LMFDB - Embedded newform 6003.2.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} + \cdots - 5 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 54x^8 - 188x^7 - 77x^6 + 342x^5 + 13x^4 - 215x^3 + 9x^2 + 37*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.613795$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.613795 q^{2} -1.62326 q^{4} +0.782514 q^{5} +3.32687 q^{7} +2.22394 q^{8} +O(q^{10})$	\(q-0.613795 q^{2} -1.62326 q^{4} +0.782514 q^{5} +3.32687 q^{7} +2.22394 q^{8} -0.480303 q^{10} +0.406317 q^{11} -5.42384 q^{13} -2.04202 q^{14} +1.88147 q^{16} -2.46092 q^{17} -6.37074 q^{19} -1.27022 q^{20} -0.249396 q^{22} +1.00000 q^{23} -4.38767 q^{25} +3.32913 q^{26} -5.40036 q^{28} +1.00000 q^{29} -5.30243 q^{31} -5.60271 q^{32} +1.51050 q^{34} +2.60332 q^{35} +11.4958 q^{37} +3.91033 q^{38} +1.74026 q^{40} +5.46253 q^{41} -8.54868 q^{43} -0.659557 q^{44} -0.613795 q^{46} +9.66382 q^{47} +4.06807 q^{49} +2.69313 q^{50} +8.80428 q^{52} +0.700276 q^{53} +0.317949 q^{55} +7.39875 q^{56} -0.613795 q^{58} +12.5798 q^{59} +4.97086 q^{61} +3.25461 q^{62} -0.324031 q^{64} -4.24423 q^{65} +10.1731 q^{67} +3.99470 q^{68} -1.59791 q^{70} -1.31238 q^{71} +1.43516 q^{73} -7.05606 q^{74} +10.3413 q^{76} +1.35177 q^{77} +11.5359 q^{79} +1.47228 q^{80} -3.35287 q^{82} -16.8324 q^{83} -1.92570 q^{85} +5.24713 q^{86} +0.903624 q^{88} -1.33781 q^{89} -18.0444 q^{91} -1.62326 q^{92} -5.93160 q^{94} -4.98519 q^{95} +4.13945 q^{97} -2.49696 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8 + 6 * q^11 - 15 * q^13 + 8 * q^14 + 17 * q^16 + 18 * q^17 - 6 * q^19 + 39 * q^20 - 5 * q^22 + 12 * q^23 + 14 * q^25 + 3 * q^26 - 19 * q^28 + 12 * q^29 + 16 * q^31 + 21 * q^32 - 7 * q^34 + 11 * q^35 - q^37 + 24 * q^38 + 30 * q^40 - 3 * q^41 - 23 * q^43 - 23 * q^44 + 3 * q^46 + 35 * q^47 + 3 * q^49 + 2 * q^50 + 45 * q^53 + 17 * q^55 + 17 * q^56 + 3 * q^58 + 11 * q^59 + 4 * q^61 + 7 * q^62 + 15 * q^64 - 5 * q^65 - 19 * q^67 - q^68 + 14 * q^70 - 19 * q^71 + 10 * q^73 + 15 * q^74 - 4 * q^76 + 39 * q^77 + 17 * q^79 + 90 * q^80 - 3 * q^82 + 12 * q^83 + 14 * q^85 - 17 * q^86 - 2 * q^88 + 20 * q^89 + 11 * q^91 + 11 * q^92 + 13 * q^94 - 12 * q^95 - 12 * 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 $

$2$	−0.613795	−0.434018	−0.217009	−	0.976170i	$-0.569630\pi$
$2$	−0.613795	−0.434018	−0.217009	+	0.976170i	$0.569630\pi$
$3$	0	0
$3$	0	0
$4$	−1.62326	−0.811628
$5$	0.782514	0.349951	0.174975	−	0.984573i	$-0.444015\pi$
$5$	0.782514	0.349951	0.174975	+	0.984573i	$0.444015\pi$
$6$	0	0
$7$	3.32687	1.25744	0.628719	−	0.777632i	$-0.283579\pi$
$7$	3.32687	1.25744	0.628719	+	0.777632i	$0.283579\pi$
$8$	2.22394	0.786280
$9$	0	0
$10$	−0.480303	−0.151885
$11$	0.406317	0.122509	0.0612547	−	0.998122i	$-0.480490\pi$
$11$	0.406317	0.122509	0.0612547	+	0.998122i	$0.480490\pi$
$12$	0	0
$13$	−5.42384	−1.50430	−0.752152	−	0.658990i	$-0.770984\pi$
$13$	−5.42384	−1.50430	−0.752152	+	0.658990i	$0.770984\pi$
$14$	−2.04202	−0.545752
$15$	0	0
$16$	1.88147	0.470368
$17$	−2.46092	−0.596861	−0.298430	−	0.954431i	$-0.596463\pi$
$17$	−2.46092	−0.596861	−0.298430	+	0.954431i	$0.596463\pi$
$18$	0	0
$19$	−6.37074	−1.46155	−0.730774	−	0.682620i	$-0.760841\pi$
$19$	−6.37074	−1.46155	−0.730774	+	0.682620i	$0.760841\pi$
$20$	−1.27022	−0.284030
$21$	0	0
$22$	−0.249396	−0.0531713
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.38767	−0.877534
$26$	3.32913	0.652895
$27$	0	0
$28$	−5.40036	−1.02057
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.30243	−0.952345	−0.476173	−	0.879352i	$-0.657976\pi$
$31$	−5.30243	−0.952345	−0.476173	+	0.879352i	$0.657976\pi$
$32$	−5.60271	−0.990428
$33$	0	0
$34$	1.51050	0.259048
$35$	2.60332	0.440042
$36$	0	0
$37$	11.4958	1.88990	0.944949	−	0.327217i	$-0.106111\pi$
$37$	11.4958	1.88990	0.944949	+	0.327217i	$0.106111\pi$
$38$	3.91033	0.634339
$39$	0	0
$40$	1.74026	0.275159
$41$	5.46253	0.853104	0.426552	−	0.904463i	$-0.359728\pi$
$41$	5.46253	0.853104	0.426552	+	0.904463i	$0.359728\pi$
$42$	0	0
$43$	−8.54868	−1.30366	−0.651830	−	0.758365i	$-0.725999\pi$
$43$	−8.54868	−1.30366	−0.651830	+	0.758365i	$0.725999\pi$
$44$	−0.659557	−0.0994320
$45$	0	0
$46$	−0.613795	−0.0904991
$47$	9.66382	1.40961	0.704807	−	0.709400i	$-0.251034\pi$
$47$	9.66382	1.40961	0.704807	+	0.709400i	$0.251034\pi$
$48$	0	0
$49$	4.06807	0.581153
$50$	2.69313	0.380866
$51$	0	0
$52$	8.80428	1.22093
$53$	0.700276	0.0961904	0.0480952	−	0.998843i	$-0.484685\pi$
$53$	0.700276	0.0961904	0.0480952	+	0.998843i	$0.484685\pi$
$54$	0	0
$55$	0.317949	0.0428722
$56$	7.39875	0.988699
$57$	0	0
$58$	−0.613795	−0.0805952
$59$	12.5798	1.63775	0.818876	−	0.573970i	$-0.194598\pi$
$59$	12.5798	1.63775	0.818876	+	0.573970i	$0.194598\pi$
$60$	0	0
$61$	4.97086	0.636453	0.318227	−	0.948015i	$-0.396913\pi$
$61$	4.97086	0.636453	0.318227	+	0.948015i	$0.396913\pi$
$62$	3.25461	0.413335
$63$	0	0
$64$	−0.324031	−0.0405039
$65$	−4.24423	−0.526432
$66$	0	0
$67$	10.1731	1.24285	0.621424	−	0.783475i	$-0.286555\pi$
