LMFDB - Embedded newform 4023.2.a.f.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.31586 q^{2} +3.36319 q^{4} +3.59358 q^{5} -2.32336 q^{7} -3.15696 q^{8} +O(q^{10})$	\(q-2.31586 q^{2} +3.36319 q^{4} +3.59358 q^{5} -2.32336 q^{7} -3.15696 q^{8} -8.32222 q^{10} -4.55753 q^{11} +4.40585 q^{13} +5.38056 q^{14} +0.584684 q^{16} +7.70558 q^{17} +5.86870 q^{19} +12.0859 q^{20} +10.5546 q^{22} +2.38305 q^{23} +7.91381 q^{25} -10.2033 q^{26} -7.81389 q^{28} +9.40291 q^{29} -2.61071 q^{31} +4.95988 q^{32} -17.8450 q^{34} -8.34916 q^{35} -2.44274 q^{37} -13.5911 q^{38} -11.3448 q^{40} +5.40833 q^{41} -7.83824 q^{43} -15.3278 q^{44} -5.51880 q^{46} -3.31884 q^{47} -1.60202 q^{49} -18.3273 q^{50} +14.8177 q^{52} -4.63380 q^{53} -16.3778 q^{55} +7.33474 q^{56} -21.7758 q^{58} +11.0777 q^{59} -1.60532 q^{61} +6.04604 q^{62} -12.6557 q^{64} +15.8328 q^{65} -8.69763 q^{67} +25.9153 q^{68} +19.3355 q^{70} +11.8869 q^{71} -2.10774 q^{73} +5.65704 q^{74} +19.7376 q^{76} +10.5888 q^{77} +7.44801 q^{79} +2.10111 q^{80} -12.5249 q^{82} -8.85699 q^{83} +27.6906 q^{85} +18.1522 q^{86} +14.3879 q^{88} -10.8952 q^{89} -10.2364 q^{91} +8.01465 q^{92} +7.68597 q^{94} +21.0896 q^{95} -0.682104 q^{97} +3.71005 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.31586	−1.63756	−0.818779	−	0.574109i	$-0.805349\pi$
$2$	−2.31586	−1.63756	−0.818779	+	0.574109i	$0.805349\pi$
$3$	0	0
$3$	0	0
$4$	3.36319	1.68160
$5$	3.59358	1.60710	0.803549	−	0.595239i	$-0.202943\pi$
$5$	3.59358	1.60710	0.803549	+	0.595239i	$0.202943\pi$
$6$	0	0
$7$	−2.32336	−0.878146	−0.439073	−	0.898451i	$-0.644693\pi$
$7$	−2.32336	−0.878146	−0.439073	+	0.898451i	$0.644693\pi$
$8$	−3.15696	−1.11615
$9$	0	0
$10$	−8.32222	−2.63172
$11$	−4.55753	−1.37415	−0.687073	−	0.726588i	$-0.741105\pi$
$11$	−4.55753	−1.37415	−0.687073	+	0.726588i	$0.741105\pi$
$12$	0	0
$13$	4.40585	1.22196	0.610982	−	0.791645i	$-0.290775\pi$
$13$	4.40585	1.22196	0.610982	+	0.791645i	$0.290775\pi$
$14$	5.38056	1.43801
$15$	0	0
$16$	0.584684	0.146171
$17$	7.70558	1.86888	0.934439	−	0.356125i	$-0.115902\pi$
$17$	7.70558	1.86888	0.934439	+	0.356125i	$0.115902\pi$
$18$	0	0
$19$	5.86870	1.34637	0.673186	−	0.739473i	$-0.264925\pi$
$19$	5.86870	1.34637	0.673186	+	0.739473i	$0.264925\pi$
$20$	12.0859	2.70249
$21$	0	0
$22$	10.5546	2.25024
$23$	2.38305	0.496900	0.248450	−	0.968645i	$-0.420079\pi$
$23$	2.38305	0.496900	0.248450	+	0.968645i	$0.420079\pi$
$24$	0	0
$25$	7.91381	1.58276
$26$	−10.2033	−2.00104
$27$	0	0
$28$	−7.81389	−1.47669
$29$	9.40291	1.74608	0.873039	−	0.487651i	$-0.162146\pi$
$29$	9.40291	1.74608	0.873039	+	0.487651i	$0.162146\pi$
$30$	0	0
$31$	−2.61071	−0.468898	−0.234449	−	0.972128i	$-0.575329\pi$
$31$	−2.61071	−0.468898	−0.234449	+	0.972128i	$0.575329\pi$
$32$	4.95988	0.876791
$33$	0	0
$34$	−17.8450	−3.06039
$35$	−8.34916	−1.41127
$36$	0	0
$37$	−2.44274	−0.401585	−0.200792	−	0.979634i	$-0.564352\pi$
$37$	−2.44274	−0.401585	−0.200792	+	0.979634i	$0.564352\pi$
$38$	−13.5911	−2.20476
$39$	0	0
$40$	−11.3448	−1.79377
$41$	5.40833	0.844640	0.422320	−	0.906447i	$-0.361216\pi$
$41$	5.40833	0.844640	0.422320	+	0.906447i	$0.361216\pi$
$42$	0	0
$43$	−7.83824	−1.19532	−0.597660	−	0.801750i	$-0.703903\pi$
$43$	−7.83824	−1.19532	−0.597660	+	0.801750i	$0.703903\pi$
$44$	−15.3278	−2.31076
$45$	0	0
$46$	−5.51880	−0.813702
$47$	−3.31884	−0.484103	−0.242052	−	0.970263i	$-0.577820\pi$
$47$	−3.31884	−0.484103	−0.242052	+	0.970263i	$0.577820\pi$
$48$	0	0
$49$	−1.60202	−0.228860
$50$	−18.3273	−2.59186
$51$	0	0
$52$	14.8177	2.05485
$53$	−4.63380	−0.636502	−0.318251	−	0.948006i	$-0.603095\pi$
$53$	−4.63380	−0.636502	−0.318251	+	0.948006i	$0.603095\pi$
$54$	0	0
$55$	−16.3778	−2.20839
$56$	7.33474	0.980146
$57$	0	0
$58$	−21.7758	−2.85930
$59$	11.0777	1.44219	0.721094	−	0.692837i	$-0.243640\pi$
$59$	11.0777	1.44219	0.721094	+	0.692837i	$0.243640\pi$
$60$	0	0
$61$	−1.60532	−0.205541	−0.102770	−	0.994705i	$-0.532771\pi$
$61$	−1.60532	−0.205541	−0.102770	+	0.994705i	$0.532771\pi$
$62$	6.04604	0.767848
$63$	0	0
$64$	−12.6557	−1.58197
$65$	15.8328	1.96381
$66$	0	0
$67$	−8.69763	−1.06258	−0.531292	−	0.847189i	$-0.678293\pi$
$67$	−8.69763	−1.06258	−0.531292	+	0.847189i	$0.678293\pi$
$68$	25.9153	3.14270
$69$	0	0
$70$	19.3355	2.31103
$71$	11.8869	1.41072	0.705360	−	0.708849i	$-0.250785\pi$
$71$	11.8869	1.41072	0.705360	+	0.708849i	$0.250785\pi$
$72$	0	0
$73$	−2.10774	−0.246693	−0.123346	−	0.992364i	$-0.539363\pi$
$73$	−2.10774	−0.246693	−0.123346	+	0.992364i	$0.539363\pi$
$74$	5.65704	0.657618
$75$	0	0
$76$	19.7376	2.26406
$77$	10.5888	1.20670
$78$	0	0
$79$	7.44801	0.837967	0.418983	−	0.907994i	$-0.362387\pi$
$79$	7.44801	0.837967	0.418983	+	0.907994i	$0.362387\pi$
$80$	2.10111	0.234911
$81$	0	0
$82$	−12.5249	−1.38315
