LMFDB - Embedded newform 309.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,2,Mod(1,309)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("309.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	309.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:	$2.46737742246$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.81509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 3x^{2} + 5x - 2 $ x^5 - x^4 - 5x^3 + 3x^2 + 5*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.370865$ of defining polynomial
Character	$\chi$	$=$	309.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.199126 q^{2} -1.00000 q^{3} -1.96035 q^{4} +3.39280 q^{5} +0.199126 q^{6} -4.89666 q^{7} +0.788610 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.199126 q^{2} -1.00000 q^{3} -1.96035 q^{4} +3.39280 q^{5} +0.199126 q^{6} -4.89666 q^{7} +0.788610 q^{8} +1.00000 q^{9} -0.675596 q^{10} -2.74173 q^{11} +1.96035 q^{12} -0.0240404 q^{13} +0.975054 q^{14} -3.39280 q^{15} +3.76366 q^{16} -6.00723 q^{17} -0.199126 q^{18} -7.22106 q^{19} -6.65107 q^{20} +4.89666 q^{21} +0.545951 q^{22} +3.00723 q^{23} -0.788610 q^{24} +6.51109 q^{25} +0.00478708 q^{26} -1.00000 q^{27} +9.59915 q^{28} -7.29491 q^{29} +0.675596 q^{30} -6.65107 q^{31} -2.32667 q^{32} +2.74173 q^{33} +1.19620 q^{34} -16.6134 q^{35} -1.96035 q^{36} +1.94522 q^{37} +1.43790 q^{38} +0.0240404 q^{39} +2.67560 q^{40} -3.76213 q^{41} -0.975054 q^{42} +10.3058 q^{43} +5.37475 q^{44} +3.39280 q^{45} -0.598819 q^{46} +5.72492 q^{47} -3.76366 q^{48} +16.9772 q^{49} -1.29653 q^{50} +6.00723 q^{51} +0.0471276 q^{52} +2.58494 q^{53} +0.199126 q^{54} -9.30214 q^{55} -3.86155 q^{56} +7.22106 q^{57} +1.45261 q^{58} +9.00666 q^{59} +6.65107 q^{60} +1.30627 q^{61} +1.32440 q^{62} -4.89666 q^{63} -7.06403 q^{64} -0.0815643 q^{65} -0.545951 q^{66} +6.05704 q^{67} +11.7763 q^{68} -3.00723 q^{69} +3.30816 q^{70} -11.0997 q^{71} +0.788610 q^{72} -4.93330 q^{73} -0.387346 q^{74} -6.51109 q^{75} +14.1558 q^{76} +13.4253 q^{77} -0.00478708 q^{78} -5.86987 q^{79} +12.7694 q^{80} +1.00000 q^{81} +0.749139 q^{82} +1.16526 q^{83} -9.59915 q^{84} -20.3813 q^{85} -2.05215 q^{86} +7.29491 q^{87} -2.16216 q^{88} +1.65568 q^{89} -0.675596 q^{90} +0.117718 q^{91} -5.89522 q^{92} +6.65107 q^{93} -1.13998 q^{94} -24.4996 q^{95} +2.32667 q^{96} +5.34706 q^{97} -3.38062 q^{98} -2.74173 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} - 12 q^{11} - 2 q^{12} + q^{13} - 8 q^{14} + 5 q^{15} - 4 q^{16} - 10 q^{17} - 2 q^{18} - 16 q^{19} - 13 q^{20} + 2 q^{21} + 4 q^{22} - 5 q^{23} + 6 q^{24} + 12 q^{25} - 10 q^{26} - 5 q^{27} + 7 q^{28} - 16 q^{29} + q^{30} - 13 q^{31} + 11 q^{32} + 12 q^{33} - 2 q^{34} - 22 q^{35} + 2 q^{36} + 4 q^{37} + 21 q^{38} - q^{39} + 11 q^{40} - 20 q^{41} + 8 q^{42} + 7 q^{43} + 2 q^{44} - 5 q^{45} + 8 q^{46} + 8 q^{47} + 4 q^{48} + 23 q^{49} + 13 q^{50} + 10 q^{51} + 31 q^{52} - 8 q^{53} + 2 q^{54} - 6 q^{55} + 5 q^{56} + 16 q^{57} + 20 q^{58} - 19 q^{59} + 13 q^{60} - 19 q^{61} + 9 q^{62} - 2 q^{63} - 16 q^{64} + q^{65} - 4 q^{66} + 11 q^{67} + 15 q^{68} + 5 q^{69} + 44 q^{70} - 10 q^{71} - 6 q^{72} + 20 q^{73} + 44 q^{74} - 12 q^{75} - 8 q^{76} + 8 q^{77} + 10 q^{78} - 14 q^{79} + 45 q^{80} + 5 q^{81} - 3 q^{82} - 11 q^{83} - 7 q^{84} + 12 q^{85} - 11 q^{86} + 16 q^{87} + 30 q^{88} + 22 q^{89} - q^{90} - 42 q^{91} - 21 q^{92} + 13 q^{93} - 6 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} - 11 q^{98} - 12 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9 - q^10 - 12 * q^11 - 2 * q^12 + q^13 - 8 * q^14 + 5 * q^15 - 4 * q^16 - 10 * q^17 - 2 * q^18 - 16 * q^19 - 13 * q^20 + 2 * q^21 + 4 * q^22 - 5 * q^23 + 6 * q^24 + 12 * q^25 - 10 * q^26 - 5 * q^27 + 7 * q^28 - 16 * q^29 + q^30 - 13 * q^31 + 11 * q^32 + 12 * q^33 - 2 * q^34 - 22 * q^35 + 2 * q^36 + 4 * q^37 + 21 * q^38 - q^39 + 11 * q^40 - 20 * q^41 + 8 * q^42 + 7 * q^43 + 2 * q^44 - 5 * q^45 + 8 * q^46 + 8 * q^47 + 4 * q^48 + 23 * q^49 + 13 * q^50 + 10 * q^51 + 31 * q^52 - 8 * q^53 + 2 * q^54 - 6 * q^55 + 5 * q^56 + 16 * q^57 + 20 * q^58 - 19 * q^59 + 13 * q^60 - 19 * q^61 + 9 * q^62 - 2 * q^63 - 16 * q^64 + q^65 - 4 * q^66 + 11 * q^67 + 15 * q^68 + 5 * q^69 + 44 * q^70 - 10 * q^71 - 6 * q^72 + 20 * q^73 + 44 * q^74 - 12 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 - 14 * q^79 + 45 * q^80 + 5 * q^81 - 3 * q^82 - 11 * q^83 - 7 * q^84 + 12 * q^85 - 11 * q^86 + 16 * q^87 + 30 * q^88 + 22 * q^89 - q^90 - 42 * q^91 - 21 * q^92 + 13 * q^93 - 6 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 - 11 * q^98 - 12 * 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 $

$2$	−0.199126	−0.140804	−0.0704018	−	0.997519i	$-0.522428\pi$
$2$	−0.199126	−0.140804	−0.0704018	+	0.997519i	$0.522428\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.96035	−0.980174
$5$	3.39280	1.51731	0.758653	−	0.651495i	$-0.225858\pi$
$5$	3.39280	1.51731	0.758653	+	0.651495i	$0.225858\pi$
$6$	0.199126	0.0812930
$7$	−4.89666	−1.85076	−0.925381	−	0.379038i	$-0.876255\pi$
$7$	−4.89666	−1.85076	−0.925381	+	0.379038i	$0.876255\pi$
$8$	0.788610	0.278816
$9$	1.00000	0.333333
$10$	−0.675596	−0.213642
$11$	−2.74173	−0.826663	−0.413331	−	0.910581i	$-0.635635\pi$
$11$	−2.74173	−0.826663	−0.413331	+	0.910581i	$0.635635\pi$
$12$	1.96035	0.565904
$13$	−0.0240404	−0.00666761	−0.00333381	−	0.999994i	$-0.501061\pi$
$13$	−0.0240404	−0.00666761	−0.00333381	+	0.999994i	$0.501061\pi$
$14$	0.975054	0.260594
$15$	−3.39280	−0.876017
$16$	3.76366	0.940916
$17$	−6.00723	−1.45697	−0.728484	−	0.685063i	$-0.759775\pi$
$17$	−6.00723	−1.45697	−0.728484	+	0.685063i	$0.759775\pi$
$18$	−0.199126	−0.0469346
$19$	−7.22106	−1.65662	−0.828312	−	0.560266i	$-0.810699\pi$
$19$	−7.22106	−1.65662	−0.828312	+	0.560266i	$0.810699\pi$
$20$	−6.65107	−1.48722
$21$	4.89666	1.06854
$22$	0.545951	0.116397
$23$	3.00723	0.627051	0.313525	−	0.949580i	$-0.398490\pi$
$23$	3.00723	0.627051	0.313525	+	0.949580i	$0.398490\pi$
$24$	−0.788610	−0.160974
$25$	6.51109	1.30222
$26$	0.00478708	0.000938824 0
$27$	−1.00000	−0.192450
$28$	9.59915	1.81407
$29$	−7.29491	−1.35463	−0.677315	−	0.735693i	$-0.736857\pi$
$29$	−7.29491	−1.35463	−0.677315	+	0.735693i	$0.736857\pi$
$30$	0.675596	0.123346
