LMFDB - Embedded newform 175.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$175 = 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	175.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:	$1.39738203537$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	175.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} +7.00000 q^{17} -4.00000 q^{18} -1.00000 q^{21} -6.00000 q^{22} +6.00000 q^{23} +2.00000 q^{26} -5.00000 q^{27} -2.00000 q^{28} -5.00000 q^{29} +2.00000 q^{31} -8.00000 q^{32} -3.00000 q^{33} +14.0000 q^{34} -4.00000 q^{36} +2.00000 q^{37} +1.00000 q^{39} +2.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -6.00000 q^{44} +12.0000 q^{46} -3.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} +7.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -10.0000 q^{54} -10.0000 q^{58} +10.0000 q^{59} -8.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} -8.00000 q^{64} -6.00000 q^{66} +2.00000 q^{67} +14.0000 q^{68} +6.00000 q^{69} -8.00000 q^{71} +6.00000 q^{73} +4.00000 q^{74} +3.00000 q^{77} +2.00000 q^{78} -5.00000 q^{79} +1.00000 q^{81} +4.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} -5.00000 q^{87} -1.00000 q^{91} +12.0000 q^{92} +2.00000 q^{93} -6.00000 q^{94} -8.00000 q^{96} +7.00000 q^{97} +2.00000 q^{98} +6.00000 q^{99} +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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−4.00000	−0.942809
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−6.00000	−1.27920
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	−5.00000	−0.962250
$28$	−2.00000	−0.377964
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−8.00000	−1.41421
$33$	−3.00000	−0.522233
$34$	14.0000	2.40098
$35$	0	0
$36$	−4.00000	−0.666667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	12.0000	1.76930
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−4.00000	−0.577350
$49$	1.00000	0.142857
$50$	0	0
$51$	7.00000	0.980196
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−10.0000	−1.36083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−10.0000	−1.31306
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.00000	0.508001
$63$	2.00000	0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	14.0000	1.69775
$69$	6.00000	0.722315
$70$	0	0
$71$	−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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	0	0
$77$	3.00000	0.341882
$78$	2.00000	0.226455
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−8.00000	−0.862662
$87$	−5.00000	−0.536056
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	12.0000	1.25109
$93$	2.00000	0.207390
$94$	−6.00000	−0.618853
$95$	0	0
$96$	−8.00000	−0.816497
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	2.00000	0.202031
$99$	6.00000	0.603023
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	14.0000	1.38621
$103$	−19.0000	−1.87213	−0.936063	−	0.351833i	$-0.885559\pi$
$103$	−19.0000	−1.87213	−0.936063	+	0.351833i	$0.885559\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−10.0000	−0.962250
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−10.0000	−0.928477
$117$	−2.00000	−0.184900
$118$	20.0000	1.84115
$119$	−7.00000	−0.641689
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−16.0000	−1.44857
$123$	2.00000	0.180334
$124$	4.00000	0.359211
$125$	0	0
$126$	4.00000	0.356348
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	22.0000	1.92215	0.961074	−	0.276289i	$-0.0891049\pi$
$131$	22.0000	1.92215	0.961074	+	0.276289i	$0.0891049\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	12.0000	1.02151
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	−16.0000	−1.34269
$143$	−3.00000	−0.250873
$144$	8.00000	0.666667
$145$	0	0
$146$	12.0000	0.993127
$147$	1.00000	0.0824786
$148$	4.00000	0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	−14.0000	−1.13183
$154$	6.00000	0.483494
$155$	0	0
$156$	2.00000	0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−10.0000	−0.795557
$159$	6.00000	0.475831
$160$	0	0
$161$	−6.00000	−0.472866
$162$	2.00000	0.157135
