LMFDB - Embedded newform 6003.2.a.p.1.9

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			1.9
Root			$1.14192$ 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)

$f(q)$	$=$	$q+1.14192 q^{2} -0.696014 q^{4} +4.45694 q^{5} -4.52974 q^{7} -3.07864 q^{8} +O(q^{10})$	\(q+1.14192 q^{2} -0.696014 q^{4} +4.45694 q^{5} -4.52974 q^{7} -3.07864 q^{8} +5.08948 q^{10} +2.39257 q^{11} +4.69508 q^{13} -5.17261 q^{14} -2.12354 q^{16} +0.991786 q^{17} -1.33795 q^{19} -3.10209 q^{20} +2.73213 q^{22} +1.00000 q^{23} +14.8643 q^{25} +5.36141 q^{26} +3.15276 q^{28} +1.00000 q^{29} +3.21228 q^{31} +3.73236 q^{32} +1.13254 q^{34} -20.1888 q^{35} +4.71234 q^{37} -1.52783 q^{38} -13.7213 q^{40} -7.59343 q^{41} -8.56990 q^{43} -1.66526 q^{44} +1.14192 q^{46} -0.810896 q^{47} +13.5185 q^{49} +16.9739 q^{50} -3.26784 q^{52} -6.18794 q^{53} +10.6636 q^{55} +13.9454 q^{56} +1.14192 q^{58} -8.95489 q^{59} -1.56132 q^{61} +3.66817 q^{62} +8.50914 q^{64} +20.9257 q^{65} +6.18798 q^{67} -0.690297 q^{68} -23.0540 q^{70} +15.1839 q^{71} +5.61542 q^{73} +5.38113 q^{74} +0.931231 q^{76} -10.8377 q^{77} -11.4605 q^{79} -9.46447 q^{80} -8.67111 q^{82} +1.73600 q^{83} +4.42033 q^{85} -9.78616 q^{86} -7.36587 q^{88} +14.5530 q^{89} -21.2675 q^{91} -0.696014 q^{92} -0.925980 q^{94} -5.96315 q^{95} +8.70253 q^{97} +15.4371 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.14192	0.807461	0.403730	−	0.914878i	$-0.367713\pi$
$2$	1.14192	0.807461	0.403730	+	0.914878i	$0.367713\pi$
$3$	0	0
$3$	0	0
$4$	−0.696014	−0.348007
$5$	4.45694	1.99320	0.996602	−	0.0823694i	$-0.0262487\pi$
$5$	4.45694	1.99320	0.996602	+	0.0823694i	$0.0262487\pi$
$6$	0	0
$7$	−4.52974	−1.71208	−0.856040	−	0.516909i	$-0.827082\pi$
$7$	−4.52974	−1.71208	−0.856040	+	0.516909i	$0.827082\pi$
$8$	−3.07864	−1.08846
$9$	0	0
$10$	5.08948	1.60943
$11$	2.39257	0.721388	0.360694	−	0.932684i	$-0.382540\pi$
$11$	2.39257	0.721388	0.360694	+	0.932684i	$0.382540\pi$
$12$	0	0
$13$	4.69508	1.30218	0.651090	−	0.759001i	$-0.274312\pi$
$13$	4.69508	1.30218	0.651090	+	0.759001i	$0.274312\pi$
$14$	−5.17261	−1.38244
$15$	0	0
$16$	−2.12354	−0.530884
$17$	0.991786	0.240543	0.120272	−	0.992741i	$-0.461623\pi$
$17$	0.991786	0.240543	0.120272	+	0.992741i	$0.461623\pi$
$18$	0	0
$19$	−1.33795	−0.306946	−0.153473	−	0.988153i	$-0.549046\pi$
$19$	−1.33795	−0.306946	−0.153473	+	0.988153i	$0.549046\pi$
$20$	−3.10209	−0.693649
$21$	0	0
$22$	2.73213	0.582493
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	14.8643	2.97286
$26$	5.36141	1.05146
$27$	0	0
$28$	3.15276	0.595816
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.21228	0.576942	0.288471	−	0.957489i	$-0.406853\pi$
$31$	3.21228	0.576942	0.288471	+	0.957489i	$0.406853\pi$
$32$	3.73236	0.659795
$33$	0	0
$34$	1.13254	0.194229
$35$	−20.1888	−3.41252
$36$	0	0
$37$	4.71234	0.774704	0.387352	−	0.921932i	$-0.373390\pi$
$37$	4.71234	0.774704	0.387352	+	0.921932i	$0.373390\pi$
$38$	−1.52783	−0.247847
$39$	0	0
$40$	−13.7213	−2.16953
$41$	−7.59343	−1.18589	−0.592947	−	0.805241i	$-0.702036\pi$
$41$	−7.59343	−1.18589	−0.592947	+	0.805241i	$0.702036\pi$
$42$	0	0
$43$	−8.56990	−1.30690	−0.653449	−	0.756971i	$-0.726678\pi$
$43$	−8.56990	−1.30690	−0.653449	+	0.756971i	$0.726678\pi$
$44$	−1.66526	−0.251048
$45$	0	0
$46$	1.14192	0.168367
$47$	−0.810896	−0.118281	−0.0591407	−	0.998250i	$-0.518836\pi$
$47$	−0.810896	−0.118281	−0.0591407	+	0.998250i	$0.518836\pi$
$48$	0	0
$49$	13.5185	1.93122
$50$	16.9739	2.40047
$51$	0	0
$52$	−3.26784	−0.453168
$53$	−6.18794	−0.849979	−0.424989	−	0.905198i	$-0.639722\pi$
$53$	−6.18794	−0.849979	−0.424989	+	0.905198i	$0.639722\pi$
$54$	0	0
$55$	10.6636	1.43787
$56$	13.9454	1.86354
$57$	0	0
$58$	1.14192	0.149942
$59$	−8.95489	−1.16583	−0.582914	−	0.812534i	$-0.698087\pi$
$59$	−8.95489	−1.16583	−0.582914	+	0.812534i	$0.698087\pi$
$60$	0	0
$61$	−1.56132	−0.199907	−0.0999535	−	0.994992i	$-0.531869\pi$
$61$	−1.56132	−0.199907	−0.0999535	+	0.994992i	$0.531869\pi$
$62$	3.66817	0.465858
$63$	0	0
$64$	8.50914	1.06364
$65$	20.9257	2.59551
$66$	0	0
$67$	6.18798	0.755983	0.377991	−	0.925809i	$-0.376615\pi$
$67$	6.18798	0.755983	0.377991	+	0.925809i	$0.376615\pi$
$68$	−0.690297	−0.0837108
$69$	0	0
$70$	−23.0540	−2.75548
$71$	15.1839	1.80200	0.900998	−	0.433822i	$-0.142835\pi$
$71$	15.1839	1.80200	0.900998	+	0.433822i	$0.142835\pi$
$72$	0	0
$73$	5.61542	0.657235	0.328618	−	0.944463i	$-0.393417\pi$
$73$	5.61542	0.657235	0.328618	+	0.944463i	$0.393417\pi$
$74$	5.38113	0.625543
$75$	0	0
$76$	0.931231	0.106819
$77$	−10.8377	−1.23507
$78$	0	0
$79$	−11.4605	−1.28941	−0.644703	−	0.764433i	$-0.723019\pi$
$79$	−11.4605	−1.28941	−0.644703	+	0.764433i	$0.723019\pi$
$80$	−9.46447	−1.05816
$81$	0	0
$82$	−8.67111	−0.957564
$83$	1.73600	0.190550	0.0952752	−	0.995451i	$-0.469627\pi$
