LMFDB - Embedded newform 5265.2.a.bl.1.1

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:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 25 x^{13} + 24 x^{12} + 244 x^{11} - 226 x^{10} - 1170 x^{9} + 1051 x^{8} + 2842 x^{7} + \cdots + 144 $ x^15 - x^14 - 25x^13 + 24x^12 + 244x^11 - 226x^10 - 1170x^9 + 1051x^8 + 2842x^7 - 2478x^6 - 3252x^5 + 2697x^4 + 1497x^3 - 1107x^2 - 225*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.66230$ 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)

$f(q)$	$=$	$q-2.66230 q^{2} +5.08784 q^{4} +1.00000 q^{5} +1.44674 q^{7} -8.22077 q^{8} +O(q^{10})$	\(q-2.66230 q^{2} +5.08784 q^{4} +1.00000 q^{5} +1.44674 q^{7} -8.22077 q^{8} -2.66230 q^{10} -6.36434 q^{11} +1.00000 q^{13} -3.85165 q^{14} +11.7105 q^{16} -3.28767 q^{17} -2.89517 q^{19} +5.08784 q^{20} +16.9438 q^{22} +1.22736 q^{23} +1.00000 q^{25} -2.66230 q^{26} +7.36078 q^{28} +9.47975 q^{29} +2.40673 q^{31} -14.7352 q^{32} +8.75276 q^{34} +1.44674 q^{35} -4.81693 q^{37} +7.70782 q^{38} -8.22077 q^{40} -5.51892 q^{41} -7.21025 q^{43} -32.3808 q^{44} -3.26760 q^{46} -8.00214 q^{47} -4.90695 q^{49} -2.66230 q^{50} +5.08784 q^{52} +0.623139 q^{53} -6.36434 q^{55} -11.8933 q^{56} -25.2379 q^{58} +9.87657 q^{59} +4.48864 q^{61} -6.40745 q^{62} +15.8087 q^{64} +1.00000 q^{65} -3.26424 q^{67} -16.7271 q^{68} -3.85165 q^{70} +9.19831 q^{71} +9.55989 q^{73} +12.8241 q^{74} -14.7302 q^{76} -9.20754 q^{77} +15.6457 q^{79} +11.7105 q^{80} +14.6930 q^{82} +1.05295 q^{83} -3.28767 q^{85} +19.1959 q^{86} +52.3198 q^{88} -2.96850 q^{89} +1.44674 q^{91} +6.24462 q^{92} +21.3041 q^{94} -2.89517 q^{95} -11.5357 q^{97} +13.0638 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7}+O(q^{10}) $ 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7	$ 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7} + q^{10} + 9 q^{11} + 15 q^{13} + 3 q^{14} + 33 q^{16} - 3 q^{17} + 15 q^{19} + 21 q^{20} + 10 q^{22} - 6 q^{23} + 15 q^{25} + q^{26} + 35 q^{28} + 8 q^{29} + 22 q^{31} + 21 q^{32} + 9 q^{34} + 10 q^{35} + 4 q^{37} - 14 q^{38} + 13 q^{41} + 24 q^{43} - 5 q^{44} - 3 q^{46} - q^{47} + 37 q^{49} + q^{50} + 21 q^{52} - 7 q^{53} + 9 q^{55} + 17 q^{56} + 22 q^{58} + 19 q^{59} + 16 q^{61} - 13 q^{62} + 36 q^{64} + 15 q^{65} + 11 q^{67} - 28 q^{68} + 3 q^{70} + 28 q^{71} + 26 q^{73} + 8 q^{74} + 18 q^{76} - 24 q^{77} + 44 q^{79} + 33 q^{80} + 35 q^{82} - 3 q^{83} - 3 q^{85} + 40 q^{86} + 37 q^{88} + 4 q^{89} + 10 q^{91} - 74 q^{92} + 2 q^{94} + 15 q^{95} + 33 q^{97} - 3 q^{98}+O(q^{100}) $ 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7 + q^10 + 9 * q^11 + 15 * q^13 + 3 * q^14 + 33 * q^16 - 3 * q^17 + 15 * q^19 + 21 * q^20 + 10 * q^22 - 6 * q^23 + 15 * q^25 + q^26 + 35 * q^28 + 8 * q^29 + 22 * q^31 + 21 * q^32 + 9 * q^34 + 10 * q^35 + 4 * q^37 - 14 * q^38 + 13 * q^41 + 24 * q^43 - 5 * q^44 - 3 * q^46 - q^47 + 37 * q^49 + q^50 + 21 * q^52 - 7 * q^53 + 9 * q^55 + 17 * q^56 + 22 * q^58 + 19 * q^59 + 16 * q^61 - 13 * q^62 + 36 * q^64 + 15 * q^65 + 11 * q^67 - 28 * q^68 + 3 * q^70 + 28 * q^71 + 26 * q^73 + 8 * q^74 + 18 * q^76 - 24 * q^77 + 44 * q^79 + 33 * q^80 + 35 * q^82 - 3 * q^83 - 3 * q^85 + 40 * q^86 + 37 * q^88 + 4 * q^89 + 10 * q^91 - 74 * q^92 + 2 * q^94 + 15 * q^95 + 33 * q^97 - 3 * 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 $

$2$	−2.66230	−1.88253	−0.941265	−	0.337668i	$-0.890362\pi$
$2$	−2.66230	−1.88253	−0.941265	+	0.337668i	$0.890362\pi$
$3$	0	0
$3$	0	0
$4$	5.08784	2.54392
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.44674	0.546816	0.273408	−	0.961898i	$-0.411849\pi$
$7$	1.44674	0.546816	0.273408	+	0.961898i	$0.411849\pi$
$8$	−8.22077	−2.90648
$9$	0	0
$10$	−2.66230	−0.841893
$11$	−6.36434	−1.91892	−0.959461	−	0.281842i	$-0.909054\pi$
$11$	−6.36434	−1.91892	−0.959461	+	0.281842i	$0.909054\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.85165	−1.02940
$15$	0	0
$16$	11.7105	2.92762
$17$	−3.28767	−0.797376	−0.398688	−	0.917087i	$-0.630534\pi$
$17$	−3.28767	−0.797376	−0.398688	+	0.917087i	$0.630534\pi$
$18$	0	0
$19$	−2.89517	−0.664198	−0.332099	−	0.943244i	$-0.607757\pi$
$19$	−2.89517	−0.664198	−0.332099	+	0.943244i	$0.607757\pi$
$20$	5.08784	1.13768
$21$	0	0
$22$	16.9438	3.61243
$23$	1.22736	0.255923	0.127961	−	0.991779i	$-0.459157\pi$
$23$	1.22736	0.255923	0.127961	+	0.991779i	$0.459157\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.66230	−0.522120
$27$	0	0
$28$	7.36078	1.39106
$29$	9.47975	1.76035	0.880173	−	0.474654i	$-0.157427\pi$
$29$	9.47975	1.76035	0.880173	+	0.474654i	$0.157427\pi$
$30$	0	0
$31$	2.40673	0.432262	0.216131	−	0.976364i	$-0.430656\pi$
$31$	2.40673	0.432262	0.216131	+	0.976364i	$0.430656\pi$
$32$	−14.7352	−2.60485
$33$	0	0
$34$	8.75276	1.50109
$35$	1.44674	0.244543
$36$	0	0
$37$	−4.81693	−0.791898	−0.395949	−	0.918272i	$-0.629584\pi$
$37$	−4.81693	−0.791898	−0.395949	+	0.918272i	$0.629584\pi$
$38$	7.70782	1.25037
$39$	0	0
$40$	−8.22077	−1.29982
$41$	−5.51892	−0.861910	−0.430955	−	0.902373i	$-0.641823\pi$
$41$	−5.51892	−0.861910	−0.430955	+	0.902373i	$0.641823\pi$
$42$	0	0
$43$	−7.21025	−1.09955	−0.549777	−	0.835312i	$-0.685287\pi$
$43$	−7.21025	−1.09955	−0.549777	+	0.835312i	$0.685287\pi$
$44$	−32.3808	−4.88159
$45$	0	0
$46$	−3.26760	−0.481782
