LMFDB - Embedded newform 1421.2.b.b.1275.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1421,2,Mod(1275,1421)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1421.1275"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1421, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1421 = 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1421.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3467421272$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1275.1
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	1421.1275
Dual form			1421.2.b.b.1275.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.23607i q^{2} -2.23607i q^{3} -3.00000 q^{4} +3.00000 q^{5} -5.00000 q^{6} +2.23607i q^{8} -2.00000 q^{9} -6.70820i q^{10} -2.23607i q^{11} +6.70820i q^{12} +1.00000 q^{13} -6.70820i q^{15} -1.00000 q^{16} -4.47214i q^{17} +4.47214i q^{18} -9.00000 q^{20} -5.00000 q^{22} +6.00000 q^{23} +5.00000 q^{24} +4.00000 q^{25} -2.23607i q^{26} -2.23607i q^{27} +(-3.00000 - 4.47214i) q^{29} -15.0000 q^{30} +6.70820i q^{31} +6.70820i q^{32} -5.00000 q^{33} -10.0000 q^{34} +6.00000 q^{36} -2.23607i q^{39} +6.70820i q^{40} -4.47214i q^{41} +6.70820i q^{43} +6.70820i q^{44} -6.00000 q^{45} -13.4164i q^{46} +2.23607i q^{47} +2.23607i q^{48} -8.94427i q^{50} -10.0000 q^{51} -3.00000 q^{52} -9.00000 q^{53} -5.00000 q^{54} -6.70820i q^{55} +(-10.0000 + 6.70820i) q^{58} -6.00000 q^{59} +20.1246i q^{60} +13.4164i q^{61} +15.0000 q^{62} +13.0000 q^{64} +3.00000 q^{65} +11.1803i q^{66} +8.00000 q^{67} +13.4164i q^{68} -13.4164i q^{69} -4.47214i q^{72} -8.94427i q^{75} -5.00000 q^{78} +6.70820i q^{79} -3.00000 q^{80} -11.0000 q^{81} -10.0000 q^{82} +6.00000 q^{83} -13.4164i q^{85} +15.0000 q^{86} +(-10.0000 + 6.70820i) q^{87} +5.00000 q^{88} -4.47214i q^{89} +13.4164i q^{90} -18.0000 q^{92} +15.0000 q^{93} +5.00000 q^{94} +15.0000 q^{96} +13.4164i q^{97} +4.47214i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{4} + 6 q^{5} - 10 q^{6} - 4 q^{9} + 2 q^{13} - 2 q^{16} - 18 q^{20} - 10 q^{22} + 12 q^{23} + 10 q^{24} + 8 q^{25} - 6 q^{29} - 30 q^{30} - 10 q^{33} - 20 q^{34} + 12 q^{36} - 12 q^{45} - 20 q^{51}+ \cdots + 30 q^{96}+O(q^{100}) $ 2 * q - 6 * q^4 + 6 * q^5 - 10 * q^6 - 4 * q^9 + 2 * q^13 - 2 * q^16 - 18 * q^20 - 10 * q^22 + 12 * q^23 + 10 * q^24 + 8 * q^25 - 6 * q^29 - 30 * q^30 - 10 * q^33 - 20 * q^34 + 12 * q^36 - 12 * q^45 - 20 * q^51 - 6 * q^52 - 18 * q^53 - 10 * q^54 - 20 * q^58 - 12 * q^59 + 30 * q^62 + 26 * q^64 + 6 * q^65 + 16 * q^67 - 10 * q^78 - 6 * q^80 - 22 * q^81 - 20 * q^82 + 12 * q^83 + 30 * q^86 - 20 * q^87 + 10 * q^88 - 36 * q^92 + 30 * q^93 + 10 * q^94 + 30 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1421\mathbb{Z}\right)^\times$.

$n$	$785$	$1277$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	2.23607i		−	1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$		−	2.23607i		−	1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$		−	2.23607i		−	1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$		−	2.23607i		−	1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$	−3.00000			−1.50000
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$	−5.00000			−2.04124
$7$		0			0
$7$		0			0
$8$			2.23607i			0.790569i
$9$	−2.00000			−0.666667
$10$		−	6.70820i		−	2.12132i
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$			6.70820i			1.93649i
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$		−	6.70820i		−	1.73205i
$16$	−1.00000			−0.250000
$17$		−	4.47214i		−	1.08465i	−0.840168	−	0.542326i	$-0.817544\pi$
$17$		−	4.47214i		−	1.08465i	0.840168	−	0.542326i	$-0.182456\pi$
$18$			4.47214i			1.05409i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−9.00000			−2.01246
$21$		0			0
$22$	−5.00000			−1.06600
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$	5.00000			1.02062
$25$	4.00000			0.800000
$26$		−	2.23607i		−	0.438529i
$27$		−	2.23607i		−	0.430331i
$28$		0			0
$29$	−3.00000	−	4.47214i	−0.557086	−	0.830455i
$29$	−3.00000	−	4.47214i	−0.557086	−	0.830455i
$30$	−15.0000			−2.73861
$31$			6.70820i			1.20483i	0.798183	+	0.602414i	$0.205795\pi$
$31$			6.70820i			1.20483i	−0.798183	+	0.602414i	$0.794205\pi$
$32$			6.70820i			1.18585i
$33$	−5.00000			−0.870388
$34$	−10.0000			−1.71499
$35$		0			0
$36$	6.00000			1.00000
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		−	2.23607i		−	0.358057i
$40$			6.70820i			1.06066i
$41$		−	4.47214i		−	0.698430i	−0.937043	−	0.349215i	$-0.886448\pi$
$41$		−	4.47214i		−	0.698430i	0.937043	−	0.349215i	$-0.113552\pi$
$42$		0			0
$43$			6.70820i			1.02299i	0.859286	+	0.511496i	$0.170908\pi$
$43$			6.70820i			1.02299i	−0.859286	+	0.511496i	$0.829092\pi$
$44$			6.70820i			1.01130i
$45$	−6.00000			−0.894427
$46$		−	13.4164i		−	1.97814i
$47$			2.23607i			0.326164i	0.986613	+	0.163082i	$0.0521435\pi$
$47$			2.23607i			0.326164i	−0.986613	+	0.163082i	$0.947856\pi$
$48$			2.23607i			0.322749i
$49$		0			0
$50$		−	8.94427i		−	1.26491i
$51$	−10.0000			−1.40028
$52$	−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	$0.712106\pi$
$54$	−5.00000			−0.680414
$55$		−	6.70820i		−	0.904534i
$56$		0			0
$57$		0			0
$58$	−10.0000	+	6.70820i	−1.31306	+	0.880830i
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$			20.1246i			2.59808i
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$	15.0000			1.90500
$63$		0			0
$64$	13.0000			1.62500
$65$	3.00000			0.372104
$66$			11.1803i			1.37620i
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$			13.4164i			1.62698i
$69$		−	13.4164i		−	1.61515i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		−	4.47214i		−	0.527046i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	8.94427i		−	1.03280i
$76$		0			0
$77$		0			0
$78$	−5.00000			−0.566139
