LMFDB - Embedded newform 6003.2.a.t.1.16

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			1.16
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)

$f(q)$	$=$	$q+1.14621 q^{2} -0.686192 q^{4} +0.461250 q^{5} -3.03232 q^{7} -3.07895 q^{8} +O(q^{10})$	\(q+1.14621 q^{2} -0.686192 q^{4} +0.461250 q^{5} -3.03232 q^{7} -3.07895 q^{8} +0.528691 q^{10} +1.24022 q^{11} +3.02812 q^{13} -3.47569 q^{14} -2.15675 q^{16} +1.56446 q^{17} -0.758514 q^{19} -0.316506 q^{20} +1.42156 q^{22} +1.00000 q^{23} -4.78725 q^{25} +3.47087 q^{26} +2.08076 q^{28} +1.00000 q^{29} +5.58933 q^{31} +3.68580 q^{32} +1.79321 q^{34} -1.39866 q^{35} +11.3773 q^{37} -0.869420 q^{38} -1.42017 q^{40} +4.22535 q^{41} -3.44184 q^{43} -0.851030 q^{44} +1.14621 q^{46} -13.0870 q^{47} +2.19499 q^{49} -5.48721 q^{50} -2.07787 q^{52} -10.3082 q^{53} +0.572051 q^{55} +9.33638 q^{56} +1.14621 q^{58} +1.58780 q^{59} -11.2127 q^{61} +6.40657 q^{62} +8.53823 q^{64} +1.39672 q^{65} -4.97364 q^{67} -1.07352 q^{68} -1.60316 q^{70} +3.12927 q^{71} -13.6069 q^{73} +13.0408 q^{74} +0.520487 q^{76} -3.76075 q^{77} +5.95073 q^{79} -0.994802 q^{80} +4.84316 q^{82} +1.97198 q^{83} +0.721608 q^{85} -3.94508 q^{86} -3.81858 q^{88} -3.14877 q^{89} -9.18224 q^{91} -0.686192 q^{92} -15.0005 q^{94} -0.349864 q^{95} -14.1283 q^{97} +2.51593 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.14621	0.810496	0.405248	−	0.914207i	$-0.367185\pi$
$2$	1.14621	0.810496	0.405248	+	0.914207i	$0.367185\pi$
$3$	0	0
$3$	0	0
$4$	−0.686192	−0.343096
$5$	0.461250	0.206277	0.103139	−	0.994667i	$-0.467112\pi$
$5$	0.461250	0.206277	0.103139	+	0.994667i	$0.467112\pi$
$6$	0	0
$7$	−3.03232	−1.14611	−0.573056	−	0.819517i	$-0.694242\pi$
$7$	−3.03232	−1.14611	−0.573056	+	0.819517i	$0.694242\pi$
$8$	−3.07895	−1.08857
$9$	0	0
$10$	0.528691	0.167187
$11$	1.24022	0.373941	0.186970	−	0.982366i	$-0.440133\pi$
$11$	1.24022	0.373941	0.186970	+	0.982366i	$0.440133\pi$
$12$	0	0
$13$	3.02812	0.839849	0.419924	−	0.907559i	$-0.362057\pi$
$13$	3.02812	0.839849	0.419924	+	0.907559i	$0.362057\pi$
$14$	−3.47569	−0.928918
$15$	0	0
$16$	−2.15675	−0.539189
$17$	1.56446	0.379438	0.189719	−	0.981838i	$-0.439242\pi$
$17$	1.56446	0.379438	0.189719	+	0.981838i	$0.439242\pi$
$18$	0	0
$19$	−0.758514	−0.174015	−0.0870075	−	0.996208i	$-0.527730\pi$
$19$	−0.758514	−0.174015	−0.0870075	+	0.996208i	$0.527730\pi$
$20$	−0.316506	−0.0707729
$21$	0	0
$22$	1.42156	0.303077
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.78725	−0.957450
$26$	3.47087	0.680694
$27$	0	0
$28$	2.08076	0.393226
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.58933	1.00387	0.501936	−	0.864905i	$-0.332621\pi$
$31$	5.58933	1.00387	0.501936	+	0.864905i	$0.332621\pi$
$32$	3.68580	0.651564
$33$	0	0
$34$	1.79321	0.307533
$35$	−1.39866	−0.236416
$36$	0	0
$37$	11.3773	1.87042	0.935209	−	0.354097i	$-0.115212\pi$
$37$	11.3773	1.87042	0.935209	+	0.354097i	$0.115212\pi$
$38$	−0.869420	−0.141038
$39$	0	0
$40$	−1.42017	−0.224548
$41$	4.22535	0.659889	0.329944	−	0.944000i	$-0.392970\pi$
$41$	4.22535	0.659889	0.329944	+	0.944000i	$0.392970\pi$
$42$	0	0
$43$	−3.44184	−0.524875	−0.262438	−	0.964949i	$-0.584526\pi$
$43$	−3.44184	−0.524875	−0.262438	+	0.964949i	$0.584526\pi$
$44$	−0.851030	−0.128298
$45$	0	0
$46$	1.14621	0.169000
$47$	−13.0870	−1.90894	−0.954468	−	0.298312i	$-0.903576\pi$
$47$	−13.0870	−1.90894	−0.954468	+	0.298312i	$0.903576\pi$
$48$	0	0
$49$	2.19499	0.313571
$50$	−5.48721	−0.776009
$51$	0	0
$52$	−2.07787	−0.288149
$53$	−10.3082	−1.41594	−0.707970	−	0.706242i	$-0.750389\pi$
$53$	−10.3082	−1.41594	−0.707970	+	0.706242i	$0.750389\pi$
$54$	0	0
$55$	0.572051	0.0771354
$56$	9.33638	1.24763
$57$	0	0
$58$	1.14621	0.150505
$59$	1.58780	0.206714	0.103357	−	0.994644i	$-0.467042\pi$
$59$	1.58780	0.206714	0.103357	+	0.994644i	$0.467042\pi$
$60$	0	0
$61$	−11.2127	−1.43564	−0.717819	−	0.696230i	$-0.754860\pi$
$61$	−11.2127	−1.43564	−0.717819	+	0.696230i	$0.754860\pi$
$62$	6.40657	0.813635
$63$	0	0
$64$	8.53823	1.06728
$65$	1.39672	0.173242
$66$	0	0
$67$	−4.97364	−0.607627	−0.303813	−	0.952732i	$-0.598260\pi$
$67$	−4.97364	−0.607627	−0.303813	+	0.952732i	$0.598260\pi$
$68$	−1.07352	−0.130184
$69$	0	0
$70$	−1.60316	−0.191615
$71$	3.12927	0.371376	0.185688	−	0.982609i	$-0.440549\pi$
$71$	3.12927	0.371376	0.185688	+	0.982609i	$0.440549\pi$
$72$	0	0
$73$	−13.6069	−1.59256	−0.796281	−	0.604927i	$-0.793202\pi$
$73$	−13.6069	−1.59256	−0.796281	+	0.604927i	$0.793202\pi$
$74$	13.0408	1.51597
$75$	0	0
$76$	0.520487	0.0597039
$77$	−3.76075	−0.428578
$78$	0	0
$79$	5.95073	0.669509	0.334755	−	0.942305i	$-0.391347\pi$
$79$	5.95073	0.669509	0.334755	+	0.942305i	$0.391347\pi$
$80$	−0.994802	−0.111222
$81$	0	0
$82$	4.84316	0.534837
$83$	1.97198	0.216453	0.108226	−	0.994126i	$-0.465483\pi$
$83$	1.97198	0.216453	0.108226	+	0.994126i	$0.465483\pi$
