LMFDB - Embedded newform 7360.2.a.cs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2255384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 2 $ x^5 - x^4 - 8x^3 + 5x^2 + 13*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.26789$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26789 q^{3} +1.00000 q^{5} +0.504137 q^{7} -1.39245 q^{9} -4.19785 q^{11} -3.16448 q^{13} -1.26789 q^{15} +0.400727 q^{17} +5.32241 q^{19} -0.639190 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56915 q^{27} -8.27114 q^{29} -1.19785 q^{31} +5.32241 q^{33} +0.504137 q^{35} +3.05452 q^{37} +4.01222 q^{39} +5.83704 q^{41} -7.80606 q^{43} -1.39245 q^{45} +3.70199 q^{47} -6.74585 q^{49} -0.508078 q^{51} -11.8898 q^{53} -4.19785 q^{55} -6.74824 q^{57} -3.82655 q^{59} +1.66207 q^{61} -0.701987 q^{63} -3.16448 q^{65} -1.26134 q^{67} -1.26789 q^{69} +6.28011 q^{71} -6.17081 q^{73} -1.26789 q^{75} -2.11629 q^{77} +9.76375 q^{79} -2.88371 q^{81} +5.02510 q^{83} +0.400727 q^{85} +10.4869 q^{87} -7.37455 q^{89} -1.59533 q^{91} +1.51874 q^{93} +5.32241 q^{95} -7.23189 q^{97} +5.84531 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 5 q^{5} + 4 q^{7} + 2 q^{9} + 3 q^{11} - q^{13} + q^{15} - 4 q^{17} + q^{19} + 4 q^{21} + 5 q^{23} + 5 q^{25} + 4 q^{27} + 5 q^{29} + 18 q^{31} + q^{33} + 4 q^{35} - 3 q^{37} + 20 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 5 * q + q^3 + 5 * q^5 + 4 * q^7 + 2 * q^9 + 3 * q^11 - q^13 + q^15 - 4 * q^17 + q^19 + 4 * q^21 + 5 * q^23 + 5 * q^25 + 4 * q^27 + 5 * q^29 + 18 * q^31 + q^33 + 4 * q^35 - 3 * q^37 + 20 * q^39 - 2 * q^41 - 12 * q^43 + 2 * q^45 - 4 * q^47 + 5 * q^49 - q^51 - 3 * q^53 + 3 * q^55 + 10 * q^57 + 5 * q^59 - q^61 + 19 * q^63 - q^65 - 3 * q^67 + q^69 + 24 * q^71 - 8 * q^73 + q^75 - 6 * q^77 + 40 * q^79 - 19 * q^81 + 13 * q^83 - 4 * q^85 + 12 * q^87 + 2 * q^89 - 17 * q^91 + 4 * q^93 + q^95 + 9 * q^97 + 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
$3$	−1.26789	−0.732017	−0.366008	−	0.930612i	$-0.619276\pi$
$3$	−1.26789	−0.732017	−0.366008	+	0.930612i	$0.619276\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.504137	0.190546	0.0952729	−	0.995451i	$-0.469628\pi$
$7$	0.504137	0.190546	0.0952729	+	0.995451i	$0.469628\pi$
$8$	0	0
$9$	−1.39245	−0.464151
$10$	0	0
$11$	−4.19785	−1.26570	−0.632850	−	0.774275i	$-0.718115\pi$
$11$	−4.19785	−1.26570	−0.632850	+	0.774275i	$0.718115\pi$
$12$	0	0
$13$	−3.16448	−0.877669	−0.438835	−	0.898568i	$-0.644609\pi$
$13$	−3.16448	−0.877669	−0.438835	+	0.898568i	$0.644609\pi$
$14$	0	0
$15$	−1.26789	−0.327368
$16$	0	0
$17$	0.400727	0.0971907	0.0485953	−	0.998819i	$-0.484526\pi$
$17$	0.400727	0.0971907	0.0485953	+	0.998819i	$0.484526\pi$
$18$	0	0
$19$	5.32241	1.22105	0.610523	−	0.791999i	$-0.290959\pi$
$19$	5.32241	1.22105	0.610523	+	0.791999i	$0.290959\pi$
$20$	0	0
$21$	−0.639190	−0.139483
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.56915	1.07178
$28$	0	0
$29$	−8.27114	−1.53591	−0.767956	−	0.640503i	$-0.778726\pi$
$29$	−8.27114	−1.53591	−0.767956	+	0.640503i	$0.778726\pi$
$30$	0	0
$31$	−1.19785	−0.215140	−0.107570	−	0.994197i	$-0.534307\pi$
$31$	−1.19785	−0.215140	−0.107570	+	0.994197i	$0.534307\pi$
$32$	0	0
$33$	5.32241	0.926513
$34$	0	0
$35$	0.504137	0.0852146
$36$	0	0
$37$	3.05452	0.502161	0.251080	−	0.967966i	$-0.419214\pi$
$37$	3.05452	0.502161	0.251080	+	0.967966i	$0.419214\pi$
$38$	0	0
$39$	4.01222	0.642469
$40$	0	0
$41$	5.83704	0.911593	0.455796	−	0.890084i	$-0.349355\pi$
$41$	5.83704	0.911593	0.455796	+	0.890084i	$0.349355\pi$
$42$	0	0
$43$	−7.80606	−1.19041	−0.595207	−	0.803573i	$-0.702930\pi$
$43$	−7.80606	−1.19041	−0.595207	+	0.803573i	$0.702930\pi$
$44$	0	0
$45$	−1.39245	−0.207575
$46$	0	0
$47$	3.70199	0.539990	0.269995	−	0.962862i	$-0.412978\pi$
$47$	3.70199	0.539990	0.269995	+	0.962862i	$0.412978\pi$
$48$	0	0
$49$	−6.74585	−0.963692
$50$	0	0
$51$	−0.508078	−0.0711452
$52$	0	0
$53$	−11.8898	−1.63319	−0.816597	−	0.577208i	$-0.804142\pi$
$53$	−11.8898	−1.63319	−0.816597	+	0.577208i	$0.804142\pi$
$54$	0	0
$55$	−4.19785	−0.566038
$56$	0	0
$57$	−6.74824	−0.893826
$58$	0	0
$59$	−3.82655	−0.498174	−0.249087	−	0.968481i	$-0.580131\pi$
$59$	−3.82655	−0.498174	−0.249087	+	0.968481i	$0.580131\pi$
$60$	0	0
$61$	1.66207	0.212806	0.106403	−	0.994323i	$-0.466067\pi$
$61$	1.66207	0.212806	0.106403	+	0.994323i	$0.466067\pi$
$62$	0	0
$63$	−0.701987	−0.0884421
$64$	0	0
$65$	−3.16448	−0.392506
$66$	0	0
$67$	−1.26134	−0.154097	−0.0770487	−	0.997027i	$-0.524550\pi$
$67$	−1.26134	−0.154097	−0.0770487	+	0.997027i	$0.524550\pi$
$68$	0	0
$69$	−1.26789	−0.152636
$70$	0	0
$71$	6.28011	0.745311	0.372656	−	0.927970i	$-0.378447\pi$
