LMFDB - Embedded newform 4023.2.a.e.1.7

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			1.7
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)

$f(q)$	$=$	$q-1.79652 q^{2} +1.22747 q^{4} +3.24604 q^{5} -3.00506 q^{7} +1.38786 q^{8} +O(q^{10})$	\(q-1.79652 q^{2} +1.22747 q^{4} +3.24604 q^{5} -3.00506 q^{7} +1.38786 q^{8} -5.83156 q^{10} +5.41489 q^{11} -3.18649 q^{13} +5.39864 q^{14} -4.94826 q^{16} -6.34045 q^{17} +6.14822 q^{19} +3.98441 q^{20} -9.72794 q^{22} -3.52983 q^{23} +5.53676 q^{25} +5.72459 q^{26} -3.68862 q^{28} +4.26264 q^{29} -3.29515 q^{31} +6.11390 q^{32} +11.3907 q^{34} -9.75454 q^{35} -2.56990 q^{37} -11.0454 q^{38} +4.50505 q^{40} -11.6526 q^{41} -3.66734 q^{43} +6.64662 q^{44} +6.34140 q^{46} -12.1447 q^{47} +2.03039 q^{49} -9.94688 q^{50} -3.91133 q^{52} +12.2837 q^{53} +17.5769 q^{55} -4.17061 q^{56} -7.65790 q^{58} +4.43120 q^{59} +8.21656 q^{61} +5.91980 q^{62} -1.08720 q^{64} -10.3435 q^{65} -4.82603 q^{67} -7.78272 q^{68} +17.5242 q^{70} -12.4319 q^{71} -4.98096 q^{73} +4.61686 q^{74} +7.54676 q^{76} -16.2721 q^{77} -9.73144 q^{79} -16.0622 q^{80} +20.9341 q^{82} -16.6230 q^{83} -20.5814 q^{85} +6.58844 q^{86} +7.51512 q^{88} +1.09097 q^{89} +9.57561 q^{91} -4.33276 q^{92} +21.8182 q^{94} +19.9574 q^{95} -5.97193 q^{97} -3.64763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.79652	−1.27033	−0.635164	−	0.772377i	$-0.719068\pi$
$2$	−1.79652	−1.27033	−0.635164	+	0.772377i	$0.719068\pi$
$3$	0	0
$3$	0	0
$4$	1.22747	0.613735
$5$	3.24604	1.45167	0.725836	−	0.687868i	$-0.241453\pi$
$5$	3.24604	1.45167	0.725836	+	0.687868i	$0.241453\pi$
$6$	0	0
$7$	−3.00506	−1.13581	−0.567903	−	0.823095i	$-0.692245\pi$
$7$	−3.00506	−1.13581	−0.567903	+	0.823095i	$0.692245\pi$
$8$	1.38786	0.490683
$9$	0	0
$10$	−5.83156	−1.84410
$11$	5.41489	1.63265	0.816325	−	0.577592i	$-0.196008\pi$
$11$	5.41489	1.63265	0.816325	+	0.577592i	$0.196008\pi$
$12$	0	0
$13$	−3.18649	−0.883774	−0.441887	−	0.897071i	$-0.645691\pi$
$13$	−3.18649	−0.883774	−0.441887	+	0.897071i	$0.645691\pi$
$14$	5.39864	1.44285
$15$	0	0
$16$	−4.94826	−1.23706
$17$	−6.34045	−1.53779	−0.768893	−	0.639377i	$-0.779192\pi$
$17$	−6.34045	−1.53779	−0.768893	+	0.639377i	$0.779192\pi$
$18$	0	0
$19$	6.14822	1.41050	0.705249	−	0.708959i	$-0.250835\pi$
$19$	6.14822	1.41050	0.705249	+	0.708959i	$0.250835\pi$
$20$	3.98441	0.890942
$21$	0	0
$22$	−9.72794	−2.07400
$23$	−3.52983	−0.736021	−0.368010	−	0.929822i	$-0.619961\pi$
$23$	−3.52983	−0.736021	−0.368010	+	0.929822i	$0.619961\pi$
$24$	0	0
$25$	5.53676	1.10735
$26$	5.72459	1.12268
$27$	0	0
$28$	−3.68862	−0.697084
$29$	4.26264	0.791552	0.395776	−	0.918347i	$-0.370476\pi$
$29$	4.26264	0.791552	0.395776	+	0.918347i	$0.370476\pi$
$30$	0	0
$31$	−3.29515	−0.591827	−0.295913	−	0.955215i	$-0.595624\pi$
$31$	−3.29515	−0.591827	−0.295913	+	0.955215i	$0.595624\pi$
$32$	6.11390	1.08079
$33$	0	0
$34$	11.3907	1.95349
$35$	−9.75454	−1.64882
$36$	0	0
$37$	−2.56990	−0.422489	−0.211244	−	0.977433i	$-0.567752\pi$
$37$	−2.56990	−0.422489	−0.211244	+	0.977433i	$0.567752\pi$
$38$	−11.0454	−1.79180
$39$	0	0
$40$	4.50505	0.712311
$41$	−11.6526	−1.81984	−0.909918	−	0.414788i	$-0.863856\pi$
$41$	−11.6526	−1.81984	−0.909918	+	0.414788i	$0.863856\pi$
$42$	0	0
$43$	−3.66734	−0.559265	−0.279632	−	0.960107i	$-0.590213\pi$
$43$	−3.66734	−0.559265	−0.279632	+	0.960107i	$0.590213\pi$
$44$	6.64662	1.00202
$45$	0	0
$46$	6.34140	0.934988
$47$	−12.1447	−1.77149	−0.885746	−	0.464170i	$-0.846353\pi$
$47$	−12.1447	−1.77149	−0.885746	+	0.464170i	$0.846353\pi$
$48$	0	0
$49$	2.03039	0.290056
$50$	−9.94688	−1.40670
$51$	0	0
$52$	−3.91133	−0.542403
$53$	12.2837	1.68730	0.843649	−	0.536896i	$-0.180403\pi$
$53$	12.2837	1.68730	0.843649	+	0.536896i	$0.180403\pi$
$54$	0	0
$55$	17.5769	2.37007
$56$	−4.17061	−0.557321
$57$	0	0
$58$	−7.65790	−1.00553
$59$	4.43120	0.576893	0.288447	−	0.957496i	$-0.406861\pi$
$59$	4.43120	0.576893	0.288447	+	0.957496i	$0.406861\pi$
$60$	0	0
$61$	8.21656	1.05202	0.526011	−	0.850478i	$-0.323687\pi$
$61$	8.21656	1.05202	0.526011	+	0.850478i	$0.323687\pi$
$62$	5.91980	0.751815
$63$	0	0
$64$	−1.08720	−0.135901
$65$	−10.3435	−1.28295
$66$	0	0
$67$	−4.82603	−0.589593	−0.294797	−	0.955560i	$-0.595252\pi$
$67$	−4.82603	−0.589593	−0.294797	+	0.955560i	$0.595252\pi$
$68$	−7.78272	−0.943793
$69$	0	0
$70$	17.5242	2.09454
$71$	−12.4319	−1.47539	−0.737697	−	0.675131i	$-0.764087\pi$
$71$	−12.4319	−1.47539	−0.737697	+	0.675131i	$0.764087\pi$
$72$	0	0
$73$	−4.98096	−0.582977	−0.291489	−	0.956574i	$-0.594151\pi$
$73$	−4.98096	−0.582977	−0.291489	+	0.956574i	$0.594151\pi$
$74$	4.61686	0.536699
$75$	0	0
$76$	7.54676	0.865673
$77$	−16.2721	−1.85438
$78$	0	0
$79$	−9.73144	−1.09487	−0.547436	−	0.836847i	$-0.684396\pi$
$79$	−9.73144	−1.09487	−0.547436	+	0.836847i	$0.684396\pi$
