LMFDB - Embedded newform 7245.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7245,2,Mod(1,7245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7245, 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("7245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7245.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:	$57.8516162644$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2415)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} -1.00000 q^{10} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} +2.00000 q^{29} -4.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} -6.00000 q^{37} -4.00000 q^{38} +3.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{46} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +3.00000 q^{56} +2.00000 q^{58} +8.00000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} -2.00000 q^{65} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +4.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{89} -2.00000 q^{91} -1.00000 q^{92} +4.00000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	3.00000	0.474342
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	2.00000	0.262613
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	1.00000	0.119523
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−1.00000	−0.104257
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−2.00000	−0.185695
$117$	0	0
$118$	8.00000	0.736460
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−2.00000	−0.181071
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	4.00000	0.345547
$135$	0	0
$136$	−18.0000	−1.54349
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−8.00000	−0.671345
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	−2.00000	−0.165521
$147$	0	0
$148$	6.00000	0.493197
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−5.00000	−0.395285
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	4.00000	0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	−3.00000	−0.221163
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	4.00000	0.290191
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	18.0000	1.26648
$203$	−2.00000	−0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	4.00000	0.273434
$215$	−4.00000	−0.272798
$216$	0	0
$217$	4.00000	0.271538
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	−6.00000	−0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−8.00000	−0.509028
$248$	12.0000	0.762001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	2.00000	0.124035
$261$	0	0
$262$	−16.0000	−0.988483
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	4.00000	0.245256
$267$	0	0
$268$	−4.00000	−0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	−3.00000	−0.179284
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	19.0000	1.11765
$290$	−2.00000	−0.117444
$291$	0	0
$292$	2.00000	0.117041
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	18.0000	1.04623
$297$	0	0
$298$	18.0000	1.04271
$299$	2.00000	0.115663
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	0	0
$320$	−7.00000	−0.391312
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	−24.0000	−1.33540
$324$	0	0
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	−8.00000	−0.437741
$335$	−4.00000	−0.218543
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	6.00000	0.325396
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−18.0000	−0.967686
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	0	0
$364$	2.00000	0.104828
$365$	2.00000	0.104685
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	6.00000	0.311925
$371$	2.00000	0.103835
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	4.00000	0.204658
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	−6.00000	−0.305392
$387$	0	0
$388$	14.0000	0.710742
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−14.0000	−0.705310
$395$	−4.00000	−0.201262
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−18.0000	−0.895533
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	16.0000	0.788263
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−4.00000	−0.196352
$416$	10.0000	0.490290
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	0	0
$427$	2.00000	0.0967868
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−4.00000	−0.192897
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−6.00000	−0.287348
$437$	−4.00000	−0.191346
$438$	0	0
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	16.0000	0.757622
$447$	0	0
$448$	−7.00000	−0.330719
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	−20.0000	−0.938647
$455$	2.00000	0.0937614
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	1.00000	0.0466252
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−2.00000	−0.0926482
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	−24.0000	−1.10469
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	6.00000	0.275010
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−30.0000	−1.36646
$483$	0	0
$484$	11.0000	0.500000
$485$	14.0000	0.635707
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	−8.00000	−0.359937
$495$	0	0
$496$	4.00000	0.179605
$497$	8.00000	0.358849
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	0	0
$508$	16.0000	0.709885
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−6.00000	−0.264649
$515$	16.0000	0.705044
$516$	0	0
$517$	0	0
$518$	6.00000	0.263625
$519$	0	0
$520$	6.00000	0.263117
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.00000	0.0868744
$531$	0	0
$532$	−4.00000	−0.173422
$533$	−4.00000	−0.173259
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−12.0000	−0.518321
$537$	0	0
$538$	10.0000	0.431131
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	30.0000	1.28624
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	−4.00000	−0.170097
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−2.00000	−0.0843649
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−12.0000	−0.504398
$567$	0	0
$568$	24.0000	1.00702
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	2.00000	0.0834784
