LMFDB - Embedded newform 288.8.d.d.145.7

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			145.7
Root			$7.97707 + 3.91414i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.8.d.d.145.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-23.0228i q^{5} +1547.56 q^{7} +O(q^{10})$	$q-23.0228i q^{5} +1547.56 q^{7} +822.251i q^{11} +5929.75i q^{13} +12727.5 q^{17} +43898.2i q^{19} -77667.6 q^{23} +77594.9 q^{25} +151275. i q^{29} +46441.4 q^{31} -35629.3i q^{35} -32422.8i q^{37} -188739. q^{41} -847931. i q^{43} -1.23337e6 q^{47} +1.57141e6 q^{49} -1.00098e6i q^{53} +18930.6 q^{55} +2.08716e6i q^{59} +252465. i q^{61} +136520. q^{65} +4.04209e6i q^{67} -2.06238e6 q^{71} +1.18939e6 q^{73} +1.27249e6i q^{77} +3.44398e6 q^{79} +4.07362e6i q^{83} -293023. i q^{85} +1.21525e7 q^{89} +9.17666e6i q^{91} +1.01066e6 q^{95} -1.02206e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 23.0228i	− 0.0823690i	−0.999152	−	0.0411845i	$-0.986887\pi$
$5$	− 23.0228i	− 0.0823690i	0.999152	−	0.0411845i	$-0.0131131\pi$
$6$	0	0
$7$	1547.56	1.70532	0.852658	−	0.522469i	$-0.174989\pi$
$7$	1547.56	1.70532	0.852658	+	0.522469i	$0.174989\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	822.251i	0.186265i	0.995654	+	0.0931323i	$0.0296880\pi$
$11$	822.251i	0.186265i	−0.995654	+	0.0931323i	$0.970312\pi$
$12$	0	0
$13$	5929.75i	0.748574i	0.927313	+	0.374287i	$0.122113\pi$
$13$	5929.75i	0.748574i	−0.927313	+	0.374287i	$0.877887\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12727.5	0.628306	0.314153	−	0.949372i	$-0.398279\pi$
$17$	12727.5	0.628306	0.314153	+	0.949372i	$0.398279\pi$
$18$	0	0
$19$	43898.2i	1.46828i	0.678998	+	0.734140i	$0.262414\pi$
$19$	43898.2i	1.46828i	−0.678998	+	0.734140i	$0.737586\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−77667.6	−1.33104	−0.665522	−	0.746378i	$-0.731791\pi$
$23$	−77667.6	−1.33104	−0.665522	+	0.746378i	$0.731791\pi$
$24$	0	0
$25$	77594.9	0.993215
$26$	0	0
$27$	0	0
$28$	0	0
$29$	151275.i	1.15179i	0.817523	+	0.575896i	$0.195347\pi$
$29$	151275.i	1.15179i	−0.817523	+	0.575896i	$0.804653\pi$
$30$	0	0
$31$	46441.4	0.279988	0.139994	−	0.990152i	$-0.455292\pi$
$31$	46441.4	0.279988	0.139994	+	0.990152i	$0.455292\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 35629.3i	− 0.140465i
$36$	0	0
$37$	− 32422.8i	− 0.105231i	−0.998615	−	0.0526156i	$-0.983244\pi$
$37$	− 32422.8i	− 0.105231i	0.998615	−	0.0526156i	$-0.0167558\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−188739.	−0.427679	−0.213840	−	0.976869i	$-0.568597\pi$
$41$	−188739.	−0.427679	−0.213840	+	0.976869i	$0.568597\pi$
$42$	0	0
$43$	− 847931.i	− 1.62637i	−0.582002	−	0.813187i	$-0.697730\pi$
$43$	− 847931.i	− 1.62637i	0.582002	−	0.813187i	$-0.302270\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.23337e6	−1.73281	−0.866405	−	0.499342i	$-0.833575\pi$
$47$	−1.23337e6	−1.73281	−0.866405	+	0.499342i	$0.833575\pi$
$48$	0	0
$49$	1.57141e6	1.90811
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.00098e6i	− 0.923547i	−0.886998	−	0.461773i	$-0.847213\pi$
$53$	− 1.00098e6i	− 0.923547i	0.886998	−	0.461773i	$-0.152787\pi$
$54$	0	0
$55$	18930.6	0.0153424
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.08716e6i	1.32304i	0.749927	+	0.661520i	$0.230088\pi$
$59$	2.08716e6i	1.32304i	−0.749927	+	0.661520i	$0.769912\pi$
$60$	0	0
$61$	252465.i	0.142412i	0.997462	+	0.0712060i	$0.0226848\pi$
$61$	252465.i	0.142412i	−0.997462	+	0.0712060i	$0.977315\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	136520.	0.0616593
$66$	0	0
$67$	4.04209e6i	1.64189i	0.571007	+	0.820946i	$0.306553\pi$
$67$	4.04209e6i	1.64189i	−0.571007	+	0.820946i	$0.693447\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.06238e6	−0.683857	−0.341928	−	0.939726i	$-0.611080\pi$
$71$	−2.06238e6	−0.683857	−0.341928	+	0.939726i	$0.611080\pi$
$72$	0	0
$73$	1.18939e6	0.357845	0.178922	−	0.983863i	$-0.442739\pi$
$73$	1.18939e6	0.357845	0.178922	+	0.983863i	$0.442739\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.27249e6i	0.317640i
$78$	0	0
$79$	3.44398e6	0.785898	0.392949	−	0.919560i	$-0.371455\pi$
$79$	3.44398e6	0.785898	0.392949	+	0.919560i	$0.371455\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.07362e6i	0.782001i	0.920390	+	0.391001i	$0.127871\pi$
$83$	4.07362e6i	0.782001i	−0.920390	+	0.391001i	$0.872129\pi$
$84$	0	0
$85$	− 293023.i	− 0.0517530i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.21525e7	1.82726	0.913629	−	0.406549i	$-0.133268\pi$
$89$	1.21525e7	1.82726	0.913629	+	0.406549i	$0.133268\pi$
$90$	0	0
$91$	9.17666e6i	1.27656i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.01066e6	0.120941
$96$	0	0
