LMFDB - Embedded newform 6045.2.a.bg.1.10

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 27 x^{14} + 51 x^{13} + 294 x^{12} - 517 x^{11} - 1657 x^{10} + 2678 x^{9} + \cdots - 428 $ x^16 - 2x^15 - 27x^14 + 51x^13 + 294x^12 - 517x^11 - 1657x^10 + 2678x^9 + 5124x^8 - 7583x^7 - 8358x^6 + 11579x^5 + 6027x^4 - 8572x^3 - 780x^2 + 2178*x - 428
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.733028$ 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)

$f(q)$	$=$	$q+0.733028 q^{2} -1.00000 q^{3} -1.46267 q^{4} -1.00000 q^{5} -0.733028 q^{6} -2.61356 q^{7} -2.53823 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.733028 q^{2} -1.00000 q^{3} -1.46267 q^{4} -1.00000 q^{5} -0.733028 q^{6} -2.61356 q^{7} -2.53823 q^{8} +1.00000 q^{9} -0.733028 q^{10} +5.62541 q^{11} +1.46267 q^{12} -1.00000 q^{13} -1.91581 q^{14} +1.00000 q^{15} +1.06474 q^{16} -4.98877 q^{17} +0.733028 q^{18} -6.33054 q^{19} +1.46267 q^{20} +2.61356 q^{21} +4.12358 q^{22} -2.74538 q^{23} +2.53823 q^{24} +1.00000 q^{25} -0.733028 q^{26} -1.00000 q^{27} +3.82278 q^{28} +5.32259 q^{29} +0.733028 q^{30} +1.00000 q^{31} +5.85695 q^{32} -5.62541 q^{33} -3.65691 q^{34} +2.61356 q^{35} -1.46267 q^{36} -11.2849 q^{37} -4.64046 q^{38} +1.00000 q^{39} +2.53823 q^{40} +5.87780 q^{41} +1.91581 q^{42} -10.4737 q^{43} -8.22812 q^{44} -1.00000 q^{45} -2.01244 q^{46} -3.09699 q^{47} -1.06474 q^{48} -0.169291 q^{49} +0.733028 q^{50} +4.98877 q^{51} +1.46267 q^{52} -14.4563 q^{53} -0.733028 q^{54} -5.62541 q^{55} +6.63383 q^{56} +6.33054 q^{57} +3.90161 q^{58} +8.08658 q^{59} -1.46267 q^{60} -9.72246 q^{61} +0.733028 q^{62} -2.61356 q^{63} +2.16382 q^{64} +1.00000 q^{65} -4.12358 q^{66} +6.40652 q^{67} +7.29693 q^{68} +2.74538 q^{69} +1.91581 q^{70} +2.18901 q^{71} -2.53823 q^{72} +8.62448 q^{73} -8.27214 q^{74} -1.00000 q^{75} +9.25949 q^{76} -14.7024 q^{77} +0.733028 q^{78} -12.2422 q^{79} -1.06474 q^{80} +1.00000 q^{81} +4.30859 q^{82} +9.46434 q^{83} -3.82278 q^{84} +4.98877 q^{85} -7.67750 q^{86} -5.32259 q^{87} -14.2786 q^{88} -11.3138 q^{89} -0.733028 q^{90} +2.61356 q^{91} +4.01559 q^{92} -1.00000 q^{93} -2.27018 q^{94} +6.33054 q^{95} -5.85695 q^{96} +6.18383 q^{97} -0.124095 q^{98} +5.62541 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.733028	0.518329	0.259164	−	0.965833i	$-0.416553\pi$
$2$	0.733028	0.518329	0.259164	+	0.965833i	$0.416553\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.46267	−0.731335
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.733028	−0.299257
$7$	−2.61356	−0.987834	−0.493917	−	0.869509i	$-0.664435\pi$
$7$	−2.61356	−0.987834	−0.493917	+	0.869509i	$0.664435\pi$
$8$	−2.53823	−0.897401
$9$	1.00000	0.333333
$10$	−0.733028	−0.231804
$11$	5.62541	1.69613	0.848063	−	0.529896i	$-0.177769\pi$
$11$	5.62541	1.69613	0.848063	+	0.529896i	$0.177769\pi$
$12$	1.46267	0.422237
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.91581	−0.512023
$15$	1.00000	0.258199
$16$	1.06474	0.266186
$17$	−4.98877	−1.20996	−0.604978	−	0.796242i	$-0.706818\pi$
$17$	−4.98877	−1.20996	−0.604978	+	0.796242i	$0.706818\pi$
$18$	0.733028	0.172776
$19$	−6.33054	−1.45233	−0.726163	−	0.687523i	$-0.758698\pi$
$19$	−6.33054	−1.45233	−0.726163	+	0.687523i	$0.758698\pi$
$20$	1.46267	0.327063
$21$	2.61356	0.570326
$22$	4.12358	0.879151
$23$	−2.74538	−0.572452	−0.286226	−	0.958162i	$-0.592401\pi$
$23$	−2.74538	−0.572452	−0.286226	+	0.958162i	$0.592401\pi$
$24$	2.53823	0.518115
$25$	1.00000	0.200000
$26$	−0.733028	−0.143759
$27$	−1.00000	−0.192450
$28$	3.82278	0.722438
$29$	5.32259	0.988380	0.494190	−	0.869354i	$-0.335465\pi$
$29$	5.32259	0.988380	0.494190	+	0.869354i	$0.335465\pi$
$30$	0.733028	0.133832
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.85695	1.03537
$33$	−5.62541	−0.979258
$34$	−3.65691	−0.627155
$35$	2.61356	0.441773
$36$	−1.46267	−0.243778
$37$	−11.2849	−1.85523	−0.927613	−	0.373544i	$-0.878143\pi$
$37$	−11.2849	−1.85523	−0.927613	+	0.373544i	$0.878143\pi$
$38$	−4.64046	−0.752782
$39$	1.00000	0.160128
$40$	2.53823	0.401330
$41$	5.87780	0.917958	0.458979	−	0.888447i	$-0.348215\pi$
$41$	5.87780	0.917958	0.458979	+	0.888447i	$0.348215\pi$
$42$	1.91581	0.295617
$43$	−10.4737	−1.59722	−0.798611	−	0.601848i	$-0.794431\pi$
$43$	−10.4737	−1.59722	−0.798611	+	0.601848i	$0.794431\pi$
$44$	−8.22812	−1.24044
$45$	−1.00000	−0.149071
$46$	−2.01244	−0.296718
$47$	−3.09699	−0.451742	−0.225871	−	0.974157i	$-0.572523\pi$
$47$	−3.09699	−0.451742	−0.225871	+	0.974157i	$0.572523\pi$
$48$	−1.06474	−0.153683
$49$	−0.169291	−0.0241844
$50$	0.733028	0.103666
$51$	4.98877	0.698568
$52$	1.46267	0.202836
$53$	−14.4563	−1.98572	−0.992862	−	0.119272i	$-0.961944\pi$
$53$	−14.4563	−1.98572	−0.992862	+	0.119272i	$0.961944\pi$
$54$	−0.733028	−0.0997525
$55$	−5.62541	−0.758530
$56$	6.63383	0.886483
$57$	6.33054	0.838500
$58$	3.90161	0.512306
$59$	8.08658	1.05278	0.526392	−	0.850242i	$-0.323545\pi$
$59$	8.08658	1.05278	0.526392	+	0.850242i	$0.323545\pi$
$60$	−1.46267	−0.188830
$61$	−9.72246	−1.24483	−0.622417	−	0.782686i	$-0.713849\pi$
$61$	−9.72246	−1.24483	−0.622417	+	0.782686i	$0.713849\pi$
$62$	0.733028	0.0930946
$63$	−2.61356	−0.329278
$64$	2.16382	0.270478
$65$	1.00000	0.124035
$66$	−4.12358	−0.507578
$67$	6.40652	0.782681	0.391340	−	0.920246i	$-0.372011\pi$
$67$	6.40652	0.782681	0.391340	+	0.920246i	$0.372011\pi$
$68$	7.29693	0.884883
$69$	2.74538	0.330505
$70$	1.91581	0.228984
$71$	2.18901	0.259787	0.129894	−	0.991528i	$-0.458536\pi$
$71$	2.18901	0.259787	0.129894	+	0.991528i	$0.458536\pi$
$72$	−2.53823	−0.299134
$73$	8.62448	1.00942	0.504710	−	0.863289i	$-0.331600\pi$
$73$	8.62448	1.00942	0.504710	+	0.863289i	$0.331600\pi$
$74$	−8.27214	−0.961617
$75$	−1.00000	−0.115470
$76$	9.25949	1.06214
$77$	−14.7024	−1.67549
$78$	0.733028	0.0829991
$79$	−12.2422	−1.37736	−0.688679	−	0.725066i	$-0.741809\pi$
