LMFDB - Embedded newform 8712.2.a.cc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.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:	$69.5656702409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 31 $ x^4 - 2x^3 - 11x^2 + 12*x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.41309$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41309 q^{5} +1.12667 q^{7} +O(q^{10})$	$q-3.41309 q^{5} +1.12667 q^{7} -7.01386 q^{13} +4.49137 q^{17} +5.53975 q^{19} -6.03112 q^{23} +6.64915 q^{25} -1.51930 q^{29} -3.29512 q^{31} -3.84540 q^{35} +7.26719 q^{37} -9.48599 q^{41} +1.96888 q^{43} -6.08147 q^{47} -5.73063 q^{49} -1.17702 q^{53} -3.46147 q^{59} +2.99462 q^{61} +23.9389 q^{65} +11.1512 q^{67} -16.2307 q^{71} +4.18890 q^{73} +2.31435 q^{79} -0.567681 q^{83} -15.3294 q^{85} +13.9581 q^{89} -7.90227 q^{91} -18.9076 q^{95} -10.1040 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 5 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 5 * q^7	$ 4 q - 2 q^{5} + 5 q^{7} - 2 q^{13} + 13 q^{17} + 11 q^{19} - 8 q^{23} + 6 q^{25} - 11 q^{29} + 2 q^{31} - 5 q^{35} + 4 q^{37} + 6 q^{41} + 24 q^{43} - 5 q^{47} + 17 q^{49} - 2 q^{53} + 4 q^{59} - 27 q^{61} + 21 q^{65} + 19 q^{67} - 17 q^{71} - 15 q^{73} + 7 q^{79} - q^{83} - 4 q^{85} - 6 q^{89} + 50 q^{91} - 33 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 5 * q^7 - 2 * q^13 + 13 * q^17 + 11 * q^19 - 8 * q^23 + 6 * q^25 - 11 * q^29 + 2 * q^31 - 5 * q^35 + 4 * q^37 + 6 * q^41 + 24 * q^43 - 5 * q^47 + 17 * q^49 - 2 * q^53 + 4 * q^59 - 27 * q^61 + 21 * q^65 + 19 * q^67 - 17 * q^71 - 15 * q^73 + 7 * q^79 - q^83 - 4 * q^85 - 6 * q^89 + 50 * q^91 - 33 * q^95 + 8 * 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
$5$	−3.41309	−1.52638	−0.763189	−	0.646175i	$-0.776368\pi$
$5$	−3.41309	−1.52638	−0.763189	+	0.646175i	$0.776368\pi$
$6$	0	0
$7$	1.12667	0.425839	0.212920	−	0.977070i	$-0.431703\pi$
$7$	1.12667	0.425839	0.212920	+	0.977070i	$0.431703\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−7.01386	−1.94529	−0.972647	−	0.232288i	$-0.925379\pi$
$13$	−7.01386	−1.94529	−0.972647	+	0.232288i	$0.925379\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.49137	1.08932	0.544658	−	0.838658i	$-0.316659\pi$
$17$	4.49137	1.08932	0.544658	+	0.838658i	$0.316659\pi$
$18$	0	0
$19$	5.53975	1.27091	0.635453	−	0.772140i	$-0.280813\pi$
$19$	5.53975	1.27091	0.635453	+	0.772140i	$0.280813\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.03112	−1.25758	−0.628788	−	0.777577i	$-0.716449\pi$
$23$	−6.03112	−1.25758	−0.628788	+	0.777577i	$0.716449\pi$
$24$	0	0
$25$	6.64915	1.32983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.51930	−0.282127	−0.141063	−	0.990001i	$-0.545052\pi$
$29$	−1.51930	−0.282127	−0.141063	+	0.990001i	$0.545052\pi$
$30$	0	0
$31$	−3.29512	−0.591821	−0.295910	−	0.955216i	$-0.595623\pi$
$31$	−3.29512	−0.591821	−0.295910	+	0.955216i	$0.595623\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.84540	−0.649992
$36$	0	0
$37$	7.26719	1.19472	0.597359	−	0.801974i	$-0.296217\pi$
$37$	7.26719	1.19472	0.597359	+	0.801974i	$0.296217\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.48599	−1.48146	−0.740732	−	0.671801i	$-0.765521\pi$
$41$	−9.48599	−1.48146	−0.740732	+	0.671801i	$0.765521\pi$
$42$	0	0
$43$	1.96888	0.300251	0.150126	−	0.988667i	$-0.452032\pi$
$43$	1.96888	0.300251	0.150126	+	0.988667i	$0.452032\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.08147	−0.887074	−0.443537	−	0.896256i	$-0.646277\pi$
$47$	−6.08147	−0.887074	−0.443537	+	0.896256i	$0.646277\pi$
$48$	0	0
$49$	−5.73063	−0.818661
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.17702	−0.161676	−0.0808379	−	0.996727i	$-0.525760\pi$
$53$	−1.17702	−0.161676	−0.0808379	+	0.996727i	$0.525760\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.46147	−0.450645	−0.225322	−	0.974284i	$-0.572344\pi$
$59$	−3.46147	−0.450645	−0.225322	+	0.974284i	$0.572344\pi$
$60$	0	0
$61$	2.99462	0.383422	0.191711	−	0.981451i	$-0.438596\pi$
$61$	2.99462	0.383422	0.191711	+	0.981451i	$0.438596\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.9389	2.96925
$66$	0	0
$67$	11.1512	1.36233	0.681167	−	0.732128i	$-0.261473\pi$
$67$	11.1512	1.36233	0.681167	+	0.732128i	$0.261473\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.2307	−1.92623	−0.963114	−	0.269092i	$-0.913276\pi$
$71$	−16.2307	−1.92623	−0.963114	+	0.269092i	$0.913276\pi$
$72$	0	0
$73$	4.18890	0.490274	0.245137	−	0.969488i	$-0.421167\pi$
