LMFDB - Embedded newform 6003.2.a.t.1.22

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:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q+2.60337 q^{2} +4.77753 q^{4} -2.74879 q^{5} -2.14366 q^{7} +7.23092 q^{8} +O(q^{10})$	\(q+2.60337 q^{2} +4.77753 q^{4} -2.74879 q^{5} -2.14366 q^{7} +7.23092 q^{8} -7.15611 q^{10} +0.935134 q^{11} -3.80816 q^{13} -5.58073 q^{14} +9.26970 q^{16} +0.395185 q^{17} +3.48996 q^{19} -13.1324 q^{20} +2.43450 q^{22} +1.00000 q^{23} +2.55585 q^{25} -9.91405 q^{26} -10.2414 q^{28} +1.00000 q^{29} -6.61322 q^{31} +9.67060 q^{32} +1.02881 q^{34} +5.89247 q^{35} -9.78131 q^{37} +9.08566 q^{38} -19.8763 q^{40} -9.64119 q^{41} +7.07145 q^{43} +4.46762 q^{44} +2.60337 q^{46} -6.39436 q^{47} -2.40473 q^{49} +6.65382 q^{50} -18.1936 q^{52} -11.1647 q^{53} -2.57049 q^{55} -15.5006 q^{56} +2.60337 q^{58} -12.9239 q^{59} +9.19445 q^{61} -17.2166 q^{62} +6.63673 q^{64} +10.4678 q^{65} -0.642802 q^{67} +1.88800 q^{68} +15.3403 q^{70} +10.7985 q^{71} -1.05682 q^{73} -25.4643 q^{74} +16.6734 q^{76} -2.00461 q^{77} +3.20278 q^{79} -25.4805 q^{80} -25.0996 q^{82} -0.860006 q^{83} -1.08628 q^{85} +18.4096 q^{86} +6.76188 q^{88} +2.79612 q^{89} +8.16340 q^{91} +4.77753 q^{92} -16.6469 q^{94} -9.59318 q^{95} +11.7179 q^{97} -6.26039 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * 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 $

$2$	2.60337	1.84086	0.920430	−	0.390908i	$-0.127839\pi$
$2$	2.60337	1.84086	0.920430	+	0.390908i	$0.127839\pi$
$3$	0	0
$3$	0	0
$4$	4.77753	2.38876
$5$	−2.74879	−1.22930	−0.614648	−	0.788801i	$-0.710702\pi$
$5$	−2.74879	−1.22930	−0.614648	+	0.788801i	$0.710702\pi$
$6$	0	0
$7$	−2.14366	−0.810227	−0.405113	−	0.914266i	$-0.632768\pi$
$7$	−2.14366	−0.810227	−0.405113	+	0.914266i	$0.632768\pi$
$8$	7.23092	2.55652
$9$	0	0
$10$	−7.15611	−2.26296
$11$	0.935134	0.281953	0.140977	−	0.990013i	$-0.454976\pi$
$11$	0.935134	0.281953	0.140977	+	0.990013i	$0.454976\pi$
$12$	0	0
$13$	−3.80816	−1.05619	−0.528097	−	0.849184i	$-0.677094\pi$
$13$	−3.80816	−1.05619	−0.528097	+	0.849184i	$0.677094\pi$
$14$	−5.58073	−1.49151
$15$	0	0
$16$	9.26970	2.31743
$17$	0.395185	0.0958463	0.0479232	−	0.998851i	$-0.484740\pi$
$17$	0.395185	0.0958463	0.0479232	+	0.998851i	$0.484740\pi$
$18$	0	0
$19$	3.48996	0.800653	0.400326	−	0.916373i	$-0.368897\pi$
$19$	3.48996	0.800653	0.400326	+	0.916373i	$0.368897\pi$
$20$	−13.1324	−2.93650
$21$	0	0
$22$	2.43450	0.519036
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.55585	0.511170
$26$	−9.91405	−1.94430
$27$	0	0
$28$	−10.2414	−1.93544
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−6.61322	−1.18777	−0.593885	−	0.804550i	$-0.702406\pi$
$31$	−6.61322	−1.18777	−0.593885	+	0.804550i	$0.702406\pi$
$32$	9.67060	1.70954
$33$	0	0
$34$	1.02881	0.176440
$35$	5.89247	0.996009
$36$	0	0
$37$	−9.78131	−1.60804	−0.804018	−	0.594604i	$-0.797309\pi$
$37$	−9.78131	−1.60804	−0.804018	+	0.594604i	$0.797309\pi$
$38$	9.08566	1.47389
$39$	0	0
$40$	−19.8763	−3.14272
$41$	−9.64119	−1.50570	−0.752851	−	0.658191i	$-0.771322\pi$
$41$	−9.64119	−1.50570	−0.752851	+	0.658191i	$0.771322\pi$
$42$	0	0
$43$	7.07145	1.07839	0.539193	−	0.842182i	$-0.318729\pi$
$43$	7.07145	1.07839	0.539193	+	0.842182i	$0.318729\pi$
$44$	4.46762	0.673520
$45$	0	0
$46$	2.60337	0.383846
$47$	−6.39436	−0.932713	−0.466356	−	0.884597i	$-0.654434\pi$
$47$	−6.39436	−0.932713	−0.466356	+	0.884597i	$0.654434\pi$
$48$	0	0
$49$	−2.40473	−0.343532
$50$	6.65382	0.940992
$51$	0	0
$52$	−18.1936	−2.52300
$53$	−11.1647	−1.53359	−0.766797	−	0.641889i	$-0.778151\pi$
$53$	−11.1647	−1.53359	−0.766797	+	0.641889i	$0.778151\pi$
$54$	0	0
$55$	−2.57049	−0.346604
$56$	−15.5006	−2.07136
$57$	0	0
$58$	2.60337	0.341839
$59$	−12.9239	−1.68255	−0.841277	−	0.540604i	$-0.818196\pi$
$59$	−12.9239	−1.68255	−0.841277	+	0.540604i	$0.818196\pi$
$60$	0	0
$61$	9.19445	1.17723	0.588614	−	0.808414i	$-0.299674\pi$
$61$	9.19445	1.17723	0.588614	+	0.808414i	$0.299674\pi$
$62$	−17.2166	−2.18652
$63$	0	0
$64$	6.63673	0.829591
$65$	10.4678	1.29838
$66$	0	0
$67$	−0.642802	−0.0785307	−0.0392654	−	0.999229i	$-0.512502\pi$
$67$	−0.642802	−0.0785307	−0.0392654	+	0.999229i	$0.512502\pi$
$68$	1.88800	0.228954
$69$	0	0
$70$	15.3403	1.83351
$71$	10.7985	1.28154	0.640772	−	0.767731i	$-0.278614\pi$
$71$	10.7985	1.28154	0.640772	+	0.767731i	$0.278614\pi$
$72$	0	0
$73$	−1.05682	−0.123692	−0.0618458	−	0.998086i	$-0.519699\pi$
$73$	−1.05682	−0.123692	−0.0618458	+	0.998086i	$0.519699\pi$
$74$	−25.4643	−2.96017
$75$	0	0
$76$	16.6734	1.91257
$77$	−2.00461	−0.228446
$78$	0	0
$79$	3.20278	0.360341	0.180170	−	0.983635i	$-0.442335\pi$
$79$	3.20278	0.360341	0.180170	+	0.983635i	$0.442335\pi$
$80$	−25.4805	−2.84880
$81$	0	0
$82$	−25.0996	−2.77178