$80$	−16.0622	−1.79581
$81$	0	0
$82$	20.9341	2.31179
$83$	−16.6230	−1.82461	−0.912304	−	0.409513i	$-0.865699\pi$
$83$	−16.6230	−1.82461	−0.912304	+	0.409513i	$0.865699\pi$
$84$	0	0
$85$	−20.5814	−2.23236
$86$	6.58844	0.710450
$87$	0	0
$88$	7.51512	0.801115
$89$	1.09097	0.115643	0.0578215	−	0.998327i	$-0.481585\pi$
$89$	1.09097	0.115643	0.0578215	+	0.998327i	$0.481585\pi$
$90$	0	0
$91$	9.57561	1.00380
$92$	−4.33276	−0.451722
$93$	0	0
$94$	21.8182	2.25038
$95$	19.9574	2.04758
$96$	0	0
$97$	−5.97193	−0.606358	−0.303179	−	0.952934i	$-0.598048\pi$
$97$	−5.97193	−0.606358	−0.303179	+	0.952934i	$0.598048\pi$
$98$	−3.64763	−0.368467
$99$	0	0
$100$	6.79621	0.679621
$101$	7.68456	0.764643	0.382321	−	0.924029i	$-0.375125\pi$
$101$	7.68456	0.764643	0.382321	+	0.924029i	$0.375125\pi$
$102$	0	0
$103$	−4.26148	−0.419896	−0.209948	−	0.977713i	$-0.567329\pi$
$103$	−4.26148	−0.419896	−0.209948	+	0.977713i	$0.567329\pi$
$104$	−4.42241	−0.433653
$105$	0	0
$106$	−22.0679	−2.14342
$107$	11.8510	1.14568	0.572840	−	0.819667i	$-0.305842\pi$
$107$	11.8510	1.14568	0.572840	+	0.819667i	$0.305842\pi$
$108$	0	0
$109$	8.08426	0.774331	0.387166	−	0.922010i	$-0.373454\pi$
$109$	8.08426	0.774331	0.387166	+	0.922010i	$0.373454\pi$
$110$	−31.5773	−3.01077
$111$	0	0
$112$	14.8698	1.40507
$113$	−17.8952	−1.68344	−0.841720	−	0.539914i	$-0.818457\pi$
$113$	−17.8952	−1.68344	−0.841720	+	0.539914i	$0.818457\pi$
$114$	0	0
$115$	−11.4580	−1.06846
$116$	5.23226	0.485803
$117$	0	0
$118$	−7.96073	−0.732844
$119$	19.0535	1.74663
$120$	0	0
$121$	18.3210	1.66555
$122$	−14.7612	−1.33641
$123$	0	0
$124$	−4.04470	−0.363225
$125$	1.74234	0.155840
$126$	0	0
$127$	12.1507	1.07820	0.539101	−	0.842241i	$-0.318764\pi$
$127$	12.1507	1.07820	0.539101	+	0.842241i	$0.318764\pi$
$128$	−10.2746	−0.908157
$129$	0	0
$130$	18.5822	1.62977
$131$	−8.20659	−0.717013	−0.358506	−	0.933527i	$-0.616714\pi$
$131$	−8.20659	−0.717013	−0.358506	+	0.933527i	$0.616714\pi$
$132$	0	0
$133$	−18.4758	−1.60205
$134$	8.67004	0.748977
$135$	0	0
$136$	−8.79968	−0.754566
$137$	−1.16975	−0.0999389	−0.0499694	−	0.998751i	$-0.515912\pi$
$137$	−1.16975	−0.0999389	−0.0499694	+	0.998751i	$0.515912\pi$
$138$	0	0
$139$	14.6724	1.24449	0.622247	−	0.782821i	$-0.286220\pi$
$139$	14.6724	1.24449	0.622247	+	0.782821i	$0.286220\pi$
$140$	−11.9734	−1.01194
$141$	0	0
$142$	22.3341	1.87424
$143$	−17.2545	−1.44289
$144$	0	0
$145$	13.8367	1.14907
$146$	8.94838	0.740573
$147$	0	0
$148$	−3.15447	−0.259296
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−16.7185	−1.36054	−0.680268	−	0.732964i	$-0.738137\pi$
$151$	−16.7185	−1.36054	−0.680268	+	0.732964i	$0.738137\pi$
$152$	8.53289	0.692108
$153$	0	0
$154$	29.2330	2.35567
$155$	−10.6962	−0.859139
$156$	0	0
$157$	−6.12561	−0.488877	−0.244439	−	0.969665i	$-0.578604\pi$
$157$	−6.12561	−0.488877	−0.244439	+	0.969665i	$0.578604\pi$
$158$	17.4827	1.39085
$159$	0	0
$160$	19.8459	1.56896
$161$	10.6074	0.835977
$162$	0	0
$163$	24.6667	1.93204	0.966022	−	0.258460i	$-0.0832149\pi$
$163$	24.6667	1.93204	0.966022	+	0.258460i	$0.0832149\pi$
$164$	−14.3033	−1.11690
$165$	0	0
$166$	29.8634	2.31785
$167$	22.7213	1.75823	0.879114	−	0.476612i	$-0.158135\pi$
$167$	22.7213	1.75823	0.879114	+	0.476612i	$0.158135\pi$
$168$	0	0
$169$	−2.84626	−0.218943
$170$	36.9747	2.83583
$171$	0	0
$172$	−4.50156	−0.343240
$173$	3.13846	0.238612	0.119306	−	0.992858i	$-0.461933\pi$
$173$	3.13846	0.238612	0.119306	+	0.992858i	$0.461933\pi$
$174$	0	0
$175$	−16.6383	−1.25774
$176$	−26.7943	−2.01969
$177$	0	0
$178$	−1.95995	−0.146905
$179$	−4.04206	−0.302118	−0.151059	−	0.988525i	$-0.548268\pi$
$179$	−4.04206	−0.302118	−0.151059	+	0.988525i	$0.548268\pi$
$180$	0	0
$181$	−2.80354	−0.208385	−0.104193	−	0.994557i	$-0.533226\pi$
$181$	−2.80354	−0.208385	−0.104193	+	0.994557i	$0.533226\pi$
$182$	−17.2027	−1.27515
$183$	0	0
$184$	−4.89892	−0.361153
$185$	−8.34199	−0.613315
$186$	0	0
$187$	−34.3329	−2.51067
$188$	−14.9073	−1.08723
$189$	0	0
$190$	−35.8537	−2.60110
$191$	8.53558	0.617613	0.308806	−	0.951125i	$-0.400070\pi$
$191$	8.53558	0.617613	0.308806	+	0.951125i	$0.400070\pi$
$192$	0	0
$193$	−13.2729	−0.955407	−0.477704	−	0.878521i	$-0.658531\pi$
$193$	−13.2729	−0.955407	−0.477704	+	0.878521i	$0.658531\pi$
$194$	10.7287	0.770274
$195$	0	0
$196$	2.49225	0.178018
$197$	−17.6240	−1.25566	−0.627829	−	0.778351i	$-0.716056\pi$
$197$	−17.6240	−1.25566	−0.627829	+	0.778351i	$0.716056\pi$
$198$	0	0
$199$	2.84947	0.201994	0.100997	−	0.994887i	$-0.467797\pi$
$199$	2.84947	0.201994	0.100997	+	0.994887i	$0.467797\pi$
$200$	7.68426	0.543359
$201$	0	0
$202$	−13.8054	−0.971347
$203$	−12.8095	−0.899050
$204$	0	0
$205$	−37.8249	−2.64181
$206$	7.65581	0.533406
$207$	0	0
$208$	15.7676	1.09329
$209$	33.2919	2.30285
$210$	0	0
