LMFDB - Embedded newform 4005.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.217002$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.217002 q^{2} -1.95291 q^{4} +1.00000 q^{5} -2.89958 q^{7} +0.857790 q^{8} +O(q^{10})$	\(q-0.217002 q^{2} -1.95291 q^{4} +1.00000 q^{5} -2.89958 q^{7} +0.857790 q^{8} -0.217002 q^{10} -3.07479 q^{11} +0.0854942 q^{13} +0.629215 q^{14} +3.71968 q^{16} +2.13219 q^{17} +2.44475 q^{19} -1.95291 q^{20} +0.667236 q^{22} -2.98507 q^{23} +1.00000 q^{25} -0.0185524 q^{26} +5.66262 q^{28} +6.39489 q^{29} +7.97229 q^{31} -2.52276 q^{32} -0.462689 q^{34} -2.89958 q^{35} -1.01070 q^{37} -0.530516 q^{38} +0.857790 q^{40} +4.08276 q^{41} -1.92193 q^{43} +6.00479 q^{44} +0.647767 q^{46} -8.73550 q^{47} +1.40756 q^{49} -0.217002 q^{50} -0.166962 q^{52} +6.19276 q^{53} -3.07479 q^{55} -2.48723 q^{56} -1.38770 q^{58} -11.6153 q^{59} +10.7793 q^{61} -1.73000 q^{62} -6.89191 q^{64} +0.0854942 q^{65} -4.48372 q^{67} -4.16397 q^{68} +0.629215 q^{70} -9.54118 q^{71} +1.45652 q^{73} +0.219325 q^{74} -4.77437 q^{76} +8.91560 q^{77} -12.8611 q^{79} +3.71968 q^{80} -0.885968 q^{82} -4.61282 q^{83} +2.13219 q^{85} +0.417062 q^{86} -2.63752 q^{88} -1.00000 q^{89} -0.247897 q^{91} +5.82958 q^{92} +1.89562 q^{94} +2.44475 q^{95} +2.35298 q^{97} -0.305442 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.217002	−0.153444	−0.0767218	−	0.997053i	$-0.524445\pi$
$2$	−0.217002	−0.153444	−0.0767218	+	0.997053i	$0.524445\pi$
$3$	0	0
$3$	0	0
$4$	−1.95291	−0.976455
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.89958	−1.09594	−0.547969	−	0.836499i	$-0.684599\pi$
$7$	−2.89958	−1.09594	−0.547969	+	0.836499i	$0.684599\pi$
$8$	0.857790	0.303274
$9$	0	0
$10$	−0.217002	−0.0686221
$11$	−3.07479	−0.927085	−0.463542	−	0.886075i	$-0.653422\pi$
$11$	−3.07479	−0.927085	−0.463542	+	0.886075i	$0.653422\pi$
$12$	0	0
$13$	0.0854942	0.0237118	0.0118559	−	0.999930i	$-0.496226\pi$
$13$	0.0854942	0.0237118	0.0118559	+	0.999930i	$0.496226\pi$
$14$	0.629215	0.168165
$15$	0	0
$16$	3.71968	0.929920
$17$	2.13219	0.517132	0.258566	−	0.965994i	$-0.416750\pi$
$17$	2.13219	0.517132	0.258566	+	0.965994i	$0.416750\pi$
$18$	0	0
$19$	2.44475	0.560864	0.280432	−	0.959874i	$-0.409522\pi$
$19$	2.44475	0.560864	0.280432	+	0.959874i	$0.409522\pi$
$20$	−1.95291	−0.436684
$21$	0	0
$22$	0.667236	0.142255
$23$	−2.98507	−0.622431	−0.311215	−	0.950339i	$-0.600736\pi$
$23$	−2.98507	−0.622431	−0.311215	+	0.950339i	$0.600736\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.0185524	−0.00363843
$27$	0	0
$28$	5.66262	1.07013
$29$	6.39489	1.18750	0.593750	−	0.804649i	$-0.297647\pi$
$29$	6.39489	1.18750	0.593750	+	0.804649i	$0.297647\pi$
$30$	0	0
$31$	7.97229	1.43186	0.715932	−	0.698170i	$-0.246002\pi$
$31$	7.97229	1.43186	0.715932	+	0.698170i	$0.246002\pi$
$32$	−2.52276	−0.445965
$33$	0	0
$34$	−0.462689	−0.0793506
$35$	−2.89958	−0.490118
$36$	0	0
$37$	−1.01070	−0.166158	−0.0830792	−	0.996543i	$-0.526475\pi$
$37$	−1.01070	−0.166158	−0.0830792	+	0.996543i	$0.526475\pi$
$38$	−0.530516	−0.0860610
$39$	0	0
$40$	0.857790	0.135628
$41$	4.08276	0.637621	0.318810	−	0.947819i	$-0.396717\pi$
$41$	4.08276	0.637621	0.318810	+	0.947819i	$0.396717\pi$
$42$	0	0
$43$	−1.92193	−0.293091	−0.146546	−	0.989204i	$-0.546816\pi$
$43$	−1.92193	−0.293091	−0.146546	+	0.989204i	$0.546816\pi$
$44$	6.00479	0.905256
$45$	0	0
$46$	0.647767	0.0955080
$47$	−8.73550	−1.27420	−0.637102	−	0.770780i	$-0.719867\pi$
$47$	−8.73550	−1.27420	−0.637102	+	0.770780i	$0.719867\pi$
$48$	0	0
$49$	1.40756	0.201079
$50$	−0.217002	−0.0306887
$51$	0	0
$52$	−0.166962	−0.0231535
$53$	6.19276	0.850641	0.425321	−	0.905043i	$-0.360161\pi$
$53$	6.19276	0.850641	0.425321	+	0.905043i	$0.360161\pi$
$54$	0	0
$55$	−3.07479	−0.414605
$56$	−2.48723	−0.332370
$57$	0	0
$58$	−1.38770	−0.182214
$59$	−11.6153	−1.51219	−0.756094	−	0.654463i	$-0.772895\pi$
$59$	−11.6153	−1.51219	−0.756094	+	0.654463i	$0.772895\pi$
$60$	0	0
$61$	10.7793	1.38014	0.690072	−	0.723740i	$-0.257579\pi$
$61$	10.7793	1.38014	0.690072	+	0.723740i	$0.257579\pi$
$62$	−1.73000	−0.219711
$63$	0	0
$64$	−6.89191	−0.861489
$65$	0.0854942	0.0106042
$66$	0	0
$67$	−4.48372	−0.547774	−0.273887	−	0.961762i	$-0.588309\pi$
$67$	−4.48372	−0.547774	−0.273887	+	0.961762i	$0.588309\pi$
$68$	−4.16397	−0.504956
$69$	0	0
$70$	0.629215	0.0752055
$71$	−9.54118	−1.13233	−0.566165	−	0.824292i	$-0.691574\pi$
$71$	−9.54118	−1.13233	−0.566165	+	0.824292i	$0.691574\pi$
$72$	0	0
$73$	1.45652	0.170473	0.0852363	−	0.996361i	$-0.472835\pi$
$73$	1.45652	0.170473	0.0852363	+	0.996361i	$0.472835\pi$
$74$	0.219325	0.0254960
$75$	0	0
$76$	−4.77437	−0.547658
$77$	8.91560	1.01603
$78$	0	0
$79$	−12.8611	−1.44699	−0.723493	−	0.690332i	$-0.757464\pi$
$79$	−12.8611	−1.44699	−0.723493	+	0.690332i	$0.757464\pi$
$80$	3.71968	0.415873
$81$	0	0
$82$	−0.885968	−0.0978388