$67$	10.1731	1.24285	0.621424	+	0.783475i	$0.286555\pi$
$68$	3.99470	0.484429
$69$	0	0
$70$	−1.59791	−0.190986
$71$	−1.31238	−0.155750	−0.0778752	−	0.996963i	$-0.524814\pi$
$71$	−1.31238	−0.155750	−0.0778752	+	0.996963i	$0.524814\pi$
$72$	0	0
$73$	1.43516	0.167972	0.0839862	−	0.996467i	$-0.473235\pi$
$73$	1.43516	0.167972	0.0839862	+	0.996467i	$0.473235\pi$
$74$	−7.05606	−0.820251
$75$	0	0
$76$	10.3413	1.18623
$77$	1.35177	0.154048
$78$	0	0
$79$	11.5359	1.29790	0.648948	−	0.760833i	$-0.275209\pi$
$79$	11.5359	1.29790	0.648948	+	0.760833i	$0.275209\pi$
$80$	1.47228	0.164606
$81$	0	0
$82$	−3.35287	−0.370263
$83$	−16.8324	−1.84760	−0.923800	−	0.382875i	$-0.874934\pi$
$83$	−16.8324	−1.84760	−0.923800	+	0.382875i	$0.874934\pi$
$84$	0	0
$85$	−1.92570	−0.208872
$86$	5.24713	0.565813
$87$	0	0
$88$	0.903624	0.0963266
$89$	−1.33781	−0.141808	−0.0709039	−	0.997483i	$-0.522588\pi$
$89$	−1.33781	−0.141808	−0.0709039	+	0.997483i	$0.522588\pi$
$90$	0	0
$91$	−18.0444	−1.89157
$92$	−1.62326	−0.169236
$93$	0	0
$94$	−5.93160	−0.611798
$95$	−4.98519	−0.511470
$96$	0	0
$97$	4.13945	0.420297	0.210149	−	0.977669i	$-0.432605\pi$
$97$	4.13945	0.420297	0.210149	+	0.977669i	$0.432605\pi$
$98$	−2.49696	−0.252231
$99$	0	0
$100$	7.12231	0.712231
$101$	1.87667	0.186736	0.0933680	−	0.995632i	$-0.470237\pi$
$101$	1.87667	0.186736	0.0933680	+	0.995632i	$0.470237\pi$
$102$	0	0
$103$	19.9333	1.96409	0.982045	−	0.188648i	$-0.0604106\pi$
$103$	19.9333	1.96409	0.982045	+	0.188648i	$0.0604106\pi$
$104$	−12.0623	−1.18280
$105$	0	0
$106$	−0.429826	−0.0417484
$107$	6.14202	0.593772	0.296886	−	0.954913i	$-0.404052\pi$
$107$	6.14202	0.593772	0.296886	+	0.954913i	$0.404052\pi$
$108$	0	0
$109$	−4.56435	−0.437185	−0.218593	−	0.975816i	$-0.570147\pi$
$109$	−4.56435	−0.437185	−0.218593	+	0.975816i	$0.570147\pi$
$110$	−0.195155	−0.0186073
$111$	0	0
$112$	6.25941	0.591459
$113$	−5.80276	−0.545878	−0.272939	−	0.962031i	$-0.587996\pi$
$113$	−5.80276	−0.545878	−0.272939	+	0.962031i	$0.587996\pi$
$114$	0	0
$115$	0.782514	0.0729698
$116$	−1.62326	−0.150716
$117$	0	0
$118$	−7.72142	−0.710815
$119$	−8.18716	−0.750516
$120$	0	0
$121$	−10.8349	−0.984991
$122$	−3.05109	−0.276232
$123$	0	0
$124$	8.60721	0.772950
$125$	−7.34598	−0.657045
$126$	0	0
$127$	−10.3131	−0.915137	−0.457568	−	0.889175i	$-0.651279\pi$
$127$	−10.3131	−0.915137	−0.457568	+	0.889175i	$0.651279\pi$
$128$	11.4043	1.00801
$129$	0	0
$130$	2.60509	0.228481
$131$	5.32995	0.465680	0.232840	−	0.972515i	$-0.425198\pi$
$131$	5.32995	0.465680	0.232840	+	0.972515i	$0.425198\pi$
$132$	0	0
$133$	−21.1946	−1.83781
$134$	−6.24422	−0.539419
$135$	0	0
$136$	−5.47293	−0.469299
$137$	12.9517	1.10654	0.553269	−	0.833003i	$-0.313380\pi$
$137$	12.9517	1.10654	0.553269	+	0.833003i	$0.313380\pi$
$138$	0	0
$139$	12.1081	1.02699	0.513497	−	0.858092i	$-0.328350\pi$
$139$	12.1081	1.02699	0.513497	+	0.858092i	$0.328350\pi$
$140$	−4.22586	−0.357150
$141$	0	0
$142$	0.805530	0.0675986
$143$	−2.20380	−0.184291
$144$	0	0
$145$	0.782514	0.0649843
$146$	−0.880892	−0.0729031
$147$	0	0
$148$	−18.6606	−1.53389
$149$	18.5599	1.52049	0.760244	−	0.649638i	$-0.225079\pi$
$149$	18.5599	1.52049	0.760244	+	0.649638i	$0.225079\pi$
$150$	0	0
$151$	4.33700	0.352940	0.176470	−	0.984306i	$-0.443532\pi$
$151$	4.33700	0.352940	0.176470	+	0.984306i	$0.443532\pi$
$152$	−14.1681	−1.14919
$153$	0	0
$154$	−0.829707	−0.0668597
$155$	−4.14923	−0.333274
$156$	0	0
$157$	3.93902	0.314368	0.157184	−	0.987569i	$-0.449758\pi$
$157$	3.93902	0.314368	0.157184	+	0.987569i	$0.449758\pi$
$158$	−7.08070	−0.563310
$159$	0	0
$160$	−4.38420	−0.346601
$161$	3.32687	0.262194
$162$	0	0
$163$	−5.95030	−0.466064	−0.233032	−	0.972469i	$-0.574865\pi$
$163$	−5.95030	−0.466064	−0.233032	+	0.972469i	$0.574865\pi$
$164$	−8.86709	−0.692403
$165$	0	0
$166$	10.3317	0.801893
$167$	−12.2376	−0.946972	−0.473486	−	0.880801i	$-0.657005\pi$
$167$	−12.2376	−0.946972	−0.473486	+	0.880801i	$0.657005\pi$
$168$	0	0
$169$	16.4181	1.26293
$170$	1.18199	0.0906543
$171$	0	0
$172$	13.8767	1.05809
$173$	17.3723	1.32079	0.660396	−	0.750918i	$-0.270389\pi$
$173$	17.3723	1.32079	0.660396	+	0.750918i	$0.270389\pi$
$174$	0	0
$175$	−14.5972	−1.10345
$176$	0.764475	0.0576245
$177$	0	0
$178$	0.821142	0.0615472
$179$	25.2647	1.88837	0.944186	−	0.329414i	$-0.106851\pi$
$179$	25.2647	1.88837	0.944186	+	0.329414i	$0.106851\pi$
$180$	0	0
$181$	−16.5386	−1.22931	−0.614654	−	0.788797i	$-0.710704\pi$
$181$	−16.5386	−1.22931	−0.614654	+	0.788797i	$0.710704\pi$
$182$	11.0756	0.820976
$183$	0	0
$184$	2.22394	0.163951
$185$	8.99563	0.661372
$186$	0	0
$187$	−0.999914	−0.0731210
$188$	−15.6869	−1.14408
$189$	0	0
$190$	3.05988	0.221987
$191$	−11.0993	−0.803117	−0.401558	−	0.915833i	$-0.631531\pi$
$191$	−11.0993	−0.803117	−0.401558	+	0.915833i	$0.631531\pi$
$192$	0	0
$193$	17.4310	1.25471	0.627356	−	0.778733i	$-0.284137\pi$
$193$	17.4310	1.25471	0.627356	+	0.778733i	$0.284137\pi$
$194$	−2.54077	−0.182417
$195$	0	0
$196$	−6.60352	−0.471680
$197$	−10.5560	−0.752085	−0.376043	−	0.926602i	$-0.622715\pi$
