LMFDB - Embedded newform 9360.2.a.cn.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1170)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -4.60555 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{13} -2.60555 q^{17} +0.605551 q^{19} +2.60555 q^{23} +1.00000 q^{25} +2.60555 q^{29} -2.00000 q^{31} -4.60555 q^{35} -3.21110 q^{37} -11.2111 q^{41} +9.21110 q^{43} -5.21110 q^{47} +14.2111 q^{49} -5.21110 q^{59} +7.21110 q^{61} +1.00000 q^{65} -7.21110 q^{67} +12.0000 q^{71} -0.605551 q^{73} -13.2111 q^{79} -2.60555 q^{85} -11.2111 q^{89} -4.60555 q^{91} +0.605551 q^{95} -0.605551 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} + 2 q^{13} + 2 q^{17} - 6 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{29} - 4 q^{31} - 2 q^{35} + 8 q^{37} - 8 q^{41} + 4 q^{43} + 4 q^{47} + 14 q^{49} + 4 q^{59} + 2 q^{65} + 24 q^{71} + 6 q^{73} - 12 q^{79} + 2 q^{85} - 8 q^{89} - 2 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 + 2 * q^13 + 2 * q^17 - 6 * q^19 - 2 * q^23 + 2 * q^25 - 2 * q^29 - 4 * q^31 - 2 * q^35 + 8 * q^37 - 8 * q^41 + 4 * q^43 + 4 * q^47 + 14 * q^49 + 4 * q^59 + 2 * q^65 + 24 * q^71 + 6 * q^73 - 12 * q^79 + 2 * q^85 - 8 * q^89 - 2 * q^91 - 6 * q^95 + 6 * 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
$7$	−4.60555	−1.74073	−0.870367	−	0.492403i	$-0.836119\pi$
$7$	−4.60555	−1.74073	−0.870367	+	0.492403i	$0.836119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.60555	−0.631939	−0.315970	−	0.948769i	$-0.602330\pi$
$17$	−2.60555	−0.631939	−0.315970	+	0.948769i	$0.602330\pi$
$18$	0	0
$19$	0.605551	0.138923	0.0694615	−	0.997585i	$-0.477872\pi$
$19$	0.605551	0.138923	0.0694615	+	0.997585i	$0.477872\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.60555	0.543295	0.271647	−	0.962397i	$-0.412432\pi$
$23$	2.60555	0.543295	0.271647	+	0.962397i	$0.412432\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.60555	0.483839	0.241919	−	0.970296i	$-0.422223\pi$
$29$	2.60555	0.483839	0.241919	+	0.970296i	$0.422223\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.60555	−0.778480
$36$	0	0
$37$	−3.21110	−0.527902	−0.263951	−	0.964536i	$-0.585026\pi$
$37$	−3.21110	−0.527902	−0.263951	+	0.964536i	$0.585026\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.2111	−1.75088	−0.875440	−	0.483327i	$-0.839428\pi$
$41$	−11.2111	−1.75088	−0.875440	+	0.483327i	$0.839428\pi$
$42$	0	0
$43$	9.21110	1.40468	0.702340	−	0.711842i	$-0.252139\pi$
$43$	9.21110	1.40468	0.702340	+	0.711842i	$0.252139\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.21110	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−7.21110	−0.880976	−0.440488	−	0.897758i	$-0.645195\pi$
$67$	−7.21110	−0.880976	−0.440488	+	0.897758i	$0.645195\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−0.605551	−0.0708744	−0.0354372	−	0.999372i	$-0.511282\pi$
$73$	−0.605551	−0.0708744	−0.0354372	+	0.999372i	$0.511282\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.2111	−1.48637	−0.743183	−	0.669089i	$-0.766685\pi$
$79$	−13.2111	−1.48637	−0.743183	+	0.669089i	$0.766685\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2.60555	−0.282612
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.2111	−1.18837	−0.594187	−	0.804327i	$-0.702526\pi$
$89$	−11.2111	−1.18837	−0.594187	+	0.804327i	$0.702526\pi$
$90$	0	0
$91$	−4.60555	−0.482793
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.605551	0.0621282
$96$	0	0
$97$	−0.605551	−0.0614844	−0.0307422	−	0.999527i	$-0.509787\pi$
$97$	−0.605551	−0.0614844	−0.0307422	+	0.999527i	$0.509787\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.81665	0.777786	0.388893	−	0.921283i	$-0.372858\pi$
