LMFDB - Embedded newform 4023.2.a.f.1.2

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.38405 q^{2} +3.68368 q^{4} +0.998141 q^{5} -3.46708 q^{7} -4.01398 q^{8} +O(q^{10})$	\(q-2.38405 q^{2} +3.68368 q^{4} +0.998141 q^{5} -3.46708 q^{7} -4.01398 q^{8} -2.37961 q^{10} -1.23725 q^{11} -0.762635 q^{13} +8.26569 q^{14} +2.20214 q^{16} -2.76285 q^{17} -6.45519 q^{19} +3.67683 q^{20} +2.94966 q^{22} +4.57326 q^{23} -4.00371 q^{25} +1.81816 q^{26} -12.7716 q^{28} -6.92170 q^{29} +1.57611 q^{31} +2.77793 q^{32} +6.58676 q^{34} -3.46064 q^{35} +1.00069 q^{37} +15.3895 q^{38} -4.00651 q^{40} +2.61346 q^{41} -11.3267 q^{43} -4.55763 q^{44} -10.9029 q^{46} +3.40145 q^{47} +5.02067 q^{49} +9.54505 q^{50} -2.80930 q^{52} +12.4193 q^{53} -1.23495 q^{55} +13.9168 q^{56} +16.5017 q^{58} -10.4463 q^{59} -5.39005 q^{61} -3.75751 q^{62} -11.0270 q^{64} -0.761217 q^{65} -0.0539558 q^{67} -10.1775 q^{68} +8.25032 q^{70} +5.95774 q^{71} -11.3804 q^{73} -2.38569 q^{74} -23.7789 q^{76} +4.28964 q^{77} +8.94972 q^{79} +2.19805 q^{80} -6.23062 q^{82} -7.61618 q^{83} -2.75771 q^{85} +27.0035 q^{86} +4.96628 q^{88} +13.3813 q^{89} +2.64412 q^{91} +16.8464 q^{92} -8.10921 q^{94} -6.44319 q^{95} +17.4286 q^{97} -11.9695 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.38405	−1.68578	−0.842888	−	0.538089i	$-0.819146\pi$
$2$	−2.38405	−1.68578	−0.842888	+	0.538089i	$0.819146\pi$
$3$	0	0
$3$	0	0
$4$	3.68368	1.84184
$5$	0.998141	0.446382	0.223191	−	0.974775i	$-0.428353\pi$
$5$	0.998141	0.446382	0.223191	+	0.974775i	$0.428353\pi$
$6$	0	0
$7$	−3.46708	−1.31043	−0.655217	−	0.755441i	$-0.727423\pi$
$7$	−3.46708	−1.31043	−0.655217	+	0.755441i	$0.727423\pi$
$8$	−4.01398	−1.41915
$9$	0	0
$10$	−2.37961	−0.752500
$11$	−1.23725	−0.373044	−0.186522	−	0.982451i	$-0.559722\pi$
$11$	−1.23725	−0.373044	−0.186522	+	0.982451i	$0.559722\pi$
$12$	0	0
$13$	−0.762635	−0.211517	−0.105758	−	0.994392i	$-0.533727\pi$
$13$	−0.762635	−0.211517	−0.105758	+	0.994392i	$0.533727\pi$
$14$	8.26569	2.20910
$15$	0	0
$16$	2.20214	0.550536
$17$	−2.76285	−0.670089	−0.335045	−	0.942202i	$-0.608751\pi$
$17$	−2.76285	−0.670089	−0.335045	+	0.942202i	$0.608751\pi$
$18$	0	0
$19$	−6.45519	−1.48092	−0.740461	−	0.672099i	$-0.765393\pi$
$19$	−6.45519	−1.48092	−0.740461	+	0.672099i	$0.765393\pi$
$20$	3.67683	0.822165
$21$	0	0
$22$	2.94966	0.628869
$23$	4.57326	0.953591	0.476795	−	0.879014i	$-0.341798\pi$
$23$	4.57326	0.953591	0.476795	+	0.879014i	$0.341798\pi$
$24$	0	0
$25$	−4.00371	−0.800743
$26$	1.81816	0.356570
$27$	0	0
$28$	−12.7716	−2.41361
$29$	−6.92170	−1.28533	−0.642664	−	0.766148i	$-0.722171\pi$
$29$	−6.92170	−1.28533	−0.642664	+	0.766148i	$0.722171\pi$
$30$	0	0
$31$	1.57611	0.283077	0.141539	−	0.989933i	$-0.454795\pi$
$31$	1.57611	0.283077	0.141539	+	0.989933i	$0.454795\pi$
$32$	2.77793	0.491074
$33$	0	0
$34$	6.58676	1.12962
$35$	−3.46064	−0.584955
$36$	0	0
$37$	1.00069	0.164512	0.0822561	−	0.996611i	$-0.473787\pi$
$37$	1.00069	0.164512	0.0822561	+	0.996611i	$0.473787\pi$
$38$	15.3895	2.49650
$39$	0	0
$40$	−4.00651	−0.633485
$41$	2.61346	0.408154	0.204077	−	0.978955i	$-0.434581\pi$
$41$	2.61346	0.408154	0.204077	+	0.978955i	$0.434581\pi$
$42$	0	0
$43$	−11.3267	−1.72731	−0.863656	−	0.504081i	$-0.831831\pi$
$43$	−11.3267	−1.72731	−0.863656	+	0.504081i	$0.831831\pi$
$44$	−4.55763	−0.687088
$45$	0	0
$46$	−10.9029	−1.60754
$47$	3.40145	0.496152	0.248076	−	0.968741i	$-0.420202\pi$
$47$	3.40145	0.496152	0.248076	+	0.968741i	$0.420202\pi$
$48$	0	0
$49$	5.02067	0.717239
$50$	9.54505	1.34987
$51$	0	0
$52$	−2.80930	−0.389580
$53$	12.4193	1.70593	0.852963	−	0.521971i	$-0.174803\pi$
$53$	12.4193	1.70593	0.852963	+	0.521971i	$0.174803\pi$
$54$	0	0
$55$	−1.23495	−0.166520
$56$	13.9168	1.85971
$57$	0	0
$58$	16.5017	2.16677
$59$	−10.4463	−1.36000	−0.679999	−	0.733213i	$-0.738020\pi$
$59$	−10.4463	−1.36000	−0.679999	+	0.733213i	$0.738020\pi$
$60$	0	0
$61$	−5.39005	−0.690125	−0.345063	−	0.938580i	$-0.612142\pi$
$61$	−5.39005	−0.690125	−0.345063	+	0.938580i	$0.612142\pi$
$62$	−3.75751	−0.477204
$63$	0	0
$64$	−11.0270	−1.37838
$65$	−0.761217	−0.0944174
$66$	0	0
$67$	−0.0539558	−0.00659175	−0.00329587	−	0.999995i	$-0.501049\pi$
$67$	−0.0539558	−0.00659175	−0.00329587	+	0.999995i	$0.501049\pi$
$68$	−10.1775	−1.23420
$69$	0	0
$70$	8.25032	0.986102
$71$	5.95774	0.707053	0.353527	−	0.935424i	$-0.384982\pi$
$71$	5.95774	0.707053	0.353527	+	0.935424i	$0.384982\pi$
$72$	0	0
$73$	−11.3804	−1.33197	−0.665986	−	0.745964i	$-0.731989\pi$
$73$	−11.3804	−1.33197	−0.665986	+	0.745964i	$0.731989\pi$
$74$	−2.38569	−0.277331
$75$	0	0
$76$	−23.7789	−2.72762
$77$	4.28964	0.488850
$78$	0	0
$79$	8.94972	1.00692	0.503461	−	0.864018i	$-0.332060\pi$
$79$	8.94972	1.00692	0.503461	+	0.864018i	$0.332060\pi$
$80$	2.19805	0.245749
$81$	0	0
