LMFDB - Embedded newform 1521.2.a.l.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 39)
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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{7} +1.73205 q^{8} -3.46410 q^{11} -6.00000 q^{14} -5.00000 q^{16} -6.00000 q^{17} +3.46410 q^{19} +6.00000 q^{22} -5.00000 q^{25} +3.46410 q^{28} -6.00000 q^{29} -3.46410 q^{31} +5.19615 q^{32} +10.3923 q^{34} -6.92820 q^{37} -6.00000 q^{38} +6.92820 q^{41} +4.00000 q^{43} -3.46410 q^{44} +3.46410 q^{47} +5.00000 q^{49} +8.66025 q^{50} -6.00000 q^{53} +6.00000 q^{56} +10.3923 q^{58} +10.3923 q^{59} -2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -10.3923 q^{67} -6.00000 q^{68} -3.46410 q^{71} +12.0000 q^{74} +3.46410 q^{76} -12.0000 q^{77} -8.00000 q^{79} -12.0000 q^{82} +3.46410 q^{83} -6.92820 q^{86} -6.00000 q^{88} -6.92820 q^{89} -6.00000 q^{94} -13.8564 q^{97} -8.66025 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 12 q^{14} - 10 q^{16} - 12 q^{17} + 12 q^{22} - 10 q^{25} - 12 q^{29} - 12 q^{38} + 8 q^{43} + 10 q^{49} - 12 q^{53} + 12 q^{56} - 4 q^{61} + 12 q^{62} + 2 q^{64} - 12 q^{68} + 24 q^{74} - 24 q^{77} - 16 q^{79} - 24 q^{82} - 12 q^{88} - 12 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 12 * q^14 - 10 * q^16 - 12 * q^17 + 12 * q^22 - 10 * q^25 - 12 * q^29 - 12 * q^38 + 8 * q^43 + 10 * q^49 - 12 * q^53 + 12 * q^56 - 4 * q^61 + 12 * q^62 + 2 * q^64 - 12 * q^68 + 24 * q^74 - 24 * q^77 - 16 * q^79 - 24 * q^82 - 12 * q^88 - 12 * q^94

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
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	0	0
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	3.46410	0.654654
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	10.3923	1.78227
$35$	0	0
$36$	0	0
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	8.66025	1.22474
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	10.3923	1.36458
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−10.3923	−1.26962	−0.634811	−	0.772667i	$-0.718922\pi$
$67$	−10.3923	−1.26962	−0.634811	+	0.772667i	$0.718922\pi$
$68$	−6.00000	−0.727607
$69$	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
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	3.46410	0.397360
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	−12.0000	−1.32518
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	0	0
$86$	−6.92820	−0.747087
$87$	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
$92$	0	0
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0	0
$96$	0	0
$97$	−13.8564	−1.40690	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564	−1.40690	−0.703452	+	0.710742i	$0.748359\pi$
$98$	−8.66025	−0.874818
$99$	0	0
$100$	−5.00000	−0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	10.3923	1.00939
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	0	0
$112$	−17.3205	−1.63663
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−18.0000	−1.65703
$119$	−20.7846	−1.90532
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.46410	0.313625
$123$	0	0
$124$	−3.46410	−0.311086
$125$	0	0
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	18.0000	1.55496
$135$	0	0
$136$	−10.3923	−0.891133
$137$	−20.7846	−1.77575	−0.887875	−	0.460086i	$-0.847819\pi$
$137$	−20.7846	−1.77575	−0.887875	+	0.460086i	$0.847819\pi$
$138$	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
$146$	0	0
$147$	0	0
$148$	−6.92820	−0.569495
$149$	−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
$151$	−10.3923	−0.845714	−0.422857	−	0.906196i	$-0.638973\pi$
$151$	−10.3923	−0.845714	−0.422857	+	0.906196i	$0.638973\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	20.7846	1.67487
