LMFDB - Embedded newform 6038.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2136727404$
Analytic rank:	$2$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.381966 q^{3} +1.00000 q^{4} -3.61803 q^{5} -0.381966 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.381966 q^{3} +1.00000 q^{4} -3.61803 q^{5} -0.381966 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} -3.61803 q^{10} -4.61803 q^{11} -0.381966 q^{12} -3.00000 q^{13} -3.00000 q^{14} +1.38197 q^{15} +1.00000 q^{16} +1.47214 q^{17} -2.85410 q^{18} -3.00000 q^{19} -3.61803 q^{20} +1.14590 q^{21} -4.61803 q^{22} -7.47214 q^{23} -0.381966 q^{24} +8.09017 q^{25} -3.00000 q^{26} +2.23607 q^{27} -3.00000 q^{28} -4.76393 q^{29} +1.38197 q^{30} +5.09017 q^{31} +1.00000 q^{32} +1.76393 q^{33} +1.47214 q^{34} +10.8541 q^{35} -2.85410 q^{36} +4.85410 q^{37} -3.00000 q^{38} +1.14590 q^{39} -3.61803 q^{40} -11.1803 q^{41} +1.14590 q^{42} -4.76393 q^{43} -4.61803 q^{44} +10.3262 q^{45} -7.47214 q^{46} -8.56231 q^{47} -0.381966 q^{48} +2.00000 q^{49} +8.09017 q^{50} -0.562306 q^{51} -3.00000 q^{52} -9.00000 q^{53} +2.23607 q^{54} +16.7082 q^{55} -3.00000 q^{56} +1.14590 q^{57} -4.76393 q^{58} -14.2361 q^{59} +1.38197 q^{60} -4.61803 q^{61} +5.09017 q^{62} +8.56231 q^{63} +1.00000 q^{64} +10.8541 q^{65} +1.76393 q^{66} -4.85410 q^{67} +1.47214 q^{68} +2.85410 q^{69} +10.8541 q^{70} +5.94427 q^{71} -2.85410 q^{72} -12.7082 q^{73} +4.85410 q^{74} -3.09017 q^{75} -3.00000 q^{76} +13.8541 q^{77} +1.14590 q^{78} +4.85410 q^{79} -3.61803 q^{80} +7.70820 q^{81} -11.1803 q^{82} +15.9443 q^{83} +1.14590 q^{84} -5.32624 q^{85} -4.76393 q^{86} +1.81966 q^{87} -4.61803 q^{88} +7.79837 q^{89} +10.3262 q^{90} +9.00000 q^{91} -7.47214 q^{92} -1.94427 q^{93} -8.56231 q^{94} +10.8541 q^{95} -0.381966 q^{96} +18.2361 q^{97} +2.00000 q^{98} +13.1803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 5 * q^5 - 3 * q^6 - 6 * q^7 + 2 * q^8 + q^9	\( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 2 q^{8} + q^{9} - 5 q^{10} - 7 q^{11} - 3 q^{12} - 6 q^{13} - 6 q^{14} + 5 q^{15} + 2 q^{16} - 6 q^{17} + q^{18} - 6 q^{19} - 5 q^{20} + 9 q^{21} - 7 q^{22} - 6 q^{23} - 3 q^{24} + 5 q^{25} - 6 q^{26} - 6 q^{28} - 14 q^{29} + 5 q^{30} - q^{31} + 2 q^{32} + 8 q^{33} - 6 q^{34} + 15 q^{35} + q^{36} + 3 q^{37} - 6 q^{38} + 9 q^{39} - 5 q^{40} + 9 q^{42} - 14 q^{43} - 7 q^{44} + 5 q^{45} - 6 q^{46} + 3 q^{47} - 3 q^{48} + 4 q^{49} + 5 q^{50} + 19 q^{51} - 6 q^{52} - 18 q^{53} + 20 q^{55} - 6 q^{56} + 9 q^{57} - 14 q^{58} - 24 q^{59} + 5 q^{60} - 7 q^{61} - q^{62} - 3 q^{63} + 2 q^{64} + 15 q^{65} + 8 q^{66} - 3 q^{67} - 6 q^{68} - q^{69} + 15 q^{70} - 6 q^{71} + q^{72} - 12 q^{73} + 3 q^{74} + 5 q^{75} - 6 q^{76} + 21 q^{77} + 9 q^{78} + 3 q^{79} - 5 q^{80} + 2 q^{81} + 14 q^{83} + 9 q^{84} + 5 q^{85} - 14 q^{86} + 26 q^{87} - 7 q^{88} - 9 q^{89} + 5 q^{90} + 18 q^{91} - 6 q^{92} + 14 q^{93} + 3 q^{94} + 15 q^{95} - 3 q^{96} + 32 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 5 * q^5 - 3 * q^6 - 6 * q^7 + 2 * q^8 + q^9 - 5 * q^10 - 7 * q^11 - 3 * q^12 - 6 * q^13 - 6 * q^14 + 5 * q^15 + 2 * q^16 - 6 * q^17 + q^18 - 6 * q^19 - 5 * q^20 + 9 * q^21 - 7 * q^22 - 6 * q^23 - 3 * q^24 + 5 * q^25 - 6 * q^26 - 6 * q^28 - 14 * q^29 + 5 * q^30 - q^31 + 2 * q^32 + 8 * q^33 - 6 * q^34 + 15 * q^35 + q^36 + 3 * q^37 - 6 * q^38 + 9 * q^39 - 5 * q^40 + 9 * q^42 - 14 * q^43 - 7 * q^44 + 5 * q^45 - 6 * q^46 + 3 * q^47 - 3 * q^48 + 4 * q^49 + 5 * q^50 + 19 * q^51 - 6 * q^52 - 18 * q^53 + 20 * q^55 - 6 * q^56 + 9 * q^57 - 14 * q^58 - 24 * q^59 + 5 * q^60 - 7 * q^61 - q^62 - 3 * q^63 + 2 * q^64 + 15 * q^65 + 8 * q^66 - 3 * q^67 - 6 * q^68 - q^69 + 15 * q^70 - 6 * q^71 + q^72 - 12 * q^73 + 3 * q^74 + 5 * q^75 - 6 * q^76 + 21 * q^77 + 9 * q^78 + 3 * q^79 - 5 * q^80 + 2 * q^81 + 14 * q^83 + 9 * q^84 + 5 * q^85 - 14 * q^86 + 26 * q^87 - 7 * q^88 - 9 * q^89 + 5 * q^90 + 18 * q^91 - 6 * q^92 + 14 * q^93 + 3 * q^94 + 15 * q^95 - 3 * q^96 + 32 * q^97 + 4 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	1.00000	0.500000
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	−0.381966	−0.155937
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	−2.85410	−0.951367
$10$	−3.61803	−1.14412
$11$	−4.61803	−1.39239	−0.696195	−	0.717853i	$-0.745125\pi$
$11$	−4.61803	−1.39239	−0.696195	+	0.717853i	$0.745125\pi$
$12$	−0.381966	−0.110264
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−3.00000	−0.801784
$15$	1.38197	0.356822
$16$	1.00000	0.250000
$17$	1.47214	0.357045	0.178523	−	0.983936i	$-0.442868\pi$
$17$	1.47214	0.357045	0.178523	+	0.983936i	$0.442868\pi$
$18$	−2.85410	−0.672718
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−3.61803	−0.809017
$21$	1.14590	0.250055
$22$	−4.61803	−0.984568
$23$	−7.47214	−1.55805	−0.779024	−	0.626994i	$-0.784285\pi$
$23$	−7.47214	−1.55805	−0.779024	+	0.626994i	$0.784285\pi$
$24$	−0.381966	−0.0779685
$25$	8.09017	1.61803
$26$	−3.00000	−0.588348
$27$	2.23607	0.430331
$28$	−3.00000	−0.566947
$29$	−4.76393	−0.884640	−0.442320	−	0.896857i	$-0.645844\pi$
$29$	−4.76393	−0.884640	−0.442320	+	0.896857i	$0.645844\pi$
$30$	1.38197	0.252311
$31$	5.09017	0.914222	0.457111	−	0.889410i	$-0.348884\pi$
$31$	5.09017	0.914222	0.457111	+	0.889410i	$0.348884\pi$
$32$	1.00000	0.176777
$33$	1.76393	0.307061
$34$	1.47214	0.252469
$35$	10.8541	1.83468
$36$	−2.85410	−0.475684
$37$	4.85410	0.798009	0.399005	−	0.916949i	$-0.369356\pi$
$37$	4.85410	0.798009	0.399005	+	0.916949i	$0.369356\pi$
$38$	−3.00000	−0.486664
$39$	1.14590	0.183491
$40$	−3.61803	−0.572061
$41$	−11.1803	−1.74608	−0.873038	−	0.487652i	$-0.837853\pi$
$41$	−11.1803	−1.74608	−0.873038	+	0.487652i	$0.837853\pi$
$42$	1.14590	0.176816
$43$	−4.76393	−0.726493	−0.363246	−	0.931693i	$-0.618332\pi$
$43$	−4.76393	−0.726493	−0.363246	+	0.931693i	$0.618332\pi$
$44$	−4.61803	−0.696195
$45$	10.3262	1.53934
$46$	−7.47214	−1.10171
$47$	−8.56231	−1.24894	−0.624470	−	0.781049i	$-0.714685\pi$
