LMFDB - Embedded newform 8496.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8496,2,Mod(1,8496)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8496 = 2^{4} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8496.a (trivial)

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:	yes
Analytic conductor:	$67.8409015573$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	8496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.357926 q^{5} -5.11491 q^{7} +O(q^{10})$	$q+0.357926 q^{5} -5.11491 q^{7} -4.58774 q^{11} -2.58774 q^{13} -2.18869 q^{17} -0.527166 q^{19} -5.70265 q^{23} -4.87189 q^{25} -2.06058 q^{29} -5.83076 q^{31} -1.83076 q^{35} -7.70265 q^{37} -2.39905 q^{41} -7.15604 q^{43} +8.77643 q^{47} +19.1623 q^{49} +8.10170 q^{53} -1.64207 q^{55} +1.00000 q^{59} +9.00624 q^{61} -0.926221 q^{65} -14.4791 q^{67} +4.12811 q^{71} -11.2361 q^{73} +23.4659 q^{77} +3.87189 q^{79} -0.737534 q^{83} -0.783389 q^{85} +8.54661 q^{89} +13.2361 q^{91} -0.188687 q^{95} -4.10170 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} - 9 q^{7}+O(q^{10}) $ 3 * q + 2 * q^5 - 9 * q^7	$ 3 q + 2 q^{5} - 9 q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 7 q^{19} + q^{23} - q^{25} + 11 q^{29} - 13 q^{31} - q^{35} - 5 q^{37} + q^{41} - 6 q^{43} + 11 q^{47} + 14 q^{49} - 2 q^{53} - 4 q^{55} + 3 q^{59} - q^{61} - 10 q^{67} + 26 q^{71} + 7 q^{73} + 17 q^{77} - 2 q^{79} - 3 q^{83} - 35 q^{85} + 23 q^{89} - q^{91} + 3 q^{95} + 14 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 - 9 * q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 7 * q^19 + q^23 - q^25 + 11 * q^29 - 13 * q^31 - q^35 - 5 * q^37 + q^41 - 6 * q^43 + 11 * q^47 + 14 * q^49 - 2 * q^53 - 4 * q^55 + 3 * q^59 - q^61 - 10 * q^67 + 26 * q^71 + 7 * q^73 + 17 * q^77 - 2 * q^79 - 3 * q^83 - 35 * q^85 + 23 * q^89 - q^91 + 3 * q^95 + 14 * q^97

Display 100 coefficients 10 coefficients

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
$5$	0.357926	0.160070	0.0800348	−	0.996792i	$-0.474497\pi$
$5$	0.357926	0.160070	0.0800348	+	0.996792i	$0.474497\pi$
$6$	0	0
$7$	−5.11491	−1.93325	−0.966627	−	0.256189i	$-0.917533\pi$
$7$	−5.11491	−1.93325	−0.966627	+	0.256189i	$0.917533\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.58774	−1.38326	−0.691628	−	0.722254i	$-0.743106\pi$
$11$	−4.58774	−1.38326	−0.691628	+	0.722254i	$0.743106\pi$
$12$	0	0
$13$	−2.58774	−0.717710	−0.358855	−	0.933393i	$-0.616833\pi$
$13$	−2.58774	−0.717710	−0.358855	+	0.933393i	$0.616833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.18869	−0.530834	−0.265417	−	0.964134i	$-0.585510\pi$
$17$	−2.18869	−0.530834	−0.265417	+	0.964134i	$0.585510\pi$
$18$	0	0
$19$	−0.527166	−0.120940	−0.0604701	−	0.998170i	$-0.519260\pi$
$19$	−0.527166	−0.120940	−0.0604701	+	0.998170i	$0.519260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.70265	−1.18908	−0.594542	−	0.804064i	$-0.702667\pi$
$23$	−5.70265	−1.18908	−0.594542	+	0.804064i	$0.702667\pi$
$24$	0	0
$25$	−4.87189	−0.974378
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.06058	−0.382639	−0.191320	−	0.981528i	$-0.561277\pi$
$29$	−2.06058	−0.382639	−0.191320	+	0.981528i	$0.561277\pi$
$30$	0	0
$31$	−5.83076	−1.04724	−0.523618	−	0.851953i	$-0.675418\pi$
$31$	−5.83076	−1.04724	−0.523618	+	0.851953i	$0.675418\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.83076	−0.309455
$36$	0	0
$37$	−7.70265	−1.26631	−0.633154	−	0.774026i	$-0.718240\pi$
$37$	−7.70265	−1.26631	−0.633154	+	0.774026i	$0.718240\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.39905	−0.374669	−0.187335	−	0.982296i	$-0.559985\pi$
$41$	−2.39905	−0.374669	−0.187335	+	0.982296i	$0.559985\pi$
$42$	0	0
$43$	−7.15604	−1.09129	−0.545643	−	0.838018i	$-0.683714\pi$
$43$	−7.15604	−1.09129	−0.545643	+	0.838018i	$0.683714\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.77643	1.28017	0.640087	−	0.768303i	$-0.278898\pi$
$47$	8.77643	1.28017	0.640087	+	0.768303i	$0.278898\pi$
$48$	0	0
$49$	19.1623	2.73747
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.10170	1.11285	0.556427	−	0.830896i	$-0.312172\pi$
$53$	8.10170	1.11285	0.556427	+	0.830896i	$0.312172\pi$
$54$	0	0
$55$	−1.64207	−0.221417
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	9.00624	1.15313	0.576566	−	0.817051i	$-0.304393\pi$
$61$	9.00624	1.15313	0.576566	+	0.817051i	$0.304393\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.926221	−0.114884
$66$	0	0
$67$	−14.4791	−1.76890	−0.884450	−	0.466634i	$-0.845466\pi$
$67$	−14.4791	−1.76890	−0.884450	+	0.466634i	$0.845466\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.12811	0.489917	0.244958	−	0.969534i	$-0.421226\pi$