$73$	4.18890	0.490274	0.245137	+	0.969488i	$0.421167\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.31435	0.260385	0.130192	−	0.991489i	$-0.458440\pi$
$79$	2.31435	0.260385	0.130192	+	0.991489i	$0.458440\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.567681	−0.0623111	−0.0311556	−	0.999515i	$-0.509919\pi$
$83$	−0.567681	−0.0623111	−0.0311556	+	0.999515i	$0.509919\pi$
$84$	0	0
$85$	−15.3294	−1.66271
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.9581	1.47956	0.739779	−	0.672850i	$-0.234930\pi$
$89$	13.9581	1.47956	0.739779	+	0.672850i	$0.234930\pi$
$90$	0	0
$91$	−7.90227	−0.828383
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−18.9076	−1.93988
$96$	0	0
$97$	−10.1040	−1.02591	−0.512954	−	0.858416i	$-0.671449\pi$
$97$	−10.1040	−1.02591	−0.512954	+	0.858416i	$0.671449\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.09555	0.407522	0.203761	−	0.979021i	$-0.434683\pi$
$101$	4.09555	0.407522	0.203761	+	0.979021i	$0.434683\pi$
$102$	0	0
$103$	−1.03968	−0.102443	−0.0512216	−	0.998687i	$-0.516311\pi$
$103$	−1.03968	−0.102443	−0.0512216	+	0.998687i	$0.516311\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.75537	−0.169698	−0.0848489	−	0.996394i	$-0.527041\pi$
$107$	−1.75537	−0.169698	−0.0848489	+	0.996394i	$0.527041\pi$
$108$	0	0
$109$	1.11797	0.107082	0.0535409	−	0.998566i	$-0.482949\pi$
$109$	1.11797	0.107082	0.0535409	+	0.998566i	$0.482949\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.659822	−0.0620709	−0.0310354	−	0.999518i	$-0.509880\pi$
$113$	−0.659822	−0.0620709	−0.0310354	+	0.999518i	$0.509880\pi$
$114$	0	0
$115$	20.5847	1.91954
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.06027	0.463874
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.62870	−0.503446
$126$	0	0
$127$	−19.5171	−1.73186	−0.865932	−	0.500162i	$-0.833274\pi$
$127$	−19.5171	−1.73186	−0.865932	+	0.500162i	$0.833274\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.41309	−0.385573	−0.192787	−	0.981241i	$-0.561752\pi$
$131$	−4.41309	−0.385573	−0.192787	+	0.981241i	$0.561752\pi$
$132$	0	0
$133$	6.24144	0.541202
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.04838	−0.260441	−0.130220	−	0.991485i	$-0.541568\pi$
$137$	−3.04838	−0.260441	−0.130220	+	0.991485i	$0.541568\pi$
$138$	0	0
$139$	3.66520	0.310878	0.155439	−	0.987845i	$-0.450321\pi$
$139$	3.66520	0.310878	0.155439	+	0.987845i	$0.450321\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.18550	0.430632
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.30687	0.188986	0.0944931	−	0.995526i	$-0.469877\pi$
$149$	2.30687	0.188986	0.0944931	+	0.995526i	$0.469877\pi$
$150$	0	0
$151$	1.26597	0.103023	0.0515116	−	0.998672i	$-0.483596\pi$
$151$	1.26597	0.103023	0.0515116	+	0.998672i	$0.483596\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.2465	0.903342
$156$	0	0
$157$	−10.3991	−0.829942	−0.414971	−	0.909835i	$-0.636208\pi$
$157$	−10.3991	−0.829942	−0.414971	+	0.909835i	$0.636208\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.79505	−0.535525
$162$	0	0
$163$	0.564359	0.0442040	0.0221020	−	0.999756i	$-0.492964\pi$
$163$	0.564359	0.0442040	0.0221020	+	0.999756i	$0.492964\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.11259	0.318242	0.159121	−	0.987259i	$-0.449134\pi$
$167$	4.11259	0.318242	0.159121	+	0.987259i	$0.449134\pi$
$168$	0	0
$169$	36.1942	2.78417
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.15635	0.620116	0.310058	−	0.950718i	$-0.399652\pi$
$173$	8.15635	0.620116	0.310058	+	0.950718i	$0.399652\pi$
$174$	0	0
$175$	7.49137	0.566294
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.7479	−1.17705	−0.588526	−	0.808478i	$-0.700292\pi$
$179$	−15.7479	−1.17705	−0.588526	+	0.808478i	$0.700292\pi$
$180$	0	0
$181$	25.8055	1.91811	0.959054	−	0.283223i	$-0.0914037\pi$
$181$	25.8055	1.91811	0.959054	+	0.283223i	$0.0914037\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.8035	−1.82359
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.28445	0.454727	0.227363	−	0.973810i	$-0.426989\pi$
$191$	6.28445	0.454727	0.227363	+	0.973810i	$0.426989\pi$
$192$	0	0
$193$	21.2285	1.52806	0.764031	−	0.645180i	$-0.223218\pi$
$193$	21.2285	1.52806	0.764031	+	0.645180i	$0.223218\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.9336	0.778987	0.389493	−	0.921029i	$-0.372650\pi$
$197$	10.9336	0.778987	0.389493	+	0.921029i	$0.372650\pi$
$198$	0	0
$199$	13.7094	0.971835	0.485918	−	0.874005i	$-0.338486\pi$
