LMFDB - Embedded newform 1521.2.a.v.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:	$6$
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			1.1
Root			$-1.09456$ 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)

$f(q)$	$=$	$q-2.45945 q^{2} +4.04892 q^{4} +3.33722 q^{5} -3.69202 q^{7} -5.03922 q^{8} +O(q^{10})$	$q-2.45945 q^{2} +4.04892 q^{4} +3.33722 q^{5} -3.69202 q^{7} -5.03922 q^{8} -8.20775 q^{10} -0.270334 q^{11} +9.08036 q^{14} +4.29590 q^{16} +4.04114 q^{17} -7.15883 q^{19} +13.5121 q^{20} +0.664874 q^{22} +2.79656 q^{23} +6.13706 q^{25} -14.9487 q^{28} -7.83578 q^{29} -2.14914 q^{31} -0.487125 q^{32} -9.93900 q^{34} -12.3211 q^{35} -4.37867 q^{37} +17.6068 q^{38} -16.8170 q^{40} -7.61898 q^{41} -3.89008 q^{43} -1.09456 q^{44} -6.87800 q^{46} +0.877769 q^{47} +6.63102 q^{49} -15.0938 q^{50} -6.83770 q^{53} -0.902165 q^{55} +18.6049 q^{56} +19.2717 q^{58} +3.40399 q^{59} +3.06100 q^{61} +5.28572 q^{62} -7.39373 q^{64} +5.00969 q^{67} +16.3622 q^{68} +30.3032 q^{70} -7.59516 q^{71} +0.405813 q^{73} +10.7691 q^{74} -28.9855 q^{76} +0.998079 q^{77} -12.3937 q^{79} +14.3364 q^{80} +18.7385 q^{82} +6.98772 q^{83} +13.4862 q^{85} +9.56748 q^{86} +1.36227 q^{88} +13.3918 q^{89} +11.3230 q^{92} -2.15883 q^{94} -23.8906 q^{95} +10.9758 q^{97} -16.3087 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.45945	−1.73910	−0.869549	−	0.493847i	$-0.835590\pi$
$2$	−2.45945	−1.73910	−0.869549	+	0.493847i	$0.835590\pi$
$3$	0	0
$3$	0	0
$4$	4.04892	2.02446
$5$	3.33722	1.49245	0.746226	−	0.665693i	$-0.231864\pi$
$5$	3.33722	1.49245	0.746226	+	0.665693i	$0.231864\pi$
$6$	0	0
$7$	−3.69202	−1.39545	−0.697726	−	0.716364i	$-0.745805\pi$
$7$	−3.69202	−1.39545	−0.697726	+	0.716364i	$0.745805\pi$
$8$	−5.03922	−1.78163
$9$	0	0
$10$	−8.20775	−2.59552
$11$	−0.270334	−0.0815088	−0.0407544	−	0.999169i	$-0.512976\pi$
$11$	−0.270334	−0.0815088	−0.0407544	+	0.999169i	$0.512976\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	9.08036	2.42683
$15$	0	0
$16$	4.29590	1.07397
$17$	4.04114	0.980120	0.490060	−	0.871689i	$-0.336975\pi$
$17$	4.04114	0.980120	0.490060	+	0.871689i	$0.336975\pi$
$18$	0	0
$19$	−7.15883	−1.64235	−0.821175	−	0.570677i	$-0.806681\pi$
$19$	−7.15883	−1.64235	−0.821175	+	0.570677i	$0.806681\pi$
$20$	13.5121	3.02141
$21$	0	0
$22$	0.664874	0.141752
$23$	2.79656	0.583122	0.291561	−	0.956552i	$-0.405825\pi$
$23$	2.79656	0.583122	0.291561	+	0.956552i	$0.405825\pi$
$24$	0	0
$25$	6.13706	1.22741
$26$	0	0
$27$	0	0
$28$	−14.9487	−2.82504
$29$	−7.83578	−1.45507	−0.727533	−	0.686072i	$-0.759333\pi$
$29$	−7.83578	−1.45507	−0.727533	+	0.686072i	$0.759333\pi$
$30$	0	0
$31$	−2.14914	−0.385998	−0.192999	−	0.981199i	$-0.561821\pi$
$31$	−2.14914	−0.385998	−0.192999	+	0.981199i	$0.561821\pi$
$32$	−0.487125	−0.0861124
$33$	0	0
$34$	−9.93900	−1.70452
$35$	−12.3211	−2.08265
$36$	0	0
$37$	−4.37867	−0.719848	−0.359924	−	0.932982i	$-0.617197\pi$
$37$	−4.37867	−0.719848	−0.359924	+	0.932982i	$0.617197\pi$
$38$	17.6068	2.85620
$39$	0	0
$40$	−16.8170	−2.65900
$41$	−7.61898	−1.18989	−0.594943	−	0.803768i	$-0.702825\pi$
$41$	−7.61898	−1.18989	−0.594943	+	0.803768i	$0.702825\pi$
$42$	0	0
$43$	−3.89008	−0.593232	−0.296616	−	0.954997i	$-0.595858\pi$
$43$	−3.89008	−0.593232	−0.296616	+	0.954997i	$0.595858\pi$
$44$	−1.09456	−0.165011
$45$	0	0
$46$	−6.87800	−1.01411
$47$	0.877769	0.128036	0.0640179	−	0.997949i	$-0.479609\pi$
$47$	0.877769	0.128036	0.0640179	+	0.997949i	$0.479609\pi$
$48$	0	0
$49$	6.63102	0.947289
$50$	−15.0938	−2.13459
$51$	0	0
$52$	0	0
$53$	−6.83770	−0.939230	−0.469615	−	0.882871i	$-0.655607\pi$
$53$	−6.83770	−0.939230	−0.469615	+	0.882871i	$0.655607\pi$
$54$	0	0
$55$	−0.902165	−0.121648
$56$	18.6049	2.48619
$57$	0	0
$58$	19.2717	2.53050
$59$	3.40399	0.443162	0.221581	−	0.975142i	$-0.428878\pi$
$59$	3.40399	0.443162	0.221581	+	0.975142i	$0.428878\pi$
$60$	0	0
$61$	3.06100	0.391921	0.195960	−	0.980612i	$-0.437218\pi$
$61$	3.06100	0.391921	0.195960	+	0.980612i	$0.437218\pi$
$62$	5.28572	0.671288
$63$	0	0
$64$	−7.39373	−0.924216
$65$	0	0
$66$	0	0
$67$	5.00969	0.612031	0.306015	−	0.952027i	$-0.401004\pi$
$67$	5.00969	0.612031	0.306015	+	0.952027i	$0.401004\pi$
$68$	16.3622	1.98421
$69$	0	0
$70$	30.3032	3.62192
$71$	−7.59516	−0.901379	−0.450690	−	0.892681i	$-0.648822\pi$
$71$	−7.59516	−0.901379	−0.450690	+	0.892681i	$0.648822\pi$
$72$	0	0
$73$	0.405813	0.0474968	0.0237484	−	0.999718i	$-0.492440\pi$
$73$	0.405813	0.0474968	0.0237484	+	0.999718i	$0.492440\pi$
$74$	10.7691	1.25189
$75$	0	0
$76$	−28.9855	−3.32487
$77$	0.998079	0.113742
$78$	0	0
$79$	−12.3937	−1.39440	−0.697202	−	0.716875i	$-0.745572\pi$
$79$	−12.3937	−1.39440	−0.697202	+	0.716875i	$0.745572\pi$
$80$	14.3364	1.60285
$81$	0	0
$82$	18.7385	2.06933