$97$	−1.02206e7	−1.13704	−0.568520	−	0.822670i	$-0.692484\pi$
$97$	−1.02206e7	−1.13704	−0.568520	+	0.822670i	$0.692484\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 279077.i	− 0.0269525i	−0.999909	−	0.0134762i	$-0.995710\pi$
$101$	− 279077.i	− 0.0269525i	0.999909	−	0.0134762i	$-0.00428975\pi$
$102$	0	0
$103$	−1.44624e7	−1.30410	−0.652050	−	0.758176i	$-0.726091\pi$
$103$	−1.44624e7	−1.30410	−0.652050	+	0.758176i	$0.726091\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.94615e6i	− 0.627067i	−0.949577	−	0.313533i	$-0.898487\pi$
$107$	− 7.94615e6i	− 0.627067i	0.949577	−	0.313533i	$-0.101513\pi$
$108$	0	0
$109$	9.83074e6i	0.727099i	0.931575	+	0.363549i	$0.118435\pi$
$109$	9.83074e6i	0.727099i	−0.931575	+	0.363549i	$0.881565\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.22077e6	−0.275180	−0.137590	−	0.990489i	$-0.543936\pi$
$113$	−4.22077e6	−0.275180	−0.137590	+	0.990489i	$0.543936\pi$
$114$	0	0
$115$	1.78813e6i	0.109637i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.96966e7	1.07146
$120$	0	0
$121$	1.88111e7	0.965306
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.58512e6i	− 0.164179i
$126$	0	0
$127$	−3.80026e6	−0.164627	−0.0823133	−	0.996607i	$-0.526231\pi$
$127$	−3.80026e6	−0.164627	−0.0823133	+	0.996607i	$0.526231\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.53929e7i	1.37552i	0.725938	+	0.687760i	$0.241406\pi$
$131$	3.53929e7i	1.37552i	−0.725938	+	0.687760i	$0.758594\pi$
$132$	0	0
$133$	6.79352e7i	2.50388i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.89492e7	0.961866	0.480933	−	0.876757i	$-0.340298\pi$
$137$	2.89492e7	0.961866	0.480933	+	0.876757i	$0.340298\pi$
$138$	0	0
$139$	2.59027e7i	0.818075i	0.912518	+	0.409037i	$0.134135\pi$
$139$	2.59027e7i	0.818075i	−0.912518	+	0.409037i	$0.865865\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.87575e6	−0.139433
$144$	0	0
$145$	3.48278e6	0.0948720
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.04725e7i	1.74529i	0.488354	+	0.872646i	$0.337597\pi$
$149$	7.04725e7i	1.74529i	−0.488354	+	0.872646i	$0.662403\pi$
$150$	0	0
$151$	4.74033e7	1.12044	0.560220	−	0.828344i	$-0.310716\pi$
$151$	4.74033e7	1.12044	0.560220	+	0.828344i	$0.310716\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.06921e6i	− 0.0230623i
$156$	0	0
$157$	1.94071e7i	0.400233i	0.979772	+	0.200116i	$0.0641321\pi$
$157$	1.94071e7i	0.400233i	−0.979772	+	0.200116i	$0.935868\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.20195e8	−2.26985
$162$	0	0
$163$	6.50685e7i	1.17683i	0.808559	+	0.588416i	$0.200248\pi$
$163$	6.50685e7i	1.17683i	−0.808559	+	0.588416i	$0.799752\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.88123e7	0.644855	0.322428	−	0.946594i	$-0.395501\pi$
$167$	3.88123e7	0.644855	0.322428	+	0.946594i	$0.395501\pi$
$168$	0	0
$169$	2.75866e7	0.439637
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.04034e8i	1.52762i	0.645443	+	0.763808i	$0.276673\pi$
$173$	1.04034e8i	1.52762i	−0.645443	+	0.763808i	$0.723327\pi$
$174$	0	0
$175$	1.20083e8	1.69375
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.19980e7i	− 1.19893i	−0.800402	−	0.599464i	$-0.795381\pi$
$179$	− 9.19980e7i	− 1.19893i	0.800402	−	0.599464i	$-0.204619\pi$
$180$	0	0
$181$	− 2.05182e7i	− 0.257196i	−0.991697	−	0.128598i	$-0.958952\pi$
$181$	− 2.05182e7i	− 0.257196i	0.991697	−	0.128598i	$-0.0410476\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−746466.	−0.00866780
$186$	0	0
$187$	1.04652e7i	0.117031i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.98622e7	−0.517791	−0.258896	−	0.965905i	$-0.583359\pi$
$191$	−4.98622e7	−0.517791	−0.258896	+	0.965905i	$0.583359\pi$
$192$	0	0
$193$	−4.33752e7	−0.434302	−0.217151	−	0.976138i	$-0.569676\pi$
$193$	−4.33752e7	−0.434302	−0.217151	+	0.976138i	$0.569676\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.73761e8i	− 1.61928i	−0.586928	−	0.809639i	$-0.699663\pi$
$197$	− 1.73761e8i	− 1.61928i	0.586928	−	0.809639i	$-0.300337\pi$
$198$	0	0
$199$	2.22783e7	0.200399	0.100200	−	0.994967i	$-0.468052\pi$
$199$	2.22783e7	0.200399	0.100200	+	0.994967i	$0.468052\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.34107e8i	1.96417i
$204$	0	0
$205$	4.34531e6i	0.0352275i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.60953e7	−0.273488
$210$	0	0
$211$	2.68398e7i	0.196694i	0.995152	+	0.0983469i	$0.0313555\pi$
$211$	2.68398e7i	0.196694i	−0.995152	+	0.0983469i	$0.968645\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.95218e7	−0.133963
$216$	0	0
$217$	7.18710e7	0.477468
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.54708e7i	0.470334i
$222$	0	0
$223$	−1.71672e8	−1.03665	−0.518325	−	0.855183i	$-0.673444\pi$
