783i	−0.998969	−	0.0453892i	$-0.985547\pi$
$421$	− 1.38985e8i	− 0.0907783i	0.998969	−	0.0453892i	$-0.0144528\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2.31228e9i	− 1.46110i
$426$	0	0
$427$	−1.35468e9	−0.842050
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.50706e9	1.50832	0.754161	−	0.656690i	$-0.228044\pi$
$431$	2.50706e9	1.50832	0.754161	+	0.656690i	$0.228044\pi$
$432$	0	0
$433$	−7.58853e8	−0.449211	−0.224605	−	0.974450i	$-0.572109\pi$
$433$	−7.58853e8	−0.449211	−0.224605	+	0.974450i	$0.572109\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.16513e8	0.238750
$438$	0	0
$439$	2.63015e9i	1.48373i	0.670548	+	0.741866i	$0.266059\pi$
$439$	2.63015e9i	1.48373i	−0.670548	+	0.741866i	$0.733941\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.63076e9i	− 0.891206i	−0.895231	−	0.445603i	$-0.852989\pi$
$443$	− 1.63076e9i	− 0.891206i	0.895231	−	0.445603i	$-0.147011\pi$
$444$	0	0
$445$	8.65397e8i	0.465538i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.68143e9i	0.876628i	0.898822	+	0.438314i	$0.144424\pi$
$449$	1.68143e9i	0.876628i	−0.898822	+	0.438314i	$0.855576\pi$
$450$	0	0
$451$	3.91258e7	0.0200838
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.44891e9	−3.70726
$456$	0	0
$457$	6.30718e8	0.309121	0.154561	−	0.987983i	$-0.450604\pi$
$457$	6.30718e8	0.309121	0.154561	+	0.987983i	$0.450604\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.44175e9	0.685390	0.342695	−	0.939447i	$-0.388660\pi$
$461$	1.44175e9	0.685390	0.342695	+	0.939447i	$0.388660\pi$
$462$	0	0
$463$	− 1.16830e9i	− 0.547044i	−0.961866	−	0.273522i	$-0.911811\pi$
$463$	− 1.16830e9i	− 0.547044i	0.961866	−	0.273522i	$-0.0881887\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.97005e8i	− 0.407555i	−0.979017	−	0.203777i	$-0.934678\pi$
$467$	− 8.97005e8i	− 0.407555i	0.979017	−	0.203777i	$-0.0653218\pi$
$468$	0	0
$469$	2.26656e9i	1.01452i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.16921e8i	0.0942513i
$474$	0	0
$475$	9.39579e9	4.02259
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.31795e8	0.345813	0.172907	−	0.984938i	$-0.444684\pi$
$479$	8.31795e8	0.345813	0.172907	+	0.984938i	$0.444684\pi$
$480$	0	0
$481$	3.93829e9	1.61362
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.24969e7	−0.0368155
$486$	0	0
$487$	1.00377e9i	0.393805i	0.980423	+	0.196902i	$0.0630882\pi$
$487$	1.00377e9i	0.393805i	−0.980423	+	0.196902i	$0.936912\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 6.35648e7i	− 0.0242344i	−0.999927	−	0.0121172i	$-0.996143\pi$
$491$	− 6.35648e7i	− 0.0242344i	0.999927	−	0.0121172i	$-0.00385711\pi$
$492$	0	0
$493$	1.26374e9i	0.474999i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.08641e8i	0.0396960i
$498$	0	0
$499$	3.66659e9	1.32102	0.660512	−	0.750816i	$-0.270339\pi$
$499$	3.66659e9	1.32102	0.660512	+	0.750816i	$0.270339\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.44858e9	−0.857880	−0.428940	−	0.903333i	$-0.641113\pi$
$503$	−2.44858e9	−0.857880	−0.428940	+	0.903333i	$0.641113\pi$
$504$	0	0
$505$	−9.36996e8	−0.323756
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.07396e9	−1.36932	−0.684659	−	0.728863i	$-0.740049\pi$
$509$	−4.07396e9	−1.36932	−0.684659	+	0.728863i	$0.740049\pi$
$510$	0	0
$511$	− 3.83649e9i	− 1.27192i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.41196e9i	− 1.42333i
$516$	0	0
$517$	6.86217e8i	0.218396i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.19867e9i	0.990917i	0.868632	+	0.495459i	$0.165000\pi$
