LMFDB - Embedded newform 177.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,2,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("177.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	177.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:	$1.41335211578$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} -0.618034 q^{6} +1.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} -0.618034 q^{6} +1.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{10} +4.23607 q^{11} -1.61803 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.85410 q^{16} +3.85410 q^{17} -0.618034 q^{18} -5.85410 q^{19} -1.61803 q^{20} +1.61803 q^{21} -2.61803 q^{22} -1.85410 q^{23} +2.23607 q^{24} -4.00000 q^{25} +0.618034 q^{26} +1.00000 q^{27} -2.61803 q^{28} +3.61803 q^{29} -0.618034 q^{30} +2.85410 q^{31} -5.61803 q^{32} +4.23607 q^{33} -2.38197 q^{34} +1.61803 q^{35} -1.61803 q^{36} -3.38197 q^{37} +3.61803 q^{38} -1.00000 q^{39} +2.23607 q^{40} -6.09017 q^{41} -1.00000 q^{42} -9.94427 q^{43} -6.85410 q^{44} +1.00000 q^{45} +1.14590 q^{46} +4.38197 q^{47} +1.85410 q^{48} -4.38197 q^{49} +2.47214 q^{50} +3.85410 q^{51} +1.61803 q^{52} -9.94427 q^{53} -0.618034 q^{54} +4.23607 q^{55} +3.61803 q^{56} -5.85410 q^{57} -2.23607 q^{58} -1.00000 q^{59} -1.61803 q^{60} +2.85410 q^{61} -1.76393 q^{62} +1.61803 q^{63} -0.236068 q^{64} -1.00000 q^{65} -2.61803 q^{66} +11.9443 q^{67} -6.23607 q^{68} -1.85410 q^{69} -1.00000 q^{70} -4.70820 q^{71} +2.23607 q^{72} +0.381966 q^{73} +2.09017 q^{74} -4.00000 q^{75} +9.47214 q^{76} +6.85410 q^{77} +0.618034 q^{78} +13.9443 q^{79} +1.85410 q^{80} +1.00000 q^{81} +3.76393 q^{82} +3.14590 q^{83} -2.61803 q^{84} +3.85410 q^{85} +6.14590 q^{86} +3.61803 q^{87} +9.47214 q^{88} -8.09017 q^{89} -0.618034 q^{90} -1.61803 q^{91} +3.00000 q^{92} +2.85410 q^{93} -2.70820 q^{94} -5.85410 q^{95} -5.61803 q^{96} -8.70820 q^{97} +2.70820 q^{98} +4.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + q^{18} - 5 q^{19} - q^{20} + q^{21} - 3 q^{22} + 3 q^{23} - 8 q^{25} - q^{26} + 2 q^{27} - 3 q^{28} + 5 q^{29} + q^{30} - q^{31} - 9 q^{32} + 4 q^{33} - 7 q^{34} + q^{35} - q^{36} - 9 q^{37} + 5 q^{38} - 2 q^{39} - q^{41} - 2 q^{42} - 2 q^{43} - 7 q^{44} + 2 q^{45} + 9 q^{46} + 11 q^{47} - 3 q^{48} - 11 q^{49} - 4 q^{50} + q^{51} + q^{52} - 2 q^{53} + q^{54} + 4 q^{55} + 5 q^{56} - 5 q^{57} - 2 q^{59} - q^{60} - q^{61} - 8 q^{62} + q^{63} + 4 q^{64} - 2 q^{65} - 3 q^{66} + 6 q^{67} - 8 q^{68} + 3 q^{69} - 2 q^{70} + 4 q^{71} + 3 q^{73} - 7 q^{74} - 8 q^{75} + 10 q^{76} + 7 q^{77} - q^{78} + 10 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} + 13 q^{83} - 3 q^{84} + q^{85} + 19 q^{86} + 5 q^{87} + 10 q^{88} - 5 q^{89} + q^{90} - q^{91} + 6 q^{92} - q^{93} + 8 q^{94} - 5 q^{95} - 9 q^{96} - 4 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^9 + q^10 + 4 * q^11 - q^12 - 2 * q^13 - 2 * q^14 + 2 * q^15 - 3 * q^16 + q^17 + q^18 - 5 * q^19 - q^20 + q^21 - 3 * q^22 + 3 * q^23 - 8 * q^25 - q^26 + 2 * q^27 - 3 * q^28 + 5 * q^29 + q^30 - q^31 - 9 * q^32 + 4 * q^33 - 7 * q^34 + q^35 - q^36 - 9 * q^37 + 5 * q^38 - 2 * q^39 - q^41 - 2 * q^42 - 2 * q^43 - 7 * q^44 + 2 * q^45 + 9 * q^46 + 11 * q^47 - 3 * q^48 - 11 * q^49 - 4 * q^50 + q^51 + q^52 - 2 * q^53 + q^54 + 4 * q^55 + 5 * q^56 - 5 * q^57 - 2 * q^59 - q^60 - q^61 - 8 * q^62 + q^63 + 4 * q^64 - 2 * q^65 - 3 * q^66 + 6 * q^67 - 8 * q^68 + 3 * q^69 - 2 * q^70 + 4 * q^71 + 3 * q^73 - 7 * q^74 - 8 * q^75 + 10 * q^76 + 7 * q^77 - q^78 + 10 * q^79 - 3 * q^80 + 2 * q^81 + 12 * q^82 + 13 * q^83 - 3 * q^84 + q^85 + 19 * q^86 + 5 * q^87 + 10 * q^88 - 5 * q^89 + q^90 - q^91 + 6 * q^92 - q^93 + 8 * q^94 - 5 * q^95 - 9 * q^96 - 4 * q^97 - 8 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−0.618034	−0.252311
$7$	1.61803	0.611559	0.305780	−	0.952102i	$-0.401083\pi$
$7$	1.61803	0.611559	0.305780	+	0.952102i	$0.401083\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	−0.618034	−0.195440
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	−1.61803	−0.467086
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.85410	0.463525
$17$	3.85410	0.934757	0.467379	−	0.884057i	$-0.345199\pi$
$17$	3.85410	0.934757	0.467379	+	0.884057i	$0.345199\pi$
$18$	−0.618034	−0.145672
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	−1.61803	−0.361803
$21$	1.61803	0.353084
$22$	−2.61803	−0.558167
$23$	−1.85410	−0.386607	−0.193303	−	0.981139i	$-0.561920\pi$
$23$	−1.85410	−0.386607	−0.193303	+	0.981139i	$0.561920\pi$
$24$	2.23607	0.456435
$25$	−4.00000	−0.800000
$26$	0.618034	0.121206
$27$	1.00000	0.192450
$28$	−2.61803	−0.494762
$29$	3.61803	0.671852	0.335926	−	0.941888i	$-0.390951\pi$
$29$	3.61803	0.671852	0.335926	+	0.941888i	$0.390951\pi$
$30$	−0.618034	−0.112837
$31$	2.85410	0.512612	0.256306	−	0.966596i	$-0.417495\pi$
$31$	2.85410	0.512612	0.256306	+	0.966596i	$0.417495\pi$
$32$	−5.61803	−0.993137
$33$	4.23607	0.737405
$34$	−2.38197	−0.408504
$35$	1.61803	0.273498
$36$	−1.61803	−0.269672
$37$	−3.38197	−0.555992	−0.277996	−	0.960582i	$-0.589670\pi$
$37$	−3.38197	−0.555992	−0.277996	+	0.960582i	$0.589670\pi$
$38$	3.61803	0.586923
$39$	−1.00000	−0.160128
$40$	2.23607	0.353553
$41$	−6.09017	−0.951125	−0.475562	−	0.879682i	$-0.657755\pi$
$41$	−6.09017	−0.951125	−0.475562	+	0.879682i	$0.657755\pi$
$42$	−1.00000	−0.154303
$43$	−9.94427	−1.51649	−0.758244	−	0.651971i	$-0.773942\pi$
$43$	−9.94427	−1.51649	−0.758244	+	0.651971i	$0.773942\pi$
$44$	−6.85410	−1.03329
$45$	1.00000	0.149071
$46$	1.14590	0.168953
$47$	4.38197	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$47$	4.38197	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$48$	1.85410	0.267617
$49$	−4.38197	−0.625995
$50$	2.47214	0.349613
$51$	3.85410	0.539682
$52$	1.61803	0.224381
$53$	−9.94427	−1.36595	−0.682975	−	0.730441i	$-0.739314\pi$
$53$	−9.94427	−1.36595	−0.682975	+	0.730441i	$0.739314\pi$
$54$	−0.618034	−0.0841038
