LMFDB - Embedded newform 8470.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
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	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +4.00000 q^{34} +1.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -3.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} -10.0000 q^{43} +2.00000 q^{45} -3.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} +3.00000 q^{57} +2.00000 q^{58} +3.00000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -2.00000 q^{67} +4.00000 q^{68} +3.00000 q^{69} +1.00000 q^{70} -12.0000 q^{71} -2.00000 q^{72} -6.00000 q^{73} +8.00000 q^{74} -1.00000 q^{75} -3.00000 q^{76} -1.00000 q^{78} +17.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -3.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -10.0000 q^{86} -2.00000 q^{87} +8.00000 q^{89} +2.00000 q^{90} -1.00000 q^{91} -3.00000 q^{92} -4.00000 q^{94} +3.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−2.00000	−0.471405
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	1.00000	0.169031
$36$	−2.00000	−0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−3.00000	−0.486664
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	−3.00000	−0.442326
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−4.00000	−0.560112
$52$	1.00000	0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	3.00000	0.397360
$58$	2.00000	0.262613
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	4.00000	0.485071
$69$	3.00000	0.361158
$70$	1.00000	0.119523
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−2.00000	−0.235702
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	8.00000	0.929981
$75$	−1.00000	−0.115470
$76$	−3.00000	−0.344124
$77$	0	0
$78$	−1.00000	−0.113228
$79$	17.0000	1.91265	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	17.0000	1.91265	0.956325	+	0.292306i	$0.0944227\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	1.00000	0.109109
$85$	−4.00000	−0.433861
$86$	−10.0000	−1.07833
$87$	−2.00000	−0.214423
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	2.00000	0.210819
$91$	−1.00000	−0.104828
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−4.00000	−0.412568
$95$	3.00000	0.307794
$96$	−1.00000	−0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	−4.00000	−0.396059
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	1.00000	0.0980581
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	5.00000	0.481125
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	−1.00000	−0.0944911
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	3.00000	0.280976
$115$	3.00000	0.279751
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	3.00000	0.276172
$119$	−4.00000	−0.366679
$120$	1.00000	0.0912871
$121$	0	0
$122$	−14.0000	−1.26750
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	1.00000	0.0883883
$129$	10.0000	0.880451
$130$	−1.00000	−0.0877058
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	−2.00000	−0.172774
$135$	−5.00000	−0.430331
$136$	4.00000	0.342997
$137$	−7.00000	−0.598050	−0.299025	−	0.954245i	$-0.596661\pi$
$137$	−7.00000	−0.598050	−0.299025	+	0.954245i	$0.596661\pi$
$138$	3.00000	0.255377
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	1.00000	0.0845154
$141$	4.00000	0.336861
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−2.00000	−0.166091
$146$	−6.00000	−0.496564
$147$	−1.00000	−0.0824786
$148$	8.00000	0.657596
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	−1.00000	−0.0816497
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	−3.00000	−0.243332
$153$	−8.00000	−0.646762
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	17.0000	1.35245
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	3.00000	0.236433
$162$	1.00000	0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−3.00000	−0.232845
