LMFDB - Embedded newform 510.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(1,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("510.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 510 = 2 \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	510.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:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	510.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} +8.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} +4.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -2.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} -6.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} -2.00000 q^{67} +1.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +2.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -8.00000 q^{77} +4.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -16.0000 q^{83} -2.00000 q^{84} -1.00000 q^{85} -6.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +8.00000 q^{91} +8.00000 q^{92} -4.00000 q^{93} -8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	−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
$21$	−2.00000	−0.436436
$22$	4.00000	0.852803
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	−4.00000	−0.640513
$40$	1.00000	0.158114
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	2.00000	0.308607
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−4.00000	−0.603023
$45$	−1.00000	−0.149071
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	4.00000	0.554700
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	−2.00000	−0.267261
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	1.00000	0.129099
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−4.00000	−0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−4.00000	−0.492366
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	1.00000	0.121268
$69$	−8.00000	−0.963087
$70$	2.00000	0.239046
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	−8.00000	−0.911685
$78$	4.00000	0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−8.00000	−0.883452
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−2.00000	−0.218218
$85$	−1.00000	−0.108465
$86$	−6.00000	−0.646997
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	8.00000	0.838628
$92$	8.00000	0.834058
$93$	−4.00000	−0.414781
$94$	−8.00000	−0.825137
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	3.00000	0.303046
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	1.00000	0.0990148
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−4.00000	−0.392232
$105$	2.00000	0.195180
$106$	2.00000	0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	2.00000	0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	4.00000	0.369800
$118$	6.00000	0.552345
$119$	2.00000	0.183340
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	−8.00000	−0.721336
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	4.00000	0.350823
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	4.00000	0.348155
$133$	−8.00000	−0.693688
$134$	2.00000	0.172774
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	8.00000	0.681005
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−2.00000	−0.169031
$141$	−8.00000	−0.673722
$142$	−2.00000	−0.167836
$143$	−16.0000	−1.33799
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−4.00000	−0.331042
$147$	3.00000	0.247436
$148$	6.00000	0.493197
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	1.00000	0.0816497
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000	0.324443
$153$	1.00000	0.0808452
$154$	8.00000	0.644658
$155$	−4.00000	−0.321288
$156$	−4.00000	−0.320256
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	1.00000	0.0790569
$161$	16.0000	1.26098
$162$	−1.00000	−0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	8.00000	0.624695
$165$	−4.00000	−0.311400
$166$	16.0000	1.24184
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	1.00000	0.0766965
$171$	−4.00000	−0.305888
$172$	6.00000	0.457496
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	2.00000	0.151620
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	6.00000	0.450988
$178$	2.00000	0.149906
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	−1.00000	−0.0745356
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	−8.00000	−0.592999
$183$	−14.0000	−1.03491
$184$	−8.00000	−0.589768
$185$	−6.00000	−0.441129
$186$	4.00000	0.293294
$187$	−4.00000	−0.292509
$188$	8.00000	0.583460
$189$	−2.00000	−0.145479
