LMFDB - Embedded newform 6003.2.a.m.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:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 $ x^11 - 2x^10 - 18x^9 + 30x^8 + 124x^7 - 152x^6 - 408x^5 + 285x^4 + 634x^3 - 93x^2 - 369x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.05971$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.05971 q^{2} -0.877023 q^{4} +1.30384 q^{5} +0.720797 q^{7} -3.04880 q^{8} +O(q^{10})$	\(q+1.05971 q^{2} -0.877023 q^{4} +1.30384 q^{5} +0.720797 q^{7} -3.04880 q^{8} +1.38169 q^{10} -4.26781 q^{11} +6.49237 q^{13} +0.763833 q^{14} -1.47678 q^{16} -1.60761 q^{17} +0.0676063 q^{19} -1.14350 q^{20} -4.52262 q^{22} -1.00000 q^{23} -3.30000 q^{25} +6.88000 q^{26} -0.632156 q^{28} +1.00000 q^{29} -3.80124 q^{31} +4.53264 q^{32} -1.70359 q^{34} +0.939804 q^{35} -3.80081 q^{37} +0.0716428 q^{38} -3.97515 q^{40} +0.467762 q^{41} +0.971410 q^{43} +3.74296 q^{44} -1.05971 q^{46} -7.97915 q^{47} -6.48045 q^{49} -3.49703 q^{50} -5.69396 q^{52} -13.3791 q^{53} -5.56454 q^{55} -2.19756 q^{56} +1.05971 q^{58} +4.90374 q^{59} +7.29114 q^{61} -4.02820 q^{62} +7.75683 q^{64} +8.46502 q^{65} +13.6988 q^{67} +1.40991 q^{68} +0.995916 q^{70} +0.843676 q^{71} -3.56146 q^{73} -4.02774 q^{74} -0.0592923 q^{76} -3.07622 q^{77} +3.50616 q^{79} -1.92549 q^{80} +0.495690 q^{82} +8.90808 q^{83} -2.09606 q^{85} +1.02941 q^{86} +13.0117 q^{88} -15.1294 q^{89} +4.67968 q^{91} +0.877023 q^{92} -8.45555 q^{94} +0.0881479 q^{95} +0.847630 q^{97} -6.86737 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * 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 $

$2$	1.05971	0.749325	0.374663	−	0.927161i	$-0.377759\pi$
$2$	1.05971	0.749325	0.374663	+	0.927161i	$0.377759\pi$
$3$	0	0
$3$	0	0
$4$	−0.877023	−0.438512
$5$	1.30384	0.583095	0.291548	−	0.956556i	$-0.405830\pi$
$5$	1.30384	0.583095	0.291548	+	0.956556i	$0.405830\pi$
$6$	0	0
$7$	0.720797	0.272436	0.136218	−	0.990679i	$-0.456505\pi$
$7$	0.720797	0.272436	0.136218	+	0.990679i	$0.456505\pi$
$8$	−3.04880	−1.07791
$9$	0	0
$10$	1.38169	0.436928
$11$	−4.26781	−1.28679	−0.643396	−	0.765534i	$-0.722475\pi$
$11$	−4.26781	−1.28679	−0.643396	+	0.765534i	$0.722475\pi$
$12$	0	0
$13$	6.49237	1.80066	0.900330	−	0.435208i	$-0.143325\pi$
$13$	6.49237	1.80066	0.900330	+	0.435208i	$0.143325\pi$
$14$	0.763833	0.204143
$15$	0	0
$16$	−1.47678	−0.369196
$17$	−1.60761	−0.389902	−0.194951	−	0.980813i	$-0.562455\pi$
$17$	−1.60761	−0.389902	−0.194951	+	0.980813i	$0.562455\pi$
$18$	0	0
$19$	0.0676063	0.0155100	0.00775498	−	0.999970i	$-0.497531\pi$
$19$	0.0676063	0.0155100	0.00775498	+	0.999970i	$0.497531\pi$
$20$	−1.14350	−0.255694
$21$	0	0
$22$	−4.52262	−0.964226
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.30000	−0.660000
$26$	6.88000	1.34928
$27$	0	0
$28$	−0.632156	−0.119466
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.80124	−0.682723	−0.341362	−	0.939932i	$-0.610888\pi$
$31$	−3.80124	−0.682723	−0.341362	+	0.939932i	$0.610888\pi$
$32$	4.53264	0.801265
$33$	0	0
$34$	−1.70359	−0.292163
$35$	0.939804	0.158856
$36$	0	0
$37$	−3.80081	−0.624849	−0.312425	−	0.949943i	$-0.601141\pi$
$37$	−3.80081	−0.624849	−0.312425	+	0.949943i	$0.601141\pi$
$38$	0.0716428	0.0116220
$39$	0	0
$40$	−3.97515	−0.628526
$41$	0.467762	0.0730521	0.0365261	−	0.999333i	$-0.488371\pi$
$41$	0.467762	0.0730521	0.0365261	+	0.999333i	$0.488371\pi$
$42$	0	0
$43$	0.971410	0.148139	0.0740693	−	0.997253i	$-0.476401\pi$
$43$	0.971410	0.148139	0.0740693	+	0.997253i	$0.476401\pi$
$44$	3.74296	0.564273
$45$	0	0
$46$	−1.05971	−0.156245
$47$	−7.97915	−1.16388	−0.581939	−	0.813232i	$-0.697706\pi$
$47$	−7.97915	−1.16388	−0.581939	+	0.813232i	$0.697706\pi$
$48$	0	0
$49$	−6.48045	−0.925779
$50$	−3.49703	−0.494555
$51$	0	0
$52$	−5.69396	−0.789610
$53$	−13.3791	−1.83776	−0.918881	−	0.394535i	$-0.870906\pi$
$53$	−13.3791	−1.83776	−0.918881	+	0.394535i	$0.870906\pi$
$54$	0	0
$55$	−5.56454	−0.750322
$56$	−2.19756	−0.293662
$57$	0	0
$58$	1.05971	0.139146
$59$	4.90374	0.638413	0.319206	−	0.947685i	$-0.396584\pi$
$59$	4.90374	0.638413	0.319206	+	0.947685i	$0.396584\pi$
$60$	0	0
$61$	7.29114	0.933534	0.466767	−	0.884380i	$-0.345419\pi$
$61$	7.29114	0.933534	0.466767	+	0.884380i	$0.345419\pi$
$62$	−4.02820	−0.511582
$63$	0	0
$64$	7.75683	0.969604
$65$	8.46502	1.04996
$66$	0	0
$67$	13.6988	1.67358	0.836788	−	0.547526i	$-0.184430\pi$
$67$	13.6988	1.67358	0.836788	+	0.547526i	$0.184430\pi$
$68$	1.40991	0.170977
$69$	0	0
$70$	0.995916	0.119035
$71$	0.843676	0.100126	0.0500630	−	0.998746i	$-0.484058\pi$
$71$	0.843676	0.100126	0.0500630	+	0.998746i	$0.484058\pi$
$72$	0	0
$73$	−3.56146	−0.416837	−0.208419	−	0.978040i	$-0.566832\pi$
$73$	−3.56146	−0.416837	−0.208419	+	0.978040i	$0.566832\pi$
$74$	−4.02774	−0.468215
$75$	0	0
$76$	−0.0592923	−0.00680130
$77$	−3.07622	−0.350568
$78$	0	0
