LMFDB - Embedded newform 8624.2.a.co.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,0,3,0,-6,0,0,0,0,0,-3,0,-3,0,-9,0,3,0,9,0,0,0,0,0,3,0,6,0, -3,0,9,0,-3,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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			1.3
Root			$1.87939$ 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)

$f(q)$	$=$	$q+2.87939 q^{3} -2.34730 q^{5} +5.29086 q^{9} -1.00000 q^{11} +0.184793 q^{13} -6.75877 q^{15} +3.92127 q^{17} +0.773318 q^{19} -8.35504 q^{23} +0.509800 q^{25} +6.59627 q^{27} -8.17024 q^{29} +2.65270 q^{31} -2.87939 q^{33} -6.82295 q^{37} +0.532089 q^{39} -0.426022 q^{41} -1.18479 q^{43} -12.4192 q^{45} -7.68004 q^{47} +11.2909 q^{51} +6.55438 q^{53} +2.34730 q^{55} +2.22668 q^{57} +0.204393 q^{59} -14.5817 q^{61} -0.433763 q^{65} -3.75877 q^{67} -24.0574 q^{69} +9.96585 q^{71} -0.120615 q^{73} +1.46791 q^{75} -0.327696 q^{79} +3.12061 q^{81} +3.35504 q^{83} -9.20439 q^{85} -23.5253 q^{87} +4.65270 q^{89} +7.63816 q^{93} -1.81521 q^{95} -13.1206 q^{97} -5.29086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	2.87939	1.66241	0.831207	−	0.555963i	$-0.187650\pi$
$3$	2.87939	1.66241	0.831207	+	0.555963i	$0.187650\pi$
$4$	0	0
$5$	−2.34730	−1.04974	−0.524871	−	0.851182i	$-0.675887\pi$
$5$	−2.34730	−1.04974	−0.524871	+	0.851182i	$0.675887\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.29086	1.76362
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.184793	0.0512522	0.0256261	−	0.999672i	$-0.491842\pi$
$13$	0.184793	0.0512522	0.0256261	+	0.999672i	$0.491842\pi$
$14$	0	0
$15$	−6.75877	−1.74511
$16$	0	0
$17$	3.92127	0.951049	0.475524	−	0.879703i	$-0.342258\pi$
$17$	3.92127	0.951049	0.475524	+	0.879703i	$0.342258\pi$
$18$	0	0
$19$	0.773318	0.177411	0.0887057	−	0.996058i	$-0.471727\pi$
$19$	0.773318	0.177411	0.0887057	+	0.996058i	$0.471727\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.35504	−1.74215	−0.871073	−	0.491154i	$-0.836575\pi$
$23$	−8.35504	−1.74215	−0.871073	+	0.491154i	$0.836575\pi$
$24$	0	0
$25$	0.509800	0.101960
$26$	0	0
$27$	6.59627	1.26945
$28$	0	0
$29$	−8.17024	−1.51718	−0.758588	−	0.651570i	$-0.774111\pi$
$29$	−8.17024	−1.51718	−0.758588	+	0.651570i	$0.774111\pi$
$30$	0	0
$31$	2.65270	0.476440	0.238220	−	0.971211i	$-0.423436\pi$
$31$	2.65270	0.476440	0.238220	+	0.971211i	$0.423436\pi$
$32$	0	0
$33$	−2.87939	−0.501237
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.82295	−1.12169	−0.560843	−	0.827922i	$-0.689523\pi$
$37$	−6.82295	−1.12169	−0.560843	+	0.827922i	$0.689523\pi$
$38$	0	0
$39$	0.532089	0.0852024
$40$	0	0
$41$	−0.426022	−0.0665335	−0.0332667	−	0.999447i	$-0.510591\pi$
$41$	−0.426022	−0.0665335	−0.0332667	+	0.999447i	$0.510591\pi$
$42$	0	0
$43$	−1.18479	−0.180679	−0.0903396	−	0.995911i	$-0.528795\pi$
$43$	−1.18479	−0.180679	−0.0903396	+	0.995911i	$0.528795\pi$
$44$	0	0
$45$	−12.4192	−1.85135
$46$	0	0
$47$	−7.68004	−1.12025	−0.560125	−	0.828408i	$-0.689247\pi$
$47$	−7.68004	−1.12025	−0.560125	+	0.828408i	$0.689247\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	11.2909	1.58104
$52$	0	0
$53$	6.55438	0.900313	0.450157	−	0.892950i	$-0.351368\pi$
$53$	6.55438	0.900313	0.450157	+	0.892950i	$0.351368\pi$
$54$	0	0
$55$	2.34730	0.316509
$56$	0	0
$57$	2.22668	0.294931
$58$	0	0
$59$	0.204393	0.0266097	0.0133048	−	0.999911i	$-0.495765\pi$
$59$	0.204393	0.0266097	0.0133048	+	0.999911i	$0.495765\pi$
$60$	0	0
$61$	−14.5817	−1.86700	−0.933499	−	0.358580i	$-0.883261\pi$
$61$	−14.5817	−1.86700	−0.933499	+	0.358580i	$0.883261\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.433763	−0.0538017
$66$	0	0
$67$	−3.75877	−0.459207	−0.229603	−	0.973284i	$-0.573743\pi$
$67$	−3.75877	−0.459207	−0.229603	+	0.973284i	$0.573743\pi$
$68$	0	0
$69$	−24.0574	−2.89617
$70$	0	0
$71$	9.96585	1.18273	0.591365	−	0.806404i	$-0.298589\pi$
$71$	9.96585	1.18273	0.591365	+	0.806404i	$0.298589\pi$
$72$	0	0
$73$	−0.120615	−0.0141169	−0.00705844	−	0.999975i	$-0.502247\pi$
$73$	−0.120615	−0.0141169	−0.00705844	+	0.999975i	$0.502247\pi$
$74$	0	0
$75$	1.46791	0.169500
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.327696	−0.0368687	−0.0184343	−	0.999830i	$-0.505868\pi$
$79$	−0.327696	−0.0368687	−0.0184343	+	0.999830i	$0.505868\pi$
$80$	0	0
$81$	3.12061	0.346735
$82$	0	0
$83$	3.35504	0.368263	0.184132	−	0.982902i	$-0.441053\pi$
$83$	3.35504	0.368263	0.184132	+	0.982902i	$0.441053\pi$
$84$	0	0
$85$	−9.20439	−0.998357
$86$	0	0
$87$	−23.5253	−2.52217
$88$	0	0
$89$	4.65270	0.493186	0.246593	−	0.969119i	$-0.420689\pi$
$89$	4.65270	0.493186	0.246593	+	0.969119i	$0.420689\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.63816	0.792040
$94$	0	0
$95$	−1.81521	−0.186236
