LMFDB - Embedded newform 5265.2.a.bf.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 $ x^8 - 3x^7 - 8x^6 + 31x^5 - x^4 - 70x^3 + 66x^2 - 19x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.494096$ 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)

$f(q)$	$=$	$q+0.494096 q^{2} -1.75587 q^{4} -1.00000 q^{5} -4.28539 q^{7} -1.85576 q^{8} +O(q^{10})$	\(q+0.494096 q^{2} -1.75587 q^{4} -1.00000 q^{5} -4.28539 q^{7} -1.85576 q^{8} -0.494096 q^{10} +4.48530 q^{11} -1.00000 q^{13} -2.11739 q^{14} +2.59482 q^{16} +1.57196 q^{17} +1.86934 q^{19} +1.75587 q^{20} +2.21617 q^{22} -5.25780 q^{23} +1.00000 q^{25} -0.494096 q^{26} +7.52458 q^{28} -0.750308 q^{29} +10.2013 q^{31} +4.99361 q^{32} +0.776699 q^{34} +4.28539 q^{35} +2.78257 q^{37} +0.923636 q^{38} +1.85576 q^{40} -6.36016 q^{41} +7.59324 q^{43} -7.87560 q^{44} -2.59786 q^{46} -8.27021 q^{47} +11.3645 q^{49} +0.494096 q^{50} +1.75587 q^{52} +0.0752567 q^{53} -4.48530 q^{55} +7.95265 q^{56} -0.370724 q^{58} +11.4407 q^{59} -11.9600 q^{61} +5.04044 q^{62} -2.72231 q^{64} +1.00000 q^{65} -12.8975 q^{67} -2.76016 q^{68} +2.11739 q^{70} +14.4466 q^{71} +9.37293 q^{73} +1.37486 q^{74} -3.28232 q^{76} -19.2212 q^{77} -6.09918 q^{79} -2.59482 q^{80} -3.14253 q^{82} +0.612178 q^{83} -1.57196 q^{85} +3.75179 q^{86} -8.32364 q^{88} -16.2485 q^{89} +4.28539 q^{91} +9.23201 q^{92} -4.08628 q^{94} -1.86934 q^{95} -14.9940 q^{97} +5.61517 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8}+O(q^{10}) $ 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8	$ 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} - 8 q^{13} + 10 q^{14} + 11 q^{16} - 2 q^{17} - 10 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 8 q^{25} - 3 q^{26} - 34 q^{28} + 14 q^{29} - 31 q^{31} + q^{32} - 7 q^{34} + 11 q^{35} + q^{37} + 9 q^{38} + 6 q^{40} - 12 q^{41} + 15 q^{43} + 16 q^{44} - 32 q^{46} - 18 q^{47} + 17 q^{49} + 3 q^{50} - 9 q^{52} + 2 q^{53} - 6 q^{55} + 16 q^{56} - 42 q^{58} + 24 q^{59} - 9 q^{61} - 20 q^{62} - 30 q^{64} + 8 q^{65} - 18 q^{67} - 14 q^{68} - 10 q^{70} + 10 q^{71} + 6 q^{73} - 37 q^{74} - 53 q^{76} - 34 q^{77} - 3 q^{79} - 11 q^{80} - 34 q^{82} - 10 q^{83} + 2 q^{85} + 60 q^{86} - 14 q^{88} - 13 q^{89} + 11 q^{91} + 5 q^{92} + 17 q^{94} + 10 q^{95} - 34 q^{97} - 30 q^{98}+O(q^{100}) $ 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 - 8 * q^13 + 10 * q^14 + 11 * q^16 - 2 * q^17 - 10 * q^19 - 9 * q^20 + 3 * q^22 + 6 * q^23 + 8 * q^25 - 3 * q^26 - 34 * q^28 + 14 * q^29 - 31 * q^31 + q^32 - 7 * q^34 + 11 * q^35 + q^37 + 9 * q^38 + 6 * q^40 - 12 * q^41 + 15 * q^43 + 16 * q^44 - 32 * q^46 - 18 * q^47 + 17 * q^49 + 3 * q^50 - 9 * q^52 + 2 * q^53 - 6 * q^55 + 16 * q^56 - 42 * q^58 + 24 * q^59 - 9 * q^61 - 20 * q^62 - 30 * q^64 + 8 * q^65 - 18 * q^67 - 14 * q^68 - 10 * q^70 + 10 * q^71 + 6 * q^73 - 37 * q^74 - 53 * q^76 - 34 * q^77 - 3 * q^79 - 11 * q^80 - 34 * q^82 - 10 * q^83 + 2 * q^85 + 60 * q^86 - 14 * q^88 - 13 * q^89 + 11 * q^91 + 5 * q^92 + 17 * q^94 + 10 * q^95 - 34 * q^97 - 30 * 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 $

$2$	0.494096	0.349379	0.174689	−	0.984624i	$-0.444108\pi$
$2$	0.494096	0.349379	0.174689	+	0.984624i	$0.444108\pi$
$3$	0	0
$3$	0	0
$4$	−1.75587	−0.877935
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.28539	−1.61972	−0.809862	−	0.586621i	$-0.800458\pi$
$7$	−4.28539	−1.61972	−0.809862	+	0.586621i	$0.800458\pi$
$8$	−1.85576	−0.656110
$9$	0	0
$10$	−0.494096	−0.156247
$11$	4.48530	1.35237	0.676185	−	0.736732i	$-0.263632\pi$
$11$	4.48530	1.35237	0.676185	+	0.736732i	$0.263632\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.11739	−0.565897
$15$	0	0
$16$	2.59482	0.648704
$17$	1.57196	0.381256	0.190628	−	0.981662i	$-0.438948\pi$
$17$	1.57196	0.381256	0.190628	+	0.981662i	$0.438948\pi$
$18$	0	0
$19$	1.86934	0.428857	0.214429	−	0.976740i	$-0.431211\pi$
$19$	1.86934	0.428857	0.214429	+	0.976740i	$0.431211\pi$
$20$	1.75587	0.392624
$21$	0	0
$22$	2.21617	0.472489
$23$	−5.25780	−1.09633	−0.548163	−	0.836371i	$-0.684673\pi$
$23$	−5.25780	−1.09633	−0.548163	+	0.836371i	$0.684673\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.494096	−0.0969002
$27$	0	0
$28$	7.52458	1.42201
$29$	−0.750308	−0.139329	−0.0696644	−	0.997570i	$-0.522193\pi$
$29$	−0.750308	−0.139329	−0.0696644	+	0.997570i	$0.522193\pi$
$30$	0	0
$31$	10.2013	1.83222	0.916108	−	0.400931i	$-0.131313\pi$
$31$	10.2013	1.83222	0.916108	+	0.400931i	$0.131313\pi$
$32$	4.99361	0.882753
$33$	0	0
$34$	0.776699	0.133203
$35$	4.28539	0.724362
$36$	0	0
$37$	2.78257	0.457452	0.228726	−	0.973491i	$-0.426544\pi$
$37$	2.78257	0.457452	0.228726	+	0.973491i	$0.426544\pi$
$38$	0.923636	0.149833
$39$	0	0
$40$	1.85576	0.293421
$41$	−6.36016	−0.993289	−0.496645	−	0.867954i	$-0.665435\pi$
$41$	−6.36016	−0.993289	−0.496645	+	0.867954i	$0.665435\pi$
$42$	0	0
$43$	7.59324	1.15796	0.578979	−	0.815342i	$-0.303451\pi$
$43$	7.59324	1.15796	0.578979	+	0.815342i	$0.303451\pi$
$44$	−7.87560	−1.18729
$45$	0	0
$46$	−2.59786	−0.383033
$47$	−8.27021	−1.20633	−0.603167	−	0.797615i	$-0.706095\pi$
$47$	−8.27021	−1.20633	−0.603167	+	0.797615i	$0.706095\pi$
$48$	0	0
$49$	11.3645	1.62350
$50$	0.494096	0.0698757
$51$	0	0
$52$	1.75587	0.243495
