LMFDB - Embedded newform 510.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("510.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.07237050309$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	2.00000	0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	−2.00000	−0.436436
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	−2.00000	−0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	4.00000	0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	1.00000	0.140028
$52$	4.00000	0.554700
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	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
$65$	−4.00000	−0.496139
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−1.00000	−0.121268
$69$	−4.00000	−0.481543
$70$	−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	$0.615871\pi$
$72$	1.00000	0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−4.00000	−0.452911
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−2.00000	−0.218218
$85$	1.00000	0.108465
$86$	10.0000	1.07833
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−1.00000	−0.105409
$91$	8.00000	0.838628
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	1.00000	0.0990148
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	4.00000	0.392232
$105$	2.00000	0.195180
$106$	2.00000	0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	−4.00000	−0.373002
$116$	2.00000	0.185695
$117$	4.00000	0.369800
$118$	−2.00000	−0.184115
$119$	−2.00000	−0.183340
$120$	1.00000	0.0912871
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	4.00000	0.360668
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	−10.0000	−0.880451
$130$	−4.00000	−0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	2.00000	0.172774
$135$	1.00000	0.0860663
$136$	−1.00000	−0.0857493
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−4.00000	−0.340503
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−4.00000	−0.331042
$147$	3.00000	0.247436
$148$	−2.00000	−0.164399
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	4.00000	0.324443
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−12.0000	−0.954669
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	8.00000	0.630488
$162$	1.00000	0.0785674
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	8.00000	0.620920
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	1.00000	0.0766965
$171$	4.00000	0.305888
$172$	10.0000	0.762493
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	−2.00000	−0.151620
$175$	2.00000	0.151186
$176$	0	0
$177$	2.00000	0.150329
$178$	−10.0000	−0.749532
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−1.00000	−0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	8.00000	0.592999
$183$	14.0000	1.03491
$184$	4.00000	0.294884
$185$	2.00000	0.147043
$186$	0	0
$187$	0	0
$188$	−8.00000	−0.583460
$189$	−2.00000	−0.145479
$190$	−4.00000	−0.290191
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	8.00000	0.574367
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	−2.00000	−0.141069
$202$	−4.00000	−0.281439
$203$	4.00000	0.280745
$204$	1.00000	0.0700140
$205$	4.00000	0.279372
$206$	8.00000	0.557386
$207$	4.00000	0.278019
$208$	4.00000	0.277350
$209$	0	0
$210$	2.00000	0.138013
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	2.00000	0.137361
$213$	6.00000	0.411113
$214$	−12.0000	−0.820303
$215$	−10.0000	−0.681994
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000	0.406371
$219$	4.00000	0.270295
$220$	0	0
$221$	−4.00000	−0.269069
$222$	2.00000	0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	2.00000	0.131306
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	4.00000	0.261488
$235$	8.00000	0.521862
$236$	−2.00000	−0.130189
$237$	12.0000	0.779484
$238$	−2.00000	−0.129641
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	3.00000	0.191663
$246$	4.00000	0.255031
$247$	16.0000	1.01806
$248$	0	0
$249$	−8.00000	−0.506979
$250$	−1.00000	−0.0632456
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	16.0000	1.00393
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−10.0000	−0.622573
$259$	−4.00000	−0.248548
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	12.0000	0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	8.00000	0.490511
$267$	10.0000	0.611990
$268$	2.00000	0.122169
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	1.00000	0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−1.00000	−0.0606339
$273$	−8.00000	−0.484182
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−4.00000	−0.240772
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	−2.00000	−0.119523
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	8.00000	0.476393
