LMFDB - Embedded newform 5265.2.a.bh.1.3

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 20 x^{11} + 17 x^{10} + 150 x^{9} - 109 x^{8} - 522 x^{7} + 334 x^{6} + 846 x^{5} + \cdots - 2 $ x^13 - x^12 - 20x^11 + 17x^10 + 150x^9 - 109x^8 - 522x^7 + 334x^6 + 846x^5 - 490x^4 - 512x^3 + 275x^2 - 3*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.75512$ 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)

$f(q)$	$=$	$q-1.75512 q^{2} +1.08045 q^{4} -1.00000 q^{5} -3.67006 q^{7} +1.61392 q^{8} +O(q^{10})$	\(q-1.75512 q^{2} +1.08045 q^{4} -1.00000 q^{5} -3.67006 q^{7} +1.61392 q^{8} +1.75512 q^{10} -1.37563 q^{11} -1.00000 q^{13} +6.44140 q^{14} -4.99353 q^{16} +0.562928 q^{17} -4.54287 q^{19} -1.08045 q^{20} +2.41440 q^{22} -7.31959 q^{23} +1.00000 q^{25} +1.75512 q^{26} -3.96533 q^{28} -8.81747 q^{29} +6.04332 q^{31} +5.53642 q^{32} -0.988007 q^{34} +3.67006 q^{35} -0.137665 q^{37} +7.97329 q^{38} -1.61392 q^{40} -8.27929 q^{41} -2.35296 q^{43} -1.48630 q^{44} +12.8468 q^{46} +1.42175 q^{47} +6.46933 q^{49} -1.75512 q^{50} -1.08045 q^{52} -5.26501 q^{53} +1.37563 q^{55} -5.92317 q^{56} +15.4757 q^{58} -9.06229 q^{59} -5.00376 q^{61} -10.6068 q^{62} +0.269971 q^{64} +1.00000 q^{65} +6.27745 q^{67} +0.608217 q^{68} -6.44140 q^{70} -3.03434 q^{71} +8.25544 q^{73} +0.241619 q^{74} -4.90836 q^{76} +5.04865 q^{77} +4.27764 q^{79} +4.99353 q^{80} +14.5312 q^{82} +0.238030 q^{83} -0.562928 q^{85} +4.12973 q^{86} -2.22015 q^{88} -2.59479 q^{89} +3.67006 q^{91} -7.90848 q^{92} -2.49535 q^{94} +4.54287 q^{95} -12.1099 q^{97} -11.3545 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + q^{2} + 15 q^{4} - 13 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 13 * q + q^2 + 15 * q^4 - 13 * q^5 + 10 * q^7 + 6 * q^8	$ 13 q + q^{2} + 15 q^{4} - 13 q^{5} + 10 q^{7} + 6 q^{8} - q^{10} - 11 q^{11} - 13 q^{13} - 15 q^{14} + 19 q^{16} - 3 q^{17} + 15 q^{19} - 15 q^{20} + 12 q^{22} - 6 q^{23} + 13 q^{25} - q^{26} + 5 q^{28} - 14 q^{29} + 30 q^{31} + 43 q^{32} + 19 q^{34} - 10 q^{35} + 16 q^{37} + 2 q^{38} - 6 q^{40} - 17 q^{41} + 6 q^{43} - 23 q^{44} + 23 q^{46} + 21 q^{47} + 29 q^{49} + q^{50} - 15 q^{52} - q^{53} + 11 q^{55} - 37 q^{56} + 14 q^{58} - 13 q^{59} + 22 q^{61} + 57 q^{62} + 42 q^{64} + 13 q^{65} + 35 q^{67} + 4 q^{68} + 15 q^{70} - 12 q^{71} + 32 q^{73} + 28 q^{74} + 54 q^{76} - 4 q^{77} + 12 q^{79} - 19 q^{80} + 23 q^{82} + 3 q^{83} + 3 q^{85} + 40 q^{86} + 29 q^{88} - 6 q^{89} - 10 q^{91} + 16 q^{92} + 44 q^{94} - 15 q^{95} + 33 q^{97} - 35 q^{98}+O(q^{100}) $ 13 * q + q^2 + 15 * q^4 - 13 * q^5 + 10 * q^7 + 6 * q^8 - q^10 - 11 * q^11 - 13 * q^13 - 15 * q^14 + 19 * q^16 - 3 * q^17 + 15 * q^19 - 15 * q^20 + 12 * q^22 - 6 * q^23 + 13 * q^25 - q^26 + 5 * q^28 - 14 * q^29 + 30 * q^31 + 43 * q^32 + 19 * q^34 - 10 * q^35 + 16 * q^37 + 2 * q^38 - 6 * q^40 - 17 * q^41 + 6 * q^43 - 23 * q^44 + 23 * q^46 + 21 * q^47 + 29 * q^49 + q^50 - 15 * q^52 - q^53 + 11 * q^55 - 37 * q^56 + 14 * q^58 - 13 * q^59 + 22 * q^61 + 57 * q^62 + 42 * q^64 + 13 * q^65 + 35 * q^67 + 4 * q^68 + 15 * q^70 - 12 * q^71 + 32 * q^73 + 28 * q^74 + 54 * q^76 - 4 * q^77 + 12 * q^79 - 19 * q^80 + 23 * q^82 + 3 * q^83 + 3 * q^85 + 40 * q^86 + 29 * q^88 - 6 * q^89 - 10 * q^91 + 16 * q^92 + 44 * q^94 - 15 * q^95 + 33 * q^97 - 35 * 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 $

$2$	−1.75512	−1.24106	−0.620529	−	0.784183i	$-0.713082\pi$
$2$	−1.75512	−1.24106	−0.620529	+	0.784183i	$0.713082\pi$
$3$	0	0
$3$	0	0
$4$	1.08045	0.540227
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.67006	−1.38715	−0.693576	−	0.720384i	$-0.743966\pi$
$7$	−3.67006	−1.38715	−0.693576	+	0.720384i	$0.743966\pi$
$8$	1.61392	0.570606
$9$	0	0
$10$	1.75512	0.555018
$11$	−1.37563	−0.414768	−0.207384	−	0.978260i	$-0.566495\pi$
$11$	−1.37563	−0.414768	−0.207384	+	0.978260i	$0.566495\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	6.44140	1.72154
$15$	0	0
$16$	−4.99353	−1.24838
$17$	0.562928	0.136530	0.0682650	−	0.997667i	$-0.478254\pi$
$17$	0.562928	0.136530	0.0682650	+	0.997667i	$0.478254\pi$
$18$	0	0
$19$	−4.54287	−1.04221	−0.521103	−	0.853494i	$-0.674479\pi$
$19$	−4.54287	−1.04221	−0.521103	+	0.853494i	$0.674479\pi$
$20$	−1.08045	−0.241597
$21$	0	0
$22$	2.41440	0.514752
$23$	−7.31959	−1.52624	−0.763120	−	0.646257i	$-0.776334\pi$
$23$	−7.31959	−1.52624	−0.763120	+	0.646257i	$0.776334\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.75512	0.344208
$27$	0	0
$28$	−3.96533	−0.749376
$29$	−8.81747	−1.63736	−0.818682	−	0.574247i	$-0.805295\pi$
$29$	−8.81747	−1.63736	−0.818682	+	0.574247i	$0.805295\pi$
$30$	0	0
$31$	6.04332	1.08541	0.542706	−	0.839923i	$-0.317400\pi$
$31$	6.04332	1.08541	0.542706	+	0.839923i	$0.317400\pi$
$32$	5.53642	0.978709
$33$	0	0
$34$	−0.988007	−0.169442
$35$	3.67006	0.620353
$36$	0	0
$37$	−0.137665	−0.0226320	−0.0113160	−	0.999936i	$-0.503602\pi$
$37$	−0.137665	−0.0226320	−0.0113160	+	0.999936i	$0.503602\pi$
$38$	7.97329	1.29344
$39$	0	0
$40$	−1.61392	−0.255183
$41$	−8.27929	−1.29301	−0.646504	−	0.762911i	$-0.723770\pi$
$41$	−8.27929	−1.29301	−0.646504	+	0.762911i	$0.723770\pi$
$42$	0	0
$43$	−2.35296	−0.358823	−0.179411	−	0.983774i	$-0.557419\pi$
$43$	−2.35296	−0.358823	−0.179411	+	0.983774i	$0.557419\pi$
$44$	−1.48630	−0.224069
$45$	0	0
$46$	12.8468	1.89415
