LMFDB - Embedded newform 6003.2.a.k.1.7

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 $ x^10 - x^9 - 17x^8 + 23x^7 + 69x^6 - 88x^5 - 106x^4 + 101x^3 + 60x^2 - 23x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.473620$ 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)

$f(q)$	$=$	$q-0.161383 q^{2} -1.97396 q^{4} -2.08087 q^{5} +3.66994 q^{7} +0.641331 q^{8} +O(q^{10})$	\(q-0.161383 q^{2} -1.97396 q^{4} -2.08087 q^{5} +3.66994 q^{7} +0.641331 q^{8} +0.335818 q^{10} -3.77836 q^{11} +3.51085 q^{13} -0.592268 q^{14} +3.84441 q^{16} -3.02517 q^{17} -1.97558 q^{19} +4.10754 q^{20} +0.609764 q^{22} +1.00000 q^{23} -0.669990 q^{25} -0.566593 q^{26} -7.24430 q^{28} -1.00000 q^{29} -3.58155 q^{31} -1.90309 q^{32} +0.488212 q^{34} -7.63666 q^{35} +5.34560 q^{37} +0.318827 q^{38} -1.33452 q^{40} +12.0312 q^{41} +3.68702 q^{43} +7.45831 q^{44} -0.161383 q^{46} +0.138067 q^{47} +6.46848 q^{49} +0.108125 q^{50} -6.93026 q^{52} -3.01823 q^{53} +7.86226 q^{55} +2.35365 q^{56} +0.161383 q^{58} -9.43156 q^{59} -10.8624 q^{61} +0.578003 q^{62} -7.38169 q^{64} -7.30561 q^{65} +4.57266 q^{67} +5.97155 q^{68} +1.23243 q^{70} +3.28228 q^{71} +1.53135 q^{73} -0.862691 q^{74} +3.89971 q^{76} -13.8664 q^{77} +7.31941 q^{79} -7.99971 q^{80} -1.94164 q^{82} -5.82693 q^{83} +6.29498 q^{85} -0.595024 q^{86} -2.42318 q^{88} -6.06380 q^{89} +12.8846 q^{91} -1.97396 q^{92} -0.0222817 q^{94} +4.11093 q^{95} -13.5506 q^{97} -1.04391 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * 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 $

$2$	−0.161383	−0.114115	−0.0570577	−	0.998371i	$-0.518172\pi$
$2$	−0.161383	−0.114115	−0.0570577	+	0.998371i	$0.518172\pi$
$3$	0	0
$3$	0	0
$4$	−1.97396	−0.986978
$5$	−2.08087	−0.930592	−0.465296	−	0.885155i	$-0.654052\pi$
$5$	−2.08087	−0.930592	−0.465296	+	0.885155i	$0.654052\pi$
$6$	0	0
$7$	3.66994	1.38711	0.693554	−	0.720405i	$-0.256044\pi$
$7$	3.66994	1.38711	0.693554	+	0.720405i	$0.256044\pi$
$8$	0.641331	0.226745
$9$	0	0
$10$	0.335818	0.106195
$11$	−3.77836	−1.13922	−0.569609	−	0.821916i	$-0.692905\pi$
$11$	−3.77836	−1.13922	−0.569609	+	0.821916i	$0.692905\pi$
$12$	0	0
$13$	3.51085	0.973735	0.486867	−	0.873476i	$-0.338140\pi$
$13$	3.51085	0.973735	0.486867	+	0.873476i	$0.338140\pi$
$14$	−0.592268	−0.158290
$15$	0	0
$16$	3.84441	0.961103
$17$	−3.02517	−0.733712	−0.366856	−	0.930278i	$-0.619566\pi$
$17$	−3.02517	−0.733712	−0.366856	+	0.930278i	$0.619566\pi$
$18$	0	0
$19$	−1.97558	−0.453230	−0.226615	−	0.973984i	$-0.572766\pi$
$19$	−1.97558	−0.453230	−0.226615	+	0.973984i	$0.572766\pi$
$20$	4.10754	0.918474
$21$	0	0
$22$	0.609764	0.130002
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−0.669990	−0.133998
$26$	−0.566593	−0.111118
$27$	0	0
$28$	−7.24430	−1.36904
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.58155	−0.643266	−0.321633	−	0.946864i	$-0.604232\pi$
$31$	−3.58155	−0.643266	−0.321633	+	0.946864i	$0.604232\pi$
$32$	−1.90309	−0.336421
$33$	0	0
$34$	0.488212	0.0837277
$35$	−7.63666	−1.29083
$36$	0	0
$37$	5.34560	0.878811	0.439406	−	0.898289i	$-0.355189\pi$
$37$	5.34560	0.878811	0.439406	+	0.898289i	$0.355189\pi$
$38$	0.318827	0.0517205
$39$	0	0
$40$	−1.33452	−0.211007
$41$	12.0312	1.87896	0.939480	−	0.342604i	$-0.111309\pi$
$41$	12.0312	1.87896	0.939480	+	0.342604i	$0.111309\pi$
$42$	0	0
$43$	3.68702	0.562266	0.281133	−	0.959669i	$-0.409290\pi$
$43$	3.68702	0.562266	0.281133	+	0.959669i	$0.409290\pi$
$44$	7.45831	1.12438
$45$	0	0
$46$	−0.161383	−0.0237947
$47$	0.138067	0.0201392	0.0100696	−	0.999949i	$-0.496795\pi$
$47$	0.138067	0.0201392	0.0100696	+	0.999949i	$0.496795\pi$
$48$	0	0
$49$	6.46848	0.924068
$50$	0.108125	0.0152912
$51$	0	0
$52$	−6.93026	−0.961054
$53$	−3.01823	−0.414586	−0.207293	−	0.978279i	$-0.566465\pi$
$53$	−3.01823	−0.414586	−0.207293	+	0.978279i	$0.566465\pi$
$54$	0	0
$55$	7.86226	1.06015
$56$	2.35365	0.314519
$57$	0	0
$58$	0.161383	0.0211907
$59$	−9.43156	−1.22788	−0.613942	−	0.789351i	$-0.710417\pi$
$59$	−9.43156	−1.22788	−0.613942	+	0.789351i	$0.710417\pi$
$60$	0	0
$61$	−10.8624	−1.39079	−0.695394	−	0.718628i	$-0.744770\pi$
$61$	−10.8624	−1.39079	−0.695394	+	0.718628i	$0.744770\pi$
$62$	0.578003	0.0734065
$63$	0	0
$64$	−7.38169	−0.922712
$65$	−7.30561	−0.906150
$66$	0	0
$67$	4.57266	0.558639	0.279320	−	0.960198i	$-0.409891\pi$
$67$	4.57266	0.558639	0.279320	+	0.960198i	$0.409891\pi$
$68$	5.97155	0.724157
$69$	0	0
$70$	1.23243	0.147304
$71$	3.28228	0.389535	0.194768	−	0.980849i	$-0.437605\pi$
$71$	3.28228	0.389535	0.194768	+	0.980849i	$0.437605\pi$
$72$	0	0
$73$	1.53135	0.179231	0.0896153	−	0.995976i	$-0.471436\pi$
$73$	1.53135	0.179231	0.0896153	+	0.995976i	$0.471436\pi$
$74$	−0.862691	−0.100286
$75$	0	0
$76$	3.89971	0.447328
$77$	−13.8664	−1.58022
$78$	0	0
$79$	7.31941	0.823498	0.411749	−	0.911297i	$-0.364918\pi$
