LMFDB - Embedded newform 943.1.n.b.620.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [943,1,Mod(367,943)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(943, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("943.367");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 943 = 23 \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	943.n (of order $20$, degree $8$, minimal)

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:	no
Analytic conductor:	$0.470618306913$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 $ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{60}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{60} - \cdots)$

Embedding invariants

Embedding label			620.2
Root			$-0.207912 + 0.978148i$ of defining polynomial
Character	$\chi$	$=$	943.620
Dual form			943.1.n.b.689.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.122881 + 0.169131i) q^{2} +(0.770236 + 0.770236i) q^{3} +(0.295511 - 0.909491i) q^{4} +(-0.0356234 + 0.224918i) q^{6} +(0.388960 - 0.126381i) q^{8} +0.186527i q^{9} +O(q^{10})$	\(q+(0.122881 + 0.169131i) q^{2} +(0.770236 + 0.770236i) q^{3} +(0.295511 - 0.909491i) q^{4} +(-0.0356234 + 0.224918i) q^{6} +(0.388960 - 0.126381i) q^{8} +0.186527i q^{9} +(0.928136 - 0.472909i) q^{12} +(0.103383 + 0.0163743i) q^{13} +(-0.704489 - 0.511841i) q^{16} +(-0.0315474 + 0.0229205i) q^{18} +(-0.809017 + 0.587785i) q^{23} +(0.396934 + 0.202248i) q^{24} +(0.809017 + 0.587785i) q^{25} +(0.00993440 + 0.0194974i) q^{26} +(0.626566 - 0.626566i) q^{27} +(-1.49452 + 0.761497i) q^{29} +(0.128496 + 0.395472i) q^{31} -0.591023i q^{32} +(0.169644 + 0.0551208i) q^{36} +(0.0670174 + 0.0922415i) q^{39} +(-0.104528 - 0.994522i) q^{41} +(-0.198825 - 0.0646021i) q^{46} +(-0.112122 + 0.707912i) q^{47} +(-0.148384 - 0.936861i) q^{48} +(-0.951057 + 0.309017i) q^{49} +0.209057i q^{50} +(0.0454432 - 0.0891873i) q^{52} +(0.182964 + 0.0289787i) q^{54} +(-0.312440 - 0.159196i) q^{58} +(-0.500000 + 0.363271i) q^{59} +(-0.0510966 + 0.0703285i) q^{62} +(-0.604528 + 0.439216i) q^{64} +(-1.07587 - 0.170401i) q^{69} +(-0.705634 + 1.38488i) q^{71} +(0.0235734 + 0.0725515i) q^{72} -1.48629i q^{73} +(0.170401 + 1.07587i) q^{75} +(-0.00736573 + 0.0226694i) q^{78} +1.15173 q^{81} +(0.155360 - 0.139886i) q^{82} +(-1.73767 - 0.564602i) q^{87} +(0.295511 + 0.909491i) q^{92} +(-0.205634 + 0.403579i) q^{93} +(-0.133507 + 0.0680253i) q^{94} +(0.455227 - 0.455227i) q^{96} +(-0.169131 - 0.122881i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{3} + 10 q^{4} - 12 q^{6}+O(q^{10}) $ 16 * q - 2 * q^3 + 10 * q^4 - 12 * q^6	$ 16 q - 2 q^{3} + 10 q^{4} - 12 q^{6} + 2 q^{13} - 6 q^{16} + 6 q^{18} - 4 q^{23} - 22 q^{24} + 4 q^{25} + 12 q^{26} + 6 q^{27} - 8 q^{29} - 10 q^{36} + 2 q^{41} + 2 q^{47} - 18 q^{48} - 6 q^{54} + 2 q^{58} - 8 q^{59} - 10 q^{62} - 6 q^{64} - 2 q^{69} - 2 q^{71} + 16 q^{72} + 2 q^{75} - 2 q^{78} - 12 q^{81} + 10 q^{92} + 6 q^{93} + 18 q^{94} - 10 q^{96} + 6 q^{98}+O(q^{100}) $ 16 * q - 2 * q^3 + 10 * q^4 - 12 * q^6 + 2 * q^13 - 6 * q^16 + 6 * q^18 - 4 * q^23 - 22 * q^24 + 4 * q^25 + 12 * q^26 + 6 * q^27 - 8 * q^29 - 10 * q^36 + 2 * q^41 + 2 * q^47 - 18 * q^48 - 6 * q^54 + 2 * q^58 - 8 * q^59 - 10 * q^62 - 6 * q^64 - 2 * q^69 - 2 * q^71 + 16 * q^72 + 2 * q^75 - 2 * q^78 - 12 * q^81 + 10 * q^92 + 6 * q^93 + 18 * q^94 - 10 * q^96 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/943\mathbb{Z}\right)^\times$.

