LMFDB - Embedded newform 288.8.f.a.143.8

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			143.8
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-238.250 q^{5} +737.398i q^{7} +O(q^{10})$	$q-238.250 q^{5} +737.398i q^{7} -1701.82i q^{11} -4482.93i q^{13} +19959.6i q^{17} +8090.28 q^{19} -110629. q^{23} -21362.2 q^{25} -19262.3 q^{29} -70291.5i q^{31} -175685. i q^{35} +441575. i q^{37} +382570. i q^{41} +329066. q^{43} -224714. q^{47} +279787. q^{49} +1.27611e6 q^{53} +405457. i q^{55} -2.48468e6i q^{59} -2.93707e6i q^{61} +1.06805e6i q^{65} -2.94806e6 q^{67} -1.63172e6 q^{71} +6.26023e6 q^{73} +1.25492e6 q^{77} +448216. i q^{79} -3.63088e6i q^{83} -4.75536e6i q^{85} -1.41664e6i q^{89} +3.30570e6 q^{91} -1.92750e6 q^{95} +1.10470e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−238.250	−0.852387	−0.426194	−	0.904632i	$-0.640146\pi$
$5$	−238.250	−0.852387	−0.426194	+	0.904632i	$0.640146\pi$
$6$	0	0
$7$	737.398i	0.812567i	0.913747	+	0.406283i	$0.133175\pi$
$7$	737.398i	0.812567i	−0.913747	+	0.406283i	$0.866825\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1701.82i	− 0.385512i	−0.981247	−	0.192756i	$-0.938257\pi$
$11$	− 1701.82i	− 0.385512i	0.981247	−	0.192756i	$-0.0617426\pi$
$12$	0	0
$13$	− 4482.93i	− 0.565926i	−0.959131	−	0.282963i	$-0.908683\pi$
$13$	− 4482.93i	− 0.565926i	0.959131	−	0.282963i	$-0.0913174\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	19959.6i	0.985327i	0.870220	+	0.492664i	$0.163977\pi$
$17$	19959.6i	0.985327i	−0.870220	+	0.492664i	$0.836023\pi$
$18$	0	0
$19$	8090.28	0.270599	0.135299	−	0.990805i	$-0.456800\pi$
$19$	8090.28	0.270599	0.135299	+	0.990805i	$0.456800\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−110629.	−1.89593	−0.947963	−	0.318381i	$-0.896861\pi$
$23$	−110629.	−1.89593	−0.947963	+	0.318381i	$0.896861\pi$
$24$	0	0
$25$	−21362.2	−0.273436
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−19262.3	−0.146661	−0.0733305	−	0.997308i	$-0.523363\pi$
$29$	−19262.3	−0.146661	−0.0733305	+	0.997308i	$0.523363\pi$
$30$	0	0
$31$	− 70291.5i	− 0.423777i	−0.977294	−	0.211888i	$-0.932039\pi$
$31$	− 70291.5i	− 0.423777i	0.977294	−	0.211888i	$-0.0679613\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 175685.i	− 0.692622i
$36$	0	0
$37$	441575.i	1.43317i	0.697499	+	0.716586i	$0.254296\pi$
$37$	441575.i	1.43317i	−0.697499	+	0.716586i	$0.745704\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	382570.i	0.866898i	0.901178	+	0.433449i	$0.142703\pi$
$41$	382570.i	0.866898i	−0.901178	+	0.433449i	$0.857297\pi$
$42$	0	0
$43$	329066.	0.631166	0.315583	−	0.948898i	$-0.397800\pi$
$43$	329066.	0.631166	0.315583	+	0.948898i	$0.397800\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−224714.	−0.315709	−0.157854	−	0.987462i	$-0.550458\pi$
$47$	−224714.	−0.315709	−0.157854	+	0.987462i	$0.550458\pi$
$48$	0	0
$49$	279787.	0.339735
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.27611e6	1.17740	0.588698	−	0.808353i	$-0.299641\pi$
$53$	1.27611e6	1.17740	0.588698	+	0.808353i	$0.299641\pi$
$54$	0	0
$55$	405457.i	0.328606i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.48468e6i	− 1.57503i	−0.616298	−	0.787513i	$-0.711368\pi$
$59$	− 2.48468e6i	− 1.57503i	0.616298	−	0.787513i	$-0.288632\pi$
$60$	0	0
$61$	− 2.93707e6i	− 1.65676i	−0.560166	−	0.828380i	$-0.689263\pi$
$61$	− 2.93707e6i	− 1.65676i	0.560166	−	0.828380i	$-0.310737\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.06805e6i	0.482388i
$66$	0	0
$67$	−2.94806e6	−1.19750	−0.598749	−	0.800937i	$-0.704335\pi$
$67$	−2.94806e6	−1.19750	−0.598749	+	0.800937i	$0.704335\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.63172e6	−0.541057	−0.270528	−	0.962712i	$-0.587198\pi$
$71$	−1.63172e6	−0.541057	−0.270528	+	0.962712i	$0.587198\pi$
$72$	0	0
$73$	6.26023e6	1.88348	0.941738	−	0.336346i	$-0.109191\pi$
$73$	6.26023e6	1.88348	0.941738	+	0.336346i	$0.109191\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.25492e6	0.313254
$78$	0	0
$79$	448216.i	0.102280i	0.998691	+	0.0511402i	$0.0162855\pi$
$79$	448216.i	0.102280i	−0.998691	+	0.0511402i	$0.983714\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.63088e6i	− 0.697009i	−0.937307	−	0.348505i	$-0.886690\pi$
$83$	− 3.63088e6i	− 0.697009i	0.937307	−	0.348505i	$-0.113310\pi$
$84$	0	0
$85$	− 4.75536e6i	− 0.839880i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.41664e6i	− 0.213008i	−0.994312	−	0.106504i	$-0.966034\pi$
$89$	− 1.41664e6i	− 0.213008i	0.994312	−	0.106504i	$-0.0339657\pi$
$90$	0	0
$91$	3.30570e6	0.459853
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.92750e6	−0.230655
$96$	0	0
$97$	1.10470e7	1.22898	0.614488	−	0.788927i	$-0.289363\pi$
