LMFDB - Embedded newform 4023.2.a.e.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.51618 q^{2} +0.298802 q^{4} +2.07566 q^{5} -1.53996 q^{7} +2.57932 q^{8} +O(q^{10})$	\(q-1.51618 q^{2} +0.298802 q^{4} +2.07566 q^{5} -1.53996 q^{7} +2.57932 q^{8} -3.14707 q^{10} +1.76571 q^{11} +1.58507 q^{13} +2.33486 q^{14} -4.50832 q^{16} +5.51225 q^{17} -2.21769 q^{19} +0.620212 q^{20} -2.67713 q^{22} -4.68413 q^{23} -0.691637 q^{25} -2.40325 q^{26} -0.460145 q^{28} -10.7325 q^{29} -4.78451 q^{31} +1.67678 q^{32} -8.35756 q^{34} -3.19644 q^{35} +7.22756 q^{37} +3.36242 q^{38} +5.35380 q^{40} -6.55487 q^{41} +6.33612 q^{43} +0.527597 q^{44} +7.10198 q^{46} -0.411483 q^{47} -4.62851 q^{49} +1.04865 q^{50} +0.473623 q^{52} -12.2529 q^{53} +3.66501 q^{55} -3.97206 q^{56} +16.2724 q^{58} -10.0757 q^{59} -8.70662 q^{61} +7.25418 q^{62} +6.47434 q^{64} +3.29007 q^{65} +4.28517 q^{67} +1.64707 q^{68} +4.84638 q^{70} +9.09329 q^{71} -1.01772 q^{73} -10.9583 q^{74} -0.662651 q^{76} -2.71913 q^{77} +4.72727 q^{79} -9.35774 q^{80} +9.93836 q^{82} +12.6775 q^{83} +11.4415 q^{85} -9.60669 q^{86} +4.55433 q^{88} +4.70749 q^{89} -2.44095 q^{91} -1.39963 q^{92} +0.623882 q^{94} -4.60317 q^{95} -6.40690 q^{97} +7.01766 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.51618	−1.07210	−0.536051	−	0.844186i	$-0.680084\pi$
$2$	−1.51618	−1.07210	−0.536051	+	0.844186i	$0.680084\pi$
$3$	0	0
$3$	0	0
$4$	0.298802	0.149401
$5$	2.07566	0.928263	0.464132	−	0.885766i	$-0.346366\pi$
$5$	2.07566	0.928263	0.464132	+	0.885766i	$0.346366\pi$
$6$	0	0
$7$	−1.53996	−0.582052	−0.291026	−	0.956715i	$-0.593997\pi$
$7$	−1.53996	−0.582052	−0.291026	+	0.956715i	$0.593997\pi$
$8$	2.57932	0.911928
$9$	0	0
$10$	−3.14707	−0.995192
$11$	1.76571	0.532381	0.266190	−	0.963920i	$-0.414235\pi$
$11$	1.76571	0.532381	0.266190	+	0.963920i	$0.414235\pi$
$12$	0	0
$13$	1.58507	0.439620	0.219810	−	0.975543i	$-0.429456\pi$
$13$	1.58507	0.439620	0.219810	+	0.975543i	$0.429456\pi$
$14$	2.33486	0.624018
$15$	0	0
$16$	−4.50832	−1.12708
$17$	5.51225	1.33692	0.668458	−	0.743750i	$-0.266955\pi$
$17$	5.51225	1.33692	0.668458	+	0.743750i	$0.266955\pi$
$18$	0	0
$19$	−2.21769	−0.508774	−0.254387	−	0.967103i	$-0.581874\pi$
$19$	−2.21769	−0.508774	−0.254387	+	0.967103i	$0.581874\pi$
$20$	0.620212	0.138684
$21$	0	0
$22$	−2.67713	−0.570766
$23$	−4.68413	−0.976708	−0.488354	−	0.872646i	$-0.662402\pi$
$23$	−4.68413	−0.976708	−0.488354	+	0.872646i	$0.662402\pi$
$24$	0	0
$25$	−0.691637	−0.138327
$26$	−2.40325	−0.471317
$27$	0	0
$28$	−0.460145	−0.0869592
$29$	−10.7325	−1.99298	−0.996489	−	0.0837297i	$-0.973317\pi$
$29$	−10.7325	−1.99298	−0.996489	+	0.0837297i	$0.973317\pi$
$30$	0	0
$31$	−4.78451	−0.859324	−0.429662	−	0.902990i	$-0.641367\pi$
$31$	−4.78451	−0.859324	−0.429662	+	0.902990i	$0.641367\pi$
$32$	1.67678	0.296416
$33$	0	0
$34$	−8.35756	−1.43331
$35$	−3.19644	−0.540297
$36$	0	0
$37$	7.22756	1.18820	0.594102	−	0.804390i	$-0.297508\pi$
$37$	7.22756	1.18820	0.594102	+	0.804390i	$0.297508\pi$
$38$	3.36242	0.545457
$39$	0	0
$40$	5.35380	0.846509
$41$	−6.55487	−1.02370	−0.511849	−	0.859075i	$-0.671039\pi$
$41$	−6.55487	−1.02370	−0.511849	+	0.859075i	$0.671039\pi$
$42$	0	0
$43$	6.33612	0.966249	0.483124	−	0.875552i	$-0.339502\pi$
$43$	6.33612	0.966249	0.483124	+	0.875552i	$0.339502\pi$
$44$	0.527597	0.0795383
$45$	0	0
$46$	7.10198	1.04713
$47$	−0.411483	−0.0600209	−0.0300105	−	0.999550i	$-0.509554\pi$
$47$	−0.411483	−0.0600209	−0.0300105	+	0.999550i	$0.509554\pi$
$48$	0	0
$49$	−4.62851	−0.661216
$50$	1.04865	0.148301
$51$	0	0
$52$	0.473623	0.0656797
$53$	−12.2529	−1.68307	−0.841534	−	0.540204i	$-0.818347\pi$
$53$	−12.2529	−1.68307	−0.841534	+	0.540204i	$0.818347\pi$
$54$	0	0
$55$	3.66501	0.494189
$56$	−3.97206	−0.530789
$57$	0	0
$58$	16.2724	2.13667
$59$	−10.0757	−1.31175	−0.655874	−	0.754871i	$-0.727700\pi$
$59$	−10.0757	−1.31175	−0.655874	+	0.754871i	$0.727700\pi$
$60$	0	0
$61$	−8.70662	−1.11477	−0.557385	−	0.830254i	$-0.688195\pi$
$61$	−8.70662	−1.11477	−0.557385	+	0.830254i	$0.688195\pi$
$62$	7.25418	0.921282
$63$	0	0
$64$	6.47434	0.809292
$65$	3.29007	0.408083
$66$	0	0
$67$	4.28517	0.523517	0.261758	−	0.965133i	$-0.415698\pi$
$67$	4.28517	0.523517	0.261758	+	0.965133i	$0.415698\pi$
$68$	1.64707	0.199737
$69$	0	0
$70$	4.84638	0.579253
$71$	9.09329	1.07917	0.539587	−	0.841930i	$-0.318580\pi$
$71$	9.09329	1.07917	0.539587	+	0.841930i	$0.318580\pi$
$72$	0	0
$73$	−1.01772	−0.119116	−0.0595578	−	0.998225i	$-0.518969\pi$
$73$	−1.01772	−0.119116	−0.0595578	+	0.998225i	$0.518969\pi$
$74$	−10.9583	−1.27387
$75$	0	0
$76$	−0.662651	−0.0760113
$77$	−2.71913	−0.309873
$78$	0	0
$79$	4.72727	0.531859	0.265929	−	0.963992i	$-0.414321\pi$
$79$	4.72727	0.531859	0.265929	+	0.963992i	$0.414321\pi$
$80$	−9.35774	−1.04623
$81$	0	0
$82$	9.93836	1.09751