$96$	0	0
$97$	−13.1206	−1.33220	−0.666098	−	0.745864i	$-0.732037\pi$
$97$	−13.1206	−1.33220	−0.666098	+	0.745864i	$0.732037\pi$
$98$	0	0
$99$	−5.29086	−0.531751
$100$	0	0
$101$	5.90167	0.587239	0.293619	−	0.955922i	$-0.405140\pi$
$101$	5.90167	0.587239	0.293619	+	0.955922i	$0.405140\pi$
$102$	0	0
$103$	0.921274	0.0907759	0.0453879	−	0.998969i	$-0.485548\pi$
$103$	0.921274	0.0907759	0.0453879	+	0.998969i	$0.485548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.3473	1.00031	0.500155	−	0.865936i	$-0.333276\pi$
$107$	10.3473	1.00031	0.500155	+	0.865936i	$0.333276\pi$
$108$	0	0
$109$	0.475652	0.0455592	0.0227796	−	0.999741i	$-0.492748\pi$
$109$	0.475652	0.0455592	0.0227796	+	0.999741i	$0.492748\pi$
$110$	0	0
$111$	−19.6459	−1.86471
$112$	0	0
$113$	−3.14290	−0.295659	−0.147830	−	0.989013i	$-0.547229\pi$
$113$	−3.14290	−0.295659	−0.147830	+	0.989013i	$0.547229\pi$
$114$	0	0
$115$	19.6117	1.82880
$116$	0	0
$117$	0.977711	0.0903894
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.22668	−0.110606
$124$	0	0
$125$	10.5398	0.942711
$126$	0	0
$127$	0.445622	0.0395426	0.0197713	−	0.999805i	$-0.493706\pi$
$127$	0.445622	0.0395426	0.0197713	+	0.999805i	$0.493706\pi$
$128$	0	0
$129$	−3.41147	−0.300364
$130$	0	0
$131$	−12.0865	−1.05600	−0.528000	−	0.849245i	$-0.677058\pi$
$131$	−12.0865	−1.05600	−0.528000	+	0.849245i	$0.677058\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−15.4834	−1.33260
$136$	0	0
$137$	−16.3482	−1.39672	−0.698362	−	0.715745i	$-0.746087\pi$
$137$	−16.3482	−1.39672	−0.698362	+	0.715745i	$0.746087\pi$
$138$	0	0
$139$	−20.7297	−1.75827	−0.879134	−	0.476575i	$-0.841878\pi$
$139$	−20.7297	−1.75827	−0.879134	+	0.476575i	$0.841878\pi$
$140$	0	0
$141$	−22.1138	−1.86232
$142$	0	0
$143$	−0.184793	−0.0154531
$144$	0	0
$145$	19.1780	1.59264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.714193	−0.0585090	−0.0292545	−	0.999572i	$-0.509313\pi$
$149$	−0.714193	−0.0585090	−0.0292545	+	0.999572i	$0.509313\pi$
$150$	0	0
$151$	9.54488	0.776751	0.388376	−	0.921501i	$-0.373036\pi$
$151$	9.54488	0.776751	0.388376	+	0.921501i	$0.373036\pi$
$152$	0	0
$153$	20.7469	1.67729
$154$	0	0
$155$	−6.22668	−0.500139
$156$	0	0
$157$	−9.24123	−0.737530	−0.368765	−	0.929523i	$-0.620219\pi$
$157$	−9.24123	−0.737530	−0.368765	+	0.929523i	$0.620219\pi$
$158$	0	0
$159$	18.8726	1.49669
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.80335	0.689531	0.344766	−	0.938689i	$-0.387958\pi$
$163$	8.80335	0.689531	0.344766	+	0.938689i	$0.387958\pi$
$164$	0	0
$165$	6.75877	0.526170
$166$	0	0
$167$	8.38238	0.648648	0.324324	−	0.945946i	$-0.394863\pi$
$167$	8.38238	0.648648	0.324324	+	0.945946i	$0.394863\pi$
$168$	0	0
$169$	−12.9659	−0.997373
$170$	0	0
$171$	4.09152	0.312886
$172$	0	0
$173$	−5.44562	−0.414023	−0.207012	−	0.978339i	$-0.566374\pi$
$173$	−5.44562	−0.414023	−0.207012	+	0.978339i	$0.566374\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.588526	0.0442363
$178$	0	0
$179$	10.0915	0.754276	0.377138	−	0.926157i	$-0.376908\pi$
$179$	10.0915	0.754276	0.377138	+	0.926157i	$0.376908\pi$
$180$	0	0
$181$	−19.8794	−1.47762	−0.738812	−	0.673912i	$-0.764613\pi$
$181$	−19.8794	−1.47762	−0.738812	+	0.673912i	$0.764613\pi$
$182$	0	0
$183$	−41.9864	−3.10372
$184$	0	0
$185$	16.0155	1.17748
$186$	0	0
$187$	−3.92127	−0.286752
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25.3851	1.83680	0.918399	−	0.395654i	$-0.129482\pi$
$191$	25.3851	1.83680	0.918399	+	0.395654i	$0.129482\pi$
$192$	0	0
$193$	−7.73143	−0.556520	−0.278260	−	0.960506i	$-0.589758\pi$
$193$	−7.73143	−0.556520	−0.278260	+	0.960506i	$0.589758\pi$
$194$	0	0
$195$	−1.24897	−0.0894406
$196$	0	0
$197$	−5.37464	−0.382927	−0.191464	−	0.981500i	$-0.561323\pi$
$197$	−5.37464	−0.382927	−0.191464	+	0.981500i	$0.561323\pi$
$198$	0	0
$199$	8.78106	0.622473	0.311236	−	0.950333i	$-0.399257\pi$
$199$	8.78106	0.622473	0.311236	+	0.950333i	$0.399257\pi$
$200$	0	0
$201$	−10.8229	−0.763392
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.00000	0.0698430
$206$	0	0
$207$	−44.2053	−3.07248
$208$	0	0
$209$	−0.773318	−0.0534916
$210$	0	0
$211$	−4.24123	−0.291978	−0.145989	−	0.989286i	$-0.546636\pi$
$211$	−4.24123	−0.291978	−0.145989	+	0.989286i	$0.546636\pi$
$212$	0	0
$213$	28.6955	1.96619
$214$	0	0
$215$	2.78106	0.189667
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.347296	−0.0234681
$220$	0	0
$221$	0.724622	0.0487434
$222$	0	0
$223$	−18.0351	−1.20772	−0.603859	−	0.797091i	$-0.706371\pi$
$223$	−18.0351	−1.20772	−0.603859	+	0.797091i	$0.706371\pi$
$224$	0	0
$225$	2.69728	0.179819
$226$	0	0
$227$	18.2053	1.20833	0.604165	−	0.796859i	$-0.293507\pi$
$227$	18.2053	1.20833	0.604165	+	0.796859i	$0.293507\pi$