$47$	−8.56231	−1.24894	−0.624470	+	0.781049i	$0.714685\pi$
$48$	−0.381966	−0.0551320
$49$	2.00000	0.285714
$50$	8.09017	1.14412
$51$	−0.562306	−0.0787386
$52$	−3.00000	−0.416025
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	2.23607	0.304290
$55$	16.7082	2.25293
$56$	−3.00000	−0.400892
$57$	1.14590	0.151778
$58$	−4.76393	−0.625535
$59$	−14.2361	−1.85338	−0.926689	−	0.375829i	$-0.877358\pi$
$59$	−14.2361	−1.85338	−0.926689	+	0.375829i	$0.877358\pi$
$60$	1.38197	0.178411
$61$	−4.61803	−0.591279	−0.295639	−	0.955300i	$-0.595533\pi$
$61$	−4.61803	−0.591279	−0.295639	+	0.955300i	$0.595533\pi$
$62$	5.09017	0.646452
$63$	8.56231	1.07875
$64$	1.00000	0.125000
$65$	10.8541	1.34629
$66$	1.76393	0.217125
$67$	−4.85410	−0.593023	−0.296511	−	0.955029i	$-0.595823\pi$
$67$	−4.85410	−0.593023	−0.296511	+	0.955029i	$0.595823\pi$
$68$	1.47214	0.178523
$69$	2.85410	0.343594
$70$	10.8541	1.29731
$71$	5.94427	0.705455	0.352728	−	0.935726i	$-0.385254\pi$
$71$	5.94427	0.705455	0.352728	+	0.935726i	$0.385254\pi$
$72$	−2.85410	−0.336359
$73$	−12.7082	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$73$	−12.7082	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$74$	4.85410	0.564278
$75$	−3.09017	−0.356822
$76$	−3.00000	−0.344124
$77$	13.8541	1.57882
$78$	1.14590	0.129747
$79$	4.85410	0.546129	0.273065	−	0.961996i	$-0.411963\pi$
$79$	4.85410	0.546129	0.273065	+	0.961996i	$0.411963\pi$
$80$	−3.61803	−0.404508
$81$	7.70820	0.856467
$82$	−11.1803	−1.23466
$83$	15.9443	1.75011	0.875056	−	0.484022i	$-0.160825\pi$
$83$	15.9443	1.75011	0.875056	+	0.484022i	$0.160825\pi$
$84$	1.14590	0.125028
$85$	−5.32624	−0.577712
$86$	−4.76393	−0.513708
$87$	1.81966	0.195088
$88$	−4.61803	−0.492284
$89$	7.79837	0.826626	0.413313	−	0.910589i	$-0.364372\pi$
$89$	7.79837	0.826626	0.413313	+	0.910589i	$0.364372\pi$
$90$	10.3262	1.08848
$91$	9.00000	0.943456
$92$	−7.47214	−0.779024
$93$	−1.94427	−0.201612
$94$	−8.56231	−0.883134
$95$	10.8541	1.11361
$96$	−0.381966	−0.0389842
$97$	18.2361	1.85159	0.925796	−	0.378023i	$-0.123396\pi$
$97$	18.2361	1.85159	0.925796	+	0.378023i	$0.123396\pi$
$98$	2.00000	0.202031
$99$	13.1803	1.32467
$100$	8.09017	0.809017
$101$	0.763932	0.0760141	0.0380070	−	0.999277i	$-0.487899\pi$
$101$	0.763932	0.0760141	0.0380070	+	0.999277i	$0.487899\pi$
$102$	−0.562306	−0.0556766
$103$	11.8541	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$103$	11.8541	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$104$	−3.00000	−0.294174
$105$	−4.14590	−0.404598
$106$	−9.00000	−0.874157
$107$	−8.23607	−0.796211	−0.398105	−	0.917340i	$-0.630332\pi$
$107$	−8.23607	−0.796211	−0.398105	+	0.917340i	$0.630332\pi$
$108$	2.23607	0.215166
$109$	−0.527864	−0.0505602	−0.0252801	−	0.999680i	$-0.508048\pi$
$109$	−0.527864	−0.0505602	−0.0252801	+	0.999680i	$0.508048\pi$
$110$	16.7082	1.59306
$111$	−1.85410	−0.175984
$112$	−3.00000	−0.283473
$113$	−6.61803	−0.622572	−0.311286	−	0.950316i	$-0.600760\pi$
$113$	−6.61803	−0.622572	−0.311286	+	0.950316i	$0.600760\pi$
$114$	1.14590	0.107323
$115$	27.0344	2.52097
$116$	−4.76393	−0.442320
$117$	8.56231	0.791585
$118$	−14.2361	−1.31054
$119$	−4.41641	−0.404851
$120$	1.38197	0.126156
$121$	10.3262	0.938749
$122$	−4.61803	−0.418097
$123$	4.27051	0.385059
$124$	5.09017	0.457111
$125$	−11.1803	−1.00000
$126$	8.56231	0.762791
$127$	14.4164	1.27925	0.639625	−	0.768687i	$-0.279090\pi$
$127$	14.4164	1.27925	0.639625	+	0.768687i	$0.279090\pi$
$128$	1.00000	0.0883883
$129$	1.81966	0.160212
$130$	10.8541	0.951968
$131$	19.3820	1.69341	0.846705	−	0.532062i	$-0.178583\pi$
$131$	19.3820	1.69341	0.846705	+	0.532062i	$0.178583\pi$
$132$	1.76393	0.153531
$133$	9.00000	0.780399
$134$	−4.85410	−0.419331
$135$	−8.09017	−0.696291
$136$	1.47214	0.126235
$137$	−11.5623	−0.987834	−0.493917	−	0.869509i	$-0.664435\pi$
$137$	−11.5623	−0.987834	−0.493917	+	0.869509i	$0.664435\pi$
$138$	2.85410	0.242957
$139$	−6.52786	−0.553686	−0.276843	−	0.960915i	$-0.589288\pi$
$139$	−6.52786	−0.553686	−0.276843	+	0.960915i	$0.589288\pi$
$140$	10.8541	0.917339
$141$	3.27051	0.275427
$142$	5.94427	0.498832
$143$	13.8541	1.15854
$144$	−2.85410	−0.237842
$145$	17.2361	1.43138
$146$	−12.7082	−1.05174
$147$	−0.763932	−0.0630081
$148$	4.85410	0.399005
$149$	2.23607	0.183186	0.0915929	−	0.995797i	$-0.470804\pi$
$149$	2.23607	0.183186	0.0915929	+	0.995797i	$0.470804\pi$
$150$	−3.09017	−0.252311
$151$	14.7984	1.20427	0.602137	−	0.798393i	$-0.294316\pi$
$151$	14.7984	1.20427	0.602137	+	0.798393i	$0.294316\pi$
$152$	−3.00000	−0.243332
$153$	−4.20163	−0.339681
$154$	13.8541	1.11640
$155$	−18.4164	−1.47924
$156$	1.14590	0.0917453
$157$	−3.09017	−0.246622	−0.123311	−	0.992368i	$-0.539351\pi$
$157$	−3.09017	−0.246622	−0.123311	+	0.992368i	$0.539351\pi$
$158$	4.85410	0.386172
$159$	3.43769	0.272627
$160$	−3.61803	−0.286031
$161$	22.4164	1.76666
$162$	7.70820	0.605614
$163$	−5.85410	−0.458529	−0.229264	−	0.973364i	$-0.573632\pi$
$163$	−5.85410	−0.458529	−0.229264	+	0.973364i	$0.573632\pi$
$164$	−11.1803	−0.873038
$165$	−6.38197	−0.496835
$166$	15.9443	1.23752
$167$	−0.763932	−0.0591148	−0.0295574	−	0.999563i	$-0.509410\pi$
$167$	−0.763932	−0.0591148	−0.0295574	+	0.999563i	$0.509410\pi$
$168$	1.14590	0.0884080
$169$	−4.00000	−0.307692
$170$	−5.32624	−0.408504
$171$	8.56231	0.654776
$172$	−4.76393	−0.363246
$173$	−23.3262	−1.77346	−0.886731	−	0.462287i	$-0.847029\pi$
$173$	−23.3262	−1.77346	−0.886731	+	0.462287i	$0.847029\pi$
$174$	1.81966	0.137948
$175$	−24.2705	−1.83468
$176$	−4.61803	−0.348097
$177$	5.43769	0.408722
$178$	7.79837	0.584513
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	10.3262	0.769672
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	9.00000	0.667124
$183$	1.76393	0.130394
$184$	−7.47214	−0.550853
$185$	−17.5623	−1.29121
$186$	−1.94427	−0.142561
$187$	−6.79837	−0.497146
$188$	−8.56231	−0.624470
$189$	−6.70820	−0.487950
$190$	10.8541	0.787439
