LMFDB - Embedded newform 6042.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 14x^{5} + 29x^{4} + 48x^{3} - 14x^{2} - 35x - 10 $ x^7 - 3x^6 - 14x^5 + 29x^4 + 48x^3 - 14x^2 - 35x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.982384$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.0176159 q^{5} -1.00000 q^{6} +0.282315 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.0176159 q^{5} -1.00000 q^{6} +0.282315 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.0176159 q^{10} -2.15752 q^{11} +1.00000 q^{12} -0.142412 q^{13} -0.282315 q^{14} +0.0176159 q^{15} +1.00000 q^{16} +4.89293 q^{17} -1.00000 q^{18} -1.00000 q^{19} +0.0176159 q^{20} +0.282315 q^{21} +2.15752 q^{22} +1.73616 q^{23} -1.00000 q^{24} -4.99969 q^{25} +0.142412 q^{26} +1.00000 q^{27} +0.282315 q^{28} -6.91392 q^{29} -0.0176159 q^{30} -0.562432 q^{31} -1.00000 q^{32} -2.15752 q^{33} -4.89293 q^{34} +0.00497324 q^{35} +1.00000 q^{36} -3.30866 q^{37} +1.00000 q^{38} -0.142412 q^{39} -0.0176159 q^{40} -5.03315 q^{41} -0.282315 q^{42} +1.91745 q^{43} -2.15752 q^{44} +0.0176159 q^{45} -1.73616 q^{46} -5.14135 q^{47} +1.00000 q^{48} -6.92030 q^{49} +4.99969 q^{50} +4.89293 q^{51} -0.142412 q^{52} +1.00000 q^{53} -1.00000 q^{54} -0.0380067 q^{55} -0.282315 q^{56} -1.00000 q^{57} +6.91392 q^{58} +10.4982 q^{59} +0.0176159 q^{60} -14.9435 q^{61} +0.562432 q^{62} +0.282315 q^{63} +1.00000 q^{64} -0.00250872 q^{65} +2.15752 q^{66} -14.1900 q^{67} +4.89293 q^{68} +1.73616 q^{69} -0.00497324 q^{70} +10.4259 q^{71} -1.00000 q^{72} +5.18767 q^{73} +3.30866 q^{74} -4.99969 q^{75} -1.00000 q^{76} -0.609100 q^{77} +0.142412 q^{78} +6.71722 q^{79} +0.0176159 q^{80} +1.00000 q^{81} +5.03315 q^{82} +3.07084 q^{83} +0.282315 q^{84} +0.0861935 q^{85} -1.91745 q^{86} -6.91392 q^{87} +2.15752 q^{88} -16.3526 q^{89} -0.0176159 q^{90} -0.0402051 q^{91} +1.73616 q^{92} -0.562432 q^{93} +5.14135 q^{94} -0.0176159 q^{95} -1.00000 q^{96} +11.1739 q^{97} +6.92030 q^{98} -2.15752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{3} + 7 q^{4} + 4 q^{5} - 7 q^{6} - 9 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^3 + 7 * q^4 + 4 * q^5 - 7 * q^6 - 9 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} + 7 q^{3} + 7 q^{4} + 4 q^{5} - 7 q^{6} - 9 q^{7} - 7 q^{8} + 7 q^{9} - 4 q^{10} - 2 q^{11} + 7 q^{12} - 7 q^{13} + 9 q^{14} + 4 q^{15} + 7 q^{16} - 10 q^{17} - 7 q^{18} - 7 q^{19} + 4 q^{20} - 9 q^{21} + 2 q^{22} + 2 q^{23} - 7 q^{24} + 3 q^{25} + 7 q^{26} + 7 q^{27} - 9 q^{28} - 6 q^{29} - 4 q^{30} + 3 q^{31} - 7 q^{32} - 2 q^{33} + 10 q^{34} + 7 q^{36} - 3 q^{37} + 7 q^{38} - 7 q^{39} - 4 q^{40} - 12 q^{41} + 9 q^{42} - 26 q^{43} - 2 q^{44} + 4 q^{45} - 2 q^{46} + 17 q^{47} + 7 q^{48} - 3 q^{50} - 10 q^{51} - 7 q^{52} + 7 q^{53} - 7 q^{54} - 23 q^{55} + 9 q^{56} - 7 q^{57} + 6 q^{58} + 7 q^{59} + 4 q^{60} - 18 q^{61} - 3 q^{62} - 9 q^{63} + 7 q^{64} - 23 q^{65} + 2 q^{66} - 3 q^{67} - 10 q^{68} + 2 q^{69} + 2 q^{71} - 7 q^{72} + q^{73} + 3 q^{74} + 3 q^{75} - 7 q^{76} - 23 q^{77} + 7 q^{78} - 18 q^{79} + 4 q^{80} + 7 q^{81} + 12 q^{82} - 17 q^{83} - 9 q^{84} - 3 q^{85} + 26 q^{86} - 6 q^{87} + 2 q^{88} - 13 q^{89} - 4 q^{90} - 8 q^{91} + 2 q^{92} + 3 q^{93} - 17 q^{94} - 4 q^{95} - 7 q^{96} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^3 + 7 * q^4 + 4 * q^5 - 7 * q^6 - 9 * q^7 - 7 * q^8 + 7 * q^9 - 4 * q^10 - 2 * q^11 + 7 * q^12 - 7 * q^13 + 9 * q^14 + 4 * q^15 + 7 * q^16 - 10 * q^17 - 7 * q^18 - 7 * q^19 + 4 * q^20 - 9 * q^21 + 2 * q^22 + 2 * q^23 - 7 * q^24 + 3 * q^25 + 7 * q^26 + 7 * q^27 - 9 * q^28 - 6 * q^29 - 4 * q^30 + 3 * q^31 - 7 * q^32 - 2 * q^33 + 10 * q^34 + 7 * q^36 - 3 * q^37 + 7 * q^38 - 7 * q^39 - 4 * q^40 - 12 * q^41 + 9 * q^42 - 26 * q^43 - 2 * q^44 + 4 * q^45 - 2 * q^46 + 17 * q^47 + 7 * q^48 - 3 * q^50 - 10 * q^51 - 7 * q^52 + 7 * q^53 - 7 * q^54 - 23 * q^55 + 9 * q^56 - 7 * q^57 + 6 * q^58 + 7 * q^59 + 4 * q^60 - 18 * q^61 - 3 * q^62 - 9 * q^63 + 7 * q^64 - 23 * q^65 + 2 * q^66 - 3 * q^67 - 10 * q^68 + 2 * q^69 + 2 * q^71 - 7 * q^72 + q^73 + 3 * q^74 + 3 * q^75 - 7 * q^76 - 23 * q^77 + 7 * q^78 - 18 * q^79 + 4 * q^80 + 7 * q^81 + 12 * q^82 - 17 * q^83 - 9 * q^84 - 3 * q^85 + 26 * q^86 - 6 * q^87 + 2 * q^88 - 13 * q^89 - 4 * q^90 - 8 * q^91 + 2 * q^92 + 3 * q^93 - 17 * q^94 - 4 * q^95 - 7 * q^96 - 20 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0.0176159	0.00787807	0.00393904	−	0.999992i	$-0.498746\pi$
$5$	0.0176159	0.00787807	0.00393904	+	0.999992i	$0.498746\pi$
$6$	−1.00000	−0.408248
$7$	0.282315	0.106705	0.0533526	−	0.998576i	$-0.483009\pi$
$7$	0.282315	0.106705	0.0533526	+	0.998576i	$0.483009\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.0176159	−0.00557064
$11$	−2.15752	−0.650516	−0.325258	−	0.945625i	$-0.605451\pi$
$11$	−2.15752	−0.650516	−0.325258	+	0.945625i	$0.605451\pi$
$12$	1.00000	0.288675
$13$	−0.142412	−0.0394980	−0.0197490	−	0.999805i	$-0.506287\pi$
$13$	−0.142412	−0.0394980	−0.0197490	+	0.999805i	$0.506287\pi$
$14$	−0.282315	−0.0754519
$15$	0.0176159	0.00454841
$16$	1.00000	0.250000
$17$	4.89293	1.18671	0.593355	−	0.804941i	$-0.297803\pi$
$17$	4.89293	1.18671	0.593355	+	0.804941i	$0.297803\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.0176159	0.00393904
$21$	0.282315	0.0616062
$22$	2.15752	0.459985
$23$	1.73616	0.362015	0.181007	−	0.983482i	$-0.442064\pi$
$23$	1.73616	0.362015	0.181007	+	0.983482i	$0.442064\pi$
$24$	−1.00000	−0.204124
$25$	−4.99969	−0.999938
$26$	0.142412	0.0279293
$27$	1.00000	0.192450
$28$	0.282315	0.0533526
$29$	−6.91392	−1.28388	−0.641941	−	0.766754i	$-0.721871\pi$
$29$	−6.91392	−1.28388	−0.641941	+	0.766754i	$0.721871\pi$
$30$	−0.0176159	−0.00321621
$31$	−0.562432	−0.101016	−0.0505079	−	0.998724i	$-0.516084\pi$
$31$	−0.562432	−0.101016	−0.0505079	+	0.998724i	$0.516084\pi$
$32$	−1.00000	−0.176777
$33$	−2.15752	−0.375576
$34$	−4.89293	−0.839131
$35$	0.00497324	0.000840631 0
$36$	1.00000	0.166667
$37$	−3.30866	−0.543940	−0.271970	−	0.962306i	$-0.587675\pi$
$37$	−3.30866	−0.543940	−0.271970	+	0.962306i	$0.587675\pi$
$38$	1.00000	0.162221
$39$	−0.142412	−0.0228042
$40$	−0.0176159	−0.00278532
$41$	−5.03315	−0.786046	−0.393023	−	0.919529i	$-0.628571\pi$
$41$	−5.03315	−0.786046	−0.393023	+	0.919529i	$0.628571\pi$