$84$	0	0
$85$	0.721608	0.0782694
$86$	−3.94508	−0.425409
$87$	0	0
$88$	−3.81858	−0.407062
$89$	−3.14877	−0.333769	−0.166885	−	0.985976i	$-0.553371\pi$
$89$	−3.14877	−0.333769	−0.166885	+	0.985976i	$0.553371\pi$
$90$	0	0
$91$	−9.18224	−0.962560
$92$	−0.686192	−0.0715405
$93$	0	0
$94$	−15.0005	−1.54719
$95$	−0.349864	−0.0358953
$96$	0	0
$97$	−14.1283	−1.43451	−0.717256	−	0.696809i	$-0.754602\pi$
$97$	−14.1283	−1.43451	−0.717256	+	0.696809i	$0.754602\pi$
$98$	2.51593	0.254148
$99$	0	0
$100$	3.28497	0.328497
$101$	−14.3966	−1.43252	−0.716259	−	0.697835i	$-0.754147\pi$
$101$	−14.3966	−1.43252	−0.716259	+	0.697835i	$0.754147\pi$
$102$	0	0
$103$	10.1143	0.996595	0.498298	−	0.867006i	$-0.333959\pi$
$103$	10.1143	0.996595	0.498298	+	0.867006i	$0.333959\pi$
$104$	−9.32343	−0.914238
$105$	0	0
$106$	−11.8154	−1.14761
$107$	−16.8285	−1.62687	−0.813435	−	0.581656i	$-0.802405\pi$
$107$	−16.8285	−1.62687	−0.813435	+	0.581656i	$0.802405\pi$
$108$	0	0
$109$	16.9409	1.62265	0.811323	−	0.584598i	$-0.198748\pi$
$109$	16.9409	1.62265	0.811323	+	0.584598i	$0.198748\pi$
$110$	0.655694	0.0625179
$111$	0	0
$112$	6.53998	0.617970
$113$	−14.4070	−1.35530	−0.677651	−	0.735384i	$-0.737002\pi$
$113$	−14.4070	−1.35530	−0.677651	+	0.735384i	$0.737002\pi$
$114$	0	0
$115$	0.461250	0.0430117
$116$	−0.686192	−0.0637114
$117$	0	0
$118$	1.81996	0.167541
$119$	−4.74396	−0.434878
$120$	0	0
$121$	−9.46185	−0.860168
$122$	−12.8522	−1.16358
$123$	0	0
$124$	−3.83535	−0.344425
$125$	−4.51436	−0.403777
$126$	0	0
$127$	2.54119	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$127$	2.54119	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$128$	2.41504	0.213461
$129$	0	0
$130$	1.60094	0.140412
$131$	−3.71190	−0.324310	−0.162155	−	0.986765i	$-0.551844\pi$
$131$	−3.71190	−0.324310	−0.162155	+	0.986765i	$0.551844\pi$
$132$	0	0
$133$	2.30006	0.199441
$134$	−5.70086	−0.492479
$135$	0	0
$136$	−4.81691	−0.413047
$137$	−8.98216	−0.767398	−0.383699	−	0.923458i	$-0.625350\pi$
$137$	−8.98216	−0.767398	−0.383699	+	0.923458i	$0.625350\pi$
$138$	0	0
$139$	2.13746	0.181297	0.0906487	−	0.995883i	$-0.471106\pi$
$139$	2.13746	0.181297	0.0906487	+	0.995883i	$0.471106\pi$
$140$	0.959749	0.0811136
$141$	0	0
$142$	3.58681	0.300999
$143$	3.75554	0.314054
$144$	0	0
$145$	0.461250	0.0383047
$146$	−15.5964	−1.29076
$147$	0	0
$148$	−7.80702	−0.641733
$149$	17.6412	1.44523	0.722614	−	0.691252i	$-0.242941\pi$
$149$	17.6412	1.44523	0.722614	+	0.691252i	$0.242941\pi$
$150$	0	0
$151$	−5.41797	−0.440908	−0.220454	−	0.975397i	$-0.570754\pi$
$151$	−5.41797	−0.440908	−0.220454	+	0.975397i	$0.570754\pi$
$152$	2.33543	0.189428
$153$	0	0
$154$	−4.31063	−0.347360
$155$	2.57807	0.207076
$156$	0	0
$157$	−19.1808	−1.53080	−0.765399	−	0.643557i	$-0.777458\pi$
$157$	−19.1808	−1.53080	−0.765399	+	0.643557i	$0.777458\pi$
$158$	6.82081	0.542634
$159$	0	0
$160$	1.70007	0.134403
$161$	−3.03232	−0.238981
$162$	0	0
$163$	−15.2529	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$163$	−15.2529	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$164$	−2.89940	−0.226405
$165$	0	0
$166$	2.26031	0.175434
$167$	9.37038	0.725102	0.362551	−	0.931964i	$-0.381906\pi$
$167$	9.37038	0.725102	0.362551	+	0.931964i	$0.381906\pi$
$168$	0	0
$169$	−3.83050	−0.294654
$170$	0.827118	0.0634370
$171$	0	0
$172$	2.36176	0.180083
$173$	5.08480	0.386591	0.193295	−	0.981141i	$-0.438082\pi$
$173$	5.08480	0.386591	0.193295	+	0.981141i	$0.438082\pi$
$174$	0	0
$175$	14.5165	1.09734
$176$	−2.67485	−0.201625
$177$	0	0
$178$	−3.60917	−0.270519
$179$	13.8992	1.03887	0.519437	−	0.854509i	$-0.326142\pi$
$179$	13.8992	1.03887	0.519437	+	0.854509i	$0.326142\pi$
$180$	0	0
$181$	14.4692	1.07549	0.537744	−	0.843108i	$-0.319277\pi$
$181$	14.4692	1.07549	0.537744	+	0.843108i	$0.319277\pi$
$182$	−10.5248	−0.780151
$183$	0	0
$184$	−3.07895	−0.226983
$185$	5.24778	0.385824
$186$	0	0
$187$	1.94028	0.141887
$188$	8.98021	0.654949
$189$	0	0
$190$	−0.401019	−0.0290930
$191$	12.1024	0.875699	0.437849	−	0.899048i	$-0.355740\pi$
$191$	12.1024	0.875699	0.437849	+	0.899048i	$0.355740\pi$
$192$	0	0
$193$	−4.52408	−0.325650	−0.162825	−	0.986655i	$-0.552061\pi$
$193$	−4.52408	−0.325650	−0.162825	+	0.986655i	$0.552061\pi$
$194$	−16.1941	−1.16267
$195$	0	0
$196$	−1.50619	−0.107585
$197$	−26.4478	−1.88433	−0.942165	−	0.335149i	$-0.891213\pi$
$197$	−26.4478	−1.88433	−0.942165	+	0.335149i	$0.891213\pi$
$198$	0	0
$199$	−2.65055	−0.187892	−0.0939462	−	0.995577i	$-0.529948\pi$
$199$	−2.65055	−0.187892	−0.0939462	+	0.995577i	$0.529948\pi$
$200$	14.7397	1.04226
$201$	0	0
$202$	−16.5016	−1.16105
$203$	−3.03232	−0.212827
$204$	0	0
$205$	1.94894	0.136120
$206$	11.5932	0.807737
$207$	0	0
$208$	−6.53091	−0.452837
$209$	−0.940725	−0.0650713
$210$	0	0
$211$	16.6333	1.14509	0.572543	−	0.819875i	$-0.305957\pi$