$223$	−1.71672e8	−1.03665	−0.518325	+	0.855183i	$0.673444\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.37729e7i	0.248379i	0.992259	+	0.124189i	$0.0396331\pi$
$227$	4.37729e7i	0.248379i	−0.992259	+	0.124189i	$0.960367\pi$
$228$	0	0
$229$	2.12326e8i	1.16836i	0.811623	+	0.584182i	$0.198585\pi$
$229$	2.12326e8i	1.16836i	−0.811623	+	0.584182i	$0.801415\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.65211e8	1.37355	0.686776	−	0.726869i	$-0.259025\pi$
$233$	2.65211e8	1.37355	0.686776	+	0.726869i	$0.259025\pi$
$234$	0	0
$235$	2.83957e7i	0.142730i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.71661e8	1.28716	0.643582	−	0.765377i	$-0.277447\pi$
$239$	2.71661e8	1.28716	0.643582	+	0.765377i	$0.277447\pi$
$240$	0	0
$241$	−1.86499e7	−0.0858255	−0.0429127	−	0.999079i	$-0.513664\pi$
$241$	−1.86499e7	−0.0858255	−0.0429127	+	0.999079i	$0.513664\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.61783e7i	− 0.157169i
$246$	0	0
$247$	−2.60305e8	−1.09912
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.00250e8i	− 1.19847i	−0.800575	−	0.599233i	$-0.795472\pi$
$251$	− 3.00250e8i	− 1.19847i	0.800575	−	0.599233i	$-0.204528\pi$
$252$	0	0
$253$	− 6.38623e7i	− 0.247926i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.04543e8	0.751656	0.375828	−	0.926690i	$-0.377358\pi$
$257$	2.04543e8	0.751656	0.375828	+	0.926690i	$0.377358\pi$
$258$	0	0
$259$	− 5.01763e7i	− 0.179453i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.95180e8	−1.33952	−0.669761	−	0.742577i	$-0.733603\pi$
$263$	−3.95180e8	−1.33952	−0.669761	+	0.742577i	$0.733603\pi$
$264$	0	0
$265$	−2.30454e7	−0.0760717
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.33472e6i	− 0.0229747i	−0.999934	−	0.0114874i	$-0.996343\pi$
$269$	− 7.33472e6i	− 0.0229747i	0.999934	−	0.0114874i	$-0.00365662\pi$
$270$	0	0
$271$	−1.42509e8	−0.434961	−0.217480	−	0.976065i	$-0.569784\pi$
$271$	−1.42509e8	−0.434961	−0.217480	+	0.976065i	$0.569784\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.38026e7i	0.185001i
$276$	0	0
$277$	− 2.28463e8i	− 0.645857i	−0.946423	−	0.322929i	$-0.895333\pi$
$277$	− 2.28463e8i	− 0.645857i	0.946423	−	0.322929i	$-0.104667\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.09407e8	0.831874	0.415937	−	0.909393i	$-0.363454\pi$
$281$	3.09407e8	0.831874	0.415937	+	0.909393i	$0.363454\pi$
$282$	0	0
$283$	− 1.98595e8i	− 0.520854i	−0.965494	−	0.260427i	$-0.916137\pi$
$283$	− 1.98595e8i	− 0.520854i	0.965494	−	0.260427i	$-0.0838633\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.92086e8	−0.729329
$288$	0	0
$289$	−2.48350e8	−0.605231
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 8.02996e8i	− 1.86499i	−0.361183	−	0.932495i	$-0.617627\pi$
$293$	− 8.02996e8i	− 1.86499i	0.361183	−	0.932495i	$-0.382373\pi$
$294$	0	0
$295$	4.80523e7	0.108978
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.60550e8i	− 0.996385i
$300$	0	0
$301$	− 1.31223e9i	− 2.77348i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.81246e6	0.0117303
$306$	0	0
$307$	− 6.21250e8i	− 1.22541i	−0.790311	−	0.612706i	$-0.790081\pi$
$307$	− 6.21250e8i	− 1.22541i	0.790311	−	0.612706i	$-0.209919\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.07457e8	0.202570	0.101285	−	0.994857i	$-0.467705\pi$
$311$	1.07457e8	0.202570	0.101285	+	0.994857i	$0.467705\pi$
$312$	0	0
$313$	−1.42815e8	−0.263250	−0.131625	−	0.991300i	$-0.542019\pi$
$313$	−1.42815e8	−0.263250	−0.131625	+	0.991300i	$0.542019\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.50102e8i	0.617288i	0.951178	+	0.308644i	$0.0998751\pi$
$317$	3.50102e8i	0.617288i	−0.951178	+	0.308644i	$0.900125\pi$
$318$	0	0
$319$	−1.24386e8	−0.214538
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.58713e8i	0.922529i
$324$	0	0
$325$	4.60119e8i	0.743495i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.90872e9	−2.95499
$330$	0	0
$331$	− 2.59141e8i	− 0.392770i	−0.980527	−	0.196385i	$-0.937080\pi$
$331$	− 2.59141e8i	− 0.392770i	0.980527	−	0.196385i	$-0.0629202\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.30604e7	0.135241
$336$	0	0
$337$	9.80216e8	1.39514	0.697569	−	0.716518i	$-0.254265\pi$
$337$	9.80216e8	1.39514	0.697569	+	0.716518i	$0.254265\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.81865e7i	0.0521518i
$342$	0	0
$343$	1.15737e9	1.54861
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.55857e8i	0.328734i	0.986399	+	0.164367i	$0.0525581\pi$
$347$	2.55857e8i	0.328734i	−0.986399	+	0.164367i	$0.947442\pi$
$348$	0	0
$349$	− 5.57027e8i	− 0.701435i	−0.936481	−	0.350717i	$-0.885938\pi$
$349$	− 5.57027e8i	− 0.701435i	0.936481	−	0.350717i	$-0.114062\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.91812e8	0.474095	0.237048	−	0.971498i	$-0.423820\pi$