$42$	−0.282315	−0.0435622
$43$	1.91745	0.292409	0.146204	−	0.989254i	$-0.453294\pi$
$43$	1.91745	0.292409	0.146204	+	0.989254i	$0.453294\pi$
$44$	−2.15752	−0.325258
$45$	0.0176159	0.00262602
$46$	−1.73616	−0.255983
$47$	−5.14135	−0.749944	−0.374972	−	0.927036i	$-0.622348\pi$
$47$	−5.14135	−0.749944	−0.374972	+	0.927036i	$0.622348\pi$
$48$	1.00000	0.144338
$49$	−6.92030	−0.988614
$50$	4.99969	0.707063
$51$	4.89293	0.685148
$52$	−0.142412	−0.0197490
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	−0.0380067	−0.00512482
$56$	−0.282315	−0.0377260
$57$	−1.00000	−0.132453
$58$	6.91392	0.907842
$59$	10.4982	1.36675	0.683374	−	0.730068i	$-0.260512\pi$
$59$	10.4982	1.36675	0.683374	+	0.730068i	$0.260512\pi$
$60$	0.0176159	0.00227420
$61$	−14.9435	−1.91332	−0.956659	−	0.291212i	$-0.905942\pi$
$61$	−14.9435	−1.91332	−0.956659	+	0.291212i	$0.905942\pi$
$62$	0.562432	0.0714289
$63$	0.282315	0.0355684
$64$	1.00000	0.125000
$65$	−0.00250872	−0.000311168 0
$66$	2.15752	0.265572
$67$	−14.1900	−1.73358	−0.866791	−	0.498672i	$-0.833821\pi$
$67$	−14.1900	−1.73358	−0.866791	+	0.498672i	$0.833821\pi$
$68$	4.89293	0.593355
$69$	1.73616	0.209009
$70$	−0.00497324	−0.000594416 0
$71$	10.4259	1.23732	0.618661	−	0.785658i	$-0.287675\pi$
$71$	10.4259	1.23732	0.618661	+	0.785658i	$0.287675\pi$
$72$	−1.00000	−0.117851
$73$	5.18767	0.607170	0.303585	−	0.952804i	$-0.401816\pi$
$73$	5.18767	0.607170	0.303585	+	0.952804i	$0.401816\pi$
$74$	3.30866	0.384624
$75$	−4.99969	−0.577314
$76$	−1.00000	−0.114708
$77$	−0.609100	−0.0694134
$78$	0.142412	0.0161250
$79$	6.71722	0.755747	0.377873	−	0.925857i	$-0.376655\pi$
$79$	6.71722	0.755747	0.377873	+	0.925857i	$0.376655\pi$
$80$	0.0176159	0.00196952
$81$	1.00000	0.111111
$82$	5.03315	0.555818
$83$	3.07084	0.337069	0.168534	−	0.985696i	$-0.446097\pi$
$83$	3.07084	0.337069	0.168534	+	0.985696i	$0.446097\pi$
$84$	0.282315	0.0308031
$85$	0.0861935	0.00934900
$86$	−1.91745	−0.206764
$87$	−6.91392	−0.741250
$88$	2.15752	0.229992
$89$	−16.3526	−1.73337	−0.866687	−	0.498852i	$-0.833755\pi$
$89$	−16.3526	−1.73337	−0.866687	+	0.498852i	$0.833755\pi$
$90$	−0.0176159	−0.00185688
$91$	−0.0402051	−0.00421464
$92$	1.73616	0.181007
$93$	−0.562432	−0.0583215
$94$	5.14135	0.530290
$95$	−0.0176159	−0.00180735
$96$	−1.00000	−0.102062
$97$	11.1739	1.13454	0.567270	−	0.823532i	$-0.308000\pi$
$97$	11.1739	1.13454	0.567270	+	0.823532i	$0.308000\pi$
$98$	6.92030	0.699056
$99$	−2.15752	−0.216839
$100$	−4.99969	−0.499969
$101$	7.26379	0.722775	0.361387	−	0.932416i	$-0.382303\pi$
$101$	7.26379	0.722775	0.361387	+	0.932416i	$0.382303\pi$
$102$	−4.89293	−0.484473
$103$	0.574957	0.0566522	0.0283261	−	0.999599i	$-0.490982\pi$
$103$	0.574957	0.0566522	0.0283261	+	0.999599i	$0.490982\pi$
$104$	0.142412	0.0139646
$105$	0.00497324	0.000485338 0
$106$	−1.00000	−0.0971286
$107$	−1.94101	−0.187644	−0.0938221	−	0.995589i	$-0.529908\pi$
$107$	−1.94101	−0.187644	−0.0938221	+	0.995589i	$0.529908\pi$
$108$	1.00000	0.0962250
$109$	−10.2409	−0.980904	−0.490452	−	0.871468i	$-0.663168\pi$
$109$	−10.2409	−0.980904	−0.490452	+	0.871468i	$0.663168\pi$
$110$	0.0380067	0.00362379
$111$	−3.30866	−0.314044
$112$	0.282315	0.0266763
$113$	12.7912	1.20329	0.601646	−	0.798763i	$-0.294512\pi$
$113$	12.7912	1.20329	0.601646	+	0.798763i	$0.294512\pi$
$114$	1.00000	0.0936586
$115$	0.0305841	0.00285198
$116$	−6.91392	−0.641941
$117$	−0.142412	−0.0131660
$118$	−10.4982	−0.966437
$119$	1.38135	0.126628
$120$	−0.0176159	−0.00160811
$121$	−6.34511	−0.576828
$122$	14.9435	1.35292
$123$	−5.03315	−0.453824
$124$	−0.562432	−0.0505079
$125$	−0.176154	−0.0157557
$126$	−0.282315	−0.0251506
$127$	1.33645	0.118591	0.0592954	−	0.998240i	$-0.481115\pi$
$127$	1.33645	0.118591	0.0592954	+	0.998240i	$0.481115\pi$
$128$	−1.00000	−0.0883883
$129$	1.91745	0.168822
$130$	0.00250872	0.000220029 0
$131$	−17.2686	−1.50877	−0.754384	−	0.656433i	$-0.772064\pi$
$131$	−17.2686	−1.50877	−0.754384	+	0.656433i	$0.772064\pi$
$132$	−2.15752	−0.187788
$133$	−0.282315	−0.0244798
$134$	14.1900	1.22583
$135$	0.0176159	0.00151614
$136$	−4.89293	−0.419566
$137$	−8.96141	−0.765625	−0.382813	−	0.923826i	$-0.625044\pi$
$137$	−8.96141	−0.765625	−0.382813	+	0.923826i	$0.625044\pi$
$138$	−1.73616	−0.147792
$139$	−10.9502	−0.928781	−0.464391	−	0.885631i	$-0.653727\pi$
$139$	−10.9502	−0.928781	−0.464391	+	0.885631i	$0.653727\pi$
$140$	0.00497324	0.000420315 0
$141$	−5.14135	−0.432980
$142$	−10.4259	−0.874919
$143$	0.307257	0.0256941
$144$	1.00000	0.0833333
$145$	−0.121795	−0.0101145
$146$	−5.18767	−0.429334
$147$	−6.92030	−0.570777
$148$	−3.30866	−0.271970
$149$	19.9498	1.63435	0.817176	−	0.576388i	$-0.195538\pi$
$149$	19.9498	1.63435	0.817176	+	0.576388i	$0.195538\pi$
$150$	4.99969	0.408223
$151$	−3.59920	−0.292899	−0.146449	−	0.989218i	$-0.546785\pi$
$151$	−3.59920	−0.292899	−0.146449	+	0.989218i	$0.546785\pi$
$152$	1.00000	0.0811107
$153$	4.89293	0.395570
$154$	0.609100	0.0490827
$155$	−0.00990775	−0.000795810 0
$156$	−0.142412	−0.0114021
$157$	2.57186	0.205257	0.102628	−	0.994720i	$-0.467275\pi$
$157$	2.57186	0.205257	0.102628	+	0.994720i	$0.467275\pi$
$158$	−6.71722	−0.534394
$159$	1.00000	0.0793052
$160$	−0.0176159	−0.00139266
$161$	0.490145	0.0386288
$162$	−1.00000	−0.0785674
$163$	−7.83837	−0.613948	−0.306974	−	0.951718i	$-0.599317\pi$
$163$	−7.83837	−0.613948	−0.306974	+	0.951718i	$0.599317\pi$
$164$	−5.03315	−0.393023
$165$	−0.0380067	−0.00295881
$166$	−3.07084	−0.238344
$167$	−2.52646	−0.195503	−0.0977517	−	0.995211i	$-0.531165\pi$
$167$	−2.52646	−0.195503	−0.0977517	+	0.995211i	$0.531165\pi$
$168$	−0.282315	−0.0217811
$169$	−12.9797	−0.998440
$170$	−0.0861935	−0.00661074
$171$	−1.00000	−0.0764719
$172$	1.91745	0.146204
$173$	−9.64553	−0.733336	−0.366668	−	0.930352i	$-0.619502\pi$
$173$	−9.64553	−0.733336	−0.366668	+	0.930352i	$0.619502\pi$
$174$	6.91392	0.524143
$175$	−1.41149	−0.106698
$176$	−2.15752	−0.162629
$177$	10.4982	0.789093
$178$	16.3526	1.22568
$179$	−11.5004	−0.859580	−0.429790	−	0.902929i	$-0.641412\pi$
$179$	−11.5004	−0.859580	−0.429790	+	0.902929i	$0.641412\pi$
$180$	0.0176159	0.00131301
$181$	−7.30332	−0.542852	−0.271426	−	0.962459i	$-0.587495\pi$
$181$	−7.30332	−0.542852	−0.271426	+	0.962459i	$0.587495\pi$
