LMFDB - Embedded newform 6762.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
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	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} +6.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} -2.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} +2.00000 q^{43} -6.00000 q^{44} +2.00000 q^{45} -1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} -2.00000 q^{51} -6.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -12.0000 q^{55} +4.00000 q^{57} -2.00000 q^{58} +8.00000 q^{59} +2.00000 q^{60} -2.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +6.00000 q^{66} +6.00000 q^{67} -2.00000 q^{68} +1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -8.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +6.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +2.00000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} +1.00000 q^{92} +6.00000 q^{93} +6.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000	0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−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	$0.348268\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	6.00000	1.27920
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−2.00000	−0.365148
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.00000	−0.648886
$39$	−6.00000	−0.960769
$40$	−2.00000	−0.316228
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−6.00000	−0.904534
$45$	2.00000	0.298142
$46$	−1.00000	−0.147442
$47$	−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
$49$	0	0
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	−6.00000	−0.832050
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−1.00000	−0.136083
$55$	−12.0000	−1.61808
$56$	0	0
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	2.00000	0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	6.00000	0.738549
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	−2.00000	−0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−8.00000	−0.929981
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	6.00000	0.679366
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−2.00000	−0.215666
$87$	2.00000	0.214423
$88$	6.00000	0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	1.00000	0.104257
$93$	6.00000	0.622171
$94$	6.00000	0.618853
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	2.00000	0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−4.00000	−0.388514
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	1.00000	0.0962250
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	12.0000	1.14416
$111$	8.00000	0.759326
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−4.00000	−0.374634
$115$	2.00000	0.186501
$116$	2.00000	0.185695
$117$	−6.00000	−0.554700
$118$	−8.00000	−0.736460
$119$	0	0
$120$	−2.00000	−0.182574
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	0	0
$124$	6.00000	0.538816
$125$	−12.0000	−1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	12.0000	1.05247
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−6.00000	−0.518321
$135$	2.00000	0.172133
$136$	2.00000	0.171499
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−1.00000	−0.0851257
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	8.00000	0.671345
$143$	36.0000	3.01047
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−4.00000	−0.331042
$147$	0	0
$148$	8.00000	0.657596
$149$	−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
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	−2.00000	−0.161690
$154$	0	0
$155$	12.0000	0.963863
$156$	−6.00000	−0.480384
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−4.00000	−0.318223
$159$	4.00000	0.317221
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−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	$0.550070\pi$
$164$	0	0
$165$	−12.0000	−0.934199
$166$	−12.0000	−0.931381
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	4.00000	0.306786
$171$	4.00000	0.305888
$172$	2.00000	0.152499
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−6.00000	−0.452267
$177$	8.00000	0.601317
$178$	−6.00000	−0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	2.00000	0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	−1.00000	−0.0737210
$185$	16.0000	1.17634
$186$	−6.00000	−0.439941
$187$	12.0000	0.877527
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	10.0000	0.717958
$195$	−12.0000	−0.859338
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	6.00000	0.426401
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	1.00000	0.0707107