$211$	−14.9734	−1.03081	−0.515405	−	0.856947i	$-0.672358\pi$
$211$	−14.9734	−1.03081	−0.515405	+	0.856947i	$0.672358\pi$
$212$	15.0779	1.03555
$213$	0	0
$214$	−21.2905	−1.45539
$215$	−11.9043	−0.811869
$216$	0	0
$217$	9.90214	0.672201
$218$	−14.5235	−0.983655
$219$	0	0
$220$	21.5752	1.45460
$221$	20.2038	1.35906
$222$	0	0
$223$	10.8668	0.727696	0.363848	−	0.931458i	$-0.381463\pi$
$223$	10.8668	0.727696	0.363848	+	0.931458i	$0.381463\pi$
$224$	−18.3726	−1.22757
$225$	0	0
$226$	32.1491	2.13852
$227$	20.0955	1.33379	0.666893	−	0.745153i	$-0.267624\pi$
$227$	20.0955	1.33379	0.666893	+	0.745153i	$0.267624\pi$
$228$	0	0
$229$	−24.6376	−1.62810	−0.814051	−	0.580794i	$-0.802742\pi$
$229$	−24.6376	−1.62810	−0.814051	+	0.580794i	$0.802742\pi$
$230$	20.5844	1.35730
$231$	0	0
$232$	5.91595	0.388401
$233$	−7.37781	−0.483337	−0.241668	−	0.970359i	$-0.577695\pi$
$233$	−7.37781	−0.483337	−0.241668	+	0.970359i	$0.577695\pi$
$234$	0	0
$235$	−39.4223	−2.57163
$236$	5.43917	0.354060
$237$	0	0
$238$	−34.2298	−2.21879
$239$	21.5544	1.39424	0.697118	−	0.716956i	$-0.254465\pi$
$239$	21.5544	1.39424	0.697118	+	0.716956i	$0.254465\pi$
$240$	0	0
$241$	29.3404	1.88998	0.944991	−	0.327096i	$-0.106070\pi$
$241$	29.3404	1.88998	0.944991	+	0.327096i	$0.106070\pi$
$242$	−32.9140	−2.11579
$243$	0	0
$244$	10.0856	0.645663
$245$	6.59073	0.421066
$246$	0	0
$247$	−19.5913	−1.24656
$248$	−4.57322	−0.290400
$249$	0	0
$250$	−3.13015	−0.197968
$251$	8.02228	0.506362	0.253181	−	0.967419i	$-0.418523\pi$
$251$	8.02228	0.506362	0.253181	+	0.967419i	$0.418523\pi$
$252$	0	0
$253$	−19.1136	−1.20166
$254$	−21.8290	−1.36967
$255$	0	0
$256$	20.6329	1.28956
$257$	−25.6233	−1.59834	−0.799168	−	0.601108i	$-0.794726\pi$
$257$	−25.6233	−1.59834	−0.799168	+	0.601108i	$0.794726\pi$
$258$	0	0
$259$	7.72270	0.479865
$260$	−12.6963	−0.787392
$261$	0	0
$262$	14.7433	0.910842
$263$	−13.1695	−0.812066	−0.406033	−	0.913858i	$-0.633088\pi$
$263$	−13.1695	−0.812066	−0.406033	+	0.913858i	$0.633088\pi$
$264$	0	0
$265$	39.8734	2.44940
$266$	33.1920	2.03513
$267$	0	0
$268$	−5.92381	−0.361854
$269$	−6.82102	−0.415885	−0.207943	−	0.978141i	$-0.566677\pi$
$269$	−6.82102	−0.415885	−0.207943	+	0.978141i	$0.566677\pi$
$270$	0	0
$271$	2.53105	0.153751	0.0768753	−	0.997041i	$-0.475506\pi$
$271$	2.53105	0.153751	0.0768753	+	0.997041i	$0.475506\pi$
$272$	31.3742	1.90234
$273$	0	0
$274$	2.10148	0.126955
$275$	29.9809	1.80792
$276$	0	0
$277$	−1.32984	−0.0799026	−0.0399513	−	0.999202i	$-0.512720\pi$
$277$	−1.32984	−0.0799026	−0.0399513	+	0.999202i	$0.512720\pi$
$278$	−26.3592	−1.58092
$279$	0	0
$280$	−13.5380	−0.809048
$281$	2.52043	0.150356	0.0751780	−	0.997170i	$-0.476047\pi$
$281$	2.52043	0.150356	0.0751780	+	0.997170i	$0.476047\pi$
$282$	0	0
$283$	8.56538	0.509159	0.254580	−	0.967052i	$-0.418063\pi$
$283$	8.56538	0.509159	0.254580	+	0.967052i	$0.418063\pi$
$284$	−15.2598	−0.905502
$285$	0	0
$286$	30.9980	1.83295
$287$	35.0169	2.06698
$288$	0	0
$289$	23.2014	1.36479
$290$	−24.8578	−1.45970
$291$	0	0
$292$	−6.11398	−0.357794
$293$	1.63825	0.0957075	0.0478538	−	0.998854i	$-0.484762\pi$
$293$	1.63825	0.0957075	0.0478538	+	0.998854i	$0.484762\pi$
$294$	0	0
$295$	14.3839	0.837460
$296$	−3.56666	−0.207308
$297$	0	0
$298$	1.79652	0.104069
$299$	11.2478	0.650476
$300$	0	0
$301$	11.0206	0.635217
$302$	30.0351	1.72833
$303$	0	0
$304$	−30.4230	−1.74488
$305$	26.6713	1.52719
$306$	0	0
$307$	−5.27381	−0.300992	−0.150496	−	0.988611i	$-0.548087\pi$
$307$	−5.27381	−0.300992	−0.150496	+	0.988611i	$0.548087\pi$
$308$	−19.9735	−1.13810
$309$	0	0
$310$	19.2159	1.09139
$311$	−27.4397	−1.55596	−0.777982	−	0.628287i	$-0.783756\pi$
$311$	−27.4397	−1.55596	−0.777982	+	0.628287i	$0.783756\pi$
$312$	0	0
$313$	−1.09475	−0.0618788	−0.0309394	−	0.999521i	$-0.509850\pi$
$313$	−1.09475	−0.0618788	−0.0309394	+	0.999521i	$0.509850\pi$
$314$	11.0048	0.621035
$315$	0	0
$316$	−11.9451	−0.671962
$317$	−29.1164	−1.63534	−0.817671	−	0.575686i	$-0.804735\pi$
$317$	−29.1164	−1.63534	−0.817671	+	0.575686i	$0.804735\pi$
$318$	0	0
$319$	23.0817	1.29233
$320$	−3.52911	−0.197283
$321$	0	0
$322$	−19.0563	−1.06197
$323$	−38.9825	−2.16905
$324$	0	0
$325$	−17.6429	−0.978649
$326$	−44.3141	−2.45433
$327$	0	0
$328$	−16.1723	−0.892963
$329$	36.4957	2.01207
$330$	0	0
$331$	−21.8249	−1.19960	−0.599801	−	0.800149i	$-0.704754\pi$
$331$	−21.8249	−1.19960	−0.599801	+	0.800149i	$0.704754\pi$
$332$	−20.4042	−1.11983
$333$	0	0
$334$	−40.8192	−2.23353
$335$	−15.6655	−0.855896
$336$	0	0
$337$	28.1233	1.53197	0.765986	−	0.642857i	$-0.222251\pi$
$337$	28.1233	1.53197	0.765986	+	0.642857i	$0.222251\pi$
$338$	5.11335	0.278129
$339$	0	0
$340$	−25.2630	−1.37008
$341$	−17.8429	−0.966247
$342$	0	0
$343$	14.9340	0.806359
$344$	−5.08977	−0.274422
$345$	0	0
$346$	−5.63829	−0.303116
