LMFDB - Embedded newform 6003.2.a.n.1.10

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			1.10
Root			$1.99588$ 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)

$f(q)$	$=$	$q+1.99588 q^{2} +1.98352 q^{4} -2.38535 q^{5} -0.101071 q^{7} -0.0328940 q^{8} +O(q^{10})$	\(q+1.99588 q^{2} +1.98352 q^{4} -2.38535 q^{5} -0.101071 q^{7} -0.0328940 q^{8} -4.76086 q^{10} +3.79046 q^{11} -3.75296 q^{13} -0.201725 q^{14} -4.03269 q^{16} +4.24895 q^{17} -3.15538 q^{19} -4.73138 q^{20} +7.56528 q^{22} +1.00000 q^{23} +0.689882 q^{25} -7.49045 q^{26} -0.200476 q^{28} +1.00000 q^{29} +0.905162 q^{31} -7.98296 q^{32} +8.48038 q^{34} +0.241089 q^{35} +6.40721 q^{37} -6.29774 q^{38} +0.0784637 q^{40} +3.03191 q^{41} +12.2360 q^{43} +7.51844 q^{44} +1.99588 q^{46} +6.38858 q^{47} -6.98978 q^{49} +1.37692 q^{50} -7.44408 q^{52} +7.07607 q^{53} -9.04156 q^{55} +0.00332463 q^{56} +1.99588 q^{58} +5.17172 q^{59} +12.4431 q^{61} +1.80659 q^{62} -7.86761 q^{64} +8.95212 q^{65} -14.1515 q^{67} +8.42787 q^{68} +0.481184 q^{70} -14.6000 q^{71} +6.65371 q^{73} +12.7880 q^{74} -6.25875 q^{76} -0.383105 q^{77} +7.81560 q^{79} +9.61937 q^{80} +6.05131 q^{82} +9.08675 q^{83} -10.1352 q^{85} +24.4216 q^{86} -0.124683 q^{88} +16.0435 q^{89} +0.379316 q^{91} +1.98352 q^{92} +12.7508 q^{94} +7.52667 q^{95} -9.31858 q^{97} -13.9507 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.99588	1.41130	0.705649	−	0.708562i	$-0.250656\pi$
$2$	1.99588	1.41130	0.705649	+	0.708562i	$0.250656\pi$
$3$	0	0
$3$	0	0
$4$	1.98352	0.991759
$5$	−2.38535	−1.06676	−0.533380	−	0.845876i	$-0.679078\pi$
$5$	−2.38535	−1.06676	−0.533380	+	0.845876i	$0.679078\pi$
$6$	0	0
$7$	−0.101071	−0.0382012	−0.0191006	−	0.999818i	$-0.506080\pi$
$7$	−0.101071	−0.0382012	−0.0191006	+	0.999818i	$0.506080\pi$
$8$	−0.0328940	−0.0116298
$9$	0	0
$10$	−4.76086	−1.50551
$11$	3.79046	1.14287	0.571433	−	0.820649i	$-0.306388\pi$
$11$	3.79046	1.14287	0.571433	+	0.820649i	$0.306388\pi$
$12$	0	0
$13$	−3.75296	−1.04088	−0.520442	−	0.853897i	$-0.674233\pi$
$13$	−3.75296	−1.04088	−0.520442	+	0.853897i	$0.674233\pi$
$14$	−0.201725	−0.0539133
$15$	0	0
$16$	−4.03269	−1.00817
$17$	4.24895	1.03052	0.515261	−	0.857033i	$-0.327695\pi$
$17$	4.24895	1.03052	0.515261	+	0.857033i	$0.327695\pi$
$18$	0	0
$19$	−3.15538	−0.723893	−0.361947	−	0.932199i	$-0.617888\pi$
$19$	−3.15538	−0.723893	−0.361947	+	0.932199i	$0.617888\pi$
$20$	−4.73138	−1.05797
$21$	0	0
$22$	7.56528	1.61292
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.689882	0.137976
$26$	−7.49045	−1.46900
$27$	0	0
$28$	−0.200476	−0.0378864
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.905162	0.162572	0.0812860	−	0.996691i	$-0.474097\pi$
$31$	0.905162	0.162572	0.0812860	+	0.996691i	$0.474097\pi$
$32$	−7.98296	−1.41120
$33$	0	0
$34$	8.48038	1.45437
$35$	0.241089	0.0407515
$36$	0	0
$37$	6.40721	1.05334	0.526670	−	0.850070i	$-0.323440\pi$
$37$	6.40721	1.05334	0.526670	+	0.850070i	$0.323440\pi$
$38$	−6.29774	−1.02163
$39$	0	0
$40$	0.0784637	0.0124062
$41$	3.03191	0.473505	0.236752	−	0.971570i	$-0.423917\pi$
$41$	3.03191	0.473505	0.236752	+	0.971570i	$0.423917\pi$
$42$	0	0
$43$	12.2360	1.86597	0.932987	−	0.359910i	$-0.117193\pi$
$43$	12.2360	1.86597	0.932987	+	0.359910i	$0.117193\pi$
$44$	7.51844	1.13345
$45$	0	0
$46$	1.99588	0.294276
$47$	6.38858	0.931869	0.465935	−	0.884819i	$-0.345718\pi$
$47$	6.38858	0.931869	0.465935	+	0.884819i	$0.345718\pi$
$48$	0	0
$49$	−6.98978	−0.998541
$50$	1.37692	0.194726
$51$	0	0
$52$	−7.44408	−1.03231
$53$	7.07607	0.971974	0.485987	−	0.873966i	$-0.338460\pi$
$53$	7.07607	0.971974	0.485987	+	0.873966i	$0.338460\pi$
$54$	0	0
$55$	−9.04156	−1.21916
$56$	0.00332463	0.000444273 0
$57$	0	0
$58$	1.99588	0.262071
$59$	5.17172	0.673300	0.336650	−	0.941630i	$-0.390706\pi$
$59$	5.17172	0.673300	0.336650	+	0.941630i	$0.390706\pi$
$60$	0	0
$61$	12.4431	1.59318	0.796588	−	0.604522i	$-0.206636\pi$
$61$	12.4431	1.59318	0.796588	+	0.604522i	$0.206636\pi$
$62$	1.80659	0.229437
$63$	0	0
$64$	−7.86761	−0.983452
$65$	8.95212	1.11037
$66$	0	0
$67$	−14.1515	−1.72888	−0.864439	−	0.502737i	$-0.832326\pi$
$67$	−14.1515	−1.72888	−0.864439	+	0.502737i	$0.832326\pi$
$68$	8.42787	1.02203
$69$	0	0
$70$	0.481184	0.0575125
$71$	−14.6000	−1.73270	−0.866352	−	0.499434i	$-0.833541\pi$
$71$	−14.6000	−1.73270	−0.866352	+	0.499434i	$0.833541\pi$
$72$	0	0
$73$	6.65371	0.778758	0.389379	−	0.921078i	$-0.372690\pi$
$73$	6.65371	0.778758	0.389379	+	0.921078i	$0.372690\pi$
$74$	12.7880	1.48657
$75$	0	0
$76$	−6.25875	−0.717928
$77$	−0.383105	−0.0436589
$78$	0	0
$79$	7.81560	0.879323	0.439662	−	0.898163i	$-0.355098\pi$
$79$	7.81560	0.879323	0.439662	+	0.898163i	$0.355098\pi$
$80$	9.61937	1.07548
$81$	0	0
$82$	6.05131	0.668256
$83$	9.08675	0.997401	0.498700	−	0.866775i	$-0.333811\pi$