$71$	4.12811	0.489917	0.244958	+	0.969534i	$0.421226\pi$
$72$	0	0
$73$	−11.2361	−1.31508	−0.657541	−	0.753419i	$-0.728403\pi$
$73$	−11.2361	−1.31508	−0.657541	+	0.753419i	$0.728403\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.4659	2.67418
$78$	0	0
$79$	3.87189	0.435622	0.217811	−	0.975991i	$-0.430108\pi$
$79$	3.87189	0.435622	0.217811	+	0.975991i	$0.430108\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.737534	−0.0809549	−0.0404775	−	0.999180i	$-0.512888\pi$
$83$	−0.737534	−0.0809549	−0.0404775	+	0.999180i	$0.512888\pi$
$84$	0	0
$85$	−0.783389	−0.0849704
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.54661	0.905939	0.452970	−	0.891526i	$-0.350365\pi$
$89$	8.54661	0.905939	0.452970	+	0.891526i	$0.350365\pi$
$90$	0	0
$91$	13.2361	1.38752
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.188687	−0.0193588
$96$	0	0
$97$	−4.10170	−0.416465	−0.208232	−	0.978079i	$-0.566771\pi$
$97$	−4.10170	−0.416465	−0.208232	+	0.978079i	$0.566771\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.22982	0.818897	0.409449	−	0.912333i	$-0.365721\pi$
$101$	8.22982	0.818897	0.409449	+	0.912333i	$0.365721\pi$
$102$	0	0
$103$	−4.79811	−0.472772	−0.236386	−	0.971659i	$-0.575963\pi$
$103$	−4.79811	−0.472772	−0.236386	+	0.971659i	$0.575963\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.0411	1.35741	0.678704	−	0.734412i	$-0.262542\pi$
$107$	14.0411	1.35741	0.678704	+	0.734412i	$0.262542\pi$
$108$	0	0
$109$	−9.81131	−0.939753	−0.469877	−	0.882732i	$-0.655702\pi$
$109$	−9.81131	−0.939753	−0.469877	+	0.882732i	$0.655702\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.2772	−1.24901	−0.624506	−	0.781020i	$-0.714700\pi$
$113$	−13.2772	−1.24901	−0.624506	+	0.781020i	$0.714700\pi$
$114$	0	0
$115$	−2.04113	−0.190336
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.1949	1.02624
$120$	0	0
$121$	10.0474	0.913397
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.53341	−0.316038
$126$	0	0
$127$	0.587741	0.0521536	0.0260768	−	0.999660i	$-0.491699\pi$
$127$	0.587741	0.0521536	0.0260768	+	0.999660i	$0.491699\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.08226	−0.706150	−0.353075	−	0.935595i	$-0.614864\pi$
$131$	−8.08226	−0.706150	−0.353075	+	0.935595i	$0.614864\pi$
$132$	0	0
$133$	2.69641	0.233808
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.10170	−0.692175	−0.346088	−	0.938202i	$-0.612490\pi$
$137$	−8.10170	−0.692175	−0.346088	+	0.938202i	$0.612490\pi$
$138$	0	0
$139$	−4.96511	−0.421136	−0.210568	−	0.977579i	$-0.567531\pi$
$139$	−4.96511	−0.421136	−0.210568	+	0.977579i	$0.567531\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.8719	0.992777
$144$	0	0
$145$	−0.737534	−0.0612489
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.9325	1.79678	0.898389	−	0.439201i	$-0.144738\pi$
$149$	21.9325	1.79678	0.898389	+	0.439201i	$0.144738\pi$
$150$	0	0
$151$	−1.49228	−0.121440	−0.0607200	−	0.998155i	$-0.519340\pi$
$151$	−1.49228	−0.121440	−0.0607200	+	0.998155i	$0.519340\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.08698	−0.167630
$156$	0	0
$157$	−18.7089	−1.49313	−0.746566	−	0.665311i	$-0.768299\pi$
$157$	−18.7089	−1.49313	−0.746566	+	0.665311i	$0.768299\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	29.1685	2.29880
$162$	0	0
$163$	−3.34472	−0.261979	−0.130989	−	0.991384i	$-0.541815\pi$
$163$	−3.34472	−0.261979	−0.130989	+	0.991384i	$0.541815\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.3991	1.26900	0.634498	−	0.772924i	$-0.281207\pi$
$167$	16.3991	1.26900	0.634498	+	0.772924i	$0.281207\pi$
$168$	0	0
$169$	−6.30359	−0.484892
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−25.2097	−1.91665	−0.958327	−	0.285673i	$-0.907783\pi$
$173$	−25.2097	−1.91665	−0.958327	+	0.285673i	$0.907783\pi$
$174$	0	0
$175$	24.9193	1.88372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.0738	−1.12667	−0.563334	−	0.826230i	$-0.690481\pi$
$179$	−15.0738	−1.12667	−0.563334	+	0.826230i	$0.690481\pi$
$180$	0	0
$181$	14.7764	1.09832	0.549162	−	0.835716i	$-0.314947\pi$
$181$	14.7764	1.09832	0.549162	+	0.835716i	$0.314947\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.75698	−0.202697
$186$	0	0
$187$	10.0411	0.734280
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.9930	−1.44665	−0.723323	−	0.690510i	$-0.757386\pi$
$191$	−19.9930	−1.44665	−0.723323	+	0.690510i	$0.757386\pi$
$192$	0	0
$193$	5.28415	0.380361	0.190181	−	0.981749i	$-0.439093\pi$
$193$	5.28415	0.380361	0.190181	+	0.981749i	$0.439093\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.12115	−0.436114	−0.218057	−	0.975936i	$-0.569972\pi$