$79$			6.70820i			0.754732i	0.926064	+	0.377366i	$0.123170\pi$
$79$			6.70820i			0.754732i	−0.926064	+	0.377366i	$0.876830\pi$
$80$	−3.00000			−0.335410
$81$	−11.0000			−1.22222
$82$	−10.0000			−1.10432
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		−	13.4164i		−	1.45521i
$86$	15.0000			1.61749
$87$	−10.0000	+	6.70820i	−1.07211	+	0.719195i
$88$	5.00000			0.533002
$89$		−	4.47214i		−	0.474045i	−0.971504	−	0.237023i	$-0.923828\pi$
$89$		−	4.47214i		−	0.474045i	0.971504	−	0.237023i	$-0.0761716\pi$
$90$			13.4164i			1.41421i
$91$		0			0
$92$	−18.0000			−1.87663
$93$	15.0000			1.55543
$94$	5.00000			0.515711
$95$		0			0
$96$	15.0000			1.53093
$97$			13.4164i			1.36223i	0.732177	+	0.681115i	$0.238505\pi$
$97$			13.4164i			1.36223i	−0.732177	+	0.681115i	$0.761495\pi$
$98$		0			0
$99$			4.47214i			0.449467i
$100$	−12.0000			−1.20000
$101$		−	17.8885i		−	1.77998i	−0.455983	−	0.889988i	$-0.650712\pi$
$101$		−	17.8885i		−	1.77998i	0.455983	−	0.889988i	$-0.349288\pi$
$102$			22.3607i			2.21404i
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$			2.23607i			0.219265i
$105$		0			0
$106$			20.1246i			1.95468i
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$			6.70820i			0.645497i
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	−15.0000			−1.43019
$111$		0			0
$112$		0			0
$113$		−	8.94427i		−	0.841406i	−0.907198	−	0.420703i	$-0.861783\pi$
$113$		−	8.94427i		−	0.841406i	0.907198	−	0.420703i	$-0.138217\pi$
$114$		0			0
$115$	18.0000			1.67851
$116$	9.00000	+	13.4164i	0.835629	+	1.24568i
$117$	−2.00000			−0.184900
$118$			13.4164i			1.23508i
$119$		0			0
$120$	15.0000			1.36931
$121$	6.00000			0.545455
$122$	30.0000			2.71607
$123$	−10.0000			−0.901670
$124$		−	20.1246i		−	1.80724i
$125$	−3.00000			−0.268328
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	15.6525i		−	1.38350i
$129$	15.0000			1.32068
$130$		−	6.70820i		−	0.588348i
$131$			8.94427i			0.781465i	0.920504	+	0.390732i	$0.127778\pi$
$131$			8.94427i			0.781465i	−0.920504	+	0.390732i	$0.872222\pi$
$132$	15.0000			1.30558
$133$		0			0
$134$		−	17.8885i		−	1.54533i
$135$		−	6.70820i		−	0.577350i
$136$	10.0000			0.857493
$137$		−	8.94427i		−	0.764161i	−0.924129	−	0.382080i	$-0.875208\pi$
$137$		−	8.94427i		−	0.764161i	0.924129	−	0.382080i	$-0.124792\pi$
$138$	−30.0000			−2.55377
$139$	10.0000			0.848189			0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000			0.848189			0.424094	+	0.905618i	$0.360592\pi$
$140$		0			0
$141$	5.00000			0.421076
$142$		0			0
$143$		−	2.23607i		−	0.186989i
$144$	2.00000			0.166667
$145$	−9.00000	−	13.4164i	−0.747409	−	1.11417i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.0000			1.22885			0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000			1.22885			0.614424	+	0.788976i	$0.289388\pi$
$150$	−20.0000			−1.63299
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$		0			0
$153$			8.94427i			0.723102i
$154$		0			0
$155$			20.1246i			1.61645i
$156$			6.70820i			0.537086i
$157$		−	13.4164i		−	1.07075i	−0.844616	−	0.535373i	$-0.820171\pi$
$157$		−	13.4164i		−	1.07075i	0.844616	−	0.535373i	$-0.179829\pi$
$158$	15.0000			1.19334
$159$			20.1246i			1.59599i
$160$			20.1246i			1.59099i
$161$		0			0
$162$			24.5967i			1.93250i
$163$			20.1246i			1.57628i	0.615495	+	0.788141i	$0.288956\pi$
$163$			20.1246i			1.57628i	−0.615495	+	0.788141i	$0.711044\pi$
$164$			13.4164i			1.04765i
$165$	−15.0000			−1.16775
$166$		−	13.4164i		−	1.04132i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$	−30.0000			−2.30089
$171$		0			0
$172$		−	20.1246i		−	1.53449i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$	15.0000	+	22.3607i	1.13715	+	1.69516i
$175$		0			0
$176$			2.23607i			0.168550i
$177$			13.4164i			1.00844i
$178$	−10.0000			−0.749532
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	18.0000			1.34164
$181$	−5.00000			−0.371647			−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000			−0.371647			−0.185824	+	0.982583i	$0.559495\pi$
$182$		0			0
$183$	30.0000			2.21766
$184$			13.4164i			0.989071i
$185$		0			0
$186$		−	33.5410i		−	2.45935i
$187$	−10.0000			−0.731272
$188$		−	6.70820i		−	0.489246i
$189$		0			0
$190$		0			0
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$		−	29.0689i		−	2.09787i
$193$			13.4164i			0.965734i	0.875694	+	0.482867i	$0.160405\pi$
$193$			13.4164i			0.965734i	−0.875694	+	0.482867i	$0.839595\pi$
$194$	30.0000			2.15387
$195$		−	6.70820i		−	0.480384i
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$	10.0000			0.710669
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$			8.94427i			0.632456i
$201$		−	17.8885i		−	1.26176i
$202$	−40.0000			−2.81439
$203$		0			0
$204$	30.0000			2.10042
$205$		−	13.4164i		−	0.937043i
$206$		−	8.94427i		−	0.623177i
$207$	−12.0000			−0.834058
$208$	−1.00000			−0.0693375
$209$		0			0
$210$		0			0
$211$		−	6.70820i		−	0.461812i	−0.972976	−	0.230906i	$-0.925831\pi$
$211$		−	6.70820i		−	0.461812i	0.972976	−	0.230906i	$-0.0741690\pi$
$212$	27.0000			1.85437
$213$		0			0
$214$		−	40.2492i		−	2.75138i
$215$			20.1246i			1.37249i
$216$	5.00000			0.340207
$217$		0			0
$218$		−	11.1803i		−	0.757228i
$219$		0			0
$220$			20.1246i			1.35680i
$221$		−	4.47214i		−	0.300828i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	−8.00000			−0.533333
$226$	−20.0000			−1.33038
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$		−	13.4164i		−	0.886581i	−0.896378	−	0.443291i	$-0.853811\pi$
$229$		−	13.4164i		−	0.886581i	0.896378	−	0.443291i	$-0.146189\pi$
$230$		−	40.2492i		−	2.65396i
$231$		0			0
$232$	10.0000	−	6.70820i	0.656532	−	0.440415i
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$			4.47214i			0.292353i
$235$			6.70820i			0.437595i
$236$	18.0000			1.17170
$237$	15.0000			0.974355