$83$	1.73600	0.190550	0.0952752	+	0.995451i	$0.469627\pi$
$84$	0	0
$85$	4.42033	0.479452
$86$	−9.78616	−1.05527
$87$	0	0
$88$	−7.36587	−0.785204
$89$	14.5530	1.54262	0.771310	−	0.636460i	$-0.219602\pi$
$89$	14.5530	1.54262	0.771310	+	0.636460i	$0.219602\pi$
$90$	0	0
$91$	−21.2675	−2.22944
$92$	−0.696014	−0.0725645
$93$	0	0
$94$	−0.925980	−0.0955076
$95$	−5.96315	−0.611807
$96$	0	0
$97$	8.70253	0.883608	0.441804	−	0.897112i	$-0.354339\pi$
$97$	8.70253	0.883608	0.441804	+	0.897112i	$0.354339\pi$
$98$	15.4371	1.55938
$99$	0	0
$100$	−10.3458	−1.03458
$101$	12.9467	1.28825	0.644125	−	0.764920i	$-0.277222\pi$
$101$	12.9467	1.28825	0.644125	+	0.764920i	$0.277222\pi$
$102$	0	0
$103$	−11.0179	−1.08563	−0.542815	−	0.839852i	$-0.682642\pi$
$103$	−11.0179	−1.08563	−0.542815	+	0.839852i	$0.682642\pi$
$104$	−14.4544	−1.41737
$105$	0	0
$106$	−7.06614	−0.686325
$107$	2.55940	0.247426	0.123713	−	0.992318i	$-0.460520\pi$
$107$	2.55940	0.247426	0.123713	+	0.992318i	$0.460520\pi$
$108$	0	0
$109$	13.8622	1.32776	0.663879	−	0.747840i	$-0.268909\pi$
$109$	13.8622	1.32776	0.663879	+	0.747840i	$0.268909\pi$
$110$	12.1769	1.16103
$111$	0	0
$112$	9.61906	0.908916
$113$	9.59634	0.902748	0.451374	−	0.892335i	$-0.350934\pi$
$113$	9.59634	0.902748	0.451374	+	0.892335i	$0.350934\pi$
$114$	0	0
$115$	4.45694	0.415612
$116$	−0.696014	−0.0646233
$117$	0	0
$118$	−10.2258	−0.941360
$119$	−4.49253	−0.411830
$120$	0	0
$121$	−5.27559	−0.479599
$122$	−1.78291	−0.161417
$123$	0	0
$124$	−2.23579	−0.200780
$125$	43.9646	3.93231
$126$	0	0
$127$	11.0495	0.980484	0.490242	−	0.871586i	$-0.336908\pi$
$127$	11.0495	0.980484	0.490242	+	0.871586i	$0.336908\pi$
$128$	2.25205	0.199055
$129$	0	0
$130$	23.8955	2.09577
$131$	4.74091	0.414215	0.207108	−	0.978318i	$-0.433595\pi$
$131$	4.74091	0.414215	0.207108	+	0.978318i	$0.433595\pi$
$132$	0	0
$133$	6.06055	0.525517
$134$	7.06620	0.610426
$135$	0	0
$136$	−3.05335	−0.261823
$137$	−11.8740	−1.01446	−0.507232	−	0.861810i	$-0.669331\pi$
$137$	−11.8740	−1.01446	−0.507232	+	0.861810i	$0.669331\pi$
$138$	0	0
$139$	20.9389	1.77601	0.888005	−	0.459834i	$-0.152091\pi$
$139$	20.9389	1.77601	0.888005	+	0.459834i	$0.152091\pi$
$140$	14.0517	1.18758
$141$	0	0
$142$	17.3388	1.45504
$143$	11.2333	0.939377
$144$	0	0
$145$	4.45694	0.370129
$146$	6.41237	0.530692
$147$	0	0
$148$	−3.27986	−0.269602
$149$	−19.1285	−1.56707	−0.783535	−	0.621347i	$-0.786586\pi$
$149$	−19.1285	−1.56707	−0.783535	+	0.621347i	$0.786586\pi$
$150$	0	0
$151$	6.69685	0.544982	0.272491	−	0.962158i	$-0.412153\pi$
$151$	6.69685	0.544982	0.272491	+	0.962158i	$0.412153\pi$
$152$	4.11906	0.334100
$153$	0	0
$154$	−12.3758	−0.997274
$155$	14.3169	1.14996
$156$	0	0
$157$	17.4125	1.38967	0.694835	−	0.719169i	$-0.255477\pi$
$157$	17.4125	1.38967	0.694835	+	0.719169i	$0.255477\pi$
$158$	−13.0870	−1.04114
$159$	0	0
$160$	16.6349	1.31511
$161$	−4.52974	−0.356993
$162$	0	0
$163$	12.7707	1.00028	0.500140	−	0.865945i	$-0.333282\pi$
$163$	12.7707	1.00028	0.500140	+	0.865945i	$0.333282\pi$
$164$	5.28514	0.412700
$165$	0	0
$166$	1.98237	0.153862
$167$	22.3562	1.72997	0.864987	−	0.501794i	$-0.167326\pi$
$167$	22.3562	1.72997	0.864987	+	0.501794i	$0.167326\pi$
$168$	0	0
$169$	9.04374	0.695672
$170$	5.04767	0.387139
$171$	0	0
$172$	5.96477	0.454809
$173$	0.211264	0.0160621	0.00803107	−	0.999968i	$-0.497444\pi$
$173$	0.211264	0.0160621	0.00803107	+	0.999968i	$0.497444\pi$
$174$	0	0
$175$	−67.3314	−5.08978
$176$	−5.08072	−0.382973
$177$	0	0
$178$	16.6184	1.24561
$179$	13.0602	0.976165	0.488082	−	0.872798i	$-0.337697\pi$
$179$	13.0602	0.976165	0.488082	+	0.872798i	$0.337697\pi$
$180$	0	0
$181$	−9.67384	−0.719051	−0.359526	−	0.933135i	$-0.617061\pi$
$181$	−9.67384	−0.719051	−0.359526	+	0.933135i	$0.617061\pi$
$182$	−24.2858	−1.80018
$183$	0	0
$184$	−3.07864	−0.226960
$185$	21.0026	1.54414
$186$	0	0
$187$	2.37292	0.173525
$188$	0.564395	0.0411628
$189$	0	0
$190$	−6.80946	−0.494010
$191$	−11.5153	−0.833221	−0.416611	−	0.909085i	$-0.636782\pi$
$191$	−11.5153	−0.833221	−0.416611	+	0.909085i	$0.636782\pi$
$192$	0	0
$193$	16.4176	1.18176	0.590881	−	0.806759i	$-0.298780\pi$
$193$	16.4176	1.18176	0.590881	+	0.806759i	$0.298780\pi$
$194$	9.93761	0.713479
$195$	0	0
$196$	−9.40909	−0.672078
$197$	15.6804	1.11718	0.558591	−	0.829443i	$-0.311342\pi$
$197$	15.6804	1.11718	0.558591	+	0.829443i	$0.311342\pi$
$198$	0	0
$199$	14.0461	0.995701	0.497850	−	0.867263i	$-0.334123\pi$
$199$	14.0461	0.995701	0.497850	+	0.867263i	$0.334123\pi$
$200$	−45.7618	−3.23585
$201$	0	0
$202$	14.7842	1.04021
$203$	−4.52974	−0.317925
$204$	0	0
$205$	−33.8435	−2.36373
$206$	−12.5816	−0.876604
$207$	0	0
$208$	−9.97017	−0.691307
$209$	−3.20114	−0.221427
$210$	0	0
$211$	−0.853143	−0.0587328	−0.0293664	−	0.999569i	$-0.509349\pi$