$71$	6.28011	0.745311	0.372656	+	0.927970i	$0.378447\pi$
$72$	0	0
$73$	−6.17081	−0.722239	−0.361120	−	0.932520i	$-0.617605\pi$
$73$	−6.17081	−0.722239	−0.361120	+	0.932520i	$0.617605\pi$
$74$	0	0
$75$	−1.26789	−0.146403
$76$	0	0
$77$	−2.11629	−0.241174
$78$	0	0
$79$	9.76375	1.09851	0.549254	−	0.835655i	$-0.314912\pi$
$79$	9.76375	1.09851	0.549254	+	0.835655i	$0.314912\pi$
$80$	0	0
$81$	−2.88371	−0.320412
$82$	0	0
$83$	5.02510	0.551576	0.275788	−	0.961218i	$-0.411061\pi$
$83$	5.02510	0.551576	0.275788	+	0.961218i	$0.411061\pi$
$84$	0	0
$85$	0.400727	0.0434650
$86$	0	0
$87$	10.4869	1.12431
$88$	0	0
$89$	−7.37455	−0.781700	−0.390850	−	0.920454i	$-0.627819\pi$
$89$	−7.37455	−0.781700	−0.390850	+	0.920454i	$0.627819\pi$
$90$	0	0
$91$	−1.59533	−0.167236
$92$	0	0
$93$	1.51874	0.157486
$94$	0	0
$95$	5.32241	0.546068
$96$	0	0
$97$	−7.23189	−0.734287	−0.367143	−	0.930164i	$-0.619664\pi$
$97$	−7.23189	−0.734287	−0.367143	+	0.930164i	$0.619664\pi$
$98$	0	0
$99$	5.84531	0.587476
$100$	0	0
$101$	15.4373	1.53607	0.768037	−	0.640406i	$-0.221234\pi$
$101$	15.4373	1.53607	0.768037	+	0.640406i	$0.221234\pi$
$102$	0	0
$103$	16.7465	1.65008	0.825041	−	0.565072i	$-0.191152\pi$
$103$	16.7465	1.65008	0.825041	+	0.565072i	$0.191152\pi$
$104$	0	0
$105$	−0.639190	−0.0623786
$106$	0	0
$107$	6.29538	0.608597	0.304299	−	0.952577i	$-0.401578\pi$
$107$	6.29538	0.608597	0.304299	+	0.952577i	$0.401578\pi$
$108$	0	0
$109$	−10.1906	−0.976083	−0.488042	−	0.872820i	$-0.662289\pi$
$109$	−10.1906	−0.976083	−0.488042	+	0.872820i	$0.662289\pi$
$110$	0	0
$111$	−3.87280	−0.367590
$112$	0	0
$113$	14.2596	1.34143	0.670713	−	0.741717i	$-0.265988\pi$
$113$	14.2596	1.34143	0.670713	+	0.741717i	$0.265988\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	4.40639	0.407371
$118$	0	0
$119$	0.202021	0.0185193
$120$	0	0
$121$	6.62195	0.601995
$122$	0	0
$123$	−7.40073	−0.667301
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.41188	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$127$	9.41188	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$128$	0	0
$129$	9.89723	0.871403
$130$	0	0
$131$	11.8191	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$131$	11.8191	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$132$	0	0
$133$	2.68322	0.232665
$134$	0	0
$135$	5.56915	0.479316
$136$	0	0
$137$	14.3436	1.22545	0.612727	−	0.790295i	$-0.290073\pi$
$137$	14.3436	1.22545	0.612727	+	0.790295i	$0.290073\pi$
$138$	0	0
$139$	−20.1016	−1.70500	−0.852499	−	0.522729i	$-0.824914\pi$
$139$	−20.1016	−1.70500	−0.852499	+	0.522729i	$0.824914\pi$
$140$	0	0
$141$	−4.69371	−0.395282
$142$	0	0
$143$	13.2840	1.11087
$144$	0	0
$145$	−8.27114	−0.686881
$146$	0	0
$147$	8.55299	0.705439
$148$	0	0
$149$	−4.01461	−0.328889	−0.164445	−	0.986386i	$-0.552583\pi$
$149$	−4.01461	−0.328889	−0.164445	+	0.986386i	$0.552583\pi$
$150$	0	0
$151$	10.8414	0.882258	0.441129	−	0.897444i	$-0.354578\pi$
$151$	10.8414	0.882258	0.441129	+	0.897444i	$0.354578\pi$
$152$	0	0
$153$	−0.557994	−0.0451112
$154$	0	0
$155$	−1.19785	−0.0962137
$156$	0	0
$157$	1.25789	0.100391	0.0501953	−	0.998739i	$-0.484016\pi$
$157$	1.25789	0.100391	0.0501953	+	0.998739i	$0.484016\pi$
$158$	0	0
$159$	15.0750	1.19553
$160$	0	0
$161$	0.504137	0.0397315
$162$	0	0
$163$	−12.7624	−0.999628	−0.499814	−	0.866133i	$-0.666598\pi$
$163$	−12.7624	−0.999628	−0.499814	+	0.866133i	$0.666598\pi$
$164$	0	0
$165$	5.32241	0.414349
$166$	0	0
$167$	−10.9315	−0.845903	−0.422952	−	0.906152i	$-0.639006\pi$
$167$	−10.9315	−0.845903	−0.422952	+	0.906152i	$0.639006\pi$
$168$	0	0
$169$	−2.98606	−0.229697
$170$	0	0
$171$	−7.41122	−0.566750
$172$	0	0
$173$	11.7173	0.890847	0.445423	−	0.895320i	$-0.353053\pi$
$173$	11.7173	0.890847	0.445423	+	0.895320i	$0.353053\pi$
$174$	0	0
$175$	0.504137	0.0381091
$176$	0	0
$177$	4.85165	0.364672
$178$	0	0
$179$	7.44919	0.556779	0.278389	−	0.960468i	$-0.410199\pi$
$179$	7.44919	0.556779	0.278389	+	0.960468i	$0.410199\pi$
$180$	0	0
$181$	−11.9249	−0.886373	−0.443187	−	0.896429i	$-0.646152\pi$
$181$	−11.9249	−0.886373	−0.443187	+	0.896429i	$0.646152\pi$
$182$	0	0
$183$	−2.10732	−0.155778
$184$	0	0
$185$	3.05452	0.224573
$186$	0	0
$187$	−1.68219	−0.123014
$188$	0	0
$189$	2.80761	0.204224
$190$	0	0
$191$	3.92017	0.283653	0.141827	−	0.989892i	$-0.454702\pi$
$191$	3.92017	0.283653	0.141827	+	0.989892i	$0.454702\pi$
$192$	0	0
$193$	−2.96749	−0.213604	−0.106802	−	0.994280i	$-0.534061\pi$
$193$	−2.96749	−0.213604	−0.106802	+	0.994280i	$0.534061\pi$
$194$	0	0
$195$	4.01222	0.287321
$196$	0	0
$197$	−6.35578	−0.452831	−0.226415	−	0.974031i	$-0.572701\pi$
$197$	−6.35578	−0.452831	−0.226415	+	0.974031i	$0.572701\pi$