$353$	3.91812e8	0.474095	0.237048	+	0.971498i	$0.423820\pi$
$354$	0	0
$355$	4.74819e7i	0.0563286i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.48200e7	0.0853468	0.0426734	−	0.999089i	$-0.486413\pi$
$359$	7.48200e7	0.0853468	0.0426734	+	0.999089i	$0.486413\pi$
$360$	0	0
$361$	−1.03318e9	−1.15584
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.73832e7i	− 0.0294753i
$366$	0	0
$367$	3.50805e8	0.370454	0.185227	−	0.982696i	$-0.440698\pi$
$367$	3.50805e8	0.370454	0.185227	+	0.982696i	$0.440698\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.54908e9i	− 1.57494i
$372$	0	0
$373$	9.55235e8i	0.953080i	0.879153	+	0.476540i	$0.158109\pi$
$373$	9.55235e8i	0.953080i	−0.879153	+	0.476540i	$0.841891\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.97023e8	−0.862201
$378$	0	0
$379$	− 1.31982e9i	− 1.24531i	−0.782495	−	0.622657i	$-0.786053\pi$
$379$	− 1.31982e9i	− 1.24531i	0.782495	−	0.622657i	$-0.213947\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.80265e8	0.163951	0.0819757	−	0.996634i	$-0.473877\pi$
$383$	1.80265e8	0.163951	0.0819757	+	0.996634i	$0.473877\pi$
$384$	0	0
$385$	2.92962e7	0.0261637
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5.78963e8i	− 0.498686i	−0.968415	−	0.249343i	$-0.919785\pi$
$389$	− 5.78963e8i	− 0.498686i	0.968415	−	0.249343i	$-0.0802146\pi$
$390$	0	0
$391$	−9.88513e8	−0.836303
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 7.92903e7i	− 0.0647336i
$396$	0	0
$397$	− 1.02114e9i	− 0.819069i	−0.912295	−	0.409535i	$-0.865691\pi$
$397$	− 1.02114e9i	− 0.819069i	0.912295	−	0.409535i	$-0.134309\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.05638e8	0.236702	0.118351	−	0.992972i	$-0.462239\pi$
$401$	3.05638e8	0.236702	0.118351	+	0.992972i	$0.462239\pi$
$402$	0	0
$403$	2.75386e8i	0.209592i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.66597e7	0.0196009
$408$	0	0
$409$	−3.62609e8	−0.262064	−0.131032	−	0.991378i	$-0.541829\pi$
$409$	−3.62609e8	−0.262064	−0.131032	+	0.991378i	$0.541829\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.23001e9i	2.25620i
$414$	0	0
$415$	9.37864e7	0.0644127
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.59204e9i	− 1.05731i	−0.848836	−	0.528657i	$-0.822696\pi$
$419$	− 1.59204e9i	− 1.05731i	0.848836	−	0.528657i	$-0.177304\pi$
$420$	0	0
$421$	− 5.67404e8i	− 0.370600i	−0.982682	−	0.185300i	$-0.940674\pi$
$421$	− 5.67404e8i	− 0.370600i	0.982682	−	0.185300i	$-0.0593257\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.87588e8	0.624043
$426$	0	0
$427$	3.90705e8i	0.242857i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.60448e9	0.965304	0.482652	−	0.875812i	$-0.339674\pi$
$431$	1.60448e9	0.965304	0.482652	+	0.875812i	$0.339674\pi$
$432$	0	0
$433$	−1.07590e9	−0.636892	−0.318446	−	0.947941i	$-0.603161\pi$
$433$	−1.07590e9	−0.636892	−0.318446	+	0.947941i	$0.603161\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.40947e9i	− 1.95434i
$438$	0	0
$439$	−2.44305e9	−1.37818	−0.689091	−	0.724675i	$-0.741990\pi$
$439$	−2.44305e9	−1.37818	−0.689091	+	0.724675i	$0.741990\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.39191e8i	0.294666i	0.989087	+	0.147333i	$0.0470689\pi$
$443$	5.39191e8i	0.294666i	−0.989087	+	0.147333i	$0.952931\pi$
$444$	0	0
$445$	− 2.79785e8i	− 0.150510i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.64716e9	1.38012	0.690061	−	0.723752i	$-0.257584\pi$
$449$	2.64716e9	1.38012	0.690061	+	0.723752i	$0.257584\pi$
$450$	0	0
$451$	− 1.55191e8i	− 0.0796615i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.11273e8	0.105149
$456$	0	0
$457$	1.19757e9	0.586941	0.293470	−	0.955968i	$-0.405190\pi$
$457$	1.19757e9	0.586941	0.293470	+	0.955968i	$0.405190\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.44286e8i	0.401362i	0.979657	+	0.200681i	$0.0643155\pi$
$461$	8.44286e8i	0.401362i	−0.979657	+	0.200681i	$0.935685\pi$
$462$	0	0
$463$	4.02775e8	0.188595	0.0942973	−	0.995544i	$-0.469940\pi$
$463$	4.02775e8	0.188595	0.0942973	+	0.995544i	$0.469940\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.09444e9i	1.86031i	0.367168	+	0.930155i	$0.380328\pi$
$467$	4.09444e9i	1.86031i	−0.367168	+	0.930155i	$0.619672\pi$
$468$	0	0
$469$	6.25539e9i	2.79994i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.97212e8	0.302936
$474$	0	0
$475$	3.40628e9i	1.45832i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.63610e9	−0.680198	−0.340099	−	0.940390i	$-0.610461\pi$
$479$	−1.63610e9	−0.680198	−0.340099	+	0.940390i	$0.610461\pi$
$480$	0	0
$481$	1.92259e8	0.0787734
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.35307e8i	0.0936568i
$486$	0	0
