LMFDB - Embedded newform 117.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, 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("117.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.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:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.73205 q^{8} +3.46410 q^{11} +1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} +6.92820 q^{17} +2.00000 q^{19} -6.00000 q^{22} -6.92820 q^{23} -5.00000 q^{25} -1.73205 q^{26} +2.00000 q^{28} -6.92820 q^{29} +2.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} +2.00000 q^{37} -3.46410 q^{38} -6.92820 q^{41} +8.00000 q^{43} +3.46410 q^{44} +12.0000 q^{46} +10.3923 q^{47} -3.00000 q^{49} +8.66025 q^{50} +1.00000 q^{52} +3.46410 q^{56} +12.0000 q^{58} -3.46410 q^{59} -10.0000 q^{61} -3.46410 q^{62} +1.00000 q^{64} +14.0000 q^{67} +6.92820 q^{68} -3.46410 q^{71} -10.0000 q^{73} -3.46410 q^{74} +2.00000 q^{76} +6.92820 q^{77} -4.00000 q^{79} +12.0000 q^{82} -10.3923 q^{83} -13.8564 q^{86} +6.00000 q^{88} -6.92820 q^{89} +2.00000 q^{91} -6.92820 q^{92} -18.0000 q^{94} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 4 * q^7	$ 2 q + 2 q^{4} + 4 q^{7} + 2 q^{13} - 10 q^{16} + 4 q^{19} - 12 q^{22} - 10 q^{25} + 4 q^{28} + 4 q^{31} - 24 q^{34} + 4 q^{37} + 16 q^{43} + 24 q^{46} - 6 q^{49} + 2 q^{52} + 24 q^{58} - 20 q^{61} + 2 q^{64} + 28 q^{67} - 20 q^{73} + 4 q^{76} - 8 q^{79} + 24 q^{82} + 12 q^{88} + 4 q^{91} - 36 q^{94} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^7 + 2 * q^13 - 10 * q^16 + 4 * q^19 - 12 * q^22 - 10 * q^25 + 4 * q^28 + 4 * q^31 - 24 * q^34 + 4 * q^37 + 16 * q^43 + 24 * q^46 - 6 * q^49 + 2 * q^52 + 24 * q^58 - 20 * q^61 + 2 * q^64 + 28 * q^67 - 20 * q^73 + 4 * q^76 - 8 * q^79 + 24 * q^82 + 12 * q^88 + 4 * q^91 - 36 * q^94 - 20 * q^97

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−1.73205	−0.339683
$27$	0	0
$28$	2.00000	0.377964
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−12.0000	−2.05798
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−3.46410	−0.561951
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	12.0000	1.76930
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	8.66025	1.22474
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	3.46410	0.462910
$57$	0	0
$58$	12.0000	1.57568
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−3.46410	−0.439941
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−3.46410	−0.402694
$75$	0	0
$76$	2.00000	0.229416
$77$	6.92820	0.789542
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	12.0000	1.32518
$83$	−10.3923	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$84$	0	0
$85$	0	0
$86$	−13.8564	−1.49417
$87$	0	0
$88$	6.00000	0.639602
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−6.92820	−0.722315
$93$	0	0
$94$	−18.0000	−1.85656
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	5.19615	0.524891
$99$	0	0
$100$	−5.00000	−0.500000
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	1.73205	0.169842
$105$	0	0
$106$	0	0
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	−10.0000	−0.944911
$113$	6.92820	0.651751	0.325875	−	0.945413i	$-0.394341\pi$
$113$	6.92820	0.651751	0.325875	+	0.945413i	$0.394341\pi$
$114$	0	0
$115$	0	0
$116$	−6.92820	−0.643268
$117$	0	0
$118$	6.00000	0.552345
$119$	13.8564	1.27021
$120$	0	0
$121$	1.00000	0.0909091
$122$	17.3205	1.56813