$198$	0	0
$199$	21.5569	1.52813	0.764065	−	0.645139i	$-0.223201\pi$
$199$	21.5569	1.52813	0.764065	+	0.645139i	$0.223201\pi$
$200$	0	0
$201$	1.59924	0.112802
$202$	0	0
$203$	−4.16978	−0.292661
$204$	0	0
$205$	5.83704	0.407677
$206$	0	0
$207$	−1.39245	−0.0967822
$208$	0	0
$209$	−22.3427	−1.54548
$210$	0	0
$211$	−10.7026	−0.736799	−0.368400	−	0.929668i	$-0.620094\pi$
$211$	−10.7026	−0.736799	−0.368400	+	0.929668i	$0.620094\pi$
$212$	0	0
$213$	−7.96249	−0.545580
$214$	0	0
$215$	−7.80606	−0.532369
$216$	0	0
$217$	−0.603880	−0.0409941
$218$	0	0
$219$	7.82392	0.528691
$220$	0	0
$221$	−1.26809	−0.0853012
$222$	0	0
$223$	13.8599	0.928129	0.464064	−	0.885801i	$-0.346391\pi$
$223$	13.8599	0.928129	0.464064	+	0.885801i	$0.346391\pi$
$224$	0	0
$225$	−1.39245	−0.0928303
$226$	0	0
$227$	0.0896727	0.00595179	0.00297589	−	0.999996i	$-0.499053\pi$
$227$	0.0896727	0.00595179	0.00297589	+	0.999996i	$0.499053\pi$
$228$	0	0
$229$	18.2485	1.20590	0.602948	−	0.797781i	$-0.293993\pi$
$229$	18.2485	1.20590	0.602948	+	0.797781i	$0.293993\pi$
$230$	0	0
$231$	2.68322	0.176543
$232$	0	0
$233$	5.73141	0.375477	0.187739	−	0.982219i	$-0.439884\pi$
$233$	5.73141	0.375477	0.187739	+	0.982219i	$0.439884\pi$
$234$	0	0
$235$	3.70199	0.241491
$236$	0	0
$237$	−12.3794	−0.804126
$238$	0	0
$239$	2.27752	0.147321	0.0736604	−	0.997283i	$-0.476532\pi$
$239$	2.27752	0.147321	0.0736604	+	0.997283i	$0.476532\pi$
$240$	0	0
$241$	7.84337	0.505236	0.252618	−	0.967566i	$-0.418708\pi$
$241$	7.84337	0.505236	0.252618	+	0.967566i	$0.418708\pi$
$242$	0	0
$243$	−13.0512	−0.837236
$244$	0	0
$245$	−6.74585	−0.430976
$246$	0	0
$247$	−16.8427	−1.07167
$248$	0	0
$249$	−6.37127	−0.403763
$250$	0	0
$251$	16.2620	1.02645	0.513224	−	0.858255i	$-0.328451\pi$
$251$	16.2620	1.02645	0.513224	+	0.858255i	$0.328451\pi$
$252$	0	0
$253$	−4.19785	−0.263917
$254$	0	0
$255$	−0.508078	−0.0318171
$256$	0	0
$257$	11.5913	0.723048	0.361524	−	0.932363i	$-0.382257\pi$
$257$	11.5913	0.723048	0.361524	+	0.932363i	$0.382257\pi$
$258$	0	0
$259$	1.53990	0.0956846
$260$	0	0
$261$	11.5172	0.712895
$262$	0	0
$263$	−15.6108	−0.962601	−0.481300	−	0.876556i	$-0.659835\pi$
$263$	−15.6108	−0.962601	−0.481300	+	0.876556i	$0.659835\pi$
$264$	0	0
$265$	−11.8898	−0.730387
$266$	0	0
$267$	9.35012	0.572218
$268$	0	0
$269$	22.7301	1.38588	0.692940	−	0.720995i	$-0.256315\pi$
$269$	22.7301	1.38588	0.692940	+	0.720995i	$0.256315\pi$
$270$	0	0
$271$	31.9275	1.93946	0.969728	−	0.244186i	$-0.0785209\pi$
$271$	31.9275	1.93946	0.969728	+	0.244186i	$0.0785209\pi$
$272$	0	0
$273$	2.02270	0.122420
$274$	0	0
$275$	−4.19785	−0.253140
$276$	0	0
$277$	−16.5665	−0.995385	−0.497693	−	0.867353i	$-0.665819\pi$
$277$	−16.5665	−0.995385	−0.497693	+	0.867353i	$0.665819\pi$
$278$	0	0
$279$	1.66795	0.0998577
$280$	0	0
$281$	2.21492	0.132131	0.0660654	−	0.997815i	$-0.478955\pi$
$281$	2.21492	0.132131	0.0660654	+	0.997815i	$0.478955\pi$
$282$	0	0
$283$	19.6097	1.16567	0.582837	−	0.812589i	$-0.301943\pi$
$283$	19.6097	1.16567	0.582837	+	0.812589i	$0.301943\pi$
$284$	0	0
$285$	−6.74824	−0.399731
$286$	0	0
$287$	2.94267	0.173700
$288$	0	0
$289$	−16.8394	−0.990554
$290$	0	0
$291$	9.16924	0.537510
$292$	0	0
$293$	9.09183	0.531151	0.265575	−	0.964090i	$-0.414438\pi$
$293$	9.09183	0.531151	0.265575	+	0.964090i	$0.414438\pi$
$294$	0	0
$295$	−3.82655	−0.222790
$296$	0	0
$297$	−23.3785	−1.35656
$298$	0	0
$299$	−3.16448	−0.183007
$300$	0	0
$301$	−3.93532	−0.226828
$302$	0	0
$303$	−19.5729	−1.12443
$304$	0	0
$305$	1.66207	0.0951698
$306$	0	0
$307$	28.6274	1.63385	0.816926	−	0.576742i	$-0.195676\pi$
$307$	28.6274	1.63385	0.816926	+	0.576742i	$0.195676\pi$
$308$	0	0
$309$	−21.2327	−1.20789
$310$	0	0
$311$	11.1398	0.631681	0.315841	−	0.948812i	$-0.397713\pi$
$311$	11.1398	0.631681	0.315841	+	0.948812i	$0.397713\pi$
$312$	0	0
$313$	−23.4550	−1.32575	−0.662877	−	0.748728i	$-0.730665\pi$
$313$	−23.4550	−1.32575	−0.662877	+	0.748728i	$0.730665\pi$
$314$	0	0
$315$	−0.701987	−0.0395525
$316$	0	0
$317$	28.5665	1.60446	0.802228	−	0.597018i	$-0.203648\pi$
$317$	28.5665	1.60446	0.802228	+	0.597018i	$0.203648\pi$
$318$	0	0
$319$	34.7210	1.94400
$320$	0	0
$321$	−7.98185	−0.445503
$322$	0	0
$323$	2.13284	0.118674
$324$	0	0
$325$	−3.16448	−0.175534
$326$	0	0
$327$	12.9206	0.714509
$328$	0	0
$329$	1.86631	0.102893
$330$	0	0
$331$	0.671893	0.0369306	0.0184653	−	0.999830i	$-0.494122\pi$
$331$	0.671893	0.0369306	0.0184653	+	0.999830i	$0.494122\pi$
$332$	0	0
$333$	−4.25328	−0.233079
$334$	0	0
$335$	−1.26134	−0.0689145
$336$	0	0
$337$	−7.32241	−0.398877	−0.199439	−	0.979910i	$-0.563912\pi$
$337$	−7.32241	−0.398877	−0.199439	+	0.979910i	$0.563912\pi$