$79$	7.31941	0.823498	0.411749	+	0.911297i	$0.364918\pi$
$80$	−7.99971	−0.894395
$81$	0	0
$82$	−1.94164	−0.214418
$83$	−5.82693	−0.639588	−0.319794	−	0.947487i	$-0.603614\pi$
$83$	−5.82693	−0.639588	−0.319794	+	0.947487i	$0.603614\pi$
$84$	0	0
$85$	6.29498	0.682786
$86$	−0.595024	−0.0641631
$87$	0	0
$88$	−2.42318	−0.258311
$89$	−6.06380	−0.642761	−0.321381	−	0.946950i	$-0.604147\pi$
$89$	−6.06380	−0.642761	−0.321381	+	0.946950i	$0.604147\pi$
$90$	0	0
$91$	12.8846	1.35067
$92$	−1.97396	−0.205799
$93$	0	0
$94$	−0.0222817	−0.00229819
$95$	4.11093	0.421772
$96$	0	0
$97$	−13.5506	−1.37585	−0.687926	−	0.725781i	$-0.741479\pi$
$97$	−13.5506	−1.37585	−0.687926	+	0.725781i	$0.741479\pi$
$98$	−1.04391	−0.105450
$99$	0	0
$100$	1.32253	0.132253
$101$	5.39674	0.536996	0.268498	−	0.963280i	$-0.413473\pi$
$101$	5.39674	0.536996	0.268498	+	0.963280i	$0.413473\pi$
$102$	0	0
$103$	7.28928	0.718234	0.359117	−	0.933293i	$-0.383078\pi$
$103$	7.28928	0.718234	0.359117	+	0.933293i	$0.383078\pi$
$104$	2.25162	0.220789
$105$	0	0
$106$	0.487093	0.0473106
$107$	5.92929	0.573207	0.286603	−	0.958049i	$-0.407474\pi$
$107$	5.92929	0.573207	0.286603	+	0.958049i	$0.407474\pi$
$108$	0	0
$109$	16.9282	1.62143	0.810715	−	0.585441i	$-0.199078\pi$
$109$	16.9282	1.62143	0.810715	+	0.585441i	$0.199078\pi$
$110$	−1.26884	−0.120979
$111$	0	0
$112$	14.1088	1.33315
$113$	−6.48668	−0.610215	−0.305108	−	0.952318i	$-0.598692\pi$
$113$	−6.48668	−0.610215	−0.305108	+	0.952318i	$0.598692\pi$
$114$	0	0
$115$	−2.08087	−0.194042
$116$	1.97396	0.183277
$117$	0	0
$118$	1.52210	0.140120
$119$	−11.1022	−1.01774
$120$	0	0
$121$	3.27598	0.297816
$122$	1.75301	0.158710
$123$	0	0
$124$	7.06982	0.634889
$125$	11.7985	1.05529
$126$	0	0
$127$	8.62092	0.764983	0.382492	−	0.923959i	$-0.375066\pi$
$127$	8.62092	0.764983	0.382492	+	0.923959i	$0.375066\pi$
$128$	4.99746	0.441717
$129$	0	0
$130$	1.17901	0.103406
$131$	4.57907	0.400075	0.200037	−	0.979788i	$-0.435894\pi$
$131$	4.57907	0.400075	0.200037	+	0.979788i	$0.435894\pi$
$132$	0	0
$133$	−7.25028	−0.628679
$134$	−0.737952	−0.0637493
$135$	0	0
$136$	−1.94013	−0.166365
$137$	0.365639	0.0312387	0.0156193	−	0.999878i	$-0.495028\pi$
$137$	0.365639	0.0312387	0.0156193	+	0.999878i	$0.495028\pi$
$138$	0	0
$139$	−12.2183	−1.03634	−0.518172	−	0.855277i	$-0.673387\pi$
$139$	−12.2183	−1.03634	−0.518172	+	0.855277i	$0.673387\pi$
$140$	15.0744	1.27402
$141$	0	0
$142$	−0.529706	−0.0444519
$143$	−13.2652	−1.10930
$144$	0	0
$145$	2.08087	0.172807
$146$	−0.247134	−0.0204530
$147$	0	0
$148$	−10.5520	−0.867367
$149$	−2.72205	−0.222999	−0.111499	−	0.993764i	$-0.535565\pi$
$149$	−2.72205	−0.222999	−0.111499	+	0.993764i	$0.535565\pi$
$150$	0	0
$151$	−14.3704	−1.16945	−0.584724	−	0.811232i	$-0.698797\pi$
$151$	−14.3704	−1.16945	−0.584724	+	0.811232i	$0.698797\pi$
$152$	−1.26700	−0.102767
$153$	0	0
$154$	2.23780	0.180327
$155$	7.45273	0.598618
$156$	0	0
$157$	−9.48437	−0.756936	−0.378468	−	0.925614i	$-0.623549\pi$
$157$	−9.48437	−0.756936	−0.378468	+	0.925614i	$0.623549\pi$
$158$	−1.18123	−0.0939738
$159$	0	0
$160$	3.96007	0.313071
$161$	3.66994	0.289232
$162$	0	0
$163$	20.6148	1.61468	0.807339	−	0.590088i	$-0.200907\pi$
$163$	20.6148	1.61468	0.807339	+	0.590088i	$0.200907\pi$
$164$	−23.7491	−1.85449
$165$	0	0
$166$	0.940370	0.0729869
$167$	−13.3426	−1.03248	−0.516241	−	0.856443i	$-0.672669\pi$
$167$	−13.3426	−1.03248	−0.516241	+	0.856443i	$0.672669\pi$
$168$	0	0
$169$	−0.673934	−0.0518411
$170$	−1.01591	−0.0779164
$171$	0	0
$172$	−7.27802	−0.554944
$173$	−9.60604	−0.730334	−0.365167	−	0.930942i	$-0.618988\pi$
$173$	−9.60604	−0.730334	−0.365167	+	0.930942i	$0.618988\pi$
$174$	0	0
$175$	−2.45883	−0.185870
$176$	−14.5256	−1.09490
$177$	0	0
$178$	0.978597	0.0733489
$179$	−10.3216	−0.771473	−0.385737	−	0.922609i	$-0.626053\pi$
$179$	−10.3216	−0.771473	−0.385737	+	0.922609i	$0.626053\pi$
$180$	0	0
$181$	9.34390	0.694527	0.347263	−	0.937768i	$-0.387111\pi$
$181$	9.34390	0.694527	0.347263	+	0.937768i	$0.387111\pi$
$182$	−2.07936	−0.154133
$183$	0	0
$184$	0.641331	0.0472795
$185$	−11.1235	−0.817815
$186$	0	0
$187$	11.4302	0.835857
$188$	−0.272538	−0.0198769
$189$	0	0
$190$	−0.663436	−0.0481307
$191$	0.626325	0.0453193	0.0226597	−	0.999743i	$-0.492787\pi$
$191$	0.626325	0.0453193	0.0226597	+	0.999743i	$0.492787\pi$
$192$	0	0
$193$	−0.516976	−0.0372128	−0.0186064	−	0.999827i	$-0.505923\pi$
$193$	−0.516976	−0.0372128	−0.0186064	+	0.999827i	$0.505923\pi$
$194$	2.18684	0.157006
$195$	0	0
$196$	−12.7685	−0.912035
$197$	0.0598105	0.00426132	0.00213066	−	0.999998i	$-0.499322\pi$
$197$	0.0598105	0.00426132	0.00213066	+	0.999998i	$0.499322\pi$
$198$	0	0
$199$	−5.54830	−0.393308	−0.196654	−	0.980473i	$-0.563008\pi$
$199$	−5.54830	−0.393308	−0.196654	+	0.980473i	$0.563008\pi$
$200$	−0.429685	−0.0303833
$201$	0	0
$202$	−0.870945	−0.0612795
$203$	−3.66994	−0.257579
