LMFDB - Embedded newform 4680.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.70156 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.70156 q^{7} +5.70156 q^{11} -1.00000 q^{13} -5.70156 q^{17} -2.00000 q^{19} -0.298438 q^{23} +1.00000 q^{25} +3.40312 q^{29} +3.40312 q^{31} -3.70156 q^{35} -0.298438 q^{37} -4.29844 q^{41} +4.00000 q^{43} -11.4031 q^{47} +6.70156 q^{49} -4.29844 q^{53} +5.70156 q^{55} -10.8062 q^{59} -3.70156 q^{61} -1.00000 q^{65} -4.00000 q^{67} +16.5078 q^{71} +0.596876 q^{73} -21.1047 q^{77} -1.70156 q^{79} -4.00000 q^{83} -5.70156 q^{85} +11.1047 q^{89} +3.70156 q^{91} -2.00000 q^{95} -1.10469 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - q^7	$ 2 q + 2 q^{5} - q^{7} + 5 q^{11} - 2 q^{13} - 5 q^{17} - 4 q^{19} - 7 q^{23} + 2 q^{25} - 6 q^{29} - 6 q^{31} - q^{35} - 7 q^{37} - 15 q^{41} + 8 q^{43} - 10 q^{47} + 7 q^{49} - 15 q^{53} + 5 q^{55} + 4 q^{59} - q^{61} - 2 q^{65} - 8 q^{67} + q^{71} + 14 q^{73} - 23 q^{77} + 3 q^{79} - 8 q^{83} - 5 q^{85} + 3 q^{89} + q^{91} - 4 q^{95} + 17 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 + 5 * q^11 - 2 * q^13 - 5 * q^17 - 4 * q^19 - 7 * q^23 + 2 * q^25 - 6 * q^29 - 6 * q^31 - q^35 - 7 * q^37 - 15 * q^41 + 8 * q^43 - 10 * q^47 + 7 * q^49 - 15 * q^53 + 5 * q^55 + 4 * q^59 - q^61 - 2 * q^65 - 8 * q^67 + q^71 + 14 * q^73 - 23 * q^77 + 3 * q^79 - 8 * q^83 - 5 * q^85 + 3 * q^89 + q^91 - 4 * q^95 + 17 * 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
$7$	−3.70156	−1.39906	−0.699529	−	0.714604i	$-0.746607\pi$
$7$	−3.70156	−1.39906	−0.699529	+	0.714604i	$0.746607\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.70156	1.71909	0.859543	−	0.511064i	$-0.170748\pi$
$11$	5.70156	1.71909	0.859543	+	0.511064i	$0.170748\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.70156	−1.38283	−0.691416	−	0.722457i	$-0.743013\pi$
$17$	−5.70156	−1.38283	−0.691416	+	0.722457i	$0.743013\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.298438	−0.0622286	−0.0311143	−	0.999516i	$-0.509906\pi$
$23$	−0.298438	−0.0622286	−0.0311143	+	0.999516i	$0.509906\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.40312	0.631944	0.315972	−	0.948768i	$-0.397669\pi$
$29$	3.40312	0.631944	0.315972	+	0.948768i	$0.397669\pi$
$30$	0	0
$31$	3.40312	0.611219	0.305610	−	0.952157i	$-0.401140\pi$
$31$	3.40312	0.611219	0.305610	+	0.952157i	$0.401140\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.70156	−0.625678
$36$	0	0
$37$	−0.298438	−0.0490629	−0.0245314	−	0.999699i	$-0.507809\pi$
$37$	−0.298438	−0.0490629	−0.0245314	+	0.999699i	$0.507809\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.29844	−0.671303	−0.335652	−	0.941986i	$-0.608956\pi$
$41$	−4.29844	−0.671303	−0.335652	+	0.941986i	$0.608956\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.4031	−1.66332	−0.831658	−	0.555288i	$-0.812608\pi$
$47$	−11.4031	−1.66332	−0.831658	+	0.555288i	$0.812608\pi$
$48$	0	0
$49$	6.70156	0.957366
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.29844	−0.590436	−0.295218	−	0.955430i	$-0.595392\pi$
$53$	−4.29844	−0.590436	−0.295218	+	0.955430i	$0.595392\pi$
$54$	0	0
$55$	5.70156	0.768798
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.8062	−1.40685	−0.703427	−	0.710768i	$-0.748348\pi$
$59$	−10.8062	−1.40685	−0.703427	+	0.710768i	$0.748348\pi$
$60$	0	0
$61$	−3.70156	−0.473936	−0.236968	−	0.971517i	$-0.576154\pi$
$61$	−3.70156	−0.473936	−0.236968	+	0.971517i	$0.576154\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.5078	1.95912	0.979558	−	0.201160i	$-0.0644712\pi$
$71$	16.5078	1.95912	0.979558	+	0.201160i	$0.0644712\pi$
$72$	0	0
$73$	0.596876	0.0698590	0.0349295	−	0.999390i	$-0.488879\pi$
$73$	0.596876	0.0698590	0.0349295	+	0.999390i	$0.488879\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−21.1047	−2.40510
$78$	0	0
$79$	−1.70156	−0.191441	−0.0957203	−	0.995408i	$-0.530515\pi$
$79$	−1.70156	−0.191441	−0.0957203	+	0.995408i	$0.530515\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−5.70156	−0.618421