$211$	16.6333	1.14509	0.572543	+	0.819875i	$0.305957\pi$
$212$	7.07341	0.485804
$213$	0	0
$214$	−19.2890	−1.31857
$215$	−1.58755	−0.108270
$216$	0	0
$217$	−16.9487	−1.15055
$218$	19.4179	1.31515
$219$	0	0
$220$	−0.392537	−0.0264649
$221$	4.73738	0.318671
$222$	0	0
$223$	2.74786	0.184010	0.0920052	−	0.995759i	$-0.470672\pi$
$223$	2.74786	0.184010	0.0920052	+	0.995759i	$0.470672\pi$
$224$	−11.1765	−0.746765
$225$	0	0
$226$	−16.5136	−1.09847
$227$	−14.8829	−0.987814	−0.493907	−	0.869515i	$-0.664432\pi$
$227$	−14.8829	−0.987814	−0.493907	+	0.869515i	$0.664432\pi$
$228$	0	0
$229$	−18.8535	−1.24588	−0.622939	−	0.782271i	$-0.714061\pi$
$229$	−18.8535	−1.24588	−0.622939	+	0.782271i	$0.714061\pi$
$230$	0.528691	0.0348608
$231$	0	0
$232$	−3.07895	−0.202143
$233$	0.877423	0.0574819	0.0287409	−	0.999587i	$-0.490850\pi$
$233$	0.877423	0.0574819	0.0287409	+	0.999587i	$0.490850\pi$
$234$	0	0
$235$	−6.03638	−0.393770
$236$	−1.08954	−0.0709229
$237$	0	0
$238$	−5.43760	−0.352467
$239$	−16.3766	−1.05931	−0.529655	−	0.848213i	$-0.677679\pi$
$239$	−16.3766	−1.05931	−0.529655	+	0.848213i	$0.677679\pi$
$240$	0	0
$241$	−22.2101	−1.43068	−0.715339	−	0.698778i	$-0.753727\pi$
$241$	−22.2101	−1.43068	−0.715339	+	0.698778i	$0.753727\pi$
$242$	−10.8453	−0.697163
$243$	0	0
$244$	7.69407	0.492562
$245$	1.01244	0.0646824
$246$	0	0
$247$	−2.29687	−0.146146
$248$	−17.2093	−1.09279
$249$	0	0
$250$	−5.17443	−0.327260
$251$	26.0956	1.64714	0.823571	−	0.567214i	$-0.191979\pi$
$251$	26.0956	1.64714	0.823571	+	0.567214i	$0.191979\pi$
$252$	0	0
$253$	1.24022	0.0779720
$254$	2.91275	0.182762
$255$	0	0
$256$	−14.3083	−0.894269
$257$	−10.7098	−0.668061	−0.334031	−	0.942562i	$-0.608409\pi$
$257$	−10.7098	−0.668061	−0.334031	+	0.942562i	$0.608409\pi$
$258$	0	0
$259$	−34.4997	−2.14371
$260$	−0.958417	−0.0594385
$261$	0	0
$262$	−4.25463	−0.262852
$263$	7.99789	0.493171	0.246585	−	0.969121i	$-0.420691\pi$
$263$	7.99789	0.493171	0.246585	+	0.969121i	$0.420691\pi$
$264$	0	0
$265$	−4.75465	−0.292076
$266$	2.63636	0.161646
$267$	0	0
$268$	3.41287	0.208474
$269$	−7.58114	−0.462230	−0.231115	−	0.972926i	$-0.574237\pi$
$269$	−7.58114	−0.462230	−0.231115	+	0.972926i	$0.574237\pi$
$270$	0	0
$271$	2.48450	0.150923	0.0754613	−	0.997149i	$-0.475957\pi$
$271$	2.48450	0.150923	0.0754613	+	0.997149i	$0.475957\pi$
$272$	−3.37417	−0.204589
$273$	0	0
$274$	−10.2955	−0.621973
$275$	−5.93725	−0.358029
$276$	0	0
$277$	−1.86945	−0.112324	−0.0561621	−	0.998422i	$-0.517886\pi$
$277$	−1.86945	−0.112324	−0.0561621	+	0.998422i	$0.517886\pi$
$278$	2.44999	0.146941
$279$	0	0
$280$	4.30640	0.257357
$281$	19.7896	1.18055	0.590273	−	0.807204i	$-0.299020\pi$
$281$	19.7896	1.18055	0.590273	+	0.807204i	$0.299020\pi$
$282$	0	0
$283$	1.71272	0.101811	0.0509054	−	0.998703i	$-0.483789\pi$
$283$	1.71272	0.101811	0.0509054	+	0.998703i	$0.483789\pi$
$284$	−2.14728	−0.127418
$285$	0	0
$286$	4.30465	0.254539
$287$	−12.8126	−0.756306
$288$	0	0
$289$	−14.5525	−0.856027
$290$	0.528691	0.0310458
$291$	0	0
$292$	9.33692	0.546402
$293$	10.0342	0.586202	0.293101	−	0.956081i	$-0.405313\pi$
$293$	10.0342	0.586202	0.293101	+	0.956081i	$0.405313\pi$
$294$	0	0
$295$	0.732373	0.0426404
$296$	−35.0302	−2.03609
$297$	0	0
$298$	20.2207	1.17135
$299$	3.02812	0.175121
$300$	0	0
$301$	10.4368	0.601565
$302$	−6.21015	−0.357354
$303$	0	0
$304$	1.63593	0.0938269
$305$	−5.17185	−0.296139
$306$	0	0
$307$	−9.23516	−0.527079	−0.263539	−	0.964649i	$-0.584890\pi$
$307$	−9.23516	−0.527079	−0.263539	+	0.964649i	$0.584890\pi$
$308$	2.58060	0.147043
$309$	0	0
$310$	2.95503	0.167834
$311$	26.4506	1.49988	0.749938	−	0.661509i	$-0.230083\pi$
$311$	26.4506	1.49988	0.749938	+	0.661509i	$0.230083\pi$
$312$	0	0
$313$	−33.1867	−1.87582	−0.937912	−	0.346874i	$-0.887243\pi$
$313$	−33.1867	−1.87582	−0.937912	+	0.346874i	$0.887243\pi$
$314$	−21.9853	−1.24070
$315$	0	0
$316$	−4.08334	−0.229706
$317$	12.6620	0.711171	0.355586	−	0.934644i	$-0.384281\pi$
$317$	12.6620	0.711171	0.355586	+	0.934644i	$0.384281\pi$
$318$	0	0
$319$	1.24022	0.0694390
$320$	3.93825	0.220155
$321$	0	0
$322$	−3.47569	−0.193693
$323$	−1.18667	−0.0660279
$324$	0	0
$325$	−14.4964	−0.804113
$326$	−17.4831	−0.968301
$327$	0	0
$328$	−13.0097	−0.718338
$329$	39.6841	2.18785
$330$	0	0
$331$	21.9555	1.20678	0.603391	−	0.797446i	$-0.293816\pi$
$331$	21.9555	1.20678	0.603391	+	0.797446i	$0.293816\pi$
$332$	−1.35316	−0.0742641
$333$	0	0
$334$	10.7405	0.587692
$335$	−2.29409	−0.125339
$336$	0	0
$337$	2.06225	0.112338	0.0561690	−	0.998421i	$-0.482111\pi$
$337$	2.06225	0.112338	0.0561690	+	0.998421i	$0.482111\pi$
$338$	−4.39057	−0.238816
$339$	0	0
$340$	−0.495162	−0.0268539
$341$	6.93200	0.375389
$342$	0	0
$343$	14.5703	0.786724
$344$	10.5973	0.571366
$345$	0	0
$346$	5.82828	0.313330
$347$	−7.14692	−0.383667	−0.191833	−	0.981428i	$-0.561443\pi$
