LMFDB - Embedded newform 507.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.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:	$4.04841538248$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} -3.00000 q^{24} -4.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +7.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} -1.00000 q^{37} -6.00000 q^{38} +3.00000 q^{40} -9.00000 q^{41} -2.00000 q^{42} +6.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} +7.00000 q^{51} -9.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -6.00000 q^{56} -6.00000 q^{57} +1.00000 q^{58} +1.00000 q^{60} +1.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} +7.00000 q^{68} +6.00000 q^{69} +2.00000 q^{70} -6.00000 q^{71} +3.00000 q^{72} -11.0000 q^{73} +1.00000 q^{74} +4.00000 q^{75} -6.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +14.0000 q^{83} -2.00000 q^{84} -7.00000 q^{85} -6.00000 q^{86} +1.00000 q^{87} +6.00000 q^{88} +14.0000 q^{89} -1.00000 q^{90} +6.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} +6.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−3.00000	−0.612372
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	1.00000	0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−5.00000	−0.883883
$33$	−2.00000	−0.348155
$34$	7.00000	1.20049
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−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	$0.348747\pi$
$44$	−2.00000	−0.301511
$45$	1.00000	0.149071
$46$	6.00000	0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	7.00000	0.980196
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	−6.00000	−0.801784
$57$	−6.00000	−0.794719
$58$	1.00000	0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	4.00000	0.508001
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	2.00000	0.246183
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	7.00000	0.848875
$69$	6.00000	0.722315
$70$	2.00000	0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	3.00000	0.353553
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	1.00000	0.116248
$75$	4.00000	0.461880
$76$	−6.00000	−0.688247
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	−2.00000	−0.218218
$85$	−7.00000	−0.759257
$86$	−6.00000	−0.646997
$87$	1.00000	0.107211
$88$	6.00000	0.639602
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	4.00000	0.414781
$94$	6.00000	0.618853
$95$	6.00000	0.615587
$96$	5.00000	0.510310
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	3.00000	0.303046
$99$	2.00000	0.201008
$100$	4.00000	0.400000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	−7.00000	−0.693103
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	9.00000	0.874157
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−2.00000	−0.190693
$111$	1.00000	0.0949158
$112$	2.00000	0.188982
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	6.00000	0.561951
$115$	−6.00000	−0.559503
$116$	1.00000	0.0928477
$117$	0	0
$118$	0	0
$119$	14.0000	1.28338
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	−1.00000	−0.0905357
$123$	9.00000	0.811503
$124$	4.00000	0.359211
$125$	−9.00000	−0.804984
$126$	2.00000	0.178174
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	3.00000	0.265165
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	2.00000	0.174078
$133$	−12.0000	−1.04053
$134$	−2.00000	−0.172774
$135$	−1.00000	−0.0860663
$136$	−21.0000	−1.80074
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	−6.00000	−0.510754
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	2.00000	0.169031
$141$	6.00000	0.505291
$142$	6.00000	0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−1.00000	−0.0830455
