LMFDB - Embedded newform 5148.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 572)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.58774 q^{5} -4.75698 q^{7} +O(q^{10})$	$q-3.58774 q^{5} -4.75698 q^{7} -1.00000 q^{11} -1.00000 q^{13} +6.22982 q^{17} +5.34472 q^{19} +4.83076 q^{23} +7.87189 q^{25} +5.28415 q^{29} -7.35793 q^{31} +17.0668 q^{35} -7.58774 q^{37} +6.98680 q^{41} -7.66152 q^{43} +5.17548 q^{47} +15.6289 q^{49} -0.169240 q^{53} +3.58774 q^{55} -1.21037 q^{59} -10.1212 q^{61} +3.58774 q^{65} +12.9930 q^{67} -9.24926 q^{71} +5.49228 q^{73} +4.75698 q^{77} -7.74378 q^{79} -3.47283 q^{83} -22.3510 q^{85} +4.04737 q^{89} +4.75698 q^{91} -19.1755 q^{95} +3.58774 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5} - 7 q^{7}+O(q^{10}) $ 3 * q + q^5 - 7 * q^7	$ 3 q + q^{5} - 7 q^{7} - 3 q^{11} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 10 q^{23} + 10 q^{25} + 14 q^{29} - 23 q^{31} + 6 q^{35} - 11 q^{37} + q^{41} - 14 q^{43} - 8 q^{47} + 26 q^{49} - 5 q^{53} - q^{55} + q^{59} + 4 q^{61} - q^{65} - 9 q^{67} - 7 q^{71} + 3 q^{73} + 7 q^{77} + 4 q^{79} - 5 q^{83} - 20 q^{85} - 25 q^{89} + 7 q^{91} - 34 q^{95} - q^{97}+O(q^{100}) $ 3 * q + q^5 - 7 * q^7 - 3 * q^11 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 10 * q^23 + 10 * q^25 + 14 * q^29 - 23 * q^31 + 6 * q^35 - 11 * q^37 + q^41 - 14 * q^43 - 8 * q^47 + 26 * q^49 - 5 * q^53 - q^55 + q^59 + 4 * q^61 - q^65 - 9 * q^67 - 7 * q^71 + 3 * q^73 + 7 * q^77 + 4 * q^79 - 5 * q^83 - 20 * q^85 - 25 * q^89 + 7 * q^91 - 34 * q^95 - 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
$5$	−3.58774	−1.60449	−0.802243	−	0.596997i	$-0.796360\pi$
$5$	−3.58774	−1.60449	−0.802243	+	0.596997i	$0.796360\pi$
$6$	0	0
$7$	−4.75698	−1.79797	−0.898985	−	0.437980i	$-0.855694\pi$
$7$	−4.75698	−1.79797	−0.898985	+	0.437980i	$0.855694\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.22982	1.51095	0.755476	−	0.655176i	$-0.227406\pi$
$17$	6.22982	1.51095	0.755476	+	0.655176i	$0.227406\pi$
$18$	0	0
$19$	5.34472	1.22616	0.613082	−	0.790019i	$-0.289930\pi$
$19$	5.34472	1.22616	0.613082	+	0.790019i	$0.289930\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.83076	1.00728	0.503642	−	0.863913i	$-0.331993\pi$
$23$	4.83076	1.00728	0.503642	+	0.863913i	$0.331993\pi$
$24$	0	0
$25$	7.87189	1.57438
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.28415	0.981242	0.490621	−	0.871373i	$-0.336770\pi$
$29$	5.28415	0.981242	0.490621	+	0.871373i	$0.336770\pi$
$30$	0	0
$31$	−7.35793	−1.32152	−0.660761	−	0.750596i	$-0.729766\pi$
$31$	−7.35793	−1.32152	−0.660761	+	0.750596i	$0.729766\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	17.0668	2.88482
$36$	0	0
$37$	−7.58774	−1.24742	−0.623709	−	0.781657i	$-0.714375\pi$
$37$	−7.58774	−1.24742	−0.623709	+	0.781657i	$0.714375\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.98680	1.09115	0.545577	−	0.838061i	$-0.316310\pi$
$41$	6.98680	1.09115	0.545577	+	0.838061i	$0.316310\pi$
$42$	0	0
$43$	−7.66152	−1.16837	−0.584185	−	0.811620i	$-0.698586\pi$
$43$	−7.66152	−1.16837	−0.584185	+	0.811620i	$0.698586\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.17548	0.754922	0.377461	−	0.926026i	$-0.376797\pi$
$47$	5.17548	0.754922	0.377461	+	0.926026i	$0.376797\pi$
$48$	0	0
$49$	15.6289	2.23270
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.169240	−0.0232469	−0.0116234	−	0.999932i	$-0.503700\pi$
$53$	−0.169240	−0.0232469	−0.0116234	+	0.999932i	$0.503700\pi$
$54$	0	0
$55$	3.58774	0.483771
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.21037	−0.157577	−0.0787883	−	0.996891i	$-0.525105\pi$
$59$	−1.21037	−0.157577	−0.0787883	+	0.996891i	$0.525105\pi$
$60$	0	0
$61$	−10.1212	−1.29588	−0.647940	−	0.761691i	$-0.724369\pi$
$61$	−10.1212	−1.29588	−0.647940	+	0.761691i	$0.724369\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.58774	0.445005
$66$	0	0
$67$	12.9930	1.58735	0.793676	−	0.608340i	$-0.208164\pi$
$67$	12.9930	1.58735	0.793676	+	0.608340i	$0.208164\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.24926	−1.09769	−0.548843	−	0.835926i	$-0.684931\pi$
$71$	−9.24926	−1.09769	−0.548843	+	0.835926i	$0.684931\pi$
$72$	0	0
$73$	5.49228	0.642823	0.321411	−	0.946940i	$-0.395843\pi$
$73$	5.49228	0.642823	0.321411	+	0.946940i	$0.395843\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.75698	0.542108
$78$	0	0
$79$	−7.74378	−0.871243	−0.435622	−	0.900130i	$-0.643471\pi$
$79$	−7.74378	−0.871243	−0.435622	+	0.900130i	$0.643471\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.47283	−0.381193	−0.190597	−	0.981668i	$-0.561042\pi$
