LMFDB - Embedded newform 6762.2.a.x.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:	no (minimal twist has level 966)
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -3.00000 q^{20} +1.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -7.00000 q^{29} +3.00000 q^{30} +7.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} +2.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +11.0000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +2.00000 q^{57} -7.00000 q^{58} +15.0000 q^{59} +3.00000 q^{60} -2.00000 q^{61} +7.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} -1.00000 q^{66} +2.00000 q^{67} -2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -2.00000 q^{76} +6.00000 q^{78} -11.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +13.0000 q^{83} +6.00000 q^{85} +2.00000 q^{86} +7.00000 q^{87} +1.00000 q^{88} +8.00000 q^{89} -3.00000 q^{90} -1.00000 q^{92} -7.00000 q^{93} +6.00000 q^{94} +6.00000 q^{95} -1.00000 q^{96} +1.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	−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
$15$	3.00000	0.774597
$16$	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
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	1.00000	0.213201
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	3.00000	0.547723
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	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	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	6.00000	0.960769
$40$	−3.00000	−0.474342
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	1.00000	0.150756
$45$	−3.00000	−0.447214
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	−1.00000	−0.136083
$55$	−3.00000	−0.404520
$56$	0	0
$57$	2.00000	0.264906
$58$	−7.00000	−0.919145
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	3.00000	0.387298
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	7.00000	0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	18.0000	2.23263
$66$	−1.00000	−0.123091
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−2.00000	−0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	−2.00000	−0.229416
$77$	0	0
$78$	6.00000	0.679366
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	13.0000	1.42694	0.713468	−	0.700688i	$-0.247124\pi$
$83$	13.0000	1.42694	0.713468	+	0.700688i	$0.247124\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	2.00000	0.215666
$87$	7.00000	0.750479
$88$	1.00000	0.106600
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	−1.00000	−0.104257
$93$	−7.00000	−0.725866
$94$	6.00000	0.618853
$95$	6.00000	0.615587
$96$	−1.00000	−0.102062
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	4.00000	0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	11.0000	1.06841
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	−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	$0.733911\pi$
$110$	−3.00000	−0.286039
$111$	2.00000	0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	2.00000	0.187317
$115$	3.00000	0.279751
$116$	−7.00000	−0.649934
$117$	−6.00000	−0.554700
$118$	15.0000	1.38086
$119$	0	0
$120$	3.00000	0.273861
$121$	−10.0000	−0.909091
$122$	−2.00000	−0.181071
$123$	10.0000	0.901670
$124$	7.00000	0.628619
$125$	3.00000	0.268328
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	1.00000	0.0883883
$129$	−2.00000	−0.176090
$130$	18.0000	1.57870
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	2.00000	0.172774
$135$	3.00000	0.258199
$136$	−2.00000	−0.171499
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	1.00000	0.0851257
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	21.0000	1.74396
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−2.00000	−0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−4.00000	−0.326599