$201$	6.00000	0.423207
$202$	−6.00000	−0.422159
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	1.00000	0.0695048
$208$	−6.00000	−0.416025
$209$	−24.0000	−1.66011
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	4.00000	0.274721
$213$	−8.00000	−0.548151
$214$	−6.00000	−0.410152
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−12.0000	−0.812743
$219$	4.00000	0.270295
$220$	−12.0000	−0.809040
$221$	12.0000	0.807207
$222$	−8.00000	−0.536925
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−18.0000	−1.19734
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	4.00000	0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	−2.00000	−0.131306
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	6.00000	0.392232
$235$	−12.0000	−0.782794
$236$	8.00000	0.520756
$237$	4.00000	0.259828
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	2.00000	0.129099
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−25.0000	−1.60706
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	−6.00000	−0.381000
$249$	12.0000	0.760469
$250$	12.0000	0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−8.00000	−0.501965
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	−12.0000	−0.744208
$261$	2.00000	0.123797
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	6.00000	0.369274
$265$	8.00000	0.491436
$266$	0	0
$267$	6.00000	0.367194
$268$	6.00000	0.366508
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−2.00000	−0.121716
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	6.00000	0.361814
$276$	1.00000	0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−4.00000	−0.239904
$279$	6.00000	0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	6.00000	0.357295
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−8.00000	−0.474713
$285$	8.00000	0.473879
$286$	−36.0000	−2.12872
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−10.0000	−0.586210
$292$	4.00000	0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−8.00000	−0.464991
$297$	−6.00000	−0.348155
$298$	12.0000	0.695141
$299$	−6.00000	−0.346989
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	4.00000	0.229416
$305$	−4.00000	−0.229039
$306$	2.00000	0.114332
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−12.0000	−0.681554
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	6.00000	0.339683
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	4.00000	0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−4.00000	−0.224309
$319$	−12.0000	−0.671871
$320$	2.00000	0.111803
$321$	6.00000	0.334887
$322$	0	0
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	6.00000	0.332820
$326$	4.00000	0.221540
$327$	12.0000	0.663602
$328$	0	0
$329$	0	0
$330$	12.0000	0.660578
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	12.0000	0.658586
$333$	8.00000	0.438397
$334$	−6.00000	−0.328305
$335$	12.0000	0.655630
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−23.0000	−1.25104
$339$	18.0000	0.977626
$340$	−4.00000	−0.216930
$341$	−36.0000	−1.94951
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−2.00000	−0.107833
$345$	2.00000	0.107676
$346$	−10.0000	−0.537603
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	2.00000	0.107211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	6.00000	0.319801
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	−8.00000	−0.425195
$355$	−16.0000	−0.849192
$356$	6.00000	0.317999
$357$	0	0
$358$	24.0000	1.26844
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	25.0000	1.31216
$364$	0	0
$365$	8.00000	0.418739
$366$	2.00000	0.104542
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−16.0000	−0.831800
$371$	0	0
$372$	6.00000	0.311086
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	−12.0000	−0.620505
$375$	−12.0000	−0.619677
$376$	6.00000	0.309426
$377$	−12.0000	−0.618031
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	8.00000	0.410391
$381$	8.00000	0.409852
$382$	16.0000	0.818631
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	2.00000	0.101666
$388$	−10.0000	−0.507673
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	12.0000	0.607644
$391$	−2.00000	−0.101144
$392$	0	0
$393$	0	0
$394$	18.0000	0.906827
$395$	8.00000	0.402524
$396$	−6.00000	−0.301511
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	−6.00000	−0.299253
$403$	−36.0000	−1.79329
$404$	6.00000	0.298511
$405$	2.00000	0.0993808
$406$	0	0
$407$	−48.0000	−2.37927
$408$	2.00000	0.0990148
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	24.0000	1.17811
$416$	6.00000	0.294174
$417$	4.00000	0.195881
$418$	24.0000	1.17388