$83$	12.6775	1.39154	0.695769	−	0.718265i	$-0.255064\pi$
$83$	12.6775	1.39154	0.695769	+	0.718265i	$0.255064\pi$
$84$	0	0
$85$	11.4415	1.24101
$86$	−9.60669	−1.03592
$87$	0	0
$88$	4.55433	0.485493
$89$	4.70749	0.498993	0.249496	−	0.968376i	$-0.419735\pi$
$89$	4.70749	0.498993	0.249496	+	0.968376i	$0.419735\pi$
$90$	0	0
$91$	−2.44095	−0.255881
$92$	−1.39963	−0.145921
$93$	0	0
$94$	0.623882	0.0643485
$95$	−4.60317	−0.472276
$96$	0	0
$97$	−6.40690	−0.650522	−0.325261	−	0.945624i	$-0.605452\pi$
$97$	−6.40690	−0.650522	−0.325261	+	0.945624i	$0.605452\pi$
$98$	7.01766	0.708890
$99$	0	0
$100$	−0.206663	−0.0206663
$101$	−13.4216	−1.33550	−0.667751	−	0.744385i	$-0.732743\pi$
$101$	−13.4216	−1.33550	−0.667751	+	0.744385i	$0.732743\pi$
$102$	0	0
$103$	−9.72260	−0.957996	−0.478998	−	0.877816i	$-0.659000\pi$
$103$	−9.72260	−0.957996	−0.478998	+	0.877816i	$0.659000\pi$
$104$	4.08841	0.400902
$105$	0	0
$106$	18.5776	1.80442
$107$	−4.97002	−0.480470	−0.240235	−	0.970715i	$-0.577225\pi$
$107$	−4.97002	−0.480470	−0.240235	+	0.970715i	$0.577225\pi$
$108$	0	0
$109$	8.77486	0.840480	0.420240	−	0.907413i	$-0.361946\pi$
$109$	8.77486	0.840480	0.420240	+	0.907413i	$0.361946\pi$
$110$	−5.55681	−0.529821
$111$	0	0
$112$	6.94265	0.656019
$113$	0.473087	0.0445043	0.0222522	−	0.999752i	$-0.492916\pi$
$113$	0.473087	0.0445043	0.0222522	+	0.999752i	$0.492916\pi$
$114$	0	0
$115$	−9.72265	−0.906642
$116$	−3.20690	−0.297753
$117$	0	0
$118$	15.2766	1.40633
$119$	−8.48866	−0.778154
$120$	0	0
$121$	−7.88228	−0.716571
$122$	13.2008	1.19515
$123$	0	0
$124$	−1.42962	−0.128384
$125$	−11.8139	−1.05667
$126$	0	0
$127$	−5.03808	−0.447058	−0.223529	−	0.974697i	$-0.571758\pi$
$127$	−5.03808	−0.447058	−0.223529	+	0.974697i	$0.571758\pi$
$128$	−13.1698	−1.16406
$129$	0	0
$130$	−4.98834	−0.437506
$131$	−4.19204	−0.366261	−0.183130	−	0.983089i	$-0.558623\pi$
$131$	−4.19204	−0.366261	−0.183130	+	0.983089i	$0.558623\pi$
$132$	0	0
$133$	3.41517	0.296133
$134$	−6.49709	−0.561263
$135$	0	0
$136$	14.2179	1.21917
$137$	−0.909674	−0.0777187	−0.0388593	−	0.999245i	$-0.512372\pi$
$137$	−0.909674	−0.0777187	−0.0388593	+	0.999245i	$0.512372\pi$
$138$	0	0
$139$	−6.64475	−0.563600	−0.281800	−	0.959473i	$-0.590932\pi$
$139$	−6.64475	−0.563600	−0.281800	+	0.959473i	$0.590932\pi$
$140$	−0.955104	−0.0807210
$141$	0	0
$142$	−13.7871	−1.15698
$143$	2.79877	0.234045
$144$	0	0
$145$	−22.2770	−1.85001
$146$	1.54305	0.127704
$147$	0	0
$148$	2.15961	0.177519
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	1.08232	0.0880776	0.0440388	−	0.999030i	$-0.485977\pi$
$151$	1.08232	0.0880776	0.0440388	+	0.999030i	$0.485977\pi$
$152$	−5.72014	−0.463965
$153$	0	0
$154$	4.12268	0.332215
$155$	−9.93102	−0.797679
$156$	0	0
$157$	12.0506	0.961741	0.480870	−	0.876792i	$-0.340321\pi$
$157$	12.0506	0.961741	0.480870	+	0.876792i	$0.340321\pi$
$158$	−7.16739	−0.570207
$159$	0	0
$160$	3.48043	0.275152
$161$	7.21338	0.568494
$162$	0	0
$163$	22.6627	1.77508	0.887541	−	0.460728i	$-0.152412\pi$
$163$	22.6627	1.77508	0.887541	+	0.460728i	$0.152412\pi$
$164$	−1.95861	−0.152942
$165$	0	0
$166$	−19.2214	−1.49187
$167$	20.0537	1.55180	0.775901	−	0.630854i	$-0.217295\pi$
$167$	20.0537	1.55180	0.775901	+	0.630854i	$0.217295\pi$
$168$	0	0
$169$	−10.4875	−0.806734
$170$	−17.3474	−1.33049
$171$	0	0
$172$	1.89325	0.144359
$173$	−3.37036	−0.256244	−0.128122	−	0.991758i	$-0.540895\pi$
$173$	−3.37036	−0.256244	−0.128122	+	0.991758i	$0.540895\pi$
$174$	0	0
$175$	1.06510	0.0805137
$176$	−7.96037	−0.600036
$177$	0	0
$178$	−7.13740	−0.534971
$179$	−12.5615	−0.938887	−0.469444	−	0.882962i	$-0.655546\pi$
$179$	−12.5615	−0.938887	−0.469444	+	0.882962i	$0.655546\pi$
$180$	0	0
$181$	12.9871	0.965324	0.482662	−	0.875807i	$-0.339670\pi$
$181$	12.9871	0.965324	0.482662	+	0.875807i	$0.339670\pi$
$182$	3.70093	0.274331
$183$	0	0
$184$	−12.0819	−0.890687
$185$	15.0020	1.10297
$186$	0	0
$187$	9.73301	0.711748
$188$	−0.122952	−0.00896719
$189$	0	0
$190$	6.97924	0.506327
$191$	−3.11329	−0.225269	−0.112635	−	0.993636i	$-0.535929\pi$
$191$	−3.11329	−0.225269	−0.112635	+	0.993636i	$0.535929\pi$
$192$	0	0
$193$	−15.3799	−1.10707	−0.553536	−	0.832825i	$-0.686722\pi$
$193$	−15.3799	−1.10707	−0.553536	+	0.832825i	$0.686722\pi$
$194$	9.71402	0.697426
$195$	0	0
$196$	−1.38301	−0.0987864
$197$	−23.7763	−1.69399	−0.846996	−	0.531599i	$-0.821591\pi$
$197$	−23.7763	−1.69399	−0.846996	+	0.531599i	$0.821591\pi$
$198$	0	0
$199$	10.6933	0.758026	0.379013	−	0.925391i	$-0.376264\pi$
$199$	10.6933	0.758026	0.379013	+	0.925391i	$0.376264\pi$
$200$	−1.78395	−0.126145
$201$	0	0
$202$	20.3496	1.43179
$203$	16.5277	1.16002
$204$	0	0
$205$	−13.6057	−0.950261
$206$	14.7412	1.02707
$207$	0	0
$208$	−7.14601	−0.495487
$209$	−3.91580	−0.270861
$210$	0	0
$211$	11.6735	0.803635	0.401817	−	0.915720i	$-0.368379\pi$