$228$	0	0
$229$	−19.2412	−1.27150	−0.635748	−	0.771897i	$-0.719308\pi$
$229$	−19.2412	−1.27150	−0.635748	+	0.771897i	$0.719308\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.32501	0.610902	0.305451	−	0.952208i	$-0.401193\pi$
$233$	9.32501	0.610902	0.305451	+	0.952208i	$0.401193\pi$
$234$	0	0
$235$	18.0273	1.17597
$236$	0	0
$237$	−0.943563	−0.0612910
$238$	0	0
$239$	−11.1898	−0.723811	−0.361905	−	0.932215i	$-0.617874\pi$
$239$	−11.1898	−0.723811	−0.361905	+	0.932215i	$0.617874\pi$
$240$	0	0
$241$	−20.2044	−1.30148	−0.650740	−	0.759301i	$-0.725541\pi$
$241$	−20.2044	−1.30148	−0.650740	+	0.759301i	$0.725541\pi$
$242$	0	0
$243$	−10.8033	−0.693035
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.142903	0.00909273
$248$	0	0
$249$	9.66044	0.612206
$250$	0	0
$251$	−5.89218	−0.371911	−0.185956	−	0.982558i	$-0.559538\pi$
$251$	−5.89218	−0.371911	−0.185956	+	0.982558i	$0.559538\pi$
$252$	0	0
$253$	8.35504	0.525277
$254$	0	0
$255$	−26.5030	−1.65968
$256$	0	0
$257$	2.73648	0.170697	0.0853485	−	0.996351i	$-0.472800\pi$
$257$	2.73648	0.170697	0.0853485	+	0.996351i	$0.472800\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−43.2276	−2.67572
$262$	0	0
$263$	−6.19759	−0.382160	−0.191080	−	0.981575i	$-0.561199\pi$
$263$	−6.19759	−0.382160	−0.191080	+	0.981575i	$0.561199\pi$
$264$	0	0
$265$	−15.3851	−0.945097
$266$	0	0
$267$	13.3969	0.819879
$268$	0	0
$269$	4.20945	0.256654	0.128327	−	0.991732i	$-0.459039\pi$
$269$	4.20945	0.256654	0.128327	+	0.991732i	$0.459039\pi$
$270$	0	0
$271$	25.5817	1.55398	0.776989	−	0.629514i	$-0.216746\pi$
$271$	25.5817	1.55398	0.776989	+	0.629514i	$0.216746\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.509800	−0.0307421
$276$	0	0
$277$	−15.3577	−0.922756	−0.461378	−	0.887204i	$-0.652645\pi$
$277$	−15.3577	−0.922756	−0.461378	+	0.887204i	$0.652645\pi$
$278$	0	0
$279$	14.0351	0.840258
$280$	0	0
$281$	19.2540	1.14860	0.574299	−	0.818645i	$-0.305275\pi$
$281$	19.2540	1.14860	0.574299	+	0.818645i	$0.305275\pi$
$282$	0	0
$283$	−27.0651	−1.60885	−0.804427	−	0.594052i	$-0.797527\pi$
$283$	−27.0651	−1.60885	−0.804427	+	0.594052i	$0.797527\pi$
$284$	0	0
$285$	−5.22668	−0.309602
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.62361	−0.0955063
$290$	0	0
$291$	−37.7793	−2.21466
$292$	0	0
$293$	−4.03508	−0.235732	−0.117866	−	0.993030i	$-0.537605\pi$
$293$	−4.03508	−0.235732	−0.117866	+	0.993030i	$0.537605\pi$
$294$	0	0
$295$	−0.479771	−0.0279333
$296$	0	0
$297$	−6.59627	−0.382754
$298$	0	0
$299$	−1.54395	−0.0892888
$300$	0	0
$301$	0	0
$302$	0	0
$303$	16.9932	0.976233
$304$	0	0
$305$	34.2276	1.95987
$306$	0	0
$307$	2.92902	0.167168	0.0835839	−	0.996501i	$-0.473363\pi$
$307$	2.92902	0.167168	0.0835839	+	0.996501i	$0.473363\pi$
$308$	0	0
$309$	2.65270	0.150907
$310$	0	0
$311$	15.6236	0.885934	0.442967	−	0.896538i	$-0.353926\pi$
$311$	15.6236	0.885934	0.442967	+	0.896538i	$0.353926\pi$
$312$	0	0
$313$	21.9760	1.24215	0.621077	−	0.783749i	$-0.286695\pi$
$313$	21.9760	1.24215	0.621077	+	0.783749i	$0.286695\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.01186	−0.169163	−0.0845814	−	0.996417i	$-0.526955\pi$
$317$	−3.01186	−0.169163	−0.0845814	+	0.996417i	$0.526955\pi$
$318$	0	0
$319$	8.17024	0.457446
$320$	0	0
$321$	29.7939	1.66293
$322$	0	0
$323$	3.03239	0.168727
$324$	0	0
$325$	0.0942073	0.00522568
$326$	0	0
$327$	1.36959	0.0757382
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−27.4688	−1.50982	−0.754912	−	0.655826i	$-0.772321\pi$
$331$	−27.4688	−1.50982	−0.754912	+	0.655826i	$0.772321\pi$
$332$	0	0
$333$	−36.0993	−1.97823
$334$	0	0
$335$	8.82295	0.482049
$336$	0	0
$337$	17.2422	0.939240	0.469620	−	0.882869i	$-0.344391\pi$
$337$	17.2422	0.939240	0.469620	+	0.882869i	$0.344391\pi$
$338$	0	0
$339$	−9.04963	−0.491508
$340$	0	0
$341$	−2.65270	−0.143652
$342$	0	0
$343$	0	0
$344$	0	0
$345$	56.4698	3.04023
$346$	0	0
$347$	11.4311	0.613652	0.306826	−	0.951766i	$-0.400733\pi$
$347$	11.4311	0.613652	0.306826	+	0.951766i	$0.400733\pi$
$348$	0	0
$349$	−28.3209	−1.51598	−0.757991	−	0.652265i	$-0.773819\pi$
$349$	−28.3209	−1.51598	−0.757991	+	0.652265i	$0.773819\pi$
$350$	0	0
$351$	1.21894	0.0650622
$352$	0	0
$353$	−16.4361	−0.874807	−0.437403	−	0.899265i	$-0.644102\pi$
$353$	−16.4361	−0.874807	−0.437403	+	0.899265i	$0.644102\pi$
$354$	0	0
$355$	−23.3928	−1.24156
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.8631	−1.31222	−0.656112	−	0.754664i	$-0.727800\pi$
$359$	−24.8631	−1.31222	−0.656112	+	0.754664i	$0.727800\pi$
$360$	0	0
$361$	−18.4020	−0.968525
$362$	0	0
$363$	2.87939	0.151129
$364$	0	0
$365$	0.283119	0.0148191
$366$	0	0
$367$	4.29860	0.224385	0.112193	−	0.993686i	$-0.464213\pi$
