LMFDB - Embedded newform 462.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, 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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	462.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:	$3.68908857338$
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	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -4.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +6.00000 q^{57} -6.00000 q^{58} -2.00000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +16.0000 q^{71} -1.00000 q^{72} +12.0000 q^{73} -10.0000 q^{74} -5.00000 q^{75} +6.00000 q^{76} +1.00000 q^{77} -6.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} -8.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -6.00000 q^{91} -4.00000 q^{92} -2.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−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	$0.258380\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−6.00000	−0.973329
$39$	6.00000	0.960769
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	1.00000	0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	−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
$49$	1.00000	0.142857
$50$	5.00000	0.707107
$51$	4.00000	0.560112
$52$	6.00000	0.832050
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	6.00000	0.794719
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	2.00000	0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.00000	0.485071
$69$	−4.00000	−0.481543
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	−1.00000	−0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−10.0000	−1.16248
$75$	−5.00000	−0.577350
$76$	6.00000	0.688247
$77$	1.00000	0.113961
$78$	−6.00000	−0.679366
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−8.00000	−0.862662
$87$	6.00000	0.643268
$88$	1.00000	0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	−4.00000	−0.417029
$93$	−2.00000	−0.207390
$94$	6.00000	0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−1.00000	−0.101015
$99$	−1.00000	−0.100504
$100$	−5.00000	−0.500000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−4.00000	−0.396059
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	−4.00000	−0.360668
$124$	−2.00000	−0.179605
$125$	0	0
$126$	1.00000	0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	−1.00000	−0.0870388
$133$	−6.00000	−0.520266
$134$	4.00000	0.345547
$135$	0	0
$136$	−4.00000	−0.342997
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	4.00000	0.340503
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−16.0000	−1.34269
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	−12.0000	−0.993127
$147$	1.00000	0.0824786
$148$	10.0000	0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	5.00000	0.408248
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−6.00000	−0.486664
$153$	4.00000	0.323381
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	6.00000	0.480384
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	16.0000	1.27289
$159$	−10.0000	−0.793052
$160$	0	0
$161$	4.00000	0.315244
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	2.00000	0.155230
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	1.00000	0.0771517
$169$	23.0000	1.76923
$170$	0	0
$171$	6.00000	0.458831
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−0.454859
$175$	5.00000	0.377964
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	6.00000	0.444750
$183$	−2.00000	−0.147844
$184$	4.00000	0.294884
$185$	0	0
$186$	2.00000	0.146647
$187$	−4.00000	−0.292509
$188$	−6.00000	−0.437595
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−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
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	1.00000	0.0710669
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	5.00000	0.353553
$201$	−4.00000	−0.282138
$202$	14.0000	0.985037
$203$	−6.00000	−0.421117
$204$	4.00000	0.280056
$205$	0	0
$206$	14.0000	0.975426
$207$	−4.00000	−0.278019
$208$	6.00000	0.416025
$209$	−6.00000	−0.415029
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−10.0000	−0.686803
$213$	16.0000	1.09630
$214$	−8.00000	−0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	2.00000	0.135769
$218$	14.0000	0.948200
$219$	12.0000	0.810885
$220$	0	0
$221$	24.0000	1.61441
$222$	−10.0000	−0.671156
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	1.00000	0.0668153
$225$	−5.00000	−0.333333
