LMFDB - Embedded newform 6003.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} + \cdots - 5 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 54x^8 - 188x^7 - 77x^6 + 342x^5 + 13x^4 - 215x^3 + 9x^2 + 37*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.724122$ 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)

$f(q)$	$=$	$q-0.724122 q^{2} -1.47565 q^{4} +1.17886 q^{5} -4.09140 q^{7} +2.51679 q^{8} +O(q^{10})$	\(q-0.724122 q^{2} -1.47565 q^{4} +1.17886 q^{5} -4.09140 q^{7} +2.51679 q^{8} -0.853636 q^{10} +3.08591 q^{11} +2.42919 q^{13} +2.96267 q^{14} +1.12883 q^{16} -0.237174 q^{17} -2.02456 q^{19} -1.73958 q^{20} -2.23457 q^{22} +1.00000 q^{23} -3.61030 q^{25} -1.75903 q^{26} +6.03746 q^{28} +1.00000 q^{29} +0.00576952 q^{31} -5.85100 q^{32} +0.171743 q^{34} -4.82317 q^{35} +1.67676 q^{37} +1.46603 q^{38} +2.96694 q^{40} +7.50127 q^{41} +6.04004 q^{43} -4.55372 q^{44} -0.724122 q^{46} -9.56040 q^{47} +9.73954 q^{49} +2.61430 q^{50} -3.58463 q^{52} +10.8627 q^{53} +3.63784 q^{55} -10.2972 q^{56} -0.724122 q^{58} -11.1467 q^{59} -13.5738 q^{61} -0.00417784 q^{62} +1.97917 q^{64} +2.86367 q^{65} +0.167888 q^{67} +0.349985 q^{68} +3.49256 q^{70} +6.07713 q^{71} +14.9046 q^{73} -1.21418 q^{74} +2.98754 q^{76} -12.6257 q^{77} +5.39171 q^{79} +1.33073 q^{80} -5.43183 q^{82} -4.61525 q^{83} -0.279594 q^{85} -4.37373 q^{86} +7.76659 q^{88} +2.09933 q^{89} -9.93880 q^{91} -1.47565 q^{92} +6.92289 q^{94} -2.38667 q^{95} -7.70879 q^{97} -7.05262 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8 + 6 * q^11 - 15 * q^13 + 8 * q^14 + 17 * q^16 + 18 * q^17 - 6 * q^19 + 39 * q^20 - 5 * q^22 + 12 * q^23 + 14 * q^25 + 3 * q^26 - 19 * q^28 + 12 * q^29 + 16 * q^31 + 21 * q^32 - 7 * q^34 + 11 * q^35 - q^37 + 24 * q^38 + 30 * q^40 - 3 * q^41 - 23 * q^43 - 23 * q^44 + 3 * q^46 + 35 * q^47 + 3 * q^49 + 2 * q^50 + 45 * q^53 + 17 * q^55 + 17 * q^56 + 3 * q^58 + 11 * q^59 + 4 * q^61 + 7 * q^62 + 15 * q^64 - 5 * q^65 - 19 * q^67 - q^68 + 14 * q^70 - 19 * q^71 + 10 * q^73 + 15 * q^74 - 4 * q^76 + 39 * q^77 + 17 * q^79 + 90 * q^80 - 3 * q^82 + 12 * q^83 + 14 * q^85 - 17 * q^86 - 2 * q^88 + 20 * q^89 + 11 * q^91 + 11 * q^92 + 13 * q^94 - 12 * q^95 - 12 * q^97 - 75 * 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 $

$2$	−0.724122	−0.512031	−0.256016	−	0.966673i	$-0.582410\pi$
$2$	−0.724122	−0.512031	−0.256016	+	0.966673i	$0.582410\pi$
$3$	0	0
$3$	0	0
$4$	−1.47565	−0.737824
$5$	1.17886	0.527201	0.263600	−	0.964632i	$-0.415090\pi$
$5$	1.17886	0.527201	0.263600	+	0.964632i	$0.415090\pi$
$6$	0	0
$7$	−4.09140	−1.54640	−0.773202	−	0.634160i	$-0.781346\pi$
$7$	−4.09140	−1.54640	−0.773202	+	0.634160i	$0.781346\pi$
$8$	2.51679	0.889820
$9$	0	0
$10$	−0.853636	−0.269943
$11$	3.08591	0.930437	0.465218	−	0.885196i	$-0.345976\pi$
$11$	3.08591	0.930437	0.465218	+	0.885196i	$0.345976\pi$
$12$	0	0
$13$	2.42919	0.673737	0.336868	−	0.941552i	$-0.390632\pi$
$13$	2.42919	0.673737	0.336868	+	0.941552i	$0.390632\pi$
$14$	2.96267	0.791807
$15$	0	0
$16$	1.12883	0.282208
$17$	−0.237174	−0.0575230	−0.0287615	−	0.999586i	$-0.509156\pi$
$17$	−0.237174	−0.0575230	−0.0287615	+	0.999586i	$0.509156\pi$
$18$	0	0
$19$	−2.02456	−0.464466	−0.232233	−	0.972660i	$-0.574603\pi$
$19$	−2.02456	−0.464466	−0.232233	+	0.972660i	$0.574603\pi$
$20$	−1.73958	−0.388981
$21$	0	0
$22$	−2.23457	−0.476413
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.61030	−0.722060
$26$	−1.75903	−0.344974
$27$	0	0
$28$	6.03746	1.14097
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.00576952	0.00103624	0.000518118	−	1.00000i	$-0.499835\pi$
$31$	0.00576952	0.00103624	0.000518118		1.00000i	$0.499835\pi$
$32$	−5.85100	−1.03432
$33$	0	0
$34$	0.171743	0.0294536
$35$	−4.82317	−0.815265
$36$	0	0
$37$	1.67676	0.275658	0.137829	−	0.990456i	$-0.455988\pi$
$37$	1.67676	0.275658	0.137829	+	0.990456i	$0.455988\pi$
$38$	1.46603	0.237821
$39$	0	0
$40$	2.96694	0.469114
$41$	7.50127	1.17150	0.585751	−	0.810491i	$-0.300800\pi$
$41$	7.50127	1.17150	0.585751	+	0.810491i	$0.300800\pi$
$42$	0	0
$43$	6.04004	0.921098	0.460549	−	0.887634i	$-0.347653\pi$
$43$	6.04004	0.921098	0.460549	+	0.887634i	$0.347653\pi$
$44$	−4.55372	−0.686498
$45$	0	0
$46$	−0.724122	−0.106766
$47$	−9.56040	−1.39453	−0.697264	−	0.716815i	$-0.745599\pi$
$47$	−9.56040	−1.39453	−0.697264	+	0.716815i	$0.745599\pi$
$48$	0	0
$49$	9.73954	1.39136
$50$	2.61430	0.369717
$51$	0	0
$52$	−3.58463	−0.497099
$53$	10.8627	1.49210	0.746051	−	0.665889i	$-0.231948\pi$
$53$	10.8627	1.49210	0.746051	+	0.665889i	$0.231948\pi$
$54$	0	0
$55$	3.63784	0.490527
$56$	−10.2972	−1.37602
$57$	0	0
$58$	−0.724122	−0.0950819
$59$	−11.1467	−1.45117	−0.725587	−	0.688130i	$-0.758432\pi$
$59$	−11.1467	−1.45117	−0.725587	+	0.688130i	$0.758432\pi$
$60$	0	0
$61$	−13.5738	−1.73795	−0.868976	−	0.494854i	$-0.835221\pi$
$61$	−13.5738	−1.73795	−0.868976	+	0.494854i	$0.835221\pi$
$62$	−0.00417784	−0.000530586 0
$63$	0	0
$64$	1.97917	0.247397
$65$	2.86367	0.355194
$66$	0	0
$67$	0.167888	0.0205108	0.0102554	−	0.999947i	$-0.496736\pi$
$67$	0.167888	0.0205108	0.0102554	+	0.999947i	$0.496736\pi$
$68$	0.349985	0.0424419
