LMFDB - Embedded newform 4005.2.a.q.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 $ x^9 - 4x^8 - 6x^7 + 31x^6 + 13x^5 - 75x^4 - 17x^3 + 52x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.88084$ 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)

$f(q)$	$=$	$q+1.88084 q^{2} +1.53757 q^{4} +1.00000 q^{5} -0.422231 q^{7} -0.869761 q^{8} +O(q^{10})$	\(q+1.88084 q^{2} +1.53757 q^{4} +1.00000 q^{5} -0.422231 q^{7} -0.869761 q^{8} +1.88084 q^{10} +2.50234 q^{11} -3.68651 q^{13} -0.794149 q^{14} -4.71102 q^{16} -3.71870 q^{17} -3.03752 q^{19} +1.53757 q^{20} +4.70651 q^{22} -7.79305 q^{23} +1.00000 q^{25} -6.93375 q^{26} -0.649208 q^{28} +0.762071 q^{29} -7.82727 q^{31} -7.12117 q^{32} -6.99428 q^{34} -0.422231 q^{35} -2.51736 q^{37} -5.71310 q^{38} -0.869761 q^{40} +7.74431 q^{41} -2.16484 q^{43} +3.84752 q^{44} -14.6575 q^{46} -2.09181 q^{47} -6.82172 q^{49} +1.88084 q^{50} -5.66827 q^{52} -2.90205 q^{53} +2.50234 q^{55} +0.367240 q^{56} +1.43334 q^{58} +11.9329 q^{59} +14.1168 q^{61} -14.7219 q^{62} -3.97175 q^{64} -3.68651 q^{65} +5.23589 q^{67} -5.71775 q^{68} -0.794149 q^{70} -15.6464 q^{71} -2.37064 q^{73} -4.73476 q^{74} -4.67040 q^{76} -1.05657 q^{77} +8.01078 q^{79} -4.71102 q^{80} +14.5658 q^{82} +10.2870 q^{83} -3.71870 q^{85} -4.07172 q^{86} -2.17644 q^{88} -1.00000 q^{89} +1.55656 q^{91} -11.9823 q^{92} -3.93437 q^{94} -3.03752 q^{95} +11.1772 q^{97} -12.8306 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * 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 $

$2$	1.88084	1.32996	0.664978	−	0.746863i	$-0.268441\pi$
$2$	1.88084	1.32996	0.664978	+	0.746863i	$0.268441\pi$
$3$	0	0
$3$	0	0
$4$	1.53757	0.768784
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.422231	−0.159588	−0.0797941	−	0.996811i	$-0.525426\pi$
$7$	−0.422231	−0.159588	−0.0797941	+	0.996811i	$0.525426\pi$
$8$	−0.869761	−0.307507
$9$	0	0
$10$	1.88084	0.594775
$11$	2.50234	0.754485	0.377242	−	0.926115i	$-0.376872\pi$
$11$	2.50234	0.754485	0.377242	+	0.926115i	$0.376872\pi$
$12$	0	0
$13$	−3.68651	−1.02245	−0.511227	−	0.859446i	$-0.670809\pi$
$13$	−3.68651	−1.02245	−0.511227	+	0.859446i	$0.670809\pi$
$14$	−0.794149	−0.212245
$15$	0	0
$16$	−4.71102	−1.17776
$17$	−3.71870	−0.901916	−0.450958	−	0.892545i	$-0.648918\pi$
$17$	−3.71870	−0.901916	−0.450958	+	0.892545i	$0.648918\pi$
$18$	0	0
$19$	−3.03752	−0.696856	−0.348428	−	0.937336i	$-0.613284\pi$
$19$	−3.03752	−0.696856	−0.348428	+	0.937336i	$0.613284\pi$
$20$	1.53757	0.343811
$21$	0	0
$22$	4.70651	1.00343
$23$	−7.79305	−1.62496	−0.812481	−	0.582987i	$-0.801884\pi$
$23$	−7.79305	−1.62496	−0.812481	+	0.582987i	$0.801884\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.93375	−1.35982
$27$	0	0
$28$	−0.649208	−0.122689
$29$	0.762071	0.141513	0.0707565	−	0.997494i	$-0.477459\pi$
$29$	0.762071	0.141513	0.0707565	+	0.997494i	$0.477459\pi$
$30$	0	0
$31$	−7.82727	−1.40582	−0.702909	−	0.711279i	$-0.748116\pi$
$31$	−7.82727	−1.40582	−0.702909	+	0.711279i	$0.748116\pi$
$32$	−7.12117	−1.25886
$33$	0	0
$34$	−6.99428	−1.19951
$35$	−0.422231	−0.0713700
$36$	0	0
$37$	−2.51736	−0.413852	−0.206926	−	0.978357i	$-0.566346\pi$
$37$	−2.51736	−0.413852	−0.206926	+	0.978357i	$0.566346\pi$
$38$	−5.71310	−0.926788
$39$	0	0
$40$	−0.869761	−0.137521
$41$	7.74431	1.20946	0.604729	−	0.796431i	$-0.293281\pi$
$41$	7.74431	1.20946	0.604729	+	0.796431i	$0.293281\pi$
$42$	0	0
$43$	−2.16484	−0.330135	−0.165067	−	0.986282i	$-0.552784\pi$
$43$	−2.16484	−0.330135	−0.165067	+	0.986282i	$0.552784\pi$
$44$	3.84752	0.580036
$45$	0	0
$46$	−14.6575	−2.16113
$47$	−2.09181	−0.305122	−0.152561	−	0.988294i	$-0.548752\pi$
$47$	−2.09181	−0.305122	−0.152561	+	0.988294i	$0.548752\pi$
$48$	0	0
$49$	−6.82172	−0.974532
$50$	1.88084	0.265991
$51$	0	0
$52$	−5.66827	−0.786047
$53$	−2.90205	−0.398628	−0.199314	−	0.979936i	$-0.563871\pi$
$53$	−2.90205	−0.398628	−0.199314	+	0.979936i	$0.563871\pi$
$54$	0	0
$55$	2.50234	0.337416
$56$	0.367240	0.0490745
$57$	0	0
$58$	1.43334	0.188206
$59$	11.9329	1.55353	0.776765	−	0.629791i	$-0.216859\pi$
$59$	11.9329	1.55353	0.776765	+	0.629791i	$0.216859\pi$
$60$	0	0
$61$	14.1168	1.80747	0.903736	−	0.428091i	$-0.140814\pi$
$61$	14.1168	1.80747	0.903736	+	0.428091i	$0.140814\pi$
$62$	−14.7219	−1.86968
$63$	0	0
$64$	−3.97175	−0.496469
$65$	−3.68651	−0.457256
$66$	0	0
$67$	5.23589	0.639665	0.319833	−	0.947474i	$-0.396373\pi$
$67$	5.23589	0.639665	0.319833	+	0.947474i	$0.396373\pi$
$68$	−5.71775	−0.693379
$69$	0	0
$70$	−0.794149	−0.0949190
$71$	−15.6464	−1.85689	−0.928445	−	0.371469i	$-0.878854\pi$
$71$	−15.6464	−1.85689	−0.928445	+	0.371469i	$0.878854\pi$
$72$	0	0
$73$	−2.37064	−0.277463	−0.138731	−	0.990330i	$-0.544302\pi$
$73$	−2.37064	−0.277463	−0.138731	+	0.990330i	$0.544302\pi$
$74$	−4.73476	−0.550405
$75$	0	0
$76$	−4.67040	−0.535732
$77$	−1.05657	−0.120407
