LMFDB - Embedded newform 8649.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8768.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 6x + 7 $ x^4 - 2x^3 - 5x^2 + 6*x + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.854912$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.854912 q^{2} -1.26913 q^{4} -3.62324 q^{5} -1.00000 q^{7} +2.79481 q^{8} +O(q^{10})$	$q-0.854912 q^{2} -1.26913 q^{4} -3.62324 q^{5} -1.00000 q^{7} +2.79481 q^{8} +3.09755 q^{10} -4.03746 q^{11} +4.62324 q^{13} +0.854912 q^{14} +0.148931 q^{16} -1.17157 q^{17} -7.12788 q^{19} +4.59835 q^{20} +3.45167 q^{22} +3.50464 q^{23} +8.12788 q^{25} -3.95246 q^{26} +1.26913 q^{28} -8.62868 q^{29} -5.71695 q^{32} +1.00159 q^{34} +3.62324 q^{35} +7.61940 q^{37} +6.09371 q^{38} -10.1263 q^{40} -0.666934 q^{41} -4.86588 q^{43} +5.12404 q^{44} -2.99616 q^{46} +4.41037 q^{47} -6.00000 q^{49} -6.94862 q^{50} -5.86747 q^{52} -1.17157 q^{53} +14.6287 q^{55} -2.79481 q^{56} +7.37676 q^{58} +7.04289 q^{59} +7.41965 q^{61} +4.58963 q^{64} -16.7511 q^{65} +13.9126 q^{67} +1.48687 q^{68} -3.09755 q^{70} -4.03361 q^{71} -11.0390 q^{73} -6.51392 q^{74} +9.04617 q^{76} +4.03746 q^{77} +9.07875 q^{79} -0.539612 q^{80} +0.570170 q^{82} +6.37052 q^{83} +4.24489 q^{85} +4.15990 q^{86} -11.2839 q^{88} -5.70439 q^{89} -4.62324 q^{91} -4.44783 q^{92} -3.77048 q^{94} +25.8260 q^{95} +8.07650 q^{97} +5.12947 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 6 q^{4} - 4 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 - 4 * q^7 + 12 * q^8	$ 4 q + 2 q^{2} + 6 q^{4} - 4 q^{5} - 4 q^{7} + 12 q^{8} - 10 q^{10} + 8 q^{13} - 2 q^{14} + 2 q^{16} - 16 q^{17} - 8 q^{19} - 10 q^{20} - 8 q^{22} + 4 q^{23} + 12 q^{25} + 12 q^{26} - 6 q^{28} - 8 q^{29} + 6 q^{32} - 8 q^{34} + 4 q^{35} + 24 q^{37} + 6 q^{38} - 32 q^{40} - 24 q^{41} + 8 q^{43} + 4 q^{44} - 16 q^{46} + 16 q^{47} - 24 q^{49} - 4 q^{50} + 16 q^{52} - 16 q^{53} + 32 q^{55} - 12 q^{56} + 40 q^{58} - 4 q^{59} + 8 q^{61} + 20 q^{64} - 36 q^{65} - 8 q^{67} - 40 q^{68} + 10 q^{70} - 4 q^{71} - 16 q^{73} + 4 q^{74} - 10 q^{76} - 54 q^{80} - 30 q^{82} - 12 q^{83} + 8 q^{85} - 4 q^{86} - 8 q^{88} - 28 q^{89} - 8 q^{91} - 8 q^{94} + 40 q^{95} - 12 q^{97} - 12 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 - 4 * q^7 + 12 * q^8 - 10 * q^10 + 8 * q^13 - 2 * q^14 + 2 * q^16 - 16 * q^17 - 8 * q^19 - 10 * q^20 - 8 * q^22 + 4 * q^23 + 12 * q^25 + 12 * q^26 - 6 * q^28 - 8 * q^29 + 6 * q^32 - 8 * q^34 + 4 * q^35 + 24 * q^37 + 6 * q^38 - 32 * q^40 - 24 * q^41 + 8 * q^43 + 4 * q^44 - 16 * q^46 + 16 * q^47 - 24 * q^49 - 4 * q^50 + 16 * q^52 - 16 * q^53 + 32 * q^55 - 12 * q^56 + 40 * q^58 - 4 * q^59 + 8 * q^61 + 20 * q^64 - 36 * q^65 - 8 * q^67 - 40 * q^68 + 10 * q^70 - 4 * q^71 - 16 * q^73 + 4 * q^74 - 10 * q^76 - 54 * q^80 - 30 * q^82 - 12 * q^83 + 8 * q^85 - 4 * q^86 - 8 * q^88 - 28 * q^89 - 8 * q^91 - 8 * q^94 + 40 * q^95 - 12 * q^97 - 12 * 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 $

$2$	−0.854912	−0.604514	−0.302257	−	0.953226i	$-0.597740\pi$
$2$	−0.854912	−0.604514	−0.302257	+	0.953226i	$0.597740\pi$
$3$	0	0
$3$	0	0
$4$	−1.26913	−0.634563
$5$	−3.62324	−1.62036	−0.810181	−	0.586179i	$-0.800631\pi$
$5$	−3.62324	−1.62036	−0.810181	+	0.586179i	$0.800631\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	2.79481	0.988116
$9$	0	0
$10$	3.09755	0.979532
$11$	−4.03746	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$11$	−4.03746	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$12$	0	0
$13$	4.62324	1.28226	0.641128	−	0.767434i	$-0.278467\pi$
$13$	4.62324	1.28226	0.641128	+	0.767434i	$0.278467\pi$
$14$	0.854912	0.228485
$15$	0	0
$16$	0.148931	0.0372327
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	−7.12788	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$19$	−7.12788	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$20$	4.59835	1.02822
$21$	0	0
$22$	3.45167	0.735898
$23$	3.50464	0.730768	0.365384	−	0.930857i	$-0.380938\pi$
$23$	3.50464	0.730768	0.365384	+	0.930857i	$0.380938\pi$
$24$	0	0
$25$	8.12788	1.62558
$26$	−3.95246	−0.775142
$27$	0	0
$28$	1.26913	0.239842
$29$	−8.62868	−1.60230	−0.801152	−	0.598460i	$-0.795779\pi$
$29$	−8.62868	−1.60230	−0.801152	+	0.598460i	$0.795779\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−5.71695	−1.01062
$33$	0	0
$34$	1.00159	0.171772
$35$	3.62324	0.612440
$36$	0	0
$37$	7.61940	1.25262	0.626311	−	0.779574i	$-0.284564\pi$
$37$	7.61940	1.25262	0.626311	+	0.779574i	$0.284564\pi$
$38$	6.09371	0.988530
$39$	0	0
$40$	−10.1263	−1.60111
$41$	−0.666934	−0.104158	−0.0520788	−	0.998643i	$-0.516585\pi$
$41$	−0.666934	−0.104158	−0.0520788	+	0.998643i	$0.516585\pi$
$42$	0	0
$43$	−4.86588	−0.742040	−0.371020	−	0.928625i	$-0.620992\pi$
$43$	−4.86588	−0.742040	−0.371020	+	0.928625i	$0.620992\pi$
$44$	5.12404	0.772478
$45$	0	0
$46$	−2.99616	−0.441759
$47$	4.41037	0.643319	0.321659	−	0.946855i	$-0.395759\pi$
$47$	4.41037	0.643319	0.321659	+	0.946855i	$0.395759\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−6.94862	−0.982684
$51$	0	0
$52$	−5.86747	−0.813672
$53$	−1.17157	−0.160928	−0.0804640	−	0.996758i	$-0.525640\pi$