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	−2.00000	−0.162221
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−21.0000	−1.68676
$156$	6.00000	0.480384
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−11.0000	−0.875113
$159$	−11.0000	−0.872357
$160$	−3.00000	−0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−10.0000	−0.780869
$165$	3.00000	0.233550
$166$	13.0000	1.00900
$167$	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
$170$	6.00000	0.460179
$171$	−2.00000	−0.152944
$172$	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	$0.375875\pi$
$174$	7.00000	0.530669
$175$	0	0
$176$	1.00000	0.0753778
$177$	−15.0000	−1.12747
$178$	8.00000	0.599625
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−3.00000	−0.223607
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−1.00000	−0.0737210
$185$	6.00000	0.441129
$186$	−7.00000	−0.513265
$187$	−2.00000	−0.146254
$188$	6.00000	0.437595
$189$	0	0
$190$	6.00000	0.435286
$191$	26.0000	1.88129	0.940647	−	0.339387i	$-0.110219\pi$
$191$	26.0000	1.88129	0.940647	+	0.339387i	$0.110219\pi$
$192$	−1.00000	−0.0721688
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	1.00000	0.0717958
$195$	−18.0000	−1.28901
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	1.00000	0.0710669
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	4.00000	0.282843
$201$	−2.00000	−0.141069
$202$	−10.0000	−0.703598
$203$	0	0
$204$	2.00000	0.140028
$205$	30.0000	2.09529
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	−6.00000	−0.416025
$209$	−2.00000	−0.138343
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	11.0000	0.755483
$213$	0	0
$214$	−17.0000	−1.16210
$215$	−6.00000	−0.409197
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	10.0000	0.675737
$220$	−3.00000	−0.202260
$221$	12.0000	0.807207
$222$	2.00000	0.134231
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−6.00000	−0.399114
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	2.00000	0.132453
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	−7.00000	−0.459573
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−6.00000	−0.392232
$235$	−18.0000	−1.17419
$236$	15.0000	0.976417
$237$	11.0000	0.714527
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	3.00000	0.193649
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	10.0000	0.637577
$247$	12.0000	0.763542
$248$	7.00000	0.444500
$249$	−13.0000	−0.823842
$250$	3.00000	0.189737
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	17.0000	1.06667
$255$	−6.00000	−0.375735
$256$	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
$260$	18.0000	1.11631
$261$	−7.00000	−0.433289
$262$	7.00000	0.432461
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	−1.00000	−0.0615457
$265$	−33.0000	−2.02717
$266$	0	0
$267$	−8.00000	−0.489592
$268$	2.00000	0.122169
$269$	29.0000	1.76816	0.884081	−	0.467334i	$-0.154786\pi$
$269$	29.0000	1.76816	0.884081	+	0.467334i	$0.154786\pi$
$270$	3.00000	0.182574
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	4.00000	0.241209
$276$	1.00000	0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−2.00000	−0.119952
$279$	7.00000	0.419079
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	−6.00000	−0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	−6.00000	−0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	21.0000	1.23316
$291$	−1.00000	−0.0586210
$292$	−10.0000	−0.585206
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	−45.0000	−2.62000
$296$	−2.00000	−0.116248
$297$	−1.00000	−0.0580259
$298$	10.0000	0.579284
$299$	6.00000	0.346989
$300$	−4.00000	−0.230940
$301$	0	0
$302$	19.0000	1.09333
$303$	10.0000	0.574485
$304$	−2.00000	−0.114708
$305$	6.00000	0.343559
$306$	−2.00000	−0.114332