$83$	−0.860006	−0.0943979	−0.0471990	−	0.998886i	$-0.515029\pi$
$83$	−0.860006	−0.0943979	−0.0471990	+	0.998886i	$0.515029\pi$
$84$	0	0
$85$	−1.08628	−0.117824
$86$	18.4096	1.98516
$87$	0	0
$88$	6.76188	0.720819
$89$	2.79612	0.296388	0.148194	−	0.988958i	$-0.452654\pi$
$89$	2.79612	0.296388	0.148194	+	0.988958i	$0.452654\pi$
$90$	0	0
$91$	8.16340	0.855757
$92$	4.77753	0.498091
$93$	0	0
$94$	−16.6469	−1.71699
$95$	−9.59318	−0.984240
$96$	0	0
$97$	11.7179	1.18977	0.594886	−	0.803810i	$-0.297197\pi$
$97$	11.7179	1.18977	0.594886	+	0.803810i	$0.297197\pi$
$98$	−6.26039	−0.632395
$99$	0	0
$100$	12.2106	1.22106
$101$	−0.878967	−0.0874605	−0.0437303	−	0.999043i	$-0.513924\pi$
$101$	−0.878967	−0.0874605	−0.0437303	+	0.999043i	$0.513924\pi$
$102$	0	0
$103$	−13.6962	−1.34953	−0.674763	−	0.738035i	$-0.735754\pi$
$103$	−13.6962	−1.34953	−0.674763	+	0.738035i	$0.735754\pi$
$104$	−27.5365	−2.70018
$105$	0	0
$106$	−29.0659	−2.82313
$107$	−11.9809	−1.15824	−0.579120	−	0.815242i	$-0.696604\pi$
$107$	−11.9809	−1.15824	−0.579120	+	0.815242i	$0.696604\pi$
$108$	0	0
$109$	3.15475	0.302171	0.151085	−	0.988521i	$-0.451723\pi$
$109$	3.15475	0.302171	0.151085	+	0.988521i	$0.451723\pi$
$110$	−6.69192	−0.638050
$111$	0	0
$112$	−19.8711	−1.87764
$113$	12.5297	1.17870	0.589350	−	0.807878i	$-0.299384\pi$
$113$	12.5297	1.17870	0.589350	+	0.807878i	$0.299384\pi$
$114$	0	0
$115$	−2.74879	−0.256326
$116$	4.77753	0.443582
$117$	0	0
$118$	−33.6458	−3.09734
$119$	−0.847141	−0.0776573
$120$	0	0
$121$	−10.1255	−0.920502
$122$	23.9365	2.16711
$123$	0	0
$124$	−31.5948	−2.83730
$125$	6.71846	0.600917
$126$	0	0
$127$	−18.8886	−1.67609	−0.838044	−	0.545602i	$-0.816301\pi$
$127$	−18.8886	−1.67609	−0.838044	+	0.545602i	$0.816301\pi$
$128$	−2.06335	−0.182376
$129$	0	0
$130$	27.2516	2.39013
$131$	−9.90165	−0.865111	−0.432556	−	0.901607i	$-0.642388\pi$
$131$	−9.90165	−0.865111	−0.432556	+	0.901607i	$0.642388\pi$
$132$	0	0
$133$	−7.48129	−0.648710
$134$	−1.67345	−0.144564
$135$	0	0
$136$	2.85755	0.245033
$137$	13.8131	1.18013	0.590066	−	0.807355i	$-0.299102\pi$
$137$	13.8131	1.18013	0.590066	+	0.807355i	$0.299102\pi$
$138$	0	0
$139$	−11.2141	−0.951171	−0.475586	−	0.879669i	$-0.657764\pi$
$139$	−11.2141	−0.951171	−0.475586	+	0.879669i	$0.657764\pi$
$140$	28.1514	2.37923
$141$	0	0
$142$	28.1124	2.35914
$143$	−3.56114	−0.297797
$144$	0	0
$145$	−2.74879	−0.228275
$146$	−2.75130	−0.227699
$147$	0	0
$148$	−46.7304	−3.84122
$149$	13.0421	1.06845	0.534226	−	0.845342i	$-0.320603\pi$
$149$	13.0421	1.06845	0.534226	+	0.845342i	$0.320603\pi$
$150$	0	0
$151$	−0.0305867	−0.00248911	−0.00124456	−	0.999999i	$-0.500396\pi$
$151$	−0.0305867	−0.00248911	−0.00124456	+	0.999999i	$0.500396\pi$
$152$	25.2357	2.04688
$153$	0	0
$154$	−5.21873	−0.420537
$155$	18.1784	1.46012
$156$	0	0
$157$	−23.8854	−1.90626	−0.953130	−	0.302562i	$-0.902158\pi$
$157$	−23.8854	−1.90626	−0.953130	+	0.302562i	$0.902158\pi$
$158$	8.33801	0.663336
$159$	0	0
$160$	−26.5825	−2.10153
$161$	−2.14366	−0.168944
$162$	0	0
$163$	−15.5516	−1.21809	−0.609047	−	0.793134i	$-0.708448\pi$
$163$	−15.5516	−1.21809	−0.609047	+	0.793134i	$0.708448\pi$
$164$	−46.0610	−3.59676
$165$	0	0
$166$	−2.23891	−0.173773
$167$	6.50316	0.503229	0.251615	−	0.967827i	$-0.419038\pi$
$167$	6.50316	0.503229	0.251615	+	0.967827i	$0.419038\pi$
$168$	0	0
$169$	1.50209	0.115546
$170$	−2.82799	−0.216897
$171$	0	0
$172$	33.7840	2.57601
$173$	6.05728	0.460527	0.230263	−	0.973128i	$-0.426041\pi$
$173$	6.05728	0.460527	0.230263	+	0.973128i	$0.426041\pi$
$174$	0	0
$175$	−5.47887	−0.414164
$176$	8.66841	0.653406
$177$	0	0
$178$	7.27933	0.545609
$179$	6.92430	0.517546	0.258773	−	0.965938i	$-0.416682\pi$
$179$	6.92430	0.517546	0.258773	+	0.965938i	$0.416682\pi$
$180$	0	0
$181$	8.90515	0.661914	0.330957	−	0.943646i	$-0.392628\pi$
$181$	8.90515	0.661914	0.330957	+	0.943646i	$0.392628\pi$
$182$	21.2523	1.57533
$183$	0	0
$184$	7.23092	0.533071
$185$	26.8868	1.97675
$186$	0	0
$187$	0.369550	0.0270242
$188$	−30.5492	−2.22803
$189$	0	0
$190$	−24.9746	−1.81185
$191$	−5.21206	−0.377132	−0.188566	−	0.982061i	$-0.560384\pi$
$191$	−5.21206	−0.377132	−0.188566	+	0.982061i	$0.560384\pi$
$192$	0	0
$193$	−11.3259	−0.815258	−0.407629	−	0.913148i	$-0.633644\pi$
$193$	−11.3259	−0.815258	−0.407629	+	0.913148i	$0.633644\pi$
$194$	30.5060	2.19020
$195$	0	0
$196$	−11.4886	−0.820618
$197$	16.8530	1.20073	0.600364	−	0.799727i	$-0.295022\pi$
$197$	16.8530	1.20073	0.600364	+	0.799727i	$0.295022\pi$
$198$	0	0
$199$	−1.64734	−0.116777	−0.0583885	−	0.998294i	$-0.518596\pi$
$199$	−1.64734	−0.116777	−0.0583885	+	0.998294i	$0.518596\pi$
$200$	18.4812	1.30682
$201$	0	0
$202$	−2.28828	−0.161002
$203$	−2.14366	−0.150455
$204$	0	0
$205$	26.5016	1.85095
$206$	−35.6562	−2.48429
$207$	0	0
$208$	−35.3005	−2.44765
$209$	3.26358	0.225747
$210$	0	0