$83$	−8.85699	−0.972181	−0.486091	−	0.873908i	$-0.661578\pi$
$83$	−8.85699	−0.972181	−0.486091	+	0.873908i	$0.661578\pi$
$84$	0	0
$85$	27.6906	3.00347
$86$	18.1522	1.95741
$87$	0	0
$88$	14.3879	1.53376
$89$	−10.8952	−1.15488	−0.577442	−	0.816431i	$-0.695949\pi$
$89$	−10.8952	−1.15488	−0.577442	+	0.816431i	$0.695949\pi$
$90$	0	0
$91$	−10.2364	−1.07306
$92$	8.01465	0.835585
$93$	0	0
$94$	7.68597	0.792747
$95$	21.0896	2.16375
$96$	0	0
$97$	−0.682104	−0.0692572	−0.0346286	−	0.999400i	$-0.511025\pi$
$97$	−0.682104	−0.0692572	−0.0346286	+	0.999400i	$0.511025\pi$
$98$	3.71005	0.374771
$99$	0	0
$100$	26.6157	2.66157
$101$	−17.0610	−1.69764	−0.848819	−	0.528684i	$-0.822686\pi$
$101$	−17.0610	−1.69764	−0.848819	+	0.528684i	$0.822686\pi$
$102$	0	0
$103$	0.827931	0.0815785	0.0407892	−	0.999168i	$-0.487013\pi$
$103$	0.827931	0.0815785	0.0407892	+	0.999168i	$0.487013\pi$
$104$	−13.9091	−1.36390
$105$	0	0
$106$	10.7312	1.04231
$107$	17.6190	1.70329	0.851644	−	0.524120i	$-0.175606\pi$
$107$	17.6190	1.70329	0.851644	+	0.524120i	$0.175606\pi$
$108$	0	0
$109$	−4.77118	−0.456996	−0.228498	−	0.973544i	$-0.573381\pi$
$109$	−4.77118	−0.456996	−0.228498	+	0.973544i	$0.573381\pi$
$110$	37.9287	3.61636
$111$	0	0
$112$	−1.35843	−0.128360
$113$	9.60110	0.903195	0.451598	−	0.892222i	$-0.350854\pi$
$113$	9.60110	0.903195	0.451598	+	0.892222i	$0.350854\pi$
$114$	0	0
$115$	8.56367	0.798566
$116$	31.6238	2.93620
$117$	0	0
$118$	−25.6543	−2.36167
$119$	−17.9028	−1.64115
$120$	0	0
$121$	9.77106	0.888278
$122$	3.71770	0.336585
$123$	0	0
$124$	−8.78034	−0.788498
$125$	10.4710	0.936555
$126$	0	0
$127$	−1.72283	−0.152876	−0.0764381	−	0.997074i	$-0.524355\pi$
$127$	−1.72283	−0.152876	−0.0764381	+	0.997074i	$0.524355\pi$
$128$	19.3891	1.71377
$129$	0	0
$130$	−36.6664	−3.21586
$131$	−15.0106	−1.31148	−0.655742	−	0.754985i	$-0.727644\pi$
$131$	−15.0106	−1.31148	−0.655742	+	0.754985i	$0.727644\pi$
$132$	0	0
$133$	−13.6351	−1.18231
$134$	20.1425	1.74004
$135$	0	0
$136$	−24.3262	−2.08596
$137$	22.2416	1.90022	0.950112	−	0.311908i	$-0.100968\pi$
$137$	22.2416	1.90022	0.950112	+	0.311908i	$0.100968\pi$
$138$	0	0
$139$	−21.2231	−1.80012	−0.900060	−	0.435765i	$-0.856478\pi$
$139$	−21.2231	−1.80012	−0.900060	+	0.435765i	$0.856478\pi$
$140$	−28.0798	−2.37318
$141$	0	0
$142$	−27.5285	−2.31014
$143$	−20.0798	−1.67916
$144$	0	0
$145$	33.7901	2.80612
$146$	4.88124	0.403974
$147$	0	0
$148$	−8.21542	−0.675303
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	16.0827	1.30879	0.654396	−	0.756152i	$-0.272923\pi$
$151$	16.0827	1.30879	0.654396	+	0.756152i	$0.272923\pi$
$152$	−18.5273	−1.50276
$153$	0	0
$154$	−24.5220	−1.97604
$155$	−9.38181	−0.753565
$156$	0	0
$157$	19.5962	1.56395	0.781973	−	0.623313i	$-0.214214\pi$
$157$	19.5962	1.56395	0.781973	+	0.623313i	$0.214214\pi$
$158$	−17.2485	−1.37222
$159$	0	0
$160$	17.8237	1.40909
$161$	−5.53667	−0.436350
$162$	0	0
$163$	7.94114	0.621998	0.310999	−	0.950410i	$-0.399336\pi$
$163$	7.94114	0.621998	0.310999	+	0.950410i	$0.399336\pi$
$164$	18.1893	1.42034
$165$	0	0
$166$	20.5115	1.59200
$167$	−5.95995	−0.461195	−0.230598	−	0.973049i	$-0.574068\pi$
$167$	−5.95995	−0.461195	−0.230598	+	0.973049i	$0.574068\pi$
$168$	0	0
$169$	6.41152	0.493194
$170$	−64.1275	−4.91835
$171$	0	0
$172$	−26.3615	−2.01005
$173$	−15.3908	−1.17014	−0.585069	−	0.810984i	$-0.698933\pi$
$173$	−15.3908	−1.17014	−0.585069	+	0.810984i	$0.698933\pi$
$174$	0	0
$175$	−18.3866	−1.38990
$176$	−2.66471	−0.200860
$177$	0	0
$178$	25.2316	1.89119
$179$	19.9945	1.49446	0.747231	−	0.664565i	$-0.231383\pi$
$179$	19.9945	1.49446	0.747231	+	0.664565i	$0.231383\pi$
$180$	0	0
$181$	2.99259	0.222438	0.111219	−	0.993796i	$-0.464525\pi$
$181$	2.99259	0.222438	0.111219	+	0.993796i	$0.464525\pi$
$182$	23.7059	1.75720
$183$	0	0
$184$	−7.52319	−0.554617
$185$	−8.77819	−0.645385
$186$	0	0
$187$	−35.1184	−2.56811
$188$	−11.1619	−0.814066
$189$	0	0
$190$	−48.8406	−3.54327
$191$	−5.49432	−0.397555	−0.198777	−	0.980045i	$-0.563697\pi$
$191$	−5.49432	−0.397555	−0.198777	+	0.980045i	$0.563697\pi$
$192$	0	0
$193$	20.0051	1.44000	0.719998	−	0.693976i	$-0.244143\pi$
$193$	20.0051	1.44000	0.719998	+	0.693976i	$0.244143\pi$
$194$	1.57966	0.113413
$195$	0	0
$196$	−5.38790	−0.384850
$197$	6.01395	0.428476	0.214238	−	0.976781i	$-0.431273\pi$
$197$	6.01395	0.428476	0.214238	+	0.976781i	$0.431273\pi$
$198$	0	0
$199$	−9.77245	−0.692751	−0.346375	−	0.938096i	$-0.612588\pi$
$199$	−9.77245	−0.692751	−0.346375	+	0.938096i	$0.612588\pi$
$200$	−24.9836	−1.76661
$201$	0	0
$202$	39.5109	2.77998
$203$	−21.8463	−1.53331
$204$	0	0
$205$	19.4353	1.35742
$206$	−1.91737	−0.133590
$207$	0	0
$208$	2.57603	0.178616
$209$	−26.7468	−1.85011
$210$	0	0
$211$	−18.1444	−1.24911	−0.624555	−	0.780981i	$-0.714720\pi$
