LMFDB - Embedded newform 4023.2.a.f.1.5

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			1.5
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)

$f(q)$	$=$	$q-1.77959 q^{2} +1.16693 q^{4} -0.742067 q^{5} +3.55470 q^{7} +1.48252 q^{8} +O(q^{10})$	\(q-1.77959 q^{2} +1.16693 q^{4} -0.742067 q^{5} +3.55470 q^{7} +1.48252 q^{8} +1.32057 q^{10} +2.30516 q^{11} -4.43488 q^{13} -6.32590 q^{14} -4.97213 q^{16} +1.82232 q^{17} -4.78718 q^{19} -0.865939 q^{20} -4.10223 q^{22} +3.20627 q^{23} -4.44934 q^{25} +7.89225 q^{26} +4.14809 q^{28} +10.2423 q^{29} -6.12999 q^{31} +5.88330 q^{32} -3.24297 q^{34} -2.63783 q^{35} +8.55909 q^{37} +8.51921 q^{38} -1.10013 q^{40} -2.95823 q^{41} +1.43939 q^{43} +2.68995 q^{44} -5.70583 q^{46} +0.237636 q^{47} +5.63592 q^{49} +7.91798 q^{50} -5.17518 q^{52} +0.970483 q^{53} -1.71058 q^{55} +5.26993 q^{56} -18.2271 q^{58} +3.07208 q^{59} +11.6017 q^{61} +10.9089 q^{62} -0.525572 q^{64} +3.29097 q^{65} +4.04825 q^{67} +2.12651 q^{68} +4.69424 q^{70} +9.80290 q^{71} -9.11492 q^{73} -15.2316 q^{74} -5.58630 q^{76} +8.19415 q^{77} +11.1608 q^{79} +3.68966 q^{80} +5.26442 q^{82} -14.2038 q^{83} -1.35228 q^{85} -2.56152 q^{86} +3.41745 q^{88} +0.992314 q^{89} -15.7647 q^{91} +3.74148 q^{92} -0.422893 q^{94} +3.55241 q^{95} +1.71313 q^{97} -10.0296 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.77959	−1.25836	−0.629179	−	0.777261i	$-0.716609\pi$
$2$	−1.77959	−1.25836	−0.629179	+	0.777261i	$0.716609\pi$
$3$	0	0
$3$	0	0
$4$	1.16693	0.583464
$5$	−0.742067	−0.331862	−0.165931	−	0.986137i	$-0.553063\pi$
$5$	−0.742067	−0.331862	−0.165931	+	0.986137i	$0.553063\pi$
$6$	0	0
$7$	3.55470	1.34355	0.671776	−	0.740754i	$-0.265532\pi$
$7$	3.55470	1.34355	0.671776	+	0.740754i	$0.265532\pi$
$8$	1.48252	0.524151
$9$	0	0
$10$	1.32057	0.417602
$11$	2.30516	0.695031	0.347516	−	0.937674i	$-0.387025\pi$
$11$	2.30516	0.695031	0.347516	+	0.937674i	$0.387025\pi$
$12$	0	0
$13$	−4.43488	−1.23001	−0.615007	−	0.788522i	$-0.710847\pi$
$13$	−4.43488	−1.23001	−0.615007	+	0.788522i	$0.710847\pi$
$14$	−6.32590	−1.69067
$15$	0	0
$16$	−4.97213	−1.24303
$17$	1.82232	0.441977	0.220988	−	0.975276i	$-0.429072\pi$
$17$	1.82232	0.441977	0.220988	+	0.975276i	$0.429072\pi$
$18$	0	0
$19$	−4.78718	−1.09826	−0.549128	−	0.835738i	$-0.685040\pi$
$19$	−4.78718	−1.09826	−0.549128	+	0.835738i	$0.685040\pi$
$20$	−0.865939	−0.193630
$21$	0	0
$22$	−4.10223	−0.874598
$23$	3.20627	0.668553	0.334276	−	0.942475i	$-0.391508\pi$
$23$	3.20627	0.668553	0.334276	+	0.942475i	$0.391508\pi$
$24$	0	0
$25$	−4.44934	−0.889867
$26$	7.89225	1.54780
$27$	0	0
$28$	4.14809	0.783915
$29$	10.2423	1.90195	0.950976	−	0.309266i	$-0.100083\pi$
$29$	10.2423	1.90195	0.950976	+	0.309266i	$0.100083\pi$
$30$	0	0
$31$	−6.12999	−1.10098	−0.550490	−	0.834842i	$-0.685559\pi$
$31$	−6.12999	−1.10098	−0.550490	+	0.834842i	$0.685559\pi$
$32$	5.88330	1.04003
$33$	0	0
$34$	−3.24297	−0.556165
$35$	−2.63783	−0.445874
$36$	0	0
$37$	8.55909	1.40711	0.703553	−	0.710643i	$-0.251596\pi$
$37$	8.55909	1.40711	0.703553	+	0.710643i	$0.251596\pi$
$38$	8.51921	1.38200
$39$	0	0
$40$	−1.10013	−0.173946
$41$	−2.95823	−0.461997	−0.230999	−	0.972954i	$-0.574199\pi$
$41$	−2.95823	−0.461997	−0.230999	+	0.972954i	$0.574199\pi$
$42$	0	0
$43$	1.43939	0.219505	0.109753	−	0.993959i	$-0.464994\pi$
$43$	1.43939	0.219505	0.109753	+	0.993959i	$0.464994\pi$
$44$	2.68995	0.405526
$45$	0	0
$46$	−5.70583	−0.841279
$47$	0.237636	0.0346627	0.0173314	−	0.999850i	$-0.494483\pi$
$47$	0.237636	0.0346627	0.0173314	+	0.999850i	$0.494483\pi$
$48$	0	0
$49$	5.63592	0.805132
$50$	7.91798	1.11977
$51$	0	0
$52$	−5.17518	−0.717669
$53$	0.970483	0.133306	0.0666530	−	0.997776i	$-0.478768\pi$
$53$	0.970483	0.133306	0.0666530	+	0.997776i	$0.478768\pi$
$54$	0	0
$55$	−1.71058	−0.230655
$56$	5.26993	0.704224
$57$	0	0
$58$	−18.2271	−2.39334
$59$	3.07208	0.399950	0.199975	−	0.979801i	$-0.435914\pi$
$59$	3.07208	0.399950	0.199975	+	0.979801i	$0.435914\pi$
$60$	0	0
$61$	11.6017	1.48545	0.742726	−	0.669596i	$-0.233533\pi$
$61$	11.6017	1.48545	0.742726	+	0.669596i	$0.233533\pi$
$62$	10.9089	1.38543
$63$	0	0
$64$	−0.525572	−0.0656965
$65$	3.29097	0.408195
$66$	0	0
$67$	4.04825	0.494573	0.247287	−	0.968942i	$-0.420461\pi$
$67$	4.04825	0.494573	0.247287	+	0.968942i	$0.420461\pi$
$68$	2.12651	0.257878
$69$	0	0
$70$	4.69424	0.561069
$71$	9.80290	1.16339	0.581695	−	0.813407i	$-0.302390\pi$
$71$	9.80290	1.16339	0.581695	+	0.813407i	$0.302390\pi$
$72$	0	0
$73$	−9.11492	−1.06682	−0.533410	−	0.845857i	$-0.679090\pi$
$73$	−9.11492	−1.06682	−0.533410	+	0.845857i	$0.679090\pi$
$74$	−15.2316	−1.77064
$75$	0	0
$76$	−5.58630	−0.640793
$77$	8.19415	0.933810
$78$	0	0
$79$	11.1608	1.25568	0.627842	−	0.778341i	$-0.283939\pi$
$79$	11.1608	1.25568	0.627842	+	0.778341i	$0.283939\pi$
$80$	3.68966	0.412516