$83$	6.98772	0.767002	0.383501	−	0.923540i	$-0.374718\pi$
$83$	6.98772	0.767002	0.383501	+	0.923540i	$0.374718\pi$
$84$	0	0
$85$	13.4862	1.46278
$86$	9.56748	1.03169
$87$	0	0
$88$	1.36227	0.145219
$89$	13.3918	1.41953	0.709766	−	0.704438i	$-0.248801\pi$
$89$	13.3918	1.41953	0.709766	+	0.704438i	$0.248801\pi$
$90$	0	0
$91$	0	0
$92$	11.3230	1.18051
$93$	0	0
$94$	−2.15883	−0.222667
$95$	−23.8906	−2.45113
$96$	0	0
$97$	10.9758	1.11443	0.557214	−	0.830369i	$-0.311870\pi$
$97$	10.9758	1.11443	0.557214	+	0.830369i	$0.311870\pi$
$98$	−16.3087	−1.64743
$99$	0	0
$100$	24.8485	2.48485
$101$	17.3603	1.72742	0.863708	−	0.503992i	$-0.168136\pi$
$101$	17.3603	1.72742	0.863708	+	0.503992i	$0.168136\pi$
$102$	0	0
$103$	−6.71379	−0.661530	−0.330765	−	0.943713i	$-0.607307\pi$
$103$	−6.71379	−0.661530	−0.330765	+	0.943713i	$0.607307\pi$
$104$	0	0
$105$	0	0
$106$	16.8170	1.63341
$107$	−5.72991	−0.553931	−0.276966	−	0.960880i	$-0.589329\pi$
$107$	−5.72991	−0.553931	−0.276966	+	0.960880i	$0.589329\pi$
$108$	0	0
$109$	−10.8509	−1.03932	−0.519662	−	0.854372i	$-0.673942\pi$
$109$	−10.8509	−1.03932	−0.519662	+	0.854372i	$0.673942\pi$
$110$	2.21883	0.211558
$111$	0	0
$112$	−15.8605	−1.49868
$113$	−7.52839	−0.708211	−0.354106	−	0.935205i	$-0.615215\pi$
$113$	−7.52839	−0.708211	−0.354106	+	0.935205i	$0.615215\pi$
$114$	0	0
$115$	9.33273	0.870282
$116$	−31.7264	−2.94572
$117$	0	0
$118$	−8.37196	−0.770702
$119$	−14.9200	−1.36771
$120$	0	0
$121$	−10.9269	−0.993356
$122$	−7.52839	−0.681588
$123$	0	0
$124$	−8.70171	−0.781437
$125$	3.79463	0.339402
$126$	0	0
$127$	3.79225	0.336508	0.168254	−	0.985744i	$-0.446187\pi$
$127$	3.79225	0.336508	0.168254	+	0.985744i	$0.446187\pi$
$128$	19.1588	1.69341
$129$	0	0
$130$	0	0
$131$	−14.8103	−1.29398	−0.646990	−	0.762499i	$-0.723972\pi$
$131$	−14.8103	−1.29398	−0.646990	+	0.762499i	$0.723972\pi$
$132$	0	0
$133$	26.4306	2.29182
$134$	−12.3211	−1.06438
$135$	0	0
$136$	−20.3642	−1.74622
$137$	−21.3479	−1.82388	−0.911938	−	0.410328i	$-0.865414\pi$
$137$	−21.3479	−1.82388	−0.911938	+	0.410328i	$0.865414\pi$
$138$	0	0
$139$	−6.63102	−0.562436	−0.281218	−	0.959644i	$-0.590738\pi$
$139$	−6.63102	−0.562436	−0.281218	+	0.959644i	$0.590738\pi$
$140$	−49.8871	−4.21623
$141$	0	0
$142$	18.6799	1.56759
$143$	0	0
$144$	0	0
$145$	−26.1497	−2.17162
$146$	−0.998079	−0.0826016
$147$	0	0
$148$	−17.7289	−1.45730
$149$	17.2294	1.41149	0.705744	−	0.708467i	$-0.250613\pi$
$149$	17.2294	1.41149	0.705744	+	0.708467i	$0.250613\pi$
$150$	0	0
$151$	−20.2838	−1.65067	−0.825337	−	0.564641i	$-0.809015\pi$
$151$	−20.2838	−1.65067	−0.825337	+	0.564641i	$0.809015\pi$
$152$	36.0749	2.92606
$153$	0	0
$154$	−2.45473	−0.197808
$155$	−7.17218	−0.576083
$156$	0	0
$157$	7.86592	0.627769	0.313884	−	0.949461i	$-0.398370\pi$
$157$	7.86592	0.627769	0.313884	+	0.949461i	$0.398370\pi$
$158$	30.4818	2.42500
$159$	0	0
$160$	−1.62565	−0.128519
$161$	−10.3249	−0.813720
$162$	0	0
$163$	10.5700	0.827908	0.413954	−	0.910298i	$-0.364147\pi$
$163$	10.5700	0.827908	0.413954	+	0.910298i	$0.364147\pi$
$164$	−30.8486	−2.40887
$165$	0	0
$166$	−17.1860	−1.33389
$167$	−17.6439	−1.36532	−0.682662	−	0.730734i	$-0.739178\pi$
$167$	−17.6439	−1.36532	−0.682662	+	0.730734i	$0.739178\pi$
$168$	0	0
$169$	0	0
$170$	−33.1687	−2.54392
$171$	0	0
$172$	−15.7506	−1.20097
$173$	−3.35045	−0.254730	−0.127365	−	0.991856i	$-0.540652\pi$
$173$	−3.35045	−0.254730	−0.127365	+	0.991856i	$0.540652\pi$
$174$	0	0
$175$	−22.6582	−1.71280
$176$	−1.16133	−0.0875383
$177$	0	0
$178$	−32.9366	−2.46870
$179$	−20.9573	−1.56642	−0.783210	−	0.621757i	$-0.786419\pi$
$179$	−20.9573	−1.56642	−0.783210	+	0.621757i	$0.786419\pi$
$180$	0	0
$181$	−0.770479	−0.0572692	−0.0286346	−	0.999590i	$-0.509116\pi$
$181$	−0.770479	−0.0572692	−0.0286346	+	0.999590i	$0.509116\pi$
$182$	0	0
$183$	0	0
$184$	−14.0925	−1.03891
$185$	−14.6126	−1.07434
$186$	0	0
$187$	−1.09246	−0.0798884
$188$	3.55401	0.259203
$189$	0	0
$190$	58.7579	4.26275
$191$	−4.28765	−0.310243	−0.155122	−	0.987895i	$-0.549577\pi$
$191$	−4.28765	−0.310243	−0.155122	+	0.987895i	$0.549577\pi$
$192$	0	0
$193$	21.1957	1.52570	0.762849	−	0.646577i	$-0.223800\pi$
$193$	21.1957	1.52570	0.762849	+	0.646577i	$0.223800\pi$
$194$	−26.9946	−1.93810
$195$	0	0
$196$	26.8485	1.91775
$197$	−8.74326	−0.622931	−0.311466	−	0.950257i	$-0.600820\pi$
$197$	−8.74326	−0.622931	−0.311466	+	0.950257i	$0.600820\pi$
$198$	0	0
$199$	−6.47219	−0.458801	−0.229401	−	0.973332i	$-0.573677\pi$
$199$	−6.47219	−0.458801	−0.229401	+	0.973332i	$0.573677\pi$
$200$	−30.9260	−2.18680
$201$	0	0
$202$	−42.6969	−3.00415
$203$	28.9298	2.03048
$204$	0	0
$205$	−25.4263	−1.77585
$206$	16.5123	1.15046
$207$	0	0
$208$	0	0
$209$	1.93528	0.133866
$210$	0	0
