LMFDB - Embedded newform 1305.2.a.t.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,2,Mod(1,1305)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1305.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,1,0,13,7,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.28209$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.28209 q^{2} -0.356256 q^{4} +1.00000 q^{5} +3.04834 q^{7} -3.02092 q^{8} +1.28209 q^{10} +2.81651 q^{11} +4.80431 q^{13} +3.90823 q^{14} -3.16057 q^{16} -2.80431 q^{17} -3.76326 q^{19} -0.356256 q^{20} +3.61101 q^{22} +1.89841 q^{23} +1.00000 q^{25} +6.15954 q^{26} -1.08599 q^{28} -1.00000 q^{29} +7.76326 q^{31} +1.98972 q^{32} -3.59537 q^{34} +3.04834 q^{35} -6.31524 q^{37} -4.82483 q^{38} -3.02092 q^{40} +8.46358 q^{41} +8.36690 q^{43} -1.00340 q^{44} +2.43393 q^{46} -5.72482 q^{47} +2.29236 q^{49} +1.28209 q^{50} -1.71157 q^{52} +8.67157 q^{53} +2.81651 q^{55} -9.20879 q^{56} -1.28209 q^{58} +2.09668 q^{59} +12.8139 q^{61} +9.95317 q^{62} +8.87214 q^{64} +4.80431 q^{65} -4.17668 q^{67} +0.999054 q^{68} +3.90823 q^{70} -11.5265 q^{71} +12.5846 q^{73} -8.09668 q^{74} +1.34069 q^{76} +8.58568 q^{77} +1.86976 q^{79} -3.16057 q^{80} +10.8510 q^{82} -14.5485 q^{83} -2.80431 q^{85} +10.7271 q^{86} -8.50846 q^{88} -16.5340 q^{89} +14.6452 q^{91} -0.676322 q^{92} -7.33971 q^{94} -3.76326 q^{95} +1.26612 q^{97} +2.93900 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 13 q^{4} + 7 q^{5} + 10 q^{7} + q^{10} - 3 q^{11} + 6 q^{13} - 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} + 13 q^{20} + 9 q^{22} + 11 q^{23} + 7 q^{25} + 3 q^{26} + 25 q^{28} - 7 q^{29}+ \cdots - 20 q^{98}+O(q^{100}) $ 7 * q + q^2 + 13 * q^4 + 7 * q^5 + 10 * q^7 + q^10 - 3 * q^11 + 6 * q^13 - 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 + 13 * q^20 + 9 * q^22 + 11 * q^23 + 7 * q^25 + 3 * q^26 + 25 * q^28 - 7 * q^29 + 18 * q^31 + q^32 - q^34 + 10 * q^35 + 13 * q^37 + 12 * q^38 - 13 * q^41 + 9 * q^43 - 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^50 - q^52 + 5 * q^53 - 3 * q^55 - 30 * q^56 - q^58 - 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 + 6 * q^65 + 14 * q^67 + 27 * q^68 - 9 * q^70 - 8 * q^71 + 3 * q^73 - 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 21 * q^80 - 20 * q^82 + 17 * q^83 + 8 * q^85 + 4 * q^86 + 26 * q^88 - 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 + 10 * q^95 + 13 * q^97 - 20 * q^98

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 $

$2$	1.28209	0.906572	0.453286	−	0.891365i	$-0.350252\pi$
$2$	1.28209	0.906572	0.453286	+	0.891365i	$0.350252\pi$
$3$	0	0
$3$	0	0
$4$	−0.356256	−0.178128
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.04834	1.15216	0.576082	−	0.817392i	$-0.304581\pi$
$7$	3.04834	1.15216	0.576082	+	0.817392i	$0.304581\pi$
$8$	−3.02092	−1.06806
$9$	0	0
$10$	1.28209	0.405431
$11$	2.81651	0.849210	0.424605	−	0.905379i	$-0.360413\pi$
$11$	2.81651	0.849210	0.424605	+	0.905379i	$0.360413\pi$
$12$	0	0
$13$	4.80431	1.33248	0.666238	−	0.745739i	$-0.267903\pi$
$13$	4.80431	1.33248	0.666238	+	0.745739i	$0.267903\pi$
$14$	3.90823	1.04452
$15$	0	0
$16$	−3.16057	−0.790142
$17$	−2.80431	−0.680146	−0.340073	−	0.940399i	$-0.610452\pi$
$17$	−2.80431	−0.680146	−0.340073	+	0.940399i	$0.610452\pi$
$18$	0	0
$19$	−3.76326	−0.863352	−0.431676	−	0.902029i	$-0.642078\pi$
$19$	−3.76326	−0.863352	−0.431676	+	0.902029i	$0.642078\pi$
$20$	−0.356256	−0.0796613
$21$	0	0
$22$	3.61101	0.769870
$23$	1.89841	0.395847	0.197923	−	0.980217i	$-0.436580\pi$
$23$	1.89841	0.395847	0.197923	+	0.980217i	$0.436580\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.15954	1.20799
$27$	0	0
$28$	−1.08599	−0.205233
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.76326	1.39432	0.697162	−	0.716914i	$-0.254446\pi$
$31$	7.76326	1.39432	0.697162	+	0.716914i	$0.254446\pi$
$32$	1.98972	0.351737
$33$	0	0
$34$	−3.59537	−0.616601
$35$	3.04834	0.515263
$36$	0	0
$37$	−6.31524	−1.03822	−0.519109	−	0.854708i	$-0.673736\pi$
$37$	−6.31524	−1.03822	−0.519109	+	0.854708i	$0.673736\pi$
$38$	−4.82483	−0.782690
$39$	0	0
$40$	−3.02092	−0.477650
$41$	8.46358	1.32179	0.660894	−	0.750479i	$-0.270177\pi$
$41$	8.46358	1.32179	0.660894	+	0.750479i	$0.270177\pi$
$42$	0	0
$43$	8.36690	1.27594	0.637970	−	0.770061i	$-0.279774\pi$
$43$	8.36690	1.27594	0.637970	+	0.770061i	$0.279774\pi$
$44$	−1.00340	−0.151268
$45$	0	0
$46$	2.43393	0.358863
$47$	−5.72482	−0.835051	−0.417526	−	0.908665i	$-0.637103\pi$
$47$	−5.72482	−0.835051	−0.417526	+	0.908665i	$0.637103\pi$
$48$	0	0
$49$	2.29236	0.327480
$50$	1.28209	0.181314
$51$	0	0
$52$	−1.71157	−0.237351
$53$	8.67157	1.19113	0.595566	−	0.803306i	$-0.296928\pi$
$53$	8.67157	1.19113	0.595566	+	0.803306i	$0.296928\pi$
$54$	0	0
$55$	2.81651	0.379778
$56$	−9.20879	−1.23058
$57$	0	0
$58$	−1.28209	−0.168346
$59$	2.09668	0.272964	0.136482	−	0.990643i	$-0.456420\pi$
$59$	2.09668	0.272964	0.136482	+	0.990643i	$0.456420\pi$
$60$	0	0
$61$	12.8139	1.64065	0.820323	−	0.571901i	$-0.193794\pi$
$61$	12.8139	1.64065	0.820323	+	0.571901i	$0.193794\pi$