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	13.8564	1.10236
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.46410	−0.271329	−0.135665	−	0.990755i	$-0.543317\pi$
$163$	−3.46410	−0.271329	−0.135665	+	0.990755i	$0.543317\pi$
$164$	6.92820	0.541002
$165$	0	0
$166$	−6.00000	−0.465690
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−17.3205	−1.30931
$176$	17.3205	1.30558
$177$	0	0
$178$	12.0000	0.899438
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.7846	1.51992
$188$	3.46410	0.252646
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	24.0000	1.72310
$195$	0	0
$196$	5.00000	0.357143
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	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
$202$	−10.3923	−0.731200
$203$	−20.7846	−1.45879
$204$	0	0
$205$	0	0
$206$	13.8564	0.965422
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	20.7846	1.42081
$215$	0	0
$216$	0	0
$217$	−12.0000	−0.814613
$218$	−12.0000	−0.812743
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3.46410	−0.231973	−0.115987	−	0.993251i	$-0.537003\pi$
$223$	−3.46410	−0.231973	−0.115987	+	0.993251i	$0.537003\pi$
$224$	18.0000	1.20268
$225$	0	0
$226$	−10.3923	−0.691286
$227$	17.3205	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$228$	0	0
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	0	0
$231$	0	0
$232$	−10.3923	−0.682288
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	10.3923	0.676481
$237$	0	0
$238$	36.0000	2.33353
$239$	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
$241$	−13.8564	−0.892570	−0.446285	−	0.894891i	$-0.647253\pi$
$241$	−13.8564	−0.892570	−0.446285	+	0.894891i	$0.647253\pi$
$242$	−1.73205	−0.111340
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−6.00000	−0.381000
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−13.8564	−0.869428
$255$	0	0
$256$	19.0000	1.18750
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	0	0
$262$	−20.7846	−1.28408
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	−20.7846	−1.27439
$267$	0	0
$268$	−10.3923	−0.634811
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−10.3923	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$271$	−10.3923	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$272$	30.0000	1.81902
$273$	0	0
$274$	36.0000	2.17484
$275$	17.3205	1.04447
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	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
$287$	24.0000	1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	0	0
$295$	0	0
$296$	−12.0000	−0.697486
$297$	0	0
$298$	24.0000	1.39028
$299$	0	0
$300$	0	0
$301$	13.8564	0.798670
$302$	18.0000	1.03578
$303$	0	0
$304$	−17.3205	−0.993399
$305$	0	0
$306$	0	0
$307$	10.3923	0.593120	0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923	0.593120	0.296560	+	0.955014i	$0.404160\pi$
$308$	−12.0000	−0.683763
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−24.2487	−1.36843
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−13.8564	−0.778253	−0.389127	−	0.921184i	$-0.627223\pi$
$317$	−13.8564	−0.778253	−0.389127	+	0.921184i	$0.627223\pi$
$318$	0	0
$319$	20.7846	1.16371
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.7846	−1.15649
$324$	0	0
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	12.0000	0.662589
$329$	12.0000	0.661581
$330$	0	0
$331$	3.46410	0.190404	0.0952021	−	0.995458i	$-0.469650\pi$
$331$	3.46410	0.190404	0.0952021	+	0.995458i	$0.469650\pi$
$332$	3.46410	0.190117
$333$	0	0
$334$	−30.0000	−1.64153
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	−6.92820	−0.374088
$344$	6.92820	0.373544
$345$	0	0
$346$	31.1769	1.67608
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	6.92820	0.370858	0.185429	−	0.982658i	$-0.440632\pi$
