LMFDB - Embedded newform 1521.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$1$
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:	no (minimal twist has level 117)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +3.46410 q^{11} +3.46410 q^{14} -5.00000 q^{16} -6.92820 q^{17} -2.00000 q^{19} -6.00000 q^{22} +6.92820 q^{23} -5.00000 q^{25} -2.00000 q^{28} +6.92820 q^{29} -2.00000 q^{31} +5.19615 q^{32} +12.0000 q^{34} -2.00000 q^{37} +3.46410 q^{38} -6.92820 q^{41} +8.00000 q^{43} +3.46410 q^{44} -12.0000 q^{46} +10.3923 q^{47} -3.00000 q^{49} +8.66025 q^{50} -3.46410 q^{56} -12.0000 q^{58} -3.46410 q^{59} -10.0000 q^{61} +3.46410 q^{62} +1.00000 q^{64} -14.0000 q^{67} -6.92820 q^{68} -3.46410 q^{71} +10.0000 q^{73} +3.46410 q^{74} -2.00000 q^{76} -6.92820 q^{77} -4.00000 q^{79} +12.0000 q^{82} -10.3923 q^{83} -13.8564 q^{86} +6.00000 q^{88} -6.92820 q^{89} +6.92820 q^{92} -18.0000 q^{94} +10.0000 q^{97} +5.19615 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 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} - 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} - 36 q^{94} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 4 * q^7 - 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 - 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 - 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	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	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
$13$	0	0
$13$	0	0
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	6.92820	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	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	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	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
$52$	0	0
$53$	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	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−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	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	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
$91$	0	0
$92$	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	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	5.19615	0.524891
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	10.0000	0.944911
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	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	$-0.862376\pi$
$131$	−20.7846	−1.81596	−0.907980	+	0.419014i	$0.862376\pi$
$132$	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
$143$	0	0
$144$	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	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	−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	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	−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
$169$	0	0
$170$	0	0
$171$	0	0
$172$	8.00000	0.609994
$173$	13.8564	1.05348	0.526742	−	0.850026i	$-0.323414\pi$
$173$	13.8564	1.05348	0.526742	+	0.850026i	$0.323414\pi$
$174$	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	$-0.673263\pi$
$179$	−13.8564	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	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	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	−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
$208$	0	0
$209$	−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
$221$	0	0
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−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	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	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	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−1.73205	−0.111340
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−3.46410	−0.219971
$249$	0	0
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	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	$-0.167737\pi$
$257$	27.7128	1.72868	0.864339	+	0.502910i	$0.167737\pi$
$258$	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	$-0.431479\pi$
$263$	6.92820	0.427211	0.213606	+	0.976920i	$0.431479\pi$
$264$	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	$-0.567740\pi$
$269$	−6.92820	−0.422420	−0.211210	+	0.977441i	$0.567740\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	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
$286$	0	0
$287$	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
$299$	0	0
$300$	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	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	−6.92820	−0.394771
$309$	0	0
$310$	0	0
$311$	−20.7846	−1.17859	−0.589294	−	0.807919i	$-0.700594\pi$
$311$	−20.7846	−1.17859	−0.589294	+	0.807919i	$0.700594\pi$
$312$	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
$325$	0	0