$211$	28.6574	1.97286	0.986428	−	0.164192i	$-0.0525017\pi$
$211$	28.6574	1.97286	0.986428	+	0.164192i	$0.0525017\pi$
$212$	−53.3398	−3.66339
$213$	0	0
$214$	−31.1908	−2.13216
$215$	−19.4379	−1.32566
$216$	0	0
$217$	14.1765	0.962363
$218$	8.21299	0.556254
$219$	0	0
$220$	−12.2806	−0.827956
$221$	−1.50493	−0.101232
$222$	0	0
$223$	−12.4082	−0.830916	−0.415458	−	0.909612i	$-0.636379\pi$
$223$	−12.4082	−0.830916	−0.415458	+	0.909612i	$0.636379\pi$
$224$	−20.7305	−1.38511
$225$	0	0
$226$	32.6195	2.16982
$227$	−15.3135	−1.01639	−0.508197	−	0.861241i	$-0.669688\pi$
$227$	−15.3135	−1.01639	−0.508197	+	0.861241i	$0.669688\pi$
$228$	0	0
$229$	15.4958	1.02399	0.511995	−	0.858989i	$-0.328907\pi$
$229$	15.4958	1.02399	0.511995	+	0.858989i	$0.328907\pi$
$230$	−7.15611	−0.471860
$231$	0	0
$232$	7.23092	0.474733
$233$	5.64312	0.369693	0.184847	−	0.982767i	$-0.440821\pi$
$233$	5.64312	0.369693	0.184847	+	0.982767i	$0.440821\pi$
$234$	0	0
$235$	17.5768	1.14658
$236$	−61.7445	−4.01922
$237$	0	0
$238$	−2.20542	−0.142956
$239$	6.58266	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$239$	6.58266	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$240$	0	0
$241$	−6.01793	−0.387649	−0.193825	−	0.981036i	$-0.562089\pi$
$241$	−6.01793	−0.387649	−0.193825	+	0.981036i	$0.562089\pi$
$242$	−26.3605	−1.69452
$243$	0	0
$244$	43.9267	2.81212
$245$	6.61009	0.422303
$246$	0	0
$247$	−13.2903	−0.845644
$248$	−47.8197	−3.03655
$249$	0	0
$250$	17.4906	1.10620
$251$	16.8183	1.06156	0.530779	−	0.847510i	$-0.321899\pi$
$251$	16.8183	1.06156	0.530779	+	0.847510i	$0.321899\pi$
$252$	0	0
$253$	0.935134	0.0587913
$254$	−49.1739	−3.08544
$255$	0	0
$256$	−18.6451	−1.16532
$257$	16.3640	1.02076	0.510378	−	0.859950i	$-0.329505\pi$
$257$	16.3640	1.02076	0.510378	+	0.859950i	$0.329505\pi$
$258$	0	0
$259$	20.9678	1.30287
$260$	50.0104	3.10151
$261$	0	0
$262$	−25.7776	−1.59255
$263$	26.9113	1.65942	0.829711	−	0.558193i	$-0.188505\pi$
$263$	26.9113	1.65942	0.829711	+	0.558193i	$0.188505\pi$
$264$	0	0
$265$	30.6895	1.88524
$266$	−19.4766	−1.19418
$267$	0	0
$268$	−3.07100	−0.187591
$269$	−3.47264	−0.211730	−0.105865	−	0.994380i	$-0.533761\pi$
$269$	−3.47264	−0.211730	−0.105865	+	0.994380i	$0.533761\pi$
$270$	0	0
$271$	2.00248	0.121642	0.0608211	−	0.998149i	$-0.480628\pi$
$271$	2.00248	0.121642	0.0608211	+	0.998149i	$0.480628\pi$
$272$	3.66324	0.222117
$273$	0	0
$274$	35.9606	2.17246
$275$	2.39006	0.144126
$276$	0	0
$277$	−10.2554	−0.616187	−0.308093	−	0.951356i	$-0.599691\pi$
$277$	−10.2554	−0.616187	−0.308093	+	0.951356i	$0.599691\pi$
$278$	−29.1945	−1.75097
$279$	0	0
$280$	42.6080	2.54631
$281$	14.4436	0.861632	0.430816	−	0.902440i	$-0.358226\pi$
$281$	14.4436	0.861632	0.430816	+	0.902440i	$0.358226\pi$
$282$	0	0
$283$	−16.6941	−0.992364	−0.496182	−	0.868219i	$-0.665265\pi$
$283$	−16.6941	−0.992364	−0.496182	+	0.868219i	$0.665265\pi$
$284$	51.5900	3.06130
$285$	0	0
$286$	−9.27096	−0.548203
$287$	20.6674	1.21996
$288$	0	0
$289$	−16.8438	−0.990813
$290$	−7.15611	−0.420222
$291$	0	0
$292$	−5.04899	−0.295470
$293$	−10.2934	−0.601346	−0.300673	−	0.953727i	$-0.597211\pi$
$293$	−10.2934	−0.601346	−0.300673	+	0.953727i	$0.597211\pi$
$294$	0	0
$295$	35.5252	2.06836
$296$	−70.7279	−4.11097
$297$	0	0
$298$	33.9534	1.96687
$299$	−3.80816	−0.220232
$300$	0	0
$301$	−15.1588	−0.873737
$302$	−0.0796286	−0.00458211
$303$	0	0
$304$	32.3509	1.85545
$305$	−25.2736	−1.44716
$306$	0	0
$307$	11.9005	0.679199	0.339600	−	0.940570i	$-0.389708\pi$
$307$	11.9005	0.679199	0.339600	+	0.940570i	$0.389708\pi$
$308$	−9.57706	−0.545704
$309$	0	0
$310$	47.3250	2.68788
$311$	28.7455	1.63001	0.815003	−	0.579457i	$-0.196735\pi$
$311$	28.7455	1.63001	0.815003	+	0.579457i	$0.196735\pi$
$312$	0	0
$313$	21.6771	1.22526	0.612631	−	0.790369i	$-0.290111\pi$
$313$	21.6771	1.22526	0.612631	+	0.790369i	$0.290111\pi$
$314$	−62.1824	−3.50916
$315$	0	0
$316$	15.3013	0.860768
$317$	13.0706	0.734117	0.367058	−	0.930198i	$-0.380365\pi$
$317$	13.0706	0.734117	0.367058	+	0.930198i	$0.380365\pi$
$318$	0	0
$319$	0.935134	0.0523574
$320$	−18.2430	−1.01981
$321$	0	0
$322$	−5.58073	−0.311002
$323$	1.37918	0.0767396
$324$	0	0
$325$	−9.73309	−0.539895
$326$	−40.4865	−2.24234
$327$	0	0
$328$	−69.7147	−3.84935
$329$	13.7073	0.755709
$330$	0	0
$331$	21.4831	1.18082	0.590409	−	0.807104i	$-0.298967\pi$
$331$	21.4831	1.18082	0.590409	+	0.807104i	$0.298967\pi$
$332$	−4.10870	−0.225494
$333$	0	0
$334$	16.9301	0.926375
$335$	1.76693	0.0965376
$336$	0	0
$337$	−8.82458	−0.480706	−0.240353	−	0.970686i	$-0.577263\pi$
$337$	−8.82458	−0.480706	−0.240353	+	0.970686i	$0.577263\pi$
$338$	3.91050	0.212703
$339$	0	0
$340$	−5.18973	−0.281453
$341$	−6.18424	−0.334896
$342$	0	0
$343$	20.1605	1.08857
$344$	51.1331	2.75691
$345$	0	0
$346$	15.7693	0.847764