$79$	3.50616	0.394474	0.197237	−	0.980356i	$-0.436803\pi$
$79$	3.50616	0.394474	0.197237	+	0.980356i	$0.436803\pi$
$80$	−1.92549	−0.215276
$81$	0	0
$82$	0.495690	0.0547398
$83$	8.90808	0.977789	0.488894	−	0.872343i	$-0.337400\pi$
$83$	8.90808	0.977789	0.488894	+	0.872343i	$0.337400\pi$
$84$	0	0
$85$	−2.09606	−0.227350
$86$	1.02941	0.111004
$87$	0	0
$88$	13.0117	1.38705
$89$	−15.1294	−1.60371	−0.801856	−	0.597517i	$-0.796154\pi$
$89$	−15.1294	−1.60371	−0.801856	+	0.597517i	$0.796154\pi$
$90$	0	0
$91$	4.67968	0.490564
$92$	0.877023	0.0914360
$93$	0	0
$94$	−8.45555	−0.872124
$95$	0.0881479	0.00904378
$96$	0	0
$97$	0.847630	0.0860637	0.0430319	−	0.999074i	$-0.486298\pi$
$97$	0.847630	0.0860637	0.0430319	+	0.999074i	$0.486298\pi$
$98$	−6.86737	−0.693709
$99$	0	0
$100$	2.89418	0.289418
$101$	4.21499	0.419407	0.209704	−	0.977765i	$-0.432750\pi$
$101$	4.21499	0.419407	0.209704	+	0.977765i	$0.432750\pi$
$102$	0	0
$103$	−3.25548	−0.320772	−0.160386	−	0.987054i	$-0.551274\pi$
$103$	−3.25548	−0.320772	−0.160386	+	0.987054i	$0.551274\pi$
$104$	−19.7939	−1.94095
$105$	0	0
$106$	−14.1779	−1.37708
$107$	−9.91828	−0.958836	−0.479418	−	0.877587i	$-0.659152\pi$
$107$	−9.91828	−0.958836	−0.479418	+	0.877587i	$0.659152\pi$
$108$	0	0
$109$	−14.9445	−1.43142	−0.715711	−	0.698396i	$-0.753897\pi$
$109$	−14.9445	−1.43142	−0.715711	+	0.698396i	$0.753897\pi$
$110$	−5.89677	−0.562235
$111$	0	0
$112$	−1.06446	−0.100582
$113$	−9.05430	−0.851757	−0.425879	−	0.904780i	$-0.640035\pi$
$113$	−9.05430	−0.851757	−0.425879	+	0.904780i	$0.640035\pi$
$114$	0	0
$115$	−1.30384	−0.121584
$116$	−0.877023	−0.0814296
$117$	0	0
$118$	5.19653	0.478379
$119$	−1.15876	−0.106223
$120$	0	0
$121$	7.21416	0.655833
$122$	7.72646	0.699521
$123$	0	0
$124$	3.33378	0.299382
$125$	−10.8219	−0.967938
$126$	0	0
$127$	−16.8234	−1.49284	−0.746420	−	0.665476i	$-0.768229\pi$
$127$	−16.8234	−1.49284	−0.746420	+	0.665476i	$0.768229\pi$
$128$	−0.845320	−0.0747164
$129$	0	0
$130$	8.97043	0.786759
$131$	−20.0112	−1.74838	−0.874192	−	0.485581i	$-0.838608\pi$
$131$	−20.0112	−1.74838	−0.874192	+	0.485581i	$0.838608\pi$
$132$	0	0
$133$	0.0487304	0.00422547
$134$	14.5167	1.25405
$135$	0	0
$136$	4.90127	0.420280
$137$	17.5827	1.50219	0.751097	−	0.660192i	$-0.229525\pi$
$137$	17.5827	1.50219	0.751097	+	0.660192i	$0.229525\pi$
$138$	0	0
$139$	−11.2785	−0.956634	−0.478317	−	0.878187i	$-0.658753\pi$
$139$	−11.2785	−0.956634	−0.478317	+	0.878187i	$0.658753\pi$
$140$	−0.824230	−0.0696602
$141$	0	0
$142$	0.894049	0.0750269
$143$	−27.7082	−2.31707
$144$	0	0
$145$	1.30384	0.108278
$146$	−3.77410	−0.312347
$147$	0	0
$148$	3.33340	0.274004
$149$	−8.68520	−0.711519	−0.355760	−	0.934578i	$-0.615778\pi$
$149$	−8.68520	−0.711519	−0.355760	+	0.934578i	$0.615778\pi$
$150$	0	0
$151$	−3.77830	−0.307474	−0.153737	−	0.988112i	$-0.549131\pi$
$151$	−3.77830	−0.307474	−0.153737	+	0.988112i	$0.549131\pi$
$152$	−0.206118	−0.0167184
$153$	0	0
$154$	−3.25989	−0.262689
$155$	−4.95621	−0.398093
$156$	0	0
$157$	−22.4357	−1.79056	−0.895280	−	0.445503i	$-0.853025\pi$
$157$	−22.4357	−1.79056	−0.895280	+	0.445503i	$0.853025\pi$
$158$	3.71550	0.295589
$159$	0	0
$160$	5.90984	0.467214
$161$	−0.720797	−0.0568068
$162$	0	0
$163$	9.58430	0.750701	0.375350	−	0.926883i	$-0.377522\pi$
$163$	9.58430	0.750701	0.375350	+	0.926883i	$0.377522\pi$
$164$	−0.410238	−0.0320342
$165$	0	0
$166$	9.43994	0.732682
$167$	22.7911	1.76363	0.881816	−	0.471594i	$-0.156321\pi$
$167$	22.7911	1.76363	0.881816	+	0.471594i	$0.156321\pi$
$168$	0	0
$169$	29.1509	2.24238
$170$	−2.22121	−0.170359
$171$	0	0
$172$	−0.851949	−0.0649605
$173$	−3.11958	−0.237177	−0.118589	−	0.992943i	$-0.537837\pi$
$173$	−3.11958	−0.237177	−0.118589	+	0.992943i	$0.537837\pi$
$174$	0	0
$175$	−2.37863	−0.179808
$176$	6.30262	0.475078
$177$	0	0
$178$	−16.0327	−1.20170
$179$	4.43515	0.331499	0.165749	−	0.986168i	$-0.446996\pi$
$179$	4.43515	0.331499	0.165749	+	0.986168i	$0.446996\pi$
$180$	0	0
$181$	7.57878	0.563326	0.281663	−	0.959513i	$-0.409114\pi$
$181$	7.57878	0.563326	0.281663	+	0.959513i	$0.409114\pi$
$182$	4.95909	0.367592
$183$	0	0
$184$	3.04880	0.224760
$185$	−4.95565	−0.364347
$186$	0	0
$187$	6.86095	0.501722
$188$	6.99790	0.510374
$189$	0	0
$190$	0.0934108	0.00677673
$191$	−23.8728	−1.72738	−0.863688	−	0.504026i	$-0.831851\pi$
$191$	−23.8728	−1.72738	−0.863688	+	0.504026i	$0.831851\pi$
$192$	0	0
$193$	−23.7901	−1.71245	−0.856224	−	0.516605i	$-0.827196\pi$
$193$	−23.7901	−1.71245	−0.856224	+	0.516605i	$0.827196\pi$
$194$	0.898238	0.0644897
$195$	0	0
$196$	5.68351	0.405965
$197$	−10.2899	−0.733125	−0.366562	−	0.930393i	$-0.619465\pi$
$197$	−10.2899	−0.733125	−0.366562	+	0.930393i	$0.619465\pi$
$198$	0	0
$199$	6.69895	0.474876	0.237438	−	0.971403i	$-0.423692\pi$
$199$	6.69895	0.474876	0.237438	+	0.971403i	$0.423692\pi$
$200$	10.0610	0.711423
$201$	0	0
