LMFDB - Embedded newform 7569.2.a.r.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q+0.445042 q^{2} -1.80194 q^{4} -0.692021 q^{5} +0.356896 q^{7} -1.69202 q^{8} +O(q^{10})$	\(q+0.445042 q^{2} -1.80194 q^{4} -0.692021 q^{5} +0.356896 q^{7} -1.69202 q^{8} -0.307979 q^{10} -4.93900 q^{11} -5.65279 q^{13} +0.158834 q^{14} +2.85086 q^{16} -4.49396 q^{17} -2.35690 q^{19} +1.24698 q^{20} -2.19806 q^{22} -2.29590 q^{23} -4.52111 q^{25} -2.51573 q^{26} -0.643104 q^{28} -6.69202 q^{31} +4.65279 q^{32} -2.00000 q^{34} -0.246980 q^{35} -4.93900 q^{37} -1.04892 q^{38} +1.17092 q^{40} -3.10992 q^{41} -3.40581 q^{43} +8.89977 q^{44} -1.02177 q^{46} +6.44504 q^{47} -6.87263 q^{49} -2.01208 q^{50} +10.1860 q^{52} +4.69202 q^{53} +3.41789 q^{55} -0.603875 q^{56} +12.4940 q^{59} -1.64310 q^{61} -2.97823 q^{62} -3.63102 q^{64} +3.91185 q^{65} -2.32304 q^{67} +8.09783 q^{68} -0.109916 q^{70} +7.33513 q^{71} +5.62565 q^{73} -2.19806 q^{74} +4.24698 q^{76} -1.76271 q^{77} -4.66487 q^{79} -1.97285 q^{80} -1.38404 q^{82} -4.45473 q^{83} +3.10992 q^{85} -1.51573 q^{86} +8.35690 q^{88} +5.67994 q^{89} -2.01746 q^{91} +4.13706 q^{92} +2.86831 q^{94} +1.63102 q^{95} -0.180604 q^{97} -3.05861 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - q^{4} + 3 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q + q^2 - q^4 + 3 * q^5 - 3 * q^7	$ 3 q + q^{2} - q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{10} - 5 q^{11} + q^{13} - 8 q^{14} - 5 q^{16} - 4 q^{17} - 3 q^{19} - q^{20} - 11 q^{22} + 7 q^{23} + 2 q^{25} + 5 q^{26} - 6 q^{28} - 15 q^{31} - 4 q^{32} - 6 q^{34} + 4 q^{35} - 5 q^{37} + 6 q^{38} + 14 q^{40} - 10 q^{41} + 3 q^{43} + 4 q^{44} + 19 q^{47} - 4 q^{49} - 25 q^{50} + 16 q^{52} + 9 q^{53} + 16 q^{55} + 7 q^{56} + 28 q^{59} - 9 q^{61} - 12 q^{62} + 4 q^{64} + 8 q^{65} + 13 q^{67} + 6 q^{68} - q^{70} + 21 q^{71} + 5 q^{73} - 11 q^{74} + 8 q^{76} + 12 q^{77} - 15 q^{79} - 12 q^{80} + 6 q^{82} + 9 q^{83} + 10 q^{85} + 8 q^{86} + 21 q^{88} - 7 q^{89} - 22 q^{91} + 7 q^{92} + 11 q^{94} - 10 q^{95} + 11 q^{97} + 22 q^{98}+O(q^{100}) $ 3 * q + q^2 - q^4 + 3 * q^5 - 3 * q^7 - 6 * q^10 - 5 * q^11 + q^13 - 8 * q^14 - 5 * q^16 - 4 * q^17 - 3 * q^19 - q^20 - 11 * q^22 + 7 * q^23 + 2 * q^25 + 5 * q^26 - 6 * q^28 - 15 * q^31 - 4 * q^32 - 6 * q^34 + 4 * q^35 - 5 * q^37 + 6 * q^38 + 14 * q^40 - 10 * q^41 + 3 * q^43 + 4 * q^44 + 19 * q^47 - 4 * q^49 - 25 * q^50 + 16 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 + 28 * q^59 - 9 * q^61 - 12 * q^62 + 4 * q^64 + 8 * q^65 + 13 * q^67 + 6 * q^68 - q^70 + 21 * q^71 + 5 * q^73 - 11 * q^74 + 8 * q^76 + 12 * q^77 - 15 * q^79 - 12 * q^80 + 6 * q^82 + 9 * q^83 + 10 * q^85 + 8 * q^86 + 21 * q^88 - 7 * q^89 - 22 * q^91 + 7 * q^92 + 11 * q^94 - 10 * q^95 + 11 * q^97 + 22 * 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 $

$2$	0.445042	0.314692	0.157346	−	0.987544i	$-0.449706\pi$
$2$	0.445042	0.314692	0.157346	+	0.987544i	$0.449706\pi$
$3$	0	0
$3$	0	0
$4$	−1.80194	−0.900969
$5$	−0.692021	−0.309481	−0.154741	−	0.987955i	$-0.549454\pi$
$5$	−0.692021	−0.309481	−0.154741	+	0.987955i	$0.549454\pi$
$6$	0	0
$7$	0.356896	0.134894	0.0674470	−	0.997723i	$-0.478515\pi$
$7$	0.356896	0.134894	0.0674470	+	0.997723i	$0.478515\pi$
$8$	−1.69202	−0.598220
$9$	0	0
$10$	−0.307979	−0.0973914
$11$	−4.93900	−1.48916	−0.744582	−	0.667531i	$-0.767351\pi$
$11$	−4.93900	−1.48916	−0.744582	+	0.667531i	$0.767351\pi$
$12$	0	0
$13$	−5.65279	−1.56780	−0.783901	−	0.620885i	$-0.786773\pi$
$13$	−5.65279	−1.56780	−0.783901	+	0.620885i	$0.786773\pi$
$14$	0.158834	0.0424501
$15$	0	0
$16$	2.85086	0.712714
$17$	−4.49396	−1.08995	−0.544973	−	0.838454i	$-0.683460\pi$
$17$	−4.49396	−1.08995	−0.544973	+	0.838454i	$0.683460\pi$
$18$	0	0
$19$	−2.35690	−0.540709	−0.270354	−	0.962761i	$-0.587141\pi$
$19$	−2.35690	−0.540709	−0.270354	+	0.962761i	$0.587141\pi$
$20$	1.24698	0.278833
$21$	0	0
$22$	−2.19806	−0.468628
$23$	−2.29590	−0.478728	−0.239364	−	0.970930i	$-0.576939\pi$
$23$	−2.29590	−0.478728	−0.239364	+	0.970930i	$0.576939\pi$
$24$	0	0
$25$	−4.52111	−0.904221
$26$	−2.51573	−0.493375
$27$	0	0
$28$	−0.643104	−0.121535
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−6.69202	−1.20192	−0.600961	−	0.799278i	$-0.705215\pi$
$31$	−6.69202	−1.20192	−0.600961	+	0.799278i	$0.705215\pi$
$32$	4.65279	0.822505
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−0.246980	−0.0417472
$36$	0	0
$37$	−4.93900	−0.811967	−0.405983	−	0.913880i	$-0.633071\pi$
$37$	−4.93900	−0.811967	−0.405983	+	0.913880i	$0.633071\pi$
$38$	−1.04892	−0.170157
$39$	0	0
$40$	1.17092	0.185138
$41$	−3.10992	−0.485687	−0.242844	−	0.970065i	$-0.578080\pi$
$41$	−3.10992	−0.485687	−0.242844	+	0.970065i	$0.578080\pi$
$42$	0	0
$43$	−3.40581	−0.519382	−0.259691	−	0.965692i	$-0.583621\pi$
$43$	−3.40581	−0.519382	−0.259691	+	0.965692i	$0.583621\pi$
$44$	8.89977	1.34169
$45$	0	0
$46$	−1.02177	−0.150652
$47$	6.44504	0.940106	0.470053	−	0.882638i	$-0.344235\pi$
$47$	6.44504	0.940106	0.470053	+	0.882638i	$0.344235\pi$
$48$	0	0
$49$	−6.87263	−0.981804
$50$	−2.01208	−0.284551
$51$	0	0
$52$	10.1860	1.41254
$53$	4.69202	0.644499	0.322249	−	0.946655i	$-0.395561\pi$
$53$	4.69202	0.644499	0.322249	+	0.946655i	$0.395561\pi$
$54$	0	0
$55$	3.41789	0.460869
$56$	−0.603875	−0.0806963
$57$	0	0
$58$	0	0
$59$	12.4940	1.62657	0.813287	−	0.581862i	$-0.197676\pi$
$59$	12.4940	1.62657	0.813287	+	0.581862i	$0.197676\pi$
$60$	0	0
$61$	−1.64310	−0.210378	−0.105189	−	0.994452i	$-0.533545\pi$