$69$	0	0
$70$	3.49256	0.417441
$71$	6.07713	0.721222	0.360611	−	0.932716i	$-0.382568\pi$
$71$	6.07713	0.721222	0.360611	+	0.932716i	$0.382568\pi$
$72$	0	0
$73$	14.9046	1.74445	0.872227	−	0.489101i	$-0.162675\pi$
$73$	14.9046	1.74445	0.872227	+	0.489101i	$0.162675\pi$
$74$	−1.21418	−0.141145
$75$	0	0
$76$	2.98754	0.342694
$77$	−12.6257	−1.43883
$78$	0	0
$79$	5.39171	0.606615	0.303307	−	0.952893i	$-0.401909\pi$
$79$	5.39171	0.606615	0.303307	+	0.952893i	$0.401909\pi$
$80$	1.33073	0.148780
$81$	0	0
$82$	−5.43183	−0.599846
$83$	−4.61525	−0.506590	−0.253295	−	0.967389i	$-0.581514\pi$
$83$	−4.61525	−0.506590	−0.253295	+	0.967389i	$0.581514\pi$
$84$	0	0
$85$	−0.279594	−0.0303262
$86$	−4.37373	−0.471631
$87$	0	0
$88$	7.76659	0.827922
$89$	2.09933	0.222529	0.111264	−	0.993791i	$-0.464510\pi$
$89$	2.09933	0.222529	0.111264	+	0.993791i	$0.464510\pi$
$90$	0	0
$91$	−9.93880	−1.04187
$92$	−1.47565	−0.153847
$93$	0	0
$94$	6.92289	0.714042
$95$	−2.38667	−0.244867
$96$	0	0
$97$	−7.70879	−0.782709	−0.391355	−	0.920240i	$-0.627993\pi$
$97$	−7.70879	−0.782709	−0.391355	+	0.920240i	$0.627993\pi$
$98$	−7.05262	−0.712422
$99$	0	0
$100$	5.32753	0.532753
$101$	6.99073	0.695603	0.347802	−	0.937568i	$-0.386928\pi$
$101$	6.99073	0.695603	0.347802	+	0.937568i	$0.386928\pi$
$102$	0	0
$103$	−16.6495	−1.64052	−0.820262	−	0.571988i	$-0.806172\pi$
$103$	−16.6495	−1.64052	−0.820262	+	0.571988i	$0.806172\pi$
$104$	6.11377	0.599505
$105$	0	0
$106$	−7.86589	−0.764003
$107$	4.79679	0.463723	0.231862	−	0.972749i	$-0.425518\pi$
$107$	4.79679	0.463723	0.231862	+	0.972749i	$0.425518\pi$
$108$	0	0
$109$	−6.13707	−0.587825	−0.293912	−	0.955832i	$-0.594957\pi$
$109$	−6.13707	−0.587825	−0.293912	+	0.955832i	$0.594957\pi$
$110$	−2.63424	−0.251165
$111$	0	0
$112$	−4.61850	−0.436407
$113$	−2.49867	−0.235055	−0.117527	−	0.993070i	$-0.537497\pi$
$113$	−2.49867	−0.235055	−0.117527	+	0.993070i	$0.537497\pi$
$114$	0	0
$115$	1.17886	0.109929
$116$	−1.47565	−0.137010
$117$	0	0
$118$	8.07155	0.743047
$119$	0.970372	0.0889538
$120$	0	0
$121$	−1.47716	−0.134287
$122$	9.82911	0.889886
$123$	0	0
$124$	−0.00851378	−0.000764560 0
$125$	−10.1503	−0.907871
$126$	0	0
$127$	−10.8385	−0.961760	−0.480880	−	0.876786i	$-0.659683\pi$
$127$	−10.8385	−0.961760	−0.480880	+	0.876786i	$0.659683\pi$
$128$	10.2688	0.907645
$129$	0	0
$130$	−2.07365	−0.181871
$131$	−7.42926	−0.649097	−0.324549	−	0.945869i	$-0.605212\pi$
$131$	−7.42926	−0.649097	−0.324549	+	0.945869i	$0.605212\pi$
$132$	0	0
$133$	8.28329	0.718252
$134$	−0.121571	−0.0105022
$135$	0	0
$136$	−0.596917	−0.0511852
$137$	7.37902	0.630433	0.315216	−	0.949020i	$-0.397923\pi$
$137$	7.37902	0.630433	0.315216	+	0.949020i	$0.397923\pi$
$138$	0	0
$139$	12.8095	1.08649	0.543245	−	0.839574i	$-0.317195\pi$
$139$	12.8095	1.08649	0.543245	+	0.839574i	$0.317195\pi$
$140$	7.11730	0.601522
$141$	0	0
$142$	−4.40058	−0.369288
$143$	7.49627	0.626870
$144$	0	0
$145$	1.17886	0.0978987
$146$	−10.7928	−0.893215
$147$	0	0
$148$	−2.47431	−0.203387
$149$	9.58288	0.785060	0.392530	−	0.919739i	$-0.371600\pi$
$149$	9.58288	0.785060	0.392530	+	0.919739i	$0.371600\pi$
$150$	0	0
$151$	−12.3299	−1.00339	−0.501695	−	0.865045i	$-0.667290\pi$
$151$	−12.3299	−1.00339	−0.501695	+	0.865045i	$0.667290\pi$
$152$	−5.09540	−0.413292
$153$	0	0
$154$	9.14254	0.736727
$155$	0.00680144	0.000546305 0
$156$	0	0
$157$	−2.50048	−0.199560	−0.0997800	−	0.995010i	$-0.531814\pi$
$157$	−2.50048	−0.199560	−0.0997800	+	0.995010i	$0.531814\pi$
$158$	−3.90426	−0.310606
$159$	0	0
$160$	−6.89748	−0.545294
$161$	−4.09140	−0.322447
$162$	0	0
$163$	−2.50468	−0.196182	−0.0980909	−	0.995177i	$-0.531274\pi$
$163$	−2.50468	−0.196182	−0.0980909	+	0.995177i	$0.531274\pi$
$164$	−11.0692	−0.864362
$165$	0	0
$166$	3.34200	0.259390
$167$	−12.7653	−0.987809	−0.493905	−	0.869516i	$-0.664431\pi$
$167$	−12.7653	−0.987809	−0.493905	+	0.869516i	$0.664431\pi$
$168$	0	0
$169$	−7.09902	−0.546079
$170$	0.202460	0.0155280
$171$	0	0
$172$	−8.91298	−0.679608
$173$	18.4750	1.40462	0.702312	−	0.711869i	$-0.252151\pi$
$173$	18.4750	1.40462	0.702312	+	0.711869i	$0.252151\pi$
$174$	0	0
$175$	14.7712	1.11660
$176$	3.48347	0.262576
$177$	0	0
$178$	−1.52017	−0.113942
$179$	−7.18991	−0.537399	−0.268700	−	0.963224i	$-0.586594\pi$
$179$	−7.18991	−0.537399	−0.268700	+	0.963224i	$0.586594\pi$
$180$	0	0
$181$	13.9824	1.03930	0.519651	−	0.854379i	$-0.326062\pi$
$181$	13.9824	1.03930	0.519651	+	0.854379i	$0.326062\pi$
$182$	7.19690	0.533470
$183$	0	0
$184$	2.51679	0.185540
$185$	1.97666	0.145327
$186$	0	0
$187$	−0.731896	−0.0535216
$188$	14.1078	1.02892
$189$	0	0
$190$	1.72824	0.125380
$191$	12.2623	0.887270	0.443635	−	0.896208i	$-0.353689\pi$
$191$	12.2623	0.887270	0.443635	+	0.896208i	$0.353689\pi$
$192$	0	0
$193$	−0.989143	−0.0712001	−0.0356001	−	0.999366i	$-0.511334\pi$
$193$	−0.989143	−0.0712001	−0.0356001	+	0.999366i	$0.511334\pi$
$194$	5.58210	0.400772
$195$	0	0
$196$	−14.3721	−1.02658
$197$	20.0897	1.43133	0.715664	−	0.698444i	$-0.246124\pi$
$197$	20.0897	1.43133	0.715664	+	0.698444i	$0.246124\pi$
$198$	0	0
$199$	13.1054	0.929015	0.464508	−	0.885569i	$-0.346231\pi$
