LMFDB - Embedded newform 5265.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 7x^{5} + 33x^{4} - 14x^{3} - 38x^{2} + 7x + 9 $ x^8 - x^7 - 10x^6 + 7x^5 + 33x^4 - 14x^3 - 38x^2 + 7x + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.633210$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.633210 q^{2} -1.59905 q^{4} +1.00000 q^{5} -1.79265 q^{7} +2.27895 q^{8} +O(q^{10})$	\(q-0.633210 q^{2} -1.59905 q^{4} +1.00000 q^{5} -1.79265 q^{7} +2.27895 q^{8} -0.633210 q^{10} -4.05578 q^{11} +1.00000 q^{13} +1.13512 q^{14} +1.75503 q^{16} +7.09008 q^{17} -1.65636 q^{19} -1.59905 q^{20} +2.56816 q^{22} -8.14713 q^{23} +1.00000 q^{25} -0.633210 q^{26} +2.86652 q^{28} +1.63302 q^{29} +4.34556 q^{31} -5.66921 q^{32} -4.48951 q^{34} -1.79265 q^{35} +0.729336 q^{37} +1.04882 q^{38} +2.27895 q^{40} +0.879250 q^{41} +5.53962 q^{43} +6.48538 q^{44} +5.15884 q^{46} -9.35239 q^{47} -3.78642 q^{49} -0.633210 q^{50} -1.59905 q^{52} +5.39044 q^{53} -4.05578 q^{55} -4.08535 q^{56} -1.03404 q^{58} +12.9024 q^{59} -5.09533 q^{61} -2.75165 q^{62} +0.0797303 q^{64} +1.00000 q^{65} +4.37066 q^{67} -11.3374 q^{68} +1.13512 q^{70} -2.80901 q^{71} -5.00513 q^{73} -0.461823 q^{74} +2.64859 q^{76} +7.27058 q^{77} +6.82293 q^{79} +1.75503 q^{80} -0.556750 q^{82} +6.51536 q^{83} +7.09008 q^{85} -3.50774 q^{86} -9.24293 q^{88} +7.59275 q^{89} -1.79265 q^{91} +13.0276 q^{92} +5.92203 q^{94} -1.65636 q^{95} -12.5443 q^{97} +2.39760 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8	$ 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - q^{10} - 9 q^{11} + 8 q^{13} + 3 q^{14} - 13 q^{16} + 6 q^{17} - 11 q^{19} + 5 q^{20} - 4 q^{22} - 3 q^{23} + 8 q^{25} - q^{26} - 13 q^{28} - 8 q^{29} - 18 q^{31} - 3 q^{32} - 9 q^{34} - 6 q^{35} - 18 q^{37} + 8 q^{38} - 6 q^{40} + 17 q^{41} - 17 q^{43} + 5 q^{44} + 3 q^{46} - 11 q^{47} - 16 q^{49} - q^{50} + 5 q^{52} + 10 q^{53} - 9 q^{55} + q^{56} - 10 q^{58} - 7 q^{59} - 21 q^{61} - 29 q^{62} - 10 q^{64} + 8 q^{65} - 13 q^{67} + 16 q^{68} + 3 q^{70} - 34 q^{71} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 18 q^{77} - 37 q^{79} - 13 q^{80} + q^{82} - 3 q^{83} + 6 q^{85} + 2 q^{86} - 19 q^{88} + 14 q^{89} - 6 q^{91} + 14 q^{92} - 44 q^{94} - 11 q^{95} - 17 q^{97} + 45 q^{98}+O(q^{100}) $ 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - q^10 - 9 * q^11 + 8 * q^13 + 3 * q^14 - 13 * q^16 + 6 * q^17 - 11 * q^19 + 5 * q^20 - 4 * q^22 - 3 * q^23 + 8 * q^25 - q^26 - 13 * q^28 - 8 * q^29 - 18 * q^31 - 3 * q^32 - 9 * q^34 - 6 * q^35 - 18 * q^37 + 8 * q^38 - 6 * q^40 + 17 * q^41 - 17 * q^43 + 5 * q^44 + 3 * q^46 - 11 * q^47 - 16 * q^49 - q^50 + 5 * q^52 + 10 * q^53 - 9 * q^55 + q^56 - 10 * q^58 - 7 * q^59 - 21 * q^61 - 29 * q^62 - 10 * q^64 + 8 * q^65 - 13 * q^67 + 16 * q^68 + 3 * q^70 - 34 * q^71 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 18 * q^77 - 37 * q^79 - 13 * q^80 + q^82 - 3 * q^83 + 6 * q^85 + 2 * q^86 - 19 * q^88 + 14 * q^89 - 6 * q^91 + 14 * q^92 - 44 * q^94 - 11 * q^95 - 17 * q^97 + 45 * q^98

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 $

$2$	−0.633210	−0.447747	−0.223874	−	0.974618i	$-0.571870\pi$
$2$	−0.633210	−0.447747	−0.223874	+	0.974618i	$0.571870\pi$
$3$	0	0
$3$	0	0
$4$	−1.59905	−0.799523
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.79265	−0.677556	−0.338778	−	0.940866i	$-0.610014\pi$
$7$	−1.79265	−0.677556	−0.338778	+	0.940866i	$0.610014\pi$
$8$	2.27895	0.805731
$9$	0	0
$10$	−0.633210	−0.200239
$11$	−4.05578	−1.22286	−0.611432	−	0.791297i	$-0.709406\pi$
$11$	−4.05578	−1.22286	−0.611432	+	0.791297i	$0.709406\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.13512	0.303374
$15$	0	0
$16$	1.75503	0.438759
$17$	7.09008	1.71960	0.859798	−	0.510634i	$-0.170589\pi$
$17$	7.09008	1.71960	0.859798	+	0.510634i	$0.170589\pi$
$18$	0	0
$19$	−1.65636	−0.379994	−0.189997	−	0.981785i	$-0.560848\pi$
$19$	−1.65636	−0.379994	−0.189997	+	0.981785i	$0.560848\pi$
$20$	−1.59905	−0.357557
$21$	0	0
$22$	2.56816	0.547534
$23$	−8.14713	−1.69879	−0.849397	−	0.527755i	$-0.823034\pi$
$23$	−8.14713	−1.69879	−0.849397	+	0.527755i	$0.823034\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.633210	−0.124183
$27$	0	0
$28$	2.86652	0.541721
$29$	1.63302	0.303244	0.151622	−	0.988439i	$-0.451550\pi$
$29$	1.63302	0.303244	0.151622	+	0.988439i	$0.451550\pi$
$30$	0	0
$31$	4.34556	0.780486	0.390243	−	0.920712i	$-0.372391\pi$
$31$	4.34556	0.780486	0.390243	+	0.920712i	$0.372391\pi$
$32$	−5.66921	−1.00218
$33$	0	0
$34$	−4.48951	−0.769944
$35$	−1.79265	−0.303012
$36$	0	0
$37$	0.729336	0.119902	0.0599510	−	0.998201i	$-0.480906\pi$
$37$	0.729336	0.119902	0.0599510	+	0.998201i	$0.480906\pi$
$38$	1.04882	0.170141
$39$	0	0
$40$	2.27895	0.360334
$41$	0.879250	0.137316	0.0686579	−	0.997640i	$-0.478128\pi$
$41$	0.879250	0.137316	0.0686579	+	0.997640i	$0.478128\pi$
$42$	0	0
$43$	5.53962	0.844784	0.422392	−	0.906413i	$-0.361191\pi$
$43$	5.53962	0.844784	0.422392	+	0.906413i	$0.361191\pi$
$44$	6.48538	0.977708
$45$	0	0
$46$	5.15884	0.760630
$47$	−9.35239	−1.36419	−0.682093	−	0.731265i	$-0.738930\pi$
$47$	−9.35239	−1.36419	−0.682093	+	0.731265i	$0.738930\pi$
$48$	0	0
$49$	−3.78642	−0.540918
$50$	−0.633210	−0.0895494
$51$	0	0
$52$	−1.59905	−0.221748