$97$	1.10470e7	1.22898	0.614488	+	0.788927i	$0.289363\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.13145e7	−1.09273	−0.546364	−	0.837548i	$-0.683988\pi$
$101$	−1.13145e7	−1.09273	−0.546364	+	0.837548i	$0.683988\pi$
$102$	0	0
$103$	1.00479e7i	0.906034i	0.891502	+	0.453017i	$0.149652\pi$
$103$	1.00479e7i	0.906034i	−0.891502	+	0.453017i	$0.850348\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.33501e7i	− 1.05352i	−0.850014	−	0.526760i	$-0.823407\pi$
$107$	− 1.33501e7i	− 1.05352i	0.850014	−	0.526760i	$-0.176593\pi$
$108$	0	0
$109$	5.57713e6i	0.412494i	0.978500	+	0.206247i	$0.0661251\pi$
$109$	5.57713e6i	0.412494i	−0.978500	+	0.206247i	$0.933875\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.54059e6i	− 0.296031i	−0.988985	−	0.148016i	$-0.952711\pi$
$113$	− 4.54059e6i	− 0.296031i	0.988985	−	0.148016i	$-0.0472886\pi$
$114$	0	0
$115$	2.63573e7	1.61606
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.47182e7	−0.800644
$120$	0	0
$121$	1.65910e7	0.851380
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.37028e7	1.08546
$126$	0	0
$127$	− 3.64412e7i	− 1.57863i	−0.613990	−	0.789314i	$-0.710436\pi$
$127$	− 3.64412e7i	− 1.57863i	0.613990	−	0.789314i	$-0.289564\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.32094e7i	− 0.902017i	−0.892520	−	0.451009i	$-0.851064\pi$
$131$	− 2.32094e7i	− 0.902017i	0.892520	−	0.451009i	$-0.148936\pi$
$132$	0	0
$133$	5.96576e6i	0.219880i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.20151e7i	− 1.39599i	−0.716101	−	0.697996i	$-0.754075\pi$
$137$	− 4.20151e7i	− 1.39599i	0.716101	−	0.697996i	$-0.245925\pi$
$138$	0	0
$139$	4.66398e7	1.47301	0.736504	−	0.676433i	$-0.236475\pi$
$139$	4.66398e7	1.47301	0.736504	+	0.676433i	$0.236475\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.62911e6	−0.218171
$144$	0	0
$145$	4.58923e6	0.125012
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.42435e7	−1.09571	−0.547857	−	0.836572i	$-0.684556\pi$
$149$	−4.42435e7	−1.09571	−0.547857	+	0.836572i	$0.684556\pi$
$150$	0	0
$151$	− 4.69078e7i	− 1.10873i	−0.832274	−	0.554365i	$-0.812961\pi$
$151$	− 4.69078e7i	− 1.10873i	0.832274	−	0.554365i	$-0.187039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.67469e7i	0.361222i
$156$	0	0
$157$	− 6.21526e7i	− 1.28177i	−0.767636	−	0.640886i	$-0.778567\pi$
$157$	− 6.21526e7i	− 1.28177i	0.767636	−	0.640886i	$-0.221433\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.15776e7i	− 1.54057i
$162$	0	0
$163$	9.54624e7	1.72654	0.863269	−	0.504745i	$-0.168413\pi$
$163$	9.54624e7	1.72654	0.863269	+	0.504745i	$0.168413\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.14834e8	1.90793	0.953963	−	0.299923i	$-0.0969611\pi$
$167$	1.14834e8	1.90793	0.953963	+	0.299923i	$0.0969611\pi$
$168$	0	0
$169$	4.26519e7	0.679728
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.67601e7	−0.833454	−0.416727	−	0.909032i	$-0.636823\pi$
$173$	−5.67601e7	−0.833454	−0.416727	+	0.909032i	$0.636823\pi$
$174$	0	0
$175$	− 1.57524e7i	− 0.222185i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.12835e7i	− 0.147048i	−0.997293	−	0.0735241i	$-0.976575\pi$
$179$	− 1.12835e7i	− 0.147048i	0.997293	−	0.0735241i	$-0.0234246\pi$
$180$	0	0
$181$	8.60787e7i	1.07900i	0.841986	+	0.539499i	$0.181386\pi$
$181$	8.60787e7i	1.07900i	−0.841986	+	0.539499i	$0.818614\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.05205e8i	− 1.22162i
$186$	0	0
$187$	3.39676e7	0.379856
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.04440e7	−0.939210	−0.469605	−	0.882877i	$-0.655604\pi$
$191$	−9.04440e7	−0.939210	−0.469605	+	0.882877i	$0.655604\pi$
$192$	0	0
$193$	1.25636e8	1.25795	0.628977	−	0.777424i	$-0.283474\pi$
$193$	1.25636e8	1.25795	0.628977	+	0.777424i	$0.283474\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.98131e7	−0.371017	−0.185509	−	0.982643i	$-0.559393\pi$
$197$	−3.98131e7	−0.371017	−0.185509	+	0.982643i	$0.559393\pi$
$198$	0	0
$199$	1.82217e8i	1.63910i	0.573011	+	0.819548i	$0.305775\pi$
$199$	1.82217e8i	1.63910i	−0.573011	+	0.819548i	$0.694225\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.42040e7i	− 0.119172i
$204$	0	0
$205$	− 9.11472e7i	− 0.738933i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.37682e7i	− 0.104319i
$210$	0	0
$211$	−3.29195e7	−0.241249	−0.120624	−	0.992698i	$-0.538490\pi$
$211$	−3.29195e7	−0.241249	−0.120624	+	0.992698i	$0.538490\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.83998e7	−0.537998
$216$	0	0
$217$	5.18329e7	0.344347
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.94774e7	0.557622
$222$	0	0
$223$	− 1.71254e8i	− 1.03412i	−0.855948	−	0.517062i	$-0.827026\pi$
