LMFDB - Embedded newform 6045.2.a.bi.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 21 x^{16} + 97 x^{15} + 156 x^{14} - 935 x^{13} - 411 x^{12} + 4582 x^{11} + \cdots - 112 $ x^18 - 4x^17 - 21x^16 + 97x^15 + 156x^14 - 935x^13 - 411x^12 + 4582x^11 - 446x^10 - 12159x^9 + 4398x^8 + 17347x^7 - 7839x^6 - 12832x^5 + 4744x^4 + 5064x^3 - 746x^2 - 892*x - 112
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.43391$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43391 q^{2} +1.00000 q^{3} +3.92391 q^{4} +1.00000 q^{5} +2.43391 q^{6} +2.58128 q^{7} +4.68261 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.43391 q^{2} +1.00000 q^{3} +3.92391 q^{4} +1.00000 q^{5} +2.43391 q^{6} +2.58128 q^{7} +4.68261 q^{8} +1.00000 q^{9} +2.43391 q^{10} +3.79890 q^{11} +3.92391 q^{12} -1.00000 q^{13} +6.28259 q^{14} +1.00000 q^{15} +3.54922 q^{16} +3.54472 q^{17} +2.43391 q^{18} -1.13331 q^{19} +3.92391 q^{20} +2.58128 q^{21} +9.24617 q^{22} -6.61420 q^{23} +4.68261 q^{24} +1.00000 q^{25} -2.43391 q^{26} +1.00000 q^{27} +10.1287 q^{28} +2.89921 q^{29} +2.43391 q^{30} +1.00000 q^{31} -0.726737 q^{32} +3.79890 q^{33} +8.62752 q^{34} +2.58128 q^{35} +3.92391 q^{36} -2.76848 q^{37} -2.75837 q^{38} -1.00000 q^{39} +4.68261 q^{40} +3.58360 q^{41} +6.28259 q^{42} -5.74884 q^{43} +14.9065 q^{44} +1.00000 q^{45} -16.0984 q^{46} -1.33520 q^{47} +3.54922 q^{48} -0.337004 q^{49} +2.43391 q^{50} +3.54472 q^{51} -3.92391 q^{52} -11.8502 q^{53} +2.43391 q^{54} +3.79890 q^{55} +12.0871 q^{56} -1.13331 q^{57} +7.05641 q^{58} -5.21698 q^{59} +3.92391 q^{60} +1.05100 q^{61} +2.43391 q^{62} +2.58128 q^{63} -8.86725 q^{64} -1.00000 q^{65} +9.24617 q^{66} +12.5455 q^{67} +13.9091 q^{68} -6.61420 q^{69} +6.28259 q^{70} -1.91917 q^{71} +4.68261 q^{72} -2.02562 q^{73} -6.73821 q^{74} +1.00000 q^{75} -4.44700 q^{76} +9.80601 q^{77} -2.43391 q^{78} -12.8857 q^{79} +3.54922 q^{80} +1.00000 q^{81} +8.72215 q^{82} +5.08496 q^{83} +10.1287 q^{84} +3.54472 q^{85} -13.9921 q^{86} +2.89921 q^{87} +17.7888 q^{88} -1.94744 q^{89} +2.43391 q^{90} -2.58128 q^{91} -25.9535 q^{92} +1.00000 q^{93} -3.24975 q^{94} -1.13331 q^{95} -0.726737 q^{96} -3.56679 q^{97} -0.820236 q^{98} +3.79890 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9} + 4 q^{10} + 6 q^{11} + 22 q^{12} - 18 q^{13} + 5 q^{14} + 18 q^{15} + 30 q^{16} + 18 q^{17} + 4 q^{18} + 12 q^{19} + 22 q^{20} + 8 q^{21} + 7 q^{22} + 32 q^{23} + 9 q^{24} + 18 q^{25} - 4 q^{26} + 18 q^{27} + 10 q^{28} + 7 q^{29} + 4 q^{30} + 18 q^{31} + 22 q^{32} + 6 q^{33} + 15 q^{34} + 8 q^{35} + 22 q^{36} + 3 q^{37} + 32 q^{38} - 18 q^{39} + 9 q^{40} + 4 q^{41} + 5 q^{42} + 14 q^{43} - 5 q^{44} + 18 q^{45} + 10 q^{46} + 23 q^{47} + 30 q^{48} + 28 q^{49} + 4 q^{50} + 18 q^{51} - 22 q^{52} + 35 q^{53} + 4 q^{54} + 6 q^{55} - 7 q^{56} + 12 q^{57} - 6 q^{58} + 28 q^{59} + 22 q^{60} + 19 q^{61} + 4 q^{62} + 8 q^{63} + 43 q^{64} - 18 q^{65} + 7 q^{66} + 34 q^{67} + 55 q^{68} + 32 q^{69} + 5 q^{70} - 8 q^{71} + 9 q^{72} + 22 q^{74} + 18 q^{75} + 2 q^{76} + 21 q^{77} - 4 q^{78} + 4 q^{79} + 30 q^{80} + 18 q^{81} + 29 q^{82} + 11 q^{83} + 10 q^{84} + 18 q^{85} - 22 q^{86} + 7 q^{87} - 31 q^{88} + 17 q^{89} + 4 q^{90} - 8 q^{91} + 33 q^{92} + 18 q^{93} - 14 q^{94} + 12 q^{95} + 22 q^{96} + 32 q^{97} + 20 q^{98} + 6 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9 + 4 * q^10 + 6 * q^11 + 22 * q^12 - 18 * q^13 + 5 * q^14 + 18 * q^15 + 30 * q^16 + 18 * q^17 + 4 * q^18 + 12 * q^19 + 22 * q^20 + 8 * q^21 + 7 * q^22 + 32 * q^23 + 9 * q^24 + 18 * q^25 - 4 * q^26 + 18 * q^27 + 10 * q^28 + 7 * q^29 + 4 * q^30 + 18 * q^31 + 22 * q^32 + 6 * q^33 + 15 * q^34 + 8 * q^35 + 22 * q^36 + 3 * q^37 + 32 * q^38 - 18 * q^39 + 9 * q^40 + 4 * q^41 + 5 * q^42 + 14 * q^43 - 5 * q^44 + 18 * q^45 + 10 * q^46 + 23 * q^47 + 30 * q^48 + 28 * q^49 + 4 * q^50 + 18 * q^51 - 22 * q^52 + 35 * q^53 + 4 * q^54 + 6 * q^55 - 7 * q^56 + 12 * q^57 - 6 * q^58 + 28 * q^59 + 22 * q^60 + 19 * q^61 + 4 * q^62 + 8 * q^63 + 43 * q^64 - 18 * q^65 + 7 * q^66 + 34 * q^67 + 55 * q^68 + 32 * q^69 + 5 * q^70 - 8 * q^71 + 9 * q^72 + 22 * q^74 + 18 * q^75 + 2 * q^76 + 21 * q^77 - 4 * q^78 + 4 * q^79 + 30 * q^80 + 18 * q^81 + 29 * q^82 + 11 * q^83 + 10 * q^84 + 18 * q^85 - 22 * q^86 + 7 * q^87 - 31 * q^88 + 17 * q^89 + 4 * q^90 - 8 * q^91 + 33 * q^92 + 18 * q^93 - 14 * q^94 + 12 * q^95 + 22 * q^96 + 32 * q^97 + 20 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.43391	1.72103	0.860516	−	0.509423i	$-0.170141\pi$
$2$	2.43391	1.72103	0.860516	+	0.509423i	$0.170141\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.92391	1.96195
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.43391	0.993639
$7$	2.58128	0.975631	0.487816	−	0.872947i	$-0.337794\pi$
$7$	2.58128	0.975631	0.487816	+	0.872947i	$0.337794\pi$
$8$	4.68261	1.65555
$9$	1.00000	0.333333
$10$	2.43391	0.769669
$11$	3.79890	1.14541	0.572706	−	0.819761i	$-0.305894\pi$
$11$	3.79890	1.14541	0.572706	+	0.819761i	$0.305894\pi$
$12$	3.92391	1.13273
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	6.28259	1.67909
$15$	1.00000	0.258199
$16$	3.54922	0.887305
$17$	3.54472	0.859721	0.429860	−	0.902895i	$-0.358563\pi$
$17$	3.54472	0.859721	0.429860	+	0.902895i	$0.358563\pi$
$18$	2.43391	0.573677
$19$	−1.13331	−0.259999	−0.130000	−	0.991514i	$-0.541498\pi$
$19$	−1.13331	−0.259999	−0.130000	+	0.991514i	$0.541498\pi$
$20$	3.92391	0.877412
$21$	2.58128	0.563281
$22$	9.24617	1.97129
$23$	−6.61420	−1.37916	−0.689579	−	0.724211i	$-0.742204\pi$
$23$	−6.61420	−1.37916	−0.689579	+	0.724211i	$0.742204\pi$
$24$	4.68261	0.955833
$25$	1.00000	0.200000
$26$	−2.43391	−0.477329
$27$	1.00000	0.192450
$28$	10.1287	1.91414
$29$	2.89921	0.538370	0.269185	−	0.963089i	$-0.413246\pi$
$29$	2.89921	0.538370	0.269185	+	0.963089i	$0.413246\pi$
$30$	2.43391	0.444369
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−0.726737	−0.128470
$33$	3.79890	0.661303
$34$	8.62752	1.47961
$35$	2.58128	0.436316
$36$	3.92391	0.653984
$37$	−2.76848	−0.455135	−0.227567	−	0.973762i	$-0.573077\pi$
$37$	−2.76848	−0.455135	−0.227567	+	0.973762i	$0.573077\pi$
