LMFDB - Embedded newform 8624.2.a.co.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.532089 q^{3} -0.120615 q^{5} -2.71688 q^{9} +O(q^{10})$	$q-0.532089 q^{3} -0.120615 q^{5} -2.71688 q^{9} -1.00000 q^{11} +1.22668 q^{13} +0.0641778 q^{15} -6.17024 q^{17} +6.41147 q^{19} +2.02229 q^{23} -4.98545 q^{25} +3.04189 q^{27} +3.24897 q^{29} +4.87939 q^{31} +0.532089 q^{33} +2.36959 q^{37} -0.652704 q^{39} -8.29086 q^{41} -2.22668 q^{43} +0.327696 q^{45} +9.23442 q^{47} +3.28312 q^{51} +9.68004 q^{53} +0.120615 q^{55} -3.41147 q^{57} -9.74422 q^{59} +1.43376 q^{61} -0.147956 q^{65} +3.06418 q^{67} -1.07604 q^{69} +8.49525 q^{71} -3.53209 q^{73} +2.65270 q^{75} -9.09152 q^{79} +6.53209 q^{81} -7.02229 q^{83} +0.744223 q^{85} -1.72874 q^{87} +6.87939 q^{89} -2.59627 q^{93} -0.773318 q^{95} -16.5321 q^{97} +2.71688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) $ 3 * q + 3 * q^3 - 6 * q^5	$ 3 q + 3 q^{3} - 6 q^{5} - 3 q^{11} - 3 q^{13} - 9 q^{15} + 3 q^{17} + 9 q^{19} + 3 q^{25} + 6 q^{27} - 3 q^{29} + 9 q^{31} - 3 q^{33} - 3 q^{39} - 9 q^{41} - 3 q^{45} - 3 q^{47} + 18 q^{51} + 9 q^{53} + 6 q^{55} - 12 q^{61} + 15 q^{65} - 21 q^{69} + 9 q^{71} - 6 q^{73} + 9 q^{75} + 3 q^{79} + 15 q^{81} - 15 q^{83} - 27 q^{85} - 24 q^{87} + 15 q^{89} + 6 q^{93} - 9 q^{95} - 45 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^5 - 3 * q^11 - 3 * q^13 - 9 * q^15 + 3 * q^17 + 9 * q^19 + 3 * q^25 + 6 * q^27 - 3 * q^29 + 9 * q^31 - 3 * q^33 - 3 * q^39 - 9 * q^41 - 3 * q^45 - 3 * q^47 + 18 * q^51 + 9 * q^53 + 6 * q^55 - 12 * q^61 + 15 * q^65 - 21 * q^69 + 9 * q^71 - 6 * q^73 + 9 * q^75 + 3 * q^79 + 15 * q^81 - 15 * q^83 - 27 * q^85 - 24 * q^87 + 15 * q^89 + 6 * q^93 - 9 * q^95 - 45 * 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
$3$	−0.532089	−0.307202	−0.153601	−	0.988133i	$-0.549087\pi$
$3$	−0.532089	−0.307202	−0.153601	+	0.988133i	$0.549087\pi$
$4$	0	0
$5$	−0.120615	−0.0539406	−0.0269703	−	0.999636i	$-0.508586\pi$
$5$	−0.120615	−0.0539406	−0.0269703	+	0.999636i	$0.508586\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.71688	−0.905627
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.22668	0.340220	0.170110	−	0.985425i	$-0.445588\pi$
$13$	1.22668	0.340220	0.170110	+	0.985425i	$0.445588\pi$
$14$	0	0
$15$	0.0641778	0.0165706
$16$	0	0
$17$	−6.17024	−1.49650	−0.748252	−	0.663415i	$-0.769107\pi$
$17$	−6.17024	−1.49650	−0.748252	+	0.663415i	$0.769107\pi$
$18$	0	0
$19$	6.41147	1.47089	0.735447	−	0.677583i	$-0.236972\pi$
$19$	6.41147	1.47089	0.735447	+	0.677583i	$0.236972\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.02229	0.421676	0.210838	−	0.977521i	$-0.432381\pi$
$23$	2.02229	0.421676	0.210838	+	0.977521i	$0.432381\pi$
$24$	0	0
$25$	−4.98545	−0.997090
$26$	0	0
$27$	3.04189	0.585412
$28$	0	0
$29$	3.24897	0.603319	0.301659	−	0.953416i	$-0.402459\pi$
$29$	3.24897	0.603319	0.301659	+	0.953416i	$0.402459\pi$
$30$	0	0
$31$	4.87939	0.876363	0.438182	−	0.898886i	$-0.355623\pi$
$31$	4.87939	0.876363	0.438182	+	0.898886i	$0.355623\pi$
$32$	0	0
$33$	0.532089	0.0926248
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.36959	0.389557	0.194779	−	0.980847i	$-0.437601\pi$
$37$	2.36959	0.389557	0.194779	+	0.980847i	$0.437601\pi$
$38$	0	0
$39$	−0.652704	−0.104516
$40$	0	0
$41$	−8.29086	−1.29481	−0.647407	−	0.762144i	$-0.724147\pi$
$41$	−8.29086	−1.29481	−0.647407	+	0.762144i	$0.724147\pi$
$42$	0	0
$43$	−2.22668	−0.339566	−0.169783	−	0.985481i	$-0.554307\pi$
$43$	−2.22668	−0.339566	−0.169783	+	0.985481i	$0.554307\pi$
$44$	0	0
$45$	0.327696	0.0488500
$46$	0	0
$47$	9.23442	1.34698	0.673489	−	0.739197i	$-0.264795\pi$
$47$	9.23442	1.34698	0.673489	+	0.739197i	$0.264795\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.28312	0.459729
$52$	0	0
$53$	9.68004	1.32966	0.664828	−	0.746996i	$-0.268505\pi$
$53$	9.68004	1.32966	0.664828	+	0.746996i	$0.268505\pi$
$54$	0	0
$55$	0.120615	0.0162637
$56$	0	0
$57$	−3.41147	−0.451861
$58$	0	0
$59$	−9.74422	−1.26859	−0.634295	−	0.773091i	$-0.718709\pi$
$59$	−9.74422	−1.26859	−0.634295	+	0.773091i	$0.718709\pi$
$60$	0	0
$61$	1.43376	0.183575	0.0917873	−	0.995779i	$-0.470742\pi$
$61$	1.43376	0.183575	0.0917873	+	0.995779i	$0.470742\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.147956	−0.0183517
$66$	0	0
$67$	3.06418	0.374349	0.187174	−	0.982327i	$-0.440067\pi$
$67$	3.06418	0.374349	0.187174	+	0.982327i	$0.440067\pi$
$68$	0	0
$69$	−1.07604	−0.129540
$70$	0	0
$71$	8.49525	1.00820	0.504100	−	0.863645i	$-0.331824\pi$
$71$	8.49525	1.00820	0.504100	+	0.863645i	$0.331824\pi$
$72$	0	0
$73$	−3.53209	−0.413400	−0.206700	−	0.978404i	$-0.566272\pi$
$73$	−3.53209	−0.413400	−0.206700	+	0.978404i	$0.566272\pi$
$74$	0	0
$75$	2.65270	0.306308
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.09152	−1.02288	−0.511438	−	0.859320i	$-0.670887\pi$
$79$	−9.09152	−1.02288	−0.511438	+	0.859320i	$0.670887\pi$
$80$	0	0
