LMFDB - Embedded newform 9360.2.a.da.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	9360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +1.14399 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +1.14399 q^{7} +3.81681 q^{11} +1.00000 q^{13} -7.34565 q^{17} -5.52884 q^{19} -6.67282 q^{23} +1.00000 q^{25} -2.85601 q^{29} +0.183190 q^{31} -1.14399 q^{35} +6.48963 q^{37} +7.34565 q^{41} -2.67282 q^{43} -2.85601 q^{47} -5.69129 q^{49} +13.6336 q^{53} -3.81681 q^{55} +5.16246 q^{59} +5.14399 q^{61} -1.00000 q^{65} +2.48963 q^{67} +3.81681 q^{71} +12.7776 q^{73} +4.36638 q^{77} -9.34565 q^{79} +11.8353 q^{83} +7.34565 q^{85} +14.9793 q^{89} +1.14399 q^{91} +5.52884 q^{95} +7.71203 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7	$ 3 q - 3 q^{5} + 2 q^{7} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 10 q^{23} + 3 q^{25} - 10 q^{29} + 12 q^{31} - 2 q^{35} - 2 q^{37} + 2 q^{41} + 2 q^{43} - 10 q^{47} + 23 q^{49} + 18 q^{53} - 16 q^{59} + 14 q^{61} - 3 q^{65} - 14 q^{67} + 14 q^{73} + 36 q^{77} - 8 q^{79} - 6 q^{83} + 2 q^{85} + 2 q^{89} + 2 q^{91} + 8 q^{95} + 26 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 + 3 * q^13 - 2 * q^17 - 8 * q^19 - 10 * q^23 + 3 * q^25 - 10 * q^29 + 12 * q^31 - 2 * q^35 - 2 * q^37 + 2 * q^41 + 2 * q^43 - 10 * q^47 + 23 * q^49 + 18 * q^53 - 16 * q^59 + 14 * q^61 - 3 * q^65 - 14 * q^67 + 14 * q^73 + 36 * q^77 - 8 * q^79 - 6 * q^83 + 2 * q^85 + 2 * q^89 + 2 * q^91 + 8 * q^95 + 26 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.14399	0.432386	0.216193	−	0.976351i	$-0.430636\pi$
$7$	1.14399	0.432386	0.216193	+	0.976351i	$0.430636\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.81681	1.15081	0.575406	−	0.817868i	$-0.304844\pi$
$11$	3.81681	1.15081	0.575406	+	0.817868i	$0.304844\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.34565	−1.78158	−0.890791	−	0.454414i	$-0.849849\pi$
$17$	−7.34565	−1.78158	−0.890791	+	0.454414i	$0.849849\pi$
$18$	0	0
$19$	−5.52884	−1.26840	−0.634201	−	0.773168i	$-0.718671\pi$
$19$	−5.52884	−1.26840	−0.634201	+	0.773168i	$0.718671\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.67282	−1.39138	−0.695690	−	0.718342i	$-0.744901\pi$
$23$	−6.67282	−1.39138	−0.695690	+	0.718342i	$0.744901\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.85601	−0.530348	−0.265174	−	0.964201i	$-0.585429\pi$
$29$	−2.85601	−0.530348	−0.265174	+	0.964201i	$0.585429\pi$
$30$	0	0
$31$	0.183190	0.0329019	0.0164509	−	0.999865i	$-0.494763\pi$
$31$	0.183190	0.0329019	0.0164509	+	0.999865i	$0.494763\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.14399	−0.193369
$36$	0	0
$37$	6.48963	1.06689	0.533445	−	0.845835i	$-0.320897\pi$
$37$	6.48963	1.06689	0.533445	+	0.845835i	$0.320897\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.34565	1.14720	0.573599	−	0.819136i	$-0.305547\pi$
$41$	7.34565	1.14720	0.573599	+	0.819136i	$0.305547\pi$
$42$	0	0
$43$	−2.67282	−0.407602	−0.203801	−	0.979012i	$-0.565330\pi$
$43$	−2.67282	−0.407602	−0.203801	+	0.979012i	$0.565330\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.85601	−0.416592	−0.208296	−	0.978066i	$-0.566792\pi$
$47$	−2.85601	−0.416592	−0.208296	+	0.978066i	$0.566792\pi$
$48$	0	0
$49$	−5.69129	−0.813042
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.6336	1.87272	0.936361	−	0.351039i	$-0.114171\pi$
$53$	13.6336	1.87272	0.936361	+	0.351039i	$0.114171\pi$
$54$	0	0
$55$	−3.81681	−0.514659
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.16246	0.672095	0.336047	−	0.941845i	$-0.390910\pi$
$59$	5.16246	0.672095	0.336047	+	0.941845i	$0.390910\pi$
$60$	0	0
$61$	5.14399	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$61$	5.14399	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.48963	0.304157	0.152079	−	0.988368i	$-0.451403\pi$
$67$	2.48963	0.304157	0.152079	+	0.988368i	$0.451403\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.81681	0.452972	0.226486	−	0.974014i	$-0.427276\pi$
$71$	3.81681	0.452972	0.226486	+	0.974014i	$0.427276\pi$
$72$	0	0
$73$	12.7776	1.49551	0.747753	−	0.663977i	$-0.231133\pi$
$73$	12.7776	1.49551	0.747753	+	0.663977i	$0.231133\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.36638	0.497595
$78$	0	0
$79$	−9.34565	−1.05147	−0.525734	−	0.850649i	$-0.676209\pi$
$79$	−9.34565	−1.05147	−0.525734	+	0.850649i	$0.676209\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.8353	1.29909	0.649545	−	0.760323i	$-0.274959\pi$