$199$	13.1054	0.929015	0.464508	+	0.885569i	$0.346231\pi$
$200$	−9.08637	−0.642503
$201$	0	0
$202$	−5.06214	−0.356171
$203$	−4.09140	−0.287160
$204$	0	0
$205$	8.84292	0.617616
$206$	12.0563	0.840000
$207$	0	0
$208$	2.74215	0.190134
$209$	−6.24761	−0.432156
$210$	0	0
$211$	15.3424	1.05621	0.528106	−	0.849179i	$-0.322902\pi$
$211$	15.3424	1.05621	0.528106	+	0.849179i	$0.322902\pi$
$212$	−16.0295	−1.10091
$213$	0	0
$214$	−3.47346	−0.237441
$215$	7.12034	0.485603
$216$	0	0
$217$	−0.0236054	−0.00160244
$218$	4.44399	0.300985
$219$	0	0
$220$	−5.36818	−0.361922
$221$	−0.576140	−0.0387554
$222$	0	0
$223$	23.9433	1.60336	0.801679	−	0.597754i	$-0.203940\pi$
$223$	23.9433	1.60336	0.801679	+	0.597754i	$0.203940\pi$
$224$	23.9388	1.59948
$225$	0	0
$226$	1.80934	0.120355
$227$	8.81659	0.585178	0.292589	−	0.956238i	$-0.405483\pi$
$227$	8.81659	0.585178	0.292589	+	0.956238i	$0.405483\pi$
$228$	0	0
$229$	14.8752	0.982981	0.491491	−	0.870883i	$-0.336452\pi$
$229$	14.8752	0.982981	0.491491	+	0.870883i	$0.336452\pi$
$230$	−0.853636	−0.0562871
$231$	0	0
$232$	2.51679	0.165236
$233$	−12.7248	−0.833630	−0.416815	−	0.908991i	$-0.636854\pi$
$233$	−12.7248	−0.833630	−0.416815	+	0.908991i	$0.636854\pi$
$234$	0	0
$235$	−11.2703	−0.735196
$236$	16.4486	1.07071
$237$	0	0
$238$	−0.702667	−0.0455472
$239$	8.33010	0.538829	0.269415	−	0.963024i	$-0.413170\pi$
$239$	8.33010	0.538829	0.269415	+	0.963024i	$0.413170\pi$
$240$	0	0
$241$	11.1245	0.716594	0.358297	−	0.933608i	$-0.383357\pi$
$241$	11.1245	0.716594	0.358297	+	0.933608i	$0.383357\pi$
$242$	1.06964	0.0687594
$243$	0	0
$244$	20.0302	1.28230
$245$	11.4815	0.733528
$246$	0	0
$247$	−4.91805	−0.312928
$248$	0.0145207	0.000922064 0
$249$	0	0
$250$	7.35006	0.464858
$251$	20.0438	1.26516	0.632578	−	0.774497i	$-0.281997\pi$
$251$	20.0438	1.26516	0.632578	+	0.774497i	$0.281997\pi$
$252$	0	0
$253$	3.08591	0.194009
$254$	7.84838	0.492451
$255$	0	0
$256$	−11.3942	−0.712139
$257$	6.94240	0.433055	0.216527	−	0.976277i	$-0.430527\pi$
$257$	6.94240	0.433055	0.216527	+	0.976277i	$0.430527\pi$
$258$	0	0
$259$	−6.86029	−0.426278
$260$	−4.22577	−0.262071
$261$	0	0
$262$	5.37969	0.332358
$263$	13.1040	0.808028	0.404014	−	0.914753i	$-0.367615\pi$
$263$	13.1040	0.808028	0.404014	+	0.914753i	$0.367615\pi$
$264$	0	0
$265$	12.8055	0.786637
$266$	−5.99811	−0.367768
$267$	0	0
$268$	−0.247743	−0.0151333
$269$	26.4529	1.61286	0.806430	−	0.591329i	$-0.201397\pi$
$269$	26.4529	1.61286	0.806430	+	0.591329i	$0.201397\pi$
$270$	0	0
$271$	−13.9142	−0.845229	−0.422614	−	0.906310i	$-0.638888\pi$
$271$	−13.9142	−0.845229	−0.422614	+	0.906310i	$0.638888\pi$
$272$	−0.267729	−0.0162334
$273$	0	0
$274$	−5.34331	−0.322801
$275$	−11.1411	−0.671831
$276$	0	0
$277$	20.4584	1.22923	0.614614	−	0.788828i	$-0.289312\pi$
$277$	20.4584	1.22923	0.614614	+	0.788828i	$0.289312\pi$
$278$	−9.27566	−0.556317
$279$	0	0
$280$	−12.1389	−0.725439
$281$	−18.0185	−1.07489	−0.537446	−	0.843298i	$-0.680611\pi$
$281$	−18.0185	−1.07489	−0.537446	+	0.843298i	$0.680611\pi$
$282$	0	0
$283$	30.6847	1.82402	0.912009	−	0.410169i	$-0.134530\pi$
$283$	30.6847	1.82402	0.912009	+	0.410169i	$0.134530\pi$
$284$	−8.96770	−0.532135
$285$	0	0
$286$	−5.42821	−0.320977
$287$	−30.6907	−1.81161
$288$	0	0
$289$	−16.9437	−0.996691
$290$	−0.853636	−0.0501272
$291$	0	0
$292$	−21.9940	−1.28710
$293$	6.47983	0.378556	0.189278	−	0.981924i	$-0.439385\pi$
$293$	6.47983	0.378556	0.189278	+	0.981924i	$0.439385\pi$
$294$	0	0
$295$	−13.1403	−0.765060
$296$	4.22005	0.245286
$297$	0	0
$298$	−6.93917	−0.401976
$299$	2.42919	0.140484
$300$	0	0
$301$	−24.7122	−1.42439
$302$	8.92832	0.513767
$303$	0	0
$304$	−2.28539	−0.131076
$305$	−16.0016	−0.916249
$306$	0	0
$307$	12.7332	0.726724	0.363362	−	0.931648i	$-0.381629\pi$
$307$	12.7332	0.726724	0.363362	+	0.931648i	$0.381629\pi$
$308$	18.6311	1.06160
$309$	0	0
$310$	−0.00492507	−0.000279725 0
$311$	30.2599	1.71588	0.857941	−	0.513748i	$-0.171743\pi$
$311$	30.2599	1.71588	0.857941	+	0.513748i	$0.171743\pi$
$312$	0	0
$313$	13.0447	0.737332	0.368666	−	0.929562i	$-0.379815\pi$
$313$	13.0447	0.737332	0.368666	+	0.929562i	$0.379815\pi$
$314$	1.81065	0.102181
$315$	0	0
$316$	−7.95627	−0.447575
$317$	−8.93249	−0.501699	−0.250849	−	0.968026i	$-0.580710\pi$
$317$	−8.93249	−0.501699	−0.250849	+	0.968026i	$0.580710\pi$
$318$	0	0
$319$	3.08591	0.172778
$320$	2.33316	0.130428
$321$	0	0
$322$	2.96267	0.165103
$323$	0.480172	0.0267175
$324$	0	0
$325$	−8.77011	−0.486478
$326$	1.81370	0.100451
$327$	0	0
$328$	18.8791	1.04243
$329$	39.1154	2.15650
$330$	0	0
$331$	−19.6340	−1.07918	−0.539590	−	0.841928i	$-0.681421\pi$
$331$	−19.6340	−1.07918	−0.539590	+	0.841928i	$0.681421\pi$
$332$	6.81048	0.373774
$333$	0	0
$334$	9.24364	0.505789
$335$	0.197916	0.0108133
$336$	0	0
$337$	23.2024	1.26392	0.631958	−	0.775003i	$-0.282252\pi$
$337$	23.2024	1.26392	0.631958	+	0.775003i	$0.282252\pi$
$338$	5.14056	0.279609
$339$	0	0
$340$	0.412582	0.0223754
$341$	0.0178042	0.000964153 0
$342$	0	0
$343$	−11.2086	−0.605206
$344$	15.2015	0.819612
$345$	0	0
$346$	−13.3781	−0.719212
$347$	29.0651	1.56030	0.780148	−	0.625595i	$-0.215144\pi$
