LMFDB - Embedded newform 6045.2.a.y.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 78 x^{8} - 252 x^{7} - 149 x^{6} + 583 x^{5} + 18 x^{4} + \cdots - 26 $ x^12 - 3x^11 - 15x^10 + 46x^9 + 78x^8 - 252x^7 - 149x^6 + 583x^5 + 18x^4 - 511x^3 + 134x^2 + 98*x - 26
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			1.11
Root			$-2.20326$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20326 q^{2} +1.00000 q^{3} +2.85433 q^{4} -1.00000 q^{5} +2.20326 q^{6} -3.44879 q^{7} +1.88232 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.20326 q^{2} +1.00000 q^{3} +2.85433 q^{4} -1.00000 q^{5} +2.20326 q^{6} -3.44879 q^{7} +1.88232 q^{8} +1.00000 q^{9} -2.20326 q^{10} +0.997786 q^{11} +2.85433 q^{12} +1.00000 q^{13} -7.59857 q^{14} -1.00000 q^{15} -1.56144 q^{16} -4.89150 q^{17} +2.20326 q^{18} -1.09286 q^{19} -2.85433 q^{20} -3.44879 q^{21} +2.19838 q^{22} +6.11250 q^{23} +1.88232 q^{24} +1.00000 q^{25} +2.20326 q^{26} +1.00000 q^{27} -9.84401 q^{28} -5.57228 q^{29} -2.20326 q^{30} -1.00000 q^{31} -7.20489 q^{32} +0.997786 q^{33} -10.7772 q^{34} +3.44879 q^{35} +2.85433 q^{36} -2.46649 q^{37} -2.40785 q^{38} +1.00000 q^{39} -1.88232 q^{40} -11.0266 q^{41} -7.59857 q^{42} +9.82536 q^{43} +2.84801 q^{44} -1.00000 q^{45} +13.4674 q^{46} -9.17479 q^{47} -1.56144 q^{48} +4.89416 q^{49} +2.20326 q^{50} -4.89150 q^{51} +2.85433 q^{52} +4.56218 q^{53} +2.20326 q^{54} -0.997786 q^{55} -6.49172 q^{56} -1.09286 q^{57} -12.2772 q^{58} -6.88653 q^{59} -2.85433 q^{60} -7.15737 q^{61} -2.20326 q^{62} -3.44879 q^{63} -12.7513 q^{64} -1.00000 q^{65} +2.19838 q^{66} -13.5984 q^{67} -13.9620 q^{68} +6.11250 q^{69} +7.59857 q^{70} +0.442078 q^{71} +1.88232 q^{72} +4.55430 q^{73} -5.43430 q^{74} +1.00000 q^{75} -3.11939 q^{76} -3.44116 q^{77} +2.20326 q^{78} +0.866725 q^{79} +1.56144 q^{80} +1.00000 q^{81} -24.2945 q^{82} -12.2593 q^{83} -9.84401 q^{84} +4.89150 q^{85} +21.6478 q^{86} -5.57228 q^{87} +1.87815 q^{88} +4.73997 q^{89} -2.20326 q^{90} -3.44879 q^{91} +17.4471 q^{92} -1.00000 q^{93} -20.2144 q^{94} +1.09286 q^{95} -7.20489 q^{96} -10.6691 q^{97} +10.7831 q^{98} +0.997786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 24 q^{11} + 15 q^{12} + 12 q^{13} - 13 q^{14} - 12 q^{15} + 21 q^{16} - 9 q^{17} - 3 q^{18} - 2 q^{19} - 15 q^{20} + 2 q^{21} + 9 q^{22} + 7 q^{23} - 12 q^{24} + 12 q^{25} - 3 q^{26} + 12 q^{27} - 16 q^{28} - 24 q^{29} + 3 q^{30} - 12 q^{31} - 57 q^{32} - 24 q^{33} - 15 q^{34} - 2 q^{35} + 15 q^{36} - 13 q^{37} - 22 q^{38} + 12 q^{39} + 12 q^{40} - 46 q^{41} - 13 q^{42} - 5 q^{43} - 49 q^{44} - 12 q^{45} - 2 q^{46} - 39 q^{47} + 21 q^{48} - 3 q^{50} - 9 q^{51} + 15 q^{52} - 12 q^{53} - 3 q^{54} + 24 q^{55} + q^{56} - 2 q^{57} - 8 q^{58} - 31 q^{59} - 15 q^{60} + 2 q^{61} + 3 q^{62} + 2 q^{63} + 18 q^{64} - 12 q^{65} + 9 q^{66} + 5 q^{67} - 5 q^{68} + 7 q^{69} + 13 q^{70} - 31 q^{71} - 12 q^{72} + 3 q^{73} + 16 q^{74} + 12 q^{75} - 2 q^{76} + 2 q^{77} - 3 q^{78} + 5 q^{79} - 21 q^{80} + 12 q^{81} + 25 q^{82} + 2 q^{83} - 16 q^{84} + 9 q^{85} - 26 q^{86} - 24 q^{87} + 27 q^{88} - 13 q^{89} + 3 q^{90} + 2 q^{91} + 5 q^{92} - 12 q^{93} - 12 q^{94} + 2 q^{95} - 57 q^{96} - 28 q^{97} - 29 q^{98} - 24 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 24 * q^11 + 15 * q^12 + 12 * q^13 - 13 * q^14 - 12 * q^15 + 21 * q^16 - 9 * q^17 - 3 * q^18 - 2 * q^19 - 15 * q^20 + 2 * q^21 + 9 * q^22 + 7 * q^23 - 12 * q^24 + 12 * q^25 - 3 * q^26 + 12 * q^27 - 16 * q^28 - 24 * q^29 + 3 * q^30 - 12 * q^31 - 57 * q^32 - 24 * q^33 - 15 * q^34 - 2 * q^35 + 15 * q^36 - 13 * q^37 - 22 * q^38 + 12 * q^39 + 12 * q^40 - 46 * q^41 - 13 * q^42 - 5 * q^43 - 49 * q^44 - 12 * q^45 - 2 * q^46 - 39 * q^47 + 21 * q^48 - 3 * q^50 - 9 * q^51 + 15 * q^52 - 12 * q^53 - 3 * q^54 + 24 * q^55 + q^56 - 2 * q^57 - 8 * q^58 - 31 * q^59 - 15 * q^60 + 2 * q^61 + 3 * q^62 + 2 * q^63 + 18 * q^64 - 12 * q^65 + 9 * q^66 + 5 * q^67 - 5 * q^68 + 7 * q^69 + 13 * q^70 - 31 * q^71 - 12 * q^72 + 3 * q^73 + 16 * q^74 + 12 * q^75 - 2 * q^76 + 2 * q^77 - 3 * q^78 + 5 * q^79 - 21 * q^80 + 12 * q^81 + 25 * q^82 + 2 * q^83 - 16 * q^84 + 9 * q^85 - 26 * q^86 - 24 * q^87 + 27 * q^88 - 13 * q^89 + 3 * q^90 + 2 * q^91 + 5 * q^92 - 12 * q^93 - 12 * q^94 + 2 * q^95 - 57 * q^96 - 28 * q^97 - 29 * q^98 - 24 * 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 $

$2$	2.20326	1.55794	0.778968	−	0.627063i	$-0.215743\pi$
$2$	2.20326	1.55794	0.778968	+	0.627063i	$0.215743\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.85433	1.42717
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.20326	0.899475
$7$	−3.44879	−1.30352	−0.651760	−	0.758425i	$-0.725969\pi$
$7$	−3.44879	−1.30352	−0.651760	+	0.758425i	$0.725969\pi$
$8$	1.88232	0.665500
$9$	1.00000	0.333333
$10$	−2.20326	−0.696731
$11$	0.997786	0.300844	0.150422	−	0.988622i	$-0.451937\pi$
$11$	0.997786	0.300844	0.150422	+	0.988622i	$0.451937\pi$
$12$	2.85433	0.823975
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−7.59857	−2.03080
$15$	−1.00000	−0.258199
$16$	−1.56144	−0.390361
$17$	−4.89150	−1.18636	−0.593181	−	0.805069i	$-0.702128\pi$
$17$	−4.89150	−1.18636	−0.593181	+	0.805069i	$0.702128\pi$
$18$	2.20326	0.519312
$19$	−1.09286	−0.250720	−0.125360	−	0.992111i	$-0.540009\pi$
$19$	−1.09286	−0.250720	−0.125360	+	0.992111i	$0.540009\pi$
$20$	−2.85433	−0.638249
$21$	−3.44879	−0.752588
$22$	2.19838	0.468696
$23$	6.11250	1.27454	0.637272	−	0.770639i	$-0.280063\pi$
$23$	6.11250	1.27454	0.637272	+	0.770639i	$0.280063\pi$
$24$	1.88232	0.384227
$25$	1.00000	0.200000
$26$	2.20326	0.432094
$27$	1.00000	0.192450
$28$	−9.84401	−1.86034
$29$	−5.57228	−1.03475	−0.517374	−	0.855760i	$-0.673090\pi$
$29$	−5.57228	−1.03475	−0.517374	+	0.855760i	$0.673090\pi$
$30$	−2.20326	−0.402258
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−7.20489	−1.27366
$33$	0.997786	0.173692
$34$	−10.7772	−1.84828
$35$	3.44879	0.582952
$36$	2.85433	0.475722
$37$	−2.46649	−0.405488	−0.202744	−	0.979232i	$-0.564986\pi$
$37$	−2.46649	−0.405488	−0.202744	+	0.979232i	$0.564986\pi$
$38$	−2.40785	−0.390605
$39$	1.00000	0.160128