$53$	0.0752567	0.0103373	0.00516865	−	0.999987i	$-0.498355\pi$
$53$	0.0752567	0.0103373	0.00516865	+	0.999987i	$0.498355\pi$
$54$	0	0
$55$	−4.48530	−0.604798
$56$	7.95265	1.06272
$57$	0	0
$58$	−0.370724	−0.0486785
$59$	11.4407	1.48946	0.744728	−	0.667368i	$-0.232579\pi$
$59$	11.4407	1.48946	0.744728	+	0.667368i	$0.232579\pi$
$60$	0	0
$61$	−11.9600	−1.53133	−0.765663	−	0.643241i	$-0.777589\pi$
$61$	−11.9600	−1.53133	−0.765663	+	0.643241i	$0.777589\pi$
$62$	5.04044	0.640137
$63$	0	0
$64$	−2.72231	−0.340289
$65$	1.00000	0.124035
$66$	0	0
$67$	−12.8975	−1.57568	−0.787839	−	0.615881i	$-0.788800\pi$
$67$	−12.8975	−1.57568	−0.787839	+	0.615881i	$0.788800\pi$
$68$	−2.76016	−0.334718
$69$	0	0
$70$	2.11739	0.253077
$71$	14.4466	1.71450	0.857250	−	0.514900i	$-0.172171\pi$
$71$	14.4466	1.71450	0.857250	+	0.514900i	$0.172171\pi$
$72$	0	0
$73$	9.37293	1.09702	0.548509	−	0.836144i	$-0.315196\pi$
$73$	9.37293	1.09702	0.548509	+	0.836144i	$0.315196\pi$
$74$	1.37486	0.159824
$75$	0	0
$76$	−3.28232	−0.376508
$77$	−19.2212	−2.19046
$78$	0	0
$79$	−6.09918	−0.686212	−0.343106	−	0.939297i	$-0.611479\pi$
$79$	−6.09918	−0.686212	−0.343106	+	0.939297i	$0.611479\pi$
$80$	−2.59482	−0.290109
$81$	0	0
$82$	−3.14253	−0.347034
$83$	0.612178	0.0671953	0.0335976	−	0.999435i	$-0.489304\pi$
$83$	0.612178	0.0671953	0.0335976	+	0.999435i	$0.489304\pi$
$84$	0	0
$85$	−1.57196	−0.170503
$86$	3.75179	0.404566
$87$	0	0
$88$	−8.32364	−0.887303
$89$	−16.2485	−1.72234	−0.861169	−	0.508319i	$-0.830267\pi$
$89$	−16.2485	−1.72234	−0.861169	+	0.508319i	$0.830267\pi$
$90$	0	0
$91$	4.28539	0.449230
$92$	9.23201	0.962503
$93$	0	0
$94$	−4.08628	−0.421467
$95$	−1.86934	−0.191791
$96$	0	0
$97$	−14.9940	−1.52241	−0.761207	−	0.648509i	$-0.775393\pi$
$97$	−14.9940	−1.52241	−0.761207	+	0.648509i	$0.775393\pi$
$98$	5.61517	0.567218
$99$	0	0
$100$	−1.75587	−0.175587
$101$	10.8897	1.08357	0.541785	−	0.840517i	$-0.317749\pi$
$101$	10.8897	1.08357	0.541785	+	0.840517i	$0.317749\pi$
$102$	0	0
$103$	−1.53733	−0.151478	−0.0757390	−	0.997128i	$-0.524132\pi$
$103$	−1.53733	−0.151478	−0.0757390	+	0.997128i	$0.524132\pi$
$104$	1.85576	0.181972
$105$	0	0
$106$	0.0371840	0.00361163
$107$	−5.43803	−0.525715	−0.262857	−	0.964835i	$-0.584665\pi$
$107$	−5.43803	−0.525715	−0.262857	+	0.964835i	$0.584665\pi$
$108$	0	0
$109$	1.44466	0.138374	0.0691868	−	0.997604i	$-0.477960\pi$
$109$	1.44466	0.138374	0.0691868	+	0.997604i	$0.477960\pi$
$110$	−2.21617	−0.211303
$111$	0	0
$112$	−11.1198	−1.05072
$113$	4.46478	0.420011	0.210006	−	0.977700i	$-0.432652\pi$
$113$	4.46478	0.420011	0.210006	+	0.977700i	$0.432652\pi$
$114$	0	0
$115$	5.25780	0.490292
$116$	1.31744	0.122322
$117$	0	0
$118$	5.65282	0.520384
$119$	−6.73645	−0.617530
$120$	0	0
$121$	9.11794	0.828903
$122$	−5.90941	−0.535013
$123$	0	0
$124$	−17.9122	−1.60857
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.92097	−0.347930	−0.173965	−	0.984752i	$-0.555658\pi$
$127$	−3.92097	−0.347930	−0.173965	+	0.984752i	$0.555658\pi$
$128$	−11.3323	−1.00164
$129$	0	0
$130$	0.494096	0.0433351
$131$	0.774428	0.0676620	0.0338310	−	0.999428i	$-0.489229\pi$
$131$	0.774428	0.0676620	0.0338310	+	0.999428i	$0.489229\pi$
$132$	0	0
$133$	−8.01086	−0.694630
$134$	−6.37259	−0.550508
$135$	0	0
$136$	−2.91718	−0.250146
$137$	4.70810	0.402240	0.201120	−	0.979567i	$-0.435542\pi$
$137$	4.70810	0.402240	0.201120	+	0.979567i	$0.435542\pi$
$138$	0	0
$139$	−6.96691	−0.590926	−0.295463	−	0.955354i	$-0.595474\pi$
$139$	−6.96691	−0.590926	−0.295463	+	0.955354i	$0.595474\pi$
$140$	−7.52458	−0.635943
$141$	0	0
$142$	7.13803	0.599010
$143$	−4.48530	−0.375080
$144$	0	0
$145$	0.750308	0.0623097
$146$	4.63113	0.383275
$147$	0	0
$148$	−4.88583	−0.401613
$149$	−8.23725	−0.674822	−0.337411	−	0.941357i	$-0.609551\pi$
$149$	−8.23725	−0.674822	−0.337411	+	0.941357i	$0.609551\pi$
$150$	0	0
$151$	−8.77188	−0.713846	−0.356923	−	0.934134i	$-0.616174\pi$
$151$	−8.77188	−0.713846	−0.356923	+	0.934134i	$0.616174\pi$
$152$	−3.46905	−0.281377
$153$	0	0
$154$	−9.49714	−0.765301
$155$	−10.2013	−0.819392
$156$	0	0
$157$	−13.3937	−1.06894	−0.534468	−	0.845189i	$-0.679488\pi$
$157$	−13.3937	−1.06894	−0.534468	+	0.845189i	$0.679488\pi$
$158$	−3.01358	−0.239748
$159$	0	0
$160$	−4.99361	−0.394779
$161$	22.5317	1.77575
$162$	0	0
$163$	−0.242444	−0.0189897	−0.00949486	−	0.999955i	$-0.503022\pi$
$163$	−0.242444	−0.0189897	−0.00949486	+	0.999955i	$0.503022\pi$
$164$	11.1676	0.872043
$165$	0	0
$166$	0.302475	0.0234766
$167$	5.08610	0.393575	0.196787	−	0.980446i	$-0.436949\pi$
$167$	5.08610	0.393575	0.196787	+	0.980446i	$0.436949\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.776699	−0.0595701
$171$	0	0
$172$	−13.3327	−1.01661
$173$	−23.1711	−1.76166	−0.880831	−	0.473430i	$-0.843016\pi$
$173$	−23.1711	−1.76166	−0.880831	+	0.473430i	$0.843016\pi$
$174$	0	0
$175$	−4.28539	−0.323945
$176$	11.6385	0.877287
$177$	0	0
$178$	−8.02832	−0.601748
$179$	13.9237	1.04070	0.520352	−	0.853952i	$-0.325801\pi$
$179$	13.9237	1.04070	0.520352	+	0.853952i	$0.325801\pi$
$180$	0	0
$181$	13.6884	1.01745	0.508725	−	0.860929i	$-0.330117\pi$
$181$	13.6884	1.01745	0.508725	+	0.860929i	$0.330117\pi$
$182$	2.11739	0.156951
$183$	0	0
$184$	9.75721	0.719311
$185$	−2.78257	−0.204579
$186$	0	0
$187$	7.05071	0.515599
