LMFDB - Embedded newform 288.8.d.d.145.1

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			145.1
Root			$-5.80663 + 4.20354i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.8.d.d.145.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-468.400i q^{5} -81.2421 q^{7} +O(q^{10})$	$q-468.400i q^{5} -81.2421 q^{7} -3394.58i q^{11} +14492.5i q^{13} -5178.19 q^{17} -46067.9i q^{19} -67535.4 q^{23} -141274. q^{25} -25387.3i q^{29} -123808. q^{31} +38053.8i q^{35} +104623. i q^{37} +160655. q^{41} -378823. i q^{43} +683674. q^{47} -816943. q^{49} +1.74261e6i q^{53} -1.59002e6 q^{55} -1.48196e6i q^{59} +444587. i q^{61} +6.78830e6 q^{65} +2.48180e6i q^{67} -510618. q^{71} +4.80319e6 q^{73} +275783. i q^{77} -1.57434e6 q^{79} +7.91066e6i q^{83} +2.42547e6i q^{85} -6.35526e6 q^{89} -1.17740e6i q^{91} -2.15782e7 q^{95} -3.68231e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 468.400i	− 1.67580i	−0.545824	−	0.837900i	$-0.683783\pi$
$5$	− 468.400i	− 1.67580i	0.545824	−	0.837900i	$-0.316217\pi$
$6$	0	0
$7$	−81.2421	−0.0895237	−0.0447618	−	0.998998i	$-0.514253\pi$
$7$	−81.2421	−0.0895237	−0.0447618	+	0.998998i	$0.514253\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3394.58i	− 0.768974i	−0.923130	−	0.384487i	$-0.874378\pi$
$11$	− 3394.58i	− 0.768974i	0.923130	−	0.384487i	$-0.125622\pi$
$12$	0	0
$13$	14492.5i	1.82954i	0.403975	+	0.914770i	$0.367628\pi$
$13$	14492.5i	1.82954i	−0.403975	+	0.914770i	$0.632372\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5178.19	−0.255627	−0.127814	−	0.991798i	$-0.540796\pi$
$17$	−5178.19	−0.255627	−0.127814	+	0.991798i	$0.540796\pi$
$18$	0	0
$19$	− 46067.9i	− 1.54085i	−0.637529	−	0.770426i	$-0.720044\pi$
$19$	− 46067.9i	− 1.54085i	0.637529	−	0.770426i	$-0.279956\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−67535.4	−1.15740	−0.578701	−	0.815540i	$-0.696440\pi$
$23$	−67535.4	−1.15740	−0.578701	+	0.815540i	$0.696440\pi$
$24$	0	0
$25$	−141274.	−1.80831
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 25387.3i	− 0.193296i	−0.995319	−	0.0966480i	$-0.969188\pi$
$29$	− 25387.3i	− 0.193296i	0.995319	−	0.0966480i	$-0.0308121\pi$
$30$	0	0
$31$	−123808.	−0.746419	−0.373210	−	0.927747i	$-0.621743\pi$
$31$	−123808.	−0.746419	−0.373210	+	0.927747i	$0.621743\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	38053.8i	0.150024i
$36$	0	0
$37$	104623.i	0.339562i	0.985482	+	0.169781i	$0.0543061\pi$
$37$	104623.i	0.339562i	−0.985482	+	0.169781i	$0.945694\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	160655.	0.364042	0.182021	−	0.983295i	$-0.441736\pi$
$41$	160655.	0.364042	0.182021	+	0.983295i	$0.441736\pi$
$42$	0	0
$43$	− 378823.i	− 0.726603i	−0.931672	−	0.363301i	$-0.881650\pi$
$43$	− 378823.i	− 0.726603i	0.931672	−	0.363301i	$-0.118350\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	683674.	0.960520	0.480260	−	0.877126i	$-0.340542\pi$
$47$	683674.	0.960520	0.480260	+	0.877126i	$0.340542\pi$
$48$	0	0
$49$	−816943.	−0.991986
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.74261e6i	1.60781i	0.594757	+	0.803906i	$0.297248\pi$
$53$	1.74261e6i	1.60781i	−0.594757	+	0.803906i	$0.702752\pi$
$54$	0	0
$55$	−1.59002e6	−1.28865
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.48196e6i	− 0.939406i	−0.882824	−	0.469703i	$-0.844361\pi$
$59$	− 1.48196e6i	− 0.939406i	0.882824	−	0.469703i	$-0.155639\pi$
$60$	0	0
$61$	444587.i	0.250785i	0.992107	+	0.125393i	$0.0400191\pi$
$61$	444587.i	0.250785i	−0.992107	+	0.125393i	$0.959981\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.78830e6	3.06594
$66$	0	0
$67$	2.48180e6i	1.00810i	0.863674	+	0.504051i	$0.168158\pi$
$67$	2.48180e6i	1.00810i	−0.863674	+	0.504051i	$0.831842\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−510618.	−0.169313	−0.0846567	−	0.996410i	$-0.526979\pi$
$71$	−510618.	−0.169313	−0.0846567	+	0.996410i	$0.526979\pi$
$72$	0	0
$73$	4.80319e6	1.44511	0.722553	−	0.691315i	$-0.242968\pi$
$73$	4.80319e6	1.44511	0.722553	+	0.691315i	$0.242968\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	275783.i	0.0688414i
$78$	0	0
$79$	−1.57434e6	−0.359255	−0.179627	−	0.983735i	$-0.557489\pi$
$79$	−1.57434e6	−0.359255	−0.179627	+	0.983735i	$0.557489\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.91066e6i	1.51859i	0.650749	+	0.759293i	$0.274455\pi$
$83$	7.91066e6i	1.51859i	−0.650749	+	0.759293i	$0.725545\pi$
$84$	0	0
$85$	2.42547e6i	0.428380i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.35526e6	−0.955583	−0.477791	−	0.878473i	$-0.658563\pi$
$89$	−6.35526e6	−0.955583	−0.477791	+	0.878473i	$0.658563\pi$
$90$	0	0
$91$	− 1.17740e6i	− 0.163787i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.15782e7	−2.58216
