LMFDB - Embedded newform 1521.2.a.v.1.2

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:	$6$
Coefficient field:	6.6.1997632.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 19x^{2} - 13 $ x^6 - 8x^4 + 19x^2 - 13
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			1.2
Root			$1.59842$ 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)

$f(q)$	$=$	$q-1.28184 q^{2} -0.356896 q^{4} +2.16889 q^{5} +1.04892 q^{7} +3.02115 q^{8} +O(q^{10})$	$q-1.28184 q^{2} -0.356896 q^{4} +2.16889 q^{5} +1.04892 q^{7} +3.02115 q^{8} -2.78017 q^{10} -4.47868 q^{11} -1.34454 q^{14} -3.15883 q^{16} +1.67661 q^{17} -6.13706 q^{19} -0.774069 q^{20} +5.74094 q^{22} -6.78847 q^{23} -0.295897 q^{25} -0.374354 q^{28} +9.80963 q^{29} -7.91185 q^{31} -1.99320 q^{32} -2.14914 q^{34} +2.27499 q^{35} -10.1250 q^{37} +7.86671 q^{38} +6.55257 q^{40} +7.32415 q^{41} -0.506041 q^{43} +1.59842 q^{44} +8.70171 q^{46} +0.887058 q^{47} -5.89977 q^{49} +0.379291 q^{50} +5.11186 q^{53} -9.71379 q^{55} +3.16894 q^{56} -12.5743 q^{58} -10.3801 q^{59} +10.8509 q^{61} +10.1417 q^{62} +8.87263 q^{64} -1.77479 q^{67} -0.598377 q^{68} -2.91617 q^{70} -1.36003 q^{71} -3.66487 q^{73} +12.9786 q^{74} +2.19029 q^{76} -4.69777 q^{77} +3.87263 q^{79} -6.85118 q^{80} -9.38835 q^{82} +4.95165 q^{83} +3.63640 q^{85} +0.648661 q^{86} -13.5308 q^{88} +4.81076 q^{89} +2.42278 q^{92} -1.13706 q^{94} -13.3106 q^{95} -13.4155 q^{97} +7.56254 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} - 12 q^{7}+O(q^{10}) $ 6 * q + 6 * q^4 - 12 * q^7	$ 6 q + 6 q^{4} - 12 q^{7} - 14 q^{10} - 2 q^{16} - 26 q^{19} + 6 q^{22} + 26 q^{25} - 26 q^{28} - 40 q^{31} - 40 q^{34} - 12 q^{37} - 42 q^{40} - 22 q^{43} - 2 q^{46} + 10 q^{49} - 42 q^{55} + 12 q^{58} + 38 q^{61} + 20 q^{64} - 14 q^{67} + 28 q^{70} - 24 q^{73} - 54 q^{76} - 10 q^{79} + 2 q^{82} - 14 q^{85} - 6 q^{88} + 4 q^{94} - 10 q^{97}+O(q^{100}) $ 6 * q + 6 * q^4 - 12 * q^7 - 14 * q^10 - 2 * q^16 - 26 * q^19 + 6 * q^22 + 26 * q^25 - 26 * q^28 - 40 * q^31 - 40 * q^34 - 12 * q^37 - 42 * q^40 - 22 * q^43 - 2 * q^46 + 10 * q^49 - 42 * q^55 + 12 * q^58 + 38 * q^61 + 20 * q^64 - 14 * q^67 + 28 * q^70 - 24 * q^73 - 54 * q^76 - 10 * q^79 + 2 * q^82 - 14 * q^85 - 6 * q^88 + 4 * q^94 - 10 * 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 $

$2$	−1.28184	−0.906395	−0.453198	−	0.891410i	$-0.649717\pi$
$2$	−1.28184	−0.906395	−0.453198	+	0.891410i	$0.649717\pi$
$3$	0	0
$3$	0	0
$4$	−0.356896	−0.178448
$5$	2.16889	0.969959	0.484980	−	0.874525i	$-0.338827\pi$
$5$	2.16889	0.969959	0.484980	+	0.874525i	$0.338827\pi$
$6$	0	0
$7$	1.04892	0.396453	0.198227	−	0.980156i	$-0.436482\pi$
$7$	1.04892	0.396453	0.198227	+	0.980156i	$0.436482\pi$
$8$	3.02115	1.06814
$9$	0	0
$10$	−2.78017	−0.879166
$11$	−4.47868	−1.35037	−0.675187	−	0.737647i	$-0.735937\pi$
$11$	−4.47868	−1.35037	−0.675187	+	0.737647i	$0.735937\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−1.34454	−0.359343
$15$	0	0
$16$	−3.15883	−0.789708
$17$	1.67661	0.406639	0.203319	−	0.979112i	$-0.434827\pi$
$17$	1.67661	0.406639	0.203319	+	0.979112i	$0.434827\pi$
$18$	0	0
$19$	−6.13706	−1.40794	−0.703969	−	0.710230i	$-0.748591\pi$
$19$	−6.13706	−1.40794	−0.703969	+	0.710230i	$0.748591\pi$
$20$	−0.774069	−0.173087
$21$	0	0
$22$	5.74094	1.22397
$23$	−6.78847	−1.41549	−0.707747	−	0.706466i	$-0.750288\pi$
$23$	−6.78847	−1.41549	−0.707747	+	0.706466i	$0.750288\pi$
$24$	0	0
$25$	−0.295897	−0.0591794
$26$	0	0
$27$	0	0
$28$	−0.374354	−0.0707463
$29$	9.80963	1.82160	0.910801	−	0.412846i	$-0.135465\pi$
$29$	9.80963	1.82160	0.910801	+	0.412846i	$0.135465\pi$
$30$	0	0
$31$	−7.91185	−1.42101	−0.710505	−	0.703692i	$-0.751534\pi$
$31$	−7.91185	−1.42101	−0.710505	+	0.703692i	$0.751534\pi$
$32$	−1.99320	−0.352352
$33$	0	0
$34$	−2.14914	−0.368575
$35$	2.27499	0.384544
$36$	0	0
$37$	−10.1250	−1.66454	−0.832268	−	0.554373i	$-0.812958\pi$
$37$	−10.1250	−1.66454	−0.832268	+	0.554373i	$0.812958\pi$
$38$	7.86671	1.27615
$39$	0	0
$40$	6.55257	1.03605
$41$	7.32415	1.14384	0.571920	−	0.820310i	$-0.306199\pi$
$41$	7.32415	1.14384	0.571920	+	0.820310i	$0.306199\pi$
$42$	0	0
$43$	−0.506041	−0.0771705	−0.0385852	−	0.999255i	$-0.512285\pi$
$43$	−0.506041	−0.0771705	−0.0385852	+	0.999255i	$0.512285\pi$
$44$	1.59842	0.240971
$45$	0	0
$46$	8.70171	1.28300
$47$	0.887058	0.129391	0.0646954	−	0.997905i	$-0.479392\pi$
$47$	0.887058	0.129391	0.0646954	+	0.997905i	$0.479392\pi$
$48$	0	0
$49$	−5.89977	−0.842825
$50$	0.379291	0.0536399
$51$	0	0
$52$	0	0
$53$	5.11186	0.702168	0.351084	−	0.936344i	$-0.385813\pi$
$53$	5.11186	0.702168	0.351084	+	0.936344i	$0.385813\pi$
$54$	0	0
$55$	−9.71379	−1.30981
$56$	3.16894	0.423468
$57$	0	0
$58$	−12.5743	−1.65109
$59$	−10.3801	−1.35137	−0.675687	−	0.737189i	$-0.736153\pi$
$59$	−10.3801	−1.35137	−0.675687	+	0.737189i	$0.736153\pi$
$60$	0	0
$61$	10.8509	1.38931	0.694655	−	0.719343i	$-0.255557\pi$
$61$	10.8509	1.38931	0.694655	+	0.719343i	$0.255557\pi$
$62$	10.1417	1.28800
$63$	0	0