$190$	−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	$0.453773\pi$
$192$	−1.00000	−0.0721688
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	4.00000	0.284268
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−1.00000	−0.0707107
$201$	2.00000	0.141069
$202$	−8.00000	−0.562878
$203$	4.00000	0.280745
$204$	−1.00000	−0.0700140
$205$	−8.00000	−0.558744
$206$	12.0000	0.836080
$207$	8.00000	0.556038
$208$	4.00000	0.277350
$209$	16.0000	1.10674
$210$	−2.00000	−0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	−2.00000	−0.137038
$214$	−12.0000	−0.820303
$215$	−6.00000	−0.409197
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	14.0000	0.948200
$219$	−4.00000	−0.270295
$220$	4.00000	0.269680
$221$	4.00000	0.269069
$222$	6.00000	0.402694
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	−2.00000	−0.133631
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	4.00000	0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	8.00000	0.527504
$231$	8.00000	0.526361
$232$	−2.00000	−0.131306
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−4.00000	−0.261488
$235$	−8.00000	−0.521862
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−2.00000	−0.129641
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	3.00000	0.191663
$246$	8.00000	0.510061
$247$	−16.0000	−1.01806
$248$	−4.00000	−0.254000
$249$	16.0000	1.01396
$250$	1.00000	0.0632456
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	2.00000	0.125988
$253$	−32.0000	−2.01182
$254$	4.00000	0.250982
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	6.00000	0.373544
$259$	12.0000	0.745644
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	16.0000	0.988483
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	2.00000	0.122859
$266$	8.00000	0.490511
$267$	2.00000	0.122398
$268$	−2.00000	−0.122169
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	−1.00000	−0.0608581
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	1.00000	0.0606339
$273$	−8.00000	−0.484182
$274$	6.00000	0.362473
$275$	−4.00000	−0.241209
$276$	−8.00000	−0.481543
$277$	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
$280$	2.00000	0.119523
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	8.00000	0.476393
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	2.00000	0.118678
$285$	−4.00000	−0.236940
$286$	16.0000	0.946100
$287$	16.0000	0.944450
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	2.00000	0.117444
$291$	0	0
$292$	4.00000	0.234082
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−3.00000	−0.174964
$295$	6.00000	0.349334
$296$	−6.00000	−0.348743
$297$	4.00000	0.232104
$298$	−16.0000	−0.926855
$299$	32.0000	1.85061
$300$	−1.00000	−0.0577350
$301$	12.0000	0.691669
$302$	8.00000	0.460348
$303$	−8.00000	−0.459588
$304$	−4.00000	−0.229416
$305$	−14.0000	−0.801638
$306$	−1.00000	−0.0571662
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	−8.00000	−0.455842
$309$	12.0000	0.682656
$310$	4.00000	0.227185
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	4.00000	0.226455
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	4.00000	0.225733
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−2.00000	−0.112154
$319$	−8.00000	−0.447914
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	−16.0000	−0.891645
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	8.00000	0.443079
$327$	14.0000	0.774202
$328$	−8.00000	−0.441726
$329$	16.0000	0.882109
$330$	4.00000	0.220193
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−16.0000	−0.878114
$333$	6.00000	0.328798
$334$	12.0000	0.656611
$335$	2.00000	0.109272
$336$	−2.00000	−0.109109
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	−3.00000	−0.163178
$339$	−6.00000	−0.325875
$340$	−1.00000	−0.0542326
$341$	−16.0000	−0.866449
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	8.00000	0.430706
$346$	−22.0000	−1.18273
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−2.00000	−0.107211
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−2.00000	−0.106904
$351$	−4.00000	−0.213504
$352$	4.00000	0.213201
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−6.00000	−0.318896
$355$	−2.00000	−0.106149
$356$	−2.00000	−0.106000
$357$	−2.00000	−0.105851
$358$	−6.00000	−0.317110
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−26.0000	−1.36653
$363$	−5.00000	−0.262432
$364$	8.00000	0.419314
$365$	−4.00000	−0.209370
$366$	14.0000	0.731792
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	8.00000	0.417029
$369$	8.00000	0.416463