$202$	4.46665	0.314273
$203$	0.720797	0.0505900
$204$	0	0
$205$	0.609887	0.0425964
$206$	−3.44986	−0.240363
$207$	0	0
$208$	−9.58783	−0.664796
$209$	−0.288531	−0.0199581
$210$	0	0
$211$	19.4730	1.34058	0.670288	−	0.742101i	$-0.266170\pi$
$211$	19.4730	1.34058	0.670288	+	0.742101i	$0.266170\pi$
$212$	11.7338	0.805880
$213$	0	0
$214$	−10.5105	−0.718480
$215$	1.26656	0.0863789
$216$	0	0
$217$	−2.73992	−0.185998
$218$	−15.8368	−1.07260
$219$	0	0
$220$	4.88023	0.329025
$221$	−10.4372	−0.702081
$222$	0	0
$223$	5.45268	0.365138	0.182569	−	0.983193i	$-0.441559\pi$
$223$	5.45268	0.365138	0.182569	+	0.983193i	$0.441559\pi$
$224$	3.26711	0.218293
$225$	0	0
$226$	−9.59490	−0.638243
$227$	−12.7976	−0.849407	−0.424704	−	0.905332i	$-0.639622\pi$
$227$	−12.7976	−0.849407	−0.424704	+	0.905332i	$0.639622\pi$
$228$	0	0
$229$	−17.9481	−1.18604	−0.593021	−	0.805187i	$-0.702065\pi$
$229$	−17.9481	−1.18604	−0.593021	+	0.805187i	$0.702065\pi$
$230$	−1.38169	−0.0911058
$231$	0	0
$232$	−3.04880	−0.200163
$233$	−24.8070	−1.62516	−0.812579	−	0.582851i	$-0.801937\pi$
$233$	−24.8070	−1.62516	−0.812579	+	0.582851i	$0.801937\pi$
$234$	0	0
$235$	−10.4035	−0.678652
$236$	−4.30070	−0.279952
$237$	0	0
$238$	−1.22794	−0.0795957
$239$	−6.47432	−0.418789	−0.209394	−	0.977831i	$-0.567149\pi$
$239$	−6.47432	−0.418789	−0.209394	+	0.977831i	$0.567149\pi$
$240$	0	0
$241$	−1.06870	−0.0688413	−0.0344207	−	0.999407i	$-0.510959\pi$
$241$	−1.06870	−0.0688413	−0.0344207	+	0.999407i	$0.510959\pi$
$242$	7.64489	0.491432
$243$	0	0
$244$	−6.39450	−0.409366
$245$	−8.44948	−0.539817
$246$	0	0
$247$	0.438925	0.0279282
$248$	11.5892	0.735916
$249$	0	0
$250$	−11.4680	−0.725300
$251$	10.6611	0.672920	0.336460	−	0.941698i	$-0.390770\pi$
$251$	10.6611	0.672920	0.336460	+	0.941698i	$0.390770\pi$
$252$	0	0
$253$	4.26781	0.268315
$254$	−17.8279	−1.11862
$255$	0	0
$256$	−16.4095	−1.02559
$257$	−26.4765	−1.65156	−0.825781	−	0.563991i	$-0.809265\pi$
$257$	−26.4765	−1.65156	−0.825781	+	0.563991i	$0.809265\pi$
$258$	0	0
$259$	−2.73961	−0.170231
$260$	−7.42402	−0.460418
$261$	0	0
$262$	−21.2059	−1.31011
$263$	−3.04107	−0.187520	−0.0937602	−	0.995595i	$-0.529889\pi$
$263$	−3.04107	−0.187520	−0.0937602	+	0.995595i	$0.529889\pi$
$264$	0	0
$265$	−17.4442	−1.07159
$266$	0.0516399	0.00316625
$267$	0	0
$268$	−12.0142	−0.733883
$269$	30.1319	1.83718	0.918589	−	0.395215i	$-0.129330\pi$
$269$	30.1319	1.83718	0.918589	+	0.395215i	$0.129330\pi$
$270$	0	0
$271$	18.0654	1.09740	0.548699	−	0.836020i	$-0.315123\pi$
$271$	18.0654	1.09740	0.548699	+	0.836020i	$0.315123\pi$
$272$	2.37409	0.143950
$273$	0	0
$274$	18.6325	1.12563
$275$	14.0838	0.849283
$276$	0	0
$277$	15.0741	0.905712	0.452856	−	0.891584i	$-0.350405\pi$
$277$	15.0741	0.905712	0.452856	+	0.891584i	$0.350405\pi$
$278$	−11.9519	−0.716830
$279$	0	0
$280$	−2.86527	−0.171233
$281$	−17.4085	−1.03850	−0.519252	−	0.854621i	$-0.673789\pi$
$281$	−17.4085	−1.03850	−0.519252	+	0.854621i	$0.673789\pi$
$282$	0	0
$283$	27.6295	1.64240	0.821202	−	0.570638i	$-0.193304\pi$
$283$	27.6295	1.64240	0.821202	+	0.570638i	$0.193304\pi$
$284$	−0.739924	−0.0439064
$285$	0	0
$286$	−29.3625	−1.73624
$287$	0.337161	0.0199020
$288$	0	0
$289$	−14.4156	−0.847977
$290$	1.38169	0.0811355
$291$	0	0
$292$	3.12348	0.182788
$293$	−9.65321	−0.563946	−0.281973	−	0.959422i	$-0.590989\pi$
$293$	−9.65321	−0.563946	−0.281973	+	0.959422i	$0.590989\pi$
$294$	0	0
$295$	6.39370	0.372256
$296$	11.5879	0.673533
$297$	0	0
$298$	−9.20376	−0.533159
$299$	−6.49237	−0.375464
$300$	0	0
$301$	0.700189	0.0403582
$302$	−4.00389	−0.230398
$303$	0	0
$304$	−0.0998399	−0.00572621
$305$	9.50648	0.544339
$306$	0	0
$307$	23.4872	1.34048	0.670242	−	0.742143i	$-0.266190\pi$
$307$	23.4872	1.34048	0.670242	+	0.742143i	$0.266190\pi$
$308$	2.69792	0.153728
$309$	0	0
$310$	−5.25213	−0.298301
$311$	−10.2227	−0.579679	−0.289839	−	0.957075i	$-0.593602\pi$
$311$	−10.2227	−0.579679	−0.289839	+	0.957075i	$0.593602\pi$
$312$	0	0
$313$	4.95500	0.280073	0.140036	−	0.990146i	$-0.455278\pi$
$313$	4.95500	0.280073	0.140036	+	0.990146i	$0.455278\pi$
$314$	−23.7752	−1.34171
$315$	0	0
$316$	−3.07498	−0.172981
$317$	−6.20695	−0.348617	−0.174309	−	0.984691i	$-0.555769\pi$
$317$	−6.20695	−0.348617	−0.174309	+	0.984691i	$0.555769\pi$
$318$	0	0
$319$	−4.26781	−0.238951
$320$	10.1137	0.565372
$321$	0	0
$322$	−0.763833	−0.0425667
$323$	−0.108684	−0.00604736
$324$	0	0
$325$	−21.4248	−1.18844
$326$	10.1565	0.562519
$327$	0	0
$328$	−1.42611	−0.0787439
$329$	−5.75135	−0.317082
$330$	0	0
$331$	4.70041	0.258358	0.129179	−	0.991621i	$-0.458766\pi$
$331$	4.70041	0.258358	0.129179	+	0.991621i	$0.458766\pi$
$332$	−7.81259	−0.428772
$333$	0	0
$334$	24.1519	1.32153
$335$	17.8611	0.975855
$336$	0	0
$337$	−9.18682	−0.500438	−0.250219	−	0.968189i	$-0.580503\pi$
$337$	−9.18682	−0.500438	−0.250219	+	0.968189i	$0.580503\pi$