$367$	4.29860	0.224385	0.112193	+	0.993686i	$0.464213\pi$
$368$	0	0
$369$	−2.25402	−0.117340
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.72967	−0.141337	−0.0706686	−	0.997500i	$-0.522513\pi$
$373$	−2.72967	−0.141337	−0.0706686	+	0.997500i	$0.522513\pi$
$374$	0	0
$375$	30.3482	1.56718
$376$	0	0
$377$	−1.50980	−0.0777587
$378$	0	0
$379$	16.4584	0.845412	0.422706	−	0.906267i	$-0.361080\pi$
$379$	16.4584	0.845412	0.422706	+	0.906267i	$0.361080\pi$
$380$	0	0
$381$	1.28312	0.0657362
$382$	0	0
$383$	−7.85978	−0.401616	−0.200808	−	0.979631i	$-0.564357\pi$
$383$	−7.85978	−0.401616	−0.200808	+	0.979631i	$0.564357\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.26857	−0.318649
$388$	0	0
$389$	9.64321	0.488930	0.244465	−	0.969658i	$-0.421388\pi$
$389$	9.64321	0.488930	0.244465	+	0.969658i	$0.421388\pi$
$390$	0	0
$391$	−32.7624	−1.65687
$392$	0	0
$393$	−34.8016	−1.75551
$394$	0	0
$395$	0.769200	0.0387026
$396$	0	0
$397$	9.43376	0.473467	0.236733	−	0.971575i	$-0.423923\pi$
$397$	9.43376	0.473467	0.236733	+	0.971575i	$0.423923\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.2199	−0.909857	−0.454929	−	0.890528i	$-0.650335\pi$
$401$	−18.2199	−0.909857	−0.454929	+	0.890528i	$0.650335\pi$
$402$	0	0
$403$	0.490200	0.0244186
$404$	0	0
$405$	−7.32501	−0.363983
$406$	0	0
$407$	6.82295	0.338201
$408$	0	0
$409$	−6.41147	−0.317027	−0.158513	−	0.987357i	$-0.550670\pi$
$409$	−6.41147	−0.317027	−0.158513	+	0.987357i	$0.550670\pi$
$410$	0	0
$411$	−47.0729	−2.32193
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7.87527	−0.386582
$416$	0	0
$417$	−59.6887	−2.92297
$418$	0	0
$419$	1.62092	0.0791871	0.0395935	−	0.999216i	$-0.487394\pi$
$419$	1.62092	0.0791871	0.0395935	+	0.999216i	$0.487394\pi$
$420$	0	0
$421$	31.1489	1.51810	0.759052	−	0.651030i	$-0.225663\pi$
$421$	31.1489	1.51810	0.759052	+	0.651030i	$0.225663\pi$
$422$	0	0
$423$	−40.6340	−1.97569
$424$	0	0
$425$	1.99907	0.0969690
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.532089	−0.0256895
$430$	0	0
$431$	−34.2080	−1.64774	−0.823871	−	0.566777i	$-0.808190\pi$
$431$	−34.2080	−1.64774	−0.823871	+	0.566777i	$0.808190\pi$
$432$	0	0
$433$	20.4979	0.985068	0.492534	−	0.870293i	$-0.336071\pi$
$433$	20.4979	0.985068	0.492534	+	0.870293i	$0.336071\pi$
$434$	0	0
$435$	55.2208	2.64764
$436$	0	0
$437$	−6.46110	−0.309077
$438$	0	0
$439$	31.3218	1.49491	0.747455	−	0.664313i	$-0.231276\pi$
$439$	31.3218	1.49491	0.747455	+	0.664313i	$0.231276\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.3979	1.39673	0.698367	−	0.715740i	$-0.253910\pi$
$443$	29.3979	1.39673	0.698367	+	0.715740i	$0.253910\pi$
$444$	0	0
$445$	−10.9213	−0.517718
$446$	0	0
$447$	−2.05644	−0.0972661
$448$	0	0
$449$	−6.90074	−0.325666	−0.162833	−	0.986654i	$-0.552063\pi$
$449$	−6.90074	−0.325666	−0.162833	+	0.986654i	$0.552063\pi$
$450$	0	0
$451$	0.426022	0.0200606
$452$	0	0
$453$	27.4834	1.29128
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.3746	0.578861	0.289431	−	0.957199i	$-0.406534\pi$
$457$	12.3746	0.578861	0.289431	+	0.957199i	$0.406534\pi$
$458$	0	0
$459$	25.8658	1.20731
$460$	0	0
$461$	−20.0942	−0.935881	−0.467940	−	0.883760i	$-0.655004\pi$
$461$	−20.0942	−0.935881	−0.467940	+	0.883760i	$0.655004\pi$
$462$	0	0
$463$	−8.69190	−0.403947	−0.201974	−	0.979391i	$-0.564735\pi$
$463$	−8.69190	−0.403947	−0.201974	+	0.979391i	$0.564735\pi$
$464$	0	0
$465$	−17.9290	−0.831438
$466$	0	0
$467$	4.76920	0.220692	0.110346	−	0.993893i	$-0.464804\pi$
$467$	4.76920	0.220692	0.110346	+	0.993893i	$0.464804\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−26.6091	−1.22608
$472$	0	0
$473$	1.18479	0.0544768
$474$	0	0
$475$	0.394238	0.0180889
$476$	0	0
$477$	34.6783	1.58781
$478$	0	0
$479$	−39.6141	−1.81001	−0.905007	−	0.425396i	$-0.860135\pi$
$479$	−39.6141	−1.81001	−0.905007	+	0.425396i	$0.860135\pi$
$480$	0	0
$481$	−1.26083	−0.0574889
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.7980	1.39846
$486$	0	0
$487$	−5.15570	−0.233627	−0.116813	−	0.993154i	$-0.537268\pi$
$487$	−5.15570	−0.233627	−0.116813	+	0.993154i	$0.537268\pi$
$488$	0	0
$489$	25.3482	1.14629
$490$	0	0
$491$	−10.1925	−0.459983	−0.229991	−	0.973193i	$-0.573870\pi$
$491$	−10.1925	−0.459983	−0.229991	+	0.973193i	$0.573870\pi$
$492$	0	0
$493$	−32.0378	−1.44291
$494$	0	0
$495$	12.4192	0.558202
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.88981	−0.263664	−0.131832	−	0.991272i	$-0.542086\pi$
$499$	−5.88981	−0.263664	−0.131832	+	0.991272i	$0.542086\pi$
$500$	0	0
$501$	24.1361	1.07832
$502$	0	0
$503$	−12.8348	−0.572276	−0.286138	−	0.958188i	$-0.592372\pi$
$503$	−12.8348	−0.572276	−0.286138	+	0.958188i	$0.592372\pi$
$504$	0	0
$505$	−13.8530	−0.616449
