LMFDB - Embedded newform 7569.2.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.183172$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.183172 q^{2} -1.96645 q^{4} -3.26086 q^{5} -0.215332 q^{7} +0.726543 q^{8} +0.597298 q^{10} -2.92503 q^{11} -3.58448 q^{13} +0.0394428 q^{14} +3.79981 q^{16} +7.11447 q^{17} -1.38197 q^{19} +6.41230 q^{20} +0.535785 q^{22} -6.71760 q^{23} +5.63318 q^{25} +0.656577 q^{26} +0.423439 q^{28} -6.41785 q^{31} -2.14910 q^{32} -1.30317 q^{34} +0.702166 q^{35} +6.11468 q^{37} +0.253138 q^{38} -2.36915 q^{40} +6.50285 q^{41} +6.24776 q^{43} +5.75193 q^{44} +1.23048 q^{46} +9.04988 q^{47} -6.95363 q^{49} -1.03184 q^{50} +7.04870 q^{52} +4.53691 q^{53} +9.53811 q^{55} -0.156448 q^{56} +12.0737 q^{59} -1.58215 q^{61} +1.17557 q^{62} -7.20597 q^{64} +11.6885 q^{65} -7.18411 q^{67} -13.9902 q^{68} -0.128617 q^{70} -13.4403 q^{71} +13.8844 q^{73} -1.12004 q^{74} +2.71756 q^{76} +0.629853 q^{77} +3.35979 q^{79} -12.3906 q^{80} -1.19114 q^{82} -5.23502 q^{83} -23.1993 q^{85} -1.14441 q^{86} -2.12516 q^{88} +12.9834 q^{89} +0.771853 q^{91} +13.2098 q^{92} -1.65769 q^{94} +4.50639 q^{95} -3.92913 q^{97} +1.27371 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 + 12 * q^10 + 2 * q^13 - 2 * q^16 - 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 - 10 * q^31 - 36 * q^34 - 18 * q^37 + 10 * q^40 - 28 * q^43 - 26 * q^46 + 4 * q^49 + 44 * q^52 - 14 * q^55 - 54 * q^61 - 44 * q^64 + 38 * q^67 - 70 * q^70 + 16 * q^73 - 20 * q^76 - 14 * q^79 + 2 * q^82 - 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 + 6 * q^97

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 $

$2$	−0.183172	−0.129522	−0.0647611	−	0.997901i	$-0.520629\pi$
$2$	−0.183172	−0.129522	−0.0647611	+	0.997901i	$0.520629\pi$
$3$	0	0
$3$	0	0
$4$	−1.96645	−0.983224
$5$	−3.26086	−1.45830	−0.729149	−	0.684354i	$-0.760084\pi$
$5$	−3.26086	−1.45830	−0.729149	+	0.684354i	$0.760084\pi$
$6$	0	0
$7$	−0.215332	−0.0813877	−0.0406939	−	0.999172i	$-0.512957\pi$
$7$	−0.215332	−0.0813877	−0.0406939	+	0.999172i	$0.512957\pi$
$8$	0.726543	0.256872
$9$	0	0
$10$	0.597298	0.188882
$11$	−2.92503	−0.881931	−0.440965	−	0.897524i	$-0.645364\pi$
$11$	−2.92503	−0.881931	−0.440965	+	0.897524i	$0.645364\pi$
$12$	0	0
$13$	−3.58448	−0.994156	−0.497078	−	0.867706i	$-0.665594\pi$
$13$	−3.58448	−0.994156	−0.497078	+	0.867706i	$0.665594\pi$
$14$	0.0394428	0.0105415
$15$	0	0
$16$	3.79981	0.949953
$17$	7.11447	1.72551	0.862756	−	0.505620i	$-0.168736\pi$
$17$	7.11447	1.72551	0.862756	+	0.505620i	$0.168736\pi$
$18$	0	0
$19$	−1.38197	−0.317045	−0.158522	−	0.987355i	$-0.550673\pi$
$19$	−1.38197	−0.317045	−0.158522	+	0.987355i	$0.550673\pi$
$20$	6.41230	1.43383
$21$	0	0
$22$	0.535785	0.114230
$23$	−6.71760	−1.40072	−0.700358	−	0.713791i	$-0.746976\pi$
$23$	−6.71760	−1.40072	−0.700358	+	0.713791i	$0.746976\pi$
$24$	0	0
$25$	5.63318	1.12664
$26$	0.656577	0.128765
$27$	0	0
$28$	0.423439	0.0800224
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−6.41785	−1.15268	−0.576340	−	0.817210i	$-0.695519\pi$
$31$	−6.41785	−1.15268	−0.576340	+	0.817210i	$0.695519\pi$
$32$	−2.14910	−0.379912
$33$	0	0
$34$	−1.30317	−0.223492
$35$	0.702166	0.118688
$36$	0	0
$37$	6.11468	1.00525	0.502623	−	0.864506i	$-0.332368\pi$
$37$	6.11468	1.00525	0.502623	+	0.864506i	$0.332368\pi$
$38$	0.253138	0.0410643
$39$	0	0
$40$	−2.36915	−0.374596
$41$	6.50285	1.01557	0.507787	−	0.861483i	$-0.330464\pi$
$41$	6.50285	1.01557	0.507787	+	0.861483i	$0.330464\pi$
$42$	0	0
$43$	6.24776	0.952774	0.476387	−	0.879236i	$-0.341946\pi$
$43$	6.24776	0.952774	0.476387	+	0.879236i	$0.341946\pi$
$44$	5.75193	0.867136
$45$	0	0
$46$	1.23048	0.181424
$47$	9.04988	1.32006	0.660030	−	0.751239i	$-0.270543\pi$
$47$	9.04988	1.32006	0.660030	+	0.751239i	$0.270543\pi$
$48$	0	0
$49$	−6.95363	−0.993376
$50$	−1.03184	−0.145924
$51$	0	0
$52$	7.04870	0.977478
$53$	4.53691	0.623193	0.311597	−	0.950214i	$-0.399136\pi$
$53$	4.53691	0.623193	0.311597	+	0.950214i	$0.399136\pi$
$54$	0	0
$55$	9.53811	1.28612
$56$	−0.156448	−0.0209062
$57$	0	0
$58$	0	0
$59$	12.0737	1.57186	0.785930	−	0.618315i	$-0.212185\pi$
$59$	12.0737	1.57186	0.785930	+	0.618315i	$0.212185\pi$
$60$	0	0
$61$	−1.58215	−0.202574	−0.101287	−	0.994857i	$-0.532296\pi$
$61$	−1.58215	−0.202574	−0.101287	+	0.994857i	$0.532296\pi$
$62$	1.17557	0.149298
$63$	0	0
$64$	−7.20597	−0.900746
$65$	11.6885	1.44978
$66$	0	0
$67$	−7.18411	−0.877679	−0.438839	−	0.898566i	$-0.644610\pi$
$67$	−7.18411	−0.877679	−0.438839	+	0.898566i	$0.644610\pi$
$68$	−13.9902	−1.69656
$69$	0	0
$70$	−0.128617	−0.0153727
$71$	−13.4403	−1.59507	−0.797534	−	0.603273i	$-0.793863\pi$
$71$	−13.4403	−1.59507	−0.797534	+	0.603273i	$0.793863\pi$
$72$	0	0
$73$	13.8844	1.62505	0.812523	−	0.582930i	$-0.198094\pi$
$73$	13.8844	1.62505	0.812523	+	0.582930i	$0.198094\pi$
$74$	−1.12004	−0.130202
$75$	0	0
$76$	2.71756	0.311726
