LMFDB - Embedded newform 8649.2.a.bv.1.1

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:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.64601 q^{2} +5.00135 q^{4} +0.929477 q^{5} -4.25556 q^{7} -7.94161 q^{8} +O(q^{10})$	$q-2.64601 q^{2} +5.00135 q^{4} +0.929477 q^{5} -4.25556 q^{7} -7.94161 q^{8} -2.45940 q^{10} +2.19895 q^{11} -0.809055 q^{13} +11.2603 q^{14} +11.0108 q^{16} +3.81225 q^{17} +6.57751 q^{19} +4.64864 q^{20} -5.81845 q^{22} +8.19106 q^{23} -4.13607 q^{25} +2.14076 q^{26} -21.2836 q^{28} +4.71763 q^{29} -13.2516 q^{32} -10.0872 q^{34} -3.95545 q^{35} +0.972822 q^{37} -17.4041 q^{38} -7.38154 q^{40} +7.25260 q^{41} +7.56575 q^{43} +10.9977 q^{44} -21.6736 q^{46} +5.15194 q^{47} +11.1098 q^{49} +10.9441 q^{50} -4.04637 q^{52} +8.77847 q^{53} +2.04388 q^{55} +33.7960 q^{56} -12.4829 q^{58} +2.45952 q^{59} +5.42872 q^{61} +13.0420 q^{64} -0.751998 q^{65} -7.74953 q^{67} +19.0664 q^{68} +10.4661 q^{70} -2.70475 q^{71} +4.36054 q^{73} -2.57409 q^{74} +32.8965 q^{76} -9.35779 q^{77} -6.32176 q^{79} +10.2343 q^{80} -19.1904 q^{82} -5.00046 q^{83} +3.54339 q^{85} -20.0190 q^{86} -17.4632 q^{88} -4.33501 q^{89} +3.44298 q^{91} +40.9664 q^{92} -13.6321 q^{94} +6.11365 q^{95} -11.6334 q^{97} -29.3967 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} + 16 q^{11} - 32 q^{13} + 24 q^{14} + 48 q^{16} + 32 q^{17} + 32 q^{19} + 24 q^{20} - 32 q^{22} + 32 q^{23} + 40 q^{25} + 16 q^{26} + 8 q^{28} + 48 q^{29} + 48 q^{32} + 48 q^{35} - 64 q^{37} + 24 q^{38} - 32 q^{43} + 48 q^{44} - 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} - 64 q^{52} + 80 q^{53} + 48 q^{56} - 32 q^{58} - 32 q^{61} + 56 q^{64} + 16 q^{65} - 16 q^{67} + 80 q^{68} + 8 q^{70} - 32 q^{73} + 56 q^{76} + 96 q^{77} - 32 q^{79} + 72 q^{80} + 8 q^{82} + 48 q^{83} - 96 q^{85} - 32 q^{86} - 96 q^{88} - 16 q^{89} + 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 + 16 * q^11 - 32 * q^13 + 24 * q^14 + 48 * q^16 + 32 * q^17 + 32 * q^19 + 24 * q^20 - 32 * q^22 + 32 * q^23 + 40 * q^25 + 16 * q^26 + 8 * q^28 + 48 * q^29 + 48 * q^32 + 48 * q^35 - 64 * q^37 + 24 * q^38 - 32 * q^43 + 48 * q^44 - 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 - 64 * q^52 + 80 * q^53 + 48 * q^56 - 32 * q^58 - 32 * q^61 + 56 * q^64 + 16 * q^65 - 16 * q^67 + 80 * q^68 + 8 * q^70 - 32 * q^73 + 56 * q^76 + 96 * q^77 - 32 * q^79 + 72 * q^80 + 8 * q^82 + 48 * q^83 - 96 * q^85 - 32 * q^86 - 96 * q^88 - 16 * q^89 + 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * 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 $

$2$	−2.64601	−1.87101	−0.935505	−	0.353314i	$-0.885055\pi$
$2$	−2.64601	−1.87101	−0.935505	+	0.353314i	$0.885055\pi$
$3$	0	0
$3$	0	0
$4$	5.00135	2.50068
$5$	0.929477	0.415675	0.207837	−	0.978163i	$-0.433358\pi$
$5$	0.929477	0.415675	0.207837	+	0.978163i	$0.433358\pi$
$6$	0	0
$7$	−4.25556	−1.60845	−0.804226	−	0.594324i	$-0.797420\pi$
$7$	−4.25556	−1.60845	−0.804226	+	0.594324i	$0.797420\pi$
$8$	−7.94161	−2.80778
$9$	0	0
$10$	−2.45940	−0.777731
$11$	2.19895	0.663009	0.331505	−	0.943454i	$-0.392444\pi$
$11$	2.19895	0.663009	0.331505	+	0.943454i	$0.392444\pi$
$12$	0	0
$13$	−0.809055	−0.224391	−0.112196	−	0.993686i	$-0.535788\pi$
$13$	−0.809055	−0.224391	−0.112196	+	0.993686i	$0.535788\pi$
$14$	11.2603	3.00943
$15$	0	0
$16$	11.0108	2.75271
$17$	3.81225	0.924605	0.462303	−	0.886722i	$-0.347023\pi$
$17$	3.81225	0.924605	0.462303	+	0.886722i	$0.347023\pi$
$18$	0	0
$19$	6.57751	1.50898	0.754492	−	0.656309i	$-0.227883\pi$
$19$	6.57751	1.50898	0.754492	+	0.656309i	$0.227883\pi$
$20$	4.64864	1.03947
$21$	0	0
$22$	−5.81845	−1.24050
$23$	8.19106	1.70795	0.853977	−	0.520311i	$-0.174184\pi$
$23$	8.19106	1.70795	0.853977	+	0.520311i	$0.174184\pi$
$24$	0	0
$25$	−4.13607	−0.827215
$26$	2.14076	0.419838
$27$	0	0
$28$	−21.2836	−4.02222
$29$	4.71763	0.876041	0.438021	−	0.898965i	$-0.355680\pi$
$29$	4.71763	0.876041	0.438021	+	0.898965i	$0.355680\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−13.2516	−2.34257
$33$	0	0
$34$	−10.0872	−1.72995
$35$	−3.95545	−0.668593
$36$	0	0
$37$	0.972822	0.159931	0.0799655	−	0.996798i	$-0.474519\pi$
$37$	0.972822	0.159931	0.0799655	+	0.996798i	$0.474519\pi$
$38$	−17.4041	−2.82333
$39$	0	0
$40$	−7.38154	−1.16712
$41$	7.25260	1.13267	0.566333	−	0.824177i	$-0.308362\pi$
$41$	7.25260	1.13267	0.566333	+	0.824177i	$0.308362\pi$
$42$	0	0
$43$	7.56575	1.15377	0.576883	−	0.816827i	$-0.304269\pi$
$43$	7.56575	1.15377	0.576883	+	0.816827i	$0.304269\pi$
$44$	10.9977	1.65797
$45$	0	0
$46$	−21.6736	−3.19560
$47$	5.15194	0.751487	0.375744	−	0.926724i	$-0.377387\pi$
$47$	5.15194	0.751487	0.375744	+	0.926724i	$0.377387\pi$
$48$	0	0
$49$	11.1098	1.58712
$50$	10.9441	1.54773
$51$	0	0
$52$	−4.04637	−0.561130
$53$	8.77847	1.20582	0.602908	−	0.797811i	$-0.294009\pi$
$53$	8.77847	1.20582	0.602908	+	0.797811i	$0.294009\pi$
$54$	0	0
$55$	2.04388	0.275596
$56$	33.7960	4.51618
$57$	0	0
$58$	−12.4829	−1.63908
$59$	2.45952	0.320202	0.160101	−	0.987101i	$-0.448818\pi$
$59$	2.45952	0.320202	0.160101	+	0.987101i	$0.448818\pi$
