LMFDB - Embedded newform 288.8.d.d.145.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(145,288)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("288.145");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.d (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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 $ x^14 - 6x^13 - 52x^12 + 300x^11 - 1005x^10 - 23250x^9 + 349930x^8 + 2867784x^7 - 20463993x^6 - 78987210x^5 + 94608296x^4 - 6477861300x^3 - 21982316987x^2 + 613270661346*x + 3813237677250
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			145.9
Root			$1.24645 + 7.99620i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.8.d.d.145.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+76.0929i q^{5} +222.735 q^{7} +O(q^{10})$	$q+76.0929i q^{5} +222.735 q^{7} -1904.05i q^{11} +309.880i q^{13} -16744.7 q^{17} +27044.6i q^{19} +77339.4 q^{23} +72334.9 q^{25} -156325. i q^{29} -265401. q^{31} +16948.5i q^{35} -113359. i q^{37} +694133. q^{41} +900118. i q^{43} -77128.1 q^{47} -773932. q^{49} -1.89845e6i q^{53} +144885. q^{55} +704858. i q^{59} +1.42087e6i q^{61} -23579.6 q^{65} +1.87192e6i q^{67} +3.31385e6 q^{71} -39036.0 q^{73} -424099. i q^{77} +2.43321e6 q^{79} +6.00752e6i q^{83} -1.27415e6i q^{85} -2.38556e6 q^{89} +69021.1i q^{91} -2.05790e6 q^{95} +1.31891e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 1372 q^{7}+O(q^{10}) $ 14 * q - 1372 * q^7	$ 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) $ 14 * q - 1372 * q^7 + 2908 * q^17 - 143416 * q^23 - 202626 * q^25 + 89468 * q^31 + 441284 * q^41 - 1056408 * q^47 + 2158134 * q^49 - 4757504 * q^55 + 2520464 * q^65 + 5172696 * q^71 - 5446196 * q^73 + 14373548 * q^79 + 11952620 * q^89 - 69327376 * q^95 + 133732 * 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
$5$	76.0929i	0.272238i	0.990692	+	0.136119i	$0.0434630\pi$
$5$	76.0929i	0.272238i	−0.990692	+	0.136119i	$0.956537\pi$
$6$	0	0
$7$	222.735	0.245440	0.122720	−	0.992441i	$-0.460838\pi$
$7$	222.735	0.245440	0.122720	+	0.992441i	$0.460838\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1904.05i	− 0.431324i	−0.976468	−	0.215662i	$-0.930809\pi$
$11$	− 1904.05i	− 0.431324i	0.976468	−	0.215662i	$-0.0691910\pi$
$12$	0	0
$13$	309.880i	0.0391193i	0.999809	+	0.0195597i	$0.00622643\pi$
$13$	309.880i	0.0391193i	−0.999809	+	0.0195597i	$0.993774\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−16744.7	−0.826619	−0.413309	−	0.910591i	$-0.635627\pi$
$17$	−16744.7	−0.826619	−0.413309	+	0.910591i	$0.635627\pi$
$18$	0	0
$19$	27044.6i	0.904572i	0.891873	+	0.452286i	$0.149391\pi$
$19$	27044.6i	0.904572i	−0.891873	+	0.452286i	$0.850609\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	77339.4	1.32542	0.662710	−	0.748876i	$-0.269406\pi$
$23$	77339.4	1.32542	0.662710	+	0.748876i	$0.269406\pi$
$24$	0	0
$25$	72334.9	0.925886
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 156325.i	− 1.19024i	−0.803636	−	0.595121i	$-0.797104\pi$
$29$	− 156325.i	− 1.19024i	0.803636	−	0.595121i	$-0.202896\pi$
$30$	0	0
$31$	−265401.	−1.60006	−0.800031	−	0.599959i	$-0.795184\pi$
$31$	−265401.	−1.60006	−0.800031	+	0.599959i	$0.795184\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16948.5i	0.0668181i
$36$	0	0
$37$	− 113359.i	− 0.367918i	−0.982934	−	0.183959i	$-0.941109\pi$
$37$	− 113359.i	− 0.367918i	0.982934	−	0.183959i	$-0.0588913\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	694133.	1.57289	0.786447	−	0.617658i	$-0.211918\pi$
$41$	694133.	1.57289	0.786447	+	0.617658i	$0.211918\pi$
$42$	0	0
$43$	900118.i	1.72647i	0.504800	+	0.863236i	$0.331566\pi$
$43$	900118.i	1.72647i	−0.504800	+	0.863236i	$0.668434\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−77128.1	−0.108360	−0.0541801	−	0.998531i	$-0.517255\pi$
$47$	−77128.1	−0.108360	−0.0541801	+	0.998531i	$0.517255\pi$
$48$	0	0
$49$	−773932.	−0.939759
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.89845e6i	− 1.75159i	−0.482682	−	0.875796i	$-0.660337\pi$
$53$	− 1.89845e6i	− 1.75159i	0.482682	−	0.875796i	$-0.339663\pi$
$54$	0	0
$55$	144885.	0.117423
$56$	0	0
$57$	0	0
$58$	0	0
$59$	704858.i	0.446807i	0.974726	+	0.223403i	$0.0717167\pi$
$59$	704858.i	0.446807i	−0.974726	+	0.223403i	$0.928283\pi$
$60$	0	0
$61$	1.42087e6i	0.801492i	0.916189	+	0.400746i	$0.131249\pi$
$61$	1.42087e6i	0.801492i	−0.916189	+	0.400746i	$0.868751\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−23579.6	−0.0106498
$66$	0	0
$67$	1.87192e6i	0.760370i	0.924911	+	0.380185i	$0.124140\pi$
$67$	1.87192e6i	0.760370i	−0.924911	+	0.380185i	$0.875860\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.31385e6	1.09883	0.549413	−	0.835551i	$-0.314851\pi$
$71$	3.31385e6	1.09883	0.549413	+	0.835551i	$0.314851\pi$
$72$	0	0
$73$	−39036.0	−0.0117445	−0.00587227	−	0.999983i	$-0.501869\pi$
