LMFDB - Embedded newform 522.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [522,2,Mod(1,522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(522, 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("522.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 522 = 2 \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	522.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$4.16819098551$
Analytic rank:	$1$
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	$\chi$	$=$	522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -5.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -5.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} -4.00000 q^{11} -6.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -5.00000 q^{19} +3.00000 q^{20} +4.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} +6.00000 q^{26} -5.00000 q^{28} -1.00000 q^{29} -1.00000 q^{32} +1.00000 q^{34} -15.0000 q^{35} +1.00000 q^{37} +5.00000 q^{38} -3.00000 q^{40} +7.00000 q^{41} +1.00000 q^{43} -4.00000 q^{44} -6.00000 q^{46} -13.0000 q^{47} +18.0000 q^{49} -4.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} -12.0000 q^{55} +5.00000 q^{56} +1.00000 q^{58} -13.0000 q^{59} -2.00000 q^{61} +1.00000 q^{64} -18.0000 q^{65} -4.00000 q^{67} -1.00000 q^{68} +15.0000 q^{70} +10.0000 q^{71} -12.0000 q^{73} -1.00000 q^{74} -5.00000 q^{76} +20.0000 q^{77} +8.00000 q^{79} +3.00000 q^{80} -7.00000 q^{82} +12.0000 q^{83} -3.00000 q^{85} -1.00000 q^{86} +4.00000 q^{88} +6.00000 q^{89} +30.0000 q^{91} +6.00000 q^{92} +13.0000 q^{94} -15.0000 q^{95} -12.0000 q^{97} -18.0000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	4.00000	0.852803
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	6.00000	1.17670
$27$	0	0
$28$	−5.00000	−0.944911
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	−15.0000	−2.53546
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	−3.00000	−0.474342
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	0	0
$49$	18.0000	2.57143
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	5.00000	0.668153
$57$	0	0
$58$	1.00000	0.131306
$59$	−13.0000	−1.69246	−0.846228	−	0.532821i	$-0.821132\pi$
$59$	−13.0000	−1.69246	−0.846228	+	0.532821i	$0.821132\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−18.0000	−2.23263
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	15.0000	1.79284
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−5.00000	−0.573539
$77$	20.0000	2.27921
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−7.00000	−0.773021
$83$	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
$85$	−3.00000	−0.325396
$86$	−1.00000	−0.107833
$87$	0	0
$88$	4.00000	0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	30.0000	3.14485
$92$	6.00000	0.625543
$93$	0	0
$94$	13.0000	1.34085
$95$	−15.0000	−1.53897
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−18.0000	−1.81827
$99$	0	0
$100$	4.00000	0.400000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	−5.00000	−0.483368	−0.241684	−	0.970355i	$-0.577700\pi$
$107$	−5.00000	−0.483368	−0.241684	+	0.970355i	$0.577700\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	12.0000	1.14416
$111$	0	0
$112$	−5.00000	−0.472456
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	13.0000	1.19675
$119$	5.00000	0.458349
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	0	0
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	18.0000	1.57870
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	25.0000	2.16777
$134$	4.00000	0.345547
$135$	0	0
$136$	1.00000	0.0857493
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−15.0000	−1.26773
$141$	0	0
$142$	−10.0000	−0.839181
$143$	24.0000	2.00698
$144$	0	0
$145$	−3.00000	−0.249136
$146$	12.0000	0.993127
$147$	0	0
$148$	1.00000	0.0821995
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	5.00000	0.405554
$153$	0	0
$154$	−20.0000	−1.61165
$155$	0	0
$156$	0	0
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−3.00000	−0.237171
$161$	−30.0000	−2.36433
$162$	0	0
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−12.0000	−0.931381
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	3.00000	0.230089
$171$	0	0
$172$	1.00000	0.0762493
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	−20.0000	−1.51186
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	−30.0000	−2.22375
$183$	0	0
$184$	−6.00000	−0.442326
$185$	3.00000	0.220564
$186$	0	0
$187$	4.00000	0.292509
$188$	−13.0000	−0.948122
$189$	0	0