$60$	0	0
$61$	5.42872	0.695077	0.347538	−	0.937666i	$-0.387018\pi$
$61$	5.42872	0.695077	0.347538	+	0.937666i	$0.387018\pi$
$62$	0	0
$63$	0	0
$64$	13.0420	1.63025
$65$	−0.751998	−0.0932738
$66$	0	0
$67$	−7.74953	−0.946756	−0.473378	−	0.880859i	$-0.656966\pi$
$67$	−7.74953	−0.946756	−0.473378	+	0.880859i	$0.656966\pi$
$68$	19.0664	2.31214
$69$	0	0
$70$	10.4661	1.25094
$71$	−2.70475	−0.320994	−0.160497	−	0.987036i	$-0.551310\pi$
$71$	−2.70475	−0.320994	−0.160497	+	0.987036i	$0.551310\pi$
$72$	0	0
$73$	4.36054	0.510362	0.255181	−	0.966893i	$-0.417865\pi$
$73$	4.36054	0.510362	0.255181	+	0.966893i	$0.417865\pi$
$74$	−2.57409	−0.299232
$75$	0	0
$76$	32.8965	3.77348
$77$	−9.35779	−1.06642
$78$	0	0
$79$	−6.32176	−0.711254	−0.355627	−	0.934628i	$-0.615733\pi$
$79$	−6.32176	−0.711254	−0.355627	+	0.934628i	$0.615733\pi$
$80$	10.2343	1.14423
$81$	0	0
$82$	−19.1904	−2.11923
$83$	−5.00046	−0.548871	−0.274436	−	0.961605i	$-0.588491\pi$
$83$	−5.00046	−0.548871	−0.274436	+	0.961605i	$0.588491\pi$
$84$	0	0
$85$	3.54339	0.384335
$86$	−20.0190	−2.15871
$87$	0	0
$88$	−17.4632	−1.86159
$89$	−4.33501	−0.459510	−0.229755	−	0.973248i	$-0.573793\pi$
$89$	−4.33501	−0.459510	−0.229755	+	0.973248i	$0.573793\pi$
$90$	0	0
$91$	3.44298	0.360923
$92$	40.9664	4.27104
$93$	0	0
$94$	−13.6321	−1.40604
$95$	6.11365	0.627247
$96$	0	0
$97$	−11.6334	−1.18119	−0.590595	−	0.806968i	$-0.701107\pi$
$97$	−11.6334	−1.18119	−0.590595	+	0.806968i	$0.701107\pi$
$98$	−29.3967	−2.96951
$99$	0	0
$100$	−20.6860	−2.06860
$101$	0.711229	0.0707699	0.0353849	−	0.999374i	$-0.488734\pi$
$101$	0.711229	0.0707699	0.0353849	+	0.999374i	$0.488734\pi$
$102$	0	0
$103$	3.65541	0.360178	0.180089	−	0.983650i	$-0.442361\pi$
$103$	3.65541	0.360178	0.180089	+	0.983650i	$0.442361\pi$
$104$	6.42519	0.630042
$105$	0	0
$106$	−23.2279	−2.25609
$107$	−9.05006	−0.874902	−0.437451	−	0.899242i	$-0.644119\pi$
$107$	−9.05006	−0.874902	−0.437451	+	0.899242i	$0.644119\pi$
$108$	0	0
$109$	−3.25630	−0.311897	−0.155948	−	0.987765i	$-0.549843\pi$
$109$	−3.25630	−0.311897	−0.155948	+	0.987765i	$0.549843\pi$
$110$	−5.40811	−0.515643
$111$	0	0
$112$	−46.8573	−4.42760
$113$	−1.80928	−0.170202	−0.0851012	−	0.996372i	$-0.527121\pi$
$113$	−1.80928	−0.170202	−0.0851012	+	0.996372i	$0.527121\pi$
$114$	0	0
$115$	7.61340	0.709953
$116$	23.5945	2.19070
$117$	0	0
$118$	−6.50790	−0.599100
$119$	−16.2233	−1.48718
$120$	0	0
$121$	−6.16460	−0.560419
$122$	−14.3644	−1.30050
$123$	0	0
$124$	0	0
$125$	−8.49177	−0.759527
$126$	0	0
$127$	−15.9236	−1.41299	−0.706497	−	0.707717i	$-0.749725\pi$
$127$	−15.9236	−1.41299	−0.706497	+	0.707717i	$0.749725\pi$
$128$	−8.00618	−0.707653
$129$	0	0
$130$	1.98979	0.174516
$131$	−7.80583	−0.681999	−0.340999	−	0.940064i	$-0.610765\pi$
$131$	−7.80583	−0.681999	−0.340999	+	0.940064i	$0.610765\pi$
$132$	0	0
$133$	−27.9910	−2.42713
$134$	20.5053	1.77139
$135$	0	0
$136$	−30.2754	−2.59609
$137$	−3.91472	−0.334457	−0.167228	−	0.985918i	$-0.553482\pi$
$137$	−3.91472	−0.334457	−0.167228	+	0.985918i	$0.553482\pi$
$138$	0	0
$139$	21.4974	1.82339	0.911694	−	0.410870i	$-0.134775\pi$
$139$	21.4974	1.82339	0.911694	+	0.410870i	$0.134775\pi$
$140$	−19.7826	−1.67193
$141$	0	0
$142$	7.15678	0.600584
$143$	−1.77907	−0.148774
$144$	0	0
$145$	4.38492	0.364148
$146$	−11.5380	−0.954893
$147$	0	0
$148$	4.86543	0.399936
$149$	2.21104	0.181136	0.0905679	−	0.995890i	$-0.471132\pi$
$149$	2.21104	0.181136	0.0905679	+	0.995890i	$0.471132\pi$
$150$	0	0
$151$	16.7803	1.36556	0.682782	−	0.730622i	$-0.260770\pi$
$151$	16.7803	1.36556	0.682782	+	0.730622i	$0.260770\pi$
$152$	−52.2360	−4.23690
$153$	0	0
$154$	24.7608	1.99528
$155$	0	0
$156$	0	0
$157$	7.62325	0.608402	0.304201	−	0.952608i	$-0.401611\pi$
$157$	7.62325	0.608402	0.304201	+	0.952608i	$0.401611\pi$
$158$	16.7274	1.33076
$159$	0	0
$160$	−12.3170	−0.973745
$161$	−34.8576	−2.74716
$162$	0	0
$163$	−15.2915	−1.19772	−0.598859	−	0.800854i	$-0.704379\pi$
$163$	−15.2915	−1.19772	−0.598859	+	0.800854i	$0.704379\pi$
$164$	36.2728	2.83243
$165$	0	0
$166$	13.2312	1.02694
$167$	−10.3625	−0.801871	−0.400935	−	0.916106i	$-0.631315\pi$
$167$	−10.3625	−0.801871	−0.400935	+	0.916106i	$0.631315\pi$
$168$	0	0
$169$	−12.3454	−0.949649
$170$	−9.37585	−0.719095
$171$	0	0
$172$	37.8390	2.88520
$173$	−12.0922	−0.919353	−0.459677	−	0.888086i	$-0.652035\pi$
$173$	−12.0922	−0.919353	−0.459677	+	0.888086i	$0.652035\pi$
$174$	0	0
$175$	17.6013	1.33053
$176$	24.2123	1.82507
$177$	0	0
$178$	11.4705	0.859748
$179$	9.02181	0.674322	0.337161	−	0.941447i	$-0.390533\pi$
$179$	9.02181	0.674322	0.337161	+	0.941447i	$0.390533\pi$
$180$	0	0
$181$	18.2137	1.35381	0.676907	−	0.736068i	$-0.263320\pi$
$181$	18.2137	1.35381	0.676907	+	0.736068i	$0.263320\pi$
$182$	−9.11016	−0.675290
$183$	0	0
$184$	−65.0501	−4.79556
$185$	0.904215	0.0664792
$186$	0	0
$187$	8.38295	0.613022
$188$	25.7667	1.87923
$189$	0	0
$190$	−16.1768	−1.17358
$191$	18.4115	1.33221	0.666103	−	0.745860i	$-0.267961\pi$
$191$	18.4115	1.33221	0.666103	+	0.745860i	$0.267961\pi$
$192$	0	0