$62$	9.95317	1.26405
$63$	0	0
$64$	8.87214	1.10902
$65$	4.80431	0.595902
$66$	0	0
$67$	−4.17668	−0.510263	−0.255131	−	0.966906i	$-0.582119\pi$
$67$	−4.17668	−0.510263	−0.255131	+	0.966906i	$0.582119\pi$
$68$	0.999054	0.121153
$69$	0	0
$70$	3.90823	0.467123
$71$	−11.5265	−1.36795	−0.683973	−	0.729507i	$-0.739750\pi$
$71$	−11.5265	−1.36795	−0.683973	+	0.729507i	$0.739750\pi$
$72$	0	0
$73$	12.5846	1.47292	0.736461	−	0.676480i	$-0.236496\pi$
$73$	12.5846	1.47292	0.736461	+	0.676480i	$0.236496\pi$
$74$	−8.09668	−0.941219
$75$	0	0
$76$	1.34069	0.153787
$77$	8.58568	0.978429
$78$	0	0
$79$	1.86976	0.210364	0.105182	−	0.994453i	$-0.466457\pi$
$79$	1.86976	0.210364	0.105182	+	0.994453i	$0.466457\pi$
$80$	−3.16057	−0.353362
$81$	0	0
$82$	10.8510	1.19830
$83$	−14.5485	−1.59690	−0.798450	−	0.602061i	$-0.794347\pi$
$83$	−14.5485	−1.59690	−0.798450	+	0.602061i	$0.794347\pi$
$84$	0	0
$85$	−2.80431	−0.304170
$86$	10.7271	1.15673
$87$	0	0
$88$	−8.50846	−0.907005
$89$	−16.5340	−1.75260	−0.876301	−	0.481764i	$-0.839996\pi$
$89$	−16.5340	−1.75260	−0.876301	+	0.481764i	$0.839996\pi$
$90$	0	0
$91$	14.6452	1.53523
$92$	−0.676322	−0.0705114
$93$	0	0
$94$	−7.33971	−0.757034
$95$	−3.76326	−0.386103
$96$	0	0
$97$	1.26612	0.128555	0.0642775	−	0.997932i	$-0.479526\pi$
$97$	1.26612	0.128555	0.0642775	+	0.997932i	$0.479526\pi$
$98$	2.93900	0.296884
$99$	0	0
$100$	−0.356256	−0.0356256
$101$	−3.05791	−0.304273	−0.152137	−	0.988359i	$-0.548615\pi$
$101$	−3.05791	−0.304273	−0.152137	+	0.988359i	$0.548615\pi$
$102$	0	0
$103$	−10.4587	−1.03052	−0.515262	−	0.857033i	$-0.672305\pi$
$103$	−10.4587	−1.03052	−0.515262	+	0.857033i	$0.672305\pi$
$104$	−14.5135	−1.42316
$105$	0	0
$106$	11.1177	1.07985
$107$	4.83547	0.467462	0.233731	−	0.972301i	$-0.424906\pi$
$107$	4.83547	0.467462	0.233731	+	0.972301i	$0.424906\pi$
$108$	0	0
$109$	−11.0923	−1.06245	−0.531224	−	0.847232i	$-0.678268\pi$
$109$	−11.0923	−1.06245	−0.531224	+	0.847232i	$0.678268\pi$
$110$	3.61101	0.344296
$111$	0	0
$112$	−9.63448	−0.910373
$113$	−16.0956	−1.51414	−0.757072	−	0.653331i	$-0.773371\pi$
$113$	−16.0956	−1.51414	−0.757072	+	0.653331i	$0.773371\pi$
$114$	0	0
$115$	1.89841	0.177028
$116$	0.356256	0.0330775
$117$	0	0
$118$	2.68812	0.247461
$119$	−8.54849	−0.783639
$120$	0	0
$121$	−3.06727	−0.278843
$122$	16.4285	1.48736
$123$	0	0
$124$	−2.76571	−0.248368
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.73388	0.242593	0.121296	−	0.992616i	$-0.461295\pi$
$127$	2.73388	0.242593	0.121296	+	0.992616i	$0.461295\pi$
$128$	7.39539	0.653666
$129$	0	0
$130$	6.15954	0.540228
$131$	−15.2833	−1.33530	−0.667652	−	0.744474i	$-0.732701\pi$
$131$	−15.2833	−1.33530	−0.667652	+	0.744474i	$0.732701\pi$
$132$	0	0
$133$	−11.4717	−0.994722
$134$	−5.35486	−0.462590
$135$	0	0
$136$	8.47161	0.726435
$137$	−7.22502	−0.617275	−0.308637	−	0.951180i	$-0.599873\pi$
$137$	−7.22502	−0.617275	−0.308637	+	0.951180i	$0.599873\pi$
$138$	0	0
$139$	22.0819	1.87296	0.936481	−	0.350717i	$-0.114062\pi$
$139$	22.0819	1.87296	0.936481	+	0.350717i	$0.114062\pi$
$140$	−1.08599	−0.0917828
$141$	0	0
$142$	−14.7780	−1.24014
$143$	13.5314	1.13155
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	16.1346	1.33531
$147$	0	0
$148$	2.24984	0.184936
$149$	4.19335	0.343533	0.171766	−	0.985138i	$-0.445053\pi$
$149$	4.19335	0.343533	0.171766	+	0.985138i	$0.445053\pi$
$150$	0	0
$151$	8.14710	0.663002	0.331501	−	0.943455i	$-0.392445\pi$
$151$	8.14710	0.663002	0.331501	+	0.943455i	$0.392445\pi$
$152$	11.3685	0.922109
$153$	0	0
$154$	11.0076	0.887015
$155$	7.76326	0.623560
$156$	0	0
$157$	−22.5435	−1.79917	−0.899585	−	0.436745i	$-0.856131\pi$
$157$	−22.5435	−1.79917	−0.899585	+	0.436745i	$0.856131\pi$
$158$	2.39719	0.190710
$159$	0	0
$160$	1.98972	0.157301
$161$	5.78701	0.456080
$162$	0	0
$163$	4.27339	0.334718	0.167359	−	0.985896i	$-0.446476\pi$
$163$	4.27339	0.334718	0.167359	+	0.985896i	$0.446476\pi$
$164$	−3.01520	−0.235448
$165$	0	0
$166$	−18.6524	−1.44770
$167$	7.46121	0.577366	0.288683	−	0.957425i	$-0.406783\pi$
$167$	7.46121	0.577366	0.288683	+	0.957425i	$0.406783\pi$
$168$	0	0
$169$	10.0814	0.775494
$170$	−3.59537	−0.275752
$171$	0	0
$172$	−2.98076	−0.227281
$173$	−19.6934	−1.49726	−0.748632	−	0.662986i	$-0.769289\pi$
$173$	−19.6934	−1.49726	−0.748632	+	0.662986i	$0.769289\pi$
$174$	0	0
$175$	3.04834	0.230433
$176$	−8.90178	−0.670997
$177$	0	0
$178$	−21.1980	−1.58886
$179$	0.183602	0.0137231	0.00686153	−	0.999976i	$-0.497816\pi$
$179$	0.183602	0.0137231	0.00686153	+	0.999976i	$0.497816\pi$
$180$	0	0
$181$	−6.12292	−0.455113	−0.227556	−	0.973765i	$-0.573074\pi$
$181$	−6.12292	−0.455113	−0.227556	+	0.973765i	$0.573074\pi$
$182$	18.7764	1.39180
$183$	0	0
$184$	−5.73496	−0.422787
$185$	−6.31524	−0.464305
$186$	0	0
$187$	−7.89838	−0.577587
$188$	2.03950	0.148746
$189$	0	0
$190$	−4.82483	−0.350030
$191$	−4.31671	−0.312346	−0.156173	−	0.987730i	$-0.549916\pi$
$191$	−4.31671	−0.312346	−0.156173	+	0.987730i	$0.549916\pi$
$192$	0	0
$193$	7.37818	0.531093	0.265547	−	0.964098i	$-0.414448\pi$
$193$	7.37818	0.531093	0.265547	+	0.964098i	$0.414448\pi$