$199$	13.7094	0.971835	0.485918	+	0.874005i	$0.338486\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.71174	−0.120141
$204$	0	0
$205$	32.3765	2.26127
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	21.3798	1.47184	0.735922	−	0.677066i	$-0.236749\pi$
$211$	21.3798	1.47184	0.735922	+	0.677066i	$0.236749\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.71996	−0.458297
$216$	0	0
$217$	−3.71249	−0.252021
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−31.5018	−2.11904
$222$	0	0
$223$	24.3207	1.62864	0.814318	−	0.580419i	$-0.197111\pi$
$223$	24.3207	1.62864	0.814318	+	0.580419i	$0.197111\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.4216	1.15631	0.578155	−	0.815927i	$-0.303773\pi$
$227$	17.4216	1.15631	0.578155	+	0.815927i	$0.303773\pi$
$228$	0	0
$229$	13.7295	0.907274	0.453637	−	0.891187i	$-0.350126\pi$
$229$	13.7295	0.907274	0.453637	+	0.891187i	$0.350126\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.20692	0.406629	0.203314	−	0.979114i	$-0.434829\pi$
$233$	6.20692	0.406629	0.203314	+	0.979114i	$0.434829\pi$
$234$	0	0
$235$	20.7566	1.35401
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.63311	−0.623114	−0.311557	−	0.950227i	$-0.600850\pi$
$239$	−9.63311	−0.623114	−0.311557	+	0.950227i	$0.600850\pi$
$240$	0	0
$241$	−12.0675	−0.777338	−0.388669	−	0.921377i	$-0.627065\pi$
$241$	−12.0675	−0.777338	−0.388669	+	0.921377i	$0.627065\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	19.5591	1.24959
$246$	0	0
$247$	−38.8550	−2.47229
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.9032	1.50875	0.754377	−	0.656442i	$-0.227939\pi$
$251$	23.9032	1.50875	0.754377	+	0.656442i	$0.227939\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.3873	0.647939	0.323970	−	0.946067i	$-0.394982\pi$
$257$	10.3873	0.647939	0.323970	+	0.946067i	$0.394982\pi$
$258$	0	0
$259$	8.18769	0.508758
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.30382	−0.0803967	−0.0401984	−	0.999192i	$-0.512799\pi$
$263$	−1.30382	−0.0803967	−0.0401984	+	0.999192i	$0.512799\pi$
$264$	0	0
$265$	4.01726	0.246778
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.97880	0.547447	0.273723	−	0.961808i	$-0.411745\pi$
$269$	8.97880	0.547447	0.273723	+	0.961808i	$0.411745\pi$
$270$	0	0
$271$	−27.0010	−1.64019	−0.820097	−	0.572224i	$-0.806081\pi$
$271$	−27.0010	−1.64019	−0.820097	+	0.572224i	$0.806081\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.8347	−1.49217	−0.746085	−	0.665851i	$-0.768069\pi$
$277$	−24.8347	−1.49217	−0.746085	+	0.665851i	$0.768069\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.8949	−1.78338	−0.891691	−	0.452645i	$-0.850480\pi$
$281$	−29.8949	−1.78338	−0.891691	+	0.452645i	$0.850480\pi$
$282$	0	0
$283$	−9.51711	−0.565734	−0.282867	−	0.959159i	$-0.591285\pi$
$283$	−9.51711	−0.565734	−0.282867	+	0.959159i	$0.591285\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.6875	−0.630865
$288$	0	0
$289$	3.17240	0.186611
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.8135	−0.631733	−0.315867	−	0.948804i	$-0.602295\pi$
$293$	−10.8135	−0.631733	−0.315867	+	0.948804i	$0.602295\pi$
$294$	0	0
$295$	11.8143	0.687854
$296$	0	0
$297$	0	0
$298$	0	0
$299$	42.3014	2.44635
$300$	0	0
$301$	2.21827	0.127859
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.2209	−0.585248
$306$	0	0
$307$	22.1996	1.26700	0.633498	−	0.773744i	$-0.281618\pi$
$307$	22.1996	1.26700	0.633498	+	0.773744i	$0.281618\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.0267	1.58925	0.794625	−	0.607100i	$-0.207667\pi$
$311$	28.0267	1.58925	0.794625	+	0.607100i	$0.207667\pi$
$312$	0	0
$313$	1.88312	0.106440	0.0532200	−	0.998583i	$-0.483052\pi$
$313$	1.88312	0.106440	0.0532200	+	0.998583i	$0.483052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.40636	0.303651	0.151826	−	0.988407i	$-0.451485\pi$
$317$	5.40636	0.303651	0.151826	+	0.988407i	$0.451485\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.8811	1.38442
$324$	0	0
$325$	−46.6362	−2.58691
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.85178	−0.377751
$330$	0	0
$331$	17.9023	0.983998	0.491999	−	0.870596i	$-0.336266\pi$
$331$	17.9023	0.983998	0.491999	+	0.870596i	$0.336266\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−38.0600	−2.07944
$336$	0	0
$337$	−30.1629	−1.64308	−0.821540	−	0.570151i	$-0.806885\pi$
$337$	−30.1629	−1.64308	−0.821540	+	0.570151i	$0.806885\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−14.3432	−0.774457
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.39793	−0.504507	−0.252254	−	0.967661i	$-0.581172\pi$
