LMFDB - Embedded newform 8034.2.a.w.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
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} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} + \cdots + 768 $ x^12 - 6x^11 - 8x^10 + 94x^9 - 7x^8 - 580x^7 + 180x^6 + 1787x^5 - 308x^4 - 2790x^3 - 352x^2 + 1768*x + 768
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.61063$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.40749 q^{5} -1.00000 q^{6} -0.998879 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.40749 q^{5} -1.00000 q^{6} -0.998879 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.40749 q^{10} +4.83053 q^{11} +1.00000 q^{12} +1.00000 q^{13} +0.998879 q^{14} +1.40749 q^{15} +1.00000 q^{16} -2.95365 q^{17} -1.00000 q^{18} +1.88860 q^{19} +1.40749 q^{20} -0.998879 q^{21} -4.83053 q^{22} -2.84489 q^{23} -1.00000 q^{24} -3.01896 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.998879 q^{28} -5.72981 q^{29} -1.40749 q^{30} -10.1479 q^{31} -1.00000 q^{32} +4.83053 q^{33} +2.95365 q^{34} -1.40592 q^{35} +1.00000 q^{36} -4.75750 q^{37} -1.88860 q^{38} +1.00000 q^{39} -1.40749 q^{40} -4.77235 q^{41} +0.998879 q^{42} +9.75344 q^{43} +4.83053 q^{44} +1.40749 q^{45} +2.84489 q^{46} -2.46105 q^{47} +1.00000 q^{48} -6.00224 q^{49} +3.01896 q^{50} -2.95365 q^{51} +1.00000 q^{52} -8.38795 q^{53} -1.00000 q^{54} +6.79894 q^{55} +0.998879 q^{56} +1.88860 q^{57} +5.72981 q^{58} -8.26441 q^{59} +1.40749 q^{60} -6.11324 q^{61} +10.1479 q^{62} -0.998879 q^{63} +1.00000 q^{64} +1.40749 q^{65} -4.83053 q^{66} -4.55266 q^{67} -2.95365 q^{68} -2.84489 q^{69} +1.40592 q^{70} -3.66759 q^{71} -1.00000 q^{72} -13.4965 q^{73} +4.75750 q^{74} -3.01896 q^{75} +1.88860 q^{76} -4.82512 q^{77} -1.00000 q^{78} -11.1098 q^{79} +1.40749 q^{80} +1.00000 q^{81} +4.77235 q^{82} +1.76274 q^{83} -0.998879 q^{84} -4.15724 q^{85} -9.75344 q^{86} -5.72981 q^{87} -4.83053 q^{88} -0.860992 q^{89} -1.40749 q^{90} -0.998879 q^{91} -2.84489 q^{92} -10.1479 q^{93} +2.46105 q^{94} +2.65819 q^{95} -1.00000 q^{96} -5.09415 q^{97} +6.00224 q^{98} +4.83053 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9 + 4 * q^10 - 9 * q^11 + 12 * q^12 + 12 * q^13 - 4 * q^15 + 12 * q^16 - 20 * q^17 - 12 * q^18 + 4 * q^19 - 4 * q^20 + 9 * q^22 - 30 * q^23 - 12 * q^24 + 14 * q^25 - 12 * q^26 + 12 * q^27 - 29 * q^29 + 4 * q^30 + 6 * q^31 - 12 * q^32 - 9 * q^33 + 20 * q^34 - 22 * q^35 + 12 * q^36 + 7 * q^37 - 4 * q^38 + 12 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 9 * q^44 - 4 * q^45 + 30 * q^46 - 16 * q^47 + 12 * q^48 + 10 * q^49 - 14 * q^50 - 20 * q^51 + 12 * q^52 - 9 * q^53 - 12 * q^54 - 20 * q^55 + 4 * q^57 + 29 * q^58 - 29 * q^59 - 4 * q^60 - 26 * q^61 - 6 * q^62 + 12 * q^64 - 4 * q^65 + 9 * q^66 + 12 * q^67 - 20 * q^68 - 30 * q^69 + 22 * q^70 - 35 * q^71 - 12 * q^72 + 18 * q^73 - 7 * q^74 + 14 * q^75 + 4 * q^76 - 25 * q^77 - 12 * q^78 - 37 * q^79 - 4 * q^80 + 12 * q^81 + 8 * q^82 - 24 * q^83 - 17 * q^85 + 8 * q^86 - 29 * q^87 + 9 * q^88 + 15 * q^89 + 4 * q^90 - 30 * q^92 + 6 * q^93 + 16 * q^94 - 54 * q^95 - 12 * q^96 - 11 * q^97 - 10 * q^98 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.40749	0.629450	0.314725	−	0.949183i	$-0.398088\pi$
$5$	1.40749	0.629450	0.314725	+	0.949183i	$0.398088\pi$
$6$	−1.00000	−0.408248
$7$	−0.998879	−0.377541	−0.188770	−	0.982021i	$-0.560450\pi$
$7$	−0.998879	−0.377541	−0.188770	+	0.982021i	$0.560450\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.40749	−0.445089
$11$	4.83053	1.45646	0.728230	−	0.685333i	$-0.240343\pi$
$11$	4.83053	1.45646	0.728230	+	0.685333i	$0.240343\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.998879	0.266962
$15$	1.40749	0.363413
$16$	1.00000	0.250000
$17$	−2.95365	−0.716365	−0.358183	−	0.933652i	$-0.616603\pi$
$17$	−2.95365	−0.716365	−0.358183	+	0.933652i	$0.616603\pi$
$18$	−1.00000	−0.235702
$19$	1.88860	0.433274	0.216637	−	0.976252i	$-0.430491\pi$
$19$	1.88860	0.433274	0.216637	+	0.976252i	$0.430491\pi$
$20$	1.40749	0.314725
$21$	−0.998879	−0.217973
$22$	−4.83053	−1.02987
$23$	−2.84489	−0.593200	−0.296600	−	0.955002i	$-0.595853\pi$
$23$	−2.84489	−0.593200	−0.296600	+	0.955002i	$0.595853\pi$
$24$	−1.00000	−0.204124
$25$	−3.01896	−0.603792
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−0.998879	−0.188770
$29$	−5.72981	−1.06400	−0.532000	−	0.846744i	$-0.678559\pi$
$29$	−5.72981	−1.06400	−0.532000	+	0.846744i	$0.678559\pi$
$30$	−1.40749	−0.256972
$31$	−10.1479	−1.82261	−0.911305	−	0.411731i	$-0.864924\pi$
$31$	−10.1479	−1.82261	−0.911305	+	0.411731i	$0.864924\pi$
$32$	−1.00000	−0.176777
$33$	4.83053	0.840888
$34$	2.95365	0.506547
$35$	−1.40592	−0.237643
$36$	1.00000	0.166667
$37$	−4.75750	−0.782128	−0.391064	−	0.920363i	$-0.627893\pi$
$37$	−4.75750	−0.782128	−0.391064	+	0.920363i	$0.627893\pi$
$38$	−1.88860	−0.306371
$39$	1.00000	0.160128
$40$	−1.40749	−0.222544
$41$	−4.77235	−0.745315	−0.372658	−	0.927969i	$-0.621553\pi$
$41$	−4.77235	−0.745315	−0.372658	+	0.927969i	$0.621553\pi$
$42$	0.998879	0.154130
$43$	9.75344	1.48739	0.743693	−	0.668521i	$-0.233072\pi$
$43$	9.75344	1.48739	0.743693	+	0.668521i	$0.233072\pi$
$44$	4.83053	0.728230
$45$	1.40749	0.209817
$46$	2.84489	0.419456
$47$	−2.46105	−0.358981	−0.179491	−	0.983760i	$-0.557445\pi$
$47$	−2.46105	−0.358981	−0.179491	+	0.983760i	$0.557445\pi$
$48$	1.00000	0.144338
$49$	−6.00224	−0.857463
$50$	3.01896	0.426946
$51$	−2.95365	−0.413594
$52$	1.00000	0.138675
$53$	−8.38795	−1.15217	−0.576087	−	0.817389i	$-0.695421\pi$
$53$	−8.38795	−1.15217	−0.576087	+	0.817389i	$0.695421\pi$
$54$	−1.00000	−0.136083
$55$	6.79894	0.916769
$56$	0.998879	0.133481
$57$	1.88860	0.250151
$58$	5.72981	0.752362
$59$	−8.26441	−1.07593	−0.537967	−	0.842966i	$-0.680808\pi$
$59$	−8.26441	−1.07593	−0.537967	+	0.842966i	$0.680808\pi$
