LMFDB - Embedded newform 6003.2.a.p.1.8

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} + \cdots - 64 $ x^14 - 2x^13 - 18x^12 + 34x^11 + 124x^10 - 216x^9 - 420x^8 + 647x^7 + 750x^6 - 939x^5 - 717x^4 + 604x^3 + 352x^2 - 128*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.630703$ 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)

$f(q)$	$=$	$q+0.630703 q^{2} -1.60221 q^{4} -2.56371 q^{5} +2.03354 q^{7} -2.27193 q^{8} +O(q^{10})$	\(q+0.630703 q^{2} -1.60221 q^{4} -2.56371 q^{5} +2.03354 q^{7} -2.27193 q^{8} -1.61694 q^{10} +5.50828 q^{11} +1.27535 q^{13} +1.28256 q^{14} +1.77152 q^{16} +4.88133 q^{17} +1.34419 q^{19} +4.10761 q^{20} +3.47409 q^{22} +1.00000 q^{23} +1.57260 q^{25} +0.804364 q^{26} -3.25816 q^{28} +1.00000 q^{29} -8.27265 q^{31} +5.66115 q^{32} +3.07867 q^{34} -5.21339 q^{35} +5.65206 q^{37} +0.847782 q^{38} +5.82456 q^{40} -5.78265 q^{41} -8.25433 q^{43} -8.82544 q^{44} +0.630703 q^{46} +4.49789 q^{47} -2.86473 q^{49} +0.991843 q^{50} -2.04338 q^{52} -3.73860 q^{53} -14.1216 q^{55} -4.62005 q^{56} +0.630703 q^{58} -6.53633 q^{59} +2.18990 q^{61} -5.21758 q^{62} +0.0274685 q^{64} -3.26961 q^{65} -0.298090 q^{67} -7.82094 q^{68} -3.28810 q^{70} +5.07739 q^{71} +6.18250 q^{73} +3.56477 q^{74} -2.15367 q^{76} +11.2013 q^{77} +2.32543 q^{79} -4.54166 q^{80} -3.64713 q^{82} +7.21327 q^{83} -12.5143 q^{85} -5.20603 q^{86} -12.5144 q^{88} +4.05929 q^{89} +2.59346 q^{91} -1.60221 q^{92} +2.83683 q^{94} -3.44610 q^{95} +3.90092 q^{97} -1.80679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * 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 $

$2$	0.630703	0.445974	0.222987	−	0.974821i	$-0.428419\pi$
$2$	0.630703	0.445974	0.222987	+	0.974821i	$0.428419\pi$
$3$	0	0
$3$	0	0
$4$	−1.60221	−0.801107
$5$	−2.56371	−1.14653	−0.573263	−	0.819372i	$-0.694322\pi$
$5$	−2.56371	−1.14653	−0.573263	+	0.819372i	$0.694322\pi$
$6$	0	0
$7$	2.03354	0.768605	0.384302	−	0.923207i	$-0.374442\pi$
$7$	2.03354	0.768605	0.384302	+	0.923207i	$0.374442\pi$
$8$	−2.27193	−0.803247
$9$	0	0
$10$	−1.61694	−0.511321
$11$	5.50828	1.66081	0.830404	−	0.557162i	$-0.188110\pi$
$11$	5.50828	1.66081	0.830404	+	0.557162i	$0.188110\pi$
$12$	0	0
$13$	1.27535	0.353717	0.176859	−	0.984236i	$-0.443406\pi$
$13$	1.27535	0.353717	0.176859	+	0.984236i	$0.443406\pi$
$14$	1.28256	0.342778
$15$	0	0
$16$	1.77152	0.442880
$17$	4.88133	1.18390	0.591948	−	0.805976i	$-0.298359\pi$
$17$	4.88133	1.18390	0.591948	+	0.805976i	$0.298359\pi$
$18$	0	0
$19$	1.34419	0.308378	0.154189	−	0.988041i	$-0.450724\pi$
$19$	1.34419	0.308378	0.154189	+	0.988041i	$0.450724\pi$
$20$	4.10761	0.918489
$21$	0	0
$22$	3.47409	0.740678
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.57260	0.314520
$26$	0.804364	0.157749
$27$	0	0
$28$	−3.25816	−0.615735
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.27265	−1.48581	−0.742906	−	0.669396i	$-0.766553\pi$
$31$	−8.27265	−1.48581	−0.742906	+	0.669396i	$0.766553\pi$
$32$	5.66115	1.00076
$33$	0	0
$34$	3.07867	0.527987
$35$	−5.21339	−0.881224
$36$	0	0
$37$	5.65206	0.929193	0.464597	−	0.885522i	$-0.346199\pi$
$37$	5.65206	0.929193	0.464597	+	0.885522i	$0.346199\pi$
$38$	0.847782	0.137528
$39$	0	0
$40$	5.82456	0.920943
$41$	−5.78265	−0.903097	−0.451549	−	0.892246i	$-0.649128\pi$
$41$	−5.78265	−0.903097	−0.451549	+	0.892246i	$0.649128\pi$
$42$	0	0
$43$	−8.25433	−1.25877	−0.629387	−	0.777092i	$-0.716694\pi$
$43$	−8.25433	−1.25877	−0.629387	+	0.777092i	$0.716694\pi$
$44$	−8.82544	−1.33049
$45$	0	0
$46$	0.630703	0.0929920
$47$	4.49789	0.656085	0.328042	−	0.944663i	$-0.393611\pi$
$47$	4.49789	0.656085	0.328042	+	0.944663i	$0.393611\pi$
$48$	0	0
$49$	−2.86473	−0.409247
$50$	0.991843	0.140268
$51$	0	0
$52$	−2.04338	−0.283365
$53$	−3.73860	−0.513537	−0.256768	−	0.966473i	$-0.582658\pi$
$53$	−3.73860	−0.513537	−0.256768	+	0.966473i	$0.582658\pi$
$54$	0	0
$55$	−14.1216	−1.90416
$56$	−4.62005	−0.617380
$57$	0	0
$58$	0.630703	0.0828153
$59$	−6.53633	−0.850958	−0.425479	−	0.904968i	$-0.639894\pi$
$59$	−6.53633	−0.850958	−0.425479	+	0.904968i	$0.639894\pi$
$60$	0	0
$61$	2.18990	0.280388	0.140194	−	0.990124i	$-0.455227\pi$
$61$	2.18990	0.280388	0.140194	+	0.990124i	$0.455227\pi$
$62$	−5.21758	−0.662633
$63$	0	0
$64$	0.0274685	0.00343357
$65$	−3.26961	−0.405546
$66$	0	0
$67$	−0.298090	−0.0364175	−0.0182088	−	0.999834i	$-0.505796\pi$
$67$	−0.298090	−0.0364175	−0.0182088	+	0.999834i	$0.505796\pi$
$68$	−7.82094	−0.948428
$69$	0	0
$70$	−3.28810	−0.393003
$71$	5.07739	0.602576	0.301288	−	0.953533i	$-0.402583\pi$
$71$	5.07739	0.602576	0.301288	+	0.953533i	$0.402583\pi$
$72$	0	0
$73$	6.18250	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$73$	6.18250	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$74$	3.56477	0.414396
$75$	0	0
$76$	−2.15367	−0.247043
$77$	11.2013	1.27650
$78$	0	0