$55$	4.23607	0.571191
$56$	3.61803	0.483480
$57$	−5.85410	−0.775395
$58$	−2.23607	−0.293610
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−1.61803	−0.208887
$61$	2.85410	0.365430	0.182715	−	0.983166i	$-0.441511\pi$
$61$	2.85410	0.365430	0.182715	+	0.983166i	$0.441511\pi$
$62$	−1.76393	−0.224020
$63$	1.61803	0.203853
$64$	−0.236068	−0.0295085
$65$	−1.00000	−0.124035
$66$	−2.61803	−0.322258
$67$	11.9443	1.45923	0.729613	−	0.683861i	$-0.239700\pi$
$67$	11.9443	1.45923	0.729613	+	0.683861i	$0.239700\pi$
$68$	−6.23607	−0.756234
$69$	−1.85410	−0.223208
$70$	−1.00000	−0.119523
$71$	−4.70820	−0.558761	−0.279381	−	0.960180i	$-0.590129\pi$
$71$	−4.70820	−0.558761	−0.279381	+	0.960180i	$0.590129\pi$
$72$	2.23607	0.263523
$73$	0.381966	0.0447057	0.0223529	−	0.999750i	$-0.492884\pi$
$73$	0.381966	0.0447057	0.0223529	+	0.999750i	$0.492884\pi$
$74$	2.09017	0.242977
$75$	−4.00000	−0.461880
$76$	9.47214	1.08653
$77$	6.85410	0.781097
$78$	0.618034	0.0699786
$79$	13.9443	1.56885	0.784427	−	0.620222i	$-0.212957\pi$
$79$	13.9443	1.56885	0.784427	+	0.620222i	$0.212957\pi$
$80$	1.85410	0.207295
$81$	1.00000	0.111111
$82$	3.76393	0.415657
$83$	3.14590	0.345307	0.172654	−	0.984983i	$-0.444766\pi$
$83$	3.14590	0.345307	0.172654	+	0.984983i	$0.444766\pi$
$84$	−2.61803	−0.285651
$85$	3.85410	0.418036
$86$	6.14590	0.662729
$87$	3.61803	0.387894
$88$	9.47214	1.00973
$89$	−8.09017	−0.857556	−0.428778	−	0.903410i	$-0.641056\pi$
$89$	−8.09017	−0.857556	−0.428778	+	0.903410i	$0.641056\pi$
$90$	−0.618034	−0.0651465
$91$	−1.61803	−0.169616
$92$	3.00000	0.312772
$93$	2.85410	0.295957
$94$	−2.70820	−0.279330
$95$	−5.85410	−0.600618
$96$	−5.61803	−0.573388
$97$	−8.70820	−0.884184	−0.442092	−	0.896970i	$-0.645764\pi$
$97$	−8.70820	−0.884184	−0.442092	+	0.896970i	$0.645764\pi$
$98$	2.70820	0.273570
$99$	4.23607	0.425741
$100$	6.47214	0.647214
$101$	9.23607	0.919023	0.459512	−	0.888172i	$-0.348024\pi$
$101$	9.23607	0.919023	0.459512	+	0.888172i	$0.348024\pi$
$102$	−2.38197	−0.235850
$103$	11.2361	1.10712	0.553561	−	0.832808i	$-0.313268\pi$
$103$	11.2361	1.10712	0.553561	+	0.832808i	$0.313268\pi$
$104$	−2.23607	−0.219265
$105$	1.61803	0.157904
$106$	6.14590	0.596942
$107$	12.7984	1.23727	0.618633	−	0.785680i	$-0.287687\pi$
$107$	12.7984	1.23727	0.618633	+	0.785680i	$0.287687\pi$
$108$	−1.61803	−0.155695
$109$	−14.7984	−1.41743	−0.708714	−	0.705496i	$-0.750724\pi$
$109$	−14.7984	−1.41743	−0.708714	+	0.705496i	$0.750724\pi$
$110$	−2.61803	−0.249620
$111$	−3.38197	−0.321002
$112$	3.00000	0.283473
$113$	12.4164	1.16804	0.584019	−	0.811740i	$-0.301479\pi$
$113$	12.4164	1.16804	0.584019	+	0.811740i	$0.301479\pi$
$114$	3.61803	0.338860
$115$	−1.85410	−0.172896
$116$	−5.85410	−0.543540
$117$	−1.00000	−0.0924500
$118$	0.618034	0.0568946
$119$	6.23607	0.571659
$120$	2.23607	0.204124
$121$	6.94427	0.631297
$122$	−1.76393	−0.159699
$123$	−6.09017	−0.549132
$124$	−4.61803	−0.414712
$125$	−9.00000	−0.804984
$126$	−1.00000	−0.0890871
$127$	−14.8885	−1.32114	−0.660572	−	0.750762i	$-0.729686\pi$
$127$	−14.8885	−1.32114	−0.660572	+	0.750762i	$0.729686\pi$
$128$	11.3820	1.00603
$129$	−9.94427	−0.875544
$130$	0.618034	0.0542052
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	−6.85410	−0.596573
$133$	−9.47214	−0.821338
$134$	−7.38197	−0.637705
$135$	1.00000	0.0860663
$136$	8.61803	0.738990
$137$	−12.5279	−1.07033	−0.535164	−	0.844748i	$-0.679750\pi$
$137$	−12.5279	−1.07033	−0.535164	+	0.844748i	$0.679750\pi$
$138$	1.14590	0.0975453
$139$	−21.1803	−1.79649	−0.898246	−	0.439492i	$-0.855158\pi$
$139$	−21.1803	−1.79649	−0.898246	+	0.439492i	$0.855158\pi$
$140$	−2.61803	−0.221264
$141$	4.38197	0.369028
$142$	2.90983	0.244188
$143$	−4.23607	−0.354238
$144$	1.85410	0.154508
$145$	3.61803	0.300461
$146$	−0.236068	−0.0195371
$147$	−4.38197	−0.361418
$148$	5.47214	0.449807
$149$	−8.61803	−0.706017	−0.353008	−	0.935620i	$-0.614841\pi$
$149$	−8.61803	−0.706017	−0.353008	+	0.935620i	$0.614841\pi$
$150$	2.47214	0.201849
$151$	10.0902	0.821126	0.410563	−	0.911832i	$-0.365332\pi$
$151$	10.0902	0.821126	0.410563	+	0.911832i	$0.365332\pi$
$152$	−13.0902	−1.06175
$153$	3.85410	0.311586
$154$	−4.23607	−0.341352
$155$	2.85410	0.229247
$156$	1.61803	0.129546
$157$	13.6525	1.08959	0.544793	−	0.838570i	$-0.316608\pi$
$157$	13.6525	1.08959	0.544793	+	0.838570i	$0.316608\pi$
$158$	−8.61803	−0.685614
$159$	−9.94427	−0.788632
$160$	−5.61803	−0.444145
$161$	−3.00000	−0.236433
$162$	−0.618034	−0.0485573
$163$	4.85410	0.380203	0.190101	−	0.981764i	$-0.439118\pi$
$163$	4.85410	0.380203	0.190101	+	0.981764i	$0.439118\pi$
$164$	9.85410	0.769476
$165$	4.23607	0.329777
$166$	−1.94427	−0.150905
$167$	12.2705	0.949521	0.474760	−	0.880115i	$-0.342535\pi$
$167$	12.2705	0.949521	0.474760	+	0.880115i	$0.342535\pi$
$168$	3.61803	0.279137
$169$	−12.0000	−0.923077
$170$	−2.38197	−0.182688
$171$	−5.85410	−0.447674
$172$	16.0902	1.22686
$173$	13.1459	0.999464	0.499732	−	0.866180i	$-0.333432\pi$
$173$	13.1459	0.999464	0.499732	+	0.866180i	$0.333432\pi$
$174$	−2.23607	−0.169516
$175$	−6.47214	−0.489247
$176$	7.85410	0.592025
$177$	−1.00000	−0.0751646
$178$	5.00000	0.374766
$179$	12.8885	0.963335	0.481667	−	0.876354i	$-0.340031\pi$
$179$	12.8885	0.963335	0.481667	+	0.876354i	$0.340031\pi$
$180$	−1.61803	−0.120601
$181$	−17.1459	−1.27444	−0.637222	−	0.770680i	$-0.719917\pi$
$181$	−17.1459	−1.27444	−0.637222	+	0.770680i	$0.719917\pi$
$182$	1.00000	0.0741249
$183$	2.85410	0.210981
$184$	−4.14590	−0.305640
$185$	−3.38197	−0.248647
$186$	−1.76393	−0.129338
$187$	16.3262	1.19389
$188$	−7.09017	−0.517104
$189$	1.61803	0.117695
$190$	3.61803	0.262480
$191$	−7.47214	−0.540665	−0.270332	−	0.962767i	$-0.587134\pi$
$191$	−7.47214	−0.540665	−0.270332	+	0.962767i	$0.587134\pi$
$192$	−0.236068	−0.0170367
$193$	−7.05573	−0.507882	−0.253941	−	0.967220i	$-0.581727\pi$
$193$	−7.05573	−0.507882	−0.253941	+	0.967220i	$0.581727\pi$
