LMFDB - Embedded newform 6003.2.a.v.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.89039 q^{2} +1.57357 q^{4} -2.79891 q^{5} +1.18730 q^{7} +0.806117 q^{8} +O(q^{10})$	\(q-1.89039 q^{2} +1.57357 q^{4} -2.79891 q^{5} +1.18730 q^{7} +0.806117 q^{8} +5.29102 q^{10} -6.19722 q^{11} +6.24610 q^{13} -2.24445 q^{14} -4.67102 q^{16} -7.04887 q^{17} -8.14253 q^{19} -4.40428 q^{20} +11.7152 q^{22} +1.00000 q^{23} +2.83388 q^{25} -11.8076 q^{26} +1.86830 q^{28} -1.00000 q^{29} -1.41381 q^{31} +7.21780 q^{32} +13.3251 q^{34} -3.32313 q^{35} +1.97391 q^{37} +15.3925 q^{38} -2.25625 q^{40} -1.66473 q^{41} -9.37822 q^{43} -9.75177 q^{44} -1.89039 q^{46} -1.54634 q^{47} -5.59033 q^{49} -5.35713 q^{50} +9.82869 q^{52} -12.0893 q^{53} +17.3455 q^{55} +0.957100 q^{56} +1.89039 q^{58} +8.99058 q^{59} +3.67607 q^{61} +2.67265 q^{62} -4.30242 q^{64} -17.4823 q^{65} -4.61614 q^{67} -11.0919 q^{68} +6.28201 q^{70} +7.09033 q^{71} -1.79948 q^{73} -3.73147 q^{74} -12.8128 q^{76} -7.35795 q^{77} -0.494512 q^{79} +13.0737 q^{80} +3.14698 q^{82} +7.66496 q^{83} +19.7291 q^{85} +17.7285 q^{86} -4.99569 q^{88} -1.70660 q^{89} +7.41598 q^{91} +1.57357 q^{92} +2.92318 q^{94} +22.7902 q^{95} -4.20759 q^{97} +10.5679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * 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 $

$2$	−1.89039	−1.33671	−0.668353	−	0.743844i	$-0.733001\pi$
$2$	−1.89039	−1.33671	−0.668353	+	0.743844i	$0.733001\pi$
$3$	0	0
$3$	0	0
$4$	1.57357	0.786785
$5$	−2.79891	−1.25171	−0.625854	−	0.779940i	$-0.715250\pi$
$5$	−2.79891	−1.25171	−0.625854	+	0.779940i	$0.715250\pi$
$6$	0	0
$7$	1.18730	0.448756	0.224378	−	0.974502i	$-0.427965\pi$
$7$	1.18730	0.448756	0.224378	+	0.974502i	$0.427965\pi$
$8$	0.806117	0.285005
$9$	0	0
$10$	5.29102	1.67317
$11$	−6.19722	−1.86853	−0.934267	−	0.356575i	$-0.883944\pi$
$11$	−6.19722	−1.86853	−0.934267	+	0.356575i	$0.883944\pi$
$12$	0	0
$13$	6.24610	1.73236	0.866179	−	0.499734i	$-0.166569\pi$
$13$	6.24610	1.73236	0.866179	+	0.499734i	$0.166569\pi$
$14$	−2.24445	−0.599855
$15$	0	0
$16$	−4.67102	−1.16775
$17$	−7.04887	−1.70960	−0.854801	−	0.518955i	$-0.826321\pi$
$17$	−7.04887	−1.70960	−0.854801	+	0.518955i	$0.826321\pi$
$18$	0	0
$19$	−8.14253	−1.86802	−0.934012	−	0.357241i	$-0.883717\pi$
$19$	−8.14253	−1.86802	−0.934012	+	0.357241i	$0.883717\pi$
$20$	−4.40428	−0.984826
$21$	0	0
$22$	11.7152	2.49768
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.83388	0.566775
$26$	−11.8076	−2.31565
$27$	0	0
$28$	1.86830	0.353075
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.41381	−0.253927	−0.126964	−	0.991907i	$-0.540523\pi$
$31$	−1.41381	−0.253927	−0.126964	+	0.991907i	$0.540523\pi$
$32$	7.21780	1.27594
$33$	0	0
$34$	13.3251	2.28524
$35$	−3.32313	−0.561712
$36$	0	0
$37$	1.97391	0.324510	0.162255	−	0.986749i	$-0.448123\pi$
$37$	1.97391	0.324510	0.162255	+	0.986749i	$0.448123\pi$
$38$	15.3925	2.49700
$39$	0	0
$40$	−2.25625	−0.356744
$41$	−1.66473	−0.259987	−0.129993	−	0.991515i	$-0.541496\pi$
$41$	−1.66473	−0.259987	−0.129993	+	0.991515i	$0.541496\pi$
$42$	0	0
$43$	−9.37822	−1.43017	−0.715083	−	0.699040i	$-0.753611\pi$
$43$	−9.37822	−1.43017	−0.715083	+	0.699040i	$0.753611\pi$
$44$	−9.75177	−1.47013
$45$	0	0
$46$	−1.89039	−0.278723
$47$	−1.54634	−0.225556	−0.112778	−	0.993620i	$-0.535975\pi$
$47$	−1.54634	−0.225556	−0.112778	+	0.993620i	$0.535975\pi$
$48$	0	0
$49$	−5.59033	−0.798618
$50$	−5.35713	−0.757612
$51$	0	0
$52$	9.82869	1.36299
$53$	−12.0893	−1.66059	−0.830296	−	0.557322i	$-0.811829\pi$
$53$	−12.0893	−1.66059	−0.830296	+	0.557322i	$0.811829\pi$
$54$	0	0
$55$	17.3455	2.33886
$56$	0.957100	0.127898
$57$	0	0
$58$	1.89039	0.248220
$59$	8.99058	1.17047	0.585237	−	0.810863i	$-0.301002\pi$
$59$	8.99058	1.17047	0.585237	+	0.810863i	$0.301002\pi$
$60$	0	0
$61$	3.67607	0.470673	0.235337	−	0.971914i	$-0.424381\pi$
$61$	3.67607	0.470673	0.235337	+	0.971914i	$0.424381\pi$
$62$	2.67265	0.339426
$63$	0	0
$64$	−4.30242	−0.537803
$65$	−17.4823	−2.16841
$66$	0	0
$67$	−4.61614	−0.563951	−0.281975	−	0.959422i	$-0.590990\pi$
$67$	−4.61614	−0.563951	−0.281975	+	0.959422i	$0.590990\pi$
$68$	−11.0919	−1.34509
$69$	0	0
$70$	6.28201	0.750844
$71$	7.09033	0.841467	0.420734	−	0.907184i	$-0.361773\pi$
$71$	7.09033	0.841467	0.420734	+	0.907184i	$0.361773\pi$
$72$	0	0
$73$	−1.79948	−0.210613	−0.105306	−	0.994440i	$-0.533582\pi$
$73$	−1.79948	−0.210613	−0.105306	+	0.994440i	$0.533582\pi$
$74$	−3.73147	−0.433774
$75$	0	0
$76$	−12.8128	−1.46973
$77$	−7.35795	−0.838516
$78$	0	0
$79$	−0.494512	−0.0556369	−0.0278185	−	0.999613i	$-0.508856\pi$
$79$	−0.494512	−0.0556369	−0.0278185	+	0.999613i	$0.508856\pi$
$80$	13.0737	1.46169
$81$	0	0
$82$	3.14698	0.347526
$83$	7.66496	0.841338	0.420669	−	0.907214i	$-0.361795\pi$