$n$	$47$	$534$
$\chi(n)$	$e\left(\frac{11}{20}\right)$	$-1$

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 $

$2$	0.122881	+	0.169131i	0.122881	+	0.169131i	0.866025	−	0.500000i	$-0.166667\pi$
$2$	0.122881	+	0.169131i	0.122881	+	0.169131i	−0.743145	+	0.669131i	$0.766667\pi$
$3$	0.770236	+	0.770236i	0.770236	+	0.770236i	0.978148	−	0.207912i	$-0.0666667\pi$
$3$	0.770236	+	0.770236i	0.770236	+	0.770236i	−0.207912	+	0.978148i	$0.566667\pi$
$4$	0.295511	−	0.909491i	0.295511	−	0.909491i
$5$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$5$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$6$	−0.0356234	+	0.224918i	−0.0356234	+	0.224918i
$7$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$7$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$8$	0.388960	−	0.126381i	0.388960	−	0.126381i
$9$			0.186527i			0.186527i
$10$		0			0
$11$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$11$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$12$	0.928136	−	0.472909i	0.928136	−	0.472909i
$13$	0.103383	+	0.0163743i	0.103383	+	0.0163743i	0.207912	−	0.978148i	$-0.433333\pi$
$13$	0.103383	+	0.0163743i	0.103383	+	0.0163743i	−0.104528	+	0.994522i	$0.533333\pi$
$14$		0			0
$15$		0			0
$16$	−0.704489	−	0.511841i	−0.704489	−	0.511841i
$17$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$17$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$18$	−0.0315474	+	0.0229205i	−0.0315474	+	0.0229205i
$19$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$19$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$23$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$24$	0.396934	+	0.202248i	0.396934	+	0.202248i
$25$	0.809017	+	0.587785i	0.809017	+	0.587785i
$26$	0.00993440	+	0.0194974i	0.00993440	+	0.0194974i
$27$	0.626566	−	0.626566i	0.626566	−	0.626566i
$28$		0			0
$29$	−1.49452	+	0.761497i	−1.49452	+	0.761497i	−0.994522	−	0.104528i	$-0.966667\pi$
$29$	−1.49452	+	0.761497i	−1.49452	+	0.761497i	−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	0.128496	+	0.395472i	0.128496	+	0.395472i	0.994522	−	0.104528i	$-0.0333333\pi$
$31$	0.128496	+	0.395472i	0.128496	+	0.395472i	−0.866025	+	0.500000i	$0.833333\pi$
$32$		−	0.591023i		−	0.591023i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.169644	+	0.0551208i	0.169644	+	0.0551208i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$		0			0
$39$	0.0670174	+	0.0922415i	0.0670174	+	0.0922415i
$40$		0			0
$41$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$41$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$42$		0			0
$43$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$43$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$44$		0			0
$45$		0			0
$46$	−0.198825	−	0.0646021i	−0.198825	−	0.0646021i
$47$	−0.112122	+	0.707912i	−0.112122	+	0.707912i	0.866025	+	0.500000i	$0.166667\pi$
$47$	−0.112122	+	0.707912i	−0.112122	+	0.707912i	−0.978148	+	0.207912i	$0.933333\pi$
$48$	−0.148384	−	0.936861i	−0.148384	−	0.936861i
$49$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$50$			0.209057i			0.209057i
$51$		0			0
$52$	0.0454432	−	0.0891873i	0.0454432	−	0.0891873i
$53$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$53$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$54$	0.182964	+	0.0289787i	0.182964	+	0.0289787i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.312440	−	0.159196i	−0.312440	−	0.159196i
$59$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$59$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$61$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$62$	−0.0510966	+	0.0703285i	−0.0510966	+	0.0703285i
$63$		0			0
$64$	−0.604528	+	0.439216i	−0.604528	+	0.439216i
$65$		0			0
$66$		0			0
$67$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$67$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$68$		0			0
$69$	−1.07587	−	0.170401i	−1.07587	−	0.170401i
$70$		0			0
$71$	−0.705634	+	1.38488i	−0.705634	+	1.38488i	0.207912	+	0.978148i	$0.433333\pi$
$71$	−0.705634	+	1.38488i	−0.705634	+	1.38488i	−0.913545	+	0.406737i	$0.866667\pi$
$72$	0.0235734	+	0.0725515i	0.0235734	+	0.0725515i
$73$		−	1.48629i		−	1.48629i	−0.669131	−	0.743145i	$-0.733333\pi$
$73$		−	1.48629i		−	1.48629i	0.669131	−	0.743145i	$-0.266667\pi$
$74$		0			0
$75$	0.170401	+	1.07587i	0.170401	+	1.07587i
$76$		0			0
$77$		0			0
$78$	−0.00736573	+	0.0226694i	−0.00736573	+	0.0226694i
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$		0			0
$81$	1.15173			1.15173
$82$	0.155360	−	0.139886i	0.155360	−	0.139886i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.73767	−	0.564602i	−1.73767	−	0.564602i
$88$		0			0
$89$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$89$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$90$		0			0
$91$		0			0
$92$	0.295511	+	0.909491i	0.295511	+	0.909491i
$93$	−0.205634	+	0.403579i	−0.205634	+	0.403579i
$94$	−0.133507	+	0.0680253i	−0.133507	+	0.0680253i
$95$		0			0
$96$	0.455227	−	0.455227i	0.455227	−	0.455227i
$97$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$97$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$98$	−0.169131	−	0.122881i	−0.169131	−	0.122881i
$99$		0			0
$100$	0.773659	−	0.562096i	0.773659	−	0.562096i
$101$	0.896802	−	0.142040i	0.896802	−	0.142040i	0.309017	−	0.951057i	$-0.400000\pi$
$101$	0.896802	−	0.142040i	0.896802	−	0.142040i	0.587785	+	0.809017i	$0.300000\pi$
$102$		0			0
$103$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$103$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$104$	0.0422814	−	0.00669671i	0.0422814	−	0.00669671i
$105$		0			0
$106$		0			0
$107$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$107$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$108$	−0.384699	−	0.755014i	−0.384699	−	0.755014i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$113$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$114$		0			0
$115$		0			0
$116$	0.250926	+	1.58428i	0.250926	+	1.58428i
$117$	−0.00305424	+	0.0192837i	−0.00305424	+	0.0192837i
$118$	−0.122881	−	0.0399263i	−0.122881	−	0.0399263i
$119$		0			0
$120$		0			0
$121$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$122$		0			0