$347$	−7.14692	−0.383667	−0.191833	+	0.981428i	$0.561443\pi$
$348$	0	0
$349$	−14.7917	−0.791783	−0.395892	−	0.918297i	$-0.629564\pi$
$349$	−14.7917	−0.791783	−0.395892	+	0.918297i	$0.629564\pi$
$350$	16.6390	0.889393
$351$	0	0
$352$	4.57121	0.243646
$353$	35.3093	1.87933	0.939663	−	0.342101i	$-0.111138\pi$
$353$	35.3093	1.87933	0.939663	+	0.342101i	$0.111138\pi$
$354$	0	0
$355$	1.44337	0.0766063
$356$	2.16066	0.114515
$357$	0	0
$358$	15.9315	0.842004
$359$	−34.6918	−1.83097	−0.915483	−	0.402358i	$-0.868191\pi$
$359$	−34.6918	−1.83097	−0.915483	+	0.402358i	$0.868191\pi$
$360$	0	0
$361$	−18.4247	−0.969719
$362$	16.5848	0.871679
$363$	0	0
$364$	6.30078	0.330251
$365$	−6.27615	−0.328509
$366$	0	0
$367$	−6.78620	−0.354237	−0.177118	−	0.984190i	$-0.556678\pi$
$367$	−6.78620	−0.354237	−0.177118	+	0.984190i	$0.556678\pi$
$368$	−2.15675	−0.112429
$369$	0	0
$370$	6.01508	0.312709
$371$	31.2578	1.62282
$372$	0	0
$373$	−31.1069	−1.61065	−0.805326	−	0.592832i	$-0.798010\pi$
$373$	−31.1069	−1.61065	−0.805326	+	0.592832i	$0.798010\pi$
$374$	2.22398	0.114999
$375$	0	0
$376$	40.2943	2.07802
$377$	3.02812	0.155956
$378$	0	0
$379$	−37.0062	−1.90088	−0.950441	−	0.310904i	$-0.899368\pi$
$379$	−37.0062	−1.90088	−0.950441	+	0.310904i	$0.899368\pi$
$380$	0.240074	0.0123155
$381$	0	0
$382$	13.8719	0.709750
$383$	−0.396982	−0.0202848	−0.0101424	−	0.999949i	$-0.503228\pi$
$383$	−0.396982	−0.0202848	−0.0101424	+	0.999949i	$0.503228\pi$
$384$	0	0
$385$	−1.73465	−0.0884057
$386$	−5.18556	−0.263938
$387$	0	0
$388$	9.69474	0.492176
$389$	−4.09069	−0.207406	−0.103703	−	0.994608i	$-0.533069\pi$
$389$	−4.09069	−0.207406	−0.103703	+	0.994608i	$0.533069\pi$
$390$	0	0
$391$	1.56446	0.0791183
$392$	−6.75828	−0.341345
$393$	0	0
$394$	−30.3149	−1.52724
$395$	2.74477	0.138104
$396$	0	0
$397$	3.63819	0.182596	0.0912978	−	0.995824i	$-0.470898\pi$
$397$	3.63819	0.182596	0.0912978	+	0.995824i	$0.470898\pi$
$398$	−3.03810	−0.152286
$399$	0	0
$400$	10.3249	0.516246
$401$	−17.3710	−0.867466	−0.433733	−	0.901042i	$-0.642804\pi$
$401$	−17.3710	−0.867466	−0.433733	+	0.901042i	$0.642804\pi$
$402$	0	0
$403$	16.9251	0.843101
$404$	9.87885	0.491491
$405$	0	0
$406$	−3.47569	−0.172496
$407$	14.1104	0.699425
$408$	0	0
$409$	13.3595	0.660586	0.330293	−	0.943878i	$-0.392852\pi$
$409$	13.3595	0.660586	0.330293	+	0.943878i	$0.392852\pi$
$410$	2.23390	0.110325
$411$	0	0
$412$	−6.94038	−0.341928
$413$	−4.81473	−0.236917
$414$	0	0
$415$	0.909574	0.0446492
$416$	11.1610	0.547215
$417$	0	0
$418$	−1.07827	−0.0527400
$419$	5.58865	0.273023	0.136512	−	0.990638i	$-0.456411\pi$
$419$	5.58865	0.273023	0.136512	+	0.990638i	$0.456411\pi$
$420$	0	0
$421$	19.4449	0.947687	0.473843	−	0.880609i	$-0.342866\pi$
$421$	19.4449	0.947687	0.473843	+	0.880609i	$0.342866\pi$
$422$	19.0654	0.928088
$423$	0	0
$424$	31.7385	1.54136
$425$	−7.48948	−0.363293
$426$	0	0
$427$	34.0005	1.64540
$428$	11.5476	0.558173
$429$	0	0
$430$	−1.81967	−0.0877522
$431$	−2.38531	−0.114896	−0.0574481	−	0.998348i	$-0.518296\pi$
$431$	−2.38531	−0.114896	−0.0574481	+	0.998348i	$0.518296\pi$
$432$	0	0
$433$	5.06699	0.243504	0.121752	−	0.992561i	$-0.461149\pi$
$433$	5.06699	0.243504	0.121752	+	0.992561i	$0.461149\pi$
$434$	−19.4268	−0.932516
$435$	0	0
$436$	−11.6247	−0.556724
$437$	−0.758514	−0.0362846
$438$	0	0
$439$	−15.6379	−0.746354	−0.373177	−	0.927760i	$-0.621732\pi$
$439$	−15.6379	−0.746354	−0.373177	+	0.927760i	$0.621732\pi$
$440$	−1.76132	−0.0839676
$441$	0	0
$442$	5.43005	0.258281
$443$	25.6358	1.21800	0.608998	−	0.793172i	$-0.291572\pi$
$443$	25.6358	1.21800	0.608998	+	0.793172i	$0.291572\pi$
$444$	0	0
$445$	−1.45237	−0.0688490
$446$	3.14964	0.149140
$447$	0	0
$448$	−25.8907	−1.22322
$449$	6.29801	0.297221	0.148611	−	0.988896i	$-0.452520\pi$
$449$	6.29801	0.297221	0.148611	+	0.988896i	$0.452520\pi$
$450$	0	0
$451$	5.24037	0.246759
$452$	9.88601	0.464999
$453$	0	0
$454$	−17.0590	−0.800620
$455$	−4.23530	−0.198554
$456$	0	0
$457$	4.88547	0.228533	0.114266	−	0.993450i	$-0.463548\pi$
$457$	4.88547	0.228533	0.114266	+	0.993450i	$0.463548\pi$
$458$	−21.6102	−1.00978
$459$	0	0
$460$	−0.316506	−0.0147572
$461$	−9.79885	−0.456378	−0.228189	−	0.973617i	$-0.573280\pi$
$461$	−9.79885	−0.456378	−0.228189	+	0.973617i	$0.573280\pi$
$462$	0	0
$463$	−6.06469	−0.281850	−0.140925	−	0.990020i	$-0.545008\pi$
$463$	−6.06469	−0.281850	−0.140925	+	0.990020i	$0.545008\pi$
$464$	−2.15675	−0.100125
$465$	0	0
$466$	1.00571	0.0465888
$467$	25.8676	1.19701	0.598505	−	0.801119i	$-0.295762\pi$
$467$	25.8676	1.19701	0.598505	+	0.801119i	$0.295762\pi$
$468$	0	0
$469$	15.0817	0.696408
$470$	−6.91898	−0.319149
$471$	0	0
$472$	−4.88877	−0.225024
$473$	−4.26864	−0.196272
$474$	0	0
$475$	3.63119	0.166611
$476$	3.25527	0.149205
$477$	0	0
$478$	−18.7710	−0.858567
$479$	−13.4965	−0.616673	−0.308336	−	0.951277i	$-0.599772\pi$