$81$	0	0
$82$	5.26442	0.581358
$83$	−14.2038	−1.55907	−0.779536	−	0.626357i	$-0.784545\pi$
$83$	−14.2038	−1.55907	−0.779536	+	0.626357i	$0.784545\pi$
$84$	0	0
$85$	−1.35228	−0.146675
$86$	−2.56152	−0.276216
$87$	0	0
$88$	3.41745	0.364301
$89$	0.992314	0.105185	0.0525925	−	0.998616i	$-0.483252\pi$
$89$	0.992314	0.105185	0.0525925	+	0.998616i	$0.483252\pi$
$90$	0	0
$91$	−15.7647	−1.65259
$92$	3.74148	0.390077
$93$	0	0
$94$	−0.422893	−0.0436181
$95$	3.55241	0.364470
$96$	0	0
$97$	1.71313	0.173942	0.0869712	−	0.996211i	$-0.472281\pi$
$97$	1.71313	0.173942	0.0869712	+	0.996211i	$0.472281\pi$
$98$	−10.0296	−1.01314
$99$	0	0
$100$	−5.19206	−0.519206
$101$	−11.8413	−1.17826	−0.589129	−	0.808039i	$-0.700529\pi$
$101$	−11.8413	−1.17826	−0.589129	+	0.808039i	$0.700529\pi$
$102$	0	0
$103$	8.67023	0.854303	0.427152	−	0.904180i	$-0.359517\pi$
$103$	8.67023	0.854303	0.427152	+	0.904180i	$0.359517\pi$
$104$	−6.57480	−0.644713
$105$	0	0
$106$	−1.72706	−0.167747
$107$	−17.7261	−1.71364	−0.856822	−	0.515612i	$-0.827565\pi$
$107$	−17.7261	−1.71364	−0.856822	+	0.515612i	$0.827565\pi$
$108$	0	0
$109$	6.45751	0.618517	0.309259	−	0.950978i	$-0.399919\pi$
$109$	6.45751	0.618517	0.309259	+	0.950978i	$0.399919\pi$
$110$	3.04413	0.290246
$111$	0	0
$112$	−17.6745	−1.67008
$113$	0.498202	0.0468669	0.0234335	−	0.999725i	$-0.492540\pi$
$113$	0.498202	0.0468669	0.0234335	+	0.999725i	$0.492540\pi$
$114$	0	0
$115$	−2.37926	−0.221868
$116$	11.9521	1.10972
$117$	0	0
$118$	−5.46703	−0.503281
$119$	6.47779	0.593818
$120$	0	0
$121$	−5.68625	−0.516932
$122$	−20.6463	−1.86923
$123$	0	0
$124$	−7.15327	−0.642382
$125$	7.01204	0.627176
$126$	0	0
$127$	−13.6468	−1.21096	−0.605478	−	0.795862i	$-0.707018\pi$
$127$	−13.6468	−1.21096	−0.605478	+	0.795862i	$0.707018\pi$
$128$	−10.8313	−0.957361
$129$	0	0
$130$	−5.85657	−0.513656
$131$	17.1036	1.49435	0.747173	−	0.664629i	$-0.231411\pi$
$131$	17.1036	1.49435	0.747173	+	0.664629i	$0.231411\pi$
$132$	0	0
$133$	−17.0170	−1.47556
$134$	−7.20422	−0.622350
$135$	0	0
$136$	2.70162	0.231662
$137$	5.54101	0.473400	0.236700	−	0.971583i	$-0.423934\pi$
$137$	5.54101	0.473400	0.236700	+	0.971583i	$0.423934\pi$
$138$	0	0
$139$	−5.96224	−0.505711	−0.252855	−	0.967504i	$-0.581370\pi$
$139$	−5.96224	−0.505711	−0.252855	+	0.967504i	$0.581370\pi$
$140$	−3.07816	−0.260152
$141$	0	0
$142$	−17.4451	−1.46396
$143$	−10.2231	−0.854898
$144$	0	0
$145$	−7.60049	−0.631186
$146$	16.2208	1.34244
$147$	0	0
$148$	9.98785	0.820996
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	13.4659	1.09584	0.547918	−	0.836532i	$-0.315420\pi$
$151$	13.4659	1.09584	0.547918	+	0.836532i	$0.315420\pi$
$152$	−7.09711	−0.575652
$153$	0	0
$154$	−14.5822	−1.17507
$155$	4.54887	0.365374
$156$	0	0
$157$	−0.109651	−0.00875110	−0.00437555	−	0.999990i	$-0.501393\pi$
$157$	−0.109651	−0.00875110	−0.00437555	+	0.999990i	$0.501393\pi$
$158$	−19.8615	−1.58010
$159$	0	0
$160$	−4.36580	−0.345147
$161$	11.3973	0.898235
$162$	0	0
$163$	7.59855	0.595164	0.297582	−	0.954696i	$-0.403820\pi$
$163$	7.59855	0.595164	0.297582	+	0.954696i	$0.403820\pi$
$164$	−3.45204	−0.269559
$165$	0	0
$166$	25.2769	1.96187
$167$	14.7944	1.14482	0.572411	−	0.819967i	$-0.306008\pi$
$167$	14.7944	1.14482	0.572411	+	0.819967i	$0.306008\pi$
$168$	0	0
$169$	6.66813	0.512933
$170$	2.40650	0.184570
$171$	0	0
$172$	1.67967	0.128073
$173$	7.53301	0.572724	0.286362	−	0.958122i	$-0.407554\pi$
$173$	7.53301	0.572724	0.286362	+	0.958122i	$0.407554\pi$
$174$	0	0
$175$	−15.8161	−1.19558
$176$	−11.4616	−0.863947
$177$	0	0
$178$	−1.76591	−0.132360
$179$	7.55052	0.564352	0.282176	−	0.959363i	$-0.408944\pi$
$179$	7.55052	0.564352	0.282176	+	0.959363i	$0.408944\pi$
$180$	0	0
$181$	−23.2057	−1.72487	−0.862433	−	0.506171i	$-0.831061\pi$
$181$	−23.2057	−1.72487	−0.862433	+	0.506171i	$0.831061\pi$
$182$	28.0546	2.07955
$183$	0	0
$184$	4.75336	0.350423
$185$	−6.35142	−0.466965
$186$	0	0
$187$	4.20073	0.307187
$188$	0.277304	0.0202245
$189$	0	0
$190$	−6.32182	−0.458633
$191$	14.2601	1.03183	0.515913	−	0.856641i	$-0.327453\pi$
$191$	14.2601	1.03183	0.515913	+	0.856641i	$0.327453\pi$
$192$	0	0
$193$	17.9093	1.28914	0.644571	−	0.764544i	$-0.277036\pi$
$193$	17.9093	1.28914	0.644571	+	0.764544i	$0.277036\pi$
$194$	−3.04867	−0.218882
$195$	0	0
$196$	6.57672	0.469766
$197$	0.152616	0.0108734	0.00543671	−	0.999985i	$-0.498269\pi$
$197$	0.152616	0.0108734	0.00543671	+	0.999985i	$0.498269\pi$
$198$	0	0
$199$	10.5060	0.744753	0.372377	−	0.928082i	$-0.378543\pi$
$199$	10.5060	0.744753	0.372377	+	0.928082i	$0.378543\pi$
$200$	−6.59624	−0.466425
$201$	0	0
$202$	21.0727	1.48267
$203$	36.4084	2.55537
$204$	0	0
$205$	2.19520	0.153319
$206$	−15.4294	−1.07502
$207$	0	0
$208$	22.0508	1.52895
$209$	−11.0352	−0.763322
$210$	0	0
$211$	−10.5515	−0.726395	−0.363197	−	0.931712i	$-0.618315\pi$