$40$	−1.88232	−0.297621
$41$	−11.0266	−1.72207	−0.861035	−	0.508545i	$-0.830183\pi$
$41$	−11.0266	−1.72207	−0.861035	+	0.508545i	$0.830183\pi$
$42$	−7.59857	−1.17248
$43$	9.82536	1.49835	0.749177	−	0.662370i	$-0.230449\pi$
$43$	9.82536	1.49835	0.749177	+	0.662370i	$0.230449\pi$
$44$	2.84801	0.429354
$45$	−1.00000	−0.149071
$46$	13.4674	1.98566
$47$	−9.17479	−1.33828	−0.669140	−	0.743136i	$-0.733337\pi$
$47$	−9.17479	−1.33828	−0.669140	+	0.743136i	$0.733337\pi$
$48$	−1.56144	−0.225375
$49$	4.89416	0.699166
$50$	2.20326	0.311587
$51$	−4.89150	−0.684947
$52$	2.85433	0.395825
$53$	4.56218	0.626664	0.313332	−	0.949644i	$-0.398555\pi$
$53$	4.56218	0.626664	0.313332	+	0.949644i	$0.398555\pi$
$54$	2.20326	0.299825
$55$	−0.997786	−0.134541
$56$	−6.49172	−0.867493
$57$	−1.09286	−0.144753
$58$	−12.2772	−1.61207
$59$	−6.88653	−0.896550	−0.448275	−	0.893896i	$-0.647961\pi$
$59$	−6.88653	−0.896550	−0.448275	+	0.893896i	$0.647961\pi$
$60$	−2.85433	−0.368493
$61$	−7.15737	−0.916407	−0.458203	−	0.888847i	$-0.651507\pi$
$61$	−7.15737	−0.916407	−0.458203	+	0.888847i	$0.651507\pi$
$62$	−2.20326	−0.279814
$63$	−3.44879	−0.434507
$64$	−12.7513	−1.59392
$65$	−1.00000	−0.124035
$66$	2.19838	0.270602
$67$	−13.5984	−1.66131	−0.830653	−	0.556790i	$-0.812033\pi$
$67$	−13.5984	−1.66131	−0.830653	+	0.556790i	$0.812033\pi$
$68$	−13.9620	−1.69314
$69$	6.11250	0.735858
$70$	7.59857	0.908203
$71$	0.442078	0.0524650	0.0262325	−	0.999656i	$-0.491649\pi$
$71$	0.442078	0.0524650	0.0262325	+	0.999656i	$0.491649\pi$
$72$	1.88232	0.221833
$73$	4.55430	0.533040	0.266520	−	0.963829i	$-0.414126\pi$
$73$	4.55430	0.533040	0.266520	+	0.963829i	$0.414126\pi$
$74$	−5.43430	−0.631725
$75$	1.00000	0.115470
$76$	−3.11939	−0.357819
$77$	−3.44116	−0.392156
$78$	2.20326	0.249470
$79$	0.866725	0.0975141	0.0487571	−	0.998811i	$-0.484474\pi$
$79$	0.866725	0.0975141	0.0487571	+	0.998811i	$0.484474\pi$
$80$	1.56144	0.174575
$81$	1.00000	0.111111
$82$	−24.2945	−2.68288
$83$	−12.2593	−1.34563	−0.672815	−	0.739810i	$-0.734915\pi$
$83$	−12.2593	−1.34563	−0.672815	+	0.739810i	$0.734915\pi$
$84$	−9.84401	−1.07407
$85$	4.89150	0.530558
$86$	21.6478	2.33434
$87$	−5.57228	−0.597412
$88$	1.87815	0.200211
$89$	4.73997	0.502436	0.251218	−	0.967931i	$-0.419169\pi$
$89$	4.73997	0.502436	0.251218	+	0.967931i	$0.419169\pi$
$90$	−2.20326	−0.232244
$91$	−3.44879	−0.361532
$92$	17.4471	1.81899
$93$	−1.00000	−0.103695
$94$	−20.2144	−2.08496
$95$	1.09286	0.112125
$96$	−7.20489	−0.735346
$97$	−10.6691	−1.08328	−0.541640	−	0.840610i	$-0.682197\pi$
$97$	−10.6691	−1.08328	−0.541640	+	0.840610i	$0.682197\pi$
$98$	10.7831	1.08926
$99$	0.997786	0.100281
$100$	2.85433	0.285433
$101$	−18.4987	−1.84069	−0.920344	−	0.391111i	$-0.872091\pi$
$101$	−18.4987	−1.84069	−0.920344	+	0.391111i	$0.872091\pi$
$102$	−10.7772	−1.06710
$103$	−13.7529	−1.35512	−0.677558	−	0.735469i	$-0.736962\pi$
$103$	−13.7529	−1.35512	−0.677558	+	0.735469i	$0.736962\pi$
$104$	1.88232	0.184576
$105$	3.44879	0.336568
$106$	10.0516	0.976302
$107$	2.23698	0.216257	0.108129	−	0.994137i	$-0.465514\pi$
$107$	2.23698	0.216257	0.108129	+	0.994137i	$0.465514\pi$
$108$	2.85433	0.274658
$109$	12.1576	1.16448	0.582242	−	0.813015i	$-0.302175\pi$
$109$	12.1576	1.16448	0.582242	+	0.813015i	$0.302175\pi$
$110$	−2.19838	−0.209607
$111$	−2.46649	−0.234109
$112$	5.38509	0.508843
$113$	7.84384	0.737887	0.368943	−	0.929452i	$-0.379720\pi$
$113$	7.84384	0.737887	0.368943	+	0.929452i	$0.379720\pi$
$114$	−2.40785	−0.225516
$115$	−6.11250	−0.569993
$116$	−15.9052	−1.47676
$117$	1.00000	0.0924500
$118$	−15.1728	−1.39677
$119$	16.8698	1.54645
$120$	−1.88232	−0.171831
$121$	−10.0044	−0.909493
$122$	−15.7695	−1.42770
$123$	−11.0266	−0.994238
$124$	−2.85433	−0.256327
$125$	−1.00000	−0.0894427
$126$	−7.59857	−0.676934
$127$	14.0169	1.24380	0.621900	−	0.783097i	$-0.286361\pi$
$127$	14.0169	1.24380	0.621900	+	0.783097i	$0.286361\pi$
$128$	−13.6847	−1.20956
$129$	9.82536	0.865075
$130$	−2.20326	−0.193238
$131$	18.1379	1.58471	0.792356	−	0.610059i	$-0.208854\pi$
$131$	18.1379	1.58471	0.792356	+	0.610059i	$0.208854\pi$
$132$	2.84801	0.247888
$133$	3.76905	0.326818
$134$	−29.9607	−2.58821
$135$	−1.00000	−0.0860663
$136$	−9.20736	−0.789524
$137$	−10.1866	−0.870303	−0.435152	−	0.900357i	$-0.643305\pi$
$137$	−10.1866	−0.870303	−0.435152	+	0.900357i	$0.643305\pi$
$138$	13.4674	1.14642
$139$	10.5644	0.896063	0.448032	−	0.894018i	$-0.352125\pi$
$139$	10.5644	0.896063	0.448032	+	0.894018i	$0.352125\pi$
$140$	9.84401	0.831970
$141$	−9.17479	−0.772656
$142$	0.974012	0.0817372
$143$	0.997786	0.0834390
$144$	−1.56144	−0.130120
$145$	5.57228	0.462753
$146$	10.0343	0.830443
$147$	4.89416	0.403664
$148$	−7.04018	−0.578699
$149$	9.04227	0.740771	0.370386	−	0.928878i	$-0.379226\pi$
$149$	9.04227	0.740771	0.370386	+	0.928878i	$0.379226\pi$
$150$	2.20326	0.179895
$151$	14.3837	1.17053	0.585263	−	0.810843i	$-0.300991\pi$
$151$	14.3837	1.17053	0.585263	+	0.810843i	$0.300991\pi$
$152$	−2.05711	−0.166854
$153$	−4.89150	−0.395454
$154$	−7.58174	−0.610954
$155$	1.00000	0.0803219
$156$	2.85433	0.228530
$157$	−2.45563	−0.195981	−0.0979904	−	0.995187i	$-0.531241\pi$
$157$	−2.45563	−0.195981	−0.0979904	+	0.995187i	$0.531241\pi$
$158$	1.90962	0.151921
$159$	4.56218	0.361804
$160$	7.20489	0.569597
$161$	−21.0807	−1.66139
$162$	2.20326	0.173104
$163$	−16.4572	−1.28903	−0.644515	−	0.764592i	$-0.722941\pi$
$163$	−16.4572	−1.28903	−0.644515	+	0.764592i	$0.722941\pi$
$164$	−31.4737	−2.45768
$165$	−0.997786	−0.0776775
$166$	−27.0103	−2.09641
$167$	2.30711	0.178530	0.0892649	−	0.996008i	$-0.471548\pi$
$167$	2.30711	0.178530	0.0892649	+	0.996008i	$0.471548\pi$
$168$	−6.49172	−0.500847
$169$	1.00000	0.0769231
$170$	10.7772	0.826575
$171$	−1.09286	−0.0835732
$172$	28.0449	2.13840
$173$	16.2935	1.23877	0.619387	−	0.785086i	$-0.287381\pi$
$173$	16.2935	1.23877	0.619387	+	0.785086i	$0.287381\pi$
$174$	−12.2772	−0.930730
$175$	−3.44879	−0.260704
$176$	−1.55799	−0.117438
$177$	−6.88653	−0.517623
$178$	10.4434	0.782763
$179$	8.69507	0.649900	0.324950	−	0.945731i	$-0.394653\pi$