$347$	−9.39793	−0.504507	−0.252254	+	0.967661i	$0.581172\pi$
$348$	0	0
$349$	−9.75672	−0.522265	−0.261133	−	0.965303i	$-0.584096\pi$
$349$	−9.75672	−0.522265	−0.261133	+	0.965303i	$0.584096\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.6632	0.886894	0.443447	−	0.896301i	$-0.353755\pi$
$353$	16.6632	0.886894	0.443447	+	0.896301i	$0.353755\pi$
$354$	0	0
$355$	55.3967	2.94015
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.3393	1.49569	0.747847	−	0.663872i	$-0.231088\pi$
$359$	28.3393	1.49569	0.747847	+	0.663872i	$0.231088\pi$
$360$	0	0
$361$	11.6888	0.615202
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.2971	−0.748344
$366$	0	0
$367$	22.9924	1.20020	0.600098	−	0.799927i	$-0.295128\pi$
$367$	22.9924	1.20020	0.600098	+	0.799927i	$0.295128\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.32610	−0.0688479
$372$	0	0
$373$	31.2842	1.61983	0.809916	−	0.586546i	$-0.199513\pi$
$373$	31.2842	1.61983	0.809916	+	0.586546i	$0.199513\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.6562	0.548820
$378$	0	0
$379$	−2.71568	−0.139495	−0.0697476	−	0.997565i	$-0.522219\pi$
$379$	−2.71568	−0.139495	−0.0697476	+	0.997565i	$0.522219\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.9754	1.68496	0.842482	−	0.538725i	$-0.181094\pi$
$383$	32.9754	1.68496	0.842482	+	0.538725i	$0.181094\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.51514	−0.127523	−0.0637614	−	0.997965i	$-0.520310\pi$
$389$	−2.51514	−0.127523	−0.0637614	+	0.997965i	$0.520310\pi$
$390$	0	0
$391$	−27.0880	−1.36990
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.89908	−0.397446
$396$	0	0
$397$	6.18374	0.310353	0.155177	−	0.987887i	$-0.450405\pi$
$397$	6.18374	0.310353	0.155177	+	0.987887i	$0.450405\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.8969	0.943666	0.471833	−	0.881688i	$-0.343593\pi$
$401$	18.8969	0.943666	0.471833	+	0.881688i	$0.343593\pi$
$402$	0	0
$403$	23.1115	1.15127
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	16.3347	0.807700	0.403850	−	0.914825i	$-0.367672\pi$
$409$	16.3347	0.807700	0.403850	+	0.914825i	$0.367672\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.89991	−0.191902
$414$	0	0
$415$	1.93755	0.0951103
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.2738	0.697320	0.348660	−	0.937249i	$-0.386637\pi$
$419$	14.2738	0.697320	0.348660	+	0.937249i	$0.386637\pi$
$420$	0	0
$421$	4.02264	0.196051	0.0980257	−	0.995184i	$-0.468747\pi$
$421$	4.02264	0.196051	0.0980257	+	0.995184i	$0.468747\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.8638	1.44861
$426$	0	0
$427$	3.37394	0.163276
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.5362	−0.700185	−0.350092	−	0.936715i	$-0.613850\pi$
$431$	−14.5362	−0.700185	−0.350092	+	0.936715i	$0.613850\pi$
$432$	0	0
$433$	4.02636	0.193495	0.0967473	−	0.995309i	$-0.469156\pi$
$433$	4.02636	0.193495	0.0967473	+	0.995309i	$0.469156\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−33.4109	−1.59826
$438$	0	0
$439$	−11.9183	−0.568830	−0.284415	−	0.958701i	$-0.591799\pi$
$439$	−11.9183	−0.568830	−0.284415	+	0.958701i	$0.591799\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.3035	1.53478	0.767392	−	0.641178i	$-0.221554\pi$
$443$	32.3035	1.53478	0.767392	+	0.641178i	$0.221554\pi$
$444$	0	0
$445$	−47.6403	−2.25837
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.7083	−0.788515	−0.394258	−	0.919000i	$-0.628998\pi$
$449$	−16.7083	−0.788515	−0.394258	+	0.919000i	$0.628998\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	26.9711	1.26443
$456$	0	0
$457$	−28.6418	−1.33981	−0.669904	−	0.742448i	$-0.733665\pi$
$457$	−28.6418	−1.33981	−0.669904	+	0.742448i	$0.733665\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.66860	−0.170864	−0.0854319	−	0.996344i	$-0.527227\pi$
$461$	−3.66860	−0.170864	−0.0854319	+	0.996344i	$0.527227\pi$
$462$	0	0
$463$	−34.3251	−1.59522	−0.797612	−	0.603171i	$-0.793904\pi$
$463$	−34.3251	−1.59522	−0.797612	+	0.603171i	$0.793904\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.7578	1.28448	0.642239	−	0.766504i	$-0.278006\pi$
$467$	27.7578	1.28448	0.642239	+	0.766504i	$0.278006\pi$
$468$	0	0
$469$	12.5637	0.580136
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	36.8347	1.69009
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.87993	0.131587	0.0657936	−	0.997833i	$-0.479042\pi$
$479$	2.87993	0.131587	0.0657936	+	0.997833i	$0.479042\pi$
$480$	0	0
$481$	−50.9710	−2.32408
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.4859	1.56592