$521$	3.19867e9i	0.990917i	−0.868632	+	0.495459i	$0.835000\pi$
$522$	0	0
$523$	−5.00703e9	−1.53047	−0.765235	−	0.643752i	$-0.777377\pi$
$523$	−5.00703e9	−1.53047	−0.765235	+	0.643752i	$0.777377\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.76584e9	−0.823172
$528$	0	0
$529$	−3.32598e9	−0.976844
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.37813e8	0.125240
$534$	0	0
$535$	1.14008e10i	3.21882i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.64198e8i	0.237713i
$540$	0	0
$541$	4.70918e9i	1.27866i	0.768932	+	0.639330i	$0.220788\pi$
$541$	4.70918e9i	1.27866i	−0.768932	+	0.639330i	$0.779212\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.91952e9i	− 1.03716i
$546$	0	0
$547$	2.77317e9	0.724472	0.362236	−	0.932086i	$-0.382013\pi$
$547$	2.77317e9	0.724472	0.362236	+	0.932086i	$0.382013\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.13511e9	−1.30773
$552$	0	0
$553$	−1.59809e9	−0.401849
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.23669e9	0.303227	0.151614	−	0.988440i	$-0.451553\pi$
$557$	1.23669e9	0.303227	0.151614	+	0.988440i	$0.451553\pi$
$558$	0	0
$559$	2.42731e9i	0.587739i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.49885e9i	1.06248i	0.847220	+	0.531242i	$0.178275\pi$
$563$	4.49885e9i	1.06248i	−0.847220	+	0.531242i	$0.821725\pi$
$564$	0	0
$565$	7.00161e9i	1.63316i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.60760e9i	1.27610i	0.769995	+	0.638049i	$0.220258\pi$
$569$	5.60760e9i	1.27610i	−0.769995	+	0.638049i	$0.779742\pi$
$570$	0	0
$571$	−3.27460e9	−0.736091	−0.368046	−	0.929808i	$-0.619973\pi$
$571$	−3.27460e9	−0.736091	−0.368046	+	0.929808i	$0.619973\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.77851e9	0.390139
$576$	0	0
$577$	−6.54667e9	−1.41875	−0.709373	−	0.704833i	$-0.751022\pi$
$577$	−6.54667e9	−1.41875	−0.709373	+	0.704833i	$0.751022\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.83160e9	0.387448
$582$	0	0
$583$	− 7.55564e8i	− 0.157918i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.23096e9i	0.659323i	0.944099	+	0.329662i	$0.106935\pi$
$587$	3.23096e9i	0.659323i	−0.944099	+	0.329662i	$0.893065\pi$
$588$	0	0
$589$	− 1.12388e10i	− 2.26630i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.14696e8i	0.0816655i	0.999166	+	0.0408328i	$0.0130011\pi$
$593$	4.14696e8i	0.0816655i	−0.999166	+	0.0408328i	$0.986999\pi$
$594$	0	0
$595$	−7.99480e9	−1.55596
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.31761e9	1.58127	0.790633	−	0.612291i	$-0.209752\pi$
$599$	8.31761e9	1.58127	0.790633	+	0.612291i	$0.209752\pi$
$600$	0	0
$601$	4.33439e9	0.814456	0.407228	−	0.913327i	$-0.366495\pi$
$601$	4.33439e9	0.814456	0.407228	+	0.913327i	$0.366495\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.79508e9	−1.79831
$606$	0	0
$607$	− 3.06356e9i	− 0.555989i	−0.960583	−	0.277994i	$-0.910330\pi$
$607$	− 3.06356e9i	− 0.555989i	0.960583	−	0.277994i	$-0.0896697\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.67867e9i	1.36189i
$612$	0	0
$613$	5.72765e9i	1.00430i	0.864779	+	0.502152i	$0.167458\pi$
$613$	5.72765e9i	1.00430i	−0.864779	+	0.502152i	$0.832542\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.41667e9i	1.09980i	0.835232	+	0.549898i	$0.185333\pi$
$617$	6.41667e9i	1.09980i	−0.835232	+	0.549898i	$0.814667\pi$
$618$	0	0
$619$	9.67083e9	1.63888	0.819439	−	0.573167i	$-0.194285\pi$
$619$	9.67083e9	1.63888	0.819439	+	0.573167i	$0.194285\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.15256e9	0.356654
