LMFDB - Embedded newform 6045.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 17 x^{11} + 14 x^{10} + 106 x^{9} - 68 x^{8} - 299 x^{7} + 141 x^{6} + 380 x^{5} + \cdots - 4 $ x^13 - x^12 - 17x^11 + 14x^10 + 106x^9 - 68x^8 - 299x^7 + 141x^6 + 380x^5 - 123x^4 - 186x^3 + 32x^2 + 20*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.41287$ 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)

$f(q)$	$=$	$q-2.41287 q^{2} -1.00000 q^{3} +3.82195 q^{4} +1.00000 q^{5} +2.41287 q^{6} -3.00058 q^{7} -4.39612 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.41287 q^{2} -1.00000 q^{3} +3.82195 q^{4} +1.00000 q^{5} +2.41287 q^{6} -3.00058 q^{7} -4.39612 q^{8} +1.00000 q^{9} -2.41287 q^{10} -0.554196 q^{11} -3.82195 q^{12} -1.00000 q^{13} +7.24001 q^{14} -1.00000 q^{15} +2.96338 q^{16} -2.65209 q^{17} -2.41287 q^{18} -1.86402 q^{19} +3.82195 q^{20} +3.00058 q^{21} +1.33720 q^{22} -2.18577 q^{23} +4.39612 q^{24} +1.00000 q^{25} +2.41287 q^{26} -1.00000 q^{27} -11.4681 q^{28} +3.24925 q^{29} +2.41287 q^{30} +1.00000 q^{31} +1.64199 q^{32} +0.554196 q^{33} +6.39914 q^{34} -3.00058 q^{35} +3.82195 q^{36} +4.86840 q^{37} +4.49764 q^{38} +1.00000 q^{39} -4.39612 q^{40} +1.71295 q^{41} -7.24001 q^{42} +8.05186 q^{43} -2.11811 q^{44} +1.00000 q^{45} +5.27397 q^{46} -10.0569 q^{47} -2.96338 q^{48} +2.00348 q^{49} -2.41287 q^{50} +2.65209 q^{51} -3.82195 q^{52} -0.842891 q^{53} +2.41287 q^{54} -0.554196 q^{55} +13.1909 q^{56} +1.86402 q^{57} -7.84002 q^{58} +10.4439 q^{59} -3.82195 q^{60} +4.60848 q^{61} -2.41287 q^{62} -3.00058 q^{63} -9.88866 q^{64} -1.00000 q^{65} -1.33720 q^{66} +0.667461 q^{67} -10.1361 q^{68} +2.18577 q^{69} +7.24001 q^{70} +13.1475 q^{71} -4.39612 q^{72} +2.66659 q^{73} -11.7468 q^{74} -1.00000 q^{75} -7.12419 q^{76} +1.66291 q^{77} -2.41287 q^{78} -9.44794 q^{79} +2.96338 q^{80} +1.00000 q^{81} -4.13312 q^{82} +10.8723 q^{83} +11.4681 q^{84} -2.65209 q^{85} -19.4281 q^{86} -3.24925 q^{87} +2.43631 q^{88} +12.8222 q^{89} -2.41287 q^{90} +3.00058 q^{91} -8.35388 q^{92} -1.00000 q^{93} +24.2660 q^{94} -1.86402 q^{95} -1.64199 q^{96} -13.0354 q^{97} -4.83413 q^{98} -0.554196 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{2} - 13 q^{3} + 9 q^{4} + 13 q^{5} + q^{6} - 11 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) $ 13 * q - q^2 - 13 * q^3 + 9 * q^4 + 13 * q^5 + q^6 - 11 * q^7 - 6 * q^8 + 13 * q^9	\( 13 q - q^{2} - 13 q^{3} + 9 q^{4} + 13 q^{5} + q^{6} - 11 q^{7} - 6 q^{8} + 13 q^{9} - q^{10} + 4 q^{11} - 9 q^{12} - 13 q^{13} + q^{14} - 13 q^{15} + 9 q^{16} + 3 q^{17} - q^{18} - 2 q^{19} + 9 q^{20} + 11 q^{21} - 17 q^{22} - 9 q^{23} + 6 q^{24} + 13 q^{25} + q^{26} - 13 q^{27} - 30 q^{28} + 7 q^{29} + q^{30} + 13 q^{31} - 13 q^{32} - 4 q^{33} - 13 q^{34} - 11 q^{35} + 9 q^{36} - 7 q^{37} - 26 q^{38} + 13 q^{39} - 6 q^{40} - 13 q^{41} - q^{42} - 24 q^{43} - q^{44} + 13 q^{45} + 22 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} - q^{50} - 3 q^{51} - 9 q^{52} - 18 q^{53} + q^{54} + 4 q^{55} + 19 q^{56} + 2 q^{57} + 20 q^{58} - 26 q^{59} - 9 q^{60} - 12 q^{61} - q^{62} - 11 q^{63} - 2 q^{64} - 13 q^{65} + 17 q^{66} - 30 q^{67} + 23 q^{68} + 9 q^{69} + q^{70} - 3 q^{71} - 6 q^{72} - 5 q^{73} - 36 q^{74} - 13 q^{75} + 16 q^{76} - q^{78} - 9 q^{79} + 9 q^{80} + 13 q^{81} + 7 q^{82} - 35 q^{83} + 30 q^{84} + 3 q^{85} - 14 q^{86} - 7 q^{87} - 29 q^{88} + q^{89} - q^{90} + 11 q^{91} - 63 q^{92} - 13 q^{93} + 24 q^{94} - 2 q^{95} + 13 q^{96} - 33 q^{97} + q^{98} + 4 q^{99}+O(q^{100}) \) 13 * q - q^2 - 13 * q^3 + 9 * q^4 + 13 * q^5 + q^6 - 11 * q^7 - 6 * q^8 + 13 * q^9 - q^10 + 4 * q^11 - 9 * q^12 - 13 * q^13 + q^14 - 13 * q^15 + 9 * q^16 + 3 * q^17 - q^18 - 2 * q^19 + 9 * q^20 + 11 * q^21 - 17 * q^22 - 9 * q^23 + 6 * q^24 + 13 * q^25 + q^26 - 13 * q^27 - 30 * q^28 + 7 * q^29 + q^30 + 13 * q^31 - 13 * q^32 - 4 * q^33 - 13 * q^34 - 11 * q^35 + 9 * q^36 - 7 * q^37 - 26 * q^38 + 13 * q^39 - 6 * q^40 - 13 * q^41 - q^42 - 24 * q^43 - q^44 + 13 * q^45 + 22 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 - q^50 - 3 * q^51 - 9 * q^52 - 18 * q^53 + q^54 + 4 * q^55 + 19 * q^56 + 2 * q^57 + 20 * q^58 - 26 * q^59 - 9 * q^60 - 12 * q^61 - q^62 - 11 * q^63 - 2 * q^64 - 13 * q^65 + 17 * q^66 - 30 * q^67 + 23 * q^68 + 9 * q^69 + q^70 - 3 * q^71 - 6 * q^72 - 5 * q^73 - 36 * q^74 - 13 * q^75 + 16 * q^76 - q^78 - 9 * q^79 + 9 * q^80 + 13 * q^81 + 7 * q^82 - 35 * q^83 + 30 * q^84 + 3 * q^85 - 14 * q^86 - 7 * q^87 - 29 * q^88 + q^89 - q^90 + 11 * q^91 - 63 * q^92 - 13 * q^93 + 24 * q^94 - 2 * q^95 + 13 * q^96 - 33 * q^97 + q^98 + 4 * 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 $

$2$	−2.41287	−1.70616	−0.853079	−	0.521782i	$-0.825267\pi$
$2$	−2.41287	−1.70616	−0.853079	+	0.521782i	$0.825267\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.82195	1.91097
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.41287	0.985050
$7$	−3.00058	−1.13411	−0.567056	−	0.823679i	$-0.691918\pi$
$7$	−3.00058	−1.13411	−0.567056	+	0.823679i	$0.691918\pi$
$8$	−4.39612	−1.55426
$9$	1.00000	0.333333
$10$	−2.41287	−0.763017
$11$	−0.554196	−0.167096	−0.0835482	−	0.996504i	$-0.526625\pi$
$11$	−0.554196	−0.167096	−0.0835482	+	0.996504i	$0.526625\pi$
$12$	−3.82195	−1.10330
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	7.24001	1.93497
$15$	−1.00000	−0.258199
$16$	2.96338	0.740845
$17$	−2.65209	−0.643226	−0.321613	−	0.946871i	$-0.604225\pi$
$17$	−2.65209	−0.643226	−0.321613	+	0.946871i	$0.604225\pi$
$18$	−2.41287	−0.568719
$19$	−1.86402	−0.427636	−0.213818	−	0.976874i	$-0.568590\pi$
$19$	−1.86402	−0.427636	−0.213818	+	0.976874i	$0.568590\pi$
$20$	3.82195	0.854613
$21$	3.00058	0.654780
$22$	1.33720	0.285093
$23$	−2.18577	−0.455764	−0.227882	−	0.973689i	$-0.573180\pi$
$23$	−2.18577	−0.455764	−0.227882	+	0.973689i	$0.573180\pi$
$24$	4.39612	0.897355
$25$	1.00000	0.200000
$26$	2.41287	0.473203
$27$	−1.00000	−0.192450
$28$	−11.4681	−2.16726
$29$	3.24925	0.603370	0.301685	−	0.953408i	$-0.402451\pi$
$29$	3.24925	0.603370	0.301685	+	0.953408i	$0.402451\pi$
$30$	2.41287	0.440528
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.64199	0.290265
$33$	0.554196	0.0964732