$81$	6.53209	0.725788
$82$	0	0
$83$	−7.02229	−0.770796	−0.385398	−	0.922750i	$-0.625936\pi$
$83$	−7.02229	−0.770796	−0.385398	+	0.922750i	$0.625936\pi$
$84$	0	0
$85$	0.744223	0.0807223
$86$	0	0
$87$	−1.72874	−0.185340
$88$	0	0
$89$	6.87939	0.729213	0.364607	−	0.931162i	$-0.381203\pi$
$89$	6.87939	0.729213	0.364607	+	0.931162i	$0.381203\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.59627	−0.269220
$94$	0	0
$95$	−0.773318	−0.0793408
$96$	0	0
$97$	−16.5321	−1.67858	−0.839290	−	0.543685i	$-0.817029\pi$
$97$	−16.5321	−1.67858	−0.839290	+	0.543685i	$0.817029\pi$
$98$	0	0
$99$	2.71688	0.273057
$100$	0	0
$101$	6.80066	0.676691	0.338345	−	0.941022i	$-0.390133\pi$
$101$	6.80066	0.676691	0.338345	+	0.941022i	$0.390133\pi$
$102$	0	0
$103$	−9.17024	−0.903571	−0.451786	−	0.892127i	$-0.649213\pi$
$103$	−9.17024	−0.903571	−0.451786	+	0.892127i	$0.649213\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.12061	0.785049	0.392525	−	0.919741i	$-0.371602\pi$
$107$	8.12061	0.785049	0.392525	+	0.919741i	$0.371602\pi$
$108$	0	0
$109$	−6.49020	−0.621648	−0.310824	−	0.950467i	$-0.600605\pi$
$109$	−6.49020	−0.621648	−0.310824	+	0.950467i	$0.600605\pi$
$110$	0	0
$111$	−1.26083	−0.119673
$112$	0	0
$113$	−10.8648	−1.02208	−0.511039	−	0.859558i	$-0.670739\pi$
$113$	−10.8648	−1.02208	−0.511039	+	0.859558i	$0.670739\pi$
$114$	0	0
$115$	−0.243918	−0.0227455
$116$	0	0
$117$	−3.33275	−0.308113
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.41147	0.397769
$124$	0	0
$125$	1.20439	0.107724
$126$	0	0
$127$	−2.68004	−0.237816	−0.118908	−	0.992905i	$-0.537939\pi$
$127$	−2.68004	−0.237816	−0.118908	+	0.992905i	$0.537939\pi$
$128$	0	0
$129$	1.18479	0.104315
$130$	0	0
$131$	−14.0273	−1.22557	−0.612787	−	0.790248i	$-0.709952\pi$
$131$	−14.0273	−1.22557	−0.612787	+	0.790248i	$0.709952\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.366897	−0.0315774
$136$	0	0
$137$	14.6408	1.25085	0.625426	−	0.780284i	$-0.284925\pi$
$137$	14.6408	1.25085	0.625426	+	0.780284i	$0.284925\pi$
$138$	0	0
$139$	11.0155	0.934321	0.467160	−	0.884173i	$-0.345277\pi$
$139$	11.0155	0.934321	0.467160	+	0.884173i	$0.345277\pi$
$140$	0	0
$141$	−4.91353	−0.413794
$142$	0	0
$143$	−1.22668	−0.102580
$144$	0	0
$145$	−0.391874	−0.0325433
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.7297	1.20670	0.603351	−	0.797476i	$-0.293832\pi$
$149$	14.7297	1.20670	0.603351	+	0.797476i	$0.293832\pi$
$150$	0	0
$151$	−23.2422	−1.89142	−0.945710	−	0.325011i	$-0.894632\pi$
$151$	−23.2422	−1.89142	−0.945710	+	0.325011i	$0.894632\pi$
$152$	0	0
$153$	16.7638	1.35527
$154$	0	0
$155$	−0.588526	−0.0472715
$156$	0	0
$157$	−16.0642	−1.28206	−0.641030	−	0.767515i	$-0.721493\pi$
$157$	−16.0642	−1.28206	−0.641030	+	0.767515i	$0.721493\pi$
$158$	0	0
$159$	−5.15064	−0.408473
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.6013	0.830359	0.415180	−	0.909739i	$-0.363719\pi$
$163$	10.6013	0.830359	0.415180	+	0.909739i	$0.363719\pi$
$164$	0	0
$165$	−0.0641778	−0.00499623
$166$	0	0
$167$	−21.1361	−1.63556	−0.817780	−	0.575531i	$-0.804795\pi$
$167$	−21.1361	−1.63556	−0.817780	+	0.575531i	$0.804795\pi$
$168$	0	0
$169$	−11.4953	−0.884250
$170$	0	0
$171$	−17.4192	−1.33208
$172$	0	0
$173$	−2.31996	−0.176383	−0.0881915	−	0.996104i	$-0.528109\pi$
$173$	−2.31996	−0.176383	−0.0881915	+	0.996104i	$0.528109\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.18479	0.389713
$178$	0	0
$179$	−11.4192	−0.853512	−0.426756	−	0.904367i	$-0.640344\pi$
$179$	−11.4192	−0.853512	−0.426756	+	0.904367i	$0.640344\pi$
$180$	0	0
$181$	−16.4679	−1.22405	−0.612025	−	0.790838i	$-0.709645\pi$
$181$	−16.4679	−1.22405	−0.612025	+	0.790838i	$0.709645\pi$
$182$	0	0
$183$	−0.762889	−0.0563944
$184$	0	0
$185$	−0.285807	−0.0210129
$186$	0	0
$187$	6.17024	0.451213
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.1676	0.808056	0.404028	−	0.914747i	$-0.367610\pi$
$191$	11.1676	0.808056	0.404028	+	0.914747i	$0.367610\pi$
$192$	0	0
$193$	−20.0496	−1.44320	−0.721602	−	0.692308i	$-0.756594\pi$
$193$	−20.0496	−1.44320	−0.721602	+	0.692308i	$0.756594\pi$
$194$	0	0
$195$	0.0787257	0.00563766
$196$	0	0
$197$	15.9932	1.13947	0.569734	−	0.821829i	$-0.307046\pi$
$197$	15.9932	1.13947	0.569734	+	0.821829i	$0.307046\pi$
$198$	0	0
$199$	6.26857	0.444367	0.222184	−	0.975005i	$-0.428682\pi$
$199$	6.26857	0.444367	0.222184	+	0.975005i	$0.428682\pi$
$200$	0	0
$201$	−1.63041	−0.115001
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.00000	0.0698430
$206$	0	0
$207$	−5.49432	−0.381882
$208$	0	0
$209$	−6.41147	−0.443491
$210$	0	0
$211$	−11.0642	−0.761689	−0.380845	−	0.924639i	$-0.624367\pi$
$211$	−11.0642	−0.761689	−0.380845	+	0.924639i	$0.624367\pi$
$212$	0	0
$213$	−4.52023	−0.309721
$214$	0	0
$215$	0.268571	0.0183164
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.87939	0.126997
$220$	0	0
$221$	−7.56893	−0.509141
$222$	0	0
$223$	9.25671	0.619875	0.309938	−	0.950757i	$-0.399692\pi$