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−8.00000	−0.620920
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	12.0000	0.904534
$177$	10.0000	0.751646
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	−2.00000	−0.148250
$183$	−8.00000	−0.591377
$184$	0	0
$185$	0	0
$186$	4.00000	0.293294
$187$	−21.0000	−1.53567
$188$	−6.00000	−0.437595
$189$	5.00000	0.363696
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−8.00000	−0.577350
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	2.00000	0.142857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	12.0000	0.852803
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	24.0000	1.68863
$203$	5.00000	0.350931
$204$	14.0000	0.980196
$205$	0	0
$206$	−38.0000	−2.64759
$207$	−12.0000	−0.834058
$208$	−4.00000	−0.277350
$209$	0	0
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	12.0000	0.824163
$213$	−8.00000	−0.548151
$214$	−16.0000	−1.09374
$215$	0	0
$216$	0	0
$217$	−2.00000	−0.135769
$218$	10.0000	0.677285
$219$	6.00000	0.405442
$220$	0	0
$221$	7.00000	0.470871
$222$	4.00000	0.268462
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	8.00000	0.534522
$225$	0	0
$226$	12.0000	0.798228
$227$	17.0000	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$227$	17.0000	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	20.0000	1.30189
$237$	−5.00000	−0.324785
$238$	−14.0000	−0.907485
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−4.00000	−0.257130
$243$	16.0000	1.02640
$244$	−16.0000	−1.02430
$245$	0	0
$246$	4.00000	0.255031
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	4.00000	0.251976
$253$	−18.0000	−1.13165
$254$	4.00000	0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	−8.00000	−0.498058
$259$	−2.00000	−0.124274
$260$	0	0
$261$	10.0000	0.618984
$262$	44.0000	2.71833
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	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
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−28.0000	−1.69775
$273$	−1.00000	−0.0605228
$274$	24.0000	1.44989
$275$	0	0
$276$	12.0000	0.722315
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−20.0000	−1.19952
$279$	−4.00000	−0.239474
$280$	0	0
$281$	7.00000	0.417585	0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000	0.417585	0.208792	+	0.977960i	$0.433047\pi$
$282$	−6.00000	−0.357295
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−2.00000	−0.118056
$288$	16.0000	0.942809
$289$	32.0000	1.88235
$290$	0	0
$291$	7.00000	0.410347
$292$	12.0000	0.702247
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	0	0
$297$	15.0000	0.870388
$298$	20.0000	1.15857
$299$	6.00000	0.346989
$300$	0	0
$301$	4.00000	0.230556
$302$	−26.0000	−1.49613
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	−28.0000	−1.60065
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	6.00000	0.341882
$309$	−19.0000	−1.08087
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	21.0000	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$313$	21.0000	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	−10.0000	−0.562544
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	12.0000	0.672927
$319$	15.0000	0.839839
$320$	0	0
$321$	−8.00000	−0.446516
$322$	−12.0000	−0.668734
$323$	0	0
$324$	2.00000	0.111111
$325$	0	0
$326$	−28.0000	−1.55078
$327$	5.00000	0.276501
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−8.00000	−0.439057
$333$	−4.00000	−0.219199
$334$	−6.00000	−0.328305
$335$	0	0
$336$	4.00000	0.218218
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−24.0000	−1.30543
$339$	6.00000	0.325875
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−10.0000	−0.536056
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	24.0000	1.27920
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	20.0000	1.06299
$355$	0	0
$356$	0	0
$357$	−7.00000	−0.370479
$358$	−40.0000	−2.11407
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−36.0000	−1.89212
$363$	−2.00000	−0.104973
$364$	−2.00000	−0.104828
$365$	0	0
$366$	−16.0000	−0.836333