$73$	−39036.0	−0.0117445	−0.00587227	+	0.999983i	$0.501869\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 424099.i	− 0.105864i
$78$	0	0
$79$	2.43321e6	0.555246	0.277623	−	0.960690i	$-0.410453\pi$
$79$	2.43321e6	0.555246	0.277623	+	0.960690i	$0.410453\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00752e6i	1.15324i	0.817011	+	0.576622i	$0.195630\pi$
$83$	6.00752e6i	1.15324i	−0.817011	+	0.576622i	$0.804370\pi$
$84$	0	0
$85$	− 1.27415e6i	− 0.225037i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.38556e6	−0.358695	−0.179347	−	0.983786i	$-0.557399\pi$
$89$	−2.38556e6	−0.358695	−0.179347	+	0.983786i	$0.557399\pi$
$90$	0	0
$91$	69021.1i	0.00960145i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.05790e6	−0.246259
$96$	0	0
$97$	1.31891e7	1.46729	0.733644	−	0.679534i	$-0.237818\pi$
$97$	1.31891e7	1.46729	0.733644	+	0.679534i	$0.237818\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.80684e6i	0.271077i	0.990772	+	0.135538i	$0.0432764\pi$
$101$	2.80684e6i	0.271077i	−0.990772	+	0.135538i	$0.956724\pi$
$102$	0	0
$103$	8.89213e6	0.801818	0.400909	−	0.916118i	$-0.368694\pi$
$103$	8.89213e6	0.801818	0.400909	+	0.916118i	$0.368694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.27764e7i	1.00824i	0.863634	+	0.504120i	$0.168183\pi$
$107$	1.27764e7i	1.00824i	−0.863634	+	0.504120i	$0.831817\pi$
$108$	0	0
$109$	1.01385e6i	0.0749859i	0.999297	+	0.0374929i	$0.0119372\pi$
$109$	1.01385e6i	0.0749859i	−0.999297	+	0.0374929i	$0.988063\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	204553.	0.0133362	0.00666810	−	0.999978i	$-0.497877\pi$
$113$	204553.	0.0133362	0.00666810	+	0.999978i	$0.497877\pi$
$114$	0	0
$115$	5.88498e6i	0.360830i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.72962e6	−0.202885
$120$	0	0
$121$	1.58618e7	0.813959
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.14489e7i	0.524300i
$126$	0	0
$127$	3.96071e7	1.71577	0.857886	−	0.513840i	$-0.171777\pi$
$127$	3.96071e7	1.71577	0.857886	+	0.513840i	$0.171777\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.23545e6i	0.0480150i	0.999712	+	0.0240075i	$0.00764256\pi$
$131$	1.23545e6i	0.0480150i	−0.999712	+	0.0240075i	$0.992357\pi$
$132$	0	0
$133$	6.02378e6i	0.222018i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.09363e7	1.02789	0.513944	−	0.857824i	$-0.328184\pi$
$137$	3.09363e7	1.02789	0.513944	+	0.857824i	$0.328184\pi$
$138$	0	0
$139$	3.41178e7i	1.07753i	0.842456	+	0.538764i	$0.181109\pi$
$139$	3.41178e7i	1.07753i	−0.842456	+	0.538764i	$0.818891\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	590026.	0.0168731
$144$	0	0
$145$	1.18952e7	0.324029
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.27261e7i	1.30579i	0.757448	+	0.652896i	$0.226446\pi$
$149$	5.27261e7i	1.30579i	−0.757448	+	0.652896i	$0.773554\pi$
$150$	0	0
$151$	−3.22362e7	−0.761946	−0.380973	−	0.924586i	$-0.624411\pi$
$151$	−3.22362e7	−0.761946	−0.380973	+	0.924586i	$0.624411\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.01951e7i	− 0.435598i
$156$	0	0
$157$	9.02694e7i	1.86162i	0.365501	+	0.930811i	$0.380898\pi$
$157$	9.02694e7i	1.86162i	−0.365501	+	0.930811i	$0.619102\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.72262e7	0.325311
$162$	0	0
$163$	1.76210e7i	0.318695i	0.987223	+	0.159347i	$0.0509390\pi$
$163$	1.76210e7i	0.318695i	−0.987223	+	0.159347i	$0.949061\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.34525e7	1.55269	0.776343	−	0.630311i	$-0.217073\pi$
$167$	9.34525e7	1.55269	0.776343	+	0.630311i	$0.217073\pi$
$168$	0	0
$169$	6.26525e7	0.998470
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 9.82824e6i	− 0.144316i	−0.997393	−	0.0721580i	$-0.977011\pi$
$173$	− 9.82824e6i	− 0.144316i	0.997393	−	0.0721580i	$-0.0229886\pi$
$174$	0	0
$175$	1.61115e7	0.227250
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.66345e7i	− 1.12903i	−0.825423	−	0.564515i	$-0.809063\pi$
$179$	− 8.66345e7i	− 1.12903i	0.825423	−	0.564515i	$-0.190937\pi$
$180$	0	0
$181$	1.00562e8i	1.26055i	0.776373	+	0.630273i	$0.217057\pi$
$181$	1.00562e8i	1.26055i	−0.776373	+	0.630273i	$0.782943\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.62583e6	0.100161
$186$	0	0
$187$	3.18827e7i	0.356541i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.52134e7	0.365671	0.182836	−	0.983143i	$-0.441472\pi$
$191$	3.52134e7	0.365671	0.182836	+	0.983143i	$0.441472\pi$
$192$	0	0
$193$	−5.71809e7	−0.572533	−0.286266	−	0.958150i	$-0.592414\pi$
$193$	−5.71809e7	−0.572533	−0.286266	+	0.958150i	$0.592414\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.33298e7i	0.403789i	0.979407	+	0.201895i	$0.0647099\pi$
$197$	4.33298e7i	0.403789i	−0.979407	+	0.201895i	$0.935290\pi$
$198$	0	0
$199$	1.11161e8	0.999923	0.499961	−	0.866048i	$-0.333348\pi$
$199$	1.11161e8	0.999923	0.499961	+	0.866048i	$0.333348\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.48190e7i	− 0.292133i