$79$	2.32543	0.261631	0.130816	−	0.991407i	$-0.458240\pi$
$79$	2.32543	0.261631	0.130816	+	0.991407i	$0.458240\pi$
$80$	−4.54166	−0.507773
$81$	0	0
$82$	−3.64713	−0.402758
$83$	7.21327	0.791759	0.395880	−	0.918302i	$-0.370440\pi$
$83$	7.21327	0.791759	0.395880	+	0.918302i	$0.370440\pi$
$84$	0	0
$85$	−12.5143	−1.35737
$86$	−5.20603	−0.561381
$87$	0	0
$88$	−12.5144	−1.33404
$89$	4.05929	0.430284	0.215142	−	0.976583i	$-0.430979\pi$
$89$	4.05929	0.430284	0.215142	+	0.976583i	$0.430979\pi$
$90$	0	0
$91$	2.59346	0.271869
$92$	−1.60221	−0.167042
$93$	0	0
$94$	2.83683	0.292597
$95$	−3.44610	−0.353563
$96$	0	0
$97$	3.90092	0.396078	0.198039	−	0.980194i	$-0.436543\pi$
$97$	3.90092	0.396078	0.198039	+	0.980194i	$0.436543\pi$
$98$	−1.80679	−0.182514
$99$	0	0
$100$	−2.51964	−0.251964
$101$	12.8977	1.28337	0.641683	−	0.766970i	$-0.278237\pi$
$101$	12.8977	1.28337	0.641683	+	0.766970i	$0.278237\pi$
$102$	0	0
$103$	4.68294	0.461424	0.230712	−	0.973022i	$-0.425895\pi$
$103$	4.68294	0.461424	0.230712	+	0.973022i	$0.425895\pi$
$104$	−2.89749	−0.284122
$105$	0	0
$106$	−2.35795	−0.229024
$107$	9.59086	0.927184	0.463592	−	0.886049i	$-0.346560\pi$
$107$	9.59086	0.927184	0.463592	+	0.886049i	$0.346560\pi$
$108$	0	0
$109$	−5.04547	−0.483269	−0.241634	−	0.970367i	$-0.577683\pi$
$109$	−5.04547	−0.483269	−0.241634	+	0.970367i	$0.577683\pi$
$110$	−8.90654	−0.849205
$111$	0	0
$112$	3.60245	0.340399
$113$	2.10834	0.198336	0.0991678	−	0.995071i	$-0.468382\pi$
$113$	2.10834	0.198336	0.0991678	+	0.995071i	$0.468382\pi$
$114$	0	0
$115$	−2.56371	−0.239067
$116$	−1.60221	−0.148762
$117$	0	0
$118$	−4.12248	−0.379505
$119$	9.92637	0.909948
$120$	0	0
$121$	19.3411	1.75828
$122$	1.38118	0.125046
$123$	0	0
$124$	13.2546	1.19029
$125$	8.78686	0.785920
$126$	0	0
$127$	−7.18709	−0.637751	−0.318876	−	0.947797i	$-0.603305\pi$
$127$	−7.18709	−0.637751	−0.318876	+	0.947797i	$0.603305\pi$
$128$	−11.3050	−0.999229
$129$	0	0
$130$	−2.06215	−0.180863
$131$	8.68477	0.758792	0.379396	−	0.925234i	$-0.376132\pi$
$131$	8.68477	0.758792	0.379396	+	0.925234i	$0.376132\pi$
$132$	0	0
$133$	2.73345	0.237020
$134$	−0.188006	−0.0162413
$135$	0	0
$136$	−11.0900	−0.950962
$137$	22.0622	1.88490	0.942450	−	0.334347i	$-0.108516\pi$
$137$	22.0622	1.88490	0.942450	+	0.334347i	$0.108516\pi$
$138$	0	0
$139$	−11.4252	−0.969072	−0.484536	−	0.874771i	$-0.661012\pi$
$139$	−11.4252	−0.969072	−0.484536	+	0.874771i	$0.661012\pi$
$140$	8.35297	0.705955
$141$	0	0
$142$	3.20233	0.268733
$143$	7.02496	0.587456
$144$	0	0
$145$	−2.56371	−0.212904
$146$	3.89932	0.322710
$147$	0	0
$148$	−9.05581	−0.744383
$149$	19.0001	1.55655	0.778275	−	0.627924i	$-0.216095\pi$
$149$	19.0001	1.55655	0.778275	+	0.627924i	$0.216095\pi$
$150$	0	0
$151$	16.0475	1.30593	0.652964	−	0.757389i	$-0.273525\pi$
$151$	16.0475	1.30593	0.652964	+	0.757389i	$0.273525\pi$
$152$	−3.05389	−0.247703
$153$	0	0
$154$	7.06468	0.569288
$155$	21.2087	1.70352
$156$	0	0
$157$	−18.4530	−1.47271	−0.736357	−	0.676593i	$-0.763455\pi$
$157$	−18.4530	−1.47271	−0.736357	+	0.676593i	$0.763455\pi$
$158$	1.46666	0.116681
$159$	0	0
$160$	−14.5135	−1.14740
$161$	2.03354	0.160265
$162$	0	0
$163$	−17.3102	−1.35584	−0.677919	−	0.735136i	$-0.737118\pi$
$163$	−17.3102	−1.35584	−0.677919	+	0.735136i	$0.737118\pi$
$164$	9.26504	0.723478
$165$	0	0
$166$	4.54943	0.353104
$167$	4.12893	0.319506	0.159753	−	0.987157i	$-0.448930\pi$
$167$	4.12893	0.319506	0.159753	+	0.987157i	$0.448930\pi$
$168$	0	0
$169$	−11.3735	−0.874884
$170$	−7.89281	−0.605351
$171$	0	0
$172$	13.2252	1.00841
$173$	15.4036	1.17112	0.585559	−	0.810630i	$-0.300875\pi$
$173$	15.4036	1.17112	0.585559	+	0.810630i	$0.300875\pi$
$174$	0	0
$175$	3.19794	0.241741
$176$	9.75801	0.735538
$177$	0	0
$178$	2.56020	0.191895
$179$	−13.6173	−1.01781	−0.508903	−	0.860824i	$-0.669949\pi$
$179$	−13.6173	−1.01781	−0.508903	+	0.860824i	$0.669949\pi$
$180$	0	0
$181$	9.80678	0.728932	0.364466	−	0.931217i	$-0.381252\pi$
$181$	9.80678	0.728932	0.364466	+	0.931217i	$0.381252\pi$
$182$	1.63570	0.121246
$183$	0	0
$184$	−2.27193	−0.167489
$185$	−14.4902	−1.06534
$186$	0	0
$187$	26.8877	1.96623
$188$	−7.20658	−0.525594
$189$	0	0
$190$	−2.17347	−0.157680
$191$	16.6177	1.20242	0.601208	−	0.799092i	$-0.294686\pi$
$191$	16.6177	1.20242	0.601208	+	0.799092i	$0.294686\pi$
$192$	0	0
$193$	−7.47464	−0.538036	−0.269018	−	0.963135i	$-0.586699\pi$
$193$	−7.47464	−0.538036	−0.269018	+	0.963135i	$0.586699\pi$
$194$	2.46032	0.176641
$195$	0	0
$196$	4.58991	0.327851
$197$	−12.0577	−0.859074	−0.429537	−	0.903049i	$-0.641323\pi$
$197$	−12.0577	−0.859074	−0.429537	+	0.903049i	$0.641323\pi$
$198$	0	0
$199$	11.5826	0.821071	0.410536	−	0.911845i	$-0.365342\pi$
$199$	11.5826	0.821071	0.410536	+	0.911845i	$0.365342\pi$
$200$	−3.57283	−0.252637
$201$	0	0
$202$	8.13459	0.572348
$203$	2.03354	0.142726