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	−8.00000	−0.389434
$423$	−6.00000	−0.291730
$424$	−4.00000	−0.194257
$425$	2.00000	0.0970143
$426$	8.00000	0.387601
$427$	0	0
$428$	6.00000	0.290021
$429$	36.0000	1.73810
$430$	−4.00000	−0.192897
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	1.00000	0.0481125
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	12.0000	0.574696
$437$	4.00000	0.191346
$438$	−4.00000	−0.191127
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	12.0000	0.572078
$441$	0	0
$442$	−12.0000	−0.570782
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	8.00000	0.379663
$445$	12.0000	0.568855
$446$	−10.0000	−0.473514
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−38.0000	−1.79333	−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000	−1.79333	−0.896665	+	0.442709i	$0.854018\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	18.0000	0.846649
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−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	$0.792648\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	2.00000	0.0932505
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	2.00000	0.0928477
$465$	12.0000	0.556487
$466$	−10.0000	−0.463241
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	12.0000	0.553519
$471$	18.0000	0.829396
$472$	−8.00000	−0.368230
$473$	−12.0000	−0.551761
$474$	−4.00000	−0.183726
$475$	−4.00000	−0.183533
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	−2.00000	−0.0912871
$481$	−48.0000	−2.18861
$482$	14.0000	0.637683
$483$	0	0
$484$	25.0000	1.13636
$485$	−20.0000	−0.908153
$486$	−1.00000	−0.0453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	2.00000	0.0905357
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	24.0000	1.07981
$495$	−12.0000	−0.539360
$496$	6.00000	0.269408
$497$	0	0
$498$	−12.0000	−0.537733
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−12.0000	−0.536656
$501$	6.00000	0.268060
$502$	12.0000	0.535586
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	6.00000	0.266733
$507$	23.0000	1.02147
$508$	8.00000	0.354943
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	12.0000	0.529297
$515$	−16.0000	−0.705044
$516$	2.00000	0.0880451
$517$	36.0000	1.58328
$518$	0	0
$519$	10.0000	0.438951
$520$	12.0000	0.526235
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	−2.00000	−0.0875376
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−12.0000	−0.522728
$528$	−6.00000	−0.261116
$529$	1.00000	0.0434783
$530$	−8.00000	−0.347498
$531$	8.00000	0.347170
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	12.0000	0.518805
$536$	−6.00000	−0.259161
$537$	−24.0000	−1.03568
$538$	6.00000	0.258678
$539$	0	0
$540$	2.00000	0.0860663
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−26.0000	−1.11680
$543$	−2.00000	−0.0858282
$544$	2.00000	0.0857493
$545$	24.0000	1.02805
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	18.0000	0.768922
$549$	−2.00000	−0.0853579
$550$	−6.00000	−0.255841
$551$	8.00000	0.340811
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	16.0000	0.679162
$556$	4.00000	0.169638
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	−6.00000	−0.254000
$559$	−12.0000	−0.507546
$560$	0	0
$561$	12.0000	0.506640
$562$	18.0000	0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−6.00000	−0.252646
$565$	36.0000	1.51453
$566$	28.0000	1.17693
$567$	0	0
$568$	8.00000	0.335673
$569$	−46.0000	−1.92842	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000	−1.92842	−0.964210	+	0.265139i	$0.914582\pi$
$570$	−8.00000	−0.335083
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	36.0000	1.50524
$573$	−16.0000	−0.668410
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	13.0000	0.540729
$579$	14.0000	0.581820
$580$	4.00000	0.166091
$581$	0	0
$582$	10.0000	0.414513
$583$	−24.0000	−0.993978
$584$	−4.00000	−0.165521
$585$	−12.0000	−0.496139
$586$	−6.00000	−0.247858
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	−16.0000	−0.658710
$591$	−18.0000	−0.740421
$592$	8.00000	0.328798
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−12.0000	−0.491539
$597$	20.0000	0.818546
$598$	6.00000	0.245358
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	1.00000	0.0408248
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	50.0000	2.03279
$606$	−6.00000	−0.243733
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	4.00000	0.161955
$611$	36.0000	1.45640
$612$	−2.00000	−0.0808452
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	8.00000	0.321807
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	12.0000	0.481932