$83$	7.66496	0.841338	0.420669	+	0.907214i	$0.361795\pi$
$84$	0	0
$85$	19.7291	2.13993
$86$	17.7285	1.91171
$87$	0	0
$88$	−4.99569	−0.532542
$89$	−1.70660	−0.180899	−0.0904496	−	0.995901i	$-0.528830\pi$
$89$	−1.70660	−0.180899	−0.0904496	+	0.995901i	$0.528830\pi$
$90$	0	0
$91$	7.41598	0.777406
$92$	1.57357	0.164056
$93$	0	0
$94$	2.92318	0.301503
$95$	22.7902	2.33822
$96$	0	0
$97$	−4.20759	−0.427216	−0.213608	−	0.976920i	$-0.568521\pi$
$97$	−4.20759	−0.427216	−0.213608	+	0.976920i	$0.568521\pi$
$98$	10.5679	1.06752
$99$	0	0
$100$	4.45930	0.445930
$101$	−1.92707	−0.191750	−0.0958751	−	0.995393i	$-0.530565\pi$
$101$	−1.92707	−0.191750	−0.0958751	+	0.995393i	$0.530565\pi$
$102$	0	0
$103$	−10.7993	−1.06409	−0.532044	−	0.846717i	$-0.678576\pi$
$103$	−10.7993	−1.06409	−0.532044	+	0.846717i	$0.678576\pi$
$104$	5.03509	0.493731
$105$	0	0
$106$	22.8535	2.21973
$107$	11.3516	1.09740	0.548702	−	0.836018i	$-0.315122\pi$
$107$	11.3516	1.09740	0.548702	+	0.836018i	$0.315122\pi$
$108$	0	0
$109$	−6.24340	−0.598009	−0.299005	−	0.954252i	$-0.596655\pi$
$109$	−6.24340	−0.598009	−0.299005	+	0.954252i	$0.596655\pi$
$110$	−32.7896	−3.12637
$111$	0	0
$112$	−5.54588	−0.524037
$113$	−6.63407	−0.624080	−0.312040	−	0.950069i	$-0.601012\pi$
$113$	−6.63407	−0.624080	−0.312040	+	0.950069i	$0.601012\pi$
$114$	0	0
$115$	−2.79891	−0.260999
$116$	−1.57357	−0.146102
$117$	0	0
$118$	−16.9957	−1.56458
$119$	−8.36911	−0.767195
$120$	0	0
$121$	27.4056	2.49142
$122$	−6.94921	−0.629152
$123$	0	0
$124$	−2.22473	−0.199786
$125$	6.06278	0.542271
$126$	0	0
$127$	−19.8314	−1.75975	−0.879875	−	0.475205i	$-0.842374\pi$
$127$	−19.8314	−1.75975	−0.879875	+	0.475205i	$0.842374\pi$
$128$	−6.30235	−0.557054
$129$	0	0
$130$	33.0483	2.89852
$131$	−7.83058	−0.684161	−0.342081	−	0.939671i	$-0.611132\pi$
$131$	−7.83058	−0.684161	−0.342081	+	0.939671i	$0.611132\pi$
$132$	0	0
$133$	−9.66760	−0.838287
$134$	8.72629	0.753837
$135$	0	0
$136$	−5.68222	−0.487246
$137$	−11.7599	−1.00472	−0.502358	−	0.864660i	$-0.667534\pi$
$137$	−11.7599	−1.00472	−0.502358	+	0.864660i	$0.667534\pi$
$138$	0	0
$139$	−4.32570	−0.366901	−0.183450	−	0.983029i	$-0.558727\pi$
$139$	−4.32570	−0.366901	−0.183450	+	0.983029i	$0.558727\pi$
$140$	−5.22918	−0.441947
$141$	0	0
$142$	−13.4035	−1.12480
$143$	−38.7085	−3.23697
$144$	0	0
$145$	2.79891	0.232436
$146$	3.40171	0.281528
$147$	0	0
$148$	3.10609	0.255319
$149$	2.52296	0.206689	0.103345	−	0.994646i	$-0.467046\pi$
$149$	2.52296	0.206689	0.103345	+	0.994646i	$0.467046\pi$
$150$	0	0
$151$	2.50084	0.203516	0.101758	−	0.994809i	$-0.467553\pi$
$151$	2.50084	0.203516	0.101758	+	0.994809i	$0.467553\pi$
$152$	−6.56383	−0.532397
$153$	0	0
$154$	13.9094	1.12085
$155$	3.95711	0.317843
$156$	0	0
$157$	9.02393	0.720188	0.360094	−	0.932916i	$-0.382745\pi$
$157$	9.02393	0.720188	0.360094	+	0.932916i	$0.382745\pi$
$158$	0.934820	0.0743703
$159$	0	0
$160$	−20.2020	−1.59710
$161$	1.18730	0.0935721
$162$	0	0
$163$	−9.03880	−0.707973	−0.353987	−	0.935250i	$-0.615174\pi$
$163$	−9.03880	−0.707973	−0.353987	+	0.935250i	$0.615174\pi$
$164$	−2.61956	−0.204554
$165$	0	0
$166$	−14.4897	−1.12462
$167$	−14.6448	−1.13325	−0.566623	−	0.823978i	$-0.691750\pi$
$167$	−14.6448	−1.13325	−0.566623	+	0.823978i	$0.691750\pi$
$168$	0	0
$169$	26.0138	2.00106
$170$	−37.2957	−2.86045
$171$	0	0
$172$	−14.7573	−1.12523
$173$	−0.298962	−0.0227297	−0.0113648	−	0.999935i	$-0.503618\pi$
$173$	−0.298962	−0.0227297	−0.0113648	+	0.999935i	$0.503618\pi$
$174$	0	0
$175$	3.36465	0.254344
$176$	28.9473	2.18199
$177$	0	0
$178$	3.22614	0.241809
$179$	−16.5506	−1.23705	−0.618525	−	0.785765i	$-0.712270\pi$
$179$	−16.5506	−1.23705	−0.618525	+	0.785765i	$0.712270\pi$
$180$	0	0
$181$	13.2195	0.982600	0.491300	−	0.870991i	$-0.336522\pi$
$181$	13.2195	0.982600	0.491300	+	0.870991i	$0.336522\pi$
$182$	−14.0191	−1.03916
$183$	0	0
$184$	0.806117	0.0594277
$185$	−5.52480	−0.406192
$186$	0	0
$187$	43.6835	3.19445
$188$	−2.43327	−0.177464
$189$	0	0
$190$	−43.0823	−3.12552
$191$	14.1600	1.02458	0.512289	−	0.858813i	$-0.328797\pi$
$191$	14.1600	1.02458	0.512289	+	0.858813i	$0.328797\pi$
$192$	0	0
$193$	−11.9840	−0.862629	−0.431315	−	0.902202i	$-0.641950\pi$
$193$	−11.9840	−0.862629	−0.431315	+	0.902202i	$0.641950\pi$
$194$	7.95397	0.571062
$195$	0	0
$196$	−8.79677	−0.628341
$197$	−6.08094	−0.433249	−0.216625	−	0.976255i	$-0.569505\pi$
$197$	−6.08094	−0.433249	−0.216625	+	0.976255i	$0.569505\pi$
$198$	0	0
$199$	−15.2478	−1.08089	−0.540445	−	0.841379i	$-0.681744\pi$
$199$	−15.2478	−1.08089	−0.540445	+	0.841379i	$0.681744\pi$
$200$	2.28444	0.161534
$201$	0	0
$202$	3.64291	0.256314
$203$	−1.18730	−0.0833319
$204$	0	0
$205$	4.65941	0.325428
$206$	20.4149	1.42237
$207$	0	0
$208$	−29.1757	−2.02297
$209$	50.4611	3.49047
$210$	0	0
$211$	18.4077	1.26724	0.633620	−	0.773644i	$-0.281568\pi$