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.1047	1.17709	0.588547	−	0.808463i	$-0.299700\pi$
$89$	11.1047	1.17709	0.588547	+	0.808463i	$0.299700\pi$
$90$	0	0
$91$	3.70156	0.388029
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−1.10469	−0.112164	−0.0560820	−	0.998426i	$-0.517861\pi$
$97$	−1.10469	−0.112164	−0.0560820	+	0.998426i	$0.517861\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.8062	−1.87129	−0.935646	−	0.352940i	$-0.885182\pi$
$101$	−18.8062	−1.87129	−0.935646	+	0.352940i	$0.885182\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.10469	0.106794	0.0533970	−	0.998573i	$-0.482995\pi$
$107$	1.10469	0.106794	0.0533970	+	0.998573i	$0.482995\pi$
$108$	0	0
$109$	−14.8062	−1.41818	−0.709091	−	0.705117i	$-0.750894\pi$
$109$	−14.8062	−1.41818	−0.709091	+	0.705117i	$0.750894\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.4031	−1.82529	−0.912646	−	0.408750i	$-0.865965\pi$
$113$	−19.4031	−1.82529	−0.912646	+	0.408750i	$0.865965\pi$
$114$	0	0
$115$	−0.298438	−0.0278295
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.1047	1.93466
$120$	0	0
$121$	21.5078	1.95526
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.596876	−0.0529642	−0.0264821	−	0.999649i	$-0.508430\pi$
$127$	−0.596876	−0.0529642	−0.0264821	+	0.999649i	$0.508430\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.8062	−1.11889	−0.559444	−	0.828868i	$-0.688985\pi$
$131$	−12.8062	−1.11889	−0.559444	+	0.828868i	$0.688985\pi$
$132$	0	0
$133$	7.40312	0.641932
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	1.70156	0.144325	0.0721623	−	0.997393i	$-0.477010\pi$
$139$	1.70156	0.144325	0.0721623	+	0.997393i	$0.477010\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.70156	−0.476789
$144$	0	0
$145$	3.40312	0.282614
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.70156	−0.630937	−0.315468	−	0.948936i	$-0.602162\pi$
$149$	−7.70156	−0.630937	−0.315468	+	0.948936i	$0.602162\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.40312	0.273346
$156$	0	0
$157$	−16.8062	−1.34128	−0.670642	−	0.741781i	$-0.733981\pi$
$157$	−16.8062	−1.34128	−0.670642	+	0.741781i	$0.733981\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.10469	0.0870615
$162$	0	0
$163$	17.7016	1.38649	0.693247	−	0.720700i	$-0.256180\pi$
$163$	17.7016	1.38649	0.693247	+	0.720700i	$0.256180\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.80625	−0.217154	−0.108577	−	0.994088i	$-0.534629\pi$
$167$	−2.80625	−0.217154	−0.108577	+	0.994088i	$0.534629\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−3.70156	−0.279812
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	0	0
$181$	−7.70156	−0.572453	−0.286226	−	0.958162i	$-0.592401\pi$
$181$	−7.70156	−0.572453	−0.286226	+	0.958162i	$0.592401\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.298438	−0.0219416
$186$	0	0
$187$	−32.5078	−2.37721
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−2.89531	−0.208409	−0.104205	−	0.994556i	$-0.533230\pi$
$193$	−2.89531	−0.208409	−0.104205	+	0.994556i	$0.533230\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.2094	−1.72485	−0.862423	−	0.506188i	$-0.831054\pi$
$197$	−24.2094	−1.72485	−0.862423	+	0.506188i	$0.831054\pi$
$198$	0	0
$199$	−22.8062	−1.61669	−0.808346	−	0.588708i	$-0.799637\pi$
$199$	−22.8062	−1.61669	−0.808346	+	0.588708i	$0.799637\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.5969	−0.884127
$204$	0	0
$205$	−4.29844	−0.300216
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.4031	−0.788771
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−12.5969	−0.855132
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.70156	0.383529
$222$	0	0
$223$	−9.40312	−0.629680	−0.314840	−	0.949145i	$-0.601951\pi$
$223$	−9.40312	−0.629680	−0.314840	+	0.949145i	$0.601951\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.4031	1.55332	0.776660	−	0.629920i	$-0.216912\pi$
$227$	23.4031	1.55332	0.776660	+	0.629920i	$0.216912\pi$
$228$	0	0