$194$	5.38197	0.386403
$195$	−1.00000	−0.0716115
$196$	7.09017	0.506441
$197$	−9.23607	−0.658043	−0.329021	−	0.944323i	$-0.606719\pi$
$197$	−9.23607	−0.658043	−0.329021	+	0.944323i	$0.606719\pi$
$198$	−2.61803	−0.186056
$199$	10.8541	0.769427	0.384713	−	0.923036i	$-0.374300\pi$
$199$	10.8541	0.769427	0.384713	+	0.923036i	$0.374300\pi$
$200$	−8.94427	−0.632456
$201$	11.9443	0.842484
$202$	−5.70820	−0.401628
$203$	5.85410	0.410877
$204$	−6.23607	−0.436612
$205$	−6.09017	−0.425356
$206$	−6.94427	−0.483830
$207$	−1.85410	−0.128869
$208$	−1.85410	−0.128559
$209$	−24.7984	−1.71534
$210$	−1.00000	−0.0690066
$211$	−0.437694	−0.0301321	−0.0150661	−	0.999887i	$-0.504796\pi$
$211$	−0.437694	−0.0301321	−0.0150661	+	0.999887i	$0.504796\pi$
$212$	16.0902	1.10508
$213$	−4.70820	−0.322601
$214$	−7.90983	−0.540705
$215$	−9.94427	−0.678194
$216$	2.23607	0.152145
$217$	4.61803	0.313493
$218$	9.14590	0.619438
$219$	0.381966	0.0258109
$220$	−6.85410	−0.462103
$221$	−3.85410	−0.259255
$222$	2.09017	0.140283
$223$	9.52786	0.638033	0.319016	−	0.947749i	$-0.396647\pi$
$223$	9.52786	0.638033	0.319016	+	0.947749i	$0.396647\pi$
$224$	−9.09017	−0.607363
$225$	−4.00000	−0.266667
$226$	−7.67376	−0.510451
$227$	25.5623	1.69663	0.848315	−	0.529492i	$-0.177617\pi$
$227$	25.5623	1.69663	0.848315	+	0.529492i	$0.177617\pi$
$228$	9.47214	0.627308
$229$	−22.5623	−1.49096	−0.745480	−	0.666528i	$-0.767779\pi$
$229$	−22.5623	−1.49096	−0.745480	+	0.666528i	$0.767779\pi$
$230$	1.14590	0.0755583
$231$	6.85410	0.450967
$232$	8.09017	0.531146
$233$	15.7082	1.02908	0.514539	−	0.857467i	$-0.327963\pi$
$233$	15.7082	1.02908	0.514539	+	0.857467i	$0.327963\pi$
$234$	0.618034	0.0404021
$235$	4.38197	0.285848
$236$	1.61803	0.105325
$237$	13.9443	0.905778
$238$	−3.85410	−0.249824
$239$	−2.23607	−0.144639	−0.0723196	−	0.997382i	$-0.523040\pi$
$239$	−2.23607	−0.144639	−0.0723196	+	0.997382i	$0.523040\pi$
$240$	1.85410	0.119682
$241$	−11.4164	−0.735395	−0.367698	−	0.929945i	$-0.619854\pi$
$241$	−11.4164	−0.735395	−0.367698	+	0.929945i	$0.619854\pi$
$242$	−4.29180	−0.275887
$243$	1.00000	0.0641500
$244$	−4.61803	−0.295639
$245$	−4.38197	−0.279954
$246$	3.76393	0.239980
$247$	5.85410	0.372488
$248$	6.38197	0.405255
$249$	3.14590	0.199363
$250$	5.56231	0.351791
$251$	−25.3607	−1.60075	−0.800376	−	0.599498i	$-0.795367\pi$
$251$	−25.3607	−1.60075	−0.800376	+	0.599498i	$0.795367\pi$
$252$	−2.61803	−0.164921
$253$	−7.85410	−0.493783
$254$	9.20163	0.577361
$255$	3.85410	0.241353
$256$	−6.56231	−0.410144
$257$	−6.47214	−0.403721	−0.201860	−	0.979414i	$-0.564699\pi$
$257$	−6.47214	−0.403721	−0.201860	+	0.979414i	$0.564699\pi$
$258$	6.14590	0.382627
$259$	−5.47214	−0.340022
$260$	1.61803	0.100346
$261$	3.61803	0.223951
$262$	8.76393	0.541438
$263$	29.3262	1.80833	0.904167	−	0.427180i	$-0.140493\pi$
$263$	29.3262	1.80833	0.904167	+	0.427180i	$0.140493\pi$
$264$	9.47214	0.582970
$265$	−9.94427	−0.610872
$266$	5.85410	0.358938
$267$	−8.09017	−0.495110
$268$	−19.3262	−1.18054
$269$	28.4164	1.73258	0.866289	−	0.499542i	$-0.166498\pi$
$269$	28.4164	1.73258	0.866289	+	0.499542i	$0.166498\pi$
$270$	−0.618034	−0.0376124
$271$	18.7082	1.13644	0.568221	−	0.822876i	$-0.307632\pi$
$271$	18.7082	1.13644	0.568221	+	0.822876i	$0.307632\pi$
$272$	7.14590	0.433284
$273$	−1.61803	−0.0979279
$274$	7.74265	0.467750
$275$	−16.9443	−1.02178
$276$	3.00000	0.180579
$277$	1.94427	0.116820	0.0584100	−	0.998293i	$-0.481397\pi$
$277$	1.94427	0.116820	0.0584100	+	0.998293i	$0.481397\pi$
$278$	13.0902	0.785096
$279$	2.85410	0.170871
$280$	3.61803	0.216219
$281$	19.2361	1.14753	0.573764	−	0.819021i	$-0.305483\pi$
$281$	19.2361	1.14753	0.573764	+	0.819021i	$0.305483\pi$
$282$	−2.70820	−0.161271
$283$	20.9098	1.24296	0.621480	−	0.783430i	$-0.286532\pi$
$283$	20.9098	1.24296	0.621480	+	0.783430i	$0.286532\pi$
$284$	7.61803	0.452047
$285$	−5.85410	−0.346767
$286$	2.61803	0.154808
$287$	−9.85410	−0.581669
$288$	−5.61803	−0.331046
$289$	−2.14590	−0.126229
$290$	−2.23607	−0.131306
$291$	−8.70820	−0.510484
$292$	−0.618034	−0.0361677
$293$	28.7984	1.68242	0.841209	−	0.540709i	$-0.181844\pi$
$293$	28.7984	1.68242	0.841209	+	0.540709i	$0.181844\pi$
$294$	2.70820	0.157946
$295$	−1.00000	−0.0582223
$296$	−7.56231	−0.439550
$297$	4.23607	0.245802
$298$	5.32624	0.308541
$299$	1.85410	0.107225
$300$	6.47214	0.373669
$301$	−16.0902	−0.927422
$302$	−6.23607	−0.358845
$303$	9.23607	0.530598
$304$	−10.8541	−0.622525
$305$	2.85410	0.163425
$306$	−2.38197	−0.136168
$307$	−6.47214	−0.369384	−0.184692	−	0.982796i	$-0.559129\pi$
$307$	−6.47214	−0.369384	−0.184692	+	0.982796i	$0.559129\pi$
$308$	−11.0902	−0.631921
$309$	11.2361	0.639198
$310$	−1.76393	−0.100185
$311$	−0.437694	−0.0248194	−0.0124097	−	0.999923i	$-0.503950\pi$
$311$	−0.437694	−0.0248194	−0.0124097	+	0.999923i	$0.503950\pi$
$312$	−2.23607	−0.126592
$313$	−16.9787	−0.959694	−0.479847	−	0.877352i	$-0.659308\pi$
$313$	−16.9787	−0.959694	−0.479847	+	0.877352i	$0.659308\pi$
$314$	−8.43769	−0.476167
$315$	1.61803	0.0911659
$316$	−22.5623	−1.26923
$317$	−23.8328	−1.33858	−0.669292	−	0.742999i	$-0.733403\pi$
$317$	−23.8328	−1.33858	−0.669292	+	0.742999i	$0.733403\pi$
$318$	6.14590	0.344645
$319$	15.3262	0.858105
$320$	−0.236068	−0.0131966
$321$	12.7984	0.714336
$322$	1.85410	0.103325
$323$	−22.5623	−1.25540
$324$	−1.61803	−0.0898908
$325$	4.00000	0.221880
$326$	−3.00000	−0.166155
$327$	−14.7984	−0.818352
$328$	−13.6180	−0.751930
$329$	7.09017	0.390894
$330$	−2.61803	−0.144118
$331$	21.5967	1.18706	0.593532	−	0.804810i	$-0.297733\pi$
$331$	21.5967	1.18706	0.593532	+	0.804810i	$0.297733\pi$
$332$	−5.09017	−0.279359
$333$	−3.38197	−0.185331
$334$	−7.58359	−0.414956
$335$	11.9443	0.652585
$336$	3.00000	0.163663
$337$	19.3820	1.05580	0.527901	−	0.849306i	$-0.322979\pi$
$337$	19.3820	1.05580	0.527901	+	0.849306i	$0.322979\pi$
$338$	7.41641	0.403399
$339$	12.4164	0.674367