$182$	0.0402051	0.00298020
$183$	−14.9435	−1.10465
$184$	−1.73616	−0.127992
$185$	−0.0582851	−0.00428520
$186$	0.562432	0.0412395
$187$	−10.5566	−0.771975
$188$	−5.14135	−0.374972
$189$	0.282315	0.0205354
$190$	0.0176159	0.00127799
$191$	−3.90679	−0.282686	−0.141343	−	0.989961i	$-0.545142\pi$
$191$	−3.90679	−0.282686	−0.141343	+	0.989961i	$0.545142\pi$
$192$	1.00000	0.0721688
$193$	18.0679	1.30056	0.650279	−	0.759696i	$-0.274652\pi$
$193$	18.0679	1.30056	0.650279	+	0.759696i	$0.274652\pi$
$194$	−11.1739	−0.802241
$195$	−0.00250872	−0.000179653 0
$196$	−6.92030	−0.494307
$197$	1.90425	0.135672	0.0678360	−	0.997696i	$-0.478391\pi$
$197$	1.90425	0.135672	0.0678360	+	0.997696i	$0.478391\pi$
$198$	2.15752	0.153328
$199$	−8.50280	−0.602747	−0.301374	−	0.953506i	$-0.597445\pi$
$199$	−8.50280	−0.602747	−0.301374	+	0.953506i	$0.597445\pi$
$200$	4.99969	0.353531
$201$	−14.1900	−1.00088
$202$	−7.26379	−0.511079
$203$	−1.95190	−0.136997
$204$	4.89293	0.342574
$205$	−0.0886635	−0.00619253
$206$	−0.574957	−0.0400591
$207$	1.73616	0.120672
$208$	−0.142412	−0.00987449
$209$	2.15752	0.149239
$210$	−0.00497324	−0.000343186 0
$211$	17.4217	1.19936	0.599680	−	0.800240i	$-0.295295\pi$
$211$	17.4217	1.19936	0.599680	+	0.800240i	$0.295295\pi$
$212$	1.00000	0.0686803
$213$	10.4259	0.714368
$214$	1.94101	0.132684
$215$	0.0337776	0.00230362
$216$	−1.00000	−0.0680414
$217$	−0.158783	−0.0107789
$218$	10.2409	0.693604
$219$	5.18767	0.350550
$220$	−0.0380067	−0.00256241
$221$	−0.696812	−0.0468727
$222$	3.30866	0.222063
$223$	−27.7931	−1.86116	−0.930582	−	0.366083i	$-0.880699\pi$
$223$	−27.7931	−1.86116	−0.930582	+	0.366083i	$0.880699\pi$
$224$	−0.282315	−0.0188630
$225$	−4.99969	−0.333313
$226$	−12.7912	−0.850856
$227$	−8.24034	−0.546931	−0.273465	−	0.961882i	$-0.588170\pi$
$227$	−8.24034	−0.546931	−0.273465	+	0.961882i	$0.588170\pi$
$228$	−1.00000	−0.0662266
$229$	−6.53464	−0.431821	−0.215911	−	0.976413i	$-0.569272\pi$
$229$	−6.53464	−0.431821	−0.215911	+	0.976413i	$0.569272\pi$
$230$	−0.0305841	−0.00201665
$231$	−0.609100	−0.0400759
$232$	6.91392	0.453921
$233$	7.13971	0.467738	0.233869	−	0.972268i	$-0.424861\pi$
$233$	7.13971	0.467738	0.233869	+	0.972268i	$0.424861\pi$
$234$	0.142412	0.00930976
$235$	−0.0905696	−0.00590811
$236$	10.4982	0.683374
$237$	6.71722	0.436330
$238$	−1.38135	−0.0895396
$239$	3.03785	0.196502	0.0982512	−	0.995162i	$-0.468675\pi$
$239$	3.03785	0.196502	0.0982512	+	0.995162i	$0.468675\pi$
$240$	0.0176159	0.00113710
$241$	7.41576	0.477691	0.238845	−	0.971058i	$-0.423231\pi$
$241$	7.41576	0.477691	0.238845	+	0.971058i	$0.423231\pi$
$242$	6.34511	0.407879
$243$	1.00000	0.0641500
$244$	−14.9435	−0.956659
$245$	−0.121907	−0.00778837
$246$	5.03315	0.320902
$247$	0.142412	0.00906146
$248$	0.562432	0.0357145
$249$	3.07084	0.194607
$250$	0.176154	0.0111409
$251$	10.2999	0.650124	0.325062	−	0.945693i	$-0.394615\pi$
$251$	10.2999	0.650124	0.325062	+	0.945693i	$0.394615\pi$
$252$	0.282315	0.0177842
$253$	−3.74580	−0.235497
$254$	−1.33645	−0.0838563
$255$	0.0861935	0.00539764
$256$	1.00000	0.0625000
$257$	25.0322	1.56146	0.780732	−	0.624866i	$-0.214846\pi$
$257$	25.0322	1.56146	0.780732	+	0.624866i	$0.214846\pi$
$258$	−1.91745	−0.119375
$259$	−0.934085	−0.0580412
$260$	−0.00250872	−0.000155584 0
$261$	−6.91392	−0.427961
$262$	17.2686	1.06686
$263$	−11.7373	−0.723754	−0.361877	−	0.932226i	$-0.617864\pi$
$263$	−11.7373	−0.723754	−0.361877	+	0.932226i	$0.617864\pi$
$264$	2.15752	0.132786
$265$	0.0176159	0.00108214
$266$	0.282315	0.0173099
$267$	−16.3526	−1.00076
$268$	−14.1900	−0.866791
$269$	−1.79863	−0.109664	−0.0548322	−	0.998496i	$-0.517462\pi$
$269$	−1.79863	−0.109664	−0.0548322	+	0.998496i	$0.517462\pi$
$270$	−0.0176159	−0.00107207
$271$	−10.5079	−0.638312	−0.319156	−	0.947702i	$-0.603399\pi$
$271$	−10.5079	−0.638312	−0.319156	+	0.947702i	$0.603399\pi$
$272$	4.89293	0.296678
$273$	−0.0402051	−0.00243332
$274$	8.96141	0.541379
$275$	10.7869	0.650476
$276$	1.73616	0.104505
$277$	−20.8279	−1.25143	−0.625714	−	0.780052i	$-0.715192\pi$
$277$	−20.8279	−1.25143	−0.625714	+	0.780052i	$0.715192\pi$
$278$	10.9502	0.656747
$279$	−0.562432	−0.0336719
$280$	−0.00497324	−0.000297208 0
$281$	−18.0996	−1.07973	−0.539866	−	0.841751i	$-0.681525\pi$
$281$	−18.0996	−1.07973	−0.539866	+	0.841751i	$0.681525\pi$
$282$	5.14135	0.306163
$283$	−23.3464	−1.38780	−0.693899	−	0.720072i	$-0.744109\pi$
$283$	−23.3464	−1.38780	−0.693899	+	0.720072i	$0.744109\pi$
$284$	10.4259	0.618661
$285$	−0.0176159	−0.00104348
$286$	−0.307257	−0.0181685
$287$	−1.42093	−0.0838751
$288$	−1.00000	−0.0589256
$289$	6.94080	0.408283
$290$	0.121795	0.00715205
$291$	11.1739	0.655027
$292$	5.18767	0.303585
$293$	16.3860	0.957280	0.478640	−	0.878011i	$-0.341130\pi$
$293$	16.3860	0.957280	0.478640	+	0.878011i	$0.341130\pi$
$294$	6.92030	0.403600
$295$	0.184935	0.0107673
$296$	3.30866	0.192312
$297$	−2.15752	−0.125192
$298$	−19.9498	−1.15566
$299$	−0.247250	−0.0142989
$300$	−4.99969	−0.288657
$301$	0.541325	0.0312015
$302$	3.59920	0.207111
$303$	7.26379	0.417294
$304$	−1.00000	−0.0573539
$305$	−0.263243	−0.0150733
$306$	−4.89293	−0.279710
$307$	2.53537	0.144701	0.0723507	−	0.997379i	$-0.476950\pi$
$307$	2.53537	0.144701	0.0723507	+	0.997379i	$0.476950\pi$
$308$	−0.609100	−0.0347067
$309$	0.574957	0.0327082
$310$	0.00990775	0.000562722 0
$311$	27.5664	1.56315	0.781573	−	0.623814i	$-0.214418\pi$
$311$	27.5664	1.56315	0.781573	+	0.623814i	$0.214418\pi$
$312$	0.142412	0.00806249
$313$	−25.3513	−1.43294	−0.716471	−	0.697617i	$-0.754244\pi$
$313$	−25.3513	−1.43294	−0.716471	+	0.697617i	$0.754244\pi$
$314$	−2.57186	−0.145139
$315$	0.00497324	0.000280210 0
$316$	6.71722	0.377873
$317$	30.0234	1.68628	0.843141	−	0.537693i	$-0.180704\pi$
$317$	30.0234	1.68628	0.843141	+	0.537693i	$0.180704\pi$
$318$	−1.00000	−0.0560772
$319$	14.9169	0.835187
$320$	0.0176159	0.000984759 0
$321$	−1.94101	−0.108336
$322$	−0.490145	−0.0273147
$323$	−4.89293	−0.272250
$324$	1.00000	0.0555556
$325$	0.712016	0.0394955
$326$	7.83837	0.434127
$327$	−10.2409	−0.566325
$328$	5.03315	0.277909
$329$	−1.45148	−0.0800228
$330$	0.0380067	0.00209220
$331$	−12.6956	−0.697816	−0.348908	−	0.937157i	$-0.613447\pi$
$331$	−12.6956	−0.697816	−0.348908	+	0.937157i	$0.613447\pi$