$53$	−1.17157	−0.160928	−0.0804640	+	0.996758i	$0.525640\pi$
$54$	0	0
$55$	14.6287	1.97253
$56$	−2.79481	−0.373473
$57$	0	0
$58$	7.37676	0.968616
$59$	7.04289	0.916906	0.458453	−	0.888719i	$-0.348404\pi$
$59$	7.04289	0.916906	0.458453	+	0.888719i	$0.348404\pi$
$60$	0	0
$61$	7.41965	0.949989	0.474994	−	0.879989i	$-0.342450\pi$
$61$	7.41965	0.949989	0.474994	+	0.879989i	$0.342450\pi$
$62$	0	0
$63$	0	0
$64$	4.58963	0.573704
$65$	−16.7511	−2.07772
$66$	0	0
$67$	13.9126	1.69970	0.849848	−	0.527028i	$-0.176694\pi$
$67$	13.9126	1.69970	0.849848	+	0.527028i	$0.176694\pi$
$68$	1.48687	0.180310
$69$	0	0
$70$	−3.09755	−0.370228
$71$	−4.03361	−0.478702	−0.239351	−	0.970933i	$-0.576935\pi$
$71$	−4.03361	−0.478702	−0.239351	+	0.970933i	$0.576935\pi$
$72$	0	0
$73$	−11.0390	−1.29202	−0.646011	−	0.763328i	$-0.723564\pi$
$73$	−11.0390	−1.29202	−0.646011	+	0.763328i	$0.723564\pi$
$74$	−6.51392	−0.757227
$75$	0	0
$76$	9.04617	1.03767
$77$	4.03746	0.460111
$78$	0	0
$79$	9.07875	1.02144	0.510720	−	0.859747i	$-0.329379\pi$
$79$	9.07875	1.02144	0.510720	+	0.859747i	$0.329379\pi$
$80$	−0.539612	−0.0603304
$81$	0	0
$82$	0.570170	0.0629648
$83$	6.37052	0.699256	0.349628	−	0.936889i	$-0.386308\pi$
$83$	6.37052	0.699256	0.349628	+	0.936889i	$0.386308\pi$
$84$	0	0
$85$	4.24489	0.460423
$86$	4.15990	0.448574
$87$	0	0
$88$	−11.2839	−1.20287
$89$	−5.70439	−0.604664	−0.302332	−	0.953203i	$-0.597765\pi$
$89$	−5.70439	−0.604664	−0.302332	+	0.953203i	$0.597765\pi$
$90$	0	0
$91$	−4.62324	−0.484647
$92$	−4.44783	−0.463718
$93$	0	0
$94$	−3.77048	−0.388895
$95$	25.8260	2.64969
$96$	0	0
$97$	8.07650	0.820045	0.410022	−	0.912076i	$-0.365521\pi$
$97$	8.07650	0.820045	0.410022	+	0.912076i	$0.365521\pi$
$98$	5.12947	0.518155
$99$	0	0
$100$	−10.3153	−1.03153
$101$	12.9370	1.28727	0.643637	−	0.765331i	$-0.277425\pi$
$101$	12.9370	1.28727	0.643637	+	0.765331i	$0.277425\pi$
$102$	0	0
$103$	13.3230	1.31275	0.656376	−	0.754434i	$-0.272088\pi$
$103$	13.3230	1.31275	0.656376	+	0.754434i	$0.272088\pi$
$104$	12.9211	1.26702
$105$	0	0
$106$	1.00159	0.0972832
$107$	−9.15990	−0.885521	−0.442761	−	0.896640i	$-0.646001\pi$
$107$	−9.15990	−0.885521	−0.442761	+	0.896640i	$0.646001\pi$
$108$	0	0
$109$	17.7940	1.70436	0.852179	−	0.523251i	$-0.175281\pi$
$109$	17.7940	1.70436	0.852179	+	0.523251i	$0.175281\pi$
$110$	−12.5062	−1.19242
$111$	0	0
$112$	−0.148931	−0.0140726
$113$	−10.4970	−0.987470	−0.493735	−	0.869612i	$-0.664369\pi$
$113$	−10.4970	−0.987470	−0.493735	+	0.869612i	$0.664369\pi$
$114$	0	0
$115$	−12.6982	−1.18411
$116$	10.9509	1.01676
$117$	0	0
$118$	−6.02105	−0.554283
$119$	1.17157	0.107398
$120$	0	0
$121$	5.30104	0.481913
$122$	−6.34315	−0.574281
$123$	0	0
$124$	0	0
$125$	−11.3331	−1.01366
$126$	0	0
$127$	8.79097	0.780073	0.390036	−	0.920799i	$-0.372462\pi$
$127$	8.79097	0.780073	0.390036	+	0.920799i	$0.372462\pi$
$128$	7.51017	0.663812
$129$	0	0
$130$	14.3207	1.25601
$131$	0.117012	0.0102233	0.00511167	−	0.999987i	$-0.498373\pi$
$131$	0.117012	0.0102233	0.00511167	+	0.999987i	$0.498373\pi$
$132$	0	0
$133$	7.12788	0.618066
$134$	−11.8941	−1.02749
$135$	0	0
$136$	−3.27433	−0.280771
$137$	−12.7511	−1.08940	−0.544701	−	0.838630i	$-0.683357\pi$
$137$	−12.7511	−1.08940	−0.544701	+	0.838630i	$0.683357\pi$
$138$	0	0
$139$	5.04673	0.428058	0.214029	−	0.976827i	$-0.431341\pi$
$139$	5.04673	0.428058	0.214029	+	0.976827i	$0.431341\pi$
$140$	−4.59835	−0.388631
$141$	0	0
$142$	3.44838	0.289382
$143$	−18.6661	−1.56094
$144$	0	0
$145$	31.2638	2.59632
$146$	9.43741	0.781046
$147$	0	0
$148$	−9.66997	−0.794867
$149$	7.06225	0.578562	0.289281	−	0.957244i	$-0.406584\pi$
$149$	7.06225	0.578562	0.289281	+	0.957244i	$0.406584\pi$
$150$	0	0
$151$	9.20134	0.748795	0.374397	−	0.927268i	$-0.377850\pi$
$151$	9.20134	0.748795	0.374397	+	0.927268i	$0.377850\pi$
$152$	−19.9211	−1.61581
$153$	0	0
$154$	−3.45167	−0.278143
$155$	0	0
$156$	0	0
$157$	−11.6661	−0.931059	−0.465529	−	0.885032i	$-0.654136\pi$
$157$	−11.6661	−0.931059	−0.465529	+	0.885032i	$0.654136\pi$
$158$	−7.76153	−0.617475
$159$	0	0
$160$	20.7139	1.63758
$161$	−3.50464	−0.276204
$162$	0	0
$163$	12.9126	1.01139	0.505697	−	0.862711i	$-0.331235\pi$
$163$	12.9126	1.01139	0.505697	+	0.862711i	$0.331235\pi$
$164$	0.846423	0.0660946
$165$	0	0
$166$	−5.44623	−0.422710
$167$	13.9227	1.07737	0.538685	−	0.842507i	$-0.318921\pi$
$167$	13.9227	1.07737	0.538685	+	0.842507i	$0.318921\pi$
$168$	0	0
$169$	8.37436	0.644182
$170$	−3.62901	−0.278332
$171$	0	0
$172$	6.17542	0.470871
$173$	6.90334	0.524851	0.262426	−	0.964952i	$-0.415478\pi$
$173$	6.90334	0.524851	0.262426	+	0.964952i	$0.415478\pi$
$174$	0	0
$175$	−8.12788	−0.614410
$176$	−0.601301	−0.0453248
$177$	0	0
$178$	4.87675	0.365528
$179$	−17.0765	−1.27636	−0.638179	−	0.769888i	$-0.720312\pi$
$179$	−17.0765	−1.27636	−0.638179	+	0.769888i	$0.720312\pi$
$180$	0	0
$181$	−11.0842	−0.823881	−0.411941	−	0.911211i	$-0.635149\pi$
$181$	−11.0842	−0.823881	−0.411941	+	0.911211i	$0.635149\pi$
$182$	3.95246	0.292976
$183$	0	0
$184$	9.79481	0.722083
$185$	−27.6069	−2.02970
$186$	0	0
$187$	4.73017	0.345905
$188$	−5.59731	−0.408226
$189$	0	0
$190$	−22.0790	−1.60178