$197$	−6.12115	−0.436114	−0.218057	+	0.975936i	$0.569972\pi$
$198$	0	0
$199$	18.7500	1.32915	0.664577	−	0.747220i	$-0.268612\pi$
$199$	18.7500	1.32915	0.664577	+	0.747220i	$0.268612\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.5397	0.739739
$204$	0	0
$205$	−0.858685	−0.0599732
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.41850	0.167291
$210$	0	0
$211$	−5.32528	−0.366607	−0.183304	−	0.983056i	$-0.558679\pi$
$211$	−5.32528	−0.366607	−0.183304	+	0.983056i	$0.558679\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.56133	−0.174682
$216$	0	0
$217$	29.8238	2.02457
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.66376	0.380985
$222$	0	0
$223$	11.0279	0.738484	0.369242	−	0.929333i	$-0.379617\pi$
$223$	11.0279	0.738484	0.369242	+	0.929333i	$0.379617\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.1344	−0.739013	−0.369507	−	0.929228i	$-0.620473\pi$
$227$	−11.1344	−0.739013	−0.369507	+	0.929228i	$0.620473\pi$
$228$	0	0
$229$	25.4270	1.68026	0.840131	−	0.542383i	$-0.182478\pi$
$229$	25.4270	1.68026	0.840131	+	0.542383i	$0.182478\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.1212	0.663059	0.331529	−	0.943445i	$-0.392435\pi$
$233$	10.1212	0.663059	0.331529	+	0.943445i	$0.392435\pi$
$234$	0	0
$235$	3.14132	0.204917
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.9387	0.707566	0.353783	−	0.935328i	$-0.384895\pi$
$239$	10.9387	0.707566	0.353783	+	0.935328i	$0.384895\pi$
$240$	0	0
$241$	−10.2034	−0.657259	−0.328630	−	0.944459i	$-0.606587\pi$
$241$	−10.2034	−0.657259	−0.328630	+	0.944459i	$0.606587\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.85868	0.438185
$246$	0	0
$247$	1.36417	0.0868000
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.2104	0.770712	0.385356	−	0.922768i	$-0.374079\pi$
$251$	12.2104	0.770712	0.385356	+	0.922768i	$0.374079\pi$
$252$	0	0
$253$	26.1623	1.64481
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.52645	0.594244	0.297122	−	0.954840i	$-0.403973\pi$
$257$	9.52645	0.594244	0.297122	+	0.954840i	$0.403973\pi$
$258$	0	0
$259$	39.3983	2.44809
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.0995	−0.992736	−0.496368	−	0.868112i	$-0.665333\pi$
$263$	−16.0995	−0.992736	−0.496368	+	0.868112i	$0.665333\pi$
$264$	0	0
$265$	2.89981	0.178134
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−28.3051	−1.72579	−0.862897	−	0.505381i	$-0.831352\pi$
$269$	−28.3051	−1.72579	−0.862897	+	0.505381i	$0.831352\pi$
$270$	0	0
$271$	−1.38585	−0.0841845	−0.0420922	−	0.999114i	$-0.513402\pi$
$271$	−1.38585	−0.0841845	−0.0420922	+	0.999114i	$0.513402\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.3510	1.34781
$276$	0	0
$277$	−1.15604	−0.0694595	−0.0347297	−	0.999397i	$-0.511057\pi$
$277$	−1.15604	−0.0694595	−0.0347297	+	0.999397i	$0.511057\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.5140	1.16411	0.582053	−	0.813151i	$-0.302250\pi$
$281$	19.5140	1.16411	0.582053	+	0.813151i	$0.302250\pi$
$282$	0	0
$283$	2.04113	0.121332	0.0606662	−	0.998158i	$-0.480677\pi$
$283$	2.04113	0.121332	0.0606662	+	0.998158i	$0.480677\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.2709	0.724331
$288$	0	0
$289$	−12.2097	−0.718215
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.73057	0.159522	0.0797609	−	0.996814i	$-0.474584\pi$
$293$	2.73057	0.159522	0.0797609	+	0.996814i	$0.474584\pi$
$294$	0	0
$295$	0.357926	0.0208393
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.7570	0.853418
$300$	0	0
$301$	36.6025	2.10973
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.22357	0.184581
$306$	0	0
$307$	28.2423	1.61187	0.805937	−	0.592002i	$-0.201662\pi$
$307$	28.2423	1.61187	0.805937	+	0.592002i	$0.201662\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.5272	1.16399	0.581994	−	0.813193i	$-0.302273\pi$
$311$	20.5272	1.16399	0.581994	+	0.813193i	$0.302273\pi$
$312$	0	0
$313$	−6.86565	−0.388069	−0.194035	−	0.980995i	$-0.562157\pi$
$313$	−6.86565	−0.388069	−0.194035	+	0.980995i	$0.562157\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.6002	−1.60635	−0.803174	−	0.595744i	$-0.796857\pi$
$317$	−28.6002	−1.60635	−0.803174	+	0.595744i	$0.796857\pi$
$318$	0	0
$319$	9.45339	0.529288
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.15380	0.0641992
$324$	0	0
$325$	12.6072	0.699321
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−44.8906	−2.47490
$330$	0	0
$331$	−21.2966	−1.17057	−0.585284	−	0.810828i	$-0.699017\pi$
$331$	−21.2966	−1.17057	−0.585284	+	0.810828i	$0.699017\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.18244	−0.283147
$336$	0	0
$337$	−11.3253	−0.616927	−0.308464	−	0.951236i	$-0.599815\pi$