$34$	6.39914	1.09744
$35$	−3.00058	−0.507191
$36$	3.82195	0.636991
$37$	4.86840	0.800359	0.400180	−	0.916437i	$-0.368948\pi$
$37$	4.86840	0.800359	0.400180	+	0.916437i	$0.368948\pi$
$38$	4.49764	0.729614
$39$	1.00000	0.160128
$40$	−4.39612	−0.695088
$41$	1.71295	0.267517	0.133759	−	0.991014i	$-0.457295\pi$
$41$	1.71295	0.267517	0.133759	+	0.991014i	$0.457295\pi$
$42$	−7.24001	−1.11716
$43$	8.05186	1.22790	0.613949	−	0.789346i	$-0.289580\pi$
$43$	8.05186	1.22790	0.613949	+	0.789346i	$0.289580\pi$
$44$	−2.11811	−0.319317
$45$	1.00000	0.149071
$46$	5.27397	0.777605
$47$	−10.0569	−1.46695	−0.733475	−	0.679717i	$-0.762103\pi$
$47$	−10.0569	−1.46695	−0.733475	+	0.679717i	$0.762103\pi$
$48$	−2.96338	−0.427727
$49$	2.00348	0.286211
$50$	−2.41287	−0.341231
$51$	2.65209	0.371367
$52$	−3.82195	−0.530009
$53$	−0.842891	−0.115780	−0.0578900	−	0.998323i	$-0.518437\pi$
$53$	−0.842891	−0.115780	−0.0578900	+	0.998323i	$0.518437\pi$
$54$	2.41287	0.328350
$55$	−0.554196	−0.0747278
$56$	13.1909	1.76271
$57$	1.86402	0.246896
$58$	−7.84002	−1.02944
$59$	10.4439	1.35968	0.679840	−	0.733360i	$-0.262049\pi$
$59$	10.4439	1.35968	0.679840	+	0.733360i	$0.262049\pi$
$60$	−3.82195	−0.493411
$61$	4.60848	0.590056	0.295028	−	0.955489i	$-0.404671\pi$
$61$	4.60848	0.590056	0.295028	+	0.955489i	$0.404671\pi$
$62$	−2.41287	−0.306435
$63$	−3.00058	−0.378038
$64$	−9.88866	−1.23608
$65$	−1.00000	−0.124035
$66$	−1.33720	−0.164598
$67$	0.667461	0.0815433	0.0407716	−	0.999168i	$-0.487018\pi$
$67$	0.667461	0.0815433	0.0407716	+	0.999168i	$0.487018\pi$
$68$	−10.1361	−1.22919
$69$	2.18577	0.263135
$70$	7.24001	0.865347
$71$	13.1475	1.56032	0.780158	−	0.625582i	$-0.215139\pi$
$71$	13.1475	1.56032	0.780158	+	0.625582i	$0.215139\pi$
$72$	−4.39612	−0.518088
$73$	2.66659	0.312100	0.156050	−	0.987749i	$-0.450124\pi$
$73$	2.66659	0.312100	0.156050	+	0.987749i	$0.450124\pi$
$74$	−11.7468	−1.36554
$75$	−1.00000	−0.115470
$76$	−7.12419	−0.817200
$77$	1.66291	0.189506
$78$	−2.41287	−0.273204
$79$	−9.44794	−1.06298	−0.531488	−	0.847066i	$-0.678367\pi$
$79$	−9.44794	−1.06298	−0.531488	+	0.847066i	$0.678367\pi$
$80$	2.96338	0.331316
$81$	1.00000	0.111111
$82$	−4.13312	−0.456426
$83$	10.8723	1.19339	0.596693	−	0.802470i	$-0.296481\pi$
$83$	10.8723	1.19339	0.596693	+	0.802470i	$0.296481\pi$
$84$	11.4681	1.25127
$85$	−2.65209	−0.287659
$86$	−19.4281	−2.09499
$87$	−3.24925	−0.348356
$88$	2.43631	0.259712
$89$	12.8222	1.35915	0.679575	−	0.733606i	$-0.262164\pi$
$89$	12.8222	1.35915	0.679575	+	0.733606i	$0.262164\pi$
$90$	−2.41287	−0.254339
$91$	3.00058	0.314546
$92$	−8.35388	−0.870953
$93$	−1.00000	−0.103695
$94$	24.2660	2.50285
$95$	−1.86402	−0.191244
$96$	−1.64199	−0.167585
$97$	−13.0354	−1.32354	−0.661772	−	0.749705i	$-0.730195\pi$
$97$	−13.0354	−1.32354	−0.661772	+	0.749705i	$0.730195\pi$
$98$	−4.83413	−0.488321
$99$	−0.554196	−0.0556988
$100$	3.82195	0.382195
$101$	7.75543	0.771694	0.385847	−	0.922563i	$-0.373909\pi$
$101$	7.75543	0.771694	0.385847	+	0.922563i	$0.373909\pi$
$102$	−6.39914	−0.633610
$103$	−9.58699	−0.944634	−0.472317	−	0.881429i	$-0.656582\pi$
$103$	−9.58699	−0.944634	−0.472317	+	0.881429i	$0.656582\pi$
$104$	4.39612	0.431075
$105$	3.00058	0.292827
$106$	2.03379	0.197539
$107$	1.24801	0.120650	0.0603250	−	0.998179i	$-0.480786\pi$
$107$	1.24801	0.120650	0.0603250	+	0.998179i	$0.480786\pi$
$108$	−3.82195	−0.367767
$109$	1.61976	0.155145	0.0775727	−	0.996987i	$-0.475283\pi$
$109$	1.61976	0.155145	0.0775727	+	0.996987i	$0.475283\pi$
$110$	1.33720	0.127497
$111$	−4.86840	−0.462088
$112$	−8.89186	−0.840202
$113$	−13.5034	−1.27030	−0.635149	−	0.772390i	$-0.719061\pi$
$113$	−13.5034	−1.27030	−0.635149	+	0.772390i	$0.719061\pi$
$114$	−4.49764	−0.421243
$115$	−2.18577	−0.203824
$116$	12.4185	1.15302
$117$	−1.00000	−0.0924500
$118$	−25.1998	−2.31983
$119$	7.95780	0.729490
$120$	4.39612	0.401309
$121$	−10.6929	−0.972079
$122$	−11.1197	−1.00673
$123$	−1.71295	−0.154451
$124$	3.82195	0.343221
$125$	1.00000	0.0894427
$126$	7.24001	0.644991
$127$	−18.1185	−1.60776	−0.803879	−	0.594793i	$-0.797234\pi$
$127$	−18.1185	−1.60776	−0.803879	+	0.594793i	$0.797234\pi$
$128$	20.5761	1.81869
$129$	−8.05186	−0.708927
$130$	2.41287	0.211623
$131$	−1.94904	−0.170289	−0.0851443	−	0.996369i	$-0.527135\pi$
$131$	−1.94904	−0.170289	−0.0851443	+	0.996369i	$0.527135\pi$
$132$	2.11811	0.184358
$133$	5.59314	0.484987
$134$	−1.61050	−0.139126
$135$	−1.00000	−0.0860663
$136$	11.6589	0.999742
$137$	3.37421	0.288278	0.144139	−	0.989557i	$-0.453959\pi$
$137$	3.37421	0.288278	0.144139	+	0.989557i	$0.453959\pi$
$138$	−5.27397	−0.448950
$139$	5.07317	0.430301	0.215151	−	0.976581i	$-0.430976\pi$
$139$	5.07317	0.430301	0.215151	+	0.976581i	$0.430976\pi$
$140$	−11.4681	−0.969227
$141$	10.0569	0.846944
$142$	−31.7231	−2.66214
$143$	0.554196	0.0463442
$144$	2.96338	0.246948
$145$	3.24925	0.269835
$146$	−6.43413	−0.532492
$147$	−2.00348	−0.165244
$148$	18.6067	1.52947
$149$	16.3126	1.33638	0.668189	−	0.743991i	$-0.267070\pi$
$149$	16.3126	1.33638	0.668189	+	0.743991i	$0.267070\pi$
$150$	2.41287	0.197010
$151$	−9.36426	−0.762053	−0.381027	−	0.924564i	$-0.624429\pi$
$151$	−9.36426	−0.762053	−0.381027	+	0.924564i	$0.624429\pi$
$152$	8.19446	0.664659
$153$	−2.65209	−0.214409
$154$	−4.01239	−0.323327
$155$	1.00000	0.0803219
$156$	3.82195	0.306001
$157$	10.5942	0.845510	0.422755	−	0.906244i	$-0.361063\pi$
$157$	10.5942	0.845510	0.422755	+	0.906244i	$0.361063\pi$
$158$	22.7966	1.81360
$159$	0.842891	0.0668456
$160$	1.64199	0.129810
$161$	6.55857	0.516888
$162$	−2.41287	−0.189573
$163$	−4.71693	−0.369458	−0.184729	−	0.982789i	$-0.559141\pi$
$163$	−4.71693	−0.369458	−0.184729	+	0.982789i	$0.559141\pi$
$164$	6.54678	0.511218
$165$	0.554196	0.0431441
$166$	−26.2334	−2.03610
$167$	−15.7963	−1.22235	−0.611176	−	0.791495i	$-0.709303\pi$
$167$	−15.7963	−1.22235	−0.611176	+	0.791495i	$0.709303\pi$
$168$	−13.1909	−1.01770
$169$	1.00000	0.0769231
$170$	6.39914	0.490792
$171$	−1.86402	−0.142545
$172$	30.7738	2.34648
$173$	14.3366	1.08999	0.544997	−	0.838438i	$-0.316531\pi$
$173$	14.3366	1.08999	0.544997	+	0.838438i	$0.316531\pi$
$174$	7.84002	0.594350
$175$	−3.00058	−0.226823