$61$	−1.64310	−0.210378	−0.105189	+	0.994452i	$0.533545\pi$
$62$	−2.97823	−0.378236
$63$	0	0
$64$	−3.63102	−0.453878
$65$	3.91185	0.485206
$66$	0	0
$67$	−2.32304	−0.283805	−0.141902	−	0.989881i	$-0.545322\pi$
$67$	−2.32304	−0.283805	−0.141902	+	0.989881i	$0.545322\pi$
$68$	8.09783	0.982007
$69$	0	0
$70$	−0.109916	−0.0131375
$71$	7.33513	0.870519	0.435260	−	0.900305i	$-0.356657\pi$
$71$	7.33513	0.870519	0.435260	+	0.900305i	$0.356657\pi$
$72$	0	0
$73$	5.62565	0.658432	0.329216	−	0.944255i	$-0.393216\pi$
$73$	5.62565	0.658432	0.329216	+	0.944255i	$0.393216\pi$
$74$	−2.19806	−0.255520
$75$	0	0
$76$	4.24698	0.487162
$77$	−1.76271	−0.200879
$78$	0	0
$79$	−4.66487	−0.524839	−0.262420	−	0.964954i	$-0.584520\pi$
$79$	−4.66487	−0.524839	−0.262420	+	0.964954i	$0.584520\pi$
$80$	−1.97285	−0.220572
$81$	0	0
$82$	−1.38404	−0.152842
$83$	−4.45473	−0.488970	−0.244485	−	0.969653i	$-0.578619\pi$
$83$	−4.45473	−0.488970	−0.244485	+	0.969653i	$0.578619\pi$
$84$	0	0
$85$	3.10992	0.337318
$86$	−1.51573	−0.163445
$87$	0	0
$88$	8.35690	0.890848
$89$	5.67994	0.602072	0.301036	−	0.953613i	$-0.402667\pi$
$89$	5.67994	0.602072	0.301036	+	0.953613i	$0.402667\pi$
$90$	0	0
$91$	−2.01746	−0.211487
$92$	4.13706	0.431319
$93$	0	0
$94$	2.86831	0.295844
$95$	1.63102	0.167339
$96$	0	0
$97$	−0.180604	−0.0183375	−0.00916877	−	0.999958i	$-0.502919\pi$
$97$	−0.180604	−0.0183375	−0.00916877	+	0.999958i	$0.502919\pi$
$98$	−3.05861	−0.308966
$99$	0	0
$100$	8.14675	0.814675
$101$	−3.22952	−0.321349	−0.160675	−	0.987007i	$-0.551367\pi$
$101$	−3.22952	−0.321349	−0.160675	+	0.987007i	$0.551367\pi$
$102$	0	0
$103$	−13.6528	−1.34525	−0.672625	−	0.739984i	$-0.734833\pi$
$103$	−13.6528	−1.34525	−0.672625	+	0.739984i	$0.734833\pi$
$104$	9.56465	0.937891
$105$	0	0
$106$	2.08815	0.202819
$107$	−16.2446	−1.57042	−0.785212	−	0.619227i	$-0.787446\pi$
$107$	−16.2446	−1.57042	−0.785212	+	0.619227i	$0.787446\pi$
$108$	0	0
$109$	5.46681	0.523626	0.261813	−	0.965119i	$-0.415680\pi$
$109$	5.46681	0.523626	0.261813	+	0.965119i	$0.415680\pi$
$110$	1.52111	0.145032
$111$	0	0
$112$	1.01746	0.0961408
$113$	10.6746	1.00418	0.502089	−	0.864816i	$-0.332565\pi$
$113$	10.6746	1.00418	0.502089	+	0.864816i	$0.332565\pi$
$114$	0	0
$115$	1.58881	0.148157
$116$	0	0
$117$	0	0
$118$	5.56033	0.511870
$119$	−1.60388	−0.147027
$120$	0	0
$121$	13.3937	1.21761
$122$	−0.731250	−0.0662043
$123$	0	0
$124$	12.0586	1.08289
$125$	6.58881	0.589321
$126$	0	0
$127$	−1.03684	−0.0920043	−0.0460021	−	0.998941i	$-0.514648\pi$
$127$	−1.03684	−0.0920043	−0.0460021	+	0.998941i	$0.514648\pi$
$128$	−10.9215	−0.965337
$129$	0	0
$130$	1.74094	0.152690
$131$	−8.02177	−0.700865	−0.350433	−	0.936588i	$-0.613965\pi$
$131$	−8.02177	−0.700865	−0.350433	+	0.936588i	$0.613965\pi$
$132$	0	0
$133$	−0.841166	−0.0729384
$134$	−1.03385	−0.0893112
$135$	0	0
$136$	7.60388	0.652027
$137$	−17.2325	−1.47227	−0.736136	−	0.676834i	$-0.763352\pi$
$137$	−17.2325	−1.47227	−0.736136	+	0.676834i	$0.763352\pi$
$138$	0	0
$139$	−16.6679	−1.41375	−0.706875	−	0.707339i	$-0.749896\pi$
$139$	−16.6679	−1.41375	−0.706875	+	0.707339i	$0.749896\pi$
$140$	0.445042	0.0376129
$141$	0	0
$142$	3.26444	0.273946
$143$	27.9191	2.33472
$144$	0	0
$145$	0	0
$146$	2.50365	0.207203
$147$	0	0
$148$	8.89977	0.731557
$149$	−18.6069	−1.52433	−0.762167	−	0.647381i	$-0.775864\pi$
$149$	−18.6069	−1.52433	−0.762167	+	0.647381i	$0.775864\pi$
$150$	0	0
$151$	−7.52781	−0.612605	−0.306302	−	0.951934i	$-0.599092\pi$
$151$	−7.52781	−0.612605	−0.306302	+	0.951934i	$0.599092\pi$
$152$	3.98792	0.323463
$153$	0	0
$154$	−0.784479	−0.0632151
$155$	4.63102	0.371973
$156$	0	0
$157$	−18.2392	−1.45565	−0.727824	−	0.685764i	$-0.759468\pi$
$157$	−18.2392	−1.45565	−0.727824	+	0.685764i	$0.759468\pi$
$158$	−2.07606	−0.165163
$159$	0	0
$160$	−3.21983	−0.254550
$161$	−0.819396	−0.0645775
$162$	0	0
$163$	12.6407	0.990097	0.495048	−	0.868865i	$-0.335150\pi$
$163$	12.6407	0.990097	0.495048	+	0.868865i	$0.335150\pi$
$164$	5.60388	0.437589
$165$	0	0
$166$	−1.98254	−0.153875
$167$	0.796561	0.0616397	0.0308199	−	0.999525i	$-0.490188\pi$
$167$	0.796561	0.0616397	0.0308199	+	0.999525i	$0.490188\pi$
$168$	0	0
$169$	18.9541	1.45801
$170$	1.38404	0.106151
$171$	0	0
$172$	6.13706	0.467947
$173$	9.15346	0.695924	0.347962	−	0.937509i	$-0.386874\pi$
$173$	9.15346	0.695924	0.347962	+	0.937509i	$0.386874\pi$
$174$	0	0
$175$	−1.61356	−0.121974
$176$	−14.0804	−1.06135
$177$	0	0
$178$	2.52781	0.189467
$179$	−3.40581	−0.254562	−0.127281	−	0.991867i	$-0.540625\pi$
$179$	−3.40581	−0.254562	−0.127281	+	0.991867i	$0.540625\pi$
$180$	0	0
$181$	12.6746	0.942093	0.471046	−	0.882108i	$-0.343876\pi$
$181$	12.6746	0.942093	0.471046	+	0.882108i	$0.343876\pi$
$182$	−0.897853	−0.0665533
$183$	0	0
$184$	3.88471	0.286384
$185$	3.41789	0.251289
$186$	0	0
$187$	22.1957	1.62311
$188$	−11.6136	−0.847006
$189$	0	0
$190$	0.725873	0.0526604
$191$	10.6703	0.772072	0.386036	−	0.922484i	$-0.373844\pi$
$191$	10.6703	0.772072	0.386036	+	0.922484i	$0.373844\pi$
$192$	0	0
$193$	22.7289	1.63606	0.818029	−	0.575176i	$-0.195067\pi$
$193$	22.7289	1.63606	0.818029	+	0.575176i	$0.195067\pi$
$194$	−0.0803763	−0.00577068
$195$	0	0
$196$	12.3840	0.884574
$197$	19.5579	1.39345	0.696723	−	0.717340i	$-0.254641\pi$
$197$	19.5579	1.39345	0.696723	+	0.717340i	$0.254641\pi$
$198$	0	0
$199$	−0.875018	−0.0620284	−0.0310142	−	0.999519i	$-0.509874\pi$
$199$	−0.875018	−0.0620284	−0.0310142	+	0.999519i	$0.509874\pi$
$200$	7.64981	0.540923
$201$	0	0