$211$	−18.1444	−1.24911	−0.624555	+	0.780981i	$0.714720\pi$
$212$	−15.5844	−1.07034
$213$	0	0
$214$	−40.8030	−2.78923
$215$	−28.1673	−1.92100
$216$	0	0
$217$	6.06562	0.411761
$218$	11.0494	0.748358
$219$	0	0
$220$	−55.0818	−3.71362
$221$	33.9496	2.28370
$222$	0	0
$223$	2.39060	0.160086	0.0800432	−	0.996791i	$-0.474494\pi$
$223$	2.39060	0.160086	0.0800432	+	0.996791i	$0.474494\pi$
$224$	−11.5236	−0.769950
$225$	0	0
$226$	−22.2348	−1.47903
$227$	14.5189	0.963651	0.481826	−	0.876267i	$-0.339974\pi$
$227$	14.5189	0.963651	0.481826	+	0.876267i	$0.339974\pi$
$228$	0	0
$229$	−4.81430	−0.318138	−0.159069	−	0.987267i	$-0.550849\pi$
$229$	−4.81430	−0.318138	−0.159069	+	0.987267i	$0.550849\pi$
$230$	−19.8322	−1.30770
$231$	0	0
$232$	−29.6846	−1.94889
$233$	−5.61764	−0.368024	−0.184012	−	0.982924i	$-0.558908\pi$
$233$	−5.61764	−0.368024	−0.184012	+	0.982924i	$0.558908\pi$
$234$	0	0
$235$	−11.9265	−0.778001
$236$	37.2563	2.42518
$237$	0	0
$238$	41.4603	2.68747
$239$	−10.2624	−0.663817	−0.331909	−	0.943312i	$-0.607693\pi$
$239$	−10.2624	−0.663817	−0.331909	+	0.943312i	$0.607693\pi$
$240$	0	0
$241$	6.55471	0.422226	0.211113	−	0.977462i	$-0.432291\pi$
$241$	6.55471	0.422226	0.211113	+	0.977462i	$0.432291\pi$
$242$	−22.6284	−1.45461
$243$	0	0
$244$	−5.39901	−0.345636
$245$	−5.75698	−0.367800
$246$	0	0
$247$	25.8566	1.64522
$248$	8.24193	0.523363
$249$	0	0
$250$	−24.2494	−1.53366
$251$	12.4505	0.785869	0.392934	−	0.919566i	$-0.371460\pi$
$251$	12.4505	0.785869	0.392934	+	0.919566i	$0.371460\pi$
$252$	0	0
$253$	−10.8608	−0.682813
$254$	3.98982	0.250344
$255$	0	0
$256$	−19.5910	−1.22443
$257$	−18.7476	−1.16944	−0.584722	−	0.811233i	$-0.698797\pi$
$257$	−18.7476	−1.16944	−0.584722	+	0.811233i	$0.698797\pi$
$258$	0	0
$259$	5.67536	0.352650
$260$	53.2487	3.30234
$261$	0	0
$262$	34.7624	2.14763
$263$	30.5678	1.88489	0.942445	−	0.334360i	$-0.108520\pi$
$263$	30.5678	1.88489	0.942445	+	0.334360i	$0.108520\pi$
$264$	0	0
$265$	−16.6519	−1.02292
$266$	31.5769	1.93610
$267$	0	0
$268$	−29.2518	−1.78684
$269$	18.3118	1.11649	0.558245	−	0.829676i	$-0.311475\pi$
$269$	18.3118	1.11649	0.558245	+	0.829676i	$0.311475\pi$
$270$	0	0
$271$	12.7578	0.774981	0.387490	−	0.921874i	$-0.373342\pi$
$271$	12.7578	0.774981	0.387490	+	0.921874i	$0.373342\pi$
$272$	4.50533	0.273176
$273$	0	0
$274$	−51.5083	−3.11173
$275$	−36.0674	−2.17495
$276$	0	0
$277$	−0.937002	−0.0562990	−0.0281495	−	0.999604i	$-0.508961\pi$
$277$	−0.937002	−0.0562990	−0.0281495	+	0.999604i	$0.508961\pi$
$278$	49.1497	2.94780
$279$	0	0
$280$	26.3580	1.57519
$281$	−3.29164	−0.196363	−0.0981814	−	0.995169i	$-0.531303\pi$
$281$	−3.29164	−0.196363	−0.0981814	+	0.995169i	$0.531303\pi$
$282$	0	0
$283$	1.28351	0.0762966	0.0381483	−	0.999272i	$-0.487854\pi$
$283$	1.28351	0.0762966	0.0381483	+	0.999272i	$0.487854\pi$
$284$	39.9781	2.37226
$285$	0	0
$286$	46.5019	2.74972
$287$	−12.5655	−0.741717
$288$	0	0
$289$	42.3759	2.49270
$290$	−78.2531	−4.59518
$291$	0	0
$292$	−7.08875	−0.414838
$293$	−5.41606	−0.316409	−0.158205	−	0.987406i	$-0.550571\pi$
$293$	−5.41606	−0.316409	−0.158205	+	0.987406i	$0.550571\pi$
$294$	0	0
$295$	39.8084	2.31774
$296$	7.71165	0.448230
$297$	0	0
$298$	−2.31586	−0.134154
$299$	10.4994	0.607193
$300$	0	0
$301$	18.2110	1.04967
$302$	−37.2452	−2.14322
$303$	0	0
$304$	3.43134	0.196801
$305$	−5.76886	−0.330324
$306$	0	0
$307$	−3.02642	−0.172727	−0.0863635	−	0.996264i	$-0.527525\pi$
$307$	−3.02642	−0.172727	−0.0863635	+	0.996264i	$0.527525\pi$
$308$	35.6120	2.02918
$309$	0	0
$310$	21.7269	1.23401
$311$	−4.60545	−0.261151	−0.130576	−	0.991438i	$-0.541683\pi$
$311$	−4.60545	−0.261151	−0.130576	+	0.991438i	$0.541683\pi$
$312$	0	0
$313$	11.8549	0.670080	0.335040	−	0.942204i	$-0.391250\pi$
$313$	11.8549	0.670080	0.335040	+	0.942204i	$0.391250\pi$
$314$	−45.3819	−2.56105
$315$	0	0
$316$	25.0491	1.40912
$317$	5.57979	0.313392	0.156696	−	0.987647i	$-0.449916\pi$
$317$	5.57979	0.313392	0.156696	+	0.987647i	$0.449916\pi$
$318$	0	0
$319$	−42.8540	−2.39937
$320$	−45.4794	−2.54238
$321$	0	0
$322$	12.8221	0.714549
$323$	45.2217	2.51621
$324$	0	0
$325$	34.8671	1.93408
$326$	−18.3905	−1.01856
$327$	0	0
$328$	−17.0739	−0.942748
$329$	7.71085	0.425113
$330$	0	0
$331$	6.13785	0.337367	0.168683	−	0.985670i	$-0.446048\pi$
$331$	6.13785	0.337367	0.168683	+	0.985670i	$0.446048\pi$
$332$	−29.7878	−1.63482
$333$	0	0
$334$	13.8024	0.755234
$335$	−31.2556	−1.70768
$336$	0	0
$337$	−0.214563	−0.0116880	−0.00584400	−	0.999983i	$-0.501860\pi$
$337$	−0.214563	−0.0116880	−0.00584400	+	0.999983i	$0.501860\pi$
$338$	−14.8482	−0.807633
$339$	0	0
$340$	93.1289	5.05062
$341$	11.8984	0.644335
$342$	0	0
$343$	19.9855	1.07912
$344$	24.7450	1.33416
$345$	0	0
$346$	35.6428	1.91617
$347$	7.89072	0.423596	0.211798	−	0.977313i	$-0.432068\pi$
