LMFDB - Embedded newform 6042.2.a.bg.1.2

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 34 x^{10} + 169 x^{9} + 453 x^{8} - 2095 x^{7} - 3056 x^{6} + 11545 x^{5} + \cdots + 7848 $ x^12 - 5x^11 - 34x^10 + 169x^9 + 453x^8 - 2095x^7 - 3056x^6 + 11545x^5 + 11014x^4 - 26766x^3 - 19370x^2 + 18296*x + 7848
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.85168$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.85168 q^{5} +1.00000 q^{6} +3.66552 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.85168 q^{5} +1.00000 q^{6} +3.66552 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.85168 q^{10} +6.42808 q^{11} +1.00000 q^{12} +3.41087 q^{13} +3.66552 q^{14} -2.85168 q^{15} +1.00000 q^{16} +2.55551 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.85168 q^{20} +3.66552 q^{21} +6.42808 q^{22} -4.12954 q^{23} +1.00000 q^{24} +3.13210 q^{25} +3.41087 q^{26} +1.00000 q^{27} +3.66552 q^{28} +6.91431 q^{29} -2.85168 q^{30} -0.101446 q^{31} +1.00000 q^{32} +6.42808 q^{33} +2.55551 q^{34} -10.4529 q^{35} +1.00000 q^{36} -0.395117 q^{37} -1.00000 q^{38} +3.41087 q^{39} -2.85168 q^{40} -0.901021 q^{41} +3.66552 q^{42} -5.12097 q^{43} +6.42808 q^{44} -2.85168 q^{45} -4.12954 q^{46} -0.0659721 q^{47} +1.00000 q^{48} +6.43604 q^{49} +3.13210 q^{50} +2.55551 q^{51} +3.41087 q^{52} +1.00000 q^{53} +1.00000 q^{54} -18.3308 q^{55} +3.66552 q^{56} -1.00000 q^{57} +6.91431 q^{58} -13.4101 q^{59} -2.85168 q^{60} +11.5650 q^{61} -0.101446 q^{62} +3.66552 q^{63} +1.00000 q^{64} -9.72672 q^{65} +6.42808 q^{66} -12.0497 q^{67} +2.55551 q^{68} -4.12954 q^{69} -10.4529 q^{70} -13.8596 q^{71} +1.00000 q^{72} +16.0108 q^{73} -0.395117 q^{74} +3.13210 q^{75} -1.00000 q^{76} +23.5622 q^{77} +3.41087 q^{78} -7.52279 q^{79} -2.85168 q^{80} +1.00000 q^{81} -0.901021 q^{82} +6.50925 q^{83} +3.66552 q^{84} -7.28750 q^{85} -5.12097 q^{86} +6.91431 q^{87} +6.42808 q^{88} +2.30147 q^{89} -2.85168 q^{90} +12.5026 q^{91} -4.12954 q^{92} -0.101446 q^{93} -0.0659721 q^{94} +2.85168 q^{95} +1.00000 q^{96} -4.40257 q^{97} +6.43604 q^{98} +6.42808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{3} + 12 q^{4} + 5 q^{5} + 12 q^{6} + 6 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^3 + 12 * q^4 + 5 * q^5 + 12 * q^6 + 6 * q^7 + 12 * q^8 + 12 * q^9	\( 12 q + 12 q^{2} + 12 q^{3} + 12 q^{4} + 5 q^{5} + 12 q^{6} + 6 q^{7} + 12 q^{8} + 12 q^{9} + 5 q^{10} + 7 q^{11} + 12 q^{12} + 9 q^{13} + 6 q^{14} + 5 q^{15} + 12 q^{16} + 25 q^{17} + 12 q^{18} - 12 q^{19} + 5 q^{20} + 6 q^{21} + 7 q^{22} + 22 q^{23} + 12 q^{24} + 33 q^{25} + 9 q^{26} + 12 q^{27} + 6 q^{28} + q^{29} + 5 q^{30} + 23 q^{31} + 12 q^{32} + 7 q^{33} + 25 q^{34} + 5 q^{35} + 12 q^{36} - q^{37} - 12 q^{38} + 9 q^{39} + 5 q^{40} - 15 q^{41} + 6 q^{42} + 2 q^{43} + 7 q^{44} + 5 q^{45} + 22 q^{46} + 11 q^{47} + 12 q^{48} + 36 q^{49} + 33 q^{50} + 25 q^{51} + 9 q^{52} + 12 q^{53} + 12 q^{54} - 4 q^{55} + 6 q^{56} - 12 q^{57} + q^{58} + 3 q^{59} + 5 q^{60} + 16 q^{61} + 23 q^{62} + 6 q^{63} + 12 q^{64} + 7 q^{65} + 7 q^{66} - 2 q^{67} + 25 q^{68} + 22 q^{69} + 5 q^{70} - 4 q^{71} + 12 q^{72} + 35 q^{73} - q^{74} + 33 q^{75} - 12 q^{76} + 11 q^{77} + 9 q^{78} + 4 q^{79} + 5 q^{80} + 12 q^{81} - 15 q^{82} + 39 q^{83} + 6 q^{84} + 10 q^{85} + 2 q^{86} + q^{87} + 7 q^{88} + 11 q^{89} + 5 q^{90} - 18 q^{91} + 22 q^{92} + 23 q^{93} + 11 q^{94} - 5 q^{95} + 12 q^{96} - 21 q^{97} + 36 q^{98} + 7 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^3 + 12 * q^4 + 5 * q^5 + 12 * q^6 + 6 * q^7 + 12 * q^8 + 12 * q^9 + 5 * q^10 + 7 * q^11 + 12 * q^12 + 9 * q^13 + 6 * q^14 + 5 * q^15 + 12 * q^16 + 25 * q^17 + 12 * q^18 - 12 * q^19 + 5 * q^20 + 6 * q^21 + 7 * q^22 + 22 * q^23 + 12 * q^24 + 33 * q^25 + 9 * q^26 + 12 * q^27 + 6 * q^28 + q^29 + 5 * q^30 + 23 * q^31 + 12 * q^32 + 7 * q^33 + 25 * q^34 + 5 * q^35 + 12 * q^36 - q^37 - 12 * q^38 + 9 * q^39 + 5 * q^40 - 15 * q^41 + 6 * q^42 + 2 * q^43 + 7 * q^44 + 5 * q^45 + 22 * q^46 + 11 * q^47 + 12 * q^48 + 36 * q^49 + 33 * q^50 + 25 * q^51 + 9 * q^52 + 12 * q^53 + 12 * q^54 - 4 * q^55 + 6 * q^56 - 12 * q^57 + q^58 + 3 * q^59 + 5 * q^60 + 16 * q^61 + 23 * q^62 + 6 * q^63 + 12 * q^64 + 7 * q^65 + 7 * q^66 - 2 * q^67 + 25 * q^68 + 22 * q^69 + 5 * q^70 - 4 * q^71 + 12 * q^72 + 35 * q^73 - q^74 + 33 * q^75 - 12 * q^76 + 11 * q^77 + 9 * q^78 + 4 * q^79 + 5 * q^80 + 12 * q^81 - 15 * q^82 + 39 * q^83 + 6 * q^84 + 10 * q^85 + 2 * q^86 + q^87 + 7 * q^88 + 11 * q^89 + 5 * q^90 - 18 * q^91 + 22 * q^92 + 23 * q^93 + 11 * q^94 - 5 * q^95 + 12 * q^96 - 21 * q^97 + 36 * q^98 + 7 * 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
$5$	−2.85168	−1.27531	−0.637656	−	0.770321i	$-0.720096\pi$
$5$	−2.85168	−1.27531	−0.637656	+	0.770321i	$0.720096\pi$
$6$	1.00000	0.408248
$7$	3.66552	1.38544	0.692718	−	0.721208i	$-0.256413\pi$
$7$	3.66552	1.38544	0.692718	+	0.721208i	$0.256413\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.85168	−0.901782
$11$	6.42808	1.93814	0.969069	−	0.246790i	$-0.0793758\pi$
$11$	6.42808	1.93814	0.969069	+	0.246790i	$0.0793758\pi$
$12$	1.00000	0.288675
$13$	3.41087	0.946005	0.473002	−	0.881061i	$-0.343170\pi$
$13$	3.41087	0.946005	0.473002	+	0.881061i	$0.343170\pi$
$14$	3.66552	0.979651
$15$	−2.85168	−0.736302
$16$	1.00000	0.250000
$17$	2.55551	0.619801	0.309901	−	0.950769i	$-0.399704\pi$
$17$	2.55551	0.619801	0.309901	+	0.950769i	$0.399704\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.85168	−0.637656
$21$	3.66552	0.799882
$22$	6.42808	1.37047
$23$	−4.12954	−0.861069	−0.430535	−	0.902574i	$-0.641675\pi$
$23$	−4.12954	−0.861069	−0.430535	+	0.902574i	$0.641675\pi$
$24$	1.00000	0.204124
$25$	3.13210	0.626420
$26$	3.41087	0.668926
$27$	1.00000	0.192450
$28$	3.66552	0.692718
$29$	6.91431	1.28396	0.641978	−	0.766723i	$-0.278114\pi$
$29$	6.91431	1.28396	0.641978	+	0.766723i	$0.278114\pi$
$30$	−2.85168	−0.520644
$31$	−0.101446	−0.0182202	−0.00911010	−	0.999959i	$-0.502900\pi$
$31$	−0.101446	−0.0182202	−0.00911010	+	0.999959i	$0.502900\pi$
$32$	1.00000	0.176777
$33$	6.42808	1.11898
$34$	2.55551	0.438266
$35$	−10.4529	−1.76686
$36$	1.00000	0.166667
