LMFDB - Embedded newform 4023.2.a.f.1.12

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			1.12
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)

$f(q)$	$=$	$q+0.0533591 q^{2} -1.99715 q^{4} -0.751347 q^{5} +2.49440 q^{7} -0.213285 q^{8} +O(q^{10})$	\(q+0.0533591 q^{2} -1.99715 q^{4} -0.751347 q^{5} +2.49440 q^{7} -0.213285 q^{8} -0.0400912 q^{10} -5.03822 q^{11} -1.59535 q^{13} +0.133099 q^{14} +3.98292 q^{16} +7.25388 q^{17} -4.33797 q^{19} +1.50055 q^{20} -0.268835 q^{22} +3.74466 q^{23} -4.43548 q^{25} -0.0851264 q^{26} -4.98171 q^{28} -2.31148 q^{29} -2.25158 q^{31} +0.639095 q^{32} +0.387061 q^{34} -1.87416 q^{35} -2.39508 q^{37} -0.231470 q^{38} +0.160251 q^{40} +0.239600 q^{41} +5.91983 q^{43} +10.0621 q^{44} +0.199812 q^{46} +0.831302 q^{47} -0.777947 q^{49} -0.236673 q^{50} +3.18615 q^{52} +3.11218 q^{53} +3.78545 q^{55} -0.532018 q^{56} -0.123339 q^{58} +2.31760 q^{59} -5.74915 q^{61} -0.120142 q^{62} -7.93175 q^{64} +1.19866 q^{65} -7.75641 q^{67} -14.4871 q^{68} -0.100004 q^{70} +7.51884 q^{71} +11.7011 q^{73} -0.127799 q^{74} +8.66359 q^{76} -12.5674 q^{77} +0.660921 q^{79} -2.99256 q^{80} +0.0127848 q^{82} +1.59900 q^{83} -5.45018 q^{85} +0.315877 q^{86} +1.07458 q^{88} +2.41526 q^{89} -3.97944 q^{91} -7.47867 q^{92} +0.0443576 q^{94} +3.25932 q^{95} +11.5257 q^{97} -0.0415105 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0533591	0.0377306	0.0188653	−	0.999822i	$-0.493995\pi$
$2$	0.0533591	0.0377306	0.0188653	+	0.999822i	$0.493995\pi$
$3$	0	0
$3$	0	0
$4$	−1.99715	−0.998576
$5$	−0.751347	−0.336012	−0.168006	−	0.985786i	$-0.553733\pi$
$5$	−0.751347	−0.336012	−0.168006	+	0.985786i	$0.553733\pi$
$6$	0	0
$7$	2.49440	0.942796	0.471398	−	0.881921i	$-0.343749\pi$
$7$	2.49440	0.942796	0.471398	+	0.881921i	$0.343749\pi$
$8$	−0.213285	−0.0754075
$9$	0	0
$10$	−0.0400912	−0.0126780
$11$	−5.03822	−1.51908	−0.759541	−	0.650460i	$-0.774576\pi$
$11$	−5.03822	−1.51908	−0.759541	+	0.650460i	$0.774576\pi$
$12$	0	0
$13$	−1.59535	−0.442470	−0.221235	−	0.975221i	$-0.571009\pi$
$13$	−1.59535	−0.442470	−0.221235	+	0.975221i	$0.571009\pi$
$14$	0.133099	0.0355723
$15$	0	0
$16$	3.98292	0.995731
$17$	7.25388	1.75932	0.879662	−	0.475599i	$-0.157769\pi$
$17$	7.25388	1.75932	0.879662	+	0.475599i	$0.157769\pi$
$18$	0	0
$19$	−4.33797	−0.995199	−0.497599	−	0.867407i	$-0.665785\pi$
$19$	−4.33797	−0.995199	−0.497599	+	0.867407i	$0.665785\pi$
$20$	1.50055	0.335534
$21$	0	0
$22$	−0.268835	−0.0573159
$23$	3.74466	0.780817	0.390408	−	0.920642i	$-0.372334\pi$
$23$	3.74466	0.780817	0.390408	+	0.920642i	$0.372334\pi$
$24$	0	0
$25$	−4.43548	−0.887096
$26$	−0.0851264	−0.0166947
$27$	0	0
$28$	−4.98171	−0.941454
$29$	−2.31148	−0.429231	−0.214615	−	0.976699i	$-0.568850\pi$
$29$	−2.31148	−0.429231	−0.214615	+	0.976699i	$0.568850\pi$
$30$	0	0
$31$	−2.25158	−0.404395	−0.202197	−	0.979345i	$-0.564808\pi$
$31$	−2.25158	−0.404395	−0.202197	+	0.979345i	$0.564808\pi$
$32$	0.639095	0.112977
$33$	0	0
$34$	0.387061	0.0663804
$35$	−1.87416	−0.316791
$36$	0	0
$37$	−2.39508	−0.393748	−0.196874	−	0.980429i	$-0.563079\pi$
$37$	−2.39508	−0.393748	−0.196874	+	0.980429i	$0.563079\pi$
$38$	−0.231470	−0.0375494
$39$	0	0
$40$	0.160251	0.0253379
$41$	0.239600	0.0374192	0.0187096	−	0.999825i	$-0.494044\pi$
$41$	0.239600	0.0374192	0.0187096	+	0.999825i	$0.494044\pi$
$42$	0	0
$43$	5.91983	0.902765	0.451383	−	0.892331i	$-0.350931\pi$
$43$	5.91983	0.902765	0.451383	+	0.892331i	$0.350931\pi$
$44$	10.0621	1.51692
$45$	0	0
$46$	0.199812	0.0294607
$47$	0.831302	0.121258	0.0606290	−	0.998160i	$-0.480689\pi$
$47$	0.831302	0.121258	0.0606290	+	0.998160i	$0.480689\pi$
$48$	0	0
$49$	−0.777947	−0.111135
$50$	−0.236673	−0.0334706
$51$	0	0
$52$	3.18615	0.441840
$53$	3.11218	0.427491	0.213745	−	0.976889i	$-0.431434\pi$
$53$	3.11218	0.427491	0.213745	+	0.976889i	$0.431434\pi$
$54$	0	0
$55$	3.78545	0.510430
$56$	−0.532018	−0.0710939
$57$	0	0
$58$	−0.123339	−0.0161951
$59$	2.31760	0.301726	0.150863	−	0.988555i	$-0.451795\pi$
$59$	2.31760	0.301726	0.150863	+	0.988555i	$0.451795\pi$
$60$	0	0
$61$	−5.74915	−0.736104	−0.368052	−	0.929805i	$-0.619975\pi$
$61$	−5.74915	−0.736104	−0.368052	+	0.929805i	$0.619975\pi$
$62$	−0.120142	−0.0152581
$63$	0	0
$64$	−7.93175	−0.991469
$65$	1.19866	0.148675
$66$	0	0
$67$	−7.75641	−0.947596	−0.473798	−	0.880634i	$-0.657117\pi$
$67$	−7.75641	−0.947596	−0.473798	+	0.880634i	$0.657117\pi$
$68$	−14.4871	−1.75682
$69$	0	0
$70$	−0.100004	−0.0119527
$71$	7.51884	0.892322	0.446161	−	0.894953i	$-0.352791\pi$
$71$	7.51884	0.892322	0.446161	+	0.894953i	$0.352791\pi$
$72$	0	0
$73$	11.7011	1.36951	0.684756	−	0.728772i	$-0.259909\pi$
$73$	11.7011	1.36951	0.684756	+	0.728772i	$0.259909\pi$
$74$	−0.127799	−0.0148564
$75$	0	0
$76$	8.66359	0.993782
$77$	−12.5674	−1.43218
$78$	0	0
$79$	0.660921	0.0743594	0.0371797	−	0.999309i	$-0.488163\pi$