$83$	11.8353	1.29909	0.649545	+	0.760323i	$0.274959\pi$
$84$	0	0
$85$	7.34565	0.796747
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.9793	1.58780	0.793900	−	0.608049i	$-0.208048\pi$
$89$	14.9793	1.58780	0.793900	+	0.608049i	$0.208048\pi$
$90$	0	0
$91$	1.14399	0.119922
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.52884	0.567247
$96$	0	0
$97$	7.71203	0.783038	0.391519	−	0.920170i	$-0.371950\pi$
$97$	7.71203	0.783038	0.391519	+	0.920170i	$0.371950\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	4.38485	0.432052	0.216026	−	0.976388i	$-0.430690\pi$
$103$	4.38485	0.432052	0.216026	+	0.976388i	$0.430690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.384851	0.0372049	0.0186025	−	0.999827i	$-0.494078\pi$
$107$	0.384851	0.0372049	0.0186025	+	0.999827i	$0.494078\pi$
$108$	0	0
$109$	−12.6913	−1.21561	−0.607803	−	0.794088i	$-0.707949\pi$
$109$	−12.6913	−1.21561	−0.607803	+	0.794088i	$0.707949\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.05767	−0.0994976	−0.0497488	−	0.998762i	$-0.515842\pi$
$113$	−1.05767	−0.0994976	−0.0497488	+	0.998762i	$0.515842\pi$
$114$	0	0
$115$	6.67282	0.622244
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.40332	−0.770331
$120$	0	0
$121$	3.56804	0.324367
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.96080	−0.795142	−0.397571	−	0.917571i	$-0.630147\pi$
$127$	−8.96080	−0.795142	−0.397571	+	0.917571i	$0.630147\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.97927	−0.784522	−0.392261	−	0.919854i	$-0.628307\pi$
$131$	−8.97927	−0.784522	−0.392261	+	0.919854i	$0.628307\pi$
$132$	0	0
$133$	−6.32492	−0.548440
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.4033	1.23056	0.615279	−	0.788309i	$-0.289043\pi$
$137$	14.4033	1.23056	0.615279	+	0.788309i	$0.289043\pi$
$138$	0	0
$139$	−21.3456	−1.81051	−0.905257	−	0.424864i	$-0.860322\pi$
$139$	−21.3456	−1.81051	−0.905257	+	0.424864i	$0.860322\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.81681	0.319178
$144$	0	0
$145$	2.85601	0.237179
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4033	1.17997	0.589983	−	0.807416i	$-0.299134\pi$
$149$	14.4033	1.17997	0.589983	+	0.807416i	$0.299134\pi$
$150$	0	0
$151$	22.5081	1.83168	0.915842	−	0.401539i	$-0.131525\pi$
$151$	22.5081	1.83168	0.915842	+	0.401539i	$0.131525\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.183190	−0.0147142
$156$	0	0
$157$	16.6913	1.33211	0.666055	−	0.745902i	$-0.267982\pi$
$157$	16.6913	1.33211	0.666055	+	0.745902i	$0.267982\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.63362	−0.601614
$162$	0	0
$163$	23.4689	1.83823	0.919113	−	0.393994i	$-0.128907\pi$
$163$	23.4689	1.83823	0.919113	+	0.393994i	$0.128907\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.14399	−0.707583	−0.353791	−	0.935324i	$-0.615108\pi$
$167$	−9.14399	−0.707583	−0.353791	+	0.935324i	$0.615108\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0369	0.763095	0.381547	−	0.924349i	$-0.375391\pi$
$173$	10.0369	0.763095	0.381547	+	0.924349i	$0.375391\pi$
$174$	0	0
$175$	1.14399	0.0864773
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.9793	−1.56806	−0.784032	−	0.620720i	$-0.786840\pi$
$179$	−20.9793	−1.56806	−0.784032	+	0.620720i	$0.786840\pi$
$180$	0	0
$181$	13.5473	1.00696	0.503482	−	0.864006i	$-0.332052\pi$
$181$	13.5473	1.00696	0.503482	+	0.864006i	$0.332052\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.48963	−0.477127
$186$	0	0
$187$	−28.0369	−2.05026
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−3.71203	−0.267198	−0.133599	−	0.991036i	$-0.542653\pi$
$193$	−3.71203	−0.267198	−0.133599	+	0.991036i	$0.542653\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.69129	−0.619229	−0.309615	−	0.950862i	$-0.600200\pi$
$197$	−8.69129	−0.619229	−0.309615	+	0.950862i	$0.600200\pi$
$198$	0	0
$199$	24.9793	1.77073	0.885367	−	0.464893i	$-0.153907\pi$
$199$	24.9793	1.77073	0.885367	+	0.464893i	$0.153907\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.26724	−0.229315
$204$	0	0
$205$	−7.34565	−0.513042
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−21.1025	−1.45969
$210$	0	0
$211$	−16.9793	−1.16890	−0.584451	−	0.811429i	$-0.698690\pi$
$211$	−16.9793	−1.16890	−0.584451	+	0.811429i	$0.698690\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.67282	0.182285
$216$	0	0
$217$	0.209567	0.0142263
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.34565	−0.494122