$204$	0	0
$205$	5.28186e7i	0.428202i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.14943e7	0.390164
$210$	0	0
$211$	− 1.03551e7i	− 0.0758866i	−0.999280	−	0.0379433i	$-0.987919\pi$
$211$	− 1.03551e7i	− 0.0758866i	0.999280	−	0.0379433i	$-0.0120806\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.84925e7	−0.470012
$216$	0	0
$217$	−5.91141e7	−0.392719
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5.18883e6i	− 0.0323368i
$222$	0	0
$223$	5.25751e7	0.317478	0.158739	−	0.987321i	$-0.449257\pi$
$223$	5.25751e7	0.317478	0.158739	+	0.987321i	$0.449257\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.54693e8i	1.44520i	0.691268	+	0.722599i	$0.257053\pi$
$227$	2.54693e8i	1.44520i	−0.691268	+	0.722599i	$0.742947\pi$
$228$	0	0
$229$	− 2.45399e8i	− 1.35036i	−0.737655	−	0.675178i	$-0.764067\pi$
$229$	− 2.45399e8i	− 1.35036i	0.737655	−	0.675178i	$-0.235933\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.96863e7	0.101957	0.0509785	−	0.998700i	$-0.483766\pi$
$233$	1.96863e7	0.101957	0.0509785	+	0.998700i	$0.483766\pi$
$234$	0	0
$235$	− 5.86890e6i	− 0.0294998i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.45425e7	−0.211048	−0.105524	−	0.994417i	$-0.533652\pi$
$239$	−4.45425e7	−0.211048	−0.105524	+	0.994417i	$0.533652\pi$
$240$	0	0
$241$	2.03884e8	0.938258	0.469129	−	0.883130i	$-0.344568\pi$
$241$	2.03884e8	0.938258	0.469129	+	0.883130i	$0.344568\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.88907e7i	− 0.255838i
$246$	0	0
$247$	−8.38058e6	−0.0353863
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.38230e8i	1.35006i	0.737788	+	0.675032i	$0.235870\pi$
$251$	3.38230e8i	1.35006i	−0.737788	+	0.675032i	$0.764130\pi$
$252$	0	0
$253$	− 1.47258e8i	− 0.571686i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.61266e8	−0.960100	−0.480050	−	0.877241i	$-0.659381\pi$
$257$	−2.61266e8	−0.960100	−0.480050	+	0.877241i	$0.659381\pi$
$258$	0	0
$259$	− 2.52491e7i	− 0.0903018i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.68095e8	−1.24771	−0.623856	−	0.781539i	$-0.714435\pi$
$263$	−3.68095e8	−1.24771	−0.623856	+	0.781539i	$0.714435\pi$
$264$	0	0
$265$	1.44458e8	0.476850
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.64845e8i	− 1.45605i	−0.685552	−	0.728024i	$-0.740439\pi$
$269$	− 4.64845e8i	− 1.45605i	0.685552	−	0.728024i	$-0.259561\pi$
$270$	0	0
$271$	−4.00194e8	−1.22146	−0.610728	−	0.791840i	$-0.709123\pi$
$271$	−4.00194e8	−1.22146	−0.610728	+	0.791840i	$0.709123\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.37729e8i	− 0.399357i
$276$	0	0
$277$	− 5.83810e8i	− 1.65041i	−0.564833	−	0.825205i	$-0.691059\pi$
$277$	− 5.83810e8i	− 1.65041i	0.564833	−	0.825205i	$-0.308941\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.91897e8	−0.515935	−0.257968	−	0.966154i	$-0.583053\pi$
$281$	−1.91897e8	−0.515935	−0.257968	+	0.966154i	$0.583053\pi$
$282$	0	0
$283$	1.24399e8i	0.326261i	0.986605	+	0.163130i	$0.0521591\pi$
$283$	1.24399e8i	0.326261i	−0.986605	+	0.163130i	$0.947841\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.54608e8	0.386051
$288$	0	0
$289$	−1.29955e8	−0.316702
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.68768e8i	1.08873i	0.838848	+	0.544366i	$0.183230\pi$
$293$	4.68768e8i	1.08873i	−0.838848	+	0.544366i	$0.816770\pi$
$294$	0	0
$295$	−5.36347e7	−0.121638
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.39659e7i	0.0518495i
$300$	0	0
$301$	2.00488e8i	0.423745i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.08118e8	−0.218197
$306$	0	0
$307$	− 5.76962e8i	− 1.13805i	−0.822319	−	0.569027i	$-0.807320\pi$
$307$	− 5.76962e8i	− 1.13805i	0.822319	−	0.569027i	$-0.192680\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.77008e8	0.710705	0.355352	−	0.934732i	$-0.384361\pi$
$311$	3.77008e8	0.710705	0.355352	+	0.934732i	$0.384361\pi$
$312$	0	0
$313$	−9.90160e8	−1.82516	−0.912579	−	0.408901i	$-0.865912\pi$
$313$	−9.90160e8	−1.82516	−0.912579	+	0.408901i	$0.865912\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.82475e8i	− 1.73226i	−0.499815	−	0.866132i	$-0.666599\pi$
$317$	− 9.82475e8i	− 1.73226i	0.499815	−	0.866132i	$-0.333401\pi$
$318$	0	0
$319$	−2.97651e8	−0.513380
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4.52853e8i	− 0.747736i
$324$	0	0
$325$	2.24151e7i	0.0362201i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.71791e7	−0.0265959
$330$	0	0
$331$	− 5.42246e8i	− 0.821861i	−0.911667	−	0.410930i	$-0.865204\pi$
$331$	− 5.42246e8i	− 0.821861i	0.911667	−	0.410930i	$-0.134796\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.42440e8	−0.207002
$336$	0	0
$337$	−1.00930e9	−1.43653	−0.718267	−	0.695768i	$-0.755064\pi$
$337$	−1.00930e9	−1.43653	−0.718267	+	0.695768i	$0.755064\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.05337e8i	0.690146i
$342$	0	0
$343$	−3.55814e8	−0.476095
