LMFDB - Embedded newform 8670.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
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	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} -4.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +5.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} +3.00000 q^{42} -2.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +5.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} -4.00000 q^{58} -11.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -5.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +1.00000 q^{69} -3.00000 q^{70} -10.0000 q^{71} +1.00000 q^{72} +16.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +3.00000 q^{77} +5.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -2.00000 q^{83} +3.00000 q^{84} -2.00000 q^{86} -4.00000 q^{87} +1.00000 q^{88} -5.00000 q^{89} -1.00000 q^{90} +15.0000 q^{91} +1.00000 q^{92} +8.00000 q^{93} +9.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} -4.00000 q^{97} +2.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	3.00000	0.801784
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	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
$21$	3.00000	0.654654
$22$	1.00000	0.213201
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	5.00000	0.980581
$27$	1.00000	0.192450
$28$	3.00000	0.566947
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−1.00000	−0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	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
$39$	5.00000	0.800641
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	3.00000	0.462910
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	0	0
$52$	5.00000	0.693375
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	3.00000	0.400892
$57$	−4.00000	−0.529813
$58$	−4.00000	−0.525226
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000	1.01600
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	−5.00000	−0.620174
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	−3.00000	−0.358569
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000	0.117851
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	3.00000	0.341882
$78$	5.00000	0.566139
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−4.00000	−0.428845
$88$	1.00000	0.106600
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	−1.00000	−0.105409
$91$	15.0000	1.57243
$92$	1.00000	0.104257
$93$	8.00000	0.829561
$94$	9.00000	0.928279
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	2.00000	0.202031
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	5.00000	0.490290
$105$	−3.00000	−0.292770
$106$	−3.00000	−0.291386
$107$	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
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−1.00000	−0.0953463
$111$	6.00000	0.569495
$112$	3.00000	0.283473
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−4.00000	−0.374634
$115$	−1.00000	−0.0932505
$116$	−4.00000	−0.371391
$117$	5.00000	0.462250
$118$	−11.0000	−1.01263
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	−2.00000	−0.181071
$123$	2.00000	0.180334
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	3.00000	0.267261
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	−2.00000	−0.176090
$130$	−5.00000	−0.438529
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	1.00000	0.0870388
$133$	−12.0000	−1.04053
$134$	4.00000	0.345547
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	1.00000	0.0851257
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−3.00000	−0.253546
$141$	9.00000	0.757937
$142$	−10.0000	−0.839181
$143$	5.00000	0.418121
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	16.0000	1.32417
$147$	2.00000	0.164957
$148$	6.00000	0.493197
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	3.00000	0.241747
$155$	−8.00000	−0.642575
$156$	5.00000	0.400320
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	−8.00000	−0.636446
$159$	−3.00000	−0.237915
$160$	−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	$0.424498\pi$
$164$	2.00000	0.156174