$37$	−0.395117	−0.0649568	−0.0324784	−	0.999472i	$-0.510340\pi$
$37$	−0.395117	−0.0649568	−0.0324784	+	0.999472i	$0.510340\pi$
$38$	−1.00000	−0.162221
$39$	3.41087	0.546176
$40$	−2.85168	−0.450891
$41$	−0.901021	−0.140716	−0.0703580	−	0.997522i	$-0.522414\pi$
$41$	−0.901021	−0.140716	−0.0703580	+	0.997522i	$0.522414\pi$
$42$	3.66552	0.565602
$43$	−5.12097	−0.780941	−0.390471	−	0.920615i	$-0.627688\pi$
$43$	−5.12097	−0.780941	−0.390471	+	0.920615i	$0.627688\pi$
$44$	6.42808	0.969069
$45$	−2.85168	−0.425104
$46$	−4.12954	−0.608868
$47$	−0.0659721	−0.00962302	−0.00481151	−	0.999988i	$-0.501532\pi$
$47$	−0.0659721	−0.00962302	−0.00481151	+	0.999988i	$0.501532\pi$
$48$	1.00000	0.144338
$49$	6.43604	0.919434
$50$	3.13210	0.442946
$51$	2.55551	0.357843
$52$	3.41087	0.473002
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	−18.3308	−2.47173
$56$	3.66552	0.489826
$57$	−1.00000	−0.132453
$58$	6.91431	0.907894
$59$	−13.4101	−1.74584	−0.872922	−	0.487860i	$-0.837778\pi$
$59$	−13.4101	−1.74584	−0.872922	+	0.487860i	$0.837778\pi$
$60$	−2.85168	−0.368151
$61$	11.5650	1.48075	0.740374	−	0.672195i	$-0.234648\pi$
$61$	11.5650	1.48075	0.740374	+	0.672195i	$0.234648\pi$
$62$	−0.101446	−0.0128836
$63$	3.66552	0.461812
$64$	1.00000	0.125000
$65$	−9.72672	−1.20645
$66$	6.42808	0.791242
$67$	−12.0497	−1.47210	−0.736050	−	0.676927i	$-0.763311\pi$
$67$	−12.0497	−1.47210	−0.736050	+	0.676927i	$0.763311\pi$
$68$	2.55551	0.309901
$69$	−4.12954	−0.497138
$70$	−10.4529	−1.24936
$71$	−13.8596	−1.64484	−0.822418	−	0.568884i	$-0.807375\pi$
$71$	−13.8596	−1.64484	−0.822418	+	0.568884i	$0.807375\pi$
$72$	1.00000	0.117851
$73$	16.0108	1.87393	0.936963	−	0.349428i	$-0.113624\pi$
$73$	16.0108	1.87393	0.936963	+	0.349428i	$0.113624\pi$
$74$	−0.395117	−0.0459314
$75$	3.13210	0.361664
$76$	−1.00000	−0.114708
$77$	23.5622	2.68517
$78$	3.41087	0.386205
$79$	−7.52279	−0.846380	−0.423190	−	0.906041i	$-0.639090\pi$
$79$	−7.52279	−0.846380	−0.423190	+	0.906041i	$0.639090\pi$
$80$	−2.85168	−0.318828
$81$	1.00000	0.111111
$82$	−0.901021	−0.0995012
$83$	6.50925	0.714483	0.357242	−	0.934012i	$-0.383717\pi$
$83$	6.50925	0.714483	0.357242	+	0.934012i	$0.383717\pi$
$84$	3.66552	0.399941
$85$	−7.28750	−0.790440
$86$	−5.12097	−0.552209
$87$	6.91431	0.741292
$88$	6.42808	0.685235
$89$	2.30147	0.243955	0.121978	−	0.992533i	$-0.461076\pi$
$89$	2.30147	0.243955	0.121978	+	0.992533i	$0.461076\pi$
$90$	−2.85168	−0.300594
$91$	12.5026	1.31063
$92$	−4.12954	−0.430535
$93$	−0.101446	−0.0105194
$94$	−0.0659721	−0.00680450
$95$	2.85168	0.292577
$96$	1.00000	0.102062
$97$	−4.40257	−0.447014	−0.223507	−	0.974702i	$-0.571750\pi$
$97$	−4.40257	−0.447014	−0.223507	+	0.974702i	$0.571750\pi$
$98$	6.43604	0.650138
$99$	6.42808	0.646046
$100$	3.13210	0.313210
$101$	−9.98369	−0.993415	−0.496707	−	0.867918i	$-0.665458\pi$
$101$	−9.98369	−0.993415	−0.496707	+	0.867918i	$0.665458\pi$
$102$	2.55551	0.253033
$103$	6.88862	0.678756	0.339378	−	0.940650i	$-0.389783\pi$
$103$	6.88862	0.678756	0.339378	+	0.940650i	$0.389783\pi$
$104$	3.41087	0.334463
$105$	−10.4529	−1.02010
$106$	1.00000	0.0971286
$107$	−8.52187	−0.823841	−0.411920	−	0.911220i	$-0.635142\pi$
$107$	−8.52187	−0.823841	−0.411920	+	0.911220i	$0.635142\pi$
$108$	1.00000	0.0962250
$109$	5.06168	0.484821	0.242410	−	0.970174i	$-0.422062\pi$
$109$	5.06168	0.484821	0.242410	+	0.970174i	$0.422062\pi$
$110$	−18.3308	−1.74778
$111$	−0.395117	−0.0375028
$112$	3.66552	0.346359
$113$	−17.7796	−1.67256	−0.836281	−	0.548300i	$-0.815275\pi$
$113$	−17.7796	−1.67256	−0.836281	+	0.548300i	$0.815275\pi$
$114$	−1.00000	−0.0936586
$115$	11.7761	1.09813
$116$	6.91431	0.641978
$117$	3.41087	0.315335
$118$	−13.4101	−1.23450
$119$	9.36726	0.858695
$120$	−2.85168	−0.260322
$121$	30.3202	2.75638
$122$	11.5650	1.04705
$123$	−0.901021	−0.0812424
$124$	−0.101446	−0.00911010
$125$	5.32666	0.476431
$126$	3.66552	0.326550
$127$	1.03585	0.0919167	0.0459583	−	0.998943i	$-0.485366\pi$
$127$	1.03585	0.0919167	0.0459583	+	0.998943i	$0.485366\pi$
$128$	1.00000	0.0883883
$129$	−5.12097	−0.450877
$130$	−9.72672	−0.853090
$131$	12.3822	1.08184	0.540918	−	0.841075i	$-0.318077\pi$
$131$	12.3822	1.08184	0.540918	+	0.841075i	$0.318077\pi$
$132$	6.42808	0.559492
$133$	−3.66552	−0.317841
$134$	−12.0497	−1.04093
$135$	−2.85168	−0.245434
$136$	2.55551	0.219133
$137$	5.20204	0.444440	0.222220	−	0.974997i	$-0.428670\pi$
$137$	5.20204	0.444440	0.222220	+	0.974997i	$0.428670\pi$
$138$	−4.12954	−0.351530
$139$	−17.9890	−1.52580	−0.762902	−	0.646514i	$-0.776226\pi$
$139$	−17.9890	−1.52580	−0.762902	+	0.646514i	$0.776226\pi$
$140$	−10.4529	−0.883432
$141$	−0.0659721	−0.00555585
$142$	−13.8596	−1.16307
$143$	21.9253	1.83349
$144$	1.00000	0.0833333
$145$	−19.7174	−1.63744
$146$	16.0108	1.32507
$147$	6.43604	0.530835
$148$	−0.395117	−0.0324784
$149$	8.58619	0.703408	0.351704	−	0.936111i	$-0.385602\pi$
$149$	8.58619	0.703408	0.351704	+	0.936111i	$0.385602\pi$
$150$	3.13210	0.255735
$151$	7.64379	0.622043	0.311022	−	0.950403i	$-0.399329\pi$
$151$	7.64379	0.622043	0.311022	+	0.950403i	$0.399329\pi$
$152$	−1.00000	−0.0811107
$153$	2.55551	0.206600
$154$	23.5622	1.89870
$155$	0.289291	0.0232364
$156$	3.41087	0.273088
$157$	−5.92830	−0.473130	−0.236565	−	0.971616i	$-0.576022\pi$
$157$	−5.92830	−0.473130	−0.236565	+	0.971616i	$0.576022\pi$
$158$	−7.52279	−0.598481
$159$	1.00000	0.0793052
$160$	−2.85168	−0.225445
$161$	−15.1369	−1.19296
$162$	1.00000	0.0785674
$163$	13.5202	1.05898	0.529490	−	0.848316i	$-0.322383\pi$
$163$	13.5202	1.05898	0.529490	+	0.848316i	$0.322383\pi$
$164$	−0.901021	−0.0703580
$165$	−18.3308	−1.42705
$166$	6.50925	0.505216
$167$	−6.41672	−0.496540	−0.248270	−	0.968691i	$-0.579862\pi$
$167$	−6.41672	−0.496540	−0.248270	+	0.968691i	$0.579862\pi$
$168$	3.66552	0.282801
$169$	−1.36597	−0.105075
$170$	−7.28750	−0.558926
$171$	−1.00000	−0.0764719
$172$	−5.12097	−0.390471
$173$	−16.9237	−1.28669	−0.643344	−	0.765578i	$-0.722453\pi$
$173$	−16.9237	−1.28669	−0.643344	+	0.765578i	$0.722453\pi$
$174$	6.91431	0.524173
$175$	11.4808	0.867865
$176$	6.42808	0.484535
$177$	−13.4101	−1.00796
$178$	2.30147	0.172502
$179$	11.1068	0.830158	0.415079	−	0.909785i	$-0.363754\pi$