$211$	−18.1793	−1.25151	−0.625756	−	0.780018i	$-0.715210\pi$
$211$	−18.1793	−1.25151	−0.625756	+	0.780018i	$0.715210\pi$
$212$	−27.6853	−1.90143
$213$	0	0
$214$	14.0925	0.963341
$215$	−12.9821	−0.885371
$216$	0	0
$217$	7.93469	0.538642
$218$	26.6872	1.80748
$219$	0	0
$220$	−3.65279	−0.246271
$221$	0	0
$222$	0	0
$223$	−4.92931	−0.330091	−0.165046	−	0.986286i	$-0.552777\pi$
$223$	−4.92931	−0.330091	−0.165046	+	0.986286i	$0.552777\pi$
$224$	1.79848	0.120166
$225$	0	0
$226$	18.5157	1.23165
$227$	20.9143	1.38813	0.694067	−	0.719911i	$-0.255817\pi$
$227$	20.9143	1.38813	0.694067	+	0.719911i	$0.255817\pi$
$228$	0	0
$229$	−0.686645	−0.0453748	−0.0226874	−	0.999743i	$-0.507222\pi$
$229$	−0.686645	−0.0453748	−0.0226874	+	0.999743i	$0.507222\pi$
$230$	−22.9534	−1.51350
$231$	0	0
$232$	39.4862	2.59240
$233$	16.1157	1.05578	0.527889	−	0.849314i	$-0.322984\pi$
$233$	16.1157	1.05578	0.527889	+	0.849314i	$0.322984\pi$
$234$	0	0
$235$	2.92931	0.191087
$236$	13.7825	0.897163
$237$	0	0
$238$	36.6950	2.37858
$239$	23.7406	1.53565	0.767826	−	0.640658i	$-0.221338\pi$
$239$	23.7406	1.53565	0.767826	+	0.640658i	$0.221338\pi$
$240$	0	0
$241$	−9.30798	−0.599580	−0.299790	−	0.954005i	$-0.596917\pi$
$241$	−9.30798	−0.599580	−0.299790	+	0.954005i	$0.596917\pi$
$242$	26.8743	1.72754
$243$	0	0
$244$	12.3937	0.793427
$245$	22.1292	1.41378
$246$	0	0
$247$	0	0
$248$	10.8300	0.687706
$249$	0	0
$250$	−9.33273	−0.590254
$251$	−2.04498	−0.129078	−0.0645391	−	0.997915i	$-0.520558\pi$
$251$	−2.04498	−0.129078	−0.0645391	+	0.997915i	$0.520558\pi$
$252$	0	0
$253$	−0.756004	−0.0475296
$254$	−9.32686	−0.585220
$255$	0	0
$256$	−32.3327	−2.02080
$257$	10.7691	0.671760	0.335880	−	0.941905i	$-0.390966\pi$
$257$	10.7691	0.671760	0.335880	+	0.941905i	$0.390966\pi$
$258$	0	0
$259$	16.1661	1.00451
$260$	0	0
$261$	0	0
$262$	36.4252	2.25036
$263$	14.3661	0.885851	0.442925	−	0.896558i	$-0.353941\pi$
$263$	14.3661	0.885851	0.442925	+	0.896558i	$0.353941\pi$
$264$	0	0
$265$	−22.8189	−1.40176
$266$	−65.0048	−3.98570
$267$	0	0
$268$	20.2838	1.23903
$269$	21.0941	1.28613	0.643064	−	0.765812i	$-0.277663\pi$
$269$	21.0941	1.28613	0.643064	+	0.765812i	$0.277663\pi$
$270$	0	0
$271$	−24.2567	−1.47349	−0.736744	−	0.676172i	$-0.763638\pi$
$271$	−24.2567	−1.47349	−0.736744	+	0.676172i	$0.763638\pi$
$272$	17.3603	1.05262
$273$	0	0
$274$	52.5042	3.17190
$275$	−1.65906	−0.100045
$276$	0	0
$277$	−25.3207	−1.52137	−0.760685	−	0.649121i	$-0.775137\pi$
$277$	−25.3207	−1.52137	−0.760685	+	0.649121i	$0.775137\pi$
$278$	16.3087	0.978131
$279$	0	0
$280$	62.0887	3.71051
$281$	−6.60768	−0.394181	−0.197091	−	0.980385i	$-0.563149\pi$
$281$	−6.60768	−0.394181	−0.197091	+	0.980385i	$0.563149\pi$
$282$	0	0
$283$	23.3002	1.38505	0.692527	−	0.721392i	$-0.256497\pi$
$283$	23.3002	1.38505	0.692527	+	0.721392i	$0.256497\pi$
$284$	−30.7522	−1.82480
$285$	0	0
$286$	0	0
$287$	28.1295	1.66043
$288$	0	0
$289$	−0.669186	−0.0393639
$290$	64.3141	3.77665
$291$	0	0
$292$	1.64310	0.0961554
$293$	6.98183	0.407883	0.203942	−	0.978983i	$-0.434625\pi$
$293$	6.98183	0.407883	0.203942	+	0.978983i	$0.434625\pi$
$294$	0	0
$295$	11.3599	0.661398
$296$	22.0651	1.28251
$297$	0	0
$298$	−42.3749	−2.45471
$299$	0	0
$300$	0	0
$301$	14.3623	0.827828
$302$	49.8871	2.87068
$303$	0	0
$304$	−30.7536	−1.76384
$305$	10.2152	0.584923
$306$	0	0
$307$	−29.0103	−1.65570	−0.827852	−	0.560946i	$-0.810437\pi$
$307$	−29.0103	−1.65570	−0.827852	+	0.560946i	$0.810437\pi$
$308$	4.04114	0.230265
$309$	0	0
$310$	17.6396	1.00186
$311$	−2.90626	−0.164799	−0.0823995	−	0.996599i	$-0.526258\pi$
$311$	−2.90626	−0.164799	−0.0823995	+	0.996599i	$0.526258\pi$
$312$	0	0
$313$	−25.1239	−1.42009	−0.710044	−	0.704157i	$-0.751325\pi$
$313$	−25.1239	−1.42009	−0.710044	+	0.704157i	$0.751325\pi$
$314$	−19.3459	−1.09175
$315$	0	0
$316$	−50.1812	−2.82291
$317$	−5.44309	−0.305714	−0.152857	−	0.988248i	$-0.548847\pi$
$317$	−5.44309	−0.305714	−0.152857	+	0.988248i	$0.548847\pi$
$318$	0	0
$319$	2.11828	0.118601
$320$	−24.6745	−1.37935
$321$	0	0
$322$	25.3937	1.41514
$323$	−28.9298	−1.60970
$324$	0	0
$325$	0	0
$326$	−25.9965	−1.43981
$327$	0	0
$328$	38.3937	2.11994
$329$	−3.24074	−0.178668
$330$	0	0
$331$	−26.4306	−1.45276	−0.726378	−	0.687296i	$-0.758798\pi$
$331$	−26.4306	−1.45276	−0.726378	+	0.687296i	$0.758798\pi$
$332$	28.2927	1.55276
$333$	0	0
$334$	43.3943	2.37443
$335$	16.7185	0.913427
$336$	0	0
$337$	3.86161	0.210355	0.105178	−	0.994453i	$-0.466459\pi$
$337$	3.86161	0.210355	0.105178	+	0.994453i	$0.466459\pi$
$338$	0	0
$339$	0	0
$340$	54.6045	2.96134
$341$	0.580987	0.0314622
$342$	0	0
$343$	1.36227	0.0735558
$344$	19.6030	1.05692
$345$	0	0
$346$	8.24027	0.443000
$347$	−25.1961	−1.35260	−0.676299	−	0.736628i	$-0.736417\pi$
$347$	−25.1961	−1.35260	−0.676299	+	0.736628i	$0.736417\pi$