$326$	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	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−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	$0.482652\pi$
$338$	0	0
$339$	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	$-0.767003\pi$
$347$	−27.7128	−1.48770	−0.743851	+	0.668346i	$0.767003\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	−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
$364$	0	0
$365$	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
$377$	0	0
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	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	$-0.323350\pi$
$389$	20.7846	1.05382	0.526911	+	0.849921i	$0.323350\pi$
$390$	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	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	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
$403$	0	0
$404$	−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	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	6.92820	0.340915
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	12.0000	0.586939
$419$	6.92820	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−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
$442$	0	0
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	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	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	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	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	−34.6410	−1.60817
$465$	0	0
$466$	0	0
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	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
$481$	0	0
$482$	−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	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−17.3205	−0.784063
$489$	0	0
$490$	0	0
$491$	6.92820	0.312665	0.156333	−	0.987704i	$-0.450033\pi$
$491$	6.92820	0.312665	0.156333	+	0.987704i	$0.450033\pi$
$492$	0	0
$493$	−48.0000	−2.16181
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	6.92820	0.310772
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	0	0
$502$	12.0000	0.535586
$503$	−6.92820	−0.308913	−0.154457	−	0.988000i	$-0.549363\pi$
$503$	−6.92820	−0.308913	−0.154457	+	0.988000i	$0.549363\pi$
$504$	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	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	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
$533$	0	0
$534$	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	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−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
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−36.0000	−1.51857
$563$	13.8564	0.583978	0.291989	−	0.956422i	$-0.405683\pi$
$563$	13.8564	0.583978	0.291989	+	0.956422i	$0.405683\pi$
$564$	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	$-0.593803\pi$
$569$	−13.8564	−0.580891	−0.290445	+	0.956892i	$0.593803\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	−34.6410	−1.44463
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−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
$598$	0	0
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	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
$611$	0	0
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	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	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	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	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	−6.92820	−0.275589
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−41.5692	−1.64574
$639$	0	0
$640$	0	0
$641$	34.6410	1.36824	0.684119	−	0.729370i	$-0.260187\pi$
$641$	34.6410	1.36824	0.684119	+	0.729370i	$0.260187\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−13.8564	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$647$	−13.8564	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	−14.0000	−0.548282
$653$	−34.6410	−1.35561	−0.677804	−	0.735243i	$-0.737068\pi$
$653$	−34.6410	−1.35561	−0.677804	+	0.735243i	$0.737068\pi$
$654$	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	$-0.413014\pi$
$659$	13.8564	0.539769	0.269884	+	0.962893i	$0.413014\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−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
$676$	0	0
$677$	−41.5692	−1.59763	−0.798817	−	0.601574i	$-0.794541\pi$
$677$	−41.5692	−1.59763	−0.798817	+	0.601574i	$0.794541\pi$
$678$	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	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	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	$-0.628394\pi$
$701$	−20.7846	−0.785024	−0.392512	+	0.919747i	$0.628394\pi$
$702$	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	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	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	$-0.276463\pi$