$38$	−2.75837	−0.447467
$39$	−1.00000	−0.160128
$40$	4.68261	0.740385
$41$	3.58360	0.559665	0.279832	−	0.960049i	$-0.409721\pi$
$41$	3.58360	0.559665	0.279832	+	0.960049i	$0.409721\pi$
$42$	6.28259	0.969425
$43$	−5.74884	−0.876690	−0.438345	−	0.898807i	$-0.644435\pi$
$43$	−5.74884	−0.876690	−0.438345	+	0.898807i	$0.644435\pi$
$44$	14.9065	2.24724
$45$	1.00000	0.149071
$46$	−16.0984	−2.37357
$47$	−1.33520	−0.194759	−0.0973793	−	0.995247i	$-0.531046\pi$
$47$	−1.33520	−0.194759	−0.0973793	+	0.995247i	$0.531046\pi$
$48$	3.54922	0.512286
$49$	−0.337004	−0.0481434
$50$	2.43391	0.344206
$51$	3.54472	0.496360
$52$	−3.92391	−0.544148
$53$	−11.8502	−1.62775	−0.813876	−	0.581038i	$-0.802647\pi$
$53$	−11.8502	−1.62775	−0.813876	+	0.581038i	$0.802647\pi$
$54$	2.43391	0.331213
$55$	3.79890	0.512243
$56$	12.0871	1.61521
$57$	−1.13331	−0.150111
$58$	7.05641	0.926552
$59$	−5.21698	−0.679193	−0.339596	−	0.940571i	$-0.610290\pi$
$59$	−5.21698	−0.679193	−0.339596	+	0.940571i	$0.610290\pi$
$60$	3.92391	0.506574
$61$	1.05100	0.134566	0.0672832	−	0.997734i	$-0.478567\pi$
$61$	1.05100	0.134566	0.0672832	+	0.997734i	$0.478567\pi$
$62$	2.43391	0.309107
$63$	2.58128	0.325210
$64$	−8.86725	−1.10841
$65$	−1.00000	−0.124035
$66$	9.24617	1.13812
$67$	12.5455	1.53268	0.766338	−	0.642438i	$-0.222077\pi$
$67$	12.5455	1.53268	0.766338	+	0.642438i	$0.222077\pi$
$68$	13.9091	1.68673
$69$	−6.61420	−0.796257
$70$	6.28259	0.750913
$71$	−1.91917	−0.227764	−0.113882	−	0.993494i	$-0.536329\pi$
$71$	−1.91917	−0.227764	−0.113882	+	0.993494i	$0.536329\pi$
$72$	4.68261	0.551851
$73$	−2.02562	−0.237081	−0.118540	−	0.992949i	$-0.537822\pi$
$73$	−2.02562	−0.237081	−0.118540	+	0.992949i	$0.537822\pi$
$74$	−6.73821	−0.783302
$75$	1.00000	0.115470
$76$	−4.44700	−0.510106
$77$	9.80601	1.11750
$78$	−2.43391	−0.275586
$79$	−12.8857	−1.44976	−0.724879	−	0.688876i	$-0.758105\pi$
$79$	−12.8857	−1.44976	−0.724879	+	0.688876i	$0.758105\pi$
$80$	3.54922	0.396815
$81$	1.00000	0.111111
$82$	8.72215	0.963201
$83$	5.08496	0.558147	0.279073	−	0.960270i	$-0.409973\pi$
$83$	5.08496	0.558147	0.279073	+	0.960270i	$0.409973\pi$
$84$	10.1287	1.10513
$85$	3.54472	0.384479
$86$	−13.9921	−1.50881
$87$	2.89921	0.310828
$88$	17.7888	1.89629
$89$	−1.94744	−0.206429	−0.103214	−	0.994659i	$-0.532913\pi$
$89$	−1.94744	−0.206429	−0.103214	+	0.994659i	$0.532913\pi$
$90$	2.43391	0.256556
$91$	−2.58128	−0.270591
$92$	−25.9535	−2.70584
$93$	1.00000	0.103695
$94$	−3.24975	−0.335186
$95$	−1.13331	−0.116275
$96$	−0.726737	−0.0741723
$97$	−3.56679	−0.362153	−0.181076	−	0.983469i	$-0.557958\pi$
$97$	−3.56679	−0.362153	−0.181076	+	0.983469i	$0.557958\pi$
$98$	−0.820236	−0.0828564
$99$	3.79890	0.381804
$100$	3.92391	0.392391
$101$	−13.3042	−1.32382	−0.661911	−	0.749582i	$-0.730254\pi$
$101$	−13.3042	−1.32382	−0.661911	+	0.749582i	$0.730254\pi$
$102$	8.62752	0.854252
$103$	1.65339	0.162914	0.0814568	−	0.996677i	$-0.474043\pi$
$103$	1.65339	0.162914	0.0814568	+	0.996677i	$0.474043\pi$
$104$	−4.68261	−0.459167
$105$	2.58128	0.251907
$106$	−28.8423	−2.80141
$107$	13.2252	1.27852	0.639262	−	0.768989i	$-0.279240\pi$
$107$	13.2252	1.27852	0.639262	+	0.768989i	$0.279240\pi$
$108$	3.92391	0.377578
$109$	15.9023	1.52317	0.761583	−	0.648068i	$-0.224423\pi$
$109$	15.9023	1.52317	0.761583	+	0.648068i	$0.224423\pi$
$110$	9.24617	0.881587
$111$	−2.76848	−0.262772
$112$	9.16153	0.865683
$113$	1.40484	0.132157	0.0660783	−	0.997814i	$-0.478951\pi$
$113$	1.40484	0.132157	0.0660783	+	0.997814i	$0.478951\pi$
$114$	−2.75837	−0.258345
$115$	−6.61420	−0.616778
$116$	11.3762	1.05626
$117$	−1.00000	−0.0924500
$118$	−12.6976	−1.16891
$119$	9.14991	0.838771
$120$	4.68261	0.427462
$121$	3.43163	0.311966
$122$	2.55803	0.231593
$123$	3.58360	0.323122
$124$	3.92391	0.352377
$125$	1.00000	0.0894427
$126$	6.28259	0.559698
$127$	8.41750	0.746932	0.373466	−	0.927644i	$-0.378169\pi$
$127$	8.41750	0.746932	0.373466	+	0.927644i	$0.378169\pi$
$128$	−20.1286	−1.77913
$129$	−5.74884	−0.506157
$130$	−2.43391	−0.213468
$131$	−3.95539	−0.345584	−0.172792	−	0.984958i	$-0.555279\pi$
$131$	−3.95539	−0.345584	−0.172792	+	0.984958i	$0.555279\pi$
$132$	14.9065	1.29745
$133$	−2.92539	−0.253663
$134$	30.5346	2.63778
$135$	1.00000	0.0860663
$136$	16.5985	1.42331
$137$	11.1141	0.949541	0.474770	−	0.880110i	$-0.342531\pi$
$137$	11.1141	0.949541	0.474770	+	0.880110i	$0.342531\pi$
$138$	−16.0984	−1.37038
$139$	3.76455	0.319305	0.159652	−	0.987173i	$-0.448963\pi$
$139$	3.76455	0.319305	0.159652	+	0.987173i	$0.448963\pi$
$140$	10.1287	0.856031
$141$	−1.33520	−0.112444
$142$	−4.67109	−0.391989
$143$	−3.79890	−0.317680
$144$	3.54922	0.295768
$145$	2.89921	0.240766
$146$	−4.93017	−0.408024
$147$	−0.337004	−0.0277956
$148$	−10.8632	−0.892953
$149$	−5.92448	−0.485353	−0.242676	−	0.970107i	$-0.578025\pi$
$149$	−5.92448	−0.485353	−0.242676	+	0.970107i	$0.578025\pi$
$150$	2.43391	0.198728
$151$	12.1898	0.991990	0.495995	−	0.868325i	$-0.334803\pi$
$151$	12.1898	0.991990	0.495995	+	0.868325i	$0.334803\pi$
$152$	−5.30684	−0.430442
$153$	3.54472	0.286574
$154$	23.8669	1.92325
$155$	1.00000	0.0803219
$156$	−3.92391	−0.314164
$157$	−8.04812	−0.642310	−0.321155	−	0.947027i	$-0.604071\pi$
$157$	−8.04812	−0.642310	−0.321155	+	0.947027i	$0.604071\pi$
$158$	−31.3627	−2.49508
$159$	−11.8502	−0.939783
$160$	−0.726737	−0.0574536
$161$	−17.0731	−1.34555
$162$	2.43391	0.191226
$163$	8.69888	0.681349	0.340674	−	0.940181i	$-0.389345\pi$
$163$	8.69888	0.681349	0.340674	+	0.940181i	$0.389345\pi$
$164$	14.0617	1.09804
$165$	3.79890	0.295744
$166$	12.3763	0.960589
$167$	−16.1763	−1.25176	−0.625878	−	0.779921i	$-0.715259\pi$
$167$	−16.1763	−1.25176	−0.625878	+	0.779921i	$0.715259\pi$
$168$	12.0871	0.932541
$169$	1.00000	0.0769231
$170$	8.62752	0.661701
$171$	−1.13331	−0.0866664
$172$	−22.5579	−1.72002
$173$	−9.62250	−0.731585	−0.365793	−	0.930696i	$-0.619202\pi$
$173$	−9.62250	−0.731585	−0.365793	+	0.930696i	$0.619202\pi$
$174$	7.05641	0.534945
$175$	2.58128	0.195126
$176$	13.4831	1.01633
$177$	−5.21698	−0.392132
$178$	−4.73990	−0.355271
$179$	−16.7335	−1.25072	−0.625362	−	0.780335i	$-0.715049\pi$
