LMFDB - Embedded newform 6003.2.a.m.1.6

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			1.6
Root			$-0.467085$ 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)

$f(q)$	$=$	$q+0.467085 q^{2} -1.78183 q^{4} +0.0105419 q^{5} -1.85912 q^{7} -1.76644 q^{8} +O(q^{10})$	\(q+0.467085 q^{2} -1.78183 q^{4} +0.0105419 q^{5} -1.85912 q^{7} -1.76644 q^{8} +0.00492395 q^{10} +1.01164 q^{11} +2.69792 q^{13} -0.868368 q^{14} +2.73859 q^{16} +3.50932 q^{17} -7.48241 q^{19} -0.0187838 q^{20} +0.472521 q^{22} -1.00000 q^{23} -4.99989 q^{25} +1.26016 q^{26} +3.31264 q^{28} +1.00000 q^{29} +9.06687 q^{31} +4.81203 q^{32} +1.63915 q^{34} -0.0195986 q^{35} -3.80689 q^{37} -3.49493 q^{38} -0.0186215 q^{40} -0.921766 q^{41} +5.98292 q^{43} -1.80257 q^{44} -0.467085 q^{46} +8.75713 q^{47} -3.54367 q^{49} -2.33537 q^{50} -4.80723 q^{52} +9.78399 q^{53} +0.0106645 q^{55} +3.28402 q^{56} +0.467085 q^{58} -13.5157 q^{59} -1.50283 q^{61} +4.23500 q^{62} -3.22954 q^{64} +0.0284411 q^{65} -7.98178 q^{67} -6.25302 q^{68} -0.00915421 q^{70} -3.19615 q^{71} +12.3494 q^{73} -1.77814 q^{74} +13.3324 q^{76} -1.88076 q^{77} +9.57497 q^{79} +0.0288698 q^{80} -0.430543 q^{82} -15.0096 q^{83} +0.0369947 q^{85} +2.79453 q^{86} -1.78699 q^{88} -9.07903 q^{89} -5.01575 q^{91} +1.78183 q^{92} +4.09033 q^{94} -0.0788785 q^{95} +1.28711 q^{97} -1.65520 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.467085	0.330279	0.165140	−	0.986270i	$-0.447193\pi$
$2$	0.467085	0.330279	0.165140	+	0.986270i	$0.447193\pi$
$3$	0	0
$3$	0	0
$4$	−1.78183	−0.890916
$5$	0.0105419	0.00471446	0.00235723	−	0.999997i	$-0.499250\pi$
$5$	0.0105419	0.00471446	0.00235723	+	0.999997i	$0.499250\pi$
$6$	0	0
$7$	−1.85912	−0.702682	−0.351341	−	0.936248i	$-0.614274\pi$
$7$	−1.85912	−0.702682	−0.351341	+	0.936248i	$0.614274\pi$
$8$	−1.76644	−0.624530
$9$	0	0
$10$	0.00492395	0.00155709
$11$	1.01164	0.305020	0.152510	−	0.988302i	$-0.451264\pi$
$11$	1.01164	0.305020	0.152510	+	0.988302i	$0.451264\pi$
$12$	0	0
$13$	2.69792	0.748268	0.374134	−	0.927375i	$-0.377940\pi$
$13$	2.69792	0.748268	0.374134	+	0.927375i	$0.377940\pi$
$14$	−0.868368	−0.232081
$15$	0	0
$16$	2.73859	0.684646
$17$	3.50932	0.851135	0.425568	−	0.904927i	$-0.360074\pi$
$17$	3.50932	0.851135	0.425568	+	0.904927i	$0.360074\pi$
$18$	0	0
$19$	−7.48241	−1.71658	−0.858292	−	0.513162i	$-0.828474\pi$
$19$	−7.48241	−1.71658	−0.858292	+	0.513162i	$0.828474\pi$
$20$	−0.0187838	−0.00420019
$21$	0	0
$22$	0.472521	0.100742
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.99989	−0.999978
$26$	1.26016	0.247137
$27$	0	0
$28$	3.31264	0.626030
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.06687	1.62846	0.814229	−	0.580544i	$-0.197160\pi$
$31$	9.06687	1.62846	0.814229	+	0.580544i	$0.197160\pi$
$32$	4.81203	0.850655
$33$	0	0
$34$	1.63915	0.281112
$35$	−0.0195986	−0.00331276
$36$	0	0
$37$	−3.80689	−0.625850	−0.312925	−	0.949778i	$-0.601309\pi$
$37$	−3.80689	−0.625850	−0.312925	+	0.949778i	$0.601309\pi$
$38$	−3.49493	−0.566952
$39$	0	0
$40$	−0.0186215	−0.00294432
$41$	−0.921766	−0.143956	−0.0719778	−	0.997406i	$-0.522931\pi$
$41$	−0.921766	−0.143956	−0.0719778	+	0.997406i	$0.522931\pi$
$42$	0	0
$43$	5.98292	0.912386	0.456193	−	0.889881i	$-0.349213\pi$
$43$	5.98292	0.912386	0.456193	+	0.889881i	$0.349213\pi$
$44$	−1.80257	−0.271747
$45$	0	0
$46$	−0.467085	−0.0688680
$47$	8.75713	1.27736	0.638679	−	0.769473i	$-0.279481\pi$
$47$	8.75713	1.27736	0.638679	+	0.769473i	$0.279481\pi$
$48$	0	0
$49$	−3.54367	−0.506239
$50$	−2.33537	−0.330272
$51$	0	0
$52$	−4.80723	−0.666643
$53$	9.78399	1.34393	0.671967	−	0.740581i	$-0.265449\pi$
$53$	9.78399	1.34393	0.671967	+	0.740581i	$0.265449\pi$
$54$	0	0
$55$	0.0106645	0.00143800
$56$	3.28402	0.438846
$57$	0	0
$58$	0.467085	0.0613313
$59$	−13.5157	−1.75960	−0.879799	−	0.475346i	$-0.842323\pi$
$59$	−13.5157	−1.75960	−0.879799	+	0.475346i	$0.842323\pi$
$60$	0	0
$61$	−1.50283	−0.192418	−0.0962088	−	0.995361i	$-0.530672\pi$
$61$	−1.50283	−0.192418	−0.0962088	+	0.995361i	$0.530672\pi$
$62$	4.23500	0.537846
$63$	0	0
$64$	−3.22954	−0.403693
$65$	0.0284411	0.00352768
$66$	0	0
$67$	−7.98178	−0.975130	−0.487565	−	0.873087i	$-0.662115\pi$
$67$	−7.98178	−0.975130	−0.487565	+	0.873087i	$0.662115\pi$
$68$	−6.25302	−0.758290
$69$	0	0
$70$	−0.00915421	−0.00109414
$71$	−3.19615	−0.379313	−0.189656	−	0.981851i	$-0.560737\pi$
$71$	−3.19615	−0.379313	−0.189656	+	0.981851i	$0.560737\pi$
$72$	0	0
$73$	12.3494	1.44539	0.722694	−	0.691168i	$-0.242903\pi$
$73$	12.3494	1.44539	0.722694	+	0.691168i	$0.242903\pi$
$74$	−1.77814	−0.206705
$75$	0	0
$76$	13.3324	1.52933
$77$	−1.88076	−0.214332
$78$	0	0
$79$	9.57497	1.07727	0.538634	−	0.842540i	$-0.318941\pi$
$79$	9.57497	1.07727	0.538634	+	0.842540i	$0.318941\pi$
$80$	0.0288698	0.00322774
$81$	0	0
$82$	−0.430543	−0.0475455
$83$	−15.0096	−1.64752	−0.823760	−	0.566939i	$-0.808128\pi$