$211$	11.6735	0.803635	0.401817	+	0.915720i	$0.368379\pi$
$212$	−3.66120	−0.251452
$213$	0	0
$214$	7.53545	0.515113
$215$	13.1516	0.896933
$216$	0	0
$217$	7.36798	0.500171
$218$	−13.3043	−0.901079
$219$	0	0
$220$	1.09511	0.0738324
$221$	8.73731	0.587735
$222$	0	0
$223$	26.7669	1.79244	0.896221	−	0.443609i	$-0.146302\pi$
$223$	26.7669	1.79244	0.896221	+	0.443609i	$0.146302\pi$
$224$	−2.58219	−0.172530
$225$	0	0
$226$	−0.717286	−0.0477131
$227$	13.0129	0.863698	0.431849	−	0.901946i	$-0.357861\pi$
$227$	13.0129	0.863698	0.431849	+	0.901946i	$0.357861\pi$
$228$	0	0
$229$	28.5375	1.88581	0.942904	−	0.333065i	$-0.108083\pi$
$229$	28.5375	1.88581	0.942904	+	0.333065i	$0.108083\pi$
$230$	14.7413	0.972012
$231$	0	0
$232$	−27.6826	−1.81745
$233$	−22.5804	−1.47929	−0.739644	−	0.672998i	$-0.765006\pi$
$233$	−22.5804	−1.47929	−0.739644	+	0.672998i	$0.765006\pi$
$234$	0	0
$235$	−0.854098	−0.0557152
$236$	−3.01065	−0.195977
$237$	0	0
$238$	12.8703	0.834260
$239$	−9.29123	−0.601000	−0.300500	−	0.953782i	$-0.597154\pi$
$239$	−9.29123	−0.601000	−0.300500	+	0.953782i	$0.597154\pi$
$240$	0	0
$241$	−11.7908	−0.759509	−0.379755	−	0.925087i	$-0.623992\pi$
$241$	−11.7908	−0.759509	−0.379755	+	0.925087i	$0.623992\pi$
$242$	11.9510	0.768236
$243$	0	0
$244$	−2.60156	−0.166548
$245$	−9.60721	−0.613782
$246$	0	0
$247$	−3.51520	−0.223667
$248$	−12.3408	−0.783642
$249$	0	0
$250$	17.9120	1.13285
$251$	−8.10510	−0.511590	−0.255795	−	0.966731i	$-0.582337\pi$
$251$	−8.10510	−0.511590	−0.255795	+	0.966731i	$0.582337\pi$
$252$	0	0
$253$	−8.27079	−0.519980
$254$	7.63864	0.479291
$255$	0	0
$256$	7.01916	0.438697
$257$	−11.4420	−0.713733	−0.356866	−	0.934155i	$-0.616155\pi$
$257$	−11.4420	−0.713733	−0.356866	+	0.934155i	$0.616155\pi$
$258$	0	0
$259$	−11.1302	−0.691596
$260$	0.983080	0.0609680
$261$	0	0
$262$	6.35590	0.392669
$263$	2.12121	0.130799	0.0653997	−	0.997859i	$-0.479168\pi$
$263$	2.12121	0.130799	0.0653997	+	0.997859i	$0.479168\pi$
$264$	0	0
$265$	−25.4329	−1.56233
$266$	−5.17801	−0.317484
$267$	0	0
$268$	1.28042	0.0782140
$269$	−8.15224	−0.497051	−0.248525	−	0.968625i	$-0.579946\pi$
$269$	−8.15224	−0.497051	−0.248525	+	0.968625i	$0.579946\pi$
$270$	0	0
$271$	22.8868	1.39027	0.695137	−	0.718877i	$-0.255344\pi$
$271$	22.8868	1.39027	0.695137	+	0.718877i	$0.255344\pi$
$272$	−24.8510	−1.50681
$273$	0	0
$274$	1.37923	0.0833223
$275$	−1.22123	−0.0736428
$276$	0	0
$277$	−30.5333	−1.83457	−0.917283	−	0.398235i	$-0.869623\pi$
$277$	−30.5333	−1.83457	−0.917283	+	0.398235i	$0.869623\pi$
$278$	10.0746	0.604237
$279$	0	0
$280$	−8.24465	−0.492712
$281$	−28.2447	−1.68494	−0.842468	−	0.538747i	$-0.818898\pi$
$281$	−28.2447	−1.68494	−0.842468	+	0.538747i	$0.818898\pi$
$282$	0	0
$283$	−22.7693	−1.35349	−0.676747	−	0.736216i	$-0.736611\pi$
$283$	−22.7693	−1.35349	−0.676747	+	0.736216i	$0.736611\pi$
$284$	2.71709	0.161230
$285$	0	0
$286$	−4.24344	−0.250920
$287$	10.0943	0.595845
$288$	0	0
$289$	13.3849	0.787344
$290$	33.7760	1.98340
$291$	0	0
$292$	−0.304098	−0.0177960
$293$	14.8757	0.869046	0.434523	−	0.900661i	$-0.356917\pi$
$293$	14.8757	0.869046	0.434523	+	0.900661i	$0.356917\pi$
$294$	0	0
$295$	−20.9138	−1.21765
$296$	18.6422	1.08356
$297$	0	0
$298$	1.51618	0.0878300
$299$	−7.42468	−0.429380
$300$	0	0
$301$	−9.75739	−0.562407
$302$	−1.64098	−0.0944281
$303$	0	0
$304$	9.99807	0.573429
$305$	−18.0720	−1.03480
$306$	0	0
$307$	−28.0042	−1.59829	−0.799143	−	0.601141i	$-0.794713\pi$
$307$	−28.0042	−1.59829	−0.799143	+	0.601141i	$0.794713\pi$
$308$	−0.812481	−0.0462954
$309$	0	0
$310$	15.0572	0.855192
$311$	21.9444	1.24435	0.622176	−	0.782877i	$-0.286249\pi$
$311$	21.9444	1.24435	0.622176	+	0.782877i	$0.286249\pi$
$312$	0	0
$313$	−23.4606	−1.32607	−0.663036	−	0.748588i	$-0.730732\pi$
$313$	−23.4606	−1.32607	−0.663036	+	0.748588i	$0.730732\pi$
$314$	−18.2708	−1.03108
$315$	0	0
$316$	1.41252	0.0794603
$317$	1.63603	0.0918883	0.0459442	−	0.998944i	$-0.485370\pi$
$317$	1.63603	0.0918883	0.0459442	+	0.998944i	$0.485370\pi$
$318$	0	0
$319$	−18.9505	−1.06102
$320$	13.4385	0.751236
$321$	0	0
$322$	−10.9368	−0.609483
$323$	−12.2245	−0.680188
$324$	0	0
$325$	−1.09629	−0.0608115
$326$	−34.3608	−1.90307
$327$	0	0
$328$	−16.9071	−0.933539
$329$	0.633668	0.0349353
$330$	0	0
$331$	−0.537670	−0.0295530	−0.0147765	−	0.999891i	$-0.504704\pi$
$331$	−0.537670	−0.0295530	−0.0147765	+	0.999891i	$0.504704\pi$
$332$	3.78807	0.207897
$333$	0	0
$334$	−30.4050	−1.66369
$335$	8.89455	0.485961
$336$	0	0
$337$	9.22513	0.502525	0.251262	−	0.967919i	$-0.419154\pi$
$337$	9.22513	0.502525	0.251262	+	0.967919i	$0.419154\pi$
$338$	15.9010	0.864901
$339$	0	0
$340$	3.41876	0.185408
$341$	−8.44805	−0.457487
$342$	0	0
$343$	17.9075	0.966913
$344$	16.3429	0.881149
$345$	0	0
$346$	5.11007	0.274719