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	−4.00000	−0.306786
$171$	6.00000	0.458831
$172$	−10.0000	−0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−2.00000	−0.151620
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−3.00000	−0.225494
$178$	8.00000	0.599625
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	2.00000	0.149071
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−1.00000	−0.0741249
$183$	14.0000	1.03491
$184$	−3.00000	−0.221163
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	−4.00000	−0.291730
$189$	−5.00000	−0.363696
$190$	3.00000	0.217643
$191$	−25.0000	−1.80894	−0.904468	−	0.426541i	$-0.859732\pi$
$191$	−25.0000	−1.80894	−0.904468	+	0.426541i	$0.859732\pi$
$192$	−1.00000	−0.0721688
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	−2.00000	−0.143592
$195$	1.00000	0.0716115
$196$	1.00000	0.0714286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	1.00000	0.0707107
$201$	2.00000	0.141069
$202$	17.0000	1.19612
$203$	−2.00000	−0.140372
$204$	−4.00000	−0.280056
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	6.00000	0.417029
$208$	1.00000	0.0693375
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000	0.412082
$213$	12.0000	0.822226
$214$	−8.00000	−0.546869
$215$	10.0000	0.681994
$216$	5.00000	0.340207
$217$	0	0
$218$	−4.00000	−0.270914
$219$	6.00000	0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	−8.00000	−0.536925
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−1.00000	−0.0668153
$225$	−2.00000	−0.133333
$226$	−3.00000	−0.199557
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	3.00000	0.198680
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	2.00000	0.131306
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	−2.00000	−0.130744
$235$	4.00000	0.260931
$236$	3.00000	0.195283
$237$	−17.0000	−1.10427
$238$	−4.00000	−0.259281
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	1.00000	0.0645497
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	−14.0000	−0.896258
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	−3.00000	−0.190885
$248$	0	0
$249$	3.00000	0.190117
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−17.0000	−1.06667
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	10.0000	0.622573
$259$	−8.00000	−0.497096
$260$	−1.00000	−0.0620174
$261$	−4.00000	−0.247594
$262$	−15.0000	−0.926703
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	3.00000	0.183942
$267$	−8.00000	−0.489592
$268$	−2.00000	−0.122169
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	−5.00000	−0.304290
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	4.00000	0.242536
$273$	1.00000	0.0605228
$274$	−7.00000	−0.422885
$275$	0	0
$276$	3.00000	0.180579
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−3.00000	−0.179928
$279$	0	0
$280$	1.00000	0.0597614
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	4.00000	0.238197
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−12.0000	−0.712069
$285$	−3.00000	−0.177705
$286$	0	0
$287$	−6.00000	−0.354169
$288$	−2.00000	−0.117851
$289$	−1.00000	−0.0588235
$290$	−2.00000	−0.117444
$291$	2.00000	0.117242
$292$	−6.00000	−0.351123
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	−1.00000	−0.0583212
$295$	−3.00000	−0.174667
$296$	8.00000	0.464991
$297$	0	0
$298$	−16.0000	−0.926855
$299$	−3.00000	−0.173494
$300$	−1.00000	−0.0577350
$301$	10.0000	0.576390
$302$	3.00000	0.172631
$303$	−17.0000	−0.976624
$304$	−3.00000	−0.172062
$305$	14.0000	0.801638
$306$	−8.00000	−0.457330
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−1.00000	−0.0566139
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	−13.0000	−0.733632
$315$	−2.00000	−0.112687
$316$	17.0000	0.956325
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	8.00000	0.446516
$322$	3.00000	0.167183