$194$	1.62327	0.116544
$195$	0	0
$196$	−0.816668	−0.0583334
$197$	−14.3327	−1.02116	−0.510581	−	0.859830i	$-0.670570\pi$
$197$	−14.3327	−1.02116	−0.510581	+	0.859830i	$0.670570\pi$
$198$	0	0
$199$	−13.3001	−0.942820	−0.471410	−	0.881914i	$-0.656255\pi$
$199$	−13.3001	−0.942820	−0.471410	+	0.881914i	$0.656255\pi$
$200$	−3.02092	−0.213611
$201$	0	0
$202$	−3.92050	−0.275846
$203$	−3.04834	−0.213951
$204$	0	0
$205$	8.46358	0.591122
$206$	−13.4089	−0.934243
$207$	0	0
$208$	−15.1844	−1.05285
$209$	−10.5993	−0.733167
$210$	0	0
$211$	−20.6904	−1.42439	−0.712194	−	0.701983i	$-0.752298\pi$
$211$	−20.6904	−1.42439	−0.712194	+	0.701983i	$0.752298\pi$
$212$	−3.08930	−0.212174
$213$	0	0
$214$	6.19948	0.423788
$215$	8.36690	0.570618
$216$	0	0
$217$	23.6650	1.60649
$218$	−14.2213	−0.963185
$219$	0	0
$220$	−1.00340	−0.0676492
$221$	−13.4728	−0.906279
$222$	0	0
$223$	6.36488	0.426224	0.213112	−	0.977028i	$-0.431640\pi$
$223$	6.36488	0.426224	0.213112	+	0.977028i	$0.431640\pi$
$224$	6.06535	0.405258
$225$	0	0
$226$	−20.6359	−1.37268
$227$	6.48836	0.430647	0.215324	−	0.976543i	$-0.430919\pi$
$227$	6.48836	0.430647	0.215324	+	0.976543i	$0.430919\pi$
$228$	0	0
$229$	−15.3481	−1.01423	−0.507117	−	0.861877i	$-0.669289\pi$
$229$	−15.3481	−1.01423	−0.507117	+	0.861877i	$0.669289\pi$
$230$	2.43393	0.160489
$231$	0	0
$232$	3.02092	0.198333
$233$	−1.44656	−0.0947671	−0.0473835	−	0.998877i	$-0.515088\pi$
$233$	−1.44656	−0.0947671	−0.0473835	+	0.998877i	$0.515088\pi$
$234$	0	0
$235$	−5.72482	−0.373446
$236$	−0.746953	−0.0486225
$237$	0	0
$238$	−10.9599	−0.710425
$239$	13.5463	0.876239	0.438119	−	0.898917i	$-0.355645\pi$
$239$	13.5463	0.876239	0.438119	+	0.898917i	$0.355645\pi$
$240$	0	0
$241$	15.8692	1.02223	0.511114	−	0.859513i	$-0.329233\pi$
$241$	15.8692	1.02223	0.511114	+	0.859513i	$0.329233\pi$
$242$	−3.93250	−0.252791
$243$	0	0
$244$	−4.56501	−0.292245
$245$	2.29236	0.146454
$246$	0	0
$247$	−18.0799	−1.15040
$248$	−23.4522	−1.48922
$249$	0	0
$250$	1.28209	0.0810862
$251$	−12.3597	−0.780140	−0.390070	−	0.920785i	$-0.627549\pi$
$251$	−12.3597	−0.780140	−0.390070	+	0.920785i	$0.627549\pi$
$252$	0	0
$253$	5.34690	0.336157
$254$	3.50507	0.219928
$255$	0	0
$256$	−8.26275	−0.516422
$257$	15.9491	0.994876	0.497438	−	0.867499i	$-0.334274\pi$
$257$	15.9491	0.994876	0.497438	+	0.867499i	$0.334274\pi$
$258$	0	0
$259$	−19.2510	−1.19620
$260$	−1.71157	−0.106147
$261$	0	0
$262$	−19.5944	−1.21055
$263$	−6.58678	−0.406158	−0.203079	−	0.979162i	$-0.565095\pi$
$263$	−6.58678	−0.406158	−0.203079	+	0.979162i	$0.565095\pi$
$264$	0	0
$265$	8.67157	0.532691
$266$	−14.7077	−0.901787
$267$	0	0
$268$	1.48797	0.0908921
$269$	5.72949	0.349333	0.174666	−	0.984628i	$-0.444115\pi$
$269$	5.72949	0.349333	0.174666	+	0.984628i	$0.444115\pi$
$270$	0	0
$271$	12.1264	0.736629	0.368315	−	0.929701i	$-0.379935\pi$
$271$	12.1264	0.736629	0.368315	+	0.929701i	$0.379935\pi$
$272$	8.86323	0.537412
$273$	0	0
$274$	−9.26309	−0.559604
$275$	2.81651	0.169842
$276$	0	0
$277$	4.75183	0.285510	0.142755	−	0.989758i	$-0.454404\pi$
$277$	4.75183	0.285510	0.142755	+	0.989758i	$0.454404\pi$
$278$	28.3109	1.69797
$279$	0	0
$280$	−9.20879	−0.550331
$281$	−5.31853	−0.317277	−0.158638	−	0.987337i	$-0.550710\pi$
$281$	−5.31853	−0.317277	−0.158638	+	0.987337i	$0.550710\pi$
$282$	0	0
$283$	29.9181	1.77844	0.889222	−	0.457476i	$-0.151246\pi$
$283$	29.9181	1.77844	0.889222	+	0.457476i	$0.151246\pi$
$284$	4.10640	0.243670
$285$	0	0
$286$	17.3484	1.02583
$287$	25.7998	1.52292
$288$	0	0
$289$	−9.13583	−0.537402
$290$	−1.28209	−0.0752867
$291$	0	0
$292$	−4.48336	−0.262369
$293$	−14.3954	−0.840989	−0.420495	−	0.907295i	$-0.638143\pi$
$293$	−14.3954	−0.840989	−0.420495	+	0.907295i	$0.638143\pi$
$294$	0	0
$295$	2.09668	0.122073
$296$	19.0778	1.10888
$297$	0	0
$298$	5.37623	0.311437
$299$	9.12058	0.527457
$300$	0	0
$301$	25.5051	1.47009
$302$	10.4453	0.601058
$303$	0	0
$304$	11.8941	0.682171
$305$	12.8139	0.733719
$306$	0	0
$307$	6.92723	0.395358	0.197679	−	0.980267i	$-0.436660\pi$
$307$	6.92723	0.395358	0.197679	+	0.980267i	$0.436660\pi$
$308$	−3.05870	−0.174286
$309$	0	0
$310$	9.95317	0.565302
$311$	−30.3284	−1.71977	−0.859883	−	0.510492i	$-0.829463\pi$
$311$	−30.3284	−1.71977	−0.859883	+	0.510492i	$0.829463\pi$
$312$	0	0
$313$	0.579687	0.0327658	0.0163829	−	0.999866i	$-0.494785\pi$
$313$	0.579687	0.0327658	0.0163829	+	0.999866i	$0.494785\pi$
$314$	−28.9028	−1.63108
$315$	0	0
$316$	−0.666113	−0.0374718
$317$	32.7502	1.83943	0.919716	−	0.392585i	$-0.128419\pi$
$317$	32.7502	1.83943	0.919716	+	0.392585i	$0.128419\pi$
$318$	0	0
$319$	−2.81651	−0.157694
$320$	8.87214	0.495967
$321$	0	0
$322$	7.41944	0.413469
$323$	10.5534	0.587205
$324$	0	0
$325$	4.80431	0.266495
$326$	5.47885	0.303446
$327$	0	0
$328$	−25.5678	−1.41175
$329$	−17.4512	−0.962115
$330$	0	0
$331$	−23.6139	−1.29794	−0.648969	−	0.760814i	$-0.724800\pi$
$331$	−23.6139	−1.29794	−0.648969	+	0.760814i	$0.724800\pi$
$332$	5.18298	0.284453
$333$	0	0
$334$	9.56592	0.523424
$335$	−4.17668	−0.228196
$336$	0	0