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$			6.70820i			0.433013i
$241$	25.0000			1.61039			0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000			1.61039			0.805196	+	0.593009i	$0.202060\pi$
$242$		−	13.4164i		−	0.862439i
$243$			17.8885i			1.14755i
$244$		−	40.2492i		−	2.57669i
$245$		0			0
$246$			22.3607i			1.42566i
$247$		0			0
$248$	−15.0000			−0.952501
$249$		−	13.4164i		−	0.850230i
$250$			6.70820i			0.424264i
$251$			15.6525i			0.987976i	0.869469	+	0.493988i	$0.164461\pi$
$251$			15.6525i			0.987976i	−0.869469	+	0.493988i	$0.835539\pi$
$252$		0			0
$253$		−	13.4164i		−	0.843482i
$254$		0			0
$255$	−30.0000			−1.87867
$256$	−9.00000			−0.562500
$257$	3.00000			0.187135			0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000			0.187135			0.0935674	+	0.995613i	$0.470173\pi$
$258$		−	33.5410i		−	2.08817i
$259$		0			0
$260$	−9.00000			−0.558156
$261$	6.00000	+	8.94427i	0.371391	+	0.553637i
$262$	20.0000			1.23560
$263$			11.1803i			0.689409i	0.938711	+	0.344705i	$0.112021\pi$
$263$			11.1803i			0.689409i	−0.938711	+	0.344705i	$0.887979\pi$
$264$		−	11.1803i		−	0.688102i
$265$	−27.0000			−1.65860
$266$		0			0
$267$	−10.0000			−0.611990
$268$	−24.0000			−1.46603
$269$		−	4.47214i		−	0.272671i	−0.990663	−	0.136335i	$-0.956467\pi$
$269$		−	4.47214i		−	0.272671i	0.990663	−	0.136335i	$-0.0435325\pi$
$270$	−15.0000			−0.912871
$271$		−	20.1246i		−	1.22248i	−0.791444	−	0.611242i	$-0.790670\pi$
$271$		−	20.1246i		−	1.22248i	0.791444	−	0.611242i	$-0.209330\pi$
$272$			4.47214i			0.271163i
$273$		0			0
$274$	−20.0000			−1.20824
$275$		−	8.94427i		−	0.539360i
$276$			40.2492i			2.42272i
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		−	22.3607i		−	1.34110i
$279$		−	13.4164i		−	0.803219i
$280$		0			0
$281$	3.00000			0.178965			0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000			0.178965			0.0894825	+	0.995988i	$0.471479\pi$
$282$		−	11.1803i		−	0.665780i
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$		0			0
$285$		0			0
$286$	−5.00000			−0.295656
$287$		0			0
$288$		−	13.4164i		−	0.790569i
$289$	−3.00000			−0.176471
$290$	−30.0000	+	20.1246i	−1.76166	+	1.18176i
$291$	30.0000			1.75863
$292$		0			0
$293$			8.94427i			0.522530i	0.965267	+	0.261265i	$0.0841396\pi$
$293$			8.94427i			0.522530i	−0.965267	+	0.261265i	$0.915860\pi$
$294$		0			0
$295$	−18.0000			−1.04800
$296$		0			0
$297$	−5.00000			−0.290129
$298$		−	33.5410i		−	1.94298i
$299$	6.00000			0.346989
$300$			26.8328i			1.54919i
$301$		0			0
$302$			22.3607i			1.28671i
$303$	−40.0000			−2.29794
$304$		0			0
$305$			40.2492i			2.30466i
$306$	20.0000			1.14332
$307$			20.1246i			1.14857i	0.818655	+	0.574286i	$0.194720\pi$
$307$			20.1246i			1.14857i	−0.818655	+	0.574286i	$0.805280\pi$
$308$		0			0
$309$		−	8.94427i		−	0.508822i
$310$	45.0000			2.55583
$311$			8.94427i			0.507183i	0.967311	+	0.253592i	$0.0816119\pi$
$311$			8.94427i			0.507183i	−0.967311	+	0.253592i	$0.918388\pi$
$312$	5.00000			0.283069
$313$	−29.0000			−1.63918			−0.819588	−	0.572953i	$-0.805798\pi$
$313$	−29.0000			−1.63918			−0.819588	+	0.572953i	$0.805798\pi$
$314$	−30.0000			−1.69300
$315$		0			0
$316$		−	20.1246i		−	1.13210i
$317$		−	22.3607i		−	1.25590i	−0.778253	−	0.627950i	$-0.783894\pi$
$317$		−	22.3607i		−	1.25590i	0.778253	−	0.627950i	$-0.216106\pi$
$318$	45.0000			2.52347
$319$	−10.0000	+	6.70820i	−0.559893	+	0.375587i
$320$	39.0000			2.18017
$321$		−	40.2492i		−	2.24649i
$322$		0			0
$323$		0			0
$324$	33.0000			1.83333
$325$	4.00000			0.221880
$326$	45.0000			2.49232
$327$		−	11.1803i		−	0.618274i
$328$	10.0000			0.552158
$329$		0			0
$330$			33.5410i			1.84637i
$331$		−	33.5410i		−	1.84358i	−0.387688	−	0.921791i	$-0.626726\pi$
$331$		−	33.5410i		−	1.84358i	0.387688	−	0.921791i	$-0.373274\pi$
$332$	−18.0000			−0.987878
$333$		0			0
$334$		−	26.8328i		−	1.46823i
$335$	24.0000			1.31126
$336$		0			0
$337$			13.4164i			0.730838i	0.930843	+	0.365419i	$0.119074\pi$
$337$			13.4164i			0.730838i	−0.930843	+	0.365419i	$0.880926\pi$
$338$			26.8328i			1.45951i
$339$	−20.0000			−1.08625
$340$			40.2492i			2.18282i
$341$	15.0000			0.812296
$342$		0			0
$343$		0			0
$344$	−15.0000			−0.808746
$345$		−	40.2492i		−	2.16695i
$346$		−	13.4164i		−	0.721271i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$	30.0000	−	20.1246i	1.60817	−	1.07879i
$349$	1.00000			0.0535288			0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000			0.0535288			0.0267644	+	0.999642i	$0.491480\pi$
$350$		0			0
$351$		−	2.23607i		−	0.119352i
$352$	15.0000			0.799503
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$	30.0000			1.59448
$355$		0			0
$356$			13.4164i			0.711068i
$357$		0			0
$358$		0			0
$359$		−	2.23607i		−	0.118015i	−0.998258	−	0.0590076i	$-0.981206\pi$
$359$		−	2.23607i		−	0.118015i	0.998258	−	0.0590076i	$-0.0187936\pi$
$360$		−	13.4164i		−	0.707107i
$361$	19.0000			1.00000
$362$			11.1803i			0.587626i
$363$		−	13.4164i		−	0.704179i
$364$		0			0
$365$		0			0
$366$		−	67.0820i		−	3.50643i
$367$		−	26.8328i		−	1.40066i	−0.713818	−	0.700331i	$-0.753036\pi$
$367$		−	26.8328i		−	1.40066i	0.713818	−	0.700331i	$-0.246964\pi$
$368$	−6.00000			−0.312772
$369$			8.94427i			0.465620i
$370$		0			0
$371$		0			0
$372$	−45.0000			−2.33314
$373$	−31.0000			−1.60512			−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000			−1.60512			−0.802560	+	0.596572i	$0.796529\pi$
$374$			22.3607i			1.15624i
$375$			6.70820i			0.346410i
$376$	−5.00000			−0.257855
$377$	−3.00000	−	4.47214i	−0.154508	−	0.230327i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	−20.0000			−1.02329
$383$	−6.00000			−0.306586			−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000			−0.306586			−0.153293	+	0.988181i	$0.548988\pi$
$384$	−35.0000			−1.78609
$385$		0			0
$386$	30.0000			1.52696