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	6.00000	0.332309
$327$	4.00000	0.221201
$328$	6.00000	0.331295
$329$	4.00000	0.220527
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−3.00000	−0.164646
$333$	−16.0000	−0.876795
$334$	6.00000	0.328305
$335$	2.00000	0.109272
$336$	1.00000	0.0545545
$337$	−7.00000	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$338$	−12.0000	−0.652714
$339$	3.00000	0.162938
$340$	−4.00000	−0.216930
$341$	0	0
$342$	6.00000	0.324443
$343$	−1.00000	−0.0539949
$344$	−10.0000	−0.539164
$345$	−3.00000	−0.161515
$346$	6.00000	0.322562
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	−2.00000	−0.107211
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	−1.00000	−0.0534522
$351$	5.00000	0.266880
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	−3.00000	−0.159448
$355$	12.0000	0.636894
$356$	8.00000	0.423999
$357$	4.00000	0.211702
$358$	2.00000	0.105703
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	2.00000	0.105409
$361$	−10.0000	−0.526316
$362$	−7.00000	−0.367912
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	6.00000	0.314054
$366$	14.0000	0.731792
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−3.00000	−0.156386
$369$	−12.0000	−0.624695
$370$	−8.00000	−0.415900
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−4.00000	−0.206284
$377$	2.00000	0.103005
$378$	−5.00000	−0.257172
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	3.00000	0.153897
$381$	17.0000	0.870936
$382$	−25.0000	−1.27911
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	1.00000	0.0508987
$387$	20.0000	1.01666
$388$	−2.00000	−0.101535
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	1.00000	0.0506370
$391$	−12.0000	−0.606866
$392$	1.00000	0.0505076
$393$	15.0000	0.756650
$394$	−10.0000	−0.503793
$395$	−17.0000	−0.855363
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−14.0000	−0.701757
$399$	−3.00000	−0.150188
$400$	1.00000	0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	17.0000	0.845782
$405$	−1.00000	−0.0496904
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	−4.00000	−0.198030
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−6.00000	−0.296319
$411$	7.00000	0.345285
$412$	−8.00000	−0.394132
$413$	−3.00000	−0.147620
$414$	6.00000	0.294884
$415$	3.00000	0.147264
$416$	1.00000	0.0490290
$417$	3.00000	0.146911
$418$	0	0
$419$	−13.0000	−0.635092	−0.317546	−	0.948243i	$-0.602859\pi$
$419$	−13.0000	−0.635092	−0.317546	+	0.948243i	$0.602859\pi$
$420$	−1.00000	−0.0487950
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	−10.0000	−0.486792
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	4.00000	0.194029
$426$	12.0000	0.581402
$427$	14.0000	0.677507
$428$	−8.00000	−0.386695
$429$	0	0
$430$	10.0000	0.482243
$431$	−17.0000	−0.818861	−0.409431	−	0.912341i	$-0.634273\pi$
$431$	−17.0000	−0.818861	−0.409431	+	0.912341i	$0.634273\pi$
$432$	5.00000	0.240563
$433$	−20.0000	−0.961139	−0.480569	−	0.876957i	$-0.659570\pi$
$433$	−20.0000	−0.961139	−0.480569	+	0.876957i	$0.659570\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	−4.00000	−0.191565
$437$	9.00000	0.430528
$438$	6.00000	0.286691
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	4.00000	0.190261
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	−8.00000	−0.379663
$445$	−8.00000	−0.379236
$446$	−2.00000	−0.0947027
$447$	16.0000	0.756774
$448$	−1.00000	−0.0472456
$449$	−31.0000	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$449$	−31.0000	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	−3.00000	−0.141108
$453$	−3.00000	−0.140952
$454$	24.0000	1.12638
$455$	1.00000	0.0468807
$456$	3.00000	0.140488
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	6.00000	0.280362
$459$	20.0000	0.933520
$460$	3.00000	0.139876
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	41.0000	1.90543	0.952716	−	0.303863i	$-0.0982765\pi$