$79$	0.660921	0.0743594	0.0371797	+	0.999309i	$0.488163\pi$
$80$	−2.99256	−0.334578
$81$	0	0
$82$	0.0127848	0.00141185
$83$	1.59900	0.175513	0.0877564	−	0.996142i	$-0.472030\pi$
$83$	1.59900	0.175513	0.0877564	+	0.996142i	$0.472030\pi$
$84$	0	0
$85$	−5.45018	−0.591155
$86$	0.315877	0.0340619
$87$	0	0
$88$	1.07458	0.114550
$89$	2.41526	0.256017	0.128009	−	0.991773i	$-0.459142\pi$
$89$	2.41526	0.256017	0.128009	+	0.991773i	$0.459142\pi$
$90$	0	0
$91$	−3.97944	−0.417159
$92$	−7.47867	−0.779705
$93$	0	0
$94$	0.0443576	0.00457513
$95$	3.25932	0.334399
$96$	0	0
$97$	11.5257	1.17026	0.585129	−	0.810940i	$-0.301044\pi$
$97$	11.5257	1.17026	0.585129	+	0.810940i	$0.301044\pi$
$98$	−0.0415105	−0.00419320
$99$	0	0
$100$	8.85833	0.885833
$101$	3.28165	0.326537	0.163268	−	0.986582i	$-0.447796\pi$
$101$	3.28165	0.326537	0.163268	+	0.986582i	$0.447796\pi$
$102$	0	0
$103$	−4.24657	−0.418427	−0.209213	−	0.977870i	$-0.567090\pi$
$103$	−4.24657	−0.418427	−0.209213	+	0.977870i	$0.567090\pi$
$104$	0.340263	0.0333655
$105$	0	0
$106$	0.166063	0.0161295
$107$	13.1447	1.27075	0.635373	−	0.772205i	$-0.280846\pi$
$107$	13.1447	1.27075	0.635373	+	0.772205i	$0.280846\pi$
$108$	0	0
$109$	17.6911	1.69450	0.847252	−	0.531191i	$-0.178255\pi$
$109$	17.6911	1.69450	0.847252	+	0.531191i	$0.178255\pi$
$110$	0.201988	0.0192588
$111$	0	0
$112$	9.93503	0.938772
$113$	−13.9626	−1.31349	−0.656745	−	0.754113i	$-0.728067\pi$
$113$	−13.9626	−1.31349	−0.656745	+	0.754113i	$0.728067\pi$
$114$	0	0
$115$	−2.81354	−0.262364
$116$	4.61638	0.428620
$117$	0	0
$118$	0.123665	0.0113843
$119$	18.0941	1.65868
$120$	0	0
$121$	14.3837	1.30761
$122$	−0.306770	−0.0277736
$123$	0	0
$124$	4.49674	0.403819
$125$	7.08932	0.634088
$126$	0	0
$127$	−0.127684	−0.0113301	−0.00566505	−	0.999984i	$-0.501803\pi$
$127$	−0.127684	−0.0113301	−0.00566505	+	0.999984i	$0.501803\pi$
$128$	−1.70142	−0.150386
$129$	0	0
$130$	0.0639594	0.00560961
$131$	5.96050	0.520771	0.260385	−	0.965505i	$-0.416150\pi$
$131$	5.96050	0.520771	0.260385	+	0.965505i	$0.416150\pi$
$132$	0	0
$133$	−10.8207	−0.938270
$134$	−0.413875	−0.0357534
$135$	0	0
$136$	−1.54714	−0.132666
$137$	12.9159	1.10348	0.551739	−	0.834017i	$-0.313964\pi$
$137$	12.9159	1.10348	0.551739	+	0.834017i	$0.313964\pi$
$138$	0	0
$139$	−3.67542	−0.311745	−0.155873	−	0.987777i	$-0.549819\pi$
$139$	−3.67542	−0.311745	−0.155873	+	0.987777i	$0.549819\pi$
$140$	3.74299	0.316340
$141$	0	0
$142$	0.401199	0.0336679
$143$	8.03772	0.672148
$144$	0	0
$145$	1.73672	0.144227
$146$	0.624362	0.0516725
$147$	0	0
$148$	4.78334	0.393188
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	3.56706	0.290283	0.145142	−	0.989411i	$-0.453636\pi$
$151$	3.56706	0.290283	0.145142	+	0.989411i	$0.453636\pi$
$152$	0.925222	0.0750454
$153$	0	0
$154$	−0.670584	−0.0540372
$155$	1.69171	0.135882
$156$	0	0
$157$	0.472600	0.0377175	0.0188588	−	0.999822i	$-0.493997\pi$
$157$	0.472600	0.0377175	0.0188588	+	0.999822i	$0.493997\pi$
$158$	0.0352662	0.00280562
$159$	0	0
$160$	−0.480182	−0.0379617
$161$	9.34071	0.736151
$162$	0	0
$163$	−17.2954	−1.35468	−0.677339	−	0.735671i	$-0.736867\pi$
$163$	−17.2954	−1.35468	−0.677339	+	0.735671i	$0.736867\pi$
$164$	−0.478517	−0.0373659
$165$	0	0
$166$	0.0853211	0.00662220
$167$	12.3220	0.953507	0.476754	−	0.879037i	$-0.341813\pi$
$167$	12.3220	0.953507	0.476754	+	0.879037i	$0.341813\pi$
$168$	0	0
$169$	−10.4549	−0.804220
$170$	−0.290817	−0.0223046
$171$	0	0
$172$	−11.8228	−0.901480
$173$	18.8583	1.43377	0.716883	−	0.697193i	$-0.245568\pi$
$173$	18.8583	1.43377	0.716883	+	0.697193i	$0.245568\pi$
$174$	0	0
$175$	−11.0639	−0.836350
$176$	−20.0669	−1.51260
$177$	0	0
$178$	0.128876	0.00965968
$179$	8.00426	0.598267	0.299133	−	0.954211i	$-0.403302\pi$
$179$	8.00426	0.598267	0.299133	+	0.954211i	$0.403302\pi$
$180$	0	0
$181$	21.1325	1.57076	0.785382	−	0.619012i	$-0.212467\pi$
$181$	21.1325	1.57076	0.785382	+	0.619012i	$0.212467\pi$
$182$	−0.212340	−0.0157397
$183$	0	0
$184$	−0.798679	−0.0588794
$185$	1.79953	0.132304
$186$	0	0
$187$	−36.5467	−2.67256
$188$	−1.66024	−0.121085
$189$	0	0
$190$	0.173914	0.0126171
$191$	−1.05363	−0.0762381	−0.0381191	−	0.999273i	$-0.512137\pi$
$191$	−1.05363	−0.0762381	−0.0381191	+	0.999273i	$0.512137\pi$
$192$	0	0
$193$	−2.88661	−0.207783	−0.103892	−	0.994589i	$-0.533129\pi$
$193$	−2.88661	−0.207783	−0.103892	+	0.994589i	$0.533129\pi$
$194$	0.615001	0.0441545
$195$	0	0
$196$	1.55368	0.110977
$197$	5.26841	0.375359	0.187679	−	0.982230i	$-0.439903\pi$
$197$	5.26841	0.375359	0.187679	+	0.982230i	$0.439903\pi$
$198$	0	0
$199$	−4.56929	−0.323909	−0.161954	−	0.986798i	$-0.551780\pi$
$199$	−4.56929	−0.323909	−0.161954	+	0.986798i	$0.551780\pi$
$200$	0.946019	0.0668936
$201$	0	0
$202$	0.175106	0.0123204
$203$	−5.76576	−0.404677
$204$	0	0
$205$	−0.180023	−0.0125733
$206$	−0.226593	−0.0157875
$207$	0	0
$208$	−6.35415	−0.440581
$209$	21.8557	1.51179
$210$	0	0
