LMFDB - Embedded newform 5265.2.a.bk.1.7

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			1.7
Root			$0.601514$ 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)

$f(q)$	$=$	$q-0.601514 q^{2} -1.63818 q^{4} -1.00000 q^{5} -2.59900 q^{7} +2.18842 q^{8} +O(q^{10})$	\(q-0.601514 q^{2} -1.63818 q^{4} -1.00000 q^{5} -2.59900 q^{7} +2.18842 q^{8} +0.601514 q^{10} -5.12295 q^{11} +1.00000 q^{13} +1.56333 q^{14} +1.96000 q^{16} -5.36998 q^{17} -4.53480 q^{19} +1.63818 q^{20} +3.08152 q^{22} +0.799668 q^{23} +1.00000 q^{25} -0.601514 q^{26} +4.25763 q^{28} -3.33101 q^{29} -6.98397 q^{31} -5.55580 q^{32} +3.23011 q^{34} +2.59900 q^{35} -10.6386 q^{37} +2.72774 q^{38} -2.18842 q^{40} +4.53980 q^{41} +6.65602 q^{43} +8.39232 q^{44} -0.481011 q^{46} -5.35095 q^{47} -0.245199 q^{49} -0.601514 q^{50} -1.63818 q^{52} -9.68150 q^{53} +5.12295 q^{55} -5.68769 q^{56} +2.00365 q^{58} -0.818248 q^{59} -4.24792 q^{61} +4.20095 q^{62} -0.578115 q^{64} -1.00000 q^{65} -9.64232 q^{67} +8.79700 q^{68} -1.56333 q^{70} -1.13844 q^{71} +1.97256 q^{73} +6.39924 q^{74} +7.42882 q^{76} +13.3145 q^{77} +7.27604 q^{79} -1.96000 q^{80} -2.73075 q^{82} -16.0655 q^{83} +5.36998 q^{85} -4.00368 q^{86} -11.2111 q^{88} +4.74177 q^{89} -2.59900 q^{91} -1.31000 q^{92} +3.21867 q^{94} +4.53480 q^{95} -6.96601 q^{97} +0.147491 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.601514	−0.425334	−0.212667	−	0.977125i	$-0.568215\pi$
$2$	−0.601514	−0.425334	−0.212667	+	0.977125i	$0.568215\pi$
$3$	0	0
$3$	0	0
$4$	−1.63818	−0.819091
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.59900	−0.982330	−0.491165	−	0.871067i	$-0.663429\pi$
$7$	−2.59900	−0.982330	−0.491165	+	0.871067i	$0.663429\pi$
$8$	2.18842	0.773722
$9$	0	0
$10$	0.601514	0.190215
$11$	−5.12295	−1.54463	−0.772313	−	0.635242i	$-0.780901\pi$
$11$	−5.12295	−1.54463	−0.772313	+	0.635242i	$0.780901\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.56333	0.417818
$15$	0	0
$16$	1.96000	0.490000
$17$	−5.36998	−1.30241	−0.651206	−	0.758901i	$-0.725736\pi$
$17$	−5.36998	−1.30241	−0.651206	+	0.758901i	$0.725736\pi$
$18$	0	0
$19$	−4.53480	−1.04035	−0.520177	−	0.854059i	$-0.674134\pi$
$19$	−4.53480	−1.04035	−0.520177	+	0.854059i	$0.674134\pi$
$20$	1.63818	0.366309
$21$	0	0
$22$	3.08152	0.656983
$23$	0.799668	0.166742	0.0833712	−	0.996519i	$-0.473431\pi$
$23$	0.799668	0.166742	0.0833712	+	0.996519i	$0.473431\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.601514	−0.117967
$27$	0	0
$28$	4.25763	0.804617
$29$	−3.33101	−0.618553	−0.309276	−	0.950972i	$-0.600087\pi$
$29$	−3.33101	−0.618553	−0.309276	+	0.950972i	$0.600087\pi$
$30$	0	0
$31$	−6.98397	−1.25436	−0.627179	−	0.778875i	$-0.715791\pi$
$31$	−6.98397	−1.25436	−0.627179	+	0.778875i	$0.715791\pi$
$32$	−5.55580	−0.982136
$33$	0	0
$34$	3.23011	0.553960
$35$	2.59900	0.439311
$36$	0	0
$37$	−10.6386	−1.74897	−0.874484	−	0.485054i	$-0.838800\pi$
$37$	−10.6386	−1.74897	−0.874484	+	0.485054i	$0.838800\pi$
$38$	2.72774	0.442498
$39$	0	0
$40$	−2.18842	−0.346019
$41$	4.53980	0.708998	0.354499	−	0.935056i	$-0.384651\pi$
$41$	4.53980	0.708998	0.354499	+	0.935056i	$0.384651\pi$
$42$	0	0
$43$	6.65602	1.01503	0.507517	−	0.861642i	$-0.330564\pi$
$43$	6.65602	1.01503	0.507517	+	0.861642i	$0.330564\pi$
$44$	8.39232	1.26519
$45$	0	0
$46$	−0.481011	−0.0709212
$47$	−5.35095	−0.780516	−0.390258	−	0.920706i	$-0.627614\pi$
$47$	−5.35095	−0.780516	−0.390258	+	0.920706i	$0.627614\pi$
$48$	0	0
$49$	−0.245199	−0.0350285
$50$	−0.601514	−0.0850669
$51$	0	0
$52$	−1.63818	−0.227175
$53$	−9.68150	−1.32986	−0.664928	−	0.746907i	$-0.731538\pi$
$53$	−9.68150	−1.32986	−0.664928	+	0.746907i	$0.731538\pi$
$54$	0	0
$55$	5.12295	0.690778
$56$	−5.68769	−0.760050
$57$	0	0
$58$	2.00365	0.263092
$59$	−0.818248	−0.106527	−0.0532634	−	0.998580i	$-0.516962\pi$
$59$	−0.818248	−0.106527	−0.0532634	+	0.998580i	$0.516962\pi$
$60$	0	0
$61$	−4.24792	−0.543891	−0.271945	−	0.962313i	$-0.587667\pi$
$61$	−4.24792	−0.543891	−0.271945	+	0.962313i	$0.587667\pi$
$62$	4.20095	0.533522
$63$	0	0
$64$	−0.578115	−0.0722644
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−9.64232	−1.17800	−0.588998	−	0.808134i	$-0.700478\pi$
$67$	−9.64232	−1.17800	−0.588998	+	0.808134i	$0.700478\pi$
$68$	8.79700	1.06679
$69$	0	0
$70$	−1.56333	−0.186854
$71$	−1.13844	−0.135108	−0.0675539	−	0.997716i	$-0.521519\pi$
$71$	−1.13844	−0.135108	−0.0675539	+	0.997716i	$0.521519\pi$
$72$	0	0
$73$	1.97256	0.230871	0.115435	−	0.993315i	$-0.463174\pi$
$73$	1.97256	0.230871	0.115435	+	0.993315i	$0.463174\pi$
$74$	6.39924	0.743896
$75$	0	0
$76$	7.42882	0.852144
$77$	13.3145	1.51733
$78$	0	0
$79$	7.27604	0.818618	0.409309	−	0.912396i	$-0.365770\pi$
$79$	7.27604	0.818618	0.409309	+	0.912396i	$0.365770\pi$
$80$	−1.96000	−0.219135
$81$	0	0
$82$	−2.73075	−0.301561
$83$	−16.0655	−1.76342	−0.881709	−	0.471794i	$-0.843607\pi$