$77$	0.629853	0.0717784
$78$	0	0
$79$	3.35979	0.378006	0.189003	−	0.981977i	$-0.439474\pi$
$79$	3.35979	0.378006	0.189003	+	0.981977i	$0.439474\pi$
$80$	−12.3906	−1.38532
$81$	0	0
$82$	−1.19114	−0.131539
$83$	−5.23502	−0.574618	−0.287309	−	0.957838i	$-0.592761\pi$
$83$	−5.23502	−0.574618	−0.287309	+	0.957838i	$0.592761\pi$
$84$	0	0
$85$	−23.1993	−2.51631
$86$	−1.14441	−0.123405
$87$	0	0
$88$	−2.12516	−0.226543
$89$	12.9834	1.37624	0.688119	−	0.725598i	$-0.258437\pi$
$89$	12.9834	1.37624	0.688119	+	0.725598i	$0.258437\pi$
$90$	0	0
$91$	0.771853	0.0809121
$92$	13.2098	1.37722
$93$	0	0
$94$	−1.65769	−0.170977
$95$	4.50639	0.462346
$96$	0	0
$97$	−3.92913	−0.398942	−0.199471	−	0.979904i	$-0.563922\pi$
$97$	−3.92913	−0.398942	−0.199471	+	0.979904i	$0.563922\pi$
$98$	1.27371	0.128664
$99$	0	0
$100$	−11.0774	−1.10774
$101$	13.3415	1.32753	0.663765	−	0.747941i	$-0.268957\pi$
$101$	13.3415	1.32753	0.663765	+	0.747941i	$0.268957\pi$
$102$	0	0
$103$	6.08458	0.599531	0.299766	−	0.954013i	$-0.403091\pi$
$103$	6.08458	0.599531	0.299766	+	0.954013i	$0.403091\pi$
$104$	−2.60428	−0.255371
$105$	0	0
$106$	−0.831036	−0.0807174
$107$	−0.238072	−0.0230153	−0.0115076	−	0.999934i	$-0.503663\pi$
$107$	−0.238072	−0.0230153	−0.0115076	+	0.999934i	$0.503663\pi$
$108$	0	0
$109$	2.75112	0.263509	0.131755	−	0.991282i	$-0.457939\pi$
$109$	2.75112	0.263509	0.131755	+	0.991282i	$0.457939\pi$
$110$	−1.74712	−0.166581
$111$	0	0
$112$	−0.818220	−0.0773146
$113$	−3.14487	−0.295845	−0.147922	−	0.988999i	$-0.547259\pi$
$113$	−3.14487	−0.295845	−0.147922	+	0.988999i	$0.547259\pi$
$114$	0	0
$115$	21.9051	2.04266
$116$	0	0
$117$	0	0
$118$	−2.21156	−0.203591
$119$	−1.53197	−0.140436
$120$	0	0
$121$	−2.44417	−0.222198
$122$	0.289806	0.0262378
$123$	0	0
$124$	12.6204	1.13334
$125$	−2.06471	−0.184673
$126$	0	0
$127$	−16.5567	−1.46917	−0.734585	−	0.678516i	$-0.762623\pi$
$127$	−16.5567	−1.46917	−0.734585	+	0.678516i	$0.762623\pi$
$128$	5.61814	0.496578
$129$	0	0
$130$	−2.14100	−0.187778
$131$	−6.91349	−0.604035	−0.302017	−	0.953302i	$-0.597660\pi$
$131$	−6.91349	−0.604035	−0.302017	+	0.953302i	$0.597660\pi$
$132$	0	0
$133$	0.297581	0.0258036
$134$	1.31593	0.113679
$135$	0	0
$136$	5.16896	0.443235
$137$	8.62415	0.736811	0.368405	−	0.929665i	$-0.379904\pi$
$137$	8.62415	0.736811	0.368405	+	0.929665i	$0.379904\pi$
$138$	0	0
$139$	18.5083	1.56986	0.784928	−	0.619587i	$-0.212700\pi$
$139$	18.5083	1.56986	0.784928	+	0.619587i	$0.212700\pi$
$140$	−1.38077	−0.116697
$141$	0	0
$142$	2.46189	0.206597
$143$	10.4847	0.876777
$144$	0	0
$145$	0	0
$146$	−2.54323	−0.210479
$147$	0	0
$148$	−12.0242	−0.988382
$149$	14.4166	1.18105	0.590527	−	0.807018i	$-0.298920\pi$
$149$	14.4166	1.18105	0.590527	+	0.807018i	$0.298920\pi$
$150$	0	0
$151$	−20.4202	−1.66177	−0.830885	−	0.556444i	$-0.812165\pi$
$151$	−20.4202	−1.66177	−0.830885	+	0.556444i	$0.812165\pi$
$152$	−1.00406	−0.0814398
$153$	0	0
$154$	−0.115371	−0.00929689
$155$	20.9277	1.68095
$156$	0	0
$157$	−16.6561	−1.32930	−0.664649	−	0.747156i	$-0.731419\pi$
$157$	−16.6561	−1.32930	−0.664649	+	0.747156i	$0.731419\pi$
$158$	−0.615420	−0.0489602
$159$	0	0
$160$	7.00792	0.554025
$161$	1.44651	0.114001
$162$	0	0
$163$	−4.98140	−0.390173	−0.195087	−	0.980786i	$-0.562499\pi$
$163$	−4.98140	−0.390173	−0.195087	+	0.980786i	$0.562499\pi$
$164$	−12.7875	−0.998537
$165$	0	0
$166$	0.958909	0.0744258
$167$	10.3671	0.802230	0.401115	−	0.916028i	$-0.368623\pi$
$167$	10.3671	0.802230	0.401115	+	0.916028i	$0.368623\pi$
$168$	0	0
$169$	−0.151489	−0.0116530
$170$	4.24946	0.325918
$171$	0	0
$172$	−12.2859	−0.936790
$173$	11.1873	0.850554	0.425277	−	0.905063i	$-0.360177\pi$
$173$	11.1873	0.850554	0.425277	+	0.905063i	$0.360177\pi$
$174$	0	0
$175$	−1.21300	−0.0916943
$176$	−11.1146	−0.837793
$177$	0	0
$178$	−2.37820	−0.178253
$179$	−16.4559	−1.22997	−0.614985	−	0.788539i	$-0.710838\pi$
$179$	−16.4559	−1.22997	−0.614985	+	0.788539i	$0.710838\pi$
$180$	0	0
$181$	22.6090	1.68051	0.840256	−	0.542189i	$-0.182404\pi$
$181$	22.6090	1.68051	0.840256	+	0.542189i	$0.182404\pi$
$182$	−0.141382	−0.0104799
$183$	0	0
$184$	−4.88062	−0.359804
$185$	−19.9391	−1.46595
$186$	0	0
$187$	−20.8101	−1.52178
$188$	−17.7961	−1.29792
$189$	0	0
$190$	−0.825445	−0.0598841
$191$	6.08877	0.440568	0.220284	−	0.975436i	$-0.429302\pi$
$191$	6.08877	0.440568	0.220284	+	0.975436i	$0.429302\pi$
$192$	0	0
$193$	−11.7756	−0.847628	−0.423814	−	0.905749i	$-0.639309\pi$
$193$	−11.7756	−0.847628	−0.423814	+	0.905749i	$0.639309\pi$
$194$	0.719706	0.0516719
$195$	0	0
$196$	13.6740	0.976711
$197$	5.85070	0.416845	0.208423	−	0.978039i	$-0.433167\pi$
$197$	5.85070	0.416845	0.208423	+	0.978039i	$0.433167\pi$
$198$	0	0
$199$	7.35979	0.521722	0.260861	−	0.965376i	$-0.415994\pi$
$199$	7.35979	0.521722	0.260861	+	0.965376i	$0.415994\pi$
$200$	4.09274	0.289401
$201$	0	0
$202$	−2.44379	−0.171945