$53$	5.39044	0.740433	0.370217	−	0.928945i	$-0.379283\pi$
$53$	5.39044	0.740433	0.370217	+	0.928945i	$0.379283\pi$
$54$	0	0
$55$	−4.05578	−0.546882
$56$	−4.08535	−0.545928
$57$	0	0
$58$	−1.03404	−0.135777
$59$	12.9024	1.67975	0.839877	−	0.542777i	$-0.182627\pi$
$59$	12.9024	1.67975	0.839877	+	0.542777i	$0.182627\pi$
$60$	0	0
$61$	−5.09533	−0.652390	−0.326195	−	0.945303i	$-0.605767\pi$
$61$	−5.09533	−0.652390	−0.326195	+	0.945303i	$0.605767\pi$
$62$	−2.75165	−0.349460
$63$	0	0
$64$	0.0797303	0.00996629
$65$	1.00000	0.124035
$66$	0	0
$67$	4.37066	0.533961	0.266980	−	0.963702i	$-0.413974\pi$
$67$	4.37066	0.533961	0.266980	+	0.963702i	$0.413974\pi$
$68$	−11.3374	−1.37486
$69$	0	0
$70$	1.13512	0.135673
$71$	−2.80901	−0.333369	−0.166684	−	0.986010i	$-0.553306\pi$
$71$	−2.80901	−0.333369	−0.166684	+	0.986010i	$0.553306\pi$
$72$	0	0
$73$	−5.00513	−0.585806	−0.292903	−	0.956142i	$-0.594621\pi$
$73$	−5.00513	−0.585806	−0.292903	+	0.956142i	$0.594621\pi$
$74$	−0.461823	−0.0536858
$75$	0	0
$76$	2.64859	0.303814
$77$	7.27058	0.828559
$78$	0	0
$79$	6.82293	0.767639	0.383820	−	0.923408i	$-0.374608\pi$
$79$	6.82293	0.767639	0.383820	+	0.923408i	$0.374608\pi$
$80$	1.75503	0.196219
$81$	0	0
$82$	−0.556750	−0.0614828
$83$	6.51536	0.715154	0.357577	−	0.933884i	$-0.383603\pi$
$83$	6.51536	0.715154	0.357577	+	0.933884i	$0.383603\pi$
$84$	0	0
$85$	7.09008	0.769027
$86$	−3.50774	−0.378250
$87$	0	0
$88$	−9.24293	−0.985300
$89$	7.59275	0.804830	0.402415	−	0.915457i	$-0.368171\pi$
$89$	7.59275	0.804830	0.402415	+	0.915457i	$0.368171\pi$
$90$	0	0
$91$	−1.79265	−0.187920
$92$	13.0276	1.35822
$93$	0	0
$94$	5.92203	0.610810
$95$	−1.65636	−0.169939
$96$	0	0
$97$	−12.5443	−1.27368	−0.636839	−	0.770997i	$-0.719758\pi$
$97$	−12.5443	−1.27368	−0.636839	+	0.770997i	$0.719758\pi$
$98$	2.39760	0.242194
$99$	0	0
$100$	−1.59905	−0.159905
$101$	−5.23184	−0.520587	−0.260294	−	0.965530i	$-0.583819\pi$
$101$	−5.23184	−0.520587	−0.260294	+	0.965530i	$0.583819\pi$
$102$	0	0
$103$	−17.1718	−1.69199	−0.845994	−	0.533192i	$-0.820992\pi$
$103$	−17.1718	−1.69199	−0.845994	+	0.533192i	$0.820992\pi$
$104$	2.27895	0.223470
$105$	0	0
$106$	−3.41328	−0.331527
$107$	−4.98432	−0.481852	−0.240926	−	0.970543i	$-0.577451\pi$
$107$	−4.98432	−0.481852	−0.240926	+	0.970543i	$0.577451\pi$
$108$	0	0
$109$	−4.81126	−0.460835	−0.230417	−	0.973092i	$-0.574009\pi$
$109$	−4.81126	−0.460835	−0.230417	+	0.973092i	$0.574009\pi$
$110$	2.56816	0.244865
$111$	0	0
$112$	−3.14615	−0.297284
$113$	9.41004	0.885222	0.442611	−	0.896714i	$-0.354052\pi$
$113$	9.41004	0.885222	0.442611	+	0.896714i	$0.354052\pi$
$114$	0	0
$115$	−8.14713	−0.759724
$116$	−2.61127	−0.242450
$117$	0	0
$118$	−8.16995	−0.752105
$119$	−12.7100	−1.16512
$120$	0	0
$121$	5.44937	0.495398
$122$	3.22641	0.292106
$123$	0	0
$124$	−6.94875	−0.624016
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.51514	−0.844332	−0.422166	−	0.906519i	$-0.638730\pi$
$127$	−9.51514	−0.844332	−0.422166	+	0.906519i	$0.638730\pi$
$128$	11.2879	0.997722
$129$	0	0
$130$	−0.633210	−0.0555362
$131$	−8.09690	−0.707430	−0.353715	−	0.935353i	$-0.615082\pi$
$131$	−8.09690	−0.707430	−0.353715	+	0.935353i	$0.615082\pi$
$132$	0	0
$133$	2.96926	0.257468
$134$	−2.76755	−0.239079
$135$	0	0
$136$	16.1579	1.38553
$137$	20.5983	1.75983	0.879917	−	0.475128i	$-0.157598\pi$
$137$	20.5983	1.75983	0.879917	+	0.475128i	$0.157598\pi$
$138$	0	0
$139$	15.6087	1.32391	0.661956	−	0.749543i	$-0.269727\pi$
$139$	15.6087	1.32391	0.661956	+	0.749543i	$0.269727\pi$
$140$	2.86652	0.242265
$141$	0	0
$142$	1.77870	0.149265
$143$	−4.05578	−0.339162
$144$	0	0
$145$	1.63302	0.135615
$146$	3.16930	0.262293
$147$	0	0
$148$	−1.16624	−0.0958644
$149$	−1.57291	−0.128858	−0.0644288	−	0.997922i	$-0.520523\pi$
$149$	−1.57291	−0.128858	−0.0644288	+	0.997922i	$0.520523\pi$
$150$	0	0
$151$	11.2853	0.918385	0.459193	−	0.888337i	$-0.348139\pi$
$151$	11.2853	0.918385	0.459193	+	0.888337i	$0.348139\pi$
$152$	−3.77476	−0.306173
$153$	0	0
$154$	−4.60380	−0.370985
$155$	4.34556	0.349044
$156$	0	0
$157$	−11.8725	−0.947531	−0.473765	−	0.880651i	$-0.657106\pi$
$157$	−11.8725	−0.947531	−0.473765	+	0.880651i	$0.657106\pi$
$158$	−4.32035	−0.343708
$159$	0	0
$160$	−5.66921	−0.448190
$161$	14.6049	1.15103
$162$	0	0
$163$	−14.1027	−1.10461	−0.552306	−	0.833642i	$-0.686252\pi$
$163$	−14.1027	−1.10461	−0.552306	+	0.833642i	$0.686252\pi$
$164$	−1.40596	−0.109787
$165$	0	0
$166$	−4.12559	−0.320208
$167$	−12.8022	−0.990663	−0.495332	−	0.868704i	$-0.664953\pi$
$167$	−12.8022	−0.990663	−0.495332	+	0.868704i	$0.664953\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−4.48951	−0.344329
$171$	0	0
$172$	−8.85810	−0.675424
$173$	−20.1834	−1.53451	−0.767256	−	0.641341i	$-0.778378\pi$
$173$	−20.1834	−1.53451	−0.767256	+	0.641341i	$0.778378\pi$
$174$	0	0
$175$	−1.79265	−0.135511
$176$	−7.11804	−0.536542
$177$	0	0
$178$	−4.80781	−0.360360
$179$	−10.7902	−0.806497	−0.403249	−	0.915090i	$-0.632119\pi$
$179$	−10.7902	−0.806497	−0.403249	+	0.915090i	$0.632119\pi$
$180$	0	0
$181$	−17.0991	−1.27096	−0.635482	−	0.772116i	$-0.719199\pi$
$181$	−17.0991	−1.27096	−0.635482	+	0.772116i	$0.719199\pi$
$182$	1.13512	0.0841408
$183$	0	0
$184$	−18.5669	−1.36877
$185$	0.729336	0.0536218
$186$	0	0
$187$	−28.7558	−2.10283
