LMFDB - Embedded newform 4023.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, 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("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35944 q^{2} -0.151922 q^{4} -3.36242 q^{5} +2.75034 q^{7} -2.92541 q^{8} +O(q^{10})$	\(q+1.35944 q^{2} -0.151922 q^{4} -3.36242 q^{5} +2.75034 q^{7} -2.92541 q^{8} -4.57100 q^{10} +0.0931303 q^{11} +3.43239 q^{13} +3.73892 q^{14} -3.67308 q^{16} +3.92305 q^{17} -0.647673 q^{19} +0.510825 q^{20} +0.126605 q^{22} -9.37943 q^{23} +6.30584 q^{25} +4.66614 q^{26} -0.417837 q^{28} +4.54961 q^{29} -3.37723 q^{31} +0.857492 q^{32} +5.33316 q^{34} -9.24779 q^{35} +0.224203 q^{37} -0.880473 q^{38} +9.83644 q^{40} -10.6942 q^{41} +6.39210 q^{43} -0.0141485 q^{44} -12.7508 q^{46} +7.32483 q^{47} +0.564373 q^{49} +8.57241 q^{50} -0.521457 q^{52} -6.61967 q^{53} -0.313143 q^{55} -8.04587 q^{56} +6.18493 q^{58} -6.63325 q^{59} +0.241485 q^{61} -4.59115 q^{62} +8.51186 q^{64} -11.5411 q^{65} -9.03657 q^{67} -0.595998 q^{68} -12.5718 q^{70} -1.03162 q^{71} +9.39701 q^{73} +0.304791 q^{74} +0.0983959 q^{76} +0.256140 q^{77} +0.898603 q^{79} +12.3504 q^{80} -14.5381 q^{82} -6.81360 q^{83} -13.1909 q^{85} +8.68967 q^{86} -0.272444 q^{88} -12.7620 q^{89} +9.44025 q^{91} +1.42494 q^{92} +9.95766 q^{94} +2.17775 q^{95} -6.44130 q^{97} +0.767232 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * 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 $

$2$	1.35944	0.961269	0.480635	−	0.876921i	$-0.340406\pi$
$2$	1.35944	0.961269	0.480635	+	0.876921i	$0.340406\pi$
$3$	0	0
$3$	0	0
$4$	−0.151922	−0.0759610
$5$	−3.36242	−1.50372	−0.751859	−	0.659324i	$-0.770843\pi$
$5$	−3.36242	−1.50372	−0.751859	+	0.659324i	$0.770843\pi$
$6$	0	0
$7$	2.75034	1.03953	0.519766	−	0.854309i	$-0.326019\pi$
$7$	2.75034	1.03953	0.519766	+	0.854309i	$0.326019\pi$
$8$	−2.92541	−1.03429
$9$	0	0
$10$	−4.57100	−1.44548
$11$	0.0931303	0.0280798	0.0140399	−	0.999901i	$-0.495531\pi$
$11$	0.0931303	0.0280798	0.0140399	+	0.999901i	$0.495531\pi$
$12$	0	0
$13$	3.43239	0.951975	0.475988	−	0.879452i	$-0.342091\pi$
$13$	3.43239	0.951975	0.475988	+	0.879452i	$0.342091\pi$
$14$	3.73892	0.999269
$15$	0	0
$16$	−3.67308	−0.918269
$17$	3.92305	0.951480	0.475740	−	0.879586i	$-0.342180\pi$
$17$	3.92305	0.951480	0.475740	+	0.879586i	$0.342180\pi$
$18$	0	0
$19$	−0.647673	−0.148586	−0.0742932	−	0.997236i	$-0.523670\pi$
$19$	−0.647673	−0.148586	−0.0742932	+	0.997236i	$0.523670\pi$
$20$	0.510825	0.114224
$21$	0	0
$22$	0.126605	0.0269923
$23$	−9.37943	−1.95575	−0.977874	−	0.209197i	$-0.932915\pi$
$23$	−9.37943	−1.95575	−0.977874	+	0.209197i	$0.932915\pi$
$24$	0	0
$25$	6.30584	1.26117
$26$	4.66614	0.915105
$27$	0	0
$28$	−0.417837	−0.0789639
$29$	4.54961	0.844842	0.422421	−	0.906400i	$-0.361180\pi$
$29$	4.54961	0.844842	0.422421	+	0.906400i	$0.361180\pi$
$30$	0	0
$31$	−3.37723	−0.606569	−0.303284	−	0.952900i	$-0.598083\pi$
$31$	−3.37723	−0.606569	−0.303284	+	0.952900i	$0.598083\pi$
$32$	0.857492	0.151585
$33$	0	0
$34$	5.33316	0.914629
$35$	−9.24779	−1.56316
$36$	0	0
$37$	0.224203	0.0368588	0.0184294	−	0.999830i	$-0.494133\pi$
$37$	0.224203	0.0368588	0.0184294	+	0.999830i	$0.494133\pi$
$38$	−0.880473	−0.142832
$39$	0	0
$40$	9.83644	1.55528
$41$	−10.6942	−1.67015	−0.835073	−	0.550138i	$-0.814575\pi$
$41$	−10.6942	−1.67015	−0.835073	+	0.550138i	$0.814575\pi$
$42$	0	0
$43$	6.39210	0.974786	0.487393	−	0.873183i	$-0.337948\pi$
$43$	6.39210	0.974786	0.487393	+	0.873183i	$0.337948\pi$
$44$	−0.0141485	−0.00213297
$45$	0	0
$46$	−12.7508	−1.88000
$47$	7.32483	1.06844	0.534218	−	0.845347i	$-0.320606\pi$
$47$	7.32483	1.06844	0.534218	+	0.845347i	$0.320606\pi$
$48$	0	0
$49$	0.564373	0.0806247
$50$	8.57241	1.21232
$51$	0	0
$52$	−0.521457	−0.0723130
$53$	−6.61967	−0.909282	−0.454641	−	0.890675i	$-0.650232\pi$
$53$	−6.61967	−0.909282	−0.454641	+	0.890675i	$0.650232\pi$
$54$	0	0
$55$	−0.313143	−0.0422241
$56$	−8.04587	−1.07517
$57$	0	0
$58$	6.18493	0.812121
$59$	−6.63325	−0.863575	−0.431788	−	0.901975i	$-0.642117\pi$
$59$	−6.63325	−0.863575	−0.431788	+	0.901975i	$0.642117\pi$
$60$	0	0
$61$	0.241485	0.0309190	0.0154595	−	0.999880i	$-0.495079\pi$
$61$	0.241485	0.0309190	0.0154595	+	0.999880i	$0.495079\pi$
$62$	−4.59115	−0.583076
$63$	0	0
$64$	8.51186	1.06398
$65$	−11.5411	−1.43150
$66$	0	0
$67$	−9.03657	−1.10399	−0.551996	−	0.833846i	$-0.686134\pi$
$67$	−9.03657	−1.10399	−0.551996	+	0.833846i	$0.686134\pi$
$68$	−0.595998	−0.0722754
$69$	0	0
$70$	−12.5718	−1.50262
$71$	−1.03162	−0.122431	−0.0612154	−	0.998125i	$-0.519498\pi$
$71$	−1.03162	−0.122431	−0.0612154	+	0.998125i	$0.519498\pi$
$72$	0	0
$73$	9.39701	1.09984	0.549918	−	0.835218i	$-0.314659\pi$
$73$	9.39701	1.09984	0.549918	+	0.835218i	$0.314659\pi$
$74$	0.304791	0.0354313
$75$	0	0
$76$	0.0983959	0.0112868
$77$	0.256140	0.0291899
$78$	0	0
