LMFDB - Embedded newform 1045.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 $ x^7 - x^6 - 10x^5 + 8x^4 + 27x^3 - 16x^2 - 18*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.54354$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.54354 q^{2} +2.58921 q^{3} +0.382516 q^{4} -1.00000 q^{5} -3.99655 q^{6} -3.41636 q^{7} +2.49665 q^{8} +3.70400 q^{9} +O(q^{10})$	\(q-1.54354 q^{2} +2.58921 q^{3} +0.382516 q^{4} -1.00000 q^{5} -3.99655 q^{6} -3.41636 q^{7} +2.49665 q^{8} +3.70400 q^{9} +1.54354 q^{10} +1.00000 q^{11} +0.990413 q^{12} +7.09747 q^{13} +5.27329 q^{14} -2.58921 q^{15} -4.61871 q^{16} +1.42643 q^{17} -5.71728 q^{18} -1.00000 q^{19} -0.382516 q^{20} -8.84568 q^{21} -1.54354 q^{22} -4.13507 q^{23} +6.46435 q^{24} +1.00000 q^{25} -10.9552 q^{26} +1.82281 q^{27} -1.30681 q^{28} +2.05758 q^{29} +3.99655 q^{30} +0.497448 q^{31} +2.13586 q^{32} +2.58921 q^{33} -2.20175 q^{34} +3.41636 q^{35} +1.41684 q^{36} -1.29496 q^{37} +1.54354 q^{38} +18.3768 q^{39} -2.49665 q^{40} +9.45656 q^{41} +13.6537 q^{42} +11.8154 q^{43} +0.382516 q^{44} -3.70400 q^{45} +6.38264 q^{46} +9.33224 q^{47} -11.9588 q^{48} +4.67155 q^{49} -1.54354 q^{50} +3.69332 q^{51} +2.71489 q^{52} -6.76115 q^{53} -2.81358 q^{54} -1.00000 q^{55} -8.52947 q^{56} -2.58921 q^{57} -3.17595 q^{58} +9.58605 q^{59} -0.990413 q^{60} +0.114870 q^{61} -0.767831 q^{62} -12.6542 q^{63} +5.94063 q^{64} -7.09747 q^{65} -3.99655 q^{66} +7.87864 q^{67} +0.545630 q^{68} -10.7066 q^{69} -5.27329 q^{70} +9.99343 q^{71} +9.24761 q^{72} +3.85639 q^{73} +1.99882 q^{74} +2.58921 q^{75} -0.382516 q^{76} -3.41636 q^{77} -28.3654 q^{78} +0.636392 q^{79} +4.61871 q^{80} -6.39237 q^{81} -14.5966 q^{82} +12.5580 q^{83} -3.38361 q^{84} -1.42643 q^{85} -18.2376 q^{86} +5.32749 q^{87} +2.49665 q^{88} -1.04231 q^{89} +5.71728 q^{90} -24.2476 q^{91} -1.58173 q^{92} +1.28800 q^{93} -14.4047 q^{94} +1.00000 q^{95} +5.53020 q^{96} -17.2651 q^{97} -7.21072 q^{98} +3.70400 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9	\( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9 - q^10 + 7 * q^11 + 13 * q^12 + q^13 + 12 * q^14 - 3 * q^15 + 3 * q^16 + q^17 + 7 * q^18 - 7 * q^19 - 7 * q^20 + 5 * q^21 + q^22 - 8 * q^23 + 25 * q^24 + 7 * q^25 + 12 * q^27 + 4 * q^28 + 11 * q^29 - 8 * q^30 + 7 * q^31 + 12 * q^32 + 3 * q^33 - 14 * q^34 + q^35 + 7 * q^36 - 17 * q^37 - q^38 + 30 * q^39 - 3 * q^40 + 17 * q^41 + 33 * q^42 - 3 * q^43 + 7 * q^44 - 2 * q^45 + 18 * q^46 + 14 * q^47 - 12 * q^48 + 6 * q^49 + q^50 + 8 * q^51 - 17 * q^52 + 7 * q^53 - 27 * q^54 - 7 * q^55 + 36 * q^56 - 3 * q^57 - 15 * q^58 + 35 * q^59 - 13 * q^60 + 17 * q^61 + 46 * q^62 - 22 * q^63 + 5 * q^64 - q^65 + 8 * q^66 + 4 * q^67 - 35 * q^68 - 4 * q^69 - 12 * q^70 + 10 * q^71 + 12 * q^72 + 22 * q^73 - 11 * q^74 + 3 * q^75 - 7 * q^76 - q^77 - 41 * q^78 + 11 * q^79 - 3 * q^80 - 21 * q^81 - 14 * q^82 + 39 * q^83 + 21 * q^84 - q^85 - 24 * q^86 - 2 * q^87 + 3 * q^88 + 18 * q^89 - 7 * q^90 - 22 * q^91 - 51 * q^92 + 10 * q^93 + 14 * q^94 + 7 * q^95 - 11 * q^96 - 4 * q^97 - 26 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.54354	−1.09145	−0.545724	−	0.837965i	$-0.683745\pi$
$2$	−1.54354	−1.09145	−0.545724	+	0.837965i	$0.683745\pi$
$3$	2.58921	1.49488	0.747440	−	0.664329i	$-0.231283\pi$
$3$	2.58921	1.49488	0.747440	+	0.664329i	$0.231283\pi$
$4$	0.382516	0.191258
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.99655	−1.63158
$7$	−3.41636	−1.29126	−0.645632	−	0.763649i	$-0.723406\pi$
$7$	−3.41636	−1.29126	−0.645632	+	0.763649i	$0.723406\pi$
$8$	2.49665	0.882700
$9$	3.70400	1.23467
$10$	1.54354	0.488110
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.990413	0.285908
$13$	7.09747	1.96848	0.984242	−	0.176825i	$-0.0565826\pi$
$13$	7.09747	1.96848	0.984242	+	0.176825i	$0.0565826\pi$
$14$	5.27329	1.40935
$15$	−2.58921	−0.668531
$16$	−4.61871	−1.15468
$17$	1.42643	0.345959	0.172980	−	0.984925i	$-0.444661\pi$
$17$	1.42643	0.345959	0.172980	+	0.984925i	$0.444661\pi$
$18$	−5.71728	−1.34758
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.382516	−0.0855331
$21$	−8.84568	−1.93029
$22$	−1.54354	−0.329084
$23$	−4.13507	−0.862221	−0.431111	−	0.902299i	$-0.641878\pi$
$23$	−4.13507	−0.862221	−0.431111	+	0.902299i	$0.641878\pi$
$24$	6.46435	1.31953
$25$	1.00000	0.200000
$26$	−10.9552	−2.14850
$27$	1.82281	0.350800
$28$	−1.30681	−0.246964
$29$	2.05758	0.382082	0.191041	−	0.981582i	$-0.438814\pi$
$29$	2.05758	0.382082	0.191041	+	0.981582i	$0.438814\pi$
$30$	3.99655	0.729666
$31$	0.497448	0.0893443	0.0446721	−	0.999002i	$-0.485776\pi$
$31$	0.497448	0.0893443	0.0446721	+	0.999002i	$0.485776\pi$
$32$	2.13586	0.377571
$33$	2.58921	0.450723
$34$	−2.20175	−0.377596
$35$	3.41636	0.577471
$36$	1.41684	0.236140
$37$	−1.29496	−0.212890	−0.106445	−	0.994319i	$-0.533947\pi$
$37$	−1.29496	−0.212890	−0.106445	+	0.994319i	$0.533947\pi$
$38$	1.54354	0.250395
$39$	18.3768	2.94265
$40$	−2.49665	−0.394755
$41$	9.45656	1.47687	0.738433	−	0.674327i	$-0.235566\pi$
$41$	9.45656	1.47687	0.738433	+	0.674327i	$0.235566\pi$
$42$	13.6537	2.10681
$43$	11.8154	1.80183	0.900916	−	0.433993i	$-0.142896\pi$
$43$	11.8154	1.80183	0.900916	+	0.433993i	$0.142896\pi$
$44$	0.382516	0.0576664
$45$	−3.70400	−0.552160
$46$	6.38264	0.941069
$47$	9.33224	1.36125	0.680623	−	0.732633i	$-0.261709\pi$
$47$	9.33224	1.36125	0.680623	+	0.732633i	$0.261709\pi$
$48$	−11.9588	−1.72611
$49$	4.67155	0.667364
$50$	−1.54354	−0.218290
$51$	3.69332	0.517168
$52$	2.71489	0.376488
$53$	−6.76115	−0.928715	−0.464357	−	0.885648i	$-0.653715\pi$
$53$	−6.76115	−0.928715	−0.464357	+	0.885648i	$0.653715\pi$
$54$	−2.81358	−0.382880
$55$	−1.00000	−0.134840
$56$	−8.52947	−1.13980
$57$	−2.58921	−0.342949
$58$	−3.17595	−0.417023
$59$	9.58605	1.24800	0.623999	−	0.781425i	$-0.285507\pi$
$59$	9.58605	1.24800	0.623999	+	0.781425i	$0.285507\pi$