$338$	0	0
$339$	−18.0796	−0.981947
$340$	0	0
$341$	5.02840	0.272303
$342$	0	0
$343$	−6.92979	−0.374173
$344$	0	0
$345$	−1.26789	−0.0682609
$346$	0	0
$347$	−31.9925	−1.71745	−0.858723	−	0.512440i	$-0.828742\pi$
$347$	−31.9925	−1.71745	−0.858723	+	0.512440i	$0.828742\pi$
$348$	0	0
$349$	30.5866	1.63726	0.818631	−	0.574319i	$-0.194733\pi$
$349$	30.5866	1.63726	0.818631	+	0.574319i	$0.194733\pi$
$350$	0	0
$351$	−17.6235	−0.940671
$352$	0	0
$353$	−21.5958	−1.14943	−0.574713	−	0.818355i	$-0.694886\pi$
$353$	−21.5958	−1.14943	−0.574713	+	0.818355i	$0.694886\pi$
$354$	0	0
$355$	6.28011	0.333313
$356$	0	0
$357$	−0.256141	−0.0135564
$358$	0	0
$359$	19.5814	1.03346	0.516732	−	0.856147i	$-0.327148\pi$
$359$	19.5814	1.03346	0.516732	+	0.856147i	$0.327148\pi$
$360$	0	0
$361$	9.32809	0.490952
$362$	0	0
$363$	−8.39590	−0.440671
$364$	0	0
$365$	−6.17081	−0.322995
$366$	0	0
$367$	−11.9731	−0.624992	−0.312496	−	0.949919i	$-0.601165\pi$
$367$	−11.9731	−0.624992	−0.312496	+	0.949919i	$0.601165\pi$
$368$	0	0
$369$	−8.12781	−0.423117
$370$	0	0
$371$	−5.99410	−0.311198
$372$	0	0
$373$	−26.9803	−1.39699	−0.698493	−	0.715617i	$-0.746146\pi$
$373$	−26.9803	−1.39699	−0.698493	+	0.715617i	$0.746146\pi$
$374$	0	0
$375$	−1.26789	−0.0654736
$376$	0	0
$377$	26.1739	1.34802
$378$	0	0
$379$	−13.2269	−0.679420	−0.339710	−	0.940530i	$-0.610329\pi$
$379$	−13.2269	−0.679420	−0.339710	+	0.940530i	$0.610329\pi$
$380$	0	0
$381$	−11.9332	−0.611358
$382$	0	0
$383$	−7.06241	−0.360872	−0.180436	−	0.983587i	$-0.557751\pi$
$383$	−7.06241	−0.360872	−0.180436	+	0.983587i	$0.557751\pi$
$384$	0	0
$385$	−2.11629	−0.107856
$386$	0	0
$387$	10.8696	0.552532
$388$	0	0
$389$	33.3429	1.69055	0.845275	−	0.534331i	$-0.179437\pi$
$389$	33.3429	1.69055	0.845275	+	0.534331i	$0.179437\pi$
$390$	0	0
$391$	0.400727	0.0202657
$392$	0	0
$393$	−14.9854	−0.755912
$394$	0	0
$395$	9.76375	0.491268
$396$	0	0
$397$	9.14900	0.459175	0.229588	−	0.973288i	$-0.426262\pi$
$397$	9.14900	0.459175	0.229588	+	0.973288i	$0.426262\pi$
$398$	0	0
$399$	−3.40203	−0.170315
$400$	0	0
$401$	−11.8076	−0.589641	−0.294821	−	0.955553i	$-0.595260\pi$
$401$	−11.8076	−0.589641	−0.294821	+	0.955553i	$0.595260\pi$
$402$	0	0
$403$	3.79058	0.188822
$404$	0	0
$405$	−2.88371	−0.143293
$406$	0	0
$407$	−12.8224	−0.635584
$408$	0	0
$409$	38.1500	1.88640	0.943199	−	0.332228i	$-0.107800\pi$
$409$	38.1500	1.88640	0.943199	+	0.332228i	$0.107800\pi$
$410$	0	0
$411$	−18.1861	−0.897053
$412$	0	0
$413$	−1.92910	−0.0949250
$414$	0	0
$415$	5.02510	0.246672
$416$	0	0
$417$	25.4867	1.24809
$418$	0	0
$419$	2.87325	0.140367	0.0701837	−	0.997534i	$-0.477641\pi$
$419$	2.87325	0.140367	0.0701837	+	0.997534i	$0.477641\pi$
$420$	0	0
$421$	−14.2085	−0.692479	−0.346240	−	0.938146i	$-0.612542\pi$
$421$	−14.2085	−0.692479	−0.346240	+	0.938146i	$0.612542\pi$
$422$	0	0
$423$	−5.15485	−0.250637
$424$	0	0
$425$	0.400727	0.0194381
$426$	0	0
$427$	0.837910	0.0405493
$428$	0	0
$429$	−16.8427	−0.813172
$430$	0	0
$431$	−5.76236	−0.277563	−0.138782	−	0.990323i	$-0.544319\pi$
$431$	−5.76236	−0.277563	−0.138782	+	0.990323i	$0.544319\pi$
$432$	0	0
$433$	1.65671	0.0796163	0.0398081	−	0.999207i	$-0.487325\pi$
$433$	1.65671	0.0796163	0.0398081	+	0.999207i	$0.487325\pi$
$434$	0	0
$435$	10.4869	0.502808
$436$	0	0
$437$	5.32241	0.254606
$438$	0	0
$439$	12.5017	0.596674	0.298337	−	0.954461i	$-0.403568\pi$
$439$	12.5017	0.596674	0.298337	+	0.954461i	$0.403568\pi$
$440$	0	0
$441$	9.39328	0.447299
$442$	0	0
$443$	−2.08447	−0.0990362	−0.0495181	−	0.998773i	$-0.515769\pi$
$443$	−2.08447	−0.0990362	−0.0495181	+	0.998773i	$0.515769\pi$
$444$	0	0
$445$	−7.37455	−0.349587
$446$	0	0
$447$	5.09008	0.240753
$448$	0	0
$449$	−24.8717	−1.17377	−0.586885	−	0.809670i	$-0.699646\pi$
$449$	−24.8717	−1.17377	−0.586885	+	0.809670i	$0.699646\pi$
$450$	0	0
$451$	−24.5030	−1.15380
$452$	0	0
$453$	−13.7457	−0.645828
$454$	0	0
$455$	−1.59533	−0.0747903
$456$	0	0
$457$	10.5949	0.495609	0.247805	−	0.968810i	$-0.420291\pi$
$457$	10.5949	0.495609	0.247805	+	0.968810i	$0.420291\pi$
$458$	0	0
$459$	2.23171	0.104167
$460$	0	0
$461$	11.7443	0.546986	0.273493	−	0.961874i	$-0.411821\pi$
$461$	11.7443	0.546986	0.273493	+	0.961874i	$0.411821\pi$
$462$	0	0
$463$	−16.7927	−0.780425	−0.390212	−	0.920725i	$-0.627598\pi$
$463$	−16.7927	−0.780425	−0.390212	+	0.920725i	$0.627598\pi$
$464$	0	0
$465$	1.51874	0.0704300
$466$	0	0
$467$	25.3448	1.17282	0.586410	−	0.810015i	$-0.300541\pi$
$467$	25.3448	1.17282	0.586410	+	0.810015i	$0.300541\pi$
$468$	0	0
$469$	−0.635889	−0.0293626
$470$	0	0
$471$	−1.59487	−0.0734877
$472$	0	0
$473$	32.7687	1.50671
$474$	0	0
$475$	5.32241	0.244209