$340$	−6.23607	−0.338198
$341$	12.0902	0.654719
$342$	3.61803	0.195641
$343$	−18.4164	−0.994393
$344$	−22.2361	−1.19889
$345$	−1.85410	−0.0998215
$346$	−8.12461	−0.436782
$347$	−30.0902	−1.61532	−0.807662	−	0.589645i	$-0.799268\pi$
$347$	−30.0902	−1.61532	−0.807662	+	0.589645i	$0.799268\pi$
$348$	−5.85410	−0.313813
$349$	20.1246	1.07725	0.538623	−	0.842547i	$-0.318945\pi$
$349$	20.1246	1.07725	0.538623	+	0.842547i	$0.318945\pi$
$350$	4.00000	0.213809
$351$	−1.00000	−0.0533761
$352$	−23.7984	−1.26846
$353$	7.09017	0.377372	0.188686	−	0.982038i	$-0.439577\pi$
$353$	7.09017	0.377372	0.188686	+	0.982038i	$0.439577\pi$
$354$	0.618034	0.0328481
$355$	−4.70820	−0.249886
$356$	13.0902	0.693778
$357$	6.23607	0.330048
$358$	−7.96556	−0.420993
$359$	5.52786	0.291750	0.145875	−	0.989303i	$-0.453400\pi$
$359$	5.52786	0.291750	0.145875	+	0.989303i	$0.453400\pi$
$360$	2.23607	0.117851
$361$	15.2705	0.803711
$362$	10.5967	0.556953
$363$	6.94427	0.364480
$364$	2.61803	0.137222
$365$	0.381966	0.0199930
$366$	−1.76393	−0.0922022
$367$	11.2918	0.589427	0.294713	−	0.955586i	$-0.404776\pi$
$367$	11.2918	0.589427	0.294713	+	0.955586i	$0.404776\pi$
$368$	−3.43769	−0.179202
$369$	−6.09017	−0.317042
$370$	2.09017	0.108663
$371$	−16.0902	−0.835360
$372$	−4.61803	−0.239434
$373$	−23.5623	−1.22001	−0.610005	−	0.792398i	$-0.708833\pi$
$373$	−23.5623	−1.22001	−0.610005	+	0.792398i	$0.708833\pi$
$374$	−10.0902	−0.521750
$375$	−9.00000	−0.464758
$376$	9.79837	0.505313
$377$	−3.61803	−0.186338
$378$	−1.00000	−0.0514344
$379$	−28.4164	−1.45965	−0.729826	−	0.683633i	$-0.760399\pi$
$379$	−28.4164	−1.45965	−0.729826	+	0.683633i	$0.760399\pi$
$380$	9.47214	0.485910
$381$	−14.8885	−0.762763
$382$	4.61803	0.236279
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	11.3820	0.580834
$385$	6.85410	0.349317
$386$	4.36068	0.221953
$387$	−9.94427	−0.505496
$388$	14.0902	0.715320
$389$	13.4164	0.680239	0.340119	−	0.940382i	$-0.389532\pi$
$389$	13.4164	0.680239	0.340119	+	0.940382i	$0.389532\pi$
$390$	0.618034	0.0312954
$391$	−7.14590	−0.361384
$392$	−9.79837	−0.494893
$393$	−14.1803	−0.715304
$394$	5.70820	0.287575
$395$	13.9443	0.701612
$396$	−6.85410	−0.344432
$397$	11.2918	0.566719	0.283359	−	0.959014i	$-0.408551\pi$
$397$	11.2918	0.566719	0.283359	+	0.959014i	$0.408551\pi$
$398$	−6.70820	−0.336252
$399$	−9.47214	−0.474200
$400$	−7.41641	−0.370820
$401$	10.6180	0.530239	0.265120	−	0.964216i	$-0.414589\pi$
$401$	10.6180	0.530239	0.265120	+	0.964216i	$0.414589\pi$
$402$	−7.38197	−0.368179
$403$	−2.85410	−0.142173
$404$	−14.9443	−0.743505
$405$	1.00000	0.0496904
$406$	−3.61803	−0.179560
$407$	−14.3262	−0.710125
$408$	8.61803	0.426656
$409$	27.8885	1.37900	0.689500	−	0.724286i	$-0.257830\pi$
$409$	27.8885	1.37900	0.689500	+	0.724286i	$0.257830\pi$
$410$	3.76393	0.185887
$411$	−12.5279	−0.617954
$412$	−18.1803	−0.895681
$413$	−1.61803	−0.0796182
$414$	1.14590	0.0563178
$415$	3.14590	0.154426
$416$	5.61803	0.275447
$417$	−21.1803	−1.03721
$418$	15.3262	0.749631
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−2.61803	−0.127747
$421$	−32.5967	−1.58867	−0.794334	−	0.607481i	$-0.792180\pi$
$421$	−32.5967	−1.58867	−0.794334	+	0.607481i	$0.792180\pi$
$422$	0.270510	0.0131682
$423$	4.38197	0.213058
$424$	−22.2361	−1.07988
$425$	−15.4164	−0.747806
$426$	2.90983	0.140982
$427$	4.61803	0.223482
$428$	−20.7082	−1.00097
$429$	−4.23607	−0.204519
$430$	6.14590	0.296382
$431$	30.0902	1.44939	0.724696	−	0.689068i	$-0.241980\pi$
$431$	30.0902	1.44939	0.724696	+	0.689068i	$0.241980\pi$
$432$	1.85410	0.0892055
$433$	16.0344	0.770566	0.385283	−	0.922798i	$-0.374104\pi$
$433$	16.0344	0.770566	0.385283	+	0.922798i	$0.374104\pi$
$434$	−2.85410	−0.137001
$435$	3.61803	0.173471
$436$	23.9443	1.14672
$437$	10.8541	0.519222
$438$	−0.236068	−0.0112798
$439$	26.9098	1.28434	0.642168	−	0.766564i	$-0.278035\pi$
$439$	26.9098	1.28434	0.642168	+	0.766564i	$0.278035\pi$
$440$	9.47214	0.451566
$441$	−4.38197	−0.208665
$442$	2.38197	0.113299
$443$	16.0344	0.761819	0.380910	−	0.924612i	$-0.375611\pi$
$443$	16.0344	0.761819	0.380910	+	0.924612i	$0.375611\pi$
$444$	5.47214	0.259696
$445$	−8.09017	−0.383511
$446$	−5.88854	−0.278831
$447$	−8.61803	−0.407619
$448$	−0.381966	−0.0180462
$449$	26.7082	1.26044	0.630219	−	0.776417i	$-0.282965\pi$
$449$	26.7082	1.26044	0.630219	+	0.776417i	$0.282965\pi$
$450$	2.47214	0.116538
$451$	−25.7984	−1.21480
$452$	−20.0902	−0.944962
$453$	10.0902	0.474078
$454$	−15.7984	−0.741454
$455$	−1.61803	−0.0758546
$456$	−13.0902	−0.613003
$457$	−3.50658	−0.164031	−0.0820154	−	0.996631i	$-0.526136\pi$
$457$	−3.50658	−0.164031	−0.0820154	+	0.996631i	$0.526136\pi$
$458$	13.9443	0.651573
$459$	3.85410	0.179894
$460$	3.00000	0.139876
$461$	22.3262	1.03984	0.519918	−	0.854216i	$-0.325962\pi$
$461$	22.3262	1.03984	0.519918	+	0.854216i	$0.325962\pi$
$462$	−4.23607	−0.197080
$463$	20.9098	0.971762	0.485881	−	0.874025i	$-0.338499\pi$
$463$	20.9098	0.971762	0.485881	+	0.874025i	$0.338499\pi$
$464$	6.70820	0.311421
$465$	2.85410	0.132356
$466$	−9.70820	−0.449724
$467$	−29.8885	−1.38308	−0.691538	−	0.722340i	$-0.743067\pi$
$467$	−29.8885	−1.38308	−0.691538	+	0.722340i	$0.743067\pi$
$468$	1.61803	0.0747936
$469$	19.3262	0.892403
$470$	−2.70820	−0.124920
$471$	13.6525	0.629073
$472$	−2.23607	−0.102923
$473$	−42.1246	−1.93689
$474$	−8.61803	−0.395839
$475$	23.4164	1.07442
$476$	−10.0902	−0.462482
$477$	−9.94427	−0.455317
$478$	1.38197	0.0632097
$479$	3.09017	0.141193	0.0705967	−	0.997505i	$-0.477510\pi$
$479$	3.09017	0.141193	0.0705967	+	0.997505i	$0.477510\pi$
$480$	−5.61803	−0.256427
$481$	3.38197	0.154204
$482$	7.05573	0.321380
$483$	−3.00000	−0.136505
$484$	−11.2361	−0.510730
$485$	−8.70820	−0.395419
$486$	−0.618034	−0.0280346
$487$	33.4508	1.51580	0.757901	−	0.652369i	$-0.226225\pi$
$487$	33.4508	1.51580	0.757901	+	0.652369i	$0.226225\pi$
$488$	6.38197	0.288898