$222$	0	0
$223$	−13.5473	−0.907195	−0.453597	−	0.891207i	$-0.649860\pi$
$223$	−13.5473	−0.907195	−0.453597	+	0.891207i	$0.649860\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.91369	0.657995	0.328997	−	0.944331i	$-0.393289\pi$
$227$	9.91369	0.657995	0.328997	+	0.944331i	$0.393289\pi$
$228$	0	0
$229$	−14.0369	−0.927587	−0.463794	−	0.885943i	$-0.653512\pi$
$229$	−14.0369	−0.927587	−0.463794	+	0.885943i	$0.653512\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	2.85601	0.186306
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.1048	−0.653624	−0.326812	−	0.945089i	$-0.605974\pi$
$239$	−10.1048	−0.653624	−0.326812	+	0.945089i	$0.605974\pi$
$240$	0	0
$241$	−26.0369	−1.67719	−0.838593	−	0.544758i	$-0.816622\pi$
$241$	−26.0369	−1.67719	−0.838593	+	0.544758i	$0.816622\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.69129	0.363603
$246$	0	0
$247$	−5.52884	−0.351791
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.94233	0.311957	0.155978	−	0.987760i	$-0.450147\pi$
$251$	4.94233	0.311957	0.155978	+	0.987760i	$0.450147\pi$
$252$	0	0
$253$	−25.4689	−1.60122
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.2880	0.766503	0.383251	−	0.923644i	$-0.374804\pi$
$257$	12.2880	0.766503	0.383251	+	0.923644i	$0.374804\pi$
$258$	0	0
$259$	7.42405	0.461308
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.44252	−0.458926	−0.229463	−	0.973317i	$-0.573697\pi$
$263$	−7.44252	−0.458926	−0.229463	+	0.973317i	$0.573697\pi$
$264$	0	0
$265$	−13.6336	−0.837507
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.2672	1.29669	0.648343	−	0.761348i	$-0.275462\pi$
$269$	21.2672	1.29669	0.648343	+	0.761348i	$0.275462\pi$
$270$	0	0
$271$	19.8168	1.20379	0.601893	−	0.798577i	$-0.294413\pi$
$271$	19.8168	1.20379	0.601893	+	0.798577i	$0.294413\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.81681	0.230162
$276$	0	0
$277$	−11.9216	−0.716299	−0.358150	−	0.933664i	$-0.616592\pi$
$277$	−11.9216	−0.716299	−0.358150	+	0.933664i	$0.616592\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.3456	−1.15406	−0.577032	−	0.816721i	$-0.695789\pi$
$281$	−19.3456	−1.15406	−0.577032	+	0.816721i	$0.695789\pi$
$282$	0	0
$283$	−29.3641	−1.74552	−0.872758	−	0.488153i	$-0.837671\pi$
$283$	−29.3641	−1.74552	−0.872758	+	0.488153i	$0.837671\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.40332	0.496032
$288$	0	0
$289$	36.9585	2.17403
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.6050	1.43744	0.718719	−	0.695300i	$-0.244729\pi$
$293$	24.6050	1.43744	0.718719	+	0.695300i	$0.244729\pi$
$294$	0	0
$295$	−5.16246	−0.300570
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.67282	−0.385899
$300$	0	0
$301$	−3.05767	−0.176241
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.14399	−0.294544
$306$	0	0
$307$	15.2593	0.870896	0.435448	−	0.900214i	$-0.356590\pi$
$307$	15.2593	0.870896	0.435448	+	0.900214i	$0.356590\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.0369	0.909372	0.454686	−	0.890652i	$-0.349752\pi$
$311$	16.0369	0.909372	0.454686	+	0.890652i	$0.349752\pi$
$312$	0	0
$313$	27.9216	1.57822	0.789111	−	0.614251i	$-0.210542\pi$
$313$	27.9216	1.57822	0.789111	+	0.614251i	$0.210542\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.9137	1.23080	0.615398	−	0.788217i	$-0.288995\pi$
$317$	21.9137	1.23080	0.615398	+	0.788217i	$0.288995\pi$
$318$	0	0
$319$	−10.9009	−0.610331
$320$	0	0
$321$	0	0
$322$	0	0
$323$	40.6129	2.25976
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.26724	−0.180129
$330$	0	0
$331$	7.24086	0.397994	0.198997	−	0.980000i	$-0.436232\pi$
$331$	7.24086	0.397994	0.198997	+	0.980000i	$0.436232\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.48963	−0.136023
$336$	0	0
$337$	9.05767	0.493403	0.246701	−	0.969092i	$-0.420653\pi$
$337$	9.05767	0.493403	0.246701	+	0.969092i	$0.420653\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.699201	0.0378638
$342$	0	0
$343$	−14.5187	−0.783935
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.63588	−0.141502	−0.0707508	−	0.997494i	$-0.522539\pi$
$347$	−2.63588	−0.141502	−0.0707508	+	0.997494i	$0.522539\pi$
$348$	0	0
$349$	−17.6336	−0.943906	−0.471953	−	0.881624i	$-0.656451\pi$
$349$	−17.6336	−0.943906	−0.471953	+	0.881624i	$0.656451\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.6050	−0.670896	−0.335448	−	0.942059i	$-0.608888\pi$
$353$	−12.6050	−0.670896	−0.335448	+	0.942059i	$0.608888\pi$
$354$	0	0