$47$	−8.00214	−1.16723	−0.583616	−	0.812030i	$-0.698363\pi$
$47$	−8.00214	−1.16723	−0.583616	+	0.812030i	$0.698363\pi$
$48$	0	0
$49$	−4.90695	−0.700992
$50$	−2.66230	−0.376506
$51$	0	0
$52$	5.08784	0.705557
$53$	0.623139	0.0855948	0.0427974	−	0.999084i	$-0.486373\pi$
$53$	0.623139	0.0855948	0.0427974	+	0.999084i	$0.486373\pi$
$54$	0	0
$55$	−6.36434	−0.858168
$56$	−11.8933	−1.58931
$57$	0	0
$58$	−25.2379	−3.31390
$59$	9.87657	1.28582	0.642910	−	0.765942i	$-0.277727\pi$
$59$	9.87657	1.28582	0.642910	+	0.765942i	$0.277727\pi$
$60$	0	0
$61$	4.48864	0.574712	0.287356	−	0.957824i	$-0.407224\pi$
$61$	4.48864	0.574712	0.287356	+	0.957824i	$0.407224\pi$
$62$	−6.40745	−0.813747
$63$	0	0
$64$	15.8087	1.97609
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.26424	−0.398791	−0.199395	−	0.979919i	$-0.563898\pi$
$67$	−3.26424	−0.398791	−0.199395	+	0.979919i	$0.563898\pi$
$68$	−16.7271	−2.02846
$69$	0	0
$70$	−3.85165	−0.460361
$71$	9.19831	1.09164	0.545819	−	0.837903i	$-0.316218\pi$
$71$	9.19831	1.09164	0.545819	+	0.837903i	$0.316218\pi$
$72$	0	0
$73$	9.55989	1.11890	0.559450	−	0.828864i	$-0.311012\pi$
$73$	9.55989	1.11890	0.559450	+	0.828864i	$0.311012\pi$
$74$	12.8241	1.49077
$75$	0	0
$76$	−14.7302	−1.68967
$77$	−9.20754	−1.04930
$78$	0	0
$79$	15.6457	1.76028	0.880138	−	0.474719i	$-0.157450\pi$
$79$	15.6457	1.76028	0.880138	+	0.474719i	$0.157450\pi$
$80$	11.7105	1.30927
$81$	0	0
$82$	14.6930	1.62257
$83$	1.05295	0.115576	0.0577882	−	0.998329i	$-0.481595\pi$
$83$	1.05295	0.115576	0.0577882	+	0.998329i	$0.481595\pi$
$84$	0	0
$85$	−3.28767	−0.356598
$86$	19.1959	2.06994
$87$	0	0
$88$	52.3198	5.57731
$89$	−2.96850	−0.314660	−0.157330	−	0.987546i	$-0.550289\pi$
$89$	−2.96850	−0.314660	−0.157330	+	0.987546i	$0.550289\pi$
$90$	0	0
$91$	1.44674	0.151659
$92$	6.24462	0.651047
$93$	0	0
$94$	21.3041	2.19735
$95$	−2.89517	−0.297039
$96$	0	0
$97$	−11.5357	−1.17128	−0.585638	−	0.810573i	$-0.699156\pi$
$97$	−11.5357	−1.17128	−0.585638	+	0.810573i	$0.699156\pi$
$98$	13.0638	1.31964
$99$	0	0
$100$	5.08784	0.508784
$101$	−14.7871	−1.47137	−0.735685	−	0.677324i	$-0.763139\pi$
$101$	−14.7871	−1.47137	−0.735685	+	0.677324i	$0.763139\pi$
$102$	0	0
$103$	17.3651	1.71103	0.855517	−	0.517775i	$-0.173240\pi$
$103$	17.3651	1.71103	0.855517	+	0.517775i	$0.173240\pi$
$104$	−8.22077	−0.806113
$105$	0	0
$106$	−1.65898	−0.161135
$107$	0.863850	0.0835116	0.0417558	−	0.999128i	$-0.486705\pi$
$107$	0.863850	0.0835116	0.0417558	+	0.999128i	$0.486705\pi$
$108$	0	0
$109$	−8.42962	−0.807411	−0.403706	−	0.914889i	$-0.632278\pi$
$109$	−8.42962	−0.807411	−0.403706	+	0.914889i	$0.632278\pi$
$110$	16.9438	1.61553
$111$	0	0
$112$	16.9420	1.60087
$113$	15.5135	1.45938	0.729692	−	0.683776i	$-0.239663\pi$
$113$	15.5135	1.45938	0.729692	+	0.683776i	$0.239663\pi$
$114$	0	0
$115$	1.22736	0.114452
$116$	48.2315	4.47818
$117$	0	0
$118$	−26.2944	−2.42060
$119$	−4.75640	−0.436018
$120$	0	0
$121$	29.5049	2.68226
$122$	−11.9501	−1.08191
$123$	0	0
$124$	12.2451	1.09964
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.94619	−0.616374	−0.308187	−	0.951326i	$-0.599722\pi$
$127$	−6.94619	−0.616374	−0.308187	+	0.951326i	$0.599722\pi$
$128$	−12.6170	−1.11520
$129$	0	0
$130$	−2.66230	−0.233499
$131$	2.51141	0.219423	0.109711	−	0.993963i	$-0.465007\pi$
$131$	2.51141	0.219423	0.109711	+	0.993963i	$0.465007\pi$
$132$	0	0
$133$	−4.18856	−0.363194
$134$	8.69040	0.750736
$135$	0	0
$136$	27.0271	2.31756
$137$	−11.7412	−1.00312	−0.501560	−	0.865123i	$-0.667240\pi$
$137$	−11.7412	−1.00312	−0.501560	+	0.865123i	$0.667240\pi$
$138$	0	0
$139$	5.08722	0.431493	0.215746	−	0.976449i	$-0.430782\pi$
$139$	5.08722	0.431493	0.215746	+	0.976449i	$0.430782\pi$
$140$	7.36078	0.622099
$141$	0	0
$142$	−24.4887	−2.05504
$143$	−6.36434	−0.532213
$144$	0	0
$145$	9.47975	0.787250
$146$	−25.4513	−2.10637
$147$	0	0
$148$	−24.5078	−2.01453
$149$	14.1931	1.16274	0.581372	−	0.813638i	$-0.302516\pi$
$149$	14.1931	1.16274	0.581372	+	0.813638i	$0.302516\pi$
$150$	0	0
$151$	18.6565	1.51825	0.759123	−	0.650947i	$-0.225628\pi$
$151$	18.6565	1.51825	0.759123	+	0.650947i	$0.225628\pi$
$152$	23.8006	1.93048
$153$	0	0
$154$	24.5132	1.97533
$155$	2.40673	0.193314
$156$	0	0
$157$	−8.95652	−0.714808	−0.357404	−	0.933950i	$-0.616338\pi$
$157$	−8.95652	−0.714808	−0.357404	+	0.933950i	$0.616338\pi$
$158$	−41.6535	−3.31377
$159$	0	0
$160$	−14.7352	−1.16492
$161$	1.77567	0.139942
$162$	0	0
$163$	7.33625	0.574619	0.287310	−	0.957838i	$-0.407239\pi$
$163$	7.33625	0.574619	0.287310	+	0.957838i	$0.407239\pi$
$164$	−28.0794	−2.19263
$165$	0	0
$166$	−2.80327	−0.217576
$167$	24.3873	1.88715	0.943574	−	0.331162i	$-0.107441\pi$
$167$	24.3873	1.88715	0.943574	+	0.331162i	$0.107441\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	8.75276	0.671306
$171$	0	0
$172$	−36.6846	−2.79718
$173$	−6.46539	−0.491554	−0.245777	−	0.969326i	$-0.579043\pi$
$173$	−6.46539	−0.491554	−0.245777	+	0.969326i	$0.579043\pi$
$174$	0	0
$175$	1.44674	0.109363
$176$	−74.5294	−5.61787
$177$	0	0
$178$	7.90304	0.592358
$179$	22.9708	1.71692	0.858460	−	0.512880i	$-0.171422\pi$
$179$	22.9708	1.71692	0.858460	+	0.512880i	$0.171422\pi$
$180$	0	0
$181$	−6.61872	−0.491966	−0.245983	−	0.969274i	$-0.579111\pi$