$83$	−3.47283	−0.381193	−0.190597	+	0.981668i	$0.561042\pi$
$84$	0	0
$85$	−22.3510	−2.42430
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.04737	0.429021	0.214510	−	0.976722i	$-0.431184\pi$
$89$	4.04737	0.429021	0.214510	+	0.976722i	$0.431184\pi$
$90$	0	0
$91$	4.75698	0.498667
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−19.1755	−1.96736
$96$	0	0
$97$	3.58774	0.364280	0.182140	−	0.983273i	$-0.441698\pi$
$97$	3.58774	0.364280	0.182140	+	0.983273i	$0.441698\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.7981	−1.07445	−0.537226	−	0.843438i	$-0.680528\pi$
$101$	−10.7981	−1.07445	−0.537226	+	0.843438i	$0.680528\pi$
$102$	0	0
$103$	10.1623	1.00132	0.500660	−	0.865644i	$-0.333091\pi$
$103$	10.1623	1.00132	0.500660	+	0.865644i	$0.333091\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.17548	−0.113638	−0.0568191	−	0.998384i	$-0.518096\pi$
$107$	−1.17548	−0.113638	−0.0568191	+	0.998384i	$0.518096\pi$
$108$	0	0
$109$	2.29039	0.219380	0.109690	−	0.993966i	$-0.465014\pi$
$109$	2.29039	0.219380	0.109690	+	0.993966i	$0.465014\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.5940	0.996598	0.498299	−	0.867005i	$-0.333958\pi$
$113$	10.5940	0.996598	0.498299	+	0.867005i	$0.333958\pi$
$114$	0	0
$115$	−17.3315	−1.61617
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−29.6351	−2.71665
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.3036	−0.921581
$126$	0	0
$127$	−1.85244	−0.164378	−0.0821888	−	0.996617i	$-0.526191\pi$
$127$	−1.85244	−0.164378	−0.0821888	+	0.996617i	$0.526191\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.14756	−0.362374	−0.181187	−	0.983449i	$-0.557994\pi$
$131$	−4.14756	−0.362374	−0.181187	+	0.983449i	$0.557994\pi$
$132$	0	0
$133$	−25.4247	−2.20460
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−23.3315	−1.99335	−0.996673	−	0.0815031i	$-0.974028\pi$
$137$	−23.3315	−1.99335	−0.996673	+	0.0815031i	$0.974028\pi$
$138$	0	0
$139$	−13.4053	−1.13702	−0.568511	−	0.822675i	$-0.692480\pi$
$139$	−13.4053	−1.13702	−0.568511	+	0.822675i	$0.692480\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−18.9582	−1.57439
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.47283	0.448352	0.224176	−	0.974549i	$-0.428031\pi$
$149$	5.47283	0.448352	0.224176	+	0.974549i	$0.428031\pi$
$150$	0	0
$151$	1.89134	0.153915	0.0769574	−	0.997034i	$-0.475479\pi$
$151$	1.89134	0.153915	0.0769574	+	0.997034i	$0.475479\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	26.3983	2.12037
$156$	0	0
$157$	10.1538	0.810362	0.405181	−	0.914237i	$-0.367209\pi$
$157$	10.1538	0.810362	0.405181	+	0.914237i	$0.367209\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−22.9798	−1.81106
$162$	0	0
$163$	−22.2298	−1.74117	−0.870587	−	0.492015i	$-0.836261\pi$
$163$	−22.2298	−1.74117	−0.870587	+	0.492015i	$0.836261\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.9798	−1.31394	−0.656970	−	0.753917i	$-0.728162\pi$
$167$	−16.9798	−1.31394	−0.656970	+	0.753917i	$0.728162\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.74378	−0.588748	−0.294374	−	0.955690i	$-0.595111\pi$
$173$	−7.74378	−0.588748	−0.294374	+	0.955690i	$0.595111\pi$
$174$	0	0
$175$	−37.4464	−2.83068
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.4402	0.855079	0.427540	−	0.903997i	$-0.359380\pi$
$179$	11.4402	0.855079	0.427540	+	0.903997i	$0.359380\pi$
$180$	0	0
$181$	−10.4464	−0.776477	−0.388238	−	0.921559i	$-0.626916\pi$
$181$	−10.4464	−0.776477	−0.388238	+	0.921559i	$0.626916\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	27.2229	2.00146
$186$	0	0
$187$	−6.22982	−0.455569
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.18869	0.375440	0.187720	−	0.982223i	$-0.439890\pi$
$191$	5.18869	0.375440	0.187720	+	0.982223i	$0.439890\pi$
$192$	0	0
$193$	−0.188687	−0.0135820	−0.00679098	−	0.999977i	$-0.502162\pi$
$193$	−0.188687	−0.0135820	−0.00679098	+	0.999977i	$0.502162\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.31680	0.450053	0.225027	−	0.974353i	$-0.427753\pi$
$197$	6.31680	0.450053	0.225027	+	0.974353i	$0.427753\pi$
$198$	0	0
$199$	13.4534	0.953685	0.476843	−	0.878989i	$-0.341781\pi$
$199$	13.4534	0.953685	0.476843	+	0.878989i	$0.341781\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−25.1366	−1.76424
$204$	0	0
$205$	−25.0668	−1.75074
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.34472	−0.369702
$210$	0	0
$211$	7.06682	0.486500	0.243250	−	0.969964i	$-0.421786\pi$
