LMFDB - Embedded newform 845.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$1$
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:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} +2.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} +1.38197 q^{6} -4.23607 q^{7} -2.23607 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +2.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} +1.38197 q^{6} -4.23607 q^{7} -2.23607 q^{8} +2.00000 q^{9} -0.618034 q^{10} +0.236068 q^{11} -3.61803 q^{12} -2.61803 q^{14} -2.23607 q^{15} +1.85410 q^{16} -3.47214 q^{17} +1.23607 q^{18} -4.23607 q^{19} +1.61803 q^{20} -9.47214 q^{21} +0.145898 q^{22} +3.76393 q^{23} -5.00000 q^{24} +1.00000 q^{25} -2.23607 q^{27} +6.85410 q^{28} -7.47214 q^{29} -1.38197 q^{30} +5.61803 q^{32} +0.527864 q^{33} -2.14590 q^{34} +4.23607 q^{35} -3.23607 q^{36} -3.00000 q^{37} -2.61803 q^{38} +2.23607 q^{40} -11.9443 q^{41} -5.85410 q^{42} -6.23607 q^{43} -0.381966 q^{44} -2.00000 q^{45} +2.32624 q^{46} +4.94427 q^{47} +4.14590 q^{48} +10.9443 q^{49} +0.618034 q^{50} -7.76393 q^{51} +6.00000 q^{53} -1.38197 q^{54} -0.236068 q^{55} +9.47214 q^{56} -9.47214 q^{57} -4.61803 q^{58} +0.708204 q^{59} +3.61803 q^{60} +14.4164 q^{61} -8.47214 q^{63} -0.236068 q^{64} +0.326238 q^{66} +2.70820 q^{67} +5.61803 q^{68} +8.41641 q^{69} +2.61803 q^{70} +6.23607 q^{71} -4.47214 q^{72} +6.00000 q^{73} -1.85410 q^{74} +2.23607 q^{75} +6.85410 q^{76} -1.00000 q^{77} -1.85410 q^{80} -11.0000 q^{81} -7.38197 q^{82} -8.94427 q^{83} +15.3262 q^{84} +3.47214 q^{85} -3.85410 q^{86} -16.7082 q^{87} -0.527864 q^{88} +9.00000 q^{89} -1.23607 q^{90} -6.09017 q^{92} +3.05573 q^{94} +4.23607 q^{95} +12.5623 q^{96} +3.47214 q^{97} +6.76393 q^{98} +0.472136 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 5 * q^6 - 4 * q^7 + 4 * q^9	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{7} + 4 q^{9} + q^{10} - 4 q^{11} - 5 q^{12} - 3 q^{14} - 3 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} + 7 q^{22} + 12 q^{23} - 10 q^{24} + 2 q^{25} + 7 q^{28} - 6 q^{29} - 5 q^{30} + 9 q^{32} + 10 q^{33} - 11 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{37} - 3 q^{38} - 6 q^{41} - 5 q^{42} - 8 q^{43} - 3 q^{44} - 4 q^{45} - 11 q^{46} - 8 q^{47} + 15 q^{48} + 4 q^{49} - q^{50} - 20 q^{51} + 12 q^{53} - 5 q^{54} + 4 q^{55} + 10 q^{56} - 10 q^{57} - 7 q^{58} - 12 q^{59} + 5 q^{60} + 2 q^{61} - 8 q^{63} + 4 q^{64} - 15 q^{66} - 8 q^{67} + 9 q^{68} - 10 q^{69} + 3 q^{70} + 8 q^{71} + 12 q^{73} + 3 q^{74} + 7 q^{76} - 2 q^{77} + 3 q^{80} - 22 q^{81} - 17 q^{82} + 15 q^{84} - 2 q^{85} - q^{86} - 20 q^{87} - 10 q^{88} + 18 q^{89} + 2 q^{90} - q^{92} + 24 q^{94} + 4 q^{95} + 5 q^{96} - 2 q^{97} + 18 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 5 * q^6 - 4 * q^7 + 4 * q^9 + q^10 - 4 * q^11 - 5 * q^12 - 3 * q^14 - 3 * q^16 + 2 * q^17 - 2 * q^18 - 4 * q^19 + q^20 - 10 * q^21 + 7 * q^22 + 12 * q^23 - 10 * q^24 + 2 * q^25 + 7 * q^28 - 6 * q^29 - 5 * q^30 + 9 * q^32 + 10 * q^33 - 11 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^37 - 3 * q^38 - 6 * q^41 - 5 * q^42 - 8 * q^43 - 3 * q^44 - 4 * q^45 - 11 * q^46 - 8 * q^47 + 15 * q^48 + 4 * q^49 - q^50 - 20 * q^51 + 12 * q^53 - 5 * q^54 + 4 * q^55 + 10 * q^56 - 10 * q^57 - 7 * q^58 - 12 * q^59 + 5 * q^60 + 2 * q^61 - 8 * q^63 + 4 * q^64 - 15 * q^66 - 8 * q^67 + 9 * q^68 - 10 * q^69 + 3 * q^70 + 8 * q^71 + 12 * q^73 + 3 * q^74 + 7 * q^76 - 2 * q^77 + 3 * q^80 - 22 * q^81 - 17 * q^82 + 15 * q^84 - 2 * q^85 - q^86 - 20 * q^87 - 10 * q^88 + 18 * q^89 + 2 * q^90 - q^92 + 24 * q^94 + 4 * q^95 + 5 * q^96 - 2 * q^97 + 18 * q^98 - 8 * 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	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	−1.61803	−0.809017
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.38197	0.564185
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	−2.23607	−0.790569
$9$	2.00000	0.666667
$10$	−0.618034	−0.195440
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	−3.61803	−1.04444
$13$	0	0
$13$	0	0
$14$	−2.61803	−0.699699
$15$	−2.23607	−0.577350
$16$	1.85410	0.463525
$17$	−3.47214	−0.842117	−0.421058	−	0.907034i	$-0.638341\pi$
$17$	−3.47214	−0.842117	−0.421058	+	0.907034i	$0.638341\pi$
$18$	1.23607	0.291344
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	1.61803	0.361803
$21$	−9.47214	−2.06699
$22$	0.145898	0.0311056
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	−5.00000	−1.02062
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.23607	−0.430331
$28$	6.85410	1.29530
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	−1.38197	−0.252311
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.61803	0.993137
$33$	0.527864	0.0918893
$34$	−2.14590	−0.368018
$35$	4.23607	0.716026
$36$	−3.23607	−0.539345
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−2.61803	−0.424701
$39$	0	0
$40$	2.23607	0.353553
$41$	−11.9443	−1.86538	−0.932691	−	0.360677i	$-0.882546\pi$
$41$	−11.9443	−1.86538	−0.932691	+	0.360677i	$0.882546\pi$
$42$	−5.85410	−0.903308
$43$	−6.23607	−0.950991	−0.475496	−	0.879718i	$-0.657731\pi$
$43$	−6.23607	−0.950991	−0.475496	+	0.879718i	$0.657731\pi$
$44$	−0.381966	−0.0575835
$45$	−2.00000	−0.298142
$46$	2.32624	0.342985
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	4.14590	0.598409
$49$	10.9443	1.56347
$50$	0.618034	0.0874032
$51$	−7.76393	−1.08717
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.38197	−0.188062
$55$	−0.236068	−0.0318314
$56$	9.47214	1.26577
$57$	−9.47214	−1.25462
$58$	−4.61803	−0.606378
$59$	0.708204	0.0922003	0.0461001	−	0.998937i	$-0.485321\pi$
$59$	0.708204	0.0922003	0.0461001	+	0.998937i	$0.485321\pi$