$370$	6.00000	0.311925
$371$	−4.00000	−0.207670
$372$	−4.00000	−0.207390
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	4.00000	0.206835
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	8.00000	0.412021
$378$	2.00000	0.102869
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	4.00000	0.205196
$381$	4.00000	0.204926
$382$	−4.00000	−0.204658
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	1.00000	0.0510310
$385$	8.00000	0.407718
$386$	−24.0000	−1.22157
$387$	6.00000	0.304997
$388$	0	0
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	−4.00000	−0.202548
$391$	8.00000	0.404577
$392$	3.00000	0.151523
$393$	16.0000	0.807093
$394$	26.0000	1.30986
$395$	0	0
$396$	−4.00000	−0.201008
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	20.0000	1.00251
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	−2.00000	−0.0997509
$403$	16.0000	0.797017
$404$	8.00000	0.398015
$405$	−1.00000	−0.0496904
$406$	−4.00000	−0.198517
$407$	−24.0000	−1.18964
$408$	1.00000	0.0495074
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	8.00000	0.395092
$411$	6.00000	0.295958
$412$	−12.0000	−0.591198
$413$	−12.0000	−0.590481
$414$	−8.00000	−0.393179
$415$	16.0000	0.785409
$416$	−4.00000	−0.196116
$417$	20.0000	0.979404
$418$	−16.0000	−0.782586
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	2.00000	0.0975900
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	−4.00000	−0.194717
$423$	8.00000	0.388973
$424$	2.00000	0.0971286
$425$	1.00000	0.0485071
$426$	2.00000	0.0969003
$427$	28.0000	1.35501
$428$	12.0000	0.580042
$429$	16.0000	0.772487
$430$	6.00000	0.289346
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	−1.00000	−0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−8.00000	−0.384012
$435$	2.00000	0.0958927
$436$	−14.0000	−0.670478
$437$	−32.0000	−1.53077
$438$	4.00000	0.191127
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−4.00000	−0.190693
$441$	−3.00000	−0.142857
$442$	−4.00000	−0.190261
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−6.00000	−0.284747
$445$	2.00000	0.0948091
$446$	−28.0000	−1.32584
$447$	−16.0000	−0.756774
$448$	2.00000	0.0944911
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−1.00000	−0.0471405
$451$	−32.0000	−1.50682
$452$	6.00000	0.282216
$453$	8.00000	0.375873
$454$	−28.0000	−1.31411
$455$	−8.00000	−0.375046
$456$	−4.00000	−0.187317
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−6.00000	−0.280362
$459$	−1.00000	−0.0466760
$460$	−8.00000	−0.373002
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	−8.00000	−0.372194
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	2.00000	0.0928477
$465$	4.00000	0.185496
$466$	−18.0000	−0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	4.00000	0.184900
$469$	−4.00000	−0.184703
$470$	8.00000	0.369012
$471$	4.00000	0.184310
$472$	6.00000	0.276172
$473$	−24.0000	−1.10352
$474$	0	0
$475$	−4.00000	−0.183533
$476$	2.00000	0.0916698
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−22.0000	−1.00521	−0.502603	−	0.864517i	$-0.667624\pi$
$479$	−22.0000	−1.00521	−0.502603	+	0.864517i	$0.667624\pi$
$480$	−1.00000	−0.0456435
$481$	24.0000	1.09431
$482$	14.0000	0.637683
$483$	−16.0000	−0.728025
$484$	5.00000	0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−14.0000	−0.633750
$489$	8.00000	0.361773
$490$	−3.00000	−0.135526
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	−8.00000	−0.360668
$493$	2.00000	0.0900755
$494$	16.0000	0.719874
$495$	4.00000	0.179787
$496$	4.00000	0.179605
$497$	4.00000	0.179425
$498$	−16.0000	−0.716977
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	−26.0000	−1.16044
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−2.00000	−0.0890871
$505$	−8.00000	−0.355995
$506$	32.0000	1.42257
$507$	−3.00000	−0.133235
$508$	−4.00000	−0.177471
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	−1.00000	−0.0442807
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	18.0000	0.793946
$515$	12.0000	0.528783
$516$	−6.00000	−0.264135
$517$	−32.0000	−1.40736
$518$	−12.0000	−0.527250
$519$	−22.0000	−0.965693
$520$	4.00000	0.175412
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	−2.00000	−0.0875376
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−16.0000	−0.698963
$525$	−2.00000	−0.0872872
$526$	−8.00000	−0.348817
$527$	4.00000	0.174243
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	−2.00000	−0.0868744
$531$	−6.00000	−0.260378
$532$	−8.00000	−0.346844
$533$	32.0000	1.38607
$534$	−2.00000	−0.0865485