$176$	−1.64229	−0.123793
$177$	−10.4439	−0.785012
$178$	−30.9383	−2.31892
$179$	−3.05022	−0.227984	−0.113992	−	0.993482i	$-0.536364\pi$
$179$	−3.05022	−0.227984	−0.113992	+	0.993482i	$0.536364\pi$
$180$	3.82195	0.284871
$181$	−23.4769	−1.74503	−0.872513	−	0.488590i	$-0.837511\pi$
$181$	−23.4769	−1.74503	−0.872513	+	0.488590i	$0.837511\pi$
$182$	−7.24001	−0.536665
$183$	−4.60848	−0.340669
$184$	9.60890	0.708377
$185$	4.86840	0.357932
$186$	2.41287	0.176920
$187$	1.46978	0.107481
$188$	−38.4369	−2.80330
$189$	3.00058	0.218260
$190$	4.49764	0.326293
$191$	16.9542	1.22676	0.613382	−	0.789786i	$-0.289808\pi$
$191$	16.9542	1.22676	0.613382	+	0.789786i	$0.289808\pi$
$192$	9.88866	0.713653
$193$	−10.4486	−0.752105	−0.376053	−	0.926598i	$-0.622719\pi$
$193$	−10.4486	−0.752105	−0.376053	+	0.926598i	$0.622719\pi$
$194$	31.4527	2.25817
$195$	1.00000	0.0716115
$196$	7.65719	0.546942
$197$	−10.8532	−0.773256	−0.386628	−	0.922236i	$-0.626360\pi$
$197$	−10.8532	−0.773256	−0.386628	+	0.922236i	$0.626360\pi$
$198$	1.33720	0.0950309
$199$	0.00845397	0.000599286 0	0.000299643	−	1.00000i	$-0.499905\pi$
$199$	0.00845397	0.000599286 0	0.000299643		1.00000i	$0.499905\pi$
$200$	−4.39612	−0.310853
$201$	−0.667461	−0.0470790
$202$	−18.7129	−1.31663
$203$	−9.74963	−0.684290
$204$	10.1361	0.709671
$205$	1.71295	0.119637
$206$	23.1322	1.61169
$207$	−2.18577	−0.151921
$208$	−2.96338	−0.205474
$209$	1.03303	0.0714564
$210$	−7.24001	−0.499608
$211$	−19.6891	−1.35545	−0.677726	−	0.735315i	$-0.737034\pi$
$211$	−19.6891	−1.35545	−0.677726	+	0.735315i	$0.737034\pi$
$212$	−3.22148	−0.221252
$213$	−13.1475	−0.900849
$214$	−3.01129	−0.205848
$215$	8.05186	0.549132
$216$	4.39612	0.299118
$217$	−3.00058	−0.203693
$218$	−3.90828	−0.264702
$219$	−2.66659	−0.180191
$220$	−2.11811	−0.142803
$221$	2.65209	0.178399
$222$	11.7468	0.788394
$223$	7.14504	0.478467	0.239234	−	0.970962i	$-0.423104\pi$
$223$	7.14504	0.478467	0.239234	+	0.970962i	$0.423104\pi$
$224$	−4.92691	−0.329193
$225$	1.00000	0.0666667
$226$	32.5821	2.16733
$227$	−1.08017	−0.0716935	−0.0358467	−	0.999357i	$-0.511413\pi$
$227$	−1.08017	−0.0716935	−0.0358467	+	0.999357i	$0.511413\pi$
$228$	7.12419	0.471811
$229$	−19.7290	−1.30373	−0.651864	−	0.758336i	$-0.726013\pi$
$229$	−19.7290	−1.30373	−0.651864	+	0.758336i	$0.726013\pi$
$230$	5.27397	0.347755
$231$	−1.66291	−0.109411
$232$	−14.2841	−0.937797
$233$	26.5376	1.73854	0.869268	−	0.494341i	$-0.164591\pi$
$233$	26.5376	1.73854	0.869268	+	0.494341i	$0.164591\pi$
$234$	2.41287	0.157734
$235$	−10.0569	−0.656040
$236$	39.9160	2.59831
$237$	9.44794	0.613709
$238$	−19.2011	−1.24463
$239$	−21.2263	−1.37302	−0.686509	−	0.727121i	$-0.740858\pi$
$239$	−21.2263	−1.37302	−0.686509	+	0.727121i	$0.740858\pi$
$240$	−2.96338	−0.191285
$241$	13.6924	0.882006	0.441003	−	0.897506i	$-0.354623\pi$
$241$	13.6924	0.882006	0.441003	+	0.897506i	$0.354623\pi$
$242$	25.8005	1.65852
$243$	−1.00000	−0.0641500
$244$	17.6134	1.12758
$245$	2.00348	0.127998
$246$	4.13312	0.263518
$247$	1.86402	0.118605
$248$	−4.39612	−0.279154
$249$	−10.8723	−0.689002
$250$	−2.41287	−0.152603
$251$	−19.3207	−1.21951	−0.609757	−	0.792588i	$-0.708733\pi$
$251$	−19.3207	−1.21951	−0.609757	+	0.792588i	$0.708733\pi$
$252$	−11.4681	−0.722420
$253$	1.21134	0.0761565
$254$	43.7176	2.74309
$255$	2.65209	0.166080
$256$	−29.8701	−1.86688
$257$	−7.27349	−0.453708	−0.226854	−	0.973929i	$-0.572844\pi$
$257$	−7.27349	−0.453708	−0.226854	+	0.973929i	$0.572844\pi$
$258$	19.4281	1.20954
$259$	−14.6080	−0.907698
$260$	−3.82195	−0.237027
$261$	3.24925	0.201123
$262$	4.70279	0.290539
$263$	19.2661	1.18800	0.594000	−	0.804465i	$-0.297548\pi$
$263$	19.2661	1.18800	0.594000	+	0.804465i	$0.297548\pi$
$264$	−2.43631	−0.149945
$265$	−0.842891	−0.0517784
$266$	−13.4955	−0.827464
$267$	−12.8222	−0.784705
$268$	2.55100	0.155827
$269$	20.7730	1.26655	0.633276	−	0.773926i	$-0.281710\pi$
$269$	20.7730	1.26655	0.633276	+	0.773926i	$0.281710\pi$
$270$	2.41287	0.146843
$271$	24.6454	1.49710	0.748552	−	0.663076i	$-0.230749\pi$
$271$	24.6454	1.49710	0.748552	+	0.663076i	$0.230749\pi$
$272$	−7.85915	−0.476531
$273$	−3.00058	−0.181603
$274$	−8.14153	−0.491848
$275$	−0.554196	−0.0334193
$276$	8.35388	0.502845
$277$	21.8538	1.31307	0.656535	−	0.754296i	$-0.272022\pi$
$277$	21.8538	1.31307	0.656535	+	0.754296i	$0.272022\pi$
$278$	−12.2409	−0.734161
$279$	1.00000	0.0598684
$280$	13.1909	0.788308
$281$	11.8263	0.705501	0.352750	−	0.935717i	$-0.385247\pi$
$281$	11.8263	0.705501	0.352750	+	0.935717i	$0.385247\pi$
$282$	−24.2660	−1.44502
$283$	−18.4048	−1.09405	−0.547026	−	0.837116i	$-0.684240\pi$
$283$	−18.4048	−1.09405	−0.547026	+	0.837116i	$0.684240\pi$
$284$	50.2489	2.98172
$285$	1.86402	0.110415
$286$	−1.33720	−0.0790705
$287$	−5.13983	−0.303394
$288$	1.64199	0.0967550
$289$	−9.96643	−0.586261
$290$	−7.84002	−0.460382
$291$	13.0354	0.764148
$292$	10.1915	0.596415
$293$	19.7105	1.15150	0.575749	−	0.817627i	$-0.304711\pi$
$293$	19.7105	1.15150	0.575749	+	0.817627i	$0.304711\pi$
$294$	4.83413	0.281932
$295$	10.4439	0.608068
$296$	−21.4021	−1.24397
$297$	0.554196	0.0321577
$298$	−39.3601	−2.28007
$299$	2.18577	0.126406
$300$	−3.82195	−0.220660
$301$	−24.1602	−1.39257
$302$	22.5948	1.30018
$303$	−7.75543	−0.445538
$304$	−5.52380	−0.316812
$305$	4.60848	0.263881
$306$	6.39914	0.365815
$307$	17.2070	0.982058	0.491029	−	0.871143i	$-0.336621\pi$
$307$	17.2070	0.982058	0.491029	+	0.871143i	$0.336621\pi$
$308$	6.35555	0.362141
$309$	9.58699	0.545385
$310$	−2.41287	−0.137042
$311$	−30.6562	−1.73835	−0.869177	−	0.494501i	$-0.835351\pi$
$311$	−30.6562	−1.73835	−0.869177	+	0.494501i	$0.835351\pi$
$312$	−4.39612	−0.248881
$313$	5.32065	0.300741	0.150370	−	0.988630i	$-0.451953\pi$
$313$	5.32065	0.300741	0.150370	+	0.988630i	$0.451953\pi$
$314$	−25.5625	−1.44257
$315$	−3.00058	−0.169064
$316$	−36.1095	−2.03132
$317$	20.3404	1.14243	0.571216	−	0.820800i	$-0.306472\pi$
$317$	20.3404	1.14243	0.571216	+	0.820800i	$0.306472\pi$
$318$	−2.03379	−0.114049
$319$	−1.80072	−0.100821
$320$	−9.88866	−0.552793
$321$	−1.24801	−0.0696573
$322$	−15.8250	−0.881891
$323$	4.94355	0.275066
$324$	3.82195	0.212330
$325$	−1.00000	−0.0554700