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	20.7846	1.81596	0.907980	−	0.419014i	$-0.137624\pi$
$131$	20.7846	1.81596	0.907980	+	0.419014i	$0.137624\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−24.2487	−2.09477
$135$	0	0
$136$	12.0000	1.02899
$137$	6.92820	0.591916	0.295958	−	0.955201i	$-0.404361\pi$
$137$	6.92820	0.591916	0.295958	+	0.955201i	$0.404361\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	3.46410	0.289683
$144$	0	0
$145$	0	0
$146$	17.3205	1.43346
$147$	0	0
$148$	2.00000	0.164399
$149$	−13.8564	−1.13516	−0.567581	−	0.823318i	$-0.692120\pi$
$149$	−13.8564	−1.13516	−0.567581	+	0.823318i	$0.692120\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	3.46410	0.280976
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	6.92820	0.551178
$159$	0	0
$160$	0	0
$161$	−13.8564	−1.09204
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−6.92820	−0.541002
$165$	0	0
$166$	18.0000	1.39707
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000	0.609994
$173$	−13.8564	−1.05348	−0.526742	−	0.850026i	$-0.676586\pi$
$173$	−13.8564	−1.05348	−0.526742	+	0.850026i	$0.676586\pi$
$174$	0	0
$175$	−10.0000	−0.755929
$176$	−17.3205	−1.30558
$177$	0	0
$178$	12.0000	0.899438
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	24.0000	1.75505
$188$	10.3923	0.757937
$189$	0	0
$190$	0	0
$191$	−13.8564	−1.00261	−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564	−1.00261	−0.501307	+	0.865269i	$0.667147\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−8.66025	−0.612372
$201$	0	0
$202$	−24.0000	−1.68863
$203$	−13.8564	−0.972529
$204$	0	0
$205$	0	0
$206$	6.92820	0.482711
$207$	0	0
$208$	−5.00000	−0.346688
$209$	6.92820	0.479234
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	17.3205	1.17309
$219$	0	0
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	−12.0000	−0.798228
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	−12.0000	−0.787839
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	−3.46410	−0.225494
$237$	0	0
$238$	−24.0000	−1.55569
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−1.73205	−0.111340
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	3.46410	0.219971
$249$	0	0
$250$	0	0
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	27.7128	1.73886
$255$	0	0
$256$	19.0000	1.18750
$257$	−27.7128	−1.72868	−0.864339	−	0.502910i	$-0.832263\pi$
$257$	−27.7128	−1.72868	−0.864339	+	0.502910i	$0.832263\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	−36.0000	−2.22409
$263$	−6.92820	−0.427211	−0.213606	−	0.976920i	$-0.568521\pi$
$263$	−6.92820	−0.427211	−0.213606	+	0.976920i	$0.568521\pi$
$264$	0	0
$265$	0	0
$266$	−6.92820	−0.424795
$267$	0	0
$268$	14.0000	0.855186
$269$	6.92820	0.422420	0.211210	−	0.977441i	$-0.432260\pi$
$269$	6.92820	0.422420	0.211210	+	0.977441i	$0.432260\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	−34.6410	−2.10042
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−17.3205	−1.04447
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	20.7846	1.23991	0.619953	−	0.784639i	$-0.287152\pi$
$281$	20.7846	1.23991	0.619953	+	0.784639i	$0.287152\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−13.8564	−0.817918
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	0	0
$292$	−10.0000	−0.585206
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	0	0
$296$	3.46410	0.201347
$297$	0	0