$188$	14.9549	1.09070
$189$	0	0
$190$	1.04882	0.0760896
$191$	−5.78112	−0.418307	−0.209154	−	0.977883i	$-0.567071\pi$
$191$	−5.78112	−0.418307	−0.209154	+	0.977883i	$0.567071\pi$
$192$	0	0
$193$	9.60199	0.691167	0.345583	−	0.938388i	$-0.387681\pi$
$193$	9.60199	0.691167	0.345583	+	0.938388i	$0.387681\pi$
$194$	7.94316	0.570285
$195$	0	0
$196$	6.05466	0.432476
$197$	−11.8141	−0.841721	−0.420861	−	0.907125i	$-0.638272\pi$
$197$	−11.8141	−0.841721	−0.420861	+	0.907125i	$0.638272\pi$
$198$	0	0
$199$	−16.7437	−1.18693	−0.593465	−	0.804860i	$-0.702241\pi$
$199$	−16.7437	−1.18693	−0.593465	+	0.804860i	$0.702241\pi$
$200$	2.27895	0.161146
$201$	0	0
$202$	3.31285	0.233091
$203$	−2.92742	−0.205465
$204$	0	0
$205$	0.879250	0.0614095
$206$	10.8734	0.757583
$207$	0	0
$208$	1.75503	0.121690
$209$	6.71783	0.464682
$210$	0	0
$211$	−22.6415	−1.55871	−0.779353	−	0.626585i	$-0.784452\pi$
$211$	−22.6415	−1.55871	−0.779353	+	0.626585i	$0.784452\pi$
$212$	−8.61955	−0.591993
$213$	0	0
$214$	3.15612	0.215748
$215$	5.53962	0.377799
$216$	0	0
$217$	−7.79005	−0.528823
$218$	3.04654	0.206337
$219$	0	0
$220$	6.48538	0.437244
$221$	7.09008	0.476930
$222$	0	0
$223$	13.5497	0.907353	0.453676	−	0.891166i	$-0.350112\pi$
$223$	13.5497	0.907353	0.453676	+	0.891166i	$0.350112\pi$
$224$	10.1629	0.679036
$225$	0	0
$226$	−5.95853	−0.396356
$227$	23.7432	1.57589	0.787946	−	0.615744i	$-0.211144\pi$
$227$	23.7432	1.57589	0.787946	+	0.615744i	$0.211144\pi$
$228$	0	0
$229$	−7.48108	−0.494364	−0.247182	−	0.968969i	$-0.579504\pi$
$229$	−7.48108	−0.494364	−0.247182	+	0.968969i	$0.579504\pi$
$230$	5.15884	0.340164
$231$	0	0
$232$	3.72157	0.244333
$233$	19.4760	1.27592	0.637959	−	0.770071i	$-0.279779\pi$
$233$	19.4760	1.27592	0.637959	+	0.770071i	$0.279779\pi$
$234$	0	0
$235$	−9.35239	−0.610083
$236$	−20.6316	−1.34300
$237$	0	0
$238$	8.04809	0.521680
$239$	9.08902	0.587920	0.293960	−	0.955818i	$-0.405027\pi$
$239$	9.08902	0.587920	0.293960	+	0.955818i	$0.405027\pi$
$240$	0	0
$241$	−11.8779	−0.765124	−0.382562	−	0.923930i	$-0.624958\pi$
$241$	−11.8779	−0.765124	−0.382562	+	0.923930i	$0.624958\pi$
$242$	−3.45060	−0.221813
$243$	0	0
$244$	8.14766	0.521600
$245$	−3.78642	−0.241906
$246$	0	0
$247$	−1.65636	−0.105392
$248$	9.90332	0.628861
$249$	0	0
$250$	−0.633210	−0.0400477
$251$	−25.0409	−1.58057	−0.790285	−	0.612740i	$-0.790067\pi$
$251$	−25.0409	−1.58057	−0.790285	+	0.612740i	$0.790067\pi$
$252$	0	0
$253$	33.0430	2.07739
$254$	6.02508	0.378047
$255$	0	0
$256$	−7.30709	−0.456693
$257$	17.0855	1.06577	0.532883	−	0.846189i	$-0.321109\pi$
$257$	17.0855	1.06577	0.532883	+	0.846189i	$0.321109\pi$
$258$	0	0
$259$	−1.30744	−0.0812404
$260$	−1.59905	−0.0991686
$261$	0	0
$262$	5.12704	0.316750
$263$	−16.5587	−1.02105	−0.510527	−	0.859862i	$-0.670549\pi$
$263$	−16.5587	−1.02105	−0.510527	+	0.859862i	$0.670549\pi$
$264$	0	0
$265$	5.39044	0.331132
$266$	−1.88017	−0.115280
$267$	0	0
$268$	−6.98888	−0.426914
$269$	12.7025	0.774487	0.387244	−	0.921977i	$-0.373427\pi$
$269$	12.7025	0.774487	0.387244	+	0.921977i	$0.373427\pi$
$270$	0	0
$271$	−27.7692	−1.68686	−0.843431	−	0.537238i	$-0.819468\pi$
$271$	−27.7692	−1.68686	−0.843431	+	0.537238i	$0.819468\pi$
$272$	12.4433	0.754488
$273$	0	0
$274$	−13.0431	−0.787960
$275$	−4.05578	−0.244573
$276$	0	0
$277$	−28.5111	−1.71306	−0.856532	−	0.516095i	$-0.827385\pi$
$277$	−28.5111	−1.71306	−0.856532	+	0.516095i	$0.827385\pi$
$278$	−9.88358	−0.592777
$279$	0	0
$280$	−4.08535	−0.244146
$281$	−5.72833	−0.341723	−0.170862	−	0.985295i	$-0.554655\pi$
$281$	−5.72833	−0.341723	−0.170862	+	0.985295i	$0.554655\pi$
$282$	0	0
$283$	−20.9869	−1.24754	−0.623770	−	0.781608i	$-0.714400\pi$
$283$	−20.9869	−1.24754	−0.623770	+	0.781608i	$0.714400\pi$
$284$	4.49174	0.266536
$285$	0	0
$286$	2.56816	0.151859
$287$	−1.57618	−0.0930392
$288$	0	0
$289$	33.2692	1.95701
$290$	−1.03404	−0.0607211
$291$	0	0
$292$	8.00343	0.468365
$293$	30.4606	1.77953	0.889763	−	0.456423i	$-0.150870\pi$
$293$	30.4606	1.77953	0.889763	+	0.456423i	$0.150870\pi$
$294$	0	0
$295$	12.9024	0.751209
$296$	1.66212	0.0966088
$297$	0	0
$298$	0.995981	0.0576956
$299$	−8.14713	−0.471161
$300$	0	0
$301$	−9.93057	−0.572389
$302$	−7.14597	−0.411204
$303$	0	0
$304$	−2.90697	−0.166726
$305$	−5.09533	−0.291758
$306$	0	0
$307$	−31.0677	−1.77313	−0.886565	−	0.462604i	$-0.846915\pi$
$307$	−31.0677	−1.77313	−0.886565	+	0.462604i	$0.846915\pi$
$308$	−11.6260	−0.662452
$309$	0	0
$310$	−2.75165	−0.156283
$311$	10.7408	0.609056	0.304528	−	0.952503i	$-0.401501\pi$
$311$	10.7408	0.609056	0.304528	+	0.952503i	$0.401501\pi$
$312$	0	0
$313$	19.5032	1.10239	0.551193	−	0.834378i	$-0.314173\pi$
$313$	19.5032	1.10239	0.551193	+	0.834378i	$0.314173\pi$
$314$	7.51780	0.424254
$315$	0	0
$316$	−10.9102	−0.613745
$317$	10.6903	0.600428	0.300214	−	0.953872i	$-0.402942\pi$
$317$	10.6903	0.600428	0.300214	+	0.953872i	$0.402942\pi$
$318$	0	0
$319$	−6.62317	−0.370826
$320$	0.0797303	0.00445706
$321$	0	0
$322$	−9.24798	−0.515370
$323$	−11.7437	−0.653437
$324$	0	0
$325$	1.00000	0.0554700
$326$	8.92999	0.494587
$327$	0	0
$328$	2.00377	0.110640
$329$	16.7655	0.924313
$330$	0	0
$331$	12.6685	0.696324	0.348162	−	0.937434i	$-0.386806\pi$
$331$	12.6685	0.696324	0.348162	+	0.937434i	$0.386806\pi$