$326$	11.3813	0.630354
$327$	−1.61976	−0.0895732
$328$	−7.53031	−0.415792
$329$	30.1765	1.66369
$330$	−1.33720	−0.0736106
$331$	4.61527	0.253678	0.126839	−	0.991923i	$-0.459517\pi$
$331$	4.61527	0.253678	0.126839	+	0.991923i	$0.459517\pi$
$332$	41.5532	2.28053
$333$	4.86840	0.266786
$334$	38.1144	2.08552
$335$	0.667461	0.0364673
$336$	8.89186	0.485091
$337$	−17.6570	−0.961837	−0.480918	−	0.876765i	$-0.659697\pi$
$337$	−17.6570	−0.961837	−0.480918	+	0.876765i	$0.659697\pi$
$338$	−2.41287	−0.131243
$339$	13.5034	0.733407
$340$	−10.1361	−0.549709
$341$	−0.554196	−0.0300114
$342$	4.49764	0.243205
$343$	14.9925	0.809517
$344$	−35.3970	−1.90848
$345$	2.18577	0.117678
$346$	−34.5925	−1.85970
$347$	9.58566	0.514585	0.257293	−	0.966334i	$-0.417170\pi$
$347$	9.58566	0.514585	0.257293	+	0.966334i	$0.417170\pi$
$348$	−12.4185	−0.665699
$349$	35.0341	1.87533	0.937666	−	0.347539i	$-0.112983\pi$
$349$	35.0341	1.87533	0.937666	+	0.347539i	$0.112983\pi$
$350$	7.24001	0.386995
$351$	1.00000	0.0533761
$352$	−0.909983	−0.0485022
$353$	−31.3456	−1.66836	−0.834179	−	0.551494i	$-0.814058\pi$
$353$	−31.3456	−1.66836	−0.834179	+	0.551494i	$0.814058\pi$
$354$	25.1998	1.33935
$355$	13.1475	0.697795
$356$	49.0057	2.59730
$357$	−7.95780	−0.421171
$358$	7.35978	0.388976
$359$	−27.3506	−1.44351	−0.721756	−	0.692148i	$-0.756665\pi$
$359$	−27.3506	−1.44351	−0.721756	+	0.692148i	$0.756665\pi$
$360$	−4.39612	−0.231696
$361$	−15.5254	−0.817128
$362$	56.6468	2.97729
$363$	10.6929	0.561230
$364$	11.4681	0.601089
$365$	2.66659	0.139575
$366$	11.1197	0.581235
$367$	−4.69463	−0.245058	−0.122529	−	0.992465i	$-0.539100\pi$
$367$	−4.69463	−0.245058	−0.122529	+	0.992465i	$0.539100\pi$
$368$	−6.47726	−0.337650
$369$	1.71295	0.0891724
$370$	−11.7468	−0.610688
$371$	2.52916	0.131308
$372$	−3.82195	−0.198159
$373$	−4.20018	−0.217477	−0.108739	−	0.994070i	$-0.534681\pi$
$373$	−4.20018	−0.217477	−0.108739	+	0.994070i	$0.534681\pi$
$374$	−3.54638	−0.183379
$375$	−1.00000	−0.0516398
$376$	44.2114	2.28003
$377$	−3.24925	−0.167345
$378$	−7.24001	−0.372386
$379$	7.49683	0.385087	0.192543	−	0.981288i	$-0.438326\pi$
$379$	7.49683	0.385087	0.192543	+	0.981288i	$0.438326\pi$
$380$	−7.12419	−0.365463
$381$	18.1185	0.928239
$382$	−40.9084	−2.09305
$383$	−30.3561	−1.55112	−0.775561	−	0.631272i	$-0.782533\pi$
$383$	−30.3561	−1.55112	−0.775561	+	0.631272i	$0.782533\pi$
$384$	−20.5761	−1.05002
$385$	1.66291	0.0847497
$386$	25.2111	1.28321
$387$	8.05186	0.409299
$388$	−49.8206	−2.52926
$389$	9.29775	0.471415	0.235707	−	0.971824i	$-0.424259\pi$
$389$	9.29775	0.471415	0.235707	+	0.971824i	$0.424259\pi$
$390$	−2.41287	−0.122180
$391$	5.79684	0.293159
$392$	−8.80753	−0.444848
$393$	1.94904	0.0983162
$394$	26.1873	1.31930
$395$	−9.44794	−0.475377
$396$	−2.11811	−0.106439
$397$	23.1101	1.15986	0.579932	−	0.814665i	$-0.303079\pi$
$397$	23.1101	1.15986	0.579932	+	0.814665i	$0.303079\pi$
$398$	−0.0203983	−0.00102248
$399$	−5.59314	−0.280007
$400$	2.96338	0.148169
$401$	−29.6007	−1.47819	−0.739095	−	0.673601i	$-0.764747\pi$
$401$	−29.6007	−1.47819	−0.739095	+	0.673601i	$0.764747\pi$
$402$	1.61050	0.0803243
$403$	−1.00000	−0.0498135
$404$	29.6408	1.47469
$405$	1.00000	0.0496904
$406$	23.5246	1.16751
$407$	−2.69805	−0.133737
$408$	−11.6589	−0.577201
$409$	23.3412	1.15415	0.577073	−	0.816692i	$-0.304195\pi$
$409$	23.3412	1.15415	0.577073	+	0.816692i	$0.304195\pi$
$410$	−4.13312	−0.204120
$411$	−3.37421	−0.166437
$412$	−36.6410	−1.80517
$413$	−31.3378	−1.54203
$414$	5.27397	0.259202
$415$	10.8723	0.533698
$416$	−1.64199	−0.0805050
$417$	−5.07317	−0.248434
$418$	−2.49258	−0.121916
$419$	5.37857	0.262761	0.131380	−	0.991332i	$-0.458059\pi$
$419$	5.37857	0.262761	0.131380	+	0.991332i	$0.458059\pi$
$420$	11.4681	0.559584
$421$	21.8390	1.06437	0.532184	−	0.846629i	$-0.321371\pi$
$421$	21.8390	1.06437	0.532184	+	0.846629i	$0.321371\pi$
$422$	47.5072	2.31261
$423$	−10.0569	−0.488983
$424$	3.70545	0.179953
$425$	−2.65209	−0.128645
$426$	31.7231	1.53699
$427$	−13.8281	−0.669190
$428$	4.76984	0.230559
$429$	−0.554196	−0.0267568
$430$	−19.4281	−0.936906
$431$	−16.2902	−0.784670	−0.392335	−	0.919822i	$-0.628333\pi$
$431$	−16.2902	−0.784670	−0.392335	+	0.919822i	$0.628333\pi$
$432$	−2.96338	−0.142576
$433$	11.8222	0.568138	0.284069	−	0.958804i	$-0.408316\pi$
$433$	11.8222	0.568138	0.284069	+	0.958804i	$0.408316\pi$
$434$	7.24001	0.347532
$435$	−3.24925	−0.155790
$436$	6.19065	0.296479
$437$	4.07431	0.194901
$438$	6.43413	0.307434
$439$	−25.6164	−1.22260	−0.611302	−	0.791397i	$-0.709354\pi$
$439$	−25.6164	−1.22260	−0.611302	+	0.791397i	$0.709354\pi$
$440$	2.43631	0.116147
$441$	2.00348	0.0954037
$442$	−6.39914	−0.304376
$443$	−8.39216	−0.398724	−0.199362	−	0.979926i	$-0.563887\pi$
$443$	−8.39216	−0.398724	−0.199362	+	0.979926i	$0.563887\pi$
$444$	−18.6067	−0.883037
$445$	12.8222	0.607830
$446$	−17.2401	−0.816341
$447$	−16.3126	−0.771558
$448$	29.6717	1.40186
$449$	−15.5769	−0.735121	−0.367561	−	0.930000i	$-0.619807\pi$
$449$	−15.5769	−0.735121	−0.367561	+	0.930000i	$0.619807\pi$
$450$	−2.41287	−0.113744
$451$	−0.949308	−0.0447012
$452$	−51.6095	−2.42750
$453$	9.36426	0.439972
$454$	2.60631	0.122320
$455$	3.00058	0.140669
$456$	−8.19446	−0.383741
$457$	−10.8347	−0.506826	−0.253413	−	0.967358i	$-0.581553\pi$
$457$	−10.8347	−0.506826	−0.253413	+	0.967358i	$0.581553\pi$
$458$	47.6035	2.22436
$459$	2.65209	0.123789
$460$	−8.35388	−0.389502
$461$	−27.3479	−1.27372	−0.636859	−	0.770980i	$-0.719767\pi$
$461$	−27.3479	−1.27372	−0.636859	+	0.770980i	$0.719767\pi$
$462$	4.01239	0.186673
$463$	−11.6709	−0.542392	−0.271196	−	0.962524i	$-0.587419\pi$
$463$	−11.6709	−0.542392	−0.271196	+	0.962524i	$0.587419\pi$
$464$	9.62876	0.447004
$465$	−1.00000	−0.0463739
$466$	−64.0318	−2.96622
$467$	−16.3303	−0.755676	−0.377838	−	0.925872i	$-0.623332\pi$
$467$	−16.3303	−0.755676	−0.377838	+	0.925872i	$0.623332\pi$
$468$	−3.82195	−0.176670
$469$	−2.00277	−0.0924793
$470$	24.2660	1.11931
$471$	−10.5942	−0.488155
$472$	−45.9127	−2.11330
$473$	−4.46231	−0.205177
$474$	−22.7966	−1.04708
$475$	−1.86402	−0.0855271
$476$	30.4143	1.39404
$477$	−0.842891	−0.0385933
$478$	51.2164	2.34259