$79$	0.898603	0.101101	0.0505504	−	0.998722i	$-0.483902\pi$
$79$	0.898603	0.101101	0.0505504	+	0.998722i	$0.483902\pi$
$80$	12.3504	1.38082
$81$	0	0
$82$	−14.5381	−1.60546
$83$	−6.81360	−0.747890	−0.373945	−	0.927451i	$-0.621995\pi$
$83$	−6.81360	−0.747890	−0.373945	+	0.927451i	$0.621995\pi$
$84$	0	0
$85$	−13.1909	−1.43076
$86$	8.68967	0.937032
$87$	0	0
$88$	−0.272444	−0.0290426
$89$	−12.7620	−1.35277	−0.676387	−	0.736546i	$-0.736455\pi$
$89$	−12.7620	−1.35277	−0.676387	+	0.736546i	$0.736455\pi$
$90$	0	0
$91$	9.44025	0.989608
$92$	1.42494	0.148561
$93$	0	0
$94$	9.95766	1.02705
$95$	2.17775	0.223432
$96$	0	0
$97$	−6.44130	−0.654015	−0.327007	−	0.945022i	$-0.606040\pi$
$97$	−6.44130	−0.654015	−0.327007	+	0.945022i	$0.606040\pi$
$98$	0.767232	0.0775021
$99$	0	0
$100$	−0.957997	−0.0957997
$101$	−18.8974	−1.88036	−0.940180	−	0.340679i	$-0.889343\pi$
$101$	−18.8974	−1.88036	−0.940180	+	0.340679i	$0.889343\pi$
$102$	0	0
$103$	−7.65224	−0.753998	−0.376999	−	0.926214i	$-0.623044\pi$
$103$	−7.65224	−0.753998	−0.376999	+	0.926214i	$0.623044\pi$
$104$	−10.0412	−0.984617
$105$	0	0
$106$	−8.99905	−0.874065
$107$	10.2125	0.987279	0.493640	−	0.869667i	$-0.335666\pi$
$107$	10.2125	0.987279	0.493640	+	0.869667i	$0.335666\pi$
$108$	0	0
$109$	−20.3357	−1.94781	−0.973904	−	0.226959i	$-0.927122\pi$
$109$	−20.3357	−1.94781	−0.973904	+	0.226959i	$0.927122\pi$
$110$	−0.425699	−0.0405888
$111$	0	0
$112$	−10.1022	−0.954569
$113$	5.12073	0.481718	0.240859	−	0.970560i	$-0.422571\pi$
$113$	5.12073	0.481718	0.240859	+	0.970560i	$0.422571\pi$
$114$	0	0
$115$	31.5376	2.94089
$116$	−0.691187	−0.0641751
$117$	0	0
$118$	−9.01750	−0.830129
$119$	10.7897	0.989093
$120$	0	0
$121$	−10.9913	−0.999212
$122$	0.328284	0.0297215
$123$	0	0
$124$	0.513076	0.0460756
$125$	−4.39078	−0.392724
$126$	0	0
$127$	−11.7183	−1.03983	−0.519914	−	0.854218i	$-0.674036\pi$
$127$	−11.7183	−1.03983	−0.519914	+	0.854218i	$0.674036\pi$
$128$	9.85638	0.871189
$129$	0	0
$130$	−15.6895	−1.37606
$131$	−11.0299	−0.963686	−0.481843	−	0.876258i	$-0.660032\pi$
$131$	−11.0299	−0.963686	−0.481843	+	0.876258i	$0.660032\pi$
$132$	0	0
$133$	−1.78132	−0.154460
$134$	−12.2847	−1.06123
$135$	0	0
$136$	−11.4765	−0.984105
$137$	−7.46690	−0.637941	−0.318970	−	0.947765i	$-0.603337\pi$
$137$	−7.46690	−0.637941	−0.318970	+	0.947765i	$0.603337\pi$
$138$	0	0
$139$	−1.18716	−0.100694	−0.0503469	−	0.998732i	$-0.516033\pi$
$139$	−1.18716	−0.100694	−0.0503469	+	0.998732i	$0.516033\pi$
$140$	1.40494	0.118739
$141$	0	0
$142$	−1.40243	−0.117689
$143$	0.319660	0.0267313
$144$	0	0
$145$	−15.2977	−1.27040
$146$	12.7747	1.05724
$147$	0	0
$148$	−0.0340615	−0.00279983
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	11.0165	0.896508	0.448254	−	0.893906i	$-0.352046\pi$
$151$	11.0165	0.896508	0.448254	+	0.893906i	$0.352046\pi$
$152$	1.89471	0.153681
$153$	0	0
$154$	0.348207	0.0280593
$155$	11.3557	0.912109
$156$	0	0
$157$	−9.85825	−0.786774	−0.393387	−	0.919373i	$-0.628697\pi$
$157$	−9.85825	−0.786774	−0.393387	+	0.919373i	$0.628697\pi$
$158$	1.22160	0.0971851
$159$	0	0
$160$	−2.88325	−0.227941
$161$	−25.7966	−2.03306
$162$	0	0
$163$	−13.6116	−1.06615	−0.533073	−	0.846070i	$-0.678963\pi$
$163$	−13.6116	−1.06615	−0.533073	+	0.846070i	$0.678963\pi$
$164$	1.62468	0.126866
$165$	0	0
$166$	−9.26268	−0.718924
$167$	9.50893	0.735823	0.367912	−	0.929861i	$-0.380073\pi$
$167$	9.50893	0.735823	0.367912	+	0.929861i	$0.380073\pi$
$168$	0	0
$169$	−1.21866	−0.0937435
$170$	−17.9323	−1.37534
$171$	0	0
$172$	−0.971101	−0.0740457
$173$	−6.22942	−0.473614	−0.236807	−	0.971557i	$-0.576101\pi$
$173$	−6.22942	−0.473614	−0.236807	+	0.971557i	$0.576101\pi$
$174$	0	0
$175$	17.3432	1.31102
$176$	−0.342074	−0.0257848
$177$	0	0
$178$	−17.3492	−1.30038
$179$	5.42894	0.405778	0.202889	−	0.979202i	$-0.434967\pi$
$179$	5.42894	0.405778	0.202889	+	0.979202i	$0.434967\pi$
$180$	0	0
$181$	−13.2384	−0.984004	−0.492002	−	0.870594i	$-0.663735\pi$
$181$	−13.2384	−0.984004	−0.492002	+	0.870594i	$0.663735\pi$
$182$	12.8335	0.951280
$183$	0	0
$184$	27.4387	2.02281
$185$	−0.753865	−0.0554253
$186$	0	0
$187$	0.365355	0.0267174
$188$	−1.11280	−0.0811595
$189$	0	0
$190$	2.96052	0.214778
$191$	−20.1828	−1.46037	−0.730187	−	0.683248i	$-0.760567\pi$
$191$	−20.1828	−1.46037	−0.730187	+	0.683248i	$0.760567\pi$
$192$	0	0
$193$	12.8773	0.926927	0.463463	−	0.886116i	$-0.346607\pi$
$193$	12.8773	0.926927	0.463463	+	0.886116i	$0.346607\pi$
$194$	−8.75656	−0.628685
$195$	0	0
$196$	−0.0857407	−0.00612434
$197$	19.2434	1.37103	0.685516	−	0.728057i	$-0.259576\pi$
$197$	19.2434	1.37103	0.685516	+	0.728057i	$0.259576\pi$
$198$	0	0
$199$	5.89383	0.417803	0.208901	−	0.977937i	$-0.433011\pi$
$199$	5.89383	0.417803	0.208901	+	0.977937i	$0.433011\pi$
$200$	−18.4472	−1.30441
$201$	0	0
$202$	−25.6899	−1.80753
$203$	12.5130	0.878240
$204$	0	0
$205$	35.9582	2.51143
$206$	−10.4028	−0.724795