$355$	−3.81681	−0.202575
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.83754	0.360872	0.180436	−	0.983587i	$-0.442249\pi$
$359$	6.83754	0.360872	0.180436	+	0.983587i	$0.442249\pi$
$360$	0	0
$361$	11.5680	0.608844
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.7776	−0.668811
$366$	0	0
$367$	10.0969	0.527053	0.263526	−	0.964652i	$-0.415114\pi$
$367$	10.0969	0.527053	0.263526	+	0.964652i	$0.415114\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.5967	0.809739
$372$	0	0
$373$	1.42405	0.0737347	0.0368674	−	0.999320i	$-0.488262\pi$
$373$	1.42405	0.0737347	0.0368674	+	0.999320i	$0.488262\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.85601	−0.147092
$378$	0	0
$379$	24.4297	1.25487	0.627435	−	0.778669i	$-0.284105\pi$
$379$	24.4297	1.25487	0.627435	+	0.778669i	$0.284105\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.54731	0.283454	0.141727	−	0.989906i	$-0.454734\pi$
$383$	5.54731	0.283454	0.141727	+	0.989906i	$0.454734\pi$
$384$	0	0
$385$	−4.36638	−0.222531
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.8066	−1.76477	−0.882383	−	0.470531i	$-0.844062\pi$
$389$	−34.8066	−1.76477	−0.882383	+	0.470531i	$0.844062\pi$
$390$	0	0
$391$	49.0162	2.47886
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.34565	0.470231
$396$	0	0
$397$	15.2224	0.763990	0.381995	−	0.924164i	$-0.375237\pi$
$397$	15.2224	0.763990	0.381995	+	0.924164i	$0.375237\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.863919	0.0431421	0.0215710	−	0.999767i	$-0.493133\pi$
$401$	0.863919	0.0431421	0.0215710	+	0.999767i	$0.493133\pi$
$402$	0	0
$403$	0.183190	0.00912533
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.7697	1.22779
$408$	0	0
$409$	6.03694	0.298508	0.149254	−	0.988799i	$-0.452313\pi$
$409$	6.03694	0.298508	0.149254	+	0.988799i	$0.452313\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.90578	0.290605
$414$	0	0
$415$	−11.8353	−0.580971
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.36638	−0.213312	−0.106656	−	0.994296i	$-0.534014\pi$
$419$	−4.36638	−0.213312	−0.106656	+	0.994296i	$0.534014\pi$
$420$	0	0
$421$	34.6498	1.68873	0.844365	−	0.535769i	$-0.179978\pi$
$421$	34.6498	1.68873	0.844365	+	0.535769i	$0.179978\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.34565	−0.356316
$426$	0	0
$427$	5.88465	0.284778
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.8168	0.761869	0.380934	−	0.924602i	$-0.375602\pi$
$431$	15.8168	0.761869	0.380934	+	0.924602i	$0.375602\pi$
$432$	0	0
$433$	7.38259	0.354785	0.177392	−	0.984140i	$-0.443234\pi$
$433$	7.38259	0.354785	0.177392	+	0.984140i	$0.443234\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	36.8930	1.76483
$438$	0	0
$439$	21.3826	1.02054	0.510268	−	0.860016i	$-0.329546\pi$
$439$	21.3826	1.02054	0.510268	+	0.860016i	$0.329546\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.24877	−0.344399	−0.172200	−	0.985062i	$-0.555087\pi$
$443$	−7.24877	−0.344399	−0.172200	+	0.985062i	$0.555087\pi$
$444$	0	0
$445$	−14.9793	−0.710085
$446$	0	0
$447$	0	0
$448$	0	0
$449$	28.9009	1.36392	0.681958	−	0.731391i	$-0.261129\pi$
$449$	28.9009	1.36392	0.681958	+	0.731391i	$0.261129\pi$
$450$	0	0
$451$	28.0369	1.32021
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.14399	−0.0536309
$456$	0	0
$457$	22.9793	1.07492	0.537462	−	0.843288i	$-0.319383\pi$
$457$	22.9793	1.07492	0.537462	+	0.843288i	$0.319383\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.4817	−0.581332	−0.290666	−	0.956825i	$-0.593877\pi$
$461$	−12.4817	−0.581332	−0.290666	+	0.956825i	$0.593877\pi$
$462$	0	0
$463$	32.2017	1.49654	0.748269	−	0.663395i	$-0.230885\pi$
$463$	32.2017	1.49654	0.748269	+	0.663395i	$0.230885\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.0761	−1.25293	−0.626467	−	0.779448i	$-0.715500\pi$
$467$	−27.0761	−1.25293	−0.626467	+	0.779448i	$0.715500\pi$
$468$	0	0
$469$	2.84811	0.131513
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.2017	−0.469073
$474$	0	0
$475$	−5.52884	−0.253680
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.7961	1.13296	0.566481	−	0.824075i	$-0.308305\pi$
$479$	24.7961	1.13296	0.566481	+	0.824075i	$0.308305\pi$
$480$	0	0
$481$	6.48963	0.295902
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.71203	−0.350185
$486$	0	0
$487$	7.62571	0.345554	0.172777	−	0.984961i	$-0.444726\pi$
$487$	7.62571	0.345554	0.172777	+	0.984961i	$0.444726\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−34.3249	−1.54906	−0.774531	−	0.632536i	$-0.782014\pi$