$60$	3.61803	0.467086
$61$	14.4164	1.84583	0.922916	−	0.385002i	$-0.125799\pi$
$61$	14.4164	1.84583	0.922916	+	0.385002i	$0.125799\pi$
$62$	0	0
$63$	−8.47214	−1.06739
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0.326238	0.0401571
$67$	2.70820	0.330860	0.165430	−	0.986222i	$-0.447099\pi$
$67$	2.70820	0.330860	0.165430	+	0.986222i	$0.447099\pi$
$68$	5.61803	0.681287
$69$	8.41641	1.01322
$70$	2.61803	0.312915
$71$	6.23607	0.740085	0.370043	−	0.929015i	$-0.379343\pi$
$71$	6.23607	0.740085	0.370043	+	0.929015i	$0.379343\pi$
$72$	−4.47214	−0.527046
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−1.85410	−0.215535
$75$	2.23607	0.258199
$76$	6.85410	0.786219
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.85410	−0.207295
$81$	−11.0000	−1.22222
$82$	−7.38197	−0.815202
$83$	−8.94427	−0.981761	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427	−0.981761	−0.490881	+	0.871227i	$0.663325\pi$
$84$	15.3262	1.67223
$85$	3.47214	0.376606
$86$	−3.85410	−0.415599
$87$	−16.7082	−1.79131
$88$	−0.527864	−0.0562705
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	−1.23607	−0.130293
$91$	0	0
$92$	−6.09017	−0.634944
$93$	0	0
$94$	3.05573	0.315174
$95$	4.23607	0.434611
$96$	12.5623	1.28213
$97$	3.47214	0.352542	0.176271	−	0.984342i	$-0.443596\pi$
$97$	3.47214	0.352542	0.176271	+	0.984342i	$0.443596\pi$
$98$	6.76393	0.683260
$99$	0.472136	0.0474514
$100$	−1.61803	−0.161803
$101$	0.527864	0.0525244	0.0262622	−	0.999655i	$-0.491640\pi$
$101$	0.527864	0.0525244	0.0262622	+	0.999655i	$0.491640\pi$
$102$	−4.79837	−0.475110
$103$	12.9443	1.27544	0.637719	−	0.770270i	$-0.279878\pi$
$103$	12.9443	1.27544	0.637719	+	0.770270i	$0.279878\pi$
$104$	0	0
$105$	9.47214	0.924386
$106$	3.70820	0.360173
$107$	−5.76393	−0.557220	−0.278610	−	0.960404i	$-0.589874\pi$
$107$	−5.76393	−0.557220	−0.278610	+	0.960404i	$0.589874\pi$
$108$	3.61803	0.348145
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−0.145898	−0.0139108
$111$	−6.70820	−0.636715
$112$	−7.85410	−0.742143
$113$	1.47214	0.138487	0.0692435	−	0.997600i	$-0.477941\pi$
$113$	1.47214	0.138487	0.0692435	+	0.997600i	$0.477941\pi$
$114$	−5.85410	−0.548287
$115$	−3.76393	−0.350988
$116$	12.0902	1.12254
$117$	0	0
$118$	0.437694	0.0402930
$119$	14.7082	1.34830
$120$	5.00000	0.456435
$121$	−10.9443	−0.994934
$122$	8.90983	0.806658
$123$	−26.7082	−2.40820
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−5.23607	−0.466466
$127$	−4.23607	−0.375890	−0.187945	−	0.982180i	$-0.560183\pi$
$127$	−4.23607	−0.375890	−0.187945	+	0.982180i	$0.560183\pi$
$128$	−11.3820	−1.00603
$129$	−13.9443	−1.22772
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−0.854102	−0.0743400
$133$	17.9443	1.55597
$134$	1.67376	0.144591
$135$	2.23607	0.192450
$136$	7.76393	0.665752
$137$	−1.47214	−0.125773	−0.0628865	−	0.998021i	$-0.520031\pi$
$137$	−1.47214	−0.125773	−0.0628865	+	0.998021i	$0.520031\pi$
$138$	5.20163	0.442792
$139$	−16.7082	−1.41717	−0.708586	−	0.705625i	$-0.750666\pi$
$139$	−16.7082	−1.41717	−0.708586	+	0.705625i	$0.750666\pi$
$140$	−6.85410	−0.579277
$141$	11.0557	0.931060
$142$	3.85410	0.323429
$143$	0	0
$144$	3.70820	0.309017
$145$	7.47214	0.620527
$146$	3.70820	0.306893
$147$	24.4721	2.01843
$148$	4.85410	0.399005
$149$	−4.52786	−0.370937	−0.185469	−	0.982650i	$-0.559380\pi$
$149$	−4.52786	−0.370937	−0.185469	+	0.982650i	$0.559380\pi$
$150$	1.38197	0.112837
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	9.47214	0.768292
$153$	−6.94427	−0.561411
$154$	−0.618034	−0.0498026
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	13.4164	1.06399
$160$	−5.61803	−0.444145
$161$	−15.9443	−1.25658
$162$	−6.79837	−0.534131
$163$	−14.7082	−1.15204	−0.576018	−	0.817437i	$-0.695394\pi$
$163$	−14.7082	−1.15204	−0.576018	+	0.817437i	$0.695394\pi$
$164$	19.3262	1.50913
$165$	−0.527864	−0.0410942
$166$	−5.52786	−0.429045
$167$	17.1803	1.32945	0.664727	−	0.747086i	$-0.268548\pi$
$167$	17.1803	1.32945	0.664727	+	0.747086i	$0.268548\pi$
$168$	21.1803	1.63410
$169$	0	0
$170$	2.14590	0.164583
$171$	−8.47214	−0.647880
$172$	10.0902	0.769368
$173$	−18.8885	−1.43607	−0.718035	−	0.696007i	$-0.754958\pi$
$173$	−18.8885	−1.43607	−0.718035	+	0.696007i	$0.754958\pi$
$174$	−10.3262	−0.782830
$175$	−4.23607	−0.320217
$176$	0.437694	0.0329924
$177$	1.58359	0.119030
$178$	5.56231	0.416912
$179$	−8.23607	−0.615593	−0.307796	−	0.951452i	$-0.599592\pi$
$179$	−8.23607	−0.615593	−0.307796	+	0.951452i	$0.599592\pi$
$180$	3.23607	0.241202
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	32.2361	2.38296
$184$	−8.41641	−0.620466
$185$	3.00000	0.220564
$186$	0	0
$187$	−0.819660	−0.0599395
$188$	−8.00000	−0.583460
$189$	9.47214	0.688997
$190$	2.61803	0.189932
$191$	−27.1803	−1.96670	−0.983350	−	0.181721i	$-0.941833\pi$
$191$	−27.1803	−1.96670	−0.983350	+	0.181721i	$0.941833\pi$
$192$	−0.527864	−0.0380953
$193$	3.47214	0.249930	0.124965	−	0.992161i	$-0.460118\pi$
$193$	3.47214	0.249930	0.124965	+	0.992161i	$0.460118\pi$
$194$	2.14590	0.154067
$195$	0	0
$196$	−17.7082	−1.26487
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0.291796	0.0207370
$199$	−1.29180	−0.0915730	−0.0457865	−	0.998951i	$-0.514579\pi$
$199$	−1.29180	−0.0915730	−0.0457865	+	0.998951i	$0.514579\pi$
$200$	−2.23607	−0.158114
$201$	6.05573	0.427138
$202$	0.326238	0.0229540
$203$	31.6525	2.22157
$204$	12.5623	0.879537
$205$	11.9443	0.834224
$206$	8.00000	0.557386
$207$	7.52786	0.523223
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	5.85410	0.403971
$211$	−13.1803	−0.907372	−0.453686	−	0.891162i	$-0.649891\pi$
$211$	−13.1803	−0.907372	−0.453686	+	0.891162i	$0.649891\pi$
$212$	−9.70820	−0.666762
$213$	13.9443	0.955446