$207$	0	0
$208$	−12.6074	−0.874169
$209$	−0.0603180	−0.00417228
$210$	0	0
$211$	−9.13648	−0.628982	−0.314491	−	0.949261i	$-0.601834\pi$
$211$	−9.13648	−0.628982	−0.314491	+	0.949261i	$0.601834\pi$
$212$	1.00567	0.0690700
$213$	0	0
$214$	13.8833	0.949041
$215$	−21.4929	−1.46580
$216$	0	0
$217$	−9.28854	−0.630547
$218$	−27.6452	−1.87237
$219$	0	0
$220$	0.0475733	0.00320739
$221$	13.4655	0.905785
$222$	0	0
$223$	−5.22163	−0.349666	−0.174833	−	0.984598i	$-0.555939\pi$
$223$	−5.22163	−0.349666	−0.174833	+	0.984598i	$0.555939\pi$
$224$	2.35840	0.157577
$225$	0	0
$226$	6.96133	0.463061
$227$	−12.0534	−0.800013	−0.400006	−	0.916512i	$-0.630992\pi$
$227$	−12.0534	−0.800013	−0.400006	+	0.916512i	$0.630992\pi$
$228$	0	0
$229$	21.8570	1.44435	0.722175	−	0.691710i	$-0.243142\pi$
$229$	21.8570	1.44435	0.722175	+	0.691710i	$0.243142\pi$
$230$	42.8734	2.82699
$231$	0	0
$232$	−13.3095	−0.873811
$233$	−0.812241	−0.0532117	−0.0266058	−	0.999646i	$-0.508470\pi$
$233$	−0.812241	−0.0532117	−0.0266058	+	0.999646i	$0.508470\pi$
$234$	0	0
$235$	−24.6291	−1.60663
$236$	1.00774	0.0655981
$237$	0	0
$238$	14.6680	0.950785
$239$	9.12028	0.589942	0.294971	−	0.955506i	$-0.404690\pi$
$239$	9.12028	0.589942	0.294971	+	0.955506i	$0.404690\pi$
$240$	0	0
$241$	−1.42752	−0.0919544	−0.0459772	−	0.998942i	$-0.514640\pi$
$241$	−1.42752	−0.0919544	−0.0459772	+	0.998942i	$0.514640\pi$
$242$	−14.9421	−0.960512
$243$	0	0
$244$	−0.0366869	−0.00234864
$245$	−1.89766	−0.121237
$246$	0	0
$247$	−2.22307	−0.141451
$248$	9.87979	0.627367
$249$	0	0
$250$	−5.96901	−0.377513
$251$	−11.5125	−0.726665	−0.363332	−	0.931660i	$-0.618361\pi$
$251$	−11.5125	−0.726665	−0.363332	+	0.931660i	$0.618361\pi$
$252$	0	0
$253$	−0.873509	−0.0549170
$254$	−15.9303	−0.999556
$255$	0	0
$256$	−3.62456	−0.226535
$257$	−8.56331	−0.534165	−0.267082	−	0.963674i	$-0.586060\pi$
$257$	−8.56331	−0.534165	−0.267082	+	0.963674i	$0.586060\pi$
$258$	0	0
$259$	0.616636	0.0383159
$260$	1.75335	0.108738
$261$	0	0
$262$	−14.9945	−0.926362
$263$	8.83028	0.544498	0.272249	−	0.962227i	$-0.412232\pi$
$263$	8.83028	0.544498	0.272249	+	0.962227i	$0.412232\pi$
$264$	0	0
$265$	22.2581	1.36730
$266$	−2.42160	−0.148478
$267$	0	0
$268$	1.37285	0.0838604
$269$	14.9189	0.909620	0.454810	−	0.890589i	$-0.349707\pi$
$269$	14.9189	0.909620	0.454810	+	0.890589i	$0.349707\pi$
$270$	0	0
$271$	6.94126	0.421652	0.210826	−	0.977524i	$-0.432385\pi$
$271$	6.94126	0.421652	0.210826	+	0.977524i	$0.432385\pi$
$272$	−14.4097	−0.873715
$273$	0	0
$274$	−10.1508	−0.613233
$275$	0.587265	0.0354134
$276$	0	0
$277$	9.99195	0.600358	0.300179	−	0.953883i	$-0.402954\pi$
$277$	9.99195	0.600358	0.300179	+	0.953883i	$0.402954\pi$
$278$	−1.61388	−0.0967938
$279$	0	0
$280$	27.0536	1.61676
$281$	5.06242	0.301999	0.150999	−	0.988534i	$-0.451751\pi$
$281$	5.06242	0.301999	0.150999	+	0.988534i	$0.451751\pi$
$282$	0	0
$283$	33.2627	1.97726	0.988631	−	0.150359i	$-0.0480430\pi$
$283$	33.2627	1.97726	0.988631	+	0.150359i	$0.0480430\pi$
$284$	0.156726	0.00929997
$285$	0	0
$286$	0.434558	0.0256960
$287$	−29.4126	−1.73617
$288$	0	0
$289$	−1.60966	−0.0946857
$290$	−20.7963	−1.22120
$291$	0	0
$292$	−1.42761	−0.0835448
$293$	33.7661	1.97264	0.986319	−	0.164848i	$-0.0527135\pi$
$293$	33.7661	1.97264	0.986319	+	0.164848i	$0.0527135\pi$
$294$	0	0
$295$	22.3037	1.29857
$296$	−0.655887	−0.0381226
$297$	0	0
$298$	−1.35944	−0.0787503
$299$	−32.1939	−1.86182
$300$	0	0
$301$	17.5804	1.01332
$302$	14.9762	0.861786
$303$	0	0
$304$	2.37895	0.136442
$305$	−0.811973	−0.0464934
$306$	0	0
$307$	29.9067	1.70686	0.853432	−	0.521205i	$-0.174517\pi$
$307$	29.9067	1.70686	0.853432	+	0.521205i	$0.174517\pi$
$308$	−0.0389133	−0.00221729
$309$	0	0
$310$	15.4373	0.876782
$311$	−10.2667	−0.582172	−0.291086	−	0.956697i	$-0.594017\pi$
$311$	−10.2667	−0.582172	−0.291086	+	0.956697i	$0.594017\pi$
$312$	0	0
$313$	22.6213	1.27863	0.639316	−	0.768944i	$-0.279218\pi$
$313$	22.6213	1.27863	0.639316	+	0.768944i	$0.279218\pi$
$314$	−13.4017	−0.756302
$315$	0	0
$316$	−0.136518	−0.00767972
$317$	−11.2213	−0.630252	−0.315126	−	0.949050i	$-0.602047\pi$
$317$	−11.2213	−0.630252	−0.315126	+	0.949050i	$0.602047\pi$
$318$	0	0
$319$	0.423707	0.0237230
$320$	−28.6204	−1.59993
$321$	0	0
$322$	−35.0690	−1.95432
$323$	−2.54086	−0.141377
$324$	0	0
$325$	21.6441	1.20060
$326$	−18.5042	−1.02485
$327$	0	0
$328$	31.2848	1.72741
$329$	20.1458	1.11067
$330$	0	0
$331$	5.63806	0.309896	0.154948	−	0.987923i	$-0.450479\pi$
$331$	5.63806	0.309896	0.154948	+	0.987923i	$0.450479\pi$
$332$	1.03514	0.0568105
$333$	0	0
$334$	12.9268	0.707324
$335$	30.3847	1.66009
$336$	0	0
$337$	26.3046	1.43290	0.716452	−	0.697636i	$-0.245765\pi$
$337$	26.3046	1.43290	0.716452	+	0.697636i	$0.245765\pi$
$338$	−1.65670	−0.0901127
$339$	0	0
$340$	2.00399	0.108682
$341$	−0.314523	−0.0170324
$342$	0	0
$343$	−17.7002	−0.955719