$179$	8.69507	0.649900	0.324950	+	0.945731i	$0.394653\pi$
$180$	−2.85433	−0.212750
$181$	11.9749	0.890091	0.445045	−	0.895508i	$-0.353188\pi$
$181$	11.9749	0.890091	0.445045	+	0.895508i	$0.353188\pi$
$182$	−7.59857	−0.563243
$183$	−7.15737	−0.529088
$184$	11.5057	0.848209
$185$	2.46649	0.181340
$186$	−2.20326	−0.161551
$187$	−4.88067	−0.356910
$188$	−26.1879	−1.90995
$189$	−3.44879	−0.250863
$190$	2.40785	0.174684
$191$	3.30099	0.238851	0.119426	−	0.992843i	$-0.461895\pi$
$191$	3.30099	0.238851	0.119426	+	0.992843i	$0.461895\pi$
$192$	−12.7513	−0.920248
$193$	−7.50827	−0.540457	−0.270229	−	0.962796i	$-0.587099\pi$
$193$	−7.50827	−0.540457	−0.270229	+	0.962796i	$0.587099\pi$
$194$	−23.5067	−1.68768
$195$	−1.00000	−0.0716115
$196$	13.9696	0.997827
$197$	−10.6735	−0.760453	−0.380226	−	0.924893i	$-0.624154\pi$
$197$	−10.6735	−0.760453	−0.380226	+	0.924893i	$0.624154\pi$
$198$	2.19838	0.156232
$199$	−15.4071	−1.09218	−0.546089	−	0.837727i	$-0.683884\pi$
$199$	−15.4071	−1.09218	−0.546089	+	0.837727i	$0.683884\pi$
$200$	1.88232	0.133100
$201$	−13.5984	−0.959156
$202$	−40.7573	−2.86767
$203$	19.2176	1.34881
$204$	−13.9620	−0.977534
$205$	11.0266	0.770133
$206$	−30.3012	−2.11119
$207$	6.11250	0.424848
$208$	−1.56144	−0.108267
$209$	−1.09044	−0.0754274
$210$	7.59857	0.524351
$211$	15.3473	1.05655	0.528277	−	0.849072i	$-0.322838\pi$
$211$	15.3473	1.05655	0.528277	+	0.849072i	$0.322838\pi$
$212$	13.0220	0.894354
$213$	0.442078	0.0302907
$214$	4.92865	0.336915
$215$	−9.82536	−0.670084
$216$	1.88232	0.128076
$217$	3.44879	0.234119
$218$	26.7862	1.81419
$219$	4.55430	0.307751
$220$	−2.84801	−0.192013
$221$	−4.89150	−0.329038
$222$	−5.43430	−0.364726
$223$	0.916156	0.0613504	0.0306752	−	0.999529i	$-0.490234\pi$
$223$	0.916156	0.0613504	0.0306752	+	0.999529i	$0.490234\pi$
$224$	24.8482	1.66024
$225$	1.00000	0.0666667
$226$	17.2820	1.14958
$227$	17.5004	1.16154	0.580771	−	0.814067i	$-0.302751\pi$
$227$	17.5004	1.16154	0.580771	+	0.814067i	$0.302751\pi$
$228$	−3.11939	−0.206587
$229$	−5.72842	−0.378544	−0.189272	−	0.981925i	$-0.560613\pi$
$229$	−5.72842	−0.378544	−0.189272	+	0.981925i	$0.560613\pi$
$230$	−13.4674	−0.888014
$231$	−3.44116	−0.226411
$232$	−10.4888	−0.688624
$233$	−7.39663	−0.484569	−0.242285	−	0.970205i	$-0.577897\pi$
$233$	−7.39663	−0.484569	−0.242285	+	0.970205i	$0.577897\pi$
$234$	2.20326	0.144031
$235$	9.17479	0.598497
$236$	−19.6565	−1.27953
$237$	0.866725	0.0562998
$238$	37.1684	2.40927
$239$	−1.87724	−0.121429	−0.0607143	−	0.998155i	$-0.519338\pi$
$239$	−1.87724	−0.121429	−0.0607143	+	0.998155i	$0.519338\pi$
$240$	1.56144	0.100791
$241$	14.8021	0.953486	0.476743	−	0.879043i	$-0.341817\pi$
$241$	14.8021	0.953486	0.476743	+	0.879043i	$0.341817\pi$
$242$	−22.0423	−1.41693
$243$	1.00000	0.0641500
$244$	−20.4295	−1.30787
$245$	−4.89416	−0.312677
$246$	−24.2945	−1.54896
$247$	−1.09286	−0.0695371
$248$	−1.88232	−0.119527
$249$	−12.2593	−0.776900
$250$	−2.20326	−0.139346
$251$	15.0292	0.948632	0.474316	−	0.880355i	$-0.342695\pi$
$251$	15.0292	0.948632	0.474316	+	0.880355i	$0.342695\pi$
$252$	−9.84401	−0.620114
$253$	6.09896	0.383439
$254$	30.8828	1.93776
$255$	4.89150	0.306318
$256$	−4.64814	−0.290509
$257$	18.0754	1.12751	0.563756	−	0.825942i	$-0.309356\pi$
$257$	18.0754	1.12751	0.563756	+	0.825942i	$0.309356\pi$
$258$	21.6478	1.34773
$259$	8.50640	0.528562
$260$	−2.85433	−0.177018
$261$	−5.57228	−0.344916
$262$	39.9623	2.46888
$263$	23.0274	1.41993	0.709965	−	0.704237i	$-0.248711\pi$
$263$	23.0274	1.41993	0.709965	+	0.704237i	$0.248711\pi$
$264$	1.87815	0.115592
$265$	−4.56218	−0.280252
$266$	8.30418	0.509162
$267$	4.73997	0.290081
$268$	−38.8143	−2.37096
$269$	−6.99797	−0.426674	−0.213337	−	0.976979i	$-0.568433\pi$
$269$	−6.99797	−0.426674	−0.213337	+	0.976979i	$0.568433\pi$
$270$	−2.20326	−0.134086
$271$	15.2145	0.924216	0.462108	−	0.886824i	$-0.347093\pi$
$271$	15.2145	0.924216	0.462108	+	0.886824i	$0.347093\pi$
$272$	7.63779	0.463109
$273$	−3.44879	−0.208730
$274$	−22.4438	−1.35588
$275$	0.997786	0.0601687
$276$	17.4471	1.05019
$277$	12.1109	0.727674	0.363837	−	0.931463i	$-0.381466\pi$
$277$	12.1109	0.727674	0.363837	+	0.931463i	$0.381466\pi$
$278$	23.2761	1.39601
$279$	−1.00000	−0.0598684
$280$	6.49172	0.387955
$281$	2.86741	0.171055	0.0855276	−	0.996336i	$-0.472742\pi$
$281$	2.86741	0.171055	0.0855276	+	0.996336i	$0.472742\pi$
$282$	−20.2144	−1.20375
$283$	−24.5298	−1.45815	−0.729074	−	0.684435i	$-0.760049\pi$
$283$	−24.5298	−1.45815	−0.729074	+	0.684435i	$0.760049\pi$
$284$	1.26184	0.0748764
$285$	1.09286	0.0647355
$286$	2.19838	0.129993
$287$	38.0285	2.24475
$288$	−7.20489	−0.424552
$289$	6.92676	0.407457
$290$	12.2772	0.720940
$291$	−10.6691	−0.625432
$292$	12.9995	0.760738
$293$	−27.8340	−1.62608	−0.813040	−	0.582209i	$-0.802189\pi$
$293$	−27.8340	−1.62608	−0.813040	+	0.582209i	$0.802189\pi$
$294$	10.7831	0.628883
$295$	6.88653	0.400949
$296$	−4.64271	−0.269852
$297$	0.997786	0.0578974
$298$	19.9224	1.15408
$299$	6.11250	0.353495
$300$	2.85433	0.164795
$301$	−33.8856	−1.95314
$302$	31.6909	1.82361
$303$	−18.4987	−1.06272
$304$	1.70644	0.0978711
$305$	7.15737	0.409830
$306$	−10.7772	−0.616093
$307$	6.66051	0.380135	0.190068	−	0.981771i	$-0.439129\pi$
$307$	6.66051	0.380135	0.190068	+	0.981771i	$0.439129\pi$
$308$	−9.82221	−0.559672
$309$	−13.7529	−0.782377
$310$	2.20326	0.125137
$311$	−22.5984	−1.28144	−0.640719	−	0.767775i	$-0.721364\pi$
$311$	−22.5984	−1.28144	−0.640719	+	0.767775i	$0.721364\pi$
$312$	1.88232	0.106565
$313$	−11.5622	−0.653534	−0.326767	−	0.945105i	$-0.605959\pi$
$313$	−11.5622	−0.653534	−0.326767	+	0.945105i	$0.605959\pi$
$314$	−5.41038	−0.305326
$315$	3.44879	0.194317
$316$	2.47392	0.139169
$317$	20.3608	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$317$	20.3608	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$318$	10.0516	0.563668
$319$	−5.55995	−0.311297
$320$	12.7513	0.712821
$321$	2.23698	0.124856
$322$	−46.4462	−2.58835
$323$	5.34573	0.297444
$324$	2.85433	0.158574
$325$	1.00000	0.0554700
$326$	−36.2595	−2.00823
$327$	12.1576	0.672316
$328$	−20.7556	−1.14604
$329$	31.6419	1.74448