$191$	18.4096	1.33207	0.666035	−	0.745921i	$-0.267990\pi$
$191$	18.4096	1.33207	0.666035	+	0.745921i	$0.267990\pi$
$192$	0	0
$193$	−6.54753	−0.471301	−0.235651	−	0.971838i	$-0.575722\pi$
$193$	−6.54753	−0.471301	−0.235651	+	0.971838i	$0.575722\pi$
$194$	−6.90470	−0.495728
$195$	0	0
$196$	7.61475	0.543911
$197$	−12.1599	−0.866357	−0.433179	−	0.901308i	$-0.642608\pi$
$197$	−12.1599	−0.866357	−0.433179	+	0.901308i	$0.642608\pi$
$198$	0	0
$199$	−9.96254	−0.706226	−0.353113	−	0.935581i	$-0.614877\pi$
$199$	−9.96254	−0.706226	−0.353113	+	0.935581i	$0.614877\pi$
$200$	22.7159	1.60626
$201$	0	0
$202$	−11.0600	−0.778176
$203$	8.62868	0.605614
$204$	0	0
$205$	2.41646	0.168773
$206$	−11.3900	−0.793577
$207$	0	0
$208$	0.688542	0.0477418
$209$	28.7785	1.99065
$210$	0	0
$211$	0.464932	0.0320073	0.0160036	−	0.999872i	$-0.494906\pi$
$211$	0.464932	0.0320073	0.0160036	+	0.999872i	$0.494906\pi$
$212$	1.48687	0.102119
$213$	0	0
$214$	7.83091	0.535310
$215$	17.6303	1.20237
$216$	0	0
$217$	0	0
$218$	−15.2123	−1.03031
$219$	0	0
$220$	−18.5656	−1.25169
$221$	−5.41646	−0.364351
$222$	0	0
$223$	−13.9555	−0.934530	−0.467265	−	0.884117i	$-0.654761\pi$
$223$	−13.9555	−0.934530	−0.467265	+	0.884117i	$0.654761\pi$
$224$	5.71695	0.381980
$225$	0	0
$226$	8.97397	0.596940
$227$	−18.1202	−1.20268	−0.601340	−	0.798993i	$-0.705366\pi$
$227$	−18.1202	−1.20268	−0.601340	+	0.798993i	$0.705366\pi$
$228$	0	0
$229$	17.4462	1.15288	0.576440	−	0.817140i	$-0.304442\pi$
$229$	17.4462	1.15288	0.576440	+	0.817140i	$0.304442\pi$
$230$	10.8558	0.715810
$231$	0	0
$232$	−24.1155	−1.58326
$233$	19.2457	1.26083	0.630413	−	0.776260i	$-0.282885\pi$
$233$	19.2457	1.26083	0.630413	+	0.776260i	$0.282885\pi$
$234$	0	0
$235$	−15.9798	−1.04241
$236$	−8.93831	−0.581834
$237$	0	0
$238$	−1.00159	−0.0649235
$239$	−9.79017	−0.633273	−0.316637	−	0.948547i	$-0.602554\pi$
$239$	−9.79017	−0.633273	−0.316637	+	0.948547i	$0.602554\pi$
$240$	0	0
$241$	21.7615	1.40178	0.700892	−	0.713268i	$-0.252786\pi$
$241$	21.7615	1.40178	0.700892	+	0.713268i	$0.252786\pi$
$242$	−4.53193	−0.291323
$243$	0	0
$244$	−9.41646	−0.602827
$245$	21.7394	1.38888
$246$	0	0
$247$	−32.9539	−2.09681
$248$	0	0
$249$	0	0
$250$	9.68877	0.612772
$251$	−4.02659	−0.254156	−0.127078	−	0.991893i	$-0.540560\pi$
$251$	−4.02659	−0.254156	−0.127078	+	0.991893i	$0.540560\pi$
$252$	0	0
$253$	−14.1498	−0.889592
$254$	−7.51551	−0.471565
$255$	0	0
$256$	−15.5998	−0.974987
$257$	−16.6997	−1.04170	−0.520851	−	0.853648i	$-0.674385\pi$
$257$	−16.6997	−1.04170	−0.520851	+	0.853648i	$0.674385\pi$
$258$	0	0
$259$	−7.61940	−0.473446
$260$	21.2593	1.31844
$261$	0	0
$262$	−0.100035	−0.00618016
$263$	−6.16230	−0.379983	−0.189992	−	0.981786i	$-0.560846\pi$
$263$	−6.16230	−0.379983	−0.189992	+	0.981786i	$0.560846\pi$
$264$	0	0
$265$	4.24489	0.260762
$266$	−6.09371	−0.373629
$267$	0	0
$268$	−17.6569	−1.07856
$269$	18.2271	1.11133	0.555664	−	0.831407i	$-0.312464\pi$
$269$	18.2271	1.11133	0.555664	+	0.831407i	$0.312464\pi$
$270$	0	0
$271$	2.75577	0.167401	0.0837005	−	0.996491i	$-0.473326\pi$
$271$	2.75577	0.167401	0.0837005	+	0.996491i	$0.473326\pi$
$272$	−0.174483	−0.0105796
$273$	0	0
$274$	10.9011	0.658559
$275$	−32.8160	−1.97888
$276$	0	0
$277$	4.06179	0.244049	0.122025	−	0.992527i	$-0.461061\pi$
$277$	4.06179	0.244049	0.122025	+	0.992527i	$0.461061\pi$
$278$	−4.31451	−0.258767
$279$	0	0
$280$	10.1263	0.605161
$281$	−3.09586	−0.184683	−0.0923417	−	0.995727i	$-0.529435\pi$
$281$	−3.09586	−0.184683	−0.0923417	+	0.995727i	$0.529435\pi$
$282$	0	0
$283$	8.49297	0.504854	0.252427	−	0.967616i	$-0.418771\pi$
$283$	8.49297	0.504854	0.252427	+	0.967616i	$0.418771\pi$
$284$	5.11916	0.303766
$285$	0	0
$286$	15.9579	0.943610
$287$	0.666934	0.0393679
$288$	0	0
$289$	−15.6274	−0.919260
$290$	−26.7278	−1.56951
$291$	0	0
$292$	14.0099	0.819870
$293$	9.47262	0.553396	0.276698	−	0.960957i	$-0.410760\pi$
$293$	9.47262	0.553396	0.276698	+	0.960957i	$0.410760\pi$
$294$	0	0
$295$	−25.5181	−1.48572
$296$	21.2948	1.23774
$297$	0	0
$298$	−6.03760	−0.349749
$299$	16.2028	0.937032
$300$	0	0
$301$	4.86588	0.280465
$302$	−7.86634	−0.452657
$303$	0	0
$304$	−1.06156	−0.0608846
$305$	−26.8832	−1.53933
$306$	0	0
$307$	−22.4416	−1.28081	−0.640405	−	0.768038i	$-0.721233\pi$
$307$	−22.4416	−1.28081	−0.640405	+	0.768038i	$0.721233\pi$
$308$	−5.12404	−0.291969
$309$	0	0
$310$	0	0
$311$	8.73416	0.495269	0.247634	−	0.968854i	$-0.420347\pi$
$311$	8.73416	0.495269	0.247634	+	0.968854i	$0.420347\pi$
$312$	0	0
$313$	−32.0656	−1.81246	−0.906228	−	0.422789i	$-0.861051\pi$
$313$	−32.0656	−1.81246	−0.906228	+	0.422789i	$0.861051\pi$
$314$	9.97352	0.562838
$315$	0	0
$316$	−11.5221	−0.648167
$317$	21.8790	1.22885	0.614424	−	0.788976i	$-0.289389\pi$
$317$	21.8790	1.22885	0.614424	+	0.788976i	$0.289389\pi$
$318$	0	0
$319$	34.8379	1.95055
$320$	−16.6293	−0.929608
$321$	0	0
$322$	2.99616	0.166969
$323$	8.35083	0.464653
$324$	0	0
$325$	37.5772	2.08441
$326$	−11.0391	−0.611402
$327$	0	0
$328$	−1.86396	−0.102920
$329$	−4.41037	−0.243152
$330$	0	0
$331$	19.6732	1.08133	0.540667	−	0.841237i	$-0.318172\pi$
$331$	19.6732	1.08133	0.540667	+	0.841237i	$0.318172\pi$
$332$	−8.08499	−0.443722
$333$	0	0