$188$	14.5214	1.05908
$189$	0	0
$190$	−0.923636	−0.0670076
$191$	−7.84336	−0.567525	−0.283763	−	0.958895i	$-0.591583\pi$
$191$	−7.84336	−0.567525	−0.283763	+	0.958895i	$0.591583\pi$
$192$	0	0
$193$	−20.6518	−1.48655	−0.743274	−	0.668987i	$-0.766728\pi$
$193$	−20.6518	−1.48655	−0.743274	+	0.668987i	$0.766728\pi$
$194$	−7.40850	−0.531899
$195$	0	0
$196$	−19.9546	−1.42533
$197$	4.38617	0.312502	0.156251	−	0.987717i	$-0.450059\pi$
$197$	4.38617	0.312502	0.156251	+	0.987717i	$0.450059\pi$
$198$	0	0
$199$	−13.3916	−0.949304	−0.474652	−	0.880174i	$-0.657426\pi$
$199$	−13.3916	−0.949304	−0.474652	+	0.880174i	$0.657426\pi$
$200$	−1.85576	−0.131222
$201$	0	0
$202$	5.38058	0.378576
$203$	3.21536	0.225674
$204$	0	0
$205$	6.36016	0.444213
$206$	−0.759591	−0.0529232
$207$	0	0
$208$	−2.59482	−0.179918
$209$	8.38458	0.579973
$210$	0	0
$211$	−6.06986	−0.417866	−0.208933	−	0.977930i	$-0.566999\pi$
$211$	−6.06986	−0.417866	−0.208933	+	0.977930i	$0.566999\pi$
$212$	−0.132141	−0.00907547
$213$	0	0
$214$	−2.68691	−0.183673
$215$	−7.59324	−0.517855
$216$	0	0
$217$	−43.7167	−2.96768
$218$	0.713802	0.0483448
$219$	0	0
$220$	7.87560	0.530973
$221$	−1.57196	−0.105741
$222$	0	0
$223$	−22.9881	−1.53940	−0.769700	−	0.638406i	$-0.779594\pi$
$223$	−22.9881	−1.53940	−0.769700	+	0.638406i	$0.779594\pi$
$224$	−21.3995	−1.42982
$225$	0	0
$226$	2.20603	0.146743
$227$	−18.8346	−1.25010	−0.625050	−	0.780585i	$-0.714921\pi$
$227$	−18.8346	−1.25010	−0.625050	+	0.780585i	$0.714921\pi$
$228$	0	0
$229$	−5.40926	−0.357454	−0.178727	−	0.983899i	$-0.557198\pi$
$229$	−5.40926	−0.357454	−0.178727	+	0.983899i	$0.557198\pi$
$230$	2.59786	0.171298
$231$	0	0
$232$	1.39239	0.0914150
$233$	−7.45934	−0.488677	−0.244339	−	0.969690i	$-0.578571\pi$
$233$	−7.45934	−0.488677	−0.244339	+	0.969690i	$0.578571\pi$
$234$	0	0
$235$	8.27021	0.539489
$236$	−20.0884	−1.30765
$237$	0	0
$238$	−3.32845	−0.215752
$239$	7.68073	0.496825	0.248412	−	0.968654i	$-0.420091\pi$
$239$	7.68073	0.496825	0.248412	+	0.968654i	$0.420091\pi$
$240$	0	0
$241$	29.2801	1.88610	0.943048	−	0.332657i	$-0.107945\pi$
$241$	29.2801	1.88610	0.943048	+	0.332657i	$0.107945\pi$
$242$	4.50514	0.289601
$243$	0	0
$244$	21.0003	1.34440
$245$	−11.3645	−0.726053
$246$	0	0
$247$	−1.86934	−0.118944
$248$	−18.9312	−1.20214
$249$	0	0
$250$	−0.494096	−0.0312494
$251$	−26.6402	−1.68152	−0.840758	−	0.541412i	$-0.817890\pi$
$251$	−26.6402	−1.68152	−0.840758	+	0.541412i	$0.817890\pi$
$252$	0	0
$253$	−23.5828	−1.48264
$254$	−1.93733	−0.121559
$255$	0	0
$256$	−0.154621	−0.00966380
$257$	−0.712283	−0.0444310	−0.0222155	−	0.999753i	$-0.507072\pi$
$257$	−0.712283	−0.0444310	−0.0222155	+	0.999753i	$0.507072\pi$
$258$	0	0
$259$	−11.9244	−0.740946
$260$	−1.75587	−0.108894
$261$	0	0
$262$	0.382641	0.0236397
$263$	−13.2030	−0.814134	−0.407067	−	0.913398i	$-0.633448\pi$
$263$	−13.2030	−0.814134	−0.407067	+	0.913398i	$0.633448\pi$
$264$	0	0
$265$	−0.0752567	−0.00462298
$266$	−3.95813	−0.242689
$267$	0	0
$268$	22.6463	1.38334
$269$	−18.3263	−1.11737	−0.558687	−	0.829379i	$-0.688695\pi$
$269$	−18.3263	−1.11737	−0.558687	+	0.829379i	$0.688695\pi$
$270$	0	0
$271$	−28.9102	−1.75617	−0.878084	−	0.478507i	$-0.841178\pi$
$271$	−28.9102	−1.75617	−0.878084	+	0.478507i	$0.841178\pi$
$272$	4.07894	0.247322
$273$	0	0
$274$	2.32625	0.140534
$275$	4.48530	0.270474
$276$	0	0
$277$	26.1305	1.57003	0.785014	−	0.619478i	$-0.212656\pi$
$277$	26.1305	1.57003	0.785014	+	0.619478i	$0.212656\pi$
$278$	−3.44232	−0.206457
$279$	0	0
$280$	−7.95265	−0.475261
$281$	27.6778	1.65112	0.825559	−	0.564315i	$-0.190860\pi$
$281$	27.6778	1.65112	0.825559	+	0.564315i	$0.190860\pi$
$282$	0	0
$283$	26.1441	1.55411	0.777053	−	0.629436i	$-0.216714\pi$
$283$	26.1441	1.55411	0.777053	+	0.629436i	$0.216714\pi$
$284$	−25.3664	−1.50522
$285$	0	0
$286$	−2.21617	−0.131045
$287$	27.2557	1.60885
$288$	0	0
$289$	−14.5289	−0.854644
$290$	0.370724	0.0217697
$291$	0	0
$292$	−16.4576	−0.963111
$293$	4.01651	0.234647	0.117323	−	0.993094i	$-0.462569\pi$
$293$	4.01651	0.234647	0.117323	+	0.993094i	$0.462569\pi$
$294$	0	0
$295$	−11.4407	−0.666105
$296$	−5.16379	−0.300139
$297$	0	0
$298$	−4.06999	−0.235768
$299$	5.25780	0.304066
$300$	0	0
$301$	−32.5400	−1.87557
$302$	−4.33415	−0.249402
$303$	0	0
$304$	4.85060	0.278201
$305$	11.9600	0.684830
$306$	0	0
$307$	−7.51331	−0.428807	−0.214404	−	0.976745i	$-0.568781\pi$
$307$	−7.51331	−0.428807	−0.214404	+	0.976745i	$0.568781\pi$
$308$	33.7500	1.92308
$309$	0	0
$310$	−5.04044	−0.286278
$311$	21.0401	1.19307	0.596537	−	0.802586i	$-0.296543\pi$
$311$	21.0401	1.19307	0.596537	+	0.802586i	$0.296543\pi$
$312$	0	0
$313$	−6.50014	−0.367410	−0.183705	−	0.982981i	$-0.558809\pi$
$313$	−6.50014	−0.367410	−0.183705	+	0.982981i	$0.558809\pi$
$314$	−6.61779	−0.373463
$315$	0	0
$316$	10.7094	0.602449
$317$	−11.4558	−0.643422	−0.321711	−	0.946838i	$-0.604258\pi$
$317$	−11.4558	−0.643422	−0.321711	+	0.946838i	$0.604258\pi$
$318$	0	0
$319$	−3.36536	−0.188424
$320$	2.72231	0.152182
$321$	0	0
$322$	11.1328	0.620408
$323$	2.93853	0.163504
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−0.119791	−0.00663460
$327$	0	0
$328$	11.8029	0.651707
$329$	35.4410	1.95393
$330$	0	0
$331$	−24.3594	−1.33891	−0.669456	−	0.742852i	$-0.733473\pi$
