LMFDB - Embedded newform 7105.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-2,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1015)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{12} +2.00000 q^{13} -2.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} +8.00000 q^{23} -6.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{29} +2.00000 q^{30} -2.00000 q^{31} -5.00000 q^{32} -1.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} -4.00000 q^{39} +3.00000 q^{40} -8.00000 q^{41} +8.00000 q^{43} +1.00000 q^{45} -8.00000 q^{46} -6.00000 q^{47} +2.00000 q^{48} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -4.00000 q^{54} +12.0000 q^{57} +1.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +4.00000 q^{61} +2.00000 q^{62} +7.00000 q^{64} +2.00000 q^{65} -12.0000 q^{67} -16.0000 q^{69} +8.00000 q^{71} +3.00000 q^{72} +2.00000 q^{74} -2.00000 q^{75} +6.00000 q^{76} +4.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +8.00000 q^{82} -8.00000 q^{86} +2.00000 q^{87} +12.0000 q^{89} -1.00000 q^{90} -8.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} -6.00000 q^{95} +10.0000 q^{96} -8.00000 q^{97} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.00000	0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−6.00000	−1.22474
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	4.00000	0.769800
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	2.00000	0.365148
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.00000	0.973329
$39$	−4.00000	−0.640513
$40$	3.00000	0.474342
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−8.00000	−1.17954
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	2.00000	0.288675
$49$	0	0
$50$	−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	$0.543862\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	0	0
$57$	12.0000	1.58944
$58$	1.00000	0.131306
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	3.00000	0.353553
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	2.00000	0.232495
$75$	−2.00000	−0.230940
$76$	6.00000	0.688247
$77$	0	0
$78$	4.00000	0.452911
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	8.00000	0.883452
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	2.00000	0.214423
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−8.00000	−0.834058
$93$	4.00000	0.414781
$94$	6.00000	0.618853
$95$	−6.00000	−0.615587
$96$	10.0000	1.02062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	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	$0.438066\pi$
$108$	−4.00000	−0.384900
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−12.0000	−1.12390
$115$	8.00000	0.746004
$116$	1.00000	0.0928477
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	−6.00000	−0.547723
$121$	−11.0000	−1.00000
$122$	−4.00000	−0.362143
$123$	16.0000	1.44267
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	3.00000	0.265165
$129$	−16.0000	−1.40872
$130$	−2.00000	−0.175412
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	4.00000	0.344265
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	16.0000	1.36201
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	−8.00000	−0.671345
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	2.00000	0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	2.00000	0.163299
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	−18.0000	−1.45999
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	4.00000	0.320256
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−16.0000	−1.27289
$159$	4.00000	0.317221
$160$	−5.00000	−0.395285
$161$	0	0
$162$	11.0000	0.864242
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	0	0
$167$	−4.00000	−0.309529	−0.154765	−	0.987951i	$-0.549462\pi$
$167$	−4.00000	−0.309529	−0.154765	+	0.987951i	$0.549462\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	−8.00000	−0.609994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	0	0
$177$	24.0000	1.80395
$178$	−12.0000	−0.899438
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−1.00000	−0.0745356
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	24.0000	1.76930
$185$	−2.00000	−0.147043
$186$	−4.00000	−0.293294
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	6.00000	0.435286
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−14.0000	−1.01036
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	8.00000	0.574367
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	3.00000	0.212132
$201$	24.0000	1.69283
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	−8.00000	−0.558744
$206$	16.0000	1.11477
$207$	8.00000	0.556038
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	2.00000	0.137361
$213$	−16.0000	−1.09630
$214$	−4.00000	−0.273434
$215$	8.00000	0.545595
$216$	12.0000	0.816497
$217$	0	0
$218$	14.0000	0.948200
$219$	0	0
$220$	0	0
$221$	0	0
$222$	−4.00000	−0.268462
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−12.0000	−0.794719
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	−2.00000	−0.130744
$235$	−6.00000	−0.391397
$236$	12.0000	0.781133
$237$	−32.0000	−2.07862
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	11.0000	0.707107
$243$	10.0000	0.641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	−16.0000	−1.02012