$211$	7.06682	0.486500	0.243250	+	0.969964i	$0.421786\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.4876	1.87464
$216$	0	0
$217$	35.0015	2.37606
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.22982	−0.419063
$222$	0	0
$223$	−16.2772	−1.09000	−0.545001	−	0.838436i	$-0.683471\pi$
$223$	−16.2772	−1.09000	−0.545001	+	0.838436i	$0.683471\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.9519	−1.19151	−0.595755	−	0.803166i	$-0.703147\pi$
$227$	−17.9519	−1.19151	−0.595755	+	0.803166i	$0.703147\pi$
$228$	0	0
$229$	16.6157	1.09799	0.548997	−	0.835824i	$-0.315010\pi$
$229$	16.6157	1.09799	0.548997	+	0.835824i	$0.315010\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.377373	−0.0247225	−0.0123613	−	0.999924i	$-0.503935\pi$
$233$	−0.377373	−0.0247225	−0.0123613	+	0.999924i	$0.503935\pi$
$234$	0	0
$235$	−18.5683	−1.21126
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.6678	1.07815	0.539074	−	0.842258i	$-0.318774\pi$
$239$	16.6678	1.07815	0.539074	+	0.842258i	$0.318774\pi$
$240$	0	0
$241$	−22.3921	−1.44240	−0.721201	−	0.692726i	$-0.756409\pi$
$241$	−22.3921	−1.44240	−0.721201	+	0.692726i	$0.756409\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−56.0723	−3.58233
$246$	0	0
$247$	−5.34472	−0.340077
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7111	1.24416	0.622078	−	0.782956i	$-0.286289\pi$
$251$	19.7111	1.24416	0.622078	+	0.782956i	$0.286289\pi$
$252$	0	0
$253$	−4.83076	−0.303707
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.5676	−1.47011	−0.735053	−	0.678010i	$-0.762843\pi$
$257$	−23.5676	−1.47011	−0.735053	+	0.678010i	$0.762843\pi$
$258$	0	0
$259$	36.0947	2.24282
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.66152	0.595755	0.297877	−	0.954604i	$-0.403721\pi$
$263$	9.66152	0.595755	0.297877	+	0.954604i	$0.403721\pi$
$264$	0	0
$265$	0.607188	0.0372993
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.70265	0.225754	0.112877	−	0.993609i	$-0.463993\pi$
$269$	3.70265	0.225754	0.112877	+	0.993609i	$0.463993\pi$
$270$	0	0
$271$	−7.26247	−0.441163	−0.220582	−	0.975369i	$-0.570796\pi$
$271$	−7.26247	−0.441163	−0.220582	+	0.975369i	$0.570796\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.87189	−0.474693
$276$	0	0
$277$	18.2423	1.09607	0.548037	−	0.836454i	$-0.315375\pi$
$277$	18.2423	1.09607	0.548037	+	0.836454i	$0.315375\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.46587	0.326067	0.163033	−	0.986621i	$-0.447872\pi$
$281$	5.46587	0.326067	0.163033	+	0.986621i	$0.447872\pi$
$282$	0	0
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33.2361	−1.96186
$288$	0	0
$289$	21.8106	1.28298
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.10866	0.240031	0.120015	−	0.992772i	$-0.461706\pi$
$293$	4.10866	0.240031	0.120015	+	0.992772i	$0.461706\pi$
$294$	0	0
$295$	4.34249	0.252829
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.83076	−0.279370
$300$	0	0
$301$	36.4457	2.10070
$302$	0	0
$303$	0	0
$304$	0	0
$305$	36.3121	2.07922
$306$	0	0
$307$	−11.7827	−0.672473	−0.336236	−	0.941778i	$-0.609154\pi$
$307$	−11.7827	−0.672473	−0.336236	+	0.941778i	$0.609154\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.45339	−0.0824140	−0.0412070	−	0.999151i	$-0.513120\pi$
$311$	−1.45339	−0.0824140	−0.0412070	+	0.999151i	$0.513120\pi$
$312$	0	0
$313$	−30.3183	−1.71369	−0.856846	−	0.515572i	$-0.827579\pi$
$313$	−30.3183	−1.71369	−0.856846	+	0.515572i	$0.827579\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.2772	1.13888	0.569440	−	0.822033i	$-0.307160\pi$
$317$	20.2772	1.13888	0.569440	+	0.822033i	$0.307160\pi$
$318$	0	0
$319$	−5.28415	−0.295855
$320$	0	0
$321$	0	0
$322$	0	0
$323$	33.2966	1.85267
$324$	0	0
$325$	−7.87189	−0.436654
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.6197	−1.35733
$330$	0	0
$331$	−13.9651	−0.767592	−0.383796	−	0.923418i	$-0.625383\pi$
$331$	−13.9651	−0.767592	−0.383796	+	0.923418i	$0.625383\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−46.6157	−2.54689
$336$	0	0
$337$	−4.12115	−0.224493	−0.112247	−	0.993680i	$-0.535805\pi$
$337$	−4.12115	−0.224493	−0.112247	+	0.993680i	$0.535805\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.35793	0.398454
$342$	0	0
$343$	−41.0474	−2.21635
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.4442	−1.15118	−0.575592	−	0.817737i	$-0.695228\pi$
$347$	−21.4442	−1.15118	−0.575592	+	0.817737i	$0.695228\pi$
$348$	0	0
$349$	2.29735	0.122974	0.0614872	−	0.998108i	$-0.480416\pi$