$334$	−11.9027	−0.651286
$335$	−50.4088	−2.75413
$336$	0	0
$337$	−11.0711	−0.603079	−0.301540	−	0.953454i	$-0.597501\pi$
$337$	−11.0711	−0.603079	−0.301540	+	0.953454i	$0.597501\pi$
$338$	−7.15934	−0.389417
$339$	0	0
$340$	−5.38730	−0.292167
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	−13.5992	−0.733222
$345$	0	0
$346$	−5.90175	−0.317280
$347$	9.64677	0.517866	0.258933	−	0.965895i	$-0.416629\pi$
$347$	9.64677	0.517866	0.258933	+	0.965895i	$0.416629\pi$
$348$	0	0
$349$	−24.1530	−1.29288	−0.646440	−	0.762965i	$-0.723743\pi$
$349$	−24.1530	−1.29288	−0.646440	+	0.762965i	$0.723743\pi$
$350$	6.94862	0.371419
$351$	0	0
$352$	23.0819	1.23027
$353$	−20.6343	−1.09825	−0.549127	−	0.835739i	$-0.685040\pi$
$353$	−20.6343	−1.09825	−0.549127	+	0.835739i	$0.685040\pi$
$354$	0	0
$355$	14.6148	0.775671
$356$	7.23959	0.383697
$357$	0	0
$358$	14.5989	0.771576
$359$	−37.4331	−1.97564	−0.987822	−	0.155590i	$-0.950272\pi$
$359$	−37.4331	−1.97564	−0.987822	+	0.155590i	$0.950272\pi$
$360$	0	0
$361$	31.8067	1.67404
$362$	9.47600	0.498048
$363$	0	0
$364$	5.86747	0.307539
$365$	39.9971	2.09355
$366$	0	0
$367$	−5.89949	−0.307951	−0.153976	−	0.988075i	$-0.549208\pi$
$367$	−5.89949	−0.307951	−0.153976	+	0.988075i	$0.549208\pi$
$368$	0.521948	0.0272084
$369$	0	0
$370$	23.6015	1.22698
$371$	1.17157	0.0608250
$372$	0	0
$373$	−6.55091	−0.339194	−0.169597	−	0.985514i	$-0.554247\pi$
$373$	−6.55091	−0.339194	−0.169597	+	0.985514i	$0.554247\pi$
$374$	−4.04388	−0.209104
$375$	0	0
$376$	12.3262	0.635674
$377$	−39.8925	−2.05457
$378$	0	0
$379$	−23.2902	−1.19634	−0.598168	−	0.801371i	$-0.704104\pi$
$379$	−23.2902	−1.19634	−0.598168	+	0.801371i	$0.704104\pi$
$380$	−32.7765	−1.68140
$381$	0	0
$382$	−15.7386	−0.805255
$383$	4.72904	0.241643	0.120821	−	0.992674i	$-0.461447\pi$
$383$	4.72904	0.241643	0.120821	+	0.992674i	$0.461447\pi$
$384$	0	0
$385$	−14.6287	−0.745546
$386$	5.59756	0.284908
$387$	0	0
$388$	−10.2501	−0.520370
$389$	33.1567	1.68111	0.840556	−	0.541725i	$-0.182229\pi$
$389$	33.1567	1.68111	0.840556	+	0.541725i	$0.182229\pi$
$390$	0	0
$391$	−4.10594	−0.207646
$392$	−16.7689	−0.846957
$393$	0	0
$394$	10.3956	0.523725
$395$	−32.8945	−1.65510
$396$	0	0
$397$	−4.18244	−0.209911	−0.104955	−	0.994477i	$-0.533470\pi$
$397$	−4.18244	−0.209911	−0.104955	+	0.994477i	$0.533470\pi$
$398$	8.51710	0.426924
$399$	0	0
$400$	1.21049	0.0605245
$401$	35.3774	1.76666	0.883332	−	0.468749i	$-0.155295\pi$
$401$	35.3774	1.76666	0.883332	+	0.468749i	$0.155295\pi$
$402$	0	0
$403$	0	0
$404$	−16.4186	−0.816857
$405$	0	0
$406$	−7.37676	−0.366102
$407$	−30.7630	−1.52486
$408$	0	0
$409$	−1.90075	−0.0939862	−0.0469931	−	0.998895i	$-0.514964\pi$
$409$	−1.90075	−0.0939862	−0.0469931	+	0.998895i	$0.514964\pi$
$410$	−2.06586	−0.102026
$411$	0	0
$412$	−16.9085	−0.833024
$413$	−7.04289	−0.346558
$414$	0	0
$415$	−23.0819	−1.13305
$416$	−26.4308	−1.29588
$417$	0	0
$418$	−24.6031	−1.20338
$419$	1.63750	0.0799970	0.0399985	−	0.999200i	$-0.487265\pi$
$419$	1.63750	0.0799970	0.0399985	+	0.999200i	$0.487265\pi$
$420$	0	0
$421$	−30.8033	−1.50126	−0.750630	−	0.660723i	$-0.770250\pi$
$421$	−30.8033	−1.50126	−0.750630	+	0.660723i	$0.770250\pi$
$422$	−0.397476	−0.0193488
$423$	0	0
$424$	−3.27433	−0.159015
$425$	−9.52240	−0.461904
$426$	0	0
$427$	−7.41965	−0.359062
$428$	11.6251	0.561919
$429$	0	0
$430$	−15.0723	−0.726852
$431$	−28.1684	−1.35682	−0.678411	−	0.734682i	$-0.737331\pi$
$431$	−28.1684	−1.35682	−0.678411	+	0.734682i	$0.737331\pi$
$432$	0	0
$433$	5.06338	0.243331	0.121665	−	0.992571i	$-0.461177\pi$
$433$	5.06338	0.243331	0.121665	+	0.992571i	$0.461177\pi$
$434$	0	0
$435$	0	0
$436$	−22.5828	−1.08152
$437$	−24.9806	−1.19499
$438$	0	0
$439$	−29.8191	−1.42319	−0.711595	−	0.702590i	$-0.752027\pi$
$439$	−29.8191	−1.42319	−0.711595	+	0.702590i	$0.752027\pi$
$440$	40.8844	1.94909
$441$	0	0
$442$	4.63060	0.220255
$443$	−21.9806	−1.04433	−0.522166	−	0.852844i	$-0.674876\pi$
$443$	−21.9806	−1.04433	−0.522166	+	0.852844i	$0.674876\pi$
$444$	0	0
$445$	20.6684	0.979775
$446$	11.9307	0.564936
$447$	0	0
$448$	−4.58963	−0.216840
$449$	−19.8675	−0.937604	−0.468802	−	0.883303i	$-0.655314\pi$
$449$	−19.8675	−0.937604	−0.468802	+	0.883303i	$0.655314\pi$
$450$	0	0
$451$	2.69272	0.126795
$452$	13.3220	0.626612
$453$	0	0
$454$	15.4912	0.727037
$455$	16.7511	0.785305
$456$	0	0
$457$	−3.90844	−0.182829	−0.0914145	−	0.995813i	$-0.529139\pi$
$457$	−3.90844	−0.182829	−0.0914145	+	0.995813i	$0.529139\pi$
$458$	−14.9150	−0.696932
$459$	0	0
$460$	16.1155	0.751391
$461$	26.6263	1.24011	0.620055	−	0.784559i	$-0.287110\pi$
$461$	26.6263	1.24011	0.620055	+	0.784559i	$0.287110\pi$
$462$	0	0
$463$	−7.06020	−0.328115	−0.164058	−	0.986451i	$-0.552458\pi$
$463$	−7.06020	−0.328115	−0.164058	+	0.986451i	$0.552458\pi$
$464$	−1.28507	−0.0596581
$465$	0	0
$466$	−16.4534	−0.762187
$467$	−5.57027	−0.257761	−0.128881	−	0.991660i	$-0.541138\pi$
$467$	−5.57027	−0.257761	−0.128881	+	0.991660i	$0.541138\pi$
$468$	0	0
$469$	−13.9126	−0.642425
$470$	13.6614	0.630151
$471$	0	0
$472$	19.6836	0.906010
$473$	19.6458	0.903314
$474$	0	0
$475$	−57.9346	−2.65822
$476$	−1.48687	−0.0681507
$477$	0	0
$478$	8.36973	0.382823