$298$	24.0000	1.39028
$299$	−6.92820	−0.400668
$300$	0	0
$301$	16.0000	0.922225
$302$	−24.2487	−1.39536
$303$	0	0
$304$	−10.0000	−0.573539
$305$	0	0
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	6.92820	0.394771
$309$	0	0
$310$	0	0
$311$	20.7846	1.17859	0.589294	−	0.807919i	$-0.299406\pi$
$311$	20.7846	1.17859	0.589294	+	0.807919i	$0.299406\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−3.46410	−0.195491
$315$	0	0
$316$	−4.00000	−0.225018
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	24.0000	1.33747
$323$	13.8564	0.770991
$324$	0	0
$325$	−5.00000	−0.277350
$326$	−24.2487	−1.34301
$327$	0	0
$328$	−12.0000	−0.662589
$329$	20.7846	1.14589
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−10.3923	−0.570352
$333$	0	0
$334$	30.0000	1.64153
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−1.73205	−0.0942111
$339$	0	0
$340$	0	0
$341$	6.92820	0.375183
$342$	0	0
$343$	−20.0000	−1.07990
$344$	13.8564	0.747087
$345$	0	0
$346$	24.0000	1.29025
$347$	27.7128	1.48770	0.743851	−	0.668346i	$-0.232997\pi$
$347$	27.7128	1.48770	0.743851	+	0.668346i	$0.232997\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	17.3205	0.925820
$351$	0	0
$352$	18.0000	0.959403
$353$	−6.92820	−0.368751	−0.184376	−	0.982856i	$-0.559026\pi$
$353$	−6.92820	−0.368751	−0.184376	+	0.982856i	$0.559026\pi$
$354$	0	0
$355$	0	0
$356$	−6.92820	−0.367194
$357$	0	0
$358$	−24.0000	−1.26844
$359$	10.3923	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	17.3205	0.910346
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	34.6410	1.80579
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−41.5692	−2.14949
$375$	0	0
$376$	18.0000	0.928279
$377$	−6.92820	−0.356821
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	0	0
$382$	24.0000	1.22795
$383$	17.3205	0.885037	0.442518	−	0.896759i	$-0.354085\pi$
$383$	17.3205	0.885037	0.442518	+	0.896759i	$0.354085\pi$
$384$	0	0
$385$	0	0
$386$	−45.0333	−2.29214
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−20.7846	−1.05382	−0.526911	−	0.849921i	$-0.676650\pi$
$389$	−20.7846	−1.05382	−0.526911	+	0.849921i	$0.676650\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	−5.19615	−0.262445
$393$	0	0
$394$	−24.0000	−1.20910
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	27.7128	1.38912
$399$	0	0
$400$	25.0000	1.25000
$401$	34.6410	1.72989	0.864945	−	0.501867i	$-0.167353\pi$
$401$	34.6410	1.72989	0.864945	+	0.501867i	$0.167353\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	13.8564	0.689382
$405$	0	0
$406$	24.0000	1.19110
$407$	6.92820	0.343418
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	−6.92820	−0.340915
$414$	0	0
$415$	0	0
$416$	5.19615	0.254762
$417$	0	0
$418$	−12.0000	−0.586939
$419$	−6.92820	−0.338465	−0.169232	−	0.985576i	$-0.554129\pi$
$419$	−6.92820	−0.338465	−0.169232	+	0.985576i	$0.554129\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−13.8564	−0.674519
$423$	0	0
$424$	0	0
$425$	−34.6410	−1.68034
$426$	0	0
$427$	−20.0000	−0.967868
$428$	−6.92820	−0.334887
$429$	0	0
$430$	0	0
$431$	−38.1051	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$431$	−38.1051	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−6.92820	−0.332564
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−13.8564	−0.662842
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−20.7846	−0.987507	−0.493753	−	0.869602i	$-0.664375\pi$