$146$	11.0000	0.910366
$147$	3.00000	0.247436
$148$	1.00000	0.0821995
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	−4.00000	−0.326599
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	18.0000	1.45999
$153$	−7.00000	−0.565916
$154$	4.00000	0.322329
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	4.00000	0.318223
$159$	9.00000	0.713746
$160$	−5.00000	−0.395285
$161$	12.0000	0.945732
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	9.00000	0.702782
$165$	−2.00000	−0.155700
$166$	−14.0000	−1.08661
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	6.00000	0.462910
$169$	0	0
$170$	7.00000	0.536875
$171$	6.00000	0.458831
$172$	−6.00000	−0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−1.00000	−0.0758098
$175$	8.00000	0.604743
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−14.0000	−1.04934
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	−1.00000	−0.0745356
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	−18.0000	−1.32698
$185$	−1.00000	−0.0735215
$186$	−4.00000	−0.293294
$187$	−14.0000	−1.02378
$188$	6.00000	0.437595
$189$	2.00000	0.145479
$190$	−6.00000	−0.435286
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−7.00000	−0.505181
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−2.00000	−0.142134
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−12.0000	−0.848528
$201$	−2.00000	−0.141069
$202$	−3.00000	−0.211079
$203$	2.00000	0.140372
$204$	−7.00000	−0.490098
$205$	−9.00000	−0.628587
$206$	−6.00000	−0.418040
$207$	−6.00000	−0.417029
$208$	0	0
$209$	12.0000	0.830057
$210$	−2.00000	−0.138013
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	9.00000	0.618123
$213$	6.00000	0.411113
$214$	6.00000	0.410152
$215$	6.00000	0.409197
$216$	−3.00000	−0.204124
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	11.0000	0.743311
$220$	−2.00000	−0.134840
$221$	0	0
$222$	−1.00000	−0.0671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	10.0000	0.668153
$225$	−4.00000	−0.266667
$226$	15.0000	0.997785
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	6.00000	0.397360
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	6.00000	0.395628
$231$	4.00000	0.263181
$232$	−3.00000	−0.196960
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	4.00000	0.259828
$238$	−14.0000	−0.907485
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	1.00000	0.0645497
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	−3.00000	−0.191663
$246$	−9.00000	−0.573819
$247$	0	0
$248$	−12.0000	−0.762001
$249$	−14.0000	−0.887214
$250$	9.00000	0.569210
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000	0.125988
$253$	−12.0000	−0.754434
$254$	−20.0000	−1.25491
$255$	7.00000	0.438357
$256$	−17.0000	−1.06250
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	6.00000	0.373544
$259$	2.00000	0.124274
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	8.00000	0.494242
$263$	−30.0000	−1.84988	−0.924940	−	0.380114i	$-0.875885\pi$
$263$	−30.0000	−1.84988	−0.924940	+	0.380114i	$0.875885\pi$
$264$	−6.00000	−0.369274
$265$	−9.00000	−0.552866
$266$	12.0000	0.735767
$267$	−14.0000	−0.856786
$268$	−2.00000	−0.122169
$269$	−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
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	−3.00000	−0.181237
$275$	−8.00000	−0.482418
$276$	−6.00000	−0.361158
$277$	−31.0000	−1.86261	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−31.0000	−1.86261	−0.931305	+	0.364241i	$0.881328\pi$
$278$	−12.0000	−0.719712
$279$	−4.00000	−0.239474
$280$	−6.00000	−0.358569
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	−6.00000	−0.357295