$83$	9.08675	0.997401	0.498700	+	0.866775i	$0.333811\pi$
$84$	0	0
$85$	−10.1352	−1.09932
$86$	24.4216	2.63344
$87$	0	0
$88$	−0.124683	−0.0132913
$89$	16.0435	1.70061	0.850303	−	0.526293i	$-0.176419\pi$
$89$	16.0435	1.70061	0.850303	+	0.526293i	$0.176419\pi$
$90$	0	0
$91$	0.379316	0.0397631
$92$	1.98352	0.206796
$93$	0	0
$94$	12.7508	1.31514
$95$	7.52667	0.772220
$96$	0	0
$97$	−9.31858	−0.946158	−0.473079	−	0.881020i	$-0.656858\pi$
$97$	−9.31858	−0.946158	−0.473079	+	0.881020i	$0.656858\pi$
$98$	−13.9507	−1.40924
$99$	0	0
$100$	1.36839	0.136839
$101$	9.68764	0.963957	0.481978	−	0.876183i	$-0.339918\pi$
$101$	9.68764	0.963957	0.481978	+	0.876183i	$0.339918\pi$
$102$	0	0
$103$	−1.84644	−0.181935	−0.0909674	−	0.995854i	$-0.528996\pi$
$103$	−1.84644	−0.181935	−0.0909674	+	0.995854i	$0.528996\pi$
$104$	0.123450	0.0121053
$105$	0	0
$106$	14.1230	1.37174
$107$	−7.65270	−0.739815	−0.369907	−	0.929069i	$-0.620611\pi$
$107$	−7.65270	−0.739815	−0.369907	+	0.929069i	$0.620611\pi$
$108$	0	0
$109$	4.37503	0.419052	0.209526	−	0.977803i	$-0.432808\pi$
$109$	4.37503	0.419052	0.209526	+	0.977803i	$0.432808\pi$
$110$	−18.0458	−1.72060
$111$	0	0
$112$	0.407588	0.0385134
$113$	7.66927	0.721464	0.360732	−	0.932669i	$-0.382527\pi$
$113$	7.66927	0.721464	0.360732	+	0.932669i	$0.382527\pi$
$114$	0	0
$115$	−2.38535	−0.222435
$116$	1.98352	0.184165
$117$	0	0
$118$	10.3221	0.950226
$119$	−0.429445	−0.0393672
$120$	0	0
$121$	3.36756	0.306142
$122$	24.8349	2.24845
$123$	0	0
$124$	1.79541	0.161232
$125$	10.2811	0.919572
$126$	0	0
$127$	15.1863	1.34757	0.673783	−	0.738929i	$-0.264668\pi$
$127$	15.1863	1.34757	0.673783	+	0.738929i	$0.264668\pi$
$128$	0.263143	0.0232588
$129$	0	0
$130$	17.8673	1.56707
$131$	−7.73579	−0.675879	−0.337940	−	0.941168i	$-0.609730\pi$
$131$	−7.73579	−0.675879	−0.337940	+	0.941168i	$0.609730\pi$
$132$	0	0
$133$	0.318917	0.0276536
$134$	−28.2446	−2.43996
$135$	0	0
$136$	−0.139765	−0.0119848
$137$	−15.5715	−1.33036	−0.665182	−	0.746681i	$-0.731646\pi$
$137$	−15.5715	−1.33036	−0.665182	+	0.746681i	$0.731646\pi$
$138$	0	0
$139$	−2.71921	−0.230640	−0.115320	−	0.993328i	$-0.536789\pi$
$139$	−2.71921	−0.230640	−0.115320	+	0.993328i	$0.536789\pi$
$140$	0.478205	0.0404157
$141$	0	0
$142$	−29.1398	−2.44536
$143$	−14.2254	−1.18959
$144$	0	0
$145$	−2.38535	−0.198092
$146$	13.2800	1.09906
$147$	0	0
$148$	12.7088	1.04466
$149$	−19.2804	−1.57952	−0.789758	−	0.613419i	$-0.789794\pi$
$149$	−19.2804	−1.57952	−0.789758	+	0.613419i	$0.789794\pi$
$150$	0	0
$151$	8.39612	0.683267	0.341633	−	0.939833i	$-0.389020\pi$
$151$	8.39612	0.683267	0.341633	+	0.939833i	$0.389020\pi$
$152$	0.103793	0.00841873
$153$	0	0
$154$	−0.764630	−0.0616156
$155$	−2.15913	−0.173425
$156$	0	0
$157$	−11.2647	−0.899020	−0.449510	−	0.893275i	$-0.648401\pi$
$157$	−11.2647	−0.899020	−0.449510	+	0.893275i	$0.648401\pi$
$158$	15.5990	1.24099
$159$	0	0
$160$	19.0421	1.50541
$161$	−0.101071	−0.00796551
$162$	0	0
$163$	11.5920	0.907956	0.453978	−	0.891013i	$-0.350005\pi$
$163$	11.5920	0.907956	0.453978	+	0.891013i	$0.350005\pi$
$164$	6.01385	0.469603
$165$	0	0
$166$	18.1360	1.40763
$167$	19.3746	1.49925	0.749625	−	0.661863i	$-0.230234\pi$
$167$	19.3746	1.49925	0.749625	+	0.661863i	$0.230234\pi$
$168$	0	0
$169$	1.08474	0.0834414
$170$	−20.2286	−1.55147
$171$	0	0
$172$	24.2704	1.85060
$173$	0.949545	0.0721926	0.0360963	−	0.999348i	$-0.488508\pi$
$173$	0.949545	0.0721926	0.0360963	+	0.999348i	$0.488508\pi$
$174$	0	0
$175$	−0.0697270	−0.00527087
$176$	−15.2857	−1.15221
$177$	0	0
$178$	32.0208	2.40006
$179$	−7.74510	−0.578896	−0.289448	−	0.957194i	$-0.593472\pi$
$179$	−7.74510	−0.578896	−0.289448	+	0.957194i	$0.593472\pi$
$180$	0	0
$181$	11.1415	0.828138	0.414069	−	0.910246i	$-0.364107\pi$
$181$	11.1415	0.828138	0.414069	+	0.910246i	$0.364107\pi$
$182$	0.757067	0.0561175
$183$	0	0
$184$	−0.0328940	−0.00242498
$185$	−15.2834	−1.12366
$186$	0	0
$187$	16.1055	1.17775
$188$	12.6719	0.924190
$189$	0	0
$190$	15.0223	1.08983
$191$	7.51527	0.543786	0.271893	−	0.962328i	$-0.412350\pi$
$191$	7.51527	0.543786	0.271893	+	0.962328i	$0.412350\pi$
$192$	0	0
$193$	−12.4284	−0.894614	−0.447307	−	0.894380i	$-0.647617\pi$
$193$	−12.4284	−0.894614	−0.447307	+	0.894380i	$0.647617\pi$
$194$	−18.5987	−1.33531
$195$	0	0
$196$	−13.8644	−0.990312
$197$	19.7716	1.40867	0.704335	−	0.709868i	$-0.251245\pi$
$197$	19.7716	1.40867	0.704335	+	0.709868i	$0.251245\pi$
$198$	0	0
$199$	20.2924	1.43849	0.719244	−	0.694758i	$-0.244489\pi$
$199$	20.2924	1.43849	0.719244	+	0.694758i	$0.244489\pi$
$200$	−0.0226930	−0.00160464
$201$	0	0
$202$	19.3353	1.36043
$203$	−0.101071	−0.00709379
$204$	0	0
$205$	−7.23216	−0.505116
$206$	−3.68526	−0.256764
$207$	0	0
$208$	15.1345	1.04939
$209$	−11.9603	−0.827313
$210$	0	0
$211$	−3.32728	−0.229059	−0.114530	−	0.993420i	$-0.536536\pi$