$190$	15.0000	1.08821
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	18.0000	1.28571
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−8.00000	−0.562878
$203$	5.00000	0.350931
$204$	0	0
$205$	21.0000	1.46670
$206$	3.00000	0.209020
$207$	0	0
$208$	−6.00000	−0.416025
$209$	20.0000	1.38343
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	5.00000	0.341793
$215$	3.00000	0.204598
$216$	0	0
$217$	0	0
$218$	−4.00000	−0.270914
$219$	0	0
$220$	−12.0000	−0.809040
$221$	6.00000	0.403604
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	−9.00000	−0.598671
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	1.00000	0.0656532
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−39.0000	−2.54408
$236$	−13.0000	−0.846228
$237$	0	0
$238$	−5.00000	−0.324102
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−2.00000	−0.128037
$245$	54.0000	3.44993
$246$	0	0
$247$	30.0000	1.90885
$248$	0	0
$249$	0	0
$250$	3.00000	0.189737
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	−18.0000	−1.11631
$261$	0	0
$262$	6.00000	0.370681
$263$	−5.00000	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$263$	−5.00000	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−25.0000	−1.53285
$267$	0	0
$268$	−4.00000	−0.244339
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	14.0000	0.845771
$275$	−16.0000	−0.964836
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	0	0
$280$	15.0000	0.896421
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	−24.0000	−1.41915
$287$	−35.0000	−2.06598
$288$	0	0
$289$	−16.0000	−0.941176
$290$	3.00000	0.176166
$291$	0	0
$292$	−12.0000	−0.702247
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	−39.0000	−2.27067
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−21.0000	−1.21650
$299$	−36.0000	−2.08193
$300$	0	0
$301$	−5.00000	−0.288195
$302$	5.00000	0.287718
$303$	0	0
$304$	−5.00000	−0.286770
$305$	−6.00000	−0.343559
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	20.0000	1.13961
$309$	0	0
$310$	0	0
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	23.0000	1.29797
$315$	0	0
$316$	8.00000	0.450035
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	3.00000	0.167705
$321$	0	0
$322$	30.0000	1.67183
$323$	5.00000	0.278207
$324$	0	0
$325$	−24.0000	−1.33128
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	−7.00000	−0.386510
$329$	65.0000	3.58357
$330$	0	0
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−2.00000	−0.109435
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	−3.00000	−0.162698
$341$	0	0
$342$	0	0
$343$	−55.0000	−2.96972
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	11.0000	0.591364
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	20.0000	1.06904
$351$	0	0
$352$	4.00000	0.213201
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	30.0000	1.59223
$356$	6.00000	0.317999
$357$	0	0
$358$	8.00000	0.422813
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	4.00000	0.210235
$363$	0	0
$364$	30.0000	1.57243
$365$	−36.0000	−1.88433
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−3.00000	−0.155963
$371$	10.0000	0.519174
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	13.0000	0.670424
$377$	6.00000	0.309016
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−15.0000	−0.769484
$381$	0	0
$382$	−9.00000	−0.460480
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	60.0000	3.05788
$386$	26.0000	1.32337
$387$	0	0
$388$	−12.0000	−0.609208
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	−18.0000	−0.909137
$393$	0	0
$394$	−11.0000	−0.554172
$395$	24.0000	1.20757
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	4.00000	0.200000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	0	0
$404$	8.00000	0.398015
$405$	0	0
$406$	−5.00000	−0.248146
$407$	−4.00000	−0.198273
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	−21.0000	−1.03712
$411$	0	0
$412$	−3.00000	−0.147799
$413$	65.0000	3.19844
$414$	0	0
$415$	36.0000	1.76717
$416$	6.00000	0.294174
$417$	0	0
$418$	−20.0000	−0.978232
$419$	1.00000	0.0488532	0.0244266	−	0.999702i	$-0.492224\pi$
$419$	1.00000	0.0488532	0.0244266	+	0.999702i	$0.492224\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	1.00000	0.0486792
$423$	0	0
$424$	2.00000	0.0971286
$425$	−4.00000	−0.194029
$426$	0	0
$427$	10.0000	0.483934
$428$	−5.00000	−0.241684
$429$	0	0
$430$	−3.00000	−0.144673
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	0	0
$436$	4.00000	0.191565
$437$	−30.0000	−1.43509