$83$	−15.0096	−1.64752	−0.823760	+	0.566939i	$0.808128\pi$
$84$	0	0
$85$	0.0369947	0.00401264
$86$	2.79453	0.301342
$87$	0	0
$88$	−1.78699	−0.190494
$89$	−9.07903	−0.962376	−0.481188	−	0.876617i	$-0.659795\pi$
$89$	−9.07903	−0.962376	−0.481188	+	0.876617i	$0.659795\pi$
$90$	0	0
$91$	−5.01575	−0.525794
$92$	1.78183	0.185769
$93$	0	0
$94$	4.09033	0.421885
$95$	−0.0788785	−0.00809276
$96$	0	0
$97$	1.28711	0.130686	0.0653432	−	0.997863i	$-0.479186\pi$
$97$	1.28711	0.130686	0.0653432	+	0.997863i	$0.479186\pi$
$98$	−1.65520	−0.167200
$99$	0	0
$100$	8.90896	0.890896
$101$	−17.9301	−1.78411	−0.892054	−	0.451929i	$-0.850736\pi$
$101$	−17.9301	−1.78411	−0.892054	+	0.451929i	$0.850736\pi$
$102$	0	0
$103$	16.4218	1.61809	0.809044	−	0.587748i	$-0.199985\pi$
$103$	16.4218	1.61809	0.809044	+	0.587748i	$0.199985\pi$
$104$	−4.76570	−0.467316
$105$	0	0
$106$	4.56996	0.443874
$107$	−3.00179	−0.290194	−0.145097	−	0.989417i	$-0.546349\pi$
$107$	−3.00179	−0.290194	−0.145097	+	0.989417i	$0.546349\pi$
$108$	0	0
$109$	−13.8887	−1.33030	−0.665150	−	0.746710i	$-0.731632\pi$
$109$	−13.8887	−1.33030	−0.665150	+	0.746710i	$0.731632\pi$
$110$	0.00498125	0.000474943 0
$111$	0	0
$112$	−5.09136	−0.481088
$113$	2.10899	0.198397	0.0991984	−	0.995068i	$-0.468372\pi$
$113$	2.10899	0.198397	0.0991984	+	0.995068i	$0.468372\pi$
$114$	0	0
$115$	−0.0105419	−0.000983033 0
$116$	−1.78183	−0.165439
$117$	0	0
$118$	−6.31300	−0.581159
$119$	−6.52425	−0.598077
$120$	0	0
$121$	−9.97659	−0.906963
$122$	−0.701949	−0.0635515
$123$	0	0
$124$	−16.1556	−1.45082
$125$	−0.105417	−0.00942882
$126$	0	0
$127$	17.6790	1.56876	0.784380	−	0.620281i	$-0.212981\pi$
$127$	17.6790	1.56876	0.784380	+	0.620281i	$0.212981\pi$
$128$	−11.1325	−0.983986
$129$	0	0
$130$	0.0132844	0.00116512
$131$	4.77362	0.417073	0.208536	−	0.978015i	$-0.433130\pi$
$131$	4.77362	0.417073	0.208536	+	0.978015i	$0.433130\pi$
$132$	0	0
$133$	13.9107	1.20621
$134$	−3.72817	−0.322065
$135$	0	0
$136$	−6.19900	−0.531560
$137$	−3.89537	−0.332804	−0.166402	−	0.986058i	$-0.553215\pi$
$137$	−3.89537	−0.332804	−0.166402	+	0.986058i	$0.553215\pi$
$138$	0	0
$139$	3.17159	0.269011	0.134505	−	0.990913i	$-0.457055\pi$
$139$	3.17159	0.269011	0.134505	+	0.990913i	$0.457055\pi$
$140$	0.0349214	0.00295139
$141$	0	0
$142$	−1.49287	−0.125279
$143$	2.72931	0.228237
$144$	0	0
$145$	0.0105419	0.000875453 0
$146$	5.76823	0.477382
$147$	0	0
$148$	6.78324	0.557579
$149$	−1.96821	−0.161242	−0.0806211	−	0.996745i	$-0.525690\pi$
$149$	−1.96821	−0.161242	−0.0806211	+	0.996745i	$0.525690\pi$
$150$	0	0
$151$	−2.94743	−0.239859	−0.119929	−	0.992782i	$-0.538267\pi$
$151$	−2.94743	−0.239859	−0.119929	+	0.992782i	$0.538267\pi$
$152$	13.2172	1.07206
$153$	0	0
$154$	−0.878473	−0.0707894
$155$	0.0955816	0.00767730
$156$	0	0
$157$	−6.42396	−0.512688	−0.256344	−	0.966586i	$-0.582518\pi$
$157$	−6.42396	−0.512688	−0.256344	+	0.966586i	$0.582518\pi$
$158$	4.47233	0.355799
$159$	0	0
$160$	0.0507277	0.00401038
$161$	1.85912	0.146519
$162$	0	0
$163$	12.2619	0.960425	0.480212	−	0.877152i	$-0.340560\pi$
$163$	12.2619	0.960425	0.480212	+	0.877152i	$0.340560\pi$
$164$	1.64243	0.128252
$165$	0	0
$166$	−7.01077	−0.544142
$167$	−5.89625	−0.456266	−0.228133	−	0.973630i	$-0.573262\pi$
$167$	−5.89625	−0.456266	−0.228133	+	0.973630i	$0.573262\pi$
$168$	0	0
$169$	−5.72124	−0.440095
$170$	0.0172797	0.00132529
$171$	0	0
$172$	−10.6605	−0.812859
$173$	1.40065	0.106490	0.0532449	−	0.998581i	$-0.483044\pi$
$173$	1.40065	0.106490	0.0532449	+	0.998581i	$0.483044\pi$
$174$	0	0
$175$	9.29540	0.702666
$176$	2.77045	0.208831
$177$	0	0
$178$	−4.24068	−0.317853
$179$	−17.5946	−1.31508	−0.657539	−	0.753420i	$-0.728403\pi$
$179$	−17.5946	−1.31508	−0.657539	+	0.753420i	$0.728403\pi$
$180$	0	0
$181$	−21.6515	−1.60934	−0.804672	−	0.593719i	$-0.797659\pi$
$181$	−21.6515	−1.60934	−0.804672	+	0.593719i	$0.797659\pi$
$182$	−2.34279	−0.173659
$183$	0	0
$184$	1.76644	0.130224
$185$	−0.0401317	−0.00295054
$186$	0	0
$187$	3.55016	0.259613
$188$	−15.6037	−1.13802
$189$	0	0
$190$	−0.0368430	−0.00267287
$191$	−9.02504	−0.653029	−0.326514	−	0.945192i	$-0.605874\pi$
$191$	−9.02504	−0.653029	−0.326514	+	0.945192i	$0.605874\pi$
$192$	0	0
$193$	2.31723	0.166798	0.0833988	−	0.996516i	$-0.473422\pi$
$193$	2.31723	0.166798	0.0833988	+	0.996516i	$0.473422\pi$
$194$	0.601191	0.0431630
$195$	0	0
$196$	6.31422	0.451016
$197$	−21.2889	−1.51677	−0.758384	−	0.651808i	$-0.774011\pi$
$197$	−21.2889	−1.51677	−0.758384	+	0.651808i	$0.774011\pi$
$198$	0	0
$199$	−5.92224	−0.419816	−0.209908	−	0.977721i	$-0.567317\pi$
$199$	−5.92224	−0.419816	−0.209908	+	0.977721i	$0.567317\pi$
$200$	8.83199	0.624516
$201$	0	0
$202$	−8.37487	−0.589254
$203$	−1.85912	−0.130485
$204$	0	0
$205$	−0.00971712	−0.000678673 0
$206$	7.67039	0.534421
$207$	0	0
$208$	7.38848	0.512299
$209$	−7.56948	−0.523592
$210$	0	0
$211$	7.72273	0.531655	0.265828	−	0.964021i	$-0.414355\pi$