$79$	−12.2422	−1.37736	−0.688679	+	0.725066i	$0.741809\pi$
$80$	−1.06474	−0.119042
$81$	1.00000	0.111111
$82$	4.30859	0.475804
$83$	9.46434	1.03885	0.519423	−	0.854517i	$-0.326147\pi$
$83$	9.46434	1.03885	0.519423	+	0.854517i	$0.326147\pi$
$84$	−3.82278	−0.417099
$85$	4.98877	0.541108
$86$	−7.67750	−0.827886
$87$	−5.32259	−0.570642
$88$	−14.2786	−1.52210
$89$	−11.3138	−1.19926	−0.599631	−	0.800276i	$-0.704686\pi$
$89$	−11.3138	−1.19926	−0.599631	+	0.800276i	$0.704686\pi$
$90$	−0.733028	−0.0772679
$91$	2.61356	0.273976
$92$	4.01559	0.418654
$93$	−1.00000	−0.103695
$94$	−2.27018	−0.234151
$95$	6.33054	0.649500
$96$	−5.85695	−0.597773
$97$	6.18383	0.627873	0.313937	−	0.949444i	$-0.398352\pi$
$97$	6.18383	0.627873	0.313937	+	0.949444i	$0.398352\pi$
$98$	−0.124095	−0.0125355
$99$	5.62541	0.565375
$100$	−1.46267	−0.146267
$101$	15.4095	1.53330	0.766652	−	0.642062i	$-0.221921\pi$
$101$	15.4095	1.53330	0.766652	+	0.642062i	$0.221921\pi$
$102$	3.65691	0.362088
$103$	−14.3380	−1.41276	−0.706380	−	0.707832i	$-0.749673\pi$
$103$	−14.3380	−1.41276	−0.706380	+	0.707832i	$0.749673\pi$
$104$	2.53823	0.248894
$105$	−2.61356	−0.255058
$106$	−10.5969	−1.02926
$107$	−5.63323	−0.544585	−0.272292	−	0.962215i	$-0.587782\pi$
$107$	−5.63323	−0.544585	−0.272292	+	0.962215i	$0.587782\pi$
$108$	1.46267	0.140746
$109$	14.5713	1.39567	0.697837	−	0.716257i	$-0.254146\pi$
$109$	14.5713	1.39567	0.697837	+	0.716257i	$0.254146\pi$
$110$	−4.12358	−0.393168
$111$	11.2849	1.07111
$112$	−2.78278	−0.262948
$113$	−8.79554	−0.827415	−0.413708	−	0.910410i	$-0.635766\pi$
$113$	−8.79554	−0.827415	−0.413708	+	0.910410i	$0.635766\pi$
$114$	4.64046	0.434619
$115$	2.74538	0.256008
$116$	−7.78519	−0.722837
$117$	−1.00000	−0.0924500
$118$	5.92769	0.545688
$119$	13.0385	1.19523
$120$	−2.53823	−0.231708
$121$	20.6453	1.87684
$122$	−7.12684	−0.645233
$123$	−5.87780	−0.529983
$124$	−1.46267	−0.131352
$125$	−1.00000	−0.0894427
$126$	−1.91581	−0.170674
$127$	13.1001	1.16245	0.581223	−	0.813744i	$-0.302574\pi$
$127$	13.1001	1.16245	0.581223	+	0.813744i	$0.302574\pi$
$128$	−10.1278	−0.895177
$129$	10.4737	0.922156
$130$	0.733028	0.0642908
$131$	17.9810	1.57101	0.785504	−	0.618857i	$-0.212404\pi$
$131$	17.9810	1.57101	0.785504	+	0.618857i	$0.212404\pi$
$132$	8.22812	0.716166
$133$	16.5453	1.43466
$134$	4.69616	0.405686
$135$	1.00000	0.0860663
$136$	12.6627	1.08582
$137$	16.7193	1.42843	0.714213	−	0.699928i	$-0.246785\pi$
$137$	16.7193	1.42843	0.714213	+	0.699928i	$0.246785\pi$
$138$	2.01244	0.171311
$139$	−8.98230	−0.761869	−0.380934	−	0.924602i	$-0.624398\pi$
$139$	−8.98230	−0.761869	−0.380934	+	0.924602i	$0.624398\pi$
$140$	−3.82278	−0.323084
$141$	3.09699	0.260813
$142$	1.60460	0.134655
$143$	−5.62541	−0.470421
$144$	1.06474	0.0887287
$145$	−5.32259	−0.442017
$146$	6.32198	0.523211
$147$	0.169291	0.0139629
$148$	16.5061	1.35679
$149$	11.0295	0.903573	0.451786	−	0.892126i	$-0.350787\pi$
$149$	11.0295	0.903573	0.451786	+	0.892126i	$0.350787\pi$
$150$	−0.733028	−0.0598515
$151$	−14.6085	−1.18882	−0.594412	−	0.804161i	$-0.702615\pi$
$151$	−14.6085	−1.18882	−0.594412	+	0.804161i	$0.702615\pi$
$152$	16.0684	1.30332
$153$	−4.98877	−0.403318
$154$	−10.7772	−0.868455
$155$	−1.00000	−0.0803219
$156$	−1.46267	−0.117107
$157$	−7.34233	−0.585982	−0.292991	−	0.956115i	$-0.594651\pi$
$157$	−7.34233	−0.585982	−0.292991	+	0.956115i	$0.594651\pi$
$158$	−8.97389	−0.713925
$159$	14.4563	1.14646
$160$	−5.85695	−0.463033
$161$	7.17523	0.565488
$162$	0.733028	0.0575921
$163$	−11.9035	−0.932355	−0.466177	−	0.884691i	$-0.654369\pi$
$163$	−11.9035	−0.932355	−0.466177	+	0.884691i	$0.654369\pi$
$164$	−8.59728	−0.671335
$165$	5.62541	0.437938
$166$	6.93762	0.538464
$167$	1.67149	0.129344	0.0646720	−	0.997907i	$-0.479400\pi$
$167$	1.67149	0.129344	0.0646720	+	0.997907i	$0.479400\pi$
$168$	−6.63383	−0.511811
$169$	1.00000	0.0769231
$170$	3.65691	0.280472
$171$	−6.33054	−0.484108
$172$	15.3195	1.16810
$173$	10.9990	0.836236	0.418118	−	0.908393i	$-0.362690\pi$
$173$	10.9990	0.836236	0.418118	+	0.908393i	$0.362690\pi$
$174$	−3.90161	−0.295780
$175$	−2.61356	−0.197567
$176$	5.98963	0.451485
$177$	−8.08658	−0.607825
$178$	−8.29335	−0.621613
$179$	13.1469	0.982643	0.491322	−	0.870978i	$-0.336514\pi$
$179$	13.1469	0.982643	0.491322	+	0.870978i	$0.336514\pi$
$180$	1.46267	0.109021
$181$	−1.92166	−0.142836	−0.0714179	−	0.997446i	$-0.522752\pi$
$181$	−1.92166	−0.142836	−0.0714179	+	0.997446i	$0.522752\pi$
$182$	1.91581	0.142010
$183$	9.72246	0.718705
$184$	6.96843	0.513719
$185$	11.2849	0.829682
$186$	−0.733028	−0.0537482
$187$	−28.0639	−2.05224
$188$	4.52987	0.330375
$189$	2.61356	0.190109
$190$	4.64046	0.336654
$191$	−10.8977	−0.788532	−0.394266	−	0.918996i	$-0.629001\pi$
$191$	−10.8977	−0.788532	−0.394266	+	0.918996i	$0.629001\pi$
$192$	−2.16382	−0.156160
$193$	14.4564	1.04059	0.520296	−	0.853986i	$-0.325822\pi$
$193$	14.4564	1.04059	0.520296	+	0.853986i	$0.325822\pi$
$194$	4.53292	0.325445
$195$	−1.00000	−0.0716115
$196$	0.247617	0.0176869
$197$	−6.02738	−0.429433	−0.214716	−	0.976676i	$-0.568883\pi$
$197$	−6.02738	−0.429433	−0.214716	+	0.976676i	$0.568883\pi$
$198$	4.12358	0.293050
$199$	−6.74965	−0.478470	−0.239235	−	0.970962i	$-0.576897\pi$
$199$	−6.74965	−0.478470	−0.239235	+	0.970962i	$0.576897\pi$
$200$	−2.53823	−0.179480
$201$	−6.40652	−0.451881
$202$	11.2956	0.794756
$203$	−13.9109	−0.976355
$204$	−7.29693	−0.510887
$205$	−5.87780	−0.410523
$206$	−10.5101	−0.732275
$207$	−2.74538	−0.190817
$208$	−1.06474	−0.0738267
$209$	−35.6119	−2.46333
$210$	−1.91581	−0.132204
$211$	1.32189	0.0910024	0.0455012	−	0.998964i	$-0.485512\pi$
$211$	1.32189	0.0910024	0.0455012	+	0.998964i	$0.485512\pi$
$212$	21.1448	1.45223
$213$	−2.18901	−0.149988
$214$	−4.12931	−0.282274
$215$	10.4737	0.714299
$216$	2.53823	0.172705
$217$	−2.61356	−0.177420
$218$	10.6811	0.723418