$223$	− 1.71254e8i	− 1.03412i	0.855948	−	0.517062i	$-0.172974\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.45746e7i	− 0.139443i	−0.997566	−	0.0697214i	$-0.977789\pi$
$227$	− 2.45746e7i	− 0.139443i	0.997566	−	0.0697214i	$-0.0222110\pi$
$228$	0	0
$229$	− 7.85931e7i	− 0.432474i	−0.976341	−	0.216237i	$-0.930622\pi$
$229$	− 7.85931e7i	− 0.432474i	0.976341	−	0.216237i	$-0.0693784\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.01184e8i	1.55986i	0.625866	+	0.779931i	$0.284746\pi$
$233$	3.01184e8i	1.55986i	−0.625866	+	0.779931i	$0.715254\pi$
$234$	0	0
$235$	5.35379e7	0.269106
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.18967e8	−0.563681	−0.281840	−	0.959461i	$-0.590945\pi$
$239$	−1.18967e8	−0.563681	−0.281840	+	0.959461i	$0.590945\pi$
$240$	0	0
$241$	−9.68834e7	−0.445851	−0.222925	−	0.974835i	$-0.571561\pi$
$241$	−9.68834e7	−0.445851	−0.222925	+	0.974835i	$0.571561\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.66590e7	−0.289586
$246$	0	0
$247$	− 3.62681e7i	− 0.153139i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.13129e8i	− 1.24987i	−0.780677	−	0.624935i	$-0.785126\pi$
$251$	− 3.13129e8i	− 1.24987i	0.780677	−	0.624935i	$-0.214874\pi$
$252$	0	0
$253$	1.88270e8i	0.730903i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.31292e8i	0.482472i	0.970466	+	0.241236i	$0.0775528\pi$
$257$	1.31292e8i	0.482472i	−0.970466	+	0.241236i	$0.922447\pi$
$258$	0	0
$259$	−3.25617e8	−1.16455
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.72319e8	0.923068	0.461534	−	0.887123i	$-0.347299\pi$
$263$	2.72319e8	0.923068	0.461534	+	0.887123i	$0.347299\pi$
$264$	0	0
$265$	−3.04033e8	−1.00360
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.42080e8	1.69797	0.848985	−	0.528416i	$-0.177214\pi$
$269$	5.42080e8	1.69797	0.848985	+	0.528416i	$0.177214\pi$
$270$	0	0
$271$	− 4.54506e7i	− 0.138723i	−0.997592	−	0.0693613i	$-0.977904\pi$
$271$	− 4.54506e7i	− 0.138723i	0.997592	−	0.0693613i	$-0.0220961\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.63545e7i	0.105413i
$276$	0	0
$277$	− 1.51529e8i	− 0.428368i	−0.976793	−	0.214184i	$-0.931291\pi$
$277$	− 1.51529e8i	− 0.428368i	0.976793	−	0.214184i	$-0.0687092\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.01196e8i	0.540938i	0.962729	+	0.270469i	$0.0871788\pi$
$281$	2.01196e8i	0.540938i	−0.962729	+	0.270469i	$0.912821\pi$
$282$	0	0
$283$	−3.07402e8	−0.806222	−0.403111	−	0.915151i	$-0.632071\pi$
$283$	−3.07402e8	−0.806222	−0.403111	+	0.915151i	$0.632071\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.82107e8	−0.704412
$288$	0	0
$289$	1.19534e7	0.0291306
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.38990e8	1.01957	0.509786	−	0.860301i	$-0.329724\pi$
$293$	4.38990e8	1.01957	0.509786	+	0.860301i	$0.329724\pi$
$294$	0	0
$295$	5.91973e8i	1.34253i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.95941e8i	1.07295i
$300$	0	0
$301$	2.42653e8i	0.512865i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.99755e8i	1.41220i
$306$	0	0
$307$	−4.94522e8	−0.975442	−0.487721	−	0.873000i	$-0.662172\pi$
$307$	−4.94522e8	−0.975442	−0.487721	+	0.873000i	$0.662172\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.88359e7	−0.0732103	−0.0366051	−	0.999330i	$-0.511654\pi$
$311$	−3.88359e7	−0.0732103	−0.0366051	+	0.999330i	$0.511654\pi$
$312$	0	0
$313$	−4.59451e7	−0.0846905	−0.0423452	−	0.999103i	$-0.513483\pi$
$313$	−4.59451e7	−0.0846905	−0.0423452	+	0.999103i	$0.513483\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.40916e8	−1.48267	−0.741336	−	0.671134i	$-0.765808\pi$
$317$	−8.40916e8	−1.48267	−0.741336	+	0.671134i	$0.765808\pi$
$318$	0	0
$319$	3.27809e7i	0.0565396i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.61479e8i	0.266628i
$324$	0	0
$325$	9.57650e7i	0.154744i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.65703e8i	− 0.256535i
$330$	0	0
$331$	−8.63353e8	−1.30855	−0.654275	−	0.756257i	$-0.727026\pi$
$331$	−8.63353e8	−1.30855	−0.654275	+	0.756257i	$0.727026\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.02375e8	1.02073
$336$	0	0
$337$	−1.07596e9	−1.53141	−0.765703	−	0.643194i	$-0.777609\pi$
$337$	−1.07596e9	−1.53141	−0.765703	+	0.643194i	$0.777609\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.19623e8	−0.163371
$342$	0	0
$343$	8.13593e8i	1.08862i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.40774e9i	1.80871i	0.426785	+	0.904353i	$0.359646\pi$
$347$	1.40774e9i	1.80871i	−0.426785	+	0.904353i	$0.640354\pi$
$348$	0	0
$349$	− 4.13124e8i	− 0.520225i	−0.965578	−	0.260112i	$-0.916240\pi$
$349$	− 4.13124e8i	− 0.520225i	0.965578	−	0.260112i	$-0.0837596\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 5.35837e8i	− 0.648367i	−0.945994	−	0.324183i	$-0.894910\pi$