$204$	0	0
$205$	−25.0354	−1.74855
$206$	−1.17637	−0.0819615
$207$	0	0
$208$	13.4971	0.935859
$209$	7.46446	0.516328
$210$	0	0
$211$	−10.0930	−0.694833	−0.347416	−	0.937711i	$-0.612941\pi$
$211$	−10.0930	−0.694833	−0.347416	+	0.937711i	$0.612941\pi$
$212$	5.95785	0.409187
$213$	0	0
$214$	−0.956890	−0.0654117
$215$	−7.67220	−0.523240
$216$	0	0
$217$	−13.1441	−0.892279
$218$	−2.73194	−0.185030
$219$	0	0
$220$	−15.5198	−1.04634
$221$	−10.6209	−0.714440
$222$	0	0
$223$	−2.65301	−0.177659	−0.0888293	−	0.996047i	$-0.528313\pi$
$223$	−2.65301	−0.177659	−0.0888293	+	0.996047i	$0.528313\pi$
$224$	−6.98422	−0.466653
$225$	0	0
$226$	1.04684	0.0696349
$227$	2.02512	0.134412	0.0672059	−	0.997739i	$-0.478592\pi$
$227$	2.02512	0.134412	0.0672059	+	0.997739i	$0.478592\pi$
$228$	0	0
$229$	−1.74269	−0.115160	−0.0575800	−	0.998341i	$-0.518338\pi$
$229$	−1.74269	−0.115160	−0.0575800	+	0.998341i	$0.518338\pi$
$230$	0.335818	0.0221432
$231$	0	0
$232$	−0.641331	−0.0421054
$233$	−10.8194	−0.708802	−0.354401	−	0.935093i	$-0.615315\pi$
$233$	−10.8194	−0.708802	−0.354401	+	0.935093i	$0.615315\pi$
$234$	0	0
$235$	−0.287299	−0.0187413
$236$	18.6175	1.21189
$237$	0	0
$238$	1.79171	0.116139
$239$	−13.5515	−0.876572	−0.438286	−	0.898836i	$-0.644414\pi$
$239$	−13.5515	−0.876572	−0.438286	+	0.898836i	$0.644414\pi$
$240$	0	0
$241$	−17.0852	−1.10055	−0.550277	−	0.834982i	$-0.685478\pi$
$241$	−17.0852	−1.10055	−0.550277	+	0.834982i	$0.685478\pi$
$242$	−0.528689	−0.0339854
$243$	0	0
$244$	21.4419	1.37268
$245$	−13.4600	−0.859931
$246$	0	0
$247$	−6.93598	−0.441326
$248$	−2.29696	−0.145857
$249$	0	0
$250$	−1.90408	−0.120425
$251$	−14.0710	−0.888151	−0.444076	−	0.895989i	$-0.646468\pi$
$251$	−14.0710	−0.888151	−0.444076	+	0.895989i	$0.646468\pi$
$252$	0	0
$253$	−3.77836	−0.237543
$254$	−1.39127	−0.0872963
$255$	0	0
$256$	13.9569	0.872305
$257$	−2.08470	−0.130040	−0.0650201	−	0.997884i	$-0.520711\pi$
$257$	−2.08470	−0.130040	−0.0650201	+	0.997884i	$0.520711\pi$
$258$	0	0
$259$	19.6180	1.21901
$260$	14.4210	0.894350
$261$	0	0
$262$	−0.738986	−0.0456547
$263$	22.5597	1.39109	0.695545	−	0.718483i	$-0.255163\pi$
$263$	22.5597	1.39109	0.695545	+	0.718483i	$0.255163\pi$
$264$	0	0
$265$	6.28054	0.385810
$266$	1.17008	0.0717419
$267$	0	0
$268$	−9.02623	−0.551365
$269$	−21.7650	−1.32704	−0.663518	−	0.748160i	$-0.730938\pi$
$269$	−21.7650	−1.32704	−0.663518	+	0.748160i	$0.730938\pi$
$270$	0	0
$271$	−19.0602	−1.15782	−0.578911	−	0.815391i	$-0.696522\pi$
$271$	−19.0602	−1.15782	−0.578911	+	0.815391i	$0.696522\pi$
$272$	−11.6300	−0.705172
$273$	0	0
$274$	−0.0590081	−0.00356481
$275$	2.53146	0.152653
$276$	0	0
$277$	−5.11613	−0.307399	−0.153699	−	0.988118i	$-0.549119\pi$
$277$	−5.11613	−0.307399	−0.153699	+	0.988118i	$0.549119\pi$
$278$	1.97183	0.118263
$279$	0	0
$280$	−4.89763	−0.292689
$281$	7.01306	0.418364	0.209182	−	0.977877i	$-0.432920\pi$
$281$	7.01306	0.418364	0.209182	+	0.977877i	$0.432920\pi$
$282$	0	0
$283$	−18.5054	−1.10003	−0.550014	−	0.835155i	$-0.685378\pi$
$283$	−18.5054	−1.10003	−0.550014	+	0.835155i	$0.685378\pi$
$284$	−6.47907	−0.384462
$285$	0	0
$286$	2.14079	0.126588
$287$	44.1539	2.60632
$288$	0	0
$289$	−7.84835	−0.461667
$290$	−0.335818	−0.0197199
$291$	0	0
$292$	−3.02281	−0.176897
$293$	−7.86714	−0.459603	−0.229802	−	0.973237i	$-0.573808\pi$
$293$	−7.86714	−0.459603	−0.229802	+	0.973237i	$0.573808\pi$
$294$	0	0
$295$	19.6258	1.14266
$296$	3.42830	0.199266
$297$	0	0
$298$	0.439294	0.0254476
$299$	3.51085	0.203038
$300$	0	0
$301$	13.5312	0.779923
$302$	2.31915	0.133452
$303$	0	0
$304$	−7.59496	−0.435601
$305$	22.6032	1.29426
$306$	0	0
$307$	−1.52335	−0.0869424	−0.0434712	−	0.999055i	$-0.513842\pi$
$307$	−1.52335	−0.0869424	−0.0434712	+	0.999055i	$0.513842\pi$
$308$	27.3716	1.55964
$309$	0	0
$310$	−1.20275	−0.0683115
$311$	−15.6060	−0.884936	−0.442468	−	0.896784i	$-0.645897\pi$
$311$	−15.6060	−0.884936	−0.442468	+	0.896784i	$0.645897\pi$
$312$	0	0
$313$	−8.42755	−0.476353	−0.238177	−	0.971222i	$-0.576550\pi$
$313$	−8.42755	−0.476353	−0.238177	+	0.971222i	$0.576550\pi$
$314$	1.53062	0.0863780
$315$	0	0
$316$	−14.4482	−0.812774
$317$	8.28338	0.465241	0.232621	−	0.972568i	$-0.425270\pi$
$317$	8.28338	0.465241	0.232621	+	0.972568i	$0.425270\pi$
$318$	0	0
$319$	3.77836	0.211547
$320$	15.3603	0.858668
$321$	0	0
$322$	−0.592268	−0.0330058
$323$	5.97648	0.332540
$324$	0	0
$325$	−2.35224	−0.130479
$326$	−3.32689	−0.184259
$327$	0	0
$328$	7.71599	0.426044
$329$	0.506698	0.0279352
$330$	0	0
$331$	−21.2067	−1.16563	−0.582814	−	0.812605i	$-0.698048\pi$
$331$	−21.2067	−1.16563	−0.582814	+	0.812605i	$0.698048\pi$
$332$	11.5021	0.631260
$333$	0	0
$334$	2.15328	0.117822
$335$	−9.51510	−0.519865
$336$	0	0
$337$	−13.8277	−0.753242	−0.376621	−	0.926367i	$-0.622914\pi$
$337$	−13.8277	−0.753242	−0.376621	+	0.926367i	$0.622914\pi$
$338$	0.108762	0.00591587