$347$	13.3911	0.718872	0.359436	−	0.933170i	$-0.382969\pi$
$347$	13.3911	0.718872	0.359436	+	0.933170i	$0.382969\pi$
$348$	0	0
$349$	−18.9966	−1.01687	−0.508433	−	0.861102i	$-0.669775\pi$
$349$	−18.9966	−1.01687	−0.508433	+	0.861102i	$0.669775\pi$
$350$	−1.61488	−0.0863188
$351$	0	0
$352$	2.96071	0.157806
$353$	−16.5494	−0.880833	−0.440417	−	0.897793i	$-0.645169\pi$
$353$	−16.5494	−0.880833	−0.440417	+	0.897793i	$0.645169\pi$
$354$	0	0
$355$	18.8746	1.00176
$356$	1.40661	0.0745500
$357$	0	0
$358$	19.0454	1.00658
$359$	−4.09813	−0.216291	−0.108146	−	0.994135i	$-0.534491\pi$
$359$	−4.09813	−0.216291	−0.108146	+	0.994135i	$0.534491\pi$
$360$	0	0
$361$	−14.0818	−0.741149
$362$	−19.6908	−1.03492
$363$	0	0
$364$	−0.729362	−0.0382290
$365$	−2.11245	−0.110571
$366$	0	0
$367$	−4.61545	−0.240924	−0.120462	−	0.992718i	$-0.538438\pi$
$367$	−4.61545	−0.240924	−0.120462	+	0.992718i	$0.538438\pi$
$368$	21.1175	1.10083
$369$	0	0
$370$	−22.7457	−1.18249
$371$	18.8691	0.979633
$372$	0	0
$373$	−27.1685	−1.40673	−0.703366	−	0.710828i	$-0.748321\pi$
$373$	−27.1685	−1.40673	−0.703366	+	0.710828i	$0.748321\pi$
$374$	−14.7570	−0.763066
$375$	0	0
$376$	−1.06135	−0.0547348
$377$	−17.0118	−0.876152
$378$	0	0
$379$	−7.90556	−0.406081	−0.203041	−	0.979170i	$-0.565082\pi$
$379$	−7.90556	−0.406081	−0.203041	+	0.979170i	$0.565082\pi$
$380$	−1.37544	−0.0705585
$381$	0	0
$382$	4.72030	0.241512
$383$	−19.7095	−1.00711	−0.503554	−	0.863964i	$-0.667975\pi$
$383$	−19.7095	−1.00711	−0.503554	+	0.863964i	$0.667975\pi$
$384$	0	0
$385$	−5.64398	−0.287644
$386$	23.3188	1.18689
$387$	0	0
$388$	−1.91440	−0.0971887
$389$	27.8880	1.41398	0.706989	−	0.707225i	$-0.250053\pi$
$389$	27.8880	1.41398	0.706989	+	0.707225i	$0.250053\pi$
$390$	0	0
$391$	−25.8201	−1.30578
$392$	−11.9384	−0.602981
$393$	0	0
$394$	36.0492	1.81613
$395$	9.81219	0.493705
$396$	0	0
$397$	10.6763	0.535826	0.267913	−	0.963443i	$-0.413666\pi$
$397$	10.6763	0.535826	0.267913	+	0.963443i	$0.413666\pi$
$398$	−16.2129	−0.812681
$399$	0	0
$400$	3.11812	0.155906
$401$	7.88305	0.393661	0.196830	−	0.980438i	$-0.436935\pi$
$401$	7.88305	0.393661	0.196830	+	0.980438i	$0.436935\pi$
$402$	0	0
$403$	−7.58380	−0.377776
$404$	−4.01041	−0.199525
$405$	0	0
$406$	−25.0589	−1.24365
$407$	12.7618	0.632576
$408$	0	0
$409$	28.4088	1.40472	0.702362	−	0.711820i	$-0.252129\pi$
$409$	28.4088	1.40472	0.702362	+	0.711820i	$0.252129\pi$
$410$	20.6286	1.01878
$411$	0	0
$412$	−2.90513	−0.143126
$413$	15.5163	0.763505
$414$	0	0
$415$	26.3142	1.29171
$416$	2.65782	0.130310
$417$	0	0
$418$	5.93705	0.290391
$419$	9.70591	0.474165	0.237082	−	0.971490i	$-0.423809\pi$
$419$	9.70591	0.474165	0.237082	+	0.971490i	$0.423809\pi$
$420$	0	0
$421$	−8.14597	−0.397010	−0.198505	−	0.980100i	$-0.563609\pi$
$421$	−8.14597	−0.397010	−0.198505	+	0.980100i	$0.563609\pi$
$422$	−17.6991	−0.861578
$423$	0	0
$424$	−31.6042	−1.53484
$425$	−3.81247	−0.184932
$426$	0	0
$427$	13.4079	0.648853
$428$	−1.48505	−0.0717828
$429$	0	0
$430$	−19.9402	−0.961603
$431$	14.2980	0.688712	0.344356	−	0.938839i	$-0.388097\pi$
$431$	14.2980	0.688712	0.344356	+	0.938839i	$0.388097\pi$
$432$	0	0
$433$	−9.77958	−0.469977	−0.234988	−	0.971998i	$-0.575505\pi$
$433$	−9.77958	−0.469977	−0.234988	+	0.971998i	$0.575505\pi$
$434$	−11.1712	−0.536234
$435$	0	0
$436$	2.62195	0.125569
$437$	10.3880	0.496923
$438$	0	0
$439$	−14.0609	−0.671088	−0.335544	−	0.942025i	$-0.608920\pi$
$439$	−14.0609	−0.671088	−0.335544	+	0.942025i	$0.608920\pi$
$440$	9.45323	0.450665
$441$	0	0
$442$	−13.2473	−0.630111
$443$	−29.0517	−1.38029	−0.690145	−	0.723671i	$-0.742453\pi$
$443$	−29.0517	−1.38029	−0.690145	+	0.723671i	$0.742453\pi$
$444$	0	0
$445$	9.77114	0.463197
$446$	−40.5834	−1.92168
$447$	0	0
$448$	−9.97025	−0.471050
$449$	−18.1944	−0.858648	−0.429324	−	0.903150i	$-0.641248\pi$
$449$	−18.1944	−0.858648	−0.429324	+	0.903150i	$0.641248\pi$
$450$	0	0
$451$	−11.5740	−0.544997
$452$	0.141360	0.00664899
$453$	0	0
$454$	−19.7299	−0.925971
$455$	−5.06659	−0.237525
$456$	0	0
$457$	27.2115	1.27290	0.636451	−	0.771317i	$-0.280402\pi$
$457$	27.2115	1.27290	0.636451	+	0.771317i	$0.280402\pi$
$458$	−43.2679	−2.02178
$459$	0	0
$460$	−2.90515	−0.135453
$461$	14.5906	0.679554	0.339777	−	0.940506i	$-0.389648\pi$
$461$	14.5906	0.679554	0.339777	+	0.940506i	$0.389648\pi$
$462$	0	0
$463$	−41.0451	−1.90753	−0.953763	−	0.300560i	$-0.902826\pi$
$463$	−41.0451	−1.90753	−0.953763	+	0.300560i	$0.902826\pi$
$464$	48.3856	2.24625
$465$	0	0
$466$	34.2359	1.58595
$467$	−3.81322	−0.176455	−0.0882274	−	0.996100i	$-0.528120\pi$
$467$	−3.81322	−0.176455	−0.0882274	+	0.996100i	$0.528120\pi$
$468$	0	0
$469$	−6.59901	−0.304714
$470$	1.29497	0.0597323
$471$	0	0
$472$	−25.9885	−1.19622
$473$	11.1877	0.514412
$474$	0	0
$475$	1.53384	0.0703773
$476$	−2.53643	−0.116257
$477$	0	0