$219$	−8.62448	−0.582788
$220$	8.22812	0.554740
$221$	4.98877	0.335581
$222$	8.27214	0.555190
$223$	27.4608	1.83891	0.919456	−	0.393193i	$-0.128630\pi$
$223$	27.4608	1.83891	0.919456	+	0.393193i	$0.128630\pi$
$224$	−15.3075	−1.02278
$225$	1.00000	0.0666667
$226$	−6.44738	−0.428873
$227$	19.8265	1.31593	0.657967	−	0.753047i	$-0.271417\pi$
$227$	19.8265	1.31593	0.657967	+	0.753047i	$0.271417\pi$
$228$	−9.25949	−0.613225
$229$	0.372697	0.0246285	0.0123143	−	0.999924i	$-0.496080\pi$
$229$	0.372697	0.0246285	0.0123143	+	0.999924i	$0.496080\pi$
$230$	2.01244	0.132697
$231$	14.7024	0.967345
$232$	−13.5100	−0.886973
$233$	−12.9525	−0.848548	−0.424274	−	0.905534i	$-0.639471\pi$
$233$	−12.9525	−0.848548	−0.424274	+	0.905534i	$0.639471\pi$
$234$	−0.733028	−0.0479195
$235$	3.09699	0.202025
$236$	−11.8280	−0.769937
$237$	12.2422	0.795218
$238$	9.55756	0.619525
$239$	13.8368	0.895029	0.447514	−	0.894277i	$-0.352309\pi$
$239$	13.8368	0.895029	0.447514	+	0.894277i	$0.352309\pi$
$240$	1.06474	0.0687290
$241$	3.75060	0.241597	0.120799	−	0.992677i	$-0.461454\pi$
$241$	3.75060	0.241597	0.120799	+	0.992677i	$0.461454\pi$
$242$	15.1335	0.972821
$243$	−1.00000	−0.0641500
$244$	14.2208	0.910391
$245$	0.169291	0.0108156
$246$	−4.30859	−0.274706
$247$	6.33054	0.402803
$248$	−2.53823	−0.161178
$249$	−9.46434	−0.599778
$250$	−0.733028	−0.0463608
$251$	29.3235	1.85088	0.925441	−	0.378892i	$-0.123695\pi$
$251$	29.3235	1.85088	0.925441	+	0.378892i	$0.123695\pi$
$252$	3.82278	0.240813
$253$	−15.4439	−0.970951
$254$	9.60274	0.602530
$255$	−4.98877	−0.312409
$256$	−11.7516	−0.734474
$257$	22.0390	1.37475	0.687377	−	0.726300i	$-0.258762\pi$
$257$	22.0390	1.37475	0.687377	+	0.726300i	$0.258762\pi$
$258$	7.67750	0.477980
$259$	29.4938	1.83265
$260$	−1.46267	−0.0907110
$261$	5.32259	0.329460
$262$	13.1806	0.814299
$263$	−11.3961	−0.702715	−0.351357	−	0.936241i	$-0.614280\pi$
$263$	−11.3961	−0.702715	−0.351357	+	0.936241i	$0.614280\pi$
$264$	14.2786	0.878788
$265$	14.4563	0.888042
$266$	12.1281	0.743624
$267$	11.3138	0.692395
$268$	−9.37062	−0.572402
$269$	−3.87259	−0.236116	−0.118058	−	0.993007i	$-0.537667\pi$
$269$	−3.87259	−0.236116	−0.118058	+	0.993007i	$0.537667\pi$
$270$	0.733028	0.0446107
$271$	−22.6857	−1.37806	−0.689028	−	0.724735i	$-0.741962\pi$
$271$	−22.6857	−1.37806	−0.689028	+	0.724735i	$0.741962\pi$
$272$	−5.31177	−0.322073
$273$	−2.61356	−0.158180
$274$	12.2557	0.740395
$275$	5.62541	0.339225
$276$	−4.01559	−0.241710
$277$	3.97394	0.238771	0.119385	−	0.992848i	$-0.461908\pi$
$277$	3.97394	0.238771	0.119385	+	0.992848i	$0.461908\pi$
$278$	−6.58428	−0.394899
$279$	1.00000	0.0598684
$280$	−6.63383	−0.396447
$281$	27.6414	1.64895	0.824474	−	0.565900i	$-0.191471\pi$
$281$	27.6414	1.64895	0.824474	+	0.565900i	$0.191471\pi$
$282$	2.27018	0.135187
$283$	28.2959	1.68202	0.841010	−	0.541020i	$-0.181962\pi$
$283$	28.2959	1.68202	0.841010	+	0.541020i	$0.181962\pi$
$284$	−3.20180	−0.189992
$285$	−6.33054	−0.374989
$286$	−4.12358	−0.243833
$287$	−15.3620	−0.906790
$288$	5.85695	0.345124
$289$	7.88786	0.463992
$290$	−3.90161	−0.229110
$291$	−6.18383	−0.362503
$292$	−12.6148	−0.738224
$293$	2.25580	0.131785	0.0658927	−	0.997827i	$-0.479010\pi$
$293$	2.25580	0.131785	0.0658927	+	0.997827i	$0.479010\pi$
$294$	0.124095	0.00723736
$295$	−8.08658	−0.470819
$296$	28.6437	1.66488
$297$	−5.62541	−0.326419
$298$	8.08494	0.468348
$299$	2.74538	0.158770
$300$	1.46267	0.0844473
$301$	27.3736	1.57779
$302$	−10.7085	−0.616202
$303$	−15.4095	−0.885254
$304$	−6.74041	−0.386589
$305$	9.72246	0.556707
$306$	−3.65691	−0.209052
$307$	−0.882007	−0.0503388	−0.0251694	−	0.999683i	$-0.508013\pi$
$307$	−0.882007	−0.0503388	−0.0251694	+	0.999683i	$0.508013\pi$
$308$	21.5047	1.22534
$309$	14.3380	0.815658
$310$	−0.733028	−0.0416332
$311$	28.9076	1.63920	0.819601	−	0.572935i	$-0.194195\pi$
$311$	28.9076	1.63920	0.819601	+	0.572935i	$0.194195\pi$
$312$	−2.53823	−0.143699
$313$	7.69166	0.434758	0.217379	−	0.976087i	$-0.430249\pi$
$313$	7.69166	0.434758	0.217379	+	0.976087i	$0.430249\pi$
$314$	−5.38213	−0.303731
$315$	2.61356	0.147258
$316$	17.9063	1.00731
$317$	−12.3762	−0.695116	−0.347558	−	0.937659i	$-0.612989\pi$
$317$	−12.3762	−0.695116	−0.347558	+	0.937659i	$0.612989\pi$
$318$	10.5969	0.594242
$319$	29.9418	1.67642
$320$	−2.16382	−0.120961
$321$	5.63323	0.314416
$322$	5.25964	0.293109
$323$	31.5816	1.75725
$324$	−1.46267	−0.0812595
$325$	−1.00000	−0.0554700
$326$	−8.72560	−0.483266
$327$	−14.5713	−0.805792
$328$	−14.9192	−0.823777
$329$	8.09417	0.446246
$330$	4.12358	0.226996
$331$	11.4111	0.627209	0.313605	−	0.949554i	$-0.398463\pi$
$331$	11.4111	0.627209	0.313605	+	0.949554i	$0.398463\pi$
$332$	−13.8432	−0.759745
$333$	−11.2849	−0.618408
$334$	1.22525	0.0670427
$335$	−6.40652	−0.350026
$336$	2.78278	0.151813
$337$	−12.7263	−0.693246	−0.346623	−	0.938005i	$-0.612672\pi$
$337$	−12.7263	−0.693246	−0.346623	+	0.938005i	$0.612672\pi$
$338$	0.733028	0.0398715
$339$	8.79554	0.477708
$340$	−7.29693	−0.395732
$341$	5.62541	0.304633
$342$	−4.64046	−0.250927
$343$	18.7374	1.01172
$344$	26.5846	1.43335
$345$	−2.74538	−0.147806
$346$	8.06255	0.433445
$347$	3.91922	0.210395	0.105197	−	0.994451i	$-0.466453\pi$
$347$	3.91922	0.210395	0.105197	+	0.994451i	$0.466453\pi$
$348$	7.78519	0.417330
$349$	−33.6034	−1.79875	−0.899374	−	0.437180i	$-0.855977\pi$
$349$	−33.6034	−1.79875	−0.899374	+	0.437180i	$0.855977\pi$
$350$	−1.91581	−0.102405
$351$	1.00000	0.0533761
$352$	32.9478	1.75612
$353$	1.51406	0.0805853	0.0402927	−	0.999188i	$-0.487171\pi$
$353$	1.51406	0.0805853	0.0402927	+	0.999188i	$0.487171\pi$
$354$	−5.92769	−0.315053
$355$	−2.18901	−0.116180
$356$	16.5484	0.877063
$357$	−13.0385	−0.690069
$358$	9.63702	0.509332
$359$	−18.2705	−0.964281	−0.482141	−	0.876094i	$-0.660141\pi$
$359$	−18.2705	−0.964281	−0.482141	+	0.876094i	$0.660141\pi$