$193$	11.3842	0.819451	0.409725	−	0.912209i	$-0.365625\pi$
$193$	11.3842	0.819451	0.409725	+	0.912209i	$0.365625\pi$
$194$	30.7820	2.21002
$195$	0	0
$196$	55.5642	3.96887
$197$	−4.66910	−0.332660	−0.166330	−	0.986070i	$-0.553192\pi$
$197$	−4.66910	−0.332660	−0.166330	+	0.986070i	$0.553192\pi$
$198$	0	0
$199$	−23.5663	−1.67057	−0.835284	−	0.549819i	$-0.814697\pi$
$199$	−23.5663	−1.67057	−0.835284	+	0.549819i	$0.814697\pi$
$200$	32.8471	2.32264
$201$	0	0
$202$	−1.88192	−0.132411
$203$	−20.0762	−1.40907
$204$	0	0
$205$	6.74112	0.470820
$206$	−9.67225	−0.673897
$207$	0	0
$208$	−8.90837	−0.617684
$209$	14.4636	1.00047
$210$	0	0
$211$	26.7056	1.83849	0.919245	−	0.393686i	$-0.128800\pi$
$211$	26.7056	1.83849	0.919245	+	0.393686i	$0.128800\pi$
$212$	43.9043	3.01536
$213$	0	0
$214$	23.9465	1.63695
$215$	7.03219	0.479592
$216$	0	0
$217$	0	0
$218$	8.61619	0.583562
$219$	0	0
$220$	10.2222	0.689177
$221$	−3.08431	−0.207473
$222$	0	0
$223$	0.845480	0.0566175	0.0283088	−	0.999599i	$-0.490988\pi$
$223$	0.845480	0.0566175	0.0283088	+	0.999599i	$0.490988\pi$
$224$	56.3928	3.76790
$225$	0	0
$226$	4.78736	0.318450
$227$	16.5053	1.09549	0.547747	−	0.836644i	$-0.315486\pi$
$227$	16.5053	1.09549	0.547747	+	0.836644i	$0.315486\pi$
$228$	0	0
$229$	−2.34857	−0.155198	−0.0775988	−	0.996985i	$-0.524725\pi$
$229$	−2.34857	−0.155198	−0.0775988	+	0.996985i	$0.524725\pi$
$230$	−20.1451	−1.32833
$231$	0	0
$232$	−37.4655	−2.45973
$233$	−0.843522	−0.0552610	−0.0276305	−	0.999618i	$-0.508796\pi$
$233$	−0.843522	−0.0552610	−0.0276305	+	0.999618i	$0.508796\pi$
$234$	0	0
$235$	4.78861	0.312374
$236$	12.3009	0.800721
$237$	0	0
$238$	42.9268	2.78253
$239$	−14.8226	−0.958796	−0.479398	−	0.877598i	$-0.659145\pi$
$239$	−14.8226	−0.958796	−0.479398	+	0.877598i	$0.659145\pi$
$240$	0	0
$241$	−6.90580	−0.444842	−0.222421	−	0.974951i	$-0.571396\pi$
$241$	−6.90580	−0.444842	−0.222421	+	0.974951i	$0.571396\pi$
$242$	16.3116	1.04855
$243$	0	0
$244$	27.1510	1.73816
$245$	10.3263	0.659725
$246$	0	0
$247$	−5.32157	−0.338603
$248$	0	0
$249$	0	0
$250$	22.4693	1.42108
$251$	−28.3252	−1.78787	−0.893935	−	0.448196i	$-0.852067\pi$
$251$	−28.3252	−1.78787	−0.893935	+	0.448196i	$0.852067\pi$
$252$	0	0
$253$	18.0117	1.13239
$254$	42.1340	2.64372
$255$	0	0
$256$	−4.89963	−0.306227
$257$	−7.01549	−0.437614	−0.218807	−	0.975768i	$-0.570217\pi$
$257$	−7.01549	−0.437614	−0.218807	+	0.975768i	$0.570217\pi$
$258$	0	0
$259$	−4.13991	−0.257241
$260$	−3.76101	−0.233248
$261$	0	0
$262$	20.6543	1.27603
$263$	10.1290	0.624583	0.312292	−	0.949986i	$-0.398903\pi$
$263$	10.1290	0.624583	0.312292	+	0.949986i	$0.398903\pi$
$264$	0	0
$265$	8.15939	0.501227
$266$	74.0645	4.54118
$267$	0	0
$268$	−38.7582	−2.36753
$269$	29.9905	1.82855	0.914277	−	0.405090i	$-0.132760\pi$
$269$	29.9905	1.82855	0.914277	+	0.405090i	$0.132760\pi$
$270$	0	0
$271$	18.1165	1.10050	0.550251	−	0.835000i	$-0.314532\pi$
$271$	18.1165	1.10050	0.550251	+	0.835000i	$0.314532\pi$
$272$	41.9760	2.54517
$273$	0	0
$274$	10.3584	0.625772
$275$	−9.09503	−0.548451
$276$	0	0
$277$	−14.1154	−0.848115	−0.424057	−	0.905635i	$-0.639395\pi$
$277$	−14.1154	−0.848115	−0.424057	+	0.905635i	$0.639395\pi$
$278$	−56.8824	−3.41158
$279$	0	0
$280$	31.4126	1.87726
$281$	24.7248	1.47496	0.737480	−	0.675369i	$-0.236016\pi$
$281$	24.7248	1.47496	0.737480	+	0.675369i	$0.236016\pi$
$282$	0	0
$283$	20.7596	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$283$	20.7596	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$284$	−13.5274	−0.802703
$285$	0	0
$286$	4.70744	0.278357
$287$	−30.8639	−1.82184
$288$	0	0
$289$	−2.46679	−0.145105
$290$	−11.6025	−0.681325
$291$	0	0
$292$	21.8086	1.27625
$293$	8.54265	0.499067	0.249533	−	0.968366i	$-0.419723\pi$
$293$	8.54265	0.499067	0.249533	+	0.968366i	$0.419723\pi$
$294$	0	0
$295$	2.28606	0.133100
$296$	−7.72577	−0.449051
$297$	0	0
$298$	−5.85044	−0.338907
$299$	−6.62701	−0.383250
$300$	0	0
$301$	−32.1965	−1.85578
$302$	−44.4009	−2.55498
$303$	0	0
$304$	72.4239	4.15380
$305$	5.04587	0.288926
$306$	0	0
$307$	−4.47395	−0.255342	−0.127671	−	0.991817i	$-0.540750\pi$
$307$	−4.47395	−0.255342	−0.127671	+	0.991817i	$0.540750\pi$
$308$	−46.8016	−2.66677
$309$	0	0
$310$	0	0
$311$	33.5486	1.90236	0.951182	−	0.308630i	$-0.0998705\pi$
$311$	33.5486	1.90236	0.951182	+	0.308630i	$0.0998705\pi$
$312$	0	0
$313$	−24.6513	−1.39337	−0.696686	−	0.717376i	$-0.745343\pi$
$313$	−24.6513	−1.39337	−0.696686	+	0.717376i	$0.745343\pi$
$314$	−20.1712	−1.13833
$315$	0	0
$316$	−31.6174	−1.77862
$317$	−0.782598	−0.0439551	−0.0219775	−	0.999758i	$-0.506996\pi$
$317$	−0.782598	−0.0439551	−0.0219775	+	0.999758i	$0.506996\pi$
$318$	0	0
$319$	10.3738	0.580823
$320$	12.1223	0.677655
$321$	0	0
$322$	92.2334	5.13996
$323$	25.0751	1.39522
$324$	0	0
$325$	3.34631	0.185620
$326$	40.4613	2.24094
$327$	0	0
$328$	−57.5973	−3.18028
$329$	−21.9244	−1.20873
$330$	0	0
$331$	33.2152	1.82568	0.912838	−	0.408323i	$-0.133886\pi$
$331$	33.2152	1.82568	0.912838	+	0.408323i	$0.133886\pi$
$332$	−25.0091	−1.37255
$333$	0	0
$334$	27.4191	1.50031
$335$	−7.20301	−0.393542