$82$	−6.23062	−0.688056
$83$	−7.61618	−0.835985	−0.417992	−	0.908451i	$-0.637266\pi$
$83$	−7.61618	−0.835985	−0.417992	+	0.908451i	$0.637266\pi$
$84$	0	0
$85$	−2.75771	−0.299116
$86$	27.0035	2.91186
$87$	0	0
$88$	4.96628	0.529407
$89$	13.3813	1.41842	0.709209	−	0.704998i	$-0.249052\pi$
$89$	13.3813	1.41842	0.709209	+	0.704998i	$0.249052\pi$
$90$	0	0
$91$	2.64412	0.277179
$92$	16.8464	1.75636
$93$	0	0
$94$	−8.10921	−0.836401
$95$	−6.44319	−0.661057
$96$	0	0
$97$	17.4286	1.76961	0.884805	−	0.465962i	$-0.154292\pi$
$97$	17.4286	1.76961	0.884805	+	0.465962i	$0.154292\pi$
$98$	−11.9695	−1.20910
$99$	0	0
$100$	−14.7484	−1.47484
$101$	18.6299	1.85374	0.926870	−	0.375382i	$-0.122488\pi$
$101$	18.6299	1.85374	0.926870	+	0.375382i	$0.122488\pi$
$102$	0	0
$103$	9.82804	0.968385	0.484193	−	0.874961i	$-0.339113\pi$
$103$	9.82804	0.968385	0.484193	+	0.874961i	$0.339113\pi$
$104$	3.06120	0.300175
$105$	0	0
$106$	−29.6083	−2.87581
$107$	−7.15620	−0.691816	−0.345908	−	0.938268i	$-0.612429\pi$
$107$	−7.15620	−0.691816	−0.345908	+	0.938268i	$0.612429\pi$
$108$	0	0
$109$	8.83204	0.845956	0.422978	−	0.906140i	$-0.360985\pi$
$109$	8.83204	0.845956	0.422978	+	0.906140i	$0.360985\pi$
$110$	2.94417	0.280716
$111$	0	0
$112$	−7.63502	−0.721441
$113$	10.5939	0.996593	0.498297	−	0.867007i	$-0.333959\pi$
$113$	10.5939	0.996593	0.498297	+	0.867007i	$0.333959\pi$
$114$	0	0
$115$	4.56476	0.425666
$116$	−25.4973	−2.36737
$117$	0	0
$118$	24.9046	2.29265
$119$	9.57903	0.878108
$120$	0	0
$121$	−9.46922	−0.860838
$122$	12.8501	1.16340
$123$	0	0
$124$	5.80587	0.521383
$125$	−8.98698	−0.803820
$126$	0	0
$127$	13.0413	1.15723	0.578614	−	0.815602i	$-0.303594\pi$
$127$	13.0413	1.15723	0.578614	+	0.815602i	$0.303594\pi$
$128$	20.7331	1.83256
$129$	0	0
$130$	1.81478	0.159167
$131$	−10.2340	−0.894147	−0.447073	−	0.894497i	$-0.647534\pi$
$131$	−10.2340	−0.894147	−0.447073	+	0.894497i	$0.647534\pi$
$132$	0	0
$133$	22.3807	1.94065
$134$	0.128633	0.0111122
$135$	0	0
$136$	11.0900	0.950960
$137$	−4.32807	−0.369772	−0.184886	−	0.982760i	$-0.559192\pi$
$137$	−4.32807	−0.369772	−0.184886	+	0.982760i	$0.559192\pi$
$138$	0	0
$139$	5.57468	0.472838	0.236419	−	0.971651i	$-0.424026\pi$
$139$	5.57468	0.472838	0.236419	+	0.971651i	$0.424026\pi$
$140$	−12.7479	−1.07739
$141$	0	0
$142$	−14.2035	−1.19193
$143$	0.943568	0.0789052
$144$	0	0
$145$	−6.90883	−0.573747
$146$	27.1313	2.24541
$147$	0	0
$148$	3.68622	0.303005
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	7.33640	0.597028	0.298514	−	0.954405i	$-0.403509\pi$
$151$	7.33640	0.597028	0.298514	+	0.954405i	$0.403509\pi$
$152$	25.9110	2.10166
$153$	0	0
$154$	−10.2267	−0.824091
$155$	1.57318	0.126361
$156$	0	0
$157$	−9.43710	−0.753163	−0.376581	−	0.926384i	$-0.622900\pi$
$157$	−9.43710	−0.753163	−0.376581	+	0.926384i	$0.622900\pi$
$158$	−21.3366	−1.69745
$159$	0	0
$160$	2.77277	0.219207
$161$	−15.8559	−1.24962
$162$	0	0
$163$	−17.4412	−1.36610	−0.683049	−	0.730372i	$-0.739347\pi$
$163$	−17.4412	−1.36610	−0.683049	+	0.730372i	$0.739347\pi$
$164$	9.62716	0.751755
$165$	0	0
$166$	18.1573	1.40928
$167$	9.08512	0.703028	0.351514	−	0.936183i	$-0.385667\pi$
$167$	9.08512	0.703028	0.351514	+	0.936183i	$0.385667\pi$
$168$	0	0
$169$	−12.4184	−0.955261
$170$	6.57452	0.504242
$171$	0	0
$172$	−41.7241	−3.18143
$173$	−13.1527	−0.999983	−0.499992	−	0.866030i	$-0.666664\pi$
$173$	−13.1527	−0.999983	−0.499992	+	0.866030i	$0.666664\pi$
$174$	0	0
$175$	13.8812	1.04932
$176$	−2.72460	−0.205374
$177$	0	0
$178$	−31.9017	−2.39114
$179$	−17.1074	−1.27867	−0.639335	−	0.768928i	$-0.720790\pi$
$179$	−17.1074	−1.27867	−0.639335	+	0.768928i	$0.720790\pi$
$180$	0	0
$181$	−7.43609	−0.552720	−0.276360	−	0.961054i	$-0.589128\pi$
$181$	−7.43609	−0.552720	−0.276360	+	0.961054i	$0.589128\pi$
$182$	−6.30371	−0.467262
$183$	0	0
$184$	−18.3570	−1.35329
$185$	0.998829	0.0734353
$186$	0	0
$187$	3.41833	0.249973
$188$	12.5298	0.913833
$189$	0	0
$190$	15.3609	1.11439
$191$	1.33487	0.0965876	0.0482938	−	0.998833i	$-0.484622\pi$
$191$	1.33487	0.0965876	0.0482938	+	0.998833i	$0.484622\pi$
$192$	0	0
$193$	3.61588	0.260276	0.130138	−	0.991496i	$-0.458458\pi$
$193$	3.61588	0.260276	0.130138	+	0.991496i	$0.458458\pi$
$194$	−41.5507	−2.98317
$195$	0	0
$196$	18.4945	1.32104
$197$	13.6180	0.970244	0.485122	−	0.874447i	$-0.338775\pi$
$197$	13.6180	0.970244	0.485122	+	0.874447i	$0.338775\pi$
$198$	0	0
$199$	14.6310	1.03717	0.518583	−	0.855027i	$-0.326460\pi$
$199$	14.6310	1.03717	0.518583	+	0.855027i	$0.326460\pi$
$200$	16.0708	1.13638
$201$	0	0
$202$	−44.4145	−3.12499
$203$	23.9981	1.68434
$204$	0	0
$205$	2.60860	0.182193
$206$	−23.4305	−1.63248
$207$	0	0
$208$	−1.67943	−0.116448
$209$	7.98667	0.552449
$210$	0	0
$211$	4.68973	0.322854	0.161427	−	0.986885i	$-0.448390\pi$
$211$	4.68973	0.322854	0.161427	+	0.986885i	$0.448390\pi$