$211$	12.5318	0.862726	0.431363	−	0.902179i	$-0.358033\pi$
$211$	12.5318	0.862726	0.431363	+	0.902179i	$0.358033\pi$
$212$	−6.21550	−0.426882
$213$	0	0
$214$	0.701390	0.0479460
$215$	−4.44784	−0.303340
$216$	0	0
$217$	−5.61634	−0.381262
$218$	0.943984	0.0639347
$219$	0	0
$220$	−7.56013	−0.509704
$221$	−11.5725	−0.778448
$222$	0	0
$223$	−7.45390	−0.499150	−0.249575	−	0.968355i	$-0.580291\pi$
$223$	−7.45390	−0.499150	−0.249575	+	0.968355i	$0.580291\pi$
$224$	1.59416	0.106514
$225$	0	0
$226$	−0.745032	−0.0495588
$227$	13.2058	0.876499	0.438250	−	0.898853i	$-0.355599\pi$
$227$	13.2058	0.876499	0.438250	+	0.898853i	$0.355599\pi$
$228$	0	0
$229$	−2.05916	−0.136073	−0.0680366	−	0.997683i	$-0.521673\pi$
$229$	−2.05916	−0.136073	−0.0680366	+	0.997683i	$0.521673\pi$
$230$	−0.150128	−0.00989916
$231$	0	0
$232$	0.493003	0.0323672
$233$	4.59460	0.301002	0.150501	−	0.988610i	$-0.451911\pi$
$233$	4.59460	0.301002	0.150501	+	0.988610i	$0.451911\pi$
$234$	0	0
$235$	−0.624596	−0.0407442
$236$	−4.62860	−0.301296
$237$	0	0
$238$	0.965486	0.0625832
$239$	19.2261	1.24363	0.621816	−	0.783163i	$-0.286395\pi$
$239$	19.2261	1.24363	0.621816	+	0.783163i	$0.286395\pi$
$240$	0	0
$241$	−3.65585	−0.235494	−0.117747	−	0.993044i	$-0.537567\pi$
$241$	−3.65585	−0.235494	−0.117747	+	0.993044i	$0.537567\pi$
$242$	0.767502	0.0493369
$243$	0	0
$244$	11.4819	0.735056
$245$	0.584508	0.0373428
$246$	0	0
$247$	6.92057	0.440345
$248$	0.480226	0.0304944
$249$	0	0
$250$	0.378280	0.0239245
$251$	−2.13963	−0.135052	−0.0675262	−	0.997718i	$-0.521511\pi$
$251$	−2.13963	−0.135052	−0.0675262	+	0.997718i	$0.521511\pi$
$252$	0	0
$253$	−18.8665	−1.18612
$254$	−0.00681310	−0.000427492 0
$255$	0	0
$256$	15.7727	0.985794
$257$	25.7647	1.60716	0.803580	−	0.595197i	$-0.202926\pi$
$257$	25.7647	1.60716	0.803580	+	0.595197i	$0.202926\pi$
$258$	0	0
$259$	−5.97429	−0.371225
$260$	−2.39391	−0.148464
$261$	0	0
$262$	0.318047	0.0196490
$263$	3.89310	0.240059	0.120029	−	0.992770i	$-0.461701\pi$
$263$	3.89310	0.240059	0.120029	+	0.992770i	$0.461701\pi$
$264$	0	0
$265$	−2.33833	−0.143642
$266$	−0.577381	−0.0354015
$267$	0	0
$268$	15.4907	0.946247
$269$	11.0235	0.672114	0.336057	−	0.941842i	$-0.390907\pi$
$269$	11.0235	0.672114	0.336057	+	0.941842i	$0.390907\pi$
$270$	0	0
$271$	24.1109	1.46463	0.732316	−	0.680965i	$-0.238439\pi$
$271$	24.1109	1.46463	0.732316	+	0.680965i	$0.238439\pi$
$272$	28.8917	1.75181
$273$	0	0
$274$	0.689180	0.0416349
$275$	22.3469	1.34757
$276$	0	0
$277$	−11.9886	−0.720328	−0.360164	−	0.932889i	$-0.617279\pi$
$277$	−11.9886	−0.720328	−0.360164	+	0.932889i	$0.617279\pi$
$278$	−0.196117	−0.0117623
$279$	0	0
$280$	0.399730	0.0238884
$281$	26.8841	1.60377	0.801885	−	0.597479i	$-0.203831\pi$
$281$	26.8841	1.60377	0.801885	+	0.597479i	$0.203831\pi$
$282$	0	0
$283$	−6.68259	−0.397239	−0.198620	−	0.980077i	$-0.563646\pi$
$283$	−6.68259	−0.397239	−0.198620	+	0.980077i	$0.563646\pi$
$284$	−15.0163	−0.891052
$285$	0	0
$286$	0.428886	0.0253605
$287$	0.597659	0.0352787
$288$	0	0
$289$	35.6188	2.09522
$290$	0.0926700	0.00544177
$291$	0	0
$292$	−23.3689	−1.36756
$293$	−18.7631	−1.09615	−0.548077	−	0.836428i	$-0.684640\pi$
$293$	−18.7631	−1.09615	−0.548077	+	0.836428i	$0.684640\pi$
$294$	0	0
$295$	−1.74132	−0.101384
$296$	0.510833	0.0296916
$297$	0	0
$298$	0.0533591	0.00309101
$299$	−5.97404	−0.345488
$300$	0	0
$301$	14.7664	0.851124
$302$	0.190335	0.0109526
$303$	0	0
$304$	−17.2778	−0.990951
$305$	4.31961	0.247340
$306$	0	0
$307$	9.47330	0.540670	0.270335	−	0.962766i	$-0.412866\pi$
$307$	9.47330	0.540670	0.270335	+	0.962766i	$0.412866\pi$
$308$	25.0990	1.43015
$309$	0	0
$310$	0.0902684	0.00512690
$311$	−35.1268	−1.99186	−0.995928	−	0.0901551i	$-0.971264\pi$
$311$	−35.1268	−1.99186	−0.995928	+	0.0901551i	$0.971264\pi$
$312$	0	0
$313$	13.1227	0.741737	0.370868	−	0.928685i	$-0.379060\pi$
$313$	13.1227	0.741737	0.370868	+	0.928685i	$0.379060\pi$
$314$	0.0252175	0.00142311
$315$	0	0
$316$	−1.31996	−0.0742535
$317$	13.0388	0.732334	0.366167	−	0.930549i	$-0.380670\pi$
$317$	13.0388	0.732334	0.366167	+	0.930549i	$0.380670\pi$
$318$	0	0
$319$	11.6458	0.652037
$320$	5.95949	0.333146
$321$	0	0
$322$	0.498412	0.0277754
$323$	−31.4671	−1.75088
$324$	0	0
$325$	7.07613	0.392513
$326$	−0.922865	−0.0511128
$327$	0	0
$328$	−0.0511029	−0.00282169
$329$	2.07360	0.114322
$330$	0	0
$331$	30.3259	1.66686	0.833432	−	0.552623i	$-0.186373\pi$
$331$	30.3259	1.66686	0.833432	+	0.552623i	$0.186373\pi$
$332$	−3.19344	−0.175263
$333$	0	0
$334$	0.657493	0.0359764
$335$	5.82775	0.318404
$336$	0	0
$337$	−8.61775	−0.469439	−0.234720	−	0.972063i	$-0.575417\pi$
$337$	−8.61775	−0.469439	−0.234720	+	0.972063i	$0.575417\pi$
$338$	−0.557862	−0.0303437
$339$	0	0
$340$	10.8848	0.590313
$341$	11.3439	0.614309
$342$	0	0
$343$	−19.4013	−1.04757
$344$	−1.26261	−0.0680752
$345$	0	0
$346$	1.00626	0.0540969