$247$	−12.0000	−0.763542
$248$	−6.00000	−0.381000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	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	$0.559920\pi$
$258$	16.0000	0.996116
$259$	0	0
$260$	−2.00000	−0.124035
$261$	−1.00000	−0.0618984
$262$	6.00000	0.370681
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−24.0000	−1.46878
$268$	12.0000	0.733017
$269$	32.0000	1.95107	0.975537	−	0.219834i	$-0.0705517\pi$
$269$	32.0000	1.95107	0.975537	+	0.219834i	$0.0705517\pi$
$270$	−4.00000	−0.243432
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	16.0000	0.963087
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−4.00000	−0.239904
$279$	−2.00000	−0.119737
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−12.0000	−0.714590
$283$	−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
$285$	12.0000	0.710819
$286$	0	0
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−17.0000	−1.00000
$290$	1.00000	0.0587220
$291$	16.0000	0.937937
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−18.0000	−1.04271
$299$	16.0000	0.925304
$300$	2.00000	0.115470
$301$	0	0
$302$	24.0000	1.38104
$303$	−24.0000	−1.37876
$304$	6.00000	0.344124
$305$	4.00000	0.229039
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	32.0000	1.82042
$310$	2.00000	0.113592
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	−12.0000	−0.679366
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	−16.0000	−0.900070
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	−4.00000	−0.224309
$319$	0	0
$320$	7.00000	0.391312
$321$	−8.00000	−0.446516
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	28.0000	1.54840
$328$	−24.0000	−1.32518
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	4.00000	0.218870
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	6.00000	0.324443
$343$	0	0
$344$	24.0000	1.29399
$345$	−16.0000	−0.861411
$346$	−2.00000	−0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−2.00000	−0.107211
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−24.0000	−1.27559
$355$	8.00000	0.424596
$356$	−12.0000	−0.635999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	3.00000	0.158114
$361$	17.0000	0.894737
$362$	−18.0000	−0.946059
$363$	22.0000	1.15470
$364$	0	0
$365$	0	0
$366$	8.00000	0.418167
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	−8.00000	−0.417029
$369$	−8.00000	−0.416463
$370$	2.00000	0.103975
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	−18.0000	−0.928279
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	6.00000	0.307794
$381$	8.00000	0.409852
$382$	8.00000	0.409316
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	−14.0000	−0.712581
$387$	8.00000	0.406663
$388$	8.00000	0.406138
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	22.0000	1.10834
$395$	16.0000	0.805047
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−24.0000	−1.19701
$403$	−4.00000	−0.199254
$404$	−12.0000	−0.597022
$405$	−11.0000	−0.546594
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	8.00000	0.395092
$411$	−36.0000	−1.77575
$412$	16.0000	0.788263
$413$	0	0
$414$	−8.00000	−0.393179
$415$	0	0
$416$	−10.0000	−0.490290
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−8.00000	−0.389434
$423$	−6.00000	−0.291730
$424$	−6.00000	−0.291386
$425$	0	0
$426$	16.0000	0.775203
$427$	0	0
$428$	−4.00000	−0.193347
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−4.00000	−0.192450
$433$	36.0000	1.73005	0.865025	−	0.501729i	$-0.167303\pi$
$433$	36.0000	1.73005	0.865025	+	0.501729i	$0.167303\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	14.0000	0.670478
$437$	−48.0000	−2.29615
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−4.00000	−0.189832
$445$	12.0000	0.568855
$446$	−4.00000	−0.189405
$447$	−36.0000	−1.70274
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	48.0000	2.25524
$454$	−12.0000	−0.563188
$455$	0	0
$456$	36.0000	1.68585
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	−8.00000	−0.373002
$461$	−16.0000	−0.745194	−0.372597	−	0.927993i	$-0.621533\pi$
$461$	−16.0000	−0.745194	−0.372597	+	0.927993i	$0.621533\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	1.00000	0.0464238
$465$	4.00000	0.185496
$466$	22.0000	1.01913
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	6.00000	0.276759
$471$	−16.0000	−0.737241
$472$	−36.0000	−1.65703
$473$	0	0
$474$	32.0000	1.46981
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−24.0000	−1.09773
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	10.0000	0.456435
$481$	−4.00000	−0.182384
$482$	−18.0000	−0.819878
$483$	0	0
$484$	11.0000	0.500000
$485$	−8.00000	−0.363261
$486$	−10.0000	−0.453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	12.0000	0.543214
$489$	−8.00000	−0.361773
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−16.0000	−0.721336
$493$	0	0
$494$	12.0000	0.539906
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	18.0000	0.803379
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	18.0000	0.799408
$508$	4.00000	0.177471
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	−24.0000	−1.05963
$514$	6.00000	0.264649
$515$	−16.0000	−0.705044
$516$	16.0000	0.704361