$203$	0	0
$204$	0	0
$205$	−21.2048	−1.48101
$206$	−1.11452	−0.0776526
$207$	0	0
$208$	−13.6204	−0.944402
$209$	4.04230	0.279612
$210$	0	0
$211$	−8.96086	−0.616891	−0.308445	−	0.951242i	$-0.599809\pi$
$211$	−8.96086	−0.616891	−0.308445	+	0.951242i	$0.599809\pi$
$212$	−8.92161	−0.612738
$213$	0	0
$214$	0.0436081	0.00298099
$215$	−20.3730	−1.38943
$216$	0	0
$217$	1.38197	0.0938140
$218$	−0.503928	−0.0341303
$219$	0	0
$220$	−18.7562	−1.26454
$221$	−25.5017	−1.71543
$222$	0	0
$223$	−24.3679	−1.63180	−0.815898	−	0.578196i	$-0.803757\pi$
$223$	−24.3679	−1.63180	−0.815898	+	0.578196i	$0.803757\pi$
$224$	0.462770	0.0309201
$225$	0	0
$226$	0.576053	0.0383185
$227$	−13.5151	−0.897030	−0.448515	−	0.893775i	$-0.648047\pi$
$227$	−13.5151	−0.897030	−0.448515	+	0.893775i	$0.648047\pi$
$228$	0	0
$229$	21.5775	1.42588	0.712939	−	0.701226i	$-0.247364\pi$
$229$	21.5775	1.42588	0.712939	+	0.701226i	$0.247364\pi$
$230$	−4.01241	−0.264570
$231$	0	0
$232$	0	0
$233$	−27.6676	−1.81256	−0.906281	−	0.422675i	$-0.861091\pi$
$233$	−27.6676	−1.81256	−0.906281	+	0.422675i	$0.861091\pi$
$234$	0	0
$235$	−29.5103	−1.92504
$236$	−23.7423	−1.54549
$237$	0	0
$238$	0.280614	0.0181895
$239$	−25.3247	−1.63812	−0.819058	−	0.573711i	$-0.805503\pi$
$239$	−25.3247	−1.63812	−0.819058	+	0.573711i	$0.805503\pi$
$240$	0	0
$241$	20.8831	1.34520	0.672598	−	0.740008i	$-0.265178\pi$
$241$	20.8831	1.34520	0.672598	+	0.740008i	$0.265178\pi$
$242$	0.447705	0.0287795
$243$	0	0
$244$	3.11122	0.199175
$245$	22.6748	1.44864
$246$	0	0
$247$	4.95363	0.315192
$248$	−4.66284	−0.296091
$249$	0	0
$250$	0.378197	0.0239193
$251$	9.97440	0.629578	0.314789	−	0.949162i	$-0.398066\pi$
$251$	9.97440	0.629578	0.314789	+	0.949162i	$0.398066\pi$
$252$	0	0
$253$	19.6492	1.23534
$254$	3.03273	0.190290
$255$	0	0
$256$	13.3829	0.836428
$257$	−5.04570	−0.314742	−0.157371	−	0.987540i	$-0.550302\pi$
$257$	−5.04570	−0.314742	−0.157371	+	0.987540i	$0.550302\pi$
$258$	0	0
$259$	−1.31668	−0.0818147
$260$	−22.9848	−1.42546
$261$	0	0
$262$	1.26636	0.0782359
$263$	−15.1575	−0.934652	−0.467326	−	0.884085i	$-0.654783\pi$
$263$	−15.1575	−0.934652	−0.467326	+	0.884085i	$0.654783\pi$
$264$	0	0
$265$	−14.7942	−0.908802
$266$	−0.0545085	−0.00334213
$267$	0	0
$268$	14.1272	0.862955
$269$	−7.83550	−0.477739	−0.238869	−	0.971052i	$-0.576777\pi$
$269$	−7.83550	−0.477739	−0.238869	+	0.971052i	$0.576777\pi$
$270$	0	0
$271$	−18.6054	−1.13020	−0.565099	−	0.825023i	$-0.691162\pi$
$271$	−18.6054	−1.13020	−0.565099	+	0.825023i	$0.691162\pi$
$272$	27.0337	1.63916
$273$	0	0
$274$	−1.57970	−0.0954333
$275$	−16.4772	−0.993615
$276$	0	0
$277$	−1.37292	−0.0824908	−0.0412454	−	0.999149i	$-0.513133\pi$
$277$	−1.37292	−0.0824908	−0.0412454	+	0.999149i	$0.513133\pi$
$278$	−3.39021	−0.203331
$279$	0	0
$280$	0.510153	0.0304875
$281$	−9.28350	−0.553807	−0.276904	−	0.960898i	$-0.589308\pi$
$281$	−9.28350	−0.553807	−0.276904	+	0.960898i	$0.589308\pi$
$282$	0	0
$283$	−15.7164	−0.934245	−0.467123	−	0.884193i	$-0.654709\pi$
$283$	−15.7164	−0.934245	−0.467123	+	0.884193i	$0.654709\pi$
$284$	26.4296	1.56831
$285$	0	0
$286$	−1.92051	−0.113562
$287$	−1.40027	−0.0826553
$288$	0	0
$289$	33.6157	1.97739
$290$	0	0
$291$	0	0
$292$	−27.3029	−1.59778
$293$	26.3891	1.54167	0.770834	−	0.637036i	$-0.219840\pi$
$293$	26.3891	1.54167	0.770834	+	0.637036i	$0.219840\pi$
$294$	0	0
$295$	−39.3705	−2.29224
$296$	4.44257	0.258219
$297$	0	0
$298$	−2.64072	−0.152973
$299$	24.0791	1.39253
$300$	0	0
$301$	−1.34534	−0.0775441
$302$	3.74041	0.215236
$303$	0	0
$304$	−5.25121	−0.301178
$305$	5.15917	0.295413
$306$	0	0
$307$	−9.16136	−0.522866	−0.261433	−	0.965222i	$-0.584195\pi$
$307$	−9.16136	−0.522866	−0.261433	+	0.965222i	$0.584195\pi$
$308$	−1.23857	−0.0705742
$309$	0	0
$310$	−3.83337	−0.217721
$311$	−14.1374	−0.801657	−0.400828	−	0.916153i	$-0.631278\pi$
$311$	−14.1374	−0.801657	−0.400828	+	0.916153i	$0.631278\pi$
$312$	0	0
$313$	−12.7558	−0.720999	−0.360500	−	0.932759i	$-0.617394\pi$
$313$	−12.7558	−0.720999	−0.360500	+	0.932759i	$0.617394\pi$
$314$	3.05092	0.172174
$315$	0	0
$316$	−6.60685	−0.371664
$317$	−17.4394	−0.979493	−0.489747	−	0.871865i	$-0.662911\pi$
$317$	−17.4394	−0.979493	−0.489747	+	0.871865i	$0.662911\pi$
$318$	0	0
$319$	0	0
$320$	23.4976	1.31356
$321$	0	0
$322$	−0.264961	−0.0147657
$323$	−9.83195	−0.547065
$324$	0	0
$325$	−20.1920	−1.12005
$326$	0.912453	0.0505361
$327$	0	0
$328$	4.72459	0.260872
$329$	−1.94873	−0.107437
$330$	0	0
$331$	−24.4053	−1.34144	−0.670720	−	0.741711i	$-0.734015\pi$
$331$	−24.4053	−1.34144	−0.670720	+	0.741711i	$0.734015\pi$
$332$	10.2944	0.564978
$333$	0	0
$334$	−1.89896	−0.103907
$335$	23.4263	1.27992
$336$	0	0
$337$	−0.660946	−0.0360040	−0.0180020	−	0.999838i	$-0.505731\pi$
$337$	−0.660946	−0.0360040	−0.0180020	+	0.999838i	$0.505731\pi$
$338$	0.0277485	0.00150932
$339$	0	0
$340$	45.6201	2.47410
$341$	18.7724	1.01658