$179$	−16.7335	−1.25072	−0.625362	+	0.780335i	$0.715049\pi$
$180$	3.92391	0.292471
$181$	3.09054	0.229718	0.114859	−	0.993382i	$-0.463358\pi$
$181$	3.09054	0.229718	0.114859	+	0.993382i	$0.463358\pi$
$182$	−6.28259	−0.465697
$183$	1.05100	0.0776920
$184$	−30.9717	−2.28327
$185$	−2.76848	−0.203542
$186$	2.43391	0.178463
$187$	13.4660	0.984734
$188$	−5.23919	−0.382107
$189$	2.58128	0.187760
$190$	−2.75837	−0.200113
$191$	0.160108	0.0115850	0.00579249	−	0.999983i	$-0.498156\pi$
$191$	0.160108	0.0115850	0.00579249	+	0.999983i	$0.498156\pi$
$192$	−8.86725	−0.639939
$193$	14.5485	1.04722	0.523612	−	0.851957i	$-0.324584\pi$
$193$	14.5485	1.04722	0.523612	+	0.851957i	$0.324584\pi$
$194$	−8.68124	−0.623276
$195$	−1.00000	−0.0716115
$196$	−1.32237	−0.0944551
$197$	5.37136	0.382694	0.191347	−	0.981522i	$-0.438714\pi$
$197$	5.37136	0.382694	0.191347	+	0.981522i	$0.438714\pi$
$198$	9.24617	0.657097
$199$	−10.3233	−0.731801	−0.365900	−	0.930654i	$-0.619239\pi$
$199$	−10.3233	−0.731801	−0.365900	+	0.930654i	$0.619239\pi$
$200$	4.68261	0.331110
$201$	12.5455	0.884891
$202$	−32.3813	−2.27834
$203$	7.48367	0.525250
$204$	13.9091	0.973835
$205$	3.58360	0.250290
$206$	4.02420	0.280379
$207$	−6.61420	−0.459719
$208$	−3.54922	−0.246094
$209$	−4.30533	−0.297806
$210$	6.28259	0.433540
$211$	−0.783008	−0.0539045	−0.0269523	−	0.999637i	$-0.508580\pi$
$211$	−0.783008	−0.0539045	−0.0269523	+	0.999637i	$0.508580\pi$
$212$	−46.4991	−3.19357
$213$	−1.91917	−0.131500
$214$	32.1888	2.20038
$215$	−5.74884	−0.392068
$216$	4.68261	0.318611
$217$	2.58128	0.175229
$218$	38.7048	2.62142
$219$	−2.02562	−0.136879
$220$	14.9065	1.00500
$221$	−3.54472	−0.238444
$222$	−6.73821	−0.452239
$223$	−18.2092	−1.21938	−0.609689	−	0.792640i	$-0.708706\pi$
$223$	−18.2092	−1.21938	−0.609689	+	0.792640i	$0.708706\pi$
$224$	−1.87591	−0.125340
$225$	1.00000	0.0666667
$226$	3.41926	0.227446
$227$	0.663529	0.0440400	0.0220200	−	0.999758i	$-0.492990\pi$
$227$	0.663529	0.0440400	0.0220200	+	0.999758i	$0.492990\pi$
$228$	−4.44700	−0.294510
$229$	−8.04004	−0.531301	−0.265650	−	0.964069i	$-0.585587\pi$
$229$	−8.04004	−0.531301	−0.265650	+	0.964069i	$0.585587\pi$
$230$	−16.0984	−1.06149
$231$	9.80601	0.645188
$232$	13.5759	0.891299
$233$	25.8290	1.69211	0.846056	−	0.533093i	$-0.178971\pi$
$233$	25.8290	1.69211	0.846056	+	0.533093i	$0.178971\pi$
$234$	−2.43391	−0.159110
$235$	−1.33520	−0.0870987
$236$	−20.4709	−1.33254
$237$	−12.8857	−0.837019
$238$	22.2700	1.44355
$239$	−2.81795	−0.182278	−0.0911391	−	0.995838i	$-0.529051\pi$
$239$	−2.81795	−0.182278	−0.0911391	+	0.995838i	$0.529051\pi$
$240$	3.54922	0.229101
$241$	8.84427	0.569709	0.284855	−	0.958571i	$-0.408055\pi$
$241$	8.84427	0.569709	0.284855	+	0.958571i	$0.408055\pi$
$242$	8.35227	0.536904
$243$	1.00000	0.0641500
$244$	4.12401	0.264013
$245$	−0.337004	−0.0215304
$246$	8.72215	0.556104
$247$	1.13331	0.0721108
$248$	4.68261	0.297346
$249$	5.08496	0.322246
$250$	2.43391	0.153934
$251$	29.9380	1.88967	0.944834	−	0.327550i	$-0.106223\pi$
$251$	29.9380	1.88967	0.944834	+	0.327550i	$0.106223\pi$
$252$	10.1287	0.638048
$253$	−25.1267	−1.57970
$254$	20.4874	1.28549
$255$	3.54472	0.221979
$256$	−31.2566	−1.95354
$257$	6.72775	0.419665	0.209833	−	0.977737i	$-0.432708\pi$
$257$	6.72775	0.419665	0.209833	+	0.977737i	$0.432708\pi$
$258$	−13.9921	−0.871113
$259$	−7.14621	−0.444044
$260$	−3.92391	−0.243350
$261$	2.89921	0.179457
$262$	−9.62704	−0.594761
$263$	13.9524	0.860344	0.430172	−	0.902747i	$-0.358453\pi$
$263$	13.9524	0.860344	0.430172	+	0.902747i	$0.358453\pi$
$264$	17.7888	1.09482
$265$	−11.8502	−0.727953
$266$	−7.12012	−0.436563
$267$	−1.94744	−0.119182
$268$	49.2273	3.00704
$269$	−19.0549	−1.16180	−0.580898	−	0.813977i	$-0.697298\pi$
$269$	−19.0549	−1.16180	−0.580898	+	0.813977i	$0.697298\pi$
$270$	2.43391	0.148123
$271$	7.09729	0.431130	0.215565	−	0.976490i	$-0.430841\pi$
$271$	7.09729	0.431130	0.215565	+	0.976490i	$0.430841\pi$
$272$	12.5810	0.762835
$273$	−2.58128	−0.156226
$274$	27.0507	1.63419
$275$	3.79890	0.229082
$276$	−25.9535	−1.56222
$277$	−12.3007	−0.739080	−0.369540	−	0.929215i	$-0.620485\pi$
$277$	−12.3007	−0.739080	−0.369540	+	0.929215i	$0.620485\pi$
$278$	9.16256	0.549534
$279$	1.00000	0.0598684
$280$	12.0871	0.722343
$281$	−14.2718	−0.851383	−0.425692	−	0.904868i	$-0.639969\pi$
$281$	−14.2718	−0.851383	−0.425692	+	0.904868i	$0.639969\pi$
$282$	−3.24975	−0.193520
$283$	−15.3869	−0.914659	−0.457329	−	0.889297i	$-0.651194\pi$
$283$	−15.3869	−0.914659	−0.457329	+	0.889297i	$0.651194\pi$
$284$	−7.53065	−0.446862
$285$	−1.13331	−0.0671315
$286$	−9.24617	−0.546737
$287$	9.25027	0.546026
$288$	−0.726737	−0.0428234
$289$	−4.43496	−0.260880
$290$	7.05641	0.414367
$291$	−3.56679	−0.209089
$292$	−7.94834	−0.465142
$293$	−0.931356	−0.0544104	−0.0272052	−	0.999630i	$-0.508661\pi$
$293$	−0.931356	−0.0544104	−0.0272052	+	0.999630i	$0.508661\pi$
$294$	−0.820236	−0.0478372
$295$	−5.21698	−0.303744
$296$	−12.9637	−0.753499
$297$	3.79890	0.220434
$298$	−14.4196	−0.835308
$299$	6.61420	0.382509
$300$	3.92391	0.226547
$301$	−14.8394	−0.855326
$302$	29.6688	1.70725
$303$	−13.3042	−0.764309
$304$	−4.02237	−0.230699
$305$	1.05100	0.0601799
$306$	8.62752	0.493202
$307$	30.9480	1.76629	0.883147	−	0.469097i	$-0.155421\pi$
$307$	30.9480	1.76629	0.883147	+	0.469097i	$0.155421\pi$
$308$	38.4779	2.19248
$309$	1.65339	0.0940582
$310$	2.43391	0.138237
$311$	7.92631	0.449460	0.224730	−	0.974421i	$-0.427850\pi$
$311$	7.92631	0.449460	0.224730	+	0.974421i	$0.427850\pi$
$312$	−4.68261	−0.265100
$313$	−16.5024	−0.932770	−0.466385	−	0.884582i	$-0.654444\pi$
$313$	−16.5024	−0.932770	−0.466385	+	0.884582i	$0.654444\pi$
$314$	−19.5884	−1.10544
$315$	2.58128	0.145439
$316$	−50.5624	−2.84436
$317$	19.5248	1.09662	0.548312	−	0.836274i	$-0.315271\pi$
$317$	19.5248	1.09662	0.548312	+	0.836274i	$0.315271\pi$
$318$	−28.8423	−1.61740
$319$	11.0138	0.616655
$320$	−8.86725	−0.495695
$321$	13.2252	0.738156
$322$	−41.5543	−2.31573
$323$	−4.01727	−0.223527
$324$	3.92391	0.217995
$325$	−1.00000	−0.0554700
$326$	21.1723	1.17262
$327$	15.9023	0.879400
$328$	16.7806	0.926554
