LMFDB - Embedded newform 8214.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8214,2,Mod(1,8214)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8214, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8214.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8214 = 2 \cdot 3 \cdot 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8214.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:	$65.5891202203$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 222)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8214.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +3.00000 q^{13} -1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +5.00000 q^{19} -4.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -5.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +3.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -4.00000 q^{29} -4.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +5.00000 q^{38} +3.00000 q^{39} -4.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{45} -5.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +11.0000 q^{50} -3.00000 q^{51} +3.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -1.00000 q^{56} +5.00000 q^{57} -4.00000 q^{58} +12.0000 q^{59} -4.00000 q^{60} -10.0000 q^{61} +10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} -1.00000 q^{66} +14.0000 q^{67} -3.00000 q^{68} -5.00000 q^{69} +4.00000 q^{70} +1.00000 q^{72} -11.0000 q^{73} +11.0000 q^{75} +5.00000 q^{76} +1.00000 q^{77} +3.00000 q^{78} +10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -9.00000 q^{83} -1.00000 q^{84} +12.0000 q^{85} -4.00000 q^{86} -4.00000 q^{87} -1.00000 q^{88} -11.0000 q^{89} -4.00000 q^{90} -3.00000 q^{91} -5.00000 q^{92} +10.0000 q^{93} +2.00000 q^{94} -20.0000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	−1.00000	−0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−4.00000	−0.894427
$21$	−1.00000	−0.218218
$22$	−1.00000	−0.213201
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	1.00000	0.204124
$25$	11.0000	2.20000
$26$	3.00000	0.588348
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−4.00000	−0.730297
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.00000	−0.514496
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	0	0
$37$	0	0
$38$	5.00000	0.811107
$39$	3.00000	0.480384
$40$	−4.00000	−0.632456
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−1.00000	−0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	−4.00000	−0.596285
$46$	−5.00000	−0.737210
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	11.0000	1.55563
$51$	−3.00000	−0.420084
$52$	3.00000	0.416025
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	−1.00000	−0.133631
$57$	5.00000	0.662266
$58$	−4.00000	−0.525226
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	−4.00000	−0.516398
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	10.0000	1.27000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	−1.00000	−0.123091
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	−3.00000	−0.363803
$69$	−5.00000	−0.601929
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	11.0000	1.27017
$76$	5.00000	0.573539
$77$	1.00000	0.113961
$78$	3.00000	0.339683
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−1.00000	−0.109109
$85$	12.0000	1.30158
$86$	−4.00000	−0.431331
$87$	−4.00000	−0.428845
$88$	−1.00000	−0.106600
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	−4.00000	−0.421637
$91$	−3.00000	−0.314485
$92$	−5.00000	−0.521286
$93$	10.0000	1.03695
$94$	2.00000	0.206284
$95$	−20.0000	−2.05196
$96$	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	$0.669494\pi$
$98$	−6.00000	−0.606092
$99$	−1.00000	−0.100504