$479$	19.9060	0.909528	0.454764	−	0.890612i	$-0.349724\pi$
$479$	19.9060	0.909528	0.454764	+	0.890612i	$0.349724\pi$
$480$	−1.64199	−0.0749461
$481$	−4.86840	−0.221980
$482$	−33.0380	−1.50484
$483$	−6.55857	−0.298425
$484$	−40.8676	−1.85762
$485$	−13.0354	−0.591907
$486$	2.41287	0.109450
$487$	−19.1341	−0.867049	−0.433525	−	0.901142i	$-0.642730\pi$
$487$	−19.1341	−0.867049	−0.433525	+	0.901142i	$0.642730\pi$
$488$	−20.2594	−0.917102
$489$	4.71693	0.213307
$490$	−4.83413	−0.218384
$491$	−1.73579	−0.0783351	−0.0391676	−	0.999233i	$-0.512471\pi$
$491$	−1.73579	−0.0783351	−0.0391676	+	0.999233i	$0.512471\pi$
$492$	−6.54678	−0.295152
$493$	−8.61729	−0.388103
$494$	−4.49764	−0.202358
$495$	−0.554196	−0.0249093
$496$	2.96338	0.133060
$497$	−39.4500	−1.76957
$498$	26.2334	1.17555
$499$	−7.22655	−0.323505	−0.161752	−	0.986831i	$-0.551715\pi$
$499$	−7.22655	−0.323505	−0.161752	+	0.986831i	$0.551715\pi$
$500$	3.82195	0.170923
$501$	15.7963	0.705725
$502$	46.6185	2.08068
$503$	13.2007	0.588589	0.294295	−	0.955715i	$-0.404915\pi$
$503$	13.2007	0.588589	0.294295	+	0.955715i	$0.404915\pi$
$504$	13.1909	0.587570
$505$	7.75543	0.345112
$506$	−2.92282	−0.129935
$507$	−1.00000	−0.0444116
$508$	−69.2480	−3.07238
$509$	−13.9322	−0.617536	−0.308768	−	0.951137i	$-0.599917\pi$
$509$	−13.9322	−0.617536	−0.308768	+	0.951137i	$0.599917\pi$
$510$	−6.39914	−0.283359
$511$	−8.00130	−0.353957
$512$	30.9206	1.36651
$513$	1.86402	0.0822985
$514$	17.5500	0.774097
$515$	−9.58699	−0.422453
$516$	−30.7738	−1.35474
$517$	5.57350	0.245122
$518$	35.2472	1.54867
$519$	−14.3366	−0.629309
$520$	4.39612	0.192783
$521$	−31.1221	−1.36348	−0.681742	−	0.731592i	$-0.738777\pi$
$521$	−31.1221	−1.36348	−0.681742	+	0.731592i	$0.738777\pi$
$522$	−7.84002	−0.343148
$523$	−16.5528	−0.723804	−0.361902	−	0.932216i	$-0.617872\pi$
$523$	−16.5528	−0.723804	−0.361902	+	0.932216i	$0.617872\pi$
$524$	−7.44914	−0.325417
$525$	3.00058	0.130956
$526$	−46.4867	−2.02692
$527$	−2.65209	−0.115527
$528$	1.64229	0.0714717
$529$	−18.2224	−0.792279
$530$	2.03379	0.0883421
$531$	10.4439	0.453227
$532$	21.3767	0.926797
$533$	−1.71295	−0.0741959
$534$	30.9383	1.33883
$535$	1.24801	0.0539563
$536$	−2.93424	−0.126740
$537$	3.05022	0.131626
$538$	−50.1226	−2.16094
$539$	−1.11032	−0.0478249
$540$	−3.82195	−0.164470
$541$	5.38033	0.231319	0.115659	−	0.993289i	$-0.463102\pi$
$541$	5.38033	0.231319	0.115659	+	0.993289i	$0.463102\pi$
$542$	−59.4663	−2.55430
$543$	23.4769	1.00749
$544$	−4.35469	−0.186706
$545$	1.61976	0.0693831
$546$	7.24001	0.309844
$547$	−20.8298	−0.890619	−0.445309	−	0.895377i	$-0.646906\pi$
$547$	−20.8298	−0.890619	−0.445309	+	0.895377i	$0.646906\pi$
$548$	12.8960	0.550892
$549$	4.60848	0.196685
$550$	1.33720	0.0570186
$551$	−6.05667	−0.258023
$552$	−9.60890	−0.408982
$553$	28.3493	1.20553
$554$	−52.7305	−2.24030
$555$	−4.86840	−0.206652
$556$	19.3894	0.822294
$557$	13.0836	0.554371	0.277185	−	0.960816i	$-0.410598\pi$
$557$	13.0836	0.554371	0.277185	+	0.960816i	$0.410598\pi$
$558$	−2.41287	−0.102145
$559$	−8.05186	−0.340557
$560$	−8.89186	−0.375750
$561$	−1.46978	−0.0620540
$562$	−28.5354	−1.20370
$563$	1.34241	0.0565756	0.0282878	−	0.999600i	$-0.490995\pi$
$563$	1.34241	0.0565756	0.0282878	+	0.999600i	$0.490995\pi$
$564$	38.4369	1.61849
$565$	−13.5034	−0.568094
$566$	44.4084	1.86662
$567$	−3.00058	−0.126013
$568$	−57.7978	−2.42514
$569$	−11.9954	−0.502873	−0.251436	−	0.967874i	$-0.580903\pi$
$569$	−11.9954	−0.502873	−0.251436	+	0.967874i	$0.580903\pi$
$570$	−4.49764	−0.188385
$571$	−37.1680	−1.55543	−0.777716	−	0.628616i	$-0.783622\pi$
$571$	−37.1680	−1.55543	−0.777716	+	0.628616i	$0.783622\pi$
$572$	2.11811	0.0885625
$573$	−16.9542	−0.708273
$574$	12.4017	0.517639
$575$	−2.18577	−0.0911528
$576$	−9.88866	−0.412028
$577$	−17.4232	−0.725339	−0.362669	−	0.931918i	$-0.618135\pi$
$577$	−17.4232	−0.725339	−0.362669	+	0.931918i	$0.618135\pi$
$578$	24.0477	1.00025
$579$	10.4486	0.434228
$580$	12.4185	0.515648
$581$	−32.6231	−1.35343
$582$	−31.4527	−1.30376
$583$	0.467127	0.0193464
$584$	−11.7226	−0.485086
$585$	−1.00000	−0.0413449
$586$	−47.5588	−1.96464
$587$	18.9134	0.780640	0.390320	−	0.920679i	$-0.372364\pi$
$587$	18.9134	0.780640	0.390320	+	0.920679i	$0.372364\pi$
$588$	−7.65719	−0.315777
$589$	−1.86402	−0.0768056
$590$	−25.1998	−1.03746
$591$	10.8532	0.446440
$592$	14.4269	0.592942
$593$	−33.8140	−1.38857	−0.694287	−	0.719698i	$-0.744280\pi$
$593$	−33.8140	−1.38857	−0.694287	+	0.719698i	$0.744280\pi$
$594$	−1.33720	−0.0548661
$595$	7.95780	0.326238
$596$	62.3458	2.55378
$597$	−0.00845397	−0.000345998 0
$598$	−5.27397	−0.215669
$599$	15.3946	0.629005	0.314503	−	0.949257i	$-0.398162\pi$
$599$	15.3946	0.629005	0.314503	+	0.949257i	$0.398162\pi$
$600$	4.39612	0.179471
$601$	−26.1874	−1.06821	−0.534103	−	0.845419i	$-0.679351\pi$
$601$	−26.1874	−1.06821	−0.534103	+	0.845419i	$0.679351\pi$
$602$	58.2956	2.37595
$603$	0.667461	0.0271811
$604$	−35.7897	−1.45626
$605$	−10.6929	−0.434727
$606$	18.7129	0.760158
$607$	−22.8924	−0.929175	−0.464587	−	0.885527i	$-0.653797\pi$
$607$	−22.8924	−0.929175	−0.464587	+	0.885527i	$0.653797\pi$
$608$	−3.06070	−0.124128
$609$	9.74963	0.395075
$610$	−11.1197	−0.450222
$611$	10.0569	0.406859
$612$	−10.1361	−0.409729
$613$	42.1221	1.70130	0.850648	−	0.525736i	$-0.176210\pi$
$613$	42.1221	1.70130	0.850648	+	0.525736i	$0.176210\pi$
$614$	−41.5184	−1.67555
$615$	−1.71295	−0.0690726
$616$	−7.31035	−0.294543
$617$	6.84982	0.275764	0.137882	−	0.990449i	$-0.455971\pi$
$617$	6.84982	0.275764	0.137882	+	0.990449i	$0.455971\pi$
$618$	−23.1322	−0.930512
$619$	−26.2496	−1.05506	−0.527530	−	0.849536i	$-0.676882\pi$
$619$	−26.2496	−1.05506	−0.527530	+	0.849536i	$0.676882\pi$
$620$	3.82195	0.153493
$621$	2.18577	0.0877118
$622$	73.9695	2.96591
$623$	−38.4740	−1.54143
$624$	2.96338	0.118630
$625$	1.00000	0.0400000
$626$	−12.8380	−0.513111
$627$	−1.03303	−0.0412554
$628$	40.4905	1.61575
$629$	−12.9114	−0.514812
$630$	7.24001	0.288449
$631$	−37.5118	−1.49332	−0.746662	−	0.665204i	$-0.768345\pi$
$631$	−37.5118	−1.49332	−0.746662	+	0.665204i	$0.768345\pi$
$632$	41.5343	1.65214
$633$	19.6891	0.782570
$634$	−49.0788	−1.94917