$349$	2.29735	0.122974	0.0614872	+	0.998108i	$0.480416\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.6421	−0.566420	−0.283210	−	0.959058i	$-0.591399\pi$
$353$	−10.6421	−0.566420	−0.283210	+	0.959058i	$0.591399\pi$
$354$	0	0
$355$	33.1840	1.76122
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.1623	0.853012	0.426506	−	0.904485i	$-0.359744\pi$
$359$	16.1623	0.853012	0.426506	+	0.904485i	$0.359744\pi$
$360$	0	0
$361$	9.56606	0.503477
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.7049	−1.03140
$366$	0	0
$367$	22.5676	1.17802	0.589009	−	0.808126i	$-0.299518\pi$
$367$	22.5676	1.17802	0.589009	+	0.808126i	$0.299518\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.805070	0.0417972
$372$	0	0
$373$	−7.43171	−0.384799	−0.192400	−	0.981317i	$-0.561627\pi$
$373$	−7.43171	−0.384799	−0.192400	+	0.981317i	$0.561627\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.28415	−0.272147
$378$	0	0
$379$	−2.68097	−0.137712	−0.0688560	−	0.997627i	$-0.521935\pi$
$379$	−2.68097	−0.137712	−0.0688560	+	0.997627i	$0.521935\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.5768	−0.744838	−0.372419	−	0.928065i	$-0.621472\pi$
$383$	−14.5768	−0.744838	−0.372419	+	0.928065i	$0.621472\pi$
$384$	0	0
$385$	−17.0668	−0.869806
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.39058	−0.222611	−0.111306	−	0.993786i	$-0.535503\pi$
$389$	−4.39058	−0.222611	−0.111306	+	0.993786i	$0.535503\pi$
$390$	0	0
$391$	30.0947	1.52196
$392$	0	0
$393$	0	0
$394$	0	0
$395$	27.7827	1.39790
$396$	0	0
$397$	−12.4721	−0.625958	−0.312979	−	0.949760i	$-0.601327\pi$
$397$	−12.4721	−0.625958	−0.312979	+	0.949760i	$0.601327\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.3664	−1.06699	−0.533494	−	0.845804i	$-0.679121\pi$
$401$	−21.3664	−1.06699	−0.533494	+	0.845804i	$0.679121\pi$
$402$	0	0
$403$	7.35793	0.366524
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.58774	0.376110
$408$	0	0
$409$	30.2423	1.49539	0.747693	−	0.664045i	$-0.231162\pi$
$409$	30.2423	1.49539	0.747693	+	0.664045i	$0.231162\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.75770	0.283318
$414$	0	0
$415$	12.4596	0.611619
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−31.9714	−1.56190	−0.780952	−	0.624592i	$-0.785265\pi$
$419$	−31.9714	−1.56190	−0.780952	+	0.624592i	$0.785265\pi$
$420$	0	0
$421$	−4.26871	−0.208044	−0.104022	−	0.994575i	$-0.533171\pi$
$421$	−4.26871	−0.208044	−0.104022	+	0.994575i	$0.533171\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	49.0404	2.37881
$426$	0	0
$427$	48.1461	2.32995
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.5008	1.08382	0.541912	−	0.840435i	$-0.317701\pi$
$431$	22.5008	1.08382	0.541912	+	0.840435i	$0.317701\pi$
$432$	0	0
$433$	12.7291	0.611719	0.305860	−	0.952077i	$-0.401056\pi$
$433$	12.7291	0.611719	0.305860	+	0.952077i	$0.401056\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.8191	1.23509
$438$	0	0
$439$	18.9193	0.902967	0.451484	−	0.892279i	$-0.350895\pi$
$439$	18.9193	0.902967	0.451484	+	0.892279i	$0.350895\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.25150	0.154483	0.0772416	−	0.997012i	$-0.475389\pi$
$443$	3.25150	0.154483	0.0772416	+	0.997012i	$0.475389\pi$
$444$	0	0
$445$	−14.5209	−0.688358
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.8844	−1.45752	−0.728762	−	0.684767i	$-0.759904\pi$
$449$	−30.8844	−1.45752	−0.728762	+	0.684767i	$0.759904\pi$
$450$	0	0
$451$	−6.98680	−0.328995
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.0668	−0.800105
$456$	0	0
$457$	1.87412	0.0876678	0.0438339	−	0.999039i	$-0.486043\pi$
$457$	1.87412	0.0876678	0.0438339	+	0.999039i	$0.486043\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.48131	−0.301865	−0.150932	−	0.988544i	$-0.548228\pi$
$461$	−6.48131	−0.301865	−0.150932	+	0.988544i	$0.548228\pi$
$462$	0	0
$463$	−34.9541	−1.62446	−0.812228	−	0.583339i	$-0.801746\pi$
$463$	−34.9541	−1.62446	−0.812228	+	0.583339i	$0.801746\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.58774	−0.351119	−0.175559	−	0.984469i	$-0.556173\pi$
$467$	−7.58774	−0.351119	−0.175559	+	0.984469i	$0.556173\pi$
$468$	0	0
$469$	−61.8076	−2.85401
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.66152	0.352277
$474$	0	0
$475$	42.0731	1.93044
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.2702	1.42877	0.714387	−	0.699751i	$-0.246706\pi$
$479$	31.2702	1.42877	0.714387	+	0.699751i	$0.246706\pi$
$480$	0	0
$481$	7.58774	0.345971
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.8719	−0.584482
$486$	0	0
$487$	−41.4916	−1.88016	−0.940081	−	0.340951i	$-0.889251\pi$