$330$	−2.19838	−0.121017
$331$	16.7884	0.922776	0.461388	−	0.887199i	$-0.347352\pi$
$331$	16.7884	0.922776	0.461388	+	0.887199i	$0.347352\pi$
$332$	−34.9921	−1.92044
$333$	−2.46649	−0.135163
$334$	5.08316	0.278138
$335$	13.5984	0.742959
$336$	5.38509	0.293781
$337$	−11.2884	−0.614917	−0.307459	−	0.951561i	$-0.599479\pi$
$337$	−11.2884	−0.614917	−0.307459	+	0.951561i	$0.599479\pi$
$338$	2.20326	0.119841
$339$	7.84384	0.426019
$340$	13.9620	0.757194
$341$	−0.997786	−0.0540331
$342$	−2.40785	−0.130202
$343$	7.26260	0.392143
$344$	18.4945	0.997154
$345$	−6.11250	−0.329086
$346$	35.8988	1.92993
$347$	−22.6617	−1.21654	−0.608272	−	0.793729i	$-0.708137\pi$
$347$	−22.6617	−1.21654	−0.608272	+	0.793729i	$0.708137\pi$
$348$	−15.9052	−0.852606
$349$	−27.0705	−1.44905	−0.724526	−	0.689247i	$-0.757941\pi$
$349$	−27.0705	−1.44905	−0.724526	+	0.689247i	$0.757941\pi$
$350$	−7.59857	−0.406161
$351$	1.00000	0.0533761
$352$	−7.18894	−0.383172
$353$	23.5148	1.25156	0.625782	−	0.779998i	$-0.284780\pi$
$353$	23.5148	1.25156	0.625782	+	0.779998i	$0.284780\pi$
$354$	−15.1728	−0.806425
$355$	−0.442078	−0.0234631
$356$	13.5295	0.717060
$357$	16.8698	0.892842
$358$	19.1575	1.01250
$359$	−4.85244	−0.256102	−0.128051	−	0.991768i	$-0.540872\pi$
$359$	−4.85244	−0.256102	−0.128051	+	0.991768i	$0.540872\pi$
$360$	−1.88232	−0.0992069
$361$	−17.8057	−0.937140
$362$	26.3839	1.38670
$363$	−10.0044	−0.525096
$364$	−9.84401	−0.515966
$365$	−4.55430	−0.238383
$366$	−15.7695	−0.824285
$367$	4.99024	0.260489	0.130244	−	0.991482i	$-0.458424\pi$
$367$	4.99024	0.260489	0.130244	+	0.991482i	$0.458424\pi$
$368$	−9.54431	−0.497532
$369$	−11.0266	−0.574023
$370$	5.43430	0.282516
$371$	−15.7340	−0.816869
$372$	−2.85433	−0.147990
$373$	−5.63236	−0.291633	−0.145816	−	0.989312i	$-0.546581\pi$
$373$	−5.63236	−0.291633	−0.145816	+	0.989312i	$0.546581\pi$
$374$	−10.7534	−0.556043
$375$	−1.00000	−0.0516398
$376$	−17.2699	−0.890625
$377$	−5.57228	−0.286987
$378$	−7.59857	−0.390828
$379$	−21.8222	−1.12093	−0.560466	−	0.828178i	$-0.689378\pi$
$379$	−21.8222	−1.12093	−0.560466	+	0.828178i	$0.689378\pi$
$380$	3.11939	0.160021
$381$	14.0169	0.718108
$382$	7.27293	0.372115
$383$	−8.18811	−0.418393	−0.209196	−	0.977874i	$-0.567085\pi$
$383$	−8.18811	−0.418393	−0.209196	+	0.977874i	$0.567085\pi$
$384$	−13.6847	−0.698342
$385$	3.44116	0.175378
$386$	−16.5426	−0.841998
$387$	9.82536	0.499451
$388$	−30.4531	−1.54602
$389$	−10.7783	−0.546480	−0.273240	−	0.961946i	$-0.588095\pi$
$389$	−10.7783	−0.546480	−0.273240	+	0.961946i	$0.588095\pi$
$390$	−2.20326	−0.111566
$391$	−29.8993	−1.51207
$392$	9.21237	0.465295
$393$	18.1379	0.914934
$394$	−23.5164	−1.18474
$395$	−0.866725	−0.0436097
$396$	2.84801	0.143118
$397$	11.7236	0.588393	0.294196	−	0.955745i	$-0.404948\pi$
$397$	11.7236	0.588393	0.294196	+	0.955745i	$0.404948\pi$
$398$	−33.9457	−1.70154
$399$	3.76905	0.188689
$400$	−1.56144	−0.0780721
$401$	2.89906	0.144772	0.0723861	−	0.997377i	$-0.476939\pi$
$401$	2.89906	0.144772	0.0723861	+	0.997377i	$0.476939\pi$
$402$	−29.9607	−1.49430
$403$	−1.00000	−0.0498135
$404$	−52.8014	−2.62697
$405$	−1.00000	−0.0496904
$406$	42.3414	2.10137
$407$	−2.46103	−0.121989
$408$	−9.20736	−0.455832
$409$	27.7177	1.37055	0.685276	−	0.728283i	$-0.259682\pi$
$409$	27.7177	1.37055	0.685276	+	0.728283i	$0.259682\pi$
$410$	24.2945	1.19982
$411$	−10.1866	−0.502470
$412$	−39.2555	−1.93398
$413$	23.7502	1.16867
$414$	13.4674	0.661886
$415$	12.2593	0.601784
$416$	−7.20489	−0.353249
$417$	10.5644	0.517342
$418$	−2.40252	−0.117511
$419$	4.51075	0.220365	0.110182	−	0.993911i	$-0.464857\pi$
$419$	4.51075	0.220365	0.110182	+	0.993911i	$0.464857\pi$
$420$	9.84401	0.480338
$421$	28.8587	1.40649	0.703244	−	0.710948i	$-0.251734\pi$
$421$	28.8587	1.40649	0.703244	+	0.710948i	$0.251734\pi$
$422$	33.8141	1.64604
$423$	−9.17479	−0.446093
$424$	8.58747	0.417045
$425$	−4.89150	−0.237273
$426$	0.974012	0.0471910
$427$	24.6843	1.19456
$428$	6.38510	0.308635
$429$	0.997786	0.0481736
$430$	−21.6478	−1.04395
$431$	−27.2184	−1.31106	−0.655531	−	0.755168i	$-0.727555\pi$
$431$	−27.2184	−1.31106	−0.655531	+	0.755168i	$0.727555\pi$
$432$	−1.56144	−0.0751249
$433$	14.3891	0.691496	0.345748	−	0.938327i	$-0.387625\pi$
$433$	14.3891	0.691496	0.345748	+	0.938327i	$0.387625\pi$
$434$	7.59857	0.364743
$435$	5.57228	0.267171
$436$	34.7018	1.66191
$437$	−6.68011	−0.319553
$438$	10.0343	0.479457
$439$	−23.2165	−1.10806	−0.554032	−	0.832496i	$-0.686911\pi$
$439$	−23.2165	−1.10806	−0.554032	+	0.832496i	$0.686911\pi$
$440$	−1.87815	−0.0895373
$441$	4.89416	0.233055
$442$	−10.7772	−0.512620
$443$	−35.6086	−1.69182	−0.845908	−	0.533328i	$-0.820941\pi$
$443$	−35.6086	−1.69182	−0.845908	+	0.533328i	$0.820941\pi$
$444$	−7.04018	−0.334112
$445$	−4.73997	−0.224696
$446$	2.01853	0.0955800
$447$	9.04227	0.427685
$448$	43.9767	2.07770
$449$	−25.5265	−1.20467	−0.602335	−	0.798243i	$-0.705763\pi$
$449$	−25.5265	−1.20467	−0.602335	+	0.798243i	$0.705763\pi$
$450$	2.20326	0.103862
$451$	−11.0022	−0.518074
$452$	22.3890	1.05309
$453$	14.3837	0.675804
$454$	38.5578	1.80961
$455$	3.44879	0.161682
$456$	−2.05711	−0.0963331
$457$	−14.1887	−0.663720	−0.331860	−	0.943329i	$-0.607676\pi$
$457$	−14.1887	−0.663720	−0.331860	+	0.943329i	$0.607676\pi$
$458$	−12.6212	−0.589748
$459$	−4.89150	−0.228316
$460$	−17.4471	−0.813476
$461$	−41.0885	−1.91368	−0.956841	−	0.290611i	$-0.906141\pi$
$461$	−41.0885	−1.91368	−0.956841	+	0.290611i	$0.906141\pi$
$462$	−7.58174	−0.352735
$463$	39.5224	1.83676	0.918380	−	0.395700i	$-0.129498\pi$
$463$	39.5224	1.83676	0.918380	+	0.395700i	$0.129498\pi$
$464$	8.70080	0.403925
$465$	1.00000	0.0463739
$466$	−16.2967	−0.754928
$467$	−12.9839	−0.600822	−0.300411	−	0.953810i	$-0.597124\pi$
$467$	−12.9839	−0.600822	−0.300411	+	0.953810i	$0.597124\pi$
$468$	2.85433	0.131942
$469$	46.8980	2.16555
$470$	20.2144	0.932421
$471$	−2.45563	−0.113150
$472$	−12.9626	−0.596654
$473$	9.80361	0.450770
$474$	1.90962	0.0877116
$475$	−1.09286	−0.0501439
$476$	48.1519	2.20704
$477$	4.56218	0.208888
$478$	−4.13604	−0.189178
$479$	15.4665	0.706682	0.353341	−	0.935495i	$-0.385046\pi$