$181$	−6.61872	−0.491966	−0.245983	+	0.969274i	$0.579111\pi$
$182$	−3.85165	−0.285503
$183$	0	0
$184$	−10.0899	−0.743834
$185$	−4.81693	−0.354148
$186$	0	0
$187$	20.9238	1.53010
$188$	−40.7136	−2.96935
$189$	0	0
$190$	7.70782	0.559184
$191$	17.7054	1.28112	0.640558	−	0.767910i	$-0.278703\pi$
$191$	17.7054	1.28112	0.640558	+	0.767910i	$0.278703\pi$
$192$	0	0
$193$	9.44696	0.680007	0.340004	−	0.940424i	$-0.389572\pi$
$193$	9.44696	0.680007	0.340004	+	0.940424i	$0.389572\pi$
$194$	30.7116	2.20496
$195$	0	0
$196$	−24.9658	−1.78327
$197$	−0.754304	−0.0537419	−0.0268710	−	0.999639i	$-0.508554\pi$
$197$	−0.754304	−0.0537419	−0.0268710	+	0.999639i	$0.508554\pi$
$198$	0	0
$199$	11.9275	0.845515	0.422758	−	0.906243i	$-0.361062\pi$
$199$	11.9275	0.845515	0.422758	+	0.906243i	$0.361062\pi$
$200$	−8.22077	−0.581296
$201$	0	0
$202$	39.3676	2.76990
$203$	13.7147	0.962585
$204$	0	0
$205$	−5.51892	−0.385458
$206$	−46.2311	−3.22107
$207$	0	0
$208$	11.7105	0.811975
$209$	18.4259	1.27454
$210$	0	0
$211$	−15.5765	−1.07233	−0.536164	−	0.844114i	$-0.680127\pi$
$211$	−15.5765	−1.07233	−0.536164	+	0.844114i	$0.680127\pi$
$212$	3.17043	0.217746
$213$	0	0
$214$	−2.29983	−0.157213
$215$	−7.21025	−0.491735
$216$	0	0
$217$	3.48192	0.236368
$218$	22.4422	1.51998
$219$	0	0
$220$	−32.3808	−2.18311
$221$	−3.28767	−0.221152
$222$	0	0
$223$	−12.4847	−0.836039	−0.418020	−	0.908438i	$-0.637276\pi$
$223$	−12.4847	−0.836039	−0.418020	+	0.908438i	$0.637276\pi$
$224$	−21.3180	−1.42437
$225$	0	0
$226$	−41.3015	−2.74734
$227$	−10.2169	−0.678119	−0.339059	−	0.940765i	$-0.610109\pi$
$227$	−10.2169	−0.678119	−0.339059	+	0.940765i	$0.610109\pi$
$228$	0	0
$229$	10.9124	0.721110	0.360555	−	0.932738i	$-0.382587\pi$
$229$	10.9124	0.721110	0.360555	+	0.932738i	$0.382587\pi$
$230$	−3.26760	−0.215459
$231$	0	0
$232$	−77.9308	−5.11641
$233$	−28.2905	−1.85337	−0.926685	−	0.375839i	$-0.877354\pi$
$233$	−28.2905	−1.85337	−0.926685	+	0.375839i	$0.877354\pi$
$234$	0	0
$235$	−8.00214	−0.522002
$236$	50.2504	3.27102
$237$	0	0
$238$	12.6630	0.820817
$239$	−0.462699	−0.0299295	−0.0149648	−	0.999888i	$-0.504764\pi$
$239$	−0.462699	−0.0299295	−0.0149648	+	0.999888i	$0.504764\pi$
$240$	0	0
$241$	9.83869	0.633766	0.316883	−	0.948465i	$-0.397364\pi$
$241$	9.83869	0.633766	0.316883	+	0.948465i	$0.397364\pi$
$242$	−78.5508	−5.04944
$243$	0	0
$244$	22.8375	1.46202
$245$	−4.90695	−0.313493
$246$	0	0
$247$	−2.89517	−0.184216
$248$	−19.7852	−1.25636
$249$	0	0
$250$	−2.66230	−0.168379
$251$	16.4559	1.03869	0.519343	−	0.854566i	$-0.326177\pi$
$251$	16.4559	1.03869	0.519343	+	0.854566i	$0.326177\pi$
$252$	0	0
$253$	−7.81135	−0.491095
$254$	18.4928	1.16034
$255$	0	0
$256$	1.97296	0.123310
$257$	20.2069	1.26047	0.630237	−	0.776403i	$-0.282958\pi$
$257$	20.2069	1.26047	0.630237	+	0.776403i	$0.282958\pi$
$258$	0	0
$259$	−6.96884	−0.433023
$260$	5.08784	0.315535
$261$	0	0
$262$	−6.68613	−0.413070
$263$	−4.53943	−0.279913	−0.139957	−	0.990158i	$-0.544696\pi$
$263$	−4.53943	−0.279913	−0.139957	+	0.990158i	$0.544696\pi$
$264$	0	0
$265$	0.623139	0.0382791
$266$	11.1512	0.683724
$267$	0	0
$268$	−16.6080	−1.01449
$269$	27.2598	1.66206	0.831028	−	0.556230i	$-0.187753\pi$
$269$	27.2598	1.66206	0.831028	+	0.556230i	$0.187753\pi$
$270$	0	0
$271$	31.1821	1.89418	0.947088	−	0.320974i	$-0.104010\pi$
$271$	31.1821	1.89418	0.947088	+	0.320974i	$0.104010\pi$
$272$	−38.5001	−2.33441
$273$	0	0
$274$	31.2587	1.88840
$275$	−6.36434	−0.383784
$276$	0	0
$277$	−3.54173	−0.212802	−0.106401	−	0.994323i	$-0.533933\pi$
$277$	−3.54173	−0.212802	−0.106401	+	0.994323i	$0.533933\pi$
$278$	−13.5437	−0.812298
$279$	0	0
$280$	−11.8933	−0.710761
$281$	−21.1160	−1.25967	−0.629837	−	0.776727i	$-0.716878\pi$
$281$	−21.1160	−1.25967	−0.629837	+	0.776727i	$0.716878\pi$
$282$	0	0
$283$	−8.72407	−0.518592	−0.259296	−	0.965798i	$-0.583491\pi$
$283$	−8.72407	−0.518592	−0.259296	+	0.965798i	$0.583491\pi$
$284$	46.7996	2.77704
$285$	0	0
$286$	16.9438	1.00191
$287$	−7.98443	−0.471306
$288$	0	0
$289$	−6.19124	−0.364191
$290$	−25.2379	−1.48202
$291$	0	0
$292$	48.6392	2.84640
$293$	−23.4778	−1.37159	−0.685793	−	0.727796i	$-0.740545\pi$
$293$	−23.4778	−1.37159	−0.685793	+	0.727796i	$0.740545\pi$
$294$	0	0
$295$	9.87657	0.575036
$296$	39.5989	2.30164
$297$	0	0
$298$	−37.7863	−2.18890
$299$	1.22736	0.0709801
$300$	0	0
$301$	−10.4314	−0.601253
$302$	−49.6693	−2.85815
$303$	0	0
$304$	−33.9038	−1.94452
$305$	4.48864	0.257019
$306$	0	0
$307$	−5.57485	−0.318173	−0.159087	−	0.987265i	$-0.550855\pi$
$307$	−5.57485	−0.318173	−0.159087	+	0.987265i	$0.550855\pi$
$308$	−46.8465	−2.66933
$309$	0	0
$310$	−6.40745	−0.363919
$311$	10.1906	0.577856	0.288928	−	0.957351i	$-0.406701\pi$
$311$	10.1906	0.577856	0.288928	+	0.957351i	$0.406701\pi$
$312$	0	0
$313$	−3.54499	−0.200375	−0.100187	−	0.994969i	$-0.531944\pi$
$313$	−3.54499	−0.200375	−0.100187	+	0.994969i	$0.531944\pi$
$314$	23.8449	1.34565
$315$	0	0
$316$	79.6027	4.47800
$317$	4.74339	0.266415	0.133208	−	0.991088i	$-0.457472\pi$
$317$	4.74339	0.266415	0.133208	+	0.991088i	$0.457472\pi$
$318$	0	0
$319$	−60.3324	−3.37796
$320$	15.8087	0.883734
$321$	0	0
$322$	−4.72737	−0.263446
$323$	9.51837	0.529616
$324$	0	0
$325$	1.00000	0.0554700
$326$	−19.5313	−1.08174
$327$	0	0
$328$	45.3697	2.50512