$463$	41.0000	1.90543	0.952716	+	0.303863i	$0.0982765\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−19.0000	−0.880158
$467$	11.0000	0.509019	0.254510	−	0.967070i	$-0.418086\pi$
$467$	11.0000	0.509019	0.254510	+	0.967070i	$0.418086\pi$
$468$	−2.00000	−0.0924500
$469$	2.00000	0.0923514
$470$	4.00000	0.184506
$471$	13.0000	0.599008
$472$	3.00000	0.138086
$473$	0	0
$474$	−17.0000	−0.780836
$475$	−3.00000	−0.137649
$476$	−4.00000	−0.183340
$477$	−12.0000	−0.549442
$478$	15.0000	0.686084
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	1.00000	0.0456435
$481$	8.00000	0.364769
$482$	−10.0000	−0.455488
$483$	−3.00000	−0.136505
$484$	0	0
$485$	2.00000	0.0908153
$486$	−16.0000	−0.725775
$487$	3.00000	0.135943	0.0679715	−	0.997687i	$-0.478347\pi$
$487$	3.00000	0.135943	0.0679715	+	0.997687i	$0.478347\pi$
$488$	−14.0000	−0.633750
$489$	−6.00000	−0.271329
$490$	−1.00000	−0.0451754
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−6.00000	−0.270501
$493$	8.00000	0.360302
$494$	−3.00000	−0.134976
$495$	0	0
$496$	0	0
$497$	12.0000	0.538274
$498$	3.00000	0.134433
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	−6.00000	−0.268060
$502$	−12.0000	−0.535586
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	2.00000	0.0890871
$505$	−17.0000	−0.756490
$506$	0	0
$507$	12.0000	0.532939
$508$	−17.0000	−0.754253
$509$	29.0000	1.28540	0.642701	−	0.766117i	$-0.277814\pi$
$509$	29.0000	1.28540	0.642701	+	0.766117i	$0.277814\pi$
$510$	4.00000	0.177123
$511$	6.00000	0.265424
$512$	1.00000	0.0441942
$513$	−15.0000	−0.662266
$514$	14.0000	0.617514
$515$	8.00000	0.352522
$516$	10.0000	0.440225
$517$	0	0
$518$	−8.00000	−0.351500
$519$	−6.00000	−0.263371
$520$	−1.00000	−0.0438529
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	−4.00000	−0.175075
$523$	17.0000	0.743358	0.371679	−	0.928361i	$-0.378782\pi$
$523$	17.0000	0.743358	0.371679	+	0.928361i	$0.378782\pi$
$524$	−15.0000	−0.655278
$525$	1.00000	0.0436436
$526$	−23.0000	−1.00285
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−6.00000	−0.260623
$531$	−6.00000	−0.260378
$532$	3.00000	0.130066
$533$	6.00000	0.259889
$534$	−8.00000	−0.346194
$535$	8.00000	0.345870
$536$	−2.00000	−0.0863868
$537$	−2.00000	−0.0863064
$538$	17.0000	0.732922
$539$	0	0
$540$	−5.00000	−0.215166
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	20.0000	0.859074
$543$	7.00000	0.300399
$544$	4.00000	0.171499
$545$	4.00000	0.171341
$546$	1.00000	0.0427960
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−7.00000	−0.299025
$549$	28.0000	1.19501
$550$	0	0
$551$	−6.00000	−0.255609
$552$	3.00000	0.127688
$553$	−17.0000	−0.722914
$554$	22.0000	0.934690
$555$	8.00000	0.339581
$556$	−3.00000	−0.127228
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	1.00000	0.0422577
$561$	0	0
$562$	−1.00000	−0.0421825
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	4.00000	0.168430
$565$	3.00000	0.126211
$566$	1.00000	0.0420331
$567$	−1.00000	−0.0419961
$568$	−12.0000	−0.503509
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	−3.00000	−0.125656
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	25.0000	1.04439
$574$	−6.00000	−0.250435
$575$	−3.00000	−0.125109
$576$	−2.00000	−0.0833333
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−1.00000	−0.0415945
$579$	−1.00000	−0.0415586
$580$	−2.00000	−0.0830455
$581$	3.00000	0.124461
$582$	2.00000	0.0829027
$583$	0	0
$584$	−6.00000	−0.248282
$585$	2.00000	0.0826898
$586$	−21.0000	−0.867502
$587$	−11.0000	−0.454019	−0.227009	−	0.973893i	$-0.572895\pi$
$587$	−11.0000	−0.454019	−0.227009	+	0.973893i	$0.572895\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	−3.00000	−0.123508
$591$	10.0000	0.411345
$592$	8.00000	0.328798
$593$	40.0000	1.64260	0.821302	−	0.570494i	$-0.193248\pi$