$332$	3.07084	0.168534
$333$	−3.30866	−0.181313
$334$	2.52646	0.138242
$335$	−0.249969	−0.0136573
$336$	0.282315	0.0154016
$337$	−16.8494	−0.917845	−0.458923	−	0.888476i	$-0.651764\pi$
$337$	−16.8494	−0.917845	−0.458923	+	0.888476i	$0.651764\pi$
$338$	12.9797	0.706004
$339$	12.7912	0.694721
$340$	0.0861935	0.00467450
$341$	1.21346	0.0657124
$342$	1.00000	0.0540738
$343$	−3.92991	−0.212195
$344$	−1.91745	−0.103382
$345$	0.0305841	0.00164659
$346$	9.64553	0.518547
$347$	−11.0665	−0.594082	−0.297041	−	0.954865i	$-0.596000\pi$
$347$	−11.0665	−0.594082	−0.297041	+	0.954865i	$0.596000\pi$
$348$	−6.91392	−0.370625
$349$	18.5029	0.990436	0.495218	−	0.868769i	$-0.335088\pi$
$349$	18.5029	0.990436	0.495218	+	0.868769i	$0.335088\pi$
$350$	1.41149	0.0754472
$351$	−0.142412	−0.00760139
$352$	2.15752	0.114996
$353$	−1.60512	−0.0854321	−0.0427161	−	0.999087i	$-0.513601\pi$
$353$	−1.60512	−0.0854321	−0.0427161	+	0.999087i	$0.513601\pi$
$354$	−10.4982	−0.557973
$355$	0.183661	0.00974772
$356$	−16.3526	−0.866687
$357$	1.38135	0.0731088
$358$	11.5004	0.607815
$359$	−29.8127	−1.57345	−0.786727	−	0.617301i	$-0.788226\pi$
$359$	−29.8127	−1.57345	−0.786727	+	0.617301i	$0.788226\pi$
$360$	−0.0176159	−0.000928440 0
$361$	1.00000	0.0526316
$362$	7.30332	0.383854
$363$	−6.34511	−0.333032
$364$	−0.0402051	−0.00210732
$365$	0.0913854	0.00478333
$366$	14.9435	0.781109
$367$	−12.5665	−0.655966	−0.327983	−	0.944684i	$-0.606369\pi$
$367$	−12.5665	−0.655966	−0.327983	+	0.944684i	$0.606369\pi$
$368$	1.73616	0.0905037
$369$	−5.03315	−0.262015
$370$	0.0582851	0.00303010
$371$	0.282315	0.0146571
$372$	−0.562432	−0.0291607
$373$	−3.22759	−0.167118	−0.0835592	−	0.996503i	$-0.526629\pi$
$373$	−3.22759	−0.167118	−0.0835592	+	0.996503i	$0.526629\pi$
$374$	10.5566	0.545869
$375$	−0.176154	−0.00909653
$376$	5.14135	0.265145
$377$	0.984625	0.0507108
$378$	−0.282315	−0.0145207
$379$	−10.3821	−0.533292	−0.266646	−	0.963795i	$-0.585915\pi$
$379$	−10.3821	−0.533292	−0.266646	+	0.963795i	$0.585915\pi$
$380$	−0.0176159	−0.000903677 0
$381$	1.33645	0.0684684
$382$	3.90679	0.199889
$383$	15.5054	0.792290	0.396145	−	0.918188i	$-0.370348\pi$
$383$	15.5054	0.792290	0.396145	+	0.918188i	$0.370348\pi$
$384$	−1.00000	−0.0510310
$385$	−0.0107299	−0.000546844 0
$386$	−18.0679	−0.919633
$387$	1.91745	0.0974695
$388$	11.1739	0.567270
$389$	−21.0052	−1.06500	−0.532502	−	0.846428i	$-0.678748\pi$
$389$	−21.0052	−1.06500	−0.532502	+	0.846428i	$0.678748\pi$
$390$	0.00250872	0.000127034 0
$391$	8.49493	0.429607
$392$	6.92030	0.349528
$393$	−17.2686	−0.871088
$394$	−1.90425	−0.0959345
$395$	0.118330	0.00595383
$396$	−2.15752	−0.108419
$397$	35.1115	1.76220	0.881098	−	0.472934i	$-0.156805\pi$
$397$	35.1115	1.76220	0.881098	+	0.472934i	$0.156805\pi$
$398$	8.50280	0.426207
$399$	−0.282315	−0.0141334
$400$	−4.99969	−0.249984
$401$	16.5759	0.827763	0.413881	−	0.910331i	$-0.364173\pi$
$401$	16.5759	0.827763	0.413881	+	0.910331i	$0.364173\pi$
$402$	14.1900	0.707732
$403$	0.0800970	0.00398992
$404$	7.26379	0.361387
$405$	0.0176159	0.000875342 0
$406$	1.95190	0.0968714
$407$	7.13850	0.353842
$408$	−4.89293	−0.242236
$409$	−13.3143	−0.658352	−0.329176	−	0.944269i	$-0.606771\pi$
$409$	−13.3143	−0.658352	−0.329176	+	0.944269i	$0.606771\pi$
$410$	0.0886635	0.00437878
$411$	−8.96141	−0.442034
$412$	0.574957	0.0283261
$413$	2.96380	0.145839
$414$	−1.73616	−0.0853277
$415$	0.0540957	0.00265545
$416$	0.142412	0.00698232
$417$	−10.9502	−0.536232
$418$	−2.15752	−0.105528
$419$	−27.6831	−1.35241	−0.676203	−	0.736715i	$-0.736376\pi$
$419$	−27.6831	−1.35241	−0.676203	+	0.736715i	$0.736376\pi$
$420$	0.00497324	0.000242669 0
$421$	−14.3844	−0.701052	−0.350526	−	0.936553i	$-0.613997\pi$
$421$	−14.3844	−0.701052	−0.350526	+	0.936553i	$0.613997\pi$
$422$	−17.4217	−0.848075
$423$	−5.14135	−0.249981
$424$	−1.00000	−0.0485643
$425$	−24.4632	−1.18664
$426$	−10.4259	−0.505135
$427$	−4.21877	−0.204161
$428$	−1.94101	−0.0938221
$429$	0.307257	0.0148345
$430$	−0.0337776	−0.00162890
$431$	−39.2041	−1.88839	−0.944197	−	0.329381i	$-0.893160\pi$
$431$	−39.2041	−1.88839	−0.944197	+	0.329381i	$0.893160\pi$
$432$	1.00000	0.0481125
$433$	19.7298	0.948152	0.474076	−	0.880484i	$-0.342782\pi$
$433$	19.7298	0.948152	0.474076	+	0.880484i	$0.342782\pi$
$434$	0.158783	0.00762183
$435$	−0.121795	−0.00583962
$436$	−10.2409	−0.490452
$437$	−1.73616	−0.0830519
$438$	−5.18767	−0.247876
$439$	11.6044	0.553849	0.276925	−	0.960892i	$-0.410685\pi$
$439$	11.6044	0.553849	0.276925	+	0.960892i	$0.410685\pi$
$440$	0.0380067	0.00181190
$441$	−6.92030	−0.329538
$442$	0.696812	0.0331440
$443$	−29.8289	−1.41721	−0.708606	−	0.705604i	$-0.750676\pi$
$443$	−29.8289	−1.41721	−0.708606	+	0.705604i	$0.750676\pi$
$444$	−3.30866	−0.157022
$445$	−0.288066	−0.0136557
$446$	27.7931	1.31604
$447$	19.9498	0.943593
$448$	0.282315	0.0133381
$449$	8.94810	0.422287	0.211143	−	0.977455i	$-0.432281\pi$
$449$	8.94810	0.422287	0.211143	+	0.977455i	$0.432281\pi$
$450$	4.99969	0.235688
$451$	10.8591	0.511336
$452$	12.7912	0.601646
$453$	−3.59920	−0.169105
$454$	8.24034	0.386738
$455$	−0.000708249 0	−3.32032e−5 0
$456$	1.00000	0.0468293
$457$	5.81701	0.272108	0.136054	−	0.990701i	$-0.456558\pi$
$457$	5.81701	0.272108	0.136054	+	0.990701i	$0.456558\pi$
$458$	6.53464	0.305344
$459$	4.89293	0.228383
$460$	0.0305841	0.00142599
$461$	4.82072	0.224524	0.112262	−	0.993679i	$-0.464190\pi$
$461$	4.82072	0.224524	0.112262	+	0.993679i	$0.464190\pi$
$462$	0.609100	0.0283379
$463$	38.6833	1.79777	0.898883	−	0.438190i	$-0.144380\pi$
$463$	38.6833	1.79777	0.898883	+	0.438190i	$0.144380\pi$
$464$	−6.91392	−0.320971
$465$	−0.00990775	−0.000459461 0
$466$	−7.13971	−0.330740
$467$	9.99090	0.462324	0.231162	−	0.972915i	$-0.425747\pi$
$467$	9.99090	0.462324	0.231162	+	0.972915i	$0.425747\pi$
$468$	−0.142412	−0.00658299
$469$	−4.00605	−0.184982
$470$	0.0905696	0.00417766
$471$	2.57186	0.118505
$472$	−10.4982	−0.483219
$473$	−4.13694	−0.190217
$474$	−6.71722	−0.308532
$475$	4.99969	0.229401
$476$	1.38135	0.0633141
$477$	1.00000	0.0457869
$478$	−3.03785	−0.138948
$479$	12.8715	0.588113	0.294057	−	0.955788i	$-0.404995\pi$
$479$	12.8715	0.588113	0.294057	+	0.955788i	$0.404995\pi$
