LMFDB - Embedded newform 9464.2.a.br.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,-4,0,-15,0,16,0,-15,0,0,0,-8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-2.57537$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.57537 q^{3} -3.57643 q^{5} -1.00000 q^{7} +3.63254 q^{9} -1.05933 q^{11} -9.21063 q^{15} -2.94347 q^{17} +2.74342 q^{19} -2.57537 q^{21} +2.27885 q^{23} +7.79083 q^{25} +1.62902 q^{27} +8.96305 q^{29} +0.354625 q^{31} -2.72817 q^{33} +3.57643 q^{35} +5.93395 q^{37} -11.0770 q^{41} -4.68820 q^{43} -12.9915 q^{45} +7.27967 q^{47} +1.00000 q^{49} -7.58052 q^{51} +11.3686 q^{53} +3.78863 q^{55} +7.06533 q^{57} -6.73376 q^{59} -10.8691 q^{61} -3.63254 q^{63} -0.482358 q^{67} +5.86890 q^{69} -7.81356 q^{71} -12.2692 q^{73} +20.0643 q^{75} +1.05933 q^{77} +0.649730 q^{79} -6.70229 q^{81} -12.6355 q^{83} +10.5271 q^{85} +23.0832 q^{87} +15.4274 q^{89} +0.913291 q^{93} -9.81164 q^{95} -3.60537 q^{97} -3.84806 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

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
$3$	2.57537	1.48689	0.743446	−	0.668796i	$-0.233190\pi$
$3$	2.57537	1.48689	0.743446	+	0.668796i	$0.233190\pi$
$4$	0	0
$5$	−3.57643	−1.59943	−0.799714	−	0.600382i	$-0.795015\pi$
$5$	−3.57643	−1.59943	−0.799714	+	0.600382i	$0.795015\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.63254	1.21085
$10$	0	0
$11$	−1.05933	−0.319401	−0.159700	−	0.987166i	$-0.551053\pi$
$11$	−1.05933	−0.319401	−0.159700	+	0.987166i	$0.551053\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−9.21063	−2.37817
$16$	0	0
$17$	−2.94347	−0.713896	−0.356948	−	0.934124i	$-0.616183\pi$
$17$	−2.94347	−0.713896	−0.356948	+	0.934124i	$0.616183\pi$
$18$	0	0
$19$	2.74342	0.629384	0.314692	−	0.949194i	$-0.398099\pi$
$19$	2.74342	0.629384	0.314692	+	0.949194i	$0.398099\pi$
$20$	0	0
$21$	−2.57537	−0.561992
$22$	0	0
$23$	2.27885	0.475174	0.237587	−	0.971366i	$-0.423644\pi$
$23$	2.27885	0.475174	0.237587	+	0.971366i	$0.423644\pi$
$24$	0	0
$25$	7.79083	1.55817
$26$	0	0
$27$	1.62902	0.313504
$28$	0	0
$29$	8.96305	1.66440	0.832198	−	0.554479i	$-0.187082\pi$
$29$	8.96305	1.66440	0.832198	+	0.554479i	$0.187082\pi$
$30$	0	0
$31$	0.354625	0.0636926	0.0318463	−	0.999493i	$-0.489861\pi$
$31$	0.354625	0.0636926	0.0318463	+	0.999493i	$0.489861\pi$
$32$	0	0
$33$	−2.72817	−0.474914
$34$	0	0
$35$	3.57643	0.604527
$36$	0	0
$37$	5.93395	0.975535	0.487767	−	0.872974i	$-0.337811\pi$
$37$	5.93395	0.975535	0.487767	+	0.872974i	$0.337811\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.0770	−1.72994	−0.864971	−	0.501822i	$-0.832663\pi$
$41$	−11.0770	−1.72994	−0.864971	+	0.501822i	$0.832663\pi$
$42$	0	0
$43$	−4.68820	−0.714944	−0.357472	−	0.933924i	$-0.616361\pi$
$43$	−4.68820	−0.714944	−0.357472	+	0.933924i	$0.616361\pi$
$44$	0	0
$45$	−12.9915	−1.93666
$46$	0	0
$47$	7.27967	1.06185	0.530924	−	0.847419i	$-0.321845\pi$
$47$	7.27967	1.06185	0.530924	+	0.847419i	$0.321845\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.58052	−1.06149
$52$	0	0
$53$	11.3686	1.56160	0.780799	−	0.624782i	$-0.214812\pi$
$53$	11.3686	1.56160	0.780799	+	0.624782i	$0.214812\pi$
$54$	0	0
$55$	3.78863	0.510858
$56$	0	0
$57$	7.06533	0.935825
$58$	0	0
$59$	−6.73376	−0.876660	−0.438330	−	0.898814i	$-0.644430\pi$
$59$	−6.73376	−0.876660	−0.438330	+	0.898814i	$0.644430\pi$
$60$	0	0
$61$	−10.8691	−1.39165	−0.695824	−	0.718212i	$-0.744961\pi$
$61$	−10.8691	−1.39165	−0.695824	+	0.718212i	$0.744961\pi$
$62$	0	0
$63$	−3.63254	−0.457657
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.482358	−0.0589294	−0.0294647	−	0.999566i	$-0.509380\pi$
$67$	−0.482358	−0.0589294	−0.0294647	+	0.999566i	$0.509380\pi$
$68$	0	0
$69$	5.86890	0.706532
$70$	0	0
$71$	−7.81356	−0.927299	−0.463650	−	0.886019i	$-0.653460\pi$
$71$	−7.81356	−0.927299	−0.463650	+	0.886019i	$0.653460\pi$
$72$	0	0
$73$	−12.2692	−1.43600	−0.718002	−	0.696041i	$-0.754943\pi$
$73$	−12.2692	−1.43600	−0.718002	+	0.696041i	$0.754943\pi$
$74$	0	0
$75$	20.0643	2.31682
$76$	0	0
$77$	1.05933	0.120722
$78$	0	0
$79$	0.649730	0.0731003	0.0365502	−	0.999332i	$-0.488363\pi$
$79$	0.649730	0.0731003	0.0365502	+	0.999332i	$0.488363\pi$
$80$	0	0
$81$	−6.70229	−0.744699
$82$	0	0
$83$	−12.6355	−1.38692	−0.693462	−	0.720493i	$-0.743916\pi$
$83$	−12.6355	−1.38692	−0.693462	+	0.720493i	$0.743916\pi$
$84$	0	0
$85$	10.5271	1.14182
$86$	0	0
$87$	23.0832	2.47478
$88$	0	0