$487$	−41.4916	−1.88016	−0.940081	+	0.340951i	$0.889251\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.25774	0.0567610	0.0283805	−	0.999597i	$-0.490965\pi$
$491$	1.25774	0.0567610	0.0283805	+	0.999597i	$0.490965\pi$
$492$	0	0
$493$	32.9193	1.48261
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.9986	1.97361
$498$	0	0
$499$	−15.7049	−0.703047	−0.351524	−	0.936179i	$-0.614336\pi$
$499$	−15.7049	−0.703047	−0.351524	+	0.936179i	$0.614336\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.7283	1.10258	0.551291	−	0.834313i	$-0.314135\pi$
$503$	24.7283	1.10258	0.551291	+	0.834313i	$0.314135\pi$
$504$	0	0
$505$	38.7408	1.72394
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.8455	1.10126	0.550628	−	0.834751i	$-0.314388\pi$
$509$	24.8455	1.10126	0.550628	+	0.834751i	$0.314388\pi$
$510$	0	0
$511$	−26.1267	−1.15578
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−36.4596	−1.60660
$516$	0	0
$517$	−5.17548	−0.227617
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.62887	0.246605	0.123303	−	0.992369i	$-0.460651\pi$
$521$	5.62887	0.246605	0.123303	+	0.992369i	$0.460651\pi$
$522$	0	0
$523$	5.95664	0.260466	0.130233	−	0.991483i	$-0.458428\pi$
$523$	5.95664	0.260466	0.130233	+	0.991483i	$0.458428\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.8385	−1.99676
$528$	0	0
$529$	0.336245	0.0146193
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.98680	−0.302632
$534$	0	0
$535$	4.21733	0.182331
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−15.6289	−0.673183
$540$	0	0
$541$	23.1468	0.995160	0.497580	−	0.867418i	$-0.334222\pi$
$541$	23.1468	0.995160	0.497580	+	0.867418i	$0.334222\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.21733	−0.351992
$546$	0	0
$547$	−25.5529	−1.09256	−0.546281	−	0.837602i	$-0.683957\pi$
$547$	−25.5529	−1.09256	−0.546281	+	0.837602i	$0.683957\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	28.2423	1.20316
$552$	0	0
$553$	36.8370	1.56647
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.6483	1.72232	0.861162	−	0.508330i	$-0.169737\pi$
$557$	40.6483	1.72232	0.861162	+	0.508330i	$0.169737\pi$
$558$	0	0
$559$	7.66152	0.324048
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.0404	−1.64536	−0.822679	−	0.568507i	$-0.807521\pi$
$563$	−39.0404	−1.64536	−0.822679	+	0.568507i	$0.807521\pi$
$564$	0	0
$565$	−38.0085	−1.59903
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.9317	0.625971	0.312986	−	0.949758i	$-0.398671\pi$
$569$	14.9317	0.625971	0.312986	+	0.949758i	$0.398671\pi$
$570$	0	0
$571$	15.3664	0.643064	0.321532	−	0.946899i	$-0.395802\pi$
$571$	15.3664	0.643064	0.321532	+	0.946899i	$0.395802\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.0272	1.58584
$576$	0	0
$577$	−8.15604	−0.339540	−0.169770	−	0.985484i	$-0.554303\pi$
$577$	−8.15604	−0.339540	−0.169770	+	0.985484i	$0.554303\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.5202	0.685374
$582$	0	0
$583$	0.169240	0.00700919
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.16004	−0.336801	−0.168401	−	0.985719i	$-0.553860\pi$
$587$	−8.16004	−0.336801	−0.168401	+	0.985719i	$0.553860\pi$
$588$	0	0
$589$	−39.3261	−1.62040
$590$	0	0
$591$	0	0
$592$	0	0
$593$	37.4031	1.53596	0.767980	−	0.640474i	$-0.221262\pi$
$593$	37.4031	1.53596	0.767980	+	0.640474i	$0.221262\pi$
$594$	0	0
$595$	106.323	4.35882
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.22357	−0.295147	−0.147574	−	0.989051i	$-0.547146\pi$
$599$	−7.22357	−0.295147	−0.147574	+	0.989051i	$0.547146\pi$
$600$	0	0
$601$	32.8106	1.33837	0.669186	−	0.743095i	$-0.266643\pi$
$601$	32.8106	1.33837	0.669186	+	0.743095i	$0.266643\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.58774	−0.145862
$606$	0	0
$607$	14.7981	0.600637	0.300318	−	0.953839i	$-0.402907\pi$
$607$	14.7981	0.600637	0.300318	+	0.953839i	$0.402907\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.17548	−0.209378
$612$	0	0
$613$	−34.8976	−1.40950	−0.704750	−	0.709456i	$-0.748941\pi$
$613$	−34.8976	−1.40950	−0.704750	+	0.709456i	$0.748941\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.3121	−0.978767	−0.489384	−	0.872069i	$-0.662778\pi$
$617$	−24.3121	−0.978767	−0.489384	+	0.872069i	$0.662778\pi$
$618$	0	0
$619$	−48.3983	−1.94529	−0.972647	−	0.232289i	$-0.925378\pi$
$619$	−48.3983	−1.94529	−0.972647	+	0.232289i	$0.925378\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19.2533	−0.771366
$624$	0	0
$625$	−2.39281	−0.0957125
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−47.2702	−1.88479
$630$	0	0