$47$	1.42175	0.207384	0.103692	−	0.994609i	$-0.466934\pi$
$47$	1.42175	0.207384	0.103692	+	0.994609i	$0.466934\pi$
$48$	0	0
$49$	6.46933	0.924190
$50$	−1.75512	−0.248212
$51$	0	0
$52$	−1.08045	−0.149832
$53$	−5.26501	−0.723205	−0.361603	−	0.932332i	$-0.617770\pi$
$53$	−5.26501	−0.723205	−0.361603	+	0.932332i	$0.617770\pi$
$54$	0	0
$55$	1.37563	0.185490
$56$	−5.92317	−0.791517
$57$	0	0
$58$	15.4757	2.03206
$59$	−9.06229	−1.17981	−0.589905	−	0.807473i	$-0.700835\pi$
$59$	−9.06229	−1.17981	−0.589905	+	0.807473i	$0.700835\pi$
$60$	0	0
$61$	−5.00376	−0.640666	−0.320333	−	0.947305i	$-0.603795\pi$
$61$	−5.00376	−0.640666	−0.320333	+	0.947305i	$0.603795\pi$
$62$	−10.6068	−1.34706
$63$	0	0
$64$	0.269971	0.0337463
$65$	1.00000	0.124035
$66$	0	0
$67$	6.27745	0.766912	0.383456	−	0.923559i	$-0.374734\pi$
$67$	6.27745	0.766912	0.383456	+	0.923559i	$0.374734\pi$
$68$	0.608217	0.0737571
$69$	0	0
$70$	−6.44140	−0.769895
$71$	−3.03434	−0.360110	−0.180055	−	0.983657i	$-0.557628\pi$
$71$	−3.03434	−0.360110	−0.180055	+	0.983657i	$0.557628\pi$
$72$	0	0
$73$	8.25544	0.966226	0.483113	−	0.875558i	$-0.339506\pi$
$73$	8.25544	0.966226	0.483113	+	0.875558i	$0.339506\pi$
$74$	0.241619	0.0280877
$75$	0	0
$76$	−4.90836	−0.563027
$77$	5.04865	0.575347
$78$	0	0
$79$	4.27764	0.481272	0.240636	−	0.970615i	$-0.422644\pi$
$79$	4.27764	0.481272	0.240636	+	0.970615i	$0.422644\pi$
$80$	4.99353	0.558293
$81$	0	0
$82$	14.5312	1.60470
$83$	0.238030	0.0261272	0.0130636	−	0.999915i	$-0.495842\pi$
$83$	0.238030	0.0261272	0.0130636	+	0.999915i	$0.495842\pi$
$84$	0	0
$85$	−0.562928	−0.0610581
$86$	4.12973	0.445320
$87$	0	0
$88$	−2.22015	−0.236669
$89$	−2.59479	−0.275048	−0.137524	−	0.990498i	$-0.543914\pi$
$89$	−2.59479	−0.275048	−0.137524	+	0.990498i	$0.543914\pi$
$90$	0	0
$91$	3.67006	0.384727
$92$	−7.90848	−0.824516
$93$	0	0
$94$	−2.49535	−0.257375
$95$	4.54287	0.466089
$96$	0	0
$97$	−12.1099	−1.22957	−0.614785	−	0.788695i	$-0.710757\pi$
$97$	−12.1099	−1.22957	−0.614785	+	0.788695i	$0.710757\pi$
$98$	−11.3545	−1.14697
$99$	0	0
$100$	1.08045	0.108045
$101$	−11.7807	−1.17223	−0.586114	−	0.810229i	$-0.699343\pi$
$101$	−11.7807	−1.17223	−0.586114	+	0.810229i	$0.699343\pi$
$102$	0	0
$103$	4.92388	0.485164	0.242582	−	0.970131i	$-0.422006\pi$
$103$	4.92388	0.485164	0.242582	+	0.970131i	$0.422006\pi$
$104$	−1.61392	−0.158258
$105$	0	0
$106$	9.24074	0.897540
$107$	−19.0229	−1.83902	−0.919509	−	0.393070i	$-0.871413\pi$
$107$	−19.0229	−1.83902	−0.919509	+	0.393070i	$0.871413\pi$
$108$	0	0
$109$	−15.8493	−1.51808	−0.759042	−	0.651042i	$-0.774332\pi$
$109$	−15.8493	−1.51808	−0.759042	+	0.651042i	$0.774332\pi$
$110$	−2.41440	−0.230204
$111$	0	0
$112$	18.3265	1.73169
$113$	13.2606	1.24745	0.623725	−	0.781644i	$-0.285618\pi$
$113$	13.2606	1.24745	0.623725	+	0.781644i	$0.285618\pi$
$114$	0	0
$115$	7.31959	0.682555
$116$	−9.52687	−0.884547
$117$	0	0
$118$	15.9054	1.46421
$119$	−2.06598	−0.189388
$120$	0	0
$121$	−9.10764	−0.827967
$122$	8.78221	0.795104
$123$	0	0
$124$	6.52952	0.586369
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.1635	−1.43428	−0.717140	−	0.696929i	$-0.754549\pi$
$127$	−16.1635	−1.43428	−0.717140	+	0.696929i	$0.754549\pi$
$128$	−11.5467	−1.02059
$129$	0	0
$130$	−1.75512	−0.153934
$131$	−11.6490	−1.01778	−0.508888	−	0.860833i	$-0.669943\pi$
$131$	−11.6490	−1.01778	−0.508888	+	0.860833i	$0.669943\pi$
$132$	0	0
$133$	16.6726	1.44570
$134$	−11.0177	−0.951783
$135$	0	0
$136$	0.908519	0.0779048
$137$	−0.893641	−0.0763489	−0.0381745	−	0.999271i	$-0.512154\pi$
$137$	−0.893641	−0.0763489	−0.0381745	+	0.999271i	$0.512154\pi$
$138$	0	0
$139$	−1.02751	−0.0871519	−0.0435759	−	0.999050i	$-0.513875\pi$
$139$	−1.02751	−0.0871519	−0.0435759	+	0.999050i	$0.513875\pi$
$140$	3.96533	0.335131
$141$	0	0
$142$	5.32564	0.446918
$143$	1.37563	0.115036
$144$	0	0
$145$	8.81747	0.732251
$146$	−14.4893	−1.19914
$147$	0	0
$148$	−0.148741	−0.0122264
$149$	1.60139	0.131191	0.0655953	−	0.997846i	$-0.479105\pi$
$149$	1.60139	0.131191	0.0655953	+	0.997846i	$0.479105\pi$
$150$	0	0
$151$	12.2265	0.994980	0.497490	−	0.867470i	$-0.334255\pi$
$151$	12.2265	0.994980	0.497490	+	0.867470i	$0.334255\pi$
$152$	−7.33182	−0.594689
$153$	0	0
$154$	−8.86099	−0.714039
$155$	−6.04332	−0.485411
$156$	0	0
$157$	−9.98438	−0.796841	−0.398420	−	0.917203i	$-0.630441\pi$
$157$	−9.98438	−0.796841	−0.398420	+	0.917203i	$0.630441\pi$
$158$	−7.50777	−0.597286
$159$	0	0
$160$	−5.53642	−0.437692
$161$	26.8633	2.11713
$162$	0	0
$163$	20.6364	1.61637	0.808183	−	0.588931i	$-0.200451\pi$
$163$	20.6364	1.61637	0.808183	+	0.588931i	$0.200451\pi$
$164$	−8.94539	−0.698517
$165$	0	0
$166$	−0.417772	−0.0324254
$167$	−18.2729	−1.41400	−0.706998	−	0.707215i	$-0.749951\pi$
$167$	−18.2729	−1.41400	−0.706998	+	0.707215i	$0.749951\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.988007	0.0757767
$171$	0	0
$172$	−2.54226	−0.193846
$173$	16.0260	1.21844	0.609218	−	0.793003i	$-0.291483\pi$
$173$	16.0260	1.21844	0.609218	+	0.793003i	$0.291483\pi$
$174$	0	0
$175$	−3.67006	−0.277430
$176$	6.86925	0.517789
$177$	0	0
$178$	4.55418	0.341350
$179$	−5.87432	−0.439067	−0.219534	−	0.975605i	$-0.570454\pi$
$179$	−5.87432	−0.439067	−0.219534	+	0.975605i	$0.570454\pi$
$180$	0	0
$181$	−1.09159	−0.0811373	−0.0405686	−	0.999177i	$-0.512917\pi$