$624$	0	0
$625$	1.83682e10	3.00944
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.22691e9	0.677244
$630$	0	0
$631$	2.36943e9i	0.375441i	0.982223	+	0.187720i	$0.0601098\pi$
$631$	2.36943e9i	0.375441i	−0.982223	+	0.187720i	$0.939890\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.73429e10i	− 2.68790i
$636$	0	0
$637$	9.67026e9i	1.48235i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.10766e9i	0.915949i	0.888965	+	0.457975i	$0.151425\pi$
$641$	6.10766e9i	0.915949i	−0.888965	+	0.457975i	$0.848575\pi$
$642$	0	0
$643$	−9.52741e9	−1.41331	−0.706653	−	0.707560i	$-0.749796\pi$
$643$	−9.52741e9	−1.41331	−0.706653	+	0.707560i	$0.749796\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.20878e9	−0.756086	−0.378043	−	0.925788i	$-0.623403\pi$
$647$	−5.20878e9	−0.756086	−0.378043	+	0.925788i	$0.623403\pi$
$648$	0	0
$649$	1.14408e9	0.164286
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.90741e9	−1.11132	−0.555659	−	0.831410i	$-0.687534\pi$
$653$	−7.90741e9	−1.11132	−0.555659	+	0.831410i	$0.687534\pi$
$654$	0	0
$655$	− 1.52900e10i	− 2.12600i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.05768e10i	1.43964i	0.694158	+	0.719822i	$0.255777\pi$
$659$	1.05768e10i	1.43964i	−0.694158	+	0.719822i	$0.744223\pi$
$660$	0	0
$661$	− 6.08761e9i	− 0.819864i	−0.912116	−	0.409932i	$-0.865552\pi$
$661$	− 6.08761e9i	− 0.819864i	0.912116	−	0.409932i	$-0.134448\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.24863e10i	− 4.28375i
$666$	0	0
$667$	−9.72017e8	−0.126833
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.92114e8	0.126775
$672$	0	0
$673$	−4.42313e9	−0.559341	−0.279671	−	0.960096i	$-0.590225\pi$
$673$	−4.42313e9	−0.559341	−0.279671	+	0.960096i	$0.590225\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.04814e10	1.29825	0.649125	−	0.760682i	$-0.275135\pi$
$677$	1.04814e10	1.29825	0.649125	+	0.760682i	$0.275135\pi$
$678$	0	0
$679$	2.30074e8i	0.0282048i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.89413e9i	− 0.467668i	−0.972277	−	0.233834i	$-0.924873\pi$
$683$	− 3.89413e9i	− 0.467668i	0.972277	−	0.233834i	$-0.0751273\pi$
$684$	0	0
$685$	− 2.64018e10i	− 3.13846i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 8.45466e9i	− 0.984756i
$690$	0	0
$691$	7.00398e9	0.807554	0.403777	−	0.914857i	$-0.367697\pi$
$691$	7.00398e9	0.807554	0.403777	+	0.914857i	$0.367697\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.86252e10	2.10453
$696$	0	0
$697$	4.69897e8	0.0525640
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.72569e9	0.518146	0.259073	−	0.965858i	$-0.416583\pi$
$701$	4.72569e9	0.518146	0.259073	+	0.965858i	$0.416583\pi$
$702$	0	0
$703$	1.71757e10i	1.86454i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.33065e9i	0.248033i
$708$	0	0
$709$	− 1.64991e10i	− 1.73859i	−0.494292	−	0.869296i	$-0.664573\pi$
$709$	− 1.64991e10i	− 1.73859i	0.494292	−	0.869296i	$-0.335427\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.12738e9i	− 0.219802i
$714$	0	0
$715$	5.45531e9	0.558146
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.64417e9	−0.766971	−0.383486	−	0.923547i	$-0.625276\pi$
$719$	−7.64417e9	−0.766971	−0.383486	+	0.923547i	$0.625276\pi$
$720$	0	0
$721$	−1.09742e10	−1.09043
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.19270e10	−2.13696
$726$	0	0
$727$	− 5.71439e9i	− 0.551568i	−0.961220	−	0.275784i	$-0.911062\pi$
$727$	− 5.71439e9i	− 0.551568i	0.961220	−	0.275784i	$-0.0889375\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.60520e9i	0.246678i