$438$	0	0
$439$	13.0000	0.620456	0.310228	−	0.950662i	$-0.399595\pi$
$439$	13.0000	0.620456	0.310228	+	0.950662i	$0.399595\pi$
$440$	12.0000	0.572078
$441$	0	0
$442$	−6.00000	−0.285391
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	−8.00000	−0.378811
$447$	0	0
$448$	−5.00000	−0.236228
$449$	−29.0000	−1.36859	−0.684297	−	0.729203i	$-0.739891\pi$
$449$	−29.0000	−1.36859	−0.684297	+	0.729203i	$0.739891\pi$
$450$	0	0
$451$	−28.0000	−1.31847
$452$	9.00000	0.423324
$453$	0	0
$454$	−15.0000	−0.703985
$455$	90.0000	4.21927
$456$	0	0
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	1.00000	0.0467269
$459$	0	0
$460$	18.0000	0.839254
$461$	−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
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−34.0000	−1.57333	−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000	−1.57333	−0.786666	+	0.617379i	$0.788195\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	39.0000	1.79894
$471$	0	0
$472$	13.0000	0.598374
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−20.0000	−0.917663
$476$	5.00000	0.229175
$477$	0	0
$478$	22.0000	1.00626
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	19.0000	0.865426
$483$	0	0
$484$	5.00000	0.227273
$485$	−36.0000	−1.63468
$486$	0	0
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	−54.0000	−2.43947
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	1.00000	0.0450377
$494$	−30.0000	−1.34976
$495$	0	0
$496$	0	0
$497$	−50.0000	−2.24281
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	16.0000	0.714115
$503$	17.0000	0.757993	0.378996	−	0.925398i	$-0.376269\pi$
$503$	17.0000	0.757993	0.378996	+	0.925398i	$0.376269\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	24.0000	1.06693
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	−27.0000	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$510$	0	0
$511$	60.0000	2.65424
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	−9.00000	−0.396587
$516$	0	0
$517$	52.0000	2.28696
$518$	5.00000	0.219687
$519$	0	0
$520$	18.0000	0.789352
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	5.00000	0.218010
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	6.00000	0.260623
$531$	0	0
$532$	25.0000	1.08389
$533$	−42.0000	−1.81922
$534$	0	0
$535$	−15.0000	−0.648507
$536$	4.00000	0.172774
$537$	0	0
$538$	−2.00000	−0.0862261
$539$	−72.0000	−3.10126
$540$	0	0
$541$	37.0000	1.59075	0.795377	−	0.606115i	$-0.207273\pi$
$541$	37.0000	1.59075	0.795377	+	0.606115i	$0.207273\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	1.00000	0.0428746
$545$	12.0000	0.514024
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−14.0000	−0.598050
$549$	0	0
$550$	16.0000	0.682242
$551$	5.00000	0.213007
$552$	0	0
$553$	−40.0000	−1.70097
$554$	−8.00000	−0.339887
$555$	0	0
$556$	0	0
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	−15.0000	−0.633866
$561$	0	0
$562$	10.0000	0.421825
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	27.0000	1.13590
$566$	−18.0000	−0.756596
$567$	0	0
$568$	−10.0000	−0.419591
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	24.0000	1.00349
$573$	0	0
$574$	35.0000	1.46087
$575$	24.0000	1.00087
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	−3.00000	−0.124568
$581$	−60.0000	−2.48922
$582$	0	0
$583$	8.00000	0.331326
$584$	12.0000	0.496564
$585$	0	0
$586$	16.0000	0.660954
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	0	0
$589$	0	0
$590$	39.0000	1.60560
$591$	0	0
$592$	1.00000	0.0410997
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	15.0000	0.614940
$596$	21.0000	0.860194
$597$	0	0
$598$	36.0000	1.47215
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	5.00000	0.203785
$603$	0	0
$604$	−5.00000	−0.203447
$605$	15.0000	0.609837
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	6.00000	0.242933
$611$	78.0000	3.15554
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	−20.0000	−0.805823
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	0	0
$622$	−3.00000	−0.120289
$623$	−30.0000	−1.20192
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	−23.0000	−0.917800
$629$	−1.00000	−0.0398726
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	12.0000	0.476581
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−108.000	−4.27912
$638$	−4.00000	−0.158362
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	0	0
$643$	−46.0000	−1.81406	−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000	−1.81406	−0.907031	+	0.421063i	$0.861657\pi$