$204$	0	0
$205$	14.8250	1.03542
$206$	2.95354	0.205783
$207$	0	0
$208$	2.25930	0.156654
$209$	7.40415	0.512156
$210$	0	0
$211$	21.3501	1.46980	0.734900	−	0.678175i	$-0.237229\pi$
$211$	21.3501	1.46980	0.734900	+	0.678175i	$0.237229\pi$
$212$	5.99004	0.411398
$213$	0	0
$214$	6.04898	0.413500
$215$	21.1617	1.44322
$216$	0	0
$217$	−16.8227	−1.14200
$218$	−3.18219	−0.215525
$219$	0	0
$220$	22.6259	1.52543
$221$	6.22538	0.418765
$222$	0	0
$223$	−24.5876	−1.64651	−0.823255	−	0.567672i	$-0.807844\pi$
$223$	−24.5876	−1.64651	−0.823255	+	0.567672i	$0.807844\pi$
$224$	11.5122	0.769189
$225$	0	0
$226$	1.32973	0.0884525
$227$	−4.38899	−0.291308	−0.145654	−	0.989336i	$-0.546529\pi$
$227$	−4.38899	−0.291308	−0.145654	+	0.989336i	$0.546529\pi$
$228$	0	0
$229$	11.2247	0.741747	0.370873	−	0.928683i	$-0.379058\pi$
$229$	11.2247	0.741747	0.370873	+	0.928683i	$0.379058\pi$
$230$	−1.61694	−0.106618
$231$	0	0
$232$	−2.27193	−0.149159
$233$	−2.47999	−0.162469	−0.0812346	−	0.996695i	$-0.525886\pi$
$233$	−2.47999	−0.162469	−0.0812346	+	0.996695i	$0.525886\pi$
$234$	0	0
$235$	−11.5313	−0.752218
$236$	10.4726	0.681709
$237$	0	0
$238$	6.26059	0.405814
$239$	−8.12142	−0.525331	−0.262665	−	0.964887i	$-0.584602\pi$
$239$	−8.12142	−0.525331	−0.262665	+	0.964887i	$0.584602\pi$
$240$	0	0
$241$	25.1081	1.61736	0.808679	−	0.588251i	$-0.200183\pi$
$241$	25.1081	1.61736	0.808679	+	0.588251i	$0.200183\pi$
$242$	12.1985	0.784149
$243$	0	0
$244$	−3.50869	−0.224621
$245$	7.34433	0.469212
$246$	0	0
$247$	1.71430	0.109078
$248$	18.7948	1.19347
$249$	0	0
$250$	5.54189	0.350500
$251$	24.3400	1.53633	0.768163	−	0.640255i	$-0.221171\pi$
$251$	24.3400	1.53633	0.768163	+	0.640255i	$0.221171\pi$
$252$	0	0
$253$	5.50828	0.346302
$254$	−4.53292	−0.284421
$255$	0	0
$256$	−7.18502	−0.449064
$257$	−10.2478	−0.639241	−0.319620	−	0.947546i	$-0.603555\pi$
$257$	−10.2478	−0.639241	−0.319620	+	0.947546i	$0.603555\pi$
$258$	0	0
$259$	11.4937	0.714182
$260$	5.23862	0.324885
$261$	0	0
$262$	5.47751	0.338402
$263$	−22.3363	−1.37732	−0.688658	−	0.725087i	$-0.741800\pi$
$263$	−22.3363	−1.37732	−0.688658	+	0.725087i	$0.741800\pi$
$264$	0	0
$265$	9.58469	0.588783
$266$	1.72400	0.105705
$267$	0	0
$268$	0.477605	0.0291744
$269$	−32.2145	−1.96416	−0.982078	−	0.188476i	$-0.939645\pi$
$269$	−32.2145	−1.96416	−0.982078	+	0.188476i	$0.939645\pi$
$270$	0	0
$271$	7.14973	0.434315	0.217158	−	0.976137i	$-0.430321\pi$
$271$	7.14973	0.434315	0.217158	+	0.976137i	$0.430321\pi$
$272$	8.64737	0.524324
$273$	0	0
$274$	13.9147	0.840617
$275$	8.66231	0.522357
$276$	0	0
$277$	−4.82843	−0.290112	−0.145056	−	0.989423i	$-0.546336\pi$
$277$	−4.82843	−0.290112	−0.145056	+	0.989423i	$0.546336\pi$
$278$	−7.20590	−0.432181
$279$	0	0
$280$	11.8444	0.707841
$281$	1.28074	0.0764026	0.0382013	−	0.999270i	$-0.487837\pi$
$281$	1.28074	0.0764026	0.0382013	+	0.999270i	$0.487837\pi$
$282$	0	0
$283$	−10.9093	−0.648492	−0.324246	−	0.945973i	$-0.605111\pi$
$283$	−10.9093	−0.648492	−0.324246	+	0.945973i	$0.605111\pi$
$284$	−8.13507	−0.482728
$285$	0	0
$286$	4.43066	0.261990
$287$	−11.7592	−0.694125
$288$	0	0
$289$	6.82739	0.401611
$290$	−1.61694	−0.0949498
$291$	0	0
$292$	−9.90569	−0.579686
$293$	19.4378	1.13557	0.567785	−	0.823177i	$-0.307801\pi$
$293$	19.4378	1.13557	0.567785	+	0.823177i	$0.307801\pi$
$294$	0	0
$295$	16.7573	0.975645
$296$	−12.8411	−0.746372
$297$	0	0
$298$	11.9834	0.694181
$299$	1.27535	0.0737551
$300$	0	0
$301$	−16.7855	−0.967499
$302$	10.1212	0.582410
$303$	0	0
$304$	2.38125	0.136574
$305$	−5.61427	−0.321472
$306$	0	0
$307$	2.07266	0.118293	0.0591465	−	0.998249i	$-0.481162\pi$
$307$	2.07266	0.118293	0.0591465	+	0.998249i	$0.481162\pi$
$308$	−17.9469	−1.02262
$309$	0	0
$310$	13.3764	0.759726
$311$	−4.15061	−0.235360	−0.117680	−	0.993052i	$-0.537546\pi$
$311$	−4.15061	−0.235360	−0.117680	+	0.993052i	$0.537546\pi$
$312$	0	0
$313$	20.5932	1.16400	0.581998	−	0.813190i	$-0.302271\pi$
$313$	20.5932	1.16400	0.581998	+	0.813190i	$0.302271\pi$
$314$	−11.6384	−0.656792
$315$	0	0
$316$	−3.72584	−0.209595
$317$	−22.4744	−1.26229	−0.631143	−	0.775667i	$-0.717414\pi$
$317$	−22.4744	−1.26229	−0.631143	+	0.775667i	$0.717414\pi$
$318$	0	0
$319$	5.50828	0.308404
$320$	−0.0704213	−0.00393667
$321$	0	0
$322$	1.28256	0.0714741
$323$	6.56142	0.365087
$324$	0	0
$325$	2.00561	0.111251
$326$	−10.9176	−0.604669
$327$	0	0
$328$	13.1377	0.725410
$329$	9.14662	0.504270
$330$	0	0
$331$	−23.6817	−1.30166	−0.650831	−	0.759223i	$-0.725579\pi$
$331$	−23.6817	−1.30166	−0.650831	+	0.759223i	$0.725579\pi$
$332$	−11.5572	−0.634284
$333$	0	0
$334$	2.60413	0.142492
$335$	0.764217	0.0417536
$336$	0	0
$337$	12.6607	0.689675	0.344837	−	0.938662i	$-0.387934\pi$
$337$	12.6607	0.689675	0.344837	+	0.938662i	$0.387934\pi$
$338$	−7.17329	−0.390176
$339$	0	0
$340$	20.0506	1.08740
$341$	−45.5680	−2.46765