$347$	16.1111	0.864890	0.432445	−	0.901660i	$-0.357651\pi$
$347$	16.1111	0.864890	0.432445	+	0.901660i	$0.357651\pi$
$348$	0	0
$349$	−0.251371	−0.0134556	−0.00672779	−	0.999977i	$-0.502142\pi$
$349$	−0.251371	−0.0134556	−0.00672779	+	0.999977i	$0.502142\pi$
$350$	29.8910	1.59774
$351$	0	0
$352$	33.1061	1.76456
$353$	−27.1586	−1.44550	−0.722752	−	0.691107i	$-0.757123\pi$
$353$	−27.1586	−1.44550	−0.722752	+	0.691107i	$0.757123\pi$
$354$	0	0
$355$	−40.3544	−2.14179
$356$	1.33914	0.0709741
$357$	0	0
$358$	7.26162	0.383789
$359$	−23.2810	−1.22872	−0.614362	−	0.789024i	$-0.710587\pi$
$359$	−23.2810	−1.22872	−0.614362	+	0.789024i	$0.710587\pi$
$360$	0	0
$361$	18.8006	0.989507
$362$	5.03660	0.264718
$363$	0	0
$364$	11.7538	0.616065
$365$	−16.1684	−0.846292
$366$	0	0
$367$	−13.4926	−0.704308	−0.352154	−	0.935942i	$-0.614551\pi$
$367$	−13.4926	−0.704308	−0.352154	+	0.935942i	$0.614551\pi$
$368$	17.4665	0.910505
$369$	0	0
$370$	14.9865	0.779112
$371$	−36.9133	−1.91644
$372$	0	0
$373$	7.27491	0.376680	0.188340	−	0.982104i	$-0.439689\pi$
$373$	7.27491	0.376680	0.188340	+	0.982104i	$0.439689\pi$
$374$	61.6795	3.18937
$375$	0	0
$376$	−16.8552	−0.869242
$377$	−13.5829	−0.699553
$378$	0	0
$379$	−13.8350	−0.710657	−0.355329	−	0.934741i	$-0.615631\pi$
$379$	−13.8350	−0.710657	−0.355329	+	0.934741i	$0.615631\pi$
$380$	24.4971	1.25667
$381$	0	0
$382$	−15.3343	−0.784571
$383$	4.47882	0.228857	0.114428	−	0.993431i	$-0.463496\pi$
$383$	4.47882	0.228857	0.114428	+	0.993431i	$0.463496\pi$
$384$	0	0
$385$	−52.8198	−2.69194
$386$	23.8451	1.21368
$387$	0	0
$388$	−7.33037	−0.372143
$389$	0.320061	0.0162277	0.00811386	−	0.999967i	$-0.497417\pi$
$389$	0.320061	0.0162277	0.00811386	+	0.999967i	$0.497417\pi$
$390$	0	0
$391$	22.3807	1.13184
$392$	2.81791	0.142326
$393$	0	0
$394$	31.6618	1.59510
$395$	−31.5886	−1.58940
$396$	0	0
$397$	37.9068	1.90249	0.951243	−	0.308442i	$-0.0998076\pi$
$397$	37.9068	1.90249	0.951243	+	0.308442i	$0.0998076\pi$
$398$	−5.11912	−0.256598
$399$	0	0
$400$	−27.3973	−1.36987
$401$	−29.6830	−1.48230	−0.741148	−	0.671341i	$-0.765718\pi$
$401$	−29.6830	−1.48230	−0.741148	+	0.671341i	$0.765718\pi$
$402$	0	0
$403$	10.5000	0.523041
$404$	9.43257	0.469288
$405$	0	0
$406$	23.0125	1.14209
$407$	−13.9157	−0.689776
$408$	0	0
$409$	−28.0672	−1.38784	−0.693918	−	0.720054i	$-0.744117\pi$
$409$	−28.0672	−1.38784	−0.693918	+	0.720054i	$0.744117\pi$
$410$	67.9530	3.35596
$411$	0	0
$412$	−5.23084	−0.257705
$413$	−13.3160	−0.655239
$414$	0	0
$415$	−53.9588	−2.64873
$416$	−19.4819	−0.955179
$417$	0	0
$418$	−59.8095	−2.92538
$419$	−6.61054	−0.322946	−0.161473	−	0.986877i	$-0.551624\pi$
$419$	−6.61054	−0.322946	−0.161473	+	0.986877i	$0.551624\pi$
$420$	0	0
$421$	−5.38873	−0.262631	−0.131315	−	0.991341i	$-0.541920\pi$
$421$	−5.38873	−0.262631	−0.131315	+	0.991341i	$0.541920\pi$
$422$	26.8999	1.30947
$423$	0	0
$424$	17.0481	0.827929
$425$	−35.1056	−1.70287
$426$	0	0
$427$	−24.6913	−1.19489
$428$	14.5468	0.703144
$429$	0	0
$430$	21.3863	1.03134
$431$	−37.6722	−1.81461	−0.907303	−	0.420477i	$-0.861863\pi$
$431$	−37.6722	−1.81461	−0.907303	+	0.420477i	$0.861863\pi$
$432$	0	0
$433$	−29.8809	−1.43598	−0.717992	−	0.696051i	$-0.754939\pi$
$433$	−29.8809	−1.43598	−0.717992	+	0.696051i	$0.754939\pi$
$434$	−17.7893	−0.853916
$435$	0	0
$436$	9.92318	0.475234
$437$	−21.7022	−1.03816
$438$	0	0
$439$	−12.1698	−0.580832	−0.290416	−	0.956900i	$-0.593794\pi$
$439$	−12.1698	−0.580832	−0.290416	+	0.956900i	$0.593794\pi$
$440$	24.3944	1.16296
$441$	0	0
$442$	−36.2965	−1.72645
$443$	18.0902	0.859490	0.429745	−	0.902950i	$-0.358603\pi$
$443$	18.0902	0.859490	0.429745	+	0.902950i	$0.358603\pi$
$444$	0	0
$445$	3.54134	0.167876
$446$	−19.5224	−0.924413
$447$	0	0
$448$	3.26712	0.154357
$449$	−36.4852	−1.72184	−0.860921	−	0.508738i	$-0.830112\pi$
$449$	−36.4852	−1.72184	−0.860921	+	0.508738i	$0.830112\pi$
$450$	0	0
$451$	−63.0977	−2.97116
$452$	−21.9658	−1.03319
$453$	0	0
$454$	−36.1019	−1.69435
$455$	31.0828	1.45718
$456$	0	0
$457$	−26.6555	−1.24689	−0.623447	−	0.781866i	$-0.714268\pi$
$457$	−26.6555	−1.24689	−0.623447	+	0.781866i	$0.714268\pi$
$458$	44.2619	2.06822
$459$	0	0
$460$	−14.0643	−0.655752
$461$	2.14614	0.0999559	0.0499780	−	0.998750i	$-0.484085\pi$
$461$	2.14614	0.0999559	0.0499780	+	0.998750i	$0.484085\pi$
$462$	0	0
$463$	12.5148	0.581613	0.290806	−	0.956782i	$-0.406076\pi$
$463$	12.5148	0.581613	0.290806	+	0.956782i	$0.406076\pi$
$464$	−21.0926	−0.979201
$465$	0	0
$466$	13.2544	0.613997
$467$	22.1735	1.02607	0.513034	−	0.858368i	$-0.328521\pi$
$467$	22.1735	1.02607	0.513034	+	0.858368i	$0.328521\pi$
$468$	0	0
$469$	14.5025	0.669664
$470$	70.8228	3.26681
$471$	0	0
$472$	6.14990	0.283072
$473$	−19.8583	−0.913084
$474$	0	0
$475$	34.0412	1.56192
$476$	23.3875	1.07197
$477$	0	0
$478$	−38.7228	−1.77114