$338$	30.8914	1.68027
$339$	0	0
$340$	1.83830	0.0996956
$341$	16.2230	0.878523
$342$	0	0
$343$	−9.71667	−0.524651
$344$	−2.96163	−0.159681
$345$	0	0
$346$	−3.30584	−0.177723
$347$	19.5451	1.04923	0.524617	−	0.851339i	$-0.324209\pi$
$347$	19.5451	1.04923	0.524617	+	0.851339i	$0.324209\pi$
$348$	0	0
$349$	35.7643	1.91442	0.957209	−	0.289398i	$-0.0934551\pi$
$349$	35.7643	1.91442	0.957209	+	0.289398i	$0.0934551\pi$
$350$	−2.52065	−0.134734
$351$	0	0
$352$	−19.3444	−1.03106
$353$	−17.5958	−0.936532	−0.468266	−	0.883588i	$-0.655121\pi$
$353$	−17.5958	−0.936532	−0.468266	+	0.883588i	$0.655121\pi$
$354$	0	0
$355$	1.10002	0.0583830
$356$	13.2688	0.703246
$357$	0	0
$358$	4.69995	0.248400
$359$	−5.45515	−0.287912	−0.143956	−	0.989584i	$-0.545982\pi$
$359$	−5.45515	−0.287912	−0.143956	+	0.989584i	$0.545982\pi$
$360$	0	0
$361$	−18.9954	−0.999759
$362$	8.03128	0.422115
$363$	0	0
$364$	−4.10419	−0.215118
$365$	−4.64358	−0.243056
$366$	0	0
$367$	7.87986	0.411326	0.205663	−	0.978623i	$-0.434065\pi$
$367$	7.87986	0.411326	0.205663	+	0.978623i	$0.434065\pi$
$368$	1.47678	0.0769827
$369$	0	0
$370$	−5.25153	−0.273014
$371$	−9.64362	−0.500672
$372$	0	0
$373$	−12.3211	−0.637961	−0.318980	−	0.947761i	$-0.603340\pi$
$373$	−12.3211	−0.637961	−0.318980	+	0.947761i	$0.603340\pi$
$374$	7.27059	0.375953
$375$	0	0
$376$	24.3268	1.25456
$377$	6.49237	0.334374
$378$	0	0
$379$	21.0221	1.07983	0.539916	−	0.841719i	$-0.318456\pi$
$379$	21.0221	1.07983	0.539916	+	0.841719i	$0.318456\pi$
$380$	−0.0773078	−0.00396580
$381$	0	0
$382$	−25.2982	−1.29437
$383$	26.7325	1.36597	0.682983	−	0.730434i	$-0.260682\pi$
$383$	26.7325	1.36597	0.682983	+	0.730434i	$0.260682\pi$
$384$	0	0
$385$	−4.01090	−0.204414
$386$	−25.2105	−1.28318
$387$	0	0
$388$	−0.743391	−0.0377400
$389$	−18.2034	−0.922949	−0.461474	−	0.887154i	$-0.652679\pi$
$389$	−18.2034	−0.922949	−0.461474	+	0.887154i	$0.652679\pi$
$390$	0	0
$391$	1.60761	0.0813002
$392$	19.7576	0.997909
$393$	0	0
$394$	−10.9043	−0.549349
$395$	4.57147	0.230016
$396$	0	0
$397$	−14.9475	−0.750193	−0.375096	−	0.926986i	$-0.622390\pi$
$397$	−14.9475	−0.750193	−0.375096	+	0.926986i	$0.622390\pi$
$398$	7.09892	0.355837
$399$	0	0
$400$	4.87339	0.243669
$401$	−5.93728	−0.296494	−0.148247	−	0.988950i	$-0.547363\pi$
$401$	−5.93728	−0.296494	−0.148247	+	0.988950i	$0.547363\pi$
$402$	0	0
$403$	−24.6791	−1.22935
$404$	−3.69665	−0.183915
$405$	0	0
$406$	0.763833	0.0379084
$407$	16.2211	0.804051
$408$	0	0
$409$	28.1625	1.39255	0.696274	−	0.717776i	$-0.254840\pi$
$409$	28.1625	1.39255	0.696274	+	0.717776i	$0.254840\pi$
$410$	0.646301	0.0319185
$411$	0	0
$412$	2.85514	0.140662
$413$	3.53460	0.173926
$414$	0	0
$415$	11.6147	0.570144
$416$	29.4276	1.44281
$417$	0	0
$418$	−0.305758	−0.0149551
$419$	15.7587	0.769862	0.384931	−	0.922945i	$-0.374225\pi$
$419$	15.7587	0.769862	0.384931	+	0.922945i	$0.374225\pi$
$420$	0	0
$421$	−37.5081	−1.82803	−0.914017	−	0.405676i	$-0.867036\pi$
$421$	−37.5081	−1.82803	−0.914017	+	0.405676i	$0.867036\pi$
$422$	20.6357	1.00453
$423$	0	0
$424$	40.7902	1.98095
$425$	5.30510	0.257335
$426$	0	0
$427$	5.25543	0.254328
$428$	8.69856	0.420461
$429$	0	0
$430$	1.34218	0.0647259
$431$	−10.7335	−0.517013	−0.258507	−	0.966009i	$-0.583230\pi$
$431$	−10.7335	−0.517013	−0.258507	+	0.966009i	$0.583230\pi$
$432$	0	0
$433$	29.9000	1.43690	0.718452	−	0.695576i	$-0.244851\pi$
$433$	29.9000	1.43690	0.718452	+	0.695576i	$0.244851\pi$
$434$	−2.90351	−0.139373
$435$	0	0
$436$	13.1067	0.627696
$437$	−0.0676063	−0.00323405
$438$	0	0
$439$	27.6423	1.31930	0.659648	−	0.751575i	$-0.270705\pi$
$439$	27.6423	1.31930	0.659648	+	0.751575i	$0.270705\pi$
$440$	16.9652	0.808782
$441$	0	0
$442$	−11.0603	−0.526087
$443$	−16.5336	−0.785534	−0.392767	−	0.919638i	$-0.628482\pi$
$443$	−16.5336	−0.785534	−0.392767	+	0.919638i	$0.628482\pi$
$444$	0	0
$445$	−19.7263	−0.935117
$446$	5.77823	0.273607
$447$	0	0
$448$	5.59110	0.264155
$449$	−15.6126	−0.736805	−0.368403	−	0.929666i	$-0.620095\pi$
$449$	−15.6126	−0.736805	−0.368403	+	0.929666i	$0.620095\pi$
$450$	0	0
$451$	−1.99632	−0.0940029
$452$	7.94084	0.373506
$453$	0	0
$454$	−13.5617	−0.636482
$455$	6.10156	0.286045
$456$	0	0
$457$	−39.1168	−1.82981	−0.914903	−	0.403674i	$-0.867733\pi$
$457$	−39.1168	−1.82981	−0.914903	+	0.403674i	$0.867733\pi$
$458$	−19.0197	−0.888731
$459$	0	0
$460$	1.14350	0.0533159
$461$	−27.3047	−1.27171	−0.635854	−	0.771810i	$-0.719352\pi$
$461$	−27.3047	−1.27171	−0.635854	+	0.771810i	$0.719352\pi$
$462$	0	0
$463$	25.1953	1.17093	0.585463	−	0.810699i	$-0.300913\pi$
$463$	25.1953	1.17093	0.585463	+	0.810699i	$0.300913\pi$
$464$	−1.47678	−0.0685580
$465$	0	0
$466$	−26.2881	−1.21777
$467$	−10.5362	−0.487556	−0.243778	−	0.969831i	$-0.578387\pi$
$467$	−10.5362	−0.487556	−0.243778	+	0.969831i	$0.578387\pi$
$468$	0	0
$469$	9.87407	0.455942