$223$	9.25671	0.619875	0.309938	+	0.950757i	$0.399692\pi$
$224$	0	0
$225$	13.5449	0.902992
$226$	0	0
$227$	−20.5057	−1.36101	−0.680505	−	0.732743i	$-0.738240\pi$
$227$	−20.5057	−1.36101	−0.680505	+	0.732743i	$0.738240\pi$
$228$	0	0
$229$	−26.0642	−1.72237	−0.861185	−	0.508292i	$-0.830277\pi$
$229$	−26.0642	−1.72237	−0.861185	+	0.508292i	$0.830277\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.78787	0.182639	0.0913196	−	0.995822i	$-0.470892\pi$
$233$	2.78787	0.182639	0.0913196	+	0.995822i	$0.470892\pi$
$234$	0	0
$235$	−1.11381	−0.0726568
$236$	0	0
$237$	4.83750	0.314229
$238$	0	0
$239$	11.2199	0.725753	0.362877	−	0.931837i	$-0.381795\pi$
$239$	11.2199	0.725753	0.362877	+	0.931837i	$0.381795\pi$
$240$	0	0
$241$	−10.2558	−0.660633	−0.330316	−	0.943870i	$-0.607155\pi$
$241$	−10.2558	−0.660633	−0.330316	+	0.943870i	$0.607155\pi$
$242$	0	0
$243$	−12.6013	−0.808375
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.86484	0.500428
$248$	0	0
$249$	3.73648	0.236790
$250$	0	0
$251$	29.1215	1.83814	0.919068	−	0.394099i	$-0.128943\pi$
$251$	29.1215	1.83814	0.919068	+	0.394099i	$0.128943\pi$
$252$	0	0
$253$	−2.02229	−0.127140
$254$	0	0
$255$	−0.395993	−0.0247980
$256$	0	0
$257$	−8.39693	−0.523786	−0.261893	−	0.965097i	$-0.584347\pi$
$257$	−8.39693	−0.523786	−0.261893	+	0.965097i	$0.584347\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.82707	−0.546382
$262$	0	0
$263$	24.3628	1.50227	0.751137	−	0.660147i	$-0.229506\pi$
$263$	24.3628	1.50227	0.751137	+	0.660147i	$0.229506\pi$
$264$	0	0
$265$	−1.16756	−0.0717224
$266$	0	0
$267$	−3.66044	−0.224016
$268$	0	0
$269$	−29.1908	−1.77979	−0.889897	−	0.456162i	$-0.849224\pi$
$269$	−29.1908	−1.77979	−0.889897	+	0.456162i	$0.849224\pi$
$270$	0	0
$271$	9.56624	0.581108	0.290554	−	0.956859i	$-0.406160\pi$
$271$	9.56624	0.581108	0.290554	+	0.956859i	$0.406160\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.98545	0.300634
$276$	0	0
$277$	−20.2814	−1.21859	−0.609295	−	0.792944i	$-0.708547\pi$
$277$	−20.2814	−1.21859	−0.609295	+	0.792944i	$0.708547\pi$
$278$	0	0
$279$	−13.2567	−0.793659
$280$	0	0
$281$	−5.52528	−0.329611	−0.164805	−	0.986326i	$-0.552700\pi$
$281$	−5.52528	−0.329611	−0.164805	+	0.986326i	$0.552700\pi$
$282$	0	0
$283$	4.06687	0.241750	0.120875	−	0.992668i	$-0.461430\pi$
$283$	4.06687	0.241750	0.120875	+	0.992668i	$0.461430\pi$
$284$	0	0
$285$	0.411474	0.0243736
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.0719	1.23952
$290$	0	0
$291$	8.79654	0.515662
$292$	0	0
$293$	23.2567	1.35867	0.679336	−	0.733828i	$-0.262268\pi$
$293$	23.2567	1.35867	0.679336	+	0.733828i	$0.262268\pi$
$294$	0	0
$295$	1.17530	0.0684284
$296$	0	0
$297$	−3.04189	−0.176508
$298$	0	0
$299$	2.48070	0.143463
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.61856	−0.207881
$304$	0	0
$305$	−0.172933	−0.00990211
$306$	0	0
$307$	−15.3131	−0.873968	−0.436984	−	0.899469i	$-0.643953\pi$
$307$	−15.3131	−0.873968	−0.436984	+	0.899469i	$0.643953\pi$
$308$	0	0
$309$	4.87939	0.277579
$310$	0	0
$311$	−7.07192	−0.401012	−0.200506	−	0.979693i	$-0.564259\pi$
$311$	−7.07192	−0.401012	−0.200506	+	0.979693i	$0.564259\pi$
$312$	0	0
$313$	−26.3979	−1.49210	−0.746048	−	0.665893i	$-0.768051\pi$
$313$	−26.3979	−1.49210	−0.746048	+	0.665893i	$0.768051\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.171999	−0.00966044	−0.00483022	−	0.999988i	$-0.501538\pi$
$317$	−0.171999	−0.00966044	−0.00483022	+	0.999988i	$0.501538\pi$
$318$	0	0
$319$	−3.24897	−0.181907
$320$	0	0
$321$	−4.32089	−0.241168
$322$	0	0
$323$	−39.5604	−2.20120
$324$	0	0
$325$	−6.11556	−0.339230
$326$	0	0
$327$	3.45336	0.190971
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.108755	0.00597773	0.00298886	−	0.999996i	$-0.499049\pi$
$331$	0.108755	0.00597773	0.00298886	+	0.999996i	$0.499049\pi$
$332$	0	0
$333$	−6.43788	−0.352794
$334$	0	0
$335$	−0.369585	−0.0201926
$336$	0	0
$337$	−4.69728	−0.255877	−0.127939	−	0.991782i	$-0.540836\pi$
$337$	−4.69728	−0.255877	−0.127939	+	0.991782i	$0.540836\pi$
$338$	0	0
$339$	5.78106	0.313984
$340$	0	0
$341$	−4.87939	−0.264234
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.129786	0.00698744
$346$	0	0
$347$	−4.15570	−0.223089	−0.111545	−	0.993759i	$-0.535580\pi$
$347$	−4.15570	−0.223089	−0.111545	+	0.993759i	$0.535580\pi$
$348$	0	0
$349$	−16.4730	−0.881778	−0.440889	−	0.897562i	$-0.645337\pi$
$349$	−16.4730	−0.881778	−0.440889	+	0.897562i	$0.645337\pi$
$350$	0	0
$351$	3.73143	0.199169
$352$	0	0
$353$	22.6023	1.20300	0.601498	−	0.798874i	$-0.294571\pi$
$353$	22.6023	1.20300	0.601498	+	0.798874i	$0.294571\pi$
$354$	0	0
$355$	−1.02465	−0.0543829
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.0729	1.85107	0.925537	−	0.378657i	$-0.123614\pi$
$359$	35.0729	1.85107	0.925537	+	0.378657i	$0.123614\pi$
$360$	0	0
$361$	22.1070	1.16353
$362$	0	0
$363$	−0.532089	−0.0279274
$364$	0	0