$211$	−10.5515	−0.726395	−0.363197	+	0.931712i	$0.618315\pi$
$212$	1.13248	0.0777793
$213$	0	0
$214$	31.5451	2.15638
$215$	−1.06812	−0.0728455
$216$	0	0
$217$	−21.7903	−1.47922
$218$	−11.4917	−0.778316
$219$	0	0
$220$	−1.99613	−0.134579
$221$	−8.08175	−0.543637
$222$	0	0
$223$	14.1450	0.947219	0.473609	−	0.880735i	$-0.342951\pi$
$223$	14.1450	0.947219	0.473609	+	0.880735i	$0.342951\pi$
$224$	20.9134	1.39733
$225$	0	0
$226$	−0.886594	−0.0589753
$227$	19.7852	1.31319	0.656596	−	0.754243i	$-0.271996\pi$
$227$	19.7852	1.31319	0.656596	+	0.754243i	$0.271996\pi$
$228$	0	0
$229$	−1.66156	−0.109799	−0.0548996	−	0.998492i	$-0.517484\pi$
$229$	−1.66156	−0.109799	−0.0548996	+	0.998492i	$0.517484\pi$
$230$	4.23411	0.279189
$231$	0	0
$232$	15.1845	0.996909
$233$	−12.0769	−0.791184	−0.395592	−	0.918426i	$-0.629461\pi$
$233$	−12.0769	−0.791184	−0.395592	+	0.918426i	$0.629461\pi$
$234$	0	0
$235$	−0.176342	−0.0115033
$236$	3.58489	0.233357
$237$	0	0
$238$	−11.5278	−0.747236
$239$	−6.55305	−0.423882	−0.211941	−	0.977282i	$-0.567978\pi$
$239$	−6.55305	−0.423882	−0.211941	+	0.977282i	$0.567978\pi$
$240$	0	0
$241$	3.22610	0.207811	0.103906	−	0.994587i	$-0.466866\pi$
$241$	3.22610	0.207811	0.103906	+	0.994587i	$0.466866\pi$
$242$	10.1192	0.650485
$243$	0	0
$244$	13.5384	0.866708
$245$	−4.18223	−0.267193
$246$	0	0
$247$	21.2306	1.35087
$248$	−9.08786	−0.577079
$249$	0	0
$250$	−12.4785	−0.789212
$251$	−11.8777	−0.749714	−0.374857	−	0.927083i	$-0.622308\pi$
$251$	−11.8777	−0.749714	−0.374857	+	0.927083i	$0.622308\pi$
$252$	0	0
$253$	7.39095	0.464665
$254$	24.2856	1.52381
$255$	0	0
$256$	20.3264	1.27040
$257$	−16.2464	−1.01343	−0.506713	−	0.862115i	$-0.669140\pi$
$257$	−16.2464	−1.01343	−0.506713	+	0.862115i	$0.669140\pi$
$258$	0	0
$259$	30.4250	1.89052
$260$	3.84033	0.238167
$261$	0	0
$262$	−30.4373	−1.88042
$263$	25.5080	1.57289	0.786445	−	0.617660i	$-0.211919\pi$
$263$	25.5080	1.57289	0.786445	+	0.617660i	$0.211919\pi$
$264$	0	0
$265$	−0.720163	−0.0442393
$266$	30.2833	1.85679
$267$	0	0
$268$	4.72402	0.288566
$269$	29.6335	1.80679	0.903394	−	0.428811i	$-0.141067\pi$
$269$	29.6335	1.80679	0.903394	+	0.428811i	$0.141067\pi$
$270$	0	0
$271$	−18.3094	−1.11222	−0.556110	−	0.831109i	$-0.687707\pi$
$271$	−18.3094	−1.11222	−0.556110	+	0.831109i	$0.687707\pi$
$272$	−9.06080	−0.549392
$273$	0	0
$274$	−9.86070	−0.595707
$275$	−10.2564	−0.618486
$276$	0	0
$277$	10.8951	0.654624	0.327312	−	0.944916i	$-0.393857\pi$
$277$	10.8951	0.654624	0.327312	+	0.944916i	$0.393857\pi$
$278$	10.6103	0.636365
$279$	0	0
$280$	−3.91064	−0.233705
$281$	24.3158	1.45056	0.725280	−	0.688454i	$-0.241710\pi$
$281$	24.3158	1.45056	0.725280	+	0.688454i	$0.241710\pi$
$282$	0	0
$283$	−24.9895	−1.48547	−0.742734	−	0.669586i	$-0.766472\pi$
$283$	−24.9895	−1.48547	−0.742734	+	0.669586i	$0.766472\pi$
$284$	11.4393	0.678797
$285$	0	0
$286$	18.1929	1.07577
$287$	−10.5156	−0.620717
$288$	0	0
$289$	−13.6792	−0.804657
$290$	13.5257	0.794258
$291$	0	0
$292$	−10.6365	−0.622452
$293$	2.54253	0.148536	0.0742680	−	0.997238i	$-0.476338\pi$
$293$	2.54253	0.148536	0.0742680	+	0.997238i	$0.476338\pi$
$294$	0	0
$295$	−2.27969	−0.132728
$296$	12.6890	0.737536
$297$	0	0
$298$	−1.77959	−0.103089
$299$	−14.2194	−0.822329
$300$	0	0
$301$	5.11661	0.294917
$302$	−23.9637	−1.37895
$303$	0	0
$304$	23.8025	1.36517
$305$	−8.60927	−0.492965
$306$	0	0
$307$	9.66717	0.551735	0.275867	−	0.961196i	$-0.411035\pi$
$307$	9.66717	0.551735	0.275867	+	0.961196i	$0.411035\pi$
$308$	9.56199	0.544845
$309$	0	0
$310$	−8.09510	−0.459771
$311$	12.2303	0.693517	0.346758	−	0.937955i	$-0.387282\pi$
$311$	12.2303	0.693517	0.346758	+	0.937955i	$0.387282\pi$
$312$	0	0
$313$	13.1203	0.741602	0.370801	−	0.928712i	$-0.379083\pi$
$313$	13.1203	0.741602	0.370801	+	0.928712i	$0.379083\pi$
$314$	0.195133	0.0110120
$315$	0	0
$316$	13.0238	0.732646
$317$	5.10119	0.286511	0.143256	−	0.989686i	$-0.454243\pi$
$317$	5.10119	0.286511	0.143256	+	0.989686i	$0.454243\pi$
$318$	0	0
$319$	23.6102	1.32192
$320$	0.390009	0.0218022
$321$	0	0
$322$	−20.2825	−1.13030
$323$	−8.72376	−0.485403
$324$	0	0
$325$	19.7323	1.09455
$326$	−13.5223	−0.748930
$327$	0	0
$328$	−4.38564	−0.242156
$329$	0.844724	0.0465712
$330$	0	0
$331$	−29.4911	−1.62098	−0.810489	−	0.585753i	$-0.800799\pi$
$331$	−29.4911	−1.62098	−0.810489	+	0.585753i	$0.800799\pi$
$332$	−16.5749	−0.909663
$333$	0	0
$334$	−26.3279	−1.44060
$335$	−3.00408	−0.164130
$336$	0	0
$337$	22.5238	1.22695	0.613475	−	0.789714i	$-0.289771\pi$
$337$	22.5238	1.22695	0.613475	+	0.789714i	$0.289771\pi$
$338$	−11.8665	−0.645453
$339$	0	0
$340$	−1.57801	−0.0855798
$341$	−14.1306	−0.765215
$342$	0	0
$343$	−4.84890	−0.261816
$344$	2.13393	0.115054
$345$	0	0
$346$	−13.4056	−0.720692
$347$	12.2437	0.657279	0.328639	−	0.944456i	$-0.393410\pi$