$337$	−11.3253	−0.616927	−0.308464	+	0.951236i	$0.599815\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.7500	1.44859
$342$	0	0
$343$	−62.2089	−3.35897
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.58998	0.139037	0.0695186	−	0.997581i	$-0.477854\pi$
$347$	2.58998	0.139037	0.0695186	+	0.997581i	$0.477854\pi$
$348$	0	0
$349$	−26.0319	−1.39346	−0.696729	−	0.717335i	$-0.745362\pi$
$349$	−26.0319	−1.39346	−0.696729	+	0.717335i	$0.745362\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.9325	−0.741550	−0.370775	−	0.928723i	$-0.620908\pi$
$353$	−13.9325	−0.741550	−0.370775	+	0.928723i	$0.620908\pi$
$354$	0	0
$355$	1.47756	0.0784207
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.89134	−0.310933	−0.155466	−	0.987841i	$-0.549688\pi$
$359$	−5.89134	−0.310933	−0.155466	+	0.987841i	$0.549688\pi$
$360$	0	0
$361$	−18.7221	−0.985373
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.02168	−0.210504
$366$	0	0
$367$	0.926221	0.0483483	0.0241742	−	0.999708i	$-0.492304\pi$
$367$	0.926221	0.0483483	0.0241742	+	0.999708i	$0.492304\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−41.4395	−2.15143
$372$	0	0
$373$	10.1498	0.525536	0.262768	−	0.964859i	$-0.415365\pi$
$373$	10.1498	0.525536	0.262768	+	0.964859i	$0.415365\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.33224	0.274624
$378$	0	0
$379$	28.0753	1.44213	0.721066	−	0.692867i	$-0.243653\pi$
$379$	28.0753	1.44213	0.721066	+	0.692867i	$0.243653\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.8176	1.77909	0.889547	−	0.456844i	$-0.151020\pi$
$383$	34.8176	1.77909	0.889547	+	0.456844i	$0.151020\pi$
$384$	0	0
$385$	8.39905	0.428055
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.78267	−0.0903850	−0.0451925	−	0.998978i	$-0.514390\pi$
$389$	−1.78267	−0.0903850	−0.0451925	+	0.998978i	$0.514390\pi$
$390$	0	0
$391$	12.4813	0.631207
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.38585	0.0697297
$396$	0	0
$397$	0.817557	0.0410320	0.0205160	−	0.999790i	$-0.493469\pi$
$397$	0.817557	0.0410320	0.0205160	+	0.999790i	$0.493469\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.35168	0.267250	0.133625	−	0.991032i	$-0.457338\pi$
$401$	5.35168	0.267250	0.133625	+	0.991032i	$0.457338\pi$
$402$	0	0
$403$	15.0885	0.751612
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.3378	1.75163
$408$	0	0
$409$	−38.4068	−1.89909	−0.949547	−	0.313624i	$-0.898457\pi$
$409$	−38.4068	−1.89909	−0.949547	+	0.313624i	$0.898457\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.11491	−0.251688
$414$	0	0
$415$	−0.263983	−0.0129584
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.71585	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$419$	8.71585	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$420$	0	0
$421$	−0.357926	−0.0174443	−0.00872213	−	0.999962i	$-0.502776\pi$
$421$	−0.357926	−0.0174443	−0.00872213	+	0.999962i	$0.502776\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.6630	0.517233
$426$	0	0
$427$	−46.0661	−2.22929
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.2361	−0.637558	−0.318779	−	0.947829i	$-0.603273\pi$
$431$	−13.2361	−0.637558	−0.318779	+	0.947829i	$0.603273\pi$
$432$	0	0
$433$	−27.4031	−1.31691	−0.658454	−	0.752621i	$-0.728789\pi$
$433$	−27.4031	−1.31691	−0.658454	+	0.752621i	$0.728789\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.00624	0.143808
$438$	0	0
$439$	−32.5955	−1.55570	−0.777849	−	0.628451i	$-0.783689\pi$
$439$	−32.5955	−1.55570	−0.777849	+	0.628451i	$0.783689\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.8392	−1.46522	−0.732608	−	0.680651i	$-0.761697\pi$
$443$	−30.8392	−1.46522	−0.732608	+	0.680651i	$0.761697\pi$
$444$	0	0
$445$	3.05906	0.145013
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.0863	−1.65582	−0.827912	−	0.560859i	$-0.810471\pi$
$449$	−35.0863	−1.65582	−0.827912	+	0.560859i	$0.810471\pi$
$450$	0	0
$451$	11.0062	0.518264
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.73753	0.222099
$456$	0	0
$457$	10.6414	0.497782	0.248891	−	0.968532i	$-0.419934\pi$
$457$	10.6414	0.497782	0.248891	+	0.968532i	$0.419934\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.4046	1.32293	0.661467	−	0.749975i	$-0.269934\pi$
$461$	28.4046	1.32293	0.661467	+	0.749975i	$0.269934\pi$
$462$	0	0
$463$	9.58150	0.445290	0.222645	−	0.974900i	$-0.428531\pi$
$463$	9.58150	0.445290	0.222645	+	0.974900i	$0.428531\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.1336	1.20932	0.604660	−	0.796484i	$-0.293309\pi$
$467$	26.1336	1.20932	0.604660	+	0.796484i	$0.293309\pi$
$468$	0	0
$469$	74.0591	3.41973
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.8300	1.50953
$474$	0	0