$31$	−6.65107	−1.19457	−0.597284	−	0.802030i	$-0.703753\pi$
$31$	−6.65107	−1.19457	−0.597284	+	0.802030i	$0.703753\pi$
$32$	−2.32667	−0.411300
$33$	2.74173	0.477274
$34$	1.19620	0.205146
$35$	−16.6134	−2.80817
$36$	−1.96035	−0.326725
$37$	1.94522	0.319793	0.159896	−	0.987134i	$-0.448884\pi$
$37$	1.94522	0.319793	0.159896	+	0.987134i	$0.448884\pi$
$38$	1.43790	0.233259
$39$	0.0240404	0.00384955
$40$	2.67560	0.423049
$41$	−3.76213	−0.587546	−0.293773	−	0.955875i	$-0.594911\pi$
$41$	−3.76213	−0.587546	−0.293773	+	0.955875i	$0.594911\pi$
$42$	−0.975054	−0.150454
$43$	10.3058	1.57162	0.785808	−	0.618470i	$-0.212247\pi$
$43$	10.3058	1.57162	0.785808	+	0.618470i	$0.212247\pi$
$44$	5.37475	0.810274
$45$	3.39280	0.505769
$46$	−0.598819	−0.0882910
$47$	5.72492	0.835065	0.417533	−	0.908662i	$-0.362895\pi$
$47$	5.72492	0.835065	0.417533	+	0.908662i	$0.362895\pi$
$48$	−3.76366	−0.543238
$49$	16.9772	2.42532
$50$	−1.29653	−0.183357
$51$	6.00723	0.841180
$52$	0.0471276	0.00653542
$53$	2.58494	0.355068	0.177534	−	0.984115i	$-0.443188\pi$
$53$	2.58494	0.355068	0.177534	+	0.984115i	$0.443188\pi$
$54$	0.199126	0.0270977
$55$	−9.30214	−1.25430
$56$	−3.86155	−0.516022
$57$	7.22106	0.956453
$58$	1.45261	0.190737
$59$	9.00666	1.17257	0.586284	−	0.810106i	$-0.300590\pi$
$59$	9.00666	1.17257	0.586284	+	0.810106i	$0.300590\pi$
$60$	6.65107	0.858649
$61$	1.30627	0.167250	0.0836252	−	0.996497i	$-0.473350\pi$
$61$	1.30627	0.167250	0.0836252	+	0.996497i	$0.473350\pi$
$62$	1.32440	0.168199
$63$	−4.89666	−0.616921
$64$	−7.06403	−0.883003
$65$	−0.0815643	−0.0101168
$66$	−0.545951	−0.0672019
$67$	6.05704	0.739985	0.369992	−	0.929035i	$-0.379360\pi$
$67$	6.05704	0.739985	0.369992	+	0.929035i	$0.379360\pi$
$68$	11.7763	1.42808
$69$	−3.00723	−0.362028
$70$	3.30816	0.395401
$71$	−11.0997	−1.31729	−0.658644	−	0.752455i	$-0.728870\pi$
$71$	−11.0997	−1.31729	−0.658644	+	0.752455i	$0.728870\pi$
$72$	0.788610	0.0929386
$73$	−4.93330	−0.577399	−0.288699	−	0.957420i	$-0.593223\pi$
$73$	−4.93330	−0.577399	−0.288699	+	0.957420i	$0.593223\pi$
$74$	−0.387346	−0.0450280
$75$	−6.51109	−0.751836
$76$	14.1558	1.62378
$77$	13.4253	1.52996
$78$	−0.00478708	−0.000542030 0
$79$	−5.86987	−0.660412	−0.330206	−	0.943909i	$-0.607118\pi$
$79$	−5.86987	−0.660412	−0.330206	+	0.943909i	$0.607118\pi$
$80$	12.7694	1.42766
$81$	1.00000	0.111111
$82$	0.749139	0.0827286
$83$	1.16526	0.127904	0.0639522	−	0.997953i	$-0.479629\pi$
$83$	1.16526	0.127904	0.0639522	+	0.997953i	$0.479629\pi$
$84$	−9.59915	−1.04735
$85$	−20.3813	−2.21066
$86$	−2.05215	−0.221289
$87$	7.29491	0.782096
$88$	−2.16216	−0.230487
$89$	1.65568	0.175502	0.0877511	−	0.996142i	$-0.472032\pi$
$89$	1.65568	0.175502	0.0877511	+	0.996142i	$0.472032\pi$
$90$	−0.675596	−0.0712141
$91$	0.117718	0.0123402
$92$	−5.89522	−0.614619
$93$	6.65107	0.689684
$94$	−1.13998	−0.117580
$95$	−24.4996	−2.51361
$96$	2.32667	0.237464
$97$	5.34706	0.542912	0.271456	−	0.962451i	$-0.412495\pi$
$97$	5.34706	0.542912	0.271456	+	0.962451i	$0.412495\pi$
$98$	−3.38062	−0.341494
$99$	−2.74173	−0.275554
$100$	−12.7640	−1.27640
$101$	−1.64433	−0.163617	−0.0818083	−	0.996648i	$-0.526070\pi$
$101$	−1.64433	−0.163617	−0.0818083	+	0.996648i	$0.526070\pi$
$102$	−1.19620	−0.118441
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−0.0189585	−0.00185904
$105$	16.6134	1.62130
$106$	−0.514729	−0.0499949
$107$	1.49566	0.144591	0.0722953	−	0.997383i	$-0.476968\pi$
$107$	1.49566	0.144591	0.0722953	+	0.997383i	$0.476968\pi$
$108$	1.96035	0.188635
$109$	−7.92070	−0.758665	−0.379333	−	0.925260i	$-0.623846\pi$
$109$	−7.92070	−0.758665	−0.379333	+	0.925260i	$0.623846\pi$
$110$	1.85230	0.176610
$111$	−1.94522	−0.184632
$112$	−18.4294	−1.74141
$113$	−13.5296	−1.27276	−0.636378	−	0.771377i	$-0.719568\pi$
$113$	−13.5296	−1.27276	−0.636378	+	0.771377i	$0.719568\pi$
$114$	−1.43790	−0.134672
$115$	10.2029	0.951428
$116$	14.3006	1.32777
$117$	−0.0240404	−0.00222254
$118$	−1.79346	−0.165102
$119$	29.4153	2.69650
$120$	−2.67560	−0.244247
$121$	−3.48292	−0.316629
$122$	−0.260112	−0.0235495
$123$	3.76213	0.339220
$124$	13.0384	1.17088
$125$	5.12681	0.458556
$126$	0.975054	0.0868647
$127$	2.31633	0.205541	0.102770	−	0.994705i	$-0.467229\pi$
$127$	2.31633	0.205541	0.102770	+	0.994705i	$0.467229\pi$
$128$	6.05997	0.535630
$129$	−10.3058	−0.907373
$130$	0.0162416	0.00142448
$131$	−13.2030	−1.15355	−0.576776	−	0.816902i	$-0.695689\pi$
$131$	−13.2030	−1.15355	−0.576776	+	0.816902i	$0.695689\pi$
$132$	−5.37475	−0.467812
$133$	35.3591	3.06602
$134$	−1.20612	−0.104193
$135$	−3.39280	−0.292006
$136$	−4.73736	−0.406225
$137$	−18.1779	−1.55304	−0.776522	−	0.630090i	$-0.783018\pi$
$137$	−18.1779	−1.55304	−0.776522	+	0.630090i	$0.783018\pi$
$138$	0.598819	0.0509748
$139$	−0.0490534	−0.00416065	−0.00208033	−	0.999998i	$-0.500662\pi$
$139$	−0.0490534	−0.00416065	−0.00208033	+	0.999998i	$0.500662\pi$
$140$	32.5680	2.75250
$141$	−5.72492	−0.482125
$142$	2.21024	0.185479
$143$	0.0659123	0.00551187
$144$	3.76366	0.313639
$145$	−24.7502	−2.05539
$146$	0.982350	0.0812999
$147$	−16.9772	−1.40026
$148$	−3.81332	−0.313453
$149$	19.4222	1.59113	0.795564	−	0.605869i	$-0.207174\pi$
$149$	19.4222	1.59113	0.795564	+	0.605869i	$0.207174\pi$
$150$	1.29653	0.105861
$151$	13.1155	1.06732	0.533660	−	0.845699i	$-0.320816\pi$
$151$	13.1155	1.06732	0.533660	+	0.845699i	$0.320816\pi$
$152$	−5.69460	−0.461893
$153$	−6.00723	−0.485656
$154$	−2.67333	−0.215423
$155$	−22.5657	−1.81252
$156$	−0.0471276	−0.00377323
$157$	−11.3257	−0.903889	−0.451944	−	0.892046i	$-0.649269\pi$
$157$	−11.3257	−0.903889	−0.451944	+	0.892046i	$0.649269\pi$
$158$	1.16885	0.0929884
$159$	−2.58494	−0.204999
$160$	−7.89391	−0.624068
$161$	−14.7254	−1.16052
$162$	−0.199126	−0.0156449
$163$	24.8113	1.94337	0.971687	−	0.236273i	$-0.0759260\pi$
$163$	24.8113	1.94337	0.971687	+	0.236273i	$0.0759260\pi$
$164$	7.37508	0.575897
$165$	9.30214	0.724171
$166$	−0.232035	−0.0180094
$167$	−3.89045	−0.301052	−0.150526	−	0.988606i	$-0.548097\pi$
$167$	−3.89045	−0.301052	−0.150526	+	0.988606i	$0.548097\pi$
$168$	3.86155	0.297925
$169$	−12.9994	−0.999956
$170$	4.05846	0.311270
$171$	−7.22106	−0.552208
$172$	−20.2029	−1.54046
$173$	−24.3477	−1.85112	−0.925561	−	0.378599i	$-0.876406\pi$