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.43859e9i	− 1.84835i	−0.381963	−	0.924177i	$-0.624752\pi$
$347$	− 1.43859e9i	− 1.84835i	0.381963	−	0.924177i	$-0.375248\pi$
$348$	0	0
$349$	9.68176e8i	1.21917i	0.792720	+	0.609586i	$0.208664\pi$
$349$	9.68176e8i	1.21917i	−0.792720	+	0.609586i	$0.791336\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.13480e9	1.37312	0.686560	−	0.727073i	$-0.259120\pi$
$353$	1.13480e9	1.37312	0.686560	+	0.727073i	$0.259120\pi$
$354$	0	0
$355$	2.52160e8i	0.299142i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.32262e8	−0.835287	−0.417644	−	0.908611i	$-0.637144\pi$
$359$	−7.32262e8	−0.835287	−0.417644	+	0.908611i	$0.637144\pi$
$360$	0	0
$361$	1.62460e8	0.181749
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.97036e6i	− 0.00319731i
$366$	0	0
$367$	2.89628e8	0.305850	0.152925	−	0.988238i	$-0.451131\pi$
$367$	2.89628e8	0.305850	0.152925	+	0.988238i	$0.451131\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 4.22851e8i	− 0.429911i
$372$	0	0
$373$	− 1.36758e9i	− 1.36450i	−0.731119	−	0.682250i	$-0.761002\pi$
$373$	− 1.36758e9i	− 1.36450i	0.731119	−	0.682250i	$-0.238998\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.84419e7	0.0465615
$378$	0	0
$379$	− 4.85906e8i	− 0.458474i	−0.973371	−	0.229237i	$-0.926377\pi$
$379$	− 4.85906e8i	− 0.458474i	0.973371	−	0.229237i	$-0.0736231\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.78877e8	0.253639	0.126820	−	0.991926i	$-0.459523\pi$
$383$	2.78877e8	0.253639	0.126820	+	0.991926i	$0.459523\pi$
$384$	0	0
$385$	3.22709e7	0.0288203
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.08906e8i	0.524477i	0.965003	+	0.262239i	$0.0844608\pi$
$389$	6.08906e8i	0.524477i	−0.965003	+	0.262239i	$0.915539\pi$
$390$	0	0
$391$	−1.29502e9	−1.09562
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.85150e8i	0.151159i
$396$	0	0
$397$	1.45195e9i	1.16462i	0.812965	+	0.582312i	$0.197852\pi$
$397$	1.45195e9i	1.16462i	−0.812965	+	0.582312i	$0.802148\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.05655e9	−1.59270	−0.796348	−	0.604838i	$-0.793238\pi$
$401$	−2.05655e9	−1.59270	−0.796348	+	0.604838i	$0.793238\pi$
$402$	0	0
$403$	− 8.22424e7i	− 0.0625933i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.15842e8	−0.158692
$408$	0	0
$409$	1.10177e9	0.796267	0.398133	−	0.917328i	$-0.369658\pi$
$409$	1.10177e9	0.796267	0.398133	+	0.917328i	$0.369658\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.56997e8i	0.109664i
$414$	0	0
$415$	−4.57129e8	−0.313957
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.78809e8i	0.317990i	0.987279	+	0.158995i	$0.0508253\pi$
$419$	4.78809e8i	0.317990i	−0.987279	+	0.158995i	$0.949175\pi$
$420$	0	0
$421$	− 2.23427e9i	− 1.45931i	−0.683814	−	0.729657i	$-0.739680\pi$
$421$	− 2.23427e9i	− 1.45931i	0.683814	−	0.729657i	$-0.260320\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.21122e9	−0.765355
$426$	0	0
$427$	3.16477e8i	0.196718i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.45204e8	0.388174	0.194087	−	0.980984i	$-0.437826\pi$
$431$	6.45204e8	0.388174	0.194087	+	0.980984i	$0.437826\pi$
$432$	0	0
$433$	1.83504e9	1.08627	0.543135	−	0.839646i	$-0.317237\pi$
$433$	1.83504e9	1.08627	0.543135	+	0.839646i	$0.317237\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.09162e9i	1.19894i
$438$	0	0
$439$	7.60899e8	0.429241	0.214621	−	0.976698i	$-0.431149\pi$
$439$	7.60899e8	0.429241	0.214621	+	0.976698i	$0.431149\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.78414e9i	0.975027i	0.873115	+	0.487514i	$0.162096\pi$
$443$	1.78414e9i	0.975027i	−0.873115	+	0.487514i	$0.837904\pi$
$444$	0	0
$445$	− 1.81524e8i	− 0.0976503i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.75964e9	0.917408	0.458704	−	0.888589i	$-0.348314\pi$
$449$	1.75964e9	0.917408	0.458704	+	0.888589i	$0.348314\pi$
$450$	0	0
$451$	− 1.32166e9i	− 0.678428i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.25201e6	−0.00261388
$456$	0	0
$457$	−1.97532e8	−0.0968122	−0.0484061	−	0.998828i	$-0.515414\pi$
$457$	−1.97532e8	−0.0968122	−0.0484061	+	0.998828i	$0.515414\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.82074e9i	− 0.865555i	−0.901501	−	0.432777i	$-0.857534\pi$
$461$	− 1.82074e9i	− 0.865555i	0.901501	−	0.432777i	$-0.142466\pi$
$462$	0	0
$463$	3.64174e9	1.70520	0.852601	−	0.522562i	$-0.175024\pi$
$463$	3.64174e9	1.70520	0.852601	+	0.522562i	$0.175024\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.35686e9i	0.616488i	0.951307	+	0.308244i	$0.0997414\pi$
$467$	1.35686e9i	0.616488i	−0.951307	+	0.308244i	$0.900259\pi$
$468$	0	0
$469$	4.16942e8i	0.186625i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.71387e9	0.744670
$474$	0	0
$475$	1.95627e9i	0.837531i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.20672e9	−1.33317	−0.666587	−	0.745427i	$-0.732246\pi$