$344$	−18.6995	−1.00821
$345$	0	0
$346$	−8.46853	−0.455271
$347$	−12.1904	−0.654418	−0.327209	−	0.944952i	$-0.606108\pi$
$347$	−12.1904	−0.654418	−0.327209	+	0.944952i	$0.606108\pi$
$348$	0	0
$349$	17.2633	0.924086	0.462043	−	0.886858i	$-0.347117\pi$
$349$	17.2633	0.924086	0.462043	+	0.886858i	$0.347117\pi$
$350$	23.5771	1.26025
$351$	0	0
$352$	0.0798585	0.00425647
$353$	−20.9581	−1.11548	−0.557742	−	0.830014i	$-0.688332\pi$
$353$	−20.9581	−1.11548	−0.557742	+	0.830014i	$0.688332\pi$
$354$	0	0
$355$	3.46874	0.184101
$356$	1.93884	0.102758
$357$	0	0
$358$	7.38033	0.390062
$359$	30.4761	1.60847	0.804233	−	0.594314i	$-0.202576\pi$
$359$	30.4761	1.60847	0.804233	+	0.594314i	$0.202576\pi$
$360$	0	0
$361$	−18.5805	−0.977922
$362$	−17.9968	−0.945893
$363$	0	0
$364$	−1.43418	−0.0751716
$365$	−31.5967	−1.65384
$366$	0	0
$367$	−21.8277	−1.13940	−0.569699	−	0.821854i	$-0.692940\pi$
$367$	−21.8277	−1.13940	−0.569699	+	0.821854i	$0.692940\pi$
$368$	34.4514	1.79590
$369$	0	0
$370$	−1.02483	−0.0532786
$371$	−18.2063	−0.945226
$372$	0	0
$373$	−34.4483	−1.78367	−0.891834	−	0.452363i	$-0.850581\pi$
$373$	−34.4483	−1.78367	−0.891834	+	0.452363i	$0.850581\pi$
$374$	0.496678	0.0256826
$375$	0	0
$376$	−21.4281	−1.10507
$377$	15.6161	0.804269
$378$	0	0
$379$	13.1081	0.673317	0.336659	−	0.941627i	$-0.390703\pi$
$379$	13.1081	0.673317	0.336659	+	0.941627i	$0.390703\pi$
$380$	−0.330848	−0.0169721
$381$	0	0
$382$	−27.4373	−1.40381
$383$	3.79187	0.193756	0.0968779	−	0.995296i	$-0.469114\pi$
$383$	3.79187	0.193756	0.0968779	+	0.995296i	$0.469114\pi$
$384$	0	0
$385$	−0.861249	−0.0438933
$386$	17.5059	0.891026
$387$	0	0
$388$	0.978576	0.0496797
$389$	−5.14526	−0.260875	−0.130437	−	0.991457i	$-0.541638\pi$
$389$	−5.14526	−0.260875	−0.130437	+	0.991457i	$0.541638\pi$
$390$	0	0
$391$	−36.7960	−1.86085
$392$	−1.65102	−0.0833892
$393$	0	0
$394$	26.1602	1.31793
$395$	−3.02148	−0.152027
$396$	0	0
$397$	0.536761	0.0269393	0.0134696	−	0.999909i	$-0.495712\pi$
$397$	0.536761	0.0269393	0.0134696	+	0.999909i	$0.495712\pi$
$398$	8.01231	0.401621
$399$	0	0
$400$	−23.1618	−1.15809
$401$	−18.3312	−0.915415	−0.457707	−	0.889103i	$-0.651329\pi$
$401$	−18.3312	−0.915415	−0.457707	+	0.889103i	$0.651329\pi$
$402$	0	0
$403$	−11.5920	−0.577438
$404$	2.87093	0.142834
$405$	0	0
$406$	17.0107	0.844225
$407$	0.0208801	0.00103499
$408$	0	0
$409$	−12.8697	−0.636366	−0.318183	−	0.948029i	$-0.603073\pi$
$409$	−12.8697	−0.636366	−0.318183	+	0.948029i	$0.603073\pi$
$410$	48.8830	2.41416
$411$	0	0
$412$	1.16254	0.0572744
$413$	−18.2437	−0.897713
$414$	0	0
$415$	22.9102	1.12462
$416$	2.94325	0.144305
$417$	0	0
$418$	−0.0819987	−0.00401069
$419$	23.0031	1.12378	0.561888	−	0.827213i	$-0.310075\pi$
$419$	23.0031	1.12378	0.561888	+	0.827213i	$0.310075\pi$
$420$	0	0
$421$	27.9999	1.36463	0.682316	−	0.731057i	$-0.260973\pi$
$421$	27.9999	1.36463	0.682316	+	0.731057i	$0.260973\pi$
$422$	−12.4205	−0.604621
$423$	0	0
$424$	19.3652	0.940459
$425$	24.7381	1.19998
$426$	0	0
$427$	0.664166	0.0321412
$428$	−1.55150	−0.0749948
$429$	0	0
$430$	−29.2183	−1.40903
$431$	18.8948	0.910132	0.455066	−	0.890458i	$-0.349616\pi$
$431$	18.8948	0.910132	0.455066	+	0.890458i	$0.349616\pi$
$432$	0	0
$433$	−24.6951	−1.18677	−0.593386	−	0.804918i	$-0.702209\pi$
$433$	−24.6951	−1.18677	−0.593386	+	0.804918i	$0.702209\pi$
$434$	−12.6272	−0.606126
$435$	0	0
$436$	3.08945	0.147958
$437$	6.07481	0.290597
$438$	0	0
$439$	−26.0861	−1.24502	−0.622510	−	0.782612i	$-0.713887\pi$
$439$	−26.0861	−1.24502	−0.622510	+	0.782612i	$0.713887\pi$
$440$	0.916071	0.0436719
$441$	0	0
$442$	18.3055	0.870704
$443$	1.62310	0.0771158	0.0385579	−	0.999256i	$-0.487724\pi$
$443$	1.62310	0.0771158	0.0385579	+	0.999256i	$0.487724\pi$
$444$	0	0
$445$	42.9113	2.03419
$446$	−7.09850	−0.336124
$447$	0	0
$448$	23.4105	1.10604
$449$	−29.1850	−1.37732	−0.688662	−	0.725082i	$-0.741802\pi$
$449$	−29.1850	−1.37732	−0.688662	+	0.725082i	$0.741802\pi$
$450$	0	0
$451$	−0.995950	−0.0468974
$452$	−0.777953	−0.0365918
$453$	0	0
$454$	−16.3859	−0.769028
$455$	−31.7421	−1.48809
$456$	0	0
$457$	8.59320	0.401973	0.200986	−	0.979594i	$-0.435585\pi$
$457$	8.59320	0.401973	0.200986	+	0.979594i	$0.435585\pi$
$458$	29.7133	1.38841
$459$	0	0
$460$	−4.79125	−0.223393
$461$	−1.38246	−0.0643875	−0.0321937	−	0.999482i	$-0.510249\pi$
$461$	−1.38246	−0.0643875	−0.0321937	+	0.999482i	$0.510249\pi$
$462$	0	0
$463$	−1.94371	−0.0903317	−0.0451658	−	0.998980i	$-0.514382\pi$
$463$	−1.94371	−0.0903317	−0.0451658	+	0.998980i	$0.514382\pi$
$464$	−16.7111	−0.775792
$465$	0	0
$466$	−1.10419	−0.0511508
$467$	−19.4192	−0.898615	−0.449308	−	0.893377i	$-0.648329\pi$
$467$	−19.4192	−0.898615	−0.449308	+	0.893377i	$0.648329\pi$
$468$	0	0
$469$	−24.8536	−1.14763
$470$	−33.4818	−1.54440
$471$	0	0
$472$	19.4050	0.893186
$473$	0.595298	0.0273718
$474$	0	0
$475$	−4.08412	−0.187392