$197$	−10.5560	−0.752085	−0.376043	+	0.926602i	$0.622715\pi$
$198$	0	0
$199$	−8.61580	−0.610758	−0.305379	−	0.952231i	$-0.598783\pi$
$199$	−8.61580	−0.610758	−0.305379	+	0.952231i	$0.598783\pi$
$200$	−9.75790	−0.689988
$201$	0	0
$202$	−1.15189	−0.0810469
$203$	3.32687	0.233501
$204$	0	0
$205$	4.27451	0.298545
$206$	−12.2350	−0.852451
$207$	0	0
$208$	−10.2048	−0.707576
$209$	−2.58854	−0.179053
$210$	0	0
$211$	−1.41739	−0.0975769	−0.0487884	−	0.998809i	$-0.515536\pi$
$211$	−1.41739	−0.0975769	−0.0487884	+	0.998809i	$0.515536\pi$
$212$	−1.13673	−0.0780708
$213$	0	0
$214$	−3.76994	−0.257708
$215$	−6.68946	−0.456217
$216$	0	0
$217$	−17.6405	−1.19752
$218$	2.80157	0.189747
$219$	0	0
$220$	−0.516113	−0.0347963
$221$	13.3476	0.897859
$222$	0	0
$223$	18.2650	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$223$	18.2650	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$224$	−18.6395	−1.24540
$225$	0	0
$226$	3.56171	0.236921
$227$	19.0713	1.26580	0.632902	−	0.774232i	$-0.281863\pi$
$227$	19.0713	1.26580	0.632902	+	0.774232i	$0.281863\pi$
$228$	0	0
$229$	−21.2005	−1.40097	−0.700484	−	0.713668i	$-0.747033\pi$
$229$	−21.2005	−1.40097	−0.700484	+	0.713668i	$0.747033\pi$
$230$	−0.480303	−0.0316702
$231$	0	0
$232$	2.22394	0.146009
$233$	−11.1553	−0.730810	−0.365405	−	0.930849i	$-0.619070\pi$
$233$	−11.1553	−0.730810	−0.365405	+	0.930849i	$0.619070\pi$
$234$	0	0
$235$	7.56208	0.493295
$236$	−20.4203	−1.32925
$237$	0	0
$238$	5.02524	0.325738
$239$	10.6447	0.688545	0.344273	−	0.938870i	$-0.388126\pi$
$239$	10.6447	0.688545	0.344273	+	0.938870i	$0.388126\pi$
$240$	0	0
$241$	−3.85136	−0.248088	−0.124044	−	0.992277i	$-0.539586\pi$
$241$	−3.85136	−0.248088	−0.124044	+	0.992277i	$0.539586\pi$
$242$	6.65041	0.427504
$243$	0	0
$244$	−8.06897	−0.516563
$245$	3.18332	0.203375
$246$	0	0
$247$	34.5539	2.19861
$248$	−11.7923	−0.748810
$249$	0	0
$250$	4.50893	0.285170
$251$	25.8855	1.63388	0.816939	−	0.576723i	$-0.195669\pi$
$251$	25.8855	1.63388	0.816939	+	0.576723i	$0.195669\pi$
$252$	0	0
$253$	0.406317	0.0255450
$254$	6.33011	0.397186
$255$	0	0
$256$	−6.35184	−0.396990
$257$	19.4756	1.21486	0.607428	−	0.794375i	$-0.292201\pi$
$257$	19.4756	1.21486	0.607428	+	0.794375i	$0.292201\pi$
$258$	0	0
$259$	38.2451	2.37643
$260$	6.88948	0.427267
$261$	0	0
$262$	−3.27149	−0.202114
$263$	−29.2990	−1.80666	−0.903328	−	0.428950i	$-0.858884\pi$
$263$	−29.2990	−1.80666	−0.903328	+	0.428950i	$0.858884\pi$
$264$	0	0
$265$	0.547976	0.0336619
$266$	13.0091	0.797642
$267$	0	0
$268$	−16.5136	−1.00873
$269$	26.0816	1.59023	0.795113	−	0.606462i	$-0.207412\pi$
$269$	26.0816	1.59023	0.795113	+	0.606462i	$0.207412\pi$
$270$	0	0
$271$	−8.80807	−0.535052	−0.267526	−	0.963551i	$-0.586206\pi$
$271$	−8.80807	−0.535052	−0.267526	+	0.963551i	$0.586206\pi$
$272$	−4.63015	−0.280744
$273$	0	0
$274$	−7.94968	−0.480257
$275$	−1.78279	−0.107506
$276$	0	0
$277$	1.02134	0.0613662	0.0306831	−	0.999529i	$-0.490232\pi$
$277$	1.02134	0.0613662	0.0306831	+	0.999529i	$0.490232\pi$
$278$	−7.43187	−0.445734
$279$	0	0
$280$	5.78962	0.345996
$281$	−8.58849	−0.512346	−0.256173	−	0.966631i	$-0.582462\pi$
$281$	−8.58849	−0.512346	−0.256173	+	0.966631i	$0.582462\pi$
$282$	0	0
$283$	−0.533628	−0.0317209	−0.0158604	−	0.999874i	$-0.505049\pi$
$283$	−0.533628	−0.0317209	−0.0158604	+	0.999874i	$0.505049\pi$
$284$	2.13032	0.126411
$285$	0	0
$286$	1.35268	0.0799858
$287$	18.1731	1.07273
$288$	0	0
$289$	−10.9439	−0.643757
$290$	−0.480303	−0.0282044
$291$	0	0
$292$	−2.32963	−0.136331
$293$	25.8572	1.51060	0.755298	−	0.655382i	$-0.227492\pi$
$293$	25.8572	1.51060	0.755298	+	0.655382i	$0.227492\pi$
$294$	0	0
$295$	9.84388	0.573133
$296$	25.5659	1.48599
$297$	0	0
$298$	−11.3920	−0.659920
$299$	−5.42384	−0.313669
$300$	0	0
$301$	−28.4403	−1.63927
$302$	−2.66203	−0.153182
$303$	0	0
$304$	−11.9864	−0.687465
$305$	3.88977	0.222727
$306$	0	0
$307$	−3.32497	−0.189766	−0.0948830	−	0.995488i	$-0.530248\pi$
$307$	−3.32497	−0.189766	−0.0948830	+	0.995488i	$0.530248\pi$
$308$	−2.19426	−0.125030
$309$	0	0
$310$	2.54678	0.144647
$311$	−5.86357	−0.332492	−0.166246	−	0.986084i	$-0.553165\pi$
$311$	−5.86357	−0.332492	−0.166246	+	0.986084i	$0.553165\pi$
$312$	0	0
$313$	5.58694	0.315793	0.157896	−	0.987456i	$-0.449529\pi$
$313$	5.58694	0.315793	0.157896	+	0.987456i	$0.449529\pi$
$314$	−2.41775	−0.136442
$315$	0	0
$316$	−18.7258	−1.05341
$317$	19.4677	1.09342	0.546708	−	0.837324i	$-0.315881\pi$
$317$	19.4677	1.09342	0.546708	+	0.837324i	$0.315881\pi$
$318$	0	0
$319$	0.406317	0.0227494
$320$	−0.253559	−0.0141744
$321$	0	0
$322$	−2.04202	−0.113797
$323$	15.6779	0.872340
$324$	0	0
$325$	23.7980	1.32008
$326$	3.65226	0.202280
$327$	0	0
$328$	12.1483	0.670779
$329$	32.1503	1.77250
$330$	0	0
$331$	17.3533	0.953822	0.476911	−	0.878952i	$-0.341756\pi$
$331$	17.3533	0.953822	0.476911	+	0.878952i	$0.341756\pi$
$332$	27.3234	1.49956
$333$	0	0
$334$	7.51136	0.411003
$335$	7.96063	0.434936
$336$	0	0
$337$	−2.86585	−0.156113	−0.0780564	−	0.996949i	$-0.524871\pi$