$479$	−13.4965	−0.616673	−0.308336	+	0.951277i	$0.599772\pi$
$480$	0	0
$481$	34.4518	1.57087
$482$	−25.4575	−1.15956
$483$	0	0
$484$	6.49265	0.295121
$485$	−6.51668	−0.295907
$486$	0	0
$487$	−30.5042	−1.38228	−0.691139	−	0.722722i	$-0.742891\pi$
$487$	−30.5042	−1.38228	−0.691139	+	0.722722i	$0.742891\pi$
$488$	34.5234	1.56280
$489$	0	0
$490$	1.16047	0.0524249
$491$	−35.2597	−1.59125	−0.795623	−	0.605792i	$-0.792856\pi$
$491$	−35.2597	−1.59125	−0.795623	+	0.605792i	$0.792856\pi$
$492$	0	0
$493$	1.56446	0.0704599
$494$	−2.63271	−0.118451
$495$	0	0
$496$	−12.0548	−0.541277
$497$	−9.48896	−0.425638
$498$	0	0
$499$	18.0864	0.809658	0.404829	−	0.914392i	$-0.367331\pi$
$499$	18.0864	0.809658	0.404829	+	0.914392i	$0.367331\pi$
$500$	3.09772	0.138534
$501$	0	0
$502$	29.9112	1.33500
$503$	41.9591	1.87086	0.935432	−	0.353506i	$-0.115011\pi$
$503$	41.9591	1.87086	0.935432	+	0.353506i	$0.115011\pi$
$504$	0	0
$505$	−6.64044	−0.295495
$506$	1.42156	0.0631960
$507$	0	0
$508$	−1.74375	−0.0773663
$509$	0.134948	0.00598148	0.00299074	−	0.999996i	$-0.499048\pi$
$509$	0.134948	0.00598148	0.00299074	+	0.999996i	$0.499048\pi$
$510$	0	0
$511$	41.2604	1.82525
$512$	−21.2305	−0.938263
$513$	0	0
$514$	−12.2758	−0.541461
$515$	4.66523	0.205575
$516$	0	0
$517$	−16.2308	−0.713829
$518$	−39.5440	−1.73747
$519$	0	0
$520$	−4.30043	−0.188586
$521$	12.9612	0.567840	0.283920	−	0.958848i	$-0.408365\pi$
$521$	12.9612	0.567840	0.283920	+	0.958848i	$0.408365\pi$
$522$	0	0
$523$	23.8876	1.04453	0.522267	−	0.852782i	$-0.325087\pi$
$523$	23.8876	1.04453	0.522267	+	0.852782i	$0.325087\pi$
$524$	2.54708	0.111270
$525$	0	0
$526$	9.16729	0.399713
$527$	8.74430	0.380908
$528$	0	0
$529$	1.00000	0.0434783
$530$	−5.44985	−0.236726
$531$	0	0
$532$	−1.57828	−0.0684273
$533$	12.7949	0.554207
$534$	0	0
$535$	−7.76212	−0.335586
$536$	15.3136	0.661447
$537$	0	0
$538$	−8.68961	−0.374636
$539$	2.72228	0.117257
$540$	0	0
$541$	−32.1719	−1.38318	−0.691588	−	0.722292i	$-0.743089\pi$
$541$	−32.1719	−1.38318	−0.691588	+	0.722292i	$0.743089\pi$
$542$	2.84777	0.122322
$543$	0	0
$544$	5.76630	0.247228
$545$	7.81399	0.334715
$546$	0	0
$547$	−22.1022	−0.945022	−0.472511	−	0.881325i	$-0.656652\pi$
$547$	−22.1022	−0.945022	−0.472511	+	0.881325i	$0.656652\pi$
$548$	6.16349	0.263291
$549$	0	0
$550$	−6.80536	−0.290181
$551$	−0.758514	−0.0323138
$552$	0	0
$553$	−18.0445	−0.767332
$554$	−2.14279	−0.0910383
$555$	0	0
$556$	−1.46671	−0.0622024
$557$	21.0377	0.891394	0.445697	−	0.895184i	$-0.352956\pi$
$557$	21.0377	0.891394	0.445697	+	0.895184i	$0.352956\pi$
$558$	0	0
$559$	−10.4223	−0.440816
$560$	3.01656	0.127473
$561$	0	0
$562$	22.6831	0.956828
$563$	10.5927	0.446428	0.223214	−	0.974769i	$-0.428345\pi$
$563$	10.5927	0.446428	0.223214	+	0.974769i	$0.428345\pi$
$564$	0	0
$565$	−6.64524	−0.279568
$566$	1.96315	0.0825172
$567$	0	0
$568$	−9.63487	−0.404270
$569$	28.7234	1.20415	0.602074	−	0.798440i	$-0.294341\pi$
$569$	28.7234	1.20415	0.602074	+	0.798440i	$0.294341\pi$
$570$	0	0
$571$	3.70160	0.154907	0.0774535	−	0.996996i	$-0.475321\pi$
$571$	3.70160	0.154907	0.0774535	+	0.996996i	$0.475321\pi$
$572$	−2.57702	−0.107751
$573$	0	0
$574$	−14.6860	−0.612983
$575$	−4.78725	−0.199642
$576$	0	0
$577$	−31.7741	−1.32277	−0.661387	−	0.750045i	$-0.730032\pi$
$577$	−31.7741	−1.32277	−0.661387	+	0.750045i	$0.730032\pi$
$578$	−16.6802	−0.693806
$579$	0	0
$580$	−0.316506	−0.0131422
$581$	−5.97968	−0.248079
$582$	0	0
$583$	−12.7844	−0.529478
$584$	41.8949	1.73362
$585$	0	0
$586$	11.5013	0.475114
$587$	5.55049	0.229093	0.114547	−	0.993418i	$-0.463458\pi$
$587$	5.55049	0.229093	0.114547	+	0.993418i	$0.463458\pi$
$588$	0	0
$589$	−4.23958	−0.174689
$590$	0.839456	0.0345599
$591$	0	0
$592$	−24.5381	−1.00851
$593$	45.2481	1.85812	0.929058	−	0.369934i	$-0.120620\pi$
$593$	45.2481	1.85812	0.929058	+	0.369934i	$0.120620\pi$
$594$	0	0
$595$	−2.18815	−0.0897054
$596$	−12.1053	−0.495852
$597$	0	0
$598$	3.47087	0.141935
$599$	−8.33358	−0.340501	−0.170251	−	0.985401i	$-0.554458\pi$
$599$	−8.33358	−0.340501	−0.170251	+	0.985401i	$0.554458\pi$
$600$	0	0
$601$	−20.2780	−0.827158	−0.413579	−	0.910468i	$-0.635721\pi$
$601$	−20.2780	−0.827158	−0.413579	+	0.910468i	$0.635721\pi$
$602$	11.9628	0.487566
$603$	0	0
$604$	3.71777	0.151274
$605$	−4.36427	−0.177433
$606$	0	0
$607$	46.1404	1.87278	0.936390	−	0.350962i	$-0.114145\pi$
$607$	46.1404	1.87278	0.936390	+	0.350962i	$0.114145\pi$
$608$	−2.79573	−0.113382
$609$	0	0
$610$	−5.92805	−0.240020
$611$	−39.6290	−1.60322
$612$	0	0
$613$	39.2871	1.58679	0.793396	−	0.608706i	$-0.208311\pi$
$613$	39.2871	1.58679	0.793396	+	0.608706i	$0.208311\pi$
$614$	−10.5855	−0.427195
$615$	0	0
$616$	11.5792	0.466538
$617$	38.1156	1.53448	0.767238	−	0.641362i	$-0.221630\pi$
$617$	38.1156	1.53448	0.767238	+	0.641362i	$0.221630\pi$