$83$	−16.0655	−1.76342	−0.881709	+	0.471794i	$0.843607\pi$
$84$	0	0
$85$	5.36998	0.582456
$86$	−4.00368	−0.431728
$87$	0	0
$88$	−11.2111	−1.19511
$89$	4.74177	0.502626	0.251313	−	0.967906i	$-0.419138\pi$
$89$	4.74177	0.502626	0.251313	+	0.967906i	$0.419138\pi$
$90$	0	0
$91$	−2.59900	−0.272449
$92$	−1.31000	−0.136577
$93$	0	0
$94$	3.21867	0.331980
$95$	4.53480	0.465260
$96$	0	0
$97$	−6.96601	−0.707291	−0.353645	−	0.935380i	$-0.615058\pi$
$97$	−6.96601	−0.707291	−0.353645	+	0.935380i	$0.615058\pi$
$98$	0.147491	0.0148988
$99$	0	0
$100$	−1.63818	−0.163818
$101$	−12.2480	−1.21872	−0.609360	−	0.792893i	$-0.708574\pi$
$101$	−12.2480	−1.21872	−0.609360	+	0.792893i	$0.708574\pi$
$102$	0	0
$103$	16.8171	1.65704	0.828521	−	0.559958i	$-0.189183\pi$
$103$	16.8171	1.65704	0.828521	+	0.559958i	$0.189183\pi$
$104$	2.18842	0.214592
$105$	0	0
$106$	5.82355	0.565634
$107$	−5.10514	−0.493532	−0.246766	−	0.969075i	$-0.579368\pi$
$107$	−5.10514	−0.493532	−0.246766	+	0.969075i	$0.579368\pi$
$108$	0	0
$109$	−6.15946	−0.589969	−0.294985	−	0.955502i	$-0.595315\pi$
$109$	−6.15946	−0.589969	−0.294985	+	0.955502i	$0.595315\pi$
$110$	−3.08152	−0.293812
$111$	0	0
$112$	−5.09404	−0.481342
$113$	−3.17392	−0.298577	−0.149289	−	0.988794i	$-0.547698\pi$
$113$	−3.17392	−0.298577	−0.149289	+	0.988794i	$0.547698\pi$
$114$	0	0
$115$	−0.799668	−0.0745694
$116$	5.45680	0.506651
$117$	0	0
$118$	0.492187	0.0453095
$119$	13.9566	1.27940
$120$	0	0
$121$	15.2446	1.38587
$122$	2.55518	0.231335
$123$	0	0
$124$	11.4410	1.02743
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.2855	1.26763	0.633817	−	0.773483i	$-0.281487\pi$
$127$	14.2855	1.26763	0.633817	+	0.773483i	$0.281487\pi$
$128$	11.4593	1.01287
$129$	0	0
$130$	0.601514	0.0527562
$131$	−17.1294	−1.49660	−0.748302	−	0.663358i	$-0.769131\pi$
$131$	−17.1294	−1.49660	−0.748302	+	0.663358i	$0.769131\pi$
$132$	0	0
$133$	11.7859	1.02197
$134$	5.79999	0.501043
$135$	0	0
$136$	−11.7517	−1.00770
$137$	−0.244279	−0.0208702	−0.0104351	−	0.999946i	$-0.503322\pi$
$137$	−0.244279	−0.0208702	−0.0104351	+	0.999946i	$0.503322\pi$
$138$	0	0
$139$	2.95224	0.250406	0.125203	−	0.992131i	$-0.460042\pi$
$139$	2.95224	0.250406	0.125203	+	0.992131i	$0.460042\pi$
$140$	−4.25763	−0.359836
$141$	0	0
$142$	0.684786	0.0574660
$143$	−5.12295	−0.428402
$144$	0	0
$145$	3.33101	0.276625
$146$	−1.18652	−0.0981973
$147$	0	0
$148$	17.4279	1.43256
$149$	−22.1153	−1.81176	−0.905878	−	0.423539i	$-0.860788\pi$
$149$	−22.1153	−1.81176	−0.905878	+	0.423539i	$0.860788\pi$
$150$	0	0
$151$	21.6695	1.76344	0.881718	−	0.471777i	$-0.156387\pi$
$151$	21.6695	1.76344	0.881718	+	0.471777i	$0.156387\pi$
$152$	−9.92402	−0.804944
$153$	0	0
$154$	−8.00888	−0.645374
$155$	6.98397	0.560966
$156$	0	0
$157$	−10.6211	−0.847656	−0.423828	−	0.905743i	$-0.639314\pi$
$157$	−10.6211	−0.847656	−0.423828	+	0.905743i	$0.639314\pi$
$158$	−4.37664	−0.348186
$159$	0	0
$160$	5.55580	0.439224
$161$	−2.07834	−0.163796
$162$	0	0
$163$	−15.6209	−1.22352	−0.611761	−	0.791042i	$-0.709539\pi$
$163$	−15.6209	−1.22352	−0.611761	+	0.791042i	$0.709539\pi$
$164$	−7.43702	−0.580734
$165$	0	0
$166$	9.66362	0.750042
$167$	24.3041	1.88071	0.940355	−	0.340195i	$-0.110493\pi$
$167$	24.3041	1.88071	0.940355	+	0.340195i	$0.110493\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−3.23011	−0.247738
$171$	0	0
$172$	−10.9038	−0.831404
$173$	−13.6935	−1.04110	−0.520549	−	0.853832i	$-0.674273\pi$
$173$	−13.6935	−1.04110	−0.520549	+	0.853832i	$0.674273\pi$
$174$	0	0
$175$	−2.59900	−0.196466
$176$	−10.0410	−0.756868
$177$	0	0
$178$	−2.85224	−0.213784
$179$	−4.97769	−0.372050	−0.186025	−	0.982545i	$-0.559561\pi$
$179$	−4.97769	−0.372050	−0.186025	+	0.982545i	$0.559561\pi$
$180$	0	0
$181$	24.4686	1.81874	0.909370	−	0.415989i	$-0.136565\pi$
$181$	24.4686	1.81874	0.909370	+	0.415989i	$0.136565\pi$
$182$	1.56333	0.115882
$183$	0	0
$184$	1.75001	0.129012
$185$	10.6386	0.782162
$186$	0	0
$187$	27.5101	2.01174
$188$	8.76582	0.639313
$189$	0	0
$190$	−2.72774	−0.197891
$191$	−12.2527	−0.886573	−0.443287	−	0.896380i	$-0.646188\pi$
$191$	−12.2527	−0.886573	−0.443287	+	0.896380i	$0.646188\pi$
$192$	0	0
$193$	5.62938	0.405212	0.202606	−	0.979260i	$-0.435059\pi$
$193$	5.62938	0.405212	0.202606	+	0.979260i	$0.435059\pi$
$194$	4.19015	0.300835
$195$	0	0
$196$	0.401681	0.0286915
$197$	8.58427	0.611604	0.305802	−	0.952095i	$-0.401075\pi$
$197$	8.58427	0.611604	0.305802	+	0.952095i	$0.401075\pi$
$198$	0	0
$199$	14.3987	1.02069	0.510347	−	0.859969i	$-0.329517\pi$
$199$	14.3987	1.02069	0.510347	+	0.859969i	$0.329517\pi$
$200$	2.18842	0.154744
$201$	0	0
$202$	7.36733	0.518364
$203$	8.65729	0.607623
$204$	0	0
$205$	−4.53980	−0.317073
$206$	−10.1157	−0.704797
$207$	0	0
$208$	1.96000	0.135902
$209$	23.2315	1.60696
$210$	0	0
$211$	15.0809	1.03821	0.519107	−	0.854709i	$-0.326265\pi$
$211$	15.0809	1.03821	0.519107	+	0.854709i	$0.326265\pi$