$329$	−3.44652	−0.190013
$330$	9.24617	0.508985
$331$	−6.03635	−0.331788	−0.165894	−	0.986144i	$-0.553051\pi$
$331$	−6.03635	−0.331788	−0.165894	+	0.986144i	$0.553051\pi$
$332$	19.9529	1.09506
$333$	−2.76848	−0.151712
$334$	−39.3715	−2.15431
$335$	12.5455	0.685433
$336$	9.16153	0.499802
$337$	12.6598	0.689624	0.344812	−	0.938672i	$-0.387943\pi$
$337$	12.6598	0.689624	0.344812	+	0.938672i	$0.387943\pi$
$338$	2.43391	0.132387
$339$	1.40484	0.0763006
$340$	13.9091	0.754329
$341$	3.79890	0.205722
$342$	−2.75837	−0.149156
$343$	−18.9388	−1.02260
$344$	−26.9196	−1.45141
$345$	−6.61420	−0.356097
$346$	−23.4203	−1.25908
$347$	13.8340	0.742649	0.371325	−	0.928503i	$-0.378904\pi$
$347$	13.8340	0.742649	0.371325	+	0.928503i	$0.378904\pi$
$348$	11.3762	0.609830
$349$	1.73884	0.0930781	0.0465390	−	0.998916i	$-0.485181\pi$
$349$	1.73884	0.0930781	0.0465390	+	0.998916i	$0.485181\pi$
$350$	6.28259	0.335819
$351$	−1.00000	−0.0533761
$352$	−2.76080	−0.147151
$353$	9.35678	0.498011	0.249005	−	0.968502i	$-0.419896\pi$
$353$	9.35678	0.498011	0.249005	+	0.968502i	$0.419896\pi$
$354$	−12.6976	−0.674872
$355$	−1.91917	−0.101859
$356$	−7.64159	−0.405003
$357$	9.14991	0.484264
$358$	−40.7279	−2.15254
$359$	24.4396	1.28987	0.644936	−	0.764237i	$-0.276884\pi$
$359$	24.4396	1.28987	0.644936	+	0.764237i	$0.276884\pi$
$360$	4.68261	0.246795
$361$	−17.7156	−0.932400
$362$	7.52208	0.395352
$363$	3.43163	0.180114
$364$	−10.1287	−0.530888
$365$	−2.02562	−0.106026
$366$	2.55803	0.133710
$367$	25.4648	1.32925	0.664625	−	0.747177i	$-0.268591\pi$
$367$	25.4648	1.32925	0.664625	+	0.747177i	$0.268591\pi$
$368$	−23.4753	−1.22373
$369$	3.58360	0.186555
$370$	−6.73821	−0.350303
$371$	−30.5887	−1.58809
$372$	3.92391	0.203445
$373$	−4.39001	−0.227306	−0.113653	−	0.993521i	$-0.536255\pi$
$373$	−4.39001	−0.227306	−0.113653	+	0.993521i	$0.536255\pi$
$374$	32.7751	1.69476
$375$	1.00000	0.0516398
$376$	−6.25221	−0.322433
$377$	−2.89921	−0.149317
$378$	6.28259	0.323142
$379$	−13.6835	−0.702872	−0.351436	−	0.936212i	$-0.614307\pi$
$379$	−13.6835	−0.702872	−0.351436	+	0.936212i	$0.614307\pi$
$380$	−4.44700	−0.228126
$381$	8.41750	0.431242
$382$	0.389687	0.0199381
$383$	18.9814	0.969905	0.484952	−	0.874541i	$-0.338837\pi$
$383$	18.9814	0.969905	0.484952	+	0.874541i	$0.338837\pi$
$384$	−20.1286	−1.02718
$385$	9.80601	0.499761
$386$	35.4097	1.80231
$387$	−5.74884	−0.292230
$388$	−13.9957	−0.710526
$389$	7.72050	0.391445	0.195722	−	0.980659i	$-0.437295\pi$
$389$	7.72050	0.391445	0.195722	+	0.980659i	$0.437295\pi$
$390$	−2.43391	−0.123246
$391$	−23.4455	−1.18569
$392$	−1.57806	−0.0797039
$393$	−3.95539	−0.199523
$394$	13.0734	0.658628
$395$	−12.8857	−0.648352
$396$	14.9065	0.749081
$397$	−24.8076	−1.24506	−0.622528	−	0.782598i	$-0.713894\pi$
$397$	−24.8076	−1.24506	−0.622528	+	0.782598i	$0.713894\pi$
$398$	−25.1260	−1.25945
$399$	−2.92539	−0.146453
$400$	3.54922	0.177461
$401$	−26.0151	−1.29913	−0.649565	−	0.760306i	$-0.725049\pi$
$401$	−26.0151	−1.29913	−0.649565	+	0.760306i	$0.725049\pi$
$402$	30.5346	1.52293
$403$	−1.00000	−0.0498135
$404$	−52.2046	−2.59728
$405$	1.00000	0.0496904
$406$	18.2146	0.903973
$407$	−10.5172	−0.521316
$408$	16.5985	0.821750
$409$	26.7814	1.32426	0.662128	−	0.749390i	$-0.269653\pi$
$409$	26.7814	1.32426	0.662128	+	0.749390i	$0.269653\pi$
$410$	8.72215	0.430756
$411$	11.1141	0.548218
$412$	6.48775	0.319629
$413$	−13.4665	−0.662642
$414$	−16.0984	−0.791191
$415$	5.08496	0.249611
$416$	0.726737	0.0356312
$417$	3.76455	0.184351
$418$	−10.4788	−0.512533
$419$	−13.4194	−0.655582	−0.327791	−	0.944750i	$-0.606304\pi$
$419$	−13.4194	−0.655582	−0.327791	+	0.944750i	$0.606304\pi$
$420$	10.1287	0.494229
$421$	24.9819	1.21754	0.608771	−	0.793346i	$-0.291663\pi$
$421$	24.9819	1.21754	0.608771	+	0.793346i	$0.291663\pi$
$422$	−1.90577	−0.0927714
$423$	−1.33520	−0.0649196
$424$	−55.4899	−2.69483
$425$	3.54472	0.171944
$426$	−4.67109	−0.226315
$427$	2.71292	0.131287
$428$	51.8942	2.50840
$429$	−3.79890	−0.183413
$430$	−13.9921	−0.674761
$431$	−17.8955	−0.861997	−0.430999	−	0.902353i	$-0.641839\pi$
$431$	−17.8955	−0.861997	−0.430999	+	0.902353i	$0.641839\pi$
$432$	3.54922	0.170762
$433$	2.43377	0.116960	0.0584799	−	0.998289i	$-0.481375\pi$
$433$	2.43377	0.116960	0.0584799	+	0.998289i	$0.481375\pi$
$434$	6.28259	0.301574
$435$	2.89921	0.139006
$436$	62.3992	2.98838
$437$	7.49594	0.358580
$438$	−4.93017	−0.235573
$439$	−11.2509	−0.536977	−0.268488	−	0.963283i	$-0.586524\pi$
$439$	−11.2509	−0.536977	−0.268488	+	0.963283i	$0.586524\pi$
$440$	17.7888	0.848045
$441$	−0.337004	−0.0160478
$442$	−8.62752	−0.410369
$443$	−16.2863	−0.773784	−0.386892	−	0.922125i	$-0.626451\pi$
$443$	−16.2863	−0.773784	−0.386892	+	0.922125i	$0.626451\pi$
$444$	−10.8632	−0.515546
$445$	−1.94744	−0.0923177
$446$	−44.3195	−2.09859
$447$	−5.92448	−0.280219
$448$	−22.8888	−1.08140
$449$	−0.773083	−0.0364840	−0.0182420	−	0.999834i	$-0.505807\pi$
$449$	−0.773083	−0.0364840	−0.0182420	+	0.999834i	$0.505807\pi$
$450$	2.43391	0.114735
$451$	13.6137	0.641046
$452$	5.51247	0.259285
$453$	12.1898	0.572726
$454$	1.61497	0.0757942
$455$	−2.58128	−0.121012
$456$	−5.30684	−0.248516
$457$	−5.28836	−0.247379	−0.123689	−	0.992321i	$-0.539473\pi$
$457$	−5.28836	−0.247379	−0.123689	+	0.992321i	$0.539473\pi$
$458$	−19.5687	−0.914386
$459$	3.54472	0.165453
$460$	−25.9535	−1.21009
$461$	31.8688	1.48428	0.742140	−	0.670245i	$-0.233811\pi$
$461$	31.8688	1.48428	0.742140	+	0.670245i	$0.233811\pi$
$462$	23.8669	1.11039
$463$	−22.7160	−1.05570	−0.527851	−	0.849337i	$-0.677002\pi$
$463$	−22.7160	−1.05570	−0.527851	+	0.849337i	$0.677002\pi$
$464$	10.2899	0.477698
$465$	1.00000	0.0463739
$466$	62.8653	2.91218
$467$	−30.0782	−1.39185	−0.695927	−	0.718112i	$-0.745006\pi$
$467$	−30.0782	−1.39185	−0.695927	+	0.718112i	$0.745006\pi$
$468$	−3.92391	−0.181383
$469$	32.3834	1.49533
$470$	−3.24975	−0.149900
$471$	−8.04812	−0.370838
$472$	−24.4291	−1.12444
$473$	−21.8393	−1.00417
$474$	−31.3627	−1.44054
$475$	−1.13331	−0.0519998
$476$	35.9034	1.64563
$477$	−11.8502	−0.542584
$478$	−6.85863	−0.313707