$181$	−1.09159	−0.0811373	−0.0405686	+	0.999177i	$0.512917\pi$
$182$	−6.44140	−0.477468
$183$	0	0
$184$	−11.8132	−0.870882
$185$	0.137665	0.0101214
$186$	0	0
$187$	−0.774381	−0.0566283
$188$	1.53613	0.112034
$189$	0	0
$190$	−7.97329	−0.578443
$191$	26.4024	1.91041	0.955205	−	0.295946i	$-0.0956349\pi$
$191$	26.4024	1.91041	0.955205	+	0.295946i	$0.0956349\pi$
$192$	0	0
$193$	5.03133	0.362163	0.181082	−	0.983468i	$-0.442040\pi$
$193$	5.03133	0.362163	0.181082	+	0.983468i	$0.442040\pi$
$194$	21.2543	1.52597
$195$	0	0
$196$	6.98981	0.499272
$197$	22.9176	1.63281	0.816405	−	0.577479i	$-0.195964\pi$
$197$	22.9176	1.63281	0.816405	+	0.577479i	$0.195964\pi$
$198$	0	0
$199$	21.3955	1.51669	0.758345	−	0.651853i	$-0.226008\pi$
$199$	21.3955	1.51669	0.758345	+	0.651853i	$0.226008\pi$
$200$	1.61392	0.114121
$201$	0	0
$202$	20.6766	1.45480
$203$	32.3606	2.27127
$204$	0	0
$205$	8.27929	0.578251
$206$	−8.64200	−0.602117
$207$	0	0
$208$	4.99353	0.346239
$209$	6.24931	0.432274
$210$	0	0
$211$	−27.1988	−1.87245	−0.936223	−	0.351407i	$-0.885703\pi$
$211$	−27.1988	−1.87245	−0.936223	+	0.351407i	$0.885703\pi$
$212$	−5.68860	−0.390695
$213$	0	0
$214$	33.3876	2.28233
$215$	2.35296	0.160470
$216$	0	0
$217$	−22.1793	−1.50563
$218$	27.8174	1.88403
$219$	0	0
$220$	1.48630	0.100207
$221$	−0.562928	−0.0378666
$222$	0	0
$223$	22.1594	1.48391	0.741953	−	0.670452i	$-0.233900\pi$
$223$	22.1594	1.48391	0.741953	+	0.670452i	$0.233900\pi$
$224$	−20.3190	−1.35762
$225$	0	0
$226$	−23.2739	−1.54816
$227$	9.67623	0.642234	0.321117	−	0.947040i	$-0.395942\pi$
$227$	9.67623	0.642234	0.321117	+	0.947040i	$0.395942\pi$
$228$	0	0
$229$	−3.87339	−0.255961	−0.127980	−	0.991777i	$-0.540849\pi$
$229$	−3.87339	−0.255961	−0.127980	+	0.991777i	$0.540849\pi$
$230$	−12.8468	−0.847091
$231$	0	0
$232$	−14.2307	−0.934289
$233$	−9.91014	−0.649235	−0.324617	−	0.945845i	$-0.605236\pi$
$233$	−9.91014	−0.649235	−0.324617	+	0.945845i	$0.605236\pi$
$234$	0	0
$235$	−1.42175	−0.0927448
$236$	−9.79138	−0.637364
$237$	0	0
$238$	3.62604	0.235041
$239$	−12.4092	−0.802687	−0.401344	−	0.915928i	$-0.631457\pi$
$239$	−12.4092	−0.802687	−0.401344	+	0.915928i	$0.631457\pi$
$240$	0	0
$241$	−11.9701	−0.771062	−0.385531	−	0.922695i	$-0.625982\pi$
$241$	−11.9701	−0.771062	−0.385531	+	0.922695i	$0.625982\pi$
$242$	15.9850	1.02756
$243$	0	0
$244$	−5.40633	−0.346105
$245$	−6.46933	−0.413310
$246$	0	0
$247$	4.54287	0.289056
$248$	9.75342	0.619343
$249$	0	0
$250$	1.75512	0.111004
$251$	−19.4893	−1.23016	−0.615078	−	0.788466i	$-0.710875\pi$
$251$	−19.4893	−1.23016	−0.615078	+	0.788466i	$0.710875\pi$
$252$	0	0
$253$	10.0691	0.633036
$254$	28.3690	1.78003
$255$	0	0
$256$	19.7259	1.23287
$257$	−21.3186	−1.32982	−0.664910	−	0.746924i	$-0.731530\pi$
$257$	−21.3186	−1.32982	−0.664910	+	0.746924i	$0.731530\pi$
$258$	0	0
$259$	0.505240	0.0313941
$260$	1.08045	0.0670069
$261$	0	0
$262$	20.4454	1.26312
$263$	30.3813	1.87339	0.936695	−	0.350147i	$-0.113868\pi$
$263$	30.3813	1.87339	0.936695	+	0.350147i	$0.113868\pi$
$264$	0	0
$265$	5.26501	0.323427
$266$	−29.2624	−1.79420
$267$	0	0
$268$	6.78249	0.414306
$269$	−13.1806	−0.803635	−0.401817	−	0.915720i	$-0.631621\pi$
$269$	−13.1806	−0.803635	−0.401817	+	0.915720i	$0.631621\pi$
$270$	0	0
$271$	30.0810	1.82729	0.913646	−	0.406511i	$-0.133255\pi$
$271$	30.0810	1.82729	0.913646	+	0.406511i	$0.133255\pi$
$272$	−2.81100	−0.170442
$273$	0	0
$274$	1.56845	0.0947535
$275$	−1.37563	−0.0829537
$276$	0	0
$277$	−20.5067	−1.23213	−0.616064	−	0.787696i	$-0.711274\pi$
$277$	−20.5067	−1.23213	−0.616064	+	0.787696i	$0.711274\pi$
$278$	1.80340	0.108161
$279$	0	0
$280$	5.92317	0.353977
$281$	−27.1964	−1.62240	−0.811200	−	0.584768i	$-0.801185\pi$
$281$	−27.1964	−1.62240	−0.811200	+	0.584768i	$0.801185\pi$
$282$	0	0
$283$	−5.27364	−0.313485	−0.156743	−	0.987639i	$-0.550099\pi$
$283$	−5.27364	−0.313485	−0.156743	+	0.987639i	$0.550099\pi$
$284$	−3.27847	−0.194541
$285$	0	0
$286$	−2.41440	−0.142766
$287$	30.3855	1.79360
$288$	0	0
$289$	−16.6831	−0.981360
$290$	−15.4757	−0.908767
$291$	0	0
$292$	8.91961	0.521981
$293$	−16.6150	−0.970661	−0.485330	−	0.874331i	$-0.661301\pi$
$293$	−16.6150	−0.970661	−0.485330	+	0.874331i	$0.661301\pi$
$294$	0	0
$295$	9.06229	0.527627
$296$	−0.222180	−0.0129140
$297$	0	0
$298$	−2.81063	−0.162815
$299$	7.31959	0.423303
$300$	0	0
$301$	8.63550	0.497742
$302$	−21.4590	−1.23483
$303$	0	0
$304$	22.6849	1.30107
$305$	5.00376	0.286514
$306$	0	0
$307$	−14.2267	−0.811962	−0.405981	−	0.913882i	$-0.633070\pi$
$307$	−14.2267	−0.811962	−0.405981	+	0.913882i	$0.633070\pi$
$308$	5.45483	0.310818
$309$	0	0
$310$	10.6068	0.602424
$311$	10.5315	0.597189	0.298594	−	0.954380i	$-0.403482\pi$
$311$	10.5315	0.597189	0.298594	+	0.954380i	$0.403482\pi$
$312$	0	0
$313$	5.10664	0.288644	0.144322	−	0.989531i	$-0.453900\pi$
$313$	5.10664	0.288644	0.144322	+	0.989531i	$0.453900\pi$
$314$	17.5238	0.988926
$315$	0	0
$316$	4.62178	0.259996
$317$	22.6895	1.27437	0.637183	−	0.770712i	$-0.280099\pi$
$317$	22.6895	1.27437	0.637183	+	0.770712i	$0.280099\pi$
$318$	0	0
$319$	12.1296	0.679127
$320$	−0.269971	−0.0150918
$321$	0	0
$322$	−47.1484	−2.62748
$323$	−2.55731	−0.142292
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−36.2194	−2.00601
$327$	0	0
$328$	−13.3621	−0.737798
$329$	−5.21791	−0.287673
$330$	0	0
$331$	18.8159	1.03422	0.517109	−	0.855920i	$-0.327008\pi$