$347$	25.1783	1.35164	0.675820	−	0.737067i	$-0.263790\pi$
$347$	25.1783	1.35164	0.675820	+	0.737067i	$0.263790\pi$
$348$	0	0
$349$	−12.0798	−0.646619	−0.323310	−	0.946293i	$-0.604795\pi$
$349$	−12.0798	−0.646619	−0.323310	+	0.946293i	$0.604795\pi$
$350$	−14.2635	−0.762417
$351$	0	0
$352$	9.04330	0.482010
$353$	7.37130	0.392335	0.196167	−	0.980570i	$-0.437150\pi$
$353$	7.37130	0.392335	0.196167	+	0.980570i	$0.437150\pi$
$354$	0	0
$355$	−29.6828	−1.57540
$356$	13.3585	0.708001
$357$	0	0
$358$	18.0265	0.952730
$359$	−15.0167	−0.792549	−0.396275	−	0.918132i	$-0.629697\pi$
$359$	−15.0167	−0.792549	−0.396275	+	0.918132i	$0.629697\pi$
$360$	0	0
$361$	−6.82015	−0.358955
$362$	23.1834	1.21849
$363$	0	0
$364$	39.0008	2.04420
$365$	2.90498	0.152054
$366$	0	0
$367$	−35.0361	−1.82887	−0.914435	−	0.404733i	$-0.867365\pi$
$367$	−35.0361	−1.82887	−0.914435	+	0.404733i	$0.867365\pi$
$368$	9.26970	0.483217
$369$	0	0
$370$	69.9961	3.63893
$371$	23.9334	1.24256
$372$	0	0
$373$	−14.3222	−0.741576	−0.370788	−	0.928718i	$-0.620912\pi$
$373$	−14.3222	−0.741576	−0.370788	+	0.928718i	$0.620912\pi$
$374$	0.962076	0.0497477
$375$	0	0
$376$	−46.2371	−2.38450
$377$	−3.80816	−0.196130
$378$	0	0
$379$	24.7951	1.27364	0.636818	−	0.771014i	$-0.280250\pi$
$379$	24.7951	1.27364	0.636818	+	0.771014i	$0.280250\pi$
$380$	−45.8317	−2.35111
$381$	0	0
$382$	−13.5689	−0.694246
$383$	5.09661	0.260425	0.130212	−	0.991486i	$-0.458434\pi$
$383$	5.09661	0.260425	0.130212	+	0.991486i	$0.458434\pi$
$384$	0	0
$385$	5.51025	0.280828
$386$	−29.4856	−1.50078
$387$	0	0
$388$	55.9826	2.84208
$389$	1.25884	0.0638256	0.0319128	−	0.999491i	$-0.489840\pi$
$389$	1.25884	0.0638256	0.0319128	+	0.999491i	$0.489840\pi$
$390$	0	0
$391$	0.395185	0.0199853
$392$	−17.3884	−0.878247
$393$	0	0
$394$	43.8746	2.21037
$395$	−8.80376	−0.442965
$396$	0	0
$397$	22.5810	1.13331	0.566653	−	0.823956i	$-0.308238\pi$
$397$	22.5810	1.13331	0.566653	+	0.823956i	$0.308238\pi$
$398$	−4.28864	−0.214970
$399$	0	0
$400$	23.6920	1.18460
$401$	35.3959	1.76759	0.883793	−	0.467879i	$-0.154982\pi$
$401$	35.3959	1.76759	0.883793	+	0.467879i	$0.154982\pi$
$402$	0	0
$403$	25.1842	1.25451
$404$	−4.19929	−0.208922
$405$	0	0
$406$	−5.58073	−0.276967
$407$	−9.14683	−0.453391
$408$	0	0
$409$	27.7789	1.37358	0.686789	−	0.726857i	$-0.259020\pi$
$409$	27.7789	1.37358	0.686789	+	0.726857i	$0.259020\pi$
$410$	68.9935	3.40734
$411$	0	0
$412$	−65.4339	−3.22370
$413$	27.7045	1.36325
$414$	0	0
$415$	2.36398	0.116043
$416$	−36.8272	−1.80560
$417$	0	0
$418$	8.49631	0.415568
$419$	−18.8066	−0.918762	−0.459381	−	0.888239i	$-0.651929\pi$
$419$	−18.8066	−0.918762	−0.459381	+	0.888239i	$0.651929\pi$
$420$	0	0
$421$	−28.2932	−1.37893	−0.689464	−	0.724320i	$-0.742154\pi$
$421$	−28.2932	−1.37893	−0.689464	+	0.724320i	$0.742154\pi$
$422$	74.6058	3.63175
$423$	0	0
$424$	−80.7314	−3.92066
$425$	1.01003	0.0489938
$426$	0	0
$427$	−19.7098	−0.953822
$428$	−57.2392	−2.76676
$429$	0	0
$430$	−50.6041	−2.44035
$431$	−2.26835	−0.109262	−0.0546312	−	0.998507i	$-0.517398\pi$
$431$	−2.26835	−0.109262	−0.0546312	+	0.998507i	$0.517398\pi$
$432$	0	0
$433$	19.6124	0.942511	0.471255	−	0.881997i	$-0.343801\pi$
$433$	19.6124	0.942511	0.471255	+	0.881997i	$0.343801\pi$
$434$	36.9066	1.77157
$435$	0	0
$436$	15.0719	0.721814
$437$	3.48996	0.166948
$438$	0	0
$439$	−10.4449	−0.498508	−0.249254	−	0.968438i	$-0.580185\pi$
$439$	−10.4449	−0.498508	−0.249254	+	0.968438i	$0.580185\pi$
$440$	−18.5870	−0.886100
$441$	0	0
$442$	−3.91788	−0.186354
$443$	−12.0910	−0.574462	−0.287231	−	0.957861i	$-0.592735\pi$
$443$	−12.0910	−0.574462	−0.287231	+	0.957861i	$0.592735\pi$
$444$	0	0
$445$	−7.68595	−0.364349
$446$	−32.3032	−1.52960
$447$	0	0
$448$	−14.2269	−0.672157
$449$	−29.3970	−1.38733	−0.693664	−	0.720298i	$-0.744005\pi$
$449$	−29.3970	−1.38733	−0.693664	+	0.720298i	$0.744005\pi$
$450$	0	0
$451$	−9.01580	−0.424538
$452$	59.8612	2.81563
$453$	0	0
$454$	−39.8667	−1.87104
$455$	−22.4395	−1.05198
$456$	0	0
$457$	−30.2374	−1.41445	−0.707223	−	0.706990i	$-0.750052\pi$
$457$	−30.2374	−1.41445	−0.707223	+	0.706990i	$0.750052\pi$
$458$	40.3412	1.88502
$459$	0	0
$460$	−13.1324	−0.612302
$461$	−27.6595	−1.28823	−0.644115	−	0.764929i	$-0.722774\pi$
$461$	−27.6595	−1.28823	−0.644115	+	0.764929i	$0.722774\pi$
$462$	0	0
$463$	−1.90846	−0.0886937	−0.0443468	−	0.999016i	$-0.514121\pi$
$463$	−1.90846	−0.0886937	−0.0443468	+	0.999016i	$0.514121\pi$
$464$	9.26970	0.430335
$465$	0	0
$466$	14.6911	0.680553
$467$	−11.9120	−0.551221	−0.275610	−	0.961269i	$-0.588880\pi$
$467$	−11.9120	−0.551221	−0.275610	+	0.961269i	$0.588880\pi$
$468$	0	0
$469$	1.37795	0.0636277
$470$	45.7588	2.11069
$471$	0	0
$472$	−93.4520	−4.30148
$473$	6.61275	0.304054
$474$	0	0
$475$	8.91983	0.409270
$476$	−4.04724	−0.185505