$337$	−14.1181	−0.769064	−0.384532	−	0.923112i	$-0.625637\pi$
$337$	−14.1181	−0.769064	−0.384532	+	0.923112i	$0.625637\pi$
$338$	12.9253	0.703041
$339$	0	0
$340$	0.999054	0.0541813
$341$	21.8653	1.18407
$342$	0	0
$343$	−14.3505	−0.774853
$344$	−25.2758	−1.36278
$345$	0	0
$346$	−25.2487	−1.35738
$347$	32.6549	1.75301	0.876503	−	0.481397i	$-0.159870\pi$
$347$	32.6549	1.75301	0.876503	+	0.481397i	$0.159870\pi$
$348$	0	0
$349$	−8.18813	−0.438300	−0.219150	−	0.975691i	$-0.570328\pi$
$349$	−8.18813	−0.438300	−0.219150	+	0.975691i	$0.570328\pi$
$350$	3.90823	0.208904
$351$	0	0
$352$	5.60408	0.298698
$353$	24.4423	1.30093	0.650465	−	0.759536i	$-0.274574\pi$
$353$	24.4423	1.30093	0.650465	+	0.759536i	$0.274574\pi$
$354$	0	0
$355$	−11.5265	−0.611764
$356$	5.89034	0.312188
$357$	0	0
$358$	0.235394	0.0124409
$359$	−1.23943	−0.0654148	−0.0327074	−	0.999465i	$-0.510413\pi$
$359$	−1.23943	−0.0654148	−0.0327074	+	0.999465i	$0.510413\pi$
$360$	0	0
$361$	−4.83785	−0.254624
$362$	−7.85011	−0.412592
$363$	0	0
$364$	−5.21743	−0.273468
$365$	12.5846	0.658711
$366$	0	0
$367$	−7.35985	−0.384181	−0.192091	−	0.981377i	$-0.561527\pi$
$367$	−7.35985	−0.384181	−0.192091	+	0.981377i	$0.561527\pi$
$368$	−6.00007	−0.312775
$369$	0	0
$370$	−8.09668	−0.420926
$371$	26.4339	1.37238
$372$	0	0
$373$	−34.0997	−1.76561	−0.882807	−	0.469735i	$-0.844349\pi$
$373$	−34.0997	−1.76561	−0.882807	+	0.469735i	$0.844349\pi$
$374$	−10.1264	−0.523624
$375$	0	0
$376$	17.2942	0.891883
$377$	−4.80431	−0.247435
$378$	0	0
$379$	−29.5074	−1.51570	−0.757848	−	0.652432i	$-0.773749\pi$
$379$	−29.5074	−1.51570	−0.757848	+	0.652432i	$0.773749\pi$
$380$	1.34069	0.0687757
$381$	0	0
$382$	−5.53439	−0.283164
$383$	12.4518	0.636256	0.318128	−	0.948048i	$-0.396946\pi$
$383$	12.4518	0.636256	0.318128	+	0.948048i	$0.396946\pi$
$384$	0	0
$385$	8.58568	0.437567
$386$	9.45947	0.481474
$387$	0	0
$388$	−0.451063	−0.0228992
$389$	−19.9464	−1.01132	−0.505660	−	0.862733i	$-0.668751\pi$
$389$	−19.9464	−1.01132	−0.505660	+	0.862733i	$0.668751\pi$
$390$	0	0
$391$	−5.32375	−0.269234
$392$	−6.92505	−0.349768
$393$	0	0
$394$	−18.3757	−0.925756
$395$	1.86976	0.0940778
$396$	0	0
$397$	9.76342	0.490012	0.245006	−	0.969522i	$-0.421210\pi$
$397$	9.76342	0.490012	0.245006	+	0.969522i	$0.421210\pi$
$398$	−17.0519	−0.854734
$399$	0	0
$400$	−3.16057	−0.158028
$401$	17.3956	0.868697	0.434348	−	0.900745i	$-0.356979\pi$
$401$	17.3956	0.868697	0.434348	+	0.900745i	$0.356979\pi$
$402$	0	0
$403$	37.2971	1.85790
$404$	1.08940	0.0541996
$405$	0	0
$406$	−3.90823	−0.193962
$407$	−17.7869	−0.881665
$408$	0	0
$409$	7.31195	0.361552	0.180776	−	0.983524i	$-0.442139\pi$
$409$	7.31195	0.361552	0.180776	+	0.983524i	$0.442139\pi$
$410$	10.8510	0.535894
$411$	0	0
$412$	3.72596	0.183565
$413$	6.39137	0.314499
$414$	0	0
$415$	−14.5485	−0.714156
$416$	9.55926	0.468681
$417$	0	0
$418$	−13.5892	−0.664668
$419$	19.6377	0.959367	0.479683	−	0.877442i	$-0.340752\pi$
$419$	19.6377	0.959367	0.479683	+	0.877442i	$0.340752\pi$
$420$	0	0
$421$	−3.10395	−0.151277	−0.0756386	−	0.997135i	$-0.524100\pi$
$421$	−3.10395	−0.151277	−0.0756386	+	0.997135i	$0.524100\pi$
$422$	−26.5269	−1.29131
$423$	0	0
$424$	−26.1962	−1.27220
$425$	−2.80431	−0.136029
$426$	0	0
$427$	39.0609	1.89029
$428$	−1.72266	−0.0832681
$429$	0	0
$430$	10.7271	0.517306
$431$	−10.7572	−0.518155	−0.259077	−	0.965857i	$-0.583418\pi$
$431$	−10.7572	−0.518155	−0.259077	+	0.965857i	$0.583418\pi$
$432$	0	0
$433$	−32.3805	−1.55611	−0.778053	−	0.628198i	$-0.783793\pi$
$433$	−32.3805	−1.55611	−0.778053	+	0.628198i	$0.783793\pi$
$434$	30.3406	1.45640
$435$	0	0
$436$	3.95169	0.189252
$437$	−7.14423	−0.341755
$438$	0	0
$439$	26.1669	1.24888	0.624438	−	0.781074i	$-0.285328\pi$
$439$	26.1669	1.24888	0.624438	+	0.781074i	$0.285328\pi$
$440$	−8.50846	−0.405625
$441$	0	0
$442$	−17.2733	−0.821606
$443$	−4.50747	−0.214156	−0.107078	−	0.994251i	$-0.534150\pi$
$443$	−4.50747	−0.214156	−0.107078	+	0.994251i	$0.534150\pi$
$444$	0	0
$445$	−16.5340	−0.783787
$446$	8.16033	0.386403
$447$	0	0
$448$	27.0453	1.27777
$449$	−14.1591	−0.668210	−0.334105	−	0.942536i	$-0.608434\pi$
$449$	−14.1591	−0.668210	−0.334105	+	0.942536i	$0.608434\pi$
$450$	0	0
$451$	23.8378	1.12248
$452$	5.73415	0.269712
$453$	0	0
$454$	8.31863	0.390413
$455$	14.6452	0.686576
$456$	0	0
$457$	24.9467	1.16696	0.583480	−	0.812128i	$-0.301691\pi$
$457$	24.9467	1.16696	0.583480	+	0.812128i	$0.301691\pi$
$458$	−19.6776	−0.919476
$459$	0	0
$460$	−0.676322	−0.0315337
$461$	−9.42297	−0.438871	−0.219436	−	0.975627i	$-0.570422\pi$
$461$	−9.42297	−0.438871	−0.219436	+	0.975627i	$0.570422\pi$
$462$	0	0
$463$	1.83425	0.0852448	0.0426224	−	0.999091i	$-0.486429\pi$
$463$	1.83425	0.0852448	0.0426224	+	0.999091i	$0.486429\pi$
$464$	3.16057	0.146726
$465$	0	0
$466$	−1.85461	−0.0859131
$467$	36.6207	1.69460	0.847301	−	0.531113i	$-0.178226\pi$
$467$	36.6207	1.69460	0.847301	+	0.531113i	$0.178226\pi$
$468$	0	0
$469$	−12.7319	−0.587906
$470$	−7.33971	−0.338556
$471$	0	0
$472$	−6.33389	−0.291541
$473$	23.5655	1.08354
$474$	0	0
$475$	−3.76326	−0.172670