$470$	−11.0247	−0.508531
$471$	0	0
$472$	−14.9505	−0.688154
$473$	−4.14579	−0.190624
$474$	0	0
$475$	−0.223101	−0.0102366
$476$	1.01626	0.0465801
$477$	0	0
$478$	−6.86087	−0.313809
$479$	−22.6259	−1.03380	−0.516902	−	0.856045i	$-0.672915\pi$
$479$	−22.6259	−1.03380	−0.516902	+	0.856045i	$0.672915\pi$
$480$	0	0
$481$	−24.6763	−1.12514
$482$	−1.13251	−0.0515845
$483$	0	0
$484$	−6.32699	−0.287590
$485$	1.10517	0.0501834
$486$	0	0
$487$	−1.33179	−0.0603490	−0.0301745	−	0.999545i	$-0.509606\pi$
$487$	−1.33179	−0.0603490	−0.0301745	+	0.999545i	$0.509606\pi$
$488$	−22.2292	−1.00627
$489$	0	0
$490$	−8.95396	−0.404499
$491$	37.5483	1.69453	0.847265	−	0.531170i	$-0.178247\pi$
$491$	37.5483	1.69453	0.847265	+	0.531170i	$0.178247\pi$
$492$	0	0
$493$	−1.60761	−0.0724030
$494$	0.465132	0.0209273
$495$	0	0
$496$	5.61361	0.252059
$497$	0.608119	0.0272779
$498$	0	0
$499$	−34.2219	−1.53198	−0.765992	−	0.642850i	$-0.777752\pi$
$499$	−34.2219	−1.53198	−0.765992	+	0.642850i	$0.777752\pi$
$500$	9.49104	0.424452
$501$	0	0
$502$	11.2976	0.504236
$503$	−28.4032	−1.26643	−0.633217	−	0.773974i	$-0.718266\pi$
$503$	−28.4032	−1.26643	−0.633217	+	0.773974i	$0.718266\pi$
$504$	0	0
$505$	5.49568	0.244554
$506$	4.52262	0.201055
$507$	0	0
$508$	14.7546	0.654627
$509$	6.08138	0.269553	0.134776	−	0.990876i	$-0.456968\pi$
$509$	6.08138	0.269553	0.134776	+	0.990876i	$0.456968\pi$
$510$	0	0
$511$	−2.56709	−0.113561
$512$	−15.6986	−0.693785
$513$	0	0
$514$	−28.0573	−1.23756
$515$	−4.24463	−0.187041
$516$	0	0
$517$	34.0535	1.49767
$518$	−2.90318	−0.127559
$519$	0	0
$520$	−25.8081	−1.13176
$521$	38.6489	1.69324	0.846619	−	0.532200i	$-0.178634\pi$
$521$	38.6489	1.69324	0.846619	+	0.532200i	$0.178634\pi$
$522$	0	0
$523$	2.97908	0.130266	0.0651331	−	0.997877i	$-0.479253\pi$
$523$	2.97908	0.130266	0.0651331	+	0.997877i	$0.479253\pi$
$524$	17.5503	0.766686
$525$	0	0
$526$	−3.22264	−0.140514
$527$	6.11090	0.266195
$528$	0	0
$529$	1.00000	0.0434783
$530$	−18.4857	−0.802970
$531$	0	0
$532$	−0.0427377	−0.00185292
$533$	3.03688	0.131542
$534$	0	0
$535$	−12.9319	−0.559093
$536$	−41.7649	−1.80397
$537$	0	0
$538$	31.9310	1.37664
$539$	27.6573	1.19128
$540$	0	0
$541$	26.1703	1.12515	0.562574	−	0.826747i	$-0.309811\pi$
$541$	26.1703	1.12515	0.562574	+	0.826747i	$0.309811\pi$
$542$	19.1441	0.822308
$543$	0	0
$544$	−7.28670	−0.312415
$545$	−19.4852	−0.834656
$546$	0	0
$547$	11.0205	0.471204	0.235602	−	0.971850i	$-0.424294\pi$
$547$	11.0205	0.471204	0.235602	+	0.971850i	$0.424294\pi$
$548$	−15.4205	−0.658729
$549$	0	0
$550$	14.9246	0.636389
$551$	0.0676063	0.00288013
$552$	0	0
$553$	2.52723	0.107469
$554$	15.9741	0.678673
$555$	0	0
$556$	9.89155	0.419495
$557$	26.2157	1.11080	0.555398	−	0.831585i	$-0.312566\pi$
$557$	26.2157	1.11080	0.555398	+	0.831585i	$0.312566\pi$
$558$	0	0
$559$	6.30675	0.266747
$560$	−1.38789	−0.0586489
$561$	0	0
$562$	−18.4479	−0.778177
$563$	39.8761	1.68058	0.840288	−	0.542140i	$-0.182386\pi$
$563$	39.8761	1.68058	0.840288	+	0.542140i	$0.182386\pi$
$564$	0	0
$565$	−11.8054	−0.496656
$566$	29.2791	1.23069
$567$	0	0
$568$	−2.57220	−0.107927
$569$	15.2160	0.637886	0.318943	−	0.947774i	$-0.396672\pi$
$569$	15.2160	0.637886	0.318943	+	0.947774i	$0.396672\pi$
$570$	0	0
$571$	−11.9807	−0.501377	−0.250688	−	0.968068i	$-0.580657\pi$
$571$	−11.9807	−0.501377	−0.250688	+	0.968068i	$0.580657\pi$
$572$	24.3007	1.01606
$573$	0	0
$574$	0.357292	0.0149131
$575$	3.30000	0.137620
$576$	0	0
$577$	−21.5518	−0.897212	−0.448606	−	0.893730i	$-0.648079\pi$
$577$	−21.5518	−0.897212	−0.448606	+	0.893730i	$0.648079\pi$
$578$	−15.2763	−0.635410
$579$	0	0
$580$	−1.14350	−0.0474812
$581$	6.42092	0.266384
$582$	0	0
$583$	57.0994	2.36482
$584$	10.8582	0.449314
$585$	0	0
$586$	−10.2296	−0.422579
$587$	21.9895	0.907604	0.453802	−	0.891102i	$-0.350067\pi$
$587$	21.9895	0.907604	0.453802	+	0.891102i	$0.350067\pi$
$588$	0	0
$589$	−0.256988	−0.0105890
$590$	6.77544	0.278940
$591$	0	0
$592$	5.61297	0.230692
$593$	18.0058	0.739411	0.369706	−	0.929149i	$-0.379459\pi$
$593$	18.0058	0.739411	0.369706	+	0.929149i	$0.379459\pi$
$594$	0	0
$595$	−1.51084	−0.0619382
$596$	7.61712	0.312009
$597$	0	0
$598$	−6.88000	−0.281344
$599$	−42.1452	−1.72201	−0.861004	−	0.508599i	$-0.830164\pi$
$599$	−42.1452	−1.72201	−0.861004	+	0.508599i	$0.830164\pi$
$600$	0	0
$601$	−1.18690	−0.0484145	−0.0242073	−	0.999707i	$-0.507706\pi$
$601$	−1.18690	−0.0484145	−0.0242073	+	0.999707i	$0.507706\pi$
$602$	0.741995	0.0302414
$603$	0	0
$604$	3.31366	0.134831
$605$	9.40612	0.382413
$606$	0	0
$607$	−21.2693	−0.863296	−0.431648	−	0.902042i	$-0.642068\pi$
$607$	−21.2693	−0.863296	−0.431648	+	0.902042i	$0.642068\pi$
$608$	0.306435	0.0124276
$609$	0	0
$610$	10.0741	0.407887
$611$	−51.8036	−2.09575
$612$	0	0
$613$	3.41464	0.137916	0.0689580	−	0.997620i	$-0.478033\pi$
$613$	3.41464	0.137916	0.0689580	+	0.997620i	$0.478033\pi$