$179$	11.1068	0.830158	0.415079	+	0.909785i	$0.363754\pi$
$180$	−2.85168	−0.212552
$181$	−16.6248	−1.23571	−0.617856	−	0.786291i	$-0.711999\pi$
$181$	−16.6248	−1.23571	−0.617856	+	0.786291i	$0.711999\pi$
$182$	12.5026	0.926755
$183$	11.5650	0.854911
$184$	−4.12954	−0.304434
$185$	1.12675	0.0828402
$186$	−0.101446	−0.00743836
$187$	16.4270	1.20126
$188$	−0.0659721	−0.00481151
$189$	3.66552	0.266627
$190$	2.85168	0.206883
$191$	−16.3443	−1.18263	−0.591316	−	0.806440i	$-0.701392\pi$
$191$	−16.3443	−1.18263	−0.591316	+	0.806440i	$0.701392\pi$
$192$	1.00000	0.0721688
$193$	12.1749	0.876366	0.438183	−	0.898886i	$-0.355622\pi$
$193$	12.1749	0.876366	0.438183	+	0.898886i	$0.355622\pi$
$194$	−4.40257	−0.316086
$195$	−9.72672	−0.696545
$196$	6.43604	0.459717
$197$	8.59545	0.612401	0.306200	−	0.951967i	$-0.400942\pi$
$197$	8.59545	0.612401	0.306200	+	0.951967i	$0.400942\pi$
$198$	6.42808	0.456824
$199$	9.86121	0.699043	0.349521	−	0.936928i	$-0.386344\pi$
$199$	9.86121	0.699043	0.349521	+	0.936928i	$0.386344\pi$
$200$	3.13210	0.221473
$201$	−12.0497	−0.849918
$202$	−9.98369	−0.702450
$203$	25.3446	1.77884
$204$	2.55551	0.178921
$205$	2.56943	0.179457
$206$	6.88862	0.479953
$207$	−4.12954	−0.287023
$208$	3.41087	0.236501
$209$	−6.42808	−0.444639
$210$	−10.4529	−0.721319
$211$	19.9384	1.37262	0.686309	−	0.727310i	$-0.259230\pi$
$211$	19.9384	1.37262	0.686309	+	0.727310i	$0.259230\pi$
$212$	1.00000	0.0686803
$213$	−13.8596	−0.949647
$214$	−8.52187	−0.582543
$215$	14.6034	0.995944
$216$	1.00000	0.0680414
$217$	−0.371851	−0.0252429
$218$	5.06168	0.342820
$219$	16.0108	1.08191
$220$	−18.3308	−1.23587
$221$	8.71650	0.586335
$222$	−0.395117	−0.0265185
$223$	4.81773	0.322619	0.161310	−	0.986904i	$-0.448428\pi$
$223$	4.81773	0.322619	0.161310	+	0.986904i	$0.448428\pi$
$224$	3.66552	0.244913
$225$	3.13210	0.208807
$226$	−17.7796	−1.18268
$227$	−25.4457	−1.68889	−0.844446	−	0.535641i	$-0.820070\pi$
$227$	−25.4457	−1.68889	−0.844446	+	0.535641i	$0.820070\pi$
$228$	−1.00000	−0.0662266
$229$	25.3646	1.67614	0.838071	−	0.545560i	$-0.183683\pi$
$229$	25.3646	1.67614	0.838071	+	0.545560i	$0.183683\pi$
$230$	11.7761	0.776496
$231$	23.5622	1.55028
$232$	6.91431	0.453947
$233$	−15.6794	−1.02719	−0.513596	−	0.858032i	$-0.671687\pi$
$233$	−15.6794	−1.02719	−0.513596	+	0.858032i	$0.671687\pi$
$234$	3.41087	0.222975
$235$	0.188132	0.0122724
$236$	−13.4101	−0.872922
$237$	−7.52279	−0.488658
$238$	9.36726	0.607189
$239$	10.5130	0.680027	0.340014	−	0.940420i	$-0.389568\pi$
$239$	10.5130	0.680027	0.340014	+	0.940420i	$0.389568\pi$
$240$	−2.85168	−0.184075
$241$	22.2562	1.43364	0.716822	−	0.697256i	$-0.245596\pi$
$241$	22.2562	1.43364	0.716822	+	0.697256i	$0.245596\pi$
$242$	30.3202	1.94905
$243$	1.00000	0.0641500
$244$	11.5650	0.740374
$245$	−18.3535	−1.17256
$246$	−0.901021	−0.0574470
$247$	−3.41087	−0.217028
$248$	−0.101446	−0.00644181
$249$	6.50925	0.412507
$250$	5.32666	0.336887
$251$	−14.7464	−0.930783	−0.465391	−	0.885105i	$-0.654086\pi$
$251$	−14.7464	−0.930783	−0.465391	+	0.885105i	$0.654086\pi$
$252$	3.66552	0.230906
$253$	−26.5450	−1.66887
$254$	1.03585	0.0649949
$255$	−7.28750	−0.456361
$256$	1.00000	0.0625000
$257$	−13.3407	−0.832168	−0.416084	−	0.909326i	$-0.636598\pi$
$257$	−13.3407	−0.832168	−0.416084	+	0.909326i	$0.636598\pi$
$258$	−5.12097	−0.318818
$259$	−1.44831	−0.0899935
$260$	−9.72672	−0.603226
$261$	6.91431	0.427985
$262$	12.3822	0.764974
$263$	−16.8026	−1.03609	−0.518047	−	0.855352i	$-0.673341\pi$
$263$	−16.8026	−1.03609	−0.518047	+	0.855352i	$0.673341\pi$
$264$	6.42808	0.395621
$265$	−2.85168	−0.175178
$266$	−3.66552	−0.224747
$267$	2.30147	0.140848
$268$	−12.0497	−0.736050
$269$	13.3153	0.811847	0.405923	−	0.913907i	$-0.366950\pi$
$269$	13.3153	0.811847	0.405923	+	0.913907i	$0.366950\pi$
$270$	−2.85168	−0.173548
$271$	−14.6959	−0.892714	−0.446357	−	0.894855i	$-0.647279\pi$
$271$	−14.6959	−0.892714	−0.446357	+	0.894855i	$0.647279\pi$
$272$	2.55551	0.154950
$273$	12.5026	0.756692
$274$	5.20204	0.314267
$275$	20.1334	1.21409
$276$	−4.12954	−0.248569
$277$	20.5575	1.23518	0.617592	−	0.786499i	$-0.288109\pi$
$277$	20.5575	1.23518	0.617592	+	0.786499i	$0.288109\pi$
$278$	−17.9890	−1.07891
$279$	−0.101446	−0.00607340
$280$	−10.4529	−0.624680
$281$	−8.81637	−0.525940	−0.262970	−	0.964804i	$-0.584702\pi$
$281$	−8.81637	−0.525940	−0.262970	+	0.964804i	$0.584702\pi$
$282$	−0.0659721	−0.00392858
$283$	25.0119	1.48680	0.743402	−	0.668845i	$-0.233211\pi$
$283$	25.0119	1.48680	0.743402	+	0.668845i	$0.233211\pi$
$284$	−13.8596	−0.822418
$285$	2.85168	0.168919
$286$	21.9253	1.29647
$287$	−3.30271	−0.194953
$288$	1.00000	0.0589256
$289$	−10.4694	−0.615846
$290$	−19.7174	−1.15785
$291$	−4.40257	−0.258083
$292$	16.0108	0.936963
$293$	23.9893	1.40147	0.700735	−	0.713422i	$-0.252856\pi$
$293$	23.9893	1.40147	0.700735	+	0.713422i	$0.252856\pi$
$294$	6.43604	0.375357
$295$	38.2413	2.22649
$296$	−0.395117	−0.0229657
$297$	6.42808	0.372995
$298$	8.58619	0.497384
$299$	−14.0853	−0.814576
$300$	3.13210	0.180832
$301$	−18.7710	−1.08194
$302$	7.64379	0.439851
$303$	−9.98369	−0.573548
$304$	−1.00000	−0.0573539
$305$	−32.9798	−1.88842
$306$	2.55551	0.146089
$307$	−18.1303	−1.03475	−0.517377	−	0.855758i	$-0.673091\pi$
$307$	−18.1303	−1.03475	−0.517377	+	0.855758i	$0.673091\pi$
$308$	23.5622	1.34258
$309$	6.88862	0.391880
$310$	0.289291	0.0164306
$311$	6.36450	0.360898	0.180449	−	0.983584i	$-0.442245\pi$
$311$	6.36450	0.360898	0.180449	+	0.983584i	$0.442245\pi$
$312$	3.41087	0.193102
$313$	30.9977	1.75209	0.876047	−	0.482226i	$-0.160172\pi$
$313$	30.9977	1.75209	0.876047	+	0.482226i	$0.160172\pi$
$314$	−5.92830	−0.334553
$315$	−10.4529	−0.588954
$316$	−7.52279	−0.423190
$317$	−8.45267	−0.474749	−0.237375	−	0.971418i	$-0.576287\pi$
$317$	−8.45267	−0.474749	−0.237375	+	0.971418i	$0.576287\pi$
$318$	1.00000	0.0560772
$319$	44.4457	2.48848
$320$	−2.85168	−0.159414
$321$	−8.52187	−0.475645
$322$	−15.1369	−0.843548
$323$	−2.55551	−0.142192
$324$	1.00000	0.0555556
$325$	10.6832	0.592597
$326$	13.5202	0.748812
$327$	5.06168	0.279911
$328$	−0.901021	−0.0497506
$329$	−0.241822	−0.0133321