$337$	−2.86585	−0.156113	−0.0780564	+	0.996949i	$0.524871\pi$
$338$	−10.0773	−0.548134
$339$	0	0
$340$	3.12591	0.169526
$341$	−2.15447	−0.116671
$342$	0	0
$343$	−9.75416	−0.526675
$344$	−19.0117	−1.02504
$345$	0	0
$346$	−10.6630	−0.573248
$347$	−23.4269	−1.25762	−0.628811	−	0.777558i	$-0.716458\pi$
$347$	−23.4269	−1.25762	−0.628811	+	0.777558i	$0.716458\pi$
$348$	0	0
$349$	16.6318	0.890277	0.445139	−	0.895462i	$-0.353154\pi$
$349$	16.6318	0.890277	0.445139	+	0.895462i	$0.353154\pi$
$350$	8.95970	0.478916
$351$	0	0
$352$	−2.27648	−0.121337
$353$	−15.4514	−0.822396	−0.411198	−	0.911546i	$-0.634889\pi$
$353$	−15.4514	−0.822396	−0.411198	+	0.911546i	$0.634889\pi$
$354$	0	0
$355$	−1.02695	−0.0545050
$356$	2.17161	0.115095
$357$	0	0
$358$	−15.5073	−0.819588
$359$	−2.32726	−0.122828	−0.0614142	−	0.998112i	$-0.519561\pi$
$359$	−2.32726	−0.122828	−0.0614142	+	0.998112i	$0.519561\pi$
$360$	0	0
$361$	21.5863	1.13612
$362$	10.1513	0.533542
$363$	0	0
$364$	29.2907	1.53525
$365$	1.12303	0.0587821
$366$	0	0
$367$	12.9453	0.675737	0.337868	−	0.941193i	$-0.390294\pi$
$367$	12.9453	0.675737	0.337868	+	0.941193i	$0.390294\pi$
$368$	1.88147	0.0980785
$369$	0	0
$370$	−5.52147	−0.287048
$371$	2.32973	0.120954
$372$	0	0
$373$	−7.16993	−0.371245	−0.185623	−	0.982621i	$-0.559430\pi$
$373$	−7.16993	−0.371245	−0.185623	+	0.982621i	$0.559430\pi$
$374$	0.613742	0.0317359
$375$	0	0
$376$	21.4917	1.10835
$377$	−5.42384	−0.279342
$378$	0	0
$379$	0.571864	0.0293747	0.0146873	−	0.999892i	$-0.495325\pi$
$379$	0.571864	0.0293747	0.0146873	+	0.999892i	$0.495325\pi$
$380$	8.09224	0.415123
$381$	0	0
$382$	6.81269	0.348567
$383$	−4.38199	−0.223909	−0.111955	−	0.993713i	$-0.535711\pi$
$383$	−4.38199	−0.223909	−0.111955	+	0.993713i	$0.535711\pi$
$384$	0	0
$385$	1.05778	0.0539092
$386$	−10.6991	−0.544568
$387$	0	0
$388$	−6.71938	−0.341125
$389$	−24.4697	−1.24066	−0.620330	−	0.784341i	$-0.713001\pi$
$389$	−24.4697	−1.24066	−0.620330	+	0.784341i	$0.713001\pi$
$390$	0	0
$391$	−2.46092	−0.124454
$392$	9.04712	0.456949
$393$	0	0
$394$	6.47923	0.326419
$395$	9.02704	0.454200
$396$	0	0
$397$	−24.6960	−1.23946	−0.619729	−	0.784816i	$-0.712758\pi$
$397$	−24.6960	−1.23946	−0.619729	+	0.784816i	$0.712758\pi$
$398$	5.28834	0.265080
$399$	0	0
$400$	−8.25528	−0.412764
$401$	38.7110	1.93313	0.966567	−	0.256414i	$-0.0825409\pi$
$401$	38.7110	1.93313	0.966567	+	0.256414i	$0.0825409\pi$
$402$	0	0
$403$	28.7596	1.43262
$404$	−3.04632	−0.151560
$405$	0	0
$406$	−2.04202	−0.101344
$407$	4.67095	0.231530
$408$	0	0
$409$	31.9145	1.57807	0.789036	−	0.614347i	$-0.210581\pi$
$409$	31.9145	1.57807	0.789036	+	0.614347i	$0.210581\pi$
$410$	−2.62367	−0.129574
$411$	0	0
$412$	−32.3569	−1.59411
$413$	41.8514	2.05937
$414$	0	0
$415$	−13.1716	−0.646569
$416$	30.3882	1.48990
$417$	0	0
$418$	1.58883	0.0777124
$419$	−0.0535481	−0.00261600	−0.00130800	−	0.999999i	$-0.500416\pi$
$419$	−0.0535481	−0.00261600	−0.00130800	+	0.999999i	$0.500416\pi$
$420$	0	0
$421$	17.1862	0.837604	0.418802	−	0.908078i	$-0.362450\pi$
$421$	17.1862	0.837604	0.418802	+	0.908078i	$0.362450\pi$
$422$	0.869984	0.0423502
$423$	0	0
$424$	1.55737	0.0756326
$425$	10.7977	0.523766
$426$	0	0
$427$	16.5374	0.800301
$428$	−9.97007	−0.481922
$429$	0	0
$430$	4.10595	0.198007
$431$	20.1267	0.969468	0.484734	−	0.874662i	$-0.338916\pi$
$431$	20.1267	0.969468	0.484734	+	0.874662i	$0.338916\pi$
$432$	0	0
$433$	20.7426	0.996827	0.498414	−	0.866939i	$-0.333916\pi$
$433$	20.7426	0.996827	0.498414	+	0.866939i	$0.333916\pi$
$434$	10.8277	0.519744
$435$	0	0
$436$	7.40911	0.354832
$437$	−6.37074	−0.304754
$438$	0	0
$439$	−33.4136	−1.59474	−0.797371	−	0.603489i	$-0.793777\pi$
$439$	−33.4136	−1.59474	−0.797371	+	0.603489i	$0.793777\pi$
$440$	0.707098	0.0337096
$441$	0	0
$442$	−8.19271	−0.389687
$443$	30.7428	1.46064	0.730318	−	0.683108i	$-0.239372\pi$
$443$	30.7428	1.46064	0.730318	+	0.683108i	$0.239372\pi$
$444$	0	0
$445$	−1.04686	−0.0496258
$446$	−11.2110	−0.530856
$447$	0	0
$448$	−1.07801	−0.0509311
$449$	33.6514	1.58811	0.794054	−	0.607847i	$-0.207967\pi$
$449$	33.6514	1.58811	0.794054	+	0.607847i	$0.207967\pi$
$450$	0	0
$451$	2.21952	0.104513
$452$	9.41937	0.443050
$453$	0	0
$454$	−11.7058	−0.549383
$455$	−14.1200	−0.661956
$456$	0	0
$457$	−19.8530	−0.928686	−0.464343	−	0.885655i	$-0.653710\pi$
$457$	−19.8530	−0.928686	−0.464343	+	0.885655i	$0.653710\pi$
$458$	13.0128	0.608046
$459$	0	0
$460$	−1.27022	−0.0592243
$461$	−35.5887	−1.65753	−0.828765	−	0.559597i	$-0.810956\pi$
$461$	−35.5887	−1.65753	−0.828765	+	0.559597i	$0.810956\pi$
$462$	0	0
$463$	26.3284	1.22358	0.611792	−	0.791019i	$-0.290449\pi$
$463$	26.3284	1.22358	0.611792	+	0.791019i	$0.290449\pi$
$464$	1.88147	0.0873451
$465$	0	0
$466$	6.84708	0.317185
$467$	19.7164	0.912365	0.456183	−	0.889886i	$-0.349216\pi$
$467$	19.7164	0.912365	0.456183	+	0.889886i	$0.349216\pi$
$468$	0	0
$469$	33.8447	1.56280
$470$	−4.64156	−0.214099
$471$	0	0
$472$	27.9767	1.28773
$473$	−3.47348	−0.159711
$474$	0	0