$211$	−3.32728	−0.229059	−0.114530	+	0.993420i	$0.536536\pi$
$212$	14.0355	0.963964
$213$	0	0
$214$	−15.2738	−1.04410
$215$	−29.1871	−1.99055
$216$	0	0
$217$	−0.0914856	−0.00621045
$218$	8.73201	0.591407
$219$	0	0
$220$	−17.9341	−1.20912
$221$	−15.9462	−1.07265
$222$	0	0
$223$	1.03293	0.0691700	0.0345850	−	0.999402i	$-0.488989\pi$
$223$	1.03293	0.0691700	0.0345850	+	0.999402i	$0.488989\pi$
$224$	0.806845	0.0539096
$225$	0	0
$226$	15.3069	1.01820
$227$	22.0373	1.46267	0.731335	−	0.682018i	$-0.238898\pi$
$227$	22.0373	1.46267	0.731335	+	0.682018i	$0.238898\pi$
$228$	0	0
$229$	−14.2895	−0.944274	−0.472137	−	0.881525i	$-0.656517\pi$
$229$	−14.2895	−0.944274	−0.472137	+	0.881525i	$0.656517\pi$
$230$	−4.76086	−0.313922
$231$	0	0
$232$	−0.0328940	−0.00215960
$233$	23.6623	1.55017	0.775086	−	0.631856i	$-0.217707\pi$
$233$	23.6623	1.55017	0.775086	+	0.631856i	$0.217707\pi$
$234$	0	0
$235$	−15.2390	−0.994081
$236$	10.2582	0.667752
$237$	0	0
$238$	−0.857119	−0.0555588
$239$	−3.21212	−0.207775	−0.103887	−	0.994589i	$-0.533128\pi$
$239$	−3.21212	−0.207775	−0.103887	+	0.994589i	$0.533128\pi$
$240$	0	0
$241$	−3.02979	−0.195166	−0.0975830	−	0.995227i	$-0.531111\pi$
$241$	−3.02979	−0.195166	−0.0975830	+	0.995227i	$0.531111\pi$
$242$	6.72123	0.432057
$243$	0	0
$244$	24.6811	1.58005
$245$	16.6731	1.06520
$246$	0	0
$247$	11.8420	0.753490
$248$	−0.0297744	−0.00189068
$249$	0	0
$250$	20.5199	1.29779
$251$	15.7102	0.991621	0.495810	−	0.868431i	$-0.334871\pi$
$251$	15.7102	0.991621	0.495810	+	0.868431i	$0.334871\pi$
$252$	0	0
$253$	3.79046	0.238304
$254$	30.3100	1.90182
$255$	0	0
$256$	16.2604	1.01628
$257$	24.1473	1.50627	0.753133	−	0.657868i	$-0.228541\pi$
$257$	24.1473	1.50627	0.753133	+	0.657868i	$0.228541\pi$
$258$	0	0
$259$	−0.647583	−0.0402389
$260$	17.7567	1.10122
$261$	0	0
$262$	−15.4397	−0.953866
$263$	−25.1690	−1.55199	−0.775993	−	0.630742i	$-0.782751\pi$
$263$	−25.1690	−1.55199	−0.775993	+	0.630742i	$0.782751\pi$
$264$	0	0
$265$	−16.8789	−1.03686
$266$	0.636518	0.0390275
$267$	0	0
$268$	−28.0697	−1.71463
$269$	14.3961	0.877747	0.438873	−	0.898549i	$-0.355378\pi$
$269$	14.3961	0.877747	0.438873	+	0.898549i	$0.355378\pi$
$270$	0	0
$271$	18.2784	1.11033	0.555167	−	0.831739i	$-0.312654\pi$
$271$	18.2784	1.11033	0.555167	+	0.831739i	$0.312654\pi$
$272$	−17.1347	−1.03894
$273$	0	0
$274$	−31.0788	−1.87754
$275$	2.61497	0.157688
$276$	0	0
$277$	−8.55734	−0.514160	−0.257080	−	0.966390i	$-0.582760\pi$
$277$	−8.55734	−0.514160	−0.257080	+	0.966390i	$0.582760\pi$
$278$	−5.42720	−0.325502
$279$	0	0
$280$	−0.00793040	−0.000473932 0
$281$	5.09111	0.303710	0.151855	−	0.988403i	$-0.451475\pi$
$281$	5.09111	0.303710	0.151855	+	0.988403i	$0.451475\pi$
$282$	0	0
$283$	−2.13085	−0.126666	−0.0633328	−	0.997992i	$-0.520173\pi$
$283$	−2.13085	−0.126666	−0.0633328	+	0.997992i	$0.520173\pi$
$284$	−28.9594	−1.71843
$285$	0	0
$286$	−28.3922	−1.67887
$287$	−0.306438	−0.0180885
$288$	0	0
$289$	1.05358	0.0619752
$290$	−4.76086	−0.279567
$291$	0	0
$292$	13.1978	0.772341
$293$	−32.5815	−1.90343	−0.951715	−	0.306984i	$-0.900680\pi$
$293$	−32.5815	−1.90343	−0.951715	+	0.306984i	$0.900680\pi$
$294$	0	0
$295$	−12.3363	−0.718249
$296$	−0.210759	−0.0122501
$297$	0	0
$298$	−38.4814	−2.22917
$299$	−3.75296	−0.217040
$300$	0	0
$301$	−1.23670	−0.0712825
$302$	16.7576	0.964292
$303$	0	0
$304$	12.7247	0.729809
$305$	−29.6811	−1.69954
$306$	0	0
$307$	−23.6868	−1.35188	−0.675940	−	0.736957i	$-0.736262\pi$
$307$	−23.6868	−1.35188	−0.675940	+	0.736957i	$0.736262\pi$
$308$	−0.759896	−0.0432991
$309$	0	0
$310$	−4.30935	−0.244754
$311$	28.1927	1.59866	0.799331	−	0.600891i	$-0.205187\pi$
$311$	28.1927	1.59866	0.799331	+	0.600891i	$0.205187\pi$
$312$	0	0
$313$	−6.81109	−0.384986	−0.192493	−	0.981298i	$-0.561657\pi$
$313$	−6.81109	−0.384986	−0.192493	+	0.981298i	$0.561657\pi$
$314$	−22.4829	−1.26878
$315$	0	0
$316$	15.5024	0.872077
$317$	−15.9924	−0.898221	−0.449111	−	0.893476i	$-0.648259\pi$
$317$	−15.9924	−0.898221	−0.449111	+	0.893476i	$0.648259\pi$
$318$	0	0
$319$	3.79046	0.212225
$320$	18.7670	1.04911
$321$	0	0
$322$	−0.201725	−0.0112417
$323$	−13.4070	−0.745988
$324$	0	0
$325$	−2.58910	−0.143618
$326$	23.1362	1.28140
$327$	0	0
$328$	−0.0997318	−0.00550676
$329$	−0.645699	−0.0355986
$330$	0	0
$331$	24.5049	1.34691	0.673456	−	0.739227i	$-0.264809\pi$
$331$	24.5049	1.34691	0.673456	+	0.739227i	$0.264809\pi$
$332$	18.0237	0.989181
$333$	0	0
$334$	38.6693	2.11589
$335$	33.7562	1.84430
$336$	0	0
$337$	−7.49449	−0.408251	−0.204126	−	0.978945i	$-0.565435\pi$
$337$	−7.49449	−0.408251	−0.204126	+	0.978945i	$0.565435\pi$
$338$	2.16500	0.117761
$339$	0	0
$340$	−20.1034	−1.09026
$341$	3.43098	0.185798
$342$	0	0
$343$	1.41396	0.0763467
$344$	−0.402492	−0.0217009
$345$	0	0
$346$	1.89517	0.101885
$347$	16.9776	0.911406	0.455703	−	0.890132i	$-0.349388\pi$