$101$	7.81665	0.777786	0.388893	+	0.921283i	$0.372858\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	11.3944	1.09139	0.545695	−	0.837984i	$-0.316266\pi$
$109$	11.3944	1.09139	0.545695	+	0.837984i	$0.316266\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.6056	1.37397	0.686987	−	0.726669i	$-0.258933\pi$
$113$	14.6056	1.37397	0.686987	+	0.726669i	$0.258933\pi$
$114$	0	0
$115$	2.60555	0.242969
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.2111	−1.17230	−0.586148	−	0.810204i	$-0.699356\pi$
$127$	−13.2111	−1.17230	−0.586148	+	0.810204i	$0.699356\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.39445	0.820797	0.410398	−	0.911906i	$-0.365390\pi$
$131$	9.39445	0.820797	0.410398	+	0.911906i	$0.365390\pi$
$132$	0	0
$133$	−2.78890	−0.241828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.6333	−1.84826	−0.924129	−	0.382080i	$-0.875208\pi$
$137$	−21.6333	−1.84826	−0.924129	+	0.382080i	$0.875208\pi$
$138$	0	0
$139$	14.4222	1.22328	0.611638	−	0.791138i	$-0.290511\pi$
$139$	14.4222	1.22328	0.611638	+	0.791138i	$0.290511\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.60555	0.216379
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	20.4222	1.66194	0.830968	−	0.556321i	$-0.187787\pi$
$151$	20.4222	1.66194	0.830968	+	0.556321i	$0.187787\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−3.21110	−0.256274	−0.128137	−	0.991756i	$-0.540900\pi$
$157$	−3.21110	−0.256274	−0.128137	+	0.991756i	$0.540900\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.4222	−0.806494	−0.403247	−	0.915091i	$-0.632119\pi$
$167$	−10.4222	−0.806494	−0.403247	+	0.915091i	$0.632119\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	−4.60555	−0.348147
$176$	0	0
$177$	0	0
$178$	0	0
$179$	25.0278	1.87066	0.935331	−	0.353773i	$-0.115102\pi$
$179$	25.0278	1.87066	0.935331	+	0.353773i	$0.115102\pi$
$180$	0	0
$181$	−4.78890	−0.355956	−0.177978	−	0.984034i	$-0.556956\pi$
$181$	−4.78890	−0.355956	−0.177978	+	0.984034i	$0.556956\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.21110	−0.236085
$186$	0	0
$187$	0	0
$188$	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
$193$	−7.39445	−0.532264	−0.266132	−	0.963937i	$-0.585746\pi$
$193$	−7.39445	−0.532264	−0.266132	+	0.963937i	$0.585746\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.42221	0.315069	0.157535	−	0.987513i	$-0.449645\pi$
$197$	4.42221	0.315069	0.157535	+	0.987513i	$0.449645\pi$
$198$	0	0
$199$	9.21110	0.652958	0.326479	−	0.945204i	$-0.394138\pi$
$199$	9.21110	0.652958	0.326479	+	0.945204i	$0.394138\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−11.2111	−0.783017
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.7889	0.742738	0.371369	−	0.928485i	$-0.378888\pi$
$211$	10.7889	0.742738	0.371369	+	0.928485i	$0.378888\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.21110	0.628192
$216$	0	0
$217$	9.21110	0.625290
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.60555	−0.175268
$222$	0	0
$223$	24.6056	1.64771	0.823855	−	0.566801i	$-0.191819\pi$
$223$	24.6056	1.64771	0.823855	+	0.566801i	$0.191819\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.4222	1.48821	0.744107	−	0.668060i	$-0.232875\pi$
$227$	22.4222	1.48821	0.744107	+	0.668060i	$0.232875\pi$
$228$	0	0
$229$	21.8167	1.44169	0.720843	−	0.693099i	$-0.243755\pi$
$229$	21.8167	1.44169	0.720843	+	0.693099i	$0.243755\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.8167	1.29823	0.649116	−	0.760689i	$-0.275139\pi$
$233$	19.8167	1.29823	0.649116	+	0.760689i	$0.275139\pi$
$234$	0	0
$235$	−5.21110	−0.339935
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.4222	0.674156	0.337078	−	0.941477i	$-0.390561\pi$
$239$	10.4222	0.674156	0.337078	+	0.941477i	$0.390561\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.2111	0.907914