$307$	6.00000	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$307$	6.00000	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$308$	0	0
$309$	0	0
$310$	−21.0000	−1.19272
$311$	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
$313$	27.0000	1.52613	0.763065	−	0.646322i	$-0.223694\pi$
$313$	27.0000	1.52613	0.763065	+	0.646322i	$0.223694\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	−11.0000	−0.618798
$317$	−15.0000	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$317$	−15.0000	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$318$	−11.0000	−0.616849
$319$	−7.00000	−0.391925
$320$	−3.00000	−0.167705
$321$	17.0000	0.948847
$322$	0	0
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	−24.0000	−1.33128
$326$	10.0000	0.553849
$327$	14.0000	0.774202
$328$	−10.0000	−0.552158
$329$	0	0
$330$	3.00000	0.165145
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	13.0000	0.713468
$333$	−2.00000	−0.109599
$334$	6.00000	0.328305
$335$	−6.00000	−0.327815
$336$	0	0
$337$	35.0000	1.90657	0.953286	−	0.302070i	$-0.0976776\pi$
$337$	35.0000	1.90657	0.953286	+	0.302070i	$0.0976776\pi$
$338$	23.0000	1.25104
$339$	6.00000	0.325875
$340$	6.00000	0.325396
$341$	7.00000	0.379071
$342$	−2.00000	−0.108148
$343$	0	0
$344$	2.00000	0.107833
$345$	−3.00000	−0.161515
$346$	10.0000	0.537603
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	7.00000	0.375239
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	1.00000	0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−15.0000	−0.797241
$355$	0	0
$356$	8.00000	0.423999
$357$	0	0
$358$	4.00000	0.211407
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	−3.00000	−0.158114
$361$	−15.0000	−0.789474
$362$	−16.0000	−0.840941
$363$	10.0000	0.524864
$364$	0	0
$365$	30.0000	1.57027
$366$	2.00000	0.104542
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	−1.00000	−0.0521286
$369$	−10.0000	−0.520579
$370$	6.00000	0.311925
$371$	0	0
$372$	−7.00000	−0.362933
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−2.00000	−0.103418
$375$	−3.00000	−0.154919
$376$	6.00000	0.309426
$377$	42.0000	2.16311
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	6.00000	0.307794
$381$	−17.0000	−0.870936
$382$	26.0000	1.33028
$383$	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
$386$	−23.0000	−1.17067
$387$	2.00000	0.101666
$388$	1.00000	0.0507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−18.0000	−0.911465
$391$	2.00000	0.101144
$392$	0	0
$393$	−7.00000	−0.353103
$394$	−2.00000	−0.100759
$395$	33.0000	1.66041
$396$	1.00000	0.0502519
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	4.00000	0.200000
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	−2.00000	−0.0997509
$403$	−42.0000	−2.09217
$404$	−10.0000	−0.497519
$405$	−3.00000	−0.149071
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	2.00000	0.0990148
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	30.0000	1.48159
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−39.0000	−1.91443
$416$	−6.00000	−0.294174
$417$	2.00000	0.0979404
$418$	−2.00000	−0.0978232
$419$	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
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	6.00000	0.292075
$423$	6.00000	0.291730
$424$	11.0000	0.534207
$425$	−8.00000	−0.388057
$426$	0	0
$427$	0	0
$428$	−17.0000	−0.821726
$429$	6.00000	0.289683
$430$	−6.00000	−0.289346
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	−21.0000	−1.00687
$436$	−14.0000	−0.670478
$437$	2.00000	0.0956730
$438$	10.0000	0.477818
$439$	−29.0000	−1.38409	−0.692047	−	0.721852i	$-0.743291\pi$
$439$	−29.0000	−1.38409	−0.692047	+	0.721852i	$0.743291\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	12.0000	0.570782
$443$	11.0000	0.522626	0.261313	−	0.965254i	$-0.415845\pi$
$443$	11.0000	0.522626	0.261313	+	0.965254i	$0.415845\pi$
$444$	2.00000	0.0949158