$83$	−4.61282	−0.506322	−0.253161	−	0.967424i	$-0.581470\pi$
$83$	−4.61282	−0.506322	−0.253161	+	0.967424i	$0.581470\pi$
$84$	0	0
$85$	2.13219	0.231268
$86$	0.417062	0.0449730
$87$	0	0
$88$	−2.63752	−0.281161
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−0.247897	−0.0259867
$92$	5.82958	0.607776
$93$	0	0
$94$	1.89562	0.195518
$95$	2.44475	0.250826
$96$	0	0
$97$	2.35298	0.238909	0.119454	−	0.992840i	$-0.461886\pi$
$97$	2.35298	0.238909	0.119454	+	0.992840i	$0.461886\pi$
$98$	−0.305442	−0.0308543
$99$	0	0
$100$	−1.95291	−0.195291
$101$	−12.5683	−1.25059	−0.625296	−	0.780388i	$-0.715022\pi$
$101$	−12.5683	−1.25059	−0.625296	+	0.780388i	$0.715022\pi$
$102$	0	0
$103$	−2.14879	−0.211727	−0.105863	−	0.994381i	$-0.533761\pi$
$103$	−2.14879	−0.211727	−0.105863	+	0.994381i	$0.533761\pi$
$104$	0.0733360	0.00719119
$105$	0	0
$106$	−1.34384	−0.130526
$107$	2.32018	0.224301	0.112150	−	0.993691i	$-0.464226\pi$
$107$	2.32018	0.224301	0.112150	+	0.993691i	$0.464226\pi$
$108$	0	0
$109$	17.7443	1.69960	0.849798	−	0.527108i	$-0.176724\pi$
$109$	17.7443	1.69960	0.849798	+	0.527108i	$0.176724\pi$
$110$	0.667236	0.0636185
$111$	0	0
$112$	−10.7855	−1.01913
$113$	−5.47285	−0.514842	−0.257421	−	0.966299i	$-0.582873\pi$
$113$	−5.47285	−0.514842	−0.257421	+	0.966299i	$0.582873\pi$
$114$	0	0
$115$	−2.98507	−0.278359
$116$	−12.4886	−1.15954
$117$	0	0
$118$	2.52055	0.232036
$119$	−6.18245	−0.566744
$120$	0	0
$121$	−1.54566	−0.140514
$122$	−2.33912	−0.211774
$123$	0	0
$124$	−15.5692	−1.39815
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.94804	−0.439067	−0.219534	−	0.975605i	$-0.570454\pi$
$127$	−4.94804	−0.439067	−0.219534	+	0.975605i	$0.570454\pi$
$128$	6.54107	0.578155
$129$	0	0
$130$	−0.0185524	−0.00162715
$131$	−4.76457	−0.416282	−0.208141	−	0.978099i	$-0.566741\pi$
$131$	−4.76457	−0.416282	−0.208141	+	0.978099i	$0.566741\pi$
$132$	0	0
$133$	−7.08874	−0.614672
$134$	0.972977	0.0840524
$135$	0	0
$136$	1.82897	0.156833
$137$	−16.4896	−1.40881	−0.704403	−	0.709801i	$-0.748785\pi$
$137$	−16.4896	−1.40881	−0.704403	+	0.709801i	$0.748785\pi$
$138$	0	0
$139$	15.7987	1.34003	0.670013	−	0.742349i	$-0.266289\pi$
$139$	15.7987	1.34003	0.670013	+	0.742349i	$0.266289\pi$
$140$	5.66262	0.478578
$141$	0	0
$142$	2.07046	0.173749
$143$	−0.262877	−0.0219829
$144$	0	0
$145$	6.39489	0.531066
$146$	−0.316068	−0.0261579
$147$	0	0
$148$	1.97381	0.162246
$149$	−20.2146	−1.65604	−0.828022	−	0.560696i	$-0.810534\pi$
$149$	−20.2146	−1.65604	−0.828022	+	0.560696i	$0.810534\pi$
$150$	0	0
$151$	−11.1576	−0.907990	−0.453995	−	0.891004i	$-0.650002\pi$
$151$	−11.1576	−0.907990	−0.453995	+	0.891004i	$0.650002\pi$
$152$	2.09708	0.170096
$153$	0	0
$154$	−1.93470	−0.155903
$155$	7.97229	0.640349
$156$	0	0
$157$	−10.4281	−0.832253	−0.416127	−	0.909307i	$-0.636613\pi$
$157$	−10.4281	−0.832253	−0.416127	+	0.909307i	$0.636613\pi$
$158$	2.79088	0.222031
$159$	0	0
$160$	−2.52276	−0.199441
$161$	8.65545	0.682145
$162$	0	0
$163$	21.8735	1.71326	0.856631	−	0.515930i	$-0.172553\pi$
$163$	21.8735	1.71326	0.856631	+	0.515930i	$0.172553\pi$
$164$	−7.97327	−0.622608
$165$	0	0
$166$	1.00099	0.0776919
$167$	−9.51097	−0.735981	−0.367990	−	0.929830i	$-0.619954\pi$
$167$	−9.51097	−0.735981	−0.367990	+	0.929830i	$0.619954\pi$
$168$	0	0
$169$	−12.9927	−0.999438
$170$	−0.462689	−0.0354867
$171$	0	0
$172$	3.75335	0.286190
$173$	11.7120	0.890447	0.445224	−	0.895419i	$-0.353124\pi$
$173$	11.7120	0.890447	0.445224	+	0.895419i	$0.353124\pi$
$174$	0	0
$175$	−2.89958	−0.219188
$176$	−11.4372	−0.862114
$177$	0	0
$178$	0.217002	0.0162650
$179$	−10.5402	−0.787808	−0.393904	−	0.919152i	$-0.628876\pi$
$179$	−10.5402	−0.787808	−0.393904	+	0.919152i	$0.628876\pi$
$180$	0	0
$181$	−14.4072	−1.07088	−0.535439	−	0.844574i	$-0.679854\pi$
$181$	−14.4072	−1.07088	−0.535439	+	0.844574i	$0.679854\pi$
$182$	0.0537942	0.00398749
$183$	0	0
$184$	−2.56056	−0.188767
$185$	−1.01070	−0.0743083
$186$	0	0
$187$	−6.55604	−0.479425
$188$	17.0596	1.24420
$189$	0	0
$190$	−0.530516	−0.0384876
$191$	13.9019	1.00590	0.502952	−	0.864314i	$-0.332247\pi$
$191$	13.9019	1.00590	0.502952	+	0.864314i	$0.332247\pi$
$192$	0	0
$193$	17.4291	1.25457	0.627287	−	0.778788i	$-0.284165\pi$
$193$	17.4291	1.25457	0.627287	+	0.778788i	$0.284165\pi$
$194$	−0.510601	−0.0366590
$195$	0	0
$196$	−2.74883	−0.196345
$197$	−15.6196	−1.11285	−0.556426	−	0.830897i	$-0.687828\pi$
$197$	−15.6196	−1.11285	−0.556426	+	0.830897i	$0.687828\pi$
$198$	0	0
$199$	14.9293	1.05831	0.529155	−	0.848525i	$-0.322509\pi$
$199$	14.9293	1.05831	0.529155	+	0.848525i	$0.322509\pi$
$200$	0.857790	0.0606549
$201$	0	0
$202$	2.72735	0.191895
$203$	−18.5425	−1.30143
$204$	0	0
$205$	4.08276	0.285153
$206$	0.466292	0.0324881
$207$	0	0
$208$	0.318011	0.0220501
$209$	−7.51709	−0.519968
$210$	0	0
$211$	−7.92872	−0.545835	−0.272918	−	0.962037i	$-0.587989\pi$