$329$	−11.5770	−0.638261
$330$	0	0
$331$	−12.9858	−0.713764	−0.356882	−	0.934149i	$-0.616160\pi$
$331$	−12.9858	−0.713764	−0.356882	+	0.934149i	$0.616160\pi$
$332$	5.35725	0.294017
$333$	0	0
$334$	−64.9264	−3.55261
$335$	−3.26424	−0.178345
$336$	0	0
$337$	−29.2234	−1.59190	−0.795949	−	0.605363i	$-0.793028\pi$
$337$	−29.2234	−1.59190	−0.795949	+	0.605363i	$0.793028\pi$
$338$	−2.66230	−0.144810
$339$	0	0
$340$	−16.7271	−0.907156
$341$	−15.3173	−0.829477
$342$	0	0
$343$	−17.2262	−0.930130
$344$	59.2738	3.19583
$345$	0	0
$346$	17.2128	0.925366
$347$	24.1448	1.29616	0.648080	−	0.761573i	$-0.275572\pi$
$347$	24.1448	1.29616	0.648080	+	0.761573i	$0.275572\pi$
$348$	0	0
$349$	−14.8861	−0.796837	−0.398419	−	0.917204i	$-0.630441\pi$
$349$	−14.8861	−0.796837	−0.398419	+	0.917204i	$0.630441\pi$
$350$	−3.85165	−0.205879
$351$	0	0
$352$	93.7801	4.99850
$353$	−4.54082	−0.241683	−0.120842	−	0.992672i	$-0.538559\pi$
$353$	−4.54082	−0.241683	−0.120842	+	0.992672i	$0.538559\pi$
$354$	0	0
$355$	9.19831	0.488196
$356$	−15.1033	−0.800471
$357$	0	0
$358$	−61.1552	−3.23216
$359$	7.63511	0.402966	0.201483	−	0.979492i	$-0.435424\pi$
$359$	7.63511	0.402966	0.201483	+	0.979492i	$0.435424\pi$
$360$	0	0
$361$	−10.6180	−0.558840
$362$	17.6210	0.926140
$363$	0	0
$364$	7.36078	0.385810
$365$	9.55989	0.500388
$366$	0	0
$367$	−3.87746	−0.202402	−0.101201	−	0.994866i	$-0.532268\pi$
$367$	−3.87746	−0.202402	−0.101201	+	0.994866i	$0.532268\pi$
$368$	14.3730	0.749243
$369$	0	0
$370$	12.8241	0.666694
$371$	0.901520	0.0468046
$372$	0	0
$373$	13.0213	0.674215	0.337108	−	0.941466i	$-0.390551\pi$
$373$	13.0213	0.674215	0.337108	+	0.941466i	$0.390551\pi$
$374$	−55.7056	−2.88047
$375$	0	0
$376$	65.7837	3.39254
$377$	9.47975	0.488232
$378$	0	0
$379$	9.24190	0.474724	0.237362	−	0.971421i	$-0.423717\pi$
$379$	9.24190	0.474724	0.237362	+	0.971421i	$0.423717\pi$
$380$	−14.7302	−0.755643
$381$	0	0
$382$	−47.1370	−2.41174
$383$	15.1249	0.772848	0.386424	−	0.922321i	$-0.373710\pi$
$383$	15.1249	0.772848	0.386424	+	0.922321i	$0.373710\pi$
$384$	0	0
$385$	−9.20754	−0.469260
$386$	−25.1507	−1.28013
$387$	0	0
$388$	−58.6920	−2.97963
$389$	18.0127	0.913278	0.456639	−	0.889652i	$-0.349053\pi$
$389$	18.0127	0.913278	0.456639	+	0.889652i	$0.349053\pi$
$390$	0	0
$391$	−4.03516	−0.204067
$392$	40.3389	2.03742
$393$	0	0
$394$	2.00818	0.101171
$395$	15.6457	0.787219
$396$	0	0
$397$	9.57454	0.480533	0.240266	−	0.970707i	$-0.422765\pi$
$397$	9.57454	0.480533	0.240266	+	0.970707i	$0.422765\pi$
$398$	−31.7545	−1.59171
$399$	0	0
$400$	11.7105	0.585523
$401$	−0.243799	−0.0121748	−0.00608738	−	0.999981i	$-0.501938\pi$
$401$	−0.243799	−0.0121748	−0.00608738	+	0.999981i	$0.501938\pi$
$402$	0	0
$403$	2.40673	0.119888
$404$	−75.2343	−3.74305
$405$	0	0
$406$	−36.5127	−1.81210
$407$	30.6566	1.51959
$408$	0	0
$409$	19.2034	0.949547	0.474774	−	0.880108i	$-0.342530\pi$
$409$	19.2034	0.949547	0.474774	+	0.880108i	$0.342530\pi$
$410$	14.6930	0.725636
$411$	0	0
$412$	88.3509	4.35274
$413$	14.2888	0.703107
$414$	0	0
$415$	1.05295	0.0516874
$416$	−14.7352	−0.722455
$417$	0	0
$418$	−49.0552	−2.39937
$419$	5.45925	0.266702	0.133351	−	0.991069i	$-0.457426\pi$
$419$	5.45925	0.266702	0.133351	+	0.991069i	$0.457426\pi$
$420$	0	0
$421$	−5.15516	−0.251247	−0.125624	−	0.992078i	$-0.540093\pi$
$421$	−5.15516	−0.251247	−0.125624	+	0.992078i	$0.540093\pi$
$422$	41.4693	2.01869
$423$	0	0
$424$	−5.12268	−0.248779
$425$	−3.28767	−0.159475
$426$	0	0
$427$	6.49389	0.314262
$428$	4.39514	0.212447
$429$	0	0
$430$	19.1959	0.925707
$431$	−14.8501	−0.715306	−0.357653	−	0.933855i	$-0.616423\pi$
$431$	−14.8501	−0.715306	−0.357653	+	0.933855i	$0.616423\pi$
$432$	0	0
$433$	26.7260	1.28437	0.642185	−	0.766549i	$-0.278028\pi$
$433$	26.7260	1.28437	0.642185	+	0.766549i	$0.278028\pi$
$434$	−9.26991	−0.444970
$435$	0	0
$436$	−42.8886	−2.05399
$437$	−3.55342	−0.169983
$438$	0	0
$439$	−4.65750	−0.222291	−0.111145	−	0.993804i	$-0.535452\pi$
$439$	−4.65750	−0.222291	−0.111145	+	0.993804i	$0.535452\pi$
$440$	52.3198	2.49425
$441$	0	0
$442$	8.75276	0.416326
$443$	−0.741780	−0.0352430	−0.0176215	−	0.999845i	$-0.505609\pi$
$443$	−0.741780	−0.0352430	−0.0176215	+	0.999845i	$0.505609\pi$
$444$	0	0
$445$	−2.96850	−0.140720
$446$	33.2381	1.57387
$447$	0	0
$448$	22.8711	1.08056
$449$	8.35094	0.394105	0.197053	−	0.980393i	$-0.436863\pi$
$449$	8.35094	0.394105	0.197053	+	0.980393i	$0.436863\pi$
$450$	0	0
$451$	35.1243	1.65394
$452$	78.9301	3.71256
$453$	0	0
$454$	27.2004	1.27658
$455$	1.44674	0.0678242
$456$	0	0
$457$	21.5819	1.00956	0.504779	−	0.863248i	$-0.331574\pi$
$457$	21.5819	1.00956	0.504779	+	0.863248i	$0.331574\pi$
$458$	−29.0520	−1.35751
$459$	0	0
$460$	6.24462	0.291157
$461$	−1.03502	−0.0482056	−0.0241028	−	0.999709i	$-0.507673\pi$
$461$	−1.03502	−0.0482056	−0.0241028	+	0.999709i	$0.507673\pi$
$462$	0	0
$463$	−13.3990	−0.622704	−0.311352	−	0.950295i	$-0.600782\pi$
$463$	−13.3990	−0.622704	−0.311352	+	0.950295i	$0.600782\pi$
$464$	111.012	5.15362
$465$	0	0
$466$	75.3177	3.48903
$467$	33.3566	1.54356	0.771779	−	0.635891i	$-0.219367\pi$
$467$	33.3566	1.54356	0.771779	+	0.635891i	$0.219367\pi$
$468$	0	0
$469$	−4.72251	−0.218065
$470$	21.3041	0.982684
$471$	0	0
$472$	−81.1930	−3.73721
$473$	45.8885	2.10996
$474$	0	0
