LMFDB - Embedded newform 6003.2.a.p.1.3

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			1.3
Root			$-1.86019$ 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)

$f(q)$	$=$	$q-1.86019 q^{2} +1.46029 q^{4} -0.608499 q^{5} +0.188796 q^{7} +1.00396 q^{8} +O(q^{10})$	\(q-1.86019 q^{2} +1.46029 q^{4} -0.608499 q^{5} +0.188796 q^{7} +1.00396 q^{8} +1.13192 q^{10} +1.80879 q^{11} -2.52247 q^{13} -0.351195 q^{14} -4.78813 q^{16} -0.572412 q^{17} -5.70723 q^{19} -0.888584 q^{20} -3.36468 q^{22} +1.00000 q^{23} -4.62973 q^{25} +4.69225 q^{26} +0.275696 q^{28} +1.00000 q^{29} +5.66327 q^{31} +6.89889 q^{32} +1.06479 q^{34} -0.114882 q^{35} -2.76093 q^{37} +10.6165 q^{38} -0.610909 q^{40} -5.30448 q^{41} -2.88895 q^{43} +2.64136 q^{44} -1.86019 q^{46} +7.76654 q^{47} -6.96436 q^{49} +8.61215 q^{50} -3.68353 q^{52} -9.05100 q^{53} -1.10065 q^{55} +0.189544 q^{56} -1.86019 q^{58} +1.78309 q^{59} -4.85481 q^{61} -10.5347 q^{62} -3.25695 q^{64} +1.53492 q^{65} +6.01571 q^{67} -0.835888 q^{68} +0.213702 q^{70} +12.3521 q^{71} +3.12254 q^{73} +5.13585 q^{74} -8.33421 q^{76} +0.341492 q^{77} +6.66648 q^{79} +2.91357 q^{80} +9.86731 q^{82} +17.9397 q^{83} +0.348312 q^{85} +5.37399 q^{86} +1.81596 q^{88} +12.9637 q^{89} -0.476231 q^{91} +1.46029 q^{92} -14.4472 q^{94} +3.47284 q^{95} -13.5092 q^{97} +12.9550 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.86019	−1.31535	−0.657675	−	0.753302i	$-0.728460\pi$
$2$	−1.86019	−1.31535	−0.657675	+	0.753302i	$0.728460\pi$
$3$	0	0
$3$	0	0
$4$	1.46029	0.730145
$5$	−0.608499	−0.272129	−0.136064	−	0.990700i	$-0.543445\pi$
$5$	−0.608499	−0.272129	−0.136064	+	0.990700i	$0.543445\pi$
$6$	0	0
$7$	0.188796	0.0713581	0.0356790	−	0.999363i	$-0.488641\pi$
$7$	0.188796	0.0713581	0.0356790	+	0.999363i	$0.488641\pi$
$8$	1.00396	0.354954
$9$	0	0
$10$	1.13192	0.357945
$11$	1.80879	0.545371	0.272685	−	0.962103i	$-0.412088\pi$
$11$	1.80879	0.545371	0.272685	+	0.962103i	$0.412088\pi$
$12$	0	0
$13$	−2.52247	−0.699606	−0.349803	−	0.936823i	$-0.613751\pi$
$13$	−2.52247	−0.699606	−0.349803	+	0.936823i	$0.613751\pi$
$14$	−0.351195	−0.0938608
$15$	0	0
$16$	−4.78813	−1.19703
$17$	−0.572412	−0.138830	−0.0694152	−	0.997588i	$-0.522113\pi$
$17$	−0.572412	−0.138830	−0.0694152	+	0.997588i	$0.522113\pi$
$18$	0	0
$19$	−5.70723	−1.30933	−0.654664	−	0.755920i	$-0.727190\pi$
$19$	−5.70723	−1.30933	−0.654664	+	0.755920i	$0.727190\pi$
$20$	−0.888584	−0.198693
$21$	0	0
$22$	−3.36468	−0.717353
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.62973	−0.925946
$26$	4.69225	0.920227
$27$	0	0
$28$	0.275696	0.0521017
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.66327	1.01715	0.508577	−	0.861017i	$-0.330172\pi$
$31$	5.66327	1.01715	0.508577	+	0.861017i	$0.330172\pi$
$32$	6.89889	1.21956
$33$	0	0
$34$	1.06479	0.182611
$35$	−0.114882	−0.0194186
$36$	0	0
$37$	−2.76093	−0.453895	−0.226947	−	0.973907i	$-0.572875\pi$
$37$	−2.76093	−0.453895	−0.226947	+	0.973907i	$0.572875\pi$
$38$	10.6165	1.72222
$39$	0	0
$40$	−0.610909	−0.0965932
$41$	−5.30448	−0.828420	−0.414210	−	0.910181i	$-0.635942\pi$
$41$	−5.30448	−0.828420	−0.414210	+	0.910181i	$0.635942\pi$
$42$	0	0
$43$	−2.88895	−0.440561	−0.220281	−	0.975437i	$-0.570697\pi$
$43$	−2.88895	−0.440561	−0.220281	+	0.975437i	$0.570697\pi$
$44$	2.64136	0.398200
$45$	0	0
$46$	−1.86019	−0.274269
$47$	7.76654	1.13287	0.566433	−	0.824108i	$-0.308323\pi$
$47$	7.76654	1.13287	0.566433	+	0.824108i	$0.308323\pi$
$48$	0	0
$49$	−6.96436	−0.994908
$50$	8.61215	1.21794
$51$	0	0
$52$	−3.68353	−0.510814
$53$	−9.05100	−1.24325	−0.621625	−	0.783315i	$-0.713527\pi$
$53$	−9.05100	−1.24325	−0.621625	+	0.783315i	$0.713527\pi$
$54$	0	0
$55$	−1.10065	−0.148411
$56$	0.189544	0.0253288
$57$	0	0
$58$	−1.86019	−0.244254
$59$	1.78309	0.232138	0.116069	−	0.993241i	$-0.462971\pi$
$59$	1.78309	0.232138	0.116069	+	0.993241i	$0.462971\pi$
$60$	0	0
$61$	−4.85481	−0.621595	−0.310797	−	0.950476i	$-0.600596\pi$
$61$	−4.85481	−0.621595	−0.310797	+	0.950476i	$0.600596\pi$
$62$	−10.5347	−1.33791
$63$	0	0
$64$	−3.25695	−0.407119
$65$	1.53492	0.190383
$66$	0	0
$67$	6.01571	0.734936	0.367468	−	0.930036i	$-0.380225\pi$
$67$	6.01571	0.734936	0.367468	+	0.930036i	$0.380225\pi$
$68$	−0.835888	−0.101366
$69$	0	0
$70$	0.213702	0.0255422
$71$	12.3521	1.46592	0.732962	−	0.680270i	$-0.238137\pi$
$71$	12.3521	1.46592	0.732962	+	0.680270i	$0.238137\pi$
$72$	0	0
$73$	3.12254	0.365465	0.182733	−	0.983163i	$-0.441506\pi$
$73$	3.12254	0.365465	0.182733	+	0.983163i	$0.441506\pi$
$74$	5.13585	0.597030
$75$	0	0
$76$	−8.33421	−0.955999
$77$	0.341492	0.0389166
$78$	0	0
$79$	6.66648	0.750038	0.375019	−	0.927017i	$-0.377636\pi$
$79$	6.66648	0.750038	0.375019	+	0.927017i	$0.377636\pi$
$80$	2.91357	0.325747
$81$	0	0
$82$	9.86731	1.08966
$83$	17.9397	1.96913	0.984567	−	0.175008i	$-0.0559951\pi$