$621$	1.00000	0.0401286
$622$	−6.00000	−0.240578
$623$	0	0
$624$	−6.00000	−0.240192
$625$	−19.0000	−0.760000
$626$	−34.0000	−1.35891
$627$	−24.0000	−0.958468
$628$	18.0000	0.718278
$629$	−16.0000	−0.637962
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−4.00000	−0.159111
$633$	8.00000	0.317971
$634$	18.0000	0.714871
$635$	16.0000	0.634941
$636$	4.00000	0.158610
$637$	0	0
$638$	12.0000	0.475085
$639$	−8.00000	−0.316475
$640$	−2.00000	−0.0790569
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	−6.00000	−0.236801
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	8.00000	0.314756
$647$	−30.0000	−1.17942	−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000	−1.17942	−0.589711	+	0.807614i	$0.700758\pi$
$648$	−1.00000	−0.0392837
$649$	−48.0000	−1.88416
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	−12.0000	−0.467099
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	12.0000	0.466393
$663$	12.0000	0.466041
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−8.00000	−0.309994
$667$	2.00000	0.0774403
$668$	6.00000	0.232147
$669$	10.0000	0.386622
$670$	−12.0000	−0.463600
$671$	12.0000	0.463255
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	18.0000	0.693334
$675$	−1.00000	−0.0384900
$676$	23.0000	0.884615
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	−18.0000	−0.691286
$679$	0	0
$680$	4.00000	0.153393
$681$	4.00000	0.153280
$682$	36.0000	1.37851
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	4.00000	0.152944
$685$	36.0000	1.37549
$686$	0	0
$687$	14.0000	0.534133
$688$	2.00000	0.0762493
$689$	−24.0000	−0.914327
$690$	−2.00000	−0.0761387
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−12.0000	−0.455514
$695$	8.00000	0.303457
$696$	−2.00000	−0.0758098
$697$	0	0
$698$	−10.0000	−0.378506
$699$	10.0000	0.378235
$700$	0	0
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	6.00000	0.226455
$703$	32.0000	1.20690
$704$	−6.00000	−0.226134
$705$	−12.0000	−0.451946
$706$	16.0000	0.602168
$707$	0	0
$708$	8.00000	0.300658
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	16.0000	0.600469
$711$	4.00000	0.150012
$712$	−6.00000	−0.224860
$713$	6.00000	0.224702
$714$	0	0
$715$	72.0000	2.69265
$716$	−24.0000	−0.896922
$717$	0	0
$718$	0	0
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	3.00000	0.111648
$723$	−14.0000	−0.520666
$724$	−2.00000	−0.0743294
$725$	−2.00000	−0.0742781
$726$	−25.0000	−0.927837
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	−4.00000	−0.147945
$732$	−2.00000	−0.0739221
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−36.0000	−1.32608
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	16.0000	0.588172
$741$	−24.0000	−0.881662
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	−6.00000	−0.219971
$745$	−24.0000	−0.879292
$746$	8.00000	0.292901
$747$	12.0000	0.439057
$748$	12.0000	0.438763
$749$	0	0
$750$	12.0000	0.438178
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−6.00000	−0.218797
$753$	−12.0000	−0.437304
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	44.0000	1.59921	0.799604	−	0.600528i	$-0.205043\pi$
$757$	44.0000	1.59921	0.799604	+	0.600528i	$0.205043\pi$
$758$	−26.0000	−0.944363
$759$	−6.00000	−0.217786
$760$	−8.00000	−0.290191
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	−16.0000	−0.578860
$765$	−4.00000	−0.144620
$766$	0	0
$767$	−48.0000	−1.73318
$768$	1.00000	0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	14.0000	0.503871
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−2.00000	−0.0718885
$775$	−6.00000	−0.215526
$776$	10.0000	0.358979
$777$	0	0
$778$	24.0000	0.860442
$779$	0	0
$780$	−12.0000	−0.429669
$781$	48.0000	1.71758
$782$	2.00000	0.0715199
$783$	2.00000	0.0714742
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	−18.0000	−0.641223
$789$	16.0000	0.569615
$790$	−8.00000	−0.284627
$791$	0	0
$792$	6.00000	0.213201
$793$	12.0000	0.426132
$794$	−26.0000	−0.922705
$795$	8.00000	0.283731
$796$	20.0000	0.708881
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	−2.00000	−0.0706225
$803$	−24.0000	−0.846942
$804$	6.00000	0.211604
$805$	0	0
$806$	36.0000	1.26805
$807$	−6.00000	−0.211210
$808$	−6.00000	−0.211079
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	−2.00000	−0.0702728
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	26.0000	0.911860
$814$	48.0000	1.68240
$815$	−8.00000	−0.280228
$816$	−2.00000	−0.0700140
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	−18.0000	−0.627822
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	8.00000	0.278693