$60$	1.40749	0.181707
$61$	−6.11324	−0.782720	−0.391360	−	0.920238i	$-0.627995\pi$
$61$	−6.11324	−0.782720	−0.391360	+	0.920238i	$0.627995\pi$
$62$	10.1479	1.28878
$63$	−0.998879	−0.125847
$64$	1.00000	0.125000
$65$	1.40749	0.174578
$66$	−4.83053	−0.594597
$67$	−4.55266	−0.556196	−0.278098	−	0.960553i	$-0.589704\pi$
$67$	−4.55266	−0.556196	−0.278098	+	0.960553i	$0.589704\pi$
$68$	−2.95365	−0.358183
$69$	−2.84489	−0.342484
$70$	1.40592	0.168039
$71$	−3.66759	−0.435262	−0.217631	−	0.976031i	$-0.569833\pi$
$71$	−3.66759	−0.435262	−0.217631	+	0.976031i	$0.569833\pi$
$72$	−1.00000	−0.117851
$73$	−13.4965	−1.57964	−0.789821	−	0.613337i	$-0.789827\pi$
$73$	−13.4965	−1.57964	−0.789821	+	0.613337i	$0.789827\pi$
$74$	4.75750	0.553048
$75$	−3.01896	−0.348600
$76$	1.88860	0.216637
$77$	−4.82512	−0.549873
$78$	−1.00000	−0.113228
$79$	−11.1098	−1.24995	−0.624977	−	0.780643i	$-0.714892\pi$
$79$	−11.1098	−1.24995	−0.624977	+	0.780643i	$0.714892\pi$
$80$	1.40749	0.157363
$81$	1.00000	0.111111
$82$	4.77235	0.527018
$83$	1.76274	0.193486	0.0967429	−	0.995309i	$-0.469158\pi$
$83$	1.76274	0.193486	0.0967429	+	0.995309i	$0.469158\pi$
$84$	−0.998879	−0.108987
$85$	−4.15724	−0.450916
$86$	−9.75344	−1.05174
$87$	−5.72981	−0.614301
$88$	−4.83053	−0.514936
$89$	−0.860992	−0.0912650	−0.0456325	−	0.998958i	$-0.514530\pi$
$89$	−0.860992	−0.0912650	−0.0456325	+	0.998958i	$0.514530\pi$
$90$	−1.40749	−0.148363
$91$	−0.998879	−0.104711
$92$	−2.84489	−0.296600
$93$	−10.1479	−1.05228
$94$	2.46105	0.253838
$95$	2.65819	0.272725
$96$	−1.00000	−0.102062
$97$	−5.09415	−0.517232	−0.258616	−	0.965980i	$-0.583266\pi$
$97$	−5.09415	−0.517232	−0.258616	+	0.965980i	$0.583266\pi$
$98$	6.00224	0.606318
$99$	4.83053	0.485487
$100$	−3.01896	−0.301896
$101$	−5.83551	−0.580655	−0.290327	−	0.956927i	$-0.593764\pi$
$101$	−5.83551	−0.580655	−0.290327	+	0.956927i	$0.593764\pi$
$102$	2.95365	0.292455
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−1.40592	−0.137203
$106$	8.38795	0.814710
$107$	13.0054	1.25728	0.628642	−	0.777695i	$-0.283611\pi$
$107$	13.0054	1.25728	0.628642	+	0.777695i	$0.283611\pi$
$108$	1.00000	0.0962250
$109$	−1.24549	−0.119297	−0.0596484	−	0.998219i	$-0.518998\pi$
$109$	−1.24549	−0.119297	−0.0596484	+	0.998219i	$0.518998\pi$
$110$	−6.79894	−0.648254
$111$	−4.75750	−0.451562
$112$	−0.998879	−0.0943852
$113$	9.96193	0.937139	0.468570	−	0.883427i	$-0.344770\pi$
$113$	9.96193	0.937139	0.468570	+	0.883427i	$0.344770\pi$
$114$	−1.88860	−0.176883
$115$	−4.00416	−0.373390
$116$	−5.72981	−0.532000
$117$	1.00000	0.0924500
$118$	8.26441	0.760801
$119$	2.95034	0.270457
$120$	−1.40749	−0.128486
$121$	12.3340	1.12128
$122$	6.11324	0.553466
$123$	−4.77235	−0.430308
$124$	−10.1479	−0.911305
$125$	−11.2866	−1.00951
$126$	0.998879	0.0889872
$127$	3.66487	0.325204	0.162602	−	0.986692i	$-0.448011\pi$
$127$	3.66487	0.325204	0.162602	+	0.986692i	$0.448011\pi$
$128$	−1.00000	−0.0883883
$129$	9.75344	0.858743
$130$	−1.40749	−0.123445
$131$	11.9908	1.04764	0.523822	−	0.851827i	$-0.324506\pi$
$131$	11.9908	1.04764	0.523822	+	0.851827i	$0.324506\pi$
$132$	4.83053	0.420444
$133$	−1.88648	−0.163579
$134$	4.55266	0.393290
$135$	1.40749	0.121138
$136$	2.95365	0.253273
$137$	11.6193	0.992702	0.496351	−	0.868122i	$-0.334673\pi$
$137$	11.6193	0.992702	0.496351	+	0.868122i	$0.334673\pi$
$138$	2.84489	0.242173
$139$	10.1782	0.863301	0.431650	−	0.902041i	$-0.357931\pi$
$139$	10.1782	0.863301	0.431650	+	0.902041i	$0.357931\pi$
$140$	−1.40592	−0.118822
$141$	−2.46105	−0.207258
$142$	3.66759	0.307777
$143$	4.83053	0.403949
$144$	1.00000	0.0833333
$145$	−8.06468	−0.669735
$146$	13.4965	1.11698
$147$	−6.00224	−0.495056
$148$	−4.75750	−0.391064
$149$	17.0885	1.39995	0.699973	−	0.714169i	$-0.253195\pi$
$149$	17.0885	1.39995	0.699973	+	0.714169i	$0.253195\pi$
$150$	3.01896	0.246497
$151$	13.2289	1.07655	0.538277	−	0.842768i	$-0.319076\pi$
$151$	13.2289	1.07655	0.538277	+	0.842768i	$0.319076\pi$
$152$	−1.88860	−0.153186
$153$	−2.95365	−0.238788
$154$	4.82512	0.388819
$155$	−14.2831	−1.14724
$156$	1.00000	0.0800641
$157$	9.61127	0.767062	0.383531	−	0.923528i	$-0.374708\pi$
$157$	9.61127	0.767062	0.383531	+	0.923528i	$0.374708\pi$
$158$	11.1098	0.883852
$159$	−8.38795	−0.665208
$160$	−1.40749	−0.111272
$161$	2.84170	0.223957
$162$	−1.00000	−0.0785674
$163$	−7.98844	−0.625703	−0.312851	−	0.949802i	$-0.601284\pi$
$163$	−7.98844	−0.625703	−0.312851	+	0.949802i	$0.601284\pi$
$164$	−4.77235	−0.372658
$165$	6.79894	0.529297
$166$	−1.76274	−0.136815
$167$	3.41720	0.264431	0.132215	−	0.991221i	$-0.457791\pi$
$167$	3.41720	0.264431	0.132215	+	0.991221i	$0.457791\pi$
$168$	0.998879	0.0770652
$169$	1.00000	0.0769231
$170$	4.15724	0.318846
$171$	1.88860	0.144425
$172$	9.75344	0.743693
$173$	−24.4518	−1.85904	−0.929519	−	0.368773i	$-0.879778\pi$
$173$	−24.4518	−1.85904	−0.929519	+	0.368773i	$0.879778\pi$
$174$	5.72981	0.434376
$175$	3.01558	0.227956
$176$	4.83053	0.364115
$177$	−8.26441	−0.621191
$178$	0.860992	0.0645341
$179$	−11.6219	−0.868664	−0.434332	−	0.900753i	$-0.643015\pi$
$179$	−11.6219	−0.868664	−0.434332	+	0.900753i	$0.643015\pi$
$180$	1.40749	0.104908
$181$	16.1688	1.20181	0.600907	−	0.799319i	$-0.294806\pi$
$181$	16.1688	1.20181	0.600907	+	0.799319i	$0.294806\pi$
$182$	0.998879	0.0740418
$183$	−6.11324	−0.451903
$184$	2.84489	0.209728
$185$	−6.69615	−0.492311
$186$	10.1479	0.744078
$187$	−14.2677	−1.04336
$188$	−2.46105	−0.179491
$189$	−0.998879	−0.0726578
$190$	−2.65819	−0.192845
$191$	−15.3479	−1.11054	−0.555269	−	0.831671i	$-0.687385\pi$
$191$	−15.3479	−1.11054	−0.555269	+	0.831671i	$0.687385\pi$
$192$	1.00000	0.0721688
$193$	5.62711	0.405048	0.202524	−	0.979277i	$-0.435086\pi$
$193$	5.62711	0.405048	0.202524	+	0.979277i	$0.435086\pi$