$342$	0	0
$343$	3.00466	0.162236
$344$	4.53926	0.244741
$345$	0	0
$346$	−2.04920	−0.110166
$347$	24.1703	1.29753	0.648765	−	0.760989i	$-0.275286\pi$
$347$	24.1703	1.29753	0.648765	+	0.760989i	$0.275286\pi$
$348$	0	0
$349$	2.29727	0.122970	0.0614849	−	0.998108i	$-0.480416\pi$
$349$	2.29727	0.122970	0.0614849	+	0.998108i	$0.480416\pi$
$350$	0.222188	0.0118765
$351$	0	0
$352$	6.28621	0.335056
$353$	23.1694	1.23318	0.616592	−	0.787283i	$-0.288513\pi$
$353$	23.1694	1.23318	0.616592	+	0.787283i	$0.288513\pi$
$354$	0	0
$355$	43.8268	2.32609
$356$	−25.5312	−1.35315
$357$	0	0
$358$	3.01426	0.159308
$359$	28.0233	1.47901	0.739506	−	0.673150i	$-0.235059\pi$
$359$	28.0233	1.47901	0.739506	+	0.673150i	$0.235059\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	−4.14134	−0.217664
$363$	0	0
$364$	−1.51781	−0.0795548
$365$	−45.2750	−2.36980
$366$	0	0
$367$	−20.8182	−1.08670	−0.543351	−	0.839506i	$-0.682845\pi$
$367$	−20.8182	−1.08670	−0.543351	+	0.839506i	$0.682845\pi$
$368$	−25.5256	−1.33062
$369$	0	0
$370$	3.65228	0.189873
$371$	−0.976941	−0.0507203
$372$	0	0
$373$	−4.60107	−0.238234	−0.119117	−	0.992880i	$-0.538006\pi$
$373$	−4.60107	−0.238234	−0.119117	+	0.992880i	$0.538006\pi$
$374$	3.81182	0.197105
$375$	0	0
$376$	6.57512	0.339086
$377$	0	0
$378$	0	0
$379$	3.56029	0.182880	0.0914399	−	0.995811i	$-0.470853\pi$
$379$	3.56029	0.182880	0.0914399	+	0.995811i	$0.470853\pi$
$380$	−8.86158	−0.454590
$381$	0	0
$382$	−1.11529	−0.0570634
$383$	−19.8155	−1.01252	−0.506262	−	0.862380i	$-0.668973\pi$
$383$	−19.8155	−1.01252	−0.506262	+	0.862380i	$0.668973\pi$
$384$	0	0
$385$	−2.05386	−0.104674
$386$	2.15697	0.109787
$387$	0	0
$388$	7.72642	0.392250
$389$	−29.6983	−1.50576	−0.752882	−	0.658155i	$-0.771337\pi$
$389$	−29.6983	−1.50576	−0.752882	+	0.658155i	$0.771337\pi$
$390$	0	0
$391$	−47.7922	−2.41695
$392$	−5.05211	−0.255170
$393$	0	0
$394$	−1.07169	−0.0539907
$395$	−10.9558	−0.551246
$396$	0	0
$397$	−3.28011	−0.164624	−0.0823119	−	0.996607i	$-0.526230\pi$
$397$	−3.28011	−0.164624	−0.0823119	+	0.996607i	$0.526230\pi$
$398$	−1.34811	−0.0675745
$399$	0	0
$400$	21.4050	1.07025
$401$	25.8859	1.29268	0.646339	−	0.763050i	$-0.276299\pi$
$401$	25.8859	1.29268	0.646339	+	0.763050i	$0.276299\pi$
$402$	0	0
$403$	23.0047	1.14594
$404$	−26.2354	−1.30526
$405$	0	0
$406$	0	0
$407$	−17.8856	−0.886558
$408$	0	0
$409$	−30.7354	−1.51977	−0.759883	−	0.650060i	$-0.774744\pi$
$409$	−30.7354	−1.51977	−0.759883	+	0.650060i	$0.774744\pi$
$410$	3.88414	0.191824
$411$	0	0
$412$	−11.9650	−0.589474
$413$	−2.59985	−0.127930
$414$	0	0
$415$	17.0706	0.837965
$416$	7.70343	0.377692
$417$	0	0
$418$	−0.740436	−0.0362159
$419$	9.64429	0.471155	0.235577	−	0.971856i	$-0.424302\pi$
$419$	9.64429	0.471155	0.235577	+	0.971856i	$0.424302\pi$
$420$	0	0
$421$	−29.4135	−1.43352	−0.716762	−	0.697318i	$-0.754377\pi$
$421$	−29.4135	−1.43352	−0.716762	+	0.697318i	$0.754377\pi$
$422$	1.64138	0.0799011
$423$	0	0
$424$	3.29626	0.160081
$425$	40.0771	1.94402
$426$	0	0
$427$	0.340688	0.0164870
$428$	0.468156	0.0226292
$429$	0	0
$430$	3.73177	0.179962
$431$	10.2903	0.495668	0.247834	−	0.968803i	$-0.420281\pi$
$431$	10.2903	0.495668	0.247834	+	0.968803i	$0.420281\pi$
$432$	0	0
$433$	−26.7403	−1.28506	−0.642529	−	0.766262i	$-0.722115\pi$
$433$	−26.7403	−1.28506	−0.642529	+	0.766262i	$0.722115\pi$
$434$	−0.253138	−0.0121510
$435$	0	0
$436$	−5.40993	−0.259089
$437$	9.28350	0.444090
$438$	0	0
$439$	−1.28671	−0.0614112	−0.0307056	−	0.999528i	$-0.509775\pi$
$439$	−1.28671	−0.0614112	−0.0307056	+	0.999528i	$0.509775\pi$
$440$	6.92985	0.330367
$441$	0	0
$442$	4.67120	0.222186
$443$	29.3364	1.39382	0.696908	−	0.717161i	$-0.254559\pi$
$443$	29.3364	1.39382	0.696908	+	0.717161i	$0.254559\pi$
$444$	0	0
$445$	−42.3370	−2.00697
$446$	4.46352	0.211354
$447$	0	0
$448$	1.55167	0.0733097
$449$	26.7561	1.26270	0.631349	−	0.775499i	$-0.282502\pi$
$449$	26.7561	1.26270	0.631349	+	0.775499i	$0.282502\pi$
$450$	0	0
$451$	−19.0211	−0.895666
$452$	6.18423	0.290882
$453$	0	0
$454$	2.47559	0.116185
$455$	−2.51690	−0.117994
$456$	0	0
$457$	29.0329	1.35810	0.679052	−	0.734090i	$-0.262391\pi$
$457$	29.0329	1.35810	0.679052	+	0.734090i	$0.262391\pi$
$458$	−3.95239	−0.184683
$459$	0	0
$460$	−43.0753	−2.00840
$461$	−23.9508	−1.11550	−0.557751	−	0.830008i	$-0.688336\pi$
$461$	−23.9508	−1.11550	−0.557751	+	0.830008i	$0.688336\pi$
$462$	0	0
$463$	22.8040	1.05979	0.529895	−	0.848063i	$-0.322231\pi$
$463$	22.8040	1.05979	0.529895	+	0.848063i	$0.322231\pi$
$464$	0	0
$465$	0	0
$466$	5.06793	0.234767
$467$	8.10901	0.375240	0.187620	−	0.982242i	$-0.439923\pi$
$467$	8.10901	0.375240	0.187620	+	0.982242i	$0.439923\pi$
$468$	0	0
$469$	1.54697	0.0714323
$470$	5.40547	0.249336
$471$	0	0
$472$	8.77205	0.403766
$473$	−18.2749	−0.840281
$474$	0	0
$475$	−7.78486	−0.357194
$476$	3.01254	0.138080
$477$	0	0
$478$	4.63877	0.212172