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−6.00000	−0.356034
$285$	4.00000	0.236940
$286$	0	0
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	−2.00000	−0.117444
$291$	−8.00000	−0.468968
$292$	−4.00000	−0.234082
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	3.00000	0.174964
$295$	2.00000	0.116445
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−12.0000	−0.695141
$299$	16.0000	0.925304
$300$	−1.00000	−0.0577350
$301$	20.0000	1.15278
$302$	8.00000	0.460348
$303$	4.00000	0.229794
$304$	4.00000	0.229416
$305$	14.0000	0.801638
$306$	−1.00000	−0.0571662
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	−4.00000	−0.226455
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	4.00000	0.225733
$315$	−2.00000	−0.112687
$316$	−12.0000	−0.675053
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−2.00000	−0.112154
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	8.00000	0.445823
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	24.0000	1.32924
$327$	−6.00000	−0.331801
$328$	−4.00000	−0.220863
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	8.00000	0.439057
$333$	−2.00000	−0.109599
$334$	−8.00000	−0.437741
$335$	−2.00000	−0.109272
$336$	−2.00000	−0.109109
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	3.00000	0.163178
$339$	6.00000	0.325875
$340$	1.00000	0.0542326
$341$	0	0
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	10.0000	0.539164
$345$	4.00000	0.215353
$346$	−2.00000	−0.107521
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−2.00000	−0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	2.00000	0.106904
$351$	−4.00000	−0.213504
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	2.00000	0.106299
$355$	6.00000	0.318447
$356$	−10.0000	−0.529999
$357$	2.00000	0.105851
$358$	18.0000	0.951330
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	11.0000	0.577350
$364$	8.00000	0.419314
$365$	4.00000	0.209370
$366$	14.0000	0.731792
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	4.00000	0.208514
$369$	−4.00000	−0.208232
$370$	2.00000	0.103975
$371$	4.00000	0.207670
$372$	0	0
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	8.00000	0.412021
$378$	−2.00000	−0.102869
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−4.00000	−0.205196
$381$	−16.0000	−0.819705
$382$	12.0000	0.613973
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−8.00000	−0.407189
$387$	10.0000	0.508329
$388$	8.00000	0.406138
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	4.00000	0.202548
$391$	−4.00000	−0.202289
$392$	−3.00000	−0.151523
$393$	−12.0000	−0.605320
$394$	6.00000	0.302276
$395$	12.0000	0.603786
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	16.0000	0.802008
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−4.00000	−0.199007
$405$	−1.00000	−0.0496904
$406$	4.00000	0.198517
$407$	0	0
$408$	1.00000	0.0495074
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	4.00000	0.197546
$411$	18.0000	0.887875
$412$	8.00000	0.394132
$413$	−4.00000	−0.196827
$414$	4.00000	0.196589
$415$	−8.00000	−0.392705
$416$	4.00000	0.196116
$417$	12.0000	0.587643
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	2.00000	0.0975900
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	12.0000	0.584151
$423$	−8.00000	−0.388973
$424$	2.00000	0.0971286
$425$	−1.00000	−0.0485071
$426$	6.00000	0.290701
$427$	−28.0000	−1.35501
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−10.0000	−0.482243
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	−1.00000	−0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	6.00000	0.287348
$437$	16.0000	0.765384
$438$	4.00000	0.191127
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−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	$0.560864\pi$
$444$	2.00000	0.0949158
$445$	10.0000	0.474045
$446$	−16.0000	−0.757622
$447$	12.0000	0.567581
$448$	2.00000	0.0944911
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	−12.0000	−0.563188
$455$	−8.00000	−0.375046
$456$	−4.00000	−0.187317
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	−2.00000	−0.0934539
$459$	1.00000	0.0466760
$460$	−4.00000	−0.186501
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	14.0000	0.648537
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	4.00000	0.184900
$469$	4.00000	0.184703
$470$	8.00000	0.369012
$471$	−4.00000	−0.184310
$472$	−2.00000	−0.0920575
$473$	0	0
$474$	12.0000	0.551178
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	2.00000	0.0915737
$478$	−24.0000	−1.09773
$479$	2.00000	0.0913823	0.0456912	−	0.998956i	$-0.485451\pi$
$479$	2.00000	0.0913823	0.0456912	+	0.998956i	$0.485451\pi$