$83$	17.9397	1.96913	0.984567	+	0.175008i	$0.0559951\pi$
$84$	0	0
$85$	0.348312	0.0377798
$86$	5.37399	0.579492
$87$	0	0
$88$	1.81596	0.193581
$89$	12.9637	1.37414	0.687072	−	0.726589i	$-0.258896\pi$
$89$	12.9637	1.37414	0.687072	+	0.726589i	$0.258896\pi$
$90$	0	0
$91$	−0.476231	−0.0499225
$92$	1.46029	0.152246
$93$	0	0
$94$	−14.4472	−1.49012
$95$	3.47284	0.356306
$96$	0	0
$97$	−13.5092	−1.37166	−0.685828	−	0.727764i	$-0.740560\pi$
$97$	−13.5092	−1.37166	−0.685828	+	0.727764i	$0.740560\pi$
$98$	12.9550	1.30865
$99$	0	0
$100$	−6.76075	−0.676075
$101$	−15.1995	−1.51240	−0.756201	−	0.654339i	$-0.772947\pi$
$101$	−15.1995	−1.51240	−0.756201	+	0.654339i	$0.772947\pi$
$102$	0	0
$103$	6.36554	0.627215	0.313607	−	0.949553i	$-0.398462\pi$
$103$	6.36554	0.627215	0.313607	+	0.949553i	$0.398462\pi$
$104$	−2.53246	−0.248328
$105$	0	0
$106$	16.8365	1.63531
$107$	−1.41398	−0.136695	−0.0683475	−	0.997662i	$-0.521773\pi$
$107$	−1.41398	−0.136695	−0.0683475	+	0.997662i	$0.521773\pi$
$108$	0	0
$109$	10.6935	1.02425	0.512126	−	0.858910i	$-0.328858\pi$
$109$	10.6935	1.02425	0.512126	+	0.858910i	$0.328858\pi$
$110$	2.04741	0.195212
$111$	0	0
$112$	−0.903979	−0.0854180
$113$	−12.9102	−1.21449	−0.607243	−	0.794517i	$-0.707724\pi$
$113$	−12.9102	−1.21449	−0.607243	+	0.794517i	$0.707724\pi$
$114$	0	0
$115$	−0.608499	−0.0567428
$116$	1.46029	0.135584
$117$	0	0
$118$	−3.31687	−0.305343
$119$	−0.108069	−0.00990667
$120$	0	0
$121$	−7.72828	−0.702571
$122$	9.03085	0.817614
$123$	0	0
$124$	8.27002	0.742669
$125$	5.85968	0.524106
$126$	0	0
$127$	−9.69089	−0.859928	−0.429964	−	0.902846i	$-0.641474\pi$
$127$	−9.69089	−0.859928	−0.429964	+	0.902846i	$0.641474\pi$
$128$	−7.73925	−0.684060
$129$	0	0
$130$	−2.85523	−0.250420
$131$	13.5559	1.18439	0.592193	−	0.805796i	$-0.298262\pi$
$131$	13.5559	1.18439	0.592193	+	0.805796i	$0.298262\pi$
$132$	0	0
$133$	−1.07750	−0.0934312
$134$	−11.1903	−0.966698
$135$	0	0
$136$	−0.574680	−0.0492784
$137$	9.36429	0.800045	0.400022	−	0.916505i	$-0.369002\pi$
$137$	9.36429	0.800045	0.400022	+	0.916505i	$0.369002\pi$
$138$	0	0
$139$	19.2469	1.63250	0.816249	−	0.577700i	$-0.196050\pi$
$139$	19.2469	1.63250	0.816249	+	0.577700i	$0.196050\pi$
$140$	−0.167761	−0.0141784
$141$	0	0
$142$	−22.9772	−1.92820
$143$	−4.56261	−0.381545
$144$	0	0
$145$	−0.608499	−0.0505331
$146$	−5.80849	−0.480714
$147$	0	0
$148$	−4.03176	−0.331409
$149$	0.321494	0.0263378	0.0131689	−	0.999913i	$-0.495808\pi$
$149$	0.321494	0.0263378	0.0131689	+	0.999913i	$0.495808\pi$
$150$	0	0
$151$	−18.1531	−1.47727	−0.738637	−	0.674103i	$-0.764530\pi$
$151$	−18.1531	−1.47727	−0.738637	+	0.674103i	$0.764530\pi$
$152$	−5.72984	−0.464751
$153$	0	0
$154$	−0.635238	−0.0511889
$155$	−3.44609	−0.276797
$156$	0	0
$157$	−7.10067	−0.566696	−0.283348	−	0.959017i	$-0.591445\pi$
$157$	−7.10067	−0.566696	−0.283348	+	0.959017i	$0.591445\pi$
$158$	−12.4009	−0.986562
$159$	0	0
$160$	−4.19797	−0.331878
$161$	0.188796	0.0148792
$162$	0	0
$163$	18.7152	1.46589	0.732945	−	0.680288i	$-0.238145\pi$
$163$	18.7152	1.46589	0.732945	+	0.680288i	$0.238145\pi$
$164$	−7.74607	−0.604866
$165$	0	0
$166$	−33.3711	−2.59010
$167$	−7.09180	−0.548780	−0.274390	−	0.961618i	$-0.588476\pi$
$167$	−7.09180	−0.548780	−0.274390	+	0.961618i	$0.588476\pi$
$168$	0	0
$169$	−6.63717	−0.510551
$170$	−0.647925	−0.0496936
$171$	0	0
$172$	−4.21871	−0.321673
$173$	−22.6949	−1.72546	−0.862730	−	0.505665i	$-0.831247\pi$
$173$	−22.6949	−1.72546	−0.862730	+	0.505665i	$0.831247\pi$
$174$	0	0
$175$	−0.874073	−0.0660737
$176$	−8.66073	−0.652827
$177$	0	0
$178$	−24.1148	−1.80748
$179$	−15.4108	−1.15185	−0.575927	−	0.817501i	$-0.695359\pi$
$179$	−15.4108	−1.15185	−0.575927	+	0.817501i	$0.695359\pi$
$180$	0	0
$181$	24.9612	1.85535	0.927676	−	0.373387i	$-0.121804\pi$
$181$	24.9612	1.85535	0.927676	+	0.373387i	$0.121804\pi$
$182$	0.885877	0.0656656
$183$	0	0
$184$	1.00396	0.0740130
$185$	1.68002	0.123518
$186$	0	0
$187$	−1.03537	−0.0757140
$188$	11.3414	0.827157
$189$	0	0
$190$	−6.46013	−0.468667
$191$	−2.23870	−0.161986	−0.0809932	−	0.996715i	$-0.525809\pi$
$191$	−2.23870	−0.161986	−0.0809932	+	0.996715i	$0.525809\pi$
$192$	0	0
$193$	23.4157	1.68550	0.842748	−	0.538309i	$-0.180937\pi$
$193$	23.4157	1.68550	0.842748	+	0.538309i	$0.180937\pi$
$194$	25.1297	1.80421
$195$	0	0
$196$	−10.1700	−0.726427
$197$	12.6453	0.900938	0.450469	−	0.892792i	$-0.351257\pi$
$197$	12.6453	0.900938	0.450469	+	0.892792i	$0.351257\pi$
$198$	0	0
$199$	−22.3193	−1.58217	−0.791087	−	0.611703i	$-0.790485\pi$
$199$	−22.3193	−1.58217	−0.791087	+	0.611703i	$0.790485\pi$
$200$	−4.64807	−0.328668
$201$	0	0
$202$	28.2738	1.98934
$203$	0.188796	0.0132509
$204$	0	0
$205$	3.22777	0.225437
$206$	−11.8411	−0.825007
$207$	0	0
$208$	12.0779	0.837452
$209$	−10.3232	−0.714069
$210$	0	0