$173$	−24.3477	−1.85112	−0.925561	+	0.378599i	$0.876406\pi$
$174$	−1.45261	−0.110122
$175$	−31.8826	−2.41009
$176$	−10.3190	−0.777820
$177$	−9.00666	−0.676982
$178$	−0.329691	−0.0247114
$179$	−1.03074	−0.0770408	−0.0385204	−	0.999258i	$-0.512264\pi$
$179$	−1.03074	−0.0770408	−0.0385204	+	0.999258i	$0.512264\pi$
$180$	−6.65107	−0.495741
$181$	11.0183	0.818985	0.409493	−	0.912313i	$-0.365706\pi$
$181$	11.0183	0.818985	0.409493	+	0.912313i	$0.365706\pi$
$182$	−0.0234407	−0.00173754
$183$	−1.30627	−0.0965621
$184$	2.37153	0.174832
$185$	6.59975	0.485224
$186$	−1.32440	−0.0971100
$187$	16.4702	1.20442
$188$	−11.2228	−0.818509
$189$	4.89666	0.356179
$190$	4.87852	0.353925
$191$	−3.92558	−0.284045	−0.142023	−	0.989863i	$-0.545361\pi$
$191$	−3.92558	−0.284045	−0.142023	+	0.989863i	$0.545361\pi$
$192$	7.06403	0.509802
$193$	3.86140	0.277950	0.138975	−	0.990296i	$-0.455619\pi$
$193$	3.86140	0.277950	0.138975	+	0.990296i	$0.455619\pi$
$194$	−1.06474	−0.0764440
$195$	0.0815643	0.00584094
$196$	−33.2813	−2.37724
$197$	−16.2740	−1.15947	−0.579736	−	0.814804i	$-0.696844\pi$
$197$	−16.2740	−1.15947	−0.579736	+	0.814804i	$0.696844\pi$
$198$	0.545951	0.0387990
$199$	−9.78436	−0.693595	−0.346797	−	0.937940i	$-0.612731\pi$
$199$	−9.78436	−0.693595	−0.346797	+	0.937940i	$0.612731\pi$
$200$	5.13471	0.363079
$201$	−6.05704	−0.427230
$202$	0.327429	0.0230378
$203$	35.7207	2.50710
$204$	−11.7763	−0.824503
$205$	−12.7641	−0.891487
$206$	0.199126	0.0138738
$207$	3.00723	0.209017
$208$	−0.0904801	−0.00627366
$209$	19.7982	1.36947
$210$	−3.30816	−0.228285
$211$	−16.8600	−1.16069	−0.580346	−	0.814370i	$-0.697083\pi$
$211$	−16.8600	−1.16069	−0.580346	+	0.814370i	$0.697083\pi$
$212$	−5.06738	−0.348029
$213$	11.0997	0.760537
$214$	−0.297825	−0.0203589
$215$	34.9654	2.38462
$216$	−0.788610	−0.0536581
$217$	32.5680	2.21086
$218$	1.57722	0.106823
$219$	4.93330	0.333361
$220$	18.2354	1.22943
$221$	0.144416	0.00971449
$222$	0.387346	0.0259969
$223$	24.1053	1.61421	0.807105	−	0.590408i	$-0.201033\pi$
$223$	24.1053	1.61421	0.807105	+	0.590408i	$0.201033\pi$
$224$	11.3929	0.761219
$225$	6.51109	0.434072
$226$	2.69410	0.179209
$227$	13.7424	0.912118	0.456059	−	0.889949i	$-0.349261\pi$
$227$	13.7424	0.912118	0.456059	+	0.889949i	$0.349261\pi$
$228$	−14.1558	−0.937491
$229$	2.90397	0.191900	0.0959499	−	0.995386i	$-0.469411\pi$
$229$	2.90397	0.191900	0.0959499	+	0.995386i	$0.469411\pi$
$230$	−2.03167	−0.133964
$231$	−13.4253	−0.883321
$232$	−5.75284	−0.377692
$233$	22.6618	1.48462	0.742311	−	0.670056i	$-0.233730\pi$
$233$	22.6618	1.48462	0.742311	+	0.670056i	$0.233730\pi$
$234$	0.00478708	0.000312941 0
$235$	19.4235	1.26705
$236$	−17.6562	−1.14932
$237$	5.86987	0.381289
$238$	−5.85737	−0.379677
$239$	−12.5767	−0.813521	−0.406761	−	0.913535i	$-0.633342\pi$
$239$	−12.5767	−0.813521	−0.406761	+	0.913535i	$0.633342\pi$
$240$	−12.7694	−0.824258
$241$	−3.29256	−0.212093	−0.106046	−	0.994361i	$-0.533819\pi$
$241$	−3.29256	−0.212093	−0.106046	+	0.994361i	$0.533819\pi$
$242$	0.693541	0.0445825
$243$	−1.00000	−0.0641500
$244$	−2.56074	−0.163935
$245$	57.6004	3.67995
$246$	−0.749139	−0.0477634
$247$	0.173597	0.0110457
$248$	−5.24510	−0.333064
$249$	−1.16526	−0.0738456
$250$	−1.02088	−0.0645664
$251$	−25.4371	−1.60558	−0.802788	−	0.596264i	$-0.796651\pi$
$251$	−25.4371	−1.60558	−0.802788	+	0.596264i	$0.796651\pi$
$252$	9.59915	0.604690
$253$	−8.24501	−0.518359
$254$	−0.461242	−0.0289409
$255$	20.3813	1.27633
$256$	12.9214	0.807585
$257$	−22.3799	−1.39602	−0.698010	−	0.716088i	$-0.745931\pi$
$257$	−22.3799	−1.39602	−0.698010	+	0.716088i	$0.745931\pi$
$258$	2.05215	0.127762
$259$	−9.52509	−0.591861
$260$	0.159894	0.00991624
$261$	−7.29491	−0.451544
$262$	2.62907	0.162424
$263$	−9.30178	−0.573572	−0.286786	−	0.957995i	$-0.592587\pi$
$263$	−9.30178	−0.573572	−0.286786	+	0.957995i	$0.592587\pi$
$264$	2.16216	0.133072
$265$	8.77017	0.538747
$266$	−7.04092	−0.431707
$267$	−1.65568	−0.101326
$268$	−11.8739	−0.725314
$269$	−23.4021	−1.42685	−0.713425	−	0.700732i	$-0.752857\pi$
$269$	−23.4021	−1.42685	−0.713425	+	0.700732i	$0.752857\pi$
$270$	0.675596	0.0411155
$271$	14.2086	0.863114	0.431557	−	0.902086i	$-0.357964\pi$
$271$	14.2086	0.863114	0.431557	+	0.902086i	$0.357964\pi$
$272$	−22.6092	−1.37088
$273$	−0.117718	−0.00712460
$274$	3.61970	0.218674
$275$	−17.8516	−1.07649
$276$	5.89522	0.354850
$277$	2.83194	0.170155	0.0850774	−	0.996374i	$-0.472886\pi$
$277$	2.83194	0.170155	0.0850774	+	0.996374i	$0.472886\pi$
$278$	0.00976782	0.000585835 0
$279$	−6.65107	−0.398189
$280$	−13.1015	−0.782963
$281$	20.7643	1.23869	0.619347	−	0.785117i	$-0.287397\pi$
$281$	20.7643	1.23869	0.619347	+	0.785117i	$0.287397\pi$
$282$	1.13998	0.0678850
$283$	−14.6198	−0.869060	−0.434530	−	0.900657i	$-0.643085\pi$
$283$	−14.6198	−0.869060	−0.434530	+	0.900657i	$0.643085\pi$
$284$	21.7592	1.29117
$285$	24.4996	1.45123
$286$	−0.0131249	−0.000776091 0
$287$	18.4218	1.08741
$288$	−2.32667	−0.137100
$289$	19.0868	1.12275
$290$	4.92841	0.289406
$291$	−5.34706	−0.313450
$292$	9.67098	0.565951
$293$	15.4409	0.902066	0.451033	−	0.892507i	$-0.351056\pi$
$293$	15.4409	0.902066	0.451033	+	0.892507i	$0.351056\pi$
$294$	3.38062	0.197162
$295$	30.5578	1.77914
$296$	1.53402	0.0891633
$297$	2.74173	0.159091
$298$	−3.86747	−0.224037
$299$	−0.0722950	−0.00418093
$300$	12.7640	0.736930
$301$	−50.4639	−2.90869
$302$	−2.61163	−0.150283
$303$	1.64433	0.0944641
$304$	−27.1776	−1.55874
$305$	4.43190	0.253770
$306$	1.19620	0.0683821
$307$	−18.7065	−1.06764	−0.533819	−	0.845599i	$-0.679244\pi$
$307$	−18.7065	−1.06764	−0.533819	+	0.845599i	$0.679244\pi$
$308$	−26.3183	−1.49962
$309$	1.00000	0.0568880
$310$	4.49344	0.255210
$311$	15.9999	0.907272	0.453636	−	0.891187i	$-0.350127\pi$
$311$	15.9999	0.907272	0.453636	+	0.891187i	$0.350127\pi$
$312$	0.0189585	0.00107331
$313$	0.0144940	0.000819249 0	0.000409624	−	1.00000i	$-0.499870\pi$
$313$	0.0144940	0.000819249 0	0.000409624		1.00000i	$0.499870\pi$
$314$	2.25525	0.127271
$315$	−16.6134	−0.936058
$316$	11.5070	0.647319
$317$	7.49521	0.420973	0.210487	−	0.977597i	$-0.432495\pi$
$317$	7.49521	0.420973	0.210487	+	0.977597i	$0.432495\pi$
$318$	0.514729	0.0288646
$319$	20.0007	1.11982
$320$	−23.9668	−1.33979
$321$	−1.49566	−0.0834794