$211$	−7.92872	−0.545835	−0.272918	+	0.962037i	$0.587989\pi$
$212$	−12.0939	−0.830613
$213$	0	0
$214$	−0.503485	−0.0344175
$215$	−1.92193	−0.131074
$216$	0	0
$217$	−23.1163	−1.56923
$218$	−3.85055	−0.260792
$219$	0	0
$220$	6.00479	0.404843
$221$	0.182290	0.0122621
$222$	0	0
$223$	−20.6302	−1.38150	−0.690751	−	0.723093i	$-0.742720\pi$
$223$	−20.6302	−1.38150	−0.690751	+	0.723093i	$0.742720\pi$
$224$	7.31493	0.488749
$225$	0	0
$226$	1.18762	0.0789993
$227$	−14.5245	−0.964027	−0.482013	−	0.876164i	$-0.660094\pi$
$227$	−14.5245	−0.964027	−0.482013	+	0.876164i	$0.660094\pi$
$228$	0	0
$229$	−3.90028	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$229$	−3.90028	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$230$	0.647767	0.0427125
$231$	0	0
$232$	5.48547	0.360139
$233$	−6.62860	−0.434254	−0.217127	−	0.976143i	$-0.569669\pi$
$233$	−6.62860	−0.434254	−0.217127	+	0.976143i	$0.569669\pi$
$234$	0	0
$235$	−8.73550	−0.569841
$236$	22.6837	1.47658
$237$	0	0
$238$	1.34160	0.0869633
$239$	19.2676	1.24632	0.623159	−	0.782095i	$-0.285849\pi$
$239$	19.2676	1.24632	0.623159	+	0.782095i	$0.285849\pi$
$240$	0	0
$241$	19.6370	1.26493	0.632465	−	0.774589i	$-0.282043\pi$
$241$	19.6370	1.26493	0.632465	+	0.774589i	$0.282043\pi$
$242$	0.335411	0.0215610
$243$	0	0
$244$	−21.0510	−1.34765
$245$	1.40756	0.0899254
$246$	0	0
$247$	0.209012	0.0132991
$248$	6.83854	0.434248
$249$	0	0
$250$	−0.217002	−0.0137244
$251$	15.7920	0.996783	0.498392	−	0.866952i	$-0.333924\pi$
$251$	15.7920	0.996783	0.498392	+	0.866952i	$0.333924\pi$
$252$	0	0
$253$	9.17848	0.577046
$254$	1.07373	0.0673721
$255$	0	0
$256$	12.3644	0.772775
$257$	−20.2283	−1.26181	−0.630903	−	0.775862i	$-0.717315\pi$
$257$	−20.2283	−1.26181	−0.630903	+	0.775862i	$0.717315\pi$
$258$	0	0
$259$	2.93061	0.182099
$260$	−0.166962	−0.0103546
$261$	0	0
$262$	1.03392	0.0638759
$263$	−6.98711	−0.430843	−0.215422	−	0.976521i	$-0.569113\pi$
$263$	−6.98711	−0.430843	−0.215422	+	0.976521i	$0.569113\pi$
$264$	0	0
$265$	6.19276	0.380418
$266$	1.53827	0.0943175
$267$	0	0
$268$	8.75631	0.534877
$269$	10.8371	0.660749	0.330375	−	0.943850i	$-0.392825\pi$
$269$	10.8371	0.660749	0.330375	+	0.943850i	$0.392825\pi$
$270$	0	0
$271$	−13.3936	−0.813601	−0.406801	−	0.913517i	$-0.633356\pi$
$271$	−13.3936	−0.813601	−0.406801	+	0.913517i	$0.633356\pi$
$272$	7.93106	0.480891
$273$	0	0
$274$	3.57829	0.216172
$275$	−3.07479	−0.185417
$276$	0	0
$277$	−20.2556	−1.21704	−0.608520	−	0.793539i	$-0.708236\pi$
$277$	−20.2556	−1.21704	−0.608520	+	0.793539i	$0.708236\pi$
$278$	−3.42834	−0.205618
$279$	0	0
$280$	−2.48723	−0.148640
$281$	−17.8054	−1.06218	−0.531090	−	0.847315i	$-0.678218\pi$
$281$	−17.8054	−1.06218	−0.531090	+	0.847315i	$0.678218\pi$
$282$	0	0
$283$	12.8718	0.765150	0.382575	−	0.923924i	$-0.375037\pi$
$283$	12.8718	0.765150	0.382575	+	0.923924i	$0.375037\pi$
$284$	18.6331	1.10567
$285$	0	0
$286$	0.0570448	0.00337313
$287$	−11.8383	−0.698793
$288$	0	0
$289$	−12.4538	−0.732575
$290$	−1.38770	−0.0814888
$291$	0	0
$292$	−2.84445	−0.166459
$293$	−23.9695	−1.40031	−0.700156	−	0.713990i	$-0.746886\pi$
$293$	−23.9695	−1.40031	−0.700156	+	0.713990i	$0.746886\pi$
$294$	0	0
$295$	−11.6153	−0.676271
$296$	−0.866970	−0.0503916
$297$	0	0
$298$	4.38661	0.254109
$299$	−0.255206	−0.0147590
$300$	0	0
$301$	5.57278	0.321210
$302$	2.42122	0.139325
$303$	0	0
$304$	9.09368	0.521558
$305$	10.7793	0.617219
$306$	0	0
$307$	24.2471	1.38385	0.691927	−	0.721967i	$-0.256762\pi$
$307$	24.2471	1.38385	0.691927	+	0.721967i	$0.256762\pi$
$308$	−17.4114	−0.992105
$309$	0	0
$310$	−1.73000	−0.0982575
$311$	2.02464	0.114807	0.0574034	−	0.998351i	$-0.481718\pi$
$311$	2.02464	0.114807	0.0574034	+	0.998351i	$0.481718\pi$
$312$	0	0
$313$	−13.7597	−0.777746	−0.388873	−	0.921291i	$-0.627136\pi$
$313$	−13.7597	−0.777746	−0.388873	+	0.921291i	$0.627136\pi$
$314$	2.26292	0.127704
$315$	0	0
$316$	25.1165	1.41292
$317$	25.9557	1.45782	0.728909	−	0.684611i	$-0.240028\pi$
$317$	25.9557	1.45782	0.728909	+	0.684611i	$0.240028\pi$
$318$	0	0
$319$	−19.6629	−1.10091
$320$	−6.89191	−0.385270
$321$	0	0
$322$	−1.87825	−0.104671
$323$	5.21267	0.290041
$324$	0	0
$325$	0.0854942	0.00474236
$326$	−4.74659	−0.262889
$327$	0	0
$328$	3.50215	0.193374
$329$	25.3293	1.39645
$330$	0	0
$331$	−14.7062	−0.808325	−0.404163	−	0.914687i	$-0.632437\pi$
$331$	−14.7062	−0.808325	−0.404163	+	0.914687i	$0.632437\pi$
$332$	9.00842	0.494401
$333$	0	0
$334$	2.06390	0.112932
$335$	−4.48372	−0.244972
$336$	0	0
$337$	−21.7701	−1.18589	−0.592947	−	0.805242i	$-0.702036\pi$
$337$	−21.7701	−1.18589	−0.592947	+	0.805242i	$0.702036\pi$
$338$	2.81944	0.153357
$339$	0	0
$340$	−4.16397	−0.225823
$341$	−24.5131	−1.32746
$342$	0	0
$343$	16.2157	0.875567
$344$	−1.64861	−0.0888871
$345$	0	0
$346$	−2.54153	−0.136633
$347$	−28.7105	−1.54126	−0.770629	−	0.637284i	$-0.780058\pi$