$479$	−3.20672e9	−1.33317	−0.666587	+	0.745427i	$0.732246\pi$
$480$	0	0
$481$	3.51277e7	0.0143927
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.00360e9i	0.399452i
$486$	0	0
$487$	2.67141e9	1.04807	0.524033	−	0.851698i	$-0.324427\pi$
$487$	2.67141e9	1.04807	0.524033	+	0.851698i	$0.324427\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.48931e8i	0.0949059i	0.998873	+	0.0474529i	$0.0151104\pi$
$491$	2.48931e8i	0.0949059i	−0.998873	+	0.0474529i	$0.984890\pi$
$492$	0	0
$493$	2.61761e9i	0.983876i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.38111e8	0.269696
$498$	0	0
$499$	4.22767e9i	1.52317i	0.648064	+	0.761586i	$0.275579\pi$
$499$	4.22767e9i	1.52317i	−0.648064	+	0.761586i	$0.724421\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.45979e9	0.511448	0.255724	−	0.966750i	$-0.417686\pi$
$503$	1.45979e9	0.511448	0.255724	+	0.966750i	$0.417686\pi$
$504$	0	0
$505$	−2.13580e8	−0.0737975
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.11582e9i	− 1.04727i	−0.851941	−	0.523637i	$-0.824575\pi$
$509$	− 3.11582e9i	− 1.04727i	0.851941	−	0.523637i	$-0.175425\pi$
$510$	0	0
$511$	−8.69470e6	−0.00288258
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.76628e8i	0.218285i
$516$	0	0
$517$	1.46856e8i	0.0467384i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.26196e8	−0.0390944	−0.0195472	−	0.999809i	$-0.506222\pi$
$521$	−1.26196e8	−0.0390944	−0.0195472	+	0.999809i	$0.506222\pi$
$522$	0	0
$523$	− 2.63497e9i	− 0.805414i	−0.915329	−	0.402707i	$-0.868069\pi$
$523$	− 2.63497e9i	− 0.805414i	0.915329	−	0.402707i	$-0.131931\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.44405e9	1.32264
$528$	0	0
$529$	2.57656e9	0.756738
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.15098e8i	0.0615306i
$534$	0	0
$535$	−9.72189e8	−0.274481
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.47361e9i	0.405341i
$540$	0	0
$541$	6.00066e9i	1.62933i	0.579933	+	0.814664i	$0.303079\pi$
$541$	6.00066e9i	1.62933i	−0.579933	+	0.814664i	$0.696921\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.71465e7	−0.0204140
$546$	0	0
$547$	3.78745e9i	0.989443i	0.869051	+	0.494722i	$0.164730\pi$
$547$	3.78745e9i	0.989443i	−0.869051	+	0.494722i	$0.835270\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.22775e9	1.07666
$552$	0	0
$553$	5.41962e8	0.136280
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.00353e9i	− 0.491251i	−0.969365	−	0.245625i	$-0.921007\pi$
$557$	− 2.00353e9i	− 0.491251i	0.969365	−	0.245625i	$-0.0789933\pi$
$558$	0	0
$559$	−2.78928e8	−0.0675384
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.27857e9i	0.301957i	0.988537	+	0.150978i	$0.0482424\pi$
$563$	1.27857e9i	0.301957i	−0.988537	+	0.150978i	$0.951758\pi$
$564$	0	0
$565$	1.55651e7i	0.00363062i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.37979e8	0.190695	0.0953477	−	0.995444i	$-0.469604\pi$
$569$	8.37979e8	0.190695	0.0953477	+	0.995444i	$0.469604\pi$
$570$	0	0
$571$	5.38249e9i	1.20992i	0.796256	+	0.604960i	$0.206811\pi$
$571$	5.38249e9i	1.20992i	−0.796256	+	0.604960i	$0.793189\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.59434e9	1.22719
$576$	0	0
$577$	−4.85813e9	−1.05282	−0.526410	−	0.850231i	$-0.676462\pi$
$577$	−4.85813e9	−1.05282	−0.526410	+	0.850231i	$0.676462\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.33808e9i	0.283053i
$582$	0	0
$583$	−3.61474e9	−0.755504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.68130e9i	− 0.955286i	−0.878554	−	0.477643i	$-0.841491\pi$
$587$	− 4.68130e9i	− 0.955286i	0.878554	−	0.477643i	$-0.158509\pi$
$588$	0	0
$589$	− 7.17767e9i	− 1.44737i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.03909e8	−0.0795412	−0.0397706	−	0.999209i	$-0.512663\pi$
$593$	−4.03909e8	−0.0795412	−0.0397706	+	0.999209i	$0.512663\pi$
$594$	0	0
$595$	− 2.83798e8i	− 0.0552331i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.14434e9	−1.73844	−0.869218	−	0.494429i	$-0.835377\pi$
$599$	−9.14434e9	−1.73844	−0.869218	+	0.494429i	$0.835377\pi$
$600$	0	0
$601$	5.20002e9	0.977111	0.488556	−	0.872533i	$-0.337524\pi$
$601$	5.20002e9	0.977111	0.488556	+	0.872533i	$0.337524\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.20697e9i	0.221591i
$606$	0	0
$607$	2.74895e9	0.498893	0.249446	−	0.968389i	$-0.419751\pi$
$607$	2.74895e9	0.498893	0.249446	+	0.968389i	$0.419751\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.39004e7i	− 0.00423898i
$612$	0	0
$613$	7.96488e9i	1.39658i	0.715813	+	0.698292i	$0.246056\pi$
$613$	7.96488e9i	1.39658i	−0.715813	+	0.698292i	$0.753944\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.53830e9	1.12064	0.560321	−	0.828276i	$-0.310678\pi$
$617$	6.53830e9	1.12064	0.560321	+	0.828276i	$0.310678\pi$
$618$	0	0
$619$	1.00603e9i	0.170488i	0.996360	+	0.0852438i	$0.0271669\pi$
$619$	1.00603e9i	0.170488i	−0.996360	+	0.0852438i	$0.972833\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.31347e8	−0.0880380