$365$	0.426022	0.0222990
$366$	0	0
$367$	−11.8598	−0.619076	−0.309538	−	0.950887i	$-0.600174\pi$
$367$	−11.8598	−0.619076	−0.309538	+	0.950887i	$0.600174\pi$
$368$	0	0
$369$	22.5253	1.17262
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.0155	1.50236	0.751182	−	0.660095i	$-0.229484\pi$
$373$	29.0155	1.50236	0.751182	+	0.660095i	$0.229484\pi$
$374$	0	0
$375$	−0.640844	−0.0330930
$376$	0	0
$377$	3.98545	0.205261
$378$	0	0
$379$	−18.2695	−0.938441	−0.469221	−	0.883081i	$-0.655465\pi$
$379$	−18.2695	−0.938441	−0.469221	+	0.883081i	$0.655465\pi$
$380$	0	0
$381$	1.42602	0.0730573
$382$	0	0
$383$	−15.4388	−0.788887	−0.394443	−	0.918920i	$-0.629063\pi$
$383$	−15.4388	−0.788887	−0.394443	+	0.918920i	$0.629063\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.04963	0.307520
$388$	0	0
$389$	−24.0428	−1.21902	−0.609510	−	0.792779i	$-0.708634\pi$
$389$	−24.0428	−1.21902	−0.609510	+	0.792779i	$0.708634\pi$
$390$	0	0
$391$	−12.4780	−0.631040
$392$	0	0
$393$	7.46379	0.376499
$394$	0	0
$395$	1.09657	0.0551745
$396$	0	0
$397$	9.14796	0.459123	0.229561	−	0.973294i	$-0.426271\pi$
$397$	9.14796	0.459123	0.229561	+	0.973294i	$0.426271\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.03003	0.401001	0.200500	−	0.979694i	$-0.435743\pi$
$401$	8.03003	0.401001	0.200500	+	0.979694i	$0.435743\pi$
$402$	0	0
$403$	5.98545	0.298157
$404$	0	0
$405$	−0.787866	−0.0391494
$406$	0	0
$407$	−2.36959	−0.117456
$408$	0	0
$409$	−1.81521	−0.0897562	−0.0448781	−	0.998992i	$-0.514290\pi$
$409$	−1.81521	−0.0897562	−0.0448781	+	0.998992i	$0.514290\pi$
$410$	0	0
$411$	−7.79023	−0.384264
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.846992	0.0415772
$416$	0	0
$417$	−5.86122	−0.287025
$418$	0	0
$419$	−36.3756	−1.77706	−0.888531	−	0.458816i	$-0.848274\pi$
$419$	−36.3756	−1.77706	−0.888531	+	0.458816i	$0.848274\pi$
$420$	0	0
$421$	−13.3432	−0.650307	−0.325153	−	0.945661i	$-0.605416\pi$
$421$	−13.3432	−0.650307	−0.325153	+	0.945661i	$0.605416\pi$
$422$	0	0
$423$	−25.0888	−1.21986
$424$	0	0
$425$	30.7615	1.49215
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.652704	0.0315128
$430$	0	0
$431$	−10.7980	−0.520120	−0.260060	−	0.965592i	$-0.583742\pi$
$431$	−10.7980	−0.520120	−0.260060	+	0.965592i	$0.583742\pi$
$432$	0	0
$433$	17.8425	0.857458	0.428729	−	0.903433i	$-0.358961\pi$
$433$	17.8425	0.857458	0.428729	+	0.903433i	$0.358961\pi$
$434$	0	0
$435$	0.208512	0.00999737
$436$	0	0
$437$	12.9659	0.620241
$438$	0	0
$439$	−9.28850	−0.443316	−0.221658	−	0.975125i	$-0.571147\pi$
$439$	−9.28850	−0.443316	−0.221658	+	0.975125i	$0.571147\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.4219	−0.780228	−0.390114	−	0.920767i	$-0.627564\pi$
$443$	−16.4219	−0.780228	−0.390114	+	0.920767i	$0.627564\pi$
$444$	0	0
$445$	−0.829755	−0.0393342
$446$	0	0
$447$	−7.83750	−0.370701
$448$	0	0
$449$	−36.5621	−1.72547	−0.862737	−	0.505654i	$-0.831251\pi$
$449$	−36.5621	−1.72547	−0.862737	+	0.505654i	$0.831251\pi$
$450$	0	0
$451$	8.29086	0.390401
$452$	0	0
$453$	12.3669	0.581047
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.99319	−0.420684	−0.210342	−	0.977628i	$-0.567458\pi$
$457$	−8.99319	−0.420684	−0.210342	+	0.977628i	$0.567458\pi$
$458$	0	0
$459$	−18.7692	−0.876071
$460$	0	0
$461$	−13.8844	−0.646663	−0.323331	−	0.946286i	$-0.604803\pi$
$461$	−13.8844	−0.646663	−0.323331	+	0.946286i	$0.604803\pi$
$462$	0	0
$463$	11.0624	0.514114	0.257057	−	0.966396i	$-0.417247\pi$
$463$	11.0624	0.514114	0.257057	+	0.966396i	$0.417247\pi$
$464$	0	0
$465$	0.313148	0.0145219
$466$	0	0
$467$	5.09657	0.235841	0.117921	−	0.993023i	$-0.462377\pi$
$467$	5.09657	0.235841	0.117921	+	0.993023i	$0.462377\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.54757	0.393851
$472$	0	0
$473$	2.22668	0.102383
$474$	0	0
$475$	−31.9641	−1.46661
$476$	0	0
$477$	−26.2995	−1.20417
$478$	0	0
$479$	18.9941	0.867864	0.433932	−	0.900946i	$-0.357126\pi$
$479$	18.9941	0.867864	0.433932	+	0.900946i	$0.357126\pi$
$480$	0	0
$481$	2.90673	0.132535
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.99401	0.0905435
$486$	0	0
$487$	18.7246	0.848494	0.424247	−	0.905547i	$-0.360539\pi$
$487$	18.7246	0.848494	0.424247	+	0.905547i	$0.360539\pi$
$488$	0	0
$489$	−5.64084	−0.255088
$490$	0	0
$491$	−3.08378	−0.139169	−0.0695845	−	0.997576i	$-0.522167\pi$
$491$	−3.08378	−0.139169	−0.0695845	+	0.997576i	$0.522167\pi$
$492$	0	0
$493$	−20.0469	−0.902869
$494$	0	0
$495$	−0.327696	−0.0147288
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.62866	−0.431038	−0.215519	−	0.976500i	$-0.569144\pi$
$499$	−9.62866	−0.431038	−0.215519	+	0.976500i	$0.569144\pi$
$500$	0	0
$501$	11.2463	0.502447
$502$	0	0
$503$	−0.802414	−0.0357779	−0.0178889	−	0.999840i	$-0.505695\pi$
$503$	−0.802414	−0.0357779	−0.0178889	+	0.999840i	$0.505695\pi$
$504$	0	0
$505$	−0.820260	−0.0365011
$506$	0	0
$507$	6.11650	0.271643
$508$	0	0
$509$	14.6486	0.649287	0.324644	−	0.945836i	$-0.394756\pi$