$78$	0	0
$79$	8.01078	0.901283	0.450641	−	0.892705i	$-0.351195\pi$
$79$	8.01078	0.901283	0.450641	+	0.892705i	$0.351195\pi$
$80$	−4.71102	−0.526708
$81$	0	0
$82$	14.5658	1.60853
$83$	10.2870	1.12915	0.564573	−	0.825383i	$-0.309041\pi$
$83$	10.2870	1.12915	0.564573	+	0.825383i	$0.309041\pi$
$84$	0	0
$85$	−3.71870	−0.403349
$86$	−4.07172	−0.439065
$87$	0	0
$88$	−2.17644	−0.232009
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	1.55656	0.163172
$92$	−11.9823	−1.24925
$93$	0	0
$94$	−3.93437	−0.405799
$95$	−3.03752	−0.311643
$96$	0	0
$97$	11.1772	1.13488	0.567439	−	0.823416i	$-0.307934\pi$
$97$	11.1772	1.13488	0.567439	+	0.823416i	$0.307934\pi$
$98$	−12.8306	−1.29608
$99$	0	0
$100$	1.53757	0.153757
$101$	−17.6823	−1.75945	−0.879727	−	0.475479i	$-0.842275\pi$
$101$	−17.6823	−1.75945	−0.879727	+	0.475479i	$0.842275\pi$
$102$	0	0
$103$	−10.7754	−1.06173	−0.530864	−	0.847457i	$-0.678133\pi$
$103$	−10.7754	−1.06173	−0.530864	+	0.847457i	$0.678133\pi$
$104$	3.20638	0.314412
$105$	0	0
$106$	−5.45831	−0.530157
$107$	−7.25908	−0.701762	−0.350881	−	0.936420i	$-0.614118\pi$
$107$	−7.25908	−0.701762	−0.350881	+	0.936420i	$0.614118\pi$
$108$	0	0
$109$	10.8479	1.03904	0.519520	−	0.854458i	$-0.326111\pi$
$109$	10.8479	1.03904	0.519520	+	0.854458i	$0.326111\pi$
$110$	4.70651	0.448748
$111$	0	0
$112$	1.98914	0.187956
$113$	−19.5388	−1.83806	−0.919030	−	0.394187i	$-0.871026\pi$
$113$	−19.5388	−1.83806	−0.919030	+	0.394187i	$0.871026\pi$
$114$	0	0
$115$	−7.79305	−0.726705
$116$	1.17174	0.108793
$117$	0	0
$118$	22.4439	2.06613
$119$	1.57015	0.143935
$120$	0	0
$121$	−4.73828	−0.430753
$122$	26.5515	2.40386
$123$	0	0
$124$	−12.0350	−1.08077
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.66620	0.236587	0.118294	−	0.992979i	$-0.462258\pi$
$127$	2.66620	0.236587	0.118294	+	0.992979i	$0.462258\pi$
$128$	6.77209	0.598574
$129$	0	0
$130$	−6.93375	−0.608130
$131$	−2.32223	−0.202894	−0.101447	−	0.994841i	$-0.532347\pi$
$131$	−2.32223	−0.202894	−0.101447	+	0.994841i	$0.532347\pi$
$132$	0	0
$133$	1.28254	0.111210
$134$	9.84788	0.850727
$135$	0	0
$136$	3.23438	0.277345
$137$	−18.1775	−1.55301	−0.776504	−	0.630113i	$-0.783009\pi$
$137$	−18.1775	−1.55301	−0.776504	+	0.630113i	$0.783009\pi$
$138$	0	0
$139$	−9.91379	−0.840877	−0.420438	−	0.907321i	$-0.638124\pi$
$139$	−9.91379	−0.840877	−0.420438	+	0.907321i	$0.638124\pi$
$140$	−0.649208	−0.0548681
$141$	0	0
$142$	−29.4285	−2.46958
$143$	−9.22492	−0.771426
$144$	0	0
$145$	0.762071	0.0632866
$146$	−4.45881	−0.369014
$147$	0	0
$148$	−3.87062	−0.318163
$149$	13.2174	1.08281	0.541405	−	0.840762i	$-0.317893\pi$
$149$	13.2174	1.08281	0.541405	+	0.840762i	$0.317893\pi$
$150$	0	0
$151$	19.1290	1.55669	0.778346	−	0.627835i	$-0.216059\pi$
$151$	19.1290	1.55669	0.778346	+	0.627835i	$0.216059\pi$
$152$	2.64192	0.214288
$153$	0	0
$154$	−1.98723	−0.160136
$155$	−7.82727	−0.628701
$156$	0	0
$157$	16.9028	1.34899	0.674497	−	0.738278i	$-0.264361\pi$
$157$	16.9028	1.34899	0.674497	+	0.738278i	$0.264361\pi$
$158$	15.0670	1.19867
$159$	0	0
$160$	−7.12117	−0.562978
$161$	3.29046	0.259325
$162$	0	0
$163$	−6.51679	−0.510434	−0.255217	−	0.966884i	$-0.582147\pi$
$163$	−6.51679	−0.510434	−0.255217	+	0.966884i	$0.582147\pi$
$164$	11.9074	0.929812
$165$	0	0
$166$	19.3482	1.50171
$167$	−24.1165	−1.86619	−0.933097	−	0.359625i	$-0.882905\pi$
$167$	−24.1165	−1.86619	−0.933097	+	0.359625i	$0.882905\pi$
$168$	0	0
$169$	0.590375	0.0454135
$170$	−6.99428	−0.536437
$171$	0	0
$172$	−3.32859	−0.253802
$173$	−8.29515	−0.630669	−0.315334	−	0.948981i	$-0.602117\pi$
$173$	−8.29515	−0.630669	−0.315334	+	0.948981i	$0.602117\pi$
$174$	0	0
$175$	−0.422231	−0.0319176
$176$	−11.7886	−0.888598
$177$	0	0
$178$	−1.88084	−0.140975
$179$	−20.9141	−1.56319	−0.781597	−	0.623784i	$-0.785595\pi$
$179$	−20.9141	−1.56319	−0.781597	+	0.623784i	$0.785595\pi$
$180$	0	0
$181$	4.28934	0.318824	0.159412	−	0.987212i	$-0.449040\pi$
$181$	4.28934	0.318824	0.159412	+	0.987212i	$0.449040\pi$
$182$	2.92764	0.217011
$183$	0	0
$184$	6.77809	0.499687
$185$	−2.51736	−0.185080
$186$	0	0
$187$	−9.30545	−0.680482
$188$	−3.21631	−0.234573
$189$	0	0
$190$	−5.71310	−0.414472
$191$	20.1724	1.45962	0.729812	−	0.683648i	$-0.239607\pi$
$191$	20.1724	1.45962	0.729812	+	0.683648i	$0.239607\pi$
$192$	0	0
$193$	16.1086	1.15952	0.579761	−	0.814786i	$-0.303146\pi$
$193$	16.1086	1.15952	0.579761	+	0.814786i	$0.303146\pi$
$194$	21.0226	1.50934
$195$	0	0
$196$	−10.4889	−0.749205
$197$	21.2908	1.51690	0.758452	−	0.651729i	$-0.225956\pi$
$197$	21.2908	1.51690	0.758452	+	0.651729i	$0.225956\pi$
$198$	0	0
$199$	15.5344	1.10120	0.550601	−	0.834768i	$-0.314398\pi$
$199$	15.5344	1.10120	0.550601	+	0.834768i	$0.314398\pi$
$200$	−0.869761	−0.0615014
$201$	0	0
$202$	−33.2576	−2.34000
$203$	−0.321770	−0.0225838
$204$	0	0
$205$	7.74431	0.540886