$226$	2.00000	0.133038
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	6.00000	0.397360
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	−6.00000	−0.393919
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	4.00000	0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	4.00000	0.255031
$247$	36.0000	2.29063
$248$	2.00000	0.127000
$249$	−2.00000	−0.126745
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	−8.00000	−0.498058
$259$	−10.0000	−0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	22.0000	1.35916
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	6.00000	0.367884
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	4.00000	0.242536
$273$	−6.00000	−0.363137
$274$	−22.0000	−1.32907
$275$	5.00000	0.301511
$276$	−4.00000	−0.240772
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	14.0000	0.839664
$279$	−2.00000	−0.119737
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	6.00000	0.357295
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	6.00000	0.354787
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−6.00000	−0.351726
$292$	12.0000	0.702247
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	−10.0000	−0.581238
$297$	−1.00000	−0.0580259
$298$	6.00000	0.347571
$299$	−24.0000	−1.38796
$300$	−5.00000	−0.288675
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−14.0000	−0.804279
$304$	6.00000	0.344124
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−30.0000	−1.71219	−0.856095	−	0.516818i	$-0.827116\pi$
$307$	−30.0000	−1.71219	−0.856095	+	0.516818i	$0.827116\pi$
$308$	1.00000	0.0569803
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−6.00000	−0.339683
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	10.0000	0.560772
$319$	−6.00000	−0.335936
$320$	0	0
$321$	8.00000	0.446516
$322$	−4.00000	−0.222911
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	−30.0000	−1.66410
$326$	4.00000	0.221540
$327$	−14.0000	−0.774202
$328$	4.00000	0.220863
$329$	6.00000	0.330791
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−2.00000	−0.109764
$333$	10.0000	0.547997
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−23.0000	−1.25104
$339$	−2.00000	−0.108625
$340$	0	0
$341$	2.00000	0.108306
$342$	−6.00000	−0.324443
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	6.00000	0.321634
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−5.00000	−0.267261
$351$	6.00000	0.320256
$352$	1.00000	0.0533002
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−4.00000	−0.211702
$358$	−4.00000	−0.211407
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−8.00000	−0.420471
$363$	1.00000	0.0524864
$364$	−6.00000	−0.314485
$365$	0	0
$366$	2.00000	0.104542
$367$	−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
$370$	0	0
$371$	10.0000	0.519174
$372$	−2.00000	−0.103695
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	6.00000	0.309426
$377$	36.0000	1.85409
$378$	1.00000	0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	12.0000	0.613973
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	8.00000	0.406663
$388$	−6.00000	−0.304604
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	−1.00000	−0.0505076
$393$	−22.0000	−1.10975
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	14.0000	0.701757
$399$	−6.00000	−0.300376
$400$	−5.00000	−0.250000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	4.00000	0.199502
$403$	−12.0000	−0.597763
$404$	−14.0000	−0.696526
$405$	0	0
$406$	6.00000	0.297775
$407$	−10.0000	−0.495682
$408$	−4.00000	−0.198030
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	−14.0000	−0.689730
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	−6.00000	−0.294174
$417$	−14.0000	−0.685583
$418$	6.00000	0.293470
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−8.00000	−0.389434
$423$	−6.00000	−0.291730
$424$	10.0000	0.485643
$425$	−20.0000	−0.970143
$426$	−16.0000	−0.775203
$427$	2.00000	0.0967868
$428$	8.00000	0.386695
$429$	−6.00000	−0.289683
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	1.00000	0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−24.0000	−1.14808
$438$	−12.0000	−0.573382
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−24.0000	−1.14156
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	26.0000	1.23114
$447$	−6.00000	−0.283790
$448$	−1.00000	−0.0472456
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	5.00000	0.235702
$451$	4.00000	0.188353