$348$	0	0
$349$	−16.5918	−0.888138	−0.444069	−	0.895993i	$-0.646466\pi$
$349$	−16.5918	−0.888138	−0.444069	+	0.895993i	$0.646466\pi$
$350$	55.7267	2.97872
$351$	0	0
$352$	0.131687	0.00701892
$353$	−11.9840	−0.637844	−0.318922	−	0.947781i	$-0.603321\pi$
$353$	−11.9840	−0.637844	−0.318922	+	0.947781i	$0.603321\pi$
$354$	0	0
$355$	−25.3467	−1.34526
$356$	54.2224	2.87378
$357$	0	0
$358$	51.5435	2.72416
$359$	−10.7097	−0.565236	−0.282618	−	0.959232i	$-0.591203\pi$
$359$	−10.7097	−0.565236	−0.282618	+	0.959232i	$0.591203\pi$
$360$	0	0
$361$	32.2489	1.69731
$362$	1.89496	0.0995968
$363$	0	0
$364$	0	0
$365$	1.35429	0.0708868
$366$	0	0
$367$	−18.1497	−0.947409	−0.473704	−	0.880684i	$-0.657083\pi$
$367$	−18.1497	−0.947409	−0.473704	+	0.880684i	$0.657083\pi$
$368$	12.0137	0.626258
$369$	0	0
$370$	35.9390	1.86838
$371$	25.2449	1.31065
$372$	0	0
$373$	−2.45473	−0.127101	−0.0635506	−	0.997979i	$-0.520242\pi$
$373$	−2.45473	−0.127101	−0.0635506	+	0.997979i	$0.520242\pi$
$374$	2.68685	0.138934
$375$	0	0
$376$	−4.42327	−0.228113
$377$	0	0
$378$	0	0
$379$	19.5080	1.00206	0.501028	−	0.865431i	$-0.332955\pi$
$379$	19.5080	1.00206	0.501028	+	0.865431i	$0.332955\pi$
$380$	−96.7312	−4.96221
$381$	0	0
$382$	10.5453	0.539543
$383$	−23.0961	−1.18016	−0.590078	−	0.807346i	$-0.700903\pi$
$383$	−23.0961	−1.18016	−0.590078	+	0.807346i	$0.700903\pi$
$384$	0	0
$385$	3.33081	0.169754
$386$	−52.1298	−2.65334
$387$	0	0
$388$	44.4403	2.25611
$389$	21.6480	1.09760	0.548798	−	0.835955i	$-0.315086\pi$
$389$	21.6480	1.09760	0.548798	+	0.835955i	$0.315086\pi$
$390$	0	0
$391$	11.3013	0.571530
$392$	−33.4152	−1.68772
$393$	0	0
$394$	21.5036	1.08334
$395$	−41.3607	−2.08108
$396$	0	0
$397$	24.7627	1.24280	0.621402	−	0.783492i	$-0.286563\pi$
$397$	24.7627	1.24280	0.621402	+	0.783492i	$0.286563\pi$
$398$	15.9181	0.797900
$399$	0	0
$400$	26.3642	1.31821
$401$	−7.35454	−0.367268	−0.183634	−	0.982995i	$-0.558786\pi$
$401$	−7.35454	−0.367268	−0.183634	+	0.982995i	$0.558786\pi$
$402$	0	0
$403$	0	0
$404$	70.2905	3.49708
$405$	0	0
$406$	−71.1517	−3.53120
$407$	1.18370	0.0586740
$408$	0	0
$409$	17.3588	0.858338	0.429169	−	0.903224i	$-0.358806\pi$
$409$	17.3588	0.858338	0.429169	+	0.903224i	$0.358806\pi$
$410$	62.5347	3.08837
$411$	0	0
$412$	−27.1836	−1.33924
$413$	−12.5676	−0.618412
$414$	0	0
$415$	23.3196	1.14471
$416$	0	0
$417$	0	0
$418$	−4.75973	−0.232806
$419$	−0.998079	−0.0487594	−0.0243797	−	0.999703i	$-0.507761\pi$
$419$	−0.998079	−0.0487594	−0.0243797	+	0.999703i	$0.507761\pi$
$420$	0	0
$421$	0.786872	0.0383498	0.0191749	−	0.999816i	$-0.493896\pi$
$421$	0.786872	0.0383498	0.0191749	+	0.999816i	$0.493896\pi$
$422$	44.7111	2.17650
$423$	0	0
$424$	34.4566	1.67336
$425$	24.8007	1.20301
$426$	0	0
$427$	−11.3013	−0.546907
$428$	−23.1999	−1.12141
$429$	0	0
$430$	31.9288	1.53975
$431$	26.5478	1.27876	0.639380	−	0.768891i	$-0.279191\pi$
$431$	26.5478	1.27876	0.639380	+	0.768891i	$0.279191\pi$
$432$	0	0
$433$	24.1118	1.15874	0.579370	−	0.815064i	$-0.303298\pi$
$433$	24.1118	1.15874	0.579370	+	0.815064i	$0.303298\pi$
$434$	−19.5150	−0.936750
$435$	0	0
$436$	−43.9342	−2.10407
$437$	−20.0201	−0.957690
$438$	0	0
$439$	5.27844	0.251926	0.125963	−	0.992035i	$-0.459798\pi$
$439$	5.27844	0.251926	0.125963	+	0.992035i	$0.459798\pi$
$440$	4.54621	0.216732
$441$	0	0
$442$	0	0
$443$	7.63809	0.362897	0.181448	−	0.983400i	$-0.441921\pi$
$443$	7.63809	0.362897	0.181448	+	0.983400i	$0.441921\pi$
$444$	0	0
$445$	44.6915	2.11858
$446$	12.1234	0.574061
$447$	0	0
$448$	27.2978	1.28970
$449$	33.3616	1.57443	0.787216	−	0.616677i	$-0.211521\pi$
$449$	33.3616	1.57443	0.787216	+	0.616677i	$0.211521\pi$
$450$	0	0
$451$	2.05967	0.0969861
$452$	−30.4818	−1.43374
$453$	0	0
$454$	−51.4379	−2.41410
$455$	0	0
$456$	0	0
$457$	−20.7342	−0.969907	−0.484953	−	0.874540i	$-0.661163\pi$
$457$	−20.7342	−0.969907	−0.484953	+	0.874540i	$0.661163\pi$
$458$	1.68877	0.0789111
$459$	0	0
$460$	37.7875	1.76185
$461$	−34.5897	−1.61100	−0.805502	−	0.592593i	$-0.798104\pi$
$461$	−34.5897	−1.61100	−0.805502	+	0.592593i	$0.798104\pi$
$462$	0	0
$463$	21.0508	0.978315	0.489158	−	0.872195i	$-0.337304\pi$
$463$	21.0508	0.978315	0.489158	+	0.872195i	$0.337304\pi$
$464$	−33.6617	−1.56270
$465$	0	0
$466$	−39.6359	−1.83610
$467$	14.0587	0.650559	0.325279	−	0.945618i	$-0.394542\pi$
$467$	14.0587	0.650559	0.325279	+	0.945618i	$0.394542\pi$
$468$	0	0
$469$	−18.4959	−0.854060
$470$	−7.20451	−0.332319
$471$	0	0
$472$	−17.1535	−0.789552
$473$	1.05162	0.0483536
$474$	0	0
$475$	−43.9342	−2.01584
$476$	−60.4098	−2.76888
$477$	0	0
$478$	−58.3889	−2.67065
$479$	28.3118	1.29360	0.646800	−	0.762660i	$-0.276107\pi$
$479$	28.3118	1.29360	0.646800	+	0.762660i	$0.276107\pi$
$480$	0	0
$481$	0	0