$246$	0	0
$247$	0.605551	0.0385303
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.60555	0.164461	0.0822305	−	0.996613i	$-0.473796\pi$
$251$	2.60555	0.164461	0.0822305	+	0.996613i	$0.473796\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.0278	1.56119	0.780594	−	0.625038i	$-0.214917\pi$
$257$	25.0278	1.56119	0.780594	+	0.625038i	$0.214917\pi$
$258$	0	0
$259$	14.7889	0.918937
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.0278	−1.54328	−0.771639	−	0.636061i	$-0.780563\pi$
$263$	−25.0278	−1.54328	−0.771639	+	0.636061i	$0.780563\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.8167	1.93990	0.969948	−	0.243313i	$-0.0782342\pi$
$269$	31.8167	1.93990	0.969948	+	0.243313i	$0.0782342\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−16.7889	−1.00875	−0.504374	−	0.863486i	$-0.668277\pi$
$277$	−16.7889	−1.00875	−0.504374	+	0.863486i	$0.668277\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.2111	0.668798	0.334399	−	0.942432i	$-0.391467\pi$
$281$	11.2111	0.668798	0.334399	+	0.942432i	$0.391467\pi$
$282$	0	0
$283$	−13.2111	−0.785319	−0.392659	−	0.919684i	$-0.628445\pi$
$283$	−13.2111	−0.785319	−0.392659	+	0.919684i	$0.628445\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	51.6333	3.04782
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	−5.21110	−0.303402
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.60555	0.150683
$300$	0	0
$301$	−42.4222	−2.44518
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.21110	0.412907
$306$	0	0
$307$	−19.2111	−1.09644	−0.548218	−	0.836336i	$-0.684694\pi$
$307$	−19.2111	−1.09644	−0.548218	+	0.836336i	$0.684694\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−3.21110	−0.181502	−0.0907511	−	0.995874i	$-0.528927\pi$
$313$	−3.21110	−0.181502	−0.0907511	+	0.995874i	$0.528927\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.4222	0.922363	0.461181	−	0.887306i	$-0.347426\pi$
$317$	16.4222	0.922363	0.461181	+	0.887306i	$0.347426\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.57779	−0.0877909
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	12.6056	0.692864	0.346432	−	0.938075i	$-0.387393\pi$
$331$	12.6056	0.692864	0.346432	+	0.938075i	$0.387393\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.21110	−0.393985
$336$	0	0
$337$	−30.8444	−1.68020	−0.840101	−	0.542430i	$-0.817504\pi$
$337$	−30.8444	−1.68020	−0.840101	+	0.542430i	$0.817504\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−33.2111	−1.79323
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.2111	−1.56813	−0.784067	−	0.620676i	$-0.786858\pi$
$347$	−29.2111	−1.56813	−0.784067	+	0.620676i	$0.786858\pi$
$348$	0	0
$349$	9.81665	0.525473	0.262737	−	0.964868i	$-0.415375\pi$
$349$	9.81665	0.525473	0.262737	+	0.964868i	$0.415375\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.6333	−1.15142	−0.575712	−	0.817652i	$-0.695275\pi$
$353$	−21.6333	−1.15142	−0.575712	+	0.817652i	$0.695275\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.57779	0.0832728	0.0416364	−	0.999133i	$-0.486743\pi$
$359$	1.57779	0.0832728	0.0416364	+	0.999133i	$0.486743\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.605551	−0.0316960
$366$	0	0
$367$	21.2111	1.10721	0.553605	−	0.832779i	$-0.313252\pi$
$367$	21.2111	1.10721	0.553605	+	0.832779i	$0.313252\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.2111	−0.787601	−0.393801	−	0.919196i	$-0.628840\pi$
$373$	−15.2111	−0.787601	−0.393801	+	0.919196i	$0.628840\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.60555	0.134193
$378$	0	0
$379$	19.3944	0.996226	0.498113	−	0.867112i	$-0.334026\pi$