$332$	−10.4184	−0.571782
$333$	0	0
$334$	8.10648	0.443567
$335$	4.37066	0.238795
$336$	0	0
$337$	−32.9844	−1.79677	−0.898387	−	0.439205i	$-0.855260\pi$
$337$	−32.9844	−1.79677	−0.898387	+	0.439205i	$0.855260\pi$
$338$	−0.633210	−0.0344421
$339$	0	0
$340$	−11.3374	−0.614854
$341$	−17.6246	−0.954428
$342$	0	0
$343$	19.3362	1.04406
$344$	12.6245	0.680669
$345$	0	0
$346$	12.7803	0.687073
$347$	−9.00386	−0.483353	−0.241676	−	0.970357i	$-0.577697\pi$
$347$	−9.00386	−0.483353	−0.241676	+	0.970357i	$0.577697\pi$
$348$	0	0
$349$	−21.2293	−1.13638	−0.568190	−	0.822898i	$-0.692356\pi$
$349$	−21.2293	−1.13638	−0.568190	+	0.822898i	$0.692356\pi$
$350$	1.13512	0.0606748
$351$	0	0
$352$	22.9931	1.22554
$353$	19.6121	1.04385	0.521923	−	0.852993i	$-0.325215\pi$
$353$	19.6121	1.04385	0.521923	+	0.852993i	$0.325215\pi$
$354$	0	0
$355$	−2.80901	−0.149087
$356$	−12.1412	−0.643480
$357$	0	0
$358$	6.83246	0.361107
$359$	16.8856	0.891187	0.445593	−	0.895235i	$-0.352993\pi$
$359$	16.8856	0.891187	0.445593	+	0.895235i	$0.352993\pi$
$360$	0	0
$361$	−16.2565	−0.855604
$362$	10.8273	0.569071
$363$	0	0
$364$	2.86652	0.150246
$365$	−5.00513	−0.261981
$366$	0	0
$367$	−19.9054	−1.03905	−0.519526	−	0.854455i	$-0.673891\pi$
$367$	−19.9054	−1.03905	−0.519526	+	0.854455i	$0.673891\pi$
$368$	−14.2985	−0.745361
$369$	0	0
$370$	−0.461823	−0.0240090
$371$	−9.66314	−0.501685
$372$	0	0
$373$	5.98204	0.309739	0.154869	−	0.987935i	$-0.450504\pi$
$373$	5.98204	0.309739	0.154869	+	0.987935i	$0.450504\pi$
$374$	18.2085	0.941537
$375$	0	0
$376$	−21.3136	−1.09917
$377$	1.63302	0.0841047
$378$	0	0
$379$	10.5304	0.540911	0.270456	−	0.962732i	$-0.412826\pi$
$379$	10.5304	0.540911	0.270456	+	0.962732i	$0.412826\pi$
$380$	2.64859	0.135870
$381$	0	0
$382$	3.66066	0.187296
$383$	−16.0807	−0.821687	−0.410844	−	0.911706i	$-0.634766\pi$
$383$	−16.0807	−0.821687	−0.410844	+	0.911706i	$0.634766\pi$
$384$	0	0
$385$	7.27058	0.370543
$386$	−6.08008	−0.309468
$387$	0	0
$388$	20.0588	1.01833
$389$	−15.1808	−0.769697	−0.384849	−	0.922980i	$-0.625746\pi$
$389$	−15.1808	−0.769697	−0.384849	+	0.922980i	$0.625746\pi$
$390$	0	0
$391$	−57.7638	−2.92124
$392$	−8.62908	−0.435834
$393$	0	0
$394$	7.48082	0.376878
$395$	6.82293	0.343299
$396$	0	0
$397$	−3.30120	−0.165683	−0.0828413	−	0.996563i	$-0.526399\pi$
$397$	−3.30120	−0.165683	−0.0828413	+	0.996563i	$0.526399\pi$
$398$	10.6023	0.531445
$399$	0	0
$400$	1.75503	0.0877517
$401$	−31.2048	−1.55829	−0.779146	−	0.626842i	$-0.784347\pi$
$401$	−31.2048	−1.55829	−0.779146	+	0.626842i	$0.784347\pi$
$402$	0	0
$403$	4.34556	0.216468
$404$	8.36594	0.416221
$405$	0	0
$406$	1.85367	0.0919962
$407$	−2.95803	−0.146624
$408$	0	0
$409$	−10.0536	−0.497120	−0.248560	−	0.968617i	$-0.579957\pi$
$409$	−10.0536	−0.497120	−0.248560	+	0.968617i	$0.579957\pi$
$410$	−0.556750	−0.0274959
$411$	0	0
$412$	27.4585	1.35278
$413$	−23.1295	−1.13813
$414$	0	0
$415$	6.51536	0.319826
$416$	−5.66921	−0.277956
$417$	0	0
$418$	−4.25380	−0.208060
$419$	−34.6259	−1.69159	−0.845793	−	0.533511i	$-0.820872\pi$
$419$	−34.6259	−1.69159	−0.845793	+	0.533511i	$0.820872\pi$
$420$	0	0
$421$	−6.62941	−0.323098	−0.161549	−	0.986865i	$-0.551649\pi$
$421$	−6.62941	−0.323098	−0.161549	+	0.986865i	$0.551649\pi$
$422$	14.3368	0.697906
$423$	0	0
$424$	12.2845	0.596590
$425$	7.09008	0.343919
$426$	0	0
$427$	9.13411	0.442031
$428$	7.97015	0.385252
$429$	0	0
$430$	−3.50774	−0.169158
$431$	−37.5004	−1.80633	−0.903165	−	0.429293i	$-0.858763\pi$
$431$	−37.5004	−1.80633	−0.903165	+	0.429293i	$0.858763\pi$
$432$	0	0
$433$	14.1428	0.679662	0.339831	−	0.940487i	$-0.389630\pi$
$433$	14.1428	0.679662	0.339831	+	0.940487i	$0.389630\pi$
$434$	4.93274	0.236779
$435$	0	0
$436$	7.69342	0.368448
$437$	13.4946	0.645532
$438$	0	0
$439$	29.6922	1.41713	0.708566	−	0.705645i	$-0.249343\pi$
$439$	29.6922	1.41713	0.708566	+	0.705645i	$0.249343\pi$
$440$	−9.24293	−0.440640
$441$	0	0
$442$	−4.48951	−0.213544
$443$	−25.4049	−1.20702	−0.603511	−	0.797355i	$-0.706232\pi$
$443$	−25.4049	−1.20702	−0.603511	+	0.797355i	$0.706232\pi$
$444$	0	0
$445$	7.59275	0.359931
$446$	−8.57979	−0.406265
$447$	0	0
$448$	−0.142928	−0.00675272
$449$	−24.8926	−1.17475	−0.587376	−	0.809314i	$-0.699839\pi$
$449$	−24.8926	−1.17475	−0.587376	+	0.809314i	$0.699839\pi$
$450$	0	0
$451$	−3.56605	−0.167919
$452$	−15.0471	−0.707755
$453$	0	0
$454$	−15.0344	−0.705601
$455$	−1.79265	−0.0840405
$456$	0	0
$457$	36.3079	1.69841	0.849207	−	0.528061i	$-0.177081\pi$
$457$	36.3079	1.69841	0.849207	+	0.528061i	$0.177081\pi$
$458$	4.73710	0.221350
$459$	0	0
$460$	13.0276	0.607416
$461$	0.709267	0.0330339	0.0165169	−	0.999864i	$-0.494742\pi$
$461$	0.709267	0.0330339	0.0165169	+	0.999864i	$0.494742\pi$
$462$	0	0
$463$	−40.2887	−1.87237	−0.936186	−	0.351504i	$-0.885670\pi$
$463$	−40.2887	−1.87237	−0.936186	+	0.351504i	$0.885670\pi$
$464$	2.86600	0.133051
$465$	0	0
$466$	−12.3324	−0.571288
$467$	−13.8688	−0.641773	−0.320887	−	0.947118i	$-0.603981\pi$
$467$	−13.8688	−0.641773	−0.320887	+	0.947118i	$0.603981\pi$
$468$	0	0
$469$	−7.83504	−0.361789
$470$	5.92203	0.273163
$471$	0	0
$472$	29.4040	1.35343
$473$	−22.4675	−1.03306
$474$	0	0
$475$	−1.65636	−0.0759989
$476$	20.3238	0.931542
$477$	0	0
$478$	−5.75526	−0.263239
$479$	−39.0559	−1.78451	−0.892255	−	0.451531i	$-0.850878\pi$