$347$	−28.7105	−1.54126	−0.770629	+	0.637284i	$0.780058\pi$
$348$	0	0
$349$	−5.52078	−0.295520	−0.147760	−	0.989023i	$-0.547206\pi$
$349$	−5.52078	−0.295520	−0.147760	+	0.989023i	$0.547206\pi$
$350$	0.629215	0.0336329
$351$	0	0
$352$	7.75695	0.413447
$353$	19.8488	1.05645	0.528223	−	0.849106i	$-0.322858\pi$
$353$	19.8488	1.05645	0.528223	+	0.849106i	$0.322858\pi$
$354$	0	0
$355$	−9.54118	−0.506394
$356$	1.95291	0.103504
$357$	0	0
$358$	2.28723	0.120884
$359$	7.11756	0.375651	0.187825	−	0.982202i	$-0.439856\pi$
$359$	7.11756	0.375651	0.187825	+	0.982202i	$0.439856\pi$
$360$	0	0
$361$	−13.0232	−0.685432
$362$	3.12639	0.164319
$363$	0	0
$364$	0.484121	0.0253748
$365$	1.45652	0.0762377
$366$	0	0
$367$	−29.6237	−1.54635	−0.773173	−	0.634195i	$-0.781332\pi$
$367$	−29.6237	−1.54635	−0.773173	+	0.634195i	$0.781332\pi$
$368$	−11.1035	−0.578810
$369$	0	0
$370$	0.219325	0.0114021
$371$	−17.9564	−0.932250
$372$	0	0
$373$	28.3346	1.46711	0.733556	−	0.679629i	$-0.237859\pi$
$373$	28.3346	1.46711	0.733556	+	0.679629i	$0.237859\pi$
$374$	1.42267	0.0735647
$375$	0	0
$376$	−7.49322	−0.386433
$377$	0.546725	0.0281578
$378$	0	0
$379$	−7.10576	−0.364999	−0.182499	−	0.983206i	$-0.558419\pi$
$379$	−7.10576	−0.364999	−0.182499	+	0.983206i	$0.558419\pi$
$380$	−4.77437	−0.244920
$381$	0	0
$382$	−3.01673	−0.154350
$383$	−13.5263	−0.691160	−0.345580	−	0.938389i	$-0.612318\pi$
$383$	−13.5263	−0.691160	−0.345580	+	0.938389i	$0.612318\pi$
$384$	0	0
$385$	8.91560	0.454381
$386$	−3.78215	−0.192506
$387$	0	0
$388$	−4.59515	−0.233283
$389$	−35.5717	−1.80356	−0.901779	−	0.432198i	$-0.857738\pi$
$389$	−35.5717	−1.80356	−0.901779	+	0.432198i	$0.857738\pi$
$390$	0	0
$391$	−6.36474	−0.321879
$392$	1.20739	0.0609822
$393$	0	0
$394$	3.38949	0.170760
$395$	−12.8611	−0.647111
$396$	0	0
$397$	−28.5754	−1.43416	−0.717078	−	0.696993i	$-0.754521\pi$
$397$	−28.5754	−1.43416	−0.717078	+	0.696993i	$0.754521\pi$
$398$	−3.23969	−0.162391
$399$	0	0
$400$	3.71968	0.185984
$401$	15.5212	0.775094	0.387547	−	0.921850i	$-0.373323\pi$
$401$	15.5212	0.775094	0.387547	+	0.921850i	$0.373323\pi$
$402$	0	0
$403$	0.681584	0.0339521
$404$	24.5448	1.22115
$405$	0	0
$406$	4.02376	0.199696
$407$	3.10770	0.154043
$408$	0	0
$409$	32.6342	1.61366	0.806830	−	0.590784i	$-0.201181\pi$
$409$	32.6342	1.61366	0.806830	+	0.590784i	$0.201181\pi$
$410$	−0.885968	−0.0437549
$411$	0	0
$412$	4.19640	0.206742
$413$	33.6796	1.65726
$414$	0	0
$415$	−4.61282	−0.226434
$416$	−0.215681	−0.0105746
$417$	0	0
$418$	1.63122	0.0797858
$419$	−12.9971	−0.634950	−0.317475	−	0.948267i	$-0.602835\pi$
$419$	−12.9971	−0.634950	−0.317475	+	0.948267i	$0.602835\pi$
$420$	0	0
$421$	−18.2329	−0.888617	−0.444309	−	0.895874i	$-0.646551\pi$
$421$	−18.2329	−0.888617	−0.444309	+	0.895874i	$0.646551\pi$
$422$	1.72055	0.0837550
$423$	0	0
$424$	5.31209	0.257978
$425$	2.13219	0.103426
$426$	0	0
$427$	−31.2553	−1.51255
$428$	−4.53111	−0.219019
$429$	0	0
$430$	0.417062	0.0201125
$431$	−8.50370	−0.409609	−0.204804	−	0.978803i	$-0.565656\pi$
$431$	−8.50370	−0.409609	−0.204804	+	0.978803i	$0.565656\pi$
$432$	0	0
$433$	−12.5444	−0.602846	−0.301423	−	0.953491i	$-0.597462\pi$
$433$	−12.5444	−0.602846	−0.301423	+	0.953491i	$0.597462\pi$
$434$	5.01628	0.240789
$435$	0	0
$436$	−34.6530	−1.65958
$437$	−7.29775	−0.349099
$438$	0	0
$439$	−11.3997	−0.544079	−0.272040	−	0.962286i	$-0.587698\pi$
$439$	−11.3997	−0.544079	−0.272040	+	0.962286i	$0.587698\pi$
$440$	−2.63752	−0.125739
$441$	0	0
$442$	−0.0395572	−0.00188155
$443$	1.45229	0.0690005	0.0345002	−	0.999405i	$-0.489016\pi$
$443$	1.45229	0.0690005	0.0345002	+	0.999405i	$0.489016\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	4.47680	0.211983
$447$	0	0
$448$	19.9836	0.944138
$449$	33.3724	1.57494	0.787471	−	0.616351i	$-0.211390\pi$
$449$	33.3724	1.57494	0.787471	+	0.616351i	$0.211390\pi$
$450$	0	0
$451$	−12.5537	−0.591128
$452$	10.6880	0.502720
$453$	0	0
$454$	3.15185	0.147924
$455$	−0.247897	−0.0116216
$456$	0	0
$457$	−17.8566	−0.835294	−0.417647	−	0.908609i	$-0.637145\pi$
$457$	−17.8566	−0.835294	−0.417647	+	0.908609i	$0.637145\pi$
$458$	0.846368	0.0395482
$459$	0	0
$460$	5.82958	0.271806
$461$	36.8041	1.71414	0.857068	−	0.515203i	$-0.172284\pi$
$461$	36.8041	1.71414	0.857068	+	0.515203i	$0.172284\pi$
$462$	0	0
$463$	−35.0568	−1.62923	−0.814615	−	0.580002i	$-0.803052\pi$
$463$	−35.0568	−1.62923	−0.814615	+	0.580002i	$0.803052\pi$
$464$	23.7869	1.10428
$465$	0	0
$466$	1.43842	0.0666335
$467$	−5.50869	−0.254912	−0.127456	−	0.991844i	$-0.540681\pi$
$467$	−5.50869	−0.254912	−0.127456	+	0.991844i	$0.540681\pi$
$468$	0	0
$469$	13.0009	0.600326
$470$	1.89562	0.0874385
$471$	0	0
$472$	−9.96352	−0.458608
$473$	5.90953	0.271720
$474$	0	0
$475$	2.44475	0.112173
$476$	12.0738	0.553400
$477$	0	0
$478$	−4.18111	−0.191239
$479$	7.72431	0.352933	0.176466	−	0.984307i	$-0.443533\pi$