$478$	14.0872	0.644333
$479$	−39.7713	−1.81720	−0.908599	−	0.417669i	$-0.862847\pi$
$479$	−39.7713	−1.81720	−0.908599	+	0.417669i	$0.862847\pi$
$480$	0	0
$481$	11.4562	0.522358
$482$	17.8769	0.814271
$483$	0	0
$484$	−2.35524	−0.107056
$485$	−13.2985	−0.603856
$486$	0	0
$487$	37.1986	1.68563	0.842815	−	0.538203i	$-0.180896\pi$
$487$	37.1986	1.68563	0.842815	+	0.538203i	$0.180896\pi$
$488$	−22.4572	−1.01659
$489$	0	0
$490$	14.5663	0.658037
$491$	−14.9831	−0.676176	−0.338088	−	0.941115i	$-0.609780\pi$
$491$	−14.9831	−0.676176	−0.338088	+	0.941115i	$0.609780\pi$
$492$	0	0
$493$	−59.1602	−2.66444
$494$	5.32968	0.239794
$495$	0	0
$496$	21.5701	0.968527
$497$	−14.0033	−0.628136
$498$	0	0
$499$	12.5772	0.563034	0.281517	−	0.959556i	$-0.409162\pi$
$499$	12.5772	0.563034	0.281517	+	0.959556i	$0.409162\pi$
$500$	−3.53002	−0.157867
$501$	0	0
$502$	12.2888	0.548476
$503$	−30.1344	−1.34363	−0.671814	−	0.740720i	$-0.734484\pi$
$503$	−30.1344	−1.34363	−0.671814	+	0.740720i	$0.734484\pi$
$504$	0	0
$505$	−27.8587	−1.23970
$506$	12.5400	0.557472
$507$	0	0
$508$	−1.50539	−0.0667909
$509$	−33.6167	−1.49003	−0.745016	−	0.667046i	$-0.767558\pi$
$509$	−33.6167	−1.49003	−0.745016	+	0.667046i	$0.767558\pi$
$510$	0	0
$511$	1.56726	0.0693314
$512$	15.6974	0.693732
$513$	0	0
$514$	17.3481	0.765194
$515$	−20.1808	−0.889272
$516$	0	0
$517$	−0.726558	−0.0319540
$518$	16.8754	0.741461
$519$	0	0
$520$	8.48615	0.372142
$521$	4.11114	0.180112	0.0900562	−	0.995937i	$-0.471295\pi$
$521$	4.11114	0.180112	0.0900562	+	0.995937i	$0.471295\pi$
$522$	0	0
$523$	−21.2045	−0.927210	−0.463605	−	0.886042i	$-0.653444\pi$
$523$	−21.2045	−0.927210	−0.463605	+	0.886042i	$0.653444\pi$
$524$	−1.25259	−0.0547197
$525$	0	0
$526$	−3.21614	−0.140230
$527$	−26.3734	−1.14884
$528$	0	0
$529$	−1.05897	−0.0460421
$530$	38.5609	1.67498
$531$	0	0
$532$	1.02046	0.0442425
$533$	−10.3899	−0.450038
$534$	0	0
$535$	−10.3161	−0.446003
$536$	11.0528	0.477410
$537$	0	0
$538$	12.3603	0.532889
$539$	−8.17259	−0.352019
$540$	0	0
$541$	20.9354	0.900085	0.450043	−	0.893007i	$-0.351409\pi$
$541$	20.9354	0.900085	0.450043	+	0.893007i	$0.351409\pi$
$542$	−34.7005	−1.49051
$543$	0	0
$544$	9.24284	0.396283
$545$	18.2136	0.780186
$546$	0	0
$547$	3.91363	0.167335	0.0836673	−	0.996494i	$-0.473337\pi$
$547$	3.91363	0.167335	0.0836673	+	0.996494i	$0.473337\pi$
$548$	−0.271813	−0.0116113
$549$	0	0
$550$	1.85160	0.0789526
$551$	23.8014	1.01397
$552$	0	0
$553$	−7.27982	−0.309569
$554$	46.2940	1.96684
$555$	0	0
$556$	−1.98547	−0.0842025
$557$	7.09091	0.300452	0.150226	−	0.988652i	$-0.452000\pi$
$557$	7.09091	0.300452	0.150226	+	0.988652i	$0.452000\pi$
$558$	0	0
$559$	10.0432	0.424782
$560$	14.4106	0.608958
$561$	0	0
$562$	42.8240	1.80642
$563$	−18.4589	−0.777951	−0.388976	−	0.921248i	$-0.627171\pi$
$563$	−18.4589	−0.777951	−0.388976	+	0.921248i	$0.627171\pi$
$564$	0	0
$565$	0.981968	0.0413117
$566$	34.5223	1.45108
$567$	0	0
$568$	23.4545	0.984130
$569$	35.5877	1.49192	0.745958	−	0.665993i	$-0.231992\pi$
$569$	35.5877	1.49192	0.745958	+	0.665993i	$0.231992\pi$
$570$	0	0
$571$	−15.0302	−0.628994	−0.314497	−	0.949258i	$-0.601836\pi$
$571$	−15.0302	−0.628994	−0.314497	+	0.949258i	$0.601836\pi$
$572$	0.836279	0.0349666
$573$	0	0
$574$	−15.3047	−0.638806
$575$	3.23971	0.135105
$576$	0	0
$577$	4.54848	0.189356	0.0946778	−	0.995508i	$-0.469818\pi$
$577$	4.54848	0.189356	0.0946778	+	0.995508i	$0.469818\pi$
$578$	−20.2938	−0.844113
$579$	0	0
$580$	−6.65643	−0.276393
$581$	−19.5229	−0.809947
$582$	0	0
$583$	−21.6351	−0.896033
$584$	−2.62504	−0.108625
$585$	0	0
$586$	−22.5542	−0.931706
$587$	22.2808	0.919626	0.459813	−	0.888016i	$-0.347916\pi$
$587$	22.2808	0.919626	0.459813	+	0.888016i	$0.347916\pi$
$588$	0	0
$589$	10.6106	0.437201
$590$	31.7091	1.30544
$591$	0	0
$592$	−32.5842	−1.33920
$593$	12.5767	0.516464	0.258232	−	0.966083i	$-0.416860\pi$
$593$	12.5767	0.516464	0.258232	+	0.966083i	$0.416860\pi$
$594$	0	0
$595$	−17.6196	−0.722332
$596$	−0.298802	−0.0122394
$597$	0	0
$598$	11.2571	0.460339
$599$	34.7209	1.41866	0.709329	−	0.704877i	$-0.248998\pi$
$599$	34.7209	1.41866	0.709329	+	0.704877i	$0.248998\pi$
$600$	0	0
$601$	−14.3706	−0.586189	−0.293094	−	0.956084i	$-0.594685\pi$
$601$	−14.3706	−0.586189	−0.293094	+	0.956084i	$0.594685\pi$
$602$	14.7940	0.602957
$603$	0	0
$604$	0.323398	0.0131589
$605$	−16.3609	−0.665166
$606$	0	0
$607$	24.7313	1.00381	0.501906	−	0.864922i	$-0.332632\pi$
$607$	24.7313	1.00381	0.501906	+	0.864922i	$0.332632\pi$
$608$	−3.71859	−0.150809
$609$	0	0
$610$	27.4004	1.10941
$611$	−0.652230	−0.0263864
$612$	0	0
$613$	5.82521	0.235278	0.117639	−	0.993056i	$-0.462467\pi$
$613$	5.82521	0.235278	0.117639	+	0.993056i	$0.462467\pi$
$614$	42.4595	1.71352
$615$	0	0
$616$	−7.01350	−0.282582
$617$	−23.5591	−0.948454	−0.474227	−	0.880403i	$-0.657272\pi$