$165$	−1.00000	−0.0778499
$166$	−2.00000	−0.155230
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	3.00000	0.231455
$169$	12.0000	0.923077
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−2.00000	−0.152499
$173$	−19.0000	−1.44454	−0.722272	−	0.691609i	$-0.756902\pi$
$173$	−19.0000	−1.44454	−0.722272	+	0.691609i	$0.756902\pi$
$174$	−4.00000	−0.303239
$175$	3.00000	0.226779
$176$	1.00000	0.0753778
$177$	−11.0000	−0.826811
$178$	−5.00000	−0.374766
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	−1.00000	−0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	15.0000	1.11187
$183$	−2.00000	−0.147844
$184$	1.00000	0.0737210
$185$	−6.00000	−0.441129
$186$	8.00000	0.586588
$187$	0	0
$188$	9.00000	0.656392
$189$	3.00000	0.218218
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	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	$0.168091\pi$
$194$	−4.00000	−0.287183
$195$	−5.00000	−0.358057
$196$	2.00000	0.142857
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	1.00000	0.0710669
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−2.00000	−0.139686
$206$	−8.00000	−0.557386
$207$	1.00000	0.0695048
$208$	5.00000	0.346688
$209$	−4.00000	−0.276686
$210$	−3.00000	−0.207020
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	−3.00000	−0.206041
$213$	−10.0000	−0.685189
$214$	12.0000	0.820303
$215$	2.00000	0.136399
$216$	1.00000	0.0680414
$217$	24.0000	1.62923
$218$	−8.00000	−0.541828
$219$	16.0000	1.08118
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	6.00000	0.402694
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	3.00000	0.200446
$225$	1.00000	0.0666667
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−4.00000	−0.264906
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	−1.00000	−0.0659380
$231$	3.00000	0.197386
$232$	−4.00000	−0.262613
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	5.00000	0.326860
$235$	−9.00000	−0.587095
$236$	−11.0000	−0.716039
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	−1.00000	−0.0645497
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	−10.0000	−0.642824
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	−2.00000	−0.127775
$246$	2.00000	0.127515
$247$	−20.0000	−1.27257
$248$	8.00000	0.508001
$249$	−2.00000	−0.126745
$250$	−1.00000	−0.0632456
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	3.00000	0.188982
$253$	1.00000	0.0628695
$254$	20.0000	1.25491
$255$	0	0
$256$	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	$0.689739\pi$
$258$	−2.00000	−0.124515
$259$	18.0000	1.11847
$260$	−5.00000	−0.310087
$261$	−4.00000	−0.247594
$262$	13.0000	0.803143
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	1.00000	0.0615457
$265$	3.00000	0.184289
$266$	−12.0000	−0.735767
$267$	−5.00000	−0.305995
$268$	4.00000	0.244339
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−1.00000	−0.0608581
$271$	−26.0000	−1.57939	−0.789694	−	0.613501i	$-0.789761\pi$
$271$	−26.0000	−1.57939	−0.789694	+	0.613501i	$0.789761\pi$
$272$	0	0
$273$	15.0000	0.907841
$274$	−12.0000	−0.724947
$275$	1.00000	0.0603023
$276$	1.00000	0.0601929
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	20.0000	1.19952
$279$	8.00000	0.478947
$280$	−3.00000	−0.179284
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	9.00000	0.535942
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	−10.0000	−0.593391
$285$	4.00000	0.236940
$286$	5.00000	0.295656
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	0	0
$290$	4.00000	0.234888
$291$	−4.00000	−0.234484
$292$	16.0000	0.936329
$293$	27.0000	1.57736	0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000	1.57736	0.788678	+	0.614806i	$0.210766\pi$
$294$	2.00000	0.116642
$295$	11.0000	0.640445
$296$	6.00000	0.348743
$297$	1.00000	0.0580259
$298$	20.0000	1.15857
$299$	5.00000	0.289157
$300$	1.00000	0.0577350
$301$	−6.00000	−0.345834
$302$	0	0
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	2.00000	0.114520
$306$	0	0
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	3.00000	0.170941
$309$	−8.00000	−0.455104