$211$	−17.1519	−1.18078	−0.590392	−	0.807117i	$-0.701027\pi$
$211$	−17.1519	−1.18078	−0.590392	+	0.807117i	$0.701027\pi$
$212$	−13.2171	−0.907753
$213$	0	0
$214$	2.63027	0.179802
$215$	1.75792	0.119889
$216$	0	0
$217$	1.06920	0.0725821
$218$	−19.8919	−1.34725
$219$	0	0
$220$	−1.60726	−0.108362
$221$	1.44389	0.0971266
$222$	0	0
$223$	18.5454	1.24189	0.620947	−	0.783852i	$-0.286748\pi$
$223$	18.5454	1.24189	0.620947	+	0.783852i	$0.286748\pi$
$224$	1.30248	0.0870257
$225$	0	0
$226$	24.0153	1.59747
$227$	−15.0455	−0.998607	−0.499303	−	0.866427i	$-0.666411\pi$
$227$	−15.0455	−0.998607	−0.499303	+	0.866427i	$0.666411\pi$
$228$	0	0
$229$	19.9007	1.31508	0.657538	−	0.753421i	$-0.271598\pi$
$229$	19.9007	1.31508	0.657538	+	0.753421i	$0.271598\pi$
$230$	1.13192	0.0746366
$231$	0	0
$232$	1.00396	0.0659133
$233$	12.9333	0.847291	0.423646	−	0.905828i	$-0.360750\pi$
$233$	12.9333	0.847291	0.423646	+	0.905828i	$0.360750\pi$
$234$	0	0
$235$	−4.72593	−0.308286
$236$	2.60382	0.169494
$237$	0	0
$238$	0.201028	0.0130307
$239$	17.6304	1.14042	0.570209	−	0.821500i	$-0.306862\pi$
$239$	17.6304	1.14042	0.570209	+	0.821500i	$0.306862\pi$
$240$	0	0
$241$	20.5561	1.32414	0.662069	−	0.749443i	$-0.269679\pi$
$241$	20.5561	1.32414	0.662069	+	0.749443i	$0.269679\pi$
$242$	14.3760	0.924126
$243$	0	0
$244$	−7.08943	−0.453854
$245$	4.23780	0.270743
$246$	0	0
$247$	14.3963	0.916014
$248$	5.68571	0.361043
$249$	0	0
$250$	−10.9001	−0.689382
$251$	13.8444	0.873852	0.436926	−	0.899497i	$-0.356067\pi$
$251$	13.8444	0.873852	0.436926	+	0.899497i	$0.356067\pi$
$252$	0	0
$253$	1.80879	0.113718
$254$	18.0269	1.13111
$255$	0	0
$256$	20.9103	1.30690
$257$	17.3425	1.08180	0.540898	−	0.841088i	$-0.318085\pi$
$257$	17.3425	1.08180	0.540898	+	0.841088i	$0.318085\pi$
$258$	0	0
$259$	−0.521252	−0.0323890
$260$	2.24142	0.139007
$261$	0	0
$262$	−25.2165	−1.55788
$263$	8.15846	0.503073	0.251536	−	0.967848i	$-0.419064\pi$
$263$	8.15846	0.503073	0.251536	+	0.967848i	$0.419064\pi$
$264$	0	0
$265$	5.50752	0.338324
$266$	2.00435	0.122895
$267$	0	0
$268$	8.78468	0.536610
$269$	−17.3726	−1.05923	−0.529613	−	0.848239i	$-0.677663\pi$
$269$	−17.3726	−1.05923	−0.529613	+	0.848239i	$0.677663\pi$
$270$	0	0
$271$	−16.4506	−0.999305	−0.499653	−	0.866226i	$-0.666539\pi$
$271$	−16.4506	−0.999305	−0.499653	+	0.866226i	$0.666539\pi$
$272$	2.74079	0.166185
$273$	0	0
$274$	−17.4193	−1.05234
$275$	−8.37421	−0.504984
$276$	0	0
$277$	17.4446	1.04814	0.524071	−	0.851675i	$-0.324413\pi$
$277$	17.4446	1.04814	0.524071	+	0.851675i	$0.324413\pi$
$278$	−35.8027	−2.14731
$279$	0	0
$280$	−0.115337	−0.00689271
$281$	−9.85250	−0.587751	−0.293875	−	0.955844i	$-0.594945\pi$
$281$	−9.85250	−0.587751	−0.293875	+	0.955844i	$0.594945\pi$
$282$	0	0
$283$	−11.8032	−0.701630	−0.350815	−	0.936445i	$-0.614095\pi$
$283$	−11.8032	−0.701630	−0.350815	+	0.936445i	$0.614095\pi$
$284$	18.0376	1.07034
$285$	0	0
$286$	8.48730	0.501865
$287$	−1.00146	−0.0591144
$288$	0	0
$289$	−16.6723	−0.980726
$290$	1.13192	0.0664687
$291$	0	0
$292$	4.55981	0.266842
$293$	−24.5605	−1.43484	−0.717418	−	0.696642i	$-0.754676\pi$
$293$	−24.5605	−1.43484	−0.717418	+	0.696642i	$0.754676\pi$
$294$	0	0
$295$	−1.08501	−0.0631715
$296$	−2.77187	−0.161112
$297$	0	0
$298$	−0.598038	−0.0346434
$299$	−2.52247	−0.145878
$300$	0	0
$301$	−0.545422	−0.0314376
$302$	33.7680	1.94313
$303$	0	0
$304$	27.3270	1.56731
$305$	2.95415	0.169154
$306$	0	0
$307$	−1.65118	−0.0942379	−0.0471189	−	0.998889i	$-0.515004\pi$
$307$	−1.65118	−0.0942379	−0.0471189	+	0.998889i	$0.515004\pi$
$308$	0.498677	0.0284148
$309$	0	0
$310$	6.41037	0.364085
$311$	−18.6331	−1.05658	−0.528292	−	0.849063i	$-0.677167\pi$
$311$	−18.6331	−1.05658	−0.528292	+	0.849063i	$0.677167\pi$
$312$	0	0
$313$	0.508102	0.0287196	0.0143598	−	0.999897i	$-0.495429\pi$
$313$	0.508102	0.0287196	0.0143598	+	0.999897i	$0.495429\pi$
$314$	13.2086	0.745403
$315$	0	0
$316$	9.73500	0.547636
$317$	33.7146	1.89360	0.946799	−	0.321825i	$-0.104296\pi$
$317$	33.7146	1.89360	0.946799	+	0.321825i	$0.104296\pi$
$318$	0	0
$319$	1.80879	0.101273
$320$	1.98185	0.110789
$321$	0	0
$322$	−0.351195	−0.0195713
$323$	3.26689	0.181775
$324$	0	0
$325$	11.6783	0.647797
$326$	−34.8138	−1.92816
$327$	0	0
$328$	−5.32549	−0.294051
$329$	1.46629	0.0808392
$330$	0	0
$331$	14.7899	0.812925	0.406462	−	0.913668i	$-0.366762\pi$
$331$	14.7899	0.812925	0.406462	+	0.913668i	$0.366762\pi$
$332$	26.1971	1.43775
$333$	0	0
$334$	13.1921	0.721838
$335$	−3.66055	−0.199997
$336$	0	0
$337$	−25.6617	−1.39788	−0.698940	−	0.715180i	$-0.746345\pi$
$337$	−25.6617	−1.39788	−0.698940	+	0.715180i	$0.746345\pi$
$338$	12.3464	0.671553
$339$	0	0
$340$	0.508637	0.0275847
$341$	10.2437	0.554726
$342$	0	0
$343$	−2.63641	−0.142353
$344$	−2.90040	−0.156379
$345$	0	0
$346$	42.2167	2.26958