$96$	0	0
$97$	−3.68231e6	−0.409656	−0.204828	−	0.978798i	$-0.565664\pi$
$97$	−3.68231e6	−0.409656	−0.204828	+	0.978798i	$0.565664\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.61014e6i	− 0.155503i	−0.996973	−	0.0777514i	$-0.975226\pi$
$101$	− 1.61014e6i	− 0.155503i	0.996973	−	0.0777514i	$-0.0247740\pi$
$102$	0	0
$103$	−392152.	−0.0353609	−0.0176805	−	0.999844i	$-0.505628\pi$
$103$	−392152.	−0.0353609	−0.0176805	+	0.999844i	$0.505628\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.62922e7i	− 1.28569i	−0.765998	−	0.642843i	$-0.777755\pi$
$107$	− 1.62922e7i	− 1.28569i	0.765998	−	0.642843i	$-0.222245\pi$
$108$	0	0
$109$	2.39639e7i	1.77241i	0.463294	+	0.886205i	$0.346667\pi$
$109$	2.39639e7i	1.77241i	−0.463294	+	0.886205i	$0.653333\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.91168e7	−1.24635	−0.623176	−	0.782081i	$-0.714158\pi$
$113$	−1.91168e7	−1.24635	−0.623176	+	0.782081i	$0.714158\pi$
$114$	0	0
$115$	3.16336e7i	1.93957i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	420687.	0.0228847
$120$	0	0
$121$	7.96400e6	0.408679
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.95790e7i	1.35456i
$126$	0	0
$127$	1.67337e7	0.724901	0.362451	−	0.932003i	$-0.381940\pi$
$127$	1.67337e7	0.724901	0.362451	+	0.932003i	$0.381940\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.21897e7i	0.862386i	0.902260	+	0.431193i	$0.141907\pi$
$131$	2.21897e7i	0.862386i	−0.902260	+	0.431193i	$0.858093\pi$
$132$	0	0
$133$	3.74265e6i	0.137943i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	582593.	0.0193572	0.00967862	−	0.999953i	$-0.496919\pi$
$137$	582593.	0.0193572	0.00967862	+	0.999953i	$0.496919\pi$
$138$	0	0
$139$	3.81869e6i	0.120604i	0.998180	+	0.0603021i	$0.0192064\pi$
$139$	3.81869e6i	0.120604i	−0.998180	+	0.0603021i	$0.980794\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.91960e7	1.40687
$144$	0	0
$145$	−1.18914e7	−0.323926
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.12196e6i	0.201145i	0.994930	+	0.100572i	$0.0320674\pi$
$149$	8.12196e6i	0.201145i	−0.994930	+	0.100572i	$0.967933\pi$
$150$	0	0
$151$	−2.00853e7	−0.474744	−0.237372	−	0.971419i	$-0.576286\pi$
$151$	−2.00853e7	−0.474744	−0.237372	+	0.971419i	$0.576286\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.79917e7i	1.25085i
$156$	0	0
$157$	7.13959e7i	1.47239i	0.676767	+	0.736197i	$0.263380\pi$
$157$	7.13959e7i	1.47239i	−0.676767	+	0.736197i	$0.736620\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.48672e6	0.103615
$162$	0	0
$163$	7.14988e7i	1.29313i	0.762859	+	0.646565i	$0.223795\pi$
$163$	7.14988e7i	1.29313i	−0.762859	+	0.646565i	$0.776205\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.80714e7	1.29713	0.648566	−	0.761158i	$-0.275369\pi$
$167$	7.80714e7	1.29713	0.648566	+	0.761158i	$0.275369\pi$
$168$	0	0
$169$	−1.47284e8	−2.34722
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4.84863e7i	− 0.711963i	−0.934493	−	0.355981i	$-0.884147\pi$
$173$	− 4.84863e7i	− 0.711963i	0.934493	−	0.355981i	$-0.115853\pi$
$174$	0	0
$175$	1.14774e7	0.161886
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.56733e7i	0.986182i	0.869978	+	0.493091i	$0.164133\pi$
$179$	7.56733e7i	0.986182i	−0.869978	+	0.493091i	$0.835867\pi$
$180$	0	0
$181$	2.01200e7i	0.252204i	0.992017	+	0.126102i	$0.0402467\pi$
$181$	2.01200e7i	0.252204i	−0.992017	+	0.126102i	$0.959753\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.90053e7	0.569038
$186$	0	0
$187$	1.75778e7i	0.196571i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.28127e7	−0.340741	−0.170371	−	0.985380i	$-0.554496\pi$
$191$	−3.28127e7	−0.340741	−0.170371	+	0.985380i	$0.554496\pi$
$192$	0	0
$193$	9.98286e7	0.999550	0.499775	−	0.866155i	$-0.333416\pi$
$193$	9.98286e7	0.999550	0.499775	+	0.866155i	$0.333416\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.59728e7i	0.801179i	0.916258	+	0.400589i	$0.131195\pi$
$197$	8.59728e7i	0.801179i	−0.916258	+	0.400589i	$0.868805\pi$
$198$	0	0
$199$	−1.62250e8	−1.45948	−0.729741	−	0.683724i	$-0.760359\pi$
$199$	−1.62250e8	−1.45948	−0.729741	+	0.683724i	$0.760359\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.06251e6i	0.0173046i
$204$	0	0
$205$	− 7.52511e7i	− 0.610062i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.56381e8	−1.18488
$210$	0	0
$211$	2.14780e8i	1.57400i	0.616953	+	0.787000i	$0.288367\pi$
$211$	2.14780e8i	1.57400i	−0.616953	+	0.787000i	$0.711633\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.77441e8	−1.21764
$216$	0	0
$217$	1.00584e7	0.0668222
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7.50450e7i	− 0.467680i
$222$	0	0
$223$	−1.26786e8	−0.765603	−0.382802	−	0.923831i	$-0.625041\pi$