$479$	5.45180	0.249099	0.124549	−	0.992213i	$-0.460251\pi$
$479$	5.45180	0.249099	0.124549	+	0.992213i	$0.460251\pi$
$480$	0	0
$481$	8.18896	0.373385
$482$	−52.7105	−2.40090
$483$	0	0
$484$	22.4885	1.02221
$485$	−19.3851	−0.880233
$486$	0	0
$487$	4.65788	0.211069	0.105534	−	0.994416i	$-0.466345\pi$
$487$	4.65788	0.211069	0.105534	+	0.994416i	$0.466345\pi$
$488$	11.4035	0.516210
$489$	0	0
$490$	−11.8404	−0.534893
$491$	−31.9116	−1.44015	−0.720076	−	0.693896i	$-0.755893\pi$
$491$	−31.9116	−1.44015	−0.720076	+	0.693896i	$0.755893\pi$
$492$	0	0
$493$	−27.0271	−1.21724
$494$	35.1960	1.58354
$495$	0	0
$496$	16.3053	0.732128
$497$	37.3586	1.67576
$498$	0	0
$499$	−5.16609	−0.231266	−0.115633	−	0.993292i	$-0.536890\pi$
$499$	−5.16609	−0.231266	−0.115633	+	0.993292i	$0.536890\pi$
$500$	2.13868	0.0956445
$501$	0	0
$502$	−14.4121	−0.643246
$503$	−2.26890	−0.101165	−0.0505825	−	0.998720i	$-0.516108\pi$
$503$	−2.26890	−0.101165	−0.0505825	+	0.998720i	$0.516108\pi$
$504$	0	0
$505$	24.9444	1.11001
$506$	34.3380	1.52651
$507$	0	0
$508$	14.9146	0.661730
$509$	−1.74384	−0.0772942	−0.0386471	−	0.999253i	$-0.512305\pi$
$509$	−1.74384	−0.0772942	−0.0386471	+	0.999253i	$0.512305\pi$
$510$	0	0
$511$	14.9681	0.662149
$512$	−16.5181	−0.730006
$513$	0	0
$514$	46.0326	2.03041
$515$	−13.8329	−0.609551
$516$	0	0
$517$	−65.7624	−2.89223
$518$	−13.8740	−0.609587
$519$	0	0
$520$	−14.3553	−0.629523
$521$	−3.73458	−0.163615	−0.0818074	−	0.996648i	$-0.526069\pi$
$521$	−3.73458	−0.163615	−0.0818074	+	0.996648i	$0.526069\pi$
$522$	0	0
$523$	−9.64876	−0.421911	−0.210955	−	0.977496i	$-0.567657\pi$
$523$	−9.64876	−0.421911	−0.210955	+	0.977496i	$0.567657\pi$
$524$	−10.0733	−0.440056
$525$	0	0
$526$	23.6592	1.03159
$527$	20.8928	0.910103
$528$	0	0
$529$	−10.5403	−0.458274
$530$	−71.6332	−3.11155
$531$	0	0
$532$	−22.6785	−0.983237
$533$	37.1311	1.60832
$534$	0	0
$535$	38.4688	1.66315
$536$	−6.69786	−0.289304
$537$	0	0
$538$	12.2541	0.528311
$539$	10.9944	0.473560
$540$	0	0
$541$	9.28419	0.399159	0.199579	−	0.979882i	$-0.436042\pi$
$541$	9.28419	0.399159	0.199579	+	0.979882i	$0.436042\pi$
$542$	−4.54708	−0.195314
$543$	0	0
$544$	−38.7649	−1.66203
$545$	26.2418	1.12408
$546$	0	0
$547$	5.77061	0.246734	0.123367	−	0.992361i	$-0.460631\pi$
$547$	5.77061	0.246734	0.123367	+	0.992361i	$0.460631\pi$
$548$	−1.43584	−0.0613360
$549$	0	0
$550$	−53.8613	−2.29665
$551$	26.2076	1.11648
$552$	0	0
$553$	29.2436	1.24356
$554$	2.38909	0.101503
$555$	0	0
$556$	18.0099	0.763790
$557$	18.1481	0.768961	0.384481	−	0.923133i	$-0.374381\pi$
$557$	18.1481	0.768961	0.384481	+	0.923133i	$0.374381\pi$
$558$	0	0
$559$	11.6860	0.494264
$560$	48.2680	2.03969
$561$	0	0
$562$	−4.52799	−0.191002
$563$	−11.9472	−0.503515	−0.251757	−	0.967790i	$-0.581009\pi$
$563$	−11.9472	−0.503515	−0.251757	+	0.967790i	$0.581009\pi$
$564$	0	0
$565$	−58.0886	−2.44380
$566$	−15.3878	−0.646800
$567$	0	0
$568$	−17.2538	−0.723952
$569$	−37.0365	−1.55265	−0.776325	−	0.630333i	$-0.782919\pi$
$569$	−37.0365	−1.55265	−0.776325	+	0.630333i	$0.782919\pi$
$570$	0	0
$571$	20.4391	0.855351	0.427675	−	0.903932i	$-0.359333\pi$
$571$	20.4391	0.855351	0.427675	+	0.903932i	$0.359333\pi$
$572$	−21.1794	−0.885555
$573$	0	0
$574$	−62.9084	−2.62575
$575$	−19.5438	−0.815034
$576$	0	0
$577$	−5.50056	−0.228991	−0.114496	−	0.993424i	$-0.536525\pi$
$577$	−5.50056	−0.228991	−0.114496	+	0.993424i	$0.536525\pi$
$578$	−41.6816	−1.73373
$579$	0	0
$580$	16.9841	0.705227
$581$	49.9531	2.07240
$582$	0	0
$583$	66.5149	2.75477
$584$	−6.91289	−0.286057
$585$	0	0
$586$	−2.94314	−0.121580
$587$	−25.2798	−1.04341	−0.521705	−	0.853126i	$-0.674704\pi$
$587$	−25.2798	−1.04341	−0.521705	+	0.853126i	$0.674704\pi$
$588$	0	0
$589$	−20.2593	−0.834771
$590$	−25.8408	−1.06385
$591$	0	0
$592$	12.7165	0.522646
$593$	−29.0051	−1.19110	−0.595548	−	0.803320i	$-0.703065\pi$
$593$	−29.0051	−1.19110	−0.595548	+	0.803320i	$0.703065\pi$
$594$	0	0
$595$	61.8482	2.53553
$596$	−1.22747	−0.0502791
$597$	0	0
$598$	−20.2068	−0.826319
$599$	−26.2356	−1.07196	−0.535978	−	0.844232i	$-0.680057\pi$
$599$	−26.2356	−1.07196	−0.535978	+	0.844232i	$0.680057\pi$
$600$	0	0
$601$	41.4569	1.69106	0.845531	−	0.533927i	$-0.179284\pi$
$601$	41.4569	1.69106	0.845531	+	0.533927i	$0.179284\pi$
$602$	−19.7987	−0.806934
$603$	0	0
$604$	−20.5215	−0.835008
$605$	59.4708	2.41783
$606$	0	0
$607$	−12.0047	−0.487255	−0.243627	−	0.969869i	$-0.578337\pi$
$607$	−12.0047	−0.487255	−0.243627	+	0.969869i	$0.578337\pi$
$608$	37.5896	1.52446
$609$	0	0
$610$	−47.9153	−1.94004
$611$	38.6991	1.56560
$612$	0	0
$613$	−17.7566	−0.717184	−0.358592	−	0.933494i	$-0.616743\pi$
$613$	−17.7566	−0.717184	−0.358592	+	0.933494i	$0.616743\pi$
$614$	9.47448	0.382359
$615$	0	0
$616$	−22.5834	−0.909911
$617$	−39.4300	−1.58739	−0.793695	−	0.608316i	$-0.791845\pi$