$214$	−3.56231	−0.243514
$215$	6.23607	0.425296
$216$	5.00000	0.340207
$217$	0	0
$218$	1.23607	0.0837171
$219$	13.4164	0.906597
$220$	0.381966	0.0257521
$221$	0	0
$222$	−4.14590	−0.278254
$223$	−0.708204	−0.0474248	−0.0237124	−	0.999719i	$-0.507549\pi$
$223$	−0.708204	−0.0474248	−0.0237124	+	0.999719i	$0.507549\pi$
$224$	−23.7984	−1.59010
$225$	2.00000	0.133333
$226$	0.909830	0.0605210
$227$	6.23607	0.413902	0.206951	−	0.978351i	$-0.433646\pi$
$227$	6.23607	0.413902	0.206951	+	0.978351i	$0.433646\pi$
$228$	15.3262	1.01500
$229$	−15.8885	−1.04994	−0.524972	−	0.851119i	$-0.675924\pi$
$229$	−15.8885	−1.04994	−0.524972	+	0.851119i	$0.675924\pi$
$230$	−2.32624	−0.153388
$231$	−2.23607	−0.147122
$232$	16.7082	1.09695
$233$	−15.8885	−1.04089	−0.520447	−	0.853894i	$-0.674235\pi$
$233$	−15.8885	−1.04089	−0.520447	+	0.853894i	$0.674235\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	−1.14590	−0.0745916
$237$	0	0
$238$	9.09017	0.589228
$239$	25.8885	1.67459	0.837295	−	0.546751i	$-0.184136\pi$
$239$	25.8885	1.67459	0.837295	+	0.546751i	$0.184136\pi$
$240$	−4.14590	−0.267617
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	−6.76393	−0.434802
$243$	−17.8885	−1.14755
$244$	−23.3262	−1.49331
$245$	−10.9443	−0.699204
$246$	−16.5066	−1.05242
$247$	0	0
$248$	0	0
$249$	−20.0000	−1.26745
$250$	−0.618034	−0.0390879
$251$	20.2361	1.27729	0.638645	−	0.769502i	$-0.279495\pi$
$251$	20.2361	1.27729	0.638645	+	0.769502i	$0.279495\pi$
$252$	13.7082	0.863536
$253$	0.888544	0.0558623
$254$	−2.61803	−0.164270
$255$	7.76393	0.486196
$256$	−6.56231	−0.410144
$257$	17.4721	1.08988	0.544941	−	0.838474i	$-0.316552\pi$
$257$	17.4721	1.08988	0.544941	+	0.838474i	$0.316552\pi$
$258$	−8.61803	−0.536535
$259$	12.7082	0.789649
$260$	0	0
$261$	−14.9443	−0.925027
$262$	−7.41641	−0.458187
$263$	3.76393	0.232094	0.116047	−	0.993244i	$-0.462978\pi$
$263$	3.76393	0.232094	0.116047	+	0.993244i	$0.462978\pi$
$264$	−1.18034	−0.0726449
$265$	−6.00000	−0.368577
$266$	11.0902	0.679982
$267$	20.1246	1.23161
$268$	−4.38197	−0.267671
$269$	−6.52786	−0.398011	−0.199005	−	0.979998i	$-0.563771\pi$
$269$	−6.52786	−0.398011	−0.199005	+	0.979998i	$0.563771\pi$
$270$	1.38197	0.0841038
$271$	−1.29180	−0.0784710	−0.0392355	−	0.999230i	$-0.512492\pi$
$271$	−1.29180	−0.0784710	−0.0392355	+	0.999230i	$0.512492\pi$
$272$	−6.43769	−0.390343
$273$	0	0
$274$	−0.909830	−0.0549648
$275$	0.236068	0.0142354
$276$	−13.6180	−0.819709
$277$	16.8885	1.01473	0.507367	−	0.861730i	$-0.330619\pi$
$277$	16.8885	1.01473	0.507367	+	0.861730i	$0.330619\pi$
$278$	−10.3262	−0.619327
$279$	0	0
$280$	−9.47214	−0.566068
$281$	15.8885	0.947831	0.473916	−	0.880570i	$-0.342840\pi$
$281$	15.8885	0.947831	0.473916	+	0.880570i	$0.342840\pi$
$282$	6.83282	0.406888
$283$	−10.7082	−0.636537	−0.318268	−	0.948001i	$-0.603101\pi$
$283$	−10.7082	−0.636537	−0.318268	+	0.948001i	$0.603101\pi$
$284$	−10.0902	−0.598741
$285$	9.47214	0.561081
$286$	0	0
$287$	50.5967	2.98663
$288$	11.2361	0.662092
$289$	−4.94427	−0.290840
$290$	4.61803	0.271180
$291$	7.76393	0.455130
$292$	−9.70820	−0.568130
$293$	29.9443	1.74936	0.874682	−	0.484698i	$-0.161071\pi$
$293$	29.9443	1.74936	0.874682	+	0.484698i	$0.161071\pi$
$294$	15.1246	0.882085
$295$	−0.708204	−0.0412332
$296$	6.70820	0.389906
$297$	−0.527864	−0.0306298
$298$	−2.79837	−0.162105
$299$	0	0
$300$	−3.61803	−0.208887
$301$	26.4164	1.52262
$302$	−4.94427	−0.284511
$303$	1.18034	0.0678088
$304$	−7.85410	−0.450464
$305$	−14.4164	−0.825481
$306$	−4.29180	−0.245346
$307$	−24.9443	−1.42364	−0.711822	−	0.702360i	$-0.752130\pi$
$307$	−24.9443	−1.42364	−0.711822	+	0.702360i	$0.752130\pi$
$308$	1.61803	0.0921960
$309$	28.9443	1.64658
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	3.88854	0.219793	0.109897	−	0.993943i	$-0.464948\pi$
$313$	3.88854	0.219793	0.109897	+	0.993943i	$0.464948\pi$
$314$	−11.1246	−0.627798
$315$	8.47214	0.477351
$316$	0	0
$317$	11.8885	0.667727	0.333864	−	0.942621i	$-0.391648\pi$
$317$	11.8885	0.667727	0.333864	+	0.942621i	$0.391648\pi$
$318$	8.29180	0.464981
$319$	−1.76393	−0.0987612
$320$	0.236068	0.0131966
$321$	−12.8885	−0.719368
$322$	−9.85410	−0.549148
$323$	14.7082	0.818386
$324$	17.7984	0.988799
$325$	0	0
$326$	−9.09017	−0.503458
$327$	4.47214	0.247310
$328$	26.7082	1.47471
$329$	−20.9443	−1.15470
$330$	−0.326238	−0.0179588
$331$	5.65248	0.310688	0.155344	−	0.987860i	$-0.450351\pi$
$331$	5.65248	0.310688	0.155344	+	0.987860i	$0.450351\pi$
$332$	14.4721	0.794262
$333$	−6.00000	−0.328798
$334$	10.6180	0.580993
$335$	−2.70820	−0.147965
$336$	−17.5623	−0.958102
$337$	−7.88854	−0.429716	−0.214858	−	0.976645i	$-0.568929\pi$
$337$	−7.88854	−0.429716	−0.214858	+	0.976645i	$0.568929\pi$
$338$	0	0
$339$	3.29180	0.178786
$340$	−5.61803	−0.304681
$341$	0	0
$342$	−5.23607	−0.283134
$343$	−16.7082	−0.902158
$344$	13.9443	0.751825
$345$	−8.41641	−0.453124
$346$	−11.6738	−0.627585
$347$	30.7082	1.64850	0.824251	−	0.566224i	$-0.191596\pi$
$347$	30.7082	1.64850	0.824251	+	0.566224i	$0.191596\pi$
$348$	27.0344	1.44920
$349$	−2.41641	−0.129347	−0.0646737	−	0.997906i	$-0.520601\pi$
$349$	−2.41641	−0.129347	−0.0646737	+	0.997906i	$0.520601\pi$
$350$	−2.61803	−0.139940
$351$	0	0
$352$	1.32624	0.0706887
$353$	19.4721	1.03640	0.518199	−	0.855260i	$-0.326603\pi$
$353$	19.4721	1.03640	0.518199	+	0.855260i	$0.326603\pi$
$354$	0.978714	0.0520180
$355$	−6.23607	−0.330976
$356$	−14.5623	−0.771801
$357$	32.8885	1.74065
$358$	−5.09017	−0.269024
$359$	−17.8885	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$359$	−17.8885	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$360$	4.47214	0.235702
$361$	−1.05573	−0.0555646
$362$	3.70820	0.194899