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	−24.0000	−1.25109
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−6.00000	−0.311504
$372$	4.00000	0.207390
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	−42.0000	−2.17177
$375$	0	0
$376$	0	0
$377$	−5.00000	−0.257513
$378$	10.0000	0.514344
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	−6.00000	−0.306987
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	32.0000	1.62876
$387$	8.00000	0.406663
$388$	14.0000	0.710742
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	42.0000	2.12403
$392$	0	0
$393$	22.0000	1.10975
$394$	4.00000	0.201517
$395$	0	0
$396$	12.0000	0.603023
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	4.00000	0.199502
$403$	2.00000	0.0996271
$404$	24.0000	1.19404
$405$	0	0
$406$	10.0000	0.496292
$407$	−6.00000	−0.297409
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	−38.0000	−1.87213
$413$	−10.0000	−0.492068
$414$	−24.0000	−1.17954
$415$	0	0
$416$	−8.00000	−0.392232
$417$	−10.0000	−0.489702
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	−26.0000	−1.26566
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	−16.0000	−0.775203
$427$	8.00000	0.387147
$428$	−16.0000	−0.773389
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−23.0000	−1.10787	−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000	−1.10787	−0.553936	+	0.832560i	$0.686875\pi$
$432$	20.0000	0.962250
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	12.0000	0.573382
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	14.0000	0.665912
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	42.0000	1.98876
$447$	10.0000	0.472984
$448$	8.00000	0.377964
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	12.0000	0.564433
$453$	−13.0000	−0.610793
$454$	34.0000	1.59570
$455$	0	0
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	−20.0000	−0.934539
$459$	−35.0000	−1.63366
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	6.00000	0.279145
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	32.0000	1.48237
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	−4.00000	−0.184900
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	12.0000	0.551761
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−14.0000	−0.641689
$477$	−12.0000	−0.549442
$478$	30.0000	1.37217
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	44.0000	2.00415
$483$	−6.00000	−0.273009
$484$	−4.00000	−0.181818
$485$	0	0
$486$	32.0000	1.45155
$487$	42.0000	1.90320	0.951601	−	0.307337i	$-0.0994378\pi$
$487$	42.0000	1.90320	0.951601	+	0.307337i	$0.0994378\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	7.00000	0.315906	0.157953	−	0.987447i	$-0.449511\pi$
$491$	7.00000	0.315906	0.157953	+	0.987447i	$0.449511\pi$
$492$	4.00000	0.180334
$493$	−35.0000	−1.57632
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	8.00000	0.358849
$498$	−8.00000	−0.358489
$499$	−35.0000	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	−36.0000	−1.60676
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	−36.0000	−1.60040
$507$	−12.0000	−0.532939
$508$	4.00000	0.177471
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	32.0000	1.41421
$513$	0	0
$514$	44.0000	1.94076
$515$	0	0
$516$	−8.00000	−0.352180
$517$	9.00000	0.395820
$518$	−4.00000	−0.175750
$519$	−9.00000	−0.395056
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	20.0000	0.875376
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	44.0000	1.92215
$525$	0	0
$526$	−48.0000	−2.09290
$527$	14.0000	0.609850
$528$	12.0000	0.522233
$529$	13.0000	0.565217
$530$	0	0
$531$	−20.0000	−0.867926
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−20.0000	−0.863064
$538$	20.0000	0.862261
$539$	−3.00000	−0.129219
$540$	0	0
$541$	27.0000	1.16082	0.580410	−	0.814324i	$-0.302892\pi$
$541$	27.0000	1.16082	0.580410	+	0.814324i	$0.302892\pi$
$542$	−16.0000	−0.687259
$543$	−18.0000	−0.772454
$544$	−56.0000	−2.40098
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	24.0000	1.02523
$549$	16.0000	0.682863
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5.00000	0.212622