$593$	40.0000	1.64260	0.821302	+	0.570494i	$0.193248\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−16.0000	−0.655386
$597$	14.0000	0.572982
$598$	−3.00000	−0.122679
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	−1.00000	−0.0408248
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	10.0000	0.407570
$603$	4.00000	0.162893
$604$	3.00000	0.122068
$605$	0	0
$606$	−17.0000	−0.690578
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−3.00000	−0.121666
$609$	2.00000	0.0810441
$610$	14.0000	0.566843
$611$	−4.00000	−0.161823
$612$	−8.00000	−0.323381
$613$	32.0000	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$614$	12.0000	0.484281
$615$	6.00000	0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	8.00000	0.321807
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	−20.0000	−0.801927
$623$	−8.00000	−0.320513
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	18.0000	0.719425
$627$	0	0
$628$	−13.0000	−0.518756
$629$	32.0000	1.27592
$630$	−2.00000	−0.0796819
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	17.0000	0.676224
$633$	10.0000	0.397464
$634$	−12.0000	−0.476581
$635$	17.0000	0.674624
$636$	−6.00000	−0.237915
$637$	1.00000	0.0396214
$638$	0	0
$639$	24.0000	0.949425
$640$	−1.00000	−0.0395285
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	8.00000	0.315735
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	3.00000	0.118217
$645$	−10.0000	−0.393750
$646$	−12.0000	−0.472134
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	6.00000	0.234978
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	4.00000	0.156412
$655$	15.0000	0.586098
$656$	6.00000	0.234261
$657$	12.0000	0.468165
$658$	4.00000	0.155936
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	−45.0000	−1.75030	−0.875149	−	0.483854i	$-0.839236\pi$
$661$	−45.0000	−1.75030	−0.875149	+	0.483854i	$0.839236\pi$
$662$	6.00000	0.233197
$663$	−4.00000	−0.155347
$664$	−3.00000	−0.116423
$665$	−3.00000	−0.116335
$666$	−16.0000	−0.619987
$667$	−6.00000	−0.232321
$668$	6.00000	0.232147
$669$	2.00000	0.0773245
$670$	2.00000	0.0772667
$671$	0	0
$672$	1.00000	0.0385758
$673$	15.0000	0.578208	0.289104	−	0.957298i	$-0.406643\pi$
$673$	15.0000	0.578208	0.289104	+	0.957298i	$0.406643\pi$
$674$	−7.00000	−0.269630
$675$	5.00000	0.192450
$676$	−12.0000	−0.461538
$677$	−23.0000	−0.883962	−0.441981	−	0.897024i	$-0.645724\pi$
$677$	−23.0000	−0.883962	−0.441981	+	0.897024i	$0.645724\pi$
$678$	3.00000	0.115214
$679$	2.00000	0.0767530
$680$	−4.00000	−0.153393
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−34.0000	−1.30097	−0.650487	−	0.759517i	$-0.725435\pi$
$683$	−34.0000	−1.30097	−0.650487	+	0.759517i	$0.725435\pi$
$684$	6.00000	0.229416
$685$	7.00000	0.267456
$686$	−1.00000	−0.0381802
$687$	−6.00000	−0.228914
$688$	−10.0000	−0.381246
$689$	6.00000	0.228582
$690$	−3.00000	−0.114208
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−14.0000	−0.531433
$695$	3.00000	0.113796
$696$	−2.00000	−0.0758098
$697$	24.0000	0.909065
$698$	15.0000	0.567758
$699$	19.0000	0.718646
$700$	−1.00000	−0.0377964
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	5.00000	0.188713
$703$	−24.0000	−0.905177
$704$	0	0
$705$	−4.00000	−0.150649
$706$	2.00000	0.0752710
$707$	−17.0000	−0.639351
$708$	−3.00000	−0.112747
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	12.0000	0.450352
$711$	−34.0000	−1.27510
$712$	8.00000	0.299813
$713$	0	0
$714$	4.00000	0.149696
$715$	0	0
$716$	2.00000	0.0747435
$717$	−15.0000	−0.560185
$718$	−24.0000	−0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	2.00000	0.0745356
$721$	8.00000	0.297936
$722$	−10.0000	−0.372161
$723$	10.0000	0.371904
$724$	−7.00000	−0.260153
$725$	2.00000	0.0742781
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	−1.00000	−0.0370625
$729$	13.0000	0.481481