$363$	−24.4721	−1.28445
$364$	0	0
$365$	−6.00000	−0.314054
$366$	19.9230	1.04139
$367$	5.65248	0.295057	0.147528	−	0.989058i	$-0.452868\pi$
$367$	5.65248	0.295057	0.147528	+	0.989058i	$0.452868\pi$
$368$	6.97871	0.363791
$369$	−23.8885	−1.24359
$370$	1.85410	0.0963902
$371$	−25.4164	−1.31955
$372$	0	0
$373$	27.9443	1.44690	0.723450	−	0.690377i	$-0.242555\pi$
$373$	27.9443	1.44690	0.723450	+	0.690377i	$0.242555\pi$
$374$	−0.506578	−0.0261945
$375$	−2.23607	−0.115470
$376$	−11.0557	−0.570156
$377$	0	0
$378$	5.85410	0.301103
$379$	−10.8197	−0.555769	−0.277884	−	0.960615i	$-0.589633\pi$
$379$	−10.8197	−0.555769	−0.277884	+	0.960615i	$0.589633\pi$
$380$	−6.85410	−0.351608
$381$	−9.47214	−0.485272
$382$	−16.7984	−0.859480
$383$	−4.23607	−0.216453	−0.108226	−	0.994126i	$-0.534517\pi$
$383$	−4.23607	−0.216453	−0.108226	+	0.994126i	$0.534517\pi$
$384$	−25.4508	−1.29878
$385$	1.00000	0.0509647
$386$	2.14590	0.109223
$387$	−12.4721	−0.633994
$388$	−5.61803	−0.285212
$389$	−0.111456	−0.00565105	−0.00282553	−	0.999996i	$-0.500899\pi$
$389$	−0.111456	−0.00565105	−0.00282553	+	0.999996i	$0.500899\pi$
$390$	0	0
$391$	−13.0689	−0.660922
$392$	−24.4721	−1.23603
$393$	−26.8328	−1.35354
$394$	−1.85410	−0.0934083
$395$	0	0
$396$	−0.763932	−0.0383890
$397$	−27.9443	−1.40248	−0.701241	−	0.712924i	$-0.747370\pi$
$397$	−27.9443	−1.40248	−0.701241	+	0.712924i	$0.747370\pi$
$398$	−0.798374	−0.0400189
$399$	40.1246	2.00874
$400$	1.85410	0.0927051
$401$	−8.88854	−0.443873	−0.221936	−	0.975061i	$-0.571238\pi$
$401$	−8.88854	−0.443873	−0.221936	+	0.975061i	$0.571238\pi$
$402$	3.74265	0.186666
$403$	0	0
$404$	−0.854102	−0.0424932
$405$	11.0000	0.546594
$406$	19.5623	0.970861
$407$	−0.708204	−0.0351044
$408$	17.3607	0.859482
$409$	−24.8885	−1.23066	−0.615330	−	0.788270i	$-0.710977\pi$
$409$	−24.8885	−1.23066	−0.615330	+	0.788270i	$0.710977\pi$
$410$	7.38197	0.364569
$411$	−3.29180	−0.162372
$412$	−20.9443	−1.03185
$413$	−3.00000	−0.147620
$414$	4.65248	0.228657
$415$	8.94427	0.439057
$416$	0	0
$417$	−37.3607	−1.82956
$418$	−0.618034	−0.0302290
$419$	17.6525	0.862380	0.431190	−	0.902261i	$-0.358094\pi$
$419$	17.6525	0.862380	0.431190	+	0.902261i	$0.358094\pi$
$420$	−15.3262	−0.747844
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−8.14590	−0.396536
$423$	9.88854	0.480797
$424$	−13.4164	−0.651558
$425$	−3.47214	−0.168423
$426$	8.61803	0.417545
$427$	−61.0689	−2.95533
$428$	9.32624	0.450801
$429$	0	0
$430$	3.85410	0.185861
$431$	−27.1803	−1.30923	−0.654615	−	0.755962i	$-0.727169\pi$
$431$	−27.1803	−1.30923	−0.654615	+	0.755962i	$0.727169\pi$
$432$	−4.14590	−0.199470
$433$	−14.5279	−0.698165	−0.349082	−	0.937092i	$-0.613507\pi$
$433$	−14.5279	−0.698165	−0.349082	+	0.937092i	$0.613507\pi$
$434$	0	0
$435$	16.7082	0.801097
$436$	−3.23607	−0.154980
$437$	−15.9443	−0.762718
$438$	8.29180	0.396197
$439$	−22.7082	−1.08380	−0.541902	−	0.840442i	$-0.682296\pi$
$439$	−22.7082	−1.08380	−0.541902	+	0.840442i	$0.682296\pi$
$440$	0.527864	0.0251649
$441$	21.8885	1.04231
$442$	0	0
$443$	0.944272	0.0448637	0.0224319	−	0.999748i	$-0.492859\pi$
$443$	0.944272	0.0448637	0.0224319	+	0.999748i	$0.492859\pi$
$444$	10.8541	0.515113
$445$	−9.00000	−0.426641
$446$	−0.437694	−0.0207254
$447$	−10.1246	−0.478878
$448$	1.00000	0.0472456
$449$	−3.94427	−0.186142	−0.0930709	−	0.995659i	$-0.529668\pi$
$449$	−3.94427	−0.186142	−0.0930709	+	0.995659i	$0.529668\pi$
$450$	1.23607	0.0582688
$451$	−2.81966	−0.132773
$452$	−2.38197	−0.112038
$453$	−17.8885	−0.840477
$454$	3.85410	0.180882
$455$	0	0
$456$	21.1803	0.991860
$457$	1.58359	0.0740773	0.0370387	−	0.999314i	$-0.488208\pi$
$457$	1.58359	0.0740773	0.0370387	+	0.999314i	$0.488208\pi$
$458$	−9.81966	−0.458843
$459$	7.76393	0.362389
$460$	6.09017	0.283956
$461$	1.58359	0.0737552	0.0368776	−	0.999320i	$-0.488259\pi$
$461$	1.58359	0.0737552	0.0368776	+	0.999320i	$0.488259\pi$
$462$	−1.38197	−0.0642949
$463$	−11.0557	−0.513803	−0.256902	−	0.966438i	$-0.582702\pi$
$463$	−11.0557	−0.513803	−0.256902	+	0.966438i	$0.582702\pi$
$464$	−13.8541	−0.643161
$465$	0	0
$466$	−9.81966	−0.454887
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	0	0
$469$	−11.4721	−0.529734
$470$	−3.05573	−0.140950
$471$	−40.2492	−1.85459
$472$	−1.58359	−0.0728907
$473$	−1.47214	−0.0676889
$474$	0	0
$475$	−4.23607	−0.194364
$476$	−23.7984	−1.09080
$477$	12.0000	0.549442
$478$	16.0000	0.731823
$479$	−32.1246	−1.46781	−0.733905	−	0.679252i	$-0.762305\pi$
$479$	−32.1246	−1.46781	−0.733905	+	0.679252i	$0.762305\pi$
$480$	−12.5623	−0.573388
$481$	0	0
$482$	−9.27051	−0.422260
$483$	−35.6525	−1.62224
$484$	17.7082	0.804918
$485$	−3.47214	−0.157662
$486$	−11.0557	−0.501498
$487$	−14.1246	−0.640047	−0.320024	−	0.947410i	$-0.603691\pi$
$487$	−14.1246	−0.640047	−0.320024	+	0.947410i	$0.603691\pi$
$488$	−32.2361	−1.45926
$489$	−32.8885	−1.48727
$490$	−6.76393	−0.305563
$491$	−0.236068	−0.0106536	−0.00532680	−	0.999986i	$-0.501696\pi$
$491$	−0.236068	−0.0106536	−0.00532680	+	0.999986i	$0.501696\pi$
$492$	43.2148	1.94827
$493$	25.9443	1.16847
$494$	0	0
$495$	−0.472136	−0.0212209
$496$	0	0
$497$	−26.4164	−1.18494
$498$	−12.3607	−0.553895
$499$	21.8885	0.979866	0.489933	−	0.871760i	$-0.337021\pi$
$499$	21.8885	0.979866	0.489933	+	0.871760i	$0.337021\pi$
$500$	1.61803	0.0723607
$501$	38.4164	1.71632
$502$	12.5066	0.558196
$503$	9.18034	0.409331	0.204666	−	0.978832i	$-0.434389\pi$
$503$	9.18034	0.409331	0.204666	+	0.978832i	$0.434389\pi$
$504$	18.9443	0.843845
$505$	−0.527864	−0.0234896
$506$	0.549150	0.0244127
$507$	0	0
$508$	6.85410	0.304102
$509$	33.3607	1.47869	0.739343	−	0.673329i	$-0.235136\pi$
$509$	33.3607	1.47869	0.739343	+	0.673329i	$0.235136\pi$