$194$	5.09415	0.365739
$195$	1.40749	0.100793
$196$	−6.00224	−0.428731
$197$	−8.91051	−0.634848	−0.317424	−	0.948284i	$-0.602818\pi$
$197$	−8.91051	−0.634848	−0.317424	+	0.948284i	$0.602818\pi$
$198$	−4.83053	−0.343291
$199$	−21.5100	−1.52480	−0.762401	−	0.647105i	$-0.775980\pi$
$199$	−21.5100	−1.52480	−0.762401	+	0.647105i	$0.775980\pi$
$200$	3.01896	0.213473
$201$	−4.55266	−0.321120
$202$	5.83551	0.410585
$203$	5.72339	0.401703
$204$	−2.95365	−0.206797
$205$	−6.71705	−0.469139
$206$	1.00000	0.0696733
$207$	−2.84489	−0.197733
$208$	1.00000	0.0693375
$209$	9.12294	0.631047
$210$	1.40592	0.0970174
$211$	25.6510	1.76589	0.882945	−	0.469476i	$-0.155557\pi$
$211$	25.6510	1.76589	0.882945	+	0.469476i	$0.155557\pi$
$212$	−8.38795	−0.576087
$213$	−3.66759	−0.251299
$214$	−13.0054	−0.889034
$215$	13.7279	0.936235
$216$	−1.00000	−0.0680414
$217$	10.1365	0.688110
$218$	1.24549	0.0843555
$219$	−13.4965	−0.912007
$220$	6.79894	0.458385
$221$	−2.95365	−0.198684
$222$	4.75750	0.319302
$223$	12.8137	0.858069	0.429034	−	0.903288i	$-0.358854\pi$
$223$	12.8137	0.858069	0.429034	+	0.903288i	$0.358854\pi$
$224$	0.998879	0.0667404
$225$	−3.01896	−0.201264
$226$	−9.96193	−0.662657
$227$	−12.3807	−0.821736	−0.410868	−	0.911695i	$-0.634774\pi$
$227$	−12.3807	−0.821736	−0.410868	+	0.911695i	$0.634774\pi$
$228$	1.88860	0.125076
$229$	12.8890	0.851726	0.425863	−	0.904788i	$-0.359971\pi$
$229$	12.8890	0.851726	0.425863	+	0.904788i	$0.359971\pi$
$230$	4.00416	0.264026
$231$	−4.82512	−0.317469
$232$	5.72981	0.376181
$233$	−6.27793	−0.411281	−0.205641	−	0.978628i	$-0.565928\pi$
$233$	−6.27793	−0.411281	−0.205641	+	0.978628i	$0.565928\pi$
$234$	−1.00000	−0.0653720
$235$	−3.46391	−0.225961
$236$	−8.26441	−0.537967
$237$	−11.1098	−0.721662
$238$	−2.95034	−0.191242
$239$	−20.3261	−1.31478	−0.657392	−	0.753549i	$-0.728341\pi$
$239$	−20.3261	−1.31478	−0.657392	+	0.753549i	$0.728341\pi$
$240$	1.40749	0.0908533
$241$	−5.81756	−0.374742	−0.187371	−	0.982289i	$-0.559997\pi$
$241$	−5.81756	−0.374742	−0.187371	+	0.982289i	$0.559997\pi$
$242$	−12.3340	−0.792862
$243$	1.00000	0.0641500
$244$	−6.11324	−0.391360
$245$	−8.44812	−0.539730
$246$	4.77235	0.304274
$247$	1.88860	0.120169
$248$	10.1479	0.644390
$249$	1.76274	0.111709
$250$	11.2866	0.713830
$251$	10.3172	0.651215	0.325607	−	0.945505i	$-0.394431\pi$
$251$	10.3172	0.651215	0.325607	+	0.945505i	$0.394431\pi$
$252$	−0.998879	−0.0629235
$253$	−13.7423	−0.863972
$254$	−3.66487	−0.229954
$255$	−4.15724	−0.260337
$256$	1.00000	0.0625000
$257$	3.22415	0.201117	0.100558	−	0.994931i	$-0.467937\pi$
$257$	3.22415	0.201117	0.100558	+	0.994931i	$0.467937\pi$
$258$	−9.75344	−0.607223
$259$	4.75217	0.295285
$260$	1.40749	0.0872890
$261$	−5.72981	−0.354667
$262$	−11.9908	−0.740797
$263$	−27.8650	−1.71823	−0.859115	−	0.511782i	$-0.828986\pi$
$263$	−27.8650	−1.71823	−0.859115	+	0.511782i	$0.828986\pi$
$264$	−4.83053	−0.297299
$265$	−11.8060	−0.725236
$266$	1.88648	0.115668
$267$	−0.860992	−0.0526918
$268$	−4.55266	−0.278098
$269$	−5.47995	−0.334119	−0.167059	−	0.985947i	$-0.553427\pi$
$269$	−5.47995	−0.334119	−0.167059	+	0.985947i	$0.553427\pi$
$270$	−1.40749	−0.0856573
$271$	25.0590	1.52223	0.761114	−	0.648618i	$-0.224653\pi$
$271$	25.0590	1.52223	0.761114	+	0.648618i	$0.224653\pi$
$272$	−2.95365	−0.179091
$273$	−0.998879	−0.0604549
$274$	−11.6193	−0.701947
$275$	−14.5832	−0.879399
$276$	−2.84489	−0.171242
$277$	4.14155	0.248841	0.124421	−	0.992230i	$-0.460293\pi$
$277$	4.14155	0.248841	0.124421	+	0.992230i	$0.460293\pi$
$278$	−10.1782	−0.610446
$279$	−10.1479	−0.607537
$280$	1.40592	0.0840195
$281$	−2.03599	−0.121457	−0.0607284	−	0.998154i	$-0.519342\pi$
$281$	−2.03599	−0.121457	−0.0607284	+	0.998154i	$0.519342\pi$
$282$	2.46105	0.146553
$283$	10.5521	0.627256	0.313628	−	0.949546i	$-0.398455\pi$
$283$	10.5521	0.627256	0.313628	+	0.949546i	$0.398455\pi$
$284$	−3.66759	−0.217631
$285$	2.65819	0.157458
$286$	−4.83053	−0.285635
$287$	4.76700	0.281387
$288$	−1.00000	−0.0589256
$289$	−8.27596	−0.486821
$290$	8.06468	0.473574
$291$	−5.09415	−0.298624
$292$	−13.4965	−0.789821
$293$	20.2309	1.18190	0.590950	−	0.806708i	$-0.298753\pi$
$293$	20.2309	1.18190	0.590950	+	0.806708i	$0.298753\pi$
$294$	6.00224	0.350058
$295$	−11.6321	−0.677248
$296$	4.75750	0.276524
$297$	4.83053	0.280296
$298$	−17.0885	−0.989912
$299$	−2.84489	−0.164524
$300$	−3.01896	−0.174300
$301$	−9.74251	−0.561549
$302$	−13.2289	−0.761238
$303$	−5.83551	−0.335241
$304$	1.88860	0.108319
$305$	−8.60434	−0.492683
$306$	2.95365	0.168849
$307$	9.28706	0.530041	0.265020	−	0.964243i	$-0.414621\pi$
$307$	9.28706	0.530041	0.265020	+	0.964243i	$0.414621\pi$
$308$	−4.82512	−0.274937
$309$	−1.00000	−0.0568880
$310$	14.2831	0.811223
$311$	18.0200	1.02182	0.510910	−	0.859634i	$-0.329309\pi$
$311$	18.0200	1.02182	0.510910	+	0.859634i	$0.329309\pi$
$312$	−1.00000	−0.0566139
$313$	26.1537	1.47829	0.739146	−	0.673545i	$-0.235229\pi$
$313$	26.1537	1.47829	0.739146	+	0.673545i	$0.235229\pi$
$314$	−9.61127	−0.542395
$315$	−1.40592	−0.0792144
$316$	−11.1098	−0.624977
$317$	7.89534	0.443447	0.221723	−	0.975110i	$-0.428832\pi$
$317$	7.89534	0.443447	0.221723	+	0.975110i	$0.428832\pi$
$318$	8.38795	0.470373
$319$	−27.6780	−1.54967
$320$	1.40749	0.0786813
$321$	13.0054	0.725893
$322$	−2.84170	−0.158362
$323$	−5.57826	−0.310383
$324$	1.00000	0.0555556
$325$	−3.01896	−0.167462
$326$	7.98844	0.442439
$327$	−1.24549	−0.0688760
$328$	4.77235	0.263509
$329$	2.45829	0.135530
$330$	−6.79894	−0.374269
$331$	21.0364	1.15627	0.578134	−	0.815942i	$-0.303781\pi$
$331$	21.0364	1.15627	0.578134	+	0.815942i	$0.303781\pi$
$332$	1.76274	0.0967429
$333$	−4.75750	−0.260709
$334$	−3.41720	−0.186981
$335$	−6.40784	−0.350098
$336$	−0.998879	−0.0544933