$506$	0	0
$507$	−37.3337	−1.65805
$508$	0	0
$509$	−32.2131	−1.42782	−0.713910	−	0.700238i	$-0.753077\pi$
$509$	−32.2131	−1.42782	−0.713910	+	0.700238i	$0.753077\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.10101	0.225215
$514$	0	0
$515$	−2.16250	−0.0952913
$516$	0	0
$517$	7.68004	0.337768
$518$	0	0
$519$	−15.6800	−0.688278
$520$	0	0
$521$	−18.5280	−0.811725	−0.405863	−	0.913934i	$-0.633029\pi$
$521$	−18.5280	−0.811725	−0.405863	+	0.913934i	$0.633029\pi$
$522$	0	0
$523$	−37.2422	−1.62849	−0.814243	−	0.580524i	$-0.802848\pi$
$523$	−37.2422	−1.62849	−0.814243	+	0.580524i	$0.802848\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.4020	0.453117
$528$	0	0
$529$	46.8066	2.03507
$530$	0	0
$531$	1.08141	0.0469294
$532$	0	0
$533$	−0.0787257	−0.00340999
$534$	0	0
$535$	−24.2882	−1.05007
$536$	0	0
$537$	29.0574	1.25392
$538$	0	0
$539$	0	0
$540$	0	0
$541$	36.1566	1.55449	0.777247	−	0.629195i	$-0.216615\pi$
$541$	36.1566	1.55449	0.777247	+	0.629195i	$0.216615\pi$
$542$	0	0
$543$	−57.2404	−2.45642
$544$	0	0
$545$	−1.11650	−0.0478254
$546$	0	0
$547$	−12.7980	−0.547202	−0.273601	−	0.961843i	$-0.588215\pi$
$547$	−12.7980	−0.547202	−0.273601	+	0.961843i	$0.588215\pi$
$548$	0	0
$549$	−77.1498	−3.29267
$550$	0	0
$551$	−6.31820	−0.269164
$552$	0	0
$553$	0	0
$554$	0	0
$555$	46.1147	1.95746
$556$	0	0
$557$	−30.6587	−1.29905	−0.649525	−	0.760340i	$-0.725032\pi$
$557$	−30.6587	−1.29905	−0.649525	+	0.760340i	$0.725032\pi$
$558$	0	0
$559$	−0.218941	−0.00926021
$560$	0	0
$561$	−11.2909	−0.476700
$562$	0	0
$563$	−14.5047	−0.611302	−0.305651	−	0.952144i	$-0.598874\pi$
$563$	−14.5047	−0.611302	−0.305651	+	0.952144i	$0.598874\pi$
$564$	0	0
$565$	7.37733	0.310366
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.0051	0.796733	0.398367	−	0.917226i	$-0.369577\pi$
$569$	19.0051	0.796733	0.398367	+	0.917226i	$0.369577\pi$
$570$	0	0
$571$	21.0009	0.878862	0.439431	−	0.898276i	$-0.355180\pi$
$571$	21.0009	0.878862	0.439431	+	0.898276i	$0.355180\pi$
$572$	0	0
$573$	73.0934	3.05352
$574$	0	0
$575$	−4.25940	−0.177629
$576$	0	0
$577$	5.18748	0.215958	0.107979	−	0.994153i	$-0.465562\pi$
$577$	5.18748	0.215958	0.107979	+	0.994153i	$0.465562\pi$
$578$	0	0
$579$	−22.2618	−0.925167
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.55438	−0.271455
$584$	0	0
$585$	−2.29498	−0.0948857
$586$	0	0
$587$	8.33813	0.344151	0.172076	−	0.985084i	$-0.444953\pi$
$587$	8.33813	0.344151	0.172076	+	0.985084i	$0.444953\pi$
$588$	0	0
$589$	2.05138	0.0845258
$590$	0	0
$591$	−15.4757	−0.636583
$592$	0	0
$593$	−6.10195	−0.250577	−0.125288	−	0.992120i	$-0.539986\pi$
$593$	−6.10195	−0.250577	−0.125288	+	0.992120i	$0.539986\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.2841	1.03481
$598$	0	0
$599$	13.8503	0.565907	0.282954	−	0.959134i	$-0.408686\pi$
$599$	13.8503	0.565907	0.282954	+	0.959134i	$0.408686\pi$
$600$	0	0
$601$	−31.2695	−1.27551	−0.637755	−	0.770239i	$-0.720137\pi$
$601$	−31.2695	−1.27551	−0.637755	+	0.770239i	$0.720137\pi$
$602$	0	0
$603$	−19.8871	−0.809866
$604$	0	0
$605$	−2.34730	−0.0954312
$606$	0	0
$607$	37.7962	1.53410	0.767051	−	0.641587i	$-0.221724\pi$
$607$	37.7962	1.53410	0.767051	+	0.641587i	$0.221724\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.41921	−0.0574153
$612$	0	0
$613$	21.1343	0.853608	0.426804	−	0.904344i	$-0.359639\pi$
$613$	21.1343	0.853608	0.426804	+	0.904344i	$0.359639\pi$
$614$	0	0
$615$	2.87939	0.116108
$616$	0	0
$617$	23.7743	0.957115	0.478558	−	0.878056i	$-0.341160\pi$
$617$	23.7743	0.957115	0.478558	+	0.878056i	$0.341160\pi$
$618$	0	0
$619$	13.5021	0.542694	0.271347	−	0.962482i	$-0.412531\pi$
$619$	13.5021	0.542694	0.271347	+	0.962482i	$0.412531\pi$
$620$	0	0
$621$	−55.1121	−2.21157
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.2891	−1.09156
$626$	0	0
$627$	−2.22668	−0.0889251
$628$	0	0
$629$	−26.7547	−1.06678
$630$	0	0
$631$	38.2354	1.52213	0.761063	−	0.648678i	$-0.224678\pi$
$631$	38.2354	1.52213	0.761063	+	0.648678i	$0.224678\pi$
$632$	0	0
$633$	−12.2121	−0.485389
$634$	0	0
$635$	−1.04601	−0.0415096
$636$	0	0
$637$	0	0
$638$	0	0
$639$	52.7279	2.08588
$640$	0	0
$641$	38.5553	1.52284	0.761422	−	0.648257i	$-0.224502\pi$
$641$	38.5553	1.52284	0.761422	+	0.648257i	$0.224502\pi$
$642$	0	0
$643$	19.2627	0.759647	0.379823	−	0.925059i	$-0.375985\pi$
$643$	19.2627	0.759647	0.379823	+	0.925059i	$0.375985\pi$
$644$	0	0
$645$	8.00774	0.315304
$646$	0	0
$647$	39.7844	1.56408	0.782042	−	0.623225i	$-0.214178\pi$
$647$	39.7844	1.56408	0.782042	+	0.623225i	$0.214178\pi$
$648$	0	0
$649$	−0.204393	−0.00802312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.76651	0.186528	0.0932640	−	0.995641i	$-0.470270\pi$