$60$	−0.990413	−0.127862
$61$	0.114870	0.0147075	0.00735377	−	0.999973i	$-0.497659\pi$
$61$	0.114870	0.0147075	0.00735377	+	0.999973i	$0.497659\pi$
$62$	−0.767831	−0.0975146
$63$	−12.6542	−1.59428
$64$	5.94063	0.742579
$65$	−7.09747	−0.880333
$66$	−3.99655	−0.491941
$67$	7.87864	0.962529	0.481264	−	0.876575i	$-0.340178\pi$
$67$	7.87864	0.962529	0.481264	+	0.876575i	$0.340178\pi$
$68$	0.545630	0.0661674
$69$	−10.7066	−1.28892
$70$	−5.27329	−0.630279
$71$	9.99343	1.18600	0.593001	−	0.805202i	$-0.297943\pi$
$71$	9.99343	1.18600	0.593001	+	0.805202i	$0.297943\pi$
$72$	9.24761	1.08984
$73$	3.85639	0.451357	0.225678	−	0.974202i	$-0.427540\pi$
$73$	3.85639	0.451357	0.225678	+	0.974202i	$0.427540\pi$
$74$	1.99882	0.232359
$75$	2.58921	0.298976
$76$	−0.382516	−0.0438776
$77$	−3.41636	−0.389331
$78$	−28.3654	−3.21175
$79$	0.636392	0.0715997	0.0357999	−	0.999359i	$-0.488602\pi$
$79$	0.636392	0.0715997	0.0357999	+	0.999359i	$0.488602\pi$
$80$	4.61871	0.516388
$81$	−6.39237	−0.710263
$82$	−14.5966	−1.61192
$83$	12.5580	1.37842	0.689209	−	0.724563i	$-0.257958\pi$
$83$	12.5580	1.37842	0.689209	+	0.724563i	$0.257958\pi$
$84$	−3.38361	−0.369182
$85$	−1.42643	−0.154718
$86$	−18.2376	−1.96661
$87$	5.32749	0.571167
$88$	2.49665	0.266144
$89$	−1.04231	−0.110485	−0.0552425	−	0.998473i	$-0.517593\pi$
$89$	−1.04231	−0.110485	−0.0552425	+	0.998473i	$0.517593\pi$
$90$	5.71728	0.602654
$91$	−24.2476	−2.54183
$92$	−1.58173	−0.164907
$93$	1.28800	0.133559
$94$	−14.4047	−1.48573
$95$	1.00000	0.102598
$96$	5.53020	0.564424
$97$	−17.2651	−1.75300	−0.876501	−	0.481399i	$-0.840129\pi$
$97$	−17.2651	−1.75300	−0.876501	+	0.481399i	$0.840129\pi$
$98$	−7.21072	−0.728392
$99$	3.70400	0.372266
$100$	0.382516	0.0382516
$101$	7.49171	0.745453	0.372727	−	0.927941i	$-0.378423\pi$
$101$	7.49171	0.745453	0.372727	+	0.927941i	$0.378423\pi$
$102$	−5.70078	−0.564461
$103$	−7.23724	−0.713106	−0.356553	−	0.934275i	$-0.616048\pi$
$103$	−7.23724	−0.713106	−0.356553	+	0.934275i	$0.616048\pi$
$104$	17.7199	1.73758
$105$	8.84568	0.863250
$106$	10.4361	1.01364
$107$	5.74785	0.555666	0.277833	−	0.960629i	$-0.410384\pi$
$107$	5.74785	0.555666	0.277833	+	0.960629i	$0.410384\pi$
$108$	0.697254	0.0670933
$109$	−11.7343	−1.12394	−0.561970	−	0.827158i	$-0.689956\pi$
$109$	−11.7343	−1.12394	−0.561970	+	0.827158i	$0.689956\pi$
$110$	1.54354	0.147171
$111$	−3.35292	−0.318245
$112$	15.7792	1.49099
$113$	−4.90651	−0.461566	−0.230783	−	0.973005i	$-0.574129\pi$
$113$	−4.90651	−0.461566	−0.230783	+	0.973005i	$0.574129\pi$
$114$	3.99655	0.374311
$115$	4.13507	0.385597
$116$	0.787055	0.0730762
$117$	26.2891	2.43042
$118$	−14.7965	−1.36212
$119$	−4.87319	−0.446725
$120$	−6.46435	−0.590112
$121$	1.00000	0.0909091
$122$	−0.177306	−0.0160525
$123$	24.4850	2.20774
$124$	0.190282	0.0170878
$125$	−1.00000	−0.0894427
$126$	19.5323	1.74008
$127$	−10.5698	−0.937919	−0.468959	−	0.883220i	$-0.655371\pi$
$127$	−10.5698	−0.937919	−0.468959	+	0.883220i	$0.655371\pi$
$128$	−13.4413	−1.18806
$129$	30.5926	2.69352
$130$	10.9552	0.960837
$131$	8.61645	0.752823	0.376411	−	0.926453i	$-0.377158\pi$
$131$	8.61645	0.752823	0.376411	+	0.926453i	$0.377158\pi$
$132$	0.990413	0.0862044
$133$	3.41636	0.296236
$134$	−12.1610	−1.05055
$135$	−1.82281	−0.156883
$136$	3.56129	0.305378
$137$	−12.2058	−1.04281	−0.521406	−	0.853309i	$-0.674592\pi$
$137$	−12.2058	−1.04281	−0.521406	+	0.853309i	$0.674592\pi$
$138$	16.5260	1.40679
$139$	−9.28127	−0.787227	−0.393614	−	0.919276i	$-0.628775\pi$
$139$	−9.28127	−0.787227	−0.393614	+	0.919276i	$0.628775\pi$
$140$	1.30681	0.110446
$141$	24.1631	2.03490
$142$	−15.4253	−1.29446
$143$	7.09747	0.593520
$144$	−17.1077	−1.42564
$145$	−2.05758	−0.170872
$146$	−5.95250	−0.492632
$147$	12.0956	0.997629
$148$	−0.495343	−0.0407169
$149$	5.06425	0.414879	0.207440	−	0.978248i	$-0.433487\pi$
$149$	5.06425	0.414879	0.207440	+	0.978248i	$0.433487\pi$
$150$	−3.99655	−0.326317
$151$	16.3009	1.32655	0.663275	−	0.748376i	$-0.269166\pi$
$151$	16.3009	1.32655	0.663275	+	0.748376i	$0.269166\pi$
$152$	−2.49665	−0.202505
$153$	5.28349	0.427145
$154$	5.27329	0.424934
$155$	−0.497448	−0.0399560
$156$	7.02943	0.562805
$157$	−8.87700	−0.708461	−0.354231	−	0.935158i	$-0.615257\pi$
$157$	−8.87700	−0.708461	−0.354231	+	0.935158i	$0.615257\pi$
$158$	−0.982297	−0.0781473
$159$	−17.5060	−1.38832
$160$	−2.13586	−0.168855
$161$	14.1269	1.11336
$162$	9.86688	0.775215
$163$	−0.0647360	−0.00507052	−0.00253526	−	0.999997i	$-0.500807\pi$
$163$	−0.0647360	−0.00507052	−0.00253526	+	0.999997i	$0.500807\pi$
$164$	3.61728	0.282462
$165$	−2.58921	−0.201570
$166$	−19.3837	−1.50447
$167$	3.68720	0.285324	0.142662	−	0.989771i	$-0.454434\pi$
$167$	3.68720	0.285324	0.142662	+	0.989771i	$0.454434\pi$
$168$	−22.0846	−1.70386
$169$	37.3741	2.87493
$170$	2.20175	0.168866
$171$	−3.70400	−0.283252
$172$	4.51958	0.344615
$173$	−23.9303	−1.81939	−0.909695	−	0.415277i	$-0.863685\pi$
$173$	−23.9303	−1.81939	−0.909695	+	0.415277i	$0.863685\pi$
$174$	−8.22320	−0.623399
$175$	−3.41636	−0.258253
$176$	−4.61871	−0.348149
$177$	24.8203	1.86561
$178$	1.60885	0.120589
$179$	16.2698	1.21607	0.608033	−	0.793912i	$-0.291959\pi$
$179$	16.2698	1.21607	0.608033	+	0.793912i	$0.291959\pi$
$180$	−1.41684	−0.105605
$181$	−5.87487	−0.436676	−0.218338	−	0.975873i	$-0.570064\pi$
$181$	−5.87487	−0.436676	−0.218338	+	0.975873i	$0.570064\pi$
$182$	37.4271	2.77428
$183$	0.297421	0.0219860
$184$	−10.3238	−0.761083
$185$	1.29496	0.0952074
$186$	−1.98807	−0.145773
$187$	1.42643	0.104311
$188$	3.56973	0.260349
$189$	−6.22739	−0.452976
$190$	−1.54354	−0.111980
$191$	4.16087	0.301070	0.150535	−	0.988605i	$-0.451900\pi$
$191$	4.16087	0.301070	0.150535	+	0.988605i	$0.451900\pi$
$192$	15.3815	1.11007
$193$	−23.8459	−1.71646	−0.858232	−	0.513262i	$-0.828437\pi$