$624$	0	0
$625$	4.77998e9	0.783152
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.89816e9i	0.304128i
$630$	0	0
$631$	2.56041e8	0.0405702	0.0202851	−	0.999794i	$-0.493543\pi$
$631$	2.56041e8	0.0405702	0.0202851	+	0.999794i	$0.493543\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.01382e9i	0.467099i
$636$	0	0
$637$	− 2.39826e8i	− 0.0367627i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.86290e9	−0.579309	−0.289655	−	0.957131i	$-0.593540\pi$
$641$	−3.86290e9	−0.579309	−0.289655	+	0.957131i	$0.593540\pi$
$642$	0	0
$643$	− 3.57223e9i	− 0.529908i	−0.964261	−	0.264954i	$-0.914643\pi$
$643$	− 3.57223e9i	− 0.529908i	0.964261	−	0.264954i	$-0.0853568\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.05669e10	−1.53385	−0.766926	−	0.641735i	$-0.778215\pi$
$647$	−1.05669e10	−1.53385	−0.766926	+	0.641735i	$0.778215\pi$
$648$	0	0
$649$	1.34209e9	0.192719
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.74392e9i	0.526176i	0.964772	+	0.263088i	$0.0847409\pi$
$653$	3.74392e9i	0.526176i	−0.964772	+	0.263088i	$0.915259\pi$
$654$	0	0
$655$	−9.40093e7	−0.0130715
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5.95673e9i	− 0.810791i	−0.914141	−	0.405396i	$-0.867134\pi$
$659$	− 5.95673e9i	− 0.810791i	0.914141	−	0.405396i	$-0.132866\pi$
$660$	0	0
$661$	− 8.24359e9i	− 1.11023i	−0.831775	−	0.555113i	$-0.812675\pi$
$661$	− 8.24359e9i	− 1.11023i	0.831775	−	0.555113i	$-0.187325\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.58367e8	−0.0604418
$666$	0	0
$667$	− 1.20901e10i	− 1.57757i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.70540e9	0.345703
$672$	0	0
$673$	−5.48262e9	−0.693323	−0.346661	−	0.937990i	$-0.612685\pi$
$673$	−5.48262e9	−0.693323	−0.346661	+	0.937990i	$0.612685\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.45217e8i	0.117077i	0.998285	+	0.0585384i	$0.0186440\pi$
$677$	9.45217e8i	0.117077i	−0.998285	+	0.0585384i	$0.981356\pi$
$678$	0	0
$679$	2.93769e9	0.360131
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.58282e8i	− 0.0430282i	−0.999769	−	0.0215141i	$-0.993151\pi$
$683$	− 3.58282e8i	− 0.0430282i	0.999769	−	0.0215141i	$-0.00684868\pi$
$684$	0	0
$685$	2.35403e9i	0.279830i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.88290e8	0.0685211
$690$	0	0
$691$	− 7.94941e9i	− 0.916562i	−0.888807	−	0.458281i	$-0.848465\pi$
$691$	− 7.94941e9i	− 0.916562i	0.888807	−	0.458281i	$-0.151535\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.59612e9	−0.293344
$696$	0	0
$697$	−1.16230e10	−1.30018
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.58910e10i	1.74236i	0.490965	+	0.871179i	$0.336644\pi$
$701$	1.58910e10i	1.74236i	−0.490965	+	0.871179i	$0.663356\pi$
$702$	0	0
$703$	3.06576e9	0.332808
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.25181e8i	0.0665331i
$708$	0	0
$709$	− 7.39228e9i	− 0.778963i	−0.921034	−	0.389481i	$-0.872654\pi$
$709$	− 7.39228e9i	− 0.778963i	0.921034	−	0.389481i	$-0.127346\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.05260e10	−2.12075
$714$	0	0
$715$	4.48968e7i	0.00459351i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.31991e9	0.935105	0.467553	−	0.883965i	$-0.345136\pi$
$719$	9.31991e9	0.935105	0.467553	+	0.883965i	$0.345136\pi$
$720$	0	0
$721$	1.98059e9	0.196798
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.13077e10i	− 1.10203i
$726$	0	0
$727$	−3.39593e9	−0.327784	−0.163892	−	0.986478i	$-0.552405\pi$
$727$	−3.39593e9	−0.327784	−0.163892	+	0.986478i	$0.552405\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.50722e10i	− 1.42713i
$732$	0	0
$733$	− 1.01101e10i	− 0.948183i	−0.880476	−	0.474092i	$-0.842777\pi$
$733$	− 1.01101e10i	− 0.948183i	0.880476	−	0.474092i	$-0.157223\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.56422e9	0.327966
$738$	0	0
$739$	− 1.67582e10i	− 1.52746i	−0.645533	−	0.763732i	$-0.723365\pi$
$739$	− 1.67582e10i	− 1.52746i	0.645533	−	0.763732i	$-0.276635\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.37271e9	−0.480544	−0.240272	−	0.970706i	$-0.577237\pi$
$743$	−5.37271e9	−0.480544	−0.240272	+	0.970706i	$0.577237\pi$
$744$	0	0
$745$	−4.01208e9	−0.355486
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.84574e9i	0.247462i
$750$	0	0
$751$	−1.32362e10	−1.14031	−0.570154	−	0.821538i	$-0.693116\pi$
$751$	−1.32362e10	−1.14031	−0.570154	+	0.821538i	$0.693116\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.45294e9i	− 0.207431i
$756$	0	0
$757$	1.58389e10i	1.32705i	0.748152	+	0.663527i	$0.230941\pi$
$757$	1.58389e10i	1.32705i	−0.748152	+	0.663527i	$0.769059\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.71988e9	−0.223719	−0.111859	−	0.993724i	$-0.535681\pi$
$761$	−2.71988e9	−0.223719	−0.111859	+	0.993724i	$0.535681\pi$
$762$	0	0
$763$	2.25819e8i	0.0184045i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.18421e8	−0.0174788