$653$	4.76651	0.186528	0.0932640	+	0.995641i	$0.470270\pi$
$654$	0	0
$655$	28.3705	1.10853
$656$	0	0
$657$	−0.638156	−0.0248968
$658$	0	0
$659$	38.3550	1.49410	0.747050	−	0.664768i	$-0.231470\pi$
$659$	38.3550	1.49410	0.747050	+	0.664768i	$0.231470\pi$
$660$	0	0
$661$	10.3628	0.403065	0.201533	−	0.979482i	$-0.435408\pi$
$661$	10.3628	0.403065	0.201533	+	0.979482i	$0.435408\pi$
$662$	0	0
$663$	2.08647	0.0810316
$664$	0	0
$665$	0	0
$666$	0	0
$667$	68.2627	2.64314
$668$	0	0
$669$	−51.9299	−2.00773
$670$	0	0
$671$	14.5817	0.562921
$672$	0	0
$673$	−25.6168	−0.987455	−0.493728	−	0.869617i	$-0.664366\pi$
$673$	−25.6168	−0.987455	−0.493728	+	0.869617i	$0.664366\pi$
$674$	0	0
$675$	3.36278	0.129433
$676$	0	0
$677$	−41.8375	−1.60795	−0.803973	−	0.594666i	$-0.797284\pi$
$677$	−41.8375	−1.60795	−0.803973	+	0.594666i	$0.797284\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	52.4201	2.00874
$682$	0	0
$683$	46.0428	1.76178	0.880890	−	0.473321i	$-0.156945\pi$
$683$	46.0428	1.76178	0.880890	+	0.473321i	$0.156945\pi$
$684$	0	0
$685$	38.3741	1.46620
$686$	0	0
$687$	−55.4029	−2.11375
$688$	0	0
$689$	1.21120	0.0461430
$690$	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
$695$	48.6587	1.84573
$696$	0	0
$697$	−1.67055	−0.0632766
$698$	0	0
$699$	26.8503	1.01557
$700$	0	0
$701$	−32.8144	−1.23938	−0.619691	−	0.784846i	$-0.712742\pi$
$701$	−32.8144	−1.23938	−0.619691	+	0.784846i	$0.712742\pi$
$702$	0	0
$703$	−5.27631	−0.199000
$704$	0	0
$705$	51.9077	1.95496
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.40198	0.277987	0.138994	−	0.990293i	$-0.455613\pi$
$709$	7.40198	0.277987	0.138994	+	0.990293i	$0.455613\pi$
$710$	0	0
$711$	−1.73379	−0.0650223
$712$	0	0
$713$	−22.1634	−0.830027
$714$	0	0
$715$	0.433763	0.0162218
$716$	0	0
$717$	−32.2199	−1.20327
$718$	0	0
$719$	−14.1370	−0.527222	−0.263611	−	0.964629i	$-0.584914\pi$
$719$	−14.1370	−0.527222	−0.263611	+	0.964629i	$0.584914\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−58.1762	−2.16360
$724$	0	0
$725$	−4.16519	−0.154691
$726$	0	0
$727$	15.2790	0.566667	0.283333	−	0.959021i	$-0.408560\pi$
$727$	15.2790	0.566667	0.283333	+	0.959021i	$0.408560\pi$
$728$	0	0
$729$	−40.4688	−1.49885
$730$	0	0
$731$	−4.64590	−0.171835
$732$	0	0
$733$	−1.34493	−0.0496762	−0.0248381	−	0.999691i	$-0.507907\pi$
$733$	−1.34493	−0.0496762	−0.0248381	+	0.999691i	$0.507907\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.75877	0.138456
$738$	0	0
$739$	25.6536	0.943684	0.471842	−	0.881683i	$-0.343589\pi$
$739$	25.6536	0.943684	0.471842	+	0.881683i	$0.343589\pi$
$740$	0	0
$741$	0.411474	0.0151159
$742$	0	0
$743$	6.08109	0.223094	0.111547	−	0.993759i	$-0.464420\pi$
$743$	6.08109	0.223094	0.111547	+	0.993759i	$0.464420\pi$
$744$	0	0
$745$	1.67642	0.0614194
$746$	0	0
$747$	17.7510	0.649476
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.8108	−0.759396	−0.379698	−	0.925111i	$-0.623972\pi$
$751$	−20.8108	−0.759396	−0.379698	+	0.925111i	$0.623972\pi$
$752$	0	0
$753$	−16.9659	−0.618270
$754$	0	0
$755$	−22.4047	−0.815389
$756$	0	0
$757$	−5.25402	−0.190961	−0.0954804	−	0.995431i	$-0.530439\pi$
$757$	−5.25402	−0.190961	−0.0954804	+	0.995431i	$0.530439\pi$
$758$	0	0
$759$	24.0574	0.873227
$760$	0	0
$761$	40.9813	1.48557	0.742786	−	0.669529i	$-0.233504\pi$
$761$	40.9813	1.48557	0.742786	+	0.669529i	$0.233504\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−48.6991	−1.76072
$766$	0	0
$767$	0.0377703	0.00136381
$768$	0	0
$769$	−47.5580	−1.71499	−0.857493	−	0.514496i	$-0.827979\pi$
$769$	−47.5580	−1.71499	−0.857493	+	0.514496i	$0.827979\pi$
$770$	0	0
$771$	7.87939	0.283769
$772$	0	0
$773$	−20.4929	−0.737078	−0.368539	−	0.929612i	$-0.620142\pi$
$773$	−20.4929	−0.737078	−0.368539	+	0.929612i	$0.620142\pi$
$774$	0	0
$775$	1.35235	0.0485778
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.329451	−0.0118038
$780$	0	0
$781$	−9.96585	−0.356606
$782$	0	0
$783$	−53.8931	−1.92598
$784$	0	0
$785$	21.6919	0.774217
$786$	0	0
$787$	−19.1206	−0.681576	−0.340788	−	0.940140i	$-0.610694\pi$
$787$	−19.1206	−0.681576	−0.340788	+	0.940140i	$0.610694\pi$
$788$	0	0
$789$	−17.8452	−0.635307
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.69459	−0.0956878
$794$	0	0
$795$	−44.2995	−1.57114
$796$	0	0
$797$	52.5144	1.86015	0.930077	−	0.367365i	$-0.119740\pi$
$797$	52.5144	1.86015	0.930077	+	0.367365i	$0.119740\pi$
$798$	0	0
$799$	−30.1156	−1.06541
$800$	0	0
$801$	24.6168	0.869792
$802$	0	0
$803$	0.120615	0.00425640
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.1206	0.426666
$808$	0	0
$809$	40.6715	1.42993	0.714967	−	0.699159i	$-0.246442\pi$
$809$	40.6715	1.42993	0.714967	+	0.699159i	$0.246442\pi$
$810$	0	0