$202$	−1.43727	−0.101126
$203$	0	0
$204$	0	0
$205$	2.15213	0.150311
$206$	−6.07606	−0.423339
$207$	0	0
$208$	−16.1153	−1.11739
$209$	11.6407	0.805205
$210$	0	0
$211$	−18.2687	−1.25767	−0.628836	−	0.777538i	$-0.716468\pi$
$211$	−18.2687	−1.25767	−0.628836	+	0.777538i	$0.716468\pi$
$212$	−8.45473	−0.580673
$213$	0	0
$214$	−7.22952	−0.494200
$215$	2.35690	0.160739
$216$	0	0
$217$	−2.38835	−0.162132
$218$	2.43296	0.164781
$219$	0	0
$220$	−6.15883	−0.415228
$221$	25.4034	1.70882
$222$	0	0
$223$	−1.81833	−0.121764	−0.0608822	−	0.998145i	$-0.519391\pi$
$223$	−1.81833	−0.121764	−0.0608822	+	0.998145i	$0.519391\pi$
$224$	1.66056	0.110951
$225$	0	0
$226$	4.75063	0.316007
$227$	13.8562	0.919670	0.459835	−	0.888004i	$-0.347908\pi$
$227$	13.8562	0.919670	0.459835	+	0.888004i	$0.347908\pi$
$228$	0	0
$229$	−12.7724	−0.844024	−0.422012	−	0.906590i	$-0.638676\pi$
$229$	−12.7724	−0.844024	−0.422012	+	0.906590i	$0.638676\pi$
$230$	0.707087	0.0466239
$231$	0	0
$232$	0	0
$233$	8.86592	0.580826	0.290413	−	0.956901i	$-0.406207\pi$
$233$	8.86592	0.580826	0.290413	+	0.956901i	$0.406207\pi$
$234$	0	0
$235$	−4.46011	−0.290945
$236$	−22.5133	−1.46549
$237$	0	0
$238$	−0.713792	−0.0462682
$239$	25.5646	1.65364	0.826820	−	0.562467i	$-0.190148\pi$
$239$	25.5646	1.65364	0.826820	+	0.562467i	$0.190148\pi$
$240$	0	0
$241$	9.73125	0.626845	0.313422	−	0.949614i	$-0.398524\pi$
$241$	9.73125	0.626845	0.313422	+	0.949614i	$0.398524\pi$
$242$	5.96077	0.383173
$243$	0	0
$244$	2.96077	0.189544
$245$	4.75600	0.303850
$246$	0	0
$247$	13.3230	0.847725
$248$	11.3230	0.719014
$249$	0	0
$250$	2.93230	0.185455
$251$	9.76032	0.616066	0.308033	−	0.951376i	$-0.400329\pi$
$251$	9.76032	0.616066	0.308033	+	0.951376i	$0.400329\pi$
$252$	0	0
$253$	11.3394	0.712904
$254$	−0.461435	−0.0289530
$255$	0	0
$256$	2.40150	0.150094
$257$	−16.3569	−1.02032	−0.510158	−	0.860081i	$-0.670413\pi$
$257$	−16.3569	−1.02032	−0.510158	+	0.860081i	$0.670413\pi$
$258$	0	0
$259$	−1.76271	−0.109529
$260$	−7.04892	−0.437155
$261$	0	0
$262$	−3.57002	−0.220557
$263$	−23.7235	−1.46285	−0.731426	−	0.681921i	$-0.761145\pi$
$263$	−23.7235	−1.46285	−0.731426	+	0.681921i	$0.761145\pi$
$264$	0	0
$265$	−3.24698	−0.199460
$266$	−0.374354	−0.0229531
$267$	0	0
$268$	4.18598	0.255699
$269$	−25.3545	−1.54589	−0.772946	−	0.634472i	$-0.781217\pi$
$269$	−25.3545	−1.54589	−0.772946	+	0.634472i	$0.781217\pi$
$270$	0	0
$271$	−1.20344	−0.0731037	−0.0365519	−	0.999332i	$-0.511637\pi$
$271$	−1.20344	−0.0731037	−0.0365519	+	0.999332i	$0.511637\pi$
$272$	−12.8116	−0.776819
$273$	0	0
$274$	−7.66919	−0.463312
$275$	22.3297	1.34653
$276$	0	0
$277$	10.6649	0.640790	0.320395	−	0.947284i	$-0.396184\pi$
$277$	10.6649	0.640790	0.320395	+	0.947284i	$0.396184\pi$
$278$	−7.41789	−0.444896
$279$	0	0
$280$	0.417895	0.0249740
$281$	16.2795	0.971154	0.485577	−	0.874194i	$-0.338610\pi$
$281$	16.2795	0.971154	0.485577	+	0.874194i	$0.338610\pi$
$282$	0	0
$283$	−5.24698	−0.311901	−0.155950	−	0.987765i	$-0.549844\pi$
$283$	−5.24698	−0.311901	−0.155950	+	0.987765i	$0.549844\pi$
$284$	−13.2174	−0.784311
$285$	0	0
$286$	12.4252	0.734717
$287$	−1.10992	−0.0655163
$288$	0	0
$289$	3.19567	0.187981
$290$	0	0
$291$	0	0
$292$	−10.1371	−0.593227
$293$	−6.76377	−0.395144	−0.197572	−	0.980288i	$-0.563306\pi$
$293$	−6.76377	−0.395144	−0.197572	+	0.980288i	$0.563306\pi$
$294$	0	0
$295$	−8.64609	−0.503395
$296$	8.35690	0.485735
$297$	0	0
$298$	−8.28083	−0.479696
$299$	12.9782	0.750550
$300$	0	0
$301$	−1.21552	−0.0700614
$302$	−3.35019	−0.192782
$303$	0	0
$304$	−6.71917	−0.385371
$305$	1.13706	0.0651081
$306$	0	0
$307$	4.51812	0.257863	0.128931	−	0.991654i	$-0.458845\pi$
$307$	4.51812	0.257863	0.128931	+	0.991654i	$0.458845\pi$
$308$	3.17629	0.180986
$309$	0	0
$310$	2.06100	0.117057
$311$	12.6165	0.715419	0.357709	−	0.933833i	$-0.383558\pi$
$311$	12.6165	0.715419	0.357709	+	0.933833i	$0.383558\pi$
$312$	0	0
$313$	19.2403	1.08752	0.543762	−	0.839239i	$-0.316999\pi$
$313$	19.2403	1.08752	0.543762	+	0.839239i	$0.316999\pi$
$314$	−8.11721	−0.458081
$315$	0	0
$316$	8.40581	0.472864
$317$	−2.86831	−0.161101	−0.0805503	−	0.996751i	$-0.525668\pi$
$317$	−2.86831	−0.161101	−0.0805503	+	0.996751i	$0.525668\pi$
$318$	0	0
$319$	0	0
$320$	2.51275	0.140467
$321$	0	0
$322$	−0.364666	−0.0203220
$323$	10.5918	0.589343
$324$	0	0
$325$	25.5569	1.41764
$326$	5.62565	0.311576
$327$	0	0
$328$	5.26205	0.290548
$329$	2.30021	0.126815
$330$	0	0
$331$	3.13408	0.172265	0.0861323	−	0.996284i	$-0.472549\pi$
$331$	3.13408	0.172265	0.0861323	+	0.996284i	$0.472549\pi$
$332$	8.02715	0.440547
$333$	0	0
$334$	0.354503	0.0193975
$335$	1.60760	0.0878324
$336$	0	0
$337$	4.98254	0.271416	0.135708	−	0.990749i	$-0.456669\pi$
$337$	4.98254	0.271416	0.135708	+	0.990749i	$0.456669\pi$
$338$	8.43535	0.458823
$339$	0	0
$340$	−5.60388	−0.303913
$341$	33.0519	1.78986
$342$	0	0
$343$	−4.95108	−0.267333
$344$	5.76271	0.310704
$345$	0	0
$346$	4.07367	0.219002
$347$	−20.1172	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$347$	−20.1172	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$348$	0	0
$349$	20.4892	1.09676	0.548380	−	0.836229i	$-0.315245\pi$
$349$	20.4892	1.09676	0.548380	+	0.836229i	$0.315245\pi$
$350$	−0.718104	−0.0383843
$351$	0	0
$352$	−22.9801	−1.22485
$353$	−17.9976	−0.957916	−0.478958	−	0.877838i	$-0.658985\pi$
$353$	−17.9976	−0.957916	−0.478958	+	0.877838i	$0.658985\pi$
$354$	0	0
$355$	−5.07606	−0.269410
$356$	−10.2349	−0.542449
$357$	0	0
$358$	−1.51573	−0.0801088