$353$	− 5.35837e8i	− 0.648367i	0.945994	−	0.324183i	$-0.105090\pi$
$354$	0	0
$355$	3.88758e8	0.461190
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.32439e7	0.0151072	0.00755362	−	0.999971i	$-0.497596\pi$
$359$	1.32439e7	0.0151072	0.00755362	+	0.999971i	$0.497596\pi$
$360$	0	0
$361$	−8.28419e8	−0.926776
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.49150e9	−1.60545
$366$	0	0
$367$	− 5.85755e8i	− 0.618565i	−0.950970	−	0.309282i	$-0.899911\pi$
$367$	− 5.85755e8i	− 0.618565i	0.950970	−	0.309282i	$-0.100089\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.41002e8i	0.956713i
$372$	0	0
$373$	8.81595e8i	0.879606i	0.898094	+	0.439803i	$0.144952\pi$
$373$	8.81595e8i	0.879606i	−0.898094	+	0.439803i	$0.855048\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.63514e7i	0.0829993i
$378$	0	0
$379$	1.38059e8	0.130265	0.0651325	−	0.997877i	$-0.479253\pi$
$379$	1.38059e8	0.130265	0.0651325	+	0.997877i	$0.479253\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.81141e8	0.528550	0.264275	−	0.964447i	$-0.414867\pi$
$383$	5.81141e8	0.528550	0.264275	+	0.964447i	$0.414867\pi$
$384$	0	0
$385$	−2.98983e8	−0.267014
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.53201e9	1.31959	0.659796	−	0.751445i	$-0.270643\pi$
$389$	1.53201e9	1.31959	0.659796	+	0.751445i	$0.270643\pi$
$390$	0	0
$391$	− 2.20811e9i	− 1.86811i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.06787e8i	− 0.0871825i
$396$	0	0
$397$	1.41811e9i	1.13748i	0.822518	+	0.568740i	$0.192569\pi$
$397$	1.41811e9i	1.13748i	−0.822518	+	0.568740i	$0.807431\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2.25678e8i	− 0.174777i	−0.996174	−	0.0873884i	$-0.972148\pi$
$401$	− 2.25678e8i	− 0.174777i	0.996174	−	0.0873884i	$-0.0278521\pi$
$402$	0	0
$403$	−3.15112e8	−0.239826
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.51479e8	0.552505
$408$	0	0
$409$	−6.63525e8	−0.479541	−0.239770	−	0.970830i	$-0.577072\pi$
$409$	−6.63525e8	−0.479541	−0.239770	+	0.970830i	$0.577072\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.83220e9	1.27981
$414$	0	0
$415$	8.65055e8i	0.594122i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.25579e9i	− 0.834006i	−0.908905	−	0.417003i	$-0.863080\pi$
$419$	− 1.25579e9i	− 0.834006i	0.908905	−	0.417003i	$-0.136920\pi$
$420$	0	0
$421$	− 1.15612e9i	− 0.755118i	−0.925986	−	0.377559i	$-0.876764\pi$
$421$	− 1.15612e9i	− 0.755118i	0.925986	−	0.377559i	$-0.123236\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.26380e8i	− 0.269424i
$426$	0	0
$427$	2.16579e9	1.34623
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.49566e8	−0.571287	−0.285644	−	0.958336i	$-0.592207\pi$
$431$	−9.49566e8	−0.571287	−0.285644	+	0.958336i	$0.592207\pi$
$432$	0	0
$433$	−3.02435e8	−0.179030	−0.0895148	−	0.995985i	$-0.528532\pi$
$433$	−3.02435e8	−0.179030	−0.0895148	+	0.995985i	$0.528532\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.95019e8	−0.513035
$438$	0	0
$439$	8.77959e8i	0.495277i	0.968852	+	0.247639i	$0.0796546\pi$
$439$	8.77959e8i	0.495277i	−0.968852	+	0.247639i	$0.920345\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 7.35244e8i	− 0.401808i	−0.979611	−	0.200904i	$-0.935612\pi$
$443$	− 7.35244e8i	− 0.401808i	0.979611	−	0.200904i	$-0.0643879\pi$
$444$	0	0
$445$	3.37515e8i	0.181565i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.48949e8i	0.286200i	0.989708	+	0.143100i	$0.0457071\pi$
$449$	5.48949e8i	0.286200i	−0.989708	+	0.143100i	$0.954293\pi$
$450$	0	0
$451$	6.51064e8	0.334200
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.87582e8	−0.391973
$456$	0	0
$457$	3.04791e9	1.49381	0.746905	−	0.664930i	$-0.231539\pi$
$457$	3.04791e9	1.49381	0.746905	+	0.664930i	$0.231539\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.10410e8	−0.195103	−0.0975516	−	0.995230i	$-0.531101\pi$
$461$	−4.10410e8	−0.195103	−0.0975516	+	0.995230i	$0.531101\pi$
$462$	0	0
$463$	6.43419e8i	0.301273i	0.988589	+	0.150636i	$0.0481323\pi$
$463$	6.43419e8i	0.301273i	−0.988589	+	0.150636i	$0.951868\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.09110e8i	− 0.413054i	−0.978441	−	0.206527i	$-0.933784\pi$
$467$	− 9.09110e8i	− 0.413054i	0.978441	−	0.206527i	$-0.0662162\pi$
$468$	0	0
$469$	− 2.17390e9i	− 0.973047i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5.60010e8i	− 0.243322i
$474$	0	0
$475$	−1.72826e8	−0.0739914
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.75601e8	−0.239302	−0.119651	−	0.992816i	$-0.538178\pi$
$479$	−5.75601e8	−0.239302	−0.119651	+	0.992816i	$0.538178\pi$
$480$	0	0
$481$	1.97955e9	0.811070
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.63194e9	−1.04756