$347$	29.0651	1.56030	0.780148	+	0.625595i	$0.215144\pi$
$348$	0	0
$349$	−0.112702	−0.00603280	−0.00301640	−	0.999995i	$-0.500960\pi$
$349$	−0.112702	−0.00603280	−0.00301640	+	0.999995i	$0.500960\pi$
$350$	−10.6961	−0.571732
$351$	0	0
$352$	−18.0556	−0.962369
$353$	1.37872	0.0733820	0.0366910	−	0.999327i	$-0.488318\pi$
$353$	1.37872	0.0733820	0.0366910	+	0.999327i	$0.488318\pi$
$354$	0	0
$355$	7.16406	0.380229
$356$	−3.09788	−0.164187
$357$	0	0
$358$	5.20637	0.275165
$359$	9.37848	0.494977	0.247489	−	0.968891i	$-0.420395\pi$
$359$	9.37848	0.494977	0.247489	+	0.968891i	$0.420395\pi$
$360$	0	0
$361$	−14.9012	−0.784271
$362$	−10.1249	−0.532155
$363$	0	0
$364$	14.6662	0.768716
$365$	17.5704	0.919677
$366$	0	0
$367$	−29.5188	−1.54087	−0.770435	−	0.637519i	$-0.779961\pi$
$367$	−29.5188	−1.54087	−0.770435	+	0.637519i	$0.779961\pi$
$368$	1.12883	0.0588444
$369$	0	0
$370$	−1.43134	−0.0744119
$371$	−44.4435	−2.30739
$372$	0	0
$373$	−29.4937	−1.52712	−0.763562	−	0.645735i	$-0.776551\pi$
$373$	−29.4937	−1.52712	−0.763562	+	0.645735i	$0.776551\pi$
$374$	0.529982	0.0274047
$375$	0	0
$376$	−24.0615	−1.24088
$377$	2.42919	0.125110
$378$	0	0
$379$	7.25592	0.372711	0.186356	−	0.982482i	$-0.440332\pi$
$379$	7.25592	0.372711	0.186356	+	0.982482i	$0.440332\pi$
$380$	3.52188	0.180669
$381$	0	0
$382$	−8.87941	−0.454310
$383$	32.0799	1.63920	0.819602	−	0.572933i	$-0.194195\pi$
$383$	32.0799	1.63920	0.819602	+	0.572933i	$0.194195\pi$
$384$	0	0
$385$	−14.8839	−0.758552
$386$	0.716260	0.0364567
$387$	0	0
$388$	11.3755	0.577501
$389$	8.07054	0.409193	0.204596	−	0.978846i	$-0.434412\pi$
$389$	8.07054	0.409193	0.204596	+	0.978846i	$0.434412\pi$
$390$	0	0
$391$	−0.237174	−0.0119944
$392$	24.5124	1.23806
$393$	0	0
$394$	−14.5474	−0.732885
$395$	6.35605	0.319808
$396$	0	0
$397$	34.8476	1.74895	0.874476	−	0.485069i	$-0.161206\pi$
$397$	34.8476	1.74895	0.874476	+	0.485069i	$0.161206\pi$
$398$	−9.48989	−0.475685
$399$	0	0
$400$	−4.07542	−0.203771
$401$	9.55698	0.477253	0.238626	−	0.971111i	$-0.423303\pi$
$401$	9.55698	0.477253	0.238626	+	0.971111i	$0.423303\pi$
$402$	0	0
$403$	0.0140153	0.000698151 0
$404$	−10.3158	−0.513233
$405$	0	0
$406$	2.96267	0.147035
$407$	5.17433	0.256482
$408$	0	0
$409$	−6.77875	−0.335188	−0.167594	−	0.985856i	$-0.553600\pi$
$409$	−6.77875	−0.335188	−0.167594	+	0.985856i	$0.553600\pi$
$410$	−6.40335	−0.316239
$411$	0	0
$412$	24.5688	1.21042
$413$	45.6055	2.24410
$414$	0	0
$415$	−5.44072	−0.267074
$416$	−14.2132	−0.696859
$417$	0	0
$418$	4.52403	0.221278
$419$	−24.9543	−1.21910	−0.609549	−	0.792748i	$-0.708650\pi$
$419$	−24.9543	−1.21910	−0.609549	+	0.792748i	$0.708650\pi$
$420$	0	0
$421$	−8.34962	−0.406935	−0.203468	−	0.979082i	$-0.565221\pi$
$421$	−8.34962	−0.406935	−0.203468	+	0.979082i	$0.565221\pi$
$422$	−11.1097	−0.540814
$423$	0	0
$424$	27.3391	1.32770
$425$	0.856267	0.0415351
$426$	0	0
$427$	55.5360	2.68757
$428$	−7.07838	−0.342146
$429$	0	0
$430$	−5.15600	−0.248644
$431$	−16.1008	−0.775550	−0.387775	−	0.921754i	$-0.626756\pi$
$431$	−16.1008	−0.775550	−0.387775	+	0.921754i	$0.626756\pi$
$432$	0	0
$433$	−19.7843	−0.950771	−0.475385	−	0.879778i	$-0.657691\pi$
$433$	−19.7843	−0.950771	−0.475385	+	0.879778i	$0.657691\pi$
$434$	0.0170932	0.000820500 0
$435$	0	0
$436$	9.05615	0.433711
$437$	−2.02456	−0.0968479
$438$	0	0
$439$	6.78432	0.323798	0.161899	−	0.986807i	$-0.448238\pi$
$439$	6.78432	0.323798	0.161899	+	0.986807i	$0.448238\pi$
$440$	9.15570	0.436481
$441$	0	0
$442$	0.417196	0.0198440
$443$	−39.0520	−1.85541	−0.927707	−	0.373308i	$-0.878223\pi$
$443$	−39.0520	−1.85541	−0.927707	+	0.373308i	$0.878223\pi$
$444$	0	0
$445$	2.47481	0.117317
$446$	−17.3378	−0.820970
$447$	0	0
$448$	−8.09758	−0.382575
$449$	−13.1416	−0.620188	−0.310094	−	0.950706i	$-0.600361\pi$
$449$	−13.1416	−0.620188	−0.310094	+	0.950706i	$0.600361\pi$
$450$	0	0
$451$	23.1482	1.09001
$452$	3.68715	0.173429
$453$	0	0
$454$	−6.38429	−0.299629
$455$	−11.7164	−0.549274
$456$	0	0
$457$	−11.4555	−0.535866	−0.267933	−	0.963438i	$-0.586341\pi$
$457$	−11.4555	−0.535866	−0.267933	+	0.963438i	$0.586341\pi$
$458$	−10.7715	−0.503317
$459$	0	0
$460$	−1.73958	−0.0811082
$461$	13.0768	0.609047	0.304523	−	0.952505i	$-0.401503\pi$
$461$	13.0768	0.609047	0.304523	+	0.952505i	$0.401503\pi$
$462$	0	0
$463$	26.8054	1.24575	0.622876	−	0.782320i	$-0.285964\pi$
$463$	26.8054	1.24575	0.622876	+	0.782320i	$0.285964\pi$
$464$	1.12883	0.0524047
$465$	0	0
$466$	9.21431	0.426845
$467$	−7.09497	−0.328316	−0.164158	−	0.986434i	$-0.552491\pi$
$467$	−7.09497	−0.328316	−0.164158	+	0.986434i	$0.552491\pi$
$468$	0	0
$469$	−0.686896	−0.0317179
$470$	8.16110	0.376443
$471$	0	0
$472$	−28.0539	−1.29128
$473$	18.6390	0.857024
$474$	0	0
$475$	7.30927	0.335372
$476$	−1.43193	−0.0656322
$477$	0	0
$478$	−6.03200	−0.275897
$479$	−34.1103	−1.55854	−0.779270	−	0.626688i	$-0.784410\pi$
$479$	−34.1103	−1.55854	−0.779270	+	0.626688i	$0.784410\pi$
$480$	0	0
$481$	4.07317	0.185721
$482$	−8.05552	−0.366919
$483$	0	0
$484$	2.17977	0.0990804
$485$	−9.08756	−0.412645
$486$	0	0
$487$	42.1047	1.90795	0.953973	−	0.299894i	$-0.0969513\pi$