$64$	8.87263	1.10908
$65$	0	0
$66$	0	0
$67$	−1.77479	−0.216825	−0.108413	−	0.994106i	$-0.534577\pi$
$67$	−1.77479	−0.216825	−0.108413	+	0.994106i	$0.534577\pi$
$68$	−0.598377	−0.0725638
$69$	0	0
$70$	−2.91617	−0.348548
$71$	−1.36003	−0.161405	−0.0807027	−	0.996738i	$-0.525716\pi$
$71$	−1.36003	−0.161405	−0.0807027	+	0.996738i	$0.525716\pi$
$72$	0	0
$73$	−3.66487	−0.428941	−0.214471	−	0.976730i	$-0.568803\pi$
$73$	−3.66487	−0.428941	−0.214471	+	0.976730i	$0.568803\pi$
$74$	12.9786	1.50873
$75$	0	0
$76$	2.19029	0.251244
$77$	−4.69777	−0.535360
$78$	0	0
$79$	3.87263	0.435704	0.217852	−	0.975982i	$-0.430095\pi$
$79$	3.87263	0.435704	0.217852	+	0.975982i	$0.430095\pi$
$80$	−6.85118	−0.765985
$81$	0	0
$82$	−9.38835	−1.03677
$83$	4.95165	0.543514	0.271757	−	0.962366i	$-0.412395\pi$
$83$	4.95165	0.543514	0.271757	+	0.962366i	$0.412395\pi$
$84$	0	0
$85$	3.63640	0.394423
$86$	0.648661	0.0699470
$87$	0	0
$88$	−13.5308	−1.44239
$89$	4.81076	0.509939	0.254970	−	0.966949i	$-0.417935\pi$
$89$	4.81076	0.509939	0.254970	+	0.966949i	$0.417935\pi$
$90$	0	0
$91$	0	0
$92$	2.42278	0.252592
$93$	0	0
$94$	−1.13706	−0.117279
$95$	−13.3106	−1.36564
$96$	0	0
$97$	−13.4155	−1.36214	−0.681069	−	0.732219i	$-0.738485\pi$
$97$	−13.4155	−1.36214	−0.681069	+	0.732219i	$0.738485\pi$
$98$	7.56254	0.763932
$99$	0	0
$100$	0.105604	0.0105604
$101$	−5.29615	−0.526986	−0.263493	−	0.964661i	$-0.584875\pi$
$101$	−5.29615	−0.526986	−0.263493	+	0.964661i	$0.584875\pi$
$102$	0	0
$103$	−7.38404	−0.727571	−0.363786	−	0.931483i	$-0.618516\pi$
$103$	−7.38404	−0.727571	−0.363786	+	0.931483i	$0.618516\pi$
$104$	0	0
$105$	0	0
$106$	−6.55257	−0.636441
$107$	−15.9997	−1.54675	−0.773376	−	0.633948i	$-0.781433\pi$
$107$	−15.9997	−1.54675	−0.773376	+	0.633948i	$0.781433\pi$
$108$	0	0
$109$	−5.08815	−0.487356	−0.243678	−	0.969856i	$-0.578354\pi$
$109$	−5.08815	−0.487356	−0.243678	+	0.969856i	$0.578354\pi$
$110$	12.4515	1.18720
$111$	0	0
$112$	−3.31336	−0.313083
$113$	−13.9090	−1.30845	−0.654225	−	0.756300i	$-0.727005\pi$
$113$	−13.9090	−1.30845	−0.654225	+	0.756300i	$0.727005\pi$
$114$	0	0
$115$	−14.7235	−1.37297
$116$	−3.50102	−0.325061
$117$	0	0
$118$	13.3056	1.22488
$119$	1.75863	0.161213
$120$	0	0
$121$	9.05861	0.823510
$122$	−13.9090	−1.25926
$123$	0	0
$124$	2.82371	0.253576
$125$	−11.4862	−1.02736
$126$	0	0
$127$	9.21983	0.818128	0.409064	−	0.912506i	$-0.365855\pi$
$127$	9.21983	0.818128	0.409064	+	0.912506i	$0.365855\pi$
$128$	−7.38685	−0.652911
$129$	0	0
$130$	0	0
$131$	−14.6552	−1.28043	−0.640215	−	0.768196i	$-0.721155\pi$
$131$	−14.6552	−1.28043	−0.640215	+	0.768196i	$0.721155\pi$
$132$	0	0
$133$	−6.43727	−0.558182
$134$	2.27499	0.196529
$135$	0	0
$136$	5.06531	0.434347
$137$	10.5837	0.904226	0.452113	−	0.891961i	$-0.350670\pi$
$137$	10.5837	0.904226	0.452113	+	0.891961i	$0.350670\pi$
$138$	0	0
$139$	5.89977	0.500412	0.250206	−	0.968193i	$-0.419502\pi$
$139$	5.89977	0.500412	0.250206	+	0.968193i	$0.419502\pi$
$140$	−0.811935	−0.0686210
$141$	0	0
$142$	1.74333	0.146297
$143$	0	0
$144$	0	0
$145$	21.2760	1.76688
$146$	4.69777	0.388790
$147$	0	0
$148$	3.61356	0.297033
$149$	−10.5403	−0.863495	−0.431748	−	0.901994i	$-0.642103\pi$
$149$	−10.5403	−0.863495	−0.431748	+	0.901994i	$0.642103\pi$
$150$	0	0
$151$	−0.633415	−0.0515466	−0.0257733	−	0.999668i	$-0.508205\pi$
$151$	−0.633415	−0.0515466	−0.0257733	+	0.999668i	$0.508205\pi$
$152$	−18.5410	−1.50388
$153$	0	0
$154$	6.02177	0.485248
$155$	−17.1600	−1.37832
$156$	0	0
$157$	−19.9095	−1.58895	−0.794474	−	0.607298i	$-0.792253\pi$
$157$	−19.9095	−1.58895	−0.794474	+	0.607298i	$0.792253\pi$
$158$	−4.96407	−0.394920
$159$	0	0
$160$	−4.32304	−0.341767
$161$	−7.12055	−0.561178
$162$	0	0
$163$	−9.75063	−0.763728	−0.381864	−	0.924219i	$-0.624718\pi$
$163$	−9.75063	−0.763728	−0.381864	+	0.924219i	$0.624718\pi$
$164$	−2.61396	−0.204116
$165$	0	0
$166$	−6.34721	−0.492639
$167$	20.3306	1.57323	0.786615	−	0.617443i	$-0.211832\pi$
$167$	20.3306	1.57323	0.786615	+	0.617443i	$0.211832\pi$
$168$	0	0
$169$	0	0
$170$	−4.66127	−0.357503
$171$	0	0
$172$	0.180604	0.0137709
$173$	17.3443	1.31866	0.659330	−	0.751854i	$-0.270840\pi$
$173$	17.3443	1.31866	0.659330	+	0.751854i	$0.270840\pi$
$174$	0	0
$175$	−0.310371	−0.0234619
$176$	14.1474	1.06640
$177$	0	0
$178$	−6.16660	−0.462206
$179$	9.47755	0.708386	0.354193	−	0.935172i	$-0.384756\pi$
$179$	9.47755	0.708386	0.354193	+	0.935172i	$0.384756\pi$
$180$	0	0
$181$	−0.786872	−0.0584878	−0.0292439	−	0.999572i	$-0.509310\pi$
$181$	−0.786872	−0.0584878	−0.0292439	+	0.999572i	$0.509310\pi$
$182$	0	0
$183$	0	0
$184$	−20.5090	−1.51195
$185$	−21.9600	−1.61453
$186$	0	0
$187$	−7.50902	−0.549114
$188$	−0.316587	−0.0230895
$189$	0	0
$190$	17.0621	1.23781
$191$	−14.8395	−1.07375	−0.536873	−	0.843663i	$-0.680395\pi$
$191$	−14.8395	−1.07375	−0.536873	+	0.843663i	$0.680395\pi$
$192$	0	0
$193$	3.57242	0.257148	0.128574	−	0.991700i	$-0.458960\pi$
$193$	3.57242	0.257148	0.128574	+	0.991700i	$0.458960\pi$
$194$	17.1965	1.23464