$486$	0	0
$487$	−7.19285	−0.325939	−0.162969	−	0.986631i	$-0.552107\pi$
$487$	−7.19285	−0.325939	−0.162969	+	0.986631i	$0.552107\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.78981	−0.396679	−0.198339	−	0.980133i	$-0.563555\pi$
$491$	−8.78981	−0.396679	−0.198339	+	0.980133i	$0.563555\pi$
$492$	0	0
$493$	−6.82374	−0.307326
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.2866	−0.820264
$498$	0	0
$499$	−30.5566	−1.36790	−0.683950	−	0.729529i	$-0.739739\pi$
$499$	−30.5566	−1.36790	−0.683950	+	0.729529i	$0.739739\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.843567	−0.0376128	−0.0188064	−	0.999823i	$-0.505987\pi$
$503$	−0.843567	−0.0376128	−0.0188064	+	0.999823i	$0.505987\pi$
$504$	0	0
$505$	−13.9784	−0.622033
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.1599	1.38114	0.690569	−	0.723267i	$-0.257360\pi$
$509$	31.1599	1.38114	0.690569	+	0.723267i	$0.257360\pi$
$510$	0	0
$511$	4.71949	0.208778
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.54853	0.156367
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.3819	−1.28724	−0.643622	−	0.765344i	$-0.722569\pi$
$521$	−29.3819	−1.28724	−0.643622	+	0.765344i	$0.722569\pi$
$522$	0	0
$523$	37.6395	1.64586	0.822930	−	0.568143i	$-0.192338\pi$
$523$	37.6395	1.64586	0.822930	+	0.568143i	$0.192338\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.7996	−0.644680
$528$	0	0
$529$	13.3744	0.581496
$530$	0	0
$531$	0	0
$532$	0	0
$533$	66.5334	2.88188
$534$	0	0
$535$	5.99122	0.259023
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	29.8208	1.28210	0.641048	−	0.767501i	$-0.278500\pi$
$541$	29.8208	1.28210	0.641048	+	0.767501i	$0.278500\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.81572	−0.163447
$546$	0	0
$547$	−11.2446	−0.480782	−0.240391	−	0.970676i	$-0.577276\pi$
$547$	−11.2446	−0.480782	−0.240391	+	0.970676i	$0.577276\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.41654	−0.358557
$552$	0	0
$553$	2.60750	0.110882
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.2745	−0.689571	−0.344785	−	0.938682i	$-0.612048\pi$
$557$	−16.2745	−0.689571	−0.344785	+	0.938682i	$0.612048\pi$
$558$	0	0
$559$	−13.8094	−0.584077
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.13196	−0.384866	−0.192433	−	0.981310i	$-0.561638\pi$
$563$	−9.13196	−0.384866	−0.192433	+	0.981310i	$0.561638\pi$
$564$	0	0
$565$	2.25203	0.0947436
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.9827	0.628109	0.314055	−	0.949405i	$-0.398313\pi$
$569$	14.9827	0.628109	0.314055	+	0.949405i	$0.398313\pi$
$570$	0	0
$571$	34.5185	1.44456	0.722278	−	0.691603i	$-0.243095\pi$
$571$	34.5185	1.44456	0.722278	+	0.691603i	$0.243095\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−40.1018	−1.67236
$576$	0	0
$577$	5.40187	0.224883	0.112441	−	0.993658i	$-0.464133\pi$
$577$	5.40187	0.224883	0.112441	+	0.993658i	$0.464133\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.639587	−0.0265345
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.27181	0.300140	0.150070	−	0.988675i	$-0.452050\pi$
$587$	7.27181	0.300140	0.150070	+	0.988675i	$0.452050\pi$
$588$	0	0
$589$	−18.2541	−0.752148
$590$	0	0
$591$	0	0
$592$	0	0
$593$	35.3724	1.45257	0.726286	−	0.687393i	$-0.241245\pi$
$593$	35.3724	1.45257	0.726286	+	0.687393i	$0.241245\pi$
$594$	0	0
$595$	−17.2711	−0.708047
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.8302	0.483371	0.241685	−	0.970355i	$-0.422300\pi$
$599$	11.8302	0.483371	0.241685	+	0.970355i	$0.422300\pi$
$600$	0	0
$601$	−12.3005	−0.501748	−0.250874	−	0.968020i	$-0.580718\pi$
$601$	−12.3005	−0.501748	−0.250874	+	0.968020i	$0.580718\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.8928	1.37567	0.687834	−	0.725868i	$-0.258562\pi$
$607$	33.8928	1.37567	0.687834	+	0.725868i	$0.258562\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	42.6546	1.72562
$612$	0	0
$613$	10.3690	0.418800	0.209400	−	0.977830i	$-0.432849\pi$
$613$	10.3690	0.418800	0.209400	+	0.977830i	$0.432849\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.0416	1.81331	0.906653	−	0.421878i	$-0.138629\pi$
$617$	45.0416	1.81331	0.906653	+	0.421878i	$0.138629\pi$
$618$	0	0
$619$	−39.9300	−1.60492	−0.802461	−	0.596705i	$-0.796476\pi$
$619$	−39.9300	−1.60492	−0.802461	+	0.596705i	$0.796476\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.7261	0.630054
$624$	0	0
$625$	−14.0345	−0.561381
$626$	0	0
$627$	0	0
$628$	0	0
$629$	32.6396	1.30143
$630$	0	0
$631$	27.1179	1.07955	0.539773	−	0.841811i	$-0.318510\pi$