$347$	16.7131	0.897208	0.448604	−	0.893731i	$-0.351921\pi$
$347$	16.7131	0.897208	0.448604	+	0.893731i	$0.351921\pi$
$348$	0	0
$349$	0.308781	0.0165287	0.00826434	−	0.999966i	$-0.497369\pi$
$349$	0.308781	0.0165287	0.00826434	+	0.999966i	$0.497369\pi$
$350$	1.62594	0.0869100
$351$	0	0
$352$	12.4786	0.665114
$353$	4.03922	0.214986	0.107493	−	0.994206i	$-0.465718\pi$
$353$	4.03922	0.214986	0.107493	+	0.994206i	$0.465718\pi$
$354$	0	0
$355$	−7.51623	−0.398920
$356$	18.9307	1.00332
$357$	0	0
$358$	28.6669	1.51509
$359$	19.6907	1.03924	0.519619	−	0.854398i	$-0.326074\pi$
$359$	19.6907	1.03924	0.519619	+	0.854398i	$0.326074\pi$
$360$	0	0
$361$	13.5725	0.714341
$362$	−46.4325	−2.44044
$363$	0	0
$364$	−0.695435	−0.0364507
$365$	−1.90006	−0.0994536
$366$	0	0
$367$	−35.3919	−1.84744	−0.923722	−	0.383064i	$-0.874869\pi$
$367$	−35.3919	−1.84744	−0.923722	+	0.383064i	$0.874869\pi$
$368$	−4.78813	−0.249599
$369$	0	0
$370$	−3.12516	−0.162469
$371$	−1.70879	−0.0887160
$372$	0	0
$373$	−5.73013	−0.296695	−0.148347	−	0.988935i	$-0.547395\pi$
$373$	−5.73013	−0.296695	−0.148347	+	0.988935i	$0.547395\pi$
$374$	1.92599	0.0995904
$375$	0	0
$376$	7.79731	0.402116
$377$	−2.52247	−0.129914
$378$	0	0
$379$	27.3401	1.40437	0.702184	−	0.711996i	$-0.252208\pi$
$379$	27.3401	1.40437	0.702184	+	0.711996i	$0.252208\pi$
$380$	5.07136	0.260155
$381$	0	0
$382$	4.16439	0.213069
$383$	−3.00017	−0.153301	−0.0766507	−	0.997058i	$-0.524423\pi$
$383$	−3.00017	−0.153301	−0.0766507	+	0.997058i	$0.524423\pi$
$384$	0	0
$385$	−0.207797	−0.0105903
$386$	−43.5575	−2.21702
$387$	0	0
$388$	−19.7274	−1.00151
$389$	−5.34057	−0.270778	−0.135389	−	0.990793i	$-0.543228\pi$
$389$	−5.34057	−0.270778	−0.135389	+	0.990793i	$0.543228\pi$
$390$	0	0
$391$	−0.572412	−0.0289481
$392$	−6.99194	−0.353147
$393$	0	0
$394$	−23.5225	−1.18505
$395$	−4.05655	−0.204107
$396$	0	0
$397$	38.4644	1.93047	0.965236	−	0.261380i	$-0.0841777\pi$
$397$	38.4644	1.93047	0.965236	+	0.261380i	$0.0841777\pi$
$398$	41.5181	2.08111
$399$	0	0
$400$	22.1678	1.10839
$401$	−18.3743	−0.917570	−0.458785	−	0.888547i	$-0.651715\pi$
$401$	−18.3743	−0.917570	−0.458785	+	0.888547i	$0.651715\pi$
$402$	0	0
$403$	−14.2854	−0.711607
$404$	−22.1956	−1.10427
$405$	0	0
$406$	−0.351195	−0.0174295
$407$	−4.99395	−0.247541
$408$	0	0
$409$	18.7527	0.927261	0.463630	−	0.886029i	$-0.346547\pi$
$409$	18.7527	0.927261	0.463630	+	0.886029i	$0.346547\pi$
$410$	−6.00424	−0.296528
$411$	0	0
$412$	9.29553	0.457958
$413$	0.336639	0.0165649
$414$	0	0
$415$	−10.9163	−0.535858
$416$	−17.4022	−0.853214
$417$	0	0
$418$	19.2030	0.939251
$419$	32.0502	1.56576	0.782878	−	0.622175i	$-0.213751\pi$
$419$	32.0502	1.56576	0.782878	+	0.622175i	$0.213751\pi$
$420$	0	0
$421$	−27.3186	−1.33143	−0.665714	−	0.746207i	$-0.731873\pi$
$421$	−27.3186	−1.33143	−0.665714	+	0.746207i	$0.731873\pi$
$422$	31.9057	1.55314
$423$	0	0
$424$	−9.08686	−0.441297
$425$	2.65011	0.128549
$426$	0	0
$427$	−0.916567	−0.0443558
$428$	−2.06483	−0.0998071
$429$	0	0
$430$	−3.27006	−0.157696
$431$	10.6207	0.511582	0.255791	−	0.966732i	$-0.417664\pi$
$431$	10.6207	0.511582	0.255791	+	0.966732i	$0.417664\pi$
$432$	0	0
$433$	36.5317	1.75560	0.877802	−	0.479024i	$-0.159009\pi$
$433$	36.5317	1.75560	0.877802	+	0.479024i	$0.159009\pi$
$434$	−1.98891	−0.0954709
$435$	0	0
$436$	15.6156	0.747852
$437$	−5.70723	−0.273014
$438$	0	0
$439$	14.0902	0.672488	0.336244	−	0.941775i	$-0.390843\pi$
$439$	14.0902	0.672488	0.336244	+	0.941775i	$0.390843\pi$
$440$	−1.10501	−0.0526791
$441$	0	0
$442$	−2.68590	−0.127755
$443$	−8.28736	−0.393744	−0.196872	−	0.980429i	$-0.563078\pi$
$443$	−8.28736	−0.393744	−0.196872	+	0.980429i	$0.563078\pi$
$444$	0	0
$445$	−7.88836	−0.373944
$446$	−34.4980	−1.63353
$447$	0	0
$448$	−0.614899	−0.0290512
$449$	17.4222	0.822205	0.411103	−	0.911589i	$-0.365144\pi$
$449$	17.4222	0.822205	0.411103	+	0.911589i	$0.365144\pi$
$450$	0	0
$451$	−9.59468	−0.451796
$452$	−18.8526	−0.886750
$453$	0	0
$454$	27.9875	1.31352
$455$	0.289786	0.0135854
$456$	0	0
$457$	−29.8688	−1.39720	−0.698602	−	0.715511i	$-0.746194\pi$
$457$	−29.8688	−1.39720	−0.698602	+	0.715511i	$0.746194\pi$
$458$	−37.0190	−1.72979
$459$	0	0
$460$	−0.888584	−0.0414305
$461$	16.4117	0.764371	0.382186	−	0.924086i	$-0.375172\pi$
$461$	16.4117	0.764371	0.382186	+	0.924086i	$0.375172\pi$
$462$	0	0
$463$	29.6351	1.37726	0.688629	−	0.725114i	$-0.258213\pi$
$463$	29.6351	1.37726	0.688629	+	0.725114i	$0.258213\pi$
$464$	−4.78813	−0.222284
$465$	0	0
$466$	−24.0584	−1.11448
$467$	23.5928	1.09175	0.545873	−	0.837868i	$-0.316198\pi$
$467$	23.5928	1.09175	0.545873	+	0.837868i	$0.316198\pi$
$468$	0	0
$469$	1.13574	0.0524436
$470$	8.79111	0.405504
$471$	0	0
$472$	1.79015	0.0823983
$473$	−5.22551	−0.240269
$474$	0	0
$475$	26.4229	1.21237
$476$	−0.157812	−0.00723330
$477$	0	0
$478$	−32.7959	−1.50005