$195$	0	0
$196$	2.10560	0.150400
$197$	−6.72577	−0.479191	−0.239596	−	0.970873i	$-0.577015\pi$
$197$	−6.72577	−0.479191	−0.239596	+	0.970873i	$0.577015\pi$
$198$	0	0
$199$	5.03684	0.357052	0.178526	−	0.983935i	$-0.442867\pi$
$199$	5.03684	0.357052	0.178526	+	0.983935i	$0.442867\pi$
$200$	−0.893950	−0.0632118
$201$	0	0
$202$	6.78879	0.477658
$203$	10.2895	0.722180
$204$	0	0
$205$	15.8853	1.10948
$206$	9.46513	0.659467
$207$	0	0
$208$	0	0
$209$	27.4860	1.90124
$210$	0	0
$211$	20.0834	1.38260	0.691298	−	0.722570i	$-0.257039\pi$
$211$	20.0834	1.38260	0.691298	+	0.722570i	$0.257039\pi$
$212$	−1.82440	−0.125300
$213$	0	0
$214$	20.5090	1.40197
$215$	−1.09755	−0.0748522
$216$	0	0
$217$	−8.29888	−0.563365
$218$	6.52217	0.441737
$219$	0	0
$220$	3.46681	0.233732
$221$	0	0
$222$	0	0
$223$	−3.92394	−0.262766	−0.131383	−	0.991332i	$-0.541942\pi$
$223$	−3.92394	−0.262766	−0.131383	+	0.991332i	$0.541942\pi$
$224$	−2.09070	−0.139691
$225$	0	0
$226$	17.8291	1.18597
$227$	−5.61273	−0.372530	−0.186265	−	0.982500i	$-0.559638\pi$
$227$	−5.61273	−0.372530	−0.186265	+	0.982500i	$0.559638\pi$
$228$	0	0
$229$	−11.1739	−0.738392	−0.369196	−	0.929352i	$-0.620367\pi$
$229$	−11.1739	−0.738392	−0.369196	+	0.929352i	$0.620367\pi$
$230$	18.8731	1.24445
$231$	0	0
$232$	29.6364	1.94572
$233$	−13.7612	−0.901528	−0.450764	−	0.892643i	$-0.648848\pi$
$233$	−13.7612	−0.901528	−0.450764	+	0.892643i	$0.648848\pi$
$234$	0	0
$235$	1.92394	0.125504
$236$	3.70461	0.241150
$237$	0	0
$238$	−2.25428	−0.146123
$239$	3.24713	0.210040	0.105020	−	0.994470i	$-0.466509\pi$
$239$	3.24713	0.210040	0.105020	+	0.994470i	$0.466509\pi$
$240$	0	0
$241$	−14.0489	−0.904970	−0.452485	−	0.891772i	$-0.649462\pi$
$241$	−14.0489	−0.904970	−0.452485	+	0.891772i	$0.649462\pi$
$242$	−11.6116	−0.746425
$243$	0	0
$244$	−3.87263	−0.247919
$245$	−12.7960	−0.817505
$246$	0	0
$247$	0	0
$248$	−23.9029	−1.51784
$249$	0	0
$250$	14.7235	0.931195
$251$	−11.0722	−0.698868	−0.349434	−	0.936961i	$-0.613626\pi$
$251$	−11.0722	−0.698868	−0.349434	+	0.936961i	$0.613626\pi$
$252$	0	0
$253$	30.4034	1.91145
$254$	−11.8183	−0.741547
$255$	0	0
$256$	−8.27652	−0.517282
$257$	12.9786	0.809581	0.404790	−	0.914409i	$-0.367344\pi$
$257$	12.9786	0.809581	0.404790	+	0.914409i	$0.367344\pi$
$258$	0	0
$259$	−10.6203	−0.659911
$260$	0	0
$261$	0	0
$262$	18.7855	1.16057
$263$	8.79716	0.542456	0.271228	−	0.962515i	$-0.412570\pi$
$263$	8.79716	0.542456	0.271228	+	0.962515i	$0.412570\pi$
$264$	0	0
$265$	11.0871	0.681074
$266$	8.25153	0.505934
$267$	0	0
$268$	0.633415	0.0386920
$269$	20.0991	1.22546	0.612732	−	0.790291i	$-0.290070\pi$
$269$	20.0991	1.22546	0.612732	+	0.790291i	$0.290070\pi$
$270$	0	0
$271$	−14.4233	−0.876151	−0.438076	−	0.898938i	$-0.644340\pi$
$271$	−14.4233	−0.876151	−0.438076	+	0.898938i	$0.644340\pi$
$272$	−5.29615	−0.321126
$273$	0	0
$274$	−13.5666	−0.819586
$275$	1.32523	0.0799143
$276$	0	0
$277$	10.9312	0.656794	0.328397	−	0.944540i	$-0.393492\pi$
$277$	10.9312	0.656794	0.328397	+	0.944540i	$0.393492\pi$
$278$	−7.56254	−0.453571
$279$	0	0
$280$	6.87310	0.410746
$281$	−16.8868	−1.00738	−0.503690	−	0.863884i	$-0.668025\pi$
$281$	−16.8868	−1.00738	−0.503690	+	0.863884i	$0.668025\pi$
$282$	0	0
$283$	24.2892	1.44384	0.721921	−	0.691975i	$-0.243259\pi$
$283$	24.2892	1.44384	0.721921	+	0.691975i	$0.243259\pi$
$284$	0.485388	0.0288025
$285$	0	0
$286$	0	0
$287$	7.68242	0.453479
$288$	0	0
$289$	−14.1890	−0.834645
$290$	−27.2724	−1.60149
$291$	0	0
$292$	1.30798	0.0765437
$293$	−19.3809	−1.13224	−0.566121	−	0.824322i	$-0.691556\pi$
$293$	−19.3809	−1.13224	−0.566121	+	0.824322i	$0.691556\pi$
$294$	0	0
$295$	−22.5133	−1.31078
$296$	−30.5891	−1.77796
$297$	0	0
$298$	13.5109	0.782668
$299$	0	0
$300$	0	0
$301$	−0.530795	−0.0305945
$302$	0.811935	0.0467216
$303$	0	0
$304$	19.3860	1.11186
$305$	23.5344	1.34757
$306$	0	0
$307$	30.9627	1.76713	0.883567	−	0.468305i	$-0.155135\pi$
$307$	30.9627	1.76713	0.883567	+	0.468305i	$0.155135\pi$
$308$	1.67661	0.0955340
$309$	0	0
$310$	21.9963	1.24930
$311$	23.2023	1.31568	0.657840	−	0.753157i	$-0.271470\pi$
$311$	23.2023	1.31568	0.657840	+	0.753157i	$0.271470\pi$
$312$	0	0
$313$	−30.1672	−1.70515	−0.852575	−	0.522604i	$-0.824960\pi$
$313$	−30.1672	−1.70515	−0.852575	+	0.522604i	$0.824960\pi$
$314$	25.5207	1.44021
$315$	0	0
$316$	−1.38212	−0.0777505
$317$	23.6405	1.32778	0.663890	−	0.747830i	$-0.268904\pi$
$317$	23.6405	1.32778	0.663890	+	0.747830i	$0.268904\pi$
$318$	0	0
$319$	−43.9342	−2.45984
$320$	19.2438	1.07576
$321$	0	0
$322$	9.12737	0.508649
$323$	−10.2895	−0.572522
$324$	0	0
$325$	0	0
$326$	12.4987	0.692239
$327$	0	0
$328$	22.1274	1.22178
$329$	0.930451	0.0512974
$330$	0	0
$331$	6.43727	0.353824	0.176912	−	0.984227i	$-0.443389\pi$
$331$	6.43727	0.353824	0.176912	+	0.984227i	$0.443389\pi$
$332$	−1.76722	−0.0969890
$333$	0	0
$334$	−26.0605	−1.42597
$335$	−3.84933	−0.210312
$336$	0	0
$337$	−32.3575	−1.76262	−0.881312	−	0.472534i	$-0.843339\pi$