$123$	0.685505	−	0.846528i	0.685505	−	0.846528i
$124$	0.397650			0.397650
$125$		0			0
$126$		0			0
$127$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$127$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$128$	−0.710666	−	0.230909i	−0.710666	−	0.230909i
$129$		0			0
$130$		0			0
$131$	−1.27276	+	0.413545i	−1.27276	+	0.413545i	−0.866025	−	0.500000i	$-0.833333\pi$
$131$	−1.27276	+	0.413545i	−1.27276	+	0.413545i	−0.406737	+	0.913545i	$0.633333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$	−0.103383	−	0.202901i	−0.103383	−	0.202901i
$139$	−1.60917	−	1.16913i	−1.60917	−	1.16913i	−0.866025	−	0.500000i	$-0.833333\pi$
$139$	−1.60917	−	1.16913i	−1.60917	−	1.16913i	−0.743145	−	0.669131i	$-0.766667\pi$
$140$		0			0
$141$	−0.631620	+	0.458898i	−0.631620	+	0.458898i
$142$	−0.320935	+	0.0508311i	−0.320935	+	0.0508311i
$143$		0			0
$144$	0.0954720	−	0.131406i	0.0954720	−	0.131406i
$145$		0			0
$146$	0.251377	−	0.182636i	0.251377	−	0.182636i
$147$	−0.970554	−	0.494522i	−0.970554	−	0.494522i
$148$		0			0
$149$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$149$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$150$	−0.161023	+	0.161023i	−0.161023	+	0.161023i
$151$	1.97267	+	0.312440i	1.97267	+	0.312440i	0.994522	+	0.104528i	$0.0333333\pi$
$151$	1.97267	+	0.312440i	1.97267	+	0.312440i	0.978148	+	0.207912i	$0.0666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.103697	−	0.0336933i	0.103697	−	0.0336933i
$157$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$157$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.141526	+	0.194794i	0.141526	+	0.194794i
$163$	1.48629			1.48629			0.743145	−	0.669131i	$-0.233333\pi$
$163$	1.48629			1.48629			0.743145	+	0.669131i	$0.233333\pi$
$164$	−0.935398	−	0.198825i	−0.935398	−	0.198825i
$165$		0			0
$166$		0			0
$167$	0.642040	+	0.642040i	0.642040	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$167$	0.642040	+	0.642040i	0.642040	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	−0.940637	−	0.305631i	−0.940637	−	0.305631i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.61803i		−	1.61803i	−0.587785	−	0.809017i	$-0.700000\pi$
$173$		−	1.61803i		−	1.61803i	0.587785	−	0.809017i	$-0.300000\pi$
$174$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$175$		0			0
$176$		0			0
$177$	−0.664923	−	0.105313i	−0.664923	−	0.105313i
$178$		0			0
$179$	−0.235003	−	0.461219i	−0.235003	−	0.461219i	0.743145	−	0.669131i	$-0.233333\pi$
$179$	−0.235003	−	0.461219i	−0.235003	−	0.461219i	−0.978148	+	0.207912i	$0.933333\pi$
$180$		0			0
$181$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$181$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$182$		0			0
$183$		0			0
$184$	−0.240391	+	0.330869i	−0.240391	+	0.330869i
$185$		0			0
$186$	−0.0935260	+	0.0148131i	−0.0935260	+	0.0148131i
$187$		0			0
$188$	0.610706	+	0.311170i	0.610706	+	0.311170i
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	−0.803929	−	0.127330i	−0.803929	−	0.127330i
$193$	0.0932634	−	0.0475201i	0.0932634	−	0.0475201i	−0.406737	−	0.913545i	$-0.633333\pi$
$193$	0.0932634	−	0.0475201i	0.0932634	−	0.0475201i	0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$			0.956295i			0.956295i
$197$	0.773659	−	0.251377i	0.773659	−	0.251377i	0.104528	−	0.994522i	$-0.466667\pi$
$197$	0.773659	−	0.251377i	0.773659	−	0.251377i	0.669131	+	0.743145i	$0.266667\pi$
$198$		0			0
$199$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$199$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$200$	0.388960	+	0.126381i	0.388960	+	0.126381i
$201$		0			0
$202$	0.134223	+	0.134223i	0.134223	+	0.134223i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.109638	−	0.150903i	−0.109638	−	0.150903i
$208$	−0.0644513	−	0.0644513i	−0.0644513	−	0.0644513i
$209$		0			0
$210$		0			0
$211$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i	0.951057	+	0.309017i	$0.100000\pi$
$211$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i	−1.00000			$\pi$
$212$		0			0
$213$	−1.61019	+	0.523183i	−1.61019	+	0.523183i
$214$		0			0
$215$		0			0
$216$	0.164523	−	0.322895i	0.164523	−	0.322895i
$217$		0			0
$218$		0			0
$219$	1.14479	−	1.14479i	1.14479	−	1.14479i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.587785	+	0.809017i	$0.700000\pi$
$223$	−1.53884	+	1.11803i	−1.53884	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$224$		0			0
$225$	−0.109638	+	0.150903i	−0.109638	+	0.150903i
$226$		0			0
$227$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$227$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$228$		0			0
$229$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$229$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$230$		0			0
$231$		0			0
$232$	−0.485071	+	0.485071i	−0.485071	+	0.485071i
$233$	0.511265	+	0.0809764i	0.511265	+	0.0809764i	0.406737	−	0.913545i	$-0.366667\pi$
$233$	0.511265	+	0.0809764i	0.511265	+	0.0809764i	0.104528	+	0.994522i	$0.466667\pi$
$234$	−0.00363678	+	0.00185303i	−0.00363678	+	0.00185303i
$235$		0			0
$236$	0.182636	+	0.562096i	0.182636	+	0.562096i
$237$		0			0
$238$		0			0
$239$	0.262394	+	1.65669i	0.262394	+	1.65669i	0.669131	+	0.743145i	$0.266667\pi$
$239$	0.262394	+	1.65669i	0.262394	+	1.65669i	−0.406737	+	0.913545i	$0.633333\pi$
$240$		0			0
$241$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$241$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$242$	0.0646021	−	0.198825i	0.0646021	−	0.198825i
$243$	0.260541	+	0.260541i	0.260541	+	0.260541i
$244$		0			0
$245$		0			0
$246$	0.227409	+	0.0119180i	0.227409	+	0.0119180i
$247$		0			0
$248$	0.0999601	+	0.137583i	0.0999601	+	0.137583i
$249$		0			0
$250$		0			0
$251$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$251$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$252$		0			0
$253$		0			0
$254$	0.198825	−	0.0646021i	0.198825	−	0.0646021i
$255$		0			0
$256$	0.182636	+	0.562096i	0.182636	+	0.562096i
$257$	0.906737	−	1.77957i	0.906737	−	1.77957i	0.406737	−	0.913545i	$-0.366667\pi$
$257$	0.906737	−	1.77957i	0.906737	−	1.77957i	0.500000	−	0.866025i	$-0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.142040	−	0.278768i	−0.142040	−	0.278768i