$322$	2.93221	0.163406
$323$	43.3786	2.41365
$324$	−1.96035	−0.108908
$325$	−0.156529	−0.00868268
$326$	−4.94059	−0.273634
$327$	7.92070	0.438016
$328$	−2.96685	−0.163817
$329$	−28.0330	−1.54551
$330$	−1.85230	−0.101966
$331$	−25.8309	−1.41980	−0.709898	−	0.704304i	$-0.751259\pi$
$331$	−25.8309	−1.41980	−0.709898	+	0.704304i	$0.751259\pi$
$332$	−2.28433	−0.125369
$333$	1.94522	0.106598
$334$	0.774691	0.0423892
$335$	20.5503	1.12278
$336$	18.4294	1.00540
$337$	24.4289	1.33073	0.665364	−	0.746519i	$-0.268276\pi$
$337$	24.4289	1.33073	0.665364	+	0.746519i	$0.268276\pi$
$338$	2.58853	0.140797
$339$	13.5296	0.734826
$340$	39.9545	2.16684
$341$	18.2354	0.987504
$342$	1.43790	0.0777530
$343$	−48.8552	−2.63793
$344$	8.12724	0.438192
$345$	−10.2029	−0.549307
$346$	4.84827	0.260645
$347$	15.3724	0.825236	0.412618	−	0.910904i	$-0.364614\pi$
$347$	15.3724	0.825236	0.412618	+	0.910904i	$0.364614\pi$
$348$	−14.3006	−0.766591
$349$	26.4947	1.41823	0.709115	−	0.705093i	$-0.249095\pi$
$349$	26.4947	1.41823	0.709115	+	0.705093i	$0.249095\pi$
$350$	6.34866	0.339350
$351$	0.0240404	0.00128318
$352$	6.37909	0.340007
$353$	−9.57496	−0.509624	−0.254812	−	0.966991i	$-0.582014\pi$
$353$	−9.57496	−0.509624	−0.254812	+	0.966991i	$0.582014\pi$
$354$	1.79346	0.0953215
$355$	−37.6589	−1.99873
$356$	−3.24572	−0.172023
$357$	−29.4153	−1.55682
$358$	0.205247	0.0108476
$359$	−6.16966	−0.325622	−0.162811	−	0.986657i	$-0.552056\pi$
$359$	−6.16966	−0.325622	−0.162811	+	0.986657i	$0.552056\pi$
$360$	2.67560	0.141016
$361$	33.1437	1.74441
$362$	−2.19404	−0.115316
$363$	3.48292	0.182806
$364$	−0.230768	−0.0120955
$365$	−16.7377	−0.876090
$366$	0.260112	0.0135963
$367$	−0.844934	−0.0441052	−0.0220526	−	0.999757i	$-0.507020\pi$
$367$	−0.844934	−0.0441052	−0.0220526	+	0.999757i	$0.507020\pi$
$368$	11.3182	0.590002
$369$	−3.76213	−0.195849
$370$	−1.31419	−0.0683213
$371$	−12.6575	−0.657147
$372$	−13.0384	−0.676010
$373$	12.6313	0.654026	0.327013	−	0.945020i	$-0.393958\pi$
$373$	12.6313	0.654026	0.327013	+	0.945020i	$0.393958\pi$
$374$	−3.27965	−0.169587
$375$	−5.12681	−0.264748
$376$	4.51473	0.232829
$377$	0.175373	0.00903215
$378$	−0.975054	−0.0501514
$379$	−30.9101	−1.58774	−0.793872	−	0.608086i	$-0.791938\pi$
$379$	−30.9101	−1.58774	−0.793872	+	0.608086i	$0.791938\pi$
$380$	48.0278	2.46377
$381$	−2.31633	−0.118669
$382$	0.781687	0.0399946
$383$	−9.46834	−0.483810	−0.241905	−	0.970300i	$-0.577772\pi$
$383$	−9.46834	−0.483810	−0.241905	+	0.970300i	$0.577772\pi$
$384$	−6.05997	−0.309246
$385$	45.5494	2.32141
$386$	−0.768907	−0.0391363
$387$	10.3058	0.523872
$388$	−10.4821	−0.532148
$389$	−16.0102	−0.811752	−0.405876	−	0.913928i	$-0.633033\pi$
$389$	−16.0102	−0.811752	−0.405876	+	0.913928i	$0.633033\pi$
$390$	−0.0162416	−0.000822426 0
$391$	−18.0651	−0.913592
$392$	13.3884	0.676218
$393$	13.2030	0.666004
$394$	3.24058	0.163258
$395$	−19.9153	−1.00205
$396$	5.37475	0.270091
$397$	−15.4821	−0.777026	−0.388513	−	0.921443i	$-0.627011\pi$
$397$	−15.4821	−0.777026	−0.388513	+	0.921443i	$0.627011\pi$
$398$	1.94832	0.0976607
$399$	−35.3591	−1.77017
$400$	24.5055	1.22528
$401$	0.0408518	0.00204004	0.00102002	−	0.999999i	$-0.499675\pi$
$401$	0.0408518	0.00204004	0.00102002	+	0.999999i	$0.499675\pi$
$402$	1.20612	0.0601556
$403$	0.159894	0.00796491
$404$	3.22345	0.160373
$405$	3.39280	0.168590
$406$	−7.11293	−0.353009
$407$	−5.33328	−0.264361
$408$	4.73736	0.234534
$409$	5.63658	0.278711	0.139355	−	0.990242i	$-0.455497\pi$
$409$	5.63658	0.278711	0.139355	+	0.990242i	$0.455497\pi$
$410$	2.54168	0.125525
$411$	18.1779	0.896650
$412$	1.96035	0.0965794
$413$	−44.1025	−2.17014
$414$	−0.598819	−0.0294303
$415$	3.95351	0.194070
$416$	0.0559340	0.00274239
$417$	0.0490534	0.00240215
$418$	−3.94234	−0.192826
$419$	14.8716	0.726523	0.363262	−	0.931687i	$-0.381663\pi$
$419$	14.8716	0.726523	0.363262	+	0.931687i	$0.381663\pi$
$420$	−32.5680	−1.58916
$421$	5.18935	0.252914	0.126457	−	0.991972i	$-0.459639\pi$
$421$	5.18935	0.252914	0.126457	+	0.991972i	$0.459639\pi$
$422$	3.35728	0.163430
$423$	5.72492	0.278355
$424$	2.03851	0.0989986
$425$	−39.1136	−1.89729
$426$	−2.21024	−0.107086
$427$	−6.39634	−0.309541
$428$	−2.93201	−0.141724
$429$	−0.0659123	−0.00318228
$430$	−6.96255	−0.335764
$431$	−32.3305	−1.55731	−0.778654	−	0.627454i	$-0.784097\pi$
$431$	−32.3305	−1.55731	−0.778654	+	0.627454i	$0.784097\pi$
$432$	−3.76366	−0.181079
$433$	−21.5744	−1.03680	−0.518401	−	0.855138i	$-0.673472\pi$
$433$	−21.5744	−1.03680	−0.518401	+	0.855138i	$0.673472\pi$
$434$	−6.48515	−0.311297
$435$	24.7502	1.18668
$436$	15.5273	0.743624
$437$	−21.7154	−1.03879
$438$	−0.982350	−0.0469385
$439$	−20.3807	−0.972715	−0.486358	−	0.873760i	$-0.661675\pi$
$439$	−20.3807	−0.972715	−0.486358	+	0.873760i	$0.661675\pi$
$440$	−7.33576	−0.349719
$441$	16.9772	0.808440
$442$	−0.0287571	−0.00136784
$443$	−12.9390	−0.614751	−0.307376	−	0.951588i	$-0.599451\pi$
$443$	−12.9390	−0.614751	−0.307376	+	0.951588i	$0.599451\pi$
$444$	3.81332	0.180972
$445$	5.61740	0.266291
$446$	−4.80000	−0.227287
$447$	−19.4222	−0.918639
$448$	34.5901	1.63423
$449$	8.59780	0.405755	0.202878	−	0.979204i	$-0.434971\pi$
$449$	8.59780	0.405755	0.202878	+	0.979204i	$0.434971\pi$
$450$	−1.29653	−0.0611190
$451$	10.3147	0.485702
$452$	26.5227	1.24752
$453$	−13.1155	−0.616218
$454$	−2.73649	−0.128430
$455$	0.399392	0.0187238
$456$	5.69460	0.266674
$457$	19.3161	0.903570	0.451785	−	0.892127i	$-0.350787\pi$
$457$	19.3161	0.903570	0.451785	+	0.892127i	$0.350787\pi$
$458$	−0.578257	−0.0270202
$459$	6.00723	0.280393
$460$	−20.0013	−0.932565
$461$	23.5956	1.09896	0.549478	−	0.835508i	$-0.314827\pi$
$461$	23.5956	1.09896	0.549478	+	0.835508i	$0.314827\pi$
$462$	2.67333	0.124375
$463$	31.7197	1.47414	0.737070	−	0.675817i	$-0.236209\pi$
$463$	31.7197	1.47414	0.737070	+	0.675817i	$0.236209\pi$
$464$	−27.4556	−1.27459
$465$	22.5657	1.04646
$466$	−4.51256	−0.209040
$467$	−40.7501	−1.88569	−0.942846	−	0.333230i	$-0.891861\pi$
$467$	−40.7501	−1.88569	−0.942846	+	0.333230i	$0.891861\pi$
$468$	0.0471276	0.00217847
$469$	−29.6592	−1.36954
$470$	−3.86773	−0.178405
$471$	11.3257	0.521860
$472$	7.10274	0.326930
$473$	−28.2557	−1.29920
$474$	−1.16885	−0.0536869
$475$	−47.0170	−2.15729