$614$	24.8895	1.00446
$615$	0	0
$616$	9.37878	0.377882
$617$	3.56041	0.143337	0.0716684	−	0.997429i	$-0.477168\pi$
$617$	3.56041	0.143337	0.0716684	+	0.997429i	$0.477168\pi$
$618$	0	0
$619$	−19.6807	−0.791033	−0.395516	−	0.918459i	$-0.629434\pi$
$619$	−19.6807	−0.791033	−0.395516	+	0.918459i	$0.629434\pi$
$620$	4.34671	0.174568
$621$	0	0
$622$	−10.8331	−0.434368
$623$	−10.9052	−0.436908
$624$	0	0
$625$	2.39000	0.0956000
$626$	5.25084	0.209866
$627$	0	0
$628$	19.6766	0.785182
$629$	6.11021	0.243630
$630$	0	0
$631$	−39.8399	−1.58600	−0.793000	−	0.609221i	$-0.791482\pi$
$631$	−39.8399	−1.58600	−0.793000	+	0.609221i	$0.791482\pi$
$632$	−10.6896	−0.425208
$633$	0	0
$634$	−6.57755	−0.261228
$635$	−21.9351	−0.870467
$636$	0	0
$637$	−42.0735	−1.66701
$638$	−4.52262	−0.179052
$639$	0	0
$640$	−1.10216	−0.0435668
$641$	−5.19938	−0.205363	−0.102682	−	0.994714i	$-0.532742\pi$
$641$	−5.19938	−0.205363	−0.102682	+	0.994714i	$0.532742\pi$
$642$	0	0
$643$	−9.54077	−0.376251	−0.188126	−	0.982145i	$-0.560241\pi$
$643$	−9.54077	−0.376251	−0.188126	+	0.982145i	$0.560241\pi$
$644$	0.632156	0.0249104
$645$	0	0
$646$	−0.115173	−0.00453144
$647$	38.0573	1.49619	0.748093	−	0.663594i	$-0.230970\pi$
$647$	38.0573	1.49619	0.748093	+	0.663594i	$0.230970\pi$
$648$	0	0
$649$	−20.9282	−0.821505
$650$	−22.7040	−0.890525
$651$	0	0
$652$	−8.40566	−0.329191
$653$	7.10248	0.277942	0.138971	−	0.990296i	$-0.455621\pi$
$653$	7.10248	0.277942	0.138971	+	0.990296i	$0.455621\pi$
$654$	0	0
$655$	−26.0914	−1.01947
$656$	−0.690783	−0.0269706
$657$	0	0
$658$	−6.09474	−0.237598
$659$	−17.2293	−0.671157	−0.335579	−	0.942012i	$-0.608932\pi$
$659$	−17.2293	−0.671157	−0.335579	+	0.942012i	$0.608932\pi$
$660$	0	0
$661$	6.28912	0.244618	0.122309	−	0.992492i	$-0.460970\pi$
$661$	6.28912	0.244618	0.122309	+	0.992492i	$0.460970\pi$
$662$	4.98105	0.193594
$663$	0	0
$664$	−27.1589	−1.05397
$665$	0.0635367	0.00246385
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−19.9884	−0.773373
$669$	0	0
$670$	18.9275	0.731233
$671$	−31.1172	−1.20126
$672$	0	0
$673$	−33.1852	−1.27920	−0.639598	−	0.768709i	$-0.720899\pi$
$673$	−33.1852	−1.27920	−0.639598	+	0.768709i	$0.720899\pi$
$674$	−9.73533	−0.374991
$675$	0	0
$676$	−25.5660	−0.983308
$677$	27.3019	1.04930	0.524649	−	0.851318i	$-0.324196\pi$
$677$	27.3019	1.04930	0.524649	+	0.851318i	$0.324196\pi$
$678$	0	0
$679$	0.610969	0.0234468
$680$	6.39047	0.245063
$681$	0	0
$682$	17.1916	0.658299
$683$	5.32782	0.203863	0.101932	−	0.994791i	$-0.467498\pi$
$683$	5.32782	0.203863	0.101932	+	0.994791i	$0.467498\pi$
$684$	0	0
$685$	22.9251	0.875922
$686$	−10.2968	−0.393134
$687$	0	0
$688$	−1.43456	−0.0546922
$689$	−86.8621	−3.30918
$690$	0	0
$691$	−2.75150	−0.104672	−0.0523360	−	0.998630i	$-0.516667\pi$
$691$	−2.75150	−0.104672	−0.0523360	+	0.998630i	$0.516667\pi$
$692$	2.73594	0.104005
$693$	0	0
$694$	20.7120	0.786217
$695$	−14.7054	−0.557808
$696$	0	0
$697$	−0.751977	−0.0284832
$698$	37.8996	1.43452
$699$	0	0
$700$	2.08611	0.0788477
$701$	37.4534	1.41460	0.707298	−	0.706916i	$-0.249914\pi$
$701$	37.4534	1.41460	0.707298	+	0.706916i	$0.249914\pi$
$702$	0	0
$703$	−0.256959	−0.00969139
$704$	−33.1047	−1.24768
$705$	0	0
$706$	−18.6464	−0.701767
$707$	3.03815	0.114262
$708$	0	0
$709$	29.7909	1.11882	0.559411	−	0.828891i	$-0.311028\pi$
$709$	29.7909	1.11882	0.559411	+	0.828891i	$0.311028\pi$
$710$	1.16570	0.0437478
$711$	0	0
$712$	46.1265	1.72866
$713$	3.80124	0.142358
$714$	0	0
$715$	−36.1270	−1.35107
$716$	−3.88973	−0.145366
$717$	0	0
$718$	−5.78086	−0.215740
$719$	2.28034	0.0850422	0.0425211	−	0.999096i	$-0.486461\pi$
$719$	2.28034	0.0850422	0.0425211	+	0.999096i	$0.486461\pi$
$720$	0	0
$721$	−2.34654	−0.0873898
$722$	−20.1296	−0.749145
$723$	0	0
$724$	−6.64677	−0.247025
$725$	−3.30000	−0.122559
$726$	0	0
$727$	17.1529	0.636164	0.318082	−	0.948063i	$-0.396961\pi$
$727$	17.1529	0.636164	0.318082	+	0.948063i	$0.396961\pi$
$728$	−14.2674	−0.528785
$729$	0	0
$730$	−4.92082	−0.182128
$731$	−1.56164	−0.0577595
$732$	0	0
$733$	7.35753	0.271757	0.135878	−	0.990726i	$-0.456614\pi$
$733$	7.35753	0.271757	0.135878	+	0.990726i	$0.456614\pi$
$734$	8.35034	0.308217
$735$	0	0
$736$	−4.53264	−0.167075
$737$	−58.4639	−2.15354
$738$	0	0
$739$	24.1145	0.887067	0.443534	−	0.896258i	$-0.353725\pi$
$739$	24.1145	0.887067	0.443534	+	0.896258i	$0.353725\pi$
$740$	4.34622	0.159770
$741$	0	0
$742$	−10.2194	−0.375166
$743$	−3.82636	−0.140376	−0.0701878	−	0.997534i	$-0.522360\pi$
$743$	−3.82636	−0.140376	−0.0701878	+	0.997534i	$0.522360\pi$
$744$	0	0
$745$	−11.3241	−0.414883
$746$	−13.0567	−0.478040
$747$	0	0
$748$	−6.01721	−0.220011
$749$	−7.14906	−0.261221
$750$	0	0
$751$	19.0615	0.695564	0.347782	−	0.937575i	$-0.386935\pi$
$751$	19.0615	0.695564	0.347782	+	0.937575i	$0.386935\pi$
$752$	11.7835	0.429699
$753$	0	0