$191$	−22.3607	−1.61796	−0.808981	−	0.587835i	$-0.799981\pi$
$191$	−22.3607	−1.61796	−0.808981	+	0.587835i	$0.799981\pi$
$192$	−0.381966	−0.0275660
$193$	9.85410	0.709314	0.354657	−	0.934997i	$-0.384598\pi$
$193$	9.85410	0.709314	0.354657	+	0.934997i	$0.384598\pi$
$194$	18.2361	1.30927
$195$	−4.14590	−0.296894
$196$	2.00000	0.142857
$197$	21.7984	1.55307	0.776535	−	0.630074i	$-0.216976\pi$
$197$	21.7984	1.55307	0.776535	+	0.630074i	$0.216976\pi$
$198$	13.1803	0.936686
$199$	−12.2705	−0.869833	−0.434917	−	0.900471i	$-0.643222\pi$
$199$	−12.2705	−0.869833	−0.434917	+	0.900471i	$0.643222\pi$
$200$	8.09017	0.572061
$201$	1.85410	0.130778
$202$	0.763932	0.0537501
$203$	14.2918	1.00309
$204$	−0.562306	−0.0393693
$205$	40.4508	2.82521
$206$	11.8541	0.825914
$207$	21.3262	1.48228
$208$	−3.00000	−0.208013
$209$	13.8541	0.958308
$210$	−4.14590	−0.286094
$211$	11.1459	0.767315	0.383658	−	0.923475i	$-0.374664\pi$
$211$	11.1459	0.767315	0.383658	+	0.923475i	$0.374664\pi$
$212$	−9.00000	−0.618123
$213$	−2.27051	−0.155573
$214$	−8.23607	−0.563006
$215$	17.2361	1.17549
$216$	2.23607	0.152145
$217$	−15.2705	−1.03663
$218$	−0.527864	−0.0357515
$219$	4.85410	0.328010
$220$	16.7082	1.12647
$221$	−4.41641	−0.297080
$222$	−1.85410	−0.124439
$223$	−28.4164	−1.90290	−0.951452	−	0.307798i	$-0.900408\pi$
$223$	−28.4164	−1.90290	−0.951452	+	0.307798i	$0.900408\pi$
$224$	−3.00000	−0.200446
$225$	−23.0902	−1.53934
$226$	−6.61803	−0.440225
$227$	7.14590	0.474290	0.237145	−	0.971474i	$-0.423788\pi$
$227$	7.14590	0.474290	0.237145	+	0.971474i	$0.423788\pi$
$228$	1.14590	0.0758890
$229$	−22.5967	−1.49324	−0.746618	−	0.665253i	$-0.768323\pi$
$229$	−22.5967	−1.49324	−0.746618	+	0.665253i	$0.768323\pi$
$230$	27.0344	1.78260
$231$	−5.29180	−0.348175
$232$	−4.76393	−0.312767
$233$	10.8541	0.711076	0.355538	−	0.934662i	$-0.384298\pi$
$233$	10.8541	0.711076	0.355538	+	0.934662i	$0.384298\pi$
$234$	8.56231	0.559735
$235$	30.9787	2.02083
$236$	−14.2361	−0.926689
$237$	−1.85410	−0.120437
$238$	−4.41641	−0.286273
$239$	−9.70820	−0.627972	−0.313986	−	0.949428i	$-0.601664\pi$
$239$	−9.70820	−0.627972	−0.313986	+	0.949428i	$0.601664\pi$
$240$	1.38197	0.0892055
$241$	16.4164	1.05747	0.528737	−	0.848786i	$-0.322666\pi$
$241$	16.4164	1.05747	0.528737	+	0.848786i	$0.322666\pi$
$242$	10.3262	0.663796
$243$	−9.65248	−0.619207
$244$	−4.61803	−0.295639
$245$	−7.23607	−0.462295
$246$	4.27051	0.272278
$247$	9.00000	0.572656
$248$	5.09017	0.323226
$249$	−6.09017	−0.385949
$250$	−11.1803	−0.707107
$251$	−16.9443	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$251$	−16.9443	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$252$	8.56231	0.539375
$253$	34.5066	2.16941
$254$	14.4164	0.904566
$255$	2.03444	0.127402
$256$	1.00000	0.0625000
$257$	−28.4164	−1.77257	−0.886283	−	0.463143i	$-0.846722\pi$
$257$	−28.4164	−1.77257	−0.886283	+	0.463143i	$0.846722\pi$
$258$	1.81966	0.113287
$259$	−14.5623	−0.904858
$260$	10.8541	0.673143
$261$	13.5967	0.841618
$262$	19.3820	1.19742
$263$	−26.3607	−1.62547	−0.812735	−	0.582634i	$-0.802022\pi$
$263$	−26.3607	−1.62547	−0.812735	+	0.582634i	$0.802022\pi$
$264$	1.76393	0.108563
$265$	32.5623	2.00029
$266$	9.00000	0.551825
$267$	−2.97871	−0.182294
$268$	−4.85410	−0.296511
$269$	14.5623	0.887879	0.443940	−	0.896057i	$-0.353580\pi$
$269$	14.5623	0.887879	0.443940	+	0.896057i	$0.353580\pi$
$270$	−8.09017	−0.492352
$271$	−4.38197	−0.266185	−0.133093	−	0.991104i	$-0.542491\pi$
$271$	−4.38197	−0.266185	−0.133093	+	0.991104i	$0.542491\pi$
$272$	1.47214	0.0892614
$273$	−3.43769	−0.208059
$274$	−11.5623	−0.698504
$275$	−37.3607	−2.25293
$276$	2.85410	0.171797
$277$	−0.437694	−0.0262985	−0.0131492	−	0.999914i	$-0.504186\pi$
$277$	−0.437694	−0.0262985	−0.0131492	+	0.999914i	$0.504186\pi$
$278$	−6.52786	−0.391515
$279$	−14.5279	−0.869760
$280$	10.8541	0.648657
$281$	19.3607	1.15496	0.577481	−	0.816404i	$-0.304036\pi$
$281$	19.3607	1.15496	0.577481	+	0.816404i	$0.304036\pi$
$282$	3.27051	0.194756
$283$	−2.70820	−0.160986	−0.0804930	−	0.996755i	$-0.525649\pi$
$283$	−2.70820	−0.160986	−0.0804930	+	0.996755i	$0.525649\pi$
$284$	5.94427	0.352728
$285$	−4.14590	−0.245582
$286$	13.8541	0.819210
$287$	33.5410	1.97986
$288$	−2.85410	−0.168180
$289$	−14.8328	−0.872519
$290$	17.2361	1.01214
$291$	−6.96556	−0.408328
$292$	−12.7082	−0.743691
$293$	−20.9098	−1.22157	−0.610783	−	0.791798i	$-0.709145\pi$
$293$	−20.9098	−1.22157	−0.610783	+	0.791798i	$0.709145\pi$
$294$	−0.763932	−0.0445534
$295$	51.5066	2.99883
$296$	4.85410	0.282139
$297$	−10.3262	−0.599189
$298$	2.23607	0.129532
$299$	22.4164	1.29637
$300$	−3.09017	−0.178411
$301$	14.2918	0.823765
$302$	14.7984	0.851551
$303$	−0.291796	−0.0167632
$304$	−3.00000	−0.172062
$305$	16.7082	0.956709
$306$	−4.20163	−0.240191
$307$	15.0902	0.861241	0.430621	−	0.902533i	$-0.358295\pi$
$307$	15.0902	0.861241	0.430621	+	0.902533i	$0.358295\pi$
$308$	13.8541	0.789411
$309$	−4.52786	−0.257581
$310$	−18.4164	−1.04598
$311$	29.2361	1.65783	0.828913	−	0.559378i	$-0.188960\pi$
$311$	29.2361	1.65783	0.828913	+	0.559378i	$0.188960\pi$
$312$	1.14590	0.0648737
$313$	−9.74265	−0.550687	−0.275343	−	0.961346i	$-0.588792\pi$
$313$	−9.74265	−0.550687	−0.275343	+	0.961346i	$0.588792\pi$
$314$	−3.09017	−0.174388
$315$	−30.9787	−1.74545
$316$	4.85410	0.273065
$317$	−8.38197	−0.470778	−0.235389	−	0.971901i	$-0.575636\pi$
$317$	−8.38197	−0.470778	−0.235389	+	0.971901i	$0.575636\pi$
$318$	3.43769	0.192776
$319$	22.0000	1.23176
$320$	−3.61803	−0.202254
$321$	3.14590	0.175587
$322$	22.4164	1.24922
$323$	−4.41641	−0.245736
$324$	7.70820	0.428234
$325$	−24.2705	−1.34629
$326$	−5.85410	−0.324229
$327$	0.201626	0.0111500
$328$	−11.1803	−0.617331
$329$	25.6869	1.41617
$330$	−6.38197	−0.351316
$331$	−25.5623	−1.40503	−0.702516	−	0.711668i	$-0.747940\pi$
$331$	−25.5623	−1.40503	−0.702516	+	0.711668i	$0.747940\pi$
$332$	15.9443	0.875056
$333$	−13.8541	−0.759200
$334$	−0.763932	−0.0418005