$339$	0	0
$340$	−12.4260	−0.673895
$341$	13.5324	0.732819
$342$	0	0
$343$	−1.95065	−0.105325
$344$	2.36460	0.127491
$345$	0	0
$346$	1.55026	0.0833423
$347$	−33.5066	−1.79873	−0.899365	−	0.437198i	$-0.855971\pi$
$347$	−33.5066	−1.79873	−0.899365	+	0.437198i	$0.855971\pi$
$348$	0	0
$349$	−20.9451	−1.12117	−0.560584	−	0.828098i	$-0.689423\pi$
$349$	−20.9451	−1.12117	−0.560584	+	0.828098i	$0.689423\pi$
$350$	0.396814	0.0212106
$351$	0	0
$352$	7.19054	0.383257
$353$	−33.6831	−1.79277	−0.896386	−	0.443275i	$-0.853817\pi$
$353$	−33.6831	−1.79277	−0.896386	+	0.443275i	$0.853817\pi$
$354$	0	0
$355$	−6.82999	−0.362498
$356$	11.9697	0.634391
$357$	0	0
$358$	1.66574	0.0880370
$359$	1.33233	0.0703176	0.0351588	−	0.999382i	$-0.488806\pi$
$359$	1.33233	0.0703176	0.0351588	+	0.999382i	$0.488806\pi$
$360$	0	0
$361$	−15.0971	−0.794583
$362$	−1.50795	−0.0792562
$363$	0	0
$364$	−25.4337	−1.33309
$365$	−3.18653	−0.166791
$366$	0	0
$367$	14.0175	0.731708	0.365854	−	0.930672i	$-0.380777\pi$
$367$	14.0175	0.731708	0.365854	+	0.930672i	$0.380777\pi$
$368$	3.84441	0.200404
$369$	0	0
$370$	1.79515	0.0933252
$371$	−11.0767	−0.575075
$372$	0	0
$373$	−17.5045	−0.906347	−0.453173	−	0.891422i	$-0.649708\pi$
$373$	−17.5045	−0.906347	−0.453173	+	0.891422i	$0.649708\pi$
$374$	−1.84464	−0.0953841
$375$	0	0
$376$	0.0885467	0.00456645
$377$	−3.51085	−0.180818
$378$	0	0
$379$	−26.0033	−1.33570	−0.667851	−	0.744295i	$-0.732786\pi$
$379$	−26.0033	−1.33570	−0.667851	+	0.744295i	$0.732786\pi$
$380$	−8.11479	−0.416280
$381$	0	0
$382$	−0.101079	−0.00517163
$383$	19.8921	1.01644	0.508219	−	0.861228i	$-0.330304\pi$
$383$	19.8921	1.01644	0.508219	+	0.861228i	$0.330304\pi$
$384$	0	0
$385$	28.8540	1.47054
$386$	0.0834314	0.00424655
$387$	0	0
$388$	26.7482	1.35793
$389$	6.92622	0.351173	0.175587	−	0.984464i	$-0.443818\pi$
$389$	6.92622	0.351173	0.175587	+	0.984464i	$0.443818\pi$
$390$	0	0
$391$	−3.02517	−0.152989
$392$	4.14843	0.209528
$393$	0	0
$394$	−0.00965242	−0.000486282 0
$395$	−15.2307	−0.766341
$396$	0	0
$397$	−23.2007	−1.16441	−0.582205	−	0.813042i	$-0.697810\pi$
$397$	−23.2007	−1.16441	−0.582205	+	0.813042i	$0.697810\pi$
$398$	0.895403	0.0448825
$399$	0	0
$400$	−2.57572	−0.128786
$401$	−12.9918	−0.648778	−0.324389	−	0.945924i	$-0.605159\pi$
$401$	−12.9918	−0.648778	−0.324389	+	0.945924i	$0.605159\pi$
$402$	0	0
$403$	−12.5743	−0.626370
$404$	−10.6529	−0.530003
$405$	0	0
$406$	0.592268	0.0293938
$407$	−20.1976	−1.00116
$408$	0	0
$409$	−32.0911	−1.58680	−0.793402	−	0.608699i	$-0.791692\pi$
$409$	−32.0911	−1.58680	−0.793402	+	0.608699i	$0.791692\pi$
$410$	4.04029	0.199536
$411$	0	0
$412$	−14.3887	−0.708881
$413$	−34.6133	−1.70321
$414$	0	0
$415$	12.1251	0.595196
$416$	−6.68145	−0.327585
$417$	0	0
$418$	−1.20464	−0.0589209
$419$	−28.0069	−1.36823	−0.684113	−	0.729376i	$-0.739811\pi$
$419$	−28.0069	−1.36823	−0.684113	+	0.729376i	$0.739811\pi$
$420$	0	0
$421$	−23.7699	−1.15847	−0.579237	−	0.815159i	$-0.696650\pi$
$421$	−23.7699	−1.15847	−0.579237	+	0.815159i	$0.696650\pi$
$422$	1.62885	0.0792911
$423$	0	0
$424$	−1.93568	−0.0940051
$425$	2.02683	0.0983159
$426$	0	0
$427$	−39.8644	−1.92917
$428$	−11.7042	−0.565742
$429$	0	0
$430$	1.23817	0.0597097
$431$	−3.63777	−0.175225	−0.0876126	−	0.996155i	$-0.527924\pi$
$431$	−3.63777	−0.175225	−0.0876126	+	0.996155i	$0.527924\pi$
$432$	0	0
$433$	39.2350	1.88552	0.942758	−	0.333478i	$-0.108222\pi$
$433$	39.2350	1.88552	0.942758	+	0.333478i	$0.108222\pi$
$434$	2.12124	0.101823
$435$	0	0
$436$	−33.4156	−1.60032
$437$	−1.97558	−0.0945050
$438$	0	0
$439$	28.2038	1.34609	0.673046	−	0.739601i	$-0.264986\pi$
$439$	28.2038	1.34609	0.673046	+	0.739601i	$0.264986\pi$
$440$	5.04231	0.240383
$441$	0	0
$442$	1.71404	0.0815286
$443$	24.3212	1.15553	0.577767	−	0.816202i	$-0.303924\pi$
$443$	24.3212	1.15553	0.577767	+	0.816202i	$0.303924\pi$
$444$	0	0
$445$	12.6180	0.598149
$446$	0.428152	0.0202736
$447$	0	0
$448$	−27.0904	−1.27990
$449$	−40.9877	−1.93433	−0.967164	−	0.254152i	$-0.918204\pi$
$449$	−40.9877	−1.93433	−0.967164	+	0.254152i	$0.918204\pi$
$450$	0	0
$451$	−45.4582	−2.14054
$452$	12.8044	0.602269
$453$	0	0
$454$	−0.326821	−0.0153384
$455$	−26.8112	−1.25693
$456$	0	0
$457$	9.80944	0.458866	0.229433	−	0.973324i	$-0.426313\pi$
$457$	9.80944	0.458866	0.229433	+	0.973324i	$0.426313\pi$
$458$	0.281241	0.0131415
$459$	0	0
$460$	4.10754	0.191515
$461$	24.6215	1.14674	0.573369	−	0.819297i	$-0.305636\pi$
$461$	24.6215	1.14674	0.573369	+	0.819297i	$0.305636\pi$
$462$	0	0
$463$	10.1126	0.469972	0.234986	−	0.971999i	$-0.424495\pi$
$463$	10.1126	0.469972	0.234986	+	0.971999i	$0.424495\pi$
$464$	−3.84441	−0.178472
$465$	0	0
$466$	1.74607	0.0808852
$467$	−0.219921	−0.0101767	−0.00508836	−	0.999987i	$-0.501620\pi$
$467$	−0.219921	−0.0101767	−0.00508836	+	0.999987i	$0.501620\pi$
$468$	0	0
$469$	16.7814	0.774893