$347$	7.89072	0.423596	0.211798	+	0.977313i	$0.432068\pi$
$348$	0	0
$349$	−18.0684	−0.967179	−0.483590	−	0.875295i	$-0.660667\pi$
$349$	−18.0684	−0.967179	−0.483590	+	0.875295i	$0.660667\pi$
$350$	42.5807	2.27604
$351$	0	0
$352$	−22.6048	−1.20484
$353$	−6.82910	−0.363476	−0.181738	−	0.983347i	$-0.558172\pi$
$353$	−6.82910	−0.363476	−0.181738	+	0.983347i	$0.558172\pi$
$354$	0	0
$355$	42.7167	2.26717
$356$	−36.6425	−1.94205
$357$	0	0
$358$	−46.3045	−2.44727
$359$	13.4409	0.709382	0.354691	−	0.934983i	$-0.384586\pi$
$359$	13.4409	0.709382	0.354691	+	0.934983i	$0.384586\pi$
$360$	0	0
$361$	15.4417	0.812720
$362$	−6.93042	−0.364255
$363$	0	0
$364$	−34.4269	−1.80446
$365$	−7.57435	−0.396459
$366$	0	0
$367$	36.1852	1.88885	0.944425	−	0.328727i	$-0.106620\pi$
$367$	36.1852	1.88885	0.944425	+	0.328727i	$0.106620\pi$
$368$	1.39333	0.0726324
$369$	0	0
$370$	20.3290	1.05686
$371$	10.7660	0.558942
$372$	0	0
$373$	−15.0351	−0.778487	−0.389244	−	0.921135i	$-0.627264\pi$
$373$	−15.0351	−0.778487	−0.389244	+	0.921135i	$0.627264\pi$
$374$	81.3291	4.20543
$375$	0	0
$376$	10.4775	0.540334
$377$	41.4278	2.13364
$378$	0	0
$379$	20.8823	1.07265	0.536326	−	0.844011i	$-0.319812\pi$
$379$	20.8823	1.07265	0.536326	+	0.844011i	$0.319812\pi$
$380$	70.9286	3.63856
$381$	0	0
$382$	12.7241	0.651019
$383$	−5.31915	−0.271796	−0.135898	−	0.990723i	$-0.543392\pi$
$383$	−5.31915	−0.271796	−0.135898	+	0.990723i	$0.543392\pi$
$384$	0	0
$385$	38.0515	1.93929
$386$	−46.3289	−2.35808
$387$	0	0
$388$	−2.29405	−0.116463
$389$	−37.1272	−1.88242	−0.941212	−	0.337817i	$-0.890311\pi$
$389$	−37.1272	−1.88242	−0.941212	+	0.337817i	$0.890311\pi$
$390$	0	0
$391$	18.3628	0.928645
$392$	5.05751	0.255443
$393$	0	0
$394$	−13.9274	−0.701654
$395$	26.7650	1.34669
$396$	0	0
$397$	−12.6692	−0.635850	−0.317925	−	0.948116i	$-0.602986\pi$
$397$	−12.6692	−0.635850	−0.317925	+	0.948116i	$0.602986\pi$
$398$	22.6316	1.13442
$399$	0	0
$400$	4.62708	0.231354
$401$	−21.5555	−1.07643	−0.538216	−	0.842807i	$-0.680902\pi$
$401$	−21.5555	−1.07643	−0.538216	+	0.842807i	$0.680902\pi$
$402$	0	0
$403$	−11.5024	−0.572976
$404$	−57.3796	−2.85474
$405$	0	0
$406$	50.5929	2.51088
$407$	11.1329	0.551836
$408$	0	0
$409$	−5.71506	−0.282591	−0.141296	−	0.989967i	$-0.545127\pi$
$409$	−5.71506	−0.282591	−0.141296	+	0.989967i	$0.545127\pi$
$410$	−45.0093	−2.22285
$411$	0	0
$412$	2.78449	0.137182
$413$	−25.7373	−1.26645
$414$	0	0
$415$	−31.8283	−1.56239
$416$	21.8525	1.07141
$417$	0	0
$418$	61.9417	3.02967
$419$	−38.3475	−1.87340	−0.936698	−	0.350138i	$-0.886135\pi$
$419$	−38.3475	−1.87340	−0.936698	+	0.350138i	$0.886135\pi$
$420$	0	0
$421$	31.2030	1.52074	0.760371	−	0.649489i	$-0.225017\pi$
$421$	31.2030	1.52074	0.760371	+	0.649489i	$0.225017\pi$
$422$	42.0198	2.04549
$423$	0	0
$424$	14.6287	0.710435
$425$	60.9805	2.95799
$426$	0	0
$427$	3.72974	0.180495
$428$	59.2560	2.86424
$429$	0	0
$430$	65.2315	3.14574
$431$	0.901384	0.0434181	0.0217091	−	0.999764i	$-0.493089\pi$
$431$	0.901384	0.0434181	0.0217091	+	0.999764i	$0.493089\pi$
$432$	0	0
$433$	−32.1947	−1.54718	−0.773589	−	0.633688i	$-0.781540\pi$
$433$	−32.1947	−1.54718	−0.773589	+	0.633688i	$0.781540\pi$
$434$	−14.0471	−0.674283
$435$	0	0
$436$	−16.0464	−0.768483
$437$	13.9854	0.669012
$438$	0	0
$439$	−0.589792	−0.0281492	−0.0140746	−	0.999901i	$-0.504480\pi$
$439$	−0.589792	−0.0281492	−0.0140746	+	0.999901i	$0.504480\pi$
$440$	51.7042	2.46490
$441$	0	0
$442$	−78.6225	−3.73969
$443$	9.24759	0.439366	0.219683	−	0.975571i	$-0.429498\pi$
$443$	9.24759	0.439366	0.219683	+	0.975571i	$0.429498\pi$
$444$	0	0
$445$	−39.1526	−1.85601
$446$	−5.53629	−0.262151
$447$	0	0
$448$	29.4038	1.38920
$449$	25.3402	1.19588	0.597938	−	0.801542i	$-0.295987\pi$
$449$	25.3402	1.19588	0.597938	+	0.801542i	$0.295987\pi$
$450$	0	0
$451$	−24.6486	−1.16066
$452$	32.2903	1.51881
$453$	0	0
$454$	−33.6236	−1.57804
$455$	−36.7852	−1.72451
$456$	0	0
$457$	21.1041	0.987206	0.493603	−	0.869687i	$-0.335680\pi$
$457$	21.1041	0.987206	0.493603	+	0.869687i	$0.335680\pi$
$458$	11.1492	0.520970
$459$	0	0
$460$	28.8013	1.34287
$461$	−3.77363	−0.175755	−0.0878776	−	0.996131i	$-0.528008\pi$
$461$	−3.77363	−0.175755	−0.0878776	+	0.996131i	$0.528008\pi$
$462$	0	0
$463$	32.7736	1.52312	0.761559	−	0.648096i	$-0.224435\pi$
$463$	32.7736	1.52312	0.761559	+	0.648096i	$0.224435\pi$
$464$	5.49773	0.255226
$465$	0	0
$466$	13.0096	0.602660
$467$	22.1999	1.02729	0.513646	−	0.858002i	$-0.328295\pi$
$467$	22.1999	1.02729	0.513646	+	0.858002i	$0.328295\pi$
$468$	0	0
$469$	20.2077	0.933104
$470$	27.6201	1.27402
$471$	0	0
$472$	−34.9717	−1.60970
$473$	35.7230	1.64254
$474$	0	0
$475$	46.4438	2.13099
$476$	−60.2106	−2.75975
$477$	0	0
$478$	23.7662	1.08704
$479$	−38.0409	−1.73813	−0.869066	−	0.494696i	$-0.835280\pi$