$223$	−1.26786e8	−0.765603	−0.382802	+	0.923831i	$0.625041\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.99430e8i	− 1.13162i	−0.824536	−	0.565810i	$-0.808564\pi$
$227$	− 1.99430e8i	− 1.13162i	0.824536	−	0.565810i	$-0.191436\pi$
$228$	0	0
$229$	3.66567e7i	0.201711i	0.994901	+	0.100856i	$0.0321580\pi$
$229$	3.66567e7i	0.201711i	−0.994901	+	0.100856i	$0.967842\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.86007e7	−0.510663	−0.255331	−	0.966854i	$-0.582185\pi$
$233$	−9.86007e7	−0.510663	−0.255331	+	0.966854i	$0.582185\pi$
$234$	0	0
$235$	− 3.20233e8i	− 1.60964i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.07872e8	−0.511110	−0.255555	−	0.966795i	$-0.582258\pi$
$239$	−1.07872e8	−0.511110	−0.255555	+	0.966795i	$0.582258\pi$
$240$	0	0
$241$	−9.47834e7	−0.436187	−0.218093	−	0.975928i	$-0.569984\pi$
$241$	−9.47834e7	−0.436187	−0.218093	+	0.975928i	$0.569984\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.82656e8i	1.66237i
$246$	0	0
$247$	6.67640e8	2.81905
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.76729e8i	− 0.705422i	−0.935732	−	0.352711i	$-0.885260\pi$
$251$	− 1.76729e8i	− 0.705422i	0.935732	−	0.352711i	$-0.114740\pi$
$252$	0	0
$253$	2.29254e8i	0.890012i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.65413e8	1.71030	0.855151	−	0.518380i	$-0.173465\pi$
$257$	4.65413e8	1.71030	0.855151	+	0.518380i	$0.173465\pi$
$258$	0	0
$259$	− 8.49975e6i	− 0.0303988i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.23925e8	0.420061	0.210031	−	0.977695i	$-0.432644\pi$
$263$	1.23925e8	0.420061	0.210031	+	0.977695i	$0.432644\pi$
$264$	0	0
$265$	8.16240e8	2.69437
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.78420e8i	− 1.49857i	−0.662250	−	0.749283i	$-0.730398\pi$
$269$	− 4.78420e8i	− 1.49857i	0.662250	−	0.749283i	$-0.269602\pi$
$270$	0	0
$271$	1.18688e8	0.362255	0.181128	−	0.983460i	$-0.442025\pi$
$271$	1.18688e8	0.362255	0.181128	+	0.983460i	$0.442025\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.79566e8i	1.39054i
$276$	0	0
$277$	− 1.89410e8i	− 0.535455i	−0.963495	−	0.267728i	$-0.913727\pi$
$277$	− 1.89410e8i	− 0.535455i	0.963495	−	0.267728i	$-0.0862727\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.40783e6	−0.0252940	−0.0126470	−	0.999920i	$-0.504026\pi$
$281$	−9.40783e6	−0.0252940	−0.0126470	+	0.999920i	$0.504026\pi$
$282$	0	0
$283$	− 5.76835e8i	− 1.51286i	−0.654075	−	0.756430i	$-0.726942\pi$
$283$	− 5.76835e8i	− 1.51286i	0.654075	−	0.756430i	$-0.273058\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.30520e7	−0.0325904
$288$	0	0
$289$	−3.83525e8	−0.934655
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.91338e8i	− 0.676643i	−0.941031	−	0.338322i	$-0.890141\pi$
$293$	− 2.91338e8i	− 0.676643i	0.941031	−	0.338322i	$-0.109859\pi$
$294$	0	0
$295$	−6.94149e8	−1.57426
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.78758e8i	− 2.11751i
$300$	0	0
$301$	3.07764e7i	0.0650481i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.08245e8	0.420266
$306$	0	0
$307$	2.68815e8i	0.530236i	0.964216	+	0.265118i	$0.0854108\pi$
$307$	2.68815e8i	0.530236i	−0.964216	+	0.265118i	$0.914589\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.66482e8	−1.63342	−0.816711	−	0.577047i	$-0.804205\pi$
$311$	−8.66482e8	−1.63342	−0.816711	+	0.577047i	$0.804205\pi$
$312$	0	0
$313$	−4.30400e8	−0.793353	−0.396677	−	0.917958i	$-0.629837\pi$
$313$	−4.30400e8	−0.793353	−0.396677	+	0.917958i	$0.629837\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.18718e8i	0.561953i	0.959715	+	0.280976i	$0.0906583\pi$
$317$	3.18718e8i	0.561953i	−0.959715	+	0.280976i	$0.909342\pi$
$318$	0	0
$319$	−8.61791e7	−0.148640
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.38548e8i	0.393883i
$324$	0	0
$325$	− 2.04741e9i	− 3.30837i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.55431e7	−0.0859893
$330$	0	0
$331$	− 1.06114e9i	− 1.60832i	−0.594411	−	0.804161i	$-0.702615\pi$
$331$	− 1.06114e9i	− 1.60832i	0.594411	−	0.804161i	$-0.297385\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.16247e9	1.68938
$336$	0	0
$337$	6.46738e8	0.920499	0.460250	−	0.887790i	$-0.347760\pi$
$337$	6.46738e8	0.920499	0.460250	+	0.887790i	$0.347760\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.20276e8i	0.573977i
$342$	0	0
$343$	1.33276e8	0.178330
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.65479e7i	0.0726546i	0.999340	+	0.0363273i	$0.0115659\pi$
$347$	5.65479e7i	0.0726546i	−0.999340	+	0.0363273i	$0.988434\pi$
$348$	0	0
$349$	2.17863e8i	0.274343i	0.990547	+	0.137172i	$0.0438012\pi$
$349$	2.17863e8i	0.274343i	−0.990547	+	0.137172i	$0.956199\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.33462e8	−0.887496	−0.443748	−	0.896152i	$-0.646351\pi$