$347$	12.2437	0.657279	0.328639	+	0.944456i	$0.393410\pi$
$348$	0	0
$349$	25.7654	1.37919	0.689594	−	0.724196i	$-0.257789\pi$
$349$	25.7654	1.37919	0.689594	+	0.724196i	$0.257789\pi$
$350$	28.1461	1.50447
$351$	0	0
$352$	13.5619	0.722853
$353$	0.336401	0.0179048	0.00895241	−	0.999960i	$-0.497150\pi$
$353$	0.336401	0.0179048	0.00895241	+	0.999960i	$0.497150\pi$
$354$	0	0
$355$	−7.27441	−0.386085
$356$	1.15796	0.0613717
$357$	0	0
$358$	−13.4368	−0.710157
$359$	25.7411	1.35857	0.679283	−	0.733877i	$-0.262291\pi$
$359$	25.7411	1.35857	0.679283	+	0.733877i	$0.262291\pi$
$360$	0	0
$361$	3.91714	0.206165
$362$	41.2966	2.17050
$363$	0	0
$364$	−18.3962	−0.964225
$365$	6.76388	0.354037
$366$	0	0
$367$	7.71692	0.402820	0.201410	−	0.979507i	$-0.435448\pi$
$367$	7.71692	0.402820	0.201410	+	0.979507i	$0.435448\pi$
$368$	−15.9420	−0.831034
$369$	0	0
$370$	11.3029	0.587610
$371$	3.44978	0.179104
$372$	0	0
$373$	16.6747	0.863385	0.431692	−	0.902021i	$-0.357917\pi$
$373$	16.6747	0.863385	0.431692	+	0.902021i	$0.357917\pi$
$374$	−7.47555	−0.386552
$375$	0	0
$376$	0.352300	0.0181685
$377$	−45.4234	−2.33943
$378$	0	0
$379$	8.79318	0.451675	0.225838	−	0.974165i	$-0.427488\pi$
$379$	8.79318	0.451675	0.225838	+	0.974165i	$0.427488\pi$
$380$	4.14541	0.212655
$381$	0	0
$382$	−25.3771	−1.29841
$383$	−14.7284	−0.752585	−0.376292	−	0.926501i	$-0.622801\pi$
$383$	−14.7284	−0.752585	−0.376292	+	0.926501i	$0.622801\pi$
$384$	0	0
$385$	−6.08061	−0.309896
$386$	−31.8712	−1.62220
$387$	0	0
$388$	1.99911	0.101489
$389$	0.405542	0.0205618	0.0102809	−	0.999947i	$-0.496727\pi$
$389$	0.405542	0.0205618	0.0102809	+	0.999947i	$0.496727\pi$
$390$	0	0
$391$	5.84283	0.295485
$392$	8.35538	0.422010
$393$	0	0
$394$	−0.271593	−0.0136827
$395$	−8.28203	−0.416714
$396$	0	0
$397$	0.560497	0.0281305	0.0140653	−	0.999901i	$-0.495523\pi$
$397$	0.560497	0.0281305	0.0140653	+	0.999901i	$0.495523\pi$
$398$	−18.6964	−0.937166
$399$	0	0
$400$	22.1227	1.10614
$401$	−13.2404	−0.661193	−0.330596	−	0.943772i	$-0.607250\pi$
$401$	−13.2404	−0.661193	−0.330596	+	0.943772i	$0.607250\pi$
$402$	0	0
$403$	27.1858	1.35422
$404$	−13.8180	−0.687471
$405$	0	0
$406$	−64.7919	−3.21557
$407$	19.7301	0.977982
$408$	0	0
$409$	−24.3462	−1.20384	−0.601921	−	0.798556i	$-0.705598\pi$
$409$	−24.3462	−1.20384	−0.601921	+	0.798556i	$0.705598\pi$
$410$	−3.90655	−0.192931
$411$	0	0
$412$	10.1175	0.498455
$413$	10.9203	0.537354
$414$	0	0
$415$	10.5402	0.517398
$416$	−26.0917	−1.27925
$417$	0	0
$418$	19.6381	0.960532
$419$	26.9851	1.31831	0.659155	−	0.752007i	$-0.270914\pi$
$419$	26.9851	1.31831	0.659155	+	0.752007i	$0.270914\pi$
$420$	0	0
$421$	−29.6218	−1.44368	−0.721840	−	0.692060i	$-0.756703\pi$
$421$	−29.6218	−1.44368	−0.721840	+	0.692060i	$0.756703\pi$
$422$	18.7773	0.914065
$423$	0	0
$424$	1.43876	0.0698725
$425$	−8.10810	−0.393301
$426$	0	0
$427$	41.2408	1.99578
$428$	−20.6851	−0.999851
$429$	0	0
$430$	1.90082	0.0916657
$431$	32.3255	1.55706	0.778532	−	0.627605i	$-0.215965\pi$
$431$	32.3255	1.55706	0.778532	+	0.627605i	$0.215965\pi$
$432$	0	0
$433$	−0.115024	−0.00552769	−0.00276385	−	0.999996i	$-0.500880\pi$
$433$	−0.115024	−0.00552769	−0.00276385	+	0.999996i	$0.500880\pi$
$434$	38.7778	1.86139
$435$	0	0
$436$	7.53545	0.360883
$437$	−15.3490	−0.734242
$438$	0	0
$439$	25.8814	1.23525	0.617625	−	0.786472i	$-0.288095\pi$
$439$	25.8814	1.23525	0.617625	+	0.786472i	$0.288095\pi$
$440$	−2.53597	−0.120898
$441$	0	0
$442$	14.3822	0.684090
$443$	17.3997	0.826685	0.413343	−	0.910576i	$-0.364361\pi$
$443$	17.3997	0.826685	0.413343	+	0.910576i	$0.364361\pi$
$444$	0	0
$445$	−0.736363	−0.0349070
$446$	−25.1722	−1.19194
$447$	0	0
$448$	−1.86825	−0.0882666
$449$	3.77399	0.178105	0.0890527	−	0.996027i	$-0.471616\pi$
$449$	3.77399	0.178105	0.0890527	+	0.996027i	$0.471616\pi$
$450$	0	0
$451$	−6.81917	−0.321102
$452$	0.581366	0.0273452
$453$	0	0
$454$	−35.2095	−1.65246
$455$	11.6984	0.548431
$456$	0	0
$457$	−35.4805	−1.65971	−0.829854	−	0.557980i	$-0.811577\pi$
$457$	−35.4805	−1.65971	−0.829854	+	0.557980i	$0.811577\pi$
$458$	2.95690	0.138167
$459$	0	0
$460$	−2.77643	−0.129452
$461$	−1.91538	−0.0892082	−0.0446041	−	0.999005i	$-0.514203\pi$
$461$	−1.91538	−0.0892082	−0.0446041	+	0.999005i	$0.514203\pi$
$462$	0	0
$463$	15.4831	0.719563	0.359781	−	0.933037i	$-0.382851\pi$
$463$	15.4831	0.719563	0.359781	+	0.933037i	$0.382851\pi$
$464$	−50.9262	−2.36419
$465$	0	0
$466$	21.4919	0.995593
$467$	−19.9190	−0.921740	−0.460870	−	0.887468i	$-0.652463\pi$
$467$	−19.9190	−0.921740	−0.460870	+	0.887468i	$0.652463\pi$
$468$	0	0
$469$	14.3903	0.664484
$470$	0.313815	0.0144752
$471$	0	0
$472$	4.55442	0.209634
$473$	3.31802	0.152563
$474$	0	0
$475$	21.2998	0.977302
$476$	7.55912	0.346472
$477$	0	0
$478$	11.6617	0.533395
$479$	−11.7314	−0.536020	−0.268010	−	0.963416i	$-0.586366\pi$