$491$	−34.3249	−1.54906	−0.774531	+	0.632536i	$0.782014\pi$
$492$	0	0
$493$	20.9793	0.944859
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.36638	0.195859
$498$	0	0
$499$	8.58651	0.384385	0.192193	−	0.981357i	$-0.438440\pi$
$499$	8.58651	0.384385	0.192193	+	0.981357i	$0.438440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	40.2280	1.79368	0.896840	−	0.442356i	$-0.145857\pi$
$503$	40.2280	1.79368	0.896840	+	0.442356i	$0.145857\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.3664	0.459482	0.229741	−	0.973252i	$-0.426212\pi$
$509$	10.3664	0.459482	0.229741	+	0.973252i	$0.426212\pi$
$510$	0	0
$511$	14.6174	0.646636
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.38485	−0.193220
$516$	0	0
$517$	−10.9009	−0.479419
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.8145	−1.43763	−0.718816	−	0.695201i	$-0.755316\pi$
$521$	−32.8145	−1.43763	−0.718816	+	0.695201i	$0.755316\pi$
$522$	0	0
$523$	25.9401	1.13428	0.567140	−	0.823621i	$-0.308050\pi$
$523$	25.9401	1.13428	0.567140	+	0.823621i	$0.308050\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.34565	−0.0586173
$528$	0	0
$529$	21.5266	0.935938
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.34565	0.318175
$534$	0	0
$535$	−0.384851	−0.0166385
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−21.7226	−0.935658
$540$	0	0
$541$	6.61289	0.284310	0.142155	−	0.989844i	$-0.454597\pi$
$541$	6.61289	0.284310	0.142155	+	0.989844i	$0.454597\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.6913	0.543635
$546$	0	0
$547$	−8.38485	−0.358510	−0.179255	−	0.983803i	$-0.557369\pi$
$547$	−8.38485	−0.358510	−0.179255	+	0.983803i	$0.557369\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.7904	0.672695
$552$	0	0
$553$	−10.6913	−0.454640
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.4482	−1.71384	−0.856922	−	0.515446i	$-0.827626\pi$
$557$	−40.4482	−1.71384	−0.856922	+	0.515446i	$0.827626\pi$
$558$	0	0
$559$	−2.67282	−0.113048
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.7512	−0.705980	−0.352990	−	0.935627i	$-0.614835\pi$
$563$	−16.7512	−0.705980	−0.352990	+	0.935627i	$0.614835\pi$
$564$	0	0
$565$	1.05767	0.0444967
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.4112	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$569$	12.4112	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$570$	0	0
$571$	−12.0369	−0.503730	−0.251865	−	0.967762i	$-0.581044\pi$
$571$	−12.0369	−0.503730	−0.251865	+	0.967762i	$0.581044\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.67282	−0.278276
$576$	0	0
$577$	−16.6050	−0.691274	−0.345637	−	0.938368i	$-0.612337\pi$
$577$	−16.6050	−0.691274	−0.345637	+	0.938368i	$0.612337\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.5394	0.561709
$582$	0	0
$583$	52.0369	2.15515
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.8515	−1.02573	−0.512865	−	0.858469i	$-0.671416\pi$
$587$	−24.8515	−1.02573	−0.512865	+	0.858469i	$0.671416\pi$
$588$	0	0
$589$	−1.01283	−0.0417328
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.5552	0.638776	0.319388	−	0.947624i	$-0.396523\pi$
$593$	15.5552	0.638776	0.319388	+	0.947624i	$0.396523\pi$
$594$	0	0
$595$	8.40332	0.344503
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.193755	−0.00791662	−0.00395831	−	0.999992i	$-0.501260\pi$
$599$	−0.193755	−0.00791662	−0.00395831	+	0.999992i	$0.501260\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.56804	−0.145061
$606$	0	0
$607$	16.9608	0.688418	0.344209	−	0.938893i	$-0.388147\pi$
$607$	16.9608	0.688418	0.344209	+	0.938893i	$0.388147\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.85601	−0.115542
$612$	0	0
$613$	−26.8066	−1.08271	−0.541355	−	0.840794i	$-0.682089\pi$
$613$	−26.8066	−1.08271	−0.541355	+	0.840794i	$0.682089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.287973	0.0115934	0.00579668	−	0.999983i	$-0.498155\pi$
$617$	0.287973	0.0115934	0.00579668	+	0.999983i	$0.498155\pi$
$618$	0	0
$619$	−12.0106	−0.482745	−0.241373	−	0.970432i	$-0.577598\pi$
$619$	−12.0106	−0.482745	−0.241373	+	0.970432i	$0.577598\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.1361	0.686543
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−47.6706	−1.90075
$630$	0	0
$631$	20.3928	0.811823	0.405911	−	0.913912i	$-0.366954\pi$
$631$	20.3928	0.811823	0.405911	+	0.913912i	$0.366954\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.96080	0.355598