$379$	19.3944	0.996226	0.498113	+	0.867112i	$0.334026\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.4222	0.532550	0.266275	−	0.963897i	$-0.414207\pi$
$383$	10.4222	0.532550	0.266275	+	0.963897i	$0.414207\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.8167	1.00474	0.502372	−	0.864652i	$-0.332461\pi$
$389$	19.8167	1.00474	0.502372	+	0.864652i	$0.332461\pi$
$390$	0	0
$391$	−6.78890	−0.343329
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.2111	−0.664723
$396$	0	0
$397$	20.7889	1.04336	0.521682	−	0.853140i	$-0.325305\pi$
$397$	20.7889	1.04336	0.521682	+	0.853140i	$0.325305\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.63331	−0.481064	−0.240532	−	0.970641i	$-0.577322\pi$
$401$	−9.63331	−0.481064	−0.240532	+	0.970641i	$0.577322\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	29.6333	1.46527	0.732636	−	0.680620i	$-0.238290\pi$
$409$	29.6333	1.46527	0.732636	+	0.680620i	$0.238290\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	24.0000	1.18096
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−35.4500	−1.73184	−0.865922	−	0.500179i	$-0.833267\pi$
$419$	−35.4500	−1.73184	−0.865922	+	0.500179i	$0.833267\pi$
$420$	0	0
$421$	27.0278	1.31725	0.658626	−	0.752470i	$-0.271138\pi$
$421$	27.0278	1.31725	0.658626	+	0.752470i	$0.271138\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.60555	−0.126388
$426$	0	0
$427$	−33.2111	−1.60720
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	34.8444	1.67452	0.837258	−	0.546808i	$-0.184157\pi$
$433$	34.8444	1.67452	0.837258	+	0.546808i	$0.184157\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.57779	0.0754762
$438$	0	0
$439$	24.8444	1.18576	0.592880	−	0.805291i	$-0.297991\pi$
$439$	24.8444	1.18576	0.592880	+	0.805291i	$0.297991\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.63331	−0.172624	−0.0863118	−	0.996268i	$-0.527508\pi$
$443$	−3.63331	−0.172624	−0.0863118	+	0.996268i	$0.527508\pi$
$444$	0	0
$445$	−11.2111	−0.531457
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.6333	1.58725	0.793627	−	0.608405i	$-0.208190\pi$
$449$	33.6333	1.58725	0.793627	+	0.608405i	$0.208190\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.60555	−0.215912
$456$	0	0
$457$	−24.6056	−1.15100	−0.575500	−	0.817802i	$-0.695192\pi$
$457$	−24.6056	−1.15100	−0.575500	+	0.817802i	$0.695192\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.2111	0.522153	0.261077	−	0.965318i	$-0.415922\pi$
$461$	11.2111	0.522153	0.261077	+	0.965318i	$0.415922\pi$
$462$	0	0
$463$	−33.8167	−1.57159	−0.785797	−	0.618485i	$-0.787747\pi$
$463$	−33.8167	−1.57159	−0.785797	+	0.618485i	$0.787747\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.4222	1.59287	0.796435	−	0.604724i	$-0.206717\pi$
$467$	34.4222	1.59287	0.796435	+	0.604724i	$0.206717\pi$
$468$	0	0
$469$	33.2111	1.53355
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.605551	0.0277846
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.4222	0.476203	0.238101	−	0.971240i	$-0.423475\pi$
$479$	10.4222	0.476203	0.238101	+	0.971240i	$0.423475\pi$
$480$	0	0
$481$	−3.21110	−0.146414
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.605551	−0.0274967
$486$	0	0
$487$	−39.0278	−1.76852	−0.884258	−	0.466998i	$-0.845335\pi$
$487$	−39.0278	−1.76852	−0.884258	+	0.466998i	$0.845335\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.2389	0.823108	0.411554	−	0.911385i	$-0.364986\pi$
$491$	18.2389	0.823108	0.411554	+	0.911385i	$0.364986\pi$
$492$	0	0
$493$	−6.78890	−0.305757
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−55.2666	−2.47905
$498$	0	0
$499$	−6.18335	−0.276805	−0.138402	−	0.990376i	$-0.544197\pi$
$499$	−6.18335	−0.276805	−0.138402	+	0.990376i	$0.544197\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.2389	1.34828	0.674142	−	0.738602i	$-0.264514\pi$