$335$	17.5623	0.959531
$336$	1.14590	0.0625139
$337$	2.38197	0.129754	0.0648770	−	0.997893i	$-0.479335\pi$
$337$	2.38197	0.129754	0.0648770	+	0.997893i	$0.479335\pi$
$338$	−4.00000	−0.217571
$339$	2.52786	0.137295
$340$	−5.32624	−0.288856
$341$	−23.5066	−1.27295
$342$	8.56231	0.462996
$343$	15.0000	0.809924
$344$	−4.76393	−0.256854
$345$	−10.3262	−0.555946
$346$	−23.3262	−1.25403
$347$	−24.3262	−1.30590	−0.652950	−	0.757401i	$-0.726469\pi$
$347$	−24.3262	−1.30590	−0.652950	+	0.757401i	$0.726469\pi$
$348$	1.81966	0.0975440
$349$	−9.29180	−0.497378	−0.248689	−	0.968583i	$-0.580000\pi$
$349$	−9.29180	−0.497378	−0.248689	+	0.968583i	$0.580000\pi$
$350$	−24.2705	−1.29731
$351$	−6.70820	−0.358057
$352$	−4.61803	−0.246142
$353$	−29.1803	−1.55311	−0.776556	−	0.630048i	$-0.783035\pi$
$353$	−29.1803	−1.55311	−0.776556	+	0.630048i	$0.783035\pi$
$354$	5.43769	0.289010
$355$	−21.5066	−1.14145
$356$	7.79837	0.413313
$357$	1.68692	0.0892812
$358$	−9.00000	−0.475665
$359$	34.5967	1.82595	0.912973	−	0.408019i	$-0.133780\pi$
$359$	34.5967	1.82595	0.912973	+	0.408019i	$0.133780\pi$
$360$	10.3262	0.544241
$361$	−10.0000	−0.526316
$362$	−18.0000	−0.946059
$363$	−3.94427	−0.207021
$364$	9.00000	0.471728
$365$	45.9787	2.40664
$366$	1.76393	0.0922022
$367$	−30.8885	−1.61237	−0.806184	−	0.591664i	$-0.798471\pi$
$367$	−30.8885	−1.61237	−0.806184	+	0.591664i	$0.798471\pi$
$368$	−7.47214	−0.389512
$369$	31.9098	1.66116
$370$	−17.5623	−0.913021
$371$	27.0000	1.40177
$372$	−1.94427	−0.100806
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	−6.79837	−0.351536
$375$	4.27051	0.220528
$376$	−8.56231	−0.441567
$377$	14.2918	0.736065
$378$	−6.70820	−0.345033
$379$	−30.8328	−1.58378	−0.791888	−	0.610667i	$-0.790901\pi$
$379$	−30.8328	−1.58378	−0.791888	+	0.610667i	$0.790901\pi$
$380$	10.8541	0.556804
$381$	−5.50658	−0.282111
$382$	−22.3607	−1.14407
$383$	−15.7082	−0.802652	−0.401326	−	0.915935i	$-0.631451\pi$
$383$	−15.7082	−0.802652	−0.401326	+	0.915935i	$0.631451\pi$
$384$	−0.381966	−0.0194921
$385$	−50.1246	−2.55459
$386$	9.85410	0.501561
$387$	13.5967	0.691162
$388$	18.2361	0.925796
$389$	−22.7639	−1.15418	−0.577089	−	0.816682i	$-0.695811\pi$
$389$	−22.7639	−1.15418	−0.577089	+	0.816682i	$0.695811\pi$
$390$	−4.14590	−0.209936
$391$	−11.0000	−0.556294
$392$	2.00000	0.101015
$393$	−7.40325	−0.373445
$394$	21.7984	1.09819
$395$	−17.5623	−0.883656
$396$	13.1803	0.662337
$397$	11.5623	0.580295	0.290148	−	0.956982i	$-0.406296\pi$
$397$	11.5623	0.580295	0.290148	+	0.956982i	$0.406296\pi$
$398$	−12.2705	−0.615065
$399$	−3.43769	−0.172100
$400$	8.09017	0.404508
$401$	−31.9443	−1.59522	−0.797610	−	0.603173i	$-0.793903\pi$
$401$	−31.9443	−1.59522	−0.797610	+	0.603173i	$0.793903\pi$
$402$	1.85410	0.0924742
$403$	−15.2705	−0.760678
$404$	0.763932	0.0380070
$405$	−27.8885	−1.38579
$406$	14.2918	0.709290
$407$	−22.4164	−1.11114
$408$	−0.562306	−0.0278383
$409$	−6.58359	−0.325538	−0.162769	−	0.986664i	$-0.552042\pi$
$409$	−6.58359	−0.325538	−0.162769	+	0.986664i	$0.552042\pi$
$410$	40.4508	1.99773
$411$	4.41641	0.217845
$412$	11.8541	0.584010
$413$	42.7082	2.10153
$414$	21.3262	1.04813
$415$	−57.6869	−2.83174
$416$	−3.00000	−0.147087
$417$	2.49342	0.122103
$418$	13.8541	0.677626
$419$	16.3262	0.797589	0.398794	−	0.917040i	$-0.369429\pi$
$419$	16.3262	0.797589	0.398794	+	0.917040i	$0.369429\pi$
$420$	−4.14590	−0.202299
$421$	−40.3951	−1.96874	−0.984369	−	0.176119i	$-0.943646\pi$
$421$	−40.3951	−1.96874	−0.984369	+	0.176119i	$0.943646\pi$
$422$	11.1459	0.542574
$423$	24.4377	1.18820
$424$	−9.00000	−0.437079
$425$	11.9098	0.577712
$426$	−2.27051	−0.110007
$427$	13.8541	0.670447
$428$	−8.23607	−0.398105
$429$	−5.29180	−0.255490
$430$	17.2361	0.831197
$431$	−29.2361	−1.40825	−0.704126	−	0.710075i	$-0.748661\pi$
$431$	−29.2361	−1.40825	−0.704126	+	0.710075i	$0.748661\pi$
$432$	2.23607	0.107583
$433$	32.4164	1.55783	0.778917	−	0.627128i	$-0.215770\pi$
$433$	32.4164	1.55783	0.778917	+	0.627128i	$0.215770\pi$
$434$	−15.2705	−0.733008
$435$	−6.58359	−0.315659
$436$	−0.527864	−0.0252801
$437$	22.4164	1.07232
$438$	4.85410	0.231938
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	16.7082	0.796532
$441$	−5.70820	−0.271819
$442$	−4.41641	−0.210067
$443$	13.4721	0.640080	0.320040	−	0.947404i	$-0.396304\pi$
$443$	13.4721	0.640080	0.320040	+	0.947404i	$0.396304\pi$
$444$	−1.85410	−0.0879918
$445$	−28.2148	−1.33751
$446$	−28.4164	−1.34556
$447$	−0.854102	−0.0403976
$448$	−3.00000	−0.141737
$449$	27.3607	1.29123	0.645615	−	0.763663i	$-0.276601\pi$
$449$	27.3607	1.29123	0.645615	+	0.763663i	$0.276601\pi$
$450$	−23.0902	−1.08848
$451$	51.6312	2.43122
$452$	−6.61803	−0.311286
$453$	−5.65248	−0.265576
$454$	7.14590	0.335374
$455$	−32.5623	−1.52654
$456$	1.14590	0.0536616
$457$	−11.9656	−0.559725	−0.279863	−	0.960040i	$-0.590289\pi$
$457$	−11.9656	−0.559725	−0.279863	+	0.960040i	$0.590289\pi$
$458$	−22.5967	−1.05588
$459$	3.29180	0.153648
$460$	27.0344	1.26049
$461$	−4.61803	−0.215083	−0.107542	−	0.994201i	$-0.534298\pi$
$461$	−4.61803	−0.215083	−0.107542	+	0.994201i	$0.534298\pi$
$462$	−5.29180	−0.246197
$463$	−18.3607	−0.853293	−0.426647	−	0.904418i	$-0.640305\pi$
$463$	−18.3607	−0.853293	−0.426647	+	0.904418i	$0.640305\pi$
$464$	−4.76393	−0.221160
$465$	7.03444	0.326214
$466$	10.8541	0.502807
$467$	7.20163	0.333252	0.166626	−	0.986020i	$-0.446713\pi$
$467$	7.20163	0.333252	0.166626	+	0.986020i	$0.446713\pi$
$468$	8.56231	0.395793
$469$	14.5623	0.672425
$470$	30.9787	1.42894
$471$	1.18034	0.0543872
$472$	−14.2361	−0.655268
$473$	22.0000	1.01156
$474$	−1.85410	−0.0851617
$475$	−24.2705	−1.11361
$476$	−4.41641	−0.202426
$477$	25.6869	1.17612
$478$	−9.70820	−0.444043
$479$	−7.14590	−0.326504	−0.163252	−	0.986584i	$-0.552198\pi$
$479$	−7.14590	−0.326504	−0.163252	+	0.986584i	$0.552198\pi$
$480$	1.38197	0.0630778
$481$	−14.5623	−0.663984
$482$	16.4164	0.747747
$483$	−8.56231	−0.389598
$484$	10.3262	0.469374