$479$	−17.2501	−0.788177	−0.394088	−	0.919073i	$-0.628940\pi$
$479$	−17.2501	−0.788177	−0.394088	+	0.919073i	$0.628940\pi$
$480$	0	0
$481$	6.96436	0.317547
$482$	−38.2382	−1.74170
$483$	0	0
$484$	−11.2855	−0.512978
$485$	8.22035	0.373267
$486$	0	0
$487$	−11.3646	−0.514981	−0.257491	−	0.966281i	$-0.582896\pi$
$487$	−11.3646	−0.514981	−0.257491	+	0.966281i	$0.582896\pi$
$488$	−4.87404	−0.220638
$489$	0	0
$490$	−7.88310	−0.356122
$491$	33.1127	1.49435	0.747177	−	0.664625i	$-0.231409\pi$
$491$	33.1127	1.49435	0.747177	+	0.664625i	$0.231409\pi$
$492$	0	0
$493$	−0.572412	−0.0257802
$494$	−26.7798	−1.20488
$495$	0	0
$496$	−27.1165	−1.21757
$497$	2.33202	0.104605
$498$	0	0
$499$	13.4818	0.603530	0.301765	−	0.953382i	$-0.402424\pi$
$499$	13.4818	0.603530	0.301765	+	0.953382i	$0.402424\pi$
$500$	8.55683	0.382673
$501$	0	0
$502$	−25.7532	−1.14942
$503$	21.4616	0.956924	0.478462	−	0.878108i	$-0.341195\pi$
$503$	21.4616	0.956924	0.478462	+	0.878108i	$0.341195\pi$
$504$	0	0
$505$	9.24885	0.411568
$506$	−3.36468	−0.149578
$507$	0	0
$508$	−14.1515	−0.627872
$509$	−15.9887	−0.708685	−0.354342	−	0.935116i	$-0.615295\pi$
$509$	−15.9887	−0.708685	−0.354342	+	0.935116i	$0.615295\pi$
$510$	0	0
$511$	0.589521	0.0260789
$512$	−23.4186	−1.03497
$513$	0	0
$514$	−32.2603	−1.42294
$515$	−3.87342	−0.170683
$516$	0	0
$517$	14.0480	0.617832
$518$	0.969626	0.0426029
$519$	0	0
$520$	1.54100	0.0675772
$521$	31.7139	1.38941	0.694706	−	0.719294i	$-0.255534\pi$
$521$	31.7139	1.38941	0.694706	+	0.719294i	$0.255534\pi$
$522$	0	0
$523$	36.8610	1.61182	0.805909	−	0.592039i	$-0.201677\pi$
$523$	36.8610	1.61182	0.805909	+	0.592039i	$0.201677\pi$
$524$	19.7956	0.864773
$525$	0	0
$526$	−15.1763	−0.661716
$527$	−3.24173	−0.141212
$528$	0	0
$529$	1.00000	0.0434783
$530$	−10.2450	−0.445015
$531$	0	0
$532$	−1.57346	−0.0682183
$533$	13.3804	0.579568
$534$	0	0
$535$	0.860407	0.0371986
$536$	6.03954	0.260868
$537$	0	0
$538$	32.3162	1.39325
$539$	−12.5971	−0.542594
$540$	0	0
$541$	31.3819	1.34922	0.674608	−	0.738177i	$-0.264313\pi$
$541$	31.3819	1.34922	0.674608	+	0.738177i	$0.264313\pi$
$542$	30.6012	1.31444
$543$	0	0
$544$	−3.94901	−0.169312
$545$	−6.50699	−0.278729
$546$	0	0
$547$	13.7723	0.588862	0.294431	−	0.955673i	$-0.404870\pi$
$547$	13.7723	0.588862	0.294431	+	0.955673i	$0.404870\pi$
$548$	13.6746	0.584149
$549$	0	0
$550$	15.5776	0.664230
$551$	−5.70723	−0.243136
$552$	0	0
$553$	1.25860	0.0535213
$554$	−32.4501	−1.37867
$555$	0	0
$556$	28.1060	1.19196
$557$	18.8128	0.797124	0.398562	−	0.917141i	$-0.369509\pi$
$557$	18.8128	0.797124	0.398562	+	0.917141i	$0.369509\pi$
$558$	0	0
$559$	7.28728	0.308219
$560$	0.550070	0.0232447
$561$	0	0
$562$	18.3275	0.773098
$563$	28.7847	1.21313	0.606566	−	0.795033i	$-0.292547\pi$
$563$	28.7847	1.21313	0.606566	+	0.795033i	$0.292547\pi$
$564$	0	0
$565$	7.85581	0.330496
$566$	21.9562	0.922889
$567$	0	0
$568$	12.4010	0.520335
$569$	16.5715	0.694712	0.347356	−	0.937733i	$-0.387079\pi$
$569$	16.5715	0.694712	0.347356	+	0.937733i	$0.387079\pi$
$570$	0	0
$571$	−41.7242	−1.74610	−0.873051	−	0.487629i	$-0.837862\pi$
$571$	−41.7242	−1.74610	−0.873051	+	0.487629i	$0.837862\pi$
$572$	−6.66273	−0.278583
$573$	0	0
$574$	1.86291	0.0777562
$575$	−4.62973	−0.193073
$576$	0	0
$577$	−16.9453	−0.705440	−0.352720	−	0.935729i	$-0.614743\pi$
$577$	−16.9453	−0.705440	−0.352720	+	0.935729i	$0.614743\pi$
$578$	31.0137	1.29000
$579$	0	0
$580$	−0.888584	−0.0368965
$581$	3.38693	0.140514
$582$	0	0
$583$	−16.3714	−0.678032
$584$	3.13490	0.129723
$585$	0	0
$586$	45.6870	1.88731
$587$	−3.22610	−0.133155	−0.0665776	−	0.997781i	$-0.521208\pi$
$587$	−3.22610	−0.133155	−0.0665776	+	0.997781i	$0.521208\pi$
$588$	0	0
$589$	−32.3216	−1.33179
$590$	2.01831	0.0830926
$591$	0	0
$592$	13.2197	0.543327
$593$	−4.05889	−0.166679	−0.0833393	−	0.996521i	$-0.526559\pi$
$593$	−4.05889	−0.166679	−0.0833393	+	0.996521i	$0.526559\pi$
$594$	0	0
$595$	0.0657598	0.00269589
$596$	0.469474	0.0192304
$597$	0	0
$598$	4.69225	0.191881
$599$	11.8716	0.485059	0.242529	−	0.970144i	$-0.422023\pi$
$599$	11.8716	0.485059	0.242529	+	0.970144i	$0.422023\pi$
$600$	0	0
$601$	−39.4427	−1.60890	−0.804450	−	0.594021i	$-0.797540\pi$
$601$	−39.4427	−1.60890	−0.804450	+	0.594021i	$0.797540\pi$
$602$	1.01459	0.0413514
$603$	0	0
$604$	−26.5087	−1.07862
$605$	4.70265	0.191190
$606$	0	0
$607$	−7.26279	−0.294788	−0.147394	−	0.989078i	$-0.547088\pi$
$607$	−7.26279	−0.294788	−0.147394	+	0.989078i	$0.547088\pi$
$608$	−39.3736	−1.59681
$609$	0	0
$610$	−5.49526	−0.222496
$611$	−19.5908	−0.792560
$612$	0	0
$613$	4.52434	0.182736	0.0913682	−	0.995817i	$-0.470876\pi$
$613$	4.52434	0.182736	0.0913682	+	0.995817i	$0.470876\pi$
$614$	3.07150	0.123956
$615$	0	0
$616$	0.342845	0.0138136
$617$	23.5563	0.948342	0.474171	−	0.880433i	$-0.342748\pi$