$443$	−20.7846	−0.987507	−0.493753	+	0.869602i	$0.664375\pi$
$444$	0	0
$445$	0	0
$446$	−3.46410	−0.164030
$447$	0	0
$448$	2.00000	0.0944911
$449$	6.92820	0.326962	0.163481	−	0.986546i	$-0.447728\pi$
$449$	6.92820	0.326962	0.163481	+	0.986546i	$0.447728\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	6.92820	0.325875
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−24.2487	−1.13307
$459$	0	0
$460$	0	0
$461$	27.7128	1.29071	0.645357	−	0.763881i	$-0.276709\pi$
$461$	27.7128	1.29071	0.645357	+	0.763881i	$0.276709\pi$
$462$	0	0
$463$	2.00000	0.0929479	0.0464739	−	0.998920i	$-0.485202\pi$
$463$	2.00000	0.0929479	0.0464739	+	0.998920i	$0.485202\pi$
$464$	34.6410	1.60817
$465$	0	0
$466$	0	0
$467$	−20.7846	−0.961797	−0.480899	−	0.876776i	$-0.659689\pi$
$467$	−20.7846	−0.961797	−0.480899	+	0.876776i	$0.659689\pi$
$468$	0	0
$469$	28.0000	1.29292
$470$	0	0
$471$	0	0
$472$	−6.00000	−0.276172
$473$	27.7128	1.27424
$474$	0	0
$475$	−10.0000	−0.458831
$476$	13.8564	0.635107
$477$	0	0
$478$	−18.0000	−0.823301
$479$	3.46410	0.158279	0.0791394	−	0.996864i	$-0.474783\pi$
$479$	3.46410	0.158279	0.0791394	+	0.996864i	$0.474783\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	17.3205	0.788928
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−17.3205	−0.784063
$489$	0	0
$490$	0	0
$491$	−6.92820	−0.312665	−0.156333	−	0.987704i	$-0.549967\pi$
$491$	−6.92820	−0.312665	−0.156333	+	0.987704i	$0.549967\pi$
$492$	0	0
$493$	−48.0000	−2.16181
$494$	−3.46410	−0.155857
$495$	0	0
$496$	−10.0000	−0.449013
$497$	−6.92820	−0.310772
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	0	0
$501$	0	0
$502$	−12.0000	−0.535586
$503$	6.92820	0.308913	0.154457	−	0.988000i	$-0.450637\pi$
$503$	6.92820	0.308913	0.154457	+	0.988000i	$0.450637\pi$
$504$	0	0
$505$	0	0
$506$	41.5692	1.84798
$507$	0	0
$508$	−16.0000	−0.709885
$509$	13.8564	0.614174	0.307087	−	0.951681i	$-0.400646\pi$
$509$	13.8564	0.614174	0.307087	+	0.951681i	$0.400646\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	−8.66025	−0.382733
$513$	0	0
$514$	48.0000	2.11719
$515$	0	0
$516$	0	0
$517$	36.0000	1.58328
$518$	−6.92820	−0.304408
$519$	0	0
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	20.7846	0.907980
$525$	0	0
$526$	12.0000	0.523225
$527$	13.8564	0.603595
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	4.00000	0.173422
$533$	−6.92820	−0.300094
$534$	0	0
$535$	0	0
$536$	24.2487	1.04738
$537$	0	0
$538$	−12.0000	−0.517357
$539$	−10.3923	−0.447628
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	38.1051	1.63675
$543$	0	0
$544$	36.0000	1.54349
$545$	0	0
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	6.92820	0.295958
$549$	0	0
$550$	30.0000	1.27920
$551$	−13.8564	−0.590303
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−3.46410	−0.147176
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−27.7128	−1.17423	−0.587115	−	0.809504i	$-0.699736\pi$
$557$	−27.7128	−1.17423	−0.587115	+	0.809504i	$0.699736\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−36.0000	−1.51857
$563$	−13.8564	−0.583978	−0.291989	−	0.956422i	$-0.594317\pi$
$563$	−13.8564	−0.583978	−0.291989	+	0.956422i	$0.594317\pi$
$564$	0	0
$565$	0	0
$566$	6.92820	0.291214
$567$	0	0
$568$	−6.00000	−0.251754
$569$	13.8564	0.580891	0.290445	−	0.956892i	$-0.406197\pi$
$569$	13.8564	0.580891	0.290445	+	0.956892i	$0.406197\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	3.46410	0.144841