$480$	1.00000	0.0456435
$481$	−8.00000	−0.364769
$482$	−30.0000	−1.36646
$483$	−8.00000	−0.364013
$484$	−11.0000	−0.500000
$485$	−8.00000	−0.363261
$486$	−1.00000	−0.0453609
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−14.0000	−0.633750
$489$	−24.0000	−1.08532
$490$	3.00000	0.135526
$491$	−14.0000	−0.631811	−0.315906	−	0.948791i	$-0.602308\pi$
$491$	−14.0000	−0.631811	−0.315906	+	0.948791i	$0.602308\pi$
$492$	4.00000	0.180334
$493$	−2.00000	−0.0900755
$494$	16.0000	0.719874
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	−8.00000	−0.358489
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	−2.00000	−0.0892644
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	2.00000	0.0890871
$505$	4.00000	0.177998
$506$	0	0
$507$	−3.00000	−0.133235
$508$	16.0000	0.709885
$509$	4.00000	0.177297	0.0886484	−	0.996063i	$-0.471745\pi$
$509$	4.00000	0.177297	0.0886484	+	0.996063i	$0.471745\pi$
$510$	−1.00000	−0.0442807
$511$	−8.00000	−0.353899
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−14.0000	−0.617514
$515$	−8.00000	−0.352522
$516$	−10.0000	−0.440225
$517$	0	0
$518$	−4.00000	−0.175750
$519$	2.00000	0.0877903
$520$	−4.00000	−0.175412
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	2.00000	0.0875376
$523$	−18.0000	−0.787085	−0.393543	−	0.919306i	$-0.628751\pi$
$523$	−18.0000	−0.787085	−0.393543	+	0.919306i	$0.628751\pi$
$524$	12.0000	0.524222
$525$	−2.00000	−0.0872872
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−2.00000	−0.0868744
$531$	−2.00000	−0.0867926
$532$	8.00000	0.346844
$533$	−16.0000	−0.693037
$534$	10.0000	0.432742
$535$	12.0000	0.518805
$536$	2.00000	0.0863868
$537$	−18.0000	−0.776757
$538$	10.0000	0.431131
$539$	0	0
$540$	1.00000	0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−8.00000	−0.343629
$543$	10.0000	0.429141
$544$	−1.00000	−0.0428746
$545$	−6.00000	−0.257012
$546$	−8.00000	−0.342368
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	−18.0000	−0.768922
$549$	−14.0000	−0.597505
$550$	0	0
$551$	8.00000	0.340811
$552$	−4.00000	−0.170251
$553$	−24.0000	−1.02058
$554$	10.0000	0.424859
$555$	−2.00000	−0.0848953
$556$	−12.0000	−0.508913
$557$	−46.0000	−1.94908	−0.974541	−	0.224208i	$-0.928020\pi$
$557$	−46.0000	−1.94908	−0.974541	+	0.224208i	$0.928020\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	22.0000	0.928014
$563$	−32.0000	−1.34864	−0.674320	−	0.738440i	$-0.735563\pi$
$563$	−32.0000	−1.34864	−0.674320	+	0.738440i	$0.735563\pi$
$564$	8.00000	0.336861
$565$	6.00000	0.252422
$566$	−28.0000	−1.17693
$567$	2.00000	0.0839921
$568$	−6.00000	−0.251754
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	4.00000	0.167542
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	−8.00000	−0.333914
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	1.00000	0.0415945
$579$	8.00000	0.332469
$580$	−2.00000	−0.0830455
$581$	16.0000	0.663792
$582$	−8.00000	−0.331611
$583$	0	0
$584$	−4.00000	−0.165521
$585$	−4.00000	−0.165380
$586$	22.0000	0.908812
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	2.00000	0.0823387
$591$	−6.00000	−0.246807
$592$	−2.00000	−0.0821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	−12.0000	−0.491539
$597$	−16.0000	−0.654836
$598$	16.0000	0.654289
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	−1.00000	−0.0408248
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	20.0000	0.815139
$603$	2.00000	0.0814463
$604$	8.00000	0.325515
$605$	11.0000	0.447214
$606$	4.00000	0.162489
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	4.00000	0.162221
$609$	−4.00000	−0.162088
$610$	14.0000	0.566843
$611$	−32.0000	−1.29458
$612$	−1.00000	−0.0404226
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	10.0000	0.403567
$615$	−4.00000	−0.161296
$616$	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
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−10.0000	−0.400963
$623$	−20.0000	−0.801283
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	4.00000	0.159617
$629$	2.00000	0.0797452
$630$	−2.00000	−0.0796819
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−12.0000	−0.477334
$633$	−12.0000	−0.476957
$634$	30.0000	1.19145
$635$	−16.0000	−0.634941
$636$	−2.00000	−0.0793052
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−6.00000	−0.237356
$640$	−1.00000	−0.0395285
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	12.0000	0.473602
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	8.00000	0.315244
$645$	10.0000	0.393750
$646$	−4.00000	−0.157378
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	4.00000	0.156893
$651$	0	0
$652$	24.0000	0.939913
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−6.00000	−0.234619