$212$	45.7489	3.14205
$213$	0	0
$214$	17.0607	1.16625
$215$	−11.3057	−0.771042
$216$	0	0
$217$	−5.46449	−0.370954
$218$	−21.0560	−1.42609
$219$	0	0
$220$	−4.54915	−0.306704
$221$	2.10705	0.141735
$222$	0	0
$223$	27.5091	1.84215	0.921074	−	0.389387i	$-0.127313\pi$
$223$	27.5091	1.84215	0.921074	+	0.389387i	$0.127313\pi$
$224$	−9.63133	−0.643520
$225$	0	0
$226$	−25.2564	−1.68003
$227$	−4.37023	−0.290062	−0.145031	−	0.989427i	$-0.546328\pi$
$227$	−4.37023	−0.290062	−0.145031	+	0.989427i	$0.546328\pi$
$228$	0	0
$229$	−14.9242	−0.986217	−0.493108	−	0.869968i	$-0.664139\pi$
$229$	−14.9242	−0.986217	−0.493108	+	0.869968i	$0.664139\pi$
$230$	−10.8826	−0.717577
$231$	0	0
$232$	27.7835	1.82408
$233$	19.2204	1.25917	0.629585	−	0.776932i	$-0.283225\pi$
$233$	19.2204	1.25917	0.629585	+	0.776932i	$0.283225\pi$
$234$	0	0
$235$	3.39512	0.221473
$236$	−38.4810	−2.50490
$237$	0	0
$238$	−22.8369	−1.48029
$239$	−18.3842	−1.18917	−0.594586	−	0.804032i	$-0.702684\pi$
$239$	−18.3842	−1.18917	−0.594586	+	0.804032i	$0.702684\pi$
$240$	0	0
$241$	11.3226	0.729352	0.364676	−	0.931135i	$-0.381180\pi$
$241$	11.3226	0.729352	0.364676	+	0.931135i	$0.381180\pi$
$242$	22.5751	1.45118
$243$	0	0
$244$	−19.8552	−1.27110
$245$	5.01134	0.320163
$246$	0	0
$247$	4.92296	0.313240
$248$	−6.32645	−0.401730
$249$	0	0
$250$	21.4254	1.35506
$251$	−16.3196	−1.03009	−0.515043	−	0.857164i	$-0.672224\pi$
$251$	−16.3196	−1.03009	−0.515043	+	0.857164i	$0.672224\pi$
$252$	0	0
$253$	−5.65825	−0.355731
$254$	−31.0911	−1.95083
$255$	0	0
$256$	−27.3746	−1.71091
$257$	8.52759	0.531936	0.265968	−	0.963982i	$-0.414308\pi$
$257$	8.52759	0.531936	0.265968	+	0.963982i	$0.414308\pi$
$258$	0	0
$259$	−3.46947	−0.215583
$260$	−2.80408	−0.173902
$261$	0	0
$262$	24.3983	1.50733
$263$	−5.13009	−0.316335	−0.158168	−	0.987412i	$-0.550559\pi$
$263$	−5.13009	−0.316335	−0.158168	+	0.987412i	$0.550559\pi$
$264$	0	0
$265$	12.3962	0.761495
$266$	−53.3566	−3.27150
$267$	0	0
$268$	−0.198756	−0.0121409
$269$	−2.76401	−0.168525	−0.0842623	−	0.996444i	$-0.526853\pi$
$269$	−2.76401	−0.168525	−0.0842623	+	0.996444i	$0.526853\pi$
$270$	0	0
$271$	−4.41409	−0.268137	−0.134068	−	0.990972i	$-0.542804\pi$
$271$	−4.41409	−0.268137	−0.134068	+	0.990972i	$0.542804\pi$
$272$	−6.08419	−0.368908
$273$	0	0
$274$	10.3183	0.623352
$275$	4.95359	0.298712
$276$	0	0
$277$	−10.4869	−0.630096	−0.315048	−	0.949076i	$-0.602021\pi$
$277$	−10.4869	−0.630096	−0.315048	+	0.949076i	$0.602021\pi$
$278$	−13.2903	−0.797099
$279$	0	0
$280$	13.8909	0.830141
$281$	−10.0782	−0.601214	−0.300607	−	0.953748i	$-0.597189\pi$
$281$	−10.0782	−0.601214	−0.300607	+	0.953748i	$0.597189\pi$
$282$	0	0
$283$	−5.64659	−0.335655	−0.167827	−	0.985816i	$-0.553675\pi$
$283$	−5.64659	−0.335655	−0.167827	+	0.985816i	$0.553675\pi$
$284$	21.9464	1.30228
$285$	0	0
$286$	−2.24951	−0.133016
$287$	−9.06109	−0.534859
$288$	0	0
$289$	−9.36666	−0.550980
$290$	16.4710	0.967210
$291$	0	0
$292$	−41.9217	−2.45328
$293$	3.57507	0.208858	0.104429	−	0.994532i	$-0.466699\pi$
$293$	3.57507	0.208858	0.104429	+	0.994532i	$0.466699\pi$
$294$	0	0
$295$	−10.4269	−0.607079
$296$	−4.01674	−0.233468
$297$	0	0
$298$	−2.38405	−0.138104
$299$	−3.48773	−0.201701
$300$	0	0
$301$	39.2708	2.26353
$302$	−17.4903	−1.00645
$303$	0	0
$304$	−14.2153	−0.815301
$305$	−5.38003	−0.308060
$306$	0	0
$307$	−25.4645	−1.45334	−0.726669	−	0.686988i	$-0.758932\pi$
$307$	−25.4645	−1.45334	−0.726669	+	0.686988i	$0.758932\pi$
$308$	15.8017	0.900384
$309$	0	0
$310$	−3.75053	−0.213016
$311$	−16.0458	−0.909873	−0.454937	−	0.890524i	$-0.650338\pi$
$311$	−16.0458	−0.909873	−0.454937	+	0.890524i	$0.650338\pi$
$312$	0	0
$313$	17.7027	1.00061	0.500307	−	0.865848i	$-0.333220\pi$
$313$	17.7027	1.00061	0.500307	+	0.865848i	$0.333220\pi$
$314$	22.4985	1.26966
$315$	0	0
$316$	32.9679	1.85459
$317$	26.3887	1.48214	0.741068	−	0.671430i	$-0.234320\pi$
$317$	26.3887	1.48214	0.741068	+	0.671430i	$0.234320\pi$
$318$	0	0
$319$	8.56386	0.479484
$320$	−11.0065	−0.615283
$321$	0	0
$322$	37.8012	2.10658
$323$	17.8347	0.992350
$324$	0	0
$325$	3.05337	0.169371
$326$	41.5806	2.30294
$327$	0	0
$328$	−10.4904	−0.579234
$329$	−11.7931	−0.650175
$330$	0	0
$331$	7.96364	0.437721	0.218861	−	0.975756i	$-0.429766\pi$
$331$	7.96364	0.437721	0.218861	+	0.975756i	$0.429766\pi$
$332$	−28.0556	−1.53975
$333$	0	0
$334$	−21.6594	−1.18515
$335$	−0.0538555	−0.00294244
$336$	0	0
$337$	5.88539	0.320598	0.160299	−	0.987069i	$-0.448754\pi$
$337$	5.88539	0.320598	0.160299	+	0.987069i	$0.448754\pi$
$338$	29.6060	1.61036
$339$	0	0
$340$	−10.1585	−0.550924
$341$	−1.95003	−0.105600
$342$	0	0
$343$	6.86250	0.370540
$344$	45.4653	2.45132
$345$	0	0
$346$	31.3567	1.68575
$347$	29.4521	1.58107	0.790536	−	0.612416i	$-0.209802\pi$