$229$	−12.5969	−0.832425	−0.416212	−	0.909267i	$-0.636643\pi$
$229$	−12.5969	−0.832425	−0.416212	+	0.909267i	$0.636643\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.10469	−0.596468	−0.298234	−	0.954493i	$-0.596398\pi$
$233$	−9.10469	−0.596468	−0.298234	+	0.954493i	$0.596398\pi$
$234$	0	0
$235$	−11.4031	−0.743858
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.1047	−0.847672	−0.423836	−	0.905739i	$-0.639317\pi$
$239$	−13.1047	−0.847672	−0.423836	+	0.905739i	$0.639317\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.70156	0.428147
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.2094	−1.27560	−0.637802	−	0.770200i	$-0.720156\pi$
$251$	−20.2094	−1.27560	−0.637802	+	0.770200i	$0.720156\pi$
$252$	0	0
$253$	−1.70156	−0.106976
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.2094	1.13587	0.567935	−	0.823074i	$-0.307743\pi$
$257$	18.2094	1.13587	0.567935	+	0.823074i	$0.307743\pi$
$258$	0	0
$259$	1.10469	0.0686419
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.4031	−1.07312	−0.536561	−	0.843861i	$-0.680277\pi$
$263$	−17.4031	−1.07312	−0.536561	+	0.843861i	$0.680277\pi$
$264$	0	0
$265$	−4.29844	−0.264051
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.8062	−0.658869	−0.329434	−	0.944179i	$-0.606858\pi$
$269$	−10.8062	−0.658869	−0.329434	+	0.944179i	$0.606858\pi$
$270$	0	0
$271$	−19.4031	−1.17866	−0.589328	−	0.807894i	$-0.700607\pi$
$271$	−19.4031	−1.17866	−0.589328	+	0.807894i	$0.700607\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.70156	0.343817
$276$	0	0
$277$	−16.8062	−1.00979	−0.504895	−	0.863181i	$-0.668469\pi$
$277$	−16.8062	−1.00979	−0.504895	+	0.863181i	$0.668469\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.9109	0.939193
$288$	0	0
$289$	15.5078	0.912224
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.59688	−0.151711	−0.0758556	−	0.997119i	$-0.524169\pi$
$293$	−2.59688	−0.151711	−0.0758556	+	0.997119i	$0.524169\pi$
$294$	0	0
$295$	−10.8062	−0.629164
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.298438	0.0172591
$300$	0	0
$301$	−14.8062	−0.853418
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.70156	−0.211951
$306$	0	0
$307$	−33.1047	−1.88938	−0.944692	−	0.327959i	$-0.893639\pi$
$307$	−33.1047	−1.88938	−0.944692	+	0.327959i	$0.893639\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−8.80625	−0.497759	−0.248879	−	0.968535i	$-0.580062\pi$
$313$	−8.80625	−0.497759	−0.248879	+	0.968535i	$0.580062\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.2094	1.35973	0.679867	−	0.733336i	$-0.262038\pi$
$317$	24.2094	1.35973	0.679867	+	0.733336i	$0.262038\pi$
$318$	0	0
$319$	19.4031	1.08637
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.4031	0.634487
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	42.2094	2.32708
$330$	0	0
$331$	24.2094	1.33067	0.665334	−	0.746546i	$-0.268289\pi$
$331$	24.2094	1.33067	0.665334	+	0.746546i	$0.268289\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	32.2094	1.75456	0.877278	−	0.479982i	$-0.159357\pi$
$337$	32.2094	1.75456	0.877278	+	0.479982i	$0.159357\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.4031	1.05074
$342$	0	0
$343$	1.10469	0.0596475
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.1047	0.703496	0.351748	−	0.936095i	$-0.385587\pi$
$347$	13.1047	0.703496	0.351748	+	0.936095i	$0.385587\pi$
$348$	0	0
$349$	−9.19375	−0.492130	−0.246065	−	0.969253i	$-0.579138\pi$
$349$	−9.19375	−0.492130	−0.246065	+	0.969253i	$0.579138\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.8062	0.681608	0.340804	−	0.940134i	$-0.389301\pi$
$353$	12.8062	0.681608	0.340804	+	0.940134i	$0.389301\pi$
$354$	0	0
$355$	16.5078	0.876144
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.6125	−1.56289	−0.781444	−	0.623975i	$-0.785516\pi$
$359$	−29.6125	−1.56289	−0.781444	+	0.623975i	$0.785516\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.596876	0.0312419
$366$	0	0
$367$	26.8062	1.39927	0.699637	−	0.714498i	$-0.253345\pi$