$89$	15.4274	1.63530	0.817650	−	0.575715i	$-0.195276\pi$
$89$	15.4274	1.63530	0.817650	+	0.575715i	$0.195276\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.913291	0.0947039
$94$	0	0
$95$	−9.81164	−1.00665
$96$	0	0
$97$	−3.60537	−0.366070	−0.183035	−	0.983106i	$-0.558592\pi$
$97$	−3.60537	−0.366070	−0.183035	+	0.983106i	$0.558592\pi$
$98$	0	0
$99$	−3.84806	−0.386745
$100$	0	0
$101$	−6.06007	−0.603000	−0.301500	−	0.953466i	$-0.597487\pi$
$101$	−6.06007	−0.603000	−0.301500	+	0.953466i	$0.597487\pi$
$102$	0	0
$103$	−17.6546	−1.73956	−0.869779	−	0.493442i	$-0.835738\pi$
$103$	−17.6546	−1.73956	−0.869779	+	0.493442i	$0.835738\pi$
$104$	0	0
$105$	9.21063	0.898865
$106$	0	0
$107$	5.54775	0.536321	0.268161	−	0.963374i	$-0.413584\pi$
$107$	5.54775	0.536321	0.268161	+	0.963374i	$0.413584\pi$
$108$	0	0
$109$	−4.54906	−0.435721	−0.217860	−	0.975980i	$-0.569908\pi$
$109$	−4.54906	−0.435721	−0.217860	+	0.975980i	$0.569908\pi$
$110$	0	0
$111$	15.2821	1.45051
$112$	0	0
$113$	16.0328	1.50823	0.754117	−	0.656740i	$-0.228065\pi$
$113$	16.0328	1.50823	0.754117	+	0.656740i	$0.228065\pi$
$114$	0	0
$115$	−8.15016	−0.760006
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.94347	0.269827
$120$	0	0
$121$	−9.87781	−0.897983
$122$	0	0
$123$	−28.5275	−2.57224
$124$	0	0
$125$	−9.98122	−0.892747
$126$	0	0
$127$	12.2334	1.08554	0.542771	−	0.839881i	$-0.317375\pi$
$127$	12.2334	1.08554	0.542771	+	0.839881i	$0.317375\pi$
$128$	0	0
$129$	−12.0739	−1.06304
$130$	0	0
$131$	−8.30601	−0.725700	−0.362850	−	0.931848i	$-0.618196\pi$
$131$	−8.30601	−0.725700	−0.362850	+	0.931848i	$0.618196\pi$
$132$	0	0
$133$	−2.74342	−0.237885
$134$	0	0
$135$	−5.82606	−0.501427
$136$	0	0
$137$	−3.57641	−0.305554	−0.152777	−	0.988261i	$-0.548822\pi$
$137$	−3.57641	−0.305554	−0.152777	+	0.988261i	$0.548822\pi$
$138$	0	0
$139$	−18.5745	−1.57547	−0.787735	−	0.616015i	$-0.788746\pi$
$139$	−18.5745	−1.57547	−0.787735	+	0.616015i	$0.788746\pi$
$140$	0	0
$141$	18.7478	1.57885
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−32.0557	−2.66208
$146$	0	0
$147$	2.57537	0.212413
$148$	0	0
$149$	−17.5119	−1.43463	−0.717316	−	0.696748i	$-0.754630\pi$
$149$	−17.5119	−1.43463	−0.717316	+	0.696748i	$0.754630\pi$
$150$	0	0
$151$	4.34030	0.353208	0.176604	−	0.984282i	$-0.443489\pi$
$151$	4.34030	0.353208	0.176604	+	0.984282i	$0.443489\pi$
$152$	0	0
$153$	−10.6923	−0.864417
$154$	0	0
$155$	−1.26829	−0.101872
$156$	0	0
$157$	−13.7386	−1.09646	−0.548230	−	0.836327i	$-0.684698\pi$
$157$	−13.7386	−1.09646	−0.548230	+	0.836327i	$0.684698\pi$
$158$	0	0
$159$	29.2784	2.32193
$160$	0	0
$161$	−2.27885	−0.179599
$162$	0	0
$163$	−17.3389	−1.35809	−0.679044	−	0.734097i	$-0.737606\pi$
$163$	−17.3389	−1.35809	−0.679044	+	0.734097i	$0.737606\pi$
$164$	0	0
$165$	9.75712	0.759591
$166$	0	0
$167$	−9.77388	−0.756326	−0.378163	−	0.925739i	$-0.623444\pi$
$167$	−9.77388	−0.756326	−0.378163	+	0.925739i	$0.623444\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	9.96557	0.762086
$172$	0	0
$173$	−11.8034	−0.897399	−0.448700	−	0.893683i	$-0.648113\pi$
$173$	−11.8034	−0.897399	−0.448700	+	0.893683i	$0.648113\pi$
$174$	0	0
$175$	−7.79083	−0.588932
$176$	0	0
$177$	−17.3419	−1.30350
$178$	0	0
$179$	22.2553	1.66344	0.831719	−	0.555196i	$-0.187357\pi$
$179$	22.2553	1.66344	0.831719	+	0.555196i	$0.187357\pi$
$180$	0	0
$181$	−11.6707	−0.867478	−0.433739	−	0.901039i	$-0.642806\pi$
$181$	−11.6707	−0.867478	−0.433739	+	0.901039i	$0.642806\pi$
$182$	0	0
$183$	−27.9920	−2.06923
$184$	0	0
$185$	−21.2223	−1.56030
$186$	0	0
$187$	3.11811	0.228019
$188$	0	0
$189$	−1.62902	−0.118493
$190$	0	0
$191$	7.31107	0.529010	0.264505	−	0.964384i	$-0.414791\pi$
$191$	7.31107	0.529010	0.264505	+	0.964384i	$0.414791\pi$
$192$	0	0
$193$	−22.7522	−1.63774	−0.818871	−	0.573977i	$-0.805400\pi$
$193$	−22.7522	−1.63774	−0.818871	+	0.573977i	$0.805400\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.4627	−1.31541	−0.657705	−	0.753275i	$-0.728473\pi$
$197$	−18.4627	−1.31541	−0.657705	+	0.753275i	$0.728473\pi$
$198$	0	0
$199$	−19.7953	−1.40325	−0.701626	−	0.712546i	$-0.747542\pi$
$199$	−19.7953	−1.40325	−0.701626	+	0.712546i	$0.747542\pi$
$200$	0	0
$201$	−1.24225	−0.0876216
$202$	0	0
$203$	−8.96305	−0.629082
$204$	0	0
$205$	39.6162	2.76692
$206$	0	0
$207$	8.27802	0.575362
$208$	0	0
$209$	−2.90619	−0.201026
$210$	0	0
$211$	−0.531926	−0.0366193	−0.0183096	−	0.999832i	$-0.505828\pi$