$503$	30.2389	1.34828	0.674142	+	0.738602i	$0.264514\pi$
$504$	0	0
$505$	7.81665	0.347837
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−33.6333	−1.49077	−0.745385	−	0.666634i	$-0.767734\pi$
$509$	−33.6333	−1.49077	−0.745385	+	0.666634i	$0.767734\pi$
$510$	0	0
$511$	2.78890	0.123374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.78890	−0.297427	−0.148713	−	0.988880i	$-0.547513\pi$
$521$	−6.78890	−0.297427	−0.148713	+	0.988880i	$0.547513\pi$
$522$	0	0
$523$	−9.57779	−0.418808	−0.209404	−	0.977829i	$-0.567152\pi$
$523$	−9.57779	−0.418808	−0.209404	+	0.977829i	$0.567152\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.21110	0.226999
$528$	0	0
$529$	−16.2111	−0.704831
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.2111	−0.485607
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	27.0278	1.16201	0.581007	−	0.813899i	$-0.302659\pi$
$541$	27.0278	1.16201	0.581007	+	0.813899i	$0.302659\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.3944	0.488085
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.57779	0.0672163
$552$	0	0
$553$	60.8444	2.58737
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.8444	1.64589	0.822945	−	0.568121i	$-0.192329\pi$
$557$	38.8444	1.64589	0.822945	+	0.568121i	$0.192329\pi$
$558$	0	0
$559$	9.21110	0.389588
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.6333	−1.16461	−0.582303	−	0.812972i	$-0.697848\pi$
$563$	−27.6333	−1.16461	−0.582303	+	0.812972i	$0.697848\pi$
$564$	0	0
$565$	14.6056	0.614460
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.2111	−1.72766	−0.863830	−	0.503784i	$-0.831941\pi$
$569$	−41.2111	−1.72766	−0.863830	+	0.503784i	$0.831941\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.60555	0.108659
$576$	0	0
$577$	6.18335	0.257416	0.128708	−	0.991683i	$-0.458917\pi$
$577$	6.18335	0.257416	0.128708	+	0.991683i	$0.458917\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.4222	−0.925463	−0.462732	−	0.886498i	$-0.653131\pi$
$587$	−22.4222	−0.925463	−0.462732	+	0.886498i	$0.653131\pi$
$588$	0	0
$589$	−1.21110	−0.0499026
$590$	0	0
$591$	0	0
$592$	0	0
$593$	35.2111	1.44595	0.722973	−	0.690876i	$-0.242775\pi$
$593$	35.2111	1.44595	0.722973	+	0.690876i	$0.242775\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.8444	−0.851680	−0.425840	−	0.904799i	$-0.640021\pi$
$599$	−20.8444	−0.851680	−0.425840	+	0.904799i	$0.640021\pi$
$600$	0	0
$601$	−25.6333	−1.04560	−0.522802	−	0.852454i	$-0.675113\pi$
$601$	−25.6333	−1.04560	−0.522802	+	0.852454i	$0.675113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−26.7889	−1.08733	−0.543664	−	0.839303i	$-0.682963\pi$
$607$	−26.7889	−1.08733	−0.543664	+	0.839303i	$0.682963\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.21110	−0.210819
$612$	0	0
$613$	−37.6333	−1.51999	−0.759997	−	0.649926i	$-0.774800\pi$
$613$	−37.6333	−1.51999	−0.759997	+	0.649926i	$0.774800\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.8444	1.56382	0.781909	−	0.623393i	$-0.214246\pi$
$617$	38.8444	1.56382	0.781909	+	0.623393i	$0.214246\pi$
$618$	0	0
$619$	−4.60555	−0.185113	−0.0925564	−	0.995707i	$-0.529504\pi$
$619$	−4.60555	−0.185113	−0.0925564	+	0.995707i	$0.529504\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	51.6333	2.06864
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.36669	0.333602
$630$	0	0
$631$	−8.78890	−0.349880	−0.174940	−	0.984579i	$-0.555973\pi$
$631$	−8.78890	−0.349880	−0.174940	+	0.984579i	$0.555973\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.2111	−0.524267
$636$	0	0
$637$	14.2111	0.563064
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	24.0555	0.948657	0.474328	−	0.880348i	$-0.342691\pi$