$336$	0	0
$337$	−2.93606	−0.159937	−0.0799687	−	0.996797i	$-0.525482\pi$
$337$	−2.93606	−0.159937	−0.0799687	+	0.996797i	$0.525482\pi$
$338$	32.6661	1.77680
$339$	0	0
$340$	17.7218	0.961098
$341$	0	0
$342$	0	0
$343$	−17.4896	−0.944351
$344$	−60.0842	−3.23952
$345$	0	0
$346$	31.9961	1.72012
$347$	−9.07409	−0.487123	−0.243561	−	0.969886i	$-0.578316\pi$
$347$	−9.07409	−0.487123	−0.243561	+	0.969886i	$0.578316\pi$
$348$	0	0
$349$	−10.2852	−0.550555	−0.275277	−	0.961365i	$-0.588770\pi$
$349$	−10.2852	−0.550555	−0.275277	+	0.961365i	$0.588770\pi$
$350$	−46.5732	−2.48944
$351$	0	0
$352$	−29.1395	−1.55314
$353$	−4.46693	−0.237751	−0.118875	−	0.992909i	$-0.537929\pi$
$353$	−4.46693	−0.237751	−0.118875	+	0.992909i	$0.537929\pi$
$354$	0	0
$355$	−2.51400	−0.133429
$356$	−21.6809	−1.14909
$357$	0	0
$358$	−23.8718	−1.26166
$359$	−24.2972	−1.28236	−0.641179	−	0.767391i	$-0.721554\pi$
$359$	−24.2972	−1.28236	−0.641179	+	0.767391i	$0.721554\pi$
$360$	0	0
$361$	24.2637	1.27704
$362$	−48.1936	−2.53300
$363$	0	0
$364$	17.2196	0.902551
$365$	4.05302	0.212145
$366$	0	0
$367$	4.95262	0.258525	0.129262	−	0.991610i	$-0.458739\pi$
$367$	4.95262	0.258525	0.129262	+	0.991610i	$0.458739\pi$
$368$	90.1904	4.70150
$369$	0	0
$370$	−2.39256	−0.124383
$371$	−37.3574	−1.93950
$372$	0	0
$373$	−4.71560	−0.244164	−0.122082	−	0.992520i	$-0.538957\pi$
$373$	−4.71560	−0.244164	−0.122082	+	0.992520i	$0.538957\pi$
$374$	−22.1813	−1.14697
$375$	0	0
$376$	−40.9147	−2.11001
$377$	−3.81682	−0.196576
$378$	0	0
$379$	−8.00073	−0.410970	−0.205485	−	0.978660i	$-0.565877\pi$
$379$	−8.00073	−0.410970	−0.205485	+	0.978660i	$0.565877\pi$
$380$	30.5765	1.56854
$381$	0	0
$382$	−48.7169	−2.49257
$383$	26.4831	1.35322	0.676611	−	0.736341i	$-0.263448\pi$
$383$	26.4831	1.35322	0.676611	+	0.736341i	$0.263448\pi$
$384$	0	0
$385$	−8.69785	−0.443283
$386$	−30.1226	−1.53320
$387$	0	0
$388$	−58.1826	−2.95378
$389$	24.9984	1.26747	0.633734	−	0.773551i	$-0.281521\pi$
$389$	24.9984	1.26747	0.633734	+	0.773551i	$0.281521\pi$
$390$	0	0
$391$	31.2263	1.57918
$392$	−88.2299	−4.45628
$393$	0	0
$394$	12.3545	0.622410
$395$	−5.87593	−0.295650
$396$	0	0
$397$	−28.9313	−1.45202	−0.726009	−	0.687685i	$-0.758627\pi$
$397$	−28.9313	−1.45202	−0.726009	+	0.687685i	$0.758627\pi$
$398$	62.3565	3.12565
$399$	0	0
$400$	−45.5416	−2.27708
$401$	36.0114	1.79832	0.899161	−	0.437618i	$-0.144178\pi$
$401$	36.0114	1.79832	0.899161	+	0.437618i	$0.144178\pi$
$402$	0	0
$403$	0	0
$404$	3.55711	0.176973
$405$	0	0
$406$	53.1217	2.63638
$407$	2.13919	0.106036
$408$	0	0
$409$	5.19057	0.256657	0.128329	−	0.991732i	$-0.459039\pi$
$409$	5.19057	0.256657	0.128329	+	0.991732i	$0.459039\pi$
$410$	−17.8371	−0.880909
$411$	0	0
$412$	18.2820	0.900690
$413$	−10.4666	−0.515029
$414$	0	0
$415$	−4.64781	−0.228152
$416$	10.7212	0.525651
$417$	0	0
$418$	−38.2709	−1.87189
$419$	−16.9602	−0.828561	−0.414280	−	0.910149i	$-0.635967\pi$
$419$	−16.9602	−0.828561	−0.414280	+	0.910149i	$0.635967\pi$
$420$	0	0
$421$	19.3011	0.940677	0.470338	−	0.882486i	$-0.344132\pi$
$421$	19.3011	0.940677	0.470338	+	0.882486i	$0.344132\pi$
$422$	−70.6632	−3.43983
$423$	0	0
$424$	−69.7152	−3.38567
$425$	−15.7677	−0.764847
$426$	0	0
$427$	−23.1023	−1.11800
$428$	−45.2626	−2.18785
$429$	0	0
$430$	−18.6072	−0.897320
$431$	−11.7117	−0.564131	−0.282065	−	0.959395i	$-0.591020\pi$
$431$	−11.7117	−0.564131	−0.282065	+	0.959395i	$0.591020\pi$
$432$	0	0
$433$	−16.3318	−0.784858	−0.392429	−	0.919782i	$-0.628365\pi$
$433$	−16.3318	−0.784858	−0.392429	+	0.919782i	$0.628365\pi$
$434$	0	0
$435$	0	0
$436$	−16.2859	−0.779953
$437$	53.8768	2.57728
$438$	0	0
$439$	17.0731	0.814854	0.407427	−	0.913238i	$-0.366426\pi$
$439$	17.0731	0.814854	0.407427	+	0.913238i	$0.366426\pi$
$440$	−16.2317	−0.773814
$441$	0	0
$442$	8.16112	0.388185
$443$	23.1546	1.10011	0.550055	−	0.835128i	$-0.314607\pi$
$443$	23.1546	1.10011	0.550055	+	0.835128i	$0.314607\pi$
$444$	0	0
$445$	−4.02929	−0.191007
$446$	−2.23715	−0.105932
$447$	0	0
$448$	−55.5012	−2.62218
$449$	18.8132	0.887848	0.443924	−	0.896064i	$-0.353586\pi$
$449$	18.8132	0.887848	0.443924	+	0.896064i	$0.353586\pi$
$450$	0	0
$451$	15.9481	0.750968
$452$	−9.04883	−0.425621
$453$	0	0
$454$	−43.6731	−2.04968
$455$	3.20017	0.150026
$456$	0	0
$457$	−23.5997	−1.10395	−0.551973	−	0.833862i	$-0.686125\pi$
$457$	−23.5997	−1.10395	−0.551973	+	0.833862i	$0.686125\pi$
$458$	6.21432	0.290376
$459$	0	0
$460$	38.0773	1.77536
$461$	−0.361982	−0.0168592	−0.00842959	−	0.999964i	$-0.502683\pi$
$461$	−0.361982	−0.0168592	−0.00842959	+	0.999964i	$0.502683\pi$
$462$	0	0
$463$	−2.34741	−0.109093	−0.0545467	−	0.998511i	$-0.517371\pi$
$463$	−2.34741	−0.109093	−0.0545467	+	0.998511i	$0.517371\pi$
$464$	51.9450	2.41149
$465$	0	0
$466$	2.23197	0.103394
$467$	−32.9912	−1.52665	−0.763326	−	0.646014i	$-0.776435\pi$
$467$	−32.9912	−1.52665	−0.763326	+	0.646014i	$0.776435\pi$
$468$	0	0
$469$	32.9786	1.52281
$470$	−12.6707	−0.584455
$471$	0	0
$472$	−19.5325	−0.899056
$473$	16.6367	0.764958
$474$	0	0
$475$	−27.2051	−1.24825
$476$	−81.1382	−3.71897