$347$	21.9405	1.17783	0.588915	−	0.808195i	$-0.299555\pi$
$347$	21.9405	1.17783	0.588915	+	0.808195i	$0.299555\pi$
$348$	0	0
$349$	6.26534	0.335376	0.167688	−	0.985840i	$-0.446370\pi$
$349$	6.26534	0.335376	0.167688	+	0.985840i	$0.446370\pi$
$350$	−0.590359	−0.0315560
$351$	0	0
$352$	−3.21990	−0.171621
$353$	−24.2620	−1.29134	−0.645668	−	0.763618i	$-0.723421\pi$
$353$	−24.2620	−1.29134	−0.645668	+	0.763618i	$0.723421\pi$
$354$	0	0
$355$	−5.64926	−0.299831
$356$	−4.82364	−0.255653
$357$	0	0
$358$	0.427100	0.0225730
$359$	37.1053	1.95834	0.979172	−	0.203033i	$-0.0650799\pi$
$359$	37.1053	1.95834	0.979172	+	0.203033i	$0.0650799\pi$
$360$	0	0
$361$	−0.182007	−0.00957930
$362$	1.12761	0.0592658
$363$	0	0
$364$	7.94755	0.416565
$365$	−8.79160	−0.460173
$366$	0	0
$367$	9.68645	0.505629	0.252814	−	0.967515i	$-0.418644\pi$
$367$	9.68645	0.505629	0.252814	+	0.967515i	$0.418644\pi$
$368$	14.9147	0.777484
$369$	0	0
$370$	0.0960216	0.00499192
$371$	7.76303	0.403037
$372$	0	0
$373$	−18.4186	−0.953680	−0.476840	−	0.878990i	$-0.658218\pi$
$373$	−18.4186	−0.953680	−0.476840	+	0.878990i	$0.658218\pi$
$374$	−1.95010	−0.100837
$375$	0	0
$376$	−0.177304	−0.00914375
$377$	3.68761	0.189922
$378$	0	0
$379$	−18.8043	−0.965913	−0.482957	−	0.875644i	$-0.660437\pi$
$379$	−18.8043	−0.965913	−0.482957	+	0.875644i	$0.660437\pi$
$380$	−6.50936	−0.333923
$381$	0	0
$382$	−0.0562209	−0.00287651
$383$	−6.78430	−0.346661	−0.173331	−	0.984864i	$-0.555453\pi$
$383$	−6.78430	−0.346661	−0.173331	+	0.984864i	$0.555453\pi$
$384$	0	0
$385$	9.44245	0.481232
$386$	−0.154027	−0.00783978
$387$	0	0
$388$	−23.0186	−1.16859
$389$	9.25867	0.469433	0.234716	−	0.972064i	$-0.424584\pi$
$389$	9.25867	0.469433	0.234716	+	0.972064i	$0.424584\pi$
$390$	0	0
$391$	27.1634	1.37371
$392$	0.165924	0.00838043
$393$	0	0
$394$	0.281118	0.0141625
$395$	−0.496581	−0.0249857
$396$	0	0
$397$	19.0678	0.956984	0.478492	−	0.878092i	$-0.341184\pi$
$397$	19.0678	0.956984	0.478492	+	0.878092i	$0.341184\pi$
$398$	−0.243814	−0.0122213
$399$	0	0
$400$	−17.6662	−0.883309
$401$	30.8033	1.53824	0.769121	−	0.639104i	$-0.220695\pi$
$401$	30.8033	1.53824	0.769121	+	0.639104i	$0.220695\pi$
$402$	0	0
$403$	3.59205	0.178933
$404$	−6.55396	−0.326072
$405$	0	0
$406$	−0.307656	−0.0152687
$407$	12.0669	0.598136
$408$	0	0
$409$	13.5040	0.667730	0.333865	−	0.942621i	$-0.391647\pi$
$409$	13.5040	0.667730	0.333865	+	0.942621i	$0.391647\pi$
$410$	−0.00960585	−0.000474399 0
$411$	0	0
$412$	8.48105	0.417831
$413$	5.78103	0.284466
$414$	0	0
$415$	−1.20140	−0.0589745
$416$	−1.01958	−0.0499889
$417$	0	0
$418$	1.16620	0.0570407
$419$	−18.4729	−0.902460	−0.451230	−	0.892408i	$-0.649015\pi$
$419$	−18.4729	−0.902460	−0.451230	+	0.892408i	$0.649015\pi$
$420$	0	0
$421$	−23.4545	−1.14310	−0.571551	−	0.820566i	$-0.693658\pi$
$421$	−23.4545	−1.14310	−0.571551	+	0.820566i	$0.693658\pi$
$422$	0.668687	0.0325512
$423$	0	0
$424$	−0.663780	−0.0322360
$425$	−32.1744	−1.56069
$426$	0	0
$427$	−14.3407	−0.693996
$428$	−26.2520	−1.26894
$429$	0	0
$430$	−0.237333	−0.0114452
$431$	−10.3112	−0.496673	−0.248336	−	0.968674i	$-0.579884\pi$
$431$	−10.3112	−0.496673	−0.248336	+	0.968674i	$0.579884\pi$
$432$	0	0
$433$	18.8262	0.904731	0.452365	−	0.891833i	$-0.350580\pi$
$433$	18.8262	0.904731	0.452365	+	0.891833i	$0.350580\pi$
$434$	−0.299683	−0.0143852
$435$	0	0
$436$	−35.3319	−1.69209
$437$	−16.2442	−0.777068
$438$	0	0
$439$	21.7852	1.03975	0.519876	−	0.854242i	$-0.325978\pi$
$439$	21.7852	1.03975	0.519876	+	0.854242i	$0.325978\pi$
$440$	−0.807379	−0.0384903
$441$	0	0
$442$	−0.617496	−0.0293713
$443$	−16.8159	−0.798946	−0.399473	−	0.916745i	$-0.630807\pi$
$443$	−16.8159	−0.798946	−0.399473	+	0.916745i	$0.630807\pi$
$444$	0	0
$445$	−1.81470	−0.0860249
$446$	−0.397733	−0.0188332
$447$	0	0
$448$	−19.7850	−0.934753
$449$	−16.1220	−0.760846	−0.380423	−	0.924813i	$-0.624222\pi$
$449$	−16.1220	−0.760846	−0.380423	+	0.924813i	$0.624222\pi$
$450$	0	0
$451$	−1.20716	−0.0568428
$452$	27.8854	1.31162
$453$	0	0
$454$	0.704649	0.0330708
$455$	2.98994	0.140171
$456$	0	0
$457$	13.3122	0.622719	0.311359	−	0.950292i	$-0.399216\pi$
$457$	13.3122	0.622719	0.311359	+	0.950292i	$0.399216\pi$
$458$	−0.109875	−0.00513412
$459$	0	0
$460$	5.61907	0.261991
$461$	3.13800	0.146151	0.0730757	−	0.997326i	$-0.476719\pi$
$461$	3.13800	0.146151	0.0730757	+	0.997326i	$0.476719\pi$
$462$	0	0
$463$	3.08431	0.143340	0.0716700	−	0.997428i	$-0.477167\pi$
$463$	3.08431	0.143340	0.0716700	+	0.997428i	$0.477167\pi$
$464$	−9.20645	−0.427399
$465$	0	0
$466$	0.245164	0.0113570
$467$	−5.33704	−0.246969	−0.123484	−	0.992347i	$-0.539407\pi$
$467$	−5.33704	−0.246969	−0.123484	+	0.992347i	$0.539407\pi$
$468$	0	0
$469$	−19.3476	−0.893390
$470$	−0.0333279	−0.00153730
$471$	0	0
$472$	−0.494308	−0.0227524
$473$	−29.8254	−1.37137
$474$	0	0
$475$	19.2410	0.882837
$476$	−36.1367	−1.65632
$477$	0	0
$478$	1.02589	0.0469230