$617$	−23.5591	−0.948454	−0.474227	+	0.880403i	$0.657272\pi$
$618$	0	0
$619$	−41.1802	−1.65517	−0.827587	−	0.561338i	$-0.810287\pi$
$619$	−41.1802	−1.65517	−0.827587	+	0.561338i	$0.810287\pi$
$620$	−2.96741	−0.119174
$621$	0	0
$622$	−33.2717	−1.33407
$623$	−7.24936	−0.290440
$624$	0	0
$625$	−21.0635	−0.842538
$626$	35.5705	1.42168
$627$	0	0
$628$	3.60074	0.143685
$629$	39.8401	1.58853
$630$	0	0
$631$	3.69732	0.147188	0.0735940	−	0.997288i	$-0.476553\pi$
$631$	3.69732	0.147188	0.0735940	+	0.997288i	$0.476553\pi$
$632$	12.1931	0.485017
$633$	0	0
$634$	−2.48051	−0.0985136
$635$	−10.4573	−0.414987
$636$	0	0
$637$	−7.33652	−0.290684
$638$	28.7323	1.13752
$639$	0	0
$640$	−27.3361	−1.08055
$641$	13.9214	0.549862	0.274931	−	0.961464i	$-0.411345\pi$
$641$	13.9214	0.549862	0.274931	+	0.961464i	$0.411345\pi$
$642$	0	0
$643$	−18.8015	−0.741458	−0.370729	−	0.928741i	$-0.620892\pi$
$643$	−18.8015	−0.741458	−0.370729	+	0.928741i	$0.620892\pi$
$644$	2.15538	0.0849337
$645$	0	0
$646$	18.5345	0.729230
$647$	−33.7042	−1.32505	−0.662525	−	0.749040i	$-0.730515\pi$
$647$	−33.7042	−1.32505	−0.662525	+	0.749040i	$0.730515\pi$
$648$	0	0
$649$	−17.7908	−0.698349
$650$	1.66218	0.0651961
$651$	0	0
$652$	6.77167	0.265199
$653$	−0.0317984	−0.00124437	−0.000622184	−	1.00000i	$-0.500198\pi$
$653$	−0.0317984	−0.00124437	−0.000622184		1.00000i	$0.500198\pi$
$654$	0	0
$655$	−8.70126	−0.339986
$656$	29.5514	1.15379
$657$	0	0
$658$	−0.960755	−0.0374541
$659$	12.5821	0.490127	0.245064	−	0.969507i	$-0.421191\pi$
$659$	12.5821	0.490127	0.245064	+	0.969507i	$0.421191\pi$
$660$	0	0
$661$	5.35269	0.208195	0.104098	−	0.994567i	$-0.466805\pi$
$661$	5.35269	0.208195	0.104098	+	0.994567i	$0.466805\pi$
$662$	0.815204	0.0316838
$663$	0	0
$664$	32.6994	1.26898
$665$	7.08872	0.274889
$666$	0	0
$667$	50.2724	1.94656
$668$	5.99209	0.231841
$669$	0	0
$670$	−13.4857	−0.521000
$671$	−15.3733	−0.593481
$672$	0	0
$673$	3.96015	0.152653	0.0763263	−	0.997083i	$-0.475681\pi$
$673$	3.96015	0.152653	0.0763263	+	0.997083i	$0.475681\pi$
$674$	−13.9870	−0.538757
$675$	0	0
$676$	−3.13370	−0.120527
$677$	−21.3303	−0.819790	−0.409895	−	0.912133i	$-0.634435\pi$
$677$	−21.3303	−0.819790	−0.409895	+	0.912133i	$0.634435\pi$
$678$	0	0
$679$	9.86640	0.378638
$680$	29.5114	1.13171
$681$	0	0
$682$	12.8088	0.490473
$683$	−40.1497	−1.53629	−0.768143	−	0.640278i	$-0.778819\pi$
$683$	−40.1497	−1.53629	−0.768143	+	0.640278i	$0.778819\pi$
$684$	0	0
$685$	−1.88817	−0.0721434
$686$	−27.1510	−1.03663
$687$	0	0
$688$	−28.5652	−1.08904
$689$	−19.4218	−0.739910
$690$	0	0
$691$	44.2291	1.68255	0.841277	−	0.540605i	$-0.181804\pi$
$691$	44.2291	1.68255	0.841277	+	0.540605i	$0.181804\pi$
$692$	−1.00707	−0.0382831
$693$	0	0
$694$	−20.3033	−0.770703
$695$	−13.7922	−0.523170
$696$	0	0
$697$	−36.1320	−1.36860
$698$	28.8023	1.09018
$699$	0	0
$700$	0.318253	0.0120288
$701$	−40.8658	−1.54348	−0.771741	−	0.635937i	$-0.780614\pi$
$701$	−40.8658	−1.54348	−0.771741	+	0.635937i	$0.780614\pi$
$702$	0	0
$703$	−16.0285	−0.604526
$704$	11.4318	0.430852
$705$	0	0
$706$	25.0918	0.944343
$707$	20.6688	0.777331
$708$	0	0
$709$	23.6669	0.888831	0.444415	−	0.895821i	$-0.353411\pi$
$709$	23.6669	0.888831	0.444415	+	0.895821i	$0.353411\pi$
$710$	−28.6173	−1.07399
$711$	0	0
$712$	12.1421	0.455045
$713$	22.4113	0.839308
$714$	0	0
$715$	5.80930	0.217255
$716$	−3.75339	−0.140271
$717$	0	0
$718$	6.21351	0.231886
$719$	2.88350	0.107536	0.0537682	−	0.998553i	$-0.482877\pi$
$719$	2.88350	0.107536	0.0537682	+	0.998553i	$0.482877\pi$
$720$	0	0
$721$	14.9724	0.557603
$722$	21.3506	0.794587
$723$	0	0
$724$	3.88057	0.144220
$725$	7.42300	0.275683
$726$	0	0
$727$	24.9146	0.924032	0.462016	−	0.886872i	$-0.347126\pi$
$727$	24.9146	0.924032	0.462016	+	0.886872i	$0.347126\pi$
$728$	−6.29601	−0.233346
$729$	0	0
$730$	3.20285	0.118543
$731$	34.9262	1.29179
$732$	0	0
$733$	30.3913	1.12253	0.561265	−	0.827636i	$-0.310315\pi$
$733$	30.3913	1.12253	0.561265	+	0.827636i	$0.310315\pi$
$734$	6.99785	0.258295
$735$	0	0
$736$	−7.85426	−0.289512
$737$	7.56635	0.278710
$738$	0	0
$739$	−35.0681	−1.29000	−0.645000	−	0.764183i	$-0.723142\pi$
$739$	−35.0681	−1.29000	−0.645000	+	0.764183i	$0.723142\pi$
$740$	4.48262	0.164784
$741$	0	0
$742$	−28.6089	−1.05027
$743$	48.6067	1.78321	0.891604	−	0.452815i	$-0.149580\pi$
$743$	48.6067	1.78321	0.891604	+	0.452815i	$0.149580\pi$
$744$	0	0
$745$	−2.07566	−0.0760463
$746$	41.1924	1.50816
$747$	0	0
$748$	2.90824	0.106336
$749$	7.65365	0.279658
$750$	0	0
$751$	47.8464	1.74594	0.872971	−	0.487772i	$-0.162190\pi$
$751$	47.8464	1.74594	0.872971	+	0.487772i	$0.162190\pi$
$752$	1.85510	0.0676484
$753$	0	0
$754$	25.7930	0.939324
$755$	2.24652	0.0817591
$756$	0	0
$757$	12.5049	0.454498	0.227249	−	0.973837i	$-0.427027\pi$