$636$	0	0
$637$	−5.69129	−0.225497
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.2672	−0.840006	−0.420003	−	0.907523i	$-0.637971\pi$
$641$	−21.2672	−0.840006	−0.420003	+	0.907523i	$0.637971\pi$
$642$	0	0
$643$	23.6627	0.933164	0.466582	−	0.884478i	$-0.345485\pi$
$643$	23.6627	0.933164	0.466582	+	0.884478i	$0.345485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.8458	−1.56650	−0.783251	−	0.621706i	$-0.786440\pi$
$647$	−39.8458	−1.56650	−0.783251	+	0.621706i	$0.786440\pi$
$648$	0	0
$649$	19.7041	0.773454
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.7120	0.458327	0.229164	−	0.973388i	$-0.426401\pi$
$653$	11.7120	0.458327	0.229164	+	0.973388i	$0.426401\pi$
$654$	0	0
$655$	8.97927	0.350849
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.1730	0.824784	0.412392	−	0.911007i	$-0.364693\pi$
$659$	21.1730	0.824784	0.412392	+	0.911007i	$0.364693\pi$
$660$	0	0
$661$	2.57595	0.100193	0.0500963	−	0.998744i	$-0.484047\pi$
$661$	2.57595	0.100193	0.0500963	+	0.998744i	$0.484047\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.32492	0.245270
$666$	0	0
$667$	19.0577	0.737916
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.6336	0.757948
$672$	0	0
$673$	−16.7282	−0.644826	−0.322413	−	0.946599i	$-0.604494\pi$
$673$	−16.7282	−0.644826	−0.322413	+	0.946599i	$0.604494\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.9423	0.420548	0.210274	−	0.977643i	$-0.432564\pi$
$677$	10.9423	0.420548	0.210274	+	0.977643i	$0.432564\pi$
$678$	0	0
$679$	8.82245	0.338575
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.89296	0.263752	0.131876	−	0.991266i	$-0.457900\pi$
$683$	6.89296	0.263752	0.131876	+	0.991266i	$0.457900\pi$
$684$	0	0
$685$	−14.4033	−0.550323
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.6336	0.519400
$690$	0	0
$691$	−43.4504	−1.65293	−0.826466	−	0.562986i	$-0.809652\pi$
$691$	−43.4504	−1.65293	−0.826466	+	0.562986i	$0.809652\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.3456	0.809687
$696$	0	0
$697$	−53.9585	−2.04383
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.1153	−1.21298	−0.606490	−	0.795091i	$-0.707423\pi$
$701$	−32.1153	−1.21298	−0.606490	+	0.795091i	$0.707423\pi$
$702$	0	0
$703$	−35.8801	−1.35324
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.86392	−0.258144
$708$	0	0
$709$	1.23030	0.0462048	0.0231024	−	0.999733i	$-0.492646\pi$
$709$	1.23030	0.0462048	0.0231024	+	0.999733i	$0.492646\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.22239	−0.0457790
$714$	0	0
$715$	−3.81681	−0.142741
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.82738	0.291912	0.145956	−	0.989291i	$-0.453374\pi$
$719$	7.82738	0.291912	0.145956	+	0.989291i	$0.453374\pi$
$720$	0	0
$721$	5.01621	0.186813
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.85601	−0.106070
$726$	0	0
$727$	27.2857	1.01197	0.505986	−	0.862542i	$-0.331129\pi$
$727$	27.2857	1.01197	0.505986	+	0.862542i	$0.331129\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.6336	0.726176
$732$	0	0
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.50246	0.350028
$738$	0	0
$739$	−30.5450	−1.12362	−0.561809	−	0.827267i	$-0.689894\pi$
$739$	−30.5450	−1.12362	−0.561809	+	0.827267i	$0.689894\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.4112	−1.33580	−0.667899	−	0.744252i	$-0.732806\pi$
$743$	−36.4112	−1.33580	−0.667899	+	0.744252i	$0.732806\pi$
$744$	0	0
$745$	−14.4033	−0.527697
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.440264	0.0160869
$750$	0	0
$751$	13.1730	0.480690	0.240345	−	0.970687i	$-0.422739\pi$
$751$	13.1730	0.480690	0.240345	+	0.970687i	$0.422739\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.5081	−0.819154
$756$	0	0
$757$	33.8801	1.23139	0.615697	−	0.787983i	$-0.288874\pi$
$757$	33.8801	1.23139	0.615697	+	0.787983i	$0.288874\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33.2672	−1.20594	−0.602968	−	0.797765i	$-0.706015\pi$
$761$	−33.2672	−1.20594	−0.602968	+	0.797765i	$0.706015\pi$
$762$	0	0
$763$	−14.5187	−0.525611
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.16246	0.186406
$768$	0	0
$769$	29.8432	1.07617	0.538086	−	0.842890i	$-0.319147\pi$
$769$	29.8432	1.07617	0.538086	+	0.842890i	$0.319147\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.7776	0.603449	0.301724	−	0.953395i	$-0.402438\pi$
$773$	16.7776	0.603449	0.301724	+	0.953395i	$0.402438\pi$
$774$	0	0