$479$	−18.4023	−0.840823	−0.420412	−	0.907333i	$-0.638114\pi$
$479$	−18.4023	−0.840823	−0.420412	+	0.907333i	$0.638114\pi$
$480$	0	0
$481$	3.82098	0.174222
$482$	−0.195073	−0.00888532
$483$	0	0
$484$	−28.7265	−1.30575
$485$	−8.65980	−0.393221
$486$	0	0
$487$	−12.3328	−0.558851	−0.279425	−	0.960167i	$-0.590144\pi$
$487$	−12.3328	−0.558851	−0.279425	+	0.960167i	$0.590144\pi$
$488$	1.22621	0.0555077
$489$	0	0
$490$	0.0311888	0.00140897
$491$	−16.9355	−0.764289	−0.382144	−	0.924103i	$-0.624814\pi$
$491$	−16.9355	−0.764289	−0.382144	+	0.924103i	$0.624814\pi$
$492$	0	0
$493$	−16.7672	−0.755157
$494$	0.369276	0.0166145
$495$	0	0
$496$	−8.96786	−0.402669
$497$	18.7550	0.841278
$498$	0	0
$499$	−27.8115	−1.24501	−0.622506	−	0.782615i	$-0.713885\pi$
$499$	−27.8115	−1.24501	−0.622506	+	0.782615i	$0.713885\pi$
$500$	−14.1584	−0.633185
$501$	0	0
$502$	−0.114169	−0.00509561
$503$	−15.5859	−0.694940	−0.347470	−	0.937691i	$-0.612959\pi$
$503$	−15.5859	−0.694940	−0.347470	+	0.937691i	$0.612959\pi$
$504$	0	0
$505$	−2.46566	−0.109720
$506$	−1.00670	−0.0447532
$507$	0	0
$508$	0.255004	0.0113140
$509$	5.87665	0.260478	0.130239	−	0.991483i	$-0.458426\pi$
$509$	5.87665	0.260478	0.130239	+	0.991483i	$0.458426\pi$
$510$	0	0
$511$	29.1873	1.29117
$512$	4.24446	0.187580
$513$	0	0
$514$	1.37478	0.0606391
$515$	3.19065	0.140597
$516$	0	0
$517$	−4.18829	−0.184201
$518$	−0.318783	−0.0140065
$519$	0	0
$520$	−0.255656	−0.0112112
$521$	−24.1563	−1.05831	−0.529154	−	0.848526i	$-0.677491\pi$
$521$	−24.1563	−1.05831	−0.529154	+	0.848526i	$0.677491\pi$
$522$	0	0
$523$	−1.85233	−0.0809967	−0.0404983	−	0.999180i	$-0.512895\pi$
$523$	−1.85233	−0.0809967	−0.0404983	+	0.999180i	$0.512895\pi$
$524$	−11.9040	−0.520030
$525$	0	0
$526$	0.207732	0.00905756
$527$	−16.3327	−0.711462
$528$	0	0
$529$	−8.97748	−0.390325
$530$	−0.124771	−0.00541971
$531$	0	0
$532$	21.6105	0.936934
$533$	−0.382245	−0.0165569
$534$	0	0
$535$	−9.87623	−0.426987
$536$	1.65432	0.0714558
$537$	0	0
$538$	0.588203	0.0253592
$539$	3.91947	0.168823
$540$	0	0
$541$	−28.7195	−1.23475	−0.617374	−	0.786670i	$-0.711803\pi$
$541$	−28.7195	−1.23475	−0.617374	+	0.786670i	$0.711803\pi$
$542$	1.28654	0.0552614
$543$	0	0
$544$	4.63592	0.198763
$545$	−13.2922	−0.569375
$546$	0	0
$547$	−36.2231	−1.54879	−0.774393	−	0.632705i	$-0.781945\pi$
$547$	−36.2231	−1.54879	−0.774393	+	0.632705i	$0.781945\pi$
$548$	−25.7950	−1.10191
$549$	0	0
$550$	1.19241	0.0508446
$551$	10.0271	0.427170
$552$	0	0
$553$	1.64860	0.0701058
$554$	−0.639703	−0.0271784
$555$	0	0
$556$	7.34038	0.311301
$557$	16.7714	0.710628	0.355314	−	0.934747i	$-0.384374\pi$
$557$	16.7714	0.710628	0.355314	+	0.934747i	$0.384374\pi$
$558$	0	0
$559$	−9.44418	−0.399446
$560$	−7.46465	−0.315439
$561$	0	0
$562$	1.43451	0.0605112
$563$	−11.4428	−0.482257	−0.241128	−	0.970493i	$-0.577517\pi$
$563$	−11.4428	−0.482257	−0.241128	+	0.970493i	$0.577517\pi$
$564$	0	0
$565$	10.4907	0.441349
$566$	−0.356577	−0.0149881
$567$	0	0
$568$	−1.60365	−0.0672878
$569$	29.3133	1.22888	0.614439	−	0.788964i	$-0.289382\pi$
$569$	29.3133	1.22888	0.614439	+	0.788964i	$0.289382\pi$
$570$	0	0
$571$	−2.90555	−0.121593	−0.0607967	−	0.998150i	$-0.519364\pi$
$571$	−2.90555	−0.121593	−0.0607967	+	0.998150i	$0.519364\pi$
$572$	−16.0526	−0.671191
$573$	0	0
$574$	0.0318906	0.00133109
$575$	−16.6094	−0.692659
$576$	0	0
$577$	5.06463	0.210843	0.105422	−	0.994428i	$-0.466381\pi$
$577$	5.06463	0.210843	0.105422	+	0.994428i	$0.466381\pi$
$578$	1.90059	0.0790540
$579$	0	0
$580$	−3.46850	−0.144022
$581$	3.98855	0.165473
$582$	0	0
$583$	−15.6799	−0.649393
$584$	−2.49567	−0.103272
$585$	0	0
$586$	−1.00119	−0.0413586
$587$	−12.4429	−0.513571	−0.256786	−	0.966468i	$-0.582663\pi$
$587$	−12.4429	−0.513571	−0.256786	+	0.966468i	$0.582663\pi$
$588$	0	0
$589$	9.76727	0.402453
$590$	−0.0929154	−0.00382527
$591$	0	0
$592$	−9.53942	−0.392068
$593$	8.09885	0.332580	0.166290	−	0.986077i	$-0.446821\pi$
$593$	8.09885	0.332580	0.166290	+	0.986077i	$0.446821\pi$
$594$	0	0
$595$	−13.5950	−0.557339
$596$	−1.99715	−0.0818066
$597$	0	0
$598$	−0.318770	−0.0130355
$599$	7.61642	0.311198	0.155599	−	0.987820i	$-0.450269\pi$
$599$	7.61642	0.311198	0.155599	+	0.987820i	$0.450269\pi$
$600$	0	0
$601$	17.5571	0.716167	0.358084	−	0.933690i	$-0.383430\pi$
$601$	17.5571	0.716167	0.358084	+	0.933690i	$0.383430\pi$
$602$	0.787924	0.0321134
$603$	0	0
$604$	−7.12396	−0.289870
$605$	−10.8071	−0.439373
$606$	0	0
$607$	9.43436	0.382929	0.191464	−	0.981500i	$-0.438676\pi$
$607$	9.43436	0.382929	0.191464	+	0.981500i	$0.438676\pi$
$608$	−2.77237	−0.112435
$609$	0	0
$610$	0.230491	0.00933229
$611$	−1.32622	−0.0536530
$612$	0	0
$613$	−30.6704	−1.23877	−0.619383	−	0.785089i	$-0.712617\pi$
$613$	−30.6704	−1.23877	−0.619383	+	0.785089i	$0.712617\pi$
$614$	0.505487	0.0203998
$615$	0	0
$616$	2.68043	0.107997
$617$	−47.4357	−1.90969	−0.954844	−	0.297107i	$-0.903978\pi$