$387$		−	13.4164i		−	0.681994i
$388$		−	40.2492i		−	2.04334i
$389$			17.8885i			0.906985i	0.891260	+	0.453493i	$0.149822\pi$
$389$			17.8885i			0.906985i	−0.891260	+	0.453493i	$0.850178\pi$
$390$	−15.0000			−0.759555
$391$		−	26.8328i		−	1.35699i
$392$		0			0
$393$	20.0000			1.00887
$394$			40.2492i			2.02773i
$395$			20.1246i			1.01258i
$396$		−	13.4164i		−	0.674200i
$397$	7.00000			0.351320			0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000			0.351320			0.175660	+	0.984451i	$0.443794\pi$
$398$			31.3050i			1.56918i
$399$		0			0
$400$	−4.00000			−0.200000
$401$	15.0000			0.749064			0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000			0.749064			0.374532	+	0.927214i	$0.377803\pi$
$402$	−40.0000			−1.99502
$403$			6.70820i			0.334159i
$404$			53.6656i			2.66996i
$405$	−33.0000			−1.63978
$406$		0			0
$407$		0			0
$408$		−	22.3607i		−	1.10702i
$409$			26.8328i			1.32680i	0.748266	+	0.663399i	$0.230887\pi$
$409$			26.8328i			1.32680i	−0.748266	+	0.663399i	$0.769113\pi$
$410$	−30.0000			−1.48159
$411$	−20.0000			−0.986527
$412$	−12.0000			−0.591198
$413$		0			0
$414$			26.8328i			1.31876i
$415$	18.0000			0.883585
$416$			6.70820i			0.328897i
$417$		−	22.3607i		−	1.09501i
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$		−	13.4164i		−	0.653876i	−0.945046	−	0.326938i	$-0.893983\pi$
$421$		−	13.4164i		−	0.653876i	0.945046	−	0.326938i	$-0.106017\pi$
$422$	−15.0000			−0.730189
$423$		−	4.47214i		−	0.217443i
$424$		−	20.1246i		−	0.977338i
$425$		−	17.8885i		−	0.867722i
$426$		0			0
$427$		0			0
$428$	−54.0000			−2.61019
$429$	−5.00000			−0.241402
$430$	45.0000			2.17009
$431$	18.0000			0.867029			0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000			0.867029			0.433515	+	0.901146i	$0.357273\pi$
$432$			2.23607i			0.107583i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−30.0000	+	20.1246i	−1.43839	+	0.964901i
$436$	−15.0000			−0.718370
$437$		0			0
$438$		0			0
$439$	34.0000			1.62273			0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000			1.62273			0.811366	+	0.584539i	$0.198725\pi$
$440$	15.0000			0.715097
$441$		0			0
$442$	−10.0000			−0.475651
$443$			17.8885i			0.849910i	0.905214	+	0.424955i	$0.139710\pi$
$443$			17.8885i			0.849910i	−0.905214	+	0.424955i	$0.860290\pi$
$444$		0			0
$445$		−	13.4164i		−	0.635999i
$446$		−	35.7771i		−	1.69409i
$447$		−	33.5410i		−	1.58644i
$448$		0			0
$449$		−	35.7771i		−	1.68843i	−0.536009	−	0.844213i	$-0.680069\pi$
$449$		−	35.7771i		−	1.68843i	0.536009	−	0.844213i	$-0.319931\pi$
$450$			17.8885i			0.843274i
$451$	−10.0000			−0.470882
$452$			26.8328i			1.26211i
$453$			22.3607i			1.05060i
$454$			26.8328i			1.25933i
$455$		0			0
$456$		0			0
$457$	2.00000			0.0935561			0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000			0.0935561			0.0467780	+	0.998905i	$0.485105\pi$
$458$	−30.0000			−1.40181
$459$	−10.0000			−0.466760
$460$	−54.0000			−2.51776
$461$			8.94427i			0.416576i	0.978068	+	0.208288i	$0.0667892\pi$
$461$			8.94427i			0.416576i	−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$	3.00000	+	4.47214i	0.139272	+	0.207614i
$465$	45.0000			2.08683
$466$			20.1246i			0.932255i
$467$			2.23607i			0.103473i	0.998661	+	0.0517364i	$0.0164756\pi$
$467$			2.23607i			0.103473i	−0.998661	+	0.0517364i	$0.983524\pi$
$468$	6.00000			0.277350
$469$		0			0
$470$	15.0000			0.691898
$471$	−30.0000			−1.38233
$472$		−	13.4164i		−	0.617540i
$473$	15.0000			0.689701
$474$		−	33.5410i		−	1.54059i
$475$		0			0
$476$		0			0
$477$	18.0000			0.824163
$478$		−	13.4164i		−	0.613652i
$479$		−	11.1803i		−	0.510843i	−0.966830	−	0.255421i	$-0.917786\pi$
$479$		−	11.1803i		−	0.510843i	0.966830	−	0.255421i	$-0.0822142\pi$
$480$	45.0000			2.05396
$481$		0			0
$482$		−	55.9017i		−	2.54625i
$483$		0			0
$484$	−18.0000			−0.818182
$485$			40.2492i			1.82762i
$486$	40.0000			1.81444
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$	−30.0000			−1.35804
$489$	45.0000			2.03497
$490$		0			0
$491$			24.5967i			1.11004i	0.831838	+	0.555018i	$0.187289\pi$
$491$			24.5967i			1.11004i	−0.831838	+	0.555018i	$0.812711\pi$
$492$	30.0000			1.35250
$493$	−20.0000	+	13.4164i	−0.900755	+	0.604245i
$494$		0			0
$495$			13.4164i			0.603023i
$496$		−	6.70820i		−	0.301207i
$497$		0			0
$498$	−30.0000			−1.34433
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$	9.00000			0.402492
$501$		−	26.8328i		−	1.19880i
$502$	35.0000			1.56213
$503$			15.6525i			0.697909i	0.937140	+	0.348955i	$0.113463\pi$
$503$			15.6525i			0.697909i	−0.937140	+	0.348955i	$0.886537\pi$
$504$		0			0
$505$		−	53.6656i		−	2.38809i
$506$	−30.0000			−1.33366
$507$			26.8328i			1.19169i
$508$		0			0
$509$	−15.0000			−0.664863			−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000			−0.664863			−0.332432	+	0.943127i	$0.607869\pi$
$510$			67.0820i			2.97044i
$511$		0			0
$512$		−	11.1803i		−	0.494106i
$513$		0			0
$514$		−	6.70820i		−	0.295886i
$515$	12.0000			0.528783
$516$	−45.0000			−1.98101
$517$	5.00000			0.219900
$518$		0			0
$519$		−	13.4164i		−	0.588915i
$520$			6.70820i			0.294174i
$521$	27.0000			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$522$	20.0000	−	13.4164i	0.875376	−	0.587220i
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$		−	26.8328i		−	1.17220i
$525$		0			0
$526$	25.0000			1.09005
$527$	30.0000			1.30682
$528$	5.00000			0.217597
$529$	13.0000			0.565217
$530$			60.3738i			2.62247i
$531$	12.0000			0.520756
$532$		0			0
$533$		−	4.47214i		−	0.193710i
$534$			22.3607i			0.967641i
$535$	54.0000			2.33462
$536$			17.8885i			0.772667i
$537$		0			0
$538$	−10.0000			−0.431131
$539$		0			0
$540$			20.1246i			0.866025i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−45.0000			−1.93292
$543$			11.1803i			0.479794i