$342$	0	0
$343$	−20.0603	−1.08315
$344$	18.7532	1.01111
$345$	0	0
$346$	9.71512	0.522288
$347$	19.4322	1.04318	0.521588	−	0.853198i	$-0.325340\pi$
$347$	19.4322	1.04318	0.521588	+	0.853198i	$0.325340\pi$
$348$	0	0
$349$	−5.89400	−0.315498	−0.157749	−	0.987479i	$-0.550424\pi$
$349$	−5.89400	−0.315498	−0.157749	+	0.987479i	$0.550424\pi$
$350$	2.01695	0.107810
$351$	0	0
$352$	31.1832	1.66207
$353$	−5.81336	−0.309414	−0.154707	−	0.987960i	$-0.549443\pi$
$353$	−5.81336	−0.309414	−0.154707	+	0.987960i	$0.549443\pi$
$354$	0	0
$355$	−13.0170	−0.690868
$356$	−6.50385	−0.344703
$357$	0	0
$358$	−8.58848	−0.453916
$359$	1.40728	0.0742734	0.0371367	−	0.999310i	$-0.488176\pi$
$359$	1.40728	0.0742734	0.0371367	+	0.999310i	$0.488176\pi$
$360$	0	0
$361$	−17.1932	−0.904903
$362$	6.18516	0.325085
$363$	0	0
$364$	−4.15528	−0.217796
$365$	−15.8501	−0.829633
$366$	0	0
$367$	28.0108	1.46215	0.731076	−	0.682297i	$-0.239019\pi$
$367$	28.0108	1.46215	0.731076	+	0.682297i	$0.239019\pi$
$368$	1.77152	0.0923468
$369$	0	0
$370$	−9.13903	−0.475116
$371$	−7.60259	−0.394707
$372$	0	0
$373$	0.409526	0.0212045	0.0106022	−	0.999944i	$-0.496625\pi$
$373$	0.409526	0.0212045	0.0106022	+	0.999944i	$0.496625\pi$
$374$	16.9582	0.876886
$375$	0	0
$376$	−10.2189	−0.526998
$377$	1.27535	0.0656836
$378$	0	0
$379$	15.1347	0.777418	0.388709	−	0.921361i	$-0.372921\pi$
$379$	15.1347	0.777418	0.388709	+	0.921361i	$0.372921\pi$
$380$	5.52139	0.283241
$381$	0	0
$382$	10.4808	0.536247
$383$	34.6829	1.77221	0.886106	−	0.463483i	$-0.153400\pi$
$383$	34.6829	1.77221	0.886106	+	0.463483i	$0.153400\pi$
$384$	0	0
$385$	−28.7168	−1.46354
$386$	−4.71428	−0.239950
$387$	0	0
$388$	−6.25011	−0.317301
$389$	12.3127	0.624276	0.312138	−	0.950037i	$-0.398955\pi$
$389$	12.3127	0.624276	0.312138	+	0.950037i	$0.398955\pi$
$390$	0	0
$391$	4.88133	0.246860
$392$	6.50845	0.328726
$393$	0	0
$394$	−7.60481	−0.383125
$395$	−5.96173	−0.299967
$396$	0	0
$397$	34.5021	1.73161	0.865806	−	0.500379i	$-0.166806\pi$
$397$	34.5021	1.73161	0.865806	+	0.500379i	$0.166806\pi$
$398$	7.30520	0.366177
$399$	0	0
$400$	2.78589	0.139294
$401$	−34.8805	−1.74185	−0.870925	−	0.491415i	$-0.836480\pi$
$401$	−34.8805	−1.74185	−0.870925	+	0.491415i	$0.836480\pi$
$402$	0	0
$403$	−10.5505	−0.525557
$404$	−20.6648	−1.02811
$405$	0	0
$406$	1.28256	0.0636522
$407$	31.1331	1.54321
$408$	0	0
$409$	2.53399	0.125298	0.0626488	−	0.998036i	$-0.480045\pi$
$409$	2.53399	0.125298	0.0626488	+	0.998036i	$0.480045\pi$
$410$	9.35018	0.461772
$411$	0	0
$412$	−7.50307	−0.369650
$413$	−13.2919	−0.654050
$414$	0	0
$415$	−18.4927	−0.907772
$416$	7.21993	0.353986
$417$	0	0
$418$	4.66982	0.228408
$419$	39.2261	1.91632	0.958160	−	0.286232i	$-0.0924029\pi$
$419$	39.2261	1.91632	0.958160	+	0.286232i	$0.0924029\pi$
$420$	0	0
$421$	22.8734	1.11478	0.557390	−	0.830251i	$-0.311803\pi$
$421$	22.8734	1.11478	0.557390	+	0.830251i	$0.311803\pi$
$422$	13.4656	0.655493
$423$	0	0
$424$	8.49383	0.412497
$425$	7.67638	0.372359
$426$	0	0
$427$	4.45325	0.215508
$428$	−15.3666	−0.742774
$429$	0	0
$430$	13.3467	0.643637
$431$	18.4185	0.887190	0.443595	−	0.896227i	$-0.353703\pi$
$431$	18.4185	0.887190	0.443595	+	0.896227i	$0.353703\pi$
$432$	0	0
$433$	−23.2329	−1.11650	−0.558251	−	0.829672i	$-0.688527\pi$
$433$	−23.2329	−1.11650	−0.558251	+	0.829672i	$0.688527\pi$
$434$	−10.6101	−0.509303
$435$	0	0
$436$	8.08393	0.387150
$437$	1.34419	0.0643012
$438$	0	0
$439$	−17.3498	−0.828061	−0.414030	−	0.910263i	$-0.635879\pi$
$439$	−17.3498	−0.828061	−0.414030	+	0.910263i	$0.635879\pi$
$440$	32.0833	1.52951
$441$	0	0
$442$	3.92637	0.186758
$443$	22.0460	1.04744	0.523718	−	0.851891i	$-0.324544\pi$
$443$	22.0460	1.04744	0.523718	+	0.851891i	$0.324544\pi$
$444$	0	0
$445$	−10.4068	−0.493331
$446$	−15.5075	−0.734301
$447$	0	0
$448$	0.0558583	0.00263906
$449$	16.1034	0.759966	0.379983	−	0.924993i	$-0.375930\pi$
$449$	16.1034	0.759966	0.379983	+	0.924993i	$0.375930\pi$
$450$	0	0
$451$	−31.8524	−1.49987
$452$	−3.37800	−0.158888
$453$	0	0
$454$	−2.76815	−0.129916
$455$	−6.64888	−0.311704
$456$	0	0
$457$	−14.4370	−0.675333	−0.337666	−	0.941266i	$-0.609638\pi$
$457$	−14.4370	−0.675333	−0.337666	+	0.941266i	$0.609638\pi$
$458$	7.07943	0.330800
$459$	0	0
$460$	4.10761	0.191518
$461$	25.3097	1.17879	0.589396	−	0.807844i	$-0.299366\pi$
$461$	25.3097	1.17879	0.589396	+	0.807844i	$0.299366\pi$
$462$	0	0
$463$	14.3449	0.666662	0.333331	−	0.942810i	$-0.391827\pi$
$463$	14.3449	0.666662	0.333331	+	0.942810i	$0.391827\pi$
$464$	1.77152	0.0822407
$465$	0	0
$466$	−1.56413	−0.0724571
$467$	−27.5488	−1.27481	−0.637404	−	0.770530i	$-0.719992\pi$
$467$	−27.5488	−1.27481	−0.637404	+	0.770530i	$0.719992\pi$
$468$	0	0
$469$	−0.606178	−0.0279907
$470$	−7.27281	−0.335470
$471$	0	0
$472$	14.8501	0.683530
$473$	−45.4671	−2.09058
$474$	0	0
$475$	2.11387	0.0969909