$618$	0	0
$619$	−37.6762	−1.51433	−0.757167	−	0.653222i	$-0.773417\pi$
$619$	−37.6762	−1.51433	−0.757167	+	0.653222i	$0.773417\pi$
$620$	−1.76906	−0.0710470
$621$	0	0
$622$	30.3180	1.21564
$623$	9.54811	0.382537
$624$	0	0
$625$	21.8540	0.874160
$626$	−38.0391	−1.52035
$627$	0	0
$628$	13.1617	0.525211
$629$	17.7994	0.709708
$630$	0	0
$631$	−24.2007	−0.963414	−0.481707	−	0.876332i	$-0.659983\pi$
$631$	−24.2007	−0.963414	−0.481707	+	0.876332i	$0.659983\pi$
$632$	−18.3220	−0.728810
$633$	0	0
$634$	14.5134	0.576402
$635$	1.17212	0.0465143
$636$	0	0
$637$	6.64670	0.263352
$638$	1.42156	0.0562801
$639$	0	0
$640$	1.11394	0.0440322
$641$	34.1062	1.34711	0.673557	−	0.739135i	$-0.264766\pi$
$641$	34.1062	1.34711	0.673557	+	0.739135i	$0.264766\pi$
$642$	0	0
$643$	−0.586521	−0.0231301	−0.0115651	−	0.999933i	$-0.503681\pi$
$643$	−0.586521	−0.0231301	−0.0115651	+	0.999933i	$0.503681\pi$
$644$	2.08076	0.0819934
$645$	0	0
$646$	−1.36018	−0.0535154
$647$	−30.9753	−1.21776	−0.608882	−	0.793261i	$-0.708382\pi$
$647$	−30.9753	−1.21776	−0.608882	+	0.793261i	$0.708382\pi$
$648$	0	0
$649$	1.96922	0.0772988
$650$	−16.6159	−0.651730
$651$	0	0
$652$	10.4664	0.409898
$653$	29.7907	1.16580	0.582900	−	0.812544i	$-0.301918\pi$
$653$	29.7907	1.16580	0.582900	+	0.812544i	$0.301918\pi$
$654$	0	0
$655$	−1.71211	−0.0668977
$656$	−9.11305	−0.355805
$657$	0	0
$658$	45.4865	1.77325
$659$	1.60411	0.0624871	0.0312436	−	0.999512i	$-0.490053\pi$
$659$	1.60411	0.0624871	0.0312436	+	0.999512i	$0.490053\pi$
$660$	0	0
$661$	−3.63035	−0.141204	−0.0706021	−	0.997505i	$-0.522492\pi$
$661$	−3.63035	−0.141204	−0.0706021	+	0.997505i	$0.522492\pi$
$662$	25.1657	0.978091
$663$	0	0
$664$	−6.07163	−0.235625
$665$	1.06090	0.0411400
$666$	0	0
$667$	1.00000	0.0387202
$668$	−6.42988	−0.248780
$669$	0	0
$670$	−2.62952	−0.101587
$671$	−13.9062	−0.536844
$672$	0	0
$673$	−20.6602	−0.796390	−0.398195	−	0.917301i	$-0.630363\pi$
$673$	−20.6602	−0.796390	−0.398195	+	0.917301i	$0.630363\pi$
$674$	2.36378	0.0910495
$675$	0	0
$676$	2.62846	0.101095
$677$	−19.9369	−0.766238	−0.383119	−	0.923699i	$-0.625150\pi$
$677$	−19.9369	−0.766238	−0.383119	+	0.923699i	$0.625150\pi$
$678$	0	0
$679$	42.8416	1.64411
$680$	−2.22180	−0.0852020
$681$	0	0
$682$	7.94556	0.304251
$683$	32.8353	1.25641	0.628203	−	0.778049i	$-0.283791\pi$
$683$	32.8353	1.25641	0.628203	+	0.778049i	$0.283791\pi$
$684$	0	0
$685$	−4.14302	−0.158297
$686$	16.7007	0.637637
$687$	0	0
$688$	7.42320	0.283007
$689$	−31.2145	−1.18918
$690$	0	0
$691$	40.8340	1.55340	0.776699	−	0.629872i	$-0.216892\pi$
$691$	40.8340	1.55340	0.776699	+	0.629872i	$0.216892\pi$
$692$	−3.48915	−0.132638
$693$	0	0
$694$	−8.19190	−0.310960
$695$	0.985904	0.0373975
$696$	0	0
$697$	6.61041	0.250387
$698$	−16.9545	−0.641737
$699$	0	0
$700$	−9.96111	−0.376495
$701$	−23.9089	−0.903028	−0.451514	−	0.892264i	$-0.649116\pi$
$701$	−23.9089	−0.903028	−0.451514	+	0.892264i	$0.649116\pi$
$702$	0	0
$703$	−8.62984	−0.325481
$704$	10.5893	0.399099
$705$	0	0
$706$	40.4721	1.52319
$707$	43.6552	1.64182
$708$	0	0
$709$	−7.52488	−0.282603	−0.141301	−	0.989967i	$-0.545129\pi$
$709$	−7.52488	−0.282603	−0.141301	+	0.989967i	$0.545129\pi$
$710$	1.65442	0.0620891
$711$	0	0
$712$	9.69493	0.363333
$713$	5.58933	0.209322
$714$	0	0
$715$	1.73224	0.0647821
$716$	−9.53752	−0.356434
$717$	0	0
$718$	−39.7643	−1.48399
$719$	−42.2000	−1.57379	−0.786897	−	0.617084i	$-0.788314\pi$
$719$	−42.2000	−1.57379	−0.786897	+	0.617084i	$0.788314\pi$
$720$	0	0
$721$	−30.6700	−1.14221
$722$	−21.1186	−0.785953
$723$	0	0
$724$	−9.92866	−0.368996
$725$	−4.78725	−0.177794
$726$	0	0
$727$	26.6149	0.987093	0.493547	−	0.869719i	$-0.335700\pi$
$727$	26.6149	0.987093	0.493547	+	0.869719i	$0.335700\pi$
$728$	28.2717	1.04782
$729$	0	0
$730$	−7.19382	−0.266255
$731$	−5.38463	−0.199158
$732$	0	0
$733$	−24.3128	−0.898015	−0.449007	−	0.893528i	$-0.648222\pi$
$733$	−24.3128	−0.898015	−0.449007	+	0.893528i	$0.648222\pi$
$734$	−7.77844	−0.287107
$735$	0	0
$736$	3.68580	0.135860
$737$	−6.16841	−0.227216
$738$	0	0
$739$	−17.2467	−0.634429	−0.317214	−	0.948354i	$-0.602748\pi$
$739$	−17.2467	−0.634429	−0.317214	+	0.948354i	$0.602748\pi$
$740$	−3.60098	−0.132375
$741$	0	0
$742$	35.8282	1.31529
$743$	−27.2488	−0.999660	−0.499830	−	0.866124i	$-0.666604\pi$
$743$	−27.2488	−0.999660	−0.499830	+	0.866124i	$0.666604\pi$
$744$	0	0
$745$	8.13702	0.298117
$746$	−35.6551	−1.30543
$747$	0	0
$748$	−1.33141	−0.0486810
$749$	51.0294	1.86457
$750$	0	0
$751$	−48.9353	−1.78567	−0.892836	−	0.450381i	$-0.851288\pi$
$751$	−48.9353	−1.78567	−0.892836	+	0.450381i	$0.851288\pi$
$752$	28.2255	1.02928
$753$	0	0
$754$	3.47087	0.126402
$755$	−2.49903	−0.0909492
$756$	0	0
$757$	−0.573928	−0.0208598	−0.0104299	−	0.999946i	$-0.503320\pi$
$757$	−0.573928	−0.0208598	−0.0104299	+	0.999946i	$0.503320\pi$