$487$	9.60936e8	0.377002	0.188501	−	0.982073i	$-0.439637\pi$
$487$	9.60936e8	0.377002	0.188501	+	0.982073i	$0.439637\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.59019e8i	0.327505i	0.986501	+	0.163752i	$0.0523598\pi$
$491$	8.59019e8i	0.327505i	−0.986501	+	0.163752i	$0.947640\pi$
$492$	0	0
$493$	1.92535e9i	0.723678i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.19167e9	−1.16619
$498$	0	0
$499$	2.55267e9i	0.919692i	0.887999	+	0.459846i	$0.152095\pi$
$499$	2.55267e9i	0.919692i	−0.887999	+	0.459846i	$0.847905\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.78799e8	0.307894	0.153947	−	0.988079i	$-0.450801\pi$
$503$	8.78799e8	0.307894	0.153947	+	0.988079i	$0.450801\pi$
$504$	0	0
$505$	−6.42514e6	−0.00222005
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.06476e9i	0.357882i	0.983860	+	0.178941i	$0.0572672\pi$
$509$	1.06476e9i	0.357882i	−0.983860	+	0.178941i	$0.942733\pi$
$510$	0	0
$511$	1.84066e9	0.610239
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.32966e8i	0.107418i
$516$	0	0
$517$	− 1.01414e9i	− 0.322761i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.73221e9	0.536623	0.268312	−	0.963332i	$-0.413534\pi$
$521$	1.73221e9	0.536623	0.268312	+	0.963332i	$0.413534\pi$
$522$	0	0
$523$	3.62695e9i	1.10863i	0.832307	+	0.554314i	$0.187019\pi$
$523$	3.62695e9i	1.10863i	−0.832307	+	0.554314i	$0.812981\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.91082e8	0.175918
$528$	0	0
$529$	2.62743e9	0.771679
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.11918e9i	− 0.320150i
$534$	0	0
$535$	−1.82943e8	−0.0516509
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.29209e9i	0.355412i
$540$	0	0
$541$	− 5.49767e9i	− 1.49275i	−0.665524	−	0.746377i	$-0.731792\pi$
$541$	− 5.49767e9i	− 1.49275i	0.665524	−	0.746377i	$-0.268208\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.26332e8	0.0598904
$546$	0	0
$547$	− 1.46920e9i	− 0.383819i	−0.981413	−	0.191909i	$-0.938532\pi$
$547$	− 1.46920e9i	− 0.383819i	0.981413	−	0.191909i	$-0.0614680\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.64069e9	−1.69115
$552$	0	0
$553$	5.32978e9	1.34020
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.20902e8i	− 0.127721i	−0.997959	−	0.0638606i	$-0.979659\pi$
$557$	− 5.20902e8i	− 0.127721i	0.997959	−	0.0638606i	$-0.0203413\pi$
$558$	0	0
$559$	5.02802e9	1.21746
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.40447e9i	0.804026i	0.915634	+	0.402013i	$0.131689\pi$
$563$	3.40447e9i	0.804026i	−0.915634	+	0.402013i	$0.868311\pi$
$564$	0	0
$565$	9.71742e7i	0.0226663i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.26573e9	0.743168	0.371584	−	0.928399i	$-0.378815\pi$
$569$	3.26573e9	0.743168	0.371584	+	0.928399i	$0.378815\pi$
$570$	0	0
$571$	6.34353e8i	0.142595i	0.997455	+	0.0712975i	$0.0227140\pi$
$571$	6.34353e8i	0.142595i	−0.997455	+	0.0712975i	$0.977286\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.02661e9	−1.32201
$576$	0	0
$577$	−3.49229e9	−0.756823	−0.378412	−	0.925637i	$-0.623530\pi$
$577$	−3.49229e9	−0.756823	−0.378412	+	0.925637i	$0.623530\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.30419e9i	1.33356i
$582$	0	0
$583$	8.23055e8	0.172024
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.78287e8i	− 0.0567884i	−0.999597	−	0.0283942i	$-0.990961\pi$
$587$	− 2.78287e8i	− 0.0567884i	0.999597	−	0.0283942i	$-0.00903937\pi$
$588$	0	0
$589$	2.03869e9i	0.411101i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.01241e9	−1.18401	−0.592007	−	0.805933i	$-0.701664\pi$
$593$	−6.01241e9	−1.18401	−0.592007	+	0.805933i	$0.701664\pi$
$594$	0	0
$595$	− 4.53471e8i	− 0.0882552i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.23203e8	0.0804552	0.0402276	−	0.999191i	$-0.487192\pi$
$599$	4.23203e8	0.0804552	0.0402276	+	0.999191i	$0.487192\pi$
$600$	0	0
$601$	−1.35293e9	−0.254222	−0.127111	−	0.991888i	$-0.540571\pi$
$601$	−1.35293e9	−0.254222	−0.127111	+	0.991888i	$0.540571\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.33084e8i	− 0.0795113i
$606$	0	0
$607$	−8.20042e9	−1.48825	−0.744124	−	0.668041i	$-0.767133\pi$
$607$	−8.20042e9	−1.48825	−0.744124	+	0.668041i	$0.767133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 7.31358e9i	− 1.29714i
$612$	0	0
$613$	8.80586e9i	1.54405i	0.635595	+	0.772023i	$0.280755\pi$
$613$	8.80586e9i	1.54405i	−0.635595	+	0.772023i	$0.719245\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.80934e9	0.310115	0.155057	−	0.987905i	$-0.450444\pi$
$617$	1.80934e9	0.310115	0.155057	+	0.987905i	$0.450444\pi$
$618$	0	0
$619$	− 4.88517e9i	− 0.827871i	−0.910306	−	0.413936i	$-0.864154\pi$