$211$	−0.531926	−0.0366193	−0.0183096	+	0.999832i	$0.505828\pi$
$212$	0	0
$213$	−20.1228	−1.37879
$214$	0	0
$215$	16.7670	1.14350
$216$	0	0
$217$	−0.354625	−0.0240735
$218$	0	0
$219$	−31.5978	−2.13518
$220$	0	0
$221$	0	0
$222$	0	0
$223$	25.4414	1.70368	0.851841	−	0.523801i	$-0.175486\pi$
$223$	25.4414	1.70368	0.851841	+	0.523801i	$0.175486\pi$
$224$	0	0
$225$	28.3005	1.88670
$226$	0	0
$227$	16.4262	1.09025	0.545123	−	0.838356i	$-0.316483\pi$
$227$	16.4262	1.09025	0.545123	+	0.838356i	$0.316483\pi$
$228$	0	0
$229$	−0.517473	−0.0341956	−0.0170978	−	0.999854i	$-0.505443\pi$
$229$	−0.517473	−0.0341956	−0.0170978	+	0.999854i	$0.505443\pi$
$230$	0	0
$231$	2.72817	0.179501
$232$	0	0
$233$	−10.4011	−0.681397	−0.340699	−	0.940173i	$-0.610664\pi$
$233$	−10.4011	−0.681397	−0.340699	+	0.940173i	$0.610664\pi$
$234$	0	0
$235$	−26.0352	−1.69835
$236$	0	0
$237$	1.67330	0.108692
$238$	0	0
$239$	9.33883	0.604079	0.302039	−	0.953295i	$-0.402333\pi$
$239$	9.33883	0.604079	0.302039	+	0.953295i	$0.402333\pi$
$240$	0	0
$241$	20.1488	1.29790	0.648950	−	0.760831i	$-0.275208\pi$
$241$	20.1488	1.29790	0.648950	+	0.760831i	$0.275208\pi$
$242$	0	0
$243$	−22.1479	−1.42079
$244$	0	0
$245$	−3.57643	−0.228490
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−32.5411	−2.06221
$250$	0	0
$251$	−1.32683	−0.0837487	−0.0418744	−	0.999123i	$-0.513333\pi$
$251$	−1.32683	−0.0837487	−0.0418744	+	0.999123i	$0.513333\pi$
$252$	0	0
$253$	−2.41406	−0.151771
$254$	0	0
$255$	27.1112	1.69777
$256$	0	0
$257$	1.99093	0.124191	0.0620954	−	0.998070i	$-0.480222\pi$
$257$	1.99093	0.124191	0.0620954	+	0.998070i	$0.480222\pi$
$258$	0	0
$259$	−5.93395	−0.368717
$260$	0	0
$261$	32.5586	2.01533
$262$	0	0
$263$	−26.2445	−1.61831	−0.809153	−	0.587598i	$-0.800074\pi$
$263$	−26.2445	−1.61831	−0.809153	+	0.587598i	$0.800074\pi$
$264$	0	0
$265$	−40.6590	−2.49766
$266$	0	0
$267$	39.7313	2.43151
$268$	0	0
$269$	21.1137	1.28732	0.643662	−	0.765310i	$-0.277414\pi$
$269$	21.1137	1.28732	0.643662	+	0.765310i	$0.277414\pi$
$270$	0	0
$271$	−4.35953	−0.264823	−0.132411	−	0.991195i	$-0.542272\pi$
$271$	−4.35953	−0.264823	−0.132411	+	0.991195i	$0.542272\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.25309	−0.497680
$276$	0	0
$277$	−12.9722	−0.779427	−0.389713	−	0.920936i	$-0.627426\pi$
$277$	−12.9722	−0.779427	−0.389713	+	0.920936i	$0.627426\pi$
$278$	0	0
$279$	1.28819	0.0771218
$280$	0	0
$281$	24.7109	1.47413	0.737064	−	0.675823i	$-0.236212\pi$
$281$	24.7109	1.47413	0.737064	+	0.675823i	$0.236212\pi$
$282$	0	0
$283$	−4.70178	−0.279492	−0.139746	−	0.990187i	$-0.544629\pi$
$283$	−4.70178	−0.279492	−0.139746	+	0.990187i	$0.544629\pi$
$284$	0	0
$285$	−25.2686	−1.49678
$286$	0	0
$287$	11.0770	0.653857
$288$	0	0
$289$	−8.33600	−0.490353
$290$	0	0
$291$	−9.28518	−0.544307
$292$	0	0
$293$	1.93786	0.113211	0.0566055	−	0.998397i	$-0.481972\pi$
$293$	1.93786	0.113211	0.0566055	+	0.998397i	$0.481972\pi$
$294$	0	0
$295$	24.0828	1.40215
$296$	0	0
$297$	−1.72567	−0.100134
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.68820	0.270224
$302$	0	0
$303$	−15.6069	−0.896595
$304$	0	0
$305$	38.8726	2.22584
$306$	0	0
$307$	5.46464	0.311884	0.155942	−	0.987766i	$-0.450159\pi$
$307$	5.46464	0.311884	0.155942	+	0.987766i	$0.450159\pi$
$308$	0	0
$309$	−45.4671	−2.58653
$310$	0	0
$311$	24.5589	1.39261	0.696303	−	0.717748i	$-0.254827\pi$
$311$	24.5589	1.39261	0.696303	+	0.717748i	$0.254827\pi$
$312$	0	0
$313$	−21.5737	−1.21942	−0.609710	−	0.792624i	$-0.708714\pi$
$313$	−21.5737	−1.21942	−0.609710	+	0.792624i	$0.708714\pi$
$314$	0	0
$315$	12.9915	0.731988
$316$	0	0
$317$	−6.31679	−0.354786	−0.177393	−	0.984140i	$-0.556766\pi$
$317$	−6.31679	−0.354786	−0.177393	+	0.984140i	$0.556766\pi$
$318$	0	0
$319$	−9.49485	−0.531609
$320$	0	0
$321$	14.2875	0.797451
$322$	0	0
$323$	−8.07517	−0.449314
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.7155	−0.647869
$328$	0	0
$329$	−7.27967	−0.401341
$330$	0	0
$331$	10.2865	0.565398	0.282699	−	0.959209i	$-0.408770\pi$
$331$	10.2865	0.565398	0.282699	+	0.959209i	$0.408770\pi$
$332$	0	0
$333$	21.5553	1.18122
$334$	0	0
$335$	1.72512	0.0942533
$336$	0	0
$337$	−13.3905	−0.729429	−0.364715	−	0.931119i	$-0.618833\pi$
$337$	−13.3905	−0.729429	−0.364715	+	0.931119i	$0.618833\pi$
$338$	0	0
$339$	41.2903	2.24258
$340$	0	0
$341$	−0.375666	−0.0203435
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−20.9897	−1.13005
$346$	0	0