$479$	−39.0559	−1.78451	−0.892255	+	0.451531i	$0.850878\pi$
$480$	0	0
$481$	0.729336	0.0332548
$482$	7.52122	0.342582
$483$	0	0
$484$	−8.71379	−0.396082
$485$	−12.5443	−0.569606
$486$	0	0
$487$	−10.3472	−0.468875	−0.234437	−	0.972131i	$-0.575325\pi$
$487$	−10.3472	−0.468875	−0.234437	+	0.972131i	$0.575325\pi$
$488$	−11.6120	−0.525651
$489$	0	0
$490$	2.39760	0.108313
$491$	1.08529	0.0489784	0.0244892	−	0.999700i	$-0.492204\pi$
$491$	1.08529	0.0489784	0.0244892	+	0.999700i	$0.492204\pi$
$492$	0	0
$493$	11.5782	0.521457
$494$	1.04882	0.0471887
$495$	0	0
$496$	7.62661	0.342445
$497$	5.03557	0.225876
$498$	0	0
$499$	−42.4200	−1.89898	−0.949489	−	0.313800i	$-0.898398\pi$
$499$	−42.4200	−1.89898	−0.949489	+	0.313800i	$0.898398\pi$
$500$	−1.59905	−0.0715115
$501$	0	0
$502$	15.8562	0.707695
$503$	7.96629	0.355199	0.177600	−	0.984103i	$-0.443167\pi$
$503$	7.96629	0.355199	0.177600	+	0.984103i	$0.443167\pi$
$504$	0	0
$505$	−5.23184	−0.232814
$506$	−20.9232	−0.930148
$507$	0	0
$508$	15.2151	0.675063
$509$	−12.8734	−0.570603	−0.285301	−	0.958438i	$-0.592094\pi$
$509$	−12.8734	−0.570603	−0.285301	+	0.958438i	$0.592094\pi$
$510$	0	0
$511$	8.97243	0.396917
$512$	−17.9489	−0.793239
$513$	0	0
$514$	−10.8187	−0.477194
$515$	−17.1718	−0.756680
$516$	0	0
$517$	37.9313	1.66821
$518$	0.827884	0.0363751
$519$	0	0
$520$	2.27895	0.0999386
$521$	−7.60942	−0.333375	−0.166687	−	0.986010i	$-0.553307\pi$
$521$	−7.60942	−0.333375	−0.166687	+	0.986010i	$0.553307\pi$
$522$	0	0
$523$	−34.6906	−1.51692	−0.758458	−	0.651722i	$-0.774047\pi$
$523$	−34.6906	−1.51692	−0.758458	+	0.651722i	$0.774047\pi$
$524$	12.9473	0.565606
$525$	0	0
$526$	10.4851	0.457174
$527$	30.8103	1.34212
$528$	0	0
$529$	43.3757	1.88590
$530$	−3.41328	−0.148263
$531$	0	0
$532$	−4.74798	−0.205851
$533$	0.879250	0.0380846
$534$	0	0
$535$	−4.98432	−0.215491
$536$	9.96052	0.430229
$537$	0	0
$538$	−8.04337	−0.346774
$539$	15.3569	0.661469
$540$	0	0
$541$	−0.0126565	−0.000544145 0	−0.000272073	−	1.00000i	$-0.500087\pi$
$541$	−0.0126565	−0.000544145 0	−0.000272073		1.00000i	$0.500087\pi$
$542$	17.5838	0.755287
$543$	0	0
$544$	−40.1951	−1.72335
$545$	−4.81126	−0.206092
$546$	0	0
$547$	−8.55713	−0.365876	−0.182938	−	0.983124i	$-0.558561\pi$
$547$	−8.55713	−0.365876	−0.182938	+	0.983124i	$0.558561\pi$
$548$	−32.9376	−1.40703
$549$	0	0
$550$	2.56816	0.109507
$551$	−2.70486	−0.115231
$552$	0	0
$553$	−12.2311	−0.520119
$554$	18.0535	0.767019
$555$	0	0
$556$	−24.9590	−1.05850
$557$	−13.3178	−0.564292	−0.282146	−	0.959371i	$-0.591046\pi$
$557$	−13.3178	−0.564292	−0.282146	+	0.959371i	$0.591046\pi$
$558$	0	0
$559$	5.53962	0.234301
$560$	−3.14615	−0.132949
$561$	0	0
$562$	3.62723	0.153006
$563$	30.5350	1.28690	0.643449	−	0.765489i	$-0.277503\pi$
$563$	30.5350	1.28690	0.643449	+	0.765489i	$0.277503\pi$
$564$	0	0
$565$	9.41004	0.395883
$566$	13.2891	0.558583
$567$	0	0
$568$	−6.40161	−0.268606
$569$	24.2606	1.01706	0.508529	−	0.861045i	$-0.330189\pi$
$569$	24.2606	1.01706	0.508529	+	0.861045i	$0.330189\pi$
$570$	0	0
$571$	27.0508	1.13204	0.566020	−	0.824392i	$-0.308483\pi$
$571$	27.0508	1.13204	0.566020	+	0.824392i	$0.308483\pi$
$572$	6.48538	0.271167
$573$	0	0
$574$	0.998055	0.0416580
$575$	−8.14713	−0.339759
$576$	0	0
$577$	21.8806	0.910903	0.455451	−	0.890261i	$-0.349478\pi$
$577$	21.8806	0.910903	0.455451	+	0.890261i	$0.349478\pi$
$578$	−21.0664	−0.876246
$579$	0	0
$580$	−2.61127	−0.108427
$581$	−11.6797	−0.484557
$582$	0	0
$583$	−21.8624	−0.905450
$584$	−11.4065	−0.472002
$585$	0	0
$586$	−19.2879	−0.796777
$587$	26.9205	1.11113	0.555564	−	0.831474i	$-0.312502\pi$
$587$	26.9205	1.11113	0.555564	+	0.831474i	$0.312502\pi$
$588$	0	0
$589$	−7.19780	−0.296580
$590$	−8.16995	−0.336352
$591$	0	0
$592$	1.28001	0.0526081
$593$	−13.3259	−0.547231	−0.273616	−	0.961839i	$-0.588220\pi$
$593$	−13.3259	−0.547231	−0.273616	+	0.961839i	$0.588220\pi$
$594$	0	0
$595$	−12.7100	−0.521059
$596$	2.51515	0.103025
$597$	0	0
$598$	5.15884	0.210961
$599$	27.3605	1.11792	0.558960	−	0.829194i	$-0.311200\pi$
$599$	27.3605	1.11792	0.558960	+	0.829194i	$0.311200\pi$
$600$	0	0
$601$	−36.7731	−1.50001	−0.750003	−	0.661434i	$-0.769948\pi$
$601$	−36.7731	−1.50001	−0.750003	+	0.661434i	$0.769948\pi$
$602$	6.28814	0.256285
$603$	0	0
$604$	−18.0457	−0.734270
$605$	5.44937	0.221549
$606$	0	0
$607$	29.6541	1.20362	0.601811	−	0.798639i	$-0.294446\pi$
$607$	29.6541	1.20362	0.601811	+	0.798639i	$0.294446\pi$
$608$	9.39024	0.380824
$609$	0	0
$610$	3.22641	0.130634
$611$	−9.35239	−0.378357
$612$	0	0
$613$	−9.69185	−0.391450	−0.195725	−	0.980659i	$-0.562706\pi$
$613$	−9.69185	−0.391450	−0.195725	+	0.980659i	$0.562706\pi$
$614$	19.6724	0.793914
$615$	0	0
$616$	16.5693	0.667596
$617$	−22.6919	−0.913541	−0.456770	−	0.889585i	$-0.650994\pi$
$617$	−22.6919	−0.913541	−0.456770	+	0.889585i	$0.650994\pi$
$618$	0	0
$619$	41.7011	1.67611	0.838055	−	0.545586i	$-0.183693\pi$
$619$	41.7011	1.67611	0.838055	+	0.545586i	$0.183693\pi$
$620$	−6.94875	−0.279068
$621$	0	0
$622$	−6.80119	−0.272703
$623$	−13.6111	−0.545318
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.3496	−0.493590
$627$	0	0
$628$	18.9847	0.757572
$629$	5.17105	0.206183
$630$	0	0
$631$	8.75145	0.348390	0.174195	−	0.984711i	$-0.444268\pi$