$331$	−24.3594	−1.33891	−0.669456	+	0.742852i	$0.733473\pi$
$332$	−1.07490	−0.0589931
$333$	0	0
$334$	2.51302	0.137507
$335$	12.8975	0.704665
$336$	0	0
$337$	11.6002	0.631902	0.315951	−	0.948776i	$-0.397677\pi$
$337$	11.6002	0.631902	0.315951	+	0.948776i	$0.397677\pi$
$338$	0.494096	0.0268753
$339$	0	0
$340$	2.76016	0.149690
$341$	45.7561	2.47783
$342$	0	0
$343$	−18.7037	−1.00990
$344$	−14.0912	−0.759748
$345$	0	0
$346$	−11.4487	−0.615487
$347$	−12.1820	−0.653964	−0.326982	−	0.945031i	$-0.606032\pi$
$347$	−12.1820	−0.653964	−0.326982	+	0.945031i	$0.606032\pi$
$348$	0	0
$349$	7.91102	0.423467	0.211734	−	0.977327i	$-0.432089\pi$
$349$	7.91102	0.423467	0.211734	+	0.977327i	$0.432089\pi$
$350$	−2.11739	−0.113179
$351$	0	0
$352$	22.3978	1.19381
$353$	21.2588	1.13149	0.565747	−	0.824579i	$-0.308588\pi$
$353$	21.2588	1.13149	0.565747	+	0.824579i	$0.308588\pi$
$354$	0	0
$355$	−14.4466	−0.766748
$356$	28.5302	1.51210
$357$	0	0
$358$	6.87962	0.363599
$359$	−7.59638	−0.400922	−0.200461	−	0.979702i	$-0.564244\pi$
$359$	−7.59638	−0.400922	−0.200461	+	0.979702i	$0.564244\pi$
$360$	0	0
$361$	−15.5056	−0.816082
$362$	6.76337	0.355475
$363$	0	0
$364$	−7.52458	−0.394395
$365$	−9.37293	−0.490602
$366$	0	0
$367$	21.6141	1.12825	0.564124	−	0.825690i	$-0.309214\pi$
$367$	21.6141	1.12825	0.564124	+	0.825690i	$0.309214\pi$
$368$	−13.6430	−0.711191
$369$	0	0
$370$	−1.37486	−0.0714754
$371$	−0.322504	−0.0167436
$372$	0	0
$373$	−12.0134	−0.622028	−0.311014	−	0.950405i	$-0.600669\pi$
$373$	−12.0134	−0.622028	−0.311014	+	0.950405i	$0.600669\pi$
$374$	3.48373	0.180139
$375$	0	0
$376$	15.3475	0.791488
$377$	0.750308	0.0386429
$378$	0	0
$379$	−3.44005	−0.176704	−0.0883518	−	0.996089i	$-0.528160\pi$
$379$	−3.44005	−0.176704	−0.0883518	+	0.996089i	$0.528160\pi$
$380$	3.28232	0.168380
$381$	0	0
$382$	−3.87537	−0.198281
$383$	26.8988	1.37446	0.687232	−	0.726438i	$-0.258826\pi$
$383$	26.8988	1.37446	0.687232	+	0.726438i	$0.258826\pi$
$384$	0	0
$385$	19.2212	0.979606
$386$	−10.2040	−0.519368
$387$	0	0
$388$	26.3276	1.33658
$389$	3.33515	0.169099	0.0845494	−	0.996419i	$-0.473055\pi$
$389$	3.33515	0.169099	0.0845494	+	0.996419i	$0.473055\pi$
$390$	0	0
$391$	−8.26504	−0.417981
$392$	−21.0898	−1.06520
$393$	0	0
$394$	2.16719	0.109181
$395$	6.09918	0.306883
$396$	0	0
$397$	0.768687	0.0385793	0.0192897	−	0.999814i	$-0.493860\pi$
$397$	0.768687	0.0385793	0.0192897	+	0.999814i	$0.493860\pi$
$398$	−6.61672	−0.331666
$399$	0	0
$400$	2.59482	0.129741
$401$	29.6031	1.47831	0.739153	−	0.673537i	$-0.235226\pi$
$401$	29.6031	1.47831	0.739153	+	0.673537i	$0.235226\pi$
$402$	0	0
$403$	−10.2013	−0.508165
$404$	−19.1210	−0.951303
$405$	0	0
$406$	1.58870	0.0788457
$407$	12.4807	0.618644
$408$	0	0
$409$	−16.2011	−0.801092	−0.400546	−	0.916277i	$-0.631180\pi$
$409$	−16.2011	−0.801092	−0.400546	+	0.916277i	$0.631180\pi$
$410$	3.14253	0.155198
$411$	0	0
$412$	2.69936	0.132988
$413$	−49.0279	−2.41251
$414$	0	0
$415$	−0.612178	−0.0300506
$416$	−4.99361	−0.244832
$417$	0	0
$418$	4.14278	0.202630
$419$	11.4127	0.557546	0.278773	−	0.960357i	$-0.410072\pi$
$419$	11.4127	0.557546	0.278773	+	0.960357i	$0.410072\pi$
$420$	0	0
$421$	10.3983	0.506784	0.253392	−	0.967364i	$-0.418454\pi$
$421$	10.3983	0.506784	0.253392	+	0.967364i	$0.418454\pi$
$422$	−2.99909	−0.145994
$423$	0	0
$424$	−0.139658	−0.00678241
$425$	1.57196	0.0762512
$426$	0	0
$427$	51.2534	2.48033
$428$	9.54848	0.461543
$429$	0	0
$430$	−3.75179	−0.180927
$431$	−0.879972	−0.0423868	−0.0211934	−	0.999775i	$-0.506747\pi$
$431$	−0.879972	−0.0423868	−0.0211934	+	0.999775i	$0.506747\pi$
$432$	0	0
$433$	31.4562	1.51169	0.755845	−	0.654750i	$-0.227226\pi$
$433$	31.4562	1.51169	0.755845	+	0.654750i	$0.227226\pi$
$434$	−21.6002	−1.03684
$435$	0	0
$436$	−2.53664	−0.121483
$437$	−9.82864	−0.470167
$438$	0	0
$439$	−5.76414	−0.275108	−0.137554	−	0.990494i	$-0.543924\pi$
$439$	−5.76414	−0.275108	−0.137554	+	0.990494i	$0.543924\pi$
$440$	8.32364	0.396814
$441$	0	0
$442$	−0.776699	−0.0369438
$443$	4.12846	0.196149	0.0980746	−	0.995179i	$-0.468732\pi$
$443$	4.12846	0.196149	0.0980746	+	0.995179i	$0.468732\pi$
$444$	0	0
$445$	16.2485	0.770253
$446$	−11.3583	−0.537833
$447$	0	0
$448$	11.6661	0.551174
$449$	−30.3402	−1.43184	−0.715921	−	0.698181i	$-0.753993\pi$
$449$	−30.3402	−1.43184	−0.715921	+	0.698181i	$0.753993\pi$
$450$	0	0
$451$	−28.5272	−1.34329
$452$	−7.83957	−0.368742
$453$	0	0
$454$	−9.30612	−0.436758
$455$	−4.28539	−0.200902
$456$	0	0
$457$	−8.46282	−0.395874	−0.197937	−	0.980215i	$-0.563424\pi$
$457$	−8.46282	−0.395874	−0.197937	+	0.980215i	$0.563424\pi$
$458$	−2.67269	−0.124887
$459$	0	0
$460$	−9.23201	−0.430444
$461$	17.6308	0.821149	0.410575	−	0.911827i	$-0.365328\pi$
$461$	17.6308	0.821149	0.410575	+	0.911827i	$0.365328\pi$
$462$	0	0
$463$	−1.25439	−0.0582962	−0.0291481	−	0.999575i	$-0.509279\pi$
$463$	−1.25439	−0.0582962	−0.0291481	+	0.999575i	$0.509279\pi$
$464$	−1.94691	−0.0903831
$465$	0	0
$466$	−3.68563	−0.170733
$467$	11.3158	0.523634	0.261817	−	0.965117i	$-0.415678\pi$
$467$	11.3158	0.523634	0.261817	+	0.965117i	$0.415678\pi$
$468$	0	0
$469$	55.2707	2.55216
$470$	4.08628	0.188486
$471$	0	0
$472$	−21.2312	−0.977247
$473$	34.0580	1.56599
$474$	0	0
$475$	1.86934	0.0857714