$211$	7.72273	0.531655	0.265828	+	0.964021i	$0.414355\pi$
$212$	−17.4334	−1.19733
$213$	0	0
$214$	−1.40209	−0.0958452
$215$	0.0630710	0.00430141
$216$	0	0
$217$	−16.8564	−1.14429
$218$	−6.48723	−0.439371
$219$	0	0
$220$	−0.0190024	−0.00128114
$221$	9.46786	0.636877
$222$	0	0
$223$	−10.9381	−0.732469	−0.366234	−	0.930523i	$-0.619353\pi$
$223$	−10.9381	−0.732469	−0.366234	+	0.930523i	$0.619353\pi$
$224$	−8.94614	−0.597739
$225$	0	0
$226$	0.985077	0.0655263
$227$	−21.4894	−1.42630	−0.713152	−	0.701010i	$-0.752733\pi$
$227$	−21.4894	−1.42630	−0.713152	+	0.701010i	$0.752733\pi$
$228$	0	0
$229$	−10.3996	−0.687228	−0.343614	−	0.939111i	$-0.611651\pi$
$229$	−10.3996	−0.687228	−0.343614	+	0.939111i	$0.611651\pi$
$230$	−0.00492395	−0.000324675 0
$231$	0	0
$232$	−1.76644	−0.115972
$233$	−9.36159	−0.613298	−0.306649	−	0.951823i	$-0.599208\pi$
$233$	−9.36159	−0.613298	−0.306649	+	0.951823i	$0.599208\pi$
$234$	0	0
$235$	0.0923164	0.00602206
$236$	24.0827	1.56765
$237$	0	0
$238$	−3.04738	−0.197532
$239$	−12.7171	−0.822603	−0.411302	−	0.911499i	$-0.634926\pi$
$239$	−12.7171	−0.822603	−0.411302	+	0.911499i	$0.634926\pi$
$240$	0	0
$241$	−1.69211	−0.108999	−0.0544993	−	0.998514i	$-0.517356\pi$
$241$	−1.69211	−0.108999	−0.0544993	+	0.998514i	$0.517356\pi$
$242$	−4.65992	−0.299551
$243$	0	0
$244$	2.67779	0.171428
$245$	−0.0373568	−0.00238664
$246$	0	0
$247$	−20.1869	−1.28446
$248$	−16.0161	−1.01702
$249$	0	0
$250$	−0.0492389	−0.00311414
$251$	−8.84699	−0.558417	−0.279209	−	0.960230i	$-0.590072\pi$
$251$	−8.84699	−0.558417	−0.279209	+	0.960230i	$0.590072\pi$
$252$	0	0
$253$	−1.01164	−0.0636011
$254$	8.25761	0.518129
$255$	0	0
$256$	1.25924	0.0787026
$257$	4.57937	0.285653	0.142827	−	0.989748i	$-0.454381\pi$
$257$	4.57937	0.285653	0.142827	+	0.989748i	$0.454381\pi$
$258$	0	0
$259$	7.07748	0.439773
$260$	−0.0506772	−0.00314286
$261$	0	0
$262$	2.22969	0.137750
$263$	−5.65466	−0.348681	−0.174341	−	0.984685i	$-0.555779\pi$
$263$	−5.65466	−0.348681	−0.174341	+	0.984685i	$0.555779\pi$
$264$	0	0
$265$	0.103141	0.00633593
$266$	6.49749	0.398387
$267$	0	0
$268$	14.2222	0.868759
$269$	−14.8896	−0.907837	−0.453919	−	0.891043i	$-0.649974\pi$
$269$	−14.8896	−0.907837	−0.453919	+	0.891043i	$0.649974\pi$
$270$	0	0
$271$	11.4241	0.693965	0.346983	−	0.937872i	$-0.387206\pi$
$271$	11.4241	0.693965	0.346983	+	0.937872i	$0.387206\pi$
$272$	9.61057	0.582727
$273$	0	0
$274$	−1.81947	−0.109918
$275$	−5.05807	−0.305013
$276$	0	0
$277$	−4.30028	−0.258379	−0.129189	−	0.991620i	$-0.541238\pi$
$277$	−4.30028	−0.258379	−0.129189	+	0.991620i	$0.541238\pi$
$278$	1.48140	0.0888487
$279$	0	0
$280$	0.0346197	0.00206892
$281$	−25.7614	−1.53679	−0.768397	−	0.639973i	$-0.778946\pi$
$281$	−25.7614	−1.53679	−0.768397	+	0.639973i	$0.778946\pi$
$282$	0	0
$283$	−26.5143	−1.57611	−0.788056	−	0.615604i	$-0.788912\pi$
$283$	−26.5143	−1.57611	−0.788056	+	0.615604i	$0.788912\pi$
$284$	5.69499	0.337936
$285$	0	0
$286$	1.27482	0.0753818
$287$	1.71367	0.101155
$288$	0	0
$289$	−4.68467	−0.275569
$290$	0.00492395	0.000289144 0
$291$	0	0
$292$	−22.0046	−1.28772
$293$	9.65842	0.564251	0.282126	−	0.959377i	$-0.408961\pi$
$293$	9.65842	0.564251	0.282126	+	0.959377i	$0.408961\pi$
$294$	0	0
$295$	−0.142481	−0.00829555
$296$	6.72464	0.390862
$297$	0	0
$298$	−0.919323	−0.0532550
$299$	−2.69792	−0.156025
$300$	0	0
$301$	−11.1230	−0.641117
$302$	−1.37670	−0.0792203
$303$	0	0
$304$	−20.4912	−1.17525
$305$	−0.0158426	−0.000907145 0
$306$	0	0
$307$	−9.87759	−0.563744	−0.281872	−	0.959452i	$-0.590955\pi$
$307$	−9.87759	−0.563744	−0.281872	+	0.959452i	$0.590955\pi$
$308$	3.35119	0.190952
$309$	0	0
$310$	0.0446448	0.00253565
$311$	24.0414	1.36326	0.681632	−	0.731695i	$-0.261270\pi$
$311$	24.0414	1.36326	0.681632	+	0.731695i	$0.261270\pi$
$312$	0	0
$313$	−16.8236	−0.950924	−0.475462	−	0.879736i	$-0.657719\pi$
$313$	−16.8236	−0.950924	−0.475462	+	0.879736i	$0.657719\pi$
$314$	−3.00054	−0.169330
$315$	0	0
$316$	−17.0610	−0.959755
$317$	−7.39291	−0.415227	−0.207614	−	0.978211i	$-0.566570\pi$
$317$	−7.39291	−0.415227	−0.207614	+	0.978211i	$0.566570\pi$
$318$	0	0
$319$	1.01164	0.0566408
$320$	−0.0340454	−0.00190319
$321$	0	0
$322$	0.868368	0.0483923
$323$	−26.2582	−1.46104
$324$	0	0
$325$	−13.4893	−0.748251
$326$	5.72735	0.317208
$327$	0	0
$328$	1.62824	0.0899046
$329$	−16.2806	−0.897577
$330$	0	0
$331$	6.93138	0.380983	0.190492	−	0.981689i	$-0.438992\pi$
$331$	6.93138	0.380983	0.190492	+	0.981689i	$0.438992\pi$
$332$	26.7446	1.46780
$333$	0	0
$334$	−2.75405	−0.150695
$335$	−0.0841428	−0.00459721
$336$	0	0
$337$	4.06977	0.221694	0.110847	−	0.993837i	$-0.464644\pi$
$337$	4.06977	0.221694	0.110847	+	0.993837i	$0.464644\pi$
$338$	−2.67231	−0.145354
$339$	0	0
$340$	−0.0659184	−0.00357493
$341$	9.17238	0.496712
$342$	0	0
$343$	19.6020	1.05841
$344$	−10.5685	−0.569813
$345$	0	0
$346$	0.654225	0.0351714