$479$	33.0197	1.50871	0.754354	−	0.656468i	$-0.227950\pi$
$479$	33.0197	1.50871	0.754354	+	0.656468i	$0.227950\pi$
$480$	−0.726737	−0.0331709
$481$	2.76848	0.126232
$482$	21.5261	0.980488
$483$	−17.0731	−0.776853
$484$	13.4654	0.612063
$485$	−3.56679	−0.161960
$486$	2.43391	0.110404
$487$	−35.4636	−1.60701	−0.803504	−	0.595300i	$-0.797033\pi$
$487$	−35.4636	−1.60701	−0.803504	+	0.595300i	$0.797033\pi$
$488$	4.92141	0.222782
$489$	8.69888	0.393377
$490$	−0.820236	−0.0370545
$491$	−23.4637	−1.05890	−0.529451	−	0.848341i	$-0.677602\pi$
$491$	−23.4637	−1.05890	−0.529451	+	0.848341i	$0.677602\pi$
$492$	14.0617	0.633951
$493$	10.2769	0.462848
$494$	2.75837	0.124105
$495$	3.79890	0.170748
$496$	3.54922	0.159365
$497$	−4.95392	−0.222214
$498$	12.3763	0.554596
$499$	−34.1226	−1.52754	−0.763770	−	0.645489i	$-0.776654\pi$
$499$	−34.1226	−1.52754	−0.763770	+	0.645489i	$0.776654\pi$
$500$	3.92391	0.175482
$501$	−16.1763	−0.722702
$502$	72.8662	3.25218
$503$	5.12389	0.228463	0.114232	−	0.993454i	$-0.463559\pi$
$503$	5.12389	0.228463	0.114232	+	0.993454i	$0.463559\pi$
$504$	12.0871	0.538403
$505$	−13.3042	−0.592031
$506$	−61.1560	−2.71872
$507$	1.00000	0.0444116
$508$	33.0295	1.46545
$509$	10.6560	0.472317	0.236159	−	0.971715i	$-0.424111\pi$
$509$	10.6560	0.472317	0.236159	+	0.971715i	$0.424111\pi$
$510$	8.62752	0.382033
$511$	−5.22869	−0.231304
$512$	−35.8186	−1.58297
$513$	−1.13331	−0.0500368
$514$	16.3747	0.722258
$515$	1.65339	0.0728571
$516$	−22.5579	−0.993056
$517$	−5.07228	−0.223079
$518$	−17.3932	−0.764214
$519$	−9.62250	−0.422381
$520$	−4.68261	−0.205346
$521$	11.6490	0.510352	0.255176	−	0.966895i	$-0.417867\pi$
$521$	11.6490	0.510352	0.255176	+	0.966895i	$0.417867\pi$
$522$	7.05641	0.308851
$523$	10.7277	0.469088	0.234544	−	0.972105i	$-0.424640\pi$
$523$	10.7277	0.469088	0.234544	+	0.972105i	$0.424640\pi$
$524$	−15.5206	−0.678019
$525$	2.58128	0.112656
$526$	33.9589	1.48068
$527$	3.54472	0.154410
$528$	13.4831	0.586778
$529$	20.7477	0.902074
$530$	−28.8423	−1.25283
$531$	−5.21698	−0.226398
$532$	−11.4789	−0.497675
$533$	−3.58360	−0.155223
$534$	−4.73990	−0.205116
$535$	13.2252	0.571773
$536$	58.7456	2.53742
$537$	−16.7335	−0.722106
$538$	−46.3778	−1.99949
$539$	−1.28024	−0.0551440
$540$	3.92391	0.168858
$541$	31.3003	1.34570	0.672852	−	0.739777i	$-0.265069\pi$
$541$	31.3003	1.34570	0.672852	+	0.739777i	$0.265069\pi$
$542$	17.2741	0.741988
$543$	3.09054	0.132628
$544$	−2.57608	−0.110449
$545$	15.9023	0.681180
$546$	−6.28259	−0.268870
$547$	−11.2855	−0.482534	−0.241267	−	0.970459i	$-0.577563\pi$
$547$	−11.2855	−0.482534	−0.241267	+	0.970459i	$0.577563\pi$
$548$	43.6106	1.86295
$549$	1.05100	0.0448555
$550$	9.24617	0.394258
$551$	−3.28570	−0.139976
$552$	−30.9717	−1.31824
$553$	−33.2617	−1.41443
$554$	−29.9389	−1.27198
$555$	−2.76848	−0.117515
$556$	14.7717	0.626461
$557$	−9.03049	−0.382634	−0.191317	−	0.981528i	$-0.561276\pi$
$557$	−9.03049	−0.382634	−0.191317	+	0.981528i	$0.561276\pi$
$558$	2.43391	0.103036
$559$	5.74884	0.243150
$560$	9.16153	0.387145
$561$	13.4660	0.568536
$562$	−34.7362	−1.46526
$563$	28.7174	1.21029	0.605147	−	0.796114i	$-0.293114\pi$
$563$	28.7174	1.21029	0.605147	+	0.796114i	$0.293114\pi$
$564$	−5.23919	−0.220610
$565$	1.40484	0.0591022
$566$	−37.4504	−1.57416
$567$	2.58128	0.108403
$568$	−8.98673	−0.377075
$569$	−31.6280	−1.32591	−0.662957	−	0.748658i	$-0.730699\pi$
$569$	−31.6280	−1.32591	−0.662957	+	0.748658i	$0.730699\pi$
$570$	−2.75837	−0.115535
$571$	3.09074	0.129343	0.0646717	−	0.997907i	$-0.479400\pi$
$571$	3.09074	0.129343	0.0646717	+	0.997907i	$0.479400\pi$
$572$	−14.9065	−0.623273
$573$	0.160108	0.00668859
$574$	22.5143	0.939729
$575$	−6.61420	−0.275831
$576$	−8.86725	−0.369469
$577$	42.1242	1.75365	0.876826	−	0.480808i	$-0.159657\pi$
$577$	42.1242	1.75365	0.876826	+	0.480808i	$0.159657\pi$
$578$	−10.7943	−0.448983
$579$	14.5485	0.604615
$580$	11.3762	0.472372
$581$	13.1257	0.544546
$582$	−8.68124	−0.359849
$583$	−45.0178	−1.86445
$584$	−9.48519	−0.392500
$585$	−1.00000	−0.0413449
$586$	−2.26683	−0.0936420
$587$	25.8104	1.06531	0.532654	−	0.846333i	$-0.321195\pi$
$587$	25.8104	1.06531	0.532654	+	0.846333i	$0.321195\pi$
$588$	−1.32237	−0.0545337
$589$	−1.13331	−0.0466972
$590$	−12.6976	−0.522754
$591$	5.37136	0.220948
$592$	−9.82594	−0.403843
$593$	−27.6866	−1.13695	−0.568476	−	0.822700i	$-0.692467\pi$
$593$	−27.6866	−1.13695	−0.568476	+	0.822700i	$0.692467\pi$
$594$	9.24617	0.379375
$595$	9.14991	0.375110
$596$	−23.2471	−0.952239
$597$	−10.3233	−0.422505
$598$	16.0984	0.658311
$599$	14.3266	0.585370	0.292685	−	0.956209i	$-0.405451\pi$
$599$	14.3266	0.585370	0.292685	+	0.956209i	$0.405451\pi$
$600$	4.68261	0.191167
$601$	−13.7066	−0.559103	−0.279551	−	0.960131i	$-0.590186\pi$
$601$	−13.7066	−0.559103	−0.279551	+	0.960131i	$0.590186\pi$
$602$	−36.1176	−1.47204
$603$	12.5455	0.510892
$604$	47.8315	1.94624
$605$	3.43163	0.139516
$606$	−32.3813	−1.31540
$607$	−19.8250	−0.804673	−0.402336	−	0.915492i	$-0.631802\pi$
$607$	−19.8250	−0.804673	−0.402336	+	0.915492i	$0.631802\pi$
$608$	0.823618	0.0334021
$609$	7.48367	0.303253
$610$	2.55803	0.103572
$611$	1.33520	0.0540163
$612$	13.9091	0.562244
$613$	−17.8990	−0.722935	−0.361467	−	0.932385i	$-0.617724\pi$
$613$	−17.8990	−0.722935	−0.361467	+	0.932385i	$0.617724\pi$
$614$	75.3245	3.03985
$615$	3.58360	0.144505
$616$	45.9177	1.85008
$617$	17.2529	0.694574	0.347287	−	0.937759i	$-0.387103\pi$
$617$	17.2529	0.694574	0.347287	+	0.937759i	$0.387103\pi$
$618$	4.02420	0.161877
$619$	−40.8405	−1.64152	−0.820758	−	0.571275i	$-0.806449\pi$
$619$	−40.8405	−1.64152	−0.820758	+	0.571275i	$0.806449\pi$
$620$	3.92391	0.157588
$621$	−6.61420	−0.265419
$622$	19.2919	0.773535
$623$	−5.02690	−0.201398
$624$	−3.54922	−0.142083
$625$	1.00000	0.0400000
$626$	−40.1653	−1.60533
$627$	−4.30533	−0.171938
$628$	−31.5801	−1.26018
$629$	−9.81347	−0.391289
$630$	6.28259	0.250304
$631$	−43.1110	−1.71622	−0.858111	−	0.513464i	$-0.828362\pi$
$631$	−43.1110	−1.71622	−0.858111	+	0.513464i	$0.828362\pi$
$632$	−60.3389	−2.40015
$633$	−0.783008	−0.0311218
$634$	47.5216	1.88732
$635$	8.41750	0.334038