$347$	−22.0127	−1.18171	−0.590853	−	0.806779i	$-0.701209\pi$
$347$	−22.0127	−1.18171	−0.590853	+	0.806779i	$0.701209\pi$
$348$	0	0
$349$	−12.9009	−0.690568	−0.345284	−	0.938498i	$-0.612217\pi$
$349$	−12.9009	−0.690568	−0.345284	+	0.938498i	$0.612217\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.60083	0.244878	0.122439	−	0.992476i	$-0.460928\pi$
$353$	4.60083	0.244878	0.122439	+	0.992476i	$0.460928\pi$
$354$	0	0
$355$	27.9446	1.48315
$356$	0	0
$357$	7.58052	0.401204
$358$	0	0
$359$	26.3157	1.38889	0.694445	−	0.719546i	$-0.255650\pi$
$359$	26.3157	1.38889	0.694445	+	0.719546i	$0.255650\pi$
$360$	0	0
$361$	−11.4736	−0.603876
$362$	0	0
$363$	−25.4390	−1.33520
$364$	0	0
$365$	43.8800	2.29678
$366$	0	0
$367$	25.2539	1.31824	0.659122	−	0.752036i	$-0.270928\pi$
$367$	25.2539	1.31824	0.659122	+	0.752036i	$0.270928\pi$
$368$	0	0
$369$	−40.2377	−2.09469
$370$	0	0
$371$	−11.3686	−0.590229
$372$	0	0
$373$	9.94332	0.514846	0.257423	−	0.966299i	$-0.417127\pi$
$373$	9.94332	0.514846	0.257423	+	0.966299i	$0.417127\pi$
$374$	0	0
$375$	−25.7053	−1.32742
$376$	0	0
$377$	0	0
$378$	0	0
$379$	12.0770	0.620355	0.310178	−	0.950679i	$-0.399611\pi$
$379$	12.0770	0.620355	0.310178	+	0.950679i	$0.399611\pi$
$380$	0	0
$381$	31.5057	1.61408
$382$	0	0
$383$	35.1539	1.79628	0.898140	−	0.439709i	$-0.144918\pi$
$383$	35.1539	1.79628	0.898140	+	0.439709i	$0.144918\pi$
$384$	0	0
$385$	−3.78863	−0.193086
$386$	0	0
$387$	−17.0301	−0.865687
$388$	0	0
$389$	16.0772	0.815144	0.407572	−	0.913173i	$-0.366376\pi$
$389$	16.0772	0.815144	0.407572	+	0.913173i	$0.366376\pi$
$390$	0	0
$391$	−6.70773	−0.339225
$392$	0	0
$393$	−21.3911	−1.07904
$394$	0	0
$395$	−2.32371	−0.116919
$396$	0	0
$397$	10.2166	0.512758	0.256379	−	0.966576i	$-0.417470\pi$
$397$	10.2166	0.512758	0.256379	+	0.966576i	$0.417470\pi$
$398$	0	0
$399$	−7.06533	−0.353709
$400$	0	0
$401$	−12.9365	−0.646019	−0.323009	−	0.946396i	$-0.604695\pi$
$401$	−12.9365	−0.646019	−0.323009	+	0.946396i	$0.604695\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	23.9703	1.19109
$406$	0	0
$407$	−6.28602	−0.311587
$408$	0	0
$409$	−11.4617	−0.566745	−0.283373	−	0.959010i	$-0.591453\pi$
$409$	−11.4617	−0.566745	−0.283373	+	0.959010i	$0.591453\pi$
$410$	0	0
$411$	−9.21059	−0.454325
$412$	0	0
$413$	6.73376	0.331347
$414$	0	0
$415$	45.1899	2.21828
$416$	0	0
$417$	−47.8363	−2.34255
$418$	0	0
$419$	−24.3559	−1.18986	−0.594932	−	0.803776i	$-0.702821\pi$
$419$	−24.3559	−1.18986	−0.594932	+	0.803776i	$0.702821\pi$
$420$	0	0
$421$	9.85062	0.480090	0.240045	−	0.970762i	$-0.422838\pi$
$421$	9.85062	0.480090	0.240045	+	0.970762i	$0.422838\pi$
$422$	0	0
$423$	26.4437	1.28573
$424$	0	0
$425$	−22.9321	−1.11237
$426$	0	0
$427$	10.8691	0.525994
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.7790	−1.81975	−0.909875	−	0.414883i	$-0.863822\pi$
$431$	−37.7790	−1.81975	−0.909875	+	0.414883i	$0.863822\pi$
$432$	0	0
$433$	7.54725	0.362698	0.181349	−	0.983419i	$-0.441954\pi$
$433$	7.54725	0.362698	0.181349	+	0.983419i	$0.441954\pi$
$434$	0	0
$435$	−82.5553	−3.95822
$436$	0	0
$437$	6.25186	0.299067
$438$	0	0
$439$	14.8551	0.708997	0.354499	−	0.935057i	$-0.384652\pi$
$439$	14.8551	0.708997	0.354499	+	0.935057i	$0.384652\pi$
$440$	0	0
$441$	3.63254	0.172978
$442$	0	0
$443$	−7.56304	−0.359331	−0.179665	−	0.983728i	$-0.557501\pi$
$443$	−7.56304	−0.359331	−0.179665	+	0.983728i	$0.557501\pi$
$444$	0	0
$445$	−55.1750	−2.61554
$446$	0	0
$447$	−45.0997	−2.13314
$448$	0	0
$449$	−1.83155	−0.0864363	−0.0432181	−	0.999066i	$-0.513761\pi$
$449$	−1.83155	−0.0864363	−0.0432181	+	0.999066i	$0.513761\pi$
$450$	0	0
$451$	11.7343	0.552545
$452$	0	0
$453$	11.1779	0.525183
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.0406	−1.35846	−0.679232	−	0.733923i	$-0.737687\pi$
$457$	−29.0406	−1.35846	−0.679232	+	0.733923i	$0.737687\pi$
$458$	0	0
$459$	−4.79495	−0.223809
$460$	0	0
$461$	21.7793	1.01436	0.507181	−	0.861840i	$-0.330688\pi$
$461$	21.7793	1.01436	0.507181	+	0.861840i	$0.330688\pi$
$462$	0	0
$463$	27.4979	1.27794	0.638969	−	0.769233i	$-0.279361\pi$
$463$	27.4979	1.27794	0.638969	+	0.769233i	$0.279361\pi$
$464$	0	0
$465$	−3.26632	−0.151472
$466$	0	0
$467$	−33.1236	−1.53278	−0.766388	−	0.642377i	$-0.777948\pi$
$467$	−33.1236	−1.53278	−0.766388	+	0.642377i	$0.777948\pi$
$468$	0	0
$469$	0.482358	0.0222732
$470$	0	0
$471$	−35.3820	−1.63032
$472$	0	0
$473$	4.96637	0.228354
$474$	0	0