$470$	0.0463654	0.00213867
$471$	0	0
$472$	−6.04875	−0.278416
$473$	−13.9309	−0.640543
$474$	0	0
$475$	1.32362	0.0607319
$476$	21.9152	1.00448
$477$	0	0
$478$	2.18698	0.100030
$479$	16.9614	0.774984	0.387492	−	0.921873i	$-0.373341\pi$
$479$	16.9614	0.774984	0.387492	+	0.921873i	$0.373341\pi$
$480$	0	0
$481$	18.7676	0.855729
$482$	2.75727	0.125590
$483$	0	0
$484$	−6.46664	−0.293938
$485$	28.1969	1.28036
$486$	0	0
$487$	28.0531	1.27121	0.635603	−	0.772016i	$-0.280752\pi$
$487$	28.0531	1.27121	0.635603	+	0.772016i	$0.280752\pi$
$488$	−6.96639	−0.315354
$489$	0	0
$490$	2.17223	0.0981313
$491$	−37.9620	−1.71320	−0.856601	−	0.515980i	$-0.827428\pi$
$491$	−37.9620	−1.71320	−0.856601	+	0.515980i	$0.827428\pi$
$492$	0	0
$493$	3.02517	0.136247
$494$	1.11935	0.0503620
$495$	0	0
$496$	−13.7690	−0.618244
$497$	12.0458	0.540327
$498$	0	0
$499$	5.03700	0.225487	0.112744	−	0.993624i	$-0.464036\pi$
$499$	5.03700	0.225487	0.112744	+	0.993624i	$0.464036\pi$
$500$	−23.2897	−1.04155
$501$	0	0
$502$	2.27082	0.101352
$503$	19.3933	0.864706	0.432353	−	0.901704i	$-0.357683\pi$
$503$	19.3933	0.864706	0.432353	+	0.901704i	$0.357683\pi$
$504$	0	0
$505$	−11.2299	−0.499724
$506$	0.609764	0.0271073
$507$	0	0
$508$	−17.0173	−0.755021
$509$	26.5066	1.17488	0.587442	−	0.809266i	$-0.300135\pi$
$509$	26.5066	1.17488	0.587442	+	0.809266i	$0.300135\pi$
$510$	0	0
$511$	5.61996	0.248612
$512$	−12.2473	−0.541260
$513$	0	0
$514$	0.336436	0.0148396
$515$	−15.1680	−0.668383
$516$	0	0
$517$	−0.521667	−0.0229429
$518$	−3.16603	−0.139107
$519$	0	0
$520$	−4.68531	−0.205465
$521$	14.1503	0.619936	0.309968	−	0.950747i	$-0.399682\pi$
$521$	14.1503	0.619936	0.309968	+	0.950747i	$0.399682\pi$
$522$	0	0
$523$	−2.42020	−0.105828	−0.0529139	−	0.998599i	$-0.516851\pi$
$523$	−2.42020	−0.105828	−0.0529139	+	0.998599i	$0.516851\pi$
$524$	−9.03887	−0.394865
$525$	0	0
$526$	−3.64076	−0.158745
$527$	10.8348	0.471971
$528$	0	0
$529$	1.00000	0.0434783
$530$	−1.01358	−0.0440269
$531$	0	0
$532$	14.3117	0.620492
$533$	42.2398	1.82961
$534$	0	0
$535$	−12.3381	−0.533422
$536$	2.93259	0.126668
$537$	0	0
$538$	3.51251	0.151435
$539$	−24.4402	−1.05271
$540$	0	0
$541$	21.1409	0.908919	0.454459	−	0.890768i	$-0.349833\pi$
$541$	21.1409	0.908919	0.454459	+	0.890768i	$0.349833\pi$
$542$	3.07600	0.132125
$543$	0	0
$544$	5.75716	0.246836
$545$	−35.2254	−1.50889
$546$	0	0
$547$	19.9428	0.852692	0.426346	−	0.904560i	$-0.359801\pi$
$547$	19.9428	0.852692	0.426346	+	0.904560i	$0.359801\pi$
$548$	−0.721755	−0.0308319
$549$	0	0
$550$	−0.408536	−0.0174200
$551$	1.97558	0.0841627
$552$	0	0
$553$	26.8618	1.14228
$554$	0.825660	0.0350789
$555$	0	0
$556$	24.1184	1.02285
$557$	−11.3239	−0.479809	−0.239904	−	0.970797i	$-0.577116\pi$
$557$	−11.3239	−0.479809	−0.239904	+	0.970797i	$0.577116\pi$
$558$	0	0
$559$	12.9446	0.547497
$560$	−29.3585	−1.24062
$561$	0	0
$562$	−1.13179	−0.0477418
$563$	5.82702	0.245580	0.122790	−	0.992433i	$-0.460816\pi$
$563$	5.82702	0.245580	0.122790	+	0.992433i	$0.460816\pi$
$564$	0	0
$565$	13.4979	0.567862
$566$	2.98646	0.125530
$567$	0	0
$568$	2.10503	0.0883250
$569$	17.5619	0.736235	0.368117	−	0.929779i	$-0.380002\pi$
$569$	17.5619	0.736235	0.368117	+	0.929779i	$0.380002\pi$
$570$	0	0
$571$	−5.56698	−0.232971	−0.116485	−	0.993192i	$-0.537163\pi$
$571$	−5.56698	−0.232971	−0.116485	+	0.993192i	$0.537163\pi$
$572$	26.1850	1.09485
$573$	0	0
$574$	−7.12570	−0.297421
$575$	−0.669990	−0.0279405
$576$	0	0
$577$	14.3262	0.596407	0.298203	−	0.954502i	$-0.403613\pi$
$577$	14.3262	0.596407	0.298203	+	0.954502i	$0.403613\pi$
$578$	1.26659	0.0526833
$579$	0	0
$580$	−4.10754	−0.170556
$581$	−21.3845	−0.887178
$582$	0	0
$583$	11.4040	0.472303
$584$	0.982100	0.0406396
$585$	0	0
$586$	1.26963	0.0524478
$587$	−41.9005	−1.72942	−0.864709	−	0.502272i	$-0.832497\pi$
$587$	−41.9005	−1.72942	−0.864709	+	0.502272i	$0.832497\pi$
$588$	0	0
$589$	7.07565	0.291547
$590$	−3.16728	−0.130395
$591$	0	0
$592$	20.5507	0.844628
$593$	42.7475	1.75543	0.877714	−	0.479185i	$-0.159068\pi$
$593$	42.7475	1.75543	0.877714	+	0.479185i	$0.159068\pi$
$594$	0	0
$595$	23.1022	0.947098
$596$	5.37320	0.220095
$597$	0	0
$598$	−0.566593	−0.0231697
$599$	−5.56292	−0.227295	−0.113647	−	0.993521i	$-0.536253\pi$
$599$	−5.56292	−0.227295	−0.113647	+	0.993521i	$0.536253\pi$
$600$	0	0
$601$	11.5568	0.471413	0.235706	−	0.971824i	$-0.424260\pi$
$601$	11.5568	0.471413	0.235706	+	0.971824i	$0.424260\pi$
$602$	−2.18371	−0.0890012
$603$	0	0
$604$	28.3666	1.15422
$605$	−6.81688	−0.277146
$606$	0	0
$607$	−1.34211	−0.0544745	−0.0272372	−	0.999629i	$-0.508671\pi$
$607$	−1.34211	−0.0544745	−0.0272372	+	0.999629i	$0.508671\pi$
$608$	3.75971	0.152476
$609$	0	0
$610$	−3.64779	−0.147695
$611$	0.484733	0.0196102
$612$	0	0
$613$	−33.3794	−1.34818	−0.674090	−	0.738649i	$-0.735464\pi$
$613$	−33.3794	−1.34818	−0.674090	+	0.738649i	$0.735464\pi$