$617$	−39.4300	−1.58739	−0.793695	+	0.608316i	$0.791845\pi$
$618$	0	0
$619$	27.6488	1.11130	0.555650	−	0.831417i	$-0.312470\pi$
$619$	27.6488	1.11130	0.555650	+	0.831417i	$0.312470\pi$
$620$	−13.1293	−0.527284
$621$	0	0
$622$	49.2959	1.97658
$623$	−3.27844	−0.131348
$624$	0	0
$625$	−22.0281	−0.881123
$626$	1.96673	0.0786064
$627$	0	0
$628$	−7.51901	−0.300041
$629$	16.2943	0.649697
$630$	0	0
$631$	22.6530	0.901803	0.450902	−	0.892574i	$-0.351103\pi$
$631$	22.6530	0.901803	0.450902	+	0.892574i	$0.351103\pi$
$632$	−13.5059	−0.537236
$633$	0	0
$634$	52.3081	2.07742
$635$	39.4417	1.56520
$636$	0	0
$637$	−6.46983	−0.256344
$638$	−41.4667	−1.64168
$639$	0	0
$640$	−33.3518	−1.31835
$641$	−22.4317	−0.885999	−0.443000	−	0.896522i	$-0.646086\pi$
$641$	−22.4317	−0.885999	−0.443000	+	0.896522i	$0.646086\pi$
$642$	0	0
$643$	−1.47986	−0.0583598	−0.0291799	−	0.999574i	$-0.509290\pi$
$643$	−1.47986	−0.0583598	−0.0291799	+	0.999574i	$0.509290\pi$
$644$	13.0202	0.513068
$645$	0	0
$646$	70.0327	2.75540
$647$	25.7327	1.01166	0.505829	−	0.862634i	$-0.331187\pi$
$647$	25.7327	1.01166	0.505829	+	0.862634i	$0.331187\pi$
$648$	0	0
$649$	23.9945	0.941866
$650$	31.6957	1.24321
$651$	0	0
$652$	30.2776	1.18576
$653$	11.6909	0.457500	0.228750	−	0.973485i	$-0.426536\pi$
$653$	11.6909	0.457500	0.228750	+	0.973485i	$0.426536\pi$
$654$	0	0
$655$	−26.6389	−1.04087
$656$	57.6602	2.25125
$657$	0	0
$658$	−65.5651	−2.55599
$659$	9.23411	0.359710	0.179855	−	0.983693i	$-0.442437\pi$
$659$	9.23411	0.359710	0.179855	+	0.983693i	$0.442437\pi$
$660$	0	0
$661$	−5.78917	−0.225173	−0.112586	−	0.993642i	$-0.535914\pi$
$661$	−5.78917	−0.225173	−0.112586	+	0.993642i	$0.535914\pi$
$662$	39.2087	1.52389
$663$	0	0
$664$	−23.0704	−0.895305
$665$	−59.9731	−2.32566
$666$	0	0
$667$	−15.0464	−0.582599
$668$	27.8897	1.07909
$669$	0	0
$670$	28.1433	1.08727
$671$	44.4918	1.71759
$672$	0	0
$673$	−12.5015	−0.481899	−0.240950	−	0.970538i	$-0.577459\pi$
$673$	−12.5015	−0.481899	−0.240950	+	0.970538i	$0.577459\pi$
$674$	−50.5239	−1.94611
$675$	0	0
$676$	−3.49369	−0.134373
$677$	4.30310	0.165382	0.0826908	−	0.996575i	$-0.473649\pi$
$677$	4.30310	0.165382	0.0826908	+	0.996575i	$0.473649\pi$
$678$	0	0
$679$	17.9460	0.688705
$680$	−28.5641	−1.09538
$681$	0	0
$682$	32.0550	1.22745
$683$	38.5137	1.47369	0.736844	−	0.676063i	$-0.236315\pi$
$683$	38.5137	1.47369	0.736844	+	0.676063i	$0.236315\pi$
$684$	0	0
$685$	−3.79707	−0.145078
$686$	−26.8291	−1.02434
$687$	0	0
$688$	18.1470	0.691847
$689$	−39.1420	−1.49119
$690$	0	0
$691$	−7.69533	−0.292744	−0.146372	−	0.989230i	$-0.546760\pi$
$691$	−7.69533	−0.292744	−0.146372	+	0.989230i	$0.546760\pi$
$692$	3.85236	0.146445
$693$	0	0
$694$	−28.9439	−1.09869
$695$	47.6271	1.80660
$696$	0	0
$697$	73.8830	2.79852
$698$	0.451592	0.0170930
$699$	0	0
$700$	−20.4230	−0.771918
$701$	−5.51106	−0.208150	−0.104075	−	0.994569i	$-0.533188\pi$
$701$	−5.51106	−0.208150	−0.104075	+	0.994569i	$0.533188\pi$
$702$	0	0
$703$	−15.8003	−0.595920
$704$	−5.88709	−0.221878
$705$	0	0
$706$	48.7908	1.83627
$707$	−23.0926	−0.868486
$708$	0	0
$709$	−24.7474	−0.929409	−0.464704	−	0.885466i	$-0.653839\pi$
$709$	−24.7474	−0.929409	−0.464704	+	0.885466i	$0.653839\pi$
$710$	72.4973	2.72078
$711$	0	0
$712$	1.51412	0.0567441
$713$	11.6313	0.435597
$714$	0	0
$715$	−56.0088	−2.09461
$716$	−4.96151	−0.185420
$717$	0	0
$718$	41.8247	1.56088
$719$	30.6882	1.14448	0.572238	−	0.820088i	$-0.306076\pi$
$719$	30.6882	1.14448	0.572238	+	0.820088i	$0.306076\pi$
$720$	0	0
$721$	12.8060	0.476920
$722$	−33.7757	−1.25700
$723$	0	0
$724$	−3.44126	−0.127893
$725$	23.6012	0.876527
$726$	0	0
$727$	24.2941	0.901019	0.450510	−	0.892772i	$-0.351242\pi$
$727$	24.2941	0.901019	0.450510	+	0.892772i	$0.351242\pi$
$728$	13.2896	0.492546
$729$	0	0
$730$	29.0468	1.07507
$731$	23.2526	0.860030
$732$	0	0
$733$	37.0946	1.37012	0.685061	−	0.728486i	$-0.259776\pi$
$733$	37.0946	1.37012	0.685061	+	0.728486i	$0.259776\pi$
$734$	24.2397	0.894702
$735$	0	0
$736$	−21.5810	−0.795487
$737$	−26.1324	−0.962600
$738$	0	0
$739$	−24.5842	−0.904344	−0.452172	−	0.891931i	$-0.649351\pi$
$739$	−24.5842	−0.904344	−0.452172	+	0.891931i	$0.649351\pi$
$740$	−10.2395	−0.376413
$741$	0	0
$742$	66.3153	2.43451
$743$	−0.816894	−0.0299689	−0.0149845	−	0.999888i	$-0.504770\pi$
$743$	−0.816894	−0.0299689	−0.0149845	+	0.999888i	$0.504770\pi$
$744$	0	0
$745$	−3.24604	−0.118926
$746$	−13.0695	−0.478508
$747$	0	0
$748$	−42.1426	−1.54088
$749$	−35.6130	−1.30127
$750$	0	0
$751$	30.2171	1.10264	0.551320	−	0.834294i	$-0.314125\pi$
$751$	30.2171	1.10264	0.551320	+	0.834294i	$0.314125\pi$
$752$	60.0953	2.19145
$753$	0	0
$754$	24.4018	0.888663
$755$	−54.2690	−1.97505
$756$	0	0
$757$	2.38706	0.0867591	0.0433796	−	0.999059i	$-0.486188\pi$
$757$	2.38706	0.0867591	0.0433796	+	0.999059i	$0.486188\pi$