$349$	6.92820	0.370858	0.185429	+	0.982658i	$0.440632\pi$
$350$	30.0000	1.60357
$351$	0	0
$352$	−18.0000	−0.959403
$353$	−34.6410	−1.84376	−0.921878	−	0.387481i	$-0.873345\pi$
$353$	−34.6410	−1.84376	−0.921878	+	0.387481i	$0.873345\pi$
$354$	0	0
$355$	0	0
$356$	−6.92820	−0.367194
$357$	0	0
$358$	20.7846	1.09850
$359$	17.3205	0.914141	0.457071	−	0.889430i	$-0.348899\pi$
$359$	17.3205	0.914141	0.457071	+	0.889430i	$0.348899\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	−17.3205	−0.910346
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.7846	−1.07908
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	6.00000	0.309426
$377$	0	0
$378$	0	0
$379$	17.3205	0.889695	0.444847	−	0.895606i	$-0.353258\pi$
$379$	17.3205	0.889695	0.444847	+	0.895606i	$0.353258\pi$
$380$	0	0
$381$	0	0
$382$	41.5692	2.12687
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−13.8564	−0.703452
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	0	0
$392$	8.66025	0.437409
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.6410	1.73858	0.869291	−	0.494300i	$-0.164576\pi$
$397$	34.6410	1.73858	0.869291	+	0.494300i	$0.164576\pi$
$398$	27.7128	1.38912
$399$	0	0
$400$	25.0000	1.25000
$401$	6.92820	0.345978	0.172989	−	0.984924i	$-0.444657\pi$
$401$	6.92820	0.345978	0.172989	+	0.984924i	$0.444657\pi$
$402$	0	0
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	36.0000	1.78665
$407$	24.0000	1.18964
$408$	0	0
$409$	27.7128	1.37031	0.685155	−	0.728397i	$-0.259734\pi$
$409$	27.7128	1.37031	0.685155	+	0.728397i	$0.259734\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	36.0000	1.77144
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	20.7846	1.01661
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−34.6410	−1.68830	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	−34.6410	−1.68830	−0.844150	+	0.536107i	$0.819894\pi$
$422$	34.6410	1.68630
$423$	0	0
$424$	−10.3923	−0.504695
$425$	30.0000	1.45521
$426$	0	0
$427$	−6.92820	−0.335279
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	24.2487	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$431$	24.2487	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	20.7846	0.997693
$435$	0	0
$436$	6.92820	0.331801
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	0	0
$446$	6.00000	0.284108
$447$	0	0
$448$	3.46410	0.163663
$449$	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
$452$	6.00000	0.282216
$453$	0	0
$454$	−30.0000	−1.40797
$455$	0	0
$456$	0	0
$457$	−27.7128	−1.29635	−0.648175	−	0.761491i	$-0.724468\pi$
$457$	−27.7128	−1.29635	−0.648175	+	0.761491i	$0.724468\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	0	0
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	17.3205	0.804952	0.402476	−	0.915430i	$-0.368150\pi$
$463$	17.3205	0.804952	0.402476	+	0.915430i	$0.368150\pi$
$464$	30.0000	1.39272
$465$	0	0
$466$	10.3923	0.481414
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	0	0
$472$	18.0000	0.828517
$473$	−13.8564	−0.637118
$474$	0	0
$475$	−17.3205	−0.794719
$476$	−20.7846	−0.952661
$477$	0	0
$478$	−18.0000	−0.823301
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	0	0
$482$	24.0000	1.09317
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	38.1051	1.72671	0.863354	−	0.504599i	$-0.168360\pi$
$487$	38.1051	1.72671	0.863354	+	0.504599i	$0.168360\pi$
$488$	−3.46410	−0.156813
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	0	0
$496$	17.3205	0.777714
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−10.3923	−0.465223	−0.232612	−	0.972570i	$-0.574727\pi$
$499$	−10.3923	−0.465223	−0.232612	+	0.972570i	$0.574727\pi$
$500$	0	0
$501$	0	0