$310$	−8.00000	−0.454369
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	5.00000	0.283069
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	−3.00000	−0.169300
$315$	−3.00000	−0.169031
$316$	−8.00000	−0.450035
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	−3.00000	−0.168232
$319$	−4.00000	−0.223957
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	3.00000	0.167183
$323$	0	0
$324$	1.00000	0.0555556
$325$	5.00000	0.277350
$326$	6.00000	0.332309
$327$	−8.00000	−0.442401
$328$	2.00000	0.110432
$329$	27.0000	1.48856
$330$	−1.00000	−0.0550482
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	−2.00000	−0.109764
$333$	6.00000	0.328798
$334$	−9.00000	−0.492458
$335$	−4.00000	−0.218543
$336$	3.00000	0.163663
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	−4.00000	−0.216295
$343$	−15.0000	−0.809924
$344$	−2.00000	−0.107833
$345$	−1.00000	−0.0538382
$346$	−19.0000	−1.02145
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	−4.00000	−0.214423
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	3.00000	0.160357
$351$	5.00000	0.266880
$352$	1.00000	0.0533002
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	−11.0000	−0.584643
$355$	10.0000	0.530745
$356$	−5.00000	−0.264999
$357$	0	0
$358$	−15.0000	−0.792775
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	−14.0000	−0.735824
$363$	−10.0000	−0.524864
$364$	15.0000	0.786214
$365$	−16.0000	−0.837478
$366$	−2.00000	−0.104542
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	1.00000	0.0521286
$369$	2.00000	0.104116
$370$	−6.00000	−0.311925
$371$	−9.00000	−0.467257
$372$	8.00000	0.414781
$373$	−3.00000	−0.155334	−0.0776671	−	0.996979i	$-0.524747\pi$
$373$	−3.00000	−0.155334	−0.0776671	+	0.996979i	$0.524747\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	9.00000	0.464140
$377$	−20.0000	−1.03005
$378$	3.00000	0.154303
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	4.00000	0.205196
$381$	20.0000	1.02463
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	−3.00000	−0.152894
$386$	24.0000	1.22157
$387$	−2.00000	−0.101666
$388$	−4.00000	−0.203069
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	−5.00000	−0.253185
$391$	0	0
$392$	2.00000	0.101015
$393$	13.0000	0.655763
$394$	15.0000	0.755689
$395$	8.00000	0.402524
$396$	1.00000	0.0502519
$397$	27.0000	1.35509	0.677546	−	0.735481i	$-0.263044\pi$
$397$	27.0000	1.35509	0.677546	+	0.735481i	$0.263044\pi$
$398$	−4.00000	−0.200502
$399$	−12.0000	−0.600751
$400$	1.00000	0.0500000
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	4.00000	0.199502
$403$	40.0000	1.99254
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	−12.0000	−0.595550
$407$	6.00000	0.297409
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	−2.00000	−0.0987730
$411$	−12.0000	−0.591916
$412$	−8.00000	−0.394132
$413$	−33.0000	−1.62382
$414$	1.00000	0.0491473
$415$	2.00000	0.0981761
$416$	5.00000	0.245145
$417$	20.0000	0.979404
$418$	−4.00000	−0.195646
$419$	−29.0000	−1.41674	−0.708371	−	0.705840i	$-0.750570\pi$
$419$	−29.0000	−1.41674	−0.708371	+	0.705840i	$0.750570\pi$
$420$	−3.00000	−0.146385
$421$	36.0000	1.75453	0.877266	−	0.480004i	$-0.159365\pi$
$421$	36.0000	1.75453	0.877266	+	0.480004i	$0.159365\pi$
$422$	9.00000	0.438113
$423$	9.00000	0.437595
$424$	−3.00000	−0.145693
$425$	0	0
$426$	−10.0000	−0.484502
$427$	−6.00000	−0.290360
$428$	12.0000	0.580042
$429$	5.00000	0.241402
$430$	2.00000	0.0964486
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	1.00000	0.0481125
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	24.0000	1.15204
$435$	4.00000	0.191785
$436$	−8.00000	−0.383131
$437$	−4.00000	−0.191346
$438$	16.0000	0.764510
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	−1.00000	−0.0476731
$441$	2.00000	0.0952381
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	6.00000	0.284747
$445$	5.00000	0.237023
$446$	−9.00000	−0.426162
$447$	20.0000	0.945968
$448$	3.00000	0.141737
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	1.00000	0.0471405
$451$	2.00000	0.0941763
$452$	0	0