$262$	−0.226341	−	0.164446i	−0.226341	−	0.164446i
$263$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$263$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.20243	+	0.873619i	−1.20243	+	0.873619i	−0.994522	−	0.104528i	$-0.966667\pi$
$269$	−1.20243	+	0.873619i	−1.20243	+	0.873619i	−0.207912	+	0.978148i	$0.566667\pi$
$270$		0			0
$271$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$271$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−0.472909	+	0.928136i	−0.472909	+	0.928136i
$277$	−0.128496	−	0.395472i	−0.128496	−	0.395472i	0.866025	−	0.500000i	$-0.166667\pi$
$277$	−0.128496	−	0.395472i	−0.128496	−	0.395472i	−0.994522	+	0.104528i	$0.966667\pi$
$278$		−	0.415823i		−	0.415823i
$279$	−0.0737660	+	0.0239680i	−0.0737660	+	0.0239680i
$280$		0			0
$281$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$281$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$282$	−0.155228	−	0.0504365i	−0.155228	−	0.0504365i
$283$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$283$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$284$	1.05102	+	1.05102i	1.05102	+	1.05102i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.110242			0.110242
$289$	0.587785	+	0.809017i	0.587785	+	0.809017i
$290$		0			0
$291$		0			0
$292$	−1.35177	−	0.439216i	−1.35177	−	0.439216i
$293$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$293$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$294$	−0.0356234	−	0.224918i	−0.0356234	−	0.224918i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.0932634	+	0.0475201i	−0.0932634	+	0.0475201i
$300$	1.02885	+	0.162953i	1.02885	+	0.162953i
$301$		0			0
$302$	0.189560	+	0.372032i	0.189560	+	0.372032i
$303$	0.800153	+	0.581345i	0.800153	+	0.581345i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.690983	−	0.951057i	0.690983	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$307$	0.690983	−	0.951057i	0.690983	−	0.951057i	1.00000			$0$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.12146	+	0.571411i	1.12146	+	0.571411i	0.913545	−	0.406737i	$-0.133333\pi$
$311$	1.12146	+	0.571411i	1.12146	+	0.571411i	0.207912	+	0.978148i	$0.433333\pi$
$312$	0.0377247	+	0.0274086i	0.0377247	+	0.0274086i
$313$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$313$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.142040	+	0.278768i	−0.142040	+	0.278768i	−0.951057	−	0.309017i	$-0.900000\pi$
$317$	−0.142040	+	0.278768i	−0.142040	+	0.278768i	0.809017	+	0.587785i	$0.200000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.340351	−	1.04749i	0.340351	−	1.04749i
$325$	0.0740142	+	0.0740142i	0.0740142	+	0.0740142i
$326$	0.182636	+	0.251377i	0.182636	+	0.251377i
$327$		0			0
$328$	−0.166346	−	0.373619i	−0.166346	−	0.373619i
$329$		0			0
$330$		0			0
$331$	1.32028	+	1.32028i	1.32028	+	1.32028i	0.913545	+	0.406737i	$0.133333\pi$
$331$	1.32028	+	1.32028i	1.32028	+	1.32028i	0.406737	+	0.913545i	$0.366667\pi$
$332$		0			0
$333$		0			0
$334$	−0.0296943	+	0.187483i	−0.0296943	+	0.187483i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−0.0638943	−	0.196647i	−0.0638943	−	0.196647i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.273659	−	0.198825i	0.273659	−	0.198825i
$347$	1.76007	−	0.278768i	1.76007	−	0.278768i	0.809017	−	0.587785i	$-0.200000\pi$
$347$	1.76007	−	0.278768i	1.76007	−	0.278768i	0.951057	+	0.309017i	$0.100000\pi$
$348$	−1.02700	+	1.41355i	−1.02700	+	1.41355i
$349$	0.786610	−	1.08268i	0.786610	−	1.08268i	−0.207912	−	0.978148i	$-0.566667\pi$
$349$	0.786610	−	1.08268i	0.786610	−	1.08268i	0.994522	−	0.104528i	$-0.0333333\pi$
$350$		0			0
$351$	0.0750360	−	0.0545169i	0.0750360	−	0.0545169i
$352$		0			0
$353$	1.60917	+	1.16913i	1.60917	+	1.16913i	0.866025	+	0.500000i	$0.166667\pi$
$353$	1.60917	+	1.16913i	1.60917	+	1.16913i	0.743145	+	0.669131i	$0.233333\pi$
$354$	−0.0638943	−	0.125400i	−0.0638943	−	0.125400i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.0491290	−	0.0964210i	0.0491290	−	0.0964210i
$359$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$359$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$360$		0			0
$361$	0.951057	−	0.309017i	0.951057	−	0.309017i
$362$		0			0
$363$	0.170401	−	1.07587i	0.170401	−	1.07587i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$367$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$368$	0.870796			0.870796
$369$	0.185505	−	0.0194974i	0.185505	−	0.0194974i
$370$		0			0
$371$		0			0
$372$	0.306284	+	0.306284i	0.306284	+	0.306284i
$373$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$373$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$374$		0			0
$375$		0			0
$376$	0.0458554	+	0.289520i	0.0458554	+	0.289520i
$377$	−0.166977	+	0.0542543i	−0.166977	+	0.0542543i
$378$		0			0
$379$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$379$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$380$		0			0
$381$	0.970554	−	0.494522i	0.970554	−	0.494522i
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$	−0.369526	−	0.725235i	−0.369526	−	0.725235i
$385$		0			0
$386$	0.0194974	+	0.00993440i	0.0194974	+	0.00993440i
$387$		0			0
$388$		0			0
$389$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$389$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$390$		0			0
$391$		0			0
$392$	−0.330869	+	0.240391i	−0.330869	+	0.240391i
$393$	−1.29885	−	0.661799i	−1.29885	−	0.661799i
$394$	0.137583	+	0.0999601i	0.137583	+	0.0999601i
$395$		0			0
$396$		0			0
$397$	−0.511265	−	0.0809764i	−0.511265	−	0.0809764i	−0.104528	−	0.994522i	$-0.533333\pi$
$397$	−0.511265	−	0.0809764i	−0.511265	−	0.0809764i	−0.406737	+	0.913545i	$0.633333\pi$
$398$		0			0
$399$		0			0
$400$	−0.269091	−	0.828176i	−0.269091	−	0.828176i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	0.00680881	+	0.0429892i	0.00680881	+	0.0429892i
$404$	0.135832	−	0.857608i	0.135832	−	0.857608i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$409$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.0120500	−	0.0370862i	0.0120500	−	0.0370862i
$415$		0			0