$347$	16.9776	0.911406	0.455703	+	0.890132i	$0.349388\pi$
$348$	0	0
$349$	19.9023	1.06535	0.532673	−	0.846321i	$-0.321187\pi$
$349$	19.9023	1.06535	0.532673	+	0.846321i	$0.321187\pi$
$350$	−0.139166	−0.00743876
$351$	0	0
$352$	−30.2591	−1.61281
$353$	−3.01163	−0.160293	−0.0801466	−	0.996783i	$-0.525539\pi$
$353$	−3.01163	−0.160293	−0.0801466	+	0.996783i	$0.525539\pi$
$354$	0	0
$355$	34.8261	1.84838
$356$	31.8226	1.68659
$357$	0	0
$358$	−15.4583	−0.816995
$359$	−24.3074	−1.28290	−0.641449	−	0.767166i	$-0.721666\pi$
$359$	−24.3074	−1.28290	−0.641449	+	0.767166i	$0.721666\pi$
$360$	0	0
$361$	−9.04359	−0.475979
$362$	22.2370	1.16875
$363$	0	0
$364$	0.752380	0.0394354
$365$	−15.8714	−0.830748
$366$	0	0
$367$	−16.3633	−0.854158	−0.427079	−	0.904214i	$-0.640457\pi$
$367$	−16.3633	−0.854158	−0.427079	+	0.904214i	$0.640457\pi$
$368$	−4.03269	−0.210219
$369$	0	0
$370$	−30.5038	−1.58582
$371$	−0.715185	−0.0371306
$372$	0	0
$373$	−29.5059	−1.52776	−0.763878	−	0.645361i	$-0.776707\pi$
$373$	−29.5059	−1.52776	−0.763878	+	0.645361i	$0.776707\pi$
$374$	32.1445	1.66215
$375$	0	0
$376$	−0.210146	−0.0108375
$377$	−3.75296	−0.193287
$378$	0	0
$379$	9.66507	0.496461	0.248231	−	0.968701i	$-0.420151\pi$
$379$	9.66507	0.496461	0.248231	+	0.968701i	$0.420151\pi$
$380$	14.9293	0.765857
$381$	0	0
$382$	14.9995	0.767443
$383$	0.241583	0.0123443	0.00617217	−	0.999981i	$-0.498035\pi$
$383$	0.241583	0.0123443	0.00617217	+	0.999981i	$0.498035\pi$
$384$	0	0
$385$	0.913838	0.0465735
$386$	−24.8055	−1.26257
$387$	0	0
$388$	−18.4836	−0.938362
$389$	−5.24122	−0.265740	−0.132870	−	0.991133i	$-0.542419\pi$
$389$	−5.24122	−0.265740	−0.132870	+	0.991133i	$0.542419\pi$
$390$	0	0
$391$	4.24895	0.214879
$392$	0.229922	0.0116128
$393$	0	0
$394$	39.4617	1.98805
$395$	−18.6429	−0.938027
$396$	0	0
$397$	5.68323	0.285233	0.142617	−	0.989778i	$-0.454448\pi$
$397$	5.68323	0.285233	0.142617	+	0.989778i	$0.454448\pi$
$398$	40.5010	2.03013
$399$	0	0
$400$	−2.78208	−0.139104
$401$	−9.60065	−0.479434	−0.239717	−	0.970843i	$-0.577055\pi$
$401$	−9.60065	−0.479434	−0.239717	+	0.970843i	$0.577055\pi$
$402$	0	0
$403$	−3.39704	−0.169219
$404$	19.2156	0.956013
$405$	0	0
$406$	−0.201725	−0.0100114
$407$	24.2863	1.20383
$408$	0	0
$409$	23.5780	1.16585	0.582927	−	0.812524i	$-0.301907\pi$
$409$	23.5780	1.16585	0.582927	+	0.812524i	$0.301907\pi$
$410$	−14.4345	−0.712868
$411$	0	0
$412$	−3.66244	−0.180436
$413$	−0.522710	−0.0257209
$414$	0	0
$415$	−21.6751	−1.06399
$416$	29.9598	1.46890
$417$	0	0
$418$	−23.8713	−1.16758
$419$	−3.37343	−0.164803	−0.0824014	−	0.996599i	$-0.526259\pi$
$419$	−3.37343	−0.164803	−0.0824014	+	0.996599i	$0.526259\pi$
$420$	0	0
$421$	6.68947	0.326025	0.163012	−	0.986624i	$-0.447879\pi$
$421$	6.68947	0.326025	0.163012	+	0.986624i	$0.447879\pi$
$422$	−6.64083	−0.323270
$423$	0	0
$424$	−0.232761	−0.0113039
$425$	2.93127	0.142188
$426$	0	0
$427$	−1.25764	−0.0608613
$428$	−15.1793	−0.733718
$429$	0	0
$430$	−58.2539	−2.80925
$431$	28.7762	1.38610	0.693049	−	0.720890i	$-0.256267\pi$
$431$	28.7762	1.38610	0.693049	+	0.720890i	$0.256267\pi$
$432$	0	0
$433$	−8.04010	−0.386383	−0.193191	−	0.981161i	$-0.561884\pi$
$433$	−8.04010	−0.386383	−0.193191	+	0.981161i	$0.561884\pi$
$434$	−0.182594	−0.00876479
$435$	0	0
$436$	8.67795	0.415599
$437$	−3.15538	−0.150942
$438$	0	0
$439$	4.86855	0.232363	0.116182	−	0.993228i	$-0.462935\pi$
$439$	4.86855	0.232363	0.116182	+	0.993228i	$0.462935\pi$
$440$	0.297413	0.0141786
$441$	0	0
$442$	−31.8265	−1.51383
$443$	−22.4060	−1.06454	−0.532270	−	0.846575i	$-0.678661\pi$
$443$	−22.4060	−1.06454	−0.532270	+	0.846575i	$0.678661\pi$
$444$	0	0
$445$	−38.2693	−1.81414
$446$	2.06160	0.0976194
$447$	0	0
$448$	0.795187	0.0375691
$449$	−39.2309	−1.85142	−0.925709	−	0.378236i	$-0.876531\pi$
$449$	−39.2309	−1.85142	−0.925709	+	0.378236i	$0.876531\pi$
$450$	0	0
$451$	11.4923	0.541152
$452$	15.2121	0.715519
$453$	0	0
$454$	43.9838	2.06426
$455$	−0.904799	−0.0424176
$456$	0	0
$457$	−16.6677	−0.779683	−0.389841	−	0.920882i	$-0.627470\pi$
$457$	−16.6677	−0.779683	−0.389841	+	0.920882i	$0.627470\pi$
$458$	−28.5200	−1.33265
$459$	0	0
$460$	−4.73138	−0.220602
$461$	22.9378	1.06832	0.534159	−	0.845384i	$-0.320628\pi$
$461$	22.9378	1.06832	0.534159	+	0.845384i	$0.320628\pi$
$462$	0	0
$463$	−34.8710	−1.62059	−0.810295	−	0.586022i	$-0.800693\pi$
$463$	−34.8710	−1.62059	−0.810295	+	0.586022i	$0.800693\pi$
$464$	−4.03269	−0.187213
$465$	0	0
$466$	47.2271	2.18775
$467$	−22.6484	−1.04804	−0.524020	−	0.851706i	$-0.675568\pi$
$467$	−22.6484	−1.04804	−0.524020	+	0.851706i	$0.675568\pi$
$468$	0	0
$469$	1.43030	0.0660453
$470$	−30.4151	−1.40294
$471$	0	0
$472$	−0.170119	−0.00783034
$473$	46.3801	2.13256
$474$	0	0
$475$	−2.17684	−0.0998802
$476$	−0.851813	−0.0390428
$477$	0	0
$478$	−6.41100	−0.293232