$479$	−11.7314	−0.536020	−0.268010	+	0.963416i	$0.586366\pi$
$480$	0	0
$481$	−37.9585	−1.73076
$482$	−5.74112	−0.261501
$483$	0	0
$484$	−6.63545	−0.301611
$485$	−1.27126	−0.0577249
$486$	0	0
$487$	16.1442	0.731562	0.365781	−	0.930701i	$-0.380802\pi$
$487$	16.1442	0.731562	0.365781	+	0.930701i	$0.380802\pi$
$488$	17.1998	0.778601
$489$	0	0
$490$	7.44264	0.336224
$491$	18.2264	0.822546	0.411273	−	0.911512i	$-0.365084\pi$
$491$	18.2264	0.822546	0.411273	+	0.911512i	$0.365084\pi$
$492$	0	0
$493$	18.6647	0.840618
$494$	−37.7816	−1.69988
$495$	0	0
$496$	30.4792	1.36855
$497$	34.8464	1.56308
$498$	0	0
$499$	10.6630	0.477339	0.238670	−	0.971101i	$-0.423289\pi$
$499$	10.6630	0.477339	0.238670	+	0.971101i	$0.423289\pi$
$500$	8.18255	0.365935
$501$	0	0
$502$	21.1374	0.943408
$503$	35.1653	1.56794	0.783971	−	0.620798i	$-0.213191\pi$
$503$	35.1653	1.56794	0.783971	+	0.620798i	$0.213191\pi$
$504$	0	0
$505$	8.78706	0.391019
$506$	−13.1528	−0.584715
$507$	0	0
$508$	−15.9248	−0.706549
$509$	−36.6079	−1.62262	−0.811308	−	0.584619i	$-0.801244\pi$
$509$	−36.6079	−1.62262	−0.811308	+	0.584619i	$0.801244\pi$
$510$	0	0
$511$	−32.4008	−1.43333
$512$	−14.5100	−0.641255
$513$	0	0
$514$	28.9120	1.27525
$515$	−6.43389	−0.283511
$516$	0	0
$517$	0.547788	0.0240917
$518$	−54.1440	−2.37895
$519$	0	0
$520$	4.87894	0.213956
$521$	20.9791	0.919113	0.459556	−	0.888149i	$-0.348008\pi$
$521$	20.9791	0.919113	0.459556	+	0.888149i	$0.348008\pi$
$522$	0	0
$523$	−20.2167	−0.884014	−0.442007	−	0.897012i	$-0.645733\pi$
$523$	−20.2167	−0.884014	−0.442007	+	0.897012i	$0.645733\pi$
$524$	19.9587	0.871898
$525$	0	0
$526$	−45.3937	−1.97926
$527$	−11.1708	−0.486607
$528$	0	0
$529$	−12.7199	−0.553037
$530$	1.28159	0.0556688
$531$	0	0
$532$	−19.8577	−0.860938
$533$	13.1194	0.568263
$534$	0	0
$535$	13.1539	0.568694
$536$	6.00163	0.259231
$537$	0	0
$538$	−52.7355	−2.27359
$539$	12.9917	0.559591
$540$	0	0
$541$	−11.1907	−0.481125	−0.240562	−	0.970634i	$-0.577332\pi$
$541$	−11.1907	−0.481125	−0.240562	+	0.970634i	$0.577332\pi$
$542$	32.5832	1.39957
$543$	0	0
$544$	10.7212	0.459669
$545$	−4.79190	−0.205263
$546$	0	0
$547$	15.4923	0.662402	0.331201	−	0.943560i	$-0.392546\pi$
$547$	15.4923	0.662402	0.331201	+	0.943560i	$0.392546\pi$
$548$	6.46596	0.276212
$549$	0	0
$550$	18.2522	0.778276
$551$	−49.0319	−2.08883
$552$	0	0
$553$	39.6732	1.68708
$554$	−19.3888	−0.823752
$555$	0	0
$556$	−6.95751	−0.295064
$557$	30.0746	1.27430	0.637151	−	0.770739i	$-0.280113\pi$
$557$	30.0746	1.27430	0.637151	+	0.770739i	$0.280113\pi$
$558$	0	0
$559$	−6.38352	−0.269994
$560$	13.1156	0.554237
$561$	0	0
$562$	−43.2721	−1.82532
$563$	−25.2042	−1.06223	−0.531114	−	0.847300i	$-0.678226\pi$
$563$	−25.2042	−1.06223	−0.531114	+	0.847300i	$0.678226\pi$
$564$	0	0
$565$	−0.369699	−0.0155534
$566$	44.4709	1.86925
$567$	0	0
$568$	14.5330	0.609792
$569$	−40.5699	−1.70078	−0.850389	−	0.526155i	$-0.823633\pi$
$569$	−40.5699	−1.70078	−0.850389	+	0.526155i	$0.823633\pi$
$570$	0	0
$571$	25.4667	1.06575	0.532873	−	0.846195i	$-0.321112\pi$
$571$	25.4667	1.06575	0.532873	+	0.846195i	$0.321112\pi$
$572$	−11.9296	−0.498802
$573$	0	0
$574$	18.7134	0.781084
$575$	−14.2658	−0.594923
$576$	0	0
$577$	37.8194	1.57444	0.787220	−	0.616672i	$-0.211519\pi$
$577$	37.8194	1.57444	0.787220	+	0.616672i	$0.211519\pi$
$578$	24.3433	1.01255
$579$	0	0
$580$	−8.86922	−0.368274
$581$	−50.4904	−2.09470
$582$	0	0
$583$	2.23712	0.0926518
$584$	−13.5131	−0.559175
$585$	0	0
$586$	−4.52464	−0.186911
$587$	−21.7071	−0.895949	−0.447975	−	0.894046i	$-0.647854\pi$
$587$	−21.7071	−0.895949	−0.447975	+	0.894046i	$0.647854\pi$
$588$	0	0
$589$	29.3454	1.20916
$590$	4.05690	0.167020
$591$	0	0
$592$	−42.5570	−1.74908
$593$	24.0810	0.988887	0.494443	−	0.869210i	$-0.335372\pi$
$593$	24.0810	0.988887	0.494443	+	0.869210i	$0.335372\pi$
$594$	0	0
$595$	−4.80696	−0.197066
$596$	1.16693	0.0477993
$597$	0	0
$598$	25.3047	1.03478
$599$	−30.4559	−1.24439	−0.622197	−	0.782861i	$-0.713760\pi$
$599$	−30.4559	−1.24439	−0.622197	+	0.782861i	$0.713760\pi$
$600$	0	0
$601$	1.26650	0.0516614	0.0258307	−	0.999666i	$-0.491777\pi$
$601$	1.26650	0.0516614	0.0258307	+	0.999666i	$0.491777\pi$
$602$	−9.10545	−0.371111
$603$	0	0
$604$	15.7137	0.639382
$605$	4.21958	0.171550
$606$	0	0
$607$	4.79571	0.194652	0.0973260	−	0.995253i	$-0.468971\pi$
$607$	4.79571	0.194652	0.0973260	+	0.995253i	$0.468971\pi$
$608$	−28.1644	−1.14222
$609$	0	0
$610$	15.3209	0.620327
$611$	−1.05388	−0.0426356
$612$	0	0
$613$	−2.82048	−0.113918	−0.0569591	−	0.998377i	$-0.518140\pi$
$613$	−2.82048	−0.113918	−0.0569591	+	0.998377i	$0.518140\pi$
$614$	−17.2036	−0.694280
$615$	0	0
$616$	12.1480	0.489458
$617$	42.4215	1.70783	0.853913	−	0.520415i	$-0.174223\pi$
$617$	42.4215	1.70783	0.853913	+	0.520415i	$0.174223\pi$
$618$	0	0