$554$	4.00000	0.169944
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	−8.00000	−0.338667
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−21.0000	−0.886621
$562$	14.0000	0.590554
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	22.0000	0.924729
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−6.00000	−0.250873
$573$	−3.00000	−0.125327
$574$	−4.00000	−0.166957
$575$	0	0
$576$	16.0000	0.666667
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	64.0000	2.66205
$579$	16.0000	0.664937
$580$	0	0
$581$	4.00000	0.165948
$582$	14.0000	0.580319
$583$	−18.0000	−0.745484
$584$	0	0
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	2.00000	0.0824786
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	−8.00000	−0.328798
$593$	41.0000	1.68367	0.841834	−	0.539736i	$-0.181476\pi$
$593$	41.0000	1.68367	0.841834	+	0.539736i	$0.181476\pi$
$594$	30.0000	1.23091
$595$	0	0
$596$	20.0000	0.819232
$597$	−10.0000	−0.409273
$598$	12.0000	0.490716
$599$	25.0000	1.02147	0.510736	−	0.859738i	$-0.329373\pi$
$599$	25.0000	1.02147	0.510736	+	0.859738i	$0.329373\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	−26.0000	−1.05792
$605$	0	0
$606$	24.0000	0.974933
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	−3.00000	−0.121367
$612$	−28.0000	−1.13183
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	−38.0000	−1.52858
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	24.0000	0.962312
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	42.0000	1.67866
$627$	0	0
$628$	−36.0000	−1.43656
$629$	14.0000	0.558217
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	24.0000	0.953162
$635$	0	0
$636$	12.0000	0.475831
$637$	1.00000	0.0396214
$638$	30.0000	1.18771
$639$	16.0000	0.632950
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	−16.0000	−0.631470
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	−28.0000	−1.09656
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−8.00000	−0.312348
$657$	−12.0000	−0.468165
$658$	6.00000	0.233904
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	24.0000	0.932786
$663$	7.00000	0.271857
$664$	0	0
$665$	0	0
$666$	−8.00000	−0.309994
$667$	−30.0000	−1.16160
$668$	−6.00000	−0.232147
$669$	21.0000	0.811907
$670$	0	0
$671$	24.0000	0.926510
$672$	8.00000	0.308607
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−36.0000	−1.38667
$675$	0	0
$676$	−24.0000	−0.923077
$677$	−43.0000	−1.65262	−0.826312	−	0.563212i	$-0.809565\pi$
$677$	−43.0000	−1.65262	−0.826312	+	0.563212i	$0.809565\pi$
$678$	12.0000	0.460857
$679$	−7.00000	−0.268635
$680$	0	0
$681$	17.0000	0.651441
$682$	−12.0000	−0.459504
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	0	0
$686$	−2.00000	−0.0763604
$687$	−10.0000	−0.381524
$688$	16.0000	0.609994
$689$	6.00000	0.228582
$690$	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
$693$	−6.00000	−0.227921
$694$	−36.0000	−1.36654
$695$	0	0
$696$	0	0
$697$	14.0000	0.530288
$698$	−40.0000	−1.51402
$699$	16.0000	0.605176
$700$	0	0
$701$	−13.0000	−0.491003	−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000	−0.491003	−0.245502	+	0.969396i	$0.578953\pi$
$702$	−10.0000	−0.377426
$703$	0	0
$704$	24.0000	0.904534
$705$	0	0
$706$	22.0000	0.827981
$707$	−12.0000	−0.451306
$708$	20.0000	0.751646
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	12.0000	0.449404
$714$	−14.0000	−0.523937
$715$	0	0
$716$	−40.0000	−1.49487
$717$	15.0000	0.560185
$718$	−40.0000	−1.49279
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	19.0000	0.707597
$722$	−38.0000	−1.41421
$723$	22.0000	0.818189
$724$	−36.0000	−1.33793
$725$	0	0
$726$	−4.00000	−0.148454
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−28.0000	−1.03562
$732$	−16.0000	−0.591377
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	−6.00000	−0.221464
$735$	0	0
$736$	−48.0000	−1.76930
$737$	−6.00000	−0.221013
$738$	−8.00000	−0.294484
$739$	−35.0000	−1.28750	−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000	−1.28750	−0.643748	+	0.765238i	$0.722621\pi$
$740$	0	0