$757$	12.5049	0.454498	0.227249	+	0.973837i	$0.427027\pi$
$758$	11.9863	0.435360
$759$	0	0
$760$	−11.8731	−0.430682
$761$	−42.7721	−1.55049	−0.775244	−	0.631662i	$-0.782373\pi$
$761$	−42.7721	−1.55049	−0.775244	+	0.631662i	$0.782373\pi$
$762$	0	0
$763$	−13.5130	−0.489203
$764$	−0.930256	−0.0336555
$765$	0	0
$766$	29.8832	1.07972
$767$	−15.9707	−0.576670
$768$	0	0
$769$	40.1463	1.44771	0.723856	−	0.689951i	$-0.242368\pi$
$769$	40.1463	1.44771	0.723856	+	0.689951i	$0.242368\pi$
$770$	8.55729	0.308383
$771$	0	0
$772$	−4.59556	−0.165398
$773$	−30.0788	−1.08186	−0.540930	−	0.841068i	$-0.681928\pi$
$773$	−30.0788	−1.08186	−0.540930	+	0.841068i	$0.681928\pi$
$774$	0	0
$775$	3.30915	0.118868
$776$	−16.5255	−0.593230
$777$	0	0
$778$	−42.2832	−1.51593
$779$	14.5367	0.520830
$780$	0	0
$781$	16.0561	0.574532
$782$	39.1478	1.39992
$783$	0	0
$784$	20.8668	0.745243
$785$	25.0129	0.892749
$786$	0	0
$787$	36.2874	1.29351	0.646753	−	0.762699i	$-0.276126\pi$
$787$	36.2874	1.29351	0.646753	+	0.762699i	$0.276126\pi$
$788$	−7.10442	−0.253084
$789$	0	0
$790$	−14.8771	−0.529302
$791$	−0.728537	−0.0259038
$792$	0	0
$793$	−13.8006	−0.490075
$794$	−16.1871	−0.574460
$795$	0	0
$796$	3.19517	0.113250
$797$	37.1983	1.31763	0.658816	−	0.752304i	$-0.271057\pi$
$797$	37.1983	1.31763	0.658816	+	0.752304i	$0.271057\pi$
$798$	0	0
$799$	−2.26819	−0.0802429
$800$	−1.15973	−0.0410025
$801$	0	0
$802$	−11.9521	−0.422044
$803$	−1.79700	−0.0634148
$804$	0	0
$805$	14.9725	0.527712
$806$	11.4984	0.405014
$807$	0	0
$808$	−34.6187	−1.21788
$809$	−24.6580	−0.866929	−0.433465	−	0.901171i	$-0.642709\pi$
$809$	−24.6580	−0.866929	−0.433465	+	0.901171i	$0.642709\pi$
$810$	0	0
$811$	−14.0750	−0.494241	−0.247121	−	0.968985i	$-0.579484\pi$
$811$	−14.0750	−0.494241	−0.247121	+	0.968985i	$0.579484\pi$
$812$	4.93851	0.173308
$813$	0	0
$814$	−19.3491	−0.678186
$815$	47.0401	1.64774
$816$	0	0
$817$	−14.0516	−0.491602
$818$	−43.0728	−1.50601
$819$	0	0
$820$	−4.06540	−0.141970
$821$	−9.19780	−0.321005	−0.160503	−	0.987035i	$-0.551312\pi$
$821$	−9.19780	−0.321005	−0.160503	+	0.987035i	$0.551312\pi$
$822$	0	0
$823$	−45.9985	−1.60341	−0.801704	−	0.597721i	$-0.796073\pi$
$823$	−45.9985	−1.60341	−0.801704	+	0.597721i	$0.796073\pi$
$824$	−25.0777	−0.873623
$825$	0	0
$826$	−23.5254	−0.818555
$827$	−13.0030	−0.452157	−0.226078	−	0.974109i	$-0.572591\pi$
$827$	−13.0030	−0.452157	−0.226078	+	0.974109i	$0.572591\pi$
$828$	0	0
$829$	−20.9147	−0.726397	−0.363198	−	0.931712i	$-0.618315\pi$
$829$	−20.9147	−0.726397	−0.363198	+	0.931712i	$0.618315\pi$
$830$	−39.8971	−1.38485
$831$	0	0
$832$	10.2623	0.355781
$833$	−25.5135	−0.883990
$834$	0	0
$835$	41.6247	1.44048
$836$	−1.17005	−0.0404670
$837$	0	0
$838$	−14.7159	−0.508353
$839$	20.7725	0.717147	0.358574	−	0.933501i	$-0.383263\pi$
$839$	20.7725	0.717147	0.358574	+	0.933501i	$0.383263\pi$
$840$	0	0
$841$	86.1868	2.97196
$842$	12.3508	0.425635
$843$	0	0
$844$	3.48806	0.120064
$845$	−21.7686	−0.748862
$846$	0	0
$847$	12.1384	0.417081
$848$	55.2401	1.89695
$849$	0	0
$850$	5.78040	0.198266
$851$	−33.8548	−1.16053
$852$	0	0
$853$	54.1788	1.85505	0.927524	−	0.373764i	$-0.121933\pi$
$853$	54.1788	1.85505	0.927524	+	0.373764i	$0.121933\pi$
$854$	−20.3288	−0.695636
$855$	0	0
$856$	−12.8193	−0.438154
$857$	18.4237	0.629342	0.314671	−	0.949201i	$-0.398106\pi$
$857$	18.4237	0.629342	0.314671	+	0.949201i	$0.398106\pi$
$858$	0	0
$859$	−12.3949	−0.422909	−0.211455	−	0.977388i	$-0.567820\pi$
$859$	−12.3949	−0.422909	−0.211455	+	0.977388i	$0.567820\pi$
$860$	3.92973	0.134003
$861$	0	0
$862$	−21.6784	−0.738369
$863$	42.6640	1.45230	0.726150	−	0.687536i	$-0.241308\pi$
$863$	42.6640	1.45230	0.726150	+	0.687536i	$0.241308\pi$
$864$	0	0
$865$	−6.99572	−0.237861
$866$	14.8276	0.503862
$867$	0	0
$868$	2.20157	0.0747261
$869$	8.34696	0.283151
$870$	0	0
$871$	6.79230	0.230148
$872$	22.6332	0.766457
$873$	0	0
$874$	−15.7500	−0.532752
$875$	18.1930	0.615035
$876$	0	0
$877$	37.5669	1.26854	0.634272	−	0.773110i	$-0.281300\pi$
$877$	37.5669	1.26854	0.634272	+	0.773110i	$0.281300\pi$
$878$	21.3188	0.719474
$879$	0	0
$880$	−16.5230	−0.556991
$881$	−23.7938	−0.801632	−0.400816	−	0.916159i	$-0.631273\pi$
$881$	−23.7938	−0.801632	−0.400816	+	0.916159i	$0.631273\pi$
$882$	0	0
$883$	−7.57729	−0.254996	−0.127498	−	0.991839i	$-0.540695\pi$
$883$	−7.57729	−0.254996	−0.127498	+	0.991839i	$0.540695\pi$
$884$	2.61073	0.0878082
$885$	0	0
$886$	44.0477	1.47981
$887$	14.2998	0.480140	0.240070	−	0.970756i	$-0.422830\pi$
$887$	14.2998	0.480140	0.240070	+	0.970756i	$0.422830\pi$
$888$	0	0
$889$	7.75847	0.260211
$890$	−14.8148	−0.496594
$891$	0	0
$892$	7.99799	0.267793
$893$	0.912542	0.0305371
$894$	0	0
$895$	−26.0733	−0.871535
$896$	20.2811	0.677543
$897$	0	0
$898$	27.5860	0.920558
$899$	51.3498	1.71261
$900$	0	0
$901$	−67.5411	−2.25012