$482$	22.8926	1.04273
$483$	0	0
$484$	−44.2422	−2.01101
$485$	36.6288	1.66323
$486$	0	0
$487$	−18.7211	−0.848333	−0.424167	−	0.905584i	$-0.639433\pi$
$487$	−18.7211	−0.848333	−0.424167	+	0.905584i	$0.639433\pi$
$488$	−15.4250	−0.698259
$489$	0	0
$490$	−54.4258	−2.45871
$491$	−22.7828	−1.02818	−0.514088	−	0.857738i	$-0.671869\pi$
$491$	−22.7828	−1.02818	−0.514088	+	0.857738i	$0.671869\pi$
$492$	0	0
$493$	−31.6655	−1.42614
$494$	0	0
$495$	0	0
$496$	−9.23251	−0.414552
$497$	28.0415	1.25783
$498$	0	0
$499$	−3.84787	−0.172254	−0.0861272	−	0.996284i	$-0.527449\pi$
$499$	−3.84787	−0.172254	−0.0861272	+	0.996284i	$0.527449\pi$
$500$	15.3642	0.687106
$501$	0	0
$502$	5.02954	0.224479
$503$	21.8456	0.974049	0.487025	−	0.873388i	$-0.338082\pi$
$503$	21.8456	0.974049	0.487025	+	0.873388i	$0.338082\pi$
$504$	0	0
$505$	57.9353	2.57809
$506$	1.85936	0.0826586
$507$	0	0
$508$	15.3545	0.681246
$509$	9.29715	0.412089	0.206044	−	0.978543i	$-0.433941\pi$
$509$	9.29715	0.412089	0.206044	+	0.978543i	$0.433941\pi$
$510$	0	0
$511$	−1.49827	−0.0662796
$512$	41.2033	1.82095
$513$	0	0
$514$	−26.4862	−1.16826
$515$	−22.4054	−0.987301
$516$	0	0
$517$	−0.237291	−0.0104360
$518$	−39.7599	−1.74695
$519$	0	0
$520$	0	0
$521$	−18.5440	−0.812428	−0.406214	−	0.913778i	$-0.633151\pi$
$521$	−18.5440	−0.812428	−0.406214	+	0.913778i	$0.633151\pi$
$522$	0	0
$523$	27.8635	1.21839	0.609193	−	0.793022i	$-0.291493\pi$
$523$	27.8635	1.21839	0.609193	+	0.793022i	$0.291493\pi$
$524$	−59.9656	−2.61961
$525$	0	0
$526$	−35.3327	−1.54058
$527$	−8.68500	−0.378324
$528$	0	0
$529$	−15.1793	−0.659969
$530$	56.1221	2.43779
$531$	0	0
$532$	107.015	4.63970
$533$	0	0
$534$	0	0
$535$	−19.1220	−0.826716
$536$	−25.2449	−1.09041
$537$	0	0
$538$	−51.8799	−2.23670
$539$	−1.79259	−0.0772124
$540$	0	0
$541$	17.2228	0.740467	0.370233	−	0.928939i	$-0.379278\pi$
$541$	17.2228	0.740467	0.370233	+	0.928939i	$0.379278\pi$
$542$	59.6582	2.56254
$543$	0	0
$544$	−1.96854	−0.0844005
$545$	−36.2117	−1.55114
$546$	0	0
$547$	5.01400	0.214383	0.107192	−	0.994238i	$-0.465814\pi$
$547$	5.01400	0.214383	0.107192	+	0.994238i	$0.465814\pi$
$548$	−86.4360	−3.69236
$549$	0	0
$550$	4.08038	0.173988
$551$	56.0950	2.38973
$552$	0	0
$553$	45.7579	1.94582
$554$	62.2750	2.64581
$555$	0	0
$556$	−26.8485	−1.13863
$557$	−22.2660	−0.943441	−0.471720	−	0.881748i	$-0.656367\pi$
$557$	−22.2660	−0.943441	−0.471720	+	0.881748i	$0.656367\pi$
$558$	0	0
$559$	0	0
$560$	−52.9302	−2.23671
$561$	0	0
$562$	16.2513	0.685519
$563$	16.2254	0.683821	0.341910	−	0.939733i	$-0.388926\pi$
$563$	16.2254	0.683821	0.341910	+	0.939733i	$0.388926\pi$
$564$	0	0
$565$	−25.1239	−1.05697
$566$	−57.3058	−2.40874
$567$	0	0
$568$	38.2737	1.60593
$569$	7.33071	0.307319	0.153660	−	0.988124i	$-0.450894\pi$
$569$	7.33071	0.307319	0.153660	+	0.988124i	$0.450894\pi$
$570$	0	0
$571$	25.2825	1.05804	0.529020	−	0.848610i	$-0.322560\pi$
$571$	25.2825	1.05804	0.529020	+	0.848610i	$0.322560\pi$
$572$	0	0
$573$	0	0
$574$	−69.1831	−2.88765
$575$	17.1626	0.715732
$576$	0	0
$577$	−4.26742	−0.177655	−0.0888275	−	0.996047i	$-0.528312\pi$
$577$	−4.26742	−0.177655	−0.0888275	+	0.996047i	$0.528312\pi$
$578$	1.64583	0.0684576
$579$	0	0
$580$	−105.878	−4.39635
$581$	−25.7988	−1.07032
$582$	0	0
$583$	1.84846	0.0765555
$584$	−2.04498	−0.0846219
$585$	0	0
$586$	−17.1715	−0.709348
$587$	27.4579	1.13331	0.566654	−	0.823956i	$-0.308238\pi$
$587$	27.4579	1.13331	0.566654	+	0.823956i	$0.308238\pi$
$588$	0	0
$589$	15.3854	0.633943
$590$	−27.9391	−1.15023
$591$	0	0
$592$	−18.8103	−0.773099
$593$	−1.16721	−0.0479317	−0.0239658	−	0.999713i	$-0.507629\pi$
$593$	−1.16721	−0.0479317	−0.0239658	+	0.999713i	$0.507629\pi$
$594$	0	0
$595$	−49.7913	−2.04124
$596$	69.7604	2.85750
$597$	0	0
$598$	0	0
$599$	0.197682	0.00807706	0.00403853	−	0.999992i	$-0.498714\pi$
$599$	0.197682	0.00807706	0.00403853	+	0.999992i	$0.498714\pi$
$600$	0	0
$601$	20.6340	0.841679	0.420839	−	0.907135i	$-0.361736\pi$
$601$	20.6340	0.841679	0.420839	+	0.907135i	$0.361736\pi$
$602$	−35.3234	−1.43967
$603$	0	0
$604$	−82.1275	−3.34172
$605$	−36.4656	−1.48254
$606$	0	0
$607$	21.8159	0.885482	0.442741	−	0.896650i	$-0.354006\pi$
$607$	21.8159	0.885482	0.442741	+	0.896650i	$0.354006\pi$
$608$	3.48725	0.141427
$609$	0	0
$610$	−25.1239	−1.01724
$611$	0	0
$612$	0	0
$613$	−26.1299	−1.05538	−0.527688	−	0.849438i	$-0.676941\pi$
$613$	−26.1299	−1.05538	−0.527688	+	0.849438i	$0.676941\pi$
$614$	71.3495	2.87943
$615$	0	0
$616$	−5.02954	−0.202646
$617$	1.84141	0.0741326	0.0370663	−	0.999313i	$-0.488199\pi$
$617$	1.84141	0.0741326	0.0370663	+	0.999313i	$0.488199\pi$
$618$	0	0
$619$	7.77000	0.312303	0.156151	−	0.987733i	$-0.450091\pi$
$619$	7.77000	0.312303	0.156151	+	0.987733i	$0.450091\pi$
$620$	−29.0396	−1.16626