$475$	2.56829	0.117841
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.5397	−0.847098	−0.423549	−	0.905873i	$-0.639216\pi$
$479$	−18.5397	−0.847098	−0.423549	+	0.905873i	$0.639216\pi$
$480$	0	0
$481$	19.9325	0.908842
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.46811	−0.0666633
$486$	0	0
$487$	21.5676	0.977320	0.488660	−	0.872474i	$-0.337486\pi$
$487$	21.5676	0.977320	0.488660	+	0.872474i	$0.337486\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.4659	−0.517448	−0.258724	−	0.965951i	$-0.583302\pi$
$491$	−11.4659	−0.517448	−0.258724	+	0.965951i	$0.583302\pi$
$492$	0	0
$493$	4.50995	0.203118
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−21.1149	−0.947133
$498$	0	0
$499$	17.4681	0.781980	0.390990	−	0.920395i	$-0.372133\pi$
$499$	17.4681	0.781980	0.390990	+	0.920395i	$0.372133\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.655277	0.0292174	0.0146087	−	0.999893i	$-0.495350\pi$
$503$	0.655277	0.0292174	0.0146087	+	0.999893i	$0.495350\pi$
$504$	0	0
$505$	2.94567	0.131080
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.8168	0.701069	0.350535	−	0.936550i	$-0.386000\pi$
$509$	15.8168	0.701069	0.350535	+	0.936550i	$0.386000\pi$
$510$	0	0
$511$	57.4714	2.54239
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.71737	−0.0756764
$516$	0	0
$517$	−40.2640	−1.77081
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.2470	0.536552	0.268276	−	0.963342i	$-0.413546\pi$
$521$	12.2470	0.536552	0.268276	+	0.963342i	$0.413546\pi$
$522$	0	0
$523$	2.23678	0.0978074	0.0489037	−	0.998803i	$-0.484427\pi$
$523$	2.23678	0.0978074	0.0489037	+	0.998803i	$0.484427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.7617	0.555909
$528$	0	0
$529$	9.52021	0.413922
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.20813	0.268904
$534$	0	0
$535$	5.02569	0.217280
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−87.9116	−3.78662
$540$	0	0
$541$	−11.0863	−0.476636	−0.238318	−	0.971187i	$-0.576596\pi$
$541$	−11.0863	−0.476636	−0.238318	+	0.971187i	$0.576596\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.51173	−0.150426
$546$	0	0
$547$	−12.1887	−0.521151	−0.260575	−	0.965454i	$-0.583912\pi$
$547$	−12.1887	−0.521151	−0.260575	+	0.965454i	$0.583912\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.08627	0.0462765
$552$	0	0
$553$	−19.8044	−0.842167
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.56829	0.363050	0.181525	−	0.983386i	$-0.441897\pi$
$557$	8.56829	0.363050	0.181525	+	0.983386i	$0.441897\pi$
$558$	0	0
$559$	18.5180	0.783227
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.98055	0.420630	0.210315	−	0.977634i	$-0.432551\pi$
$563$	9.98055	0.420630	0.210315	+	0.977634i	$0.432551\pi$
$564$	0	0
$565$	−4.75226	−0.199929
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.1428	−1.93441	−0.967204	−	0.254001i	$-0.918253\pi$
$569$	−46.1428	−1.93441	−0.967204	+	0.254001i	$0.918253\pi$
$570$	0	0
$571$	21.1824	0.886458	0.443229	−	0.896409i	$-0.353833\pi$
$571$	21.1824	0.886458	0.443229	+	0.896409i	$0.353833\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	27.7827	1.15862
$576$	0	0
$577$	40.8859	1.70210	0.851051	−	0.525083i	$-0.175966\pi$
$577$	40.8859	1.70210	0.851051	+	0.525083i	$0.175966\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.77242	0.156506
$582$	0	0
$583$	−37.1685	−1.53936
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.21037	−0.173780	−0.0868902	−	0.996218i	$-0.527693\pi$
$587$	−4.21037	−0.173780	−0.0868902	+	0.996218i	$0.527693\pi$
$588$	0	0
$589$	3.07378	0.126653
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.5202	0.924794	0.462397	−	0.886673i	$-0.346990\pi$
$593$	22.5202	0.924794	0.462397	+	0.886673i	$0.346990\pi$
$594$	0	0
$595$	4.00696	0.164269
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16.7981	−0.686352	−0.343176	−	0.939271i	$-0.611503\pi$
$599$	−16.7981	−0.686352	−0.343176	+	0.939271i	$0.611503\pi$
$600$	0	0
$601$	−17.0955	−0.697338	−0.348669	−	0.937246i	$-0.613366\pi$
$601$	−17.0955	−0.697338	−0.348669	+	0.937246i	$0.613366\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.59622	0.146207
$606$	0	0
$607$	7.53341	0.305772	0.152886	−	0.988244i	$-0.451143\pi$
$607$	7.53341	0.305772	0.152886	+	0.988244i	$0.451143\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.7111	−0.918794
$612$	0	0
$613$	15.1321	0.611181	0.305590	−	0.952163i	$-0.401146\pi$
$613$	15.1321	0.611181	0.305590	+	0.952163i	$0.401146\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.8672	−0.638788	−0.319394	−	0.947622i	$-0.603479\pi$
$617$	−15.8672	−0.638788	−0.319394	+	0.947622i	$0.603479\pi$
$618$	0	0