$502$	20.7846	0.927663
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	41.5692	1.84252	0.921262	−	0.388943i	$-0.127160\pi$
$509$	41.5692	1.84252	0.921262	+	0.388943i	$0.127160\pi$
$510$	0	0
$511$	0	0
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−31.1769	−1.37515
$515$	0	0
$516$	0	0
$517$	−12.0000	−0.527759
$518$	41.5692	1.82645
$519$	0	0
$520$	0	0
$521$	−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
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−41.5692	−1.81250
$527$	20.7846	0.905392
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	12.0000	0.520266
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−18.0000	−0.777482
$537$	0	0
$538$	10.3923	0.448044
$539$	−17.3205	−0.746047
$540$	0	0
$541$	−6.92820	−0.297867	−0.148933	−	0.988847i	$-0.547584\pi$
$541$	−6.92820	−0.297867	−0.148933	+	0.988847i	$0.547584\pi$
$542$	18.0000	0.773166
$543$	0	0
$544$	−31.1769	−1.33670
$545$	0	0
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−20.7846	−0.887875
$549$	0	0
$550$	−30.0000	−1.27920
$551$	−20.7846	−0.885454
$552$	0	0
$553$	−27.7128	−1.17847
$554$	−17.3205	−0.735878
$555$	0	0
$556$	−4.00000	−0.169638
$557$	13.8564	0.587115	0.293557	−	0.955941i	$-0.405161\pi$
$557$	13.8564	0.587115	0.293557	+	0.955941i	$0.405161\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	0	0
$566$	6.92820	0.291214
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	−41.5692	−1.73507
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	−32.9090	−1.36883
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	20.7846	0.860811
$584$	0	0
$585$	0	0
$586$	−48.0000	−1.98286
$587$	−10.3923	−0.428936	−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923	−0.428936	−0.214468	+	0.976731i	$0.568802\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	34.6410	1.42374
$593$	6.92820	0.284507	0.142254	−	0.989830i	$-0.454565\pi$
$593$	6.92820	0.284507	0.142254	+	0.989830i	$0.454565\pi$
$594$	0	0
$595$	0	0
$596$	−13.8564	−0.567581
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−24.0000	−0.978167
$603$	0	0
$604$	−10.3923	−0.422857
$605$	0	0
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	18.0000	0.729996
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−20.7846	−0.839482	−0.419741	−	0.907644i	$-0.637879\pi$
$613$	−20.7846	−0.839482	−0.419741	+	0.907644i	$0.637879\pi$
$614$	−18.0000	−0.726421
$615$	0	0
$616$	−20.7846	−0.837436
$617$	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
$619$	−31.1769	−1.25311	−0.626553	−	0.779379i	$-0.715535\pi$
$619$	−31.1769	−1.25311	−0.626553	+	0.779379i	$0.715535\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	25.0000	1.00000
$626$	−17.3205	−0.692267
$627$	0	0
$628$	14.0000	0.558661
$629$	41.5692	1.65747
$630$	0	0
$631$	−38.1051	−1.51694	−0.758470	−	0.651707i	$-0.774053\pi$
$631$	−38.1051	−1.51694	−0.758470	+	0.651707i	$0.774053\pi$
$632$	−13.8564	−0.551178
$633$	0	0
$634$	24.0000	0.953162
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−36.0000	−1.42525
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	−10.3923	−0.409832	−0.204916	−	0.978780i	$-0.565692\pi$
$643$	−10.3923	−0.409832	−0.204916	+	0.978780i	$0.565692\pi$
$644$	0	0
$645$	0	0
$646$	36.0000	1.41640
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	−3.46410	−0.135665
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	0	0
$656$	−34.6410	−1.35250
$657$	0	0
$658$	−20.7846	−0.810268
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	20.7846	0.808428	0.404214	−	0.914665i	$-0.367545\pi$
$661$	20.7846	0.808428	0.404214	+	0.914665i	$0.367545\pi$
$662$	−6.00000	−0.233197
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	0	0
$668$	17.3205	0.670151