$479$	−38.0409	−1.73813	−0.869066	+	0.494696i	$0.835280\pi$
$480$	0	0
$481$	−10.7624	−0.490721
$482$	−15.1798	−0.691420
$483$	0	0
$484$	32.8620	1.49373
$485$	−2.45120	−0.111303
$486$	0	0
$487$	8.71224	0.394789	0.197395	−	0.980324i	$-0.436752\pi$
$487$	8.71224	0.394789	0.197395	+	0.980324i	$0.436752\pi$
$488$	5.06794	0.229415
$489$	0	0
$490$	13.3323	0.602294
$491$	−29.3427	−1.32422	−0.662109	−	0.749408i	$-0.730338\pi$
$491$	−29.3427	−1.32422	−0.662109	+	0.749408i	$0.730338\pi$
$492$	0	0
$493$	72.4549	3.26320
$494$	−59.8803	−2.69414
$495$	0	0
$496$	−1.52644	−0.0685393
$497$	−27.6176	−1.23882
$498$	0	0
$499$	−14.8716	−0.665745	−0.332872	−	0.942972i	$-0.608018\pi$
$499$	−14.8716	−0.665745	−0.332872	+	0.942972i	$0.608018\pi$
$500$	35.2160	1.57491
$501$	0	0
$502$	−28.8336	−1.28691
$503$	−14.5015	−0.646591	−0.323295	−	0.946298i	$-0.604791\pi$
$503$	−14.5015	−0.646591	−0.323295	+	0.946298i	$0.604791\pi$
$504$	0	0
$505$	−61.3102	−2.72827
$506$	25.1521	1.11815
$507$	0	0
$508$	−5.79420	−0.257076
$509$	35.0490	1.55352	0.776759	−	0.629798i	$-0.216862\pi$
$509$	35.0490	1.55352	0.776759	+	0.629798i	$0.216862\pi$
$510$	0	0
$511$	4.89704	0.216632
$512$	6.59161	0.291311
$513$	0	0
$514$	43.4168	1.91503
$515$	2.97524	0.131105
$516$	0	0
$517$	15.1257	0.665228
$518$	−13.1433	−0.577485
$519$	0	0
$520$	−49.9835	−2.19192
$521$	24.1933	1.05993	0.529963	−	0.848021i	$-0.322206\pi$
$521$	24.1933	1.05993	0.529963	+	0.848021i	$0.322206\pi$
$522$	0	0
$523$	−23.8054	−1.04094	−0.520469	−	0.853880i	$-0.674243\pi$
$523$	−23.8054	−1.04094	−0.520469	+	0.853880i	$0.674243\pi$
$524$	−50.4836	−2.20539
$525$	0	0
$526$	−70.7906	−3.08662
$527$	−20.1171	−0.876313
$528$	0	0
$529$	−17.3211	−0.753091
$530$	38.5635	1.67509
$531$	0	0
$532$	−45.8574	−1.98817
$533$	23.8283	1.03212
$534$	0	0
$535$	63.3151	2.73735
$536$	27.4581	1.18601
$537$	0	0
$538$	−42.4075	−1.82832
$539$	7.30125	0.314487
$540$	0	0
$541$	−33.4591	−1.43852	−0.719260	−	0.694741i	$-0.755519\pi$
$541$	−33.4591	−1.43852	−0.719260	+	0.694741i	$0.755519\pi$
$542$	−29.5452	−1.26908
$543$	0	0
$544$	38.2187	1.63861
$545$	−17.1456	−0.734437
$546$	0	0
$547$	−9.37799	−0.400974	−0.200487	−	0.979696i	$-0.564252\pi$
$547$	−9.37799	−0.400974	−0.200487	+	0.979696i	$0.564252\pi$
$548$	74.8027	3.19541
$549$	0	0
$550$	83.5270	3.56160
$551$	55.1829	2.35087
$552$	0	0
$553$	−17.3044	−0.735857
$554$	2.16996	0.0921929
$555$	0	0
$556$	−71.3774	−3.02708
$557$	17.8896	0.758007	0.379003	−	0.925395i	$-0.376267\pi$
$557$	17.8896	0.758007	0.379003	+	0.925395i	$0.376267\pi$
$558$	0	0
$559$	−34.5341	−1.46064
$560$	−4.88162	−0.206286
$561$	0	0
$562$	7.62297	0.321556
$563$	41.2079	1.73671	0.868353	−	0.495946i	$-0.165179\pi$
$563$	41.2079	1.73671	0.868353	+	0.495946i	$0.165179\pi$
$564$	0	0
$565$	34.5023	1.45152
$566$	−2.97242	−0.124940
$567$	0	0
$568$	−37.5266	−1.57458
$569$	31.5897	1.32431	0.662154	−	0.749367i	$-0.269642\pi$
$569$	31.5897	1.32431	0.662154	+	0.749367i	$0.269642\pi$
$570$	0	0
$571$	−7.80228	−0.326515	−0.163258	−	0.986583i	$-0.552200\pi$
$571$	−7.80228	−0.326515	−0.163258	+	0.986583i	$0.552200\pi$
$572$	−67.5322	−2.82366
$573$	0	0
$574$	29.0999	1.21460
$575$	18.8590	0.786474
$576$	0	0
$577$	−1.48685	−0.0618982	−0.0309491	−	0.999521i	$-0.509853\pi$
$577$	−1.48685	−0.0618982	−0.0309491	+	0.999521i	$0.509853\pi$
$578$	−98.1366	−4.08194
$579$	0	0
$580$	113.643	4.71876
$581$	20.5779	0.853717
$582$	0	0
$583$	21.1187	0.874647
$584$	6.65407	0.275347
$585$	0	0
$586$	12.5428	0.518139
$587$	−1.98321	−0.0818560	−0.0409280	−	0.999162i	$-0.513031\pi$
$587$	−1.98321	−0.0818560	−0.0409280	+	0.999162i	$0.513031\pi$
$588$	0	0
$589$	−15.3215	−0.631312
$590$	−92.1906	−3.79543
$591$	0	0
$592$	−1.42823	−0.0587000
$593$	31.9620	1.31252	0.656262	−	0.754533i	$-0.272137\pi$
$593$	31.9620	1.31252	0.656262	+	0.754533i	$0.272137\pi$
$594$	0	0
$595$	−64.3351	−2.63748
$596$	3.36319	0.137762
$597$	0	0
$598$	−24.3150	−0.994314
$599$	−29.7135	−1.21406	−0.607031	−	0.794678i	$-0.707640\pi$
$599$	−29.7135	−1.21406	−0.607031	+	0.794678i	$0.707640\pi$
$600$	0	0
$601$	−7.80184	−0.318244	−0.159122	−	0.987259i	$-0.550866\pi$
$601$	−7.80184	−0.318244	−0.159122	+	0.987259i	$0.550866\pi$
$602$	−42.1741	−1.71889
$603$	0	0
$604$	54.0892	2.20086
$605$	35.1131	1.42755
$606$	0	0
$607$	−49.2642	−1.99957	−0.999785	−	0.0207189i	$-0.993404\pi$
$607$	−49.2642	−1.99957	−0.999785	+	0.0207189i	$0.993404\pi$
$608$	29.1081	1.18049
$609$	0	0
$610$	13.3598	0.540924
$611$	−14.6223	−0.591556
$612$	0	0
$613$	41.1370	1.66151	0.830753	−	0.556641i	$-0.187910\pi$
$613$	41.1370	1.66151	0.830753	+	0.556641i	$0.187910\pi$
$614$	7.00876	0.282850
$615$	0	0
$616$	−33.4283	−1.34686
$617$	49.1931	1.98044	0.990220	−	0.139517i	$-0.0445551\pi$
$617$	49.1931	1.98044	0.990220	+	0.139517i	$0.0445551\pi$