$535$	−12.0000	−0.518805
$536$	2.00000	0.0863868
$537$	−6.00000	−0.258919
$538$	30.0000	1.29339
$539$	12.0000	0.516877
$540$	1.00000	0.0430331
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	24.0000	1.03089
$543$	−26.0000	−1.11577
$544$	−1.00000	−0.0428746
$545$	14.0000	0.599694
$546$	8.00000	0.342368
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	−6.00000	−0.256307
$549$	14.0000	0.597505
$550$	4.00000	0.170561
$551$	−8.00000	−0.340811
$552$	8.00000	0.340503
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	6.00000	0.254686
$556$	−20.0000	−0.848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−4.00000	−0.169334
$559$	24.0000	1.01509
$560$	−2.00000	−0.0845154
$561$	4.00000	0.168880
$562$	2.00000	0.0843649
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−8.00000	−0.336861
$565$	−6.00000	−0.252422
$566$	20.0000	0.840663
$567$	2.00000	0.0839921
$568$	−2.00000	−0.0839181
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	4.00000	0.167542
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−16.0000	−0.668994
$573$	−4.00000	−0.167102
$574$	−16.0000	−0.667827
$575$	8.00000	0.333623
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	−1.00000	−0.0415945
$579$	−24.0000	−0.997406
$580$	−2.00000	−0.0830455
$581$	−32.0000	−1.32758
$582$	0	0
$583$	8.00000	0.331326
$584$	−4.00000	−0.165521
$585$	−4.00000	−0.165380
$586$	14.0000	0.578335
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	3.00000	0.123718
$589$	−16.0000	−0.659269
$590$	−6.00000	−0.247016
$591$	26.0000	1.06950
$592$	6.00000	0.246598
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−4.00000	−0.164122
$595$	−2.00000	−0.0819920
$596$	16.0000	0.655386
$597$	20.0000	0.818546
$598$	−32.0000	−1.30858
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	1.00000	0.0408248
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	−12.0000	−0.489083
$603$	−2.00000	−0.0814463
$604$	−8.00000	−0.325515
$605$	−5.00000	−0.203279
$606$	8.00000	0.324978
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	4.00000	0.162221
$609$	−4.00000	−0.162088
$610$	14.0000	0.566843
$611$	32.0000	1.29458
$612$	1.00000	0.0404226
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	10.0000	0.403567
$615$	8.00000	0.322591
$616$	8.00000	0.322329
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−12.0000	−0.482711
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−4.00000	−0.160644
$621$	−8.00000	−0.321029
$622$	−14.0000	−0.561349
$623$	−4.00000	−0.160257
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	−16.0000	−0.638978
$628$	−4.00000	−0.159617
$629$	6.00000	0.239236
$630$	2.00000	0.0796819
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	2.00000	0.0794301
$635$	4.00000	0.158735
$636$	2.00000	0.0793052
$637$	−12.0000	−0.475457
$638$	8.00000	0.316723
$639$	2.00000	0.0791188
$640$	1.00000	0.0395285
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	12.0000	0.473602
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	16.0000	0.630488
$645$	6.00000	0.236250
$646$	4.00000	0.157378
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	−1.00000	−0.0392837
$649$	24.0000	0.942082
$650$	−4.00000	−0.156893
$651$	−8.00000	−0.313545
$652$	−8.00000	−0.313304
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−14.0000	−0.547443
$655$	16.0000	0.625172
$656$	8.00000	0.312348
$657$	4.00000	0.156055
$658$	−16.0000	−0.623745
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	−4.00000	−0.155700
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	8.00000	0.310929
$663$	−4.00000	−0.155347
$664$	16.0000	0.620920
$665$	8.00000	0.310227
$666$	−6.00000	−0.232495
$667$	16.0000	0.619522
$668$	−12.0000	−0.464294
$669$	−28.0000	−1.08254
$670$	−2.00000	−0.0772667
$671$	−56.0000	−2.16186
$672$	2.00000	0.0771517
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	−20.0000	−0.770371
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	1.00000	0.0383482
$681$	−28.0000	−1.07296
$682$	16.0000	0.612672
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−4.00000	−0.152944
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	−6.00000	−0.228914
$688$	6.00000	0.228748
$689$	−8.00000	−0.304776
$690$	−8.00000	−0.304555
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	22.0000	0.836315
$693$	−8.00000	−0.303895
$694$	28.0000	1.06287
$695$	20.0000	0.758643
$696$	2.00000	0.0758098
$697$	8.00000	0.303022
$698$	14.0000	0.529908
$699$	−18.0000	−0.680823
$700$	2.00000	0.0755929