$479$	−1.37178	−0.0626782	−0.0313391	−	0.999509i	$-0.509977\pi$
$479$	−1.37178	−0.0626782	−0.0313391	+	0.999509i	$0.509977\pi$
$480$	0	0
$481$	35.2263	1.60618
$482$	−18.6042	−0.847398
$483$	0	0
$484$	−6.72769	−0.305804
$485$	−29.2631	−1.32877
$486$	0	0
$487$	19.0368	0.862640	0.431320	−	0.902199i	$-0.358048\pi$
$487$	19.0368	0.862640	0.431320	+	0.902199i	$0.358048\pi$
$488$	20.7365	0.938699
$489$	0	0
$490$	−18.5853	−0.839599
$491$	4.62981	0.208940	0.104470	−	0.994528i	$-0.466685\pi$
$491$	4.62981	0.208940	0.104470	+	0.994528i	$0.466685\pi$
$492$	0	0
$493$	10.1091	0.455292
$494$	28.1727	1.26755
$495$	0	0
$496$	0	0
$497$	4.03361	0.180932
$498$	0	0
$499$	−16.4126	−0.734730	−0.367365	−	0.930077i	$-0.619740\pi$
$499$	−16.4126	−0.734730	−0.367365	+	0.930077i	$0.619740\pi$
$500$	14.3831	0.643231
$501$	0	0
$502$	3.44238	0.153641
$503$	−32.7151	−1.45869	−0.729347	−	0.684144i	$-0.760176\pi$
$503$	−32.7151	−1.45869	−0.729347	+	0.684144i	$0.760176\pi$
$504$	0	0
$505$	−46.8737	−2.08585
$506$	12.0969	0.537771
$507$	0	0
$508$	−11.1568	−0.495005
$509$	21.4648	0.951410	0.475705	−	0.879605i	$-0.342193\pi$
$509$	21.4648	0.951410	0.475705	+	0.879605i	$0.342193\pi$
$510$	0	0
$511$	11.0390	0.488339
$512$	−1.68390	−0.0744184
$513$	0	0
$514$	14.2768	0.629723
$515$	−48.2724	−2.12714
$516$	0	0
$517$	−17.8067	−0.783137
$518$	6.51392	0.286205
$519$	0	0
$520$	−46.8163	−2.05303
$521$	33.4493	1.46544	0.732720	−	0.680531i	$-0.238251\pi$
$521$	33.4493	1.46544	0.732720	+	0.680531i	$0.238251\pi$
$522$	0	0
$523$	−24.7686	−1.08305	−0.541527	−	0.840684i	$-0.682154\pi$
$523$	−24.7686	−1.08305	−0.541527	+	0.840684i	$0.682154\pi$
$524$	−0.148502	−0.00648735
$525$	0	0
$526$	5.26822	0.229705
$527$	0	0
$528$	0	0
$529$	−10.7175	−0.465979
$530$	−3.62901	−0.157634
$531$	0	0
$532$	−9.04617	−0.392201
$533$	−3.08340	−0.133557
$534$	0	0
$535$	33.1885	1.43487
$536$	38.8832	1.67950
$537$	0	0
$538$	−15.5826	−0.671813
$539$	24.2247	1.04343
$540$	0	0
$541$	−10.7394	−0.461725	−0.230863	−	0.972986i	$-0.574155\pi$
$541$	−10.7394	−0.461725	−0.230863	+	0.972986i	$0.574155\pi$
$542$	−2.35594	−0.101196
$543$	0	0
$544$	6.69783	0.287167
$545$	−64.4720	−2.76168
$546$	0	0
$547$	−20.1623	−0.862077	−0.431038	−	0.902334i	$-0.641853\pi$
$547$	−20.1623	−0.862077	−0.431038	+	0.902334i	$0.641853\pi$
$548$	16.1828	0.691294
$549$	0	0
$550$	28.0548	1.19626
$551$	61.5042	2.62017
$552$	0	0
$553$	−9.07875	−0.386068
$554$	−3.47247	−0.147531
$555$	0	0
$556$	−6.40494	−0.271630
$557$	8.81066	0.373320	0.186660	−	0.982425i	$-0.440234\pi$
$557$	8.81066	0.373320	0.186660	+	0.982425i	$0.440234\pi$
$558$	0	0
$559$	−22.4961	−0.951486
$560$	0.539612	0.0228028
$561$	0	0
$562$	2.64669	0.111644
$563$	11.9806	0.504924	0.252462	−	0.967607i	$-0.418760\pi$
$563$	11.9806	0.504924	0.252462	+	0.967607i	$0.418760\pi$
$564$	0	0
$565$	38.0330	1.60006
$566$	−7.26074	−0.305192
$567$	0	0
$568$	−11.2732	−0.473013
$569$	−2.65845	−0.111448	−0.0557239	−	0.998446i	$-0.517747\pi$
$569$	−2.65845	−0.111448	−0.0557239	+	0.998446i	$0.517747\pi$
$570$	0	0
$571$	−39.9917	−1.67360	−0.836800	−	0.547508i	$-0.815576\pi$
$571$	−39.9917	−1.67360	−0.836800	+	0.547508i	$0.815576\pi$
$572$	23.6897	0.990515
$573$	0	0
$574$	−0.570170	−0.0237984
$575$	28.4853	1.18792
$576$	0	0
$577$	−22.9253	−0.954392	−0.477196	−	0.878797i	$-0.658347\pi$
$577$	−22.9253	−0.954392	−0.477196	+	0.878797i	$0.658347\pi$
$578$	13.3601	0.555705
$579$	0	0
$580$	−39.6777	−1.64753
$581$	−6.37052	−0.264294
$582$	0	0
$583$	4.73017	0.195904
$584$	−30.8521	−1.27667
$585$	0	0
$586$	−8.09825	−0.334536
$587$	−38.8035	−1.60159	−0.800795	−	0.598938i	$-0.795590\pi$
$587$	−38.8035	−1.60159	−0.800795	+	0.598938i	$0.795590\pi$
$588$	0	0
$589$	0	0
$590$	21.8157	0.898139
$591$	0	0
$592$	1.13476	0.0466384
$593$	−18.1008	−0.743312	−0.371656	−	0.928370i	$-0.621210\pi$
$593$	−18.1008	−0.743312	−0.371656	+	0.928370i	$0.621210\pi$
$594$	0	0
$595$	−4.24489	−0.174024
$596$	−8.96288	−0.367134
$597$	0	0
$598$	−13.8520	−0.566449
$599$	−28.1051	−1.14834	−0.574172	−	0.818734i	$-0.694676\pi$
$599$	−28.1051	−1.14834	−0.574172	+	0.818734i	$0.694676\pi$
$600$	0	0
$601$	45.1479	1.84162	0.920810	−	0.390011i	$-0.127529\pi$
$601$	45.1479	1.84162	0.920810	+	0.390011i	$0.127529\pi$
$602$	−4.15990	−0.169545
$603$	0	0
$604$	−11.6777	−0.475157
$605$	−19.2070	−0.780874
$606$	0	0
$607$	4.25258	0.172607	0.0863034	−	0.996269i	$-0.472495\pi$
$607$	4.25258	0.172607	0.0863034	+	0.996269i	$0.472495\pi$
$608$	40.7497	1.65262
$609$	0	0
$610$	22.9827	0.930544
$611$	20.3902	0.824900
$612$	0	0
$613$	18.8524	0.761442	0.380721	−	0.924690i	$-0.375676\pi$
$613$	18.8524	0.761442	0.380721	+	0.924690i	$0.375676\pi$
$614$	19.1856	0.774267
$615$	0	0
$616$	11.2839	0.454643
$617$	−1.12629	−0.0453427	−0.0226713	−	0.999743i	$-0.507217\pi$
$617$	−1.12629	−0.0453427	−0.0226713	+	0.999743i	$0.507217\pi$
$618$	0	0
$619$	−3.39566	−0.136483	−0.0682415	−	0.997669i	$-0.521739\pi$
$619$	−3.39566	−0.136483	−0.0682415	+	0.997669i	$0.521739\pi$
$620$	0	0
$621$	0	0
$622$	−7.46694	−0.299397
$623$	5.70439	0.228542
$624$	0	0
$625$	0.423035	0.0169214
$626$	27.4133	1.09566
$627$	0	0
$628$	14.8058	0.590815