$631$	27.1179	1.07955	0.539773	+	0.841811i	$0.318510\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	66.6136	2.64348
$636$	0	0
$637$	40.1938	1.59254
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.7714	−0.978413	−0.489207	−	0.872168i	$-0.662714\pi$
$641$	−24.7714	−0.978413	−0.489207	+	0.872168i	$0.662714\pi$
$642$	0	0
$643$	−1.40793	−0.0555232	−0.0277616	−	0.999615i	$-0.508838\pi$
$643$	−1.40793	−0.0555232	−0.0277616	+	0.999615i	$0.508838\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.29377	0.129491	0.0647457	−	0.997902i	$-0.479376\pi$
$647$	3.29377	0.129491	0.0647457	+	0.997902i	$0.479376\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.9336	0.897461	0.448731	−	0.893667i	$-0.351876\pi$
$653$	22.9336	0.897461	0.448731	+	0.893667i	$0.351876\pi$
$654$	0	0
$655$	15.0622	0.588530
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.6953	−0.845128	−0.422564	−	0.906333i	$-0.638870\pi$
$659$	−21.6953	−0.845128	−0.422564	+	0.906333i	$0.638870\pi$
$660$	0	0
$661$	24.5020	0.953016	0.476508	−	0.879170i	$-0.341902\pi$
$661$	24.5020	0.953016	0.476508	+	0.879170i	$0.341902\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.3026	−0.826079
$666$	0	0
$667$	9.16308	0.354796
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−17.1682	−0.661787	−0.330893	−	0.943668i	$-0.607350\pi$
$673$	−17.1682	−0.661787	−0.330893	+	0.943668i	$0.607350\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.7029	−1.06471	−0.532355	−	0.846521i	$-0.678693\pi$
$677$	−27.7029	−1.06471	−0.532355	+	0.846521i	$0.678693\pi$
$678$	0	0
$679$	−11.3839	−0.436872
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.1156	−0.731440	−0.365720	−	0.930725i	$-0.619177\pi$
$683$	−19.1156	−0.731440	−0.365720	+	0.930725i	$0.619177\pi$
$684$	0	0
$685$	10.4044	0.397531
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.25543	0.314507
$690$	0	0
$691$	−9.26378	−0.352411	−0.176205	−	0.984353i	$-0.556382\pi$
$691$	−9.26378	−0.352411	−0.176205	+	0.984353i	$0.556382\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.5096	−0.474517
$696$	0	0
$697$	−42.6051	−1.61378
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26.3659	−0.995827	−0.497914	−	0.867227i	$-0.665900\pi$
$701$	−26.3659	−0.995827	−0.497914	+	0.867227i	$0.665900\pi$
$702$	0	0
$703$	40.2584	1.51837
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.61431	0.173539
$708$	0	0
$709$	−12.0366	−0.452045	−0.226023	−	0.974122i	$-0.572572\pi$
$709$	−12.0366	−0.452045	−0.226023	+	0.974122i	$0.572572\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.8733	0.744259
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	48.5266	1.80974	0.904868	−	0.425692i	$-0.139969\pi$
$719$	48.5266	1.80974	0.904868	+	0.425692i	$0.139969\pi$
$720$	0	0
$721$	−1.17138	−0.0436243
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.1021	−0.375181
$726$	0	0
$727$	15.2577	0.565878	0.282939	−	0.959138i	$-0.408691\pi$
$727$	15.2577	0.565878	0.282939	+	0.959138i	$0.408691\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.84297	0.327069
$732$	0	0
$733$	−34.3325	−1.26810	−0.634051	−	0.773292i	$-0.718609\pi$
$733$	−34.3325	−1.26810	−0.634051	+	0.773292i	$0.718609\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.0935	0.923079	0.461540	−	0.887120i	$-0.347297\pi$
$739$	25.0935	0.923079	0.461540	+	0.887120i	$0.347297\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.8638	0.875478	0.437739	−	0.899102i	$-0.355779\pi$
$743$	23.8638	0.875478	0.437739	+	0.899102i	$0.355779\pi$
$744$	0	0
$745$	−7.87355	−0.288465
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.97771	−0.0722640
$750$	0	0
$751$	6.98825	0.255005	0.127502	−	0.991838i	$-0.459304\pi$
$751$	6.98825	0.255005	0.127502	+	0.991838i	$0.459304\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.32086	−0.157252
$756$	0	0
$757$	17.2390	0.626564	0.313282	−	0.949660i	$-0.398572\pi$
$757$	17.2390	0.626564	0.313282	+	0.949660i	$0.398572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.743132	0.0269385	0.0134693	−	0.999909i	$-0.495712\pi$
$761$	0.743132	0.0269385	0.0134693	+	0.999909i	$0.495712\pi$
$762$	0	0
$763$	1.25957	0.0455997
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.2782	0.876636
$768$	0	0
$769$	24.8227	0.895129	0.447564	−	0.894252i	$-0.352292\pi$
$769$	24.8227	0.895129	0.447564	+	0.894252i	$0.352292\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.619334	0.0222759	0.0111380	−	0.999938i	$-0.496455\pi$
$773$	0.619334	0.0222759	0.0111380	+	0.999938i	$0.496455\pi$