$367$	26.8062	1.39927	0.699637	+	0.714498i	$0.253345\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.9109	0.826055
$372$	0	0
$373$	−21.4031	−1.10821	−0.554106	−	0.832446i	$-0.686940\pi$
$373$	−21.4031	−1.10821	−0.554106	+	0.832446i	$0.686940\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.40312	−0.175270
$378$	0	0
$379$	−35.6125	−1.82929	−0.914646	−	0.404257i	$-0.867530\pi$
$379$	−35.6125	−1.82929	−0.914646	+	0.404257i	$0.867530\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.40312	−0.378282	−0.189141	−	0.981950i	$-0.560570\pi$
$383$	−7.40312	−0.378282	−0.189141	+	0.981950i	$0.560570\pi$
$384$	0	0
$385$	−21.1047	−1.07559
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.6125	1.29861	0.649303	−	0.760530i	$-0.275061\pi$
$389$	25.6125	1.29861	0.649303	+	0.760530i	$0.275061\pi$
$390$	0	0
$391$	1.70156	0.0860517
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.70156	−0.0856149
$396$	0	0
$397$	−16.2984	−0.817995	−0.408998	−	0.912535i	$-0.634122\pi$
$397$	−16.2984	−0.817995	−0.408998	+	0.912535i	$0.634122\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.8062	1.23876	0.619382	−	0.785089i	$-0.287383\pi$
$401$	24.8062	1.23876	0.619382	+	0.785089i	$0.287383\pi$
$402$	0	0
$403$	−3.40312	−0.169522
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.70156	−0.0843433
$408$	0	0
$409$	20.8062	1.02880	0.514401	−	0.857550i	$-0.328014\pi$
$409$	20.8062	1.02880	0.514401	+	0.857550i	$0.328014\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	40.0000	1.96827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.6125	1.15355	0.576773	−	0.816904i	$-0.304312\pi$
$419$	23.6125	1.15355	0.576773	+	0.816904i	$0.304312\pi$
$420$	0	0
$421$	25.6125	1.24828	0.624138	−	0.781314i	$-0.285450\pi$
$421$	25.6125	1.24828	0.624138	+	0.781314i	$0.285450\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.70156	−0.276566
$426$	0	0
$427$	13.7016	0.663065
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	25.4031	1.22080	0.610398	−	0.792095i	$-0.291009\pi$
$433$	25.4031	1.22080	0.610398	+	0.792095i	$0.291009\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.596876	0.0285524
$438$	0	0
$439$	−21.1047	−1.00727	−0.503636	−	0.863916i	$-0.668005\pi$
$439$	−21.1047	−1.00727	−0.503636	+	0.863916i	$0.668005\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.3141	−1.48778	−0.743888	−	0.668304i	$-0.767020\pi$
$443$	−31.3141	−1.48778	−0.743888	+	0.668304i	$0.767020\pi$
$444$	0	0
$445$	11.1047	0.526413
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.1047	−1.09038	−0.545189	−	0.838313i	$-0.683542\pi$
$449$	−23.1047	−1.09038	−0.545189	+	0.838313i	$0.683542\pi$
$450$	0	0
$451$	−24.5078	−1.15403
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.70156	0.173532
$456$	0	0
$457$	13.7016	0.640932	0.320466	−	0.947260i	$-0.396160\pi$
$457$	13.7016	0.640932	0.320466	+	0.947260i	$0.396160\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.1047	−1.82129	−0.910643	−	0.413193i	$-0.864413\pi$
$461$	−39.1047	−1.82129	−0.910643	+	0.413193i	$0.864413\pi$
$462$	0	0
$463$	15.1047	0.701974	0.350987	−	0.936380i	$-0.385846\pi$
$463$	15.1047	0.701974	0.350987	+	0.936380i	$0.385846\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.10469	0.0511188	0.0255594	−	0.999673i	$-0.491863\pi$
$467$	1.10469	0.0511188	0.0255594	+	0.999673i	$0.491863\pi$
$468$	0	0
$469$	14.8062	0.683689
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.8062	1.04863
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.2984	1.01884	0.509421	−	0.860518i	$-0.329860\pi$
$479$	22.2984	1.01884	0.509421	+	0.860518i	$0.329860\pi$
$480$	0	0
$481$	0.298438	0.0136076
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.10469	−0.0501612
$486$	0	0
$487$	21.3141	0.965832	0.482916	−	0.875667i	$-0.339578\pi$
$487$	21.3141	0.965832	0.482916	+	0.875667i	$0.339578\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.2094	−1.45359	−0.726794	−	0.686855i	$-0.758991\pi$
$491$	−32.2094	−1.45359	−0.726794	+	0.686855i	$0.758991\pi$
$492$	0	0