$100$	11.0000	1.10000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−3.00000	−0.297044
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	3.00000	0.294174
$105$	4.00000	0.390360
$106$	−11.0000	−1.06841
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	1.00000	0.0962250
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	5.00000	0.468293
$115$	20.0000	1.86501
$116$	−4.00000	−0.371391
$117$	3.00000	0.277350
$118$	12.0000	1.10469
$119$	3.00000	0.275010
$120$	−4.00000	−0.365148
$121$	−10.0000	−0.909091
$122$	−10.0000	−0.905357
$123$	−6.00000	−0.541002
$124$	10.0000	0.898027
$125$	−24.0000	−2.14663
$126$	−1.00000	−0.0890871
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	−12.0000	−1.05247
$131$	−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
$133$	−5.00000	−0.433555
$134$	14.0000	1.20942
$135$	−4.00000	−0.344265
$136$	−3.00000	−0.257248
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	−5.00000	−0.425628
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	4.00000	0.338062
$141$	2.00000	0.168430
$142$	0	0
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	16.0000	1.32873
$146$	−11.0000	−0.910366
$147$	−6.00000	−0.494872
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	11.0000	0.898146
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	5.00000	0.405554
$153$	−3.00000	−0.242536
$154$	1.00000	0.0805823
$155$	−40.0000	−3.21288
$156$	3.00000	0.240192
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	10.0000	0.795557
$159$	−11.0000	−0.872357
$160$	−4.00000	−0.316228
$161$	5.00000	0.394055
$162$	1.00000	0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	−6.00000	−0.468521
$165$	4.00000	0.311400
$166$	−9.00000	−0.698535
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	−1.00000	−0.0771517
$169$	−4.00000	−0.307692
$170$	12.0000	0.920358
$171$	5.00000	0.382360
$172$	−4.00000	−0.304997
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	−4.00000	−0.303239
$175$	−11.0000	−0.831522
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	−11.0000	−0.824485
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−4.00000	−0.298142
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	−3.00000	−0.222375
$183$	−10.0000	−0.739221
$184$	−5.00000	−0.368605
$185$	0	0
$186$	10.0000	0.733236
$187$	3.00000	0.219382
$188$	2.00000	0.145865
$189$	−1.00000	−0.0727393
$190$	−20.0000	−1.45095
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	1.00000	0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−10.0000	−0.717958
$195$	−12.0000	−0.859338
$196$	−6.00000	−0.428571
$197$	7.00000	0.498729	0.249365	−	0.968410i	$-0.419778\pi$
$197$	7.00000	0.498729	0.249365	+	0.968410i	$0.419778\pi$
$198$	−1.00000	−0.0710669
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	11.0000	0.777817
$201$	14.0000	0.987484
$202$	6.00000	0.422159
$203$	4.00000	0.280745
$204$	−3.00000	−0.210042
$205$	24.0000	1.67623
$206$	2.00000	0.139347
$207$	−5.00000	−0.347524
$208$	3.00000	0.208013
$209$	−5.00000	−0.345857
$210$	4.00000	0.276026
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−11.0000	−0.755483
$213$	0	0
$214$	−3.00000	−0.205076
$215$	16.0000	1.09119
$216$	1.00000	0.0680414
$217$	−10.0000	−0.678844
$218$	9.00000	0.609557
$219$	−11.0000	−0.743311
$220$	4.00000	0.269680
$221$	−9.00000	−0.605406
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−1.00000	−0.0668153
$225$	11.0000	0.733333
$226$	−14.0000	−0.931266
$227$	−16.0000	−1.06196	−0.530979	−	0.847385i	$-0.678176\pi$
$227$	−16.0000	−1.06196	−0.530979	+	0.847385i	$0.678176\pi$
$228$	5.00000	0.331133
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	20.0000	1.31876
$231$	1.00000	0.0657952
$232$	−4.00000	−0.262613
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	3.00000	0.196116