$476$	11.8283	0.542151
$477$	0	0
$478$	3.79502	0.173580
$479$	−33.3836	−1.52533	−0.762667	−	0.646791i	$-0.776111\pi$
$479$	−33.3836	−1.52533	−0.762667	+	0.646791i	$0.776111\pi$
$480$	0	0
$481$	−2.78257	−0.126874
$482$	14.4672	0.658961
$483$	0	0
$484$	−16.0099	−0.727723
$485$	14.9940	0.680844
$486$	0	0
$487$	−42.2806	−1.91592	−0.957959	−	0.286904i	$-0.907374\pi$
$487$	−42.2806	−1.91592	−0.957959	+	0.286904i	$0.907374\pi$
$488$	22.1950	1.00472
$489$	0	0
$490$	−5.61517	−0.253667
$491$	11.4657	0.517441	0.258720	−	0.965952i	$-0.416699\pi$
$491$	11.4657	0.517441	0.258720	+	0.965952i	$0.416699\pi$
$492$	0	0
$493$	−1.17945	−0.0531200
$494$	−0.923636	−0.0415563
$495$	0	0
$496$	26.4706	1.18857
$497$	−61.9094	−2.77702
$498$	0	0
$499$	6.70853	0.300315	0.150157	−	0.988662i	$-0.452022\pi$
$499$	6.70853	0.300315	0.150157	+	0.988662i	$0.452022\pi$
$500$	1.75587	0.0785249
$501$	0	0
$502$	−13.1628	−0.587485
$503$	13.1409	0.585925	0.292963	−	0.956124i	$-0.405359\pi$
$503$	13.1409	0.585925	0.292963	+	0.956124i	$0.405359\pi$
$504$	0	0
$505$	−10.8897	−0.484587
$506$	−11.6522	−0.518002
$507$	0	0
$508$	6.88471	0.305459
$509$	−31.1201	−1.37937	−0.689687	−	0.724107i	$-0.742252\pi$
$509$	−31.1201	−1.37937	−0.689687	+	0.724107i	$0.742252\pi$
$510$	0	0
$511$	−40.1666	−1.77687
$512$	22.5882	0.998267
$513$	0	0
$514$	−0.351936	−0.0155232
$515$	1.53733	0.0677431
$516$	0	0
$517$	−37.0944	−1.63141
$518$	−5.89179	−0.258871
$519$	0	0
$520$	−1.85576	−0.0813804
$521$	10.6889	0.468287	0.234144	−	0.972202i	$-0.424771\pi$
$521$	10.6889	0.468287	0.234144	+	0.972202i	$0.424771\pi$
$522$	0	0
$523$	−22.8750	−1.00025	−0.500127	−	0.865952i	$-0.666713\pi$
$523$	−22.8750	−1.00025	−0.500127	+	0.865952i	$0.666713\pi$
$524$	−1.35979	−0.0594029
$525$	0	0
$526$	−6.52357	−0.284441
$527$	16.0361	0.698544
$528$	0	0
$529$	4.64444	0.201932
$530$	−0.0371840	−0.00161517
$531$	0	0
$532$	14.0660	0.609840
$533$	6.36016	0.275489
$534$	0	0
$535$	5.43803	0.235107
$536$	23.9346	1.03382
$537$	0	0
$538$	−9.05495	−0.390387
$539$	50.9733	2.19558
$540$	0	0
$541$	−13.5880	−0.584192	−0.292096	−	0.956389i	$-0.594353\pi$
$541$	−13.5880	−0.584192	−0.292096	+	0.956389i	$0.594353\pi$
$542$	−14.2844	−0.613567
$543$	0	0
$544$	7.84975	0.336555
$545$	−1.44466	−0.0618825
$546$	0	0
$547$	23.9636	1.02461	0.512305	−	0.858804i	$-0.328792\pi$
$547$	23.9636	1.02461	0.512305	+	0.858804i	$0.328792\pi$
$548$	−8.26680	−0.353140
$549$	0	0
$550$	2.21617	0.0944978
$551$	−1.40259	−0.0597521
$552$	0	0
$553$	26.1374	1.11147
$554$	12.9110	0.548534
$555$	0	0
$556$	12.2330	0.518794
$557$	0.501996	0.0212702	0.0106351	−	0.999943i	$-0.496615\pi$
$557$	0.501996	0.0212702	0.0106351	+	0.999943i	$0.496615\pi$
$558$	0	0
$559$	−7.59324	−0.321160
$560$	11.1198	0.469897
$561$	0	0
$562$	13.6755	0.576866
$563$	16.9153	0.712893	0.356447	−	0.934316i	$-0.383988\pi$
$563$	16.9153	0.712893	0.356447	+	0.934316i	$0.383988\pi$
$564$	0	0
$565$	−4.46478	−0.187835
$566$	12.9177	0.542971
$567$	0	0
$568$	−26.8095	−1.12490
$569$	−24.1253	−1.01139	−0.505693	−	0.862714i	$-0.668763\pi$
$569$	−24.1253	−1.01139	−0.505693	+	0.862714i	$0.668763\pi$
$570$	0	0
$571$	−14.6557	−0.613323	−0.306661	−	0.951819i	$-0.599212\pi$
$571$	−14.6557	−0.613323	−0.306661	+	0.951819i	$0.599212\pi$
$572$	7.87560	0.329296
$573$	0	0
$574$	13.4669	0.562099
$575$	−5.25780	−0.219265
$576$	0	0
$577$	4.42835	0.184355	0.0921774	−	0.995743i	$-0.470617\pi$
$577$	4.42835	0.184355	0.0921774	+	0.995743i	$0.470617\pi$
$578$	−7.17869	−0.298594
$579$	0	0
$580$	−1.31744	−0.0547039
$581$	−2.62342	−0.108838
$582$	0	0
$583$	0.337549	0.0139798
$584$	−17.3939	−0.719765
$585$	0	0
$586$	1.98454	0.0819806
$587$	−6.81157	−0.281143	−0.140572	−	0.990071i	$-0.544894\pi$
$587$	−6.81157	−0.281143	−0.140572	+	0.990071i	$0.544894\pi$
$588$	0	0
$589$	19.0698	0.785759
$590$	−5.65282	−0.232723
$591$	0	0
$592$	7.22026	0.296751
$593$	9.71004	0.398744	0.199372	−	0.979924i	$-0.436110\pi$
$593$	9.71004	0.398744	0.199372	+	0.979924i	$0.436110\pi$
$594$	0	0
$595$	6.73645	0.276168
$596$	14.4635	0.592450
$597$	0	0
$598$	2.59786	0.106234
$599$	22.2733	0.910064	0.455032	−	0.890475i	$-0.349628\pi$
$599$	22.2733	0.910064	0.455032	+	0.890475i	$0.349628\pi$
$600$	0	0
$601$	13.5423	0.552402	0.276201	−	0.961100i	$-0.410924\pi$
$601$	13.5423	0.552402	0.276201	+	0.961100i	$0.410924\pi$
$602$	−16.0779	−0.655285
$603$	0	0
$604$	15.4023	0.626710
$605$	−9.11794	−0.370697
$606$	0	0
$607$	−4.76397	−0.193364	−0.0966818	−	0.995315i	$-0.530823\pi$
$607$	−4.76397	−0.193364	−0.0966818	+	0.995315i	$0.530823\pi$
$608$	9.33477	0.378575
$609$	0	0
$610$	5.90941	0.239265
$611$	8.27021	0.334577
$612$	0	0
$613$	−7.58211	−0.306238	−0.153119	−	0.988208i	$-0.548932\pi$
$613$	−7.58211	−0.306238	−0.153119	+	0.988208i	$0.548932\pi$
$614$	−3.71230	−0.149816
$615$	0	0
$616$	35.6700	1.43719
$617$	14.5642	0.586331	0.293166	−	0.956062i	$-0.405291\pi$
$617$	14.5642	0.586331	0.293166	+	0.956062i	$0.405291\pi$
$618$	0	0
$619$	−42.7018	−1.71633	−0.858165	−	0.513374i	$-0.828395\pi$
$619$	−42.7018	−1.71633	−0.858165	+	0.513374i	$0.828395\pi$
$620$	17.9122	0.719373
$621$	0	0
$622$	10.3958	0.416835
$623$	69.6311	2.78971
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.21169	−0.128365
$627$	0	0
$628$	23.5176	0.938456