$775$	0.183190	0.00658037
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−40.6129	−1.45511
$780$	0	0
$781$	14.5680	0.521285
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.6913	−0.595738
$786$	0	0
$787$	−20.1647	−0.718795	−0.359397	−	0.933185i	$-0.617018\pi$
$787$	−20.1647	−0.718795	−0.359397	+	0.933185i	$0.617018\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.20997	−0.0430214
$792$	0	0
$793$	5.14399	0.182668
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.30871	−0.117200	−0.0586002	−	0.998282i	$-0.518664\pi$
$797$	−3.30871	−0.117200	−0.0586002	+	0.998282i	$0.518664\pi$
$798$	0	0
$799$	20.9793	0.742193
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.7697	1.72105
$804$	0	0
$805$	7.63362	0.269050
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.6834	−0.797505	−0.398753	−	0.917058i	$-0.630557\pi$
$809$	−22.6834	−0.797505	−0.398753	+	0.917058i	$0.630557\pi$
$810$	0	0
$811$	9.93216	0.348765	0.174383	−	0.984678i	$-0.444207\pi$
$811$	9.93216	0.348765	0.174383	+	0.984678i	$0.444207\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.4689	−0.822080
$816$	0	0
$817$	14.7776	0.517003
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.4033	1.34029	0.670143	−	0.742232i	$-0.266233\pi$
$821$	38.4033	1.34029	0.670143	+	0.742232i	$0.266233\pi$
$822$	0	0
$823$	13.1176	0.457251	0.228626	−	0.973514i	$-0.426577\pi$
$823$	13.1176	0.457251	0.228626	+	0.973514i	$0.426577\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.9714	−0.590152	−0.295076	−	0.955474i	$-0.595345\pi$
$827$	−16.9714	−0.590152	−0.295076	+	0.955474i	$0.595345\pi$
$828$	0	0
$829$	31.8353	1.10569	0.552843	−	0.833286i	$-0.313543\pi$
$829$	31.8353	1.10569	0.552843	+	0.833286i	$0.313543\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	41.8062	1.44850
$834$	0	0
$835$	9.14399	0.316441
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.62306	−0.125082	−0.0625409	−	0.998042i	$-0.519920\pi$
$839$	−3.62306	−0.125082	−0.0625409	+	0.998042i	$0.519920\pi$
$840$	0	0
$841$	−20.8432	−0.718731
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	4.08179	0.140252
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43.3042	−1.48445
$852$	0	0
$853$	21.5104	0.736501	0.368250	−	0.929727i	$-0.379957\pi$
$853$	21.5104	0.736501	0.368250	+	0.929727i	$0.379957\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.6706	0.603615	0.301807	−	0.953369i	$-0.402410\pi$
$857$	17.6706	0.603615	0.301807	+	0.953369i	$0.402410\pi$
$858$	0	0
$859$	−3.63362	−0.123978	−0.0619888	−	0.998077i	$-0.519744\pi$
$859$	−3.63362	−0.123978	−0.0619888	+	0.998077i	$0.519744\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.9506	−0.474885	−0.237442	−	0.971402i	$-0.576309\pi$
$863$	−13.9506	−0.474885	−0.237442	+	0.971402i	$0.576309\pi$
$864$	0	0
$865$	−10.0369	−0.341266
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−35.6706	−1.21004
$870$	0	0
$871$	2.48963	0.0843580
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.14399	−0.0386738
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.3954	0.552376	0.276188	−	0.961104i	$-0.410929\pi$
$881$	16.3954	0.552376	0.276188	+	0.961104i	$0.410929\pi$
$882$	0	0
$883$	−44.6314	−1.50196	−0.750982	−	0.660322i	$-0.770420\pi$
$883$	−44.6314	−1.50196	−0.750982	+	0.660322i	$0.770420\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.4795	−1.59420	−0.797102	−	0.603844i	$-0.793635\pi$
$887$	−47.4795	−1.59420	−0.797102	+	0.603844i	$0.793635\pi$
$888$	0	0
$889$	−10.2510	−0.343809
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.7904	0.528407
$894$	0	0
$895$	20.9793	0.701260
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.523192	−0.0174494
$900$	0	0
$901$	−100.148	−3.33641
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.5473	−0.450328
$906$	0	0
$907$	−22.8824	−0.759797	−0.379899	−	0.925028i	$-0.624041\pi$
$907$	−22.8824	−0.759797	−0.379899	+	0.925028i	$0.624041\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.329437	0.0109147	0.00545737	−	0.999985i	$-0.498263\pi$
$911$	0.329437	0.0109147	0.00545737	+	0.999985i	$0.498263\pi$
$912$	0	0
$913$	45.1730	1.49501
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.2722	−0.339217
$918$	0	0
$919$	−36.0369	−1.18875	−0.594375	−	0.804188i	$-0.702600\pi$
$919$	−36.0369	−1.18875	−0.594375	+	0.804188i	$0.702600\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.81681	0.125632
$924$	0	0
$925$	6.48963	0.213378