$493$	−19.4031	−0.873873
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−61.1047	−2.74092
$498$	0	0
$499$	40.8062	1.82674	0.913369	−	0.407132i	$-0.133471\pi$
$499$	40.8062	1.82674	0.913369	+	0.407132i	$0.133471\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.5969	−0.829194	−0.414597	−	0.910005i	$-0.636077\pi$
$503$	−18.5969	−0.829194	−0.414597	+	0.910005i	$0.636077\pi$
$504$	0	0
$505$	−18.8062	−0.836867
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.3141	−1.12203	−0.561013	−	0.827807i	$-0.689588\pi$
$509$	−25.3141	−1.12203	−0.561013	+	0.827807i	$0.689588\pi$
$510$	0	0
$511$	−2.20937	−0.0977369
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	−65.0156	−2.85938
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.4031	−0.845213
$528$	0	0
$529$	−22.9109	−0.996128
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.29844	0.186186
$534$	0	0
$535$	1.10469	0.0477598
$536$	0	0
$537$	0	0
$538$	0	0
$539$	38.2094	1.64579
$540$	0	0
$541$	1.79063	0.0769851	0.0384925	−	0.999259i	$-0.487744\pi$
$541$	1.79063	0.0769851	0.0384925	+	0.999259i	$0.487744\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.8062	−0.634230
$546$	0	0
$547$	25.6125	1.09511	0.547556	−	0.836769i	$-0.315558\pi$
$547$	25.6125	1.09511	0.547556	+	0.836769i	$0.315558\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.80625	−0.289956
$552$	0	0
$553$	6.29844	0.267837
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.89531	−0.122023	−0.0610115	−	0.998137i	$-0.519433\pi$
$563$	−2.89531	−0.122023	−0.0610115	+	0.998137i	$0.519433\pi$
$564$	0	0
$565$	−19.4031	−0.816296
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.5969	0.611933	0.305966	−	0.952042i	$-0.401020\pi$
$569$	14.5969	0.611933	0.305966	+	0.952042i	$0.401020\pi$
$570$	0	0
$571$	−31.3141	−1.31045	−0.655226	−	0.755433i	$-0.727427\pi$
$571$	−31.3141	−1.31045	−0.655226	+	0.755433i	$0.727427\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.298438	−0.0124457
$576$	0	0
$577$	0.507811	0.0211404	0.0105702	−	0.999944i	$-0.496635\pi$
$577$	0.507811	0.0211404	0.0105702	+	0.999944i	$0.496635\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14.8062	0.614267
$582$	0	0
$583$	−24.5078	−1.01501
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.2094	−0.586484	−0.293242	−	0.956038i	$-0.594734\pi$
$587$	−14.2094	−0.586484	−0.293242	+	0.956038i	$0.594734\pi$
$588$	0	0
$589$	−6.80625	−0.280447
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.40312	−0.386140	−0.193070	−	0.981185i	$-0.561844\pi$
$593$	−9.40312	−0.386140	−0.193070	+	0.981185i	$0.561844\pi$
$594$	0	0
$595$	21.1047	0.865208
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17.1938	0.702518	0.351259	−	0.936278i	$-0.385754\pi$
$599$	17.1938	0.702518	0.351259	+	0.936278i	$0.385754\pi$
$600$	0	0
$601$	15.7016	0.640480	0.320240	−	0.947336i	$-0.396236\pi$
$601$	15.7016	0.640480	0.320240	+	0.947336i	$0.396236\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	21.5078	0.874417
$606$	0	0
$607$	9.61250	0.390159	0.195080	−	0.980787i	$-0.437503\pi$
$607$	9.61250	0.390159	0.195080	+	0.980787i	$0.437503\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.4031	0.461321
$612$	0	0
$613$	24.7172	0.998318	0.499159	−	0.866511i	$-0.333642\pi$
$613$	24.7172	0.998318	0.499159	+	0.866511i	$0.333642\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−11.6125	−0.466746	−0.233373	−	0.972387i	$-0.574976\pi$
$619$	−11.6125	−0.466746	−0.233373	+	0.972387i	$0.574976\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−41.1047	−1.64682
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.70156	0.0678457
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.596876	−0.0236863
$636$	0	0
$637$	−6.70156	−0.265526
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28.2094	−1.11420	−0.557102	−	0.830444i	$-0.688087\pi$
$641$	−28.2094	−1.11420	−0.557102	+	0.830444i	$0.688087\pi$
$642$	0	0
$643$	37.7016	1.48680	0.743402	−	0.668845i	$-0.233211\pi$