$479$	7.72431	0.352933	0.176466	+	0.984307i	$0.443533\pi$
$480$	0	0
$481$	−0.0864092	−0.00393992
$482$	−4.26127	−0.194095
$483$	0	0
$484$	3.01853	0.137206
$485$	2.35298	0.106843
$486$	0	0
$487$	27.1768	1.23150	0.615749	−	0.787943i	$-0.288854\pi$
$487$	27.1768	1.23150	0.615749	+	0.787943i	$0.288854\pi$
$488$	9.24635	0.418563
$489$	0	0
$490$	−0.305442	−0.0137985
$491$	−19.3503	−0.873266	−0.436633	−	0.899640i	$-0.643829\pi$
$491$	−19.3503	−0.873266	−0.436633	+	0.899640i	$0.643829\pi$
$492$	0	0
$493$	13.6351	0.614094
$494$	−0.0453560	−0.00204066
$495$	0	0
$496$	29.6543	1.33152
$497$	27.6654	1.24096
$498$	0	0
$499$	17.3273	0.775678	0.387839	−	0.921727i	$-0.373222\pi$
$499$	17.3273	0.775678	0.387839	+	0.921727i	$0.373222\pi$
$500$	−1.95291	−0.0873368
$501$	0	0
$502$	−3.42690	−0.152950
$503$	−27.1992	−1.21275	−0.606376	−	0.795178i	$-0.707377\pi$
$503$	−27.1992	−1.21275	−0.606376	+	0.795178i	$0.707377\pi$
$504$	0	0
$505$	−12.5683	−0.559282
$506$	−1.99175	−0.0885440
$507$	0	0
$508$	9.66307	0.428730
$509$	1.73289	0.0768091	0.0384046	−	0.999262i	$-0.487772\pi$
$509$	1.73289	0.0768091	0.0384046	+	0.999262i	$0.487772\pi$
$510$	0	0
$511$	−4.22329	−0.186827
$512$	−15.7652	−0.696732
$513$	0	0
$514$	4.38958	0.193616
$515$	−2.14879	−0.0946871
$516$	0	0
$517$	26.8598	1.18129
$518$	−0.635949	−0.0279420
$519$	0	0
$520$	0.0733360	0.00321600
$521$	4.23423	0.185505	0.0927524	−	0.995689i	$-0.470433\pi$
$521$	4.23423	0.185505	0.0927524	+	0.995689i	$0.470433\pi$
$522$	0	0
$523$	34.3472	1.50190	0.750949	−	0.660360i	$-0.229596\pi$
$523$	34.3472	1.50190	0.750949	+	0.660360i	$0.229596\pi$
$524$	9.30478	0.406481
$525$	0	0
$526$	1.51622	0.0661102
$527$	16.9984	0.740463
$528$	0	0
$529$	−14.0893	−0.612580
$530$	−1.34384	−0.0583728
$531$	0	0
$532$	13.8437	0.600199
$533$	0.349053	0.0151191
$534$	0	0
$535$	2.32018	0.100310
$536$	−3.84609	−0.166126
$537$	0	0
$538$	−2.35167	−0.101388
$539$	−4.32794	−0.186418
$540$	0	0
$541$	29.2111	1.25588	0.627942	−	0.778260i	$-0.283897\pi$
$541$	29.2111	1.25588	0.627942	+	0.778260i	$0.283897\pi$
$542$	2.90643	0.124842
$543$	0	0
$544$	−5.37899	−0.230622
$545$	17.7443	0.760083
$546$	0	0
$547$	−12.7289	−0.544248	−0.272124	−	0.962262i	$-0.587726\pi$
$547$	−12.7289	−0.544248	−0.272124	+	0.962262i	$0.587726\pi$
$548$	32.2028	1.37564
$549$	0	0
$550$	0.667236	0.0284510
$551$	15.6339	0.666026
$552$	0	0
$553$	37.2917	1.58581
$554$	4.39550	0.186747
$555$	0	0
$556$	−30.8534	−1.30848
$557$	−28.9428	−1.22635	−0.613173	−	0.789949i	$-0.710107\pi$
$557$	−28.9428	−1.22635	−0.613173	+	0.789949i	$0.710107\pi$
$558$	0	0
$559$	−0.164314	−0.00694973
$560$	−10.7855	−0.455771
$561$	0	0
$562$	3.86381	0.162985
$563$	22.9924	0.969016	0.484508	−	0.874787i	$-0.338999\pi$
$563$	22.9924	0.969016	0.484508	+	0.874787i	$0.338999\pi$
$564$	0	0
$565$	−5.47285	−0.230244
$566$	−2.79321	−0.117407
$567$	0	0
$568$	−8.18433	−0.343407
$569$	13.8257	0.579604	0.289802	−	0.957087i	$-0.406411\pi$
$569$	13.8257	0.579604	0.289802	+	0.957087i	$0.406411\pi$
$570$	0	0
$571$	−2.42140	−0.101332	−0.0506662	−	0.998716i	$-0.516134\pi$
$571$	−2.42140	−0.101332	−0.0506662	+	0.998716i	$0.516134\pi$
$572$	0.513375	0.0214653
$573$	0	0
$574$	2.56893	0.107225
$575$	−2.98507	−0.124486
$576$	0	0
$577$	44.7007	1.86091	0.930456	−	0.366403i	$-0.119411\pi$
$577$	44.7007	1.86091	0.930456	+	0.366403i	$0.119411\pi$
$578$	2.70249	0.112409
$579$	0	0
$580$	−12.4886	−0.518563
$581$	13.3752	0.554898
$582$	0	0
$583$	−19.0415	−0.788617
$584$	1.24939	0.0517000
$585$	0	0
$586$	5.20143	0.214869
$587$	−29.3215	−1.21023	−0.605113	−	0.796139i	$-0.706872\pi$
$587$	−29.3215	−1.21023	−0.605113	+	0.796139i	$0.706872\pi$
$588$	0	0
$589$	19.4902	0.803081
$590$	2.52055	0.103770
$591$	0	0
$592$	−3.75949	−0.154514
$593$	−44.0890	−1.81052	−0.905260	−	0.424858i	$-0.860324\pi$
$593$	−44.0890	−1.81052	−0.905260	+	0.424858i	$0.860324\pi$
$594$	0	0
$595$	−6.18245	−0.253456
$596$	39.4773	1.61705
$597$	0	0
$598$	0.0553803	0.00226467
$599$	18.2067	0.743907	0.371953	−	0.928251i	$-0.378688\pi$
$599$	18.2067	0.743907	0.371953	+	0.928251i	$0.378688\pi$
$600$	0	0
$601$	13.2451	0.540279	0.270139	−	0.962821i	$-0.412930\pi$
$601$	13.2451	0.540279	0.270139	+	0.962821i	$0.412930\pi$
$602$	−1.20931	−0.0492876
$603$	0	0
$604$	21.7897	0.886612
$605$	−1.54566	−0.0628399
$606$	0	0
$607$	39.4524	1.60132	0.800662	−	0.599116i	$-0.204481\pi$
$607$	39.4524	1.60132	0.800662	+	0.599116i	$0.204481\pi$
$608$	−6.16751	−0.250125
$609$	0	0
$610$	−2.33912	−0.0947084
$611$	−0.746834	−0.0302137
$612$	0	0
$613$	32.7000	1.32074	0.660370	−	0.750940i	$-0.270399\pi$
$613$	32.7000	1.32074	0.660370	+	0.750940i	$0.270399\pi$
$614$	−5.26167	−0.212344
$615$	0	0
$616$	7.64771	0.308135
$617$	9.43966	0.380026	0.190013	−	0.981782i	$-0.439147\pi$
$617$	9.43966	0.380026	0.190013	+	0.981782i	$0.439147\pi$
$618$	0	0
$619$	34.8324	1.40003	0.700016	−	0.714127i	$-0.253176\pi$