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−14.0000	−0.513610	−0.256805	−	0.966463i	$-0.582670\pi$
$743$	−14.0000	−0.513610	−0.256805	+	0.966463i	$0.582670\pi$
$744$	0	0
$745$	0	0
$746$	−48.0000	−1.75740
$747$	8.00000	0.292705
$748$	−42.0000	−1.53567
$749$	8.00000	0.292314
$750$	0	0
$751$	−33.0000	−1.20419	−0.602094	−	0.798426i	$-0.705667\pi$
$751$	−33.0000	−1.20419	−0.602094	+	0.798426i	$0.705667\pi$
$752$	12.0000	0.437595
$753$	−18.0000	−0.655956
$754$	−10.0000	−0.364179
$755$	0	0
$756$	10.0000	0.363696
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	40.0000	1.45287
$759$	−18.0000	−0.653359
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	4.00000	0.144905
$763$	−5.00000	−0.181012
$764$	−6.00000	−0.217072
$765$	0	0
$766$	32.0000	1.15621
$767$	10.0000	0.361079
$768$	16.0000	0.577350
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	32.0000	1.15171
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	16.0000	0.575108
$775$	0	0
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	10.0000	0.358517
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	84.0000	3.00383
$783$	25.0000	0.893427
$784$	−4.00000	−0.142857
$785$	0	0
$786$	44.0000	1.56943
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	4.00000	0.142494
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−8.00000	−0.284088
$794$	14.0000	0.496841
$795$	0	0
$796$	−20.0000	−0.708881
$797$	−13.0000	−0.460484	−0.230242	−	0.973133i	$-0.573952\pi$
$797$	−13.0000	−0.460484	−0.230242	+	0.973133i	$0.573952\pi$
$798$	0	0
$799$	−21.0000	−0.742927
$800$	0	0
$801$	0	0
$802$	−6.00000	−0.211867
$803$	−18.0000	−0.635206
$804$	4.00000	0.141069
$805$	0	0
$806$	4.00000	0.140894
$807$	10.0000	0.352017
$808$	0	0
$809$	45.0000	1.58212	0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000	1.58212	0.791058	+	0.611741i	$0.209531\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	10.0000	0.350931
$813$	−8.00000	−0.280572
$814$	−12.0000	−0.420600
$815$	0	0
$816$	−28.0000	−0.980196
$817$	0	0
$818$	40.0000	1.39857
$819$	2.00000	0.0698857
$820$	0	0
$821$	−23.0000	−0.802706	−0.401353	−	0.915924i	$-0.631460\pi$
$821$	−23.0000	−0.802706	−0.401353	+	0.915924i	$0.631460\pi$
$822$	24.0000	0.837096
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	0	0
$825$	0	0
$826$	−20.0000	−0.695889
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	−24.0000	−0.834058
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−8.00000	−0.277350
$833$	7.00000	0.242536
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	60.0000	2.07267
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−6.00000	−0.206774
$843$	7.00000	0.241093
$844$	−26.0000	−0.894957
$845$	0	0
$846$	12.0000	0.412568
$847$	2.00000	0.0687208
$848$	−24.0000	−0.824163
$849$	11.0000	0.377519
$850$	0	0
$851$	12.0000	0.411355
$852$	−16.0000	−0.548151
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−6.00000	−0.204837
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	−46.0000	−1.56677
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	40.0000	1.36083
$865$	0	0
$866$	52.0000	1.76703
$867$	32.0000	1.08678
$868$	−4.00000	−0.135769
$869$	15.0000	0.508840
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	60.0000	2.02490
$879$	−9.00000	−0.303562
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	−4.00000	−0.134687
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	14.0000	0.470871
$885$	0	0
$886$	−8.00000	−0.268765
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−3.00000	−0.100504
$892$	42.0000	1.40626
$893$	0	0
$894$	20.0000	0.668900
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	−10.0000	−0.333704
$899$	−10.0000	−0.333519
$900$	0	0
$901$	42.0000	1.39922
$902$	−12.0000	−0.399556
$903$	4.00000	0.133112
$904$	0	0
$905$	0	0
$906$	−26.0000	−0.863792
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	34.0000	1.12833
$909$	−24.0000	−0.796030
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	−76.0000	−2.51386
$915$	0	0
$916$	−20.0000	−0.660819
$917$	−22.0000	−0.726504
$918$	−70.0000	−2.31034