$212$	15.8601	1.08927
$213$	0	0
$214$	3.07081	0.209916
$215$	−6.65602	−0.453937
$216$	0	0
$217$	18.1513	1.23219
$218$	3.70500	0.250934
$219$	0	0
$220$	−8.39232	−0.565810
$221$	−5.36998	−0.361224
$222$	0	0
$223$	−5.00187	−0.334950	−0.167475	−	0.985876i	$-0.553561\pi$
$223$	−5.00187	−0.334950	−0.167475	+	0.985876i	$0.553561\pi$
$224$	14.4395	0.964781
$225$	0	0
$226$	1.90916	0.126995
$227$	−3.74879	−0.248816	−0.124408	−	0.992231i	$-0.539703\pi$
$227$	−3.74879	−0.248816	−0.124408	+	0.992231i	$0.539703\pi$
$228$	0	0
$229$	−15.0288	−0.993132	−0.496566	−	0.867999i	$-0.665406\pi$
$229$	−15.0288	−0.993132	−0.496566	+	0.867999i	$0.665406\pi$
$230$	0.481011	0.0317169
$231$	0	0
$232$	−7.28963	−0.478588
$233$	4.28953	0.281016	0.140508	−	0.990080i	$-0.455126\pi$
$233$	4.28953	0.281016	0.140508	+	0.990080i	$0.455126\pi$
$234$	0	0
$235$	5.35095	0.349057
$236$	1.34044	0.0872551
$237$	0	0
$238$	−8.39507	−0.544171
$239$	−17.8120	−1.15216	−0.576080	−	0.817393i	$-0.695418\pi$
$239$	−17.8120	−1.15216	−0.576080	+	0.817393i	$0.695418\pi$
$240$	0	0
$241$	−16.0930	−1.03664	−0.518322	−	0.855186i	$-0.673443\pi$
$241$	−16.0930	−1.03664	−0.518322	+	0.855186i	$0.673443\pi$
$242$	−9.16983	−0.589459
$243$	0	0
$244$	6.95887	0.445496
$245$	0.245199	0.0156652
$246$	0	0
$247$	−4.53480	−0.288542
$248$	−15.2838	−0.970524
$249$	0	0
$250$	0.601514	0.0380431
$251$	−13.8501	−0.874210	−0.437105	−	0.899411i	$-0.643996\pi$
$251$	−13.8501	−0.874210	−0.437105	+	0.899411i	$0.643996\pi$
$252$	0	0
$253$	−4.09666	−0.257555
$254$	−8.59293	−0.539168
$255$	0	0
$256$	−5.73672	−0.358545
$257$	−4.49918	−0.280651	−0.140326	−	0.990105i	$-0.544815\pi$
$257$	−4.49918	−0.280651	−0.140326	+	0.990105i	$0.544815\pi$
$258$	0	0
$259$	27.6496	1.71806
$260$	1.63818	0.101596
$261$	0	0
$262$	10.3036	0.636557
$263$	17.1405	1.05693	0.528464	−	0.848956i	$-0.322768\pi$
$263$	17.1405	1.05693	0.528464	+	0.848956i	$0.322768\pi$
$264$	0	0
$265$	9.68150	0.594730
$266$	−7.08940	−0.434679
$267$	0	0
$268$	15.7959	0.964886
$269$	1.81067	0.110398	0.0551992	−	0.998475i	$-0.482421\pi$
$269$	1.81067	0.110398	0.0551992	+	0.998475i	$0.482421\pi$
$270$	0	0
$271$	19.6697	1.19485	0.597424	−	0.801926i	$-0.296191\pi$
$271$	19.6697	1.19485	0.597424	+	0.801926i	$0.296191\pi$
$272$	−10.5252	−0.638182
$273$	0	0
$274$	0.146937	0.00887681
$275$	−5.12295	−0.308925
$276$	0	0
$277$	−12.3937	−0.744663	−0.372331	−	0.928100i	$-0.621441\pi$
$277$	−12.3937	−0.744663	−0.372331	+	0.928100i	$0.621441\pi$
$278$	−1.77581	−0.106506
$279$	0	0
$280$	5.68769	0.339905
$281$	11.5672	0.690043	0.345022	−	0.938595i	$-0.387872\pi$
$281$	11.5672	0.690043	0.345022	+	0.938595i	$0.387872\pi$
$282$	0	0
$283$	14.1887	0.843432	0.421716	−	0.906728i	$-0.361428\pi$
$283$	14.1887	0.843432	0.421716	+	0.906728i	$0.361428\pi$
$284$	1.86497	0.110666
$285$	0	0
$286$	3.08152	0.182214
$287$	−11.7989	−0.696470
$288$	0	0
$289$	11.8367	0.696274
$290$	−2.00365	−0.117658
$291$	0	0
$292$	−3.23141	−0.189104
$293$	17.5685	1.02636	0.513182	−	0.858280i	$-0.328467\pi$
$293$	17.5685	1.02636	0.513182	+	0.858280i	$0.328467\pi$
$294$	0	0
$295$	0.818248	0.0476402
$296$	−23.2816	−1.35321
$297$	0	0
$298$	13.3027	0.770602
$299$	0.799668	0.0462460
$300$	0	0
$301$	−17.2990	−0.997097
$302$	−13.0345	−0.750050
$303$	0	0
$304$	−8.88821	−0.509774
$305$	4.24792	0.243235
$306$	0	0
$307$	28.4980	1.62647	0.813233	−	0.581938i	$-0.197705\pi$
$307$	28.4980	1.62647	0.813233	+	0.581938i	$0.197705\pi$
$308$	−21.8116	−1.24283
$309$	0	0
$310$	−4.20095	−0.238598
$311$	−21.1922	−1.20170	−0.600851	−	0.799361i	$-0.705171\pi$
$311$	−21.1922	−1.20170	−0.600851	+	0.799361i	$0.705171\pi$
$312$	0	0
$313$	12.5835	0.711260	0.355630	−	0.934627i	$-0.384266\pi$
$313$	12.5835	0.711260	0.355630	+	0.934627i	$0.384266\pi$
$314$	6.38874	0.360537
$315$	0	0
$316$	−11.9195	−0.670523
$317$	−12.7237	−0.714634	−0.357317	−	0.933983i	$-0.616308\pi$
$317$	−12.7237	−0.714634	−0.357317	+	0.933983i	$0.616308\pi$
$318$	0	0
$319$	17.0646	0.955433
$320$	0.578115	0.0323176
$321$	0	0
$322$	1.25015	0.0696680
$323$	24.3518	1.35497
$324$	0	0
$325$	1.00000	0.0554700
$326$	9.39618	0.520406
$327$	0	0
$328$	9.93497	0.548567
$329$	13.9071	0.766724
$330$	0	0
$331$	−6.92761	−0.380776	−0.190388	−	0.981709i	$-0.560975\pi$
$331$	−6.92761	−0.380776	−0.190388	+	0.981709i	$0.560975\pi$
$332$	26.3182	1.44440
$333$	0	0
$334$	−14.6193	−0.799930
$335$	9.64232	0.526816
$336$	0	0
$337$	−2.23831	−0.121928	−0.0609642	−	0.998140i	$-0.519418\pi$
$337$	−2.23831	−0.121928	−0.0609642	+	0.998140i	$0.519418\pi$
$338$	−0.601514	−0.0327180
$339$	0	0
$340$	−8.79700	−0.477084
$341$	35.7785	1.93752
$342$	0	0
$343$	18.8303	1.01674
$344$	14.5661	0.785353
$345$	0	0
$346$	8.23683	0.442815
$347$	26.4547	1.42016	0.710082	−	0.704119i	$-0.248658\pi$
$347$	26.4547	1.42016	0.710082	+	0.704119i	$0.248658\pi$
$348$	0	0
$349$	16.8342	0.901115	0.450558	−	0.892747i	$-0.351225\pi$