$573$	0	0
$574$	24.0000	1.00174
$575$	34.6410	1.44463
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	−53.6936	−2.23336
$579$	0	0
$580$	0	0
$581$	−20.7846	−0.862291
$582$	0	0
$583$	0	0
$584$	−17.3205	−0.716728
$585$	0	0
$586$	24.0000	0.991431
$587$	45.0333	1.85872	0.929362	−	0.369170i	$-0.120358\pi$
$587$	45.0333	1.85872	0.929362	+	0.369170i	$0.120358\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	0	0
$595$	0	0
$596$	−13.8564	−0.567581
$597$	0	0
$598$	12.0000	0.490716
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	−27.7128	−1.12949
$603$	0	0
$604$	14.0000	0.569652
$605$	0	0
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	10.3923	0.421464
$609$	0	0
$610$	0	0
$611$	10.3923	0.420428
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−45.0333	−1.81740
$615$	0	0
$616$	12.0000	0.483494
$617$	6.92820	0.278919	0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820	0.278919	0.139459	+	0.990228i	$0.455464\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	0	0
$622$	−36.0000	−1.44347
$623$	−13.8564	−0.555145
$624$	0	0
$625$	25.0000	1.00000
$626$	−24.2487	−0.969173
$627$	0	0
$628$	2.00000	0.0798087
$629$	13.8564	0.552491
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	−6.92820	−0.275589
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.00000	−0.118864
$638$	41.5692	1.64574
$639$	0	0
$640$	0	0
$641$	−34.6410	−1.36824	−0.684119	−	0.729370i	$-0.739813\pi$
$641$	−34.6410	−1.36824	−0.684119	+	0.729370i	$0.739813\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	−24.0000	−0.944267
$647$	13.8564	0.544752	0.272376	−	0.962191i	$-0.412191\pi$
$647$	13.8564	0.544752	0.272376	+	0.962191i	$0.412191\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	8.66025	0.339683
$651$	0	0
$652$	14.0000	0.548282
$653$	34.6410	1.35561	0.677804	−	0.735243i	$-0.262932\pi$
$653$	34.6410	1.35561	0.677804	+	0.735243i	$0.262932\pi$
$654$	0	0
$655$	0	0
$656$	34.6410	1.35250
$657$	0	0
$658$	−36.0000	−1.40343
$659$	−13.8564	−0.539769	−0.269884	−	0.962893i	$-0.586986\pi$
$659$	−13.8564	−0.539769	−0.269884	+	0.962893i	$0.586986\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	17.3205	0.673181
$663$	0	0
$664$	−18.0000	−0.698535
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	−17.3205	−0.670151
$669$	0	0
$670$	0	0
$671$	−34.6410	−1.33730
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−3.46410	−0.133432
$675$	0	0
$676$	1.00000	0.0384615
$677$	41.5692	1.59763	0.798817	−	0.601574i	$-0.205459\pi$
$677$	41.5692	1.59763	0.798817	+	0.601574i	$0.205459\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	−12.0000	−0.459504
$683$	17.3205	0.662751	0.331375	−	0.943499i	$-0.392487\pi$
$683$	17.3205	0.662751	0.331375	+	0.943499i	$0.392487\pi$
$684$	0	0
$685$	0	0
$686$	34.6410	1.32260
$687$	0	0
$688$	−40.0000	−1.52499
$689$	0	0
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−13.8564	−0.526742
$693$	0	0
$694$	−48.0000	−1.82206
$695$	0	0
$696$	0	0
$697$	−48.0000	−1.81813
$698$	−45.0333	−1.70454
$699$	0	0
$700$	−10.0000	−0.377964
$701$	20.7846	0.785024	0.392512	−	0.919747i	$-0.371606\pi$
$701$	20.7846	0.785024	0.392512	+	0.919747i	$0.371606\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	3.46410	0.130558
$705$	0	0
$706$	12.0000	0.451626
$707$	27.7128	1.04225
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	−12.0000	−0.449719