$331$	18.8159	1.03422	0.517109	+	0.855920i	$0.327008\pi$
$332$	0.257181	0.0141146
$333$	0	0
$334$	32.0711	1.75485
$335$	−6.27745	−0.342974
$336$	0	0
$337$	18.0544	0.983484	0.491742	−	0.870741i	$-0.336360\pi$
$337$	18.0544	0.983484	0.491742	+	0.870741i	$0.336360\pi$
$338$	−1.75512	−0.0954660
$339$	0	0
$340$	−0.608217	−0.0329852
$341$	−8.31338	−0.450195
$342$	0	0
$343$	1.94760	0.105160
$344$	−3.79748	−0.204746
$345$	0	0
$346$	−28.1276	−1.51215
$347$	15.9493	0.856202	0.428101	−	0.903731i	$-0.359183\pi$
$347$	15.9493	0.856202	0.428101	+	0.903731i	$0.359183\pi$
$348$	0	0
$349$	15.7300	0.842005	0.421003	−	0.907059i	$-0.361678\pi$
$349$	15.7300	0.842005	0.421003	+	0.907059i	$0.361678\pi$
$350$	6.44140	0.344307
$351$	0	0
$352$	−7.61606	−0.405938
$353$	9.54196	0.507867	0.253934	−	0.967222i	$-0.418276\pi$
$353$	9.54196	0.507867	0.253934	+	0.967222i	$0.418276\pi$
$354$	0	0
$355$	3.03434	0.161046
$356$	−2.80355	−0.148588
$357$	0	0
$358$	10.3101	0.544908
$359$	−16.6365	−0.878039	−0.439020	−	0.898478i	$-0.644674\pi$
$359$	−16.6365	−0.878039	−0.439020	+	0.898478i	$0.644674\pi$
$360$	0	0
$361$	1.63768	0.0861935
$362$	1.91587	0.100696
$363$	0	0
$364$	3.96533	0.207840
$365$	−8.25544	−0.432109
$366$	0	0
$367$	−3.94737	−0.206051	−0.103026	−	0.994679i	$-0.532852\pi$
$367$	−3.94737	−0.206051	−0.103026	+	0.994679i	$0.532852\pi$
$368$	36.5506	1.90533
$369$	0	0
$370$	−0.241619	−0.0125612
$371$	19.3229	1.00320
$372$	0	0
$373$	13.5904	0.703682	0.351841	−	0.936060i	$-0.385556\pi$
$373$	13.5904	0.703682	0.351841	+	0.936060i	$0.385556\pi$
$374$	1.35913	0.0702791
$375$	0	0
$376$	2.29459	0.118334
$377$	8.81747	0.454123
$378$	0	0
$379$	15.2538	0.783534	0.391767	−	0.920064i	$-0.371864\pi$
$379$	15.2538	0.783534	0.391767	+	0.920064i	$0.371864\pi$
$380$	4.90836	0.251793
$381$	0	0
$382$	−46.3394	−2.37093
$383$	18.9439	0.967988	0.483994	−	0.875071i	$-0.339186\pi$
$383$	18.9439	0.967988	0.483994	+	0.875071i	$0.339186\pi$
$384$	0	0
$385$	−5.04865	−0.257303
$386$	−8.83061	−0.449466
$387$	0	0
$388$	−13.0841	−0.664246
$389$	−17.4313	−0.883802	−0.441901	−	0.897064i	$-0.645696\pi$
$389$	−17.4313	−0.883802	−0.441901	+	0.897064i	$0.645696\pi$
$390$	0	0
$391$	−4.12040	−0.208378
$392$	10.4410	0.527348
$393$	0	0
$394$	−40.2232	−2.02641
$395$	−4.27764	−0.215231
$396$	0	0
$397$	27.2409	1.36718	0.683591	−	0.729866i	$-0.260417\pi$
$397$	27.2409	1.36718	0.683591	+	0.729866i	$0.260417\pi$
$398$	−37.5518	−1.88230
$399$	0	0
$400$	−4.99353	−0.249676
$401$	−23.8314	−1.19008	−0.595041	−	0.803695i	$-0.702864\pi$
$401$	−23.8314	−1.19008	−0.595041	+	0.803695i	$0.702864\pi$
$402$	0	0
$403$	−6.04332	−0.301039
$404$	−12.7285	−0.633268
$405$	0	0
$406$	−56.7969	−2.81878
$407$	0.189377	0.00938705
$408$	0	0
$409$	1.03114	0.0509865	0.0254933	−	0.999675i	$-0.491884\pi$
$409$	1.03114	0.0509865	0.0254933	+	0.999675i	$0.491884\pi$
$410$	−14.5312	−0.717643
$411$	0	0
$412$	5.32002	0.262098
$413$	33.2591	1.63657
$414$	0	0
$415$	−0.238030	−0.0116845
$416$	−5.53642	−0.271445
$417$	0	0
$418$	−10.9683	−0.536478
$419$	−20.8789	−1.02000	−0.510001	−	0.860174i	$-0.670355\pi$
$419$	−20.8789	−1.02000	−0.510001	+	0.860174i	$0.670355\pi$
$420$	0	0
$421$	−38.6850	−1.88539	−0.942695	−	0.333656i	$-0.891718\pi$
$421$	−38.6850	−1.88539	−0.942695	+	0.333656i	$0.891718\pi$
$422$	47.7373	2.32381
$423$	0	0
$424$	−8.49729	−0.412665
$425$	0.562928	0.0273060
$426$	0	0
$427$	18.3641	0.888701
$428$	−20.5534	−0.993486
$429$	0	0
$430$	−4.12973	−0.199153
$431$	4.20227	0.202416	0.101208	−	0.994865i	$-0.467729\pi$
$431$	4.20227	0.202416	0.101208	+	0.994865i	$0.467729\pi$
$432$	0	0
$433$	−32.7943	−1.57599	−0.787996	−	0.615680i	$-0.788881\pi$
$433$	−32.7943	−1.57599	−0.787996	+	0.615680i	$0.788881\pi$
$434$	38.9274	1.86858
$435$	0	0
$436$	−17.1244	−0.820109
$437$	33.2520	1.59066
$438$	0	0
$439$	35.2287	1.68138	0.840688	−	0.541520i	$-0.182151\pi$
$439$	35.2287	1.68138	0.840688	+	0.541520i	$0.182151\pi$
$440$	2.22015	0.105842
$441$	0	0
$442$	0.988007	0.0469947
$443$	−32.0044	−1.52058	−0.760288	−	0.649586i	$-0.774942\pi$
$443$	−32.0044	−1.52058	−0.760288	+	0.649586i	$0.774942\pi$
$444$	0	0
$445$	2.59479	0.123005
$446$	−38.8925	−1.84161
$447$	0	0
$448$	−0.990808	−0.0468113
$449$	22.9644	1.08376	0.541879	−	0.840457i	$-0.317713\pi$
$449$	22.9644	1.08376	0.541879	+	0.840457i	$0.317713\pi$
$450$	0	0
$451$	11.3893	0.536299
$452$	14.3274	0.673906
$453$	0	0
$454$	−16.9830	−0.797050
$455$	−3.67006	−0.172055
$456$	0	0
$457$	34.1678	1.59830	0.799150	−	0.601131i	$-0.205283\pi$
$457$	34.1678	1.59830	0.799150	+	0.601131i	$0.205283\pi$
$458$	6.79827	0.317662
$459$	0	0
$460$	7.90848	0.368735
$461$	18.2234	0.848748	0.424374	−	0.905487i	$-0.360494\pi$
$461$	18.2234	0.848748	0.424374	+	0.905487i	$0.360494\pi$
$462$	0	0
$463$	40.8533	1.89862	0.949308	−	0.314349i	$-0.101786\pi$
$463$	40.8533	1.89862	0.949308	+	0.314349i	$0.101786\pi$
$464$	44.0303	2.04406
$465$	0	0
$466$	17.3935	0.805739
$467$	39.5314	1.82929	0.914646	−	0.404255i	$-0.132469\pi$
$467$	39.5314	1.82929	0.914646	+	0.404255i	$0.132469\pi$
$468$	0	0
$469$	−23.0386	−1.06382
$470$	2.49535	0.115102
$471$	0	0
$472$	−14.6258	−0.673206
$473$	3.23680	0.148828
$474$	0	0
$475$	−4.54287	−0.208441
$476$	−2.23219	−0.102312
$477$	0	0
$478$	21.7797	0.996182
$479$	31.9791	1.46116	0.730581	−	0.682826i	$-0.239249\pi$