$349$	16.8342	0.901115	0.450558	+	0.892747i	$0.351225\pi$
$350$	1.56333	0.0835637
$351$	0	0
$352$	28.4621	1.51703
$353$	32.8951	1.75083	0.875416	−	0.483371i	$-0.160588\pi$
$353$	32.8951	1.75083	0.875416	+	0.483371i	$0.160588\pi$
$354$	0	0
$355$	1.13844	0.0604220
$356$	−7.76787	−0.411696
$357$	0	0
$358$	2.99415	0.158246
$359$	−3.73239	−0.196988	−0.0984941	−	0.995138i	$-0.531403\pi$
$359$	−3.73239	−0.196988	−0.0984941	+	0.995138i	$0.531403\pi$
$360$	0	0
$361$	1.56438	0.0823355
$362$	−14.7182	−0.773572
$363$	0	0
$364$	4.25763	0.223161
$365$	−1.97256	−0.103249
$366$	0	0
$367$	−24.2975	−1.26832	−0.634160	−	0.773202i	$-0.718654\pi$
$367$	−24.2975	−1.26832	−0.634160	+	0.773202i	$0.718654\pi$
$368$	1.56735	0.0817038
$369$	0	0
$370$	−6.39924	−0.332680
$371$	25.1622	1.30636
$372$	0	0
$373$	−9.91754	−0.513511	−0.256755	−	0.966476i	$-0.582653\pi$
$373$	−9.91754	−0.513511	−0.256755	+	0.966476i	$0.582653\pi$
$374$	−16.5477	−0.855662
$375$	0	0
$376$	−11.7101	−0.603902
$377$	−3.33101	−0.171556
$378$	0	0
$379$	−15.6671	−0.804763	−0.402381	−	0.915472i	$-0.631817\pi$
$379$	−15.6671	−0.804763	−0.402381	+	0.915472i	$0.631817\pi$
$380$	−7.42882	−0.381090
$381$	0	0
$382$	7.37016	0.377090
$383$	−16.2703	−0.831374	−0.415687	−	0.909508i	$-0.636459\pi$
$383$	−16.2703	−0.831374	−0.415687	+	0.909508i	$0.636459\pi$
$384$	0	0
$385$	−13.3145	−0.678572
$386$	−3.38615	−0.172350
$387$	0	0
$388$	11.4116	0.579335
$389$	25.7980	1.30801	0.654005	−	0.756491i	$-0.273088\pi$
$389$	25.7980	1.30801	0.654005	+	0.756491i	$0.273088\pi$
$390$	0	0
$391$	−4.29420	−0.217167
$392$	−0.536598	−0.0271023
$393$	0	0
$394$	−5.16355	−0.260136
$395$	−7.27604	−0.366097
$396$	0	0
$397$	10.3951	0.521713	0.260857	−	0.965378i	$-0.415995\pi$
$397$	10.3951	0.521713	0.260857	+	0.965378i	$0.415995\pi$
$398$	−8.66099	−0.434136
$399$	0	0
$400$	1.96000	0.0980001
$401$	0.0846003	0.00422474	0.00211237	−	0.999998i	$-0.499328\pi$
$401$	0.0846003	0.00422474	0.00211237	+	0.999998i	$0.499328\pi$
$402$	0	0
$403$	−6.98397	−0.347896
$404$	20.0644	0.998243
$405$	0	0
$406$	−5.20748	−0.258443
$407$	54.5008	2.70150
$408$	0	0
$409$	−20.9394	−1.03539	−0.517694	−	0.855566i	$-0.673209\pi$
$409$	−20.9394	−1.03539	−0.517694	+	0.855566i	$0.673209\pi$
$410$	2.73075	0.134862
$411$	0	0
$412$	−27.5495	−1.35727
$413$	2.12663	0.104644
$414$	0	0
$415$	16.0655	0.788624
$416$	−5.55580	−0.272395
$417$	0	0
$418$	−13.9741	−0.683494
$419$	21.4183	1.04635	0.523176	−	0.852225i	$-0.324747\pi$
$419$	21.4183	1.04635	0.523176	+	0.852225i	$0.324747\pi$
$420$	0	0
$421$	−27.8727	−1.35843	−0.679217	−	0.733937i	$-0.737681\pi$
$421$	−27.8727	−1.35843	−0.679217	+	0.733937i	$0.737681\pi$
$422$	−9.07139	−0.441588
$423$	0	0
$424$	−21.1871	−1.02894
$425$	−5.36998	−0.260482
$426$	0	0
$427$	11.0404	0.534280
$428$	8.36314	0.404248
$429$	0	0
$430$	4.00368	0.193075
$431$	0.802560	0.0386580	0.0193290	−	0.999813i	$-0.493847\pi$
$431$	0.802560	0.0386580	0.0193290	+	0.999813i	$0.493847\pi$
$432$	0	0
$433$	−22.8580	−1.09848	−0.549242	−	0.835663i	$-0.685083\pi$
$433$	−22.8580	−1.09848	−0.549242	+	0.835663i	$0.685083\pi$
$434$	−10.9183	−0.524094
$435$	0	0
$436$	10.0903	0.483238
$437$	−3.62633	−0.173471
$438$	0	0
$439$	38.4500	1.83512	0.917559	−	0.397600i	$-0.130157\pi$
$439$	38.4500	1.83512	0.917559	+	0.397600i	$0.130157\pi$
$440$	11.2111	0.534470
$441$	0	0
$442$	3.23011	0.153641
$443$	34.5288	1.64051	0.820257	−	0.571995i	$-0.193830\pi$
$443$	34.5288	1.64051	0.820257	+	0.571995i	$0.193830\pi$
$444$	0	0
$445$	−4.74177	−0.224781
$446$	3.00869	0.142466
$447$	0	0
$448$	1.50252	0.0709874
$449$	−31.3590	−1.47992	−0.739962	−	0.672649i	$-0.765157\pi$
$449$	−31.3590	−1.47992	−0.739962	+	0.672649i	$0.765157\pi$
$450$	0	0
$451$	−23.2572	−1.09514
$452$	5.19946	0.244562
$453$	0	0
$454$	2.25495	0.105830
$455$	2.59900	0.121843
$456$	0	0
$457$	−13.0361	−0.609804	−0.304902	−	0.952384i	$-0.598624\pi$
$457$	−13.0361	−0.609804	−0.304902	+	0.952384i	$0.598624\pi$
$458$	9.04004	0.422413
$459$	0	0
$460$	1.31000	0.0610791
$461$	−40.2203	−1.87325	−0.936623	−	0.350338i	$-0.886067\pi$
$461$	−40.2203	−1.87325	−0.936623	+	0.350338i	$0.886067\pi$
$462$	0	0
$463$	−30.0786	−1.39787	−0.698935	−	0.715185i	$-0.746342\pi$
$463$	−30.0786	−1.39787	−0.698935	+	0.715185i	$0.746342\pi$
$464$	−6.52878	−0.303091
$465$	0	0
$466$	−2.58021	−0.119526
$467$	−15.9026	−0.735882	−0.367941	−	0.929849i	$-0.619937\pi$
$467$	−15.9026	−0.735882	−0.367941	+	0.929849i	$0.619937\pi$
$468$	0	0
$469$	25.0604	1.15718
$470$	−3.21867	−0.148466
$471$	0	0
$472$	−1.79067	−0.0824221
$473$	−34.0984	−1.56785
$474$	0	0
$475$	−4.53480	−0.208071
$476$	−22.8634	−1.04794
$477$	0	0
$478$	10.7141	0.490054
$479$	−19.7984	−0.904611	−0.452306	−	0.891863i	$-0.649398\pi$
$479$	−19.7984	−0.904611	−0.452306	+	0.891863i	$0.649398\pi$
$480$	0	0
$481$	−10.6386	−0.485076
$482$	9.68018	0.440920
$483$	0	0
$484$	−24.9734	−1.13516
$485$	6.96601	0.316310