$614$	0.245844	0.00992147
$615$	0	0
$616$	−8.89292	−0.358306
$617$	35.6912	1.43687	0.718436	−	0.695593i	$-0.244858\pi$
$617$	35.6912	1.43687	0.718436	+	0.695593i	$0.244858\pi$
$618$	0	0
$619$	−37.9577	−1.52565	−0.762825	−	0.646605i	$-0.776188\pi$
$619$	−37.9577	−1.52565	−0.762825	+	0.646605i	$0.776188\pi$
$620$	−14.7114	−0.590823
$621$	0	0
$622$	2.51855	0.100985
$623$	−22.2538	−0.891579
$624$	0	0
$625$	−21.2012	−0.848046
$626$	1.36007	0.0543592
$627$	0	0
$628$	18.7217	0.747078
$629$	−16.1713	−0.644794
$630$	0	0
$631$	33.9249	1.35053	0.675264	−	0.737576i	$-0.264030\pi$
$631$	33.9249	1.35053	0.675264	+	0.737576i	$0.264030\pi$
$632$	4.69416	0.186724
$633$	0	0
$634$	−1.33680	−0.0530912
$635$	−17.9390	−0.711888
$636$	0	0
$637$	22.7099	0.899797
$638$	−0.609764	−0.0241408
$639$	0	0
$640$	−10.3990	−0.411058
$641$	33.6357	1.32853	0.664266	−	0.747496i	$-0.268744\pi$
$641$	33.6357	1.32853	0.664266	+	0.747496i	$0.268744\pi$
$642$	0	0
$643$	−9.50885	−0.374992	−0.187496	−	0.982265i	$-0.560037\pi$
$643$	−9.50885	−0.374992	−0.187496	+	0.982265i	$0.560037\pi$
$644$	−7.24430	−0.285466
$645$	0	0
$646$	−0.964505	−0.0379479
$647$	−32.0048	−1.25824	−0.629120	−	0.777309i	$-0.716584\pi$
$647$	−32.0048	−1.25824	−0.629120	+	0.777309i	$0.716584\pi$
$648$	0	0
$649$	35.6358	1.39883
$650$	0.379612	0.0148896
$651$	0	0
$652$	−40.6927	−1.59365
$653$	37.8578	1.48149	0.740745	−	0.671786i	$-0.234473\pi$
$653$	37.8578	1.48149	0.740745	+	0.671786i	$0.234473\pi$
$654$	0	0
$655$	−9.52843	−0.372307
$656$	46.2529	1.80587
$657$	0	0
$658$	−0.0817727	−0.00318783
$659$	−19.1091	−0.744383	−0.372192	−	0.928156i	$-0.621394\pi$
$659$	−19.1091	−0.744383	−0.372192	+	0.928156i	$0.621394\pi$
$660$	0	0
$661$	−42.2818	−1.64457	−0.822286	−	0.569074i	$-0.807302\pi$
$661$	−42.2818	−1.64457	−0.822286	+	0.569074i	$0.807302\pi$
$662$	3.42242	0.133016
$663$	0	0
$664$	−3.73699	−0.145023
$665$	15.0869	0.585044
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	26.3377	1.01904
$669$	0	0
$670$	1.53558	0.0593246
$671$	41.0420	1.58441
$672$	0	0
$673$	31.7136	1.22247	0.611234	−	0.791450i	$-0.290673\pi$
$673$	31.7136	1.22247	0.611234	+	0.791450i	$0.290673\pi$
$674$	2.23156	0.0859565
$675$	0	0
$676$	1.33032	0.0511660
$677$	6.06261	0.233005	0.116502	−	0.993190i	$-0.462832\pi$
$677$	6.06261	0.233005	0.116502	+	0.993190i	$0.462832\pi$
$678$	0	0
$679$	−49.7298	−1.90845
$680$	4.03716	0.154818
$681$	0	0
$682$	−2.18390	−0.0836259
$683$	−25.3602	−0.970379	−0.485190	−	0.874409i	$-0.661249\pi$
$683$	−25.3602	−0.970379	−0.485190	+	0.874409i	$0.661249\pi$
$684$	0	0
$685$	−0.760847	−0.0290705
$686$	0.314803	0.0120192
$687$	0	0
$688$	14.1744	0.540395
$689$	−10.5966	−0.403697
$690$	0	0
$691$	−32.7346	−1.24528	−0.622642	−	0.782507i	$-0.713941\pi$
$691$	−32.7346	−1.24528	−0.622642	+	0.782507i	$0.713941\pi$
$692$	18.9619	0.720823
$693$	0	0
$694$	5.40742	0.205263
$695$	25.4247	0.964413
$696$	0	0
$697$	−36.3965	−1.37861
$698$	3.38020	0.127942
$699$	0	0
$700$	4.85361	0.183449
$701$	2.21763	0.0837587	0.0418793	−	0.999123i	$-0.486665\pi$
$701$	2.21763	0.0837587	0.0418793	+	0.999123i	$0.486665\pi$
$702$	0	0
$703$	−10.5607	−0.398304
$704$	27.8907	1.05117
$705$	0	0
$706$	5.43590	0.204583
$707$	19.8057	0.744871
$708$	0	0
$709$	−21.0149	−0.789232	−0.394616	−	0.918846i	$-0.629122\pi$
$709$	−21.0149	−0.789232	−0.394616	+	0.918846i	$0.629122\pi$
$710$	1.10225	0.0413666
$711$	0	0
$712$	−3.88890	−0.145743
$713$	−3.58155	−0.134130
$714$	0	0
$715$	27.6032	1.03230
$716$	20.3744	0.761427
$717$	0	0
$718$	−0.215016	−0.00802432
$719$	−29.7877	−1.11089	−0.555447	−	0.831552i	$-0.687453\pi$
$719$	−29.7877	−1.11089	−0.555447	+	0.831552i	$0.687453\pi$
$720$	0	0
$721$	26.7512	0.996268
$722$	2.43642	0.0906741
$723$	0	0
$724$	−18.4444	−0.685482
$725$	0.669990	0.0248828
$726$	0	0
$727$	52.4598	1.94563	0.972813	−	0.231590i	$-0.0743928\pi$
$727$	52.4598	1.94563	0.972813	+	0.231590i	$0.0743928\pi$
$728$	8.26330	0.306258
$729$	0	0
$730$	0.514253	0.0190334
$731$	−11.1539	−0.412541
$732$	0	0
$733$	43.1979	1.59555	0.797776	−	0.602954i	$-0.206010\pi$
$733$	43.1979	1.59555	0.797776	+	0.602954i	$0.206010\pi$
$734$	−2.26219	−0.0834991
$735$	0	0
$736$	−1.90309	−0.0701487
$737$	−17.2771	−0.636412
$738$	0	0
$739$	−7.29686	−0.268419	−0.134210	−	0.990953i	$-0.542850\pi$
$739$	−7.29686	−0.268419	−0.134210	+	0.990953i	$0.542850\pi$
$740$	21.9573	0.807165
$741$	0	0
$742$	1.78760	0.0656249
$743$	30.7316	1.12743	0.563716	−	0.825969i	$-0.309371\pi$
$743$	30.7316	1.12743	0.563716	+	0.825969i	$0.309371\pi$
$744$	0	0
$745$	5.66422	0.207521
$746$	2.82493	0.103428
$747$	0	0
$748$	−22.5626	−0.824972
$749$	21.7602	0.795099
$750$	0	0
$751$	−34.1301	−1.24543	−0.622713	−	0.782450i	$-0.713970\pi$
$751$	−34.1301	−1.24543	−0.622713	+	0.782450i	$0.713970\pi$
$752$	0.530787	0.0193558
$753$	0	0
$754$	0.566593	0.0206341
$755$	29.9029	1.08828
$756$	0	0