$353$	−7.33462e8	−0.887496	−0.443748	+	0.896152i	$0.646351\pi$
$354$	0	0
$355$	2.39173e8i	0.283736i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.21920e9	1.39073	0.695366	−	0.718656i	$-0.255242\pi$
$359$	1.21920e9	1.39073	0.695366	+	0.718656i	$0.255242\pi$
$360$	0	0
$361$	−1.22838e9	−1.37422
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.24982e9i	− 2.42171i
$366$	0	0
$367$	−6.80496e8	−0.718612	−0.359306	−	0.933220i	$-0.616987\pi$
$367$	−6.80496e8	−0.718612	−0.359306	+	0.933220i	$0.616987\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.41573e8i	− 0.143937i
$372$	0	0
$373$	− 7.23854e8i	− 0.722221i	−0.932523	−	0.361110i	$-0.882398\pi$
$373$	− 7.23854e8i	− 0.722221i	0.932523	−	0.361110i	$-0.117602\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.67925e8	0.353643
$378$	0	0
$379$	8.60978e8i	0.812372i	0.913790	+	0.406186i	$0.133141\pi$
$379$	8.60978e8i	0.812372i	−0.913790	+	0.406186i	$0.866859\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.03603e8	0.0942269	0.0471135	−	0.998890i	$-0.484998\pi$
$383$	1.03603e8	0.0942269	0.0471135	+	0.998890i	$0.484998\pi$
$384$	0	0
$385$	1.29177e8	0.115364
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.23948e8i	0.192896i	0.995338	+	0.0964481i	$0.0307482\pi$
$389$	2.23948e8i	0.192896i	−0.995338	+	0.0964481i	$0.969252\pi$
$390$	0	0
$391$	3.49711e8	0.295863
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.37420e8i	0.602039i
$396$	0	0
$397$	1.14346e9i	0.917179i	0.888648	+	0.458589i	$0.151645\pi$
$397$	1.14346e9i	0.917179i	−0.888648	+	0.458589i	$0.848355\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.48918e8	−0.270220	−0.135110	−	0.990831i	$-0.543139\pi$
$401$	−3.48918e8	−0.270220	−0.135110	+	0.990831i	$0.543139\pi$
$402$	0	0
$403$	− 1.79429e9i	− 1.36560i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.55150e8	0.261115
$408$	0	0
$409$	1.29700e8	0.0937368	0.0468684	−	0.998901i	$-0.485076\pi$
$409$	1.29700e8	0.0937368	0.0468684	+	0.998901i	$0.485076\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.20397e8i	0.0840991i
$414$	0	0
$415$	3.70536e9	2.54485
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.23819e8i	− 0.0822312i	−0.999154	−	0.0411156i	$-0.986909\pi$
$419$	− 1.23819e8i	− 0.0822312i	0.999154	−	0.0411156i	$-0.0130912\pi$
$420$	0	0
$421$	2.72698e7i	0.0178112i	0.999960	+	0.00890562i	$0.00283478\pi$
$421$	2.72698e7i	0.0178112i	−0.999960	+	0.00890562i	$0.997165\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.31543e8	0.462252
$426$	0	0
$427$	− 3.61192e7i	− 0.0224512i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.06503e9	−0.640754	−0.320377	−	0.947290i	$-0.603810\pi$
$431$	−1.06503e9	−0.640754	−0.320377	+	0.947290i	$0.603810\pi$
$432$	0	0
$433$	7.41521e8	0.438951	0.219475	−	0.975618i	$-0.429565\pi$
$433$	7.41521e8	0.438951	0.219475	+	0.975618i	$0.429565\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.11122e9i	1.78338i
$438$	0	0
$439$	1.10708e9	0.624531	0.312265	−	0.949995i	$-0.398912\pi$
$439$	1.10708e9	0.624531	0.312265	+	0.949995i	$0.398912\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.11670e9i	0.610272i	0.952309	+	0.305136i	$0.0987019\pi$
$443$	1.11670e9i	0.610272i	−0.952309	+	0.305136i	$0.901298\pi$
$444$	0	0
$445$	2.97681e9i	1.60137i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.94185e9	−1.01240	−0.506201	−	0.862416i	$-0.668950\pi$
$449$	−1.94185e9	−1.01240	−0.506201	+	0.862416i	$0.668950\pi$
$450$	0	0
$451$	− 5.45358e8i	− 0.279939i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.51495e8	−0.274475
$456$	0	0
$457$	1.89332e9	0.927936	0.463968	−	0.885852i	$-0.346425\pi$
$457$	1.89332e9	0.927936	0.463968	+	0.885852i	$0.346425\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 2.70468e9i	− 1.28577i	−0.765964	−	0.642883i	$-0.777738\pi$
$461$	− 2.70468e9i	− 1.28577i	0.765964	−	0.642883i	$-0.222262\pi$
$462$	0	0
$463$	−3.19942e9	−1.49809	−0.749046	−	0.662518i	$-0.769488\pi$
$463$	−3.19942e9	−1.49809	−0.749046	+	0.662518i	$0.769488\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.00868e9i	− 1.82134i	−0.413130	−	0.910672i	$-0.635565\pi$
$467$	− 4.00868e9i	− 1.82134i	0.413130	−	0.910672i	$-0.364435\pi$
$468$	0	0
$469$	− 2.01626e8i	− 0.0902490i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.28595e9	−0.558739
$474$	0	0
$475$	6.50820e9i	2.78633i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.36136e9	−0.981721	−0.490860	−	0.871238i	$-0.663317\pi$
$479$	−2.36136e9	−0.981721	−0.490860	+	0.871238i	$0.663317\pi$
$480$	0	0
$481$	−1.51624e9	−0.621243
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.72480e9i	0.686502i