$629$	4.37409	0.174406
$630$	0	0
$631$	−35.7936	−1.42492	−0.712461	−	0.701712i	$-0.752419\pi$
$631$	−35.7936	−1.42492	−0.712461	+	0.701712i	$0.752419\pi$
$632$	11.3186	0.450231
$633$	0	0
$634$	−5.66026	−0.224798
$635$	3.92097	0.155599
$636$	0	0
$637$	−11.3645	−0.450279
$638$	−1.66281	−0.0658313
$639$	0	0
$640$	11.3323	0.447948
$641$	11.4684	0.452973	0.226486	−	0.974014i	$-0.427276\pi$
$641$	11.4684	0.452973	0.226486	+	0.974014i	$0.427276\pi$
$642$	0	0
$643$	−28.9956	−1.14348	−0.571738	−	0.820436i	$-0.693731\pi$
$643$	−28.9956	−1.14348	−0.571738	+	0.820436i	$0.693731\pi$
$644$	−39.5627	−1.55899
$645$	0	0
$646$	1.45192	0.0571249
$647$	−11.2588	−0.442630	−0.221315	−	0.975202i	$-0.571035\pi$
$647$	−11.2588	−0.442630	−0.221315	+	0.975202i	$0.571035\pi$
$648$	0	0
$649$	51.3151	2.01430
$650$	−0.494096	−0.0193800
$651$	0	0
$652$	0.425701	0.0166717
$653$	−34.4838	−1.34946	−0.674729	−	0.738066i	$-0.735739\pi$
$653$	−34.4838	−1.34946	−0.674729	+	0.738066i	$0.735739\pi$
$654$	0	0
$655$	−0.774428	−0.0302594
$656$	−16.5034	−0.644351
$657$	0	0
$658$	17.5113	0.682660
$659$	30.4667	1.18682	0.593408	−	0.804902i	$-0.297782\pi$
$659$	30.4667	1.18682	0.593408	+	0.804902i	$0.297782\pi$
$660$	0	0
$661$	21.4816	0.835538	0.417769	−	0.908553i	$-0.362812\pi$
$661$	21.4816	0.835538	0.417769	+	0.908553i	$0.362812\pi$
$662$	−12.0359	−0.467787
$663$	0	0
$664$	−1.13606	−0.0440875
$665$	8.01086	0.310648
$666$	0	0
$667$	3.94497	0.152750
$668$	−8.93053	−0.345533
$669$	0	0
$670$	6.37259	0.246195
$671$	−53.6444	−2.07092
$672$	0	0
$673$	9.10550	0.350991	0.175496	−	0.984480i	$-0.443847\pi$
$673$	9.10550	0.350991	0.175496	+	0.984480i	$0.443847\pi$
$674$	5.73160	0.220773
$675$	0	0
$676$	−1.75587	−0.0675334
$677$	3.44575	0.132431	0.0662155	−	0.997805i	$-0.478908\pi$
$677$	3.44575	0.132431	0.0662155	+	0.997805i	$0.478908\pi$
$678$	0	0
$679$	64.2553	2.46589
$680$	2.91718	0.111869
$681$	0	0
$682$	22.6079	0.865702
$683$	36.2676	1.38774	0.693870	−	0.720100i	$-0.255904\pi$
$683$	36.2676	1.38774	0.693870	+	0.720100i	$0.255904\pi$
$684$	0	0
$685$	−4.70810	−0.179887
$686$	−9.24141	−0.352839
$687$	0	0
$688$	19.7031	0.751172
$689$	−0.0752567	−0.00286705
$690$	0	0
$691$	−38.2326	−1.45444	−0.727218	−	0.686407i	$-0.759187\pi$
$691$	−38.2326	−1.45444	−0.727218	+	0.686407i	$0.759187\pi$
$692$	40.6853	1.54662
$693$	0	0
$694$	−6.01907	−0.228481
$695$	6.96691	0.264270
$696$	0	0
$697$	−9.99791	−0.378698
$698$	3.90880	0.147950
$699$	0	0
$700$	7.52458	0.284402
$701$	−2.09523	−0.0791359	−0.0395679	−	0.999217i	$-0.512598\pi$
$701$	−2.09523	−0.0791359	−0.0395679	+	0.999217i	$0.512598\pi$
$702$	0	0
$703$	5.20159	0.196182
$704$	−12.2104	−0.460196
$705$	0	0
$706$	10.5039	0.395320
$707$	−46.6667	−1.75508
$708$	0	0
$709$	−19.8774	−0.746511	−0.373255	−	0.927729i	$-0.621759\pi$
$709$	−19.8774	−0.746511	−0.373255	+	0.927729i	$0.621759\pi$
$710$	−7.13803	−0.267885
$711$	0	0
$712$	30.1533	1.13004
$713$	−53.6366	−2.00871
$714$	0	0
$715$	4.48530	0.167741
$716$	−24.4481	−0.913669
$717$	0	0
$718$	−3.75334	−0.140073
$719$	−15.9997	−0.596687	−0.298344	−	0.954459i	$-0.596434\pi$
$719$	−15.9997	−0.596687	−0.298344	+	0.954459i	$0.596434\pi$
$720$	0	0
$721$	6.58807	0.245353
$722$	−7.66123	−0.285121
$723$	0	0
$724$	−24.0350	−0.893254
$725$	−0.750308	−0.0278658
$726$	0	0
$727$	−30.5724	−1.13387	−0.566934	−	0.823763i	$-0.691871\pi$
$727$	−30.5724	−1.13387	−0.566934	+	0.823763i	$0.691871\pi$
$728$	−7.95265	−0.294745
$729$	0	0
$730$	−4.63113	−0.171406
$731$	11.9363	0.441479
$732$	0	0
$733$	27.6616	1.02170	0.510852	−	0.859669i	$-0.329330\pi$
$733$	27.6616	1.02170	0.510852	+	0.859669i	$0.329330\pi$
$734$	10.6795	0.394186
$735$	0	0
$736$	−26.2554	−0.967786
$737$	−57.8491	−2.13090
$738$	0	0
$739$	−2.03805	−0.0749710	−0.0374855	−	0.999297i	$-0.511935\pi$
$739$	−2.03805	−0.0749710	−0.0374855	+	0.999297i	$0.511935\pi$
$740$	4.88583	0.179607
$741$	0	0
$742$	−0.159348	−0.00584984
$743$	8.46498	0.310550	0.155275	−	0.987871i	$-0.450374\pi$
$743$	8.46498	0.310550	0.155275	+	0.987871i	$0.450374\pi$
$744$	0	0
$745$	8.23725	0.301790
$746$	−5.93575	−0.217323
$747$	0	0
$748$	−12.3801	−0.452662
$749$	23.3041	0.851512
$750$	0	0
$751$	−18.6979	−0.682298	−0.341149	−	0.940009i	$-0.610816\pi$
$751$	−18.6979	−0.682298	−0.341149	+	0.940009i	$0.610816\pi$
$752$	−21.4597	−0.782553
$753$	0	0
$754$	0.370724	0.0135010
$755$	8.77188	0.319241
$756$	0	0
$757$	−4.02963	−0.146459	−0.0732296	−	0.997315i	$-0.523331\pi$
$757$	−4.02963	−0.146459	−0.0732296	+	0.997315i	$0.523331\pi$
$758$	−1.69972	−0.0617365
$759$	0	0
$760$	3.46905	0.125836
$761$	16.4332	0.595704	0.297852	−	0.954612i	$-0.403730\pi$
$761$	16.4332	0.595704	0.297852	+	0.954612i	$0.403730\pi$
$762$	0	0
$763$	−6.19094	−0.224127
$764$	13.7719	0.498250
$765$	0	0
$766$	13.2906	0.480208
$767$	−11.4407	−0.413101
$768$	0	0
$769$	−27.0670	−0.976062	−0.488031	−	0.872826i	$-0.662285\pi$
$769$	−27.0670	−0.976062	−0.488031	+	0.872826i	$0.662285\pi$
$770$	9.49714	0.342253
$771$	0	0
$772$	36.2618	1.30509
$773$	−50.3630	−1.81143	−0.905716	−	0.423886i	$-0.860666\pi$
$773$	−50.3630	−1.81143	−0.905716	+	0.423886i	$0.860666\pi$
$774$	0	0
$775$	10.2013	0.366443
$776$	27.8253	0.998872
$777$	0	0
$778$	1.64788	0.0590795
$779$	−11.8893	−0.425979
$780$	0	0