$347$	29.4521	1.58107	0.790536	+	0.612416i	$0.209802\pi$
$348$	0	0
$349$	−2.91696	−0.156141	−0.0780707	−	0.996948i	$-0.524876\pi$
$349$	−2.91696	−0.156141	−0.0780707	+	0.996948i	$0.524876\pi$
$350$	−33.0935	−1.76892
$351$	0	0
$352$	−3.43699	−0.183192
$353$	14.1278	0.751946	0.375973	−	0.926631i	$-0.377309\pi$
$353$	14.1278	0.751946	0.375973	+	0.926631i	$0.377309\pi$
$354$	0	0
$355$	5.94666	0.315616
$356$	49.2926	2.61250
$357$	0	0
$358$	40.7850	2.15555
$359$	21.9246	1.15714	0.578569	−	0.815633i	$-0.303611\pi$
$359$	21.9246	1.15714	0.578569	+	0.815633i	$0.303611\pi$
$360$	0	0
$361$	22.6695	1.19313
$362$	17.7280	0.931762
$363$	0	0
$364$	9.74010	0.510520
$365$	−11.3592	−0.594568
$366$	0	0
$367$	22.2116	1.15943	0.579717	−	0.814818i	$-0.303163\pi$
$367$	22.2116	1.15943	0.579717	+	0.814818i	$0.303163\pi$
$368$	10.0710	0.524986
$369$	0	0
$370$	−2.38125	−0.123796
$371$	−43.0589	−2.23551
$372$	0	0
$373$	23.4742	1.21545	0.607725	−	0.794148i	$-0.292082\pi$
$373$	23.4742	1.21545	0.607725	+	0.794148i	$0.292082\pi$
$374$	−8.14946	−0.421398
$375$	0	0
$376$	−13.6533	−0.704116
$377$	5.27873	0.271869
$378$	0	0
$379$	−4.61345	−0.236977	−0.118489	−	0.992955i	$-0.537805\pi$
$379$	−4.61345	−0.236977	−0.118489	+	0.992955i	$0.537805\pi$
$380$	−23.7347	−1.21756
$381$	0	0
$382$	−3.18238	−0.162825
$383$	7.55684	0.386136	0.193068	−	0.981185i	$-0.438156\pi$
$383$	7.55684	0.386136	0.193068	+	0.981185i	$0.438156\pi$
$384$	0	0
$385$	4.28167	0.218214
$386$	−8.62042	−0.438768
$387$	0	0
$388$	64.2015	3.25934
$389$	−18.6228	−0.944215	−0.472108	−	0.881541i	$-0.656507\pi$
$389$	−18.6228	−0.944215	−0.472108	+	0.881541i	$0.656507\pi$
$390$	0	0
$391$	−12.6352	−0.638991
$392$	−20.1528	−1.01787
$393$	0	0
$394$	−32.4660	−1.63561
$395$	8.93308	0.449472
$396$	0	0
$397$	21.0308	1.05550	0.527752	−	0.849398i	$-0.323035\pi$
$397$	21.0308	1.05550	0.527752	+	0.849398i	$0.323035\pi$
$398$	−34.8811	−1.74843
$399$	0	0
$400$	−8.81676	−0.440838
$401$	9.19974	0.459413	0.229707	−	0.973260i	$-0.426223\pi$
$401$	9.19974	0.459413	0.229707	+	0.973260i	$0.426223\pi$
$402$	0	0
$403$	−1.20199	−0.0598756
$404$	68.6265	3.41429
$405$	0	0
$406$	−57.2126	−2.83942
$407$	−1.23810	−0.0613703
$408$	0	0
$409$	10.9723	0.542548	0.271274	−	0.962502i	$-0.412555\pi$
$409$	10.9723	0.542548	0.271274	+	0.962502i	$0.412555\pi$
$410$	−6.21903	−0.307136
$411$	0	0
$412$	36.2033	1.78361
$413$	36.2183	1.78219
$414$	0	0
$415$	−7.60202	−0.373169
$416$	−2.11855	−0.103870
$417$	0	0
$418$	−19.0406	−0.931306
$419$	−36.7714	−1.79640	−0.898200	−	0.439588i	$-0.855124\pi$
$419$	−36.7714	−1.79640	−0.898200	+	0.439588i	$0.855124\pi$
$420$	0	0
$421$	39.3962	1.92005	0.960026	−	0.279910i	$-0.0903047\pi$
$421$	39.3962	1.92005	0.960026	+	0.279910i	$0.0903047\pi$
$422$	−11.1805	−0.544260
$423$	0	0
$424$	−49.8509	−2.42097
$425$	11.0617	0.536569
$426$	0	0
$427$	18.6878	0.904364
$428$	−26.3612	−1.27421
$429$	0	0
$430$	26.9533	1.29980
$431$	15.6398	0.753342	0.376671	−	0.926347i	$-0.377069\pi$
$431$	15.6398	0.753342	0.376671	+	0.926347i	$0.377069\pi$
$432$	0	0
$433$	22.4318	1.07800	0.539001	−	0.842305i	$-0.318802\pi$
$433$	22.4318	1.07800	0.539001	+	0.842305i	$0.318802\pi$
$434$	13.0276	0.625345
$435$	0	0
$436$	32.5344	1.55812
$437$	−29.5213	−1.41219
$438$	0	0
$439$	−33.4139	−1.59476	−0.797380	−	0.603478i	$-0.793781\pi$
$439$	−33.4139	−1.59476	−0.797380	+	0.603478i	$0.793781\pi$
$440$	4.95705	0.236318
$441$	0	0
$442$	−5.02330	−0.238934
$443$	12.7494	0.605741	0.302870	−	0.953032i	$-0.402055\pi$
$443$	12.7494	0.605741	0.302870	+	0.953032i	$0.402055\pi$
$444$	0	0
$445$	13.3565	0.633157
$446$	−65.5831	−3.10545
$447$	0	0
$448$	38.2316	1.80627
$449$	29.9188	1.41196	0.705978	−	0.708234i	$-0.250508\pi$
$449$	29.9188	1.41196	0.705978	+	0.708234i	$0.250508\pi$
$450$	0	0
$451$	−3.23350	−0.152260
$452$	39.0247	1.83557
$453$	0	0
$454$	10.4188	0.488980
$455$	2.63920	0.123728
$456$	0	0
$457$	12.4980	0.584633	0.292317	−	0.956322i	$-0.405574\pi$
$457$	12.4980	0.584633	0.292317	+	0.956322i	$0.405574\pi$
$458$	35.5799	1.66254
$459$	0	0
$460$	16.8151	0.784009
$461$	−21.2948	−0.991799	−0.495899	−	0.868380i	$-0.665162\pi$
$461$	−21.2948	−0.991799	−0.495899	+	0.868380i	$0.665162\pi$
$462$	0	0
$463$	21.1301	0.982000	0.491000	−	0.871160i	$-0.336631\pi$
$463$	21.1301	0.982000	0.491000	+	0.871160i	$0.336631\pi$
$464$	−15.2426	−0.707619
$465$	0	0
$466$	−45.8223	−2.12268
$467$	2.70466	0.125157	0.0625784	−	0.998040i	$-0.480068\pi$
$467$	2.70466	0.125157	0.0625784	+	0.998040i	$0.480068\pi$
$468$	0	0
$469$	0.187069	0.00863805
$470$	−8.09413	−0.373355
$471$	0	0
$472$	41.9313	1.93005
$473$	14.0140	0.644364
$474$	0	0
$475$	25.8447	1.18584
$476$	35.2861	1.61734
$477$	0	0
$478$	43.8287	2.00468
$479$	1.78132	0.0813907	0.0406954	−	0.999172i	$-0.487043\pi$
$479$	1.78132	0.0813907	0.0406954	+	0.999172i	$0.487043\pi$