$475$	21.3735	0.980685
$476$	0	0
$477$	41.2969	1.89085
$478$	0	0
$479$	−41.2448	−1.88452	−0.942262	−	0.334877i	$-0.891305\pi$
$479$	−41.2448	−1.88452	−0.942262	+	0.334877i	$0.891305\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−5.86890	−0.267044
$484$	0	0
$485$	12.8944	0.585503
$486$	0	0
$487$	2.67905	0.121399	0.0606997	−	0.998156i	$-0.480667\pi$
$487$	2.67905	0.121399	0.0606997	+	0.998156i	$0.480667\pi$
$488$	0	0
$489$	−44.6541	−2.01933
$490$	0	0
$491$	−15.2033	−0.686117	−0.343059	−	0.939314i	$-0.611463\pi$
$491$	−15.2033	−0.686117	−0.343059	+	0.939314i	$0.611463\pi$
$492$	0	0
$493$	−26.3824	−1.18820
$494$	0	0
$495$	13.7623	0.618570
$496$	0	0
$497$	7.81356	0.350486
$498$	0	0
$499$	−4.05625	−0.181583	−0.0907914	−	0.995870i	$-0.528940\pi$
$499$	−4.05625	−0.181583	−0.0907914	+	0.995870i	$0.528940\pi$
$500$	0	0
$501$	−25.1714	−1.12457
$502$	0	0
$503$	−10.0159	−0.446588	−0.223294	−	0.974751i	$-0.571681\pi$
$503$	−10.0159	−0.446588	−0.223294	+	0.974751i	$0.571681\pi$
$504$	0	0
$505$	21.6734	0.964454
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.1836	0.894621	0.447311	−	0.894379i	$-0.352382\pi$
$509$	20.1836	0.894621	0.447311	+	0.894379i	$0.352382\pi$
$510$	0	0
$511$	12.2692	0.542759
$512$	0	0
$513$	4.46907	0.197314
$514$	0	0
$515$	63.1403	2.78229
$516$	0	0
$517$	−7.71159	−0.339155
$518$	0	0
$519$	−30.3983	−1.33434
$520$	0	0
$521$	−36.1789	−1.58502	−0.792512	−	0.609856i	$-0.791227\pi$
$521$	−36.1789	−1.58502	−0.792512	+	0.609856i	$0.791227\pi$
$522$	0	0
$523$	−14.9033	−0.651675	−0.325837	−	0.945426i	$-0.605646\pi$
$523$	−14.9033	−0.651675	−0.325837	+	0.945426i	$0.605646\pi$
$524$	0	0
$525$	−20.0643	−0.875677
$526$	0	0
$527$	−1.04383	−0.0454698
$528$	0	0
$529$	−17.8068	−0.774210
$530$	0	0
$531$	−24.4606	−1.06150
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−19.8411	−0.857807
$536$	0	0
$537$	57.3156	2.47335
$538$	0	0
$539$	−1.05933	−0.0456287
$540$	0	0
$541$	−28.6548	−1.23197	−0.615983	−	0.787760i	$-0.711241\pi$
$541$	−28.6548	−1.23197	−0.615983	+	0.787760i	$0.711241\pi$
$542$	0	0
$543$	−30.0564	−1.28985
$544$	0	0
$545$	16.2694	0.696904
$546$	0	0
$547$	−21.8783	−0.935449	−0.467724	−	0.883874i	$-0.654926\pi$
$547$	−21.8783	−0.935449	−0.467724	+	0.883874i	$0.654926\pi$
$548$	0	0
$549$	−39.4825	−1.68507
$550$	0	0
$551$	24.5894	1.04754
$552$	0	0
$553$	−0.649730	−0.0276293
$554$	0	0
$555$	−54.6554	−2.31999
$556$	0	0
$557$	32.9482	1.39606	0.698030	−	0.716069i	$-0.254060\pi$
$557$	32.9482	1.39606	0.698030	+	0.716069i	$0.254060\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.03029	0.339039
$562$	0	0
$563$	31.8406	1.34192	0.670960	−	0.741493i	$-0.265882\pi$
$563$	31.8406	1.34192	0.670960	+	0.741493i	$0.265882\pi$
$564$	0	0
$565$	−57.3400	−2.41231
$566$	0	0
$567$	6.70229	0.281470
$568$	0	0
$569$	8.19684	0.343629	0.171815	−	0.985129i	$-0.445037\pi$
$569$	8.19684	0.343629	0.171815	+	0.985129i	$0.445037\pi$
$570$	0	0
$571$	−17.6032	−0.736671	−0.368336	−	0.929693i	$-0.620072\pi$
$571$	−17.6032	−0.736671	−0.368336	+	0.929693i	$0.620072\pi$
$572$	0	0
$573$	18.8287	0.786581
$574$	0	0
$575$	17.7542	0.740400
$576$	0	0
$577$	16.9601	0.706057	0.353028	−	0.935613i	$-0.385152\pi$
$577$	16.9601	0.706057	0.353028	+	0.935613i	$0.385152\pi$
$578$	0	0
$579$	−58.5955	−2.43514
$580$	0	0
$581$	12.6355	0.524208
$582$	0	0
$583$	−12.0431	−0.498776
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.6288	0.892715	0.446357	−	0.894855i	$-0.352721\pi$
$587$	21.6288	0.892715	0.446357	+	0.894855i	$0.352721\pi$
$588$	0	0
$589$	0.972886	0.0400871
$590$	0	0
$591$	−47.5482	−1.95587
$592$	0	0
$593$	−39.5099	−1.62248	−0.811238	−	0.584717i	$-0.801206\pi$
$593$	−39.5099	−1.62248	−0.811238	+	0.584717i	$0.801206\pi$
$594$	0	0
$595$	−10.5271	−0.431569
$596$	0	0
$597$	−50.9802	−2.08648
$598$	0	0
$599$	45.8876	1.87492	0.937459	−	0.348097i	$-0.113172\pi$
$599$	45.8876	1.87492	0.937459	+	0.348097i	$0.113172\pi$
$600$	0	0
$601$	12.3263	0.502801	0.251401	−	0.967883i	$-0.419109\pi$
$601$	12.3263	0.502801	0.251401	+	0.967883i	$0.419109\pi$
$602$	0	0
$603$	−1.75218	−0.0713544
$604$	0	0
$605$	35.3273	1.43626
$606$	0	0
$607$	−31.3785	−1.27361	−0.636807	−	0.771023i	$-0.719745\pi$
$607$	−31.3785	−1.27361	−0.636807	+	0.771023i	$0.719745\pi$
$608$	0	0
$609$	−23.0832	−0.935377
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−14.1252	−0.570513	−0.285256	−	0.958451i	$-0.592079\pi$