$359$	−23.6286	−1.24707	−0.623536	−	0.781795i	$-0.714304\pi$
$359$	−23.6286	−1.24707	−0.623536	+	0.781795i	$0.714304\pi$
$360$	0	0
$361$	−13.4450	−0.707634
$362$	5.64071	0.296469
$363$	0	0
$364$	3.63533	0.190543
$365$	−3.89307	−0.203772
$366$	0	0
$367$	−29.6431	−1.54736	−0.773679	−	0.633578i	$-0.781586\pi$
$367$	−29.6431	−1.54736	−0.773679	+	0.633578i	$0.781586\pi$
$368$	−6.54527	−0.341196
$369$	0	0
$370$	1.52111	0.0790786
$371$	1.67456	0.0869390
$372$	0	0
$373$	−25.1879	−1.30418	−0.652090	−	0.758142i	$-0.726108\pi$
$373$	−25.1879	−1.30418	−0.652090	+	0.758142i	$0.726108\pi$
$374$	9.87800	0.510779
$375$	0	0
$376$	−10.9051	−0.562390
$377$	0	0
$378$	0	0
$379$	−26.9071	−1.38212	−0.691062	−	0.722796i	$-0.742857\pi$
$379$	−26.9071	−1.38212	−0.691062	+	0.722796i	$0.742857\pi$
$380$	−2.93900	−0.150768
$381$	0	0
$382$	4.74871	0.242965
$383$	19.6692	1.00505	0.502524	−	0.864563i	$-0.332405\pi$
$383$	19.6692	1.00505	0.502524	+	0.864563i	$0.332405\pi$
$384$	0	0
$385$	1.21983	0.0621684
$386$	10.1153	0.514855
$387$	0	0
$388$	0.325437	0.0165216
$389$	−24.8552	−1.26021	−0.630103	−	0.776511i	$-0.716988\pi$
$389$	−24.8552	−1.26021	−0.630103	+	0.776511i	$0.716988\pi$
$390$	0	0
$391$	10.3177	0.521787
$392$	11.6286	0.587334
$393$	0	0
$394$	8.70410	0.438506
$395$	3.22819	0.162428
$396$	0	0
$397$	−3.99462	−0.200484	−0.100242	−	0.994963i	$-0.531962\pi$
$397$	−3.99462	−0.200484	−0.100242	+	0.994963i	$0.531962\pi$
$398$	−0.389420	−0.0195198
$399$	0	0
$400$	−12.8890	−0.644451
$401$	24.9071	1.24380	0.621900	−	0.783097i	$-0.286361\pi$
$401$	24.9071	1.24380	0.621900	+	0.783097i	$0.286361\pi$
$402$	0	0
$403$	37.8286	1.88438
$404$	5.81940	0.289526
$405$	0	0
$406$	0	0
$407$	24.3937	1.20915
$408$	0	0
$409$	0.283224	0.0140045	0.00700227	−	0.999975i	$-0.497771\pi$
$409$	0.283224	0.0140045	0.00700227	+	0.999975i	$0.497771\pi$
$410$	0.957787	0.0473017
$411$	0	0
$412$	24.6015	1.21203
$413$	4.45904	0.219415
$414$	0	0
$415$	3.08277	0.151327
$416$	−26.3013	−1.28953
$417$	0	0
$418$	5.18060	0.253392
$419$	26.4886	1.29405	0.647026	−	0.762468i	$-0.276013\pi$
$419$	26.4886	1.29405	0.647026	+	0.762468i	$0.276013\pi$
$420$	0	0
$421$	−17.5821	−0.856899	−0.428450	−	0.903566i	$-0.640940\pi$
$421$	−17.5821	−0.856899	−0.428450	+	0.903566i	$0.640940\pi$
$422$	−8.13036	−0.395780
$423$	0	0
$424$	−7.93900	−0.385552
$425$	20.3177	0.985552
$426$	0	0
$427$	−0.586417	−0.0283787
$428$	29.2717	1.41490
$429$	0	0
$430$	1.04892	0.0505833
$431$	27.7952	1.33885	0.669425	−	0.742880i	$-0.266541\pi$
$431$	27.7952	1.33885	0.669425	+	0.742880i	$0.266541\pi$
$432$	0	0
$433$	5.87561	0.282364	0.141182	−	0.989984i	$-0.454910\pi$
$433$	5.87561	0.282364	0.141182	+	0.989984i	$0.454910\pi$
$434$	−1.06292	−0.0510217
$435$	0	0
$436$	−9.85086	−0.471770
$437$	5.41119	0.258852
$438$	0	0
$439$	−15.7409	−0.751274	−0.375637	−	0.926767i	$-0.622576\pi$
$439$	−15.7409	−0.751274	−0.375637	+	0.926767i	$0.622576\pi$
$440$	−5.78315	−0.275701
$441$	0	0
$442$	11.3056	0.537752
$443$	−6.73855	−0.320158	−0.160079	−	0.987104i	$-0.551175\pi$
$443$	−6.73855	−0.320158	−0.160079	+	0.987104i	$0.551175\pi$
$444$	0	0
$445$	−3.93064	−0.186330
$446$	−0.809234	−0.0383183
$447$	0	0
$448$	−1.29590	−0.0612254
$449$	−12.3201	−0.581420	−0.290710	−	0.956811i	$-0.593891\pi$
$449$	−12.3201	−0.581420	−0.290710	+	0.956811i	$0.593891\pi$
$450$	0	0
$451$	15.3599	0.723268
$452$	−19.2349	−0.904733
$453$	0	0
$454$	6.16660	0.289413
$455$	1.39612	0.0654513
$456$	0	0
$457$	13.6625	0.639104	0.319552	−	0.947569i	$-0.396468\pi$
$457$	13.6625	0.639104	0.319552	+	0.947569i	$0.396468\pi$
$458$	−5.68425	−0.265608
$459$	0	0
$460$	−2.86294	−0.133485
$461$	−11.6082	−0.540647	−0.270324	−	0.962770i	$-0.587131\pi$
$461$	−11.6082	−0.540647	−0.270324	+	0.962770i	$0.587131\pi$
$462$	0	0
$463$	−7.24267	−0.336595	−0.168298	−	0.985736i	$-0.553827\pi$
$463$	−7.24267	−0.336595	−0.168298	+	0.985736i	$0.553827\pi$
$464$	0	0
$465$	0	0
$466$	3.94571	0.182781
$467$	2.06339	0.0954824	0.0477412	−	0.998860i	$-0.484798\pi$
$467$	2.06339	0.0954824	0.0477412	+	0.998860i	$0.484798\pi$
$468$	0	0
$469$	−0.829085	−0.0382836
$470$	−1.98493	−0.0915582
$471$	0	0
$472$	−21.1400	−0.973050
$473$	16.8213	0.773445
$474$	0	0
$475$	10.6558	0.488921
$476$	2.89008	0.132467
$477$	0	0
$478$	11.3773	0.520387
$479$	3.88902	0.177694	0.0888469	−	0.996045i	$-0.471682\pi$
$479$	3.88902	0.177694	0.0888469	+	0.996045i	$0.471682\pi$
$480$	0	0
$481$	27.9191	1.27300
$482$	4.33081	0.197263
$483$	0	0
$484$	−24.1347	−1.09703
$485$	0.124982	0.00567513
$486$	0	0
$487$	9.84548	0.446141	0.223071	−	0.974802i	$-0.428392\pi$
$487$	9.84548	0.446141	0.223071	+	0.974802i	$0.428392\pi$
$488$	2.78017	0.125852
$489$	0	0
$490$	2.11662	0.0956192
$491$	−7.79118	−0.351611	−0.175806	−	0.984425i	$-0.556253\pi$
$491$	−7.79118	−0.351611	−0.175806	+	0.984425i	$0.556253\pi$
$492$	0	0
$493$	0	0
$494$	5.92931	0.266772
$495$	0	0
$496$	−19.0780	−0.856627
$497$	2.61788	0.117428
$498$	0	0
$499$	−20.5652	−0.920626	−0.460313	−	0.887757i	$-0.652263\pi$
$499$	−20.5652	−0.920626	−0.460313	+	0.887757i	$0.652263\pi$
$500$	−11.8726	−0.530960
$501$	0	0
$502$	4.34375	0.193871
$503$	−8.23490	−0.367176	−0.183588	−	0.983003i	$-0.558771\pi$
$503$	−8.23490	−0.367176	−0.183588	+	0.983003i	$0.558771\pi$
$504$	0	0
$505$	2.23490	0.0994517
$506$	5.04652	0.224345
$507$	0	0
$508$	1.86831	0.0828930
$509$	−7.91617	−0.350878	−0.175439	−	0.984490i	$-0.556134\pi$
$509$	−7.91617	−0.350878	−0.175439	+	0.984490i	$0.556134\pi$
$510$	0	0