$487$	42.1047	1.90795	0.953973	+	0.299894i	$0.0969513\pi$
$488$	−34.1625	−1.54647
$489$	0	0
$490$	−8.31402	−0.375589
$491$	2.34913	0.106015	0.0530073	−	0.998594i	$-0.483119\pi$
$491$	2.34913	0.106015	0.0530073	+	0.998594i	$0.483119\pi$
$492$	0	0
$493$	−0.237174	−0.0106818
$494$	3.56127	0.160229
$495$	0	0
$496$	0.00651281	0.000292434 0
$497$	−24.8639	−1.11530
$498$	0	0
$499$	7.43721	0.332935	0.166468	−	0.986047i	$-0.446764\pi$
$499$	7.43721	0.332935	0.166468	+	0.986047i	$0.446764\pi$
$500$	14.9783	0.669849
$501$	0	0
$502$	−14.5142	−0.647799
$503$	28.8324	1.28557	0.642786	−	0.766046i	$-0.277779\pi$
$503$	28.8324	1.28557	0.642786	+	0.766046i	$0.277779\pi$
$504$	0	0
$505$	8.24106	0.366722
$506$	−2.23457	−0.0993390
$507$	0	0
$508$	15.9938	0.709610
$509$	−23.9116	−1.05986	−0.529932	−	0.848040i	$-0.677783\pi$
$509$	−23.9116	−1.05986	−0.529932	+	0.848040i	$0.677783\pi$
$510$	0	0
$511$	−60.9808	−2.69763
$512$	−12.2869	−0.543007
$513$	0	0
$514$	−5.02714	−0.221738
$515$	−19.6274	−0.864885
$516$	0	0
$517$	−29.5025	−1.29752
$518$	4.96769	0.218268
$519$	0	0
$520$	7.20726	0.316059
$521$	29.8605	1.30821	0.654106	−	0.756403i	$-0.273045\pi$
$521$	29.8605	1.30821	0.654106	+	0.756403i	$0.273045\pi$
$522$	0	0
$523$	−26.7914	−1.17151	−0.585753	−	0.810489i	$-0.699201\pi$
$523$	−26.7914	−1.17151	−0.585753	+	0.810489i	$0.699201\pi$
$524$	10.9630	0.478920
$525$	0	0
$526$	−9.48890	−0.413736
$527$	−0.00136838	−5.96075e−5 0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.27275	−0.402783
$531$	0	0
$532$	−12.2232	−0.529944
$533$	18.2220	0.789284
$534$	0	0
$535$	5.65473	0.244475
$536$	0.422539	0.0182509
$537$	0	0
$538$	−19.1551	−0.825835
$539$	30.0554	1.29458
$540$	0	0
$541$	17.5034	0.752531	0.376266	−	0.926512i	$-0.377208\pi$
$541$	17.5034	0.752531	0.376266	+	0.926512i	$0.377208\pi$
$542$	10.0756	0.432784
$543$	0	0
$544$	1.38770	0.0594972
$545$	−7.23472	−0.309901
$546$	0	0
$547$	−21.8166	−0.932812	−0.466406	−	0.884571i	$-0.654451\pi$
$547$	−21.8166	−0.932812	−0.466406	+	0.884571i	$0.654451\pi$
$548$	−10.8888	−0.465148
$549$	0	0
$550$	8.06748	0.343998
$551$	−2.02456	−0.0862492
$552$	0	0
$553$	−22.0596	−0.938071
$554$	−14.8144	−0.629404
$555$	0	0
$556$	−18.9024	−0.801639
$557$	25.5295	1.08172	0.540860	−	0.841113i	$-0.318099\pi$
$557$	25.5295	1.08172	0.540860	+	0.841113i	$0.318099\pi$
$558$	0	0
$559$	14.6724	0.620578
$560$	−5.44454	−0.230074
$561$	0	0
$562$	13.0476	0.550379
$563$	−40.7060	−1.71555	−0.857777	−	0.514022i	$-0.828155\pi$
$563$	−40.7060	−1.71555	−0.857777	+	0.514022i	$0.828155\pi$
$564$	0	0
$565$	−2.94557	−0.123921
$566$	−22.2195	−0.933955
$567$	0	0
$568$	15.2949	0.641758
$569$	23.8159	0.998413	0.499206	−	0.866483i	$-0.333625\pi$
$569$	23.8159	0.998413	0.499206	+	0.866483i	$0.333625\pi$
$570$	0	0
$571$	43.1369	1.80522	0.902612	−	0.430455i	$-0.141647\pi$
$571$	43.1369	1.80522	0.902612	+	0.430455i	$0.141647\pi$
$572$	−11.0619	−0.462519
$573$	0	0
$574$	22.2238	0.927603
$575$	−3.61030	−0.150560
$576$	0	0
$577$	−23.6828	−0.985929	−0.492964	−	0.870050i	$-0.664087\pi$
$577$	−23.6828	−0.985929	−0.492964	+	0.870050i	$0.664087\pi$
$578$	12.2693	0.510337
$579$	0	0
$580$	−1.73958	−0.0722320
$581$	18.8828	0.783392
$582$	0	0
$583$	33.5212	1.38831
$584$	37.5118	1.55225
$585$	0	0
$586$	−4.69219	−0.193833
$587$	16.4654	0.679599	0.339799	−	0.940498i	$-0.389641\pi$
$587$	16.4654	0.679599	0.339799	+	0.940498i	$0.389641\pi$
$588$	0	0
$589$	−0.0116807	−0.000481297 0
$590$	9.51520	0.391735
$591$	0	0
$592$	1.89278	0.0777927
$593$	3.09974	0.127291	0.0636456	−	0.997973i	$-0.479727\pi$
$593$	3.09974	0.127291	0.0636456	+	0.997973i	$0.479727\pi$
$594$	0	0
$595$	1.14393	0.0468965
$596$	−14.1410	−0.579236
$597$	0	0
$598$	−1.75903	−0.0719321
$599$	19.8538	0.811204	0.405602	−	0.914050i	$-0.367062\pi$
$599$	19.8538	0.811204	0.405602	+	0.914050i	$0.367062\pi$
$600$	0	0
$601$	32.3423	1.31927	0.659634	−	0.751587i	$-0.270711\pi$
$601$	32.3423	1.31927	0.659634	+	0.751587i	$0.270711\pi$
$602$	17.8947	0.729332
$603$	0	0
$604$	18.1945	0.740325
$605$	−1.74136	−0.0707964
$606$	0	0
$607$	41.1837	1.67159	0.835796	−	0.549039i	$-0.185006\pi$
$607$	41.1837	1.67159	0.835796	+	0.549039i	$0.185006\pi$
$608$	11.8457	0.480407
$609$	0	0
$610$	11.5871	0.469148
$611$	−23.2240	−0.939544
$612$	0	0
$613$	−4.93002	−0.199122	−0.0995608	−	0.995031i	$-0.531744\pi$
$613$	−4.93002	−0.199122	−0.0995608	+	0.995031i	$0.531744\pi$
$614$	−9.22041	−0.372106
$615$	0	0
$616$	−31.7762	−1.28030
$617$	3.79379	0.152732	0.0763660	−	0.997080i	$-0.475668\pi$
$617$	3.79379	0.152732	0.0763660	+	0.997080i	$0.475668\pi$
$618$	0	0
$619$	−5.02514	−0.201977	−0.100989	−	0.994888i	$-0.532201\pi$
$619$	−5.02514	−0.201977	−0.100989	+	0.994888i	$0.532201\pi$
$620$	−0.0100365	−0.000403076 0
$621$	0	0
$622$	−21.9119	−0.878586
$623$	−8.58921	−0.344120
$624$	0	0
$625$	6.08574	0.243430
$626$	−9.44597	−0.377537
$627$	0	0
$628$	3.68983	0.147240
$629$	−0.397683	−0.0158567
$630$	0	0
$631$	28.4900	1.13417	0.567084	−	0.823660i	$-0.308071\pi$
$631$	28.4900	1.13417	0.567084	+	0.823660i	$0.308071\pi$
$632$	13.5698	0.539778
$633$	0	0
$634$	6.46821	0.256886
$635$	−12.7770	−0.507041
$636$	0	0