$635$	−18.1185	−0.719011
$636$	3.22148	0.127740
$637$	−2.00348	−0.0793807
$638$	4.34491	0.172017
$639$	13.1475	0.520105
$640$	20.5761	0.813342
$641$	48.8945	1.93122	0.965608	−	0.260003i	$-0.0837237\pi$
$641$	48.8945	1.93122	0.965608	+	0.260003i	$0.0837237\pi$
$642$	3.01129	0.118846
$643$	15.9703	0.629806	0.314903	−	0.949124i	$-0.398028\pi$
$643$	15.9703	0.629806	0.314903	+	0.949124i	$0.398028\pi$
$644$	25.0665	0.987758
$645$	−8.05186	−0.317042
$646$	−11.9281	−0.469306
$647$	−5.19732	−0.204328	−0.102164	−	0.994768i	$-0.532577\pi$
$647$	−5.19732	−0.204328	−0.102164	+	0.994768i	$0.532577\pi$
$648$	−4.39612	−0.172696
$649$	−5.78797	−0.227198
$650$	2.41287	0.0946406
$651$	3.00058	0.117602
$652$	−18.0278	−0.706025
$653$	−17.1197	−0.669944	−0.334972	−	0.942228i	$-0.608727\pi$
$653$	−17.1197	−0.669944	−0.334972	+	0.942228i	$0.608727\pi$
$654$	3.90828	0.152826
$655$	−1.94904	−0.0761554
$656$	5.07611	0.198189
$657$	2.66659	0.104033
$658$	−72.8121	−2.83851
$659$	−20.6200	−0.803240	−0.401620	−	0.915806i	$-0.631553\pi$
$659$	−20.6200	−0.803240	−0.401620	+	0.915806i	$0.631553\pi$
$660$	2.11811	0.0824472
$661$	−33.7698	−1.31349	−0.656747	−	0.754111i	$-0.728068\pi$
$661$	−33.7698	−1.31349	−0.656747	+	0.754111i	$0.728068\pi$
$662$	−11.1361	−0.432815
$663$	−2.65209	−0.102999
$664$	−47.7958	−1.85484
$665$	5.59314	0.216893
$666$	−11.7468	−0.455180
$667$	−7.10210	−0.274994
$668$	−60.3725	−2.33588
$669$	−7.14504	−0.276243
$670$	−1.61050	−0.0622189
$671$	−2.55400	−0.0985962
$672$	4.92691	0.190060
$673$	−36.7034	−1.41481	−0.707406	−	0.706807i	$-0.750135\pi$
$673$	−36.7034	−1.41481	−0.707406	+	0.706807i	$0.750135\pi$
$674$	42.6040	1.64104
$675$	−1.00000	−0.0384900
$676$	3.82195	0.146998
$677$	21.5551	0.828430	0.414215	−	0.910179i	$-0.364056\pi$
$677$	21.5551	0.828430	0.414215	+	0.910179i	$0.364056\pi$
$678$	−32.5821	−1.25131
$679$	39.1137	1.50105
$680$	11.6589	0.447098
$681$	1.08017	0.0413922
$682$	1.33720	0.0512042
$683$	33.7338	1.29079	0.645394	−	0.763850i	$-0.276693\pi$
$683$	33.7338	1.29079	0.645394	+	0.763850i	$0.276693\pi$
$684$	−7.12419	−0.272400
$685$	3.37421	0.128922
$686$	−36.1749	−1.38116
$687$	19.7290	0.752708
$688$	23.8607	0.909682
$689$	0.842891	0.0321116
$690$	−5.27397	−0.200777
$691$	−0.0792708	−0.00301560	−0.00150780	−	0.999999i	$-0.500480\pi$
$691$	−0.0792708	−0.00301560	−0.00150780	+	0.999999i	$0.500480\pi$
$692$	54.7939	2.08295
$693$	1.66291	0.0631687
$694$	−23.1290	−0.877963
$695$	5.07317	0.192436
$696$	14.2841	0.541437
$697$	−4.54288	−0.172074
$698$	−84.5327	−3.19961
$699$	−26.5376	−1.00374
$700$	−11.4681	−0.433452
$701$	−13.7603	−0.519718	−0.259859	−	0.965647i	$-0.583676\pi$
$701$	−13.7603	−0.519718	−0.259859	+	0.965647i	$0.583676\pi$
$702$	−2.41287	−0.0910679
$703$	−9.07479	−0.342262
$704$	5.48026	0.206545
$705$	10.0569	0.378765
$706$	75.6329	2.84648
$707$	−23.2708	−0.875188
$708$	−39.9160	−1.50014
$709$	2.86212	0.107489	0.0537446	−	0.998555i	$-0.482884\pi$
$709$	2.86212	0.107489	0.0537446	+	0.998555i	$0.482884\pi$
$710$	−31.7231	−1.19055
$711$	−9.44794	−0.354325
$712$	−56.3679	−2.11248
$713$	−2.18577	−0.0818576
$714$	19.2011	0.718585
$715$	0.554196	0.0207258
$716$	−11.6578	−0.435671
$717$	21.2263	0.792712
$718$	65.9936	2.46286
$719$	−19.8580	−0.740579	−0.370290	−	0.928916i	$-0.620742\pi$
$719$	−19.8580	−0.740579	−0.370290	+	0.928916i	$0.620742\pi$
$720$	2.96338	0.110439
$721$	28.7665	1.07132
$722$	37.4609	1.39415
$723$	−13.6924	−0.509226
$724$	−89.7276	−3.33470
$725$	3.24925	0.120674
$726$	−25.8005	−0.957547
$727$	−33.6103	−1.24654	−0.623268	−	0.782008i	$-0.714196\pi$
$727$	−33.6103	−1.24654	−0.623268	+	0.782008i	$0.714196\pi$
$728$	−13.1909	−0.488888
$729$	1.00000	0.0370370
$730$	−6.43413	−0.238138
$731$	−21.3542	−0.789815
$732$	−17.6134	−0.651009
$733$	−35.5187	−1.31191	−0.655956	−	0.754799i	$-0.727734\pi$
$733$	−35.5187	−1.31191	−0.655956	+	0.754799i	$0.727734\pi$
$734$	11.3275	0.418107
$735$	−2.00348	−0.0738994
$736$	−3.58900	−0.132292
$737$	−0.369904	−0.0136256
$738$	−4.13312	−0.152142
$739$	24.9388	0.917387	0.458694	−	0.888594i	$-0.348318\pi$
$739$	24.9388	0.917387	0.458694	+	0.888594i	$0.348318\pi$
$740$	18.6067	0.683998
$741$	−1.86402	−0.0684765
$742$	−6.10254	−0.224031
$743$	−10.8063	−0.396445	−0.198223	−	0.980157i	$-0.563517\pi$
$743$	−10.8063	−0.396445	−0.198223	+	0.980157i	$0.563517\pi$
$744$	4.39612	0.161170
$745$	16.3126	0.597647
$746$	10.1345	0.371050
$747$	10.8723	0.397795
$748$	5.61741	0.205393
$749$	−3.74476	−0.136831
$750$	2.41287	0.0881056
$751$	36.5224	1.33272	0.666361	−	0.745629i	$-0.267851\pi$
$751$	36.5224	1.33272	0.666361	+	0.745629i	$0.267851\pi$
$752$	−29.8024	−1.08678
$753$	19.3207	0.704087
$754$	7.84002	0.285517
$755$	−9.36426	−0.340800
$756$	11.4681	0.417089
$757$	−34.1631	−1.24168	−0.620839	−	0.783938i	$-0.713208\pi$
$757$	−34.1631	−1.24168	−0.620839	+	0.783938i	$0.713208\pi$
$758$	−18.0889	−0.657018
$759$	−1.21134	−0.0439690
$760$	8.19446	0.297244
$761$	−26.0934	−0.945885	−0.472942	−	0.881093i	$-0.656808\pi$
$761$	−26.0934	−0.945885	−0.472942	+	0.881093i	$0.656808\pi$
$762$	−43.7176	−1.58372
$763$	−4.86023	−0.175952
$764$	64.7981	2.34431
$765$	−2.65209	−0.0958864
$766$	73.2453	2.64646
$767$	−10.4439	−0.377108
$768$	29.8701	1.07785
$769$	1.76216	0.0635450	0.0317725	−	0.999495i	$-0.489885\pi$
$769$	1.76216	0.0635450	0.0317725	+	0.999495i	$0.489885\pi$
$770$	−4.01239	−0.144596
$771$	7.27349	0.261948
$772$	−39.9339	−1.43725
$773$	1.69900	0.0611088	0.0305544	−	0.999533i	$-0.490273\pi$
$773$	1.69900	0.0611088	0.0305544	+	0.999533i	$0.490273\pi$
$774$	−19.4281	−0.698329
$775$	1.00000	0.0359211
$776$	57.3052	2.05714
$777$	14.6080	0.524059
$778$	−22.4343	−0.804308
$779$	−3.19296	−0.114400
$780$	3.82195	0.136848
$781$	−7.28627	−0.260723
$782$	−13.9870	−0.500175
$783$	−3.24925	−0.116119
$784$	5.93707	0.212038
$785$	10.5942	0.378123
$786$	−4.70279	−0.167743
$787$	−31.7816	−1.13289	−0.566445	−	0.824100i	$-0.691682\pi$
$787$	−31.7816	−1.13289	−0.566445	+	0.824100i	$0.691682\pi$
$788$	−41.4802	−1.47767
$789$	−19.2661	−0.685893
$790$	22.7966	0.811068
$791$	40.5182	1.44066
$792$	2.43631	0.0865706
$793$	−4.60848	−0.163652
$794$	−55.7618	−1.97891