$235$	−8.00000	−0.521862
$236$	12.0000	0.781133
$237$	10.0000	0.649570
$238$	3.00000	0.194461
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−4.00000	−0.258199
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−10.0000	−0.642824
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	24.0000	1.53330
$246$	−6.00000	−0.382546
$247$	15.0000	0.954427
$248$	10.0000	0.635001
$249$	−9.00000	−0.570352
$250$	−24.0000	−1.51789
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	−1.00000	−0.0629941
$253$	5.00000	0.314347
$254$	−3.00000	−0.188237
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−12.0000	−0.744208
$261$	−4.00000	−0.247594
$262$	−22.0000	−1.35916
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	−1.00000	−0.0615457
$265$	44.0000	2.70290
$266$	−5.00000	−0.306570
$267$	−11.0000	−0.673189
$268$	14.0000	0.855186
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	−4.00000	−0.243432
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−3.00000	−0.181902
$273$	−3.00000	−0.181568
$274$	20.0000	1.20824
$275$	−11.0000	−0.663325
$276$	−5.00000	−0.300965
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	−14.0000	−0.839664
$279$	10.0000	0.598684
$280$	4.00000	0.239046
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	2.00000	0.119098
$283$	−21.0000	−1.24832	−0.624160	−	0.781296i	$-0.714559\pi$
$283$	−21.0000	−1.24832	−0.624160	+	0.781296i	$0.714559\pi$
$284$	0	0
$285$	−20.0000	−1.18470
$286$	−3.00000	−0.177394
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	16.0000	0.939552
$291$	−10.0000	−0.586210
$292$	−11.0000	−0.643726
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	−6.00000	−0.349927
$295$	−48.0000	−2.79467
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	−10.0000	−0.579284
$299$	−15.0000	−0.867472
$300$	11.0000	0.635085
$301$	4.00000	0.230556
$302$	5.00000	0.287718
$303$	6.00000	0.344691
$304$	5.00000	0.286770
$305$	40.0000	2.29039
$306$	−3.00000	−0.171499
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	1.00000	0.0569803
$309$	2.00000	0.113776
$310$	−40.0000	−2.27185
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	3.00000	0.169842
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	10.0000	0.562544
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−11.0000	−0.616849
$319$	4.00000	0.223957
$320$	−4.00000	−0.223607
$321$	−3.00000	−0.167444
$322$	5.00000	0.278639
$323$	−15.0000	−0.834622
$324$	1.00000	0.0555556
$325$	33.0000	1.83051
$326$	−9.00000	−0.498464
$327$	9.00000	0.497701
$328$	−6.00000	−0.331295
$329$	−2.00000	−0.110264
$330$	4.00000	0.220193
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−9.00000	−0.493939
$333$	0	0
$334$	17.0000	0.930199
$335$	−56.0000	−3.05961
$336$	−1.00000	−0.0545545
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	−4.00000	−0.217571
$339$	−14.0000	−0.760376
$340$	12.0000	0.650791
$341$	−10.0000	−0.541530
$342$	5.00000	0.270369
$343$	13.0000	0.701934
$344$	−4.00000	−0.215666
$345$	20.0000	1.07676
$346$	−13.0000	−0.698884
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−4.00000	−0.214423
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−11.0000	−0.587975
$351$	3.00000	0.160128
$352$	−1.00000	−0.0533002
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−11.0000	−0.582999
$357$	3.00000	0.158777
$358$	−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	$0.449388\pi$
$360$	−4.00000	−0.210819
$361$	6.00000	0.315789
$362$	−16.0000	−0.840941
$363$	−10.0000	−0.524864
$364$	−3.00000	−0.157243
$365$	44.0000	2.30307
$366$	−10.0000	−0.522708
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	−5.00000	−0.260643
$369$	−6.00000	−0.312348
$370$	0	0
$371$	11.0000	0.571092
$372$	10.0000	0.518476