$774$	0	0
$775$	−21.9097	−0.787021
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−52.5500	−1.88280
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	35.4932	1.26681
$786$	0	0
$787$	17.4375	0.621579	0.310789	−	0.950479i	$-0.399407\pi$
$787$	17.4375	0.621579	0.310789	+	0.950479i	$0.399407\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.743399	−0.0264322
$792$	0	0
$793$	−21.0039	−0.745869
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.2649	−0.682397	−0.341198	−	0.939991i	$-0.610833\pi$
$797$	−19.2649	−0.682397	−0.341198	+	0.939991i	$0.610833\pi$
$798$	0	0
$799$	−27.3141	−0.966305
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	23.1921	0.817414
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.0701	0.529835	0.264917	−	0.964271i	$-0.414655\pi$
$809$	15.0701	0.529835	0.264917	+	0.964271i	$0.414655\pi$
$810$	0	0
$811$	19.2058	0.674407	0.337204	−	0.941432i	$-0.390519\pi$
$811$	19.2058	0.674407	0.337204	+	0.941432i	$0.390519\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.92621	−0.0674721
$816$	0	0
$817$	10.9071	0.381591
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.0149	−0.384424	−0.192212	−	0.981353i	$-0.561566\pi$
$821$	−11.0149	−0.384424	−0.192212	+	0.981353i	$0.561566\pi$
$822$	0	0
$823$	52.6429	1.83502	0.917509	−	0.397715i	$-0.130197\pi$
$823$	52.6429	1.83502	0.917509	+	0.397715i	$0.130197\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.3144	0.393439	0.196719	−	0.980460i	$-0.436971\pi$
$827$	11.3144	0.393439	0.196719	+	0.980460i	$0.436971\pi$
$828$	0	0
$829$	10.0085	0.347609	0.173804	−	0.984780i	$-0.444394\pi$
$829$	10.0085	0.347609	0.173804	+	0.984780i	$0.444394\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−25.7384	−0.891781
$834$	0	0
$835$	−14.0366	−0.485758
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.47948	−0.0856012	−0.0428006	−	0.999084i	$-0.513628\pi$
$839$	−2.47948	−0.0856012	−0.0428006	+	0.999084i	$0.513628\pi$
$840$	0	0
$841$	−26.6917	−0.920404
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−123.534	−4.24970
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43.8293	−1.50245
$852$	0	0
$853$	21.7087	0.743293	0.371647	−	0.928374i	$-0.378793\pi$
$853$	21.7087	0.743293	0.371647	+	0.928374i	$0.378793\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.8959	−0.952905	−0.476453	−	0.879200i	$-0.658078\pi$
$857$	−27.8959	−0.952905	−0.476453	+	0.879200i	$0.658078\pi$
$858$	0	0
$859$	−38.9369	−1.32851	−0.664255	−	0.747506i	$-0.731251\pi$
$859$	−38.9369	−1.32851	−0.664255	+	0.747506i	$0.731251\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.9617	−0.713543	−0.356772	−	0.934192i	$-0.616123\pi$
$863$	−20.9617	−0.713543	−0.356772	+	0.934192i	$0.616123\pi$
$864$	0	0
$865$	−27.8383	−0.946531
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−78.2129	−2.65014
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.34166	−0.214387
$876$	0	0
$877$	6.07596	0.205171	0.102585	−	0.994724i	$-0.467289\pi$
$877$	6.07596	0.205171	0.102585	+	0.994724i	$0.467289\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.25951	0.311961	0.155980	−	0.987760i	$-0.450146\pi$
$881$	9.25951	0.311961	0.155980	+	0.987760i	$0.450146\pi$
$882$	0	0
$883$	−15.3022	−0.514962	−0.257481	−	0.966283i	$-0.582892\pi$
$883$	−15.3022	−0.514962	−0.257481	+	0.966283i	$0.582892\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.00340	0.134421	0.0672106	−	0.997739i	$-0.478590\pi$
$887$	4.00340	0.134421	0.0672106	+	0.997739i	$0.478590\pi$
$888$	0	0
$889$	−21.9892	−0.737496
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−33.6898	−1.12739
$894$	0	0
$895$	53.7489	1.79663
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.00627	0.166969
$900$	0	0
$901$	−5.28642	−0.176116
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−88.0764	−2.92776
$906$	0	0
$907$	−0.143174	−0.00475403	−0.00237701	−	0.999997i	$-0.500757\pi$
$907$	−0.143174	−0.00475403	−0.00237701	+	0.999997i	$0.500757\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.6215	0.650089	0.325044	−	0.945699i	$-0.394621\pi$
$911$	19.6215	0.650089	0.325044	+	0.945699i	$0.394621\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.97207	−0.164192
$918$	0	0
$919$	9.27969	0.306109	0.153054	−	0.988218i	$-0.451089\pi$
$919$	9.27969	0.306109	0.153054	+	0.988218i	$0.451089\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	113.840	3.74708
$924$	0	0
$925$	48.3206	1.58877