$636$	−46.4991	−1.84381
$637$	0.337004	0.0133526
$638$	26.8066	1.06128
$639$	−1.91917	−0.0759213
$640$	−20.1286	−0.795653
$641$	−22.9807	−0.907682	−0.453841	−	0.891083i	$-0.649947\pi$
$641$	−22.9807	−0.907682	−0.453841	+	0.891083i	$0.649947\pi$
$642$	32.1888	1.27039
$643$	36.7530	1.44940	0.724698	−	0.689066i	$-0.241979\pi$
$643$	36.7530	1.44940	0.724698	+	0.689066i	$0.241979\pi$
$644$	−66.9932	−2.63990
$645$	−5.74884	−0.226360
$646$	−9.77765	−0.384697
$647$	21.6522	0.851235	0.425618	−	0.904903i	$-0.360057\pi$
$647$	21.6522	0.851235	0.425618	+	0.904903i	$0.360057\pi$
$648$	4.68261	0.183950
$649$	−19.8188	−0.777955
$650$	−2.43391	−0.0954657
$651$	2.58128	0.101168
$652$	34.1336	1.33677
$653$	15.2716	0.597623	0.298811	−	0.954312i	$-0.403410\pi$
$653$	15.2716	0.597623	0.298811	+	0.954312i	$0.403410\pi$
$654$	38.7048	1.51348
$655$	−3.95539	−0.154550
$656$	12.7190	0.496593
$657$	−2.02562	−0.0790270
$658$	−8.38850	−0.327018
$659$	13.5945	0.529567	0.264784	−	0.964308i	$-0.414699\pi$
$659$	13.5945	0.529567	0.264784	+	0.964308i	$0.414699\pi$
$660$	14.9065	0.580235
$661$	−20.6708	−0.804000	−0.402000	−	0.915640i	$-0.631685\pi$
$661$	−20.6708	−0.804000	−0.402000	+	0.915640i	$0.631685\pi$
$662$	−14.6919	−0.571017
$663$	−3.54472	−0.137666
$664$	23.8109	0.924041
$665$	−2.92539	−0.113442
$666$	−6.73821	−0.261101
$667$	−19.1760	−0.742497
$668$	−63.4741	−2.45589
$669$	−18.2092	−0.704009
$670$	30.5346	1.17965
$671$	3.99263	0.154134
$672$	−1.87591	−0.0723648
$673$	−15.8404	−0.610601	−0.305301	−	0.952256i	$-0.598757\pi$
$673$	−15.8404	−0.610601	−0.305301	+	0.952256i	$0.598757\pi$
$674$	30.8128	1.18687
$675$	1.00000	0.0384900
$676$	3.92391	0.150919
$677$	−6.41753	−0.246646	−0.123323	−	0.992367i	$-0.539355\pi$
$677$	−6.41753	−0.246646	−0.123323	+	0.992367i	$0.539355\pi$
$678$	3.41926	0.131316
$679$	−9.20688	−0.353327
$680$	16.5985	0.636525
$681$	0.663529	0.0254265
$682$	9.24617	0.354054
$683$	27.3324	1.04584	0.522922	−	0.852380i	$-0.324842\pi$
$683$	27.3324	1.04584	0.522922	+	0.852380i	$0.324842\pi$
$684$	−4.44700	−0.170035
$685$	11.1141	0.424647
$686$	−46.0954	−1.75993
$687$	−8.04004	−0.306747
$688$	−20.4039	−0.777892
$689$	11.8502	0.451457
$690$	−16.0984	−0.612854
$691$	−1.07733	−0.0409837	−0.0204918	−	0.999790i	$-0.506523\pi$
$691$	−1.07733	−0.0409837	−0.0204918	+	0.999790i	$0.506523\pi$
$692$	−37.7578	−1.43534
$693$	9.80601	0.372500
$694$	33.6707	1.27812
$695$	3.76455	0.142797
$696$	13.5759	0.514592
$697$	12.7029	0.481155
$698$	4.23218	0.160190
$699$	25.8290	0.976942
$700$	10.1287	0.382829
$701$	−18.9748	−0.716668	−0.358334	−	0.933594i	$-0.616655\pi$
$701$	−18.9748	−0.716668	−0.358334	+	0.933594i	$0.616655\pi$
$702$	−2.43391	−0.0918619
$703$	3.13754	0.118335
$704$	−33.6858	−1.26958
$705$	−1.33520	−0.0502865
$706$	22.7735	0.857093
$707$	−34.3420	−1.29156
$708$	−20.4709	−0.769345
$709$	−6.45188	−0.242305	−0.121153	−	0.992634i	$-0.538659\pi$
$709$	−6.45188	−0.242305	−0.121153	+	0.992634i	$0.538659\pi$
$710$	−4.67109	−0.175303
$711$	−12.8857	−0.483253
$712$	−9.11912	−0.341753
$713$	−6.61420	−0.247704
$714$	22.2700	0.833435
$715$	−3.79890	−0.142071
$716$	−65.6608	−2.45386
$717$	−2.81795	−0.105238
$718$	59.4837	2.21991
$719$	−24.9895	−0.931953	−0.465976	−	0.884797i	$-0.654297\pi$
$719$	−24.9895	−0.931953	−0.465976	+	0.884797i	$0.654297\pi$
$720$	3.54922	0.132272
$721$	4.26786	0.158944
$722$	−43.1182	−1.60469
$723$	8.84427	0.328922
$724$	12.1270	0.450696
$725$	2.89921	0.107674
$726$	8.35227	0.309982
$727$	20.4505	0.758468	0.379234	−	0.925301i	$-0.376188\pi$
$727$	20.4505	0.758468	0.379234	+	0.925301i	$0.376188\pi$
$728$	−12.0871	−0.447978
$729$	1.00000	0.0370370
$730$	−4.93017	−0.182474
$731$	−20.3780	−0.753708
$732$	4.12401	0.152428
$733$	−0.243568	−0.00899639	−0.00449820	−	0.999990i	$-0.501432\pi$
$733$	−0.243568	−0.00899639	−0.00449820	+	0.999990i	$0.501432\pi$
$734$	61.9789	2.28768
$735$	−0.337004	−0.0124306
$736$	4.80679	0.177181
$737$	47.6590	1.75554
$738$	8.72215	0.321067
$739$	14.4050	0.529895	0.264948	−	0.964263i	$-0.414645\pi$
$739$	14.4050	0.529895	0.264948	+	0.964263i	$0.414645\pi$
$740$	−10.8632	−0.399341
$741$	1.13331	0.0416332
$742$	−74.4501	−2.73315
$743$	−36.3289	−1.33278	−0.666389	−	0.745604i	$-0.732161\pi$
$743$	−36.3289	−1.33278	−0.666389	+	0.745604i	$0.732161\pi$
$744$	4.68261	0.171673
$745$	−5.92448	−0.217056
$746$	−10.6849	−0.391201
$747$	5.08496	0.186049
$748$	52.8394	1.93200
$749$	34.1378	1.24737
$750$	2.43391	0.0888737
$751$	−5.31308	−0.193877	−0.0969385	−	0.995290i	$-0.530905\pi$
$751$	−5.31308	−0.193877	−0.0969385	+	0.995290i	$0.530905\pi$
$752$	−4.73891	−0.172810
$753$	29.9380	1.09100
$754$	−7.05641	−0.256979
$755$	12.1898	0.443632
$756$	10.1287	0.368377
$757$	−34.5576	−1.25602	−0.628009	−	0.778206i	$-0.716130\pi$
$757$	−34.5576	−1.25602	−0.628009	+	0.778206i	$0.716130\pi$
$758$	−33.3043	−1.20967
$759$	−25.1267	−0.912041
$760$	−5.30684	−0.192499
$761$	−45.1097	−1.63523	−0.817613	−	0.575769i	$-0.804703\pi$
$761$	−45.1097	−1.63523	−0.817613	+	0.575769i	$0.804703\pi$
$762$	20.4874	0.742181
$763$	41.0483	1.48605
$764$	0.628247	0.0227292
$765$	3.54472	0.128160
$766$	46.1990	1.66924
$767$	5.21698	0.188374
$768$	−31.2566	−1.12788
$769$	53.7151	1.93701	0.968507	−	0.248985i	$-0.0800971\pi$
$769$	53.7151	1.93701	0.968507	+	0.248985i	$0.0800971\pi$
$770$	23.8669	0.860104
$771$	6.72775	0.242294
$772$	57.0870	2.05461
$773$	16.9822	0.610806	0.305403	−	0.952223i	$-0.401209\pi$
$773$	16.9822	0.610806	0.305403	+	0.952223i	$0.401209\pi$
$774$	−13.9921	−0.502937
$775$	1.00000	0.0359211
$776$	−16.7019	−0.599562
$777$	−7.14621	−0.256369
$778$	18.7910	0.673689
$779$	−4.06133	−0.145512
$780$	−3.92391	−0.140498
$781$	−7.29074	−0.260883
$782$	−57.0642	−2.04061
$783$	2.89921	0.103609
$784$	−1.19610	−0.0427179
$785$	−8.04812	−0.287250
$786$	−9.62704	−0.343385
$787$	−2.27860	−0.0812232	−0.0406116	−	0.999175i	$-0.512931\pi$
$787$	−2.27860	−0.0812232	−0.0406116	+	0.999175i	$0.512931\pi$
$788$	21.0767	0.750827
$789$	13.9524	0.496720
$790$	−31.3627	−1.11583
$791$	3.62629	0.128936
$792$	17.7888	0.632096
$793$	−1.05100	−0.0373220
$794$	−60.3793	−2.14278