$629$	−8.92668	−0.355930
$630$	0	0
$631$	−40.7078	−1.62055	−0.810275	−	0.586050i	$-0.800682\pi$
$631$	−40.7078	−1.62055	−0.810275	+	0.586050i	$0.800682\pi$
$632$	25.3734	1.00930
$633$	0	0
$634$	−18.7046	−0.742855
$635$	−31.8518	−1.26400
$636$	0	0
$637$	−27.7394	−1.09908
$638$	−29.7833	−1.17913
$639$	0	0
$640$	−27.2112	−1.07562
$641$	29.1360	1.15080	0.575402	−	0.817871i	$-0.304846\pi$
$641$	29.1360	1.15080	0.575402	+	0.817871i	$0.304846\pi$
$642$	0	0
$643$	−8.74329	−0.344802	−0.172401	−	0.985027i	$-0.555152\pi$
$643$	−8.74329	−0.344802	−0.172401	+	0.985027i	$0.555152\pi$
$644$	4.44783	0.175269
$645$	0	0
$646$	−7.13923	−0.280889
$647$	20.1708	0.792995	0.396497	−	0.918036i	$-0.370226\pi$
$647$	20.1708	0.792995	0.396497	+	0.918036i	$0.370226\pi$
$648$	0	0
$649$	−28.4354	−1.11619
$650$	−32.1252	−1.26005
$651$	0	0
$652$	−16.3877	−0.641793
$653$	31.3323	1.22613	0.613063	−	0.790034i	$-0.289937\pi$
$653$	31.3323	1.22613	0.613063	+	0.790034i	$0.289937\pi$
$654$	0	0
$655$	−0.423961	−0.0165655
$656$	−0.0993270	−0.00387807
$657$	0	0
$658$	3.77048	0.146989
$659$	19.4643	0.758223	0.379111	−	0.925351i	$-0.376230\pi$
$659$	19.4643	0.758223	0.379111	+	0.925351i	$0.376230\pi$
$660$	0	0
$661$	−18.9126	−0.735615	−0.367808	−	0.929902i	$-0.619892\pi$
$661$	−18.9126	−0.735615	−0.367808	+	0.929902i	$0.619892\pi$
$662$	−16.8188	−0.653682
$663$	0	0
$664$	17.8044	0.690946
$665$	−25.8260	−1.00149
$666$	0	0
$667$	−30.2404	−1.17091
$668$	−17.6696	−0.683659
$669$	0	0
$670$	43.0951	1.66491
$671$	−29.9565	−1.15646
$672$	0	0
$673$	7.85342	0.302727	0.151364	−	0.988478i	$-0.451634\pi$
$673$	7.85342	0.302727	0.151364	+	0.988478i	$0.451634\pi$
$674$	9.46479	0.364570
$675$	0	0
$676$	−10.6281	−0.408774
$677$	19.4523	0.747613	0.373807	−	0.927507i	$-0.378052\pi$
$677$	19.4523	0.747613	0.373807	+	0.927507i	$0.378052\pi$
$678$	0	0
$679$	−8.07650	−0.309948
$680$	11.8637	0.454952
$681$	0	0
$682$	0	0
$683$	46.9507	1.79652	0.898260	−	0.439465i	$-0.144832\pi$
$683$	46.9507	1.79652	0.898260	+	0.439465i	$0.144832\pi$
$684$	0	0
$685$	46.2004	1.76523
$686$	−11.1139	−0.424329
$687$	0	0
$688$	−0.724679	−0.0276281
$689$	−5.41646	−0.206351
$690$	0	0
$691$	3.32459	0.126473	0.0632367	−	0.997999i	$-0.479858\pi$
$691$	3.32459	0.126473	0.0632367	+	0.997999i	$0.479858\pi$
$692$	−8.76120	−0.333051
$693$	0	0
$694$	−8.24714	−0.313057
$695$	−18.2855	−0.693610
$696$	0	0
$697$	0.781362	0.0295962
$698$	20.6487	0.781565
$699$	0	0
$700$	10.3153	0.389882
$701$	31.8059	1.20129	0.600646	−	0.799515i	$-0.294910\pi$
$701$	31.8059	1.20129	0.600646	+	0.799515i	$0.294910\pi$
$702$	0	0
$703$	−54.3102	−2.04835
$704$	−18.5304	−0.698392
$705$	0	0
$706$	17.6405	0.663910
$707$	−12.9370	−0.486544
$708$	0	0
$709$	−15.1437	−0.568735	−0.284367	−	0.958715i	$-0.591784\pi$
$709$	−15.1437	−0.568735	−0.284367	+	0.958715i	$0.591784\pi$
$710$	−12.4943	−0.468904
$711$	0	0
$712$	−15.9427	−0.597478
$713$	0	0
$714$	0	0
$715$	67.6319	2.52929
$716$	21.6722	0.809929
$717$	0	0
$718$	32.0020	1.19430
$719$	1.93391	0.0721227	0.0360614	−	0.999350i	$-0.488519\pi$
$719$	1.93391	0.0721227	0.0360614	+	0.999350i	$0.488519\pi$
$720$	0	0
$721$	−13.3230	−0.496174
$722$	−27.1919	−1.01198
$723$	0	0
$724$	14.0672	0.522804
$725$	−70.1328	−2.60467
$726$	0	0
$727$	−15.5351	−0.576164	−0.288082	−	0.957606i	$-0.593018\pi$
$727$	−15.5351	−0.576164	−0.288082	+	0.957606i	$0.593018\pi$
$728$	−12.9211	−0.478888
$729$	0	0
$730$	−34.1940	−1.26558
$731$	5.70074	0.210849
$732$	0	0
$733$	−16.3742	−0.604793	−0.302397	−	0.953182i	$-0.597787\pi$
$733$	−16.3742	−0.604793	−0.302397	+	0.953182i	$0.597787\pi$
$734$	5.04355	0.186161
$735$	0	0
$736$	−20.0358	−0.738531
$737$	−56.1716	−2.06911
$738$	0	0
$739$	−15.0592	−0.553962	−0.276981	−	0.960875i	$-0.589334\pi$
$739$	−15.0592	−0.553962	−0.276981	+	0.960875i	$0.589334\pi$
$740$	35.0367	1.28797
$741$	0	0
$742$	−1.00159	−0.0367696
$743$	2.59925	0.0953574	0.0476787	−	0.998863i	$-0.484818\pi$
$743$	2.59925	0.0953574	0.0476787	+	0.998863i	$0.484818\pi$
$744$	0	0
$745$	−25.5882	−0.937480
$746$	5.60046	0.205047
$747$	0	0
$748$	−6.00318	−0.219498
$749$	9.15990	0.334695
$750$	0	0
$751$	12.8875	0.470271	0.235136	−	0.971963i	$-0.424447\pi$
$751$	12.8875	0.470271	0.235136	+	0.971963i	$0.424447\pi$
$752$	0.656839	0.0239525
$753$	0	0
$754$	34.1045	1.24201
$755$	−33.3387	−1.21332
$756$	0	0
$757$	26.0592	0.947137	0.473569	−	0.880757i	$-0.342966\pi$
$757$	26.0592	0.947137	0.473569	+	0.880757i	$0.342966\pi$
$758$	19.9110	0.723201
$759$	0	0
$760$	72.1790	2.61821
$761$	−7.06210	−0.256001	−0.128001	−	0.991774i	$-0.540856\pi$
$761$	−7.06210	−0.256001	−0.128001	+	0.991774i	$0.540856\pi$
$762$	0	0
$763$	−17.7940	−0.644186
$764$	−23.3641	−0.845282
$765$	0	0
$766$	−4.04291	−0.146076
$767$	32.5610	1.17571
$768$	0	0
$769$	25.6915	0.926458	0.463229	−	0.886239i	$-0.346691\pi$
$769$	25.6915	0.926458	0.463229	+	0.886239i	$0.346691\pi$
$770$	12.5062	0.450693
$771$	0	0
$772$	8.30963	0.299070
$773$	−21.4757	−0.772426	−0.386213	−	0.922410i	$-0.626217\pi$
$773$	−21.4757	−0.772426	−0.386213	+	0.922410i	$0.626217\pi$
$774$	0	0
$775$	0	0
$776$	22.5723	0.810299
$777$	0	0
$778$	−28.3461	−1.01626
$779$	4.75383	0.170324
$780$	0	0