$486$	0	0
$487$	− 4.63509e9i	− 1.81847i	−0.416279	−	0.909237i	$-0.636666\pi$
$487$	− 4.63509e9i	− 1.81847i	0.416279	−	0.909237i	$-0.363334\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 1.75783e9i	− 0.670181i	−0.942186	−	0.335091i	$-0.891233\pi$
$491$	− 1.75783e9i	− 0.670181i	0.942186	−	0.335091i	$-0.108767\pi$
$492$	0	0
$493$	− 3.84467e8i	− 0.144509i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.20323e9i	− 0.439645i
$498$	0	0
$499$	−9.64456e8	−0.347481	−0.173740	−	0.984791i	$-0.555585\pi$
$499$	−9.64456e8	−0.347481	−0.173740	+	0.984791i	$0.555585\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.53727e9	−1.23931	−0.619656	−	0.784874i	$-0.712728\pi$
$503$	−3.53727e9	−1.23931	−0.619656	+	0.784874i	$0.712728\pi$
$504$	0	0
$505$	2.69568e9	0.931427
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.22175e9	−0.410649	−0.205324	−	0.978694i	$-0.565825\pi$
$509$	−1.22175e9	−0.410649	−0.205324	+	0.978694i	$0.565825\pi$
$510$	0	0
$511$	4.61628e9i	1.53045i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.39390e9i	− 0.772292i
$516$	0	0
$517$	3.82421e8i	0.121710i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.68053e8i	0.0830404i	0.999138	+	0.0415202i	$0.0132201\pi$
$521$	2.68053e8i	0.0830404i	−0.999138	+	0.0415202i	$0.986780\pi$
$522$	0	0
$523$	3.14297e9	0.960691	0.480346	−	0.877079i	$-0.340511\pi$
$523$	3.14297e9	0.960691	0.480346	+	0.877079i	$0.340511\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.40299e9	0.417559
$528$	0	0
$529$	8.83394e9	2.59454
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.71503e9	0.490600
$534$	0	0
$535$	3.18066e9i	0.898007i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.76145e8i	− 0.130972i
$540$	0	0
$541$	− 6.12302e9i	− 1.66255i	−0.555859	−	0.831276i	$-0.687611\pi$
$541$	− 6.12302e9i	− 1.66255i	0.555859	−	0.831276i	$-0.312389\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.32875e9i	− 0.351605i
$546$	0	0
$547$	−1.33232e9	−0.348058	−0.174029	−	0.984740i	$-0.555679\pi$
$547$	−1.33232e9	−0.348058	−0.174029	+	0.984740i	$0.555679\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.55837e8	−0.0396863
$552$	0	0
$553$	−3.30513e8	−0.0831096
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.96837e9	1.21821	0.609103	−	0.793091i	$-0.291530\pi$
$557$	4.96837e9	1.21821	0.609103	+	0.793091i	$0.291530\pi$
$558$	0	0
$559$	− 1.47518e9i	− 0.357193i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 7.63988e9i	− 1.80429i	−0.431428	−	0.902147i	$-0.641990\pi$
$563$	− 7.63988e9i	− 1.80429i	0.431428	−	0.902147i	$-0.358010\pi$
$564$	0	0
$565$	1.08179e9i	0.252333i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.81831e8i	0.109648i	0.998496	+	0.0548242i	$0.0174598\pi$
$569$	4.81831e8i	0.109648i	−0.998496	+	0.0548242i	$0.982540\pi$
$570$	0	0
$571$	4.95723e9	1.11433	0.557164	−	0.830403i	$-0.311890\pi$
$571$	4.95723e9	1.11433	0.557164	+	0.830403i	$0.311890\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.36327e9	0.518414
$576$	0	0
$577$	−3.49388e9	−0.757168	−0.378584	−	0.925567i	$-0.623589\pi$
$577$	−3.49388e9	−0.757168	−0.378584	+	0.925567i	$0.623589\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.67740e9	0.566366
$582$	0	0
$583$	− 2.17171e9i	− 0.453901i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.05369e8i	0.143940i	0.997407	+	0.0719702i	$0.0229287\pi$
$587$	7.05369e8i	0.143940i	−0.997407	+	0.0719702i	$0.977071\pi$
$588$	0	0
$589$	− 5.68678e8i	− 0.114673i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.36729e9i	− 0.466188i	−0.972454	−	0.233094i	$-0.925115\pi$
$593$	− 2.36729e9i	− 0.466188i	0.972454	−	0.233094i	$-0.0748850\pi$
$594$	0	0
$595$	3.50660e9	0.682459
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.23088e9	−0.804334	−0.402167	−	0.915566i	$-0.631743\pi$
$599$	−4.23088e9	−0.804334	−0.402167	+	0.915566i	$0.631743\pi$
$600$	0	0
$601$	−2.07946e8	−0.0390742	−0.0195371	−	0.999809i	$-0.506219\pi$
$601$	−2.07946e8	−0.0390742	−0.0195371	+	0.999809i	$0.506219\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.95280e9	−0.725706
$606$	0	0
$607$	− 5.95951e9i	− 1.08156i	−0.841165	−	0.540779i	$-0.818130\pi$
$607$	− 5.95951e9i	− 1.08156i	0.841165	−	0.540779i	$-0.181870\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.00737e9i	0.178668i
$612$	0	0
$613$	8.16210e9i	1.43117i	0.698527	+	0.715584i	$0.253839\pi$
$613$	8.16210e9i	1.43117i	−0.698527	+	0.715584i	$0.746161\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.21778e9i	− 0.722912i	−0.932389	−	0.361456i	$-0.882280\pi$
$617$	− 4.21778e9i	− 0.722912i	0.932389	−	0.361456i	$-0.117720\pi$
$618$	0	0
$619$	−3.34483e9	−0.566835	−0.283417	−	0.958997i	$-0.591468\pi$