$479$	31.9791	1.46116	0.730581	+	0.682826i	$0.239249\pi$
$480$	0	0
$481$	0.137665	0.00627700
$482$	21.0090	0.956933
$483$	0	0
$484$	−9.84038	−0.447290
$485$	12.1099	0.549880
$486$	0	0
$487$	−36.9408	−1.67395	−0.836973	−	0.547244i	$-0.815677\pi$
$487$	−36.9408	−1.67395	−0.836973	+	0.547244i	$0.815677\pi$
$488$	−8.07565	−0.365568
$489$	0	0
$490$	11.3545	0.512942
$491$	39.8447	1.79816	0.899082	−	0.437779i	$-0.144235\pi$
$491$	39.8447	1.79816	0.899082	+	0.437779i	$0.144235\pi$
$492$	0	0
$493$	−4.96360	−0.223549
$494$	−7.97329	−0.358735
$495$	0	0
$496$	−30.1775	−1.35501
$497$	11.1362	0.499528
$498$	0	0
$499$	16.2983	0.729612	0.364806	−	0.931084i	$-0.381135\pi$
$499$	16.2983	0.729612	0.364806	+	0.931084i	$0.381135\pi$
$500$	−1.08045	−0.0483193
$501$	0	0
$502$	34.2062	1.52670
$503$	−14.6428	−0.652890	−0.326445	−	0.945216i	$-0.605851\pi$
$503$	−14.6428	−0.652890	−0.326445	+	0.945216i	$0.605851\pi$
$504$	0	0
$505$	11.7807	0.524236
$506$	−17.6724	−0.785635
$507$	0	0
$508$	−17.4639	−0.774836
$509$	−14.8958	−0.660244	−0.330122	−	0.943938i	$-0.607090\pi$
$509$	−14.8958	−0.660244	−0.330122	+	0.943938i	$0.607090\pi$
$510$	0	0
$511$	−30.2979	−1.34030
$512$	−11.5280	−0.509469
$513$	0	0
$514$	37.4168	1.65038
$515$	−4.92388	−0.216972
$516$	0	0
$517$	−1.95580	−0.0860162
$518$	−0.886757	−0.0389619
$519$	0	0
$520$	1.61392	0.0707749
$521$	8.33697	0.365249	0.182625	−	0.983183i	$-0.441541\pi$
$521$	8.33697	0.365249	0.182625	+	0.983183i	$0.441541\pi$
$522$	0	0
$523$	−37.4803	−1.63890	−0.819450	−	0.573150i	$-0.805721\pi$
$523$	−37.4803	−1.63890	−0.819450	+	0.573150i	$0.805721\pi$
$524$	−12.5862	−0.549829
$525$	0	0
$526$	−53.3228	−2.32499
$527$	3.40195	0.148191
$528$	0	0
$529$	30.5764	1.32941
$530$	−9.24074	−0.401392
$531$	0	0
$532$	18.0140	0.781004
$533$	8.27929	0.358616
$534$	0	0
$535$	19.0229	0.822434
$536$	10.1313	0.437605
$537$	0	0
$538$	23.1335	0.997358
$539$	−8.89941	−0.383325
$540$	0	0
$541$	3.76481	0.161862	0.0809309	−	0.996720i	$-0.474211\pi$
$541$	3.76481	0.161862	0.0809309	+	0.996720i	$0.474211\pi$
$542$	−52.7959	−2.26778
$543$	0	0
$544$	3.11660	0.133623
$545$	15.8493	0.678908
$546$	0	0
$547$	−19.1848	−0.820281	−0.410140	−	0.912022i	$-0.634520\pi$
$547$	−19.1848	−0.820281	−0.410140	+	0.912022i	$0.634520\pi$
$548$	−0.965537	−0.0412457
$549$	0	0
$550$	2.41440	0.102950
$551$	40.0566	1.70647
$552$	0	0
$553$	−15.6992	−0.667597
$554$	35.9918	1.52914
$555$	0	0
$556$	−1.11017	−0.0470817
$557$	12.0185	0.509242	0.254621	−	0.967041i	$-0.418049\pi$
$557$	12.0185	0.509242	0.254621	+	0.967041i	$0.418049\pi$
$558$	0	0
$559$	2.35296	0.0995196
$560$	−18.3265	−0.774438
$561$	0	0
$562$	47.7330	2.01349
$563$	18.4814	0.778897	0.389449	−	0.921048i	$-0.372666\pi$
$563$	18.4814	0.778897	0.389449	+	0.921048i	$0.372666\pi$
$564$	0	0
$565$	−13.2606	−0.557877
$566$	9.25588	0.389054
$567$	0	0
$568$	−4.89718	−0.205481
$569$	2.40327	0.100750	0.0503751	−	0.998730i	$-0.483958\pi$
$569$	2.40327	0.100750	0.0503751	+	0.998730i	$0.483958\pi$
$570$	0	0
$571$	−2.26697	−0.0948699	−0.0474349	−	0.998874i	$-0.515105\pi$
$571$	−2.26697	−0.0948699	−0.0474349	+	0.998874i	$0.515105\pi$
$572$	1.48630	0.0621455
$573$	0	0
$574$	−53.3302	−2.22596
$575$	−7.31959	−0.305248
$576$	0	0
$577$	14.8087	0.616493	0.308246	−	0.951307i	$-0.400258\pi$
$577$	14.8087	0.616493	0.308246	+	0.951307i	$0.400258\pi$
$578$	29.2809	1.21792
$579$	0	0
$580$	9.52687	0.395582
$581$	−0.873585	−0.0362424
$582$	0	0
$583$	7.24272	0.299963
$584$	13.3236	0.551334
$585$	0	0
$586$	29.1614	1.20465
$587$	−2.94040	−0.121363	−0.0606816	−	0.998157i	$-0.519327\pi$
$587$	−2.94040	−0.121363	−0.0606816	+	0.998157i	$0.519327\pi$
$588$	0	0
$589$	−27.4540	−1.13122
$590$	−15.9054	−0.654816
$591$	0	0
$592$	0.687435	0.0282534
$593$	−18.5447	−0.761540	−0.380770	−	0.924670i	$-0.624341\pi$
$593$	−18.5447	−0.761540	−0.380770	+	0.924670i	$0.624341\pi$
$594$	0	0
$595$	2.06598	0.0846968
$596$	1.73022	0.0708726
$597$	0	0
$598$	−12.8468	−0.525344
$599$	−38.8968	−1.58928	−0.794641	−	0.607080i	$-0.792341\pi$
$599$	−38.8968	−1.58928	−0.794641	+	0.607080i	$0.792341\pi$
$600$	0	0
$601$	26.7009	1.08915	0.544577	−	0.838711i	$-0.316690\pi$
$601$	26.7009	1.08915	0.544577	+	0.838711i	$0.316690\pi$
$602$	−15.1564	−0.617727
$603$	0	0
$604$	13.2102	0.537514
$605$	9.10764	0.370278
$606$	0	0
$607$	−3.77518	−0.153230	−0.0766149	−	0.997061i	$-0.524411\pi$
$607$	−3.77518	−0.153230	−0.0766149	+	0.997061i	$0.524411\pi$
$608$	−25.1512	−1.02002
$609$	0	0
$610$	−8.78221	−0.355581
$611$	−1.42175	−0.0575179
$612$	0	0
$613$	21.4506	0.866384	0.433192	−	0.901302i	$-0.357387\pi$
$613$	21.4506	0.866384	0.433192	+	0.901302i	$0.357387\pi$
$614$	24.9696	1.00769
$615$	0	0
$616$	8.14810	0.328296
$617$	−7.11594	−0.286477	−0.143238	−	0.989688i	$-0.545752\pi$
$617$	−7.11594	−0.286477	−0.143238	+	0.989688i	$0.545752\pi$
$618$	0	0
$619$	1.87582	0.0753957	0.0376978	−	0.999289i	$-0.487998\pi$
$619$	1.87582	0.0753957	0.0376978	+	0.999289i	$0.487998\pi$
$620$	−6.52952	−0.262232
$621$	0	0
$622$	−18.4841	−0.741147
$623$	9.52305	0.381533
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.96277	−0.358224
$627$	0	0
$628$	−10.7877	−0.430474
$629$	−0.0774956	−0.00308995
$630$	0	0
$631$	−3.97963	−0.158426	−0.0792132	−	0.996858i	$-0.525241\pi$
$631$	−3.97963	−0.158426	−0.0792132	+	0.996858i	$0.525241\pi$