$781$	16.2855	0.582742
$782$	3.51022	0.125525
$783$	0	0
$784$	−0.893584	−0.0319137
$785$	42.2692	1.50865
$786$	0	0
$787$	39.4984	1.40797	0.703983	−	0.710217i	$-0.251403\pi$
$787$	39.4984	1.40797	0.703983	+	0.710217i	$0.251403\pi$
$788$	15.4324	0.549758
$789$	0	0
$790$	28.1219	1.00053
$791$	10.4970	0.373229
$792$	0	0
$793$	34.3028	1.21813
$794$	3.57562	0.126894
$795$	0	0
$796$	12.6437	0.448145
$797$	1.92191	0.0680774	0.0340387	−	0.999421i	$-0.489163\pi$
$797$	1.92191	0.0680774	0.0340387	+	0.999421i	$0.489163\pi$
$798$	0	0
$799$	−5.16707	−0.182798
$800$	−46.4667	−1.64285
$801$	0	0
$802$	−30.2446	−1.06797
$803$	44.5697	1.57283
$804$	0	0
$805$	12.6982	0.447551
$806$	0	0
$807$	0	0
$808$	36.1564	1.27198
$809$	3.90414	0.137262	0.0686311	−	0.997642i	$-0.478137\pi$
$809$	3.90414	0.137262	0.0686311	+	0.997642i	$0.478137\pi$
$810$	0	0
$811$	9.90943	0.347967	0.173984	−	0.984749i	$-0.444336\pi$
$811$	9.90943	0.347967	0.173984	+	0.984749i	$0.444336\pi$
$812$	−10.9509	−0.384300
$813$	0	0
$814$	26.2996	0.921802
$815$	−46.7855	−1.63883
$816$	0	0
$817$	34.6834	1.21342
$818$	1.62498	0.0568160
$819$	0	0
$820$	−3.06680	−0.107097
$821$	−16.6983	−0.582775	−0.291387	−	0.956605i	$-0.594117\pi$
$821$	−16.6983	−0.582775	−0.291387	+	0.956605i	$0.594117\pi$
$822$	0	0
$823$	4.69716	0.163733	0.0818664	−	0.996643i	$-0.473912\pi$
$823$	4.69716	0.163733	0.0818664	+	0.996643i	$0.473912\pi$
$824$	37.2353	1.29715
$825$	0	0
$826$	6.02105	0.209499
$827$	−49.6266	−1.72569	−0.862843	−	0.505472i	$-0.831319\pi$
$827$	−49.6266	−1.72569	−0.862843	+	0.505472i	$0.831319\pi$
$828$	0	0
$829$	31.3268	1.08803	0.544013	−	0.839077i	$-0.316904\pi$
$829$	31.3268	1.08803	0.544013	+	0.839077i	$0.316904\pi$
$830$	19.7330	0.684943
$831$	0	0
$832$	21.2190	0.735635
$833$	7.02944	0.243556
$834$	0	0
$835$	−50.4453	−1.74573
$836$	−36.5235	−1.26319
$837$	0	0
$838$	−1.39992	−0.0483593
$839$	−10.0530	−0.347067	−0.173534	−	0.984828i	$-0.555518\pi$
$839$	−10.0530	−0.347067	−0.173534	+	0.984828i	$0.555518\pi$
$840$	0	0
$841$	45.4540	1.56738
$842$	26.3341	0.907533
$843$	0	0
$844$	−0.590057	−0.0203106
$845$	−30.3423	−1.04381
$846$	0	0
$847$	−5.30104	−0.182146
$848$	−0.174483	−0.00599178
$849$	0	0
$850$	8.14082	0.279228
$851$	26.7032	0.915375
$852$	0	0
$853$	27.9393	0.956625	0.478312	−	0.878190i	$-0.341249\pi$
$853$	27.9393	0.956625	0.478312	+	0.878190i	$0.341249\pi$
$854$	6.34315	0.217058
$855$	0	0
$856$	−25.6002	−0.874998
$857$	−3.49117	−0.119256	−0.0596281	−	0.998221i	$-0.518991\pi$
$857$	−3.49117	−0.119256	−0.0596281	+	0.998221i	$0.518991\pi$
$858$	0	0
$859$	42.2446	1.44137	0.720683	−	0.693265i	$-0.243828\pi$
$859$	42.2446	1.44137	0.720683	+	0.693265i	$0.243828\pi$
$860$	−22.3750	−0.762982
$861$	0	0
$862$	24.0815	0.820218
$863$	−30.4765	−1.03743	−0.518715	−	0.854947i	$-0.673590\pi$
$863$	−30.4765	−1.03743	−0.518715	+	0.854947i	$0.673590\pi$
$864$	0	0
$865$	−25.0125	−0.850449
$866$	−4.32875	−0.147097
$867$	0	0
$868$	0	0
$869$	−36.6551	−1.24344
$870$	0	0
$871$	64.3214	2.17945
$872$	49.7310	1.68410
$873$	0	0
$874$	21.3563	0.722386
$875$	11.3331	0.383128
$876$	0	0
$877$	4.81465	0.162579	0.0812896	−	0.996691i	$-0.474096\pi$
$877$	4.81465	0.162579	0.0812896	+	0.996691i	$0.474096\pi$
$878$	25.4927	0.860338
$879$	0	0
$880$	2.17866	0.0734426
$881$	2.43371	0.0819939	0.0409970	−	0.999159i	$-0.486947\pi$
$881$	2.43371	0.0819939	0.0409970	+	0.999159i	$0.486947\pi$
$882$	0	0
$883$	−27.2340	−0.916497	−0.458249	−	0.888824i	$-0.651523\pi$
$883$	−27.2340	−0.916497	−0.458249	+	0.888824i	$0.651523\pi$
$884$	6.87417	0.231203
$885$	0	0
$886$	18.7915	0.631313
$887$	4.62145	0.155173	0.0775865	−	0.996986i	$-0.475279\pi$
$887$	4.62145	0.155173	0.0775865	+	0.996986i	$0.475279\pi$
$888$	0	0
$889$	−8.79097	−0.294840
$890$	−17.6696	−0.592288
$891$	0	0
$892$	17.7113	0.593018
$893$	−31.4366	−1.05199
$894$	0	0
$895$	61.8723	2.06816
$896$	−7.51017	−0.250897
$897$	0	0
$898$	16.9849	0.566795
$899$	0	0
$900$	0	0
$901$	1.37258	0.0457274
$902$	−2.30204	−0.0766494
$903$	0	0
$904$	−29.3370	−0.975735
$905$	40.1607	1.33499
$906$	0	0
$907$	18.0831	0.600439	0.300219	−	0.953870i	$-0.402940\pi$
$907$	18.0831	0.600439	0.300219	+	0.953870i	$0.402940\pi$
$908$	22.9968	0.763176
$909$	0	0
$910$	−14.3207	−0.474728
$911$	−1.81112	−0.0600050	−0.0300025	−	0.999550i	$-0.509552\pi$
$911$	−1.81112	−0.0600050	−0.0300025	+	0.999550i	$0.509552\pi$
$912$	0	0
$913$	−25.7207	−0.851231
$914$	3.34137	0.110523
$915$	0	0
$916$	−22.1415	−0.731574
$917$	−0.117012	−0.00386406
$918$	0	0
$919$	−36.1498	−1.19247	−0.596236	−	0.802809i	$-0.703338\pi$
$919$	−36.1498	−1.19247	−0.596236	+	0.802809i	$0.703338\pi$
$920$	−35.4890	−1.17004
$921$	0	0
$922$	−22.7631	−0.749664
$923$	−18.6484	−0.613818
$924$	0	0
$925$	61.9296	2.03623
$926$	6.03585	0.198350
$927$	0	0
$928$	49.3297	1.61933
$929$	39.1397	1.28413	0.642066	−	0.766649i	$-0.278077\pi$
$929$	39.1397	1.28413	0.642066	+	0.766649i	$0.278077\pi$
$930$	0	0
$931$	42.7673	1.40164
$932$	−24.4252	−0.800074
$933$	0	0
$934$	4.76209	0.155820
$935$	−17.1386	−0.560491
$936$	0	0
$937$	−19.0765	−0.623202	−0.311601	−	0.950213i	$-0.600865\pi$
$937$	−19.0765	−0.623202	−0.311601	+	0.950213i	$0.600865\pi$