$644$	−30.0000	−1.18217
$645$	0	0
$646$	−5.00000	−0.196722
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	0	0
$649$	52.0000	2.04118
$650$	24.0000	0.941357
$651$	0	0
$652$	1.00000	0.0391630
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	7.00000	0.273304
$657$	0	0
$658$	−65.0000	−2.53396
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−20.0000	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$662$	27.0000	1.04938
$663$	0	0
$664$	−12.0000	−0.465690
$665$	75.0000	2.90838
$666$	0	0
$667$	−6.00000	−0.232321
$668$	2.00000	0.0773823
$669$	0	0
$670$	12.0000	0.463600
$671$	8.00000	0.308837
$672$	0	0
$673$	−17.0000	−0.655302	−0.327651	−	0.944799i	$-0.606257\pi$
$673$	−17.0000	−0.655302	−0.327651	+	0.944799i	$0.606257\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	23.0000	0.884615
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	60.0000	2.30259
$680$	3.00000	0.115045
$681$	0	0
$682$	0	0
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	0	0
$685$	−42.0000	−1.60474
$686$	55.0000	2.09991
$687$	0	0
$688$	1.00000	0.0381246
$689$	12.0000	0.457164
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−11.0000	−0.418157
$693$	0	0
$694$	15.0000	0.569392
$695$	0	0
$696$	0	0
$697$	−7.00000	−0.265144
$698$	4.00000	0.151402
$699$	0	0
$700$	−20.0000	−0.755929
$701$	−41.0000	−1.54855	−0.774274	−	0.632850i	$-0.781885\pi$
$701$	−41.0000	−1.54855	−0.774274	+	0.632850i	$0.781885\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−22.0000	−0.827981
$707$	−40.0000	−1.50435
$708$	0	0
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	−30.0000	−1.12588
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	72.0000	2.69265
$716$	−8.00000	−0.298974
$717$	0	0
$718$	−15.0000	−0.559795
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	15.0000	0.558629
$722$	−6.00000	−0.223297
$723$	0	0
$724$	−4.00000	−0.148659
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	−30.0000	−1.11187
$729$	0	0
$730$	36.0000	1.33242
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	−6.00000	−0.221163
$737$	16.0000	0.589368
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	3.00000	0.110282
$741$	0	0
$742$	−10.0000	−0.367112
$743$	41.0000	1.50414	0.752072	−	0.659081i	$-0.229055\pi$
$743$	41.0000	1.50414	0.752072	+	0.659081i	$0.229055\pi$
$744$	0	0
$745$	63.0000	2.30814
$746$	−14.0000	−0.512576
$747$	0	0
$748$	4.00000	0.146254
$749$	25.0000	0.913480
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−13.0000	−0.474061
$753$	0	0
$754$	−6.00000	−0.218507
$755$	−15.0000	−0.545906
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	15.0000	0.544107
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	9.00000	0.325609
$765$	0	0
$766$	−2.00000	−0.0722629
$767$	78.0000	2.81642
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	−60.0000	−2.16225
$771$	0	0
$772$	−26.0000	−0.935760
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	20.0000	0.717035
$779$	−35.0000	−1.25401
$780$	0	0
$781$	−40.0000	−1.43131
$782$	6.00000	0.214560
$783$	0	0
$784$	18.0000	0.642857
$785$	−69.0000	−2.46272
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	11.0000	0.391859
$789$	0	0
$790$	−24.0000	−0.853882
$791$	−45.0000	−1.60002
$792$	0	0
$793$	12.0000	0.426132
$794$	−10.0000	−0.354887
$795$	0	0
$796$	16.0000	0.567105
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	13.0000	0.459907
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−34.0000	−1.20058
$803$	48.0000	1.69388
$804$	0	0
$805$	−90.0000	−3.17208
$806$	0	0
$807$	0	0
$808$	−8.00000	−0.281439
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	5.00000	0.175466
$813$	0	0
$814$	4.00000	0.140200
$815$	3.00000	0.105085
$816$	0	0
$817$	−5.00000	−0.174928
$818$	−22.0000	−0.769212
$819$	0	0
$820$	21.0000	0.733352
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	56.0000	1.95204	0.976019	−	0.217687i	$-0.0698512\pi$
$823$	56.0000	1.95204	0.976019	+	0.217687i	$0.0698512\pi$
$824$	3.00000	0.104510
$825$	0	0
$826$	−65.0000	−2.26164
$827$	2.00000	0.0695468	0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000	0.0695468	0.0347734	+	0.999395i	$0.488929\pi$
$828$	0	0
$829$	−43.0000	−1.49345	−0.746726	−	0.665132i	$-0.768375\pi$
$829$	−43.0000	−1.49345	−0.746726	+	0.665132i	$0.768375\pi$
$830$	−36.0000	−1.24958
$831$	0	0
$832$	−6.00000	−0.208013
$833$	−18.0000	−0.623663
$834$	0	0
$835$	6.00000	0.207639
$836$	20.0000	0.691714
$837$	0	0
$838$	−1.00000	−0.0345444