$445$	−24.0000	−1.13771
$446$	21.0000	0.994379
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	4.00000	0.188562
$451$	−10.0000	−0.470882
$452$	−6.00000	−0.282216
$453$	−19.0000	−0.892698
$454$	−13.0000	−0.610120
$455$	0	0
$456$	2.00000	0.0936586
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	−10.0000	−0.467269
$459$	2.00000	0.0933520
$460$	3.00000	0.139876
$461$	−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
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−7.00000	−0.324967
$465$	21.0000	0.973852
$466$	18.0000	0.833834
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	−18.0000	−0.830278
$471$	−22.0000	−1.01371
$472$	15.0000	0.690431
$473$	2.00000	0.0919601
$474$	11.0000	0.505247
$475$	−8.00000	−0.367065
$476$	0	0
$477$	11.0000	0.503655
$478$	20.0000	0.914779
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	3.00000	0.136931
$481$	12.0000	0.547153
$482$	−7.00000	−0.318841
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−3.00000	−0.136223
$486$	−1.00000	−0.0453609
$487$	1.00000	0.0453143	0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000	0.0453143	0.0226572	+	0.999743i	$0.492787\pi$
$488$	−2.00000	−0.0905357
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	10.0000	0.450835
$493$	14.0000	0.630528
$494$	12.0000	0.539906
$495$	−3.00000	−0.134840
$496$	7.00000	0.314309
$497$	0	0
$498$	−13.0000	−0.582544
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	3.00000	0.134164
$501$	−6.00000	−0.268060
$502$	−15.0000	−0.669483
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	30.0000	1.33498
$506$	−1.00000	−0.0444554
$507$	−23.0000	−1.02147
$508$	17.0000	0.754253
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	6.00000	0.263880
$518$	0	0
$519$	−10.0000	−0.438951
$520$	18.0000	0.789352
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	−7.00000	−0.306382
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	7.00000	0.305796
$525$	0	0
$526$	6.00000	0.261612
$527$	−14.0000	−0.609850
$528$	−1.00000	−0.0435194
$529$	1.00000	0.0434783
$530$	−33.0000	−1.43343
$531$	15.0000	0.650945
$532$	0	0
$533$	60.0000	2.59889
$534$	−8.00000	−0.346194
$535$	51.0000	2.20492
$536$	2.00000	0.0863868
$537$	−4.00000	−0.172613
$538$	29.0000	1.25028
$539$	0	0
$540$	3.00000	0.129099
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−1.00000	−0.0429537
$543$	16.0000	0.686626
$544$	−2.00000	−0.0857493
$545$	42.0000	1.79908
$546$	0	0
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	2.00000	0.0854358
$549$	−2.00000	−0.0853579
$550$	4.00000	0.170561
$551$	14.0000	0.596420
$552$	1.00000	0.0425628
$553$	0	0
$554$	8.00000	0.339887
$555$	−6.00000	−0.254686
$556$	−2.00000	−0.0848189
$557$	−27.0000	−1.14403	−0.572013	−	0.820244i	$-0.693837\pi$
$557$	−27.0000	−1.14403	−0.572013	+	0.820244i	$0.693837\pi$
$558$	7.00000	0.296334
$559$	−12.0000	−0.507546
$560$	0	0
$561$	2.00000	0.0844401
$562$	20.0000	0.843649
$563$	41.0000	1.72794	0.863972	−	0.503540i	$-0.167969\pi$
$563$	41.0000	1.72794	0.863972	+	0.503540i	$0.167969\pi$
$564$	−6.00000	−0.252646
$565$	18.0000	0.757266
$566$	14.0000	0.588464
$567$	0	0
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	−6.00000	−0.251312
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−6.00000	−0.250873
$573$	−26.0000	−1.08617
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	−13.0000	−0.540729
$579$	23.0000	0.955847
$580$	21.0000	0.871978
$581$	0	0
$582$	−1.00000	−0.0414513
$583$	11.0000	0.455573
$584$	−10.0000	−0.413803
$585$	18.0000	0.744208
$586$	−1.00000	−0.0413096
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	−45.0000	−1.85262
$591$	2.00000	0.0822690
$592$	−2.00000	−0.0821995
$593$	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
$596$	10.0000	0.409616
$597$	−20.0000	−0.818546