$475$	27.9527	1.28256
$476$	13.2899	0.609140
$477$	0	0
$478$	−6.53363	−0.298841
$479$	16.5603	0.756658	0.378329	−	0.925671i	$-0.376499\pi$
$479$	16.5603	0.756658	0.378329	+	0.925671i	$0.376499\pi$
$480$	0	0
$481$	−62.3514	−2.84298
$482$	2.36394	0.107675
$483$	0	0
$484$	17.5878	0.799447
$485$	3.23918	0.147083
$486$	0	0
$487$	−12.7239	−0.576574	−0.288287	−	0.957544i	$-0.593086\pi$
$487$	−12.7239	−0.576574	−0.288287	+	0.957544i	$0.593086\pi$
$488$	11.0549	0.500430
$489$	0	0
$490$	−1.95391	−0.0882685
$491$	14.5816	0.658059	0.329030	−	0.944320i	$-0.393278\pi$
$491$	14.5816	0.658059	0.329030	+	0.944320i	$0.393278\pi$
$492$	0	0
$493$	−2.46092	−0.110834
$494$	−21.2090	−0.954238
$495$	0	0
$496$	−9.97638	−0.447953
$497$	−4.36611	−0.195847
$498$	0	0
$499$	32.3845	1.44973	0.724865	−	0.688891i	$-0.241902\pi$
$499$	32.3845	1.44973	0.724865	+	0.688891i	$0.241902\pi$
$500$	11.9244	0.533276
$501$	0	0
$502$	−15.8884	−0.709134
$503$	9.35364	0.417058	0.208529	−	0.978016i	$-0.433132\pi$
$503$	9.35364	0.417058	0.208529	+	0.978016i	$0.433132\pi$
$504$	0	0
$505$	1.46852	0.0653485
$506$	−0.249396	−0.0110870
$507$	0	0
$508$	16.7407	0.742750
$509$	−12.4816	−0.553235	−0.276618	−	0.960980i	$-0.589214\pi$
$509$	−12.4816	−0.553235	−0.276618	+	0.960980i	$0.589214\pi$
$510$	0	0
$511$	4.77458	0.211215
$512$	−18.9099	−0.835707
$513$	0	0
$514$	−11.9540	−0.527270
$515$	15.5981	0.687335
$516$	0	0
$517$	3.92658	0.172691
$518$	−23.4746	−1.03142
$519$	0	0
$520$	−9.43890	−0.413923
$521$	−33.1613	−1.45282	−0.726412	−	0.687260i	$-0.758814\pi$
$521$	−33.1613	−1.45282	−0.726412	+	0.687260i	$0.758814\pi$
$522$	0	0
$523$	−15.7853	−0.690242	−0.345121	−	0.938558i	$-0.612162\pi$
$523$	−15.7853	−0.690242	−0.345121	+	0.938558i	$0.612162\pi$
$524$	−8.65187	−0.377959
$525$	0	0
$526$	17.9836	0.784122
$527$	13.0489	0.568417
$528$	0	0
$529$	1.00000	0.0434783
$530$	−0.336345	−0.0146099
$531$	0	0
$532$	34.4043	1.49162
$533$	−29.6279	−1.28333
$534$	0	0
$535$	4.80622	0.207791
$536$	22.6244	0.977226
$537$	0	0
$538$	−16.0088	−0.690187
$539$	1.65293	0.0711966
$540$	0	0
$541$	−36.3339	−1.56212	−0.781059	−	0.624457i	$-0.785320\pi$
$541$	−36.3339	−1.56212	−0.781059	+	0.624457i	$0.785320\pi$
$542$	5.40635	0.232223
$543$	0	0
$544$	13.7878	0.591148
$545$	−3.57167	−0.152993
$546$	0	0
$547$	−32.9178	−1.40746	−0.703732	−	0.710466i	$-0.748484\pi$
$547$	−32.9178	−1.40746	−0.703732	+	0.710466i	$0.748484\pi$
$548$	−21.0239	−0.898096
$549$	0	0
$550$	1.09427	0.0466596
$551$	−6.37074	−0.271403
$552$	0	0
$553$	38.3786	1.63202
$554$	−0.626891	−0.0266341
$555$	0	0
$556$	−19.6545	−0.833537
$557$	12.5151	0.530281	0.265140	−	0.964210i	$-0.414582\pi$
$557$	12.5151	0.530281	0.265140	+	0.964210i	$0.414582\pi$
$558$	0	0
$559$	46.3667	1.96110
$560$	4.89808	0.206982
$561$	0	0
$562$	5.27157	0.222368
$563$	21.3237	0.898688	0.449344	−	0.893359i	$-0.351658\pi$
$563$	21.3237	0.898688	0.449344	+	0.893359i	$0.351658\pi$
$564$	0	0
$565$	−4.54074	−0.191031
$566$	0.327538	0.0137675
$567$	0	0
$568$	−2.91864	−0.122463
$569$	−17.7507	−0.744148	−0.372074	−	0.928203i	$-0.621353\pi$
$569$	−17.7507	−0.744148	−0.372074	+	0.928203i	$0.621353\pi$
$570$	0	0
$571$	20.6810	0.865475	0.432737	−	0.901520i	$-0.357548\pi$
$571$	20.6810	0.865475	0.432737	+	0.901520i	$0.357548\pi$
$572$	3.57733	0.149576
$573$	0	0
$574$	−11.1546	−0.465583
$575$	−4.38767	−0.182979
$576$	0	0
$577$	−30.1615	−1.25564	−0.627820	−	0.778358i	$-0.716053\pi$
$577$	−30.1615	−1.25564	−0.627820	+	0.778358i	$0.716053\pi$
$578$	6.71729	0.279403
$579$	0	0
$580$	−1.27022	−0.0527430
$581$	−55.9994	−2.32324
$582$	0	0
$583$	0.284535	0.0117842
$584$	3.19170	0.132073
$585$	0	0
$586$	−15.8710	−0.655626
$587$	36.4970	1.50639	0.753195	−	0.657797i	$-0.228512\pi$
$587$	36.4970	1.50639	0.753195	+	0.657797i	$0.228512\pi$
$588$	0	0
$589$	33.7804	1.39190
$590$	−6.04212	−0.248750
$591$	0	0
$592$	21.6290	0.888948
$593$	23.3409	0.958497	0.479249	−	0.877679i	$-0.340909\pi$
$593$	23.3409	0.958497	0.479249	+	0.877679i	$0.340909\pi$
$594$	0	0
$595$	−6.40657	−0.262644
$596$	−30.1275	−1.23407
$597$	0	0
$598$	3.32913	0.136138
$599$	−6.24596	−0.255203	−0.127602	−	0.991826i	$-0.540728\pi$
$599$	−6.24596	−0.255203	−0.127602	+	0.991826i	$0.540728\pi$
$600$	0	0
$601$	16.4225	0.669889	0.334944	−	0.942238i	$-0.391282\pi$
$601$	16.4225	0.669889	0.334944	+	0.942238i	$0.391282\pi$
$602$	17.4565	0.711475
$603$	0	0
$604$	−7.04006	−0.286456
$605$	−8.47847	−0.344699
$606$	0	0
$607$	−28.0848	−1.13993	−0.569963	−	0.821670i	$-0.693042\pi$
$607$	−28.0848	−1.13993	−0.569963	+	0.821670i	$0.693042\pi$
$608$	35.6934	1.44756
$609$	0	0
$610$	−2.38752	−0.0966678
$611$	−52.4150	−2.12049
$612$	0	0
$613$	15.0842	0.609245	0.304623	−	0.952473i	$-0.401470\pi$
$613$	15.0842	0.609245	0.304623	+	0.952473i	$0.401470\pi$
$614$	2.04085	0.0823619
$615$	0	0
$616$	3.00624	0.121125
$617$	−32.9052	−1.32471	−0.662356	−	0.749189i	$-0.730443\pi$
$617$	−32.9052	−1.32471	−0.662356	+	0.749189i	$0.730443\pi$
$618$	0	0
$619$	15.8196	0.635842	0.317921	−	0.948117i	$-0.397015\pi$