$211$	18.4077	1.26724	0.633620	+	0.773644i	$0.281568\pi$
$212$	−19.0234	−1.30653
$213$	0	0
$214$	−21.4590	−1.46691
$215$	26.2488	1.79015
$216$	0	0
$217$	−1.67861	−0.113951
$218$	11.8025	0.799363
$219$	0	0
$220$	27.2943	1.84018
$221$	−44.0280	−2.96164
$222$	0	0
$223$	28.3459	1.89818	0.949092	−	0.315000i	$-0.102004\pi$
$223$	28.3459	1.89818	0.949092	+	0.315000i	$0.102004\pi$
$224$	8.56968	0.572586
$225$	0	0
$226$	12.5410	0.834213
$227$	−9.50318	−0.630748	−0.315374	−	0.948967i	$-0.602130\pi$
$227$	−9.50318	−0.630748	−0.315374	+	0.948967i	$0.602130\pi$
$228$	0	0
$229$	−18.1270	−1.19787	−0.598934	−	0.800798i	$-0.704409\pi$
$229$	−18.1270	−1.19787	−0.598934	+	0.800798i	$0.704409\pi$
$230$	5.29102	0.348880
$231$	0	0
$232$	−0.806117	−0.0529242
$233$	22.6609	1.48456	0.742282	−	0.670088i	$-0.233743\pi$
$233$	22.6609	1.48456	0.742282	+	0.670088i	$0.233743\pi$
$234$	0	0
$235$	4.32805	0.282331
$236$	14.1473	0.920911
$237$	0	0
$238$	15.8209	1.02551
$239$	−22.2664	−1.44030	−0.720148	−	0.693821i	$-0.755926\pi$
$239$	−22.2664	−1.44030	−0.720148	+	0.693821i	$0.755926\pi$
$240$	0	0
$241$	−14.8982	−0.959679	−0.479839	−	0.877356i	$-0.659305\pi$
$241$	−14.8982	−0.959679	−0.479839	+	0.877356i	$0.659305\pi$
$242$	−51.8072	−3.33030
$243$	0	0
$244$	5.78456	0.370319
$245$	15.6468	0.999637
$246$	0	0
$247$	−50.8591	−3.23609
$248$	−1.13969	−0.0723707
$249$	0	0
$250$	−11.4610	−0.724858
$251$	−10.6481	−0.672100	−0.336050	−	0.941844i	$-0.609091\pi$
$251$	−10.6481	−0.672100	−0.336050	+	0.941844i	$0.609091\pi$
$252$	0	0
$253$	−6.19722	−0.389616
$254$	37.4890	2.35227
$255$	0	0
$256$	20.5187	1.28242
$257$	−22.9546	−1.43187	−0.715934	−	0.698168i	$-0.753999\pi$
$257$	−22.9546	−1.43187	−0.715934	+	0.698168i	$0.753999\pi$
$258$	0	0
$259$	2.34362	0.145626
$260$	−27.5096	−1.70607
$261$	0	0
$262$	14.8029	0.914523
$263$	−15.4750	−0.954226	−0.477113	−	0.878842i	$-0.658317\pi$
$263$	−15.4750	−0.954226	−0.477113	+	0.878842i	$0.658317\pi$
$264$	0	0
$265$	33.8368	2.07858
$266$	18.2755	1.12054
$267$	0	0
$268$	−7.26382	−0.443708
$269$	−2.15942	−0.131662	−0.0658310	−	0.997831i	$-0.520970\pi$
$269$	−2.15942	−0.131662	−0.0658310	+	0.997831i	$0.520970\pi$
$270$	0	0
$271$	−9.85502	−0.598650	−0.299325	−	0.954151i	$-0.596761\pi$
$271$	−9.85502	−0.598650	−0.299325	+	0.954151i	$0.596761\pi$
$272$	32.9254	1.99640
$273$	0	0
$274$	22.2308	1.34301
$275$	−17.5622	−1.05904
$276$	0	0
$277$	0.00974093	0.000585276 0	0.000292638	−	1.00000i	$-0.499907\pi$
$277$	0.00974093	0.000585276 0	0.000292638		1.00000i	$0.499907\pi$
$278$	8.17725	0.490439
$279$	0	0
$280$	−2.67883	−0.160091
$281$	22.9104	1.36672	0.683360	−	0.730082i	$-0.260518\pi$
$281$	22.9104	1.36672	0.683360	+	0.730082i	$0.260518\pi$
$282$	0	0
$283$	23.1119	1.37386	0.686930	−	0.726723i	$-0.258958\pi$
$283$	23.1119	1.37386	0.686930	+	0.726723i	$0.258958\pi$
$284$	11.1571	0.662054
$285$	0	0
$286$	73.1741	4.32688
$287$	−1.97652	−0.116671
$288$	0	0
$289$	32.6866	1.92274
$290$	−5.29102	−0.310699
$291$	0	0
$292$	−2.83160	−0.165707
$293$	21.4606	1.25374	0.626872	−	0.779122i	$-0.284335\pi$
$293$	21.4606	1.25374	0.626872	+	0.779122i	$0.284335\pi$
$294$	0	0
$295$	−25.1638	−1.46509
$296$	1.59121	0.0924870
$297$	0	0
$298$	−4.76938	−0.276283
$299$	6.24610	0.361222
$300$	0	0
$301$	−11.1347	−0.641796
$302$	−4.72757	−0.272041
$303$	0	0
$304$	38.0339	2.18139
$305$	−10.2890	−0.589146
$306$	0	0
$307$	−13.5522	−0.773465	−0.386733	−	0.922192i	$-0.626396\pi$
$307$	−13.5522	−0.773465	−0.386733	+	0.922192i	$0.626396\pi$
$308$	−11.5782	−0.659732
$309$	0	0
$310$	−7.48049	−0.424863
$311$	−17.2533	−0.978342	−0.489171	−	0.872188i	$-0.662701\pi$
$311$	−17.2533	−0.978342	−0.489171	+	0.872188i	$0.662701\pi$
$312$	0	0
$313$	26.2233	1.48223	0.741113	−	0.671380i	$-0.234298\pi$
$313$	26.2233	1.48223	0.741113	+	0.671380i	$0.234298\pi$
$314$	−17.0587	−0.962681
$315$	0	0
$316$	−0.778150	−0.0437743
$317$	33.1244	1.86045	0.930225	−	0.366989i	$-0.119611\pi$
$317$	33.1244	1.86045	0.930225	+	0.366989i	$0.119611\pi$
$318$	0	0
$319$	6.19722	0.346978
$320$	12.0421	0.673173
$321$	0	0
$322$	−2.24445	−0.125078
$323$	57.3957	3.19358
$324$	0	0
$325$	17.7007	0.981857
$326$	17.0868	0.946353
$327$	0	0
$328$	−1.34196	−0.0740976
$329$	−1.83596	−0.101220
$330$	0	0
$331$	28.0026	1.53916	0.769581	−	0.638549i	$-0.220465\pi$
$331$	28.0026	1.53916	0.769581	+	0.638549i	$0.220465\pi$
$332$	12.0614	0.661953
$333$	0	0
$334$	27.6843	1.51482
$335$	12.9201	0.705902
$336$	0	0
$337$	22.2724	1.21325	0.606626	−	0.794987i	$-0.292522\pi$
$337$	22.2724	1.21325	0.606626	+	0.794987i	$0.292522\pi$
$338$	−49.1762	−2.67483
$339$	0	0
$340$	31.0452	1.68366
$341$	8.76168	0.474472
$342$	0	0
$343$	−14.9485	−0.807141
$344$	−7.55995	−0.407605
$345$	0	0
$346$	0.565155	0.0303829
$347$	−28.3292	−1.52079	−0.760396	−	0.649460i	$-0.774995\pi$