$732$	0	0
$733$	1.06519e10i	0.998997i	0.866315	+	0.499499i	$0.166482\pi$
$733$	1.06519e10i	0.998997i	−0.866315	+	0.499499i	$0.833518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.65994e9i	− 0.152742i
$738$	0	0
$739$	−1.09195e10	−0.995287	−0.497643	−	0.867382i	$-0.665801\pi$
$739$	−1.09195e10	−0.995287	−0.497643	+	0.867382i	$0.665801\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.36427e9	−0.122022	−0.0610111	−	0.998137i	$-0.519433\pi$
$743$	−1.36427e9	−0.122022	−0.0610111	+	0.998137i	$0.519433\pi$
$744$	0	0
$745$	−1.19475e10	−1.05860
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.83580e10	2.46597
$750$	0	0
$751$	6.79230e8i	0.0585164i	0.999572	+	0.0292582i	$0.00931450\pi$
$751$	6.79230e8i	0.0585164i	−0.999572	+	0.0292582i	$0.990686\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 4.84798e9i	− 0.409965i
$756$	0	0
$757$	− 1.77513e9i	− 0.148729i	−0.997231	−	0.0743643i	$-0.976307\pi$
$757$	− 1.77513e9i	− 0.148729i	0.997231	−	0.0743643i	$-0.0236928\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.17859e10i	1.79196i	0.444095	+	0.895980i	$0.353525\pi$
$761$	2.17859e10i	1.79196i	−0.444095	+	0.895980i	$0.646475\pi$
$762$	0	0
$763$	−9.74927e9	−0.794577
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.28021e10	1.02447
$768$	0	0
$769$	−1.13873e10	−0.902983	−0.451491	−	0.892276i	$-0.649108\pi$
$769$	−1.13873e10	−0.902983	−0.451491	+	0.892276i	$0.649108\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.45052e10	1.90823	0.954114	−	0.299445i	$-0.0968013\pi$
$773$	2.45052e10	1.90823	0.954114	+	0.299445i	$0.0968013\pi$
$774$	0	0
$775$	− 4.79898e10i	− 3.70334i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.90939e9i	0.144715i
$780$	0	0
$781$	− 7.95647e7i	− 0.00597643i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.86374e9i	0.580210i
$786$	0	0
$787$	−1.76625e10	−1.29164	−0.645818	−	0.763492i	$-0.723483\pi$
$787$	−1.76625e10	−1.29164	−0.645818	+	0.763492i	$0.723483\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.74156e10	1.25118
$792$	0	0
$793$	1.11016e10	0.790552
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.92078e9	0.134392	0.0671960	−	0.997740i	$-0.478595\pi$
$797$	1.92078e9	0.134392	0.0671960	+	0.997740i	$0.478595\pi$
$798$	0	0
$799$	8.24140e9i	0.571593i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.80971e9i	0.191495i
$804$	0	0
$805$	− 6.14928e9i	− 0.415469i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.41023e10i	0.936422i	0.883617	+	0.468211i	$0.155101\pi$
$809$	1.41023e10i	0.936422i	−0.883617	+	0.468211i	$0.844899\pi$
$810$	0	0
$811$	1.95320e10	1.28580	0.642899	−	0.765951i	$-0.277732\pi$
$811$	1.95320e10	1.28580	0.642899	+	0.765951i	$0.277732\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.41109e10	−1.56013
$816$	0	0
$817$	−1.05860e10	−0.679135
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.50736e10	−0.950637	−0.475319	−	0.879814i	$-0.657667\pi$
$821$	−1.50736e10	−0.950637	−0.475319	+	0.879814i	$0.657667\pi$
$822$	0	0
$823$	− 2.14319e10i	− 1.34018i	−0.742282	−	0.670088i	$-0.766257\pi$
$823$	− 2.14319e10i	− 1.34018i	0.742282	−	0.670088i	$-0.233743\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.03725e9i	− 0.248208i	−0.992269	−	0.124104i	$-0.960394\pi$
$827$	− 4.03725e9i	− 0.248208i	0.992269	−	0.124104i	$-0.0396057\pi$
$828$	0	0
$829$	1.60911e10i	0.980946i	0.871456	+	0.490473i	$0.163176\pi$