$476$	−15.9042	−0.728966
$477$	0	0
$478$	−5.12220	−0.234284
$479$	36.9320	1.68747	0.843733	−	0.536764i	$-0.180353\pi$
$479$	36.9320	1.68747	0.843733	+	0.536764i	$0.180353\pi$
$480$	0	0
$481$	7.20833	0.328672
$482$	15.8358	0.721300
$483$	0	0
$484$	−30.9886	−1.40857
$485$	−10.0008	−0.454114
$486$	0	0
$487$	4.65272	0.210835	0.105418	−	0.994428i	$-0.466382\pi$
$487$	4.65272	0.210835	0.105418	+	0.994428i	$0.466382\pi$
$488$	−4.97530	−0.225221
$489$	0	0
$490$	4.63209	0.209256
$491$	−1.99458	−0.0900141	−0.0450071	−	0.998987i	$-0.514331\pi$
$491$	−1.99458	−0.0900141	−0.0450071	+	0.998987i	$0.514331\pi$
$492$	0	0
$493$	4.88133	0.219844
$494$	1.08121	0.0486462
$495$	0	0
$496$	−14.6551	−0.658035
$497$	10.3251	0.463143
$498$	0	0
$499$	−2.40425	−0.107629	−0.0538145	−	0.998551i	$-0.517138\pi$
$499$	−2.40425	−0.107629	−0.0538145	+	0.998551i	$0.517138\pi$
$500$	−14.0784	−0.629606
$501$	0	0
$502$	15.3513	0.685161
$503$	−24.2446	−1.08101	−0.540506	−	0.841340i	$-0.681767\pi$
$503$	−24.2446	−1.08101	−0.540506	+	0.841340i	$0.681767\pi$
$504$	0	0
$505$	−33.0659	−1.47141
$506$	3.47409	0.154442
$507$	0	0
$508$	11.5153	0.510907
$509$	−26.3653	−1.16862	−0.584311	−	0.811529i	$-0.698635\pi$
$509$	−26.3653	−1.16862	−0.584311	+	0.811529i	$0.698635\pi$
$510$	0	0
$511$	12.5723	0.556167
$512$	18.0784	0.798958
$513$	0	0
$514$	−6.46332	−0.285085
$515$	−12.0057	−0.529034
$516$	0	0
$517$	24.7756	1.08963
$518$	7.24909	0.318507
$519$	0	0
$520$	7.42832	0.325753
$521$	30.3000	1.32747	0.663734	−	0.747969i	$-0.268971\pi$
$521$	30.3000	1.32747	0.663734	+	0.747969i	$0.268971\pi$
$522$	0	0
$523$	−7.34838	−0.321322	−0.160661	−	0.987010i	$-0.551363\pi$
$523$	−7.34838	−0.321322	−0.160661	+	0.987010i	$0.551363\pi$
$524$	−13.9149	−0.607874
$525$	0	0
$526$	−14.0876	−0.614247
$527$	−40.3815	−1.75905
$528$	0	0
$529$	1.00000	0.0434783
$530$	6.04509	0.262582
$531$	0	0
$532$	−4.37958	−0.189879
$533$	−7.37487	−0.319441
$534$	0	0
$535$	−24.5882	−1.06304
$536$	0.677240	0.0292523
$537$	0	0
$538$	−20.3178	−0.875963
$539$	−15.7797	−0.679681
$540$	0	0
$541$	19.8410	0.853031	0.426515	−	0.904480i	$-0.359741\pi$
$541$	19.8410	0.853031	0.426515	+	0.904480i	$0.359741\pi$
$542$	4.50935	0.193693
$543$	0	0
$544$	27.6340	1.18480
$545$	12.9351	0.554080
$546$	0	0
$547$	37.5906	1.60726	0.803630	−	0.595130i	$-0.202899\pi$
$547$	37.5906	1.60726	0.803630	+	0.595130i	$0.202899\pi$
$548$	−35.3484	−1.51001
$549$	0	0
$550$	5.46334	0.232958
$551$	1.34419	0.0572643
$552$	0	0
$553$	4.72885	0.201091
$554$	−3.04531	−0.129383
$555$	0	0
$556$	18.3056	0.776331
$557$	−13.1925	−0.558985	−0.279493	−	0.960148i	$-0.590166\pi$
$557$	−13.1925	−0.558985	−0.279493	+	0.960148i	$0.590166\pi$
$558$	0	0
$559$	−10.5271	−0.445250
$560$	−9.23562	−0.390276
$561$	0	0
$562$	0.807766	0.0340736
$563$	8.81467	0.371494	0.185747	−	0.982598i	$-0.440530\pi$
$563$	8.81467	0.371494	0.185747	+	0.982598i	$0.440530\pi$
$564$	0	0
$565$	−5.40516	−0.227397
$566$	−6.88054	−0.289211
$567$	0	0
$568$	−11.5355	−0.484017
$569$	39.8436	1.67033	0.835165	−	0.549999i	$-0.185372\pi$
$569$	39.8436	1.67033	0.835165	+	0.549999i	$0.185372\pi$
$570$	0	0
$571$	0.350696	0.0146762	0.00733808	−	0.999973i	$-0.497664\pi$
$571$	0.350696	0.0146762	0.00733808	+	0.999973i	$0.497664\pi$
$572$	−11.2555	−0.470615
$573$	0	0
$574$	−7.41657	−0.309562
$575$	1.57260	0.0655819
$576$	0	0
$577$	−16.6022	−0.691159	−0.345579	−	0.938390i	$-0.612318\pi$
$577$	−16.6022	−0.691159	−0.345579	+	0.938390i	$0.612318\pi$
$578$	4.30606	0.179108
$579$	0	0
$580$	4.10761	0.170559
$581$	14.6684	0.608550
$582$	0	0
$583$	−20.5933	−0.852886
$584$	−14.0462	−0.581235
$585$	0	0
$586$	12.2595	0.506434
$587$	11.3702	0.469297	0.234649	−	0.972080i	$-0.424606\pi$
$587$	11.3702	0.469297	0.234649	+	0.972080i	$0.424606\pi$
$588$	0	0
$589$	−11.1200	−0.458191
$590$	10.5688	0.435112
$591$	0	0
$592$	10.0127	0.411521
$593$	25.1510	1.03283	0.516414	−	0.856339i	$-0.327266\pi$
$593$	25.1510	1.03283	0.516414	+	0.856339i	$0.327266\pi$
$594$	0	0
$595$	−25.4483	−1.04328
$596$	−30.4423	−1.24696
$597$	0	0
$598$	0.804364	0.0328929
$599$	−23.6593	−0.966694	−0.483347	−	0.875429i	$-0.660579\pi$
$599$	−23.6593	−0.966694	−0.483347	+	0.875429i	$0.660579\pi$
$600$	0	0
$601$	−19.3312	−0.788538	−0.394269	−	0.918995i	$-0.629002\pi$
$601$	−19.3312	−0.788538	−0.394269	+	0.918995i	$0.629002\pi$
$602$	−10.5867	−0.431480
$603$	0	0
$604$	−25.7116	−1.04619
$605$	−49.5850	−2.01592
$606$	0	0
$607$	31.0956	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$607$	31.0956	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$608$	7.60965	0.308612
$609$	0	0
$610$	−3.54094	−0.143368
$611$	5.73636	0.232068
$612$	0	0
$613$	40.7516	1.64594	0.822971	−	0.568083i	$-0.192315\pi$
$613$	40.7516	1.64594	0.822971	+	0.568083i	$0.192315\pi$
$614$	1.30723	0.0527557
$615$	0	0
$616$	−25.4485	−1.02535