$347$	−7.42107	−0.398384	−0.199192	−	0.979960i	$-0.563832\pi$
$347$	−7.42107	−0.398384	−0.199192	+	0.979960i	$0.563832\pi$
$348$	0	0
$349$	24.6446	1.31919	0.659597	−	0.751619i	$-0.270727\pi$
$349$	24.6446	1.31919	0.659597	+	0.751619i	$0.270727\pi$
$350$	4.34174	0.232076
$351$	0	0
$352$	4.86803	0.259467
$353$	21.7155	1.15580	0.577901	−	0.816107i	$-0.303872\pi$
$353$	21.7155	1.15580	0.577901	+	0.816107i	$0.303872\pi$
$354$	0	0
$355$	−0.0336933	−0.00178826
$356$	16.1773	0.857396
$357$	0	0
$358$	−8.21816	−0.434343
$359$	11.8532	0.625587	0.312794	−	0.949821i	$-0.398735\pi$
$359$	11.8532	0.625587	0.312794	+	0.949821i	$0.398735\pi$
$360$	0	0
$361$	36.9865	1.94666
$362$	−10.1131	−0.531533
$363$	0	0
$364$	8.93723	0.468438
$365$	0.130186	0.00681423
$366$	0	0
$367$	−11.8459	−0.618349	−0.309174	−	0.951005i	$-0.600053\pi$
$367$	−11.8459	−0.618349	−0.309174	+	0.951005i	$0.600053\pi$
$368$	−2.73859	−0.142759
$369$	0	0
$370$	−0.0187449	−0.000974503 0
$371$	−18.1896	−0.944358
$372$	0	0
$373$	−34.5101	−1.78687	−0.893434	−	0.449195i	$-0.851711\pi$
$373$	−34.5101	−1.78687	−0.893434	+	0.449195i	$0.851711\pi$
$374$	1.65823	0.0857449
$375$	0	0
$376$	−15.4689	−0.797749
$377$	2.69792	0.138950
$378$	0	0
$379$	−3.08382	−0.158405	−0.0792027	−	0.996859i	$-0.525237\pi$
$379$	−3.08382	−0.158405	−0.0792027	+	0.996859i	$0.525237\pi$
$380$	0.140548	0.00720997
$381$	0	0
$382$	−4.21546	−0.215682
$383$	24.4060	1.24709	0.623544	−	0.781788i	$-0.285692\pi$
$383$	24.4060	1.24709	0.623544	+	0.781788i	$0.285692\pi$
$384$	0	0
$385$	−0.0198266	−0.00101046
$386$	1.08234	0.0550898
$387$	0	0
$388$	−2.29342	−0.116431
$389$	0.324732	0.0164645	0.00823227	−	0.999966i	$-0.497380\pi$
$389$	0.324732	0.0164645	0.00823227	+	0.999966i	$0.497380\pi$
$390$	0	0
$391$	−3.50932	−0.177474
$392$	6.25967	0.316161
$393$	0	0
$394$	−9.94371	−0.500957
$395$	0.100938	0.00507874
$396$	0	0
$397$	6.99747	0.351193	0.175596	−	0.984462i	$-0.443815\pi$
$397$	6.99747	0.351193	0.175596	+	0.984462i	$0.443815\pi$
$398$	−2.76619	−0.138657
$399$	0	0
$400$	−13.6926	−0.684631
$401$	−18.5106	−0.924374	−0.462187	−	0.886782i	$-0.652935\pi$
$401$	−18.5106	−0.924374	−0.462187	+	0.886782i	$0.652935\pi$
$402$	0	0
$403$	24.4617	1.21852
$404$	31.9483	1.58949
$405$	0	0
$406$	−0.868368	−0.0430964
$407$	−3.85119	−0.190897
$408$	0	0
$409$	−10.9316	−0.540533	−0.270266	−	0.962786i	$-0.587112\pi$
$409$	−10.9316	−0.540533	−0.270266	+	0.962786i	$0.587112\pi$
$410$	−0.00453872	−0.000224152 0
$411$	0	0
$412$	−29.2609	−1.44158
$413$	25.1274	1.23644
$414$	0	0
$415$	−0.158229	−0.00776717
$416$	12.9825	0.636517
$417$	0	0
$418$	−3.53560	−0.172932
$419$	−25.7158	−1.25630	−0.628150	−	0.778092i	$-0.716188\pi$
$419$	−25.7158	−1.25630	−0.628150	+	0.778092i	$0.716188\pi$
$420$	0	0
$421$	34.0277	1.65841	0.829204	−	0.558945i	$-0.188794\pi$
$421$	34.0277	1.65841	0.829204	+	0.558945i	$0.188794\pi$
$422$	3.60718	0.175595
$423$	0	0
$424$	−17.2828	−0.839328
$425$	−17.5462	−0.851116
$426$	0	0
$427$	2.79394	0.135208
$428$	5.34869	0.258539
$429$	0	0
$430$	0.0294596	0.00142067
$431$	34.2245	1.64854	0.824269	−	0.566199i	$-0.191587\pi$
$431$	34.2245	1.64854	0.824269	+	0.566199i	$0.191587\pi$
$432$	0	0
$433$	9.01368	0.433170	0.216585	−	0.976264i	$-0.430508\pi$
$433$	9.01368	0.433170	0.216585	+	0.976264i	$0.430508\pi$
$434$	−7.87338	−0.377934
$435$	0	0
$436$	24.7474	1.18519
$437$	7.48241	0.357932
$438$	0	0
$439$	29.1400	1.39078	0.695389	−	0.718633i	$-0.255232\pi$
$439$	29.1400	1.39078	0.695389	+	0.718633i	$0.255232\pi$
$440$	−0.0188382	−0.000898077 0
$441$	0	0
$442$	4.42230	0.210347
$443$	40.2151	1.91068	0.955338	−	0.295516i	$-0.0954917\pi$
$443$	40.2151	1.91068	0.955338	+	0.295516i	$0.0954917\pi$
$444$	0	0
$445$	−0.0957099	−0.00453708
$446$	−5.10902	−0.241919
$447$	0	0
$448$	6.00411	0.283667
$449$	−5.94087	−0.280367	−0.140183	−	0.990126i	$-0.544769\pi$
$449$	−5.94087	−0.280367	−0.140183	+	0.990126i	$0.544769\pi$
$450$	0	0
$451$	−0.932492	−0.0439093
$452$	−3.75786	−0.176755
$453$	0	0
$454$	−10.0374	−0.471078
$455$	−0.0528754	−0.00247883
$456$	0	0
$457$	9.12815	0.426997	0.213498	−	0.976943i	$-0.431514\pi$
$457$	9.12815	0.426997	0.213498	+	0.976943i	$0.431514\pi$
$458$	−4.85752	−0.226977
$459$	0	0
$460$	0.0187838	0.000875799 0
$461$	−29.1021	−1.35542	−0.677709	−	0.735330i	$-0.737027\pi$
$461$	−29.1021	−1.35542	−0.677709	+	0.735330i	$0.737027\pi$
$462$	0	0
$463$	−19.7693	−0.918760	−0.459380	−	0.888240i	$-0.651928\pi$
$463$	−19.7693	−0.918760	−0.459380	+	0.888240i	$0.651928\pi$
$464$	2.73859	0.127136
$465$	0	0
$466$	−4.37266	−0.202560
$467$	−17.0583	−0.789366	−0.394683	−	0.918817i	$-0.629146\pi$
$467$	−17.0583	−0.789366	−0.394683	+	0.918817i	$0.629146\pi$
$468$	0	0
$469$	14.8391	0.685206
$470$	0.0431196	0.00198896
$471$	0	0
$472$	23.8747	1.09892
$473$	6.05254	0.278296
$474$	0	0
$475$	37.4112	1.71654
$476$	11.6251	0.532836
$477$	0	0
$478$	−5.93999	−0.271689
$479$	17.3642	0.793392	0.396696	−	0.917950i	$-0.370157\pi$