$619$	15.8196	0.635842	0.317921	+	0.948117i	$0.397015\pi$
$620$	6.73526	0.270495
$621$	0	0
$622$	3.59903	0.144308
$623$	−4.45073	−0.178315
$624$	0	0
$625$	16.1900	0.647601
$626$	−3.42924	−0.137060
$627$	0	0
$628$	−6.39405	−0.255150
$629$	−28.2902	−1.12801
$630$	0	0
$631$	34.3335	1.36679	0.683397	−	0.730047i	$-0.260502\pi$
$631$	34.3335	1.36679	0.683397	+	0.730047i	$0.260502\pi$
$632$	25.6552	1.02051
$633$	0	0
$634$	−11.9492	−0.474562
$635$	−8.07012	−0.320253
$636$	0	0
$637$	−22.0646	−0.874230
$638$	−0.249396	−0.00987366
$639$	0	0
$640$	8.92403	0.352753
$641$	−17.9225	−0.707896	−0.353948	−	0.935265i	$-0.615161\pi$
$641$	−17.9225	−0.707896	−0.353948	+	0.935265i	$0.615161\pi$
$642$	0	0
$643$	5.12069	0.201940	0.100970	−	0.994889i	$-0.467805\pi$
$643$	5.12069	0.201940	0.100970	+	0.994889i	$0.467805\pi$
$644$	−5.40036	−0.212804
$645$	0	0
$646$	−9.62299	−0.378612
$647$	−17.6799	−0.695068	−0.347534	−	0.937667i	$-0.612981\pi$
$647$	−17.6799	−0.695068	−0.347534	+	0.937667i	$0.612981\pi$
$648$	0	0
$649$	5.11140	0.200640
$650$	−14.6071	−0.572938
$651$	0	0
$652$	9.65886	0.378270
$653$	−14.0391	−0.549391	−0.274696	−	0.961531i	$-0.588577\pi$
$653$	−14.0391	−0.549391	−0.274696	+	0.961531i	$0.588577\pi$
$654$	0	0
$655$	4.17076	0.162965
$656$	10.2776	0.401273
$657$	0	0
$658$	−19.7337	−0.769299
$659$	−24.5368	−0.955820	−0.477910	−	0.878409i	$-0.658605\pi$
$659$	−24.5368	−0.955820	−0.477910	+	0.878409i	$0.658605\pi$
$660$	0	0
$661$	1.71122	0.0665587	0.0332794	−	0.999446i	$-0.489405\pi$
$661$	1.71122	0.0665587	0.0332794	+	0.999446i	$0.489405\pi$
$662$	−10.6513	−0.413976
$663$	0	0
$664$	−37.4343	−1.45273
$665$	−16.5851	−0.643142
$666$	0	0
$667$	1.00000	0.0387202
$668$	19.8647	0.768589
$669$	0	0
$670$	−4.88619	−0.188770
$671$	2.01975	0.0779714
$672$	0	0
$673$	−47.0079	−1.81202	−0.906011	−	0.423255i	$-0.860887\pi$
$673$	−47.0079	−1.81202	−0.906011	+	0.423255i	$0.860887\pi$
$674$	1.75904	0.0677558
$675$	0	0
$676$	−26.6507	−1.02503
$677$	−2.45583	−0.0943851	−0.0471926	−	0.998886i	$-0.515027\pi$
$677$	−2.45583	−0.0943851	−0.0471926	+	0.998886i	$0.515027\pi$
$678$	0	0
$679$	13.7714	0.528498
$680$	−4.28264	−0.164232
$681$	0	0
$682$	1.32240	0.0506374
$683$	−39.0901	−1.49574	−0.747870	−	0.663845i	$-0.768923\pi$
$683$	−39.0901	−1.49574	−0.747870	+	0.663845i	$0.768923\pi$
$684$	0	0
$685$	10.1349	0.387234
$686$	5.98705	0.228587
$687$	0	0
$688$	−16.0841	−0.613200
$689$	−3.79819	−0.144699
$690$	0	0
$691$	−17.6833	−0.672703	−0.336352	−	0.941736i	$-0.609193\pi$
$691$	−17.6833	−0.672703	−0.336352	+	0.941736i	$0.609193\pi$
$692$	−28.1997	−1.07199
$693$	0	0
$694$	14.3793	0.545831
$695$	9.47474	0.359397
$696$	0	0
$697$	−13.4428	−0.509184
$698$	−10.2085	−0.386397
$699$	0	0
$700$	23.6950	0.895588
$701$	−7.01496	−0.264951	−0.132476	−	0.991186i	$-0.542293\pi$
$701$	−7.01496	−0.264951	−0.132476	+	0.991186i	$0.542293\pi$
$702$	0	0
$703$	−73.2368	−2.76218
$704$	−0.131659	−0.00496210
$705$	0	0
$706$	9.48399	0.356935
$707$	6.24345	0.234809
$708$	0	0
$709$	−22.4609	−0.843536	−0.421768	−	0.906704i	$-0.638590\pi$
$709$	−22.4609	−0.843536	−0.421768	+	0.906704i	$0.638590\pi$
$710$	0.630339	0.0236562
$711$	0	0
$712$	−2.97521	−0.111501
$713$	−5.30243	−0.198578
$714$	0	0
$715$	−1.72451	−0.0644929
$716$	−41.0110	−1.53265
$717$	0	0
$718$	1.42846	0.0533098
$719$	6.64525	0.247826	0.123913	−	0.992293i	$-0.460456\pi$
$719$	6.64525	0.247826	0.123913	+	0.992293i	$0.460456\pi$
$720$	0	0
$721$	66.3156	2.46972
$722$	−13.2496	−0.493098
$723$	0	0
$724$	26.8465	0.997741
$725$	−4.38767	−0.162954
$726$	0	0
$727$	40.4518	1.50027	0.750137	−	0.661282i	$-0.229987\pi$
$727$	40.4518	1.50027	0.750137	+	0.661282i	$0.229987\pi$
$728$	−40.1296	−1.48730
$729$	0	0
$730$	−0.689310	−0.0255125
$731$	21.0376	0.778104
$732$	0	0
$733$	−31.5260	−1.16444	−0.582221	−	0.813031i	$-0.697816\pi$
$733$	−31.5260	−1.16444	−0.582221	+	0.813031i	$0.697816\pi$
$734$	−7.94573	−0.293282
$735$	0	0
$736$	−5.60271	−0.206519
$737$	4.13353	0.152260
$738$	0	0
$739$	13.7641	0.506320	0.253160	−	0.967424i	$-0.418530\pi$
$739$	13.7641	0.506320	0.253160	+	0.967424i	$0.418530\pi$
$740$	−14.6022	−0.536788
$741$	0	0
$742$	−1.42998	−0.0524961
$743$	19.2211	0.705152	0.352576	−	0.935783i	$-0.385306\pi$
$743$	19.2211	0.705152	0.352576	+	0.935783i	$0.385306\pi$
$744$	0	0
$745$	14.5234	0.532096
$746$	4.40087	0.161127
$747$	0	0
$748$	1.62312	0.0593470
$749$	20.4337	0.746632
$750$	0	0
$751$	4.53449	0.165466	0.0827329	−	0.996572i	$-0.473635\pi$
$751$	4.53449	0.165466	0.0827329	+	0.996572i	$0.473635\pi$
$752$	18.1822	0.663037
$753$	0	0
$754$	3.32913	0.121240
$755$	3.39376	0.123512
$756$	0	0
$757$	39.0396	1.41892	0.709458	−	0.704747i	$-0.248940\pi$
$757$	39.0396	1.41892	0.709458	+	0.704747i	$0.248940\pi$
$758$	−0.351007	−0.0127492
$759$	0	0
$760$	−11.0867	−0.402159
$761$	−6.73778	−0.244244	−0.122122	−	0.992515i	$-0.538970\pi$
$761$	−6.73778	−0.244244	−0.122122	+	0.992515i	$0.538970\pi$
$762$	0	0
$763$	−15.1850	−0.549734
$764$	18.0170	0.651832
$765$	0	0