$618$	0	0
$619$	28.2396	1.13504	0.567522	−	0.823358i	$-0.307902\pi$
$619$	28.2396	1.13504	0.567522	+	0.823358i	$0.307902\pi$
$620$	−31.5528	−1.26719
$621$	0	0
$622$	10.6656	0.427650
$623$	25.3133	1.01416
$624$	0	0
$625$	−1.94066	−0.0776263
$626$	−27.4543	−1.09729
$627$	0	0
$628$	65.9057	2.62993
$629$	−18.8227	−0.750512
$630$	0	0
$631$	6.25233	0.248901	0.124451	−	0.992226i	$-0.460283\pi$
$631$	6.25233	0.248901	0.124451	+	0.992226i	$0.460283\pi$
$632$	−23.5131	−0.935301
$633$	0	0
$634$	−12.9220	−0.513198
$635$	−6.19111	−0.245687
$636$	0	0
$637$	−7.05826	−0.279658
$638$	99.2438	3.92910
$639$	0	0
$640$	69.6763	2.75420
$641$	−37.9688	−1.49968	−0.749838	−	0.661621i	$-0.769869\pi$
$641$	−37.9688	−1.49968	−0.749838	+	0.661621i	$0.769869\pi$
$642$	0	0
$643$	−4.45240	−0.175585	−0.0877927	−	0.996139i	$-0.527981\pi$
$643$	−4.45240	−0.175585	−0.0877927	+	0.996139i	$0.527981\pi$
$644$	−18.6209	−0.733766
$645$	0	0
$646$	−104.727	−4.12043
$647$	−1.84325	−0.0724658	−0.0362329	−	0.999343i	$-0.511536\pi$
$647$	−1.84325	−0.0724658	−0.0362329	+	0.999343i	$0.511536\pi$
$648$	0	0
$649$	−50.4867	−1.98178
$650$	−80.7471	−3.16716
$651$	0	0
$652$	26.7076	1.04595
$653$	26.4371	1.03456	0.517282	−	0.855815i	$-0.326944\pi$
$653$	26.4371	1.03456	0.517282	+	0.855815i	$0.326944\pi$
$654$	0	0
$655$	−53.9418	−2.10768
$656$	3.16217	0.123462
$657$	0	0
$658$	−17.8572	−0.696147
$659$	10.0388	0.391056	0.195528	−	0.980698i	$-0.437358\pi$
$659$	10.0388	0.391056	0.195528	+	0.980698i	$0.437358\pi$
$660$	0	0
$661$	14.6170	0.568535	0.284268	−	0.958745i	$-0.408250\pi$
$661$	14.6170	0.568535	0.284268	+	0.958745i	$0.408250\pi$
$662$	−14.2144	−0.552458
$663$	0	0
$664$	27.9612	1.08510
$665$	−48.9988	−1.90009
$666$	0	0
$667$	22.4076	0.867625
$668$	−20.0445	−0.775544
$669$	0	0
$670$	72.3835	2.79642
$671$	7.31631	0.282443
$672$	0	0
$673$	−39.8304	−1.53535	−0.767674	−	0.640841i	$-0.778586\pi$
$673$	−39.8304	−1.53535	−0.767674	+	0.640841i	$0.778586\pi$
$674$	0.496898	0.0191398
$675$	0	0
$676$	21.5632	0.829353
$677$	2.42460	0.0931849	0.0465924	−	0.998914i	$-0.485164\pi$
$677$	2.42460	0.0931849	0.0465924	+	0.998914i	$0.485164\pi$
$678$	0	0
$679$	1.58477	0.0608179
$680$	−87.4182	−3.35233
$681$	0	0
$682$	−27.5550	−1.05514
$683$	−36.2895	−1.38858	−0.694290	−	0.719695i	$-0.744282\pi$
$683$	−36.2895	−1.38858	−0.694290	+	0.719695i	$0.744282\pi$
$684$	0	0
$685$	79.9268	3.05385
$686$	−46.2837	−1.76712
$687$	0	0
$688$	−4.58289	−0.174721
$689$	−20.4159	−0.777782
$690$	0	0
$691$	−34.1349	−1.29855	−0.649276	−	0.760552i	$-0.724928\pi$
$691$	−34.1349	−1.29855	−0.649276	+	0.760552i	$0.724928\pi$
$692$	−51.7621	−1.96770
$693$	0	0
$694$	−18.2738	−0.693663
$695$	−76.2669	−2.89297
$696$	0	0
$697$	41.6743	1.57853
$698$	41.8438	1.58381
$699$	0	0
$700$	−61.8377	−2.33724
$701$	−43.6048	−1.64693	−0.823464	−	0.567368i	$-0.807962\pi$
$701$	−43.6048	−1.64693	−0.823464	+	0.567368i	$0.807962\pi$
$702$	0	0
$703$	−14.3357	−0.540682
$704$	57.6789	2.17385
$705$	0	0
$706$	15.8152	0.595214
$707$	39.6389	1.49077
$708$	0	0
$709$	37.9963	1.42698	0.713491	−	0.700664i	$-0.247113\pi$
$709$	37.9963	1.42698	0.713491	+	0.700664i	$0.247113\pi$
$710$	−98.9257	−3.71262
$711$	0	0
$712$	34.3956	1.28903
$713$	−6.22146	−0.232995
$714$	0	0
$715$	−72.1583	−2.69857
$716$	67.2455	2.51308
$717$	0	0
$718$	−31.1272	−1.16166
$719$	48.2893	1.80089	0.900444	−	0.434972i	$-0.143242\pi$
$719$	48.2893	1.80089	0.900444	+	0.434972i	$0.143242\pi$
$720$	0	0
$721$	−1.92358	−0.0716378
$722$	−35.7607	−1.33088
$723$	0	0
$724$	10.0647	0.374051
$725$	74.4129	2.76362
$726$	0	0
$727$	−19.4164	−0.720113	−0.360056	−	0.932931i	$-0.617243\pi$
$727$	−19.4164	−0.720113	−0.360056	+	0.932931i	$0.617243\pi$
$728$	32.3158	1.19770
$729$	0	0
$730$	17.5411	0.649225
$731$	−60.3982	−2.23391
$732$	0	0
$733$	−43.6806	−1.61338	−0.806691	−	0.590974i	$-0.798744\pi$
$733$	−43.6806	−1.61338	−0.806691	+	0.590974i	$0.798744\pi$
$734$	−83.7997	−3.09310
$735$	0	0
$736$	11.8196	0.435677
$737$	39.6397	1.46015
$738$	0	0
$739$	13.0590	0.480384	0.240192	−	0.970725i	$-0.422790\pi$
$739$	13.0590	0.480384	0.240192	+	0.970725i	$0.422790\pi$
$740$	−29.5228	−1.08528
$741$	0	0
$742$	−24.9325	−0.915299
$743$	20.6818	0.758743	0.379371	−	0.925245i	$-0.376140\pi$
$743$	20.6818	0.758743	0.379371	+	0.925245i	$0.376140\pi$
$744$	0	0
$745$	3.59358	0.131659
$746$	34.8191	1.27482
$747$	0	0
$748$	−118.110	−4.31853
$749$	−40.9351	−1.49574
$750$	0	0
$751$	4.97281	0.181460	0.0907302	−	0.995876i	$-0.471080\pi$
$751$	4.97281	0.181460	0.0907302	+	0.995876i	$0.471080\pi$
$752$	−1.94048	−0.0707619
$753$	0	0
$754$	−95.9409	−3.49396
$755$	57.7945	2.10336
$756$	0	0
$757$	−29.1413	−1.05916	−0.529579	−	0.848261i	$-0.677650\pi$
$757$	−29.1413	−1.05916	−0.529579	+	0.848261i	$0.677650\pi$