$613$	−14.1252	−0.570513	−0.285256	+	0.958451i	$0.592079\pi$
$614$	0	0
$615$	102.026	4.11410
$616$	0	0
$617$	16.0488	0.646101	0.323051	−	0.946382i	$-0.395292\pi$
$617$	16.0488	0.646101	0.323051	+	0.946382i	$0.395292\pi$
$618$	0	0
$619$	−7.67934	−0.308659	−0.154329	−	0.988019i	$-0.549322\pi$
$619$	−7.67934	−0.308659	−0.154329	+	0.988019i	$0.549322\pi$
$620$	0	0
$621$	3.71229	0.148969
$622$	0	0
$623$	−15.4274	−0.618085
$624$	0	0
$625$	−3.25707	−0.130283
$626$	0	0
$627$	−7.48453	−0.298903
$628$	0	0
$629$	−17.4664	−0.696430
$630$	0	0
$631$	−1.56385	−0.0622560	−0.0311280	−	0.999515i	$-0.509910\pi$
$631$	−1.56385	−0.0622560	−0.0311280	+	0.999515i	$0.509910\pi$
$632$	0	0
$633$	−1.36991	−0.0544489
$634$	0	0
$635$	−43.7520	−1.73625
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−28.3830	−1.12282
$640$	0	0
$641$	−35.1384	−1.38788	−0.693941	−	0.720032i	$-0.744127\pi$
$641$	−35.1384	−1.38788	−0.693941	+	0.720032i	$0.744127\pi$
$642$	0	0
$643$	−13.4329	−0.529742	−0.264871	−	0.964284i	$-0.585329\pi$
$643$	−13.4329	−0.529742	−0.264871	+	0.964284i	$0.585329\pi$
$644$	0	0
$645$	43.1813	1.70026
$646$	0	0
$647$	−23.5426	−0.925555	−0.462777	−	0.886475i	$-0.653147\pi$
$647$	−23.5426	−0.925555	−0.462777	+	0.886475i	$0.653147\pi$
$648$	0	0
$649$	7.13329	0.280006
$650$	0	0
$651$	−0.913291	−0.0357947
$652$	0	0
$653$	29.2031	1.14281	0.571403	−	0.820670i	$-0.306399\pi$
$653$	29.2031	1.14281	0.571403	+	0.820670i	$0.306399\pi$
$654$	0	0
$655$	29.7059	1.16070
$656$	0	0
$657$	−44.5684	−1.73878
$658$	0	0
$659$	41.5054	1.61682	0.808411	−	0.588618i	$-0.200328\pi$
$659$	41.5054	1.61682	0.808411	+	0.588618i	$0.200328\pi$
$660$	0	0
$661$	15.8678	0.617187	0.308593	−	0.951194i	$-0.400142\pi$
$661$	15.8678	0.617187	0.308593	+	0.951194i	$0.400142\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.81164	0.380479
$666$	0	0
$667$	20.4255	0.790878
$668$	0	0
$669$	65.5210	2.53319
$670$	0	0
$671$	11.5140	0.444494
$672$	0	0
$673$	22.0283	0.849127	0.424564	−	0.905398i	$-0.360427\pi$
$673$	22.0283	0.849127	0.424564	+	0.905398i	$0.360427\pi$
$674$	0	0
$675$	12.6914	0.488492
$676$	0	0
$677$	21.2821	0.817936	0.408968	−	0.912549i	$-0.365889\pi$
$677$	21.2821	0.817936	0.408968	+	0.912549i	$0.365889\pi$
$678$	0	0
$679$	3.60537	0.138362
$680$	0	0
$681$	42.3036	1.62108
$682$	0	0
$683$	1.87884	0.0718918	0.0359459	−	0.999354i	$-0.488556\pi$
$683$	1.87884	0.0718918	0.0359459	+	0.999354i	$0.488556\pi$
$684$	0	0
$685$	12.7908	0.488711
$686$	0	0
$687$	−1.33268	−0.0508451
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29.8375	1.13507	0.567536	−	0.823349i	$-0.307897\pi$
$691$	29.8375	1.13507	0.567536	+	0.823349i	$0.307897\pi$
$692$	0	0
$693$	3.84806	0.146176
$694$	0	0
$695$	66.4304	2.51985
$696$	0	0
$697$	32.6049	1.23500
$698$	0	0
$699$	−26.7866	−1.01316
$700$	0	0
$701$	−0.215793	−0.00815037	−0.00407519	−	0.999992i	$-0.501297\pi$
$701$	−0.215793	−0.00815037	−0.00407519	+	0.999992i	$0.501297\pi$
$702$	0	0
$703$	16.2793	0.613986
$704$	0	0
$705$	−67.0503	−2.52526
$706$	0	0
$707$	6.06007	0.227913
$708$	0	0
$709$	−4.49203	−0.168702	−0.0843509	−	0.996436i	$-0.526882\pi$
$709$	−4.49203	−0.168702	−0.0843509	+	0.996436i	$0.526882\pi$
$710$	0	0
$711$	2.36017	0.0885132
$712$	0	0
$713$	0.808139	0.0302650
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0509	0.898199
$718$	0	0
$719$	−36.0410	−1.34410	−0.672051	−	0.740505i	$-0.734586\pi$
$719$	−36.0410	−1.34410	−0.672051	+	0.740505i	$0.734586\pi$
$720$	0	0
$721$	17.6546	0.657491
$722$	0	0
$723$	51.8907	1.92984
$724$	0	0
$725$	69.8296	2.59341
$726$	0	0
$727$	−20.4958	−0.760148	−0.380074	−	0.924956i	$-0.624101\pi$
$727$	−20.4958	−0.760148	−0.380074	+	0.924956i	$0.624101\pi$
$728$	0	0
$729$	−36.9323	−1.36786
$730$	0	0
$731$	13.7996	0.510396
$732$	0	0
$733$	3.82577	0.141308	0.0706540	−	0.997501i	$-0.477491\pi$
$733$	3.82577	0.141308	0.0706540	+	0.997501i	$0.477491\pi$
$734$	0	0
$735$	−9.21063	−0.339739
$736$	0	0
$737$	0.510978	0.0188221
$738$	0	0
$739$	48.2200	1.77380	0.886901	−	0.461959i	$-0.152853\pi$
$739$	48.2200	1.77380	0.886901	+	0.461959i	$0.152853\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.1501	1.28953	0.644766	−	0.764380i	$-0.276955\pi$
$743$	35.1501	1.28953	0.644766	+	0.764380i	$0.276955\pi$
$744$	0	0
$745$	62.6301	2.29459
$746$	0	0
$747$	−45.8988	−1.67935
$748$	0	0
$749$	−5.54775	−0.202710
$750$	0	0
$751$	−6.63557	−0.242136	−0.121068	−	0.992644i	$-0.538632\pi$