$617$	23.5563	0.948342	0.474171	+	0.880433i	$0.342748\pi$
$618$	0	0
$619$	7.33630	0.294871	0.147435	−	0.989072i	$-0.452898\pi$
$619$	7.33630	0.294871	0.147435	+	0.989072i	$0.452898\pi$
$620$	−5.03229	−0.202102
$621$	0	0
$622$	34.6609	1.38978
$623$	2.44748	0.0980563
$624$	0	0
$625$	19.5830	0.783322
$626$	−0.945164	−0.0377764
$627$	0	0
$628$	−10.3690	−0.413770
$629$	1.58039	0.0630144
$630$	0	0
$631$	−23.0356	−0.917031	−0.458516	−	0.888686i	$-0.651619\pi$
$631$	−23.0356	−0.917031	−0.458516	+	0.888686i	$0.651619\pi$
$632$	6.69289	0.266229
$633$	0	0
$634$	−62.7153	−2.49074
$635$	5.89690	0.234011
$636$	0	0
$637$	17.5673	0.696044
$638$	−3.36468	−0.133209
$639$	0	0
$640$	4.70932	0.186152
$641$	7.58759	0.299692	0.149846	−	0.988709i	$-0.452122\pi$
$641$	7.58759	0.299692	0.149846	+	0.988709i	$0.452122\pi$
$642$	0	0
$643$	9.33232	0.368031	0.184015	−	0.982923i	$-0.441090\pi$
$643$	9.33232	0.368031	0.184015	+	0.982923i	$0.441090\pi$
$644$	0.275696	0.0108640
$645$	0	0
$646$	−6.07702	−0.239097
$647$	18.1884	0.715059	0.357529	−	0.933902i	$-0.383619\pi$
$647$	18.1884	0.715059	0.357529	+	0.933902i	$0.383619\pi$
$648$	0	0
$649$	3.22523	0.126601
$650$	−21.7239	−0.852080
$651$	0	0
$652$	27.3297	1.07031
$653$	27.1462	1.06231	0.531156	−	0.847274i	$-0.321758\pi$
$653$	27.1462	1.06231	0.531156	+	0.847274i	$0.321758\pi$
$654$	0	0
$655$	−8.24875	−0.322305
$656$	25.3985	0.991646
$657$	0	0
$658$	−2.72757	−0.106332
$659$	39.7414	1.54810	0.774052	−	0.633122i	$-0.218227\pi$
$659$	39.7414	1.54810	0.774052	+	0.633122i	$0.218227\pi$
$660$	0	0
$661$	−48.6605	−1.89267	−0.946337	−	0.323180i	$-0.895248\pi$
$661$	−48.6605	−1.89267	−0.946337	+	0.323180i	$0.895248\pi$
$662$	−27.5119	−1.06928
$663$	0	0
$664$	18.0107	0.698952
$665$	0.655658	0.0254253
$666$	0	0
$667$	1.00000	0.0387202
$668$	−10.3561	−0.400689
$669$	0	0
$670$	6.80931	0.263066
$671$	−8.78133	−0.339000
$672$	0	0
$673$	20.1258	0.775794	0.387897	−	0.921703i	$-0.373202\pi$
$673$	20.1258	0.775794	0.387897	+	0.921703i	$0.373202\pi$
$674$	47.7355	1.83870
$675$	0	0
$676$	−9.69219	−0.372776
$677$	−12.1523	−0.467051	−0.233526	−	0.972351i	$-0.575026\pi$
$677$	−12.1523	−0.467051	−0.233526	+	0.972351i	$0.575026\pi$
$678$	0	0
$679$	−2.55049	−0.0978787
$680$	0.349692	0.0134101
$681$	0	0
$682$	−19.0551	−0.729658
$683$	39.3793	1.50681	0.753403	−	0.657560i	$-0.228411\pi$
$683$	39.3793	1.50681	0.753403	+	0.657560i	$0.228411\pi$
$684$	0	0
$685$	−5.69816	−0.217715
$686$	4.90421	0.187244
$687$	0	0
$688$	13.8327	0.527366
$689$	22.8308	0.869786
$690$	0	0
$691$	35.9003	1.36571	0.682857	−	0.730552i	$-0.260737\pi$
$691$	35.9003	1.36571	0.682857	+	0.730552i	$0.260737\pi$
$692$	−33.1411	−1.25984
$693$	0	0
$694$	−31.0895	−1.18014
$695$	−11.7117	−0.444250
$696$	0	0
$697$	3.03635	0.115010
$698$	−0.574391	−0.0217410
$699$	0	0
$700$	−1.27640	−0.0482434
$701$	−2.34897	−0.0887195	−0.0443598	−	0.999016i	$-0.514125\pi$
$701$	−2.34897	−0.0887195	−0.0443598	+	0.999016i	$0.514125\pi$
$702$	0	0
$703$	15.7573	0.594297
$704$	−5.89114	−0.222031
$705$	0	0
$706$	−7.51370	−0.282782
$707$	−2.86959	−0.107922
$708$	0	0
$709$	−24.4798	−0.919359	−0.459680	−	0.888085i	$-0.652036\pi$
$709$	−24.4798	−0.919359	−0.459680	+	0.888085i	$0.652036\pi$
$710$	13.9816	0.524719
$711$	0	0
$712$	13.0150	0.487758
$713$	5.66327	0.212091
$714$	0	0
$715$	2.77634	0.103829
$716$	−22.5042	−0.841020
$717$	0	0
$718$	−36.6284	−1.36696
$719$	−23.6371	−0.881515	−0.440757	−	0.897626i	$-0.645290\pi$
$719$	−23.6371	−0.881515	−0.440757	+	0.897626i	$0.645290\pi$
$720$	0	0
$721$	1.20179	0.0447568
$722$	−25.2473	−0.939608
$723$	0	0
$724$	36.4506	1.35468
$725$	−4.62973	−0.171944
$726$	0	0
$727$	−16.6253	−0.616599	−0.308300	−	0.951289i	$-0.599760\pi$
$727$	−16.6253	−0.616599	−0.308300	+	0.951289i	$0.599760\pi$
$728$	−0.478117	−0.0177202
$729$	0	0
$730$	3.53446	0.130816
$731$	1.65367	0.0611633
$732$	0	0
$733$	29.6937	1.09676	0.548381	−	0.836228i	$-0.315244\pi$
$733$	29.6937	1.09676	0.548381	+	0.836228i	$0.315244\pi$
$734$	65.8355	2.43003
$735$	0	0
$736$	6.89889	0.254297
$737$	10.8812	0.400813
$738$	0	0
$739$	−44.0522	−1.62049	−0.810243	−	0.586094i	$-0.800665\pi$
$739$	−44.0522	−1.62049	−0.810243	+	0.586094i	$0.800665\pi$
$740$	2.45332	0.0901859
$741$	0	0
$742$	3.17867	0.116693
$743$	19.7350	0.724008	0.362004	−	0.932177i	$-0.382093\pi$
$743$	19.7350	0.724008	0.362004	+	0.932177i	$0.382093\pi$
$744$	0	0
$745$	−0.195629	−0.00716728
$746$	10.6591	0.390257
$747$	0	0
$748$	−1.51195	−0.0552822
$749$	−0.266954	−0.00975429
$750$	0	0
$751$	11.7557	0.428971	0.214485	−	0.976727i	$-0.431193\pi$
$751$	11.7557	0.428971	0.214485	+	0.976727i	$0.431193\pi$
$752$	−37.1872	−1.35608
$753$	0	0
$754$	4.69225	0.170882
$755$	11.0461	0.402009
$756$	0	0
$757$	18.2108	0.661883	0.330942	−	0.943651i	$-0.392634\pi$
$757$	18.2108	0.661883	0.330942	+	0.943651i	$0.392634\pi$