$517$	0	0
$518$	0	0
$519$	−4.00000	−0.175581
$520$	6.00000	0.263117
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	1.00000	0.0437688
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	4.00000	0.174408
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−16.0000	−0.693037
$534$	24.0000	1.03858
$535$	4.00000	0.172935
$536$	−36.0000	−1.55496
$537$	−24.0000	−1.03568
$538$	−32.0000	−1.37962
$539$	0	0
$540$	−4.00000	−0.172133
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	2.00000	0.0859074
$543$	−36.0000	−1.54491
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−18.0000	−0.768922
$549$	4.00000	0.170716
$550$	0	0
$551$	6.00000	0.255609
$552$	−48.0000	−2.04302
$553$	0	0
$554$	22.0000	0.934690
$555$	4.00000	0.169791
$556$	−4.00000	−0.169638
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	2.00000	0.0846668
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	−12.0000	−0.505291
$565$	6.00000	0.252422
$566$	12.0000	0.504398
$567$	0	0
$568$	24.0000	1.00702
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	−12.0000	−0.502625
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	8.00000	0.333623
$576$	7.00000	0.291667
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	17.0000	0.707107
$579$	−28.0000	−1.16364
$580$	1.00000	0.0415227
$581$	0	0
$582$	−16.0000	−0.663221
$583$	0	0
$584$	0	0
$585$	2.00000	0.0826898
$586$	−12.0000	−0.495715
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	12.0000	0.494032
$591$	44.0000	1.80992
$592$	2.00000	0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	−48.0000	−1.96451
$598$	−16.0000	−0.654289
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	−6.00000	−0.244949
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	24.0000	0.976546
$605$	−11.0000	−0.447214
$606$	24.0000	0.974933
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	30.0000	1.21666
$609$	0	0
$610$	−4.00000	−0.161955
$611$	−12.0000	−0.485468
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	14.0000	0.564994
$615$	16.0000	0.645182
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−32.0000	−1.28723
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	2.00000	0.0803219
$621$	32.0000	1.28412
$622$	10.0000	0.400963
$623$	0	0
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	0	0
$628$	−8.00000	−0.319235
$629$	0	0
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	48.0000	1.90934
$633$	−16.0000	−0.635943
$634$	−26.0000	−1.03259
$635$	−4.00000	−0.158735
$636$	−4.00000	−0.158610
$637$	0	0
$638$	0	0
$639$	8.00000	0.316475
$640$	3.00000	0.118585
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	8.00000	0.315735
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−33.0000	−1.29636
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−28.0000	−1.09489
$655$	−6.00000	−0.234439
$656$	8.00000	0.312348
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	0	0
$665$	0	0
$666$	2.00000	0.0774984
$667$	−8.00000	−0.309761
$668$	4.00000	0.154765
$669$	−8.00000	−0.309298
$670$	12.0000	0.463600
$671$	0	0
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	14.0000	0.539260
$675$	4.00000	0.153960
$676$	9.00000	0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	12.0000	0.460857
$679$	0	0
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	6.00000	0.229416
$685$	18.0000	0.687745
$686$	0	0
$687$	−40.0000	−1.52610
$688$	−8.00000	−0.304997
$689$	−4.00000	−0.152388
$690$	16.0000	0.609110
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	12.0000	0.455514
$695$	4.00000	0.151729
$696$	6.00000	0.227429
$697$	0	0
$698$	−6.00000	−0.227103
$699$	44.0000	1.66423
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	−8.00000	−0.301941
$703$	12.0000	0.452589
$704$	0	0
$705$	12.0000	0.451946
$706$	10.0000	0.376355
$707$	0	0
$708$	−24.0000	−0.901975
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−8.00000	−0.300235
$711$	16.0000	0.600047
$712$	36.0000	1.34916
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−48.0000	−1.79259
$718$	24.0000	0.895672
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−17.0000	−0.632674
$723$	−36.0000	−1.33885
$724$	−18.0000	−0.668965
$725$	−1.00000	−0.0371391
$726$	−22.0000	−0.816497
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	8.00000	0.295689
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	30.0000	1.10732
$735$	0	0
$736$	−40.0000	−1.47442
$737$	0	0
$738$	8.00000	0.294484
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	2.00000	0.0735215
$741$	24.0000	0.881662
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	12.0000	0.439941
$745$	18.0000	0.659469
$746$	6.00000	0.219676
$747$	0	0
$748$	0	0
$749$	0	0
$750$	2.00000	0.0730297
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	6.00000	0.218797
$753$	36.0000	1.31191
$754$	2.00000	0.0728357
$755$	−24.0000	−0.873449
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	4.00000	0.144526
$767$	−24.0000	−0.866590
$768$	34.0000	1.22687
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	−14.0000	−0.503871
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	−8.00000	−0.287554
$775$	−2.00000	−0.0718421
$776$	−24.0000	−0.861550
$777$	0	0
$778$	30.0000	1.07555
$779$	48.0000	1.71978