$416$	0.00967758	−	0.0611019i	0.00967758	−	0.0611019i
$417$	−0.338934	−	2.13995i	−0.338934	−	2.13995i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$421$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$422$	−0.0582784	+	0.0296943i	−0.0582784	+	0.0296943i
$423$	−0.132044	−	0.0209138i	−0.132044	−	0.0209138i
$424$		0			0
$425$		0			0
$426$	−0.286348	−	0.208044i	−0.286348	−	0.208044i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$	−0.762111	+	0.120707i	−0.762111	+	0.120707i
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	0.334293	+	0.0529467i	0.334293	+	0.0529467i
$439$	1.12146	−	0.571411i	1.12146	−	0.571411i	0.207912	−	0.978148i	$-0.433333\pi$
$439$	1.12146	−	0.571411i	1.12146	−	0.571411i	0.913545	+	0.406737i	$0.133333\pi$
$440$		0			0
$441$	−0.0576399	−	0.177397i	−0.0576399	−	0.177397i
$442$		0			0
$443$	−0.198825	+	0.0646021i	−0.198825	+	0.0646021i	−0.406737	−	0.913545i	$-0.633333\pi$
$443$	−0.198825	+	0.0646021i	−0.198825	+	0.0646021i	0.207912	+	0.978148i	$0.433333\pi$
$444$		0			0
$445$		0			0
$446$	−0.378188	−	0.122881i	−0.378188	−	0.122881i
$447$		0			0
$448$		0			0
$449$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$449$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$450$	−0.0389947			−0.0389947
$451$		0			0
$452$		0			0
$453$	1.27877	+	1.76007i	1.27877	+	1.76007i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$457$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.564602	−	1.73767i	−0.564602	−	1.73767i	−0.669131	−	0.743145i	$-0.733333\pi$
$461$	−0.564602	−	1.73767i	−0.564602	−	1.73767i	0.104528	−	0.994522i	$-0.466667\pi$
$462$		0			0
$463$	0.809017	−	0.412215i	0.809017	−	0.412215i		−	1.00000i	$-0.5\pi$
$463$	0.809017	−	0.412215i	0.809017	−	0.412215i	0.809017	+	0.587785i	$0.200000\pi$
$464$	1.44264	+	0.228492i	1.44264	+	0.228492i
$465$		0			0
$466$	0.0491290	+	0.0964210i	0.0491290	+	0.0964210i
$467$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$467$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$468$	0.0166358	+	0.00847637i	0.0166358	+	0.00847637i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.148570	+	0.204489i	−0.148570	+	0.204489i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.247954	+	0.247954i	−0.247954	+	0.247954i
$479$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$479$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.909491	+	0.295511i	−0.909491	+	0.295511i
$485$		0			0
$486$	−0.0120500	+	0.0760809i	−0.0120500	+	0.0760809i
$487$	1.27276	+	0.413545i	1.27276	+	0.413545i	0.866025	−	0.500000i	$-0.166667\pi$
$487$	1.27276	+	0.413545i	1.27276	+	0.413545i	0.406737	+	0.913545i	$0.366667\pi$
$488$		0			0
$489$	1.14479	+	1.14479i	1.14479	+	1.14479i
$490$		0			0
$491$	−0.415823			−0.415823			−0.207912	−	0.978148i	$-0.566667\pi$
$491$	−0.415823			−0.415823			−0.207912	+	0.978148i	$0.566667\pi$
$492$	−0.567335	−	0.873619i	−0.567335	−	0.873619i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.111894	−	0.344375i	0.111894	−	0.344375i
$497$		0			0
$498$		0			0
$499$	0.292088	+	1.84417i	0.292088	+	1.84417i	0.500000	+	0.866025i	$0.333333\pi$
$499$	0.292088	+	1.84417i	0.292088	+	1.84417i	−0.207912	+	0.978148i	$0.566667\pi$
$500$		0			0
$501$			0.989044i			0.989044i
$502$		0			0
$503$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$503$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.489104	−	0.959920i	−0.489104	−	0.959920i
$508$	−0.773659	−	0.562096i	−0.773659	−	0.562096i
$509$	−1.72129	−	0.877042i	−1.72129	−	0.877042i	−0.978148	−	0.207912i	$-0.933333\pi$
$509$	−1.72129	−	0.877042i	−1.72129	−	0.877042i	−0.743145	−	0.669131i	$-0.766667\pi$
$510$		0			0
$511$		0			0
$512$	−0.511841	+	0.704489i	−0.511841	+	0.704489i
$513$		0			0
$514$	0.412400	−	0.0653178i	0.412400	−	0.0653178i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.24627	−	1.24627i	1.24627	−	1.24627i
$520$		0			0
$521$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$521$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$522$	0.0296943	−	0.0582784i	0.0296943	−	0.0582784i
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$			1.27977i			1.27977i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.309017	−	0.951057i	0.309017	−	0.951057i
$530$		0			0
$531$	−0.0677598	−	0.0932634i	−0.0677598	−	0.0932634i
$532$		0			0
$533$	0.00547810	−	0.104528i	0.00547810	−	0.104528i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.174240	−	0.536255i	0.174240	−	0.536255i
$538$	−0.295511	−	0.0960175i	−0.295511	−	0.0960175i
$539$		0			0
$540$		0			0
$541$	−1.64728	+	0.535233i	−1.64728	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$541$	−1.64728	+	0.535233i	−1.64728	+	0.535233i	−0.669131	+	0.743145i	$0.733333\pi$
$542$			0.245761i			0.245761i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.41228	+	1.41228i	−1.41228	+	1.41228i	−0.669131	+	0.743145i	$0.733333\pi$
$547$	−1.41228	+	1.41228i	−1.41228	+	1.41228i	−0.743145	+	0.669131i	$0.766667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−0.440005	+	0.0696899i	−0.440005	+	0.0696899i
$553$		0			0
$554$	0.0510966	−	0.0703285i	0.0510966	−	0.0703285i
$555$		0			0
$556$	−1.53884	+	1.11803i	−1.53884	+	1.11803i
$557$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$557$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$558$	−0.0131181	−	0.00953088i	−0.0131181	−	0.00953088i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$563$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$564$	0.230713	+	0.710062i	0.230713	+	0.710062i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−0.0994407	+	0.627844i	−0.0994407	+	0.627844i
$569$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$569$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−1.00000
$576$	−0.0819254	−	0.112761i	−0.0819254	−	0.112761i
$577$	1.09905	+	1.09905i	1.09905	+	1.09905i	0.994522	+	0.104528i	$0.0333333\pi$
$577$	1.09905	+	1.09905i	1.09905	+	1.09905i	0.104528	+	0.994522i	$0.466667\pi$
$578$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i
$579$	0.108436	+	0.0352331i	0.108436	+	0.0352331i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.187839	−	0.578108i	−0.187839	−	0.578108i