$337$	−32.3575	−1.76262	−0.881312	+	0.472534i	$0.843339\pi$
$338$	0	0
$339$	0	0
$340$	−1.29782	−0.0703839
$341$	35.4347	1.91890
$342$	0	0
$343$	−13.5308	−0.730594
$344$	−1.52883	−0.0824289
$345$	0	0
$346$	−22.2325	−1.19523
$347$	15.1058	0.810920	0.405460	−	0.914113i	$-0.367111\pi$
$347$	15.1058	0.810920	0.405460	+	0.914113i	$0.367111\pi$
$348$	0	0
$349$	−1.68233	−0.0900532	−0.0450266	−	0.998986i	$-0.514337\pi$
$349$	−1.68233	−0.0900532	−0.0450266	+	0.998986i	$0.514337\pi$
$350$	0.397845	0.0212657
$351$	0	0
$352$	8.92692	0.475806
$353$	−5.79532	−0.308454	−0.154227	−	0.988035i	$-0.549289\pi$
$353$	−5.79532	−0.308454	−0.154227	+	0.988035i	$0.549289\pi$
$354$	0	0
$355$	−2.94975	−0.156557
$356$	−1.71694	−0.0909976
$357$	0	0
$358$	−12.1487	−0.642077
$359$	18.3181	0.966793	0.483396	−	0.875402i	$-0.339403\pi$
$359$	18.3181	0.966793	0.483396	+	0.875402i	$0.339403\pi$
$360$	0	0
$361$	18.6635	0.982292
$362$	1.00864	0.0530130
$363$	0	0
$364$	0	0
$365$	−7.94873	−0.416055
$366$	0	0
$367$	29.2760	1.52820	0.764099	−	0.645100i	$-0.223184\pi$
$367$	29.2760	1.52820	0.764099	+	0.645100i	$0.223184\pi$
$368$	21.4437	1.11783
$369$	0	0
$370$	28.1491	1.46340
$371$	5.36192	0.278377
$372$	0	0
$373$	6.02177	0.311795	0.155898	−	0.987773i	$-0.450173\pi$
$373$	6.02177	0.311795	0.155898	+	0.987773i	$0.450173\pi$
$374$	9.62534	0.497714
$375$	0	0
$376$	2.67994	0.138207
$377$	0	0
$378$	0	0
$379$	15.0694	0.774061	0.387031	−	0.922067i	$-0.373501\pi$
$379$	15.0694	0.774061	0.387031	+	0.922067i	$0.373501\pi$
$380$	4.75051	0.243696
$381$	0	0
$382$	19.0218	0.973238
$383$	−35.0361	−1.79026	−0.895130	−	0.445805i	$-0.852918\pi$
$383$	−35.0361	−1.79026	−0.895130	+	0.445805i	$0.852918\pi$
$384$	0	0
$385$	−10.1890	−0.519278
$386$	−4.57925	−0.233078
$387$	0	0
$388$	4.78794	0.243071
$389$	9.54332	0.483866	0.241933	−	0.970293i	$-0.422219\pi$
$389$	9.54332	0.483866	0.241933	+	0.970293i	$0.422219\pi$
$390$	0	0
$391$	−11.3817	−0.575595
$392$	−17.8241	−0.900254
$393$	0	0
$394$	8.62133	0.434337
$395$	8.39932	0.422615
$396$	0	0
$397$	21.0271	1.05532	0.527661	−	0.849455i	$-0.323069\pi$
$397$	21.0271	1.05532	0.527661	+	0.849455i	$0.323069\pi$
$398$	−6.45640	−0.323630
$399$	0	0
$400$	0.934689	0.0467345
$401$	−12.5297	−0.625702	−0.312851	−	0.949802i	$-0.601284\pi$
$401$	−12.5297	−0.625702	−0.312851	+	0.949802i	$0.601284\pi$
$402$	0	0
$403$	0	0
$404$	1.89017	0.0940396
$405$	0	0
$406$	−13.1894	−0.654581
$407$	45.3466	2.24775
$408$	0	0
$409$	7.15751	0.353916	0.176958	−	0.984218i	$-0.443374\pi$
$409$	7.15751	0.353916	0.176958	+	0.984218i	$0.443374\pi$
$410$	−20.3624	−1.00562
$411$	0	0
$412$	2.63533	0.129834
$413$	−10.8879	−0.535757
$414$	0	0
$415$	10.7396	0.527187
$416$	0	0
$417$	0	0
$418$	−35.2325	−1.72328
$419$	4.69777	0.229501	0.114751	−	0.993394i	$-0.463393\pi$
$419$	4.69777	0.229501	0.114751	+	0.993394i	$0.463393\pi$
$420$	0	0
$421$	21.4426	1.04505	0.522525	−	0.852624i	$-0.324990\pi$
$421$	21.4426	1.04505	0.522525	+	0.852624i	$0.324990\pi$
$422$	−25.7436	−1.25318
$423$	0	0
$424$	15.4437	0.750013
$425$	−0.496105	−0.0240646
$426$	0	0
$427$	11.3817	0.550797
$428$	5.71024	0.276015
$429$	0	0
$430$	1.40688	0.0678457
$431$	7.28764	0.351034	0.175517	−	0.984476i	$-0.443840\pi$
$431$	7.28764	0.351034	0.175517	+	0.984476i	$0.443840\pi$
$432$	0	0
$433$	16.9594	0.815019	0.407509	−	0.913201i	$-0.366397\pi$
$433$	16.9594	0.815019	0.407509	+	0.913201i	$0.366397\pi$
$434$	10.6378	0.510631
$435$	0	0
$436$	1.81594	0.0869677
$437$	41.6613	1.99293
$438$	0	0
$439$	0.856232	0.0408657	0.0204329	−	0.999791i	$-0.493496\pi$
$439$	0.856232	0.0408657	0.0204329	+	0.999791i	$0.493496\pi$
$440$	−29.3469	−1.39906
$441$	0	0
$442$	0	0
$443$	−2.50479	−0.119006	−0.0595032	−	0.998228i	$-0.518952\pi$
$443$	−2.50479	−0.119006	−0.0595032	+	0.998228i	$0.518952\pi$
$444$	0	0
$445$	10.4340	0.494620
$446$	5.02984	0.238170
$447$	0	0
$448$	9.30665	0.439698
$449$	10.8600	0.512513	0.256257	−	0.966609i	$-0.417511\pi$
$449$	10.8600	0.512513	0.256257	+	0.966609i	$0.417511\pi$
$450$	0	0
$451$	−32.8025	−1.54461
$452$	4.96407	0.233490
$453$	0	0
$454$	7.19460	0.337660
$455$	0	0
$456$	0	0
$457$	15.8364	0.740795	0.370397	−	0.928873i	$-0.379222\pi$
$457$	15.8364	0.740795	0.370397	+	0.928873i	$0.379222\pi$
$458$	14.3231	0.669275
$459$	0	0
$460$	5.25475	0.245004
$461$	15.8364	0.737577	0.368788	−	0.929513i	$-0.379773\pi$
$461$	15.8364	0.737577	0.368788	+	0.929513i	$0.379773\pi$
$462$	0	0
$463$	6.10859	0.283890	0.141945	−	0.989875i	$-0.454664\pi$
$463$	6.10859	0.283890	0.141945	+	0.989875i	$0.454664\pi$
$464$	−30.9870	−1.43853
$465$	0	0
$466$	17.6396	0.817141
$467$	32.5158	1.50465	0.752326	−	0.658791i	$-0.228932\pi$
$467$	32.5158	1.50465	0.752326	+	0.658791i	$0.228932\pi$
$468$	0	0
$469$	−1.86161	−0.0859611
$470$	−2.46617	−0.113756
$471$	0	0
$472$	−31.3599	−1.44346
$473$	2.26640	0.104209
$474$	0	0
$475$	1.81594	0.0833210
$476$	−0.627648	−0.0287682
$477$	0	0
$478$	−4.16229	−0.190379
$479$	3.05213	0.139455	0.0697276	−	0.997566i	$-0.477787\pi$