$360$	2.53823	0.133777
$361$	21.0757	1.10925
$362$	−1.40863	−0.0740359
$363$	−20.6453	−1.08360
$364$	−3.82278	−0.200368
$365$	−8.62448	−0.451426
$366$	7.12684	0.372526
$367$	8.19477	0.427764	0.213882	−	0.976860i	$-0.431389\pi$
$367$	8.19477	0.427764	0.213882	+	0.976860i	$0.431389\pi$
$368$	−2.92313	−0.152379
$369$	5.87780	0.305986
$370$	8.27214	0.430048
$371$	37.7824	1.96156
$372$	1.46267	0.0758359
$373$	8.50557	0.440402	0.220201	−	0.975455i	$-0.429329\pi$
$373$	8.50557	0.440402	0.220201	+	0.975455i	$0.429329\pi$
$374$	−20.5716	−1.06373
$375$	1.00000	0.0516398
$376$	7.86087	0.405394
$377$	−5.32259	−0.274127
$378$	1.91581	0.0985388
$379$	−10.5582	−0.542337	−0.271168	−	0.962532i	$-0.587410\pi$
$379$	−10.5582	−0.542337	−0.271168	+	0.962532i	$0.587410\pi$
$380$	−9.25949	−0.475002
$381$	−13.1001	−0.671139
$382$	−7.98834	−0.408719
$383$	36.2370	1.85162	0.925812	−	0.377985i	$-0.123383\pi$
$383$	36.2370	1.85162	0.925812	+	0.377985i	$0.123383\pi$
$384$	10.1278	0.516830
$385$	14.7024	0.749302
$386$	10.5969	0.539369
$387$	−10.4737	−0.532407
$388$	−9.04491	−0.459186
$389$	−32.4476	−1.64516	−0.822580	−	0.568649i	$-0.807466\pi$
$389$	−32.4476	−1.64516	−0.822580	+	0.568649i	$0.807466\pi$
$390$	−0.733028	−0.0371183
$391$	13.6961	0.692641
$392$	0.429700	0.0217031
$393$	−17.9810	−0.907022
$394$	−4.41824	−0.222588
$395$	12.2422	0.615973
$396$	−8.22812	−0.413479
$397$	−0.642668	−0.0322546	−0.0161273	−	0.999870i	$-0.505134\pi$
$397$	−0.642668	−0.0322546	−0.0161273	+	0.999870i	$0.505134\pi$
$398$	−4.94768	−0.248005
$399$	−16.5453	−0.828299
$400$	1.06474	0.0532372
$401$	−16.7290	−0.835407	−0.417703	−	0.908583i	$-0.637165\pi$
$401$	−16.7290	−0.835407	−0.417703	+	0.908583i	$0.637165\pi$
$402$	−4.69616	−0.234223
$403$	−1.00000	−0.0498135
$404$	−22.5390	−1.12136
$405$	−1.00000	−0.0496904
$406$	−10.1971	−0.506073
$407$	−63.4822	−3.14669
$408$	−12.6627	−0.626896
$409$	−32.9407	−1.62881	−0.814406	−	0.580296i	$-0.802937\pi$
$409$	−32.9407	−1.62881	−0.814406	+	0.580296i	$0.802937\pi$
$410$	−4.30859	−0.212786
$411$	−16.7193	−0.824703
$412$	20.9717	1.03320
$413$	−21.1348	−1.03997
$414$	−2.01244	−0.0989062
$415$	−9.46434	−0.464586
$416$	−5.85695	−0.287161
$417$	8.98230	0.439865
$418$	−26.1045	−1.27681
$419$	21.9842	1.07400	0.536998	−	0.843583i	$-0.319558\pi$
$419$	21.9842	1.07400	0.536998	+	0.843583i	$0.319558\pi$
$420$	3.82278	0.186533
$421$	28.7484	1.40111	0.700555	−	0.713598i	$-0.252936\pi$
$421$	28.7484	1.40111	0.700555	+	0.713598i	$0.252936\pi$
$422$	0.968979	0.0471692
$423$	−3.09699	−0.150581
$424$	36.6934	1.78199
$425$	−4.98877	−0.241991
$426$	−1.60460	−0.0777433
$427$	25.4103	1.22969
$428$	8.23956	0.398274
$429$	5.62541	0.271597
$430$	7.67750	0.370242
$431$	−31.4110	−1.51302	−0.756508	−	0.653985i	$-0.773096\pi$
$431$	−31.4110	−1.51302	−0.756508	+	0.653985i	$0.773096\pi$
$432$	−1.06474	−0.0512275
$433$	30.7485	1.47768	0.738840	−	0.673881i	$-0.235374\pi$
$433$	30.7485	1.47768	0.738840	+	0.673881i	$0.235374\pi$
$434$	−1.91581	−0.0919620
$435$	5.32259	0.255199
$436$	−21.3129	−1.02070
$437$	17.3798	0.831387
$438$	−6.32198	−0.302076
$439$	29.5321	1.40949	0.704746	−	0.709460i	$-0.251061\pi$
$439$	29.5321	1.40949	0.704746	+	0.709460i	$0.251061\pi$
$440$	14.2786	0.680706
$441$	−0.169291	−0.00806147
$442$	3.65691	0.173941
$443$	−33.2144	−1.57806	−0.789031	−	0.614353i	$-0.789417\pi$
$443$	−33.2144	−1.57806	−0.789031	+	0.614353i	$0.789417\pi$
$444$	−16.5061	−0.783344
$445$	11.3138	0.536327
$446$	20.1295	0.953161
$447$	−11.0295	−0.521678
$448$	−5.65528	−0.267187
$449$	−10.5797	−0.499288	−0.249644	−	0.968338i	$-0.580314\pi$
$449$	−10.5797	−0.499288	−0.249644	+	0.968338i	$0.580314\pi$
$450$	0.733028	0.0345553
$451$	33.0650	1.55697
$452$	12.8650	0.605118
$453$	14.6085	0.686368
$454$	14.5334	0.682086
$455$	−2.61356	−0.122526
$456$	−16.0684	−0.752471
$457$	13.7746	0.644350	0.322175	−	0.946680i	$-0.395586\pi$
$457$	13.7746	0.644350	0.322175	+	0.946680i	$0.395586\pi$
$458$	0.273197	0.0127657
$459$	4.98877	0.232856
$460$	−4.01559	−0.187228
$461$	−27.6120	−1.28602	−0.643010	−	0.765858i	$-0.722315\pi$
$461$	−27.6120	−1.28602	−0.643010	+	0.765858i	$0.722315\pi$
$462$	10.7772	0.501403
$463$	14.2059	0.660206	0.330103	−	0.943945i	$-0.392916\pi$
$463$	14.2059	0.660206	0.330103	+	0.943945i	$0.392916\pi$
$464$	5.66720	0.263093
$465$	1.00000	0.0463739
$466$	−9.49456	−0.439827
$467$	−37.6150	−1.74062	−0.870308	−	0.492507i	$-0.836081\pi$
$467$	−37.6150	−1.74062	−0.870308	+	0.492507i	$0.836081\pi$
$468$	1.46267	0.0676120
$469$	−16.7438	−0.773159
$470$	2.27018	0.104715
$471$	7.34233	0.338317
$472$	−20.5256	−0.944769
$473$	−58.9188	−2.70909
$474$	8.97389	0.412185
$475$	−6.33054	−0.290465
$476$	−19.0710	−0.874117
$477$	−14.4563	−0.661908
$478$	10.1428	0.463919
$479$	40.0884	1.83169	0.915844	−	0.401534i	$-0.131523\pi$
$479$	40.0884	1.83169	0.915844	+	0.401534i	$0.131523\pi$
$480$	5.85695	0.267332
$481$	11.2849	0.514547
$482$	2.74929	0.125227
$483$	−7.17523	−0.326484
$484$	−30.1972	−1.37260
$485$	−6.18383	−0.280793
$486$	−0.733028	−0.0332508
$487$	−24.0611	−1.09031	−0.545157	−	0.838334i	$-0.683530\pi$
$487$	−24.0611	−1.09031	−0.545157	+	0.838334i	$0.683530\pi$
$488$	24.6779	1.11712
$489$	11.9035	0.538295
$490$	0.124095	0.00560604
$491$	−14.5360	−0.656001	−0.328000	−	0.944678i	$-0.606375\pi$
$491$	−14.5360	−0.656001	−0.328000	+	0.944678i	$0.606375\pi$
$492$	8.59728	0.387595
$493$	−26.5532	−1.19590
$494$	4.64046	0.208784
$495$	−5.62541	−0.252843
$496$	1.06474	0.0478084
$497$	−5.72111	−0.256627
$498$	−6.93762	−0.310882
$499$	−15.2294	−0.681763	−0.340881	−	0.940106i	$-0.610725\pi$
$499$	−15.2294	−0.681763	−0.340881	+	0.940106i	$0.610725\pi$
$500$	1.46267	0.0654126
$501$	−1.67149	−0.0746768
$502$	21.4949	0.959366
$503$	22.4448	1.00077	0.500383	−	0.865804i	$-0.333193\pi$
$503$	22.4448	1.00077	0.500383	+	0.865804i	$0.333193\pi$