$585$		0			0
$586$		0			0
$587$	−1.24314	−	0.196895i	−1.24314	−	0.196895i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.24314	−	0.196895i	−1.24314	−	0.196895i	−0.743145	+	0.669131i	$0.766667\pi$
$588$	−0.736573	+	0.736573i	−0.736573	+	0.736573i
$589$		0			0
$590$		0			0
$591$	0.789520	+	0.402280i	0.789520	+	0.402280i
$592$		0			0
$593$	0.309017	−	0.0489435i	0.309017	−	0.0489435i		−	1.00000i	$-0.5\pi$
$593$	0.309017	−	0.0489435i	0.309017	−	0.0489435i	0.309017	+	0.951057i	$0.400000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.0194974	−	0.00993440i	−0.0194974	−	0.00993440i
$599$	−0.951057	−	0.690983i	−0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$599$	−0.951057	−	0.690983i	−0.951057	−	0.690983i	−0.951057	+	0.309017i	$0.900000\pi$
$600$	0.202248	+	0.396934i	0.202248	+	0.396934i
$601$	−1.18606	+	1.18606i	−1.18606	+	1.18606i	−0.207912	+	0.978148i	$0.566667\pi$
$601$	−1.18606	+	1.18606i	−1.18606	+	1.18606i	−0.978148	+	0.207912i	$0.933333\pi$
$602$		0			0
$603$		0			0
$604$	0.867108	−	1.70180i	0.867108	−	1.70180i
$605$		0			0
$606$			0.206766i			0.206766i
$607$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$607$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.0231831	+	0.0713503i	−0.0231831	+	0.0713503i
$612$		0			0
$613$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$613$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$614$	0.245761			0.245761
$615$		0			0
$616$		0			0
$617$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$617$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$618$		0			0
$619$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$619$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$620$		0			0
$621$	−0.138616	+	0.875189i	−0.138616	+	0.875189i
$622$	0.0411622	+	0.259888i	0.0411622	+	0.259888i
$623$		0			0
$624$		−	0.0992854i		−	0.0992854i
$625$	0.309017	+	0.951057i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$		0			0
$633$	−0.275714	+	0.200318i	−0.275714	+	0.200318i
$634$	−0.0646021	+	0.0102320i	−0.0646021	+	0.0102320i
$635$		0			0
$636$		0			0
$637$	−0.103383	+	0.0163743i	−0.103383	+	0.0163743i
$638$		0			0
$639$	−0.258318	−	0.131620i	−0.258318	−	0.131620i
$640$		0			0
$641$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$641$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$642$		0			0
$643$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$643$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$647$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$648$	0.447979	−	0.145557i	0.447979	−	0.145557i
$649$		0			0
$650$	−0.00342316	+	0.0216130i	−0.00342316	+	0.0216130i
$651$		0			0
$652$	0.439216	−	1.35177i	0.439216	−	1.35177i
$653$	−0.889993	−	0.889993i	−0.889993	−	0.889993i	0.104528	−	0.994522i	$-0.466667\pi$
$653$	−0.889993	−	0.889993i	−0.889993	−	0.889993i	−0.994522	+	0.104528i	$0.966667\pi$
$654$		0			0
$655$		0			0
$656$	−0.435398	+	0.754131i	−0.435398	+	0.754131i
$657$	0.277233			0.277233
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$661$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$662$	−0.0610631	+	0.385537i	−0.0610631	+	0.385537i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.761497	−	1.49452i	0.761497	−	1.49452i
$668$	0.773659	−	0.394199i	0.773659	−	0.394199i
$669$	−2.04642	−	0.324121i	−2.04642	−	0.324121i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.77957	−	0.906737i	−1.77957	−	0.906737i	−0.913545	−	0.406737i	$-0.866667\pi$
$673$	−1.77957	−	0.906737i	−1.77957	−	0.906737i	−0.866025	−	0.500000i	$-0.833333\pi$
$674$		0			0
$675$	0.875189	−	0.138616i	0.875189	−	0.138616i
$676$	−0.555938	+	0.765183i	−0.555938	+	0.765183i
$677$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$677$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.770236	−	0.770236i	0.770236	−	0.770236i	−0.207912	−	0.978148i	$-0.566667\pi$
$683$	0.770236	−	0.770236i	0.770236	−	0.770236i	0.978148	+	0.207912i	$0.0666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.221232	−	1.39680i	0.221232	−	1.39680i	−0.587785	−	0.809017i	$-0.700000\pi$
$691$	0.221232	−	1.39680i	0.221232	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$692$	−1.47159	−	0.478148i	−1.47159	−	0.478148i
$693$		0			0
$694$	0.263427	+	0.263427i	0.263427	+	0.263427i
$695$		0			0
$696$	−0.747238			−0.747238
$697$		0			0
$698$	0.279773			0.279773
$699$	0.331424	+	0.456166i	0.331424	+	0.456166i
$700$		0			0
$701$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$	0.0184409	+	0.00599183i	0.0184409	+	0.00599183i
$703$		0			0
$704$		0			0
$705$		0			0
$706$			0.415823i			0.415823i
$707$		0			0
$708$	−0.292274	+	0.573620i	−0.292274	+	0.573620i
$709$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$709$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.336408	−	0.244415i	−0.336408	−	0.244415i
$714$		0			0
$715$		0			0
$716$	−0.488920	+	0.0774374i	−0.488920	+	0.0774374i
$717$	−1.07394	+	1.47815i	−1.07394	+	1.47815i
$718$		0			0
$719$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i	−0.309017	−	0.951057i	$-0.600000\pi$
$719$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i			1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	0.169131	+	0.122881i	0.169131	+	0.122881i
$723$		0			0
$724$		0			0
$725$	−1.65669	−	0.262394i	−1.65669	−	0.262394i
$726$	0.202901	−	0.103383i	0.202901	−	0.103383i
$727$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$727$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$728$		0			0
$729$		−	0.750379i		−	0.750379i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$733$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$734$		0			0
$735$		0			0
$736$	0.347395	+	0.478148i	0.347395	+	0.478148i
$737$		0			0
$738$	0.0260925	+	0.0289787i	0.0260925	+	0.0289787i
$739$	1.95630			1.95630			0.978148	−	0.207912i	$-0.0666667\pi$
$739$	1.95630			1.95630			0.978148	+	0.207912i	$0.0666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$743$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$744$	−0.0289787	+	0.182964i	−0.0289787	+	0.182964i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$751$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$752$	0.441327	−	0.441327i	0.441327	−	0.441327i