$193$	−23.8459	−1.71646	−0.858232	+	0.513262i	$0.828437\pi$
$194$	26.6493	1.91331
$195$	−18.3768	−1.31599
$196$	1.78694	0.127639
$197$	−12.6266	−0.899606	−0.449803	−	0.893128i	$-0.648506\pi$
$197$	−12.6266	−0.899606	−0.449803	+	0.893128i	$0.648506\pi$
$198$	−5.71728	−0.406309
$199$	−7.72351	−0.547505	−0.273752	−	0.961800i	$-0.588265\pi$
$199$	−7.72351	−0.547505	−0.273752	+	0.961800i	$0.588265\pi$
$200$	2.49665	0.176540
$201$	20.3994	1.43887
$202$	−11.5638	−0.813623
$203$	−7.02943	−0.493369
$204$	1.41275	0.0989123
$205$	−9.45656	−0.660475
$206$	11.1710	0.778318
$207$	−15.3163	−1.06456
$208$	−32.7812	−2.27297
$209$	−1.00000	−0.0691714
$210$	−13.6537	−0.942192
$211$	3.28502	0.226150	0.113075	−	0.993586i	$-0.463930\pi$
$211$	3.28502	0.226150	0.113075	+	0.993586i	$0.463930\pi$
$212$	−2.58624	−0.177624
$213$	25.8751	1.77293
$214$	−8.87204	−0.606480
$215$	−11.8154	−0.805804
$216$	4.55092	0.309651
$217$	−1.69946	−0.115367
$218$	18.1123	1.22672
$219$	9.98501	0.674724
$220$	−0.382516	−0.0257892
$221$	10.1240	0.681015
$222$	5.17537	0.347348
$223$	−14.8208	−0.992473	−0.496237	−	0.868187i	$-0.665285\pi$
$223$	−14.8208	−0.992473	−0.496237	+	0.868187i	$0.665285\pi$
$224$	−7.29689	−0.487544
$225$	3.70400	0.246934
$226$	7.57340	0.503775
$227$	−6.95916	−0.461895	−0.230948	−	0.972966i	$-0.574183\pi$
$227$	−6.95916	−0.461895	−0.230948	+	0.972966i	$0.574183\pi$
$228$	−0.990413	−0.0655917
$229$	−28.2027	−1.86368	−0.931842	−	0.362865i	$-0.881799\pi$
$229$	−28.2027	−1.86368	−0.931842	+	0.362865i	$0.881799\pi$
$230$	−6.38264	−0.420859
$231$	−8.84568	−0.582003
$232$	5.13705	0.337264
$233$	−19.3661	−1.26871	−0.634357	−	0.773040i	$-0.718735\pi$
$233$	−19.3661	−1.26871	−0.634357	+	0.773040i	$0.718735\pi$
$234$	−40.5782	−2.65268
$235$	−9.33224	−0.608768
$236$	3.66681	0.238689
$237$	1.64775	0.107033
$238$	7.52197	0.487577
$239$	11.1398	0.720574	0.360287	−	0.932841i	$-0.382679\pi$
$239$	11.1398	0.720574	0.360287	+	0.932841i	$0.382679\pi$
$240$	11.9588	0.771938
$241$	−18.7905	−1.21040	−0.605202	−	0.796072i	$-0.706907\pi$
$241$	−18.7905	−1.21040	−0.605202	+	0.796072i	$0.706907\pi$
$242$	−1.54354	−0.0992225
$243$	−22.0196	−1.41256
$244$	0.0439394	0.00281293
$245$	−4.67155	−0.298454
$246$	−37.7936	−2.40963
$247$	−7.09747	−0.451601
$248$	1.24195	0.0788642
$249$	32.5152	2.06057
$250$	1.54354	0.0976220
$251$	11.3997	0.719541	0.359770	−	0.933041i	$-0.382855\pi$
$251$	11.3997	0.719541	0.359770	+	0.933041i	$0.382855\pi$
$252$	−4.84044	−0.304919
$253$	−4.13507	−0.259970
$254$	16.3149	1.02369
$255$	−3.69332	−0.231284
$256$	8.86597	0.554123
$257$	24.3068	1.51622	0.758109	−	0.652128i	$-0.226123\pi$
$257$	24.3068	1.51622	0.758109	+	0.652128i	$0.226123\pi$
$258$	−47.2208	−2.93984
$259$	4.42406	0.274898
$260$	−2.71489	−0.168371
$261$	7.62127	0.471745
$262$	−13.2998	−0.821666
$263$	−1.75843	−0.108430	−0.0542148	−	0.998529i	$-0.517266\pi$
$263$	−1.75843	−0.108430	−0.0542148	+	0.998529i	$0.517266\pi$
$264$	6.46435	0.397853
$265$	6.76115	0.415334
$266$	−5.27329	−0.323326
$267$	−2.69877	−0.165162
$268$	3.01370	0.184091
$269$	−23.9178	−1.45829	−0.729146	−	0.684358i	$-0.760083\pi$
$269$	−23.9178	−1.45829	−0.729146	+	0.684358i	$0.760083\pi$
$270$	2.81358	0.171229
$271$	−2.95989	−0.179801	−0.0899003	−	0.995951i	$-0.528655\pi$
$271$	−2.95989	−0.179801	−0.0899003	+	0.995951i	$0.528655\pi$
$272$	−6.58825	−0.399472
$273$	−62.7820	−3.79974
$274$	18.8401	1.13817
$275$	1.00000	0.0603023
$276$	−4.09543	−0.246516
$277$	0.686083	0.0412227	0.0206114	−	0.999788i	$-0.493439\pi$
$277$	0.686083	0.0412227	0.0206114	+	0.999788i	$0.493439\pi$
$278$	14.3260	0.859217
$279$	1.84255	0.110310
$280$	8.52947	0.509733
$281$	29.3996	1.75383	0.876915	−	0.480645i	$-0.159597\pi$
$281$	29.3996	1.75383	0.876915	+	0.480645i	$0.159597\pi$
$282$	−37.2967	−2.22099
$283$	−9.90327	−0.588689	−0.294344	−	0.955699i	$-0.595101\pi$
$283$	−9.90327	−0.588689	−0.294344	+	0.955699i	$0.595101\pi$
$284$	3.82264	0.226832
$285$	2.58921	0.153372
$286$	−10.9552	−0.647796
$287$	−32.3070	−1.90702
$288$	7.91125	0.466175
$289$	−14.9653	−0.880312
$290$	3.17595	0.186498
$291$	−44.7029	−2.62053
$292$	1.47513	0.0863255
$293$	15.7487	0.920048	0.460024	−	0.887906i	$-0.347841\pi$
$293$	15.7487	0.920048	0.460024	+	0.887906i	$0.347841\pi$
$294$	−18.6701	−1.08886
$295$	−9.58605	−0.558121
$296$	−3.23307	−0.187918
$297$	1.82281	0.105770
$298$	−7.81687	−0.452819
$299$	−29.3485	−1.69727
$300$	0.990413	0.0571815
$301$	−40.3657	−2.32664
$302$	−25.1611	−1.44786
$303$	19.3976	1.11436
$304$	4.61871	0.264901
$305$	−0.114870	−0.00657741
$306$	−8.15527	−0.466206
$307$	−29.6962	−1.69485	−0.847427	−	0.530912i	$-0.821849\pi$
$307$	−29.6962	−1.69485	−0.847427	+	0.530912i	$0.821849\pi$
$308$	−1.30681	−0.0744626
$309$	−18.7387	−1.06601
$310$	0.767831	0.0436098
$311$	15.1197	0.857357	0.428679	−	0.903457i	$-0.358979\pi$
$311$	15.1197	0.857357	0.428679	+	0.903457i	$0.358979\pi$
$312$	45.8806	2.59748
$313$	−4.95630	−0.280147	−0.140073	−	0.990141i	$-0.544734\pi$
$313$	−4.95630	−0.280147	−0.140073	+	0.990141i	$0.544734\pi$
$314$	13.7020	0.773249
$315$	12.6542	0.712985
$316$	0.243430	0.0136940
$317$	15.8872	0.892315	0.446158	−	0.894954i	$-0.352792\pi$
$317$	15.8872	0.892315	0.446158	+	0.894954i	$0.352792\pi$
$318$	27.0212	1.51528
$319$	2.05758	0.115202
$320$	−5.94063	−0.332092
$321$	14.8824	0.830654
$322$	−21.8054	−1.21517
$323$	−1.42643	−0.0793685
$324$	−2.44518	−0.135843
$325$	7.09747	0.393697
$326$	0.0999226	0.00553420
$327$	−30.3825	−1.68015
$328$	23.6097	1.30363
$329$	−31.8823	−1.75773
$330$	3.99655	0.220003
$331$	25.3385	1.39273	0.696365	−	0.717688i	$-0.254800\pi$
$331$	25.3385	1.39273	0.696365	+	0.717688i	$0.254800\pi$
$332$	4.80362	0.263633
$333$	−4.79654	−0.262849
$334$	−5.69135	−0.311417
$335$	−7.87864	−0.430456
$336$	40.8557	2.22886
$337$	18.7625	1.02206	0.511029	−	0.859564i	$-0.329265\pi$