$619$	21.3634	0.858667	0.429334	−	0.903146i	$-0.358748\pi$
$619$	21.3634	0.858667	0.429334	+	0.903146i	$0.358748\pi$
$620$	5.30820	0.213183
$621$	0	0
$622$	−21.7649	−0.872692
$623$	3.52738	0.141322
$624$	0	0
$625$	17.0433	0.681731
$626$	−23.3487	−0.933201
$627$	0	0
$628$	−0.127955	−0.00510595
$629$	15.5974	0.621908
$630$	0	0
$631$	−9.10031	−0.362278	−0.181139	−	0.983458i	$-0.557978\pi$
$631$	−9.10031	−0.362278	−0.181139	+	0.983458i	$0.557978\pi$
$632$	16.5461	0.658167
$633$	0	0
$634$	−9.07801	−0.360534
$635$	10.1268	0.401870
$636$	0	0
$637$	−24.9946	−0.990323
$638$	−42.0163	−1.66344
$639$	0	0
$640$	8.03755	0.317712
$641$	43.0625	1.70087	0.850433	−	0.526084i	$-0.176340\pi$
$641$	43.0625	1.70087	0.850433	+	0.526084i	$0.176340\pi$
$642$	0	0
$643$	30.9401	1.22016	0.610079	−	0.792341i	$-0.291138\pi$
$643$	30.9401	1.22016	0.610079	+	0.792341i	$0.291138\pi$
$644$	13.2999	0.524088
$645$	0	0
$646$	15.5247	0.610811
$647$	−26.5417	−1.04346	−0.521730	−	0.853111i	$-0.674713\pi$
$647$	−26.5417	−1.04346	−0.521730	+	0.853111i	$0.674713\pi$
$648$	0	0
$649$	7.08162	0.277978
$650$	−35.1153	−1.37733
$651$	0	0
$652$	8.86697	0.347257
$653$	−40.5607	−1.58726	−0.793632	−	0.608398i	$-0.791813\pi$
$653$	−40.5607	−1.58726	−0.793632	+	0.608398i	$0.791813\pi$
$654$	0	0
$655$	−12.6920	−0.495917
$656$	14.7087	0.574278
$657$	0	0
$658$	−1.50326	−0.0586032
$659$	−0.288065	−0.0112214	−0.00561071	−	0.999984i	$-0.501786\pi$
$659$	−0.288065	−0.0112214	−0.00561071	+	0.999984i	$0.501786\pi$
$660$	0	0
$661$	5.01045	0.194884	0.0974420	−	0.995241i	$-0.468934\pi$
$661$	5.01045	0.194884	0.0974420	+	0.995241i	$0.468934\pi$
$662$	52.4820	2.03977
$663$	0	0
$664$	−21.0575	−0.817189
$665$	12.6278	0.489684
$666$	0	0
$667$	32.8396	1.27155
$668$	17.2640	0.667963
$669$	0	0
$670$	5.34601	0.206534
$671$	26.7438	1.03243
$672$	0	0
$673$	16.4351	0.633527	0.316764	−	0.948505i	$-0.397404\pi$
$673$	16.4351	0.633527	0.316764	+	0.948505i	$0.397404\pi$
$674$	−40.0831	−1.54394
$675$	0	0
$676$	7.78123	0.299278
$677$	15.0178	0.577182	0.288591	−	0.957452i	$-0.406813\pi$
$677$	15.0178	0.577182	0.288591	+	0.957452i	$0.406813\pi$
$678$	0	0
$679$	6.08968	0.233701
$680$	−2.00479	−0.0768800
$681$	0	0
$682$	25.1466	0.962914
$683$	26.1909	1.00217	0.501083	−	0.865400i	$-0.332935\pi$
$683$	26.1909	1.00217	0.501083	+	0.865400i	$0.332935\pi$
$684$	0	0
$685$	−4.11180	−0.157104
$686$	8.62903	0.329458
$687$	0	0
$688$	−7.15685	−0.272852
$689$	−4.30397	−0.163968
$690$	0	0
$691$	18.2768	0.695281	0.347641	−	0.937628i	$-0.386983\pi$
$691$	18.2768	0.695281	0.347641	+	0.937628i	$0.386983\pi$
$692$	8.79048	0.334164
$693$	0	0
$694$	−21.7888	−0.827092
$695$	4.42438	0.167826
$696$	0	0
$697$	−5.39082	−0.204192
$698$	−45.8517	−1.73551
$699$	0	0
$700$	−18.4562	−0.697580
$701$	27.1517	1.02551	0.512753	−	0.858536i	$-0.328626\pi$
$701$	27.1517	1.02551	0.512753	+	0.858536i	$0.328626\pi$
$702$	0	0
$703$	−40.9740	−1.54536
$704$	−1.21153	−0.0456611
$705$	0	0
$706$	−0.598655	−0.0225307
$707$	−42.0925	−1.58305
$708$	0	0
$709$	48.4101	1.81808	0.909040	−	0.416710i	$-0.136817\pi$
$709$	48.4101	1.81808	0.909040	+	0.416710i	$0.136817\pi$
$710$	12.9454	0.485834
$711$	0	0
$712$	1.47113	0.0551328
$713$	−19.6544	−0.736063
$714$	0	0
$715$	7.58621	0.283708
$716$	8.81092	0.329279
$717$	0	0
$718$	−45.8086	−1.70956
$719$	30.9626	1.15471	0.577356	−	0.816493i	$-0.304085\pi$
$719$	30.9626	1.15471	0.577356	+	0.816493i	$0.304085\pi$
$720$	0	0
$721$	30.8201	1.14780
$722$	−6.97088	−0.259429
$723$	0	0
$724$	−27.0794	−1.00640
$725$	−45.5715	−1.69248
$726$	0	0
$727$	−39.7361	−1.47373	−0.736865	−	0.676040i	$-0.763694\pi$
$727$	−39.7361	−1.47373	−0.736865	+	0.676040i	$0.763694\pi$
$728$	−23.3715	−0.866205
$729$	0	0
$730$	−12.0369	−0.445506
$731$	2.62303	0.0970161
$732$	0	0
$733$	−21.6355	−0.799127	−0.399563	−	0.916706i	$-0.630838\pi$
$733$	−21.6355	−0.799127	−0.399563	+	0.916706i	$0.630838\pi$
$734$	−13.7329	−0.506892
$735$	0	0
$736$	18.8634	0.695315
$737$	9.33186	0.343744
$738$	0	0
$739$	−17.7120	−0.651547	−0.325774	−	0.945448i	$-0.605625\pi$
$739$	−17.7120	−0.651547	−0.325774	+	0.945448i	$0.605625\pi$
$740$	−7.41165	−0.272458
$741$	0	0
$742$	−6.13918	−0.225376
$743$	34.4144	1.26254	0.631271	−	0.775562i	$-0.282533\pi$
$743$	34.4144	1.26254	0.631271	+	0.775562i	$0.282533\pi$
$744$	0	0
$745$	−0.742067	−0.0271872
$746$	−29.6741	−1.08645
$747$	0	0
$748$	4.90195	0.179233
$749$	−63.0110	−2.30237
$750$	0	0
$751$	−12.3882	−0.452051	−0.226025	−	0.974121i	$-0.572573\pi$
$751$	−12.3882	−0.452051	−0.226025	+	0.974121i	$0.572573\pi$
$752$	−1.18156	−0.0430869
$753$	0	0
$754$	80.8349	2.94383
$755$	−9.99257	−0.363667
$756$	0	0
$757$	−9.19668	−0.334259	−0.167130	−	0.985935i	$-0.553450\pi$
$757$	−9.19668	−0.334259	−0.167130	+	0.985935i	$0.553450\pi$
$758$	−15.6482	−0.568369
$759$	0	0
$760$	5.26653	0.191037