$713$	−13.8564	−0.518927
$714$	0	0
$715$	0	0
$716$	13.8564	0.517838
$717$	0	0
$718$	−18.0000	−0.671754
$719$	−34.6410	−1.29189	−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410	−1.29189	−0.645946	+	0.763383i	$0.723537\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	25.9808	0.966904
$723$	0	0
$724$	−10.0000	−0.371647
$725$	34.6410	1.28654
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	0	0
$731$	55.4256	2.04999
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−13.8564	−0.511449
$735$	0	0
$736$	−36.0000	−1.32698
$737$	48.4974	1.78643
$738$	0	0
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.2487	0.889599	0.444799	−	0.895630i	$-0.353275\pi$
$743$	24.2487	0.889599	0.444799	+	0.895630i	$0.353275\pi$
$744$	0	0
$745$	0	0
$746$	−45.0333	−1.64879
$747$	0	0
$748$	24.0000	0.877527
$749$	−13.8564	−0.506302
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−51.9615	−1.89484
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−3.46410	−0.125822
$759$	0	0
$760$	0	0
$761$	−34.6410	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$761$	−34.6410	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−13.8564	−0.501307
$765$	0	0
$766$	−30.0000	−1.08394
$767$	−3.46410	−0.125081
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	26.0000	0.935760
$773$	−13.8564	−0.498380	−0.249190	−	0.968455i	$-0.580164\pi$
$773$	−13.8564	−0.498380	−0.249190	+	0.968455i	$0.580164\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	−17.3205	−0.621770
$777$	0	0
$778$	36.0000	1.29066
$779$	−13.8564	−0.496457
$780$	0	0
$781$	−12.0000	−0.429394
$782$	83.1384	2.97302
$783$	0	0
$784$	15.0000	0.535714
$785$	0	0
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	13.8564	0.493614
$789$	0	0
$790$	0	0
$791$	13.8564	0.492677
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−3.46410	−0.122936
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−34.6410	−1.22705	−0.613524	−	0.789676i	$-0.710249\pi$
$797$	−34.6410	−1.22705	−0.613524	+	0.789676i	$0.710249\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	−25.9808	−0.918559
$801$	0	0
$802$	−60.0000	−2.11867
$803$	−34.6410	−1.22245
$804$	0	0
$805$	0	0
$806$	−3.46410	−0.122018
$807$	0	0
$808$	24.0000	0.844317
$809$	−6.92820	−0.243583	−0.121791	−	0.992556i	$-0.538864\pi$
$809$	−6.92820	−0.243583	−0.121791	+	0.992556i	$0.538864\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	−13.8564	−0.486265
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	16.0000	0.559769
$818$	17.3205	0.605597
$819$	0	0
$820$	0	0
$821$	−27.7128	−0.967184	−0.483592	−	0.875294i	$-0.660668\pi$
$821$	−27.7128	−0.967184	−0.483592	+	0.875294i	$0.660668\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−6.92820	−0.241355
$825$	0	0
$826$	12.0000	0.417533
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	−20.7846	−0.720144
$834$	0	0
$835$	0	0
$836$	6.92820	0.239617
$837$	0	0
$838$	12.0000	0.414533
$839$	−24.2487	−0.837158	−0.418579	−	0.908180i	$-0.637472\pi$
$839$	−24.2487	−0.837158	−0.418579	+	0.908180i	$0.637472\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	17.3205	0.596904
$843$	0	0
$844$	8.00000	0.275371
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	0	0
$850$	60.0000	2.05798
$851$	−13.8564	−0.474991
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	34.6410	1.18539
$855$	0	0
$856$	−12.0000	−0.410152