$719$	34.6410	1.29189	0.645946	+	0.763383i	$0.276463\pi$
$720$	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	$0.297782\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−55.4256	−2.04999
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−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	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	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
$754$	0	0
$755$	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
$767$	0	0
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	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	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	13.8564	0.493614
$789$	0	0
$790$	0	0
$791$	13.8564	0.492677
$792$	0	0
$793$	0	0
$794$	3.46410	0.122936
$795$	0	0
$796$	−16.0000	−0.567105
$797$	34.6410	1.22705	0.613524	−	0.789676i	$-0.289751\pi$
$797$	34.6410	1.22705	0.613524	+	0.789676i	$0.289751\pi$
$798$	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
$806$	0	0
$807$	0	0
$808$	−24.0000	−0.844317
$809$	6.92820	0.243583	0.121791	−	0.992556i	$-0.461136\pi$
$809$	6.92820	0.243583	0.121791	+	0.992556i	$0.461136\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	−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
$832$	0	0
$833$	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	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	−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
$871$	0	0
$872$	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	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	48.4974	1.63671
$879$	0	0
$880$	0	0
$881$	−13.8564	−0.466834	−0.233417	−	0.972377i	$-0.574991\pi$
$881$	−13.8564	−0.466834	−0.233417	+	0.972377i	$0.574991\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	27.7128	0.930505	0.465253	−	0.885178i	$-0.345963\pi$
$887$	27.7128	0.930505	0.465253	+	0.885178i	$0.345963\pi$
$888$	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	$-0.741785\pi$
$911$	−41.5692	−1.37725	−0.688625	+	0.725118i	$0.741785\pi$
$912$	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
$923$	0	0
$924$	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
$949$	0	0
$950$	−17.3205	−0.561951
$951$	0	0
$952$	24.0000	0.777844
$953$	−27.7128	−0.897706	−0.448853	−	0.893606i	$-0.648167\pi$
$953$	−27.7128	−0.897706	−0.448853	+	0.893606i	$0.648167\pi$
$954$	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
$962$	0	0
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	0	0
$971$	55.4256	1.77869	0.889346	−	0.457234i	$-0.151160\pi$
$971$	55.4256	1.77869	0.889346	+	0.457234i	$0.151160\pi$
$972$	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
$988$	0	0
$989$	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	1521.2.a.j.1.1		2
3.2	odd	2	inner	1521.2.a.j.1.2		2
13.5	odd	4		1521.2.b.i.1351.3		4
13.8	odd	4		1521.2.b.i.1351.2		4
13.12	even	2		117.2.a.b.1.2	yes	2
39.5	even	4		1521.2.b.i.1351.1		4
39.8	even	4		1521.2.b.i.1351.4		4
39.38	odd	2		117.2.a.b.1.1	✓	2
52.51	odd	2		1872.2.a.v.1.2		2
65.12	odd	4		2925.2.c.s.2224.4		4
65.38	odd	4		2925.2.c.s.2224.1		4
65.64	even	2		2925.2.a.y.1.1		2
91.90	odd	2		5733.2.a.t.1.2		2
104.51	odd	2		7488.2.a.cj.1.1		2
104.77	even	2		7488.2.a.cq.1.2		2
117.25	even	6		1053.2.e.i.352.1		4
117.38	odd	6		1053.2.e.i.352.2		4
117.77	odd	6		1053.2.e.i.703.2		4
117.103	even	6		1053.2.e.i.703.1		4
156.155	even	2		1872.2.a.v.1.1		2
195.38	even	4		2925.2.c.s.2224.3		4
195.77	even	4		2925.2.c.s.2224.2		4
195.194	odd	2		2925.2.a.y.1.2		2
273.272	even	2		5733.2.a.t.1.1		2
312.77	odd	2		7488.2.a.cq.1.1		2
312.155	even	2		7488.2.a.cj.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.a.b.1.1	✓	2	39.38	odd	2
117.2.a.b.1.2	yes	2	13.12	even	2
1053.2.e.i.352.1		4	117.25	even	6
1053.2.e.i.352.2		4	117.38	odd	6
1053.2.e.i.703.1		4	117.103	even	6
1053.2.e.i.703.2		4	117.77	odd	6
1521.2.a.j.1.1		2	1.1	even	1	trivial
1521.2.a.j.1.2		2	3.2	odd	2	inner
1521.2.b.i.1351.1		4	39.5	even	4
1521.2.b.i.1351.2		4	13.8	odd	4
1521.2.b.i.1351.3		4	13.5	odd	4
1521.2.b.i.1351.4		4	39.8	even	4
1872.2.a.v.1.1		2	156.155	even	2
1872.2.a.v.1.2		2	52.51	odd	2
2925.2.a.y.1.1		2	65.64	even	2
2925.2.a.y.1.2		2	195.194	odd	2
2925.2.c.s.2224.1		4	65.38	odd	4
2925.2.c.s.2224.2		4	195.77	even	4
2925.2.c.s.2224.3		4	195.38	even	4
2925.2.c.s.2224.4		4	65.12	odd	4
5733.2.a.t.1.1		2	273.272	even	2
5733.2.a.t.1.2		2	91.90	odd	2
7488.2.a.cj.1.1		2	104.51	odd	2
7488.2.a.cj.1.2		2	312.155	even	2
7488.2.a.cq.1.1		2	312.77	odd	2
7488.2.a.cq.1.2		2	104.77	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

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

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