$486$	0	0
$487$	−2.82968e9	−1.11016	−0.555080	−	0.831797i	$-0.687312\pi$
$487$	−2.82968e9	−1.11016	−0.555080	+	0.831797i	$0.687312\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.04210e9i	1.15981i	0.814682	+	0.579907i	$0.196911\pi$
$491$	3.04210e9i	1.15981i	−0.814682	+	0.579907i	$0.803089\pi$
$492$	0	0
$493$	1.31460e8i	0.0494117i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.14836e7	0.0151576
$498$	0	0
$499$	− 6.92433e8i	− 0.249474i	−0.992190	−	0.124737i	$-0.960191\pi$
$499$	− 6.92433e8i	− 0.249474i	0.992190	−	0.124737i	$-0.0398088\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.40321e9	−1.19234	−0.596171	−	0.802858i	$-0.703312\pi$
$503$	−3.40321e9	−1.19234	−0.596171	+	0.802858i	$0.703312\pi$
$504$	0	0
$505$	−7.54189e8	−0.260592
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.36640e8i	0.0795383i	0.999209	+	0.0397691i	$0.0126623\pi$
$509$	2.36640e8i	0.0795383i	−0.999209	+	0.0397691i	$0.987338\pi$
$510$	0	0
$511$	−3.90221e8	−0.129371
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.83684e8i	0.0592579i
$516$	0	0
$517$	− 2.32079e9i	− 0.738615i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.10503e8	0.0652119	0.0326059	−	0.999468i	$-0.489619\pi$
$521$	2.10503e8	0.0652119	0.0326059	+	0.999468i	$0.489619\pi$
$522$	0	0
$523$	4.88451e9i	1.49302i	0.665376	+	0.746509i	$0.268271\pi$
$523$	4.88451e9i	1.49302i	−0.665376	+	0.746509i	$0.731729\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.41101e8	0.190805
$528$	0	0
$529$	1.15621e9	0.339579
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.32830e9i	0.666030i
$534$	0	0
$535$	−7.63125e9	−2.15455
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.77318e9i	0.762811i
$540$	0	0
$541$	3.13607e9i	0.851521i	0.904836	+	0.425760i	$0.139993\pi$
$541$	3.13607e9i	0.851521i	−0.904836	+	0.425760i	$0.860007\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.12247e10	2.97020
$546$	0	0
$547$	− 2.24354e9i	− 0.586110i	−0.956096	−	0.293055i	$-0.905328\pi$
$547$	− 2.24354e9i	− 0.586110i	0.956096	−	0.293055i	$-0.0946719\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.16954e9	−0.297841
$552$	0	0
$553$	1.27902e8	0.0321618
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.59390e9i	1.12639i	0.826324	+	0.563195i	$0.190428\pi$
$557$	4.59390e9i	1.12639i	−0.826324	+	0.563195i	$0.809572\pi$
$558$	0	0
$559$	5.49010e9	1.32935
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.25935e7i	0.0218676i	0.999940	+	0.0109338i	$0.00348040\pi$
$563$	9.25935e7i	0.0218676i	−0.999940	+	0.0109338i	$0.996520\pi$
$564$	0	0
$565$	8.95432e9i	2.08864i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.80174e9	0.410013	0.205007	−	0.978761i	$-0.434278\pi$
$569$	1.80174e9	0.410013	0.205007	+	0.978761i	$0.434278\pi$
$570$	0	0
$571$	− 4.05799e9i	− 0.912188i	−0.889932	−	0.456094i	$-0.849248\pi$
$571$	− 4.05799e9i	− 0.912188i	0.889932	−	0.456094i	$-0.150752\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.54099e9	2.09294
$576$	0	0
$577$	−7.82764e9	−1.69635	−0.848175	−	0.529716i	$-0.822298\pi$
$577$	−7.82764e9	−1.69635	−0.848175	+	0.529716i	$0.822298\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.42679e8i	− 0.135949i
$582$	0	0
$583$	5.91543e9	1.23637
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.45350e9i	− 0.296607i	−0.988942	−	0.148304i	$-0.952619\pi$
$587$	− 1.45350e9i	− 0.296607i	0.988942	−	0.148304i	$-0.0473813\pi$
$588$	0	0
$589$	5.70358e9i	1.15012i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.26005e8	−0.162664	−0.0813320	−	0.996687i	$-0.525917\pi$
$593$	−8.26005e8	−0.162664	−0.0813320	+	0.996687i	$0.525917\pi$
$594$	0	0
$595$	− 1.97050e8i	− 0.0383501i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.81768e9	0.725780	0.362890	−	0.931832i	$-0.381790\pi$
$599$	3.81768e9	0.725780	0.362890	+	0.931832i	$0.381790\pi$
$600$	0	0
$601$	1.48375e9	0.278805	0.139402	−	0.990236i	$-0.455482\pi$
$601$	1.48375e9	0.278805	0.139402	+	0.990236i	$0.455482\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.73034e9i	− 0.684864i
$606$	0	0
$607$	−1.01217e10	−1.83693	−0.918463	−	0.395506i	$-0.870569\pi$
$607$	−1.01217e10	−1.83693	−0.918463	+	0.395506i	$0.870569\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.90815e9i	1.75731i
$612$	0	0
$613$	− 8.24178e9i	− 1.44514i	−0.691299	−	0.722569i	$-0.742961\pi$
$613$	− 8.24178e9i	− 1.44514i	0.691299	−	0.722569i	$-0.257039\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.98164e9	−1.53942	−0.769710	−	0.638393i	$-0.779599\pi$
$617$	−8.98164e9	−1.53942	−0.769710	+	0.638393i	$0.779599\pi$
$618$	0	0
$619$	3.83653e9i	0.650162i	0.945686	+	0.325081i	$0.105392\pi$