$511$	2.00777	0.0888185
$512$	22.9119	1.01257
$513$	0	0
$514$	−7.27950	−0.321085
$515$	9.44803	0.416330
$516$	0	0
$517$	−31.8321	−1.39997
$518$	−0.784479	−0.0344680
$519$	0	0
$520$	−6.61894	−0.290260
$521$	3.52542	0.154451	0.0772257	−	0.997014i	$-0.475394\pi$
$521$	3.52542	0.154451	0.0772257	+	0.997014i	$0.475394\pi$
$522$	0	0
$523$	10.0301	0.438587	0.219294	−	0.975659i	$-0.429625\pi$
$523$	10.0301	0.438587	0.219294	+	0.975659i	$0.429625\pi$
$524$	14.4547	0.631458
$525$	0	0
$526$	−10.5579	−0.460348
$527$	30.0737	1.31003
$528$	0	0
$529$	−17.7289	−0.770820
$530$	−1.44504	−0.0627686
$531$	0	0
$532$	1.51573	0.0657152
$533$	17.5797	0.761462
$534$	0	0
$535$	11.2416	0.486017
$536$	3.93064	0.169778
$537$	0	0
$538$	−11.2838	−0.486480
$539$	33.9439	1.46207
$540$	0	0
$541$	8.22414	0.353584	0.176792	−	0.984248i	$-0.443428\pi$
$541$	8.22414	0.353584	0.176792	+	0.984248i	$0.443428\pi$
$542$	−0.535581	−0.0230052
$543$	0	0
$544$	−20.9095	−0.896486
$545$	−3.78315	−0.162052
$546$	0	0
$547$	25.8629	1.10582	0.552910	−	0.833241i	$-0.313517\pi$
$547$	25.8629	1.10582	0.552910	+	0.833241i	$0.313517\pi$
$548$	31.0519	1.32647
$549$	0	0
$550$	9.93767	0.423744
$551$	0	0
$552$	0	0
$553$	−1.66487	−0.0707977
$554$	4.74632	0.201652
$555$	0	0
$556$	30.0344	1.27374
$557$	−23.0097	−0.974952	−0.487476	−	0.873136i	$-0.662082\pi$
$557$	−23.0097	−0.974952	−0.487476	+	0.873136i	$0.662082\pi$
$558$	0	0
$559$	19.2524	0.814288
$560$	−0.704103	−0.0297538
$561$	0	0
$562$	7.24506	0.305614
$563$	−43.1159	−1.81712	−0.908559	−	0.417757i	$-0.862816\pi$
$563$	−43.1159	−1.81712	−0.908559	+	0.417757i	$0.862816\pi$
$564$	0	0
$565$	−7.38703	−0.310775
$566$	−2.33513	−0.0981527
$567$	0	0
$568$	−12.4112	−0.520762
$569$	24.3123	1.01922	0.509612	−	0.860404i	$-0.329789\pi$
$569$	24.3123	1.01922	0.509612	+	0.860404i	$0.329789\pi$
$570$	0	0
$571$	18.4886	0.773723	0.386862	−	0.922138i	$-0.373559\pi$
$571$	18.4886	0.773723	0.386862	+	0.922138i	$0.373559\pi$
$572$	−50.3086	−2.10351
$573$	0	0
$574$	−0.493959	−0.0206175
$575$	10.3800	0.432876
$576$	0	0
$577$	−37.8582	−1.57606	−0.788028	−	0.615640i	$-0.788898\pi$
$577$	−37.8582	−1.57606	−0.788028	+	0.615640i	$0.788898\pi$
$578$	1.42221	0.0591560
$579$	0	0
$580$	0	0
$581$	−1.58987	−0.0659591
$582$	0	0
$583$	−23.1739	−0.959765
$584$	−9.51871	−0.393887
$585$	0	0
$586$	−3.01016	−0.124349
$587$	14.4354	0.595811	0.297905	−	0.954595i	$-0.403712\pi$
$587$	14.4354	0.595811	0.297905	+	0.954595i	$0.403712\pi$
$588$	0	0
$589$	15.7724	0.649890
$590$	−3.84787	−0.158414
$591$	0	0
$592$	−14.0804	−0.578700
$593$	13.0019	0.533925	0.266962	−	0.963707i	$-0.413980\pi$
$593$	13.0019	0.533925	0.266962	+	0.963707i	$0.413980\pi$
$594$	0	0
$595$	1.10992	0.0455021
$596$	33.5284	1.37338
$597$	0	0
$598$	5.77586	0.236192
$599$	−12.0935	−0.494128	−0.247064	−	0.968999i	$-0.579466\pi$
$599$	−12.0935	−0.494128	−0.247064	+	0.968999i	$0.579466\pi$
$600$	0	0
$601$	−22.3472	−0.911562	−0.455781	−	0.890092i	$-0.650640\pi$
$601$	−22.3472	−0.911562	−0.455781	+	0.890092i	$0.650640\pi$
$602$	−0.540958	−0.0220478
$603$	0	0
$604$	13.5646	0.551938
$605$	−9.26875	−0.376828
$606$	0	0
$607$	41.4687	1.68316	0.841582	−	0.540129i	$-0.181625\pi$
$607$	41.4687	1.68316	0.841582	+	0.540129i	$0.181625\pi$
$608$	−10.9661	−0.444736
$609$	0	0
$610$	0.506041	0.0204890
$611$	−36.4325	−1.47390
$612$	0	0
$613$	25.7845	1.04143	0.520713	−	0.853732i	$-0.325666\pi$
$613$	25.7845	1.04143	0.520713	+	0.853732i	$0.325666\pi$
$614$	2.01075	0.0811474
$615$	0	0
$616$	2.98254	0.120170
$617$	−8.86533	−0.356905	−0.178452	−	0.983949i	$-0.557109\pi$
$617$	−8.86533	−0.356905	−0.178452	+	0.983949i	$0.557109\pi$
$618$	0	0
$619$	29.4513	1.18375	0.591873	−	0.806031i	$-0.298389\pi$
$619$	29.4513	1.18375	0.591873	+	0.806031i	$0.298389\pi$
$620$	−8.34481	−0.335136
$621$	0	0
$622$	5.61489	0.225137
$623$	2.02715	0.0812159
$624$	0	0
$625$	18.0459	0.721837
$626$	8.56273	0.342235
$627$	0	0
$628$	32.8659	1.31149
$629$	22.1957	0.884999
$630$	0	0
$631$	23.7711	0.946311	0.473156	−	0.880979i	$-0.343115\pi$
$631$	23.7711	0.946311	0.473156	+	0.880979i	$0.343115\pi$
$632$	7.89307	0.313969
$633$	0	0
$634$	−1.27652	−0.0506971
$635$	0.717513	0.0284736
$636$	0	0
$637$	38.8495	1.53927
$638$	0	0
$639$	0	0
$640$	7.55794	0.298754
$641$	−28.4655	−1.12432	−0.562160	−	0.827029i	$-0.690029\pi$
$641$	−28.4655	−1.12432	−0.562160	+	0.827029i	$0.690029\pi$
$642$	0	0
$643$	−19.3467	−0.762961	−0.381480	−	0.924377i	$-0.624586\pi$
$643$	−19.3467	−0.762961	−0.381480	+	0.924377i	$0.624586\pi$
$644$	1.47650	0.0581823
$645$	0	0
$646$	4.71379	0.185462
$647$	−22.7144	−0.892995	−0.446497	−	0.894785i	$-0.647329\pi$
$647$	−22.7144	−0.892995	−0.446497	+	0.894785i	$0.647329\pi$
$648$	0	0
$649$	−61.7077	−2.42224
$650$	11.3739	0.446120
$651$	0	0
$652$	−22.7778	−0.892046
$653$	−22.9312	−0.897368	−0.448684	−	0.893690i	$-0.648107\pi$
$653$	−22.9312	−0.897368	−0.448684	+	0.893690i	$0.648107\pi$
$654$	0	0
$655$	5.55124	0.216905
$656$	−8.86592	−0.346156
$657$	0	0
$658$	1.02369	0.0399076
$659$	19.1782	0.747077	0.373539	−	0.927615i	$-0.378144\pi$
$659$	19.1782	0.747077	0.373539	+	0.927615i	$0.378144\pi$
$660$	0	0
$661$	−3.38644	−0.131717	−0.0658585	−	0.997829i	$-0.520979\pi$
$661$	−3.38644	−0.131717	−0.0658585	+	0.997829i	$0.520979\pi$
$662$	1.39480	0.0542103
$663$	0	0
$664$	7.53750	0.292512
$665$	0.582105	0.0225731
$666$	0	0
$667$	0	0
$668$	−1.43535	−0.0555355
$669$	0	0
$670$	0.715448	0.0276402
$671$	8.11529	0.313287
$672$	0	0
$673$	4.77048	0.183888	0.0919442	−	0.995764i	$-0.470692\pi$