$829$	1.60911e10i	0.980946i	−0.871456	+	0.490473i	$0.836824\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.03789e10i	0.622150i
$834$	0	0
$835$	1.45512e10	0.864960
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.27045e9	−0.483462	−0.241731	−	0.970343i	$-0.577715\pi$
$839$	−8.27045e9	−0.483462	−0.241731	+	0.970343i	$0.577715\pi$
$840$	0	0
$841$	−5.26605e9	−0.305281
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.79342e10	1.59272
$846$	0	0
$847$	2.43639e10i	1.37770i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.25117e9i	0.180836i
$852$	0	0
$853$	− 4.07261e9i	− 0.224673i	−0.993670	−	0.112337i	$-0.964167\pi$
$853$	− 4.07261e9i	− 0.224673i	0.993670	−	0.112337i	$-0.0358335\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.23793e9i	0.284267i	0.989847	+	0.142134i	$0.0453963\pi$
$857$	5.23793e9i	0.284267i	−0.989847	+	0.142134i	$0.954604\pi$
$858$	0	0
$859$	1.68112e10	0.904945	0.452472	−	0.891778i	$-0.350542\pi$
$859$	1.68112e10	0.904945	0.452472	+	0.891778i	$0.350542\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.03500e10	1.07777	0.538887	−	0.842378i	$-0.318845\pi$
$863$	2.03500e10	1.07777	0.538887	+	0.842378i	$0.318845\pi$
$864$	0	0
$865$	5.16397e10	2.71286
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.17038e9	0.0605004
$870$	0	0
$871$	− 1.85745e10i	− 0.952477i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 8.46117e10i	− 4.26975i
$876$	0	0
$877$	− 2.52596e10i	− 1.26453i	−0.774754	−	0.632263i	$-0.782126\pi$
$877$	− 2.52596e10i	− 1.26453i	0.774754	−	0.632263i	$-0.217874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.32056e10i	− 0.650644i	−0.945603	−	0.325322i	$-0.894527\pi$
$881$	− 1.32056e10i	− 0.650644i	0.945603	−	0.325322i	$-0.105473\pi$
$882$	0	0
$883$	−1.31436e10	−0.642470	−0.321235	−	0.947000i	$-0.604098\pi$
$883$	−1.31436e10	−0.642470	−0.321235	+	0.947000i	$0.604098\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.65129e9	−0.175677	−0.0878384	−	0.996135i	$-0.527996\pi$
$887$	−3.65129e9	−0.175677	−0.0878384	+	0.996135i	$0.527996\pi$
$888$	0	0
$889$	−4.31380e10	−2.05923
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.34883e10	−1.57367
$894$	0	0
$895$	1.37200e10i	0.639695i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.62280e10i	1.20395i
$900$	0	0
$901$	− 9.07425e9i	− 0.413308i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 4.26309e10i	− 1.91185i
$906$	0	0
$907$	−2.78144e9	−0.123778	−0.0618890	−	0.998083i	$-0.519712\pi$
$907$	−2.78144e9	−0.123778	−0.0618890	+	0.998083i	$0.519712\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.30001e10	−1.00789	−0.503947	−	0.863735i	$-0.668119\pi$
$911$	−2.30001e10	−1.00789	−0.503947	+	0.863735i	$0.668119\pi$
$912$	0	0
$913$	−1.34139e9	−0.0583322
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.80319e10	−1.62875
$918$	0	0
$919$	6.10760e9i	0.259577i	0.991542	+	0.129789i	$0.0414299\pi$
$919$	6.10760e9i	0.259577i	−0.991542	+	0.129789i	$0.958570\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 8.90318e8i	− 0.0372683i
$924$	0	0
$925$	7.33406e10i	3.04683i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.44541e10i	− 1.40989i	−0.709260	−	0.704947i	$-0.750971\pi$
$929$	− 3.44541e10i	− 1.40989i	0.709260	−	0.704947i	$-0.249029\pi$
$930$	0	0
$931$	−4.21740e10	−1.71286
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.85509e9	0.234257
$936$	0	0
$937$	3.31829e10	1.31773	0.658864	−	0.752262i	$-0.271037\pi$