$211$	−0.853143	−0.0587328	−0.0293664	+	0.999569i	$0.509349\pi$
$212$	4.30689	0.295799
$213$	0	0
$214$	2.92263	0.199787
$215$	−38.1955	−2.60491
$216$	0	0
$217$	−14.5508	−0.987771
$218$	15.8296	1.07211
$219$	0	0
$220$	−7.42198	−0.500390
$221$	4.65651	0.313231
$222$	0	0
$223$	−22.1582	−1.48382	−0.741910	−	0.670500i	$-0.766080\pi$
$223$	−22.1582	−1.48382	−0.741910	+	0.670500i	$0.766080\pi$
$224$	−16.9066	−1.12962
$225$	0	0
$226$	10.9583	0.728934
$227$	−19.6626	−1.30505	−0.652525	−	0.757767i	$-0.726290\pi$
$227$	−19.6626	−1.30505	−0.652525	+	0.757767i	$0.726290\pi$
$228$	0	0
$229$	15.2495	1.00771	0.503857	−	0.863787i	$-0.331914\pi$
$229$	15.2495	1.00771	0.503857	+	0.863787i	$0.331914\pi$
$230$	5.08948	0.335590
$231$	0	0
$232$	−3.07864	−0.202122
$233$	−8.58802	−0.562620	−0.281310	−	0.959617i	$-0.590769\pi$
$233$	−8.58802	−0.562620	−0.281310	+	0.959617i	$0.590769\pi$
$234$	0	0
$235$	−3.61412	−0.235759
$236$	6.23273	0.405716
$237$	0	0
$238$	−5.13012	−0.332536
$239$	−10.3294	−0.668156	−0.334078	−	0.942545i	$-0.608425\pi$
$239$	−10.3294	−0.668156	−0.334078	+	0.942545i	$0.608425\pi$
$240$	0	0
$241$	−16.0936	−1.03668	−0.518339	−	0.855176i	$-0.673449\pi$
$241$	−16.0936	−1.03668	−0.518339	+	0.855176i	$0.673449\pi$
$242$	−6.02432	−0.387258
$243$	0	0
$244$	1.08670	0.0695690
$245$	60.2513	3.84931
$246$	0	0
$247$	−6.28177	−0.399699
$248$	−9.88943	−0.627980
$249$	0	0
$250$	50.2042	3.17519
$251$	−3.21044	−0.202641	−0.101321	−	0.994854i	$-0.532307\pi$
$251$	−3.21044	−0.202641	−0.101321	+	0.994854i	$0.532307\pi$
$252$	0	0
$253$	2.39257	0.150420
$254$	12.6177	0.791703
$255$	0	0
$256$	−14.4466	−0.902913
$257$	4.47033	0.278852	0.139426	−	0.990233i	$-0.455474\pi$
$257$	4.47033	0.278852	0.139426	+	0.990233i	$0.455474\pi$
$258$	0	0
$259$	−21.3457	−1.32636
$260$	−14.5646	−0.903255
$261$	0	0
$262$	5.41375	0.334463
$263$	0.722442	0.0445477	0.0222738	−	0.999752i	$-0.492909\pi$
$263$	0.722442	0.0445477	0.0222738	+	0.999752i	$0.492909\pi$
$264$	0	0
$265$	−27.5793	−1.69418
$266$	6.92068	0.424334
$267$	0	0
$268$	−4.30692	−0.263087
$269$	28.3625	1.72929	0.864647	−	0.502380i	$-0.167542\pi$
$269$	28.3625	1.72929	0.864647	+	0.502380i	$0.167542\pi$
$270$	0	0
$271$	−12.6886	−0.770775	−0.385388	−	0.922755i	$-0.625932\pi$
$271$	−12.6886	−0.770775	−0.385388	+	0.922755i	$0.625932\pi$
$272$	−2.10609	−0.127701
$273$	0	0
$274$	−13.5592	−0.819140
$275$	35.5639	2.14459
$276$	0	0
$277$	−3.73989	−0.224708	−0.112354	−	0.993668i	$-0.535839\pi$
$277$	−3.73989	−0.224708	−0.112354	+	0.993668i	$0.535839\pi$
$278$	23.9105	1.43406
$279$	0	0
$280$	62.1539	3.71441
$281$	−23.6125	−1.40860	−0.704301	−	0.709901i	$-0.748740\pi$
$281$	−23.6125	−1.40860	−0.704301	+	0.709901i	$0.748740\pi$
$282$	0	0
$283$	6.38861	0.379763	0.189882	−	0.981807i	$-0.439190\pi$
$283$	6.38861	0.379763	0.189882	+	0.981807i	$0.439190\pi$
$284$	−10.5682	−0.627108
$285$	0	0
$286$	12.8276	0.758510
$287$	34.3963	2.03035
$288$	0	0
$289$	−16.0164	−0.942139
$290$	5.08948	0.298864
$291$	0	0
$292$	−3.90841	−0.228723
$293$	−1.71371	−0.100116	−0.0500579	−	0.998746i	$-0.515941\pi$
$293$	−1.71371	−0.100116	−0.0500579	+	0.998746i	$0.515941\pi$
$294$	0	0
$295$	−39.9114	−2.32373
$296$	−14.5076	−0.843237
$297$	0	0
$298$	−21.8433	−1.26535
$299$	4.69508	0.271523
$300$	0	0
$301$	38.8194	2.23751
$302$	7.64728	0.440052
$303$	0	0
$304$	2.84118	0.162953
$305$	−6.95872	−0.398455
$306$	0	0
$307$	29.7392	1.69731	0.848653	−	0.528950i	$-0.177414\pi$
$307$	29.7392	1.69731	0.848653	+	0.528950i	$0.177414\pi$
$308$	7.54321	0.429814
$309$	0	0
$310$	16.3488	0.928550
$311$	−7.92664	−0.449479	−0.224739	−	0.974419i	$-0.572153\pi$
$311$	−7.92664	−0.449479	−0.224739	+	0.974419i	$0.572153\pi$
$312$	0	0
$313$	0.663467	0.0375014	0.0187507	−	0.999824i	$-0.494031\pi$
$313$	0.663467	0.0375014	0.0187507	+	0.999824i	$0.494031\pi$
$314$	19.8837	1.12210
$315$	0	0
$316$	7.97666	0.448722
$317$	−12.9020	−0.724650	−0.362325	−	0.932052i	$-0.618017\pi$
$317$	−12.9020	−0.724650	−0.362325	+	0.932052i	$0.618017\pi$
$318$	0	0
$319$	2.39257	0.133958
$320$	37.9247	2.12006
$321$	0	0
$322$	−5.17261	−0.288258
$323$	−1.32696	−0.0738339
$324$	0	0
$325$	69.7890	3.87120
$326$	14.5832	0.807687
$327$	0	0
$328$	23.3774	1.29080
$329$	3.67315	0.202507
$330$	0	0
$331$	−29.1452	−1.60197	−0.800983	−	0.598687i	$-0.795689\pi$
$331$	−29.1452	−1.60197	−0.800983	+	0.598687i	$0.795689\pi$
$332$	−1.20828	−0.0663128
$333$	0	0
$334$	25.5290	1.39689
$335$	27.5795	1.50683
$336$	0	0
$337$	4.28685	0.233520	0.116760	−	0.993160i	$-0.462749\pi$
$337$	4.28685	0.233520	0.116760	+	0.993160i	$0.462749\pi$
$338$	10.3272	0.561728
$339$	0	0
$340$	−3.07661	−0.166853
$341$	7.68560	0.416199
$342$	0	0
$343$	−29.5272	−1.59432
$344$	26.3836	1.42251
$345$	0	0
$346$	0.241247	0.0129695
$347$	2.43021	0.130460	0.0652302	−	0.997870i	$-0.479222\pi$