$509$	14.6486	0.649287	0.324644	+	0.945836i	$0.394756\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	19.5030	0.861078
$514$	0	0
$515$	1.10607	0.0487391
$516$	0	0
$517$	−9.23442	−0.406129
$518$	0	0
$519$	1.23442	0.0541851
$520$	0	0
$521$	−12.0324	−0.527149	−0.263574	−	0.964639i	$-0.584901\pi$
$521$	−12.0324	−0.527149	−0.263574	+	0.964639i	$0.584901\pi$
$522$	0	0
$523$	−15.3027	−0.669141	−0.334571	−	0.942371i	$-0.608591\pi$
$523$	−15.3027	−0.669141	−0.334571	+	0.942371i	$0.608591\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30.1070	−1.31148
$528$	0	0
$529$	−18.9103	−0.822189
$530$	0	0
$531$	26.4739	1.14887
$532$	0	0
$533$	−10.1702	−0.440522
$534$	0	0
$535$	−0.979466	−0.0423460
$536$	0	0
$537$	6.07604	0.262200
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.4861	−0.708792	−0.354396	−	0.935095i	$-0.615314\pi$
$541$	−16.4861	−0.708792	−0.354396	+	0.935095i	$0.615314\pi$
$542$	0	0
$543$	8.76239	0.376030
$544$	0	0
$545$	0.782814	0.0335321
$546$	0	0
$547$	16.0060	0.684367	0.342183	−	0.939633i	$-0.388834\pi$
$547$	16.0060	0.684367	0.342183	+	0.939633i	$0.388834\pi$
$548$	0	0
$549$	−3.89536	−0.166250
$550$	0	0
$551$	20.8307	0.887417
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.152075	0.00645521
$556$	0	0
$557$	19.3286	0.818980	0.409490	−	0.912315i	$-0.365707\pi$
$557$	19.3286	0.818980	0.409490	+	0.912315i	$0.365707\pi$
$558$	0	0
$559$	−2.73143	−0.115527
$560$	0	0
$561$	−3.28312	−0.138613
$562$	0	0
$563$	−32.4611	−1.36807	−0.684036	−	0.729448i	$-0.739777\pi$
$563$	−32.4611	−1.36807	−0.684036	+	0.729448i	$0.739777\pi$
$564$	0	0
$565$	1.31046	0.0551315
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.44656	−0.186409	−0.0932047	−	0.995647i	$-0.529711\pi$
$569$	−4.44656	−0.186409	−0.0932047	+	0.995647i	$0.529711\pi$
$570$	0	0
$571$	−7.76146	−0.324807	−0.162403	−	0.986724i	$-0.551925\pi$
$571$	−7.76146	−0.324807	−0.162403	+	0.986724i	$0.551925\pi$
$572$	0	0
$573$	−5.94213	−0.248236
$574$	0	0
$575$	−10.0820	−0.420449
$576$	0	0
$577$	21.5303	0.896320	0.448160	−	0.893953i	$-0.352080\pi$
$577$	21.5303	0.896320	0.448160	+	0.893953i	$0.352080\pi$
$578$	0	0
$579$	10.6682	0.443355
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.68004	−0.400906
$584$	0	0
$585$	0.401979	0.0166198
$586$	0	0
$587$	24.2523	1.00100	0.500499	−	0.865737i	$-0.333150\pi$
$587$	24.2523	1.00100	0.500499	+	0.865737i	$0.333150\pi$
$588$	0	0
$589$	31.2841	1.28904
$590$	0	0
$591$	−8.50980	−0.350046
$592$	0	0
$593$	8.25847	0.339135	0.169567	−	0.985519i	$-0.445763\pi$
$593$	8.25847	0.339135	0.169567	+	0.985519i	$0.445763\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.33544	−0.136510
$598$	0	0
$599$	−14.4834	−0.591775	−0.295888	−	0.955223i	$-0.595615\pi$
$599$	−14.4834	−0.591775	−0.295888	+	0.955223i	$0.595615\pi$
$600$	0	0
$601$	9.81109	0.400203	0.200101	−	0.979775i	$-0.435873\pi$
$601$	9.81109	0.400203	0.200101	+	0.979775i	$0.435873\pi$
$602$	0	0
$603$	−8.32501	−0.339021
$604$	0	0
$605$	−0.120615	−0.00490369
$606$	0	0
$607$	−35.0711	−1.42349	−0.711746	−	0.702437i	$-0.752095\pi$
$607$	−35.0711	−1.42349	−0.711746	+	0.702437i	$0.752095\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.3277	0.458270
$612$	0	0
$613$	−35.8188	−1.44671	−0.723354	−	0.690477i	$-0.757401\pi$
$613$	−35.8188	−1.44671	−0.723354	+	0.690477i	$0.757401\pi$
$614$	0	0
$615$	−0.532089	−0.0214559
$616$	0	0
$617$	0.650015	0.0261686	0.0130843	−	0.999914i	$-0.495835\pi$
$617$	0.650015	0.0261686	0.0130843	+	0.999914i	$0.495835\pi$
$618$	0	0
$619$	16.1575	0.649423	0.324711	−	0.945813i	$-0.394733\pi$
$619$	16.1575	0.649423	0.324711	+	0.945813i	$0.394733\pi$
$620$	0	0
$621$	6.15158	0.246854
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.7820	0.991280
$626$	0	0
$627$	3.41147	0.136241
$628$	0	0
$629$	−14.6209	−0.582974
$630$	0	0
$631$	−4.31584	−0.171811	−0.0859054	−	0.996303i	$-0.527378\pi$
$631$	−4.31584	−0.171811	−0.0859054	+	0.996303i	$0.527378\pi$
$632$	0	0
$633$	5.88713	0.233992
$634$	0	0
$635$	0.323253	0.0128279
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−23.0806	−0.913054
$640$	0	0
$641$	12.9186	0.510253	0.255127	−	0.966908i	$-0.417883\pi$
$641$	12.9186	0.510253	0.255127	+	0.966908i	$0.417883\pi$
$642$	0	0
$643$	−42.4296	−1.67326	−0.836631	−	0.547767i	$-0.815478\pi$
$643$	−42.4296	−1.67326	−0.836631	+	0.547767i	$0.815478\pi$
$644$	0	0
$645$	−0.142903	−0.00562682
$646$	0	0
$647$	−30.2431	−1.18898	−0.594489	−	0.804103i	$-0.702646\pi$
$647$	−30.2431	−1.18898	−0.594489	+	0.804103i	$0.702646\pi$
$648$	0	0
$649$	9.74422	0.382494
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.2071	−0.399434	−0.199717	−	0.979854i	$-0.564002\pi$
$653$	−10.2071	−0.399434	−0.199717	+	0.979854i	$0.564002\pi$
$654$	0	0
$655$	1.69190	0.0661082
$656$	0	0
$657$	9.59627	0.374386
$658$	0	0
$659$	27.9777	1.08986	0.544928	−	0.838483i	$-0.316557\pi$