$619$	34.8324	1.40003	0.700016	+	0.714127i	$0.253176\pi$
$620$	−15.5692	−0.625272
$621$	0	0
$622$	−0.439351	−0.0176164
$623$	2.89958	0.116169
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.98589	0.119340
$627$	0	0
$628$	20.3652	0.812658
$629$	−2.15501	−0.0859258
$630$	0	0
$631$	−29.5893	−1.17793	−0.588965	−	0.808159i	$-0.700464\pi$
$631$	−29.5893	−1.17793	−0.588965	+	0.808159i	$0.700464\pi$
$632$	−11.0321	−0.438834
$633$	0	0
$634$	−5.63244	−0.223693
$635$	−4.94804	−0.196357
$636$	0	0
$637$	0.120338	0.00476796
$638$	4.26690	0.168928
$639$	0	0
$640$	6.54107	0.258559
$641$	−42.2095	−1.66718	−0.833588	−	0.552387i	$-0.813717\pi$
$641$	−42.2095	−1.66718	−0.833588	+	0.552387i	$0.813717\pi$
$642$	0	0
$643$	8.60054	0.339172	0.169586	−	0.985515i	$-0.445757\pi$
$643$	8.60054	0.339172	0.169586	+	0.985515i	$0.445757\pi$
$644$	−16.9033	−0.666084
$645$	0	0
$646$	−1.13116	−0.0445049
$647$	−7.36546	−0.289566	−0.144783	−	0.989463i	$-0.546248\pi$
$647$	−7.36546	−0.289566	−0.144783	+	0.989463i	$0.546248\pi$
$648$	0	0
$649$	35.7148	1.40193
$650$	−0.0185524	−0.000727685 0
$651$	0	0
$652$	−42.7169	−1.67292
$653$	−46.8039	−1.83158	−0.915790	−	0.401658i	$-0.868434\pi$
$653$	−46.8039	−1.83158	−0.915790	+	0.401658i	$0.868434\pi$
$654$	0	0
$655$	−4.76457	−0.186167
$656$	15.1866	0.592936
$657$	0	0
$658$	−5.49650	−0.214276
$659$	5.79198	0.225623	0.112812	−	0.993616i	$-0.464014\pi$
$659$	5.79198	0.225623	0.112812	+	0.993616i	$0.464014\pi$
$660$	0	0
$661$	−17.2823	−0.672205	−0.336102	−	0.941825i	$-0.609109\pi$
$661$	−17.2823	−0.672205	−0.336102	+	0.941825i	$0.609109\pi$
$662$	3.19127	0.124032
$663$	0	0
$664$	−3.95683	−0.153555
$665$	−7.08874	−0.274890
$666$	0	0
$667$	−19.0892	−0.739137
$668$	18.5741	0.718652
$669$	0	0
$670$	0.972977	0.0375894
$671$	−33.1440	−1.27951
$672$	0	0
$673$	−29.0530	−1.11991	−0.559955	−	0.828523i	$-0.689182\pi$
$673$	−29.0530	−1.11991	−0.559955	+	0.828523i	$0.689182\pi$
$674$	4.72416	0.181968
$675$	0	0
$676$	25.3736	0.975906
$677$	0.715159	0.0274858	0.0137429	−	0.999906i	$-0.495625\pi$
$677$	0.715159	0.0274858	0.0137429	+	0.999906i	$0.495625\pi$
$678$	0	0
$679$	−6.82264	−0.261829
$680$	1.82897	0.0701378
$681$	0	0
$682$	5.31940	0.203690
$683$	−31.4207	−1.20228	−0.601141	−	0.799143i	$-0.705287\pi$
$683$	−31.4207	−1.20228	−0.601141	+	0.799143i	$0.705287\pi$
$684$	0	0
$685$	−16.4896	−0.630037
$686$	−3.51885	−0.134350
$687$	0	0
$688$	−7.14895	−0.272551
$689$	0.529445	0.0201703
$690$	0	0
$691$	−4.95338	−0.188435	−0.0942177	−	0.995552i	$-0.530035\pi$
$691$	−4.95338	−0.188435	−0.0942177	+	0.995552i	$0.530035\pi$
$692$	−22.8725	−0.869482
$693$	0	0
$694$	6.23023	0.236496
$695$	15.7987	0.599278
$696$	0	0
$697$	8.70523	0.329734
$698$	1.19802	0.0453457
$699$	0	0
$700$	5.66262	0.214027
$701$	−17.4064	−0.657432	−0.328716	−	0.944429i	$-0.606616\pi$
$701$	−17.4064	−0.657432	−0.328716	+	0.944429i	$0.606616\pi$
$702$	0	0
$703$	−2.47091	−0.0931923
$704$	21.1912	0.798673
$705$	0	0
$706$	−4.30724	−0.162105
$707$	36.4428	1.37057
$708$	0	0
$709$	36.6600	1.37679	0.688397	−	0.725334i	$-0.258315\pi$
$709$	36.6600	1.37679	0.688397	+	0.725334i	$0.258315\pi$
$710$	2.07046	0.0777029
$711$	0	0
$712$	−0.857790	−0.0321470
$713$	−23.7979	−0.891237
$714$	0	0
$715$	−0.262877	−0.00983103
$716$	20.5840	0.769259
$717$	0	0
$718$	−1.54453	−0.0576412
$719$	7.06178	0.263360	0.131680	−	0.991292i	$-0.457963\pi$
$719$	7.06178	0.263360	0.131680	+	0.991292i	$0.457963\pi$
$720$	0	0
$721$	6.23059	0.232039
$722$	2.82606	0.105175
$723$	0	0
$724$	28.1360	1.04566
$725$	6.39489	0.237500
$726$	0	0
$727$	1.55147	0.0575408	0.0287704	−	0.999586i	$-0.490841\pi$
$727$	1.55147	0.0575408	0.0287704	+	0.999586i	$0.490841\pi$
$728$	−0.212643	−0.00788109
$729$	0	0
$730$	−0.316068	−0.0116982
$731$	−4.09791	−0.151567
$732$	0	0
$733$	−25.3100	−0.934844	−0.467422	−	0.884034i	$-0.654817\pi$
$733$	−25.3100	−0.934844	−0.467422	+	0.884034i	$0.654817\pi$
$734$	6.42841	0.237277
$735$	0	0
$736$	7.53061	0.277582
$737$	13.7865	0.507833
$738$	0	0
$739$	−37.5699	−1.38203	−0.691016	−	0.722839i	$-0.742837\pi$
$739$	−37.5699	−1.38203	−0.691016	+	0.722839i	$0.742837\pi$
$740$	1.97381	0.0725587
$741$	0	0
$742$	3.89658	0.143048
$743$	34.4271	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$743$	34.4271	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$744$	0	0
$745$	−20.2146	−0.740605
$746$	−6.14867	−0.225119
$747$	0	0
$748$	12.8034	0.468137
$749$	−6.72756	−0.245820
$750$	0	0
$751$	21.2041	0.773748	0.386874	−	0.922133i	$-0.373555\pi$
$751$	21.2041	0.773748	0.386874	+	0.922133i	$0.373555\pi$
$752$	−32.4933	−1.18491
$753$	0	0
$754$	−0.118641	−0.00432063
$755$	−11.1576	−0.406066
$756$	0	0
$757$	−5.17722	−0.188169	−0.0940846	−	0.995564i	$-0.529992\pi$
$757$	−5.17722	−0.188169	−0.0940846	+	0.995564i	$0.529992\pi$
$758$	1.54197	0.0560067
$759$	0	0
$760$	2.09708	0.0760691