$795$	0.842891	0.0298943
$796$	0.0323106	0.00114522
$797$	−22.4023	−0.793529	−0.396764	−	0.917920i	$-0.629867\pi$
$797$	−22.4023	−0.793529	−0.396764	+	0.917920i	$0.629867\pi$
$798$	13.4955	0.477737
$799$	26.6718	0.943580
$800$	1.64199	0.0580530
$801$	12.8222	0.453050
$802$	71.4228	2.52203
$803$	−1.47781	−0.0521508
$804$	−2.55100	−0.0899668
$805$	6.55857	0.231159
$806$	2.41287	0.0849898
$807$	−20.7730	−0.731244
$808$	−34.0938	−1.19942
$809$	−13.6014	−0.478201	−0.239101	−	0.970995i	$-0.576853\pi$
$809$	−13.6014	−0.478201	−0.239101	+	0.970995i	$0.576853\pi$
$810$	−2.41287	−0.0847796
$811$	−27.5504	−0.967424	−0.483712	−	0.875227i	$-0.660712\pi$
$811$	−27.5504	−0.967424	−0.483712	+	0.875227i	$0.660712\pi$
$812$	−37.2626	−1.30766
$813$	−24.6454	−0.864353
$814$	6.51004	0.228177
$815$	−4.71693	−0.165227
$816$	7.85915	0.275125
$817$	−15.0088	−0.525093
$818$	−56.3192	−1.96916
$819$	3.00058	0.104849
$820$	6.54678	0.228624
$821$	−25.7508	−0.898709	−0.449355	−	0.893354i	$-0.648346\pi$
$821$	−25.7508	−0.898709	−0.449355	+	0.893354i	$0.648346\pi$
$822$	8.14153	0.283969
$823$	13.0861	0.456154	0.228077	−	0.973643i	$-0.426756\pi$
$823$	13.0861	0.456154	0.228077	+	0.973643i	$0.426756\pi$
$824$	42.1456	1.46821
$825$	0.554196	0.0192946
$826$	75.6140	2.63095
$827$	−47.3479	−1.64645	−0.823223	−	0.567717i	$-0.807827\pi$
$827$	−47.3479	−1.64645	−0.823223	+	0.567717i	$0.807827\pi$
$828$	−8.35388	−0.290318
$829$	−41.0773	−1.42667	−0.713337	−	0.700821i	$-0.752817\pi$
$829$	−41.0773	−1.42667	−0.713337	+	0.700821i	$0.752817\pi$
$830$	−26.2334	−0.910573
$831$	−21.8538	−0.758101
$832$	9.88866	0.342828
$833$	−5.31340	−0.184098
$834$	12.2409	0.423868
$835$	−15.7963	−0.546652
$836$	3.94820	0.136551
$837$	−1.00000	−0.0345651
$838$	−12.9778	−0.448311
$839$	25.9989	0.897580	0.448790	−	0.893637i	$-0.351855\pi$
$839$	25.9989	0.897580	0.448790	+	0.893637i	$0.351855\pi$
$840$	−13.1909	−0.455130
$841$	−18.4424	−0.635944
$842$	−52.6947	−1.81598
$843$	−11.8263	−0.407321
$844$	−75.2506	−2.59023
$845$	1.00000	0.0344010
$846$	24.2660	0.834282
$847$	32.0848	1.10245
$848$	−2.49781	−0.0857751
$849$	18.4048	0.631651
$850$	6.39914	0.219489
$851$	−10.6412	−0.364775
$852$	−50.2489	−1.72150
$853$	−39.0220	−1.33609	−0.668045	−	0.744121i	$-0.732869\pi$
$853$	−39.0220	−1.33609	−0.668045	+	0.744121i	$0.732869\pi$
$854$	33.3655	1.14174
$855$	−1.86402	−0.0637482
$856$	−5.48641	−0.187522
$857$	−1.93991	−0.0662661	−0.0331330	−	0.999451i	$-0.510549\pi$
$857$	−1.93991	−0.0662661	−0.0331330	+	0.999451i	$0.510549\pi$
$858$	1.33720	0.0456514
$859$	−20.8694	−0.712056	−0.356028	−	0.934475i	$-0.615869\pi$
$859$	−20.8694	−0.712056	−0.356028	+	0.934475i	$0.615869\pi$
$860$	30.7738	1.04938
$861$	5.13983	0.175165
$862$	39.3061	1.33877
$863$	−24.6222	−0.838148	−0.419074	−	0.907952i	$-0.637645\pi$
$863$	−24.6222	−0.838148	−0.419074	+	0.907952i	$0.637645\pi$
$864$	−1.64199	−0.0558615
$865$	14.3366	0.487461
$866$	−28.5254	−0.969333
$867$	9.96643	0.338478
$868$	−11.4681	−0.389251
$869$	5.23601	0.177619
$870$	7.84002	0.265801
$871$	−0.667461	−0.0226160
$872$	−7.12068	−0.241137
$873$	−13.0354	−0.441181
$874$	−9.83079	−0.332532
$875$	−3.00058	−0.101438
$876$	−10.1915	−0.344340
$877$	3.59568	0.121418	0.0607088	−	0.998156i	$-0.480664\pi$
$877$	3.59568	0.121418	0.0607088	+	0.998156i	$0.480664\pi$
$878$	61.8091	2.08596
$879$	−19.7105	−0.664817
$880$	−1.64229	−0.0553617
$881$	24.0240	0.809390	0.404695	−	0.914452i	$-0.367378\pi$
$881$	24.0240	0.809390	0.404695	+	0.914452i	$0.367378\pi$
$882$	−4.83413	−0.162774
$883$	9.00805	0.303145	0.151573	−	0.988446i	$-0.451566\pi$
$883$	9.00805	0.303145	0.151573	+	0.988446i	$0.451566\pi$
$884$	10.1361	0.340915
$885$	−10.4439	−0.351068
$886$	20.2492	0.680285
$887$	12.4166	0.416908	0.208454	−	0.978032i	$-0.433157\pi$
$887$	12.4166	0.416908	0.208454	+	0.978032i	$0.433157\pi$
$888$	21.4021	0.718206
$889$	54.3660	1.82338
$890$	−30.9383	−1.03705
$891$	−0.554196	−0.0185663
$892$	27.3080	0.914338
$893$	18.7463	0.627320
$894$	39.3601	1.31640
$895$	−3.05022	−0.101957
$896$	−61.7402	−2.06260
$897$	−2.18577	−0.0729806
$898$	37.5851	1.25423
$899$	3.24925	0.108369
$900$	3.82195	0.127398
$901$	2.23542	0.0744727
$902$	2.29056	0.0762672
$903$	24.1602	0.804003
$904$	59.3628	1.97438
$905$	−23.4769	−0.780400
$906$	−22.5948	−0.750661
$907$	14.8177	0.492013	0.246006	−	0.969268i	$-0.420882\pi$
$907$	14.8177	0.492013	0.246006	+	0.969268i	$0.420882\pi$
$908$	−4.12835	−0.137004
$909$	7.75543	0.257231
$910$	−7.24001	−0.240004
$911$	−32.0663	−1.06241	−0.531203	−	0.847245i	$-0.678260\pi$
$911$	−32.0663	−1.06241	−0.531203	+	0.847245i	$0.678260\pi$
$912$	5.52380	0.182911
$913$	−6.02537	−0.199411
$914$	26.1428	0.864726
$915$	−4.60848	−0.152352
$916$	−75.4031	−2.49139
$917$	5.84826	0.193127
$918$	−6.39914	−0.211203
$919$	−9.64418	−0.318132	−0.159066	−	0.987268i	$-0.550848\pi$
$919$	−9.64418	−0.318132	−0.159066	+	0.987268i	$0.550848\pi$
$920$	9.60890	0.316796
$921$	−17.2070	−0.566991
$922$	65.9869	2.17316
$923$	−13.1475	−0.432754
$924$	−6.35555	−0.209082
$925$	4.86840	0.160072
$926$	28.1603	0.925406
$927$	−9.58699	−0.314878
$928$	5.33522	0.175137
$929$	13.1019	0.429860	0.214930	−	0.976629i	$-0.431048\pi$
$929$	13.1019	0.429860	0.214930	+	0.976629i	$0.431048\pi$
$930$	2.41287	0.0791212
$931$	−3.73452	−0.122394
$932$	101.425	3.32230
$933$	30.6562	1.00364
$934$	39.4029	1.28930
$935$	1.46978	0.0480668
$936$	4.39612	0.143692
$937$	−2.39284	−0.0781708	−0.0390854	−	0.999236i	$-0.512444\pi$
$937$	−2.39284	−0.0781708	−0.0390854	+	0.999236i	$0.512444\pi$
$938$	4.83242	0.157784
$939$	−5.32065	−0.173633
$940$	−38.4369	−1.25367
$941$	2.44264	0.0796278	0.0398139	−	0.999207i	$-0.487323\pi$
$941$	2.44264	0.0796278	0.0398139	+	0.999207i	$0.487323\pi$
$942$	25.5625	0.832870
$943$	−3.74410	−0.121925
$944$	30.9493	1.00731
$945$	3.00058	0.0976089
$946$	10.7670	0.350065
$947$	45.1828	1.46824	0.734122	−	0.679017i	$-0.237594\pi$
$947$	45.1828	1.46824	0.734122	+	0.679017i	$0.237594\pi$
$948$	36.1095	1.17278
$949$	−2.66659	−0.0865610
$950$	4.49764	0.145923
$951$	−20.3404	−0.659583
$952$	−34.9835	−1.13382
$953$	41.3588	1.33974	0.669871	−	0.742477i	$-0.266349\pi$