$780$	4.00000	0.143223
$781$	0	0
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	8.00000	0.285532
$786$	−12.0000	−0.428026
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	22.0000	0.783718
$789$	8.00000	0.284808
$790$	−16.0000	−0.569254
$791$	0	0
$792$	0	0
$793$	8.00000	0.284088
$794$	14.0000	0.496841
$795$	4.00000	0.141865
$796$	−24.0000	−0.850657
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	12.0000	0.423999
$802$	18.0000	0.635602
$803$	0	0
$804$	−24.0000	−0.846415
$805$	0	0
$806$	4.00000	0.140894
$807$	−64.0000	−2.25291
$808$	36.0000	1.26648
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	11.0000	0.386501
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−48.0000	−1.67931
$818$	4.00000	0.139857
$819$	0	0
$820$	8.00000	0.279372
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	36.0000	1.25564
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−48.0000	−1.67216
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	−8.00000	−0.278019
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	14.0000	0.485363
$833$	0	0
$834$	8.00000	0.277017
$835$	−4.00000	−0.138426
$836$	0	0
$837$	−8.00000	−0.276520
$838$	28.0000	0.967244
$839$	50.0000	1.72619	0.863096	−	0.505040i	$-0.168522\pi$
$839$	50.0000	1.72619	0.863096	+	0.505040i	$0.168522\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−22.0000	−0.758170
$843$	−12.0000	−0.413302
$844$	−8.00000	−0.275371
$845$	−9.00000	−0.309609
$846$	6.00000	0.206284
$847$	0	0
$848$	2.00000	0.0686803
$849$	24.0000	0.823678
$850$	0	0
$851$	−16.0000	−0.548473
$852$	16.0000	0.548151
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	12.0000	0.410152
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	−38.0000	−1.29654	−0.648272	−	0.761409i	$-0.724508\pi$
$859$	−38.0000	−1.29654	−0.648272	+	0.761409i	$0.724508\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	8.00000	0.272481
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	−20.0000	−0.680414
$865$	2.00000	0.0680020
$866$	−36.0000	−1.22333
$867$	34.0000	1.15470
$868$	0	0
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	−24.0000	−0.813209
$872$	−42.0000	−1.42230
$873$	−8.00000	−0.270759
$874$	48.0000	1.62362
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	−24.0000	−0.806296
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	12.0000	0.402694
$889$	0	0
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−4.00000	−0.133930
$893$	36.0000	1.20469
$894$	36.0000	1.20402
$895$	12.0000	0.401116
$896$	0	0
$897$	−32.0000	−1.06845
$898$	6.00000	0.200223
$899$	2.00000	0.0667037
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	18.0000	0.598340
$906$	−48.0000	−1.59469
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	−12.0000	−0.398234
$909$	12.0000	0.398015
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−12.0000	−0.397360
$913$	0	0
$914$	6.00000	0.198462
$915$	−8.00000	−0.264472
$916$	−20.0000	−0.660819
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	24.0000	0.791257
$921$	28.0000	0.922631
$922$	16.0000	0.526932
$923$	16.0000	0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	40.0000	1.31448
$927$	−16.0000	−0.525509
$928$	5.00000	0.164133
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	−4.00000	−0.131165
$931$	0	0
$932$	22.0000	0.720634
$933$	20.0000	0.654771
$934$	−6.00000	−0.196326
$935$	0	0
$936$	6.00000	0.196116
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	6.00000	0.195698
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	16.0000	0.521308
$943$	−64.0000	−2.08413
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	32.0000	1.03931
$949$	0	0
$950$	6.00000	0.194666
$951$	−52.0000	−1.68622
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	2.00000	0.0647524
$955$	−8.00000	−0.258874
$956$	−24.0000	−0.776215
$957$	0	0
$958$	10.0000	0.323085
$959$	0	0
$960$	−14.0000	−0.451848
$961$	−27.0000	−0.870968
$962$	4.00000	0.128965
$963$	4.00000	0.128898
$964$	−18.0000	−0.579741
$965$	14.0000	0.450676
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	8.00000	0.256865
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	−10.0000	−0.320750
$973$	0	0
$974$	−24.0000	−0.769010
$975$	−4.00000	−0.128103
$976$	−4.00000	−0.128037
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	8.00000	0.255812
$979$	0	0
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	38.0000	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$983$	38.0000	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$984$	48.0000	1.53018
$985$	−22.0000	−0.700978
$986$	0	0
$987$	0	0
$988$	12.0000	0.381771
$989$	64.0000	2.03508
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	10.0000	0.317500
$993$	24.0000	0.761617
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	28.0000	0.886325
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.a.1.1		1
7.6	odd	2		1015.2.a.a.1.1	✓	1
21.20	even	2		9135.2.a.m.1.1		1
35.34	odd	2		5075.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.a.1.1	✓	1	7.6	odd	2
5075.2.a.h.1.1		1	35.34	odd	2
7105.2.a.a.1.1		1	1.1	even	1	trivial
9135.2.a.m.1.1		1	21.20	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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