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	66.0000	2.24797
$863$	31.1769	1.06127	0.530637	−	0.847599i	$-0.321953\pi$
$863$	31.1769	1.06127	0.530637	+	0.847599i	$0.321953\pi$
$864$	0	0
$865$	0	0
$866$	−3.46410	−0.117715
$867$	0	0
$868$	4.00000	0.135769
$869$	−13.8564	−0.470046
$870$	0	0
$871$	14.0000	0.474372
$872$	−17.3205	−0.586546
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	48.4974	1.63671
$879$	0	0
$880$	0	0
$881$	13.8564	0.466834	0.233417	−	0.972377i	$-0.425009\pi$
$881$	13.8564	0.466834	0.233417	+	0.972377i	$0.425009\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	6.92820	0.233021
$885$	0	0
$886$	36.0000	1.20944
$887$	−27.7128	−0.930505	−0.465253	−	0.885178i	$-0.654037\pi$
$887$	−27.7128	−0.930505	−0.465253	+	0.885178i	$0.654037\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	0	0
$892$	2.00000	0.0669650
$893$	20.7846	0.695530
$894$	0	0
$895$	0	0
$896$	−24.2487	−0.810093
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−13.8564	−0.462137
$900$	0	0
$901$	0	0
$902$	41.5692	1.38410
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−3.46410	−0.114960
$909$	0	0
$910$	0	0
$911$	41.5692	1.37725	0.688625	−	0.725118i	$-0.258215\pi$
$911$	41.5692	1.37725	0.688625	+	0.725118i	$0.258215\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	17.3205	0.572911
$915$	0	0
$916$	14.0000	0.462573
$917$	41.5692	1.37274
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	0	0
$922$	−48.0000	−1.58080
$923$	−3.46410	−0.114022
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−3.46410	−0.113837
$927$	0	0
$928$	−36.0000	−1.18176
$929$	48.4974	1.59115	0.795574	−	0.605856i	$-0.207169\pi$
$929$	48.4974	1.59115	0.795574	+	0.605856i	$0.207169\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	−48.4974	−1.58350
$939$	0	0
$940$	0	0
$941$	−41.5692	−1.35512	−0.677559	−	0.735469i	$-0.736962\pi$
$941$	−41.5692	−1.35512	−0.677559	+	0.735469i	$0.736962\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	17.3205	0.563735
$945$	0	0
$946$	−48.0000	−1.56061
$947$	−17.3205	−0.562841	−0.281420	−	0.959585i	$-0.590806\pi$
$947$	−17.3205	−0.562841	−0.281420	+	0.959585i	$0.590806\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	17.3205	0.561951
$951$	0	0
$952$	24.0000	0.777844
$953$	27.7128	0.897706	0.448853	−	0.893606i	$-0.351833\pi$
$953$	27.7128	0.897706	0.448853	+	0.893606i	$0.351833\pi$
$954$	0	0
$955$	0	0
$956$	10.3923	0.336111
$957$	0	0
$958$	−6.00000	−0.193851
$959$	13.8564	0.447447
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−3.46410	−0.111687
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	0	0
$971$	−55.4256	−1.77869	−0.889346	−	0.457234i	$-0.848840\pi$
$971$	−55.4256	−1.77869	−0.889346	+	0.457234i	$0.848840\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	−3.46410	−0.110997
$975$	0	0
$976$	50.0000	1.60046
$977$	−6.92820	−0.221653	−0.110826	−	0.993840i	$-0.535350\pi$
$977$	−6.92820	−0.221653	−0.110826	+	0.993840i	$0.535350\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	−10.3923	−0.331463	−0.165732	−	0.986171i	$-0.552999\pi$
$983$	−10.3923	−0.331463	−0.165732	+	0.986171i	$0.552999\pi$
$984$	0	0
$985$	0	0
$986$	83.1384	2.64767
$987$	0	0
$988$	2.00000	0.0636285
$989$	−55.4256	−1.76243
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	10.3923	0.329956
$993$	0	0
$994$	12.0000	0.380617
$995$	0	0
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−24.2487	−0.767580