$486$	0	0
$487$	28.2257	1.27903	0.639514	−	0.768779i	$-0.279135\pi$
$487$	28.2257	1.27903	0.639514	+	0.768779i	$0.279135\pi$
$488$	−9.29622	−0.420820
$489$	0	0
$490$	−0.147491	−0.00666296
$491$	3.43024	0.154804	0.0774022	−	0.997000i	$-0.475337\pi$
$491$	3.43024	0.154804	0.0774022	+	0.997000i	$0.475337\pi$
$492$	0	0
$493$	17.8874	0.805610
$494$	2.72774	0.122727
$495$	0	0
$496$	−13.6886	−0.614636
$497$	2.95880	0.132720
$498$	0	0
$499$	35.3824	1.58393	0.791967	−	0.610564i	$-0.209057\pi$
$499$	35.3824	1.58393	0.791967	+	0.610564i	$0.209057\pi$
$500$	1.63818	0.0732617
$501$	0	0
$502$	8.33102	0.371831
$503$	−13.9869	−0.623645	−0.311822	−	0.950140i	$-0.600939\pi$
$503$	−13.9869	−0.623645	−0.311822	+	0.950140i	$0.600939\pi$
$504$	0	0
$505$	12.2480	0.545028
$506$	2.46420	0.109547
$507$	0	0
$508$	−23.4023	−1.03831
$509$	−1.99467	−0.0884124	−0.0442062	−	0.999022i	$-0.514076\pi$
$509$	−1.99467	−0.0884124	−0.0442062	+	0.999022i	$0.514076\pi$
$510$	0	0
$511$	−5.12669	−0.226791
$512$	−19.4680	−0.860371
$513$	0	0
$514$	2.70632	0.119371
$515$	−16.8171	−0.741052
$516$	0	0
$517$	27.4126	1.20561
$518$	−16.6316	−0.730751
$519$	0	0
$520$	−2.18842	−0.0959684
$521$	−13.4590	−0.589648	−0.294824	−	0.955552i	$-0.595261\pi$
$521$	−13.4590	−0.589648	−0.294824	+	0.955552i	$0.595261\pi$
$522$	0	0
$523$	−14.1211	−0.617475	−0.308737	−	0.951147i	$-0.599906\pi$
$523$	−14.1211	−0.617475	−0.308737	+	0.951147i	$0.599906\pi$
$524$	28.0611	1.22585
$525$	0	0
$526$	−10.3102	−0.449547
$527$	37.5038	1.63369
$528$	0	0
$529$	−22.3605	−0.972197
$530$	−5.82355	−0.252959
$531$	0	0
$532$	−19.3075	−0.837086
$533$	4.53980	0.196641
$534$	0	0
$535$	5.10514	0.220714
$536$	−21.1014	−0.911442
$537$	0	0
$538$	−1.08914	−0.0469562
$539$	1.25614	0.0541060
$540$	0	0
$541$	27.8421	1.19703	0.598513	−	0.801113i	$-0.295758\pi$
$541$	27.8421	1.19703	0.598513	+	0.801113i	$0.295758\pi$
$542$	−11.8316	−0.508210
$543$	0	0
$544$	29.8345	1.27914
$545$	6.15946	0.263842
$546$	0	0
$547$	42.3628	1.81130	0.905652	−	0.424022i	$-0.139382\pi$
$547$	42.3628	1.81130	0.905652	+	0.424022i	$0.139382\pi$
$548$	0.400174	0.0170946
$549$	0	0
$550$	3.08152	0.131397
$551$	15.1054	0.643514
$552$	0	0
$553$	−18.9104	−0.804153
$554$	7.45495	0.316731
$555$	0	0
$556$	−4.83631	−0.205105
$557$	0.190749	0.00808228	0.00404114	−	0.999992i	$-0.498714\pi$
$557$	0.190749	0.00808228	0.00404114	+	0.999992i	$0.498714\pi$
$558$	0	0
$559$	6.65602	0.281520
$560$	5.09404	0.215263
$561$	0	0
$562$	−6.95785	−0.293499
$563$	−27.0175	−1.13865	−0.569325	−	0.822112i	$-0.692795\pi$
$563$	−27.0175	−1.13865	−0.569325	+	0.822112i	$0.692795\pi$
$564$	0	0
$565$	3.17392	0.133528
$566$	−8.53471	−0.358740
$567$	0	0
$568$	−2.49138	−0.104536
$569$	−13.8286	−0.579724	−0.289862	−	0.957068i	$-0.593609\pi$
$569$	−13.8286	−0.579724	−0.289862	+	0.957068i	$0.593609\pi$
$570$	0	0
$571$	−43.6163	−1.82528	−0.912642	−	0.408760i	$-0.865961\pi$
$571$	−43.6163	−1.82528	−0.912642	+	0.408760i	$0.865961\pi$
$572$	8.39232	0.350900
$573$	0	0
$574$	7.09722	0.296232
$575$	0.799668	0.0333485
$576$	0	0
$577$	−15.4561	−0.643445	−0.321723	−	0.946834i	$-0.604262\pi$
$577$	−15.4561	−0.643445	−0.321723	+	0.946834i	$0.604262\pi$
$578$	−7.11991	−0.296149
$579$	0	0
$580$	−5.45680	−0.226581
$581$	41.7542	1.73226
$582$	0	0
$583$	49.5978	2.05413
$584$	4.31678	0.178630
$585$	0	0
$586$	−10.5677	−0.436548
$587$	32.9439	1.35974	0.679871	−	0.733332i	$-0.262036\pi$
$587$	32.9439	1.35974	0.679871	+	0.733332i	$0.262036\pi$
$588$	0	0
$589$	31.6709	1.30498
$590$	−0.492187	−0.0202630
$591$	0	0
$592$	−20.8516	−0.856995
$593$	−13.4309	−0.551540	−0.275770	−	0.961224i	$-0.588933\pi$
$593$	−13.4309	−0.551540	−0.275770	+	0.961224i	$0.588933\pi$
$594$	0	0
$595$	−13.9566	−0.572164
$596$	36.2289	1.48399
$597$	0	0
$598$	−0.481011	−0.0196700
$599$	23.0772	0.942907	0.471454	−	0.881891i	$-0.343729\pi$
$599$	23.0772	0.942907	0.471454	+	0.881891i	$0.343729\pi$
$600$	0	0
$601$	15.2497	0.622048	0.311024	−	0.950402i	$-0.399328\pi$
$601$	15.2497	0.622048	0.311024	+	0.950402i	$0.399328\pi$
$602$	10.4056	0.424100
$603$	0	0
$604$	−35.4985	−1.44441
$605$	−15.2446	−0.619781
$606$	0	0
$607$	−26.6312	−1.08093	−0.540463	−	0.841368i	$-0.681751\pi$
$607$	−26.6312	−1.08093	−0.540463	+	0.841368i	$0.681751\pi$
$608$	25.1944	1.02177
$609$	0	0
$610$	−2.55518	−0.103456
$611$	−5.35095	−0.216476
$612$	0	0
$613$	−6.15713	−0.248684	−0.124342	−	0.992239i	$-0.539682\pi$
$613$	−6.15713	−0.248684	−0.124342	+	0.992239i	$0.539682\pi$
$614$	−17.1419	−0.691792
$615$	0	0
$616$	29.1377	1.17399
$617$	−0.326232	−0.0131336	−0.00656680	−	0.999978i	$-0.502090\pi$
$617$	−0.326232	−0.0131336	−0.00656680	+	0.999978i	$0.502090\pi$
$618$	0	0
$619$	−20.1344	−0.809271	−0.404636	−	0.914478i	$-0.632602\pi$
$619$	−20.1344	−0.809271	−0.404636	+	0.914478i	$0.632602\pi$
$620$	−11.4410	−0.459482
$621$	0	0
$622$	12.7474	0.511125
$623$	−12.3238	−0.493745