$476$	−1.63920	−0.0751325
$477$	0	0
$478$	12.3985	0.567093
$479$	−31.6401	−1.44567	−0.722837	−	0.691018i	$-0.757162\pi$
$479$	−31.6401	−1.44567	−0.722837	+	0.691018i	$0.757162\pi$
$480$	0	0
$481$	0.769555	0.0350887
$482$	−1.94062	−0.0883929
$483$	0	0
$484$	1.66983	0.0759011
$485$	21.6583	0.983454
$486$	0	0
$487$	−1.05110	−0.0476299	−0.0238150	−	0.999716i	$-0.507581\pi$
$487$	−1.05110	−0.0476299	−0.0238150	+	0.999716i	$0.507581\pi$
$488$	−0.706442	−0.0319791
$489$	0	0
$490$	−2.57975	−0.116541
$491$	27.4500	1.23880	0.619401	−	0.785075i	$-0.287376\pi$
$491$	27.4500	1.23880	0.619401	+	0.785075i	$0.287376\pi$
$492$	0	0
$493$	17.8484	0.803850
$494$	−3.02213	−0.135972
$495$	0	0
$496$	12.4048	0.556993
$497$	−2.83731	−0.127271
$498$	0	0
$499$	−20.0010	−0.895368	−0.447684	−	0.894192i	$-0.647751\pi$
$499$	−20.0010	−0.895368	−0.447684	+	0.894192i	$0.647751\pi$
$500$	0.667057	0.0298317
$501$	0	0
$502$	−15.6506	−0.698520
$503$	30.5196	1.36080	0.680400	−	0.732841i	$-0.261806\pi$
$503$	30.5196	1.36080	0.680400	+	0.732841i	$0.261806\pi$
$504$	0	0
$505$	63.5408	2.82753
$506$	−1.18748	−0.0527901
$507$	0	0
$508$	1.78026	0.0789865
$509$	−28.8132	−1.27712	−0.638560	−	0.769572i	$-0.720470\pi$
$509$	−28.8132	−1.27712	−0.638560	+	0.769572i	$0.720470\pi$
$510$	0	0
$511$	25.8450	1.14331
$512$	−24.6401	−1.08895
$513$	0	0
$514$	−11.6413	−0.513476
$515$	25.7300	1.13380
$516$	0	0
$517$	0.682163	0.0300015
$518$	0.838280	0.0368319
$519$	0	0
$520$	33.7626	1.48059
$521$	0.0430821	0.00188746	0.000943730	−	1.00000i	$-0.499700\pi$
$521$	0.0430821	0.00188746	0.000943730		1.00000i	$0.499700\pi$
$522$	0	0
$523$	33.0562	1.44545	0.722723	−	0.691138i	$-0.242890\pi$
$523$	33.0562	1.44545	0.722723	+	0.691138i	$0.242890\pi$
$524$	1.67568	0.0732026
$525$	0	0
$526$	12.0042	0.523410
$527$	−13.2491	−0.577138
$528$	0	0
$529$	64.9738	2.82495
$530$	30.2585	1.31435
$531$	0	0
$532$	0.270622	0.0117330
$533$	−36.7066	−1.58994
$534$	0	0
$535$	−34.3387	−1.48459
$536$	26.4357	1.14185
$537$	0	0
$538$	20.2813	0.874390
$539$	0.0525602	0.00226393
$540$	0	0
$541$	37.3459	1.60563	0.802813	−	0.596230i	$-0.203335\pi$
$541$	37.3459	1.60563	0.802813	+	0.596230i	$0.203335\pi$
$542$	9.43623	0.405321
$543$	0	0
$544$	3.36399	0.144230
$545$	68.3772	2.92896
$546$	0	0
$547$	−28.8653	−1.23419	−0.617095	−	0.786888i	$-0.711691\pi$
$547$	−28.8653	−1.23419	−0.617095	+	0.786888i	$0.711691\pi$
$548$	1.13439	0.0484586
$549$	0	0
$550$	0.798351	0.0340418
$551$	−2.94666	−0.125532
$552$	0	0
$553$	2.47146	0.105097
$554$	13.5835	0.577106
$555$	0	0
$556$	0.180356	0.00764880
$557$	−13.5220	−0.572945	−0.286473	−	0.958088i	$-0.592483\pi$
$557$	−13.5220	−0.572945	−0.286473	+	0.958088i	$0.592483\pi$
$558$	0	0
$559$	21.9402	0.927972
$560$	33.9678	1.43540
$561$	0	0
$562$	6.88205	0.290302
$563$	24.8508	1.04734	0.523668	−	0.851923i	$-0.324563\pi$
$563$	24.8508	1.04734	0.523668	+	0.851923i	$0.324563\pi$
$564$	0	0
$565$	−17.2180	−0.724368
$566$	45.2187	1.90068
$567$	0	0
$568$	3.01791	0.126629
$569$	−36.4632	−1.52862	−0.764309	−	0.644851i	$-0.776920\pi$
$569$	−36.4632	−1.52862	−0.764309	+	0.644851i	$0.776920\pi$
$570$	0	0
$571$	−30.8543	−1.29121	−0.645607	−	0.763670i	$-0.723396\pi$
$571$	−30.8543	−1.29121	−0.645607	+	0.763670i	$0.723396\pi$
$572$	−0.0485634	−0.00203054
$573$	0	0
$574$	−39.9846	−1.66893
$575$	−59.1452	−2.46653
$576$	0	0
$577$	26.3239	1.09588	0.547938	−	0.836519i	$-0.315413\pi$
$577$	26.3239	1.09588	0.547938	+	0.836519i	$0.315413\pi$
$578$	−2.18823	−0.0910185
$579$	0	0
$580$	2.32406	0.0965012
$581$	−18.7397	−0.777455
$582$	0	0
$583$	−0.616492	−0.0255325
$584$	−27.4901	−1.13755
$585$	0	0
$586$	45.9030	1.89624
$587$	40.4441	1.66931	0.834653	−	0.550776i	$-0.185668\pi$
$587$	40.4441	1.66931	0.834653	+	0.550776i	$0.185668\pi$
$588$	0	0
$589$	2.18734	0.0901279
$590$	30.3206	1.24828
$591$	0	0
$592$	−0.823516	−0.0338463
$593$	12.8855	0.529146	0.264573	−	0.964366i	$-0.414769\pi$
$593$	12.8855	0.529146	0.264573	+	0.964366i	$0.414769\pi$
$594$	0	0
$595$	−36.2796	−1.48732
$596$	0.151922	0.00622297
$597$	0	0
$598$	−43.7657	−1.78971
$599$	−22.6950	−0.927291	−0.463645	−	0.886021i	$-0.653459\pi$
$599$	−22.6950	−0.927291	−0.463645	+	0.886021i	$0.653459\pi$
$600$	0	0
$601$	12.5235	0.510845	0.255422	−	0.966830i	$-0.417785\pi$
$601$	12.5235	0.510845	0.255422	+	0.966830i	$0.417785\pi$
$602$	23.8996	0.974073
$603$	0	0
$604$	−1.67365	−0.0680997
$605$	36.9574	1.50253
$606$	0	0
$607$	26.8674	1.09051	0.545256	−	0.838269i	$-0.316432\pi$
$607$	26.8674	1.09051	0.545256	+	0.838269i	$0.316432\pi$
$608$	−0.555375	−0.0225234
$609$	0	0
$610$	−1.10383	−0.0446927
$611$	25.1417	1.01712
$612$	0	0
$613$	−10.5351	−0.425507	−0.212753	−	0.977106i	$-0.568243\pi$
$613$	−10.5351	−0.425507	−0.212753	+	0.977106i	$0.568243\pi$
$614$	40.6563	1.64076
$615$	0	0
$616$	−0.749314	−0.0301907
$617$	11.0415	0.444513	0.222257	−	0.974988i	$-0.428658\pi$