$453$	0	0
$454$	18.0000	0.844782
$455$	−15.0000	−0.703211
$456$	−4.00000	−0.187317
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−24.0000	−1.12145
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	3.00000	0.139573
$463$	15.0000	0.697109	0.348555	−	0.937288i	$-0.386673\pi$
$463$	15.0000	0.697109	0.348555	+	0.937288i	$0.386673\pi$
$464$	−4.00000	−0.185695
$465$	−8.00000	−0.370991
$466$	24.0000	1.11178
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	5.00000	0.231125
$469$	12.0000	0.554109
$470$	−9.00000	−0.415139
$471$	−3.00000	−0.138233
$472$	−11.0000	−0.506316
$473$	−2.00000	−0.0919601
$474$	−8.00000	−0.367452
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−3.00000	−0.137361
$478$	−4.00000	−0.182956
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−1.00000	−0.0456435
$481$	30.0000	1.36788
$482$	−17.0000	−0.774329
$483$	3.00000	0.136505
$484$	−10.0000	−0.454545
$485$	4.00000	0.181631
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−2.00000	−0.0905357
$489$	6.00000	0.271329
$490$	−2.00000	−0.0903508
$491$	39.0000	1.76005	0.880023	−	0.474932i	$-0.157527\pi$
$491$	39.0000	1.76005	0.880023	+	0.474932i	$0.157527\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	−20.0000	−0.899843
$495$	−1.00000	−0.0449467
$496$	8.00000	0.359211
$497$	−30.0000	−1.34568
$498$	−2.00000	−0.0896221
$499$	−29.0000	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$500$	−1.00000	−0.0447214
$501$	−9.00000	−0.402090
$502$	−27.0000	−1.20507
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	3.00000	0.133631
$505$	−6.00000	−0.266996
$506$	1.00000	0.0444554
$507$	12.0000	0.532939
$508$	20.0000	0.887357
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	48.0000	2.12339
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−18.0000	−0.793946
$515$	8.00000	0.352522
$516$	−2.00000	−0.0880451
$517$	9.00000	0.395820
$518$	18.0000	0.790875
$519$	−19.0000	−0.834007
$520$	−5.00000	−0.219265
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−4.00000	−0.175075
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	13.0000	0.567908
$525$	3.00000	0.130931
$526$	−23.0000	−1.00285
$527$	0	0
$528$	1.00000	0.0435194
$529$	−22.0000	−0.956522
$530$	3.00000	0.130312
$531$	−11.0000	−0.477359
$532$	−12.0000	−0.520266
$533$	10.0000	0.433148
$534$	−5.00000	−0.216371
$535$	−12.0000	−0.518805
$536$	4.00000	0.172774
$537$	−15.0000	−0.647298
$538$	−14.0000	−0.603583
$539$	2.00000	0.0861461
$540$	−1.00000	−0.0430331
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	−26.0000	−1.11680
$543$	−14.0000	−0.600798
$544$	0	0
$545$	8.00000	0.342682
$546$	15.0000	0.641941
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−12.0000	−0.512615
$549$	−2.00000	−0.0853579
$550$	1.00000	0.0426401
$551$	16.0000	0.681623
$552$	1.00000	0.0425628
$553$	−24.0000	−1.02058
$554$	19.0000	0.807233
$555$	−6.00000	−0.254686
$556$	20.0000	0.848189
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	8.00000	0.338667
$559$	−10.0000	−0.422955
$560$	−3.00000	−0.126773
$561$	0	0
$562$	27.0000	1.13893
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	26.0000	1.09286
$567$	3.00000	0.125988
$568$	−10.0000	−0.419591
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	4.00000	0.167542
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	5.00000	0.209061
$573$	−12.0000	−0.501307
$574$	6.00000	0.250435
$575$	1.00000	0.0417029
$576$	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	$0.460142\pi$
$578$	0	0
$579$	24.0000	0.997406
$580$	4.00000	0.166091
$581$	−6.00000	−0.248922
$582$	−4.00000	−0.165805
$583$	−3.00000	−0.124247
$584$	16.0000	0.662085
$585$	−5.00000	−0.206725
$586$	27.0000	1.11536
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	2.00000	0.0824786
$589$	−32.0000	−1.31854
$590$	11.0000	0.452863
$591$	15.0000	0.617018
$592$	6.00000	0.246598
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	20.0000	0.819232
$597$	−4.00000	−0.163709
$598$	5.00000	0.204465
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	1.00000	0.0408248