$504$	6.63383	0.295494
$505$	−15.4095	−0.685715
$506$	−11.3208	−0.503272
$507$	−1.00000	−0.0444116
$508$	−19.1611	−0.850138
$509$	−21.8650	−0.969150	−0.484575	−	0.874750i	$-0.661026\pi$
$509$	−21.8650	−0.969150	−0.484575	+	0.874750i	$0.661026\pi$
$510$	−3.65691	−0.161931
$511$	−22.5406	−0.997138
$512$	11.6413	0.514478
$513$	6.33054	0.279500
$514$	16.1552	0.712575
$515$	14.3380	0.631806
$516$	−15.3195	−0.674405
$517$	−17.4218	−0.766211
$518$	21.6198	0.949918
$519$	−10.9990	−0.482801
$520$	−2.53823	−0.111309
$521$	9.06235	0.397029	0.198514	−	0.980098i	$-0.436388\pi$
$521$	9.06235	0.397029	0.198514	+	0.980098i	$0.436388\pi$
$522$	3.90161	0.170769
$523$	13.5423	0.592165	0.296082	−	0.955162i	$-0.404320\pi$
$523$	13.5423	0.592165	0.296082	+	0.955162i	$0.404320\pi$
$524$	−26.3003	−1.14893
$525$	2.61356	0.114065
$526$	−8.35367	−0.364238
$527$	−4.98877	−0.217314
$528$	−5.98963	−0.260665
$529$	−15.4629	−0.672299
$530$	10.5969	0.460298
$531$	8.08658	0.350928
$532$	−24.2003	−1.04921
$533$	−5.87780	−0.254596
$534$	8.29335	0.358888
$535$	5.63323	0.243546
$536$	−16.2612	−0.702379
$537$	−13.1469	−0.567329
$538$	−2.83871	−0.122386
$539$	−0.952331	−0.0410198
$540$	−1.46267	−0.0629433
$541$	−32.0312	−1.37713	−0.688565	−	0.725175i	$-0.741759\pi$
$541$	−32.0312	−1.37713	−0.688565	+	0.725175i	$0.741759\pi$
$542$	−16.6292	−0.714287
$543$	1.92166	0.0824663
$544$	−29.2190	−1.25276
$545$	−14.5713	−0.624164
$546$	−1.91581	−0.0819893
$547$	−1.15302	−0.0492996	−0.0246498	−	0.999696i	$-0.507847\pi$
$547$	−1.15302	−0.0492996	−0.0246498	+	0.999696i	$0.507847\pi$
$548$	−24.4548	−1.04466
$549$	−9.72246	−0.414945
$550$	4.12358	0.175830
$551$	−33.6949	−1.43545
$552$	−6.96843	−0.296596
$553$	31.9958	1.36060
$554$	2.91301	0.123762
$555$	−11.2849	−0.479017
$556$	13.1381	0.557181
$557$	29.5885	1.25370	0.626852	−	0.779138i	$-0.284343\pi$
$557$	29.5885	1.25370	0.626852	+	0.779138i	$0.284343\pi$
$558$	0.733028	0.0310315
$559$	10.4737	0.442990
$560$	2.78278	0.117594
$561$	28.0639	1.18486
$562$	20.2619	0.854697
$563$	−2.43660	−0.102690	−0.0513452	−	0.998681i	$-0.516351\pi$
$563$	−2.43660	−0.102690	−0.0513452	+	0.998681i	$0.516351\pi$
$564$	−4.52987	−0.190742
$565$	8.79554	0.370031
$566$	20.7417	0.871839
$567$	−2.61356	−0.109759
$568$	−5.55621	−0.233133
$569$	−3.29874	−0.138290	−0.0691452	−	0.997607i	$-0.522027\pi$
$569$	−3.29874	−0.138290	−0.0691452	+	0.997607i	$0.522027\pi$
$570$	−4.64046	−0.194368
$571$	9.28840	0.388707	0.194354	−	0.980932i	$-0.437739\pi$
$571$	9.28840	0.388707	0.194354	+	0.980932i	$0.437739\pi$
$572$	8.22812	0.344035
$573$	10.8977	0.455259
$574$	−11.2608	−0.470016
$575$	−2.74538	−0.114490
$576$	2.16382	0.0901593
$577$	−13.9415	−0.580393	−0.290197	−	0.956967i	$-0.593721\pi$
$577$	−13.9415	−0.580393	−0.290197	+	0.956967i	$0.593721\pi$
$578$	5.78202	0.240500
$579$	−14.4564	−0.600786
$580$	7.78519	0.323263
$581$	−24.7356	−1.02621
$582$	−4.53292	−0.187896
$583$	−81.3225	−3.36804
$584$	−21.8909	−0.905854
$585$	1.00000	0.0413449
$586$	1.65357	0.0683082
$587$	4.89155	0.201896	0.100948	−	0.994892i	$-0.467812\pi$
$587$	4.89155	0.201896	0.100948	+	0.994892i	$0.467812\pi$
$588$	−0.247617	−0.0102115
$589$	−6.33054	−0.260845
$590$	−5.92769	−0.244039
$591$	6.02738	0.247933
$592$	−12.0155	−0.493835
$593$	45.8074	1.88109	0.940543	−	0.339675i	$-0.110317\pi$
$593$	45.8074	1.88109	0.940543	+	0.339675i	$0.110317\pi$
$594$	−4.12358	−0.169193
$595$	−13.0385	−0.534525
$596$	−16.1325	−0.660814
$597$	6.74965	0.276245
$598$	2.01244	0.0822949
$599$	−42.1719	−1.72310	−0.861549	−	0.507674i	$-0.830506\pi$
$599$	−42.1719	−1.72310	−0.861549	+	0.507674i	$0.830506\pi$
$600$	2.53823	0.103623
$601$	−17.7811	−0.725305	−0.362652	−	0.931924i	$-0.618129\pi$
$601$	−17.7811	−0.725305	−0.362652	+	0.931924i	$0.618129\pi$
$602$	20.0656	0.817814
$603$	6.40652	0.260894
$604$	21.3674	0.869429
$605$	−20.6453	−0.839349
$606$	−11.2956	−0.458853
$607$	26.9065	1.09210	0.546051	−	0.837752i	$-0.316130\pi$
$607$	26.9065	1.09210	0.546051	+	0.837752i	$0.316130\pi$
$608$	−37.0777	−1.50370
$609$	13.9109	0.563699
$610$	7.12684	0.288557
$611$	3.09699	0.125291
$612$	7.29693	0.294961
$613$	30.3316	1.22508	0.612542	−	0.790438i	$-0.290147\pi$
$613$	30.3316	1.22508	0.612542	+	0.790438i	$0.290147\pi$
$614$	−0.646536	−0.0260921
$615$	5.87780	0.237016
$616$	37.3180	1.50359
$617$	3.10040	0.124817	0.0624087	−	0.998051i	$-0.480122\pi$
$617$	3.10040	0.124817	0.0624087	+	0.998051i	$0.480122\pi$
$618$	10.5101	0.422779
$619$	−17.4429	−0.701091	−0.350546	−	0.936546i	$-0.614004\pi$
$619$	−17.4429	−0.701091	−0.350546	+	0.936546i	$0.614004\pi$
$620$	1.46267	0.0587422
$621$	2.74538	0.110168
$622$	21.1901	0.849646
$623$	29.5694	1.18467
$624$	1.06474	0.0426239
$625$	1.00000	0.0400000
$626$	5.63820	0.225348
$627$	35.6119	1.42220
$628$	10.7394	0.428549
$629$	56.2978	2.24474
$630$	1.91581	0.0763279
$631$	−18.1764	−0.723593	−0.361796	−	0.932257i	$-0.617836\pi$
$631$	−18.1764	−0.723593	−0.361796	+	0.932257i	$0.617836\pi$
$632$	31.0736	1.23604
$633$	−1.32189	−0.0525403
$634$	−9.07209	−0.360299
$635$	−13.1001	−0.519862
$636$	−21.1448	−0.838445
$637$	0.169291	0.00670755
$638$	21.9481	0.868935
$639$	2.18901	0.0865958
$640$	10.1278	0.400335
$641$	19.6991	0.778069	0.389035	−	0.921223i	$-0.372809\pi$
$641$	19.6991	0.778069	0.389035	+	0.921223i	$0.372809\pi$
$642$	4.12931	0.162971
$643$	17.7747	0.700967	0.350483	−	0.936569i	$-0.386017\pi$
$643$	17.7747	0.700967	0.350483	+	0.936569i	$0.386017\pi$
$644$	−10.4950	−0.413561
$645$	−10.4737	−0.412401
$646$	23.1502	0.910833
$647$	1.39041	0.0546628	0.0273314	−	0.999626i	$-0.491299\pi$
$647$	1.39041	0.0546628	0.0273314	+	0.999626i	$0.491299\pi$
$648$	−2.53823	−0.0997112
$649$	45.4903	1.78565
$650$	−0.733028	−0.0287517
$651$	2.61356	0.102434
$652$	17.4109	0.681864
$653$	12.1549	0.475658	0.237829	−	0.971307i	$-0.423564\pi$