$283$	−18.0000	−1.06999	−0.534994	−	0.844856i	$-0.679686\pi$
$283$	−18.0000	−1.06999	−0.534994	+	0.844856i	$0.679686\pi$
$284$	6.00000	0.356034
$285$	−6.00000	−0.355409
$286$	0	0
$287$	18.0000	1.06251
$288$	−5.00000	−0.294628
$289$	32.0000	1.88235
$290$	1.00000	0.0587220
$291$	−2.00000	−0.117242
$292$	11.0000	0.643726
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−3.00000	−0.174371
$297$	−2.00000	−0.116052
$298$	3.00000	0.173785
$299$	0	0
$300$	−4.00000	−0.230940
$301$	−12.0000	−0.691669
$302$	−2.00000	−0.115087
$303$	−3.00000	−0.172345
$304$	−6.00000	−0.344124
$305$	1.00000	0.0572598
$306$	7.00000	0.400163
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	4.00000	0.227921
$309$	−6.00000	−0.341328
$310$	4.00000	0.227185
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	3.00000	0.169300
$315$	−2.00000	−0.112687
$316$	4.00000	0.225018
$317$	25.0000	1.40414	0.702070	−	0.712108i	$-0.252259\pi$
$317$	25.0000	1.40414	0.702070	+	0.712108i	$0.252259\pi$
$318$	−9.00000	−0.504695
$319$	−2.00000	−0.111979
$320$	7.00000	0.391312
$321$	6.00000	0.334887
$322$	−12.0000	−0.668734
$323$	−42.0000	−2.33694
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−2.00000	−0.110600
$328$	−27.0000	−1.49083
$329$	12.0000	0.661581
$330$	2.00000	0.110096
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−14.0000	−0.768350
$333$	−1.00000	−0.0547997
$334$	16.0000	0.875481
$335$	2.00000	0.109272
$336$	−2.00000	−0.109109
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	7.00000	0.379628
$341$	−8.00000	−0.433224
$342$	−6.00000	−0.324443
$343$	20.0000	1.07990
$344$	18.0000	0.970495
$345$	6.00000	0.323029
$346$	6.00000	0.322562
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	−1.00000	−0.0536056
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	−10.0000	−0.533002
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−14.0000	−0.741999
$357$	−14.0000	−0.740959
$358$	2.00000	0.105703
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	3.00000	0.158114
$361$	17.0000	0.894737
$362$	7.00000	0.367912
$363$	7.00000	0.367405
$364$	0	0
$365$	−11.0000	−0.575766
$366$	1.00000	0.0522708
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	6.00000	0.312772
$369$	−9.00000	−0.468521
$370$	1.00000	0.0519875
$371$	18.0000	0.934513
$372$	−4.00000	−0.207390
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	14.0000	0.723923
$375$	9.00000	0.464758
$376$	−18.0000	−0.928279
$377$	0	0
$378$	−2.00000	−0.102869
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	−6.00000	−0.307794
$381$	−20.0000	−1.02463
$382$	4.00000	0.204658
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−3.00000	−0.153093
$385$	−4.00000	−0.203859
$386$	−9.00000	−0.458088
$387$	6.00000	0.304997
$388$	−2.00000	−0.101535
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	42.0000	2.12403
$392$	−9.00000	−0.454569
$393$	8.00000	0.403547
$394$	6.00000	0.302276
$395$	−4.00000	−0.201262
$396$	−2.00000	−0.100504
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−14.0000	−0.701757
$399$	12.0000	0.600751
$400$	4.00000	0.200000
$401$	−1.00000	−0.0499376	−0.0249688	−	0.999688i	$-0.507949\pi$
$401$	−1.00000	−0.0499376	−0.0249688	+	0.999688i	$0.507949\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	−3.00000	−0.149256
$405$	1.00000	0.0496904
$406$	−2.00000	−0.0992583
$407$	−2.00000	−0.0991363
$408$	21.0000	1.03965
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	9.00000	0.444478
$411$	−3.00000	−0.147979
$412$	−6.00000	−0.295599
$413$	0	0
$414$	6.00000	0.294884
$415$	14.0000	0.687233
$416$	0	0
$417$	−12.0000	−0.587643
$418$	−12.0000	−0.586939