$480$	0	0
$481$	−0.763161	−0.0347971
$482$	−26.9936	−1.22952
$483$	0	0
$484$	−34.8816	−1.58553
$485$	17.3962	0.789922
$486$	0	0
$487$	−41.8679	−1.89722	−0.948609	−	0.316451i	$-0.897509\pi$
$487$	−41.8679	−1.89722	−0.948609	+	0.316451i	$0.897509\pi$
$488$	21.6355	0.979394
$489$	0	0
$490$	−11.9473	−0.539722
$491$	21.8464	0.985912	0.492956	−	0.870054i	$-0.335916\pi$
$491$	21.8464	0.985912	0.492956	+	0.870054i	$0.335916\pi$
$492$	0	0
$493$	19.1236	0.861284
$494$	−11.7366	−0.528053
$495$	0	0
$496$	3.47081	0.155844
$497$	−20.6560	−0.926547
$498$	0	0
$499$	−25.9249	−1.16056	−0.580279	−	0.814418i	$-0.697057\pi$
$499$	−25.9249	−1.16056	−0.580279	+	0.814418i	$0.697057\pi$
$500$	−33.1052	−1.48051
$501$	0	0
$502$	38.9068	1.73649
$503$	29.7341	1.32578	0.662889	−	0.748717i	$-0.269330\pi$
$503$	29.7341	1.32578	0.662889	+	0.748717i	$0.269330\pi$
$504$	0	0
$505$	18.5952	0.827477
$506$	13.4895	0.599683
$507$	0	0
$508$	48.0400	2.13143
$509$	−0.0992115	−0.00439747	−0.00219874	−	0.999998i	$-0.500700\pi$
$509$	−0.0992115	−0.00439747	−0.00219874	+	0.999998i	$0.500700\pi$
$510$	0	0
$511$	39.4567	1.74546
$512$	23.7961	1.05165
$513$	0	0
$514$	−20.3302	−0.896726
$515$	9.80976	0.432270
$516$	0	0
$517$	−4.20843	−0.185087
$518$	8.27139	0.363424
$519$	0	0
$520$	3.05551	0.133993
$521$	32.1553	1.40875	0.704375	−	0.709828i	$-0.251227\pi$
$521$	32.1553	1.40875	0.704375	+	0.709828i	$0.251227\pi$
$522$	0	0
$523$	13.6048	0.594896	0.297448	−	0.954738i	$-0.403865\pi$
$523$	13.6048	0.594896	0.297448	+	0.954738i	$0.403865\pi$
$524$	−37.6987	−1.64688
$525$	0	0
$526$	12.2304	0.533270
$527$	−4.35454	−0.189687
$528$	0	0
$529$	−2.08529	−0.0906650
$530$	−29.5532	−1.28371
$531$	0	0
$532$	82.4433	3.57437
$533$	−1.99312	−0.0863315
$534$	0	0
$535$	−7.14289	−0.308814
$536$	0.216577	0.00935471
$537$	0	0
$538$	6.58953	0.284095
$539$	−6.21181	−0.267562
$540$	0	0
$541$	−14.9180	−0.641375	−0.320687	−	0.947185i	$-0.603914\pi$
$541$	−14.9180	−0.641375	−0.320687	+	0.947185i	$0.603914\pi$
$542$	10.5234	0.452018
$543$	0	0
$544$	−7.67502	−0.329064
$545$	8.81562	0.377620
$546$	0	0
$547$	−19.8457	−0.848540	−0.424270	−	0.905536i	$-0.639469\pi$
$547$	−19.8457	−0.848540	−0.424270	+	0.905536i	$0.639469\pi$
$548$	−15.9432	−0.681060
$549$	0	0
$550$	−11.8096	−0.503562
$551$	44.6809	1.90347
$552$	0	0
$553$	−31.0294	−1.31951
$554$	25.0012	1.06220
$555$	0	0
$556$	20.5353	0.870893
$557$	43.1249	1.82726	0.913629	−	0.406548i	$-0.133268\pi$
$557$	43.1249	1.82726	0.913629	+	0.406548i	$0.133268\pi$
$558$	0	0
$559$	8.63818	0.365356
$560$	−7.62082	−0.322039
$561$	0	0
$562$	24.0268	1.01351
$563$	19.0889	0.804500	0.402250	−	0.915530i	$-0.368228\pi$
$563$	19.0889	0.804500	0.402250	+	0.915530i	$0.368228\pi$
$564$	0	0
$565$	10.5742	0.444862
$566$	13.4617	0.565839
$567$	0	0
$568$	−23.9142	−1.00342
$569$	−0.699429	−0.0293216	−0.0146608	−	0.999893i	$-0.504667\pi$
$569$	−0.699429	−0.0293216	−0.0146608	+	0.999893i	$0.504667\pi$
$570$	0	0
$571$	44.1116	1.84601	0.923006	−	0.384785i	$-0.125724\pi$
$571$	44.1116	1.84601	0.923006	+	0.384785i	$0.125724\pi$
$572$	3.47581	0.145331
$573$	0	0
$574$	21.6021	0.901653
$575$	−18.3100	−0.763581
$576$	0	0
$577$	−8.96112	−0.373056	−0.186528	−	0.982450i	$-0.559724\pi$
$577$	−8.96112	−0.373056	−0.186528	+	0.982450i	$0.559724\pi$
$578$	22.3306	0.928829
$579$	0	0
$580$	−25.4499	−1.05675
$581$	26.4059	1.09550
$582$	0	0
$583$	−15.3658	−0.636386
$584$	45.6805	1.89027
$585$	0	0
$586$	−8.52313	−0.352087
$587$	19.6623	0.811550	0.405775	−	0.913973i	$-0.367002\pi$
$587$	19.6623	0.811550	0.405775	+	0.913973i	$0.367002\pi$
$588$	0	0
$589$	−10.1741	−0.419215
$590$	24.8583	1.02340
$591$	0	0
$592$	2.20366	0.0905699
$593$	19.9795	0.820461	0.410230	−	0.911982i	$-0.365448\pi$
$593$	19.9795	0.820461	0.410230	+	0.911982i	$0.365448\pi$
$594$	0	0
$595$	9.56122	0.391972
$596$	3.68368	0.150889
$597$	0	0
$598$	8.31491	0.340022
$599$	−22.0729	−0.901872	−0.450936	−	0.892556i	$-0.648910\pi$
$599$	−22.0729	−0.901872	−0.450936	+	0.892556i	$0.648910\pi$
$600$	0	0
$601$	−28.2995	−1.15436	−0.577181	−	0.816617i	$-0.695847\pi$
$601$	−28.2995	−1.15436	−0.577181	+	0.816617i	$0.695847\pi$
$602$	−93.6234	−3.81580
$603$	0	0
$604$	27.0250	1.09963
$605$	−9.45161	−0.384263
$606$	0	0
$607$	13.7850	0.559514	0.279757	−	0.960071i	$-0.409746\pi$
$607$	13.7850	0.559514	0.279757	+	0.960071i	$0.409746\pi$
$608$	−17.9321	−0.727243
$609$	0	0
$610$	12.8262	0.519319
$611$	−2.59406	−0.104945
$612$	0	0
$613$	−35.0346	−1.41503	−0.707516	−	0.706697i	$-0.750184\pi$
$613$	−35.0346	−1.41503	−0.707516	+	0.706697i	$0.750184\pi$
$614$	60.7086	2.45000
$615$	0	0
$616$	−17.2185	−0.693754
$617$	−0.327044	−0.0131663	−0.00658315	−	0.999978i	$-0.502095\pi$
$617$	−0.327044	−0.0131663	−0.00658315	+	0.999978i	$0.502095\pi$
$618$	0	0