$902$	17.5482	0.584292
$903$	0	0
$904$	1.22024	0.0405847
$905$	26.9568	0.896075
$906$	0	0
$907$	−12.8784	−0.427621	−0.213811	−	0.976875i	$-0.568588\pi$
$907$	−12.8784	−0.427621	−0.213811	+	0.976875i	$0.568588\pi$
$908$	3.88829	0.129037
$909$	0	0
$910$	7.68186	0.254651
$911$	−10.5950	−0.351030	−0.175515	−	0.984477i	$-0.556159\pi$
$911$	−10.5950	−0.351030	−0.175515	+	0.984477i	$0.556159\pi$
$912$	0	0
$913$	22.3848	0.740828
$914$	−41.2576	−1.36468
$915$	0	0
$916$	8.52705	0.281742
$917$	6.45560	0.213183
$918$	0	0
$919$	32.1490	1.06050	0.530248	−	0.847843i	$-0.322099\pi$
$919$	32.1490	1.06050	0.530248	+	0.847843i	$0.322099\pi$
$920$	−25.0778	−0.826792
$921$	0	0
$922$	−22.1220	−0.728550
$923$	14.4135	0.474427
$924$	0	0
$925$	−4.99885	−0.164361
$926$	62.2317	2.04506
$927$	0	0
$928$	−17.9961	−0.590751
$929$	−9.15675	−0.300423	−0.150212	−	0.988654i	$-0.547995\pi$
$929$	−9.15675	−0.300423	−0.150212	+	0.988654i	$0.547995\pi$
$930$	0	0
$931$	10.2646	0.336409
$932$	−6.74706	−0.221007
$933$	0	0
$934$	5.78153	0.189177
$935$	20.2024	0.660690
$936$	0	0
$937$	34.0677	1.11294	0.556472	−	0.830867i	$-0.312155\pi$
$937$	34.0677	1.11294	0.556472	+	0.830867i	$0.312155\pi$
$938$	10.0053	0.326684
$939$	0	0
$940$	−0.255206	−0.00832391
$941$	43.4702	1.41709	0.708544	−	0.705666i	$-0.249352\pi$
$941$	43.4702	1.41709	0.708544	+	0.705666i	$0.249352\pi$
$942$	0	0
$943$	30.7038	0.999854
$944$	45.4246	1.47845
$945$	0	0
$946$	−16.9626	−0.551502
$947$	−48.3338	−1.57064	−0.785319	−	0.619091i	$-0.787501\pi$
$947$	−48.3338	−1.57064	−0.785319	+	0.619091i	$0.787501\pi$
$948$	0	0
$949$	−1.61317	−0.0523656
$950$	−2.32558	−0.0754516
$951$	0	0
$952$	−21.8950	−0.709621
$953$	46.6568	1.51136	0.755681	−	0.654939i	$-0.227306\pi$
$953$	46.6568	1.51136	0.755681	+	0.654939i	$0.227306\pi$
$954$	0	0
$955$	−6.46212	−0.209109
$956$	−2.77624	−0.0897900
$957$	0	0
$958$	60.3005	1.94822
$959$	1.40087	0.0452363
$960$	0	0
$961$	−8.10844	−0.261563
$962$	−17.3697	−0.560020
$963$	0	0
$964$	−3.52310	−0.113471
$965$	−31.9235	−1.02765
$966$	0	0
$967$	36.8966	1.18651	0.593257	−	0.805013i	$-0.297842\pi$
$967$	36.8966	1.18651	0.593257	+	0.805013i	$0.297842\pi$
$968$	−20.3309	−0.653461
$969$	0	0
$970$	20.1630	0.647395
$971$	−33.7338	−1.08257	−0.541284	−	0.840840i	$-0.682062\pi$
$971$	−33.7338	−1.08257	−0.541284	+	0.840840i	$0.682062\pi$
$972$	0	0
$973$	10.2327	0.328045
$974$	−56.3998	−1.80717
$975$	0	0
$976$	39.2523	1.25643
$977$	−50.0312	−1.60064	−0.800319	−	0.599574i	$-0.795337\pi$
$977$	−50.0312	−1.60064	−0.800319	+	0.599574i	$0.795337\pi$
$978$	0	0
$979$	8.31204	0.265654
$980$	−2.87066	−0.0916998
$981$	0	0
$982$	22.7170	0.724929
$983$	−36.5241	−1.16494	−0.582469	−	0.812853i	$-0.697913\pi$
$983$	−36.5241	−1.16494	−0.582469	+	0.812853i	$0.697913\pi$
$984$	0	0
$985$	−49.3515	−1.57247
$986$	89.6976	2.85655
$987$	0	0
$988$	−1.05035	−0.0334161
$989$	−29.6792	−0.943742
$990$	0	0
$991$	6.49223	0.206232	0.103116	−	0.994669i	$-0.467119\pi$
$991$	6.49223	0.206232	0.103116	+	0.994669i	$0.467119\pi$
$992$	−8.02259	−0.254717
$993$	0	0
$994$	21.2316	0.673425
$995$	22.1956	0.703648
$996$	0	0
$997$	26.1488	0.828140	0.414070	−	0.910245i	$-0.364107\pi$
$997$	26.1488	0.828140	0.414070	+	0.910245i	$0.364107\pi$
$998$	−19.0694	−0.603630
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.8	✓	25
3.2	odd	2		4023.2.a.f.1.18	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.8	✓	25	1.1	even	1	trivial
4023.2.a.f.1.18	yes	25	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.51618 q^{2} +0.298802 q^{4} +2.07566 q^{5} -1.53996 q^{7} +2.57932 q^{8} +O(q^{10})\)	\(q-1.51618 q^{2} +0.298802 q^{4} +2.07566 q^{5} -1.53996 q^{7} +2.57932 q^{8} -3.14707 q^{10} +1.76571 q^{11} +1.58507 q^{13} +2.33486 q^{14} -4.50832 q^{16} +5.51225 q^{17} -2.21769 q^{19} +0.620212 q^{20} -2.67713 q^{22} -4.68413 q^{23} -0.691637 q^{25} -2.40325 q^{26} -0.460145 q^{28} -10.7325 q^{29} -4.78451 q^{31} +1.67678 q^{32} -8.35756 q^{34} -3.19644 q^{35} +7.22756 q^{37} +3.36242 q^{38} +5.35380 q^{40} -6.55487 q^{41} +6.33612 q^{43} +0.527597 q^{44} +7.10198 q^{46} -0.411483 q^{47} -4.62851 q^{49} +1.04865 q^{50} +0.473623 q^{52} -12.2529 q^{53} +3.66501 q^{55} -3.97206 q^{56} +16.2724 q^{58} -10.0757 q^{59} -8.70662 q^{61} +7.25418 q^{62} +6.47434 q^{64} +3.29007 q^{65} +4.28517 q^{67} +1.64707 q^{68} +4.84638 q^{70} +9.09329 q^{71} -1.01772 q^{73} -10.9583 q^{74} -0.662651 q^{76} -2.71913 q^{77} +4.72727 q^{79} -9.35774 q^{80} +9.93836 q^{82} +12.6775 q^{83} +11.4415 q^{85} -9.60669 q^{86} +4.55433 q^{88} +4.70749 q^{89} -2.44095 q^{91} -1.39963 q^{92} +0.623882 q^{94} -4.60317 q^{95} -6.40690 q^{97} +7.01766 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 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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