$758$	−50.8577	−1.84723
$759$	0	0
$760$	3.48660	0.126472
$761$	15.6654	0.567870	0.283935	−	0.958844i	$-0.408360\pi$
$761$	15.6654	0.567870	0.283935	+	0.958844i	$0.408360\pi$
$762$	0	0
$763$	2.01889	0.0730887
$764$	−3.26914	−0.118274
$765$	0	0
$766$	5.58086	0.201645
$767$	−4.49777	−0.162405
$768$	0	0
$769$	−47.7618	−1.72233	−0.861167	−	0.508322i	$-0.830266\pi$
$769$	−47.7618	−1.72233	−0.861167	+	0.508322i	$0.830266\pi$
$770$	0.386541	0.0139300
$771$	0	0
$772$	34.1936	1.23066
$773$	34.3239	1.23455	0.617273	−	0.786749i	$-0.288237\pi$
$773$	34.3239	1.23455	0.617273	+	0.786749i	$0.288237\pi$
$774$	0	0
$775$	−26.2194	−0.941829
$776$	−13.5628	−0.486874
$777$	0	0
$778$	9.93445	0.356167
$779$	30.2739	1.08467
$780$	0	0
$781$	22.3423	0.799472
$782$	1.06479	0.0380769
$783$	0	0
$784$	33.3463	1.19094
$785$	4.32075	0.154214
$786$	0	0
$787$	−18.4011	−0.655929	−0.327964	−	0.944690i	$-0.606363\pi$
$787$	−18.4011	−0.655929	−0.327964	+	0.944690i	$0.606363\pi$
$788$	18.4658	0.657815
$789$	0	0
$790$	7.54593	0.268472
$791$	−2.43738	−0.0866633
$792$	0	0
$793$	12.2461	0.434871
$794$	−71.5509	−2.53925
$795$	0	0
$796$	−32.5927	−1.15522
$797$	−4.85810	−0.172083	−0.0860413	−	0.996292i	$-0.527422\pi$
$797$	−4.85810	−0.172083	−0.0860413	+	0.996292i	$0.527422\pi$
$798$	0	0
$799$	−4.44567	−0.157276
$800$	−31.9400	−1.12925
$801$	0	0
$802$	34.1797	1.20693
$803$	5.64801	0.199314
$804$	0	0
$805$	−0.114882	−0.00404906
$806$	26.5735	0.936012
$807$	0	0
$808$	−15.2597	−0.536833
$809$	28.4680	1.00088	0.500441	−	0.865771i	$-0.333171\pi$
$809$	28.4680	1.00088	0.500441	+	0.865771i	$0.333171\pi$
$810$	0	0
$811$	−21.5178	−0.755592	−0.377796	−	0.925889i	$-0.623318\pi$
$811$	−21.5178	−0.755592	−0.377796	+	0.925889i	$0.623318\pi$
$812$	0.275696	0.00967505
$813$	0	0
$814$	9.28967	0.325603
$815$	−11.3882	−0.398911
$816$	0	0
$817$	16.4879	0.576839
$818$	−34.8835	−1.21967
$819$	0	0
$820$	4.71347	0.164602
$821$	−50.2926	−1.75522	−0.877612	−	0.479371i	$-0.840865\pi$
$821$	−50.2926	−1.75522	−0.877612	+	0.479371i	$0.840865\pi$
$822$	0	0
$823$	−3.13445	−0.109260	−0.0546301	−	0.998507i	$-0.517398\pi$
$823$	−3.13445	−0.109260	−0.0546301	+	0.998507i	$0.517398\pi$
$824$	6.39075	0.222632
$825$	0	0
$826$	−0.626211	−0.0217887
$827$	44.4900	1.54707	0.773535	−	0.633753i	$-0.218487\pi$
$827$	44.4900	1.54707	0.773535	+	0.633753i	$0.218487\pi$
$828$	0	0
$829$	25.3746	0.881296	0.440648	−	0.897680i	$-0.354749\pi$
$829$	25.3746	0.881296	0.440648	+	0.897680i	$0.354749\pi$
$830$	20.3063	0.704841
$831$	0	0
$832$	8.21555	0.284823
$833$	3.98648	0.138123
$834$	0	0
$835$	4.31535	0.149339
$836$	−15.0748	−0.521374
$837$	0	0
$838$	−59.6194	−2.05952
$839$	−6.52439	−0.225247	−0.112623	−	0.993638i	$-0.535925\pi$
$839$	−6.52439	−0.225247	−0.112623	+	0.993638i	$0.535925\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	50.8177	1.75129
$843$	0	0
$844$	−25.0467	−0.862143
$845$	4.03871	0.138936
$846$	0	0
$847$	−1.45907	−0.0501341
$848$	43.3374	1.48821
$849$	0	0
$850$	−4.92970	−0.169087
$851$	−2.76093	−0.0946436
$852$	0	0
$853$	40.6632	1.39228	0.696140	−	0.717906i	$-0.254899\pi$
$853$	40.6632	1.39228	0.696140	+	0.717906i	$0.254899\pi$
$854$	1.70499	0.0583434
$855$	0	0
$856$	−1.41958	−0.0485204
$857$	−17.2520	−0.589318	−0.294659	−	0.955602i	$-0.595206\pi$
$857$	−17.2520	−0.589318	−0.294659	+	0.955602i	$0.595206\pi$
$858$	0	0
$859$	22.0917	0.753759	0.376879	−	0.926262i	$-0.376997\pi$
$859$	22.0917	0.753759	0.376879	+	0.926262i	$0.376997\pi$
$860$	2.56708	0.0875366
$861$	0	0
$862$	−19.7565	−0.672909
$863$	19.1687	0.652509	0.326254	−	0.945282i	$-0.394213\pi$
$863$	19.1687	0.652509	0.326254	+	0.945282i	$0.394213\pi$
$864$	0	0
$865$	13.8098	0.469548
$866$	−67.9558	−2.30923
$867$	0	0
$868$	1.56134	0.0529955
$869$	12.0583	0.409049
$870$	0	0
$871$	−15.1744	−0.514166
$872$	10.7359	0.363562
$873$	0	0
$874$	10.6165	0.359109
$875$	1.10628	0.0373992
$876$	0	0
$877$	45.5676	1.53871	0.769354	−	0.638822i	$-0.220578\pi$
$877$	45.5676	1.53871	0.769354	+	0.638822i	$0.220578\pi$
$878$	−26.2104	−0.884557
$879$	0	0
$880$	5.27004	0.177653
$881$	−45.1235	−1.52025	−0.760125	−	0.649777i	$-0.774862\pi$
$881$	−45.1235	−1.52025	−0.760125	+	0.649777i	$0.774862\pi$
$882$	0	0
$883$	32.2894	1.08663	0.543313	−	0.839530i	$-0.317170\pi$
$883$	32.2894	1.08663	0.543313	+	0.839530i	$0.317170\pi$
$884$	2.10850	0.0709165
$885$	0	0
$886$	15.4160	0.517911
$887$	−12.7413	−0.427811	−0.213905	−	0.976854i	$-0.568618\pi$
$887$	−12.7413	−0.427811	−0.213905	+	0.976854i	$0.568618\pi$
$888$	0	0
$889$	−1.82960	−0.0613628
$890$	14.6738	0.491868
$891$	0	0
$892$	27.0817	0.906763
$893$	−44.3255	−1.48329
$894$	0	0
$895$	9.37742	0.313453
$896$	−1.46114	−0.0488132
$897$	0	0
$898$	−32.4086	−1.08149
$899$	5.66327	0.188881
$900$	0	0
$901$	5.18090	0.172601
$902$	17.8479	0.594270
$903$	0	0