$479$	−23.1307	−1.05687	−0.528433	−	0.848975i	$-0.677220\pi$
$479$	−23.1307	−1.05687	−0.528433	+	0.848975i	$0.677220\pi$
$480$	0	0
$481$	−21.9179	−0.999372
$482$	−3.82520	−0.174233
$483$	0	0
$484$	4.80634	0.218470
$485$	12.8123	0.581777
$486$	0	0
$487$	−8.24795	−0.373750	−0.186875	−	0.982384i	$-0.559836\pi$
$487$	−8.24795	−0.373750	−0.186875	+	0.982384i	$0.559836\pi$
$488$	−1.14950	−0.0520355
$489$	0	0
$490$	−4.15339	−0.187631
$491$	39.1461	1.76664	0.883319	−	0.468772i	$-0.155303\pi$
$491$	39.1461	1.76664	0.883319	+	0.468772i	$0.155303\pi$
$492$	0	0
$493$	0	0
$494$	−0.907367	−0.0408244
$495$	0	0
$496$	−24.3866	−1.09499
$497$	2.89412	0.129819
$498$	0	0
$499$	−21.5288	−0.963759	−0.481880	−	0.876237i	$-0.660046\pi$
$499$	−21.5288	−0.963759	−0.481880	+	0.876237i	$0.660046\pi$
$500$	4.06014	0.181575
$501$	0	0
$502$	−1.82703	−0.0815444
$503$	11.5903	0.516788	0.258394	−	0.966040i	$-0.416807\pi$
$503$	11.5903	0.516788	0.258394	+	0.966040i	$0.416807\pi$
$504$	0	0
$505$	−43.5048	−1.93594
$506$	−3.59919	−0.160003
$507$	0	0
$508$	32.5579	1.44452
$509$	−1.59645	−0.0707612	−0.0353806	−	0.999374i	$-0.511264\pi$
$509$	−1.59645	−0.0707612	−0.0353806	+	0.999374i	$0.511264\pi$
$510$	0	0
$511$	−2.98975	−0.132259
$512$	−13.6877	−0.604914
$513$	0	0
$514$	0.924231	0.0407661
$515$	−19.8409	−0.874296
$516$	0	0
$517$	−26.4712	−1.16420
$518$	0.241180	0.0105968
$519$	0	0
$520$	8.49218	0.372407
$521$	9.45950	0.414428	0.207214	−	0.978296i	$-0.433560\pi$
$521$	9.45950	0.414428	0.207214	+	0.978296i	$0.433560\pi$
$522$	0	0
$523$	−31.9147	−1.39553	−0.697766	−	0.716326i	$-0.745822\pi$
$523$	−31.9147	−1.39553	−0.697766	+	0.716326i	$0.745822\pi$
$524$	13.5950	0.593901
$525$	0	0
$526$	2.77643	0.121058
$527$	−45.6596	−1.98896
$528$	0	0
$529$	22.1262	0.962007
$530$	2.70989	0.117710
$531$	0	0
$532$	−0.585178	−0.0253707
$533$	−23.3093	−1.00964
$534$	0	0
$535$	0.776318	0.0335631
$536$	−5.21956	−0.225451
$537$	0	0
$538$	1.43524	0.0618778
$539$	20.3396	0.876089
$540$	0	0
$541$	−4.49299	−0.193169	−0.0965844	−	0.995325i	$-0.530792\pi$
$541$	−4.49299	−0.193169	−0.0965844	+	0.995325i	$0.530792\pi$
$542$	3.40799	0.146386
$543$	0	0
$544$	−15.2897	−0.655542
$545$	−8.97099	−0.384275
$546$	0	0
$547$	23.3569	0.998669	0.499335	−	0.866409i	$-0.333578\pi$
$547$	23.3569	0.998669	0.499335	+	0.866409i	$0.333578\pi$
$548$	−16.9589	−0.724450
$549$	0	0
$550$	3.01817	0.128695
$551$	0	0
$552$	0	0
$553$	−0.723469	−0.0307650
$554$	0.251481	0.0106844
$555$	0	0
$556$	−36.3957	−1.54352
$557$	−4.50084	−0.190707	−0.0953534	−	0.995443i	$-0.530398\pi$
$557$	−4.50084	−0.190707	−0.0953534	+	0.995443i	$0.530398\pi$
$558$	0	0
$559$	−22.3950	−0.947207
$560$	2.66810	0.112748
$561$	0	0
$562$	1.70048	0.0717303
$563$	31.1391	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$563$	31.1391	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$564$	0	0
$565$	10.2550	0.431430
$566$	2.87881	0.121006
$567$	0	0
$568$	−9.76494	−0.409728
$569$	10.1235	0.424398	0.212199	−	0.977226i	$-0.431937\pi$
$569$	10.1235	0.424398	0.212199	+	0.977226i	$0.431937\pi$
$570$	0	0
$571$	8.61244	0.360420	0.180210	−	0.983628i	$-0.442322\pi$
$571$	8.61244	0.360420	0.180210	+	0.983628i	$0.442322\pi$
$572$	−20.6177	−0.862069
$573$	0	0
$574$	0.256490	0.0107057
$575$	−37.8415	−1.57810
$576$	0	0
$577$	2.97518	0.123858	0.0619292	−	0.998081i	$-0.480275\pi$
$577$	2.97518	0.123858	0.0619292	+	0.998081i	$0.480275\pi$
$578$	−6.15745	−0.256116
$579$	0	0
$580$	0	0
$581$	1.12727	0.0467668
$582$	0	0
$583$	−13.2706	−0.549613
$584$	10.0876	0.417428
$585$	0	0
$586$	−4.83375	−0.199680
$587$	−1.66166	−0.0685840	−0.0342920	−	0.999412i	$-0.510918\pi$
$587$	−1.66166	−0.0685840	−0.0342920	+	0.999412i	$0.510918\pi$
$588$	0	0
$589$	8.86925	0.365451
$590$	7.21159	0.296896
$591$	0	0
$592$	23.2346	0.954937
$593$	−14.5417	−0.597157	−0.298579	−	0.954385i	$-0.596512\pi$
$593$	−14.5417	−0.597157	−0.298579	+	0.954385i	$0.596512\pi$
$594$	0	0
$595$	4.99553	0.204797
$596$	−28.3495	−1.16124
$597$	0	0
$598$	−4.41062	−0.180364
$599$	−44.6964	−1.82625	−0.913123	−	0.407684i	$-0.866336\pi$
$599$	−44.6964	−1.82625	−0.913123	+	0.407684i	$0.866336\pi$
$600$	0	0
$601$	2.47245	0.100853	0.0504267	−	0.998728i	$-0.483942\pi$
$601$	2.47245	0.100853	0.0504267	+	0.998728i	$0.483942\pi$
$602$	0.246429	0.0100437
$603$	0	0
$604$	40.1552	1.63389
$605$	7.97010	0.324031
$606$	0	0
$607$	−32.7470	−1.32916	−0.664581	−	0.747216i	$-0.731390\pi$
$607$	−32.7470	−1.32916	−0.664581	+	0.747216i	$0.731390\pi$
$608$	2.96999	0.120449
$609$	0	0
$610$	−0.945016	−0.0382626
$611$	−32.4391	−1.31235
$612$	0	0
$613$	0.487087	0.0196733	0.00983663	−	0.999952i	$-0.496869\pi$
$613$	0.487087	0.0196733	0.00983663	+	0.999952i	$0.496869\pi$
$614$	1.67810	0.0677228
$615$	0	0
$616$	0.457615	0.0184378
$617$	−22.3660	−0.900420	−0.450210	−	0.892923i	$-0.648651\pi$
$617$	−22.3660	−0.900420	−0.450210	+	0.892923i	$0.648651\pi$