$479$	17.3642	0.793392	0.396696	+	0.917950i	$0.370157\pi$
$480$	0	0
$481$	−10.2707	−0.468303
$482$	−0.790362	−0.0360000
$483$	0	0
$484$	17.7766	0.808027
$485$	0.0135685	0.000616116 0
$486$	0	0
$487$	39.5581	1.79255	0.896274	−	0.443500i	$-0.146263\pi$
$487$	39.5581	1.79255	0.896274	+	0.443500i	$0.146263\pi$
$488$	2.65465	0.120171
$489$	0	0
$490$	−0.0174488	−0.000788258 0
$491$	−8.63557	−0.389718	−0.194859	−	0.980831i	$-0.562425\pi$
$491$	−8.63557	−0.389718	−0.194859	+	0.980831i	$0.562425\pi$
$492$	0	0
$493$	3.50932	0.158052
$494$	−9.42902	−0.424232
$495$	0	0
$496$	24.8304	1.11492
$497$	5.94202	0.266536
$498$	0	0
$499$	−39.3668	−1.76230	−0.881150	−	0.472837i	$-0.843230\pi$
$499$	−39.3668	−1.76230	−0.881150	+	0.472837i	$0.843230\pi$
$500$	0.187836	0.00840028
$501$	0	0
$502$	−4.13230	−0.184434
$503$	34.5468	1.54037	0.770183	−	0.637823i	$-0.220165\pi$
$503$	34.5468	1.54037	0.770183	+	0.637823i	$0.220165\pi$
$504$	0	0
$505$	−0.189016	−0.00841111
$506$	−0.472521	−0.0210061
$507$	0	0
$508$	−31.5010	−1.39763
$509$	4.87692	0.216165	0.108083	−	0.994142i	$-0.465529\pi$
$509$	4.87692	0.216165	0.108083	+	0.994142i	$0.465529\pi$
$510$	0	0
$511$	−22.9590	−1.01565
$512$	22.8532	1.00998
$513$	0	0
$514$	2.13896	0.0943454
$515$	0.173116	0.00762841
$516$	0	0
$517$	8.85904	0.389620
$518$	3.30579	0.145248
$519$	0	0
$520$	−0.0502394	−0.00220314
$521$	−28.3250	−1.24094	−0.620470	−	0.784230i	$-0.713058\pi$
$521$	−28.3250	−1.24094	−0.620470	+	0.784230i	$0.713058\pi$
$522$	0	0
$523$	9.04905	0.395687	0.197844	−	0.980234i	$-0.436606\pi$
$523$	9.04905	0.395687	0.197844	+	0.980234i	$0.436606\pi$
$524$	−8.50578	−0.371577
$525$	0	0
$526$	−2.64121	−0.115162
$527$	31.8185	1.38604
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.0481759	0.00209263
$531$	0	0
$532$	−24.7865	−1.07463
$533$	−2.48685	−0.107717
$534$	0	0
$535$	−0.0316445	−0.00136811
$536$	14.0993	0.608998
$537$	0	0
$538$	−6.95473	−0.299840
$539$	−3.58491	−0.154413
$540$	0	0
$541$	−42.1779	−1.81337	−0.906685	−	0.421808i	$-0.861396\pi$
$541$	−42.1779	−1.81337	−0.906685	+	0.421808i	$0.861396\pi$
$542$	5.33604	0.229202
$543$	0	0
$544$	16.8870	0.724022
$545$	−0.146413	−0.00627165
$546$	0	0
$547$	−31.3645	−1.34105	−0.670525	−	0.741887i	$-0.733931\pi$
$547$	−31.3645	−1.34105	−0.670525	+	0.741887i	$0.733931\pi$
$548$	6.94089	0.296500
$549$	0	0
$550$	−2.36255	−0.100740
$551$	−7.48241	−0.318761
$552$	0	0
$553$	−17.8010	−0.756977
$554$	−2.00860	−0.0853372
$555$	0	0
$556$	−5.65124	−0.239666
$557$	39.1429	1.65854	0.829269	−	0.558850i	$-0.188757\pi$
$557$	39.1429	1.65854	0.829269	+	0.558850i	$0.188757\pi$
$558$	0	0
$559$	16.1414	0.682709
$560$	−0.0536724	−0.00226807
$561$	0	0
$562$	−12.0328	−0.507571
$563$	−18.0039	−0.758775	−0.379388	−	0.925238i	$-0.623865\pi$
$563$	−18.0039	−0.758775	−0.379388	+	0.925238i	$0.623865\pi$
$564$	0	0
$565$	0.0222326	0.000935334 0
$566$	−12.3844	−0.520557
$567$	0	0
$568$	5.64580	0.236892
$569$	−41.1091	−1.72338	−0.861691	−	0.507433i	$-0.830594\pi$
$569$	−41.1091	−1.72338	−0.861691	+	0.507433i	$0.830594\pi$
$570$	0	0
$571$	−32.4790	−1.35920	−0.679601	−	0.733582i	$-0.737847\pi$
$571$	−32.4790	−1.35920	−0.679601	+	0.733582i	$0.737847\pi$
$572$	−4.86317	−0.203340
$573$	0	0
$574$	0.800432	0.0334094
$575$	4.99989	0.208510
$576$	0	0
$577$	7.17376	0.298647	0.149324	−	0.988788i	$-0.452290\pi$
$577$	7.17376	0.298647	0.149324	+	0.988788i	$0.452290\pi$
$578$	−2.18814	−0.0910147
$579$	0	0
$580$	−0.0187838	−0.000779955 0
$581$	27.9047	1.15768
$582$	0	0
$583$	9.89785	0.409927
$584$	−21.8145	−0.902689
$585$	0	0
$586$	4.51131	0.186360
$587$	−3.16686	−0.130710	−0.0653552	−	0.997862i	$-0.520818\pi$
$587$	−3.16686	−0.130710	−0.0653552	+	0.997862i	$0.520818\pi$
$588$	0	0
$589$	−67.8421	−2.79538
$590$	−0.0665507	−0.00273985
$591$	0	0
$592$	−10.4255	−0.428486
$593$	−1.38330	−0.0568054	−0.0284027	−	0.999597i	$-0.509042\pi$
$593$	−1.38330	−0.0568054	−0.0284027	+	0.999597i	$0.509042\pi$
$594$	0	0
$595$	−0.0687777	−0.00281961
$596$	3.50702	0.143653
$597$	0	0
$598$	−1.26016	−0.0515317
$599$	0.353620	0.0144485	0.00722427	−	0.999974i	$-0.497700\pi$
$599$	0.353620	0.0144485	0.00722427	+	0.999974i	$0.497700\pi$
$600$	0	0
$601$	−1.76477	−0.0719863	−0.0359931	−	0.999352i	$-0.511459\pi$
$601$	−1.76477	−0.0719863	−0.0359931	+	0.999352i	$0.511459\pi$
$602$	−5.19537	−0.211748
$603$	0	0
$604$	5.25183	0.213694
$605$	−0.105172	−0.00427584
$606$	0	0
$607$	−34.2574	−1.39047	−0.695233	−	0.718785i	$-0.744699\pi$
$607$	−34.2574	−1.39047	−0.695233	+	0.718785i	$0.744699\pi$
$608$	−36.0056	−1.46022
$609$	0	0
$610$	−0.00739985	−0.000299611 0
$611$	23.6260	0.955806
$612$	0	0
$613$	10.3428	0.417743	0.208871	−	0.977943i	$-0.433021\pi$
$613$	10.3428	0.417743	0.208871	+	0.977943i	$0.433021\pi$
$614$	−4.61368	−0.186193
$615$	0	0
$616$	3.32224	0.133857
$617$	−33.9967	−1.36866	−0.684328	−	0.729174i	$-0.739904\pi$