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−6.00000	−0.280976
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−8.00000	−0.373815
$459$	4.00000	0.186704
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	−1.00000	−0.0465242
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−22.0000	−1.01913
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	6.00000	0.277350
$469$	4.00000	0.184703
$470$	0	0
$471$	12.0000	0.552931
$472$	0	0
$473$	−8.00000	−0.367840
$474$	16.0000	0.734904
$475$	−30.0000	−1.37649
$476$	−4.00000	−0.183340
$477$	−10.0000	−0.457869
$478$	−8.00000	−0.365911
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	60.0000	2.73576
$482$	20.0000	0.910975
$483$	4.00000	0.182006
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	2.00000	0.0905357
$489$	−4.00000	−0.180886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−4.00000	−0.180334
$493$	24.0000	1.08091
$494$	−36.0000	−1.61972
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	−16.0000	−0.717698
$498$	2.00000	0.0896221
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−4.00000	−0.177822
$507$	23.0000	1.02147
$508$	−16.0000	−0.709885
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	−1.00000	−0.0441942
$513$	6.00000	0.264906
$514$	−10.0000	−0.441081
$515$	0	0
$516$	8.00000	0.352180
$517$	6.00000	0.263880
$518$	10.0000	0.439375
$519$	6.00000	0.263371
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	−6.00000	−0.262613
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	−22.0000	−0.961074
$525$	5.00000	0.218218
$526$	0	0
$527$	−8.00000	−0.348485
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−6.00000	−0.260133
$533$	−24.0000	−1.03956
$534$	6.00000	0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	4.00000	0.172613
$538$	24.0000	1.03471
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−12.0000	−0.515444
$543$	8.00000	0.343313
$544$	−4.00000	−0.171499
$545$	0	0
$546$	6.00000	0.256776
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	22.0000	0.939793
$549$	−2.00000	−0.0853579
$550$	−5.00000	−0.213201
$551$	36.0000	1.53365
$552$	4.00000	0.170251
$553$	16.0000	0.680389
$554$	6.00000	0.254916
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	2.00000	0.0846668
$559$	48.0000	2.03018
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−14.0000	−0.590554
$563$	−22.0000	−0.927189	−0.463595	−	0.886047i	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−18.0000	−0.756596
$567$	−1.00000	−0.0419961
$568$	−16.0000	−0.671345
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−6.00000	−0.250873
$573$	−12.0000	−0.501307
$574$	−4.00000	−0.166957
$575$	20.0000	0.834058
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	1.00000	0.0415945
$579$	−6.00000	−0.249351
$580$	0	0
$581$	2.00000	0.0829740
$582$	6.00000	0.248708
$583$	10.0000	0.414158
$584$	−12.0000	−0.496564
$585$	0	0
$586$	−14.0000	−0.578335
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	1.00000	0.0412393
$589$	−12.0000	−0.494451
$590$	0	0
$591$	6.00000	0.246807
$592$	10.0000	0.410997
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−14.0000	−0.572982
$598$	24.0000	0.981433
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	5.00000	0.204124
$601$	−32.0000	−1.30531	−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000	−1.30531	−0.652654	+	0.757656i	$0.726344\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	0	0
$605$	0	0
$606$	14.0000	0.568711
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−6.00000	−0.243332
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−36.0000	−1.45640
$612$	4.00000	0.161690
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	30.0000	1.21070
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	14.0000	0.563163
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	18.0000	0.721734
$623$	6.00000	0.240385
$624$	6.00000	0.240192
$625$	25.0000	1.00000
$626$	−6.00000	−0.239808
$627$	−6.00000	−0.239617
$628$	12.0000	0.478852
$629$	40.0000	1.59490
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	16.0000	0.636446
$633$	8.00000	0.317971
$634$	18.0000	0.714871
$635$	0	0
$636$	−10.0000	−0.396526
$637$	6.00000	0.237729
$638$	6.00000	0.237542
$639$	16.0000	0.632950
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	−8.00000	−0.315735
$643$	−24.0000	−0.946468	−0.473234	−	0.880937i	$-0.656913\pi$