$477$	0	0
$478$	39.2208	1.79392
$479$	29.3901	1.34287	0.671434	−	0.741064i	$-0.265679\pi$
$479$	29.3901	1.34287	0.671434	+	0.741064i	$0.265679\pi$
$480$	0	0
$481$	−0.787066	−0.0358871
$482$	18.2728	0.832303
$483$	0	0
$484$	−30.8314	−1.40143
$485$	−10.8130	−0.490991
$486$	0	0
$487$	−41.2885	−1.87096	−0.935481	−	0.353376i	$-0.885034\pi$
$487$	−41.2885	−1.87096	−0.935481	+	0.353376i	$0.885034\pi$
$488$	−43.1128	−1.95162
$489$	0	0
$490$	−27.3235	−1.23435
$491$	−11.5099	−0.519436	−0.259718	−	0.965684i	$-0.583630\pi$
$491$	−11.5099	−0.519436	−0.259718	+	0.965684i	$0.583630\pi$
$492$	0	0
$493$	17.9847	0.809992
$494$	14.0809	0.633530
$495$	0	0
$496$	0	0
$497$	11.5102	0.516304
$498$	0	0
$499$	20.7009	0.926698	0.463349	−	0.886176i	$-0.346648\pi$
$499$	20.7009	0.926698	0.463349	+	0.886176i	$0.346648\pi$
$500$	−42.4703	−1.89933
$501$	0	0
$502$	74.9487	3.34512
$503$	25.1721	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$503$	25.1721	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$504$	0	0
$505$	0.661071	0.0294173
$506$	−47.6592	−2.11871
$507$	0	0
$508$	−79.6397	−3.53344
$509$	−35.0196	−1.55222	−0.776108	−	0.630600i	$-0.782809\pi$
$509$	−35.0196	−1.55222	−0.776108	+	0.630600i	$0.782809\pi$
$510$	0	0
$511$	−18.5565	−0.820893
$512$	28.9768	1.28061
$513$	0	0
$514$	18.5630	0.818780
$515$	3.39762	0.149717
$516$	0	0
$517$	11.3289	0.498243
$518$	10.9542	0.481301
$519$	0	0
$520$	5.97207	0.261893
$521$	−12.0478	−0.527823	−0.263911	−	0.964547i	$-0.585013\pi$
$521$	−12.0478	−0.527823	−0.263911	+	0.964547i	$0.585013\pi$
$522$	0	0
$523$	11.1810	0.488912	0.244456	−	0.969660i	$-0.421391\pi$
$523$	11.1810	0.488912	0.244456	+	0.969660i	$0.421391\pi$
$524$	−39.0397	−1.70546
$525$	0	0
$526$	−26.8015	−1.16860
$527$	0	0
$528$	0	0
$529$	44.0934	1.91710
$530$	−21.5898	−0.937801
$531$	0	0
$532$	−139.993	−6.06947
$533$	−5.86775	−0.254160
$534$	0	0
$535$	−8.41182	−0.363675
$536$	61.5437	2.65828
$537$	0	0
$538$	−79.3551	−3.42124
$539$	24.4300	1.05227
$540$	0	0
$541$	18.6509	0.801864	0.400932	−	0.916108i	$-0.368686\pi$
$541$	18.6509	0.801864	0.400932	+	0.916108i	$0.368686\pi$
$542$	−47.9365	−2.05905
$543$	0	0
$544$	−50.5182	−2.16595
$545$	−3.02665	−0.129648
$546$	0	0
$547$	10.5810	0.452410	0.226205	−	0.974080i	$-0.427368\pi$
$547$	10.5810	0.452410	0.226205	+	0.974080i	$0.427368\pi$
$548$	−19.5789	−0.836369
$549$	0	0
$550$	24.0655	1.02616
$551$	31.0302	1.32193
$552$	0	0
$553$	26.9027	1.14402
$554$	37.3496	1.58683
$555$	0	0
$556$	107.516	4.55971
$557$	28.9259	1.22563	0.612814	−	0.790227i	$-0.290038\pi$
$557$	28.9259	1.22563	0.612814	+	0.790227i	$0.290038\pi$
$558$	0	0
$559$	−6.12111	−0.258895
$560$	−43.5528	−1.84044
$561$	0	0
$562$	−65.4221	−2.75966
$563$	36.4498	1.53618	0.768089	−	0.640344i	$-0.221208\pi$
$563$	36.4498	1.53618	0.768089	+	0.640344i	$0.221208\pi$
$564$	0	0
$565$	−1.68168	−0.0707488
$566$	−54.9300	−2.30888
$567$	0	0
$568$	21.4800	0.901282
$569$	−11.5596	−0.484605	−0.242302	−	0.970201i	$-0.577903\pi$
$569$	−11.5596	−0.484605	−0.242302	+	0.970201i	$0.577903\pi$
$570$	0	0
$571$	8.31547	0.347992	0.173996	−	0.984746i	$-0.444332\pi$
$571$	8.31547	0.347992	0.173996	+	0.984746i	$0.444332\pi$
$572$	−8.89778	−0.372035
$573$	0	0
$574$	81.6661	3.40868
$575$	−33.8788	−1.41284
$576$	0	0
$577$	−6.92363	−0.288234	−0.144117	−	0.989561i	$-0.546034\pi$
$577$	−6.92363	−0.288234	−0.144117	+	0.989561i	$0.546034\pi$
$578$	6.52714	0.271493
$579$	0	0
$580$	21.9306	0.910617
$581$	21.2798	0.882833
$582$	0	0
$583$	19.3035	0.799467
$584$	−34.6297	−1.43299
$585$	0	0
$586$	−22.6039	−0.933759
$587$	−17.0579	−0.704053	−0.352027	−	0.935990i	$-0.614507\pi$
$587$	−17.0579	−0.704053	−0.352027	+	0.935990i	$0.614507\pi$
$588$	0	0
$589$	0	0
$590$	−6.04894	−0.249031
$591$	0	0
$592$	10.7116	0.440243
$593$	−25.7470	−1.05730	−0.528651	−	0.848839i	$-0.677302\pi$
$593$	−25.7470	−1.05730	−0.528651	+	0.848839i	$0.677302\pi$
$594$	0	0
$595$	−15.0791	−0.618184
$596$	11.0582	0.452962
$597$	0	0
$598$	17.5351	0.717064
$599$	−16.9641	−0.693135	−0.346567	−	0.938025i	$-0.612653\pi$
$599$	−16.9641	−0.693135	−0.346567	+	0.938025i	$0.612653\pi$
$600$	0	0
$601$	−17.8906	−0.729772	−0.364886	−	0.931052i	$-0.618892\pi$
$601$	−17.8906	−0.729772	−0.364886	+	0.931052i	$0.618892\pi$
$602$	85.1923	3.47218
$603$	0	0
$604$	83.9244	3.41484
$605$	−5.72986	−0.232952
$606$	0	0
$607$	23.5578	0.956181	0.478090	−	0.878311i	$-0.341329\pi$
$607$	23.5578	0.956181	0.478090	+	0.878311i	$0.341329\pi$
$608$	−87.1622	−3.53490
$609$	0	0
$610$	−13.3514	−0.540583
$611$	−4.16820	−0.168627
$612$	0	0
$613$	−20.1491	−0.813814	−0.406907	−	0.913470i	$-0.633393\pi$
$613$	−20.1491	−0.813814	−0.406907	+	0.913470i	$0.633393\pi$
$614$	11.8381	0.477747
$615$	0	0
$616$	74.3159	2.99427
$617$	43.1404	1.73677	0.868384	−	0.495893i	$-0.165159\pi$
$617$	43.1404	1.73677	0.868384	+	0.495893i	$0.165159\pi$
$618$	0	0
$619$	−34.9607	−1.40519	−0.702596	−	0.711589i	$-0.747976\pi$
$619$	−34.9607	−1.40519	−0.702596	+	0.711589i	$0.747976\pi$
$620$	0	0
$621$	0	0
$622$	−88.7697	−3.55934
$623$	18.4479	0.739100
$624$	0	0