$618$	0	0
$619$	20.0488	0.805828	0.402914	−	0.915238i	$-0.367997\pi$
$619$	20.0488	0.805828	0.402914	+	0.915238i	$0.367997\pi$
$620$	−41.1532	−1.65275
$621$	0	0
$622$	2.58957	0.103832
$623$	−2.79574	−0.112009
$624$	0	0
$625$	−21.4332	−0.857327
$626$	2.33650	0.0933854
$627$	0	0
$628$	32.7533	1.30700
$629$	43.5027	1.73456
$630$	0	0
$631$	−4.35043	−0.173188	−0.0865939	−	0.996244i	$-0.527598\pi$
$631$	−4.35043	−0.173188	−0.0865939	+	0.996244i	$0.527598\pi$
$632$	2.44103	0.0970990
$633$	0	0
$634$	3.19441	0.126866
$635$	53.9891	2.14249
$636$	0	0
$637$	24.9252	0.987571
$638$	0	0
$639$	0	0
$640$	−18.3200	−0.724160
$641$	−47.0709	−1.85919	−0.929594	−	0.368586i	$-0.879842\pi$
$641$	−47.0709	−1.85919	−0.929594	+	0.368586i	$0.879842\pi$
$642$	0	0
$643$	27.6486	1.09036	0.545178	−	0.838320i	$-0.316462\pi$
$643$	27.6486	1.09036	0.545178	+	0.838320i	$0.316462\pi$
$644$	−2.84449	−0.112089
$645$	0	0
$646$	1.80094	0.0708570
$647$	−13.4530	−0.528891	−0.264445	−	0.964401i	$-0.585189\pi$
$647$	−13.4530	−0.528891	−0.264445	+	0.964401i	$0.585189\pi$
$648$	0	0
$649$	−35.3159	−1.38627
$650$	3.69862	0.145072
$651$	0	0
$652$	9.79566	0.383628
$653$	17.8470	0.698407	0.349203	−	0.937047i	$-0.386452\pi$
$653$	17.8470	0.698407	0.349203	+	0.937047i	$0.386452\pi$
$654$	0	0
$655$	22.5439	0.880863
$656$	24.7096	0.964748
$657$	0	0
$658$	0.356952	0.0139154
$659$	42.0430	1.63776	0.818882	−	0.573963i	$-0.194594\pi$
$659$	42.0430	1.63776	0.818882	+	0.573963i	$0.194594\pi$
$660$	0	0
$661$	2.10493	0.0818724	0.0409362	−	0.999162i	$-0.486966\pi$
$661$	2.10493	0.0818724	0.0409362	+	0.999162i	$0.486966\pi$
$662$	4.47038	0.173746
$663$	0	0
$664$	−3.80346	−0.147603
$665$	−0.970369	−0.0376293
$666$	0	0
$667$	0	0
$668$	−20.3864	−0.788772
$669$	0	0
$670$	−4.29105	−0.165778
$671$	4.62785	0.178656
$672$	0	0
$673$	20.6268	0.795106	0.397553	−	0.917579i	$-0.369859\pi$
$673$	20.6268	0.795106	0.397553	+	0.917579i	$0.369859\pi$
$674$	0.121067	0.00466332
$675$	0	0
$676$	0.297895	0.0114575
$677$	−39.0923	−1.50244	−0.751219	−	0.660053i	$-0.770534\pi$
$677$	−39.0923	−1.50244	−0.751219	+	0.660053i	$0.770534\pi$
$678$	0	0
$679$	0.846065	0.0324690
$680$	−16.8552	−0.646369
$681$	0	0
$682$	−3.43858	−0.131670
$683$	−18.1402	−0.694114	−0.347057	−	0.937844i	$-0.612819\pi$
$683$	−18.1402	−0.694114	−0.347057	+	0.937844i	$0.612819\pi$
$684$	0	0
$685$	−28.1221	−1.07449
$686$	−0.550370	−0.0210132
$687$	0	0
$688$	23.7403	0.905091
$689$	−16.2625	−0.619551
$690$	0	0
$691$	−3.25680	−0.123895	−0.0619473	−	0.998079i	$-0.519731\pi$
$691$	−3.25680	−0.123895	−0.0619473	+	0.998079i	$0.519731\pi$
$692$	−21.9992	−0.836285
$693$	0	0
$694$	−4.42732	−0.168059
$695$	−60.3530	−2.28932
$696$	0	0
$697$	46.2643	1.75239
$698$	−0.420795	−0.0159273
$699$	0	0
$700$	2.38531	0.0901561
$701$	32.3256	1.22092	0.610460	−	0.792047i	$-0.290985\pi$
$701$	32.3256	1.22092	0.610460	+	0.792047i	$0.290985\pi$
$702$	0	0
$703$	−8.45027	−0.318708
$704$	21.0777	0.794396
$705$	0	0
$706$	−4.24399	−0.159725
$707$	−2.87285	−0.108045
$708$	0	0
$709$	−33.1286	−1.24417	−0.622085	−	0.782950i	$-0.713714\pi$
$709$	−33.1286	−1.24417	−0.622085	+	0.782950i	$0.713714\pi$
$710$	−8.02785	−0.301280
$711$	0	0
$712$	9.43299	0.353516
$713$	43.1125	1.61458
$714$	0	0
$715$	−34.1892	−1.27860
$716$	32.3596	1.20934
$717$	0	0
$718$	−5.13308	−0.191565
$719$	−28.9305	−1.07892	−0.539462	−	0.842010i	$-0.681373\pi$
$719$	−28.9305	−1.07892	−0.539462	+	0.842010i	$0.681373\pi$
$720$	0	0
$721$	−1.31020	−0.0487945
$722$	3.13044	0.116503
$723$	0	0
$724$	−44.4594	−1.65232
$725$	0	0
$726$	0	0
$727$	−22.6821	−0.841231	−0.420616	−	0.907239i	$-0.638186\pi$
$727$	−22.6821	−0.841231	−0.420616	+	0.907239i	$0.638186\pi$
$728$	0.560784	0.0207840
$729$	0	0
$730$	8.29312	0.306942
$731$	44.4495	1.64402
$732$	0	0
$733$	−9.98505	−0.368806	−0.184403	−	0.982851i	$-0.559035\pi$
$733$	−9.98505	−0.368806	−0.184403	+	0.982851i	$0.559035\pi$
$734$	3.81332	0.140752
$735$	0	0
$736$	14.4368	0.532149
$737$	21.0138	0.774052
$738$	0	0
$739$	9.80051	0.360518	0.180259	−	0.983619i	$-0.442306\pi$
$739$	9.80051	0.360518	0.180259	+	0.983619i	$0.442306\pi$
$740$	39.2092	1.44136
$741$	0	0
$742$	0.178948	0.00656940
$743$	50.7572	1.86210	0.931051	−	0.364888i	$-0.118893\pi$
$743$	50.7572	1.86210	0.931051	+	0.364888i	$0.118893\pi$
$744$	0	0
$745$	−47.0104	−1.72233
$746$	0.842787	0.0308566
$747$	0	0
$748$	40.9219	1.49625
$749$	0.0512644	0.00187316
$750$	0	0
$751$	0.556715	0.0203148	0.0101574	−	0.999948i	$-0.496767\pi$
$751$	0.556715	0.0203148	0.0101574	+	0.999948i	$0.496767\pi$
$752$	34.3879	1.25400
$753$	0	0
$754$	0	0
$755$	66.5872	2.42336
$756$	0	0
$757$	38.4699	1.39821	0.699106	−	0.715018i	$-0.253582\pi$
$757$	38.4699	1.39821	0.699106	+	0.715018i	$0.253582\pi$
$758$	−0.652146	−0.0236870
$759$	0	0
$760$	3.27409	0.118764
$761$	13.4890	0.488977	0.244489	−	0.969652i	$-0.421380\pi$