$330$	−18.3308	−1.00908
$331$	−5.94713	−0.326884	−0.163442	−	0.986553i	$-0.552260\pi$
$331$	−5.94713	−0.326884	−0.163442	+	0.986553i	$0.552260\pi$
$332$	6.50925	0.357242
$333$	−0.395117	−0.0216523
$334$	−6.41672	−0.351107
$335$	34.3618	1.87739
$336$	3.66552	0.199971
$337$	28.8236	1.57012	0.785060	−	0.619420i	$-0.212632\pi$
$337$	28.8236	1.57012	0.785060	+	0.619420i	$0.212632\pi$
$338$	−1.36597	−0.0742990
$339$	−17.7796	−0.965655
$340$	−7.28750	−0.395220
$341$	−0.652101	−0.0353132
$342$	−1.00000	−0.0540738
$343$	−2.06722	−0.111619
$344$	−5.12097	−0.276104
$345$	11.7761	0.634007
$346$	−16.9237	−0.909825
$347$	−2.06253	−0.110722	−0.0553612	−	0.998466i	$-0.517631\pi$
$347$	−2.06253	−0.110722	−0.0553612	+	0.998466i	$0.517631\pi$
$348$	6.91431	0.370646
$349$	−2.82169	−0.151041	−0.0755207	−	0.997144i	$-0.524062\pi$
$349$	−2.82169	−0.151041	−0.0755207	+	0.997144i	$0.524062\pi$
$350$	11.4808	0.613673
$351$	3.41087	0.182059
$352$	6.42808	0.342618
$353$	0.146932	0.00782039	0.00391020	−	0.999992i	$-0.498755\pi$
$353$	0.146932	0.00782039	0.00391020	+	0.999992i	$0.498755\pi$
$354$	−13.4101	−0.712738
$355$	39.5233	2.09768
$356$	2.30147	0.121978
$357$	9.36726	0.495768
$358$	11.1068	0.587010
$359$	21.2661	1.12238	0.561192	−	0.827686i	$-0.310343\pi$
$359$	21.2661	1.12238	0.561192	+	0.827686i	$0.310343\pi$
$360$	−2.85168	−0.150297
$361$	1.00000	0.0526316
$362$	−16.6248	−0.873780
$363$	30.3202	1.59140
$364$	12.5026	0.655315
$365$	−45.6578	−2.38984
$366$	11.5650	0.604513
$367$	3.87367	0.202204	0.101102	−	0.994876i	$-0.467763\pi$
$367$	3.87367	0.202204	0.101102	+	0.994876i	$0.467763\pi$
$368$	−4.12954	−0.215267
$369$	−0.901021	−0.0469053
$370$	1.12675	0.0585768
$371$	3.66552	0.190304
$372$	−0.101446	−0.00525972
$373$	36.7613	1.90343	0.951714	−	0.306986i	$-0.0993204\pi$
$373$	36.7613	1.90343	0.951714	+	0.306986i	$0.0993204\pi$
$374$	16.4270	0.849420
$375$	5.32666	0.275067
$376$	−0.0659721	−0.00340225
$377$	23.5838	1.21463
$378$	3.66552	0.188534
$379$	−11.6126	−0.596498	−0.298249	−	0.954488i	$-0.596403\pi$
$379$	−11.6126	−0.596498	−0.298249	+	0.954488i	$0.596403\pi$
$380$	2.85168	0.146288
$381$	1.03585	0.0530681
$382$	−16.3443	−0.836248
$383$	7.89942	0.403642	0.201821	−	0.979422i	$-0.435314\pi$
$383$	7.89942	0.403642	0.201821	+	0.979422i	$0.435314\pi$
$384$	1.00000	0.0510310
$385$	−67.1921	−3.42442
$386$	12.1749	0.619685
$387$	−5.12097	−0.260314
$388$	−4.40257	−0.223507
$389$	11.4232	0.579178	0.289589	−	0.957151i	$-0.406481\pi$
$389$	11.4232	0.579178	0.289589	+	0.957151i	$0.406481\pi$
$390$	−9.72672	−0.492532
$391$	−10.5531	−0.533692
$392$	6.43604	0.325069
$393$	12.3822	0.624599
$394$	8.59545	0.433033
$395$	21.4526	1.07940
$396$	6.42808	0.323023
$397$	−14.5644	−0.730966	−0.365483	−	0.930818i	$-0.619096\pi$
$397$	−14.5644	−0.730966	−0.365483	+	0.930818i	$0.619096\pi$
$398$	9.86121	0.494298
$399$	−3.66552	−0.183506
$400$	3.13210	0.156605
$401$	−23.8234	−1.18969	−0.594843	−	0.803842i	$-0.702786\pi$
$401$	−23.8234	−1.18969	−0.594843	+	0.803842i	$0.702786\pi$
$402$	−12.0497	−0.600983
$403$	−0.346018	−0.0172364
$404$	−9.98369	−0.496707
$405$	−2.85168	−0.141701
$406$	25.3446	1.25783
$407$	−2.53984	−0.125895
$408$	2.55551	0.126516
$409$	9.17174	0.453513	0.226757	−	0.973951i	$-0.427188\pi$
$409$	9.17174	0.453513	0.226757	+	0.973951i	$0.427188\pi$
$410$	2.56943	0.126895
$411$	5.20204	0.256598
$412$	6.88862	0.339378
$413$	−49.1549	−2.41875
$414$	−4.12954	−0.202956
$415$	−18.5623	−0.911189
$416$	3.41087	0.167232
$417$	−17.9890	−0.880923
$418$	−6.42808	−0.314408
$419$	−39.6300	−1.93605	−0.968027	−	0.250847i	$-0.919291\pi$
$419$	−39.6300	−1.93605	−0.968027	+	0.250847i	$0.919291\pi$
$420$	−10.4529	−0.510049
$421$	−39.3959	−1.92004	−0.960020	−	0.279930i	$-0.909689\pi$
$421$	−39.3959	−1.92004	−0.960020	+	0.279930i	$0.909689\pi$
$422$	19.9384	0.970588
$423$	−0.0659721	−0.00320767
$424$	1.00000	0.0485643
$425$	8.00411	0.388256
$426$	−13.8596	−0.671502
$427$	42.3918	2.05148
$428$	−8.52187	−0.411920
$429$	21.9253	1.05856
$430$	14.6034	0.704239
$431$	−21.4467	−1.03305	−0.516525	−	0.856272i	$-0.672775\pi$
$431$	−21.4467	−1.03305	−0.516525	+	0.856272i	$0.672775\pi$
$432$	1.00000	0.0481125
$433$	16.2567	0.781247	0.390624	−	0.920550i	$-0.372259\pi$
$433$	16.2567	0.781247	0.390624	+	0.920550i	$0.372259\pi$
$434$	−0.371851	−0.0178494
$435$	−19.7174	−0.945379
$436$	5.06168	0.242410
$437$	4.12954	0.197543
$438$	16.0108	0.765027
$439$	−19.9017	−0.949856	−0.474928	−	0.880025i	$-0.657526\pi$
$439$	−19.9017	−0.949856	−0.474928	+	0.880025i	$0.657526\pi$
$440$	−18.3308	−0.873889
$441$	6.43604	0.306478
$442$	8.71650	0.414602
$443$	−11.4127	−0.542236	−0.271118	−	0.962546i	$-0.587393\pi$
$443$	−11.4127	−0.542236	−0.271118	+	0.962546i	$0.587393\pi$
$444$	−0.395117	−0.0187514
$445$	−6.56306	−0.311119
$446$	4.81773	0.228126
$447$	8.58619	0.406113
$448$	3.66552	0.173180
$449$	−7.41532	−0.349951	−0.174975	−	0.984573i	$-0.555985\pi$
$449$	−7.41532	−0.349951	−0.174975	+	0.984573i	$0.555985\pi$
$450$	3.13210	0.147649
$451$	−5.79183	−0.272727
$452$	−17.7796	−0.836281
$453$	7.64379	0.359137
$454$	−25.4457	−1.19423
$455$	−35.6535	−1.67146
$456$	−1.00000	−0.0468293
$457$	−21.8042	−1.01996	−0.509980	−	0.860186i	$-0.670347\pi$
$457$	−21.8042	−1.01996	−0.509980	+	0.860186i	$0.670347\pi$
$458$	25.3646	1.18521
$459$	2.55551	0.119281
$460$	11.7761	0.549066
$461$	−6.83876	−0.318513	−0.159256	−	0.987237i	$-0.550910\pi$
$461$	−6.83876	−0.318513	−0.159256	+	0.987237i	$0.550910\pi$
$462$	23.5622	1.09621
$463$	18.8787	0.877366	0.438683	−	0.898642i	$-0.355445\pi$
$463$	18.8787	0.877366	0.438683	+	0.898642i	$0.355445\pi$
$464$	6.91431	0.320989
$465$	0.289291	0.0134156
$466$	−15.6794	−0.726334
$467$	−18.5527	−0.858517	−0.429259	−	0.903182i	$-0.641225\pi$
$467$	−18.5527	−0.858517	−0.429259	+	0.903182i	$0.641225\pi$
$468$	3.41087	0.157667
$469$	−44.1683	−2.03950
$470$	0.188132	0.00867786
$471$	−5.92830	−0.273162
$472$	−13.4101	−0.617249
$473$	−32.9180	−1.51357
$474$	−7.52279	−0.345533
$475$	−3.13210	−0.143711
$476$	9.36726	0.429348
$477$	1.00000	0.0457869
$478$	10.5130	0.480852
$479$	−19.4822	−0.890166	−0.445083	−	0.895489i	$-0.646826\pi$