$625$	12.7875	0.511498
$626$	65.2274	2.60701
$627$	0	0
$628$	38.1266	1.52142
$629$	3.70863	0.147873
$630$	0	0
$631$	−11.9804	−0.476933	−0.238467	−	0.971151i	$-0.576645\pi$
$631$	−11.9804	−0.476933	−0.238467	+	0.971151i	$0.576645\pi$
$632$	50.2050	1.99705
$633$	0	0
$634$	2.07076	0.0822404
$635$	−14.8006	−0.587345
$636$	0	0
$637$	−8.98846	−0.356136
$638$	−27.4493	−1.08673
$639$	0	0
$640$	−7.44156	−0.294153
$641$	45.5574	1.79941	0.899704	−	0.436500i	$-0.143782\pi$
$641$	45.5574	1.79941	0.899704	+	0.436500i	$0.143782\pi$
$642$	0	0
$643$	−8.36135	−0.329739	−0.164870	−	0.986315i	$-0.552720\pi$
$643$	−8.36135	−0.329739	−0.164870	+	0.986315i	$0.552720\pi$
$644$	−174.335	−6.86976
$645$	0	0
$646$	−66.3489	−2.61046
$647$	−31.6818	−1.24554	−0.622770	−	0.782405i	$-0.713993\pi$
$647$	−31.6818	−1.24554	−0.622770	+	0.782405i	$0.713993\pi$
$648$	0	0
$649$	5.40836	0.212297
$650$	−8.85436	−0.347296
$651$	0	0
$652$	−76.4780	−2.99511
$653$	19.0921	0.747133	0.373567	−	0.927603i	$-0.378135\pi$
$653$	19.0921	0.747133	0.373567	+	0.927603i	$0.378135\pi$
$654$	0	0
$655$	−7.25534	−0.283490
$656$	79.8572	3.11790
$657$	0	0
$658$	58.0121	2.26155
$659$	8.27142	0.322209	0.161104	−	0.986937i	$-0.448494\pi$
$659$	8.27142	0.322209	0.161104	+	0.986937i	$0.448494\pi$
$660$	0	0
$661$	−16.1366	−0.627642	−0.313821	−	0.949482i	$-0.601609\pi$
$661$	−16.1366	−0.627642	−0.313821	+	0.949482i	$0.601609\pi$
$662$	−87.8878	−3.41586
$663$	0	0
$664$	39.7117	1.54111
$665$	−26.0170	−1.00890
$666$	0	0
$667$	38.6423	1.49624
$668$	−51.8263	−2.00522
$669$	0	0
$670$	19.0592	0.736322
$671$	11.9375	0.460842
$672$	0	0
$673$	21.2293	0.818328	0.409164	−	0.912461i	$-0.365820\pi$
$673$	21.2293	0.818328	0.409164	+	0.912461i	$0.365820\pi$
$674$	7.76884	0.299244
$675$	0	0
$676$	−61.7439	−2.37476
$677$	10.1797	0.391237	0.195618	−	0.980680i	$-0.437329\pi$
$677$	10.1797	0.391237	0.195618	+	0.980680i	$0.437329\pi$
$678$	0	0
$679$	49.5066	1.89989
$680$	−28.1402	−1.07913
$681$	0	0
$682$	0	0
$683$	−40.9292	−1.56611	−0.783056	−	0.621952i	$-0.786340\pi$
$683$	−40.9292	−1.56611	−0.783056	+	0.621952i	$0.786340\pi$
$684$	0	0
$685$	−3.63864	−0.139025
$686$	46.2777	1.76689
$687$	0	0
$688$	83.3053	3.17598
$689$	−7.10226	−0.270575
$690$	0	0
$691$	−42.1215	−1.60238	−0.801189	−	0.598412i	$-0.795799\pi$
$691$	−42.1215	−1.60238	−0.801189	+	0.598412i	$0.795799\pi$
$692$	−60.4774	−2.29901
$693$	0	0
$694$	24.0101	0.911411
$695$	19.9814	0.757936
$696$	0	0
$697$	27.6487	1.04727
$698$	27.2147	1.03009
$699$	0	0
$700$	88.0305	3.32724
$701$	7.38302	0.278853	0.139426	−	0.990232i	$-0.455474\pi$
$701$	7.38302	0.278853	0.139426	+	0.990232i	$0.455474\pi$
$702$	0	0
$703$	6.39875	0.241333
$704$	28.6788	1.08087
$705$	0	0
$706$	11.8195	0.444834
$707$	−3.02668	−0.113830
$708$	0	0
$709$	−34.7134	−1.30369	−0.651844	−	0.758353i	$-0.726004\pi$
$709$	−34.7134	−1.30369	−0.651844	+	0.758353i	$0.726004\pi$
$710$	6.65206	0.249647
$711$	0	0
$712$	34.4270	1.29020
$713$	0	0
$714$	0	0
$715$	−1.65361	−0.0618414
$716$	45.1213	1.68626
$717$	0	0
$718$	64.2906	2.39930
$719$	−14.2635	−0.531937	−0.265969	−	0.963982i	$-0.585692\pi$
$719$	−14.2635	−0.531937	−0.265969	+	0.963982i	$0.585692\pi$
$720$	0	0
$721$	−15.5558	−0.579330
$722$	−64.2018	−2.38935
$723$	0	0
$724$	91.0933	3.38545
$725$	−19.5124	−0.724674
$726$	0	0
$727$	9.30648	0.345158	0.172579	−	0.984996i	$-0.444790\pi$
$727$	9.30648	0.345158	0.172579	+	0.984996i	$0.444790\pi$
$728$	−27.3428	−1.01339
$729$	0	0
$730$	−10.7243	−0.396925
$731$	28.8425	1.06678
$732$	0	0
$733$	−19.3446	−0.714511	−0.357255	−	0.934007i	$-0.616287\pi$
$733$	−19.3446	−0.714511	−0.357255	+	0.934007i	$0.616287\pi$
$734$	−13.1047	−0.483702
$735$	0	0
$736$	−108.544	−4.00099
$737$	−17.0409	−0.627708
$738$	0	0
$739$	28.1155	1.03425	0.517123	−	0.855911i	$-0.327003\pi$
$739$	28.1155	1.03425	0.517123	+	0.855911i	$0.327003\pi$
$740$	4.52230	0.166243
$741$	0	0
$742$	98.8478	3.62882
$743$	−20.5287	−0.753125	−0.376563	−	0.926391i	$-0.622894\pi$
$743$	−20.5287	−0.753125	−0.376563	+	0.926391i	$0.622894\pi$
$744$	0	0
$745$	2.05511	0.0752935
$746$	12.4775	0.456834
$747$	0	0
$748$	41.9261	1.53297
$749$	38.5131	1.40724
$750$	0	0
$751$	−14.3842	−0.524887	−0.262443	−	0.964947i	$-0.584528\pi$
$751$	−14.3842	−0.524887	−0.262443	+	0.964947i	$0.584528\pi$
$752$	56.7272	2.06863
$753$	0	0
$754$	10.0993	0.367796
$755$	15.5969	0.567631
$756$	0	0
$757$	−5.48680	−0.199421	−0.0997107	−	0.995016i	$-0.531792\pi$
$757$	−5.48680	−0.199421	−0.0997107	+	0.995016i	$0.531792\pi$
$758$	21.1700	0.768929
$759$	0	0
$760$	−48.5522	−1.76117
$761$	−6.63649	−0.240573	−0.120286	−	0.992739i	$-0.538381\pi$
$761$	−6.63649	−0.240573	−0.120286	+	0.992739i	$0.538381\pi$
$762$	0	0
$763$	13.8574	0.501671
$764$	92.0822	3.33142
$765$	0	0
$766$	−70.0744	−2.53189
$767$	−1.98988	−0.0718505
$768$	0	0
$769$	−43.9483	−1.58482	−0.792408	−	0.609991i	$-0.791173\pi$
$769$	−43.9483	−1.58482	−0.792408	+	0.609991i	$0.791173\pi$
$770$	23.0146	0.829387
$771$	0	0
$772$	56.9363	2.04918
$773$	−52.4432	−1.88625	−0.943125	−	0.332439i	$-0.892128\pi$