$768$	0	0
$769$	1.90198e9	0.150822	0.0754110	−	0.997153i	$-0.475973\pi$
$769$	1.90198e9	0.150822	0.0754110	+	0.997153i	$0.475973\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.35372e10i	− 1.05415i	−0.849819	−	0.527074i	$-0.823289\pi$
$773$	− 1.35372e10i	− 1.05415i	0.849819	−	0.527074i	$-0.176711\pi$
$774$	0	0
$775$	−1.91978e10	−1.48148
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.87726e10i	1.42280i
$780$	0	0
$781$	− 6.30974e9i	− 0.473950i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.86885e9	−0.506804
$786$	0	0
$787$	2.38864e10i	1.74678i	0.487018	+	0.873392i	$0.338085\pi$
$787$	2.38864e10i	1.74678i	−0.487018	+	0.873392i	$0.661915\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.55612e7	0.00327324
$792$	0	0
$793$	−4.40298e8	−0.0313538
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.60983e10i	− 1.12636i	−0.826335	−	0.563179i	$-0.809578\pi$
$797$	− 1.60983e10i	− 1.12636i	0.826335	−	0.563179i	$-0.190422\pi$
$798$	0	0
$799$	1.29148e9	0.0895726
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.43266e7i	0.00506570i
$804$	0	0
$805$	1.31079e9i	0.0885621i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.77138e10	1.17623	0.588114	−	0.808778i	$-0.299871\pi$
$809$	1.77138e10	1.17623	0.588114	+	0.808778i	$0.299871\pi$
$810$	0	0
$811$	− 2.81456e9i	− 0.185284i	−0.995699	−	0.0926419i	$-0.970469\pi$
$811$	− 2.81456e9i	− 0.185284i	0.995699	−	0.0926419i	$-0.0295312\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.34083e9	−0.0867608
$816$	0	0
$817$	−2.43433e10	−1.56172
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.24202e10i	− 1.41396i	−0.707232	−	0.706981i	$-0.750056\pi$
$821$	− 2.24202e10i	− 1.41396i	0.707232	−	0.706981i	$-0.249944\pi$
$822$	0	0
$823$	−1.58776e9	−0.0992854	−0.0496427	−	0.998767i	$-0.515808\pi$
$823$	−1.58776e9	−0.0992854	−0.0496427	+	0.998767i	$0.515808\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.24713e9i	− 0.199632i	−0.995006	−	0.0998162i	$-0.968175\pi$
$827$	− 3.24713e9i	− 0.199632i	0.995006	−	0.0998162i	$-0.0318255\pi$
$828$	0	0
$829$	− 2.33477e10i	− 1.42332i	−0.702522	−	0.711662i	$-0.747943\pi$
$829$	− 2.33477e10i	− 1.42332i	0.702522	−	0.711662i	$-0.252057\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.29592e10	0.776822
$834$	0	0
$835$	7.11107e9i	0.422700i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.96927e10	−1.15116	−0.575582	−	0.817744i	$-0.695225\pi$
$839$	−1.96927e10	−1.15116	−0.575582	+	0.817744i	$0.695225\pi$
$840$	0	0
$841$	−7.18761e9	−0.416676
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.76741e9i	0.271821i
$846$	0	0
$847$	3.53297e9	0.199778
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 8.76714e9i	− 0.487646i
$852$	0	0
$853$	1.52908e9i	0.0843546i	0.999110	+	0.0421773i	$0.0134294\pi$
$853$	1.52908e9i	0.0843546i	−0.999110	+	0.0421773i	$0.986571\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.01180e10	0.549114	0.274557	−	0.961571i	$-0.411469\pi$
$857$	1.01180e10	0.549114	0.274557	+	0.961571i	$0.411469\pi$
$858$	0	0
$859$	1.74738e10i	0.940613i	0.882503	+	0.470306i	$0.155857\pi$
$859$	1.74738e10i	0.940613i	−0.882503	+	0.470306i	$0.844143\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.30408e10	1.22028	0.610141	−	0.792293i	$-0.291113\pi$
$863$	2.30408e10	1.22028	0.610141	+	0.792293i	$0.291113\pi$
$864$	0	0
$865$	7.47859e8	0.0392883
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.63296e9i	− 0.239491i
$870$	0	0
$871$	−5.80069e8	−0.0297452
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.55008e9i	0.128684i
$876$	0	0
$877$	− 1.47417e10i	− 0.737989i	−0.929431	−	0.368995i	$-0.879702\pi$
$877$	− 1.47417e10i	− 0.737989i	0.929431	−	0.368995i	$-0.120298\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.65578e9	−0.229391	−0.114696	−	0.993401i	$-0.536589\pi$
$881$	−4.65578e9	−0.229391	−0.114696	+	0.993401i	$0.536589\pi$
$882$	0	0
$883$	− 1.13936e10i	− 0.556927i	−0.960447	−	0.278464i	$-0.910175\pi$
$883$	− 1.13936e10i	− 0.556927i	0.960447	−	0.278464i	$-0.0898252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.37678e8	−0.0210583	−0.0105291	−	0.999945i	$-0.503352\pi$
$887$	−4.37678e8	−0.0210583	−0.0105291	+	0.999945i	$0.503352\pi$
$888$	0	0
$889$	8.82188e9	0.421119
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.08590e9i	− 0.0980197i
$894$	0	0
$895$	6.59227e9	0.307365
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.14888e10i	1.90446i
$900$	0	0
$901$	3.17888e10i	1.44790i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.65205e9	−0.343169
$906$	0	0
$907$	4.51709e9i	0.201017i	0.994936	+	0.100509i	$0.0320470\pi$
$907$	4.51709e9i	0.201017i	−0.994936	+	0.100509i	$0.967953\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.19994e10	0.525829	0.262915	−	0.964819i	$-0.415316\pi$