$479$	15.4665	0.706682	0.353341	+	0.935495i	$0.385046\pi$
$480$	7.20489	0.328857
$481$	−2.46649	−0.112462
$482$	32.6128	1.48547
$483$	−21.0807	−0.959207
$484$	−28.5560	−1.29800
$485$	10.6691	0.484458
$486$	2.20326	0.0999417
$487$	18.1763	0.823647	0.411823	−	0.911264i	$-0.364892\pi$
$487$	18.1763	0.823647	0.411823	+	0.911264i	$0.364892\pi$
$488$	−13.4724	−0.609869
$489$	−16.4572	−0.744222
$490$	−10.7831	−0.487130
$491$	−0.934470	−0.0421721	−0.0210860	−	0.999778i	$-0.506712\pi$
$491$	−0.934470	−0.0421721	−0.0210860	+	0.999778i	$0.506712\pi$
$492$	−31.4737	−1.41894
$493$	27.2568	1.22759
$494$	−2.40785	−0.108334
$495$	−0.997786	−0.0448471
$496$	1.56144	0.0701108
$497$	−1.52464	−0.0683893
$498$	−27.0103	−1.21036
$499$	10.4086	0.465955	0.232977	−	0.972482i	$-0.425153\pi$
$499$	10.4086	0.465955	0.232977	+	0.972482i	$0.425153\pi$
$500$	−2.85433	−0.127650
$501$	2.30711	0.103074
$502$	33.1131	1.47791
$503$	35.6768	1.59075	0.795374	−	0.606118i	$-0.207274\pi$
$503$	35.6768	1.59075	0.795374	+	0.606118i	$0.207274\pi$
$504$	−6.49172	−0.289164
$505$	18.4987	0.823180
$506$	13.4376	0.597373
$507$	1.00000	0.0444116
$508$	40.0090	1.77511
$509$	−4.14033	−0.183517	−0.0917585	−	0.995781i	$-0.529249\pi$
$509$	−4.14033	−0.183517	−0.0917585	+	0.995781i	$0.529249\pi$
$510$	10.7772	0.477223
$511$	−15.7068	−0.694829
$512$	17.1283	0.756971
$513$	−1.09286	−0.0482510
$514$	39.8247	1.75659
$515$	13.7529	0.606027
$516$	28.0449	1.23461
$517$	−9.15447	−0.402613
$518$	18.7418	0.823466
$519$	16.2935	0.715206
$520$	−1.88232	−0.0825451
$521$	−23.5527	−1.03186	−0.515931	−	0.856630i	$-0.672554\pi$
$521$	−23.5527	−1.03186	−0.515931	+	0.856630i	$0.672554\pi$
$522$	−12.2772	−0.537357
$523$	34.9997	1.53043	0.765216	−	0.643774i	$-0.222632\pi$
$523$	34.9997	1.53043	0.765216	+	0.643774i	$0.222632\pi$
$524$	51.7715	2.26165
$525$	−3.44879	−0.150518
$526$	50.7352	2.21216
$527$	4.89150	0.213077
$528$	−1.55799	−0.0678026
$529$	14.3626	0.624462
$530$	−10.0516	−0.436616
$531$	−6.88653	−0.298850
$532$	10.7581	0.466424
$533$	−11.0266	−0.477616
$534$	10.4434	0.451929
$535$	−2.23698	−0.0967132
$536$	−25.5965	−1.10560
$537$	8.69507	0.375220
$538$	−15.4183	−0.664730
$539$	4.88333	0.210340
$540$	−2.85433	−0.122831
$541$	−13.7678	−0.591925	−0.295962	−	0.955200i	$-0.595640\pi$
$541$	−13.7678	−0.591925	−0.295962	+	0.955200i	$0.595640\pi$
$542$	33.5215	1.43987
$543$	11.9749	0.513894
$544$	35.2427	1.51102
$545$	−12.1576	−0.520773
$546$	−7.59857	−0.325189
$547$	−11.7690	−0.503205	−0.251603	−	0.967831i	$-0.580958\pi$
$547$	−11.7690	−0.503205	−0.251603	+	0.967831i	$0.580958\pi$
$548$	−29.0761	−1.24207
$549$	−7.15737	−0.305469
$550$	2.19838	0.0937391
$551$	6.08973	0.259431
$552$	11.5057	0.489714
$553$	−2.98915	−0.127112
$554$	26.6834	1.13367
$555$	2.46649	0.104697
$556$	30.1544	1.27883
$557$	22.3395	0.946557	0.473278	−	0.880913i	$-0.343070\pi$
$557$	22.3395	0.946557	0.473278	+	0.880913i	$0.343070\pi$
$558$	−2.20326	−0.0932712
$559$	9.82536	0.415569
$560$	−5.38509	−0.227562
$561$	−4.88067	−0.206062
$562$	6.31763	0.266493
$563$	7.02381	0.296018	0.148009	−	0.988986i	$-0.452714\pi$
$563$	7.02381	0.296018	0.148009	+	0.988986i	$0.452714\pi$
$564$	−26.1879	−1.10271
$565$	−7.84384	−0.329993
$566$	−54.0455	−2.27170
$567$	−3.44879	−0.144836
$568$	0.832132	0.0349155
$569$	−35.6076	−1.49275	−0.746373	−	0.665527i	$-0.768207\pi$
$569$	−35.6076	−1.49275	−0.746373	+	0.665527i	$0.768207\pi$
$570$	2.40785	0.100854
$571$	−18.9384	−0.792547	−0.396273	−	0.918133i	$-0.629697\pi$
$571$	−18.9384	−0.792547	−0.396273	+	0.918133i	$0.629697\pi$
$572$	2.84801	0.119081
$573$	3.30099	0.137901
$574$	83.7866	3.49719
$575$	6.11250	0.254909
$576$	−12.7513	−0.531306
$577$	−3.55333	−0.147927	−0.0739636	−	0.997261i	$-0.523565\pi$
$577$	−3.55333	−0.147927	−0.0739636	+	0.997261i	$0.523565\pi$
$578$	15.2614	0.634792
$579$	−7.50827	−0.312033
$580$	15.9052	0.660426
$581$	42.2797	1.75406
$582$	−23.5067	−0.974384
$583$	4.55208	0.188528
$584$	8.57264	0.354738
$585$	−1.00000	−0.0413449
$586$	−61.3254	−2.53333
$587$	20.4543	0.844241	0.422121	−	0.906540i	$-0.361286\pi$
$587$	20.4543	0.844241	0.422121	+	0.906540i	$0.361286\pi$
$588$	13.9696	0.576096
$589$	1.09286	0.0450306
$590$	15.1728	0.624654
$591$	−10.6735	−0.439048
$592$	3.85128	0.158287
$593$	−43.9430	−1.80452	−0.902262	−	0.431188i	$-0.858095\pi$
$593$	−43.9430	−1.80452	−0.902262	+	0.431188i	$0.858095\pi$
$594$	2.19838	0.0902005
$595$	−16.8698	−0.691593
$596$	25.8097	1.05720
$597$	−15.4071	−0.630569
$598$	13.4674	0.550723
$599$	−40.4751	−1.65377	−0.826885	−	0.562371i	$-0.809889\pi$
$599$	−40.4751	−1.65377	−0.826885	+	0.562371i	$0.809889\pi$
$600$	1.88232	0.0768453
$601$	18.8977	0.770853	0.385427	−	0.922738i	$-0.374054\pi$
$601$	18.8977	0.770853	0.385427	+	0.922738i	$0.374054\pi$
$602$	−74.6587	−3.04286
$603$	−13.5984	−0.553769
$604$	41.0558	1.67054
$605$	10.0044	0.406738
$606$	−40.7573	−1.65565
$607$	−29.4046	−1.19350	−0.596749	−	0.802428i	$-0.703541\pi$
$607$	−29.4046	−1.19350	−0.596749	+	0.802428i	$0.703541\pi$
$608$	7.87395	0.319331
$609$	19.2176	0.778738
$610$	15.7695	0.638489
$611$	−9.17479	−0.371172
$612$	−13.9620	−0.564379
$613$	32.5004	1.31268	0.656339	−	0.754466i	$-0.272104\pi$
$613$	32.5004	1.31268	0.656339	+	0.754466i	$0.272104\pi$
$614$	14.6748	0.592227
$615$	11.0266	0.444637
$616$	−6.47735	−0.260980
$617$	25.3869	1.02204	0.511019	−	0.859569i	$-0.329268\pi$
$617$	25.3869	1.02204	0.511019	+	0.859569i	$0.329268\pi$
$618$	−30.3012	−1.21889
$619$	5.98922	0.240727	0.120364	−	0.992730i	$-0.461594\pi$
$619$	5.98922	0.240727	0.120364	+	0.992730i	$0.461594\pi$
$620$	2.85433	0.114633
$621$	6.11250	0.245286
$622$	−49.7901	−1.99640
$623$	−16.3472	−0.654935
$624$	−1.56144	−0.0625077
$625$	1.00000	0.0400000
$626$	−25.4745	−1.01816
$627$	−1.09044	−0.0435480
$628$	−7.00920	−0.279697
$629$	12.0648	0.481056
$630$	7.59857	0.302734
$631$	−21.3433	−0.849663	−0.424832	−	0.905272i	$-0.639667\pi$
$631$	−21.3433	−0.849663	−0.424832	+	0.905272i	$0.639667\pi$
$632$	1.63145	0.0648956
$633$	15.3473	0.610002
$634$	44.8601	1.78162
$635$	−14.0169	−0.556244