$619$	3.83653e9i	0.650162i	−0.945686	+	0.325081i	$0.894608\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.16315e8	0.0855473
$624$	0	0
$625$	2.81779e9	0.461666
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 5.41756e8i	− 0.0868013i
$630$	0	0
$631$	5.55269e9	0.879834	0.439917	−	0.898039i	$-0.355008\pi$
$631$	5.55269e9	0.879834	0.439917	+	0.898039i	$0.355008\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.83807e9i	− 1.21479i
$636$	0	0
$637$	− 1.18396e10i	− 1.81488i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.35378e9	−1.10283	−0.551413	−	0.834232i	$-0.685911\pi$
$641$	−7.35378e9	−1.10283	−0.551413	+	0.834232i	$0.685911\pi$
$642$	0	0
$643$	4.34387e9i	0.644375i	0.946676	+	0.322187i	$0.104418\pi$
$643$	4.34387e9i	0.644375i	−0.946676	+	0.322187i	$0.895582\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.96133e9	−1.44595	−0.722973	−	0.690876i	$-0.757225\pi$
$647$	−9.96133e9	−1.44595	−0.722973	+	0.690876i	$0.757225\pi$
$648$	0	0
$649$	−5.03062e9	−0.722379
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.99451e9i	0.420852i	0.977610	+	0.210426i	$0.0674850\pi$
$653$	2.99451e9i	0.420852i	−0.977610	+	0.210426i	$0.932515\pi$
$654$	0	0
$655$	1.03937e10	1.44519
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.84192e9i	− 1.06739i	−0.845677	−	0.533695i	$-0.820803\pi$
$659$	− 7.84192e9i	− 1.06739i	0.845677	−	0.533695i	$-0.179197\pi$
$660$	0	0
$661$	5.55066e9i	0.747549i	0.927520	+	0.373775i	$0.121937\pi$
$661$	5.55066e9i	0.747549i	−0.927520	+	0.373775i	$0.878063\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.75306e9	0.231164
$666$	0	0
$667$	1.71454e9i	0.223721i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.50919e9	0.192847
$672$	0	0
$673$	9.93043e9	1.25578	0.627892	−	0.778300i	$-0.283918\pi$
$673$	9.93043e9	1.25578	0.627892	+	0.778300i	$0.283918\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.68747e9i	− 1.07605i	−0.842929	−	0.538026i	$-0.819170\pi$
$677$	− 8.68747e9i	− 1.07605i	0.842929	−	0.538026i	$-0.180830\pi$
$678$	0	0
$679$	2.99159e8	0.0366739
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.19143e8i	0.110385i	0.998476	+	0.0551925i	$0.0175773\pi$
$683$	9.19143e8i	0.110385i	−0.998476	+	0.0551925i	$0.982423\pi$
$684$	0	0
$685$	− 2.72887e8i	− 0.0324389i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.52548e10	−2.94156
$690$	0	0
$691$	− 2.66387e9i	− 0.307142i	−0.988138	−	0.153571i	$-0.950923\pi$
$691$	− 2.66387e9i	− 0.307142i	0.988138	−	0.153571i	$-0.0490774\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.78868e9	0.202109
$696$	0	0
$697$	−8.31904e8	−0.0930590
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 9.10026e9i	− 0.997794i	−0.866661	−	0.498897i	$-0.833739\pi$
$701$	− 9.10026e9i	− 0.997794i	0.866661	−	0.498897i	$-0.166261\pi$
$702$	0	0
$703$	4.81974e9	0.523215
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.30811e8i	0.0139212i
$708$	0	0
$709$	− 4.42773e9i	− 0.466573i	−0.972408	−	0.233286i	$-0.925052\pi$
$709$	− 4.42773e9i	− 0.466573i	0.972408	−	0.233286i	$-0.0749480\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.36142e9	0.863907
$714$	0	0
$715$	− 2.30434e10i	− 2.35763i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.91984e9	−0.292960	−0.146480	−	0.989214i	$-0.546794\pi$
$719$	−2.91984e9	−0.292960	−0.146480	+	0.989214i	$0.546794\pi$
$720$	0	0
$721$	3.18592e7	0.00316564
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.58656e9i	0.349539i
$726$	0	0
$727$	−1.29199e10	−1.24706	−0.623530	−	0.781799i	$-0.714302\pi$
$727$	−1.29199e10	−1.24706	−0.623530	+	0.781799i	$0.714302\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.96162e9i	0.185739i
$732$	0	0
$733$	− 1.48967e10i	− 1.39710i	−0.715563	−	0.698548i	$-0.753830\pi$
$733$	− 1.48967e10i	− 1.39710i	0.715563	−	0.698548i	$-0.246170\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.42466e9	0.775204
$738$	0	0
$739$	− 3.02167e9i	− 0.275418i	−0.990473	−	0.137709i	$-0.956026\pi$
$739$	− 3.02167e9i	− 0.275418i	0.990473	−	0.137709i	$-0.0439738\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.52499e10	−1.36397	−0.681985	−	0.731366i	$-0.738883\pi$
$743$	−1.52499e10	−1.36397	−0.681985	+	0.731366i	$0.738883\pi$
$744$	0	0
$745$	3.80433e9	0.337078
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.32361e9i	0.115099i
$750$	0	0
$751$	1.30306e9	0.112260	0.0561298	−	0.998423i	$-0.482124\pi$
$751$	1.30306e9	0.112260	0.0561298	+	0.998423i	$0.482124\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.40797e9i	0.795576i
$756$	0	0
$757$	− 3.51770e9i	− 0.294729i	−0.989082	−	0.147364i	$-0.952921\pi$
$757$	− 3.51770e9i	− 0.294729i	0.989082	−	0.147364i	$-0.0470790\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.23448e10	1.01541	0.507703	−	0.861532i	$-0.330495\pi$