$337$	18.7625	1.02206	0.511029	+	0.859564i	$0.329265\pi$
$338$	−57.6884	−3.13784
$339$	−12.7040	−0.689986
$340$	−0.545630	−0.0295910
$341$	0.497448	0.0269383
$342$	5.71728	0.309155
$343$	7.95485	0.429522
$344$	29.4990	1.59048
$345$	10.7066	0.576422
$346$	36.9374	1.98577
$347$	−31.9168	−1.71338	−0.856691	−	0.515830i	$-0.827484\pi$
$347$	−31.9168	−1.71338	−0.856691	+	0.515830i	$0.827484\pi$
$348$	2.03785	0.109240
$349$	33.0084	1.76690	0.883449	−	0.468528i	$-0.155215\pi$
$349$	33.0084	1.76690	0.883449	+	0.468528i	$0.155215\pi$
$350$	5.27329	0.281869
$351$	12.9373	0.690545
$352$	2.13586	0.113842
$353$	24.9667	1.32884	0.664422	−	0.747357i	$-0.268678\pi$
$353$	24.9667	1.32884	0.664422	+	0.747357i	$0.268678\pi$
$354$	−38.3111	−2.03621
$355$	−9.99343	−0.530396
$356$	−0.398701	−0.0211311
$357$	−12.6177	−0.667800
$358$	−25.1132	−1.32727
$359$	26.3644	1.39146	0.695731	−	0.718303i	$-0.255081\pi$
$359$	26.3644	1.39146	0.695731	+	0.718303i	$0.255081\pi$
$360$	−9.24761	−0.487392
$361$	1.00000	0.0526316
$362$	9.06810	0.476609
$363$	2.58921	0.135898
$364$	−9.27507	−0.486146
$365$	−3.85639	−0.201853
$366$	−0.459082	−0.0239966
$367$	13.2277	0.690478	0.345239	−	0.938515i	$-0.387798\pi$
$367$	13.2277	0.690478	0.345239	+	0.938515i	$0.387798\pi$
$368$	19.0987	0.995588
$369$	35.0271	1.82344
$370$	−1.99882	−0.103914
$371$	23.0985	1.19922
$372$	0.492679	0.0255442
$373$	−20.9582	−1.08517	−0.542587	−	0.840000i	$-0.682555\pi$
$373$	−20.9582	−1.08517	−0.542587	+	0.840000i	$0.682555\pi$
$374$	−2.20175	−0.113850
$375$	−2.58921	−0.133706
$376$	23.2993	1.20157
$377$	14.6036	0.752123
$378$	9.61222	0.494399
$379$	−13.5007	−0.693483	−0.346741	−	0.937961i	$-0.612712\pi$
$379$	−13.5007	−0.693483	−0.346741	+	0.937961i	$0.612712\pi$
$380$	0.382516	0.0196226
$381$	−27.3674	−1.40208
$382$	−6.42246	−0.328602
$383$	18.7031	0.955682	0.477841	−	0.878446i	$-0.341419\pi$
$383$	18.7031	0.955682	0.477841	+	0.878446i	$0.341419\pi$
$384$	−34.8024	−1.77600
$385$	3.41636	0.174114
$386$	36.8071	1.87343
$387$	43.7643	2.22466
$388$	−6.60416	−0.335275
$389$	−13.3203	−0.675366	−0.337683	−	0.941260i	$-0.609643\pi$
$389$	−13.3203	−0.675366	−0.337683	+	0.941260i	$0.609643\pi$
$390$	28.3654	1.43634
$391$	−5.89837	−0.298293
$392$	11.6632	0.589082
$393$	22.3098	1.12538
$394$	19.4896	0.981873
$395$	−0.636392	−0.0320204
$396$	1.41684	0.0711988
$397$	−7.98214	−0.400612	−0.200306	−	0.979733i	$-0.564194\pi$
$397$	−7.98214	−0.400612	−0.200306	+	0.979733i	$0.564194\pi$
$398$	11.9215	0.597573
$399$	8.84568	0.442838
$400$	−4.61871	−0.230936
$401$	−25.8366	−1.29022	−0.645109	−	0.764091i	$-0.723188\pi$
$401$	−25.8366	−1.29022	−0.645109	+	0.764091i	$0.723188\pi$
$402$	−31.4873	−1.57045
$403$	3.53062	0.175873
$404$	2.86570	0.142574
$405$	6.39237	0.317639
$406$	10.8502	0.538487
$407$	−1.29496	−0.0641888
$408$	9.22092	0.456504
$409$	33.6176	1.66229	0.831143	−	0.556059i	$-0.187687\pi$
$409$	33.6176	1.66229	0.831143	+	0.556059i	$0.187687\pi$
$410$	14.5966	0.720873
$411$	−31.6034	−1.55888
$412$	−2.76836	−0.136387
$413$	−32.7494	−1.61149
$414$	23.6413	1.16191
$415$	−12.5580	−0.616447
$416$	15.1592	0.743243
$417$	−24.0312	−1.17681
$418$	1.54354	0.0754970
$419$	−7.06804	−0.345296	−0.172648	−	0.984984i	$-0.555232\pi$
$419$	−7.06804	−0.345296	−0.172648	+	0.984984i	$0.555232\pi$
$420$	3.38361	0.165103
$421$	20.8952	1.01837	0.509184	−	0.860658i	$-0.329947\pi$
$421$	20.8952	1.01837	0.509184	+	0.860658i	$0.329947\pi$
$422$	−5.07056	−0.246831
$423$	34.5666	1.68069
$424$	−16.8802	−0.819776
$425$	1.42643	0.0691918
$426$	−39.9392	−1.93506
$427$	−0.392436	−0.0189913
$428$	2.19864	0.106275
$429$	18.3768	0.887242
$430$	18.2376	0.879493
$431$	9.30871	0.448385	0.224192	−	0.974545i	$-0.428026\pi$
$431$	9.30871	0.448385	0.224192	+	0.974545i	$0.428026\pi$
$432$	−8.41904	−0.405061
$433$	−2.69400	−0.129465	−0.0647327	−	0.997903i	$-0.520619\pi$
$433$	−2.69400	−0.129465	−0.0647327	+	0.997903i	$0.520619\pi$
$434$	2.62319	0.125917
$435$	−5.32749	−0.255434
$436$	−4.48854	−0.214962
$437$	4.13507	0.197807
$438$	−15.4123	−0.736426
$439$	−19.2674	−0.919581	−0.459790	−	0.888027i	$-0.652075\pi$
$439$	−19.2674	−0.919581	−0.459790	+	0.888027i	$0.652075\pi$
$440$	−2.49665	−0.119023
$441$	17.3034	0.823972
$442$	−15.6268	−0.743293
$443$	−11.9554	−0.568019	−0.284009	−	0.958821i	$-0.591665\pi$
$443$	−11.9554	−0.568019	−0.284009	+	0.958821i	$0.591665\pi$
$444$	−1.28255	−0.0608669
$445$	1.04231	0.0494104
$446$	22.8765	1.08323
$447$	13.1124	0.620195
$448$	−20.2954	−0.958866
$449$	−5.38577	−0.254170	−0.127085	−	0.991892i	$-0.540562\pi$
$449$	−5.38577	−0.254170	−0.127085	+	0.991892i	$0.540562\pi$
$450$	−5.71728	−0.269515
$451$	9.45656	0.445292
$452$	−1.87682	−0.0882781
$453$	42.2065	1.98303
$454$	10.7417	0.504135
$455$	24.2476	1.13674
$456$	−6.46435	−0.302721
$457$	28.4123	1.32907	0.664536	−	0.747256i	$-0.268629\pi$
$457$	28.4123	1.32907	0.664536	+	0.747256i	$0.268629\pi$
$458$	43.5319	2.03411
$459$	2.60011	0.121363
$460$	1.58173	0.0737485
$461$	−23.0660	−1.07429	−0.537146	−	0.843489i	$-0.680498\pi$
$461$	−23.0660	−1.07429	−0.537146	+	0.843489i	$0.680498\pi$
$462$	13.6537	0.635226
$463$	19.3608	0.899774	0.449887	−	0.893085i	$-0.351464\pi$
$463$	19.3608	0.899774	0.449887	+	0.893085i	$0.351464\pi$
$464$	−9.50335	−0.441182
$465$	−1.28800	−0.0597294
$466$	29.8923	1.38473
$467$	−41.6789	−1.92867	−0.964334	−	0.264688i	$-0.914731\pi$
$467$	−41.6789	−1.92867	−0.964334	+	0.264688i	$0.914731\pi$
$468$	10.0560	0.464838
$469$	−26.9163	−1.24288
$470$	14.4047	0.664438
$471$	−22.9844	−1.05907
$472$	23.9330	1.10161
$473$	11.8154	0.543273
$474$	−2.54337	−0.116821
$475$	−1.00000	−0.0458831
$476$	−1.86407	−0.0854396
$477$	−25.0433	−1.14665
$478$	−17.1947	−0.786469
$479$	−2.99573	−0.136878	−0.0684392	−	0.997655i	$-0.521802\pi$