$337$	26.1538	1.42469	0.712344	−	0.701831i	$-0.247634\pi$
$337$	26.1538	1.42469	0.712344	+	0.701831i	$0.247634\pi$
$338$	−1.00000	−0.0543928
$339$	9.96193	0.541058
$340$	−4.15724	−0.225458
$341$	−49.0196	−2.65456
$342$	−1.88860	−0.102124
$343$	12.9877	0.701268
$344$	−9.75344	−0.525870
$345$	−4.00416	−0.215577
$346$	24.4518	1.31454
$347$	−20.9493	−1.12462	−0.562308	−	0.826928i	$-0.690087\pi$
$347$	−20.9493	−1.12462	−0.562308	+	0.826928i	$0.690087\pi$
$348$	−5.72981	−0.307150
$349$	33.7856	1.80850	0.904251	−	0.427002i	$-0.140430\pi$
$349$	33.7856	1.80850	0.904251	+	0.427002i	$0.140430\pi$
$350$	−3.01558	−0.161189
$351$	1.00000	0.0533761
$352$	−4.83053	−0.257468
$353$	−8.70939	−0.463554	−0.231777	−	0.972769i	$-0.574454\pi$
$353$	−8.70939	−0.463554	−0.231777	+	0.972769i	$0.574454\pi$
$354$	8.26441	0.439249
$355$	−5.16210	−0.273976
$356$	−0.860992	−0.0456325
$357$	2.95034	0.156148
$358$	11.6219	0.614238
$359$	−26.5369	−1.40056	−0.700281	−	0.713867i	$-0.746942\pi$
$359$	−26.5369	−1.40056	−0.700281	+	0.713867i	$0.746942\pi$
$360$	−1.40749	−0.0741814
$361$	−15.4332	−0.812273
$362$	−16.1688	−0.849811
$363$	12.3340	0.647369
$364$	−0.998879	−0.0523555
$365$	−18.9962	−0.994306
$366$	6.11324	0.319544
$367$	9.83164	0.513207	0.256604	−	0.966517i	$-0.417396\pi$
$367$	9.83164	0.513207	0.256604	+	0.966517i	$0.417396\pi$
$368$	−2.84489	−0.148300
$369$	−4.77235	−0.248438
$370$	6.69615	0.348116
$371$	8.37855	0.434992
$372$	−10.1479	−0.526142
$373$	−36.6010	−1.89513	−0.947565	−	0.319563i	$-0.896464\pi$
$373$	−36.6010	−1.89513	−0.947565	+	0.319563i	$0.896464\pi$
$374$	14.2677	0.737765
$375$	−11.2866	−0.582839
$376$	2.46105	0.126919
$377$	−5.72981	−0.295100
$378$	0.998879	0.0513768
$379$	−23.6372	−1.21416	−0.607080	−	0.794640i	$-0.707659\pi$
$379$	−23.6372	−1.21416	−0.607080	+	0.794640i	$0.707659\pi$
$380$	2.65819	0.136362
$381$	3.66487	0.187757
$382$	15.3479	0.785269
$383$	29.2380	1.49399	0.746996	−	0.664829i	$-0.231496\pi$
$383$	29.2380	1.49399	0.746996	+	0.664829i	$0.231496\pi$
$384$	−1.00000	−0.0510310
$385$	−6.79132	−0.346118
$386$	−5.62711	−0.286412
$387$	9.75344	0.495795
$388$	−5.09415	−0.258616
$389$	9.48255	0.480784	0.240392	−	0.970676i	$-0.422724\pi$
$389$	9.48255	0.480784	0.240392	+	0.970676i	$0.422724\pi$
$390$	−1.40749	−0.0712712
$391$	8.40279	0.424948
$392$	6.00224	0.303159
$393$	11.9908	0.604858
$394$	8.91051	0.448905
$395$	−15.6370	−0.786784
$396$	4.83053	0.242743
$397$	6.03291	0.302783	0.151392	−	0.988474i	$-0.451625\pi$
$397$	6.03291	0.302783	0.151392	+	0.988474i	$0.451625\pi$
$398$	21.5100	1.07820
$399$	−1.88648	−0.0944422
$400$	−3.01896	−0.150948
$401$	−10.9056	−0.544601	−0.272300	−	0.962212i	$-0.587784\pi$
$401$	−10.9056	−0.544601	−0.272300	+	0.962212i	$0.587784\pi$
$402$	4.55266	0.227066
$403$	−10.1479	−0.505501
$404$	−5.83551	−0.290327
$405$	1.40749	0.0699389
$406$	−5.72339	−0.284047
$407$	−22.9812	−1.13914
$408$	2.95365	0.146227
$409$	−20.6537	−1.02126	−0.510631	−	0.859800i	$-0.670588\pi$
$409$	−20.6537	−1.02126	−0.510631	+	0.859800i	$0.670588\pi$
$410$	6.71705	0.331731
$411$	11.6193	0.573137
$412$	−1.00000	−0.0492665
$413$	8.25515	0.406209
$414$	2.84489	0.139819
$415$	2.48104	0.121790
$416$	−1.00000	−0.0490290
$417$	10.1782	0.498427
$418$	−9.12294	−0.446217
$419$	27.0666	1.32229	0.661145	−	0.750258i	$-0.270071\pi$
$419$	27.0666	1.32229	0.661145	+	0.750258i	$0.270071\pi$
$420$	−1.40592	−0.0686017
$421$	−25.4862	−1.24212	−0.621062	−	0.783762i	$-0.713298\pi$
$421$	−25.4862	−1.24212	−0.621062	+	0.783762i	$0.713298\pi$
$422$	−25.6510	−1.24867
$423$	−2.46105	−0.119660
$424$	8.38795	0.407355
$425$	8.91695	0.432536
$426$	3.66759	0.177695
$427$	6.10638	0.295509
$428$	13.0054	0.628642
$429$	4.83053	0.233220
$430$	−13.7279	−0.662018
$431$	30.4313	1.46582	0.732911	−	0.680324i	$-0.238161\pi$
$431$	30.4313	1.46582	0.732911	+	0.680324i	$0.238161\pi$
$432$	1.00000	0.0481125
$433$	37.2374	1.78952	0.894758	−	0.446551i	$-0.147348\pi$
$433$	37.2374	1.78952	0.894758	+	0.446551i	$0.147348\pi$
$434$	−10.1365	−0.486567
$435$	−8.06468	−0.386672
$436$	−1.24549	−0.0596484
$437$	−5.37285	−0.257018
$438$	13.4965	0.644886
$439$	−28.7436	−1.37186	−0.685929	−	0.727668i	$-0.740604\pi$
$439$	−28.7436	−1.37186	−0.685929	+	0.727668i	$0.740604\pi$
$440$	−6.79894	−0.324127
$441$	−6.00224	−0.285821
$442$	2.95365	0.140491
$443$	−9.15518	−0.434976	−0.217488	−	0.976063i	$-0.569786\pi$
$443$	−9.15518	−0.434976	−0.217488	+	0.976063i	$0.569786\pi$
$444$	−4.75750	−0.225781
$445$	−1.21184	−0.0574468
$446$	−12.8137	−0.606746
$447$	17.0885	0.808260
$448$	−0.998879	−0.0471926
$449$	35.7192	1.68569	0.842847	−	0.538154i	$-0.180878\pi$
$449$	35.7192	1.68569	0.842847	+	0.538154i	$0.180878\pi$
$450$	3.01896	0.142315
$451$	−23.0530	−1.08552
$452$	9.96193	0.468570
$453$	13.2289	0.621548
$454$	12.3807	0.581055
$455$	−1.40592	−0.0659104
$456$	−1.88860	−0.0884417
$457$	22.8348	1.06817	0.534083	−	0.845432i	$-0.320657\pi$
$457$	22.8348	1.06817	0.534083	+	0.845432i	$0.320657\pi$
$458$	−12.8890	−0.602261
$459$	−2.95365	−0.137865
$460$	−4.00416	−0.186695
$461$	−24.7047	−1.15061	−0.575305	−	0.817939i	$-0.695117\pi$
$461$	−24.7047	−1.15061	−0.575305	+	0.817939i	$0.695117\pi$
$462$	4.82512	0.224485
$463$	9.59246	0.445799	0.222900	−	0.974841i	$-0.428448\pi$
$463$	9.59246	0.445799	0.222900	+	0.974841i	$0.428448\pi$
$464$	−5.72981	−0.266000
$465$	−14.2831	−0.662361
$466$	6.27793	0.290820
$467$	−20.3961	−0.943817	−0.471908	−	0.881648i	$-0.656435\pi$
$467$	−20.3961	−0.943817	−0.471908	+	0.881648i	$0.656435\pi$
$468$	1.00000	0.0462250
$469$	4.54756	0.209987
$470$	3.46391	0.159778
$471$	9.61127	0.442864
$472$	8.26441	0.380400
$473$	47.1143	2.16632
$474$	11.1098	0.510292
$475$	−5.70161	−0.261608
$476$	2.95034	0.135229
$477$	−8.38795	−0.384058
$478$	20.3261	0.929693