$758$	−42.4171	−1.54066
$759$	0	0
$760$	1.07722	0.0390747
$761$	−47.1634	−1.70967	−0.854836	−	0.518898i	$-0.826342\pi$
$761$	−47.1634	−1.70967	−0.854836	+	0.518898i	$0.826342\pi$
$762$	0	0
$763$	−51.3704	−1.85973
$764$	−8.30457	−0.300449
$765$	0	0
$766$	−0.455027	−0.0164408
$767$	4.80805	0.173609
$768$	0	0
$769$	−40.2948	−1.45307	−0.726534	−	0.687130i	$-0.758870\pi$
$769$	−40.2948	−1.45307	−0.726534	+	0.687130i	$0.758870\pi$
$770$	−1.98828	−0.0716525
$771$	0	0
$772$	3.10439	0.111729
$773$	2.89284	0.104048	0.0520241	−	0.998646i	$-0.483433\pi$
$773$	2.89284	0.104048	0.0520241	+	0.998646i	$0.483433\pi$
$774$	0	0
$775$	−26.7575	−0.961158
$776$	43.5004	1.56157
$777$	0	0
$778$	−4.68881	−0.168102
$779$	−3.20499	−0.114831
$780$	0	0
$781$	3.88098	0.138872
$782$	1.79321	0.0641251
$783$	0	0
$784$	−4.73407	−0.169074
$785$	−8.84715	−0.315768
$786$	0	0
$787$	18.6559	0.665010	0.332505	−	0.943101i	$-0.392106\pi$
$787$	18.6559	0.665010	0.332505	+	0.943101i	$0.392106\pi$
$788$	18.1483	0.646507
$789$	0	0
$790$	3.14610	0.111933
$791$	43.6869	1.55333
$792$	0	0
$793$	−33.9534	−1.20572
$794$	4.17015	0.147993
$795$	0	0
$796$	1.81879	0.0644652
$797$	−21.1459	−0.749026	−0.374513	−	0.927222i	$-0.622190\pi$
$797$	−21.1459	−0.749026	−0.374513	+	0.927222i	$0.622190\pi$
$798$	0	0
$799$	−20.4742	−0.724324
$800$	−17.6448	−0.623840
$801$	0	0
$802$	−19.9109	−0.703077
$803$	−16.8755	−0.595524
$804$	0	0
$805$	−1.39866	−0.0492962
$806$	19.3998	0.683330
$807$	0	0
$808$	44.3265	1.55940
$809$	−3.28804	−0.115601	−0.0578007	−	0.998328i	$-0.518409\pi$
$809$	−3.28804	−0.115601	−0.0578007	+	0.998328i	$0.518409\pi$
$810$	0	0
$811$	−2.39301	−0.0840299	−0.0420149	−	0.999117i	$-0.513378\pi$
$811$	−2.39301	−0.0840299	−0.0420149	+	0.999117i	$0.513378\pi$
$812$	2.08076	0.0730203
$813$	0	0
$814$	16.1735	0.566881
$815$	−7.03541	−0.246440
$816$	0	0
$817$	2.61068	0.0913362
$818$	15.3129	0.535403
$819$	0	0
$820$	−1.33735	−0.0467022
$821$	−33.4973	−1.16906	−0.584532	−	0.811370i	$-0.698722\pi$
$821$	−33.4973	−1.16906	−0.584532	+	0.811370i	$0.698722\pi$
$822$	0	0
$823$	−45.2775	−1.57828	−0.789138	−	0.614216i	$-0.789472\pi$
$823$	−45.2775	−1.57828	−0.789138	+	0.614216i	$0.789472\pi$
$824$	−31.1416	−1.08487
$825$	0	0
$826$	−5.51871	−0.192021
$827$	32.4045	1.12682	0.563408	−	0.826179i	$-0.309490\pi$
$827$	32.4045	1.12682	0.563408	+	0.826179i	$0.309490\pi$
$828$	0	0
$829$	15.1470	0.526076	0.263038	−	0.964786i	$-0.415276\pi$
$829$	15.1470	0.526076	0.263038	+	0.964786i	$0.415276\pi$
$830$	1.04257	0.0361880
$831$	0	0
$832$	25.8548	0.896353
$833$	3.43399	0.118981
$834$	0	0
$835$	4.32208	0.149572
$836$	0.645518	0.0223257
$837$	0	0
$838$	6.40579	0.221284
$839$	41.2271	1.42332	0.711658	−	0.702526i	$-0.247944\pi$
$839$	41.2271	1.42332	0.711658	+	0.702526i	$0.247944\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	22.2880	0.768096
$843$	0	0
$844$	−11.4137	−0.392875
$845$	−1.76682	−0.0607803
$846$	0	0
$847$	28.6914	0.985848
$848$	22.2323	0.763459
$849$	0	0
$850$	−8.58455	−0.294448
$851$	11.3773	0.390009
$852$	0	0
$853$	−13.5149	−0.462741	−0.231370	−	0.972866i	$-0.574321\pi$
$853$	−13.5149	−0.462741	−0.231370	+	0.972866i	$0.574321\pi$
$854$	38.9719	1.33359
$855$	0	0
$856$	51.8141	1.77097
$857$	−44.7861	−1.52986	−0.764932	−	0.644111i	$-0.777227\pi$
$857$	−44.7861	−1.52986	−0.764932	+	0.644111i	$0.777227\pi$
$858$	0	0
$859$	41.0825	1.40172	0.700859	−	0.713300i	$-0.252800\pi$
$859$	41.0825	1.40172	0.700859	+	0.713300i	$0.252800\pi$
$860$	1.08936	0.0371469
$861$	0	0
$862$	−2.73407	−0.0931229
$863$	−35.5358	−1.20965	−0.604827	−	0.796357i	$-0.706758\pi$
$863$	−35.5358	−1.20965	−0.604827	+	0.796357i	$0.706758\pi$
$864$	0	0
$865$	2.34536	0.0797448
$866$	5.80786	0.197359
$867$	0	0
$868$	11.6300	0.394749
$869$	7.38022	0.250357
$870$	0	0
$871$	−15.0608	−0.510315
$872$	−52.1603	−1.76637
$873$	0	0
$874$	−0.869420	−0.0294086
$875$	13.6890	0.462773
$876$	0	0
$877$	26.6815	0.900970	0.450485	−	0.892784i	$-0.351251\pi$
$877$	26.6815	0.900970	0.450485	+	0.892784i	$0.351251\pi$
$878$	−17.9243	−0.604917
$879$	0	0
$880$	−1.23377	−0.0415905
$881$	47.6041	1.60382	0.801911	−	0.597444i	$-0.203817\pi$
$881$	47.6041	1.60382	0.801911	+	0.597444i	$0.203817\pi$
$882$	0	0
$883$	9.91948	0.333817	0.166909	−	0.985972i	$-0.446622\pi$
$883$	9.91948	0.333817	0.166909	+	0.985972i	$0.446622\pi$
$884$	−3.25076	−0.109335
$885$	0	0
$886$	29.3842	0.987180
$887$	−50.8435	−1.70716	−0.853578	−	0.520965i	$-0.825572\pi$
$887$	−50.8435	−1.70716	−0.853578	+	0.520965i	$0.825572\pi$
$888$	0	0
$889$	−7.70572	−0.258442
$890$	−1.66473	−0.0558018
$891$	0	0
$892$	−1.88556	−0.0631333
$893$	9.92668	0.332184
$894$	0	0
$895$	6.41100	0.214296
$896$	−7.32318	−0.244650
$897$	0	0
$898$	7.21887	0.240897
$899$	5.58933	0.186414
$900$	0	0
$901$	−16.1268	−0.537262
$902$	6.00659	0.199997
$903$	0	0