$643$	−24.0000	−0.946468	−0.473234	+	0.880937i	$0.656913\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−34.0000	−1.33668	−0.668339	−	0.743857i	$-0.732994\pi$
$647$	−34.0000	−1.33668	−0.668339	+	0.743857i	$0.732994\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	30.0000	1.17670
$651$	2.00000	0.0783862
$652$	−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	$0.537456\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−4.00000	−0.156174
$657$	12.0000	0.468165
$658$	−6.00000	−0.233904
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	4.00000	0.155464
$663$	24.0000	0.932083
$664$	2.00000	0.0776151
$665$	0	0
$666$	−10.0000	−0.387492
$667$	−24.0000	−0.929284
$668$	12.0000	0.464294
$669$	−26.0000	−1.00522
$670$	0	0
$671$	2.00000	0.0772091
$672$	1.00000	0.0385758
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−22.0000	−0.847408
$675$	−5.00000	−0.192450
$676$	23.0000	0.884615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	2.00000	0.0768095
$679$	6.00000	0.230259
$680$	0	0
$681$	18.0000	0.689761
$682$	−2.00000	−0.0765840
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	1.00000	0.0381802
$687$	8.00000	0.305219
$688$	8.00000	0.304997
$689$	−60.0000	−2.28582
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	6.00000	0.228086
$693$	1.00000	0.0379869
$694$	−16.0000	−0.607352
$695$	0	0
$696$	−6.00000	−0.227429
$697$	−16.0000	−0.606043
$698$	−26.0000	−0.984115
$699$	22.0000	0.832116
$700$	5.00000	0.188982
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	−6.00000	−0.226455
$703$	60.0000	2.26294
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	14.0000	0.526524
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	6.00000	0.224860
$713$	8.00000	0.299602
$714$	4.00000	0.149696
$715$	0	0
$716$	4.00000	0.149487
$717$	8.00000	0.298765
$718$	−8.00000	−0.298557
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	−17.0000	−0.632674
$723$	−20.0000	−0.743808
$724$	8.00000	0.297318
$725$	−30.0000	−1.11417
$726$	−1.00000	−0.0371135
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.0000	1.18356
$732$	−2.00000	−0.0739221
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	4.00000	0.147442
$737$	4.00000	0.147342
$738$	4.00000	0.147242
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	36.0000	1.32249
$742$	−10.0000	−0.367112
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	10.0000	0.366126
$747$	−2.00000	−0.0731762
$748$	−4.00000	−0.146254
$749$	−8.00000	−0.292314
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−36.0000	−1.31104
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	20.0000	0.726433
$759$	4.00000	0.145191
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	16.0000	0.579619
$763$	14.0000	0.506834
$764$	−12.0000	−0.434145
$765$	0	0
$766$	34.0000	1.22847
$767$	0	0
$768$	1.00000	0.0360844
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	10.0000	0.360141
$772$	−6.00000	−0.215945
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	−8.00000	−0.287554
$775$	10.0000	0.359211
$776$	6.00000	0.215387
$777$	−10.0000	−0.358748
$778$	30.0000	1.07555
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−16.0000	−0.572525
$782$	16.0000	0.572159
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	0	0
$786$	22.0000	0.784714
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	0	0
$791$	2.00000	0.0711118
$792$	1.00000	0.0355335
$793$	−12.0000	−0.426132
$794$	4.00000	0.141955
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	6.00000	0.212398
$799$	−24.0000	−0.849059
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	−30.0000	−1.05934
$803$	−12.0000	−0.423471
$804$	−4.00000	−0.141069
$805$	0	0
$806$	12.0000	0.422682
$807$	−24.0000	−0.844840
$808$	14.0000	0.492518
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	−6.00000	−0.210559
$813$	12.0000	0.420858
$814$	10.0000	0.350500
$815$	0	0
$816$	4.00000	0.140028
$817$	48.0000	1.67931
$818$	−16.0000	−0.559427
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−22.0000	−0.767338
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	14.0000	0.487713
$825$	5.00000	0.174078
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−4.00000	−0.139010
$829$	24.0000	0.833554	0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000	0.833554	0.416777	+	0.909009i	$0.363160\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	6.00000	0.208013
$833$	4.00000	0.138592
$834$	14.0000	0.484780
$835$	0	0