$811$	−20.5648	−0.722128	−0.361064	−	0.932541i	$-0.617586\pi$
$811$	−20.5648	−0.722128	−0.361064	+	0.932541i	$0.617586\pi$
$812$	0	0
$813$	73.6596	2.58336
$814$	0	0
$815$	−20.6641	−0.723831
$816$	0	0
$817$	−0.916222	−0.0320545
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.36278	0.222062	0.111031	−	0.993817i	$-0.464585\pi$
$821$	6.36278	0.222062	0.111031	+	0.993817i	$0.464585\pi$
$822$	0	0
$823$	2.50887	0.0874536	0.0437268	−	0.999044i	$-0.486077\pi$
$823$	2.50887	0.0874536	0.0437268	+	0.999044i	$0.486077\pi$
$824$	0	0
$825$	−1.46791	−0.0511061
$826$	0	0
$827$	41.2645	1.43491	0.717453	−	0.696607i	$-0.245308\pi$
$827$	41.2645	1.43491	0.717453	+	0.696607i	$0.245308\pi$
$828$	0	0
$829$	46.7779	1.62466	0.812331	−	0.583196i	$-0.198198\pi$
$829$	46.7779	1.62466	0.812331	+	0.583196i	$0.198198\pi$
$830$	0	0
$831$	−44.2208	−1.53400
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19.6759	−0.680913
$836$	0	0
$837$	17.4979	0.604817
$838$	0	0
$839$	16.5439	0.571161	0.285580	−	0.958355i	$-0.407814\pi$
$839$	16.5439	0.571161	0.285580	+	0.958355i	$0.407814\pi$
$840$	0	0
$841$	37.7529	1.30182
$842$	0	0
$843$	55.4397	1.90945
$844$	0	0
$845$	30.4347	1.04699
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−77.9309	−2.67458
$850$	0	0
$851$	57.0060	1.95414
$852$	0	0
$853$	−13.2163	−0.452516	−0.226258	−	0.974067i	$-0.572649\pi$
$853$	−13.2163	−0.452516	−0.226258	+	0.974067i	$0.572649\pi$
$854$	0	0
$855$	−9.60401	−0.328450
$856$	0	0
$857$	50.7211	1.73260	0.866300	−	0.499524i	$-0.166492\pi$
$857$	50.7211	1.73260	0.866300	+	0.499524i	$0.166492\pi$
$858$	0	0
$859$	55.1124	1.88041	0.940205	−	0.340609i	$-0.110633\pi$
$859$	55.1124	1.88041	0.940205	+	0.340609i	$0.110633\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.5202	−0.358113	−0.179056	−	0.983839i	$-0.557304\pi$
$863$	−10.5202	−0.358113	−0.179056	+	0.983839i	$0.557304\pi$
$864$	0	0
$865$	12.7825	0.434618
$866$	0	0
$867$	−4.67499	−0.158771
$868$	0	0
$869$	0.327696	0.0111163
$870$	0	0
$871$	−0.694593	−0.0235354
$872$	0	0
$873$	−69.4193	−2.34949
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.6536	−1.33901	−0.669504	−	0.742808i	$-0.733493\pi$
$877$	−39.6536	−1.33901	−0.669504	+	0.742808i	$0.733493\pi$
$878$	0	0
$879$	−11.6186	−0.391884
$880$	0	0
$881$	−25.0077	−0.842532	−0.421266	−	0.906937i	$-0.638414\pi$
$881$	−25.0077	−0.842532	−0.421266	+	0.906937i	$0.638414\pi$
$882$	0	0
$883$	−50.9341	−1.71407	−0.857034	−	0.515260i	$-0.827695\pi$
$883$	−50.9341	−1.71407	−0.857034	+	0.515260i	$0.827695\pi$
$884$	0	0
$885$	−1.38144	−0.0464368
$886$	0	0
$887$	32.0360	1.07566	0.537832	−	0.843052i	$-0.319243\pi$
$887$	32.0360	1.07566	0.537832	+	0.843052i	$0.319243\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.12061	−0.104545
$892$	0	0
$893$	−5.93912	−0.198745
$894$	0	0
$895$	−23.6878	−0.791795
$896$	0	0
$897$	−4.44562	−0.148435
$898$	0	0
$899$	−21.6732	−0.722843
$900$	0	0
$901$	25.7015	0.856242
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46.6628	1.55112
$906$	0	0
$907$	16.4730	0.546976	0.273488	−	0.961875i	$-0.411823\pi$
$907$	16.4730	0.546976	0.273488	+	0.961875i	$0.411823\pi$
$908$	0	0
$909$	31.2249	1.03567
$910$	0	0
$911$	45.3979	1.50410	0.752049	−	0.659107i	$-0.229066\pi$
$911$	45.3979	1.50410	0.752049	+	0.659107i	$0.229066\pi$
$912$	0	0
$913$	−3.35504	−0.111036
$914$	0	0
$915$	98.5545	3.25811
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12.3209	0.406429	0.203214	−	0.979134i	$-0.434861\pi$
$919$	12.3209	0.406429	0.203214	+	0.979134i	$0.434861\pi$
$920$	0	0
$921$	8.43376	0.277902
$922$	0	0
$923$	1.84161	0.0606175
$924$	0	0
$925$	−3.47834	−0.114367
$926$	0	0
$927$	4.87433	0.160094
$928$	0	0
$929$	−5.93313	−0.194660	−0.0973299	−	0.995252i	$-0.531030\pi$
$929$	−5.93313	−0.194660	−0.0973299	+	0.995252i	$0.531030\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	44.9864	1.47279
$934$	0	0
$935$	9.20439	0.301016
$936$	0	0
$937$	45.0242	1.47088	0.735438	−	0.677593i	$-0.236977\pi$
$937$	45.0242	1.47088	0.735438	+	0.677593i	$0.236977\pi$
$938$	0	0
$939$	63.2772	2.06497
$940$	0	0
$941$	6.06242	0.197629	0.0988147	−	0.995106i	$-0.468495\pi$
$941$	6.06242	0.197629	0.0988147	+	0.995106i	$0.468495\pi$
$942$	0	0
$943$	3.55943	0.115911
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.9426	0.615553	0.307776	−	0.951459i	$-0.400415\pi$
$947$	18.9426	0.615553	0.307776	+	0.951459i	$0.400415\pi$
$948$	0	0
$949$	−0.0222887	−0.000723522 0
$950$	0	0
$951$	−8.67230	−0.281219
$952$	0	0
$953$	−12.4970	−0.404818	−0.202409	−	0.979301i	$-0.564877\pi$
$953$	−12.4970	−0.404818	−0.202409	+	0.979301i	$0.564877\pi$
$954$	0	0
$955$	−59.5863	−1.92817
$956$	0	0
$957$	23.5253	0.760464