$476$	−57.6643	−2.64304
$477$	2.58494	0.118356
$478$	2.50436	0.114547
$479$	−6.18584	−0.282638	−0.141319	−	0.989964i	$-0.545134\pi$
$479$	−6.18584	−0.282638	−0.141319	+	0.989964i	$0.545134\pi$
$480$	7.89391	0.360306
$481$	−0.0467640	−0.00213225
$482$	0.655637	0.0298634
$483$	14.7254	0.670027
$484$	6.82773	0.310351
$485$	18.1415	0.823764
$486$	0.199126	0.00903256
$487$	2.04236	0.0925481	0.0462740	−	0.998929i	$-0.485265\pi$
$487$	2.04236	0.0925481	0.0462740	+	0.998929i	$0.485265\pi$
$488$	1.03014	0.0466321
$489$	−24.8113	−1.12201
$490$	−11.4698	−0.518151
$491$	−39.2323	−1.77053	−0.885265	−	0.465087i	$-0.846023\pi$
$491$	−39.2323	−1.77053	−0.885265	+	0.465087i	$0.846023\pi$
$492$	−7.37508	−0.332494
$493$	43.8222	1.97365
$494$	−0.0345678	−0.00155528
$495$	−9.30214	−0.418100
$496$	−25.0324	−1.12399
$497$	54.3513	2.43799
$498$	0.232035	0.0103977
$499$	1.90446	0.0852551	0.0426276	−	0.999091i	$-0.486427\pi$
$499$	1.90446	0.0852551	0.0426276	+	0.999091i	$0.486427\pi$
$500$	−10.0503	−0.449465
$501$	3.89045	0.173812
$502$	5.06520	0.226071
$503$	−0.312116	−0.0139166	−0.00695828	−	0.999976i	$-0.502215\pi$
$503$	−0.312116	−0.0139166	−0.00695828	+	0.999976i	$0.502215\pi$
$504$	−3.86155	−0.172007
$505$	−5.57887	−0.248256
$506$	1.64180	0.0729869
$507$	12.9994	0.577325
$508$	−4.54081	−0.201466
$509$	−17.4965	−0.775517	−0.387758	−	0.921761i	$-0.626750\pi$
$509$	−17.4965	−0.775517	−0.387758	+	0.921761i	$0.626750\pi$
$510$	−4.05846	−0.179712
$511$	24.1567	1.06863
$512$	−14.6929	−0.649341
$513$	7.22106	0.318818
$514$	4.45643	0.196565
$515$	−3.39280	−0.149505
$516$	20.2029	0.889384
$517$	−15.6962	−0.690317
$518$	1.89670	0.0833361
$519$	24.3477	1.06875
$520$	−0.0643224	−0.00282073
$521$	−2.74128	−0.120098	−0.0600488	−	0.998195i	$-0.519126\pi$
$521$	−2.74128	−0.120098	−0.0600488	+	0.998195i	$0.519126\pi$
$522$	1.45261	0.0635790
$523$	24.3692	1.06559	0.532796	−	0.846244i	$-0.321141\pi$
$523$	24.3692	1.06559	0.532796	+	0.846244i	$0.321141\pi$
$524$	25.8825	1.13068
$525$	31.8826	1.39147
$526$	1.85223	0.0807611
$527$	39.9545	1.74045
$528$	10.3190	0.449075
$529$	−13.9566	−0.606808
$530$	−1.74637	−0.0758576
$531$	9.00666	0.390856
$532$	−69.3161	−3.00523
$533$	0.0904431	0.00391753
$534$	0.329691	0.0142671
$535$	5.07446	0.219388
$536$	4.77664	0.206319
$537$	1.03074	0.0444795
$538$	4.65997	0.200906
$539$	−46.5470	−2.00492
$540$	6.65107	0.286216
$541$	−30.3726	−1.30582	−0.652910	−	0.757436i	$-0.726452\pi$
$541$	−30.3726	−1.30582	−0.652910	+	0.757436i	$0.726452\pi$
$542$	−2.82932	−0.121530
$543$	−11.0183	−0.472841
$544$	13.9768	0.599251
$545$	−26.8733	−1.15113
$546$	0.0234407	0.00100317
$547$	−23.7602	−1.01591	−0.507957	−	0.861382i	$-0.669599\pi$
$547$	−23.7602	−1.01591	−0.507957	+	0.861382i	$0.669599\pi$
$548$	35.6350	1.52225
$549$	1.30627	0.0557501
$550$	3.55473	0.151574
$551$	52.6770	2.24412
$552$	−2.37153	−0.100939
$553$	28.7427	1.22227
$554$	−0.563914	−0.0239584
$555$	−6.59975	−0.280144
$556$	0.0961617	0.00407816
$557$	15.7067	0.665514	0.332757	−	0.943013i	$-0.392021\pi$
$557$	15.7067	0.665514	0.332757	+	0.943013i	$0.392021\pi$
$558$	1.32440	0.0560665
$559$	−0.247755	−0.0104789
$560$	−62.5272	−2.64225
$561$	−16.4702	−0.695372
$562$	−4.13472	−0.174413
$563$	23.4004	0.986210	0.493105	−	0.869970i	$-0.335862\pi$
$563$	23.4004	0.986210	0.493105	+	0.869970i	$0.335862\pi$
$564$	11.2228	0.472567
$565$	−45.9032	−1.93116
$566$	2.91120	0.122367
$567$	−4.89666	−0.205640
$568$	−8.75331	−0.367281
$569$	−21.5381	−0.902924	−0.451462	−	0.892290i	$-0.649097\pi$
$569$	−21.5381	−0.902924	−0.451462	+	0.892290i	$0.649097\pi$
$570$	−4.87852	−0.204339
$571$	43.9801	1.84051	0.920254	−	0.391321i	$-0.127982\pi$
$571$	43.9801	1.84051	0.920254	+	0.391321i	$0.127982\pi$
$572$	−0.129211	−0.00540259
$573$	3.92558	0.163994
$574$	−3.66828	−0.153111
$575$	19.5803	0.816556
$576$	−7.06403	−0.294334
$577$	13.0976	0.545261	0.272631	−	0.962119i	$-0.412106\pi$
$577$	13.0976	0.545261	0.272631	+	0.962119i	$0.412106\pi$
$578$	−3.80069	−0.158088
$579$	−3.86140	−0.160474
$580$	48.5190	2.01464
$581$	−5.70590	−0.236721
$582$	1.06474	0.0441350
$583$	−7.08720	−0.293522
$584$	−3.89045	−0.160988
$585$	−0.0815643	−0.00337227
$586$	−3.07469	−0.127014
$587$	−11.1159	−0.458803	−0.229401	−	0.973332i	$-0.573677\pi$
$587$	−11.1159	−0.458803	−0.229401	+	0.973332i	$0.573677\pi$
$588$	33.2813	1.37250
$589$	48.0278	1.97895
$590$	−6.08486	−0.250510
$591$	16.2740	0.669422
$592$	7.32117	0.300898
$593$	−11.3299	−0.465263	−0.232632	−	0.972565i	$-0.574734\pi$
$593$	−11.3299	−0.465263	−0.232632	+	0.972565i	$0.574734\pi$
$594$	−0.545951	−0.0224006
$595$	99.8003	4.09142
$596$	−38.0743	−1.55958
$597$	9.78436	0.400447
$598$	0.0143959	0.000588690 0
$599$	29.4810	1.20456	0.602280	−	0.798285i	$-0.294259\pi$
$599$	29.4810	1.20456	0.602280	+	0.798285i	$0.294259\pi$
$600$	−5.13471	−0.209624
$601$	−0.170413	−0.00695131	−0.00347565	−	0.999994i	$-0.501106\pi$
$601$	−0.170413	−0.00695131	−0.00347565	+	0.999994i	$0.501106\pi$
$602$	10.0487	0.409554
$603$	6.05704	0.246662
$604$	−25.7109	−1.04616
$605$	−11.8168	−0.480423
$606$	−0.327429	−0.0133009
$607$	−16.4055	−0.665878	−0.332939	−	0.942948i	$-0.608040\pi$
$607$	−16.4055	−0.665878	−0.332939	+	0.942948i	$0.608040\pi$
$608$	16.8010	0.681370
$609$	−35.7207	−1.44747
$610$	−0.882509	−0.0357318
$611$	−0.137629	−0.00556789
$612$	11.7763	0.476027
$613$	−18.2199	−0.735893	−0.367947	−	0.929847i	$-0.619939\pi$
$613$	−18.2199	−0.735893	−0.367947	+	0.929847i	$0.619939\pi$
$614$	3.72496	0.150327
$615$	12.7641	0.514700
$616$	10.5873	0.426576
$617$	−3.18363	−0.128168	−0.0640841	−	0.997945i	$-0.520413\pi$
$617$	−3.18363	−0.128168	−0.0640841	+	0.997945i	$0.520413\pi$
$618$	−0.199126	−0.00801004
$619$	−21.2018	−0.852173	−0.426087	−	0.904682i	$-0.640108\pi$
$619$	−21.2018	−0.852173	−0.426087	+	0.904682i	$0.640108\pi$
$620$	44.2367	1.77659
$621$	−3.00723	−0.120676
$622$	−3.18601	−0.127747
$623$	−8.10732	−0.324813
$624$	0.0904801	0.00362210
$625$	−15.1612	−0.606447
$626$	−0.00288614	−0.000115353 0
$627$	−19.7982	−0.790664
$628$	22.2023	0.885969
$629$	−11.6854	−0.465928
$630$	3.30816	0.131800
$631$	22.2903	0.887365	0.443682	−	0.896184i	$-0.353672\pi$
$631$	22.2903	0.887365	0.443682	+	0.896184i	$0.353672\pi$
$632$	−4.62904	−0.184133