$476$	0	0
$477$	16.5561	0.758050
$478$	0	0
$479$	23.4215	1.07016	0.535078	−	0.844803i	$-0.320282\pi$
$479$	23.4215	1.07016	0.535078	+	0.844803i	$0.320282\pi$
$480$	0	0
$481$	−9.66598	−0.440731
$482$	0	0
$483$	−0.639190	−0.0290842
$484$	0	0
$485$	−7.23189	−0.328383
$486$	0	0
$487$	−25.9132	−1.17424	−0.587119	−	0.809501i	$-0.699738\pi$
$487$	−25.9132	−1.17424	−0.587119	+	0.809501i	$0.699738\pi$
$488$	0	0
$489$	16.1813	0.731744
$490$	0	0
$491$	17.5545	0.792224	0.396112	−	0.918202i	$-0.370359\pi$
$491$	17.5545	0.792224	0.396112	+	0.918202i	$0.370359\pi$
$492$	0	0
$493$	−3.31447	−0.149276
$494$	0	0
$495$	5.84531	0.262727
$496$	0	0
$497$	3.16603	0.142016
$498$	0	0
$499$	36.9650	1.65478	0.827390	−	0.561628i	$-0.189825\pi$
$499$	36.9650	1.65478	0.827390	+	0.561628i	$0.189825\pi$
$500$	0	0
$501$	13.8599	0.619216
$502$	0	0
$503$	25.0868	1.11856	0.559282	−	0.828978i	$-0.311077\pi$
$503$	25.0868	1.11856	0.559282	+	0.828978i	$0.311077\pi$
$504$	0	0
$505$	15.4373	0.686953
$506$	0	0
$507$	3.78600	0.168142
$508$	0	0
$509$	18.3097	0.811564	0.405782	−	0.913970i	$-0.366999\pi$
$509$	18.3097	0.811564	0.405782	+	0.913970i	$0.366999\pi$
$510$	0	0
$511$	−3.11093	−0.137620
$512$	0	0
$513$	29.6413	1.30870
$514$	0	0
$515$	16.7465	0.737939
$516$	0	0
$517$	−15.5404	−0.683466
$518$	0	0
$519$	−14.8562	−0.652115
$520$	0	0
$521$	32.2129	1.41127	0.705636	−	0.708574i	$-0.250661\pi$
$521$	32.2129	1.41127	0.705636	+	0.708574i	$0.250661\pi$
$522$	0	0
$523$	−26.2049	−1.14586	−0.572929	−	0.819605i	$-0.694193\pi$
$523$	−26.2049	−1.14586	−0.572929	+	0.819605i	$0.694193\pi$
$524$	0	0
$525$	−0.639190	−0.0278965
$526$	0	0
$527$	−0.480011	−0.0209096
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.32830	0.231228
$532$	0	0
$533$	−18.4712	−0.800077
$534$	0	0
$535$	6.29538	0.272173
$536$	0	0
$537$	−9.44476	−0.407572
$538$	0	0
$539$	28.3181	1.21974
$540$	0	0
$541$	28.3643	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$541$	28.3643	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$542$	0	0
$543$	15.1195	0.648840
$544$	0	0
$545$	−10.1906	−0.436518
$546$	0	0
$547$	29.5775	1.26464	0.632321	−	0.774706i	$-0.282102\pi$
$547$	29.5775	1.26464	0.632321	+	0.774706i	$0.282102\pi$
$548$	0	0
$549$	−2.31436	−0.0987743
$550$	0	0
$551$	−44.0224	−1.87542
$552$	0	0
$553$	4.92227	0.209316
$554$	0	0
$555$	−3.87280	−0.164391
$556$	0	0
$557$	26.6439	1.12894	0.564470	−	0.825454i	$-0.309081\pi$
$557$	26.6439	1.12894	0.564470	+	0.825454i	$0.309081\pi$
$558$	0	0
$559$	24.7021	1.04479
$560$	0	0
$561$	2.13284	0.0900484
$562$	0	0
$563$	43.7187	1.84252	0.921262	−	0.388942i	$-0.127159\pi$
$563$	43.7187	1.84252	0.921262	+	0.388942i	$0.127159\pi$
$564$	0	0
$565$	14.2596	0.599904
$566$	0	0
$567$	−1.45378	−0.0610532
$568$	0	0
$569$	−20.5148	−0.860025	−0.430013	−	0.902823i	$-0.641491\pi$
$569$	−20.5148	−0.860025	−0.430013	+	0.902823i	$0.641491\pi$
$570$	0	0
$571$	−15.5314	−0.649970	−0.324985	−	0.945719i	$-0.605359\pi$
$571$	−15.5314	−0.649970	−0.324985	+	0.945719i	$0.605359\pi$
$572$	0	0
$573$	−4.97034	−0.207639
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	29.4587	1.22638	0.613192	−	0.789934i	$-0.289885\pi$
$577$	29.4587	1.22638	0.613192	+	0.789934i	$0.289885\pi$
$578$	0	0
$579$	3.76245	0.156362
$580$	0	0
$581$	2.53334	0.105100
$582$	0	0
$583$	49.9118	2.06713
$584$	0	0
$585$	4.40639	0.182182
$586$	0	0
$587$	29.6993	1.22582	0.612911	−	0.790152i	$-0.289999\pi$
$587$	29.6993	1.22582	0.612911	+	0.790152i	$0.289999\pi$
$588$	0	0
$589$	−6.37546	−0.262696
$590$	0	0
$591$	8.05844	0.331480
$592$	0	0
$593$	−13.0357	−0.535314	−0.267657	−	0.963514i	$-0.586249\pi$
$593$	−13.0357	−0.535314	−0.267657	+	0.963514i	$0.586249\pi$
$594$	0	0
$595$	0.202021	0.00828207
$596$	0	0
$597$	−27.3318	−1.11862
$598$	0	0
$599$	33.5204	1.36961	0.684803	−	0.728728i	$-0.259888\pi$
$599$	33.5204	1.36961	0.684803	+	0.728728i	$0.259888\pi$
$600$	0	0
$601$	21.1257	0.861736	0.430868	−	0.902415i	$-0.358208\pi$
$601$	21.1257	0.861736	0.430868	+	0.902415i	$0.358208\pi$
$602$	0	0
$603$	1.75636	0.0715246
$604$	0	0
$605$	6.62195	0.269220
$606$	0	0
$607$	−8.11565	−0.329404	−0.164702	−	0.986343i	$-0.552666\pi$
$607$	−8.11565	−0.329404	−0.164702	+	0.986343i	$0.552666\pi$
$608$	0	0
$609$	5.28683	0.214233
$610$	0	0
$611$	−11.7149	−0.473933
$612$	0	0
$613$	−11.4463	−0.462311	−0.231156	−	0.972917i	$-0.574251\pi$
$613$	−11.4463	−0.462311	−0.231156	+	0.972917i	$0.574251\pi$
$614$	0	0
$615$	−7.40073	−0.298426
$616$	0	0
$617$	−16.1789	−0.651336	−0.325668	−	0.945484i	$-0.605589\pi$
$617$	−16.1789	−0.651336	−0.325668	+	0.945484i	$0.605589\pi$
$618$	0	0
$619$	28.8789	1.16074	0.580371	−	0.814352i	$-0.302907\pi$