$480$	−0.0176159	−0.000804053 0
$481$	0.471193	0.0214845
$482$	−7.41576	−0.337779
$483$	0.490145	0.0223024
$484$	−6.34511	−0.288414
$485$	0.196839	0.00893799
$486$	−1.00000	−0.0453609
$487$	22.9649	1.04064	0.520320	−	0.853971i	$-0.325813\pi$
$487$	22.9649	1.04064	0.520320	+	0.853971i	$0.325813\pi$
$488$	14.9435	0.676460
$489$	−7.83837	−0.354463
$490$	0.121907	0.00550721
$491$	13.3040	0.600402	0.300201	−	0.953876i	$-0.402946\pi$
$491$	13.3040	0.600402	0.300201	+	0.953876i	$0.402946\pi$
$492$	−5.03315	−0.226912
$493$	−33.8294	−1.52360
$494$	−0.142412	−0.00640742
$495$	−0.0380067	−0.00170827
$496$	−0.562432	−0.0252539
$497$	2.94338	0.132029
$498$	−3.07084	−0.137608
$499$	−19.9587	−0.893472	−0.446736	−	0.894666i	$-0.647414\pi$
$499$	−19.9587	−0.893472	−0.446736	+	0.894666i	$0.647414\pi$
$500$	−0.176154	−0.00787783
$501$	−2.52646	−0.112874
$502$	−10.2999	−0.459707
$503$	−14.7937	−0.659620	−0.329810	−	0.944047i	$-0.606985\pi$
$503$	−14.7937	−0.659620	−0.329810	+	0.944047i	$0.606985\pi$
$504$	−0.282315	−0.0125753
$505$	0.127958	0.00569407
$506$	3.74580	0.166521
$507$	−12.9797	−0.576450
$508$	1.33645	0.0592954
$509$	−13.4673	−0.596927	−0.298464	−	0.954421i	$-0.596474\pi$
$509$	−13.4673	−0.596927	−0.298464	+	0.954421i	$0.596474\pi$
$510$	−0.0861935	−0.00381671
$511$	1.46456	0.0647882
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	−25.0322	−1.10412
$515$	0.0101284	0.000446310 0
$516$	1.91745	0.0844111
$517$	11.0926	0.487851
$518$	0.934085	0.0410413
$519$	−9.64553	−0.423392
$520$	0.00250872	0.000110014 0
$521$	−24.0096	−1.05188	−0.525940	−	0.850522i	$-0.676286\pi$
$521$	−24.0096	−1.05188	−0.525940	+	0.850522i	$0.676286\pi$
$522$	6.91392	0.302614
$523$	3.79860	0.166101	0.0830506	−	0.996545i	$-0.473534\pi$
$523$	3.79860	0.166101	0.0830506	+	0.996545i	$0.473534\pi$
$524$	−17.2686	−0.754384
$525$	−1.41149	−0.0616024
$526$	11.7373	0.511771
$527$	−2.75194	−0.119876
$528$	−2.15752	−0.0938940
$529$	−19.9857	−0.868945
$530$	−0.0176159	−0.000765186 0
$531$	10.4982	0.455583
$532$	−0.282315	−0.0122399
$533$	0.716780	0.0310472
$534$	16.3526	0.707647
$535$	−0.0341926	−0.00147827
$536$	14.1900	0.612914
$537$	−11.5004	−0.496279
$538$	1.79863	0.0775445
$539$	14.9307	0.643110
$540$	0.0176159	0.000758068 0
$541$	4.70964	0.202483	0.101242	−	0.994862i	$-0.467718\pi$
$541$	4.70964	0.202483	0.101242	+	0.994862i	$0.467718\pi$
$542$	10.5079	0.451355
$543$	−7.30332	−0.313416
$544$	−4.89293	−0.209783
$545$	−0.180403	−0.00772763
$546$	0.0402051	0.00172062
$547$	−0.545434	−0.0233211	−0.0116605	−	0.999932i	$-0.503712\pi$
$547$	−0.545434	−0.0233211	−0.0116605	+	0.999932i	$0.503712\pi$
$548$	−8.96141	−0.382813
$549$	−14.9435	−0.637772
$550$	−10.7869	−0.459956
$551$	6.91392	0.294543
$552$	−1.73616	−0.0738960
$553$	1.89637	0.0806420
$554$	20.8279	0.884894
$555$	−0.0582851	−0.00247406
$556$	−10.9502	−0.464391
$557$	24.1931	1.02509	0.512546	−	0.858660i	$-0.328702\pi$
$557$	24.1931	1.02509	0.512546	+	0.858660i	$0.328702\pi$
$558$	0.562432	0.0238096
$559$	−0.273068	−0.0115495
$560$	0.00497324	0.000210158 0
$561$	−10.5566	−0.445700
$562$	18.0996	0.763486
$563$	−2.59791	−0.109489	−0.0547445	−	0.998500i	$-0.517434\pi$
$563$	−2.59791	−0.109489	−0.0547445	+	0.998500i	$0.517434\pi$
$564$	−5.14135	−0.216490
$565$	0.225328	0.00947962
$566$	23.3464	0.981322
$567$	0.282315	0.0118561
$568$	−10.4259	−0.437459
$569$	6.30279	0.264227	0.132113	−	0.991235i	$-0.457824\pi$
$569$	6.30279	0.264227	0.132113	+	0.991235i	$0.457824\pi$
$570$	0.0176159	0.000737849 0
$571$	−39.9175	−1.67050	−0.835249	−	0.549872i	$-0.814676\pi$
$571$	−39.9175	−1.67050	−0.835249	+	0.549872i	$0.814676\pi$
$572$	0.307257	0.0128470
$573$	−3.90679	−0.163209
$574$	1.42093	0.0593086
$575$	−8.68028	−0.361992
$576$	1.00000	0.0416667
$577$	−18.6044	−0.774510	−0.387255	−	0.921973i	$-0.626577\pi$
$577$	−18.6044	−0.774510	−0.387255	+	0.921973i	$0.626577\pi$
$578$	−6.94080	−0.288699
$579$	18.0679	0.750877
$580$	−0.121795	−0.00505726
$581$	0.866946	0.0359670
$582$	−11.1739	−0.463174
$583$	−2.15752	−0.0893553
$584$	−5.18767	−0.214667
$585$	−0.00250872	−0.000103723 0
$586$	−16.3860	−0.676899
$587$	−9.68225	−0.399629	−0.199815	−	0.979834i	$-0.564034\pi$
$587$	−9.68225	−0.399629	−0.199815	+	0.979834i	$0.564034\pi$
$588$	−6.92030	−0.285388
$589$	0.562432	0.0231746
$590$	−0.184935	−0.00761366
$591$	1.90425	0.0783302
$592$	−3.30866	−0.135985
$593$	14.9810	0.615197	0.307598	−	0.951516i	$-0.400475\pi$
$593$	14.9810	0.615197	0.307598	+	0.951516i	$0.400475\pi$
$594$	2.15752	0.0885241
$595$	0.0243337	0.000997586 0
$596$	19.9498	0.817176
$597$	−8.50280	−0.347996
$598$	0.247250	0.0101108
$599$	−28.1739	−1.15115	−0.575577	−	0.817748i	$-0.695222\pi$
$599$	−28.1739	−1.15115	−0.575577	+	0.817748i	$0.695222\pi$
$600$	4.99969	0.204111
$601$	18.6322	0.760024	0.380012	−	0.924981i	$-0.375920\pi$
$601$	18.6322	0.760024	0.380012	+	0.924981i	$0.375920\pi$
$602$	−0.541325	−0.0220628
$603$	−14.1900	−0.577861
$604$	−3.59920	−0.146449
$605$	−0.111775	−0.00454430
$606$	−7.26379	−0.295071
$607$	−0.991856	−0.0402582	−0.0201291	−	0.999797i	$-0.506408\pi$
$607$	−0.991856	−0.0402582	−0.0201291	+	0.999797i	$0.506408\pi$
$608$	1.00000	0.0405554
$609$	−1.95190	−0.0790952
$610$	0.263243	0.0106584
$611$	0.732190	0.0296212
$612$	4.89293	0.197785
$613$	25.8279	1.04318	0.521589	−	0.853197i	$-0.325339\pi$
$613$	25.8279	1.04318	0.521589	+	0.853197i	$0.325339\pi$
$614$	−2.53537	−0.102319
$615$	−0.0886635	−0.00357526
$616$	0.609100	0.0245414
$617$	20.7056	0.833577	0.416788	−	0.909003i	$-0.363156\pi$
$617$	20.7056	0.833577	0.416788	+	0.909003i	$0.363156\pi$
$618$	−0.574957	−0.0231282
$619$	6.18007	0.248398	0.124199	−	0.992257i	$-0.460364\pi$
$619$	6.18007	0.248398	0.124199	+	0.992257i	$0.460364\pi$
$620$	−0.00990775	−0.000397905 0
$621$	1.73616	0.0696698
$622$	−27.5664	−1.10531
$623$	−4.61659	−0.184960
$624$	−0.142412	−0.00570104
$625$	24.9953	0.999814
$626$	25.3513	1.01324
$627$	2.15752	0.0861630
$628$	2.57186	0.102628
$629$	−16.1891	−0.645500
$630$	−0.00497324	−0.000198139 0
$631$	21.2310	0.845193	0.422597	−	0.906318i	$-0.361119\pi$
$631$	21.2310	0.845193	0.422597	+	0.906318i	$0.361119\pi$
$632$	−6.71722	−0.267197
$633$	17.4217	0.692450
$634$	−30.0234	−1.19238
$635$	0.0235428	0.000934266 0