$937$	3.31829e10	1.31773	0.658864	+	0.752262i	$0.271037\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.21421e10	0.866274	0.433137	−	0.901328i	$-0.357407\pi$
$941$	2.21421e10	0.866274	0.433137	+	0.901328i	$0.357407\pi$
$942$	0	0
$943$	3.61426e8i	0.0140355i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.78393e10i	1.06521i	0.846365	+	0.532603i	$0.178786\pi$
$947$	2.78393e10i	1.06521i	−0.846365	+	0.532603i	$0.821214\pi$
$948$	0	0
$949$	3.14402e10i	1.19414i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.27491e10i	1.59993i	0.600044	+	0.799967i	$0.295150\pi$
$953$	4.27491e10i	1.59993i	−0.600044	+	0.799967i	$0.704850\pi$
$954$	0	0
$955$	6.06942e10	2.25494
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.56709e10	−2.40441
$960$	0	0
$961$	−2.98906e10	−1.08643
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.63250e9	0.201769
$966$	0	0
$967$	3.47491e10i	1.23581i	0.786254	+	0.617904i	$0.212018\pi$
$967$	3.47491e10i	1.23581i	−0.786254	+	0.617904i	$0.787982\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.10930e10i	− 0.388849i	−0.980917	−	0.194425i	$-0.937716\pi$
$971$	− 1.10930e10i	− 0.388849i	0.980917	−	0.194425i	$-0.0622840\pi$
$972$	0	0
$973$	− 4.63278e10i	− 1.61230i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.68888e10i	1.60856i	0.594248	+	0.804282i	$0.297450\pi$
$977$	4.68888e10i	1.60856i	−0.594248	+	0.804282i	$0.702550\pi$
$978$	0	0
$979$	−1.57645e9	−0.0536960
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.96515e10	1.33144	0.665721	−	0.746200i	$-0.268124\pi$
$983$	3.96515e10	1.33144	0.665721	+	0.746200i	$0.268124\pi$
$984$	0	0
$985$	2.72838e10	0.909657
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.00381e9	−0.0658674
$990$	0	0
$991$	1.80562e10i	0.589344i	0.955598	+	0.294672i	$0.0952104\pi$
$991$	1.80562e10i	0.589344i	−0.955598	+	0.294672i	$0.904790\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.11717e11i	3.59534i
$996$	0	0
$997$	− 2.59063e10i	− 0.827891i	−0.910302	−	0.413945i	$-0.864150\pi$
$997$	− 2.59063e10i	− 0.827891i	0.910302	−	0.413945i	$-0.135850\pi$
$998$	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.143.2		28
3.2	odd	2	inner	288.8.f.a.143.28		28
4.3	odd	2		72.8.f.a.35.3	✓	28
8.3	odd	2	inner	288.8.f.a.143.27		28
8.5	even	2		72.8.f.a.35.25	yes	28
12.11	even	2		72.8.f.a.35.26	yes	28
24.5	odd	2		72.8.f.a.35.4	yes	28
24.11	even	2	inner	288.8.f.a.143.1		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.3	✓	28	4.3	odd	2
72.8.f.a.35.4	yes	28	24.5	odd	2
72.8.f.a.35.25	yes	28	8.5	even	2
72.8.f.a.35.26	yes	28	12.11	even	2
288.8.f.a.143.1		28	24.11	even	2	inner
288.8.f.a.143.2		28	1.1	even	1	trivial
288.8.f.a.143.27		28	8.3	odd	2	inner
288.8.f.a.143.28		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)

\(f(q)\)	\(=\)	\(q-527.660 q^{5} +1312.48i q^{7} +O(q^{10})\)	\(q-527.660 q^{5} +1312.48i q^{7} -961.214i q^{11} -10755.9i q^{13} -11544.1i q^{17} +46908.6 q^{19} +8879.25 q^{23} +200300. q^{25} -109471. q^{29} -239590. i q^{31} -692545. i q^{35} +366153. i q^{37} +40704.6i q^{41} -225674. q^{43} -713906. q^{47} -899069. q^{49} +786052. q^{53} +507194. i q^{55} +1.19024e6i q^{59} +1.03215e6i q^{61} +5.67543e6i q^{65} +1.72692e6 q^{67} +82775.2 q^{71} -2.92308e6 q^{73} +1.26158e6 q^{77} +1.21761e6i q^{79} -1.39552e6i q^{83} +6.09135e6i q^{85} -1.64007e6i q^{89} +1.41169e7 q^{91} -2.47518e7 q^{95} +175296. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 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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