$761$	1.23448e10	1.01541	0.507703	+	0.861532i	$0.330495\pi$
$762$	0	0
$763$	− 1.94687e9i	− 0.158673i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.14773e10	1.71868
$768$	0	0
$769$	1.18155e10	0.936940	0.468470	−	0.883479i	$-0.344805\pi$
$769$	1.18155e10	0.936940	0.468470	+	0.883479i	$0.344805\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.62582e10i	− 1.26603i	−0.774138	−	0.633017i	$-0.781816\pi$
$773$	− 1.62582e10i	− 1.26603i	0.774138	−	0.633017i	$-0.218184\pi$
$774$	0	0
$775$	1.74908e10	1.34975
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 7.40106e9i	− 0.560935i
$780$	0	0
$781$	1.73333e9i	0.130198i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.34418e10	2.46744
$786$	0	0
$787$	1.80876e10i	1.32273i	0.750066	+	0.661363i	$0.230022\pi$
$787$	1.80876e10i	1.32273i	−0.750066	+	0.661363i	$0.769978\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.55309e9	0.111578
$792$	0	0
$793$	−6.44318e9	−0.458822
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.07805e9i	0.565201i	0.959238	+	0.282600i	$0.0911970\pi$
$797$	8.07805e9i	0.565201i	−0.959238	+	0.282600i	$0.908803\pi$
$798$	0	0
$799$	−3.54019e9	−0.245535
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.63048e10i	− 1.11125i
$804$	0	0
$805$	− 2.56998e9i	− 0.173638i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.18309e9	−0.410568	−0.205284	−	0.978702i	$-0.565812\pi$
$809$	−6.18309e9	−0.410568	−0.205284	+	0.978702i	$0.565812\pi$
$810$	0	0
$811$	− 3.48351e9i	− 0.229321i	−0.993405	−	0.114660i	$-0.963422\pi$
$811$	− 3.48351e9i	− 0.229321i	0.993405	−	0.114660i	$-0.0365780\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.34901e10	2.16703
$816$	0	0
$817$	−1.74516e10	−1.11959
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.22317e10i	− 1.40208i	−0.713123	−	0.701039i	$-0.752720\pi$
$821$	− 2.22317e10i	− 1.40208i	0.713123	−	0.701039i	$-0.247280\pi$
$822$	0	0
$823$	−2.30533e10	−1.44156	−0.720781	−	0.693163i	$-0.756217\pi$
$823$	−2.30533e10	−1.44156	−0.720781	+	0.693163i	$0.756217\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.93422e9i	− 0.241874i	−0.992660	−	0.120937i	$-0.961410\pi$
$827$	− 3.93422e9i	− 0.241874i	0.992660	−	0.120937i	$-0.0385899\pi$
$828$	0	0
$829$	− 2.39962e10i	− 1.46286i	−0.681917	−	0.731430i	$-0.738853\pi$
$829$	− 2.39962e10i	− 1.46286i	0.681917	−	0.731430i	$-0.261147\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.23028e9	0.253578
$834$	0	0
$835$	− 3.65687e10i	− 2.17373i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.16432e10	−1.26518	−0.632592	−	0.774485i	$-0.718009\pi$
$839$	−2.16432e10	−1.26518	−0.632592	+	0.774485i	$0.718009\pi$
$840$	0	0
$841$	1.66054e10	0.962637
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.89881e10i	3.93347i
$846$	0	0
$847$	−6.47012e8	−0.0365864
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 7.06573e9i	− 0.393010i
$852$	0	0
$853$	2.96549e10i	1.63597i	0.575239	+	0.817985i	$0.304909\pi$
$853$	2.96549e10i	1.63597i	−0.575239	+	0.817985i	$0.695091\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.12112e10	−0.608442	−0.304221	−	0.952602i	$-0.598396\pi$
$857$	−1.12112e10	−0.608442	−0.304221	+	0.952602i	$0.598396\pi$
$858$	0	0
$859$	− 3.49012e10i	− 1.87873i	−0.342917	−	0.939366i	$-0.611415\pi$
$859$	− 3.49012e10i	− 1.87873i	0.342917	−	0.939366i	$-0.388585\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.84338e10	−0.976286	−0.488143	−	0.872764i	$-0.662326\pi$
$863$	−1.84338e10	−0.976286	−0.488143	+	0.872764i	$0.662326\pi$
$864$	0	0
$865$	−2.27110e10	−1.19311
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.34421e9i	0.276258i
$870$	0	0
$871$	−3.59675e10	−1.84436
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 2.40306e9i	− 0.121265i
$876$	0	0
$877$	5.90107e9i	0.295415i	0.989031	+	0.147707i	$0.0471894\pi$
$877$	5.90107e9i	0.295415i	−0.989031	+	0.147707i	$0.952811\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.16251e9	−0.205088	−0.102544	−	0.994728i	$-0.532698\pi$
$881$	−4.16251e9	−0.205088	−0.102544	+	0.994728i	$0.532698\pi$
$882$	0	0
$883$	3.49741e10i	1.70956i	0.518990	+	0.854780i	$0.326308\pi$
$883$	3.49741e10i	1.70956i	−0.518990	+	0.854780i	$0.673692\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.16854e9	−0.441131	−0.220566	−	0.975372i	$-0.570790\pi$
$887$	−9.16854e9	−0.441131	−0.220566	+	0.975372i	$0.570790\pi$
$888$	0	0
$889$	−1.35948e9	−0.0648958
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.14954e10i	− 1.48002i
$894$	0	0
$895$	3.54454e10	1.65264
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.14315e9i	0.144280i
$900$	0	0
$901$	− 9.02358e9i	− 0.411000i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.42420e9	0.422644