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	3.00000	0.155126
$375$	−24.0000	−1.23935
$376$	2.00000	0.103142
$377$	−12.0000	−0.618031
$378$	−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	$0.671713\pi$
$380$	−20.0000	−1.02598
$381$	−3.00000	−0.153695
$382$	−9.00000	−0.460480
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	1.00000	0.0510310
$385$	−4.00000	−0.203859
$386$	−18.0000	−0.916176
$387$	−4.00000	−0.203331
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−12.0000	−0.607644
$391$	15.0000	0.758583
$392$	−6.00000	−0.303046
$393$	−22.0000	−1.10975
$394$	7.00000	0.352655
$395$	−40.0000	−2.01262
$396$	−1.00000	−0.0502519
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	−16.0000	−0.802008
$399$	−5.00000	−0.250313
$400$	11.0000	0.550000
$401$	−19.0000	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$401$	−19.0000	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$402$	14.0000	0.698257
$403$	30.0000	1.49441
$404$	6.00000	0.298511
$405$	−4.00000	−0.198762
$406$	4.00000	0.198517
$407$	0	0
$408$	−3.00000	−0.148522
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	24.0000	1.18528
$411$	20.0000	0.986527
$412$	2.00000	0.0985329
$413$	−12.0000	−0.590481
$414$	−5.00000	−0.245737
$415$	36.0000	1.76717
$416$	3.00000	0.147087
$417$	−14.0000	−0.685583
$418$	−5.00000	−0.244558
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	4.00000	0.195180
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	2.00000	0.0972433
$424$	−11.0000	−0.534207
$425$	−33.0000	−1.60074
$426$	0	0
$427$	10.0000	0.483934
$428$	−3.00000	−0.145010
$429$	−3.00000	−0.144841
$430$	16.0000	0.771589
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	1.00000	0.0481125
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	−10.0000	−0.480015
$435$	16.0000	0.767141
$436$	9.00000	0.431022
$437$	−25.0000	−1.19591
$438$	−11.0000	−0.525600
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	4.00000	0.190693
$441$	−6.00000	−0.285714
$442$	−9.00000	−0.428086
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	44.0000	2.08580
$446$	0	0
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	11.0000	0.518545
$451$	6.00000	0.282529
$452$	−14.0000	−0.658505
$453$	5.00000	0.234920
$454$	−16.0000	−0.750917
$455$	12.0000	0.562569
$456$	5.00000	0.234146
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−8.00000	−0.373815
$459$	−3.00000	−0.140028
$460$	20.0000	0.932505
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	1.00000	0.0465242
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−4.00000	−0.185695
$465$	−40.0000	−1.85496
$466$	10.0000	0.463241
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	3.00000	0.138675
$469$	−14.0000	−0.646460
$470$	−8.00000	−0.369012
$471$	0	0
$472$	12.0000	0.552345
$473$	4.00000	0.183920
$474$	10.0000	0.459315
$475$	55.0000	2.52357
$476$	3.00000	0.137505
$477$	−11.0000	−0.503655
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	−4.00000	−0.182574
$481$	0	0
$482$	−28.0000	−1.27537
$483$	5.00000	0.227508
$484$	−10.0000	−0.454545
$485$	40.0000	1.81631
$486$	1.00000	0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−10.0000	−0.452679
$489$	−9.00000	−0.406994
$490$	24.0000	1.08421
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	−6.00000	−0.270501
$493$	12.0000	0.540453
$494$	15.0000	0.674882
$495$	4.00000	0.179787
$496$	10.0000	0.449013
$497$	0	0
$498$	−9.00000	−0.403300
$499$	15.0000	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$499$	15.0000	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$500$	−24.0000	−1.07331
$501$	17.0000	0.759504
$502$	6.00000	0.267793
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−1.00000	−0.0445435
$505$	−24.0000	−1.06799