$489$	4.85410	0.219510
$490$	2.70820	0.122344
$491$	−29.9098	−1.34981	−0.674906	−	0.737904i	$-0.735816\pi$
$491$	−29.9098	−1.34981	−0.674906	+	0.737904i	$0.735816\pi$
$492$	9.85410	0.444257
$493$	13.9443	0.628018
$494$	−3.61803	−0.162783
$495$	4.23607	0.190397
$496$	5.29180	0.237609
$497$	−7.61803	−0.341716
$498$	−1.94427	−0.0871249
$499$	−23.9443	−1.07189	−0.535946	−	0.844252i	$-0.680045\pi$
$499$	−23.9443	−1.07189	−0.535946	+	0.844252i	$0.680045\pi$
$500$	14.5623	0.651246
$501$	12.2705	0.548206
$502$	15.6738	0.699554
$503$	−21.3262	−0.950890	−0.475445	−	0.879745i	$-0.657713\pi$
$503$	−21.3262	−0.950890	−0.475445	+	0.879745i	$0.657713\pi$
$504$	3.61803	0.161160
$505$	9.23607	0.411000
$506$	4.85410	0.215791
$507$	−12.0000	−0.532939
$508$	24.0902	1.06883
$509$	44.2705	1.96226	0.981128	−	0.193360i	$-0.0619385\pi$
$509$	44.2705	1.96226	0.981128	+	0.193360i	$0.0619385\pi$
$510$	−2.38197	−0.105475
$511$	0.618034	0.0273402
$512$	−18.7082	−0.826794
$513$	−5.85410	−0.258465
$514$	4.00000	0.176432
$515$	11.2361	0.495120
$516$	16.0902	0.708330
$517$	18.5623	0.816369
$518$	3.38197	0.148595
$519$	13.1459	0.577041
$520$	−2.23607	−0.0980581
$521$	−26.0902	−1.14303	−0.571516	−	0.820591i	$-0.693644\pi$
$521$	−26.0902	−1.14303	−0.571516	+	0.820591i	$0.693644\pi$
$522$	−2.23607	−0.0978700
$523$	34.7771	1.52070	0.760348	−	0.649516i	$-0.225028\pi$
$523$	34.7771	1.52070	0.760348	+	0.649516i	$0.225028\pi$
$524$	22.9443	1.00233
$525$	−6.47214	−0.282467
$526$	−18.1246	−0.790271
$527$	11.0000	0.479168
$528$	7.85410	0.341806
$529$	−19.5623	−0.850535
$530$	6.14590	0.266961
$531$	−1.00000	−0.0433963
$532$	15.3262	0.664477
$533$	6.09017	0.263795
$534$	5.00000	0.216371
$535$	12.7984	0.553322
$536$	26.7082	1.15362
$537$	12.8885	0.556182
$538$	−17.5623	−0.757165
$539$	−18.5623	−0.799535
$540$	−1.61803	−0.0696291
$541$	14.8885	0.640108	0.320054	−	0.947399i	$-0.396299\pi$
$541$	14.8885	0.640108	0.320054	+	0.947399i	$0.396299\pi$
$542$	−11.5623	−0.496644
$543$	−17.1459	−0.735801
$544$	−21.6525	−0.928342
$545$	−14.7984	−0.633893
$546$	1.00000	0.0427960
$547$	−25.0902	−1.07278	−0.536389	−	0.843971i	$-0.680212\pi$
$547$	−25.0902	−1.07278	−0.536389	+	0.843971i	$0.680212\pi$
$548$	20.2705	0.865913
$549$	2.85410	0.121810
$550$	10.4721	0.446533
$551$	−21.1803	−0.902313
$552$	−4.14590	−0.176461
$553$	22.5623	0.959447
$554$	−1.20163	−0.0510522
$555$	−3.38197	−0.143556
$556$	34.2705	1.45339
$557$	32.4721	1.37589	0.687944	−	0.725764i	$-0.258513\pi$
$557$	32.4721	1.37589	0.687944	+	0.725764i	$0.258513\pi$
$558$	−1.76393	−0.0746732
$559$	9.94427	0.420598
$560$	3.00000	0.126773
$561$	16.3262	0.689294
$562$	−11.8885	−0.501488
$563$	−32.7082	−1.37849	−0.689243	−	0.724530i	$-0.742057\pi$
$563$	−32.7082	−1.37849	−0.689243	+	0.724530i	$0.742057\pi$
$564$	−7.09017	−0.298550
$565$	12.4164	0.522362
$566$	−12.9230	−0.543194
$567$	1.61803	0.0679510
$568$	−10.5279	−0.441739
$569$	13.6180	0.570898	0.285449	−	0.958394i	$-0.407857\pi$
$569$	13.6180	0.570898	0.285449	+	0.958394i	$0.407857\pi$
$570$	3.61803	0.151543
$571$	−3.65248	−0.152851	−0.0764257	−	0.997075i	$-0.524351\pi$
$571$	−3.65248	−0.152851	−0.0764257	+	0.997075i	$0.524351\pi$
$572$	6.85410	0.286584
$573$	−7.47214	−0.312153
$574$	6.09017	0.254199
$575$	7.41641	0.309286
$576$	−0.236068	−0.00983617
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	1.32624	0.0551642
$579$	−7.05573	−0.293226
$580$	−5.85410	−0.243078
$581$	5.09017	0.211176
$582$	5.38197	0.223090
$583$	−42.1246	−1.74462
$584$	0.854102	0.0353430
$585$	−1.00000	−0.0413449
$586$	−17.7984	−0.735244
$587$	9.83282	0.405844	0.202922	−	0.979195i	$-0.434956\pi$
$587$	9.83282	0.405844	0.202922	+	0.979195i	$0.434956\pi$
$588$	7.09017	0.292394
$589$	−16.7082	−0.688450
$590$	0.618034	0.0254441
$591$	−9.23607	−0.379921
$592$	−6.27051	−0.257716
$593$	2.49342	0.102393	0.0511963	−	0.998689i	$-0.483697\pi$
$593$	2.49342	0.102393	0.0511963	+	0.998689i	$0.483697\pi$
$594$	−2.61803	−0.107419
$595$	6.23607	0.255654
$596$	13.9443	0.571180
$597$	10.8541	0.444229
$598$	−1.14590	−0.0468593
$599$	−45.1246	−1.84374	−0.921871	−	0.387497i	$-0.873340\pi$
$599$	−45.1246	−1.84374	−0.921871	+	0.387497i	$0.873340\pi$
$600$	−8.94427	−0.365148
$601$	14.0344	0.572477	0.286238	−	0.958158i	$-0.407595\pi$
$601$	14.0344	0.572477	0.286238	+	0.958158i	$0.407595\pi$
$602$	9.94427	0.405298
$603$	11.9443	0.486408
$604$	−16.3262	−0.664305
$605$	6.94427	0.282325
$606$	−5.70820	−0.231880
$607$	11.2918	0.458320	0.229160	−	0.973389i	$-0.426402\pi$
$607$	11.2918	0.458320	0.229160	+	0.973389i	$0.426402\pi$
$608$	32.8885	1.33381
$609$	5.85410	0.237220
$610$	−1.76393	−0.0714195
$611$	−4.38197	−0.177275
$612$	−6.23607	−0.252078
$613$	20.1803	0.815076	0.407538	−	0.913188i	$-0.366387\pi$
$613$	20.1803	0.815076	0.407538	+	0.913188i	$0.366387\pi$
$614$	4.00000	0.161427
$615$	−6.09017	−0.245579
$616$	15.3262	0.617512
$617$	−11.2705	−0.453734	−0.226867	−	0.973926i	$-0.572848\pi$
$617$	−11.2705	−0.453734	−0.226867	+	0.973926i	$0.572848\pi$
$618$	−6.94427	−0.279340
$619$	−6.70820	−0.269625	−0.134813	−	0.990871i	$-0.543043\pi$
$619$	−6.70820	−0.269625	−0.134813	+	0.990871i	$0.543043\pi$
$620$	−4.61803	−0.185465
$621$	−1.85410	−0.0744025
$622$	0.270510	0.0108465
$623$	−13.0902	−0.524447
$624$	−1.85410	−0.0742235
$625$	11.0000	0.440000
$626$	10.4934	0.419402
$627$	−24.7984	−0.990352
$628$	−22.0902	−0.881494
$629$	−13.0344	−0.519717
$630$	−1.00000	−0.0398410
$631$	−0.639320	−0.0254509	−0.0127255	−	0.999919i	$-0.504051\pi$
$631$	−0.639320	−0.0254509	−0.0127255	+	0.999919i	$0.504051\pi$
$632$	31.1803	1.24029
$633$	−0.437694	−0.0173968
$634$	14.7295	0.584983
$635$	−14.8885	−0.590834
$636$	16.0902	0.638017
$637$	4.38197	0.173620
$638$	−9.47214	−0.375005
$639$	−4.70820	−0.186254
$640$	11.3820	0.449912
$641$	−30.2361	−1.19425	−0.597126	−	0.802147i	$-0.703691\pi$
$641$	−30.2361	−1.19425	−0.597126	+	0.802147i	$0.703691\pi$