$836$	−6.00000	−0.207514
$837$	−2.00000	−0.0691301
$838$	−4.00000	−0.138178
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	14.0000	0.482186
$844$	8.00000	0.275371
$845$	0	0
$846$	6.00000	0.206284
$847$	−1.00000	−0.0343604
$848$	−10.0000	−0.343401
$849$	18.0000	0.617758
$850$	20.0000	0.685994
$851$	−40.0000	−1.37118
$852$	16.0000	0.548151
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−8.00000	−0.273434
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	6.00000	0.204837
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	−40.0000	−1.36241
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	−1.00000	−0.0339618
$868$	2.00000	0.0678844
$869$	16.0000	0.542763
$870$	0	0
$871$	−24.0000	−0.813209
$872$	14.0000	0.474100
$873$	−6.00000	−0.203069
$874$	24.0000	0.811812
$875$	0	0
$876$	12.0000	0.405442
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	20.0000	0.674967
$879$	14.0000	0.472208
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	−1.00000	−0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	4.00000	0.134383
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−10.0000	−0.335578
$889$	16.0000	0.536623
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−26.0000	−0.870544
$893$	−36.0000	−1.20469
$894$	6.00000	0.200670
$895$	0	0
$896$	1.00000	0.0334077
$897$	−24.0000	−0.801337
$898$	34.0000	1.13459
$899$	−12.0000	−0.400222
$900$	−5.00000	−0.166667
$901$	−40.0000	−1.33259
$902$	−4.00000	−0.133185
$903$	−8.00000	−0.266223
$904$	2.00000	0.0665190
$905$	0	0
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	18.0000	0.597351
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	6.00000	0.198680
$913$	2.00000	0.0661903
$914$	−10.0000	−0.330771
$915$	0	0
$916$	8.00000	0.264327
$917$	22.0000	0.726504
$918$	−4.00000	−0.132020
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	−2.00000	−0.0658665
$923$	96.0000	3.15988
$924$	1.00000	0.0328976
$925$	−50.0000	−1.64399
$926$	−16.0000	−0.525793
$927$	−14.0000	−0.459820
$928$	−6.00000	−0.196960
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	22.0000	0.720634
$933$	−18.0000	−0.589294
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−6.00000	−0.196116
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	−4.00000	−0.130605
$939$	6.00000	0.195803
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−12.0000	−0.390981
$943$	16.0000	0.521032
$944$	0	0
$945$	0	0
$946$	8.00000	0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−16.0000	−0.519656
$949$	72.0000	2.33722
$950$	30.0000	0.973329
$951$	−18.0000	−0.583690
$952$	4.00000	0.129641
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	8.00000	0.258738
$957$	−6.00000	−0.193952
$958$	24.0000	0.775405
$959$	−22.0000	−0.710417
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−60.0000	−1.93448
$963$	8.00000	0.257796
$964$	−20.0000	−0.644157
$965$	0	0
$966$	−4.00000	−0.128698
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−1.00000	−0.0321412
$969$	24.0000	0.770991
$970$	0	0
$971$	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
$973$	14.0000	0.448819
$974$	−20.0000	−0.640841
$975$	−30.0000	−0.960769
$976$	−2.00000	−0.0640184
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	4.00000	0.127906
$979$	6.00000	0.191761
$980$	0	0
$981$	−14.0000	−0.446986
$982$	−12.0000	−0.382935
$983$	10.0000	0.318950	0.159475	−	0.987202i	$-0.449020\pi$
$983$	10.0000	0.318950	0.159475	+	0.987202i	$0.449020\pi$
$984$	4.00000	0.127515
$985$	0	0
$986$	−24.0000	−0.764316
$987$	6.00000	0.190982
$988$	36.0000	1.14531
$989$	−32.0000	−1.01754
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	2.00000	0.0635001
$993$	−4.00000	−0.126936
$994$	16.0000	0.507489
$995$	0	0
$996$	−2.00000	−0.0633724
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−36.0000	−1.13956
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.2.a.d.1.1	✓	1
3.2	odd	2		1386.2.a.j.1.1		1
4.3	odd	2		3696.2.a.j.1.1		1
7.6	odd	2		3234.2.a.b.1.1		1
11.10	odd	2		5082.2.a.ba.1.1		1
21.20	even	2		9702.2.a.bp.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.d.1.1	✓	1	1.1	even	1	trivial
1386.2.a.j.1.1		1	3.2	odd	2
3234.2.a.b.1.1		1	7.6	odd	2
3696.2.a.j.1.1		1	4.3	odd	2
5082.2.a.ba.1.1		1	11.10	odd	2
9702.2.a.bp.1.1		1	21.20	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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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