$477$	0	0
$478$	17.1371	0.783832
$479$	−7.24640	−0.331097	−0.165548	−	0.986202i	$-0.552939\pi$
$479$	−7.24640	−0.331097	−0.165548	+	0.986202i	$0.552939\pi$
$480$	0	0
$481$	37.2488	1.69840
$482$	−15.6669	−0.713607
$483$	0	0
$484$	−48.3750	−2.19886
$485$	−32.2101	−1.46258
$486$	0	0
$487$	1.55045	0.0702577	0.0351289	−	0.999383i	$-0.488816\pi$
$487$	1.55045	0.0702577	0.0351289	+	0.999383i	$0.488816\pi$
$488$	66.4844	3.00961
$489$	0	0
$490$	17.2085	0.777401
$491$	16.6954	0.753454	0.376727	−	0.926324i	$-0.377050\pi$
$491$	16.6954	0.753454	0.376727	+	0.926324i	$0.377050\pi$
$492$	0	0
$493$	0.395185	0.0177982
$494$	−34.5997	−1.55671
$495$	0	0
$496$	−61.3026	−2.75257
$497$	−23.1483	−1.03834
$498$	0	0
$499$	6.18980	0.277093	0.138547	−	0.990356i	$-0.455757\pi$
$499$	6.18980	0.277093	0.138547	+	0.990356i	$0.455757\pi$
$500$	32.0976	1.43545
$501$	0	0
$502$	43.7841	1.95418
$503$	−21.5398	−0.960413	−0.480206	−	0.877156i	$-0.659438\pi$
$503$	−21.5398	−0.960413	−0.480206	+	0.877156i	$0.659438\pi$
$504$	0	0
$505$	2.41610	0.107515
$506$	2.43450	0.108227
$507$	0	0
$508$	−90.2406	−4.00378
$509$	9.20262	0.407899	0.203949	−	0.978981i	$-0.434622\pi$
$509$	9.20262	0.407899	0.203949	+	0.978981i	$0.434622\pi$
$510$	0	0
$511$	2.26547	0.100218
$512$	−44.4134	−1.96281
$513$	0	0
$514$	42.6014	1.87907
$515$	37.6480	1.65897
$516$	0	0
$517$	−5.97958	−0.262982
$518$	54.5869	2.39841
$519$	0	0
$520$	75.6921	3.31932
$521$	1.02791	0.0450336	0.0225168	−	0.999746i	$-0.492832\pi$
$521$	1.02791	0.0450336	0.0225168	+	0.999746i	$0.492832\pi$
$522$	0	0
$523$	17.8505	0.780546	0.390273	−	0.920699i	$-0.372381\pi$
$523$	17.8505	0.780546	0.390273	+	0.920699i	$0.372381\pi$
$524$	−47.3054	−2.06655
$525$	0	0
$526$	70.0600	3.05476
$527$	−2.61344	−0.113843
$528$	0	0
$529$	1.00000	0.0434783
$530$	79.8962	3.47047
$531$	0	0
$532$	−35.7421	−1.54961
$533$	36.7152	1.59031
$534$	0	0
$535$	32.9331	1.42382
$536$	−4.64805	−0.200765
$537$	0	0
$538$	−9.04055	−0.389766
$539$	−2.24874	−0.0968601
$540$	0	0
$541$	−12.6863	−0.545427	−0.272714	−	0.962095i	$-0.587921\pi$
$541$	−12.6863	−0.545427	−0.272714	+	0.962095i	$0.587921\pi$
$542$	5.21320	0.223926
$543$	0	0
$544$	3.82167	0.163853
$545$	−8.67176	−0.371457
$546$	0	0
$547$	14.2782	0.610493	0.305247	−	0.952273i	$-0.401261\pi$
$547$	14.2782	0.610493	0.305247	+	0.952273i	$0.401261\pi$
$548$	65.9924	2.81906
$549$	0	0
$550$	6.22221	0.265316
$551$	3.48996	0.148677
$552$	0	0
$553$	−6.86566	−0.291958
$554$	−26.6986	−1.13431
$555$	0	0
$556$	−53.5759	−2.27212
$557$	9.77754	0.414288	0.207144	−	0.978310i	$-0.433583\pi$
$557$	9.77754	0.414288	0.207144	+	0.978310i	$0.433583\pi$
$558$	0	0
$559$	−26.9292	−1.13898
$560$	54.6214	2.30818
$561$	0	0
$562$	37.6019	1.58614
$563$	−22.0378	−0.928782	−0.464391	−	0.885630i	$-0.653727\pi$
$563$	−22.0378	−0.928782	−0.464391	+	0.885630i	$0.653727\pi$
$564$	0	0
$565$	−34.4417	−1.44897
$566$	−43.4610	−1.82680
$567$	0	0
$568$	78.0830	3.27629
$569$	9.95929	0.417515	0.208758	−	0.977967i	$-0.433058\pi$
$569$	9.95929	0.417515	0.208758	+	0.977967i	$0.433058\pi$
$570$	0	0
$571$	−19.3496	−0.809755	−0.404877	−	0.914371i	$-0.632686\pi$
$571$	−19.3496	−0.809755	−0.404877	+	0.914371i	$0.632686\pi$
$572$	−17.0134	−0.711368
$573$	0	0
$574$	53.8049	2.24577
$575$	2.55585	0.106586
$576$	0	0
$577$	−21.7692	−0.906262	−0.453131	−	0.891444i	$-0.649693\pi$
$577$	−21.7692	−0.906262	−0.453131	+	0.891444i	$0.649693\pi$
$578$	−43.8507	−1.82395
$579$	0	0
$580$	−13.1324	−0.545294
$581$	1.84356	0.0764837
$582$	0	0
$583$	−10.4405	−0.432402
$584$	−7.64180	−0.316220
$585$	0	0
$586$	−26.7975	−1.10699
$587$	−3.87620	−0.159988	−0.0799939	−	0.996795i	$-0.525490\pi$
$587$	−3.87620	−0.159988	−0.0799939	+	0.996795i	$0.525490\pi$
$588$	0	0
$589$	−23.0799	−0.950991
$590$	92.4852	3.80756
$591$	0	0
$592$	−90.6698	−3.72650
$593$	−18.8991	−0.776092	−0.388046	−	0.921640i	$-0.626850\pi$
$593$	−18.8991	−0.776092	−0.388046	+	0.921640i	$0.626850\pi$
$594$	0	0
$595$	2.32861	0.0954638
$596$	62.3091	2.55228
$597$	0	0
$598$	−9.91405	−0.405416
$599$	−12.2026	−0.498584	−0.249292	−	0.968428i	$-0.580198\pi$
$599$	−12.2026	−0.498584	−0.249292	+	0.968428i	$0.580198\pi$
$600$	0	0
$601$	−23.5426	−0.960322	−0.480161	−	0.877180i	$-0.659422\pi$
$601$	−23.5426	−0.960322	−0.480161	+	0.877180i	$0.659422\pi$
$602$	−39.4638	−1.60843
$603$	0	0
$604$	−0.146129	−0.00594590
$605$	27.8330	1.13157
$606$	0	0
$607$	−16.5073	−0.670011	−0.335006	−	0.942216i	$-0.608738\pi$
$607$	−16.5073	−0.670011	−0.335006	+	0.942216i	$0.608738\pi$
$608$	33.7500	1.36875
$609$	0	0
$610$	−65.7965	−2.66402
$611$	24.3507	0.985126
$612$	0	0
$613$	−1.67276	−0.0675620	−0.0337810	−	0.999429i	$-0.510755\pi$
$613$	−1.67276	−0.0675620	−0.0337810	+	0.999429i	$0.510755\pi$
$614$	30.9815	1.25031
$615$	0	0
$616$	−14.4952	−0.584027