$479$	−6.02670	−0.275367	−0.137684	−	0.990476i	$-0.543966\pi$
$479$	−6.02670	−0.275367	−0.137684	+	0.990476i	$0.543966\pi$
$480$	−1.40749	−0.0642430
$481$	−4.75750	−0.216923
$482$	5.81756	0.264983
$483$	2.84170	0.129302
$484$	12.3340	0.560638
$485$	−7.16998	−0.325572
$486$	−1.00000	−0.0453609
$487$	23.6141	1.07006	0.535029	−	0.844834i	$-0.320301\pi$
$487$	23.6141	1.07006	0.535029	+	0.844834i	$0.320301\pi$
$488$	6.11324	0.276733
$489$	−7.98844	−0.361250
$490$	8.44812	0.381647
$491$	−18.4702	−0.833550	−0.416775	−	0.909010i	$-0.636840\pi$
$491$	−18.4702	−0.833550	−0.416775	+	0.909010i	$0.636840\pi$
$492$	−4.77235	−0.215154
$493$	16.9239	0.762212
$494$	−1.88860	−0.0849721
$495$	6.79894	0.305590
$496$	−10.1479	−0.455653
$497$	3.66347	0.164329
$498$	−1.76274	−0.0789902
$499$	24.8331	1.11168	0.555841	−	0.831288i	$-0.312396\pi$
$499$	24.8331	1.11168	0.555841	+	0.831288i	$0.312396\pi$
$500$	−11.2866	−0.504754
$501$	3.41720	0.152669
$502$	−10.3172	−0.460478
$503$	−21.4279	−0.955423	−0.477712	−	0.878517i	$-0.658534\pi$
$503$	−21.4279	−0.955423	−0.477712	+	0.878517i	$0.658534\pi$
$504$	0.998879	0.0444936
$505$	−8.21344	−0.365493
$506$	13.7423	0.610920
$507$	1.00000	0.0444116
$508$	3.66487	0.162602
$509$	−44.6685	−1.97989	−0.989947	−	0.141437i	$-0.954828\pi$
$509$	−44.6685	−1.97989	−0.989947	+	0.141437i	$0.954828\pi$
$510$	4.15724	0.184086
$511$	13.4813	0.596379
$512$	−1.00000	−0.0441942
$513$	1.88860	0.0833837
$514$	−3.22415	−0.142211
$515$	−1.40749	−0.0620216
$516$	9.75344	0.429371
$517$	−11.8882	−0.522841
$518$	−4.75217	−0.208798
$519$	−24.4518	−1.07332
$520$	−1.40749	−0.0617227
$521$	23.6255	1.03505	0.517525	−	0.855668i	$-0.326853\pi$
$521$	23.6255	1.03505	0.517525	+	0.855668i	$0.326853\pi$
$522$	5.72981	0.250787
$523$	−14.6631	−0.641175	−0.320587	−	0.947219i	$-0.603880\pi$
$523$	−14.6631	−0.641175	−0.320587	+	0.947219i	$0.603880\pi$
$524$	11.9908	0.523822
$525$	3.01558	0.131611
$526$	27.8650	1.21497
$527$	29.9732	1.30565
$528$	4.83053	0.210222
$529$	−14.9066	−0.648114
$530$	11.8060	0.512819
$531$	−8.26441	−0.358645
$532$	−1.88648	−0.0817894
$533$	−4.77235	−0.206713
$534$	0.860992	0.0372588
$535$	18.3051	0.791398
$536$	4.55266	0.196645
$537$	−11.6219	−0.501523
$538$	5.47995	0.236258
$539$	−28.9940	−1.24886
$540$	1.40749	0.0605689
$541$	−14.5293	−0.624663	−0.312331	−	0.949973i	$-0.601110\pi$
$541$	−14.5293	−0.624663	−0.312331	+	0.949973i	$0.601110\pi$
$542$	−25.0590	−1.07638
$543$	16.1688	0.693868
$544$	2.95365	0.126637
$545$	−1.75303	−0.0750914
$546$	0.998879	0.0427481
$547$	7.36417	0.314869	0.157435	−	0.987529i	$-0.449678\pi$
$547$	7.36417	0.314869	0.157435	+	0.987529i	$0.449678\pi$
$548$	11.6193	0.496351
$549$	−6.11324	−0.260907
$550$	14.5832	0.621829
$551$	−10.8213	−0.461004
$552$	2.84489	0.121086
$553$	11.0974	0.471909
$554$	−4.14155	−0.175957
$555$	−6.69615	−0.284236
$556$	10.1782	0.431650
$557$	31.0943	1.31751	0.658753	−	0.752359i	$-0.271084\pi$
$557$	31.0943	1.31751	0.658753	+	0.752359i	$0.271084\pi$
$558$	10.1479	0.429593
$559$	9.75344	0.412527
$560$	−1.40592	−0.0594108
$561$	−14.2677	−0.602383
$562$	2.03599	0.0858830
$563$	−14.1457	−0.596171	−0.298086	−	0.954539i	$-0.596348\pi$
$563$	−14.1457	−0.596171	−0.298086	+	0.954539i	$0.596348\pi$
$564$	−2.46105	−0.103629
$565$	14.0213	0.589883
$566$	−10.5521	−0.443537
$567$	−0.998879	−0.0419490
$568$	3.66759	0.153888
$569$	−29.1790	−1.22325	−0.611623	−	0.791149i	$-0.709483\pi$
$569$	−29.1790	−1.22325	−0.611623	+	0.791149i	$0.709483\pi$
$570$	−2.65819	−0.111339
$571$	−28.3905	−1.18810	−0.594052	−	0.804426i	$-0.702473\pi$
$571$	−28.3905	−1.18810	−0.594052	+	0.804426i	$0.702473\pi$
$572$	4.83053	0.201975
$573$	−15.3479	−0.641169
$574$	−4.76700	−0.198971
$575$	8.58860	0.358169
$576$	1.00000	0.0416667
$577$	24.1055	1.00352	0.501762	−	0.865006i	$-0.332685\pi$
$577$	24.1055	1.00352	0.501762	+	0.865006i	$0.332685\pi$
$578$	8.27596	0.344234
$579$	5.62711	0.233855
$580$	−8.06468	−0.334868
$581$	−1.76076	−0.0730488
$582$	5.09415	0.211159
$583$	−40.5182	−1.67809
$584$	13.4965	0.558488
$585$	1.40749	0.0581927
$586$	−20.2309	−0.835729
$587$	−20.9165	−0.863318	−0.431659	−	0.902037i	$-0.642072\pi$
$587$	−20.9165	−0.863318	−0.431659	+	0.902037i	$0.642072\pi$
$588$	−6.00224	−0.247528
$589$	−19.1653	−0.789690
$590$	11.6321	0.478886
$591$	−8.91051	−0.366529
$592$	−4.75750	−0.195532
$593$	31.4689	1.29227	0.646137	−	0.763222i	$-0.276384\pi$
$593$	31.4689	1.29227	0.646137	+	0.763222i	$0.276384\pi$
$594$	−4.83053	−0.198199
$595$	4.15258	0.170239
$596$	17.0885	0.699973
$597$	−21.5100	−0.880345
$598$	2.84489	0.116336
$599$	−23.6003	−0.964284	−0.482142	−	0.876093i	$-0.660141\pi$
$599$	−23.6003	−0.964284	−0.482142	+	0.876093i	$0.660141\pi$
$600$	3.01896	0.123249
$601$	31.7300	1.29429	0.647147	−	0.762365i	$-0.275962\pi$
$601$	31.7300	1.29429	0.647147	+	0.762365i	$0.275962\pi$
$602$	9.74251	0.397075
$603$	−4.55266	−0.185399
$604$	13.2289	0.538277
$605$	17.3601	0.705787
$606$	5.83551	0.237051
$607$	−14.8494	−0.602718	−0.301359	−	0.953511i	$-0.597440\pi$
$607$	−14.8494	−0.602718	−0.301359	+	0.953511i	$0.597440\pi$
$608$	−1.88860	−0.0765928
$609$	5.72339	0.231924
$610$	8.60434	0.348380
$611$	−2.46105	−0.0995634
$612$	−2.95365	−0.119394
$613$	23.6567	0.955484	0.477742	−	0.878500i	$-0.341455\pi$
$613$	23.6567	0.955484	0.477742	+	0.878500i	$0.341455\pi$
$614$	−9.28706	−0.374795
$615$	−6.71705	−0.270858
$616$	4.82512	0.194409
$617$	−9.47246	−0.381347	−0.190673	−	0.981654i	$-0.561067\pi$
$617$	−9.47246	−0.381347	−0.190673	+	0.981654i	$0.561067\pi$
$618$	1.00000	0.0402259
$619$	26.5081	1.06545	0.532726	−	0.846288i	$-0.321168\pi$
$619$	26.5081	1.06545	0.532726	+	0.846288i	$0.321168\pi$
$620$	−14.2831	−0.573621
$621$	−2.84489	−0.114161
$622$	−18.0200	−0.722536
$623$	0.860027	0.0344562
$624$	1.00000	0.0400320