$598$	6.00000	0.245358
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−4.00000	−0.163299
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	19.0000	0.773099
$605$	30.0000	1.21967
$606$	10.0000	0.406222
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	6.00000	0.242933
$611$	−36.0000	−1.45640
$612$	−2.00000	−0.0808452
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	6.00000	0.242140
$615$	−30.0000	−1.20972
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	−21.0000	−0.843380
$621$	1.00000	0.0401286
$622$	6.00000	0.240578
$623$	0	0
$624$	6.00000	0.240192
$625$	−29.0000	−1.16000
$626$	27.0000	1.07914
$627$	2.00000	0.0798723
$628$	22.0000	0.877896
$629$	4.00000	0.159490
$630$	0	0
$631$	−21.0000	−0.835997	−0.417998	−	0.908448i	$-0.637268\pi$
$631$	−21.0000	−0.835997	−0.417998	+	0.908448i	$0.637268\pi$
$632$	−11.0000	−0.437557
$633$	−6.00000	−0.238479
$634$	−15.0000	−0.595726
$635$	−51.0000	−2.02387
$636$	−11.0000	−0.436178
$637$	0	0
$638$	−7.00000	−0.277133
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	17.0000	0.670936
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	4.00000	0.157378
$647$	22.0000	0.864909	0.432455	−	0.901656i	$-0.357648\pi$
$647$	22.0000	0.864909	0.432455	+	0.901656i	$0.357648\pi$
$648$	1.00000	0.0392837
$649$	15.0000	0.588802
$650$	−24.0000	−0.941357
$651$	0	0
$652$	10.0000	0.391630
$653$	−37.0000	−1.44792	−0.723961	−	0.689841i	$-0.757680\pi$
$653$	−37.0000	−1.44792	−0.723961	+	0.689841i	$0.757680\pi$
$654$	14.0000	0.547443
$655$	−21.0000	−0.820538
$656$	−10.0000	−0.390434
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	3.00000	0.116775
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	−14.0000	−0.544125
$663$	−12.0000	−0.466041
$664$	13.0000	0.504498
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	7.00000	0.271041
$668$	6.00000	0.232147
$669$	−21.0000	−0.811907
$670$	−6.00000	−0.231800
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−7.00000	−0.269830	−0.134915	−	0.990857i	$-0.543076\pi$
$673$	−7.00000	−0.269830	−0.134915	+	0.990857i	$0.543076\pi$
$674$	35.0000	1.34815
$675$	−4.00000	−0.153960
$676$	23.0000	0.884615
$677$	−23.0000	−0.883962	−0.441981	−	0.897024i	$-0.645724\pi$
$677$	−23.0000	−0.883962	−0.441981	+	0.897024i	$0.645724\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	6.00000	0.230089
$681$	13.0000	0.498161
$682$	7.00000	0.268044
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−2.00000	−0.0764719
$685$	−6.00000	−0.229248
$686$	0	0
$687$	10.0000	0.381524
$688$	2.00000	0.0762493
$689$	−66.0000	−2.51440
$690$	−3.00000	−0.114208
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−12.0000	−0.455514
$695$	6.00000	0.227593
$696$	7.00000	0.265334
$697$	20.0000	0.757554
$698$	−24.0000	−0.908413
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	6.00000	0.226455
$703$	4.00000	0.150863
$704$	1.00000	0.0376889
$705$	18.0000	0.677919
$706$	14.0000	0.526897
$707$	0	0
$708$	−15.0000	−0.563735
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	8.00000	0.299813
$713$	−7.00000	−0.262152
$714$	0	0
$715$	18.0000	0.673162
$716$	4.00000	0.149487
$717$	−20.0000	−0.746914
$718$	6.00000	0.223918
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	−15.0000	−0.558242
$723$	7.00000	0.260333
$724$	−16.0000	−0.594635
$725$	−28.0000	−1.03989
$726$	10.0000	0.371135
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	30.0000	1.11035
$731$	−4.00000	−0.147945
$732$	2.00000	0.0739221
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	11.0000	0.406017
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	2.00000	0.0736709
$738$	−10.0000	−0.368105
$739$	−14.0000	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$739$	−14.0000	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$740$	6.00000	0.220564