$795$	−11.8502	−0.420284
$796$	−40.5077	−1.43576
$797$	35.3836	1.25335	0.626676	−	0.779280i	$-0.284415\pi$
$797$	35.3836	1.25335	0.626676	+	0.779280i	$0.284415\pi$
$798$	−7.12012	−0.252050
$799$	−4.73290	−0.167438
$800$	−0.726737	−0.0256940
$801$	−1.94744	−0.0688096
$802$	−63.3182	−2.23585
$803$	−7.69513	−0.271555
$804$	49.2273	1.73611
$805$	−17.0731	−0.601748
$806$	−2.43391	−0.0857307
$807$	−19.0549	−0.670763
$808$	−62.2986	−2.19166
$809$	37.5442	1.31998	0.659991	−	0.751273i	$-0.270560\pi$
$809$	37.5442	1.31998	0.659991	+	0.751273i	$0.270560\pi$
$810$	2.43391	0.0855188
$811$	54.3498	1.90848	0.954240	−	0.299043i	$-0.0966673\pi$
$811$	54.3498	1.90848	0.954240	+	0.299043i	$0.0966673\pi$
$812$	29.3652	1.03052
$813$	7.09729	0.248913
$814$	−25.5978	−0.897202
$815$	8.69888	0.304708
$816$	12.5810	0.440423
$817$	6.51522	0.227939
$818$	65.1835	2.27909
$819$	−2.58128	−0.0901972
$820$	14.0617	0.491056
$821$	−23.9455	−0.835705	−0.417852	−	0.908515i	$-0.637217\pi$
$821$	−23.9455	−0.835705	−0.417852	+	0.908515i	$0.637217\pi$
$822$	27.0507	0.943500
$823$	−48.1735	−1.67922	−0.839611	−	0.543189i	$-0.817217\pi$
$823$	−48.1735	−1.67922	−0.839611	+	0.543189i	$0.817217\pi$
$824$	7.74218	0.269712
$825$	3.79890	0.132261
$826$	−32.7761	−1.14043
$827$	−22.1978	−0.771893	−0.385947	−	0.922521i	$-0.626125\pi$
$827$	−22.1978	−0.771893	−0.385947	+	0.922521i	$0.626125\pi$
$828$	−25.9535	−0.901947
$829$	−5.14127	−0.178564	−0.0892819	−	0.996006i	$-0.528457\pi$
$829$	−5.14127	−0.178564	−0.0892819	+	0.996006i	$0.528457\pi$
$830$	12.3763	0.429588
$831$	−12.3007	−0.426708
$832$	8.86725	0.307417
$833$	−1.19458	−0.0413899
$834$	9.16256	0.317273
$835$	−16.1763	−0.559802
$836$	−16.8937	−0.584281
$837$	1.00000	0.0345651
$838$	−32.6617	−1.12828
$839$	−10.0007	−0.345264	−0.172632	−	0.984986i	$-0.555227\pi$
$839$	−10.0007	−0.345264	−0.172632	+	0.984986i	$0.555227\pi$
$840$	12.0871	0.417045
$841$	−20.5946	−0.710158
$842$	60.8035	2.09543
$843$	−14.2718	−0.491546
$844$	−3.07245	−0.105758
$845$	1.00000	0.0344010
$846$	−3.24975	−0.111729
$847$	8.85799	0.304364
$848$	−42.0590	−1.44431
$849$	−15.3869	−0.528078
$850$	8.62752	0.295921
$851$	18.3113	0.627702
$852$	−7.53065	−0.257996
$853$	46.0361	1.57625	0.788123	−	0.615517i	$-0.211053\pi$
$853$	46.0361	1.57625	0.788123	+	0.615517i	$0.211053\pi$
$854$	6.60299	0.225950
$855$	−1.13331	−0.0387584
$856$	61.9282	2.11666
$857$	−4.43595	−0.151529	−0.0757645	−	0.997126i	$-0.524140\pi$
$857$	−4.43595	−0.151529	−0.0757645	+	0.997126i	$0.524140\pi$
$858$	−9.24617	−0.315659
$859$	−6.39437	−0.218173	−0.109087	−	0.994032i	$-0.534793\pi$
$859$	−6.39437	−0.218173	−0.109087	+	0.994032i	$0.534793\pi$
$860$	−22.5579	−0.769218
$861$	9.25027	0.315248
$862$	−43.5561	−1.48353
$863$	−41.5460	−1.41424	−0.707121	−	0.707092i	$-0.750006\pi$
$863$	−41.5460	−1.41424	−0.707121	+	0.707092i	$0.750006\pi$
$864$	−0.726737	−0.0247241
$865$	−9.62250	−0.327175
$866$	5.92358	0.201291
$867$	−4.43496	−0.150619
$868$	10.1287	0.343790
$869$	−48.9516	−1.66057
$870$	7.05641	0.239235
$871$	−12.5455	−0.425088
$872$	74.4643	2.52168
$873$	−3.56679	−0.120718
$874$	18.2444	0.617127
$875$	2.58128	0.0872631
$876$	−7.94834	−0.268550
$877$	−16.7993	−0.567271	−0.283635	−	0.958932i	$-0.591541\pi$
$877$	−16.7993	−0.567271	−0.283635	+	0.958932i	$0.591541\pi$
$878$	−27.3837	−0.924155
$879$	−0.931356	−0.0314139
$880$	13.4831	0.454516
$881$	24.0474	0.810178	0.405089	−	0.914277i	$-0.367241\pi$
$881$	24.0474	0.810178	0.405089	+	0.914277i	$0.367241\pi$
$882$	−0.820236	−0.0276188
$883$	−16.9042	−0.568870	−0.284435	−	0.958695i	$-0.591806\pi$
$883$	−16.9042	−0.568870	−0.284435	+	0.958695i	$0.591806\pi$
$884$	−13.9091	−0.467815
$885$	−5.21698	−0.175367
$886$	−39.6393	−1.33171
$887$	−5.01375	−0.168345	−0.0841725	−	0.996451i	$-0.526825\pi$
$887$	−5.01375	−0.168345	−0.0841725	+	0.996451i	$0.526825\pi$
$888$	−12.9637	−0.435033
$889$	21.7279	0.728731
$890$	−4.73990	−0.158882
$891$	3.79890	0.127268
$892$	−71.4512	−2.39236
$893$	1.51319	0.0506371
$894$	−14.4196	−0.482265
$895$	−16.7335	−0.559341
$896$	−51.9575	−1.73578
$897$	6.61420	0.220842
$898$	−1.88161	−0.0627902
$899$	2.89921	0.0966941
$900$	3.92391	0.130797
$901$	−42.0057	−1.39941
$902$	33.1346	1.10326
$903$	−14.8394	−0.493823
$904$	6.57833	0.218792
$905$	3.09054	0.102733
$906$	29.6688	0.985680
$907$	−35.9584	−1.19398	−0.596990	−	0.802249i	$-0.703637\pi$
$907$	−35.9584	−1.19398	−0.596990	+	0.802249i	$0.703637\pi$
$908$	2.60363	0.0864044
$909$	−13.3042	−0.441274
$910$	−6.28259	−0.208266
$911$	39.0878	1.29504	0.647518	−	0.762050i	$-0.275807\pi$
$911$	39.0878	1.29504	0.647518	+	0.762050i	$0.275807\pi$
$912$	−4.02237	−0.133194
$913$	19.3173	0.639308
$914$	−12.8714	−0.425747
$915$	1.05100	0.0347449
$916$	−31.5483	−1.04239
$917$	−10.2099	−0.337162
$918$	8.62752	0.284751
$919$	18.7584	0.618783	0.309391	−	0.950935i	$-0.399875\pi$
$919$	18.7584	0.618783	0.309391	+	0.950935i	$0.399875\pi$
$920$	−30.9717	−1.02111
$921$	30.9480	1.01977
$922$	77.5658	2.55449
$923$	1.91917	0.0631703
$924$	38.4779	1.26583
$925$	−2.76848	−0.0910269
$926$	−55.2886	−1.81690
$927$	1.65339	0.0543045
$928$	−2.10696	−0.0691645
$929$	46.0424	1.51060	0.755301	−	0.655378i	$-0.227491\pi$
$929$	46.0424	1.51060	0.755301	+	0.655378i	$0.227491\pi$
$930$	2.43391	0.0798110
$931$	0.381930	0.0125172
$932$	101.350	3.31985
$933$	7.92631	0.259496
$934$	−73.2076	−2.39543
$935$	13.4660	0.440386
$936$	−4.68261	−0.153056
$937$	−42.1719	−1.37769	−0.688847	−	0.724906i	$-0.741883\pi$
$937$	−42.1719	−1.37769	−0.688847	+	0.724906i	$0.741883\pi$
$938$	78.8182	2.57351
$939$	−16.5024	−0.538535
$940$	−5.23919	−0.170884
$941$	−7.99505	−0.260631	−0.130316	−	0.991473i	$-0.541599\pi$
$941$	−7.99505	−0.260631	−0.130316	+	0.991473i	$0.541599\pi$
$942$	−19.5884	−0.638224
$943$	−23.7027	−0.771865
$944$	−18.5162	−0.602651
$945$	2.58128	0.0839690
$946$	−53.1547	−1.72821
$947$	32.5740	1.05851	0.529256	−	0.848462i	$-0.322471\pi$
$947$	32.5740	1.05851	0.529256	+	0.848462i	$0.322471\pi$
$948$	−50.5624	−1.64219
$949$	2.02562	0.0657544
$950$	−2.75837	−0.0894934
$951$	19.5248	0.633136
$952$	42.8454	1.38863