$633$	16.8600	0.670126
$634$	−1.49250	−0.0592745
$635$	7.85884	0.311868
$636$	5.06738	0.200934
$637$	−0.408140	−0.0161711
$638$	−3.98266	−0.157675
$639$	−11.0997	−0.439096
$640$	20.5602	0.812715
$641$	15.9480	0.629907	0.314954	−	0.949107i	$-0.398011\pi$
$641$	15.9480	0.629907	0.314954	+	0.949107i	$0.398011\pi$
$642$	0.297825	0.0117542
$643$	−18.0174	−0.710537	−0.355268	−	0.934764i	$-0.615611\pi$
$643$	−18.0174	−0.710537	−0.355268	+	0.934764i	$0.615611\pi$
$644$	28.8669	1.13751
$645$	−34.9654	−1.37676
$646$	−8.63782	−0.339850
$647$	−32.7308	−1.28678	−0.643391	−	0.765538i	$-0.722473\pi$
$647$	−32.7308	−1.28678	−0.643391	+	0.765538i	$0.722473\pi$
$648$	0.788610	0.0309795
$649$	−24.6938	−0.969317
$650$	0.0311691	0.00122255
$651$	−32.5680	−1.27644
$652$	−48.6389	−1.90484
$653$	26.7022	1.04494	0.522469	−	0.852658i	$-0.325011\pi$
$653$	26.7022	1.04494	0.522469	+	0.852658i	$0.325011\pi$
$654$	−1.57722	−0.0616742
$655$	−44.7952	−1.75029
$656$	−14.1594	−0.552831
$657$	−4.93330	−0.192466
$658$	5.58210	0.217613
$659$	2.03490	0.0792684	0.0396342	−	0.999214i	$-0.487381\pi$
$659$	2.03490	0.0792684	0.0396342	+	0.999214i	$0.487381\pi$
$660$	−18.2354	−0.709813
$661$	−10.5628	−0.410846	−0.205423	−	0.978673i	$-0.565857\pi$
$661$	−10.5628	−0.410846	−0.205423	+	0.978673i	$0.565857\pi$
$662$	5.14362	0.199913
$663$	−0.144416	−0.00560866
$664$	0.918940	0.0356618
$665$	119.966	4.65209
$666$	−0.387346	−0.0150093
$667$	−21.9375	−0.849422
$668$	7.62663	0.295083
$669$	−24.1053	−0.931965
$670$	−4.09211	−0.158092
$671$	−3.58143	−0.138260
$672$	−11.3929	−0.439490
$673$	33.9602	1.30907	0.654534	−	0.756032i	$-0.272865\pi$
$673$	33.9602	1.30907	0.654534	+	0.756032i	$0.272865\pi$
$674$	−4.86444	−0.187371
$675$	−6.51109	−0.250612
$676$	25.4834	0.980131
$677$	25.8880	0.994957	0.497479	−	0.867476i	$-0.334259\pi$
$677$	25.8880	0.994957	0.497479	+	0.867476i	$0.334259\pi$
$678$	−2.69410	−0.103466
$679$	−26.1827	−1.00480
$680$	−16.0729	−0.616368
$681$	−13.7424	−0.526612
$682$	−3.63116	−0.139044
$683$	−40.3420	−1.54365	−0.771823	−	0.635838i	$-0.780655\pi$
$683$	−40.3420	−1.54365	−0.771823	+	0.635838i	$0.780655\pi$
$684$	14.1558	0.541260
$685$	−61.6740	−2.35644
$686$	9.72836	0.371430
$687$	−2.90397	−0.110793
$688$	38.7875	1.47876
$689$	−0.0621429	−0.00236746
$690$	2.03167	0.0773444
$691$	−28.6680	−1.09058	−0.545291	−	0.838247i	$-0.683581\pi$
$691$	−28.6680	−1.09058	−0.545291	+	0.838247i	$0.683581\pi$
$692$	47.7300	1.81442
$693$	13.4253	0.509985
$694$	−3.06106	−0.116196
$695$	−0.166428	−0.00631298
$696$	5.75284	0.218061
$697$	22.6000	0.856035
$698$	−5.27580	−0.199692
$699$	−22.6618	−0.857147
$700$	62.5009	2.36231
$701$	4.54450	0.171643	0.0858217	−	0.996311i	$-0.472648\pi$
$701$	4.54450	0.171643	0.0858217	+	0.996311i	$0.472648\pi$
$702$	−0.00478708	−0.000180677 0
$703$	−14.0466	−0.529777
$704$	19.3677	0.729946
$705$	−19.4235	−0.731531
$706$	1.90663	0.0717569
$707$	8.05170	0.302815
$708$	17.6562	0.663560
$709$	−15.8288	−0.594465	−0.297232	−	0.954805i	$-0.596064\pi$
$709$	−15.8288	−0.594465	−0.297232	+	0.954805i	$0.596064\pi$
$710$	7.49889	0.281428
$711$	−5.86987	−0.220137
$712$	1.30569	0.0489328
$713$	−20.0013	−0.749054
$714$	5.85737	0.219207
$715$	0.223627	0.00836319
$716$	2.02060	0.0755134
$717$	12.5767	0.469687
$718$	1.22854	0.0458488
$719$	39.8942	1.48780	0.743901	−	0.668290i	$-0.232973\pi$
$719$	39.8942	1.48780	0.743901	+	0.668290i	$0.232973\pi$
$720$	12.7694	0.475886
$721$	4.89666	0.182361
$722$	−6.59979	−0.245619
$723$	3.29256	0.122452
$724$	−21.5998	−0.802748
$725$	−47.4978	−1.76402
$726$	−0.693541	−0.0257397
$727$	34.6983	1.28689	0.643445	−	0.765493i	$-0.277505\pi$
$727$	34.6983	1.28689	0.643445	+	0.765493i	$0.277505\pi$
$728$	0.0928334	0.00344063
$729$	1.00000	0.0370370
$730$	3.33292	0.123357
$731$	−61.9092	−2.28979
$732$	2.56074	0.0946477
$733$	45.0560	1.66418	0.832090	−	0.554640i	$-0.187144\pi$
$733$	45.0560	1.66418	0.832090	+	0.554640i	$0.187144\pi$
$734$	0.168249	0.00621017
$735$	−57.6004	−2.12462
$736$	−6.99682	−0.257906
$737$	−16.6068	−0.611718
$738$	0.749139	0.0275762
$739$	44.5954	1.64047	0.820234	−	0.572028i	$-0.193843\pi$
$739$	44.5954	1.64047	0.820234	+	0.572028i	$0.193843\pi$
$740$	−12.9378	−0.475604
$741$	−0.173597	−0.00637726
$742$	2.52045	0.0925287
$743$	41.5182	1.52315	0.761577	−	0.648075i	$-0.224426\pi$
$743$	41.5182	1.52315	0.761577	+	0.648075i	$0.224426\pi$
$744$	5.24510	0.192295
$745$	65.8956	2.41423
$746$	−2.51523	−0.0920893
$747$	1.16526	0.0426348
$748$	−32.2873	−1.18054
$749$	−7.32371	−0.267603
$750$	1.02088	0.0372774
$751$	3.06698	0.111916	0.0559578	−	0.998433i	$-0.482179\pi$
$751$	3.06698	0.111916	0.0559578	+	0.998433i	$0.482179\pi$
$752$	21.5467	0.785726
$753$	25.4371	0.926980
$754$	−0.0349213	−0.00127176
$755$	44.4981	1.61945
$756$	−9.59915	−0.349118
$757$	−18.1165	−0.658454	−0.329227	−	0.944251i	$-0.606788\pi$
$757$	−18.1165	−0.658454	−0.329227	+	0.944251i	$0.606788\pi$
$758$	6.15501	0.223560
$759$	8.24501	0.299275
$760$	−19.3206	−0.700833
$761$	13.3264	0.483083	0.241542	−	0.970390i	$-0.422347\pi$
$761$	13.3264	0.483083	0.241542	+	0.970390i	$0.422347\pi$
$762$	0.461242	0.0167090
$763$	38.7849	1.40411
$764$	7.69551	0.278414
$765$	−20.3813	−0.736888
$766$	1.88540	0.0681222
$767$	−0.216524	−0.00781822
$768$	−12.9214	−0.466259
$769$	10.2975	0.371337	0.185669	−	0.982612i	$-0.440555\pi$
$769$	10.2975	0.371337	0.185669	+	0.982612i	$0.440555\pi$
$770$	−9.07009	−0.326863
$771$	22.3799	0.805992
$772$	−7.56969	−0.272439
$773$	−34.7159	−1.24864	−0.624322	−	0.781167i	$-0.714624\pi$
$773$	−34.7159	−1.24864	−0.624322	+	0.781167i	$0.714624\pi$
$774$	−2.05215	−0.0737631
$775$	−43.3057	−1.55559
$776$	4.21675	0.151372
$777$	9.52509	0.341711
$778$	3.18806	0.114298
$779$	27.1666	0.973343
$780$	−0.159894	−0.00572514
$781$	30.4323	1.08895
$782$	3.59724	0.128637
$783$	7.29491	0.260699
$784$	63.8967	2.28202
$785$	−38.4258	−1.37148
$786$	−2.62907	−0.0937758
$787$	20.6909	0.737549	0.368775	−	0.929519i	$-0.379777\pi$
$787$	20.6909	0.737549	0.368775	+	0.929519i	$0.379777\pi$
$788$	31.9027	1.13649
$789$	9.30178	0.331152
$790$	3.96566	0.141092
$791$	66.2498	2.35557
$792$	−2.16216	−0.0768289
$793$	−0.0314032	−0.00111516
$794$	3.08290	0.109408
$795$	−8.77017	−0.311046