$476$	3.04545	0.139588
$477$	0	0
$478$	17.3675	0.794373
$479$	−42.4972	−1.94175	−0.970874	−	0.239592i	$-0.922986\pi$
$479$	−42.4972	−1.94175	−0.970874	+	0.239592i	$0.922986\pi$
$480$	0	0
$481$	−30.3404	−1.38340
$482$	20.3457	0.926723
$483$	0	0
$484$	1.09273	0.0496697
$485$	1.26612	0.0574915
$486$	0	0
$487$	25.6410	1.16191	0.580953	−	0.813937i	$-0.302680\pi$
$487$	25.6410	1.16191	0.580953	+	0.813937i	$0.302680\pi$
$488$	−38.7096	−1.75230
$489$	0	0
$490$	2.93900	0.132771
$491$	−22.4285	−1.01218	−0.506092	−	0.862480i	$-0.668910\pi$
$491$	−22.4285	−1.01218	−0.506092	+	0.862480i	$0.668910\pi$
$492$	0	0
$493$	2.80431	0.126300
$494$	−23.1800	−1.04292
$495$	0	0
$496$	−24.5363	−1.10171
$497$	−35.1367	−1.57610
$498$	0	0
$499$	34.3650	1.53839	0.769195	−	0.639014i	$-0.220657\pi$
$499$	34.3650	1.53839	0.769195	+	0.639014i	$0.220657\pi$
$500$	−0.356256	−0.0159323
$501$	0	0
$502$	−15.8462	−0.707253
$503$	9.91136	0.441926	0.220963	−	0.975282i	$-0.429080\pi$
$503$	9.91136	0.441926	0.220963	+	0.975282i	$0.429080\pi$
$504$	0	0
$505$	−3.05791	−0.136075
$506$	6.85519	0.304750
$507$	0	0
$508$	−0.973962	−0.0432126
$509$	−38.8937	−1.72393	−0.861967	−	0.506964i	$-0.830768\pi$
$509$	−38.8937	−1.72393	−0.861967	+	0.506964i	$0.830768\pi$
$510$	0	0
$511$	38.3623	1.69705
$512$	−25.3843	−1.12184
$513$	0	0
$514$	20.4481	0.901926
$515$	−10.4587	−0.460864
$516$	0	0
$517$	−16.1240	−0.709134
$518$	−24.6814	−1.08444
$519$	0	0
$520$	−14.5135	−0.636457
$521$	21.0434	0.921928	0.460964	−	0.887419i	$-0.347504\pi$
$521$	21.0434	0.921928	0.460964	+	0.887419i	$0.347504\pi$
$522$	0	0
$523$	13.4413	0.587746	0.293873	−	0.955845i	$-0.405056\pi$
$523$	13.4413	0.587746	0.293873	+	0.955845i	$0.405056\pi$
$524$	5.44475	0.237855
$525$	0	0
$526$	−8.44482	−0.368211
$527$	−21.7706	−0.948343
$528$	0	0
$529$	−19.3960	−0.843305
$530$	11.1177	0.482922
$531$	0	0
$532$	4.08686	0.177188
$533$	40.6617	1.76125
$534$	0	0
$535$	4.83547	0.209055
$536$	12.6174	0.544990
$537$	0	0
$538$	7.34569	0.316695
$539$	6.45646	0.278099
$540$	0	0
$541$	20.2537	0.870777	0.435388	−	0.900243i	$-0.356611\pi$
$541$	20.2537	0.870777	0.435388	+	0.900243i	$0.356611\pi$
$542$	15.5471	0.667807
$543$	0	0
$544$	−5.57981	−0.239232
$545$	−11.0923	−0.475141
$546$	0	0
$547$	9.62331	0.411463	0.205732	−	0.978608i	$-0.434043\pi$
$547$	9.62331	0.411463	0.205732	+	0.978608i	$0.434043\pi$
$548$	2.57396	0.109954
$549$	0	0
$550$	3.61101	0.153974
$551$	3.76326	0.160320
$552$	0	0
$553$	5.69966	0.242374
$554$	6.09225	0.258835
$555$	0	0
$556$	−7.86681	−0.333627
$557$	25.3720	1.07504	0.537522	−	0.843249i	$-0.319360\pi$
$557$	25.3720	1.07504	0.537522	+	0.843249i	$0.319360\pi$
$558$	0	0
$559$	40.1972	1.70016
$560$	−9.63448	−0.407131
$561$	0	0
$562$	−6.81881	−0.287634
$563$	−23.2414	−0.979509	−0.489755	−	0.871860i	$-0.662914\pi$
$563$	−23.2414	−0.979509	−0.489755	+	0.871860i	$0.662914\pi$
$564$	0	0
$565$	−16.0956	−0.677146
$566$	38.3575	1.61229
$567$	0	0
$568$	34.8207	1.46105
$569$	−6.19363	−0.259651	−0.129825	−	0.991537i	$-0.541442\pi$
$569$	−6.19363	−0.259651	−0.129825	+	0.991537i	$0.541442\pi$
$570$	0	0
$571$	32.5435	1.36190	0.680951	−	0.732329i	$-0.261567\pi$
$571$	32.5435	1.36190	0.680951	+	0.732329i	$0.261567\pi$
$572$	−4.82064	−0.201561
$573$	0	0
$574$	33.0776	1.38063
$575$	1.89841	0.0791694
$576$	0	0
$577$	12.3229	0.513010	0.256505	−	0.966543i	$-0.417429\pi$
$577$	12.3229	0.513010	0.256505	+	0.966543i	$0.417429\pi$
$578$	−11.7129	−0.487193
$579$	0	0
$580$	0.356256	0.0147927
$581$	−44.3486	−1.83989
$582$	0	0
$583$	24.4236	1.01152
$584$	−38.0172	−1.57316
$585$	0	0
$586$	−18.4562	−0.762417
$587$	−9.91125	−0.409081	−0.204541	−	0.978858i	$-0.565570\pi$
$587$	−9.91125	−0.409081	−0.204541	+	0.978858i	$0.565570\pi$
$588$	0	0
$589$	−29.2152	−1.20379
$590$	2.68812	0.110668
$591$	0	0
$592$	19.9597	0.820340
$593$	4.59660	0.188760	0.0943799	−	0.995536i	$-0.469913\pi$
$593$	4.59660	0.188760	0.0943799	+	0.995536i	$0.469913\pi$
$594$	0	0
$595$	−8.54849	−0.350454
$596$	−1.49391	−0.0611928
$597$	0	0
$598$	11.6934	0.478177
$599$	5.96837	0.243861	0.121930	−	0.992539i	$-0.461092\pi$
$599$	5.96837	0.243861	0.121930	+	0.992539i	$0.461092\pi$
$600$	0	0
$601$	−20.5147	−0.836810	−0.418405	−	0.908261i	$-0.637411\pi$
$601$	−20.5147	−0.836810	−0.418405	+	0.908261i	$0.637411\pi$
$602$	32.6998	1.33274
$603$	0	0
$604$	−2.90245	−0.118099
$605$	−3.06727	−0.124702
$606$	0	0
$607$	22.4002	0.909195	0.454597	−	0.890697i	$-0.349783\pi$
$607$	22.4002	0.909195	0.454597	+	0.890697i	$0.349783\pi$
$608$	−7.48785	−0.303673
$609$	0	0
$610$	16.4285	0.665169
$611$	−27.5038	−1.11269
$612$	0	0
$613$	−13.0623	−0.527580	−0.263790	−	0.964580i	$-0.584973\pi$
$613$	−13.0623	−0.527580	−0.263790	+	0.964580i	$0.584973\pi$
$614$	8.88130	0.358420
$615$	0	0
$616$	−25.9367	−1.04502
$617$	−36.4129	−1.46593	−0.732965	−	0.680266i	$-0.761864\pi$
$617$	−36.4129	−1.46593	−0.732965	+	0.680266i	$0.761864\pi$
$618$	0	0
$619$	19.1399	0.769298	0.384649	−	0.923063i	$-0.374323\pi$
$619$	19.1399	0.769298	0.384649	+	0.923063i	$0.374323\pi$
$620$	−2.76571	−0.111074
$621$	0	0