$659$	27.9777	1.08986	0.544928	+	0.838483i	$0.316557\pi$
$660$	0	0
$661$	−8.16519	−0.317589	−0.158795	−	0.987312i	$-0.550761\pi$
$661$	−8.16519	−0.317589	−0.158795	+	0.987312i	$0.550761\pi$
$662$	0	0
$663$	4.02734	0.156409
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.57036	0.254405
$668$	0	0
$669$	−4.92539	−0.190427
$670$	0	0
$671$	−1.43376	−0.0553498
$672$	0	0
$673$	17.6905	0.681918	0.340959	−	0.940078i	$-0.389248\pi$
$673$	17.6905	0.681918	0.340959	+	0.940078i	$0.389248\pi$
$674$	0	0
$675$	−15.1652	−0.583709
$676$	0	0
$677$	−45.1061	−1.73357	−0.866783	−	0.498685i	$-0.833817\pi$
$677$	−45.1061	−1.73357	−0.866783	+	0.498685i	$0.833817\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.9108	0.418104
$682$	0	0
$683$	10.6004	0.405612	0.202806	−	0.979219i	$-0.434994\pi$
$683$	10.6004	0.405612	0.202806	+	0.979219i	$0.434994\pi$
$684$	0	0
$685$	−1.76590	−0.0674716
$686$	0	0
$687$	13.8685	0.529115
$688$	0	0
$689$	11.8743	0.452376
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.32863	−0.0503978
$696$	0	0
$697$	51.1566	1.93770
$698$	0	0
$699$	−1.48339	−0.0561071
$700$	0	0
$701$	41.0533	1.55056	0.775280	−	0.631618i	$-0.217609\pi$
$701$	41.0533	1.55056	0.775280	+	0.631618i	$0.217609\pi$
$702$	0	0
$703$	15.1925	0.572997
$704$	0	0
$705$	0.592645	0.0223203
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−33.1070	−1.24336	−0.621680	−	0.783272i	$-0.713549\pi$
$709$	−33.1070	−1.24336	−0.621680	+	0.783272i	$0.713549\pi$
$710$	0	0
$711$	24.7006	0.926344
$712$	0	0
$713$	9.86753	0.369542
$714$	0	0
$715$	0.147956	0.00553324
$716$	0	0
$717$	−5.96997	−0.222953
$718$	0	0
$719$	27.5152	1.02614	0.513071	−	0.858346i	$-0.328508\pi$
$719$	27.5152	1.02614	0.513071	+	0.858346i	$0.328508\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.45699	0.202947
$724$	0	0
$725$	−16.1976	−0.601563
$726$	0	0
$727$	10.1111	0.375001	0.187500	−	0.982265i	$-0.439961\pi$
$727$	10.1111	0.375001	0.187500	+	0.982265i	$0.439961\pi$
$728$	0	0
$729$	−12.8912	−0.477454
$730$	0	0
$731$	13.7392	0.508162
$732$	0	0
$733$	−37.8708	−1.39879	−0.699395	−	0.714735i	$-0.746547\pi$
$733$	−37.8708	−1.39879	−0.699395	+	0.714735i	$0.746547\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.06418	−0.112870
$738$	0	0
$739$	−0.882074	−0.0324476	−0.0162238	−	0.999868i	$-0.505164\pi$
$739$	−0.882074	−0.0324476	−0.0162238	+	0.999868i	$0.505164\pi$
$740$	0	0
$741$	−4.18479	−0.153732
$742$	0	0
$743$	−22.5800	−0.828379	−0.414189	−	0.910191i	$-0.635935\pi$
$743$	−22.5800	−0.828379	−0.414189	+	0.910191i	$0.635935\pi$
$744$	0	0
$745$	−1.77662	−0.0650902
$746$	0	0
$747$	19.0787	0.698054
$748$	0	0
$749$	0	0
$750$	0	0
$751$	39.5954	1.44486	0.722429	−	0.691445i	$-0.243026\pi$
$751$	39.5954	1.44486	0.722429	+	0.691445i	$0.243026\pi$
$752$	0	0
$753$	−15.4953	−0.564678
$754$	0	0
$755$	2.80335	0.102024
$756$	0	0
$757$	19.5253	0.709658	0.354829	−	0.934931i	$-0.384539\pi$
$757$	19.5253	0.709658	0.354829	+	0.934931i	$0.384539\pi$
$758$	0	0
$759$	1.07604	0.0390577
$760$	0	0
$761$	23.2094	0.841342	0.420671	−	0.907213i	$-0.361795\pi$
$761$	23.2094	0.841342	0.420671	+	0.907213i	$0.361795\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.02196	−0.0731043
$766$	0	0
$767$	−11.9531	−0.431600
$768$	0	0
$769$	−37.2222	−1.34227	−0.671134	−	0.741336i	$-0.734193\pi$
$769$	−37.2222	−1.34227	−0.671134	+	0.741336i	$0.734193\pi$
$770$	0	0
$771$	4.46791	0.160908
$772$	0	0
$773$	−41.2891	−1.48507	−0.742533	−	0.669810i	$-0.766376\pi$
$773$	−41.2891	−1.48507	−0.742533	+	0.669810i	$0.766376\pi$
$774$	0	0
$775$	−24.3259	−0.873814
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−53.1566	−1.90453
$780$	0	0
$781$	−8.49525	−0.303984
$782$	0	0
$783$	9.88301	0.353190
$784$	0	0
$785$	1.93758	0.0691551
$786$	0	0
$787$	−22.5321	−0.803182	−0.401591	−	0.915819i	$-0.631543\pi$
$787$	−22.5321	−0.803182	−0.401591	+	0.915819i	$0.631543\pi$
$788$	0	0
$789$	−12.9632	−0.461501
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.75877	0.0624558
$794$	0	0
$795$	0.621244	0.0220332
$796$	0	0
$797$	4.79528	0.169858	0.0849288	−	0.996387i	$-0.472934\pi$
$797$	4.79528	0.169858	0.0849288	+	0.996387i	$0.472934\pi$
$798$	0	0
$799$	−56.9786	−2.01576
$800$	0	0
$801$	−18.6905	−0.660395
$802$	0	0
$803$	3.53209	0.124645
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.5321	0.546755
$808$	0	0
$809$	−40.9181	−1.43860	−0.719302	−	0.694698i	$-0.755538\pi$
$809$	−40.9181	−1.43860	−0.719302	+	0.694698i	$0.755538\pi$
$810$	0	0
$811$	−30.8408	−1.08297	−0.541483	−	0.840711i	$-0.682137\pi$
$811$	−30.8408	−1.08297	−0.541483	+	0.840711i	$0.682137\pi$
$812$	0	0
$813$	−5.09009	−0.178517
$814$	0	0
$815$	−1.27868	−0.0447901
$816$	0	0
$817$	−14.2763	−0.499465
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.1652	−0.424568	−0.212284	−	0.977208i	$-0.568090\pi$
$821$	−12.1652	−0.424568	−0.212284	+	0.977208i	$0.568090\pi$
$822$	0	0
$823$	25.7760	0.898495	0.449248	−	0.893407i	$-0.351692\pi$