$506$	5.00000	0.222277
$507$	−4.00000	−0.177646
$508$	−3.00000	−0.133103
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	12.0000	0.531369
$511$	11.0000	0.486611
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	−27.0000	−1.19092
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	−2.00000	−0.0879599
$518$	0	0
$519$	−13.0000	−0.570637
$520$	−12.0000	−0.526235
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	−4.00000	−0.175075
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−22.0000	−0.961074
$525$	−11.0000	−0.480079
$526$	18.0000	0.784837
$527$	−30.0000	−1.30682
$528$	−1.00000	−0.0435194
$529$	2.00000	0.0869565
$530$	44.0000	1.91124
$531$	12.0000	0.520756
$532$	−5.00000	−0.216777
$533$	−18.0000	−0.779667
$534$	−11.0000	−0.476017
$535$	12.0000	0.518805
$536$	14.0000	0.604708
$537$	−4.00000	−0.172613
$538$	27.0000	1.16405
$539$	6.00000	0.258438
$540$	−4.00000	−0.172133
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	8.00000	0.343629
$543$	−16.0000	−0.686626
$544$	−3.00000	−0.128624
$545$	−36.0000	−1.54207
$546$	−3.00000	−0.128388
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	20.0000	0.854358
$549$	−10.0000	−0.426790
$550$	−11.0000	−0.469042
$551$	−20.0000	−0.852029
$552$	−5.00000	−0.212814
$553$	−10.0000	−0.425243
$554$	5.00000	0.212430
$555$	0	0
$556$	−14.0000	−0.593732
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	10.0000	0.423334
$559$	−12.0000	−0.507546
$560$	4.00000	0.169031
$561$	3.00000	0.126660
$562$	9.00000	0.379642
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	2.00000	0.0842152
$565$	56.0000	2.35594
$566$	−21.0000	−0.882696
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	−20.0000	−0.837708
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−3.00000	−0.125436
$573$	−9.00000	−0.375980
$574$	6.00000	0.250435
$575$	−55.0000	−2.29366
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−8.00000	−0.332756
$579$	−18.0000	−0.748054
$580$	16.0000	0.664364
$581$	9.00000	0.373383
$582$	−10.0000	−0.414513
$583$	11.0000	0.455573
$584$	−11.0000	−0.455183
$585$	−12.0000	−0.496139
$586$	21.0000	0.867502
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−6.00000	−0.247436
$589$	50.0000	2.06021
$590$	−48.0000	−1.97613
$591$	7.00000	0.287942
$592$	0	0
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	−1.00000	−0.0410305
$595$	−12.0000	−0.491952
$596$	−10.0000	−0.409616
$597$	−16.0000	−0.654836
$598$	−15.0000	−0.613396
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	11.0000	0.449073
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	4.00000	0.163028
$603$	14.0000	0.570124
$604$	5.00000	0.203447
$605$	40.0000	1.62623
$606$	6.00000	0.243733
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	5.00000	0.202777
$609$	4.00000	0.162088
$610$	40.0000	1.61955
$611$	6.00000	0.242734
$612$	−3.00000	−0.121268
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	32.0000	1.29141
$615$	24.0000	0.967773
$616$	1.00000	0.0402911
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	2.00000	0.0804518
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	−40.0000	−1.60644
$621$	−5.00000	−0.200643
$622$	16.0000	0.641542
$623$	11.0000	0.440706
$624$	3.00000	0.120096
$625$	41.0000	1.64000
$626$	14.0000	0.559553
$627$	−5.00000	−0.199681
$628$	0	0
$629$	0	0
$630$	4.00000	0.159364
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−18.0000	−0.714871
$635$	12.0000	0.476205
$636$	−11.0000	−0.436178
$637$	−18.0000	−0.713186
$638$	4.00000	0.158362
$639$	0	0
$640$	−4.00000	−0.158114
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	−3.00000	−0.118401
$643$	41.0000	1.61688	0.808441	−	0.588577i	$-0.200312\pi$
$643$	41.0000	1.61688	0.808441	+	0.588577i	$0.200312\pi$