$622$	−38.8836	−1.55909
$623$	−50.4012	−2.01928
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.743208	0.0297046
$627$	0	0
$628$	8.03128	0.320483
$629$	17.7099	0.706140
$630$	0	0
$631$	−10.4468	−0.415879	−0.207939	−	0.978142i	$-0.566676\pi$
$631$	−10.4468	−0.415879	−0.207939	+	0.978142i	$0.566676\pi$
$632$	−5.64840	−0.224681
$633$	0	0
$634$	41.9885	1.66758
$635$	2.73388	0.108491
$636$	0	0
$637$	11.0132	0.436360
$638$	−3.61101	−0.142961
$639$	0	0
$640$	7.39539	0.292328
$641$	19.4186	0.766988	0.383494	−	0.923543i	$-0.374721\pi$
$641$	19.4186	0.766988	0.383494	+	0.923543i	$0.374721\pi$
$642$	0	0
$643$	−12.2156	−0.481738	−0.240869	−	0.970558i	$-0.577432\pi$
$643$	−12.2156	−0.481738	−0.240869	+	0.970558i	$0.577432\pi$
$644$	−2.06166	−0.0812407
$645$	0	0
$646$	13.5303	0.532343
$647$	12.6785	0.498445	0.249223	−	0.968446i	$-0.419825\pi$
$647$	12.6785	0.498445	0.249223	+	0.968446i	$0.419825\pi$
$648$	0	0
$649$	5.90531	0.231804
$650$	6.15954	0.241597
$651$	0	0
$652$	−1.52242	−0.0596226
$653$	39.7406	1.55517	0.777585	−	0.628777i	$-0.216444\pi$
$653$	39.7406	1.55517	0.777585	+	0.628777i	$0.216444\pi$
$654$	0	0
$655$	−15.2833	−0.597166
$656$	−26.7497	−1.04440
$657$	0	0
$658$	−22.3739	−0.872226
$659$	16.8212	0.655259	0.327630	−	0.944806i	$-0.393750\pi$
$659$	16.8212	0.655259	0.327630	+	0.944806i	$0.393750\pi$
$660$	0	0
$661$	−32.0658	−1.24721	−0.623607	−	0.781738i	$-0.714333\pi$
$661$	−32.0658	−1.24721	−0.623607	+	0.781738i	$0.714333\pi$
$662$	−30.2751	−1.17667
$663$	0	0
$664$	43.9498	1.70558
$665$	−11.4717	−0.444853
$666$	0	0
$667$	−1.89841	−0.0735069
$668$	−2.65810	−0.102845
$669$	0	0
$670$	−5.35486	−0.206876
$671$	36.0903	1.39325
$672$	0	0
$673$	33.9106	1.30716	0.653578	−	0.756859i	$-0.273267\pi$
$673$	33.9106	1.30716	0.653578	+	0.756859i	$0.273267\pi$
$674$	−18.1007	−0.697211
$675$	0	0
$676$	−3.59157	−0.138137
$677$	21.8353	0.839199	0.419600	−	0.907709i	$-0.362170\pi$
$677$	21.8353	0.839199	0.419600	+	0.907709i	$0.362170\pi$
$678$	0	0
$679$	3.85956	0.148116
$680$	8.47161	0.324872
$681$	0	0
$682$	28.0332	1.07345
$683$	51.5851	1.97385	0.986925	−	0.161183i	$-0.0515309\pi$
$683$	51.5851	1.97385	0.986925	+	0.161183i	$0.0515309\pi$
$684$	0	0
$685$	−7.22502	−0.276054
$686$	−18.3985	−0.702459
$687$	0	0
$688$	−26.4442	−1.00817
$689$	41.6610	1.58716
$690$	0	0
$691$	−33.7654	−1.28449	−0.642247	−	0.766498i	$-0.721998\pi$
$691$	−33.7654	−1.28449	−0.642247	+	0.766498i	$0.721998\pi$
$692$	7.01590	0.266705
$693$	0	0
$694$	41.8664	1.58923
$695$	22.0819	0.837614
$696$	0	0
$697$	−23.7345	−0.899009
$698$	−10.4979	−0.397351
$699$	0	0
$700$	−1.08599	−0.0410465
$701$	−4.56010	−0.172233	−0.0861164	−	0.996285i	$-0.527446\pi$
$701$	−4.56010	−0.172233	−0.0861164	+	0.996285i	$0.527446\pi$
$702$	0	0
$703$	23.7659	0.896348
$704$	24.9885	0.941788
$705$	0	0
$706$	31.3371	1.17939
$707$	−9.32154	−0.350573
$708$	0	0
$709$	−21.3972	−0.803587	−0.401794	−	0.915730i	$-0.631613\pi$
$709$	−21.3972	−0.803587	−0.401794	+	0.915730i	$0.631613\pi$
$710$	−14.7780	−0.554608
$711$	0	0
$712$	49.9480	1.87188
$713$	14.7379	0.551938
$714$	0	0
$715$	13.5314	0.506046
$716$	−0.0654093	−0.00244446
$717$	0	0
$718$	−1.58906	−0.0593032
$719$	23.6194	0.880855	0.440428	−	0.897788i	$-0.354827\pi$
$719$	23.6194	0.880855	0.440428	+	0.897788i	$0.354827\pi$
$720$	0	0
$721$	−31.8815	−1.18733
$722$	−6.20254	−0.230835
$723$	0	0
$724$	2.18133	0.0810684
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−4.17101	−0.154694	−0.0773471	−	0.997004i	$-0.524645\pi$
$727$	−4.17101	−0.154694	−0.0773471	+	0.997004i	$0.524645\pi$
$728$	−44.2419	−1.63971
$729$	0	0
$730$	16.1346	0.597168
$731$	−23.4634	−0.867826
$732$	0	0
$733$	−10.0875	−0.372591	−0.186296	−	0.982494i	$-0.559648\pi$
$733$	−10.0875	−0.372591	−0.186296	+	0.982494i	$0.559648\pi$
$734$	−9.43596	−0.348288
$735$	0	0
$736$	3.77732	0.139234
$737$	−11.7637	−0.433320
$738$	0	0
$739$	0.903562	0.0332381	0.0166190	−	0.999862i	$-0.494710\pi$
$739$	0.903562	0.0332381	0.0166190	+	0.999862i	$0.494710\pi$
$740$	2.24984	0.0827058
$741$	0	0
$742$	33.8905	1.24416
$743$	−14.5745	−0.534688	−0.267344	−	0.963601i	$-0.586146\pi$
$743$	−14.5745	−0.534688	−0.267344	+	0.963601i	$0.586146\pi$
$744$	0	0
$745$	4.19335	0.153632
$746$	−43.7187	−1.60066
$747$	0	0
$748$	2.81385	0.102884
$749$	14.7401	0.538593
$750$	0	0
$751$	−12.3037	−0.448969	−0.224485	−	0.974478i	$-0.572070\pi$
$751$	−12.3037	−0.448969	−0.224485	+	0.974478i	$0.572070\pi$
$752$	18.0937	0.659809
$753$	0	0
$754$	−6.15954	−0.224317
$755$	8.14710	0.296503
$756$	0	0
$757$	52.5511	1.91000	0.955001	−	0.296603i	$-0.0958539\pi$
$757$	52.5511	1.91000	0.955001	+	0.296603i	$0.0958539\pi$
$758$	−37.8311	−1.37409
$759$	0	0
$760$	11.3685	0.412380
$761$	−17.2173	−0.624125	−0.312062	−	0.950062i	$-0.601020\pi$
$761$	−17.2173	−0.624125	−0.312062	+	0.950062i	$0.601020\pi$
$762$	0	0
$763$	−33.8130	−1.22411
$764$	1.53785	0.0556376
$765$	0	0
$766$	15.9643	0.576812
$767$	10.0731	0.363718
$768$	0	0
$769$	−53.9886	−1.94688	−0.973439	−	0.228946i	$-0.926472\pi$
$769$	−53.9886	−1.94688	−0.973439	+	0.228946i	$0.926472\pi$