$631$	8.75145	0.348390	0.174195	+	0.984711i	$0.444268\pi$
$632$	15.5491	0.618511
$633$	0	0
$634$	−6.76922	−0.268840
$635$	−9.51514	−0.377597
$636$	0	0
$637$	−3.78642	−0.150024
$638$	4.19386	0.166036
$639$	0	0
$640$	11.2879	0.446195
$641$	2.10529	0.0831541	0.0415770	−	0.999135i	$-0.486762\pi$
$641$	2.10529	0.0831541	0.0415770	+	0.999135i	$0.486762\pi$
$642$	0	0
$643$	21.6215	0.852671	0.426335	−	0.904565i	$-0.359804\pi$
$643$	21.6215	0.852671	0.426335	+	0.904565i	$0.359804\pi$
$644$	−23.3539	−0.920273
$645$	0	0
$646$	7.43623	0.292575
$647$	42.5198	1.67163	0.835814	−	0.549013i	$-0.184996\pi$
$647$	42.5198	1.67163	0.835814	+	0.549013i	$0.184996\pi$
$648$	0	0
$649$	−52.3295	−2.05411
$650$	−0.633210	−0.0248365
$651$	0	0
$652$	22.5509	0.883162
$653$	−6.91515	−0.270611	−0.135305	−	0.990804i	$-0.543202\pi$
$653$	−6.91515	−0.270611	−0.135305	+	0.990804i	$0.543202\pi$
$654$	0	0
$655$	−8.09690	−0.316372
$656$	1.54311	0.0602485
$657$	0	0
$658$	−10.6161	−0.413858
$659$	−32.1531	−1.25251	−0.626254	−	0.779619i	$-0.715413\pi$
$659$	−32.1531	−1.25251	−0.626254	+	0.779619i	$0.715413\pi$
$660$	0	0
$661$	−1.77716	−0.0691235	−0.0345617	−	0.999403i	$-0.511004\pi$
$661$	−1.77716	−0.0691235	−0.0345617	+	0.999403i	$0.511004\pi$
$662$	−8.02182	−0.311777
$663$	0	0
$664$	14.8482	0.576222
$665$	2.96926	0.115143
$666$	0	0
$667$	−13.3044	−0.515149
$668$	20.4713	0.792057
$669$	0	0
$670$	−2.76755	−0.106920
$671$	20.6655	0.797784
$672$	0	0
$673$	−20.3418	−0.784120	−0.392060	−	0.919940i	$-0.628237\pi$
$673$	−20.3418	−0.784120	−0.392060	+	0.919940i	$0.628237\pi$
$674$	20.8860	0.804501
$675$	0	0
$676$	−1.59905	−0.0615017
$677$	−14.7425	−0.566601	−0.283300	−	0.959031i	$-0.591429\pi$
$677$	−14.7425	−0.566601	−0.283300	+	0.959031i	$0.591429\pi$
$678$	0	0
$679$	22.4874	0.862988
$680$	16.1579	0.619629
$681$	0	0
$682$	11.1601	0.427342
$683$	15.1539	0.579847	0.289923	−	0.957050i	$-0.406370\pi$
$683$	15.1539	0.579847	0.289923	+	0.957050i	$0.406370\pi$
$684$	0	0
$685$	20.5983	0.787021
$686$	−12.2439	−0.467474
$687$	0	0
$688$	9.72222	0.370656
$689$	5.39044	0.205359
$690$	0	0
$691$	−15.6321	−0.594674	−0.297337	−	0.954773i	$-0.596099\pi$
$691$	−15.6321	−0.594674	−0.297337	+	0.954773i	$0.596099\pi$
$692$	32.2741	1.22688
$693$	0	0
$694$	5.70134	0.216420
$695$	15.6087	0.592071
$696$	0	0
$697$	6.23395	0.236128
$698$	13.4426	0.508810
$699$	0	0
$700$	2.86652	0.108344
$701$	−8.91657	−0.336774	−0.168387	−	0.985721i	$-0.553856\pi$
$701$	−8.91657	−0.336774	−0.168387	+	0.985721i	$0.553856\pi$
$702$	0	0
$703$	−1.20804	−0.0455621
$704$	−0.323369	−0.0121874
$705$	0	0
$706$	−12.4186	−0.467379
$707$	9.37883	0.352727
$708$	0	0
$709$	−39.9110	−1.49889	−0.749444	−	0.662068i	$-0.769679\pi$
$709$	−39.9110	−1.49889	−0.749444	+	0.662068i	$0.769679\pi$
$710$	1.77870	0.0667533
$711$	0	0
$712$	17.3035	0.648477
$713$	−35.4038	−1.32588
$714$	0	0
$715$	−4.05578	−0.151678
$716$	17.2540	0.644813
$717$	0	0
$718$	−10.6921	−0.399026
$719$	34.4119	1.28335	0.641673	−	0.766978i	$-0.278241\pi$
$719$	34.4119	1.28335	0.641673	+	0.766978i	$0.278241\pi$
$720$	0	0
$721$	30.7830	1.14642
$722$	10.2938	0.383094
$723$	0	0
$724$	27.3422	1.01616
$725$	1.63302	0.0606488
$726$	0	0
$727$	−6.96162	−0.258192	−0.129096	−	0.991632i	$-0.541208\pi$
$727$	−6.96162	−0.258192	−0.129096	+	0.991632i	$0.541208\pi$
$728$	−4.08535	−0.151413
$729$	0	0
$730$	3.16930	0.117301
$731$	39.2763	1.45269
$732$	0	0
$733$	25.3146	0.935016	0.467508	−	0.883989i	$-0.345152\pi$
$733$	25.3146	0.935016	0.467508	+	0.883989i	$0.345152\pi$
$734$	12.6043	0.465233
$735$	0	0
$736$	46.1878	1.70250
$737$	−17.7264	−0.652962
$738$	0	0
$739$	45.7042	1.68126	0.840628	−	0.541613i	$-0.182186\pi$
$739$	45.7042	1.68126	0.840628	+	0.541613i	$0.182186\pi$
$740$	−1.16624	−0.0428719
$741$	0	0
$742$	6.11880	0.224628
$743$	−15.6774	−0.575149	−0.287575	−	0.957758i	$-0.592849\pi$
$743$	−15.6774	−0.575149	−0.287575	+	0.957758i	$0.592849\pi$
$744$	0	0
$745$	−1.57291	−0.0576269
$746$	−3.78789	−0.138685
$747$	0	0
$748$	45.9818	1.68126
$749$	8.93512	0.326482
$750$	0	0
$751$	0.604969	0.0220756	0.0110378	−	0.999939i	$-0.496486\pi$
$751$	0.604969	0.0220756	0.0110378	+	0.999939i	$0.496486\pi$
$752$	−16.4138	−0.598549
$753$	0	0
$754$	−1.03404	−0.0376576
$755$	11.2853	0.410714
$756$	0	0
$757$	35.7570	1.29961	0.649806	−	0.760100i	$-0.274850\pi$
$757$	35.7570	1.29961	0.649806	+	0.760100i	$0.274850\pi$
$758$	−6.66797	−0.242191
$759$	0	0
$760$	−3.77476	−0.136925
$761$	−7.68235	−0.278485	−0.139242	−	0.990258i	$-0.544467\pi$
$761$	−7.68235	−0.278485	−0.139242	+	0.990258i	$0.544467\pi$
$762$	0	0
$763$	8.62488	0.312241
$764$	9.24427	0.334446
$765$	0	0
$766$	10.1825	0.367908
$767$	12.9024	0.465880
$768$	0	0
$769$	−3.52957	−0.127280	−0.0636398	−	0.997973i	$-0.520271\pi$
$769$	−3.52957	−0.127280	−0.0636398	+	0.997973i	$0.520271\pi$
$770$	−4.60380	−0.165910
$771$	0	0
$772$	−15.3540	−0.552603
$773$	37.6062	1.35260	0.676300	−	0.736626i	$-0.263582\pi$
$773$	37.6062	1.35260	0.676300	+	0.736626i	$0.263582\pi$
$774$	0	0
$775$	4.34556	0.156097
$776$	−28.5878	−1.02624
$777$	0	0
$778$	9.61264	0.344630
$779$	−1.45635	−0.0521793
$780$	0	0
$781$	11.3928	0.407665
$782$	36.5766	1.30798
$783$	0	0
$784$	−6.64531	−0.237332
$785$	−11.8725	−0.423749
$786$	0	0