$643$	24.0555	0.948657	0.474328	+	0.880348i	$0.342691\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.02776	−0.0404053	−0.0202026	−	0.999796i	$-0.506431\pi$
$647$	−1.02776	−0.0404053	−0.0202026	+	0.999796i	$0.506431\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.2111	−0.673522	−0.336761	−	0.941590i	$-0.609332\pi$
$653$	−17.2111	−0.673522	−0.336761	+	0.941590i	$0.609332\pi$
$654$	0	0
$655$	9.39445	0.367071
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.81665	0.304494	0.152247	−	0.988343i	$-0.451349\pi$
$659$	7.81665	0.304494	0.152247	+	0.988343i	$0.451349\pi$
$660$	0	0
$661$	15.0278	0.584512	0.292256	−	0.956340i	$-0.405594\pi$
$661$	15.0278	0.584512	0.292256	+	0.956340i	$0.405594\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.78890	−0.108149
$666$	0	0
$667$	6.78890	0.262867
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.78890	−0.184598	−0.0922992	−	0.995731i	$-0.529422\pi$
$673$	−4.78890	−0.184598	−0.0922992	+	0.995731i	$0.529422\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.21110	−0.200279	−0.100139	−	0.994973i	$-0.531929\pi$
$677$	−5.21110	−0.200279	−0.100139	+	0.994973i	$0.531929\pi$
$678$	0	0
$679$	2.78890	0.107028
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.4222	−1.31713	−0.658565	−	0.752524i	$-0.728836\pi$
$683$	−34.4222	−1.31713	−0.658565	+	0.752524i	$0.728836\pi$
$684$	0	0
$685$	−21.6333	−0.826566
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.18335	0.0830584	0.0415292	−	0.999137i	$-0.486777\pi$
$691$	2.18335	0.0830584	0.0415292	+	0.999137i	$0.486777\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.4222	0.547065
$696$	0	0
$697$	29.2111	1.10645
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.39445	0.354823	0.177412	−	0.984137i	$-0.443228\pi$
$701$	9.39445	0.354823	0.177412	+	0.984137i	$0.443228\pi$
$702$	0	0
$703$	−1.94449	−0.0733377
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36.0000	−1.35392
$708$	0	0
$709$	−33.4500	−1.25624	−0.628120	−	0.778117i	$-0.716175\pi$
$709$	−33.4500	−1.25624	−0.628120	+	0.778117i	$0.716175\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.21110	−0.195157
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.78890	−0.253183	−0.126592	−	0.991955i	$-0.540404\pi$
$719$	−6.78890	−0.253183	−0.126592	+	0.991955i	$0.540404\pi$
$720$	0	0
$721$	−18.4222	−0.686079
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.60555	0.0967677
$726$	0	0
$727$	0.366692	0.0135999	0.00679993	−	0.999977i	$-0.497835\pi$
$727$	0.366692	0.0135999	0.00679993	+	0.999977i	$0.497835\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−1.63331	−0.0603276	−0.0301638	−	0.999545i	$-0.509603\pi$
$733$	−1.63331	−0.0603276	−0.0301638	+	0.999545i	$0.509603\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	19.3944	0.713436	0.356718	−	0.934212i	$-0.383896\pi$
$739$	19.3944	0.713436	0.356718	+	0.934212i	$0.383896\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.0555	−0.955884	−0.477942	−	0.878391i	$-0.658617\pi$
$743$	−26.0555	−0.955884	−0.477942	+	0.878391i	$0.658617\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−55.2666	−2.01940
$750$	0	0
$751$	−51.2666	−1.87075	−0.935373	−	0.353664i	$-0.884936\pi$
$751$	−51.2666	−1.87075	−0.935373	+	0.353664i	$0.884936\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.4222	0.743240
$756$	0	0
$757$	22.8444	0.830294	0.415147	−	0.909754i	$-0.363730\pi$
$757$	22.8444	0.830294	0.415147	+	0.909754i	$0.363730\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.6333	1.21921	0.609603	−	0.792707i	$-0.291329\pi$
$761$	33.6333	1.21921	0.609603	+	0.792707i	$0.291329\pi$
$762$	0	0
$763$	−52.4777	−1.89982
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.21110	−0.188162
$768$	0	0