$796$	19.1808	0.679844
$797$	17.7981	0.630440	0.315220	−	0.949019i	$-0.397922\pi$
$797$	17.7981	0.630440	0.315220	+	0.949019i	$0.397922\pi$
$798$	7.04092	0.249246
$799$	−34.3909	−1.21666
$800$	−15.1491	−0.535602
$801$	1.65568	0.0585007
$802$	−0.00813467	−0.000287245 0
$803$	13.5258	0.477314
$804$	11.8739	0.418760
$805$	−49.9602	−1.76087
$806$	−0.0318392	−0.00112149
$807$	23.4021	0.823792
$808$	−1.29673	−0.0456189
$809$	−32.5503	−1.14441	−0.572204	−	0.820111i	$-0.693911\pi$
$809$	−32.5503	−1.14441	−0.572204	+	0.820111i	$0.693911\pi$
$810$	−0.675596	−0.0237380
$811$	11.2165	0.393864	0.196932	−	0.980417i	$-0.436902\pi$
$811$	11.2165	0.393864	0.196932	+	0.980417i	$0.436902\pi$
$812$	−70.0250	−2.45739
$813$	−14.2086	−0.498319
$814$	1.06200	0.0372230
$815$	84.1799	2.94869
$816$	22.6092	0.791480
$817$	−74.4187	−2.60358
$818$	−1.12239	−0.0392435
$819$	0.117718	0.00411339
$820$	25.0222	0.873812
$821$	44.6736	1.55912	0.779560	−	0.626328i	$-0.215443\pi$
$821$	44.6736	1.55912	0.779560	+	0.626328i	$0.215443\pi$
$822$	−3.61970	−0.126252
$823$	13.7236	0.478373	0.239187	−	0.970974i	$-0.423119\pi$
$823$	13.7236	0.478373	0.239187	+	0.970974i	$0.423119\pi$
$824$	−0.788610	−0.0274725
$825$	17.8516	0.621514
$826$	8.78198	0.305564
$827$	35.7863	1.24441	0.622206	−	0.782853i	$-0.286237\pi$
$827$	35.7863	1.24441	0.622206	+	0.782853i	$0.286237\pi$
$828$	−5.89522	−0.204873
$829$	1.60440	0.0557230	0.0278615	−	0.999612i	$-0.491130\pi$
$829$	1.60440	0.0557230	0.0278615	+	0.999612i	$0.491130\pi$
$830$	−0.787248	−0.0273258
$831$	−2.83194	−0.0982389
$832$	0.169822	0.00588752
$833$	−101.986	−3.53361
$834$	−0.00976782	−0.000338232 0
$835$	−13.1995	−0.456788
$836$	−38.8114	−1.34232
$837$	6.65107	0.229895
$838$	−2.96132	−0.102297
$839$	−9.71407	−0.335367	−0.167683	−	0.985841i	$-0.553629\pi$
$839$	−9.71407	−0.335367	−0.167683	+	0.985841i	$0.553629\pi$
$840$	13.1015	0.452044
$841$	24.2157	0.835024
$842$	−1.03334	−0.0356112
$843$	−20.7643	−0.715161
$844$	33.0515	1.13768
$845$	−44.1044	−1.51724
$846$	−1.13998	−0.0391934
$847$	17.0547	0.586005
$848$	9.72883	0.334089
$849$	14.6198	0.501752
$850$	7.78855	0.267145
$851$	5.84973	0.200526
$852$	−21.7592	−0.745458
$853$	11.7452	0.402147	0.201074	−	0.979576i	$-0.435557\pi$
$853$	11.7452	0.402147	0.201074	+	0.979576i	$0.435557\pi$
$854$	1.27368	0.0435845
$855$	−24.4996	−0.837869
$856$	1.17949	0.0403141
$857$	7.99773	0.273197	0.136599	−	0.990626i	$-0.456383\pi$
$857$	7.99773	0.273197	0.136599	+	0.990626i	$0.456383\pi$
$858$	0.0131249	0.000448076 0
$859$	−53.7169	−1.83280	−0.916399	−	0.400265i	$-0.868918\pi$
$859$	−53.7169	−1.83280	−0.916399	+	0.400265i	$0.868918\pi$
$860$	−68.5445	−2.33735
$861$	−18.4218	−0.627815
$862$	6.43787	0.219275
$863$	−53.5899	−1.82422	−0.912110	−	0.409945i	$-0.865548\pi$
$863$	−53.5899	−1.82422	−0.912110	+	0.409945i	$0.865548\pi$
$864$	2.32667	0.0791548
$865$	−82.6069	−2.80872
$866$	4.29604	0.145985
$867$	−19.0868	−0.648222
$868$	−63.8446	−2.16703
$869$	16.0936	0.545938
$870$	−4.92841	−0.167089
$871$	−0.145614	−0.00493393
$872$	−6.24634	−0.211528
$873$	5.34706	0.180971
$874$	4.32411	0.146265
$875$	−25.1042	−0.848678
$876$	−9.67098	−0.326752
$877$	−10.8880	−0.367661	−0.183830	−	0.982958i	$-0.558850\pi$
$877$	−10.8880	−0.367661	−0.183830	+	0.982958i	$0.558850\pi$
$878$	4.05833	0.136962
$879$	−15.4409	−0.520808
$880$	−35.0101	−1.18019
$881$	−16.2636	−0.547934	−0.273967	−	0.961739i	$-0.588336\pi$
$881$	−16.2636	−0.547934	−0.273967	+	0.961739i	$0.588336\pi$
$882$	−3.38062	−0.113831
$883$	−28.5268	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$883$	−28.5268	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$884$	−0.283106	−0.00952190
$885$	−30.5578	−1.02719
$886$	2.57650	0.0865592
$887$	38.3471	1.28757	0.643785	−	0.765207i	$-0.277363\pi$
$887$	38.3471	1.28757	0.643785	+	0.765207i	$0.277363\pi$
$888$	−1.53402	−0.0514785
$889$	−11.3423	−0.380407
$890$	−1.11857	−0.0374947
$891$	−2.74173	−0.0918514
$892$	−47.2548	−1.58221
$893$	−41.3400	−1.38339
$894$	3.86747	0.129348
$895$	−3.49708	−0.116894
$896$	−29.6736	−0.991325
$897$	0.0722950	0.00241386
$898$	−1.71205	−0.0571319
$899$	48.5190	1.61820
$900$	−12.7640	−0.425467
$901$	−15.5283	−0.517323
$902$	−2.05394	−0.0683886
$903$	50.4639	1.67933
$904$	−10.6696	−0.354865
$905$	37.3830	1.24265
$906$	2.61163	0.0867657
$907$	−25.2518	−0.838471	−0.419235	−	0.907878i	$-0.637702\pi$
$907$	−25.2518	−0.838471	−0.419235	+	0.907878i	$0.637702\pi$
$908$	−26.9400	−0.894035
$909$	−1.64433	−0.0545389
$910$	−0.0795296	−0.00263638
$911$	48.3881	1.60317	0.801585	−	0.597881i	$-0.203990\pi$
$911$	48.3881	1.60317	0.801585	+	0.597881i	$0.203990\pi$
$912$	27.1776	0.899942
$913$	−3.19484	−0.105734
$914$	−3.84635	−0.127226
$915$	−4.43190	−0.146514
$916$	−5.69279	−0.188095
$917$	64.6506	2.13495
$918$	−1.19620	−0.0394804
$919$	15.7866	0.520752	0.260376	−	0.965507i	$-0.416153\pi$
$919$	15.7866	0.520752	0.260376	+	0.965507i	$0.416153\pi$
$920$	8.04613	0.265273
$921$	18.7065	0.616401
$922$	−4.69851	−0.154737
$923$	0.266841	0.00878316
$924$	26.3183	0.865808
$925$	12.6655	0.416440
$926$	−6.31623	−0.207564
$927$	−1.00000	−0.0328443
$928$	16.9728	0.557160
$929$	−16.8237	−0.551967	−0.275984	−	0.961162i	$-0.589004\pi$
$929$	−16.8237	−0.551967	−0.275984	+	0.961162i	$0.589004\pi$
$930$	−4.49344	−0.147346
$931$	−122.594	−4.01785
$932$	−44.4250	−1.45519
$933$	−15.9999	−0.523814
$934$	8.11443	0.265512
$935$	55.8801	1.82747
$936$	−0.0189585	−0.000619679 0
$937$	12.7017	0.414947	0.207473	−	0.978241i	$-0.433476\pi$
$937$	12.7017	0.414947	0.207473	+	0.978241i	$0.433476\pi$
$938$	5.90594	0.192836
$939$	−0.0144940	−0.000472993 0
$940$	−38.0768	−1.24193
$941$	−23.4678	−0.765030	−0.382515	−	0.923949i	$-0.624942\pi$
$941$	−23.4678	−0.765030	−0.382515	+	0.923949i	$0.624942\pi$
$942$	−2.25525	−0.0734799
$943$	−11.3136	−0.368421
$944$	33.8980	1.10329
$945$	16.6134	0.540433
$946$	5.62645	0.182932
$947$	−18.2031	−0.591523	−0.295761	−	0.955262i	$-0.595573\pi$
$947$	−18.2031	−0.591523	−0.295761	+	0.955262i	$0.595573\pi$
$948$	−11.5070	−0.373730
$949$	0.118599	0.00384987
$950$	9.36232	0.303754
$951$	−7.49521	−0.243049
$952$	23.1972	0.751827
$953$	1.26598	0.0410090	0.0205045	−	0.999790i	$-0.493473\pi$