$636$	13.0220	0.516355
$637$	4.89416	0.193914
$638$	−12.2500	−0.484981
$639$	0.442078	0.0174883
$640$	13.6847	0.540934
$641$	20.6267	0.814707	0.407354	−	0.913271i	$-0.366452\pi$
$641$	20.6267	0.814707	0.407354	+	0.913271i	$0.366452\pi$
$642$	4.92865	0.194518
$643$	−10.7764	−0.424979	−0.212490	−	0.977163i	$-0.568157\pi$
$643$	−10.7764	−0.424979	−0.212490	+	0.977163i	$0.568157\pi$
$644$	−60.1715	−2.37109
$645$	−9.82536	−0.386873
$646$	11.7780	0.463400
$647$	32.0791	1.26116	0.630581	−	0.776124i	$-0.282817\pi$
$647$	32.0791	1.26116	0.630581	+	0.776124i	$0.282817\pi$
$648$	1.88232	0.0739444
$649$	−6.87128	−0.269722
$650$	2.20326	0.0864188
$651$	3.44879	0.135169
$652$	−46.9744	−1.83966
$653$	29.8640	1.16867	0.584335	−	0.811513i	$-0.301356\pi$
$653$	29.8640	1.16867	0.584335	+	0.811513i	$0.301356\pi$
$654$	26.7862	1.04743
$655$	−18.1379	−0.708705
$656$	17.2174	0.672228
$657$	4.55430	0.177680
$658$	69.7152	2.71778
$659$	−36.1055	−1.40647	−0.703236	−	0.710956i	$-0.748262\pi$
$659$	−36.1055	−1.40647	−0.703236	+	0.710956i	$0.748262\pi$
$660$	−2.84801	−0.110859
$661$	35.6492	1.38659	0.693296	−	0.720653i	$-0.256158\pi$
$661$	35.6492	1.38659	0.693296	+	0.720653i	$0.256158\pi$
$662$	36.9892	1.43763
$663$	−4.89150	−0.189970
$664$	−23.0759	−0.895517
$665$	−3.76905	−0.146158
$666$	−5.43430	−0.210575
$667$	−34.0606	−1.31883
$668$	6.58528	0.254792
$669$	0.916156	0.0354206
$670$	29.9607	1.15748
$671$	−7.14152	−0.275695
$672$	24.8482	0.958539
$673$	−5.99147	−0.230954	−0.115477	−	0.993310i	$-0.536840\pi$
$673$	−5.99147	−0.230954	−0.115477	+	0.993310i	$0.536840\pi$
$674$	−24.8712	−0.958002
$675$	1.00000	0.0384900
$676$	2.85433	0.109782
$677$	49.1735	1.88989	0.944946	−	0.327227i	$-0.106114\pi$
$677$	49.1735	1.88989	0.944946	+	0.327227i	$0.106114\pi$
$678$	17.2820	0.663711
$679$	36.7954	1.41208
$680$	9.20736	0.353086
$681$	17.5004	0.670616
$682$	−2.19838	−0.0841802
$683$	−20.6768	−0.791176	−0.395588	−	0.918428i	$-0.629459\pi$
$683$	−20.6768	−0.791176	−0.395588	+	0.918428i	$0.629459\pi$
$684$	−3.11939	−0.119273
$685$	10.1866	0.389212
$686$	16.0014	0.610935
$687$	−5.72842	−0.218553
$688$	−15.3417	−0.584898
$689$	4.56218	0.173805
$690$	−13.4674	−0.512695
$691$	−24.0994	−0.916783	−0.458392	−	0.888750i	$-0.651574\pi$
$691$	−24.0994	−0.916783	−0.458392	+	0.888750i	$0.651574\pi$
$692$	46.5072	1.76794
$693$	−3.44116	−0.130719
$694$	−49.9295	−1.89530
$695$	−10.5644	−0.400732
$696$	−10.4888	−0.397577
$697$	53.9368	2.04300
$698$	−59.6433	−2.25753
$699$	−7.39663	−0.279766
$700$	−9.84401	−0.372068
$701$	−32.0544	−1.21068	−0.605338	−	0.795968i	$-0.706962\pi$
$701$	−32.0544	−1.21068	−0.605338	+	0.795968i	$0.706962\pi$
$702$	2.20326	0.0831565
$703$	2.69553	0.101664
$704$	−12.7231	−0.479520
$705$	9.17479	0.345542
$706$	51.8090	1.94986
$707$	63.7981	2.39937
$708$	−19.6565	−0.738735
$709$	19.1155	0.717896	0.358948	−	0.933358i	$-0.383136\pi$
$709$	19.1155	0.717896	0.358948	+	0.933358i	$0.383136\pi$
$710$	−0.974012	−0.0365540
$711$	0.866725	0.0325047
$712$	8.92213	0.334371
$713$	−6.11250	−0.228915
$714$	37.1684	1.39099
$715$	−0.997786	−0.0373151
$716$	24.8186	0.927516
$717$	−1.87724	−0.0701068
$718$	−10.6912	−0.398991
$719$	−8.03233	−0.299555	−0.149778	−	0.988720i	$-0.547856\pi$
$719$	−8.03233	−0.299555	−0.149778	+	0.988720i	$0.547856\pi$
$720$	1.56144	0.0581915
$721$	47.4310	1.76642
$722$	−39.2304	−1.46000
$723$	14.8021	0.550495
$724$	34.1805	1.27031
$725$	−5.57228	−0.206949
$726$	−22.0423	−0.818067
$727$	−7.45381	−0.276447	−0.138223	−	0.990401i	$-0.544139\pi$
$727$	−7.45381	−0.276447	−0.138223	+	0.990401i	$0.544139\pi$
$728$	−6.49172	−0.240599
$729$	1.00000	0.0370370
$730$	−10.0343	−0.371386
$731$	−48.0608	−1.77759
$732$	−20.4295	−0.755097
$733$	15.3554	0.567164	0.283582	−	0.958948i	$-0.408477\pi$
$733$	15.3554	0.567164	0.283582	+	0.958948i	$0.408477\pi$
$734$	10.9948	0.405825
$735$	−4.89416	−0.180524
$736$	−44.0399	−1.62333
$737$	−13.5683	−0.499794
$738$	−24.2945	−0.894292
$739$	−27.6105	−1.01567	−0.507833	−	0.861455i	$-0.669553\pi$
$739$	−27.6105	−1.01567	−0.507833	+	0.861455i	$0.669553\pi$
$740$	7.04018	0.258802
$741$	−1.09286	−0.0401473
$742$	−34.6660	−1.27263
$743$	−8.90749	−0.326784	−0.163392	−	0.986561i	$-0.552244\pi$
$743$	−8.90749	−0.326784	−0.163392	+	0.986561i	$0.552244\pi$
$744$	−1.88232	−0.0690091
$745$	−9.04227	−0.331283
$746$	−12.4095	−0.454345
$747$	−12.2593	−0.448544
$748$	−13.9311	−0.509370
$749$	−7.71489	−0.281896
$750$	−2.20326	−0.0804515
$751$	−27.5099	−1.00385	−0.501925	−	0.864911i	$-0.667375\pi$
$751$	−27.5099	−1.00385	−0.501925	+	0.864911i	$0.667375\pi$
$752$	14.3259	0.522412
$753$	15.0292	0.547693
$754$	−12.2772	−0.447108
$755$	−14.3837	−0.523475
$756$	−9.84401	−0.358023
$757$	48.0571	1.74667	0.873333	−	0.487124i	$-0.161954\pi$
$757$	48.0571	1.74667	0.873333	+	0.487124i	$0.161954\pi$
$758$	−48.0799	−1.74634
$759$	6.09896	0.221378
$760$	2.05711	0.0746193
$761$	−44.2130	−1.60272	−0.801360	−	0.598182i	$-0.795890\pi$
$761$	−44.2130	−1.60272	−0.801360	+	0.598182i	$0.795890\pi$
$762$	30.8828	1.11877
$763$	−41.9289	−1.51793
$764$	9.42214	0.340881
$765$	4.89150	0.176853
$766$	−18.0405	−0.651830
$767$	−6.88653	−0.248658
$768$	−4.64814	−0.167725
$769$	48.1177	1.73517	0.867584	−	0.497290i	$-0.165672\pi$
$769$	48.1177	1.73517	0.867584	+	0.497290i	$0.165672\pi$
$770$	7.58174	0.273227
$771$	18.0754	0.650969
$772$	−21.4311	−0.771323
$773$	−20.7528	−0.746425	−0.373213	−	0.927746i	$-0.621744\pi$
$773$	−20.7528	−0.746425	−0.373213	+	0.927746i	$0.621744\pi$
$774$	21.6478	0.778114
$775$	−1.00000	−0.0359211
$776$	−20.0826	−0.720923
$777$	8.50640	0.305165
$778$	−23.7473	−0.851381
$779$	12.0506	0.431757
$780$	−2.85433	−0.102202
$781$	0.441100	0.0157838
$782$	−65.8758	−2.35571
$783$	−5.57228	−0.199137
$784$	−7.64195	−0.272927
$785$	2.45563	0.0876453
$786$	39.9623	1.42541
$787$	−39.7028	−1.41525	−0.707627	−	0.706587i	$-0.750234\pi$
$787$	−39.7028	−1.41525	−0.707627	+	0.706587i	$0.750234\pi$
$788$	−30.4656	−1.08529
$789$	23.0274	0.819797
$790$	−1.90962	−0.0679411
$791$	−27.0518	−0.961851
$792$	1.87815	0.0667372
$793$	−7.15737	−0.254166