$631$	11.6266	0.462849	0.231425	−	0.972853i	$-0.425661\pi$
$631$	11.6266	0.462849	0.231425	+	0.972853i	$0.425661\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.64608	0.263742
$636$	0	0
$637$	−15.6289	−0.619238
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.8851	0.785414	0.392707	−	0.919664i	$-0.371539\pi$
$641$	19.8851	0.785414	0.392707	+	0.919664i	$0.371539\pi$
$642$	0	0
$643$	11.1406	0.439342	0.219671	−	0.975574i	$-0.429502\pi$
$643$	11.1406	0.439342	0.219671	+	0.975574i	$0.429502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.55357	−0.218334	−0.109167	−	0.994023i	$-0.534818\pi$
$647$	−5.55357	−0.218334	−0.109167	+	0.994023i	$0.534818\pi$
$648$	0	0
$649$	1.21037	0.0475111
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.0753	1.29434	0.647168	−	0.762347i	$-0.275953\pi$
$653$	33.0753	1.29434	0.647168	+	0.762347i	$0.275953\pi$
$654$	0	0
$655$	14.8804	0.581424
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31.5404	−1.22864	−0.614319	−	0.789058i	$-0.710569\pi$
$659$	−31.5404	−1.22864	−0.614319	+	0.789058i	$0.710569\pi$
$660$	0	0
$661$	13.6825	0.532187	0.266093	−	0.963947i	$-0.414267\pi$
$661$	13.6825	0.532187	0.266093	+	0.963947i	$0.414267\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	91.2174	3.53726
$666$	0	0
$667$	25.5264	0.988388
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.1212	0.390723
$672$	0	0
$673$	−49.9986	−1.92730	−0.963652	−	0.267162i	$-0.913914\pi$
$673$	−49.9986	−1.92730	−0.963652	+	0.267162i	$0.913914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.4487	−0.978071	−0.489036	−	0.872264i	$-0.662651\pi$
$677$	−25.4487	−0.978071	−0.489036	+	0.872264i	$0.662651\pi$
$678$	0	0
$679$	−17.0668	−0.654964
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.7019	−0.945193	−0.472597	−	0.881279i	$-0.656683\pi$
$683$	−24.7019	−0.945193	−0.472597	+	0.881279i	$0.656683\pi$
$684$	0	0
$685$	83.7075	3.19830
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.169240	0.00644752
$690$	0	0
$691$	18.3470	0.697951	0.348976	−	0.937132i	$-0.386530\pi$
$691$	18.3470	0.697951	0.348976	+	0.937132i	$0.386530\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.0947	1.82434
$696$	0	0
$697$	43.5264	1.64868
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.7563	−0.444028	−0.222014	−	0.975043i	$-0.571263\pi$
$701$	−11.7563	−0.444028	−0.222014	+	0.975043i	$0.571263\pi$
$702$	0	0
$703$	−40.5544	−1.52954
$704$	0	0
$705$	0	0
$706$	0	0
$707$	51.3664	1.93183
$708$	0	0
$709$	−19.7478	−0.741644	−0.370822	−	0.928704i	$-0.620924\pi$
$709$	−19.7478	−0.741644	−0.370822	+	0.928704i	$0.620924\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−35.5444	−1.33115
$714$	0	0
$715$	−3.58774	−0.134174
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.3594	−1.20680	−0.603402	−	0.797437i	$-0.706188\pi$
$719$	−32.3594	−1.20680	−0.603402	+	0.797437i	$0.706188\pi$
$720$	0	0
$721$	−48.3418	−1.80034
$722$	0	0
$723$	0	0
$724$	0	0
$725$	41.5962	1.54484
$726$	0	0
$727$	−6.10242	−0.226326	−0.113163	−	0.993576i	$-0.536098\pi$
$727$	−6.10242	−0.226326	−0.113163	+	0.993576i	$0.536098\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−47.7299	−1.76535
$732$	0	0
$733$	−38.4744	−1.42108	−0.710541	−	0.703656i	$-0.751550\pi$
$733$	−38.4744	−1.42108	−0.710541	+	0.703656i	$0.751550\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.9930	−0.478605
$738$	0	0
$739$	−34.3487	−1.26354	−0.631769	−	0.775157i	$-0.717671\pi$
$739$	−34.3487	−1.26354	−0.631769	+	0.775157i	$0.717671\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.7967	−1.34994	−0.674970	−	0.737846i	$-0.735843\pi$
$743$	−36.7967	−1.34994	−0.674970	+	0.737846i	$0.735843\pi$
$744$	0	0
$745$	−19.6351	−0.719375
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.59175	0.204318
$750$	0	0
$751$	−8.45891	−0.308670	−0.154335	−	0.988019i	$-0.549324\pi$
$751$	−8.45891	−0.308670	−0.154335	+	0.988019i	$0.549324\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.78562	−0.246954
$756$	0	0
$757$	−31.0257	−1.12765	−0.563824	−	0.825895i	$-0.690670\pi$
$757$	−31.0257	−1.12765	−0.563824	+	0.825895i	$0.690670\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.7959	1.04385	0.521925	−	0.852991i	$-0.325214\pi$
$761$	28.7959	1.04385	0.521925	+	0.852991i	$0.325214\pi$
$762$	0	0
$763$	−10.8953	−0.394438
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.21037	0.0437039
$768$	0	0
$769$	−25.7849	−0.929827	−0.464914	−	0.885356i	$-0.653915\pi$
$769$	−25.7849	−0.929827	−0.464914	+	0.885356i	$0.653915\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.3246	−0.659089	−0.329544	−	0.944140i	$-0.606895\pi$