$347$	2.43021	0.130460	0.0652302	+	0.997870i	$0.479222\pi$
$348$	0	0
$349$	−13.4575	−0.720362	−0.360181	−	0.932882i	$-0.617285\pi$
$349$	−13.4575	−0.720362	−0.360181	+	0.932882i	$0.617285\pi$
$350$	−76.8872	−4.10980
$351$	0	0
$352$	8.92995	0.475968
$353$	−20.7927	−1.10668	−0.553341	−	0.832955i	$-0.686647\pi$
$353$	−20.7927	−1.10668	−0.553341	+	0.832955i	$0.686647\pi$
$354$	0	0
$355$	67.6737	3.59175
$356$	−10.1291	−0.536843
$357$	0	0
$358$	14.9137	0.788215
$359$	−10.8660	−0.573485	−0.286742	−	0.958008i	$-0.592572\pi$
$359$	−10.8660	−0.573485	−0.286742	+	0.958008i	$0.592572\pi$
$360$	0	0
$361$	−17.2099	−0.905784
$362$	−11.0468	−0.580606
$363$	0	0
$364$	14.8025	0.775859
$365$	25.0276	1.31000
$366$	0	0
$367$	−14.9307	−0.779375	−0.389687	−	0.920947i	$-0.627417\pi$
$367$	−14.9307	−0.779375	−0.389687	+	0.920947i	$0.627417\pi$
$368$	−2.12354	−0.110697
$369$	0	0
$370$	23.9834	1.24684
$371$	28.0297	1.45523
$372$	0	0
$373$	−22.7620	−1.17857	−0.589286	−	0.807924i	$-0.700591\pi$
$373$	−22.7620	−1.17857	−0.589286	+	0.807924i	$0.700591\pi$
$374$	2.70969	0.140115
$375$	0	0
$376$	2.49646	0.128745
$377$	4.69508	0.241809
$378$	0	0
$379$	17.4424	0.895954	0.447977	−	0.894045i	$-0.352145\pi$
$379$	17.4424	0.895954	0.447977	+	0.894045i	$0.352145\pi$
$380$	4.15044	0.212913
$381$	0	0
$382$	−13.1496	−0.672793
$383$	−31.0136	−1.58472	−0.792360	−	0.610053i	$-0.791148\pi$
$383$	−31.0136	−1.58472	−0.792360	+	0.610053i	$0.791148\pi$
$384$	0	0
$385$	−48.3031	−2.46175
$386$	18.7476	0.954227
$387$	0	0
$388$	−6.05708	−0.307502
$389$	2.32335	0.117798	0.0588992	−	0.998264i	$-0.481241\pi$
$389$	2.32335	0.117798	0.0588992	+	0.998264i	$0.481241\pi$
$390$	0	0
$391$	0.991786	0.0501568
$392$	−41.6187	−2.10206
$393$	0	0
$394$	17.9058	0.902081
$395$	−51.0787	−2.57005
$396$	0	0
$397$	−15.7319	−0.789562	−0.394781	−	0.918775i	$-0.629180\pi$
$397$	−15.7319	−0.789562	−0.394781	+	0.918775i	$0.629180\pi$
$398$	16.0395	0.803989
$399$	0	0
$400$	−31.5649	−1.57824
$401$	35.7406	1.78480	0.892399	−	0.451247i	$-0.149020\pi$
$401$	35.7406	1.78480	0.892399	+	0.451247i	$0.149020\pi$
$402$	0	0
$403$	15.0819	0.751282
$404$	−9.01112	−0.448320
$405$	0	0
$406$	−5.17261	−0.256712
$407$	11.2746	0.558862
$408$	0	0
$409$	−15.6308	−0.772895	−0.386447	−	0.922311i	$-0.626298\pi$
$409$	−15.6308	−0.772895	−0.386447	+	0.922311i	$0.626298\pi$
$410$	−38.6466	−1.90862
$411$	0	0
$412$	7.66864	0.377807
$413$	40.5633	1.99599
$414$	0	0
$415$	7.73723	0.379806
$416$	17.5237	0.859171
$417$	0	0
$418$	−3.65545	−0.178794
$419$	−27.0072	−1.31939	−0.659695	−	0.751534i	$-0.729314\pi$
$419$	−27.0072	−1.31939	−0.659695	+	0.751534i	$0.729314\pi$
$420$	0	0
$421$	−32.6173	−1.58967	−0.794835	−	0.606825i	$-0.792443\pi$
$421$	−32.6173	−1.58967	−0.794835	+	0.606825i	$0.792443\pi$
$422$	−0.974223	−0.0474244
$423$	0	0
$424$	19.0504	0.925170
$425$	14.7422	0.715102
$426$	0	0
$427$	7.07238	0.342257
$428$	−1.78138	−0.0861061
$429$	0	0
$430$	−43.6163	−2.10336
$431$	34.4130	1.65762	0.828809	−	0.559532i	$-0.189019\pi$
$431$	34.4130	1.65762	0.828809	+	0.559532i	$0.189019\pi$
$432$	0	0
$433$	−4.55750	−0.219019	−0.109510	−	0.993986i	$-0.534928\pi$
$433$	−4.55750	−0.219019	−0.109510	+	0.993986i	$0.534928\pi$
$434$	−16.6158	−0.797586
$435$	0	0
$436$	−9.64829	−0.462069
$437$	−1.33795	−0.0640027
$438$	0	0
$439$	9.10572	0.434592	0.217296	−	0.976106i	$-0.430276\pi$
$439$	9.10572	0.434592	0.217296	+	0.976106i	$0.430276\pi$
$440$	−32.8292	−1.56507
$441$	0	0
$442$	5.31737	0.252922
$443$	15.8245	0.751844	0.375922	−	0.926651i	$-0.377326\pi$
$443$	15.8245	0.751844	0.375922	+	0.926651i	$0.377326\pi$
$444$	0	0
$445$	64.8621	3.07476
$446$	−25.3029	−1.19813
$447$	0	0
$448$	−38.5442	−1.82104
$449$	5.22266	0.246473	0.123236	−	0.992377i	$-0.460673\pi$
$449$	5.22266	0.246473	0.123236	+	0.992377i	$0.460673\pi$
$450$	0	0
$451$	−18.1678	−0.855490
$452$	−6.67919	−0.314163
$453$	0	0
$454$	−22.4531	−1.05378
$455$	−94.7878	−4.44372
$456$	0	0
$457$	−10.6132	−0.496462	−0.248231	−	0.968701i	$-0.579849\pi$
$457$	−10.6132	−0.496462	−0.248231	+	0.968701i	$0.579849\pi$
$458$	17.4137	0.813690
$459$	0	0
$460$	−3.10209	−0.144636
$461$	−34.8299	−1.62219	−0.811094	−	0.584915i	$-0.801128\pi$
$461$	−34.8299	−1.62219	−0.811094	+	0.584915i	$0.801128\pi$
$462$	0	0
$463$	−30.3894	−1.41231	−0.706157	−	0.708055i	$-0.749573\pi$
$463$	−30.3894	−1.41231	−0.706157	+	0.708055i	$0.749573\pi$
$464$	−2.12354	−0.0985827
$465$	0	0
$466$	−9.80685	−0.454294
$467$	−1.63370	−0.0755985	−0.0377992	−	0.999285i	$-0.512035\pi$
$467$	−1.63370	−0.0755985	−0.0377992	+	0.999285i	$0.512035\pi$
$468$	0	0
$469$	−28.0300	−1.29430
$470$	−4.12704	−0.190366
$471$	0	0
$472$	27.5689	1.26896
$473$	−20.5041	−0.942780
$474$	0	0
$475$	−19.8877	−0.912509
$476$	3.12687	0.143320
$477$	0	0
$478$	−11.7954	−0.539510