$754$	6.88000	0.250555
$755$	−4.92630	−0.179286
$756$	0	0
$757$	46.5969	1.69359	0.846797	−	0.531916i	$-0.178528\pi$
$757$	46.5969	1.69359	0.846797	+	0.531916i	$0.178528\pi$
$758$	22.2772	0.809146
$759$	0	0
$760$	−0.268745	−0.00974841
$761$	13.6355	0.494288	0.247144	−	0.968979i	$-0.420508\pi$
$761$	13.6355	0.494288	0.247144	+	0.968979i	$0.420508\pi$
$762$	0	0
$763$	−10.7719	−0.389971
$764$	20.9370	0.757475
$765$	0	0
$766$	28.3286	1.02355
$767$	31.8369	1.14956
$768$	0	0
$769$	−7.50359	−0.270586	−0.135293	−	0.990806i	$-0.543198\pi$
$769$	−7.50359	−0.270586	−0.135293	+	0.990806i	$0.543198\pi$
$770$	−4.25038	−0.153173
$771$	0	0
$772$	20.8645	0.750929
$773$	54.1536	1.94777	0.973885	−	0.227043i	$-0.0729058\pi$
$773$	54.1536	1.94777	0.973885	+	0.227043i	$0.0729058\pi$
$774$	0	0
$775$	12.5441	0.450597
$776$	−2.58425	−0.0927692
$777$	0	0
$778$	−19.2902	−0.691589
$779$	0.0316237	0.00113304
$780$	0	0
$781$	−3.60065	−0.128841
$782$	1.70359	0.0609203
$783$	0	0
$784$	9.57022	0.341794
$785$	−29.2525	−1.04407
$786$	0	0
$787$	−29.3341	−1.04565	−0.522823	−	0.852441i	$-0.675121\pi$
$787$	−29.3341	−1.04565	−0.522823	+	0.852441i	$0.675121\pi$
$788$	9.02448	0.321484
$789$	0	0
$790$	4.84442	0.172357
$791$	−6.52632	−0.232049
$792$	0	0
$793$	47.3368	1.68098
$794$	−15.8399	−0.562139
$795$	0	0
$796$	−5.87514	−0.208239
$797$	26.7775	0.948507	0.474253	−	0.880388i	$-0.342718\pi$
$797$	26.7775	0.948507	0.474253	+	0.880388i	$0.342718\pi$
$798$	0	0
$799$	12.8273	0.453798
$800$	−14.9577	−0.528835
$801$	0	0
$802$	−6.29177	−0.222170
$803$	15.1996	0.536383
$804$	0	0
$805$	−0.939804	−0.0331237
$806$	−26.1526	−0.921185
$807$	0	0
$808$	−12.8507	−0.452085
$809$	−44.6884	−1.57116	−0.785579	−	0.618761i	$-0.787635\pi$
$809$	−44.6884	−1.57116	−0.785579	+	0.618761i	$0.787635\pi$
$810$	0	0
$811$	44.3620	1.55776	0.778880	−	0.627174i	$-0.215788\pi$
$811$	44.3620	1.55776	0.778880	+	0.627174i	$0.215788\pi$
$812$	−0.632156	−0.0221843
$813$	0	0
$814$	17.1896	0.602496
$815$	12.4964	0.437730
$816$	0	0
$817$	0.0656735	0.00229762
$818$	29.8440	1.04347
$819$	0	0
$820$	−0.534885	−0.0186790
$821$	21.3612	0.745510	0.372755	−	0.927930i	$-0.378413\pi$
$821$	21.3612	0.745510	0.372755	+	0.927930i	$0.378413\pi$
$822$	0	0
$823$	16.2985	0.568131	0.284065	−	0.958805i	$-0.408317\pi$
$823$	16.2985	0.568131	0.284065	+	0.958805i	$0.408317\pi$
$824$	9.92532	0.345765
$825$	0	0
$826$	3.74564	0.130327
$827$	−5.01177	−0.174276	−0.0871381	−	0.996196i	$-0.527772\pi$
$827$	−5.01177	−0.174276	−0.0871381	+	0.996196i	$0.527772\pi$
$828$	0	0
$829$	−19.5777	−0.679962	−0.339981	−	0.940432i	$-0.610421\pi$
$829$	−19.5777	−0.679962	−0.339981	+	0.940432i	$0.610421\pi$
$830$	12.3082	0.427223
$831$	0	0
$832$	50.3602	1.74593
$833$	10.4180	0.360963
$834$	0	0
$835$	29.7160	1.02837
$836$	0.253048	0.00875185
$837$	0	0
$838$	16.6996	0.576877
$839$	−43.9097	−1.51593	−0.757965	−	0.652295i	$-0.773806\pi$
$839$	−43.9097	−1.51593	−0.757965	+	0.652295i	$0.773806\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−39.7476	−1.36979
$843$	0	0
$844$	−17.0783	−0.587859
$845$	38.0081	1.30752
$846$	0	0
$847$	5.19995	0.178672
$848$	19.7580	0.678494
$849$	0	0
$850$	5.62185	0.192828
$851$	3.80081	0.130290
$852$	0	0
$853$	26.4519	0.905696	0.452848	−	0.891588i	$-0.350408\pi$
$853$	26.4519	0.905696	0.452848	+	0.891588i	$0.350408\pi$
$854$	5.56921	0.190574
$855$	0	0
$856$	30.2388	1.03354
$857$	26.0097	0.888475	0.444238	−	0.895909i	$-0.353475\pi$
$857$	26.0097	0.888475	0.444238	+	0.895909i	$0.353475\pi$
$858$	0	0
$859$	12.0451	0.410973	0.205486	−	0.978660i	$-0.434122\pi$
$859$	12.0451	0.410973	0.205486	+	0.978660i	$0.434122\pi$
$860$	−1.11081	−0.0378782
$861$	0	0
$862$	−11.3743	−0.387411
$863$	26.1020	0.888523	0.444262	−	0.895897i	$-0.353466\pi$
$863$	26.1020	0.888523	0.444262	+	0.895897i	$0.353466\pi$
$864$	0	0
$865$	−4.06743	−0.138297
$866$	31.6853	1.07671
$867$	0	0
$868$	2.40298	0.0815623
$869$	−14.9636	−0.507606
$870$	0	0
$871$	88.9378	3.01354
$872$	45.5628	1.54295
$873$	0	0
$874$	−0.0716428	−0.00242336
$875$	−7.80038	−0.263701
$876$	0	0
$877$	−7.72870	−0.260980	−0.130490	−	0.991450i	$-0.541655\pi$
$877$	−7.72870	−0.260980	−0.130490	+	0.991450i	$0.541655\pi$
$878$	29.2927	0.988582
$879$	0	0
$880$	8.21762	0.277016
$881$	14.2550	0.480262	0.240131	−	0.970741i	$-0.422810\pi$
$881$	14.2550	0.480262	0.240131	+	0.970741i	$0.422810\pi$
$882$	0	0
$883$	18.3998	0.619203	0.309601	−	0.950866i	$-0.399804\pi$
$883$	18.3998	0.619203	0.309601	+	0.950866i	$0.399804\pi$
$884$	9.15365	0.307871
$885$	0	0
$886$	−17.5207	−0.588621
$887$	−0.871823	−0.0292730	−0.0146365	−	0.999893i	$-0.504659\pi$
$887$	−0.871823	−0.0292730	−0.0146365	+	0.999893i	$0.504659\pi$
$888$	0	0
$889$	−12.1263	−0.406703
$890$	−20.9041	−0.700707
$891$	0	0
$892$	−4.78212	−0.160117
$893$	−0.539441	−0.0180517
$894$	0	0
$895$	5.78273	0.193295