$347$	−28.3292	−1.52079	−0.760396	+	0.649460i	$0.774995\pi$
$348$	0	0
$349$	−25.3069	−1.35465	−0.677323	−	0.735685i	$-0.736860\pi$
$349$	−25.3069	−1.35465	−0.677323	+	0.735685i	$0.736860\pi$
$350$	−6.36050	−0.339983
$351$	0	0
$352$	−44.7304	−2.38414
$353$	25.9803	1.38279	0.691396	−	0.722476i	$-0.256996\pi$
$353$	25.9803	1.38279	0.691396	+	0.722476i	$0.256996\pi$
$354$	0	0
$355$	−19.8452	−1.05327
$356$	−2.68545	−0.142329
$357$	0	0
$358$	31.2871	1.65357
$359$	7.27200	0.383802	0.191901	−	0.981414i	$-0.438535\pi$
$359$	7.27200	0.383802	0.191901	+	0.981414i	$0.438535\pi$
$360$	0	0
$361$	47.3008	2.48951
$362$	−24.9900	−1.31345
$363$	0	0
$364$	11.6696	0.611652
$365$	5.03657	0.263626
$366$	0	0
$367$	−8.20991	−0.428554	−0.214277	−	0.976773i	$-0.568740\pi$
$367$	−8.20991	−0.428554	−0.214277	+	0.976773i	$0.568740\pi$
$368$	−4.67102	−0.243494
$369$	0	0
$370$	10.4440	0.542959
$371$	−14.3536	−0.745201
$372$	0	0
$373$	−10.1481	−0.525446	−0.262723	−	0.964871i	$-0.584621\pi$
$373$	−10.1481	−0.525446	−0.262723	+	0.964871i	$0.584621\pi$
$374$	−82.5787	−4.27004
$375$	0	0
$376$	−1.24653	−0.0642848
$377$	−6.24610	−0.321691
$378$	0	0
$379$	13.1925	0.677655	0.338828	−	0.940848i	$-0.389970\pi$
$379$	13.1925	0.677655	0.338828	+	0.940848i	$0.389970\pi$
$380$	35.8620	1.83968
$381$	0	0
$382$	−26.7678	−1.36956
$383$	12.9409	0.661249	0.330624	−	0.943762i	$-0.392741\pi$
$383$	12.9409	0.661249	0.330624	+	0.943762i	$0.392741\pi$
$384$	0	0
$385$	20.5942	1.04958
$386$	22.6545	1.15308
$387$	0	0
$388$	−6.62093	−0.336127
$389$	−14.2605	−0.723034	−0.361517	−	0.932366i	$-0.617741\pi$
$389$	−14.2605	−0.723034	−0.361517	+	0.932366i	$0.617741\pi$
$390$	0	0
$391$	−7.04887	−0.356477
$392$	−4.50646	−0.227611
$393$	0	0
$394$	11.4953	0.579127
$395$	1.38409	0.0696413
$396$	0	0
$397$	1.76112	0.0883881	0.0441940	−	0.999023i	$-0.485928\pi$
$397$	1.76112	0.0883881	0.0441940	+	0.999023i	$0.485928\pi$
$398$	28.8243	1.44483
$399$	0	0
$400$	−13.2371	−0.661854
$401$	26.2404	1.31038	0.655191	−	0.755464i	$-0.272588\pi$
$401$	26.2404	1.31038	0.655191	+	0.755464i	$0.272588\pi$
$402$	0	0
$403$	−8.83079	−0.439893
$404$	−3.03238	−0.150866
$405$	0	0
$406$	2.24445	0.111390
$407$	−12.2328	−0.606357
$408$	0	0
$409$	−23.8524	−1.17943	−0.589714	−	0.807612i	$-0.700759\pi$
$409$	−23.8524	−1.17943	−0.589714	+	0.807612i	$0.700759\pi$
$410$	−8.80810	−0.435001
$411$	0	0
$412$	−16.9935	−0.837209
$413$	10.6745	0.525257
$414$	0	0
$415$	−21.4535	−1.05311
$416$	45.0832	2.21038
$417$	0	0
$418$	−95.3911	−4.66573
$419$	9.23032	0.450931	0.225465	−	0.974251i	$-0.427610\pi$
$419$	9.23032	0.450931	0.225465	+	0.974251i	$0.427610\pi$
$420$	0	0
$421$	27.0970	1.32063	0.660313	−	0.750991i	$-0.270424\pi$
$421$	27.0970	1.32063	0.660313	+	0.750991i	$0.270424\pi$
$422$	−34.7978	−1.69393
$423$	0	0
$424$	−9.74539	−0.473278
$425$	−19.9756	−0.968960
$426$	0	0
$427$	4.36459	0.211217
$428$	17.8626	0.863422
$429$	0	0
$430$	−49.6204	−2.39291
$431$	3.22106	0.155153	0.0775764	−	0.996986i	$-0.475282\pi$
$431$	3.22106	0.155153	0.0775764	+	0.996986i	$0.475282\pi$
$432$	0	0
$433$	−2.53860	−0.121997	−0.0609987	−	0.998138i	$-0.519429\pi$
$433$	−2.53860	−0.121997	−0.0609987	+	0.998138i	$0.519429\pi$
$434$	3.17322	0.152320
$435$	0	0
$436$	−9.82443	−0.470505
$437$	−8.14253	−0.389510
$438$	0	0
$439$	−39.0461	−1.86357	−0.931783	−	0.363015i	$-0.881747\pi$
$439$	−39.0461	−1.86357	−0.931783	+	0.363015i	$0.881747\pi$
$440$	13.9825	0.666588
$441$	0	0
$442$	83.2300	3.95885
$443$	3.13047	0.148733	0.0743665	−	0.997231i	$-0.476307\pi$
$443$	3.13047	0.148733	0.0743665	+	0.997231i	$0.476307\pi$
$444$	0	0
$445$	4.77661	0.226433
$446$	−53.5848	−2.53732
$447$	0	0
$448$	−5.10826	−0.241342
$449$	−16.5324	−0.780214	−0.390107	−	0.920770i	$-0.627562\pi$
$449$	−16.5324	−0.780214	−0.390107	+	0.920770i	$0.627562\pi$
$450$	0	0
$451$	10.3167	0.485794
$452$	−10.4392	−0.491017
$453$	0	0
$454$	17.9647	0.843125
$455$	−20.7566	−0.973086
$456$	0	0
$457$	−14.8804	−0.696078	−0.348039	−	0.937480i	$-0.613152\pi$
$457$	−14.8804	−0.696078	−0.348039	+	0.937480i	$0.613152\pi$
$458$	34.2672	1.60120
$459$	0	0
$460$	−4.40428	−0.205350
$461$	12.6272	0.588106	0.294053	−	0.955789i	$-0.404996\pi$
$461$	12.6272	0.588106	0.294053	+	0.955789i	$0.404996\pi$
$462$	0	0
$463$	2.90878	0.135183	0.0675913	−	0.997713i	$-0.478469\pi$
$463$	2.90878	0.135183	0.0675913	+	0.997713i	$0.478469\pi$
$464$	4.67102	0.216847
$465$	0	0
$466$	−42.8379	−1.98443
$467$	30.7812	1.42438	0.712192	−	0.701985i	$-0.247703\pi$
$467$	30.7812	1.42438	0.712192	+	0.701985i	$0.247703\pi$
$468$	0	0
$469$	−5.48072	−0.253076
$470$	−8.18170	−0.377394
$471$	0	0
$472$	7.24746	0.333591
$473$	58.1190	2.67231
$474$	0	0
$475$	−23.0749	−1.05875
$476$	−13.1694	−0.603617
$477$	0	0
$478$	42.0922	1.92525