$206$	−20.2668	−1.41205
$207$	0	0
$208$	17.3672	1.20420
$209$	−7.60092	−0.525767
$210$	0	0
$211$	8.46831	0.582982	0.291491	−	0.956574i	$-0.405849\pi$
$211$	8.46831	0.582982	0.291491	+	0.956574i	$0.405849\pi$
$212$	−4.46211	−0.306459
$213$	0	0
$214$	−13.6532	−0.933313
$215$	−2.16484	−0.147641
$216$	0	0
$217$	3.30491	0.224352
$218$	20.4032	1.38188
$219$	0	0
$220$	3.84752	0.259400
$221$	13.7090	0.922168
$222$	0	0
$223$	−26.4235	−1.76945	−0.884723	−	0.466117i	$-0.845653\pi$
$223$	−26.4235	−1.76945	−0.884723	+	0.466117i	$0.845653\pi$
$224$	3.00677	0.200899
$225$	0	0
$226$	−36.7495	−2.44454
$227$	−5.62560	−0.373384	−0.186692	−	0.982418i	$-0.559777\pi$
$227$	−5.62560	−0.373384	−0.186692	+	0.982418i	$0.559777\pi$
$228$	0	0
$229$	−17.9929	−1.18900	−0.594502	−	0.804094i	$-0.702651\pi$
$229$	−17.9929	−1.18900	−0.594502	+	0.804094i	$0.702651\pi$
$230$	−14.6575	−0.966486
$231$	0	0
$232$	−0.662820	−0.0435162
$233$	−23.6841	−1.55160	−0.775798	−	0.630982i	$-0.782652\pi$
$233$	−23.6841	−1.55160	−0.775798	+	0.630982i	$0.782652\pi$
$234$	0	0
$235$	−2.09181	−0.136455
$236$	18.3476	1.19433
$237$	0	0
$238$	2.95320	0.191427
$239$	10.2803	0.664977	0.332488	−	0.943107i	$-0.392112\pi$
$239$	10.2803	0.664977	0.332488	+	0.943107i	$0.392112\pi$
$240$	0	0
$241$	−0.693622	−0.0446801	−0.0223400	−	0.999750i	$-0.507112\pi$
$241$	−0.693622	−0.0446801	−0.0223400	+	0.999750i	$0.507112\pi$
$242$	−8.91196	−0.572883
$243$	0	0
$244$	21.7056	1.38956
$245$	−6.82172	−0.435824
$246$	0	0
$247$	11.1979	0.712503
$248$	6.80785	0.432299
$249$	0	0
$250$	1.88084	0.118955
$251$	−8.00598	−0.505333	−0.252666	−	0.967553i	$-0.581308\pi$
$251$	−8.00598	−0.505333	−0.252666	+	0.967553i	$0.581308\pi$
$252$	0	0
$253$	−19.5009	−1.22601
$254$	5.01471	0.314651
$255$	0	0
$256$	20.6807	1.29255
$257$	−14.2659	−0.889884	−0.444942	−	0.895559i	$-0.646776\pi$
$257$	−14.2659	−0.889884	−0.444942	+	0.895559i	$0.646776\pi$
$258$	0	0
$259$	1.06291	0.0660459
$260$	−5.66827	−0.351531
$261$	0	0
$262$	−4.36774	−0.269840
$263$	15.3339	0.945527	0.472764	−	0.881189i	$-0.343256\pi$
$263$	15.3339	0.945527	0.472764	+	0.881189i	$0.343256\pi$
$264$	0	0
$265$	−2.90205	−0.178272
$266$	2.41225	0.147904
$267$	0	0
$268$	8.05053	0.491765
$269$	9.45411	0.576427	0.288214	−	0.957566i	$-0.406939\pi$
$269$	9.45411	0.576427	0.288214	+	0.957566i	$0.406939\pi$
$270$	0	0
$271$	−9.57347	−0.581547	−0.290773	−	0.956792i	$-0.593913\pi$
$271$	−9.57347	−0.581547	−0.290773	+	0.956792i	$0.593913\pi$
$272$	17.5188	1.06224
$273$	0	0
$274$	−34.1890	−2.06543
$275$	2.50234	0.150897
$276$	0	0
$277$	27.7763	1.66891	0.834457	−	0.551072i	$-0.185781\pi$
$277$	27.7763	1.66891	0.834457	+	0.551072i	$0.185781\pi$
$278$	−18.6463	−1.11833
$279$	0	0
$280$	0.367240	0.0219468
$281$	−16.1214	−0.961724	−0.480862	−	0.876796i	$-0.659676\pi$
$281$	−16.1214	−0.961724	−0.480862	+	0.876796i	$0.659676\pi$
$282$	0	0
$283$	−7.03188	−0.418002	−0.209001	−	0.977915i	$-0.567021\pi$
$283$	−7.03188	−0.418002	−0.209001	+	0.977915i	$0.567021\pi$
$284$	−24.0575	−1.42755
$285$	0	0
$286$	−17.3506	−1.02596
$287$	−3.26989	−0.193015
$288$	0	0
$289$	−3.17131	−0.186547
$290$	1.43334	0.0841684
$291$	0	0
$292$	−3.64503	−0.213309
$293$	10.4095	0.608127	0.304063	−	0.952652i	$-0.401657\pi$
$293$	10.4095	0.608127	0.304063	+	0.952652i	$0.401657\pi$
$294$	0	0
$295$	11.9329	0.694760
$296$	2.18950	0.127262
$297$	0	0
$298$	24.8598	1.44009
$299$	28.7292	1.66145
$300$	0	0
$301$	0.914061	0.0526856
$302$	35.9786	2.07033
$303$	0	0
$304$	14.3098	0.820725
$305$	14.1168	0.808326
$306$	0	0
$307$	−12.2769	−0.700678	−0.350339	−	0.936623i	$-0.613934\pi$
$307$	−12.2769	−0.700678	−0.350339	+	0.936623i	$0.613934\pi$
$308$	−1.62454	−0.0925669
$309$	0	0
$310$	−14.7219	−0.836145
$311$	−10.3468	−0.586712	−0.293356	−	0.956003i	$-0.594772\pi$
$311$	−10.3468	−0.586712	−0.293356	+	0.956003i	$0.594772\pi$
$312$	0	0
$313$	−7.58516	−0.428739	−0.214369	−	0.976753i	$-0.568770\pi$
$313$	−7.58516	−0.428739	−0.214369	+	0.976753i	$0.568770\pi$
$314$	31.7916	1.79410
$315$	0	0
$316$	12.3171	0.692892
$317$	30.2656	1.69989	0.849943	−	0.526875i	$-0.176637\pi$
$317$	30.2656	1.69989	0.849943	+	0.526875i	$0.176637\pi$
$318$	0	0
$319$	1.90696	0.106769
$320$	−3.97175	−0.222028
$321$	0	0
$322$	6.18884	0.344891
$323$	11.2956	0.628505
$324$	0	0
$325$	−3.68651	−0.204491
$326$	−12.2571	−0.678856
$327$	0	0
$328$	−6.73570	−0.371917
$329$	0.883228	0.0486939
$330$	0	0
$331$	9.66560	0.531270	0.265635	−	0.964074i	$-0.414418\pi$
$331$	9.66560	0.531270	0.265635	+	0.964074i	$0.414418\pi$
$332$	15.8170	0.868069
$333$	0	0
$334$	−45.3594	−2.48196
$335$	5.23589	0.286067
$336$	0	0
$337$	−6.06052	−0.330138	−0.165069	−	0.986282i	$-0.552785\pi$
$337$	−6.06052	−0.330138	−0.165069	+	0.986282i	$0.552785\pi$
$338$	1.11040	0.0603979
$339$	0	0
$340$	−5.71775	−0.310088
$341$	−19.5865	−1.06067
$342$	0	0
$343$	5.83595	0.315112