$479$	0.577663	0.0263941	0.0131970	−	0.999913i	$-0.495799\pi$
$479$	0.577663	0.0263941	0.0131970	+	0.999913i	$0.495799\pi$
$480$	0	0
$481$	−24.0460	−1.09641
$482$	−6.04709	−0.275437
$483$	0	0
$484$	6.67962	0.303619
$485$	22.2280	1.00932
$486$	0	0
$487$	−1.41447	−0.0640958	−0.0320479	−	0.999486i	$-0.510203\pi$
$487$	−1.41447	−0.0640958	−0.0320479	+	0.999486i	$0.510203\pi$
$488$	−0.409304	−0.0185283
$489$	0	0
$490$	33.2774	1.50332
$491$	−26.8889	−1.21348	−0.606739	−	0.794901i	$-0.707523\pi$
$491$	−26.8889	−1.21348	−0.606739	+	0.794901i	$0.707523\pi$
$492$	0	0
$493$	4.24895	0.191363
$494$	23.6352	1.06340
$495$	0	0
$496$	−3.65024	−0.163901
$497$	1.47564	0.0661914
$498$	0	0
$499$	7.71201	0.345237	0.172619	−	0.984989i	$-0.444777\pi$
$499$	7.71201	0.345237	0.172619	+	0.984989i	$0.444777\pi$
$500$	20.3928	0.911994
$501$	0	0
$502$	31.3557	1.39947
$503$	−35.1399	−1.56681	−0.783406	−	0.621510i	$-0.786520\pi$
$503$	−35.1399	−1.56681	−0.783406	+	0.621510i	$0.786520\pi$
$504$	0	0
$505$	−23.1084	−1.02831
$506$	7.56528	0.336318
$507$	0	0
$508$	30.1223	1.33646
$509$	−25.6109	−1.13519	−0.567593	−	0.823310i	$-0.692125\pi$
$509$	−25.6109	−1.13519	−0.567593	+	0.823310i	$0.692125\pi$
$510$	0	0
$511$	−0.672497	−0.0297495
$512$	31.9275	1.41101
$513$	0	0
$514$	48.1950	2.12579
$515$	4.40439	0.194081
$516$	0	0
$517$	24.2156	1.06500
$518$	−1.29250	−0.0567890
$519$	0	0
$520$	−0.294472	−0.0129134
$521$	−13.0594	−0.572141	−0.286070	−	0.958209i	$-0.592349\pi$
$521$	−13.0594	−0.572141	−0.286070	+	0.958209i	$0.592349\pi$
$522$	0	0
$523$	18.6499	0.815503	0.407751	−	0.913093i	$-0.366313\pi$
$523$	18.6499	0.815503	0.407751	+	0.913093i	$0.366313\pi$
$524$	−15.3441	−0.670309
$525$	0	0
$526$	−50.2342	−2.19031
$527$	3.84599	0.167534
$528$	0	0
$529$	1.00000	0.0434783
$530$	−33.6882	−1.46332
$531$	0	0
$532$	0.632578	0.0274257
$533$	−11.3786	−0.492864
$534$	0	0
$535$	18.2544	0.789205
$536$	0.465500	0.0201065
$537$	0	0
$538$	28.7329	1.23876
$539$	−26.4945	−1.14120
$540$	0	0
$541$	44.3431	1.90646	0.953229	−	0.302249i	$-0.0977373\pi$
$541$	44.3431	1.90646	0.953229	+	0.302249i	$0.0977373\pi$
$542$	36.4814	1.56701
$543$	0	0
$544$	−33.9192	−1.45427
$545$	−10.4360	−0.447028
$546$	0	0
$547$	−25.6174	−1.09532	−0.547660	−	0.836701i	$-0.684481\pi$
$547$	−25.6174	−1.09532	−0.547660	+	0.836701i	$0.684481\pi$
$548$	−30.8864	−1.31940
$549$	0	0
$550$	5.21915	0.222545
$551$	−3.15538	−0.134424
$552$	0	0
$553$	−0.789930	−0.0335912
$554$	−17.0794	−0.725633
$555$	0	0
$556$	−5.39360	−0.228740
$557$	24.9875	1.05875	0.529377	−	0.848387i	$-0.322426\pi$
$557$	24.9875	1.05875	0.529377	+	0.848387i	$0.322426\pi$
$558$	0	0
$559$	−45.9213	−1.94226
$560$	−0.972238	−0.0410846
$561$	0	0
$562$	10.1612	0.428625
$563$	7.02084	0.295893	0.147947	−	0.988995i	$-0.452734\pi$
$563$	7.02084	0.295893	0.147947	+	0.988995i	$0.452734\pi$
$564$	0	0
$565$	−18.2939	−0.769629
$566$	−4.25290	−0.178763
$567$	0	0
$568$	0.480254	0.0201510
$569$	13.6405	0.571841	0.285921	−	0.958253i	$-0.407701\pi$
$569$	13.6405	0.571841	0.285921	+	0.958253i	$0.407701\pi$
$570$	0	0
$571$	−5.65592	−0.236693	−0.118346	−	0.992972i	$-0.537759\pi$
$571$	−5.65592	−0.236693	−0.118346	+	0.992972i	$0.537759\pi$
$572$	−28.2164	−1.17979
$573$	0	0
$574$	−0.611612	−0.0255282
$575$	0.689882	0.0287701
$576$	0	0
$577$	−35.5333	−1.47927	−0.739636	−	0.673007i	$-0.765002\pi$
$577$	−35.5333	−1.47927	−0.739636	+	0.673007i	$0.765002\pi$
$578$	2.10281	0.0874655
$579$	0	0
$580$	−4.73138	−0.196460
$581$	−0.918406	−0.0381019
$582$	0	0
$583$	26.8216	1.11084
$584$	−0.218867	−0.00905680
$585$	0	0
$586$	−65.0285	−2.68630
$587$	−34.4318	−1.42115	−0.710577	−	0.703619i	$-0.751566\pi$
$587$	−34.4318	−1.42115	−0.710577	+	0.703619i	$0.751566\pi$
$588$	0	0
$589$	−2.85613	−0.117685
$590$	−24.6218	−1.01366
$591$	0	0
$592$	−25.8383	−1.06195
$593$	−34.0692	−1.39905	−0.699526	−	0.714607i	$-0.746606\pi$
$593$	−34.0692	−1.39905	−0.699526	+	0.714607i	$0.746606\pi$
$594$	0	0
$595$	1.02438	0.0419953
$596$	−38.2431	−1.56650
$597$	0	0
$598$	−7.49045	−0.306307
$599$	16.7419	0.684055	0.342027	−	0.939690i	$-0.388886\pi$
$599$	16.7419	0.684055	0.342027	+	0.939690i	$0.388886\pi$
$600$	0	0
$601$	6.88580	0.280878	0.140439	−	0.990089i	$-0.455149\pi$
$601$	6.88580	0.280878	0.140439	+	0.990089i	$0.455149\pi$
$602$	−2.46831	−0.100601
$603$	0	0
$604$	16.6539	0.677636
$605$	−8.03280	−0.326580
$606$	0	0
$607$	11.0359	0.447932	0.223966	−	0.974597i	$-0.428100\pi$
$607$	11.0359	0.447932	0.223966	+	0.974597i	$0.428100\pi$
$608$	25.1893	1.02156
$609$	0	0
$610$	−59.2398	−2.39855
$611$	−23.9761	−0.969969
$612$	0	0
$613$	−27.2076	−1.09891	−0.549453	−	0.835524i	$-0.685164\pi$
$613$	−27.2076	−1.09891	−0.549453	+	0.835524i	$0.685164\pi$
$614$	−47.2760	−1.90790
$615$	0	0
$616$	0.0126019	0.000507744 0
$617$	36.1376	1.45484	0.727422	−	0.686191i	$-0.240719\pi$