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	4.00000	0.150970
$703$	−24.0000	−0.905177
$704$	−4.00000	−0.150756
$705$	8.00000	0.301297
$706$	10.0000	0.376355
$707$	16.0000	0.601742
$708$	6.00000	0.225494
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	2.00000	0.0750587
$711$	0	0
$712$	2.00000	0.0749532
$713$	32.0000	1.19841
$714$	2.00000	0.0748481
$715$	16.0000	0.598366
$716$	6.00000	0.224231
$717$	0	0
$718$	−36.0000	−1.34351
$719$	10.0000	0.372937	0.186469	−	0.982461i	$-0.440296\pi$
$719$	10.0000	0.372937	0.186469	+	0.982461i	$0.440296\pi$
$720$	−1.00000	−0.0372678
$721$	−24.0000	−0.893807
$722$	3.00000	0.111648
$723$	14.0000	0.520666
$724$	26.0000	0.966282
$725$	2.00000	0.0742781
$726$	5.00000	0.185567
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	6.00000	0.221918
$732$	−14.0000	−0.517455
$733$	48.0000	1.77292	0.886460	−	0.462805i	$-0.153157\pi$
$733$	48.0000	1.77292	0.886460	+	0.462805i	$0.153157\pi$
$734$	−22.0000	−0.812035
$735$	−3.00000	−0.110657
$736$	−8.00000	−0.294884
$737$	8.00000	0.294684
$738$	−8.00000	−0.294484
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−6.00000	−0.220564
$741$	16.0000	0.587775
$742$	4.00000	0.146845
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	4.00000	0.146647
$745$	−16.0000	−0.586195
$746$	16.0000	0.585802
$747$	−16.0000	−0.585409
$748$	−4.00000	−0.146254
$749$	24.0000	0.876941
$750$	−1.00000	−0.0365148
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	8.00000	0.291730
$753$	−26.0000	−0.947493
$754$	−8.00000	−0.291343
$755$	8.00000	0.291150
$756$	−2.00000	−0.0727393
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	4.00000	0.145287
$759$	32.0000	1.16153
$760$	−4.00000	−0.145095
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−4.00000	−0.144905
$763$	−28.0000	−1.01367
$764$	4.00000	0.144715
$765$	−1.00000	−0.0361551
$766$	−8.00000	−0.289052
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	−8.00000	−0.288300
$771$	18.0000	0.648254
$772$	24.0000	0.863779
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−6.00000	−0.215666
$775$	4.00000	0.143684
$776$	0	0
$777$	−12.0000	−0.430498
$778$	−24.0000	−0.860442
$779$	−32.0000	−1.14652
$780$	4.00000	0.143223
$781$	−8.00000	−0.286263
$782$	−8.00000	−0.286079
$783$	−2.00000	−0.0714742
$784$	−3.00000	−0.107143
$785$	4.00000	0.142766
$786$	−16.0000	−0.570701
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	−26.0000	−0.926212
$789$	−8.00000	−0.284808
$790$	0	0
$791$	12.0000	0.426671
$792$	4.00000	0.142134
$793$	56.0000	1.98862
$794$	−6.00000	−0.212932
$795$	−2.00000	−0.0709327
$796$	−20.0000	−0.708881
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	−8.00000	−0.283197
$799$	8.00000	0.283020
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	12.0000	0.423735
$803$	−16.0000	−0.564628
$804$	2.00000	0.0705346
$805$	−16.0000	−0.563926
$806$	−16.0000	−0.563576
$807$	30.0000	1.05605
$808$	−8.00000	−0.281439
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	1.00000	0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	4.00000	0.140372
$813$	24.0000	0.841717
$814$	24.0000	0.841200
$815$	8.00000	0.280228
$816$	−1.00000	−0.0350070
$817$	−24.0000	−0.839654
$818$	26.0000	0.909069
$819$	8.00000	0.279543
$820$	−8.00000	−0.279372
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−6.00000	−0.209274
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	12.0000	0.418040
$825$	4.00000	0.139262
$826$	12.0000	0.417533
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	8.00000	0.278019
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	−16.0000	−0.555368
$831$	−2.00000	−0.0693792
$832$	4.00000	0.138675
$833$	−3.00000	−0.103944
$834$	−20.0000	−0.692543
$835$	12.0000	0.415277
$836$	16.0000	0.553372
$837$	−4.00000	−0.138260
$838$	0	0
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	18.0000	0.620321
$843$	2.00000	0.0688837
$844$	4.00000	0.137686
$845$	−3.00000	−0.103203
$846$	−8.00000	−0.275046
$847$	10.0000	0.343604
$848$	−2.00000	−0.0686803
$849$	20.0000	0.686398
$850$	−1.00000	−0.0342997
$851$	48.0000	1.64542
$852$	−2.00000	−0.0685189
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	−28.0000	−0.958140
$855$	4.00000	0.136797
$856$	−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	$0.622616\pi$
$858$	−16.0000	−0.546231
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	−6.00000	−0.204598
$861$	−16.0000	−0.545279
$862$	−14.0000	−0.476842
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	1.00000	0.0340207
$865$	−22.0000	−0.748022