$617$	−19.8476	−0.799034	−0.399517	−	0.916726i	$-0.630822\pi$
$617$	−19.8476	−0.799034	−0.399517	+	0.916726i	$0.630822\pi$
$618$	0	0
$619$	−4.54279	−0.182590	−0.0912950	−	0.995824i	$-0.529101\pi$
$619$	−4.54279	−0.182590	−0.0912950	+	0.995824i	$0.529101\pi$
$620$	−33.9808	−1.36470
$621$	0	0
$622$	−2.61780	−0.104964
$623$	8.25471	0.330718
$624$	0	0
$625$	−30.3899	−1.21560
$626$	12.9882	0.519112
$627$	0	0
$628$	29.5657	1.17980
$629$	27.5896	1.10007
$630$	0	0
$631$	25.8076	1.02738	0.513692	−	0.857975i	$-0.328277\pi$
$631$	25.8076	1.02738	0.513692	+	0.857975i	$0.328277\pi$
$632$	−5.28321	−0.210155
$633$	0	0
$634$	−14.1746	−0.562947
$635$	18.4256	0.731198
$636$	0	0
$637$	−3.65352	−0.144758
$638$	3.47409	0.137540
$639$	0	0
$640$	28.9827	1.14564
$641$	11.9141	0.470578	0.235289	−	0.971925i	$-0.424396\pi$
$641$	11.9141	0.470578	0.235289	+	0.971925i	$0.424396\pi$
$642$	0	0
$643$	34.2158	1.34934	0.674669	−	0.738121i	$-0.264286\pi$
$643$	34.2158	1.34934	0.674669	+	0.738121i	$0.264286\pi$
$644$	−3.25816	−0.128390
$645$	0	0
$646$	4.13830	0.162819
$647$	20.4743	0.804928	0.402464	−	0.915436i	$-0.368154\pi$
$647$	20.4743	0.804928	0.402464	+	0.915436i	$0.368154\pi$
$648$	0	0
$649$	−36.0039	−1.41328
$650$	1.26494	0.0496151
$651$	0	0
$652$	27.7346	1.08617
$653$	−37.8601	−1.48158	−0.740790	−	0.671737i	$-0.765549\pi$
$653$	−37.8601	−1.48158	−0.740790	+	0.671737i	$0.765549\pi$
$654$	0	0
$655$	−22.2652	−0.869974
$656$	−10.2441	−0.399963
$657$	0	0
$658$	5.76880	0.224891
$659$	−4.47194	−0.174202	−0.0871010	−	0.996199i	$-0.527760\pi$
$659$	−4.47194	−0.174202	−0.0871010	+	0.996199i	$0.527760\pi$
$660$	0	0
$661$	−14.2256	−0.553313	−0.276656	−	0.960969i	$-0.589226\pi$
$661$	−14.2256	−0.553313	−0.276656	+	0.960969i	$0.589226\pi$
$662$	−14.9361	−0.580508
$663$	0	0
$664$	−16.3880	−0.635978
$665$	−7.00777	−0.271750
$666$	0	0
$667$	1.00000	0.0387202
$668$	−6.61543	−0.255959
$669$	0	0
$670$	0.481994	0.0186210
$671$	12.0626	0.465671
$672$	0	0
$673$	−33.5178	−1.29202	−0.646009	−	0.763330i	$-0.723563\pi$
$673$	−33.5178	−1.29202	−0.646009	+	0.763330i	$0.723563\pi$
$674$	7.98516	0.307577
$675$	0	0
$676$	18.2228	0.700876
$677$	−48.4870	−1.86351	−0.931753	−	0.363092i	$-0.881721\pi$
$677$	−48.4870	−1.86351	−0.931753	+	0.363092i	$0.881721\pi$
$678$	0	0
$679$	7.93266	0.304428
$680$	28.4316	1.09030
$681$	0	0
$682$	−28.7399	−1.10051
$683$	8.50718	0.325518	0.162759	−	0.986666i	$-0.447961\pi$
$683$	8.50718	0.325518	0.162759	+	0.986666i	$0.447961\pi$
$684$	0	0
$685$	−56.5610	−2.16109
$686$	−12.6521	−0.483059
$687$	0	0
$688$	−14.6227	−0.557485
$689$	−4.76801	−0.181647
$690$	0	0
$691$	−3.00888	−0.114463	−0.0572316	−	0.998361i	$-0.518227\pi$
$691$	−3.00888	−0.114463	−0.0572316	+	0.998361i	$0.518227\pi$
$692$	−24.6799	−0.938190
$693$	0	0
$694$	12.2559	0.465229
$695$	29.2909	1.11107
$696$	0	0
$697$	−28.2270	−1.06917
$698$	−3.71736	−0.140704
$699$	0	0
$700$	−5.12378	−0.193661
$701$	−27.6729	−1.04519	−0.522596	−	0.852580i	$-0.675036\pi$
$701$	−27.6729	−1.04519	−0.522596	+	0.852580i	$0.675036\pi$
$702$	0	0
$703$	7.59742	0.286542
$704$	0.151304	0.00570250
$705$	0	0
$706$	−3.66650	−0.137991
$707$	26.2279	0.986401
$708$	0	0
$709$	17.4190	0.654186	0.327093	−	0.944992i	$-0.393931\pi$
$709$	17.4190	0.654186	0.327093	+	0.944992i	$0.393931\pi$
$710$	−8.20983	−0.308109
$711$	0	0
$712$	−9.22241	−0.345624
$713$	−8.27265	−0.309813
$714$	0	0
$715$	−18.0099	−0.673533
$716$	21.8179	0.815372
$717$	0	0
$718$	0.887576	0.0331240
$719$	52.2397	1.94821	0.974107	−	0.226090i	$-0.0725942\pi$
$719$	52.2397	1.94821	0.974107	+	0.226090i	$0.0725942\pi$
$720$	0	0
$721$	9.52293	0.354652
$722$	−10.8438	−0.403563
$723$	0	0
$724$	−15.7126	−0.583953
$725$	1.57260	0.0584049
$726$	0	0
$727$	2.19295	0.0813319	0.0406659	−	0.999173i	$-0.487052\pi$
$727$	2.19295	0.0813319	0.0406659	+	0.999173i	$0.487052\pi$
$728$	−5.89215	−0.218378
$729$	0	0
$730$	−9.99671	−0.369995
$731$	−40.2921	−1.49026
$732$	0	0
$733$	−8.83403	−0.326292	−0.163146	−	0.986602i	$-0.552164\pi$
$733$	−8.83403	−0.326292	−0.163146	+	0.986602i	$0.552164\pi$
$734$	17.6665	0.652082
$735$	0	0
$736$	5.66115	0.208673
$737$	−1.64197	−0.0604826
$738$	0	0
$739$	19.1084	0.702914	0.351457	−	0.936204i	$-0.385686\pi$
$739$	19.1084	0.702914	0.351457	+	0.936204i	$0.385686\pi$
$740$	23.2165	0.853454
$741$	0	0
$742$	−4.79497	−0.176029
$743$	−14.9500	−0.548463	−0.274232	−	0.961664i	$-0.588423\pi$
$743$	−14.9500	−0.548463	−0.274232	+	0.961664i	$0.588423\pi$
$744$	0	0
$745$	−48.7107	−1.78462
$746$	0.258289	0.00945664
$747$	0	0
$748$	−43.0799	−1.57516
$749$	19.5034	0.712638
$750$	0	0
$751$	−29.6741	−1.08282	−0.541412	−	0.840757i	$-0.682110\pi$
$751$	−29.6741	−1.08282	−0.541412	+	0.840757i	$0.682110\pi$
$752$	7.96809	0.290566
$753$	0	0
$754$	0.804364	0.0292932
$755$	−41.1412	−1.49728
$756$	0	0
$757$	20.3856	0.740928	0.370464	−	0.928847i	$-0.379199\pi$