$344$	1.88289	0.101519
$345$	0	0
$346$	−15.6019	−0.838762
$347$	−5.60902	−0.301108	−0.150554	−	0.988602i	$-0.548106\pi$
$347$	−5.60902	−0.301108	−0.150554	+	0.988602i	$0.548106\pi$
$348$	0	0
$349$	−4.49739	−0.240740	−0.120370	−	0.992729i	$-0.538408\pi$
$349$	−4.49739	−0.240740	−0.120370	+	0.992729i	$0.538408\pi$
$350$	−0.794149	−0.0424491
$351$	0	0
$352$	−17.8196	−0.949787
$353$	−28.3157	−1.50710	−0.753548	−	0.657393i	$-0.771659\pi$
$353$	−28.3157	−1.50710	−0.753548	+	0.657393i	$0.771659\pi$
$354$	0	0
$355$	−15.6464	−0.830427
$356$	−1.53757	−0.0814910
$357$	0	0
$358$	−39.3361	−2.07898
$359$	22.9683	1.21222	0.606111	−	0.795380i	$-0.292729\pi$
$359$	22.9683	1.21222	0.606111	+	0.795380i	$0.292729\pi$
$360$	0	0
$361$	−9.77345	−0.514392
$362$	8.06758	0.424022
$363$	0	0
$364$	2.39332	0.125444
$365$	−2.37064	−0.124085
$366$	0	0
$367$	4.86501	0.253952	0.126976	−	0.991906i	$-0.459473\pi$
$367$	4.86501	0.253952	0.126976	+	0.991906i	$0.459473\pi$
$368$	36.7132	1.91381
$369$	0	0
$370$	−4.73476	−0.246149
$371$	1.22534	0.0636163
$372$	0	0
$373$	20.5203	1.06250	0.531251	−	0.847215i	$-0.321722\pi$
$373$	20.5203	1.06250	0.531251	+	0.847215i	$0.321722\pi$
$374$	−17.5021	−0.905011
$375$	0	0
$376$	1.81938	0.0938272
$377$	−2.80938	−0.144691
$378$	0	0
$379$	26.1624	1.34387	0.671937	−	0.740608i	$-0.265462\pi$
$379$	26.1624	1.34387	0.671937	+	0.740608i	$0.265462\pi$
$380$	−4.67040	−0.239587
$381$	0	0
$382$	37.9411	1.94124
$383$	13.3637	0.682854	0.341427	−	0.939908i	$-0.389090\pi$
$383$	13.3637	0.682854	0.341427	+	0.939908i	$0.389090\pi$
$384$	0	0
$385$	−1.05657	−0.0538476
$386$	30.2977	1.54211
$387$	0	0
$388$	17.1858	0.872476
$389$	32.0296	1.62397	0.811983	−	0.583681i	$-0.198388\pi$
$389$	32.0296	1.62397	0.811983	+	0.583681i	$0.198388\pi$
$390$	0	0
$391$	28.9800	1.46558
$392$	5.93327	0.299675
$393$	0	0
$394$	40.0446	2.01742
$395$	8.01078	0.403066
$396$	0	0
$397$	12.0221	0.603372	0.301686	−	0.953407i	$-0.402451\pi$
$397$	12.0221	0.603372	0.301686	+	0.953407i	$0.402451\pi$
$398$	29.2177	1.46455
$399$	0	0
$400$	−4.71102	−0.235551
$401$	−11.7668	−0.587604	−0.293802	−	0.955866i	$-0.594921\pi$
$401$	−11.7668	−0.587604	−0.293802	+	0.955866i	$0.594921\pi$
$402$	0	0
$403$	28.8553	1.43739
$404$	−27.1877	−1.35264
$405$	0	0
$406$	−0.605198	−0.0300355
$407$	−6.29930	−0.312245
$408$	0	0
$409$	−3.14084	−0.155304	−0.0776522	−	0.996981i	$-0.524742\pi$
$409$	−3.14084	−0.155304	−0.0776522	+	0.996981i	$0.524742\pi$
$410$	14.5658	0.719355
$411$	0	0
$412$	−16.5679	−0.816240
$413$	−5.03843	−0.247925
$414$	0	0
$415$	10.2870	0.504969
$416$	26.2523	1.28712
$417$	0	0
$418$	−14.2961	−0.699247
$419$	27.5735	1.34705	0.673527	−	0.739163i	$-0.264779\pi$
$419$	27.5735	1.34705	0.673527	+	0.739163i	$0.264779\pi$
$420$	0	0
$421$	5.41991	0.264150	0.132075	−	0.991240i	$-0.457836\pi$
$421$	5.41991	0.264150	0.132075	+	0.991240i	$0.457836\pi$
$422$	15.9276	0.775341
$423$	0	0
$424$	2.52409	0.122581
$425$	−3.71870	−0.180383
$426$	0	0
$427$	−5.96055	−0.288451
$428$	−11.1613	−0.539504
$429$	0	0
$430$	−4.07172	−0.196356
$431$	−23.5399	−1.13388	−0.566940	−	0.823759i	$-0.691873\pi$
$431$	−23.5399	−1.13388	−0.566940	+	0.823759i	$0.691873\pi$
$432$	0	0
$433$	20.3290	0.976947	0.488474	−	0.872579i	$-0.337554\pi$
$433$	20.3290	0.976947	0.488474	+	0.872579i	$0.337554\pi$
$434$	6.21602	0.298378
$435$	0	0
$436$	16.6794	0.798797
$437$	23.6716	1.13236
$438$	0	0
$439$	−2.41644	−0.115330	−0.0576651	−	0.998336i	$-0.518366\pi$
$439$	−2.41644	−0.115330	−0.0576651	+	0.998336i	$0.518366\pi$
$440$	−2.17644	−0.103758
$441$	0	0
$442$	25.7845	1.22644
$443$	1.63857	0.0778508	0.0389254	−	0.999242i	$-0.487607\pi$
$443$	1.63857	0.0778508	0.0389254	+	0.999242i	$0.487607\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−49.6984	−2.35329
$447$	0	0
$448$	1.67699	0.0792305
$449$	−8.07410	−0.381040	−0.190520	−	0.981683i	$-0.561017\pi$
$449$	−8.07410	−0.381040	−0.190520	+	0.981683i	$0.561017\pi$
$450$	0	0
$451$	19.3789	0.912518
$452$	−30.0423	−1.41307
$453$	0	0
$454$	−10.5809	−0.496584
$455$	1.55656	0.0729726
$456$	0	0
$457$	5.04202	0.235856	0.117928	−	0.993022i	$-0.462375\pi$
$457$	5.04202	0.235856	0.117928	+	0.993022i	$0.462375\pi$
$458$	−33.8418	−1.58132
$459$	0	0
$460$	−11.9823	−0.558680
$461$	−9.87768	−0.460049	−0.230025	−	0.973185i	$-0.573881\pi$
$461$	−9.87768	−0.460049	−0.230025	+	0.973185i	$0.573881\pi$
$462$	0	0
$463$	−13.5121	−0.627961	−0.313980	−	0.949429i	$-0.601663\pi$
$463$	−13.5121	−0.627961	−0.313980	+	0.949429i	$0.601663\pi$
$464$	−3.59013	−0.166668
$465$	0	0
$466$	−44.5460	−2.06355
$467$	5.55710	0.257152	0.128576	−	0.991700i	$-0.458959\pi$
$467$	5.55710	0.257152	0.128576	+	0.991700i	$0.458959\pi$
$468$	0	0
$469$	−2.21075	−0.102083
$470$	−3.93437	−0.181479
$471$	0	0
$472$	−10.3788	−0.477721
$473$	−5.41717	−0.249082
$474$	0	0
$475$	−3.03752	−0.139371