$644$	5.00000	0.197028
$645$	16.0000	0.629999
$646$	−15.0000	−0.590167
$647$	11.0000	0.432455	0.216227	−	0.976343i	$-0.430625\pi$
$647$	11.0000	0.432455	0.216227	+	0.976343i	$0.430625\pi$
$648$	1.00000	0.0392837
$649$	−12.0000	−0.471041
$650$	33.0000	1.29437
$651$	−10.0000	−0.391931
$652$	−9.00000	−0.352467
$653$	20.0000	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$654$	9.00000	0.351928
$655$	88.0000	3.43844
$656$	−6.00000	−0.234261
$657$	−11.0000	−0.429151
$658$	−2.00000	−0.0779681
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	4.00000	0.155700
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	−28.0000	−1.08825
$663$	−9.00000	−0.349531
$664$	−9.00000	−0.349268
$665$	20.0000	0.775567
$666$	0	0
$667$	20.0000	0.774403
$668$	17.0000	0.657750
$669$	0	0
$670$	−56.0000	−2.16347
$671$	10.0000	0.386046
$672$	−1.00000	−0.0385758
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	5.00000	0.192593
$675$	11.0000	0.423390
$676$	−4.00000	−0.153846
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	−14.0000	−0.537667
$679$	10.0000	0.383765
$680$	12.0000	0.460179
$681$	−16.0000	−0.613121
$682$	−10.0000	−0.382920
$683$	22.0000	0.841807	0.420903	−	0.907106i	$-0.361713\pi$
$683$	22.0000	0.841807	0.420903	+	0.907106i	$0.361713\pi$
$684$	5.00000	0.191180
$685$	−80.0000	−3.05664
$686$	13.0000	0.496342
$687$	−8.00000	−0.305219
$688$	−4.00000	−0.152499
$689$	−33.0000	−1.25720
$690$	20.0000	0.761387
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−13.0000	−0.494186
$693$	1.00000	0.0379869
$694$	−18.0000	−0.683271
$695$	56.0000	2.12420
$696$	−4.00000	−0.151620
$697$	18.0000	0.681799
$698$	−30.0000	−1.13552
$699$	10.0000	0.378235
$700$	−11.0000	−0.415761
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	3.00000	0.113228
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−8.00000	−0.301297
$706$	26.0000	0.978523
$707$	−6.00000	−0.225653
$708$	12.0000	0.450988
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	−11.0000	−0.412242
$713$	−50.0000	−1.87251
$714$	3.00000	0.112272
$715$	12.0000	0.448775
$716$	−4.00000	−0.149487
$717$	0	0
$718$	6.00000	0.223918
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−4.00000	−0.149071
$721$	−2.00000	−0.0744839
$722$	6.00000	0.223297
$723$	−28.0000	−1.04133
$724$	−16.0000	−0.594635
$725$	−44.0000	−1.63412
$726$	−10.0000	−0.371135
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	−3.00000	−0.111187
$729$	1.00000	0.0370370
$730$	44.0000	1.62851
$731$	12.0000	0.443836
$732$	−10.0000	−0.369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	15.0000	0.553660
$735$	24.0000	0.885253
$736$	−5.00000	−0.184302
$737$	−14.0000	−0.515697
$738$	−6.00000	−0.220863
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	0	0
$741$	15.0000	0.551039
$742$	11.0000	0.403823
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	10.0000	0.366618
$745$	40.0000	1.46549
$746$	−28.0000	−1.02515
$747$	−9.00000	−0.329293
$748$	3.00000	0.109691
$749$	3.00000	0.109618
$750$	−24.0000	−0.876356
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	2.00000	0.0729325
$753$	6.00000	0.218652
$754$	−12.0000	−0.437014
$755$	−20.0000	−0.727875
$756$	−1.00000	−0.0363696
$757$	−15.0000	−0.545184	−0.272592	−	0.962130i	$-0.587881\pi$
$757$	−15.0000	−0.545184	−0.272592	+	0.962130i	$0.587881\pi$
$758$	−20.0000	−0.726433
$759$	5.00000	0.181489
$760$	−20.0000	−0.725476
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−3.00000	−0.108679
$763$	−9.00000	−0.325822
$764$	−9.00000	−0.325609
$765$	12.0000	0.433861
$766$	21.0000	0.758761
$767$	36.0000	1.29988
$768$	1.00000	0.0360844
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	−4.00000	−0.144150
$771$	−27.0000	−0.972381
$772$	−18.0000	−0.647834