$781$	64.7976	2.31864
$782$	−4.08372	−0.146034
$783$	0	0
$784$	29.4889	1.05317
$785$	13.3937	0.478043
$786$	0	0
$787$	−35.9512	−1.28152	−0.640760	−	0.767741i	$-0.721381\pi$
$787$	−35.9512	−1.28152	−0.640760	+	0.767741i	$0.721381\pi$
$788$	−7.70154	−0.274356
$789$	0	0
$790$	3.01358	0.107218
$791$	−19.1333	−0.680302
$792$	0	0
$793$	11.9600	0.424714
$794$	0.379805	0.0134788
$795$	0	0
$796$	23.5139	0.833427
$797$	−32.2790	−1.14338	−0.571690	−	0.820470i	$-0.693712\pi$
$797$	−32.2790	−1.14338	−0.571690	+	0.820470i	$0.693712\pi$
$798$	0	0
$799$	−13.0004	−0.459922
$800$	4.99361	0.176551
$801$	0	0
$802$	14.6268	0.516489
$803$	42.0404	1.48357
$804$	0	0
$805$	−22.5317	−0.794138
$806$	−5.04044	−0.177542
$807$	0	0
$808$	−20.2087	−0.710941
$809$	42.4707	1.49319	0.746594	−	0.665279i	$-0.231688\pi$
$809$	42.4707	1.49319	0.746594	+	0.665279i	$0.231688\pi$
$810$	0	0
$811$	−14.1284	−0.496116	−0.248058	−	0.968745i	$-0.579792\pi$
$811$	−14.1284	−0.496116	−0.248058	+	0.968745i	$0.579792\pi$
$812$	−5.64575	−0.198127
$813$	0	0
$814$	6.16665	0.216141
$815$	0.242444	0.00849246
$816$	0	0
$817$	14.1944	0.496599
$818$	−8.00489	−0.279885
$819$	0	0
$820$	−11.1676	−0.389990
$821$	−44.1796	−1.54188	−0.770939	−	0.636909i	$-0.780213\pi$
$821$	−44.1796	−1.54188	−0.770939	+	0.636909i	$0.780213\pi$
$822$	0	0
$823$	−26.3196	−0.917443	−0.458721	−	0.888580i	$-0.651692\pi$
$823$	−26.3196	−0.917443	−0.458721	+	0.888580i	$0.651692\pi$
$824$	2.85292	0.0993863
$825$	0	0
$826$	−24.2245	−0.842878
$827$	−52.3359	−1.81990	−0.909949	−	0.414720i	$-0.863879\pi$
$827$	−52.3359	−1.81990	−0.909949	+	0.414720i	$0.863879\pi$
$828$	0	0
$829$	−44.0239	−1.52901	−0.764506	−	0.644617i	$-0.777017\pi$
$829$	−44.0239	−1.52901	−0.764506	+	0.644617i	$0.777017\pi$
$830$	−0.302475	−0.0104991
$831$	0	0
$832$	2.72231	0.0943791
$833$	17.8646	0.618971
$834$	0	0
$835$	−5.08610	−0.176012
$836$	−14.7222	−0.509179
$837$	0	0
$838$	5.63896	0.194795
$839$	7.99106	0.275882	0.137941	−	0.990440i	$-0.455952\pi$
$839$	7.99106	0.275882	0.137941	+	0.990440i	$0.455952\pi$
$840$	0	0
$841$	−28.4370	−0.980587
$842$	5.13778	0.177059
$843$	0	0
$844$	10.6579	0.366859
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−39.0739	−1.34259
$848$	0.195277	0.00670585
$849$	0	0
$850$	0.776699	0.0266405
$851$	−14.6302	−0.501517
$852$	0	0
$853$	−16.7270	−0.572723	−0.286361	−	0.958122i	$-0.592446\pi$
$853$	−16.7270	−0.572723	−0.286361	+	0.958122i	$0.592446\pi$
$854$	25.3241	0.866573
$855$	0	0
$856$	10.0917	0.344927
$857$	23.5174	0.803339	0.401669	−	0.915785i	$-0.368430\pi$
$857$	23.5174	0.803339	0.401669	+	0.915785i	$0.368430\pi$
$858$	0	0
$859$	−19.8510	−0.677307	−0.338653	−	0.940911i	$-0.609971\pi$
$859$	−19.8510	−0.677307	−0.338653	+	0.940911i	$0.609971\pi$
$860$	13.3327	0.454643
$861$	0	0
$862$	−0.434791	−0.0148090
$863$	11.3010	0.384690	0.192345	−	0.981327i	$-0.438391\pi$
$863$	11.3010	0.384690	0.192345	+	0.981327i	$0.438391\pi$
$864$	0	0
$865$	23.1711	0.787839
$866$	15.5424	0.528152
$867$	0	0
$868$	76.7608	2.60543
$869$	−27.3567	−0.928012
$870$	0	0
$871$	12.8975	0.437015
$872$	−2.68095	−0.0907883
$873$	0	0
$874$	−4.85629	−0.164266
$875$	4.28539	0.144872
$876$	0	0
$877$	47.2377	1.59510	0.797552	−	0.603250i	$-0.206128\pi$
$877$	47.2377	1.59510	0.797552	+	0.603250i	$0.206128\pi$
$878$	−2.84804	−0.0961167
$879$	0	0
$880$	−11.6385	−0.392335
$881$	5.64812	0.190290	0.0951450	−	0.995463i	$-0.469669\pi$
$881$	5.64812	0.190290	0.0951450	+	0.995463i	$0.469669\pi$
$882$	0	0
$883$	16.3032	0.548645	0.274323	−	0.961638i	$-0.411546\pi$
$883$	16.3032	0.548645	0.274323	+	0.961638i	$0.411546\pi$
$884$	2.76016	0.0928341
$885$	0	0
$886$	2.03986	0.0685303
$887$	−37.6918	−1.26557	−0.632783	−	0.774329i	$-0.718087\pi$
$887$	−37.6918	−1.26557	−0.632783	+	0.774329i	$0.718087\pi$
$888$	0	0
$889$	16.8029	0.563550
$890$	8.02832	0.269110
$891$	0	0
$892$	40.3642	1.35149
$893$	−15.4599	−0.517345
$894$	0	0
$895$	−13.9237	−0.465417
$896$	48.5633	1.62238
$897$	0	0
$898$	−14.9910	−0.500255
$899$	−7.65416	−0.255280
$900$	0	0
$901$	0.118300	0.00394116
$902$	−14.0952	−0.469318
$903$	0	0
$904$	−8.28556	−0.275574
$905$	−13.6884	−0.455017
$906$	0	0
$907$	23.8528	0.792019	0.396009	−	0.918246i	$-0.370395\pi$
$907$	23.8528	0.792019	0.396009	+	0.918246i	$0.370395\pi$
$908$	33.0712	1.09751
$909$	0	0
$910$	−2.11739	−0.0701908
$911$	46.7610	1.54926	0.774630	−	0.632414i	$-0.217936\pi$
$911$	46.7610	1.54926	0.774630	+	0.632414i	$0.217936\pi$
$912$	0	0
$913$	2.74580	0.0908729
$914$	−4.18145	−0.138310
$915$	0	0
$916$	9.49795	0.313821
$917$	−3.31872	−0.109594
$918$	0	0
$919$	−12.3909	−0.408738	−0.204369	−	0.978894i	$-0.565514\pi$
$919$	−12.3909	−0.408738	−0.204369	+	0.978894i	$0.565514\pi$
$920$	−9.75721	−0.321686
$921$	0	0
$922$	8.71132	0.286892
$923$	−14.4466	−0.475517
$924$	0	0
$925$	2.78257	0.0914904
$926$	−0.619787	−0.0203675
$927$	0	0
$928$	−3.74675	−0.122993
$929$	10.2213	0.335349	0.167674	−	0.985842i	$-0.446374\pi$
$929$	10.2213	0.335349	0.167674	+	0.985842i	$0.446374\pi$
$930$	0	0
$931$	21.2442	0.696251
$932$	13.0976	0.429027
$933$	0	0
$934$	5.59111	0.182947
$935$	−7.05071	−0.230583
$936$	0	0
$937$	23.8520	0.779212	0.389606	−	0.920982i	$-0.372611\pi$
$937$	23.8520	0.779212	0.389606	+	0.920982i	$0.372611\pi$