$766$	2.68964	0.0971808
$767$	−68.2309	−2.46368
$768$	0	0
$769$	32.8445	1.18440	0.592202	−	0.805789i	$-0.298259\pi$
$769$	32.8445	1.18440	0.592202	+	0.805789i	$0.298259\pi$
$770$	−0.649257	−0.0233976
$771$	0	0
$772$	−28.2950	−1.01836
$773$	43.3837	1.56040	0.780201	−	0.625528i	$-0.215117\pi$
$773$	43.3837	1.56040	0.780201	+	0.625528i	$0.215117\pi$
$774$	0	0
$775$	23.2653	0.835716
$776$	9.20586	0.330471
$777$	0	0
$778$	15.0193	0.538470
$779$	−34.8004	−1.24685
$780$	0	0
$781$	−0.533242	−0.0190809
$782$	1.51050	0.0540153
$783$	0	0
$784$	7.65396	0.273356
$785$	3.08234	0.110014
$786$	0	0
$787$	36.6658	1.30699	0.653497	−	0.756929i	$-0.273301\pi$
$787$	36.6658	1.30699	0.653497	+	0.756929i	$0.273301\pi$
$788$	17.1351	0.610413
$789$	0	0
$790$	−5.54075	−0.197131
$791$	−19.3050	−0.686408
$792$	0	0
$793$	−26.9611	−0.957418
$794$	15.1583	0.537948
$795$	0	0
$796$	13.9857	0.495708
$797$	11.1622	0.395386	0.197693	−	0.980264i	$-0.436655\pi$
$797$	11.1622	0.395386	0.197693	+	0.980264i	$0.436655\pi$
$798$	0	0
$799$	−23.7819	−0.841342
$800$	24.5828	0.869135
$801$	0	0
$802$	−23.7606	−0.839016
$803$	0.583129	0.0205782
$804$	0	0
$805$	2.60332	0.0917551
$806$	−17.6525	−0.621782
$807$	0	0
$808$	4.17360	0.146827
$809$	−8.46324	−0.297552	−0.148776	−	0.988871i	$-0.547533\pi$
$809$	−8.46324	−0.297552	−0.148776	+	0.988871i	$0.547533\pi$
$810$	0	0
$811$	10.0341	0.352346	0.176173	−	0.984359i	$-0.443628\pi$
$811$	10.0341	0.352346	0.176173	+	0.984359i	$0.443628\pi$
$812$	−5.40036	−0.189516
$813$	0	0
$814$	−2.86700	−0.100488
$815$	−4.65620	−0.163099
$816$	0	0
$817$	54.4614	1.90536
$818$	−19.5890	−0.684912
$819$	0	0
$820$	−6.93862	−0.242307
$821$	35.6671	1.24479	0.622396	−	0.782703i	$-0.286160\pi$
$821$	35.6671	1.24479	0.622396	+	0.782703i	$0.286160\pi$
$822$	0	0
$823$	22.6271	0.788732	0.394366	−	0.918953i	$-0.370964\pi$
$823$	22.6271	0.788732	0.394366	+	0.918953i	$0.370964\pi$
$824$	44.3304	1.54432
$825$	0	0
$826$	−25.6882	−0.893806
$827$	−46.9972	−1.63425	−0.817127	−	0.576457i	$-0.804435\pi$
$827$	−46.9972	−1.63425	−0.817127	+	0.576457i	$0.804435\pi$
$828$	0	0
$829$	−19.5954	−0.680575	−0.340287	−	0.940321i	$-0.610524\pi$
$829$	−19.5954	−0.680575	−0.340287	+	0.940321i	$0.610524\pi$
$830$	8.08467	0.280623
$831$	0	0
$832$	1.75749	0.0609301
$833$	−10.0112	−0.346867
$834$	0	0
$835$	−9.57607	−0.331394
$836$	4.20187	0.145325
$837$	0	0
$838$	0.0328676	0.00113539
$839$	8.04882	0.277876	0.138938	−	0.990301i	$-0.455631\pi$
$839$	8.04882	0.277876	0.138938	+	0.990301i	$0.455631\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−10.5488	−0.363536
$843$	0	0
$844$	2.30078	0.0791961
$845$	12.8474	0.441963
$846$	0	0
$847$	−36.0463	−1.23857
$848$	1.31755	0.0452449
$849$	0	0
$850$	−6.62757	−0.227324
$851$	11.4958	0.394071
$852$	0	0
$853$	−7.46746	−0.255681	−0.127841	−	0.991795i	$-0.540805\pi$
$853$	−7.46746	−0.255681	−0.127841	+	0.991795i	$0.540805\pi$
$854$	−10.1506	−0.347345
$855$	0	0
$856$	13.6595	0.466871
$857$	−8.18956	−0.279750	−0.139875	−	0.990169i	$-0.544670\pi$
$857$	−8.18956	−0.279750	−0.139875	+	0.990169i	$0.544670\pi$
$858$	0	0
$859$	25.0580	0.854966	0.427483	−	0.904023i	$-0.359400\pi$
$859$	25.0580	0.854966	0.427483	+	0.904023i	$0.359400\pi$
$860$	10.8587	0.370279
$861$	0	0
$862$	−12.3537	−0.420767
$863$	−20.5562	−0.699740	−0.349870	−	0.936798i	$-0.613774\pi$
$863$	−20.5562	−0.699740	−0.349870	+	0.936798i	$0.613774\pi$
$864$	0	0
$865$	13.5941	0.462212
$866$	−12.7317	−0.432641
$867$	0	0
$868$	28.6351	0.971938
$869$	4.68726	0.159004
$870$	0	0
$871$	−55.1775	−1.86962
$872$	−10.1508	−0.343750
$873$	0	0
$874$	3.91033	0.132269
$875$	−24.4391	−0.826194
$876$	0	0
$877$	−56.8970	−1.92127	−0.960637	−	0.277805i	$-0.910393\pi$
$877$	−56.8970	−1.92127	−0.960637	+	0.277805i	$0.910393\pi$
$878$	20.5091	0.692147
$879$	0	0
$880$	0.598212	0.0201657
$881$	9.50818	0.320339	0.160169	−	0.987090i	$-0.448796\pi$
$881$	9.50818	0.320339	0.160169	+	0.987090i	$0.448796\pi$
$882$	0	0
$883$	−13.1230	−0.441625	−0.220813	−	0.975316i	$-0.570871\pi$
$883$	−13.1230	−0.441625	−0.220813	+	0.975316i	$0.570871\pi$
$884$	−21.6666	−0.728728
$885$	0	0
$886$	−18.8698	−0.633943
$887$	11.1163	0.373250	0.186625	−	0.982431i	$-0.440245\pi$
$887$	11.1163	0.373250	0.186625	+	0.982431i	$0.440245\pi$
$888$	0	0
$889$	−34.3102	−1.15073
$890$	0.642555	0.0215385
$891$	0	0
$892$	−29.6488	−0.992717
$893$	−61.5657	−2.06022
$894$	0	0
$895$	19.7700	0.660837
$896$	37.9407	1.26751
$897$	0	0
$898$	−20.6551	−0.689268
$899$	−5.30243	−0.176846
$900$	0	0
$901$	−1.72332	−0.0574122
$902$	−1.36233	−0.0453607
$903$	0	0
$904$	−12.9050	−0.429213
$905$	−12.9417	−0.430197
$906$	0	0
$907$	46.8834	1.55674	0.778369	−	0.627807i	$-0.216047\pi$
$907$	46.8834	1.55674	0.778369	+	0.627807i	$0.216047\pi$
$908$	−30.9576	−1.02736
$909$	0	0
$910$	8.66679	0.287301
$911$	−58.9023	−1.95152	−0.975760	−	0.218843i	$-0.929772\pi$
$911$	−58.9023	−1.95152	−0.975760	+	0.218843i	$0.929772\pi$
$912$	0	0
$913$	−6.83931	−0.226348
$914$	12.1857	0.403067
$915$	0	0
$916$	34.4138	1.13706