$770$	11.0076	0.396685
$771$	0	0
$772$	−2.62852	−0.0946026
$773$	4.54822	0.163588	0.0817940	−	0.996649i	$-0.473935\pi$
$773$	4.54822	0.163588	0.0817940	+	0.996649i	$0.473935\pi$
$774$	0	0
$775$	7.76326	0.278865
$776$	−3.82485	−0.137304
$777$	0	0
$778$	−25.5729	−0.916834
$779$	−31.8507	−1.14117
$780$	0	0
$781$	−32.4646	−1.16167
$782$	−6.82550	−0.244079
$783$	0	0
$784$	−7.24517	−0.258756
$785$	−22.5435	−0.804614
$786$	0	0
$787$	6.17181	0.220001	0.110001	−	0.993932i	$-0.464915\pi$
$787$	6.17181	0.220001	0.110001	+	0.993932i	$0.464915\pi$
$788$	5.10611	0.181898
$789$	0	0
$790$	2.39719	0.0852882
$791$	−49.0647	−1.74454
$792$	0	0
$793$	61.5617	2.18612
$794$	12.5175	0.444231
$795$	0	0
$796$	4.73825	0.167943
$797$	−37.0901	−1.31380	−0.656900	−	0.753978i	$-0.728133\pi$
$797$	−37.0901	−1.31380	−0.656900	+	0.753978i	$0.728133\pi$
$798$	0	0
$799$	16.0542	0.567957
$800$	1.98972	0.0703474
$801$	0	0
$802$	22.3027	0.787536
$803$	35.4448	1.25082
$804$	0	0
$805$	5.78701	0.203965
$806$	47.8181	1.68432
$807$	0	0
$808$	9.23771	0.324982
$809$	−9.68286	−0.340431	−0.170216	−	0.985407i	$-0.554446\pi$
$809$	−9.68286	−0.340431	−0.170216	+	0.985407i	$0.554446\pi$
$810$	0	0
$811$	37.0679	1.30163	0.650815	−	0.759236i	$-0.274427\pi$
$811$	37.0679	1.30163	0.650815	+	0.759236i	$0.274427\pi$
$812$	1.08599	0.0381107
$813$	0	0
$814$	−22.8044	−0.799293
$815$	4.27339	0.149690
$816$	0	0
$817$	−31.4869	−1.10159
$818$	9.37454	0.327773
$819$	0	0
$820$	−3.01520	−0.105295
$821$	15.8261	0.552336	0.276168	−	0.961109i	$-0.410935\pi$
$821$	15.8261	0.552336	0.276168	+	0.961109i	$0.410935\pi$
$822$	0	0
$823$	−3.74633	−0.130589	−0.0652944	−	0.997866i	$-0.520799\pi$
$823$	−3.74633	−0.130589	−0.0652944	+	0.997866i	$0.520799\pi$
$824$	31.5948	1.10066
$825$	0	0
$826$	8.19429	0.285116
$827$	12.5732	0.437212	0.218606	−	0.975813i	$-0.429849\pi$
$827$	12.5732	0.437212	0.218606	+	0.975813i	$0.429849\pi$
$828$	0	0
$829$	−22.7218	−0.789160	−0.394580	−	0.918861i	$-0.629110\pi$
$829$	−22.7218	−0.789160	−0.394580	+	0.918861i	$0.629110\pi$
$830$	−18.6524	−0.647433
$831$	0	0
$832$	42.6245	1.47774
$833$	−6.42850	−0.222734
$834$	0	0
$835$	7.46121	0.258206
$836$	3.77605	0.130598
$837$	0	0
$838$	25.1773	0.869735
$839$	−24.4758	−0.844998	−0.422499	−	0.906363i	$-0.638847\pi$
$839$	−24.4758	−0.844998	−0.422499	+	0.906363i	$0.638847\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−3.97953	−0.137144
$843$	0	0
$844$	7.37109	0.253723
$845$	10.0814	0.346812
$846$	0	0
$847$	−9.35007	−0.321272
$848$	−27.4071	−0.941164
$849$	0	0
$850$	−3.59537	−0.123320
$851$	−11.9889	−0.410975
$852$	0	0
$853$	−51.3695	−1.75886	−0.879429	−	0.476030i	$-0.842075\pi$
$853$	−51.3695	−1.75886	−0.879429	+	0.476030i	$0.842075\pi$
$854$	50.0795	1.71368
$855$	0	0
$856$	−14.6076	−0.499276
$857$	−31.9427	−1.09114	−0.545572	−	0.838064i	$-0.683687\pi$
$857$	−31.9427	−1.09114	−0.545572	+	0.838064i	$0.683687\pi$
$858$	0	0
$859$	14.6511	0.499888	0.249944	−	0.968260i	$-0.419588\pi$
$859$	14.6511	0.499888	0.249944	+	0.968260i	$0.419588\pi$
$860$	−2.98076	−0.101643
$861$	0	0
$862$	−13.7916	−0.469744
$863$	35.4628	1.20717	0.603583	−	0.797300i	$-0.293739\pi$
$863$	35.4628	1.20717	0.603583	+	0.797300i	$0.293739\pi$
$864$	0	0
$865$	−19.6934	−0.669596
$866$	−41.5146	−1.41072
$867$	0	0
$868$	−8.43082	−0.286161
$869$	5.26620	0.178643
$870$	0	0
$871$	−20.0661	−0.679913
$872$	33.5089	1.13476
$873$	0	0
$874$	−9.15952	−0.309825
$875$	3.04834	0.103053
$876$	0	0
$877$	−6.35693	−0.214658	−0.107329	−	0.994224i	$-0.534230\pi$
$877$	−6.35693	−0.214658	−0.107329	+	0.994224i	$0.534230\pi$
$878$	33.5482	1.13220
$879$	0	0
$880$	−8.90178	−0.300079
$881$	26.3799	0.888762	0.444381	−	0.895838i	$-0.353424\pi$
$881$	26.3799	0.888762	0.444381	+	0.895838i	$0.353424\pi$
$882$	0	0
$883$	22.7479	0.765528	0.382764	−	0.923846i	$-0.374972\pi$
$883$	22.7479	0.765528	0.382764	+	0.923846i	$0.374972\pi$
$884$	4.79977	0.161434
$885$	0	0
$886$	−5.77896	−0.194148
$887$	29.6259	0.994741	0.497370	−	0.867538i	$-0.334299\pi$
$887$	29.6259	0.994741	0.497370	+	0.867538i	$0.334299\pi$
$888$	0	0
$889$	8.33379	0.279506
$890$	−21.1980	−0.710559
$891$	0	0
$892$	−2.26753	−0.0759225
$893$	21.5440	0.720943
$894$	0	0
$895$	0.183602	0.00613714
$896$	22.5436	0.753130
$897$	0	0
$898$	−18.1532	−0.605780
$899$	−7.76326	−0.258919
$900$	0	0
$901$	−24.3178	−0.810144
$902$	30.5620	1.01760
$903$	0	0
$904$	48.6235	1.61719
$905$	−6.12292	−0.203533
$906$	0	0
$907$	−20.6308	−0.685034	−0.342517	−	0.939512i	$-0.611279\pi$
$907$	−20.6308	−0.685034	−0.342517	+	0.939512i	$0.611279\pi$
$908$	−2.31152	−0.0767104
$909$	0	0
$910$	18.7764	0.622430
$911$	50.8968	1.68629	0.843143	−	0.537690i	$-0.180703\pi$
$911$	50.8968	1.68629	0.843143	+	0.537690i	$0.180703\pi$
$912$	0	0
$913$	−40.9759	−1.35610
$914$	31.9839	1.05793
$915$	0	0
$916$	5.46787	0.180664
$917$	−46.5885	−1.53849
$918$	0	0
$919$	26.6896	0.880408	0.440204	−	0.897898i	$-0.354906\pi$
$919$	26.6896	0.880408	0.440204	+	0.897898i	$0.354906\pi$
$920$	−5.73496	−0.189076
$921$	0	0
$922$	−12.0810	−0.397868
$923$	−55.3770	−1.82276
$924$	0	0