$636$	1.00000	0.0396526
$637$	0.985533	0.0390482
$638$	−14.9169	−0.590566
$639$	10.4259	0.412441
$640$	−0.0176159	−0.000696330 0
$641$	24.9499	0.985462	0.492731	−	0.870182i	$-0.335999\pi$
$641$	24.9499	0.985462	0.492731	+	0.870182i	$0.335999\pi$
$642$	1.94101	0.0766054
$643$	−14.7513	−0.581736	−0.290868	−	0.956763i	$-0.593944\pi$
$643$	−14.7513	−0.581736	−0.290868	+	0.956763i	$0.593944\pi$
$644$	0.490145	0.0193144
$645$	0.0337776	0.00132999
$646$	4.89293	0.192510
$647$	−0.642024	−0.0252406	−0.0126203	−	0.999920i	$-0.504017\pi$
$647$	−0.642024	−0.0252406	−0.0126203	+	0.999920i	$0.504017\pi$
$648$	−1.00000	−0.0392837
$649$	−22.6501	−0.889092
$650$	−0.712016	−0.0279275
$651$	−0.158783	−0.00622320
$652$	−7.83837	−0.306974
$653$	1.05293	0.0412043	0.0206021	−	0.999788i	$-0.493442\pi$
$653$	1.05293	0.0412043	0.0206021	+	0.999788i	$0.493442\pi$
$654$	10.2409	0.400452
$655$	−0.304203	−0.0118862
$656$	−5.03315	−0.196511
$657$	5.18767	0.202390
$658$	1.45148	0.0565847
$659$	13.5413	0.527494	0.263747	−	0.964592i	$-0.415042\pi$
$659$	13.5413	0.527494	0.263747	+	0.964592i	$0.415042\pi$
$660$	−0.0380067	−0.00147941
$661$	−41.5017	−1.61423	−0.807114	−	0.590395i	$-0.798972\pi$
$661$	−41.5017	−1.61423	−0.807114	+	0.590395i	$0.798972\pi$
$662$	12.6956	0.493430
$663$	−0.696812	−0.0270619
$664$	−3.07084	−0.119172
$665$	−0.00497324	−0.000192854 0
$666$	3.30866	0.128208
$667$	−12.0037	−0.464785
$668$	−2.52646	−0.0977517
$669$	−27.7931	−1.07454
$670$	0.249969	0.00965716
$671$	32.2409	1.24464
$672$	−0.282315	−0.0108905
$673$	−2.68007	−0.103309	−0.0516545	−	0.998665i	$-0.516449\pi$
$673$	−2.68007	−0.103309	−0.0516545	+	0.998665i	$0.516449\pi$
$674$	16.8494	0.649015
$675$	−4.99969	−0.192438
$676$	−12.9797	−0.499220
$677$	−22.0745	−0.848393	−0.424196	−	0.905570i	$-0.639443\pi$
$677$	−22.0745	−0.848393	−0.424196	+	0.905570i	$0.639443\pi$
$678$	−12.7912	−0.491242
$679$	3.15457	0.121061
$680$	−0.0861935	−0.00330537
$681$	−8.24034	−0.315771
$682$	−1.21346	−0.0464657
$683$	38.6068	1.47725	0.738623	−	0.674118i	$-0.235476\pi$
$683$	38.6068	1.47725	0.738623	+	0.674118i	$0.235476\pi$
$684$	−1.00000	−0.0382360
$685$	−0.157863	−0.00603165
$686$	3.92991	0.150045
$687$	−6.53464	−0.249312
$688$	1.91745	0.0731021
$689$	−0.142412	−0.00542546
$690$	−0.0305841	−0.00116432
$691$	4.93344	0.187677	0.0938385	−	0.995587i	$-0.470086\pi$
$691$	4.93344	0.187677	0.0938385	+	0.995587i	$0.470086\pi$
$692$	−9.64553	−0.366668
$693$	−0.609100	−0.0231378
$694$	11.0665	0.420079
$695$	−0.192897	−0.00731701
$696$	6.91392	0.262071
$697$	−24.6269	−0.932809
$698$	−18.5029	−0.700344
$699$	7.13971	0.270048
$700$	−1.41149	−0.0533492
$701$	−7.09040	−0.267801	−0.133900	−	0.990995i	$-0.542750\pi$
$701$	−7.09040	−0.267801	−0.133900	+	0.990995i	$0.542750\pi$
$702$	0.142412	0.00537499
$703$	3.30866	0.124789
$704$	−2.15752	−0.0813146
$705$	−0.0905696	−0.00341105
$706$	1.60512	0.0604096
$707$	2.05068	0.0771237
$708$	10.4982	0.394546
$709$	11.1790	0.419838	0.209919	−	0.977719i	$-0.432680\pi$
$709$	11.1790	0.419838	0.209919	+	0.977719i	$0.432680\pi$
$710$	−0.183661	−0.00689268
$711$	6.71722	0.251916
$712$	16.3526	0.612841
$713$	−0.976473	−0.0365692
$714$	−1.38135	−0.0516957
$715$	0.00541260	0.000202420 0
$716$	−11.5004	−0.429790
$717$	3.03785	0.113451
$718$	29.8127	1.11260
$719$	28.5447	1.06454	0.532268	−	0.846576i	$-0.321340\pi$
$719$	28.5447	1.06454	0.532268	+	0.846576i	$0.321340\pi$
$720$	0.0176159	0.000656506 0
$721$	0.162319	0.00604508
$722$	−1.00000	−0.0372161
$723$	7.41576	0.275795
$724$	−7.30332	−0.271426
$725$	34.5675	1.28380
$726$	6.34511	0.235489
$727$	7.30209	0.270819	0.135410	−	0.990790i	$-0.456765\pi$
$727$	7.30209	0.270819	0.135410	+	0.990790i	$0.456765\pi$
$728$	0.0402051	0.00149010
$729$	1.00000	0.0370370
$730$	−0.0913854	−0.00338233
$731$	9.38196	0.347004
$732$	−14.9435	−0.552327
$733$	−20.9509	−0.773839	−0.386919	−	0.922114i	$-0.626461\pi$
$733$	−20.9509	−0.773839	−0.386919	+	0.922114i	$0.626461\pi$
$734$	12.5665	0.463838
$735$	−0.121907	−0.00449662
$736$	−1.73616	−0.0639958
$737$	30.6151	1.12772
$738$	5.03315	0.185273
$739$	−29.4336	−1.08273	−0.541367	−	0.840787i	$-0.682093\pi$
$739$	−29.4336	−1.08273	−0.541367	+	0.840787i	$0.682093\pi$
$740$	−0.0582851	−0.00214260
$741$	0.142412	0.00523163
$742$	−0.282315	−0.0103641
$743$	−37.6301	−1.38051	−0.690257	−	0.723564i	$-0.742502\pi$
$743$	−37.6301	−1.38051	−0.690257	+	0.723564i	$0.742502\pi$
$744$	0.562432	0.0206198
$745$	0.351434	0.0128755
$746$	3.22759	0.118171
$747$	3.07084	0.112356
$748$	−10.5566	−0.385987
$749$	−0.547976	−0.0200226
$750$	0.176154	0.00643222
$751$	15.8126	0.577009	0.288505	−	0.957478i	$-0.406842\pi$
$751$	15.8126	0.577009	0.288505	+	0.957478i	$0.406842\pi$
$752$	−5.14135	−0.187486
$753$	10.2999	0.375349
$754$	−0.984625	−0.0358579
$755$	−0.0634031	−0.00230748
$756$	0.282315	0.0102677
$757$	5.32041	0.193373	0.0966867	−	0.995315i	$-0.469176\pi$
$757$	5.32041	0.193373	0.0966867	+	0.995315i	$0.469176\pi$
$758$	10.3821	0.377094
$759$	−3.74580	−0.135964
$760$	0.0176159	0.000638996 0
$761$	−6.82372	−0.247360	−0.123680	−	0.992322i	$-0.539470\pi$
$761$	−6.82372	−0.247360	−0.123680	+	0.992322i	$0.539470\pi$
$762$	−1.33645	−0.0484145
$763$	−2.89117	−0.104667
$764$	−3.90679	−0.141343
$765$	0.0861935	0.00311633
$766$	−15.5054	−0.560234
$767$	−1.49507	−0.0539838
$768$	1.00000	0.0360844
$769$	3.99104	0.143921	0.0719603	−	0.997407i	$-0.477075\pi$
$769$	3.99104	0.143921	0.0719603	+	0.997407i	$0.477075\pi$
$770$	0.0107299	0.000386677 0
$771$	25.0322	0.901512
$772$	18.0679	0.650279
$773$	−31.0572	−1.11705	−0.558524	−	0.829488i	$-0.688632\pi$
$773$	−31.0572	−1.11705	−0.558524	+	0.829488i	$0.688632\pi$
$774$	−1.91745	−0.0689213
$775$	2.81199	0.101009
$776$	−11.1739	−0.401120
$777$	−0.934085	−0.0335101
$778$	21.0052	0.753072
$779$	5.03315	0.180331
$780$	−0.00250872	−8.98264e−5 0
$781$	−22.4940	−0.804899
$782$	−8.49493	−0.303778
$783$	−6.91392	−0.247083
$784$	−6.92030	−0.247154
$785$	0.0453057	0.00161703
$786$	17.2686	0.615952
$787$	−1.14588	−0.0408464	−0.0204232	−	0.999791i	$-0.506501\pi$
$787$	−1.14588	−0.0408464	−0.0204232	+	0.999791i	$0.506501\pi$
$788$	1.90425	0.0678360
$789$	−11.7373	−0.417859
$790$	−0.118330	−0.00420999
$791$	3.61114	0.128397
$792$	2.15752	0.0766641