$625$	−0.791061	−0.0316424
$626$	−26.1537	−1.04531
$627$	9.12294	0.364335
$628$	9.61127	0.383531
$629$	14.0520	0.560289
$630$	1.40592	0.0560130
$631$	38.7613	1.54306	0.771531	−	0.636192i	$-0.219491\pi$
$631$	38.7613	1.54306	0.771531	+	0.636192i	$0.219491\pi$
$632$	11.1098	0.441926
$633$	25.6510	1.01954
$634$	−7.89534	−0.313564
$635$	5.15828	0.204700
$636$	−8.38795	−0.332604
$637$	−6.00224	−0.237817
$638$	27.6780	1.09578
$639$	−3.66759	−0.145087
$640$	−1.40749	−0.0556361
$641$	21.2063	0.837600	0.418800	−	0.908079i	$-0.362451\pi$
$641$	21.2063	0.837600	0.418800	+	0.908079i	$0.362451\pi$
$642$	−13.0054	−0.513284
$643$	−15.7977	−0.622999	−0.311500	−	0.950246i	$-0.600831\pi$
$643$	−15.7977	−0.622999	−0.311500	+	0.950246i	$0.600831\pi$
$644$	2.84170	0.111979
$645$	13.7279	0.540536
$646$	5.57826	0.219474
$647$	−31.0349	−1.22011	−0.610054	−	0.792360i	$-0.708852\pi$
$647$	−31.0349	−1.22011	−0.610054	+	0.792360i	$0.708852\pi$
$648$	−1.00000	−0.0392837
$649$	−39.9215	−1.56706
$650$	3.01896	0.118413
$651$	10.1365	0.397280
$652$	−7.98844	−0.312851
$653$	−35.0335	−1.37097	−0.685483	−	0.728089i	$-0.740409\pi$
$653$	−35.0335	−1.37097	−0.685483	+	0.728089i	$0.740409\pi$
$654$	1.24549	0.0487027
$655$	16.8770	0.659440
$656$	−4.77235	−0.186329
$657$	−13.4965	−0.526547
$658$	−2.45829	−0.0958342
$659$	−49.0836	−1.91202	−0.956012	−	0.293327i	$-0.905237\pi$
$659$	−49.0836	−1.91202	−0.956012	+	0.293327i	$0.905237\pi$
$660$	6.79894	0.264648
$661$	−24.0650	−0.936020	−0.468010	−	0.883723i	$-0.655029\pi$
$661$	−24.0650	−0.936020	−0.468010	+	0.883723i	$0.655029\pi$
$662$	−21.0364	−0.817605
$663$	−2.95365	−0.114710
$664$	−1.76274	−0.0684076
$665$	−2.65521	−0.102965
$666$	4.75750	0.184349
$667$	16.3007	0.631164
$668$	3.41720	0.132215
$669$	12.8137	0.495406
$670$	6.40784	0.247556
$671$	−29.5302	−1.14000
$672$	0.998879	0.0385326
$673$	9.91676	0.382263	0.191131	−	0.981564i	$-0.438784\pi$
$673$	9.91676	0.382263	0.191131	+	0.981564i	$0.438784\pi$
$674$	−26.1538	−1.00741
$675$	−3.01896	−0.116200
$676$	1.00000	0.0384615
$677$	17.8633	0.686543	0.343272	−	0.939236i	$-0.388465\pi$
$677$	17.8633	0.686543	0.343272	+	0.939236i	$0.388465\pi$
$678$	−9.96193	−0.382585
$679$	5.08844	0.195276
$680$	4.15724	0.159423
$681$	−12.3807	−0.474429
$682$	49.0196	1.87706
$683$	3.17969	0.121668	0.0608338	−	0.998148i	$-0.480624\pi$
$683$	3.17969	0.121668	0.0608338	+	0.998148i	$0.480624\pi$
$684$	1.88860	0.0722124
$685$	16.3541	0.624857
$686$	−12.9877	−0.495871
$687$	12.8890	0.491744
$688$	9.75344	0.371846
$689$	−8.38795	−0.319555
$690$	4.00416	0.152436
$691$	22.1457	0.842462	0.421231	−	0.906953i	$-0.361598\pi$
$691$	22.1457	0.842462	0.421231	+	0.906953i	$0.361598\pi$
$692$	−24.4518	−0.929519
$693$	−4.82512	−0.183291
$694$	20.9493	0.795224
$695$	14.3257	0.543405
$696$	5.72981	0.217188
$697$	14.0958	0.533918
$698$	−33.7856	−1.27880
$699$	−6.27793	−0.237453
$700$	3.01558	0.113978
$701$	−21.3476	−0.806289	−0.403144	−	0.915136i	$-0.632083\pi$
$701$	−21.3476	−0.806289	−0.403144	+	0.915136i	$0.632083\pi$
$702$	−1.00000	−0.0377426
$703$	−8.98501	−0.338876
$704$	4.83053	0.182057
$705$	−3.46391	−0.130458
$706$	8.70939	0.327782
$707$	5.82897	0.219221
$708$	−8.26441	−0.310596
$709$	16.7407	0.628709	0.314355	−	0.949306i	$-0.398212\pi$
$709$	16.7407	0.628709	0.314355	+	0.949306i	$0.398212\pi$
$710$	5.16210	0.193730
$711$	−11.1098	−0.416652
$712$	0.860992	0.0322670
$713$	28.8695	1.08117
$714$	−2.95034	−0.110414
$715$	6.79894	0.254266
$716$	−11.6219	−0.434332
$717$	−20.3261	−0.759091
$718$	26.5369	0.990347
$719$	13.5812	0.506494	0.253247	−	0.967402i	$-0.418502\pi$
$719$	13.5812	0.506494	0.253247	+	0.967402i	$0.418502\pi$
$720$	1.40749	0.0524542
$721$	0.998879	0.0372002
$722$	15.4332	0.574364
$723$	−5.81756	−0.216357
$724$	16.1688	0.600907
$725$	17.2981	0.642435
$726$	−12.3340	−0.457759
$727$	20.6522	0.765947	0.382974	−	0.923759i	$-0.374900\pi$
$727$	20.6522	0.765947	0.382974	+	0.923759i	$0.374900\pi$
$728$	0.998879	0.0370209
$729$	1.00000	0.0370370
$730$	18.9962	0.703081
$731$	−28.8082	−1.06551
$732$	−6.11324	−0.225952
$733$	−24.6696	−0.911194	−0.455597	−	0.890186i	$-0.650574\pi$
$733$	−24.6696	−0.911194	−0.455597	+	0.890186i	$0.650574\pi$
$734$	−9.83164	−0.362892
$735$	−8.44812	−0.311613
$736$	2.84489	0.104864
$737$	−21.9918	−0.810077
$738$	4.77235	0.175673
$739$	30.0209	1.10434	0.552169	−	0.833732i	$-0.313800\pi$
$739$	30.0209	1.10434	0.552169	+	0.833732i	$0.313800\pi$
$740$	−6.69615	−0.246155
$741$	1.88860	0.0693794
$742$	−8.37855	−0.307586
$743$	−27.9466	−1.02526	−0.512631	−	0.858609i	$-0.671329\pi$
$743$	−27.9466	−1.02526	−0.512631	+	0.858609i	$0.671329\pi$
$744$	10.1479	0.372039
$745$	24.0520	0.881197
$746$	36.6010	1.34006
$747$	1.76274	0.0644953
$748$	−14.2677	−0.521679
$749$	−12.9909	−0.474676
$750$	11.2866	0.412130
$751$	−1.09991	−0.0401365	−0.0200682	−	0.999799i	$-0.506388\pi$
$751$	−1.09991	−0.0401365	−0.0200682	+	0.999799i	$0.506388\pi$
$752$	−2.46105	−0.0897453
$753$	10.3172	0.375979
$754$	5.72981	0.208668
$755$	18.6196	0.677637
$756$	−0.998879	−0.0363289
$757$	47.2731	1.71817	0.859084	−	0.511834i	$-0.171034\pi$
$757$	47.2731	1.71817	0.859084	+	0.511834i	$0.171034\pi$
$758$	23.6372	0.858541
$759$	−13.7423	−0.498814
$760$	−2.65819	−0.0964227
$761$	−48.2098	−1.74760	−0.873802	−	0.486282i	$-0.838353\pi$
$761$	−48.2098	−1.74760	−0.873802	+	0.486282i	$0.838353\pi$
$762$	−3.66487	−0.132764
$763$	1.24410	0.0450394
$764$	−15.3479	−0.555269
$765$	−4.15724	−0.150305
$766$	−29.2380	−1.05641
$767$	−8.26441	−0.298411
$768$	1.00000	0.0360844
$769$	−44.9515	−1.62099	−0.810497	−	0.585743i	$-0.800803\pi$
$769$	−44.9515	−1.62099	−0.810497	+	0.585743i	$0.800803\pi$
$770$	6.79132	0.244742
$771$	3.22415	0.116115
$772$	5.62711	0.202524
$773$	20.0462	0.721011	0.360506	−	0.932757i	$-0.382604\pi$