$938$	27.3090	0.891671
$939$	0	0
$940$	−14.5214	−0.473636
$941$	−25.3193	−0.825385	−0.412692	−	0.910870i	$-0.635412\pi$
$941$	−25.3193	−0.825385	−0.412692	+	0.910870i	$0.635412\pi$
$942$	0	0
$943$	33.4404	1.08897
$944$	29.6866	0.966216
$945$	0	0
$946$	16.8279	0.547122
$947$	−23.1019	−0.750710	−0.375355	−	0.926881i	$-0.622479\pi$
$947$	−23.1019	−0.750710	−0.375355	+	0.926881i	$0.622479\pi$
$948$	0	0
$949$	−9.37293	−0.304258
$950$	0.923636	0.0299667
$951$	0	0
$952$	12.5012	0.405167
$953$	46.4511	1.50470	0.752350	−	0.658764i	$-0.228920\pi$
$953$	46.4511	1.50470	0.752350	+	0.658764i	$0.228920\pi$
$954$	0	0
$955$	7.84336	0.253805
$956$	−13.4864	−0.436180
$957$	0	0
$958$	−16.4947	−0.532919
$959$	−20.1760	−0.651518
$960$	0	0
$961$	73.0675	2.35702
$962$	−1.37486	−0.0443272
$963$	0	0
$964$	−51.4120	−1.65587
$965$	20.6518	0.664804
$966$	0	0
$967$	−7.74255	−0.248984	−0.124492	−	0.992221i	$-0.539730\pi$
$967$	−7.74255	−0.248984	−0.124492	+	0.992221i	$0.539730\pi$
$968$	−16.9207	−0.543852
$969$	0	0
$970$	7.40850	0.237872
$971$	−2.78737	−0.0894511	−0.0447256	−	0.998999i	$-0.514241\pi$
$971$	−2.78737	−0.0894511	−0.0447256	+	0.998999i	$0.514241\pi$
$972$	0	0
$973$	29.8559	0.957136
$974$	−20.8907	−0.669381
$975$	0	0
$976$	−31.0341	−0.993378
$977$	−11.4785	−0.367231	−0.183616	−	0.982998i	$-0.558780\pi$
$977$	−11.4785	−0.367231	−0.183616	+	0.982998i	$0.558780\pi$
$978$	0	0
$979$	−72.8794	−2.32924
$980$	19.9546	0.637427
$981$	0	0
$982$	5.66517	0.180783
$983$	−28.9499	−0.923359	−0.461679	−	0.887047i	$-0.652753\pi$
$983$	−28.9499	−0.923359	−0.461679	+	0.887047i	$0.652753\pi$
$984$	0	0
$985$	−4.38617	−0.139755
$986$	−0.582764	−0.0185590
$987$	0	0
$988$	3.28232	0.104425
$989$	−39.9237	−1.26950
$990$	0	0
$991$	−5.40688	−0.171755	−0.0858776	−	0.996306i	$-0.527369\pi$
$991$	−5.40688	−0.171755	−0.0858776	+	0.996306i	$0.527369\pi$
$992$	50.9415	1.61739
$993$	0	0
$994$	−30.5892	−0.970230
$995$	13.3916	0.424542
$996$	0	0
$997$	−36.9243	−1.16941	−0.584703	−	0.811248i	$-0.698789\pi$
$997$	−36.9243	−1.16941	−0.584703	+	0.811248i	$0.698789\pi$
$998$	3.31466	0.104924
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.5	✓	16	9.4	even	3
585.2.i.e.391.5	yes	16	9.7	even	3
1755.2.i.f.586.4		16	9.5	odd	6
1755.2.i.f.1171.4		16	9.2	odd	6
5265.2.a.ba.1.5		8	3.2	odd	2
5265.2.a.bf.1.4		8	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)

\(f(q)\)	\(=\)	\(q+0.494096 q^{2} -1.75587 q^{4} -1.00000 q^{5} -4.28539 q^{7} -1.85576 q^{8} +O(q^{10})\)	\(q+0.494096 q^{2} -1.75587 q^{4} -1.00000 q^{5} -4.28539 q^{7} -1.85576 q^{8} -0.494096 q^{10} +4.48530 q^{11} -1.00000 q^{13} -2.11739 q^{14} +2.59482 q^{16} +1.57196 q^{17} +1.86934 q^{19} +1.75587 q^{20} +2.21617 q^{22} -5.25780 q^{23} +1.00000 q^{25} -0.494096 q^{26} +7.52458 q^{28} -0.750308 q^{29} +10.2013 q^{31} +4.99361 q^{32} +0.776699 q^{34} +4.28539 q^{35} +2.78257 q^{37} +0.923636 q^{38} +1.85576 q^{40} -6.36016 q^{41} +7.59324 q^{43} -7.87560 q^{44} -2.59786 q^{46} -8.27021 q^{47} +11.3645 q^{49} +0.494096 q^{50} +1.75587 q^{52} +0.0752567 q^{53} -4.48530 q^{55} +7.95265 q^{56} -0.370724 q^{58} +11.4407 q^{59} -11.9600 q^{61} +5.04044 q^{62} -2.72231 q^{64} +1.00000 q^{65} -12.8975 q^{67} -2.76016 q^{68} +2.11739 q^{70} +14.4466 q^{71} +9.37293 q^{73} +1.37486 q^{74} -3.28232 q^{76} -19.2212 q^{77} -6.09918 q^{79} -2.59482 q^{80} -3.14253 q^{82} +0.612178 q^{83} -1.57196 q^{85} +3.75179 q^{86} -8.32364 q^{88} -16.2485 q^{89} +4.28539 q^{91} +9.23201 q^{92} -4.08628 q^{94} -1.86934 q^{95} -14.9940 q^{97} +5.61517 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8	\( 8 q + 3 q^{2} + 9 q^{4} - 8 q^{5} - 11 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} - 8 q^{13} + 10 q^{14} + 11 q^{16} - 2 q^{17} - 10 q^{19} - 9 q^{20} + 3 q^{22} + 6 q^{23} + 8 q^{25} - 3 q^{26} - 34 q^{28} + 14 q^{29} - 31 q^{31} + q^{32} - 7 q^{34} + 11 q^{35} + q^{37} + 9 q^{38} + 6 q^{40} - 12 q^{41} + 15 q^{43} + 16 q^{44} - 32 q^{46} - 18 q^{47} + 17 q^{49} + 3 q^{50} - 9 q^{52} + 2 q^{53} - 6 q^{55} + 16 q^{56} - 42 q^{58} + 24 q^{59} - 9 q^{61} - 20 q^{62} - 30 q^{64} + 8 q^{65} - 18 q^{67} - 14 q^{68} - 10 q^{70} + 10 q^{71} + 6 q^{73} - 37 q^{74} - 53 q^{76} - 34 q^{77} - 3 q^{79} - 11 q^{80} - 34 q^{82} - 10 q^{83} + 2 q^{85} + 60 q^{86} - 14 q^{88} - 13 q^{89} + 11 q^{91} + 5 q^{92} + 17 q^{94} + 10 q^{95} - 34 q^{97} - 30 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 + 9 * q^4 - 8 * q^5 - 11 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 - 8 * q^13 + 10 * q^14 + 11 * q^16 - 2 * q^17 - 10 * q^19 - 9 * q^20 + 3 * q^22 + 6 * q^23 + 8 * q^25 - 3 * q^26 - 34 * q^28 + 14 * q^29 - 31 * q^31 + q^32 - 7 * q^34 + 11 * q^35 + q^37 + 9 * q^38 + 6 * q^40 - 12 * q^41 + 15 * q^43 + 16 * q^44 - 32 * q^46 - 18 * q^47 + 17 * q^49 + 3 * q^50 - 9 * q^52 + 2 * q^53 - 6 * q^55 + 16 * q^56 - 42 * q^58 + 24 * q^59 - 9 * q^61 - 20 * q^62 - 30 * q^64 + 8 * q^65 - 18 * q^67 - 14 * q^68 - 10 * q^70 + 10 * q^71 + 6 * q^73 - 37 * q^74 - 53 * q^76 - 34 * q^77 - 3 * q^79 - 11 * q^80 - 34 * q^82 - 10 * q^83 + 2 * q^85 + 60 * q^86 - 14 * q^88 - 13 * q^89 + 11 * q^91 + 5 * q^92 + 17 * q^94 + 10 * q^95 - 34 * q^97 - 30 * q^98

Introduction

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