$642$	−7.90983	−0.312176
$643$	0.506578	0.0199775	0.00998874	−	0.999950i	$-0.496820\pi$
$643$	0.506578	0.0199775	0.00998874	+	0.999950i	$0.496820\pi$
$644$	4.85410	0.191278
$645$	−9.94427	−0.391555
$646$	13.9443	0.548630
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	2.23607	0.0878410
$649$	−4.23607	−0.166280
$650$	−2.47214	−0.0969651
$651$	4.61803	0.180995
$652$	−7.85410	−0.307590
$653$	−35.3951	−1.38512	−0.692559	−	0.721361i	$-0.743517\pi$
$653$	−35.3951	−1.38512	−0.692559	+	0.721361i	$0.743517\pi$
$654$	9.14590	0.357633
$655$	−14.1803	−0.554072
$656$	−11.2918	−0.440871
$657$	0.381966	0.0149019
$658$	−4.38197	−0.170827
$659$	−14.7984	−0.576463	−0.288231	−	0.957561i	$-0.593067\pi$
$659$	−14.7984	−0.576463	−0.288231	+	0.957561i	$0.593067\pi$
$660$	−6.85410	−0.266796
$661$	−48.4508	−1.88452	−0.942260	−	0.334883i	$-0.891303\pi$
$661$	−48.4508	−1.88452	−0.942260	+	0.334883i	$0.891303\pi$
$662$	−13.3475	−0.518766
$663$	−3.85410	−0.149681
$664$	7.03444	0.272989
$665$	−9.47214	−0.367314
$666$	2.09017	0.0809924
$667$	−6.70820	−0.259743
$668$	−19.8541	−0.768178
$669$	9.52786	0.368369
$670$	−7.38197	−0.285190
$671$	12.0902	0.466736
$672$	−9.09017	−0.350661
$673$	−18.3607	−0.707752	−0.353876	−	0.935292i	$-0.615137\pi$
$673$	−18.3607	−0.707752	−0.353876	+	0.935292i	$0.615137\pi$
$674$	−11.9787	−0.461403
$675$	−4.00000	−0.153960
$676$	19.4164	0.746785
$677$	−22.6525	−0.870605	−0.435303	−	0.900284i	$-0.643359\pi$
$677$	−22.6525	−0.870605	−0.435303	+	0.900284i	$0.643359\pi$
$678$	−7.67376	−0.294709
$679$	−14.0902	−0.540731
$680$	8.61803	0.330487
$681$	25.5623	0.979550
$682$	−7.47214	−0.286123
$683$	−8.88854	−0.340111	−0.170055	−	0.985435i	$-0.554395\pi$
$683$	−8.88854	−0.340111	−0.170055	+	0.985435i	$0.554395\pi$
$684$	9.47214	0.362176
$685$	−12.5279	−0.478665
$686$	11.3820	0.434565
$687$	−22.5623	−0.860806
$688$	−18.4377	−0.702930
$689$	9.94427	0.378847
$690$	1.14590	0.0436236
$691$	18.1803	0.691613	0.345806	−	0.938306i	$-0.387605\pi$
$691$	18.1803	0.691613	0.345806	+	0.938306i	$0.387605\pi$
$692$	−21.2705	−0.808583
$693$	6.85410	0.260366
$694$	18.5967	0.705923
$695$	−21.1803	−0.803416
$696$	8.09017	0.306657
$697$	−23.4721	−0.889071
$698$	−12.4377	−0.470774
$699$	15.7082	0.594139
$700$	10.4721	0.395810
$701$	43.1033	1.62799	0.813995	−	0.580872i	$-0.197288\pi$
$701$	43.1033	1.62799	0.813995	+	0.580872i	$0.197288\pi$
$702$	0.618034	0.0233262
$703$	19.7984	0.746710
$704$	−1.00000	−0.0376889
$705$	4.38197	0.165034
$706$	−4.38197	−0.164917
$707$	14.9443	0.562037
$708$	1.61803	0.0608094
$709$	−22.7639	−0.854917	−0.427459	−	0.904035i	$-0.640591\pi$
$709$	−22.7639	−0.854917	−0.427459	+	0.904035i	$0.640591\pi$
$710$	2.90983	0.109204
$711$	13.9443	0.522951
$712$	−18.0902	−0.677958
$713$	−5.29180	−0.198179
$714$	−3.85410	−0.144236
$715$	−4.23607	−0.158420
$716$	−20.8541	−0.779354
$717$	−2.23607	−0.0835075
$718$	−3.41641	−0.127499
$719$	19.2705	0.718669	0.359334	−	0.933209i	$-0.383004\pi$
$719$	19.2705	0.718669	0.359334	+	0.933209i	$0.383004\pi$
$720$	1.85410	0.0690983
$721$	18.1803	0.677071
$722$	−9.43769	−0.351235
$723$	−11.4164	−0.424581
$724$	27.7426	1.03105
$725$	−14.4721	−0.537482
$726$	−4.29180	−0.159283
$727$	−23.3050	−0.864333	−0.432166	−	0.901794i	$-0.642251\pi$
$727$	−23.3050	−0.864333	−0.432166	+	0.901794i	$0.642251\pi$
$728$	−3.61803	−0.134093
$729$	1.00000	0.0370370
$730$	−0.236068	−0.00873727
$731$	−38.3262	−1.41755
$732$	−4.61803	−0.170687
$733$	29.2492	1.08034	0.540172	−	0.841554i	$-0.318359\pi$
$733$	29.2492	1.08034	0.540172	+	0.841554i	$0.318359\pi$
$734$	−6.97871	−0.257589
$735$	−4.38197	−0.161631
$736$	10.4164	0.383954
$737$	50.5967	1.86376
$738$	3.76393	0.138552
$739$	−21.3820	−0.786548	−0.393274	−	0.919421i	$-0.628658\pi$
$739$	−21.3820	−0.786548	−0.393274	+	0.919421i	$0.628658\pi$
$740$	5.47214	0.201160
$741$	5.85410	0.215056
$742$	9.94427	0.365066
$743$	26.6869	0.979048	0.489524	−	0.871990i	$-0.337171\pi$
$743$	26.6869	0.979048	0.489524	+	0.871990i	$0.337171\pi$
$744$	6.38197	0.233974
$745$	−8.61803	−0.315740
$746$	14.5623	0.533164
$747$	3.14590	0.115102
$748$	−26.4164	−0.965880
$749$	20.7082	0.756661
$750$	5.56231	0.203107
$751$	−37.5967	−1.37192	−0.685962	−	0.727637i	$-0.740619\pi$
$751$	−37.5967	−1.37192	−0.685962	+	0.727637i	$0.740619\pi$
$752$	8.12461	0.296274
$753$	−25.3607	−0.924195
$754$	2.23607	0.0814328
$755$	10.0902	0.367219
$756$	−2.61803	−0.0952170
$757$	−49.1591	−1.78672	−0.893358	−	0.449345i	$-0.851657\pi$
$757$	−49.1591	−1.78672	−0.893358	+	0.449345i	$0.851657\pi$
$758$	17.5623	0.637892
$759$	−7.85410	−0.285086
$760$	−13.0902	−0.474830
$761$	29.3607	1.06432	0.532162	−	0.846643i	$-0.321380\pi$
$761$	29.3607	1.06432	0.532162	+	0.846643i	$0.321380\pi$
$762$	9.20163	0.333340
$763$	−23.9443	−0.866841
$764$	12.0902	0.437407
$765$	3.85410	0.139345
$766$	−5.56231	−0.200974
$767$	1.00000	0.0361079
$768$	−6.56231	−0.236797
$769$	7.96556	0.287245	0.143623	−	0.989633i	$-0.454125\pi$
$769$	7.96556	0.287245	0.143623	+	0.989633i	$0.454125\pi$
$770$	−4.23607	−0.152657
$771$	−6.47214	−0.233088
$772$	11.4164	0.410886
$773$	−33.2361	−1.19542	−0.597709	−	0.801713i	$-0.703922\pi$
$773$	−33.2361	−1.19542	−0.597709	+	0.801713i	$0.703922\pi$
$774$	6.14590	0.220910
$775$	−11.4164	−0.410089
$776$	−19.4721	−0.699009
$777$	−5.47214	−0.196312
$778$	−8.29180	−0.297275
$779$	35.6525	1.27738
$780$	1.61803	0.0579349
$781$	−19.9443	−0.713662
$782$	4.41641	0.157930
$783$	3.61803	0.129298
$784$	−8.12461	−0.290165
$785$	13.6525	0.487278
$786$	8.76393	0.312599
$787$	5.23607	0.186646	0.0933228	−	0.995636i	$-0.470251\pi$
$787$	5.23607	0.186646	0.0933228	+	0.995636i	$0.470251\pi$
$788$	14.9443	0.532368
$789$	29.3262	1.04404
$790$	−8.61803	−0.306616
$791$	20.0902	0.714324
$792$	9.47214	0.336578
$793$	−2.85410	−0.101352
$794$	−6.97871	−0.247665
$795$	−9.94427	−0.352687
$796$	−17.5623	−0.622479
$797$	−27.1246	−0.960803	−0.480402	−	0.877049i	$-0.659509\pi$