$479$	−2.99573	−0.136878	−0.0684392	+	0.997655i	$0.521802\pi$
$480$	−5.53020	−0.252418
$481$	−9.19095	−0.419071
$482$	29.0039	1.32109
$483$	36.5775	1.66433
$484$	0.382516	0.0173871
$485$	17.2651	0.783967
$486$	33.9882	1.54173
$487$	−33.7741	−1.53045	−0.765225	−	0.643763i	$-0.777372\pi$
$487$	−33.7741	−1.53045	−0.765225	+	0.643763i	$0.777372\pi$
$488$	0.286789	0.0129823
$489$	−0.167615	−0.00757981
$490$	7.21072	0.325747
$491$	−11.8496	−0.534765	−0.267382	−	0.963591i	$-0.586159\pi$
$491$	−11.8496	−0.534765	−0.267382	+	0.963591i	$0.586159\pi$
$492$	9.36590	0.422247
$493$	2.93498	0.132185
$494$	10.9552	0.492899
$495$	−3.70400	−0.166483
$496$	−2.29757	−0.103164
$497$	−34.1412	−1.53144
$498$	−50.1886	−2.24900
$499$	22.2401	0.995602	0.497801	−	0.867291i	$-0.334141\pi$
$499$	22.2401	0.995602	0.497801	+	0.867291i	$0.334141\pi$
$500$	−0.382516	−0.0171066
$501$	9.54694	0.426526
$502$	−17.5958	−0.785341
$503$	−10.8248	−0.482653	−0.241326	−	0.970444i	$-0.577582\pi$
$503$	−10.8248	−0.482653	−0.241326	+	0.970444i	$0.577582\pi$
$504$	−31.5932	−1.40727
$505$	−7.49171	−0.333377
$506$	6.38264	0.283743
$507$	96.7694	4.29768
$508$	−4.04312	−0.179384
$509$	−25.9087	−1.14838	−0.574192	−	0.818721i	$-0.694684\pi$
$509$	−25.9087	−1.14838	−0.574192	+	0.818721i	$0.694684\pi$
$510$	5.70078	0.252435
$511$	−13.1748	−0.582821
$512$	13.1977	0.583261
$513$	−1.82281	−0.0804791
$514$	−37.5186	−1.65487
$515$	7.23724	0.318911
$516$	11.7021	0.515158
$517$	9.33224	0.410431
$518$	−6.82871	−0.300036
$519$	−61.9606	−2.71977
$520$	−17.7199	−0.777070
$521$	−23.2425	−1.01827	−0.509136	−	0.860686i	$-0.670035\pi$
$521$	−23.2425	−1.01827	−0.509136	+	0.860686i	$0.670035\pi$
$522$	−11.7637	−0.514884
$523$	−24.6216	−1.07663	−0.538314	−	0.842744i	$-0.680939\pi$
$523$	−24.6216	−1.07663	−0.538314	+	0.842744i	$0.680939\pi$
$524$	3.29593	0.143983
$525$	−8.84568	−0.386057
$526$	2.71421	0.118345
$527$	0.709573	0.0309095
$528$	−11.9588	−0.520441
$529$	−5.90121	−0.256574
$530$	−10.4361	−0.453315
$531$	35.5068	1.54086
$532$	1.30681	0.0566575
$533$	67.1176	2.90719
$534$	4.16565	0.180265
$535$	−5.74785	−0.248501
$536$	19.6702	0.849624
$537$	42.1260	1.81787
$538$	36.9180	1.59165
$539$	4.67155	0.201218
$540$	−0.697254	−0.0300050
$541$	−27.7500	−1.19306	−0.596532	−	0.802589i	$-0.703455\pi$
$541$	−27.7500	−1.19306	−0.596532	+	0.802589i	$0.703455\pi$
$542$	4.56871	0.196243
$543$	−15.2113	−0.652778
$544$	3.04665	0.130624
$545$	11.7343	0.502641
$546$	96.9065	4.14722
$547$	12.6819	0.542237	0.271119	−	0.962546i	$-0.412606\pi$
$547$	12.6819	0.542237	0.271119	+	0.962546i	$0.412606\pi$
$548$	−4.66891	−0.199446
$549$	0.425477	0.0181589
$550$	−1.54354	−0.0658168
$551$	−2.05758	−0.0876557
$552$	−26.7305	−1.13773
$553$	−2.17415	−0.0924542
$554$	−1.05900	−0.0449924
$555$	3.35292	0.142324
$556$	−3.55023	−0.150563
$557$	−26.2189	−1.11093	−0.555465	−	0.831540i	$-0.687460\pi$
$557$	−26.2189	−1.11093	−0.555465	+	0.831540i	$0.687460\pi$
$558$	−2.84405	−0.120398
$559$	83.8595	3.54688
$560$	−15.7792	−0.666793
$561$	3.69332	0.155932
$562$	−45.3794	−1.91421
$563$	43.3056	1.82511	0.912557	−	0.408950i	$-0.134105\pi$
$563$	43.3056	1.82511	0.912557	+	0.408950i	$0.134105\pi$
$564$	9.24277	0.389191
$565$	4.90651	0.206419
$566$	15.2861	0.642523
$567$	21.8387	0.917138
$568$	24.9501	1.04688
$569$	43.2165	1.81173	0.905866	−	0.423565i	$-0.139221\pi$
$569$	43.2165	1.81173	0.905866	+	0.423565i	$0.139221\pi$
$570$	−3.99655	−0.167397
$571$	−9.85278	−0.412326	−0.206163	−	0.978518i	$-0.566098\pi$
$571$	−9.85278	−0.412326	−0.206163	+	0.978518i	$0.566098\pi$
$572$	2.71489	0.113515
$573$	10.7734	0.450063
$574$	49.8672	2.08142
$575$	−4.13507	−0.172444
$576$	22.0041	0.916839
$577$	14.0965	0.586845	0.293423	−	0.955983i	$-0.405206\pi$
$577$	14.0965	0.586845	0.293423	+	0.955983i	$0.405206\pi$
$578$	23.0996	0.960815
$579$	−61.7420	−2.56591
$580$	−0.787055	−0.0326807
$581$	−42.9026	−1.77990
$582$	69.0007	2.86017
$583$	−6.76115	−0.280018
$584$	9.62807	0.398412
$585$	−26.2891	−1.08692
$586$	−24.3087	−1.00418
$587$	32.6218	1.34644	0.673222	−	0.739440i	$-0.264910\pi$
$587$	32.6218	1.34644	0.673222	+	0.739440i	$0.264910\pi$
$588$	4.62676	0.190804
$589$	−0.497448	−0.0204970
$590$	14.7965	0.609160
$591$	−32.6928	−1.34480
$592$	5.98105	0.245820
$593$	5.11512	0.210053	0.105026	−	0.994469i	$-0.466507\pi$
$593$	5.11512	0.210053	0.105026	+	0.994469i	$0.466507\pi$
$594$	−2.81358	−0.115443
$595$	4.87319	0.199781
$596$	1.93715	0.0793489
$597$	−19.9978	−0.818455
$598$	45.3006	1.85248
$599$	−23.3602	−0.954474	−0.477237	−	0.878775i	$-0.658362\pi$
$599$	−23.3602	−0.954474	−0.477237	+	0.878775i	$0.658362\pi$
$600$	6.46435	0.263906
$601$	41.3213	1.68553	0.842765	−	0.538281i	$-0.180926\pi$
$601$	41.3213	1.68553	0.842765	+	0.538281i	$0.180926\pi$
$602$	62.3061	2.53941
$603$	29.1825	1.18840
$604$	6.23536	0.253713
$605$	−1.00000	−0.0406558
$606$	−29.9410	−1.21627
$607$	−27.0455	−1.09774	−0.548872	−	0.835907i	$-0.684942\pi$
$607$	−27.0455	−1.09774	−0.548872	+	0.835907i	$0.684942\pi$
$608$	−2.13586	−0.0866208
$609$	−18.2007	−0.737528
$610$	0.177306	0.00717890
$611$	66.2353	2.67959
$612$	2.02102	0.0816947
$613$	33.1063	1.33715	0.668575	−	0.743645i	$-0.266905\pi$
$613$	33.1063	1.33715	0.668575	+	0.743645i	$0.266905\pi$
$614$	45.8373	1.84984
$615$	−24.4850	−0.987330
$616$	−8.52947	−0.343662
$617$	8.37280	0.337076	0.168538	−	0.985695i	$-0.446095\pi$
$617$	8.37280	0.337076	0.168538	+	0.985695i	$0.446095\pi$
$618$	28.9240	1.16349
$619$	−18.4901	−0.743181	−0.371590	−	0.928397i	$-0.621187\pi$
$619$	−18.4901	−0.743181	−0.371590	+	0.928397i	$0.621187\pi$
$620$	−0.190282	−0.00764189
$621$	−7.53745	−0.302467
$622$	−23.3378	−0.935761
$623$	3.56092	0.142665
$624$	−84.8773	−3.39781
$625$	1.00000	0.0400000
$626$	7.65025	0.305765
$627$	−2.58921	−0.103403
$628$	−3.39559	−0.135499
$629$	−1.84717	−0.0736513