$485$	−65.9787	−2.99594
$486$	−9.65248	−0.437845
$487$	−19.8885	−0.901236	−0.450618	−	0.892717i	$-0.648796\pi$
$487$	−19.8885	−0.901236	−0.450618	+	0.892717i	$0.648796\pi$
$488$	−4.61803	−0.209049
$489$	2.23607	0.101118
$490$	−7.23607	−0.326892
$491$	27.2705	1.23070	0.615350	−	0.788254i	$-0.289015\pi$
$491$	27.2705	1.23070	0.615350	+	0.788254i	$0.289015\pi$
$492$	4.27051	0.192529
$493$	−7.01316	−0.315857
$494$	9.00000	0.404929
$495$	−47.6869	−2.14337
$496$	5.09017	0.228555
$497$	−17.8328	−0.799911
$498$	−6.09017	−0.272907
$499$	25.6525	1.14836	0.574181	−	0.818728i	$-0.305320\pi$
$499$	25.6525	1.14836	0.574181	+	0.818728i	$0.305320\pi$
$500$	−11.1803	−0.500000
$501$	0.291796	0.0130365
$502$	−16.9443	−0.756260
$503$	18.7082	0.834158	0.417079	−	0.908870i	$-0.363054\pi$
$503$	18.7082	0.834158	0.417079	+	0.908870i	$0.363054\pi$
$504$	8.56231	0.381395
$505$	−2.76393	−0.122993
$506$	34.5066	1.53400
$507$	1.52786	0.0678548
$508$	14.4164	0.639625
$509$	−1.58359	−0.0701915	−0.0350957	−	0.999384i	$-0.511174\pi$
$509$	−1.58359	−0.0701915	−0.0350957	+	0.999384i	$0.511174\pi$
$510$	2.03444	0.0900866
$511$	38.1246	1.68653
$512$	1.00000	0.0441942
$513$	−6.70820	−0.296174
$514$	−28.4164	−1.25339
$515$	−42.8885	−1.88990
$516$	1.81966	0.0801061
$517$	39.5410	1.73901
$518$	−14.5623	−0.639831
$519$	8.90983	0.391098
$520$	10.8541	0.475984
$521$	18.6525	0.817180	0.408590	−	0.912718i	$-0.366021\pi$
$521$	18.6525	0.817180	0.408590	+	0.912718i	$0.366021\pi$
$522$	13.5967	0.595113
$523$	23.0344	1.00723	0.503613	−	0.863929i	$-0.332004\pi$
$523$	23.0344	1.00723	0.503613	+	0.863929i	$0.332004\pi$
$524$	19.3820	0.846705
$525$	9.27051	0.404598
$526$	−26.3607	−1.14938
$527$	7.49342	0.326419
$528$	1.76393	0.0767653
$529$	32.8328	1.42751
$530$	32.5623	1.41442
$531$	40.6312	1.76324
$532$	9.00000	0.390199
$533$	33.5410	1.45282
$534$	−2.97871	−0.128902
$535$	29.7984	1.28830
$536$	−4.85410	−0.209665
$537$	3.43769	0.148347
$538$	14.5623	0.627826
$539$	−9.23607	−0.397826
$540$	−8.09017	−0.348145
$541$	−18.2918	−0.786426	−0.393213	−	0.919447i	$-0.628636\pi$
$541$	−18.2918	−0.786426	−0.393213	+	0.919447i	$0.628636\pi$
$542$	−4.38197	−0.188222
$543$	6.87539	0.295051
$544$	1.47214	0.0631173
$545$	1.90983	0.0818081
$546$	−3.43769	−0.147120
$547$	−7.94427	−0.339673	−0.169836	−	0.985472i	$-0.554324\pi$
$547$	−7.94427	−0.339673	−0.169836	+	0.985472i	$0.554324\pi$
$548$	−11.5623	−0.493917
$549$	13.1803	0.562523
$550$	−37.3607	−1.59306
$551$	14.2918	0.608851
$552$	2.85410	0.121479
$553$	−14.5623	−0.619252
$554$	−0.437694	−0.0185958
$555$	6.70820	0.284747
$556$	−6.52786	−0.276843
$557$	−12.2361	−0.518459	−0.259229	−	0.965816i	$-0.583469\pi$
$557$	−12.2361	−0.518459	−0.259229	+	0.965816i	$0.583469\pi$
$558$	−14.5279	−0.615014
$559$	14.2918	0.604479
$560$	10.8541	0.458670
$561$	2.59675	0.109635
$562$	19.3607	0.816681
$563$	0.270510	0.0114006	0.00570032	−	0.999984i	$-0.498186\pi$
$563$	0.270510	0.0114006	0.00570032	+	0.999984i	$0.498186\pi$
$564$	3.27051	0.137713
$565$	23.9443	1.00734
$566$	−2.70820	−0.113834
$567$	−23.1246	−0.971142
$568$	5.94427	0.249416
$569$	−43.0344	−1.80410	−0.902049	−	0.431634i	$-0.857937\pi$
$569$	−43.0344	−1.80410	−0.902049	+	0.431634i	$0.857937\pi$
$570$	−4.14590	−0.173653
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	13.8541	0.579269
$573$	8.54102	0.356806
$574$	33.5410	1.39998
$575$	−60.4508	−2.52097
$576$	−2.85410	−0.118921
$577$	22.4164	0.933207	0.466604	−	0.884467i	$-0.345477\pi$
$577$	22.4164	0.933207	0.466604	+	0.884467i	$0.345477\pi$
$578$	−14.8328	−0.616964
$579$	−3.76393	−0.156424
$580$	17.2361	0.715689
$581$	−47.8328	−1.98444
$582$	−6.96556	−0.288732
$583$	41.5623	1.72133
$584$	−12.7082	−0.525869
$585$	−30.9787	−1.28081
$586$	−20.9098	−0.863777
$587$	−23.0557	−0.951612	−0.475806	−	0.879550i	$-0.657844\pi$
$587$	−23.0557	−0.951612	−0.475806	+	0.879550i	$0.657844\pi$
$588$	−0.763932	−0.0315040
$589$	−15.2705	−0.629210
$590$	51.5066	2.12049
$591$	−8.32624	−0.342496
$592$	4.85410	0.199502
$593$	−21.2148	−0.871187	−0.435593	−	0.900144i	$-0.643461\pi$
$593$	−21.2148	−0.871187	−0.435593	+	0.900144i	$0.643461\pi$
$594$	−10.3262	−0.423691
$595$	15.9787	0.655063
$596$	2.23607	0.0915929
$597$	4.68692	0.191823
$598$	22.4164	0.916675
$599$	−22.8541	−0.933793	−0.466897	−	0.884312i	$-0.654628\pi$
$599$	−22.8541	−0.933793	−0.466897	+	0.884312i	$0.654628\pi$
$600$	−3.09017	−0.126156
$601$	16.2705	0.663688	0.331844	−	0.943334i	$-0.392329\pi$
$601$	16.2705	0.663688	0.331844	+	0.943334i	$0.392329\pi$
$602$	14.2918	0.582490
$603$	13.8541	0.564183
$604$	14.7984	0.602137
$605$	−37.3607	−1.51893
$606$	−0.291796	−0.0118534
$607$	−20.4164	−0.828676	−0.414338	−	0.910123i	$-0.635987\pi$
$607$	−20.4164	−0.828676	−0.414338	+	0.910123i	$0.635987\pi$
$608$	−3.00000	−0.121666
$609$	−5.45898	−0.221209
$610$	16.7082	0.676495
$611$	25.6869	1.03918
$612$	−4.20163	−0.169841
$613$	−3.41641	−0.137987	−0.0689937	−	0.997617i	$-0.521979\pi$
$613$	−3.41641	−0.137987	−0.0689937	+	0.997617i	$0.521979\pi$
$614$	15.0902	0.608990
$615$	−15.4508	−0.623038
$616$	13.8541	0.558198
$617$	5.67376	0.228417	0.114209	−	0.993457i	$-0.463567\pi$
$617$	5.67376	0.228417	0.114209	+	0.993457i	$0.463567\pi$
$618$	−4.52786	−0.182137
$619$	38.1459	1.53321	0.766607	−	0.642117i	$-0.221944\pi$
$619$	38.1459	1.53321	0.766607	+	0.642117i	$0.221944\pi$
$620$	−18.4164	−0.739621
$621$	−16.7082	−0.670477
$622$	29.2361	1.17226
$623$	−23.3951	−0.937306
$624$	1.14590	0.0458726
$625$	0	0
$626$	−9.74265	−0.389394
$627$	−5.29180	−0.211334
$628$	−3.09017	−0.123311
$629$	7.14590	0.284926
$630$	−30.9787	−1.23422
$631$	−18.2705	−0.727338	−0.363669	−	0.931528i	$-0.618476\pi$
$631$	−18.2705	−0.727338	−0.363669	+	0.931528i	$0.618476\pi$
$632$	4.85410	0.193086
$633$	−4.25735	−0.169215
$634$	−8.38197	−0.332890
$635$	−52.1591	−2.06987
$636$	3.43769	0.136313
$637$	−6.00000	−0.237729
$638$	22.0000	0.870988
$639$	−16.9656	−0.671147