$643$	37.7016	1.48680	0.743402	+	0.668845i	$0.233211\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.895314	−0.0351984	−0.0175992	−	0.999845i	$-0.505602\pi$
$647$	−0.895314	−0.0351984	−0.0175992	+	0.999845i	$0.505602\pi$
$648$	0	0
$649$	−61.6125	−2.41850
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.6125	1.39362	0.696812	−	0.717253i	$-0.254601\pi$
$653$	35.6125	1.39362	0.696812	+	0.717253i	$0.254601\pi$
$654$	0	0
$655$	−12.8062	−0.500382
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.5969	0.568614	0.284307	−	0.958733i	$-0.408237\pi$
$659$	14.5969	0.568614	0.284307	+	0.958733i	$0.408237\pi$
$660$	0	0
$661$	−12.0000	−0.466746	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000	−0.466746	−0.233373	+	0.972387i	$0.574976\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.40312	0.287081
$666$	0	0
$667$	−1.01562	−0.0393250
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−21.1047	−0.814737
$672$	0	0
$673$	−28.8062	−1.11040	−0.555200	−	0.831717i	$-0.687358\pi$
$673$	−28.8062	−1.11040	−0.555200	+	0.831717i	$0.687358\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.1047	−0.580520	−0.290260	−	0.956948i	$-0.593742\pi$
$677$	−15.1047	−0.580520	−0.290260	+	0.956948i	$0.593742\pi$
$678$	0	0
$679$	4.08907	0.156924
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.29844	0.163757
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.70156	0.0645439
$696$	0	0
$697$	24.5078	0.928300
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25.6125	−0.967371	−0.483685	−	0.875242i	$-0.660702\pi$
$701$	−25.6125	−0.967371	−0.483685	+	0.875242i	$0.660702\pi$
$702$	0	0
$703$	0.596876	0.0225116
$704$	0	0
$705$	0	0
$706$	0	0
$707$	69.6125	2.61805
$708$	0	0
$709$	17.1938	0.645725	0.322862	−	0.946446i	$-0.395355\pi$
$709$	17.1938	0.645725	0.322862	+	0.946446i	$0.395355\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.01562	−0.0380353
$714$	0	0
$715$	−5.70156	−0.213226
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.8062	−1.29805	−0.649027	−	0.760765i	$-0.724824\pi$
$719$	−34.8062	−1.29805	−0.649027	+	0.760765i	$0.724824\pi$
$720$	0	0
$721$	−44.4187	−1.65424
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.40312	0.126389
$726$	0	0
$727$	3.40312	0.126215	0.0631074	−	0.998007i	$-0.479899\pi$
$727$	3.40312	0.126215	0.0631074	+	0.998007i	$0.479899\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−22.8062	−0.843520
$732$	0	0
$733$	−28.7172	−1.06069	−0.530347	−	0.847781i	$-0.677938\pi$
$733$	−28.7172	−1.06069	−0.530347	+	0.847781i	$0.677938\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.8062	−0.840079
$738$	0	0
$739$	16.8062	0.618228	0.309114	−	0.951025i	$-0.399968\pi$
$739$	16.8062	0.618228	0.309114	+	0.951025i	$0.399968\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.4031	0.858577	0.429289	−	0.903167i	$-0.358764\pi$
$743$	23.4031	0.858577	0.429289	+	0.903167i	$0.358764\pi$
$744$	0	0
$745$	−7.70156	−0.282163
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.08907	−0.149411
$750$	0	0
$751$	15.3141	0.558818	0.279409	−	0.960172i	$-0.409861\pi$
$751$	15.3141	0.558818	0.279409	+	0.960172i	$0.409861\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.5969	−1.25744	−0.628722	−	0.777630i	$-0.716422\pi$
$757$	−34.5969	−1.25744	−0.628722	+	0.777630i	$0.716422\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	54.8062	1.98412
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.8062	0.390191
$768$	0	0
$769$	38.5969	1.39184	0.695919	−	0.718120i	$-0.254997\pi$
$769$	38.5969	1.39184	0.695919	+	0.718120i	$0.254997\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	45.4031	1.63304	0.816518	−	0.577319i	$-0.195901\pi$
$773$	45.4031	1.63304	0.816518	+	0.577319i	$0.195901\pi$
$774$	0	0
$775$	3.40312	0.122244
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.59688	0.308015
$780$	0	0
$781$	94.1203	3.36789
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.8062	−0.599841
$786$	0	0
$787$	30.8062	1.09812	0.549062	−	0.835782i	$-0.314985\pi$