$621$	0	0
$622$	7.14782	0.286601
$623$	−49.4429	−1.98089
$624$	0	0
$625$	−18.0218	−0.720871
$626$	61.7911	2.46967
$627$	0	0
$628$	31.8485	1.27089
$629$	−17.6948	−0.705538
$630$	0	0
$631$	−19.1468	−0.762220	−0.381110	−	0.924530i	$-0.624458\pi$
$631$	−19.1468	−0.762220	−0.381110	+	0.924530i	$0.624458\pi$
$632$	62.4547	2.48432
$633$	0	0
$634$	13.3870	0.531667
$635$	12.6556	0.502222
$636$	0	0
$637$	0	0
$638$	−5.20981	−0.206258
$639$	0	0
$640$	63.9372	2.52734
$641$	24.9496	0.985450	0.492725	−	0.870185i	$-0.336001\pi$
$641$	24.9496	0.985450	0.492725	+	0.870185i	$0.336001\pi$
$642$	0	0
$643$	42.4161	1.67273	0.836364	−	0.548175i	$-0.184677\pi$
$643$	42.4161	1.67273	0.836364	+	0.548175i	$0.184677\pi$
$644$	−41.8048	−1.64734
$645$	0	0
$646$	71.1517	2.79942
$647$	−9.05326	−0.355921	−0.177960	−	0.984038i	$-0.556950\pi$
$647$	−9.05326	−0.355921	−0.177960	+	0.984038i	$0.556950\pi$
$648$	0	0
$649$	−0.920215	−0.0361216
$650$	0	0
$651$	0	0
$652$	42.7972	1.67607
$653$	−1.57907	−0.0617936	−0.0308968	−	0.999523i	$-0.509836\pi$
$653$	−1.57907	−0.0617936	−0.0308968	+	0.999523i	$0.509836\pi$
$654$	0	0
$655$	−49.4252	−1.93120
$656$	−32.7304	−1.27791
$657$	0	0
$658$	7.97046	0.310721
$659$	43.1591	1.68124	0.840621	−	0.541624i	$-0.182191\pi$
$659$	43.1591	1.68124	0.840621	+	0.541624i	$0.182191\pi$
$660$	0	0
$661$	−4.64848	−0.180805	−0.0904025	−	0.995905i	$-0.528815\pi$
$661$	−4.64848	−0.180805	−0.0904025	+	0.995905i	$0.528815\pi$
$662$	65.0048	2.52648
$663$	0	0
$664$	−35.2127	−1.36652
$665$	88.2047	3.42043
$666$	0	0
$667$	−21.9132	−0.848482
$668$	−71.4386	−2.76404
$669$	0	0
$670$	−41.1183	−1.58854
$671$	−0.827492	−0.0319450
$672$	0	0
$673$	−4.52243	−0.174327	−0.0871635	−	0.996194i	$-0.527780\pi$
$673$	−4.52243	−0.174327	−0.0871635	+	0.996194i	$0.527780\pi$
$674$	−9.49745	−0.365828
$675$	0	0
$676$	0	0
$677$	32.1706	1.23642	0.618208	−	0.786014i	$-0.287859\pi$
$677$	32.1706	1.23642	0.618208	+	0.786014i	$0.287859\pi$
$678$	0	0
$679$	−40.5230	−1.55513
$680$	−67.9599	−2.60614
$681$	0	0
$682$	−1.42891	−0.0547158
$683$	32.2241	1.23302	0.616511	−	0.787346i	$-0.288546\pi$
$683$	32.2241	1.23302	0.616511	+	0.787346i	$0.288546\pi$
$684$	0	0
$685$	−71.2428	−2.72205
$686$	−3.35045	−0.127921
$687$	0	0
$688$	−16.7114	−0.637116
$689$	0	0
$690$	0	0
$691$	14.7482	0.561049	0.280525	−	0.959847i	$-0.409492\pi$
$691$	14.7482	0.561049	0.280525	+	0.959847i	$0.409492\pi$
$692$	−13.5657	−0.515690
$693$	0	0
$694$	61.9687	2.35230
$695$	−22.1292	−0.839409
$696$	0	0
$697$	−30.7894	−1.16623
$698$	40.8068	1.54456
$699$	0	0
$700$	−91.7411	−3.46749
$701$	−41.5583	−1.56964	−0.784818	−	0.619726i	$-0.787244\pi$
$701$	−41.5583	−1.56964	−0.784818	+	0.619726i	$0.787244\pi$
$702$	0	0
$703$	31.3461	1.18224
$704$	1.99878	0.0753318
$705$	0	0
$706$	29.4741	1.10927
$707$	−64.0947	−2.41053
$708$	0	0
$709$	40.6034	1.52489	0.762446	−	0.647051i	$-0.223998\pi$
$709$	40.6034	1.52489	0.762446	+	0.647051i	$0.223998\pi$
$710$	62.3391	2.33955
$711$	0	0
$712$	−67.4844	−2.52908
$713$	−6.01020	−0.225084
$714$	0	0
$715$	0	0
$716$	−84.8543	−3.17115
$717$	0	0
$718$	26.3400	0.983001
$719$	42.1881	1.57335	0.786676	−	0.617366i	$-0.211800\pi$
$719$	42.1881	1.57335	0.786676	+	0.617366i	$0.211800\pi$
$720$	0	0
$721$	24.7875	0.923133
$722$	−79.3147	−2.95179
$723$	0	0
$724$	−3.11960	−0.115939
$725$	−48.0886	−1.78597
$726$	0	0
$727$	−29.1661	−1.08171	−0.540856	−	0.841115i	$-0.681900\pi$
$727$	−29.1661	−1.08171	−0.540856	+	0.841115i	$0.681900\pi$
$728$	0	0
$729$	0	0
$730$	−3.33081	−0.123279
$731$	−15.7204	−0.581439
$732$	0	0
$733$	47.7622	1.76414	0.882069	−	0.471120i	$-0.156150\pi$
$733$	47.7622	1.76414	0.882069	+	0.471120i	$0.156150\pi$
$734$	44.6385	1.64764
$735$	0	0
$736$	−1.36227	−0.0502140
$737$	−1.35429	−0.0498859
$738$	0	0
$739$	−28.7157	−1.05632	−0.528162	−	0.849143i	$-0.677119\pi$
$739$	−28.7157	−1.05632	−0.528162	+	0.849143i	$0.677119\pi$
$740$	−59.1652	−2.17495
$741$	0	0
$742$	−62.0887	−2.27935
$743$	−2.76684	−0.101506	−0.0507528	−	0.998711i	$-0.516162\pi$
$743$	−2.76684	−0.101506	−0.0507528	+	0.998711i	$0.516162\pi$
$744$	0	0
$745$	57.4984	2.10658
$746$	6.03730	0.221041
$747$	0	0
$748$	−4.42327	−0.161731
$749$	21.1550	0.772985
$750$	0	0
$751$	31.6980	1.15668	0.578338	−	0.815797i	$-0.303701\pi$
$751$	31.6980	1.15668	0.578338	+	0.815797i	$0.303701\pi$
$752$	3.77081	0.137507
$753$	0	0
$754$	0	0
$755$	−67.6916	−2.46355
$756$	0	0
$757$	−12.5138	−0.454822	−0.227411	−	0.973799i	$-0.573026\pi$
$757$	−12.5138	−0.454822	−0.227411	+	0.973799i	$0.573026\pi$
$758$	−47.9789	−1.74267
$759$	0	0
$760$	120.390	4.36701
$761$	0.883654	0.0320324	0.0160162	−	0.999872i	$-0.494902\pi$
$761$	0.883654	0.0320324	0.0160162	+	0.999872i	$0.494902\pi$
$762$	0	0
$763$	40.0616	1.45033
$764$	−17.3603	−0.628074