$640$	−3.61803	−0.143015
$641$	41.1803	1.62653	0.813263	−	0.581897i	$-0.197689\pi$
$641$	41.1803	1.62653	0.813263	+	0.581897i	$0.197689\pi$
$642$	3.14590	0.124159
$643$	32.3050	1.27398	0.636991	−	0.770871i	$-0.280179\pi$
$643$	32.3050	1.27398	0.636991	+	0.770871i	$0.280179\pi$
$644$	22.4164	0.883330
$645$	−6.58359	−0.259229
$646$	−4.41641	−0.173761
$647$	18.3262	0.720479	0.360239	−	0.932860i	$-0.382695\pi$
$647$	18.3262	0.720479	0.360239	+	0.932860i	$0.382695\pi$
$648$	7.70820	0.302807
$649$	65.7426	2.58062
$650$	−24.2705	−0.951968
$651$	5.83282	0.228606
$652$	−5.85410	−0.229264
$653$	6.90983	0.270403	0.135201	−	0.990818i	$-0.456832\pi$
$653$	6.90983	0.270403	0.135201	+	0.990818i	$0.456832\pi$
$654$	0.201626	0.00788421
$655$	−70.1246	−2.74000
$656$	−11.1803	−0.436519
$657$	36.2705	1.41505
$658$	25.6869	1.00138
$659$	−6.47214	−0.252119	−0.126059	−	0.992023i	$-0.540233\pi$
$659$	−6.47214	−0.252119	−0.126059	+	0.992023i	$0.540233\pi$
$660$	−6.38197	−0.248418
$661$	32.2918	1.25601	0.628003	−	0.778211i	$-0.283873\pi$
$661$	32.2918	1.25601	0.628003	+	0.778211i	$0.283873\pi$
$662$	−25.5623	−0.993507
$663$	1.68692	0.0655145
$664$	15.9443	0.618758
$665$	−32.5623	−1.26271
$666$	−13.8541	−0.536836
$667$	35.5967	1.37831
$668$	−0.763932	−0.0295574
$669$	10.8541	0.419644
$670$	17.5623	0.678491
$671$	21.3262	0.823290
$672$	1.14590	0.0442040
$673$	48.5410	1.87112	0.935559	−	0.353169i	$-0.114896\pi$
$673$	48.5410	1.87112	0.935559	+	0.353169i	$0.114896\pi$
$674$	2.38197	0.0917499
$675$	18.0902	0.696291
$676$	−4.00000	−0.153846
$677$	12.8885	0.495347	0.247673	−	0.968844i	$-0.420334\pi$
$677$	12.8885	0.495347	0.247673	+	0.968844i	$0.420334\pi$
$678$	2.52786	0.0970820
$679$	−54.7082	−2.09951
$680$	−5.32624	−0.204252
$681$	−2.72949	−0.104594
$682$	−23.5066	−0.900113
$683$	11.6180	0.444552	0.222276	−	0.974984i	$-0.428651\pi$
$683$	11.6180	0.444552	0.222276	+	0.974984i	$0.428651\pi$
$684$	8.56231	0.327388
$685$	41.8328	1.59835
$686$	15.0000	0.572703
$687$	8.63119	0.329300
$688$	−4.76393	−0.181623
$689$	27.0000	1.02862
$690$	−10.3262	−0.393113
$691$	−3.90983	−0.148737	−0.0743685	−	0.997231i	$-0.523694\pi$
$691$	−3.90983	−0.148737	−0.0743685	+	0.997231i	$0.523694\pi$
$692$	−23.3262	−0.886731
$693$	−39.5410	−1.50204
$694$	−24.3262	−0.923411
$695$	23.6180	0.895883
$696$	1.81966	0.0689740
$697$	−16.4590	−0.623428
$698$	−9.29180	−0.351700
$699$	−4.14590	−0.156812
$700$	−24.2705	−0.917339
$701$	2.21478	0.0836512	0.0418256	−	0.999125i	$-0.486683\pi$
$701$	2.21478	0.0836512	0.0418256	+	0.999125i	$0.486683\pi$
$702$	−6.70820	−0.253185
$703$	−14.5623	−0.549228
$704$	−4.61803	−0.174049
$705$	−11.8328	−0.445650
$706$	−29.1803	−1.09822
$707$	−2.29180	−0.0861919
$708$	5.43769	0.204361
$709$	−34.9443	−1.31236	−0.656180	−	0.754605i	$-0.727829\pi$
$709$	−34.9443	−1.31236	−0.656180	+	0.754605i	$0.727829\pi$
$710$	−21.5066	−0.807127
$711$	−13.8541	−0.519569
$712$	7.79837	0.292256
$713$	−38.0344	−1.42440
$714$	1.68692	0.0631313
$715$	−50.1246	−1.87455
$716$	−9.00000	−0.336346
$717$	3.70820	0.138485
$718$	34.5967	1.29114
$719$	−32.9443	−1.22861	−0.614307	−	0.789067i	$-0.710564\pi$
$719$	−32.9443	−1.22861	−0.614307	+	0.789067i	$0.710564\pi$
$720$	10.3262	0.384836
$721$	−35.5623	−1.32441
$722$	−10.0000	−0.372161
$723$	−6.27051	−0.233203
$724$	−18.0000	−0.668965
$725$	−38.5410	−1.43138
$726$	−3.94427	−0.146386
$727$	6.47214	0.240038	0.120019	−	0.992772i	$-0.461704\pi$
$727$	6.47214	0.240038	0.120019	+	0.992772i	$0.461704\pi$
$728$	9.00000	0.333562
$729$	−19.4377	−0.719915
$730$	45.9787	1.70175
$731$	−7.01316	−0.259391
$732$	1.76393	0.0651968
$733$	−36.8885	−1.36251	−0.681255	−	0.732046i	$-0.738565\pi$
$733$	−36.8885	−1.36251	−0.681255	+	0.732046i	$0.738565\pi$
$734$	−30.8885	−1.14012
$735$	2.76393	0.101949
$736$	−7.47214	−0.275427
$737$	22.4164	0.825719
$738$	31.9098	1.17462
$739$	−29.5279	−1.08620	−0.543100	−	0.839668i	$-0.682750\pi$
$739$	−29.5279	−1.08620	−0.543100	+	0.839668i	$0.682750\pi$
$740$	−17.5623	−0.645603
$741$	−3.43769	−0.126287
$742$	27.0000	0.991201
$743$	−17.2361	−0.632330	−0.316165	−	0.948704i	$-0.602395\pi$
$743$	−17.2361	−0.632330	−0.316165	+	0.948704i	$0.602395\pi$
$744$	−1.94427	−0.0712805
$745$	−8.09017	−0.296401
$746$	−12.0000	−0.439351
$747$	−45.5066	−1.66500
$748$	−6.79837	−0.248573
$749$	24.7082	0.902818
$750$	4.27051	0.155937
$751$	−3.43769	−0.125443	−0.0627216	−	0.998031i	$-0.519978\pi$
$751$	−3.43769	−0.125443	−0.0627216	+	0.998031i	$0.519978\pi$
$752$	−8.56231	−0.312235
$753$	6.47214	0.235858
$754$	14.2918	0.520477
$755$	−53.5410	−1.94856
$756$	−6.70820	−0.243975
$757$	−1.70820	−0.0620857	−0.0310429	−	0.999518i	$-0.509883\pi$
$757$	−1.70820	−0.0620857	−0.0310429	+	0.999518i	$0.509883\pi$
$758$	−30.8328	−1.11990
$759$	−13.1803	−0.478416
$760$	10.8541	0.393720
$761$	26.0344	0.943748	0.471874	−	0.881666i	$-0.343578\pi$
$761$	26.0344	0.943748	0.471874	+	0.881666i	$0.343578\pi$
$762$	−5.50658	−0.199482
$763$	1.58359	0.0573299
$764$	−22.3607	−0.808981
$765$	15.2016	0.549616
$766$	−15.7082	−0.567560
$767$	42.7082	1.54210
$768$	−0.381966	−0.0137830
$769$	13.4164	0.483808	0.241904	−	0.970300i	$-0.422228\pi$
$769$	13.4164	0.483808	0.241904	+	0.970300i	$0.422228\pi$
$770$	−50.1246	−1.80637
$771$	10.8541	0.390901
$772$	9.85410	0.354657
$773$	−36.7639	−1.32231	−0.661153	−	0.750251i	$-0.729933\pi$
$773$	−36.7639	−1.32231	−0.661153	+	0.750251i	$0.729933\pi$
$774$	13.5967	0.488725
$775$	41.1803	1.47924
$776$	18.2361	0.654637
$777$	5.56231	0.199547
$778$	−22.7639	−0.816127
$779$	33.5410	1.20173
$780$	−4.14590	−0.148447
$781$	−27.4508	−0.982269
$782$	−11.0000	−0.393359
$783$	−10.6525	−0.380688
$784$	2.00000	0.0714286
$785$	11.1803	0.399043
$786$	−7.40325	−0.264065
$787$	9.94427	0.354475	0.177238	−	0.984168i	$-0.443284\pi$
$787$	9.94427	0.354475	0.177238	+	0.984168i	$0.443284\pi$
$788$	21.7984	0.776535
$789$	10.0689	0.358462
$790$	−17.5623	−0.624839
$791$	19.8541	0.705931
$792$	13.1803	0.468343