$617$	−47.4357	−1.90969	−0.954844	+	0.297107i	$0.903978\pi$
$618$	0	0
$619$	29.4009	1.18172	0.590860	−	0.806774i	$-0.298788\pi$
$619$	29.4009	1.18172	0.590860	+	0.806774i	$0.298788\pi$
$620$	−3.37861	−0.135688
$621$	0	0
$622$	−1.87433	−0.0751539
$623$	6.02464	0.241372
$624$	0	0
$625$	16.8509	0.674034
$626$	0.700214	0.0279862
$627$	0	0
$628$	−0.943853	−0.0376639
$629$	−17.3736	−0.692731
$630$	0	0
$631$	−48.5857	−1.93416	−0.967082	−	0.254465i	$-0.918101\pi$
$631$	−48.5857	−1.93416	−0.967082	+	0.254465i	$0.918101\pi$
$632$	−0.140964	−0.00560726
$633$	0	0
$634$	0.695740	0.0276314
$635$	0.0959348	0.00380706
$636$	0	0
$637$	1.24110	0.0491740
$638$	0.621407	0.0246017
$639$	0	0
$640$	1.27836	0.0505315
$641$	−31.0512	−1.22645	−0.613225	−	0.789908i	$-0.710128\pi$
$641$	−31.0512	−1.22645	−0.613225	+	0.789908i	$0.710128\pi$
$642$	0	0
$643$	14.4091	0.568240	0.284120	−	0.958789i	$-0.408299\pi$
$643$	14.4091	0.568240	0.284120	+	0.958789i	$0.408299\pi$
$644$	−18.6548	−0.735103
$645$	0	0
$646$	−1.67906	−0.0660617
$647$	−31.6375	−1.24380	−0.621900	−	0.783097i	$-0.713639\pi$
$647$	−31.6375	−1.24380	−0.621900	+	0.783097i	$0.713639\pi$
$648$	0	0
$649$	−11.6766	−0.458346
$650$	0.377576	0.0148098
$651$	0	0
$652$	34.5415	1.35275
$653$	−33.6975	−1.31869	−0.659343	−	0.751842i	$-0.729166\pi$
$653$	−33.6975	−1.31869	−0.659343	+	0.751842i	$0.729166\pi$
$654$	0	0
$655$	−4.47840	−0.174986
$656$	0.954308	0.0372595
$657$	0	0
$658$	0.110646	0.00431342
$659$	25.5345	0.994683	0.497341	−	0.867555i	$-0.334310\pi$
$659$	25.5345	0.994683	0.497341	+	0.867555i	$0.334310\pi$
$660$	0	0
$661$	47.9848	1.86639	0.933196	−	0.359368i	$-0.117008\pi$
$661$	47.9848	1.86639	0.933196	+	0.359368i	$0.117008\pi$
$662$	1.61816	0.0628917
$663$	0	0
$664$	−0.341042	−0.0132350
$665$	8.13006	0.315270
$666$	0	0
$667$	−8.65572	−0.335151
$668$	−24.6090	−0.952150
$669$	0	0
$670$	0.310964	0.0120136
$671$	28.9655	1.11820
$672$	0	0
$673$	27.3029	1.05245	0.526225	−	0.850345i	$-0.323607\pi$
$673$	27.3029	1.05245	0.526225	+	0.850345i	$0.323607\pi$
$674$	−0.459836	−0.0177122
$675$	0	0
$676$	20.8800	0.803076
$677$	11.7880	0.453048	0.226524	−	0.974006i	$-0.427264\pi$
$677$	11.7880	0.453048	0.226524	+	0.974006i	$0.427264\pi$
$678$	0	0
$679$	28.7498	1.10331
$680$	1.16244	0.0445775
$681$	0	0
$682$	0.605303	0.0231782
$683$	−35.4143	−1.35509	−0.677546	−	0.735481i	$-0.736956\pi$
$683$	−35.4143	−1.35509	−0.677546	+	0.735481i	$0.736956\pi$
$684$	0	0
$685$	−9.70431	−0.370783
$686$	−1.03524	−0.0395256
$687$	0	0
$688$	23.5782	0.898911
$689$	−4.96501	−0.189152
$690$	0	0
$691$	−8.53694	−0.324761	−0.162380	−	0.986728i	$-0.551917\pi$
$691$	−8.53694	−0.324761	−0.162380	+	0.986728i	$0.551917\pi$
$692$	−37.6628	−1.43173
$693$	0	0
$694$	1.17073	0.0444402
$695$	2.76152	0.104750
$696$	0	0
$697$	1.73803	0.0658325
$698$	0.334313	0.0126539
$699$	0	0
$700$	22.0963	0.835160
$701$	−31.7059	−1.19752	−0.598758	−	0.800930i	$-0.704339\pi$
$701$	−31.7059	−1.19752	−0.598758	+	0.800930i	$0.704339\pi$
$702$	0	0
$703$	10.3898	0.391858
$704$	39.9619	1.50612
$705$	0	0
$706$	−1.29460	−0.0487229
$707$	8.18577	0.307858
$708$	0	0
$709$	−45.5642	−1.71120	−0.855599	−	0.517639i	$-0.826811\pi$
$709$	−45.5642	−1.71120	−0.855599	+	0.517639i	$0.826811\pi$
$710$	−0.301439	−0.0113128
$711$	0	0
$712$	−0.515138	−0.0193056
$713$	−8.43140	−0.315758
$714$	0	0
$715$	−6.03911	−0.225850
$716$	−15.9857	−0.597415
$717$	0	0
$718$	1.97991	0.0738895
$719$	34.1322	1.27292	0.636458	−	0.771311i	$-0.280399\pi$
$719$	34.1322	1.27292	0.636458	+	0.771311i	$0.280399\pi$
$720$	0	0
$721$	−10.5927	−0.394491
$722$	−0.00971171	−0.000361433 0
$723$	0	0
$724$	−42.2047	−1.56853
$725$	10.2525	0.380769
$726$	0	0
$727$	17.0495	0.632329	0.316165	−	0.948704i	$-0.397605\pi$
$727$	17.0495	0.632329	0.316165	+	0.948704i	$0.397605\pi$
$728$	0.848754	0.0314569
$729$	0	0
$730$	−0.469112	−0.0173626
$731$	42.9417	1.58826
$732$	0	0
$733$	−24.9997	−0.923384	−0.461692	−	0.887040i	$-0.652757\pi$
$733$	−24.9997	−0.923384	−0.461692	+	0.887040i	$0.652757\pi$
$734$	0.516861	0.0190777
$735$	0	0
$736$	2.39319	0.0882143
$737$	39.0785	1.43948
$738$	0	0
$739$	23.1477	0.851502	0.425751	−	0.904840i	$-0.360010\pi$
$739$	23.1477	0.851502	0.425751	+	0.904840i	$0.360010\pi$
$740$	−3.59395	−0.132116
$741$	0	0
$742$	0.414229	0.0152068
$743$	47.7677	1.75243	0.876214	−	0.481923i	$-0.160061\pi$
$743$	47.7677	1.75243	0.876214	+	0.481923i	$0.160061\pi$
$744$	0	0
$745$	−0.751347	−0.0275272
$746$	−0.982801	−0.0359829
$747$	0	0
$748$	72.9893	2.66875
$749$	32.7882	1.19806
$750$	0	0
$751$	31.0260	1.13216	0.566078	−	0.824352i	$-0.308460\pi$
$751$	31.0260	1.13216	0.566078	+	0.824352i	$0.308460\pi$
$752$	3.31102	0.120740
$753$	0	0
$754$	0.196768	0.00716586
$755$	−2.68010	−0.0975387
$756$	0	0
$757$	−1.51540	−0.0550782	−0.0275391	−	0.999621i	$-0.508767\pi$
$757$	−1.51540	−0.0550782	−0.0275391	+	0.999621i	$0.508767\pi$