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	−2.00000	−0.0975900
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	8.00000	0.389434
$423$	−6.00000	−0.291730
$424$	−27.0000	−1.31124
$425$	28.0000	1.35820
$426$	−6.00000	−0.290701
$427$	−2.00000	−0.0967868
$428$	6.00000	0.290021
$429$	0	0
$430$	−6.00000	−0.289346
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	1.00000	0.0481125
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	−8.00000	−0.384012
$435$	1.00000	0.0479463
$436$	−2.00000	−0.0957826
$437$	−36.0000	−1.72211
$438$	−11.0000	−0.525600
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	6.00000	0.286039
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−1.00000	−0.0474579
$445$	14.0000	0.663664
$446$	16.0000	0.757622
$447$	3.00000	0.141895
$448$	−14.0000	−0.661438
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	4.00000	0.188562
$451$	−18.0000	−0.847587
$452$	15.0000	0.705541
$453$	−2.00000	−0.0939682
$454$	−14.0000	−0.657053
$455$	0	0
$456$	−18.0000	−0.842927
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	−22.0000	−1.02799
$459$	7.00000	0.326732
$460$	6.00000	0.279751
$461$	−19.0000	−0.884918	−0.442459	−	0.896789i	$-0.645894\pi$
$461$	−19.0000	−0.884918	−0.442459	+	0.896789i	$0.645894\pi$
$462$	−4.00000	−0.186097
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	1.00000	0.0464238
$465$	4.00000	0.185496
$466$	−10.0000	−0.463241
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	6.00000	0.276759
$471$	3.00000	0.138233
$472$	0	0
$473$	12.0000	0.551761
$474$	−4.00000	−0.183726
$475$	−24.0000	−1.10120
$476$	−14.0000	−0.641689
$477$	−9.00000	−0.412082
$478$	30.0000	1.37217
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	5.00000	0.228218
$481$	0	0
$482$	7.00000	0.318841
$483$	−12.0000	−0.546019
$484$	7.00000	0.318182
$485$	2.00000	0.0908153
$486$	1.00000	0.0453609
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	3.00000	0.135804
$489$	−4.00000	−0.180886
$490$	3.00000	0.135526
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−9.00000	−0.405751
$493$	7.00000	0.315264
$494$	0	0
$495$	2.00000	0.0898933
$496$	4.00000	0.179605
$497$	12.0000	0.538274
$498$	14.0000	0.627355
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	9.00000	0.402492
$501$	16.0000	0.714827
$502$	−12.0000	−0.535586
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	−6.00000	−0.267261
$505$	3.00000	0.133498
$506$	12.0000	0.533465
$507$	0	0
$508$	−20.0000	−0.887357
$509$	−7.00000	−0.310270	−0.155135	−	0.987893i	$-0.549581\pi$
$509$	−7.00000	−0.310270	−0.155135	+	0.987893i	$0.549581\pi$
$510$	−7.00000	−0.309965
$511$	22.0000	0.973223
$512$	11.0000	0.486136
$513$	−6.00000	−0.264906
$514$	7.00000	0.308757
$515$	6.00000	0.264392
$516$	6.00000	0.264135
$517$	−12.0000	−0.527759
$518$	−2.00000	−0.0878750
$519$	6.00000	0.263371
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	1.00000	0.0437688
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	8.00000	0.349482
$525$	−8.00000	−0.349149
$526$	30.0000	1.30806
$527$	28.0000	1.21970
$528$	2.00000	0.0870388
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	0	0
$532$	12.0000	0.520266
$533$	0	0
$534$	14.0000	0.605839
$535$	−6.00000	−0.259403
$536$	6.00000	0.259161
$537$	2.00000	0.0863064
$538$	14.0000	0.603583
$539$	−6.00000	−0.258438
$540$	1.00000	0.0430331
$541$	−45.0000	−1.93470	−0.967351	−	0.253442i	$-0.918437\pi$
$541$	−45.0000	−1.93470	−0.967351	+	0.253442i	$0.918437\pi$
$542$	0	0
$543$	7.00000	0.300399
$544$	35.0000	1.50061
$545$	2.00000	0.0856706
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−3.00000	−0.128154