$753$		0			0
$754$	−0.0296943	−	0.0215742i	−0.0296943	−	0.0215742i
$755$		0			0
$756$		0			0
$757$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$757$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.406737	−	0.913545i	$-0.366667\pi$
$761$	1.40126	−	1.01807i	1.40126	−	1.01807i	0.994522	−	0.104528i	$-0.0333333\pi$
$762$	0.202901	+	0.103383i	0.202901	+	0.103383i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.0576399	+	0.0293690i	−0.0576399	+	0.0293690i
$768$	−0.292274	+	0.573620i	−0.292274	+	0.573620i
$769$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$769$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$770$		0			0
$771$	2.06909	−	0.672288i	2.06909	−	0.672288i
$772$	−0.0156587	−	0.0988649i	−0.0156587	−	0.0988649i
$773$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$773$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$774$		0			0
$775$	−0.128496	+	0.395472i	−0.128496	+	0.395472i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.459289	+	1.41355i	−0.459289	+	1.41355i
$784$	0.828176	+	0.269091i	0.828176	+	0.269091i
$785$		0			0
$786$	−0.0476735	−	0.300998i	−0.0476735	−	0.300998i
$787$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$787$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$788$		−	0.777921i		−	0.777921i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.0491290	−	0.0964210i	−0.0491290	−	0.0964210i
$795$		0			0
$796$		0			0
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	0.347395	−	0.478148i	0.347395	−	0.478148i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.00643411	+	0.00643411i	−0.00643411	+	0.00643411i
$807$	−1.59905	−	0.253265i	−1.59905	−	0.253265i
$808$	0.330869	−	0.168586i	0.330869	−	0.168586i
$809$	0.642040	−	1.26007i	0.642040	−	1.26007i	−0.309017	−	0.951057i	$-0.600000\pi$
$809$	0.642040	−	1.26007i	0.642040	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$810$		0			0
$811$		−	0.813473i		−	0.813473i	−0.913545	−	0.406737i	$-0.866667\pi$
$811$		−	0.813473i		−	0.813473i	0.913545	−	0.406737i	$-0.133333\pi$
$812$		0			0
$813$	0.200318	+	1.26476i	0.200318	+	1.26476i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.122881	−	0.169131i	−0.122881	−	0.169131i
$819$		0			0
$820$		0			0
$821$	−1.90211			−1.90211			−0.951057	−	0.309017i	$-0.900000\pi$
$821$	−1.90211			−1.90211			−0.951057	+	0.309017i	$0.900000\pi$
$822$		0			0
$823$	−0.506809	−	0.506809i	−0.506809	−	0.506809i	0.406737	−	0.913545i	$-0.366667\pi$
$823$	−0.506809	−	0.506809i	−0.506809	−	0.506809i	−0.913545	+	0.406737i	$0.866667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$827$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$828$	−0.169644	+	0.0551208i	−0.169644	+	0.0551208i
$829$		−	1.90211i		−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$829$		−	1.90211i		−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$830$		0			0
$831$	0.205634	−	0.403579i	0.205634	−	0.403579i
$832$	−0.0696899	+	0.0355088i	−0.0696899	+	0.0355088i
$833$		0			0
$834$	0.320282	−	0.320282i	0.320282	−	0.320282i
$835$		0			0
$836$		0			0
$837$	0.328301	+	0.167278i	0.328301	+	0.167278i
$838$		0			0
$839$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$839$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$840$		0			0
$841$	1.06593	−	1.46713i	1.06593	−	1.46713i
$842$		0			0
$843$		0			0
$844$	0.266585	+	0.135832i	0.266585	+	0.135832i
$845$		0			0
$846$	−0.0126885	−	0.0249027i	−0.0126885	−	0.0249027i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$			1.61906i			1.61906i
$853$	0.587785	−	0.190983i	0.587785	−	0.190983i		−	1.00000i	$-0.5\pi$
$853$	0.587785	−	0.190983i	0.587785	−	0.190983i	0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i	−0.978148	−	0.207912i	$-0.933333\pi$
$857$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i	0.913545	+	0.406737i	$0.133333\pi$
$858$		0			0
$859$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	0.406737	−	0.913545i	$-0.366667\pi$
$859$	−0.587785	−	0.809017i	−0.587785	−	0.809017i	−0.994522	+	0.104528i	$0.966667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.873619	+	1.20243i	0.873619	+	1.20243i	0.978148	+	0.207912i	$0.0666667\pi$
$863$	0.873619	+	1.20243i	0.873619	+	1.20243i	−0.104528	+	0.994522i	$0.533333\pi$
$864$	−0.370315	−	0.370315i	−0.370315	−	0.370315i
$865$		0			0
$866$		0			0
$867$	−0.170401	+	1.07587i	−0.170401	+	1.07587i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.702880	−	1.37948i	−0.702880	−	1.37948i
$877$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$877$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$878$	0.234448	+	0.119457i	0.234448	+	0.119457i
$879$		0			0
$880$		0			0
$881$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$881$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$882$	0.0229205	−	0.0315474i	0.0229205	−	0.0315474i
$883$	−1.39680	+	0.221232i	−1.39680	+	0.221232i	−0.809017	−	0.587785i	$-0.800000\pi$
$883$	−1.39680	+	0.221232i	−1.39680	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$884$		0			0
$885$		0			0
$886$	−0.0353579	−	0.0256890i	−0.0353579	−	0.0256890i
$887$	−0.877042	−	1.72129i	−0.877042	−	1.72129i	−0.669131	−	0.743145i	$-0.733333\pi$
$887$	−0.877042	−	1.72129i	−0.877042	−	1.72129i	−0.207912	−	0.978148i	$-0.566667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	0.562096	+	1.72995i	0.562096	+	1.72995i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.108436	−	0.0352331i	−0.108436	−	0.0352331i
$898$		0			0
$899$	−0.493191	−	0.493191i	−0.493191	−	0.493191i
$900$	0.104846	+	0.144308i	0.104846	+	0.144308i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.140547	+	0.432558i	−0.140547	+	0.432558i
$907$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$907$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$908$		0			0
$909$	0.0264942	+	0.167278i	0.0264942	+	0.167278i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$919$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$920$		0			0