$479$	28.9331	1.32199	0.660993	−	0.750392i	$-0.270135\pi$
$479$	28.9331	1.32199	0.660993	+	0.750392i	$0.270135\pi$
$480$	0	0
$481$	22.1248	1.00880
$482$	−18.3776	−0.837076
$483$	0	0
$484$	3.67189	0.166904
$485$	38.7866	1.76121
$486$	0	0
$487$	26.3735	1.19510	0.597548	−	0.801833i	$-0.296142\pi$
$487$	26.3735	1.19510	0.597548	+	0.801833i	$0.296142\pi$
$488$	4.80675	0.217591
$489$	0	0
$490$	68.8022	3.10817
$491$	−10.0449	−0.453322	−0.226661	−	0.973974i	$-0.572781\pi$
$491$	−10.0449	−0.453322	−0.226661	+	0.973974i	$0.572781\pi$
$492$	0	0
$493$	0.991786	0.0446678
$494$	−7.17329	−0.322742
$495$	0	0
$496$	−6.82138	−0.306289
$497$	−68.7791	−3.08516
$498$	0	0
$499$	3.73640	0.167264	0.0836322	−	0.996497i	$-0.473348\pi$
$499$	3.73640	0.167264	0.0836322	+	0.996497i	$0.473348\pi$
$500$	−30.6000	−1.36847
$501$	0	0
$502$	−3.66607	−0.163625
$503$	−36.6827	−1.63560	−0.817801	−	0.575501i	$-0.804807\pi$
$503$	−36.6827	−1.63560	−0.817801	+	0.575501i	$0.804807\pi$
$504$	0	0
$505$	57.7029	2.56774
$506$	2.73213	0.121458
$507$	0	0
$508$	−7.69060	−0.341215
$509$	−13.6620	−0.605558	−0.302779	−	0.953061i	$-0.597914\pi$
$509$	−13.6620	−0.605558	−0.302779	+	0.953061i	$0.597914\pi$
$510$	0	0
$511$	−25.4364	−1.12524
$512$	−21.0010	−0.928122
$513$	0	0
$514$	5.10477	0.225162
$515$	−49.1063	−2.16388
$516$	0	0
$517$	−1.94013	−0.0853268
$518$	−24.3751	−1.07098
$519$	0	0
$520$	−64.4225	−2.82512
$521$	−10.3068	−0.451551	−0.225776	−	0.974179i	$-0.572492\pi$
$521$	−10.3068	−0.451551	−0.225776	+	0.974179i	$0.572492\pi$
$522$	0	0
$523$	4.69213	0.205173	0.102586	−	0.994724i	$-0.467288\pi$
$523$	4.69213	0.205173	0.102586	+	0.994724i	$0.467288\pi$
$524$	−3.29974	−0.144150
$525$	0	0
$526$	0.824972	0.0359705
$527$	3.18589	0.138780
$528$	0	0
$529$	1.00000	0.0434783
$530$	−31.4934	−1.36798
$531$	0	0
$532$	−4.21823	−0.182884
$533$	−35.6517	−1.54425
$534$	0	0
$535$	11.4071	0.493171
$536$	−19.0506	−0.822859
$537$	0	0
$538$	32.3878	1.39634
$539$	32.3441	1.39316
$540$	0	0
$541$	−11.7932	−0.507031	−0.253516	−	0.967331i	$-0.581587\pi$
$541$	−11.7932	−0.507031	−0.253516	+	0.967331i	$0.581587\pi$
$542$	−14.4894	−0.622371
$543$	0	0
$544$	3.70171	0.158709
$545$	61.7830	2.64649
$546$	0	0
$547$	−8.44209	−0.360958	−0.180479	−	0.983579i	$-0.557765\pi$
$547$	−8.44209	−0.360958	−0.180479	+	0.983579i	$0.557765\pi$
$548$	8.26446	0.353040
$549$	0	0
$550$	40.6112	1.73167
$551$	−1.33795	−0.0569985
$552$	0	0
$553$	51.9130	2.20757
$554$	−4.27066	−0.181443
$555$	0	0
$556$	−14.5737	−0.618064
$557$	−22.0035	−0.932318	−0.466159	−	0.884701i	$-0.654363\pi$
$557$	−22.0035	−0.932318	−0.466159	+	0.884701i	$0.654363\pi$
$558$	0	0
$559$	−40.2363	−1.70182
$560$	42.8716	1.81166
$561$	0	0
$562$	−26.9636	−1.13739
$563$	8.46579	0.356791	0.178395	−	0.983959i	$-0.442909\pi$
$563$	8.46579	0.356791	0.178395	+	0.983959i	$0.442909\pi$
$564$	0	0
$565$	42.7703	1.79936
$566$	7.29529	0.306644
$567$	0	0
$568$	−46.7457	−1.96141
$569$	42.4822	1.78094	0.890472	−	0.455037i	$-0.150374\pi$
$569$	42.4822	1.78094	0.890472	+	0.455037i	$0.150374\pi$
$570$	0	0
$571$	−11.5098	−0.481672	−0.240836	−	0.970566i	$-0.577422\pi$
$571$	−11.5098	−0.481672	−0.240836	+	0.970566i	$0.577422\pi$
$572$	−7.81854	−0.326910
$573$	0	0
$574$	39.2779	1.63943
$575$	14.8643	0.619884
$576$	0	0
$577$	34.6809	1.44379	0.721893	−	0.692005i	$-0.243272\pi$
$577$	34.6809	1.44379	0.721893	+	0.692005i	$0.243272\pi$
$578$	−18.2894	−0.760740
$579$	0	0
$580$	−3.10209	−0.128807
$581$	−7.86361	−0.326237
$582$	0	0
$583$	−14.8051	−0.613164
$584$	−17.2879	−0.715376
$585$	0	0
$586$	−1.95692	−0.0808396
$587$	19.7081	0.813442	0.406721	−	0.913552i	$-0.366672\pi$
$587$	19.7081	0.813442	0.406721	+	0.913552i	$0.366672\pi$
$588$	0	0
$589$	−4.29786	−0.177090
$590$	−45.5757	−1.87632
$591$	0	0
$592$	−10.0068	−0.411278
$593$	4.64112	0.190588	0.0952939	−	0.995449i	$-0.469621\pi$
$593$	4.64112	0.190588	0.0952939	+	0.995449i	$0.469621\pi$
$594$	0	0
$595$	−20.0229	−0.820861
$596$	13.3137	0.545352
$597$	0	0
$598$	5.36141	0.219244
$599$	9.35393	0.382191	0.191096	−	0.981571i	$-0.438796\pi$
$599$	9.35393	0.382191	0.191096	+	0.981571i	$0.438796\pi$
$600$	0	0
$601$	−45.6524	−1.86220	−0.931099	−	0.364765i	$-0.881149\pi$
$601$	−45.6524	−1.86220	−0.931099	+	0.364765i	$0.881149\pi$
$602$	44.3287	1.80670
$603$	0	0
$604$	−4.66110	−0.189657
$605$	−23.5130	−0.955939
$606$	0	0
$607$	−18.8955	−0.766944	−0.383472	−	0.923552i	$-0.625272\pi$
$607$	−18.8955	−0.766944	−0.383472	+	0.923552i	$0.625272\pi$
$608$	−4.99371	−0.202522
$609$	0	0
$610$	−7.94632	−0.321737
$611$	−3.80722	−0.154024
$612$	0	0
$613$	27.1641	1.09715	0.548573	−	0.836103i	$-0.315171\pi$
$613$	27.1641	1.09715	0.548573	+	0.836103i	$0.315171\pi$
$614$	33.9599	1.37051
$615$	0	0
$616$	33.3654	1.34433
$617$	1.30821	0.0526665	0.0263332	−	0.999653i	$-0.491617\pi$