$793$	13.8541	0.491974
$794$	11.5623	0.410331
$795$	−12.4377	−0.441120
$796$	−12.2705	−0.434917
$797$	−13.0557	−0.462458	−0.231229	−	0.972899i	$-0.574275\pi$
$797$	−13.0557	−0.462458	−0.231229	+	0.972899i	$0.574275\pi$
$798$	−3.43769	−0.121693
$799$	−12.6049	−0.445929
$800$	8.09017	0.286031
$801$	−22.2574	−0.786425
$802$	−31.9443	−1.12799
$803$	58.6869	2.07102
$804$	1.85410	0.0653891
$805$	−81.1033	−2.85852
$806$	−15.2705	−0.537881
$807$	−5.56231	−0.195802
$808$	0.763932	0.0268750
$809$	37.8541	1.33088	0.665440	−	0.746452i	$-0.268244\pi$
$809$	37.8541	1.33088	0.665440	+	0.746452i	$0.268244\pi$
$810$	−27.8885	−0.979904
$811$	−5.05573	−0.177531	−0.0887653	−	0.996053i	$-0.528292\pi$
$811$	−5.05573	−0.177531	−0.0887653	+	0.996053i	$0.528292\pi$
$812$	14.2918	0.501544
$813$	1.67376	0.0587014
$814$	−22.4164	−0.785695
$815$	21.1803	0.741915
$816$	−0.562306	−0.0196846
$817$	14.2918	0.500007
$818$	−6.58359	−0.230190
$819$	−25.6869	−0.897574
$820$	40.4508	1.41260
$821$	−39.5066	−1.37879	−0.689395	−	0.724386i	$-0.742123\pi$
$821$	−39.5066	−1.37879	−0.689395	+	0.724386i	$0.742123\pi$
$822$	4.41641	0.154040
$823$	−7.41641	−0.258520	−0.129260	−	0.991611i	$-0.541260\pi$
$823$	−7.41641	−0.258520	−0.129260	+	0.991611i	$0.541260\pi$
$824$	11.8541	0.412957
$825$	14.2705	0.496835
$826$	42.7082	1.48601
$827$	23.0557	0.801726	0.400863	−	0.916138i	$-0.368710\pi$
$827$	23.0557	0.801726	0.400863	+	0.916138i	$0.368710\pi$
$828$	21.3262	0.741138
$829$	−20.7082	−0.719226	−0.359613	−	0.933102i	$-0.617091\pi$
$829$	−20.7082	−0.719226	−0.359613	+	0.933102i	$0.617091\pi$
$830$	−57.6869	−2.00234
$831$	0.167184	0.00579956
$832$	−3.00000	−0.104006
$833$	2.94427	0.102013
$834$	2.49342	0.0863401
$835$	2.76393	0.0956498
$836$	13.8541	0.479154
$837$	11.3820	0.393418
$838$	16.3262	0.563981
$839$	46.6525	1.61062	0.805311	−	0.592852i	$-0.201998\pi$
$839$	46.6525	1.61062	0.805311	+	0.592852i	$0.201998\pi$
$840$	−4.14590	−0.143047
$841$	−6.30495	−0.217412
$842$	−40.3951	−1.39211
$843$	−7.39512	−0.254702
$844$	11.1459	0.383658
$845$	14.4721	0.497857
$846$	24.4377	0.840185
$847$	−30.9787	−1.06444
$848$	−9.00000	−0.309061
$849$	1.03444	0.0355020
$850$	11.9098	0.408504
$851$	−36.2705	−1.24334
$852$	−2.27051	−0.0777864
$853$	33.5623	1.14915	0.574576	−	0.818451i	$-0.305167\pi$
$853$	33.5623	1.14915	0.574576	+	0.818451i	$0.305167\pi$
$854$	13.8541	0.474078
$855$	−30.9787	−1.05945
$856$	−8.23607	−0.281503
$857$	−20.9098	−0.714266	−0.357133	−	0.934054i	$-0.616246\pi$
$857$	−20.9098	−0.714266	−0.357133	+	0.934054i	$0.616246\pi$
$858$	−5.29180	−0.180659
$859$	33.9443	1.15816	0.579082	−	0.815269i	$-0.303411\pi$
$859$	33.9443	1.15816	0.579082	+	0.815269i	$0.303411\pi$
$860$	17.2361	0.587745
$861$	−12.8115	−0.436616
$862$	−29.2361	−0.995784
$863$	32.0132	1.08974	0.544870	−	0.838520i	$-0.316579\pi$
$863$	32.0132	1.08974	0.544870	+	0.838520i	$0.316579\pi$
$864$	2.23607	0.0760726
$865$	84.3951	2.86952
$866$	32.4164	1.10155
$867$	5.66563	0.192415
$868$	−15.2705	−0.518315
$869$	−22.4164	−0.760425
$870$	−6.58359	−0.223205
$871$	14.5623	0.493425
$872$	−0.527864	−0.0178757
$873$	−52.0476	−1.76154
$874$	22.4164	0.758246
$875$	33.5410	1.13389
$876$	4.85410	0.164005
$877$	5.12461	0.173046	0.0865229	−	0.996250i	$-0.472424\pi$
$877$	5.12461	0.173046	0.0865229	+	0.996250i	$0.472424\pi$
$878$	−13.0000	−0.438729
$879$	7.98684	0.269390
$880$	16.7082	0.563233
$881$	31.7214	1.06872	0.534360	−	0.845257i	$-0.320553\pi$
$881$	31.7214	1.06872	0.534360	+	0.845257i	$0.320553\pi$
$882$	−5.70820	−0.192205
$883$	−21.9656	−0.739200	−0.369600	−	0.929191i	$-0.620505\pi$
$883$	−21.9656	−0.739200	−0.369600	+	0.929191i	$0.620505\pi$
$884$	−4.41641	−0.148540
$885$	−19.6738	−0.661326
$886$	13.4721	0.452605
$887$	9.70820	0.325970	0.162985	−	0.986629i	$-0.447888\pi$
$887$	9.70820	0.325970	0.162985	+	0.986629i	$0.447888\pi$
$888$	−1.85410	−0.0622196
$889$	−43.2492	−1.45053
$890$	−28.2148	−0.945762
$891$	−35.5967	−1.19254
$892$	−28.4164	−0.951452
$893$	25.6869	0.859580
$894$	−0.854102	−0.0285654
$895$	32.5623	1.08844
$896$	−3.00000	−0.100223
$897$	−8.56231	−0.285887
$898$	27.3607	0.913038
$899$	−24.2492	−0.808757
$900$	−23.0902	−0.769672
$901$	−13.2492	−0.441396
$902$	51.6312	1.71913
$903$	−5.45898	−0.181663
$904$	−6.61803	−0.220113
$905$	65.1246	2.16482
$906$	−5.65248	−0.187791
$907$	5.41641	0.179849	0.0899244	−	0.995949i	$-0.471337\pi$
$907$	5.41641	0.179849	0.0899244	+	0.995949i	$0.471337\pi$
$908$	7.14590	0.237145
$909$	−2.18034	−0.0723173
$910$	−32.5623	−1.07943
$911$	−11.5623	−0.383076	−0.191538	−	0.981485i	$-0.561348\pi$
$911$	−11.5623	−0.383076	−0.191538	+	0.981485i	$0.561348\pi$
$912$	1.14590	0.0379445
$913$	−73.6312	−2.43684
$914$	−11.9656	−0.395785
$915$	−6.38197	−0.210981
$916$	−22.5967	−0.746618
$917$	−58.1459	−1.92015
$918$	3.29180	0.108645
$919$	−45.7214	−1.50821	−0.754104	−	0.656755i	$-0.771929\pi$
$919$	−45.7214	−1.50821	−0.754104	+	0.656755i	$0.771929\pi$
$920$	27.0344	0.891299
$921$	−5.76393	−0.189928
$922$	−4.61803	−0.152087
$923$	−17.8328	−0.586974
$924$	−5.29180	−0.174087
$925$	39.2705	1.29121
$926$	−18.3607	−0.603369
$927$	−33.8328	−1.11122
$928$	−4.76393	−0.156384
$929$	−11.8328	−0.388222	−0.194111	−	0.980980i	$-0.562182\pi$
$929$	−11.8328	−0.388222	−0.194111	+	0.980980i	$0.562182\pi$
$930$	7.03444	0.230668
$931$	−6.00000	−0.196642
$932$	10.8541	0.355538
$933$	−11.1672	−0.365597
$934$	7.20163	0.235644
$935$	24.5967	0.804400
$936$	8.56231	0.279868
$937$	44.8328	1.46462	0.732312	−	0.680969i	$-0.238441\pi$
$937$	44.8328	1.46462	0.732312	+	0.680969i	$0.238441\pi$
$938$	14.5623	0.475476
$939$	3.72136	0.121442
$940$	30.9787	1.01041
$941$	−22.7984	−0.743206	−0.371603	−	0.928392i	$-0.621192\pi$
$941$	−22.7984	−0.743206	−0.371603	+	0.928392i	$0.621192\pi$
$942$	1.18034	0.0384576
$943$	83.5410	2.72047
$944$	−14.2361	−0.463345
$945$	24.2705	0.789520
$946$	22.0000	0.715282
$947$	−40.7984	−1.32577	−0.662885	−	0.748722i	$-0.730668\pi$