$475$	−2.89517	−0.132840
$476$	−24.1998	−1.10920
$477$	0	0
$478$	1.23184	0.0563432
$479$	8.45851	0.386479	0.193240	−	0.981152i	$-0.438101\pi$
$479$	8.45851	0.386479	0.193240	+	0.981152i	$0.438101\pi$
$480$	0	0
$481$	−4.81693	−0.219633
$482$	−26.1936	−1.19308
$483$	0	0
$484$	150.116	6.82346
$485$	−11.5357	−0.523810
$486$	0	0
$487$	17.1854	0.778745	0.389372	−	0.921080i	$-0.372692\pi$
$487$	17.1854	0.778745	0.389372	+	0.921080i	$0.372692\pi$
$488$	−36.9001	−1.67039
$489$	0	0
$490$	13.0638	0.590161
$491$	11.2292	0.506768	0.253384	−	0.967366i	$-0.418456\pi$
$491$	11.2292	0.506768	0.253384	+	0.967366i	$0.418456\pi$
$492$	0	0
$493$	−31.1663	−1.40366
$494$	7.70782	0.346791
$495$	0	0
$496$	28.1840	1.26550
$497$	13.3076	0.596925
$498$	0	0
$499$	−11.0319	−0.493855	−0.246927	−	0.969034i	$-0.579421\pi$
$499$	−11.0319	−0.493855	−0.246927	+	0.969034i	$0.579421\pi$
$500$	5.08784	0.227535
$501$	0	0
$502$	−43.8105	−1.95536
$503$	−11.0781	−0.493947	−0.246973	−	0.969022i	$-0.579436\pi$
$503$	−11.0781	−0.493947	−0.246973	+	0.969022i	$0.579436\pi$
$504$	0	0
$505$	−14.7871	−0.658016
$506$	20.7962	0.924502
$507$	0	0
$508$	−35.3411	−1.56801
$509$	−13.0115	−0.576724	−0.288362	−	0.957521i	$-0.593111\pi$
$509$	−13.0115	−0.576724	−0.288362	+	0.957521i	$0.593111\pi$
$510$	0	0
$511$	13.8307	0.611833
$512$	19.9815	0.883066
$513$	0	0
$514$	−53.7969	−2.37288
$515$	17.3651	0.765197
$516$	0	0
$517$	50.9283	2.23983
$518$	18.5531	0.815178
$519$	0	0
$520$	−8.22077	−0.360504
$521$	23.0283	1.00889	0.504443	−	0.863445i	$-0.331698\pi$
$521$	23.0283	1.00889	0.504443	+	0.863445i	$0.331698\pi$
$522$	0	0
$523$	22.3493	0.977266	0.488633	−	0.872489i	$-0.337496\pi$
$523$	22.3493	0.977266	0.488633	+	0.872489i	$0.337496\pi$
$524$	12.7777	0.558194
$525$	0	0
$526$	12.0853	0.526946
$527$	−7.91254	−0.344676
$528$	0	0
$529$	−21.4936	−0.934504
$530$	−1.65898	−0.0720617
$531$	0	0
$532$	−21.3107	−0.923938
$533$	−5.51892	−0.239051
$534$	0	0
$535$	0.863850	0.0373475
$536$	26.8346	1.15908
$537$	0	0
$538$	−72.5737	−3.12887
$539$	31.2295	1.34515
$540$	0	0
$541$	−2.95715	−0.127138	−0.0635690	−	0.997977i	$-0.520248\pi$
$541$	−2.95715	−0.127138	−0.0635690	+	0.997977i	$0.520248\pi$
$542$	−83.0160	−3.56584
$543$	0	0
$544$	48.4446	2.07704
$545$	−8.42962	−0.361085
$546$	0	0
$547$	17.1716	0.734206	0.367103	−	0.930180i	$-0.380350\pi$
$547$	17.1716	0.734206	0.367103	+	0.930180i	$0.380350\pi$
$548$	−59.7375	−2.55186
$549$	0	0
$550$	16.9438	0.722486
$551$	−27.4455	−1.16922
$552$	0	0
$553$	22.6352	0.962546
$554$	9.42915	0.400606
$555$	0	0
$556$	25.8830	1.09768
$557$	−18.7370	−0.793912	−0.396956	−	0.917838i	$-0.629934\pi$
$557$	−18.7370	−0.793912	−0.396956	+	0.917838i	$0.629934\pi$
$558$	0	0
$559$	−7.21025	−0.304961
$560$	16.9420	0.715929
$561$	0	0
$562$	56.2171	2.37137
$563$	1.40583	0.0592485	0.0296243	−	0.999561i	$-0.490569\pi$
$563$	1.40583	0.0592485	0.0296243	+	0.999561i	$0.490569\pi$
$564$	0	0
$565$	15.5135	0.652656
$566$	23.2261	0.976265
$567$	0	0
$568$	−75.6172	−3.17283
$569$	19.6578	0.824096	0.412048	−	0.911162i	$-0.364814\pi$
$569$	19.6578	0.824096	0.412048	+	0.911162i	$0.364814\pi$
$570$	0	0
$571$	−9.42379	−0.394373	−0.197187	−	0.980366i	$-0.563181\pi$
$571$	−9.42379	−0.394373	−0.197187	+	0.980366i	$0.563181\pi$
$572$	−32.3808	−1.35391
$573$	0	0
$574$	21.2570	0.887248
$575$	1.22736	0.0511845
$576$	0	0
$577$	43.3327	1.80397	0.901983	−	0.431772i	$-0.142111\pi$
$577$	43.3327	1.80397	0.901983	+	0.431772i	$0.142111\pi$
$578$	16.4829	0.685600
$579$	0	0
$580$	48.2315	2.00270
$581$	1.52335	0.0631990
$582$	0	0
$583$	−3.96587	−0.164250
$584$	−78.5896	−3.25206
$585$	0	0
$586$	62.5049	2.58205
$587$	39.7705	1.64150	0.820752	−	0.571285i	$-0.193555\pi$
$587$	39.7705	1.64150	0.820752	+	0.571285i	$0.193555\pi$
$588$	0	0
$589$	−6.96792	−0.287108
$590$	−26.2944	−1.08252
$591$	0	0
$592$	−56.4085	−2.31837
$593$	5.46481	0.224413	0.112207	−	0.993685i	$-0.464208\pi$
$593$	5.46481	0.224413	0.112207	+	0.993685i	$0.464208\pi$
$594$	0	0
$595$	−4.75640	−0.194993
$596$	72.2123	2.95793
$597$	0	0
$598$	−3.26760	−0.133622
$599$	−7.58354	−0.309855	−0.154928	−	0.987926i	$-0.549514\pi$
$599$	−7.58354	−0.309855	−0.154928	+	0.987926i	$0.549514\pi$
$600$	0	0
$601$	−5.38378	−0.219609	−0.109804	−	0.993953i	$-0.535022\pi$
$601$	−5.38378	−0.219609	−0.109804	+	0.993953i	$0.535022\pi$
$602$	27.7714	1.13188
$603$	0	0
$604$	94.9215	3.86230
$605$	29.5049	1.19954
$606$	0	0
$607$	3.86876	0.157028	0.0785141	−	0.996913i	$-0.474982\pi$
$607$	3.86876	0.157028	0.0785141	+	0.996913i	$0.474982\pi$
$608$	42.6611	1.73014
$609$	0	0
$610$	−11.9501	−0.483846
$611$	−8.00214	−0.323732
$612$	0	0
$613$	−43.7711	−1.76790	−0.883950	−	0.467582i	$-0.845125\pi$
$613$	−43.7711	−1.76790	−0.883950	+	0.467582i	$0.845125\pi$
$614$	14.8419	0.598971
$615$	0	0
$616$	75.6930	3.04976
$617$	−2.56406	−0.103225	−0.0516125	−	0.998667i	$-0.516436\pi$
$617$	−2.56406	−0.103225	−0.0516125	+	0.998667i	$0.516436\pi$
$618$	0	0
$619$	25.6270	1.03004	0.515019	−	0.857179i	$-0.327785\pi$
$619$	25.6270	1.03004	0.515019	+	0.857179i	$0.327785\pi$
$620$	12.2451	0.491775
$621$	0	0
$622$	−27.1304	−1.08783
$623$	−4.29464	−0.172061
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.43784	0.377212
$627$	0	0
$628$	−45.5694	−1.81842
$629$	15.8365	0.631441