$757$	20.3856	0.740928	0.370464	+	0.928847i	$0.379199\pi$
$758$	9.54550	0.346708
$759$	0	0
$760$	7.82929	0.283998
$761$	−47.4548	−1.72024	−0.860118	−	0.510095i	$-0.829610\pi$
$761$	−47.4548	−1.72024	−0.860118	+	0.510095i	$0.829610\pi$
$762$	0	0
$763$	−10.2602	−0.371443
$764$	−26.6252	−0.963264
$765$	0	0
$766$	21.8746	0.790361
$767$	−8.33608	−0.300998
$768$	0	0
$769$	8.55949	0.308663	0.154332	−	0.988019i	$-0.450678\pi$
$769$	8.55949	0.308663	0.154332	+	0.988019i	$0.450678\pi$
$770$	−18.1118	−0.652703
$771$	0	0
$772$	11.9760	0.431025
$773$	9.90662	0.356316	0.178158	−	0.984002i	$-0.442986\pi$
$773$	9.90662	0.356316	0.178158	+	0.984002i	$0.442986\pi$
$774$	0	0
$775$	−13.0096	−0.467317
$776$	−8.86260	−0.318149
$777$	0	0
$778$	7.76562	0.278411
$779$	−7.77295	−0.278495
$780$	0	0
$781$	27.9677	1.00076
$782$	3.07867	0.110093
$783$	0	0
$784$	−5.07492	−0.181247
$785$	47.3082	1.68850
$786$	0	0
$787$	51.5526	1.83765	0.918826	−	0.394663i	$-0.129139\pi$
$787$	51.5526	1.83765	0.918826	+	0.394663i	$0.129139\pi$
$788$	19.3190	0.688210
$789$	0	0
$790$	−3.76008	−0.133778
$791$	4.28738	0.152442
$792$	0	0
$793$	2.79288	0.0991782
$794$	21.7606	0.772254
$795$	0	0
$796$	−18.5579	−0.657766
$797$	−14.1603	−0.501584	−0.250792	−	0.968041i	$-0.580691\pi$
$797$	−14.1603	−0.501584	−0.250792	+	0.968041i	$0.580691\pi$
$798$	0	0
$799$	21.9557	0.776736
$800$	8.90273	0.314759
$801$	0	0
$802$	−21.9992	−0.776820
$803$	34.0549	1.20177
$804$	0	0
$805$	−5.21339	−0.183748
$806$	−6.65422	−0.234385
$807$	0	0
$808$	−29.3025	−1.03086
$809$	28.3795	0.997770	0.498885	−	0.866668i	$-0.333743\pi$
$809$	28.3795	0.997770	0.498885	+	0.866668i	$0.333743\pi$
$810$	0	0
$811$	−30.5389	−1.07237	−0.536183	−	0.844102i	$-0.680134\pi$
$811$	−30.5389	−1.07237	−0.536183	+	0.844102i	$0.680134\pi$
$812$	−3.25816	−0.114339
$813$	0	0
$814$	19.6357	0.688233
$815$	44.3783	1.55450
$816$	0	0
$817$	−11.0954	−0.388177
$818$	1.59819	0.0558795
$819$	0	0
$820$	−23.7528	−0.829485
$821$	−27.0534	−0.944169	−0.472085	−	0.881553i	$-0.656498\pi$
$821$	−27.0534	−0.944169	−0.472085	+	0.881553i	$0.656498\pi$
$822$	0	0
$823$	16.4371	0.572963	0.286482	−	0.958086i	$-0.407514\pi$
$823$	16.4371	0.572963	0.286482	+	0.958086i	$0.407514\pi$
$824$	−10.6393	−0.370637
$825$	0	0
$826$	−8.38322	−0.291690
$827$	−9.98601	−0.347247	−0.173624	−	0.984812i	$-0.555548\pi$
$827$	−9.98601	−0.347247	−0.173624	+	0.984812i	$0.555548\pi$
$828$	0	0
$829$	25.3998	0.882170	0.441085	−	0.897465i	$-0.354594\pi$
$829$	25.3998	0.882170	0.441085	+	0.897465i	$0.354594\pi$
$830$	−11.6634	−0.404843
$831$	0	0
$832$	0.0350319	0.00121451
$833$	−13.9837	−0.484506
$834$	0	0
$835$	−10.5854	−0.366322
$836$	−11.8630	−0.410292
$837$	0	0
$838$	24.7400	0.854629
$839$	−11.5377	−0.398325	−0.199163	−	0.979966i	$-0.563822\pi$
$839$	−11.5377	−0.398325	−0.199163	+	0.979966i	$0.563822\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	14.4263	0.497163
$843$	0	0
$844$	−34.2074	−1.17747
$845$	29.1583	1.00308
$846$	0	0
$847$	39.3309	1.35143
$848$	−6.62300	−0.227435
$849$	0	0
$850$	4.84151	0.166062
$851$	5.65206	0.193750
$852$	0	0
$853$	−4.09871	−0.140337	−0.0701686	−	0.997535i	$-0.522354\pi$
$853$	−4.09871	−0.140337	−0.0701686	+	0.997535i	$0.522354\pi$
$854$	2.80868	0.0961109
$855$	0	0
$856$	−21.7897	−0.744758
$857$	−12.5364	−0.428235	−0.214117	−	0.976808i	$-0.568687\pi$
$857$	−12.5364	−0.428235	−0.214117	+	0.976808i	$0.568687\pi$
$858$	0	0
$859$	15.9042	0.542645	0.271323	−	0.962488i	$-0.412539\pi$
$859$	15.9042	0.542645	0.271323	+	0.962488i	$0.412539\pi$
$860$	−33.9056	−1.15617
$861$	0	0
$862$	11.6166	0.395664
$863$	−17.5247	−0.596549	−0.298275	−	0.954480i	$-0.596411\pi$
$863$	−17.5247	−0.596549	−0.298275	+	0.954480i	$0.596411\pi$
$864$	0	0
$865$	−39.4905	−1.34272
$866$	−14.6530	−0.497931
$867$	0	0
$868$	26.9536	0.914865
$869$	12.8091	0.434520
$870$	0	0
$871$	−0.380168	−0.0128815
$872$	11.4629	0.388184
$873$	0	0
$874$	0.847782	0.0286767
$875$	17.8684	0.604062
$876$	0	0
$877$	41.4072	1.39822	0.699111	−	0.715013i	$-0.253579\pi$
$877$	41.4072	1.39822	0.699111	+	0.715013i	$0.253579\pi$
$878$	−10.9426	−0.369294
$879$	0	0
$880$	−25.0167	−0.843313
$881$	−3.95482	−0.133241	−0.0666206	−	0.997778i	$-0.521222\pi$
$881$	−3.95482	−0.133241	−0.0666206	+	0.997778i	$0.521222\pi$
$882$	0	0
$883$	16.9860	0.571624	0.285812	−	0.958286i	$-0.407737\pi$
$883$	16.9860	0.571624	0.285812	+	0.958286i	$0.407737\pi$
$884$	−9.97440	−0.335475
$885$	0	0
$886$	13.9045	0.467130
$887$	29.5301	0.991524	0.495762	−	0.868458i	$-0.334889\pi$
$887$	29.5301	0.991524	0.495762	+	0.868458i	$0.334889\pi$
$888$	0	0
$889$	−14.6152	−0.490179
$890$	−6.56362	−0.220013
$891$	0	0
$892$	39.3947	1.31903
$893$	6.04600	0.202322
$894$	0	0
$895$	34.9108	1.16694
$896$	−22.9891	−0.768012
$897$	0	0
$898$	10.1565	0.338925
$899$	−8.27265	−0.275908