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.7855	−0.878804	−0.439402	−	0.898290i	$-0.644810\pi$
$929$	−26.7855	−0.878804	−0.439402	+	0.898290i	$0.644810\pi$
$930$	0	0
$931$	31.4662	1.03126
$932$	0	0
$933$	0	0
$934$	0	0
$935$	28.0369	0.916906
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.9631	0.455183	0.227591	−	0.973757i	$-0.426915\pi$
$941$	13.9631	0.455183	0.227591	+	0.973757i	$0.426915\pi$
$942$	0	0
$943$	−49.0162	−1.59619
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.1233	−0.588927	−0.294463	−	0.955663i	$-0.595141\pi$
$947$	−18.1233	−0.588927	−0.294463	+	0.955663i	$0.595141\pi$
$948$	0	0
$949$	12.7776	0.414779
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.97927	−0.0965080	−0.0482540	−	0.998835i	$-0.515366\pi$
$953$	−2.97927	−0.0965080	−0.0482540	+	0.998835i	$0.515366\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.4772	0.532077
$960$	0	0
$961$	−30.9664	−0.998917
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.71203	0.119494
$966$	0	0
$967$	−6.48963	−0.208693	−0.104346	−	0.994541i	$-0.533275\pi$
$967$	−6.48963	−0.208693	−0.104346	+	0.994541i	$0.533275\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.9793	1.05836	0.529178	−	0.848511i	$-0.322501\pi$
$971$	32.9793	1.05836	0.529178	+	0.848511i	$0.322501\pi$
$972$	0	0
$973$	−24.4191	−0.782841
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.7983	1.01732	0.508659	−	0.860968i	$-0.330141\pi$
$977$	31.7983	1.01732	0.508659	+	0.860968i	$0.330141\pi$
$978$	0	0
$979$	57.1730	1.82726
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.0656	0.735678	0.367839	−	0.929890i	$-0.380098\pi$
$983$	23.0656	0.735678	0.367839	+	0.929890i	$0.380098\pi$
$984$	0	0
$985$	8.69129	0.276928
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.8353	0.567129
$990$	0	0
$991$	−52.0739	−1.65418	−0.827091	−	0.562068i	$-0.810006\pi$
$991$	−52.0739	−1.65418	−0.827091	+	0.562068i	$0.810006\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.9793	−0.791896
$996$	0	0
$997$	−6.78551	−0.214899	−0.107450	−	0.994211i	$-0.534268\pi$
$997$	−6.78551	−0.214899	−0.107450	+	0.994211i	$0.534268\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.da.1.2		3
3.2	odd	2		1040.2.a.o.1.3		3
4.3	odd	2		2340.2.a.n.1.2		3
12.11	even	2		260.2.a.b.1.1	✓	3
15.14	odd	2		5200.2.a.ci.1.1		3
24.5	odd	2		4160.2.a.br.1.1		3
24.11	even	2		4160.2.a.bo.1.3		3
60.23	odd	4		1300.2.c.f.1249.2		6
60.47	odd	4		1300.2.c.f.1249.5		6
60.59	even	2		1300.2.a.i.1.3		3
156.47	odd	4		3380.2.f.h.3041.2		6
156.83	odd	4		3380.2.f.h.3041.1		6
156.155	even	2		3380.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.1	✓	3	12.11	even	2
1040.2.a.o.1.3		3	3.2	odd	2
1300.2.a.i.1.3		3	60.59	even	2
1300.2.c.f.1249.2		6	60.23	odd	4
1300.2.c.f.1249.5		6	60.47	odd	4
2340.2.a.n.1.2		3	4.3	odd	2
3380.2.a.o.1.1		3	156.155	even	2
3380.2.f.h.3041.1		6	156.83	odd	4
3380.2.f.h.3041.2		6	156.47	odd	4
4160.2.a.bo.1.3		3	24.11	even	2
4160.2.a.br.1.1		3	24.5	odd	2
5200.2.a.ci.1.1		3	15.14	odd	2
9360.2.a.da.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +1.14399 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +1.14399 q^{7} +3.81681 q^{11} +1.00000 q^{13} -7.34565 q^{17} -5.52884 q^{19} -6.67282 q^{23} +1.00000 q^{25} -2.85601 q^{29} +0.183190 q^{31} -1.14399 q^{35} +6.48963 q^{37} +7.34565 q^{41} -2.67282 q^{43} -2.85601 q^{47} -5.69129 q^{49} +13.6336 q^{53} -3.81681 q^{55} +5.16246 q^{59} +5.14399 q^{61} -1.00000 q^{65} +2.48963 q^{67} +3.81681 q^{71} +12.7776 q^{73} +4.36638 q^{77} -9.34565 q^{79} +11.8353 q^{83} +7.34565 q^{85} +14.9793 q^{89} +1.14399 q^{91} +5.52884 q^{95} +7.71203 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7	\( 3 q - 3 q^{5} + 2 q^{7} + 3 q^{13} - 2 q^{17} - 8 q^{19} - 10 q^{23} + 3 q^{25} - 10 q^{29} + 12 q^{31} - 2 q^{35} - 2 q^{37} + 2 q^{41} + 2 q^{43} - 10 q^{47} + 23 q^{49} + 18 q^{53} - 16 q^{59} + 14 q^{61} - 3 q^{65} - 14 q^{67} + 14 q^{73} + 36 q^{77} - 8 q^{79} - 6 q^{83} + 2 q^{85} + 2 q^{89} + 2 q^{91} + 8 q^{95} + 26 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 + 3 * q^13 - 2 * q^17 - 8 * q^19 - 10 * q^23 + 3 * q^25 - 10 * q^29 + 12 * q^31 - 2 * q^35 - 2 * q^37 + 2 * q^41 + 2 * q^43 - 10 * q^47 + 23 * q^49 + 18 * q^53 - 16 * q^59 + 14 * q^61 - 3 * q^65 - 14 * q^67 + 14 * q^73 + 36 * q^77 - 8 * q^79 - 6 * q^83 + 2 * q^85 + 2 * q^89 + 2 * q^91 + 8 * q^95 + 26 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more