$637$	23.6592	0.937413
$638$	−2.23457	−0.0884677
$639$	0	0
$640$	12.1055	0.478511
$641$	8.48927	0.335306	0.167653	−	0.985846i	$-0.446381\pi$
$641$	8.48927	0.335306	0.167653	+	0.985846i	$0.446381\pi$
$642$	0	0
$643$	21.3205	0.840798	0.420399	−	0.907339i	$-0.361890\pi$
$643$	21.3205	0.840798	0.420399	+	0.907339i	$0.361890\pi$
$644$	6.03746	0.237909
$645$	0	0
$646$	−0.347703	−0.0136802
$647$	19.1958	0.754666	0.377333	−	0.926078i	$-0.376841\pi$
$647$	19.1958	0.754666	0.377333	+	0.926078i	$0.376841\pi$
$648$	0	0
$649$	−34.3976	−1.35023
$650$	6.35063	0.249092
$651$	0	0
$652$	3.69603	0.144748
$653$	6.25556	0.244799	0.122399	−	0.992481i	$-0.460941\pi$
$653$	6.25556	0.244799	0.122399	+	0.992481i	$0.460941\pi$
$654$	0	0
$655$	−8.75803	−0.342205
$656$	8.46767	0.330607
$657$	0	0
$658$	−28.3243	−1.10420
$659$	−37.2600	−1.45144	−0.725722	−	0.687988i	$-0.758494\pi$
$659$	−37.2600	−1.45144	−0.725722	+	0.687988i	$0.758494\pi$
$660$	0	0
$661$	11.4199	0.444183	0.222091	−	0.975026i	$-0.428712\pi$
$661$	11.4199	0.444183	0.222091	+	0.975026i	$0.428712\pi$
$662$	14.2174	0.552574
$663$	0	0
$664$	−11.6156	−0.450774
$665$	9.76481	0.378663
$666$	0	0
$667$	1.00000	0.0387202
$668$	18.8371	0.728829
$669$	0	0
$670$	−0.143315	−0.00553675
$671$	−41.8876	−1.61705
$672$	0	0
$673$	33.3489	1.28550	0.642752	−	0.766074i	$-0.277793\pi$
$673$	33.3489	1.28550	0.642752	+	0.766074i	$0.277793\pi$
$674$	−16.8014	−0.647165
$675$	0	0
$676$	10.4757	0.402910
$677$	−12.7153	−0.488687	−0.244343	−	0.969689i	$-0.578572\pi$
$677$	−12.7153	−0.488687	−0.244343	+	0.969689i	$0.578572\pi$
$678$	0	0
$679$	31.5397	1.21038
$680$	−0.703679	−0.0269849
$681$	0	0
$682$	−0.0128924	−0.000493676 0
$683$	23.8362	0.912068	0.456034	−	0.889962i	$-0.349270\pi$
$683$	23.8362	0.912068	0.456034	+	0.889962i	$0.349270\pi$
$684$	0	0
$685$	8.69881	0.332364
$686$	8.11637	0.309884
$687$	0	0
$688$	6.81819	0.259941
$689$	26.3875	1.00528
$690$	0	0
$691$	−3.45761	−0.131534	−0.0657668	−	0.997835i	$-0.520949\pi$
$691$	−3.45761	−0.131534	−0.0657668	+	0.997835i	$0.520949\pi$
$692$	−27.2625	−1.03637
$693$	0	0
$694$	−21.0467	−0.798921
$695$	15.1006	0.572798
$696$	0	0
$697$	−1.77910	−0.0673883
$698$	0.0816100	0.00308898
$699$	0	0
$700$	−21.7970	−0.823851
$701$	15.2400	0.575606	0.287803	−	0.957690i	$-0.407075\pi$
$701$	15.2400	0.575606	0.287803	+	0.957690i	$0.407075\pi$
$702$	0	0
$703$	−3.39470	−0.128034
$704$	6.10755	0.230187
$705$	0	0
$706$	−0.998363	−0.0375739
$707$	−28.6019	−1.07568
$708$	0	0
$709$	−37.0045	−1.38973	−0.694866	−	0.719139i	$-0.744536\pi$
$709$	−37.0045	−1.38973	−0.694866	+	0.719139i	$0.744536\pi$
$710$	−5.18765	−0.194689
$711$	0	0
$712$	5.28359	0.198011
$713$	0.00576952	0.000216070 0
$714$	0	0
$715$	8.83702	0.330486
$716$	10.6098	0.396506
$717$	0	0
$718$	−6.79116	−0.253444
$719$	5.03681	0.187841	0.0939206	−	0.995580i	$-0.470060\pi$
$719$	5.03681	0.187841	0.0939206	+	0.995580i	$0.470060\pi$
$720$	0	0
$721$	68.1197	2.53691
$722$	10.7902	0.401571
$723$	0	0
$724$	−20.6331	−0.766822
$725$	−3.61030	−0.134083
$726$	0	0
$727$	8.96392	0.332453	0.166227	−	0.986088i	$-0.446842\pi$
$727$	8.96392	0.332453	0.166227	+	0.986088i	$0.446842\pi$
$728$	−25.0139	−0.927076
$729$	0	0
$730$	−12.7231	−0.470904
$731$	−1.43254	−0.0529844
$732$	0	0
$733$	20.2174	0.746746	0.373373	−	0.927681i	$-0.378201\pi$
$733$	20.2174	0.746746	0.373373	+	0.927681i	$0.378201\pi$
$734$	21.3752	0.788974
$735$	0	0
$736$	−5.85100	−0.215671
$737$	0.518087	0.0190840
$738$	0	0
$739$	−24.6664	−0.907368	−0.453684	−	0.891163i	$-0.649890\pi$
$739$	−24.6664	−0.907368	−0.453684	+	0.891163i	$0.649890\pi$
$740$	−2.91685	−0.107226
$741$	0	0
$742$	32.1825	1.18146
$743$	−4.68390	−0.171836	−0.0859179	−	0.996302i	$-0.527382\pi$
$743$	−4.68390	−0.171836	−0.0859179	+	0.996302i	$0.527382\pi$
$744$	0	0
$745$	11.2968	0.413884
$746$	21.3570	0.781935
$747$	0	0
$748$	1.08002	0.0394895
$749$	−19.6256	−0.717103
$750$	0	0
$751$	1.17784	0.0429799	0.0214899	−	0.999769i	$-0.493159\pi$
$751$	1.17784	0.0429799	0.0214899	+	0.999769i	$0.493159\pi$
$752$	−10.7921	−0.393546
$753$	0	0
$754$	−1.75903	−0.0640601
$755$	−14.5351	−0.528988
$756$	0	0
$757$	−33.9047	−1.23229	−0.616144	−	0.787634i	$-0.711306\pi$
$757$	−33.9047	−1.23229	−0.616144	+	0.787634i	$0.711306\pi$
$758$	−5.25417	−0.190840
$759$	0	0
$760$	−6.00675	−0.217888
$761$	−1.56648	−0.0567849	−0.0283924	−	0.999597i	$-0.509039\pi$
$761$	−1.56648	−0.0567849	−0.0283924	+	0.999597i	$0.509039\pi$
$762$	0	0
$763$	25.1092	0.909014
$764$	−18.0949	−0.654649
$765$	0	0
$766$	−23.2297	−0.839324
$767$	−27.0774	−0.977709
$768$	0	0
$769$	39.5943	1.42781	0.713904	−	0.700244i	$-0.246925\pi$
$769$	39.5943	1.42781	0.713904	+	0.700244i	$0.246925\pi$
$770$	10.7777	0.388403
$771$	0	0
$772$	1.45963	0.0525331
$773$	40.8881	1.47064	0.735320	−	0.677720i	$-0.237032\pi$
$773$	40.8881	1.47064	0.735320	+	0.677720i	$0.237032\pi$
$774$	0	0
$775$	−0.0208297	−0.000748225 0
$776$	−19.4014	−0.696471
$777$	0	0
$778$	−5.84405	−0.209519
$779$	−15.1868	−0.544123
$780$	0	0
$781$	18.7535	0.671052
$782$	0.171743	0.00614150
$783$	0	0
$784$	10.9943	0.392653
$785$	−2.94771	−0.105208
$786$	0	0