$758$	−48.3604	−1.75653
$759$	0	0
$760$	−66.5792	−2.41508
$761$	−1.80581	−0.0654604	−0.0327302	−	0.999464i	$-0.510420\pi$
$761$	−1.80581	−0.0654604	−0.0327302	+	0.999464i	$0.510420\pi$
$762$	0	0
$763$	11.0851	0.401309
$764$	−18.4785	−0.668527
$765$	0	0
$766$	12.3184	0.445082
$767$	48.8065	1.76230
$768$	0	0
$769$	40.6230	1.46490	0.732452	−	0.680818i	$-0.238376\pi$
$769$	40.6230	1.46490	0.732452	+	0.680818i	$0.238376\pi$
$770$	−88.1219	−3.17569
$771$	0	0
$772$	67.2809	2.42149
$773$	−6.15597	−0.221415	−0.110708	−	0.993853i	$-0.535312\pi$
$773$	−6.15597	−0.221415	−0.110708	+	0.993853i	$0.535312\pi$
$774$	0	0
$775$	−20.6607	−0.742154
$776$	2.15338	0.0773017
$777$	0	0
$778$	85.9813	3.08258
$779$	31.7399	1.13720
$780$	0	0
$781$	−54.1751	−1.93854
$782$	−42.5255	−1.52071
$783$	0	0
$784$	−0.936675	−0.0334527
$785$	70.4204	2.51341
$786$	0	0
$787$	−7.59356	−0.270681	−0.135341	−	0.990799i	$-0.543213\pi$
$787$	−7.59356	−0.270681	−0.135341	+	0.990799i	$0.543213\pi$
$788$	20.2261	0.720524
$789$	0	0
$790$	−61.9840	−2.20529
$791$	−22.3068	−0.793137
$792$	0	0
$793$	−7.07282	−0.251163
$794$	29.3401	1.04124
$795$	0	0
$796$	−32.8666	−1.16493
$797$	−44.0689	−1.56100	−0.780500	−	0.625156i	$-0.785035\pi$
$797$	−44.0689	−1.56100	−0.780500	+	0.625156i	$0.785035\pi$
$798$	0	0
$799$	−25.5736	−0.904729
$800$	39.2515	1.38775
$801$	0	0
$802$	49.9196	1.76272
$803$	9.60610	0.338992
$804$	0	0
$805$	−19.8965	−0.701258
$806$	26.6380	0.938282
$807$	0	0
$808$	53.8611	1.89483
$809$	−9.76414	−0.343289	−0.171645	−	0.985159i	$-0.554908\pi$
$809$	−9.76414	−0.343289	−0.171645	+	0.985159i	$0.554908\pi$
$810$	0	0
$811$	−10.9585	−0.384804	−0.192402	−	0.981316i	$-0.561628\pi$
$811$	−10.9585	−0.384804	−0.192402	+	0.981316i	$0.561628\pi$
$812$	−73.4734	−2.57841
$813$	0	0
$814$	−25.7821	−0.903663
$815$	28.5371	0.999611
$816$	0	0
$817$	−46.0003	−1.60935
$818$	13.2353	0.462760
$819$	0	0
$820$	65.3646	2.28263
$821$	7.87455	0.274824	0.137412	−	0.990514i	$-0.456122\pi$
$821$	7.87455	0.274824	0.137412	+	0.990514i	$0.456122\pi$
$822$	0	0
$823$	12.6038	0.439341	0.219671	−	0.975574i	$-0.429502\pi$
$823$	12.6038	0.439341	0.219671	+	0.975574i	$0.429502\pi$
$824$	−2.61375	−0.0910542
$825$	0	0
$826$	59.6040	2.07389
$827$	43.8873	1.52611	0.763055	−	0.646333i	$-0.223698\pi$
$827$	43.8873	1.52611	0.763055	+	0.646333i	$0.223698\pi$
$828$	0	0
$829$	−41.1989	−1.43090	−0.715449	−	0.698665i	$-0.753778\pi$
$829$	−41.1989	−1.43090	−0.715449	+	0.698665i	$0.753778\pi$
$830$	73.7098	2.55850
$831$	0	0
$832$	−55.7593	−1.93311
$833$	−12.3445	−0.427711
$834$	0	0
$835$	−21.4176	−0.741185
$836$	−89.9546	−3.11114
$837$	0	0
$838$	88.8072	3.06779
$839$	−17.9604	−0.620063	−0.310031	−	0.950726i	$-0.600340\pi$
$839$	−17.9604	−0.620063	−0.310031	+	0.950726i	$0.600340\pi$
$840$	0	0
$841$	59.4148	2.04878
$842$	−72.2617	−2.49030
$843$	0	0
$844$	−61.0231	−2.10050
$845$	23.0403	0.792610
$846$	0	0
$847$	−22.7016	−0.780038
$848$	−2.70931	−0.0930382
$849$	0	0
$850$	−141.222	−4.84388
$851$	−5.82117	−0.199547
$852$	0	0
$853$	−15.2591	−0.522460	−0.261230	−	0.965277i	$-0.584128\pi$
$853$	−15.2591	−0.522460	−0.261230	+	0.965277i	$0.584128\pi$
$854$	−8.63754	−0.295570
$855$	0	0
$856$	−55.6224	−1.90113
$857$	51.2214	1.74969	0.874845	−	0.484404i	$-0.160963\pi$
$857$	51.2214	1.74969	0.874845	+	0.484404i	$0.160963\pi$
$858$	0	0
$859$	−11.6665	−0.398054	−0.199027	−	0.979994i	$-0.563778\pi$
$859$	−11.6665	−0.398054	−0.199027	+	0.979994i	$0.563778\pi$
$860$	−94.7322	−3.23034
$861$	0	0
$862$	−2.08748	−0.0710997
$863$	1.43045	0.0486932	0.0243466	−	0.999704i	$-0.492249\pi$
$863$	1.43045	0.0486932	0.0243466	+	0.999704i	$0.492249\pi$
$864$	0	0
$865$	−55.3079	−1.88053
$866$	74.5583	2.53359
$867$	0	0
$868$	20.3998	0.692416
$869$	−33.9445	−1.15149
$870$	0	0
$871$	−38.3204	−1.29844
$872$	15.0624	0.510078
$873$	0	0
$874$	−32.3882	−1.09555
$875$	−24.3279	−0.822432
$876$	0	0
$877$	13.0011	0.439014	0.219507	−	0.975611i	$-0.429555\pi$
$877$	13.0011	0.439014	0.219507	+	0.975611i	$0.429555\pi$
$878$	1.36587	0.0460960
$879$	0	0
$880$	−9.57586	−0.322802
$881$	−31.0124	−1.04483	−0.522417	−	0.852690i	$-0.674970\pi$
$881$	−31.0124	−1.04483	−0.522417	+	0.852690i	$0.674970\pi$
$882$	0	0
$883$	11.7017	0.393792	0.196896	−	0.980424i	$-0.436914\pi$
$883$	11.7017	0.393792	0.196896	+	0.980424i	$0.436914\pi$
$884$	114.179	3.84026
$885$	0	0
$886$	−21.4161	−0.719488
$887$	36.4964	1.22543	0.612714	−	0.790305i	$-0.290078\pi$
$887$	36.4964	1.22543	0.612714	+	0.790305i	$0.290078\pi$
$888$	0	0
$889$	4.00274	0.134248
$890$	90.6719	3.03933
$891$	0	0
$892$	8.04005	0.269201
$893$	−19.4773	−0.651783
$894$	0	0
$895$	71.8519	2.40175
$896$	−45.0478	−1.50494
$897$	0	0
$898$	−58.6842	−1.95832
$899$	−24.5483	−0.818732
$900$	0	0
$901$	−35.7061	−1.18954
$902$	57.0827	1.90065