$617$	1.30821	0.0526665	0.0263332	+	0.999653i	$0.491617\pi$
$618$	0	0
$619$	28.4948	1.14530	0.572651	−	0.819799i	$-0.305915\pi$
$619$	28.4948	1.14530	0.572651	+	0.819799i	$0.305915\pi$
$620$	−9.96477	−0.400195
$621$	0	0
$622$	−9.05161	−0.362936
$623$	−65.9215	−2.64109
$624$	0	0
$625$	121.626	4.86504
$626$	0.757628	0.0302809
$627$	0	0
$628$	−12.1194	−0.483615
$629$	4.67364	0.186350
$630$	0	0
$631$	18.6317	0.741718	0.370859	−	0.928689i	$-0.379063\pi$
$631$	18.6317	0.741718	0.370859	+	0.928689i	$0.379063\pi$
$632$	35.2827	1.40347
$633$	0	0
$634$	−14.7331	−0.585126
$635$	49.2469	1.95430
$636$	0	0
$637$	63.4705	2.51479
$638$	2.73213	0.108166
$639$	0	0
$640$	10.0372	0.396757
$641$	42.3346	1.67212	0.836058	−	0.548642i	$-0.184855\pi$
$641$	42.3346	1.67212	0.836058	+	0.548642i	$0.184855\pi$
$642$	0	0
$643$	17.9840	0.709220	0.354610	−	0.935014i	$-0.384614\pi$
$643$	17.9840	0.709220	0.354610	+	0.935014i	$0.384614\pi$
$644$	3.15276	0.124236
$645$	0	0
$646$	−1.51528	−0.0596180
$647$	−44.6035	−1.75354	−0.876772	−	0.480907i	$-0.840307\pi$
$647$	−44.6035	−1.75354	−0.876772	+	0.480907i	$0.840307\pi$
$648$	0	0
$649$	−21.4252	−0.841014
$650$	79.6936	3.12584
$651$	0	0
$652$	−8.88860	−0.348104
$653$	1.52868	0.0598219	0.0299109	−	0.999553i	$-0.490478\pi$
$653$	1.52868	0.0598219	0.0299109	+	0.999553i	$0.490478\pi$
$654$	0	0
$655$	21.1299	0.825615
$656$	16.1249	0.629573
$657$	0	0
$658$	4.19445	0.163517
$659$	29.3700	1.14409	0.572047	−	0.820221i	$-0.306150\pi$
$659$	29.3700	1.14409	0.572047	+	0.820221i	$0.306150\pi$
$660$	0	0
$661$	−24.7688	−0.963396	−0.481698	−	0.876337i	$-0.659980\pi$
$661$	−24.7688	−0.963396	−0.481698	+	0.876337i	$0.659980\pi$
$662$	−33.2816	−1.29353
$663$	0	0
$664$	−5.34450	−0.207407
$665$	27.0115	1.04746
$666$	0	0
$667$	1.00000	0.0387202
$668$	−15.5602	−0.602043
$669$	0	0
$670$	31.4936	1.21670
$671$	−3.73558	−0.144210
$672$	0	0
$673$	5.03710	0.194166	0.0970829	−	0.995276i	$-0.469049\pi$
$673$	5.03710	0.194166	0.0970829	+	0.995276i	$0.469049\pi$
$674$	4.89525	0.188558
$675$	0	0
$676$	−6.29457	−0.242099
$677$	−3.15599	−0.121295	−0.0606473	−	0.998159i	$-0.519317\pi$
$677$	−3.15599	−0.121295	−0.0606473	+	0.998159i	$0.519317\pi$
$678$	0	0
$679$	−39.4202	−1.51281
$680$	−13.6086	−0.521866
$681$	0	0
$682$	8.77636	0.336064
$683$	17.3321	0.663196	0.331598	−	0.943421i	$-0.392412\pi$
$683$	17.3321	0.663196	0.331598	+	0.943421i	$0.392412\pi$
$684$	0	0
$685$	−52.9217	−2.02203
$686$	−33.7178	−1.28735
$687$	0	0
$688$	18.1985	0.693811
$689$	−29.0528	−1.10683
$690$	0	0
$691$	0.738408	0.0280904	0.0140452	−	0.999901i	$-0.495529\pi$
$691$	0.738408	0.0280904	0.0140452	+	0.999901i	$0.495529\pi$
$692$	−0.147043	−0.00558973
$693$	0	0
$694$	2.77511	0.105342
$695$	93.3232	3.53995
$696$	0	0
$697$	−7.53106	−0.285259
$698$	−15.3674	−0.581664
$699$	0	0
$700$	46.8636	1.77128
$701$	−37.6764	−1.42302	−0.711508	−	0.702677i	$-0.751988\pi$
$701$	−37.6764	−1.42302	−0.711508	+	0.702677i	$0.751988\pi$
$702$	0	0
$703$	−6.30487	−0.237793
$704$	20.3587	0.767299
$705$	0	0
$706$	−23.7436	−0.893603
$707$	−58.6454	−2.20559
$708$	0	0
$709$	1.43861	0.0540282	0.0270141	−	0.999635i	$-0.491400\pi$
$709$	1.43861	0.0540282	0.0270141	+	0.999635i	$0.491400\pi$
$710$	77.2781	2.90019
$711$	0	0
$712$	−44.8036	−1.67908
$713$	3.21228	0.120301
$714$	0	0
$715$	50.0662	1.87237
$716$	−9.09008	−0.339712
$717$	0	0
$718$	−12.4081	−0.463067
$719$	52.6104	1.96204	0.981019	−	0.193912i	$-0.0621175\pi$
$719$	52.6104	1.96204	0.981019	+	0.193912i	$0.0621175\pi$
$720$	0	0
$721$	49.9084	1.85869
$722$	−19.6524	−0.731385
$723$	0	0
$724$	6.73313	0.250235
$725$	14.8643	0.552046
$726$	0	0
$727$	34.3751	1.27490	0.637450	−	0.770491i	$-0.279989\pi$
$727$	34.3751	1.27490	0.637450	+	0.770491i	$0.279989\pi$
$728$	65.4748	2.42666
$729$	0	0
$730$	28.5796	1.05778
$731$	−8.49951	−0.314366
$732$	0	0
$733$	−12.6578	−0.467525	−0.233763	−	0.972294i	$-0.575104\pi$
$733$	−12.6578	−0.467525	−0.233763	+	0.972294i	$0.575104\pi$
$734$	−17.0497	−0.629315
$735$	0	0
$736$	3.73236	0.137577
$737$	14.8052	0.545357
$738$	0	0
$739$	−47.8470	−1.76008	−0.880041	−	0.474898i	$-0.842485\pi$
$739$	−47.8470	−1.76008	−0.880041	+	0.474898i	$0.842485\pi$
$740$	−14.6181	−0.537373
$741$	0	0
$742$	32.0078	1.17504
$743$	11.4065	0.418465	0.209232	−	0.977866i	$-0.432903\pi$
$743$	11.4065	0.418465	0.209232	+	0.977866i	$0.432903\pi$
$744$	0	0
$745$	−85.2547	−3.12349
$746$	−25.9925	−0.951651
$747$	0	0
$748$	−1.65159	−0.0603880
$749$	−11.5934	−0.423614
$750$	0	0
$751$	−46.5784	−1.69967	−0.849836	−	0.527048i	$-0.823299\pi$
$751$	−46.5784	−1.69967	−0.849836	+	0.527048i	$0.823299\pi$
$752$	1.72197	0.0627937
$753$	0	0
$754$	5.36141	0.195251
$755$	29.8474	1.08626
$756$	0	0
$757$	−19.3734	−0.704140	−0.352070	−	0.935974i	$-0.614522\pi$