$479$	15.2580	0.697158	0.348579	−	0.937279i	$-0.386664\pi$
$479$	15.2580	0.697158	0.348579	+	0.937279i	$0.386664\pi$
$480$	0	0
$481$	12.3293	0.562167
$482$	28.1634	1.28281
$483$	0	0
$484$	43.1246	1.96021
$485$	11.7766	0.534750
$486$	0	0
$487$	−13.0548	−0.591570	−0.295785	−	0.955255i	$-0.595581\pi$
$487$	−13.0548	−0.591570	−0.295785	+	0.955255i	$0.595581\pi$
$488$	2.96335	0.134144
$489$	0	0
$490$	−29.5785	−1.33622
$491$	32.2878	1.45713	0.728563	−	0.684978i	$-0.240188\pi$
$491$	32.2878	1.45713	0.728563	+	0.684978i	$0.240188\pi$
$492$	0	0
$493$	7.04887	0.317465
$494$	96.1434	4.32570
$495$	0	0
$496$	6.60392	0.296525
$497$	8.41833	0.377614
$498$	0	0
$499$	14.1246	0.632306	0.316153	−	0.948708i	$-0.397609\pi$
$499$	14.1246	0.632306	0.316153	+	0.948708i	$0.397609\pi$
$500$	9.54021	0.426651
$501$	0	0
$502$	20.1290	0.898401
$503$	7.12971	0.317898	0.158949	−	0.987287i	$-0.449189\pi$
$503$	7.12971	0.317898	0.158949	+	0.987287i	$0.449189\pi$
$504$	0	0
$505$	5.39368	0.240016
$506$	11.7152	0.520803
$507$	0	0
$508$	−31.2061	−1.38455
$509$	−28.2834	−1.25364	−0.626819	−	0.779165i	$-0.715644\pi$
$509$	−28.2834	−1.25364	−0.626819	+	0.779165i	$0.715644\pi$
$510$	0	0
$511$	−2.13651	−0.0945138
$512$	−26.1837	−1.15717
$513$	0	0
$514$	43.3931	1.91399
$515$	30.2263	1.33193
$516$	0	0
$517$	9.58299	0.421460
$518$	−4.43036	−0.194659
$519$	0	0
$520$	−14.0927	−0.618008
$521$	1.62872	0.0713556	0.0356778	−	0.999363i	$-0.488641\pi$
$521$	1.62872	0.0713556	0.0356778	+	0.999363i	$0.488641\pi$
$522$	0	0
$523$	−21.7338	−0.950351	−0.475175	−	0.879891i	$-0.657615\pi$
$523$	−21.7338	−0.950351	−0.475175	+	0.879891i	$0.657615\pi$
$524$	−12.3220	−0.538288
$525$	0	0
$526$	29.2537	1.27552
$527$	9.96575	0.434115
$528$	0	0
$529$	1.00000	0.0434783
$530$	−63.9647	−2.77845
$531$	0	0
$532$	−15.2127	−0.659552
$533$	−10.3981	−0.450390
$534$	0	0
$535$	−31.7722	−1.37363
$536$	−3.72115	−0.160729
$537$	0	0
$538$	4.08214	0.175994
$539$	34.6445	1.49224
$540$	0	0
$541$	−15.9206	−0.684481	−0.342241	−	0.939612i	$-0.611186\pi$
$541$	−15.9206	−0.684481	−0.342241	+	0.939612i	$0.611186\pi$
$542$	18.6298	0.800219
$543$	0	0
$544$	−50.8774	−2.18135
$545$	17.4747	0.748533
$546$	0	0
$547$	20.8157	0.890013	0.445007	−	0.895527i	$-0.353201\pi$
$547$	20.8157	0.890013	0.445007	+	0.895527i	$0.353201\pi$
$548$	−18.5050	−0.790496
$549$	0	0
$550$	33.1993	1.41562
$551$	8.14253	0.346883
$552$	0	0
$553$	−0.587133	−0.0249674
$554$	−0.0184141	−0.000782342 0
$555$	0	0
$556$	−6.80679	−0.288672
$557$	27.4076	1.16130	0.580650	−	0.814154i	$-0.302799\pi$
$557$	27.4076	1.16130	0.580650	+	0.814154i	$0.302799\pi$
$558$	0	0
$559$	−58.5774	−2.47756
$560$	15.5224	0.655941
$561$	0	0
$562$	−43.3095	−1.82690
$563$	30.0796	1.26770	0.633852	−	0.773454i	$-0.281473\pi$
$563$	30.0796	1.26770	0.633852	+	0.773454i	$0.281473\pi$
$564$	0	0
$565$	18.5681	0.781167
$566$	−43.6905	−1.83645
$567$	0	0
$568$	5.71564	0.239823
$569$	9.41414	0.394661	0.197331	−	0.980337i	$-0.436773\pi$
$569$	9.41414	0.394661	0.197331	+	0.980337i	$0.436773\pi$
$570$	0	0
$571$	17.7368	0.742262	0.371131	−	0.928580i	$-0.378970\pi$
$571$	17.7368	0.742262	0.371131	+	0.928580i	$0.378970\pi$
$572$	−60.9106	−2.54680
$573$	0	0
$574$	3.73640	0.155954
$575$	2.83388	0.118181
$576$	0	0
$577$	−25.3452	−1.05513	−0.527566	−	0.849514i	$-0.676895\pi$
$577$	−25.3452	−1.05513	−0.527566	+	0.849514i	$0.676895\pi$
$578$	−61.7904	−2.57014
$579$	0	0
$580$	4.40428	0.182878
$581$	9.10058	0.377556
$582$	0	0
$583$	74.9201	3.10287
$584$	−1.45059	−0.0600258
$585$	0	0
$586$	−40.5690	−1.67589
$587$	−15.1599	−0.625718	−0.312859	−	0.949800i	$-0.601287\pi$
$587$	−15.1599	−0.625718	−0.312859	+	0.949800i	$0.601287\pi$
$588$	0	0
$589$	11.5120	0.474342
$590$	47.5693	1.95840
$591$	0	0
$592$	−9.22019	−0.378947
$593$	−5.55550	−0.228137	−0.114069	−	0.993473i	$-0.536388\pi$
$593$	−5.55550	−0.228137	−0.114069	+	0.993473i	$0.536388\pi$
$594$	0	0
$595$	23.4243	0.960304
$596$	3.97006	0.162620
$597$	0	0
$598$	−11.8076	−0.482847
$599$	36.7131	1.50005	0.750027	−	0.661407i	$-0.230040\pi$
$599$	36.7131	1.50005	0.750027	+	0.661407i	$0.230040\pi$
$600$	0	0
$601$	6.01049	0.245173	0.122586	−	0.992458i	$-0.460881\pi$
$601$	6.01049	0.245173	0.122586	+	0.992458i	$0.460881\pi$
$602$	21.0490	0.857893
$603$	0	0
$604$	3.93525	0.160123
$605$	−76.7057	−3.11853
$606$	0	0
$607$	34.2355	1.38958	0.694789	−	0.719214i	$-0.255498\pi$
$607$	34.2355	1.38958	0.694789	+	0.719214i	$0.255498\pi$
$608$	−58.7712	−2.38349
$609$	0	0
$610$	19.4502	0.787515
$611$	−9.65858	−0.390744
$612$	0	0
$613$	4.47230	0.180634	0.0903172	−	0.995913i	$-0.471212\pi$
$613$	4.47230	0.180634	0.0903172	+	0.995913i	$0.471212\pi$
$614$	25.6189	1.03390
$615$	0	0
$616$	−5.93137	−0.238982
$617$	−38.3098	−1.54229	−0.771147	−	0.636657i	$-0.780317\pi$