$906$	0	0
$907$	6.50089e9i	0.289299i	0.989483	+	0.144650i	$0.0462055\pi$
$907$	6.50089e9i	0.289299i	−0.989483	+	0.144650i	$0.953794\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.88490e10	−1.26420	−0.632100	−	0.774887i	$-0.717807\pi$
$911$	−2.88490e10	−1.26420	−0.632100	+	0.774887i	$0.717807\pi$
$912$	0	0
$913$	2.68534e10	1.16775
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.80274e9i	− 0.0772040i
$918$	0	0
$919$	2.06389e10	0.877167	0.438584	−	0.898690i	$-0.355480\pi$
$919$	2.06389e10	0.877167	0.438584	+	0.898690i	$0.355480\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 7.40013e9i	− 0.309766i
$924$	0	0
$925$	− 1.47804e10i	− 0.614033i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.08036e10	−0.851301	−0.425651	−	0.904888i	$-0.639955\pi$
$929$	−2.08036e10	−0.851301	−0.425651	+	0.904888i	$0.639955\pi$
$930$	0	0
$931$	3.76348e10i	1.52850i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.23344e9	0.329413
$936$	0	0
$937$	−1.01845e10	−0.404437	−0.202219	−	0.979340i	$-0.564815\pi$
$937$	−1.01845e10	−0.404437	−0.202219	+	0.979340i	$0.564815\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.22264e10i	− 0.478340i	−0.970978	−	0.239170i	$-0.923125\pi$
$941$	− 1.22264e10i	− 0.478340i	0.970978	−	0.239170i	$-0.0768753\pi$
$942$	0	0
$943$	−1.08499e10	−0.421343
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.46789e10i	0.561654i	0.959758	+	0.280827i	$0.0906087\pi$
$947$	1.46789e10i	0.561654i	−0.959758	+	0.280827i	$0.909391\pi$
$948$	0	0
$949$	6.96103e10i	2.64388i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.02381e10	−1.13170	−0.565848	−	0.824510i	$-0.691451\pi$
$953$	−3.02381e10	−1.13170	−0.565848	+	0.824510i	$0.691451\pi$
$954$	0	0
$955$	1.53695e10i	0.571014i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.73311e7	−0.00173293
$960$	0	0
$961$	−1.21842e10	−0.442858
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.67598e10i	− 1.67505i
$966$	0	0
$967$	−5.85290e9	−0.208151	−0.104075	−	0.994569i	$-0.533188\pi$
$967$	−5.85290e9	−0.208151	−0.104075	+	0.994569i	$0.533188\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.40526e10i	1.19367i	0.802366	+	0.596833i	$0.203574\pi$
$971$	3.40526e10i	1.19367i	−0.802366	+	0.596833i	$0.796426\pi$
$972$	0	0
$973$	− 3.10238e8i	− 0.0107969i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.22080e10	0.418807	0.209404	−	0.977829i	$-0.432848\pi$
$977$	1.22080e10	0.418807	0.209404	+	0.977829i	$0.432848\pi$
$978$	0	0
$979$	2.15734e10i	0.734818i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.99405e10	−1.00536	−0.502680	−	0.864473i	$-0.667652\pi$
$983$	−2.99405e10	−1.00536	−0.502680	+	0.864473i	$0.667652\pi$
$984$	0	0
$985$	4.02697e10	1.34262
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.55840e10i	0.840971i
$990$	0	0
$991$	2.04298e10	0.666815	0.333408	−	0.942783i	$-0.391801\pi$
$991$	2.04298e10	0.666815	0.333408	+	0.942783i	$0.391801\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.59979e10i	2.44580i
$996$	0	0
$997$	− 2.36775e10i	− 0.756664i	−0.925670	−	0.378332i	$-0.876498\pi$
$997$	− 2.36775e10i	− 0.756664i	0.925670	−	0.378332i	$-0.123502\pi$
$998$	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.145.1		14
3.2	odd	2		96.8.d.a.49.14		14
4.3	odd	2		72.8.d.d.37.8		14
8.3	odd	2		72.8.d.d.37.7		14
8.5	even	2	inner	288.8.d.d.145.14		14
12.11	even	2		24.8.d.a.13.7	✓	14
24.5	odd	2		96.8.d.a.49.1		14
24.11	even	2		24.8.d.a.13.8	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.d.a.13.7	✓	14	12.11	even	2
24.8.d.a.13.8	yes	14	24.11	even	2
72.8.d.d.37.7		14	8.3	odd	2
72.8.d.d.37.8		14	4.3	odd	2
96.8.d.a.49.1		14	24.5	odd	2
96.8.d.a.49.14		14	3.2	odd	2
288.8.d.d.145.1		14	1.1	even	1	trivial
288.8.d.d.145.14		14	8.5	even	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.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-468.400i q^{5} -81.2421 q^{7} +O(q^{10})\)	\(q-468.400i q^{5} -81.2421 q^{7} -3394.58i q^{11} +14492.5i q^{13} -5178.19 q^{17} -46067.9i q^{19} -67535.4 q^{23} -141274. q^{25} -25387.3i q^{29} -123808. q^{31} +38053.8i q^{35} +104623. i q^{37} +160655. q^{41} -378823. i q^{43} +683674. q^{47} -816943. q^{49} +1.74261e6i q^{53} -1.59002e6 q^{55} -1.48196e6i q^{59} +444587. i q^{61} +6.78830e6 q^{65} +2.48180e6i q^{67} -510618. q^{71} +4.80319e6 q^{73} +275783. i q^{77} -1.57434e6 q^{79} +7.91066e6i q^{83} +2.42547e6i q^{85} -6.35526e6 q^{89} -1.17740e6i q^{91} -2.15782e7 q^{95} -3.68231e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 1372 q^{7}+O(q^{10}) \) 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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