$787$	11.7004	0.417073	0.208536	−	0.978015i	$-0.433130\pi$
$787$	11.7004	0.417073	0.208536	+	0.978015i	$0.433130\pi$
$788$	18.8913	0.672975
$789$	0	0
$790$	−4.32035	−0.153711
$791$	−16.8689	−0.599788
$792$	0	0
$793$	−5.09533	−0.180940
$794$	2.09035	0.0741839
$795$	0	0
$796$	26.7740	0.948978
$797$	50.5456	1.79042	0.895208	−	0.445649i	$-0.147027\pi$
$797$	50.5456	1.79042	0.895208	+	0.445649i	$0.147027\pi$
$798$	0	0
$799$	−66.3091	−2.34585
$800$	−5.66921	−0.200437
$801$	0	0
$802$	19.7592	0.697721
$803$	20.2997	0.716362
$804$	0	0
$805$	14.6049	0.514756
$806$	−2.75165	−0.0969228
$807$	0	0
$808$	−11.9231	−0.419453
$809$	35.0243	1.23139	0.615695	−	0.787985i	$-0.288875\pi$
$809$	35.0243	1.23139	0.615695	+	0.787985i	$0.288875\pi$
$810$	0	0
$811$	−14.0250	−0.492483	−0.246242	−	0.969208i	$-0.579196\pi$
$811$	−14.0250	−0.492483	−0.246242	+	0.969208i	$0.579196\pi$
$812$	4.68108	0.164274
$813$	0	0
$814$	1.87305	0.0656505
$815$	−14.1027	−0.493997
$816$	0	0
$817$	−9.17559	−0.321013
$818$	6.36606	0.222584
$819$	0	0
$820$	−1.40596	−0.0490983
$821$	22.5543	0.787149	0.393575	−	0.919293i	$-0.371238\pi$
$821$	22.5543	0.787149	0.393575	+	0.919293i	$0.371238\pi$
$822$	0	0
$823$	−15.3493	−0.535043	−0.267521	−	0.963552i	$-0.586205\pi$
$823$	−15.3493	−0.535043	−0.267521	+	0.963552i	$0.586205\pi$
$824$	−39.1337	−1.36329
$825$	0	0
$826$	14.6458	0.509593
$827$	6.03698	0.209926	0.104963	−	0.994476i	$-0.466528\pi$
$827$	6.03698	0.209926	0.104963	+	0.994476i	$0.466528\pi$
$828$	0	0
$829$	−8.49351	−0.294992	−0.147496	−	0.989063i	$-0.547121\pi$
$829$	−8.49351	−0.294992	−0.147496	+	0.989063i	$0.547121\pi$
$830$	−4.12559	−0.143201
$831$	0	0
$832$	0.0797303	0.00276415
$833$	−26.8460	−0.930160
$834$	0	0
$835$	−12.8022	−0.443038
$836$	−10.7421	−0.371524
$837$	0	0
$838$	21.9255	0.757403
$839$	−4.25540	−0.146913	−0.0734563	−	0.997298i	$-0.523403\pi$
$839$	−4.25540	−0.146913	−0.0734563	+	0.997298i	$0.523403\pi$
$840$	0	0
$841$	−26.3333	−0.908043
$842$	4.19781	0.144666
$843$	0	0
$844$	36.2048	1.24622
$845$	1.00000	0.0344010
$846$	0	0
$847$	−9.76879	−0.335660
$848$	9.46041	0.324872
$849$	0	0
$850$	−4.48951	−0.153989
$851$	−5.94199	−0.203689
$852$	0	0
$853$	7.27021	0.248927	0.124464	−	0.992224i	$-0.460279\pi$
$853$	7.27021	0.248927	0.124464	+	0.992224i	$0.460279\pi$
$854$	−5.78381	−0.197918
$855$	0	0
$856$	−11.3590	−0.388243
$857$	2.19613	0.0750183	0.0375092	−	0.999296i	$-0.488058\pi$
$857$	2.19613	0.0750183	0.0375092	+	0.999296i	$0.488058\pi$
$858$	0	0
$859$	22.4177	0.764881	0.382441	−	0.923980i	$-0.375084\pi$
$859$	22.4177	0.764881	0.382441	+	0.923980i	$0.375084\pi$
$860$	−8.85810	−0.302059
$861$	0	0
$862$	23.7456	0.808779
$863$	20.7310	0.705690	0.352845	−	0.935682i	$-0.385214\pi$
$863$	20.7310	0.705690	0.352845	+	0.935682i	$0.385214\pi$
$864$	0	0
$865$	−20.1834	−0.686255
$866$	−8.95539	−0.304317
$867$	0	0
$868$	12.4566	0.422806
$869$	−27.6723	−0.938719
$870$	0	0
$871$	4.37066	0.148094
$872$	−10.9646	−0.371309
$873$	0	0
$874$	−8.54489	−0.289035
$875$	−1.79265	−0.0606025
$876$	0	0
$877$	3.29642	0.111312	0.0556561	−	0.998450i	$-0.482275\pi$
$877$	3.29642	0.111312	0.0556561	+	0.998450i	$0.482275\pi$
$878$	−18.8014	−0.634517
$879$	0	0
$880$	−7.11804	−0.239949
$881$	−28.3286	−0.954415	−0.477207	−	0.878791i	$-0.658351\pi$
$881$	−28.3286	−0.954415	−0.477207	+	0.878791i	$0.658351\pi$
$882$	0	0
$883$	−36.8595	−1.24042	−0.620210	−	0.784436i	$-0.712952\pi$
$883$	−36.8595	−1.24042	−0.620210	+	0.784436i	$0.712952\pi$
$884$	−11.3374	−0.381316
$885$	0	0
$886$	16.0866	0.540441
$887$	−39.9397	−1.34104	−0.670521	−	0.741890i	$-0.733930\pi$
$887$	−39.9397	−1.34104	−0.670521	+	0.741890i	$0.733930\pi$
$888$	0	0
$889$	17.0573	0.572082
$890$	−4.80781	−0.161158
$891$	0	0
$892$	−21.6665	−0.725449
$893$	15.4909	0.518383
$894$	0	0
$895$	−10.7902	−0.360677
$896$	−20.2353	−0.676012
$897$	0	0
$898$	15.7622	0.525992
$899$	7.09638	0.236677
$900$	0	0
$901$	38.2186	1.27325
$902$	2.25806	0.0751851
$903$	0	0
$904$	21.4450	0.713251
$905$	−17.0991	−0.568392
$906$	0	0
$907$	−2.40079	−0.0797169	−0.0398585	−	0.999205i	$-0.512691\pi$
$907$	−2.40079	−0.0797169	−0.0398585	+	0.999205i	$0.512691\pi$
$908$	−37.9665	−1.25996
$909$	0	0
$910$	1.13512	0.0376289
$911$	−54.2370	−1.79695	−0.898476	−	0.439023i	$-0.855325\pi$
$911$	−54.2370	−1.79695	−0.898476	+	0.439023i	$0.855325\pi$
$912$	0	0
$913$	−26.4249	−0.874536
$914$	−22.9905	−0.760460
$915$	0	0
$916$	11.9626	0.395255
$917$	14.5149	0.479323
$918$	0	0
$919$	13.8459	0.456733	0.228367	−	0.973575i	$-0.426662\pi$
$919$	13.8459	0.456733	0.228367	+	0.973575i	$0.426662\pi$
$920$	−18.5669	−0.612133
$921$	0	0
$922$	−0.449115	−0.0147908
$923$	−2.80901	−0.0924598
$924$	0	0
$925$	0.729336	0.0239804
$926$	25.5112	0.838350
$927$	0	0
$928$	−9.25792	−0.303906
$929$	2.57324	0.0844253	0.0422127	−	0.999109i	$-0.486559\pi$
$929$	2.57324	0.0844253	0.0422127	+	0.999109i	$0.486559\pi$
$930$	0	0
$931$	6.27167	0.205546
$932$	−31.1431	−1.02012
$933$	0	0
$934$	8.78189	0.287352
$935$	−28.7558	−0.940415
$936$	0	0
$937$	4.68434	0.153031	0.0765154	−	0.997068i	$-0.475621\pi$
$937$	4.68434	0.153031	0.0765154	+	0.997068i	$0.475621\pi$
$938$	4.96123	0.161990
$939$	0	0
$940$	14.9549	0.487775
$941$	−56.4806	−1.84122	−0.920608	−	0.390488i	$-0.872306\pi$