$549$	1.00000	0.0426790
$550$	8.00000	0.341121
$551$	−6.00000	−0.255609
$552$	18.0000	0.766131
$553$	8.00000	0.340195
$554$	31.0000	1.31706
$555$	1.00000	0.0424476
$556$	−12.0000	−0.508913
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	2.00000	0.0845154
$561$	14.0000	0.591080
$562$	−19.0000	−0.801467
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−6.00000	−0.252646
$565$	−15.0000	−0.631055
$566$	18.0000	0.756596
$567$	−2.00000	−0.0839921
$568$	−18.0000	−0.755263
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	6.00000	0.251312
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	−18.0000	−0.751305
$575$	24.0000	1.00087
$576$	7.00000	0.291667
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−32.0000	−1.33102
$579$	−9.00000	−0.374027
$580$	1.00000	0.0415227
$581$	−28.0000	−1.16164
$582$	2.00000	0.0829027
$583$	−18.0000	−0.745484
$584$	−33.0000	−1.36555
$585$	0	0
$586$	−9.00000	−0.371787
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	−3.00000	−0.123718
$589$	−24.0000	−0.988903
$590$	0	0
$591$	6.00000	0.246807
$592$	1.00000	0.0410997
$593$	−13.0000	−0.533846	−0.266923	−	0.963718i	$-0.586007\pi$
$593$	−13.0000	−0.533846	−0.266923	+	0.963718i	$0.586007\pi$
$594$	2.00000	0.0820610
$595$	14.0000	0.573944
$596$	3.00000	0.122885
$597$	−14.0000	−0.572982
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	12.0000	0.489898
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	12.0000	0.489083
$603$	2.00000	0.0814463
$604$	−2.00000	−0.0813788
$605$	−7.00000	−0.284590
$606$	3.00000	0.121867
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−30.0000	−1.21666
$609$	−2.00000	−0.0810441
$610$	−1.00000	−0.0404888
$611$	0	0
$612$	7.00000	0.282958
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	14.0000	0.564994
$615$	9.00000	0.362915
$616$	−12.0000	−0.483494
$617$	−13.0000	−0.523360	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	−13.0000	−0.523360	−0.261680	+	0.965155i	$0.584277\pi$
$618$	6.00000	0.241355
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	4.00000	0.160644
$621$	6.00000	0.240772
$622$	−18.0000	−0.721734
$623$	−28.0000	−1.12180
$624$	0	0
$625$	11.0000	0.440000
$626$	−6.00000	−0.239808
$627$	−12.0000	−0.479234
$628$	3.00000	0.119713
$629$	7.00000	0.279108
$630$	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	$0.630328\pi$
$632$	−12.0000	−0.477334
$633$	8.00000	0.317971
$634$	−25.0000	−0.992877
$635$	20.0000	0.793676
$636$	−9.00000	−0.356873
$637$	0	0
$638$	2.00000	0.0791808
$639$	−6.00000	−0.237356
$640$	3.00000	0.118585
$641$	−31.0000	−1.22443	−0.612213	−	0.790693i	$-0.709721\pi$
$641$	−31.0000	−1.22443	−0.612213	+	0.790693i	$0.709721\pi$
$642$	−6.00000	−0.236801
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	−12.0000	−0.472866
$645$	−6.00000	−0.236250
$646$	42.0000	1.65247
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	−4.00000	−0.156652
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	2.00000	0.0782062
$655$	−8.00000	−0.312586
$656$	9.00000	0.351391
$657$	−11.0000	−0.429151
$658$	−12.0000	−0.467809
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	2.00000	0.0778499
$661$	−45.0000	−1.75030	−0.875149	−	0.483854i	$-0.839236\pi$
$661$	−45.0000	−1.75030	−0.875149	+	0.483854i	$0.839236\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	42.0000	1.62992
$665$	−12.0000	−0.465340
$666$	1.00000	0.0387492
$667$	6.00000	0.232321
$668$	16.0000	0.619059
$669$	16.0000	0.618596
$670$	−2.00000	−0.0772667
$671$	2.00000	0.0772091
$672$	−10.0000	−0.385758
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	33.0000	1.27111
$675$	4.00000	0.153960