$757$	−15.6198	−0.567710	−0.283855	−	0.958867i	$-0.591613\pi$
$757$	−15.6198	−0.567710	−0.283855	+	0.958867i	$0.591613\pi$
$758$	4.19651	0.152424
$759$	0	0
$760$	2.63646	0.0956346
$761$	9.03111	0.327377	0.163689	−	0.986512i	$-0.447661\pi$
$761$	9.03111	0.327377	0.163689	+	0.986512i	$0.447661\pi$
$762$	0	0
$763$	62.1256	2.24910
$764$	−1.23634	−0.0447292
$765$	0	0
$766$	−3.21026	−0.115991
$767$	−33.1128	−1.19563
$768$	0	0
$769$	23.4779	0.846635	0.423318	−	0.905981i	$-0.360865\pi$
$769$	23.4779	0.846635	0.423318	+	0.905981i	$0.360865\pi$
$770$	−4.65657	−0.167811
$771$	0	0
$772$	1.02049	0.0367282
$773$	21.4602	0.771871	0.385935	−	0.922526i	$-0.373879\pi$
$773$	21.4602	0.771871	0.385935	+	0.922526i	$0.373879\pi$
$774$	0	0
$775$	2.39960	0.0861963
$776$	−8.69039	−0.311967
$777$	0	0
$778$	−1.11778	−0.0400743
$779$	−23.7687	−0.851601
$780$	0	0
$781$	−12.4016	−0.443765
$782$	0.488212	0.0174584
$783$	0	0
$784$	24.8675	0.888125
$785$	19.7357	0.704398
$786$	0	0
$787$	28.7283	1.02405	0.512026	−	0.858970i	$-0.328895\pi$
$787$	28.7283	1.02405	0.512026	+	0.858970i	$0.328895\pi$
$788$	−0.118063	−0.00420583
$789$	0	0
$790$	2.45799	0.0874513
$791$	−23.8057	−0.846434
$792$	0	0
$793$	−38.1363	−1.35426
$794$	3.74421	0.132877
$795$	0	0
$796$	10.9521	0.388186
$797$	21.6103	0.765475	0.382737	−	0.923857i	$-0.374981\pi$
$797$	21.6103	0.765475	0.382737	+	0.923857i	$0.374981\pi$
$798$	0	0
$799$	−0.417676	−0.0147763
$800$	1.27505	0.0450798
$801$	0	0
$802$	2.09666	0.0740355
$803$	−5.78598	−0.204183
$804$	0	0
$805$	−7.63666	−0.269157
$806$	2.02928	0.0714784
$807$	0	0
$808$	3.46110	0.121761
$809$	38.6195	1.35779	0.678895	−	0.734235i	$-0.262459\pi$
$809$	38.6195	1.35779	0.678895	+	0.734235i	$0.262459\pi$
$810$	0	0
$811$	25.4982	0.895363	0.447681	−	0.894193i	$-0.352250\pi$
$811$	25.4982	0.895363	0.447681	+	0.894193i	$0.352250\pi$
$812$	7.24430	0.254225
$813$	0	0
$814$	3.25956	0.114247
$815$	−42.8967	−1.50261
$816$	0	0
$817$	−7.28402	−0.254836
$818$	5.17897	0.181079
$819$	0	0
$820$	49.4187	1.72578
$821$	−27.1559	−0.947747	−0.473874	−	0.880593i	$-0.657145\pi$
$821$	−27.1559	−0.947747	−0.473874	+	0.880593i	$0.657145\pi$
$822$	0	0
$823$	−15.3728	−0.535861	−0.267930	−	0.963438i	$-0.586340\pi$
$823$	−15.3728	−0.535861	−0.267930	+	0.963438i	$0.586340\pi$
$824$	4.67484	0.162856
$825$	0	0
$826$	5.58601	0.194362
$827$	4.88039	0.169708	0.0848539	−	0.996393i	$-0.472958\pi$
$827$	4.88039	0.169708	0.0848539	+	0.996393i	$0.472958\pi$
$828$	0	0
$829$	27.6304	0.959645	0.479823	−	0.877366i	$-0.340701\pi$
$829$	27.6304	0.959645	0.479823	+	0.877366i	$0.340701\pi$
$830$	−1.95679	−0.0679210
$831$	0	0
$832$	−25.9160	−0.898476
$833$	−19.5682	−0.678000
$834$	0	0
$835$	27.7642	0.960820
$836$	−14.7345	−0.509604
$837$	0	0
$838$	4.51985	0.156136
$839$	−18.3181	−0.632411	−0.316206	−	0.948691i	$-0.602409\pi$
$839$	−18.3181	−0.632411	−0.316206	+	0.948691i	$0.602409\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	3.83607	0.132200
$843$	0	0
$844$	19.9232	0.685784
$845$	1.40237	0.0482429
$846$	0	0
$847$	12.0227	0.413104
$848$	−11.6033	−0.398460
$849$	0	0
$850$	−0.327098	−0.0112194
$851$	5.34560	0.183245
$852$	0	0
$853$	−38.3460	−1.31294	−0.656472	−	0.754351i	$-0.727952\pi$
$853$	−38.3460	−1.31294	−0.656472	+	0.754351i	$0.727952\pi$
$854$	6.43346	0.220148
$855$	0	0
$856$	3.80264	0.129972
$857$	−6.81213	−0.232698	−0.116349	−	0.993208i	$-0.537119\pi$
$857$	−6.81213	−0.232698	−0.116349	+	0.993208i	$0.537119\pi$
$858$	0	0
$859$	−18.6302	−0.635656	−0.317828	−	0.948148i	$-0.602953\pi$
$859$	−18.6302	−0.635656	−0.317828	+	0.948148i	$0.602953\pi$
$860$	15.1446	0.516426
$861$	0	0
$862$	0.587076	0.0199959
$863$	15.8372	0.539105	0.269552	−	0.962986i	$-0.413124\pi$
$863$	15.8372	0.539105	0.269552	+	0.962986i	$0.413124\pi$
$864$	0	0
$865$	19.9889	0.679643
$866$	−6.33189	−0.215166
$867$	0	0
$868$	25.9458	0.880659
$869$	−27.6554	−0.938144
$870$	0	0
$871$	16.0539	0.543966
$872$	10.8566	0.367651
$873$	0	0
$874$	0.318827	0.0107845
$875$	43.2998	1.46380
$876$	0	0
$877$	−45.1404	−1.52428	−0.762141	−	0.647411i	$-0.775852\pi$
$877$	−45.1404	−1.52428	−0.762141	+	0.647411i	$0.775852\pi$
$878$	−4.55162	−0.153610
$879$	0	0
$880$	30.2258	1.01891
$881$	54.7405	1.84425	0.922127	−	0.386887i	$-0.126450\pi$
$881$	54.7405	1.84425	0.922127	+	0.386887i	$0.126450\pi$
$882$	0	0
$883$	−47.5964	−1.60175	−0.800874	−	0.598833i	$-0.795631\pi$
$883$	−47.5964	−1.60175	−0.800874	+	0.598833i	$0.795631\pi$
$884$	20.9652	0.705137
$885$	0	0
$886$	−3.92504	−0.131864
$887$	−3.05499	−0.102577	−0.0512883	−	0.998684i	$-0.516333\pi$
$887$	−3.05499	−0.102577	−0.0512883	+	0.998684i	$0.516333\pi$
$888$	0	0
$889$	31.6383	1.06111
$890$	−2.03633	−0.0682580
$891$	0	0
$892$	5.23692	0.175345
$893$	−0.272763	−0.00912767
$894$	0	0
$895$	21.4779	0.717927
$896$	18.3404	0.612709
$897$	0	0
$898$	6.61474	0.220737