$619$	− 4.88517e9i	− 0.827871i	0.910306	−	0.413936i	$-0.135846\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.88067e10	3.11605
$624$	0	0
$625$	5.97957e9	0.979692
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.12661e8i	− 0.0661175i
$630$	0	0
$631$	6.54396e9	1.03690	0.518451	−	0.855107i	$-0.326509\pi$
$631$	6.54396e9	1.03690	0.518451	+	0.855107i	$0.326509\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.74928e7i	0.0135601i
$636$	0	0
$637$	9.31805e9i	1.42836i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.73336e9	−0.859817	−0.429909	−	0.902872i	$-0.641454\pi$
$641$	−5.73336e9	−0.859817	−0.429909	+	0.902872i	$0.641454\pi$
$642$	0	0
$643$	1.01094e10i	1.49964i	0.661642	+	0.749820i	$0.269860\pi$
$643$	1.01094e10i	1.49964i	−0.661642	+	0.749820i	$0.730140\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.68401e8	−0.111538	−0.0557690	−	0.998444i	$-0.517761\pi$
$647$	−7.68401e8	−0.111538	−0.0557690	+	0.998444i	$0.517761\pi$
$648$	0	0
$649$	−1.71617e9	−0.246436
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.16051e9i	1.00635i	0.864185	+	0.503174i	$0.167834\pi$
$653$	7.16051e9i	1.00635i	−0.864185	+	0.503174i	$0.832166\pi$
$654$	0	0
$655$	8.14846e8	0.113300
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.61521e8i	0.0219851i	0.999940	+	0.0109926i	$0.00349911\pi$
$659$	1.61521e8i	0.0219851i	−0.999940	+	0.0109926i	$0.996501\pi$
$660$	0	0
$661$	− 8.84460e9i	− 1.19117i	−0.803293	−	0.595584i	$-0.796921\pi$
$661$	− 8.84460e9i	− 1.19117i	0.803293	−	0.595584i	$-0.203079\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.56406e9	0.206242
$666$	0	0
$667$	− 1.17492e10i	− 1.53309i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.07590e8	−0.0265263
$672$	0	0
$673$	−1.05989e10	−1.34032	−0.670159	−	0.742217i	$-0.733774\pi$
$673$	−1.05989e10	−1.34032	−0.670159	+	0.742217i	$0.733774\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.12081e10i	− 1.38826i	−0.719849	−	0.694131i	$-0.755789\pi$
$677$	− 1.12081e10i	− 1.38826i	0.719849	−	0.694131i	$-0.244211\pi$
$678$	0	0
$679$	−1.58170e10	−1.93901
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.01055e9i	− 0.241458i	−0.992686	−	0.120729i	$-0.961477\pi$
$683$	− 2.01055e9i	− 0.241458i	0.992686	−	0.120729i	$-0.0385232\pi$
$684$	0	0
$685$	− 6.66493e8i	− 0.0792280i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.93555e9	0.691343
$690$	0	0
$691$	− 6.29632e9i	− 0.725962i	−0.931797	−	0.362981i	$-0.881759\pi$
$691$	− 6.29632e9i	− 0.725962i	0.931797	−	0.362981i	$-0.118241\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.96353e8	0.0673840
$696$	0	0
$697$	−2.40217e9	−0.268714
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.01507e10i	− 1.11297i	−0.830859	−	0.556483i	$-0.812150\pi$
$701$	− 1.01507e10i	− 1.11297i	0.830859	−	0.556483i	$-0.187850\pi$
$702$	0	0
$703$	1.42330e9	0.154509
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.31889e8i	− 0.0459625i
$708$	0	0
$709$	− 1.54442e10i	− 1.62743i	−0.581261	−	0.813717i	$-0.697441\pi$
$709$	− 1.54442e10i	− 1.62743i	0.581261	−	0.813717i	$-0.302559\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.60699e9	−0.372676
$714$	0	0
$715$	1.12254e8i	0.0114849i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.84038e10	1.84653	0.923263	−	0.384168i	$-0.125512\pi$
$719$	1.84038e10	1.84653	0.923263	+	0.384168i	$0.125512\pi$
$720$	0	0
$721$	−2.23815e10	−2.22390
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.17382e10i	1.14398i
$726$	0	0
$727$	3.35574e9	0.323905	0.161953	−	0.986799i	$-0.448221\pi$
$727$	3.35574e9	0.323905	0.161953	+	0.986799i	$0.448221\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.07920e10i	− 1.02186i
$732$	0	0
$733$	− 1.39912e10i	− 1.31217i	−0.754687	−	0.656084i	$-0.772211\pi$
$733$	− 1.39912e10i	− 1.31217i	0.754687	−	0.656084i	$-0.227789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.32362e9	−0.305826
$738$	0	0
$739$	− 1.21885e10i	− 1.11095i	−0.831533	−	0.555476i	$-0.812536\pi$
$739$	− 1.21885e10i	− 1.11095i	0.831533	−	0.555476i	$-0.187464\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.72303e10	−1.54111	−0.770553	−	0.637376i	$-0.780020\pi$
$743$	−1.72303e10	−1.54111	−0.770553	+	0.637376i	$0.780020\pi$
$744$	0	0
$745$	1.62248e9	0.143758
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.22972e10i	− 1.06935i
$750$	0	0
$751$	1.15192e10	0.992394	0.496197	−	0.868210i	$-0.334729\pi$
$751$	1.15192e10	0.992394	0.496197	+	0.868210i	$0.334729\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.09136e9i	− 0.0922896i
$756$	0	0
$757$	− 1.10168e10i	− 0.923042i	−0.887129	−	0.461521i	$-0.847304\pi$
$757$	− 1.10168e10i	− 0.923042i	0.887129	−	0.461521i	$-0.152696\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.71338e8	−0.0469945	−0.0234972	−	0.999724i	$-0.507480\pi$