$617$	43.7944	1.76310	0.881549	−	0.472093i	$-0.156501\pi$
$617$	43.7944	1.76310	0.881549	+	0.472093i	$0.156501\pi$
$618$	0	0
$619$	8.92052	0.358546	0.179273	−	0.983799i	$-0.442625\pi$
$619$	8.92052	0.358546	0.179273	+	0.983799i	$0.442625\pi$
$620$	86.8476	3.48788
$621$	0	0
$622$	74.8350	3.00061
$623$	−5.99392	−0.240142
$624$	0	0
$625$	−31.2469	−1.24988
$626$	56.4335	2.25553
$627$	0	0
$628$	−114.113	−4.55360
$629$	−3.86542	−0.154124
$630$	0	0
$631$	−36.7446	−1.46278	−0.731390	−	0.681959i	$-0.761128\pi$
$631$	−36.7446	−1.46278	−0.731390	+	0.681959i	$0.761128\pi$
$632$	23.1590	0.921217
$633$	0	0
$634$	34.0275	1.35141
$635$	51.9207	2.06041
$636$	0	0
$637$	9.15759	0.362837
$638$	2.43450	0.0963827
$639$	0	0
$640$	5.67171	0.224194
$641$	−39.8607	−1.57440	−0.787202	−	0.616695i	$-0.788471\pi$
$641$	−39.8607	−1.57440	−0.787202	+	0.616695i	$0.788471\pi$
$642$	0	0
$643$	13.9526	0.550236	0.275118	−	0.961411i	$-0.411283\pi$
$643$	13.9526	0.550236	0.275118	+	0.961411i	$0.411283\pi$
$644$	−10.2414	−0.403567
$645$	0	0
$646$	3.59051	0.141267
$647$	6.07851	0.238971	0.119485	−	0.992836i	$-0.461876\pi$
$647$	6.07851	0.238971	0.119485	+	0.992836i	$0.461876\pi$
$648$	0	0
$649$	−12.0856	−0.474402
$650$	−25.3388	−0.993870
$651$	0	0
$652$	−74.2981	−2.90974
$653$	22.3745	0.875582	0.437791	−	0.899077i	$-0.355761\pi$
$653$	22.3745	0.875582	0.437791	+	0.899077i	$0.355761\pi$
$654$	0	0
$655$	27.2176	1.06348
$656$	−89.3710	−3.48935
$657$	0	0
$658$	35.6852	1.39115
$659$	−38.1027	−1.48427	−0.742136	−	0.670250i	$-0.766187\pi$
$659$	−38.1027	−1.48427	−0.742136	+	0.670250i	$0.766187\pi$
$660$	0	0
$661$	−6.48996	−0.252430	−0.126215	−	0.992003i	$-0.540283\pi$
$661$	−6.48996	−0.252430	−0.126215	+	0.992003i	$0.540283\pi$
$662$	55.9284	2.17372
$663$	0	0
$664$	−6.21864	−0.241330
$665$	20.5645	0.797457
$666$	0	0
$667$	1.00000	0.0387202
$668$	31.0690	1.20210
$669$	0	0
$670$	4.59996	0.177712
$671$	8.59804	0.331924
$672$	0	0
$673$	30.5785	1.17872	0.589358	−	0.807872i	$-0.299381\pi$
$673$	30.5785	1.17872	0.589358	+	0.807872i	$0.299381\pi$
$674$	−22.9736	−0.884912
$675$	0	0
$676$	7.17630	0.276011
$677$	−50.5208	−1.94167	−0.970836	−	0.239743i	$-0.922937\pi$
$677$	−50.5208	−1.94167	−0.970836	+	0.239743i	$0.922937\pi$
$678$	0	0
$679$	−25.1192	−0.963986
$680$	−7.85480	−0.301218
$681$	0	0
$682$	−16.0999	−0.616496
$683$	4.47893	0.171382	0.0856908	−	0.996322i	$-0.472690\pi$
$683$	4.47893	0.171382	0.0856908	+	0.996322i	$0.472690\pi$
$684$	0	0
$685$	−37.9693	−1.45073
$686$	52.4853	2.00390
$687$	0	0
$688$	65.5502	2.49908
$689$	42.5171	1.61977
$690$	0	0
$691$	−17.9551	−0.683044	−0.341522	−	0.939874i	$-0.610942\pi$
$691$	−17.9551	−0.683044	−0.341522	+	0.939874i	$0.610942\pi$
$692$	28.9388	1.10009
$693$	0	0
$694$	65.5483	2.48818
$695$	30.8253	1.16927
$696$	0	0
$697$	−3.81005	−0.144316
$698$	−31.4483	−1.19033
$699$	0	0
$700$	−26.1755	−0.989339
$701$	−12.6626	−0.478261	−0.239131	−	0.970987i	$-0.576862\pi$
$701$	−12.6626	−0.478261	−0.239131	+	0.970987i	$0.576862\pi$
$702$	0	0
$703$	−34.1364	−1.28748
$704$	6.20623	0.233906
$705$	0	0
$706$	19.1902	0.722233
$707$	1.88421	0.0708629
$708$	0	0
$709$	−20.5699	−0.772520	−0.386260	−	0.922390i	$-0.626233\pi$
$709$	−20.5699	−0.772520	−0.386260	+	0.922390i	$0.626233\pi$
$710$	−77.2751	−2.90008
$711$	0	0
$712$	20.2185	0.757721
$713$	−6.61322	−0.247667
$714$	0	0
$715$	9.78883	0.366081
$716$	33.0810	1.23630
$717$	0	0
$718$	−39.0939	−1.45897
$719$	42.7421	1.59401	0.797005	−	0.603972i	$-0.206416\pi$
$719$	42.7421	1.59401	0.797005	+	0.603972i	$0.206416\pi$
$720$	0	0
$721$	29.3600	1.09342
$722$	−17.7554	−0.660786
$723$	0	0
$724$	42.5446	1.58116
$725$	2.55585	0.0949219
$726$	0	0
$727$	26.8622	0.996266	0.498133	−	0.867101i	$-0.334019\pi$
$727$	26.8622	0.996266	0.498133	+	0.867101i	$0.334019\pi$
$728$	59.0289	2.18776
$729$	0	0
$730$	7.56274	0.279909
$731$	2.79453	0.103359
$732$	0	0
$733$	−17.0946	−0.631403	−0.315701	−	0.948859i	$-0.602240\pi$
$733$	−17.0946	−0.631403	−0.315701	+	0.948859i	$0.602240\pi$
$734$	−91.2119	−3.36669
$735$	0	0
$736$	9.67060	0.356463
$737$	−0.601105	−0.0221420
$738$	0	0
$739$	28.4610	1.04695	0.523477	−	0.852040i	$-0.324635\pi$
$739$	28.4610	1.04695	0.523477	+	0.852040i	$0.324635\pi$
$740$	128.452	4.72200
$741$	0	0
$742$	62.3074	2.28738
$743$	26.8306	0.984319	0.492160	−	0.870505i	$-0.336208\pi$
$743$	26.8306	0.984319	0.492160	+	0.870505i	$0.336208\pi$
$744$	0	0
$745$	−35.8501	−1.31344
$746$	−37.2860	−1.36514
$747$	0	0
$748$	1.76554	0.0645544
$749$	25.6830	0.938438
$750$	0	0
$751$	49.3690	1.80150	0.900750	−	0.434337i	$-0.143017\pi$
$751$	49.3690	1.80150	0.900750	+	0.434337i	$0.143017\pi$
$752$	−59.2738	−2.16149
$753$	0	0
$754$	−9.91405	−0.361048
$755$	0.0840766	0.00305986
$756$	0	0
$757$	−48.2396	−1.75330	−0.876648	−	0.481131i	$-0.840226\pi$