$630$	0	0
$631$	−13.7360	−0.546820	−0.273410	−	0.961898i	$-0.588152\pi$
$631$	−13.7360	−0.546820	−0.273410	+	0.961898i	$0.588152\pi$
$632$	−128.619	−5.11621
$633$	0	0
$634$	−12.6283	−0.501535
$635$	−6.94619	−0.275651
$636$	0	0
$637$	−4.90695	−0.194420
$638$	160.623	6.35912
$639$	0	0
$640$	−12.6170	−0.498733
$641$	40.4574	1.59797	0.798986	−	0.601349i	$-0.205370\pi$
$641$	40.4574	1.59797	0.798986	+	0.601349i	$0.205370\pi$
$642$	0	0
$643$	13.0379	0.514165	0.257082	−	0.966389i	$-0.417239\pi$
$643$	13.0379	0.514165	0.257082	+	0.966389i	$0.417239\pi$
$644$	9.03434	0.356003
$645$	0	0
$646$	−25.3408	−0.997019
$647$	−18.2690	−0.718228	−0.359114	−	0.933294i	$-0.616921\pi$
$647$	−18.2690	−0.718228	−0.359114	+	0.933294i	$0.616921\pi$
$648$	0	0
$649$	−62.8579	−2.46739
$650$	−2.66230	−0.104424
$651$	0	0
$652$	37.3257	1.46179
$653$	23.7357	0.928850	0.464425	−	0.885612i	$-0.346261\pi$
$653$	23.7357	0.928850	0.464425	+	0.885612i	$0.346261\pi$
$654$	0	0
$655$	2.51141	0.0981289
$656$	−64.6291	−2.52334
$657$	0	0
$658$	30.8214	1.20155
$659$	15.3329	0.597285	0.298643	−	0.954365i	$-0.403466\pi$
$659$	15.3329	0.597285	0.298643	+	0.954365i	$0.403466\pi$
$660$	0	0
$661$	9.21451	0.358403	0.179201	−	0.983812i	$-0.442649\pi$
$661$	9.21451	0.358403	0.179201	+	0.983812i	$0.442649\pi$
$662$	34.5721	1.34368
$663$	0	0
$664$	−8.65607	−0.335921
$665$	−4.18856	−0.162425
$666$	0	0
$667$	11.6351	0.450512
$668$	124.079	4.80076
$669$	0	0
$670$	8.69040	0.335739
$671$	−28.5673	−1.10283
$672$	0	0
$673$	3.57844	0.137939	0.0689694	−	0.997619i	$-0.478029\pi$
$673$	3.57844	0.137939	0.0689694	+	0.997619i	$0.478029\pi$
$674$	77.8014	2.99680
$675$	0	0
$676$	5.08784	0.195686
$677$	−51.1770	−1.96689	−0.983446	−	0.181201i	$-0.942002\pi$
$677$	−51.1770	−1.96689	−0.983446	+	0.181201i	$0.942002\pi$
$678$	0	0
$679$	−16.6892	−0.640472
$680$	27.0271	1.03644
$681$	0	0
$682$	40.7792	1.56152
$683$	19.6179	0.750657	0.375329	−	0.926892i	$-0.377530\pi$
$683$	19.6179	0.750657	0.375329	+	0.926892i	$0.377530\pi$
$684$	0	0
$685$	−11.7412	−0.448609
$686$	45.8614	1.75100
$687$	0	0
$688$	−84.4354	−3.21907
$689$	0.623139	0.0237397
$690$	0	0
$691$	−42.1642	−1.60400	−0.802000	−	0.597323i	$-0.796231\pi$
$691$	−42.1642	−1.60400	−0.802000	+	0.597323i	$0.796231\pi$
$692$	−32.8949	−1.25048
$693$	0	0
$694$	−64.2806	−2.44006
$695$	5.08722	0.192969
$696$	0	0
$697$	18.1444	0.687267
$698$	39.6314	1.50007
$699$	0	0
$700$	7.36078	0.278211
$701$	1.06508	0.0402275	0.0201138	−	0.999798i	$-0.493597\pi$
$701$	1.06508	0.0402275	0.0201138	+	0.999798i	$0.493597\pi$
$702$	0	0
$703$	13.9458	0.525978
$704$	−100.612	−3.79196
$705$	0	0
$706$	12.0890	0.454976
$707$	−21.3930	−0.804568
$708$	0	0
$709$	12.4910	0.469108	0.234554	−	0.972103i	$-0.424637\pi$
$709$	12.4910	0.469108	0.234554	+	0.972103i	$0.424637\pi$
$710$	−24.4887	−0.919043
$711$	0	0
$712$	24.4034	0.914554
$713$	2.95393	0.110626
$714$	0	0
$715$	−6.36434	−0.238013
$716$	116.872	4.36771
$717$	0	0
$718$	−20.3270	−0.758595
$719$	−53.4746	−1.99426	−0.997132	−	0.0756801i	$-0.975887\pi$
$719$	−53.4746	−1.99426	−0.997132	+	0.0756801i	$0.975887\pi$
$720$	0	0
$721$	25.1228	0.935620
$722$	28.2682	1.05203
$723$	0	0
$724$	−33.6750	−1.25152
$725$	9.47975	0.352069
$726$	0	0
$727$	−5.17588	−0.191963	−0.0959813	−	0.995383i	$-0.530599\pi$
$727$	−5.17588	−0.191963	−0.0959813	+	0.995383i	$0.530599\pi$
$728$	−11.8933	−0.440795
$729$	0	0
$730$	−25.4513	−0.941995
$731$	23.7049	0.876758
$732$	0	0
$733$	−10.1075	−0.373329	−0.186664	−	0.982424i	$-0.559768\pi$
$733$	−10.1075	−0.373329	−0.186664	+	0.982424i	$0.559768\pi$
$734$	10.3230	0.381027
$735$	0	0
$736$	−18.0855	−0.666639
$737$	20.7748	0.765248
$738$	0	0
$739$	23.8634	0.877828	0.438914	−	0.898529i	$-0.355363\pi$
$739$	23.8634	0.877828	0.438914	+	0.898529i	$0.355363\pi$
$740$	−24.5078	−0.900924
$741$	0	0
$742$	−2.40012	−0.0881110
$743$	−48.5128	−1.77976	−0.889881	−	0.456193i	$-0.849213\pi$
$743$	−48.5128	−1.77976	−0.889881	+	0.456193i	$0.849213\pi$
$744$	0	0
$745$	14.1931	0.519995
$746$	−34.6665	−1.26923
$747$	0	0
$748$	106.457	3.89246
$749$	1.24977	0.0456654
$750$	0	0
$751$	3.26158	0.119017	0.0595084	−	0.998228i	$-0.481047\pi$
$751$	3.26158	0.119017	0.0595084	+	0.998228i	$0.481047\pi$
$752$	−93.7087	−3.41721
$753$	0	0
$754$	−25.2379	−0.919112
$755$	18.6565	0.678980
$756$	0	0
$757$	27.8497	1.01221	0.506107	−	0.862471i	$-0.331084\pi$
$757$	27.8497	1.01221	0.506107	+	0.862471i	$0.331084\pi$
$758$	−24.6047	−0.893683
$759$	0	0
$760$	23.8006	0.863337
$761$	1.56427	0.0567046	0.0283523	−	0.999598i	$-0.490974\pi$
$761$	1.56427	0.0567046	0.0283523	+	0.999598i	$0.490974\pi$
$762$	0	0
$763$	−12.1955	−0.441505
$764$	90.0822	3.25906
$765$	0	0
$766$	−40.2671	−1.45491
$767$	9.87657	0.356622
$768$	0	0
$769$	−29.8539	−1.07656	−0.538280	−	0.842766i	$-0.680926\pi$
$769$	−29.8539	−1.07656	−0.538280	+	0.842766i	$0.680926\pi$
$770$	24.5132	0.883396
$771$	0	0
$772$	48.0647	1.72989
$773$	29.3989	1.05741	0.528703	−	0.848807i	$-0.322679\pi$
$773$	29.3989	1.05741	0.528703	+	0.848807i	$0.322679\pi$
$774$	0	0
$775$	2.40673	0.0864525
$776$	94.8325	3.40429
$777$	0	0
$778$	−47.9551	−1.71927
$779$	15.9782	0.572479
$780$	0	0
$781$	−58.5412	−2.09477
$782$	10.7428	0.384162
$783$	0	0
$784$	−57.4626	−2.05224