$617$	36.1376	1.45484	0.727422	+	0.686191i	$0.240719\pi$
$618$	0	0
$619$	−38.3191	−1.54017	−0.770087	−	0.637939i	$-0.779787\pi$
$619$	−38.3191	−1.54017	−0.770087	+	0.637939i	$0.779787\pi$
$620$	−4.28267	−0.171996
$621$	0	0
$622$	56.2691	2.25619
$623$	−1.62153	−0.0649652
$624$	0	0
$625$	−27.9735	−1.11894
$626$	−13.5941	−0.543329
$627$	0	0
$628$	−22.3437	−0.891611
$629$	27.2239	1.08549
$630$	0	0
$631$	−5.15108	−0.205061	−0.102531	−	0.994730i	$-0.532694\pi$
$631$	−5.15108	−0.205061	−0.102531	+	0.994730i	$0.532694\pi$
$632$	−0.257087	−0.0102264
$633$	0	0
$634$	−31.9188	−1.26766
$635$	−36.2246	−1.43753
$636$	0	0
$637$	26.2324	1.03937
$638$	7.56528	0.299512
$639$	0	0
$640$	−0.627689	−0.0248116
$641$	27.0606	1.06883	0.534415	−	0.845223i	$-0.320532\pi$
$641$	27.0606	1.06883	0.534415	+	0.845223i	$0.320532\pi$
$642$	0	0
$643$	10.5064	0.414334	0.207167	−	0.978306i	$-0.433576\pi$
$643$	10.5064	0.414334	0.207167	+	0.978306i	$0.433576\pi$
$644$	−0.200476	−0.00789987
$645$	0	0
$646$	−26.7588	−1.05281
$647$	11.9116	0.468293	0.234146	−	0.972201i	$-0.424771\pi$
$647$	11.9116	0.468293	0.234146	+	0.972201i	$0.424771\pi$
$648$	0	0
$649$	19.6032	0.769491
$650$	−5.16753	−0.202687
$651$	0	0
$652$	22.9930	0.900474
$653$	4.08257	0.159763	0.0798816	−	0.996804i	$-0.474546\pi$
$653$	4.08257	0.159763	0.0798816	+	0.996804i	$0.474546\pi$
$654$	0	0
$655$	18.4525	0.721001
$656$	−12.2268	−0.477374
$657$	0	0
$658$	−1.28874	−0.0502401
$659$	−8.67462	−0.337915	−0.168958	−	0.985623i	$-0.554040\pi$
$659$	−8.67462	−0.337915	−0.168958	+	0.985623i	$0.554040\pi$
$660$	0	0
$661$	−28.3990	−1.10459	−0.552296	−	0.833648i	$-0.686248\pi$
$661$	−28.3990	−1.10459	−0.552296	+	0.833648i	$0.686248\pi$
$662$	48.9088	1.90089
$663$	0	0
$664$	−0.298900	−0.0115996
$665$	−0.760728	−0.0294998
$666$	0	0
$667$	1.00000	0.0387202
$668$	38.4299	1.48690
$669$	0	0
$670$	67.3732	2.60285
$671$	47.1650	1.82079
$672$	0	0
$673$	4.64059	0.178882	0.0894409	−	0.995992i	$-0.471492\pi$
$673$	4.64059	0.178882	0.0894409	+	0.995992i	$0.471492\pi$
$674$	−14.9581	−0.576164
$675$	0	0
$676$	2.15160	0.0827538
$677$	46.4808	1.78640	0.893201	−	0.449658i	$-0.148454\pi$
$677$	46.4808	1.78640	0.893201	+	0.449658i	$0.148454\pi$
$678$	0	0
$679$	0.941837	0.0361444
$680$	0.333388	0.0127849
$681$	0	0
$682$	6.84781	0.262216
$683$	−43.7092	−1.67249	−0.836243	−	0.548358i	$-0.815253\pi$
$683$	−43.7092	−1.67249	−0.836243	+	0.548358i	$0.815253\pi$
$684$	0	0
$685$	37.1435	1.41918
$686$	2.82209	0.107748
$687$	0	0
$688$	−49.3440	−1.88122
$689$	−26.5563	−1.01171
$690$	0	0
$691$	5.34696	0.203408	0.101704	−	0.994815i	$-0.467571\pi$
$691$	5.34696	0.203408	0.101704	+	0.994815i	$0.467571\pi$
$692$	1.88344	0.0715977
$693$	0	0
$694$	33.8852	1.28626
$695$	6.48626	0.246038
$696$	0	0
$697$	12.8824	0.487957
$698$	39.7225	1.50352
$699$	0	0
$700$	−0.138305	−0.00522743
$701$	−4.65925	−0.175977	−0.0879887	−	0.996121i	$-0.528044\pi$
$701$	−4.65925	−0.175977	−0.0879887	+	0.996121i	$0.528044\pi$
$702$	0	0
$703$	−20.2172	−0.762505
$704$	−29.8218	−1.12395
$705$	0	0
$706$	−6.01085	−0.226221
$707$	−0.979139	−0.0368243
$708$	0	0
$709$	−23.9500	−0.899461	−0.449731	−	0.893164i	$-0.648480\pi$
$709$	−23.9500	−0.899461	−0.449731	+	0.893164i	$0.648480\pi$
$710$	69.5086	2.60861
$711$	0	0
$712$	−0.527735	−0.0197777
$713$	0.905162	0.0338986
$714$	0	0
$715$	33.9326	1.26901
$716$	−15.3626	−0.574126
$717$	0	0
$718$	−48.5146	−1.81055
$719$	11.6404	0.434115	0.217058	−	0.976159i	$-0.430354\pi$
$719$	11.6404	0.434115	0.217058	+	0.976159i	$0.430354\pi$
$720$	0	0
$721$	0.186621	0.00695013
$722$	−18.0499	−0.671747
$723$	0	0
$724$	22.0993	0.821313
$725$	0.689882	0.0256216
$726$	0	0
$727$	45.4682	1.68632	0.843160	−	0.537662i	$-0.180692\pi$
$727$	45.4682	1.68632	0.843160	+	0.537662i	$0.180692\pi$
$728$	−0.0124772	−0.000462437 0
$729$	0	0
$730$	−31.6774	−1.17243
$731$	51.9902	1.92293
$732$	0	0
$733$	−17.7104	−0.654150	−0.327075	−	0.944998i	$-0.606063\pi$
$733$	−17.7104	−0.654150	−0.327075	+	0.944998i	$0.606063\pi$
$734$	−32.6591	−1.20547
$735$	0	0
$736$	−7.98296	−0.294256
$737$	−53.6406	−1.97588
$738$	0	0
$739$	−28.6770	−1.05490	−0.527450	−	0.849586i	$-0.676852\pi$
$739$	−28.6770	−1.05490	−0.527450	+	0.849586i	$0.676852\pi$
$740$	−30.3150	−1.11440
$741$	0	0
$742$	−1.42742	−0.0524023
$743$	−0.0217181	−0.000796760 0	−0.000398380	−	1.00000i	$-0.500127\pi$
$743$	−0.0217181	−0.000796760 0	−0.000398380		1.00000i	$0.500127\pi$
$744$	0	0
$745$	45.9906	1.68496
$746$	−58.8900	−2.15612
$747$	0	0
$748$	31.9455	1.16804
$749$	0.773466	0.0282618
$750$	0	0
$751$	−18.5858	−0.678205	−0.339102	−	0.940750i	$-0.610123\pi$
$751$	−18.5858	−0.678205	−0.339102	+	0.940750i	$0.610123\pi$
$752$	−25.7631	−0.939485
$753$	0	0
$754$	−7.49045	−0.272786
$755$	−20.0277	−0.728882
$756$	0	0
$757$	−17.1901	−0.624786	−0.312393	−	0.949953i	$-0.601131\pi$