$619$	28.8789	1.16074	0.580371	+	0.814352i	$0.302907\pi$
$620$	0	0
$621$	5.56915	0.223482
$622$	0	0
$623$	−3.71778	−0.148950
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	28.3281	1.13132
$628$	0	0
$629$	1.22403	0.0488053
$630$	0	0
$631$	−38.7017	−1.54069	−0.770346	−	0.637626i	$-0.779916\pi$
$631$	−38.7017	−1.54069	−0.770346	+	0.637626i	$0.779916\pi$
$632$	0	0
$633$	13.5698	0.539349
$634$	0	0
$635$	9.41188	0.373499
$636$	0	0
$637$	21.3471	0.845803
$638$	0	0
$639$	−8.74476	−0.345937
$640$	0	0
$641$	−15.9732	−0.630903	−0.315451	−	0.948942i	$-0.602156\pi$
$641$	−15.9732	−0.630903	−0.315451	+	0.948942i	$0.602156\pi$
$642$	0	0
$643$	−7.50147	−0.295829	−0.147915	−	0.989000i	$-0.547256\pi$
$643$	−7.50147	−0.295829	−0.147915	+	0.989000i	$0.547256\pi$
$644$	0	0
$645$	9.89723	0.389703
$646$	0	0
$647$	−18.8857	−0.742473	−0.371237	−	0.928538i	$-0.621066\pi$
$647$	−18.8857	−0.742473	−0.371237	+	0.928538i	$0.621066\pi$
$648$	0	0
$649$	16.0633	0.630539
$650$	0	0
$651$	0.765654	0.0300083
$652$	0	0
$653$	5.97078	0.233655	0.116827	−	0.993152i	$-0.462728\pi$
$653$	5.97078	0.233655	0.116827	+	0.993152i	$0.462728\pi$
$654$	0	0
$655$	11.8191	0.461812
$656$	0	0
$657$	8.59258	0.335228
$658$	0	0
$659$	−44.7975	−1.74506	−0.872531	−	0.488558i	$-0.837523\pi$
$659$	−44.7975	−1.74506	−0.872531	+	0.488558i	$0.837523\pi$
$660$	0	0
$661$	16.6409	0.647254	0.323627	−	0.946185i	$-0.395098\pi$
$661$	16.6409	0.647254	0.323627	+	0.946185i	$0.395098\pi$
$662$	0	0
$663$	1.60780	0.0624419
$664$	0	0
$665$	2.68322	0.104051
$666$	0	0
$667$	−8.27114	−0.320260
$668$	0	0
$669$	−17.5729	−0.679406
$670$	0	0
$671$	−6.97712	−0.269349
$672$	0	0
$673$	−41.0272	−1.58148	−0.790741	−	0.612150i	$-0.790305\pi$
$673$	−41.0272	−1.58148	−0.790741	+	0.612150i	$0.790305\pi$
$674$	0	0
$675$	5.56915	0.214357
$676$	0	0
$677$	−7.19094	−0.276370	−0.138185	−	0.990406i	$-0.544127\pi$
$677$	−7.19094	−0.276370	−0.138185	+	0.990406i	$0.544127\pi$
$678$	0	0
$679$	−3.64586	−0.139915
$680$	0	0
$681$	−0.113695	−0.00435681
$682$	0	0
$683$	−15.6677	−0.599507	−0.299754	−	0.954017i	$-0.596904\pi$
$683$	−15.6677	−0.599507	−0.299754	+	0.954017i	$0.596904\pi$
$684$	0	0
$685$	14.3436	0.548040
$686$	0	0
$687$	−23.1371	−0.882736
$688$	0	0
$689$	37.6252	1.43340
$690$	0	0
$691$	45.1395	1.71719	0.858594	−	0.512656i	$-0.171338\pi$
$691$	45.1395	1.71719	0.858594	+	0.512656i	$0.171338\pi$
$692$	0	0
$693$	2.94684	0.111941
$694$	0	0
$695$	−20.1016	−0.762498
$696$	0	0
$697$	2.33906	0.0885983
$698$	0	0
$699$	−7.26681	−0.274856
$700$	0	0
$701$	−25.3079	−0.955867	−0.477934	−	0.878396i	$-0.658614\pi$
$701$	−25.3079	−0.955867	−0.477934	+	0.878396i	$0.658614\pi$
$702$	0	0
$703$	16.2574	0.613161
$704$	0	0
$705$	−4.69371	−0.176775
$706$	0	0
$707$	7.78253	0.292692
$708$	0	0
$709$	20.3833	0.765510	0.382755	−	0.923850i	$-0.374975\pi$
$709$	20.3833	0.765510	0.382755	+	0.923850i	$0.374975\pi$
$710$	0	0
$711$	−13.5956	−0.509874
$712$	0	0
$713$	−1.19785	−0.0448599
$714$	0	0
$715$	13.2840	0.496794
$716$	0	0
$717$	−2.88765	−0.107841
$718$	0	0
$719$	8.37523	0.312343	0.156172	−	0.987730i	$-0.450085\pi$
$719$	8.37523	0.312343	0.156172	+	0.987730i	$0.450085\pi$
$720$	0	0
$721$	8.44253	0.314416
$722$	0	0
$723$	−9.94454	−0.369841
$724$	0	0
$725$	−8.27114	−0.307182
$726$	0	0
$727$	−42.6494	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$727$	−42.6494	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$728$	0	0
$729$	25.1986	0.933283
$730$	0	0
$731$	−3.12810	−0.115697
$732$	0	0
$733$	32.5756	1.20321	0.601603	−	0.798795i	$-0.294529\pi$
$733$	32.5756	1.20321	0.601603	+	0.798795i	$0.294529\pi$
$734$	0	0
$735$	8.55299	0.315482
$736$	0	0
$737$	5.29493	0.195041
$738$	0	0
$739$	−29.2378	−1.07553	−0.537765	−	0.843095i	$-0.680731\pi$
$739$	−29.2378	−1.07553	−0.537765	+	0.843095i	$0.680731\pi$
$740$	0	0
$741$	21.3547	0.784483
$742$	0	0
$743$	−19.1316	−0.701869	−0.350935	−	0.936400i	$-0.614136\pi$
$743$	−19.1316	−0.701869	−0.350935	+	0.936400i	$0.614136\pi$
$744$	0	0
$745$	−4.01461	−0.147084
$746$	0	0
$747$	−6.99722	−0.256015
$748$	0	0
$749$	3.17373	0.115966
$750$	0	0
$751$	27.5211	1.00426	0.502130	−	0.864792i	$-0.332550\pi$
$751$	27.5211	1.00426	0.502130	+	0.864792i	$0.332550\pi$
$752$	0	0
$753$	−20.6184	−0.751377
$754$	0	0
$755$	10.8414	0.394558
$756$	0	0
$757$	−9.38998	−0.341285	−0.170642	−	0.985333i	$-0.554584\pi$
$757$	−9.38998	−0.341285	−0.170642	+	0.985333i	$0.554584\pi$
$758$	0	0
$759$	5.32241	0.193191
$760$	0	0
$761$	−37.0902	−1.34452	−0.672260	−	0.740315i	$-0.734676\pi$
$761$	−37.0902	−1.34452	−0.672260	+	0.740315i	$0.734676\pi$
$762$	0	0
$763$	−5.13746	−0.185989
$764$	0	0
$765$	−0.557994	−0.0201743