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.83365	−0.0601601	−0.0300801	−	0.999547i	$-0.509576\pi$
$929$	−1.83365	−0.0601601	−0.0300801	+	0.999547i	$0.509576\pi$
$930$	0	0
$931$	−31.7462	−1.04044
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−34.0388	−1.11200	−0.556000	−	0.831182i	$-0.687665\pi$
$937$	−34.0388	−1.11200	−0.556000	+	0.831182i	$0.687665\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−37.7723	−1.23134	−0.615671	−	0.788004i	$-0.711115\pi$
$941$	−37.7723	−1.23134	−0.615671	+	0.788004i	$0.711115\pi$
$942$	0	0
$943$	57.2112	1.86305
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.7292	−0.738600	−0.369300	−	0.929310i	$-0.620402\pi$
$947$	−22.7292	−0.738600	−0.369300	+	0.929310i	$0.620402\pi$
$948$	0	0
$949$	−29.3804	−0.953727
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.6894	1.25327	0.626637	−	0.779311i	$-0.284431\pi$
$953$	38.6894	1.25327	0.626637	+	0.779311i	$0.284431\pi$
$954$	0	0
$955$	−21.4494	−0.694085
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.43451	−0.110906
$960$	0	0
$961$	−20.1422	−0.649748
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−72.4547	−2.33240
$966$	0	0
$967$	−17.0338	−0.547769	−0.273885	−	0.961763i	$-0.588309\pi$
$967$	−17.0338	−0.547769	−0.273885	+	0.961763i	$0.588309\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.2636	−1.03539	−0.517695	−	0.855565i	$-0.673210\pi$
$971$	−32.2636	−1.03539	−0.517695	+	0.855565i	$0.673210\pi$
$972$	0	0
$973$	4.12945	0.132384
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.5479	1.48920	0.744600	−	0.667511i	$-0.232640\pi$
$977$	46.5479	1.48920	0.744600	+	0.667511i	$0.232640\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.1220	0.896954	0.448477	−	0.893794i	$-0.351967\pi$
$983$	28.1220	0.896954	0.448477	+	0.893794i	$0.351967\pi$
$984$	0	0
$985$	−37.3173	−1.18903
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.8746	−0.377589
$990$	0	0
$991$	20.6254	0.655187	0.327593	−	0.944819i	$-0.393762\pi$
$991$	20.6254	0.655187	0.327593	+	0.944819i	$0.393762\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−46.7914	−1.48339
$996$	0	0
$997$	4.46025	0.141258	0.0706288	−	0.997503i	$-0.477499\pi$
$997$	4.46025	0.141258	0.0706288	+	0.997503i	$0.477499\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.cc.1.1		4
3.2	odd	2		2904.2.a.be.1.4		4
11.2	odd	10		792.2.r.f.433.2		8
11.6	odd	10		792.2.r.f.289.2		8
11.10	odd	2		8712.2.a.bz.1.1		4
12.11	even	2		5808.2.a.cl.1.4		4
33.2	even	10		264.2.q.e.169.1	yes	8
33.17	even	10		264.2.q.e.25.1	✓	8
33.32	even	2		2904.2.a.bb.1.4		4
132.35	odd	10		528.2.y.k.433.1		8
132.83	odd	10		528.2.y.k.289.1		8
132.131	odd	2		5808.2.a.co.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.2.q.e.25.1	✓	8	33.17	even	10
264.2.q.e.169.1	yes	8	33.2	even	10
528.2.y.k.289.1		8	132.83	odd	10
528.2.y.k.433.1		8	132.35	odd	10
792.2.r.f.289.2		8	11.6	odd	10
792.2.r.f.433.2		8	11.2	odd	10
2904.2.a.bb.1.4		4	33.32	even	2
2904.2.a.be.1.4		4	3.2	odd	2
5808.2.a.cl.1.4		4	12.11	even	2
5808.2.a.co.1.4		4	132.131	odd	2
8712.2.a.bz.1.1		4	11.10	odd	2
8712.2.a.cc.1.1		4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q-3.41309 q^{5} +1.12667 q^{7} +O(q^{10})\)	\(q-3.41309 q^{5} +1.12667 q^{7} -7.01386 q^{13} +4.49137 q^{17} +5.53975 q^{19} -6.03112 q^{23} +6.64915 q^{25} -1.51930 q^{29} -3.29512 q^{31} -3.84540 q^{35} +7.26719 q^{37} -9.48599 q^{41} +1.96888 q^{43} -6.08147 q^{47} -5.73063 q^{49} -1.17702 q^{53} -3.46147 q^{59} +2.99462 q^{61} +23.9389 q^{65} +11.1512 q^{67} -16.2307 q^{71} +4.18890 q^{73} +2.31435 q^{79} -0.567681 q^{83} -15.3294 q^{85} +13.9581 q^{89} -7.90227 q^{91} -18.9076 q^{95} -10.1040 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 5 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 5 * q^7	\( 4 q - 2 q^{5} + 5 q^{7} - 2 q^{13} + 13 q^{17} + 11 q^{19} - 8 q^{23} + 6 q^{25} - 11 q^{29} + 2 q^{31} - 5 q^{35} + 4 q^{37} + 6 q^{41} + 24 q^{43} - 5 q^{47} + 17 q^{49} - 2 q^{53} + 4 q^{59} - 27 q^{61} + 21 q^{65} + 19 q^{67} - 17 q^{71} - 15 q^{73} + 7 q^{79} - q^{83} - 4 q^{85} - 6 q^{89} + 50 q^{91} - 33 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 5 * q^7 - 2 * q^13 + 13 * q^17 + 11 * q^19 - 8 * q^23 + 6 * q^25 - 11 * q^29 + 2 * q^31 - 5 * q^35 + 4 * q^37 + 6 * q^41 + 24 * q^43 - 5 * q^47 + 17 * q^49 - 2 * q^53 + 4 * q^59 - 27 * q^61 + 21 * q^65 + 19 * q^67 - 17 * q^71 - 15 * q^73 + 7 * q^79 - q^83 - 4 * q^85 - 6 * q^89 + 50 * q^91 - 33 * q^95 + 8 * q^97

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