$911$	1.19994e10	0.525829	0.262915	+	0.964819i	$0.415316\pi$
$912$	0	0
$913$	1.14386e10	0.497423
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.75179e8i	0.0117848i
$918$	0	0
$919$	−2.19473e10	−0.932774	−0.466387	−	0.884581i	$-0.654445\pi$
$919$	−2.19473e10	−0.932774	−0.466387	+	0.884581i	$0.654445\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.02690e9i	0.0429853i
$924$	0	0
$925$	− 8.19982e9i	− 0.340650i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.40255e10	−1.39235	−0.696176	−	0.717871i	$-0.745117\pi$
$929$	−3.40255e10	−1.39235	−0.696176	+	0.717871i	$0.745117\pi$
$930$	0	0
$931$	− 2.09307e10i	− 0.850080i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.42604e9	−0.0970640
$936$	0	0
$937$	2.10871e10	0.837389	0.418695	−	0.908127i	$-0.362488\pi$
$937$	2.10871e10	0.837389	0.418695	+	0.908127i	$0.362488\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.64361e10i	1.42550i	0.701417	+	0.712751i	$0.252551\pi$
$941$	3.64361e10i	1.42550i	−0.701417	+	0.712751i	$0.747449\pi$
$942$	0	0
$943$	5.36839e10	2.08475
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.18147e10i	− 1.59994i	−0.600039	−	0.799971i	$-0.704848\pi$
$947$	− 4.18147e10i	− 1.59994i	0.600039	−	0.799971i	$-0.295152\pi$
$948$	0	0
$949$	− 1.20965e7i	0 0.000459438i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.51456e10	−0.566841	−0.283421	−	0.958996i	$-0.591469\pi$
$953$	−1.51456e10	−0.566841	−0.283421	+	0.958996i	$0.591469\pi$
$954$	0	0
$955$	2.67949e9i	0.0995497i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.89059e9	0.252285
$960$	0	0
$961$	4.29251e10	1.56020
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.35106e9i	− 0.155865i
$966$	0	0
$967$	−1.40777e10	−0.500656	−0.250328	−	0.968161i	$-0.580538\pi$
$967$	−1.40777e10	−0.500656	−0.250328	+	0.968161i	$0.580538\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.43761e10i	0.503934i	0.967736	+	0.251967i	$0.0810775\pi$
$971$	1.43761e10i	0.503934i	−0.967736	+	0.251967i	$0.918922\pi$
$972$	0	0
$973$	7.59922e9i	0.264469i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.88103e10	0.988363	0.494181	−	0.869359i	$-0.335468\pi$
$977$	2.88103e10	0.988363	0.494181	+	0.869359i	$0.335468\pi$
$978$	0	0
$979$	4.54222e9i	0.154714i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.63392e10	−1.89179	−0.945897	−	0.324468i	$-0.894815\pi$
$983$	−5.63392e10	−1.89179	−0.945897	+	0.324468i	$0.894815\pi$
$984$	0	0
$985$	−3.29709e9	−0.109927
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.96146e10i	2.28830i
$990$	0	0
$991$	4.60741e10	1.50383	0.751915	−	0.659260i	$-0.229130\pi$
$991$	4.60741e10	1.50383	0.751915	+	0.659260i	$0.229130\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.45855e9i	0.272217i
$996$	0	0
$997$	4.08705e10i	1.30610i	0.757314	+	0.653051i	$0.226511\pi$
$997$	4.08705e10i	1.30610i	−0.757314	+	0.653051i	$0.773489\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.8.d.d.145.9		14
3.2	odd	2		96.8.d.a.49.10		14
4.3	odd	2		72.8.d.d.37.4		14
8.3	odd	2		72.8.d.d.37.3		14
8.5	even	2	inner	288.8.d.d.145.6		14
12.11	even	2		24.8.d.a.13.11	✓	14
24.5	odd	2		96.8.d.a.49.5		14
24.11	even	2		24.8.d.a.13.12	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.d.a.13.11	✓	14	12.11	even	2
24.8.d.a.13.12	yes	14	24.11	even	2
72.8.d.d.37.3		14	8.3	odd	2
72.8.d.d.37.4		14	4.3	odd	2
96.8.d.a.49.5		14	24.5	odd	2
96.8.d.a.49.10		14	3.2	odd	2
288.8.d.d.145.6		14	8.5	even	2	inner
288.8.d.d.145.9		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+76.0929i q^{5} +222.735 q^{7} +O(q^{10})\)	\(q+76.0929i q^{5} +222.735 q^{7} -1904.05i q^{11} +309.880i q^{13} -16744.7 q^{17} +27044.6i q^{19} +77339.4 q^{23} +72334.9 q^{25} -156325. i q^{29} -265401. q^{31} +16948.5i q^{35} -113359. i q^{37} +694133. q^{41} +900118. i q^{43} -77128.1 q^{47} -773932. q^{49} -1.89845e6i q^{53} +144885. q^{55} +704858. i q^{59} +1.42087e6i q^{61} -23579.6 q^{65} +1.87192e6i q^{67} +3.31385e6 q^{71} -39036.0 q^{73} -424099. i q^{77} +2.43321e6 q^{79} +6.00752e6i q^{83} -1.27415e6i q^{85} -2.38556e6 q^{89} +69021.1i q^{91} -2.05790e6 q^{95} +1.31891e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 1372 q^{7}+O(q^{10}) \) 14 * q - 1372 * q^7	\( 14 q - 1372 q^{7} + 2908 q^{17} - 143416 q^{23} - 202626 q^{25} + 89468 q^{31} + 441284 q^{41} - 1056408 q^{47} + 2158134 q^{49} - 4757504 q^{55} + 2520464 q^{65} + 5172696 q^{71} - 5446196 q^{73} + 14373548 q^{79} + 11952620 q^{89} - 69327376 q^{95} + 133732 q^{97}+O(q^{100}) \) 14 * q - 1372 * q^7 + 2908 * q^17 - 143416 * q^23 - 202626 * q^25 + 89468 * q^31 + 441284 * q^41 - 1056408 * q^47 + 2158134 * q^49 - 4757504 * q^55 + 2520464 * q^65 + 5172696 * q^71 - 5446196 * q^73 + 14373548 * q^79 + 11952620 * q^89 - 69327376 * q^95 + 133732 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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