$903$	0	0
$904$	−30.3103	−1.00811
$905$	10.7541	0.357479
$906$	0	0
$907$	9.21352	0.305930	0.152965	−	0.988232i	$-0.451118\pi$
$907$	9.21352	0.305930	0.152965	+	0.988232i	$0.451118\pi$
$908$	48.8298	1.62047
$909$	0	0
$910$	85.1892	2.82399
$911$	6.84427	0.226761	0.113380	−	0.993552i	$-0.463832\pi$
$911$	6.84427	0.226761	0.113380	+	0.993552i	$0.463832\pi$
$912$	0	0
$913$	40.3660	1.33592
$914$	−48.8740	−1.61661
$915$	0	0
$916$	−16.1914	−0.534980
$917$	34.8750	1.15167
$918$	0	0
$919$	8.70539	0.287164	0.143582	−	0.989638i	$-0.454138\pi$
$919$	8.70539	0.287164	0.143582	+	0.989638i	$0.454138\pi$
$920$	−27.0352	−0.891323
$921$	0	0
$922$	8.73918	0.287809
$923$	52.3721	1.72385
$924$	0	0
$925$	−19.3314	−0.635613
$926$	−75.8989	−2.49419
$927$	0	0
$928$	46.6373	1.53094
$929$	−11.7696	−0.386149	−0.193074	−	0.981184i	$-0.561846\pi$
$929$	−11.7696	−0.386149	−0.193074	+	0.981184i	$0.561846\pi$
$930$	0	0
$931$	−9.40177	−0.308131
$932$	−18.8932	−0.618867
$933$	0	0
$934$	−51.4119	−1.68225
$935$	−126.201	−4.12720
$936$	0	0
$937$	2.42969	0.0793745	0.0396872	−	0.999212i	$-0.487364\pi$
$937$	2.42969	0.0793745	0.0396872	+	0.999212i	$0.487364\pi$
$938$	−46.7981	−1.52801
$939$	0	0
$940$	−40.1112	−1.30828
$941$	−34.9693	−1.13997	−0.569984	−	0.821656i	$-0.693051\pi$
$941$	−34.9693	−1.13997	−0.569984	+	0.821656i	$0.693051\pi$
$942$	0	0
$943$	12.8883	0.419701
$944$	6.47693	0.210806
$945$	0	0
$946$	−82.7293	−2.68976
$947$	23.0587	0.749306	0.374653	−	0.927165i	$-0.377762\pi$
$947$	23.0587	0.749306	0.374653	+	0.927165i	$0.377762\pi$
$948$	0	0
$949$	−9.28641	−0.301450
$950$	−107.557	−3.48962
$951$	0	0
$952$	56.5184	1.83177
$953$	25.4937	0.825821	0.412910	−	0.910772i	$-0.364512\pi$
$953$	25.4937	0.825821	0.412910	+	0.910772i	$0.364512\pi$
$954$	0	0
$955$	−19.7443	−0.638909
$956$	−34.5143	−1.11627
$957$	0	0
$958$	88.0973	2.84629
$959$	−51.6750	−1.66867
$960$	0	0
$961$	−24.1842	−0.780134
$962$	24.9241	0.803585
$963$	0	0
$964$	22.0448	0.710014
$965$	71.8898	2.31421
$966$	0	0
$967$	23.9683	0.770770	0.385385	−	0.922756i	$-0.374069\pi$
$967$	23.9683	0.770770	0.385385	+	0.922756i	$0.374069\pi$
$968$	−30.8468	−0.991455
$969$	0	0
$970$	5.67662	0.182265
$971$	46.8212	1.50257	0.751283	−	0.659981i	$-0.229436\pi$
$971$	46.8212	1.50257	0.751283	+	0.659981i	$0.229436\pi$
$972$	0	0
$973$	49.3088	1.58077
$974$	−20.1763	−0.646491
$975$	0	0
$976$	−0.938607	−0.0300441
$977$	3.99411	0.127783	0.0638915	−	0.997957i	$-0.479649\pi$
$977$	3.99411	0.127783	0.0638915	+	0.997957i	$0.479649\pi$
$978$	0	0
$979$	49.6550	1.58698
$980$	−19.3618	−0.618491
$981$	0	0
$982$	67.9535	2.16848
$983$	8.50383	0.271230	0.135615	−	0.990762i	$-0.456699\pi$
$983$	8.50383	0.271230	0.135615	+	0.990762i	$0.456699\pi$
$984$	0	0
$985$	21.6116	0.688603
$986$	−167.795	−5.34369
$987$	0	0
$988$	86.9608	2.76659
$989$	−18.6789	−0.593954
$990$	0	0
$991$	5.49089	0.174424	0.0872119	−	0.996190i	$-0.472204\pi$
$991$	5.49089	0.174424	0.0872119	+	0.996190i	$0.472204\pi$
$992$	−12.9488	−0.411126
$993$	0	0
$994$	63.9584	2.02864
$995$	−35.1181	−1.11332
$996$	0	0
$997$	43.6321	1.38184	0.690921	−	0.722931i	$-0.257205\pi$
$997$	43.6321	1.38184	0.690921	+	0.722931i	$0.257205\pi$
$998$	34.4405	1.09020
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.f.1.3	yes	25
3.2	odd	2		4023.2.a.e.1.23	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.23	✓	25	3.2	odd	2
4023.2.a.f.1.3	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)

\(f(q)\)	\(=\)	\(q-2.31586 q^{2} +3.36319 q^{4} +3.59358 q^{5} -2.32336 q^{7} -3.15696 q^{8} +O(q^{10})\)	\(q-2.31586 q^{2} +3.36319 q^{4} +3.59358 q^{5} -2.32336 q^{7} -3.15696 q^{8} -8.32222 q^{10} -4.55753 q^{11} +4.40585 q^{13} +5.38056 q^{14} +0.584684 q^{16} +7.70558 q^{17} +5.86870 q^{19} +12.0859 q^{20} +10.5546 q^{22} +2.38305 q^{23} +7.91381 q^{25} -10.2033 q^{26} -7.81389 q^{28} +9.40291 q^{29} -2.61071 q^{31} +4.95988 q^{32} -17.8450 q^{34} -8.34916 q^{35} -2.44274 q^{37} -13.5911 q^{38} -11.3448 q^{40} +5.40833 q^{41} -7.83824 q^{43} -15.3278 q^{44} -5.51880 q^{46} -3.31884 q^{47} -1.60202 q^{49} -18.3273 q^{50} +14.8177 q^{52} -4.63380 q^{53} -16.3778 q^{55} +7.33474 q^{56} -21.7758 q^{58} +11.0777 q^{59} -1.60532 q^{61} +6.04604 q^{62} -12.6557 q^{64} +15.8328 q^{65} -8.69763 q^{67} +25.9153 q^{68} +19.3355 q^{70} +11.8869 q^{71} -2.10774 q^{73} +5.65704 q^{74} +19.7376 q^{76} +10.5888 q^{77} +7.44801 q^{79} +2.10111 q^{80} -12.5249 q^{82} -8.85699 q^{83} +27.6906 q^{85} +18.1522 q^{86} +14.3879 q^{88} -10.8952 q^{89} -10.2364 q^{91} +8.01465 q^{92} +7.68597 q^{94} +21.0896 q^{95} -0.682104 q^{97} +3.71005 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) 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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