$510$	4.79837	0.212476
$511$	−25.4164	−1.12436
$512$	18.7082	0.826794
$513$	9.47214	0.418205
$514$	10.7984	0.476296
$515$	−12.9443	−0.570393
$516$	22.5623	0.993250
$517$	1.16718	0.0513327
$518$	7.85410	0.345089
$519$	−42.2361	−1.85396
$520$	0	0
$521$	29.7771	1.30456	0.652279	−	0.757979i	$-0.273813\pi$
$521$	29.7771	1.30456	0.652279	+	0.757979i	$0.273813\pi$
$522$	−9.23607	−0.404252
$523$	5.29180	0.231394	0.115697	−	0.993285i	$-0.463090\pi$
$523$	5.29180	0.231394	0.115697	+	0.993285i	$0.463090\pi$
$524$	19.4164	0.848210
$525$	−9.47214	−0.413398
$526$	2.32624	0.101429
$527$	0	0
$528$	0.978714	0.0425930
$529$	−8.83282	−0.384035
$530$	−3.70820	−0.161074
$531$	1.41641	0.0614669
$532$	−29.0344	−1.25880
$533$	0	0
$534$	12.4377	0.538232
$535$	5.76393	0.249197
$536$	−6.05573	−0.261568
$537$	−18.4164	−0.794727
$538$	−4.03444	−0.173937
$539$	2.58359	0.111283
$540$	−3.61803	−0.155695
$541$	27.8885	1.19902	0.599511	−	0.800366i	$-0.295362\pi$
$541$	27.8885	1.19902	0.599511	+	0.800366i	$0.295362\pi$
$542$	−0.798374	−0.0342931
$543$	13.4164	0.575753
$544$	−19.5066	−0.836338
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	18.8328	0.805233	0.402617	−	0.915369i	$-0.368101\pi$
$547$	18.8328	0.805233	0.402617	+	0.915369i	$0.368101\pi$
$548$	2.38197	0.101753
$549$	28.8328	1.23055
$550$	0.145898	0.00622111
$551$	31.6525	1.34844
$552$	−18.8197	−0.801018
$553$	0	0
$554$	10.4377	0.443455
$555$	6.70820	0.284747
$556$	27.0344	1.14652
$557$	9.94427	0.421352	0.210676	−	0.977556i	$-0.432433\pi$
$557$	9.94427	0.421352	0.210676	+	0.977556i	$0.432433\pi$
$558$	0	0
$559$	0	0
$560$	7.85410	0.331896
$561$	−1.83282	−0.0773815
$562$	9.81966	0.414217
$563$	30.7082	1.29420	0.647098	−	0.762407i	$-0.275982\pi$
$563$	30.7082	1.29420	0.647098	+	0.762407i	$0.275982\pi$
$564$	−17.8885	−0.753244
$565$	−1.47214	−0.0619332
$566$	−6.61803	−0.278177
$567$	46.5967	1.95688
$568$	−13.9443	−0.585089
$569$	16.8885	0.708005	0.354002	−	0.935245i	$-0.384821\pi$
$569$	16.8885	0.708005	0.354002	+	0.935245i	$0.384821\pi$
$570$	5.85410	0.245201
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	−60.7771	−2.53900
$574$	31.2705	1.30521
$575$	3.76393	0.156967
$576$	−0.472136	−0.0196723
$577$	−27.8885	−1.16102	−0.580508	−	0.814255i	$-0.697146\pi$
$577$	−27.8885	−1.16102	−0.580508	+	0.814255i	$0.697146\pi$
$578$	−3.05573	−0.127102
$579$	7.76393	0.322658
$580$	−12.0902	−0.502017
$581$	37.8885	1.57188
$582$	4.79837	0.198899
$583$	1.41641	0.0586616
$584$	−13.4164	−0.555175
$585$	0	0
$586$	18.5066	0.764500
$587$	−10.2361	−0.422488	−0.211244	−	0.977433i	$-0.567751\pi$
$587$	−10.2361	−0.422488	−0.211244	+	0.977433i	$0.567751\pi$
$588$	−39.5967	−1.63294
$589$	0	0
$590$	−0.437694	−0.0180196
$591$	−6.70820	−0.275939
$592$	−5.56231	−0.228609
$593$	7.88854	0.323944	0.161972	−	0.986795i	$-0.448215\pi$
$593$	7.88854	0.323944	0.161972	+	0.986795i	$0.448215\pi$
$594$	−0.326238	−0.0133857
$595$	−14.7082	−0.602978
$596$	7.32624	0.300094
$597$	−2.88854	−0.118220
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−5.00000	−0.204124
$601$	43.9443	1.79252	0.896262	−	0.443525i	$-0.146272\pi$
$601$	43.9443	1.79252	0.896262	+	0.443525i	$0.146272\pi$
$602$	16.3262	0.665408
$603$	5.41641	0.220573
$604$	12.9443	0.526695
$605$	10.9443	0.444948
$606$	0.729490	0.0296335
$607$	−0.708204	−0.0287451	−0.0143726	−	0.999897i	$-0.504575\pi$
$607$	−0.708204	−0.0287451	−0.0143726	+	0.999897i	$0.504575\pi$
$608$	−23.7984	−0.965152
$609$	70.7771	2.86803
$610$	−8.90983	−0.360748
$611$	0	0
$612$	11.2361	0.454191
$613$	−39.9443	−1.61333	−0.806667	−	0.591006i	$-0.798731\pi$
$613$	−39.9443	−1.61333	−0.806667	+	0.591006i	$0.798731\pi$
$614$	−15.4164	−0.622156
$615$	26.7082	1.07698
$616$	2.23607	0.0900937
$617$	40.4164	1.62710	0.813552	−	0.581493i	$-0.197531\pi$
$617$	40.4164	1.62710	0.813552	+	0.581493i	$0.197531\pi$
$618$	17.8885	0.719583
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	−8.41641	−0.337739
$622$	−14.8328	−0.594742
$623$	−38.1246	−1.52743
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.40325	0.0960533
$627$	−2.23607	−0.0893000
$628$	29.1246	1.16220
$629$	10.4164	0.415329
$630$	5.23607	0.208610
$631$	16.1246	0.641911	0.320955	−	0.947094i	$-0.395996\pi$
$631$	16.1246	0.641911	0.320955	+	0.947094i	$0.395996\pi$
$632$	0	0
$633$	−29.4721	−1.17141
$634$	7.34752	0.291807
$635$	4.23607	0.168103
$636$	−21.7082	−0.860786
$637$	0	0
$638$	−1.09017	−0.0431602
$639$	12.4721	0.493390
$640$	11.3820	0.449912
$641$	8.88854	0.351076	0.175538	−	0.984473i	$-0.443833\pi$
$641$	8.88854	0.351076	0.175538	+	0.984473i	$0.443833\pi$
$642$	−7.96556	−0.314376
$643$	17.2918	0.681922	0.340961	−	0.940077i	$-0.389248\pi$
$643$	17.2918	0.681922	0.340961	+	0.940077i	$0.389248\pi$
$644$	25.7984	1.01660
$645$	13.9443	0.549055
$646$	9.09017	0.357648
$647$	2.59675	0.102089	0.0510443	−	0.998696i	$-0.483745\pi$
$647$	2.59675	0.102089	0.0510443	+	0.998696i	$0.483745\pi$
$648$	24.5967	0.966252
$649$	0.167184	0.00656256
$650$	0	0
$651$	0	0
$652$	23.7984	0.932016
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	2.76393	0.108078
$655$	12.0000	0.468879
$656$	−22.1459	−0.864652
$657$	12.0000	0.468165
$658$	−12.9443	−0.504620
$659$	−43.7639	−1.70480	−0.852400	−	0.522890i	$-0.824854\pi$
$659$	−43.7639	−1.70480	−0.852400	+	0.522890i	$0.824854\pi$
$660$	0.854102	0.0332459
$661$	41.3607	1.60874	0.804372	−	0.594126i	$-0.202502\pi$
$661$	41.3607	1.60874	0.804372	+	0.594126i	$0.202502\pi$
$662$	3.49342	0.135776
$663$	0	0
$664$	20.0000	0.776151
$665$	−17.9443	−0.695849
$666$	−3.70820	−0.143690
$667$	−28.1246	−1.08899
$668$	−27.7984	−1.07555
$669$	−1.58359	−0.0612252
$670$	−1.67376	−0.0646631
$671$	3.40325	0.131381