$917$	17.7320	0.585564
$918$	0	0
$919$	6.50750	0.214663	0.107331	−	0.994223i	$-0.465769\pi$
$919$	6.50750	0.214663	0.107331	+	0.994223i	$0.465769\pi$
$920$	1.74026	0.0573747
$921$	0	0
$922$	21.8441	0.719399
$923$	7.11812	0.234296
$924$	0	0
$925$	−50.4398	−1.65845
$926$	−16.1602	−0.531058
$927$	0	0
$928$	−5.60271	−0.183918
$929$	24.2544	0.795761	0.397881	−	0.917437i	$-0.369746\pi$
$929$	24.2544	0.795761	0.397881	+	0.917437i	$0.369746\pi$
$930$	0	0
$931$	−25.9166	−0.849382
$932$	18.1080	0.593146
$933$	0	0
$934$	−12.1018	−0.395983
$935$	−0.782447	−0.0255888
$936$	0	0
$937$	−21.5372	−0.703590	−0.351795	−	0.936077i	$-0.614429\pi$
$937$	−21.5372	−0.703590	−0.351795	+	0.936077i	$0.614429\pi$
$938$	−20.7737	−0.678286
$939$	0	0
$940$	−12.2752	−0.400372
$941$	9.34945	0.304783	0.152392	−	0.988320i	$-0.451302\pi$
$941$	9.34945	0.304783	0.152392	+	0.988320i	$0.451302\pi$
$942$	0	0
$943$	5.46253	0.177884
$944$	23.6686	0.770346
$945$	0	0
$946$	2.13200	0.0693173
$947$	−45.3753	−1.47450	−0.737250	−	0.675620i	$-0.763876\pi$
$947$	−45.3753	−1.47450	−0.737250	+	0.675620i	$0.763876\pi$
$948$	0	0
$949$	−7.78406	−0.252681
$950$	−17.1572	−0.556654
$951$	0	0
$952$	−18.2077	−0.590115
$953$	46.7193	1.51338	0.756692	−	0.653771i	$-0.226814\pi$
$953$	46.7193	1.51338	0.756692	+	0.653771i	$0.226814\pi$
$954$	0	0
$955$	−8.68535	−0.281051
$956$	−17.2790	−0.558843
$957$	0	0
$958$	−10.1646	−0.328404
$959$	43.0886	1.39140
$960$	0	0
$961$	−2.88419	−0.0930384
$962$	38.2710	1.23391
$963$	0	0
$964$	6.25174	0.201355
$965$	13.6400	0.439087
$966$	0	0
$967$	−25.0414	−0.805277	−0.402638	−	0.915359i	$-0.631907\pi$
$967$	−25.0414	−0.805277	−0.402638	+	0.915359i	$0.631907\pi$
$968$	−24.0961	−0.774479
$969$	0	0
$970$	−1.98819	−0.0638369
$971$	8.31839	0.266950	0.133475	−	0.991052i	$-0.457386\pi$
$971$	8.31839	0.266950	0.133475	+	0.991052i	$0.457386\pi$
$972$	0	0
$973$	40.2820	1.29138
$974$	7.80985	0.250244
$975$	0	0
$976$	9.35253	0.299367
$977$	−12.6967	−0.406202	−0.203101	−	0.979158i	$-0.565102\pi$
$977$	−12.6967	−0.406202	−0.203101	+	0.979158i	$0.565102\pi$
$978$	0	0
$979$	−0.543576	−0.0173728
$980$	−5.16735	−0.165065
$981$	0	0
$982$	−8.95012	−0.285610
$983$	−26.9842	−0.860662	−0.430331	−	0.902671i	$-0.641603\pi$
$983$	−26.9842	−0.860662	−0.430331	+	0.902671i	$0.641603\pi$
$984$	0	0
$985$	−8.26023	−0.263193
$986$	1.51050	0.0481041
$987$	0	0
$988$	−56.0898	−1.78445
$989$	−8.54868	−0.271832
$990$	0	0
$991$	−8.13084	−0.258285	−0.129142	−	0.991626i	$-0.541222\pi$
$991$	−8.13084	−0.258285	−0.129142	+	0.991626i	$0.541222\pi$
$992$	29.7080	0.943230
$993$	0	0
$994$	2.67989	0.0850011
$995$	−6.74199	−0.213735
$996$	0	0
$997$	−22.8192	−0.722691	−0.361345	−	0.932432i	$-0.617682\pi$
$997$	−22.8192	−0.722691	−0.361345	+	0.932432i	$0.617682\pi$
$998$	−19.8775	−0.629210
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.n.1.5		12
3.2	odd	2		667.2.a.b.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.b.1.8	✓	12	3.2	odd	2
6003.2.a.n.1.5		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.613795 q^{2} -1.62326 q^{4} +0.782514 q^{5} +3.32687 q^{7} +2.22394 q^{8} +O(q^{10})\)	\(q-0.613795 q^{2} -1.62326 q^{4} +0.782514 q^{5} +3.32687 q^{7} +2.22394 q^{8} -0.480303 q^{10} +0.406317 q^{11} -5.42384 q^{13} -2.04202 q^{14} +1.88147 q^{16} -2.46092 q^{17} -6.37074 q^{19} -1.27022 q^{20} -0.249396 q^{22} +1.00000 q^{23} -4.38767 q^{25} +3.32913 q^{26} -5.40036 q^{28} +1.00000 q^{29} -5.30243 q^{31} -5.60271 q^{32} +1.51050 q^{34} +2.60332 q^{35} +11.4958 q^{37} +3.91033 q^{38} +1.74026 q^{40} +5.46253 q^{41} -8.54868 q^{43} -0.659557 q^{44} -0.613795 q^{46} +9.66382 q^{47} +4.06807 q^{49} +2.69313 q^{50} +8.80428 q^{52} +0.700276 q^{53} +0.317949 q^{55} +7.39875 q^{56} -0.613795 q^{58} +12.5798 q^{59} +4.97086 q^{61} +3.25461 q^{62} -0.324031 q^{64} -4.24423 q^{65} +10.1731 q^{67} +3.99470 q^{68} -1.59791 q^{70} -1.31238 q^{71} +1.43516 q^{73} -7.05606 q^{74} +10.3413 q^{76} +1.35177 q^{77} +11.5359 q^{79} +1.47228 q^{80} -3.35287 q^{82} -16.8324 q^{83} -1.92570 q^{85} +5.24713 q^{86} +0.903624 q^{88} -1.33781 q^{89} -18.0444 q^{91} -1.62326 q^{92} -5.93160 q^{94} -4.98519 q^{95} +4.13945 q^{97} -2.49696 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8 + 6 * q^11 - 15 * q^13 + 8 * q^14 + 17 * q^16 + 18 * q^17 - 6 * q^19 + 39 * q^20 - 5 * q^22 + 12 * q^23 + 14 * q^25 + 3 * q^26 - 19 * q^28 + 12 * q^29 + 16 * q^31 + 21 * q^32 - 7 * q^34 + 11 * q^35 - q^37 + 24 * q^38 + 30 * q^40 - 3 * q^41 - 23 * q^43 - 23 * q^44 + 3 * q^46 + 35 * q^47 + 3 * q^49 + 2 * q^50 + 45 * q^53 + 17 * q^55 + 17 * q^56 + 3 * q^58 + 11 * q^59 + 4 * q^61 + 7 * q^62 + 15 * q^64 - 5 * q^65 - 19 * q^67 - q^68 + 14 * q^70 - 19 * q^71 + 10 * q^73 + 15 * q^74 - 4 * q^76 + 39 * q^77 + 17 * q^79 + 90 * q^80 - 3 * q^82 + 12 * q^83 + 14 * q^85 - 17 * q^86 - 2 * q^88 + 20 * q^89 + 11 * q^91 + 11 * q^92 + 13 * q^94 - 12 * q^95 - 12 * q^97 - 75 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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