$730$	6.00000	0.222070
$731$	−40.0000	−1.47945
$732$	14.0000	0.517455
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	−18.0000	−0.664392
$735$	1.00000	0.0368856
$736$	−3.00000	−0.110581
$737$	0	0
$738$	−12.0000	−0.441726
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	−8.00000	−0.294086
$741$	3.00000	0.110208
$742$	−6.00000	−0.220267
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	−16.0000	−0.585802
$747$	6.00000	0.219529
$748$	0	0
$749$	8.00000	0.292314
$750$	1.00000	0.0365148
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	−4.00000	−0.145865
$753$	12.0000	0.437304
$754$	2.00000	0.0728357
$755$	−3.00000	−0.109181
$756$	−5.00000	−0.181848
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	3.00000	0.108821
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	17.0000	0.615845
$763$	4.00000	0.144810
$764$	−25.0000	−0.904468
$765$	8.00000	0.289241
$766$	28.0000	1.01168
$767$	3.00000	0.108324
$768$	−1.00000	−0.0360844
$769$	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
$771$	−14.0000	−0.504198
$772$	1.00000	0.0359908
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	20.0000	0.718885
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	8.00000	0.286998
$778$	−18.0000	−0.645331
$779$	−18.0000	−0.644917
$780$	1.00000	0.0358057
$781$	0	0
$782$	−12.0000	−0.429119
$783$	10.0000	0.357371
$784$	1.00000	0.0357143
$785$	13.0000	0.463990
$786$	15.0000	0.535032
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−10.0000	−0.356235
$789$	23.0000	0.818822
$790$	−17.0000	−0.604833
$791$	3.00000	0.106668
$792$	0	0
$793$	−14.0000	−0.497155
$794$	−22.0000	−0.780751
$795$	6.00000	0.212798
$796$	−14.0000	−0.496217
$797$	−21.0000	−0.743858	−0.371929	−	0.928261i	$-0.621304\pi$
$797$	−21.0000	−0.743858	−0.371929	+	0.928261i	$0.621304\pi$
$798$	−3.00000	−0.106199
$799$	−16.0000	−0.566039
$800$	1.00000	0.0353553
$801$	−16.0000	−0.565332
$802$	−10.0000	−0.353112
$803$	0	0
$804$	2.00000	0.0705346
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−17.0000	−0.598428
$808$	17.0000	0.598058
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	−1.00000	−0.0351364
$811$	−48.0000	−1.68551	−0.842754	−	0.538299i	$-0.819067\pi$
$811$	−48.0000	−1.68551	−0.842754	+	0.538299i	$0.819067\pi$
$812$	−2.00000	−0.0701862
$813$	−20.0000	−0.701431
$814$	0	0
$815$	−6.00000	−0.210171
$816$	−4.00000	−0.140028
$817$	30.0000	1.04957
$818$	6.00000	0.209785
$819$	2.00000	0.0698857
$820$	−6.00000	−0.209529
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	7.00000	0.244153
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−3.00000	−0.104383
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	6.00000	0.208514
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	3.00000	0.104132
$831$	−22.0000	−0.763172
$832$	1.00000	0.0346688
$833$	4.00000	0.138592
$834$	3.00000	0.103882
$835$	−6.00000	−0.207639
$836$	0	0
$837$	0	0
$838$	−13.0000	−0.449078
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	−1.00000	−0.0345033
$841$	−25.0000	−0.862069
$842$	−32.0000	−1.10279
$843$	1.00000	0.0344418
$844$	−10.0000	−0.344214
$845$	12.0000	0.412813
$846$	8.00000	0.275046
$847$	0	0
$848$	6.00000	0.206041
$849$	−1.00000	−0.0343199
$850$	4.00000	0.137199
$851$	−24.0000	−0.822709
$852$	12.0000	0.411113
$853$	−43.0000	−1.47229	−0.736146	−	0.676823i	$-0.763356\pi$
$853$	−43.0000	−1.47229	−0.736146	+	0.676823i	$0.763356\pi$
$854$	14.0000	0.479070
$855$	−6.00000	−0.205196
$856$	−8.00000	−0.273434
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	10.0000	0.340997
$861$	6.00000	0.204479
$862$	−17.0000	−0.579022
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	5.00000	0.170103
$865$	−6.00000	−0.204006
$866$	−20.0000	−0.679628
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	2.00000	0.0678064
$871$	−2.00000	−0.0677674