$761$	20.6589	0.748884	0.374442	−	0.927250i	$-0.377834\pi$
$761$	20.6589	0.748884	0.374442	+	0.927250i	$0.377834\pi$
$762$	0	0
$763$	22.9545	0.831010
$764$	16.6406	0.602034
$765$	0	0
$766$	26.2104	0.947021
$767$	−13.6243	−0.491944
$768$	0	0
$769$	−36.0840	−1.30122	−0.650611	−	0.759411i	$-0.725487\pi$
$769$	−36.0840	−1.30122	−0.650611	+	0.759411i	$0.725487\pi$
$770$	10.8210	0.389961
$771$	0	0
$772$	20.8989	0.752169
$773$	24.5874	0.884346	0.442173	−	0.896930i	$-0.354208\pi$
$773$	24.5874	0.884346	0.442173	+	0.896930i	$0.354208\pi$
$774$	0	0
$775$	27.2744	0.979726
$776$	2.53976	0.0911721
$777$	0	0
$778$	−0.721697	−0.0258741
$779$	14.1616	0.507391
$780$	0	0
$781$	22.5972	0.808592
$782$	−10.3978	−0.371825
$783$	0	0
$784$	−28.0226	−1.00081
$785$	0.0813683	0.00290416
$786$	0	0
$787$	13.3785	0.476891	0.238445	−	0.971156i	$-0.423362\pi$
$787$	13.3785	0.476891	0.238445	+	0.971156i	$0.423362\pi$
$788$	0.178092	0.00634426
$789$	0	0
$790$	14.7386	0.524375
$791$	1.77096	0.0629681
$792$	0	0
$793$	−51.4523	−1.82712
$794$	−0.997453	−0.0353983
$795$	0	0
$796$	12.2598	0.434537
$797$	14.7314	0.521813	0.260906	−	0.965364i	$-0.415979\pi$
$797$	14.7314	0.521813	0.260906	+	0.965364i	$0.415979\pi$
$798$	0	0
$799$	0.433047	0.0153201
$800$	−26.1768	−0.925489
$801$	0	0
$802$	23.5624	0.832017
$803$	−21.0113	−0.741473
$804$	0	0
$805$	−8.45758	−0.298091
$806$	−48.3794	−1.70409
$807$	0	0
$808$	−17.5551	−0.617585
$809$	16.0564	0.564512	0.282256	−	0.959339i	$-0.408917\pi$
$809$	16.0564	0.564512	0.282256	+	0.959339i	$0.408917\pi$
$810$	0	0
$811$	−46.3933	−1.62909	−0.814544	−	0.580101i	$-0.803013\pi$
$811$	−46.3933	−1.62909	−0.814544	+	0.580101i	$0.803013\pi$
$812$	42.4860	1.49097
$813$	0	0
$814$	−35.1113	−1.23065
$815$	−5.63863	−0.197513
$816$	0	0
$817$	−6.89063	−0.241073
$818$	43.3262	1.51486
$819$	0	0
$820$	2.56164	0.0894564
$821$	−26.7852	−0.934808	−0.467404	−	0.884044i	$-0.654811\pi$
$821$	−26.7852	−0.934808	−0.467404	+	0.884044i	$0.654811\pi$
$822$	0	0
$823$	−8.27411	−0.288418	−0.144209	−	0.989547i	$-0.546064\pi$
$823$	−8.27411	−0.288418	−0.144209	+	0.989547i	$0.546064\pi$
$824$	12.8538	0.447784
$825$	0	0
$826$	−19.4337	−0.676183
$827$	−40.5539	−1.41020	−0.705099	−	0.709109i	$-0.749097\pi$
$827$	−40.5539	−1.41020	−0.705099	+	0.709109i	$0.749097\pi$
$828$	0	0
$829$	−19.9199	−0.691846	−0.345923	−	0.938263i	$-0.612434\pi$
$829$	−19.9199	−0.691846	−0.345923	+	0.938263i	$0.612434\pi$
$830$	−18.7572	−0.651071
$831$	0	0
$832$	2.33085	0.0808075
$833$	10.2704	0.355849
$834$	0	0
$835$	−10.9784	−0.379923
$836$	−12.8773	−0.445371
$837$	0	0
$838$	−48.0224	−1.65891
$839$	−24.2904	−0.838597	−0.419298	−	0.907848i	$-0.637724\pi$
$839$	−24.2904	−0.838597	−0.419298	+	0.907848i	$0.637724\pi$
$840$	0	0
$841$	75.9051	2.61742
$842$	52.7146	1.81667
$843$	0	0
$844$	−12.3128	−0.423826
$845$	−4.94820	−0.170223
$846$	0	0
$847$	−20.2129	−0.694525
$848$	−4.82537	−0.165704
$849$	0	0
$850$	14.4291	0.494913
$851$	27.4427	0.940725
$852$	0	0
$853$	−26.5586	−0.909350	−0.454675	−	0.890657i	$-0.650245\pi$
$853$	−26.5586	−0.909350	−0.454675	+	0.890657i	$0.650245\pi$
$854$	−73.3915	−2.51141
$855$	0	0
$856$	−26.2793	−0.898208
$857$	39.3259	1.34335	0.671673	−	0.740848i	$-0.265576\pi$
$857$	39.3259	1.34335	0.671673	+	0.740848i	$0.265576\pi$
$858$	0	0
$859$	19.5855	0.668249	0.334125	−	0.942529i	$-0.391559\pi$
$859$	19.5855	0.668249	0.334125	+	0.942529i	$0.391559\pi$
$860$	−1.24643	−0.0425027
$861$	0	0
$862$	−57.5260	−1.95934
$863$	−21.3083	−0.725342	−0.362671	−	0.931917i	$-0.618135\pi$
$863$	−21.3083	−0.725342	−0.362671	+	0.931917i	$0.618135\pi$
$864$	0	0
$865$	−5.58999	−0.190066
$866$	0.204695	0.00695581
$867$	0	0
$868$	−25.4277	−0.863074
$869$	25.7273	0.872739
$870$	0	0
$871$	−17.9535	−0.608331
$872$	9.57340	0.324196
$873$	0	0
$874$	27.3149	0.923939
$875$	24.9257	0.842643
$876$	0	0
$877$	−46.2510	−1.56178	−0.780892	−	0.624666i	$-0.785235\pi$
$877$	−46.2510	−1.56178	−0.780892	+	0.624666i	$0.785235\pi$
$878$	−46.0581	−1.55439
$879$	0	0
$880$	8.50524	0.286712
$881$	48.4243	1.63146	0.815729	−	0.578435i	$-0.196336\pi$
$881$	48.4243	1.63146	0.815729	+	0.578435i	$0.196336\pi$
$882$	0	0
$883$	44.0738	1.48320	0.741601	−	0.670842i	$-0.234067\pi$
$883$	44.0738	1.48320	0.741601	+	0.670842i	$0.234067\pi$
$884$	−9.43082	−0.317193
$885$	0	0
$886$	−30.9643	−1.04027
$887$	−15.8157	−0.531039	−0.265520	−	0.964105i	$-0.585544\pi$
$887$	−15.8157	−0.531039	−0.265520	+	0.964105i	$0.585544\pi$
$888$	0	0
$889$	−48.5102	−1.62698
$890$	1.31042	0.0439254
$891$	0	0
$892$	16.5062	0.552668
$893$	−1.13761	−0.0380685
$894$	0	0
$895$	−5.60299	−0.187287
$896$	−38.5021	−1.28626
$897$	0	0
$898$	−6.71614	−0.224120
$899$	−62.7854	−2.09401
$900$	0	0
$901$	1.76853	0.0589181
$902$	12.1353	0.404062
$903$	0	0
$904$	0.738596	0.0245653
$905$	17.2202	0.572418
$906$	0	0