$787$	30.8062	1.09812	0.549062	+	0.835782i	$0.314985\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	71.8219	2.55369
$792$	0	0
$793$	3.70156	0.131446
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.29844	0.293946	0.146973	−	0.989141i	$-0.453047\pi$
$797$	8.29844	0.293946	0.146973	+	0.989141i	$0.453047\pi$
$798$	0	0
$799$	65.0156	2.30009
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.40312	0.120094
$804$	0	0
$805$	1.10469	0.0389351
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.8062	−0.872141	−0.436071	−	0.899912i	$-0.643630\pi$
$809$	−24.8062	−0.872141	−0.436071	+	0.899912i	$0.643630\pi$
$810$	0	0
$811$	24.2094	0.850106	0.425053	−	0.905168i	$-0.360255\pi$
$811$	24.2094	0.850106	0.425053	+	0.905168i	$0.360255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.7016	0.620059
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.5078	1.20433	0.602165	−	0.798371i	$-0.294305\pi$
$821$	34.5078	1.20433	0.602165	+	0.798371i	$0.294305\pi$
$822$	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
$827$	−47.4031	−1.64837	−0.824184	−	0.566322i	$-0.808366\pi$
$827$	−47.4031	−1.64837	−0.824184	+	0.566322i	$0.808366\pi$
$828$	0	0
$829$	3.61250	0.125467	0.0627336	−	0.998030i	$-0.480018\pi$
$829$	3.61250	0.125467	0.0627336	+	0.998030i	$0.480018\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−38.2094	−1.32388
$834$	0	0
$835$	−2.80625	−0.0971142
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.1203	1.31606	0.658030	−	0.752992i	$-0.271390\pi$
$839$	38.1203	1.31606	0.658030	+	0.752992i	$0.271390\pi$
$840$	0	0
$841$	−17.4187	−0.600646
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−79.6125	−2.73552
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.0890652	0.00305311
$852$	0	0
$853$	48.7172	1.66804	0.834022	−	0.551731i	$-0.186032\pi$
$853$	48.7172	1.66804	0.834022	+	0.551731i	$0.186032\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.29844	−0.215151	−0.107575	−	0.994197i	$-0.534309\pi$
$857$	−6.29844	−0.215151	−0.107575	+	0.994197i	$0.534309\pi$
$858$	0	0
$859$	37.7016	1.28636	0.643180	−	0.765715i	$-0.277615\pi$
$859$	37.7016	1.28636	0.643180	+	0.765715i	$0.277615\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.596876	−0.0203179	−0.0101589	−	0.999948i	$-0.503234\pi$
$863$	−0.596876	−0.0203179	−0.0101589	+	0.999948i	$0.503234\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.70156	−0.329103
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.70156	−0.125136
$876$	0	0
$877$	47.6125	1.60776	0.803880	−	0.594792i	$-0.202765\pi$
$877$	47.6125	1.60776	0.803880	+	0.594792i	$0.202765\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.8062	−0.431453	−0.215727	−	0.976454i	$-0.569212\pi$
$881$	−12.8062	−0.431453	−0.215727	+	0.976454i	$0.569212\pi$
$882$	0	0
$883$	33.6125	1.13115	0.565575	−	0.824697i	$-0.308654\pi$
$883$	33.6125	1.13115	0.565575	+	0.824697i	$0.308654\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.5078	−1.69589	−0.847943	−	0.530087i	$-0.822159\pi$
$887$	−50.5078	−1.69589	−0.847943	+	0.530087i	$0.822159\pi$
$888$	0	0
$889$	2.20937	0.0741000
$890$	0	0
$891$	0	0
$892$	0	0
$893$	22.8062	0.763182
$894$	0	0
$895$	22.0000	0.735379
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.5813	0.386256
$900$	0	0
$901$	24.5078	0.816474
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.70156	−0.256009
$906$	0	0
$907$	22.2094	0.737450	0.368725	−	0.929539i	$-0.379794\pi$
$907$	22.2094	0.737450	0.368725	+	0.929539i	$0.379794\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.8062	0.623079	0.311539	−	0.950233i	$-0.399155\pi$
$911$	18.8062	0.623079	0.311539	+	0.950233i	$0.399155\pi$
$912$	0	0
$913$	−22.8062	−0.754777
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47.4031	1.56539
$918$	0	0
$919$	51.9109	1.71238	0.856192	−	0.516658i	$-0.172824\pi$
$919$	51.9109	1.71238	0.856192	+	0.516658i	$0.172824\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.5078	−0.543361
$924$	0	0
$925$	−0.298438	−0.00981258