$741$	−12.0000	−0.440831
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−7.00000	−0.256632
$745$	−30.0000	−1.09911
$746$	16.0000	0.585802
$747$	13.0000	0.475645
$748$	−2.00000	−0.0731272
$749$	0	0
$750$	−3.00000	−0.109545
$751$	47.0000	1.71505	0.857527	−	0.514439i	$-0.172000\pi$
$751$	47.0000	1.71505	0.857527	+	0.514439i	$0.172000\pi$
$752$	6.00000	0.218797
$753$	15.0000	0.546630
$754$	42.0000	1.52955
$755$	−57.0000	−2.07444
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−4.00000	−0.145287
$759$	1.00000	0.0362977
$760$	6.00000	0.217643
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	−17.0000	−0.615845
$763$	0	0
$764$	26.0000	0.940647
$765$	6.00000	0.216930
$766$	16.0000	0.578103
$767$	−90.0000	−3.24971
$768$	−1.00000	−0.0360844
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	−23.0000	−0.827788
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	2.00000	0.0718885
$775$	28.0000	1.00579
$776$	1.00000	0.0358979
$777$	0	0
$778$	6.00000	0.215110
$779$	20.0000	0.716574
$780$	−18.0000	−0.644503
$781$	0	0
$782$	2.00000	0.0715199
$783$	7.00000	0.250160
$784$	0	0
$785$	−66.0000	−2.35564
$786$	−7.00000	−0.249682
$787$	−54.0000	−1.92489	−0.962446	−	0.271473i	$-0.912489\pi$
$787$	−54.0000	−1.92489	−0.962446	+	0.271473i	$0.912489\pi$
$788$	−2.00000	−0.0712470
$789$	−6.00000	−0.213606
$790$	33.0000	1.17409
$791$	0	0
$792$	1.00000	0.0355335
$793$	12.0000	0.426132
$794$	2.00000	0.0709773
$795$	33.0000	1.17039
$796$	20.0000	0.708881
$797$	21.0000	0.743858	0.371929	−	0.928261i	$-0.378696\pi$
$797$	21.0000	0.743858	0.371929	+	0.928261i	$0.378696\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	4.00000	0.141421
$801$	8.00000	0.282666
$802$	4.00000	0.141245
$803$	−10.0000	−0.352892
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−42.0000	−1.47939
$807$	−29.0000	−1.02085
$808$	−10.0000	−0.351799
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−3.00000	−0.105409
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	1.00000	0.0350715
$814$	−2.00000	−0.0701000
$815$	−30.0000	−1.05085
$816$	2.00000	0.0700140
$817$	−4.00000	−0.139942
$818$	19.0000	0.664319
$819$	0	0
$820$	30.0000	1.04765
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	−2.00000	−0.0697580
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−23.0000	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$827$	−23.0000	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$828$	−1.00000	−0.0347524
$829$	12.0000	0.416777	0.208389	−	0.978046i	$-0.433178\pi$
$829$	12.0000	0.416777	0.208389	+	0.978046i	$0.433178\pi$
$830$	−39.0000	−1.35371
$831$	−8.00000	−0.277517
$832$	−6.00000	−0.208013
$833$	0	0
$834$	2.00000	0.0692543
$835$	−18.0000	−0.622916
$836$	−2.00000	−0.0691714
$837$	−7.00000	−0.241955
$838$	4.00000	0.138178
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	22.0000	0.758170
$843$	−20.0000	−0.688837
$844$	6.00000	0.206529
$845$	−69.0000	−2.37367
$846$	6.00000	0.206284
$847$	0	0
$848$	11.0000	0.377742
$849$	−14.0000	−0.480479
$850$	−8.00000	−0.274398
$851$	2.00000	0.0685591
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	−17.0000	−0.581048
$857$	40.0000	1.36637	0.683187	−	0.730243i	$-0.260593\pi$
$857$	40.0000	1.36637	0.683187	+	0.730243i	$0.260593\pi$
$858$	6.00000	0.204837
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	16.0000	0.544962
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	−1.00000	−0.0340207
$865$	−30.0000	−1.02003
$866$	−26.0000	−0.883516
$867$	13.0000	0.441503
$868$	0	0
$869$	−11.0000	−0.373149
$870$	−21.0000	−0.711967
$871$	−12.0000	−0.406604
$872$	−14.0000	−0.474100
$873$	1.00000	0.0338449
$874$	2.00000	0.0676510
$875$	0	0
$876$	10.0000	0.337869
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	−29.0000	−0.978703