$479$	3.05213	0.139455	0.0697276	+	0.997566i	$0.477787\pi$
$480$	0	0
$481$	0	0
$482$	18.0084	0.820261
$483$	0	0
$484$	−3.23298	−0.146954
$485$	−29.0968	−1.32122
$486$	0	0
$487$	6.37329	0.288801	0.144401	−	0.989519i	$-0.453875\pi$
$487$	6.37329	0.288801	0.144401	+	0.989519i	$0.453875\pi$
$488$	32.7821	1.48398
$489$	0	0
$490$	16.4024	0.740983
$491$	−34.4222	−1.55345	−0.776727	−	0.629838i	$-0.783121\pi$
$491$	−34.4222	−1.55345	−0.776727	+	0.629838i	$0.783121\pi$
$492$	0	0
$493$	16.4470	0.740734
$494$	0	0
$495$	0	0
$496$	24.9922	1.12218
$497$	−1.42656	−0.0639898
$498$	0	0
$499$	−32.2935	−1.44566	−0.722828	−	0.691028i	$-0.757158\pi$
$499$	−32.2935	−1.44566	−0.722828	+	0.691028i	$0.757158\pi$
$500$	4.09939	0.183330
$501$	0	0
$502$	14.1927	0.633451
$503$	2.23849	0.0998094	0.0499047	−	0.998754i	$-0.484108\pi$
$503$	2.23849	0.0998094	0.0499047	+	0.998754i	$0.484108\pi$
$504$	0	0
$505$	−11.4868	−0.511155
$506$	−38.9722	−1.73253
$507$	0	0
$508$	−3.29052	−0.145993
$509$	−3.83002	−0.169763	−0.0848814	−	0.996391i	$-0.527051\pi$
$509$	−3.83002	−0.169763	−0.0848814	+	0.996391i	$0.527051\pi$
$510$	0	0
$511$	−3.84415	−0.170055
$512$	25.3828	1.12177
$513$	0	0
$514$	−16.6364	−0.733800
$515$	−16.0152	−0.705714
$516$	0	0
$517$	−3.97285	−0.174726
$518$	13.6134	0.598140
$519$	0	0
$520$	0	0
$521$	−40.0504	−1.75464	−0.877321	−	0.479904i	$-0.840671\pi$
$521$	−40.0504	−1.75464	−0.877321	+	0.479904i	$0.840671\pi$
$522$	0	0
$523$	−18.8920	−0.826090	−0.413045	−	0.910711i	$-0.635535\pi$
$523$	−18.8920	−0.826090	−0.413045	+	0.910711i	$0.635535\pi$
$524$	5.23037	0.228490
$525$	0	0
$526$	−11.2765	−0.491680
$527$	−13.2651	−0.577838
$528$	0	0
$529$	23.0834	1.00362
$530$	−14.2118	−0.617322
$531$	0	0
$532$	2.29744	0.0996065
$533$	0	0
$534$	0	0
$535$	−34.7017	−1.50029
$536$	−5.36192	−0.231600
$537$	0	0
$538$	−25.7638	−1.11076
$539$	26.4232	1.13813
$540$	0	0
$541$	−10.2174	−0.439282	−0.219641	−	0.975581i	$-0.570489\pi$
$541$	−10.2174	−0.439282	−0.219641	+	0.975581i	$0.570489\pi$
$542$	18.4883	0.794139
$543$	0	0
$544$	−3.34183	−0.143280
$545$	−11.0357	−0.472715
$546$	0	0
$547$	6.67324	0.285327	0.142663	−	0.989771i	$-0.454433\pi$
$547$	6.67324	0.285327	0.142663	+	0.989771i	$0.454433\pi$
$548$	−3.77728	−0.161357
$549$	0	0
$550$	−1.69873	−0.0724339
$551$	−60.2023	−2.56470
$552$	0	0
$553$	4.06206	0.172737
$554$	−14.0120	−0.595315
$555$	0	0
$556$	−2.10560	−0.0892975
$557$	−16.7807	−0.711020	−0.355510	−	0.934672i	$-0.615693\pi$
$557$	−16.7807	−0.711020	−0.355510	+	0.934672i	$0.615693\pi$
$558$	0	0
$559$	0	0
$560$	−7.18632	−0.303677
$561$	0	0
$562$	21.6461	0.913085
$563$	−30.1750	−1.27173	−0.635863	−	0.771802i	$-0.719356\pi$
$563$	−30.1750	−1.27173	−0.635863	+	0.771802i	$0.719356\pi$
$564$	0	0
$565$	−30.1672	−1.26914
$566$	−31.1348	−1.30869
$567$	0	0
$568$	−4.10885	−0.172404
$569$	21.2139	0.889331	0.444665	−	0.895697i	$-0.353323\pi$
$569$	21.2139	0.889331	0.444665	+	0.895697i	$0.353323\pi$
$570$	0	0
$571$	−37.0200	−1.54924	−0.774619	−	0.632429i	$-0.782058\pi$
$571$	−37.0200	−1.54924	−0.774619	+	0.632429i	$0.782058\pi$
$572$	0	0
$573$	0	0
$574$	−9.84761	−0.411031
$575$	2.00869	0.0837681
$576$	0	0
$577$	36.0224	1.49963	0.749815	−	0.661647i	$-0.230142\pi$
$577$	36.0224	1.49963	0.749815	+	0.661647i	$0.230142\pi$
$578$	18.1879	0.756518
$579$	0	0
$580$	−7.59333	−0.315296
$581$	5.19387	0.215478
$582$	0	0
$583$	−22.8944	−0.948189
$584$	−11.0722	−0.458169
$585$	0	0
$586$	24.8431	1.02626
$587$	−6.51910	−0.269072	−0.134536	−	0.990909i	$-0.542954\pi$
$587$	−6.51910	−0.269072	−0.134536	+	0.990909i	$0.542954\pi$
$588$	0	0
$589$	48.5555	2.00070
$590$	28.8584	1.18808
$591$	0	0
$592$	31.9831	1.31450
$593$	−10.1851	−0.418252	−0.209126	−	0.977889i	$-0.567062\pi$
$593$	−10.1851	−0.418252	−0.209126	+	0.977889i	$0.567062\pi$
$594$	0	0
$595$	3.81428	0.156370
$596$	3.76179	0.154089
$597$	0	0
$598$	0	0
$599$	−7.30483	−0.298467	−0.149234	−	0.988802i	$-0.547681\pi$
$599$	−7.30483	−0.298467	−0.149234	+	0.988802i	$0.547681\pi$
$600$	0	0
$601$	−26.1051	−1.06485	−0.532425	−	0.846477i	$-0.678719\pi$
$601$	−26.1051	−1.06485	−0.532425	+	0.846477i	$0.678719\pi$
$602$	0.680392	0.0277307
$603$	0	0
$604$	0.226063	0.00919839
$605$	19.6472	0.798771
$606$	0	0
$607$	22.1183	0.897753	0.448877	−	0.893594i	$-0.351824\pi$
$607$	22.1183	0.897753	0.448877	+	0.893594i	$0.351824\pi$
$608$	12.2324	0.496090
$609$	0	0
$610$	−30.1672	−1.22143
$611$	0	0
$612$	0	0
$613$	37.2435	1.50425	0.752126	−	0.659020i	$-0.229029\pi$
$613$	37.2435	1.50425	0.752126	+	0.659020i	$0.229029\pi$
$614$	−39.6891	−1.60172
$615$	0	0
$616$	−14.1927	−0.571840
$617$	−5.95552	−0.239760	−0.119880	−	0.992788i	$-0.538251\pi$
$617$	−5.95552	−0.239760	−0.119880	+	0.992788i	$0.538251\pi$
$618$	0	0
$619$	−21.7302	−0.873410	−0.436705	−	0.899605i	$-0.643855\pi$
$619$	−21.7302	−0.873410	−0.436705	+	0.899605i	$0.643855\pi$
$620$	6.12432	0.245959
$621$	0	0
$622$	−29.7415	−1.19253
$623$	5.04609	0.202167
$624$	0	0
$625$	−23.4330	−0.937318
$626$	38.6694	1.54554
$627$	0	0