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	−6.00000	−0.244542
$603$	4.00000	0.162893
$604$	0	0
$605$	10.0000	0.406558
$606$	6.00000	0.243733
$607$	−11.0000	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$608$	−4.00000	−0.162221
$609$	−12.0000	−0.486265
$610$	2.00000	0.0809776
$611$	45.0000	1.82051
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	22.0000	0.887848
$615$	−2.00000	−0.0806478
$616$	3.00000	0.120873
$617$	20.0000	0.805170	0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000	0.805170	0.402585	+	0.915383i	$0.368112\pi$
$618$	−8.00000	−0.321807
$619$	11.0000	0.442127	0.221064	−	0.975259i	$-0.429047\pi$
$619$	11.0000	0.442127	0.221064	+	0.975259i	$0.429047\pi$
$620$	−8.00000	−0.321288
$621$	1.00000	0.0401286
$622$	4.00000	0.160385
$623$	−15.0000	−0.600962
$624$	5.00000	0.200160
$625$	1.00000	0.0400000
$626$	16.0000	0.639489
$627$	−4.00000	−0.159745
$628$	−3.00000	−0.119713
$629$	0	0
$630$	−3.00000	−0.119523
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−8.00000	−0.318223
$633$	9.00000	0.357718
$634$	−10.0000	−0.397151
$635$	−20.0000	−0.793676
$636$	−3.00000	−0.118958
$637$	10.0000	0.396214
$638$	−4.00000	−0.158362
$639$	−10.0000	−0.395594
$640$	−1.00000	−0.0395285
$641$	−39.0000	−1.54041	−0.770204	−	0.637798i	$-0.779845\pi$
$641$	−39.0000	−1.54041	−0.770204	+	0.637798i	$0.779845\pi$
$642$	12.0000	0.473602
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	3.00000	0.118217
$645$	2.00000	0.0787499
$646$	0	0
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	1.00000	0.0392837
$649$	−11.0000	−0.431788
$650$	5.00000	0.196116
$651$	24.0000	0.940634
$652$	6.00000	0.234978
$653$	19.0000	0.743527	0.371764	−	0.928327i	$-0.378753\pi$
$653$	19.0000	0.743527	0.371764	+	0.928327i	$0.378753\pi$
$654$	−8.00000	−0.312825
$655$	−13.0000	−0.507952
$656$	2.00000	0.0780869
$657$	16.0000	0.624219
$658$	27.0000	1.05257
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	−1.00000	−0.0389249
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	17.0000	0.660724
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	12.0000	0.465340
$666$	6.00000	0.232495
$667$	−4.00000	−0.154881
$668$	−9.00000	−0.348220
$669$	−9.00000	−0.347960
$670$	−4.00000	−0.154533
$671$	−2.00000	−0.0772091
$672$	3.00000	0.115728
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−24.0000	−0.924445
$675$	1.00000	0.0384900
$676$	12.0000	0.461538
$677$	51.0000	1.96009	0.980045	−	0.198778i	$-0.0636972\pi$
$677$	51.0000	1.96009	0.980045	+	0.198778i	$0.0636972\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	18.0000	0.689761
$682$	8.00000	0.306336
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	−4.00000	−0.152944
$685$	12.0000	0.458496
$686$	−15.0000	−0.572703
$687$	−24.0000	−0.915657
$688$	−2.00000	−0.0762493
$689$	−15.0000	−0.571454
$690$	−1.00000	−0.0380693
$691$	−3.00000	−0.114125	−0.0570627	−	0.998371i	$-0.518173\pi$
$691$	−3.00000	−0.114125	−0.0570627	+	0.998371i	$0.518173\pi$
$692$	−19.0000	−0.722272
$693$	3.00000	0.113961
$694$	10.0000	0.379595
$695$	−20.0000	−0.758643
$696$	−4.00000	−0.151620
$697$	0	0
$698$	−16.0000	−0.605609
$699$	24.0000	0.907763
$700$	3.00000	0.113389
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	5.00000	0.188713
$703$	−24.0000	−0.905177
$704$	1.00000	0.0376889
$705$	−9.00000	−0.338960
$706$	−12.0000	−0.451626
$707$	18.0000	0.676960
$708$	−11.0000	−0.413405
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	10.0000	0.375293
$711$	−8.00000	−0.300023
$712$	−5.00000	−0.187383
$713$	8.00000	0.299602
$714$	0	0
$715$	−5.00000	−0.186989
$716$	−15.0000	−0.560576
$717$	−4.00000	−0.149383
$718$	−22.0000	−0.821033
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−1.00000	−0.0372678
$721$	−24.0000	−0.893807
$722$	−3.00000	−0.111648
$723$	−17.0000	−0.632237
$724$	−14.0000	−0.520306
$725$	−4.00000	−0.148556
$726$	−10.0000	−0.371135
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	15.0000	0.555937
$729$	1.00000	0.0370370
$730$	−16.0000	−0.592187
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−7.00000	−0.258375