$899$	3.58155	0.119451
$900$	0	0
$901$	9.13066	0.304186
$902$	7.33620	0.244269
$903$	0	0
$904$	−4.16011	−0.138363
$905$	−19.4434	−0.646321
$906$	0	0
$907$	−24.3521	−0.808599	−0.404300	−	0.914627i	$-0.632485\pi$
$907$	−24.3521	−0.808599	−0.404300	+	0.914627i	$0.632485\pi$
$908$	−3.99749	−0.132661
$909$	0	0
$910$	4.32688	0.143435
$911$	−8.64600	−0.286455	−0.143227	−	0.989690i	$-0.545748\pi$
$911$	−8.64600	−0.286455	−0.143227	+	0.989690i	$0.545748\pi$
$912$	0	0
$913$	22.0162	0.728630
$914$	−1.58308	−0.0523637
$915$	0	0
$916$	3.43999	0.113660
$917$	16.8049	0.554947
$918$	0	0
$919$	38.3295	1.26437	0.632187	−	0.774816i	$-0.282157\pi$
$919$	38.3295	1.26437	0.632187	+	0.774816i	$0.282157\pi$
$920$	−1.33452	−0.0439980
$921$	0	0
$922$	−3.97351	−0.130860
$923$	11.5236	0.379304
$924$	0	0
$925$	−3.58150	−0.117759
$926$	−1.63201	−0.0536311
$927$	0	0
$928$	1.90309	0.0624719
$929$	−34.5501	−1.13355	−0.566776	−	0.823872i	$-0.691809\pi$
$929$	−34.5501	−1.13355	−0.566776	+	0.823872i	$0.691809\pi$
$930$	0	0
$931$	−12.7790	−0.418816
$932$	21.3570	0.699572
$933$	0	0
$934$	0.0354916	0.00116132
$935$	−23.7847	−0.777842
$936$	0	0
$937$	31.5802	1.03168	0.515839	−	0.856685i	$-0.327480\pi$
$937$	31.5802	1.03168	0.515839	+	0.856685i	$0.327480\pi$
$938$	−2.70824	−0.0884272
$939$	0	0
$940$	0.567116	0.0184973
$941$	17.7672	0.579195	0.289597	−	0.957149i	$-0.406479\pi$
$941$	17.7672	0.579195	0.289597	+	0.957149i	$0.406479\pi$
$942$	0	0
$943$	12.0312	0.391790
$944$	−36.2588	−1.18012
$945$	0	0
$946$	2.24821	0.0730958
$947$	0.409000	0.0132907	0.00664535	−	0.999978i	$-0.497885\pi$
$947$	0.409000	0.0132907	0.00664535	+	0.999978i	$0.497885\pi$
$948$	0	0
$949$	5.37633	0.174523
$950$	−0.213611	−0.00693045
$951$	0	0
$952$	−7.12018	−0.230766
$953$	15.6870	0.508151	0.254076	−	0.967184i	$-0.418229\pi$
$953$	15.6870	0.508151	0.254076	+	0.967184i	$0.418229\pi$
$954$	0	0
$955$	−1.30330	−0.0421738
$956$	26.7500	0.865157
$957$	0	0
$958$	−2.73728	−0.0884376
$959$	1.34187	0.0433314
$960$	0	0
$961$	−18.1725	−0.586209
$962$	−3.02878	−0.0976518
$963$	0	0
$964$	33.7254	1.08622
$965$	1.07576	0.0346299
$966$	0	0
$967$	48.0986	1.54675	0.773373	−	0.633951i	$-0.218568\pi$
$967$	48.0986	1.54675	0.773373	+	0.633951i	$0.218568\pi$
$968$	2.10099	0.0675283
$969$	0	0
$970$	−4.55052	−0.146108
$971$	24.7927	0.795636	0.397818	−	0.917464i	$-0.369768\pi$
$971$	24.7927	0.795636	0.397818	+	0.917464i	$0.369768\pi$
$972$	0	0
$973$	−44.8405	−1.43752
$974$	−4.52730	−0.145064
$975$	0	0
$976$	−41.7595	−1.33669
$977$	26.6260	0.851841	0.425921	−	0.904761i	$-0.359950\pi$
$977$	26.6260	0.851841	0.425921	+	0.904761i	$0.359950\pi$
$978$	0	0
$979$	22.9112	0.732245
$980$	26.5695	0.848733
$981$	0	0
$982$	6.12644	0.195503
$983$	−6.74470	−0.215123	−0.107561	−	0.994198i	$-0.534304\pi$
$983$	−6.74470	−0.215123	−0.107561	+	0.994198i	$0.534304\pi$
$984$	0	0
$985$	−0.124458	−0.00396555
$986$	−0.488212	−0.0155479
$987$	0	0
$988$	13.6913	0.435579
$989$	3.68702	0.117240
$990$	0	0
$991$	40.4589	1.28522	0.642610	−	0.766193i	$-0.277851\pi$
$991$	40.4589	1.28522	0.642610	+	0.766193i	$0.277851\pi$
$992$	6.81600	0.216408
$993$	0	0
$994$	−1.94399	−0.0616596
$995$	11.5453	0.366010
$996$	0	0
$997$	−29.3174	−0.928491	−0.464246	−	0.885706i	$-0.653675\pi$
$997$	−29.3174	−0.928491	−0.464246	+	0.885706i	$0.653675\pi$
$998$	−0.812889	−0.0257316
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.k.1.7		10
3.2	odd	2		2001.2.a.k.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.k.1.4	✓	10	3.2	odd	2
6003.2.a.k.1.7		10	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)

\(f(q)\)	\(=\)	\(q-0.161383 q^{2} -1.97396 q^{4} -2.08087 q^{5} +3.66994 q^{7} +0.641331 q^{8} +O(q^{10})\)	\(q-0.161383 q^{2} -1.97396 q^{4} -2.08087 q^{5} +3.66994 q^{7} +0.641331 q^{8} +0.335818 q^{10} -3.77836 q^{11} +3.51085 q^{13} -0.592268 q^{14} +3.84441 q^{16} -3.02517 q^{17} -1.97558 q^{19} +4.10754 q^{20} +0.609764 q^{22} +1.00000 q^{23} -0.669990 q^{25} -0.566593 q^{26} -7.24430 q^{28} -1.00000 q^{29} -3.58155 q^{31} -1.90309 q^{32} +0.488212 q^{34} -7.63666 q^{35} +5.34560 q^{37} +0.318827 q^{38} -1.33452 q^{40} +12.0312 q^{41} +3.68702 q^{43} +7.45831 q^{44} -0.161383 q^{46} +0.138067 q^{47} +6.46848 q^{49} +0.108125 q^{50} -6.93026 q^{52} -3.01823 q^{53} +7.86226 q^{55} +2.35365 q^{56} +0.161383 q^{58} -9.43156 q^{59} -10.8624 q^{61} +0.578003 q^{62} -7.38169 q^{64} -7.30561 q^{65} +4.57266 q^{67} +5.97155 q^{68} +1.23243 q^{70} +3.28228 q^{71} +1.53135 q^{73} -0.862691 q^{74} +3.89971 q^{76} -13.8664 q^{77} +7.31941 q^{79} -7.99971 q^{80} -1.94164 q^{82} -5.82693 q^{83} +6.29498 q^{85} -0.595024 q^{86} -2.42318 q^{88} -6.06380 q^{89} +12.8846 q^{91} -1.97396 q^{92} -0.0222817 q^{94} +4.11093 q^{95} -13.5506 q^{97} -1.04391 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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