$761$	−1.46516	−0.0531121	−0.0265561	−	0.999647i	$-0.508454\pi$
$761$	−1.46516	−0.0531121	−0.0265561	+	0.999647i	$0.508454\pi$
$762$	0	0
$763$	−51.4510	−1.86265
$764$	−27.1491	−0.982220
$765$	0	0
$766$	2.93523	0.106054
$767$	−0.993044	−0.0358567
$768$	0	0
$769$	14.6373	0.527836	0.263918	−	0.964545i	$-0.414985\pi$
$769$	14.6373	0.527836	0.263918	+	0.964545i	$0.414985\pi$
$770$	−1.93470	−0.0697219
$771$	0	0
$772$	−34.0375	−1.22503
$773$	9.55856	0.343797	0.171899	−	0.985115i	$-0.445010\pi$
$773$	9.55856	0.343797	0.171899	+	0.985115i	$0.445010\pi$
$774$	0	0
$775$	7.97229	0.286373
$776$	2.01836	0.0724549
$777$	0	0
$778$	7.71914	0.276744
$779$	9.98133	0.357618
$780$	0	0
$781$	29.3372	1.04977
$782$	1.38116	0.0493902
$783$	0	0
$784$	5.23565	0.186988
$785$	−10.4281	−0.372195
$786$	0	0
$787$	34.2818	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$787$	34.2818	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$788$	30.5037	1.08665
$789$	0	0
$790$	2.79088	0.0992951
$791$	15.8690	0.564235
$792$	0	0
$793$	0.921565	0.0327257
$794$	6.20091	0.220062
$795$	0	0
$796$	−29.1556	−1.03339
$797$	20.5514	0.727969	0.363985	−	0.931405i	$-0.381416\pi$
$797$	20.5514	0.727969	0.363985	+	0.931405i	$0.381416\pi$
$798$	0	0
$799$	−18.6257	−0.658931
$800$	−2.52276	−0.0891929
$801$	0	0
$802$	−3.36814	−0.118933
$803$	−4.47849	−0.158043
$804$	0	0
$805$	8.65545	0.305065
$806$	−0.147905	−0.00520973
$807$	0	0
$808$	−10.7810	−0.379273
$809$	2.35755	0.0828871	0.0414435	−	0.999141i	$-0.486804\pi$
$809$	2.35755	0.0828871	0.0414435	+	0.999141i	$0.486804\pi$
$810$	0	0
$811$	−2.02279	−0.0710298	−0.0355149	−	0.999369i	$-0.511307\pi$
$811$	−2.02279	−0.0710298	−0.0355149	+	0.999369i	$0.511307\pi$
$812$	36.2118	1.27078
$813$	0	0
$814$	−0.674377	−0.0236369
$815$	21.8735	0.766194
$816$	0	0
$817$	−4.69863	−0.164384
$818$	−7.08170	−0.247606
$819$	0	0
$820$	−7.97327	−0.278439
$821$	38.4774	1.34287	0.671435	−	0.741064i	$-0.265678\pi$
$821$	38.4774	1.34287	0.671435	+	0.741064i	$0.265678\pi$
$822$	0	0
$823$	48.6469	1.69572	0.847862	−	0.530216i	$-0.177889\pi$
$823$	48.6469	1.69572	0.847862	+	0.530216i	$0.177889\pi$
$824$	−1.84321	−0.0642113
$825$	0	0
$826$	−7.30854	−0.254297
$827$	−50.5027	−1.75615	−0.878076	−	0.478521i	$-0.841173\pi$
$827$	−50.5027	−1.75615	−0.878076	+	0.478521i	$0.841173\pi$
$828$	0	0
$829$	−13.0462	−0.453113	−0.226557	−	0.973998i	$-0.572747\pi$
$829$	−13.0462	−0.453113	−0.226557	+	0.973998i	$0.572747\pi$
$830$	1.00099	0.0347449
$831$	0	0
$832$	−0.589218	−0.0204275
$833$	3.00117	0.103985
$834$	0	0
$835$	−9.51097	−0.329141
$836$	14.6802	0.507726
$837$	0	0
$838$	2.82040	0.0974290
$839$	−56.2774	−1.94291	−0.971455	−	0.237222i	$-0.923763\pi$
$839$	−56.2774	−1.94291	−0.971455	+	0.237222i	$0.923763\pi$
$840$	0	0
$841$	11.8946	0.410158
$842$	3.95658	0.136353
$843$	0	0
$844$	15.4841	0.532984
$845$	−12.9927	−0.446962
$846$	0	0
$847$	4.48175	0.153995
$848$	23.0351	0.791028
$849$	0	0
$850$	−0.462689	−0.0158701
$851$	3.01702	0.103422
$852$	0	0
$853$	22.0873	0.756253	0.378127	−	0.925754i	$-0.376568\pi$
$853$	22.0873	0.756253	0.378127	+	0.925754i	$0.376568\pi$
$854$	6.78248	0.232092
$855$	0	0
$856$	1.99023	0.0680247
$857$	41.4385	1.41551	0.707756	−	0.706457i	$-0.249708\pi$
$857$	41.4385	1.41551	0.707756	+	0.706457i	$0.249708\pi$
$858$	0	0
$859$	48.6370	1.65947	0.829737	−	0.558154i	$-0.188490\pi$
$859$	48.6370	1.65947	0.829737	+	0.558154i	$0.188490\pi$
$860$	3.75335	0.127988
$861$	0	0
$862$	1.84532	0.0628518
$863$	39.8223	1.35557	0.677784	−	0.735261i	$-0.262940\pi$
$863$	39.8223	1.35557	0.677784	+	0.735261i	$0.262940\pi$
$864$	0	0
$865$	11.7120	0.398220
$866$	2.72216	0.0925028
$867$	0	0
$868$	45.1440	1.53229
$869$	39.5452	1.34148
$870$	0	0
$871$	−0.383332	−0.0129887
$872$	15.2209	0.515444
$873$	0	0
$874$	1.58363	0.0535670
$875$	−2.89958	−0.0980236
$876$	0	0
$877$	−22.1393	−0.747590	−0.373795	−	0.927511i	$-0.621944\pi$
$877$	−22.1393	−0.747590	−0.373795	+	0.927511i	$0.621944\pi$
$878$	2.47376	0.0834855
$879$	0	0
$880$	−11.4372	−0.385549
$881$	8.16284	0.275013	0.137507	−	0.990501i	$-0.456091\pi$
$881$	8.16284	0.275013	0.137507	+	0.990501i	$0.456091\pi$
$882$	0	0
$883$	−13.9344	−0.468929	−0.234465	−	0.972125i	$-0.575334\pi$
$883$	−13.9344	−0.468929	−0.234465	+	0.972125i	$0.575334\pi$
$884$	−0.355995	−0.0119734
$885$	0	0
$886$	−0.315150	−0.0105877
$887$	−12.6081	−0.423339	−0.211669	−	0.977341i	$-0.567890\pi$
$887$	−12.6081	−0.423339	−0.211669	+	0.977341i	$0.567890\pi$
$888$	0	0
$889$	14.3472	0.481191
$890$	0.217002	0.00727393
$891$	0	0
$892$	40.2890	1.34897
$893$	−21.3561	−0.714655
$894$	0	0
$895$	−10.5402	−0.352318
$896$	−18.9664	−0.633621
$897$	0	0
$898$	−7.24189	−0.241665
$899$	50.9819	1.70034
$900$	0	0
$901$	13.2041	0.439894
$902$	2.72417	0.0907049
$903$	0	0
$904$	−4.69455	−0.156139
$905$	−14.4072	−0.478911
$906$	0	0