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	−4.00000	−0.143777
$775$	110.000	3.95132
$776$	−10.0000	−0.358979
$777$	0	0
$778$	6.00000	0.215110
$779$	−30.0000	−1.07486
$780$	−12.0000	−0.429669
$781$	0	0
$782$	15.0000	0.536399
$783$	−4.00000	−0.142948
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−22.0000	−0.784714
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	7.00000	0.249365
$789$	18.0000	0.640817
$790$	−40.0000	−1.42314
$791$	14.0000	0.497783
$792$	−1.00000	−0.0355335
$793$	−30.0000	−1.06533
$794$	−32.0000	−1.13564
$795$	44.0000	1.56052
$796$	−16.0000	−0.567105
$797$	−4.00000	−0.141687	−0.0708436	−	0.997487i	$-0.522569\pi$
$797$	−4.00000	−0.141687	−0.0708436	+	0.997487i	$0.522569\pi$
$798$	−5.00000	−0.176998
$799$	−6.00000	−0.212265
$800$	11.0000	0.388909
$801$	−11.0000	−0.388666
$802$	−19.0000	−0.670913
$803$	11.0000	0.388182
$804$	14.0000	0.493742
$805$	−20.0000	−0.704907
$806$	30.0000	1.05670
$807$	27.0000	0.950445
$808$	6.00000	0.211079
$809$	−19.0000	−0.668004	−0.334002	−	0.942572i	$-0.608399\pi$
$809$	−19.0000	−0.668004	−0.334002	+	0.942572i	$0.608399\pi$
$810$	−4.00000	−0.140546
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	4.00000	0.140372
$813$	8.00000	0.280572
$814$	0	0
$815$	36.0000	1.26102
$816$	−3.00000	−0.105021
$817$	−20.0000	−0.699711
$818$	−10.0000	−0.349642
$819$	−3.00000	−0.104828
$820$	24.0000	0.838116
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	20.0000	0.697580
$823$	9.00000	0.313720	0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000	0.313720	0.156860	+	0.987621i	$0.449863\pi$
$824$	2.00000	0.0696733
$825$	−11.0000	−0.382971
$826$	−12.0000	−0.417533
$827$	14.0000	0.486828	0.243414	−	0.969923i	$-0.421733\pi$
$827$	14.0000	0.486828	0.243414	+	0.969923i	$0.421733\pi$
$828$	−5.00000	−0.173762
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	36.0000	1.24958
$831$	5.00000	0.173448
$832$	3.00000	0.104006
$833$	18.0000	0.623663
$834$	−14.0000	−0.484780
$835$	−68.0000	−2.35324
$836$	−5.00000	−0.172929
$837$	10.0000	0.345651
$838$	5.00000	0.172722
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	4.00000	0.138013
$841$	−13.0000	−0.448276
$842$	6.00000	0.206774
$843$	9.00000	0.309976
$844$	0	0
$845$	16.0000	0.550417
$846$	2.00000	0.0687614
$847$	10.0000	0.343604
$848$	−11.0000	−0.377742
$849$	−21.0000	−0.720718
$850$	−33.0000	−1.13189
$851$	0	0
$852$	0	0
$853$	55.0000	1.88316	0.941582	−	0.336784i	$-0.109339\pi$
$853$	55.0000	1.88316	0.941582	+	0.336784i	$0.109339\pi$
$854$	10.0000	0.342193
$855$	−20.0000	−0.683986
$856$	−3.00000	−0.102538
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	−3.00000	−0.102418
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	16.0000	0.545595
$861$	6.00000	0.204479
$862$	−15.0000	−0.510902
$863$	−22.0000	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$864$	1.00000	0.0340207
$865$	52.0000	1.76805
$866$	−25.0000	−0.849535
$867$	−8.00000	−0.271694
$868$	−10.0000	−0.339422
$869$	−10.0000	−0.339227
$870$	16.0000	0.542451
$871$	42.0000	1.42312
$872$	9.00000	0.304778
$873$	−10.0000	−0.338449
$874$	−25.0000	−0.845638
$875$	24.0000	0.811348
$876$	−11.0000	−0.371656
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	14.0000	0.472477
$879$	21.0000	0.708312
$880$	4.00000	0.134840
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	−6.00000	−0.202031
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	−9.00000	−0.302703
$885$	−48.0000	−1.61350
$886$	36.0000	1.20944
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	44.0000	1.47488
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	10.0000	0.334637
$894$	−10.0000	−0.334450
$895$	16.0000	0.534821
$896$	−1.00000	−0.0334077
$897$	−15.0000	−0.500835