$758$	−1.00338	−0.0364445
$759$	0	0
$760$	−0.695163	−0.0252162
$761$	−24.5684	−0.890602	−0.445301	−	0.895381i	$-0.646903\pi$
$761$	−24.5684	−0.890602	−0.445301	+	0.895381i	$0.646903\pi$
$762$	0	0
$763$	44.1289	1.59757
$764$	2.10426	0.0761296
$765$	0	0
$766$	−0.362004	−0.0130797
$767$	−3.69738	−0.133505
$768$	0	0
$769$	2.61614	0.0943403	0.0471702	−	0.998887i	$-0.484980\pi$
$769$	2.61614	0.0943403	0.0471702	+	0.998887i	$0.484980\pi$
$770$	0.503841	0.0181572
$771$	0	0
$772$	5.76501	0.207487
$773$	35.1199	1.26317	0.631587	−	0.775305i	$-0.282404\pi$
$773$	35.1199	1.26317	0.631587	+	0.775305i	$0.282404\pi$
$774$	0	0
$775$	9.98681	0.358737
$776$	−2.45825	−0.0882462
$777$	0	0
$778$	0.494034	0.0177120
$779$	−1.03938	−0.0372396
$780$	0	0
$781$	−37.8816	−1.35551
$782$	1.44941	0.0518309
$783$	0	0
$784$	−3.09850	−0.110661
$785$	−0.355086	−0.0126736
$786$	0	0
$787$	−21.8118	−0.777506	−0.388753	−	0.921342i	$-0.627094\pi$
$787$	−21.8118	−0.777506	−0.388753	+	0.921342i	$0.627094\pi$
$788$	−10.5218	−0.374824
$789$	0	0
$790$	−0.0264971	−0.000942725 0
$791$	−34.8284	−1.23835
$792$	0	0
$793$	9.17190	0.325704
$794$	1.01744	0.0361076
$795$	0	0
$796$	9.12558	0.323448
$797$	18.7985	0.665878	0.332939	−	0.942948i	$-0.391960\pi$
$797$	18.7985	0.665878	0.332939	+	0.942948i	$0.391960\pi$
$798$	0	0
$799$	6.03017	0.213332
$800$	−2.83469	−0.100221
$801$	0	0
$802$	1.64363	0.0580388
$803$	−58.9529	−2.08040
$804$	0	0
$805$	−7.01811	−0.247356
$806$	0.191668	0.00675123
$807$	0	0
$808$	−0.699926	−0.0246233
$809$	−22.9814	−0.807982	−0.403991	−	0.914763i	$-0.632377\pi$
$809$	−22.9814	−0.807982	−0.403991	+	0.914763i	$0.632377\pi$
$810$	0	0
$811$	−24.9458	−0.875964	−0.437982	−	0.898984i	$-0.644307\pi$
$811$	−24.9458	−0.875964	−0.437982	+	0.898984i	$0.644307\pi$
$812$	11.5151	0.404101
$813$	0	0
$814$	0.643881	0.0225680
$815$	12.9948	0.455189
$816$	0	0
$817$	−25.6800	−0.898431
$818$	0.720561	0.0251938
$819$	0	0
$820$	0.359533	0.0125554
$821$	18.7607	0.654753	0.327376	−	0.944894i	$-0.393836\pi$
$821$	18.7607	0.654753	0.327376	+	0.944894i	$0.393836\pi$
$822$	0	0
$823$	6.14645	0.214252	0.107126	−	0.994245i	$-0.465835\pi$
$823$	6.14645	0.214252	0.107126	+	0.994245i	$0.465835\pi$
$824$	0.905728	0.0315525
$825$	0	0
$826$	0.308471	0.0107331
$827$	−7.51220	−0.261225	−0.130612	−	0.991434i	$-0.541694\pi$
$827$	−7.51220	−0.261225	−0.130612	+	0.991434i	$0.541694\pi$
$828$	0	0
$829$	29.6604	1.03015	0.515075	−	0.857145i	$-0.327764\pi$
$829$	29.6604	1.03015	0.515075	+	0.857145i	$0.327764\pi$
$830$	−0.0641058	−0.00222514
$831$	0	0
$832$	12.6539	0.438695
$833$	−5.64313	−0.195523
$834$	0	0
$835$	−9.25812	−0.320390
$836$	−43.6491	−1.50964
$837$	0	0
$838$	−0.985698	−0.0340504
$839$	1.83185	0.0632424	0.0316212	−	0.999500i	$-0.489933\pi$
$839$	1.83185	0.0632424	0.0316212	+	0.999500i	$0.489933\pi$
$840$	0	0
$841$	−23.6571	−0.815761
$842$	−1.25151	−0.0431299
$843$	0	0
$844$	−25.0280	−0.861498
$845$	7.85523	0.270228
$846$	0	0
$847$	35.8788	1.23281
$848$	12.3956	0.425666
$849$	0	0
$850$	−1.71680	−0.0588857
$851$	−8.96877	−0.307445
$852$	0	0
$853$	29.0676	0.995257	0.497629	−	0.867390i	$-0.334204\pi$
$853$	29.0676	0.995257	0.497629	+	0.867390i	$0.334204\pi$
$854$	−0.765208	−0.0261849
$855$	0	0
$856$	−2.80356	−0.0958238
$857$	−56.3980	−1.92652	−0.963259	−	0.268574i	$-0.913448\pi$
$857$	−56.3980	−1.92652	−0.963259	+	0.268574i	$0.913448\pi$
$858$	0	0
$859$	−10.3923	−0.354580	−0.177290	−	0.984159i	$-0.556733\pi$
$859$	−10.3923	−0.354580	−0.177290	+	0.984159i	$0.556733\pi$
$860$	8.88302	0.302909
$861$	0	0
$862$	−0.550196	−0.0187398
$863$	37.0721	1.26195	0.630975	−	0.775803i	$-0.282655\pi$
$863$	37.0721	1.26195	0.630975	+	0.775803i	$0.282655\pi$
$864$	0	0
$865$	−14.1691	−0.481764
$866$	1.00455	0.0341360
$867$	0	0
$868$	11.2167	0.380719
$869$	−3.32987	−0.112958
$870$	0	0
$871$	12.3742	0.419283
$872$	−3.77325	−0.127778
$873$	0	0
$874$	−0.866779	−0.0293192
$875$	17.6836	0.597816
$876$	0	0
$877$	−22.0627	−0.745004	−0.372502	−	0.928031i	$-0.621500\pi$
$877$	−22.0627	−0.745004	−0.372502	+	0.928031i	$0.621500\pi$
$878$	1.16244	0.0392304
$879$	0	0
$880$	15.0772	0.508252
$881$	30.2594	1.01946	0.509732	−	0.860333i	$-0.329745\pi$
$881$	30.2594	1.01946	0.509732	+	0.860333i	$0.329745\pi$
$882$	0	0
$883$	−25.9936	−0.874755	−0.437377	−	0.899278i	$-0.644093\pi$
$883$	−25.9936	−0.874755	−0.437377	+	0.899278i	$0.644093\pi$
$884$	23.1120	0.777340
$885$	0	0
$886$	−0.897280	−0.0301447
$887$	−9.02638	−0.303076	−0.151538	−	0.988451i	$-0.548423\pi$
$887$	−9.02638	−0.303076	−0.151538	+	0.988451i	$0.548423\pi$
$888$	0	0
$889$	−0.318495	−0.0106820
$890$	−0.0968307	−0.00324577
$891$	0	0
$892$	14.8866	0.498439
$893$	−3.60617	−0.120676
$894$	0	0
$895$	−6.01398	−0.201025
$896$	−4.24403	−0.141783
$897$	0	0
$898$	−0.860258	−0.0287072
$899$	5.20447	0.173579
$900$	0	0
$901$	22.5754	0.752095
$902$	−0.0644129	−0.00214471