$794$	25.8302	0.916679
$795$	−4.56218	−0.161804
$796$	−43.9769	−1.55872
$797$	3.26115	0.115516	0.0577580	−	0.998331i	$-0.481605\pi$
$797$	3.26115	0.115516	0.0577580	+	0.998331i	$0.481605\pi$
$798$	8.30418	0.293965
$799$	44.8785	1.58769
$800$	−7.20489	−0.254731
$801$	4.73997	0.167479
$802$	6.38737	0.225546
$803$	4.54422	0.160362
$804$	−38.8143	−1.36888
$805$	21.0807	0.742998
$806$	−2.20326	−0.0776064
$807$	−6.99797	−0.246340
$808$	−34.8204	−1.22498
$809$	31.2272	1.09789	0.548944	−	0.835859i	$-0.315030\pi$
$809$	31.2272	1.09789	0.548944	+	0.835859i	$0.315030\pi$
$810$	−2.20326	−0.0774145
$811$	−21.0833	−0.740334	−0.370167	−	0.928965i	$-0.620700\pi$
$811$	−21.0833	−0.740334	−0.370167	+	0.928965i	$0.620700\pi$
$812$	54.8536	1.92498
$813$	15.2145	0.533596
$814$	−5.42227	−0.190050
$815$	16.4572	0.576472
$816$	7.63779	0.267376
$817$	−10.7378	−0.375667
$818$	61.0692	2.13523
$819$	−3.44879	−0.120511
$820$	31.4737	1.09911
$821$	12.9061	0.450425	0.225213	−	0.974310i	$-0.427692\pi$
$821$	12.9061	0.450425	0.225213	+	0.974310i	$0.427692\pi$
$822$	−22.4438	−0.782816
$823$	−35.8559	−1.24986	−0.624929	−	0.780681i	$-0.714872\pi$
$823$	−35.8559	−1.24986	−0.624929	+	0.780681i	$0.714872\pi$
$824$	−25.8874	−0.901830
$825$	0.997786	0.0347384
$826$	52.3278	1.82072
$827$	−12.4877	−0.434241	−0.217121	−	0.976145i	$-0.569667\pi$
$827$	−12.4877	−0.434241	−0.217121	+	0.976145i	$0.569667\pi$
$828$	17.4471	0.606329
$829$	−1.39767	−0.0485429	−0.0242715	−	0.999705i	$-0.507727\pi$
$829$	−1.39767	−0.0485429	−0.0242715	+	0.999705i	$0.507727\pi$
$830$	27.0103	0.937542
$831$	12.1109	0.420123
$832$	−12.7513	−0.442073
$833$	−23.9398	−0.829465
$834$	23.2761	0.805987
$835$	−2.30711	−0.0798410
$836$	−3.11249	−0.107648
$837$	−1.00000	−0.0345651
$838$	9.93834	0.343314
$839$	48.0409	1.65856	0.829278	−	0.558836i	$-0.188752\pi$
$839$	48.0409	1.65856	0.829278	+	0.558836i	$0.188752\pi$
$840$	6.49172	0.223986
$841$	2.05036	0.0707020
$842$	63.5832	2.19122
$843$	2.86741	0.0987588
$844$	43.8064	1.50788
$845$	−1.00000	−0.0344010
$846$	−20.2144	−0.694985
$847$	34.5032	1.18554
$848$	−7.12358	−0.244625
$849$	−24.5298	−0.841862
$850$	−10.7772	−0.369656
$851$	−15.0764	−0.516812
$852$	1.26184	0.0432299
$853$	−46.2225	−1.58263	−0.791314	−	0.611411i	$-0.790602\pi$
$853$	−46.2225	−1.58263	−0.791314	+	0.611411i	$0.790602\pi$
$854$	54.3857	1.86104
$855$	1.09286	0.0373751
$856$	4.21071	0.143919
$857$	−44.1373	−1.50770	−0.753851	−	0.657046i	$-0.771806\pi$
$857$	−44.1373	−1.50770	−0.753851	+	0.657046i	$0.771806\pi$
$858$	2.19838	0.0750514
$859$	−24.8411	−0.847567	−0.423783	−	0.905764i	$-0.639298\pi$
$859$	−24.8411	−0.847567	−0.423783	+	0.905764i	$0.639298\pi$
$860$	−28.0449	−0.956322
$861$	38.0285	1.29601
$862$	−59.9690	−2.04255
$863$	3.22952	0.109934	0.0549670	−	0.998488i	$-0.482495\pi$
$863$	3.22952	0.109934	0.0549670	+	0.998488i	$0.482495\pi$
$864$	−7.20489	−0.245115
$865$	−16.2935	−0.553996
$866$	31.7029	1.07731
$867$	6.92676	0.235245
$868$	9.84401	0.334127
$869$	0.864806	0.0293365
$870$	12.2772	0.416235
$871$	−13.5984	−0.460764
$872$	22.8844	0.774964
$873$	−10.6691	−0.361094
$874$	−14.7180	−0.497844
$875$	3.44879	0.116590
$876$	12.9995	0.439212
$877$	−50.6997	−1.71201	−0.856003	−	0.516971i	$-0.827059\pi$
$877$	−50.6997	−1.71201	−0.856003	+	0.516971i	$0.827059\pi$
$878$	−51.1519	−1.72629
$879$	−27.8340	−0.938817
$880$	1.55799	0.0525197
$881$	−19.1333	−0.644618	−0.322309	−	0.946635i	$-0.604459\pi$
$881$	−19.1333	−0.644618	−0.322309	+	0.946635i	$0.604459\pi$
$882$	10.7831	0.363086
$883$	18.2923	0.615587	0.307793	−	0.951453i	$-0.400409\pi$
$883$	18.2923	0.615587	0.307793	+	0.951453i	$0.400409\pi$
$884$	−13.9620	−0.469592
$885$	6.88653	0.231488
$886$	−78.4549	−2.63574
$887$	37.8932	1.27233	0.636165	−	0.771553i	$-0.280520\pi$
$887$	37.8932	1.27233	0.636165	+	0.771553i	$0.280520\pi$
$888$	−4.64271	−0.155799
$889$	−48.3414	−1.62132
$890$	−10.4434	−0.350062
$891$	0.997786	0.0334271
$892$	2.61502	0.0875572
$893$	10.0268	0.335533
$894$	19.9224	0.666306
$895$	−8.69507	−0.290644
$896$	47.1955	1.57669
$897$	6.11250	0.204090
$898$	−56.2414	−1.87680
$899$	5.57228	0.185846
$900$	2.85433	0.0951445
$901$	−22.3159	−0.743450
$902$	−24.2407	−0.807127
$903$	−33.8856	−1.12764
$904$	14.7646	0.491063
$905$	−11.9749	−0.398061
$906$	31.6909	1.05286
$907$	−18.6230	−0.618368	−0.309184	−	0.951002i	$-0.600056\pi$
$907$	−18.6230	−0.618368	−0.309184	+	0.951002i	$0.600056\pi$
$908$	49.9519	1.65771
$909$	−18.4987	−0.613562
$910$	7.59857	0.251890
$911$	4.88944	0.161994	0.0809972	−	0.996714i	$-0.474190\pi$
$911$	4.88944	0.161994	0.0809972	+	0.996714i	$0.474190\pi$
$912$	1.70644	0.0565059
$913$	−12.2321	−0.404825
$914$	−31.2613	−1.03403
$915$	7.15737	0.236615
$916$	−16.3508	−0.540246
$917$	−62.5537	−2.06570
$918$	−10.7772	−0.355701
$919$	−6.02433	−0.198724	−0.0993622	−	0.995051i	$-0.531680\pi$
$919$	−6.02433	−0.198724	−0.0993622	+	0.995051i	$0.531680\pi$
$920$	−11.5057	−0.379331
$921$	6.66051	0.219471
$922$	−90.5285	−2.98140
$923$	0.442078	0.0145512
$924$	−9.82221	−0.323127
$925$	−2.46649	−0.0810976
$926$	87.0778	2.86156
$927$	−13.7529	−0.451706
$928$	40.1477	1.31791
$929$	−57.9809	−1.90229	−0.951146	−	0.308742i	$-0.900092\pi$
$929$	−57.9809	−1.90229	−0.951146	+	0.308742i	$0.900092\pi$
$930$	2.20326	0.0722476
$931$	−5.34864	−0.175295
$932$	−21.1124	−0.691561
$933$	−22.5984	−0.739839
$934$	−28.6068	−0.936043
$935$	4.88067	0.159615
$936$	1.88232	0.0615255
$937$	35.4649	1.15859	0.579294	−	0.815118i	$-0.303328\pi$
$937$	35.4649	1.15859	0.579294	+	0.815118i	$0.303328\pi$
$938$	103.328	3.37379
$939$	−11.5622	−0.377318
$940$	26.1879	0.854155
$941$	14.8447	0.483922	0.241961	−	0.970286i	$-0.422209\pi$
$941$	14.8447	0.483922	0.241961	+	0.970286i	$0.422209\pi$
$942$	−5.41038	−0.176280
$943$	−67.4003	−2.19485
$944$	10.7529	0.349978
$945$	3.44879	0.112189
$946$	21.5999	0.702272
$947$	37.5695	1.22085	0.610423	−	0.792076i	$-0.291000\pi$
$947$	37.5695	1.22085	0.610423	+	0.792076i	$0.291000\pi$
$948$	2.47392	0.0803493
$949$	4.55430	0.147839
$950$	−2.40785	−0.0781211
$951$	20.3608	0.660245
$952$	31.7542	1.02916