$628$	7.10560	0.283544
$629$	−16.9757	−0.676865
$630$	0	0
$631$	−5.92931	−0.236042	−0.118021	−	0.993011i	$-0.537655\pi$
$631$	−5.92931	−0.236042	−0.118021	+	0.993011i	$0.537655\pi$
$632$	11.6998	0.465393
$633$	0	0
$634$	−30.3032	−1.20349
$635$	19.9968	0.793551
$636$	0	0
$637$	0	0
$638$	56.3165	2.22959
$639$	0	0
$640$	−16.0213	−0.633297
$641$	−28.2686	−1.11654	−0.558272	−	0.829658i	$-0.688535\pi$
$641$	−28.2686	−1.11654	−0.558272	+	0.829658i	$0.688535\pi$
$642$	0	0
$643$	−21.6276	−0.852908	−0.426454	−	0.904509i	$-0.640237\pi$
$643$	−21.6276	−0.852908	−0.426454	+	0.904509i	$0.640237\pi$
$644$	2.54129	0.100141
$645$	0	0
$646$	13.1894	0.518932
$647$	47.3350	1.86093	0.930466	−	0.366379i	$-0.119403\pi$
$647$	47.3350	1.86093	0.930466	+	0.366379i	$0.119403\pi$
$648$	0	0
$649$	46.4892	1.82486
$650$	0	0
$651$	0	0
$652$	3.47996	0.136286
$653$	−30.7369	−1.20283	−0.601414	−	0.798937i	$-0.705396\pi$
$653$	−30.7369	−1.20283	−0.601414	+	0.798937i	$0.705396\pi$
$654$	0	0
$655$	−31.7855	−1.24196
$656$	−23.1358	−0.903300
$657$	0	0
$658$	−1.19269	−0.0464957
$659$	−10.4900	−0.408633	−0.204317	−	0.978905i	$-0.565497\pi$
$659$	−10.4900	−0.408633	−0.204317	+	0.978905i	$0.565497\pi$
$660$	0	0
$661$	10.9148	0.424538	0.212269	−	0.977211i	$-0.431915\pi$
$661$	10.9148	0.424538	0.212269	+	0.977211i	$0.431915\pi$
$662$	−8.25153	−0.320705
$663$	0	0
$664$	14.9597	0.580549
$665$	−13.9618	−0.541414
$666$	0	0
$667$	−66.5924	−2.57847
$668$	−7.25591	−0.280740
$669$	0	0
$670$	4.93422	0.190625
$671$	−48.5975	−1.87609
$672$	0	0
$673$	−31.2597	−1.20497	−0.602486	−	0.798130i	$-0.705823\pi$
$673$	−31.2597	−1.20497	−0.602486	+	0.798130i	$0.705823\pi$
$674$	41.4770	1.59763
$675$	0	0
$676$	0	0
$677$	9.35904	0.359697	0.179849	−	0.983694i	$-0.442439\pi$
$677$	9.35904	0.359697	0.179849	+	0.983694i	$0.442439\pi$
$678$	0	0
$679$	−14.0718	−0.540024
$680$	10.9861	0.421299
$681$	0	0
$682$	−45.4215	−1.73928
$683$	16.3232	0.624590	0.312295	−	0.949985i	$-0.398902\pi$
$683$	16.3232	0.624590	0.312295	+	0.949985i	$0.398902\pi$
$684$	0	0
$685$	22.9549	0.877062
$686$	17.3443	0.662207
$687$	0	0
$688$	1.59850	0.0609422
$689$	0	0
$690$	0	0
$691$	−20.1631	−0.767042	−0.383521	−	0.923532i	$-0.625289\pi$
$691$	−20.1631	−0.767042	−0.383521	+	0.923532i	$0.625289\pi$
$692$	−6.19010	−0.235312
$693$	0	0
$694$	−19.3631	−0.735014
$695$	12.7960	0.485379
$696$	0	0
$697$	12.2798	0.465129
$698$	2.15648	0.0816238
$699$	0	0
$700$	0.110770	0.00418672
$701$	15.7041	0.593138	0.296569	−	0.955012i	$-0.404158\pi$
$701$	15.7041	0.593138	0.296569	+	0.955012i	$0.404158\pi$
$702$	0	0
$703$	62.1377	2.34357
$704$	−39.7377	−1.49767
$705$	0	0
$706$	7.42865	0.279581
$707$	−5.55522	−0.208926
$708$	0	0
$709$	8.37303	0.314456	0.157228	−	0.987562i	$-0.449744\pi$
$709$	8.37303	0.314456	0.157228	+	0.987562i	$0.449744\pi$
$710$	3.78110	0.141902
$711$	0	0
$712$	14.5340	0.544686
$713$	53.7094	2.01143
$714$	0	0
$715$	0	0
$716$	−3.38250	−0.126410
$717$	0	0
$718$	−23.4808	−0.876296
$719$	40.1982	1.49914	0.749571	−	0.661925i	$-0.230260\pi$
$719$	40.1982	1.49914	0.749571	+	0.661925i	$0.230260\pi$
$720$	0	0
$721$	−7.74525	−0.288448
$722$	−23.9236	−0.890345
$723$	0	0
$724$	0.280831	0.0104370
$725$	−2.90264	−0.107801
$726$	0	0
$727$	−2.37973	−0.0882593	−0.0441297	−	0.999026i	$-0.514051\pi$
$727$	−2.37973	−0.0882593	−0.0441297	+	0.999026i	$0.514051\pi$
$728$	0	0
$729$	0	0
$730$	10.1890	0.377110
$731$	−0.848435	−0.0313805
$732$	0	0
$733$	14.5101	0.535942	0.267971	−	0.963427i	$-0.413647\pi$
$733$	14.5101	0.535942	0.267971	+	0.963427i	$0.413647\pi$
$734$	−37.5271	−1.38515
$735$	0	0
$736$	13.5308	0.498752
$737$	7.94873	0.292795
$738$	0	0
$739$	−18.8495	−0.693391	−0.346696	−	0.937978i	$-0.612696\pi$
$739$	−18.8495	−0.693391	−0.346696	+	0.937978i	$0.612696\pi$
$740$	7.83744	0.288110
$741$	0	0
$742$	−6.87310	−0.252319
$743$	22.4368	0.823127	0.411563	−	0.911381i	$-0.364983\pi$
$743$	22.4368	0.823127	0.411563	+	0.911381i	$0.364983\pi$
$744$	0	0
$745$	−22.8608	−0.837555
$746$	−7.71892	−0.282610
$747$	0	0
$748$	2.67994	0.0979883
$749$	−16.7824	−0.613215
$750$	0	0
$751$	−41.4596	−1.51288	−0.756442	−	0.654061i	$-0.773064\pi$
$751$	−41.4596	−1.51288	−0.756442	+	0.654061i	$0.773064\pi$
$752$	−2.80207	−0.102181
$753$	0	0
$754$	0	0
$755$	−1.37381	−0.0499981
$756$	0	0
$757$	−22.3636	−0.812819	−0.406409	−	0.913691i	$-0.633219\pi$
$757$	−22.3636	−0.812819	−0.406409	+	0.913691i	$0.633219\pi$
$758$	−19.3165	−0.701605
$759$	0	0
$760$	−40.2135	−1.45870
$761$	25.2196	0.914209	0.457104	−	0.889413i	$-0.348887\pi$
$761$	25.2196	0.914209	0.457104	+	0.889413i	$0.348887\pi$
$762$	0	0
$763$	−5.33704	−0.193214
$764$	5.29615	0.191608
$765$	0	0
$766$	44.9105	1.62268
$767$	0	0
$768$	0	0
$769$	−24.8611	−0.896515	−0.448258	−	0.893904i	$-0.647955\pi$
$769$	−24.8611	−0.896515	−0.448258	+	0.893904i	$0.647955\pi$
$770$	13.0606	0.470671
$771$	0	0
$772$	−1.27498	−0.0458876
$773$	14.4099	0.518288	0.259144	−	0.965839i	$-0.416560\pi$
$773$	14.4099	0.518288	0.259144	+	0.965839i	$0.416560\pi$