$785$	−8.95652	−0.319672
$786$	0	0
$787$	−9.20360	−0.328073	−0.164037	−	0.986454i	$-0.552451\pi$
$787$	−9.20360	−0.328073	−0.164037	+	0.986454i	$0.552451\pi$
$788$	−3.83778	−0.136715
$789$	0	0
$790$	−41.6535	−1.48196
$791$	22.4439	0.798014
$792$	0	0
$793$	4.48864	0.159396
$794$	−25.4903	−0.904617
$795$	0	0
$796$	60.6850	2.15092
$797$	−37.5811	−1.33119	−0.665595	−	0.746313i	$-0.731822\pi$
$797$	−37.5811	−1.33119	−0.665595	+	0.746313i	$0.731822\pi$
$798$	0	0
$799$	26.3084	0.930723
$800$	−14.7352	−0.520969
$801$	0	0
$802$	0.649067	0.0229194
$803$	−60.8424	−2.14708
$804$	0	0
$805$	1.77567	0.0625842
$806$	−6.40745	−0.225693
$807$	0	0
$808$	121.561	4.27650
$809$	−45.1472	−1.58729	−0.793646	−	0.608380i	$-0.791820\pi$
$809$	−45.1472	−1.58729	−0.793646	+	0.608380i	$0.791820\pi$
$810$	0	0
$811$	37.3392	1.31116	0.655578	−	0.755127i	$-0.272425\pi$
$811$	37.3392	1.31116	0.655578	+	0.755127i	$0.272425\pi$
$812$	69.7783	2.44874
$813$	0	0
$814$	−81.6171	−2.86068
$815$	7.33625	0.256978
$816$	0	0
$817$	20.8749	0.730322
$818$	−51.1252	−1.78755
$819$	0	0
$820$	−28.0794	−0.980575
$821$	16.6335	0.580513	0.290256	−	0.956949i	$-0.406259\pi$
$821$	16.6335	0.580513	0.290256	+	0.956949i	$0.406259\pi$
$822$	0	0
$823$	28.5067	0.993683	0.496841	−	0.867841i	$-0.334493\pi$
$823$	28.5067	0.993683	0.496841	+	0.867841i	$0.334493\pi$
$824$	−142.754	−4.97308
$825$	0	0
$826$	−38.0411	−1.32362
$827$	−15.3543	−0.533921	−0.266961	−	0.963707i	$-0.586019\pi$
$827$	−15.3543	−0.533921	−0.266961	+	0.963707i	$0.586019\pi$
$828$	0	0
$829$	4.98146	0.173013	0.0865067	−	0.996251i	$-0.472430\pi$
$829$	4.98146	0.173013	0.0865067	+	0.996251i	$0.472430\pi$
$830$	−2.80327	−0.0973030
$831$	0	0
$832$	15.8087	0.548068
$833$	16.1324	0.558955
$834$	0	0
$835$	24.3873	0.843958
$836$	93.7480	3.24234
$837$	0	0
$838$	−14.5342	−0.502074
$839$	−15.6346	−0.539766	−0.269883	−	0.962893i	$-0.586985\pi$
$839$	−15.6346	−0.539766	−0.269883	+	0.962893i	$0.586985\pi$
$840$	0	0
$841$	60.8657	2.09882
$842$	13.7246	0.472981
$843$	0	0
$844$	−79.2507	−2.72792
$845$	1.00000	0.0344010
$846$	0	0
$847$	42.6858	1.46670
$848$	7.29725	0.250589
$849$	0	0
$850$	8.75276	0.300217
$851$	−5.91211	−0.202665
$852$	0	0
$853$	31.5193	1.07920	0.539600	−	0.841922i	$-0.318576\pi$
$853$	31.5193	1.07920	0.539600	+	0.841922i	$0.318576\pi$
$854$	−17.2887	−0.591607
$855$	0	0
$856$	−7.10151	−0.242725
$857$	10.6197	0.362763	0.181382	−	0.983413i	$-0.441943\pi$
$857$	10.6197	0.362763	0.181382	+	0.983413i	$0.441943\pi$
$858$	0	0
$859$	−50.7845	−1.73275	−0.866373	−	0.499398i	$-0.833555\pi$
$859$	−50.7845	−1.73275	−0.866373	+	0.499398i	$0.833555\pi$
$860$	−36.6846	−1.25094
$861$	0	0
$862$	39.5355	1.34658
$863$	−4.65637	−0.158505	−0.0792523	−	0.996855i	$-0.525253\pi$
$863$	−4.65637	−0.158505	−0.0792523	+	0.996855i	$0.525253\pi$
$864$	0	0
$865$	−6.46539	−0.219830
$866$	−71.1527	−2.41787
$867$	0	0
$868$	17.7154	0.601301
$869$	−99.5744	−3.37783
$870$	0	0
$871$	−3.26424	−0.110605
$872$	69.2979	2.34672
$873$	0	0
$874$	9.46028	0.319999
$875$	1.44674	0.0489087
$876$	0	0
$877$	23.8513	0.805401	0.402701	−	0.915332i	$-0.368072\pi$
$877$	23.8513	0.805401	0.402701	+	0.915332i	$0.368072\pi$
$878$	12.3997	0.418469
$879$	0	0
$880$	−74.5294	−2.51239
$881$	−21.1957	−0.714101	−0.357050	−	0.934085i	$-0.616218\pi$
$881$	−21.1957	−0.714101	−0.357050	+	0.934085i	$0.616218\pi$
$882$	0	0
$883$	37.8592	1.27406	0.637032	−	0.770838i	$-0.280162\pi$
$883$	37.8592	1.27406	0.637032	+	0.770838i	$0.280162\pi$
$884$	−16.7271	−0.562595
$885$	0	0
$886$	1.97484	0.0663460
$887$	12.6074	0.423316	0.211658	−	0.977344i	$-0.432114\pi$
$887$	12.6074	0.423316	0.211658	+	0.977344i	$0.432114\pi$
$888$	0	0
$889$	−10.0493	−0.337043
$890$	7.90304	0.264910
$891$	0	0
$892$	−63.5203	−2.12682
$893$	23.1676	0.775273
$894$	0	0
$895$	22.9708	0.767830
$896$	−18.2536	−0.609809
$897$	0	0
$898$	−22.2327	−0.741915
$899$	22.8152	0.760931
$900$	0	0
$901$	−2.04867	−0.0682512
$902$	−93.5114	−3.11359
$903$	0	0
$904$	−127.533	−4.24167
$905$	−6.61872	−0.220014
$906$	0	0
$907$	5.64367	0.187395	0.0936975	−	0.995601i	$-0.470131\pi$
$907$	5.64367	0.187395	0.0936975	+	0.995601i	$0.470131\pi$
$908$	−51.9819	−1.72508
$909$	0	0
$910$	−3.85165	−0.127681
$911$	24.2391	0.803077	0.401538	−	0.915842i	$-0.368476\pi$
$911$	24.2391	0.803077	0.401538	+	0.915842i	$0.368476\pi$
$912$	0	0
$913$	−6.70134	−0.221782
$914$	−57.4575	−1.90052
$915$	0	0
$916$	55.5204	1.83445
$917$	3.63335	0.119984
$918$	0	0
$919$	44.4178	1.46521	0.732603	−	0.680656i	$-0.238305\pi$
$919$	44.4178	1.46521	0.732603	+	0.680656i	$0.238305\pi$
$920$	−10.0899	−0.332653
$921$	0	0
$922$	2.75553	0.0907485
$923$	9.19831	0.302766
$924$	0	0
$925$	−4.81693	−0.158380
$926$	35.6722	1.17226
$927$	0	0
$928$	−139.686	−4.58543
$929$	−22.6060	−0.741679	−0.370840	−	0.928697i	$-0.620930\pi$
$929$	−22.6060	−0.741679	−0.370840	+	0.928697i	$0.620930\pi$
$930$	0	0
$931$	14.2065	0.465598
$932$	−143.937	−4.71483
$933$	0	0
$934$	−88.8052	−2.90579
$935$	20.9238	0.684283
$936$	0	0
$937$	31.1975	1.01918	0.509588	−	0.860418i	$-0.329798\pi$
$937$	31.1975	1.01918	0.509588	+	0.860418i	$0.329798\pi$
$938$	12.5727	0.410514
$939$	0	0
$940$	−40.7136	−1.32793
$941$	−32.3358	−1.05412	−0.527058	−	0.849829i	$-0.676705\pi$
$941$	−32.3358	−1.05412	−0.527058	+	0.849829i	$0.676705\pi$