$769$	22.8444	0.823791	0.411895	−	0.911231i	$-0.364867\pi$
$769$	22.8444	0.823791	0.411895	+	0.911231i	$0.364867\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.4222	0.590666	0.295333	−	0.955394i	$-0.404569\pi$
$773$	16.4222	0.590666	0.295333	+	0.955394i	$0.404569\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.78890	−0.243237
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.21110	−0.114609
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−67.2666	−2.39173
$792$	0	0
$793$	7.21110	0.256074
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.5778	0.480950	0.240475	−	0.970655i	$-0.422697\pi$
$797$	13.5778	0.480950	0.240475	+	0.970655i	$0.422697\pi$
$798$	0	0
$799$	13.5778	0.480348
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.4222	−0.788323	−0.394161	−	0.919041i	$-0.628965\pi$
$809$	−22.4222	−0.788323	−0.394161	+	0.919041i	$0.628965\pi$
$810$	0	0
$811$	−12.9722	−0.455517	−0.227759	−	0.973718i	$-0.573140\pi$
$811$	−12.9722	−0.455517	−0.227759	+	0.973718i	$0.573140\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	5.57779	0.195142
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−37.0278	−1.28294
$834$	0	0
$835$	−10.4222	−0.360675
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.5778	0.468758	0.234379	−	0.972145i	$-0.424694\pi$
$839$	13.5778	0.468758	0.234379	+	0.972145i	$0.424694\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	50.6611	1.74073
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.36669	−0.286807
$852$	0	0
$853$	−48.0555	−1.64539	−0.822695	−	0.568483i	$-0.807530\pi$
$853$	−48.0555	−1.64539	−0.822695	+	0.568483i	$0.807530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.4500	1.21095	0.605474	−	0.795865i	$-0.292984\pi$
$857$	35.4500	1.21095	0.605474	+	0.795865i	$0.292984\pi$
$858$	0	0
$859$	−47.6333	−1.62523	−0.812614	−	0.582803i	$-0.801956\pi$
$859$	−47.6333	−1.62523	−0.812614	+	0.582803i	$0.801956\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.5778	0.462194	0.231097	−	0.972931i	$-0.425769\pi$
$863$	13.5778	0.462194	0.231097	+	0.972931i	$0.425769\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−7.21110	−0.244339
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.60555	−0.155696
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.2111	1.38844	0.694219	−	0.719764i	$-0.255750\pi$
$881$	41.2111	1.38844	0.694219	+	0.719764i	$0.255750\pi$
$882$	0	0
$883$	14.4222	0.485346	0.242673	−	0.970108i	$-0.421976\pi$
$883$	14.4222	0.485346	0.242673	+	0.970108i	$0.421976\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.60555	0.0874858	0.0437429	−	0.999043i	$-0.486072\pi$
$887$	2.60555	0.0874858	0.0437429	+	0.999043i	$0.486072\pi$
$888$	0	0
$889$	60.8444	2.04066
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.15559	−0.105598
$894$	0	0
$895$	25.0278	0.836586
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.21110	−0.173800
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.78890	−0.159188
$906$	0	0
$907$	55.6333	1.84727	0.923637	−	0.383269i	$-0.125202\pi$
$907$	55.6333	1.84727	0.923637	+	0.383269i	$0.125202\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	27.6333	0.915532	0.457766	−	0.889073i	$-0.348650\pi$
$911$	27.6333	0.915532	0.457766	+	0.889073i	$0.348650\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43.2666	−1.42879
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	−3.21110	−0.105580
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.0555	1.05171	0.525854	−	0.850575i	$-0.323746\pi$
$929$	32.0555	1.05171	0.525854	+	0.850575i	$0.323746\pi$
$930$	0	0
$931$	8.60555	0.282036
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.6333	0.968078	0.484039	−	0.875046i	$-0.339169\pi$