$575$	1.00000	0.0417029
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	2.00000	0.0830455
$581$	−4.00000	−0.165948
$582$	0	0
$583$	0	0
$584$	6.00000	0.248282
$585$	0	0
$586$	10.0000	0.413096
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−8.00000	−0.329355
$591$	0	0
$592$	6.00000	0.246598
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	−18.0000	−0.737309
$597$	0	0
$598$	2.00000	0.0817861
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−6.00000	−0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	10.0000	0.399043
$629$	−36.0000	−1.43541
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	10.0000	0.397151
$635$	16.0000	0.634941
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	3.00000	0.118585
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	12.0000	0.469956
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	2.00000	0.0780869
$657$	0	0
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−4.00000	−0.155113
$666$	0	0
$667$	2.00000	0.0774403
$668$	8.00000	0.309529
$669$	0	0
$670$	−4.00000	−0.154533
$671$	0	0
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	9.00000	0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	18.0000	0.690268
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−18.0000	−0.681310
$699$	0	0
$700$	1.00000	0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	0	0
$706$	−14.0000	−0.526897
$707$	−18.0000	−0.676960
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	−18.0000	−0.674579
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	−28.0000	−1.04495
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	−3.00000	−0.111648
$723$	0	0
$724$	10.0000	0.371647
$725$	2.00000	0.0742781
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	2.00000	0.0740233
$731$	24.0000	0.887672
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	5.00000	0.184302
$737$	0	0
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	2.00000	0.0734223
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	26.0000	0.951928
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	0	0
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	−12.0000	−0.435286
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	−4.00000	−0.144715
$765$	0	0
$766$	24.0000	0.867155
$767$	16.0000	0.577727
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	6.00000	0.215945
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	42.0000	1.50771
$777$	0	0
$778$	18.0000	0.645331
$779$	8.00000	0.286630
$780$	0	0
$781$	0	0
$782$	6.00000	0.214560
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	10.0000	0.356915
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	14.0000	0.498729
$789$	0	0
$790$	−4.00000	−0.142314
$791$	6.00000	0.213335
$792$	0	0
$793$	−4.00000	−0.142044
$794$	26.0000	0.922705
$795$	0	0
$796$	16.0000	0.567105
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	0	0
$802$	30.0000	1.05934
$803$	0	0
$804$	0	0
$805$	1.00000	0.0352454
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−54.0000	−1.89971
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	−16.0000	−0.559769
$818$	10.0000	0.349642
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	48.0000	1.67216
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	14.0000	0.485363
$833$	6.00000	0.207888
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	−20.0000	−0.690889
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	0	0
$844$	20.0000	0.688428
$845$	9.00000	0.309609
$846$	0	0
$847$	11.0000	0.377964
$848$	2.00000	0.0686803
$849$	0	0
$850$	6.00000	0.205798
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	−20.0000	−0.681203
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	18.0000	0.611665
$867$	0	0
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	−18.0000	−0.609557
$873$	0	0
$874$	−4.00000	−0.135302
$875$	1.00000	0.0338062
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	12.0000	0.404980
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−20.0000	−0.671913
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	0	0
$895$	4.00000	0.133705
$896$	3.00000	0.100223
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	10.0000	0.332411
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	20.0000	0.663723
$909$	0	0
$910$	2.00000	0.0662994
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	−6.00000	−0.198246
$917$	16.0000	0.528367
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	42.0000	1.38320
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−6.00000	−0.197279
$926$	24.0000	0.788689
$927$	0	0
$928$	10.0000	0.328266
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	2.00000	0.0655122
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	−4.00000	−0.129777
$951$	0	0
$952$	18.0000	0.583383
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	0	0
$957$	0	0
$958$	24.0000	0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	0	0
$964$	30.0000	0.966235
$965$	6.00000	0.193147
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	33.0000	1.06066
$969$	0	0
$970$	14.0000	0.449513
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	1.00000	0.0319438
$981$	0	0
$982$	−28.0000	−0.893516
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	14.0000	0.446077
$986$	12.0000	0.382158
$987$	0	0
$988$	8.00000	0.254514
$989$	4.00000	0.127193
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	−20.0000	−0.635001
$993$	0	0
$994$	8.00000	0.253745
$995$	16.0000	0.507234
$996$	0	0
$997$	−54.0000	−1.71020	−0.855099	−	0.518465i	$-0.826503\pi$
$997$	−54.0000	−1.71020	−0.855099	+	0.518465i	$0.826503\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7245.2.a.l.1.1		1
3.2	odd	2		2415.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.c.1.1	✓	1	3.2	odd	2
7245.2.a.l.1.1		1	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:	\( N \)	\(=\)	\( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7245.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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