$879$	1.00000	0.0337292
$880$	−3.00000	−0.101130
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	−54.0000	−1.81724	−0.908622	−	0.417619i	$-0.862865\pi$
$883$	−54.0000	−1.81724	−0.908622	+	0.417619i	$0.862865\pi$
$884$	12.0000	0.403604
$885$	45.0000	1.51266
$886$	11.0000	0.369552
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−24.0000	−0.804482
$891$	1.00000	0.0335013
$892$	21.0000	0.703132
$893$	−12.0000	−0.401565
$894$	−10.0000	−0.334450
$895$	−12.0000	−0.401116
$896$	0	0
$897$	−6.00000	−0.200334
$898$	−6.00000	−0.200223
$899$	−49.0000	−1.63424
$900$	4.00000	0.133333
$901$	−22.0000	−0.732926
$902$	−10.0000	−0.332964
$903$	0	0
$904$	−6.00000	−0.199557
$905$	48.0000	1.59557
$906$	−19.0000	−0.631233
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	−13.0000	−0.431420
$909$	−10.0000	−0.331679
$910$	0	0
$911$	14.0000	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$912$	2.00000	0.0662266
$913$	13.0000	0.430237
$914$	−11.0000	−0.363848
$915$	−6.00000	−0.198354
$916$	−10.0000	−0.330409
$917$	0	0
$918$	2.00000	0.0660098
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	3.00000	0.0989071
$921$	−6.00000	−0.197707
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	−8.00000	−0.263038
$926$	8.00000	0.262896
$927$	0	0
$928$	−7.00000	−0.229786
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	21.0000	0.688617
$931$	0	0
$932$	18.0000	0.589610
$933$	−6.00000	−0.196431
$934$	−20.0000	−0.654420
$935$	6.00000	0.196221
$936$	−6.00000	−0.196116
$937$	1.00000	0.0326686	0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000	0.0326686	0.0163343	+	0.999867i	$0.494800\pi$
$938$	0	0
$939$	−27.0000	−0.881112
$940$	−18.0000	−0.587095
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	−22.0000	−0.716799
$943$	10.0000	0.325645
$944$	15.0000	0.488208
$945$	0	0
$946$	2.00000	0.0650256
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	11.0000	0.357263
$949$	60.0000	1.94768
$950$	−8.00000	−0.259554
$951$	15.0000	0.486408
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	11.0000	0.356138
$955$	−78.0000	−2.52402
$956$	20.0000	0.646846
$957$	7.00000	0.226278
$958$	20.0000	0.646171
$959$	0	0
$960$	3.00000	0.0968246
$961$	18.0000	0.580645
$962$	12.0000	0.386896
$963$	−17.0000	−0.547817
$964$	−7.00000	−0.225455
$965$	69.0000	2.22119
$966$	0	0
$967$	3.00000	0.0964735	0.0482367	−	0.998836i	$-0.484640\pi$
$967$	3.00000	0.0964735	0.0482367	+	0.998836i	$0.484640\pi$
$968$	−10.0000	−0.321412
$969$	−4.00000	−0.128499
$970$	−3.00000	−0.0963242
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	1.00000	0.0320421
$975$	24.0000	0.768615
$976$	−2.00000	−0.0640184
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−10.0000	−0.319765
$979$	8.00000	0.255681
$980$	0	0
$981$	−14.0000	−0.446986
$982$	−29.0000	−0.925427
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	10.0000	0.318788
$985$	6.00000	0.191176
$986$	14.0000	0.445851
$987$	0	0
$988$	12.0000	0.381771
$989$	−2.00000	−0.0635963
$990$	−3.00000	−0.0953463
$991$	−29.0000	−0.921215	−0.460608	−	0.887604i	$-0.652368\pi$
$991$	−29.0000	−0.921215	−0.460608	+	0.887604i	$0.652368\pi$
$992$	7.00000	0.222250
$993$	14.0000	0.444277
$994$	0	0
$995$	−60.0000	−1.90213
$996$	−13.0000	−0.411921
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−10.0000	−0.316544
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.x.1.1		1
7.2	even	3		966.2.i.d.277.1	✓	2
7.4	even	3		966.2.i.d.415.1	yes	2
7.6	odd	2		6762.2.a.bm.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.i.d.277.1	✓	2	7.2	even	3
966.2.i.d.415.1	yes	2	7.4	even	3
6762.2.a.x.1.1		1	1.1	even	1	trivial
6762.2.a.bm.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
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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