$735$	−2.00000	−0.0737711
$736$	1.00000	0.0368605
$737$	4.00000	0.147342
$738$	2.00000	0.0736210
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	−6.00000	−0.220564
$741$	−20.0000	−0.734718
$742$	−9.00000	−0.330400
$743$	13.0000	0.476924	0.238462	−	0.971152i	$-0.423357\pi$
$743$	13.0000	0.476924	0.238462	+	0.971152i	$0.423357\pi$
$744$	8.00000	0.293294
$745$	−20.0000	−0.732743
$746$	−3.00000	−0.109838
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	36.0000	1.31541
$750$	−1.00000	−0.0365148
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	9.00000	0.328196
$753$	−27.0000	−0.983935
$754$	−20.0000	−0.728357
$755$	0	0
$756$	3.00000	0.109109
$757$	31.0000	1.12671	0.563357	−	0.826214i	$-0.309510\pi$
$757$	31.0000	1.12671	0.563357	+	0.826214i	$0.309510\pi$
$758$	20.0000	0.726433
$759$	1.00000	0.0362977
$760$	4.00000	0.145095
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	20.0000	0.724524
$763$	−24.0000	−0.868858
$764$	−12.0000	−0.434145
$765$	0	0
$766$	0	0
$767$	−55.0000	−1.98593
$768$	1.00000	0.0360844
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	−3.00000	−0.108112
$771$	−18.0000	−0.648254
$772$	24.0000	0.863779
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−2.00000	−0.0718885
$775$	8.00000	0.287368
$776$	−4.00000	−0.143592
$777$	18.0000	0.645746
$778$	12.0000	0.430221
$779$	−8.00000	−0.286630
$780$	−5.00000	−0.179029
$781$	−10.0000	−0.357828
$782$	0	0
$783$	−4.00000	−0.142948
$784$	2.00000	0.0714286
$785$	3.00000	0.107075
$786$	13.0000	0.463695
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	15.0000	0.534353
$789$	−23.0000	−0.818822
$790$	8.00000	0.284627
$791$	0	0
$792$	1.00000	0.0355335
$793$	−10.0000	−0.355110
$794$	27.0000	0.958194
$795$	3.00000	0.106399
$796$	−4.00000	−0.141776
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	−12.0000	−0.424795
$799$	0	0
$800$	1.00000	0.0353553
$801$	−5.00000	−0.176666
$802$	−9.00000	−0.317801
$803$	16.0000	0.564628
$804$	4.00000	0.141069
$805$	−3.00000	−0.105736
$806$	40.0000	1.40894
$807$	−14.0000	−0.492823
$808$	6.00000	0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−1.00000	−0.0351364
$811$	−19.0000	−0.667180	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000	−0.667180	−0.333590	+	0.942718i	$0.608260\pi$
$812$	−12.0000	−0.421117
$813$	−26.0000	−0.911860
$814$	6.00000	0.210300
$815$	−6.00000	−0.210171
$816$	0	0
$817$	8.00000	0.279885
$818$	−11.0000	−0.384606
$819$	15.0000	0.524142
$820$	−2.00000	−0.0698430
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	−12.0000	−0.418548
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	−8.00000	−0.278693
$825$	1.00000	0.0348155
$826$	−33.0000	−1.14822
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	1.00000	0.0347524
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	2.00000	0.0694210
$831$	19.0000	0.659103
$832$	5.00000	0.173344
$833$	0	0
$834$	20.0000	0.692543
$835$	9.00000	0.311458
$836$	−4.00000	−0.138343
$837$	8.00000	0.276520
$838$	−29.0000	−1.00179
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	−3.00000	−0.103510
$841$	−13.0000	−0.448276
$842$	36.0000	1.24064
$843$	27.0000	0.929929
$844$	9.00000	0.309793
$845$	−12.0000	−0.412813
$846$	9.00000	0.309426
$847$	−30.0000	−1.03081
$848$	−3.00000	−0.103020
$849$	26.0000	0.892318
$850$	0	0
$851$	6.00000	0.205677
$852$	−10.0000	−0.342594
$853$	39.0000	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$853$	39.0000	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$854$	−6.00000	−0.205316
$855$	4.00000	0.136797
$856$	12.0000	0.410152
$857$	56.0000	1.91292	0.956462	−	0.291858i	$-0.0942733\pi$
$857$	56.0000	1.91292	0.956462	+	0.291858i	$0.0942733\pi$
$858$	5.00000	0.170697
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	2.00000	0.0681994
$861$	6.00000	0.204479
$862$	8.00000	0.272481
$863$	−35.0000	−1.19141	−0.595707	−	0.803202i	$-0.703128\pi$
$863$	−35.0000	−1.19141	−0.595707	+	0.803202i	$0.703128\pi$
$864$	1.00000	0.0340207
$865$	19.0000	0.646019
$866$	0	0
$867$	0	0