$757$	−48.2396	−1.75330	−0.876648	+	0.481131i	$0.840226\pi$
$758$	64.5507	2.34459
$759$	0	0
$760$	−69.3675	−2.51623
$761$	15.3989	0.558209	0.279104	−	0.960261i	$-0.409962\pi$
$761$	15.3989	0.558209	0.279104	+	0.960261i	$0.409962\pi$
$762$	0	0
$763$	−6.76272	−0.244827
$764$	−24.9008	−0.900878
$765$	0	0
$766$	13.2684	0.479405
$767$	49.2165	1.77710
$768$	0	0
$769$	14.9576	0.539385	0.269692	−	0.962947i	$-0.413078\pi$
$769$	14.9576	0.539385	0.269692	+	0.962947i	$0.413078\pi$
$770$	14.3452	0.516965
$771$	0	0
$772$	−54.1099	−1.94746
$773$	−31.0305	−1.11609	−0.558044	−	0.829811i	$-0.688448\pi$
$773$	−31.0305	−1.11609	−0.558044	+	0.829811i	$0.688448\pi$
$774$	0	0
$775$	−16.9024	−0.607152
$776$	84.7312	3.04167
$777$	0	0
$778$	3.27722	0.117494
$779$	−33.6474	−1.20554
$780$	0	0
$781$	10.0980	0.361336
$782$	1.02881	0.0367902
$783$	0	0
$784$	−22.2911	−0.796111
$785$	65.6559	2.34336
$786$	0	0
$787$	0.943923	0.0336472	0.0168236	−	0.999858i	$-0.494645\pi$
$787$	0.943923	0.0336472	0.0168236	+	0.999858i	$0.494645\pi$
$788$	80.5157	2.86825
$789$	0	0
$790$	−22.9194	−0.815437
$791$	−26.8595	−0.955014
$792$	0	0
$793$	−35.0140	−1.24338
$794$	58.7866	2.08626
$795$	0	0
$796$	−7.87022	−0.278952
$797$	13.6806	0.484593	0.242297	−	0.970202i	$-0.422099\pi$
$797$	13.6806	0.484593	0.242297	+	0.970202i	$0.422099\pi$
$798$	0	0
$799$	−2.52695	−0.0893971
$800$	24.7166	0.873864
$801$	0	0
$802$	92.1485	3.25388
$803$	−0.988270	−0.0348753
$804$	0	0
$805$	5.89247	0.207682
$806$	65.5638	2.30939
$807$	0	0
$808$	−6.35574	−0.223594
$809$	38.3214	1.34731	0.673654	−	0.739047i	$-0.264724\pi$
$809$	38.3214	1.34731	0.673654	+	0.739047i	$0.264724\pi$
$810$	0	0
$811$	−7.33098	−0.257426	−0.128713	−	0.991682i	$-0.541085\pi$
$811$	−7.33098	−0.257426	−0.128713	+	0.991682i	$0.541085\pi$
$812$	−10.2414	−0.359402
$813$	0	0
$814$	−23.8126	−0.834630
$815$	42.7480	1.49740
$816$	0	0
$817$	24.6791	0.863412
$818$	72.3186	2.52856
$819$	0	0
$820$	126.612	4.42149
$821$	50.3587	1.75753	0.878765	−	0.477254i	$-0.158368\pi$
$821$	50.3587	1.75753	0.878765	+	0.477254i	$0.158368\pi$
$822$	0	0
$823$	−46.2199	−1.61112	−0.805562	−	0.592512i	$-0.798136\pi$
$823$	−46.2199	−1.61112	−0.805562	+	0.592512i	$0.798136\pi$
$824$	−99.0361	−3.45009
$825$	0	0
$826$	72.1251	2.50955
$827$	43.3628	1.50787	0.753936	−	0.656948i	$-0.228153\pi$
$827$	43.3628	1.50787	0.753936	+	0.656948i	$0.228153\pi$
$828$	0	0
$829$	−15.9732	−0.554773	−0.277387	−	0.960758i	$-0.589468\pi$
$829$	−15.9732	−0.554773	−0.277387	+	0.960758i	$0.589468\pi$
$830$	6.15430	0.213619
$831$	0	0
$832$	−25.2737	−0.876209
$833$	−0.950311	−0.0329263
$834$	0	0
$835$	−17.8758	−0.618618
$836$	15.5918	0.539255
$837$	0	0
$838$	−48.9605	−1.69131
$839$	5.52503	0.190745	0.0953726	−	0.995442i	$-0.469596\pi$
$839$	5.52503	0.190745	0.0953726	+	0.995442i	$0.469596\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−73.6577	−2.53841
$843$	0	0
$844$	136.911	4.71269
$845$	−4.12894	−0.142040
$846$	0	0
$847$	21.7057	0.745816
$848$	−103.494	−3.55399
$849$	0	0
$850$	2.62949	0.0901907
$851$	−9.78131	−0.335299
$852$	0	0
$853$	24.5699	0.841259	0.420629	−	0.907233i	$-0.361809\pi$
$853$	24.5699	0.841259	0.420629	+	0.907233i	$0.361809\pi$
$854$	−51.3118	−1.75585
$855$	0	0
$856$	−86.6332	−2.96106
$857$	12.3705	0.422568	0.211284	−	0.977425i	$-0.432235\pi$
$857$	12.3705	0.422568	0.211284	+	0.977425i	$0.432235\pi$
$858$	0	0
$859$	−7.88043	−0.268877	−0.134438	−	0.990922i	$-0.542923\pi$
$859$	−7.88043	−0.268877	−0.134438	+	0.990922i	$0.542923\pi$
$860$	−92.8652	−3.16668
$861$	0	0
$862$	−5.90534	−0.201137
$863$	−51.0535	−1.73788	−0.868941	−	0.494916i	$-0.835199\pi$
$863$	−51.0535	−1.73788	−0.868941	+	0.494916i	$0.835199\pi$
$864$	0	0
$865$	−16.6502	−0.566124
$866$	51.0583	1.73503
$867$	0	0
$868$	67.7285	2.29886
$869$	2.99502	0.101599
$870$	0	0
$871$	2.44789	0.0829437
$872$	22.8118	0.772504
$873$	0	0
$874$	9.08566	0.307327
$875$	−14.4021	−0.486879
$876$	0	0
$877$	10.6497	0.359614	0.179807	−	0.983702i	$-0.442453\pi$
$877$	10.6497	0.359614	0.179807	+	0.983702i	$0.442453\pi$
$878$	−27.1919	−0.917683
$879$	0	0
$880$	−23.8276	−0.803230
$881$	−0.762404	−0.0256861	−0.0128430	−	0.999918i	$-0.504088\pi$
$881$	−0.762404	−0.0256861	−0.0128430	+	0.999918i	$0.504088\pi$
$882$	0	0
$883$	9.03933	0.304198	0.152099	−	0.988365i	$-0.451397\pi$
$883$	9.03933	0.304198	0.152099	+	0.988365i	$0.451397\pi$
$884$	−7.18983	−0.241820
$885$	0	0
$886$	−31.4774	−1.05750
$887$	−43.6806	−1.46665	−0.733325	−	0.679878i	$-0.762033\pi$
$887$	−43.6806	−1.46665	−0.733325	+	0.679878i	$0.762033\pi$
$888$	0	0
$889$	40.4906	1.35801
$890$	−20.0093	−0.670715
$891$	0	0
$892$	−59.2806	−1.98486
$893$	−22.3161	−0.746779
$894$	0	0
$895$	−19.0334	−0.636218
$896$	4.42311	0.147766
$897$	0	0
$898$	−76.5311	−2.55388
$899$	−6.61322	−0.220563
$900$	0	0