$476$	2.41421	0.110655
$477$	0	0
$478$	19.3356	0.884390
$479$	−16.7498	−0.765319	−0.382660	−	0.923889i	$-0.624992\pi$
$479$	−16.7498	−0.765319	−0.382660	+	0.923889i	$0.624992\pi$
$480$	0	0
$481$	9.28029	0.423145
$482$	−1.30459	−0.0594226
$483$	0	0
$484$	−7.28544	−0.331156
$485$	11.1772	0.507533
$486$	0	0
$487$	−12.8397	−0.581822	−0.290911	−	0.956750i	$-0.593958\pi$
$487$	−12.8397	−0.581822	−0.290911	+	0.956750i	$0.593958\pi$
$488$	−12.2782	−0.555810
$489$	0	0
$490$	−12.8306	−0.579627
$491$	−27.1850	−1.22684	−0.613421	−	0.789756i	$-0.710207\pi$
$491$	−27.1850	−1.22684	−0.613421	+	0.789756i	$0.710207\pi$
$492$	0	0
$493$	−2.83391	−0.127633
$494$	21.0614	0.947599
$495$	0	0
$496$	36.8744	1.65571
$497$	6.60640	0.296338
$498$	0	0
$499$	−0.701829	−0.0314182	−0.0157091	−	0.999877i	$-0.505001\pi$
$499$	−0.701829	−0.0314182	−0.0157091	+	0.999877i	$0.505001\pi$
$500$	1.53757	0.0687622
$501$	0	0
$502$	−15.0580	−0.672071
$503$	20.9661	0.934831	0.467415	−	0.884038i	$-0.345185\pi$
$503$	20.9661	0.934831	0.467415	+	0.884038i	$0.345185\pi$
$504$	0	0
$505$	−17.6823	−0.786852
$506$	−36.6781	−1.63054
$507$	0	0
$508$	4.09947	0.181885
$509$	−35.2465	−1.56227	−0.781136	−	0.624361i	$-0.785359\pi$
$509$	−35.2465	−1.56227	−0.781136	+	0.624361i	$0.785359\pi$
$510$	0	0
$511$	1.00096	0.0442798
$512$	25.3530	1.12046
$513$	0	0
$514$	−26.8319	−1.18351
$515$	−10.7754	−0.474819
$516$	0	0
$517$	−5.23443	−0.230210
$518$	1.99916	0.0878381
$519$	0	0
$520$	3.20638	0.140609
$521$	−25.0174	−1.09603	−0.548015	−	0.836468i	$-0.684617\pi$
$521$	−25.0174	−1.09603	−0.548015	+	0.836468i	$0.684617\pi$
$522$	0	0
$523$	0.871129	0.0380918	0.0190459	−	0.999819i	$-0.493937\pi$
$523$	0.871129	0.0380918	0.0190459	+	0.999819i	$0.493937\pi$
$524$	−3.57058	−0.155982
$525$	0	0
$526$	28.8406	1.25751
$527$	29.1072	1.26793
$528$	0	0
$529$	37.7316	1.64050
$530$	−5.45831	−0.237094
$531$	0	0
$532$	1.97199	0.0854964
$533$	−28.5495	−1.23662
$534$	0	0
$535$	−7.25908	−0.313838
$536$	−4.55397	−0.196702
$537$	0	0
$538$	17.7817	0.766623
$539$	−17.0703	−0.735269
$540$	0	0
$541$	2.64616	0.113767	0.0568836	−	0.998381i	$-0.481884\pi$
$541$	2.64616	0.113767	0.0568836	+	0.998381i	$0.481884\pi$
$542$	−18.0062	−0.773432
$543$	0	0
$544$	26.4814	1.13538
$545$	10.8479	0.464673
$546$	0	0
$547$	−42.8286	−1.83122	−0.915609	−	0.402070i	$-0.868291\pi$
$547$	−42.8286	−1.83122	−0.915609	+	0.402070i	$0.868291\pi$
$548$	−27.9491	−1.19393
$549$	0	0
$550$	4.70651	0.200686
$551$	−2.31481	−0.0986142
$552$	0	0
$553$	−3.38240	−0.143834
$554$	52.2428	2.21958
$555$	0	0
$556$	−15.2431	−0.646453
$557$	4.77534	0.202338	0.101169	−	0.994869i	$-0.467742\pi$
$557$	4.77534	0.202338	0.101169	+	0.994869i	$0.467742\pi$
$558$	0	0
$559$	7.98070	0.337548
$560$	1.98914	0.0840564
$561$	0	0
$562$	−30.3219	−1.27905
$563$	15.1856	0.639996	0.319998	−	0.947418i	$-0.396318\pi$
$563$	15.1856	0.639996	0.319998	+	0.947418i	$0.396318\pi$
$564$	0	0
$565$	−19.5388	−0.822006
$566$	−13.2259	−0.555924
$567$	0	0
$568$	13.6087	0.571007
$569$	0.959265	0.0402145	0.0201072	−	0.999798i	$-0.493599\pi$
$569$	0.959265	0.0402145	0.0201072	+	0.999798i	$0.493599\pi$
$570$	0	0
$571$	−22.1733	−0.927925	−0.463963	−	0.885855i	$-0.653573\pi$
$571$	−22.1733	−0.927925	−0.463963	+	0.885855i	$0.653573\pi$
$572$	−14.1839	−0.593060
$573$	0	0
$574$	−6.15014	−0.256702
$575$	−7.79305	−0.324992
$576$	0	0
$577$	39.3398	1.63774	0.818869	−	0.573981i	$-0.194602\pi$
$577$	39.3398	1.63774	0.818869	+	0.573981i	$0.194602\pi$
$578$	−5.96473	−0.248100
$579$	0	0
$580$	1.17174	0.0486537
$581$	−4.34349	−0.180198
$582$	0	0
$583$	−7.26193	−0.300758
$584$	2.06189	0.0853218
$585$	0	0
$586$	19.5785	0.808782
$587$	8.38322	0.346013	0.173006	−	0.984921i	$-0.444652\pi$
$587$	8.38322	0.346013	0.173006	+	0.984921i	$0.444652\pi$
$588$	0	0
$589$	23.7755	0.979653
$590$	22.4439	0.924000
$591$	0	0
$592$	11.8593	0.487416
$593$	−35.7578	−1.46840	−0.734199	−	0.678934i	$-0.762442\pi$
$593$	−35.7578	−1.46840	−0.734199	+	0.678934i	$0.762442\pi$
$594$	0	0
$595$	1.57015	0.0643697
$596$	20.3226	0.832447
$597$	0	0
$598$	54.0350	2.20966
$599$	6.59091	0.269297	0.134649	−	0.990893i	$-0.457009\pi$
$599$	6.59091	0.269297	0.134649	+	0.990893i	$0.457009\pi$
$600$	0	0
$601$	6.56082	0.267621	0.133811	−	0.991007i	$-0.457279\pi$
$601$	6.56082	0.267621	0.133811	+	0.991007i	$0.457279\pi$
$602$	1.71920	0.0700696
$603$	0	0
$604$	29.4121	1.19676
$605$	−4.73828	−0.192639
$606$	0	0
$607$	13.1813	0.535013	0.267506	−	0.963556i	$-0.413800\pi$
$607$	13.1813	0.535013	0.267506	+	0.963556i	$0.413800\pi$
$608$	21.6307	0.877241
$609$	0	0
$610$	26.5515	1.07504
$611$	7.71150	0.311974
$612$	0	0
$613$	−24.7631	−1.00017	−0.500086	−	0.865976i	$-0.666698\pi$
$613$	−24.7631	−1.00017	−0.500086	+	0.865976i	$0.666698\pi$
$614$	−23.0908	−0.931871
$615$	0	0
$616$	0.918959	0.0370259