$653$	12.1549	0.475658	0.237829	+	0.971307i	$0.423564\pi$
$654$	−10.6811	−0.417666
$655$	−17.9810	−0.702576
$656$	6.25835	0.244348
$657$	8.62448	0.336473
$658$	5.93325	0.231302
$659$	−45.1742	−1.75974	−0.879868	−	0.475218i	$-0.842369\pi$
$659$	−45.1742	−1.75974	−0.879868	+	0.475218i	$0.842369\pi$
$660$	−8.22812	−0.320279
$661$	−22.4205	−0.872058	−0.436029	−	0.899933i	$-0.643615\pi$
$661$	−22.4205	−0.872058	−0.436029	+	0.899933i	$0.643615\pi$
$662$	8.36463	0.325101
$663$	−4.98877	−0.193748
$664$	−24.0227	−0.932262
$665$	−16.5453	−0.641598
$666$	−8.27214	−0.320539
$667$	−14.6126	−0.565800
$668$	−2.44484	−0.0945938
$669$	−27.4608	−1.06170
$670$	−4.69616	−0.181428
$671$	−54.6929	−2.11139
$672$	15.3075	0.590500
$673$	−8.61808	−0.332203	−0.166101	−	0.986109i	$-0.553118\pi$
$673$	−8.61808	−0.332203	−0.166101	+	0.986109i	$0.553118\pi$
$674$	−9.32874	−0.359330
$675$	−1.00000	−0.0384900
$676$	−1.46267	−0.0562565
$677$	0.653230	0.0251056	0.0125528	−	0.999921i	$-0.496004\pi$
$677$	0.653230	0.0251056	0.0125528	+	0.999921i	$0.496004\pi$
$678$	6.44738	0.247610
$679$	−16.1618	−0.620234
$680$	−12.6627	−0.485591
$681$	−19.8265	−0.759754
$682$	4.12358	0.157900
$683$	−6.85689	−0.262372	−0.131186	−	0.991358i	$-0.541878\pi$
$683$	−6.85689	−0.262372	−0.131186	+	0.991358i	$0.541878\pi$
$684$	9.25949	0.354045
$685$	−16.7193	−0.638812
$686$	13.7350	0.524406
$687$	−0.372697	−0.0142193
$688$	−11.1518	−0.425158
$689$	14.4563	0.550741
$690$	−2.01244	−0.0766124
$691$	16.7701	0.637966	0.318983	−	0.947760i	$-0.396659\pi$
$691$	16.7701	0.637966	0.318983	+	0.947760i	$0.396659\pi$
$692$	−16.0879	−0.611569
$693$	−14.7024	−0.558497
$694$	2.87290	0.109054
$695$	8.98230	0.340718
$696$	13.5100	0.512094
$697$	−29.3230	−1.11069
$698$	−24.6322	−0.932343
$699$	12.9525	0.489909
$700$	3.82278	0.144488
$701$	11.6997	0.441891	0.220945	−	0.975286i	$-0.429086\pi$
$701$	11.6997	0.441891	0.220945	+	0.975286i	$0.429086\pi$
$702$	0.733028	0.0276664
$703$	71.4395	2.69439
$704$	12.1724	0.458764
$705$	−3.09699	−0.116639
$706$	1.10985	0.0417697
$707$	−40.2738	−1.51465
$708$	11.8280	0.444523
$709$	15.6282	0.586929	0.293465	−	0.955970i	$-0.405192\pi$
$709$	15.6282	0.586929	0.293465	+	0.955970i	$0.405192\pi$
$710$	−1.60460	−0.0602197
$711$	−12.2422	−0.459119
$712$	28.7171	1.07622
$713$	−2.74538	−0.102815
$714$	−9.55756	−0.357683
$715$	5.62541	0.210378
$716$	−19.2295	−0.718641
$717$	−13.8368	−0.516745
$718$	−13.3928	−0.499815
$719$	−12.8663	−0.479830	−0.239915	−	0.970794i	$-0.577120\pi$
$719$	−12.8663	−0.479830	−0.239915	+	0.970794i	$0.577120\pi$
$720$	−1.06474	−0.0396807
$721$	37.4732	1.39557
$722$	15.4491	0.574955
$723$	−3.75060	−0.139486
$724$	2.81075	0.104461
$725$	5.32259	0.197676
$726$	−15.1335	−0.561659
$727$	27.2919	1.01220	0.506101	−	0.862474i	$-0.331086\pi$
$727$	27.2919	1.01220	0.506101	+	0.862474i	$0.331086\pi$
$728$	−6.63383	−0.245866
$729$	1.00000	0.0370370
$730$	−6.32198	−0.233987
$731$	52.2508	1.93257
$732$	−14.2208	−0.525614
$733$	−33.1701	−1.22517	−0.612583	−	0.790406i	$-0.709869\pi$
$733$	−33.1701	−1.22517	−0.612583	+	0.790406i	$0.709869\pi$
$734$	6.00700	0.221722
$735$	−0.169291	−0.00624439
$736$	−16.0796	−0.592701
$737$	36.0393	1.32753
$738$	4.30859	0.158601
$739$	41.8986	1.54126	0.770632	−	0.637280i	$-0.219941\pi$
$739$	41.8986	1.54126	0.770632	+	0.637280i	$0.219941\pi$
$740$	−16.5061	−0.606776
$741$	−6.33054	−0.232558
$742$	27.6956	1.01674
$743$	12.2914	0.450927	0.225464	−	0.974252i	$-0.427610\pi$
$743$	12.2914	0.450927	0.225464	+	0.974252i	$0.427610\pi$
$744$	2.53823	0.0930562
$745$	−11.0295	−0.404090
$746$	6.23482	0.228273
$747$	9.46434	0.346282
$748$	41.0482	1.50087
$749$	14.7228	0.537959
$750$	0.733028	0.0267664
$751$	−6.35597	−0.231933	−0.115966	−	0.993253i	$-0.536996\pi$
$751$	−6.35597	−0.231933	−0.115966	+	0.993253i	$0.536996\pi$
$752$	−3.29750	−0.120247
$753$	−29.3235	−1.06861
$754$	−3.90161	−0.142088
$755$	14.6085	0.531658
$756$	−3.82278	−0.139033
$757$	12.0242	0.437029	0.218514	−	0.975834i	$-0.429879\pi$
$757$	12.0242	0.437029	0.218514	+	0.975834i	$0.429879\pi$
$758$	−7.73944	−0.281109
$759$	15.4439	0.560579
$760$	−16.0684	−0.582862
$761$	−24.5253	−0.889043	−0.444521	−	0.895768i	$-0.646626\pi$
$761$	−24.5253	−0.889043	−0.444521	+	0.895768i	$0.646626\pi$
$762$	−9.60274	−0.347871
$763$	−38.0829	−1.37869
$764$	15.9398	0.576681
$765$	4.98877	0.180369
$766$	26.5627	0.959750
$767$	−8.08658	−0.291990
$768$	11.7516	0.424049
$769$	0.892949	0.0322006	0.0161003	−	0.999870i	$-0.494875\pi$
$769$	0.892949	0.0322006	0.0161003	+	0.999870i	$0.494875\pi$
$770$	10.7772	0.388385
$771$	−22.0390	−0.793715
$772$	−21.1449	−0.761022
$773$	5.09659	0.183311	0.0916557	−	0.995791i	$-0.470784\pi$
$773$	5.09659	0.183311	0.0916557	+	0.995791i	$0.470784\pi$
$774$	−7.67750	−0.275962
$775$	1.00000	0.0359211
$776$	−15.6960	−0.563454
$777$	−29.4938	−1.05808
$778$	−23.7850	−0.852734
$779$	−37.2096	−1.33317
$780$	1.46267	0.0523720
$781$	12.3141	0.440632
$782$	10.0396	0.359016
$783$	−5.32259	−0.190214
$784$	−0.180252	−0.00643756
$785$	7.34233	0.262059
$786$	−13.1806	−0.470136
$787$	−49.8399	−1.77660	−0.888300	−	0.459263i	$-0.848114\pi$
$787$	−49.8399	−1.77660	−0.888300	+	0.459263i	$0.848114\pi$
$788$	8.81607	0.314059
$789$	11.3961	0.405713
$790$	8.97389	0.319277
$791$	22.9877	0.817349
$792$	−14.2786	−0.507368
$793$	9.72246	0.345255
$794$	−0.471094	−0.0167185
$795$	−14.4563	−0.512712
$796$	9.87251	0.349922
$797$	21.7534	0.770546	0.385273	−	0.922803i	$-0.374107\pi$
$797$	21.7534	0.770546	0.385273	+	0.922803i	$0.374107\pi$
$798$	−12.1281	−0.429331
$799$	15.4502	0.546587
$800$	5.85695	0.207075
$801$	−11.3138	−0.399754
$802$	−12.2628	−0.433016
$803$	48.5163	1.71210
$804$	9.37062	0.330476
$805$	−7.17523	−0.252894
$806$	−0.733028	−0.0258198
$807$	3.87259	0.136321
$808$	−39.1130	−1.37599
$809$	−20.3097	−0.714053	−0.357026	−	0.934094i	$-0.616209\pi$