$907$	−26.8870	−0.892767	−0.446384	−	0.894842i	$-0.647288\pi$
$907$	−26.8870	−0.892767	−0.446384	+	0.894842i	$0.647288\pi$
$908$	23.0879	0.766200
$909$	0	0
$910$	−20.8184	−0.690123
$911$	−40.1487	−1.33019	−0.665093	−	0.746760i	$-0.731608\pi$
$911$	−40.1487	−1.33019	−0.665093	+	0.746760i	$0.731608\pi$
$912$	0	0
$913$	−32.7421	−1.08360
$914$	63.1407	2.08851
$915$	0	0
$916$	−1.93893	−0.0640639
$917$	60.7981	2.00773
$918$	0	0
$919$	1.87257	0.0617705	0.0308852	−	0.999523i	$-0.490167\pi$
$919$	1.87257	0.0617705	0.0308852	+	0.999523i	$0.490167\pi$
$920$	−3.52731	−0.116292
$921$	0	0
$922$	3.40859	0.112256
$923$	−43.4747	−1.43099
$924$	0	0
$925$	−38.0823	−1.25214
$926$	−27.5536	−0.905467
$927$	0	0
$928$	60.2586	1.97809
$929$	−27.5276	−0.903151	−0.451576	−	0.892233i	$-0.649138\pi$
$929$	−27.5276	−0.903151	−0.451576	+	0.892233i	$0.649138\pi$
$930$	0	0
$931$	−26.9802	−0.884240
$932$	−14.0929	−0.461628
$933$	0	0
$934$	35.4475	1.15988
$935$	−3.11722	−0.101944
$936$	0	0
$937$	−23.9119	−0.781167	−0.390583	−	0.920568i	$-0.627727\pi$
$937$	−23.9119	−0.781167	−0.390583	+	0.920568i	$0.627727\pi$
$938$	−25.6089	−0.836159
$939$	0	0
$940$	−0.205778	−0.00671174
$941$	−27.3946	−0.893038	−0.446519	−	0.894774i	$-0.647336\pi$
$941$	−27.3946	−0.893038	−0.446519	+	0.894774i	$0.647336\pi$
$942$	0	0
$943$	−9.48486	−0.308870
$944$	−15.2748	−0.497152
$945$	0	0
$946$	−5.90471	−0.191979
$947$	−49.2181	−1.59937	−0.799687	−	0.600417i	$-0.795001\pi$
$947$	−49.2181	−1.59937	−0.799687	+	0.600417i	$0.795001\pi$
$948$	0	0
$949$	40.4235	1.31220
$950$	−37.9048	−1.22980
$951$	0	0
$952$	9.60348	0.311250
$953$	−20.5815	−0.666701	−0.333351	−	0.942803i	$-0.608179\pi$
$953$	−20.5815	−0.666701	−0.333351	+	0.942803i	$0.608179\pi$
$954$	0	0
$955$	−10.5820	−0.342425
$956$	−7.64695	−0.247320
$957$	0	0
$958$	20.8770	0.674505
$959$	19.6966	0.636038
$960$	0	0
$961$	6.57684	0.212156
$962$	67.5505	2.17791
$963$	0	0
$964$	3.76462	0.121250
$965$	−13.2899	−0.427818
$966$	0	0
$967$	55.2578	1.77697	0.888486	−	0.458904i	$-0.151758\pi$
$967$	55.2578	1.77697	0.888486	+	0.458904i	$0.151758\pi$
$968$	−8.42999	−0.270950
$969$	0	0
$970$	2.26232	0.0726386
$971$	−55.8857	−1.79346	−0.896729	−	0.442580i	$-0.854063\pi$
$971$	−55.8857	−1.79346	−0.896729	+	0.442580i	$0.854063\pi$
$972$	0	0
$973$	−21.1940	−0.679449
$974$	−28.7300	−0.920567
$975$	0	0
$976$	−57.6854	−1.84647
$977$	39.6786	1.26943	0.634715	−	0.772746i	$-0.281118\pi$
$977$	39.6786	1.26943	0.634715	+	0.772746i	$0.281118\pi$
$978$	0	0
$979$	2.28744	0.0731069
$980$	−4.88036	−0.155897
$981$	0	0
$982$	−32.4355	−1.03506
$983$	30.0468	0.958345	0.479173	−	0.877721i	$-0.340937\pi$
$983$	30.0468	0.958345	0.479173	+	0.877721i	$0.340937\pi$
$984$	0	0
$985$	−0.113251	−0.00360848
$986$	−33.2155	−1.05780
$987$	0	0
$988$	24.7746	0.788184
$989$	4.61507	0.146751
$990$	0	0
$991$	−39.1317	−1.24306	−0.621529	−	0.783391i	$-0.713488\pi$
$991$	−39.1317	−1.24306	−0.621529	+	0.783391i	$0.713488\pi$
$992$	−36.0646	−1.14505
$993$	0	0
$994$	−62.0122	−1.96691
$995$	−7.79618	−0.247156
$996$	0	0
$997$	21.2663	0.673510	0.336755	−	0.941592i	$-0.390671\pi$
$997$	21.2663	0.673510	0.336755	+	0.941592i	$0.390671\pi$
$998$	−18.9756	−0.600664
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.f.1.5	yes	25
3.2	odd	2		4023.2.a.e.1.21	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.21	✓	25	3.2	odd	2
4023.2.a.f.1.5	yes	25	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.77959 q^{2} +1.16693 q^{4} -0.742067 q^{5} +3.55470 q^{7} +1.48252 q^{8} +O(q^{10})\)	\(q-1.77959 q^{2} +1.16693 q^{4} -0.742067 q^{5} +3.55470 q^{7} +1.48252 q^{8} +1.32057 q^{10} +2.30516 q^{11} -4.43488 q^{13} -6.32590 q^{14} -4.97213 q^{16} +1.82232 q^{17} -4.78718 q^{19} -0.865939 q^{20} -4.10223 q^{22} +3.20627 q^{23} -4.44934 q^{25} +7.89225 q^{26} +4.14809 q^{28} +10.2423 q^{29} -6.12999 q^{31} +5.88330 q^{32} -3.24297 q^{34} -2.63783 q^{35} +8.55909 q^{37} +8.51921 q^{38} -1.10013 q^{40} -2.95823 q^{41} +1.43939 q^{43} +2.68995 q^{44} -5.70583 q^{46} +0.237636 q^{47} +5.63592 q^{49} +7.91798 q^{50} -5.17518 q^{52} +0.970483 q^{53} -1.71058 q^{55} +5.26993 q^{56} -18.2271 q^{58} +3.07208 q^{59} +11.6017 q^{61} +10.9089 q^{62} -0.525572 q^{64} +3.29097 q^{65} +4.04825 q^{67} +2.12651 q^{68} +4.69424 q^{70} +9.80290 q^{71} -9.11492 q^{73} -15.2316 q^{74} -5.58630 q^{76} +8.19415 q^{77} +11.1608 q^{79} +3.68966 q^{80} +5.26442 q^{82} -14.2038 q^{83} -1.35228 q^{85} -2.56152 q^{86} +3.41745 q^{88} +0.992314 q^{89} -15.7647 q^{91} +3.74148 q^{92} -0.422893 q^{94} +3.55241 q^{95} +1.71313 q^{97} -10.0296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 7 q^{2} + 27 q^{4} + 12 q^{5} - 2 q^{7} + 21 q^{8}+O(q^{10}) \) 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

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