$544$	30.0000			1.28624
$545$	15.0000			0.642529
$546$		0			0
$547$	38.0000			1.62476			0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000			1.62476			0.812381	+	0.583127i	$0.198171\pi$
$548$			26.8328i			1.14624i
$549$		−	26.8328i		−	1.14520i
$550$	−20.0000			−0.852803
$551$		0			0
$552$	30.0000			1.27688
$553$		0			0
$554$		−	4.47214i		−	0.190003i
$555$		0			0
$556$	−30.0000			−1.27228
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$	−30.0000			−1.27000
$559$			6.70820i			0.283727i
$560$		0			0
$561$			22.3607i			0.944069i
$562$		−	6.70820i		−	0.282969i
$563$		−	11.1803i		−	0.471195i	−0.971851	−	0.235598i	$-0.924295\pi$
$563$		−	11.1803i		−	0.471195i	0.971851	−	0.235598i	$-0.0757047\pi$
$564$	−15.0000			−0.631614
$565$		−	26.8328i		−	1.12887i
$566$			31.3050i			1.31585i
$567$		0			0
$568$		0			0
$569$			17.8885i			0.749927i	0.927040	+	0.374963i	$0.122345\pi$
$569$			17.8885i			0.749927i	−0.927040	+	0.374963i	$0.877655\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$			6.70820i			0.280484i
$573$	−20.0000			−0.835512
$574$		0			0
$575$	24.0000			1.00087
$576$	−26.0000			−1.08333
$577$		−	40.2492i		−	1.67560i	−0.545979	−	0.837799i	$-0.683842\pi$
$577$		−	40.2492i		−	1.67560i	0.545979	−	0.837799i	$-0.316158\pi$
$578$			6.70820i			0.279024i
$579$	30.0000			1.24676
$580$	27.0000	+	40.2492i	1.12111	+	1.67126i
$581$		0			0
$582$		−	67.0820i		−	2.78064i
$583$			20.1246i			0.833476i
$584$		0			0
$585$	−6.00000			−0.248069
$586$	20.0000			0.826192
$587$	42.0000			1.73353			0.866763	−	0.498721i	$-0.166197\pi$
$587$	42.0000			1.73353			0.866763	+	0.498721i	$0.166197\pi$
$588$		0			0
$589$		0			0
$590$			40.2492i			1.65703i
$591$			40.2492i			1.65563i
$592$		0			0
$593$	−9.00000			−0.369586			−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000			−0.369586			−0.184793	+	0.982777i	$0.559161\pi$
$594$			11.1803i			0.458735i
$595$		0			0
$596$	−45.0000			−1.84327
$597$			31.3050i			1.28123i
$598$		−	13.4164i		−	0.548638i
$599$		−	29.0689i		−	1.18772i	−0.804568	−	0.593861i	$-0.797603\pi$
$599$		−	29.0689i		−	1.18772i	0.804568	−	0.593861i	$-0.202397\pi$
$600$	20.0000			0.816497
$601$			13.4164i			0.547267i	0.961834	+	0.273633i	$0.0882255\pi$
$601$			13.4164i			0.547267i	−0.961834	+	0.273633i	$0.911775\pi$
$602$		0			0
$603$	−16.0000			−0.651570
$604$	30.0000			1.22068
$605$	18.0000			0.731804
$606$			89.4427i			3.63336i
$607$			6.70820i			0.272278i	0.990690	+	0.136139i	$0.0434693\pi$
$607$			6.70820i			0.272278i	−0.990690	+	0.136139i	$0.956531\pi$
$608$		0			0
$609$		0			0
$610$	90.0000			3.64399
$611$			2.23607i			0.0904616i
$612$		−	26.8328i		−	1.08465i
$613$	−31.0000			−1.25208			−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000			−1.25208			−0.626039	+	0.779792i	$0.715325\pi$
$614$	45.0000			1.81605
$615$	−30.0000			−1.20972
$616$		0			0
$617$			44.7214i			1.80041i	0.435462	+	0.900207i	$0.356585\pi$
$617$			44.7214i			1.80041i	−0.435462	+	0.900207i	$0.643415\pi$
$618$	−20.0000			−0.804518
$619$			33.5410i			1.34813i	0.738673	+	0.674064i	$0.235453\pi$
$619$			33.5410i			1.34813i	−0.738673	+	0.674064i	$0.764547\pi$
$620$		−	60.3738i		−	2.42467i
$621$		−	13.4164i		−	0.538382i
$622$	20.0000			0.801927
$623$		0			0
$624$			2.23607i			0.0895144i
$625$	−29.0000			−1.16000
$626$			64.8460i			2.59177i
$627$		0			0
$628$			40.2492i			1.60612i
$629$		0			0
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$	−15.0000			−0.596668
$633$	−15.0000			−0.596196
$634$	−50.0000			−1.98575
$635$		0			0
$636$		−	60.3738i		−	2.39398i
$637$		0			0
$638$	15.0000	+	22.3607i	0.593856	+	0.885268i
$639$		0			0
$640$		−	46.9574i		−	1.85616i
$641$		−	22.3607i		−	0.883194i	−0.897214	−	0.441597i	$-0.854412\pi$
$641$		−	22.3607i		−	0.883194i	0.897214	−	0.441597i	$-0.145588\pi$
$642$	−90.0000			−3.55202
$643$	−26.0000			−1.02534			−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000			−1.02534			−0.512670	+	0.858586i	$0.671344\pi$
$644$		0			0
$645$	45.0000			1.77187
$646$		0			0
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$		−	24.5967i		−	0.966252i
$649$			13.4164i			0.526640i
$650$		−	8.94427i		−	0.350823i
$651$		0			0
$652$		−	60.3738i		−	2.36442i
$653$		−	49.1935i		−	1.92509i	−0.271122	−	0.962545i	$-0.587395\pi$
$653$		−	49.1935i		−	1.92509i	0.271122	−	0.962545i	$-0.412605\pi$
$654$	−25.0000			−0.977577
$655$			26.8328i			1.04844i
$656$			4.47214i			0.174608i
$657$		0			0
$658$		0			0
$659$			38.0132i			1.48078i	0.672176	+	0.740391i	$0.265360\pi$
$659$			38.0132i			1.48078i	−0.672176	+	0.740391i	$0.734640\pi$
$660$	45.0000			1.75162
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$	−75.0000			−2.91496
$663$	−10.0000			−0.388368
$664$			13.4164i			0.520658i
$665$		0			0
$666$		0			0
$667$	−18.0000	−	26.8328i	−0.696963	−	1.03897i
$668$	−36.0000			−1.39288
$669$		−	35.7771i		−	1.38322i
$670$		−	53.6656i		−	2.07328i
$671$	30.0000			1.15814
$672$		0			0
$673$	11.0000			0.424019			0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000			0.424019			0.212009	+	0.977268i	$0.431999\pi$
$674$	30.0000			1.15556
$675$		−	8.94427i		−	0.344265i
$676$	36.0000			1.38462
$677$		−	17.8885i		−	0.687513i	−0.939059	−	0.343756i	$-0.888301\pi$
$677$		−	17.8885i		−	0.687513i	0.939059	−	0.343756i	$-0.111699\pi$
$678$			44.7214i			1.71751i
$679$		0			0
$680$	30.0000			1.15045
$681$			26.8328i			1.02824i
$682$		−	33.5410i		−	1.28435i
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$		−	26.8328i		−	1.02523i
$686$		0			0
$687$	−30.0000			−1.14457
$688$		−	6.70820i		−	0.255748i
$689$	−9.00000			−0.342873
$690$	−90.0000			−3.42624
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$	−18.0000			−0.684257
$693$		0			0
$694$			26.8328i			1.01856i
$695$	30.0000			1.13796