$619$	−11.0210	−0.442970	−0.221485	−	0.975164i	$-0.571090\pi$
$619$	−11.0210	−0.442970	−0.221485	+	0.975164i	$0.571090\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−43.7151	−1.75141
$624$	0	0
$625$	23.0947	0.923790
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.8587	0.672200
$630$	0	0
$631$	23.1560	0.921827	0.460914	−	0.887445i	$-0.347522\pi$
$631$	23.1560	0.921827	0.460914	+	0.887445i	$0.347522\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.210368	0.00834821
$636$	0	0
$637$	−49.5870	−1.96471
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.5459	−1.56197	−0.780984	−	0.624550i	$-0.785282\pi$
$641$	−39.5459	−1.56197	−0.780984	+	0.624550i	$0.785282\pi$
$642$	0	0
$643$	−18.6219	−0.734376	−0.367188	−	0.930147i	$-0.619680\pi$
$643$	−18.6219	−0.734376	−0.367188	+	0.930147i	$0.619680\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.24525	0.284840	0.142420	−	0.989806i	$-0.454512\pi$
$647$	7.24525	0.284840	0.142420	+	0.989806i	$0.454512\pi$
$648$	0	0
$649$	−4.58774	−0.180085
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.3183	−1.22558	−0.612790	−	0.790246i	$-0.709953\pi$
$653$	−31.3183	−1.22558	−0.612790	+	0.790246i	$0.709953\pi$
$654$	0	0
$655$	−2.89285	−0.113033
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.13587	0.316929	0.158464	−	0.987365i	$-0.449346\pi$
$659$	8.13587	0.316929	0.158464	+	0.987365i	$0.449346\pi$
$660$	0	0
$661$	−9.47979	−0.368721	−0.184361	−	0.982859i	$-0.559021\pi$
$661$	−9.47979	−0.368721	−0.184361	+	0.982859i	$0.559021\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.965115	0.0374255
$666$	0	0
$667$	11.7507	0.454990
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−41.3183	−1.59508
$672$	0	0
$673$	11.4317	0.440660	0.220330	−	0.975425i	$-0.429287\pi$
$673$	11.4317	0.440660	0.220330	+	0.975425i	$0.429287\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.83700	0.262767	0.131384	−	0.991332i	$-0.458058\pi$
$677$	6.83700	0.262767	0.131384	+	0.991332i	$0.458058\pi$
$678$	0	0
$679$	20.9798	0.805132
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.6002	−1.32394	−0.661970	−	0.749530i	$-0.730280\pi$
$683$	−34.6002	−1.32394	−0.661970	+	0.749530i	$0.730280\pi$
$684$	0	0
$685$	−2.89981	−0.110796
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−20.9651	−0.798707
$690$	0	0
$691$	42.8929	1.63172	0.815861	−	0.578249i	$-0.196264\pi$
$691$	42.8929	1.63172	0.815861	+	0.578249i	$0.196264\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.77715	−0.0674110
$696$	0	0
$697$	5.25078	0.198887
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.83228	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$701$	−8.83228	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$702$	0	0
$703$	4.06058	0.153148
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−42.0947	−1.58314
$708$	0	0
$709$	22.7981	0.856201	0.428100	−	0.903731i	$-0.359183\pi$
$709$	22.7981	0.856201	0.428100	+	0.903731i	$0.359183\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	33.2508	1.24525
$714$	0	0
$715$	4.24926	0.158913
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.6887	0.808853	0.404427	−	0.914570i	$-0.367471\pi$
$719$	21.6887	0.808853	0.404427	+	0.914570i	$0.367471\pi$
$720$	0	0
$721$	24.5419	0.913988
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.0389	0.372835
$726$	0	0
$727$	−47.7438	−1.77072	−0.885359	−	0.464907i	$-0.846088\pi$
$727$	−47.7438	−1.77072	−0.885359	+	0.464907i	$0.846088\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.6623	0.579292
$732$	0	0
$733$	37.3983	1.38134	0.690670	−	0.723170i	$-0.257316\pi$
$733$	37.3983	1.38134	0.690670	+	0.723170i	$0.257316\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	66.4263	2.44684
$738$	0	0
$739$	13.9450	0.512973	0.256487	−	0.966548i	$-0.417435\pi$
$739$	13.9450	0.512973	0.256487	+	0.966548i	$0.417435\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.4200	1.73967	0.869836	−	0.493341i	$-0.164225\pi$
$743$	47.4200	1.73967	0.869836	+	0.493341i	$0.164225\pi$
$744$	0	0
$745$	7.85021	0.287609
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−71.8191	−2.62421
$750$	0	0
$751$	−46.8929	−1.71114	−0.855572	−	0.517683i	$-0.826795\pi$
$751$	−46.8929	−1.71114	−0.855572	+	0.517683i	$0.826795\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.534127	−0.0194389
$756$	0	0
$757$	−21.7151	−0.789250	−0.394625	−	0.918842i	$-0.629125\pi$
$757$	−21.7151	−0.789250	−0.394625	+	0.918842i	$0.629125\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.6002	1.32676	0.663379	−	0.748284i	$-0.269122\pi$
$761$	36.6002	1.32676	0.663379	+	0.748284i	$0.269122\pi$
$762$	0	0
$763$	50.1840	1.81678