$947$	−40.7984	−1.32577	−0.662885	+	0.748722i	$0.730668\pi$
$948$	−1.85410	−0.0602184
$949$	38.1246	1.23758
$950$	−24.2705	−0.787439
$951$	3.20163	0.103820
$952$	−4.41641	−0.143137
$953$	−57.6525	−1.86755	−0.933773	−	0.357865i	$-0.883505\pi$
$953$	−57.6525	−1.86755	−0.933773	+	0.357865i	$0.883505\pi$
$954$	25.6869	0.831645
$955$	80.9017	2.61792
$956$	−9.70820	−0.313986
$957$	−8.40325	−0.271639
$958$	−7.14590	−0.230873
$959$	34.6869	1.12010
$960$	1.38197	0.0446028
$961$	−5.09017	−0.164199
$962$	−14.5623	−0.469508
$963$	23.5066	0.757489
$964$	16.4164	0.528737
$965$	−35.6525	−1.14769
$966$	−8.56231	−0.275488
$967$	5.16718	0.166165	0.0830827	−	0.996543i	$-0.473523\pi$
$967$	5.16718	0.166165	0.0830827	+	0.996543i	$0.473523\pi$
$968$	10.3262	0.331898
$969$	1.68692	0.0541916
$970$	−65.9787	−2.11845
$971$	49.2492	1.58048	0.790241	−	0.612796i	$-0.209955\pi$
$971$	49.2492	1.58048	0.790241	+	0.612796i	$0.209955\pi$
$972$	−9.65248	−0.309603
$973$	19.5836	0.627821
$974$	−19.8885	−0.637270
$975$	9.27051	0.296894
$976$	−4.61803	−0.147820
$977$	−18.2361	−0.583424	−0.291712	−	0.956506i	$-0.594225\pi$
$977$	−18.2361	−0.583424	−0.291712	+	0.956506i	$0.594225\pi$
$978$	2.23607	0.0715016
$979$	−36.0132	−1.15099
$980$	−7.23607	−0.231148
$981$	1.50658	0.0481013
$982$	27.2705	0.870237
$983$	31.7426	1.01243	0.506217	−	0.862406i	$-0.331044\pi$
$983$	31.7426	1.01243	0.506217	+	0.862406i	$0.331044\pi$
$984$	4.27051	0.136139
$985$	−78.8673	−2.51292
$986$	−7.01316	−0.223344
$987$	−9.81153	−0.312304
$988$	9.00000	0.286328
$989$	35.5967	1.13191
$990$	−47.6869	−1.51559
$991$	53.3050	1.69329	0.846644	−	0.532160i	$-0.178620\pi$
$991$	53.3050	1.69329	0.846644	+	0.532160i	$0.178620\pi$
$992$	5.09017	0.161613
$993$	9.76393	0.309849
$994$	−17.8328	−0.565623
$995$	44.3951	1.40742
$996$	−6.09017	−0.192974
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	25.6525	0.812015
$999$	10.8541	0.343409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.a.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.381966 q^{3} +1.00000 q^{4} -3.61803 q^{5} -0.381966 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.381966 q^{3} +1.00000 q^{4} -3.61803 q^{5} -0.381966 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} -3.61803 q^{10} -4.61803 q^{11} -0.381966 q^{12} -3.00000 q^{13} -3.00000 q^{14} +1.38197 q^{15} +1.00000 q^{16} +1.47214 q^{17} -2.85410 q^{18} -3.00000 q^{19} -3.61803 q^{20} +1.14590 q^{21} -4.61803 q^{22} -7.47214 q^{23} -0.381966 q^{24} +8.09017 q^{25} -3.00000 q^{26} +2.23607 q^{27} -3.00000 q^{28} -4.76393 q^{29} +1.38197 q^{30} +5.09017 q^{31} +1.00000 q^{32} +1.76393 q^{33} +1.47214 q^{34} +10.8541 q^{35} -2.85410 q^{36} +4.85410 q^{37} -3.00000 q^{38} +1.14590 q^{39} -3.61803 q^{40} -11.1803 q^{41} +1.14590 q^{42} -4.76393 q^{43} -4.61803 q^{44} +10.3262 q^{45} -7.47214 q^{46} -8.56231 q^{47} -0.381966 q^{48} +2.00000 q^{49} +8.09017 q^{50} -0.562306 q^{51} -3.00000 q^{52} -9.00000 q^{53} +2.23607 q^{54} +16.7082 q^{55} -3.00000 q^{56} +1.14590 q^{57} -4.76393 q^{58} -14.2361 q^{59} +1.38197 q^{60} -4.61803 q^{61} +5.09017 q^{62} +8.56231 q^{63} +1.00000 q^{64} +10.8541 q^{65} +1.76393 q^{66} -4.85410 q^{67} +1.47214 q^{68} +2.85410 q^{69} +10.8541 q^{70} +5.94427 q^{71} -2.85410 q^{72} -12.7082 q^{73} +4.85410 q^{74} -3.09017 q^{75} -3.00000 q^{76} +13.8541 q^{77} +1.14590 q^{78} +4.85410 q^{79} -3.61803 q^{80} +7.70820 q^{81} -11.1803 q^{82} +15.9443 q^{83} +1.14590 q^{84} -5.32624 q^{85} -4.76393 q^{86} +1.81966 q^{87} -4.61803 q^{88} +7.79837 q^{89} +10.3262 q^{90} +9.00000 q^{91} -7.47214 q^{92} -1.94427 q^{93} -8.56231 q^{94} +10.8541 q^{95} -0.381966 q^{96} +18.2361 q^{97} +2.00000 q^{98} +13.1803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 5 * q^5 - 3 * q^6 - 6 * q^7 + 2 * q^8 + q^9	\( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 2 q^{8} + q^{9} - 5 q^{10} - 7 q^{11} - 3 q^{12} - 6 q^{13} - 6 q^{14} + 5 q^{15} + 2 q^{16} - 6 q^{17} + q^{18} - 6 q^{19} - 5 q^{20} + 9 q^{21} - 7 q^{22} - 6 q^{23} - 3 q^{24} + 5 q^{25} - 6 q^{26} - 6 q^{28} - 14 q^{29} + 5 q^{30} - q^{31} + 2 q^{32} + 8 q^{33} - 6 q^{34} + 15 q^{35} + q^{36} + 3 q^{37} - 6 q^{38} + 9 q^{39} - 5 q^{40} + 9 q^{42} - 14 q^{43} - 7 q^{44} + 5 q^{45} - 6 q^{46} + 3 q^{47} - 3 q^{48} + 4 q^{49} + 5 q^{50} + 19 q^{51} - 6 q^{52} - 18 q^{53} + 20 q^{55} - 6 q^{56} + 9 q^{57} - 14 q^{58} - 24 q^{59} + 5 q^{60} - 7 q^{61} - q^{62} - 3 q^{63} + 2 q^{64} + 15 q^{65} + 8 q^{66} - 3 q^{67} - 6 q^{68} - q^{69} + 15 q^{70} - 6 q^{71} + q^{72} - 12 q^{73} + 3 q^{74} + 5 q^{75} - 6 q^{76} + 21 q^{77} + 9 q^{78} + 3 q^{79} - 5 q^{80} + 2 q^{81} + 14 q^{83} + 9 q^{84} + 5 q^{85} - 14 q^{86} + 26 q^{87} - 7 q^{88} - 9 q^{89} + 5 q^{90} + 18 q^{91} - 6 q^{92} + 14 q^{93} + 3 q^{94} + 15 q^{95} - 3 q^{96} + 32 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 5 * q^5 - 3 * q^6 - 6 * q^7 + 2 * q^8 + q^9 - 5 * q^10 - 7 * q^11 - 3 * q^12 - 6 * q^13 - 6 * q^14 + 5 * q^15 + 2 * q^16 - 6 * q^17 + q^18 - 6 * q^19 - 5 * q^20 + 9 * q^21 - 7 * q^22 - 6 * q^23 - 3 * q^24 + 5 * q^25 - 6 * q^26 - 6 * q^28 - 14 * q^29 + 5 * q^30 - q^31 + 2 * q^32 + 8 * q^33 - 6 * q^34 + 15 * q^35 + q^36 + 3 * q^37 - 6 * q^38 + 9 * q^39 - 5 * q^40 + 9 * q^42 - 14 * q^43 - 7 * q^44 + 5 * q^45 - 6 * q^46 + 3 * q^47 - 3 * q^48 + 4 * q^49 + 5 * q^50 + 19 * q^51 - 6 * q^52 - 18 * q^53 + 20 * q^55 - 6 * q^56 + 9 * q^57 - 14 * q^58 - 24 * q^59 + 5 * q^60 - 7 * q^61 - q^62 - 3 * q^63 + 2 * q^64 + 15 * q^65 + 8 * q^66 - 3 * q^67 - 6 * q^68 - q^69 + 15 * q^70 - 6 * q^71 + q^72 - 12 * q^73 + 3 * q^74 + 5 * q^75 - 6 * q^76 + 21 * q^77 + 9 * q^78 + 3 * q^79 - 5 * q^80 + 2 * q^81 + 14 * q^83 + 9 * q^84 + 5 * q^85 - 14 * q^86 + 26 * q^87 - 7 * q^88 - 9 * q^89 + 5 * q^90 + 18 * q^91 - 6 * q^92 + 14 * q^93 + 3 * q^94 + 15 * q^95 - 3 * q^96 + 32 * q^97 + 4 * q^98 + 4 * q^99

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