$758$	24.8548	0.902768
$759$	0	0
$760$	27.6981	1.00471
$761$	46.5455	1.68727	0.843636	−	0.536916i	$-0.180411\pi$
$761$	46.5455	1.68727	0.843636	+	0.536916i	$0.180411\pi$
$762$	0	0
$763$	−24.2937	−0.879490
$764$	10.4772	0.379051
$765$	0	0
$766$	−8.04627	−0.290724
$767$	−14.1200	−0.509844
$768$	0	0
$769$	−17.8347	−0.643137	−0.321569	−	0.946886i	$-0.604210\pi$
$769$	−17.8347	−0.643137	−0.321569	+	0.946886i	$0.604210\pi$
$770$	94.8916	3.41965
$771$	0	0
$772$	−16.2921	−0.586367
$773$	0.654674	0.0235470	0.0117735	−	0.999931i	$-0.496252\pi$
$773$	0.654674	0.0235470	0.0117735	+	0.999931i	$0.496252\pi$
$774$	0	0
$775$	−18.2445	−0.655361
$776$	−8.28822	−0.297530
$777$	0	0
$778$	−0.574994	−0.0206145
$779$	−71.6430	−2.56688
$780$	0	0
$781$	−67.3174	−2.40880
$782$	−40.2073	−1.43781
$783$	0	0
$784$	−10.0469	−0.358818
$785$	−19.8840	−0.709689
$786$	0	0
$787$	44.7561	1.59538	0.797692	−	0.603065i	$-0.206054\pi$
$787$	44.7561	1.59538	0.797692	+	0.603065i	$0.206054\pi$
$788$	−21.6329	−0.770641
$789$	0	0
$790$	56.7495	2.01906
$791$	53.7762	1.91206
$792$	0	0
$793$	−26.1820	−0.929751
$794$	−68.1001	−2.41678
$795$	0	0
$796$	3.49764	0.123971
$797$	−14.0082	−0.496197	−0.248098	−	0.968735i	$-0.579806\pi$
$797$	−14.0082	−0.496197	−0.248098	+	0.968735i	$0.579806\pi$
$798$	0	0
$799$	77.0032	2.72418
$800$	33.8512	1.19682
$801$	0	0
$802$	53.3259	1.88300
$803$	−26.9714	−0.951798
$804$	0	0
$805$	34.4319	1.21356
$806$	−18.8634	−0.664435
$807$	0	0
$808$	10.6651	0.375197
$809$	2.20651	0.0775767	0.0387883	−	0.999247i	$-0.487650\pi$
$809$	2.20651	0.0775767	0.0387883	+	0.999247i	$0.487650\pi$
$810$	0	0
$811$	30.7473	1.07968	0.539842	−	0.841767i	$-0.318484\pi$
$811$	30.7473	1.07968	0.539842	+	0.841767i	$0.318484\pi$
$812$	−15.7233	−0.551778
$813$	0	0
$814$	24.9998	0.876243
$815$	80.0690	2.80469
$816$	0	0
$817$	−22.5477	−0.788842
$818$	50.4233	1.76301
$819$	0	0
$820$	−46.4289	−1.62137
$821$	−22.3874	−0.781325	−0.390662	−	0.920534i	$-0.627754\pi$
$821$	−22.3874	−0.781325	−0.390662	+	0.920534i	$0.627754\pi$
$822$	0	0
$823$	−18.5603	−0.646971	−0.323486	−	0.946233i	$-0.604855\pi$
$823$	−18.5603	−0.646971	−0.323486	+	0.946233i	$0.604855\pi$
$824$	−5.91434	−0.206036
$825$	0	0
$826$	23.9225	0.832369
$827$	−9.83505	−0.341998	−0.170999	−	0.985271i	$-0.554700\pi$
$827$	−9.83505	−0.341998	−0.170999	+	0.985271i	$0.554700\pi$
$828$	0	0
$829$	33.7011	1.17049	0.585245	−	0.810857i	$-0.300998\pi$
$829$	33.7011	1.17049	0.585245	+	0.810857i	$0.300998\pi$
$830$	96.9378	3.36476
$831$	0	0
$832$	3.46437	0.120105
$833$	−12.8736	−0.446044
$834$	0	0
$835$	73.7542	2.55237
$836$	40.8649	1.41334
$837$	0	0
$838$	11.8759	0.410247
$839$	7.25324	0.250410	0.125205	−	0.992131i	$-0.460041\pi$
$839$	7.25324	0.250410	0.125205	+	0.992131i	$0.460041\pi$
$840$	0	0
$841$	−10.8299	−0.373445
$842$	9.68094	0.333627
$843$	0	0
$844$	−18.3794	−0.632644
$845$	−9.23905	−0.317833
$846$	0	0
$847$	−55.0558	−1.89174
$848$	−60.7830	−2.08730
$849$	0	0
$850$	63.0677	2.16321
$851$	9.07130	0.310960
$852$	0	0
$853$	36.2554	1.24136	0.620680	−	0.784064i	$-0.286857\pi$
$853$	36.2554	1.24136	0.620680	+	0.784064i	$0.286857\pi$
$854$	44.3583	1.51791
$855$	0	0
$856$	16.4476	0.562166
$857$	54.1903	1.85111	0.925553	−	0.378617i	$-0.123600\pi$
$857$	54.1903	1.85111	0.925553	+	0.378617i	$0.123600\pi$
$858$	0	0
$859$	25.9967	0.886994	0.443497	−	0.896276i	$-0.353738\pi$
$859$	25.9967	0.886994	0.443497	+	0.896276i	$0.353738\pi$
$860$	−14.6122	−0.498273
$861$	0	0
$862$	67.6787	2.30515
$863$	26.6869	0.908433	0.454217	−	0.890891i	$-0.349919\pi$
$863$	26.6869	0.908433	0.454217	+	0.890891i	$0.349919\pi$
$864$	0	0
$865$	10.1875	0.346387
$866$	53.6815	1.82417
$867$	0	0
$868$	12.1546	0.412553
$869$	−52.6947	−1.78755
$870$	0	0
$871$	15.3781	0.521067
$872$	11.2198	0.379952
$873$	0	0
$874$	38.9883	1.31880
$875$	−5.23585	−0.177004
$876$	0	0
$877$	18.4017	0.621382	0.310691	−	0.950511i	$-0.399439\pi$
$877$	18.4017	0.621382	0.310691	+	0.950511i	$0.399439\pi$
$878$	21.8632	0.737848
$879$	0	0
$880$	−86.9752	−2.93193
$881$	−24.8003	−0.835543	−0.417772	−	0.908552i	$-0.637189\pi$
$881$	−24.8003	−0.835543	−0.417772	+	0.908552i	$0.637189\pi$
$882$	0	0
$883$	2.76747	0.0931329	0.0465665	−	0.998915i	$-0.485172\pi$
$883$	2.76747	0.0931329	0.0465665	+	0.998915i	$0.485172\pi$
$884$	24.7996	0.834100
$885$	0	0
$886$	−32.4993	−1.09183
$887$	47.3313	1.58923	0.794615	−	0.607113i	$-0.207673\pi$
$887$	47.3313	1.58923	0.794615	+	0.607113i	$0.207673\pi$
$888$	0	0
$889$	−36.5136	−1.22463
$890$	−6.36207	−0.213257
$891$	0	0
$892$	13.3387	0.446613
$893$	−74.6686	−2.49869
$894$	0	0
$895$	−13.1207	−0.438576
$896$	30.8759	1.03149
$897$	0	0
$898$	65.5462	2.18731
$899$	−14.0460	−0.468462
$900$	0	0
$901$	−77.8843	−2.59470
$902$	113.356	3.77435
$903$	0	0