$630$	−19.5323	−0.778185
$631$	−35.9235	−1.43009	−0.715047	−	0.699076i	$-0.753595\pi$
$631$	−35.9235	−1.43009	−0.715047	+	0.699076i	$0.753595\pi$
$632$	1.58885	0.0632010
$633$	8.50560	0.338067
$634$	−24.5226	−0.973915
$635$	10.5698	0.419450
$636$	−6.69633	−0.265527
$637$	33.1562	1.31370
$638$	−3.17595	−0.125737
$639$	37.0157	1.46432
$640$	13.4413	0.531315
$641$	−10.1750	−0.401889	−0.200944	−	0.979603i	$-0.564401\pi$
$641$	−10.1750	−0.401889	−0.200944	+	0.979603i	$0.564401\pi$
$642$	−22.9716	−0.906615
$643$	13.6541	0.538465	0.269232	−	0.963075i	$-0.413230\pi$
$643$	13.6541	0.538465	0.269232	+	0.963075i	$0.413230\pi$
$644$	5.40376	0.212938
$645$	−30.5926	−1.20458
$646$	2.20175	0.0866265
$647$	30.3322	1.19248	0.596242	−	0.802805i	$-0.296660\pi$
$647$	30.3322	1.19248	0.596242	+	0.802805i	$0.296660\pi$
$648$	−15.9595	−0.626949
$649$	9.58605	0.376285
$650$	−10.9552	−0.429700
$651$	−4.40026	−0.172460
$652$	−0.0247625	−0.000969776 0
$653$	−14.3264	−0.560637	−0.280318	−	0.959907i	$-0.590440\pi$
$653$	−14.3264	−0.560637	−0.280318	+	0.959907i	$0.590440\pi$
$654$	46.8966	1.83380
$655$	−8.61645	−0.336672
$656$	−43.6771	−1.70530
$657$	14.2841	0.557275
$658$	49.2116	1.91847
$659$	−14.4660	−0.563515	−0.281758	−	0.959486i	$-0.590917\pi$
$659$	−14.4660	−0.563515	−0.281758	+	0.959486i	$0.590917\pi$
$660$	−0.990413	−0.0385518
$661$	−26.7251	−1.03949	−0.519743	−	0.854323i	$-0.673972\pi$
$661$	−26.7251	−1.03949	−0.519743	+	0.854323i	$0.673972\pi$
$662$	−39.1110	−1.52009
$663$	26.2132	1.01804
$664$	31.3529	1.21673
$665$	−3.41636	−0.132481
$666$	7.40365	0.286886
$667$	−8.50822	−0.329439
$668$	1.41041	0.0545705
$669$	−38.3741	−1.48363
$670$	12.1610	0.469820
$671$	0.114870	0.00443449
$672$	−18.8932	−0.728820
$673$	−15.8379	−0.610507	−0.305254	−	0.952271i	$-0.598741\pi$
$673$	−15.8379	−0.610507	−0.305254	+	0.952271i	$0.598741\pi$
$674$	−28.9606	−1.11552
$675$	1.82281	0.0701600
$676$	14.2962	0.549853
$677$	−42.1373	−1.61947	−0.809733	−	0.586798i	$-0.800388\pi$
$677$	−42.1373	−1.61947	−0.809733	+	0.586798i	$0.800388\pi$
$678$	19.6091	0.753084
$679$	58.9838	2.26359
$680$	−3.56129	−0.136569
$681$	−18.0187	−0.690479
$682$	−0.767831	−0.0294018
$683$	0.147534	0.00564521	0.00282261	−	0.999996i	$-0.499102\pi$
$683$	0.147534	0.00564521	0.00282261	+	0.999996i	$0.499102\pi$
$684$	−1.41684	−0.0541742
$685$	12.2058	0.466360
$686$	−12.2786	−0.468800
$687$	−73.0226	−2.78598
$688$	−54.5720	−2.08054
$689$	−47.9870	−1.82816
$690$	−16.5260	−0.629134
$691$	23.4055	0.890389	0.445194	−	0.895434i	$-0.353135\pi$
$691$	23.4055	0.890389	0.445194	+	0.895434i	$0.353135\pi$
$692$	−9.15373	−0.347973
$693$	−12.6542	−0.480694
$694$	49.2648	1.87007
$695$	9.28127	0.352059
$696$	13.3009	0.504169
$697$	13.4891	0.510935
$698$	−50.9497	−1.92848
$699$	−50.1428	−1.89658
$700$	−1.30681	−0.0493929
$701$	−14.2102	−0.536710	−0.268355	−	0.963320i	$-0.586480\pi$
$701$	−14.2102	−0.536710	−0.268355	+	0.963320i	$0.586480\pi$
$702$	−19.9693	−0.753693
$703$	1.29496	0.0488404
$704$	5.94063	0.223896
$705$	−24.1631	−0.910035
$706$	−38.5371	−1.45036
$707$	−25.5944	−0.962577
$708$	9.49415	0.356812
$709$	40.1288	1.50707	0.753534	−	0.657409i	$-0.228348\pi$
$709$	40.1288	1.50707	0.753534	+	0.657409i	$0.228348\pi$
$710$	15.4253	0.578900
$711$	2.35720	0.0884019
$712$	−2.60229	−0.0975250
$713$	−2.05698	−0.0770345
$714$	19.4759	0.728869
$715$	−7.09747	−0.265430
$716$	6.22347	0.232582
$717$	28.8433	1.07717
$718$	−40.6946	−1.51871
$719$	2.21517	0.0826119	0.0413060	−	0.999147i	$-0.486848\pi$
$719$	2.21517	0.0826119	0.0413060	+	0.999147i	$0.486848\pi$
$720$	17.1077	0.637567
$721$	24.7250	0.920808
$722$	−1.54354	−0.0574446
$723$	−48.6526	−1.80941
$724$	−2.24723	−0.0835177
$725$	2.05758	0.0764164
$726$	−3.99655	−0.148326
$727$	−8.24366	−0.305740	−0.152870	−	0.988246i	$-0.548852\pi$
$727$	−8.24366	−0.305740	−0.152870	+	0.988246i	$0.548852\pi$
$728$	−60.5377	−2.24368
$729$	−37.8363	−1.40134
$730$	5.95250	0.220312
$731$	16.8538	0.623361
$732$	0.113768	0.00420500
$733$	−27.7905	−1.02647	−0.513233	−	0.858249i	$-0.671552\pi$
$733$	−27.7905	−1.02647	−0.513233	+	0.858249i	$0.671552\pi$
$734$	−20.4174	−0.753621
$735$	−12.0956	−0.446153
$736$	−8.83195	−0.325550
$737$	7.87864	0.290213
$738$	−54.0657	−1.99019
$739$	19.4755	0.716416	0.358208	−	0.933642i	$-0.383388\pi$
$739$	19.4755	0.716416	0.358208	+	0.933642i	$0.383388\pi$
$740$	0.495343	0.0182092
$741$	−18.3768	−0.675090
$742$	−35.6535	−1.30888
$743$	−0.575855	−0.0211261	−0.0105630	−	0.999944i	$-0.503362\pi$
$743$	−0.575855	−0.0211261	−0.0105630	+	0.999944i	$0.503362\pi$
$744$	3.21568	0.117892
$745$	−5.06425	−0.185540
$746$	32.3498	1.18441
$747$	46.5148	1.70189
$748$	0.545630	0.0199502
$749$	−19.6368	−0.717511
$750$	3.99655	0.145933
$751$	7.29390	0.266158	0.133079	−	0.991105i	$-0.457514\pi$
$751$	7.29390	0.266158	0.133079	+	0.991105i	$0.457514\pi$
$752$	−43.1029	−1.57180
$753$	29.5161	1.07563
$754$	−22.5412	−0.820903
$755$	−16.3009	−0.593251
$756$	−2.38207	−0.0866351
$757$	−4.80190	−0.174528	−0.0872639	−	0.996185i	$-0.527812\pi$
$757$	−4.80190	−0.174528	−0.0872639	+	0.996185i	$0.527812\pi$
$758$	20.8388	0.756900
$759$	−10.7066	−0.388623
$760$	2.49665	0.0905631
$761$	−18.6691	−0.676753	−0.338377	−	0.941011i	$-0.609878\pi$
$761$	−18.6691	−0.676753	−0.338377	+	0.941011i	$0.609878\pi$
$762$	42.2427	1.53029
$763$	40.0885	1.45130
$764$	1.59160	0.0575819
$765$	−5.28349	−0.191025
$766$	−28.8689	−1.04308
$767$	68.0367	2.45666
$768$	22.9559	0.828348
$769$	−50.0562	−1.80507	−0.902536	−	0.430615i	$-0.858297\pi$
$769$	−50.0562	−1.80507	−0.902536	+	0.430615i	$0.858297\pi$
$770$	−5.27329	−0.190036
$771$	62.9355	2.26657
$772$	−9.12142	−0.328287
$773$	5.16202	0.185665	0.0928325	−	0.995682i	$-0.470408\pi$
$773$	5.16202	0.185665	0.0928325	+	0.995682i	$0.470408\pi$
$774$	−67.5520	−2.42810
$775$	0.497448	0.0178689