$672$	−53.2148	−2.05280
$673$	−9.58359	−0.369420	−0.184710	−	0.982793i	$-0.559135\pi$
$673$	−9.58359	−0.369420	−0.184710	+	0.982793i	$0.559135\pi$
$674$	−4.87539	−0.187793
$675$	−2.23607	−0.0860663
$676$	0	0
$677$	12.1115	0.465481	0.232741	−	0.972539i	$-0.425231\pi$
$677$	12.1115	0.465481	0.232741	+	0.972539i	$0.425231\pi$
$678$	2.03444	0.0781323
$679$	−14.7082	−0.564449
$680$	−7.76393	−0.297733
$681$	13.9443	0.534346
$682$	0	0
$683$	−25.7639	−0.985829	−0.492915	−	0.870078i	$-0.664069\pi$
$683$	−25.7639	−0.985829	−0.492915	+	0.870078i	$0.664069\pi$
$684$	13.7082	0.524146
$685$	1.47214	0.0562474
$686$	−10.3262	−0.394258
$687$	−35.5279	−1.35547
$688$	−11.5623	−0.440809
$689$	0	0
$690$	−5.20163	−0.198023
$691$	−38.5967	−1.46829	−0.734145	−	0.678993i	$-0.762417\pi$
$691$	−38.5967	−1.46829	−0.734145	+	0.678993i	$0.762417\pi$
$692$	30.5623	1.16180
$693$	−2.00000	−0.0759737
$694$	18.9787	0.720422
$695$	16.7082	0.633778
$696$	37.3607	1.41615
$697$	41.4721	1.57087
$698$	−1.49342	−0.0565269
$699$	−35.5279	−1.34379
$700$	6.85410	0.259061
$701$	7.88854	0.297946	0.148973	−	0.988841i	$-0.452403\pi$
$701$	7.88854	0.297946	0.148973	+	0.988841i	$0.452403\pi$
$702$	0	0
$703$	12.7082	0.479299
$704$	−0.0557281	−0.00210033
$705$	−11.0557	−0.416383
$706$	12.0344	0.452922
$707$	−2.23607	−0.0840960
$708$	−2.56231	−0.0962974
$709$	−24.3050	−0.912792	−0.456396	−	0.889777i	$-0.650860\pi$
$709$	−24.3050	−0.912792	−0.456396	+	0.889777i	$0.650860\pi$
$710$	−3.85410	−0.144642
$711$	0	0
$712$	−20.1246	−0.754202
$713$	0	0
$714$	20.3262	0.760690
$715$	0	0
$716$	13.3262	0.498025
$717$	57.8885	2.16189
$718$	−11.0557	−0.412596
$719$	−32.1246	−1.19805	−0.599023	−	0.800732i	$-0.704444\pi$
$719$	−32.1246	−1.19805	−0.599023	+	0.800732i	$0.704444\pi$
$720$	−3.70820	−0.138197
$721$	−54.8328	−2.04208
$722$	−0.652476	−0.0242826
$723$	−33.5410	−1.24740
$724$	−9.70820	−0.360803
$725$	−7.47214	−0.277508
$726$	−15.1246	−0.561327
$727$	28.9443	1.07348	0.536742	−	0.843747i	$-0.319655\pi$
$727$	28.9443	1.07348	0.536742	+	0.843747i	$0.319655\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	−3.70820	−0.137247
$731$	21.6525	0.800846
$732$	−52.1591	−1.92785
$733$	43.8885	1.62106	0.810530	−	0.585697i	$-0.199179\pi$
$733$	43.8885	1.62106	0.810530	+	0.585697i	$0.199179\pi$
$734$	3.49342	0.128945
$735$	−24.4721	−0.902668
$736$	21.1459	0.779448
$737$	0.639320	0.0235497
$738$	−14.7639	−0.543468
$739$	−27.5410	−1.01311	−0.506556	−	0.862207i	$-0.669082\pi$
$739$	−27.5410	−1.01311	−0.506556	+	0.862207i	$0.669082\pi$
$740$	−4.85410	−0.178440
$741$	0	0
$742$	−15.7082	−0.576666
$743$	38.1246	1.39866	0.699328	−	0.714801i	$-0.253483\pi$
$743$	38.1246	1.39866	0.699328	+	0.714801i	$0.253483\pi$
$744$	0	0
$745$	4.52786	0.165888
$746$	17.2705	0.632318
$747$	−17.8885	−0.654508
$748$	1.32624	0.0484921
$749$	24.4164	0.892156
$750$	−1.38197	−0.0504623
$751$	−30.7082	−1.12056	−0.560279	−	0.828304i	$-0.689306\pi$
$751$	−30.7082	−1.12056	−0.560279	+	0.828304i	$0.689306\pi$
$752$	9.16718	0.334293
$753$	45.2492	1.64897
$754$	0	0
$755$	8.00000	0.291150
$756$	−15.3262	−0.557410
$757$	−6.88854	−0.250368	−0.125184	−	0.992134i	$-0.539952\pi$
$757$	−6.88854	−0.250368	−0.125184	+	0.992134i	$0.539952\pi$
$758$	−6.68692	−0.242880
$759$	1.98684	0.0721179
$760$	−9.47214	−0.343590
$761$	37.9443	1.37548	0.687739	−	0.725958i	$-0.258603\pi$
$761$	37.9443	1.37548	0.687739	+	0.725958i	$0.258603\pi$
$762$	−5.85410	−0.212072
$763$	−8.47214	−0.306712
$764$	43.9787	1.59109
$765$	6.94427	0.251071
$766$	−2.61803	−0.0945934
$767$	0	0
$768$	−14.6738	−0.529494
$769$	−29.8328	−1.07580	−0.537899	−	0.843009i	$-0.680782\pi$
$769$	−29.8328	−1.07580	−0.537899	+	0.843009i	$0.680782\pi$
$770$	0.618034	0.0222724
$771$	39.0689	1.40703
$772$	−5.61803	−0.202197
$773$	20.0557	0.721354	0.360677	−	0.932691i	$-0.382546\pi$
$773$	20.0557	0.721354	0.360677	+	0.932691i	$0.382546\pi$
$774$	−7.70820	−0.277066
$775$	0	0
$776$	−7.76393	−0.278709
$777$	28.4164	1.01943
$778$	−0.0688837	−0.00246960
$779$	50.5967	1.81282
$780$	0	0
$781$	1.47214	0.0526772
$782$	−8.07701	−0.288833
$783$	16.7082	0.597102
$784$	20.2918	0.724707
$785$	18.0000	0.642448
$786$	−16.5836	−0.591517
$787$	−29.5410	−1.05302	−0.526512	−	0.850168i	$-0.676500\pi$
$787$	−29.5410	−1.05302	−0.526512	+	0.850168i	$0.676500\pi$
$788$	4.85410	0.172920
$789$	8.41641	0.299632
$790$	0	0
$791$	−6.23607	−0.221729
$792$	−1.05573	−0.0375137
$793$	0	0
$794$	−17.2705	−0.612907
$795$	−13.4164	−0.475831
$796$	2.09017	0.0740841
$797$	53.8328	1.90686	0.953428	−	0.301620i	$-0.0975275\pi$
$797$	53.8328	1.90686	0.953428	+	0.301620i	$0.0975275\pi$
$798$	24.7984	0.877853
$799$	−17.1672	−0.607331
$800$	5.61803	0.198627
$801$	18.0000	0.635999
$802$	−5.49342	−0.193979
$803$	1.41641	0.0499839
$804$	−9.79837	−0.345562
$805$	15.9443	0.561962
$806$	0	0
$807$	−14.5967	−0.513830
$808$	−1.18034	−0.0415242
$809$	−2.88854	−0.101556	−0.0507779	−	0.998710i	$-0.516170\pi$
$809$	−2.88854	−0.101556	−0.0507779	+	0.998710i	$0.516170\pi$
$810$	6.79837	0.238871
$811$	−31.7771	−1.11584	−0.557922	−	0.829893i	$-0.688401\pi$
$811$	−31.7771	−1.11584	−0.557922	+	0.829893i	$0.688401\pi$
$812$	−51.2148	−1.79729
$813$	−2.88854	−0.101306
$814$	−0.437694	−0.0153412
$815$	14.7082	0.515206
$816$	−14.3951	−0.503930
$817$	26.4164	0.924193
$818$	−15.3820	−0.537818
$819$	0	0
$820$	−19.3262	−0.674902
$821$	25.3607	0.885094	0.442547	−	0.896745i	$-0.354075\pi$
$821$	25.3607	0.885094	0.442547	+	0.896745i	$0.354075\pi$
$822$	−2.03444	−0.0709593
$823$	−26.5967	−0.927104	−0.463552	−	0.886070i	$-0.653425\pi$
$823$	−26.5967	−0.927104	−0.463552	+	0.886070i	$0.653425\pi$
$824$	−28.9443	−1.00832
$825$	0.527864	0.0183779