$696$	−15.0000	−	22.3607i	−0.568574	−	0.847579i
$697$	−20.0000			−0.757554
$698$		−	2.23607i		−	0.0846364i
$699$			20.1246i			0.761183i
$700$		0			0
$701$	−45.0000			−1.69963			−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000			−1.69963			−0.849813	+	0.527084i	$0.823285\pi$
$702$	−5.00000			−0.188713
$703$		0			0
$704$		−	29.0689i		−	1.09557i
$705$	15.0000			0.564933
$706$		−	13.4164i		−	0.504933i
$707$		0			0
$708$		−	40.2492i		−	1.51266i
$709$	35.0000			1.31445			0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000			1.31445			0.657226	+	0.753693i	$0.271730\pi$
$710$		0			0
$711$		−	13.4164i		−	0.503155i
$712$	10.0000			0.374766
$713$			40.2492i			1.50735i
$714$		0			0
$715$		−	6.70820i		−	0.250873i
$716$		0			0
$717$		−	13.4164i		−	0.501045i
$718$	−5.00000			−0.186598
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$	6.00000			0.223607
$721$		0			0
$722$		−	42.4853i		−	1.58114i
$723$		−	55.9017i		−	2.07901i
$724$	15.0000			0.557471
$725$	−12.0000	−	17.8885i	−0.445669	−	0.664364i
$726$	−30.0000			−1.11340
$727$		−	26.8328i		−	0.995174i	−0.867414	−	0.497587i	$-0.834220\pi$
$727$		−	26.8328i		−	0.995174i	0.867414	−	0.497587i	$-0.165780\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$	30.0000			1.10959
$732$	−90.0000			−3.32650
$733$			26.8328i			0.991093i	0.868582	+	0.495546i	$0.165032\pi$
$733$			26.8328i			0.991093i	−0.868582	+	0.495546i	$0.834968\pi$
$734$	−60.0000			−2.21464
$735$		0			0
$736$			40.2492i			1.48361i
$737$		−	17.8885i		−	0.658933i
$738$	20.0000			0.736210
$739$			20.1246i			0.740296i	0.928973	+	0.370148i	$0.120693\pi$
$739$			20.1246i			0.740296i	−0.928973	+	0.370148i	$0.879307\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	8.94427i		−	0.328134i	−0.986449	−	0.164067i	$-0.947539\pi$
$743$		−	8.94427i		−	0.328134i	0.986449	−	0.164067i	$-0.0524612\pi$
$744$			33.5410i			1.22967i
$745$	45.0000			1.64867
$746$			69.3181i			2.53792i
$747$	−12.0000			−0.439057
$748$	30.0000			1.09691
$749$		0			0
$750$	15.0000			0.547723
$751$		−	26.8328i		−	0.979143i	−0.871963	−	0.489572i	$-0.837153\pi$
$751$		−	26.8328i		−	0.979143i	0.871963	−	0.489572i	$-0.162847\pi$
$752$		−	2.23607i		−	0.0815410i
$753$	35.0000			1.27547
$754$	−10.0000	+	6.70820i	−0.364179	+	0.244298i
$755$	−30.0000			−1.09181
$756$		0			0
$757$			40.2492i			1.46288i	0.681904	+	0.731441i	$0.261152\pi$
$757$			40.2492i			1.46288i	−0.681904	+	0.731441i	$0.738848\pi$
$758$		0			0
$759$	−30.0000			−1.08893
$760$		0			0
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$		0			0
$764$			26.8328i			0.970777i
$765$			26.8328i			0.970143i
$766$			13.4164i			0.484755i
$767$	−6.00000			−0.216647
$768$			20.1246i			0.726184i
$769$		−	13.4164i		−	0.483808i	−0.970300	−	0.241904i	$-0.922228\pi$
$769$		−	13.4164i		−	0.483808i	0.970300	−	0.241904i	$-0.0777719\pi$
$770$		0			0
$771$		−	6.70820i		−	0.241590i
$772$		−	40.2492i		−	1.44860i
$773$			22.3607i			0.804258i	0.915583	+	0.402129i	$0.131730\pi$
$773$			22.3607i			0.804258i	−0.915583	+	0.402129i	$0.868270\pi$
$774$	−30.0000			−1.07833
$775$			26.8328i			0.963863i
$776$	−30.0000			−1.07694
$777$		0			0
$778$	40.0000			1.43407
$779$		0			0
$780$			20.1246i			0.720577i
$781$		0			0
$782$	−60.0000			−2.14560
$783$	−10.0000	+	6.70820i	−0.357371	+	0.239732i
$784$		0			0
$785$		−	40.2492i		−	1.43656i
$786$		−	44.7214i		−	1.59516i
$787$	−32.0000			−1.14068			−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000			−1.14068			−0.570338	+	0.821410i	$0.693188\pi$
$788$	54.0000			1.92367
$789$	25.0000			0.890024
$790$	45.0000			1.60103
$791$		0			0
$792$	−10.0000			−0.355335
$793$			13.4164i			0.476431i
$794$		−	15.6525i		−	0.555486i
$795$			60.3738i			2.14124i
$796$	42.0000			1.48865
$797$		−	4.47214i		−	0.158411i	−0.996858	−	0.0792056i	$-0.974762\pi$
$797$		−	4.47214i		−	0.158411i	0.996858	−	0.0792056i	$-0.0252384\pi$
$798$		0			0
$799$	10.0000			0.353775
$800$			26.8328i			0.948683i
$801$			8.94427i			0.316030i
$802$		−	33.5410i		−	1.18437i
$803$		0			0
$804$			53.6656i			1.89264i
$805$		0			0
$806$	15.0000			0.528352
$807$	−10.0000			−0.352017
$808$	40.0000			1.40720
$809$			44.7214i			1.57232i	0.618023	+	0.786160i	$0.287934\pi$
$809$			44.7214i			1.57232i	−0.618023	+	0.786160i	$0.712066\pi$
$810$			73.7902i			2.59272i
$811$	22.0000			0.772524			0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000			0.772524			0.386262	+	0.922389i	$0.373766\pi$
$812$		0			0
$813$	−45.0000			−1.57822
$814$		0			0
$815$			60.3738i			2.11480i
$816$	10.0000			0.350070
$817$		0			0
$818$	60.0000			2.09785
$819$		0			0
$820$			40.2492i			1.40556i
$821$	−33.0000			−1.15171			−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000			−1.15171			−0.575854	+	0.817553i	$0.695330\pi$
$822$			44.7214i			1.55984i
$823$			53.6656i			1.87067i	0.353768	+	0.935333i	$0.384900\pi$
$823$			53.6656i			1.87067i	−0.353768	+	0.935333i	$0.615100\pi$
$824$			8.94427i			0.311588i
$825$	−20.0000			−0.696311
$826$		0			0
$827$			38.0132i			1.32185i	0.750453	+	0.660923i	$0.229835\pi$
$827$			38.0132i			1.32185i	−0.750453	+	0.660923i	$0.770165\pi$
$828$	36.0000			1.25109
$829$		−	40.2492i		−	1.39791i	−0.715164	−	0.698957i	$-0.753648\pi$
$829$		−	40.2492i		−	1.39791i	0.715164	−	0.698957i	$-0.246352\pi$
$830$		−	40.2492i		−	1.39707i
$831$		−	4.47214i		−	0.155137i
$832$	13.0000			0.450694
$833$		0			0
$834$	−50.0000			−1.73136
$835$	36.0000			1.24583
$836$		0			0
$837$	15.0000			0.518476
$838$			53.6656i			1.85385i
$839$			2.23607i			0.0771976i	0.999255	+	0.0385988i	$0.0122894\pi$
$839$			2.23607i			0.0771976i	−0.999255	+	0.0385988i	$0.987711\pi$
$840$		0			0
$841$	−11.0000	+	26.8328i	−0.379310	+	0.925270i
$842$	−30.0000			−1.03387
$843$		−	6.70820i		−	0.231043i
$844$			20.1246i			0.692718i