$676$	0	0
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	−15.0000	−0.576072
$679$	−4.00000	−0.153506
$680$	−21.0000	−0.805313
$681$	−14.0000	−0.536481
$682$	8.00000	0.306336
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−6.00000	−0.229416
$685$	3.00000	0.114624
$686$	−20.0000	−0.763604
$687$	−22.0000	−0.839352
$688$	−6.00000	−0.228748
$689$	0	0
$690$	−6.00000	−0.228416
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	6.00000	0.228086
$693$	−4.00000	−0.151947
$694$	−18.0000	−0.683271
$695$	12.0000	0.455186
$696$	3.00000	0.113715
$697$	63.0000	2.38630
$698$	−26.0000	−0.984115
$699$	−10.0000	−0.378235
$700$	−8.00000	−0.302372
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	14.0000	0.527645
$705$	6.00000	0.225973
$706$	−11.0000	−0.413990
$707$	−6.00000	−0.225653
$708$	0	0
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	6.00000	0.225176
$711$	−4.00000	−0.150012
$712$	42.0000	1.57402
$713$	24.0000	0.898807
$714$	14.0000	0.523937
$715$	0	0
$716$	2.00000	0.0747435
$717$	30.0000	1.12037
$718$	−18.0000	−0.671754
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	−1.00000	−0.0372678
$721$	−12.0000	−0.446903
$722$	−17.0000	−0.632674
$723$	7.00000	0.260333
$724$	7.00000	0.260153
$725$	4.00000	0.148556
$726$	−7.00000	−0.259794
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	11.0000	0.407128
$731$	−42.0000	−1.55343
$732$	1.00000	0.0369611
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	−10.0000	−0.369107
$735$	3.00000	0.110657
$736$	30.0000	1.10581
$737$	4.00000	0.147342
$738$	9.00000	0.331295
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−18.0000	−0.660801
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	12.0000	0.439941
$745$	−3.00000	−0.109911
$746$	11.0000	0.402739
$747$	14.0000	0.512233
$748$	14.0000	0.511891
$749$	12.0000	0.438470
$750$	−9.00000	−0.328634
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	6.00000	0.218797
$753$	−12.0000	−0.437304
$754$	0	0
$755$	2.00000	0.0727875
$756$	−2.00000	−0.0727393
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−36.0000	−1.30758
$759$	12.0000	0.435572
$760$	18.0000	0.652929
$761$	−50.0000	−1.81250	−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000	−1.81250	−0.906249	+	0.422744i	$0.861067\pi$
$762$	20.0000	0.724524
$763$	−4.00000	−0.144810
$764$	4.00000	0.144715
$765$	−7.00000	−0.253086
$766$	8.00000	0.289052
$767$	0	0
$768$	17.0000	0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	4.00000	0.144150
$771$	7.00000	0.252099
$772$	−9.00000	−0.323917
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	−6.00000	−0.215666
$775$	16.0000	0.574737
$776$	6.00000	0.215387
$777$	−2.00000	−0.0717496
$778$	−19.0000	−0.681183
$779$	−54.0000	−1.93475
$780$	0	0
$781$	−12.0000	−0.429394
$782$	−42.0000	−1.50192
$783$	1.00000	0.0357371
$784$	3.00000	0.107143
$785$	−3.00000	−0.107075
$786$	−8.00000	−0.285351
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	6.00000	0.213741
$789$	30.0000	1.06803
$790$	4.00000	0.142314
$791$	30.0000	1.06668
$792$	6.00000	0.213201
$793$	0	0
$794$	−34.0000	−1.20661
$795$	9.00000	0.319197
$796$	−14.0000	−0.496217
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−12.0000	−0.424795
$799$	42.0000	1.48585
$800$	20.0000	0.707107
$801$	14.0000	0.494666
$802$	1.00000	0.0353112
$803$	−22.0000	−0.776363
$804$	2.00000	0.0705346
$805$	12.0000	0.422944
$806$	0	0
$807$	14.0000	0.492823
$808$	9.00000	0.316619
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	−1.00000	−0.0351364
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	2.00000	0.0701000
$815$	4.00000	0.140114
$816$	−7.00000	−0.245049