$953$	1.26598	0.0410090	0.0205045	+	0.999790i	$0.493473\pi$
$954$	−0.514729	−0.0166650
$955$	−13.3187	−0.430983
$956$	24.6548	0.797393
$957$	−20.0007	−0.646530
$958$	1.23177	0.0397965
$959$	89.0110	2.87431
$960$	23.9668	0.773526
$961$	13.2367	0.426991
$962$	0.00931195	0.000300229 0
$963$	1.49566	0.0481968
$964$	6.45457	0.207888
$965$	13.1009	0.421734
$966$	−2.93221	−0.0943423
$967$	44.6065	1.43445	0.717225	−	0.696842i	$-0.245412\pi$
$967$	44.6065	1.43445	0.717225	+	0.696842i	$0.245412\pi$
$968$	−2.74666	−0.0882811
$969$	−43.3786	−1.39352
$970$	−3.61246	−0.115989
$971$	15.1277	0.485472	0.242736	−	0.970092i	$-0.421955\pi$
$971$	15.1277	0.485472	0.242736	+	0.970092i	$0.421955\pi$
$972$	1.96035	0.0628782
$973$	0.240197	0.00770038
$974$	−0.406688	−0.0130311
$975$	0.156529	0.00501295
$976$	4.91635	0.157369
$977$	−15.7114	−0.502652	−0.251326	−	0.967902i	$-0.580867\pi$
$977$	−15.7114	−0.502652	−0.251326	+	0.967902i	$0.580867\pi$
$978$	4.94059	0.157983
$979$	−4.53944	−0.145081
$980$	−112.917	−3.60700
$981$	−7.92070	−0.252888
$982$	7.81219	0.249297
$983$	32.4385	1.03463	0.517313	−	0.855796i	$-0.326932\pi$
$983$	32.4385	1.03463	0.517313	+	0.855796i	$0.326932\pi$
$984$	2.96685	0.0945798
$985$	−55.2143	−1.75927
$986$	−8.72616	−0.277897
$987$	28.0330	0.892299
$988$	−0.340311	−0.0108267
$989$	30.9918	0.985483
$990$	1.85230	0.0588700
$991$	10.8935	0.346043	0.173021	−	0.984918i	$-0.444647\pi$
$991$	10.8935	0.346043	0.173021	+	0.984918i	$0.444647\pi$
$992$	15.4748	0.491326
$993$	25.8309	0.819720
$994$	−10.8228	−0.343277
$995$	−33.1964	−1.05240
$996$	2.28433	0.0723816
$997$	−5.83725	−0.184868	−0.0924338	−	0.995719i	$-0.529465\pi$
$997$	−5.83725	−0.184868	−0.0924338	+	0.995719i	$0.529465\pi$
$998$	−0.379228	−0.0120042
$999$	−1.94522	−0.0615442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.a.c.1.3	✓	5
3.2	odd	2		927.2.a.e.1.3		5
4.3	odd	2		4944.2.a.bb.1.5		5
5.4	even	2		7725.2.a.t.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.c.1.3	✓	5	1.1	even	1	trivial
927.2.a.e.1.3		5	3.2	odd	2
4944.2.a.bb.1.5		5	4.3	odd	2
7725.2.a.t.1.3		5	5.4	even	2

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:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.a (trivial)

\(f(q)\)	\(=\)	\(q-0.199126 q^{2} -1.00000 q^{3} -1.96035 q^{4} +3.39280 q^{5} +0.199126 q^{6} -4.89666 q^{7} +0.788610 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.199126 q^{2} -1.00000 q^{3} -1.96035 q^{4} +3.39280 q^{5} +0.199126 q^{6} -4.89666 q^{7} +0.788610 q^{8} +1.00000 q^{9} -0.675596 q^{10} -2.74173 q^{11} +1.96035 q^{12} -0.0240404 q^{13} +0.975054 q^{14} -3.39280 q^{15} +3.76366 q^{16} -6.00723 q^{17} -0.199126 q^{18} -7.22106 q^{19} -6.65107 q^{20} +4.89666 q^{21} +0.545951 q^{22} +3.00723 q^{23} -0.788610 q^{24} +6.51109 q^{25} +0.00478708 q^{26} -1.00000 q^{27} +9.59915 q^{28} -7.29491 q^{29} +0.675596 q^{30} -6.65107 q^{31} -2.32667 q^{32} +2.74173 q^{33} +1.19620 q^{34} -16.6134 q^{35} -1.96035 q^{36} +1.94522 q^{37} +1.43790 q^{38} +0.0240404 q^{39} +2.67560 q^{40} -3.76213 q^{41} -0.975054 q^{42} +10.3058 q^{43} +5.37475 q^{44} +3.39280 q^{45} -0.598819 q^{46} +5.72492 q^{47} -3.76366 q^{48} +16.9772 q^{49} -1.29653 q^{50} +6.00723 q^{51} +0.0471276 q^{52} +2.58494 q^{53} +0.199126 q^{54} -9.30214 q^{55} -3.86155 q^{56} +7.22106 q^{57} +1.45261 q^{58} +9.00666 q^{59} +6.65107 q^{60} +1.30627 q^{61} +1.32440 q^{62} -4.89666 q^{63} -7.06403 q^{64} -0.0815643 q^{65} -0.545951 q^{66} +6.05704 q^{67} +11.7763 q^{68} -3.00723 q^{69} +3.30816 q^{70} -11.0997 q^{71} +0.788610 q^{72} -4.93330 q^{73} -0.387346 q^{74} -6.51109 q^{75} +14.1558 q^{76} +13.4253 q^{77} -0.00478708 q^{78} -5.86987 q^{79} +12.7694 q^{80} +1.00000 q^{81} +0.749139 q^{82} +1.16526 q^{83} -9.59915 q^{84} -20.3813 q^{85} -2.05215 q^{86} +7.29491 q^{87} -2.16216 q^{88} +1.65568 q^{89} -0.675596 q^{90} +0.117718 q^{91} -5.89522 q^{92} +6.65107 q^{93} -1.13998 q^{94} -24.4996 q^{95} +2.32667 q^{96} +5.34706 q^{97} -3.38062 q^{98} -2.74173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} - 12 q^{11} - 2 q^{12} + q^{13} - 8 q^{14} + 5 q^{15} - 4 q^{16} - 10 q^{17} - 2 q^{18} - 16 q^{19} - 13 q^{20} + 2 q^{21} + 4 q^{22} - 5 q^{23} + 6 q^{24} + 12 q^{25} - 10 q^{26} - 5 q^{27} + 7 q^{28} - 16 q^{29} + q^{30} - 13 q^{31} + 11 q^{32} + 12 q^{33} - 2 q^{34} - 22 q^{35} + 2 q^{36} + 4 q^{37} + 21 q^{38} - q^{39} + 11 q^{40} - 20 q^{41} + 8 q^{42} + 7 q^{43} + 2 q^{44} - 5 q^{45} + 8 q^{46} + 8 q^{47} + 4 q^{48} + 23 q^{49} + 13 q^{50} + 10 q^{51} + 31 q^{52} - 8 q^{53} + 2 q^{54} - 6 q^{55} + 5 q^{56} + 16 q^{57} + 20 q^{58} - 19 q^{59} + 13 q^{60} - 19 q^{61} + 9 q^{62} - 2 q^{63} - 16 q^{64} + q^{65} - 4 q^{66} + 11 q^{67} + 15 q^{68} + 5 q^{69} + 44 q^{70} - 10 q^{71} - 6 q^{72} + 20 q^{73} + 44 q^{74} - 12 q^{75} - 8 q^{76} + 8 q^{77} + 10 q^{78} - 14 q^{79} + 45 q^{80} + 5 q^{81} - 3 q^{82} - 11 q^{83} - 7 q^{84} + 12 q^{85} - 11 q^{86} + 16 q^{87} + 30 q^{88} + 22 q^{89} - q^{90} - 42 q^{91} - 21 q^{92} + 13 q^{93} - 6 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} - 11 q^{98} - 12 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9 - q^10 - 12 * q^11 - 2 * q^12 + q^13 - 8 * q^14 + 5 * q^15 - 4 * q^16 - 10 * q^17 - 2 * q^18 - 16 * q^19 - 13 * q^20 + 2 * q^21 + 4 * q^22 - 5 * q^23 + 6 * q^24 + 12 * q^25 - 10 * q^26 - 5 * q^27 + 7 * q^28 - 16 * q^29 + q^30 - 13 * q^31 + 11 * q^32 + 12 * q^33 - 2 * q^34 - 22 * q^35 + 2 * q^36 + 4 * q^37 + 21 * q^38 - q^39 + 11 * q^40 - 20 * q^41 + 8 * q^42 + 7 * q^43 + 2 * q^44 - 5 * q^45 + 8 * q^46 + 8 * q^47 + 4 * q^48 + 23 * q^49 + 13 * q^50 + 10 * q^51 + 31 * q^52 - 8 * q^53 + 2 * q^54 - 6 * q^55 + 5 * q^56 + 16 * q^57 + 20 * q^58 - 19 * q^59 + 13 * q^60 - 19 * q^61 + 9 * q^62 - 2 * q^63 - 16 * q^64 + q^65 - 4 * q^66 + 11 * q^67 + 15 * q^68 + 5 * q^69 + 44 * q^70 - 10 * q^71 - 6 * q^72 + 20 * q^73 + 44 * q^74 - 12 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 - 14 * q^79 + 45 * q^80 + 5 * q^81 - 3 * q^82 - 11 * q^83 - 7 * q^84 + 12 * q^85 - 11 * q^86 + 16 * q^87 + 30 * q^88 + 22 * q^89 - q^90 - 42 * q^91 - 21 * q^92 + 13 * q^93 - 6 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 - 11 * q^98 - 12 * q^99

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