$826$	−1.85410	−0.0645125
$827$	8.94427	0.311023	0.155511	−	0.987834i	$-0.450297\pi$
$827$	8.94427	0.311023	0.155511	+	0.987834i	$0.450297\pi$
$828$	−12.1803	−0.423296
$829$	−43.2492	−1.50211	−0.751054	−	0.660241i	$-0.770454\pi$
$829$	−43.2492	−1.50211	−0.751054	+	0.660241i	$0.770454\pi$
$830$	5.52786	0.191875
$831$	37.7639	1.31002
$832$	0	0
$833$	−38.0000	−1.31662
$834$	−23.0902	−0.799547
$835$	−17.1803	−0.594550
$836$	1.61803	0.0559609
$837$	0	0
$838$	10.9098	0.376874
$839$	−30.7082	−1.06016	−0.530082	−	0.847946i	$-0.677839\pi$
$839$	−30.7082	−1.06016	−0.530082	+	0.847946i	$0.677839\pi$
$840$	−21.1803	−0.730791
$841$	26.8328	0.925270
$842$	−3.70820	−0.127793
$843$	35.5279	1.22364
$844$	21.3262	0.734079
$845$	0	0
$846$	6.11146	0.210116
$847$	46.3607	1.59297
$848$	11.1246	0.382021
$849$	−23.9443	−0.821765
$850$	−2.14590	−0.0736037
$851$	−11.2918	−0.387078
$852$	−22.5623	−0.772972
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−37.7426	−1.29153
$855$	8.47214	0.289741
$856$	12.8885	0.440521
$857$	35.8885	1.22593	0.612965	−	0.790110i	$-0.289977\pi$
$857$	35.8885	1.22593	0.612965	+	0.790110i	$0.289977\pi$
$858$	0	0
$859$	21.8885	0.746827	0.373414	−	0.927665i	$-0.378187\pi$
$859$	21.8885	0.746827	0.373414	+	0.927665i	$0.378187\pi$
$860$	−10.0902	−0.344072
$861$	113.138	3.85572
$862$	−16.7984	−0.572155
$863$	−14.8328	−0.504915	−0.252457	−	0.967608i	$-0.581239\pi$
$863$	−14.8328	−0.504915	−0.252457	+	0.967608i	$0.581239\pi$
$864$	−12.5623	−0.427378
$865$	18.8885	0.642230
$866$	−8.97871	−0.305109
$867$	−11.0557	−0.375472
$868$	0	0
$869$	0	0
$870$	10.3262	0.350092
$871$	0	0
$872$	−4.47214	−0.151446
$873$	6.94427	0.235028
$874$	−9.85410	−0.333320
$875$	4.23607	0.143205
$876$	−21.7082	−0.733452
$877$	33.7214	1.13869	0.569345	−	0.822099i	$-0.307197\pi$
$877$	33.7214	1.13869	0.569345	+	0.822099i	$0.307197\pi$
$878$	−14.0344	−0.473639
$879$	66.9574	2.25842
$880$	−0.437694	−0.0147547
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	13.5279	0.455507
$883$	−42.8328	−1.44144	−0.720720	−	0.693227i	$-0.756188\pi$
$883$	−42.8328	−1.44144	−0.720720	+	0.693227i	$0.756188\pi$
$884$	0	0
$885$	−1.58359	−0.0532319
$886$	0.583592	0.0196062
$887$	44.4853	1.49367	0.746835	−	0.665009i	$-0.231572\pi$
$887$	44.4853	1.49367	0.746835	+	0.665009i	$0.231572\pi$
$888$	15.0000	0.503367
$889$	17.9443	0.601832
$890$	−5.56231	−0.186449
$891$	−2.59675	−0.0869943
$892$	1.14590	0.0383675
$893$	−20.9443	−0.700873
$894$	−6.25735	−0.209277
$895$	8.23607	0.275301
$896$	48.2148	1.61074
$897$	0	0
$898$	−2.43769	−0.0813469
$899$	0	0
$900$	−3.23607	−0.107869
$901$	−20.8328	−0.694042
$902$	−1.74265	−0.0580238
$903$	59.0689	1.96569
$904$	−3.29180	−0.109484
$905$	−6.00000	−0.199447
$906$	−11.0557	−0.367302
$907$	20.1246	0.668227	0.334113	−	0.942533i	$-0.391563\pi$
$907$	20.1246	0.668227	0.334113	+	0.942533i	$0.391563\pi$
$908$	−10.0902	−0.334854
$909$	1.05573	0.0350163
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−17.5623	−0.581546
$913$	−2.11146	−0.0698790
$914$	0.978714	0.0323730
$915$	−32.2361	−1.06569
$916$	25.7082	0.849423
$917$	50.8328	1.67865
$918$	4.79837	0.158370
$919$	46.7082	1.54076	0.770381	−	0.637584i	$-0.220066\pi$
$919$	46.7082	1.54076	0.770381	+	0.637584i	$0.220066\pi$
$920$	8.41641	0.277481
$921$	−55.7771	−1.83792
$922$	0.978714	0.0322322
$923$	0	0
$924$	3.61803	0.119025
$925$	−3.00000	−0.0986394
$926$	−6.83282	−0.224540
$927$	25.8885	0.850291
$928$	−41.9787	−1.37802
$929$	−19.9443	−0.654350	−0.327175	−	0.944964i	$-0.606097\pi$
$929$	−19.9443	−0.654350	−0.327175	+	0.944964i	$0.606097\pi$
$930$	0	0
$931$	−46.3607	−1.51941
$932$	25.7082	0.842100
$933$	−53.6656	−1.75693
$934$	5.52786	0.180877
$935$	0.819660	0.0268058
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	−7.09017	−0.231502
$939$	8.69505	0.283752
$940$	8.00000	0.260931
$941$	11.8885	0.387555	0.193778	−	0.981045i	$-0.437926\pi$
$941$	11.8885	0.387555	0.193778	+	0.981045i	$0.437926\pi$
$942$	−24.8754	−0.810484
$943$	−44.9574	−1.46402
$944$	1.31308	0.0427372
$945$	−9.47214	−0.308129
$946$	−0.909830	−0.0295811
$947$	−39.1803	−1.27319	−0.636595	−	0.771198i	$-0.719658\pi$
$947$	−39.1803	−1.27319	−0.636595	+	0.771198i	$0.719658\pi$
$948$	0	0
$949$	0	0
$950$	−2.61803	−0.0849402
$951$	26.5836	0.862032
$952$	−32.8885	−1.06592
$953$	42.1935	1.36678	0.683391	−	0.730053i	$-0.260505\pi$
$953$	42.1935	1.36678	0.683391	+	0.730053i	$0.260505\pi$
$954$	7.41641	0.240115
$955$	27.1803	0.879535
$956$	−41.8885	−1.35477
$957$	−3.94427	−0.127500
$958$	−19.8541	−0.641457
$959$	6.23607	0.201373
$960$	0.527864	0.0170367
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−11.5279	−0.371480
$964$	24.2705	0.781700
$965$	−3.47214	−0.111772
$966$	−22.0344	−0.708947
$967$	54.8328	1.76330	0.881652	−	0.471900i	$-0.156432\pi$
$967$	54.8328	1.76330	0.881652	+	0.471900i	$0.156432\pi$
$968$	24.4721	0.786564
$969$	32.8885	1.05653
$970$	−2.14590	−0.0689006
$971$	−11.7639	−0.377523	−0.188761	−	0.982023i	$-0.560447\pi$
$971$	−11.7639	−0.377523	−0.188761	+	0.982023i	$0.560447\pi$
$972$	28.9443	0.928388
$973$	70.7771	2.26901
$974$	−8.72949	−0.279711
$975$	0	0
$976$	26.7295	0.855590
$977$	−7.58359	−0.242621	−0.121310	−	0.992615i	$-0.538710\pi$
$977$	−7.58359	−0.242621	−0.121310	+	0.992615i	$0.538710\pi$
$978$	−20.3262	−0.649961
$979$	2.12461	0.0679029
$980$	17.7082	0.565668
$981$	4.00000	0.127710
$982$	−0.145898	−0.00465579
$983$	−38.8328	−1.23857	−0.619287	−	0.785165i	$-0.712578\pi$
$983$	−38.8328	−1.23857	−0.619287	+	0.785165i	$0.712578\pi$
$984$	59.7214	1.90385
$985$	3.00000	0.0955879
$986$	16.0344	0.510641
$987$	−46.8328	−1.49070
$988$	0	0
$989$	−23.4721	−0.746371
$990$	−0.291796	−0.00927389