$617$	−33.9967	−1.36866	−0.684328	+	0.729174i	$0.739904\pi$
$618$	0	0
$619$	22.6908	0.912022	0.456011	−	0.889974i	$-0.349278\pi$
$619$	22.6908	0.912022	0.456011	+	0.889974i	$0.349278\pi$
$620$	−0.170310	−0.00683983
$621$	0	0
$622$	11.2294	0.450258
$623$	16.8790	0.676244
$624$	0	0
$625$	24.9983	0.999933
$626$	−7.85804	−0.314070
$627$	0	0
$628$	11.4464	0.456762
$629$	−13.3596	−0.532683
$630$	0	0
$631$	44.0926	1.75530	0.877649	−	0.479304i	$-0.159111\pi$
$631$	44.0926	1.75530	0.877649	+	0.479304i	$0.159111\pi$
$632$	−16.9136	−0.672786
$633$	0	0
$634$	−3.45312	−0.137141
$635$	0.186370	0.00739586
$636$	0	0
$637$	−9.56053	−0.378802
$638$	0.472521	0.0187073
$639$	0	0
$640$	−0.117358	−0.00463896
$641$	−35.7799	−1.41322	−0.706611	−	0.707603i	$-0.749777\pi$
$641$	−35.7799	−1.41322	−0.706611	+	0.707603i	$0.749777\pi$
$642$	0	0
$643$	14.1099	0.556438	0.278219	−	0.960518i	$-0.410256\pi$
$643$	14.1099	0.556438	0.278219	+	0.960518i	$0.410256\pi$
$644$	−3.31264	−0.130536
$645$	0	0
$646$	−12.2648	−0.482553
$647$	−31.5829	−1.24165	−0.620826	−	0.783948i	$-0.713203\pi$
$647$	−31.5829	−1.24165	−0.620826	+	0.783948i	$0.713203\pi$
$648$	0	0
$649$	−13.6730	−0.536713
$650$	−6.30065	−0.247132
$651$	0	0
$652$	−21.8486	−0.855657
$653$	−40.0535	−1.56742	−0.783708	−	0.621130i	$-0.786674\pi$
$653$	−40.0535	−1.56742	−0.783708	+	0.621130i	$0.786674\pi$
$654$	0	0
$655$	0.0503228	0.00196627
$656$	−2.52433	−0.0985587
$657$	0	0
$658$	−7.60441	−0.296451
$659$	−2.01244	−0.0783933	−0.0391967	−	0.999232i	$-0.512480\pi$
$659$	−2.01244	−0.0783933	−0.0391967	+	0.999232i	$0.512480\pi$
$660$	0	0
$661$	−22.8081	−0.887134	−0.443567	−	0.896241i	$-0.646287\pi$
$661$	−22.8081	−0.887134	−0.443567	+	0.896241i	$0.646287\pi$
$662$	3.23755	0.125831
$663$	0	0
$664$	26.5136	1.02893
$665$	0.146645	0.00568664
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	10.5061	0.406494
$669$	0	0
$670$	−0.0393019	−0.00151836
$671$	−1.52032	−0.0586912
$672$	0	0
$673$	5.35191	0.206301	0.103150	−	0.994666i	$-0.467108\pi$
$673$	5.35191	0.206301	0.103150	+	0.994666i	$0.467108\pi$
$674$	1.90093	0.0732211
$675$	0	0
$676$	10.1943	0.392088
$677$	30.9706	1.19030	0.595149	−	0.803615i	$-0.297093\pi$
$677$	30.9706	1.19030	0.595149	+	0.803615i	$0.297093\pi$
$678$	0	0
$679$	−2.39290	−0.0918309
$680$	−0.0653489	−0.00250602
$681$	0	0
$682$	4.28428	0.164054
$683$	48.6419	1.86123	0.930615	−	0.365999i	$-0.119273\pi$
$683$	48.6419	1.86123	0.930615	+	0.365999i	$0.119273\pi$
$684$	0	0
$685$	−0.0410644	−0.00156899
$686$	9.15579	0.349570
$687$	0	0
$688$	16.3847	0.624662
$689$	26.3964	1.00562
$690$	0	0
$691$	49.3224	1.87631	0.938156	−	0.346213i	$-0.112532\pi$
$691$	49.3224	1.87631	0.938156	+	0.346213i	$0.112532\pi$
$692$	−2.49573	−0.0948734
$693$	0	0
$694$	−3.46627	−0.131578
$695$	0.0334344	0.00126824
$696$	0	0
$697$	−3.23477	−0.122526
$698$	11.5111	0.435703
$699$	0	0
$700$	−16.5628	−0.626016
$701$	−24.7023	−0.932993	−0.466497	−	0.884523i	$-0.654484\pi$
$701$	−24.7023	−0.932993	−0.466497	+	0.884523i	$0.654484\pi$
$702$	0	0
$703$	28.4848	1.07432
$704$	−3.26712	−0.123134
$705$	0	0
$706$	10.1430	0.381737
$707$	33.3342	1.25366
$708$	0	0
$709$	21.9049	0.822655	0.411328	−	0.911488i	$-0.365065\pi$
$709$	21.9049	0.822655	0.411328	+	0.911488i	$0.365065\pi$
$710$	−0.0157377	−0.000590624 0
$711$	0	0
$712$	16.0376	0.601033
$713$	−9.06687	−0.339557
$714$	0	0
$715$	0.0287720	0.00107601
$716$	31.3505	1.17162
$717$	0	0
$718$	5.53645	0.206618
$719$	−24.6474	−0.919194	−0.459597	−	0.888128i	$-0.652006\pi$
$719$	−24.6474	−0.919194	−0.459597	+	0.888128i	$0.652006\pi$
$720$	0	0
$721$	−30.5301	−1.13700
$722$	17.2758	0.642941
$723$	0	0
$724$	38.5793	1.43379
$725$	−4.99989	−0.185691
$726$	0	0
$727$	1.09130	0.0404741	0.0202370	−	0.999795i	$-0.493558\pi$
$727$	1.09130	0.0404741	0.0202370	+	0.999795i	$0.493558\pi$
$728$	8.86002	0.328374
$729$	0	0
$730$	0.0608078	0.00225060
$731$	20.9960	0.776564
$732$	0	0
$733$	−11.3513	−0.419271	−0.209635	−	0.977780i	$-0.567228\pi$
$733$	−11.3513	−0.419271	−0.209635	+	0.977780i	$0.567228\pi$
$734$	−5.53303	−0.204228
$735$	0	0
$736$	−4.81203	−0.177374
$737$	−8.07467	−0.297434
$738$	0	0
$739$	37.8330	1.39171	0.695854	−	0.718183i	$-0.255026\pi$
$739$	37.8330	1.39171	0.695854	+	0.718183i	$0.255026\pi$
$740$	0.0715080	0.00262868
$741$	0	0
$742$	−8.49611	−0.311902
$743$	−19.9697	−0.732618	−0.366309	−	0.930493i	$-0.619379\pi$
$743$	−19.9697	−0.732618	−0.366309	+	0.930493i	$0.619379\pi$
$744$	0	0
$745$	−0.0207486	−0.000760170 0
$746$	−16.1192	−0.590165
$747$	0	0
$748$	−6.32578	−0.231294
$749$	5.58070	0.203914
$750$	0	0
$751$	−10.9630	−0.400046	−0.200023	−	0.979791i	$-0.564102\pi$
$751$	−10.9630	−0.400046	−0.200023	+	0.979791i	$0.564102\pi$
$752$	23.9821	0.874539
$753$	0	0
$754$	1.26016	0.0458922
$755$	−0.0310714	−0.00113080
$756$	0	0
$757$	30.3461	1.10295	0.551474	−	0.834192i	$-0.314066\pi$
$757$	30.3461	1.10295	0.551474	+	0.834192i	$0.314066\pi$