$904$	−24.8361	−0.826036
$905$	−9.10038	−0.302507
$906$	0	0
$907$	−2.93823	−0.0975622	−0.0487811	−	0.998809i	$-0.515534\pi$
$907$	−2.93823	−0.0975622	−0.0487811	+	0.998809i	$0.515534\pi$
$908$	24.6666	0.818591
$909$	0	0
$910$	−55.8407	−1.85110
$911$	−0.995826	−0.0329932	−0.0164966	−	0.999864i	$-0.505251\pi$
$911$	−0.995826	−0.0329932	−0.0164966	+	0.999864i	$0.505251\pi$
$912$	0	0
$913$	−90.0116	−2.97895
$914$	47.8871	1.58397
$915$	0	0
$916$	−30.2420	−0.999223
$917$	24.6613	0.814388
$918$	0	0
$919$	−5.44303	−0.179549	−0.0897745	−	0.995962i	$-0.528615\pi$
$919$	−5.44303	−0.179549	−0.0897745	+	0.995962i	$0.528615\pi$
$920$	−15.9021	−0.524276
$921$	0	0
$922$	−3.85558	−0.126977
$923$	39.6142	1.30392
$924$	0	0
$925$	−14.2289	−0.467844
$926$	−22.4831	−0.738839
$927$	0	0
$928$	26.0613	0.855506
$929$	11.4355	0.375187	0.187594	−	0.982247i	$-0.439931\pi$
$929$	11.4355	0.375187	0.187594	+	0.982247i	$0.439931\pi$
$930$	0	0
$931$	12.4833	0.409124
$932$	−9.05605	−0.296641
$933$	0	0
$934$	−39.8351	−1.30344
$935$	−111.446	−3.64467
$936$	0	0
$937$	52.0648	1.70088	0.850442	−	0.526069i	$-0.176335\pi$
$937$	52.0648	1.70088	0.850442	+	0.526069i	$0.176335\pi$
$938$	−26.0540	−0.850693
$939$	0	0
$940$	−48.3897	−1.57830
$941$	−15.0057	−0.489171	−0.244585	−	0.969628i	$-0.578652\pi$
$941$	−15.0057	−0.489171	−0.244585	+	0.969628i	$0.578652\pi$
$942$	0	0
$943$	41.1318	1.33944
$944$	−21.9267	−0.713654
$945$	0	0
$946$	35.6757	1.15992
$947$	−49.5745	−1.61095	−0.805477	−	0.592627i	$-0.798091\pi$
$947$	−49.5745	−1.61095	−0.805477	+	0.592627i	$0.798091\pi$
$948$	0	0
$949$	15.8718	0.515220
$950$	−61.1556	−1.98415
$951$	0	0
$952$	26.4436	0.857041
$953$	32.1918	1.04280	0.521398	−	0.853314i	$-0.325411\pi$
$953$	32.1918	1.04280	0.521398	+	0.853314i	$0.325411\pi$
$954$	0	0
$955$	27.7068	0.896571
$956$	26.4573	0.855692
$957$	0	0
$958$	−9.79424	−0.316438
$959$	3.51518	0.113511
$960$	0	0
$961$	−20.1420	−0.649741
$962$	−14.7116	−0.474321
$963$	0	0
$964$	36.0145	1.15995
$965$	−43.0845	−1.38694
$966$	0	0
$967$	13.6198	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$967$	13.6198	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$968$	25.4271	0.817257
$969$	0	0
$970$	34.8257	1.11819
$971$	56.5980	1.81632	0.908158	−	0.418627i	$-0.137489\pi$
$971$	56.5980	1.81632	0.908158	+	0.418627i	$0.137489\pi$
$972$	0	0
$973$	−44.0914	−1.41351
$974$	−8.36795	−0.268126
$975$	0	0
$976$	−40.6576	−1.30142
$977$	2.07020	0.0662315	0.0331158	−	0.999452i	$-0.489457\pi$
$977$	2.07020	0.0662315	0.0331158	+	0.999452i	$0.489457\pi$
$978$	0	0
$979$	5.90750	0.188804
$980$	8.08993	0.258423
$981$	0	0
$982$	57.3297	1.82947
$983$	37.2092	1.18679	0.593394	−	0.804912i	$-0.297787\pi$
$983$	37.2092	1.18679	0.593394	+	0.804912i	$0.297787\pi$
$984$	0	0
$985$	−57.2082	−1.82280
$986$	48.5546	1.54629
$987$	0	0
$988$	−24.0477	−0.765059
$989$	12.9451	0.411630
$990$	0	0
$991$	−46.1909	−1.46730	−0.733651	−	0.679526i	$-0.762185\pi$
$991$	−46.1909	−1.46730	−0.733651	+	0.679526i	$0.762185\pi$
$992$	−20.1462	−0.639644
$993$	0	0
$994$	−67.1154	−2.12877
$995$	9.24949	0.293228
$996$	0	0
$997$	−48.9611	−1.55061	−0.775307	−	0.631584i	$-0.782405\pi$
$997$	−48.9611	−1.55061	−0.775307	+	0.631584i	$0.782405\pi$
$998$	9.28096	0.293784
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.7	✓	25
3.2	odd	2		4023.2.a.f.1.19	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.7	✓	25	1.1	even	1	trivial
4023.2.a.f.1.19	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)

\(f(q)\)	\(=\)	\(q-1.79652 q^{2} +1.22747 q^{4} +3.24604 q^{5} -3.00506 q^{7} +1.38786 q^{8} +O(q^{10})\)	\(q-1.79652 q^{2} +1.22747 q^{4} +3.24604 q^{5} -3.00506 q^{7} +1.38786 q^{8} -5.83156 q^{10} +5.41489 q^{11} -3.18649 q^{13} +5.39864 q^{14} -4.94826 q^{16} -6.34045 q^{17} +6.14822 q^{19} +3.98441 q^{20} -9.72794 q^{22} -3.52983 q^{23} +5.53676 q^{25} +5.72459 q^{26} -3.68862 q^{28} +4.26264 q^{29} -3.29515 q^{31} +6.11390 q^{32} +11.3907 q^{34} -9.75454 q^{35} -2.56990 q^{37} -11.0454 q^{38} +4.50505 q^{40} -11.6526 q^{41} -3.66734 q^{43} +6.64662 q^{44} +6.34140 q^{46} -12.1447 q^{47} +2.03039 q^{49} -9.94688 q^{50} -3.91133 q^{52} +12.2837 q^{53} +17.5769 q^{55} -4.17061 q^{56} -7.65790 q^{58} +4.43120 q^{59} +8.21656 q^{61} +5.91980 q^{62} -1.08720 q^{64} -10.3435 q^{65} -4.82603 q^{67} -7.78272 q^{68} +17.5242 q^{70} -12.4319 q^{71} -4.98096 q^{73} +4.61686 q^{74} +7.54676 q^{76} -16.2721 q^{77} -9.73144 q^{79} -16.0622 q^{80} +20.9341 q^{82} -16.6230 q^{83} -20.5814 q^{85} +6.58844 q^{86} +7.51512 q^{88} +1.09097 q^{89} +9.57561 q^{91} -4.33276 q^{92} +21.8182 q^{94} +19.9574 q^{95} -5.97193 q^{97} -3.64763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 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

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