$773$	20.0462	0.721011	0.360506	+	0.932757i	$0.382604\pi$
$774$	−9.75344	−0.350580
$775$	30.6360	1.10048
$776$	5.09415	0.182869
$777$	4.75217	0.170483
$778$	−9.48255	−0.339966
$779$	−9.01305	−0.322926
$780$	1.40749	0.0503964
$781$	−17.7164	−0.633942
$782$	−8.40279	−0.300483
$783$	−5.72981	−0.204767
$784$	−6.00224	−0.214366
$785$	13.5278	0.482828
$786$	−11.9908	−0.427699
$787$	−14.6857	−0.523489	−0.261744	−	0.965137i	$-0.584298\pi$
$787$	−14.6857	−0.523489	−0.261744	+	0.965137i	$0.584298\pi$
$788$	−8.91051	−0.317424
$789$	−27.8650	−0.992021
$790$	15.6370	0.556341
$791$	−9.95076	−0.353808
$792$	−4.83053	−0.171645
$793$	−6.11324	−0.217087
$794$	−6.03291	−0.214100
$795$	−11.8060	−0.418715
$796$	−21.5100	−0.762401
$797$	37.0049	1.31078	0.655391	−	0.755290i	$-0.272504\pi$
$797$	37.0049	1.31078	0.655391	+	0.755290i	$0.272504\pi$
$798$	1.88648	0.0667807
$799$	7.26908	0.257161
$800$	3.01896	0.106736
$801$	−0.860992	−0.0304217
$802$	10.9056	0.385091
$803$	−65.1951	−2.30069
$804$	−4.55266	−0.160560
$805$	3.99967	0.140970
$806$	10.1479	0.357443
$807$	−5.47995	−0.192904
$808$	5.83551	0.205292
$809$	−34.4491	−1.21117	−0.605583	−	0.795782i	$-0.707060\pi$
$809$	−34.4491	−1.21117	−0.605583	+	0.795782i	$0.707060\pi$
$810$	−1.40749	−0.0494543
$811$	−25.1527	−0.883230	−0.441615	−	0.897205i	$-0.645594\pi$
$811$	−25.1527	−0.883230	−0.441615	+	0.897205i	$0.645594\pi$
$812$	5.72339	0.200852
$813$	25.0590	0.878859
$814$	22.9812	0.805492
$815$	−11.2437	−0.393849
$816$	−2.95365	−0.103398
$817$	18.4203	0.644446
$818$	20.6537	0.722141
$819$	−0.998879	−0.0349037
$820$	−6.71705	−0.234569
$821$	51.1856	1.78639	0.893194	−	0.449671i	$-0.148459\pi$
$821$	51.1856	1.78639	0.893194	+	0.449671i	$0.148459\pi$
$822$	−11.6193	−0.405269
$823$	9.93393	0.346275	0.173138	−	0.984898i	$-0.444609\pi$
$823$	9.93393	0.346275	0.173138	+	0.984898i	$0.444609\pi$
$824$	1.00000	0.0348367
$825$	−14.5832	−0.507721
$826$	−8.25515	−0.287233
$827$	−4.04881	−0.140791	−0.0703954	−	0.997519i	$-0.522426\pi$
$827$	−4.04881	−0.140791	−0.0703954	+	0.997519i	$0.522426\pi$
$828$	−2.84489	−0.0988666
$829$	−44.1314	−1.53275	−0.766374	−	0.642395i	$-0.777941\pi$
$829$	−44.1314	−1.53275	−0.766374	+	0.642395i	$0.777941\pi$
$830$	−2.48104	−0.0861183
$831$	4.14155	0.143669
$832$	1.00000	0.0346688
$833$	17.7285	0.614257
$834$	−10.1782	−0.352441
$835$	4.80969	0.166446
$836$	9.12294	0.315523
$837$	−10.1479	−0.350762
$838$	−27.0666	−0.935001
$839$	−25.9048	−0.894332	−0.447166	−	0.894451i	$-0.647567\pi$
$839$	−25.9048	−0.894332	−0.447166	+	0.894451i	$0.647567\pi$
$840$	1.40592	0.0485087
$841$	3.83078	0.132096
$842$	25.4862	0.878314
$843$	−2.03599	−0.0701231
$844$	25.6510	0.882945
$845$	1.40749	0.0484193
$846$	2.46105	0.0846126
$847$	−12.3202	−0.423327
$848$	−8.38795	−0.288043
$849$	10.5521	0.362147
$850$	−8.91695	−0.305849
$851$	13.5345	0.463958
$852$	−3.66759	−0.125649
$853$	9.38918	0.321479	0.160740	−	0.986997i	$-0.448612\pi$
$853$	9.38918	0.321479	0.160740	+	0.986997i	$0.448612\pi$
$854$	−6.10638	−0.208956
$855$	2.65819	0.0909082
$856$	−13.0054	−0.444517
$857$	−22.2294	−0.759342	−0.379671	−	0.925122i	$-0.623963\pi$
$857$	−22.2294	−0.759342	−0.379671	+	0.925122i	$0.623963\pi$
$858$	−4.83053	−0.164912
$859$	37.6415	1.28431	0.642156	−	0.766574i	$-0.278040\pi$
$859$	37.6415	1.28431	0.642156	+	0.766574i	$0.278040\pi$
$860$	13.7279	0.468118
$861$	4.76700	0.162459
$862$	−30.4313	−1.03649
$863$	33.2589	1.13215	0.566073	−	0.824355i	$-0.308462\pi$
$863$	33.2589	1.13215	0.566073	+	0.824355i	$0.308462\pi$
$864$	−1.00000	−0.0340207
$865$	−34.4158	−1.17017
$866$	−37.2374	−1.26538
$867$	−8.27596	−0.281066
$868$	10.1365	0.344055
$869$	−53.6664	−1.82051
$870$	8.06468	0.273418
$871$	−4.55266	−0.154261
$872$	1.24549	0.0421778
$873$	−5.09415	−0.172411
$874$	5.37285	0.181739
$875$	11.2740	0.381130
$876$	−13.4965	−0.456003
$877$	−35.4445	−1.19688	−0.598438	−	0.801169i	$-0.704212\pi$
$877$	−35.4445	−1.19688	−0.598438	+	0.801169i	$0.704212\pi$
$878$	28.7436	0.970050
$879$	20.2309	0.682370
$880$	6.79894	0.229192
$881$	−9.58825	−0.323036	−0.161518	−	0.986870i	$-0.551639\pi$
$881$	−9.58825	−0.323036	−0.161518	+	0.986870i	$0.551639\pi$
$882$	6.00224	0.202106
$883$	−31.9681	−1.07581	−0.537907	−	0.843004i	$-0.680785\pi$
$883$	−31.9681	−1.07581	−0.537907	+	0.843004i	$0.680785\pi$
$884$	−2.95365	−0.0993420
$885$	−11.6321	−0.391009
$886$	9.15518	0.307574
$887$	25.2887	0.849113	0.424556	−	0.905402i	$-0.360430\pi$
$887$	25.2887	0.849113	0.424556	+	0.905402i	$0.360430\pi$
$888$	4.75750	0.159651
$889$	−3.66076	−0.122778
$890$	1.21184	0.0406210
$891$	4.83053	0.161829
$892$	12.8137	0.429034
$893$	−4.64794	−0.155537
$894$	−17.0885	−0.571526
$895$	−16.3578	−0.546781
$896$	0.998879	0.0333702
$897$	−2.84489	−0.0949880
$898$	−35.7192	−1.19197
$899$	58.1454	1.93926
$900$	−3.01896	−0.100632
$901$	24.7751	0.825377
$902$	23.0530	0.767580
$903$	−9.74251	−0.324210
$904$	−9.96193	−0.331329
$905$	22.7574	0.756482
$906$	−13.2289	−0.439501
$907$	24.8400	0.824798	0.412399	−	0.911003i	$-0.364691\pi$
$907$	24.8400	0.824798	0.412399	+	0.911003i	$0.364691\pi$
$908$	−12.3807	−0.410868
$909$	−5.83551	−0.193552
$910$	1.40592	0.0466057
$911$	−52.5136	−1.73985	−0.869926	−	0.493182i	$-0.835834\pi$
$911$	−52.5136	−1.73985	−0.869926	+	0.493182i	$0.835834\pi$
$912$	1.88860	0.0625378
$913$	8.51497	0.281804
$914$	−22.8348	−0.755307
$915$	−8.60434	−0.284451
$916$	12.8890	0.425863
$917$	−11.9774	−0.395529
$918$	2.95365	0.0974850
$919$	−17.4881	−0.576878	−0.288439	−	0.957498i	$-0.593136\pi$
$919$	−17.4881	−0.576878	−0.288439	+	0.957498i	$0.593136\pi$
$920$	4.00416	0.132013
$921$	9.28706	0.306019
$922$	24.7047	0.813605
$923$	−3.66759	−0.120720
$924$	−4.82512	−0.158735
$925$	14.3627	0.472243
$926$	−9.59246	−0.315228