$758$	−1.44041	−0.0523180
$759$	0	0
$760$	0.139334	0.00505417
$761$	53.4315	1.93689	0.968446	−	0.249223i	$-0.0801753\pi$
$761$	53.4315	1.93689	0.968446	+	0.249223i	$0.0801753\pi$
$762$	0	0
$763$	25.8208	0.934777
$764$	16.0811	0.581793
$765$	0	0
$766$	11.3997	0.411887
$767$	−36.4643	−1.31665
$768$	0	0
$769$	40.3898	1.45649	0.728246	−	0.685315i	$-0.240336\pi$
$769$	40.3898	1.45649	0.728246	+	0.685315i	$0.240336\pi$
$770$	−0.00926074	−0.000333734 0
$771$	0	0
$772$	−4.12891	−0.148603
$773$	−21.5543	−0.775255	−0.387627	−	0.921816i	$-0.626705\pi$
$773$	−21.5543	−0.775255	−0.387627	+	0.921816i	$0.626705\pi$
$774$	0	0
$775$	−45.3333	−1.62842
$776$	−2.27360	−0.0816176
$777$	0	0
$778$	0.151677	0.00543790
$779$	6.89703	0.247112
$780$	0	0
$781$	−3.23334	−0.115698
$782$	−1.63915	−0.0586160
$783$	0	0
$784$	−9.70464	−0.346594
$785$	−0.0677205	−0.00241705
$786$	0	0
$787$	−28.6203	−1.02020	−0.510101	−	0.860114i	$-0.670392\pi$
$787$	−28.6203	−1.02020	−0.510101	+	0.860114i	$0.670392\pi$
$788$	37.9331	1.35131
$789$	0	0
$790$	0.0471466	0.00167740
$791$	−3.92086	−0.139410
$792$	0	0
$793$	−4.05451	−0.143980
$794$	3.26841	0.115992
$795$	0	0
$796$	10.5524	0.374021
$797$	41.4204	1.46719	0.733593	−	0.679589i	$-0.237842\pi$
$797$	41.4204	1.46719	0.733593	+	0.679589i	$0.237842\pi$
$798$	0	0
$799$	30.7316	1.08721
$800$	−24.0596	−0.850636
$801$	0	0
$802$	−8.64602	−0.305302
$803$	12.4931	0.440872
$804$	0	0
$805$	0.0195986	0.000690759 0
$806$	11.4257	0.402453
$807$	0	0
$808$	31.6723	1.11423
$809$	−16.4746	−0.579215	−0.289608	−	0.957145i	$-0.593525\pi$
$809$	−16.4746	−0.579215	−0.289608	+	0.957145i	$0.593525\pi$
$810$	0	0
$811$	−20.4990	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$811$	−20.4990	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$812$	3.31264	0.116251
$813$	0	0
$814$	−1.79884	−0.0630492
$815$	0.129263	0.00452788
$816$	0	0
$817$	−44.7667	−1.56619
$818$	−5.10599	−0.178527
$819$	0	0
$820$	0.0173143	0.000604640 0
$821$	−13.5365	−0.472427	−0.236213	−	0.971701i	$-0.575906\pi$
$821$	−13.5365	−0.472427	−0.236213	+	0.971701i	$0.575906\pi$
$822$	0	0
$823$	5.32001	0.185444	0.0927219	−	0.995692i	$-0.470443\pi$
$823$	5.32001	0.185444	0.0927219	+	0.995692i	$0.470443\pi$
$824$	−29.0081	−1.01055
$825$	0	0
$826$	11.7366	0.408370
$827$	−2.86383	−0.0995851	−0.0497926	−	0.998760i	$-0.515856\pi$
$827$	−2.86383	−0.0995851	−0.0497926	+	0.998760i	$0.515856\pi$
$828$	0	0
$829$	−50.6084	−1.75770	−0.878852	−	0.477095i	$-0.841690\pi$
$829$	−50.6084	−1.75770	−0.878852	+	0.477095i	$0.841690\pi$
$830$	−0.0739066	−0.00256533
$831$	0	0
$832$	−8.71304	−0.302070
$833$	−12.4359	−0.430877
$834$	0	0
$835$	−0.0621574	−0.00215105
$836$	13.4875	0.466476
$837$	0	0
$838$	−12.0115	−0.414930
$839$	19.9598	0.689089	0.344545	−	0.938770i	$-0.388033\pi$
$839$	19.9598	0.689089	0.344545	+	0.938770i	$0.388033\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	15.8938	0.547738
$843$	0	0
$844$	−13.7606	−0.473660
$845$	−0.0603125	−0.00207481
$846$	0	0
$847$	18.5477	0.637306
$848$	26.7943	0.920120
$849$	0	0
$850$	−8.19558	−0.281106
$851$	3.80689	0.130499
$852$	0	0
$853$	2.93688	0.100557	0.0502784	−	0.998735i	$-0.483989\pi$
$853$	2.93688	0.100557	0.0502784	+	0.998735i	$0.483989\pi$
$854$	1.30501	0.0446565
$855$	0	0
$856$	5.30248	0.181235
$857$	36.5423	1.24826	0.624131	−	0.781320i	$-0.285453\pi$
$857$	36.5423	1.24826	0.624131	+	0.781320i	$0.285453\pi$
$858$	0	0
$859$	15.2091	0.518929	0.259464	−	0.965753i	$-0.416454\pi$
$859$	15.2091	0.518929	0.259464	+	0.965753i	$0.416454\pi$
$860$	−0.112382	−0.00383219
$861$	0	0
$862$	15.9858	0.544478
$863$	11.0986	0.377802	0.188901	−	0.981996i	$-0.439507\pi$
$863$	11.0986	0.377802	0.188901	+	0.981996i	$0.439507\pi$
$864$	0	0
$865$	0.0147655	0.000502042 0
$866$	4.21016	0.143067
$867$	0	0
$868$	30.0353	1.01946
$869$	9.68639	0.328588
$870$	0	0
$871$	−21.5342	−0.729658
$872$	24.5336	0.830813
$873$	0	0
$874$	3.49493	0.118218
$875$	0.195984	0.00662546
$876$	0	0
$877$	−23.3671	−0.789051	−0.394526	−	0.918885i	$-0.629091\pi$
$877$	−23.3671	−0.789051	−0.394526	+	0.918885i	$0.629091\pi$
$878$	13.6109	0.459345
$879$	0	0
$880$	0.0292057	0.000984525 0
$881$	−45.1791	−1.52212	−0.761061	−	0.648680i	$-0.775321\pi$
$881$	−45.1791	−1.52212	−0.761061	+	0.648680i	$0.775321\pi$
$882$	0	0
$883$	25.5241	0.858955	0.429478	−	0.903077i	$-0.358698\pi$
$883$	25.5241	0.858955	0.429478	+	0.903077i	$0.358698\pi$
$884$	−16.8701	−0.567404
$885$	0	0
$886$	18.7839	0.631056
$887$	−15.3894	−0.516725	−0.258362	−	0.966048i	$-0.583183\pi$
$887$	−15.3894	−0.516725	−0.258362	+	0.966048i	$0.583183\pi$
$888$	0	0
$889$	−32.8674	−1.10234
$890$	−0.0447047	−0.00149850
$891$	0	0
$892$	19.4898	0.652568
$893$	−65.5245	−2.19269
$894$	0	0
$895$	−0.185479	−0.00619989
$896$	20.6967	0.691429
$897$	0	0
$898$	−2.77489	−0.0925994
$899$	9.06687	0.302397
$900$	0	0
$901$	34.3352	1.14387
$902$	−0.435553	−0.0145023