$938$	11.8941	0.388355
$939$	0	0
$940$	20.2804	0.661474
$941$	5.36893	0.175022	0.0875110	−	0.996164i	$-0.472109\pi$
$941$	5.36893	0.175022	0.0875110	+	0.996164i	$0.472109\pi$
$942$	0	0
$943$	−2.33736	−0.0761150
$944$	1.04890	0.0341389
$945$	0	0
$946$	−16.7954	−0.546066
$947$	−6.80184	−0.221030	−0.110515	−	0.993874i	$-0.535250\pi$
$947$	−6.80184	−0.221030	−0.110515	+	0.993874i	$0.535250\pi$
$948$	0	0
$949$	−51.0362	−1.65670
$950$	49.5289	1.60693
$951$	0	0
$952$	3.27433	0.106122
$953$	−16.5070	−0.534715	−0.267357	−	0.963597i	$-0.586150\pi$
$953$	−16.5070	−0.534715	−0.267357	+	0.963597i	$0.586150\pi$
$954$	0	0
$955$	−66.7023	−2.15844
$956$	12.4250	0.401852
$957$	0	0
$958$	1.17275	0.0378899
$959$	12.7511	0.411755
$960$	0	0
$961$	0	0
$962$	−30.1154	−0.970960
$963$	0	0
$964$	−27.6181	−0.889520
$965$	23.7233	0.763679
$966$	0	0
$967$	17.4440	0.560961	0.280480	−	0.959860i	$-0.409506\pi$
$967$	17.4440	0.560961	0.280480	+	0.959860i	$0.409506\pi$
$968$	14.8154	0.476186
$969$	0	0
$970$	25.0174	0.803260
$971$	0.372772	0.0119628	0.00598141	−	0.999982i	$-0.498096\pi$
$971$	0.372772	0.0119628	0.00598141	+	0.999982i	$0.498096\pi$
$972$	0	0
$973$	−5.04673	−0.161791
$974$	−16.2748	−0.521478
$975$	0	0
$976$	1.10501	0.0353706
$977$	−37.2144	−1.19060	−0.595298	−	0.803505i	$-0.702966\pi$
$977$	−37.2144	−1.19060	−0.595298	+	0.803505i	$0.702966\pi$
$978$	0	0
$979$	23.0312	0.736081
$980$	−27.5901	−0.881333
$981$	0	0
$982$	−3.95808	−0.126307
$983$	37.3806	1.19226	0.596128	−	0.802890i	$-0.296705\pi$
$983$	37.3806	1.19226	0.596128	+	0.802890i	$0.296705\pi$
$984$	0	0
$985$	44.0583	1.40381
$986$	−8.64241	−0.275230
$987$	0	0
$988$	41.8227	1.33056
$989$	−17.0532	−0.542259
$990$	0	0
$991$	−13.8486	−0.439914	−0.219957	−	0.975510i	$-0.570592\pi$
$991$	−13.8486	−0.439914	−0.219957	+	0.975510i	$0.570592\pi$
$992$	0	0
$993$	0	0
$994$	−3.44838	−0.109376
$995$	36.0967	1.14434
$996$	0	0
$997$	−25.5147	−0.808059	−0.404029	−	0.914746i	$-0.632391\pi$
$997$	−25.5147	−0.808059	−0.404029	+	0.914746i	$0.632391\pi$
$998$	14.0313	0.444155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.z.1.2		4
3.2	odd	2		2883.2.a.i.1.3	✓	4
31.30	odd	2		8649.2.a.y.1.2		4
93.92	even	2		2883.2.a.j.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.i.1.3	✓	4	3.2	odd	2
2883.2.a.j.1.3	yes	4	93.92	even	2
8649.2.a.y.1.2		4	31.30	odd	2
8649.2.a.z.1.2		4	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-0.854912 q^{2} -1.26913 q^{4} -3.62324 q^{5} -1.00000 q^{7} +2.79481 q^{8} +O(q^{10})\)	\(q-0.854912 q^{2} -1.26913 q^{4} -3.62324 q^{5} -1.00000 q^{7} +2.79481 q^{8} +3.09755 q^{10} -4.03746 q^{11} +4.62324 q^{13} +0.854912 q^{14} +0.148931 q^{16} -1.17157 q^{17} -7.12788 q^{19} +4.59835 q^{20} +3.45167 q^{22} +3.50464 q^{23} +8.12788 q^{25} -3.95246 q^{26} +1.26913 q^{28} -8.62868 q^{29} -5.71695 q^{32} +1.00159 q^{34} +3.62324 q^{35} +7.61940 q^{37} +6.09371 q^{38} -10.1263 q^{40} -0.666934 q^{41} -4.86588 q^{43} +5.12404 q^{44} -2.99616 q^{46} +4.41037 q^{47} -6.00000 q^{49} -6.94862 q^{50} -5.86747 q^{52} -1.17157 q^{53} +14.6287 q^{55} -2.79481 q^{56} +7.37676 q^{58} +7.04289 q^{59} +7.41965 q^{61} +4.58963 q^{64} -16.7511 q^{65} +13.9126 q^{67} +1.48687 q^{68} -3.09755 q^{70} -4.03361 q^{71} -11.0390 q^{73} -6.51392 q^{74} +9.04617 q^{76} +4.03746 q^{77} +9.07875 q^{79} -0.539612 q^{80} +0.570170 q^{82} +6.37052 q^{83} +4.24489 q^{85} +4.15990 q^{86} -11.2839 q^{88} -5.70439 q^{89} -4.62324 q^{91} -4.44783 q^{92} -3.77048 q^{94} +25.8260 q^{95} +8.07650 q^{97} +5.12947 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 6 q^{4} - 4 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 - 4 * q^7 + 12 * q^8	\( 4 q + 2 q^{2} + 6 q^{4} - 4 q^{5} - 4 q^{7} + 12 q^{8} - 10 q^{10} + 8 q^{13} - 2 q^{14} + 2 q^{16} - 16 q^{17} - 8 q^{19} - 10 q^{20} - 8 q^{22} + 4 q^{23} + 12 q^{25} + 12 q^{26} - 6 q^{28} - 8 q^{29} + 6 q^{32} - 8 q^{34} + 4 q^{35} + 24 q^{37} + 6 q^{38} - 32 q^{40} - 24 q^{41} + 8 q^{43} + 4 q^{44} - 16 q^{46} + 16 q^{47} - 24 q^{49} - 4 q^{50} + 16 q^{52} - 16 q^{53} + 32 q^{55} - 12 q^{56} + 40 q^{58} - 4 q^{59} + 8 q^{61} + 20 q^{64} - 36 q^{65} - 8 q^{67} - 40 q^{68} + 10 q^{70} - 4 q^{71} - 16 q^{73} + 4 q^{74} - 10 q^{76} - 54 q^{80} - 30 q^{82} - 12 q^{83} + 8 q^{85} - 4 q^{86} - 8 q^{88} - 28 q^{89} - 8 q^{91} - 8 q^{94} + 40 q^{95} - 12 q^{97} - 12 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 - 4 * q^7 + 12 * q^8 - 10 * q^10 + 8 * q^13 - 2 * q^14 + 2 * q^16 - 16 * q^17 - 8 * q^19 - 10 * q^20 - 8 * q^22 + 4 * q^23 + 12 * q^25 + 12 * q^26 - 6 * q^28 - 8 * q^29 + 6 * q^32 - 8 * q^34 + 4 * q^35 + 24 * q^37 + 6 * q^38 - 32 * q^40 - 24 * q^41 + 8 * q^43 + 4 * q^44 - 16 * q^46 + 16 * q^47 - 24 * q^49 - 4 * q^50 + 16 * q^52 - 16 * q^53 + 32 * q^55 - 12 * q^56 + 40 * q^58 - 4 * q^59 + 8 * q^61 + 20 * q^64 - 36 * q^65 - 8 * q^67 - 40 * q^68 + 10 * q^70 - 4 * q^71 - 16 * q^73 + 4 * q^74 - 10 * q^76 - 54 * q^80 - 30 * q^82 - 12 * q^83 + 8 * q^85 - 4 * q^86 - 8 * q^88 - 28 * q^89 - 8 * q^91 - 8 * q^94 + 40 * q^95 - 12 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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