$766$	0	0
$767$	12.1090	0.437232
$768$	0	0
$769$	34.0797	1.22895	0.614473	−	0.788938i	$-0.289369\pi$
$769$	34.0797	1.22895	0.614473	+	0.788938i	$0.289369\pi$
$770$	0	0
$771$	−14.6965	−0.529283
$772$	0	0
$773$	−32.7693	−1.17863	−0.589315	−	0.807903i	$-0.700602\pi$
$773$	−32.7693	−1.17863	−0.589315	+	0.807903i	$0.700602\pi$
$774$	0	0
$775$	−1.19785	−0.0430281
$776$	0	0
$777$	−1.95242	−0.0700427
$778$	0	0
$779$	31.0671	1.11310
$780$	0	0
$781$	−26.3629	−0.943340
$782$	0	0
$783$	−46.0632	−1.64616
$784$	0	0
$785$	1.25789	0.0448961
$786$	0	0
$787$	7.38411	0.263215	0.131608	−	0.991302i	$-0.457986\pi$
$787$	7.38411	0.263215	0.131608	+	0.991302i	$0.457986\pi$
$788$	0	0
$789$	19.7927	0.704640
$790$	0	0
$791$	7.18877	0.255603
$792$	0	0
$793$	−5.25959	−0.186773
$794$	0	0
$795$	15.0750	0.534656
$796$	0	0
$797$	26.1488	0.926237	0.463119	−	0.886296i	$-0.346730\pi$
$797$	26.1488	0.926237	0.463119	+	0.886296i	$0.346730\pi$
$798$	0	0
$799$	1.48349	0.0524820
$800$	0	0
$801$	10.2687	0.362827
$802$	0	0
$803$	25.9042	0.914138
$804$	0	0
$805$	0.504137	0.0177685
$806$	0	0
$807$	−28.8193	−1.01449
$808$	0	0
$809$	−10.5132	−0.369625	−0.184813	−	0.982774i	$-0.559168\pi$
$809$	−10.5132	−0.369625	−0.184813	+	0.982774i	$0.559168\pi$
$810$	0	0
$811$	−29.3074	−1.02912	−0.514562	−	0.857453i	$-0.672045\pi$
$811$	−29.3074	−1.02912	−0.514562	+	0.857453i	$0.672045\pi$
$812$	0	0
$813$	−40.4806	−1.41971
$814$	0	0
$815$	−12.7624	−0.447047
$816$	0	0
$817$	−41.5471	−1.45355
$818$	0	0
$819$	2.22142	0.0776229
$820$	0	0
$821$	−39.1661	−1.36691	−0.683454	−	0.729994i	$-0.739523\pi$
$821$	−39.1661	−1.36691	−0.683454	+	0.729994i	$0.739523\pi$
$822$	0	0
$823$	−15.3245	−0.534180	−0.267090	−	0.963672i	$-0.586062\pi$
$823$	−15.3245	−0.534180	−0.267090	+	0.963672i	$0.586062\pi$
$824$	0	0
$825$	5.32241	0.185303
$826$	0	0
$827$	22.8118	0.793243	0.396622	−	0.917982i	$-0.370182\pi$
$827$	22.8118	0.793243	0.396622	+	0.917982i	$0.370182\pi$
$828$	0	0
$829$	−19.7511	−0.685984	−0.342992	−	0.939338i	$-0.611440\pi$
$829$	−19.7511	−0.685984	−0.342992	+	0.939338i	$0.611440\pi$
$830$	0	0
$831$	21.0045	0.728639
$832$	0	0
$833$	−2.70325	−0.0936619
$834$	0	0
$835$	−10.9315	−0.378300
$836$	0	0
$837$	−6.67101	−0.230584
$838$	0	0
$839$	−12.7833	−0.441329	−0.220664	−	0.975350i	$-0.570823\pi$
$839$	−12.7833	−0.441329	−0.220664	+	0.975350i	$0.570823\pi$
$840$	0	0
$841$	39.4117	1.35902
$842$	0	0
$843$	−2.80827	−0.0967220
$844$	0	0
$845$	−2.98606	−0.102724
$846$	0	0
$847$	3.33837	0.114708
$848$	0	0
$849$	−24.8629	−0.853293
$850$	0	0
$851$	3.05452	0.104708
$852$	0	0
$853$	19.4643	0.666446	0.333223	−	0.942848i	$-0.391864\pi$
$853$	19.4643	0.666446	0.333223	+	0.942848i	$0.391864\pi$
$854$	0	0
$855$	−7.41122	−0.253458
$856$	0	0
$857$	−12.0525	−0.411705	−0.205853	−	0.978583i	$-0.565997\pi$
$857$	−12.0525	−0.411705	−0.205853	+	0.978583i	$0.565997\pi$
$858$	0	0
$859$	7.42232	0.253246	0.126623	−	0.991951i	$-0.459586\pi$
$859$	7.42232	0.253246	0.126623	+	0.991951i	$0.459586\pi$
$860$	0	0
$861$	−3.73098	−0.127151
$862$	0	0
$863$	24.4442	0.832090	0.416045	−	0.909344i	$-0.363416\pi$
$863$	24.4442	0.832090	0.416045	+	0.909344i	$0.363416\pi$
$864$	0	0
$865$	11.7173	0.398399
$866$	0	0
$867$	21.3505	0.725102
$868$	0	0
$869$	−40.9868	−1.39038
$870$	0	0
$871$	3.99149	0.135247
$872$	0	0
$873$	10.0701	0.340820
$874$	0	0
$875$	0.504137	0.0170429
$876$	0	0
$877$	35.4611	1.19744	0.598719	−	0.800960i	$-0.295677\pi$
$877$	35.4611	1.19744	0.598719	+	0.800960i	$0.295677\pi$
$878$	0	0
$879$	−11.5274	−0.388811
$880$	0	0
$881$	19.7120	0.664113	0.332056	−	0.943260i	$-0.392258\pi$
$881$	19.7120	0.664113	0.332056	+	0.943260i	$0.392258\pi$
$882$	0	0
$883$	−32.2916	−1.08670	−0.543349	−	0.839507i	$-0.682844\pi$
$883$	−32.2916	−1.08670	−0.543349	+	0.839507i	$0.682844\pi$
$884$	0	0
$885$	4.85165	0.163086
$886$	0	0
$887$	15.7766	0.529727	0.264864	−	0.964286i	$-0.414673\pi$
$887$	15.7766	0.529727	0.264864	+	0.964286i	$0.414673\pi$
$888$	0	0
$889$	4.74488	0.159138
$890$	0	0
$891$	12.1054	0.405546
$892$	0	0
$893$	19.7035	0.659353
$894$	0	0
$895$	7.44919	0.248999
$896$	0	0
$897$	4.01222	0.133964
$898$	0	0
$899$	9.90759	0.330436
$900$	0	0
$901$	−4.76458	−0.158731
$902$	0	0
$903$	4.98956	0.166042
$904$	0	0
$905$	−11.9249	−0.396398
$906$	0	0
$907$	−46.7141	−1.55112	−0.775558	−	0.631276i	$-0.782532\pi$
$907$	−46.7141	−1.55112	−0.775558	+	0.631276i	$0.782532\pi$
$908$	0	0
$909$	−21.4958	−0.712970
$910$	0	0
$911$	33.8989	1.12312	0.561560	−	0.827436i	$-0.310201\pi$
$911$	33.8989	1.12312	0.561560	+	0.827436i	$0.310201\pi$
$912$	0	0
$913$	−21.0946	−0.698129
$914$	0	0
$915$	−2.10732	−0.0696659
$916$	0	0
$917$	5.95846	0.196766