$937$	29.6333	0.968078	0.484039	+	0.875046i	$0.339169\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.6333	−0.705226	−0.352613	−	0.935769i	$-0.614707\pi$
$941$	−21.6333	−0.705226	−0.352613	+	0.935769i	$0.614707\pi$
$942$	0	0
$943$	−29.2111	−0.951244
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.5778	−0.441219	−0.220610	−	0.975362i	$-0.570805\pi$
$947$	−13.5778	−0.441219	−0.220610	+	0.975362i	$0.570805\pi$
$948$	0	0
$949$	−0.605551	−0.0196570
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.39445	0.304316	0.152158	−	0.988356i	$-0.451378\pi$
$953$	9.39445	0.304316	0.152158	+	0.988356i	$0.451378\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	99.6333	3.21733
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.39445	−0.238036
$966$	0	0
$967$	19.3944	0.623683	0.311842	−	0.950134i	$-0.399054\pi$
$967$	19.3944	0.623683	0.311842	+	0.950134i	$0.399054\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.6056	0.853813	0.426906	−	0.904296i	$-0.359603\pi$
$971$	26.6056	0.853813	0.426906	+	0.904296i	$0.359603\pi$
$972$	0	0
$973$	−66.4222	−2.12940
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.3667	0.459631	0.229816	−	0.973234i	$-0.426188\pi$
$977$	14.3667	0.459631	0.229816	+	0.973234i	$0.426188\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.6333	0.498625	0.249313	−	0.968423i	$-0.419795\pi$
$983$	15.6333	0.498625	0.249313	+	0.968423i	$0.419795\pi$
$984$	0	0
$985$	4.42221	0.140903
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−11.6333	−0.369544	−0.184772	−	0.982781i	$-0.559155\pi$
$991$	−11.6333	−0.369544	−0.184772	+	0.982781i	$0.559155\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.21110	0.292012
$996$	0	0
$997$	19.2111	0.608422	0.304211	−	0.952605i	$-0.401607\pi$
$997$	19.2111	0.608422	0.304211	+	0.952605i	$0.401607\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.cn.1.1		2
3.2	odd	2		9360.2.a.cf.1.1		2
4.3	odd	2		1170.2.a.q.1.2	yes	2
12.11	even	2		1170.2.a.p.1.2	✓	2
20.3	even	4		5850.2.e.bj.5149.1		4
20.7	even	4		5850.2.e.bj.5149.4		4
20.19	odd	2		5850.2.a.ce.1.1		2
60.23	odd	4		5850.2.e.bl.5149.3		4
60.47	odd	4		5850.2.e.bl.5149.2		4
60.59	even	2		5850.2.a.ck.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.a.p.1.2	✓	2	12.11	even	2
1170.2.a.q.1.2	yes	2	4.3	odd	2
5850.2.a.ce.1.1		2	20.19	odd	2
5850.2.a.ck.1.1		2	60.59	even	2
5850.2.e.bj.5149.1		4	20.3	even	4
5850.2.e.bj.5149.4		4	20.7	even	4
5850.2.e.bl.5149.2		4	60.47	odd	4
5850.2.e.bl.5149.3		4	60.23	odd	4
9360.2.a.cf.1.1		2	3.2	odd	2
9360.2.a.cn.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.60555 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{13} -2.60555 q^{17} +0.605551 q^{19} +2.60555 q^{23} +1.00000 q^{25} +2.60555 q^{29} -2.00000 q^{31} -4.60555 q^{35} -3.21110 q^{37} -11.2111 q^{41} +9.21110 q^{43} -5.21110 q^{47} +14.2111 q^{49} -5.21110 q^{59} +7.21110 q^{61} +1.00000 q^{65} -7.21110 q^{67} +12.0000 q^{71} -0.605551 q^{73} -13.2111 q^{79} -2.60555 q^{85} -11.2111 q^{89} -4.60555 q^{91} +0.605551 q^{95} -0.605551 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} + 2 q^{13} + 2 q^{17} - 6 q^{19} - 2 q^{23} + 2 q^{25} - 2 q^{29} - 4 q^{31} - 2 q^{35} + 8 q^{37} - 8 q^{41} + 4 q^{43} + 4 q^{47} + 14 q^{49} + 4 q^{59} + 2 q^{65} + 24 q^{71} + 6 q^{73} - 12 q^{79} + 2 q^{85} - 8 q^{89} - 2 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 + 2 * q^13 + 2 * q^17 - 6 * q^19 - 2 * q^23 + 2 * q^25 - 2 * q^29 - 4 * q^31 - 2 * q^35 + 8 * q^37 - 8 * q^41 + 4 * q^43 + 4 * q^47 + 14 * q^49 + 4 * q^59 + 2 * q^65 + 24 * q^71 + 6 * q^73 - 12 * q^79 + 2 * q^85 - 8 * q^89 - 2 * q^91 - 6 * q^95 + 6 * q^97

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