$619$	−3.34483e9	−0.566835	−0.283417	+	0.958997i	$0.591468\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.04463e9	0.173083
$624$	0	0
$625$	−3.97825e9	−0.651797
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.81365e9	−1.41214
$630$	0	0
$631$	− 9.88860e9i	− 1.56687i	−0.621475	−	0.783434i	$-0.713466\pi$
$631$	− 9.88860e9i	− 1.56687i	0.621475	−	0.783434i	$-0.286534\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.68210e9i	1.34560i
$636$	0	0
$637$	− 1.25426e9i	− 0.192265i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.30111e9i	0.645026i	0.946565	+	0.322513i	$0.104528\pi$
$641$	4.30111e9i	0.645026i	−0.946565	+	0.322513i	$0.895472\pi$
$642$	0	0
$643$	6.64317e9	0.985455	0.492728	−	0.870184i	$-0.336000\pi$
$643$	6.64317e9	0.985455	0.492728	+	0.870184i	$0.336000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.43332e9	1.36930	0.684651	−	0.728871i	$-0.259955\pi$
$647$	9.43332e9	1.36930	0.684651	+	0.728871i	$0.259955\pi$
$648$	0	0
$649$	−4.22846e9	−0.607192
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.73991e9	0.806694	0.403347	−	0.915047i	$-0.367847\pi$
$653$	5.73991e9	0.806694	0.403347	+	0.915047i	$0.367847\pi$
$654$	0	0
$655$	5.52963e9i	0.768868i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.23892e9i	0.440860i	0.975403	+	0.220430i	$0.0707461\pi$
$659$	3.23892e9i	0.440860i	−0.975403	+	0.220430i	$0.929254\pi$
$660$	0	0
$661$	1.77949e9i	0.239657i	0.992795	+	0.119829i	$0.0382345\pi$
$661$	1.77949e9i	0.239657i	−0.992795	+	0.119829i	$0.961766\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.42134e9i	− 0.187423i
$666$	0	0
$667$	2.13097e9	0.278058
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.99835e9	−0.638701
$672$	0	0
$673$	−3.44495e9	−0.435642	−0.217821	−	0.975989i	$-0.569895\pi$
$673$	−3.44495e9	−0.435642	−0.217821	+	0.975989i	$0.569895\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.11751e9	−0.633868	−0.316934	−	0.948448i	$-0.602653\pi$
$677$	−5.11751e9	−0.633868	−0.316934	+	0.948448i	$0.602653\pi$
$678$	0	0
$679$	8.14604e9i	0.998624i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.48675e10i	1.78552i	0.450530	+	0.892761i	$0.351235\pi$
$683$	1.48675e10i	1.78552i	−0.450530	+	0.892761i	$0.648765\pi$
$684$	0	0
$685$	1.00101e10i	1.18993i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5.72071e9i	− 0.666319i
$690$	0	0
$691$	−1.52586e10	−1.75931	−0.879655	−	0.475612i	$-0.842227\pi$
$691$	−1.52586e10	−1.75931	−0.879655	+	0.475612i	$0.842227\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.11119e10	−1.25557
$696$	0	0
$697$	−7.63595e9	−0.854178
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.98929e9	−0.547049	−0.273524	−	0.961865i	$-0.588189\pi$
$701$	−4.98929e9	−0.547049	−0.273524	+	0.961865i	$0.588189\pi$
$702$	0	0
$703$	3.57246e9i	0.387815i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.34332e9i	− 0.887914i
$708$	0	0
$709$	− 1.01729e10i	− 1.07197i	−0.844226	−	0.535987i	$-0.819940\pi$
$709$	− 1.01729e10i	− 1.07197i	0.844226	−	0.535987i	$-0.180060\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.77628e9i	0.803449i
$714$	0	0
$715$	1.81763e9	0.185967
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.76261e10	1.76850	0.884248	−	0.467017i	$-0.154671\pi$
$719$	1.76261e10	1.76850	0.884248	+	0.467017i	$0.154671\pi$
$720$	0	0
$721$	−7.40929e9	−0.736213
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.11484e8	0.0401024
$726$	0	0
$727$	1.77014e10i	1.70859i	0.519790	+	0.854294i	$0.326010\pi$
$727$	1.77014e10i	1.70859i	−0.519790	+	0.854294i	$0.673990\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.56803e9i	0.621905i
$732$	0	0
$733$	3.93288e9i	0.368848i	0.982847	+	0.184424i	$0.0590419\pi$
$733$	3.93288e9i	0.368848i	−0.982847	+	0.184424i	$0.940958\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.01706e9i	0.461650i
$738$	0	0
$739$	−2.06051e10	−1.87810	−0.939051	−	0.343777i	$-0.888294\pi$
$739$	−2.06051e10	−1.87810	−0.939051	+	0.343777i	$0.888294\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.69616e10	1.51707	0.758537	−	0.651630i	$-0.225915\pi$
$743$	1.69616e10	1.51707	0.758537	+	0.651630i	$0.225915\pi$
$744$	0	0
$745$	1.05410e10	0.933973
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.84437e9	0.856055
$750$	0	0
$751$	− 1.23098e10i	− 1.06050i	−0.847841	−	0.530251i	$-0.822098\pi$
$751$	− 1.23098e10i	− 1.06050i	0.847841	−	0.530251i	$-0.177902\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.11758e10i	0.945067i
$756$	0	0
$757$	− 8.82480e9i	− 0.739383i	−0.929155	−	0.369691i	$-0.879463\pi$
$757$	− 8.82480e9i	− 0.739383i	0.929155	−	0.369691i	$-0.120537\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.05695e8i	0.0662712i	0.999451	+	0.0331356i	$0.0105493\pi$