$845$	−36.0000			−1.23844
$846$	−10.0000			−0.343807
$847$		0			0
$848$	9.00000			0.309061
$849$			31.3050i			1.07438i
$850$	−40.0000			−1.37199
$851$		0			0
$852$		0			0
$853$		−	26.8328i		−	0.918738i	−0.888246	−	0.459369i	$-0.848076\pi$
$853$		−	26.8328i		−	0.918738i	0.888246	−	0.459369i	$-0.151924\pi$
$854$		0			0
$855$		0			0
$856$			40.2492i			1.37569i
$857$	−57.0000			−1.94708			−0.973541	−	0.228510i	$-0.926614\pi$
$857$	−57.0000			−1.94708			−0.973541	+	0.228510i	$0.926614\pi$
$858$			11.1803i			0.381691i
$859$			46.9574i			1.60217i	0.598553	+	0.801083i	$0.295743\pi$
$859$			46.9574i			1.60217i	−0.598553	+	0.801083i	$0.704257\pi$
$860$		−	60.3738i		−	2.05873i
$861$		0			0
$862$		−	40.2492i		−	1.37089i
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$	15.0000			0.510310
$865$	18.0000			0.612018
$866$		0			0
$867$			6.70820i			0.227823i
$868$		0			0
$869$	15.0000			0.508840
$870$	45.0000	+	67.0820i	1.52564	+	2.27429i
$871$	8.00000			0.271070
$872$			11.1803i			0.378614i
$873$		−	26.8328i		−	0.908153i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−13.0000			−0.438979			−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000			−0.438979			−0.219489	+	0.975615i	$0.570439\pi$
$878$		−	76.0263i		−	2.56576i
$879$	20.0000			0.674583
$880$			6.70820i			0.226134i
$881$			35.7771i			1.20536i	0.797983	+	0.602680i	$0.205901\pi$
$881$			35.7771i			1.20536i	−0.797983	+	0.602680i	$0.794099\pi$
$882$		0			0
$883$	−34.0000			−1.14419			−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000			−1.14419			−0.572096	+	0.820187i	$0.693869\pi$
$884$			13.4164i			0.451243i
$885$			40.2492i			1.35296i
$886$	40.0000			1.34383
$887$		−	11.1803i		−	0.375399i	−0.982226	−	0.187700i	$-0.939897\pi$
$887$		−	11.1803i		−	0.375399i	0.982226	−	0.187700i	$-0.0601031\pi$
$888$		0			0
$889$		0			0
$890$	−30.0000			−1.00560
$891$			24.5967i			0.824022i
$892$	−48.0000			−1.60716
$893$		0			0
$894$	−75.0000			−2.50838
$895$		0			0
$896$		0			0
$897$		−	13.4164i		−	0.447961i
$898$	−80.0000			−2.66963
$899$	30.0000	−	20.1246i	1.00056	−	0.671193i
$900$	24.0000			0.800000
$901$			40.2492i			1.34090i
$902$			22.3607i			0.744529i
$903$		0			0
$904$	20.0000			0.665190
$905$	−15.0000			−0.498617
$906$	50.0000			1.66114
$907$		−	53.6656i		−	1.78194i	−0.454064	−	0.890969i	$-0.650026\pi$
$907$		−	53.6656i		−	1.78194i	0.454064	−	0.890969i	$-0.349974\pi$
$908$	36.0000			1.19470
$909$			35.7771i			1.18665i
$910$		0			0
$911$			38.0132i			1.25943i	0.776825	+	0.629716i	$0.216829\pi$
$911$			38.0132i			1.25943i	−0.776825	+	0.629716i	$0.783171\pi$
$912$		0			0
$913$		−	13.4164i		−	0.444018i
$914$		−	4.47214i		−	0.147925i
$915$	90.0000			2.97531
$916$			40.2492i			1.32987i
$917$		0			0
$918$			22.3607i			0.738012i
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$			40.2492i			1.32698i
$921$	45.0000			1.48280
$922$	20.0000			0.658665
$923$		0			0
$924$		0			0
$925$		0			0
$926$		−	58.1378i		−	1.91053i
$927$	−8.00000			−0.262754
$928$	30.0000	−	20.1246i	0.984798	−	0.660623i
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		−	100.623i		−	3.29956i
$931$		0			0
$932$	27.0000			0.884414
$933$	20.0000			0.654771
$934$	5.00000			0.163605
$935$	−30.0000			−0.981105
$936$		−	4.47214i		−	0.146176i
$937$	22.0000			0.718709			0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000			0.718709			0.359354	+	0.933201i	$0.382997\pi$
$938$		0			0
$939$			64.8460i			2.11617i
$940$		−	20.1246i		−	0.656392i
$941$	3.00000			0.0977972			0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000			0.0977972			0.0488986	+	0.998804i	$0.484429\pi$
$942$			67.0820i			2.18565i
$943$		−	26.8328i		−	0.873797i
$944$	6.00000			0.195283
$945$		0			0
$946$		−	33.5410i		−	1.09051i
$947$		−	2.23607i		−	0.0726624i	−0.999340	−	0.0363312i	$-0.988433\pi$
$947$		−	2.23607i		−	0.0726624i	0.999340	−	0.0363312i	$-0.0115671\pi$
$948$	−45.0000			−1.46153
$949$		0			0
$950$		0			0
$951$	−50.0000			−1.62136
$952$		0			0
$953$	−9.00000			−0.291539			−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000			−0.291539			−0.145769	+	0.989319i	$0.546566\pi$
$954$		−	40.2492i		−	1.30312i
$955$		−	26.8328i		−	0.868290i
$956$	−18.0000			−0.582162
$957$	15.0000	+	22.3607i	0.484881	+	0.722818i
$958$	−25.0000			−0.807713
$959$		0			0
$960$		−	87.2067i		−	2.81458i
$961$	−14.0000			−0.451613
$962$		0			0
$963$	−36.0000			−1.16008
$964$	−75.0000			−2.41559
$965$			40.2492i			1.29567i
$966$		0			0
$967$		−	46.9574i		−	1.51005i	−0.655697	−	0.755025i	$-0.727625\pi$
$967$		−	46.9574i		−	1.51005i	0.655697	−	0.755025i	$-0.272375\pi$
$968$			13.4164i			0.431220i
$969$		0			0
$970$	90.0000			2.88973
$971$		−	17.8885i		−	0.574071i	−0.957920	−	0.287035i	$-0.907330\pi$
$971$		−	17.8885i		−	0.574071i	0.957920	−	0.287035i	$-0.0926697\pi$
$972$		−	53.6656i		−	1.72133i
$973$		0			0
$974$		−	4.47214i		−	0.143296i
$975$		−	8.94427i		−	0.286446i
$976$		−	13.4164i		−	0.429449i
$977$	33.0000			1.05576			0.527882	−	0.849318i	$-0.322986\pi$
$977$	33.0000			1.05576			0.527882	+	0.849318i	$0.322986\pi$
$978$		−	100.623i		−	3.21757i
$979$	−10.0000			−0.319601
$980$		0			0
$981$	−10.0000			−0.319275
$982$	55.0000			1.75512
$983$			29.0689i			0.927153i	0.886057	+	0.463577i	$0.153434\pi$
$983$			29.0689i			0.927153i	−0.886057	+	0.463577i	$0.846566\pi$
$984$		−	22.3607i		−	0.712832i
$985$	−54.0000			−1.72058
$986$	30.0000	+	44.7214i	0.955395	+	1.42422i
$987$		0			0
$988$		0			0
$989$			40.2492i			1.27985i
$990$	30.0000			0.953463
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$	−45.0000			−1.42875
$993$	−75.0000			−2.38005
$994$		0			0
$995$	−42.0000			−1.33149
$996$			40.2492i			1.27535i
$997$		−	26.8328i		−	0.849804i	−0.905239	−	0.424902i	$-0.860309\pi$