$900$	0	0
$901$	−18.2494	−0.607974
$902$	−20.0894	−0.668904
$903$	0	0
$904$	−4.78998	−0.159312
$905$	−25.1417	−0.835739
$906$	0	0
$907$	−12.0820	−0.401178	−0.200589	−	0.979676i	$-0.564286\pi$
$907$	−12.0820	−0.401178	−0.200589	+	0.979676i	$0.564286\pi$
$908$	7.03211	0.233369
$909$	0	0
$910$	−4.19347	−0.139012
$911$	−10.0917	−0.334354	−0.167177	−	0.985927i	$-0.553465\pi$
$911$	−10.0917	−0.334354	−0.167177	+	0.985927i	$0.553465\pi$
$912$	0	0
$913$	39.7327	1.31496
$914$	−9.10543	−0.301181
$915$	0	0
$916$	−17.9843	−0.594219
$917$	17.6608	0.583211
$918$	0	0
$919$	−27.2142	−0.897715	−0.448857	−	0.893603i	$-0.648169\pi$
$919$	−27.2142	−0.897715	−0.448857	+	0.893603i	$0.648169\pi$
$920$	5.82456	0.192030
$921$	0	0
$922$	15.9629	0.525711
$923$	6.47543	0.213141
$924$	0	0
$925$	8.88843	0.292250
$926$	9.04734	0.297314
$927$	0	0
$928$	5.66115	0.185836
$929$	37.1914	1.22021	0.610106	−	0.792320i	$-0.291127\pi$
$929$	37.1914	1.22021	0.610106	+	0.792320i	$0.291127\pi$
$930$	0	0
$931$	−3.85073	−0.126203
$932$	3.97347	0.130155
$933$	0	0
$934$	−17.3751	−0.568531
$935$	−68.9323	−2.25433
$936$	0	0
$937$	24.4556	0.798928	0.399464	−	0.916749i	$-0.369196\pi$
$937$	24.4556	0.798928	0.399464	+	0.916749i	$0.369196\pi$
$938$	−0.382318	−0.0124831
$939$	0	0
$940$	18.4756	0.602607
$941$	−8.74837	−0.285189	−0.142594	−	0.989781i	$-0.545544\pi$
$941$	−8.74837	−0.285189	−0.142594	+	0.989781i	$0.545544\pi$
$942$	0	0
$943$	−5.78265	−0.188309
$944$	−11.5792	−0.376872
$945$	0	0
$946$	−28.6763	−0.932345
$947$	−32.6807	−1.06198	−0.530990	−	0.847378i	$-0.678180\pi$
$947$	−32.6807	−1.06198	−0.530990	+	0.847378i	$0.678180\pi$
$948$	0	0
$949$	7.88482	0.255952
$950$	1.33322	0.0432554
$951$	0	0
$952$	−22.5520	−0.730914
$953$	5.79143	0.187603	0.0938014	−	0.995591i	$-0.470098\pi$
$953$	5.79143	0.187603	0.0938014	+	0.995591i	$0.470098\pi$
$954$	0	0
$955$	−42.6030	−1.37860
$956$	13.0123	0.420846
$957$	0	0
$958$	23.2931	0.752566
$959$	44.8643	1.44874
$960$	0	0
$961$	37.4367	1.20764
$962$	4.54631	0.146579
$963$	0	0
$964$	−40.2286	−1.29568
$965$	19.1628	0.616872
$966$	0	0
$967$	50.6111	1.62754	0.813772	−	0.581184i	$-0.197410\pi$
$967$	50.6111	1.62754	0.813772	+	0.581184i	$0.197410\pi$
$968$	−43.9416	−1.41234
$969$	0	0
$970$	−6.30754	−0.202523
$971$	−54.6997	−1.75540	−0.877698	−	0.479214i	$-0.840922\pi$
$971$	−54.6997	−1.75540	−0.877698	+	0.479214i	$0.840922\pi$
$972$	0	0
$973$	−23.2336	−0.744834
$974$	2.93449	0.0940270
$975$	0	0
$976$	3.87945	0.124178
$977$	−24.6573	−0.788856	−0.394428	−	0.918927i	$-0.629057\pi$
$977$	−24.6573	−0.788856	−0.394428	+	0.918927i	$0.629057\pi$
$978$	0	0
$979$	22.3597	0.714619
$980$	−11.7672	−0.375889
$981$	0	0
$982$	−1.25799	−0.0401440
$983$	9.83213	0.313596	0.156798	−	0.987631i	$-0.449883\pi$
$983$	9.83213	0.313596	0.156798	+	0.987631i	$0.449883\pi$
$984$	0	0
$985$	30.9124	0.984950
$986$	3.07867	0.0980448
$987$	0	0
$988$	−2.74668	−0.0873835
$989$	−8.25433	−0.262472
$990$	0	0
$991$	−54.6142	−1.73488	−0.867439	−	0.497544i	$-0.834235\pi$
$991$	−54.6142	−1.73488	−0.867439	+	0.497544i	$0.834235\pi$
$992$	−46.8327	−1.48694
$993$	0	0
$994$	6.51205	0.206550
$995$	−29.6945	−0.941379
$996$	0	0
$997$	−28.0189	−0.887366	−0.443683	−	0.896184i	$-0.646328\pi$
$997$	−28.0189	−0.887366	−0.443683	+	0.896184i	$0.646328\pi$
$998$	−1.51637	−0.0479998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.8		14
3.2	odd	2		2001.2.a.m.1.7	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.7	✓	14	3.2	odd	2
6003.2.a.p.1.8		14	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)

\(f(q)\)	\(=\)	\(q+0.630703 q^{2} -1.60221 q^{4} -2.56371 q^{5} +2.03354 q^{7} -2.27193 q^{8} +O(q^{10})\)	\(q+0.630703 q^{2} -1.60221 q^{4} -2.56371 q^{5} +2.03354 q^{7} -2.27193 q^{8} -1.61694 q^{10} +5.50828 q^{11} +1.27535 q^{13} +1.28256 q^{14} +1.77152 q^{16} +4.88133 q^{17} +1.34419 q^{19} +4.10761 q^{20} +3.47409 q^{22} +1.00000 q^{23} +1.57260 q^{25} +0.804364 q^{26} -3.25816 q^{28} +1.00000 q^{29} -8.27265 q^{31} +5.66115 q^{32} +3.07867 q^{34} -5.21339 q^{35} +5.65206 q^{37} +0.847782 q^{38} +5.82456 q^{40} -5.78265 q^{41} -8.25433 q^{43} -8.82544 q^{44} +0.630703 q^{46} +4.49789 q^{47} -2.86473 q^{49} +0.991843 q^{50} -2.04338 q^{52} -3.73860 q^{53} -14.1216 q^{55} -4.62005 q^{56} +0.630703 q^{58} -6.53633 q^{59} +2.18990 q^{61} -5.21758 q^{62} +0.0274685 q^{64} -3.26961 q^{65} -0.298090 q^{67} -7.82094 q^{68} -3.28810 q^{70} +5.07739 q^{71} +6.18250 q^{73} +3.56477 q^{74} -2.15367 q^{76} +11.2013 q^{77} +2.32543 q^{79} -4.54166 q^{80} -3.64713 q^{82} +7.21327 q^{83} -12.5143 q^{85} -5.20603 q^{86} -12.5144 q^{88} +4.05929 q^{89} +2.59346 q^{91} -1.60221 q^{92} +2.83683 q^{94} -3.44610 q^{95} +3.90092 q^{97} -1.80679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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