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	24.0000	0.790398
$923$	−8.00000	−0.263323
$924$	6.00000	0.197386
$925$	0	0
$926$	72.0000	2.36607
$927$	38.0000	1.24808
$928$	40.0000	1.31306
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	0	0
$932$	32.0000	1.04819
$933$	12.0000	0.392862
$934$	54.0000	1.76693
$935$	0	0
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	−4.00000	−0.130605
$939$	21.0000	0.685309
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	−36.0000	−1.17294
$943$	12.0000	0.390774
$944$	−40.0000	−1.30189
$945$	0	0
$946$	24.0000	0.780307
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	−10.0000	−0.324785
$949$	6.00000	0.194768
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−24.0000	−0.777029
$955$	0	0
$956$	30.0000	0.970269
$957$	15.0000	0.484881
$958$	−60.0000	−1.93851
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	4.00000	0.128965
$963$	16.0000	0.515593
$964$	44.0000	1.41714
$965$	0	0
$966$	−12.0000	−0.386094
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	32.0000	1.02640
$973$	10.0000	0.320585
$974$	84.0000	2.69153
$975$	0	0
$976$	32.0000	1.02430
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	−28.0000	−0.895341
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	14.0000	0.446758
$983$	51.0000	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$983$	51.0000	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$984$	0	0
$985$	0	0
$986$	−70.0000	−2.22925
$987$	3.00000	0.0954911
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−16.0000	−0.508001
$993$	12.0000	0.380808
$994$	16.0000	0.507489
$995$	0	0
$996$	−8.00000	−0.253490
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	−70.0000	−2.21581
$999$	−10.0000	−0.316386

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.2.a.c.1.1		1
3.2	odd	2		1575.2.a.a.1.1		1
4.3	odd	2		2800.2.a.l.1.1		1
5.2	odd	4		35.2.b.a.29.2	yes	2
5.3	odd	4		35.2.b.a.29.1	✓	2
5.4	even	2		175.2.a.a.1.1		1
7.6	odd	2		1225.2.a.i.1.1		1
15.2	even	4		315.2.d.a.64.1		2
15.8	even	4		315.2.d.a.64.2		2
15.14	odd	2		1575.2.a.k.1.1		1
20.3	even	4		560.2.g.b.449.1		2
20.7	even	4		560.2.g.b.449.2		2
20.19	odd	2		2800.2.a.w.1.1		1
35.2	odd	12		245.2.j.e.214.2		4
35.3	even	12		245.2.j.d.79.2		4
35.12	even	12		245.2.j.d.214.2		4
35.13	even	4		245.2.b.a.99.1		2
35.17	even	12		245.2.j.d.79.1		4
35.18	odd	12		245.2.j.e.79.2		4
35.23	odd	12		245.2.j.e.214.1		4
35.27	even	4		245.2.b.a.99.2		2
35.32	odd	12		245.2.j.e.79.1		4
35.33	even	12		245.2.j.d.214.1		4
35.34	odd	2		1225.2.a.a.1.1		1
40.3	even	4		2240.2.g.g.449.2		2
40.13	odd	4		2240.2.g.h.449.1		2
40.27	even	4		2240.2.g.g.449.1		2
40.37	odd	4		2240.2.g.h.449.2		2
60.23	odd	4		5040.2.t.p.1009.1		2
60.47	odd	4		5040.2.t.p.1009.2		2
105.62	odd	4		2205.2.d.b.1324.1		2
105.83	odd	4		2205.2.d.b.1324.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.b.a.29.1	✓	2	5.3	odd	4
35.2.b.a.29.2	yes	2	5.2	odd	4
175.2.a.a.1.1		1	5.4	even	2
175.2.a.c.1.1		1	1.1	even	1	trivial
245.2.b.a.99.1		2	35.13	even	4
245.2.b.a.99.2		2	35.27	even	4
245.2.j.d.79.1		4	35.17	even	12
245.2.j.d.79.2		4	35.3	even	12
245.2.j.d.214.1		4	35.33	even	12
245.2.j.d.214.2		4	35.12	even	12
245.2.j.e.79.1		4	35.32	odd	12
245.2.j.e.79.2		4	35.18	odd	12
245.2.j.e.214.1		4	35.23	odd	12
245.2.j.e.214.2		4	35.2	odd	12
315.2.d.a.64.1		2	15.2	even	4
315.2.d.a.64.2		2	15.8	even	4
560.2.g.b.449.1		2	20.3	even	4
560.2.g.b.449.2		2	20.7	even	4
1225.2.a.a.1.1		1	35.34	odd	2
1225.2.a.i.1.1		1	7.6	odd	2
1575.2.a.a.1.1		1	3.2	odd	2
1575.2.a.k.1.1		1	15.14	odd	2
2205.2.d.b.1324.1		2	105.62	odd	4
2205.2.d.b.1324.2		2	105.83	odd	4
2240.2.g.g.449.1		2	40.27	even	4
2240.2.g.g.449.2		2	40.3	even	4
2240.2.g.h.449.1		2	40.13	odd	4
2240.2.g.h.449.2		2	40.37	odd	4
2800.2.a.l.1.1		1	4.3	odd	2
2800.2.a.w.1.1		1	20.19	odd	2
5040.2.t.p.1009.1		2	60.23	odd	4
5040.2.t.p.1009.2		2	60.47	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	175.2.a.c.1.1

Level	$175$
Weight	$2$
Character	175.1
Self dual	yes
Analytic conductor	$1.397$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$