$617$	−41.1110	−1.65507	−0.827533	−	0.561417i	$-0.810256\pi$
$617$	−41.1110	−1.65507	−0.827533	+	0.561417i	$0.810256\pi$
$618$	0	0
$619$	17.0984	0.687243	0.343621	−	0.939108i	$-0.388346\pi$
$619$	17.0984	0.687243	0.343621	+	0.939108i	$0.388346\pi$
$620$	−12.0350	−0.483336
$621$	0	0
$622$	−19.4607	−0.780302
$623$	0.422231	0.0169163
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.2665	−0.570204
$627$	0	0
$628$	25.9893	1.03708
$629$	9.36130	0.373260
$630$	0	0
$631$	−40.4527	−1.61040	−0.805198	−	0.593006i	$-0.797941\pi$
$631$	−40.4527	−1.61040	−0.805198	+	0.593006i	$0.797941\pi$
$632$	−6.96746	−0.277151
$633$	0	0
$634$	56.9248	2.26077
$635$	2.66620	0.105805
$636$	0	0
$637$	25.1484	0.996414
$638$	3.58670	0.141999
$639$	0	0
$640$	6.77209	0.267691
$641$	48.1370	1.90130	0.950649	−	0.310270i	$-0.100419\pi$
$641$	48.1370	1.90130	0.950649	+	0.310270i	$0.100419\pi$
$642$	0	0
$643$	−28.1054	−1.10837	−0.554185	−	0.832394i	$-0.686970\pi$
$643$	−28.1054	−1.10837	−0.554185	+	0.832394i	$0.686970\pi$
$644$	5.05931	0.199365
$645$	0	0
$646$	21.2453	0.835885
$647$	16.5035	0.648818	0.324409	−	0.945917i	$-0.394835\pi$
$647$	16.5035	0.648818	0.324409	+	0.945917i	$0.394835\pi$
$648$	0	0
$649$	29.8602	1.17211
$650$	−6.93375	−0.271964
$651$	0	0
$652$	−10.0200	−0.392414
$653$	35.7140	1.39760	0.698798	−	0.715319i	$-0.253719\pi$
$653$	35.7140	1.39760	0.698798	+	0.715319i	$0.253719\pi$
$654$	0	0
$655$	−2.32223	−0.0907369
$656$	−36.4836	−1.42445
$657$	0	0
$658$	1.66121	0.0647608
$659$	−12.4670	−0.485644	−0.242822	−	0.970071i	$-0.578073\pi$
$659$	−12.4670	−0.485644	−0.242822	+	0.970071i	$0.578073\pi$
$660$	0	0
$661$	5.20718	0.202536	0.101268	−	0.994859i	$-0.467710\pi$
$661$	5.20718	0.202536	0.101268	+	0.994859i	$0.467710\pi$
$662$	18.1795	0.706565
$663$	0	0
$664$	−8.94723	−0.347220
$665$	1.28254	0.0497346
$666$	0	0
$667$	−5.93886	−0.229953
$668$	−37.0808	−1.43470
$669$	0	0
$670$	9.84788	0.380457
$671$	35.3251	1.36371
$672$	0	0
$673$	30.1591	1.16255	0.581275	−	0.813707i	$-0.302554\pi$
$673$	30.1591	1.16255	0.581275	+	0.813707i	$0.302554\pi$
$674$	−11.3989	−0.439069
$675$	0	0
$676$	0.907742	0.0349131
$677$	1.37503	0.0528466	0.0264233	−	0.999651i	$-0.491588\pi$
$677$	1.37503	0.0528466	0.0264233	+	0.999651i	$0.491588\pi$
$678$	0	0
$679$	−4.71938	−0.181113
$680$	3.23438	0.124033
$681$	0	0
$682$	−36.8391	−1.41064
$683$	−35.1790	−1.34609	−0.673043	−	0.739603i	$-0.735013\pi$
$683$	−35.1790	−1.34609	−0.673043	+	0.739603i	$0.735013\pi$
$684$	0	0
$685$	−18.1775	−0.694526
$686$	10.9765	0.419085
$687$	0	0
$688$	10.1986	0.388818
$689$	10.6985	0.407579
$690$	0	0
$691$	−16.7244	−0.636225	−0.318113	−	0.948053i	$-0.603049\pi$
$691$	−16.7244	−0.636225	−0.318113	+	0.948053i	$0.603049\pi$
$692$	−12.7544	−0.484848
$693$	0	0
$694$	−10.5497	−0.400461
$695$	−9.91379	−0.376051
$696$	0	0
$697$	−28.7987	−1.09083
$698$	−8.45888	−0.320173
$699$	0	0
$700$	−0.649208	−0.0245378
$701$	−21.2934	−0.804242	−0.402121	−	0.915587i	$-0.631727\pi$
$701$	−21.2934	−0.804242	−0.402121	+	0.915587i	$0.631727\pi$
$702$	0	0
$703$	7.64655	0.288395
$704$	−9.93868	−0.374578
$705$	0	0
$706$	−53.2575	−2.00437
$707$	7.46601	0.280788
$708$	0	0
$709$	−37.3572	−1.40298	−0.701489	−	0.712680i	$-0.747481\pi$
$709$	−37.3572	−1.40298	−0.701489	+	0.712680i	$0.747481\pi$
$710$	−29.4285	−1.10443
$711$	0	0
$712$	0.869761	0.0325957
$713$	60.9983	2.28440
$714$	0	0
$715$	−9.22492	−0.344992
$716$	−32.1569	−1.20176
$717$	0	0
$718$	43.1998	1.61220
$719$	−48.4669	−1.80751	−0.903756	−	0.428049i	$-0.859201\pi$
$719$	−48.4669	−1.80751	−0.903756	+	0.428049i	$0.859201\pi$
$720$	0	0
$721$	4.54969	0.169439
$722$	−18.3823	−0.684119
$723$	0	0
$724$	6.59516	0.245107
$725$	0.762071	0.0283026
$726$	0	0
$727$	24.8476	0.921545	0.460773	−	0.887518i	$-0.347572\pi$
$727$	24.8476	0.921545	0.460773	+	0.887518i	$0.347572\pi$
$728$	−1.35383	−0.0501764
$729$	0	0
$730$	−4.45881	−0.165028
$731$	8.05037	0.297754
$732$	0	0
$733$	7.84473	0.289752	0.144876	−	0.989450i	$-0.453722\pi$
$733$	7.84473	0.289752	0.144876	+	0.989450i	$0.453722\pi$
$734$	9.15032	0.337745
$735$	0	0
$736$	55.4956	2.04559
$737$	13.1020	0.482618
$738$	0	0
$739$	44.7503	1.64617	0.823084	−	0.567920i	$-0.192252\pi$
$739$	44.7503	1.64617	0.823084	+	0.567920i	$0.192252\pi$
$740$	−3.87062	−0.142287
$741$	0	0
$742$	2.30466	0.0846069
$743$	39.7620	1.45873	0.729364	−	0.684126i	$-0.239816\pi$
$743$	39.7620	1.45873	0.729364	+	0.684126i	$0.239816\pi$
$744$	0	0
$745$	13.2174	0.484247
$746$	38.5955	1.41308
$747$	0	0
$748$	−14.3078	−0.523144
$749$	3.06501	0.111993
$750$	0	0
$751$	−10.1848	−0.371649	−0.185825	−	0.982583i	$-0.559496\pi$
$751$	−10.1848	−0.371649	−0.185825	+	0.982583i	$0.559496\pi$
$752$	9.85458	0.359359
$753$	0	0
$754$	−5.28401	−0.192432
$755$	19.1290	0.696174
$756$	0	0
$757$	19.7448	0.717636	0.358818	−	0.933408i	$-0.383180\pi$
$757$	19.7448	0.717636	0.358818	+	0.933408i	$0.383180\pi$