$825$	6.00000	0.208893
$826$	0	0
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	1.00000	0.0347524
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−24.0000	−0.833052
$831$	2.00000	0.0693792
$832$	−6.00000	−0.208013
$833$	0	0
$834$	−4.00000	−0.138509
$835$	12.0000	0.415277
$836$	−24.0000	−0.830057
$837$	6.00000	0.207390
$838$	20.0000	0.690889
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	24.0000	0.827095
$843$	−18.0000	−0.619953
$844$	8.00000	0.275371
$845$	46.0000	1.58245
$846$	6.00000	0.206284
$847$	0	0
$848$	4.00000	0.137361
$849$	−28.0000	−0.960958
$850$	−2.00000	−0.0685994
$851$	8.00000	0.274236
$852$	−8.00000	−0.274075
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	−6.00000	−0.205076
$857$	16.0000	0.546550	0.273275	−	0.961936i	$-0.411893\pi$
$857$	16.0000	0.546550	0.273275	+	0.961936i	$0.411893\pi$
$858$	−36.0000	−1.22902
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	−28.0000	−0.953684
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	20.0000	0.680020
$866$	−38.0000	−1.29129
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−24.0000	−0.814144
$870$	−4.00000	−0.135613
$871$	−36.0000	−1.21981
$872$	−12.0000	−0.406371
$873$	−10.0000	−0.338449
$874$	−4.00000	−0.135302
$875$	0	0
$876$	4.00000	0.135147
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−14.0000	−0.472477
$879$	6.00000	0.202375
$880$	−12.0000	−0.404520
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	12.0000	0.403604
$885$	16.0000	0.537834
$886$	−36.0000	−1.20944
$887$	38.0000	1.27592	0.637958	−	0.770072i	$-0.279780\pi$
$887$	38.0000	1.27592	0.637958	+	0.770072i	$0.279780\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	−12.0000	−0.402241
$891$	−6.00000	−0.201008
$892$	10.0000	0.334825
$893$	−24.0000	−0.803129
$894$	12.0000	0.401340
$895$	−48.0000	−1.60446
$896$	0	0
$897$	−6.00000	−0.200334
$898$	38.0000	1.26808
$899$	12.0000	0.400222
$900$	−1.00000	−0.0333333
$901$	−8.00000	−0.266519
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−4.00000	−0.132964
$906$	0	0
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	4.00000	0.132745
$909$	6.00000	0.199007
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	4.00000	0.132453
$913$	−72.0000	−2.38285
$914$	34.0000	1.12462
$915$	−4.00000	−0.132236
$916$	14.0000	0.462573
$917$	0	0
$918$	2.00000	0.0660098
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	−2.00000	−0.0659380
$921$	−32.0000	−1.05444
$922$	30.0000	0.987997
$923$	48.0000	1.57994
$924$	0	0
$925$	−8.00000	−0.263038
$926$	−40.0000	−1.31448
$927$	−8.00000	−0.262754
$928$	−2.00000	−0.0656532
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	−12.0000	−0.393496
$931$	0	0
$932$	10.0000	0.327561
$933$	6.00000	0.196431
$934$	−12.0000	−0.392652
$935$	24.0000	0.784884
$936$	6.00000	0.196116
$937$	46.0000	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$938$	0	0
$939$	34.0000	1.10955
$940$	−12.0000	−0.391397
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	−18.0000	−0.586472
$943$	0	0
$944$	8.00000	0.260378
$945$	0	0
$946$	12.0000	0.390154
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	4.00000	0.129914
$949$	−24.0000	−0.779073
$950$	4.00000	0.129777
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	−4.00000	−0.129505
$955$	−32.0000	−1.03550
$956$	0	0
$957$	−12.0000	−0.387905
$958$	−20.0000	−0.646171
$959$	0	0
$960$	2.00000	0.0645497
$961$	5.00000	0.161290
$962$	48.0000	1.54758
$963$	6.00000	0.193347
$964$	−14.0000	−0.450910
$965$	28.0000	0.901352
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−25.0000	−0.803530
$969$	−8.00000	−0.256997
$970$	20.0000	0.642161
$971$	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
$973$	0	0
$974$	0	0
$975$	6.00000	0.192154
$976$	−2.00000	−0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	4.00000	0.127906
$979$	−36.0000	−1.15056
$980$	0	0
$981$	12.0000	0.383131
$982$	28.0000	0.893516
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	4.00000	0.127386
$987$	0	0
$988$	−24.0000	−0.763542
$989$	2.00000	0.0635963
$990$	12.0000	0.381385
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−6.00000	−0.190500
$993$	−12.0000	−0.380808
$994$	0	0
$995$	40.0000	1.26809
$996$	12.0000	0.380235
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	−28.0000	−0.886325
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.p.1.1	yes	1
7.6	odd	2		6762.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.c.1.1	✓	1	7.6	odd	2
6762.2.a.p.1.1	yes	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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