$823$	25.7760	0.898495	0.449248	+	0.893407i	$0.351692\pi$
$824$	0	0
$825$	−2.65270	−0.0923553
$826$	0	0
$827$	23.6355	0.821886	0.410943	−	0.911661i	$-0.365200\pi$
$827$	23.6355	0.821886	0.410943	+	0.911661i	$0.365200\pi$
$828$	0	0
$829$	10.1922	0.353990	0.176995	−	0.984212i	$-0.443362\pi$
$829$	10.1922	0.353990	0.176995	+	0.984212i	$0.443362\pi$
$830$	0	0
$831$	10.7915	0.374353
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.54933	0.0882230
$836$	0	0
$837$	14.8425	0.513034
$838$	0	0
$839$	12.5193	0.432214	0.216107	−	0.976370i	$-0.430664\pi$
$839$	12.5193	0.432214	0.216107	+	0.976370i	$0.430664\pi$
$840$	0	0
$841$	−18.4442	−0.636007
$842$	0	0
$843$	2.93994	0.101257
$844$	0	0
$845$	1.38650	0.0476969
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.16393	−0.0742660
$850$	0	0
$851$	4.79199	0.164267
$852$	0	0
$853$	−0.427777	−0.0146468	−0.00732340	−	0.999973i	$-0.502331\pi$
$853$	−0.427777	−0.0146468	−0.00732340	+	0.999973i	$0.502331\pi$
$854$	0	0
$855$	2.10101	0.0718532
$856$	0	0
$857$	−45.6991	−1.56105	−0.780527	−	0.625123i	$-0.785049\pi$
$857$	−45.6991	−1.56105	−0.780527	+	0.625123i	$0.785049\pi$
$858$	0	0
$859$	47.9023	1.63440	0.817202	−	0.576351i	$-0.195524\pi$
$859$	47.9023	1.63440	0.817202	+	0.576351i	$0.195524\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.1753	−0.414452	−0.207226	−	0.978293i	$-0.566444\pi$
$863$	−12.1753	−0.414452	−0.207226	+	0.978293i	$0.566444\pi$
$864$	0	0
$865$	0.279821	0.00951419
$866$	0	0
$867$	−11.2121	−0.380784
$868$	0	0
$869$	9.09152	0.308409
$870$	0	0
$871$	3.75877	0.127361
$872$	0	0
$873$	44.9157	1.52017
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.1179	−0.442961	−0.221480	−	0.975165i	$-0.571089\pi$
$877$	−13.1179	−0.442961	−0.221480	+	0.975165i	$0.571089\pi$
$878$	0	0
$879$	−12.3746	−0.417386
$880$	0	0
$881$	−16.8571	−0.567930	−0.283965	−	0.958835i	$-0.591650\pi$
$881$	−16.8571	−0.567930	−0.283965	+	0.958835i	$0.591650\pi$
$882$	0	0
$883$	−9.24030	−0.310961	−0.155480	−	0.987839i	$-0.549693\pi$
$883$	−9.24030	−0.310961	−0.155480	+	0.987839i	$0.549693\pi$
$884$	0	0
$885$	−0.625362	−0.0210213
$886$	0	0
$887$	−24.0182	−0.806451	−0.403226	−	0.915101i	$-0.632111\pi$
$887$	−24.0182	−0.806451	−0.403226	+	0.915101i	$0.632111\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.53209	−0.218833
$892$	0	0
$893$	59.2063	1.98126
$894$	0	0
$895$	1.37733	0.0460389
$896$	0	0
$897$	−1.31996	−0.0440720
$898$	0	0
$899$	15.8530	0.528726
$900$	0	0
$901$	−59.7282	−1.98984
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.98627	0.0660260
$906$	0	0
$907$	−5.79385	−0.192382	−0.0961909	−	0.995363i	$-0.530666\pi$
$907$	−5.79385	−0.192382	−0.0961909	+	0.995363i	$0.530666\pi$
$908$	0	0
$909$	−18.4766	−0.612830
$910$	0	0
$911$	−0.421903	−0.0139783	−0.00698914	−	0.999976i	$-0.502225\pi$
$911$	−0.421903	−0.0139783	−0.00698914	+	0.999976i	$0.502225\pi$
$912$	0	0
$913$	7.02229	0.232404
$914$	0	0
$915$	0.0920157	0.00304195
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0.472964	0.0156016	0.00780081	−	0.999970i	$-0.497517\pi$
$919$	0.472964	0.0156016	0.00780081	+	0.999970i	$0.497517\pi$
$920$	0	0
$921$	8.14796	0.268484
$922$	0	0
$923$	10.4210	0.343010
$924$	0	0
$925$	−11.8135	−0.388424
$926$	0	0
$927$	24.9145	0.818298
$928$	0	0
$929$	6.99825	0.229605	0.114802	−	0.993388i	$-0.463376\pi$
$929$	6.99825	0.229605	0.114802	+	0.993388i	$0.463376\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.76289	0.123191
$934$	0	0
$935$	−0.744223	−0.0243387
$936$	0	0
$937$	−8.19017	−0.267561	−0.133781	−	0.991011i	$-0.542712\pi$
$937$	−8.19017	−0.267561	−0.133781	+	0.991011i	$0.542712\pi$
$938$	0	0
$939$	14.0460	0.458374
$940$	0	0
$941$	−40.3705	−1.31604	−0.658021	−	0.753000i	$-0.728606\pi$
$941$	−40.3705	−1.31604	−0.658021	+	0.753000i	$0.728606\pi$
$942$	0	0
$943$	−16.7665	−0.545993
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.9240	1.36235	0.681173	−	0.732123i	$-0.261470\pi$
$947$	41.9240	1.36235	0.681173	+	0.732123i	$0.261470\pi$
$948$	0	0
$949$	−4.33275	−0.140647
$950$	0	0
$951$	0.0915189	0.00296770
$952$	0	0
$953$	−38.6040	−1.25051	−0.625253	−	0.780422i	$-0.715004\pi$
$953$	−38.6040	−1.25051	−0.625253	+	0.780422i	$0.715004\pi$
$954$	0	0
$955$	−1.34697	−0.0435870
$956$	0	0
$957$	1.72874	0.0558823
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7.19160	−0.231987
$962$	0	0
$963$	−22.0627	−0.710962
$964$	0	0
$965$	2.41828	0.0778472
$966$	0	0
$967$	2.04364	0.0657192	0.0328596	−	0.999460i	$-0.489539\pi$
$967$	2.04364	0.0657192	0.0328596	+	0.999460i	$0.489539\pi$
$968$	0	0
$969$	21.0496	0.676212
$970$	0	0
$971$	−9.55531	−0.306645	−0.153322	−	0.988176i	$-0.548997\pi$
$971$	−9.55531	−0.306645	−0.153322	+	0.988176i	$0.548997\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.25402	0.104212
$976$	0	0
$977$	−47.1958	−1.50993	−0.754964	−	0.655766i	$-0.772346\pi$