$901$	−4.41213	−0.146989
$902$	−23.4715	−0.781514
$903$	0	0
$904$	90.6016	3.01337
$905$	−24.4784	−0.813689
$906$	0	0
$907$	11.0837	0.368028	0.184014	−	0.982924i	$-0.441091\pi$
$907$	11.0837	0.368028	0.184014	+	0.982924i	$0.441091\pi$
$908$	−73.1607	−2.42792
$909$	0	0
$910$	−58.4182	−1.93655
$911$	7.53101	0.249514	0.124757	−	0.992187i	$-0.460185\pi$
$911$	7.53101	0.249514	0.124757	+	0.992187i	$0.460185\pi$
$912$	0	0
$913$	−0.804221	−0.0266158
$914$	−78.7191	−2.60380
$915$	0	0
$916$	74.0315	2.44607
$917$	21.2258	0.700936
$918$	0	0
$919$	−23.3625	−0.770659	−0.385329	−	0.922779i	$-0.625912\pi$
$919$	−23.3625	−0.770659	−0.385329	+	0.922779i	$0.625912\pi$
$920$	−19.8763	−0.655302
$921$	0	0
$922$	−72.0078	−2.37145
$923$	−41.1224	−1.35356
$924$	0	0
$925$	−24.9996	−0.821980
$926$	−4.96842	−0.163273
$927$	0	0
$928$	9.67060	0.317453
$929$	−39.3399	−1.29070	−0.645351	−	0.763886i	$-0.723289\pi$
$929$	−39.3399	−1.29070	−0.645351	+	0.763886i	$0.723289\pi$
$930$	0	0
$931$	−8.39241	−0.275050
$932$	26.9602	0.883109
$933$	0	0
$934$	−31.0113	−1.01472
$935$	−1.01582	−0.0332208
$936$	0	0
$937$	24.0288	0.784985	0.392493	−	0.919755i	$-0.371613\pi$
$937$	24.0288	0.784985	0.392493	+	0.919755i	$0.371613\pi$
$938$	3.58730	0.117130
$939$	0	0
$940$	83.9734	2.73891
$941$	−3.74465	−0.122072	−0.0610361	−	0.998136i	$-0.519440\pi$
$941$	−3.74465	−0.122072	−0.0610361	+	0.998136i	$0.519440\pi$
$942$	0	0
$943$	−9.64119	−0.313960
$944$	−119.801	−3.89919
$945$	0	0
$946$	17.2154	0.559721
$947$	−38.4273	−1.24872	−0.624360	−	0.781136i	$-0.714640\pi$
$947$	−38.4273	−1.24872	−0.624360	+	0.781136i	$0.714640\pi$
$948$	0	0
$949$	4.02455	0.130642
$950$	23.2216	0.753408
$951$	0	0
$952$	−6.12561	−0.198532
$953$	21.0295	0.681211	0.340606	−	0.940206i	$-0.389368\pi$
$953$	21.0295	0.681211	0.340606	+	0.940206i	$0.389368\pi$
$954$	0	0
$955$	14.3269	0.463607
$956$	31.4488	1.01713
$957$	0	0
$958$	−18.8651	−0.609502
$959$	−29.6106	−0.956175
$960$	0	0
$961$	12.7347	0.410796
$962$	96.9723	3.12651
$963$	0	0
$964$	−28.7508	−0.926002
$965$	31.1326	1.00219
$966$	0	0
$967$	−36.3360	−1.16849	−0.584244	−	0.811578i	$-0.698609\pi$
$967$	−36.3360	−1.16849	−0.584244	+	0.811578i	$0.698609\pi$
$968$	−73.2169	−2.35328
$969$	0	0
$970$	−83.8546	−2.69241
$971$	35.6304	1.14343	0.571716	−	0.820451i	$-0.306278\pi$
$971$	35.6304	1.14343	0.571716	+	0.820451i	$0.306278\pi$
$972$	0	0
$973$	24.0393	0.770664
$974$	4.03640	0.129335
$975$	0	0
$976$	85.2298	2.72814
$977$	−40.0091	−1.28001	−0.640003	−	0.768373i	$-0.721067\pi$
$977$	−40.0091	−1.28001	−0.640003	+	0.768373i	$0.721067\pi$
$978$	0	0
$979$	2.61474	0.0835676
$980$	31.5799	1.00878
$981$	0	0
$982$	43.4643	1.38700
$983$	−36.9133	−1.17735	−0.588676	−	0.808369i	$-0.700351\pi$
$983$	−36.9133	−1.17735	−0.588676	+	0.808369i	$0.700351\pi$
$984$	0	0
$985$	−46.3254	−1.47605
$986$	1.02881	0.0327640
$987$	0	0
$988$	−63.4950	−2.02004
$989$	7.07145	0.224859
$990$	0	0
$991$	33.1415	1.05277	0.526387	−	0.850245i	$-0.323546\pi$
$991$	33.1415	1.05277	0.526387	+	0.850245i	$0.323546\pi$
$992$	−63.9538	−2.03054
$993$	0	0
$994$	−60.2634	−1.91144
$995$	4.52820	0.143554
$996$	0	0
$997$	39.3035	1.24475	0.622377	−	0.782717i	$-0.286167\pi$
$997$	39.3035	1.24475	0.622377	+	0.782717i	$0.286167\pi$
$998$	16.1143	0.510090
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.22	✓	22
3.2	odd	2		6003.2.a.u.1.1	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.22	✓	22	1.1	even	1	trivial
6003.2.a.u.1.1	yes	22	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)

\(f(q)\)	\(=\)	\(q+2.60337 q^{2} +4.77753 q^{4} -2.74879 q^{5} -2.14366 q^{7} +7.23092 q^{8} +O(q^{10})\)	\(q+2.60337 q^{2} +4.77753 q^{4} -2.74879 q^{5} -2.14366 q^{7} +7.23092 q^{8} -7.15611 q^{10} +0.935134 q^{11} -3.80816 q^{13} -5.58073 q^{14} +9.26970 q^{16} +0.395185 q^{17} +3.48996 q^{19} -13.1324 q^{20} +2.43450 q^{22} +1.00000 q^{23} +2.55585 q^{25} -9.91405 q^{26} -10.2414 q^{28} +1.00000 q^{29} -6.61322 q^{31} +9.67060 q^{32} +1.02881 q^{34} +5.89247 q^{35} -9.78131 q^{37} +9.08566 q^{38} -19.8763 q^{40} -9.64119 q^{41} +7.07145 q^{43} +4.46762 q^{44} +2.60337 q^{46} -6.39436 q^{47} -2.40473 q^{49} +6.65382 q^{50} -18.1936 q^{52} -11.1647 q^{53} -2.57049 q^{55} -15.5006 q^{56} +2.60337 q^{58} -12.9239 q^{59} +9.19445 q^{61} -17.2166 q^{62} +6.63673 q^{64} +10.4678 q^{65} -0.642802 q^{67} +1.88800 q^{68} +15.3403 q^{70} +10.7985 q^{71} -1.05682 q^{73} -25.4643 q^{74} +16.6734 q^{76} -2.00461 q^{77} +3.20278 q^{79} -25.4805 q^{80} -25.0996 q^{82} -0.860006 q^{83} -1.08628 q^{85} +18.4096 q^{86} +6.76188 q^{88} +2.79612 q^{89} +8.16340 q^{91} +4.77753 q^{92} -16.6469 q^{94} -9.59318 q^{95} +11.7179 q^{97} -6.26039 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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