$757$	−19.3734	−0.704140	−0.352070	+	0.935974i	$0.614522\pi$
$758$	19.9178	0.723448
$759$	0	0
$760$	18.3584	0.665929
$761$	6.02792	0.218512	0.109256	−	0.994014i	$-0.465153\pi$
$761$	6.02792	0.218512	0.109256	+	0.994014i	$0.465153\pi$
$762$	0	0
$763$	−62.7921	−2.27323
$764$	8.01484	0.289967
$765$	0	0
$766$	−35.4151	−1.27960
$767$	−42.0439	−1.51812
$768$	0	0
$769$	−29.3710	−1.05915	−0.529573	−	0.848264i	$-0.677648\pi$
$769$	−29.3710	−1.05915	−0.529573	+	0.848264i	$0.677648\pi$
$770$	−55.1584	−1.98777
$771$	0	0
$772$	−11.4269	−0.411261
$773$	33.4941	1.20470	0.602349	−	0.798233i	$-0.294232\pi$
$773$	33.4941	1.20470	0.602349	+	0.798233i	$0.294232\pi$
$774$	0	0
$775$	47.7482	1.71517
$776$	−26.7919	−0.961774
$777$	0	0
$778$	2.65308	0.0951176
$779$	10.1596	0.364006
$780$	0	0
$781$	36.3286	1.29994
$782$	1.13254	0.0404996
$783$	0	0
$784$	−28.7071	−1.02525
$785$	77.6065	2.76989
$786$	0	0
$787$	−21.2094	−0.756034	−0.378017	−	0.925799i	$-0.623394\pi$
$787$	−21.2094	−0.756034	−0.378017	+	0.925799i	$0.623394\pi$
$788$	−10.9138	−0.388787
$789$	0	0
$790$	−58.3279	−2.07521
$791$	−43.4689	−1.54558
$792$	0	0
$793$	−7.33053	−0.260315
$794$	−17.9646	−0.637540
$795$	0	0
$796$	−9.77627	−0.346511
$797$	−44.0629	−1.56079	−0.780394	−	0.625287i	$-0.784982\pi$
$797$	−44.0629	−1.56079	−0.780394	+	0.625287i	$0.784982\pi$
$798$	0	0
$799$	−0.804236	−0.0284518
$800$	55.4790	1.96148
$801$	0	0
$802$	40.8129	1.44116
$803$	13.4353	0.474122
$804$	0	0
$805$	−20.1888	−0.711561
$806$	17.2223	0.606631
$807$	0	0
$808$	−39.8583	−1.40221
$809$	19.7882	0.695717	0.347859	−	0.937547i	$-0.386909\pi$
$809$	19.7882	0.695717	0.347859	+	0.937547i	$0.386909\pi$
$810$	0	0
$811$	−1.24609	−0.0437560	−0.0218780	−	0.999761i	$-0.506965\pi$
$811$	−1.24609	−0.0437560	−0.0218780	+	0.999761i	$0.506965\pi$
$812$	3.15276	0.110640
$813$	0	0
$814$	12.8747	0.451259
$815$	56.9183	1.99376
$816$	0	0
$817$	11.4661	0.401147
$818$	−17.8492	−0.624082
$819$	0	0
$820$	23.5555	0.822595
$821$	26.5856	0.927844	0.463922	−	0.885876i	$-0.346442\pi$
$821$	26.5856	0.927844	0.463922	+	0.885876i	$0.346442\pi$
$822$	0	0
$823$	−37.3252	−1.30108	−0.650538	−	0.759474i	$-0.725457\pi$
$823$	−37.3252	−1.30108	−0.650538	+	0.759474i	$0.725457\pi$
$824$	33.9203	1.18167
$825$	0	0
$826$	46.3201	1.61168
$827$	17.7539	0.617365	0.308683	−	0.951165i	$-0.400112\pi$
$827$	17.7539	0.617365	0.308683	+	0.951165i	$0.400112\pi$
$828$	0	0
$829$	−30.6735	−1.06534	−0.532668	−	0.846324i	$-0.678811\pi$
$829$	−30.6735	−1.06534	−0.532668	+	0.846324i	$0.678811\pi$
$830$	8.83531	0.306678
$831$	0	0
$832$	39.9511	1.38505
$833$	13.4075	0.464542
$834$	0	0
$835$	99.6402	3.44819
$836$	2.22804	0.0770583
$837$	0	0
$838$	−30.8401	−1.06535
$839$	6.08297	0.210007	0.105004	−	0.994472i	$-0.466515\pi$
$839$	6.08297	0.210007	0.105004	+	0.994472i	$0.466515\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−37.2464	−1.28360
$843$	0	0
$844$	0.593800	0.0204394
$845$	40.3074	1.38662
$846$	0	0
$847$	23.8971	0.821113
$848$	13.1403	0.451240
$849$	0	0
$850$	16.8345	0.577417
$851$	4.71234	0.161537
$852$	0	0
$853$	34.8537	1.19337	0.596683	−	0.802477i	$-0.296485\pi$
$853$	34.8537	1.19337	0.596683	+	0.802477i	$0.296485\pi$
$854$	8.07611	0.276359
$855$	0	0
$856$	−7.87946	−0.269314
$857$	−16.2958	−0.556653	−0.278327	−	0.960487i	$-0.589780\pi$
$857$	−16.2958	−0.556653	−0.278327	+	0.960487i	$0.589780\pi$
$858$	0	0
$859$	22.8974	0.781248	0.390624	−	0.920550i	$-0.372259\pi$
$859$	22.8974	0.781248	0.390624	+	0.920550i	$0.372259\pi$
$860$	26.5846	0.906528
$861$	0	0
$862$	39.2970	1.33846
$863$	−23.0828	−0.785747	−0.392874	−	0.919593i	$-0.628519\pi$
$863$	−23.0828	−0.785747	−0.392874	+	0.919593i	$0.628519\pi$
$864$	0	0
$865$	0.941592	0.0320151
$866$	−5.20430	−0.176849
$867$	0	0
$868$	10.1275	0.343751
$869$	−27.4201	−0.930162
$870$	0	0
$871$	29.0531	0.984425
$872$	−42.6767	−1.44522
$873$	0	0
$874$	−1.52783	−0.0516797
$875$	−199.148	−6.73244
$876$	0	0
$877$	−44.3054	−1.49609	−0.748044	−	0.663649i	$-0.769007\pi$
$877$	−44.3054	−1.49609	−0.748044	+	0.663649i	$0.769007\pi$
$878$	10.3980	0.350916
$879$	0	0
$880$	−22.6444	−0.763344
$881$	−19.8581	−0.669035	−0.334518	−	0.942389i	$-0.608573\pi$
$881$	−19.8581	−0.669035	−0.334518	+	0.942389i	$0.608573\pi$
$882$	0	0
$883$	−47.4387	−1.59644	−0.798220	−	0.602366i	$-0.794225\pi$
$883$	−47.4387	−1.59644	−0.798220	+	0.602366i	$0.794225\pi$
$884$	−3.24100	−0.109007
$885$	0	0
$886$	18.0703	0.607084
$887$	24.4360	0.820482	0.410241	−	0.911977i	$-0.365445\pi$
$887$	24.4360	0.820482	0.410241	+	0.911977i	$0.365445\pi$
$888$	0	0
$889$	−50.0513	−1.67867
$890$	74.0674	2.48275
$891$	0	0
$892$	15.4224	0.516380
$893$	1.08494	0.0363060
$894$	0	0
$895$	58.2085	1.94569
$896$	−10.2012	−0.340798
$897$	0	0
$898$	5.96387	0.199017
$899$	3.21228	0.107135
$900$	0	0
$901$	−6.13711	−0.204457