$673$	4.77048	0.183888	0.0919442	+	0.995764i	$0.470692\pi$
$674$	2.21744	0.0854126
$675$	0	0
$676$	−34.1540	−1.31362
$677$	43.0901	1.65609	0.828043	−	0.560665i	$-0.189454\pi$
$677$	43.0901	1.65609	0.828043	+	0.560665i	$0.189454\pi$
$678$	0	0
$679$	−0.0644568	−0.00247362
$680$	−5.26205	−0.201790
$681$	0	0
$682$	14.7095	0.563255
$683$	−23.4330	−0.896637	−0.448319	−	0.893874i	$-0.647977\pi$
$683$	−23.4330	−0.896637	−0.448319	+	0.893874i	$0.647977\pi$
$684$	0	0
$685$	11.9253	0.455641
$686$	−2.20344	−0.0841277
$687$	0	0
$688$	−9.70948	−0.370170
$689$	−26.5230	−1.01045
$690$	0	0
$691$	43.1377	1.64103	0.820517	−	0.571622i	$-0.193686\pi$
$691$	43.1377	1.64103	0.820517	+	0.571622i	$0.193686\pi$
$692$	−16.4940	−0.627006
$693$	0	0
$694$	−8.95300	−0.339851
$695$	11.5345	0.437529
$696$	0	0
$697$	13.9758	0.529373
$698$	9.11854	0.345142
$699$	0	0
$700$	2.90754	0.109895
$701$	−4.23085	−0.159797	−0.0798985	−	0.996803i	$-0.525460\pi$
$701$	−4.23085	−0.159797	−0.0798985	+	0.996803i	$0.525460\pi$
$702$	0	0
$703$	11.6407	0.439038
$704$	17.9336	0.675899
$705$	0	0
$706$	−8.00969	−0.301449
$707$	−1.15260	−0.0433481
$708$	0	0
$709$	14.3037	0.537185	0.268593	−	0.963254i	$-0.413441\pi$
$709$	14.3037	0.537185	0.268593	+	0.963254i	$0.413441\pi$
$710$	−2.25906	−0.0847811
$711$	0	0
$712$	−9.61058	−0.360172
$713$	15.3642	0.575393
$714$	0	0
$715$	−19.3207	−0.722551
$716$	6.13706	0.229353
$717$	0	0
$718$	−10.5157	−0.392444
$719$	23.4407	0.874192	0.437096	−	0.899415i	$-0.356007\pi$
$719$	23.4407	0.874192	0.437096	+	0.899415i	$0.356007\pi$
$720$	0	0
$721$	−4.87263	−0.181466
$722$	−5.98361	−0.222687
$723$	0	0
$724$	−22.8388	−0.848796
$725$	0	0
$726$	0	0
$727$	51.9976	1.92848	0.964242	−	0.265022i	$-0.0853794\pi$
$727$	51.9976	1.92848	0.964242	+	0.265022i	$0.0853794\pi$
$728$	3.41358	0.126516
$729$	0	0
$730$	−1.73258	−0.0641256
$731$	15.3056	0.566098
$732$	0	0
$733$	34.1903	1.26285	0.631424	−	0.775438i	$-0.282471\pi$
$733$	34.1903	1.26285	0.631424	+	0.775438i	$0.282471\pi$
$734$	−13.1924	−0.486941
$735$	0	0
$736$	−10.6823	−0.393756
$737$	11.4735	0.422632
$738$	0	0
$739$	39.7730	1.46307	0.731537	−	0.681802i	$-0.238804\pi$
$739$	39.7730	1.46307	0.731537	+	0.681802i	$0.238804\pi$
$740$	−6.15883	−0.226403
$741$	0	0
$742$	0.745251	0.0273590
$743$	−7.17821	−0.263343	−0.131672	−	0.991293i	$-0.542034\pi$
$743$	−7.17821	−0.263343	−0.131672	+	0.991293i	$0.542034\pi$
$744$	0	0
$745$	12.8763	0.471753
$746$	−11.2097	−0.410415
$747$	0	0
$748$	−39.9952	−1.46237
$749$	−5.79763	−0.211841
$750$	0	0
$751$	−27.1685	−0.991393	−0.495697	−	0.868496i	$-0.665087\pi$
$751$	−27.1685	−0.991393	−0.495697	+	0.868496i	$0.665087\pi$
$752$	18.3739	0.670026
$753$	0	0
$754$	0	0
$755$	5.20941	0.189590
$756$	0	0
$757$	23.6926	0.861123	0.430561	−	0.902561i	$-0.358316\pi$
$757$	23.6926	0.861123	0.430561	+	0.902561i	$0.358316\pi$
$758$	−11.9748	−0.434943
$759$	0	0
$760$	−2.75973	−0.100106
$761$	14.2543	0.516717	0.258359	−	0.966049i	$-0.416818\pi$
$761$	14.2543	0.516717	0.258359	+	0.966049i	$0.416818\pi$
$762$	0	0
$763$	1.95108	0.0706339
$764$	−19.2271	−0.695613
$765$	0	0
$766$	8.75361	0.316281
$767$	−70.6258	−2.55015
$768$	0	0
$769$	−44.7066	−1.61216	−0.806081	−	0.591805i	$-0.798415\pi$
$769$	−44.7066	−1.61216	−0.806081	+	0.591805i	$0.798415\pi$
$770$	0.542877	0.0195639
$771$	0	0
$772$	−40.9560	−1.47404
$773$	−22.9487	−0.825407	−0.412704	−	0.910865i	$-0.635415\pi$
$773$	−22.9487	−0.825407	−0.412704	+	0.910865i	$0.635415\pi$
$774$	0	0
$775$	30.2553	1.08680
$776$	0.305586	0.0109699
$777$	0	0
$778$	−11.0616	−0.396577
$779$	7.32975	0.262616
$780$	0	0
$781$	−36.2282	−1.29635
$782$	4.59179	0.164202
$783$	0	0
$784$	−19.5929	−0.699745
$785$	12.6219	0.450496
$786$	0	0
$787$	14.3327	0.510907	0.255453	−	0.966821i	$-0.417775\pi$
$787$	14.3327	0.510907	0.255453	+	0.966821i	$0.417775\pi$
$788$	−35.2422	−1.25545
$789$	0	0
$790$	1.43668	0.0511148
$791$	3.80971	0.135458
$792$	0	0
$793$	9.28813	0.329831
$794$	−1.77777	−0.0630909
$795$	0	0
$796$	1.57673	0.0558857
$797$	−10.1933	−0.361064	−0.180532	−	0.983569i	$-0.557782\pi$
$797$	−10.1933	−0.361064	−0.180532	+	0.983569i	$0.557782\pi$
$798$	0	0
$799$	−28.9638	−1.02466
$800$	−21.0358	−0.743727
$801$	0	0
$802$	11.0847	0.391414
$803$	−27.7851	−0.980514
$804$	0	0
$805$	0.567040	0.0199855
$806$	16.8353	0.592999
$807$	0	0
$808$	5.46442	0.192238
$809$	−8.97285	−0.315469	−0.157734	−	0.987482i	$-0.550419\pi$
$809$	−8.97285	−0.315469	−0.157734	+	0.987482i	$0.550419\pi$
$810$	0	0
$811$	−28.5628	−1.00298	−0.501489	−	0.865164i	$-0.667214\pi$
$811$	−28.5628	−1.00298	−0.501489	+	0.865164i	$0.667214\pi$
$812$	0	0
$813$	0	0
$814$	10.8562	0.380511
$815$	−8.74764	−0.306417
$816$	0	0
$817$	8.02715	0.280834
$818$	0.126047	0.00440712
$819$	0	0
$820$	−3.87800	−0.135426
$821$	−11.3351	−0.395599	−0.197799	−	0.980243i	$-0.563379\pi$
$821$	−11.3351	−0.395599	−0.197799	+	0.980243i	$0.563379\pi$
$822$	0	0
$823$	5.67217	0.197719	0.0988597	−	0.995101i	$-0.468480\pi$
$823$	5.67217	0.197719	0.0988597	+	0.995101i	$0.468480\pi$
$824$	23.1008	0.804755
$825$	0	0
$826$	1.98446	0.0690482
$827$	2.90323	0.100955	0.0504776	−	0.998725i	$-0.483926\pi$
$827$	2.90323	0.100955	0.0504776	+	0.998725i	$0.483926\pi$
$828$	0	0
$829$	−45.2137	−1.57034	−0.785169	−	0.619282i	$-0.787424\pi$
$829$	−45.2137	−1.57034	−0.785169	+	0.619282i	$0.787424\pi$
$830$	1.37196	0.0476215
$831$	0	0
$832$	20.5254	0.711591
$833$	30.8853	1.07011
$834$	0	0
$835$	−0.551237	−0.0190764
$836$	−20.9758	−0.725464