$632$	6.90375	0.274616
$633$	0	0
$634$	−39.8228	−1.58156
$635$	16.1635	0.641430
$636$	0	0
$637$	−6.46933	−0.256324
$638$	−21.2889	−0.842836
$639$	0	0
$640$	11.5467	0.456422
$641$	13.6512	0.539191	0.269596	−	0.962974i	$-0.413110\pi$
$641$	13.6512	0.539191	0.269596	+	0.962974i	$0.413110\pi$
$642$	0	0
$643$	38.7963	1.52998	0.764988	−	0.644045i	$-0.222745\pi$
$643$	38.7963	1.52998	0.764988	+	0.644045i	$0.222745\pi$
$644$	29.0246	1.14373
$645$	0	0
$646$	4.48839	0.176593
$647$	1.26108	0.0495780	0.0247890	−	0.999693i	$-0.492109\pi$
$647$	1.26108	0.0495780	0.0247890	+	0.999693i	$0.492109\pi$
$648$	0	0
$649$	12.4664	0.489348
$650$	1.75512	0.0688415
$651$	0	0
$652$	22.2966	0.873204
$653$	14.3548	0.561746	0.280873	−	0.959745i	$-0.409376\pi$
$653$	14.3548	0.561746	0.280873	+	0.959745i	$0.409376\pi$
$654$	0	0
$655$	11.6490	0.455163
$656$	41.3429	1.61417
$657$	0	0
$658$	9.15806	0.357019
$659$	−8.08008	−0.314755	−0.157378	−	0.987538i	$-0.550304\pi$
$659$	−8.08008	−0.314755	−0.157378	+	0.987538i	$0.550304\pi$
$660$	0	0
$661$	−34.2745	−1.33313	−0.666563	−	0.745449i	$-0.732235\pi$
$661$	−34.2745	−1.33313	−0.666563	+	0.745449i	$0.732235\pi$
$662$	−33.0243	−1.28353
$663$	0	0
$664$	0.384161	0.0149084
$665$	−16.6726	−0.646536
$666$	0	0
$667$	64.5403	2.49901
$668$	−19.7430	−0.763878
$669$	0	0
$670$	11.0177	0.425650
$671$	6.88333	0.265728
$672$	0	0
$673$	20.3714	0.785259	0.392630	−	0.919697i	$-0.371565\pi$
$673$	20.3714	0.785259	0.392630	+	0.919697i	$0.371565\pi$
$674$	−31.6876	−1.22056
$675$	0	0
$676$	1.08045	0.0415559
$677$	−2.82772	−0.108678	−0.0543390	−	0.998523i	$-0.517305\pi$
$677$	−2.82772	−0.108678	−0.0543390	+	0.998523i	$0.517305\pi$
$678$	0	0
$679$	44.4439	1.70560
$680$	−0.908519	−0.0348401
$681$	0	0
$682$	14.5910	0.558718
$683$	−12.7250	−0.486907	−0.243453	−	0.969913i	$-0.578280\pi$
$683$	−12.7250	−0.486907	−0.243453	+	0.969913i	$0.578280\pi$
$684$	0	0
$685$	0.893641	0.0341443
$686$	−3.41827	−0.130510
$687$	0	0
$688$	11.7496	0.447948
$689$	5.26501	0.200581
$690$	0	0
$691$	0.425377	0.0161821	0.00809104	−	0.999967i	$-0.497425\pi$
$691$	0.425377	0.0161821	0.00809104	+	0.999967i	$0.497425\pi$
$692$	17.3154	0.658231
$693$	0	0
$694$	−27.9929	−1.06260
$695$	1.02751	0.0389755
$696$	0	0
$697$	−4.66064	−0.176534
$698$	−27.6080	−1.04498
$699$	0	0
$700$	−3.96533	−0.149875
$701$	−4.20502	−0.158822	−0.0794108	−	0.996842i	$-0.525304\pi$
$701$	−4.20502	−0.158822	−0.0794108	+	0.996842i	$0.525304\pi$
$702$	0	0
$703$	0.625395	0.0235872
$704$	−0.371380	−0.0139969
$705$	0	0
$706$	−16.7473	−0.630293
$707$	43.2360	1.62606
$708$	0	0
$709$	5.58961	0.209922	0.104961	−	0.994476i	$-0.466528\pi$
$709$	5.58961	0.209922	0.104961	+	0.994476i	$0.466528\pi$
$710$	−5.32564	−0.199868
$711$	0	0
$712$	−4.18778	−0.156944
$713$	−44.2346	−1.65660
$714$	0	0
$715$	−1.37563	−0.0514457
$716$	−6.34693	−0.237196
$717$	0	0
$718$	29.1990	1.08970
$719$	31.4731	1.17375	0.586873	−	0.809679i	$-0.300359\pi$
$719$	31.4731	1.17375	0.586873	+	0.809679i	$0.300359\pi$
$720$	0	0
$721$	−18.0709	−0.672996
$722$	−2.87432	−0.106971
$723$	0	0
$724$	−1.17941	−0.0438325
$725$	−8.81747	−0.327473
$726$	0	0
$727$	−10.5003	−0.389434	−0.194717	−	0.980859i	$-0.562379\pi$
$727$	−10.5003	−0.389434	−0.194717	+	0.980859i	$0.562379\pi$
$728$	5.92317	0.219527
$729$	0	0
$730$	14.4893	0.536273
$731$	−1.32455	−0.0489901
$732$	0	0
$733$	25.0096	0.923749	0.461875	−	0.886945i	$-0.347177\pi$
$733$	25.0096	0.923749	0.461875	+	0.886945i	$0.347177\pi$
$734$	6.92812	0.255721
$735$	0	0
$736$	−40.5243	−1.49375
$737$	−8.63545	−0.318091
$738$	0	0
$739$	−3.30636	−0.121627	−0.0608133	−	0.998149i	$-0.519369\pi$
$739$	−3.30636	−0.121627	−0.0608133	+	0.998149i	$0.519369\pi$
$740$	0.148741	0.00546782
$741$	0	0
$742$	−33.9141	−1.24502
$743$	32.6454	1.19764	0.598821	−	0.800883i	$-0.295636\pi$
$743$	32.6454	1.19764	0.598821	+	0.800883i	$0.295636\pi$
$744$	0	0
$745$	−1.60139	−0.0586702
$746$	−23.8527	−0.873311
$747$	0	0
$748$	−0.836682	−0.0305921
$749$	69.8153	2.55100
$750$	0	0
$751$	44.7916	1.63447	0.817235	−	0.576304i	$-0.195506\pi$
$751$	44.7916	1.63447	0.817235	+	0.576304i	$0.195506\pi$
$752$	−7.09955	−0.258894
$753$	0	0
$754$	−15.4757	−0.563593
$755$	−12.2265	−0.444968
$756$	0	0
$757$	−20.5488	−0.746859	−0.373430	−	0.927659i	$-0.621818\pi$
$757$	−20.5488	−0.746859	−0.373430	+	0.927659i	$0.621818\pi$
$758$	−26.7723	−0.972412
$759$	0	0
$760$	7.33182	0.265953
$761$	−22.2290	−0.805800	−0.402900	−	0.915244i	$-0.631998\pi$
$761$	−22.2290	−0.805800	−0.402900	+	0.915244i	$0.631998\pi$
$762$	0	0
$763$	58.1677	2.10581
$764$	28.5265	1.03205
$765$	0	0
$766$	−33.2489	−1.20133
$767$	9.06229	0.327220
$768$	0	0
$769$	−49.1695	−1.77310	−0.886549	−	0.462634i	$-0.846904\pi$
$769$	−49.1695	−1.77310	−0.886549	+	0.462634i	$0.846904\pi$
$770$	8.86099	0.319328
$771$	0	0
$772$	5.43612	0.195650
$773$	−1.74867	−0.0628952	−0.0314476	−	0.999505i	$-0.510012\pi$
$773$	−1.74867	−0.0628952	−0.0314476	+	0.999505i	$0.510012\pi$
$774$	0	0
$775$	6.04332	0.217083
$776$	−19.5443	−0.701600
$777$	0	0
$778$	30.5941	1.09685
$779$	37.6117	1.34758
$780$	0	0
$781$	4.17414	0.149362
$782$	7.23181	0.258609
$783$	0	0
$784$	−32.3048	−1.15374
$785$	9.98438	0.356358
$786$	0	0
$787$	−5.40473	−0.192658	−0.0963289	−	0.995350i	$-0.530710\pi$
$787$	−5.40473	−0.192658	−0.0963289	+	0.995350i	$0.530710\pi$