$977$	−47.1958	−1.50993	−0.754964	+	0.655766i	$0.772346\pi$
$978$	0	0
$979$	−6.87939	−0.219866
$980$	0	0
$981$	17.6331	0.562982
$982$	0	0
$983$	2.05375	0.0655044	0.0327522	−	0.999464i	$-0.489573\pi$
$983$	2.05375	0.0655044	0.0327522	+	0.999464i	$0.489573\pi$
$984$	0	0
$985$	−1.92902	−0.0614635
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.50299	−0.143187
$990$	0	0
$991$	−47.6979	−1.51517	−0.757587	−	0.652735i	$-0.773622\pi$
$991$	−47.6979	−1.51517	−0.757587	+	0.652735i	$0.773622\pi$
$992$	0	0
$993$	−0.0578674	−0.00183637
$994$	0	0
$995$	−0.756082	−0.0239694
$996$	0	0
$997$	46.7205	1.47965	0.739827	−	0.672798i	$-0.234908\pi$
$997$	46.7205	1.47965	0.739827	+	0.672798i	$0.234908\pi$
$998$	0	0
$999$	7.20801	0.228051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.co.1.1		3
4.3	odd	2		539.2.a.g.1.2		3
7.3	odd	6		1232.2.q.m.177.1		6
7.5	odd	6		1232.2.q.m.529.1		6
7.6	odd	2		8624.2.a.ch.1.3		3
12.11	even	2		4851.2.a.bk.1.2		3
28.3	even	6		77.2.e.a.23.2	✓	6
28.11	odd	6		539.2.e.m.177.2		6
28.19	even	6		77.2.e.a.67.2	yes	6
28.23	odd	6		539.2.e.m.67.2		6
28.27	even	2		539.2.a.j.1.2		3
44.43	even	2		5929.2.a.u.1.2		3
84.47	odd	6		693.2.i.h.298.2		6
84.59	odd	6		693.2.i.h.100.2		6
84.83	odd	2		4851.2.a.bj.1.2		3
308.3	even	30		847.2.n.g.9.2		24
308.19	odd	30		847.2.n.f.130.2		24
308.31	even	30		847.2.n.g.807.2		24
308.47	even	30		847.2.n.g.130.2		24
308.59	even	30		847.2.n.g.632.2		24
308.75	even	30		847.2.n.g.81.2		24
308.87	odd	6		847.2.e.c.485.2		6
308.103	even	30		847.2.n.g.753.2		24
308.115	even	30		847.2.n.g.366.2		24
308.131	odd	6		847.2.e.c.606.2		6
308.159	even	30		847.2.n.g.487.2		24
308.171	odd	30		847.2.n.f.366.2		24
308.215	odd	30		847.2.n.f.487.2		24
308.227	odd	30		847.2.n.f.632.2		24
308.255	odd	30		847.2.n.f.807.2		24
308.271	odd	30		847.2.n.f.753.2		24
308.283	odd	30		847.2.n.f.9.2		24
308.299	odd	30		847.2.n.f.81.2		24
308.307	odd	2		5929.2.a.x.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.a.23.2	✓	6	28.3	even	6
77.2.e.a.67.2	yes	6	28.19	even	6
539.2.a.g.1.2		3	4.3	odd	2
539.2.a.j.1.2		3	28.27	even	2
539.2.e.m.67.2		6	28.23	odd	6
539.2.e.m.177.2		6	28.11	odd	6
693.2.i.h.100.2		6	84.59	odd	6
693.2.i.h.298.2		6	84.47	odd	6
847.2.e.c.485.2		6	308.87	odd	6
847.2.e.c.606.2		6	308.131	odd	6
847.2.n.f.9.2		24	308.283	odd	30
847.2.n.f.81.2		24	308.299	odd	30
847.2.n.f.130.2		24	308.19	odd	30
847.2.n.f.366.2		24	308.171	odd	30
847.2.n.f.487.2		24	308.215	odd	30
847.2.n.f.632.2		24	308.227	odd	30
847.2.n.f.753.2		24	308.271	odd	30
847.2.n.f.807.2		24	308.255	odd	30
847.2.n.g.9.2		24	308.3	even	30
847.2.n.g.81.2		24	308.75	even	30
847.2.n.g.130.2		24	308.47	even	30
847.2.n.g.366.2		24	308.115	even	30
847.2.n.g.487.2		24	308.159	even	30
847.2.n.g.632.2		24	308.59	even	30
847.2.n.g.753.2		24	308.103	even	30
847.2.n.g.807.2		24	308.31	even	30
1232.2.q.m.177.1		6	7.3	odd	6
1232.2.q.m.529.1		6	7.5	odd	6
4851.2.a.bj.1.2		3	84.83	odd	2
4851.2.a.bk.1.2		3	12.11	even	2
5929.2.a.u.1.2		3	44.43	even	2
5929.2.a.x.1.2		3	308.307	odd	2
8624.2.a.ch.1.3		3	7.6	odd	2
8624.2.a.co.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-0.532089 q^{3} -0.120615 q^{5} -2.71688 q^{9} +O(q^{10})\)	\(q-0.532089 q^{3} -0.120615 q^{5} -2.71688 q^{9} -1.00000 q^{11} +1.22668 q^{13} +0.0641778 q^{15} -6.17024 q^{17} +6.41147 q^{19} +2.02229 q^{23} -4.98545 q^{25} +3.04189 q^{27} +3.24897 q^{29} +4.87939 q^{31} +0.532089 q^{33} +2.36959 q^{37} -0.652704 q^{39} -8.29086 q^{41} -2.22668 q^{43} +0.327696 q^{45} +9.23442 q^{47} +3.28312 q^{51} +9.68004 q^{53} +0.120615 q^{55} -3.41147 q^{57} -9.74422 q^{59} +1.43376 q^{61} -0.147956 q^{65} +3.06418 q^{67} -1.07604 q^{69} +8.49525 q^{71} -3.53209 q^{73} +2.65270 q^{75} -9.09152 q^{79} +6.53209 q^{81} -7.02229 q^{83} +0.744223 q^{85} -1.72874 q^{87} +6.87939 q^{89} -2.59627 q^{93} -0.773318 q^{95} -16.5321 q^{97} +2.71688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) \) 3 * q + 3 * q^3 - 6 * q^5	\( 3 q + 3 q^{3} - 6 q^{5} - 3 q^{11} - 3 q^{13} - 9 q^{15} + 3 q^{17} + 9 q^{19} + 3 q^{25} + 6 q^{27} - 3 q^{29} + 9 q^{31} - 3 q^{33} - 3 q^{39} - 9 q^{41} - 3 q^{45} - 3 q^{47} + 18 q^{51} + 9 q^{53} + 6 q^{55} - 12 q^{61} + 15 q^{65} - 21 q^{69} + 9 q^{71} - 6 q^{73} + 9 q^{75} + 3 q^{79} + 15 q^{81} - 15 q^{83} - 27 q^{85} - 24 q^{87} + 15 q^{89} + 6 q^{93} - 9 q^{95} - 45 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^5 - 3 * q^11 - 3 * q^13 - 9 * q^15 + 3 * q^17 + 9 * q^19 + 3 * q^25 + 6 * q^27 - 3 * q^29 + 9 * q^31 - 3 * q^33 - 3 * q^39 - 9 * q^41 - 3 * q^45 - 3 * q^47 + 18 * q^51 + 9 * q^53 + 6 * q^55 - 12 * q^61 + 15 * q^65 - 21 * q^69 + 9 * q^71 - 6 * q^73 + 9 * q^75 + 3 * q^79 + 15 * q^81 - 15 * q^83 - 27 * q^85 - 24 * q^87 + 15 * q^89 + 6 * q^93 - 9 * q^95 - 45 * q^97

Introduction

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Database

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