$617$	−38.3098	−1.54229	−0.771147	+	0.636657i	$0.780317\pi$
$618$	0	0
$619$	40.1909	1.61541	0.807704	−	0.589589i	$-0.200710\pi$
$619$	40.1909	1.61541	0.807704	+	0.589589i	$0.200710\pi$
$620$	6.22680	0.250074
$621$	0	0
$622$	32.6154	1.30776
$623$	−2.02624	−0.0811796
$624$	0	0
$625$	−31.1385	−1.24554
$626$	−49.5722	−1.98130
$627$	0	0
$628$	14.1998	0.566634
$629$	−13.9139	−0.554782
$630$	0	0
$631$	−35.0523	−1.39541	−0.697705	−	0.716385i	$-0.745795\pi$
$631$	−35.0523	−1.39541	−0.697705	+	0.716385i	$0.745795\pi$
$632$	−0.398635	−0.0158568
$633$	0	0
$634$	−62.6180	−2.48688
$635$	55.5062	2.20269
$636$	0	0
$637$	−34.9178	−1.38349
$638$	−11.7152	−0.463808
$639$	0	0
$640$	17.6397	0.697270
$641$	3.46072	0.136690	0.0683452	−	0.997662i	$-0.478228\pi$
$641$	3.46072	0.136690	0.0683452	+	0.997662i	$0.478228\pi$
$642$	0	0
$643$	−42.0229	−1.65722	−0.828611	−	0.559825i	$-0.810868\pi$
$643$	−42.0229	−1.65722	−0.828611	+	0.559825i	$0.810868\pi$
$644$	1.86830	0.0736212
$645$	0	0
$646$	−108.500	−4.26888
$647$	41.4794	1.63072	0.815362	−	0.578951i	$-0.196538\pi$
$647$	41.4794	1.63072	0.815362	+	0.578951i	$0.196538\pi$
$648$	0	0
$649$	−55.7166	−2.18707
$650$	−33.4612	−1.31246
$651$	0	0
$652$	−14.2232	−0.557023
$653$	50.2218	1.96533	0.982666	−	0.185386i	$-0.0593535\pi$
$653$	50.2218	1.96533	0.982666	+	0.185386i	$0.0593535\pi$
$654$	0	0
$655$	21.9171	0.856371
$656$	7.77596	0.303600
$657$	0	0
$658$	3.47068	0.135301
$659$	−26.7507	−1.04206	−0.521029	−	0.853539i	$-0.674452\pi$
$659$	−26.7507	−1.04206	−0.521029	+	0.853539i	$0.674452\pi$
$660$	0	0
$661$	−0.149639	−0.00582028	−0.00291014	−	0.999996i	$-0.500926\pi$
$661$	−0.149639	−0.00582028	−0.00291014	+	0.999996i	$0.500926\pi$
$662$	−52.9358	−2.05741
$663$	0	0
$664$	6.17885	0.239786
$665$	27.0587	1.04929
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−23.0446	−0.891621
$669$	0	0
$670$	−24.4241	−0.943584
$671$	−22.7815	−0.879468
$672$	0	0
$673$	34.3044	1.32234	0.661168	−	0.750238i	$-0.270061\pi$
$673$	34.3044	1.32234	0.661168	+	0.750238i	$0.270061\pi$
$674$	−42.1034	−1.62176
$675$	0	0
$676$	40.9346	1.57441
$677$	50.0361	1.92304	0.961522	−	0.274729i	$-0.0885880\pi$
$677$	50.0361	1.92304	0.961522	+	0.274729i	$0.0885880\pi$
$678$	0	0
$679$	−4.99565	−0.191716
$680$	15.9040	0.609890
$681$	0	0
$682$	−16.5630	−0.634230
$683$	11.4762	0.439123	0.219561	−	0.975599i	$-0.429537\pi$
$683$	11.4762	0.439123	0.219561	+	0.975599i	$0.429537\pi$
$684$	0	0
$685$	32.9149	1.25761
$686$	28.2584	1.07891
$687$	0	0
$688$	43.8058	1.67008
$689$	−75.5110	−2.87674
$690$	0	0
$691$	8.63938	0.328658	0.164329	−	0.986406i	$-0.447454\pi$
$691$	8.63938	0.328658	0.164329	+	0.986406i	$0.447454\pi$
$692$	−0.470438	−0.0178834
$693$	0	0
$694$	53.5532	2.03285
$695$	12.1072	0.459253
$696$	0	0
$697$	11.7344	0.444474
$698$	47.8399	1.81077
$699$	0	0
$700$	5.29452	0.200114
$701$	−7.38440	−0.278905	−0.139452	−	0.990229i	$-0.544534\pi$
$701$	−7.38440	−0.278905	−0.139452	+	0.990229i	$0.544534\pi$
$702$	0	0
$703$	−16.0727	−0.606192
$704$	26.6631	1.00490
$705$	0	0
$706$	−49.1129	−1.84839
$707$	−2.28800	−0.0860491
$708$	0	0
$709$	−18.9785	−0.712751	−0.356376	−	0.934343i	$-0.615988\pi$
$709$	−18.9785	−0.712751	−0.356376	+	0.934343i	$0.615988\pi$
$710$	37.5151	1.40792
$711$	0	0
$712$	−1.37572	−0.0515572
$713$	−1.41381	−0.0529475
$714$	0	0
$715$	108.341	4.05174
$716$	−26.0436	−0.973294
$717$	0	0
$718$	−13.7469	−0.513030
$719$	−18.3313	−0.683643	−0.341822	−	0.939765i	$-0.611044\pi$
$719$	−18.3313	−0.683643	−0.341822	+	0.939765i	$0.611044\pi$
$720$	0	0
$721$	−12.8220	−0.477516
$722$	−89.4169	−3.32775
$723$	0	0
$724$	20.8019	0.773095
$725$	−2.83388	−0.105247
$726$	0	0
$727$	4.30446	0.159644	0.0798218	−	0.996809i	$-0.474565\pi$
$727$	4.30446	0.159644	0.0798218	+	0.996809i	$0.474565\pi$
$728$	5.97815	0.221565
$729$	0	0
$730$	−9.52107	−0.352391
$731$	66.1059	2.44502
$732$	0	0
$733$	−10.8969	−0.402488	−0.201244	−	0.979541i	$-0.564498\pi$
$733$	−10.8969	−0.402488	−0.201244	+	0.979541i	$0.564498\pi$
$734$	15.5199	0.572851
$735$	0	0
$736$	7.21780	0.266052
$737$	28.6072	1.05376
$738$	0	0
$739$	52.8394	1.94373	0.971865	−	0.235539i	$-0.0756854\pi$
$739$	52.8394	1.94373	0.971865	+	0.235539i	$0.0756854\pi$
$740$	−8.69367	−0.319586
$741$	0	0
$742$	27.1339	0.996115
$743$	−35.0915	−1.28738	−0.643692	−	0.765285i	$-0.722598\pi$
$743$	−35.0915	−1.28738	−0.643692	+	0.765285i	$0.722598\pi$
$744$	0	0
$745$	−7.06154	−0.258715
$746$	19.1838	0.702368
$747$	0	0
$748$	68.7390	2.51335
$749$	13.4778	0.492467
$750$	0	0
$751$	38.1516	1.39217	0.696085	−	0.717959i	$-0.254924\pi$
$751$	38.1516	1.39217	0.696085	+	0.717959i	$0.254924\pi$
$752$	7.22296	0.263394
$753$	0	0
$754$	11.8076	0.430006
$755$	−6.99962	−0.254742
$756$	0	0
$757$	4.57024	0.166108	0.0830541	−	0.996545i	$-0.473533\pi$
$757$	4.57024	0.166108	0.0830541	+	0.996545i	$0.473533\pi$