$617$	11.0415	0.444513	0.222257	+	0.974988i	$0.428658\pi$
$618$	0	0
$619$	−45.6196	−1.83361	−0.916803	−	0.399340i	$-0.869239\pi$
$619$	−45.6196	−1.83361	−0.916803	+	0.399340i	$0.869239\pi$
$620$	−1.72518	−0.0692847
$621$	0	0
$622$	−13.9570	−0.559624
$623$	−35.1000	−1.40625
$624$	0	0
$625$	−16.7656	−0.670623
$626$	30.7523	1.22911
$627$	0	0
$628$	1.49769	0.0597642
$629$	0.879562	0.0350704
$630$	0	0
$631$	−9.89052	−0.393735	−0.196868	−	0.980430i	$-0.563077\pi$
$631$	−9.89052	−0.393735	−0.196868	+	0.980430i	$0.563077\pi$
$632$	−2.62878	−0.104567
$633$	0	0
$634$	−15.2547	−0.605842
$635$	39.4017	1.56361
$636$	0	0
$637$	1.93715	0.0767527
$638$	0.576004	0.0228042
$639$	0	0
$640$	−33.1413	−1.31002
$641$	11.1666	0.441055	0.220527	−	0.975381i	$-0.429222\pi$
$641$	11.1666	0.441055	0.220527	+	0.975381i	$0.429222\pi$
$642$	0	0
$643$	−13.7442	−0.542019	−0.271009	−	0.962577i	$-0.587357\pi$
$643$	−13.7442	−0.542019	−0.271009	+	0.962577i	$0.587357\pi$
$644$	3.91908	0.154433
$645$	0	0
$646$	−3.45414	−0.135901
$647$	4.18281	0.164443	0.0822215	−	0.996614i	$-0.473798\pi$
$647$	4.18281	0.164443	0.0822215	+	0.996614i	$0.473798\pi$
$648$	0	0
$649$	−0.617756	−0.0242490
$650$	29.4239	1.15410
$651$	0	0
$652$	2.06791	0.0809855
$653$	7.21759	0.282446	0.141223	−	0.989978i	$-0.454896\pi$
$653$	7.21759	0.282446	0.141223	+	0.989978i	$0.454896\pi$
$654$	0	0
$655$	37.0871	1.44911
$656$	39.2805	1.53364
$657$	0	0
$658$	27.3870	1.06765
$659$	39.1806	1.52626	0.763130	−	0.646245i	$-0.223661\pi$
$659$	39.1806	1.52626	0.763130	+	0.646245i	$0.223661\pi$
$660$	0	0
$661$	−16.0457	−0.624104	−0.312052	−	0.950065i	$-0.601016\pi$
$661$	−16.0457	−0.624104	−0.312052	+	0.950065i	$0.601016\pi$
$662$	7.66461	0.297893
$663$	0	0
$664$	19.9326	0.773534
$665$	5.98955	0.232265
$666$	0	0
$667$	−42.6728	−1.65230
$668$	−1.44462	−0.0558939
$669$	0	0
$670$	41.3062	1.59580
$671$	0.0224895	0.000868199 0
$672$	0	0
$673$	−17.2843	−0.666261	−0.333131	−	0.942881i	$-0.608105\pi$
$673$	−17.2843	−0.666261	−0.333131	+	0.942881i	$0.608105\pi$
$674$	35.7596	1.37741
$675$	0	0
$676$	0.185142	0.00712085
$677$	31.3588	1.20522	0.602609	−	0.798037i	$-0.294128\pi$
$677$	31.3588	1.20522	0.602609	+	0.798037i	$0.294128\pi$
$678$	0	0
$679$	−17.7158	−0.679869
$680$	38.5889	1.47982
$681$	0	0
$682$	−0.427575	−0.0163727
$683$	−23.5307	−0.900376	−0.450188	−	0.892934i	$-0.648643\pi$
$683$	−23.5307	−0.900376	−0.450188	+	0.892934i	$0.648643\pi$
$684$	0	0
$685$	25.1068	0.959283
$686$	−24.0623	−0.918704
$687$	0	0
$688$	−23.4787	−0.895115
$689$	−22.7213	−0.865613
$690$	0	0
$691$	−31.0910	−1.18276	−0.591379	−	0.806393i	$-0.701416\pi$
$691$	−31.0910	−1.18276	−0.591379	+	0.806393i	$0.701416\pi$
$692$	0.946387	0.0359762
$693$	0	0
$694$	−16.5722	−0.629072
$695$	3.99173	0.151415
$696$	0	0
$697$	−41.9537	−1.58911
$698$	23.4685	0.888295
$699$	0	0
$700$	−2.63482	−0.0995867
$701$	13.3636	0.504737	0.252369	−	0.967631i	$-0.418790\pi$
$701$	13.3636	0.504737	0.252369	+	0.967631i	$0.418790\pi$
$702$	0	0
$703$	−0.145211	−0.00547672
$704$	0.792712	0.0298764
$705$	0	0
$706$	−28.4912	−1.07228
$707$	−51.9742	−1.95469
$708$	0	0
$709$	5.21637	0.195905	0.0979524	−	0.995191i	$-0.468771\pi$
$709$	5.21637	0.195905	0.0979524	+	0.995191i	$0.468771\pi$
$710$	4.71554	0.176971
$711$	0	0
$712$	37.3342	1.39916
$713$	31.6765	1.18630
$714$	0	0
$715$	−1.07483	−0.0401963
$716$	−0.824777	−0.0308233
$717$	0	0
$718$	41.4304	1.54617
$719$	−0.587483	−0.0219094	−0.0109547	−	0.999940i	$-0.503487\pi$
$719$	−0.587483	−0.0219094	−0.0109547	+	0.999940i	$0.503487\pi$
$720$	0	0
$721$	−21.0463	−0.783804
$722$	−25.2591	−0.940047
$723$	0	0
$724$	2.01121	0.0747460
$725$	28.6891	1.06549
$726$	0	0
$727$	−21.9077	−0.812512	−0.406256	−	0.913759i	$-0.633166\pi$
$727$	−21.9077	−0.812512	−0.406256	+	0.913759i	$0.633166\pi$
$728$	−27.6166	−1.02354
$729$	0	0
$730$	−42.9538	−1.58979
$731$	25.0765	0.927489
$732$	0	0
$733$	45.8510	1.69354	0.846772	−	0.531956i	$-0.178543\pi$
$733$	45.8510	1.69354	0.846772	+	0.531956i	$0.178543\pi$
$734$	−29.6735	−1.09527
$735$	0	0
$736$	−8.04279	−0.296461
$737$	−0.841578	−0.0309999
$738$	0	0
$739$	6.56861	0.241630	0.120815	−	0.992675i	$-0.461449\pi$
$739$	6.56861	0.241630	0.120815	+	0.992675i	$0.461449\pi$
$740$	0.114529	0.00421016
$741$	0	0
$742$	−24.7504	−0.908617
$743$	−42.2554	−1.55020	−0.775101	−	0.631838i	$-0.782301\pi$
$743$	−42.2554	−1.55020	−0.775101	+	0.631838i	$0.782301\pi$
$744$	0	0
$745$	3.36242	0.123189
$746$	−46.8305	−1.71458
$747$	0	0
$748$	−0.0555055	−0.00202948
$749$	28.0878	1.02631
$750$	0	0
$751$	27.3880	0.999402	0.499701	−	0.866198i	$-0.333443\pi$
$751$	27.3880	0.999402	0.499701	+	0.866198i	$0.333443\pi$
$752$	−26.9046	−0.981111
$753$	0	0
$754$	21.2291	0.773119
$755$	−37.0420	−1.34810
$756$	0	0
$757$	−8.55390	−0.310897	−0.155448	−	0.987844i	$-0.549682\pi$
$757$	−8.55390	−0.310897	−0.155448	+	0.987844i	$0.549682\pi$