$774$	0	0
$775$	2.34109	0.0840946
$776$	−40.5303	−1.45495
$777$	0	0
$778$	−12.2330	−0.438574
$779$	−44.9487	−1.61046
$780$	0	0
$781$	6.09113	0.217958
$782$	14.5894	0.521716
$783$	0	0
$784$	18.6364	0.665586
$785$	−43.1815	−1.54121
$786$	0	0
$787$	49.9687	1.78119	0.890595	−	0.454797i	$-0.150288\pi$
$787$	49.9687	1.78119	0.890595	+	0.454797i	$0.150288\pi$
$788$	2.40040	0.0855107
$789$	0	0
$790$	−10.7665	−0.383057
$791$	−14.5894	−0.518740
$792$	0	0
$793$	0	0
$794$	−26.9534	−0.956539
$795$	0	0
$796$	−1.79763	−0.0637152
$797$	18.1269	0.642089	0.321044	−	0.947064i	$-0.395966\pi$
$797$	18.1269	0.642089	0.321044	+	0.947064i	$0.395966\pi$
$798$	0	0
$799$	1.48725	0.0526153
$800$	0.589782	0.0208520
$801$	0	0
$802$	16.0610	0.567134
$803$	16.4138	0.579231
$804$	0	0
$805$	−15.4437	−0.544319
$806$	0	0
$807$	0	0
$808$	−16.0005	−0.562895
$809$	−23.9484	−0.841983	−0.420991	−	0.907065i	$-0.638318\pi$
$809$	−23.9484	−0.841983	−0.420991	+	0.907065i	$0.638318\pi$
$810$	0	0
$811$	−42.8112	−1.50330	−0.751651	−	0.659561i	$-0.770742\pi$
$811$	−42.8112	−1.50330	−0.751651	+	0.659561i	$0.770742\pi$
$812$	−3.67228	−0.128872
$813$	0	0
$814$	−58.1269	−2.03735
$815$	−21.1481	−0.740785
$816$	0	0
$817$	3.10560	0.108651
$818$	−9.17475	−0.320788
$819$	0	0
$820$	−5.66940	−0.197984
$821$	30.7176	1.07205	0.536026	−	0.844201i	$-0.319925\pi$
$821$	30.7176	1.07205	0.536026	+	0.844201i	$0.319925\pi$
$822$	0	0
$823$	2.46575	0.0859505	0.0429753	−	0.999076i	$-0.486316\pi$
$823$	2.46575	0.0859505	0.0429753	+	0.999076i	$0.486316\pi$
$824$	−22.3083	−0.777148
$825$	0	0
$826$	13.9565	0.485607
$827$	15.8906	0.552569	0.276284	−	0.961076i	$-0.410897\pi$
$827$	15.8906	0.552569	0.276284	+	0.961076i	$0.410897\pi$
$828$	0	0
$829$	−17.3907	−0.604006	−0.302003	−	0.953307i	$-0.597655\pi$
$829$	−17.3907	−0.604006	−0.302003	+	0.953307i	$0.597655\pi$
$830$	−13.7664	−0.477839
$831$	0	0
$832$	0	0
$833$	−9.89164	−0.342725
$834$	0	0
$835$	44.0950	1.52597
$836$	−9.80963	−0.339273
$837$	0	0
$838$	−6.02177	−0.208019
$839$	41.9190	1.44720	0.723602	−	0.690217i	$-0.242485\pi$
$839$	41.9190	1.44720	0.723602	+	0.690217i	$0.242485\pi$
$840$	0	0
$841$	67.2288	2.31823
$842$	−27.4860	−0.947229
$843$	0	0
$844$	−7.16767	−0.246721
$845$	0	0
$846$	0	0
$847$	9.50173	0.326483
$848$	−16.1475	−0.554508
$849$	0	0
$850$	0.635925	0.0218121
$851$	68.7332	2.35614
$852$	0	0
$853$	−25.7821	−0.882762	−0.441381	−	0.897320i	$-0.645511\pi$
$853$	−25.7821	−0.882762	−0.441381	+	0.897320i	$0.645511\pi$
$854$	−14.5894	−0.499239
$855$	0	0
$856$	−48.3376	−1.65215
$857$	22.9725	0.784725	0.392363	−	0.919811i	$-0.371658\pi$
$857$	22.9725	0.784725	0.392363	+	0.919811i	$0.371658\pi$
$858$	0	0
$859$	37.2506	1.27097	0.635486	−	0.772112i	$-0.280800\pi$
$859$	37.2506	1.27097	0.635486	+	0.772112i	$0.280800\pi$
$860$	0.391711	0.0133572
$861$	0	0
$862$	−9.34157	−0.318175
$863$	25.0391	0.852341	0.426171	−	0.904643i	$-0.359862\pi$
$863$	25.0391	0.852341	0.426171	+	0.904643i	$0.359862\pi$
$864$	0	0
$865$	37.6179	1.27905
$866$	−21.7392	−0.738729
$867$	0	0
$868$	2.96184	0.100531
$869$	−17.3443	−0.588364
$870$	0	0
$871$	0	0
$872$	−15.3721	−0.520564
$873$	0	0
$874$	−53.4029	−1.80638
$875$	−12.0481	−0.407301
$876$	0	0
$877$	−40.4771	−1.36681	−0.683407	−	0.730037i	$-0.739503\pi$
$877$	−40.4771	−1.36681	−0.683407	+	0.730037i	$0.739503\pi$
$878$	−1.09755	−0.0370405
$879$	0	0
$880$	30.6843	1.03437
$881$	−37.0293	−1.24755	−0.623774	−	0.781605i	$-0.714402\pi$
$881$	−37.0293	−1.24755	−0.623774	+	0.781605i	$0.714402\pi$
$882$	0	0
$883$	15.3797	0.517569	0.258785	−	0.965935i	$-0.416678\pi$
$883$	15.3797	0.517569	0.258785	+	0.965935i	$0.416678\pi$
$884$	0	0
$885$	0	0
$886$	3.21073	0.107867
$887$	30.3886	1.02035	0.510175	−	0.860071i	$-0.329581\pi$
$887$	30.3886	1.02035	0.510175	+	0.860071i	$0.329581\pi$
$888$	0	0
$889$	9.67084	0.324350
$890$	−13.3747	−0.448321
$891$	0	0
$892$	1.40044	0.0468901
$893$	−5.44393	−0.182174
$894$	0	0
$895$	20.5558	0.687105
$896$	−7.74819	−0.258849
$897$	0	0
$898$	−13.9207	−0.464539
$899$	−77.6123	−2.58852
$900$	0	0
$901$	8.57061	0.285529
$902$	42.0475	1.40003
$903$	0	0
$904$	−42.0213	−1.39761
$905$	−1.70664	−0.0567307
$906$	0	0
$907$	28.2416	0.937747	0.468874	−	0.883265i	$-0.344660\pi$
$907$	28.2416	0.937747	0.468874	+	0.883265i	$0.344660\pi$
$908$	2.00316	0.0664772
$909$	0	0
$910$	0	0
$911$	−2.43902	−0.0808084	−0.0404042	−	0.999183i	$-0.512865\pi$
$911$	−2.43902	−0.0808084	−0.0404042	+	0.999183i	$0.512865\pi$
$912$	0	0
$913$	−22.1769	−0.733948
$914$	−20.2996	−0.671453
$915$	0	0
$916$	3.98792	0.131765
$917$	−15.3721	−0.507631
$918$	0	0
$919$	−43.2140	−1.42550	−0.712749	−	0.701419i	$-0.752550\pi$
$919$	−43.2140	−1.42550	−0.712749	+	0.701419i	$0.752550\pi$
$920$	−44.4819	−1.46653
$921$	0	0
$922$	−20.2997	−0.668536
$923$	0	0
$924$	0	0
$925$	2.99595	0.0985063
$926$	−7.83021	−0.257317
$927$	0	0
$928$	−19.5526	−0.641844
$929$	−49.1642	−1.61302	−0.806512	−	0.591217i	$-0.798648\pi$