$942$	0	0
$943$	−6.77371	−0.220582
$944$	115.659	3.76439
$945$	0	0
$946$	−122.169	−3.97206
$947$	26.8768	0.873379	0.436690	−	0.899612i	$-0.356151\pi$
$947$	26.8768	0.873379	0.436690	+	0.899612i	$0.356151\pi$
$948$	0	0
$949$	9.55989	0.310327
$950$	7.70782	0.250075
$951$	0	0
$952$	39.1012	1.26728
$953$	−14.8355	−0.480570	−0.240285	−	0.970702i	$-0.577241\pi$
$953$	−14.8355	−0.480570	−0.240285	+	0.970702i	$0.577241\pi$
$954$	0	0
$955$	17.7054	0.572933
$956$	−2.35414	−0.0761384
$957$	0	0
$958$	−22.5191	−0.727559
$959$	−16.9865	−0.548522
$960$	0	0
$961$	−25.2076	−0.813149
$962$	12.8241	0.413466
$963$	0	0
$964$	50.0577	1.61225
$965$	9.44696	0.304109
$966$	0	0
$967$	−31.7460	−1.02088	−0.510440	−	0.859913i	$-0.670518\pi$
$967$	−31.7460	−1.02088	−0.510440	+	0.859913i	$0.670518\pi$
$968$	−242.553	−7.79593
$969$	0	0
$970$	30.7116	0.986089
$971$	39.4110	1.26476	0.632380	−	0.774658i	$-0.282078\pi$
$971$	39.4110	1.26476	0.632380	+	0.774658i	$0.282078\pi$
$972$	0	0
$973$	7.35988	0.235947
$974$	−45.7527	−1.46601
$975$	0	0
$976$	52.5641	1.68254
$977$	31.3080	1.00163	0.500816	−	0.865554i	$-0.333033\pi$
$977$	31.3080	1.00163	0.500816	+	0.865554i	$0.333033\pi$
$978$	0	0
$979$	18.8926	0.603809
$980$	−24.9658	−0.797503
$981$	0	0
$982$	−29.8956	−0.954006
$983$	45.8260	1.46162	0.730811	−	0.682580i	$-0.239142\pi$
$983$	45.8260	1.46162	0.730811	+	0.682580i	$0.239142\pi$
$984$	0	0
$985$	−0.754304	−0.0240341
$986$	82.9740	2.64243
$987$	0	0
$988$	−14.7302	−0.468630
$989$	−8.84959	−0.281400
$990$	0	0
$991$	29.8542	0.948350	0.474175	−	0.880431i	$-0.342746\pi$
$991$	29.8542	0.948350	0.474175	+	0.880431i	$0.342746\pi$
$992$	−35.4638	−1.12598
$993$	0	0
$994$	−35.4287	−1.12373
$995$	11.9275	0.378126
$996$	0	0
$997$	22.9670	0.727373	0.363686	−	0.931521i	$-0.381518\pi$
$997$	22.9670	0.727373	0.363686	+	0.931521i	$0.381518\pi$
$998$	29.3702	0.929696
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bl.1.1		15
3.2	odd	2		5265.2.a.bk.1.15		15
9.2	odd	6		585.2.i.h.391.1	yes	30
9.4	even	3		1755.2.i.h.586.15		30
9.5	odd	6		585.2.i.h.196.1	✓	30
9.7	even	3		1755.2.i.h.1171.15		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.h.196.1	✓	30	9.5	odd	6
585.2.i.h.391.1	yes	30	9.2	odd	6
1755.2.i.h.586.15		30	9.4	even	3
1755.2.i.h.1171.15		30	9.7	even	3
5265.2.a.bk.1.15		15	3.2	odd	2
5265.2.a.bl.1.1		15	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.66230 q^{2} +5.08784 q^{4} +1.00000 q^{5} +1.44674 q^{7} -8.22077 q^{8} +O(q^{10})\)	\(q-2.66230 q^{2} +5.08784 q^{4} +1.00000 q^{5} +1.44674 q^{7} -8.22077 q^{8} -2.66230 q^{10} -6.36434 q^{11} +1.00000 q^{13} -3.85165 q^{14} +11.7105 q^{16} -3.28767 q^{17} -2.89517 q^{19} +5.08784 q^{20} +16.9438 q^{22} +1.22736 q^{23} +1.00000 q^{25} -2.66230 q^{26} +7.36078 q^{28} +9.47975 q^{29} +2.40673 q^{31} -14.7352 q^{32} +8.75276 q^{34} +1.44674 q^{35} -4.81693 q^{37} +7.70782 q^{38} -8.22077 q^{40} -5.51892 q^{41} -7.21025 q^{43} -32.3808 q^{44} -3.26760 q^{46} -8.00214 q^{47} -4.90695 q^{49} -2.66230 q^{50} +5.08784 q^{52} +0.623139 q^{53} -6.36434 q^{55} -11.8933 q^{56} -25.2379 q^{58} +9.87657 q^{59} +4.48864 q^{61} -6.40745 q^{62} +15.8087 q^{64} +1.00000 q^{65} -3.26424 q^{67} -16.7271 q^{68} -3.85165 q^{70} +9.19831 q^{71} +9.55989 q^{73} +12.8241 q^{74} -14.7302 q^{76} -9.20754 q^{77} +15.6457 q^{79} +11.7105 q^{80} +14.6930 q^{82} +1.05295 q^{83} -3.28767 q^{85} +19.1959 q^{86} +52.3198 q^{88} -2.96850 q^{89} +1.44674 q^{91} +6.24462 q^{92} +21.3041 q^{94} -2.89517 q^{95} -11.5357 q^{97} +13.0638 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7}+O(q^{10}) \) 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7	\( 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7} + q^{10} + 9 q^{11} + 15 q^{13} + 3 q^{14} + 33 q^{16} - 3 q^{17} + 15 q^{19} + 21 q^{20} + 10 q^{22} - 6 q^{23} + 15 q^{25} + q^{26} + 35 q^{28} + 8 q^{29} + 22 q^{31} + 21 q^{32} + 9 q^{34} + 10 q^{35} + 4 q^{37} - 14 q^{38} + 13 q^{41} + 24 q^{43} - 5 q^{44} - 3 q^{46} - q^{47} + 37 q^{49} + q^{50} + 21 q^{52} - 7 q^{53} + 9 q^{55} + 17 q^{56} + 22 q^{58} + 19 q^{59} + 16 q^{61} - 13 q^{62} + 36 q^{64} + 15 q^{65} + 11 q^{67} - 28 q^{68} + 3 q^{70} + 28 q^{71} + 26 q^{73} + 8 q^{74} + 18 q^{76} - 24 q^{77} + 44 q^{79} + 33 q^{80} + 35 q^{82} - 3 q^{83} - 3 q^{85} + 40 q^{86} + 37 q^{88} + 4 q^{89} + 10 q^{91} - 74 q^{92} + 2 q^{94} + 15 q^{95} + 33 q^{97} - 3 q^{98}+O(q^{100}) \) 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7 + q^10 + 9 * q^11 + 15 * q^13 + 3 * q^14 + 33 * q^16 - 3 * q^17 + 15 * q^19 + 21 * q^20 + 10 * q^22 - 6 * q^23 + 15 * q^25 + q^26 + 35 * q^28 + 8 * q^29 + 22 * q^31 + 21 * q^32 + 9 * q^34 + 10 * q^35 + 4 * q^37 - 14 * q^38 + 13 * q^41 + 24 * q^43 - 5 * q^44 - 3 * q^46 - q^47 + 37 * q^49 + q^50 + 21 * q^52 - 7 * q^53 + 9 * q^55 + 17 * q^56 + 22 * q^58 + 19 * q^59 + 16 * q^61 - 13 * q^62 + 36 * q^64 + 15 * q^65 + 11 * q^67 - 28 * q^68 + 3 * q^70 + 28 * q^71 + 26 * q^73 + 8 * q^74 + 18 * q^76 - 24 * q^77 + 44 * q^79 + 33 * q^80 + 35 * q^82 - 3 * q^83 - 3 * q^85 + 40 * q^86 + 37 * q^88 + 4 * q^89 + 10 * q^91 - 74 * q^92 + 2 * q^94 + 15 * q^95 + 33 * q^97 - 3 * q^98

Introduction

L-functions

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