$837$	0	0
$838$	11.7885	0.407228
$839$	45.6256	1.57517	0.787586	−	0.616205i	$-0.211331\pi$
$839$	45.6256	1.57517	0.787586	+	0.616205i	$0.211331\pi$
$840$	0	0
$841$	0	0
$842$	−7.82477	−0.269659
$843$	0	0
$844$	32.9191	1.13312
$845$	−13.1166	−0.451225
$846$	0	0
$847$	4.78017	0.164248
$848$	13.3763	0.459343
$849$	0	0
$850$	9.04221	0.310145
$851$	11.3394	0.388711
$852$	0	0
$853$	−36.9288	−1.26442	−0.632210	−	0.774797i	$-0.717852\pi$
$853$	−36.9288	−1.26442	−0.632210	+	0.774797i	$0.717852\pi$
$854$	−0.260980	−0.00893056
$855$	0	0
$856$	27.4862	0.939459
$857$	−35.5502	−1.21437	−0.607185	−	0.794560i	$-0.707701\pi$
$857$	−35.5502	−1.21437	−0.607185	+	0.794560i	$0.707701\pi$
$858$	0	0
$859$	42.3812	1.44603	0.723014	−	0.690834i	$-0.242756\pi$
$859$	42.3812	1.44603	0.723014	+	0.690834i	$0.242756\pi$
$860$	−4.24698	−0.144821
$861$	0	0
$862$	12.3700	0.421325
$863$	−49.7797	−1.69452	−0.847260	−	0.531178i	$-0.821750\pi$
$863$	−49.7797	−1.69452	−0.847260	+	0.531178i	$0.821750\pi$
$864$	0	0
$865$	−6.33439	−0.215376
$866$	2.61489	0.0888577
$867$	0	0
$868$	4.30367	0.146076
$869$	23.0398	0.781572
$870$	0	0
$871$	13.1317	0.444950
$872$	−9.24996	−0.313243
$873$	0	0
$874$	2.40821	0.0814588
$875$	2.35152	0.0794959
$876$	0	0
$877$	21.8931	0.739276	0.369638	−	0.929176i	$-0.379482\pi$
$877$	21.8931	0.739276	0.369638	+	0.929176i	$0.379482\pi$
$878$	−7.00538	−0.236420
$879$	0	0
$880$	9.74392	0.328468
$881$	33.5308	1.12968	0.564841	−	0.825200i	$-0.308938\pi$
$881$	33.5308	1.12968	0.564841	+	0.825200i	$0.308938\pi$
$882$	0	0
$883$	−16.1274	−0.542729	−0.271365	−	0.962477i	$-0.587475\pi$
$883$	−16.1274	−0.542729	−0.271365	+	0.962477i	$0.587475\pi$
$884$	−45.7754	−1.53959
$885$	0	0
$886$	−2.99894	−0.100751
$887$	−52.7391	−1.77081	−0.885403	−	0.464823i	$-0.846118\pi$
$887$	−52.7391	−1.77081	−0.885403	+	0.464823i	$0.846118\pi$
$888$	0	0
$889$	−0.370042	−0.0124108
$890$	−1.74930	−0.0586367
$891$	0	0
$892$	3.27652	0.109706
$893$	−15.1903	−0.508324
$894$	0	0
$895$	2.35690	0.0787823
$896$	−3.89785	−0.130218
$897$	0	0
$898$	−5.48294	−0.182968
$899$	0	0
$900$	0	0
$901$	−21.0858	−0.702468
$902$	6.83579	0.227607
$903$	0	0
$904$	−18.0616	−0.600720
$905$	−8.77107	−0.291560
$906$	0	0
$907$	−29.8437	−0.990943	−0.495472	−	0.868624i	$-0.665005\pi$
$907$	−29.8437	−0.990943	−0.495472	+	0.868624i	$0.665005\pi$
$908$	−24.9681	−0.828594
$909$	0	0
$910$	0.621334	0.0205970
$911$	9.34050	0.309465	0.154732	−	0.987956i	$-0.450548\pi$
$911$	9.34050	0.309465	0.154732	+	0.987956i	$0.450548\pi$
$912$	0	0
$913$	22.0019	0.728157
$914$	6.08038	0.201121
$915$	0	0
$916$	23.0151	0.760439
$917$	−2.86294	−0.0945425
$918$	0	0
$919$	−18.4209	−0.607649	−0.303824	−	0.952728i	$-0.598264\pi$
$919$	−18.4209	−0.607649	−0.303824	+	0.952728i	$0.598264\pi$
$920$	−2.68830	−0.0886306
$921$	0	0
$922$	−5.16613	−0.170137
$923$	−41.4639	−1.36480
$924$	0	0
$925$	22.3297	0.734198
$926$	−3.22329	−0.105924
$927$	0	0
$928$	0	0
$929$	4.84654	0.159010	0.0795050	−	0.996834i	$-0.474666\pi$
$929$	4.84654	0.159010	0.0795050	+	0.996834i	$0.474666\pi$
$930$	0	0
$931$	16.1981	0.530870
$932$	−15.9758	−0.523306
$933$	0	0
$934$	0.918296	0.0300476
$935$	−15.3599	−0.502322
$936$	0	0
$937$	−44.7144	−1.46076	−0.730378	−	0.683044i	$-0.760656\pi$
$937$	−44.7144	−1.46076	−0.730378	+	0.683044i	$0.760656\pi$
$938$	−0.368977	−0.0120475
$939$	0	0
$940$	8.03684	0.262133
$941$	−13.5851	−0.442861	−0.221431	−	0.975176i	$-0.571073\pi$
$941$	−13.5851	−0.442861	−0.221431	+	0.975176i	$0.571073\pi$
$942$	0	0
$943$	7.14005	0.232512
$944$	35.6185	1.15928
$945$	0	0
$946$	7.48619	0.243397
$947$	−15.0116	−0.487812	−0.243906	−	0.969799i	$-0.578429\pi$
$947$	−15.0116	−0.487812	−0.243906	+	0.969799i	$0.578429\pi$
$948$	0	0
$949$	−31.8006	−1.03229
$950$	4.74227	0.153859
$951$	0	0
$952$	2.71379	0.0879545
$953$	−51.8447	−1.67942	−0.839708	−	0.543038i	$-0.817274\pi$
$953$	−51.8447	−1.67942	−0.839708	+	0.543038i	$0.817274\pi$
$954$	0	0
$955$	−7.38404	−0.238942
$956$	−46.0659	−1.48988
$957$	0	0
$958$	1.73078	0.0559188
$959$	−6.15021	−0.198601
$960$	0	0
$961$	13.7832	0.444618
$962$	12.4252	0.400604
$963$	0	0
$964$	−17.5351	−0.564768
$965$	−15.7289	−0.506330
$966$	0	0
$967$	−41.6249	−1.33857	−0.669283	−	0.743007i	$-0.733399\pi$
$967$	−41.6249	−1.33857	−0.669283	+	0.743007i	$0.733399\pi$
$968$	−22.6625	−0.728400
$969$	0	0
$970$	0.0556221	0.00178592
$971$	55.7730	1.78984	0.894920	−	0.446226i	$-0.147232\pi$
$971$	55.7730	1.78984	0.894920	+	0.446226i	$0.147232\pi$
$972$	0	0
$973$	−5.94869	−0.190706
$974$	4.38165	0.140397
$975$	0	0
$976$	−4.68425	−0.149939
$977$	−48.9584	−1.56632	−0.783159	−	0.621822i	$-0.786393\pi$
$977$	−48.9584	−1.56632	−0.783159	+	0.621822i	$0.786393\pi$
$978$	0	0
$979$	−28.0532	−0.896585
$980$	−8.57002	−0.273759
$981$	0	0
$982$	−3.46740	−0.110649
$983$	−5.17331	−0.165003	−0.0825015	−	0.996591i	$-0.526291\pi$
$983$	−5.17331	−0.165003	−0.0825015	+	0.996591i	$0.526291\pi$
$984$	0	0
$985$	−13.5345	−0.431246
$986$	0	0
$987$	0	0
$988$	−24.0073	−0.763774
$989$	7.81940	0.248642
$990$	0	0
$991$	−16.8528	−0.535346	−0.267673	−	0.963510i	$-0.586255\pi$
$991$	−16.8528	−0.535346	−0.267673	+	0.963510i	$0.586255\pi$
$992$	−31.1366	−0.988588
$993$	0	0
$994$	1.16506	0.0369536
$995$	0.605531	0.0191966
$996$	0	0
$997$	38.2097	1.21011	0.605056	−	0.796183i	$-0.293151\pi$
$997$	38.2097	1.21011	0.605056	+	0.796183i	$0.293151\pi$
$998$	−9.15239	−0.289714
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.r.1.2		3
3.2	odd	2		841.2.a.e.1.2		3