$765$	0	0
$766$	56.8039	2.05241
$767$	0	0
$768$	0	0
$769$	29.9866	1.08134	0.540672	−	0.841233i	$-0.318170\pi$
$769$	29.9866	1.08134	0.540672	+	0.841233i	$0.318170\pi$
$770$	−8.19199	−0.295219
$771$	0	0
$772$	85.8195	3.08871
$773$	−6.54825	−0.235524	−0.117762	−	0.993042i	$-0.537572\pi$
$773$	−6.54825	−0.235524	−0.117762	+	0.993042i	$0.537572\pi$
$774$	0	0
$775$	−13.1894	−0.473779
$776$	−55.3096	−1.98550
$777$	0	0
$778$	−53.2422	−1.90883
$779$	54.5430	1.95421
$780$	0	0
$781$	2.05323	0.0734703
$782$	−27.7950	−0.993946
$783$	0	0
$784$	28.4862	1.01736
$785$	26.2503	0.936915
$786$	0	0
$787$	−29.6055	−1.05532	−0.527662	−	0.849455i	$-0.676931\pi$
$787$	−29.6055	−1.05532	−0.527662	+	0.849455i	$0.676931\pi$
$788$	−35.4007	−1.26110
$789$	0	0
$790$	101.725	3.61920
$791$	27.7950	0.988275
$792$	0	0
$793$	0	0
$794$	−60.9028	−2.16136
$795$	0	0
$796$	−26.2054	−0.928824
$797$	−30.2353	−1.07099	−0.535495	−	0.844538i	$-0.679875\pi$
$797$	−30.2353	−1.07099	−0.535495	+	0.844538i	$0.679875\pi$
$798$	0	0
$799$	3.54719	0.125491
$800$	−2.98952	−0.105695
$801$	0	0
$802$	18.0881	0.638715
$803$	−0.109705	−0.00387141
$804$	0	0
$805$	−34.4566	−1.21444
$806$	0	0
$807$	0	0
$808$	−87.4825	−3.07762
$809$	−4.37562	−0.153839	−0.0769193	−	0.997037i	$-0.524508\pi$
$809$	−4.37562	−0.153839	−0.0769193	+	0.997037i	$0.524508\pi$
$810$	0	0
$811$	46.9643	1.64914	0.824571	−	0.565759i	$-0.191417\pi$
$811$	46.9643	1.64914	0.824571	+	0.565759i	$0.191417\pi$
$812$	117.135	4.11062
$813$	0	0
$814$	−2.91126	−0.102040
$815$	35.2745	1.23561
$816$	0	0
$817$	27.8485	0.974294
$818$	−42.6932	−1.49273
$819$	0	0
$820$	−102.949	−3.59513
$821$	9.00037	0.314115	0.157058	−	0.987589i	$-0.449799\pi$
$821$	9.00037	0.314115	0.157058	+	0.987589i	$0.449799\pi$
$822$	0	0
$823$	−44.3226	−1.54499	−0.772494	−	0.635022i	$-0.780991\pi$
$823$	−44.3226	−1.54499	−0.772494	+	0.635022i	$0.780991\pi$
$824$	33.8323	1.17860
$825$	0	0
$826$	30.9095	1.07548
$827$	−32.7568	−1.13907	−0.569533	−	0.821968i	$-0.692876\pi$
$827$	−32.7568	−1.13907	−0.569533	+	0.821968i	$0.692876\pi$
$828$	0	0
$829$	−11.2765	−0.391650	−0.195825	−	0.980639i	$-0.562738\pi$
$829$	−11.2765	−0.391650	−0.195825	+	0.980639i	$0.562738\pi$
$830$	−57.3535	−1.99077
$831$	0	0
$832$	0	0
$833$	26.7969	0.928457
$834$	0	0
$835$	−58.8816	−2.03768
$836$	7.83578	0.271006
$837$	0	0
$838$	2.45473	0.0847973
$839$	−36.1311	−1.24738	−0.623692	−	0.781670i	$-0.714368\pi$
$839$	−36.1311	−1.24738	−0.623692	+	0.781670i	$0.714368\pi$
$840$	0	0
$841$	32.3994	1.11722
$842$	−1.93528	−0.0666940
$843$	0	0
$844$	−73.6064	−2.53364
$845$	0	0
$846$	0	0
$847$	40.3424	1.38618
$848$	−29.3740	−1.00871
$849$	0	0
$850$	−60.9963	−2.09216
$851$	−12.2452	−0.419760
$852$	0	0
$853$	18.2597	0.625199	0.312599	−	0.949885i	$-0.398800\pi$
$853$	18.2597	0.625199	0.312599	+	0.949885i	$0.398800\pi$
$854$	27.7950	0.951124
$855$	0	0
$856$	28.8743	0.986903
$857$	−7.58927	−0.259245	−0.129622	−	0.991563i	$-0.541376\pi$
$857$	−7.58927	−0.259245	−0.129622	+	0.991563i	$0.541376\pi$
$858$	0	0
$859$	−6.05536	−0.206606	−0.103303	−	0.994650i	$-0.532941\pi$
$859$	−6.05536	−0.206606	−0.103303	+	0.994650i	$0.532941\pi$
$860$	−52.5634	−1.79240
$861$	0	0
$862$	−65.2930	−2.22389
$863$	−12.6905	−0.431991	−0.215995	−	0.976394i	$-0.569300\pi$
$863$	−12.6905	−0.431991	−0.215995	+	0.976394i	$0.569300\pi$
$864$	0	0
$865$	−11.1812	−0.380172
$866$	−59.3020	−2.01516
$867$	0	0
$868$	32.1269	1.09046
$869$	3.35045	0.113656
$870$	0	0
$871$	0	0
$872$	54.6798	1.85169
$873$	0	0
$874$	49.2385	1.66552
$875$	−14.0099	−0.473620
$876$	0	0
$877$	13.7004	0.462629	0.231315	−	0.972879i	$-0.425697\pi$
$877$	13.7004	0.462629	0.231315	+	0.972879i	$0.425697\pi$
$878$	−12.9821	−0.438124
$879$	0	0
$880$	−3.87561	−0.130647
$881$	−23.5832	−0.794540	−0.397270	−	0.917702i	$-0.630042\pi$
$881$	−23.5832	−0.794540	−0.397270	+	0.917702i	$0.630042\pi$
$882$	0	0
$883$	42.1661	1.41900	0.709502	−	0.704704i	$-0.248920\pi$
$883$	42.1661	1.41900	0.709502	+	0.704704i	$0.248920\pi$
$884$	0	0
$885$	0	0
$886$	−18.7855	−0.631113
$887$	50.0239	1.67964	0.839820	−	0.542866i	$-0.182661\pi$
$887$	50.0239	1.67964	0.839820	+	0.542866i	$0.182661\pi$
$888$	0	0
$889$	−14.0011	−0.469581
$890$	−109.917	−3.68442
$891$	0	0
$892$	−19.9584	−0.668256
$893$	−6.28380	−0.210279
$894$	0	0
$895$	−69.9391	−2.33781
$896$	−70.7347	−2.36308
$897$	0	0
$898$	−82.0514	−2.73809
$899$	16.8402	0.561653
$900$	0	0
$901$	−27.6321	−0.920558
$902$	−5.06567	−0.168668
$903$	0	0
$904$	37.9372	1.26177
$905$	−2.57126	−0.0854716
$906$	0	0
$907$	14.3375	0.476069	0.238035	−	0.971257i	$-0.423497\pi$
$907$	14.3375	0.476069	0.238035	+	0.971257i	$0.423497\pi$
$908$	84.6804	2.81022
$909$	0	0
$910$	0	0
$911$	50.2433	1.66464	0.832318	−	0.554298i	$-0.187013\pi$