$619$	−24.0165	−0.965307	−0.482653	−	0.875812i	$-0.660327\pi$
$619$	−24.0165	−0.965307	−0.482653	+	0.875812i	$0.660327\pi$
$620$	5.79508	0.232736
$621$	0	0
$622$	38.2539	1.53384
$623$	−46.3942	−1.85874
$624$	0	0
$625$	11.0483	0.441932
$626$	−42.2040	−1.68681
$627$	0	0
$628$	−34.7633	−1.38721
$629$	−2.76475	−0.110238
$630$	0	0
$631$	−24.5278	−0.976435	−0.488217	−	0.872722i	$-0.662353\pi$
$631$	−24.5278	−0.976435	−0.488217	+	0.872722i	$0.662353\pi$
$632$	−35.9240	−1.42898
$633$	0	0
$634$	−62.9118	−2.49855
$635$	13.0170	0.516566
$636$	0	0
$637$	−3.82894	−0.151708
$638$	−20.4166	−0.808303
$639$	0	0
$640$	20.6945	0.818022
$641$	21.9268	0.866057	0.433028	−	0.901380i	$-0.357445\pi$
$641$	21.9268	0.866057	0.433028	+	0.901380i	$0.357445\pi$
$642$	0	0
$643$	−19.3024	−0.761212	−0.380606	−	0.924737i	$-0.624285\pi$
$643$	−19.3024	−0.761212	−0.380606	+	0.924737i	$0.624285\pi$
$644$	−58.4080	−2.30160
$645$	0	0
$646$	−42.5188	−1.67288
$647$	20.0482	0.788175	0.394088	−	0.919073i	$-0.371061\pi$
$647$	20.0482	0.788175	0.394088	+	0.919073i	$0.371061\pi$
$648$	0	0
$649$	12.9247	0.507339
$650$	−7.27939	−0.285521
$651$	0	0
$652$	−64.2478	−2.51614
$653$	6.89943	0.269996	0.134998	−	0.990846i	$-0.456897\pi$
$653$	6.89943	0.269996	0.134998	+	0.990846i	$0.456897\pi$
$654$	0	0
$655$	−10.2149	−0.399131
$656$	5.75522	0.224704
$657$	0	0
$658$	28.1153	1.09605
$659$	−33.3445	−1.29892	−0.649459	−	0.760397i	$-0.725004\pi$
$659$	−33.3445	−1.29892	−0.649459	+	0.760397i	$0.725004\pi$
$660$	0	0
$661$	0.377199	0.0146713	0.00733566	−	0.999973i	$-0.497665\pi$
$661$	0.377199	0.0146713	0.00733566	+	0.999973i	$0.497665\pi$
$662$	−18.9857	−0.737900
$663$	0	0
$664$	30.5712	1.18639
$665$	22.3391	0.866272
$666$	0	0
$667$	−31.6547	−1.22568
$668$	33.4667	1.29487
$669$	0	0
$670$	0.128394	0.00496029
$671$	6.66883	0.257447
$672$	0	0
$673$	−10.5634	−0.407190	−0.203595	−	0.979055i	$-0.565263\pi$
$673$	−10.5634	−0.407190	−0.203595	+	0.979055i	$0.565263\pi$
$674$	−14.0311	−0.540456
$675$	0	0
$676$	−45.7454	−1.75944
$677$	−29.3172	−1.12675	−0.563376	−	0.826201i	$-0.690498\pi$
$677$	−29.3172	−1.12675	−0.563376	+	0.826201i	$0.690498\pi$
$678$	0	0
$679$	−60.4265	−2.31896
$680$	11.0694	0.424492
$681$	0	0
$682$	4.64897	0.178018
$683$	−28.7079	−1.09848	−0.549238	−	0.835666i	$-0.685082\pi$
$683$	−28.7079	−1.09848	−0.549238	+	0.835666i	$0.685082\pi$
$684$	0	0
$685$	−4.32002	−0.165059
$686$	−16.3605	−0.624648
$687$	0	0
$688$	−24.9431	−0.950948
$689$	−9.47142	−0.360832
$690$	0	0
$691$	13.4672	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$691$	13.4672	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$692$	−48.4504	−1.84181
$693$	0	0
$694$	−70.2152	−2.66533
$695$	5.56432	0.211067
$696$	0	0
$697$	−7.22060	−0.273500
$698$	6.95418	0.263220
$699$	0	0
$700$	51.1340	1.93268
$701$	24.3059	0.918023	0.459011	−	0.888430i	$-0.348204\pi$
$701$	24.3059	0.918023	0.459011	+	0.888430i	$0.348204\pi$
$702$	0	0
$703$	−6.45964	−0.243630
$704$	13.6431	0.514195
$705$	0	0
$706$	−33.6813	−1.26761
$707$	−64.5913	−2.42921
$708$	0	0
$709$	−28.6912	−1.07752	−0.538760	−	0.842460i	$-0.681107\pi$
$709$	−28.6912	−1.07752	−0.538760	+	0.842460i	$0.681107\pi$
$710$	−14.1771	−0.532058
$711$	0	0
$712$	−53.7123	−2.01295
$713$	7.20794	0.269940
$714$	0	0
$715$	0.941814	0.0352219
$716$	−63.0184	−2.35511
$717$	0	0
$718$	−52.2694	−1.95068
$719$	−25.2731	−0.942527	−0.471264	−	0.881992i	$-0.656202\pi$
$719$	−25.2731	−0.942527	−0.471264	+	0.881992i	$0.656202\pi$
$720$	0	0
$721$	−34.0746	−1.26901
$722$	−54.0452	−2.01135
$723$	0	0
$724$	−27.3922	−1.01802
$725$	27.7125	1.02922
$726$	0	0
$727$	34.5720	1.28220	0.641102	−	0.767456i	$-0.278477\pi$
$727$	34.5720	1.28220	0.641102	+	0.767456i	$0.278477\pi$
$728$	−10.6134	−0.393360
$729$	0	0
$730$	27.0809	1.00231
$731$	31.2941	1.15745
$732$	0	0
$733$	48.9688	1.80870	0.904352	−	0.426787i	$-0.140355\pi$
$733$	48.9688	1.80870	0.904352	+	0.426787i	$0.140355\pi$
$734$	−52.9534	−1.95455
$735$	0	0
$736$	12.7042	0.468284
$737$	0.0667566	0.00245901
$738$	0	0
$739$	30.6815	1.12864	0.564319	−	0.825557i	$-0.309139\pi$
$739$	30.6815	1.12864	0.564319	+	0.825557i	$0.309139\pi$
$740$	3.67937	0.135256
$741$	0	0
$742$	102.654	3.76856
$743$	0.463889	0.0170184	0.00850922	−	0.999964i	$-0.497291\pi$
$743$	0.463889	0.0170184	0.00850922	+	0.999964i	$0.497291\pi$
$744$	0	0
$745$	0.998141	0.0365691
$746$	−55.9637	−2.04898
$747$	0	0
$748$	12.5920	0.460410
$749$	24.8111	0.906579
$750$	0	0
$751$	−26.4668	−0.965789	−0.482894	−	0.875679i	$-0.660414\pi$
$751$	−26.4668	−0.965789	−0.482894	+	0.875679i	$0.660414\pi$
$752$	7.49048	0.273150
$753$	0	0
$754$	−12.5847	−0.458310
$755$	7.32276	0.266502
$756$	0	0
$757$	−19.2297	−0.698917	−0.349458	−	0.936952i	$-0.613634\pi$
$757$	−19.2297	−0.698917	−0.349458	+	0.936952i	$0.613634\pi$
$758$	10.9987	0.399490
$759$	0	0