$809$	−20.3097	−0.714053	−0.357026	+	0.934094i	$0.616209\pi$
$810$	−0.733028	−0.0257560
$811$	5.40359	0.189746	0.0948728	−	0.995489i	$-0.469756\pi$
$811$	5.40359	0.189746	0.0948728	+	0.995489i	$0.469756\pi$
$812$	20.3471	0.714043
$813$	22.6857	0.795621
$814$	−46.5342	−1.63102
$815$	11.9035	0.416962
$816$	5.31177	0.185949
$817$	66.3040	2.31968
$818$	−24.1464	−0.844260
$819$	2.61356	0.0913253
$820$	8.59728	0.300230
$821$	4.45529	0.155491	0.0777453	−	0.996973i	$-0.475228\pi$
$821$	4.45529	0.155491	0.0777453	+	0.996973i	$0.475228\pi$
$822$	−12.2557	−0.427467
$823$	40.9887	1.42878	0.714388	−	0.699750i	$-0.246705\pi$
$823$	40.9887	1.42878	0.714388	+	0.699750i	$0.246705\pi$
$824$	36.3931	1.26781
$825$	−5.62541	−0.195852
$826$	−15.4924	−0.539049
$827$	−25.6500	−0.891937	−0.445968	−	0.895049i	$-0.647141\pi$
$827$	−25.6500	−0.891937	−0.445968	+	0.895049i	$0.647141\pi$
$828$	4.01559	0.139551
$829$	−3.69255	−0.128248	−0.0641238	−	0.997942i	$-0.520425\pi$
$829$	−3.69255	−0.128248	−0.0641238	+	0.997942i	$0.520425\pi$
$830$	−6.93762	−0.240808
$831$	−3.97394	−0.137854
$832$	−2.16382	−0.0750170
$833$	0.844554	0.0292621
$834$	6.58428	0.227995
$835$	−1.67149	−0.0578444
$836$	52.0884	1.80152
$837$	−1.00000	−0.0345651
$838$	16.1150	0.556684
$839$	−6.38842	−0.220553	−0.110276	−	0.993901i	$-0.535174\pi$
$839$	−6.38842	−0.220553	−0.110276	+	0.993901i	$0.535174\pi$
$840$	6.63383	0.228889
$841$	−0.670036	−0.0231047
$842$	21.0734	0.726236
$843$	−27.6414	−0.952020
$844$	−1.93348	−0.0665533
$845$	−1.00000	−0.0344010
$846$	−2.27018	−0.0780503
$847$	−53.9577	−1.85401
$848$	−15.3922	−0.528572
$849$	−28.2959	−0.971114
$850$	−3.65691	−0.125431
$851$	30.9814	1.06203
$852$	3.20180	0.109692
$853$	−33.8912	−1.16041	−0.580206	−	0.814470i	$-0.697028\pi$
$853$	−33.8912	−1.16041	−0.580206	+	0.814470i	$0.697028\pi$
$854$	18.6264	0.637383
$855$	6.33054	0.216500
$856$	14.2985	0.488711
$857$	7.23845	0.247261	0.123630	−	0.992328i	$-0.460546\pi$
$857$	7.23845	0.247261	0.123630	+	0.992328i	$0.460546\pi$
$858$	4.12358	0.140777
$859$	45.4469	1.55063	0.775314	−	0.631576i	$-0.217592\pi$
$859$	45.4469	1.55063	0.775314	+	0.631576i	$0.217592\pi$
$860$	−15.3195	−0.522392
$861$	15.3620	0.523535
$862$	−23.0252	−0.784240
$863$	−6.62482	−0.225512	−0.112756	−	0.993623i	$-0.535968\pi$
$863$	−6.62482	−0.225512	−0.112756	+	0.993623i	$0.535968\pi$
$864$	−5.85695	−0.199258
$865$	−10.9990	−0.373976
$866$	22.5395	0.765924
$867$	−7.88786	−0.267886
$868$	3.82278	0.129754
$869$	−68.8676	−2.33617
$870$	3.90161	0.132277
$871$	−6.40652	−0.217077
$872$	−36.9853	−1.25248
$873$	6.18383	0.209291
$874$	12.7398	0.430932
$875$	2.61356	0.0883545
$876$	12.6148	0.426214
$877$	−5.95124	−0.200959	−0.100480	−	0.994939i	$-0.532038\pi$
$877$	−5.95124	−0.200959	−0.100480	+	0.994939i	$0.532038\pi$
$878$	21.6479	0.730581
$879$	−2.25580	−0.0760864
$880$	−5.98963	−0.201910
$881$	−39.0939	−1.31711	−0.658553	−	0.752534i	$-0.728831\pi$
$881$	−39.0939	−1.31711	−0.658553	+	0.752534i	$0.728831\pi$
$882$	−0.124095	−0.00417849
$883$	23.2141	0.781215	0.390608	−	0.920557i	$-0.372265\pi$
$883$	23.2141	0.781215	0.390608	+	0.920557i	$0.372265\pi$
$884$	−7.29693	−0.245422
$885$	8.08658	0.271827
$886$	−24.3471	−0.817955
$887$	44.9526	1.50936	0.754680	−	0.656093i	$-0.227792\pi$
$887$	44.9526	1.50936	0.754680	+	0.656093i	$0.227792\pi$
$888$	−28.6437	−0.961220
$889$	−34.2379	−1.14830
$890$	8.29335	0.277994
$891$	5.62541	0.188458
$892$	−40.1661	−1.34486
$893$	19.6056	0.656076
$894$	−8.08494	−0.270401
$895$	−13.1469	−0.439451
$896$	26.4696	0.884286
$897$	−2.74538	−0.0916657
$898$	−7.75523	−0.258795
$899$	5.32259	0.177518
$900$	−1.46267	−0.0487557
$901$	72.1191	2.40264
$902$	24.2376	0.807024
$903$	−27.3736	−0.910937
$904$	22.3251	0.742523
$905$	1.92166	0.0638781
$906$	10.7085	0.355764
$907$	−2.02158	−0.0671254	−0.0335627	−	0.999437i	$-0.510685\pi$
$907$	−2.02158	−0.0671254	−0.0335627	+	0.999437i	$0.510685\pi$
$908$	−28.9997	−0.962388
$909$	15.4095	0.511102
$910$	−1.91581	−0.0635086
$911$	41.8134	1.38534	0.692670	−	0.721254i	$-0.256434\pi$
$911$	41.8134	1.38534	0.692670	+	0.721254i	$0.256434\pi$
$912$	6.74041	0.223197
$913$	53.2408	1.76201
$914$	10.0972	0.333985
$915$	−9.72246	−0.321415
$916$	−0.545133	−0.0180117
$917$	−46.9945	−1.55189
$918$	3.65691	0.120696
$919$	−16.7438	−0.552326	−0.276163	−	0.961111i	$-0.589063\pi$
$919$	−16.7438	−0.552326	−0.276163	+	0.961111i	$0.589063\pi$
$920$	−6.96843	−0.229742
$921$	0.882007	0.0290631
$922$	−20.2404	−0.666581
$923$	−2.18901	−0.0720520
$924$	−21.5047	−0.707453
$925$	−11.2849	−0.371045
$926$	10.4134	0.342204
$927$	−14.3380	−0.470920
$928$	31.1742	1.02334
$929$	14.7406	0.483622	0.241811	−	0.970323i	$-0.422259\pi$
$929$	14.7406	0.483622	0.241811	+	0.970323i	$0.422259\pi$
$930$	0.733028	0.0240369
$931$	1.07170	0.0351236
$932$	18.9453	0.620573
$933$	−28.9076	−0.946393
$934$	−27.5729	−0.902212
$935$	28.0639	0.917788
$936$	2.53823	0.0829648
$937$	42.5582	1.39032	0.695158	−	0.718857i	$-0.255334\pi$
$937$	42.5582	1.39032	0.695158	+	0.718857i	$0.255334\pi$
$938$	−12.2737	−0.400751
$939$	−7.69166	−0.251008
$940$	−4.52987	−0.147748
$941$	21.4177	0.698198	0.349099	−	0.937086i	$-0.386488\pi$
$941$	21.4177	0.698198	0.349099	+	0.937086i	$0.386488\pi$
$942$	5.38213	0.175359
$943$	−16.1368	−0.525487
$944$	8.61014	0.280236
$945$	−2.61356	−0.0850192
$946$	−43.1891	−1.40420
$947$	14.7069	0.477909	0.238954	−	0.971031i	$-0.423195\pi$
$947$	14.7069	0.477909	0.238954	+	0.971031i	$0.423195\pi$
$948$	−17.9063	−0.581571
$949$	−8.62448	−0.279962
$950$	−4.64046	−0.150556
$951$	12.3762	0.401325
$952$	−33.0947	−1.07260
$953$	−54.0588	−1.75114	−0.875568	−	0.483095i	$-0.839513\pi$
$953$	−54.0588	−1.75114	−0.875568	+	0.483095i	$0.839513\pi$
$954$	−10.5969	−0.343086
$955$	10.8977	0.352642
$956$	−20.2387	−0.654566
$957$	−29.9418	−0.967880
$958$	29.3859	0.949417
$959$	−43.6969	−1.41105