$479$	−19.4822	−0.890166	−0.445083	+	0.895489i	$0.646826\pi$
$480$	−2.85168	−0.130161
$481$	−1.34769	−0.0614494
$482$	22.2562	1.01374
$483$	−15.1369	−0.688754
$484$	30.3202	1.37819
$485$	12.5547	0.570082
$486$	1.00000	0.0453609
$487$	4.07330	0.184579	0.0922894	−	0.995732i	$-0.470582\pi$
$487$	4.07330	0.184579	0.0922894	+	0.995732i	$0.470582\pi$
$488$	11.5650	0.523524
$489$	13.5202	0.611403
$490$	−18.3535	−0.829128
$491$	−0.201881	−0.00911074	−0.00455537	−	0.999990i	$-0.501450\pi$
$491$	−0.201881	−0.00911074	−0.00455537	+	0.999990i	$0.501450\pi$
$492$	−0.901021	−0.0406212
$493$	17.6696	0.795798
$494$	−3.41087	−0.153462
$495$	−18.3308	−0.823910
$496$	−0.101446	−0.00455505
$497$	−50.8028	−2.27882
$498$	6.50925	0.291687
$499$	−29.7445	−1.33155	−0.665774	−	0.746154i	$-0.731898\pi$
$499$	−29.7445	−1.33155	−0.665774	+	0.746154i	$0.731898\pi$
$500$	5.32666	0.238215
$501$	−6.41672	−0.286678
$502$	−14.7464	−0.658163
$503$	−17.0905	−0.762030	−0.381015	−	0.924569i	$-0.624425\pi$
$503$	−17.0905	−0.762030	−0.381015	+	0.924569i	$0.624425\pi$
$504$	3.66552	0.163275
$505$	28.4703	1.26691
$506$	−26.5450	−1.18007
$507$	−1.36597	−0.0606649
$508$	1.03585	0.0459583
$509$	−20.0622	−0.889241	−0.444621	−	0.895719i	$-0.646662\pi$
$509$	−20.0622	−0.889241	−0.444621	+	0.895719i	$0.646662\pi$
$510$	−7.28750	−0.322696
$511$	58.6880	2.59621
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−13.3407	−0.588431
$515$	−19.6442	−0.865626
$516$	−5.12097	−0.225438
$517$	−0.424074	−0.0186507
$518$	−1.44831	−0.0636350
$519$	−16.9237	−0.742869
$520$	−9.72672	−0.426545
$521$	9.85681	0.431835	0.215917	−	0.976412i	$-0.430726\pi$
$521$	9.85681	0.431835	0.215917	+	0.976412i	$0.430726\pi$
$522$	6.91431	0.302631
$523$	27.6648	1.20970	0.604848	−	0.796341i	$-0.293234\pi$
$523$	27.6648	1.20970	0.604848	+	0.796341i	$0.293234\pi$
$524$	12.3822	0.540918
$525$	11.4808	0.501062
$526$	−16.8026	−0.732629
$527$	−0.259245	−0.0112929
$528$	6.42808	0.279746
$529$	−5.94688	−0.258560
$530$	−2.85168	−0.123869
$531$	−13.4101	−0.581948
$532$	−3.66552	−0.158920
$533$	−3.07327	−0.133118
$534$	2.30147	0.0995943
$535$	24.3017	1.05065
$536$	−12.0497	−0.520466
$537$	11.1068	0.479292
$538$	13.3153	0.574062
$539$	41.3713	1.78199
$540$	−2.85168	−0.122717
$541$	17.0479	0.732948	0.366474	−	0.930428i	$-0.380565\pi$
$541$	17.0479	0.732948	0.366474	+	0.930428i	$0.380565\pi$
$542$	−14.6959	−0.631244
$543$	−16.6248	−0.713439
$544$	2.55551	0.109566
$545$	−14.4343	−0.618297
$546$	12.5026	0.535062
$547$	−20.6316	−0.882143	−0.441071	−	0.897472i	$-0.645401\pi$
$547$	−20.6316	−0.882143	−0.441071	+	0.897472i	$0.645401\pi$
$548$	5.20204	0.222220
$549$	11.5650	0.493583
$550$	20.1334	0.858490
$551$	−6.91431	−0.294560
$552$	−4.12954	−0.175765
$553$	−27.5749	−1.17261
$554$	20.5575	0.873406
$555$	1.12675	0.0478278
$556$	−17.9890	−0.762902
$557$	−7.64127	−0.323771	−0.161886	−	0.986810i	$-0.551758\pi$
$557$	−7.64127	−0.323771	−0.161886	+	0.986810i	$0.551758\pi$
$558$	−0.101446	−0.00429454
$559$	−17.4670	−0.738774
$560$	−10.4529	−0.441716
$561$	16.4270	0.693548
$562$	−8.81637	−0.371896
$563$	47.2148	1.98986	0.994932	−	0.100545i	$-0.0320588\pi$
$563$	47.2148	1.98986	0.994932	+	0.100545i	$0.0320588\pi$
$564$	−0.0659721	−0.00277793
$565$	50.7018	2.13304
$566$	25.0119	1.05133
$567$	3.66552	0.153937
$568$	−13.8596	−0.581537
$569$	−10.1792	−0.426736	−0.213368	−	0.976972i	$-0.568443\pi$
$569$	−10.1792	−0.426736	−0.213368	+	0.976972i	$0.568443\pi$
$570$	2.85168	0.119444
$571$	−40.3445	−1.68836	−0.844182	−	0.536056i	$-0.819913\pi$
$571$	−40.3445	−1.68836	−0.844182	+	0.536056i	$0.819913\pi$
$572$	21.9253	0.916744
$573$	−16.3443	−0.682793
$574$	−3.30271	−0.137853
$575$	−12.9341	−0.539391
$576$	1.00000	0.0416667
$577$	−39.7491	−1.65478	−0.827389	−	0.561630i	$-0.810175\pi$
$577$	−39.7491	−1.65478	−0.827389	+	0.561630i	$0.810175\pi$
$578$	−10.4694	−0.435469
$579$	12.1749	0.505970
$580$	−19.7174	−0.818722
$581$	23.8598	0.989871
$582$	−4.40257	−0.182493
$583$	6.42808	0.266224
$584$	16.0108	0.662533
$585$	−9.72672	−0.402150
$586$	23.9893	0.990989
$587$	−3.13827	−0.129530	−0.0647651	−	0.997901i	$-0.520630\pi$
$587$	−3.13827	−0.129530	−0.0647651	+	0.997901i	$0.520630\pi$
$588$	6.43604	0.265418
$589$	0.101446	0.00418000
$590$	38.2413	1.57437
$591$	8.59545	0.353570
$592$	−0.395117	−0.0162392
$593$	10.1415	0.416462	0.208231	−	0.978080i	$-0.433229\pi$
$593$	10.1415	0.416462	0.208231	+	0.978080i	$0.433229\pi$
$594$	6.42808	0.263747
$595$	−26.7125	−1.09510
$596$	8.58619	0.351704
$597$	9.86121	0.403592
$598$	−14.0853	−0.575992
$599$	14.2982	0.584210	0.292105	−	0.956386i	$-0.405644\pi$
$599$	14.2982	0.584210	0.292105	+	0.956386i	$0.405644\pi$
$600$	3.13210	0.127867
$601$	−1.02664	−0.0418776	−0.0209388	−	0.999781i	$-0.506666\pi$
$601$	−1.02664	−0.0418776	−0.0209388	+	0.999781i	$0.506666\pi$
$602$	−18.7710	−0.765050
$603$	−12.0497	−0.490700
$604$	7.64379	0.311022
$605$	−86.4635	−3.51524
$606$	−9.98369	−0.405560
$607$	−21.6386	−0.878285	−0.439142	−	0.898417i	$-0.644718\pi$
$607$	−21.6386	−0.878285	−0.439142	+	0.898417i	$0.644718\pi$
$608$	−1.00000	−0.0405554
$609$	25.3446	1.02701
$610$	−32.9798	−1.33531
$611$	−0.225022	−0.00910342
$612$	2.55551	0.103300
$613$	−37.3878	−1.51008	−0.755039	−	0.655680i	$-0.772382\pi$
$613$	−37.3878	−1.51008	−0.755039	+	0.655680i	$0.772382\pi$
$614$	−18.1303	−0.731681
$615$	2.56943	0.103609
$616$	23.5622	0.949350
$617$	19.4247	0.782011	0.391005	−	0.920388i	$-0.372127\pi$
$617$	19.4247	0.782011	0.391005	+	0.920388i	$0.372127\pi$
$618$	6.88862	0.277101
$619$	−20.4220	−0.820830	−0.410415	−	0.911899i	$-0.634616\pi$
$619$	−20.4220	−0.820830	−0.410415	+	0.911899i	$0.634616\pi$
$620$	0.289291	0.0116182
$621$	−4.12954	−0.165713
$622$	6.36450	0.255193
$623$	8.43608	0.337984
$624$	3.41087	0.136544
$625$	−30.8504	−1.23402
$626$	30.9977	1.23892
$627$	−6.42808	−0.256713
$628$	−5.92830	−0.236565
$629$	−1.00972	−0.0402603
$630$	−10.4529	−0.416454
$631$	25.9270	1.03214	0.516070	−	0.856547i	$-0.327395\pi$
$631$	25.9270	1.03214	0.516070	+	0.856547i	$0.327395\pi$
$632$	−7.52279	−0.299241
$633$	19.9384	0.792482
$634$	−8.45267	−0.335698
$635$	−2.95391	−0.117222