$758$	17.8197	0.647239
$759$	0	0
$760$	−6.37080	−0.231093
$761$	−24.3403	−0.882335	−0.441168	−	0.897425i	$-0.645436\pi$
$761$	−24.3403	−0.882335	−0.441168	+	0.897425i	$0.645436\pi$
$762$	0	0
$763$	−55.9302	−2.02481
$764$	3.06621	0.110931
$765$	0	0
$766$	5.15483	0.186251
$767$	−22.7679	−0.822102
$768$	0	0
$769$	−32.6681	−1.17804	−0.589021	−	0.808118i	$-0.700486\pi$
$769$	−32.6681	−1.17804	−0.589021	+	0.808118i	$0.700486\pi$
$770$	−1.17082	−0.0421933
$771$	0	0
$772$	−1.95634	−0.0704103
$773$	−41.2754	−1.48457	−0.742286	−	0.670083i	$-0.766259\pi$
$773$	−41.2754	−1.48457	−0.742286	+	0.670083i	$0.766259\pi$
$774$	0	0
$775$	−21.2963	−0.764985
$776$	18.8434	0.676440
$777$	0	0
$778$	−6.99467	−0.250771
$779$	6.92632	0.248161
$780$	0	0
$781$	−0.0960750	−0.00343784
$782$	−50.0220	−1.78878
$783$	0	0
$784$	−2.07299	−0.0740352
$785$	33.1475	1.18309
$786$	0	0
$787$	−25.3051	−0.902031	−0.451016	−	0.892516i	$-0.648938\pi$
$787$	−25.3051	−0.902031	−0.451016	+	0.892516i	$0.648938\pi$
$788$	−2.92349	−0.104145
$789$	0	0
$790$	−4.10752	−0.146139
$791$	14.0838	0.500761
$792$	0	0
$793$	0.828872	0.0294341
$794$	0.729695	0.0258959
$795$	0	0
$796$	−0.895403	−0.0317367
$797$	17.6004	0.623436	0.311718	−	0.950175i	$-0.399096\pi$
$797$	17.6004	0.623436	0.311718	+	0.950175i	$0.399096\pi$
$798$	0	0
$799$	28.7357	1.01660
$800$	5.40721	0.191174
$801$	0	0
$802$	−24.9201	−0.879960
$803$	0.875146	0.0308832
$804$	0	0
$805$	86.7390	3.05715
$806$	−15.7586	−0.555074
$807$	0	0
$808$	55.2826	1.94483
$809$	−31.5793	−1.11027	−0.555134	−	0.831761i	$-0.687333\pi$
$809$	−31.5793	−1.11027	−0.555134	+	0.831761i	$0.687333\pi$
$810$	0	0
$811$	−51.8384	−1.82029	−0.910146	−	0.414288i	$-0.864031\pi$
$811$	−51.8384	−1.82029	−0.910146	+	0.414288i	$0.864031\pi$
$812$	−1.90100	−0.0667120
$813$	0	0
$814$	0.0283853	0.000994904 0
$815$	45.7680	1.60318
$816$	0	0
$817$	−4.13999	−0.144840
$818$	−17.4956	−0.611719
$819$	0	0
$820$	−5.46285	−0.190771
$821$	−51.5541	−1.79925	−0.899625	−	0.436664i	$-0.856160\pi$
$821$	−51.5541	−1.79925	−0.899625	+	0.436664i	$0.856160\pi$
$822$	0	0
$823$	14.0843	0.490949	0.245475	−	0.969403i	$-0.421056\pi$
$823$	14.0843	0.490949	0.245475	+	0.969403i	$0.421056\pi$
$824$	22.3859	0.779851
$825$	0	0
$826$	−24.8012	−0.862944
$827$	5.72620	0.199119	0.0995597	−	0.995032i	$-0.468257\pi$
$827$	5.72620	0.199119	0.0995597	+	0.995032i	$0.468257\pi$
$828$	0	0
$829$	−10.4332	−0.362361	−0.181180	−	0.983450i	$-0.557992\pi$
$829$	−10.4332	−0.362361	−0.181180	+	0.983450i	$0.557992\pi$
$830$	31.1450	1.08106
$831$	0	0
$832$	29.2161	1.01288
$833$	2.21407	0.0767128
$834$	0	0
$835$	−31.9730	−1.10647
$836$	0.00916363	0.000316931 0
$837$	0	0
$838$	31.2714	1.08025
$839$	32.5312	1.12310	0.561550	−	0.827442i	$-0.310205\pi$
$839$	32.5312	1.12310	0.561550	+	0.827442i	$0.310205\pi$
$840$	0	0
$841$	−8.30101	−0.286242
$842$	38.0642	1.31178
$843$	0	0
$844$	1.38803	0.0477781
$845$	4.09766	0.140964
$846$	0	0
$847$	−30.2299	−1.03871
$848$	24.3145	0.834965
$849$	0	0
$850$	33.6300	1.15350
$851$	−2.10290	−0.0720865
$852$	0	0
$853$	−34.6253	−1.18555	−0.592774	−	0.805369i	$-0.701967\pi$
$853$	−34.6253	−1.18555	−0.592774	+	0.805369i	$0.701967\pi$
$854$	0.902894	0.0308964
$855$	0	0
$856$	−29.8757	−1.02113
$857$	−15.2429	−0.520687	−0.260343	−	0.965516i	$-0.583836\pi$
$857$	−15.2429	−0.520687	−0.260343	+	0.965516i	$0.583836\pi$
$858$	0	0
$859$	39.7967	1.35785	0.678923	−	0.734210i	$-0.262447\pi$
$859$	39.7967	1.35785	0.678923	+	0.734210i	$0.262447\pi$
$860$	3.26524	0.111344
$861$	0	0
$862$	25.6864	0.874882
$863$	−2.16351	−0.0736466	−0.0368233	−	0.999322i	$-0.511724\pi$
$863$	−2.16351	−0.0736466	−0.0368233	+	0.999322i	$0.511724\pi$
$864$	0	0
$865$	20.9459	0.712182
$866$	−33.5716	−1.14081
$867$	0	0
$868$	1.41113	0.0478970
$869$	0.0836871	0.00283889
$870$	0	0
$871$	−31.0171	−1.05097
$872$	59.4903	2.01460
$873$	0	0
$874$	8.25834	0.279342
$875$	−12.0761	−0.408248
$876$	0	0
$877$	21.5395	0.727338	0.363669	−	0.931528i	$-0.381524\pi$
$877$	21.5395	0.727338	0.363669	+	0.931528i	$0.381524\pi$
$878$	−35.4624	−1.19680
$879$	0	0
$880$	1.15020	0.0387731
$881$	−5.86866	−0.197720	−0.0988602	−	0.995101i	$-0.531520\pi$
$881$	−5.86866	−0.197720	−0.0988602	+	0.995101i	$0.531520\pi$
$882$	0	0
$883$	6.68200	0.224867	0.112434	−	0.993659i	$-0.464135\pi$
$883$	6.68200	0.224867	0.112434	+	0.993659i	$0.464135\pi$
$884$	−2.04570	−0.0688044
$885$	0	0
$886$	2.20651	0.0741291
$887$	−51.3792	−1.72515	−0.862573	−	0.505933i	$-0.831148\pi$
$887$	−51.3792	−1.72515	−0.862573	+	0.505933i	$0.831148\pi$
$888$	0	0
$889$	−32.2292	−1.08093
$890$	58.3354	1.95541
$891$	0	0
$892$	0.793281	0.0265610
$893$	−4.74409	−0.158755
$894$	0	0
$895$	−18.2544	−0.610176
$896$	27.1084	0.905628
$897$	0	0
$898$	−39.6752	−1.32398
$899$	−15.3651	−0.512455
$900$	0	0
$901$	−25.9693	−0.865163
$902$	−1.35393	−0.0450811
$903$	0	0