$761$	−5.71338e8	−0.0469945	−0.0234972	+	0.999724i	$0.507480\pi$
$762$	0	0
$763$	1.52137e10i	1.23993i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.23763e10	−0.990394
$768$	0	0
$769$	1.72893e10	1.37099	0.685495	−	0.728077i	$-0.259586\pi$
$769$	1.72893e10	1.37099	0.685495	+	0.728077i	$0.259586\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.59365e9i	0.435579i	0.975996	+	0.217790i	$0.0698847\pi$
$773$	5.59365e9i	0.435579i	−0.975996	+	0.217790i	$0.930115\pi$
$774$	0	0
$775$	3.60362e9	0.278088
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.28530e9i	− 0.627953i
$780$	0	0
$781$	− 1.69580e9i	− 0.127378i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.46808e8	0.0329668
$786$	0	0
$787$	− 1.40947e10i	− 1.03073i	−0.856970	−	0.515366i	$-0.827656\pi$
$787$	− 1.40947e10i	− 1.03073i	0.856970	−	0.515366i	$-0.172344\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.53191e9	−0.469270
$792$	0	0
$793$	−1.49705e9	−0.106606
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8.89776e9i	− 0.622553i	−0.950319	−	0.311277i	$-0.899243\pi$
$797$	− 8.89776e9i	− 0.622553i	0.950319	−	0.311277i	$-0.100757\pi$
$798$	0	0
$799$	−1.56977e10	−1.08874
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.77979e8i	0.0666538i
$804$	0	0
$805$	2.76724e9i	0.186966i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.43326e9	−0.294377	−0.147188	−	0.989108i	$-0.547022\pi$
$809$	−4.43326e9	−0.294377	−0.147188	+	0.989108i	$0.547022\pi$
$810$	0	0
$811$	− 1.27951e9i	− 0.0842308i	−0.999113	−	0.0421154i	$-0.986590\pi$
$811$	− 1.27951e9i	− 0.0842308i	0.999113	−	0.0421154i	$-0.0134097\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.49806e9	0.0969345
$816$	0	0
$817$	3.72226e10	2.38797
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.14407e9i	0.198286i	0.995073	+	0.0991429i	$0.0316101\pi$
$821$	3.14407e9i	0.198286i	−0.995073	+	0.0991429i	$0.968390\pi$
$822$	0	0
$823$	−1.15739e10	−0.723735	−0.361868	−	0.932229i	$-0.617861\pi$
$823$	−1.15739e10	−0.723735	−0.361868	+	0.932229i	$0.617861\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.09932e10i	1.90545i	0.303834	+	0.952725i	$0.401733\pi$
$827$	3.09932e10i	1.90545i	−0.303834	+	0.952725i	$0.598267\pi$
$828$	0	0
$829$	3.22167e10i	1.96399i	0.188903	+	0.981996i	$0.439507\pi$
$829$	3.22167e10i	1.96399i	−0.188903	+	0.981996i	$0.560493\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00001e10	1.19887
$834$	0	0
$835$	− 8.93571e8i	− 0.0531161i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.00994e10	−1.17494	−0.587470	−	0.809246i	$-0.699876\pi$
$839$	−2.00994e10	−1.17494	−0.587470	+	0.809246i	$0.699876\pi$
$840$	0	0
$841$	−5.63423e9	−0.326624
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 6.35121e8i	− 0.0362125i
$846$	0	0
$847$	2.91113e10	1.64615
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.51820e9i	0.140067i
$852$	0	0
$853$	− 2.34608e10i	− 1.29426i	−0.762379	−	0.647131i	$-0.775969\pi$
$853$	− 2.34608e10i	− 1.29426i	0.762379	−	0.647131i	$-0.224031\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.17254e10	−1.72176	−0.860882	−	0.508804i	$-0.830088\pi$
$857$	−3.17254e10	−1.72176	−0.860882	+	0.508804i	$0.830088\pi$
$858$	0	0
$859$	− 2.50832e10i	− 1.35023i	−0.737714	−	0.675114i	$-0.764095\pi$
$859$	− 2.50832e10i	− 1.35023i	0.737714	−	0.675114i	$-0.235905\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.09078e9	−0.428502	−0.214251	−	0.976779i	$-0.568731\pi$
$863$	−8.09078e9	−0.428502	−0.214251	+	0.976779i	$0.568731\pi$
$864$	0	0
$865$	2.39516e9	0.125828
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.83182e9i	0.146385i
$870$	0	0
$871$	−2.39686e10	−1.22908
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 5.54819e9i	− 0.279978i
$876$	0	0
$877$	− 1.60833e9i	− 0.0805152i	−0.999189	−	0.0402576i	$-0.987182\pi$
$877$	− 1.60833e9i	− 0.0805152i	0.999189	−	0.0402576i	$-0.0128179\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.47631e10	1.22008	0.610041	−	0.792370i	$-0.291153\pi$
$881$	2.47631e10	1.22008	0.610041	+	0.792370i	$0.291153\pi$
$882$	0	0
$883$	− 1.05695e10i	− 0.516643i	−0.966059	−	0.258322i	$-0.916831\pi$
$883$	− 1.05695e10i	− 0.516643i	0.966059	−	0.258322i	$-0.0831694\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.37481e10	−0.661472	−0.330736	−	0.943723i	$-0.607297\pi$
$887$	−1.37481e10	−0.661472	−0.330736	+	0.943723i	$0.607297\pi$
$888$	0	0
$889$	−5.88114e9	−0.280741
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.41427e10i	− 2.54425i
$894$	0	0
$895$	−2.11806e9	−0.0987545
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.02542e9i	0.322488i
$900$	0	0