$636$	1.00000	0.0396526
$637$	21.9525	0.869789
$638$	44.4457	1.75962
$639$	−13.8596	−0.548279
$640$	−2.85168	−0.112723
$641$	−39.0253	−1.54141	−0.770704	−	0.637194i	$-0.780095\pi$
$641$	−39.0253	−1.54141	−0.770704	+	0.637194i	$0.780095\pi$
$642$	−8.52187	−0.336332
$643$	−13.3294	−0.525659	−0.262829	−	0.964842i	$-0.584656\pi$
$643$	−13.3294	−0.525659	−0.262829	+	0.964842i	$0.584656\pi$
$644$	−15.1369	−0.596478
$645$	14.6034	0.575008
$646$	−2.55551	−0.100545
$647$	25.0723	0.985692	0.492846	−	0.870116i	$-0.335957\pi$
$647$	25.0723	0.985692	0.492846	+	0.870116i	$0.335957\pi$
$648$	1.00000	0.0392837
$649$	−86.2010	−3.38369
$650$	10.6832	0.419029
$651$	−0.371851	−0.0145740
$652$	13.5202	0.529490
$653$	31.6342	1.23794	0.618972	−	0.785413i	$-0.287550\pi$
$653$	31.6342	1.23794	0.618972	+	0.785413i	$0.287550\pi$
$654$	5.06168	0.197927
$655$	−35.3101	−1.37968
$656$	−0.901021	−0.0351790
$657$	16.0108	0.624642
$658$	−0.241822	−0.00942720
$659$	48.1453	1.87548	0.937738	−	0.347343i	$-0.112916\pi$
$659$	48.1453	1.87548	0.937738	+	0.347343i	$0.112916\pi$
$660$	−18.3308	−0.713527
$661$	−12.6457	−0.491861	−0.245930	−	0.969287i	$-0.579093\pi$
$661$	−12.6457	−0.491861	−0.245930	+	0.969287i	$0.579093\pi$
$662$	−5.94713	−0.231142
$663$	8.71650	0.338521
$664$	6.50925	0.252608
$665$	10.4529	0.405346
$666$	−0.395117	−0.0153105
$667$	−28.5530	−1.10557
$668$	−6.41672	−0.248270
$669$	4.81773	0.186264
$670$	34.3618	1.32751
$671$	74.3408	2.86990
$672$	3.66552	0.141400
$673$	−36.1704	−1.39427	−0.697134	−	0.716941i	$-0.745542\pi$
$673$	−36.1704	−1.39427	−0.697134	+	0.716941i	$0.745542\pi$
$674$	28.8236	1.11024
$675$	3.13210	0.120555
$676$	−1.36597	−0.0525373
$677$	−12.5048	−0.480598	−0.240299	−	0.970699i	$-0.577246\pi$
$677$	−12.5048	−0.480598	−0.240299	+	0.970699i	$0.577246\pi$
$678$	−17.7796	−0.682821
$679$	−16.1377	−0.619309
$680$	−7.28750	−0.279463
$681$	−25.4457	−0.975082
$682$	−0.652101	−0.0249702
$683$	−19.2017	−0.734732	−0.367366	−	0.930077i	$-0.619740\pi$
$683$	−19.2017	−0.734732	−0.367366	+	0.930077i	$0.619740\pi$
$684$	−1.00000	−0.0382360
$685$	−14.8346	−0.566800
$686$	−2.06722	−0.0789269
$687$	25.3646	0.967722
$688$	−5.12097	−0.195235
$689$	3.41087	0.129944
$690$	11.7761	0.448310
$691$	41.7360	1.58771	0.793856	−	0.608106i	$-0.208070\pi$
$691$	41.7360	1.58771	0.793856	+	0.608106i	$0.208070\pi$
$692$	−16.9237	−0.643344
$693$	23.5622	0.895056
$694$	−2.06253	−0.0782926
$695$	51.2988	1.94588
$696$	6.91431	0.262086
$697$	−2.30257	−0.0872159
$698$	−2.82169	−0.106802
$699$	−15.6794	−0.593049
$700$	11.4808	0.433933
$701$	14.6059	0.551656	0.275828	−	0.961207i	$-0.411048\pi$
$701$	14.6059	0.551656	0.275828	+	0.961207i	$0.411048\pi$
$702$	3.41087	0.128735
$703$	0.395117	0.0149021
$704$	6.42808	0.242267
$705$	0.188132	0.00708544
$706$	0.146932	0.00552985
$707$	−36.5954	−1.37631
$708$	−13.4101	−0.503982
$709$	−14.6619	−0.550640	−0.275320	−	0.961353i	$-0.588784\pi$
$709$	−14.6619	−0.550640	−0.275320	+	0.961353i	$0.588784\pi$
$710$	39.5233	1.48328
$711$	−7.52279	−0.282127
$712$	2.30147	0.0862512
$713$	0.418924	0.0156888
$714$	9.36726	0.350561
$715$	−62.5241	−2.33827
$716$	11.1068	0.415079
$717$	10.5130	0.392614
$718$	21.2661	0.793645
$719$	20.0162	0.746478	0.373239	−	0.927735i	$-0.378247\pi$
$719$	20.0162	0.746478	0.373239	+	0.927735i	$0.378247\pi$
$720$	−2.85168	−0.106276
$721$	25.2504	0.940373
$722$	1.00000	0.0372161
$723$	22.2562	0.827715
$724$	−16.6248	−0.617856
$725$	21.6563	0.804296
$726$	30.3202	1.12529
$727$	20.3739	0.755624	0.377812	−	0.925882i	$-0.376677\pi$
$727$	20.3739	0.755624	0.377812	+	0.925882i	$0.376677\pi$
$728$	12.5026	0.463378
$729$	1.00000	0.0370370
$730$	−45.6578	−1.68987
$731$	−13.0867	−0.484029
$732$	11.5650	0.427455
$733$	43.5717	1.60936	0.804679	−	0.593711i	$-0.202338\pi$
$733$	43.5717	1.60936	0.804679	+	0.593711i	$0.202338\pi$
$734$	3.87367	0.142980
$735$	−18.3535	−0.676981
$736$	−4.12954	−0.152217
$737$	−77.4562	−2.85313
$738$	−0.901021	−0.0331671
$739$	−45.8202	−1.68552	−0.842762	−	0.538287i	$-0.819072\pi$
$739$	−45.8202	−1.68552	−0.842762	+	0.538287i	$0.819072\pi$
$740$	1.12675	0.0414201
$741$	−3.41087	−0.125301
$742$	3.66552	0.134565
$743$	14.9724	0.549285	0.274643	−	0.961546i	$-0.411440\pi$
$743$	14.9724	0.549285	0.274643	+	0.961546i	$0.411440\pi$
$744$	−0.101446	−0.00371918
$745$	−24.4851	−0.897064
$746$	36.7613	1.34593
$747$	6.50925	0.238161
$748$	16.4270	0.600630
$749$	−31.2371	−1.14138
$750$	5.32666	0.194502
$751$	−28.1959	−1.02888	−0.514442	−	0.857525i	$-0.672001\pi$
$751$	−28.1959	−1.02888	−0.514442	+	0.857525i	$0.672001\pi$
$752$	−0.0659721	−0.00240575
$753$	−14.7464	−0.537388
$754$	23.5838	0.858872
$755$	−21.7977	−0.793299
$756$	3.66552	0.133314
$757$	−12.2874	−0.446592	−0.223296	−	0.974751i	$-0.571682\pi$
$757$	−12.2874	−0.446592	−0.223296	+	0.974751i	$0.571682\pi$
$758$	−11.6126	−0.421788
$759$	−26.5450	−0.963523
$760$	2.85168	0.103441
$761$	−9.30519	−0.337313	−0.168656	−	0.985675i	$-0.553943\pi$
$761$	−9.30519	−0.337313	−0.168656	+	0.985675i	$0.553943\pi$
$762$	1.03585	0.0375248
$763$	18.5537	0.671688
$764$	−16.3443	−0.591316
$765$	−7.28750	−0.263480
$766$	7.89942	0.285418
$767$	−45.7400	−1.65158
$768$	1.00000	0.0360844
$769$	−4.02236	−0.145050	−0.0725250	−	0.997367i	$-0.523106\pi$
$769$	−4.02236	−0.145050	−0.0725250	+	0.997367i	$0.523106\pi$
$770$	−67.1921	−2.42143
$771$	−13.3407	−0.480452
$772$	12.1749	0.438183
$773$	−13.0686	−0.470045	−0.235022	−	0.971990i	$-0.575516\pi$
$773$	−13.0686	−0.470045	−0.235022	+	0.971990i	$0.575516\pi$
$774$	−5.12097	−0.184070
$775$	−0.317738	−0.0114135
$776$	−4.40257	−0.158043
$777$	−1.44831	−0.0519578
$778$	11.4232	0.409540
$779$	0.901021	0.0322824
$780$	−9.72672	−0.348272
$781$	−89.0908	−3.18792
$782$	−10.5531	−0.377377
$783$	6.91431	0.247097
$784$	6.43604	0.229858
$785$	16.9056	0.603388
$786$	12.3822	0.441658
$787$	36.0772	1.28601	0.643007	−	0.765860i	$-0.277687\pi$
$787$	36.0772	1.28601	0.643007	+	0.765860i	$0.277687\pi$
$788$	8.59545	0.306200
$789$	−16.8026	−0.598189
$790$	21.4526	0.763250
$791$	−65.1714	−2.31723
$792$	6.42808	0.228412
$793$	39.4468	1.40080
$794$	−14.5644	−0.516871
$795$	−2.85168	−0.101139