$773$	−52.4432	−1.88625	−0.943125	+	0.332439i	$0.892128\pi$
$774$	0	0
$775$	0	0
$776$	92.3877	3.31653
$777$	0	0
$778$	−66.1459	−2.37144
$779$	47.7040	1.70917
$780$	0	0
$781$	−5.94761	−0.212822
$782$	−82.6250	−2.95467
$783$	0	0
$784$	122.329	4.36888
$785$	7.08564	0.252897
$786$	0	0
$787$	3.34342	0.119180	0.0595899	−	0.998223i	$-0.481021\pi$
$787$	3.34342	0.119180	0.0595899	+	0.998223i	$0.481021\pi$
$788$	−23.3518	−0.831875
$789$	0	0
$790$	15.5478	0.553165
$791$	7.69949	0.273762
$792$	0	0
$793$	−4.39213	−0.155969
$794$	76.5524	2.71674
$795$	0	0
$796$	−117.863	−4.17755
$797$	−5.23448	−0.185415	−0.0927075	−	0.995693i	$-0.529552\pi$
$797$	−5.23448	−0.185415	−0.0927075	+	0.995693i	$0.529552\pi$
$798$	0	0
$799$	19.6404	0.694829
$800$	54.8094	1.93780
$801$	0	0
$802$	−95.2864	−3.36468
$803$	9.58862	0.338375
$804$	0	0
$805$	−32.3993	−1.14193
$806$	0	0
$807$	0	0
$808$	−5.64830	−0.198706
$809$	−51.4386	−1.80848	−0.904242	−	0.427020i	$-0.859563\pi$
$809$	−51.4386	−1.80848	−0.904242	+	0.427020i	$0.859563\pi$
$810$	0	0
$811$	21.9367	0.770303	0.385152	−	0.922853i	$-0.374149\pi$
$811$	21.9367	0.770303	0.385152	+	0.922853i	$0.374149\pi$
$812$	−100.408	−3.52363
$813$	0	0
$814$	−5.66031	−0.198394
$815$	−14.2131	−0.497861
$816$	0	0
$817$	49.7638	1.74102
$818$	−13.7343	−0.480208
$819$	0	0
$820$	33.7147	1.17737
$821$	−8.65275	−0.301983	−0.150992	−	0.988535i	$-0.548247\pi$
$821$	−8.65275	−0.301983	−0.150992	+	0.988535i	$0.548247\pi$
$822$	0	0
$823$	−29.4022	−1.02490	−0.512448	−	0.858718i	$-0.671261\pi$
$823$	−29.4022	−1.02490	−0.512448	+	0.858718i	$0.671261\pi$
$824$	−29.0298	−1.01130
$825$	0	0
$826$	27.6948	0.963624
$827$	20.4659	0.711669	0.355834	−	0.934549i	$-0.384197\pi$
$827$	20.4659	0.711669	0.355834	+	0.934549i	$0.384197\pi$
$828$	0	0
$829$	22.0541	0.765970	0.382985	−	0.923755i	$-0.374896\pi$
$829$	22.0541	0.765970	0.382985	+	0.923755i	$0.374896\pi$
$830$	12.2981	0.426875
$831$	0	0
$832$	−10.5517	−0.365815
$833$	42.3534	1.46746
$834$	0	0
$835$	−9.63166	−0.333317
$836$	72.3378	2.50186
$837$	0	0
$838$	44.8769	1.55025
$839$	−20.3565	−0.702784	−0.351392	−	0.936228i	$-0.614292\pi$
$839$	−20.3565	−0.702784	−0.351392	+	0.936228i	$0.614292\pi$
$840$	0	0
$841$	−6.74401	−0.232552
$842$	−51.0708	−1.76002
$843$	0	0
$844$	133.564	4.59747
$845$	−11.4748	−0.394745
$846$	0	0
$847$	26.2339	0.901406
$848$	96.6584	3.31926
$849$	0	0
$850$	41.7215	1.43104
$851$	7.96844	0.273154
$852$	0	0
$853$	21.0177	0.719633	0.359817	−	0.933023i	$-0.382839\pi$
$853$	21.0177	0.719633	0.359817	+	0.933023i	$0.382839\pi$
$854$	61.1288	2.09178
$855$	0	0
$856$	71.8720	2.45653
$857$	29.2222	0.998211	0.499106	−	0.866541i	$-0.333662\pi$
$857$	29.2222	0.998211	0.499106	+	0.866541i	$0.333662\pi$
$858$	0	0
$859$	27.8940	0.951729	0.475865	−	0.879519i	$-0.342135\pi$
$859$	27.8940	0.951729	0.475865	+	0.879519i	$0.342135\pi$
$860$	35.1705	1.19930
$861$	0	0
$862$	30.9891	1.05549
$863$	39.2606	1.33645	0.668224	−	0.743960i	$-0.267055\pi$
$863$	39.2606	1.33645	0.668224	+	0.743960i	$0.267055\pi$
$864$	0	0
$865$	−11.2394	−0.382152
$866$	43.2142	1.46848
$867$	0	0
$868$	0	0
$869$	−13.9013	−0.471568
$870$	0	0
$871$	6.26979	0.212444
$872$	25.8602	0.875738
$873$	0	0
$874$	−142.558	−4.82211
$875$	36.1373	1.22166
$876$	0	0
$877$	−7.01238	−0.236791	−0.118396	−	0.992966i	$-0.537775\pi$
$877$	−7.01238	−0.236791	−0.118396	+	0.992966i	$0.537775\pi$
$878$	−45.1755	−1.52460
$879$	0	0
$880$	22.5048	0.758637
$881$	−28.3259	−0.954323	−0.477161	−	0.878816i	$-0.658334\pi$
$881$	−28.3259	−0.954323	−0.477161	+	0.878816i	$0.658334\pi$
$882$	0	0
$883$	25.4273	0.855697	0.427848	−	0.903851i	$-0.359272\pi$
$883$	25.4273	0.855697	0.427848	+	0.903851i	$0.359272\pi$
$884$	−15.4258	−0.518824
$885$	0	0
$886$	−61.2673	−2.05832
$887$	37.4228	1.25653	0.628267	−	0.777998i	$-0.283765\pi$
$887$	37.4228	1.25653	0.628267	+	0.777998i	$0.283765\pi$
$888$	0	0
$889$	67.7640	2.27273
$890$	10.6615	0.357376
$891$	0	0
$892$	4.22854	0.141582
$893$	33.8869	1.13398
$894$	0	0
$895$	8.38556	0.280298
$896$	34.0708	1.13823
$897$	0	0
$898$	−49.7798	−1.66117
$899$	0	0
$900$	0	0
$901$	33.4657	1.11490
$902$	−42.1988	−1.40507
$903$	0	0
$904$	14.3686	0.477891
$905$	16.9292	0.562747
$906$	0	0
$907$	−24.4906	−0.813197	−0.406599	−	0.913607i	$-0.633285\pi$
$907$	−24.4906	−0.813197	−0.406599	+	0.913607i	$0.633285\pi$
$908$	82.5488	2.73948
$909$	0	0
$910$	−8.46768	−0.280701
$911$	2.54207	0.0842225	0.0421112	−	0.999113i	$-0.486592\pi$
$911$	2.54207	0.0842225	0.0421112	+	0.999113i	$0.486592\pi$
$912$	0	0
$913$	−10.9958	−0.363907
$914$	62.4449	2.06549
$915$	0	0
$916$	−11.7460	−0.388099
$917$	33.2182	1.09696
$918$	0	0
$919$	29.3927	0.969575	0.484787	−	0.874632i	$-0.338897\pi$
$919$	29.3927	0.969575	0.484787	+	0.874632i	$0.338897\pi$
$920$	−60.4626	−1.99339
$921$	0	0
$922$	0.957806	0.0315437
$923$	2.18829	0.0720284
$924$	0	0
$925$	−4.02366	−0.132297
$926$	6.21126	0.204115
$927$	0	0
$928$	−62.5159	−2.05218
$929$	−6.36434	−0.208807	−0.104404	−	0.994535i	$-0.533293\pi$
$929$	−6.36434	−0.208807	−0.104404	+	0.994535i	$0.533293\pi$
$930$	0	0