$925$	−6.31524	−0.207644
$926$	2.35166	0.0772805
$927$	0	0
$928$	−1.98972	−0.0653159
$929$	58.7249	1.92670	0.963350	−	0.268247i	$-0.0864444\pi$
$929$	58.7249	1.92670	0.963350	+	0.268247i	$0.0864444\pi$
$930$	0	0
$931$	−8.62676	−0.282731
$932$	0.515345	0.0168807
$933$	0	0
$934$	46.9508	1.53628
$935$	−7.89838	−0.258305
$936$	0	0
$937$	−59.8471	−1.95512	−0.977559	−	0.210661i	$-0.932438\pi$
$937$	−59.8471	−1.95512	−0.977559	+	0.210661i	$0.932438\pi$
$938$	−16.3234	−0.532979
$939$	0	0
$940$	2.03950	0.0665213
$941$	22.5358	0.734645	0.367323	−	0.930094i	$-0.380274\pi$
$941$	22.5358	0.734645	0.367323	+	0.930094i	$0.380274\pi$
$942$	0	0
$943$	16.0674	0.523226
$944$	−6.62669	−0.215680
$945$	0	0
$946$	30.2130	0.982308
$947$	43.5409	1.41489	0.707444	−	0.706769i	$-0.249848\pi$
$947$	43.5409	1.41489	0.707444	+	0.706769i	$0.249848\pi$
$948$	0	0
$949$	60.4606	1.96263
$950$	−4.82483	−0.156538
$951$	0	0
$952$	25.8243	0.836972
$953$	12.5224	0.405641	0.202820	−	0.979216i	$-0.434989\pi$
$953$	12.5224	0.405641	0.202820	+	0.979216i	$0.434989\pi$
$954$	0	0
$955$	−4.31671	−0.139685
$956$	−4.82596	−0.156083
$957$	0	0
$958$	−54.4851	−1.76033
$959$	−22.0243	−0.711202
$960$	0	0
$961$	29.2682	0.944137
$962$	−38.8990	−1.25415
$963$	0	0
$964$	−5.65352	−0.182087
$965$	7.37818	0.237512
$966$	0	0
$967$	59.2236	1.90450	0.952251	−	0.305316i	$-0.0987620\pi$
$967$	59.2236	1.90450	0.952251	+	0.305316i	$0.0987620\pi$
$968$	9.26598	0.297820
$969$	0	0
$970$	1.62327	0.0521202
$971$	−51.2735	−1.64544	−0.822722	−	0.568444i	$-0.807546\pi$
$971$	−51.2735	−1.64544	−0.822722	+	0.568444i	$0.807546\pi$
$972$	0	0
$973$	67.3131	2.15796
$974$	32.8740	1.05335
$975$	0	0
$976$	−40.4991	−1.29634
$977$	48.6811	1.55745	0.778723	−	0.627368i	$-0.215868\pi$
$977$	48.6811	1.55745	0.778723	+	0.627368i	$0.215868\pi$
$978$	0	0
$979$	−46.5682	−1.48833
$980$	−0.816668	−0.0260875
$981$	0	0
$982$	−28.7552	−0.917617
$983$	−9.71972	−0.310011	−0.155006	−	0.987914i	$-0.549540\pi$
$983$	−9.71972	−0.310011	−0.155006	+	0.987914i	$0.549540\pi$
$984$	0	0
$985$	−14.3327	−0.456677
$986$	3.59537	0.114500
$987$	0	0
$988$	6.44107	0.204918
$989$	15.8838	0.505077
$990$	0	0
$991$	52.1919	1.65793	0.828966	−	0.559299i	$-0.188930\pi$
$991$	52.1919	1.65793	0.828966	+	0.559299i	$0.188930\pi$
$992$	15.4467	0.490435
$993$	0	0
$994$	−45.0483	−1.42885
$995$	−13.3001	−0.421642
$996$	0	0
$997$	8.91886	0.282463	0.141232	−	0.989977i	$-0.454894\pi$
$997$	8.91886	0.282463	0.141232	+	0.989977i	$0.454894\pi$
$998$	44.0589	1.39466
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.t.1.5	yes	7
3.2	odd	2		1305.2.a.s.1.3	✓	7
5.4	even	2		6525.2.a.bv.1.3		7
15.14	odd	2		6525.2.a.bw.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.a.s.1.3	✓	7	3.2	odd	2
1305.2.a.t.1.5	yes	7	1.1	even	1	trivial
6525.2.a.bv.1.3		7	5.4	even	2
6525.2.a.bw.1.5		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28209 q^{2} -0.356256 q^{4} +1.00000 q^{5} +3.04834 q^{7} -3.02092 q^{8} +1.28209 q^{10} +2.81651 q^{11} +4.80431 q^{13} +3.90823 q^{14} -3.16057 q^{16} -2.80431 q^{17} -3.76326 q^{19} -0.356256 q^{20} +3.61101 q^{22} +1.89841 q^{23} +1.00000 q^{25} +6.15954 q^{26} -1.08599 q^{28} -1.00000 q^{29} +7.76326 q^{31} +1.98972 q^{32} -3.59537 q^{34} +3.04834 q^{35} -6.31524 q^{37} -4.82483 q^{38} -3.02092 q^{40} +8.46358 q^{41} +8.36690 q^{43} -1.00340 q^{44} +2.43393 q^{46} -5.72482 q^{47} +2.29236 q^{49} +1.28209 q^{50} -1.71157 q^{52} +8.67157 q^{53} +2.81651 q^{55} -9.20879 q^{56} -1.28209 q^{58} +2.09668 q^{59} +12.8139 q^{61} +9.95317 q^{62} +8.87214 q^{64} +4.80431 q^{65} -4.17668 q^{67} +0.999054 q^{68} +3.90823 q^{70} -11.5265 q^{71} +12.5846 q^{73} -8.09668 q^{74} +1.34069 q^{76} +8.58568 q^{77} +1.86976 q^{79} -3.16057 q^{80} +10.8510 q^{82} -14.5485 q^{83} -2.80431 q^{85} +10.7271 q^{86} -8.50846 q^{88} -16.5340 q^{89} +14.6452 q^{91} -0.676322 q^{92} -7.33971 q^{94} -3.76326 q^{95} +1.26612 q^{97} +2.93900 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 13 q^{4} + 7 q^{5} + 10 q^{7} + q^{10} - 3 q^{11} + 6 q^{13} - 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} + 13 q^{20} + 9 q^{22} + 11 q^{23} + 7 q^{25} + 3 q^{26} + 25 q^{28} - 7 q^{29}+ \cdots - 20 q^{98}+O(q^{100}) \) 7 * q + q^2 + 13 * q^4 + 7 * q^5 + 10 * q^7 + q^10 - 3 * q^11 + 6 * q^13 - 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 + 13 * q^20 + 9 * q^22 + 11 * q^23 + 7 * q^25 + 3 * q^26 + 25 * q^28 - 7 * q^29 + 18 * q^31 + q^32 - q^34 + 10 * q^35 + 13 * q^37 + 12 * q^38 - 13 * q^41 + 9 * q^43 - 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^50 - q^52 + 5 * q^53 - 3 * q^55 - 30 * q^56 - q^58 - 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 + 6 * q^65 + 14 * q^67 + 27 * q^68 - 9 * q^70 - 8 * q^71 + 3 * q^73 - 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 21 * q^80 - 20 * q^82 + 17 * q^83 + 8 * q^85 + 4 * q^86 + 26 * q^88 - 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 + 10 * q^95 + 13 * q^97 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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