$787$	16.9993	0.605958	0.302979	−	0.952997i	$-0.402019\pi$
$787$	16.9993	0.605958	0.302979	+	0.952997i	$0.402019\pi$
$788$	−29.6452	−1.05607
$789$	0	0
$790$	−4.60256	−0.163752
$791$	10.2230	0.363489
$792$	0	0
$793$	−32.9735	−1.17092
$794$	−25.2339	−0.895518
$795$	0	0
$796$	−19.3389	−0.685450
$797$	−13.6227	−0.482539	−0.241270	−	0.970458i	$-0.577564\pi$
$797$	−13.6227	−0.482539	−0.241270	+	0.970458i	$0.577564\pi$
$798$	0	0
$799$	2.26747	0.0802174
$800$	21.1238	0.746840
$801$	0	0
$802$	−6.92042	−0.244369
$803$	45.9943	1.62310
$804$	0	0
$805$	−4.82317	−0.169994
$806$	−0.0101488	−0.000357475 0
$807$	0	0
$808$	17.5942	0.618962
$809$	25.5001	0.896537	0.448268	−	0.893899i	$-0.352041\pi$
$809$	25.5001	0.896537	0.448268	+	0.893899i	$0.352041\pi$
$810$	0	0
$811$	−26.6838	−0.936994	−0.468497	−	0.883465i	$-0.655204\pi$
$811$	−26.6838	−0.936994	−0.468497	+	0.883465i	$0.655204\pi$
$812$	6.03746	0.211873
$813$	0	0
$814$	−3.74684	−0.131327
$815$	−2.95266	−0.103427
$816$	0	0
$817$	−12.2284	−0.427819
$818$	4.90864	0.171627
$819$	0	0
$820$	−13.0490	−0.455692
$821$	−53.8480	−1.87931	−0.939653	−	0.342128i	$-0.888852\pi$
$821$	−53.8480	−1.87931	−0.939653	+	0.342128i	$0.888852\pi$
$822$	0	0
$823$	−14.9934	−0.522637	−0.261319	−	0.965253i	$-0.584157\pi$
$823$	−14.9934	−0.522637	−0.261319	+	0.965253i	$0.584157\pi$
$824$	−41.9033	−1.45977
$825$	0	0
$826$	−33.0239	−1.14905
$827$	−33.6879	−1.17144	−0.585722	−	0.810512i	$-0.699189\pi$
$827$	−33.6879	−1.17144	−0.585722	+	0.810512i	$0.699189\pi$
$828$	0	0
$829$	38.3682	1.33258	0.666292	−	0.745691i	$-0.267880\pi$
$829$	38.3682	1.33258	0.666292	+	0.745691i	$0.267880\pi$
$830$	3.93974	0.136750
$831$	0	0
$832$	4.80779	0.166680
$833$	−2.30996	−0.0800354
$834$	0	0
$835$	−15.0485	−0.520774
$836$	9.21928	0.318855
$837$	0	0
$838$	18.0700	0.624217
$839$	−4.94162	−0.170604	−0.0853019	−	0.996355i	$-0.527185\pi$
$839$	−4.94162	−0.170604	−0.0853019	+	0.996355i	$0.527185\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	6.04614	0.208364
$843$	0	0
$844$	−22.6399	−0.779298
$845$	−8.36873	−0.287893
$846$	0	0
$847$	6.04366	0.207662
$848$	12.2621	0.421083
$849$	0	0
$850$	−0.620042	−0.0212673
$851$	1.67676	0.0574786
$852$	0	0
$853$	−28.5213	−0.976551	−0.488276	−	0.872689i	$-0.662374\pi$
$853$	−28.5213	−0.976551	−0.488276	+	0.872689i	$0.662374\pi$
$854$	−40.2148	−1.37612
$855$	0	0
$856$	12.0725	0.412631
$857$	51.9724	1.77534	0.887671	−	0.460478i	$-0.152322\pi$
$857$	51.9724	1.77534	0.887671	+	0.460478i	$0.152322\pi$
$858$	0	0
$859$	17.7424	0.605363	0.302681	−	0.953092i	$-0.402118\pi$
$859$	17.7424	0.605363	0.302681	+	0.953092i	$0.402118\pi$
$860$	−10.5071	−0.358290
$861$	0	0
$862$	11.6590	0.397106
$863$	16.2492	0.553129	0.276564	−	0.960995i	$-0.410804\pi$
$863$	16.2492	0.553129	0.276564	+	0.960995i	$0.410804\pi$
$864$	0	0
$865$	21.7793	0.740519
$866$	14.3262	0.486824
$867$	0	0
$868$	0.0348333	0.00118232
$869$	16.6383	0.564417
$870$	0	0
$871$	0.407832	0.0138189
$872$	−15.4457	−0.523058
$873$	0	0
$874$	1.46603	0.0495892
$875$	41.5289	1.40393
$876$	0	0
$877$	33.8094	1.14166	0.570831	−	0.821068i	$-0.306621\pi$
$877$	33.8094	1.14166	0.570831	+	0.821068i	$0.306621\pi$
$878$	−4.91268	−0.165795
$879$	0	0
$880$	4.10651	0.138430
$881$	−5.56211	−0.187392	−0.0936962	−	0.995601i	$-0.529868\pi$
$881$	−5.56211	−0.187392	−0.0936962	+	0.995601i	$0.529868\pi$
$882$	0	0
$883$	−39.1935	−1.31897	−0.659484	−	0.751719i	$-0.729225\pi$
$883$	−39.1935	−1.31897	−0.659484	+	0.751719i	$0.729225\pi$
$884$	0.850180	0.0285946
$885$	0	0
$886$	28.2784	0.950031
$887$	21.3202	0.715861	0.357931	−	0.933748i	$-0.383482\pi$
$887$	21.3202	0.715861	0.357931	+	0.933748i	$0.383482\pi$
$888$	0	0
$889$	44.3446	1.48727
$890$	−1.79207	−0.0600702
$891$	0	0
$892$	−35.3318	−1.18300
$893$	19.3556	0.647711
$894$	0	0
$895$	−8.47587	−0.283317
$896$	−42.0139	−1.40359
$897$	0	0
$898$	9.51609	0.317556
$899$	0.00576952	0.000192424 0
$900$	0	0
$901$	−2.57634	−0.0858302
$902$	−16.7621	−0.558119
$903$	0	0
$904$	−6.28862	−0.209156
$905$	16.4832	0.547921
$906$	0	0
$907$	−12.2435	−0.406537	−0.203269	−	0.979123i	$-0.565156\pi$
$907$	−12.2435	−0.406537	−0.203269	+	0.979123i	$0.565156\pi$
$908$	−13.0102	−0.431758
$909$	0	0
$910$	8.48411	0.281246
$911$	−52.8702	−1.75167	−0.875835	−	0.482611i	$-0.839688\pi$
$911$	−52.8702	−1.75167	−0.875835	+	0.482611i	$0.839688\pi$
$912$	0	0
$913$	−14.2422	−0.471350
$914$	8.29519	0.274380
$915$	0	0
$916$	−21.9506	−0.725267
$917$	30.3961	1.00377
$918$	0	0
$919$	−49.6021	−1.63622	−0.818112	−	0.575059i	$-0.804979\pi$
$919$	−49.6021	−1.63622	−0.818112	+	0.575059i	$0.804979\pi$
$920$	2.96694	0.0978170
$921$	0	0
$922$	−9.46919	−0.311851
$923$	14.7625	0.485914
$924$	0	0
$925$	−6.05360	−0.199041
$926$	−19.4104	−0.637864
$927$	0	0
$928$	−5.85100	−0.192068
$929$	24.9510	0.818617	0.409309	−	0.912396i	$-0.365770\pi$
$929$	24.9510	0.818617	0.409309	+	0.912396i	$0.365770\pi$
$930$	0	0
$931$	−19.7183	−0.646241
$932$	18.7773	0.615072
$933$	0	0
$934$	5.13763	0.168108
$935$	−0.862800	−0.0282166
$936$	0	0
$937$	27.8390	0.909460	0.454730	−	0.890629i	$-0.349736\pi$
$937$	27.8390	0.909460	0.454730	+	0.890629i	$0.349736\pi$