$904$	−14.9802	−0.498236
$905$	44.5131	1.47967
$906$	0	0
$907$	−12.9918	−0.431385	−0.215693	−	0.976461i	$-0.569201\pi$
$907$	−12.9918	−0.431385	−0.215693	+	0.976461i	$0.569201\pi$
$908$	1.83118	0.0607698
$909$	0	0
$910$	−43.1514	−1.43046
$911$	2.50184	0.0828896	0.0414448	−	0.999141i	$-0.486804\pi$
$911$	2.50184	0.0828896	0.0414448	+	0.999141i	$0.486804\pi$
$912$	0	0
$913$	−0.634552	−0.0210006
$914$	11.6819	0.386404
$915$	0	0
$916$	−3.32056	−0.109714
$917$	−30.3359	−1.00178
$918$	0	0
$919$	−47.2092	−1.55729	−0.778645	−	0.627465i	$-0.784092\pi$
$919$	−47.2092	−1.55729	−0.778645	+	0.627465i	$0.784092\pi$
$920$	−92.2603	−3.04173
$921$	0	0
$922$	−1.87937	−0.0618937
$923$	−3.54093	−0.116551
$924$	0	0
$925$	1.41379	0.0464852
$926$	−2.64235	−0.0868331
$927$	0	0
$928$	3.90126	0.128065
$929$	−40.2330	−1.32000	−0.660001	−	0.751265i	$-0.729444\pi$
$929$	−40.2330	−1.32000	−0.660001	+	0.751265i	$0.729444\pi$
$930$	0	0
$931$	−0.365529	−0.0119797
$932$	0.123397	0.00404202
$933$	0	0
$934$	−26.3993	−0.863812
$935$	−1.22848	−0.0401754
$936$	0	0
$937$	56.6800	1.85166	0.925828	−	0.377946i	$-0.123369\pi$
$937$	56.6800	1.85166	0.925828	+	0.377946i	$0.123369\pi$
$938$	−33.7871	−1.10319
$939$	0	0
$940$	3.74171	0.122041
$941$	−17.4800	−0.569833	−0.284916	−	0.958552i	$-0.591966\pi$
$941$	−17.4800	−0.569833	−0.284916	+	0.958552i	$0.591966\pi$
$942$	0	0
$943$	100.305	3.26639
$944$	24.3644	0.792994
$945$	0	0
$946$	0.809271	0.0263117
$947$	−0.639443	−0.0207791	−0.0103896	−	0.999946i	$-0.503307\pi$
$947$	−0.639443	−0.0207791	−0.0103896	+	0.999946i	$0.503307\pi$
$948$	0	0
$949$	32.2543	1.04702
$950$	−5.55212	−0.180135
$951$	0	0
$952$	−31.5644	−1.02301
$953$	47.7540	1.54690	0.773452	−	0.633855i	$-0.218529\pi$
$953$	47.7540	1.54690	0.773452	+	0.633855i	$0.218529\pi$
$954$	0	0
$955$	67.8628	2.19599
$956$	−1.38557	−0.0448126
$957$	0	0
$958$	−43.0129	−1.38968
$959$	−20.5365	−0.663159
$960$	0	0
$961$	−19.5943	−0.632074
$962$	1.04616	0.0337297
$963$	0	0
$964$	0.216871	0.00698495
$965$	−43.2988	−1.39384
$966$	0	0
$967$	−13.3691	−0.429922	−0.214961	−	0.976623i	$-0.568962\pi$
$967$	−13.3691	−0.429922	−0.214961	+	0.976623i	$0.568962\pi$
$968$	32.1541	1.03347
$969$	0	0
$970$	29.4432	0.945364
$971$	0.167774	0.00538412	0.00269206	−	0.999996i	$-0.499143\pi$
$971$	0.167774	0.00538412	0.00269206	+	0.999996i	$0.499143\pi$
$972$	0	0
$973$	−3.26510	−0.104674
$974$	−1.42891	−0.0457852
$975$	0	0
$976$	−0.886992	−0.0283919
$977$	−1.25124	−0.0400308	−0.0200154	−	0.999800i	$-0.506372\pi$
$977$	−1.25124	−0.0400308	−0.0200154	+	0.999800i	$0.506372\pi$
$978$	0	0
$979$	−1.18853	−0.0379857
$980$	0.288296	0.00920928
$981$	0	0
$982$	37.3166	1.19082
$983$	−17.1294	−0.546344	−0.273172	−	0.961965i	$-0.588073\pi$
$983$	−17.1294	−0.546344	−0.273172	+	0.961965i	$0.588073\pi$
$984$	0	0
$985$	−64.7042	−2.06165
$986$	24.2638	0.772717
$987$	0	0
$988$	0.337733	0.0107447
$989$	−59.9542	−1.90643
$990$	0	0
$991$	16.7316	0.531497	0.265749	−	0.964042i	$-0.414381\pi$
$991$	16.7316	0.531497	0.265749	+	0.964042i	$0.414381\pi$
$992$	−2.89595	−0.0919465
$993$	0	0
$994$	−3.85715	−0.122341
$995$	−19.8175	−0.628257
$996$	0	0
$997$	20.1891	0.639396	0.319698	−	0.947520i	$-0.396419\pi$
$997$	20.1891	0.639396	0.319698	+	0.947520i	$0.396419\pi$
$998$	−27.1902	−0.860689
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.20	✓	25
3.2	odd	2		4023.2.a.f.1.6	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.20	✓	25	1.1	even	1	trivial
4023.2.a.f.1.6	yes	25	3.2	odd	2

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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35944 q^{2} -0.151922 q^{4} -3.36242 q^{5} +2.75034 q^{7} -2.92541 q^{8} +O(q^{10})\)	\(q+1.35944 q^{2} -0.151922 q^{4} -3.36242 q^{5} +2.75034 q^{7} -2.92541 q^{8} -4.57100 q^{10} +0.0931303 q^{11} +3.43239 q^{13} +3.73892 q^{14} -3.67308 q^{16} +3.92305 q^{17} -0.647673 q^{19} +0.510825 q^{20} +0.126605 q^{22} -9.37943 q^{23} +6.30584 q^{25} +4.66614 q^{26} -0.417837 q^{28} +4.54961 q^{29} -3.37723 q^{31} +0.857492 q^{32} +5.33316 q^{34} -9.24779 q^{35} +0.224203 q^{37} -0.880473 q^{38} +9.83644 q^{40} -10.6942 q^{41} +6.39210 q^{43} -0.0141485 q^{44} -12.7508 q^{46} +7.32483 q^{47} +0.564373 q^{49} +8.57241 q^{50} -0.521457 q^{52} -6.61967 q^{53} -0.313143 q^{55} -8.04587 q^{56} +6.18493 q^{58} -6.63325 q^{59} +0.241485 q^{61} -4.59115 q^{62} +8.51186 q^{64} -11.5411 q^{65} -9.03657 q^{67} -0.595998 q^{68} -12.5718 q^{70} -1.03162 q^{71} +9.39701 q^{73} +0.304791 q^{74} +0.0983959 q^{76} +0.256140 q^{77} +0.898603 q^{79} +12.3504 q^{80} -14.5381 q^{82} -6.81360 q^{83} -13.1909 q^{85} +8.68967 q^{86} -0.272444 q^{88} -12.7620 q^{89} +9.44025 q^{91} +1.42494 q^{92} +9.95766 q^{94} +2.17775 q^{95} -6.44130 q^{97} +0.767232 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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