$669$	0	0
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	−24.2487	−0.934025
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−48.0000	−1.84207
$680$	0	0
$681$	0	0
$682$	−20.7846	−0.795884
$683$	−31.1769	−1.19295	−0.596476	−	0.802631i	$-0.703433\pi$
$683$	−31.1769	−1.19295	−0.596476	+	0.802631i	$0.703433\pi$
$684$	0	0
$685$	0	0
$686$	12.0000	0.458162
$687$	0	0
$688$	−20.0000	−0.762493
$689$	0	0
$690$	0	0
$691$	45.0333	1.71315	0.856574	−	0.516024i	$-0.172588\pi$
$691$	45.0333	1.71315	0.856574	+	0.516024i	$0.172588\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	62.3538	2.36692
$695$	0	0
$696$	0	0
$697$	−41.5692	−1.57455
$698$	−12.0000	−0.454207
$699$	0	0
$700$	−17.3205	−0.654654
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−3.46410	−0.130558
$705$	0	0
$706$	60.0000	2.25813
$707$	20.7846	0.781686
$708$	0	0
$709$	6.92820	0.260194	0.130097	−	0.991501i	$-0.458471\pi$
$709$	6.92820	0.260194	0.130097	+	0.991501i	$0.458471\pi$
$710$	0	0
$711$	0	0
$712$	−12.0000	−0.449719
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	−30.0000	−1.11959
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−27.7128	−1.03208
$722$	12.1244	0.451222
$723$	0	0
$724$	10.0000	0.371647
$725$	30.0000	1.11417
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	34.6410	1.27950	0.639748	−	0.768585i	$-0.279039\pi$
$733$	34.6410	1.27950	0.639748	+	0.768585i	$0.279039\pi$
$734$	27.7128	1.02290
$735$	0	0
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	38.1051	1.40172	0.700860	−	0.713299i	$-0.252800\pi$
$739$	38.1051	1.40172	0.700860	+	0.713299i	$0.252800\pi$
$740$	0	0
$741$	0	0
$742$	36.0000	1.32160
$743$	−3.46410	−0.127086	−0.0635428	−	0.997979i	$-0.520240\pi$
$743$	−3.46410	−0.127086	−0.0635428	+	0.997979i	$0.520240\pi$
$744$	0	0
$745$	0	0
$746$	−38.1051	−1.39513
$747$	0	0
$748$	20.7846	0.759961
$749$	−41.5692	−1.51891
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−17.3205	−0.631614
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−30.0000	−1.08965
$759$	0	0
$760$	0	0
$761$	−48.4974	−1.75803	−0.879015	−	0.476794i	$-0.841799\pi$
$761$	−48.4974	−1.75803	−0.879015	+	0.476794i	$0.841799\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	−24.0000	−0.868290
$765$	0	0
$766$	6.00000	0.216789
$767$	0	0
$768$	0	0
$769$	27.7128	0.999350	0.499675	−	0.866213i	$-0.333453\pi$
$769$	27.7128	0.999350	0.499675	+	0.866213i	$0.333453\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.8564	0.498380	0.249190	−	0.968455i	$-0.419836\pi$
$773$	13.8564	0.498380	0.249190	+	0.968455i	$0.419836\pi$
$774$	0	0
$775$	17.3205	0.622171
$776$	−24.0000	−0.861550
$777$	0	0
$778$	31.1769	1.11775
$779$	24.0000	0.859889
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	0	0
$784$	−25.0000	−0.892857
$785$	0	0
$786$	0	0
$787$	10.3923	0.370446	0.185223	−	0.982697i	$-0.440699\pi$
$787$	10.3923	0.370446	0.185223	+	0.982697i	$0.440699\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	0	0
$794$	−60.0000	−2.12932
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−20.7846	−0.735307
$800$	−25.9808	−0.918559
$801$	0	0
$802$	−12.0000	−0.423735
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	10.3923	0.365600
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−38.1051	−1.33805	−0.669026	−	0.743239i	$-0.733288\pi$
$811$	−38.1051	−1.33805	−0.669026	+	0.743239i	$0.733288\pi$
$812$	−20.7846	−0.729397
$813$	0	0
$814$	−41.5692	−1.45700
$815$	0	0
$816$	0	0
$817$	13.8564	0.484774
$818$	−48.0000	−1.67828
$819$	0	0
$820$	0	0
$821$	−13.8564	−0.483592	−0.241796	−	0.970327i	$-0.577736\pi$
$821$	−13.8564	−0.483592	−0.241796	+	0.970327i	$0.577736\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−13.8564	−0.482711