$797$	−27.1246	−0.960803	−0.480402	+	0.877049i	$0.659509\pi$
$798$	5.85410	0.207233
$799$	16.8885	0.597474
$800$	22.4721	0.794510
$801$	−8.09017	−0.285852
$802$	−6.56231	−0.231723
$803$	1.61803	0.0570992
$804$	−19.3262	−0.681584
$805$	−3.00000	−0.105736
$806$	1.76393	0.0621319
$807$	28.4164	1.00030
$808$	20.6525	0.726552
$809$	22.4377	0.788867	0.394434	−	0.918924i	$-0.370941\pi$
$809$	22.4377	0.788867	0.394434	+	0.918924i	$0.370941\pi$
$810$	−0.618034	−0.0217155
$811$	−42.0689	−1.47724	−0.738619	−	0.674123i	$-0.764522\pi$
$811$	−42.0689	−1.47724	−0.738619	+	0.674123i	$0.764522\pi$
$812$	−9.47214	−0.332407
$813$	18.7082	0.656125
$814$	8.85410	0.310336
$815$	4.85410	0.170032
$816$	7.14590	0.250156
$817$	58.2148	2.03668
$818$	−17.2361	−0.602645
$819$	−1.61803	−0.0565387
$820$	9.85410	0.344120
$821$	−31.2148	−1.08940	−0.544702	−	0.838630i	$-0.683357\pi$
$821$	−31.2148	−1.08940	−0.544702	+	0.838630i	$0.683357\pi$
$822$	7.74265	0.270056
$823$	−51.2492	−1.78644	−0.893218	−	0.449624i	$-0.851558\pi$
$823$	−51.2492	−1.78644	−0.893218	+	0.449624i	$0.851558\pi$
$824$	25.1246	0.875257
$825$	−16.9443	−0.589924
$826$	1.00000	0.0347945
$827$	39.8328	1.38512	0.692561	−	0.721359i	$-0.256482\pi$
$827$	39.8328	1.38512	0.692561	+	0.721359i	$0.256482\pi$
$828$	3.00000	0.104257
$829$	−3.21478	−0.111654	−0.0558270	−	0.998440i	$-0.517780\pi$
$829$	−3.21478	−0.111654	−0.0558270	+	0.998440i	$0.517780\pi$
$830$	−1.94427	−0.0674867
$831$	1.94427	0.0674460
$832$	0.236068	0.00818418
$833$	−16.8885	−0.585153
$834$	13.0902	0.453276
$835$	12.2705	0.424639
$836$	40.1246	1.38774
$837$	2.85410	0.0986522
$838$	18.5410	0.640489
$839$	30.1246	1.04002	0.520009	−	0.854161i	$-0.325929\pi$
$839$	30.1246	1.04002	0.520009	+	0.854161i	$0.325929\pi$
$840$	3.61803	0.124834
$841$	−15.9098	−0.548615
$842$	20.1459	0.694273
$843$	19.2361	0.662525
$844$	0.708204	0.0243774
$845$	−12.0000	−0.412813
$846$	−2.70820	−0.0931100
$847$	11.2361	0.386076
$848$	−18.4377	−0.633153
$849$	20.9098	0.717624
$850$	9.52786	0.326803
$851$	6.27051	0.214950
$852$	7.61803	0.260990
$853$	13.9230	0.476714	0.238357	−	0.971178i	$-0.423391\pi$
$853$	13.9230	0.476714	0.238357	+	0.971178i	$0.423391\pi$
$854$	−2.85410	−0.0976654
$855$	−5.85410	−0.200206
$856$	28.6180	0.978144
$857$	−23.5836	−0.805600	−0.402800	−	0.915288i	$-0.631963\pi$
$857$	−23.5836	−0.805600	−0.402800	+	0.915288i	$0.631963\pi$
$858$	2.61803	0.0893782
$859$	36.8328	1.25672	0.628360	−	0.777923i	$-0.283727\pi$
$859$	36.8328	1.25672	0.628360	+	0.777923i	$0.283727\pi$
$860$	16.0902	0.548670
$861$	−9.85410	−0.335827
$862$	−18.5967	−0.633408
$863$	45.3820	1.54482	0.772410	−	0.635124i	$-0.219051\pi$
$863$	45.3820	1.54482	0.772410	+	0.635124i	$0.219051\pi$
$864$	−5.61803	−0.191129
$865$	13.1459	0.446974
$866$	−9.90983	−0.336750
$867$	−2.14590	−0.0728785
$868$	−7.47214	−0.253621
$869$	59.0689	2.00377
$870$	−2.23607	−0.0758098
$871$	−11.9443	−0.404716
$872$	−33.0902	−1.12057
$873$	−8.70820	−0.294728
$874$	−6.70820	−0.226908
$875$	−14.5623	−0.492296
$876$	−0.618034	−0.0208814
$877$	−5.41641	−0.182899	−0.0914495	−	0.995810i	$-0.529150\pi$
$877$	−5.41641	−0.182899	−0.0914495	+	0.995810i	$0.529150\pi$
$878$	−16.6312	−0.561275
$879$	28.7984	0.971345
$880$	7.85410	0.264762
$881$	16.8754	0.568546	0.284273	−	0.958743i	$-0.408248\pi$
$881$	16.8754	0.568546	0.284273	+	0.958743i	$0.408248\pi$
$882$	2.70820	0.0911900
$883$	−24.9443	−0.839442	−0.419721	−	0.907653i	$-0.637872\pi$
$883$	−24.9443	−0.839442	−0.419721	+	0.907653i	$0.637872\pi$
$884$	6.23607	0.209742
$885$	−1.00000	−0.0336146
$886$	−9.90983	−0.332927
$887$	−13.8328	−0.464460	−0.232230	−	0.972661i	$-0.574602\pi$
$887$	−13.8328	−0.464460	−0.232230	+	0.972661i	$0.574602\pi$
$888$	−7.56231	−0.253774
$889$	−24.0902	−0.807958
$890$	5.00000	0.167600
$891$	4.23607	0.141914
$892$	−15.4164	−0.516180
$893$	−25.6525	−0.858427
$894$	5.32624	0.178136
$895$	12.8885	0.430817
$896$	18.4164	0.615249
$897$	1.85410	0.0619067
$898$	−16.5066	−0.550832
$899$	10.3262	0.344399
$900$	6.47214	0.215738
$901$	−38.3262	−1.27683
$902$	15.9443	0.530886
$903$	−16.0902	−0.535447
$904$	27.7639	0.923415
$905$	−17.1459	−0.569949
$906$	−6.23607	−0.207179
$907$	27.4721	0.912197	0.456099	−	0.889929i	$-0.349246\pi$
$907$	27.4721	0.912197	0.456099	+	0.889929i	$0.349246\pi$
$908$	−41.3607	−1.37260
$909$	9.23607	0.306341
$910$	1.00000	0.0331497
$911$	12.8541	0.425875	0.212938	−	0.977066i	$-0.431697\pi$
$911$	12.8541	0.425875	0.212938	+	0.977066i	$0.431697\pi$
$912$	−10.8541	−0.359415
$913$	13.3262	0.441034
$914$	2.16718	0.0716841
$915$	2.85410	0.0943537
$916$	36.5066	1.20621
$917$	−22.9443	−0.757687
$918$	−2.38197	−0.0786166
$919$	−26.1803	−0.863610	−0.431805	−	0.901967i	$-0.642123\pi$
$919$	−26.1803	−0.863610	−0.431805	+	0.901967i	$0.642123\pi$
$920$	−4.14590	−0.136686
$921$	−6.47214	−0.213264
$922$	−13.7984	−0.454425
$923$	4.70820	0.154972
$924$	−11.0902	−0.364840
$925$	13.5279	0.444793
$926$	−12.9230	−0.424676
$927$	11.2361	0.369041
$928$	−20.3262	−0.667241
$929$	43.5410	1.42853	0.714267	−	0.699873i	$-0.246760\pi$
$929$	43.5410	1.42853	0.714267	+	0.699873i	$0.246760\pi$
$930$	−1.76393	−0.0578416
$931$	25.6525	0.840726
$932$	−25.4164	−0.832542
$933$	−0.437694	−0.0143295
$934$	18.4721	0.604427
$935$	16.3262	0.533925
$936$	−2.23607	−0.0730882
$937$	38.1246	1.24548	0.622738	−	0.782430i	$-0.286020\pi$
$937$	38.1246	1.24548	0.622738	+	0.782430i	$0.286020\pi$
$938$	−11.9443	−0.389994
$939$	−16.9787	−0.554079
$940$	−7.09017	−0.231256
$941$	−47.7214	−1.55567	−0.777836	−	0.628467i	$-0.783683\pi$
$941$	−47.7214	−1.55567	−0.777836	+	0.628467i	$0.783683\pi$
$942$	−8.43769	−0.274915
$943$	11.2918	0.367711
$944$	−1.85410	−0.0603459
$945$	1.61803	0.0526346
$946$	26.0344	0.846453
$947$	−24.5623	−0.798168	−0.399084	−	0.916914i	$-0.630672\pi$
$947$	−24.5623	−0.798168	−0.399084	+	0.916914i	$0.630672\pi$
$948$	−22.5623	−0.732790