$931$	73.0750	2.39494
$932$	−4.21875	−0.138190
$933$	0	0
$934$	87.2950	2.85638
$935$	7.79176	0.254818
$936$	0	0
$937$	−37.0141	−1.20920	−0.604599	−	0.796530i	$-0.706667\pi$
$937$	−37.0141	−1.20920	−0.604599	+	0.796530i	$0.706667\pi$
$938$	−87.2617	−2.84920
$939$	0	0
$940$	23.9495	0.781147
$941$	41.0148	1.33704	0.668521	−	0.743693i	$-0.266928\pi$
$941$	41.0148	1.33704	0.668521	+	0.743693i	$0.266928\pi$
$942$	0	0
$943$	59.4064	1.93454
$944$	27.0813	0.881422
$945$	0	0
$946$	−44.0209	−1.43124
$947$	1.14888	0.0373336	0.0186668	−	0.999826i	$-0.494058\pi$
$947$	1.14888	0.0373336	0.0186668	+	0.999826i	$0.494058\pi$
$948$	0	0
$949$	−3.52791	−0.114521
$950$	71.9848	2.33550
$951$	0	0
$952$	128.839	4.17569
$953$	42.4551	1.37526	0.687628	−	0.726064i	$-0.258652\pi$
$953$	42.4551	1.37526	0.687628	+	0.726064i	$0.258652\pi$
$954$	0	0
$955$	17.1130	0.553765
$956$	−74.1332	−2.39764
$957$	0	0
$958$	−77.7664	−2.51252
$959$	16.6593	0.537958
$960$	0	0
$961$	0	0
$962$	2.08258	0.0671451
$963$	0	0
$964$	−34.5384	−1.11241
$965$	10.5813	0.340625
$966$	0	0
$967$	−58.8653	−1.89298	−0.946490	−	0.322733i	$-0.895399\pi$
$967$	−58.8653	−1.89298	−0.946490	+	0.322733i	$0.895399\pi$
$968$	48.9569	1.57353
$969$	0	0
$970$	28.6112	0.918649
$971$	56.0174	1.79768	0.898842	−	0.438272i	$-0.144409\pi$
$971$	56.0174	1.79768	0.898842	+	0.438272i	$0.144409\pi$
$972$	0	0
$973$	−91.4837	−2.93283
$974$	109.250	3.50059
$975$	0	0
$976$	59.7748	1.91334
$977$	−59.1759	−1.89320	−0.946602	−	0.322404i	$-0.895509\pi$
$977$	−59.1759	−1.89320	−0.946602	+	0.322404i	$0.895509\pi$
$978$	0	0
$979$	−9.53249	−0.304660
$980$	51.6456	1.64976
$981$	0	0
$982$	30.4554	0.971870
$983$	24.2729	0.774184	0.387092	−	0.922041i	$-0.373480\pi$
$983$	24.2729	0.774184	0.387092	+	0.922041i	$0.373480\pi$
$984$	0	0
$985$	−4.33982	−0.138278
$986$	−47.5878	−1.51550
$987$	0	0
$988$	−26.6150	−0.846737
$989$	61.9715	1.97058
$990$	0	0
$991$	17.5051	0.556066	0.278033	−	0.960571i	$-0.410317\pi$
$991$	17.5051	0.556066	0.278033	+	0.960571i	$0.410317\pi$
$992$	0	0
$993$	0	0
$994$	−30.4561	−0.966010
$995$	−21.9043	−0.694413
$996$	0	0
$997$	−21.2560	−0.673185	−0.336592	−	0.941650i	$-0.609274\pi$
$997$	−21.2560	−0.673185	−0.336592	+	0.941650i	$0.609274\pi$
$998$	−54.7747	−1.73386
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bv.1.1		24
3.2	odd	2		2883.2.a.u.1.24	✓	24
31.30	odd	2		8649.2.a.bu.1.1		24
93.92	even	2		2883.2.a.v.1.24	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.24	✓	24	3.2	odd	2
2883.2.a.v.1.24	yes	24	93.92	even	2
8649.2.a.bu.1.1		24	31.30	odd	2
8649.2.a.bv.1.1		24	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)

\(f(q)\)	\(=\)	\(q-2.64601 q^{2} +5.00135 q^{4} +0.929477 q^{5} -4.25556 q^{7} -7.94161 q^{8} +O(q^{10})\)	\(q-2.64601 q^{2} +5.00135 q^{4} +0.929477 q^{5} -4.25556 q^{7} -7.94161 q^{8} -2.45940 q^{10} +2.19895 q^{11} -0.809055 q^{13} +11.2603 q^{14} +11.0108 q^{16} +3.81225 q^{17} +6.57751 q^{19} +4.64864 q^{20} -5.81845 q^{22} +8.19106 q^{23} -4.13607 q^{25} +2.14076 q^{26} -21.2836 q^{28} +4.71763 q^{29} -13.2516 q^{32} -10.0872 q^{34} -3.95545 q^{35} +0.972822 q^{37} -17.4041 q^{38} -7.38154 q^{40} +7.25260 q^{41} +7.56575 q^{43} +10.9977 q^{44} -21.6736 q^{46} +5.15194 q^{47} +11.1098 q^{49} +10.9441 q^{50} -4.04637 q^{52} +8.77847 q^{53} +2.04388 q^{55} +33.7960 q^{56} -12.4829 q^{58} +2.45952 q^{59} +5.42872 q^{61} +13.0420 q^{64} -0.751998 q^{65} -7.74953 q^{67} +19.0664 q^{68} +10.4661 q^{70} -2.70475 q^{71} +4.36054 q^{73} -2.57409 q^{74} +32.8965 q^{76} -9.35779 q^{77} -6.32176 q^{79} +10.2343 q^{80} -19.1904 q^{82} -5.00046 q^{83} +3.54339 q^{85} -20.0190 q^{86} -17.4632 q^{88} -4.33501 q^{89} +3.44298 q^{91} +40.9664 q^{92} -13.6321 q^{94} +6.11365 q^{95} -11.6334 q^{97} -29.3967 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} + 16 q^{11} - 32 q^{13} + 24 q^{14} + 48 q^{16} + 32 q^{17} + 32 q^{19} + 24 q^{20} - 32 q^{22} + 32 q^{23} + 40 q^{25} + 16 q^{26} + 8 q^{28} + 48 q^{29} + 48 q^{32} + 48 q^{35} - 64 q^{37} + 24 q^{38} - 32 q^{43} + 48 q^{44} - 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} - 64 q^{52} + 80 q^{53} + 48 q^{56} - 32 q^{58} - 32 q^{61} + 56 q^{64} + 16 q^{65} - 16 q^{67} + 80 q^{68} + 8 q^{70} - 32 q^{73} + 56 q^{76} + 96 q^{77} - 32 q^{79} + 72 q^{80} + 8 q^{82} + 48 q^{83} - 96 q^{85} - 32 q^{86} - 96 q^{88} - 16 q^{89} + 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 + 16 * q^11 - 32 * q^13 + 24 * q^14 + 48 * q^16 + 32 * q^17 + 32 * q^19 + 24 * q^20 - 32 * q^22 + 32 * q^23 + 40 * q^25 + 16 * q^26 + 8 * q^28 + 48 * q^29 + 48 * q^32 + 48 * q^35 - 64 * q^37 + 24 * q^38 - 32 * q^43 + 48 * q^44 - 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 - 64 * q^52 + 80 * q^53 + 48 * q^56 - 32 * q^58 - 32 * q^61 + 56 * q^64 + 16 * q^65 - 16 * q^67 + 80 * q^68 + 8 * q^70 - 32 * q^73 + 56 * q^76 + 96 * q^77 - 32 * q^79 + 72 * q^80 + 8 * q^82 + 48 * q^83 - 96 * q^85 - 32 * q^86 - 96 * q^88 - 16 * q^89 + 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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