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.a.b.1.1	✓	2
3.2	odd	2	inner	117.2.a.b.1.2	yes	2
4.3	odd	2		1872.2.a.v.1.1		2
5.2	odd	4		2925.2.c.s.2224.2		4
5.3	odd	4		2925.2.c.s.2224.3		4
5.4	even	2		2925.2.a.y.1.2		2
7.6	odd	2		5733.2.a.t.1.1		2
8.3	odd	2		7488.2.a.cj.1.2		2
8.5	even	2		7488.2.a.cq.1.1		2
9.2	odd	6		1053.2.e.i.352.1		4
9.4	even	3		1053.2.e.i.703.2		4
9.5	odd	6		1053.2.e.i.703.1		4
9.7	even	3		1053.2.e.i.352.2		4
12.11	even	2		1872.2.a.v.1.2		2
13.5	odd	4		1521.2.b.i.1351.4		4
13.8	odd	4		1521.2.b.i.1351.1		4
13.12	even	2		1521.2.a.j.1.2		2
15.2	even	4		2925.2.c.s.2224.4		4
15.8	even	4		2925.2.c.s.2224.1		4
15.14	odd	2		2925.2.a.y.1.1		2
21.20	even	2		5733.2.a.t.1.2		2
24.5	odd	2		7488.2.a.cq.1.2		2
24.11	even	2		7488.2.a.cj.1.1		2
39.5	even	4		1521.2.b.i.1351.2		4
39.8	even	4		1521.2.b.i.1351.3		4
39.38	odd	2		1521.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.a.b.1.1	✓	2	1.1	even	1	trivial
117.2.a.b.1.2	yes	2	3.2	odd	2	inner
1053.2.e.i.352.1		4	9.2	odd	6
1053.2.e.i.352.2		4	9.7	even	3
1053.2.e.i.703.1		4	9.5	odd	6
1053.2.e.i.703.2		4	9.4	even	3
1521.2.a.j.1.1		2	39.38	odd	2
1521.2.a.j.1.2		2	13.12	even	2
1521.2.b.i.1351.1		4	13.8	odd	4
1521.2.b.i.1351.2		4	39.5	even	4
1521.2.b.i.1351.3		4	39.8	even	4
1521.2.b.i.1351.4		4	13.5	odd	4
1872.2.a.v.1.1		2	4.3	odd	2
1872.2.a.v.1.2		2	12.11	even	2
2925.2.a.y.1.1		2	15.14	odd	2
2925.2.a.y.1.2		2	5.4	even	2
2925.2.c.s.2224.1		4	15.8	even	4
2925.2.c.s.2224.2		4	5.2	odd	4
2925.2.c.s.2224.3		4	5.3	odd	4
2925.2.c.s.2224.4		4	15.2	even	4
5733.2.a.t.1.1		2	7.6	odd	2
5733.2.a.t.1.2		2	21.20	even	2
7488.2.a.cj.1.1		2	24.11	even	2
7488.2.a.cj.1.2		2	8.3	odd	2
7488.2.a.cq.1.1		2	8.5	even	2
7488.2.a.cq.1.2		2	24.5	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.73205 q^{8} +3.46410 q^{11} +1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} +6.92820 q^{17} +2.00000 q^{19} -6.00000 q^{22} -6.92820 q^{23} -5.00000 q^{25} -1.73205 q^{26} +2.00000 q^{28} -6.92820 q^{29} +2.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} +2.00000 q^{37} -3.46410 q^{38} -6.92820 q^{41} +8.00000 q^{43} +3.46410 q^{44} +12.0000 q^{46} +10.3923 q^{47} -3.00000 q^{49} +8.66025 q^{50} +1.00000 q^{52} +3.46410 q^{56} +12.0000 q^{58} -3.46410 q^{59} -10.0000 q^{61} -3.46410 q^{62} +1.00000 q^{64} +14.0000 q^{67} +6.92820 q^{68} -3.46410 q^{71} -10.0000 q^{73} -3.46410 q^{74} +2.00000 q^{76} +6.92820 q^{77} -4.00000 q^{79} +12.0000 q^{82} -10.3923 q^{83} -13.8564 q^{86} +6.00000 q^{88} -6.92820 q^{89} +2.00000 q^{91} -6.92820 q^{92} -18.0000 q^{94} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 4 * q^7	\( 2 q + 2 q^{4} + 4 q^{7} + 2 q^{13} - 10 q^{16} + 4 q^{19} - 12 q^{22} - 10 q^{25} + 4 q^{28} + 4 q^{31} - 24 q^{34} + 4 q^{37} + 16 q^{43} + 24 q^{46} - 6 q^{49} + 2 q^{52} + 24 q^{58} - 20 q^{61} + 2 q^{64} + 28 q^{67} - 20 q^{73} + 4 q^{76} - 8 q^{79} + 24 q^{82} + 12 q^{88} + 4 q^{91} - 36 q^{94} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^7 + 2 * q^13 - 10 * q^16 + 4 * q^19 - 12 * q^22 - 10 * q^25 + 4 * q^28 + 4 * q^31 - 24 * q^34 + 4 * q^37 + 16 * q^43 + 24 * q^46 - 6 * q^49 + 2 * q^52 + 24 * q^58 - 20 * q^61 + 2 * q^64 + 28 * q^67 - 20 * q^73 + 4 * q^76 - 8 * q^79 + 24 * q^82 + 12 * q^88 + 4 * q^91 - 36 * q^94 - 20 * q^97

Introduction

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