$758$	49.2074	1.78729
$759$	0	0
$760$	2.64192	0.0958325
$761$	−37.4129	−1.35622	−0.678109	−	0.734962i	$-0.737200\pi$
$761$	−37.4129	−1.35622	−0.678109	+	0.734962i	$0.737200\pi$
$762$	0	0
$763$	−4.58031	−0.165818
$764$	31.0165	1.12214
$765$	0	0
$766$	25.1351	0.908166
$767$	−43.9908	−1.58841
$768$	0	0
$769$	−22.7179	−0.819228	−0.409614	−	0.912259i	$-0.634337\pi$
$769$	−22.7179	−0.819228	−0.409614	+	0.912259i	$0.634337\pi$
$770$	−1.98723	−0.0716149
$771$	0	0
$772$	24.7681	0.891422
$773$	−26.3108	−0.946335	−0.473168	−	0.880972i	$-0.656890\pi$
$773$	−26.3108	−0.946335	−0.473168	+	0.880972i	$0.656890\pi$
$774$	0	0
$775$	−7.82727	−0.281164
$776$	−9.72153	−0.348983
$777$	0	0
$778$	60.2427	2.15981
$779$	−23.5235	−0.842818
$780$	0	0
$781$	−39.1527	−1.40100
$782$	54.5067	1.94916
$783$	0	0
$784$	32.1373	1.14776
$785$	16.9028	0.603288
$786$	0	0
$787$	−15.3570	−0.547417	−0.273708	−	0.961813i	$-0.588250\pi$
$787$	−15.3570	−0.547417	−0.273708	+	0.961813i	$0.588250\pi$
$788$	32.7360	1.16617
$789$	0	0
$790$	15.0670	0.536060
$791$	8.24990	0.293333
$792$	0	0
$793$	−52.0418	−1.84806
$794$	22.6117	0.802458
$795$	0	0
$796$	23.8852	0.846587
$797$	12.3674	0.438077	0.219038	−	0.975716i	$-0.429708\pi$
$797$	12.3674	0.438077	0.219038	+	0.975716i	$0.429708\pi$
$798$	0	0
$799$	7.77882	0.275195
$800$	−7.12117	−0.251771
$801$	0	0
$802$	−22.1314	−0.781488
$803$	−5.93216	−0.209341
$804$	0	0
$805$	3.29046	0.115974
$806$	54.2723	1.91166
$807$	0	0
$808$	15.3794	0.541044
$809$	18.5968	0.653829	0.326915	−	0.945054i	$-0.393991\pi$
$809$	18.5968	0.653829	0.326915	+	0.945054i	$0.393991\pi$
$810$	0	0
$811$	−9.96219	−0.349820	−0.174910	−	0.984584i	$-0.555963\pi$
$811$	−9.96219	−0.349820	−0.174910	+	0.984584i	$0.555963\pi$
$812$	−0.494743	−0.0173621
$813$	0	0
$814$	−11.8480	−0.415272
$815$	−6.51679	−0.228273
$816$	0	0
$817$	6.57575	0.230056
$818$	−5.90742	−0.206548
$819$	0	0
$820$	11.9074	0.415825
$821$	−19.3686	−0.675969	−0.337985	−	0.941152i	$-0.609745\pi$
$821$	−19.3686	−0.675969	−0.337985	+	0.941152i	$0.609745\pi$
$822$	0	0
$823$	16.0175	0.558336	0.279168	−	0.960242i	$-0.409941\pi$
$823$	16.0175	0.558336	0.279168	+	0.960242i	$0.409941\pi$
$824$	9.37199	0.326489
$825$	0	0
$826$	−9.47650	−0.329729
$827$	−18.5112	−0.643699	−0.321849	−	0.946791i	$-0.604304\pi$
$827$	−18.5112	−0.643699	−0.321849	+	0.946791i	$0.604304\pi$
$828$	0	0
$829$	46.5134	1.61548	0.807738	−	0.589542i	$-0.200692\pi$
$829$	46.5134	1.61548	0.807738	+	0.589542i	$0.200692\pi$
$830$	19.3482	0.671587
$831$	0	0
$832$	14.6419	0.507617
$833$	25.3679	0.878946
$834$	0	0
$835$	−24.1165	−0.834587
$836$	−11.6869	−0.404201
$837$	0	0
$838$	51.8614	1.79152
$839$	32.6568	1.12744	0.563718	−	0.825967i	$-0.309370\pi$
$839$	32.6568	1.12744	0.563718	+	0.825967i	$0.309370\pi$
$840$	0	0
$841$	−28.4192	−0.979974
$842$	10.1940	0.351308
$843$	0	0
$844$	13.0206	0.448188
$845$	0.590375	0.0203095
$846$	0	0
$847$	2.00065	0.0687431
$848$	13.6716	0.469486
$849$	0	0
$850$	−6.99428	−0.239902
$851$	19.6179	0.672494
$852$	0	0
$853$	−26.9404	−0.922421	−0.461211	−	0.887291i	$-0.652585\pi$
$853$	−26.9404	−0.922421	−0.461211	+	0.887291i	$0.652585\pi$
$854$	−11.2108	−0.383627
$855$	0	0
$856$	6.31367	0.215797
$857$	−13.1776	−0.450138	−0.225069	−	0.974343i	$-0.572261\pi$
$857$	−13.1776	−0.450138	−0.225069	+	0.974343i	$0.572261\pi$
$858$	0	0
$859$	−18.0793	−0.616858	−0.308429	−	0.951247i	$-0.599803\pi$
$859$	−18.0793	−0.616858	−0.308429	+	0.951247i	$0.599803\pi$
$860$	−3.32859	−0.113504
$861$	0	0
$862$	−44.2749	−1.50801
$863$	−48.2543	−1.64260	−0.821298	−	0.570500i	$-0.806749\pi$
$863$	−48.2543	−1.64260	−0.821298	+	0.570500i	$0.806749\pi$
$864$	0	0
$865$	−8.29515	−0.282044
$866$	38.2356	1.29930
$867$	0	0
$868$	5.08153	0.172478
$869$	20.0457	0.680004
$870$	0	0
$871$	−19.3022	−0.654029
$872$	−9.43507	−0.319512
$873$	0	0
$874$	44.5225	1.50600
$875$	−0.422231	−0.0142740
$876$	0	0
$877$	14.9185	0.503764	0.251882	−	0.967758i	$-0.418951\pi$
$877$	14.9185	0.503764	0.251882	+	0.967758i	$0.418951\pi$
$878$	−4.54494	−0.153384
$879$	0	0
$880$	−11.7886	−0.397393
$881$	7.05947	0.237840	0.118920	−	0.992904i	$-0.462057\pi$
$881$	7.05947	0.237840	0.118920	+	0.992904i	$0.462057\pi$
$882$	0	0
$883$	8.15832	0.274549	0.137275	−	0.990533i	$-0.456166\pi$
$883$	8.15832	0.274549	0.137275	+	0.990533i	$0.456166\pi$
$884$	21.0786	0.708948
$885$	0	0
$886$	3.08189	0.103538
$887$	2.31911	0.0778681	0.0389341	−	0.999242i	$-0.487604\pi$
$887$	2.31911	0.0778681	0.0389341	+	0.999242i	$0.487604\pi$
$888$	0	0
$889$	−1.12575	−0.0377566
$890$	−1.88084	−0.0630460
$891$	0	0
$892$	−40.6279	−1.36032
$893$	6.35393	0.212626
$894$	0	0
$895$	−20.9141	−0.699082
$896$	−2.85939	−0.0955254
$897$	0	0
$898$	−15.1861	−0.506767
$899$	−5.96494	−0.198942
$900$	0	0
$901$	10.7919	0.359529
$902$	36.4487	1.21361
$903$	0	0