$941$	−56.4806	−1.84122	−0.920608	+	0.390488i	$0.872306\pi$
$942$	0	0
$943$	−7.16337	−0.233271
$944$	22.6442	0.737007
$945$	0	0
$946$	14.2266	0.462548
$947$	26.0306	0.845880	0.422940	−	0.906158i	$-0.360998\pi$
$947$	26.0306	0.845880	0.422940	+	0.906158i	$0.360998\pi$
$948$	0	0
$949$	−5.00513	−0.162473
$950$	1.04882	0.0340283
$951$	0	0
$952$	−28.9655	−0.938776
$953$	−17.8899	−0.579512	−0.289756	−	0.957101i	$-0.593574\pi$
$953$	−17.8899	−0.579512	−0.289756	+	0.957101i	$0.593574\pi$
$954$	0	0
$955$	−5.78112	−0.187073
$956$	−14.5337	−0.470055
$957$	0	0
$958$	24.7306	0.799010
$959$	−36.9255	−1.19239
$960$	0	0
$961$	−12.1161	−0.390842
$962$	−0.461823	−0.0148898
$963$	0	0
$964$	18.9933	0.611734
$965$	9.60199	0.309099
$966$	0	0
$967$	34.5176	1.11001	0.555006	−	0.831847i	$-0.312716\pi$
$967$	34.5176	1.11001	0.555006	+	0.831847i	$0.312716\pi$
$968$	12.4189	0.399157
$969$	0	0
$970$	7.94316	0.255039
$971$	14.6900	0.471423	0.235712	−	0.971823i	$-0.424258\pi$
$971$	14.6900	0.471423	0.235712	+	0.971823i	$0.424258\pi$
$972$	0	0
$973$	−27.9808	−0.897024
$974$	6.55193	0.209937
$975$	0	0
$976$	−8.94248	−0.286242
$977$	37.1693	1.18915	0.594576	−	0.804040i	$-0.297320\pi$
$977$	37.1693	1.18915	0.594576	+	0.804040i	$0.297320\pi$
$978$	0	0
$979$	−30.7946	−0.984198
$980$	6.05466	0.193409
$981$	0	0
$982$	−0.687216	−0.0219299
$983$	−41.4044	−1.32060	−0.660298	−	0.751004i	$-0.729570\pi$
$983$	−41.4044	−1.32060	−0.660298	+	0.751004i	$0.729570\pi$
$984$	0	0
$985$	−11.8141	−0.376429
$986$	−7.33145	−0.233481
$987$	0	0
$988$	2.64859	0.0842629
$989$	−45.1320	−1.43511
$990$	0	0
$991$	8.06215	0.256102	0.128051	−	0.991768i	$-0.459128\pi$
$991$	8.06215	0.256102	0.128051	+	0.991768i	$0.459128\pi$
$992$	−24.6359	−0.782190
$993$	0	0
$994$	−3.18857	−0.101135
$995$	−16.7437	−0.530811
$996$	0	0
$997$	30.1952	0.956293	0.478147	−	0.878280i	$-0.341309\pi$
$997$	30.1952	0.956293	0.478147	+	0.878280i	$0.341309\pi$
$998$	26.8607	0.850262
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bb.1.4		8
3.2	odd	2		5265.2.a.be.1.5		8
9.2	odd	6		1755.2.i.e.1171.4		16
9.4	even	3		585.2.i.f.196.5	✓	16
9.5	odd	6		1755.2.i.e.586.4		16
9.7	even	3		585.2.i.f.391.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.f.196.5	✓	16	9.4	even	3
585.2.i.f.391.5	yes	16	9.7	even	3
1755.2.i.e.586.4		16	9.5	odd	6
1755.2.i.e.1171.4		16	9.2	odd	6
5265.2.a.bb.1.4		8	1.1	even	1	trivial
5265.2.a.be.1.5		8	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-0.633210 q^{2} -1.59905 q^{4} +1.00000 q^{5} -1.79265 q^{7} +2.27895 q^{8} +O(q^{10})\)	\(q-0.633210 q^{2} -1.59905 q^{4} +1.00000 q^{5} -1.79265 q^{7} +2.27895 q^{8} -0.633210 q^{10} -4.05578 q^{11} +1.00000 q^{13} +1.13512 q^{14} +1.75503 q^{16} +7.09008 q^{17} -1.65636 q^{19} -1.59905 q^{20} +2.56816 q^{22} -8.14713 q^{23} +1.00000 q^{25} -0.633210 q^{26} +2.86652 q^{28} +1.63302 q^{29} +4.34556 q^{31} -5.66921 q^{32} -4.48951 q^{34} -1.79265 q^{35} +0.729336 q^{37} +1.04882 q^{38} +2.27895 q^{40} +0.879250 q^{41} +5.53962 q^{43} +6.48538 q^{44} +5.15884 q^{46} -9.35239 q^{47} -3.78642 q^{49} -0.633210 q^{50} -1.59905 q^{52} +5.39044 q^{53} -4.05578 q^{55} -4.08535 q^{56} -1.03404 q^{58} +12.9024 q^{59} -5.09533 q^{61} -2.75165 q^{62} +0.0797303 q^{64} +1.00000 q^{65} +4.37066 q^{67} -11.3374 q^{68} +1.13512 q^{70} -2.80901 q^{71} -5.00513 q^{73} -0.461823 q^{74} +2.64859 q^{76} +7.27058 q^{77} +6.82293 q^{79} +1.75503 q^{80} -0.556750 q^{82} +6.51536 q^{83} +7.09008 q^{85} -3.50774 q^{86} -9.24293 q^{88} +7.59275 q^{89} -1.79265 q^{91} +13.0276 q^{92} +5.92203 q^{94} -1.65636 q^{95} -12.5443 q^{97} +2.39760 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8	\( 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - q^{10} - 9 q^{11} + 8 q^{13} + 3 q^{14} - 13 q^{16} + 6 q^{17} - 11 q^{19} + 5 q^{20} - 4 q^{22} - 3 q^{23} + 8 q^{25} - q^{26} - 13 q^{28} - 8 q^{29} - 18 q^{31} - 3 q^{32} - 9 q^{34} - 6 q^{35} - 18 q^{37} + 8 q^{38} - 6 q^{40} + 17 q^{41} - 17 q^{43} + 5 q^{44} + 3 q^{46} - 11 q^{47} - 16 q^{49} - q^{50} + 5 q^{52} + 10 q^{53} - 9 q^{55} + q^{56} - 10 q^{58} - 7 q^{59} - 21 q^{61} - 29 q^{62} - 10 q^{64} + 8 q^{65} - 13 q^{67} + 16 q^{68} + 3 q^{70} - 34 q^{71} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 18 q^{77} - 37 q^{79} - 13 q^{80} + q^{82} - 3 q^{83} + 6 q^{85} + 2 q^{86} - 19 q^{88} + 14 q^{89} - 6 q^{91} + 14 q^{92} - 44 q^{94} - 11 q^{95} - 17 q^{97} + 45 q^{98}+O(q^{100}) \) 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - q^10 - 9 * q^11 + 8 * q^13 + 3 * q^14 - 13 * q^16 + 6 * q^17 - 11 * q^19 + 5 * q^20 - 4 * q^22 - 3 * q^23 + 8 * q^25 - q^26 - 13 * q^28 - 8 * q^29 - 18 * q^31 - 3 * q^32 - 9 * q^34 - 6 * q^35 - 18 * q^37 + 8 * q^38 - 6 * q^40 + 17 * q^41 - 17 * q^43 + 5 * q^44 + 3 * q^46 - 11 * q^47 - 16 * q^49 - q^50 + 5 * q^52 + 10 * q^53 - 9 * q^55 + q^56 - 10 * q^58 - 7 * q^59 - 21 * q^61 - 29 * q^62 - 10 * q^64 + 8 * q^65 - 13 * q^67 + 16 * q^68 + 3 * q^70 - 34 * q^71 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 18 * q^77 - 37 * q^79 - 13 * q^80 + q^82 - 3 * q^83 + 6 * q^85 + 2 * q^86 - 19 * q^88 + 14 * q^89 - 6 * q^91 + 14 * q^92 - 44 * q^94 - 11 * q^95 - 17 * q^97 + 45 * q^98

Introduction

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