$760$	25.8628	0.938143
$761$	1.25541	0.0455086	0.0227543	−	0.999741i	$-0.492756\pi$
$761$	1.25541	0.0455086	0.0227543	+	0.999741i	$0.492756\pi$
$762$	0	0
$763$	−30.6214	−1.10857
$764$	4.91722	0.177899
$765$	0	0
$766$	−18.0159	−0.650939
$767$	7.96675	0.287663
$768$	0	0
$769$	32.5758	1.17471	0.587356	−	0.809329i	$-0.300169\pi$
$769$	32.5758	1.17471	0.587356	+	0.809329i	$0.300169\pi$
$770$	−10.2077	−0.367860
$771$	0	0
$772$	13.3197	0.479388
$773$	21.1555	0.760912	0.380456	−	0.924799i	$-0.375767\pi$
$773$	21.1555	0.760912	0.380456	+	0.924799i	$0.375767\pi$
$774$	0	0
$775$	−6.31028	−0.226672
$776$	−69.9581	−2.51135
$777$	0	0
$778$	44.3977	1.59174
$779$	−16.8704	−0.604445
$780$	0	0
$781$	−7.37119	−0.263762
$782$	30.1230	1.07720
$783$	0	0
$784$	11.0562	0.394866
$785$	−9.41956	−0.336198
$786$	0	0
$787$	−23.7954	−0.848213	−0.424106	−	0.905612i	$-0.639412\pi$
$787$	−23.7954	−0.848213	−0.424106	+	0.905612i	$0.639412\pi$
$788$	50.1644	1.78703
$789$	0	0
$790$	−21.2969	−0.757710
$791$	−36.7301	−1.30597
$792$	0	0
$793$	4.11064	0.145973
$794$	−50.1384	−1.77934
$795$	0	0
$796$	53.8960	1.91029
$797$	−1.17425	−0.0415939	−0.0207970	−	0.999784i	$-0.506620\pi$
$797$	−1.17425	−0.0415939	−0.0207970	+	0.999784i	$0.506620\pi$
$798$	0	0
$799$	−9.39768	−0.332466
$800$	−11.1221	−0.393224
$801$	0	0
$802$	−21.9326	−0.774468
$803$	14.0803	0.496884
$804$	0	0
$805$	−15.8264	−0.557807
$806$	2.86561	0.100937
$807$	0	0
$808$	−74.7798	−2.63074
$809$	39.9147	1.40333	0.701663	−	0.712509i	$-0.252441\pi$
$809$	39.9147	1.40333	0.701663	+	0.712509i	$0.252441\pi$
$810$	0	0
$811$	−5.26347	−0.184826	−0.0924128	−	0.995721i	$-0.529458\pi$
$811$	−5.26347	−0.184826	−0.0924128	+	0.995721i	$0.529458\pi$
$812$	88.4014	3.10228
$813$	0	0
$814$	2.95169	0.103457
$815$	−17.4088	−0.609802
$816$	0	0
$817$	73.1163	2.55802
$818$	−26.1586	−0.914614
$819$	0	0
$820$	9.60926	0.335570
$821$	35.6781	1.24517	0.622587	−	0.782550i	$-0.286082\pi$
$821$	35.6781	1.24517	0.622587	+	0.782550i	$0.286082\pi$
$822$	0	0
$823$	−12.9686	−0.452058	−0.226029	−	0.974121i	$-0.572574\pi$
$823$	−12.9686	−0.452058	−0.226029	+	0.974121i	$0.572574\pi$
$824$	−39.4495	−1.37429
$825$	0	0
$826$	−86.3462	−3.00437
$827$	21.3652	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$827$	21.3652	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$828$	0	0
$829$	32.5877	1.13182	0.565909	−	0.824467i	$-0.308525\pi$
$829$	32.5877	1.13182	0.565909	+	0.824467i	$0.308525\pi$
$830$	18.1236	0.629079
$831$	0	0
$832$	8.40959	0.291550
$833$	−13.8714	−0.480614
$834$	0	0
$835$	9.06823	0.313819
$836$	29.4203	1.01752
$837$	0	0
$838$	87.6647	3.02833
$839$	9.62803	0.332396	0.166198	−	0.986092i	$-0.446851\pi$
$839$	9.62803	0.332396	0.166198	+	0.986092i	$0.446851\pi$
$840$	0	0
$841$	18.9100	0.652067
$842$	−93.9224	−3.23678
$843$	0	0
$844$	17.2755	0.594646
$845$	−12.3953	−0.426411
$846$	0	0
$847$	32.8306	1.12807
$848$	27.3492	0.939174
$849$	0	0
$850$	−26.3715	−0.904536
$851$	4.57641	0.156877
$852$	0	0
$853$	27.4189	0.938805	0.469403	−	0.882984i	$-0.344469\pi$
$853$	27.4189	0.938805	0.469403	+	0.882984i	$0.344469\pi$
$854$	−44.5525	−1.52455
$855$	0	0
$856$	28.7248	0.981794
$857$	19.7707	0.675355	0.337678	−	0.941262i	$-0.390359\pi$
$857$	19.7707	0.675355	0.337678	+	0.941262i	$0.390359\pi$
$858$	0	0
$859$	−34.6155	−1.18107	−0.590533	−	0.807014i	$-0.701082\pi$
$859$	−34.6155	−1.18107	−0.590533	+	0.807014i	$0.701082\pi$
$860$	−41.6466	−1.42014
$861$	0	0
$862$	−37.2860	−1.26997
$863$	39.6115	1.34839	0.674195	−	0.738553i	$-0.264491\pi$
$863$	39.6115	1.34839	0.674195	+	0.738553i	$0.264491\pi$
$864$	0	0
$865$	−13.1283	−0.446375
$866$	−53.4784	−1.81727
$867$	0	0
$868$	−20.1294	−0.683238
$869$	−11.0730	−0.375627
$870$	0	0
$871$	0.0411486	0.00139427
$872$	−35.4516	−1.20054
$873$	0	0
$874$	70.3801	2.38064
$875$	31.1586	1.05335
$876$	0	0
$877$	−3.00714	−0.101544	−0.0507720	−	0.998710i	$-0.516168\pi$
$877$	−3.00714	−0.101544	−0.0507720	+	0.998710i	$0.516168\pi$
$878$	79.6604	2.68841
$879$	0	0
$880$	−2.71953	−0.0916754
$881$	−17.3641	−0.585010	−0.292505	−	0.956264i	$-0.594489\pi$
$881$	−17.3641	−0.585010	−0.292505	+	0.956264i	$0.594489\pi$
$882$	0	0
$883$	15.4017	0.518310	0.259155	−	0.965836i	$-0.416556\pi$
$883$	15.4017	0.518310	0.259155	+	0.965836i	$0.416556\pi$
$884$	7.76169	0.261054
$885$	0	0
$886$	−30.3951	−1.02114
$887$	15.3335	0.514849	0.257424	−	0.966298i	$-0.417126\pi$
$887$	15.3335	0.514849	0.257424	+	0.966298i	$0.417126\pi$
$888$	0	0
$889$	−45.2153	−1.51647
$890$	−31.8424	−1.06736
$891$	0	0
$892$	101.335	3.39294
$893$	−21.9570	−0.734763
$894$	0	0
$895$	−17.0756	−0.570776
$896$	−71.8832	−2.40145
$897$	0	0
$898$	−71.3278	−2.38024
$899$	−10.9093	−0.363847
$900$	0	0
$901$	−34.3128	−1.14312
$902$	7.70881	0.256675
$903$	0	0
$904$	−42.5238	−1.41432
$905$	−7.42226	−0.246724