$866$	26.0000	0.883516
$867$	−1.00000	−0.0339618
$868$	8.00000	0.271538
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	−8.00000	−0.271070
$872$	14.0000	0.474100
$873$	0	0
$874$	32.0000	1.08242
$875$	−2.00000	−0.0676123
$876$	−4.00000	−0.135147
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	24.0000	0.809961
$879$	14.0000	0.472208
$880$	4.00000	0.134840
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	3.00000	0.101015
$883$	−46.0000	−1.54802	−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000	−1.54802	−0.774012	+	0.633171i	$0.781753\pi$
$884$	4.00000	0.134535
$885$	−6.00000	−0.201688
$886$	−24.0000	−0.806296
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	6.00000	0.201347
$889$	−8.00000	−0.268311
$890$	−2.00000	−0.0670402
$891$	−4.00000	−0.134005
$892$	28.0000	0.937509
$893$	−32.0000	−1.07084
$894$	16.0000	0.535120
$895$	−6.00000	−0.200558
$896$	−2.00000	−0.0668153
$897$	−32.0000	−1.06845
$898$	12.0000	0.400445
$899$	8.00000	0.266815
$900$	1.00000	0.0333333
$901$	−2.00000	−0.0666297
$902$	32.0000	1.06548
$903$	−12.0000	−0.399335
$904$	−6.00000	−0.199557
$905$	−26.0000	−0.864269
$906$	−8.00000	−0.265782
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	28.0000	0.929213
$909$	8.00000	0.265343
$910$	8.00000	0.265197
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	4.00000	0.132453
$913$	64.0000	2.11809
$914$	2.00000	0.0661541
$915$	14.0000	0.462826
$916$	6.00000	0.198246
$917$	−32.0000	−1.05673
$918$	1.00000	0.0330049
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	8.00000	0.263752
$921$	10.0000	0.329511
$922$	28.0000	0.922131
$923$	8.00000	0.263323
$924$	8.00000	0.263181
$925$	6.00000	0.197279
$926$	−4.00000	−0.131448
$927$	−12.0000	−0.394132
$928$	−2.00000	−0.0656532
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	−4.00000	−0.131165
$931$	12.0000	0.393284
$932$	18.0000	0.589610
$933$	−14.0000	−0.458339
$934$	12.0000	0.392652
$935$	4.00000	0.130814
$936$	−4.00000	−0.130744
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	4.00000	0.130605
$939$	−16.0000	−0.522140
$940$	−8.00000	−0.260931
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	−4.00000	−0.130327
$943$	64.0000	2.08413
$944$	−6.00000	−0.195283
$945$	2.00000	0.0650600
$946$	24.0000	0.780307
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	4.00000	0.129777
$951$	2.00000	0.0648544
$952$	−2.00000	−0.0648204
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	2.00000	0.0647524
$955$	−4.00000	−0.129437
$956$	0	0
$957$	8.00000	0.258603
$958$	22.0000	0.710788
$959$	−12.0000	−0.387500
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−24.0000	−0.773791
$963$	12.0000	0.386695
$964$	−14.0000	−0.450910
$965$	−24.0000	−0.772587
$966$	16.0000	0.514792
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−5.00000	−0.160706
$969$	4.00000	0.128499
$970$	0	0
$971$	−50.0000	−1.60458	−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000	−1.60458	−0.802288	+	0.596937i	$0.796384\pi$
$972$	−1.00000	−0.0320750
$973$	−40.0000	−1.28234
$974$	22.0000	0.704925
$975$	−4.00000	−0.128103
$976$	14.0000	0.448129
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−8.00000	−0.255812
$979$	8.00000	0.255681
$980$	3.00000	0.0958315
$981$	−14.0000	−0.446986
$982$	18.0000	0.574403
$983$	−52.0000	−1.65854	−0.829271	−	0.558846i	$-0.811244\pi$
$983$	−52.0000	−1.65854	−0.829271	+	0.558846i	$0.811244\pi$
$984$	8.00000	0.255031
$985$	26.0000	0.828429
$986$	−2.00000	−0.0636930
$987$	−16.0000	−0.509286
$988$	−16.0000	−0.509028
$989$	48.0000	1.52631
$990$	−4.00000	−0.127128
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−4.00000	−0.127000
$993$	8.00000	0.253872
$994$	−4.00000	−0.126872
$995$	20.0000	0.634043
$996$	16.0000	0.506979
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	−4.00000	−0.126618
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	510.2.a.a.1.1	✓	1
3.2	odd	2		1530.2.a.p.1.1		1
4.3	odd	2		4080.2.a.s.1.1		1
5.2	odd	4		2550.2.d.n.2449.1		2
5.3	odd	4		2550.2.d.n.2449.2		2
5.4	even	2		2550.2.a.bc.1.1		1
15.14	odd	2		7650.2.a.k.1.1		1
17.16	even	2		8670.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.a.1.1	✓	1	1.1	even	1	trivial
1530.2.a.p.1.1		1	3.2	odd	2
2550.2.a.bc.1.1		1	5.4	even	2
2550.2.d.n.2449.1		2	5.2	odd	4
2550.2.d.n.2449.2		2	5.3	odd	4
4080.2.a.s.1.1		1	4.3	odd	2
7650.2.a.k.1.1		1	15.14	odd	2
8670.2.a.k.1.1		1	17.16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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