$949$	−0.381966	−0.0123991
$950$	−14.4721	−0.469538
$951$	−23.8328	−0.772832
$952$	13.9443	0.451936
$953$	7.41641	0.240241	0.120121	−	0.992759i	$-0.461672\pi$
$953$	7.41641	0.240241	0.120121	+	0.992759i	$0.461672\pi$
$954$	6.14590	0.198981
$955$	−7.47214	−0.241793
$956$	3.61803	0.117016
$957$	15.3262	0.495427
$958$	−1.90983	−0.0617038
$959$	−20.2705	−0.654569
$960$	−0.236068	−0.00761906
$961$	−22.8541	−0.737229
$962$	−2.09017	−0.0673898
$963$	12.7984	0.412422
$964$	18.4721	0.594947
$965$	−7.05573	−0.227132
$966$	1.85410	0.0596548
$967$	41.3394	1.32939	0.664693	−	0.747117i	$-0.268563\pi$
$967$	41.3394	1.32939	0.664693	+	0.747117i	$0.268563\pi$
$968$	15.5279	0.499084
$969$	−22.5623	−0.724806
$970$	5.38197	0.172805
$971$	18.9098	0.606845	0.303423	−	0.952856i	$-0.401871\pi$
$971$	18.9098	0.606845	0.303423	+	0.952856i	$0.401871\pi$
$972$	−1.61803	−0.0518985
$973$	−34.2705	−1.09866
$974$	−20.6738	−0.662430
$975$	4.00000	0.128103
$976$	5.29180	0.169386
$977$	35.3607	1.13129	0.565644	−	0.824649i	$-0.308628\pi$
$977$	35.3607	1.13129	0.565644	+	0.824649i	$0.308628\pi$
$978$	−3.00000	−0.0959294
$979$	−34.2705	−1.09529
$980$	7.09017	0.226487
$981$	−14.7984	−0.472476
$982$	18.4853	0.589889
$983$	−19.5410	−0.623262	−0.311631	−	0.950203i	$-0.600875\pi$
$983$	−19.5410	−0.623262	−0.311631	+	0.950203i	$0.600875\pi$
$984$	−13.6180	−0.434127
$985$	−9.23607	−0.294286
$986$	−8.61803	−0.274454
$987$	7.09017	0.225683
$988$	−9.47214	−0.301349
$989$	18.4377	0.586285
$990$	−2.61803	−0.0832066
$991$	1.67376	0.0531688	0.0265844	−	0.999647i	$-0.491537\pi$
$991$	1.67376	0.0531688	0.0265844	+	0.999647i	$0.491537\pi$
$992$	−16.0344	−0.509094
$993$	21.5967	0.685352
$994$	4.70820	0.149335
$995$	10.8541	0.344098
$996$	−5.09017	−0.161288
$997$	−2.20163	−0.0697262	−0.0348631	−	0.999392i	$-0.511100\pi$
$997$	−2.20163	−0.0697262	−0.0348631	+	0.999392i	$0.511100\pi$
$998$	14.7984	0.468434
$999$	−3.38197	−0.107001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.a.c.1.1	✓	2
3.2	odd	2		531.2.a.a.1.2		2
4.3	odd	2		2832.2.a.m.1.1		2
5.4	even	2		4425.2.a.o.1.2		2
7.6	odd	2		8673.2.a.n.1.1		2
12.11	even	2		8496.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.c.1.1	✓	2	1.1	even	1	trivial
531.2.a.a.1.2		2	3.2	odd	2
2832.2.a.m.1.1		2	4.3	odd	2
4425.2.a.o.1.2		2	5.4	even	2
8496.2.a.ba.1.1		2	12.11	even	2
8673.2.a.n.1.1		2	7.6	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} -0.618034 q^{6} +1.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +1.00000 q^{5} -0.618034 q^{6} +1.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{10} +4.23607 q^{11} -1.61803 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.85410 q^{16} +3.85410 q^{17} -0.618034 q^{18} -5.85410 q^{19} -1.61803 q^{20} +1.61803 q^{21} -2.61803 q^{22} -1.85410 q^{23} +2.23607 q^{24} -4.00000 q^{25} +0.618034 q^{26} +1.00000 q^{27} -2.61803 q^{28} +3.61803 q^{29} -0.618034 q^{30} +2.85410 q^{31} -5.61803 q^{32} +4.23607 q^{33} -2.38197 q^{34} +1.61803 q^{35} -1.61803 q^{36} -3.38197 q^{37} +3.61803 q^{38} -1.00000 q^{39} +2.23607 q^{40} -6.09017 q^{41} -1.00000 q^{42} -9.94427 q^{43} -6.85410 q^{44} +1.00000 q^{45} +1.14590 q^{46} +4.38197 q^{47} +1.85410 q^{48} -4.38197 q^{49} +2.47214 q^{50} +3.85410 q^{51} +1.61803 q^{52} -9.94427 q^{53} -0.618034 q^{54} +4.23607 q^{55} +3.61803 q^{56} -5.85410 q^{57} -2.23607 q^{58} -1.00000 q^{59} -1.61803 q^{60} +2.85410 q^{61} -1.76393 q^{62} +1.61803 q^{63} -0.236068 q^{64} -1.00000 q^{65} -2.61803 q^{66} +11.9443 q^{67} -6.23607 q^{68} -1.85410 q^{69} -1.00000 q^{70} -4.70820 q^{71} +2.23607 q^{72} +0.381966 q^{73} +2.09017 q^{74} -4.00000 q^{75} +9.47214 q^{76} +6.85410 q^{77} +0.618034 q^{78} +13.9443 q^{79} +1.85410 q^{80} +1.00000 q^{81} +3.76393 q^{82} +3.14590 q^{83} -2.61803 q^{84} +3.85410 q^{85} +6.14590 q^{86} +3.61803 q^{87} +9.47214 q^{88} -8.09017 q^{89} -0.618034 q^{90} -1.61803 q^{91} +3.00000 q^{92} +2.85410 q^{93} -2.70820 q^{94} -5.85410 q^{95} -5.61803 q^{96} -8.70820 q^{97} +2.70820 q^{98} +4.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} + 2 q^{15} - 3 q^{16} + q^{17} + q^{18} - 5 q^{19} - q^{20} + q^{21} - 3 q^{22} + 3 q^{23} - 8 q^{25} - q^{26} + 2 q^{27} - 3 q^{28} + 5 q^{29} + q^{30} - q^{31} - 9 q^{32} + 4 q^{33} - 7 q^{34} + q^{35} - q^{36} - 9 q^{37} + 5 q^{38} - 2 q^{39} - q^{41} - 2 q^{42} - 2 q^{43} - 7 q^{44} + 2 q^{45} + 9 q^{46} + 11 q^{47} - 3 q^{48} - 11 q^{49} - 4 q^{50} + q^{51} + q^{52} - 2 q^{53} + q^{54} + 4 q^{55} + 5 q^{56} - 5 q^{57} - 2 q^{59} - q^{60} - q^{61} - 8 q^{62} + q^{63} + 4 q^{64} - 2 q^{65} - 3 q^{66} + 6 q^{67} - 8 q^{68} + 3 q^{69} - 2 q^{70} + 4 q^{71} + 3 q^{73} - 7 q^{74} - 8 q^{75} + 10 q^{76} + 7 q^{77} - q^{78} + 10 q^{79} - 3 q^{80} + 2 q^{81} + 12 q^{82} + 13 q^{83} - 3 q^{84} + q^{85} + 19 q^{86} + 5 q^{87} + 10 q^{88} - 5 q^{89} + q^{90} - q^{91} + 6 q^{92} - q^{93} + 8 q^{94} - 5 q^{95} - 9 q^{96} - 4 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^9 + q^10 + 4 * q^11 - q^12 - 2 * q^13 - 2 * q^14 + 2 * q^15 - 3 * q^16 + q^17 + q^18 - 5 * q^19 - q^20 + q^21 - 3 * q^22 + 3 * q^23 - 8 * q^25 - q^26 + 2 * q^27 - 3 * q^28 + 5 * q^29 + q^30 - q^31 - 9 * q^32 + 4 * q^33 - 7 * q^34 + q^35 - q^36 - 9 * q^37 + 5 * q^38 - 2 * q^39 - q^41 - 2 * q^42 - 2 * q^43 - 7 * q^44 + 2 * q^45 + 9 * q^46 + 11 * q^47 - 3 * q^48 - 11 * q^49 - 4 * q^50 + q^51 + q^52 - 2 * q^53 + q^54 + 4 * q^55 + 5 * q^56 - 5 * q^57 - 2 * q^59 - q^60 - q^61 - 8 * q^62 + q^63 + 4 * q^64 - 2 * q^65 - 3 * q^66 + 6 * q^67 - 8 * q^68 + 3 * q^69 - 2 * q^70 + 4 * q^71 + 3 * q^73 - 7 * q^74 - 8 * q^75 + 10 * q^76 + 7 * q^77 - q^78 + 10 * q^79 - 3 * q^80 + 2 * q^81 + 12 * q^82 + 13 * q^83 - 3 * q^84 + q^85 + 19 * q^86 + 5 * q^87 + 10 * q^88 - 5 * q^89 + q^90 - q^91 + 6 * q^92 - q^93 + 8 * q^94 - 5 * q^95 - 9 * q^96 - 4 * q^97 - 8 * q^98 + 4 * q^99

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