$655$	−12.0000	−0.468879
$656$	−4.00000	−0.156174
$657$	−4.00000	−0.156055
$658$	−16.0000	−0.623745
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	−16.0000	−0.621858
$663$	4.00000	0.155347
$664$	8.00000	0.310460
$665$	−8.00000	−0.310227
$666$	−2.00000	−0.0774984
$667$	8.00000	0.309761
$668$	−8.00000	−0.309529
$669$	16.0000	0.618596
$670$	−2.00000	−0.0772667
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	12.0000	0.462223
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	6.00000	0.230429
$679$	16.0000	0.614024
$680$	1.00000	0.0383482
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	4.00000	0.152944
$685$	18.0000	0.687745
$686$	−20.0000	−0.763604
$687$	2.00000	0.0763048
$688$	10.0000	0.381246
$689$	8.00000	0.304776
$690$	4.00000	0.152277
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−20.0000	−0.759190
$695$	12.0000	0.455186
$696$	−2.00000	−0.0758098
$697$	4.00000	0.151511
$698$	2.00000	0.0757011
$699$	−14.0000	−0.529529
$700$	2.00000	0.0755929
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	−4.00000	−0.150970
$703$	−8.00000	−0.301726
$704$	0	0
$705$	−8.00000	−0.301297
$706$	18.0000	0.677439
$707$	−8.00000	−0.300871
$708$	2.00000	0.0751646
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	6.00000	0.225176
$711$	−12.0000	−0.450035
$712$	−10.0000	−0.374766
$713$	0	0
$714$	2.00000	0.0748481
$715$	0	0
$716$	18.0000	0.672692
$717$	24.0000	0.896296
$718$	20.0000	0.746393
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	−3.00000	−0.111648
$723$	30.0000	1.11571
$724$	−10.0000	−0.371647
$725$	2.00000	0.0742781
$726$	11.0000	0.408248
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−10.0000	−0.369863
$732$	14.0000	0.517455
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	−2.00000	−0.0738213
$735$	−3.00000	−0.110657
$736$	4.00000	0.147442
$737$	0	0
$738$	−4.00000	−0.147242
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	2.00000	0.0735215
$741$	−16.0000	−0.587775
$742$	4.00000	0.146845
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	8.00000	0.292901
$747$	8.00000	0.292705
$748$	0	0
$749$	−24.0000	−0.876941
$750$	1.00000	0.0365148
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−8.00000	−0.291730
$753$	2.00000	0.0728841
$754$	8.00000	0.291343
$755$	−8.00000	−0.291150
$756$	−2.00000	−0.0727393
$757$	52.0000	1.88997	0.944986	−	0.327111i	$-0.106075\pi$
$757$	52.0000	1.88997	0.944986	+	0.327111i	$0.106075\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	−16.0000	−0.579619
$763$	12.0000	0.434429
$764$	12.0000	0.434145
$765$	1.00000	0.0361551
$766$	16.0000	0.578103
$767$	−8.00000	−0.288863
$768$	−1.00000	−0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−8.00000	−0.287926
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	8.00000	0.287183
$777$	4.00000	0.143499
$778$	12.0000	0.430221
$779$	−16.0000	−0.573259
$780$	4.00000	0.143223
$781$	0	0
$782$	−4.00000	−0.143040
$783$	−2.00000	−0.0714742
$784$	−3.00000	−0.107143
$785$	−4.00000	−0.142766
$786$	−12.0000	−0.428026
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	6.00000	0.213741
$789$	−24.0000	−0.854423
$790$	12.0000	0.426941
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−56.0000	−1.98862
$794$	−10.0000	−0.354887
$795$	2.00000	0.0709327
$796$	16.0000	0.567105
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	−8.00000	−0.283197
$799$	8.00000	0.283020
$800$	1.00000	0.0353553
$801$	−10.0000	−0.353333
$802$	−24.0000	−0.847469
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−10.0000	−0.352017
$808$	−4.00000	−0.140720
$809$	−28.0000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$809$	−28.0000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$810$	−1.00000	−0.0351364
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	4.00000	0.140372
$813$	8.00000	0.280572
$814$	0	0
$815$	−24.0000	−0.840683
$816$	1.00000	0.0350070
$817$	40.0000	1.39942
$818$	−26.0000	−0.909069
$819$	8.00000	0.279543
$820$	4.00000	0.139686
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	18.0000	0.627822
$823$	−50.0000	−1.74289	−0.871445	−	0.490493i	$-0.836817\pi$
$823$	−50.0000	−1.74289	−0.871445	+	0.490493i	$0.836817\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	4.00000	0.139010
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−8.00000	−0.277684
$831$	−10.0000	−0.346896
$832$	4.00000	0.138675
$833$	3.00000	0.103944
$834$	12.0000	0.415526
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	−4.00000	−0.138178
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	2.00000	0.0690066
$841$	−25.0000	−0.862069
$842$	−34.0000	−1.17172
$843$	−22.0000	−0.757720