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.9632	−0.773005
$962$	0	0
$963$	54.7461	1.76417
$964$	0	0
$965$	18.1480	0.584203
$966$	0	0
$967$	18.5294	0.595865	0.297933	−	0.954587i	$-0.403703\pi$
$967$	18.5294	0.595865	0.297933	+	0.954587i	$0.403703\pi$
$968$	0	0
$969$	8.73143	0.280494
$970$	0	0
$971$	41.4739	1.33096	0.665480	−	0.746415i	$-0.268227\pi$
$971$	41.4739	1.33096	0.665480	+	0.746415i	$0.268227\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.271259	0.00868724
$976$	0	0
$977$	−22.2321	−0.711267	−0.355633	−	0.934626i	$-0.615735\pi$
$977$	−22.2321	−0.711267	−0.355633	+	0.934626i	$0.615735\pi$
$978$	0	0
$979$	−4.65270	−0.148701
$980$	0	0
$981$	2.51661	0.0803491
$982$	0	0
$983$	35.4124	1.12948	0.564740	−	0.825269i	$-0.308976\pi$
$983$	35.4124	1.12948	0.564740	+	0.825269i	$0.308976\pi$
$984$	0	0
$985$	12.6159	0.401975
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.89899	0.314769
$990$	0	0
$991$	−45.5725	−1.44766	−0.723830	−	0.689979i	$-0.757620\pi$
$991$	−45.5725	−1.44766	−0.723830	+	0.689979i	$0.757620\pi$
$992$	0	0
$993$	−79.0934	−2.50995
$994$	0	0
$995$	−20.6117	−0.653436
$996$	0	0
$997$	−22.8367	−0.723245	−0.361622	−	0.932325i	$-0.617777\pi$
$997$	−22.8367	−0.723245	−0.361622	+	0.932325i	$0.617777\pi$
$998$	0	0
$999$	−45.0060	−1.42393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.co.1.3		3
4.3	odd	2		539.2.a.g.1.1		3
7.3	odd	6		1232.2.q.m.177.3		6
7.5	odd	6		1232.2.q.m.529.3		6
7.6	odd	2		8624.2.a.ch.1.1		3
12.11	even	2		4851.2.a.bk.1.3		3
28.3	even	6		77.2.e.a.23.3	✓	6
28.11	odd	6		539.2.e.m.177.3		6
28.19	even	6		77.2.e.a.67.3	yes	6
28.23	odd	6		539.2.e.m.67.3		6
28.27	even	2		539.2.a.j.1.1		3
44.43	even	2		5929.2.a.u.1.3		3
84.47	odd	6		693.2.i.h.298.1		6
84.59	odd	6		693.2.i.h.100.1		6
84.83	odd	2		4851.2.a.bj.1.3		3
308.3	even	30		847.2.n.g.9.1		24
308.19	odd	30		847.2.n.f.130.1		24
308.31	even	30		847.2.n.g.807.1		24
308.47	even	30		847.2.n.g.130.3		24
308.59	even	30		847.2.n.g.632.3		24
308.75	even	30		847.2.n.g.81.3		24
308.87	odd	6		847.2.e.c.485.1		6
308.103	even	30		847.2.n.g.753.1		24
308.115	even	30		847.2.n.g.366.3		24
308.131	odd	6		847.2.e.c.606.1		6
308.159	even	30		847.2.n.g.487.1		24
308.171	odd	30		847.2.n.f.366.1		24
308.215	odd	30		847.2.n.f.487.3		24
308.227	odd	30		847.2.n.f.632.1		24
308.255	odd	30		847.2.n.f.807.3		24
308.271	odd	30		847.2.n.f.753.3		24
308.283	odd	30		847.2.n.f.9.3		24
308.299	odd	30		847.2.n.f.81.1		24
308.307	odd	2		5929.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.a.23.3	✓	6	28.3	even	6
77.2.e.a.67.3	yes	6	28.19	even	6
539.2.a.g.1.1		3	4.3	odd	2
539.2.a.j.1.1		3	28.27	even	2
539.2.e.m.67.3		6	28.23	odd	6
539.2.e.m.177.3		6	28.11	odd	6
693.2.i.h.100.1		6	84.59	odd	6
693.2.i.h.298.1		6	84.47	odd	6
847.2.e.c.485.1		6	308.87	odd	6
847.2.e.c.606.1		6	308.131	odd	6
847.2.n.f.9.3		24	308.283	odd	30
847.2.n.f.81.1		24	308.299	odd	30
847.2.n.f.130.1		24	308.19	odd	30
847.2.n.f.366.1		24	308.171	odd	30
847.2.n.f.487.3		24	308.215	odd	30
847.2.n.f.632.1		24	308.227	odd	30
847.2.n.f.753.3		24	308.271	odd	30
847.2.n.f.807.3		24	308.255	odd	30
847.2.n.g.9.1		24	308.3	even	30
847.2.n.g.81.3		24	308.75	even	30
847.2.n.g.130.3		24	308.47	even	30
847.2.n.g.366.3		24	308.115	even	30
847.2.n.g.487.1		24	308.159	even	30
847.2.n.g.632.3		24	308.59	even	30
847.2.n.g.753.1		24	308.103	even	30
847.2.n.g.807.1		24	308.31	even	30
1232.2.q.m.177.3		6	7.3	odd	6
1232.2.q.m.529.3		6	7.5	odd	6
4851.2.a.bj.1.3		3	84.83	odd	2
4851.2.a.bk.1.3		3	12.11	even	2
5929.2.a.u.1.3		3	44.43	even	2
5929.2.a.x.1.3		3	308.307	odd	2
8624.2.a.ch.1.1		3	7.6	odd	2
8624.2.a.co.1.3		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.87939 q^{3} -2.34730 q^{5} +5.29086 q^{9} -1.00000 q^{11} +0.184793 q^{13} -6.75877 q^{15} +3.92127 q^{17} +0.773318 q^{19} -8.35504 q^{23} +0.509800 q^{25} +6.59627 q^{27} -8.17024 q^{29} +2.65270 q^{31} -2.87939 q^{33} -6.82295 q^{37} +0.532089 q^{39} -0.426022 q^{41} -1.18479 q^{43} -12.4192 q^{45} -7.68004 q^{47} +11.2909 q^{51} +6.55438 q^{53} +2.34730 q^{55} +2.22668 q^{57} +0.204393 q^{59} -14.5817 q^{61} -0.433763 q^{65} -3.75877 q^{67} -24.0574 q^{69} +9.96585 q^{71} -0.120615 q^{73} +1.46791 q^{75} -0.327696 q^{79} +3.12061 q^{81} +3.35504 q^{83} -9.20439 q^{85} -23.5253 q^{87} +4.65270 q^{89} +7.63816 q^{93} -1.81521 q^{95} -13.1206 q^{97} -5.29086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5} - 3 q^{11} - 3 q^{13} - 9 q^{15} + 3 q^{17} + 9 q^{19} + 3 q^{25} + 6 q^{27} - 3 q^{29} + 9 q^{31} - 3 q^{33} - 3 q^{39} - 9 q^{41} - 3 q^{45} - 3 q^{47} + 18 q^{51} + 9 q^{53} + 6 q^{55}+ \cdots - 45 q^{97}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more