$793$	2.12813	0.0755721
$794$	−35.1115	−1.24606
$795$	0.0176159	0.000624772 0
$796$	−8.50280	−0.301374
$797$	−34.9487	−1.23795	−0.618974	−	0.785411i	$-0.712451\pi$
$797$	−34.9487	−1.23795	−0.618974	+	0.785411i	$0.712451\pi$
$798$	0.282315	0.00999385
$799$	−25.1563	−0.889966
$800$	4.99969	0.176766
$801$	−16.3526	−0.577792
$802$	−16.5759	−0.585317
$803$	−11.1925	−0.394974
$804$	−14.1900	−0.500442
$805$	0.00863435	0.000304321 0
$806$	−0.0800970	−0.00282130
$807$	−1.79863	−0.0633148
$808$	−7.26379	−0.255539
$809$	37.3727	1.31395	0.656977	−	0.753910i	$-0.271835\pi$
$809$	37.3727	1.31395	0.656977	+	0.753910i	$0.271835\pi$
$810$	−0.0176159	−0.000618960 0
$811$	46.9939	1.65018	0.825090	−	0.565001i	$-0.191124\pi$
$811$	46.9939	1.65018	0.825090	+	0.565001i	$0.191124\pi$
$812$	−1.95190	−0.0684984
$813$	−10.5079	−0.368529
$814$	−7.13850	−0.250204
$815$	−0.138080	−0.00483673
$816$	4.89293	0.171287
$817$	−1.91745	−0.0670831
$818$	13.3143	0.465525
$819$	−0.0402051	−0.00140488
$820$	−0.0886635	−0.00309626
$821$	−29.0568	−1.01409	−0.507044	−	0.861920i	$-0.669262\pi$
$821$	−29.0568	−1.01409	−0.507044	+	0.861920i	$0.669262\pi$
$822$	8.96141	0.312565
$823$	−54.4312	−1.89735	−0.948677	−	0.316248i	$-0.897577\pi$
$823$	−54.4312	−1.89735	−0.948677	+	0.316248i	$0.897577\pi$
$824$	−0.574957	−0.0200296
$825$	10.7869	0.375553
$826$	−2.96380	−0.103124
$827$	−47.6869	−1.65824	−0.829118	−	0.559073i	$-0.811157\pi$
$827$	−47.6869	−1.65824	−0.829118	+	0.559073i	$0.811157\pi$
$828$	1.73616	0.0603358
$829$	36.0914	1.25351	0.626754	−	0.779217i	$-0.284383\pi$
$829$	36.0914	1.25351	0.626754	+	0.779217i	$0.284383\pi$
$830$	−0.0540957	−0.00187769
$831$	−20.8279	−0.722513
$832$	−0.142412	−0.00493725
$833$	−33.8606	−1.17320
$834$	10.9502	0.379173
$835$	−0.0445059	−0.00154019
$836$	2.15752	0.0746194
$837$	−0.562432	−0.0194405
$838$	27.6831	0.956296
$839$	41.4929	1.43249	0.716246	−	0.697847i	$-0.245859\pi$
$839$	41.4929	1.43249	0.716246	+	0.697847i	$0.245859\pi$
$840$	−0.00497324	−0.000171593 0
$841$	18.8023	0.648355
$842$	14.3844	0.495719
$843$	−18.0996	−0.623384
$844$	17.4217	0.599680
$845$	−0.228650	−0.00786578
$846$	5.14135	0.176763
$847$	−1.79132	−0.0615505
$848$	1.00000	0.0343401
$849$	−23.3464	−0.801246
$850$	24.4632	0.839079
$851$	−5.74437	−0.196915
$852$	10.4259	0.357184
$853$	−28.1941	−0.965346	−0.482673	−	0.875801i	$-0.660334\pi$
$853$	−28.1941	−0.965346	−0.482673	+	0.875801i	$0.660334\pi$
$854$	4.21877	0.144363
$855$	−0.0176159	−0.000602451 0
$856$	1.94101	0.0663422
$857$	−34.7559	−1.18724	−0.593620	−	0.804746i	$-0.702302\pi$
$857$	−34.7559	−1.18724	−0.593620	+	0.804746i	$0.702302\pi$
$858$	−0.307257	−0.0104896
$859$	−15.8716	−0.541530	−0.270765	−	0.962645i	$-0.587277\pi$
$859$	−15.8716	−0.541530	−0.270765	+	0.962645i	$0.587277\pi$
$860$	0.0337776	0.00115181
$861$	−1.42093	−0.0484253
$862$	39.2041	1.33530
$863$	−2.06218	−0.0701976	−0.0350988	−	0.999384i	$-0.511175\pi$
$863$	−2.06218	−0.0701976	−0.0350988	+	0.999384i	$0.511175\pi$
$864$	−1.00000	−0.0340207
$865$	−0.169915	−0.00577728
$866$	−19.7298	−0.670445
$867$	6.94080	0.235722
$868$	−0.158783	−0.00538945
$869$	−14.4925	−0.491626
$870$	0.121795	0.00412924
$871$	2.02082	0.0684730
$872$	10.2409	0.346802
$873$	11.1739	0.378180
$874$	1.73616	0.0587266
$875$	−0.0497308	−0.00168121
$876$	5.18767	0.175275
$877$	−24.9310	−0.841860	−0.420930	−	0.907093i	$-0.638296\pi$
$877$	−24.9310	−0.841860	−0.420930	+	0.907093i	$0.638296\pi$
$878$	−11.6044	−0.391631
$879$	16.3860	0.552686
$880$	−0.0380067	−0.00128120
$881$	13.5541	0.456649	0.228324	−	0.973585i	$-0.426675\pi$
$881$	13.5541	0.456649	0.228324	+	0.973585i	$0.426675\pi$
$882$	6.92030	0.233019
$883$	−20.8134	−0.700426	−0.350213	−	0.936670i	$-0.613891\pi$
$883$	−20.8134	−0.700426	−0.350213	+	0.936670i	$0.613891\pi$
$884$	−0.696812	−0.0234363
$885$	0.184935	0.00621653
$886$	29.8289	1.00212
$887$	10.9297	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$887$	10.9297	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$888$	3.30866	0.111031
$889$	0.377300	0.0126542
$890$	0.288066	0.00965601
$891$	−2.15752	−0.0722796
$892$	−27.7931	−0.930582
$893$	5.14135	0.172049
$894$	−19.9498	−0.667221
$895$	−0.202590	−0.00677183
$896$	−0.282315	−0.00943149
$897$	−0.247250	−0.00825545
$898$	−8.94810	−0.298602
$899$	3.88861	0.129692
$900$	−4.99969	−0.166656
$901$	4.89293	0.163007
$902$	−10.8591	−0.361569
$903$	0.541325	0.0180142
$904$	−12.7912	−0.425428
$905$	−0.128655	−0.00427663
$906$	3.59920	0.119575
$907$	−38.1977	−1.26833	−0.634166	−	0.773197i	$-0.718656\pi$
$907$	−38.1977	−1.26833	−0.634166	+	0.773197i	$0.718656\pi$
$908$	−8.24034	−0.273465
$909$	7.26379	0.240925
$910$	0.000708249 0	2.34782e−5 0
$911$	16.0622	0.532163	0.266081	−	0.963951i	$-0.414271\pi$
$911$	16.0622	0.532163	0.266081	+	0.963951i	$0.414271\pi$
$912$	−1.00000	−0.0331133
$913$	−6.62540	−0.219269
$914$	−5.81701	−0.192410
$915$	−0.263243	−0.00870255
$916$	−6.53464	−0.215911
$917$	−4.87520	−0.160993
$918$	−4.89293	−0.161491
$919$	−21.8980	−0.722349	−0.361174	−	0.932498i	$-0.617624\pi$
$919$	−21.8980	−0.722349	−0.361174	+	0.932498i	$0.617624\pi$
$920$	−0.0305841	−0.00100833
$921$	2.53537	0.0835434
$922$	−4.82072	−0.158762
$923$	−1.48477	−0.0488717
$924$	−0.609100	−0.0200379
$925$	16.5423	0.543907
$926$	−38.6833	−1.27121
$927$	0.574957	0.0188841
$928$	6.91392	0.226961
$929$	60.2675	1.97731	0.988656	−	0.150196i	$-0.0479905\pi$
$929$	60.2675	1.97731	0.988656	+	0.150196i	$0.0479905\pi$
$930$	0.00990775	0.000324888 0
$931$	6.92030	0.226804
$932$	7.13971	0.233869
$933$	27.5664	0.902483
$934$	−9.99090	−0.326912
$935$	−0.185964	−0.00608168
$936$	0.142412	0.00465488
$937$	20.6956	0.676095	0.338047	−	0.941129i	$-0.390234\pi$
$937$	20.6956	0.676095	0.338047	+	0.941129i	$0.390234\pi$
$938$	4.00605	0.130802
$939$	−25.3513	−0.827309
$940$	−0.0905696	−0.00295406
$941$	42.4912	1.38517	0.692587	−	0.721335i	$-0.256471\pi$
$941$	42.4912	1.38517	0.692587	+	0.721335i	$0.256471\pi$
$942$	−2.57186	−0.0837958
$943$	−8.73836	−0.284560
$944$	10.4982	0.341687
$945$	0.00497324	0.000161779 0
$946$	4.13694	0.134503
$947$	−3.16233	−0.102762	−0.0513810	−	0.998679i	$-0.516362\pi$
$947$	−3.16233	−0.102762	−0.0513810	+	0.998679i	$0.516362\pi$
$948$	6.71722	0.218165