$953$	−20.8594	−0.675701	−0.337850	−	0.941200i	$-0.609700\pi$
$953$	−20.8594	−0.675701	−0.337850	+	0.941200i	$0.609700\pi$
$954$	10.0516	0.325434
$955$	−3.30099	−0.106818
$956$	−5.35827	−0.173299
$957$	−5.55995	−0.179728
$958$	34.0766	1.10097
$959$	35.1316	1.13446
$960$	12.7513	0.411548
$961$	1.00000	0.0322581
$962$	−5.43430	−0.175209
$963$	2.23698	0.0720858
$964$	42.2501	1.36078
$965$	7.50827	0.241700
$966$	−46.4462	−1.49438
$967$	−42.2215	−1.35775	−0.678876	−	0.734253i	$-0.737532\pi$
$967$	−42.2215	−1.35775	−0.678876	+	0.734253i	$0.737532\pi$
$968$	−18.8315	−0.605267
$969$	5.34573	0.171730
$970$	23.5067	0.754755
$971$	−53.2035	−1.70738	−0.853690	−	0.520781i	$-0.825641\pi$
$971$	−53.2035	−1.70738	−0.853690	+	0.520781i	$0.825641\pi$
$972$	2.85433	0.0915528
$973$	−36.4345	−1.16804
$974$	40.0470	1.28319
$975$	1.00000	0.0320256
$976$	11.1758	0.357729
$977$	3.48263	0.111419	0.0557096	−	0.998447i	$-0.482258\pi$
$977$	3.48263	0.111419	0.0557096	+	0.998447i	$0.482258\pi$
$978$	−36.2595	−1.15945
$979$	4.72947	0.151155
$980$	−13.9696	−0.446242
$981$	12.1576	0.388162
$982$	−2.05888	−0.0657014
$983$	−17.5733	−0.560501	−0.280250	−	0.959927i	$-0.590418\pi$
$983$	−17.5733	−0.560501	−0.280250	+	0.959927i	$0.590418\pi$
$984$	−20.7556	−0.661665
$985$	10.6735	0.340085
$986$	60.0538	1.91250
$987$	31.6419	1.00717
$988$	−3.11939	−0.0992411
$989$	60.0575	1.90972
$990$	−2.19838	−0.0698690
$991$	−47.8689	−1.52060	−0.760302	−	0.649569i	$-0.774949\pi$
$991$	−47.8689	−1.52060	−0.760302	+	0.649569i	$0.774949\pi$
$992$	7.20489	0.228756
$993$	16.7884	0.532765
$994$	−3.35916	−0.106546
$995$	15.4071	0.488437
$996$	−34.9921	−1.10877
$997$	−4.51351	−0.142944	−0.0714722	−	0.997443i	$-0.522770\pi$
$997$	−4.51351	−0.142944	−0.0714722	+	0.997443i	$0.522770\pi$
$998$	22.9329	0.725928
$999$	−2.46649	−0.0780362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.y.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.y.1.11	✓	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20326 q^{2} +1.00000 q^{3} +2.85433 q^{4} -1.00000 q^{5} +2.20326 q^{6} -3.44879 q^{7} +1.88232 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.20326 q^{2} +1.00000 q^{3} +2.85433 q^{4} -1.00000 q^{5} +2.20326 q^{6} -3.44879 q^{7} +1.88232 q^{8} +1.00000 q^{9} -2.20326 q^{10} +0.997786 q^{11} +2.85433 q^{12} +1.00000 q^{13} -7.59857 q^{14} -1.00000 q^{15} -1.56144 q^{16} -4.89150 q^{17} +2.20326 q^{18} -1.09286 q^{19} -2.85433 q^{20} -3.44879 q^{21} +2.19838 q^{22} +6.11250 q^{23} +1.88232 q^{24} +1.00000 q^{25} +2.20326 q^{26} +1.00000 q^{27} -9.84401 q^{28} -5.57228 q^{29} -2.20326 q^{30} -1.00000 q^{31} -7.20489 q^{32} +0.997786 q^{33} -10.7772 q^{34} +3.44879 q^{35} +2.85433 q^{36} -2.46649 q^{37} -2.40785 q^{38} +1.00000 q^{39} -1.88232 q^{40} -11.0266 q^{41} -7.59857 q^{42} +9.82536 q^{43} +2.84801 q^{44} -1.00000 q^{45} +13.4674 q^{46} -9.17479 q^{47} -1.56144 q^{48} +4.89416 q^{49} +2.20326 q^{50} -4.89150 q^{51} +2.85433 q^{52} +4.56218 q^{53} +2.20326 q^{54} -0.997786 q^{55} -6.49172 q^{56} -1.09286 q^{57} -12.2772 q^{58} -6.88653 q^{59} -2.85433 q^{60} -7.15737 q^{61} -2.20326 q^{62} -3.44879 q^{63} -12.7513 q^{64} -1.00000 q^{65} +2.19838 q^{66} -13.5984 q^{67} -13.9620 q^{68} +6.11250 q^{69} +7.59857 q^{70} +0.442078 q^{71} +1.88232 q^{72} +4.55430 q^{73} -5.43430 q^{74} +1.00000 q^{75} -3.11939 q^{76} -3.44116 q^{77} +2.20326 q^{78} +0.866725 q^{79} +1.56144 q^{80} +1.00000 q^{81} -24.2945 q^{82} -12.2593 q^{83} -9.84401 q^{84} +4.89150 q^{85} +21.6478 q^{86} -5.57228 q^{87} +1.87815 q^{88} +4.73997 q^{89} -2.20326 q^{90} -3.44879 q^{91} +17.4471 q^{92} -1.00000 q^{93} -20.2144 q^{94} +1.09286 q^{95} -7.20489 q^{96} -10.6691 q^{97} +10.7831 q^{98} +0.997786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 12 q^{5} - 3 q^{6} + 2 q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 24 q^{11} + 15 q^{12} + 12 q^{13} - 13 q^{14} - 12 q^{15} + 21 q^{16} - 9 q^{17} - 3 q^{18} - 2 q^{19} - 15 q^{20} + 2 q^{21} + 9 q^{22} + 7 q^{23} - 12 q^{24} + 12 q^{25} - 3 q^{26} + 12 q^{27} - 16 q^{28} - 24 q^{29} + 3 q^{30} - 12 q^{31} - 57 q^{32} - 24 q^{33} - 15 q^{34} - 2 q^{35} + 15 q^{36} - 13 q^{37} - 22 q^{38} + 12 q^{39} + 12 q^{40} - 46 q^{41} - 13 q^{42} - 5 q^{43} - 49 q^{44} - 12 q^{45} - 2 q^{46} - 39 q^{47} + 21 q^{48} - 3 q^{50} - 9 q^{51} + 15 q^{52} - 12 q^{53} - 3 q^{54} + 24 q^{55} + q^{56} - 2 q^{57} - 8 q^{58} - 31 q^{59} - 15 q^{60} + 2 q^{61} + 3 q^{62} + 2 q^{63} + 18 q^{64} - 12 q^{65} + 9 q^{66} + 5 q^{67} - 5 q^{68} + 7 q^{69} + 13 q^{70} - 31 q^{71} - 12 q^{72} + 3 q^{73} + 16 q^{74} + 12 q^{75} - 2 q^{76} + 2 q^{77} - 3 q^{78} + 5 q^{79} - 21 q^{80} + 12 q^{81} + 25 q^{82} + 2 q^{83} - 16 q^{84} + 9 q^{85} - 26 q^{86} - 24 q^{87} + 27 q^{88} - 13 q^{89} + 3 q^{90} + 2 q^{91} + 5 q^{92} - 12 q^{93} - 12 q^{94} + 2 q^{95} - 57 q^{96} - 28 q^{97} - 29 q^{98} - 24 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 12 * q^5 - 3 * q^6 + 2 * q^7 - 12 * q^8 + 12 * q^9 + 3 * q^10 - 24 * q^11 + 15 * q^12 + 12 * q^13 - 13 * q^14 - 12 * q^15 + 21 * q^16 - 9 * q^17 - 3 * q^18 - 2 * q^19 - 15 * q^20 + 2 * q^21 + 9 * q^22 + 7 * q^23 - 12 * q^24 + 12 * q^25 - 3 * q^26 + 12 * q^27 - 16 * q^28 - 24 * q^29 + 3 * q^30 - 12 * q^31 - 57 * q^32 - 24 * q^33 - 15 * q^34 - 2 * q^35 + 15 * q^36 - 13 * q^37 - 22 * q^38 + 12 * q^39 + 12 * q^40 - 46 * q^41 - 13 * q^42 - 5 * q^43 - 49 * q^44 - 12 * q^45 - 2 * q^46 - 39 * q^47 + 21 * q^48 - 3 * q^50 - 9 * q^51 + 15 * q^52 - 12 * q^53 - 3 * q^54 + 24 * q^55 + q^56 - 2 * q^57 - 8 * q^58 - 31 * q^59 - 15 * q^60 + 2 * q^61 + 3 * q^62 + 2 * q^63 + 18 * q^64 - 12 * q^65 + 9 * q^66 + 5 * q^67 - 5 * q^68 + 7 * q^69 + 13 * q^70 - 31 * q^71 - 12 * q^72 + 3 * q^73 + 16 * q^74 + 12 * q^75 - 2 * q^76 + 2 * q^77 - 3 * q^78 + 5 * q^79 - 21 * q^80 + 12 * q^81 + 25 * q^82 + 2 * q^83 - 16 * q^84 + 9 * q^85 - 26 * q^86 - 24 * q^87 + 27 * q^88 - 13 * q^89 + 3 * q^90 + 2 * q^91 + 5 * q^92 - 12 * q^93 - 12 * q^94 + 2 * q^95 - 57 * q^96 - 28 * q^97 - 29 * q^98 - 24 * q^99

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