$960$	2.16382	0.0698371
$961$	1.00000	0.0322581
$962$	8.27214	0.266705
$963$	−5.63323	−0.181528
$964$	−5.48588	−0.176688
$965$	−14.4564	−0.465367
$966$	−5.25964	−0.169226
$967$	21.7294	0.698770	0.349385	−	0.936979i	$-0.386391\pi$
$967$	21.7294	0.698770	0.349385	+	0.936979i	$0.386391\pi$
$968$	−52.4025	−1.68428
$969$	−31.5816	−1.01455
$970$	−4.53292	−0.145543
$971$	57.8139	1.85534	0.927669	−	0.373404i	$-0.121810\pi$
$971$	57.8139	1.85534	0.927669	+	0.373404i	$0.121810\pi$
$972$	1.46267	0.0469152
$973$	23.4758	0.752600
$974$	−17.6375	−0.565141
$975$	1.00000	0.0320256
$976$	−10.3519	−0.331357
$977$	−19.2631	−0.616281	−0.308141	−	0.951341i	$-0.599707\pi$
$977$	−19.2631	−0.616281	−0.308141	+	0.951341i	$0.599707\pi$
$978$	8.72560	0.279014
$979$	−63.6449	−2.03410
$980$	−0.247617	−0.00790983
$981$	14.5713	0.465224
$982$	−10.6553	−0.340024
$983$	−8.28544	−0.264265	−0.132132	−	0.991232i	$-0.542182\pi$
$983$	−8.28544	−0.264265	−0.132132	+	0.991232i	$0.542182\pi$
$984$	14.9192	0.475608
$985$	6.02738	0.192048
$986$	−19.4642	−0.619867
$987$	−8.09417	−0.257640
$988$	−9.25949	−0.294584
$989$	28.7543	0.914333
$990$	−4.12358	−0.131056
$991$	−19.4988	−0.619398	−0.309699	−	0.950835i	$-0.600228\pi$
$991$	−19.4988	−0.619398	−0.309699	+	0.950835i	$0.600228\pi$
$992$	5.85695	0.185958
$993$	−11.4111	−0.362119
$994$	−4.19373	−0.133017
$995$	6.74965	0.213978
$996$	13.8432	0.438639
$997$	42.7754	1.35471	0.677355	−	0.735657i	$-0.263126\pi$
$997$	42.7754	1.35471	0.677355	+	0.735657i	$0.263126\pi$
$998$	−11.1636	−0.353377
$999$	11.2849	0.357038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bg.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bg.1.10	✓	16	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)

\(f(q)\)	\(=\)	\(q+0.733028 q^{2} -1.00000 q^{3} -1.46267 q^{4} -1.00000 q^{5} -0.733028 q^{6} -2.61356 q^{7} -2.53823 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.733028 q^{2} -1.00000 q^{3} -1.46267 q^{4} -1.00000 q^{5} -0.733028 q^{6} -2.61356 q^{7} -2.53823 q^{8} +1.00000 q^{9} -0.733028 q^{10} +5.62541 q^{11} +1.46267 q^{12} -1.00000 q^{13} -1.91581 q^{14} +1.00000 q^{15} +1.06474 q^{16} -4.98877 q^{17} +0.733028 q^{18} -6.33054 q^{19} +1.46267 q^{20} +2.61356 q^{21} +4.12358 q^{22} -2.74538 q^{23} +2.53823 q^{24} +1.00000 q^{25} -0.733028 q^{26} -1.00000 q^{27} +3.82278 q^{28} +5.32259 q^{29} +0.733028 q^{30} +1.00000 q^{31} +5.85695 q^{32} -5.62541 q^{33} -3.65691 q^{34} +2.61356 q^{35} -1.46267 q^{36} -11.2849 q^{37} -4.64046 q^{38} +1.00000 q^{39} +2.53823 q^{40} +5.87780 q^{41} +1.91581 q^{42} -10.4737 q^{43} -8.22812 q^{44} -1.00000 q^{45} -2.01244 q^{46} -3.09699 q^{47} -1.06474 q^{48} -0.169291 q^{49} +0.733028 q^{50} +4.98877 q^{51} +1.46267 q^{52} -14.4563 q^{53} -0.733028 q^{54} -5.62541 q^{55} +6.63383 q^{56} +6.33054 q^{57} +3.90161 q^{58} +8.08658 q^{59} -1.46267 q^{60} -9.72246 q^{61} +0.733028 q^{62} -2.61356 q^{63} +2.16382 q^{64} +1.00000 q^{65} -4.12358 q^{66} +6.40652 q^{67} +7.29693 q^{68} +2.74538 q^{69} +1.91581 q^{70} +2.18901 q^{71} -2.53823 q^{72} +8.62448 q^{73} -8.27214 q^{74} -1.00000 q^{75} +9.25949 q^{76} -14.7024 q^{77} +0.733028 q^{78} -12.2422 q^{79} -1.06474 q^{80} +1.00000 q^{81} +4.30859 q^{82} +9.46434 q^{83} -3.82278 q^{84} +4.98877 q^{85} -7.67750 q^{86} -5.32259 q^{87} -14.2786 q^{88} -11.3138 q^{89} -0.733028 q^{90} +2.61356 q^{91} +4.01559 q^{92} -1.00000 q^{93} -2.27018 q^{94} +6.33054 q^{95} -5.85695 q^{96} +6.18383 q^{97} -0.124095 q^{98} +5.62541 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9	\( 16 q - 2 q^{2} - 16 q^{3} + 26 q^{4} - 16 q^{5} + 2 q^{6} - 2 q^{7} - 9 q^{8} + 16 q^{9} + 2 q^{10} + 3 q^{11} - 26 q^{12} - 16 q^{13} - 5 q^{14} + 16 q^{15} + 38 q^{16} - 13 q^{17} - 2 q^{18} - 26 q^{20} + 2 q^{21} + q^{22} - 15 q^{23} + 9 q^{24} + 16 q^{25} + 2 q^{26} - 16 q^{27} + 8 q^{28} - 4 q^{29} - 2 q^{30} + 16 q^{31} - 30 q^{32} - 3 q^{33} + 29 q^{34} + 2 q^{35} + 26 q^{36} + 12 q^{37} + 16 q^{39} + 9 q^{40} - 12 q^{41} + 5 q^{42} - 7 q^{43} - 13 q^{44} - 16 q^{45} + 14 q^{46} + 17 q^{47} - 38 q^{48} + 16 q^{49} - 2 q^{50} + 13 q^{51} - 26 q^{52} - 36 q^{53} + 2 q^{54} - 3 q^{55} + 41 q^{56} + 16 q^{58} + 53 q^{59} + 26 q^{60} + 34 q^{61} - 2 q^{62} - 2 q^{63} + 79 q^{64} + 16 q^{65} - q^{66} - 13 q^{67} - 39 q^{68} + 15 q^{69} + 5 q^{70} - 11 q^{71} - 9 q^{72} + 34 q^{73} - 12 q^{74} - 16 q^{75} + 86 q^{76} - 32 q^{77} - 2 q^{78} - 7 q^{79} - 38 q^{80} + 16 q^{81} + 27 q^{82} - 28 q^{83} - 8 q^{84} + 13 q^{85} + 38 q^{86} + 4 q^{87} + 23 q^{88} - 8 q^{89} + 2 q^{90} + 2 q^{91} - 71 q^{92} - 16 q^{93} + 66 q^{94} + 30 q^{96} + 4 q^{97} + 22 q^{98} + 3 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 - 16 * q^3 + 26 * q^4 - 16 * q^5 + 2 * q^6 - 2 * q^7 - 9 * q^8 + 16 * q^9 + 2 * q^10 + 3 * q^11 - 26 * q^12 - 16 * q^13 - 5 * q^14 + 16 * q^15 + 38 * q^16 - 13 * q^17 - 2 * q^18 - 26 * q^20 + 2 * q^21 + q^22 - 15 * q^23 + 9 * q^24 + 16 * q^25 + 2 * q^26 - 16 * q^27 + 8 * q^28 - 4 * q^29 - 2 * q^30 + 16 * q^31 - 30 * q^32 - 3 * q^33 + 29 * q^34 + 2 * q^35 + 26 * q^36 + 12 * q^37 + 16 * q^39 + 9 * q^40 - 12 * q^41 + 5 * q^42 - 7 * q^43 - 13 * q^44 - 16 * q^45 + 14 * q^46 + 17 * q^47 - 38 * q^48 + 16 * q^49 - 2 * q^50 + 13 * q^51 - 26 * q^52 - 36 * q^53 + 2 * q^54 - 3 * q^55 + 41 * q^56 + 16 * q^58 + 53 * q^59 + 26 * q^60 + 34 * q^61 - 2 * q^62 - 2 * q^63 + 79 * q^64 + 16 * q^65 - q^66 - 13 * q^67 - 39 * q^68 + 15 * q^69 + 5 * q^70 - 11 * q^71 - 9 * q^72 + 34 * q^73 - 12 * q^74 - 16 * q^75 + 86 * q^76 - 32 * q^77 - 2 * q^78 - 7 * q^79 - 38 * q^80 + 16 * q^81 + 27 * q^82 - 28 * q^83 - 8 * q^84 + 13 * q^85 + 38 * q^86 + 4 * q^87 + 23 * q^88 - 8 * q^89 + 2 * q^90 + 2 * q^91 - 71 * q^92 - 16 * q^93 + 66 * q^94 + 30 * q^96 + 4 * q^97 + 22 * q^98 + 3 * q^99

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