$906$	0	0
$907$	−51.5212	−1.71073	−0.855367	−	0.518022i	$-0.826668\pi$
$907$	−51.5212	−1.71073	−0.855367	+	0.518022i	$0.826668\pi$
$908$	−16.0985	−0.534248
$909$	0	0
$910$	−6.29199	−0.208577
$911$	−24.2094	−0.802092	−0.401046	−	0.916058i	$-0.631353\pi$
$911$	−24.2094	−0.802092	−0.401046	+	0.916058i	$0.631353\pi$
$912$	0	0
$913$	9.42310	0.311859
$914$	−29.7959	−0.985561
$915$	0	0
$916$	−54.9759	−1.81645
$917$	35.4820	1.17172
$918$	0	0
$919$	15.2328	0.502483	0.251241	−	0.967924i	$-0.419161\pi$
$919$	15.2328	0.502483	0.251241	+	0.967924i	$0.419161\pi$
$920$	−18.3228	−0.604086
$921$	0	0
$922$	50.7679	1.67195
$923$	−4.54358	−0.149554
$924$	0	0
$925$	−4.00647	−0.131732
$926$	−50.3752	−1.65543
$927$	0	0
$928$	−19.2280	−0.631191
$929$	−47.5116	−1.55880	−0.779402	−	0.626524i	$-0.784477\pi$
$929$	−47.5116	−1.55880	−0.779402	+	0.626524i	$0.784477\pi$
$930$	0	0
$931$	−32.4094	−1.06217
$932$	70.8018	2.31919
$933$	0	0
$934$	−6.44804	−0.210986
$935$	3.41197	0.111583
$936$	0	0
$937$	−10.1144	−0.330424	−0.165212	−	0.986258i	$-0.552831\pi$
$937$	−10.1144	−0.330424	−0.165212	+	0.986258i	$0.552831\pi$
$938$	−0.445982	−0.0145618
$939$	0	0
$940$	12.5066	0.407919
$941$	40.3682	1.31597	0.657983	−	0.753033i	$-0.271410\pi$
$941$	40.3682	1.31597	0.657983	+	0.753033i	$0.271410\pi$
$942$	0	0
$943$	11.9520	0.389212
$944$	−23.0043	−0.748728
$945$	0	0
$946$	−33.4100	−1.08625
$947$	−10.8991	−0.354173	−0.177087	−	0.984195i	$-0.556667\pi$
$947$	−10.8991	−0.354173	−0.177087	+	0.984195i	$0.556667\pi$
$948$	0	0
$949$	8.67907	0.281735
$950$	−61.6151	−1.99906
$951$	0	0
$952$	−38.4500	−1.24617
$953$	4.62590	0.149848	0.0749239	−	0.997189i	$-0.476129\pi$
$953$	4.62590	0.149848	0.0749239	+	0.997189i	$0.476129\pi$
$954$	0	0
$955$	1.33238	0.0431150
$956$	−67.7214	−2.19027
$957$	0	0
$958$	−4.24676	−0.137207
$959$	15.0058	0.484561
$960$	0	0
$961$	−28.5159	−0.919867
$962$	1.81941	0.0586602
$963$	0	0
$964$	41.7088	1.34335
$965$	3.60915	0.116183
$966$	0	0
$967$	5.54960	0.178463	0.0892316	−	0.996011i	$-0.471559\pi$
$967$	5.54960	0.178463	0.0892316	+	0.996011i	$0.471559\pi$
$968$	38.0092	1.22166
$969$	0	0
$970$	−41.4734	−1.33163
$971$	−8.69417	−0.279009	−0.139505	−	0.990221i	$-0.544551\pi$
$971$	−8.69417	−0.279009	−0.139505	+	0.990221i	$0.544551\pi$
$972$	0	0
$973$	−19.3279	−0.619624
$974$	99.8151	3.19828
$975$	0	0
$976$	−11.8697	−0.379939
$977$	3.26411	0.104428	0.0522141	−	0.998636i	$-0.483372\pi$
$977$	3.26411	0.104428	0.0522141	+	0.998636i	$0.483372\pi$
$978$	0	0
$979$	−16.5560	−0.529133
$980$	18.4602	0.589688
$981$	0	0
$982$	−52.0828	−1.66203
$983$	−24.7325	−0.788844	−0.394422	−	0.918930i	$-0.629055\pi$
$983$	−24.7325	−0.788844	−0.394422	+	0.918930i	$0.629055\pi$
$984$	0	0
$985$	13.5927	0.433100
$986$	−45.5916	−1.45193
$987$	0	0
$988$	18.1346	0.576939
$989$	−51.8002	−1.64715
$990$	0	0
$991$	30.1691	0.958352	0.479176	−	0.877719i	$-0.340936\pi$
$991$	30.1691	0.958352	0.479176	+	0.877719i	$0.340936\pi$
$992$	4.37832	0.139012
$993$	0	0
$994$	49.2448	1.56195
$995$	14.6038	0.462972
$996$	0	0
$997$	−44.5838	−1.41198	−0.705991	−	0.708221i	$-0.749498\pi$
$997$	−44.5838	−1.41198	−0.705991	+	0.708221i	$0.749498\pi$
$998$	61.8062	1.95644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.f.1.2	yes	25
3.2	odd	2		4023.2.a.e.1.24	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.24	✓	25	3.2	odd	2
4023.2.a.f.1.2	yes	25	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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38405 q^{2} +3.68368 q^{4} +0.998141 q^{5} -3.46708 q^{7} -4.01398 q^{8} +O(q^{10})\)	\(q-2.38405 q^{2} +3.68368 q^{4} +0.998141 q^{5} -3.46708 q^{7} -4.01398 q^{8} -2.37961 q^{10} -1.23725 q^{11} -0.762635 q^{13} +8.26569 q^{14} +2.20214 q^{16} -2.76285 q^{17} -6.45519 q^{19} +3.67683 q^{20} +2.94966 q^{22} +4.57326 q^{23} -4.00371 q^{25} +1.81816 q^{26} -12.7716 q^{28} -6.92170 q^{29} +1.57611 q^{31} +2.77793 q^{32} +6.58676 q^{34} -3.46064 q^{35} +1.00069 q^{37} +15.3895 q^{38} -4.00651 q^{40} +2.61346 q^{41} -11.3267 q^{43} -4.55763 q^{44} -10.9029 q^{46} +3.40145 q^{47} +5.02067 q^{49} +9.54505 q^{50} -2.80930 q^{52} +12.4193 q^{53} -1.23495 q^{55} +13.9168 q^{56} +16.5017 q^{58} -10.4463 q^{59} -5.39005 q^{61} -3.75751 q^{62} -11.0270 q^{64} -0.761217 q^{65} -0.0539558 q^{67} -10.1775 q^{68} +8.25032 q^{70} +5.95774 q^{71} -11.3804 q^{73} -2.38569 q^{74} -23.7789 q^{76} +4.28964 q^{77} +8.94972 q^{79} +2.19805 q^{80} -6.23062 q^{82} -7.61618 q^{83} -2.75771 q^{85} +27.0035 q^{86} +4.96628 q^{88} +13.3813 q^{89} +2.64412 q^{91} +16.8464 q^{92} -8.10921 q^{94} -6.44319 q^{95} +17.4286 q^{97} -11.9695 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) 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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