$817$	36.0000	1.25948
$818$	7.00000	0.244749
$819$	0	0
$820$	9.00000	0.314294
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	3.00000	0.104637
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	18.0000	0.627060
$825$	8.00000	0.278524
$826$	0	0
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	6.00000	0.208514
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	−14.0000	−0.485947
$831$	31.0000	1.07538
$832$	0	0
$833$	21.0000	0.727607
$834$	12.0000	0.415526
$835$	−16.0000	−0.553703
$836$	−12.0000	−0.415029
$837$	4.00000	0.138260
$838$	−16.0000	−0.552711
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	6.00000	0.207020
$841$	−28.0000	−0.965517
$842$	−19.0000	−0.654783
$843$	−19.0000	−0.654395
$844$	8.00000	0.275371
$845$	0	0
$846$	6.00000	0.206284
$847$	14.0000	0.481046
$848$	9.00000	0.309061
$849$	18.0000	0.617758
$850$	−28.0000	−0.960392
$851$	6.00000	0.205677
$852$	−6.00000	−0.205557
$853$	−21.0000	−0.719026	−0.359513	−	0.933140i	$-0.617057\pi$
$853$	−21.0000	−0.719026	−0.359513	+	0.933140i	$0.617057\pi$
$854$	2.00000	0.0684386
$855$	6.00000	0.205196
$856$	−18.0000	−0.615227
$857$	−31.0000	−1.05894	−0.529470	−	0.848329i	$-0.677609\pi$
$857$	−31.0000	−1.05894	−0.529470	+	0.848329i	$0.677609\pi$
$858$	0	0
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	−6.00000	−0.204598
$861$	−18.0000	−0.613438
$862$	−30.0000	−1.02180
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	5.00000	0.170103
$865$	−6.00000	−0.204006
$866$	−19.0000	−0.645646
$867$	−32.0000	−1.08678
$868$	−8.00000	−0.271538
$869$	−8.00000	−0.271381
$870$	−1.00000	−0.0339032
$871$	0	0
$872$	6.00000	0.203186
$873$	2.00000	0.0676897
$874$	36.0000	1.21772
$875$	18.0000	0.608511
$876$	−11.0000	−0.371656
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	−14.0000	−0.472477
$879$	−9.00000	−0.303562
$880$	−2.00000	−0.0674200
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	3.00000	0.101015
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	4.00000	0.134383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	3.00000	0.100673
$889$	−40.0000	−1.34156
$890$	−14.0000	−0.469281
$891$	2.00000	0.0670025
$892$	16.0000	0.535720
$893$	−36.0000	−1.20469
$894$	−3.00000	−0.100335
$895$	−2.00000	−0.0668526
$896$	−6.00000	−0.200446
$897$	0	0
$898$	34.0000	1.13459
$899$	4.00000	0.133407
$900$	4.00000	0.133333
$901$	63.0000	2.09883
$902$	18.0000	0.599334
$903$	12.0000	0.399335
$904$	−45.0000	−1.49668
$905$	−7.00000	−0.232688
$906$	2.00000	0.0664455
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−14.0000	−0.464606
$909$	3.00000	0.0995037
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	6.00000	0.198680
$913$	28.0000	0.926665
$914$	−13.0000	−0.430002
$915$	−1.00000	−0.0330590
$916$	−22.0000	−0.726900
$917$	16.0000	0.528367
$918$	−7.00000	−0.231034
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	−18.0000	−0.593442
$921$	14.0000	0.461316
$922$	19.0000	0.625732
$923$	0	0
$924$	−4.00000	−0.131590
$925$	4.00000	0.131519
$926$	−26.0000	−0.854413
$927$	6.00000	0.197066
$928$	5.00000	0.164133
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	−4.00000	−0.131165
$931$	−18.0000	−0.589926
$932$	−10.0000	−0.327561
$933$	−18.0000	−0.589294
$934$	−6.00000	−0.196326
$935$	−14.0000	−0.457849
$936$	0	0
$937$	−49.0000	−1.60076	−0.800380	−	0.599493i	$-0.795369\pi$
$937$	−49.0000	−1.60076	−0.800380	+	0.599493i	$0.795369\pi$
$938$	4.00000	0.130605
$939$	−6.00000	−0.195803
$940$	6.00000	0.195698
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	−3.00000	−0.0977453
$943$	54.0000	1.75848
$944$	0	0
$945$	2.00000	0.0650600