$921$	1.26476	−	0.200318i	1.26476	−	0.200318i
$922$	0.224514	−	0.309017i	0.224514	−	0.309017i
$923$	−0.0956272	+	0.131620i	−0.0956272	+	0.131620i
$924$		0			0
$925$		0			0
$926$	0.169131	+	0.0861763i	0.169131	+	0.0861763i
$927$		0			0
$928$	0.450062	+	0.883297i	0.450062	+	0.883297i
$929$	0.506809	−	0.506809i	0.506809	−	0.506809i	−0.406737	−	0.913545i	$-0.633333\pi$
$929$	0.506809	−	0.506809i	0.506809	−	0.506809i	0.913545	+	0.406737i	$0.133333\pi$
$930$		0			0
$931$		0			0
$932$	0.224732	−	0.441061i	0.224732	−	0.441061i
$933$	0.423665	+	1.30391i	0.423665	+	1.30391i
$934$		0			0
$935$		0			0
$936$	0.00124912	+	0.00788660i	0.00124912	+	0.00788660i
$937$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$937$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$	0.669131	+	0.743145i	0.669131	+	0.743145i
$944$	0.538181			0.538181
$945$		0			0
$946$		0			0
$947$	−0.604528	+	1.86055i	−0.604528	+	1.86055i	−0.104528	+	0.994522i	$0.533333\pi$
$947$	−0.604528	+	1.86055i	−0.604528	+	1.86055i	−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	0.0243369	−	0.153657i	0.0243369	−	0.153657i
$950$		0			0
$951$	−0.324121	+	0.105313i	−0.324121	+	0.105313i
$952$		0			0
$953$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$953$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$954$		0			0
$955$		0			0
$956$	1.58428	+	0.250926i	1.58428	+	0.250926i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.669131	−	0.486152i	0.669131	−	0.486152i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.72129	−	0.877042i	−1.72129	−	0.877042i	−0.978148	−	0.207912i	$-0.933333\pi$
$967$	−1.72129	−	0.877042i	−1.72129	−	0.877042i	−0.743145	−	0.669131i	$-0.766667\pi$
$968$	−0.330869	−	0.240391i	−0.330869	−	0.240391i
$969$		0			0
$970$		0			0
$971$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$971$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$972$	0.313952	−	0.159967i	0.313952	−	0.159967i
$973$		0			0
$974$	0.0864545	+	0.266080i	0.0864545	+	0.266080i
$975$			0.114017i			0.114017i
$976$		0			0
$977$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$977$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$978$	−0.0529467	+	0.334293i	−0.0529467	+	0.334293i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.0510966	−	0.0703285i	−0.0510966	−	0.0703285i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	0.159649	−	0.415901i	0.159649	−	0.415901i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.278768	+	1.76007i	0.278768	+	1.76007i	0.587785	+	0.809017i	$0.300000\pi$
$991$	0.278768	+	1.76007i	0.278768	+	1.76007i	−0.309017	+	0.951057i	$0.600000\pi$
$992$	0.233733	−	0.0759444i	0.233733	−	0.0759444i
$993$			2.03386i			2.03386i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.95106	−	0.309017i	−1.95106	−	0.309017i	−0.951057	−	0.309017i	$-0.900000\pi$
$997$	−1.95106	−	0.309017i	−1.95106	−	0.309017i	−1.00000			$\pi$
$998$	−0.276014	+	0.276014i	−0.276014	+	0.276014i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	943.1.n.b.620.2	✓	16
23.22	odd	2	CM	943.1.n.b.620.2	✓	16
41.33	even	20	inner	943.1.n.b.689.2	yes	16
943.689	odd	20	inner	943.1.n.b.689.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
943.1.n.b.620.2	✓	16	1.1	even	1	trivial
943.1.n.b.620.2	✓	16	23.22	odd	2	CM
943.1.n.b.689.2	yes	16	41.33	even	20	inner
943.1.n.b.689.2	yes	16	943.689	odd	20	inner

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:	\( N \)	\(=\)	\( 943 = 23 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	943.n (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.122881 + 0.169131i) q^{2} +(0.770236 + 0.770236i) q^{3} +(0.295511 - 0.909491i) q^{4} +(-0.0356234 + 0.224918i) q^{6} +(0.388960 - 0.126381i) q^{8} +0.186527i q^{9} +O(q^{10})\)	\(q+(0.122881 + 0.169131i) q^{2} +(0.770236 + 0.770236i) q^{3} +(0.295511 - 0.909491i) q^{4} +(-0.0356234 + 0.224918i) q^{6} +(0.388960 - 0.126381i) q^{8} +0.186527i q^{9} +(0.928136 - 0.472909i) q^{12} +(0.103383 + 0.0163743i) q^{13} +(-0.704489 - 0.511841i) q^{16} +(-0.0315474 + 0.0229205i) q^{18} +(-0.809017 + 0.587785i) q^{23} +(0.396934 + 0.202248i) q^{24} +(0.809017 + 0.587785i) q^{25} +(0.00993440 + 0.0194974i) q^{26} +(0.626566 - 0.626566i) q^{27} +(-1.49452 + 0.761497i) q^{29} +(0.128496 + 0.395472i) q^{31} -0.591023i q^{32} +(0.169644 + 0.0551208i) q^{36} +(0.0670174 + 0.0922415i) q^{39} +(-0.104528 - 0.994522i) q^{41} +(-0.198825 - 0.0646021i) q^{46} +(-0.112122 + 0.707912i) q^{47} +(-0.148384 - 0.936861i) q^{48} +(-0.951057 + 0.309017i) q^{49} +0.209057i q^{50} +(0.0454432 - 0.0891873i) q^{52} +(0.182964 + 0.0289787i) q^{54} +(-0.312440 - 0.159196i) q^{58} +(-0.500000 + 0.363271i) q^{59} +(-0.0510966 + 0.0703285i) q^{62} +(-0.604528 + 0.439216i) q^{64} +(-1.07587 - 0.170401i) q^{69} +(-0.705634 + 1.38488i) q^{71} +(0.0235734 + 0.0725515i) q^{72} -1.48629i q^{73} +(0.170401 + 1.07587i) q^{75} +(-0.00736573 + 0.0226694i) q^{78} +1.15173 q^{81} +(0.155360 - 0.139886i) q^{82} +(-1.73767 - 0.564602i) q^{87} +(0.295511 + 0.909491i) q^{92} +(-0.205634 + 0.403579i) q^{93} +(-0.133507 + 0.0680253i) q^{94} +(0.455227 - 0.455227i) q^{96} +(-0.169131 - 0.122881i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{3} + 10 q^{4} - 12 q^{6}+O(q^{10}) \) 16 * q - 2 * q^3 + 10 * q^4 - 12 * q^6	\( 16 q - 2 q^{3} + 10 q^{4} - 12 q^{6} + 2 q^{13} - 6 q^{16} + 6 q^{18} - 4 q^{23} - 22 q^{24} + 4 q^{25} + 12 q^{26} + 6 q^{27} - 8 q^{29} - 10 q^{36} + 2 q^{41} + 2 q^{47} - 18 q^{48} - 6 q^{54} + 2 q^{58} - 8 q^{59} - 10 q^{62} - 6 q^{64} - 2 q^{69} - 2 q^{71} + 16 q^{72} + 2 q^{75} - 2 q^{78} - 12 q^{81} + 10 q^{92} + 6 q^{93} + 18 q^{94} - 10 q^{96} + 6 q^{98}+O(q^{100}) \) 16 * q - 2 * q^3 + 10 * q^4 - 12 * q^6 + 2 * q^13 - 6 * q^16 + 6 * q^18 - 4 * q^23 - 22 * q^24 + 4 * q^25 + 12 * q^26 + 6 * q^27 - 8 * q^29 - 10 * q^36 + 2 * q^41 + 2 * q^47 - 18 * q^48 - 6 * q^54 + 2 * q^58 - 8 * q^59 - 10 * q^62 - 6 * q^64 - 2 * q^69 - 2 * q^71 + 16 * q^72 + 2 * q^75 - 2 * q^78 - 12 * q^81 + 10 * q^92 + 6 * q^93 + 18 * q^94 - 10 * q^96 + 6 * q^98

Introduction

L-functions

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