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11.1047	0.364333	0.182166	−	0.983268i	$-0.441689\pi$
$929$	11.1047	0.364333	0.182166	+	0.983268i	$0.441689\pi$
$930$	0	0
$931$	−13.4031	−0.439270
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−32.5078	−1.06312
$936$	0	0
$937$	27.1938	0.888381	0.444191	−	0.895932i	$-0.353491\pi$
$937$	27.1938	0.888381	0.444191	+	0.895932i	$0.353491\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	48.2984	1.57448	0.787242	−	0.616644i	$-0.211508\pi$
$941$	48.2984	1.57448	0.787242	+	0.616644i	$0.211508\pi$
$942$	0	0
$943$	1.28282	0.0417743
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.6125	−1.35222	−0.676112	−	0.736799i	$-0.736337\pi$
$947$	−41.6125	−1.35222	−0.676112	+	0.736799i	$0.736337\pi$
$948$	0	0
$949$	−0.596876	−0.0193754
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39.9109	1.29284	0.646421	−	0.762981i	$-0.276265\pi$
$953$	39.9109	1.29284	0.646421	+	0.762981i	$0.276265\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	66.6281	2.15153
$960$	0	0
$961$	−19.4187	−0.626411
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.89531	−0.0932034
$966$	0	0
$967$	−49.8219	−1.60216	−0.801082	−	0.598555i	$-0.795742\pi$
$967$	−49.8219	−1.60216	−0.801082	+	0.598555i	$0.795742\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.01562	0.225142	0.112571	−	0.993644i	$-0.464091\pi$
$971$	7.01562	0.225142	0.112571	+	0.993644i	$0.464091\pi$
$972$	0	0
$973$	−6.29844	−0.201919
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.2094	0.390612	0.195306	−	0.980742i	$-0.437430\pi$
$977$	12.2094	0.390612	0.195306	+	0.980742i	$0.437430\pi$
$978$	0	0
$979$	63.3141	2.02353
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	0	0
$985$	−24.2094	−0.771375
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.19375	−0.0379591
$990$	0	0
$991$	−42.7172	−1.35696	−0.678478	−	0.734621i	$-0.737360\pi$
$991$	−42.7172	−1.35696	−0.678478	+	0.734621i	$0.737360\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22.8062	−0.723007
$996$	0	0
$997$	−38.5969	−1.22238	−0.611188	−	0.791486i	$-0.709308\pi$
$997$	−38.5969	−1.22238	−0.611188	+	0.791486i	$0.709308\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bb.1.1		2
3.2	odd	2		1560.2.a.m.1.1	✓	2
4.3	odd	2		9360.2.a.ct.1.2		2
12.11	even	2		3120.2.a.bf.1.2		2
15.14	odd	2		7800.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.m.1.1	✓	2	3.2	odd	2
3120.2.a.bf.1.2		2	12.11	even	2
4680.2.a.bb.1.1		2	1.1	even	1	trivial
7800.2.a.be.1.2		2	15.14	odd	2
9360.2.a.ct.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.70156 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.70156 q^{7} +5.70156 q^{11} -1.00000 q^{13} -5.70156 q^{17} -2.00000 q^{19} -0.298438 q^{23} +1.00000 q^{25} +3.40312 q^{29} +3.40312 q^{31} -3.70156 q^{35} -0.298438 q^{37} -4.29844 q^{41} +4.00000 q^{43} -11.4031 q^{47} +6.70156 q^{49} -4.29844 q^{53} +5.70156 q^{55} -10.8062 q^{59} -3.70156 q^{61} -1.00000 q^{65} -4.00000 q^{67} +16.5078 q^{71} +0.596876 q^{73} -21.1047 q^{77} -1.70156 q^{79} -4.00000 q^{83} -5.70156 q^{85} +11.1047 q^{89} +3.70156 q^{91} -2.00000 q^{95} -1.10469 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - q^7	\( 2 q + 2 q^{5} - q^{7} + 5 q^{11} - 2 q^{13} - 5 q^{17} - 4 q^{19} - 7 q^{23} + 2 q^{25} - 6 q^{29} - 6 q^{31} - q^{35} - 7 q^{37} - 15 q^{41} + 8 q^{43} - 10 q^{47} + 7 q^{49} - 15 q^{53} + 5 q^{55} + 4 q^{59} - q^{61} - 2 q^{65} - 8 q^{67} + q^{71} + 14 q^{73} - 23 q^{77} + 3 q^{79} - 8 q^{83} - 5 q^{85} + 3 q^{89} + q^{91} - 4 q^{95} + 17 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 + 5 * q^11 - 2 * q^13 - 5 * q^17 - 4 * q^19 - 7 * q^23 + 2 * q^25 - 6 * q^29 - 6 * q^31 - q^35 - 7 * q^37 - 15 * q^41 + 8 * q^43 - 10 * q^47 + 7 * q^49 - 15 * q^53 + 5 * q^55 + 4 * q^59 - q^61 - 2 * q^65 - 8 * q^67 + q^71 + 14 * q^73 - 23 * q^77 + 3 * q^79 - 8 * q^83 - 5 * q^85 + 3 * q^89 + q^91 - 4 * q^95 + 17 * q^97

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