$773$	−18.3246	−0.659089	−0.329544	+	0.944140i	$0.606895\pi$
$774$	0	0
$775$	−57.9208	−2.08058
$776$	0	0
$777$	0	0
$778$	0	0
$779$	37.3425	1.33793
$780$	0	0
$781$	9.24926	0.330965
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−36.4292	−1.30021
$786$	0	0
$787$	40.3921	1.43982	0.719911	−	0.694066i	$-0.244182\pi$
$787$	40.3921	1.43982	0.719911	+	0.694066i	$0.244182\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−50.3954	−1.79185
$792$	0	0
$793$	10.1212	0.359413
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.5209	0.443514	0.221757	−	0.975102i	$-0.428821\pi$
$797$	12.5209	0.443514	0.221757	+	0.975102i	$0.428821\pi$
$798$	0	0
$799$	32.2423	1.14065
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.49228	−0.193818
$804$	0	0
$805$	82.4457	2.90583
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−47.4736	−1.66908	−0.834542	−	0.550944i	$-0.814268\pi$
$809$	−47.4736	−1.66908	−0.834542	+	0.550944i	$0.814268\pi$
$810$	0	0
$811$	−42.6080	−1.49617	−0.748084	−	0.663604i	$-0.769026\pi$
$811$	−42.6080	−1.49617	−0.748084	+	0.663604i	$0.769026\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	79.7548	2.79369
$816$	0	0
$817$	−40.9487	−1.43261
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.4487	−1.23717	−0.618583	−	0.785719i	$-0.712293\pi$
$821$	−35.4487	−1.23717	−0.618583	+	0.785719i	$0.712293\pi$
$822$	0	0
$823$	14.8991	0.519350	0.259675	−	0.965696i	$-0.416385\pi$
$823$	14.8991	0.519350	0.259675	+	0.965696i	$0.416385\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.53413	0.0881202	0.0440601	−	0.999029i	$-0.485971\pi$
$827$	2.53413	0.0881202	0.0440601	+	0.999029i	$0.485971\pi$
$828$	0	0
$829$	32.9644	1.14490	0.572450	−	0.819939i	$-0.305993\pi$
$829$	32.9644	1.14490	0.572450	+	0.819939i	$0.305993\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	97.3650	3.37350
$834$	0	0
$835$	60.9193	2.10820
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34.3425	1.18563	0.592817	−	0.805337i	$-0.298016\pi$
$839$	34.3425	1.18563	0.592817	+	0.805337i	$0.298016\pi$
$840$	0	0
$841$	−1.07779	−0.0371651
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.58774	−0.123422
$846$	0	0
$847$	−4.75698	−0.163452
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−36.6546	−1.25650
$852$	0	0
$853$	25.8044	0.883524	0.441762	−	0.897132i	$-0.354354\pi$
$853$	25.8044	0.883524	0.441762	+	0.897132i	$0.354354\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.1212	−1.37051	−0.685256	−	0.728302i	$-0.740310\pi$
$857$	−40.1212	−1.37051	−0.685256	+	0.728302i	$0.740310\pi$
$858$	0	0
$859$	−14.6204	−0.498841	−0.249421	−	0.968395i	$-0.580240\pi$
$859$	−14.6204	−0.498841	−0.249421	+	0.968395i	$0.580240\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.2577	−0.995945	−0.497973	−	0.867193i	$-0.665922\pi$
$863$	−29.2577	−0.995945	−0.497973	+	0.867193i	$0.665922\pi$
$864$	0	0
$865$	27.7827	0.944639
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.74378	0.262690
$870$	0	0
$871$	−12.9930	−0.440252
$872$	0	0
$873$	0	0
$874$	0	0
$875$	49.0140	1.65698
$876$	0	0
$877$	−22.4813	−0.759140	−0.379570	−	0.925163i	$-0.623928\pi$
$877$	−22.4813	−0.759140	−0.379570	+	0.925163i	$0.623928\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	57.9170	1.95127	0.975637	−	0.219391i	$-0.0704069\pi$
$881$	57.9170	1.95127	0.975637	+	0.219391i	$0.0704069\pi$
$882$	0	0
$883$	−56.7236	−1.90890	−0.954451	−	0.298368i	$-0.903558\pi$
$883$	−56.7236	−1.90890	−0.954451	+	0.298368i	$0.903558\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.8385	−1.53911	−0.769553	−	0.638582i	$-0.779521\pi$
$887$	−45.8385	−1.53911	−0.769553	+	0.638582i	$0.779521\pi$
$888$	0	0
$889$	8.81203	0.295546
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.6615	0.925657
$894$	0	0
$895$	−41.0444	−1.37196
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38.8804	−1.29673
$900$	0	0
$901$	−1.05433	−0.0351249
$902$	0	0
$903$	0	0
$904$	0	0
$905$	37.4791	1.24585
$906$	0	0
$907$	−29.4075	−0.976461	−0.488231	−	0.872715i	$-0.662357\pi$
$907$	−29.4075	−0.976461	−0.488231	+	0.872715i	$0.662357\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.4868	−1.54018	−0.770089	−	0.637937i	$-0.779788\pi$
$911$	−46.4868	−1.54018	−0.770089	+	0.637937i	$0.779788\pi$
$912$	0	0
$913$	3.47283	0.114934
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.7299	0.651537
$918$	0	0
$919$	−19.2841	−0.636125	−0.318063	−	0.948070i	$-0.603032\pi$