$929$	−49.1642	−1.61302	−0.806512	+	0.591217i	$0.798648\pi$
$930$	0	0
$931$	36.2073	1.18665
$932$	4.91133	0.160876
$933$	0	0
$934$	−41.6799	−1.36381
$935$	−16.2863	−0.532618
$936$	0	0
$937$	1.48725	0.0485865	0.0242932	−	0.999705i	$-0.492266\pi$
$937$	1.48725	0.0485865	0.0242932	+	0.999705i	$0.492266\pi$
$938$	2.38628	0.0779147
$939$	0	0
$940$	−0.686645	−0.0223959
$941$	1.82823	0.0595985	0.0297992	−	0.999556i	$-0.490513\pi$
$941$	1.82823	0.0595985	0.0297992	+	0.999556i	$0.490513\pi$
$942$	0	0
$943$	−49.7198	−1.61910
$944$	32.7890	1.06719
$945$	0	0
$946$	−2.90515	−0.0944545
$947$	−34.1576	−1.10997	−0.554987	−	0.831859i	$-0.687277\pi$
$947$	−34.1576	−1.10997	−0.554987	+	0.831859i	$0.687277\pi$
$948$	0	0
$949$	0	0
$950$	−2.32774	−0.0755217
$951$	0	0
$952$	5.31309	0.172198
$953$	1.79513	0.0581500	0.0290750	−	0.999577i	$-0.490744\pi$
$953$	1.79513	0.0581500	0.0290750	+	0.999577i	$0.490744\pi$
$954$	0	0
$955$	−32.1852	−1.04149
$956$	−1.15889	−0.0374811
$957$	0	0
$958$	−3.91233	−0.126402
$959$	11.1014	0.358484
$960$	0	0
$961$	31.5974	1.01927
$962$	0	0
$963$	0	0
$964$	5.01400	0.161490
$965$	7.74819	0.249423
$966$	0	0
$967$	−49.1275	−1.57983	−0.789917	−	0.613214i	$-0.789876\pi$
$967$	−49.1275	−1.57983	−0.789917	+	0.613214i	$0.789876\pi$
$968$	27.3674	0.879623
$969$	0	0
$970$	37.2973	1.19755
$971$	27.6703	0.887981	0.443990	−	0.896032i	$-0.353562\pi$
$971$	27.6703	0.887981	0.443990	+	0.896032i	$0.353562\pi$
$972$	0	0
$973$	6.18837	0.198390
$974$	−8.16951	−0.261768
$975$	0	0
$976$	−34.2760	−1.09715
$977$	−44.6035	−1.42699	−0.713496	−	0.700659i	$-0.752889\pi$
$977$	−44.6035	−1.42699	−0.713496	+	0.700659i	$0.752889\pi$
$978$	0	0
$979$	−21.5459	−0.688609
$980$	4.56683	0.145882
$981$	0	0
$982$	44.1237	1.40804
$983$	32.1768	1.02628	0.513141	−	0.858304i	$-0.328482\pi$
$983$	32.1768	1.02628	0.513141	+	0.858304i	$0.328482\pi$
$984$	0	0
$985$	−14.5875	−0.464796
$986$	−21.0823	−0.671398
$987$	0	0
$988$	0	0
$989$	3.43524	0.109234
$990$	0	0
$991$	−6.97690	−0.221629	−0.110814	−	0.993841i	$-0.535346\pi$
$991$	−6.97690	−0.221629	−0.110814	+	0.993841i	$0.535346\pi$
$992$	15.7699	0.500695
$993$	0	0
$994$	1.82861	0.0580000
$995$	10.9244	0.346326
$996$	0	0
$997$	−9.40017	−0.297706	−0.148853	−	0.988859i	$-0.547558\pi$
$997$	−9.40017	−0.297706	−0.148853	+	0.988859i	$0.547558\pi$
$998$	41.3950	1.31034
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.v.1.2	✓	6
3.2	odd	2	inner	1521.2.a.v.1.5	yes	6
13.5	odd	4		1521.2.b.n.1351.10		12
13.8	odd	4		1521.2.b.n.1351.3		12
13.12	even	2		1521.2.a.w.1.5	yes	6
39.5	even	4		1521.2.b.n.1351.4		12
39.8	even	4		1521.2.b.n.1351.9		12
39.38	odd	2		1521.2.a.w.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.2.a.v.1.2	✓	6	1.1	even	1	trivial
1521.2.a.v.1.5	yes	6	3.2	odd	2	inner
1521.2.a.w.1.2	yes	6	39.38	odd	2
1521.2.a.w.1.5	yes	6	13.12	even	2
1521.2.b.n.1351.3		12	13.8	odd	4
1521.2.b.n.1351.4		12	39.5	even	4
1521.2.b.n.1351.9		12	39.8	even	4
1521.2.b.n.1351.10		12	13.5	odd	4

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)

\(f(q)\)	\(=\)	\(q-1.28184 q^{2} -0.356896 q^{4} +2.16889 q^{5} +1.04892 q^{7} +3.02115 q^{8} +O(q^{10})\)	\(q-1.28184 q^{2} -0.356896 q^{4} +2.16889 q^{5} +1.04892 q^{7} +3.02115 q^{8} -2.78017 q^{10} -4.47868 q^{11} -1.34454 q^{14} -3.15883 q^{16} +1.67661 q^{17} -6.13706 q^{19} -0.774069 q^{20} +5.74094 q^{22} -6.78847 q^{23} -0.295897 q^{25} -0.374354 q^{28} +9.80963 q^{29} -7.91185 q^{31} -1.99320 q^{32} -2.14914 q^{34} +2.27499 q^{35} -10.1250 q^{37} +7.86671 q^{38} +6.55257 q^{40} +7.32415 q^{41} -0.506041 q^{43} +1.59842 q^{44} +8.70171 q^{46} +0.887058 q^{47} -5.89977 q^{49} +0.379291 q^{50} +5.11186 q^{53} -9.71379 q^{55} +3.16894 q^{56} -12.5743 q^{58} -10.3801 q^{59} +10.8509 q^{61} +10.1417 q^{62} +8.87263 q^{64} -1.77479 q^{67} -0.598377 q^{68} -2.91617 q^{70} -1.36003 q^{71} -3.66487 q^{73} +12.9786 q^{74} +2.19029 q^{76} -4.69777 q^{77} +3.87263 q^{79} -6.85118 q^{80} -9.38835 q^{82} +4.95165 q^{83} +3.63640 q^{85} +0.648661 q^{86} -13.5308 q^{88} +4.81076 q^{89} +2.42278 q^{92} -1.13706 q^{94} -13.3106 q^{95} -13.4155 q^{97} +7.56254 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 12 q^{7}+O(q^{10}) \) 6 * q + 6 * q^4 - 12 * q^7	\( 6 q + 6 q^{4} - 12 q^{7} - 14 q^{10} - 2 q^{16} - 26 q^{19} + 6 q^{22} + 26 q^{25} - 26 q^{28} - 40 q^{31} - 40 q^{34} - 12 q^{37} - 42 q^{40} - 22 q^{43} - 2 q^{46} + 10 q^{49} - 42 q^{55} + 12 q^{58} + 38 q^{61} + 20 q^{64} - 14 q^{67} + 28 q^{70} - 24 q^{73} - 54 q^{76} - 10 q^{79} + 2 q^{82} - 14 q^{85} - 6 q^{88} + 4 q^{94} - 10 q^{97}+O(q^{100}) \) 6 * q + 6 * q^4 - 12 * q^7 - 14 * q^10 - 2 * q^16 - 26 * q^19 + 6 * q^22 + 26 * q^25 - 26 * q^28 - 40 * q^31 - 40 * q^34 - 12 * q^37 - 42 * q^40 - 22 * q^43 - 2 * q^46 + 10 * q^49 - 42 * q^55 + 12 * q^58 + 38 * q^61 + 20 * q^64 - 14 * q^67 + 28 * q^70 - 24 * q^73 - 54 * q^76 - 10 * q^79 + 2 * q^82 - 14 * q^85 - 6 * q^88 + 4 * q^94 - 10 * q^97

Introduction

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