$776$	−43.1049	−1.54737
$777$	11.4548	0.410939
$778$	20.5604	0.737126
$779$	−9.45656	−0.338816
$780$	−7.02943	−0.251694
$781$	9.99343	0.357593
$782$	9.10437	0.325572
$783$	3.75057	0.134034
$784$	−21.5765	−0.770590
$785$	8.87700	0.316834
$786$	−34.4360	−1.22829
$787$	44.5638	1.58853	0.794264	−	0.607573i	$-0.207857\pi$
$787$	44.5638	1.58853	0.794264	+	0.607573i	$0.207857\pi$
$788$	−4.82986	−0.172057
$789$	−4.55295	−0.162089
$790$	0.982297	0.0349486
$791$	16.7624	0.596004
$792$	9.24761	0.328599
$793$	0.815284	0.0289516
$794$	12.3208	0.437247
$795$	17.5060	0.620875
$796$	−2.95436	−0.104715
$797$	1.06065	0.0375703	0.0187852	−	0.999824i	$-0.494020\pi$
$797$	1.06065	0.0375703	0.0187852	+	0.999824i	$0.494020\pi$
$798$	−13.6537	−0.483334
$799$	13.3117	0.470936
$800$	2.13586	0.0755142
$801$	−3.86073	−0.136412
$802$	39.8798	1.40821
$803$	3.85639	0.136089
$804$	7.80310	0.275194
$805$	−14.1269	−0.497908
$806$	−5.44966	−0.191956
$807$	−61.9281	−2.17997
$808$	18.7042	0.658011
$809$	−17.3398	−0.609634	−0.304817	−	0.952411i	$-0.598595\pi$
$809$	−17.3398	−0.609634	−0.304817	+	0.952411i	$0.598595\pi$
$810$	−9.86688	−0.346687
$811$	37.8473	1.32900	0.664499	−	0.747289i	$-0.268645\pi$
$811$	37.8473	1.32900	0.664499	+	0.747289i	$0.268645\pi$
$812$	−2.68887	−0.0943607
$813$	−7.66377	−0.268780
$814$	1.99882	0.0700587
$815$	0.0647360	0.00226760
$816$	−17.0584	−0.597162
$817$	−11.8154	−0.413369
$818$	−51.8902	−1.81430
$819$	−89.8130	−3.13832
$820$	−3.61728	−0.126321
$821$	−47.7607	−1.66686	−0.833430	−	0.552625i	$-0.813626\pi$
$821$	−47.7607	−1.66686	−0.833430	+	0.552625i	$0.813626\pi$
$822$	48.7811	1.70143
$823$	5.93527	0.206891	0.103445	−	0.994635i	$-0.467013\pi$
$823$	5.93527	0.206891	0.103445	+	0.994635i	$0.467013\pi$
$824$	−18.0689	−0.629458
$825$	2.58921	0.0901447
$826$	50.5501	1.75886
$827$	−19.3136	−0.671599	−0.335800	−	0.941933i	$-0.609006\pi$
$827$	−19.3136	−0.671599	−0.335800	+	0.941933i	$0.609006\pi$
$828$	−5.85873	−0.203605
$829$	11.9719	0.415801	0.207901	−	0.978150i	$-0.433337\pi$
$829$	11.9719	0.415801	0.207901	+	0.978150i	$0.433337\pi$
$830$	19.3837	0.672820
$831$	1.77641	0.0616230
$832$	42.1635	1.46176
$833$	6.66361	0.230881
$834$	37.0930	1.28443
$835$	−3.68720	−0.127601
$836$	−0.382516	−0.0132296
$837$	0.906753	0.0313420
$838$	10.9098	0.376873
$839$	−32.1973	−1.11157	−0.555786	−	0.831325i	$-0.687583\pi$
$839$	−32.1973	−1.11157	−0.555786	+	0.831325i	$0.687583\pi$
$840$	22.0846	0.761991
$841$	−24.7664	−0.854013
$842$	−32.2525	−1.11150
$843$	76.1216	2.62177
$844$	1.25657	0.0432530
$845$	−37.3741	−1.28571
$846$	−53.3550	−1.83438
$847$	−3.41636	−0.117388
$848$	31.2278	1.07237
$849$	−25.6416	−0.880019
$850$	−2.20175	−0.0755193
$851$	5.35475	0.183559
$852$	9.89762	0.339087
$853$	−9.24837	−0.316658	−0.158329	−	0.987386i	$-0.550611\pi$
$853$	−9.24837	−0.316658	−0.158329	+	0.987386i	$0.550611\pi$
$854$	0.605741	0.0207280
$855$	3.70400	0.126674
$856$	14.3504	0.490486
$857$	30.4912	1.04156	0.520780	−	0.853691i	$-0.325641\pi$
$857$	30.4912	1.04156	0.520780	+	0.853691i	$0.325641\pi$
$858$	−28.3654	−0.968378
$859$	16.4204	0.560256	0.280128	−	0.959963i	$-0.409623\pi$
$859$	16.4204	0.560256	0.280128	+	0.959963i	$0.409623\pi$
$860$	−4.51958	−0.154116
$861$	−83.6497	−2.85077
$862$	−14.3684	−0.489388
$863$	1.71945	0.0585307	0.0292653	−	0.999572i	$-0.490683\pi$
$863$	1.71945	0.0585307	0.0292653	+	0.999572i	$0.490683\pi$
$864$	3.89328	0.132452
$865$	23.9303	0.813656
$866$	4.15830	0.141305
$867$	−38.7483	−1.31596
$868$	−0.650071	−0.0220649
$869$	0.636392	0.0215881
$870$	8.22320	0.278793
$871$	55.9184	1.89472
$872$	−29.2964	−0.992101
$873$	−63.9499	−2.16438
$874$	−6.38264	−0.215896
$875$	3.41636	0.115494
$876$	3.81942	0.129046
$877$	−27.1332	−0.916222	−0.458111	−	0.888895i	$-0.651474\pi$
$877$	−27.1332	−0.916222	−0.458111	+	0.888895i	$0.651474\pi$
$878$	29.7399	1.00367
$879$	40.7767	1.37536
$880$	4.61871	0.155697
$881$	1.72426	0.0580917	0.0290458	−	0.999578i	$-0.490753\pi$
$881$	1.72426	0.0580917	0.0290458	+	0.999578i	$0.490753\pi$
$882$	−26.7085	−0.899323
$883$	−47.0867	−1.58459	−0.792297	−	0.610135i	$-0.791115\pi$
$883$	−47.0867	−1.58459	−0.792297	+	0.610135i	$0.791115\pi$
$884$	3.87260	0.130250
$885$	−24.8203	−0.834325
$886$	18.4537	0.619963
$887$	50.6363	1.70020	0.850100	−	0.526621i	$-0.176541\pi$
$887$	50.6363	1.70020	0.850100	+	0.526621i	$0.176541\pi$
$888$	−8.37108	−0.280915
$889$	36.1103	1.21110
$890$	−1.60885	−0.0539288
$891$	−6.39237	−0.214152
$892$	−5.66918	−0.189818
$893$	−9.33224	−0.312291
$894$	−20.2395	−0.676910
$895$	−16.2698	−0.543841
$896$	45.9205	1.53410
$897$	−75.9895	−2.53722
$898$	8.31315	0.277414
$899$	1.02354	0.0341369
$900$	1.41684	0.0472280
$901$	−9.64428	−0.321297
$902$	−14.5966	−0.486013
$903$	−104.515	−3.47805
$904$	−12.2499	−0.407424
$905$	5.87487	0.195287
$906$	−65.1474	−2.16438
$907$	19.4163	0.644708	0.322354	−	0.946619i	$-0.395526\pi$
$907$	19.4163	0.644708	0.322354	+	0.946619i	$0.395526\pi$
$908$	−2.66199	−0.0883411
$909$	27.7493	0.920387
$910$	−37.4271	−1.24070
$911$	23.3720	0.774348	0.387174	−	0.922007i	$-0.373451\pi$
$911$	23.3720	0.774348	0.387174	+	0.922007i	$0.373451\pi$
$912$	11.9588	0.395996
$913$	12.5580	0.415608
$914$	−43.8555	−1.45061
$915$	−0.297421	−0.00983244
$916$	−10.7880	−0.356444
$917$	−29.4369	−0.972093
$918$	−4.01337	−0.132461
$919$	40.8775	1.34842	0.674212	−	0.738538i	$-0.264483\pi$
$919$	40.8775	1.34842	0.674212	+	0.738538i	$0.264483\pi$
$920$	10.3238	0.340366
$921$	−76.8898	−2.53360
$922$	35.6033	1.17253
$923$	70.9281	2.33463
$924$	−3.38361	−0.111313
$925$	−1.29496	−0.0425781
$926$	−29.8842	−0.982056
$927$	−26.8067	−0.880449
$928$	4.39470	0.144263
$929$	27.0203	0.886506	0.443253	−	0.896397i	$-0.353824\pi$
$929$	27.0203	0.886506	0.443253	+	0.896397i	$0.353824\pi$
$930$	1.98807	0.0651915
$931$	−4.67155	−0.153104