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	−1.00000	−0.0344214
$845$	69.0000	2.37367
$846$	0	0
$847$	−25.0000	−0.859010
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	4.00000	0.137199
$851$	6.00000	0.205677
$852$	0	0
$853$	−21.0000	−0.719026	−0.359513	−	0.933140i	$-0.617057\pi$
$853$	−21.0000	−0.719026	−0.359513	+	0.933140i	$0.617057\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	5.00000	0.170896
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	9.00000	0.307076	0.153538	−	0.988143i	$-0.450933\pi$
$859$	9.00000	0.307076	0.153538	+	0.988143i	$0.450933\pi$
$860$	3.00000	0.102299
$861$	0	0
$862$	−4.00000	−0.136241
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−33.0000	−1.12203
$866$	28.0000	0.951479
$867$	0	0
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	24.0000	0.813209
$872$	−4.00000	−0.135457
$873$	0	0
$874$	30.0000	1.01477
$875$	15.0000	0.507093
$876$	0	0
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	−13.0000	−0.438729
$879$	0	0
$880$	−12.0000	−0.404520
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	30.0000	1.00958	0.504790	−	0.863242i	$-0.331570\pi$
$883$	30.0000	1.00958	0.504790	+	0.863242i	$0.331570\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	24.0000	0.806296
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	−18.0000	−0.603361
$891$	0	0
$892$	8.00000	0.267860
$893$	65.0000	2.17514
$894$	0	0
$895$	−24.0000	−0.802232
$896$	5.00000	0.167038
$897$	0	0
$898$	29.0000	0.967743
$899$	0	0
$900$	0	0
$901$	2.00000	0.0666297
$902$	28.0000	0.932298
$903$	0	0
$904$	−9.00000	−0.299336
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	15.0000	0.497792
$909$	0	0
$910$	−90.0000	−2.98347
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	25.0000	0.826927
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	30.0000	0.990687
$918$	0	0
$919$	−9.00000	−0.296883	−0.148441	−	0.988921i	$-0.547426\pi$
$919$	−9.00000	−0.296883	−0.148441	+	0.988921i	$0.547426\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	30.0000	0.987997
$923$	−60.0000	−1.97492
$924$	0	0
$925$	4.00000	0.131519
$926$	16.0000	0.525793
$927$	0	0
$928$	1.00000	0.0328266
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	−90.0000	−2.94963
$932$	14.0000	0.458585
$933$	0	0
$934$	34.0000	1.11251
$935$	12.0000	0.392442
$936$	0	0
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	−20.0000	−0.653023
$939$	0	0
$940$	−39.0000	−1.27204
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	42.0000	1.36771
$944$	−13.0000	−0.423114
$945$	0	0
$946$	4.00000	0.130051
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	72.0000	2.33722
$950$	20.0000	0.648886
$951$	0	0
$952$	−5.00000	−0.162051
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	27.0000	0.873699
$956$	−22.0000	−0.711531
$957$	0	0
$958$	−28.0000	−0.904639
$959$	70.0000	2.26042
$960$	0	0
$961$	−31.0000	−1.00000
$962$	6.00000	0.193448
$963$	0	0
$964$	−19.0000	−0.611949
$965$	−78.0000	−2.51091
$966$	0	0
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	36.0000	1.15589
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	−17.0000	−0.544715
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−20.0000	−0.639857	−0.319928	−	0.947442i	$-0.603659\pi$
$977$	−20.0000	−0.639857	−0.319928	+	0.947442i	$0.603659\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	54.0000	1.72497
$981$	0	0
$982$	−8.00000	−0.255290
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	33.0000	1.05147
$986$	−1.00000	−0.0318465
$987$	0	0
$988$	30.0000	0.954427
$989$	6.00000	0.190789
$990$	0	0
$991$	−55.0000	−1.74713	−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000	−1.74713	−0.873566	+	0.486705i	$0.838199\pi$
$992$	0	0
$993$	0	0
$994$	50.0000	1.58590
$995$	48.0000	1.52170
$996$	0	0
$997$	−5.00000	−0.158352	−0.0791758	−	0.996861i	$-0.525229\pi$
$997$	−5.00000	−0.158352	−0.0791758	+	0.996861i	$0.525229\pi$
$998$	−40.0000	−1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	522.2.a.f.1.1	✓	1
3.2	odd	2		522.2.a.g.1.1	yes	1
4.3	odd	2		4176.2.a.bi.1.1		1
12.11	even	2		4176.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
522.2.a.f.1.1	✓	1	1.1	even	1	trivial
522.2.a.g.1.1	yes	1	3.2	odd	2
4176.2.a.g.1.1		1	12.11	even	2
4176.2.a.bi.1.1		1	4.3	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:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	522.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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