$919$	−19.2841	−0.636125	−0.318063	+	0.948070i	$0.603032\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.24926	0.304443
$924$	0	0
$925$	−59.7299	−1.96391
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.7672	−0.943822	−0.471911	−	0.881646i	$-0.656436\pi$
$929$	−28.7672	−0.943822	−0.471911	+	0.881646i	$0.656436\pi$
$930$	0	0
$931$	83.5320	2.73765
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.3510	0.730955
$936$	0	0
$937$	12.1645	0.397397	0.198699	−	0.980061i	$-0.436328\pi$
$937$	12.1645	0.397397	0.198699	+	0.980061i	$0.436328\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.75297	0.187542	0.0937708	−	0.995594i	$-0.470108\pi$
$941$	5.75297	0.187542	0.0937708	+	0.995594i	$0.470108\pi$
$942$	0	0
$943$	33.7515	1.09910
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.6157	0.864893	0.432446	−	0.901660i	$-0.357650\pi$
$947$	26.6157	0.864893	0.432446	+	0.901660i	$0.357650\pi$
$948$	0	0
$949$	−5.49228	−0.178287
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.0389	0.584337	0.292169	−	0.956367i	$-0.405623\pi$
$953$	18.0389	0.584337	0.292169	+	0.956367i	$0.405623\pi$
$954$	0	0
$955$	−18.6157	−0.602389
$956$	0	0
$957$	0	0
$958$	0	0
$959$	110.988	3.58398
$960$	0	0
$961$	23.1391	0.746422
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.676959	0.0217921
$966$	0	0
$967$	−55.5048	−1.78491	−0.892456	−	0.451133i	$-0.851020\pi$
$967$	−55.5048	−1.78491	−0.892456	+	0.451133i	$0.851020\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.3782	−1.19952	−0.599761	−	0.800179i	$-0.704738\pi$
$971$	−37.3782	−1.19952	−0.599761	+	0.800179i	$0.704738\pi$
$972$	0	0
$973$	63.7688	2.04433
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.1421	1.02832	0.514159	−	0.857695i	$-0.328104\pi$
$977$	32.1421	1.02832	0.514159	+	0.857695i	$0.328104\pi$
$978$	0	0
$979$	−4.04737	−0.129355
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33.5349	1.06960	0.534799	−	0.844979i	$-0.320387\pi$
$983$	33.5349	1.06960	0.534799	+	0.844979i	$0.320387\pi$
$984$	0	0
$985$	−22.6630	−0.722104
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−37.0110	−1.17688
$990$	0	0
$991$	−35.3697	−1.12356	−0.561778	−	0.827288i	$-0.689882\pi$
$991$	−35.3697	−1.12356	−0.561778	+	0.827288i	$0.689882\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−48.2673	−1.53018
$996$	0	0
$997$	−6.58078	−0.208415	−0.104208	−	0.994556i	$-0.533231\pi$
$997$	−6.58078	−0.208415	−0.104208	+	0.994556i	$0.533231\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.o.1.1		3
3.2	odd	2		572.2.a.e.1.1	✓	3
12.11	even	2		2288.2.a.w.1.3		3
24.5	odd	2		9152.2.a.cc.1.3		3
24.11	even	2		9152.2.a.bv.1.1		3
33.32	even	2		6292.2.a.p.1.1		3
39.38	odd	2		7436.2.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.e.1.1	✓	3	3.2	odd	2
2288.2.a.w.1.3		3	12.11	even	2
5148.2.a.o.1.1		3	1.1	even	1	trivial
6292.2.a.p.1.1		3	33.32	even	2
7436.2.a.l.1.1		3	39.38	odd	2
9152.2.a.bv.1.1		3	24.11	even	2
9152.2.a.cc.1.3		3	24.5	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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-3.58774 q^{5} -4.75698 q^{7} +O(q^{10})\)	\(q-3.58774 q^{5} -4.75698 q^{7} -1.00000 q^{11} -1.00000 q^{13} +6.22982 q^{17} +5.34472 q^{19} +4.83076 q^{23} +7.87189 q^{25} +5.28415 q^{29} -7.35793 q^{31} +17.0668 q^{35} -7.58774 q^{37} +6.98680 q^{41} -7.66152 q^{43} +5.17548 q^{47} +15.6289 q^{49} -0.169240 q^{53} +3.58774 q^{55} -1.21037 q^{59} -10.1212 q^{61} +3.58774 q^{65} +12.9930 q^{67} -9.24926 q^{71} +5.49228 q^{73} +4.75698 q^{77} -7.74378 q^{79} -3.47283 q^{83} -22.3510 q^{85} +4.04737 q^{89} +4.75698 q^{91} -19.1755 q^{95} +3.58774 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5} - 7 q^{7}+O(q^{10}) \) 3 * q + q^5 - 7 * q^7	\( 3 q + q^{5} - 7 q^{7} - 3 q^{11} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 10 q^{23} + 10 q^{25} + 14 q^{29} - 23 q^{31} + 6 q^{35} - 11 q^{37} + q^{41} - 14 q^{43} - 8 q^{47} + 26 q^{49} - 5 q^{53} - q^{55} + q^{59} + 4 q^{61} - q^{65} - 9 q^{67} - 7 q^{71} + 3 q^{73} + 7 q^{77} + 4 q^{79} - 5 q^{83} - 20 q^{85} - 25 q^{89} + 7 q^{91} - 34 q^{95} - q^{97}+O(q^{100}) \) 3 * q + q^5 - 7 * q^7 - 3 * q^11 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 10 * q^23 + 10 * q^25 + 14 * q^29 - 23 * q^31 + 6 * q^35 - 11 * q^37 + q^41 - 14 * q^43 - 8 * q^47 + 26 * q^49 - 5 * q^53 - q^55 + q^59 + 4 * q^61 - q^65 - 9 * q^67 - 7 * q^71 + 3 * q^73 + 7 * q^77 + 4 * q^79 - 5 * q^83 - 20 * q^85 - 25 * q^89 + 7 * q^91 - 34 * q^95 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more