$896$	−0.609304	−0.0203554
$897$	0	0
$898$	−16.5448	−0.552107
$899$	−3.80124	−0.126779
$900$	0	0
$901$	21.5083	0.716547
$902$	−2.11551	−0.0704387
$903$	0	0
$904$	27.6048	0.918120
$905$	9.88152	0.328473
$906$	0	0
$907$	55.9890	1.85908	0.929542	−	0.368716i	$-0.120202\pi$
$907$	55.9890	1.85908	0.929542	+	0.368716i	$0.120202\pi$
$908$	11.2238	0.372475
$909$	0	0
$910$	6.46586	0.214341
$911$	−53.4421	−1.77061	−0.885307	−	0.465007i	$-0.846052\pi$
$911$	−53.4421	−1.77061	−0.885307	+	0.465007i	$0.846052\pi$
$912$	0	0
$913$	−38.0179	−1.25821
$914$	−41.4523	−1.37112
$915$	0	0
$916$	15.7409	0.520093
$917$	−14.4240	−0.476322
$918$	0	0
$919$	42.9636	1.41724	0.708619	−	0.705591i	$-0.249319\pi$
$919$	42.9636	1.41724	0.708619	+	0.705591i	$0.249319\pi$
$920$	3.97515	0.131057
$921$	0	0
$922$	−28.9350	−0.952922
$923$	5.47746	0.180293
$924$	0	0
$925$	12.5427	0.412400
$926$	26.6996	0.877404
$927$	0	0
$928$	4.53264	0.148791
$929$	−12.0253	−0.394537	−0.197269	−	0.980349i	$-0.563207\pi$
$929$	−12.0253	−0.394537	−0.197269	+	0.980349i	$0.563207\pi$
$930$	0	0
$931$	−0.438120	−0.0143588
$932$	21.7563	0.712651
$933$	0	0
$934$	−11.1652	−0.365338
$935$	8.94559	0.292552
$936$	0	0
$937$	−30.0372	−0.981274	−0.490637	−	0.871364i	$-0.663236\pi$
$937$	−30.0372	−0.981274	−0.490637	+	0.871364i	$0.663236\pi$
$938$	10.4636	0.341649
$939$	0	0
$940$	9.12415	0.297597
$941$	−19.6719	−0.641286	−0.320643	−	0.947200i	$-0.603899\pi$
$941$	−19.6719	−0.641286	−0.320643	+	0.947200i	$0.603899\pi$
$942$	0	0
$943$	−0.467762	−0.0152324
$944$	−7.24177	−0.235699
$945$	0	0
$946$	−4.39332	−0.142839
$947$	18.8653	0.613041	0.306520	−	0.951864i	$-0.400835\pi$
$947$	18.8653	0.613041	0.306520	+	0.951864i	$0.400835\pi$
$948$	0	0
$949$	−23.1223	−0.750582
$950$	−0.236421	−0.00767052
$951$	0	0
$952$	3.53282	0.114499
$953$	33.1756	1.07466	0.537331	−	0.843371i	$-0.319432\pi$
$953$	33.1756	1.07466	0.537331	+	0.843371i	$0.319432\pi$
$954$	0	0
$955$	−31.1263	−1.00722
$956$	5.67813	0.183644
$957$	0	0
$958$	−23.9768	−0.774655
$959$	12.6736	0.409251
$960$	0	0
$961$	−16.5506	−0.533889
$962$	−26.1496	−0.843097
$963$	0	0
$964$	0.937279	0.0301877
$965$	−31.0185	−0.998520
$966$	0	0
$967$	−41.8683	−1.34640	−0.673198	−	0.739463i	$-0.735080\pi$
$967$	−41.8683	−1.34640	−0.673198	+	0.739463i	$0.735080\pi$
$968$	−21.9945	−0.706931
$969$	0	0
$970$	1.17116	0.0376037
$971$	−24.2057	−0.776799	−0.388400	−	0.921491i	$-0.626972\pi$
$971$	−24.2057	−0.776799	−0.388400	+	0.921491i	$0.626972\pi$
$972$	0	0
$973$	−8.12954	−0.260621
$974$	−1.41130	−0.0452210
$975$	0	0
$976$	−10.7674	−0.344657
$977$	−11.7282	−0.375218	−0.187609	−	0.982244i	$-0.560074\pi$
$977$	−11.7282	−0.375218	−0.187609	+	0.982244i	$0.560074\pi$
$978$	0	0
$979$	64.5693	2.06364
$980$	7.41039	0.236716
$981$	0	0
$982$	39.7902	1.26975
$983$	1.37537	0.0438675	0.0219338	−	0.999759i	$-0.493018\pi$
$983$	1.37537	0.0438675	0.0219338	+	0.999759i	$0.493018\pi$
$984$	0	0
$985$	−13.4164	−0.427482
$986$	−1.70359	−0.0542534
$987$	0	0
$988$	−0.384948	−0.0122468
$989$	−0.971410	−0.0308890
$990$	0	0
$991$	60.9345	1.93565	0.967824	−	0.251629i	$-0.0809662\pi$
$991$	60.9345	1.93565	0.967824	+	0.251629i	$0.0809662\pi$
$992$	−17.2297	−0.547042
$993$	0	0
$994$	0.644428	0.0204400
$995$	8.73436	0.276898
$996$	0	0
$997$	−9.78792	−0.309986	−0.154993	−	0.987916i	$-0.549536\pi$
$997$	−9.78792	−0.309986	−0.154993	+	0.987916i	$0.549536\pi$
$998$	−36.2652	−1.14795
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.8		11
3.2	odd	2		2001.2.a.l.1.4	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.4	✓	11	3.2	odd	2
6003.2.a.m.1.8		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.05971 q^{2} -0.877023 q^{4} +1.30384 q^{5} +0.720797 q^{7} -3.04880 q^{8} +O(q^{10})\)	\(q+1.05971 q^{2} -0.877023 q^{4} +1.30384 q^{5} +0.720797 q^{7} -3.04880 q^{8} +1.38169 q^{10} -4.26781 q^{11} +6.49237 q^{13} +0.763833 q^{14} -1.47678 q^{16} -1.60761 q^{17} +0.0676063 q^{19} -1.14350 q^{20} -4.52262 q^{22} -1.00000 q^{23} -3.30000 q^{25} +6.88000 q^{26} -0.632156 q^{28} +1.00000 q^{29} -3.80124 q^{31} +4.53264 q^{32} -1.70359 q^{34} +0.939804 q^{35} -3.80081 q^{37} +0.0716428 q^{38} -3.97515 q^{40} +0.467762 q^{41} +0.971410 q^{43} +3.74296 q^{44} -1.05971 q^{46} -7.97915 q^{47} -6.48045 q^{49} -3.49703 q^{50} -5.69396 q^{52} -13.3791 q^{53} -5.56454 q^{55} -2.19756 q^{56} +1.05971 q^{58} +4.90374 q^{59} +7.29114 q^{61} -4.02820 q^{62} +7.75683 q^{64} +8.46502 q^{65} +13.6988 q^{67} +1.40991 q^{68} +0.995916 q^{70} +0.843676 q^{71} -3.56146 q^{73} -4.02774 q^{74} -0.0592923 q^{76} -3.07622 q^{77} +3.50616 q^{79} -1.92549 q^{80} +0.495690 q^{82} +8.90808 q^{83} -2.09606 q^{85} +1.02941 q^{86} +13.0117 q^{88} -15.1294 q^{89} +4.67968 q^{91} +0.877023 q^{92} -8.45555 q^{94} +0.0881479 q^{95} +0.847630 q^{97} -6.86737 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more