$868$	24.0000	0.814613
$869$	−8.00000	−0.271381
$870$	4.00000	0.135613
$871$	20.0000	0.677674
$872$	−8.00000	−0.270914
$873$	−4.00000	−0.135379
$874$	−4.00000	−0.135302
$875$	−3.00000	−0.101419
$876$	16.0000	0.540590
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	26.0000	0.877457
$879$	27.0000	0.910687
$880$	−1.00000	−0.0337100
$881$	31.0000	1.04442	0.522208	−	0.852818i	$-0.325108\pi$
$881$	31.0000	1.04442	0.522208	+	0.852818i	$0.325108\pi$
$882$	2.00000	0.0673435
$883$	−54.0000	−1.81724	−0.908622	−	0.417619i	$-0.862865\pi$
$883$	−54.0000	−1.81724	−0.908622	+	0.417619i	$0.862865\pi$
$884$	0	0
$885$	11.0000	0.369761
$886$	14.0000	0.470339
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	6.00000	0.201347
$889$	60.0000	2.01234
$890$	5.00000	0.167600
$891$	1.00000	0.0335013
$892$	−9.00000	−0.301342
$893$	−36.0000	−1.20469
$894$	20.0000	0.668900
$895$	15.0000	0.501395
$896$	3.00000	0.100223
$897$	5.00000	0.166945
$898$	−2.00000	−0.0667409
$899$	−32.0000	−1.06726
$900$	1.00000	0.0333333
$901$	0	0
$902$	2.00000	0.0665927
$903$	−6.00000	−0.199667
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	18.0000	0.597351
$909$	6.00000	0.199007
$910$	−15.0000	−0.497245
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	−4.00000	−0.132453
$913$	−2.00000	−0.0661903
$914$	−16.0000	−0.529233
$915$	2.00000	0.0661180
$916$	−24.0000	−0.792982
$917$	39.0000	1.28789
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	−1.00000	−0.0329690
$921$	22.0000	0.724925
$922$	−14.0000	−0.461065
$923$	−50.0000	−1.64577
$924$	3.00000	0.0986928
$925$	6.00000	0.197279
$926$	15.0000	0.492931
$927$	−8.00000	−0.262754
$928$	−4.00000	−0.131306
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	−8.00000	−0.262330
$931$	−8.00000	−0.262189
$932$	24.0000	0.786146
$933$	4.00000	0.130954
$934$	6.00000	0.196326
$935$	0	0
$936$	5.00000	0.163430
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	12.0000	0.391814
$939$	16.0000	0.522140
$940$	−9.00000	−0.293548
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−3.00000	−0.0977453
$943$	2.00000	0.0651290
$944$	−11.0000	−0.358020
$945$	−3.00000	−0.0975900
$946$	−2.00000	−0.0650256
$947$	26.0000	0.844886	0.422443	−	0.906389i	$-0.361173\pi$
$947$	26.0000	0.844886	0.422443	+	0.906389i	$0.361173\pi$
$948$	−8.00000	−0.259828
$949$	80.0000	2.59691
$950$	−4.00000	−0.129777
$951$	−10.0000	−0.324272
$952$	0	0
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	−3.00000	−0.0971286
$955$	12.0000	0.388311
$956$	−4.00000	−0.129369
$957$	−4.00000	−0.129302
$958$	−24.0000	−0.775405
$959$	−36.0000	−1.16250
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	30.0000	0.967239
$963$	12.0000	0.386695
$964$	−17.0000	−0.547533
$965$	−24.0000	−0.772587
$966$	3.00000	0.0965234
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	4.00000	0.128432
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	1.00000	0.0320750
$973$	60.0000	1.92351
$974$	8.00000	0.256337
$975$	5.00000	0.160128
$976$	−2.00000	−0.0640184
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	6.00000	0.191859
$979$	−5.00000	−0.159801
$980$	−2.00000	−0.0638877
$981$	−8.00000	−0.255420
$982$	39.0000	1.24454
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	2.00000	0.0637577
$985$	−15.0000	−0.477940
$986$	0	0
$987$	27.0000	0.859419
$988$	−20.0000	−0.636285
$989$	−2.00000	−0.0635963
$990$	−1.00000	−0.0317821
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	8.00000	0.254000
$993$	17.0000	0.539479
$994$	−30.0000	−0.951542
$995$	4.00000	0.126809
$996$	−2.00000	−0.0633724
$997$	−35.0000	−1.10846	−0.554231	−	0.832363i	$-0.686987\pi$
$997$	−35.0000	−1.10846	−0.554231	+	0.832363i	$0.686987\pi$
$998$	−29.0000	−0.917979
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.x.1.1	yes	1
17.16	even	2		8670.2.a.q.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.q.1.1	✓	1	17.16	even	2
8670.2.a.x.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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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