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.56913	−0.302523
$627$	0	0
$628$	17.3993	0.694308
$629$	57.1288	2.27788
$630$	0	0
$631$	5.53688	0.220420	0.110210	−	0.993908i	$-0.464848\pi$
$631$	5.53688	0.220420	0.110210	+	0.993908i	$0.464848\pi$
$632$	15.9230	0.633383
$633$	0	0
$634$	7.65348	0.303958
$635$	−14.2855	−0.566903
$636$	0	0
$637$	−0.245199	−0.00971516
$638$	−10.2646	−0.406379
$639$	0	0
$640$	−11.4593	−0.452970
$641$	−26.7431	−1.05629	−0.528144	−	0.849155i	$-0.677112\pi$
$641$	−26.7431	−1.05629	−0.528144	+	0.849155i	$0.677112\pi$
$642$	0	0
$643$	22.5029	0.887429	0.443715	−	0.896168i	$-0.353660\pi$
$643$	22.5029	0.887429	0.443715	+	0.896168i	$0.353660\pi$
$644$	3.40469	0.134164
$645$	0	0
$646$	−14.6479	−0.576314
$647$	−26.4570	−1.04013	−0.520066	−	0.854126i	$-0.674093\pi$
$647$	−26.4570	−1.04013	−0.520066	+	0.854126i	$0.674093\pi$
$648$	0	0
$649$	4.19184	0.164544
$650$	−0.601514	−0.0235933
$651$	0	0
$652$	25.5899	1.00218
$653$	−44.4831	−1.74076	−0.870380	−	0.492381i	$-0.836127\pi$
$653$	−44.4831	−1.74076	−0.870380	+	0.492381i	$0.836127\pi$
$654$	0	0
$655$	17.1294	0.669302
$656$	8.89802	0.347409
$657$	0	0
$658$	−8.36531	−0.326114
$659$	25.6663	0.999816	0.499908	−	0.866079i	$-0.333367\pi$
$659$	25.6663	0.999816	0.499908	+	0.866079i	$0.333367\pi$
$660$	0	0
$661$	10.5375	0.409861	0.204931	−	0.978776i	$-0.434303\pi$
$661$	10.5375	0.409861	0.204931	+	0.978776i	$0.434303\pi$
$662$	4.16705	0.161957
$663$	0	0
$664$	−35.1580	−1.36439
$665$	−11.7859	−0.457039
$666$	0	0
$667$	−2.66370	−0.103139
$668$	−39.8146	−1.54047
$669$	0	0
$670$	−5.79999	−0.224073
$671$	21.7619	0.840108
$672$	0	0
$673$	−33.4414	−1.28907	−0.644536	−	0.764574i	$-0.722949\pi$
$673$	−33.4414	−1.28907	−0.644536	+	0.764574i	$0.722949\pi$
$674$	1.34637	0.0518603
$675$	0	0
$676$	−1.63818	−0.0630070
$677$	6.65558	0.255795	0.127897	−	0.991787i	$-0.459177\pi$
$677$	6.65558	0.255795	0.127897	+	0.991787i	$0.459177\pi$
$678$	0	0
$679$	18.1046	0.694793
$680$	11.7517	0.450659
$681$	0	0
$682$	−21.5213	−0.824092
$683$	5.46952	0.209285	0.104643	−	0.994510i	$-0.466630\pi$
$683$	5.46952	0.209285	0.104643	+	0.994510i	$0.466630\pi$
$684$	0	0
$685$	0.244279	0.00933343
$686$	−11.3267	−0.432454
$687$	0	0
$688$	13.0458	0.497367
$689$	−9.68150	−0.368836
$690$	0	0
$691$	23.4220	0.891015	0.445508	−	0.895278i	$-0.353023\pi$
$691$	23.4220	0.891015	0.445508	+	0.895278i	$0.353023\pi$
$692$	22.4325	0.852754
$693$	0	0
$694$	−15.9129	−0.604045
$695$	−2.95224	−0.111985
$696$	0	0
$697$	−24.3786	−0.923407
$698$	−10.1260	−0.383275
$699$	0	0
$700$	4.25763	0.160923
$701$	−43.7946	−1.65410	−0.827050	−	0.562129i	$-0.809983\pi$
$701$	−43.7946	−1.65410	−0.827050	+	0.562129i	$0.809983\pi$
$702$	0	0
$703$	48.2437	1.81954
$704$	2.96165	0.111621
$705$	0	0
$706$	−19.7869	−0.744689
$707$	31.8325	1.19719
$708$	0	0
$709$	−9.90275	−0.371905	−0.185953	−	0.982559i	$-0.559537\pi$
$709$	−9.90275	−0.371905	−0.185953	+	0.982559i	$0.559537\pi$
$710$	−0.684786	−0.0256996
$711$	0	0
$712$	10.3770	0.388893
$713$	−5.58486	−0.209155
$714$	0	0
$715$	5.12295	0.191587
$716$	8.15437	0.304743
$717$	0	0
$718$	2.24509	0.0837859
$719$	1.51958	0.0566708	0.0283354	−	0.999598i	$-0.490979\pi$
$719$	1.51958	0.0566708	0.0283354	+	0.999598i	$0.490979\pi$
$720$	0	0
$721$	−43.7078	−1.62776
$722$	−0.940993	−0.0350201
$723$	0	0
$724$	−40.0841	−1.48971
$725$	−3.33101	−0.123711
$726$	0	0
$727$	48.0025	1.78032	0.890158	−	0.455653i	$-0.150594\pi$
$727$	48.0025	1.78032	0.890158	+	0.455653i	$0.150594\pi$
$728$	−5.68769	−0.210800
$729$	0	0
$730$	1.18652	0.0439152
$731$	−35.7427	−1.32199
$732$	0	0
$733$	15.5043	0.572665	0.286333	−	0.958130i	$-0.407564\pi$
$733$	15.5043	0.572665	0.286333	+	0.958130i	$0.407564\pi$
$734$	14.6153	0.539460
$735$	0	0
$736$	−4.44280	−0.163764
$737$	49.3971	1.81957
$738$	0	0
$739$	−28.8573	−1.06153	−0.530766	−	0.847519i	$-0.678096\pi$
$739$	−28.8573	−1.06153	−0.530766	+	0.847519i	$0.678096\pi$
$740$	−17.4279	−0.640662
$741$	0	0
$742$	−15.1354	−0.555639
$743$	7.05785	0.258927	0.129464	−	0.991584i	$-0.458674\pi$
$743$	7.05785	0.258927	0.129464	+	0.991584i	$0.458674\pi$
$744$	0	0
$745$	22.1153	0.810242
$746$	5.96554	0.218414
$747$	0	0
$748$	−45.0666	−1.64780
$749$	13.2683	0.484812
$750$	0	0
$751$	−39.6093	−1.44536	−0.722681	−	0.691182i	$-0.757090\pi$
$751$	−39.6093	−1.44536	−0.722681	+	0.691182i	$0.757090\pi$
$752$	−10.4879	−0.382453
$753$	0	0
$754$	2.00365	0.0729685
$755$	−21.6695	−0.788632
$756$	0	0
$757$	7.29994	0.265321	0.132660	−	0.991162i	$-0.457648\pi$
$757$	7.29994	0.265321	0.132660	+	0.991162i	$0.457648\pi$
$758$	9.42395	0.342293
$759$	0	0
$760$	9.92402	0.359982
$761$	−34.7012	−1.25792	−0.628958	−	0.777439i	$-0.716518\pi$
$761$	−34.7012	−1.25792	−0.628958	+	0.777439i	$0.716518\pi$
$762$	0	0
$763$	16.0084	0.579544
$764$	20.0721	0.726184
$765$	0	0
$766$	9.78681	0.353612
$767$	−0.818248	−0.0295452
$768$	0	0