$907$	−20.2619	−0.672784	−0.336392	−	0.941722i	$-0.609207\pi$
$907$	−20.2619	−0.672784	−0.336392	+	0.941722i	$0.609207\pi$
$908$	28.3651	0.941329
$909$	0	0
$910$	0.0537942	0.00178326
$911$	−22.2785	−0.738119	−0.369059	−	0.929406i	$-0.620320\pi$
$911$	−22.2785	−0.738119	−0.369059	+	0.929406i	$0.620320\pi$
$912$	0	0
$913$	14.1834	0.469404
$914$	3.87491	0.128171
$915$	0	0
$916$	7.61689	0.251669
$917$	13.8152	0.456220
$918$	0	0
$919$	55.7446	1.83884	0.919422	−	0.393271i	$-0.128657\pi$
$919$	55.7446	1.83884	0.919422	+	0.393271i	$0.128657\pi$
$920$	−2.56056	−0.0844193
$921$	0	0
$922$	−7.98656	−0.263023
$923$	−0.815716	−0.0268496
$924$	0	0
$925$	−1.01070	−0.0332317
$926$	7.60741	0.249995
$927$	0	0
$928$	−16.1327	−0.529583
$929$	−1.15202	−0.0377967	−0.0188983	−	0.999821i	$-0.506016\pi$
$929$	−1.15202	−0.0377967	−0.0188983	+	0.999821i	$0.506016\pi$
$930$	0	0
$931$	3.44112	0.112778
$932$	12.9451	0.424030
$933$	0	0
$934$	1.19540	0.0391146
$935$	−6.55604	−0.214405
$936$	0	0
$937$	−44.8375	−1.46478	−0.732388	−	0.680887i	$-0.761594\pi$
$937$	−44.8375	−1.46478	−0.732388	+	0.680887i	$0.761594\pi$
$938$	−2.82122	−0.0921162
$939$	0	0
$940$	17.0596	0.556424
$941$	1.24362	0.0405408	0.0202704	−	0.999795i	$-0.493547\pi$
$941$	1.24362	0.0405408	0.0202704	+	0.999795i	$0.493547\pi$
$942$	0	0
$943$	−12.1873	−0.396875
$944$	−43.2053	−1.40621
$945$	0	0
$946$	−1.28238	−0.0416938
$947$	−39.5512	−1.28524	−0.642621	−	0.766185i	$-0.722153\pi$
$947$	−39.5512	−1.28524	−0.642621	+	0.766185i	$0.722153\pi$
$948$	0	0
$949$	0.124524	0.00404222
$950$	−0.530516	−0.0172122
$951$	0	0
$952$	−5.30324	−0.171879
$953$	−43.1091	−1.39644	−0.698221	−	0.715883i	$-0.746024\pi$
$953$	−43.1091	−1.39644	−0.698221	+	0.715883i	$0.746024\pi$
$954$	0	0
$955$	13.9019	0.449854
$956$	−37.6279	−1.21697
$957$	0	0
$958$	−1.67619	−0.0541553
$959$	47.8130	1.54396
$960$	0	0
$961$	32.5573	1.05024
$962$	0.0187510	0.000604555 0
$963$	0	0
$964$	−38.3493	−1.23515
$965$	17.4291	0.561062
$966$	0	0
$967$	9.21109	0.296209	0.148104	−	0.988972i	$-0.452683\pi$
$967$	9.21109	0.296209	0.148104	+	0.988972i	$0.452683\pi$
$968$	−1.32585	−0.0426144
$969$	0	0
$970$	−0.510601	−0.0163944
$971$	35.7097	1.14598	0.572990	−	0.819562i	$-0.305783\pi$
$971$	35.7097	1.14598	0.572990	+	0.819562i	$0.305783\pi$
$972$	0	0
$973$	−45.8095	−1.46858
$974$	−5.89742	−0.188965
$975$	0	0
$976$	40.0954	1.28342
$977$	27.4614	0.878569	0.439284	−	0.898348i	$-0.355232\pi$
$977$	27.4614	0.878569	0.439284	+	0.898348i	$0.355232\pi$
$978$	0	0
$979$	3.07479	0.0982708
$980$	−2.74883	−0.0878081
$981$	0	0
$982$	4.19905	0.133997
$983$	53.3643	1.70206	0.851029	−	0.525118i	$-0.175979\pi$
$983$	53.3643	1.70206	0.851029	+	0.525118i	$0.175979\pi$
$984$	0	0
$985$	−15.6196	−0.497683
$986$	−2.95885	−0.0942289
$987$	0	0
$988$	−0.408181	−0.0129860
$989$	5.73710	0.182429
$990$	0	0
$991$	0.953902	0.0303017	0.0151509	−	0.999885i	$-0.495177\pi$
$991$	0.953902	0.0303017	0.0151509	+	0.999885i	$0.495177\pi$
$992$	−20.1121	−0.638561
$993$	0	0
$994$	−6.00345	−0.190418
$995$	14.9293	0.473290
$996$	0	0
$997$	−23.3821	−0.740519	−0.370260	−	0.928928i	$-0.620731\pi$
$997$	−23.3821	−0.740519	−0.370260	+	0.928928i	$0.620731\pi$
$998$	−3.76007	−0.119023
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.p.1.4		8
3.2	odd	2		445.2.a.g.1.5	✓	8
12.11	even	2		7120.2.a.bk.1.8		8
15.14	odd	2		2225.2.a.l.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.5	✓	8	3.2	odd	2
2225.2.a.l.1.4		8	15.14	odd	2
4005.2.a.p.1.4		8	1.1	even	1	trivial
7120.2.a.bk.1.8		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-0.217002 q^{2} -1.95291 q^{4} +1.00000 q^{5} -2.89958 q^{7} +0.857790 q^{8} +O(q^{10})\)	\(q-0.217002 q^{2} -1.95291 q^{4} +1.00000 q^{5} -2.89958 q^{7} +0.857790 q^{8} -0.217002 q^{10} -3.07479 q^{11} +0.0854942 q^{13} +0.629215 q^{14} +3.71968 q^{16} +2.13219 q^{17} +2.44475 q^{19} -1.95291 q^{20} +0.667236 q^{22} -2.98507 q^{23} +1.00000 q^{25} -0.0185524 q^{26} +5.66262 q^{28} +6.39489 q^{29} +7.97229 q^{31} -2.52276 q^{32} -0.462689 q^{34} -2.89958 q^{35} -1.01070 q^{37} -0.530516 q^{38} +0.857790 q^{40} +4.08276 q^{41} -1.92193 q^{43} +6.00479 q^{44} +0.647767 q^{46} -8.73550 q^{47} +1.40756 q^{49} -0.217002 q^{50} -0.166962 q^{52} +6.19276 q^{53} -3.07479 q^{55} -2.48723 q^{56} -1.38770 q^{58} -11.6153 q^{59} +10.7793 q^{61} -1.73000 q^{62} -6.89191 q^{64} +0.0854942 q^{65} -4.48372 q^{67} -4.16397 q^{68} +0.629215 q^{70} -9.54118 q^{71} +1.45652 q^{73} +0.219325 q^{74} -4.77437 q^{76} +8.91560 q^{77} -12.8611 q^{79} +3.71968 q^{80} -0.885968 q^{82} -4.61282 q^{83} +2.13219 q^{85} +0.417062 q^{86} -2.63752 q^{88} -1.00000 q^{89} -0.247897 q^{91} +5.82958 q^{92} +1.89562 q^{94} +2.44475 q^{95} +2.35298 q^{97} -0.305442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Introduction

L-functions

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