$751$	−6.63557	−0.242136	−0.121068	+	0.992644i	$0.538632\pi$
$752$	0	0
$753$	−3.41708	−0.124525
$754$	0	0
$755$	−15.5228	−0.564931
$756$	0	0
$757$	−10.2035	−0.370851	−0.185426	−	0.982658i	$-0.559366\pi$
$757$	−10.2035	−0.370851	−0.185426	+	0.982658i	$0.559366\pi$
$758$	0	0
$759$	−6.21711	−0.225667
$760$	0	0
$761$	−44.0344	−1.59624	−0.798122	−	0.602495i	$-0.794173\pi$
$761$	−44.0344	−1.59624	−0.798122	+	0.602495i	$0.794173\pi$
$762$	0	0
$763$	4.54906	0.164687
$764$	0	0
$765$	38.2401	1.38257
$766$	0	0
$767$	0	0
$768$	0	0
$769$	19.1267	0.689728	0.344864	−	0.938653i	$-0.387925\pi$
$769$	19.1267	0.689728	0.344864	+	0.938653i	$0.387925\pi$
$770$	0	0
$771$	5.12738	0.184658
$772$	0	0
$773$	51.0736	1.83699	0.918494	−	0.395434i	$-0.129406\pi$
$773$	51.0736	1.83699	0.918494	+	0.395434i	$0.129406\pi$
$774$	0	0
$775$	2.76283	0.0992436
$776$	0	0
$777$	−15.2821	−0.548243
$778$	0	0
$779$	−30.3890	−1.08880
$780$	0	0
$781$	8.27716	0.296180
$782$	0	0
$783$	14.6009	0.521795
$784$	0	0
$785$	49.1351	1.75371
$786$	0	0
$787$	15.5313	0.553631	0.276816	−	0.960923i	$-0.410721\pi$
$787$	15.5313	0.553631	0.276816	+	0.960923i	$0.410721\pi$
$788$	0	0
$789$	−67.5894	−2.40625
$790$	0	0
$791$	−16.0328	−0.570059
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−104.712	−3.71375
$796$	0	0
$797$	18.7902	0.665583	0.332791	−	0.943000i	$-0.392009\pi$
$797$	18.7902	0.665583	0.332791	+	0.943000i	$0.392009\pi$
$798$	0	0
$799$	−21.4275	−0.758049
$800$	0	0
$801$	56.0406	1.98010
$802$	0	0
$803$	12.9972	0.458661
$804$	0	0
$805$	8.15016	0.287255
$806$	0	0
$807$	54.3756	1.91411
$808$	0	0
$809$	−14.6640	−0.515559	−0.257780	−	0.966204i	$-0.582991\pi$
$809$	−14.6640	−0.515559	−0.257780	+	0.966204i	$0.582991\pi$
$810$	0	0
$811$	20.6247	0.724230	0.362115	−	0.932133i	$-0.382055\pi$
$811$	20.6247	0.724230	0.362115	+	0.932133i	$0.382055\pi$
$812$	0	0
$813$	−11.2274	−0.393763
$814$	0	0
$815$	62.0114	2.17216
$816$	0	0
$817$	−12.8617	−0.449974
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.1298	−1.26094	−0.630469	−	0.776214i	$-0.717137\pi$
$821$	−36.1298	−1.26094	−0.630469	+	0.776214i	$0.717137\pi$
$822$	0	0
$823$	−50.7197	−1.76798	−0.883989	−	0.467508i	$-0.845152\pi$
$823$	−50.7197	−1.76798	−0.883989	+	0.467508i	$0.845152\pi$
$824$	0	0
$825$	−21.2548	−0.739996
$826$	0	0
$827$	−26.0935	−0.907358	−0.453679	−	0.891165i	$-0.649889\pi$
$827$	−26.0935	−0.907358	−0.453679	+	0.891165i	$0.649889\pi$
$828$	0	0
$829$	21.3535	0.741638	0.370819	−	0.928705i	$-0.379077\pi$
$829$	21.3535	0.741638	0.370819	+	0.928705i	$0.379077\pi$
$830$	0	0
$831$	−33.4083	−1.15892
$832$	0	0
$833$	−2.94347	−0.101985
$834$	0	0
$835$	34.9556	1.20969
$836$	0	0
$837$	0.577690	0.0199679
$838$	0	0
$839$	−39.4300	−1.36127	−0.680637	−	0.732621i	$-0.738297\pi$
$839$	−39.4300	−1.36127	−0.680637	+	0.732621i	$0.738297\pi$
$840$	0	0
$841$	51.3362	1.77021
$842$	0	0
$843$	63.6397	2.19187
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.87781	0.339406
$848$	0	0
$849$	−12.1088	−0.415574
$850$	0	0
$851$	13.5226	0.463549
$852$	0	0
$853$	1.21461	0.0415876	0.0207938	−	0.999784i	$-0.493381\pi$
$853$	1.21461	0.0415876	0.0207938	+	0.999784i	$0.493381\pi$
$854$	0	0
$855$	−35.6412	−1.21890
$856$	0	0
$857$	16.4153	0.560736	0.280368	−	0.959893i	$-0.409543\pi$
$857$	16.4153	0.560736	0.280368	+	0.959893i	$0.409543\pi$
$858$	0	0
$859$	−16.3062	−0.556361	−0.278180	−	0.960529i	$-0.589731\pi$
$859$	−16.3062	−0.556361	−0.278180	+	0.960529i	$0.589731\pi$
$860$	0	0
$861$	28.5275	0.972214
$862$	0	0
$863$	−55.2725	−1.88150	−0.940749	−	0.339102i	$-0.889877\pi$
$863$	−55.2725	−1.88150	−0.940749	+	0.339102i	$0.889877\pi$
$864$	0	0
$865$	42.2142	1.43532
$866$	0	0
$867$	−21.4683	−0.729102
$868$	0	0
$869$	−0.688280	−0.0233483
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−13.0967	−0.443255
$874$	0	0
$875$	9.98122	0.337427
$876$	0	0
$877$	42.4634	1.43389	0.716943	−	0.697131i	$-0.245541\pi$
$877$	42.4634	1.43389	0.716943	+	0.697131i	$0.245541\pi$
$878$	0	0
$879$	4.99071	0.168332
$880$	0	0
$881$	25.8651	0.871418	0.435709	−	0.900088i	$-0.356498\pi$
$881$	25.8651	0.871418	0.435709	+	0.900088i	$0.356498\pi$
$882$	0	0
$883$	12.8108	0.431117	0.215559	−	0.976491i	$-0.430843\pi$
$883$	12.8108	0.431117	0.215559	+	0.976491i	$0.430843\pi$
$884$	0	0
$885$	62.0221	2.08485
$886$	0	0
$887$	−50.8765	−1.70827	−0.854133	−	0.520054i	$-0.825912\pi$
$887$	−50.8765	−1.70827	−0.854133	+	0.520054i	$0.825912\pi$
$888$	0	0