$758$	−24.9390	−0.905826
$759$	0	0
$760$	18.3716	0.666406
$761$	−11.8747	−0.430456	−0.215228	−	0.976564i	$-0.569049\pi$
$761$	−11.8747	−0.430456	−0.215228	+	0.976564i	$0.569049\pi$
$762$	0	0
$763$	−7.41277	−0.268360
$764$	22.2817	0.806124
$765$	0	0
$766$	−24.4633	−0.883896
$767$	56.1561	2.02768
$768$	0	0
$769$	5.15809	0.186006	0.0930028	−	0.995666i	$-0.470353\pi$
$769$	5.15809	0.186006	0.0930028	+	0.995666i	$0.470353\pi$
$770$	−38.9310	−1.40298
$771$	0	0
$772$	−18.8577	−0.678704
$773$	−40.4175	−1.45372	−0.726858	−	0.686788i	$-0.759020\pi$
$773$	−40.4175	−1.45372	−0.726858	+	0.686788i	$0.759020\pi$
$774$	0	0
$775$	−4.00655	−0.143920
$776$	−3.39181	−0.121759
$777$	0	0
$778$	26.9578	0.966484
$779$	13.5551	0.485661
$780$	0	0
$781$	−43.9404	−1.57231
$782$	13.3251	0.476505
$783$	0	0
$784$	26.1125	0.932590
$785$	−25.2571	−0.901466
$786$	0	0
$787$	−18.0838	−0.644617	−0.322309	−	0.946635i	$-0.604459\pi$
$787$	−18.0838	−0.644617	−0.322309	+	0.946635i	$0.604459\pi$
$788$	−9.56879	−0.340874
$789$	0	0
$790$	−2.61647	−0.0930900
$791$	−7.87661	−0.280060
$792$	0	0
$793$	22.9611	0.815374
$794$	−3.32920	−0.118149
$795$	0	0
$796$	−23.9935	−0.850429
$797$	47.6334	1.68726	0.843631	−	0.536924i	$-0.180414\pi$
$797$	47.6334	1.68726	0.843631	+	0.536924i	$0.180414\pi$
$798$	0	0
$799$	10.8999	0.385612
$800$	20.4544	0.723171
$801$	0	0
$802$	−49.6045	−1.75160
$803$	11.1518	0.393537
$804$	0	0
$805$	−3.32313	−0.117125
$806$	16.6936	0.588008
$807$	0	0
$808$	−1.55344	−0.0546499
$809$	−10.5727	−0.371718	−0.185859	−	0.982576i	$-0.559507\pi$
$809$	−10.5727	−0.371718	−0.185859	+	0.982576i	$0.559507\pi$
$810$	0	0
$811$	40.7716	1.43169	0.715843	−	0.698261i	$-0.246043\pi$
$811$	40.7716	1.43169	0.715843	+	0.698261i	$0.246043\pi$
$812$	−1.86830	−0.0655643
$813$	0	0
$814$	23.1247	0.810522
$815$	25.2987	0.886176
$816$	0	0
$817$	76.3625	2.67158
$818$	45.0904	1.57655
$819$	0	0
$820$	7.33192	0.256042
$821$	3.49449	0.121958	0.0609792	−	0.998139i	$-0.480578\pi$
$821$	3.49449	0.121958	0.0609792	+	0.998139i	$0.480578\pi$
$822$	0	0
$823$	49.0706	1.71049	0.855247	−	0.518221i	$-0.173406\pi$
$823$	49.0706	1.71049	0.855247	+	0.518221i	$0.173406\pi$
$824$	−8.70551	−0.303271
$825$	0	0
$826$	−20.1789	−0.702115
$827$	13.5475	0.471094	0.235547	−	0.971863i	$-0.424312\pi$
$827$	13.5475	0.471094	0.235547	+	0.971863i	$0.424312\pi$
$828$	0	0
$829$	23.8237	0.827430	0.413715	−	0.910406i	$-0.364231\pi$
$829$	23.8237	0.827430	0.413715	+	0.910406i	$0.364231\pi$
$830$	40.5554	1.40770
$831$	0	0
$832$	−26.8734	−0.931667
$833$	39.4055	1.36532
$834$	0	0
$835$	40.9893	1.41849
$836$	79.4041	2.74625
$837$	0	0
$838$	−17.4489	−0.602762
$839$	−2.35344	−0.0812496	−0.0406248	−	0.999174i	$-0.512935\pi$
$839$	−2.35344	−0.0812496	−0.0406248	+	0.999174i	$0.512935\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−51.2238	−1.76529
$843$	0	0
$844$	28.9659	0.997046
$845$	−72.8102	−2.50475
$846$	0	0
$847$	32.5386	1.11804
$848$	56.4693	1.93916
$849$	0	0
$850$	37.7617	1.29522
$851$	1.97391	0.0676649
$852$	0	0
$853$	36.2124	1.23989	0.619944	−	0.784646i	$-0.287155\pi$
$853$	36.2124	1.23989	0.619944	+	0.784646i	$0.287155\pi$
$854$	−8.25078	−0.282336
$855$	0	0
$856$	9.15075	0.312766
$857$	0.0147245	0.000502980 0	0.000251490	−	1.00000i	$-0.499920\pi$
$857$	0.0147245	0.000502980 0	0.000251490		1.00000i	$0.499920\pi$
$858$	0	0
$859$	−54.1756	−1.84845	−0.924225	−	0.381849i	$-0.875287\pi$
$859$	−54.1756	−1.84845	−0.924225	+	0.381849i	$0.875287\pi$
$860$	41.3043	1.40846
$861$	0	0
$862$	−6.08905	−0.207394
$863$	−25.2505	−0.859537	−0.429769	−	0.902939i	$-0.641405\pi$
$863$	−25.2505	−0.859537	−0.429769	+	0.902939i	$0.641405\pi$
$864$	0	0
$865$	0.836767	0.0284509
$866$	4.79895	0.163075
$867$	0	0
$868$	−2.64141	−0.0896553
$869$	3.06460	0.103960
$870$	0	0
$871$	−28.8329	−0.976964
$872$	−5.03291	−0.170436
$873$	0	0
$874$	15.3925	0.520661
$875$	7.19832	0.243348
$876$	0	0
$877$	−52.7351	−1.78074	−0.890369	−	0.455239i	$-0.849554\pi$
$877$	−52.7351	−1.78074	−0.890369	+	0.455239i	$0.849554\pi$
$878$	73.8122	2.49104
$879$	0	0
$880$	−81.0209	−2.73121
$881$	−38.1438	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$881$	−38.1438	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$882$	0	0
$883$	−40.4393	−1.36089	−0.680446	−	0.732798i	$-0.738214\pi$
$883$	−40.4393	−1.36089	−0.680446	+	0.732798i	$0.738214\pi$
$884$	−69.2812	−2.33018
$885$	0	0
$886$	−5.91780	−0.198812
$887$	−45.3485	−1.52265	−0.761327	−	0.648368i	$-0.775452\pi$
$887$	−45.3485	−1.52265	−0.761327	+	0.648368i	$0.775452\pi$
$888$	0	0
$889$	−23.5457	−0.789698
$890$	−9.02965	−0.302675
$891$	0	0
$892$	44.6043	1.49346
$893$	12.5911	0.421345
$894$	0	0
$895$	46.3236	1.54843
$896$	−7.48276	−0.249982
$897$	0	0
$898$	31.2527	1.04292
$899$	1.41381	0.0471531
$900$	0	0