$761$	13.4890	0.488977	0.244489	+	0.969652i	$0.421380\pi$
$762$	0	0
$763$	−0.592403	−0.0214464
$764$	−11.9733	−0.433177
$765$	0	0
$766$	3.62965	0.131144
$767$	−43.2779	−1.56267
$768$	0	0
$769$	29.5712	1.06637	0.533183	−	0.846000i	$-0.320996\pi$
$769$	29.5712	1.06637	0.533183	+	0.846000i	$0.320996\pi$
$770$	0.376209	0.0135576
$771$	0	0
$772$	23.1561	0.833408
$773$	−17.8469	−0.641908	−0.320954	−	0.947095i	$-0.604003\pi$
$773$	−17.8469	−0.641908	−0.320954	+	0.947095i	$0.604003\pi$
$774$	0	0
$775$	−36.1529	−1.29865
$776$	−2.85468	−0.102477
$777$	0	0
$778$	5.43990	0.195030
$779$	−8.98671	−0.321982
$780$	0	0
$781$	39.3133	1.40674
$782$	8.75419	0.313049
$783$	0	0
$784$	−26.4225	−0.943661
$785$	54.3130	1.93851
$786$	0	0
$787$	−24.8647	−0.886330	−0.443165	−	0.896440i	$-0.646144\pi$
$787$	−24.8647	−0.886330	−0.443165	+	0.896440i	$0.646144\pi$
$788$	−11.5051	−0.409852
$789$	0	0
$790$	2.00679	0.0713986
$791$	0.677191	0.0240781
$792$	0	0
$793$	5.67120	0.201390
$794$	0.600824	0.0213224
$795$	0	0
$796$	−14.4726	−0.512969
$797$	16.4041	0.581062	0.290531	−	0.956866i	$-0.406168\pi$
$797$	16.4041	0.581062	0.290531	+	0.956866i	$0.406168\pi$
$798$	0	0
$799$	64.3851	2.27778
$800$	−12.1063	−0.428022
$801$	0	0
$802$	−4.74157	−0.167431
$803$	−40.6123	−1.43318
$804$	0	0
$805$	−4.71687	−0.166248
$806$	−4.21381	−0.148425
$807$	0	0
$808$	9.69318	0.341005
$809$	6.14159	0.215927	0.107963	−	0.994155i	$-0.465567\pi$
$809$	6.14159	0.215927	0.107963	+	0.994155i	$0.465567\pi$
$810$	0	0
$811$	9.01608	0.316597	0.158299	−	0.987391i	$-0.449399\pi$
$811$	9.01608	0.316597	0.158299	+	0.987391i	$0.449399\pi$
$812$	0	0
$813$	0	0
$814$	3.27615	0.114829
$815$	16.2436	0.568989
$816$	0	0
$817$	−8.63419	−0.302072
$818$	5.62986	0.196843
$819$	0	0
$820$	41.6982	1.45617
$821$	2.27437	0.0793762	0.0396881	−	0.999212i	$-0.487364\pi$
$821$	2.27437	0.0793762	0.0396881	+	0.999212i	$0.487364\pi$
$822$	0	0
$823$	0.873948	0.0304639	0.0152320	−	0.999884i	$-0.495151\pi$
$823$	0.873948	0.0304639	0.0152320	+	0.999884i	$0.495151\pi$
$824$	4.42071	0.154003
$825$	0	0
$826$	0.476219	0.0165698
$827$	−10.5733	−0.367668	−0.183834	−	0.982957i	$-0.558851\pi$
$827$	−10.5733	−0.367668	−0.183834	+	0.982957i	$0.558851\pi$
$828$	0	0
$829$	12.3245	0.428049	0.214024	−	0.976828i	$-0.431343\pi$
$829$	12.3245	0.428049	0.214024	+	0.976828i	$0.431343\pi$
$830$	−3.12686	−0.108535
$831$	0	0
$832$	25.8297	0.895483
$833$	−49.4714	−1.71408
$834$	0	0
$835$	−33.8056	−1.16989
$836$	−7.94897	−0.274921
$837$	0	0
$838$	−1.76657	−0.0610250
$839$	5.10166	0.176129	0.0880644	−	0.996115i	$-0.471932\pi$
$839$	5.10166	0.176129	0.0880644	+	0.996115i	$0.471932\pi$
$840$	0	0
$841$	0	0
$842$	5.38772	0.185673
$843$	0	0
$844$	17.6211	0.606542
$845$	0.493984	0.0169936
$846$	0	0
$847$	0.526308	0.0180842
$848$	17.2394	0.592004
$849$	0	0
$850$	−7.34100	−0.251794
$851$	−41.0760	−1.40807
$852$	0	0
$853$	−3.94074	−0.134928	−0.0674642	−	0.997722i	$-0.521491\pi$
$853$	−3.94074	−0.134928	−0.0674642	+	0.997722i	$0.521491\pi$
$854$	−0.0624044	−0.00213544
$855$	0	0
$856$	−0.172969	−0.00591197
$857$	−17.7219	−0.605369	−0.302685	−	0.953091i	$-0.597883\pi$
$857$	−17.7219	−0.605369	−0.302685	+	0.953091i	$0.597883\pi$
$858$	0	0
$859$	17.2468	0.588454	0.294227	−	0.955736i	$-0.404938\pi$
$859$	17.2468	0.588454	0.294227	+	0.955736i	$0.404938\pi$
$860$	40.0625	1.36612
$861$	0	0
$862$	−1.88490	−0.0642000
$863$	6.44660	0.219445	0.109722	−	0.993962i	$-0.465004\pi$
$863$	6.44660	0.219445	0.109722	+	0.993962i	$0.465004\pi$
$864$	0	0
$865$	−36.4801	−1.24036
$866$	4.89808	0.166444
$867$	0	0
$868$	−2.71756	−0.0922401
$869$	−9.82750	−0.333375
$870$	0	0
$871$	25.7513	0.872550
$872$	1.99880	0.0676880
$873$	0	0
$874$	−1.70048	−0.0575195
$875$	0.444597	0.0150301
$876$	0	0
$877$	−38.1105	−1.28690	−0.643450	−	0.765488i	$-0.722497\pi$
$877$	−38.1105	−1.28690	−0.643450	+	0.765488i	$0.722497\pi$
$878$	0.235689	0.00795411
$879$	0	0
$880$	36.2431	1.22175
$881$	5.81612	0.195950	0.0979750	−	0.995189i	$-0.468763\pi$
$881$	5.81612	0.195950	0.0979750	+	0.995189i	$0.468763\pi$
$882$	0	0
$883$	34.6346	1.16555	0.582774	−	0.812634i	$-0.301967\pi$
$883$	34.6346	1.16555	0.582774	+	0.812634i	$0.301967\pi$
$884$	50.1477	1.68665
$885$	0	0
$886$	−5.37362	−0.180530
$887$	4.44026	0.149089	0.0745447	−	0.997218i	$-0.476250\pi$
$887$	4.44026	0.149089	0.0745447	+	0.997218i	$0.476250\pi$
$888$	0	0
$889$	3.56519	0.119572
$890$	7.75496	0.259947
$891$	0	0
$892$	47.9182	1.60442
$893$	−12.5066	−0.418518
$894$	0	0
$895$	53.6602	1.79366
$896$	−1.20976	−0.0404154
$897$	0	0
$898$	−4.90097	−0.163547
$899$	0	0
$900$	0	0
$901$	32.2777	1.07533
$902$	3.48413	0.116009
$903$	0	0
$904$	−2.28489	−0.0759942
$905$	−73.7246	−2.45069
$906$	0	0
$907$	48.7959	1.62024	0.810121	−	0.586262i	$-0.199401\pi$
$907$	48.7959	1.62024	0.810121	+	0.586262i	$0.199401\pi$
$908$	26.5768	0.881981
$909$	0	0