$911$	50.2433	1.66464	0.832318	+	0.554298i	$0.187013\pi$
$912$	0	0
$913$	−1.88902	−0.0625174
$914$	50.9949	1.68676
$915$	0	0
$916$	−2.78017	−0.0918593
$917$	54.6798	1.80569
$918$	0	0
$919$	45.9077	1.51435	0.757177	−	0.653210i	$-0.226578\pi$
$919$	45.9077	1.51435	0.757177	+	0.653210i	$0.226578\pi$
$920$	−47.0297	−1.55052
$921$	0	0
$922$	85.0719	2.80169
$923$	0	0
$924$	0	0
$925$	−26.8722	−0.883551
$926$	−51.7736	−1.70139
$927$	0	0
$928$	3.81700	0.125299
$929$	−29.3561	−0.963142	−0.481571	−	0.876407i	$-0.659934\pi$
$929$	−29.3561	−0.963142	−0.481571	+	0.876407i	$0.659934\pi$
$930$	0	0
$931$	−47.4704	−1.55578
$932$	65.2513	2.13738
$933$	0	0
$934$	−34.5767	−1.13139
$935$	−3.64578	−0.119230
$936$	0	0
$937$	3.54719	0.115882	0.0579408	−	0.998320i	$-0.481547\pi$
$937$	3.54719	0.115882	0.0579408	+	0.998320i	$0.481547\pi$
$938$	45.4898	1.48529
$939$	0	0
$940$	11.8605	0.386848
$941$	3.58845	0.116980	0.0584900	−	0.998288i	$-0.481371\pi$
$941$	3.58845	0.116980	0.0584900	+	0.998288i	$0.481371\pi$
$942$	0	0
$943$	−21.3069	−0.693849
$944$	14.6232	0.475944
$945$	0	0
$946$	−2.58642	−0.0840917
$947$	−8.84562	−0.287444	−0.143722	−	0.989618i	$-0.545907\pi$
$947$	−8.84562	−0.287444	−0.143722	+	0.989618i	$0.545907\pi$
$948$	0	0
$949$	0	0
$950$	108.054	3.50574
$951$	0	0
$952$	75.1850	2.43676
$953$	−49.0867	−1.59008	−0.795038	−	0.606560i	$-0.792549\pi$
$953$	−49.0867	−1.59008	−0.795038	+	0.606560i	$0.792549\pi$
$954$	0	0
$955$	−14.3088	−0.463023
$956$	96.1238	3.10886
$957$	0	0
$958$	−69.6316	−2.24970
$959$	78.8170	2.54513
$960$	0	0
$961$	−26.3812	−0.851006
$962$	0	0
$963$	0	0
$964$	−37.6872	−1.21382
$965$	70.7347	2.27703
$966$	0	0
$967$	37.9014	1.21883	0.609414	−	0.792852i	$-0.291405\pi$
$967$	37.9014	1.21883	0.609414	+	0.792852i	$0.291405\pi$
$968$	55.0631	1.76980
$969$	0	0
$970$	−90.0869	−2.89252
$971$	−8.58735	−0.275581	−0.137791	−	0.990461i	$-0.544000\pi$
$971$	−8.58735	−0.275581	−0.137791	+	0.990461i	$0.544000\pi$
$972$	0	0
$973$	24.4819	0.784853
$974$	46.0437	1.47533
$975$	0	0
$976$	13.1497	0.420913
$977$	−48.0469	−1.53716	−0.768578	−	0.639756i	$-0.779035\pi$
$977$	−48.0469	−1.53716	−0.768578	+	0.639756i	$0.779035\pi$
$978$	0	0
$979$	−3.62027	−0.115704
$980$	89.5993	2.86215
$981$	0	0
$982$	56.0334	1.78810
$983$	−29.1110	−0.928498	−0.464249	−	0.885705i	$-0.653676\pi$
$983$	−29.1110	−0.928498	−0.464249	+	0.885705i	$0.653676\pi$
$984$	0	0
$985$	−29.1782	−0.929695
$986$	77.8798	2.48020
$987$	0	0
$988$	0	0
$989$	−10.8788	−0.345927
$990$	0	0
$991$	−33.1094	−1.05176	−0.525878	−	0.850560i	$-0.676263\pi$
$991$	−33.1094	−1.05176	−0.525878	+	0.850560i	$0.676263\pi$
$992$	1.04690	0.0332392
$993$	0	0
$994$	−68.9667	−2.18749
$995$	−21.5991	−0.684739
$996$	0	0
$997$	−45.3661	−1.43676	−0.718380	−	0.695651i	$-0.755116\pi$
$997$	−45.3661	−1.43676	−0.718380	+	0.695651i	$0.755116\pi$
$998$	9.46366	0.299567
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.v.1.1	✓	6
3.2	odd	2	inner	1521.2.a.v.1.6	yes	6
13.5	odd	4		1521.2.b.n.1351.11		12
13.8	odd	4		1521.2.b.n.1351.2		12
13.12	even	2		1521.2.a.w.1.6	yes	6
39.5	even	4		1521.2.b.n.1351.1		12
39.8	even	4		1521.2.b.n.1351.12		12
39.38	odd	2		1521.2.a.w.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.2.a.v.1.1	✓	6	1.1	even	1	trivial
1521.2.a.v.1.6	yes	6	3.2	odd	2	inner
1521.2.a.w.1.1	yes	6	39.38	odd	2
1521.2.a.w.1.6	yes	6	13.12	even	2
1521.2.b.n.1351.1		12	39.5	even	4
1521.2.b.n.1351.2		12	13.8	odd	4
1521.2.b.n.1351.11		12	13.5	odd	4
1521.2.b.n.1351.12		12	39.8	even	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)

\(f(q)\)	\(=\)	\(q-2.45945 q^{2} +4.04892 q^{4} +3.33722 q^{5} -3.69202 q^{7} -5.03922 q^{8} +O(q^{10})\)	\(q-2.45945 q^{2} +4.04892 q^{4} +3.33722 q^{5} -3.69202 q^{7} -5.03922 q^{8} -8.20775 q^{10} -0.270334 q^{11} +9.08036 q^{14} +4.29590 q^{16} +4.04114 q^{17} -7.15883 q^{19} +13.5121 q^{20} +0.664874 q^{22} +2.79656 q^{23} +6.13706 q^{25} -14.9487 q^{28} -7.83578 q^{29} -2.14914 q^{31} -0.487125 q^{32} -9.93900 q^{34} -12.3211 q^{35} -4.37867 q^{37} +17.6068 q^{38} -16.8170 q^{40} -7.61898 q^{41} -3.89008 q^{43} -1.09456 q^{44} -6.87800 q^{46} +0.877769 q^{47} +6.63102 q^{49} -15.0938 q^{50} -6.83770 q^{53} -0.902165 q^{55} +18.6049 q^{56} +19.2717 q^{58} +3.40399 q^{59} +3.06100 q^{61} +5.28572 q^{62} -7.39373 q^{64} +5.00969 q^{67} +16.3622 q^{68} +30.3032 q^{70} -7.59516 q^{71} +0.405813 q^{73} +10.7691 q^{74} -28.9855 q^{76} +0.998079 q^{77} -12.3937 q^{79} +14.3364 q^{80} +18.7385 q^{82} +6.98772 q^{83} +13.4862 q^{85} +9.56748 q^{86} +1.36227 q^{88} +13.3918 q^{89} +11.3230 q^{92} -2.15883 q^{94} -23.8906 q^{95} +10.9758 q^{97} -16.3087 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 12 q^{7}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more