$904$	16.9941	0.565216
$905$	4.28934	0.142583
$906$	0	0
$907$	−4.27857	−0.142068	−0.0710338	−	0.997474i	$-0.522630\pi$
$907$	−4.27857	−0.142068	−0.0710338	+	0.997474i	$0.522630\pi$
$908$	−8.64974	−0.287052
$909$	0	0
$910$	2.92764	0.0970504
$911$	53.9583	1.78772	0.893859	−	0.448348i	$-0.147987\pi$
$911$	53.9583	1.78772	0.893859	+	0.448348i	$0.147987\pi$
$912$	0	0
$913$	25.7416	0.851923
$914$	9.48325	0.313678
$915$	0	0
$916$	−27.6653	−0.914087
$917$	0.980515	0.0323795
$918$	0	0
$919$	−1.46679	−0.0483851	−0.0241925	−	0.999707i	$-0.507701\pi$
$919$	−1.46679	−0.0483851	−0.0241925	+	0.999707i	$0.507701\pi$
$920$	6.77809	0.223467
$921$	0	0
$922$	−18.5784	−0.611846
$923$	57.6808	1.89859
$924$	0	0
$925$	−2.51736	−0.0827704
$926$	−25.4141	−0.835161
$927$	0	0
$928$	−5.42683	−0.178145
$929$	14.5441	0.477176	0.238588	−	0.971121i	$-0.423315\pi$
$929$	14.5441	0.477176	0.238588	+	0.971121i	$0.423315\pi$
$930$	0	0
$931$	20.7211	0.679108
$932$	−36.4159	−1.19284
$933$	0	0
$934$	10.4520	0.342001
$935$	−9.30545	−0.304321
$936$	0	0
$937$	−10.4978	−0.342950	−0.171475	−	0.985189i	$-0.554853\pi$
$937$	−10.4978	−0.342950	−0.171475	+	0.985189i	$0.554853\pi$
$938$	−4.15808	−0.135766
$939$	0	0
$940$	−3.21631	−0.104904
$941$	−1.30296	−0.0424753	−0.0212376	−	0.999774i	$-0.506761\pi$
$941$	−1.30296	−0.0424753	−0.0212376	+	0.999774i	$0.506761\pi$
$942$	0	0
$943$	−60.3518	−1.96532
$944$	−56.2161	−1.82968
$945$	0	0
$946$	−10.1888	−0.331268
$947$	−28.1439	−0.914552	−0.457276	−	0.889325i	$-0.651175\pi$
$947$	−28.1439	−0.914552	−0.457276	+	0.889325i	$0.651175\pi$
$948$	0	0
$949$	8.73941	0.283693
$950$	−5.71310	−0.185358
$951$	0	0
$952$	−1.36565	−0.0442610
$953$	46.3006	1.49982	0.749911	−	0.661539i	$-0.230096\pi$
$953$	46.3006	1.49982	0.749911	+	0.661539i	$0.230096\pi$
$954$	0	0
$955$	20.1724	0.652764
$956$	15.8067	0.511224
$957$	0	0
$958$	−31.5038	−1.01784
$959$	7.67509	0.247842
$960$	0	0
$961$	30.2661	0.976326
$962$	17.4548	0.562764
$963$	0	0
$964$	−1.06649	−0.0343494
$965$	16.1086	0.518554
$966$	0	0
$967$	5.70924	0.183597	0.0917984	−	0.995778i	$-0.470738\pi$
$967$	5.70924	0.183597	0.0917984	+	0.995778i	$0.470738\pi$
$968$	4.12117	0.132460
$969$	0	0
$970$	21.0226	0.674996
$971$	−5.12282	−0.164399	−0.0821996	−	0.996616i	$-0.526195\pi$
$971$	−5.12282	−0.164399	−0.0821996	+	0.996616i	$0.526195\pi$
$972$	0	0
$973$	4.18590	0.134194
$974$	−24.1494	−0.773798
$975$	0	0
$976$	−66.5045	−2.12876
$977$	−51.3318	−1.64225	−0.821125	−	0.570748i	$-0.806653\pi$
$977$	−51.3318	−1.64225	−0.821125	+	0.570748i	$0.806653\pi$
$978$	0	0
$979$	−2.50234	−0.0799752
$980$	−10.4889	−0.335054
$981$	0	0
$982$	−51.1307	−1.63165
$983$	−56.6259	−1.80609	−0.903044	−	0.429549i	$-0.858673\pi$
$983$	−56.6259	−1.80609	−0.903044	+	0.429549i	$0.858673\pi$
$984$	0	0
$985$	21.2908	0.678380
$986$	−5.33014	−0.169746
$987$	0	0
$988$	17.2175	0.547761
$989$	16.8707	0.536457
$990$	0	0
$991$	−26.0156	−0.826412	−0.413206	−	0.910638i	$-0.635591\pi$
$991$	−26.0156	−0.826412	−0.413206	+	0.910638i	$0.635591\pi$
$992$	55.7393	1.76972
$993$	0	0
$994$	12.4256	0.394116
$995$	15.5344	0.492473
$996$	0	0
$997$	−24.0950	−0.763098	−0.381549	−	0.924349i	$-0.624609\pi$
$997$	−24.0950	−0.763098	−0.381549	+	0.924349i	$0.624609\pi$
$998$	−1.32003	−0.0417848
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.q.1.9		9
3.2	odd	2		1335.2.a.h.1.1	✓	9
15.14	odd	2		6675.2.a.x.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.h.1.1	✓	9	3.2	odd	2
4005.2.a.q.1.9		9	1.1	even	1	trivial
6675.2.a.x.1.9		9	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.88084 q^{2} +1.53757 q^{4} +1.00000 q^{5} -0.422231 q^{7} -0.869761 q^{8} +O(q^{10})\)	\(q+1.88084 q^{2} +1.53757 q^{4} +1.00000 q^{5} -0.422231 q^{7} -0.869761 q^{8} +1.88084 q^{10} +2.50234 q^{11} -3.68651 q^{13} -0.794149 q^{14} -4.71102 q^{16} -3.71870 q^{17} -3.03752 q^{19} +1.53757 q^{20} +4.70651 q^{22} -7.79305 q^{23} +1.00000 q^{25} -6.93375 q^{26} -0.649208 q^{28} +0.762071 q^{29} -7.82727 q^{31} -7.12117 q^{32} -6.99428 q^{34} -0.422231 q^{35} -2.51736 q^{37} -5.71310 q^{38} -0.869761 q^{40} +7.74431 q^{41} -2.16484 q^{43} +3.84752 q^{44} -14.6575 q^{46} -2.09181 q^{47} -6.82172 q^{49} +1.88084 q^{50} -5.66827 q^{52} -2.90205 q^{53} +2.50234 q^{55} +0.367240 q^{56} +1.43334 q^{58} +11.9329 q^{59} +14.1168 q^{61} -14.7219 q^{62} -3.97175 q^{64} -3.68651 q^{65} +5.23589 q^{67} -5.71775 q^{68} -0.794149 q^{70} -15.6464 q^{71} -2.37064 q^{73} -4.73476 q^{74} -4.67040 q^{76} -1.05657 q^{77} +8.01078 q^{79} -4.71102 q^{80} +14.5658 q^{82} +10.2870 q^{83} -3.71870 q^{85} -4.07172 q^{86} -2.17644 q^{88} -1.00000 q^{89} +1.55656 q^{91} -11.9823 q^{92} -3.93437 q^{94} -3.03752 q^{95} +11.1772 q^{97} -12.8306 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * q^98

Introduction

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