$825$	0	0
$826$	−62.3538	−2.16957
$827$	24.2487	0.843210	0.421605	−	0.906780i	$-0.361467\pi$
$827$	24.2487	0.843210	0.421605	+	0.906780i	$0.361467\pi$
$828$	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
$833$	−30.0000	−1.03944
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	−20.7846	−0.717992
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	60.0000	2.06774
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	3.46410	0.119028
$848$	30.0000	1.03020
$849$	0	0
$850$	−51.9615	−1.78227
$851$	0	0
$852$	0	0
$853$	20.7846	0.711651	0.355826	−	0.934552i	$-0.384200\pi$
$853$	20.7846	0.711651	0.355826	+	0.934552i	$0.384200\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	−20.7846	−0.710403
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	−42.0000	−1.43053
$863$	−31.1769	−1.06127	−0.530637	−	0.847599i	$-0.678047\pi$
$863$	−31.1769	−1.06127	−0.530637	+	0.847599i	$0.678047\pi$
$864$	0	0
$865$	0	0
$866$	58.8897	2.00115
$867$	0	0
$868$	−12.0000	−0.407307
$869$	27.7128	0.940093
$870$	0	0
$871$	0	0
$872$	12.0000	0.406371
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.4974	1.63764	0.818821	−	0.574049i	$-0.194628\pi$
$877$	48.4974	1.63764	0.818821	+	0.574049i	$0.194628\pi$
$878$	13.8564	0.467631
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	0	0
$886$	−62.3538	−2.09482
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	27.7128	0.929458
$890$	0	0
$891$	0	0
$892$	−3.46410	−0.115987
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	−42.0000	−1.40312
$897$	0	0
$898$	−12.0000	−0.400445
$899$	20.7846	0.693206
$900$	0	0
$901$	36.0000	1.19933
$902$	41.5692	1.38410
$903$	0	0
$904$	10.3923	0.345643
$905$	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	$0.239288\pi$
$908$	17.3205	0.574801
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	48.0000	1.58770
$915$	0	0
$916$	6.92820	0.228914
$917$	41.5692	1.37274
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	−24.0000	−0.790398
$923$	0	0
$924$	0	0
$925$	34.6410	1.13899
$926$	−30.0000	−0.985861
$927$	0	0
$928$	−31.1769	−1.02343
$929$	20.7846	0.681921	0.340960	−	0.940078i	$-0.389248\pi$
$929$	20.7846	0.681921	0.340960	+	0.940078i	$0.389248\pi$
$930$	0	0
$931$	17.3205	0.567657
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−20.7846	−0.680093
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	62.3538	2.03592
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	−51.9615	−1.69120
$945$	0	0
$946$	24.0000	0.780307
$947$	51.9615	1.68852	0.844261	−	0.535932i	$-0.180040\pi$
$947$	51.9615	1.68852	0.844261	+	0.535932i	$0.180040\pi$
$948$	0	0
$949$	0	0
$950$	30.0000	0.973329
$951$	0	0
$952$	−36.0000	−1.16677
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	0	0
$956$	10.3923	0.336111
$957$	0	0
$958$	18.0000	0.581554
$959$	−72.0000	−2.32500
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	0	0
$964$	−13.8564	−0.446285
$965$	0	0
$966$	0	0
$967$	10.3923	0.334194	0.167097	−	0.985940i	$-0.446561\pi$
$967$	10.3923	0.334194	0.167097	+	0.985940i	$0.446561\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−13.8564	−0.444216
$974$	−66.0000	−2.11478
$975$	0	0
$976$	10.0000	0.320092
$977$	−48.4974	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	20.7846	0.663264
$983$	−51.9615	−1.65732	−0.828658	−	0.559756i	$-0.810895\pi$
$983$	−51.9615	−1.65732	−0.828658	+	0.559756i	$0.810895\pi$
$984$	0	0
$985$	0	0
$986$	−62.3538	−1.98575
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−18.0000	−0.571501
$993$	0	0
$994$	20.7846	0.659248
$995$	0	0
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	18.0000	0.569780
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.l.1.1		2