$904$	44.3586	1.47535
$905$	6.67392	0.221849
$906$	0	0
$907$	−34.5190	−1.14618	−0.573092	−	0.819491i	$-0.694256\pi$
$907$	−34.5190	−1.14618	−0.573092	+	0.819491i	$0.694256\pi$
$908$	10.2126	0.338915
$909$	0	0
$910$	−4.85457	−0.160927
$911$	41.7823	1.38431	0.692155	−	0.721749i	$-0.256661\pi$
$911$	41.7823	1.38431	0.692155	+	0.721749i	$0.256661\pi$
$912$	0	0
$913$	2.44569	0.0809405
$914$	5.59979	0.185225
$915$	0	0
$916$	12.9372	0.427456
$917$	11.2557	0.371695
$918$	0	0
$919$	−1.38061	−0.0455422	−0.0227711	−	0.999741i	$-0.507249\pi$
$919$	−1.38061	−0.0455422	−0.0227711	+	0.999741i	$0.507249\pi$
$920$	−1.42017	−0.0468215
$921$	0	0
$922$	−11.2316	−0.369892
$923$	9.47579	0.311899
$924$	0	0
$925$	−54.4660	−1.79083
$926$	−6.95144	−0.228439
$927$	0	0
$928$	3.68580	0.120992
$929$	−35.2819	−1.15756	−0.578781	−	0.815483i	$-0.696471\pi$
$929$	−35.2819	−1.15756	−0.578781	+	0.815483i	$0.696471\pi$
$930$	0	0
$931$	−1.66493	−0.0545660
$932$	−0.602081	−0.0197218
$933$	0	0
$934$	29.6498	0.970171
$935$	0.894954	0.0292681
$936$	0	0
$937$	−29.8035	−0.973638	−0.486819	−	0.873503i	$-0.661843\pi$
$937$	−29.8035	−0.973638	−0.486819	+	0.873503i	$0.661843\pi$
$938$	17.2869	0.564436
$939$	0	0
$940$	4.14212	0.135101
$941$	−2.97294	−0.0969150	−0.0484575	−	0.998825i	$-0.515431\pi$
$941$	−2.97294	−0.0969150	−0.0484575	+	0.998825i	$0.515431\pi$
$942$	0	0
$943$	4.22535	0.137596
$944$	−3.42450	−0.111458
$945$	0	0
$946$	−4.89278	−0.159078
$947$	−53.9229	−1.75226	−0.876130	−	0.482075i	$-0.839883\pi$
$947$	−53.9229	−1.75226	−0.876130	+	0.482075i	$0.839883\pi$
$948$	0	0
$949$	−41.2032	−1.33751
$950$	4.16213	0.135037
$951$	0	0
$952$	14.6064	0.473397
$953$	29.7467	0.963591	0.481796	−	0.876284i	$-0.339985\pi$
$953$	29.7467	0.963591	0.481796	+	0.876284i	$0.339985\pi$
$954$	0	0
$955$	5.58222	0.180637
$956$	11.2375	0.363446
$957$	0	0
$958$	−15.4699	−0.499811
$959$	27.2368	0.879523
$960$	0	0
$961$	0.240574	0.00776044
$962$	39.4892	1.27318
$963$	0	0
$964$	15.2404	0.490860
$965$	−2.08673	−0.0671742
$966$	0	0
$967$	36.8210	1.18408	0.592042	−	0.805907i	$-0.298322\pi$
$967$	36.8210	1.18408	0.592042	+	0.805907i	$0.298322\pi$
$968$	29.1326	0.936357
$969$	0	0
$970$	−7.46951	−0.239832
$971$	4.91875	0.157850	0.0789251	−	0.996881i	$-0.474851\pi$
$971$	4.91875	0.157850	0.0789251	+	0.996881i	$0.474851\pi$
$972$	0	0
$973$	−6.48149	−0.207787
$974$	−34.9644	−1.12033
$975$	0	0
$976$	24.1830	0.774080
$977$	−2.00780	−0.0642351	−0.0321175	−	0.999484i	$-0.510225\pi$
$977$	−2.00780	−0.0642351	−0.0321175	+	0.999484i	$0.510225\pi$
$978$	0	0
$979$	−3.90518	−0.124810
$980$	−0.694729	−0.0221923
$981$	0	0
$982$	−40.4151	−1.28970
$983$	−17.0414	−0.543537	−0.271769	−	0.962363i	$-0.587609\pi$
$983$	−17.0414	−0.543537	−0.271769	+	0.962363i	$0.587609\pi$
$984$	0	0
$985$	−12.1991	−0.388694
$986$	1.79321	0.0571075
$987$	0	0
$988$	1.57609	0.0501423
$989$	−3.44184	−0.109444
$990$	0	0
$991$	26.4632	0.840630	0.420315	−	0.907378i	$-0.361920\pi$
$991$	26.4632	0.840630	0.420315	+	0.907378i	$0.361920\pi$
$992$	20.6011	0.654087
$993$	0	0
$994$	−10.8764	−0.344978
$995$	−1.22256	−0.0387579
$996$	0	0
$997$	34.2493	1.08469	0.542343	−	0.840157i	$-0.317537\pi$
$997$	34.2493	1.08469	0.542343	+	0.840157i	$0.317537\pi$
$998$	20.7309	0.656225
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.16	✓	22
3.2	odd	2		6003.2.a.u.1.7	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.16	✓	22	1.1	even	1	trivial
6003.2.a.u.1.7	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)

\(f(q)\)	\(=\)	\(q+1.14621 q^{2} -0.686192 q^{4} +0.461250 q^{5} -3.03232 q^{7} -3.07895 q^{8} +O(q^{10})\)	\(q+1.14621 q^{2} -0.686192 q^{4} +0.461250 q^{5} -3.03232 q^{7} -3.07895 q^{8} +0.528691 q^{10} +1.24022 q^{11} +3.02812 q^{13} -3.47569 q^{14} -2.15675 q^{16} +1.56446 q^{17} -0.758514 q^{19} -0.316506 q^{20} +1.42156 q^{22} +1.00000 q^{23} -4.78725 q^{25} +3.47087 q^{26} +2.08076 q^{28} +1.00000 q^{29} +5.58933 q^{31} +3.68580 q^{32} +1.79321 q^{34} -1.39866 q^{35} +11.3773 q^{37} -0.869420 q^{38} -1.42017 q^{40} +4.22535 q^{41} -3.44184 q^{43} -0.851030 q^{44} +1.14621 q^{46} -13.0870 q^{47} +2.19499 q^{49} -5.48721 q^{50} -2.07787 q^{52} -10.3082 q^{53} +0.572051 q^{55} +9.33638 q^{56} +1.14621 q^{58} +1.58780 q^{59} -11.2127 q^{61} +6.40657 q^{62} +8.53823 q^{64} +1.39672 q^{65} -4.97364 q^{67} -1.07352 q^{68} -1.60316 q^{70} +3.12927 q^{71} -13.6069 q^{73} +13.0408 q^{74} +0.520487 q^{76} -3.76075 q^{77} +5.95073 q^{79} -0.994802 q^{80} +4.84316 q^{82} +1.97198 q^{83} +0.721608 q^{85} -3.94508 q^{86} -3.81858 q^{88} -3.14877 q^{89} -9.18224 q^{91} -0.686192 q^{92} -15.0005 q^{94} -0.349864 q^{95} -14.1283 q^{97} +2.51593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 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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