$953$	36.8445	1.19351	0.596755	−	0.802423i	$-0.296456\pi$
$953$	36.8445	1.19351	0.596755	+	0.802423i	$0.296456\pi$
$954$	−28.8423	−0.933805
$955$	0.160108	0.00518096
$956$	−11.0574	−0.357621
$957$	11.0138	0.356026
$958$	80.3669	2.59654
$959$	28.6885	0.926402
$960$	−8.86725	−0.286189
$961$	1.00000	0.0322581
$962$	6.73821	0.217249
$963$	13.2252	0.426175
$964$	34.7041	1.11774
$965$	14.5485	0.468333
$966$	−41.5543	−1.33699
$967$	30.5310	0.981812	0.490906	−	0.871212i	$-0.336666\pi$
$967$	30.5310	0.981812	0.490906	+	0.871212i	$0.336666\pi$
$968$	16.0690	0.516477
$969$	−4.01727	−0.129053
$970$	−8.68124	−0.278738
$971$	25.5656	0.820440	0.410220	−	0.911987i	$-0.365452\pi$
$971$	25.5656	0.820440	0.410220	+	0.911987i	$0.365452\pi$
$972$	3.92391	0.125859
$973$	9.71734	0.311524
$974$	−86.3150	−2.76571
$975$	−1.00000	−0.0320256
$976$	3.73022	0.119402
$977$	3.35895	0.107462	0.0537312	−	0.998555i	$-0.482889\pi$
$977$	3.35895	0.107462	0.0537312	+	0.998555i	$0.482889\pi$
$978$	21.1723	0.677014
$979$	−7.39814	−0.236446
$980$	−1.32237	−0.0422416
$981$	15.9023	0.507722
$982$	−57.1084	−1.82240
$983$	22.2142	0.708523	0.354262	−	0.935146i	$-0.384732\pi$
$983$	22.2142	0.708523	0.354262	+	0.935146i	$0.384732\pi$
$984$	16.7806	0.534946
$985$	5.37136	0.171146
$986$	25.0130	0.796576
$987$	−3.44652	−0.109704
$988$	4.44700	0.141478
$989$	38.0240	1.20909
$990$	9.24617	0.293862
$991$	14.0262	0.445557	0.222779	−	0.974869i	$-0.428487\pi$
$991$	14.0262	0.445557	0.222779	+	0.974869i	$0.428487\pi$
$992$	−0.726737	−0.0230739
$993$	−6.03635	−0.191558
$994$	−12.0574	−0.382437
$995$	−10.3233	−0.327271
$996$	19.9529	0.632232
$997$	58.5207	1.85337	0.926684	−	0.375840i	$-0.122646\pi$
$997$	58.5207	1.85337	0.926684	+	0.375840i	$0.122646\pi$
$998$	−83.0513	−2.62894
$999$	−2.76848	−0.0875907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bi.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bi.1.16	✓	18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q+2.43391 q^{2} +1.00000 q^{3} +3.92391 q^{4} +1.00000 q^{5} +2.43391 q^{6} +2.58128 q^{7} +4.68261 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.43391 q^{2} +1.00000 q^{3} +3.92391 q^{4} +1.00000 q^{5} +2.43391 q^{6} +2.58128 q^{7} +4.68261 q^{8} +1.00000 q^{9} +2.43391 q^{10} +3.79890 q^{11} +3.92391 q^{12} -1.00000 q^{13} +6.28259 q^{14} +1.00000 q^{15} +3.54922 q^{16} +3.54472 q^{17} +2.43391 q^{18} -1.13331 q^{19} +3.92391 q^{20} +2.58128 q^{21} +9.24617 q^{22} -6.61420 q^{23} +4.68261 q^{24} +1.00000 q^{25} -2.43391 q^{26} +1.00000 q^{27} +10.1287 q^{28} +2.89921 q^{29} +2.43391 q^{30} +1.00000 q^{31} -0.726737 q^{32} +3.79890 q^{33} +8.62752 q^{34} +2.58128 q^{35} +3.92391 q^{36} -2.76848 q^{37} -2.75837 q^{38} -1.00000 q^{39} +4.68261 q^{40} +3.58360 q^{41} +6.28259 q^{42} -5.74884 q^{43} +14.9065 q^{44} +1.00000 q^{45} -16.0984 q^{46} -1.33520 q^{47} +3.54922 q^{48} -0.337004 q^{49} +2.43391 q^{50} +3.54472 q^{51} -3.92391 q^{52} -11.8502 q^{53} +2.43391 q^{54} +3.79890 q^{55} +12.0871 q^{56} -1.13331 q^{57} +7.05641 q^{58} -5.21698 q^{59} +3.92391 q^{60} +1.05100 q^{61} +2.43391 q^{62} +2.58128 q^{63} -8.86725 q^{64} -1.00000 q^{65} +9.24617 q^{66} +12.5455 q^{67} +13.9091 q^{68} -6.61420 q^{69} +6.28259 q^{70} -1.91917 q^{71} +4.68261 q^{72} -2.02562 q^{73} -6.73821 q^{74} +1.00000 q^{75} -4.44700 q^{76} +9.80601 q^{77} -2.43391 q^{78} -12.8857 q^{79} +3.54922 q^{80} +1.00000 q^{81} +8.72215 q^{82} +5.08496 q^{83} +10.1287 q^{84} +3.54472 q^{85} -13.9921 q^{86} +2.89921 q^{87} +17.7888 q^{88} -1.94744 q^{89} +2.43391 q^{90} -2.58128 q^{91} -25.9535 q^{92} +1.00000 q^{93} -3.24975 q^{94} -1.13331 q^{95} -0.726737 q^{96} -3.56679 q^{97} -0.820236 q^{98} +3.79890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 18 q^{3} + 22 q^{4} + 18 q^{5} + 4 q^{6} + 8 q^{7} + 9 q^{8} + 18 q^{9} + 4 q^{10} + 6 q^{11} + 22 q^{12} - 18 q^{13} + 5 q^{14} + 18 q^{15} + 30 q^{16} + 18 q^{17} + 4 q^{18} + 12 q^{19} + 22 q^{20} + 8 q^{21} + 7 q^{22} + 32 q^{23} + 9 q^{24} + 18 q^{25} - 4 q^{26} + 18 q^{27} + 10 q^{28} + 7 q^{29} + 4 q^{30} + 18 q^{31} + 22 q^{32} + 6 q^{33} + 15 q^{34} + 8 q^{35} + 22 q^{36} + 3 q^{37} + 32 q^{38} - 18 q^{39} + 9 q^{40} + 4 q^{41} + 5 q^{42} + 14 q^{43} - 5 q^{44} + 18 q^{45} + 10 q^{46} + 23 q^{47} + 30 q^{48} + 28 q^{49} + 4 q^{50} + 18 q^{51} - 22 q^{52} + 35 q^{53} + 4 q^{54} + 6 q^{55} - 7 q^{56} + 12 q^{57} - 6 q^{58} + 28 q^{59} + 22 q^{60} + 19 q^{61} + 4 q^{62} + 8 q^{63} + 43 q^{64} - 18 q^{65} + 7 q^{66} + 34 q^{67} + 55 q^{68} + 32 q^{69} + 5 q^{70} - 8 q^{71} + 9 q^{72} + 22 q^{74} + 18 q^{75} + 2 q^{76} + 21 q^{77} - 4 q^{78} + 4 q^{79} + 30 q^{80} + 18 q^{81} + 29 q^{82} + 11 q^{83} + 10 q^{84} + 18 q^{85} - 22 q^{86} + 7 q^{87} - 31 q^{88} + 17 q^{89} + 4 q^{90} - 8 q^{91} + 33 q^{92} + 18 q^{93} - 14 q^{94} + 12 q^{95} + 22 q^{96} + 32 q^{97} + 20 q^{98} + 6 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 18 * q^3 + 22 * q^4 + 18 * q^5 + 4 * q^6 + 8 * q^7 + 9 * q^8 + 18 * q^9 + 4 * q^10 + 6 * q^11 + 22 * q^12 - 18 * q^13 + 5 * q^14 + 18 * q^15 + 30 * q^16 + 18 * q^17 + 4 * q^18 + 12 * q^19 + 22 * q^20 + 8 * q^21 + 7 * q^22 + 32 * q^23 + 9 * q^24 + 18 * q^25 - 4 * q^26 + 18 * q^27 + 10 * q^28 + 7 * q^29 + 4 * q^30 + 18 * q^31 + 22 * q^32 + 6 * q^33 + 15 * q^34 + 8 * q^35 + 22 * q^36 + 3 * q^37 + 32 * q^38 - 18 * q^39 + 9 * q^40 + 4 * q^41 + 5 * q^42 + 14 * q^43 - 5 * q^44 + 18 * q^45 + 10 * q^46 + 23 * q^47 + 30 * q^48 + 28 * q^49 + 4 * q^50 + 18 * q^51 - 22 * q^52 + 35 * q^53 + 4 * q^54 + 6 * q^55 - 7 * q^56 + 12 * q^57 - 6 * q^58 + 28 * q^59 + 22 * q^60 + 19 * q^61 + 4 * q^62 + 8 * q^63 + 43 * q^64 - 18 * q^65 + 7 * q^66 + 34 * q^67 + 55 * q^68 + 32 * q^69 + 5 * q^70 - 8 * q^71 + 9 * q^72 + 22 * q^74 + 18 * q^75 + 2 * q^76 + 21 * q^77 - 4 * q^78 + 4 * q^79 + 30 * q^80 + 18 * q^81 + 29 * q^82 + 11 * q^83 + 10 * q^84 + 18 * q^85 - 22 * q^86 + 7 * q^87 - 31 * q^88 + 17 * q^89 + 4 * q^90 - 8 * q^91 + 33 * q^92 + 18 * q^93 - 14 * q^94 + 12 * q^95 + 22 * q^96 + 32 * q^97 + 20 * q^98 + 6 * q^99

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