$769$	22.6012	0.815019	0.407510	−	0.913201i	$-0.366397\pi$
$769$	22.6012	0.815019	0.407510	+	0.913201i	$0.366397\pi$
$770$	8.00888	0.288620
$771$	0	0
$772$	−9.22194	−0.331905
$773$	3.33276	0.119871	0.0599355	−	0.998202i	$-0.480910\pi$
$773$	3.33276	0.119871	0.0599355	+	0.998202i	$0.480910\pi$
$774$	0	0
$775$	−6.98397	−0.250872
$776$	−15.2445	−0.547246
$777$	0	0
$778$	−15.5178	−0.556341
$779$	−20.5871	−0.737608
$780$	0	0
$781$	5.83216	0.208691
$782$	2.58302	0.0923686
$783$	0	0
$784$	−0.480591	−0.0171640
$785$	10.6211	0.379084
$786$	0	0
$787$	21.4116	0.763242	0.381621	−	0.924319i	$-0.375366\pi$
$787$	21.4116	0.763242	0.381621	+	0.924319i	$0.375366\pi$
$788$	−14.0626	−0.500959
$789$	0	0
$790$	4.37664	0.155714
$791$	8.24902	0.293301
$792$	0	0
$793$	−4.24792	−0.150848
$794$	−6.25277	−0.221903
$795$	0	0
$796$	−23.5876	−0.836041
$797$	−3.23357	−0.114539	−0.0572694	−	0.998359i	$-0.518239\pi$
$797$	−3.23357	−0.114539	−0.0572694	+	0.998359i	$0.518239\pi$
$798$	0	0
$799$	28.7345	1.01655
$800$	−5.55580	−0.196427
$801$	0	0
$802$	−0.0508882	−0.00179693
$803$	−10.1053	−0.356609
$804$	0	0
$805$	2.07834	0.0732518
$806$	4.20095	0.147972
$807$	0	0
$808$	−26.8037	−0.942951
$809$	22.3591	0.786106	0.393053	−	0.919516i	$-0.371419\pi$
$809$	22.3591	0.786106	0.393053	+	0.919516i	$0.371419\pi$
$810$	0	0
$811$	−45.4604	−1.59633	−0.798166	−	0.602438i	$-0.794196\pi$
$811$	−45.4604	−1.59633	−0.798166	+	0.602438i	$0.794196\pi$
$812$	−14.1822	−0.497698
$813$	0	0
$814$	−32.7830	−1.14904
$815$	15.6209	0.547176
$816$	0	0
$817$	−30.1837	−1.05599
$818$	12.5953	0.440386
$819$	0	0
$820$	7.43702	0.259712
$821$	−30.2229	−1.05479	−0.527394	−	0.849621i	$-0.676831\pi$
$821$	−30.2229	−1.05479	−0.527394	+	0.849621i	$0.676831\pi$
$822$	0	0
$823$	4.79089	0.167000	0.0834999	−	0.996508i	$-0.473390\pi$
$823$	4.79089	0.167000	0.0834999	+	0.996508i	$0.473390\pi$
$824$	36.8029	1.28209
$825$	0	0
$826$	−1.27919	−0.0445088
$827$	22.0981	0.768426	0.384213	−	0.923244i	$-0.374473\pi$
$827$	22.0981	0.768426	0.384213	+	0.923244i	$0.374473\pi$
$828$	0	0
$829$	−16.9618	−0.589108	−0.294554	−	0.955635i	$-0.595171\pi$
$829$	−16.9618	−0.589108	−0.294554	+	0.955635i	$0.595171\pi$
$830$	−9.66362	−0.335429
$831$	0	0
$832$	−0.578115	−0.0200425
$833$	1.31672	0.0456215
$834$	0	0
$835$	−24.3041	−0.841079
$836$	−38.0575	−1.31624
$837$	0	0
$838$	−12.8834	−0.445049
$839$	−44.7354	−1.54444	−0.772220	−	0.635356i	$-0.780853\pi$
$839$	−44.7354	−1.54444	−0.772220	+	0.635356i	$0.780853\pi$
$840$	0	0
$841$	−17.9044	−0.617392
$842$	16.7658	0.577789
$843$	0	0
$844$	−24.7053	−0.850392
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−39.6207	−1.36138
$848$	−18.9758	−0.651630
$849$	0	0
$850$	3.23011	0.110792
$851$	−8.50732	−0.291627
$852$	0	0
$853$	−3.51534	−0.120363	−0.0601816	−	0.998187i	$-0.519168\pi$
$853$	−3.51534	−0.120363	−0.0601816	+	0.998187i	$0.519168\pi$
$854$	−6.64092	−0.227248
$855$	0	0
$856$	−11.1722	−0.381857
$857$	5.00168	0.170854	0.0854271	−	0.996344i	$-0.472775\pi$
$857$	5.00168	0.170854	0.0854271	+	0.996344i	$0.472775\pi$
$858$	0	0
$859$	−48.9730	−1.67094	−0.835469	−	0.549538i	$-0.814804\pi$
$859$	−48.9730	−1.67094	−0.835469	+	0.549538i	$0.814804\pi$
$860$	10.9038	0.371815
$861$	0	0
$862$	−0.482751	−0.0164426
$863$	−10.6834	−0.363666	−0.181833	−	0.983329i	$-0.558203\pi$
$863$	−10.6834	−0.363666	−0.181833	+	0.983329i	$0.558203\pi$
$864$	0	0
$865$	13.6935	0.465593
$866$	13.7494	0.467223
$867$	0	0
$868$	−29.7352	−1.00928
$869$	−37.2748	−1.26446
$870$	0	0
$871$	−9.64232	−0.326718
$872$	−13.4795	−0.456472
$873$	0	0
$874$	2.18129	0.0737832
$875$	2.59900	0.0878622
$876$	0	0
$877$	21.3594	0.721257	0.360628	−	0.932710i	$-0.382562\pi$
$877$	21.3594	0.721257	0.360628	+	0.932710i	$0.382562\pi$
$878$	−23.1282	−0.780539
$879$	0	0
$880$	10.0410	0.338482
$881$	45.2544	1.52466	0.762330	−	0.647189i	$-0.224056\pi$
$881$	45.2544	1.52466	0.762330	+	0.647189i	$0.224056\pi$
$882$	0	0
$883$	20.4837	0.689330	0.344665	−	0.938726i	$-0.387992\pi$
$883$	20.4837	0.689330	0.344665	+	0.938726i	$0.387992\pi$
$884$	8.79700	0.295875
$885$	0	0
$886$	−20.7696	−0.697767
$887$	12.3187	0.413623	0.206811	−	0.978381i	$-0.433691\pi$
$887$	12.3187	0.413623	0.206811	+	0.978381i	$0.433691\pi$
$888$	0	0
$889$	−37.1281	−1.24523
$890$	2.85224	0.0956072
$891$	0	0
$892$	8.19398	0.274355
$893$	24.2655	0.812012
$894$	0	0
$895$	4.97769	0.166386
$896$	−29.7828	−0.994974
$897$	0	0
$898$	18.8629	0.629463
$899$	23.2637	0.775887
$900$	0	0
$901$	51.9895	1.73202
$902$	13.9895	0.465799
$903$	0	0
$904$	−6.94586	−0.231016
$905$	−24.4686	−0.813365
$906$	0	0
$907$	42.6331	1.41561	0.707805	−	0.706408i	$-0.249685\pi$
$907$	42.6331	1.41561	0.707805	+	0.706408i	$0.249685\pi$
$908$	6.14120	0.203803
$909$	0	0
$910$	−1.56333	−0.0518240
$911$	−32.8835	−1.08948	−0.544739	−	0.838606i	$-0.683371\pi$
$911$	−32.8835	−1.08948	−0.544739	+	0.838606i	$0.683371\pi$