$949$	−0.738786	−0.0239820
$950$	−4.99969	−0.162211
$951$	30.0234	0.973575
$952$	−1.38135	−0.0447698
$953$	5.11312	0.165630	0.0828151	−	0.996565i	$-0.473609\pi$
$953$	5.11312	0.165630	0.0828151	+	0.996565i	$0.473609\pi$
$954$	−1.00000	−0.0323762
$955$	−0.0688217	−0.00222702
$956$	3.03785	0.0982512
$957$	14.9169	0.482195
$958$	−12.8715	−0.415859
$959$	−2.52994	−0.0816961
$960$	0.0176159	0.000568551 0
$961$	−30.6837	−0.989796
$962$	−0.471193	−0.0151919
$963$	−1.94101	−0.0625481
$964$	7.41576	0.238845
$965$	0.318283	0.0102459
$966$	−0.490145	−0.0157702
$967$	24.0659	0.773906	0.386953	−	0.922099i	$-0.373528\pi$
$967$	24.0659	0.773906	0.386953	+	0.922099i	$0.373528\pi$
$968$	6.34511	0.203940
$969$	−4.89293	−0.157184
$970$	−0.196839	−0.00632011
$971$	49.1193	1.57631	0.788156	−	0.615475i	$-0.211036\pi$
$971$	49.1193	1.57631	0.788156	+	0.615475i	$0.211036\pi$
$972$	1.00000	0.0320750
$973$	−3.09140	−0.0991057
$974$	−22.9649	−0.735844
$975$	0.712016	0.0228027
$976$	−14.9435	−0.478329
$977$	30.3107	0.969724	0.484862	−	0.874591i	$-0.338870\pi$
$977$	30.3107	0.969724	0.484862	+	0.874591i	$0.338870\pi$
$978$	7.83837	0.250643
$979$	35.2811	1.12759
$980$	−0.121907	−0.00389419
$981$	−10.2409	−0.326968
$982$	−13.3040	−0.424548
$983$	58.7984	1.87538	0.937689	−	0.347476i	$-0.112961\pi$
$983$	58.7984	1.87538	0.937689	+	0.347476i	$0.112961\pi$
$984$	5.03315	0.160451
$985$	0.0335450	0.00106883
$986$	33.8294	1.07735
$987$	−1.45148	−0.0462012
$988$	0.142412	0.00453073
$989$	3.32901	0.105856
$990$	0.0380067	0.00120793
$991$	−49.5425	−1.57377	−0.786885	−	0.617099i	$-0.788308\pi$
$991$	−49.5425	−1.57377	−0.786885	+	0.617099i	$0.788308\pi$
$992$	0.562432	0.0178572
$993$	−12.6956	−0.402884
$994$	−2.94338	−0.0933583
$995$	−0.149785	−0.00474849
$996$	3.07084	0.0973034
$997$	19.3444	0.612642	0.306321	−	0.951928i	$-0.400902\pi$
$997$	19.3444	0.612642	0.306321	+	0.951928i	$0.400902\pi$
$998$	19.9587	0.631780
$999$	−3.30866	−0.104681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.y.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.y.1.3	✓	7	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.0176159 q^{5} -1.00000 q^{6} +0.282315 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.0176159 q^{5} -1.00000 q^{6} +0.282315 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.0176159 q^{10} -2.15752 q^{11} +1.00000 q^{12} -0.142412 q^{13} -0.282315 q^{14} +0.0176159 q^{15} +1.00000 q^{16} +4.89293 q^{17} -1.00000 q^{18} -1.00000 q^{19} +0.0176159 q^{20} +0.282315 q^{21} +2.15752 q^{22} +1.73616 q^{23} -1.00000 q^{24} -4.99969 q^{25} +0.142412 q^{26} +1.00000 q^{27} +0.282315 q^{28} -6.91392 q^{29} -0.0176159 q^{30} -0.562432 q^{31} -1.00000 q^{32} -2.15752 q^{33} -4.89293 q^{34} +0.00497324 q^{35} +1.00000 q^{36} -3.30866 q^{37} +1.00000 q^{38} -0.142412 q^{39} -0.0176159 q^{40} -5.03315 q^{41} -0.282315 q^{42} +1.91745 q^{43} -2.15752 q^{44} +0.0176159 q^{45} -1.73616 q^{46} -5.14135 q^{47} +1.00000 q^{48} -6.92030 q^{49} +4.99969 q^{50} +4.89293 q^{51} -0.142412 q^{52} +1.00000 q^{53} -1.00000 q^{54} -0.0380067 q^{55} -0.282315 q^{56} -1.00000 q^{57} +6.91392 q^{58} +10.4982 q^{59} +0.0176159 q^{60} -14.9435 q^{61} +0.562432 q^{62} +0.282315 q^{63} +1.00000 q^{64} -0.00250872 q^{65} +2.15752 q^{66} -14.1900 q^{67} +4.89293 q^{68} +1.73616 q^{69} -0.00497324 q^{70} +10.4259 q^{71} -1.00000 q^{72} +5.18767 q^{73} +3.30866 q^{74} -4.99969 q^{75} -1.00000 q^{76} -0.609100 q^{77} +0.142412 q^{78} +6.71722 q^{79} +0.0176159 q^{80} +1.00000 q^{81} +5.03315 q^{82} +3.07084 q^{83} +0.282315 q^{84} +0.0861935 q^{85} -1.91745 q^{86} -6.91392 q^{87} +2.15752 q^{88} -16.3526 q^{89} -0.0176159 q^{90} -0.0402051 q^{91} +1.73616 q^{92} -0.562432 q^{93} +5.14135 q^{94} -0.0176159 q^{95} -1.00000 q^{96} +11.1739 q^{97} +6.92030 q^{98} -2.15752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{3} + 7 q^{4} + 4 q^{5} - 7 q^{6} - 9 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^3 + 7 * q^4 + 4 * q^5 - 7 * q^6 - 9 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} + 7 q^{3} + 7 q^{4} + 4 q^{5} - 7 q^{6} - 9 q^{7} - 7 q^{8} + 7 q^{9} - 4 q^{10} - 2 q^{11} + 7 q^{12} - 7 q^{13} + 9 q^{14} + 4 q^{15} + 7 q^{16} - 10 q^{17} - 7 q^{18} - 7 q^{19} + 4 q^{20} - 9 q^{21} + 2 q^{22} + 2 q^{23} - 7 q^{24} + 3 q^{25} + 7 q^{26} + 7 q^{27} - 9 q^{28} - 6 q^{29} - 4 q^{30} + 3 q^{31} - 7 q^{32} - 2 q^{33} + 10 q^{34} + 7 q^{36} - 3 q^{37} + 7 q^{38} - 7 q^{39} - 4 q^{40} - 12 q^{41} + 9 q^{42} - 26 q^{43} - 2 q^{44} + 4 q^{45} - 2 q^{46} + 17 q^{47} + 7 q^{48} - 3 q^{50} - 10 q^{51} - 7 q^{52} + 7 q^{53} - 7 q^{54} - 23 q^{55} + 9 q^{56} - 7 q^{57} + 6 q^{58} + 7 q^{59} + 4 q^{60} - 18 q^{61} - 3 q^{62} - 9 q^{63} + 7 q^{64} - 23 q^{65} + 2 q^{66} - 3 q^{67} - 10 q^{68} + 2 q^{69} + 2 q^{71} - 7 q^{72} + q^{73} + 3 q^{74} + 3 q^{75} - 7 q^{76} - 23 q^{77} + 7 q^{78} - 18 q^{79} + 4 q^{80} + 7 q^{81} + 12 q^{82} - 17 q^{83} - 9 q^{84} - 3 q^{85} + 26 q^{86} - 6 q^{87} + 2 q^{88} - 13 q^{89} - 4 q^{90} - 8 q^{91} + 2 q^{92} + 3 q^{93} - 17 q^{94} - 4 q^{95} - 7 q^{96} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^3 + 7 * q^4 + 4 * q^5 - 7 * q^6 - 9 * q^7 - 7 * q^8 + 7 * q^9 - 4 * q^10 - 2 * q^11 + 7 * q^12 - 7 * q^13 + 9 * q^14 + 4 * q^15 + 7 * q^16 - 10 * q^17 - 7 * q^18 - 7 * q^19 + 4 * q^20 - 9 * q^21 + 2 * q^22 + 2 * q^23 - 7 * q^24 + 3 * q^25 + 7 * q^26 + 7 * q^27 - 9 * q^28 - 6 * q^29 - 4 * q^30 + 3 * q^31 - 7 * q^32 - 2 * q^33 + 10 * q^34 + 7 * q^36 - 3 * q^37 + 7 * q^38 - 7 * q^39 - 4 * q^40 - 12 * q^41 + 9 * q^42 - 26 * q^43 - 2 * q^44 + 4 * q^45 - 2 * q^46 + 17 * q^47 + 7 * q^48 - 3 * q^50 - 10 * q^51 - 7 * q^52 + 7 * q^53 - 7 * q^54 - 23 * q^55 + 9 * q^56 - 7 * q^57 + 6 * q^58 + 7 * q^59 + 4 * q^60 - 18 * q^61 - 3 * q^62 - 9 * q^63 + 7 * q^64 - 23 * q^65 + 2 * q^66 - 3 * q^67 - 10 * q^68 + 2 * q^69 + 2 * q^71 - 7 * q^72 + q^73 + 3 * q^74 + 3 * q^75 - 7 * q^76 - 23 * q^77 + 7 * q^78 - 18 * q^79 + 4 * q^80 + 7 * q^81 + 12 * q^82 - 17 * q^83 - 9 * q^84 - 3 * q^85 + 26 * q^86 - 6 * q^87 + 2 * q^88 - 13 * q^89 - 4 * q^90 - 8 * q^91 + 2 * q^92 + 3 * q^93 - 17 * q^94 - 4 * q^95 - 7 * q^96 - 20 * q^97 - 2 * q^99

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