$898$	−2.00000	−0.0667409
$899$	−40.0000	−1.33407
$900$	11.0000	0.366667
$901$	33.0000	1.09939
$902$	6.00000	0.199778
$903$	4.00000	0.133112
$904$	−14.0000	−0.465633
$905$	64.0000	2.12743
$906$	5.00000	0.166114
$907$	11.0000	0.365249	0.182625	−	0.983183i	$-0.441541\pi$
$907$	11.0000	0.365249	0.182625	+	0.983183i	$0.441541\pi$
$908$	−16.0000	−0.530979
$909$	6.00000	0.199007
$910$	12.0000	0.397796
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	5.00000	0.165567
$913$	9.00000	0.297857
$914$	−28.0000	−0.926158
$915$	40.0000	1.32236
$916$	−8.00000	−0.264327
$917$	22.0000	0.726504
$918$	−3.00000	−0.0990148
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	20.0000	0.659380
$921$	32.0000	1.05444
$922$	14.0000	0.461065
$923$	0	0
$924$	1.00000	0.0328976
$925$	0	0
$926$	22.0000	0.722965
$927$	2.00000	0.0656886
$928$	−4.00000	−0.131306
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	−40.0000	−1.31165
$931$	−30.0000	−0.983210
$932$	10.0000	0.327561
$933$	16.0000	0.523816
$934$	34.0000	1.11251
$935$	−12.0000	−0.392442
$936$	3.00000	0.0980581
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−14.0000	−0.457116
$939$	14.0000	0.456873
$940$	−8.00000	−0.260931
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	12.0000	0.390567
$945$	4.00000	0.130120
$946$	4.00000	0.130051
$947$	26.0000	0.844886	0.422443	−	0.906389i	$-0.361173\pi$
$947$	26.0000	0.844886	0.422443	+	0.906389i	$0.361173\pi$
$948$	10.0000	0.324785
$949$	−33.0000	−1.07123
$950$	55.0000	1.78444
$951$	−18.0000	−0.583690
$952$	3.00000	0.0972306
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	−11.0000	−0.356138
$955$	36.0000	1.16493
$956$	0	0
$957$	4.00000	0.129302
$958$	−15.0000	−0.484628
$959$	−20.0000	−0.645834
$960$	−4.00000	−0.129099
$961$	69.0000	2.22581
$962$	0	0
$963$	−3.00000	−0.0966736
$964$	−28.0000	−0.901819
$965$	72.0000	2.31776
$966$	5.00000	0.160872
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	−10.0000	−0.321412
$969$	−15.0000	−0.481869
$970$	40.0000	1.28432
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	1.00000	0.0320750
$973$	14.0000	0.448819
$974$	2.00000	0.0640841
$975$	33.0000	1.05685
$976$	−10.0000	−0.320092
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	−9.00000	−0.287788
$979$	11.0000	0.351562
$980$	24.0000	0.766652
$981$	9.00000	0.287348
$982$	−29.0000	−0.925427
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	−6.00000	−0.191273
$985$	−28.0000	−0.892154
$986$	12.0000	0.382158
$987$	−2.00000	−0.0636607
$988$	15.0000	0.477214
$989$	20.0000	0.635963
$990$	4.00000	0.127128
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	10.0000	0.317500
$993$	−28.0000	−0.888553
$994$	0	0
$995$	64.0000	2.02894
$996$	−9.00000	−0.285176
$997$	9.00000	0.285033	0.142516	−	0.989792i	$-0.454481\pi$
$997$	9.00000	0.285033	0.142516	+	0.989792i	$0.454481\pi$
$998$	15.0000	0.474817
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8214.2.a.j.1.1		1
37.36	even	2		222.2.a.c.1.1	✓	1
111.110	odd	2		666.2.a.d.1.1		1
148.147	odd	2		1776.2.a.e.1.1		1
185.184	even	2		5550.2.a.z.1.1		1
296.147	odd	2		7104.2.a.p.1.1		1
296.221	even	2		7104.2.a.a.1.1		1
444.443	even	2		5328.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.2.a.c.1.1	✓	1	37.36	even	2
666.2.a.d.1.1		1	111.110	odd	2
1776.2.a.e.1.1		1	148.147	odd	2
5328.2.a.b.1.1		1	444.443	even	2
5550.2.a.z.1.1		1	185.184	even	2
7104.2.a.a.1.1		1	296.221	even	2
7104.2.a.p.1.1		1	296.147	odd	2
8214.2.a.j.1.1		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:	\( N \)	\(=\)	\( 8214 = 2 \cdot 3 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8214.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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