$757$	−17.1901	−0.624786	−0.312393	+	0.949953i	$0.601131\pi$
$758$	19.2903	0.700654
$759$	0	0
$760$	−0.247583	−0.00898077
$761$	17.6927	0.641359	0.320680	−	0.947188i	$-0.396089\pi$
$761$	17.6927	0.641359	0.320680	+	0.947188i	$0.396089\pi$
$762$	0	0
$763$	−0.442188	−0.0160083
$764$	14.9067	0.539305
$765$	0	0
$766$	0.482170	0.0174215
$767$	−19.4093	−0.700828
$768$	0	0
$769$	−22.5546	−0.813340	−0.406670	−	0.913575i	$-0.633310\pi$
$769$	−22.5546	−0.813340	−0.406670	+	0.913575i	$0.633310\pi$
$770$	1.82391	0.0657291
$771$	0	0
$772$	−24.6519	−0.887242
$773$	50.5024	1.81644	0.908222	−	0.418488i	$-0.137440\pi$
$773$	50.5024	1.81644	0.908222	+	0.418488i	$0.137440\pi$
$774$	0	0
$775$	0.624455	0.0224311
$776$	0.306526	0.0110036
$777$	0	0
$778$	−10.4608	−0.375039
$779$	−9.56682	−0.342767
$780$	0	0
$781$	−55.3408	−1.98025
$782$	8.48038	0.303258
$783$	0	0
$784$	28.1876	1.00670
$785$	26.8702	0.959038
$786$	0	0
$787$	23.7857	0.847869	0.423935	−	0.905693i	$-0.360649\pi$
$787$	23.7857	0.847869	0.423935	+	0.905693i	$0.360649\pi$
$788$	39.2174	1.39706
$789$	0	0
$790$	−37.2089	−1.32383
$791$	−0.775141	−0.0275608
$792$	0	0
$793$	−46.6985	−1.65831
$794$	11.3430	0.402549
$795$	0	0
$796$	40.2503	1.42663
$797$	16.5845	0.587454	0.293727	−	0.955889i	$-0.405104\pi$
$797$	16.5845	0.587454	0.293727	+	0.955889i	$0.405104\pi$
$798$	0	0
$799$	27.1447	0.960312
$800$	−5.50730	−0.194712
$801$	0	0
$802$	−19.1617	−0.676623
$803$	25.2206	0.890016
$804$	0	0
$805$	0.241089	0.00849728
$806$	−6.78007	−0.238818
$807$	0	0
$808$	−0.318666	−0.0112106
$809$	−16.0541	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$809$	−16.0541	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$810$	0	0
$811$	15.3936	0.540542	0.270271	−	0.962784i	$-0.412887\pi$
$811$	15.3936	0.540542	0.270271	+	0.962784i	$0.412887\pi$
$812$	−0.200476	−0.00703533
$813$	0	0
$814$	48.4724	1.69896
$815$	−27.6510	−0.968571
$816$	0	0
$817$	−38.6092	−1.35077
$818$	47.0587	1.64537
$819$	0	0
$820$	−14.3451	−0.500953
$821$	9.41577	0.328613	0.164306	−	0.986409i	$-0.447461\pi$
$821$	9.41577	0.328613	0.164306	+	0.986409i	$0.447461\pi$
$822$	0	0
$823$	−22.0685	−0.769258	−0.384629	−	0.923071i	$-0.625671\pi$
$823$	−22.0685	−0.769258	−0.384629	+	0.923071i	$0.625671\pi$
$824$	0.0607368	0.00211587
$825$	0	0
$826$	−1.04326	−0.0362998
$827$	55.7825	1.93975	0.969874	−	0.243606i	$-0.0783302\pi$
$827$	55.7825	1.93975	0.969874	+	0.243606i	$0.0783302\pi$
$828$	0	0
$829$	−44.1904	−1.53479	−0.767397	−	0.641172i	$-0.778449\pi$
$829$	−44.1904	−1.53479	−0.767397	+	0.641172i	$0.778449\pi$
$830$	−43.2607	−1.50160
$831$	0	0
$832$	29.5269	1.02366
$833$	−29.6992	−1.02902
$834$	0	0
$835$	−46.2151	−1.59934
$836$	−23.7235	−0.820495
$837$	0	0
$838$	−6.73295	−0.232586
$839$	2.77141	0.0956797	0.0478398	−	0.998855i	$-0.484766\pi$
$839$	2.77141	0.0956797	0.0478398	+	0.998855i	$0.484766\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	13.3514	0.460118
$843$	0	0
$844$	−6.59971	−0.227172
$845$	−2.58748	−0.0890119
$846$	0	0
$847$	−0.340362	−0.0116950
$848$	−28.5356	−0.979917
$849$	0	0
$850$	5.85046	0.200669
$851$	6.40721	0.219636
$852$	0	0
$853$	−5.58032	−0.191066	−0.0955332	−	0.995426i	$-0.530456\pi$
$853$	−5.58032	−0.191066	−0.0955332	+	0.995426i	$0.530456\pi$
$854$	−2.51009	−0.0858933
$855$	0	0
$856$	0.251728	0.00860390
$857$	47.8293	1.63382	0.816910	−	0.576766i	$-0.195685\pi$
$857$	47.8293	1.63382	0.816910	+	0.576766i	$0.195685\pi$
$858$	0	0
$859$	−24.0567	−0.820803	−0.410401	−	0.911905i	$-0.634611\pi$
$859$	−24.0567	−0.820803	−0.410401	+	0.911905i	$0.634611\pi$
$860$	−57.8932	−1.97414
$861$	0	0
$862$	57.4336	1.95620
$863$	−34.7220	−1.18195	−0.590975	−	0.806690i	$-0.701257\pi$
$863$	−34.7220	−1.18195	−0.590975	+	0.806690i	$0.701257\pi$
$864$	0	0
$865$	−2.26500	−0.0770122
$866$	−16.0470	−0.545301
$867$	0	0
$868$	−0.181463	−0.00615927
$869$	29.6247	1.00495
$870$	0	0
$871$	53.1100	1.79956
$872$	−0.143912	−0.00487349
$873$	0	0
$874$	−6.29774	−0.213024
$875$	−1.03912	−0.0351288
$876$	0	0
$877$	11.5907	0.391391	0.195695	−	0.980665i	$-0.437304\pi$
$877$	11.5907	0.391391	0.195695	+	0.980665i	$0.437304\pi$
$878$	9.71702	0.327934
$879$	0	0
$880$	36.4618	1.22913
$881$	11.8160	0.398091	0.199045	−	0.979990i	$-0.436216\pi$
$881$	11.8160	0.398091	0.199045	+	0.979990i	$0.436216\pi$
$882$	0	0
$883$	41.1493	1.38478	0.692392	−	0.721521i	$-0.256557\pi$
$883$	41.1493	1.38478	0.692392	+	0.721521i	$0.256557\pi$
$884$	−31.6295	−1.06382
$885$	0	0
$886$	−44.7195	−1.50238
$887$	−47.3816	−1.59092	−0.795459	−	0.606007i	$-0.792770\pi$
$887$	−47.3816	−1.59092	−0.795459	+	0.606007i	$0.792770\pi$
$888$	0	0
$889$	−1.53489	−0.0514787
$890$	−76.3807	−2.56029
$891$	0	0
$892$	2.04883	0.0686000
$893$	−20.1584	−0.674574
$894$	0	0
$895$	18.4748	0.617543
$896$	−0.0265962	−0.000888515 0
$897$	0	0
$898$	−78.2999	−2.61290