$903$	0	0
$904$	−3.72539	−0.123905
$905$	−0.228247	−0.00758719
$906$	0	0
$907$	−15.7921	−0.524368	−0.262184	−	0.965018i	$-0.584443\pi$
$907$	−15.7921	−0.524368	−0.262184	+	0.965018i	$0.584443\pi$
$908$	38.2905	1.27072
$909$	0	0
$910$	−0.0246973	−0.000818708 0
$911$	−2.70731	−0.0896972	−0.0448486	−	0.998994i	$-0.514281\pi$
$911$	−2.70731	−0.0896972	−0.0448486	+	0.998994i	$0.514281\pi$
$912$	0	0
$913$	−15.1843	−0.502526
$914$	4.26363	0.141028
$915$	0	0
$916$	18.5304	0.612262
$917$	−8.87473	−0.293069
$918$	0	0
$919$	23.0679	0.760940	0.380470	−	0.924793i	$-0.375762\pi$
$919$	23.0679	0.760940	0.380470	+	0.924793i	$0.375762\pi$
$920$	0.0186215	0.000613934 0
$921$	0	0
$922$	−13.5932	−0.447667
$923$	−8.62294	−0.283828
$924$	0	0
$925$	19.0340	0.625836
$926$	−9.23397	−0.303447
$927$	0	0
$928$	4.81203	0.157963
$929$	27.9038	0.915494	0.457747	−	0.889083i	$-0.348657\pi$
$929$	27.9038	0.915494	0.457747	+	0.889083i	$0.348657\pi$
$930$	0	0
$931$	26.5152	0.869000
$932$	16.6808	0.546397
$933$	0	0
$934$	−7.96770	−0.260711
$935$	0.0374252	0.00122394
$936$	0	0
$937$	−50.2667	−1.64214	−0.821070	−	0.570827i	$-0.806623\pi$
$937$	−50.2667	−1.64214	−0.821070	+	0.570827i	$0.806623\pi$
$938$	6.93113	0.226309
$939$	0	0
$940$	−0.164492	−0.00536515
$941$	0.205558	0.00670099	0.00335050	−	0.999994i	$-0.498934\pi$
$941$	0.205558	0.00670099	0.00335050	+	0.999994i	$0.498934\pi$
$942$	0	0
$943$	0.921766	0.0300168
$944$	−37.0140	−1.20470
$945$	0	0
$946$	2.82705	0.0919154
$947$	−12.3180	−0.400283	−0.200141	−	0.979767i	$-0.564140\pi$
$947$	−12.3180	−0.400283	−0.200141	+	0.979767i	$0.564140\pi$
$948$	0	0
$949$	33.3177	1.08154
$950$	17.4742	0.566939
$951$	0	0
$952$	11.5247	0.373517
$953$	12.5356	0.406068	0.203034	−	0.979172i	$-0.434920\pi$
$953$	12.5356	0.406068	0.203034	+	0.979172i	$0.434920\pi$
$954$	0	0
$955$	−0.0951406	−0.00307868
$956$	22.6598	0.732870
$957$	0	0
$958$	8.11057	0.262041
$959$	7.24197	0.233855
$960$	0	0
$961$	51.2081	1.65188
$962$	−4.79729	−0.154671
$963$	0	0
$964$	3.01506	0.0971086
$965$	0.0244279	0.000786361 0
$966$	0	0
$967$	−33.5440	−1.07870	−0.539351	−	0.842081i	$-0.681330\pi$
$967$	−33.5440	−1.07870	−0.539351	+	0.842081i	$0.681330\pi$
$968$	17.6230	0.566426
$969$	0	0
$970$	0.00633767	0.000203490 0
$971$	22.4426	0.720216	0.360108	−	0.932911i	$-0.382740\pi$
$971$	22.4426	0.720216	0.360108	+	0.932911i	$0.382740\pi$
$972$	0	0
$973$	−5.89637	−0.189029
$974$	18.4770	0.592042
$975$	0	0
$976$	−4.11563	−0.131738
$977$	7.78408	0.249035	0.124517	−	0.992217i	$-0.460262\pi$
$977$	7.78408	0.249035	0.124517	+	0.992217i	$0.460262\pi$
$978$	0	0
$979$	−9.18469	−0.293544
$980$	0.0665636	0.00212630
$981$	0	0
$982$	−4.03355	−0.128716
$983$	−33.4760	−1.06772	−0.533859	−	0.845574i	$-0.679259\pi$
$983$	−33.4760	−1.06772	−0.533859	+	0.845574i	$0.679259\pi$
$984$	0	0
$985$	−0.224424	−0.00715074
$986$	1.63915	0.0522012
$987$	0	0
$988$	35.9697	1.14435
$989$	−5.98292	−0.190246
$990$	0	0
$991$	2.35201	0.0747140	0.0373570	−	0.999302i	$-0.488106\pi$
$991$	2.35201	0.0747140	0.0373570	+	0.999302i	$0.488106\pi$
$992$	43.6300	1.38526
$993$	0	0
$994$	2.77543	0.0880314
$995$	−0.0624314	−0.00197921
$996$	0	0
$997$	28.4195	0.900055	0.450028	−	0.893015i	$-0.351414\pi$
$997$	28.4195	0.900055	0.450028	+	0.893015i	$0.351414\pi$
$998$	−18.3876	−0.582051
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.6		11
3.2	odd	2		2001.2.a.l.1.6	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.6	✓	11	3.2	odd	2
6003.2.a.m.1.6		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)

\(f(q)\)	\(=\)	\(q+0.467085 q^{2} -1.78183 q^{4} +0.0105419 q^{5} -1.85912 q^{7} -1.76644 q^{8} +O(q^{10})\)	\(q+0.467085 q^{2} -1.78183 q^{4} +0.0105419 q^{5} -1.85912 q^{7} -1.76644 q^{8} +0.00492395 q^{10} +1.01164 q^{11} +2.69792 q^{13} -0.868368 q^{14} +2.73859 q^{16} +3.50932 q^{17} -7.48241 q^{19} -0.0187838 q^{20} +0.472521 q^{22} -1.00000 q^{23} -4.99989 q^{25} +1.26016 q^{26} +3.31264 q^{28} +1.00000 q^{29} +9.06687 q^{31} +4.81203 q^{32} +1.63915 q^{34} -0.0195986 q^{35} -3.80689 q^{37} -3.49493 q^{38} -0.0186215 q^{40} -0.921766 q^{41} +5.98292 q^{43} -1.80257 q^{44} -0.467085 q^{46} +8.75713 q^{47} -3.54367 q^{49} -2.33537 q^{50} -4.80723 q^{52} +9.78399 q^{53} +0.0106645 q^{55} +3.28402 q^{56} +0.467085 q^{58} -13.5157 q^{59} -1.50283 q^{61} +4.23500 q^{62} -3.22954 q^{64} +0.0284411 q^{65} -7.98178 q^{67} -6.25302 q^{68} -0.00915421 q^{70} -3.19615 q^{71} +12.3494 q^{73} -1.77814 q^{74} +13.3324 q^{76} -1.88076 q^{77} +9.57497 q^{79} +0.0288698 q^{80} -0.430543 q^{82} -15.0096 q^{83} +0.0369947 q^{85} +2.79453 q^{86} -1.78699 q^{88} -9.07903 q^{89} -5.01575 q^{91} +1.78183 q^{92} +4.09033 q^{94} -0.0788785 q^{95} +1.28711 q^{97} -1.65520 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 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

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