$938$	0.497397	0.0162406
$939$	0	0
$940$	16.6310	0.542445
$941$	−23.5204	−0.766742	−0.383371	−	0.923595i	$-0.625237\pi$
$941$	−23.5204	−0.766742	−0.383371	+	0.923595i	$0.625237\pi$
$942$	0	0
$943$	7.50127	0.244275
$944$	−12.5827	−0.409533
$945$	0	0
$946$	−13.4969	−0.438823
$947$	24.8678	0.808096	0.404048	−	0.914738i	$-0.367603\pi$
$947$	24.8678	0.808096	0.404048	+	0.914738i	$0.367603\pi$
$948$	0	0
$949$	36.2062	1.17530
$950$	−5.29280	−0.171721
$951$	0	0
$952$	2.44222	0.0791529
$953$	−10.0494	−0.325531	−0.162766	−	0.986665i	$-0.552041\pi$
$953$	−10.0494	−0.325531	−0.162766	+	0.986665i	$0.552041\pi$
$954$	0	0
$955$	14.4555	0.467769
$956$	−12.2923	−0.397561
$957$	0	0
$958$	24.7000	0.798022
$959$	−30.1905	−0.974903
$960$	0	0
$961$	−31.0000	−0.999999
$962$	−2.94947	−0.0950948
$963$	0	0
$964$	−16.4159	−0.528720
$965$	−1.16606	−0.0375367
$966$	0	0
$967$	−35.2339	−1.13305	−0.566524	−	0.824046i	$-0.691712\pi$
$967$	−35.2339	−1.13305	−0.566524	+	0.824046i	$0.691712\pi$
$968$	−3.71771	−0.119492
$969$	0	0
$970$	6.58050	0.211287
$971$	−4.97818	−0.159758	−0.0798788	−	0.996805i	$-0.525453\pi$
$971$	−4.97818	−0.159758	−0.0798788	+	0.996805i	$0.525453\pi$
$972$	0	0
$973$	−52.4089	−1.68015
$974$	−30.4889	−0.976928
$975$	0	0
$976$	−15.3226	−0.490463
$977$	10.8940	0.348529	0.174264	−	0.984699i	$-0.444245\pi$
$977$	10.8940	0.348529	0.174264	+	0.984699i	$0.444245\pi$
$978$	0	0
$979$	6.47835	0.207049
$980$	−16.9427	−0.541214
$981$	0	0
$982$	−1.70105	−0.0542828
$983$	−0.224404	−0.00715737	−0.00357868	−	0.999994i	$-0.501139\pi$
$983$	−0.224404	−0.00715737	−0.00357868	+	0.999994i	$0.501139\pi$
$984$	0	0
$985$	23.6828	0.754597
$986$	0.171743	0.00546940
$987$	0	0
$988$	7.25731	0.230886
$989$	6.04004	0.192062
$990$	0	0
$991$	−9.77464	−0.310502	−0.155251	−	0.987875i	$-0.549619\pi$
$991$	−9.77464	−0.310502	−0.155251	+	0.987875i	$0.549619\pi$
$992$	−0.0337574	−0.00107180
$993$	0	0
$994$	18.0045	0.571069
$995$	15.4494	0.489777
$996$	0	0
$997$	−58.5358	−1.85385	−0.926923	−	0.375252i	$-0.877556\pi$
$997$	−58.5358	−1.85385	−0.926923	+	0.375252i	$0.877556\pi$
$998$	−5.38544	−0.170473
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.n.1.4		12
3.2	odd	2		667.2.a.b.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.b.1.9	✓	12	3.2	odd	2
6003.2.a.n.1.4		12	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)

\(f(q)\)	\(=\)	\(q-0.724122 q^{2} -1.47565 q^{4} +1.17886 q^{5} -4.09140 q^{7} +2.51679 q^{8} +O(q^{10})\)	\(q-0.724122 q^{2} -1.47565 q^{4} +1.17886 q^{5} -4.09140 q^{7} +2.51679 q^{8} -0.853636 q^{10} +3.08591 q^{11} +2.42919 q^{13} +2.96267 q^{14} +1.12883 q^{16} -0.237174 q^{17} -2.02456 q^{19} -1.73958 q^{20} -2.23457 q^{22} +1.00000 q^{23} -3.61030 q^{25} -1.75903 q^{26} +6.03746 q^{28} +1.00000 q^{29} +0.00576952 q^{31} -5.85100 q^{32} +0.171743 q^{34} -4.82317 q^{35} +1.67676 q^{37} +1.46603 q^{38} +2.96694 q^{40} +7.50127 q^{41} +6.04004 q^{43} -4.55372 q^{44} -0.724122 q^{46} -9.56040 q^{47} +9.73954 q^{49} +2.61430 q^{50} -3.58463 q^{52} +10.8627 q^{53} +3.63784 q^{55} -10.2972 q^{56} -0.724122 q^{58} -11.1467 q^{59} -13.5738 q^{61} -0.00417784 q^{62} +1.97917 q^{64} +2.86367 q^{65} +0.167888 q^{67} +0.349985 q^{68} +3.49256 q^{70} +6.07713 q^{71} +14.9046 q^{73} -1.21418 q^{74} +2.98754 q^{76} -12.6257 q^{77} +5.39171 q^{79} +1.33073 q^{80} -5.43183 q^{82} -4.61525 q^{83} -0.279594 q^{85} -4.37373 q^{86} +7.76659 q^{88} +2.09933 q^{89} -9.93880 q^{91} -1.47565 q^{92} +6.92289 q^{94} -2.38667 q^{95} -7.70879 q^{97} -7.05262 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 11 q^{4} + 16 q^{5} - 7 q^{7} + 9 q^{8} + 6 q^{11} - 15 q^{13} + 8 q^{14} + 17 q^{16} + 18 q^{17} - 6 q^{19} + 39 q^{20} - 5 q^{22} + 12 q^{23} + 14 q^{25} + 3 q^{26} - 19 q^{28} + 12 q^{29} + 16 q^{31} + 21 q^{32} - 7 q^{34} + 11 q^{35} - q^{37} + 24 q^{38} + 30 q^{40} - 3 q^{41} - 23 q^{43} - 23 q^{44} + 3 q^{46} + 35 q^{47} + 3 q^{49} + 2 q^{50} + 45 q^{53} + 17 q^{55} + 17 q^{56} + 3 q^{58} + 11 q^{59} + 4 q^{61} + 7 q^{62} + 15 q^{64} - 5 q^{65} - 19 q^{67} - q^{68} + 14 q^{70} - 19 q^{71} + 10 q^{73} + 15 q^{74} - 4 q^{76} + 39 q^{77} + 17 q^{79} + 90 q^{80} - 3 q^{82} + 12 q^{83} + 14 q^{85} - 17 q^{86} - 2 q^{88} + 20 q^{89} + 11 q^{91} + 11 q^{92} + 13 q^{94} - 12 q^{95} - 12 q^{97} - 75 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 11 * q^4 + 16 * q^5 - 7 * q^7 + 9 * q^8 + 6 * q^11 - 15 * q^13 + 8 * q^14 + 17 * q^16 + 18 * q^17 - 6 * q^19 + 39 * q^20 - 5 * q^22 + 12 * q^23 + 14 * q^25 + 3 * q^26 - 19 * q^28 + 12 * q^29 + 16 * q^31 + 21 * q^32 - 7 * q^34 + 11 * q^35 - q^37 + 24 * q^38 + 30 * q^40 - 3 * q^41 - 23 * q^43 - 23 * q^44 + 3 * q^46 + 35 * q^47 + 3 * q^49 + 2 * q^50 + 45 * q^53 + 17 * q^55 + 17 * q^56 + 3 * q^58 + 11 * q^59 + 4 * q^61 + 7 * q^62 + 15 * q^64 - 5 * q^65 - 19 * q^67 - q^68 + 14 * q^70 - 19 * q^71 + 10 * q^73 + 15 * q^74 - 4 * q^76 + 39 * q^77 + 17 * q^79 + 90 * q^80 - 3 * q^82 + 12 * q^83 + 14 * q^85 - 17 * q^86 - 2 * q^88 + 20 * q^89 + 11 * q^91 + 11 * q^92 + 13 * q^94 - 12 * q^95 - 12 * q^97 - 75 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more