$997$		−	26.8328i		−	0.849804i	0.905239	−	0.424902i	$-0.139691\pi$
$998$			22.3607i			0.707815i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1421.2.b.b.1275.1		2
7.6	odd	2		29.2.b.a.28.1	✓	2
21.20	even	2		261.2.c.a.28.2		2
28.27	even	2		464.2.e.a.289.1		2
29.28	even	2	inner	1421.2.b.b.1275.2		2
35.13	even	4		725.2.d.a.724.1		4
35.27	even	4		725.2.d.a.724.4		4
35.34	odd	2		725.2.c.c.376.2		2
56.13	odd	2		1856.2.e.g.1217.1		2
56.27	even	2		1856.2.e.f.1217.2		2
84.83	odd	2		4176.2.o.k.289.1		2
203.6	odd	14		841.2.e.g.196.2		12
203.13	odd	14		841.2.e.g.63.1		12
203.20	odd	14		841.2.e.g.267.1		12
203.27	even	28		841.2.d.h.605.1		12
203.34	odd	14		841.2.e.g.236.1		12
203.41	even	4		841.2.a.b.1.2		2
203.48	even	28		841.2.d.h.190.2		12
203.55	even	28		841.2.d.h.571.1		12
203.62	odd	14		841.2.e.g.651.2		12
203.69	even	28		841.2.d.h.778.1		12
203.76	even	28		841.2.d.h.778.2		12
203.83	odd	14		841.2.e.g.651.1		12
203.90	even	28		841.2.d.h.571.2		12
203.97	even	28		841.2.d.h.190.1		12
203.104	even	4		841.2.a.b.1.1		2
203.111	odd	14		841.2.e.g.236.2		12
203.118	even	28		841.2.d.h.605.2		12
203.125	odd	14		841.2.e.g.267.2		12
203.132	odd	14		841.2.e.g.63.2		12
203.139	odd	14		841.2.e.g.196.1		12
203.153	even	28		841.2.d.h.574.2		12
203.160	even	28		841.2.d.h.645.1		12
203.167	odd	14		841.2.e.g.270.1		12
203.181	odd	14		841.2.e.g.270.2		12
203.188	even	28		841.2.d.h.645.2		12
203.195	even	28		841.2.d.h.574.1		12
203.202	odd	2		29.2.b.a.28.2	yes	2
609.41	odd	4		7569.2.a.i.1.1		2
609.104	odd	4		7569.2.a.i.1.2		2
609.608	even	2		261.2.c.a.28.1		2
812.811	even	2		464.2.e.a.289.2		2
1015.202	even	4		725.2.d.a.724.2		4
1015.608	even	4		725.2.d.a.724.3		4
1015.1014	odd	2		725.2.c.c.376.1		2
1624.405	odd	2		1856.2.e.g.1217.2		2
1624.811	even	2		1856.2.e.f.1217.1		2
2436.2435	odd	2		4176.2.o.k.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.b.a.28.1	✓	2	7.6	odd	2
29.2.b.a.28.2	yes	2	203.202	odd	2
261.2.c.a.28.1		2	609.608	even	2
261.2.c.a.28.2		2	21.20	even	2
464.2.e.a.289.1		2	28.27	even	2
464.2.e.a.289.2		2	812.811	even	2
725.2.c.c.376.1		2	1015.1014	odd	2
725.2.c.c.376.2		2	35.34	odd	2
725.2.d.a.724.1		4	35.13	even	4
725.2.d.a.724.2		4	1015.202	even	4
725.2.d.a.724.3		4	1015.608	even	4
725.2.d.a.724.4		4	35.27	even	4
841.2.a.b.1.1		2	203.104	even	4
841.2.a.b.1.2		2	203.41	even	4
841.2.d.h.190.1		12	203.97	even	28
841.2.d.h.190.2		12	203.48	even	28
841.2.d.h.571.1		12	203.55	even	28
841.2.d.h.571.2		12	203.90	even	28
841.2.d.h.574.1		12	203.195	even	28
841.2.d.h.574.2		12	203.153	even	28
841.2.d.h.605.1		12	203.27	even	28
841.2.d.h.605.2		12	203.118	even	28
841.2.d.h.645.1		12	203.160	even	28
841.2.d.h.645.2		12	203.188	even	28
841.2.d.h.778.1		12	203.69	even	28
841.2.d.h.778.2		12	203.76	even	28
841.2.e.g.63.1		12	203.13	odd	14
841.2.e.g.63.2		12	203.132	odd	14
841.2.e.g.196.1		12	203.139	odd	14
841.2.e.g.196.2		12	203.6	odd	14
841.2.e.g.236.1		12	203.34	odd	14
841.2.e.g.236.2		12	203.111	odd	14
841.2.e.g.267.1		12	203.20	odd	14
841.2.e.g.267.2		12	203.125	odd	14
841.2.e.g.270.1		12	203.167	odd	14
841.2.e.g.270.2		12	203.181	odd	14
841.2.e.g.651.1		12	203.83	odd	14
841.2.e.g.651.2		12	203.62	odd	14
1421.2.b.b.1275.1		2	1.1	even	1	trivial
1421.2.b.b.1275.2		2	29.28	even	2	inner
1856.2.e.f.1217.1		2	1624.811	even	2
1856.2.e.f.1217.2		2	56.27	even	2
1856.2.e.g.1217.1		2	56.13	odd	2
1856.2.e.g.1217.2		2	1624.405	odd	2
4176.2.o.k.289.1		2	84.83	odd	2
4176.2.o.k.289.2		2	2436.2435	odd	2
7569.2.a.i.1.1		2	609.41	odd	4
7569.2.a.i.1.2		2	609.104	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1421 = 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1421.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -2.23607i q^{3} -3.00000 q^{4} +3.00000 q^{5} -5.00000 q^{6} +2.23607i q^{8} -2.00000 q^{9} -6.70820i q^{10} -2.23607i q^{11} +6.70820i q^{12} +1.00000 q^{13} -6.70820i q^{15} -1.00000 q^{16} -4.47214i q^{17} +4.47214i q^{18} -9.00000 q^{20} -5.00000 q^{22} +6.00000 q^{23} +5.00000 q^{24} +4.00000 q^{25} -2.23607i q^{26} -2.23607i q^{27} +(-3.00000 - 4.47214i) q^{29} -15.0000 q^{30} +6.70820i q^{31} +6.70820i q^{32} -5.00000 q^{33} -10.0000 q^{34} +6.00000 q^{36} -2.23607i q^{39} +6.70820i q^{40} -4.47214i q^{41} +6.70820i q^{43} +6.70820i q^{44} -6.00000 q^{45} -13.4164i q^{46} +2.23607i q^{47} +2.23607i q^{48} -8.94427i q^{50} -10.0000 q^{51} -3.00000 q^{52} -9.00000 q^{53} -5.00000 q^{54} -6.70820i q^{55} +(-10.0000 + 6.70820i) q^{58} -6.00000 q^{59} +20.1246i q^{60} +13.4164i q^{61} +15.0000 q^{62} +13.0000 q^{64} +3.00000 q^{65} +11.1803i q^{66} +8.00000 q^{67} +13.4164i q^{68} -13.4164i q^{69} -4.47214i q^{72} -8.94427i q^{75} -5.00000 q^{78} +6.70820i q^{79} -3.00000 q^{80} -11.0000 q^{81} -10.0000 q^{82} +6.00000 q^{83} -13.4164i q^{85} +15.0000 q^{86} +(-10.0000 + 6.70820i) q^{87} +5.00000 q^{88} -4.47214i q^{89} +13.4164i q^{90} -18.0000 q^{92} +15.0000 q^{93} +5.00000 q^{94} +15.0000 q^{96} +13.4164i q^{97} +4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{4} + 6 q^{5} - 10 q^{6} - 4 q^{9} + 2 q^{13} - 2 q^{16} - 18 q^{20} - 10 q^{22} + 12 q^{23} + 10 q^{24} + 8 q^{25} - 6 q^{29} - 30 q^{30} - 10 q^{33} - 20 q^{34} + 12 q^{36} - 12 q^{45} - 20 q^{51}+ \cdots + 30 q^{96}+O(q^{100}) \) 2 * q - 6 * q^4 + 6 * q^5 - 10 * q^6 - 4 * q^9 + 2 * q^13 - 2 * q^16 - 18 * q^20 - 10 * q^22 + 12 * q^23 + 10 * q^24 + 8 * q^25 - 6 * q^29 - 30 * q^30 - 10 * q^33 - 20 * q^34 + 12 * q^36 - 12 * q^45 - 20 * q^51 - 6 * q^52 - 18 * q^53 - 10 * q^54 - 20 * q^58 - 12 * q^59 + 30 * q^62 + 26 * q^64 + 6 * q^65 + 16 * q^67 - 10 * q^78 - 6 * q^80 - 22 * q^81 - 20 * q^82 + 12 * q^83 + 30 * q^86 - 20 * q^87 + 10 * q^88 - 36 * q^92 + 30 * q^93 + 10 * q^94 + 30 * q^96

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