$788$	24.7614	0.882088
$789$	0	0
$790$	7.50777	0.267115
$791$	−48.6671	−1.73040
$792$	0	0
$793$	5.00376	0.177689
$794$	−47.8111	−1.69675
$795$	0	0
$796$	23.1169	0.819356
$797$	8.08485	0.286380	0.143190	−	0.989695i	$-0.454264\pi$
$797$	8.08485	0.286380	0.143190	+	0.989695i	$0.454264\pi$
$798$	0	0
$799$	0.800343	0.0283141
$800$	5.53642	0.195742
$801$	0	0
$802$	41.8270	1.47696
$803$	−11.3564	−0.400760
$804$	0	0
$805$	−26.8633	−0.946808
$806$	10.6068	0.373607
$807$	0	0
$808$	−19.0131	−0.668880
$809$	29.4435	1.03518	0.517590	−	0.855629i	$-0.326829\pi$
$809$	29.4435	1.03518	0.517590	+	0.855629i	$0.326829\pi$
$810$	0	0
$811$	17.1377	0.601788	0.300894	−	0.953658i	$-0.402715\pi$
$811$	17.1377	0.601788	0.300894	+	0.953658i	$0.402715\pi$
$812$	34.9642	1.22700
$813$	0	0
$814$	−0.332379	−0.0116499
$815$	−20.6364	−0.722861
$816$	0	0
$817$	10.6892	0.373967
$818$	−1.80977	−0.0632773
$819$	0	0
$820$	8.94539	0.312386
$821$	−25.0135	−0.872977	−0.436488	−	0.899710i	$-0.643778\pi$
$821$	−25.0135	−0.872977	−0.436488	+	0.899710i	$0.643778\pi$
$822$	0	0
$823$	2.26110	0.0788171	0.0394086	−	0.999223i	$-0.487453\pi$
$823$	2.26110	0.0788171	0.0394086	+	0.999223i	$0.487453\pi$
$824$	7.94673	0.276837
$825$	0	0
$826$	−58.3738	−2.03108
$827$	−16.3611	−0.568930	−0.284465	−	0.958686i	$-0.591816\pi$
$827$	−16.3611	−0.568930	−0.284465	+	0.958686i	$0.591816\pi$
$828$	0	0
$829$	6.88002	0.238953	0.119476	−	0.992837i	$-0.461878\pi$
$829$	6.88002	0.238953	0.119476	+	0.992837i	$0.461878\pi$
$830$	0.417772	0.0145011
$831$	0	0
$832$	−0.269971	−0.00935955
$833$	3.64176	0.126180
$834$	0	0
$835$	18.2729	0.632358
$836$	6.75209	0.233526
$837$	0	0
$838$	36.6451	1.26588
$839$	−24.8524	−0.858002	−0.429001	−	0.903304i	$-0.641134\pi$
$839$	−24.8524	−0.858002	−0.429001	+	0.903304i	$0.641134\pi$
$840$	0	0
$841$	48.7478	1.68096
$842$	67.8968	2.33988
$843$	0	0
$844$	−29.3871	−1.01154
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	33.4256	1.14852
$848$	26.2910	0.902836
$849$	0	0
$850$	−0.988007	−0.0338884
$851$	1.00765	0.0345419
$852$	0	0
$853$	7.48700	0.256350	0.128175	−	0.991752i	$-0.459088\pi$
$853$	7.48700	0.256350	0.128175	+	0.991752i	$0.459088\pi$
$854$	−32.2312	−1.10293
$855$	0	0
$856$	−30.7015	−1.04935
$857$	6.54048	0.223419	0.111709	−	0.993741i	$-0.464367\pi$
$857$	6.54048	0.223419	0.111709	+	0.993741i	$0.464367\pi$
$858$	0	0
$859$	−17.5360	−0.598320	−0.299160	−	0.954203i	$-0.596706\pi$
$859$	−17.5360	−0.598320	−0.299160	+	0.954203i	$0.596706\pi$
$860$	2.54226	0.0866904
$861$	0	0
$862$	−7.37549	−0.251210
$863$	−11.9144	−0.405571	−0.202786	−	0.979223i	$-0.564999\pi$
$863$	−11.9144	−0.405571	−0.202786	+	0.979223i	$0.564999\pi$
$864$	0	0
$865$	−16.0260	−0.544901
$866$	57.5579	1.95590
$867$	0	0
$868$	−23.9637	−0.813382
$869$	−5.88445	−0.199616
$870$	0	0
$871$	−6.27745	−0.212703
$872$	−25.5794	−0.866227
$873$	0	0
$874$	−58.3612	−1.97410
$875$	3.67006	0.124071
$876$	0	0
$877$	−31.6658	−1.06928	−0.534638	−	0.845081i	$-0.679552\pi$
$877$	−31.6658	−1.06928	−0.534638	+	0.845081i	$0.679552\pi$
$878$	−61.8307	−2.08669
$879$	0	0
$880$	−6.86925	−0.231562
$881$	−2.16907	−0.0730778	−0.0365389	−	0.999332i	$-0.511633\pi$
$881$	−2.16907	−0.0730778	−0.0365389	+	0.999332i	$0.511633\pi$
$882$	0	0
$883$	37.5249	1.26281	0.631406	−	0.775452i	$-0.282478\pi$
$883$	37.5249	1.26281	0.631406	+	0.775452i	$0.282478\pi$
$884$	−0.608217	−0.0204566
$885$	0	0
$886$	56.1717	1.88712
$887$	−15.0634	−0.505778	−0.252889	−	0.967495i	$-0.581381\pi$
$887$	−15.0634	−0.505778	−0.252889	+	0.967495i	$0.581381\pi$
$888$	0	0
$889$	59.3211	1.98956
$890$	−4.55418	−0.152656
$891$	0	0
$892$	23.9422	0.801646
$893$	−6.45883	−0.216136
$894$	0	0
$895$	5.87432	0.196357
$896$	42.3769	1.41571
$897$	0	0
$898$	−40.3053	−1.34501
$899$	−53.2868	−1.77721
$900$	0	0
$901$	−2.96382	−0.0987392
$902$	−19.9895	−0.665578
$903$	0	0
$904$	21.4015	0.711802
$905$	1.09159	0.0362857
$906$	0	0
$907$	−56.8544	−1.88782	−0.943910	−	0.330202i	$-0.892883\pi$
$907$	−56.8544	−1.88782	−0.943910	+	0.330202i	$0.892883\pi$
$908$	10.4547	0.346952
$909$	0	0
$910$	6.44140	0.213530
$911$	8.87016	0.293882	0.146941	−	0.989145i	$-0.453057\pi$
$911$	8.87016	0.293882	0.146941	+	0.989145i	$0.453057\pi$
$912$	0	0
$913$	−0.327442	−0.0108367
$914$	−59.9686	−1.98358
$915$	0	0
$916$	−4.18501	−0.138277
$917$	42.7524	1.41181
$918$	0	0
$919$	−33.3934	−1.10155	−0.550774	−	0.834655i	$-0.685667\pi$
$919$	−33.3934	−1.10155	−0.550774	+	0.834655i	$0.685667\pi$
$920$	11.8132	0.389470
$921$	0	0
$922$	−31.9843	−1.05335
$923$	3.03434	0.0998766
$924$	0	0
$925$	−0.137665	−0.00452641
$926$	−71.7026	−2.35629
$927$	0	0
$928$	−48.8172	−1.60250
$929$	−19.1235	−0.627422	−0.313711	−	0.949519i	$-0.601572\pi$
$929$	−19.1235	−0.627422	−0.313711	+	0.949519i	$0.601572\pi$
$930$	0	0
$931$	−29.3893	−0.963196
$932$	−10.7074	−0.350734
$933$	0	0
$934$	−69.3823	−2.27026
$935$	0.774381	0.0253250
$936$	0	0
$937$	−30.5662	−0.998554	−0.499277	−	0.866442i	$-0.666401\pi$
$937$	−30.5662	−0.998554	−0.499277	+	0.866442i	$0.666401\pi$
$938$	40.4355	1.32027
$939$	0	0
$940$	−1.53613	−0.0501032
$941$	−5.04864	−0.164581	−0.0822904	−	0.996608i	$-0.526224\pi$
$941$	−5.04864	−0.164581	−0.0822904	+	0.996608i	$0.526224\pi$
$942$	0	0
$943$	60.6010	1.97344
$944$	45.2528	1.47285
$945$	0	0
$946$	−5.68099	−0.184705