$761$	8.05695e8i	0.0662712i	−0.999451	+	0.0331356i	$0.989451\pi$
$762$	0	0
$763$	−4.11257e9	−0.335179
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.11386e10	−0.891348
$768$	0	0
$769$	−1.00081e10	−0.793611	−0.396806	−	0.917903i	$-0.629881\pi$
$769$	−1.00081e10	−0.793611	−0.396806	+	0.917903i	$0.629881\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.09881e10	0.855648	0.427824	−	0.903862i	$-0.359280\pi$
$773$	1.09881e10	0.855648	0.427824	+	0.903862i	$0.359280\pi$
$774$	0	0
$775$	1.50158e9i	0.115876i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.09510e9i	0.234581i
$780$	0	0
$781$	2.77690e9i	0.208584i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.48078e10i	1.09257i
$786$	0	0
$787$	−1.23147e10	−0.900562	−0.450281	−	0.892887i	$-0.648676\pi$
$787$	−1.23147e10	−0.900562	−0.450281	+	0.892887i	$0.648676\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.34823e9	0.240545
$792$	0	0
$793$	−1.31667e10	−0.937604
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.04586e9	−0.492981	−0.246490	−	0.969145i	$-0.579277\pi$
$797$	−7.04586e9	−0.492981	−0.246490	+	0.969145i	$0.579277\pi$
$798$	0	0
$799$	− 4.48519e9i	− 0.311076i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.06538e10i	− 0.726103i
$804$	0	0
$805$	1.94358e10i	1.31316i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.50045e9i	0.498044i	0.968498	+	0.249022i	$0.0801091\pi$
$809$	7.50045e9i	0.498044i	−0.968498	+	0.249022i	$0.919891\pi$
$810$	0	0
$811$	7.77055e9	0.511539	0.255770	−	0.966738i	$-0.417671\pi$
$811$	7.77055e9	0.511539	0.255770	+	0.966738i	$0.417671\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.27439e10	−1.47168
$816$	0	0
$817$	2.66224e9	0.170793
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.32808e10	1.46824	0.734120	−	0.679019i	$-0.237595\pi$
$821$	2.32808e10	1.46824	0.734120	+	0.679019i	$0.237595\pi$
$822$	0	0
$823$	− 1.37296e10i	− 0.858537i	−0.903177	−	0.429268i	$-0.858771\pi$
$823$	− 1.37296e10i	− 0.858537i	0.903177	−	0.429268i	$-0.141229\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.49685e9i	0.399423i	0.979855	+	0.199712i	$0.0640006\pi$
$827$	6.49685e9i	0.399423i	−0.979855	+	0.199712i	$0.935999\pi$
$828$	0	0
$829$	− 1.86189e10i	− 1.13504i	−0.823359	−	0.567521i	$-0.807902\pi$
$829$	− 1.86189e10i	− 1.13504i	0.823359	−	0.567521i	$-0.192098\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.58443e9i	0.334750i
$834$	0	0
$835$	−2.73591e10	−1.62629
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.39216e10	0.813811	0.406906	−	0.913470i	$-0.366608\pi$
$839$	1.39216e10	0.813811	0.406906	+	0.913470i	$0.366608\pi$
$840$	0	0
$841$	−1.68788e10	−0.978491
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.01618e10	−0.579391
$846$	0	0
$847$	1.22342e10i	0.691803i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4.88510e10i	− 2.71719i
$852$	0	0
$853$	− 1.40948e10i	− 0.777568i	−0.921329	−	0.388784i	$-0.872895\pi$
$853$	− 1.40948e10i	− 0.777568i	0.921329	−	0.388784i	$-0.127105\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.00414e10i	1.08766i	0.839194	+	0.543832i	$0.183027\pi$
$857$	2.00414e10i	1.08766i	−0.839194	+	0.543832i	$0.816973\pi$
$858$	0	0
$859$	1.04263e10	0.561245	0.280623	−	0.959818i	$-0.409459\pi$
$859$	1.04263e10	0.561245	0.280623	+	0.959818i	$0.409459\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.98495e10	−1.05127	−0.525633	−	0.850712i	$-0.676171\pi$
$863$	−1.98495e10	−1.05127	−0.525633	+	0.850712i	$0.676171\pi$
$864$	0	0
$865$	1.35231e10	0.710426
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.62781e8	0.0394303
$870$	0	0
$871$	1.32159e10i	0.677696i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.74784e10i	0.882009i
$876$	0	0
$877$	1.64195e10i	0.821982i	0.911639	+	0.410991i	$0.134817\pi$
$877$	1.64195e10i	0.821982i	−0.911639	+	0.410991i	$0.865183\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.36845e10i	1.16694i	0.812135	+	0.583470i	$0.198305\pi$
$881$	2.36845e10i	1.16694i	−0.812135	+	0.583470i	$0.801695\pi$
$882$	0	0
$883$	1.57236e10	0.768580	0.384290	−	0.923213i	$-0.374446\pi$
$883$	1.57236e10	0.768580	0.384290	+	0.923213i	$0.374446\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.88503e10	−0.906953	−0.453477	−	0.891268i	$-0.649816\pi$
$887$	−1.88503e10	−0.906953	−0.453477	+	0.891268i	$0.649816\pi$
$888$	0	0
$889$	2.68717e10	1.28274
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.81799e9	−0.0854304
$894$	0	0
$895$	2.68830e9i	0.125342i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.35398e9i	0.0621515i
$900$	0	0
$901$	2.54706e10i	1.16012i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2.05082e10i	− 0.919724i