$910$	0.461026	0.0152829
$911$	13.4600	0.445950	0.222975	−	0.974824i	$-0.428423\pi$
$911$	13.4600	0.445950	0.222975	+	0.974824i	$0.428423\pi$
$912$	0	0
$913$	15.3126	0.506773
$914$	−5.31802	−0.175905
$915$	0	0
$916$	−42.4309	−1.40196
$917$	1.48869	0.0491610
$918$	0	0
$919$	−3.71411	−0.122517	−0.0612586	−	0.998122i	$-0.519511\pi$
$919$	−3.71411	−0.122517	−0.0612586	+	0.998122i	$0.519511\pi$
$920$	15.9150	0.524702
$921$	0	0
$922$	4.38713	0.144482
$923$	48.1765	1.58575
$924$	0	0
$925$	34.4451	1.13255
$926$	−4.17705	−0.137266
$927$	0	0
$928$	0	0
$929$	−6.54651	−0.214784	−0.107392	−	0.994217i	$-0.534250\pi$
$929$	−6.54651	−0.214784	−0.107392	+	0.994217i	$0.534250\pi$
$930$	0	0
$931$	9.60968	0.314945
$932$	54.4068	1.78216
$933$	0	0
$934$	−1.48534	−0.0486019
$935$	67.8586	2.21921
$936$	0	0
$937$	−40.1304	−1.31100	−0.655502	−	0.755193i	$-0.727543\pi$
$937$	−40.1304	−1.31100	−0.655502	+	0.755193i	$0.727543\pi$
$938$	−0.283361	−0.00925207
$939$	0	0
$940$	58.0306	1.89275
$941$	−4.22105	−0.137602	−0.0688012	−	0.997630i	$-0.521917\pi$
$941$	−4.22105	−0.137602	−0.0688012	+	0.997630i	$0.521917\pi$
$942$	0	0
$943$	−43.6835	−1.42253
$944$	45.8778	1.49319
$945$	0	0
$946$	3.34745	0.108835
$947$	10.7656	0.349836	0.174918	−	0.984583i	$-0.444034\pi$
$947$	10.7656	0.349836	0.174918	+	0.984583i	$0.444034\pi$
$948$	0	0
$949$	−49.7684	−1.61555
$950$	1.42597	0.0462646
$951$	0	0
$952$	−1.11304	−0.0360739
$953$	−37.2593	−1.20695	−0.603474	−	0.797383i	$-0.706217\pi$
$953$	−37.2593	−1.20695	−0.603474	+	0.797383i	$0.706217\pi$
$954$	0	0
$955$	−19.8546	−0.642480
$956$	49.7996	1.61063
$957$	0	0
$958$	4.23689	0.136888
$959$	−1.85705	−0.0599673
$960$	0	0
$961$	10.1888	0.328670
$962$	4.01476	0.129441
$963$	0	0
$964$	−41.0655	−1.32263
$965$	38.3986	1.23609
$966$	0	0
$967$	−13.6210	−0.438022	−0.219011	−	0.975722i	$-0.570283\pi$
$967$	−13.6210	−0.438022	−0.219011	+	0.975722i	$0.570283\pi$
$968$	−1.77580	−0.0570763
$969$	0	0
$970$	−2.34686	−0.0753531
$971$	26.5860	0.853186	0.426593	−	0.904444i	$-0.359714\pi$
$971$	26.5860	0.853186	0.426593	+	0.904444i	$0.359714\pi$
$972$	0	0
$973$	−3.98543	−0.127767
$974$	1.51079	0.0484090
$975$	0	0
$976$	−6.01188	−0.192436
$977$	14.3977	0.460624	0.230312	−	0.973117i	$-0.426025\pi$
$977$	14.3977	0.460624	0.230312	+	0.973117i	$0.426025\pi$
$978$	0	0
$979$	−37.9769	−1.21375
$980$	−44.5888	−1.42434
$981$	0	0
$982$	−7.17047	−0.228819
$983$	−20.1243	−0.641866	−0.320933	−	0.947102i	$-0.603997\pi$
$983$	−20.1243	−0.641866	−0.320933	+	0.947102i	$0.603997\pi$
$984$	0	0
$985$	−19.0783	−0.607885
$986$	0	0
$987$	0	0
$988$	−9.74106	−0.309904
$989$	−41.9699	−1.33457
$990$	0	0
$991$	23.4189	0.743925	0.371962	−	0.928248i	$-0.378685\pi$
$991$	23.4189	0.743925	0.371962	+	0.928248i	$0.378685\pi$
$992$	13.7926	0.437916
$993$	0	0
$994$	−0.530122	−0.0168144
$995$	−23.9992	−0.760826
$996$	0	0
$997$	10.3739	0.328543	0.164272	−	0.986415i	$-0.447473\pi$
$997$	10.3739	0.328543	0.164272	+	0.986415i	$0.447473\pi$
$998$	3.94347	0.124828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bg.1.4	yes	8
3.2	odd	2	inner	7569.2.a.bg.1.5	yes	8
29.28	even	2		7569.2.a.bf.1.5	yes	8
87.86	odd	2		7569.2.a.bf.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.bf.1.4	✓	8	87.86	odd	2
7569.2.a.bf.1.5	yes	8	29.28	even	2
7569.2.a.bg.1.4	yes	8	1.1	even	1	trivial
7569.2.a.bg.1.5	yes	8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-0.183172 q^{2} -1.96645 q^{4} -3.26086 q^{5} -0.215332 q^{7} +0.726543 q^{8} +0.597298 q^{10} -2.92503 q^{11} -3.58448 q^{13} +0.0394428 q^{14} +3.79981 q^{16} +7.11447 q^{17} -1.38197 q^{19} +6.41230 q^{20} +0.535785 q^{22} -6.71760 q^{23} +5.63318 q^{25} +0.656577 q^{26} +0.423439 q^{28} -6.41785 q^{31} -2.14910 q^{32} -1.30317 q^{34} +0.702166 q^{35} +6.11468 q^{37} +0.253138 q^{38} -2.36915 q^{40} +6.50285 q^{41} +6.24776 q^{43} +5.75193 q^{44} +1.23048 q^{46} +9.04988 q^{47} -6.95363 q^{49} -1.03184 q^{50} +7.04870 q^{52} +4.53691 q^{53} +9.53811 q^{55} -0.156448 q^{56} +12.0737 q^{59} -1.58215 q^{61} +1.17557 q^{62} -7.20597 q^{64} +11.6885 q^{65} -7.18411 q^{67} -13.9902 q^{68} -0.128617 q^{70} -13.4403 q^{71} +13.8844 q^{73} -1.12004 q^{74} +2.71756 q^{76} +0.629853 q^{77} +3.35979 q^{79} -12.3906 q^{80} -1.19114 q^{82} -5.23502 q^{83} -23.1993 q^{85} -1.14441 q^{86} -2.12516 q^{88} +12.9834 q^{89} +0.771853 q^{91} +13.2098 q^{92} -1.65769 q^{94} +4.50639 q^{95} -3.92913 q^{97} +1.27371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 + 12 * q^10 + 2 * q^13 - 2 * q^16 - 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 - 10 * q^31 - 36 * q^34 - 18 * q^37 + 10 * q^40 - 28 * q^43 - 26 * q^46 + 4 * q^49 + 44 * q^52 - 14 * q^55 - 54 * q^61 - 44 * q^64 + 38 * q^67 - 70 * q^70 + 16 * q^73 - 20 * q^76 - 14 * q^79 + 2 * q^82 - 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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