$889$	−12.2334	−0.410296
$890$	0	0
$891$	7.09995	0.237857
$892$	0	0
$893$	19.9712	0.668310
$894$	0	0
$895$	−79.5944	−2.66055
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.17852	0.106010
$900$	0	0
$901$	−33.4631	−1.11482
$902$	0	0
$903$	12.0739	0.401793
$904$	0	0
$905$	41.7395	1.38747
$906$	0	0
$907$	10.8474	0.360183	0.180092	−	0.983650i	$-0.442361\pi$
$907$	10.8474	0.360183	0.180092	+	0.983650i	$0.442361\pi$
$908$	0	0
$909$	−22.0134	−0.730140
$910$	0	0
$911$	21.7893	0.721912	0.360956	−	0.932583i	$-0.382451\pi$
$911$	21.7893	0.721912	0.360956	+	0.932583i	$0.382451\pi$
$912$	0	0
$913$	13.3852	0.442985
$914$	0	0
$915$	100.111	3.30958
$916$	0	0
$917$	8.30601	0.274289
$918$	0	0
$919$	−24.8038	−0.818202	−0.409101	−	0.912489i	$-0.634158\pi$
$919$	−24.8038	−0.818202	−0.409101	+	0.912489i	$0.634158\pi$
$920$	0	0
$921$	14.0735	0.463737
$922$	0	0
$923$	0	0
$924$	0	0
$925$	46.2304	1.52005
$926$	0	0
$927$	−64.1309	−2.10633
$928$	0	0
$929$	8.80169	0.288774	0.144387	−	0.989521i	$-0.453879\pi$
$929$	8.80169	0.288774	0.144387	+	0.989521i	$0.453879\pi$
$930$	0	0
$931$	2.74342	0.0899120
$932$	0	0
$933$	63.2482	2.07065
$934$	0	0
$935$	−11.1517	−0.364699
$936$	0	0
$937$	34.8853	1.13965	0.569826	−	0.821765i	$-0.307011\pi$
$937$	34.8853	1.13965	0.569826	+	0.821765i	$0.307011\pi$
$938$	0	0
$939$	−55.5604	−1.81315
$940$	0	0
$941$	−3.48623	−0.113648	−0.0568240	−	0.998384i	$-0.518097\pi$
$941$	−3.48623	−0.113648	−0.0568240	+	0.998384i	$0.518097\pi$
$942$	0	0
$943$	−25.2430	−0.822024
$944$	0	0
$945$	5.82606	0.189522
$946$	0	0
$947$	−2.70496	−0.0878994	−0.0439497	−	0.999034i	$-0.513994\pi$
$947$	−2.70496	−0.0878994	−0.0439497	+	0.999034i	$0.513994\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−16.2681	−0.527529
$952$	0	0
$953$	55.8338	1.80864	0.904318	−	0.426860i	$-0.140380\pi$
$953$	55.8338	1.80864	0.904318	+	0.426860i	$0.140380\pi$
$954$	0	0
$955$	−26.1475	−0.846113
$956$	0	0
$957$	−24.4528	−0.790445
$958$	0	0
$959$	3.57641	0.115488
$960$	0	0
$961$	−30.8742	−0.995943
$962$	0	0
$963$	20.1524	0.649402
$964$	0	0
$965$	81.3718	2.61945
$966$	0	0
$967$	−39.5972	−1.27336	−0.636681	−	0.771127i	$-0.719693\pi$
$967$	−39.5972	−1.27336	−0.636681	+	0.771127i	$0.719693\pi$
$968$	0	0
$969$	−20.7966	−0.668081
$970$	0	0
$971$	32.7975	1.05252	0.526261	−	0.850323i	$-0.323594\pi$
$971$	32.7975	1.05252	0.526261	+	0.850323i	$0.323594\pi$
$972$	0	0
$973$	18.5745	0.595471
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.78407	0.153056	0.0765280	−	0.997067i	$-0.475617\pi$
$977$	4.78407	0.153056	0.0765280	+	0.997067i	$0.475617\pi$
$978$	0	0
$979$	−16.3427	−0.522316
$980$	0	0
$981$	−16.5246	−0.527591
$982$	0	0
$983$	−50.6466	−1.61538	−0.807688	−	0.589611i	$-0.799281\pi$
$983$	−50.6466	−1.61538	−0.807688	+	0.589611i	$0.799281\pi$
$984$	0	0
$985$	66.0304	2.10390
$986$	0	0
$987$	−18.7478	−0.596750
$988$	0	0
$989$	−10.6837	−0.339723
$990$	0	0
$991$	31.0230	0.985479	0.492740	−	0.870177i	$-0.335996\pi$
$991$	31.0230	0.985479	0.492740	+	0.870177i	$0.335996\pi$
$992$	0	0
$993$	26.4916	0.840685
$994$	0	0
$995$	70.7964	2.24440
$996$	0	0
$997$	52.6281	1.66675	0.833374	−	0.552710i	$-0.186406\pi$
$997$	52.6281	1.66675	0.833374	+	0.552710i	$0.186406\pi$
$998$	0	0
$999$	9.66649	0.305834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.br.1.14	✓	15
13.12	even	2		9464.2.a.bs.1.14	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.14	✓	15	1.1	even	1	trivial
9464.2.a.bs.1.14	yes	15	13.12	even	2

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:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.57537 q^{3} -3.57643 q^{5} -1.00000 q^{7} +3.63254 q^{9} -1.05933 q^{11} -9.21063 q^{15} -2.94347 q^{17} +2.74342 q^{19} -2.57537 q^{21} +2.27885 q^{23} +7.79083 q^{25} +1.62902 q^{27} +8.96305 q^{29} +0.354625 q^{31} -2.72817 q^{33} +3.57643 q^{35} +5.93395 q^{37} -11.0770 q^{41} -4.68820 q^{43} -12.9915 q^{45} +7.27967 q^{47} +1.00000 q^{49} -7.58052 q^{51} +11.3686 q^{53} +3.78863 q^{55} +7.06533 q^{57} -6.73376 q^{59} -10.8691 q^{61} -3.63254 q^{63} -0.482358 q^{67} +5.86890 q^{69} -7.81356 q^{71} -12.2692 q^{73} +20.0643 q^{75} +1.05933 q^{77} +0.649730 q^{79} -6.70229 q^{81} -12.6355 q^{83} +10.5271 q^{85} +23.0832 q^{87} +15.4274 q^{89} +0.913291 q^{93} -9.81164 q^{95} -3.60537 q^{97} -3.84806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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