$796$	9.86121	0.349521
$797$	−35.7661	−1.26690	−0.633449	−	0.773784i	$-0.718362\pi$
$797$	−35.7661	−1.26690	−0.633449	+	0.773784i	$0.718362\pi$
$798$	−3.66552	−0.129758
$799$	−0.168592	−0.00596436
$800$	3.13210	0.110736
$801$	2.30147	0.0813184
$802$	−23.8234	−0.841235
$803$	102.919	3.63193
$804$	−12.0497	−0.424959
$805$	43.1657	1.52139
$806$	−0.346018	−0.0121880
$807$	13.3153	0.468720
$808$	−9.98369	−0.351225
$809$	−19.5743	−0.688196	−0.344098	−	0.938934i	$-0.611815\pi$
$809$	−19.5743	−0.688196	−0.344098	+	0.938934i	$0.611815\pi$
$810$	−2.85168	−0.100198
$811$	−24.6432	−0.865338	−0.432669	−	0.901553i	$-0.642428\pi$
$811$	−24.6432	−0.865338	−0.432669	+	0.901553i	$0.642428\pi$
$812$	25.3446	0.889420
$813$	−14.6959	−0.515409
$814$	−2.53984	−0.0890214
$815$	−38.5552	−1.35053
$816$	2.55551	0.0894606
$817$	5.12097	0.179160
$818$	9.17174	0.320682
$819$	12.5026	0.436877
$820$	2.56943	0.0897283
$821$	7.18798	0.250862	0.125431	−	0.992102i	$-0.459969\pi$
$821$	7.18798	0.250862	0.125431	+	0.992102i	$0.459969\pi$
$822$	5.20204	0.181442
$823$	−42.1150	−1.46804	−0.734018	−	0.679130i	$-0.762357\pi$
$823$	−42.1150	−1.46804	−0.734018	+	0.679130i	$0.762357\pi$
$824$	6.88862	0.239977
$825$	20.1334	0.700955
$826$	−49.1549	−1.71032
$827$	−56.4856	−1.96420	−0.982098	−	0.188373i	$-0.939679\pi$
$827$	−56.4856	−1.96420	−0.982098	+	0.188373i	$0.939679\pi$
$828$	−4.12954	−0.143512
$829$	−7.49043	−0.260153	−0.130077	−	0.991504i	$-0.541522\pi$
$829$	−7.49043	−0.260153	−0.130077	+	0.991504i	$0.541522\pi$
$830$	−18.5623	−0.644308
$831$	20.5575	0.713133
$832$	3.41087	0.118251
$833$	16.4473	0.569866
$834$	−17.9890	−0.622907
$835$	18.2984	0.633244
$836$	−6.42808	−0.222320
$837$	−0.101446	−0.00350648
$838$	−39.6300	−1.36900
$839$	30.1887	1.04223	0.521115	−	0.853486i	$-0.325516\pi$
$839$	30.1887	1.04223	0.521115	+	0.853486i	$0.325516\pi$
$840$	−10.4529	−0.360659
$841$	18.8077	0.648543
$842$	−39.3959	−1.35767
$843$	−8.81637	−0.303652
$844$	19.9384	0.686309
$845$	3.89532	0.134003
$846$	−0.0659721	−0.00226817
$847$	111.139	3.81879
$848$	1.00000	0.0343401
$849$	25.0119	0.858407
$850$	8.00411	0.274539
$851$	1.63165	0.0559323
$852$	−13.8596	−0.474823
$853$	−8.42694	−0.288533	−0.144266	−	0.989539i	$-0.546082\pi$
$853$	−8.42694	−0.288533	−0.144266	+	0.989539i	$0.546082\pi$
$854$	42.3918	1.45062
$855$	2.85168	0.0975255
$856$	−8.52187	−0.291272
$857$	−30.8783	−1.05478	−0.527391	−	0.849623i	$-0.676830\pi$
$857$	−30.8783	−1.05478	−0.527391	+	0.849623i	$0.676830\pi$
$858$	21.9253	0.748518
$859$	28.4143	0.969482	0.484741	−	0.874658i	$-0.338914\pi$
$859$	28.4143	0.969482	0.484741	+	0.874658i	$0.338914\pi$
$860$	14.6034	0.497972
$861$	−3.30271	−0.112556
$862$	−21.4467	−0.730476
$863$	5.36063	0.182478	0.0912390	−	0.995829i	$-0.470917\pi$
$863$	5.36063	0.182478	0.0912390	+	0.995829i	$0.470917\pi$
$864$	1.00000	0.0340207
$865$	48.2611	1.64093
$866$	16.2567	0.552425
$867$	−10.4694	−0.355559
$868$	−0.371851	−0.0126215
$869$	−48.3571	−1.64040
$870$	−19.7174	−0.668484
$871$	−41.0998	−1.39261
$872$	5.06168	0.171410
$873$	−4.40257	−0.149005
$874$	4.12954	0.139684
$875$	19.5250	0.660064
$876$	16.0108	0.540956
$877$	17.1480	0.579047	0.289524	−	0.957171i	$-0.406503\pi$
$877$	17.1480	0.579047	0.289524	+	0.957171i	$0.406503\pi$
$878$	−19.9017	−0.671649
$879$	23.9893	0.809139
$880$	−18.3308	−0.617933
$881$	36.0104	1.21322	0.606611	−	0.794999i	$-0.292529\pi$
$881$	36.0104	1.21322	0.606611	+	0.794999i	$0.292529\pi$
$882$	6.43604	0.216713
$883$	−23.7380	−0.798846	−0.399423	−	0.916767i	$-0.630789\pi$
$883$	−23.7380	−0.798846	−0.399423	+	0.916767i	$0.630789\pi$
$884$	8.71650	0.293168
$885$	38.2413	1.28547
$886$	−11.4127	−0.383419
$887$	27.0988	0.909889	0.454945	−	0.890520i	$-0.349659\pi$
$887$	27.0988	0.909889	0.454945	+	0.890520i	$0.349659\pi$
$888$	−0.395117	−0.0132592
$889$	3.79692	0.127345
$890$	−6.56306	−0.219994
$891$	6.42808	0.215349
$892$	4.81773	0.161310
$893$	0.0659721	0.00220767
$894$	8.58619	0.287165
$895$	−31.6730	−1.05871
$896$	3.66552	0.122456
$897$	−14.0853	−0.470295
$898$	−7.41532	−0.247452
$899$	−0.701428	−0.0233939
$900$	3.13210	0.104403
$901$	2.55551	0.0851363
$902$	−5.79183	−0.192847
$903$	−18.7710	−0.624661
$904$	−17.7796	−0.591340
$905$	47.4087	1.57592
$906$	7.64379	0.253948
$907$	−48.3638	−1.60589	−0.802947	−	0.596051i	$-0.796736\pi$
$907$	−48.3638	−1.60589	−0.802947	+	0.596051i	$0.796736\pi$
$908$	−25.4457	−0.844446
$909$	−9.98369	−0.331138
$910$	−35.6535	−1.18190
$911$	−20.5477	−0.680774	−0.340387	−	0.940285i	$-0.610558\pi$
$911$	−20.5477	−0.680774	−0.340387	+	0.940285i	$0.610558\pi$
$912$	−1.00000	−0.0331133
$913$	41.8420	1.38477
$914$	−21.8042	−0.721220
$915$	−32.9798	−1.09028
$916$	25.3646	0.838071
$917$	45.3872	1.49882
$918$	2.55551	0.0843443
$919$	−28.4727	−0.939226	−0.469613	−	0.882872i	$-0.655607\pi$
$919$	−28.4727	−0.939226	−0.469613	+	0.882872i	$0.655607\pi$
$920$	11.7761	0.388248
$921$	−18.1303	−0.597415
$922$	−6.83876	−0.225222
$923$	−47.2734	−1.55602
$924$	23.5622	0.775141
$925$	−1.23755	−0.0406902
$926$	18.8787	0.620392
$927$	6.88862	0.226252
$928$	6.91431	0.226973
$929$	18.8841	0.619567	0.309783	−	0.950807i	$-0.399743\pi$
$929$	18.8841	0.619567	0.309783	+	0.950807i	$0.399743\pi$
$930$	0.289291	0.00948623
$931$	−6.43604	−0.210933
$932$	−15.6794	−0.513596
$933$	6.36450	0.208364
$934$	−18.5527	−0.607063
$935$	−46.8446	−1.53198
$936$	3.41087	0.111488
$937$	20.8100	0.679835	0.339917	−	0.940455i	$-0.389601\pi$
$937$	20.8100	0.679835	0.339917	+	0.940455i	$0.389601\pi$
$938$	−44.1683	−1.44215
$939$	30.9977	1.01157
$940$	0.188132	0.00613618
$941$	39.3226	1.28188	0.640940	−	0.767591i	$-0.278545\pi$
$941$	39.3226	1.28188	0.640940	+	0.767591i	$0.278545\pi$
$942$	−5.92830	−0.193154
$943$	3.72081	0.121166
$944$	−13.4101	−0.436461
$945$	−10.4529	−0.340033
$946$	−32.9180	−1.07026
$947$	−26.4293	−0.858838	−0.429419	−	0.903105i	$-0.641282\pi$
$947$	−26.4293	−0.858838	−0.429419	+	0.903105i	$0.641282\pi$
$948$	−7.52279	−0.244329
$949$	54.6109	1.77274
$950$	−3.13210	−0.101619
$951$	−8.45267	−0.274097
$952$	9.36726	0.303595
$953$	−33.5561	−1.08699	−0.543494	−	0.839413i	$-0.682899\pi$