$872$	−4.00000	−0.135457
$873$	4.00000	0.135379
$874$	9.00000	0.304430
$875$	1.00000	0.0338062
$876$	6.00000	0.202721
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−24.0000	−0.809961
$879$	21.0000	0.708312
$880$	0	0
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	−2.00000	−0.0673435
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	4.00000	0.134535
$885$	3.00000	0.100844
$886$	−8.00000	−0.268765
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−8.00000	−0.268462
$889$	17.0000	0.570162
$890$	−8.00000	−0.268161
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	12.0000	0.401565
$894$	16.0000	0.535120
$895$	−2.00000	−0.0668526
$896$	−1.00000	−0.0334077
$897$	3.00000	0.100167
$898$	−31.0000	−1.03448
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	24.0000	0.799556
$902$	0	0
$903$	−10.0000	−0.332779
$904$	−3.00000	−0.0997785
$905$	7.00000	0.232688
$906$	−3.00000	−0.0996683
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	24.0000	0.796468
$909$	−34.0000	−1.12771
$910$	1.00000	0.0331497
$911$	−39.0000	−1.29213	−0.646064	−	0.763283i	$-0.723586\pi$
$911$	−39.0000	−1.29213	−0.646064	+	0.763283i	$0.723586\pi$
$912$	3.00000	0.0993399
$913$	0	0
$914$	5.00000	0.165385
$915$	−14.0000	−0.462826
$916$	6.00000	0.198246
$917$	15.0000	0.495344
$918$	20.0000	0.660098
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	3.00000	0.0989071
$921$	−12.0000	−0.395413
$922$	2.00000	0.0658665
$923$	−12.0000	−0.394985
$924$	0	0
$925$	8.00000	0.263038
$926$	41.0000	1.34734
$927$	16.0000	0.525509
$928$	2.00000	0.0656532
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	−19.0000	−0.622366
$933$	20.0000	0.654771
$934$	11.0000	0.359931
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	54.0000	1.76410	0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000	1.76410	0.882052	+	0.471153i	$0.156162\pi$
$938$	2.00000	0.0653023
$939$	−18.0000	−0.587408
$940$	4.00000	0.130466
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	13.0000	0.423563
$943$	−18.0000	−0.586161
$944$	3.00000	0.0976417
$945$	5.00000	0.162650
$946$	0	0
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	−17.0000	−0.552134
$949$	−6.00000	−0.194768
$950$	−3.00000	−0.0973329
$951$	12.0000	0.389127
$952$	−4.00000	−0.129641
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	−12.0000	−0.388514
$955$	25.0000	0.808981
$956$	15.0000	0.485135
$957$	0	0
$958$	22.0000	0.710788
$959$	7.00000	0.226042
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	8.00000	0.257930
$963$	16.0000	0.515593
$964$	−10.0000	−0.322078
$965$	−1.00000	−0.0321911
$966$	−3.00000	−0.0965234
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	2.00000	0.0642161
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	−16.0000	−0.513200
$973$	3.00000	0.0961756
$974$	3.00000	0.0961262
$975$	−1.00000	−0.0320256
$976$	−14.0000	−0.448129
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	−6.00000	−0.191859
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	8.00000	0.255420
$982$	6.00000	0.191468
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	−6.00000	−0.191273
$985$	10.0000	0.318626
$986$	8.00000	0.254772
$987$	−4.00000	−0.127321
$988$	−3.00000	−0.0954427
$989$	30.0000	0.953945
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	0	0
$993$	−6.00000	−0.190404
$994$	12.0000	0.380617
$995$	14.0000	0.443830
$996$	3.00000	0.0950586
$997$	27.0000	0.855099	0.427549	−	0.903992i	$-0.359377\pi$
$997$	27.0000	0.855099	0.427549	+	0.903992i	$0.359377\pi$
$998$	−4.00000	−0.126618
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.w.1.1	yes	1
11.10	odd	2		8470.2.a.g.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.g.1.1	✓	1	11.10	odd	2
8470.2.a.w.1.1	yes	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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