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.58774	−0.0934379
$768$	0	0
$769$	11.9200	0.429845	0.214923	−	0.976631i	$-0.431050\pi$
$769$	11.9200	0.429845	0.214923	+	0.976631i	$0.431050\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20.3076	−0.730414	−0.365207	−	0.930926i	$-0.619002\pi$
$773$	−20.3076	−0.730414	−0.365207	+	0.930926i	$0.619002\pi$
$774$	0	0
$775$	28.4068	1.02040
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.26470	0.0453126
$780$	0	0
$781$	−18.9387	−0.677680
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.69641	−0.239005
$786$	0	0
$787$	−18.2687	−0.651209	−0.325605	−	0.945506i	$-0.605568\pi$
$787$	−18.2687	−0.651209	−0.325605	+	0.945506i	$0.605568\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	67.9116	2.41466
$792$	0	0
$793$	−23.3058	−0.827614
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.3664	0.756837	0.378418	−	0.925635i	$-0.376468\pi$
$797$	21.3664	0.756837	0.378418	+	0.925635i	$0.376468\pi$
$798$	0	0
$799$	−19.2089	−0.679560
$800$	0	0
$801$	0	0
$802$	0	0
$803$	51.5481	1.81909
$804$	0	0
$805$	10.4402	0.367968
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.7460	0.834865	0.417433	−	0.908708i	$-0.362930\pi$
$809$	23.7460	0.834865	0.417433	+	0.908708i	$0.362930\pi$
$810$	0	0
$811$	−15.3664	−0.539587	−0.269794	−	0.962918i	$-0.586956\pi$
$811$	−15.3664	−0.539587	−0.269794	+	0.962918i	$0.586956\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.19716	−0.0419348
$816$	0	0
$817$	3.77242	0.131980
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.6009	1.03308	0.516540	−	0.856263i	$-0.327220\pi$
$821$	29.6009	1.03308	0.516540	+	0.856263i	$0.327220\pi$
$822$	0	0
$823$	19.6546	0.685115	0.342557	−	0.939497i	$-0.388707\pi$
$823$	19.6546	0.685115	0.342557	+	0.939497i	$0.388707\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.1296	−0.456562	−0.228281	−	0.973595i	$-0.573310\pi$
$827$	−13.1296	−0.456562	−0.228281	+	0.973595i	$0.573310\pi$
$828$	0	0
$829$	−3.69569	−0.128357	−0.0641783	−	0.997938i	$-0.520443\pi$
$829$	−3.69569	−0.128357	−0.0641783	+	0.997938i	$0.520443\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−41.9402	−1.45314
$834$	0	0
$835$	5.86965	0.203128
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.7911	0.717790	0.358895	−	0.933378i	$-0.383154\pi$
$839$	20.7911	0.717790	0.358895	+	0.933378i	$0.383154\pi$
$840$	0	0
$841$	−24.7540	−0.853587
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.25622	−0.0776164
$846$	0	0
$847$	−51.3914	−1.76583
$848$	0	0
$849$	0	0
$850$	0	0
$851$	43.9255	1.50575
$852$	0	0
$853$	17.5162	0.599743	0.299872	−	0.953980i	$-0.403056\pi$
$853$	17.5162	0.599743	0.299872	+	0.953980i	$0.403056\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.0628	−0.548695	−0.274348	−	0.961631i	$-0.588462\pi$
$857$	−16.0628	−0.548695	−0.274348	+	0.961631i	$0.588462\pi$
$858$	0	0
$859$	5.39281	0.184000	0.0920002	−	0.995759i	$-0.470674\pi$
$859$	5.39281	0.184000	0.0920002	+	0.995759i	$0.470674\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.55509	0.257178	0.128589	−	0.991698i	$-0.458955\pi$
$863$	7.55509	0.257178	0.128589	+	0.991698i	$0.458955\pi$
$864$	0	0
$865$	−9.02320	−0.306798
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−17.7632	−0.602576
$870$	0	0
$871$	37.4681	1.26956
$872$	0	0
$873$	0	0
$874$	0	0
$875$	18.0731	0.610981
$876$	0	0
$877$	21.0807	0.711846	0.355923	−	0.934515i	$-0.384167\pi$
$877$	21.0807	0.711846	0.355923	+	0.934515i	$0.384167\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.90677	−0.300077	−0.150038	−	0.988680i	$-0.547940\pi$
$881$	−8.90677	−0.300077	−0.150038	+	0.988680i	$0.547940\pi$
$882$	0	0
$883$	−7.94719	−0.267444	−0.133722	−	0.991019i	$-0.542693\pi$
$883$	−7.94719	−0.267444	−0.133722	+	0.991019i	$0.542693\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.1685	−0.442156	−0.221078	−	0.975256i	$-0.570957\pi$
$887$	−13.1685	−0.442156	−0.221078	+	0.975256i	$0.570957\pi$
$888$	0	0
$889$	−3.00624	−0.100826
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.62664	−0.154824
$894$	0	0
$895$	−5.39530	−0.180345
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0147	0.400713
$900$	0	0
$901$	−17.7321	−0.590742
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.28887	0.175808
$906$	0	0
$907$	23.9497	0.795236	0.397618	−	0.917551i	$-0.369837\pi$
$907$	23.9497	0.795236	0.397618	+	0.917551i	$0.369837\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.8184	−1.55116	−0.775581	−	0.631248i	$-0.782543\pi$
$911$	−46.8184	−1.55116	−0.775581	+	0.631248i	$0.782543\pi$
$912$	0	0
$913$	3.38362	0.111981
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.3400	1.36517