$899$	0.905162	0.0301889
$900$	0	0
$901$	30.0659	1.00164
$902$	22.9372	0.763726
$903$	0	0
$904$	−0.252273	−0.00839049
$905$	−26.5762	−0.883424
$906$	0	0
$907$	−2.27361	−0.0754939	−0.0377470	−	0.999287i	$-0.512018\pi$
$907$	−2.27361	−0.0754939	−0.0377470	+	0.999287i	$0.512018\pi$
$908$	43.7115	1.45062
$909$	0	0
$910$	−1.80587	−0.0598639
$911$	33.7327	1.11761	0.558807	−	0.829298i	$-0.311259\pi$
$911$	33.7327	1.11761	0.558807	+	0.829298i	$0.311259\pi$
$912$	0	0
$913$	34.4429	1.13989
$914$	−33.2667	−1.10036
$915$	0	0
$916$	−28.3434	−0.936492
$917$	0.781863	0.0258194
$918$	0	0
$919$	−5.95015	−0.196277	−0.0981387	−	0.995173i	$-0.531289\pi$
$919$	−5.95015	−0.196277	−0.0981387	+	0.995173i	$0.531289\pi$
$920$	0.0784637	0.00258687
$921$	0	0
$922$	45.7809	1.50772
$923$	54.7934	1.80355
$924$	0	0
$925$	4.42022	0.145336
$926$	−69.5981	−2.28714
$927$	0	0
$928$	−7.98296	−0.262053
$929$	−33.9194	−1.11286	−0.556429	−	0.830895i	$-0.687829\pi$
$929$	−33.9194	−1.11286	−0.556429	+	0.830895i	$0.687829\pi$
$930$	0	0
$931$	22.0554	0.722837
$932$	46.9347	1.53740
$933$	0	0
$934$	−45.2033	−1.47910
$935$	−38.4171	−1.25637
$936$	0	0
$937$	−9.09106	−0.296992	−0.148496	−	0.988913i	$-0.547443\pi$
$937$	−9.09106	−0.296992	−0.148496	+	0.988913i	$0.547443\pi$
$938$	2.85471	0.0932095
$939$	0	0
$940$	−30.2268	−0.985889
$941$	−0.863413	−0.0281465	−0.0140732	−	0.999901i	$-0.504480\pi$
$941$	−0.863413	−0.0281465	−0.0140732	+	0.999901i	$0.504480\pi$
$942$	0	0
$943$	3.03191	0.0987325
$944$	−20.8559	−0.678803
$945$	0	0
$946$	92.5688	3.00967
$947$	25.8238	0.839159	0.419580	−	0.907718i	$-0.362177\pi$
$947$	25.8238	0.839159	0.419580	+	0.907718i	$0.362177\pi$
$948$	0	0
$949$	−24.9711	−0.810597
$950$	−4.34470	−0.140961
$951$	0	0
$952$	0.0141262	0.000457833 0
$953$	16.0071	0.518522	0.259261	−	0.965807i	$-0.416521\pi$
$953$	16.0071	0.518522	0.259261	+	0.965807i	$0.416521\pi$
$954$	0	0
$955$	−17.9265	−0.580089
$956$	−6.37131	−0.206063
$957$	0	0
$958$	1.15294	0.0372499
$959$	1.57383	0.0508216
$960$	0	0
$961$	−30.1807	−0.973570
$962$	−47.9929	−1.54735
$963$	0	0
$964$	−6.00965	−0.193558
$965$	29.6460	0.954338
$966$	0	0
$967$	25.8487	0.831238	0.415619	−	0.909539i	$-0.363565\pi$
$967$	25.8487	0.831238	0.415619	+	0.909539i	$0.363565\pi$
$968$	−0.110773	−0.00356037
$969$	0	0
$970$	44.3644	1.42446
$971$	31.7760	1.01974	0.509871	−	0.860251i	$-0.329693\pi$
$971$	31.7760	1.01974	0.509871	+	0.860251i	$0.329693\pi$
$972$	0	0
$973$	0.274833	0.00881074
$974$	−2.82311	−0.0904582
$975$	0	0
$976$	−50.1792	−1.60620
$977$	−23.5914	−0.754754	−0.377377	−	0.926060i	$-0.623174\pi$
$977$	−23.5914	−0.754754	−0.377377	+	0.926060i	$0.623174\pi$
$978$	0	0
$979$	60.8121	1.94356
$980$	33.0713	1.05643
$981$	0	0
$982$	−53.6668	−1.71258
$983$	−7.42071	−0.236684	−0.118342	−	0.992973i	$-0.537758\pi$
$983$	−7.42071	−0.236684	−0.118342	+	0.992973i	$0.537758\pi$
$984$	0	0
$985$	−47.1622	−1.50271
$986$	8.48038	0.270070
$987$	0	0
$988$	23.4889	0.747280
$989$	12.2360	0.389083
$990$	0	0
$991$	−22.4288	−0.712475	−0.356237	−	0.934395i	$-0.615941\pi$
$991$	−22.4288	−0.712475	−0.356237	+	0.934395i	$0.615941\pi$
$992$	−7.22587	−0.229422
$993$	0	0
$994$	2.94519	0.0934158
$995$	−48.4043	−1.53452
$996$	0	0
$997$	2.22873	0.0705846	0.0352923	−	0.999377i	$-0.488764\pi$
$997$	2.22873	0.0705846	0.0352923	+	0.999377i	$0.488764\pi$
$998$	15.3922	0.487232
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.n.1.10		12
3.2	odd	2		667.2.a.b.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.b.1.3	✓	12	3.2	odd	2
6003.2.a.n.1.10		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)

\(f(q)\)	\(=\)	\(q+1.99588 q^{2} +1.98352 q^{4} -2.38535 q^{5} -0.101071 q^{7} -0.0328940 q^{8} +O(q^{10})\)	\(q+1.99588 q^{2} +1.98352 q^{4} -2.38535 q^{5} -0.101071 q^{7} -0.0328940 q^{8} -4.76086 q^{10} +3.79046 q^{11} -3.75296 q^{13} -0.201725 q^{14} -4.03269 q^{16} +4.24895 q^{17} -3.15538 q^{19} -4.73138 q^{20} +7.56528 q^{22} +1.00000 q^{23} +0.689882 q^{25} -7.49045 q^{26} -0.200476 q^{28} +1.00000 q^{29} +0.905162 q^{31} -7.98296 q^{32} +8.48038 q^{34} +0.241089 q^{35} +6.40721 q^{37} -6.29774 q^{38} +0.0784637 q^{40} +3.03191 q^{41} +12.2360 q^{43} +7.51844 q^{44} +1.99588 q^{46} +6.38858 q^{47} -6.98978 q^{49} +1.37692 q^{50} -7.44408 q^{52} +7.07607 q^{53} -9.04156 q^{55} +0.00332463 q^{56} +1.99588 q^{58} +5.17172 q^{59} +12.4431 q^{61} +1.80659 q^{62} -7.86761 q^{64} +8.95212 q^{65} -14.1515 q^{67} +8.42787 q^{68} +0.481184 q^{70} -14.6000 q^{71} +6.65371 q^{73} +12.7880 q^{74} -6.25875 q^{76} -0.383105 q^{77} +7.81560 q^{79} +9.61937 q^{80} +6.05131 q^{82} +9.08675 q^{83} -10.1352 q^{85} +24.4216 q^{86} -0.124683 q^{88} +16.0435 q^{89} +0.379316 q^{91} +1.98352 q^{92} +12.7508 q^{94} +7.52667 q^{95} -9.31858 q^{97} -13.9507 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more