$904$	−12.9613	−0.431086
$905$	−15.1889	−0.504895
$906$	0	0
$907$	7.26224	0.241139	0.120569	−	0.992705i	$-0.461528\pi$
$907$	7.26224	0.241139	0.120569	+	0.992705i	$0.461528\pi$
$908$	−21.9708	−0.729127
$909$	0	0
$910$	−0.539055	−0.0178695
$911$	48.7369	1.61473	0.807363	−	0.590055i	$-0.200894\pi$
$911$	48.7369	1.61473	0.807363	+	0.590055i	$0.200894\pi$
$912$	0	0
$913$	32.4491	1.07391
$914$	55.5615	1.83781
$915$	0	0
$916$	29.0608	0.960196
$917$	2.55930	0.0845154
$918$	0	0
$919$	−27.3842	−0.903320	−0.451660	−	0.892190i	$-0.649168\pi$
$919$	−27.3842	−0.903320	−0.451660	+	0.892190i	$0.649168\pi$
$920$	−0.610909	−0.0201411
$921$	0	0
$922$	−30.5289	−1.00542
$923$	−31.1577	−1.02557
$924$	0	0
$925$	12.7824	0.420282
$926$	−55.1267	−1.81158
$927$	0	0
$928$	6.89889	0.226467
$929$	−35.4357	−1.16261	−0.581305	−	0.813686i	$-0.697458\pi$
$929$	−35.4357	−1.16261	−0.581305	+	0.813686i	$0.697458\pi$
$930$	0	0
$931$	39.7472	1.30266
$932$	18.8864	0.618645
$933$	0	0
$934$	−43.8870	−1.43603
$935$	0.630024	0.0206040
$936$	0	0
$937$	−10.2302	−0.334206	−0.167103	−	0.985939i	$-0.553441\pi$
$937$	−10.2302	−0.334206	−0.167103	+	0.985939i	$0.553441\pi$
$938$	−2.11269	−0.0689817
$939$	0	0
$940$	−6.90123	−0.225093
$941$	52.3389	1.70620	0.853100	−	0.521748i	$-0.174720\pi$
$941$	52.3389	1.70620	0.853100	+	0.521748i	$0.174720\pi$
$942$	0	0
$943$	−5.30448	−0.172737
$944$	−8.53766	−0.277877
$945$	0	0
$946$	9.72041	0.316038
$947$	−55.1480	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$947$	−55.1480	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$948$	0	0
$949$	−7.87649	−0.255682
$950$	−49.1516	−1.59469
$951$	0	0
$952$	−0.108497	−0.00351641
$953$	27.1544	0.879616	0.439808	−	0.898092i	$-0.355046\pi$
$953$	27.1544	0.879616	0.439808	+	0.898092i	$0.355046\pi$
$954$	0	0
$955$	1.36224	0.0440812
$956$	25.7455	0.832670
$957$	0	0
$958$	32.0884	1.03673
$959$	1.76794	0.0570897
$960$	0	0
$961$	1.07265	0.0346016
$962$	−12.9550	−0.417686
$963$	0	0
$964$	30.0179	0.966812
$965$	−14.2484	−0.458672
$966$	0	0
$967$	9.89851	0.318315	0.159157	−	0.987253i	$-0.449122\pi$
$967$	9.89851	0.318315	0.159157	+	0.987253i	$0.449122\pi$
$968$	−7.75889	−0.249380
$969$	0	0
$970$	−15.2914	−0.490977
$971$	30.7536	0.986931	0.493466	−	0.869765i	$-0.335730\pi$
$971$	30.7536	0.986931	0.493466	+	0.869765i	$0.335730\pi$
$972$	0	0
$973$	3.63373	0.116492
$974$	21.1404	0.677381
$975$	0	0
$976$	23.2455	0.744070
$977$	−46.0303	−1.47264	−0.736320	−	0.676633i	$-0.763438\pi$
$977$	−46.0303	−1.47264	−0.736320	+	0.676633i	$0.763438\pi$
$978$	0	0
$979$	23.4485	0.749418
$980$	6.18842	0.197682
$981$	0	0
$982$	−61.5957	−1.96560
$983$	22.6809	0.723407	0.361704	−	0.932293i	$-0.382195\pi$
$983$	22.6809	0.723407	0.361704	+	0.932293i	$0.382195\pi$
$984$	0	0
$985$	−7.69463	−0.245171
$986$	1.06479	0.0339099
$987$	0	0
$988$	21.0228	0.668823
$989$	−2.88895	−0.0918633
$990$	0	0
$991$	−7.71521	−0.245082	−0.122541	−	0.992463i	$-0.539104\pi$
$991$	−7.71521	−0.245082	−0.122541	+	0.992463i	$0.539104\pi$
$992$	39.0703	1.24048
$993$	0	0
$994$	−4.33799	−0.137593
$995$	13.5813	0.430555
$996$	0	0
$997$	−19.9719	−0.632516	−0.316258	−	0.948673i	$-0.602427\pi$
$997$	−19.9719	−0.632516	−0.316258	+	0.948673i	$0.602427\pi$
$998$	−25.0787	−0.793853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.3		14
3.2	odd	2		2001.2.a.m.1.12	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.12	✓	14	3.2	odd	2
6003.2.a.p.1.3		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)

\(f(q)\)	\(=\)	\(q-1.86019 q^{2} +1.46029 q^{4} -0.608499 q^{5} +0.188796 q^{7} +1.00396 q^{8} +O(q^{10})\)	\(q-1.86019 q^{2} +1.46029 q^{4} -0.608499 q^{5} +0.188796 q^{7} +1.00396 q^{8} +1.13192 q^{10} +1.80879 q^{11} -2.52247 q^{13} -0.351195 q^{14} -4.78813 q^{16} -0.572412 q^{17} -5.70723 q^{19} -0.888584 q^{20} -3.36468 q^{22} +1.00000 q^{23} -4.62973 q^{25} +4.69225 q^{26} +0.275696 q^{28} +1.00000 q^{29} +5.66327 q^{31} +6.89889 q^{32} +1.06479 q^{34} -0.114882 q^{35} -2.76093 q^{37} +10.6165 q^{38} -0.610909 q^{40} -5.30448 q^{41} -2.88895 q^{43} +2.64136 q^{44} -1.86019 q^{46} +7.76654 q^{47} -6.96436 q^{49} +8.61215 q^{50} -3.68353 q^{52} -9.05100 q^{53} -1.10065 q^{55} +0.189544 q^{56} -1.86019 q^{58} +1.78309 q^{59} -4.85481 q^{61} -10.5347 q^{62} -3.25695 q^{64} +1.53492 q^{65} +6.01571 q^{67} -0.835888 q^{68} +0.213702 q^{70} +12.3521 q^{71} +3.12254 q^{73} +5.13585 q^{74} -8.33421 q^{76} +0.341492 q^{77} +6.66648 q^{79} +2.91357 q^{80} +9.86731 q^{82} +17.9397 q^{83} +0.348312 q^{85} +5.37399 q^{86} +1.81596 q^{88} +12.9637 q^{89} -0.476231 q^{91} +1.46029 q^{92} -14.4472 q^{94} +3.47284 q^{95} -13.5092 q^{97} +12.9550 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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Database

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