$932$	−7.40783	−0.242651
$933$	39.1480	1.28165
$934$	64.3330	2.10504
$935$	−1.42643	−0.0466491
$936$	65.6346	2.14533
$937$	13.6860	0.447102	0.223551	−	0.974692i	$-0.428235\pi$
$937$	13.6860	0.447102	0.223551	+	0.974692i	$0.428235\pi$
$938$	41.5464	1.35654
$939$	−12.8329	−0.418786
$940$	−3.56973	−0.116432
$941$	−1.39984	−0.0456334	−0.0228167	−	0.999740i	$-0.507263\pi$
$941$	−1.39984	−0.0456334	−0.0228167	+	0.999740i	$0.507263\pi$
$942$	35.4773	1.15591
$943$	−39.1035	−1.27339
$944$	−44.2752	−1.44104
$945$	6.22739	0.202577
$946$	−18.2376	−0.592954
$947$	5.42630	0.176331	0.0881655	−	0.996106i	$-0.471900\pi$
$947$	5.42630	0.176331	0.0881655	+	0.996106i	$0.471900\pi$
$948$	0.630291	0.0204709
$949$	27.3706	0.888489
$950$	1.54354	0.0500790
$951$	41.1353	1.33390
$952$	−12.1667	−0.394324
$953$	−11.5574	−0.374382	−0.187191	−	0.982324i	$-0.559938\pi$
$953$	−11.5574	−0.374382	−0.187191	+	0.982324i	$0.559938\pi$
$954$	38.6553	1.25151
$955$	−4.16087	−0.134642
$956$	4.26115	0.137815
$957$	5.32749	0.172213
$958$	4.62403	0.149396
$959$	41.6995	1.34655
$960$	−15.3815	−0.496437
$961$	−30.7525	−0.992018
$962$	14.1866	0.457394
$963$	21.2901	0.686062
$964$	−7.18766	−0.231499
$965$	23.8459	0.767626
$966$	−56.4588	−1.81653
$967$	−48.1917	−1.54974	−0.774870	−	0.632121i	$-0.782185\pi$
$967$	−48.1917	−1.54974	−0.774870	+	0.632121i	$0.782185\pi$
$968$	2.49665	0.0802454
$969$	−3.69332	−0.118646
$970$	−26.6493	−0.855658
$971$	29.1559	0.935656	0.467828	−	0.883819i	$-0.345037\pi$
$971$	29.1559	0.935656	0.467828	+	0.883819i	$0.345037\pi$
$972$	−8.42285	−0.270163
$973$	31.7082	1.01652
$974$	52.1316	1.67041
$975$	18.3768	0.588530
$976$	−0.530550	−0.0169825
$977$	39.0704	1.24997	0.624987	−	0.780635i	$-0.285104\pi$
$977$	39.0704	1.24997	0.624987	+	0.780635i	$0.285104\pi$
$978$	0.258721	0.00827297
$979$	−1.04231	−0.0333125
$980$	−1.78694	−0.0570817
$981$	−43.4638	−1.38769
$982$	18.2903	0.583668
$983$	−49.4811	−1.57820	−0.789101	−	0.614263i	$-0.789453\pi$
$983$	−49.4811	−1.57820	−0.789101	+	0.614263i	$0.789453\pi$
$984$	61.1305	1.94877
$985$	12.6266	0.402316
$986$	−4.53026	−0.144273
$987$	−82.5500	−2.62760
$988$	−2.71489	−0.0863723
$989$	−48.8575	−1.55358
$990$	5.71728	0.181707
$991$	−35.4082	−1.12478	−0.562389	−	0.826873i	$-0.690117\pi$
$991$	−35.4082	−1.12478	−0.562389	+	0.826873i	$0.690117\pi$
$992$	1.06248	0.0337338
$993$	65.6067	2.08196
$994$	52.6983	1.67149
$995$	7.72351	0.244852
$996$	12.4376	0.394100
$997$	−4.15605	−0.131623	−0.0658117	−	0.997832i	$-0.520964\pi$
$997$	−4.15605	−0.131623	−0.0658117	+	0.997832i	$0.520964\pi$
$998$	−34.3284	−1.08665
$999$	−2.36047	−0.0746819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.h.1.2	✓	7
3.2	odd	2		9405.2.a.bd.1.6		7
5.4	even	2		5225.2.a.m.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.h.1.2	✓	7	1.1	even	1	trivial
5225.2.a.m.1.6		7	5.4	even	2
9405.2.a.bd.1.6		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.54354 q^{2} +2.58921 q^{3} +0.382516 q^{4} -1.00000 q^{5} -3.99655 q^{6} -3.41636 q^{7} +2.49665 q^{8} +3.70400 q^{9} +O(q^{10})\)	\(q-1.54354 q^{2} +2.58921 q^{3} +0.382516 q^{4} -1.00000 q^{5} -3.99655 q^{6} -3.41636 q^{7} +2.49665 q^{8} +3.70400 q^{9} +1.54354 q^{10} +1.00000 q^{11} +0.990413 q^{12} +7.09747 q^{13} +5.27329 q^{14} -2.58921 q^{15} -4.61871 q^{16} +1.42643 q^{17} -5.71728 q^{18} -1.00000 q^{19} -0.382516 q^{20} -8.84568 q^{21} -1.54354 q^{22} -4.13507 q^{23} +6.46435 q^{24} +1.00000 q^{25} -10.9552 q^{26} +1.82281 q^{27} -1.30681 q^{28} +2.05758 q^{29} +3.99655 q^{30} +0.497448 q^{31} +2.13586 q^{32} +2.58921 q^{33} -2.20175 q^{34} +3.41636 q^{35} +1.41684 q^{36} -1.29496 q^{37} +1.54354 q^{38} +18.3768 q^{39} -2.49665 q^{40} +9.45656 q^{41} +13.6537 q^{42} +11.8154 q^{43} +0.382516 q^{44} -3.70400 q^{45} +6.38264 q^{46} +9.33224 q^{47} -11.9588 q^{48} +4.67155 q^{49} -1.54354 q^{50} +3.69332 q^{51} +2.71489 q^{52} -6.76115 q^{53} -2.81358 q^{54} -1.00000 q^{55} -8.52947 q^{56} -2.58921 q^{57} -3.17595 q^{58} +9.58605 q^{59} -0.990413 q^{60} +0.114870 q^{61} -0.767831 q^{62} -12.6542 q^{63} +5.94063 q^{64} -7.09747 q^{65} -3.99655 q^{66} +7.87864 q^{67} +0.545630 q^{68} -10.7066 q^{69} -5.27329 q^{70} +9.99343 q^{71} +9.24761 q^{72} +3.85639 q^{73} +1.99882 q^{74} +2.58921 q^{75} -0.382516 q^{76} -3.41636 q^{77} -28.3654 q^{78} +0.636392 q^{79} +4.61871 q^{80} -6.39237 q^{81} -14.5966 q^{82} +12.5580 q^{83} -3.38361 q^{84} -1.42643 q^{85} -18.2376 q^{86} +5.32749 q^{87} +2.49665 q^{88} -1.04231 q^{89} +5.71728 q^{90} -24.2476 q^{91} -1.58173 q^{92} +1.28800 q^{93} -14.4047 q^{94} +1.00000 q^{95} +5.53020 q^{96} -17.2651 q^{97} -7.21072 q^{98} +3.70400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9	\( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9 - q^10 + 7 * q^11 + 13 * q^12 + q^13 + 12 * q^14 - 3 * q^15 + 3 * q^16 + q^17 + 7 * q^18 - 7 * q^19 - 7 * q^20 + 5 * q^21 + q^22 - 8 * q^23 + 25 * q^24 + 7 * q^25 + 12 * q^27 + 4 * q^28 + 11 * q^29 - 8 * q^30 + 7 * q^31 + 12 * q^32 + 3 * q^33 - 14 * q^34 + q^35 + 7 * q^36 - 17 * q^37 - q^38 + 30 * q^39 - 3 * q^40 + 17 * q^41 + 33 * q^42 - 3 * q^43 + 7 * q^44 - 2 * q^45 + 18 * q^46 + 14 * q^47 - 12 * q^48 + 6 * q^49 + q^50 + 8 * q^51 - 17 * q^52 + 7 * q^53 - 27 * q^54 - 7 * q^55 + 36 * q^56 - 3 * q^57 - 15 * q^58 + 35 * q^59 - 13 * q^60 + 17 * q^61 + 46 * q^62 - 22 * q^63 + 5 * q^64 - q^65 + 8 * q^66 + 4 * q^67 - 35 * q^68 - 4 * q^69 - 12 * q^70 + 10 * q^71 + 12 * q^72 + 22 * q^73 - 11 * q^74 + 3 * q^75 - 7 * q^76 - q^77 - 41 * q^78 + 11 * q^79 - 3 * q^80 - 21 * q^81 - 14 * q^82 + 39 * q^83 + 21 * q^84 - q^85 - 24 * q^86 - 2 * q^87 + 3 * q^88 + 18 * q^89 - 7 * q^90 - 22 * q^91 - 51 * q^92 + 10 * q^93 + 14 * q^94 + 7 * q^95 - 11 * q^96 - 4 * q^97 - 26 * q^98 + 2 * q^99

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