$953$	41.3588	1.33974	0.669871	+	0.742477i	$0.266349\pi$
$954$	2.03379	0.0658463
$955$	16.9542	0.548626
$956$	−81.1260	−2.62380
$957$	1.80072	0.0582090
$958$	−48.0306	−1.55180
$959$	−10.1246	−0.326940
$960$	9.88866	0.319155
$961$	1.00000	0.0322581
$962$	11.7468	0.378732
$963$	1.24801	0.0402166
$964$	52.3317	1.68549
$965$	−10.4486	−0.336352
$966$	15.8250	0.509160
$967$	57.5051	1.84924	0.924619	−	0.380893i	$-0.124384\pi$
$967$	57.5051	1.84924	0.924619	+	0.380893i	$0.124384\pi$
$968$	47.0071	1.51087
$969$	−4.94355	−0.158810
$970$	31.4527	1.00989
$971$	−13.7109	−0.440004	−0.220002	−	0.975499i	$-0.570606\pi$
$971$	−13.7109	−0.440004	−0.220002	+	0.975499i	$0.570606\pi$
$972$	−3.82195	−0.122589
$973$	−15.2225	−0.488010
$974$	46.1681	1.47932
$975$	1.00000	0.0320256
$976$	13.6567	0.437140
$977$	45.8082	1.46554	0.732768	−	0.680479i	$-0.238228\pi$
$977$	45.8082	1.46554	0.732768	+	0.680479i	$0.238228\pi$
$978$	−11.3813	−0.363935
$979$	−7.10601	−0.227109
$980$	7.65719	0.244600
$981$	1.61976	0.0517151
$982$	4.18824	0.133652
$983$	−37.7022	−1.20251	−0.601257	−	0.799056i	$-0.705333\pi$
$983$	−37.7022	−1.20251	−0.601257	+	0.799056i	$0.705333\pi$
$984$	7.53031	0.240058
$985$	−10.8532	−0.345811
$986$	20.7924	0.662165
$987$	−30.1765	−0.960530
$988$	7.12419	0.226651
$989$	−17.5995	−0.559631
$990$	1.33720	0.0424991
$991$	15.9098	0.505390	0.252695	−	0.967546i	$-0.418683\pi$
$991$	15.9098	0.505390	0.252695	+	0.967546i	$0.418683\pi$
$992$	1.64199	0.0521331
$993$	−4.61527	−0.146461
$994$	95.1877	3.01917
$995$	0.00845397	0.000268009 0
$996$	−41.5532	−1.31666
$997$	−3.50446	−0.110987	−0.0554937	−	0.998459i	$-0.517673\pi$
$997$	−3.50446	−0.110987	−0.0554937	+	0.998459i	$0.517673\pi$
$998$	17.4367	0.551950
$999$	−4.86840	−0.154029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.ba.1.2	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.ba.1.2	✓	13	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)

\(f(q)\)	\(=\)	\(q-2.41287 q^{2} -1.00000 q^{3} +3.82195 q^{4} +1.00000 q^{5} +2.41287 q^{6} -3.00058 q^{7} -4.39612 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.41287 q^{2} -1.00000 q^{3} +3.82195 q^{4} +1.00000 q^{5} +2.41287 q^{6} -3.00058 q^{7} -4.39612 q^{8} +1.00000 q^{9} -2.41287 q^{10} -0.554196 q^{11} -3.82195 q^{12} -1.00000 q^{13} +7.24001 q^{14} -1.00000 q^{15} +2.96338 q^{16} -2.65209 q^{17} -2.41287 q^{18} -1.86402 q^{19} +3.82195 q^{20} +3.00058 q^{21} +1.33720 q^{22} -2.18577 q^{23} +4.39612 q^{24} +1.00000 q^{25} +2.41287 q^{26} -1.00000 q^{27} -11.4681 q^{28} +3.24925 q^{29} +2.41287 q^{30} +1.00000 q^{31} +1.64199 q^{32} +0.554196 q^{33} +6.39914 q^{34} -3.00058 q^{35} +3.82195 q^{36} +4.86840 q^{37} +4.49764 q^{38} +1.00000 q^{39} -4.39612 q^{40} +1.71295 q^{41} -7.24001 q^{42} +8.05186 q^{43} -2.11811 q^{44} +1.00000 q^{45} +5.27397 q^{46} -10.0569 q^{47} -2.96338 q^{48} +2.00348 q^{49} -2.41287 q^{50} +2.65209 q^{51} -3.82195 q^{52} -0.842891 q^{53} +2.41287 q^{54} -0.554196 q^{55} +13.1909 q^{56} +1.86402 q^{57} -7.84002 q^{58} +10.4439 q^{59} -3.82195 q^{60} +4.60848 q^{61} -2.41287 q^{62} -3.00058 q^{63} -9.88866 q^{64} -1.00000 q^{65} -1.33720 q^{66} +0.667461 q^{67} -10.1361 q^{68} +2.18577 q^{69} +7.24001 q^{70} +13.1475 q^{71} -4.39612 q^{72} +2.66659 q^{73} -11.7468 q^{74} -1.00000 q^{75} -7.12419 q^{76} +1.66291 q^{77} -2.41287 q^{78} -9.44794 q^{79} +2.96338 q^{80} +1.00000 q^{81} -4.13312 q^{82} +10.8723 q^{83} +11.4681 q^{84} -2.65209 q^{85} -19.4281 q^{86} -3.24925 q^{87} +2.43631 q^{88} +12.8222 q^{89} -2.41287 q^{90} +3.00058 q^{91} -8.35388 q^{92} -1.00000 q^{93} +24.2660 q^{94} -1.86402 q^{95} -1.64199 q^{96} -13.0354 q^{97} -4.83413 q^{98} -0.554196 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{2} - 13 q^{3} + 9 q^{4} + 13 q^{5} + q^{6} - 11 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \) 13 * q - q^2 - 13 * q^3 + 9 * q^4 + 13 * q^5 + q^6 - 11 * q^7 - 6 * q^8 + 13 * q^9	\( 13 q - q^{2} - 13 q^{3} + 9 q^{4} + 13 q^{5} + q^{6} - 11 q^{7} - 6 q^{8} + 13 q^{9} - q^{10} + 4 q^{11} - 9 q^{12} - 13 q^{13} + q^{14} - 13 q^{15} + 9 q^{16} + 3 q^{17} - q^{18} - 2 q^{19} + 9 q^{20} + 11 q^{21} - 17 q^{22} - 9 q^{23} + 6 q^{24} + 13 q^{25} + q^{26} - 13 q^{27} - 30 q^{28} + 7 q^{29} + q^{30} + 13 q^{31} - 13 q^{32} - 4 q^{33} - 13 q^{34} - 11 q^{35} + 9 q^{36} - 7 q^{37} - 26 q^{38} + 13 q^{39} - 6 q^{40} - 13 q^{41} - q^{42} - 24 q^{43} - q^{44} + 13 q^{45} + 22 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} - q^{50} - 3 q^{51} - 9 q^{52} - 18 q^{53} + q^{54} + 4 q^{55} + 19 q^{56} + 2 q^{57} + 20 q^{58} - 26 q^{59} - 9 q^{60} - 12 q^{61} - q^{62} - 11 q^{63} - 2 q^{64} - 13 q^{65} + 17 q^{66} - 30 q^{67} + 23 q^{68} + 9 q^{69} + q^{70} - 3 q^{71} - 6 q^{72} - 5 q^{73} - 36 q^{74} - 13 q^{75} + 16 q^{76} - q^{78} - 9 q^{79} + 9 q^{80} + 13 q^{81} + 7 q^{82} - 35 q^{83} + 30 q^{84} + 3 q^{85} - 14 q^{86} - 7 q^{87} - 29 q^{88} + q^{89} - q^{90} + 11 q^{91} - 63 q^{92} - 13 q^{93} + 24 q^{94} - 2 q^{95} + 13 q^{96} - 33 q^{97} + q^{98} + 4 q^{99}+O(q^{100}) \) 13 * q - q^2 - 13 * q^3 + 9 * q^4 + 13 * q^5 + q^6 - 11 * q^7 - 6 * q^8 + 13 * q^9 - q^10 + 4 * q^11 - 9 * q^12 - 13 * q^13 + q^14 - 13 * q^15 + 9 * q^16 + 3 * q^17 - q^18 - 2 * q^19 + 9 * q^20 + 11 * q^21 - 17 * q^22 - 9 * q^23 + 6 * q^24 + 13 * q^25 + q^26 - 13 * q^27 - 30 * q^28 + 7 * q^29 + q^30 + 13 * q^31 - 13 * q^32 - 4 * q^33 - 13 * q^34 - 11 * q^35 + 9 * q^36 - 7 * q^37 - 26 * q^38 + 13 * q^39 - 6 * q^40 - 13 * q^41 - q^42 - 24 * q^43 - q^44 + 13 * q^45 + 22 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 - q^50 - 3 * q^51 - 9 * q^52 - 18 * q^53 + q^54 + 4 * q^55 + 19 * q^56 + 2 * q^57 + 20 * q^58 - 26 * q^59 - 9 * q^60 - 12 * q^61 - q^62 - 11 * q^63 - 2 * q^64 - 13 * q^65 + 17 * q^66 - 30 * q^67 + 23 * q^68 + 9 * q^69 + q^70 - 3 * q^71 - 6 * q^72 - 5 * q^73 - 36 * q^74 - 13 * q^75 + 16 * q^76 - q^78 - 9 * q^79 + 9 * q^80 + 13 * q^81 + 7 * q^82 - 35 * q^83 + 30 * q^84 + 3 * q^85 - 14 * q^86 - 7 * q^87 - 29 * q^88 + q^89 - q^90 + 11 * q^91 - 63 * q^92 - 13 * q^93 + 24 * q^94 - 2 * q^95 + 13 * q^96 - 33 * q^97 + q^98 + 4 * q^99

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