$912$	0	0
$913$	82.3027	2.72382
$914$	7.84140	0.259370
$915$	0	0
$916$	24.6199	0.813466
$917$	44.5193	1.47016
$918$	0	0
$919$	−53.2590	−1.75685	−0.878427	−	0.477877i	$-0.841406\pi$
$919$	−53.2590	−1.75685	−0.878427	+	0.477877i	$0.841406\pi$
$920$	−1.75001	−0.0576960
$921$	0	0
$922$	24.1931	0.796756
$923$	−1.13844	−0.0374722
$924$	0	0
$925$	−10.6386	−0.349794
$926$	18.0927	0.594562
$927$	0	0
$928$	18.5064	0.607503
$929$	48.9975	1.60756	0.803778	−	0.594929i	$-0.202820\pi$
$929$	48.9975	1.60756	0.803778	+	0.594929i	$0.202820\pi$
$930$	0	0
$931$	1.11193	0.0364420
$932$	−7.02702	−0.230178
$933$	0	0
$934$	9.56560	0.312996
$935$	−27.5101	−0.899677
$936$	0	0
$937$	−51.7278	−1.68987	−0.844937	−	0.534865i	$-0.820362\pi$
$937$	−51.7278	−1.68987	−0.844937	+	0.534865i	$0.820362\pi$
$938$	−15.0742	−0.492189
$939$	0	0
$940$	−8.76582	−0.285910
$941$	−12.4024	−0.404308	−0.202154	−	0.979354i	$-0.564794\pi$
$941$	−12.4024	−0.404308	−0.202154	+	0.979354i	$0.564794\pi$
$942$	0	0
$943$	3.63033	0.118220
$944$	−1.60377	−0.0521981
$945$	0	0
$946$	20.5107	0.666859
$947$	−3.46029	−0.112444	−0.0562222	−	0.998418i	$-0.517906\pi$
$947$	−3.46029	−0.112444	−0.0562222	+	0.998418i	$0.517906\pi$
$948$	0	0
$949$	1.97256	0.0640321
$950$	2.72774	0.0884996
$951$	0	0
$952$	30.5428	0.989897
$953$	21.5313	0.697467	0.348733	−	0.937222i	$-0.386612\pi$
$953$	21.5313	0.697467	0.348733	+	0.937222i	$0.386612\pi$
$954$	0	0
$955$	12.2527	0.396488
$956$	29.1792	0.943724
$957$	0	0
$958$	11.9090	0.384762
$959$	0.634882	0.0205014
$960$	0	0
$961$	17.7759	0.573415
$962$	6.39924	0.206320
$963$	0	0
$964$	26.3633	0.849105
$965$	−5.62938	−0.181216
$966$	0	0
$967$	−51.9665	−1.67113	−0.835565	−	0.549391i	$-0.814860\pi$
$967$	−51.9665	−1.67113	−0.835565	+	0.549391i	$0.814860\pi$
$968$	33.3615	1.07228
$969$	0	0
$970$	−4.19015	−0.134538
$971$	−1.67242	−0.0536705	−0.0268352	−	0.999640i	$-0.508543\pi$
$971$	−1.67242	−0.0536705	−0.0268352	+	0.999640i	$0.508543\pi$
$972$	0	0
$973$	−7.67288	−0.245981
$974$	−16.9781	−0.544015
$975$	0	0
$976$	−8.32593	−0.266507
$977$	−24.1339	−0.772111	−0.386055	−	0.922476i	$-0.626163\pi$
$977$	−24.1339	−0.772111	−0.386055	+	0.922476i	$0.626163\pi$
$978$	0	0
$979$	−24.2918	−0.776370
$980$	−0.401681	−0.0128312
$981$	0	0
$982$	−2.06333	−0.0658436
$983$	26.0412	0.830585	0.415293	−	0.909688i	$-0.363679\pi$
$983$	26.0412	0.830585	0.415293	+	0.909688i	$0.363679\pi$
$984$	0	0
$985$	−8.58427	−0.273518
$986$	−10.7595	−0.342654
$987$	0	0
$988$	7.42882	0.236342
$989$	5.32261	0.169249
$990$	0	0
$991$	−31.4180	−0.998026	−0.499013	−	0.866594i	$-0.666304\pi$
$991$	−31.4180	−0.998026	−0.499013	+	0.866594i	$0.666304\pi$
$992$	38.8015	1.23195
$993$	0	0
$994$	−1.77976	−0.0564505
$995$	−14.3987	−0.456468
$996$	0	0
$997$	−30.4488	−0.964322	−0.482161	−	0.876083i	$-0.660148\pi$
$997$	−30.4488	−0.964322	−0.482161	+	0.876083i	$0.660148\pi$
$998$	−21.2830	−0.673701
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bk.1.7		15
3.2	odd	2		5265.2.a.bl.1.9		15
9.2	odd	6		1755.2.i.h.1171.7		30
9.4	even	3		585.2.i.h.196.9	✓	30
9.5	odd	6		1755.2.i.h.586.7		30
9.7	even	3		585.2.i.h.391.9	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.h.196.9	✓	30	9.4	even	3
585.2.i.h.391.9	yes	30	9.7	even	3
1755.2.i.h.586.7		30	9.5	odd	6
1755.2.i.h.1171.7		30	9.2	odd	6
5265.2.a.bk.1.7		15	1.1	even	1	trivial
5265.2.a.bl.1.9		15	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)

\(f(q)\)	\(=\)	\(q-0.601514 q^{2} -1.63818 q^{4} -1.00000 q^{5} -2.59900 q^{7} +2.18842 q^{8} +O(q^{10})\)	\(q-0.601514 q^{2} -1.63818 q^{4} -1.00000 q^{5} -2.59900 q^{7} +2.18842 q^{8} +0.601514 q^{10} -5.12295 q^{11} +1.00000 q^{13} +1.56333 q^{14} +1.96000 q^{16} -5.36998 q^{17} -4.53480 q^{19} +1.63818 q^{20} +3.08152 q^{22} +0.799668 q^{23} +1.00000 q^{25} -0.601514 q^{26} +4.25763 q^{28} -3.33101 q^{29} -6.98397 q^{31} -5.55580 q^{32} +3.23011 q^{34} +2.59900 q^{35} -10.6386 q^{37} +2.72774 q^{38} -2.18842 q^{40} +4.53980 q^{41} +6.65602 q^{43} +8.39232 q^{44} -0.481011 q^{46} -5.35095 q^{47} -0.245199 q^{49} -0.601514 q^{50} -1.63818 q^{52} -9.68150 q^{53} +5.12295 q^{55} -5.68769 q^{56} +2.00365 q^{58} -0.818248 q^{59} -4.24792 q^{61} +4.20095 q^{62} -0.578115 q^{64} -1.00000 q^{65} -9.64232 q^{67} +8.79700 q^{68} -1.56333 q^{70} -1.13844 q^{71} +1.97256 q^{73} +6.39924 q^{74} +7.42882 q^{76} +13.3145 q^{77} +7.27604 q^{79} -1.96000 q^{80} -2.73075 q^{82} -16.0655 q^{83} +5.36998 q^{85} -4.00368 q^{86} -11.2111 q^{88} +4.74177 q^{89} -2.59900 q^{91} -1.31000 q^{92} +3.21867 q^{94} +4.53480 q^{95} -6.96601 q^{97} +0.147491 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} + 21 q^{4} - 15 q^{5} + 10 q^{7}+O(q^{10}) \) 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

Modular forms

Varieties

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Database

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