$918$	0	0
$919$	−30.6506	−1.01107	−0.505534	−	0.862807i	$-0.668705\pi$
$919$	−30.6506	−1.01107	−0.505534	+	0.862807i	$0.668705\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.6825	−0.351618
$924$	0	0
$925$	37.5264	1.23386
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.52092	−0.246753	−0.123377	−	0.992360i	$-0.539372\pi$
$929$	−7.52092	−0.246753	−0.123377	+	0.992360i	$0.539372\pi$
$930$	0	0
$931$	−10.1017	−0.331070
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.59398	0.117536
$936$	0	0
$937$	−31.6421	−1.03370	−0.516851	−	0.856076i	$-0.672896\pi$
$937$	−31.6421	−1.03370	−0.516851	+	0.856076i	$0.672896\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.1824	0.364537	0.182269	−	0.983249i	$-0.441656\pi$
$941$	11.1824	0.364537	0.182269	+	0.983249i	$0.441656\pi$
$942$	0	0
$943$	13.6810	0.445514
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.6149	−0.994852	−0.497426	−	0.867506i	$-0.665721\pi$
$947$	−30.6149	−0.994852	−0.497426	+	0.867506i	$0.665721\pi$
$948$	0	0
$949$	29.0760	0.943847
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.17548	−0.102864	−0.0514320	−	0.998676i	$-0.516379\pi$
$953$	−3.17548	−0.102864	−0.0514320	+	0.998676i	$0.516379\pi$
$954$	0	0
$955$	−7.15604	−0.231564
$956$	0	0
$957$	0	0
$958$	0	0
$959$	41.4395	1.33815
$960$	0	0
$961$	2.99777	0.0967021
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.89134	0.0608842
$966$	0	0
$967$	−40.5566	−1.30421	−0.652106	−	0.758128i	$-0.726114\pi$
$967$	−40.5566	−1.30421	−0.652106	+	0.758128i	$0.726114\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.23205	−0.135813	−0.0679065	−	0.997692i	$-0.521632\pi$
$971$	−4.23205	−0.135813	−0.0679065	+	0.997692i	$0.521632\pi$
$972$	0	0
$973$	25.3961	0.814162
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.5055	−0.847986	−0.423993	−	0.905666i	$-0.639372\pi$
$977$	−26.5055	−0.847986	−0.423993	+	0.905666i	$0.639372\pi$
$978$	0	0
$979$	−39.2097	−1.25315
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.27567	0.0725826	0.0362913	−	0.999341i	$-0.488446\pi$
$983$	2.27567	0.0725826	0.0362913	+	0.999341i	$0.488446\pi$
$984$	0	0
$985$	−2.19092	−0.0698086
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.8084	1.29763
$990$	0	0
$991$	−43.9185	−1.39512	−0.697559	−	0.716527i	$-0.745731\pi$
$991$	−43.9185	−1.39512	−0.697559	+	0.716527i	$0.745731\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.71113	0.212757
$996$	0	0
$997$	2.81532	0.0891621	0.0445811	−	0.999006i	$-0.485805\pi$
$997$	2.81532	0.0891621	0.0445811	+	0.999006i	$0.485805\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.bl.1.2		3
3.2	odd	2		2832.2.a.t.1.2		3
4.3	odd	2		531.2.a.d.1.1		3
12.11	even	2		177.2.a.d.1.3	✓	3
60.59	even	2		4425.2.a.w.1.1		3
84.83	odd	2		8673.2.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.d.1.3	✓	3	12.11	even	2
531.2.a.d.1.1		3	4.3	odd	2
2832.2.a.t.1.2		3	3.2	odd	2
4425.2.a.w.1.1		3	60.59	even	2
8496.2.a.bl.1.2		3	1.1	even	1	trivial
8673.2.a.s.1.3		3	84.83	odd	2

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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:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q+0.357926 q^{5} -5.11491 q^{7} +O(q^{10})\)	\(q+0.357926 q^{5} -5.11491 q^{7} -4.58774 q^{11} -2.58774 q^{13} -2.18869 q^{17} -0.527166 q^{19} -5.70265 q^{23} -4.87189 q^{25} -2.06058 q^{29} -5.83076 q^{31} -1.83076 q^{35} -7.70265 q^{37} -2.39905 q^{41} -7.15604 q^{43} +8.77643 q^{47} +19.1623 q^{49} +8.10170 q^{53} -1.64207 q^{55} +1.00000 q^{59} +9.00624 q^{61} -0.926221 q^{65} -14.4791 q^{67} +4.12811 q^{71} -11.2361 q^{73} +23.4659 q^{77} +3.87189 q^{79} -0.737534 q^{83} -0.783389 q^{85} +8.54661 q^{89} +13.2361 q^{91} -0.188687 q^{95} -4.10170 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} - 9 q^{7}+O(q^{10}) \) 3 * q + 2 * q^5 - 9 * q^7	\( 3 q + 2 q^{5} - 9 q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 7 q^{19} + q^{23} - q^{25} + 11 q^{29} - 13 q^{31} - q^{35} - 5 q^{37} + q^{41} - 6 q^{43} + 11 q^{47} + 14 q^{49} - 2 q^{53} - 4 q^{55} + 3 q^{59} - q^{61} - 10 q^{67} + 26 q^{71} + 7 q^{73} + 17 q^{77} - 2 q^{79} - 3 q^{83} - 35 q^{85} + 23 q^{89} - q^{91} + 3 q^{95} + 14 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 - 9 * q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 7 * q^19 + q^23 - q^25 + 11 * q^29 - 13 * q^31 - q^35 - 5 * q^37 + q^41 - 6 * q^43 + 11 * q^47 + 14 * q^49 - 2 * q^53 - 4 * q^55 + 3 * q^59 - q^61 - 10 * q^67 + 26 * q^71 + 7 * q^73 + 17 * q^77 - 2 * q^79 - 3 * q^83 - 35 * q^85 + 23 * q^89 - q^91 + 3 * q^95 + 14 * q^97

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