$901$	85.2159	2.83895
$902$	−19.5025	−0.649364
$903$	0	0
$904$	−5.34783	−0.177866
$905$	−37.0002	−1.22993
$906$	0	0
$907$	−13.8779	−0.460809	−0.230404	−	0.973095i	$-0.574005\pi$
$907$	−13.8779	−0.460809	−0.230404	+	0.973095i	$0.574005\pi$
$908$	−14.9539	−0.496263
$909$	0	0
$910$	39.2381	1.30073
$911$	−26.2383	−0.869312	−0.434656	−	0.900597i	$-0.643130\pi$
$911$	−26.2383	−0.869312	−0.434656	+	0.900597i	$0.643130\pi$
$912$	0	0
$913$	−47.5015	−1.57207
$914$	28.1298	0.930452
$915$	0	0
$916$	−28.5242	−0.942466
$917$	−9.29723	−0.307022
$918$	0	0
$919$	38.3570	1.26528	0.632640	−	0.774446i	$-0.281971\pi$
$919$	38.3570	1.26528	0.632640	+	0.774446i	$0.281971\pi$
$920$	−2.25625	−0.0743862
$921$	0	0
$922$	−23.8703	−0.786125
$923$	44.2869	1.45772
$924$	0	0
$925$	5.59383	0.183924
$926$	−5.49873	−0.180699
$927$	0	0
$928$	−7.21780	−0.236936
$929$	33.1149	1.08646	0.543232	−	0.839583i	$-0.317201\pi$
$929$	33.1149	1.08646	0.543232	+	0.839583i	$0.317201\pi$
$930$	0	0
$931$	45.5194	1.49184
$932$	35.6585	1.16803
$933$	0	0
$934$	−58.1884	−1.90398
$935$	−122.266	−3.99852
$936$	0	0
$937$	9.57586	0.312830	0.156415	−	0.987691i	$-0.450006\pi$
$937$	9.57586	0.312830	0.156415	+	0.987691i	$0.450006\pi$
$938$	10.3607	0.338289
$939$	0	0
$940$	6.81049	0.222134
$941$	3.20500	0.104480	0.0522400	−	0.998635i	$-0.483364\pi$
$941$	3.20500	0.104480	0.0522400	+	0.998635i	$0.483364\pi$
$942$	0	0
$943$	−1.66473	−0.0542110
$944$	−41.9951	−1.36682
$945$	0	0
$946$	−109.867	−3.57210
$947$	2.13437	0.0693577	0.0346789	−	0.999399i	$-0.488959\pi$
$947$	2.13437	0.0693577	0.0346789	+	0.999399i	$0.488959\pi$
$948$	0	0
$949$	−11.2397	−0.364857
$950$	43.6206	1.41524
$951$	0	0
$952$	−6.74648	−0.218655
$953$	43.6116	1.41272	0.706359	−	0.707854i	$-0.250336\pi$
$953$	43.6116	1.41272	0.706359	+	0.707854i	$0.250336\pi$
$954$	0	0
$955$	−39.6324	−1.28247
$956$	−35.0378	−1.13320
$957$	0	0
$958$	−28.8436	−0.931895
$959$	−13.9625	−0.450873
$960$	0	0
$961$	−29.0011	−0.935521
$962$	−23.3071	−0.751452
$963$	0	0
$964$	−23.4434	−0.755061
$965$	33.5422	1.07976
$966$	0	0
$967$	−32.3397	−1.03998	−0.519988	−	0.854174i	$-0.674063\pi$
$967$	−32.3397	−1.03998	−0.519988	+	0.854174i	$0.674063\pi$
$968$	22.0921	0.710068
$969$	0	0
$970$	−22.2624	−0.714803
$971$	−44.0976	−1.41516	−0.707579	−	0.706634i	$-0.750213\pi$
$971$	−44.0976	−1.41516	−0.707579	+	0.706634i	$0.750213\pi$
$972$	0	0
$973$	−5.13589	−0.164649
$974$	24.6787	0.790756
$975$	0	0
$976$	−17.1710	−0.549630
$977$	−10.8862	−0.348281	−0.174140	−	0.984721i	$-0.555715\pi$
$977$	−10.8862	−0.348281	−0.174140	+	0.984721i	$0.555715\pi$
$978$	0	0
$979$	10.5762	0.338016
$980$	24.6213	0.786500
$981$	0	0
$982$	−61.0365	−1.94775
$983$	−31.7163	−1.01159	−0.505796	−	0.862653i	$-0.668801\pi$
$983$	−31.7163	−1.01159	−0.505796	+	0.862653i	$0.668801\pi$
$984$	0	0
$985$	17.0200	0.542302
$986$	−13.3251	−0.424358
$987$	0	0
$988$	−80.0304	−2.54610
$989$	−9.37822	−0.298210
$990$	0	0
$991$	−57.7747	−1.83527	−0.917637	−	0.397419i	$-0.869906\pi$
$991$	−57.7747	−1.83527	−0.917637	+	0.397419i	$0.869906\pi$
$992$	−10.2046	−0.323996
$993$	0	0
$994$	−15.9139	−0.504759
$995$	42.6773	1.35296
$996$	0	0
$997$	49.0709	1.55409	0.777045	−	0.629445i	$-0.216718\pi$
$997$	49.0709	1.55409	0.777045	+	0.629445i	$0.216718\pi$
$998$	−26.7011	−0.845207
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.8	✓	30
3.2	odd	2		6003.2.a.w.1.23	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.8	✓	30	1.1	even	1	trivial
6003.2.a.w.1.23	yes	30	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.89039 q^{2} +1.57357 q^{4} -2.79891 q^{5} +1.18730 q^{7} +0.806117 q^{8} +O(q^{10})\)	\(q-1.89039 q^{2} +1.57357 q^{4} -2.79891 q^{5} +1.18730 q^{7} +0.806117 q^{8} +5.29102 q^{10} -6.19722 q^{11} +6.24610 q^{13} -2.24445 q^{14} -4.67102 q^{16} -7.04887 q^{17} -8.14253 q^{19} -4.40428 q^{20} +11.7152 q^{22} +1.00000 q^{23} +2.83388 q^{25} -11.8076 q^{26} +1.86830 q^{28} -1.00000 q^{29} -1.41381 q^{31} +7.21780 q^{32} +13.3251 q^{34} -3.32313 q^{35} +1.97391 q^{37} +15.3925 q^{38} -2.25625 q^{40} -1.66473 q^{41} -9.37822 q^{43} -9.75177 q^{44} -1.89039 q^{46} -1.54634 q^{47} -5.59033 q^{49} -5.35713 q^{50} +9.82869 q^{52} -12.0893 q^{53} +17.3455 q^{55} +0.957100 q^{56} +1.89039 q^{58} +8.99058 q^{59} +3.67607 q^{61} +2.67265 q^{62} -4.30242 q^{64} -17.4823 q^{65} -4.61614 q^{67} -11.0919 q^{68} +6.28201 q^{70} +7.09033 q^{71} -1.79948 q^{73} -3.73147 q^{74} -12.8128 q^{76} -7.35795 q^{77} -0.494512 q^{79} +13.0737 q^{80} +3.14698 q^{82} +7.66496 q^{83} +19.7291 q^{85} +17.7285 q^{86} -4.99569 q^{88} -1.70660 q^{89} +7.41598 q^{91} +1.57357 q^{92} +2.92318 q^{94} +22.7902 q^{95} -4.20759 q^{97} +10.5679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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