$844$	12.0000	0.413057
$845$	−3.00000	−0.103203
$846$	−8.00000	−0.275046
$847$	−22.0000	−0.755929
$848$	2.00000	0.0686803
$849$	28.0000	0.960958
$850$	−1.00000	−0.0342997
$851$	−8.00000	−0.274236
$852$	6.00000	0.205557
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	−28.0000	−0.958140
$855$	−4.00000	−0.136797
$856$	−12.0000	−0.410152
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	−10.0000	−0.340997
$861$	8.00000	0.272639
$862$	−34.0000	−1.15804
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	−1.00000	−0.0340207
$865$	2.00000	0.0680020
$866$	−34.0000	−1.15537
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	2.00000	0.0678064
$871$	8.00000	0.271070
$872$	6.00000	0.203186
$873$	8.00000	0.270759
$874$	16.0000	0.541208
$875$	−2.00000	−0.0676123
$876$	4.00000	0.135147
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	−12.0000	−0.404980
$879$	−22.0000	−0.742042
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	−3.00000	−0.101015
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	−4.00000	−0.134535
$885$	−2.00000	−0.0672293
$886$	−8.00000	−0.268765
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	2.00000	0.0671156
$889$	32.0000	1.07325
$890$	10.0000	0.335201
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−32.0000	−1.07084
$894$	12.0000	0.401340
$895$	−18.0000	−0.601674
$896$	2.00000	0.0668153
$897$	−16.0000	−0.534224
$898$	40.0000	1.33482
$899$	0	0
$900$	1.00000	0.0333333
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	−20.0000	−0.665558
$904$	−6.00000	−0.199557
$905$	10.0000	0.332411
$906$	−8.00000	−0.265782
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−12.0000	−0.398234
$909$	−4.00000	−0.132672
$910$	−8.00000	−0.265197
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−34.0000	−1.12462
$915$	−14.0000	−0.462826
$916$	−2.00000	−0.0660819
$917$	24.0000	0.792550
$918$	1.00000	0.0330049
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−4.00000	−0.131876
$921$	−10.0000	−0.329511
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	16.0000	0.525793
$927$	8.00000	0.262754
$928$	2.00000	0.0656532
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	14.0000	0.458585
$933$	10.0000	0.327385
$934$	−36.0000	−1.17796
$935$	0	0
$936$	4.00000	0.130744
$937$	54.0000	1.76410	0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000	1.76410	0.882052	+	0.471153i	$0.156162\pi$
$938$	4.00000	0.130605
$939$	16.0000	0.522140
$940$	8.00000	0.260931
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	−4.00000	−0.130327
$943$	−16.0000	−0.521032
$944$	−2.00000	−0.0650945
$945$	2.00000	0.0650600
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	12.0000	0.389742
$949$	−16.0000	−0.519382
$950$	4.00000	0.129777
$951$	−30.0000	−0.972817
$952$	−2.00000	−0.0648204
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	2.00000	0.0647524
$955$	−12.0000	−0.388311
$956$	−24.0000	−0.776215
$957$	0	0
$958$	2.00000	0.0646171
$959$	−36.0000	−1.16250
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	−12.0000	−0.386695
$964$	−30.0000	−0.966235
$965$	8.00000	0.257529
$966$	−8.00000	−0.257396
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	−11.0000	−0.353553
$969$	4.00000	0.128499
$970$	−8.00000	−0.256865
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	−1.00000	−0.0320750
$973$	−24.0000	−0.769405
$974$	−22.0000	−0.704925
$975$	−4.00000	−0.128103
$976$	−14.0000	−0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−24.0000	−0.767435
$979$	0	0
$980$	3.00000	0.0958315
$981$	6.00000	0.191565
$982$	−14.0000	−0.446758
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	4.00000	0.127515
$985$	−6.00000	−0.191176
$986$	−2.00000	−0.0636930
$987$	16.0000	0.509286
$988$	16.0000	0.509028
$989$	40.0000	1.27193
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	16.0000	0.507745
$994$	−12.0000	−0.380617
$995$	−16.0000	−0.507234
$996$	−8.00000	−0.253490
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−12.0000	−0.379853
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	510.2.a.d.1.1	✓	1
3.2	odd	2		1530.2.a.h.1.1		1
4.3	odd	2		4080.2.a.r.1.1		1
5.2	odd	4		2550.2.d.f.2449.2		2
5.3	odd	4		2550.2.d.f.2449.1		2
5.4	even	2		2550.2.a.i.1.1		1
15.14	odd	2		7650.2.a.bn.1.1		1
17.16	even	2		8670.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.d.1.1	✓	1	1.1	even	1	trivial
1530.2.a.h.1.1		1	3.2	odd	2
2550.2.a.i.1.1		1	5.4	even	2
2550.2.d.f.2449.1		2	5.3	odd	4
2550.2.d.f.2449.2		2	5.2	odd	4
4080.2.a.r.1.1		1	4.3	odd	2
7650.2.a.bn.1.1		1	15.14	odd	2
8670.2.a.y.1.1		1	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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