$947$	16.6830	0.542123	0.271062	−	0.962562i	$-0.412625\pi$
$947$	16.6830	0.542123	0.271062	+	0.962562i	$0.412625\pi$
$948$	0	0
$949$	−8.25544	−0.267983
$950$	7.97329	0.258688
$951$	0	0
$952$	−3.33432	−0.108066
$953$	9.68585	0.313755	0.156878	−	0.987618i	$-0.449857\pi$
$953$	9.68585	0.313755	0.156878	+	0.987618i	$0.449857\pi$
$954$	0	0
$955$	−26.4024	−0.854361
$956$	−13.4076	−0.433633
$957$	0	0
$958$	−56.1272	−1.81339
$959$	3.27972	0.105908
$960$	0	0
$961$	5.52173	0.178120
$962$	−0.241619	−0.00779012
$963$	0	0
$964$	−12.9331	−0.416548
$965$	−5.03133	−0.161964
$966$	0	0
$967$	13.9456	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$967$	13.9456	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$968$	−14.6990	−0.472443
$969$	0	0
$970$	−21.2543	−0.682434
$971$	−41.3208	−1.32605	−0.663024	−	0.748598i	$-0.730727\pi$
$971$	−41.3208	−1.32605	−0.663024	+	0.748598i	$0.730727\pi$
$972$	0	0
$973$	3.77100	0.120893
$974$	64.8356	2.07747
$975$	0	0
$976$	24.9864	0.799795
$977$	−14.8970	−0.476597	−0.238299	−	0.971192i	$-0.576590\pi$
$977$	−14.8970	−0.476597	−0.238299	+	0.971192i	$0.576590\pi$
$978$	0	0
$979$	3.56948	0.114081
$980$	−6.98981	−0.223281
$981$	0	0
$982$	−69.9323	−2.23163
$983$	−17.4239	−0.555735	−0.277868	−	0.960619i	$-0.589628\pi$
$983$	−17.4239	−0.555735	−0.277868	+	0.960619i	$0.589628\pi$
$984$	0	0
$985$	−22.9176	−0.730215
$986$	8.71172	0.277438
$987$	0	0
$988$	4.90836	0.156156
$989$	17.2227	0.547650
$990$	0	0
$991$	−43.3691	−1.37766	−0.688832	−	0.724921i	$-0.741876\pi$
$991$	−43.3691	−1.37766	−0.688832	+	0.724921i	$0.741876\pi$
$992$	33.4583	1.06230
$993$	0	0
$994$	−19.5454	−0.619943
$995$	−21.3955	−0.678284
$996$	0	0
$997$	−30.5609	−0.967873	−0.483937	−	0.875103i	$-0.660793\pi$
$997$	−30.5609	−0.967873	−0.483937	+	0.875103i	$0.660793\pi$
$998$	−28.6055	−0.905491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bh.1.3		13
3.2	odd	2		5265.2.a.bg.1.11		13
9.2	odd	6		585.2.i.g.391.3	yes	26
9.4	even	3		1755.2.i.g.586.11		26
9.5	odd	6		585.2.i.g.196.3	✓	26
9.7	even	3		1755.2.i.g.1171.11		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.g.196.3	✓	26	9.5	odd	6
585.2.i.g.391.3	yes	26	9.2	odd	6
1755.2.i.g.586.11		26	9.4	even	3
1755.2.i.g.1171.11		26	9.7	even	3
5265.2.a.bg.1.11		13	3.2	odd	2
5265.2.a.bh.1.3		13	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)

\(f(q)\)	\(=\)	\(q-1.75512 q^{2} +1.08045 q^{4} -1.00000 q^{5} -3.67006 q^{7} +1.61392 q^{8} +O(q^{10})\)	\(q-1.75512 q^{2} +1.08045 q^{4} -1.00000 q^{5} -3.67006 q^{7} +1.61392 q^{8} +1.75512 q^{10} -1.37563 q^{11} -1.00000 q^{13} +6.44140 q^{14} -4.99353 q^{16} +0.562928 q^{17} -4.54287 q^{19} -1.08045 q^{20} +2.41440 q^{22} -7.31959 q^{23} +1.00000 q^{25} +1.75512 q^{26} -3.96533 q^{28} -8.81747 q^{29} +6.04332 q^{31} +5.53642 q^{32} -0.988007 q^{34} +3.67006 q^{35} -0.137665 q^{37} +7.97329 q^{38} -1.61392 q^{40} -8.27929 q^{41} -2.35296 q^{43} -1.48630 q^{44} +12.8468 q^{46} +1.42175 q^{47} +6.46933 q^{49} -1.75512 q^{50} -1.08045 q^{52} -5.26501 q^{53} +1.37563 q^{55} -5.92317 q^{56} +15.4757 q^{58} -9.06229 q^{59} -5.00376 q^{61} -10.6068 q^{62} +0.269971 q^{64} +1.00000 q^{65} +6.27745 q^{67} +0.608217 q^{68} -6.44140 q^{70} -3.03434 q^{71} +8.25544 q^{73} +0.241619 q^{74} -4.90836 q^{76} +5.04865 q^{77} +4.27764 q^{79} +4.99353 q^{80} +14.5312 q^{82} +0.238030 q^{83} -0.562928 q^{85} +4.12973 q^{86} -2.22015 q^{88} -2.59479 q^{89} +3.67006 q^{91} -7.90848 q^{92} -2.49535 q^{94} +4.54287 q^{95} -12.1099 q^{97} -11.3545 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + q^{2} + 15 q^{4} - 13 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 13 * q + q^2 + 15 * q^4 - 13 * q^5 + 10 * q^7 + 6 * q^8	\( 13 q + q^{2} + 15 q^{4} - 13 q^{5} + 10 q^{7} + 6 q^{8} - q^{10} - 11 q^{11} - 13 q^{13} - 15 q^{14} + 19 q^{16} - 3 q^{17} + 15 q^{19} - 15 q^{20} + 12 q^{22} - 6 q^{23} + 13 q^{25} - q^{26} + 5 q^{28} - 14 q^{29} + 30 q^{31} + 43 q^{32} + 19 q^{34} - 10 q^{35} + 16 q^{37} + 2 q^{38} - 6 q^{40} - 17 q^{41} + 6 q^{43} - 23 q^{44} + 23 q^{46} + 21 q^{47} + 29 q^{49} + q^{50} - 15 q^{52} - q^{53} + 11 q^{55} - 37 q^{56} + 14 q^{58} - 13 q^{59} + 22 q^{61} + 57 q^{62} + 42 q^{64} + 13 q^{65} + 35 q^{67} + 4 q^{68} + 15 q^{70} - 12 q^{71} + 32 q^{73} + 28 q^{74} + 54 q^{76} - 4 q^{77} + 12 q^{79} - 19 q^{80} + 23 q^{82} + 3 q^{83} + 3 q^{85} + 40 q^{86} + 29 q^{88} - 6 q^{89} - 10 q^{91} + 16 q^{92} + 44 q^{94} - 15 q^{95} + 33 q^{97} - 35 q^{98}+O(q^{100}) \) 13 * q + q^2 + 15 * q^4 - 13 * q^5 + 10 * q^7 + 6 * q^8 - q^10 - 11 * q^11 - 13 * q^13 - 15 * q^14 + 19 * q^16 - 3 * q^17 + 15 * q^19 - 15 * q^20 + 12 * q^22 - 6 * q^23 + 13 * q^25 - q^26 + 5 * q^28 - 14 * q^29 + 30 * q^31 + 43 * q^32 + 19 * q^34 - 10 * q^35 + 16 * q^37 + 2 * q^38 - 6 * q^40 - 17 * q^41 + 6 * q^43 - 23 * q^44 + 23 * q^46 + 21 * q^47 + 29 * q^49 + q^50 - 15 * q^52 - q^53 + 11 * q^55 - 37 * q^56 + 14 * q^58 - 13 * q^59 + 22 * q^61 + 57 * q^62 + 42 * q^64 + 13 * q^65 + 35 * q^67 + 4 * q^68 + 15 * q^70 - 12 * q^71 + 32 * q^73 + 28 * q^74 + 54 * q^76 - 4 * q^77 + 12 * q^79 - 19 * q^80 + 23 * q^82 + 3 * q^83 + 3 * q^85 + 40 * q^86 + 29 * q^88 - 6 * q^89 - 10 * q^91 + 16 * q^92 + 44 * q^94 - 15 * q^95 + 33 * q^97 - 35 * q^98

Introduction

L-functions

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