$906$	0	0
$907$	5.59421e9	0.248951	0.124475	−	0.992223i	$-0.460275\pi$
$907$	5.59421e9	0.248951	0.124475	+	0.992223i	$0.460275\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.38461e9	0.235961	0.117980	−	0.993016i	$-0.462358\pi$
$911$	5.38461e9	0.235961	0.117980	+	0.993016i	$0.462358\pi$
$912$	0	0
$913$	−6.17909e9	−0.268706
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.71146e10	0.732949
$918$	0	0
$919$	3.73542e10i	1.58758i	0.608193	+	0.793789i	$0.291895\pi$
$919$	3.73542e10i	1.58758i	−0.608193	+	0.793789i	$0.708105\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.31490e9i	0.306198i
$924$	0	0
$925$	− 9.43300e9i	− 0.391881i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 3.91715e10i	− 1.60293i	−0.598040	−	0.801467i	$-0.704053\pi$
$929$	− 3.91715e10i	− 1.60293i	0.598040	−	0.801467i	$-0.295947\pi$
$930$	0	0
$931$	2.26355e9	0.0919319
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.09275e9	−0.323784
$936$	0	0
$937$	1.34561e10	0.534355	0.267177	−	0.963647i	$-0.413909\pi$
$937$	1.34561e10	0.534355	0.267177	+	0.963647i	$0.413909\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.29393e9	−0.207116	−0.103558	−	0.994623i	$-0.533023\pi$
$941$	−5.29393e9	−0.207116	−0.103558	+	0.994623i	$0.533023\pi$
$942$	0	0
$943$	− 4.23234e10i	− 1.64357i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.20048e10i	− 0.841962i	−0.907070	−	0.420981i	$-0.861686\pi$
$947$	− 2.20048e10i	− 0.841962i	0.907070	−	0.420981i	$-0.138314\pi$
$948$	0	0
$949$	− 2.80641e10i	− 1.06591i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.42103e10i	− 0.531836i	−0.963996	−	0.265918i	$-0.914325\pi$
$953$	− 1.42103e10i	− 0.531836i	0.963996	−	0.265918i	$-0.0856750\pi$
$954$	0	0
$955$	2.15482e10	0.800571
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.09818e10	1.13434
$960$	0	0
$961$	2.25717e10	0.820413
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.99328e10	−1.07226
$966$	0	0
$967$	− 3.05306e10i	− 1.08578i	−0.839803	−	0.542892i	$-0.817329\pi$
$967$	− 3.05306e10i	− 1.08578i	0.839803	−	0.542892i	$-0.182671\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.10450e10i	1.43877i	0.694610	+	0.719387i	$0.255577\pi$
$971$	4.10450e10i	1.43877i	−0.694610	+	0.719387i	$0.744423\pi$
$972$	0	0
$973$	3.43921e10i	1.19692i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.37480e10i	− 1.50082i	−0.660975	−	0.750408i	$-0.729857\pi$
$977$	− 4.37480e10i	− 1.50082i	0.660975	−	0.750408i	$-0.270143\pi$
$978$	0	0
$979$	−2.41087e9	−0.0821172
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.01275e10	1.34742	0.673712	−	0.738994i	$-0.264699\pi$
$983$	4.01275e10	1.34742	0.673712	+	0.738994i	$0.264699\pi$
$984$	0	0
$985$	9.48545e9	0.316250
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.64042e10	−1.19664
$990$	0	0
$991$	− 8.59795e9i	− 0.280632i	−0.990107	−	0.140316i	$-0.955188\pi$
$991$	− 8.59795e9i	− 0.280632i	0.990107	−	0.140316i	$-0.0448118\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.34132e10i	− 1.39714i
$996$	0	0
$997$	− 5.27262e10i	− 1.68497i	−0.538716	−	0.842487i	$-0.681091\pi$
$997$	− 5.27262e10i	− 1.68497i	0.538716	−	0.842487i	$-0.318909\pi$
$998$	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.143.8		28
3.2	odd	2	inner	288.8.f.a.143.22		28
4.3	odd	2		72.8.f.a.35.12	yes	28
8.3	odd	2	inner	288.8.f.a.143.21		28
8.5	even	2		72.8.f.a.35.18	yes	28
12.11	even	2		72.8.f.a.35.17	yes	28
24.5	odd	2		72.8.f.a.35.11	✓	28
24.11	even	2	inner	288.8.f.a.143.7		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.11	✓	28	24.5	odd	2
72.8.f.a.35.12	yes	28	4.3	odd	2
72.8.f.a.35.17	yes	28	12.11	even	2
72.8.f.a.35.18	yes	28	8.5	even	2
288.8.f.a.143.7		28	24.11	even	2	inner
288.8.f.a.143.8		28	1.1	even	1	trivial
288.8.f.a.143.21		28	8.3	odd	2	inner
288.8.f.a.143.22		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)

\(f(q)\)	\(=\)	\(q-238.250 q^{5} +737.398i q^{7} +O(q^{10})\)	\(q-238.250 q^{5} +737.398i q^{7} -1701.82i q^{11} -4482.93i q^{13} +19959.6i q^{17} +8090.28 q^{19} -110629. q^{23} -21362.2 q^{25} -19262.3 q^{29} -70291.5i q^{31} -175685. i q^{35} +441575. i q^{37} +382570. i q^{41} +329066. q^{43} -224714. q^{47} +279787. q^{49} +1.27611e6 q^{53} +405457. i q^{55} -2.48468e6i q^{59} -2.93707e6i q^{61} +1.06805e6i q^{65} -2.94806e6 q^{67} -1.63172e6 q^{71} +6.26023e6 q^{73} +1.25492e6 q^{77} +448216. i q^{79} -3.63088e6i q^{83} -4.75536e6i q^{85} -1.41664e6i q^{89} +3.30570e6 q^{91} -1.92750e6 q^{95} +1.10470e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 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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