$927$	−1.00000	−0.0328443
$928$	5.72981	0.188090
$929$	−7.34782	−0.241074	−0.120537	−	0.992709i	$-0.538462\pi$
$929$	−7.34782	−0.241074	−0.120537	+	0.992709i	$0.538462\pi$
$930$	14.2831	0.468360
$931$	−11.3358	−0.371517
$932$	−6.27793	−0.205641
$933$	18.0200	0.589948
$934$	20.3961	0.667379
$935$	−20.0817	−0.656741
$936$	−1.00000	−0.0326860
$937$	36.0157	1.17658	0.588291	−	0.808649i	$-0.299801\pi$
$937$	36.0157	1.17658	0.588291	+	0.808649i	$0.299801\pi$
$938$	−4.54756	−0.148483
$939$	26.1537	0.853493
$940$	−3.46391	−0.112980
$941$	2.80335	0.0913865	0.0456933	−	0.998956i	$-0.485450\pi$
$941$	2.80335	0.0913865	0.0456933	+	0.998956i	$0.485450\pi$
$942$	−9.61127	−0.313152
$943$	13.5768	0.442121
$944$	−8.26441	−0.268984
$945$	−1.40592	−0.0457345
$946$	−47.1143	−1.53182
$947$	−17.8646	−0.580522	−0.290261	−	0.956947i	$-0.593742\pi$
$947$	−17.8646	−0.580522	−0.290261	+	0.956947i	$0.593742\pi$
$948$	−11.1098	−0.360831
$949$	−13.4965	−0.438114
$950$	5.70161	0.184985
$951$	7.89534	0.256024
$952$	−2.95034	−0.0956210
$953$	37.0971	1.20169	0.600846	−	0.799365i	$-0.294831\pi$
$953$	37.0971	1.20169	0.600846	+	0.799365i	$0.294831\pi$
$954$	8.38795	0.271570
$955$	−21.6021	−0.699028
$956$	−20.3261	−0.657392
$957$	−27.6780	−0.894704
$958$	6.02670	0.194714
$959$	−11.6063	−0.374786
$960$	1.40749	0.0454267
$961$	71.9792	2.32191
$962$	4.75750	0.153388
$963$	13.0054	0.419095
$964$	−5.81756	−0.187371
$965$	7.92013	0.254958
$966$	−2.84170	−0.0914301
$967$	−12.1837	−0.391800	−0.195900	−	0.980624i	$-0.562763\pi$
$967$	−12.1837	−0.391800	−0.195900	+	0.980624i	$0.562763\pi$
$968$	−12.3340	−0.396431
$969$	−5.57826	−0.179199
$970$	7.16998	0.230214
$971$	−57.5801	−1.84783	−0.923917	−	0.382592i	$-0.875031\pi$
$971$	−57.5801	−1.84783	−0.923917	+	0.382592i	$0.875031\pi$
$972$	1.00000	0.0320750
$973$	−10.1668	−0.325931
$974$	−23.6141	−0.756645
$975$	−3.01896	−0.0966842
$976$	−6.11324	−0.195680
$977$	−32.1063	−1.02717	−0.513585	−	0.858039i	$-0.671683\pi$
$977$	−32.1063	−1.02717	−0.513585	+	0.858039i	$0.671683\pi$
$978$	7.98844	0.255442
$979$	−4.15905	−0.132924
$980$	−8.44812	−0.269865
$981$	−1.24549	−0.0397656
$982$	18.4702	0.589409
$983$	42.9319	1.36931	0.684657	−	0.728865i	$-0.259952\pi$
$983$	42.9319	1.36931	0.684657	+	0.728865i	$0.259952\pi$
$984$	4.77235	0.152137
$985$	−12.5415	−0.399605
$986$	−16.9239	−0.538966
$987$	2.45829	0.0782483
$988$	1.88860	0.0600843
$989$	−27.7474	−0.882317
$990$	−6.79894	−0.216085
$991$	−26.2680	−0.834429	−0.417215	−	0.908808i	$-0.636994\pi$
$991$	−26.2680	−0.834429	−0.417215	+	0.908808i	$0.636994\pi$
$992$	10.1479	0.322195
$993$	21.0364	0.667571
$994$	−3.66347	−0.116198
$995$	−30.2752	−0.959787
$996$	1.76274	0.0558545
$997$	−60.1384	−1.90460	−0.952301	−	0.305160i	$-0.901290\pi$
$997$	−60.1384	−1.90460	−0.952301	+	0.305160i	$0.901290\pi$
$998$	−24.8331	−0.786078
$999$	−4.75750	−0.150521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.w.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.w.1.8	✓	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.40749 q^{5} -1.00000 q^{6} -0.998879 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.40749 q^{5} -1.00000 q^{6} -0.998879 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.40749 q^{10} +4.83053 q^{11} +1.00000 q^{12} +1.00000 q^{13} +0.998879 q^{14} +1.40749 q^{15} +1.00000 q^{16} -2.95365 q^{17} -1.00000 q^{18} +1.88860 q^{19} +1.40749 q^{20} -0.998879 q^{21} -4.83053 q^{22} -2.84489 q^{23} -1.00000 q^{24} -3.01896 q^{25} -1.00000 q^{26} +1.00000 q^{27} -0.998879 q^{28} -5.72981 q^{29} -1.40749 q^{30} -10.1479 q^{31} -1.00000 q^{32} +4.83053 q^{33} +2.95365 q^{34} -1.40592 q^{35} +1.00000 q^{36} -4.75750 q^{37} -1.88860 q^{38} +1.00000 q^{39} -1.40749 q^{40} -4.77235 q^{41} +0.998879 q^{42} +9.75344 q^{43} +4.83053 q^{44} +1.40749 q^{45} +2.84489 q^{46} -2.46105 q^{47} +1.00000 q^{48} -6.00224 q^{49} +3.01896 q^{50} -2.95365 q^{51} +1.00000 q^{52} -8.38795 q^{53} -1.00000 q^{54} +6.79894 q^{55} +0.998879 q^{56} +1.88860 q^{57} +5.72981 q^{58} -8.26441 q^{59} +1.40749 q^{60} -6.11324 q^{61} +10.1479 q^{62} -0.998879 q^{63} +1.00000 q^{64} +1.40749 q^{65} -4.83053 q^{66} -4.55266 q^{67} -2.95365 q^{68} -2.84489 q^{69} +1.40592 q^{70} -3.66759 q^{71} -1.00000 q^{72} -13.4965 q^{73} +4.75750 q^{74} -3.01896 q^{75} +1.88860 q^{76} -4.82512 q^{77} -1.00000 q^{78} -11.1098 q^{79} +1.40749 q^{80} +1.00000 q^{81} +4.77235 q^{82} +1.76274 q^{83} -0.998879 q^{84} -4.15724 q^{85} -9.75344 q^{86} -5.72981 q^{87} -4.83053 q^{88} -0.860992 q^{89} -1.40749 q^{90} -0.998879 q^{91} -2.84489 q^{92} -10.1479 q^{93} +2.46105 q^{94} +2.65819 q^{95} -1.00000 q^{96} -5.09415 q^{97} +6.00224 q^{98} +4.83053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9 + 4 * q^10 - 9 * q^11 + 12 * q^12 + 12 * q^13 - 4 * q^15 + 12 * q^16 - 20 * q^17 - 12 * q^18 + 4 * q^19 - 4 * q^20 + 9 * q^22 - 30 * q^23 - 12 * q^24 + 14 * q^25 - 12 * q^26 + 12 * q^27 - 29 * q^29 + 4 * q^30 + 6 * q^31 - 12 * q^32 - 9 * q^33 + 20 * q^34 - 22 * q^35 + 12 * q^36 + 7 * q^37 - 4 * q^38 + 12 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 9 * q^44 - 4 * q^45 + 30 * q^46 - 16 * q^47 + 12 * q^48 + 10 * q^49 - 14 * q^50 - 20 * q^51 + 12 * q^52 - 9 * q^53 - 12 * q^54 - 20 * q^55 + 4 * q^57 + 29 * q^58 - 29 * q^59 - 4 * q^60 - 26 * q^61 - 6 * q^62 + 12 * q^64 - 4 * q^65 + 9 * q^66 + 12 * q^67 - 20 * q^68 - 30 * q^69 + 22 * q^70 - 35 * q^71 - 12 * q^72 + 18 * q^73 - 7 * q^74 + 14 * q^75 + 4 * q^76 - 25 * q^77 - 12 * q^78 - 37 * q^79 - 4 * q^80 + 12 * q^81 + 8 * q^82 - 24 * q^83 - 17 * q^85 + 8 * q^86 - 29 * q^87 + 9 * q^88 + 15 * q^89 + 4 * q^90 - 30 * q^92 + 6 * q^93 + 16 * q^94 - 54 * q^95 - 12 * q^96 - 11 * q^97 - 10 * q^98 - 9 * q^99

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