$953$	−33.5561	−1.08699	−0.543494	+	0.839413i	$0.682899\pi$
$954$	1.00000	0.0323762
$955$	46.6088	1.50823
$956$	10.5130	0.340014
$957$	44.4457	1.43673
$958$	−19.4822	−0.629443
$959$	19.0682	0.615743
$960$	−2.85168	−0.0920377
$961$	−30.9897	−0.999668
$962$	−1.34769	−0.0434513
$963$	−8.52187	−0.274614
$964$	22.2562	0.716822
$965$	−34.7189	−1.11764
$966$	−15.1369	−0.487022
$967$	−56.0626	−1.80285	−0.901426	−	0.432934i	$-0.857478\pi$
$967$	−56.0626	−1.80285	−0.901426	+	0.432934i	$0.857478\pi$
$968$	30.3202	0.974527
$969$	−2.55551	−0.0820947
$970$	12.5547	0.403109
$971$	35.2748	1.13202	0.566010	−	0.824398i	$-0.308486\pi$
$971$	35.2748	1.13202	0.566010	+	0.824398i	$0.308486\pi$
$972$	1.00000	0.0320750
$973$	−65.9389	−2.11390
$974$	4.07330	0.130517
$975$	10.6832	0.342136
$976$	11.5650	0.370187
$977$	−27.8600	−0.891320	−0.445660	−	0.895202i	$-0.647031\pi$
$977$	−27.8600	−0.891320	−0.445660	+	0.895202i	$0.647031\pi$
$978$	13.5202	0.432327
$979$	14.7940	0.472819
$980$	−18.3535	−0.586282
$981$	5.06168	0.161607
$982$	−0.201881	−0.00644227
$983$	−8.25505	−0.263295	−0.131648	−	0.991297i	$-0.542027\pi$
$983$	−8.25505	−0.263295	−0.131648	+	0.991297i	$0.542027\pi$
$984$	−0.901021	−0.0287235
$985$	−24.5115	−0.781002
$986$	17.6696	0.562714
$987$	−0.241822	−0.00769728
$988$	−3.41087	−0.108514
$989$	21.1473	0.672444
$990$	−18.3308	−0.582592
$991$	38.8222	1.23323	0.616614	−	0.787266i	$-0.288504\pi$
$991$	38.8222	1.23323	0.616614	+	0.787266i	$0.288504\pi$
$992$	−0.101446	−0.00322091
$993$	−5.94713	−0.188727
$994$	−50.8028	−1.61137
$995$	−28.1210	−0.891497
$996$	6.50925	0.206254
$997$	−40.3936	−1.27928	−0.639639	−	0.768675i	$-0.720916\pi$
$997$	−40.3936	−1.27928	−0.639639	+	0.768675i	$0.720916\pi$
$998$	−29.7445	−0.941546
$999$	−0.395117	−0.0125009

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bg.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bg.1.2	✓	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.85168 q^{5} +1.00000 q^{6} +3.66552 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.85168 q^{5} +1.00000 q^{6} +3.66552 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.85168 q^{10} +6.42808 q^{11} +1.00000 q^{12} +3.41087 q^{13} +3.66552 q^{14} -2.85168 q^{15} +1.00000 q^{16} +2.55551 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.85168 q^{20} +3.66552 q^{21} +6.42808 q^{22} -4.12954 q^{23} +1.00000 q^{24} +3.13210 q^{25} +3.41087 q^{26} +1.00000 q^{27} +3.66552 q^{28} +6.91431 q^{29} -2.85168 q^{30} -0.101446 q^{31} +1.00000 q^{32} +6.42808 q^{33} +2.55551 q^{34} -10.4529 q^{35} +1.00000 q^{36} -0.395117 q^{37} -1.00000 q^{38} +3.41087 q^{39} -2.85168 q^{40} -0.901021 q^{41} +3.66552 q^{42} -5.12097 q^{43} +6.42808 q^{44} -2.85168 q^{45} -4.12954 q^{46} -0.0659721 q^{47} +1.00000 q^{48} +6.43604 q^{49} +3.13210 q^{50} +2.55551 q^{51} +3.41087 q^{52} +1.00000 q^{53} +1.00000 q^{54} -18.3308 q^{55} +3.66552 q^{56} -1.00000 q^{57} +6.91431 q^{58} -13.4101 q^{59} -2.85168 q^{60} +11.5650 q^{61} -0.101446 q^{62} +3.66552 q^{63} +1.00000 q^{64} -9.72672 q^{65} +6.42808 q^{66} -12.0497 q^{67} +2.55551 q^{68} -4.12954 q^{69} -10.4529 q^{70} -13.8596 q^{71} +1.00000 q^{72} +16.0108 q^{73} -0.395117 q^{74} +3.13210 q^{75} -1.00000 q^{76} +23.5622 q^{77} +3.41087 q^{78} -7.52279 q^{79} -2.85168 q^{80} +1.00000 q^{81} -0.901021 q^{82} +6.50925 q^{83} +3.66552 q^{84} -7.28750 q^{85} -5.12097 q^{86} +6.91431 q^{87} +6.42808 q^{88} +2.30147 q^{89} -2.85168 q^{90} +12.5026 q^{91} -4.12954 q^{92} -0.101446 q^{93} -0.0659721 q^{94} +2.85168 q^{95} +1.00000 q^{96} -4.40257 q^{97} +6.43604 q^{98} +6.42808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{3} + 12 q^{4} + 5 q^{5} + 12 q^{6} + 6 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^3 + 12 * q^4 + 5 * q^5 + 12 * q^6 + 6 * q^7 + 12 * q^8 + 12 * q^9	\( 12 q + 12 q^{2} + 12 q^{3} + 12 q^{4} + 5 q^{5} + 12 q^{6} + 6 q^{7} + 12 q^{8} + 12 q^{9} + 5 q^{10} + 7 q^{11} + 12 q^{12} + 9 q^{13} + 6 q^{14} + 5 q^{15} + 12 q^{16} + 25 q^{17} + 12 q^{18} - 12 q^{19} + 5 q^{20} + 6 q^{21} + 7 q^{22} + 22 q^{23} + 12 q^{24} + 33 q^{25} + 9 q^{26} + 12 q^{27} + 6 q^{28} + q^{29} + 5 q^{30} + 23 q^{31} + 12 q^{32} + 7 q^{33} + 25 q^{34} + 5 q^{35} + 12 q^{36} - q^{37} - 12 q^{38} + 9 q^{39} + 5 q^{40} - 15 q^{41} + 6 q^{42} + 2 q^{43} + 7 q^{44} + 5 q^{45} + 22 q^{46} + 11 q^{47} + 12 q^{48} + 36 q^{49} + 33 q^{50} + 25 q^{51} + 9 q^{52} + 12 q^{53} + 12 q^{54} - 4 q^{55} + 6 q^{56} - 12 q^{57} + q^{58} + 3 q^{59} + 5 q^{60} + 16 q^{61} + 23 q^{62} + 6 q^{63} + 12 q^{64} + 7 q^{65} + 7 q^{66} - 2 q^{67} + 25 q^{68} + 22 q^{69} + 5 q^{70} - 4 q^{71} + 12 q^{72} + 35 q^{73} - q^{74} + 33 q^{75} - 12 q^{76} + 11 q^{77} + 9 q^{78} + 4 q^{79} + 5 q^{80} + 12 q^{81} - 15 q^{82} + 39 q^{83} + 6 q^{84} + 10 q^{85} + 2 q^{86} + q^{87} + 7 q^{88} + 11 q^{89} + 5 q^{90} - 18 q^{91} + 22 q^{92} + 23 q^{93} + 11 q^{94} - 5 q^{95} + 12 q^{96} - 21 q^{97} + 36 q^{98} + 7 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^3 + 12 * q^4 + 5 * q^5 + 12 * q^6 + 6 * q^7 + 12 * q^8 + 12 * q^9 + 5 * q^10 + 7 * q^11 + 12 * q^12 + 9 * q^13 + 6 * q^14 + 5 * q^15 + 12 * q^16 + 25 * q^17 + 12 * q^18 - 12 * q^19 + 5 * q^20 + 6 * q^21 + 7 * q^22 + 22 * q^23 + 12 * q^24 + 33 * q^25 + 9 * q^26 + 12 * q^27 + 6 * q^28 + q^29 + 5 * q^30 + 23 * q^31 + 12 * q^32 + 7 * q^33 + 25 * q^34 + 5 * q^35 + 12 * q^36 - q^37 - 12 * q^38 + 9 * q^39 + 5 * q^40 - 15 * q^41 + 6 * q^42 + 2 * q^43 + 7 * q^44 + 5 * q^45 + 22 * q^46 + 11 * q^47 + 12 * q^48 + 36 * q^49 + 33 * q^50 + 25 * q^51 + 9 * q^52 + 12 * q^53 + 12 * q^54 - 4 * q^55 + 6 * q^56 - 12 * q^57 + q^58 + 3 * q^59 + 5 * q^60 + 16 * q^61 + 23 * q^62 + 6 * q^63 + 12 * q^64 + 7 * q^65 + 7 * q^66 - 2 * q^67 + 25 * q^68 + 22 * q^69 + 5 * q^70 - 4 * q^71 + 12 * q^72 + 35 * q^73 - q^74 + 33 * q^75 - 12 * q^76 + 11 * q^77 + 9 * q^78 + 4 * q^79 + 5 * q^80 + 12 * q^81 - 15 * q^82 + 39 * q^83 + 6 * q^84 + 10 * q^85 + 2 * q^86 + q^87 + 7 * q^88 + 11 * q^89 + 5 * q^90 - 18 * q^91 + 22 * q^92 + 23 * q^93 + 11 * q^94 - 5 * q^95 + 12 * q^96 - 21 * q^97 + 36 * q^98 + 7 * q^99

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