$946$	−12.0000	−0.390154
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	24.0000	0.778663
$951$	−25.0000	−0.810681
$952$	42.0000	1.36123
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	9.00000	0.291386
$955$	−4.00000	−0.129437
$956$	30.0000	0.970269
$957$	2.00000	0.0646508
$958$	24.0000	0.775405
$959$	−6.00000	−0.193750
$960$	−7.00000	−0.225924
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−6.00000	−0.193347
$964$	7.00000	0.225455
$965$	9.00000	0.289720
$966$	12.0000	0.386094
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	−21.0000	−0.674966
$969$	42.0000	1.34923
$970$	−2.00000	−0.0642161
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000	0.0320750
$973$	−24.0000	−0.769405
$974$	18.0000	0.576757
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−33.0000	−1.05576	−0.527882	−	0.849318i	$-0.677014\pi$
$977$	−33.0000	−1.05576	−0.527882	+	0.849318i	$0.677014\pi$
$978$	4.00000	0.127906
$979$	28.0000	0.894884
$980$	3.00000	0.0958315
$981$	2.00000	0.0638551
$982$	−6.00000	−0.191468
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	27.0000	0.860729
$985$	−6.00000	−0.191176
$986$	−7.00000	−0.222925
$987$	−12.0000	−0.381964
$988$	0	0
$989$	−36.0000	−1.14473
$990$	−2.00000	−0.0635642
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	20.0000	0.635001
$993$	−4.00000	−0.126936
$994$	−12.0000	−0.380617
$995$	14.0000	0.443830
$996$	14.0000	0.443607
$997$	−35.0000	−1.10846	−0.554231	−	0.832363i	$-0.686987\pi$
$997$	−35.0000	−1.10846	−0.554231	+	0.832363i	$0.686987\pi$
$998$	24.0000	0.759707
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.a.b.1.1		1
3.2	odd	2		1521.2.a.d.1.1		1
4.3	odd	2		8112.2.a.bc.1.1		1
13.2	odd	12		507.2.j.d.316.1		4
13.3	even	3		507.2.e.c.22.1		2
13.4	even	6		39.2.e.a.16.1	✓	2
13.5	odd	4		507.2.b.b.337.2		2
13.6	odd	12		507.2.j.d.361.2		4
13.7	odd	12		507.2.j.d.361.1		4
13.8	odd	4		507.2.b.b.337.1		2
13.9	even	3		507.2.e.c.484.1		2
13.10	even	6		39.2.e.a.22.1	yes	2
13.11	odd	12		507.2.j.d.316.2		4
13.12	even	2		507.2.a.c.1.1		1
39.5	even	4		1521.2.b.c.1351.1		2
39.8	even	4		1521.2.b.c.1351.2		2
39.17	odd	6		117.2.g.b.55.1		2
39.23	odd	6		117.2.g.b.100.1		2
39.38	odd	2		1521.2.a.a.1.1		1
52.23	odd	6		624.2.q.c.529.1		2
52.43	odd	6		624.2.q.c.289.1		2
52.51	odd	2		8112.2.a.w.1.1		1
65.4	even	6		975.2.i.f.601.1		2
65.17	odd	12		975.2.bb.d.874.2		4
65.23	odd	12		975.2.bb.d.724.2		4
65.43	odd	12		975.2.bb.d.874.1		4
65.49	even	6		975.2.i.f.451.1		2
65.62	odd	12		975.2.bb.d.724.1		4
156.23	even	6		1872.2.t.j.1153.1		2
156.95	even	6		1872.2.t.j.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.a.16.1	✓	2	13.4	even	6
39.2.e.a.22.1	yes	2	13.10	even	6
117.2.g.b.55.1		2	39.17	odd	6
117.2.g.b.100.1		2	39.23	odd	6
507.2.a.b.1.1		1	1.1	even	1	trivial
507.2.a.c.1.1		1	13.12	even	2
507.2.b.b.337.1		2	13.8	odd	4
507.2.b.b.337.2		2	13.5	odd	4
507.2.e.c.22.1		2	13.3	even	3
507.2.e.c.484.1		2	13.9	even	3
507.2.j.d.316.1		4	13.2	odd	12
507.2.j.d.316.2		4	13.11	odd	12
507.2.j.d.361.1		4	13.7	odd	12
507.2.j.d.361.2		4	13.6	odd	12
624.2.q.c.289.1		2	52.43	odd	6
624.2.q.c.529.1		2	52.23	odd	6
975.2.i.f.451.1		2	65.49	even	6
975.2.i.f.601.1		2	65.4	even	6
975.2.bb.d.724.1		4	65.62	odd	12
975.2.bb.d.724.2		4	65.23	odd	12
975.2.bb.d.874.1		4	65.43	odd	12
975.2.bb.d.874.2		4	65.17	odd	12
1521.2.a.a.1.1		1	39.38	odd	2
1521.2.a.d.1.1		1	3.2	odd	2
1521.2.b.c.1351.1		2	39.5	even	4
1521.2.b.c.1351.2		2	39.8	even	4
1872.2.t.j.289.1		2	156.95	even	6
1872.2.t.j.1153.1		2	156.23	even	6
8112.2.a.w.1.1		1	52.51	odd	2
8112.2.a.bc.1.1		1	4.3	odd	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more