$903$	0	0
$904$	2.97801	0.0990470
$905$	−15.8778	−0.527796
$906$	0	0
$907$	−6.25728	−0.207770	−0.103885	−	0.994589i	$-0.533127\pi$
$907$	−6.25728	−0.207770	−0.103885	+	0.994589i	$0.533127\pi$
$908$	−26.3740	−0.875251
$909$	0	0
$910$	0.159541	0.00528872
$911$	11.3051	0.374555	0.187277	−	0.982307i	$-0.440034\pi$
$911$	11.3051	0.374555	0.187277	+	0.982307i	$0.440034\pi$
$912$	0	0
$913$	−8.05611	−0.266618
$914$	0.710327	0.0234955
$915$	0	0
$916$	4.11246	0.135879
$917$	14.8679	0.490981
$918$	0	0
$919$	35.0497	1.15618	0.578091	−	0.815973i	$-0.303798\pi$
$919$	35.0497	1.15618	0.578091	+	0.815973i	$0.303798\pi$
$920$	0.600085	0.0197842
$921$	0	0
$922$	0.167441	0.00551438
$923$	−11.9952	−0.394826
$924$	0	0
$925$	10.6233	0.349293
$926$	0.164576	0.00540830
$927$	0	0
$928$	−1.47725	−0.0484932
$929$	18.2171	0.597683	0.298841	−	0.954303i	$-0.403400\pi$
$929$	18.2171	0.597683	0.298841	+	0.954303i	$0.403400\pi$
$930$	0	0
$931$	3.37471	0.110602
$932$	−9.17612	−0.300574
$933$	0	0
$934$	−0.284780	−0.00931828
$935$	27.4592	0.898013
$936$	0	0
$937$	−45.8731	−1.49861	−0.749305	−	0.662225i	$-0.769612\pi$
$937$	−45.8731	−1.49861	−0.749305	+	0.662225i	$0.769612\pi$
$938$	−1.03237	−0.0337081
$939$	0	0
$940$	1.24741	0.0406862
$941$	−42.4328	−1.38327	−0.691635	−	0.722247i	$-0.743109\pi$
$941$	−42.4328	−1.38327	−0.691635	+	0.722247i	$0.743109\pi$
$942$	0	0
$943$	0.897221	0.0292175
$944$	9.23083	0.300438
$945$	0	0
$946$	−1.59146	−0.0517428
$947$	−24.0986	−0.783100	−0.391550	−	0.920157i	$-0.628061\pi$
$947$	−24.0986	−0.783100	−0.391550	+	0.920157i	$0.628061\pi$
$948$	0	0
$949$	−18.6674	−0.605968
$950$	1.02668	0.0333099
$951$	0	0
$952$	−3.85919	−0.125077
$953$	47.0548	1.52425	0.762127	−	0.647427i	$-0.224155\pi$
$953$	47.0548	1.52425	0.762127	+	0.647427i	$0.224155\pi$
$954$	0	0
$955$	0.791643	0.0256170
$956$	−38.3975	−1.24186
$957$	0	0
$958$	−0.981932	−0.0317248
$959$	32.2174	1.04036
$960$	0	0
$961$	−25.9304	−0.836465
$962$	0.203884	0.00657349
$963$	0	0
$964$	7.30128	0.235158
$965$	2.16885	0.0698177
$966$	0	0
$967$	29.8671	0.960462	0.480231	−	0.877142i	$-0.340553\pi$
$967$	29.8671	0.960462	0.480231	+	0.877142i	$0.340553\pi$
$968$	−3.06782	−0.0986035
$969$	0	0
$970$	−0.462079	−0.0148365
$971$	−38.2713	−1.22819	−0.614093	−	0.789234i	$-0.710478\pi$
$971$	−38.2713	−1.22819	−0.614093	+	0.789234i	$0.710478\pi$
$972$	0	0
$973$	−9.16799	−0.293912
$974$	−0.658065	−0.0210858
$975$	0	0
$976$	−22.8985	−0.732962
$977$	50.3062	1.60944	0.804719	−	0.593656i	$-0.202316\pi$
$977$	50.3062	1.60944	0.804719	+	0.593656i	$0.202316\pi$
$978$	0	0
$979$	−12.1686	−0.388911
$980$	−1.16735	−0.0372897
$981$	0	0
$982$	−0.903664	−0.0288371
$983$	−36.6332	−1.16842	−0.584208	−	0.811604i	$-0.698595\pi$
$983$	−36.6332	−1.16842	−0.584208	+	0.811604i	$0.698595\pi$
$984$	0	0
$985$	−3.95840	−0.126125
$986$	−0.894683	−0.0284925
$987$	0	0
$988$	−13.8214	−0.439719
$989$	22.1678	0.704894
$990$	0	0
$991$	10.1594	0.322724	0.161362	−	0.986895i	$-0.448411\pi$
$991$	10.1594	0.322724	0.161362	+	0.986895i	$0.448411\pi$
$992$	−1.43897	−0.0456873
$993$	0	0
$994$	1.00075	0.0317419
$995$	3.43312	0.108837
$996$	0	0
$997$	−46.7678	−1.48115	−0.740576	−	0.671973i	$-0.765447\pi$
$997$	−46.7678	−1.48115	−0.740576	+	0.671973i	$0.765447\pi$
$998$	−1.48400	−0.0469751
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.f.1.12	yes	25
3.2	odd	2		4023.2.a.e.1.14	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.14	✓	25	3.2	odd	2
4023.2.a.f.1.12	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)

\(f(q)\)	\(=\)	\(q+0.0533591 q^{2} -1.99715 q^{4} -0.751347 q^{5} +2.49440 q^{7} -0.213285 q^{8} +O(q^{10})\)	\(q+0.0533591 q^{2} -1.99715 q^{4} -0.751347 q^{5} +2.49440 q^{7} -0.213285 q^{8} -0.0400912 q^{10} -5.03822 q^{11} -1.59535 q^{13} +0.133099 q^{14} +3.98292 q^{16} +7.25388 q^{17} -4.33797 q^{19} +1.50055 q^{20} -0.268835 q^{22} +3.74466 q^{23} -4.43548 q^{25} -0.0851264 q^{26} -4.98171 q^{28} -2.31148 q^{29} -2.25158 q^{31} +0.639095 q^{32} +0.387061 q^{34} -1.87416 q^{35} -2.39508 q^{37} -0.231470 q^{38} +0.160251 q^{40} +0.239600 q^{41} +5.91983 q^{43} +10.0621 q^{44} +0.199812 q^{46} +0.831302 q^{47} -0.777947 q^{49} -0.236673 q^{50} +3.18615 q^{52} +3.11218 q^{53} +3.78545 q^{55} -0.532018 q^{56} -0.123339 q^{58} +2.31760 q^{59} -5.74915 q^{61} -0.120142 q^{62} -7.93175 q^{64} +1.19866 q^{65} -7.75641 q^{67} -14.4871 q^{68} -0.100004 q^{70} +7.51884 q^{71} +11.7011 q^{73} -0.127799 q^{74} +8.66359 q^{76} -12.5674 q^{77} +0.660921 q^{79} -2.99256 q^{80} +0.0127848 q^{82} +1.59900 q^{83} -5.45018 q^{85} +0.315877 q^{86} +1.07458 q^{88} +2.41526 q^{89} -3.97944 q^{91} -7.47867 q^{92} +0.0443576 q^{94} +3.25932 q^{95} +11.5257 q^{97} -0.0415105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) 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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