$902$	−20.7463	−0.690775
$903$	0	0
$904$	−29.5437	−0.982608
$905$	−43.1157	−1.43322
$906$	0	0
$907$	4.34528	0.144283	0.0721414	−	0.997394i	$-0.477017\pi$
$907$	4.34528	0.144283	0.0721414	+	0.997394i	$0.477017\pi$
$908$	13.6854	0.454167
$909$	0	0
$910$	−108.240	−3.58813
$911$	43.1053	1.42814	0.714071	−	0.700073i	$-0.246849\pi$
$911$	43.1053	1.42814	0.714071	+	0.700073i	$0.246849\pi$
$912$	0	0
$913$	4.15350	0.137461
$914$	−12.1194	−0.400874
$915$	0	0
$916$	−10.6139	−0.350692
$917$	−21.4751	−0.709170
$918$	0	0
$919$	44.9126	1.48153	0.740765	−	0.671764i	$-0.234463\pi$
$919$	44.9126	1.48153	0.740765	+	0.671764i	$0.234463\pi$
$920$	−13.7213	−0.452378
$921$	0	0
$922$	−39.7730	−1.30985
$923$	71.2895	2.34652
$924$	0	0
$925$	70.0457	2.30309
$926$	−34.7023	−1.14039
$927$	0	0
$928$	3.73236	0.122521
$929$	−33.6230	−1.10313	−0.551567	−	0.834130i	$-0.685970\pi$
$929$	−33.6230	−1.10313	−0.551567	+	0.834130i	$0.685970\pi$
$930$	0	0
$931$	−18.0871	−0.592780
$932$	5.97738	0.195796
$933$	0	0
$934$	−1.86555	−0.0610428
$935$	10.5760	0.345871
$936$	0	0
$937$	12.8762	0.420648	0.210324	−	0.977632i	$-0.432548\pi$
$937$	12.8762	0.420648	0.210324	+	0.977632i	$0.432548\pi$
$938$	−32.0080	−1.04510
$939$	0	0
$940$	2.51548	0.0820458
$941$	−31.8653	−1.03878	−0.519389	−	0.854538i	$-0.673840\pi$
$941$	−31.8653	−1.03878	−0.519389	+	0.854538i	$0.673840\pi$
$942$	0	0
$943$	−7.59343	−0.247276
$944$	19.0160	0.618919
$945$	0	0
$946$	−23.4141	−0.761258
$947$	14.3355	0.465842	0.232921	−	0.972496i	$-0.425172\pi$
$947$	14.3355	0.465842	0.232921	+	0.972496i	$0.425172\pi$
$948$	0	0
$949$	26.3648	0.855839
$950$	−22.7102	−0.736815
$951$	0	0
$952$	13.8309	0.448261
$953$	−8.97116	−0.290604	−0.145302	−	0.989387i	$-0.546415\pi$
$953$	−8.97116	−0.290604	−0.145302	+	0.989387i	$0.546415\pi$
$954$	0	0
$955$	−51.3232	−1.66078
$956$	7.18944	0.232523
$957$	0	0
$958$	33.0393	1.06745
$959$	53.7861	1.73684
$960$	0	0
$961$	−20.6813	−0.667138
$962$	25.2648	0.814570
$963$	0	0
$964$	11.2013	0.360771
$965$	73.1721	2.35549
$966$	0	0
$967$	12.4438	0.400166	0.200083	−	0.979779i	$-0.435879\pi$
$967$	12.4438	0.400166	0.200083	+	0.979779i	$0.435879\pi$
$968$	16.2416	0.522026
$969$	0	0
$970$	44.2913	1.42211
$971$	−46.9890	−1.50795	−0.753974	−	0.656904i	$-0.771866\pi$
$971$	−46.9890	−1.50795	−0.753974	+	0.656904i	$0.771866\pi$
$972$	0	0
$973$	−94.8475	−3.04067
$974$	30.1164	0.964993
$975$	0	0
$976$	3.31553	0.106127
$977$	37.9297	1.21348	0.606739	−	0.794901i	$-0.292477\pi$
$977$	37.9297	1.21348	0.606739	+	0.794901i	$0.292477\pi$
$978$	0	0
$979$	34.8192	1.11283
$980$	−41.9357	−1.33959
$981$	0	0
$982$	−11.4705	−0.366040
$983$	29.5846	0.943603	0.471802	−	0.881705i	$-0.343604\pi$
$983$	29.5846	0.943603	0.471802	+	0.881705i	$0.343604\pi$
$984$	0	0
$985$	69.8866	2.22677
$986$	1.13254	0.0360675
$987$	0	0
$988$	4.37220	0.139098
$989$	−8.56990	−0.272507
$990$	0	0
$991$	−58.4080	−1.85539	−0.927696	−	0.373338i	$-0.878213\pi$
$991$	−58.4080	−1.85539	−0.927696	+	0.373338i	$0.878213\pi$
$992$	11.9894	0.380663
$993$	0	0
$994$	−78.5403	−2.49115
$995$	62.6026	1.98463
$996$	0	0
$997$	−10.4638	−0.331390	−0.165695	−	0.986177i	$-0.552987\pi$
$997$	−10.4638	−0.331390	−0.165695	+	0.986177i	$0.552987\pi$
$998$	4.26668	0.135059
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.9		14
3.2	odd	2		2001.2.a.m.1.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.6	✓	14	3.2	odd	2
6003.2.a.p.1.9		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)

\(f(q)\)	\(=\)	\(q+1.14192 q^{2} -0.696014 q^{4} +4.45694 q^{5} -4.52974 q^{7} -3.07864 q^{8} +O(q^{10})\)	\(q+1.14192 q^{2} -0.696014 q^{4} +4.45694 q^{5} -4.52974 q^{7} -3.07864 q^{8} +5.08948 q^{10} +2.39257 q^{11} +4.69508 q^{13} -5.17261 q^{14} -2.12354 q^{16} +0.991786 q^{17} -1.33795 q^{19} -3.10209 q^{20} +2.73213 q^{22} +1.00000 q^{23} +14.8643 q^{25} +5.36141 q^{26} +3.15276 q^{28} +1.00000 q^{29} +3.21228 q^{31} +3.73236 q^{32} +1.13254 q^{34} -20.1888 q^{35} +4.71234 q^{37} -1.52783 q^{38} -13.7213 q^{40} -7.59343 q^{41} -8.56990 q^{43} -1.66526 q^{44} +1.14192 q^{46} -0.810896 q^{47} +13.5185 q^{49} +16.9739 q^{50} -3.26784 q^{52} -6.18794 q^{53} +10.6636 q^{55} +13.9454 q^{56} +1.14192 q^{58} -8.95489 q^{59} -1.56132 q^{61} +3.66817 q^{62} +8.50914 q^{64} +20.9257 q^{65} +6.18798 q^{67} -0.690297 q^{68} -23.0540 q^{70} +15.1839 q^{71} +5.61542 q^{73} +5.38113 q^{74} +0.931231 q^{76} -10.8377 q^{77} -11.4605 q^{79} -9.46447 q^{80} -8.67111 q^{82} +1.73600 q^{83} +4.42033 q^{85} -9.78616 q^{86} -7.36587 q^{88} +14.5530 q^{89} -21.2675 q^{91} -0.696014 q^{92} -0.925980 q^{94} -5.96315 q^{95} +8.70253 q^{97} +15.4371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 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

Related objects

Downloads

Learn more