$901$	− 1.27399e10i	− 0.580270i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.72386e8	−0.0211849
$906$	0	0
$907$	7.78327e9i	0.346367i	0.984890	+	0.173184i	$0.0554054\pi$
$907$	7.78327e9i	0.346367i	−0.984890	+	0.173184i	$0.944595\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.79370e10	1.22424	0.612118	−	0.790767i	$-0.290318\pi$
$911$	2.79370e10	1.22424	0.612118	+	0.790767i	$0.290318\pi$
$912$	0	0
$913$	−3.34954e9	−0.145659
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.47728e10i	2.34570i
$918$	0	0
$919$	−2.31093e10	−0.982161	−0.491080	−	0.871114i	$-0.663398\pi$
$919$	−2.31093e10	−0.982161	−0.491080	+	0.871114i	$0.663398\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.22294e10i	− 0.511917i
$924$	0	0
$925$	− 2.51585e9i	− 0.104517i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.11395e10	0.865046	0.432523	−	0.901623i	$-0.357623\pi$
$929$	2.11395e10	0.865046	0.432523	+	0.901623i	$0.357623\pi$
$930$	0	0
$931$	6.89819e10i	2.80163i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.40939e8	0.00963975
$936$	0	0
$937$	−8.67640e9	−0.344549	−0.172275	−	0.985049i	$-0.555112\pi$
$937$	−8.67640e9	−0.344549	−0.172275	+	0.985049i	$0.555112\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.27192e9i	− 0.0497617i	−0.999690	−	0.0248809i	$-0.992079\pi$
$941$	− 1.27192e9i	− 0.0497617i	0.999690	−	0.0248809i	$-0.00792064\pi$
$942$	0	0
$943$	1.46589e10	0.569260
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.80803e10i	− 1.45705i	−0.685017	−	0.728527i	$-0.740205\pi$
$947$	− 3.80803e10i	− 1.45705i	0.685017	−	0.728527i	$-0.259795\pi$
$948$	0	0
$949$	7.05280e9i	0.267873i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.08569e10	0.780591	0.390296	−	0.920690i	$-0.372373\pi$
$953$	2.08569e10	0.780591	0.390296	+	0.920690i	$0.372373\pi$
$954$	0	0
$955$	1.14797e9i	0.0426500i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.48007e10	1.64029
$960$	0	0
$961$	−2.53558e10	−0.921607
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.98622e8i	0.0357730i
$966$	0	0
$967$	2.33238e10	0.829482	0.414741	−	0.909940i	$-0.363872\pi$
$967$	2.33238e10	0.829482	0.414741	+	0.909940i	$0.363872\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.90339e10i	1.01774i	0.860842	+	0.508872i	$0.169937\pi$
$971$	2.90339e10i	1.01774i	−0.860842	+	0.508872i	$0.830063\pi$
$972$	0	0
$973$	4.00860e10i	1.39508i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.99904e10	−1.37191	−0.685954	−	0.727645i	$-0.740615\pi$
$977$	−3.99904e10	−1.37191	−0.685954	+	0.727645i	$0.740615\pi$
$978$	0	0
$979$	9.99240e9i	0.340354i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.32451e9	−0.0444751	−0.0222376	−	0.999753i	$-0.507079\pi$
$983$	−1.32451e9	−0.0444751	−0.0222376	+	0.999753i	$0.507079\pi$
$984$	0	0
$985$	−4.00048e9	−0.133378
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.58567e10i	2.16478i
$990$	0	0
$991$	3.64267e10	1.18895	0.594473	−	0.804116i	$-0.297361\pi$
$991$	3.64267e10	1.18895	0.594473	+	0.804116i	$0.297361\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5.12910e8i	− 0.0165067i
$996$	0	0
$997$	2.99132e10i	0.955940i	0.878376	+	0.477970i	$0.158627\pi$
$997$	2.99132e10i	0.955940i	−0.878376	+	0.477970i	$0.841373\pi$
$998$	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.145.7		14
3.2	odd	2		96.8.d.a.49.4		14
4.3	odd	2		72.8.d.d.37.6		14
8.3	odd	2		72.8.d.d.37.5		14
8.5	even	2	inner	288.8.d.d.145.8		14
12.11	even	2		24.8.d.a.13.9	✓	14
24.5	odd	2		96.8.d.a.49.11		14
24.11	even	2		24.8.d.a.13.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.d.a.13.9	✓	14	12.11	even	2
24.8.d.a.13.10	yes	14	24.11	even	2
72.8.d.d.37.5		14	8.3	odd	2
72.8.d.d.37.6		14	4.3	odd	2
96.8.d.a.49.4		14	3.2	odd	2
96.8.d.a.49.11		14	24.5	odd	2
288.8.d.d.145.7		14	1.1	even	1	trivial
288.8.d.d.145.8		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)

\(f(q)\)	\(=\)	\(q-23.0228i q^{5} +1547.56 q^{7} +O(q^{10})\)	\(q-23.0228i q^{5} +1547.56 q^{7} +822.251i q^{11} +5929.75i q^{13} +12727.5 q^{17} +43898.2i q^{19} -77667.6 q^{23} +77594.9 q^{25} +151275. i q^{29} +46441.4 q^{31} -35629.3i q^{35} -32422.8i q^{37} -188739. q^{41} -847931. i q^{43} -1.23337e6 q^{47} +1.57141e6 q^{49} -1.00098e6i q^{53} +18930.6 q^{55} +2.08716e6i q^{59} +252465. i q^{61} +136520. q^{65} +4.04209e6i q^{67} -2.06238e6 q^{71} +1.18939e6 q^{73} +1.27249e6i q^{77} +3.44398e6 q^{79} +4.07362e6i q^{83} -293023. i q^{85} +1.21525e7 q^{89} +9.17666e6i q^{91} +1.01066e6 q^{95} -1.02206e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 1372 q^{7}+O(q^{10}) \) 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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