$918$	0	0
$919$	19.1171	0.630615	0.315308	−	0.948990i	$-0.397892\pi$
$919$	19.1171	0.630615	0.315308	+	0.948990i	$0.397892\pi$
$920$	0	0
$921$	−36.2964	−1.19601
$922$	0	0
$923$	−19.8733	−0.654137
$924$	0	0
$925$	3.05452	0.100432
$926$	0	0
$927$	−23.3187	−0.765888
$928$	0	0
$929$	30.0512	0.985949	0.492974	−	0.870044i	$-0.335910\pi$
$929$	30.0512	0.985949	0.492974	+	0.870044i	$0.335910\pi$
$930$	0	0
$931$	−35.9042	−1.17671
$932$	0	0
$933$	−14.1241	−0.462401
$934$	0	0
$935$	−1.68219	−0.0550136
$936$	0	0
$937$	−31.6714	−1.03466	−0.517329	−	0.855786i	$-0.673074\pi$
$937$	−31.6714	−1.03466	−0.517329	+	0.855786i	$0.673074\pi$
$938$	0	0
$939$	29.7384	0.970475
$940$	0	0
$941$	−24.9550	−0.813509	−0.406755	−	0.913537i	$-0.633340\pi$
$941$	−24.9550	−0.813509	−0.406755	+	0.913537i	$0.633340\pi$
$942$	0	0
$943$	5.83704	0.190080
$944$	0	0
$945$	2.80761	0.0913317
$946$	0	0
$947$	22.1419	0.719516	0.359758	−	0.933046i	$-0.382859\pi$
$947$	22.1419	0.719516	0.359758	+	0.933046i	$0.382859\pi$
$948$	0	0
$949$	19.5274	0.633887
$950$	0	0
$951$	−36.2192	−1.17449
$952$	0	0
$953$	−45.2529	−1.46589	−0.732943	−	0.680290i	$-0.761854\pi$
$953$	−45.2529	−1.46589	−0.732943	+	0.680290i	$0.761854\pi$
$954$	0	0
$955$	3.92017	0.126854
$956$	0	0
$957$	−44.0224	−1.42304
$958$	0	0
$959$	7.23112	0.233505
$960$	0	0
$961$	−29.5652	−0.953715
$962$	0	0
$963$	−8.76602	−0.282481
$964$	0	0
$965$	−2.96749	−0.0955268
$966$	0	0
$967$	23.4231	0.753236	0.376618	−	0.926369i	$-0.377087\pi$
$967$	23.4231	0.753236	0.376618	+	0.926369i	$0.377087\pi$
$968$	0	0
$969$	−2.70420	−0.0868715
$970$	0	0
$971$	20.4616	0.656643	0.328322	−	0.944566i	$-0.393517\pi$
$971$	20.4616	0.656643	0.328322	+	0.944566i	$0.393517\pi$
$972$	0	0
$973$	−10.1340	−0.324880
$974$	0	0
$975$	4.01222	0.128494
$976$	0	0
$977$	10.9208	0.349388	0.174694	−	0.984623i	$-0.444106\pi$
$977$	10.9208	0.349388	0.174694	+	0.984623i	$0.444106\pi$
$978$	0	0
$979$	30.9572	0.989398
$980$	0	0
$981$	14.1900	0.453050
$982$	0	0
$983$	−43.1629	−1.37668	−0.688342	−	0.725387i	$-0.741661\pi$
$983$	−43.1629	−1.37668	−0.688342	+	0.725387i	$0.741661\pi$
$984$	0	0
$985$	−6.35578	−0.202512
$986$	0	0
$987$	−2.36627	−0.0753193
$988$	0	0
$989$	−7.80606	−0.248218
$990$	0	0
$991$	−21.9381	−0.696887	−0.348444	−	0.937330i	$-0.613290\pi$
$991$	−21.9381	−0.696887	−0.348444	+	0.937330i	$0.613290\pi$
$992$	0	0
$993$	−0.851887	−0.0270338
$994$	0	0
$995$	21.5569	0.683401
$996$	0	0
$997$	14.5061	0.459414	0.229707	−	0.973260i	$-0.426223\pi$
$997$	14.5061	0.459414	0.229707	+	0.973260i	$0.426223\pi$
$998$	0	0
$999$	17.0111	0.538207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cs.1.2		5
4.3	odd	2		7360.2.a.cm.1.4		5
8.3	odd	2		3680.2.a.y.1.2	yes	5
8.5	even	2		3680.2.a.w.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.w.1.4	✓	5	8.5	even	2
3680.2.a.y.1.2	yes	5	8.3	odd	2
7360.2.a.cm.1.4		5	4.3	odd	2
7360.2.a.cs.1.2		5	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26789 q^{3} +1.00000 q^{5} +0.504137 q^{7} -1.39245 q^{9} -4.19785 q^{11} -3.16448 q^{13} -1.26789 q^{15} +0.400727 q^{17} +5.32241 q^{19} -0.639190 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56915 q^{27} -8.27114 q^{29} -1.19785 q^{31} +5.32241 q^{33} +0.504137 q^{35} +3.05452 q^{37} +4.01222 q^{39} +5.83704 q^{41} -7.80606 q^{43} -1.39245 q^{45} +3.70199 q^{47} -6.74585 q^{49} -0.508078 q^{51} -11.8898 q^{53} -4.19785 q^{55} -6.74824 q^{57} -3.82655 q^{59} +1.66207 q^{61} -0.701987 q^{63} -3.16448 q^{65} -1.26134 q^{67} -1.26789 q^{69} +6.28011 q^{71} -6.17081 q^{73} -1.26789 q^{75} -2.11629 q^{77} +9.76375 q^{79} -2.88371 q^{81} +5.02510 q^{83} +0.400727 q^{85} +10.4869 q^{87} -7.37455 q^{89} -1.59533 q^{91} +1.51874 q^{93} +5.32241 q^{95} -7.23189 q^{97} +5.84531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + 5 q^{5} + 4 q^{7} + 2 q^{9} + 3 q^{11} - q^{13} + q^{15} - 4 q^{17} + q^{19} + 4 q^{21} + 5 q^{23} + 5 q^{25} + 4 q^{27} + 5 q^{29} + 18 q^{31} + q^{33} + 4 q^{35} - 3 q^{37} + 20 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 5 * q + q^3 + 5 * q^5 + 4 * q^7 + 2 * q^9 + 3 * q^11 - q^13 + q^15 - 4 * q^17 + q^19 + 4 * q^21 + 5 * q^23 + 5 * q^25 + 4 * q^27 + 5 * q^29 + 18 * q^31 + q^33 + 4 * q^35 - 3 * q^37 + 20 * q^39 - 2 * q^41 - 12 * q^43 + 2 * q^45 - 4 * q^47 + 5 * q^49 - q^51 - 3 * q^53 + 3 * q^55 + 10 * q^57 + 5 * q^59 - q^61 + 19 * q^63 - q^65 - 3 * q^67 + q^69 + 24 * q^71 - 8 * q^73 + q^75 - 6 * q^77 + 40 * q^79 - 19 * q^81 + 13 * q^83 - 4 * q^85 + 12 * q^87 + 2 * q^89 - 17 * q^91 + 4 * q^93 + q^95 + 9 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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