29.7	even	7		261.2.k.a.136.1		6
29.25	even	7		261.2.k.a.190.1		6
29.28	even	2		7569.2.a.p.1.2		3
87.2	even	28		841.2.e.b.236.1		12
87.5	odd	14		841.2.d.a.605.1		6
87.8	even	28		841.2.e.c.267.1		12
87.11	even	28		841.2.e.c.63.1		12
87.14	even	28		841.2.e.b.196.2		12
87.17	even	4		841.2.b.c.840.4		6
87.20	odd	14		841.2.d.b.574.1		6
87.23	odd	14		841.2.d.e.645.1		6
87.26	even	28		841.2.e.d.270.2		12
87.32	even	28		841.2.e.d.270.1		12
87.35	odd	14		841.2.d.a.645.1		6
87.38	odd	14		841.2.d.c.574.1		6
87.41	even	4		841.2.b.c.840.3		6
87.44	even	28		841.2.e.b.196.1		12
87.47	even	28		841.2.e.c.63.2		12
87.50	even	28		841.2.e.c.267.2		12
87.53	odd	14		841.2.d.e.605.1		6
87.56	even	28		841.2.e.b.236.2		12
87.62	odd	14		841.2.d.d.190.1		6
87.65	odd	14		29.2.d.a.20.1	yes	6
87.68	even	28		841.2.e.d.651.1		12
87.71	odd	14		841.2.d.c.778.1		6
87.74	odd	14		841.2.d.b.778.1		6
87.77	even	28		841.2.e.d.651.2		12
87.80	odd	14		841.2.d.d.571.1		6
87.83	odd	14		29.2.d.a.16.1	✓	6
87.86	odd	2		841.2.a.f.1.2		3
348.83	even	14		464.2.u.f.161.1		6
348.239	even	14		464.2.u.f.49.1		6
435.83	even	28		725.2.r.b.74.2		12
435.152	even	28		725.2.r.b.49.2		12
435.239	odd	14		725.2.l.b.426.1		6
435.257	even	28		725.2.r.b.74.1		12
435.344	odd	14		725.2.l.b.451.1		6
435.413	even	28		725.2.r.b.49.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.d.a.16.1	✓	6	87.83	odd	14
29.2.d.a.20.1	yes	6	87.65	odd	14
261.2.k.a.136.1		6	29.7	even	7
261.2.k.a.190.1		6	29.25	even	7
464.2.u.f.49.1		6	348.239	even	14
464.2.u.f.161.1		6	348.83	even	14
725.2.l.b.426.1		6	435.239	odd	14
725.2.l.b.451.1		6	435.344	odd	14
725.2.r.b.49.1		12	435.413	even	28
725.2.r.b.49.2		12	435.152	even	28
725.2.r.b.74.1		12	435.257	even	28
725.2.r.b.74.2		12	435.83	even	28
841.2.a.e.1.2		3	3.2	odd	2
841.2.a.f.1.2		3	87.86	odd	2
841.2.b.c.840.3		6	87.41	even	4
841.2.b.c.840.4		6	87.17	even	4
841.2.d.a.605.1		6	87.5	odd	14
841.2.d.a.645.1		6	87.35	odd	14
841.2.d.b.574.1		6	87.20	odd	14
841.2.d.b.778.1		6	87.74	odd	14
841.2.d.c.574.1		6	87.38	odd	14
841.2.d.c.778.1		6	87.71	odd	14
841.2.d.d.190.1		6	87.62	odd	14
841.2.d.d.571.1		6	87.80	odd	14
841.2.d.e.605.1		6	87.53	odd	14
841.2.d.e.645.1		6	87.23	odd	14
841.2.e.b.196.1		12	87.44	even	28
841.2.e.b.196.2		12	87.14	even	28
841.2.e.b.236.1		12	87.2	even	28
841.2.e.b.236.2		12	87.56	even	28
841.2.e.c.63.1		12	87.11	even	28
841.2.e.c.63.2		12	87.47	even	28
841.2.e.c.267.1		12	87.8	even	28
841.2.e.c.267.2		12	87.50	even	28
841.2.e.d.270.1		12	87.32	even	28
841.2.e.d.270.2		12	87.26	even	28
841.2.e.d.651.1		12	87.68	even	28
841.2.e.d.651.2		12	87.77	even	28
7569.2.a.p.1.2		3	29.28	even	2
7569.2.a.r.1.2		3	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+0.445042 q^{2} -1.80194 q^{4} -0.692021 q^{5} +0.356896 q^{7} -1.69202 q^{8} +O(q^{10})\)	\(q+0.445042 q^{2} -1.80194 q^{4} -0.692021 q^{5} +0.356896 q^{7} -1.69202 q^{8} -0.307979 q^{10} -4.93900 q^{11} -5.65279 q^{13} +0.158834 q^{14} +2.85086 q^{16} -4.49396 q^{17} -2.35690 q^{19} +1.24698 q^{20} -2.19806 q^{22} -2.29590 q^{23} -4.52111 q^{25} -2.51573 q^{26} -0.643104 q^{28} -6.69202 q^{31} +4.65279 q^{32} -2.00000 q^{34} -0.246980 q^{35} -4.93900 q^{37} -1.04892 q^{38} +1.17092 q^{40} -3.10992 q^{41} -3.40581 q^{43} +8.89977 q^{44} -1.02177 q^{46} +6.44504 q^{47} -6.87263 q^{49} -2.01208 q^{50} +10.1860 q^{52} +4.69202 q^{53} +3.41789 q^{55} -0.603875 q^{56} +12.4940 q^{59} -1.64310 q^{61} -2.97823 q^{62} -3.63102 q^{64} +3.91185 q^{65} -2.32304 q^{67} +8.09783 q^{68} -0.109916 q^{70} +7.33513 q^{71} +5.62565 q^{73} -2.19806 q^{74} +4.24698 q^{76} -1.76271 q^{77} -4.66487 q^{79} -1.97285 q^{80} -1.38404 q^{82} -4.45473 q^{83} +3.10992 q^{85} -1.51573 q^{86} +8.35690 q^{88} +5.67994 q^{89} -2.01746 q^{91} +4.13706 q^{92} +2.86831 q^{94} +1.63102 q^{95} -0.180604 q^{97} -3.05861 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - q^{4} + 3 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q + q^2 - q^4 + 3 * q^5 - 3 * q^7	\( 3 q + q^{2} - q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{10} - 5 q^{11} + q^{13} - 8 q^{14} - 5 q^{16} - 4 q^{17} - 3 q^{19} - q^{20} - 11 q^{22} + 7 q^{23} + 2 q^{25} + 5 q^{26} - 6 q^{28} - 15 q^{31} - 4 q^{32} - 6 q^{34} + 4 q^{35} - 5 q^{37} + 6 q^{38} + 14 q^{40} - 10 q^{41} + 3 q^{43} + 4 q^{44} + 19 q^{47} - 4 q^{49} - 25 q^{50} + 16 q^{52} + 9 q^{53} + 16 q^{55} + 7 q^{56} + 28 q^{59} - 9 q^{61} - 12 q^{62} + 4 q^{64} + 8 q^{65} + 13 q^{67} + 6 q^{68} - q^{70} + 21 q^{71} + 5 q^{73} - 11 q^{74} + 8 q^{76} + 12 q^{77} - 15 q^{79} - 12 q^{80} + 6 q^{82} + 9 q^{83} + 10 q^{85} + 8 q^{86} + 21 q^{88} - 7 q^{89} - 22 q^{91} + 7 q^{92} + 11 q^{94} - 10 q^{95} + 11 q^{97} + 22 q^{98}+O(q^{100}) \) 3 * q + q^2 - q^4 + 3 * q^5 - 3 * q^7 - 6 * q^10 - 5 * q^11 + q^13 - 8 * q^14 - 5 * q^16 - 4 * q^17 - 3 * q^19 - q^20 - 11 * q^22 + 7 * q^23 + 2 * q^25 + 5 * q^26 - 6 * q^28 - 15 * q^31 - 4 * q^32 - 6 * q^34 + 4 * q^35 - 5 * q^37 + 6 * q^38 + 14 * q^40 - 10 * q^41 + 3 * q^43 + 4 * q^44 + 19 * q^47 - 4 * q^49 - 25 * q^50 + 16 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 + 28 * q^59 - 9 * q^61 - 12 * q^62 + 4 * q^64 + 8 * q^65 + 13 * q^67 + 6 * q^68 - q^70 + 21 * q^71 + 5 * q^73 - 11 * q^74 + 8 * q^76 + 12 * q^77 - 15 * q^79 - 12 * q^80 + 6 * q^82 + 9 * q^83 + 10 * q^85 + 8 * q^86 + 21 * q^88 - 7 * q^89 - 22 * q^91 + 7 * q^92 + 11 * q^94 - 10 * q^95 + 11 * q^97 + 22 * q^98

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