3.2	odd	2		507.2.a.f.1.2		2
12.11	even	2		8112.2.a.bv.1.1		2
13.5	odd	4		117.2.b.a.64.2		2
13.8	odd	4		117.2.b.a.64.1		2
13.12	even	2	inner	1521.2.a.l.1.2		2
39.2	even	12		507.2.j.c.316.1		2
39.5	even	4		39.2.b.a.25.1	✓	2
39.8	even	4		39.2.b.a.25.2	yes	2
39.11	even	12		507.2.j.a.316.1		2
39.17	odd	6		507.2.e.e.484.2		4
39.20	even	12		507.2.j.c.361.1		2
39.23	odd	6		507.2.e.e.22.2		4
39.29	odd	6		507.2.e.e.22.1		4
39.32	even	12		507.2.j.a.361.1		2
39.35	odd	6		507.2.e.e.484.1		4
39.38	odd	2		507.2.a.f.1.1		2
52.31	even	4		1872.2.c.e.1585.1		2
52.47	even	4		1872.2.c.e.1585.2		2
156.47	odd	4		624.2.c.e.337.2		2
156.83	odd	4		624.2.c.e.337.1		2
156.155	even	2		8112.2.a.bv.1.2		2
195.8	odd	4		975.2.h.f.649.3		4
195.44	even	4		975.2.b.d.376.2		2
195.47	odd	4		975.2.h.f.649.2		4
195.83	odd	4		975.2.h.f.649.1		4
195.122	odd	4		975.2.h.f.649.4		4
195.164	even	4		975.2.b.d.376.1		2
273.83	odd	4		1911.2.c.d.883.1		2
273.125	odd	4		1911.2.c.d.883.2		2
312.5	even	4		2496.2.c.k.961.2		2
312.83	odd	4		2496.2.c.d.961.1		2
312.125	even	4		2496.2.c.k.961.1		2
312.203	odd	4		2496.2.c.d.961.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.b.a.25.1	✓	2	39.5	even	4
39.2.b.a.25.2	yes	2	39.8	even	4
117.2.b.a.64.1		2	13.8	odd	4
117.2.b.a.64.2		2	13.5	odd	4
507.2.a.f.1.1		2	39.38	odd	2
507.2.a.f.1.2		2	3.2	odd	2
507.2.e.e.22.1		4	39.29	odd	6
507.2.e.e.22.2		4	39.23	odd	6
507.2.e.e.484.1		4	39.35	odd	6
507.2.e.e.484.2		4	39.17	odd	6
507.2.j.a.316.1		2	39.11	even	12
507.2.j.a.361.1		2	39.32	even	12
507.2.j.c.316.1		2	39.2	even	12
507.2.j.c.361.1		2	39.20	even	12
624.2.c.e.337.1		2	156.83	odd	4
624.2.c.e.337.2		2	156.47	odd	4
975.2.b.d.376.1		2	195.164	even	4
975.2.b.d.376.2		2	195.44	even	4
975.2.h.f.649.1		4	195.83	odd	4
975.2.h.f.649.2		4	195.47	odd	4
975.2.h.f.649.3		4	195.8	odd	4
975.2.h.f.649.4		4	195.122	odd	4
1521.2.a.l.1.1		2	1.1	even	1	trivial
1521.2.a.l.1.2		2	13.12	even	2	inner
1872.2.c.e.1585.1		2	52.31	even	4
1872.2.c.e.1585.2		2	52.47	even	4
1911.2.c.d.883.1		2	273.83	odd	4
1911.2.c.d.883.2		2	273.125	odd	4
2496.2.c.d.961.1		2	312.83	odd	4
2496.2.c.d.961.2		2	312.203	odd	4
2496.2.c.k.961.1		2	312.125	even	4
2496.2.c.k.961.2		2	312.5	even	4
8112.2.a.bv.1.1		2	12.11	even	2
8112.2.a.bv.1.2		2	156.155	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{7} +1.73205 q^{8} -3.46410 q^{11} -6.00000 q^{14} -5.00000 q^{16} -6.00000 q^{17} +3.46410 q^{19} +6.00000 q^{22} -5.00000 q^{25} +3.46410 q^{28} -6.00000 q^{29} -3.46410 q^{31} +5.19615 q^{32} +10.3923 q^{34} -6.92820 q^{37} -6.00000 q^{38} +6.92820 q^{41} +4.00000 q^{43} -3.46410 q^{44} +3.46410 q^{47} +5.00000 q^{49} +8.66025 q^{50} -6.00000 q^{53} +6.00000 q^{56} +10.3923 q^{58} +10.3923 q^{59} -2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -10.3923 q^{67} -6.00000 q^{68} -3.46410 q^{71} +12.0000 q^{74} +3.46410 q^{76} -12.0000 q^{77} -8.00000 q^{79} -12.0000 q^{82} +3.46410 q^{83} -6.92820 q^{86} -6.00000 q^{88} -6.92820 q^{89} -6.00000 q^{94} -13.8564 q^{97} -8.66025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 12 q^{14} - 10 q^{16} - 12 q^{17} + 12 q^{22} - 10 q^{25} - 12 q^{29} - 12 q^{38} + 8 q^{43} + 10 q^{49} - 12 q^{53} + 12 q^{56} - 4 q^{61} + 12 q^{62} + 2 q^{64} - 12 q^{68} + 24 q^{74} - 24 q^{77} - 16 q^{79} - 24 q^{82} - 12 q^{88} - 12 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 12 * q^14 - 10 * q^16 - 12 * q^17 + 12 * q^22 - 10 * q^25 - 12 * q^29 - 12 * q^38 + 8 * q^43 + 10 * q^49 - 12 * q^53 + 12 * q^56 - 4 * q^61 + 12 * q^62 + 2 * q^64 - 12 * q^68 + 24 * q^74 - 24 * q^77 - 16 * q^79 - 24 * q^82 - 12 * q^88 - 12 * q^94

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