$991$	−46.7082	−1.48373	−0.741867	−	0.670547i	$-0.766060\pi$
$991$	−46.7082	−1.48373	−0.741867	+	0.670547i	$0.766060\pi$
$992$	0	0
$993$	12.6393	0.401097
$994$	−16.3262	−0.517837
$995$	1.29180	0.0409527
$996$	32.3607	1.02539
$997$	−18.8885	−0.598206	−0.299103	−	0.954221i	$-0.596687\pi$
$997$	−18.8885	−0.598206	−0.299103	+	0.954221i	$0.596687\pi$
$998$	13.5279	0.428217
$999$	6.70820	0.212238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.b.1.2		2
3.2	odd	2		7605.2.a.bf.1.1		2
5.4	even	2		4225.2.a.y.1.1		2
13.2	odd	12		845.2.m.e.316.3		8
13.3	even	3		845.2.e.g.191.1		4
13.4	even	6		65.2.e.a.16.2	✓	4
13.5	odd	4		845.2.c.c.506.2		4
13.6	odd	12		845.2.m.e.361.2		8
13.7	odd	12		845.2.m.e.361.3		8
13.8	odd	4		845.2.c.c.506.3		4
13.9	even	3		845.2.e.g.146.1		4
13.10	even	6		65.2.e.a.61.2	yes	4
13.11	odd	12		845.2.m.e.316.2		8
13.12	even	2		845.2.a.e.1.1		2
39.17	odd	6		585.2.j.e.406.1		4
39.23	odd	6		585.2.j.e.451.1		4
39.38	odd	2		7605.2.a.ba.1.2		2
52.23	odd	6		1040.2.q.n.321.2		4
52.43	odd	6		1040.2.q.n.81.2		4
65.4	even	6		325.2.e.b.276.1		4
65.17	odd	12		325.2.o.a.224.2		8
65.23	odd	12		325.2.o.a.74.2		8
65.43	odd	12		325.2.o.a.224.3		8
65.49	even	6		325.2.e.b.126.1		4
65.62	odd	12		325.2.o.a.74.3		8
65.64	even	2		4225.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.a.16.2	✓	4	13.4	even	6
65.2.e.a.61.2	yes	4	13.10	even	6
325.2.e.b.126.1		4	65.49	even	6
325.2.e.b.276.1		4	65.4	even	6
325.2.o.a.74.2		8	65.23	odd	12
325.2.o.a.74.3		8	65.62	odd	12
325.2.o.a.224.2		8	65.17	odd	12
325.2.o.a.224.3		8	65.43	odd	12
585.2.j.e.406.1		4	39.17	odd	6
585.2.j.e.451.1		4	39.23	odd	6
845.2.a.b.1.2		2	1.1	even	1	trivial
845.2.a.e.1.1		2	13.12	even	2
845.2.c.c.506.2		4	13.5	odd	4
845.2.c.c.506.3		4	13.8	odd	4
845.2.e.g.146.1		4	13.9	even	3
845.2.e.g.191.1		4	13.3	even	3
845.2.m.e.316.2		8	13.11	odd	12
845.2.m.e.316.3		8	13.2	odd	12
845.2.m.e.361.2		8	13.6	odd	12
845.2.m.e.361.3		8	13.7	odd	12
1040.2.q.n.81.2		4	52.43	odd	6
1040.2.q.n.321.2		4	52.23	odd	6
4225.2.a.u.1.2		2	65.64	even	2
4225.2.a.y.1.1		2	5.4	even	2
7605.2.a.ba.1.2		2	39.38	odd	2
7605.2.a.bf.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +2.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} +1.38197 q^{6} -4.23607 q^{7} -2.23607 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +2.23607 q^{3} -1.61803 q^{4} -1.00000 q^{5} +1.38197 q^{6} -4.23607 q^{7} -2.23607 q^{8} +2.00000 q^{9} -0.618034 q^{10} +0.236068 q^{11} -3.61803 q^{12} -2.61803 q^{14} -2.23607 q^{15} +1.85410 q^{16} -3.47214 q^{17} +1.23607 q^{18} -4.23607 q^{19} +1.61803 q^{20} -9.47214 q^{21} +0.145898 q^{22} +3.76393 q^{23} -5.00000 q^{24} +1.00000 q^{25} -2.23607 q^{27} +6.85410 q^{28} -7.47214 q^{29} -1.38197 q^{30} +5.61803 q^{32} +0.527864 q^{33} -2.14590 q^{34} +4.23607 q^{35} -3.23607 q^{36} -3.00000 q^{37} -2.61803 q^{38} +2.23607 q^{40} -11.9443 q^{41} -5.85410 q^{42} -6.23607 q^{43} -0.381966 q^{44} -2.00000 q^{45} +2.32624 q^{46} +4.94427 q^{47} +4.14590 q^{48} +10.9443 q^{49} +0.618034 q^{50} -7.76393 q^{51} +6.00000 q^{53} -1.38197 q^{54} -0.236068 q^{55} +9.47214 q^{56} -9.47214 q^{57} -4.61803 q^{58} +0.708204 q^{59} +3.61803 q^{60} +14.4164 q^{61} -8.47214 q^{63} -0.236068 q^{64} +0.326238 q^{66} +2.70820 q^{67} +5.61803 q^{68} +8.41641 q^{69} +2.61803 q^{70} +6.23607 q^{71} -4.47214 q^{72} +6.00000 q^{73} -1.85410 q^{74} +2.23607 q^{75} +6.85410 q^{76} -1.00000 q^{77} -1.85410 q^{80} -11.0000 q^{81} -7.38197 q^{82} -8.94427 q^{83} +15.3262 q^{84} +3.47214 q^{85} -3.85410 q^{86} -16.7082 q^{87} -0.527864 q^{88} +9.00000 q^{89} -1.23607 q^{90} -6.09017 q^{92} +3.05573 q^{94} +4.23607 q^{95} +12.5623 q^{96} +3.47214 q^{97} +6.76393 q^{98} +0.472136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 5 * q^6 - 4 * q^7 + 4 * q^9	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 5 q^{6} - 4 q^{7} + 4 q^{9} + q^{10} - 4 q^{11} - 5 q^{12} - 3 q^{14} - 3 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} + 7 q^{22} + 12 q^{23} - 10 q^{24} + 2 q^{25} + 7 q^{28} - 6 q^{29} - 5 q^{30} + 9 q^{32} + 10 q^{33} - 11 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{37} - 3 q^{38} - 6 q^{41} - 5 q^{42} - 8 q^{43} - 3 q^{44} - 4 q^{45} - 11 q^{46} - 8 q^{47} + 15 q^{48} + 4 q^{49} - q^{50} - 20 q^{51} + 12 q^{53} - 5 q^{54} + 4 q^{55} + 10 q^{56} - 10 q^{57} - 7 q^{58} - 12 q^{59} + 5 q^{60} + 2 q^{61} - 8 q^{63} + 4 q^{64} - 15 q^{66} - 8 q^{67} + 9 q^{68} - 10 q^{69} + 3 q^{70} + 8 q^{71} + 12 q^{73} + 3 q^{74} + 7 q^{76} - 2 q^{77} + 3 q^{80} - 22 q^{81} - 17 q^{82} + 15 q^{84} - 2 q^{85} - q^{86} - 20 q^{87} - 10 q^{88} + 18 q^{89} + 2 q^{90} - q^{92} + 24 q^{94} + 4 q^{95} + 5 q^{96} - 2 q^{97} + 18 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 5 * q^6 - 4 * q^7 + 4 * q^9 + q^10 - 4 * q^11 - 5 * q^12 - 3 * q^14 - 3 * q^16 + 2 * q^17 - 2 * q^18 - 4 * q^19 + q^20 - 10 * q^21 + 7 * q^22 + 12 * q^23 - 10 * q^24 + 2 * q^25 + 7 * q^28 - 6 * q^29 - 5 * q^30 + 9 * q^32 + 10 * q^33 - 11 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^37 - 3 * q^38 - 6 * q^41 - 5 * q^42 - 8 * q^43 - 3 * q^44 - 4 * q^45 - 11 * q^46 - 8 * q^47 + 15 * q^48 + 4 * q^49 - q^50 - 20 * q^51 + 12 * q^53 - 5 * q^54 + 4 * q^55 + 10 * q^56 - 10 * q^57 - 7 * q^58 - 12 * q^59 + 5 * q^60 + 2 * q^61 - 8 * q^63 + 4 * q^64 - 15 * q^66 - 8 * q^67 + 9 * q^68 - 10 * q^69 + 3 * q^70 + 8 * q^71 + 12 * q^73 + 3 * q^74 + 7 * q^76 - 2 * q^77 + 3 * q^80 - 22 * q^81 - 17 * q^82 + 15 * q^84 - 2 * q^85 - q^86 - 20 * q^87 - 10 * q^88 + 18 * q^89 + 2 * q^90 - q^92 + 24 * q^94 + 4 * q^95 + 5 * q^96 - 2 * q^97 + 18 * q^98 - 8 * q^99

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