LMFDB - Newform orbit 621.1.k.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [621,1,Mod(22,621)] 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("621.22"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(621, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([14, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 621 = 3^{3} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	621.k (of order $18$, degree $6$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.309919372845$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$3$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{54})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{9} + 1 $ x^18 - x^9 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} + \cdots)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + (\zeta_{54}^{20} - \zeta_{54}) q^{2} - \zeta_{54}^{5} q^{3} + ( - \zeta_{54}^{21} + \cdots + \zeta_{54}^{2}) q^{4} + ( - \zeta_{54}^{25} + \zeta_{54}^{6}) q^{6} + (\zeta_{54}^{22} + \cdots - \zeta_{54}^{3}) q^{8} + \cdots + ( - \zeta_{54}^{13} - \zeta_{54}^{5}) q^{98} +O(q^{100}) $ q + (z^20 - z) * q^2 - z^5 * q^3 + (-z^21 - z^13 + z^2) * q^4 + (-z^25 + z^6) * q^6 + (z^22 + z^14 + z^6 - z^3) * q^8 + z^10 * q^9 + (z^26 + z^18 - z^7) * q^12 + (-z^17 + z^16) * q^13 + (z^26 - z^23 - z^15 - z^7 + z^4) * q^16 + (-z^11 - z^3) * q^18 - z^21 * q^23 + (-z^19 - z^11 + z^8 + 1) * q^24 + z^24 * q^25 + (z^18 - z^17 + z^10 - z^9) * q^26 - z^15 * q^27 + (-z^13 + z^8) * q^29 + (-z^23 - z^19) * q^31 + (z^24 - z^19 + z^16 + z^8 - z^5 + 1) * q^32 + (-z^23 + z^12 + z^4) * q^36 + (z^22 - z^21) * q^39 + (z^4 + z^2) * q^41 + (z^22 + z^14) * q^46 + (-z^25 + z^14) * q^47 + (z^20 + z^12 - z^9 + z^4 - z) * q^48 + z^12 * q^49 + (-z^25 - z^17) * q^50 + (-z^19 + z^18 - z^11 + z^10 - z^3 + z^2) * q^52 + (z^16 + z^8) * q^54 + (z^14 - z^9 + z^6 - z) * q^58 + (-z^15 - z^9) * q^59 + (z^24 + z^20 + z^16 + z^12) * q^62 + (-z^25 + z^20 - z^17 + z^12 - z^9 + z^6 - z) * q^64 + z^26 * q^69 + (-z^11 - z^7) * q^71 + (z^24 + z^16 - z^13 - z^5) * q^72 + (-z^25 - z^11) * q^73 + z^2 * q^75 + (-z^23 + z^22 - z^15 + z^14) * q^78 + z^20 * q^81 + (z^24 + z^22 - z^5 - z^3) * q^82 + (z^18 - z^13) * q^87 + (-z^23 - z^15 - z^7) * q^92 + (z^24 - z) * q^93 + (z^26 + z^18 - z^15 - z^7) * q^94 + (z^24 - z^21 - z^13 + z^10 - z^5 + z^2) * q^96 + (-z^13 - z^5) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 9 q^{12} + 18 q^{24} - 18 q^{26} + 18 q^{32} - 9 q^{48} - 9 q^{52} - 9 q^{58} - 9 q^{59} - 9 q^{64} - 9 q^{87} - 9 q^{94}+O(q^{100}) $ 18 * q - 9 * q^12 + 18 * q^24 - 18 * q^26 + 18 * q^32 - 9 * q^48 - 9 * q^52 - 9 * q^58 - 9 * q^59 - 9 * q^64 - 9 * q^87 - 9 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/621\mathbb{Z}\right)^\times$.

$n$	$28$	$461$
$\chi(n)$	$-1$	$-\zeta_{54}^{21}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

22.1

0.686242	−	0.727374i
0.286803	+	0.957990i
−0.973045	−	0.230616i
0.835488	−	0.549509i
−0.893633	−	0.448799i
0.0581448	+	0.998308i
0.835488	+	0.549509i
−0.893633	+	0.448799i
0.0581448	−	0.998308i
0.686242	+	0.727374i
0.286803	−	0.957990i
−0.973045	+	0.230616i
0.993238	−	0.116093i
−0.396080	+	0.918216i
−0.597159	−	0.802123i
0.993238	+	0.116093i
−0.396080	−	0.918216i
−0.597159	+	0.802123i

−1.52173

1.27688i

0.597159

−

0.802123i

0.511583

−

2.90133i

0.115503

1.98312i

1.93293

3.34793i

−0.286803

−

0.957990i

22.2

0.606829

−

0.509190i

−0.993238

−

0.116093i

−0.0646810

0.366824i

−0.661840

0.435299i

0.543613

0.941565i

0.973045

0.230616i

22.3

0.914900

−

0.767692i

0.396080

0.918216i

0.0740425

−

0.419916i

1.06728

0.536009i

0.342534

0.593286i

−0.686242

0.727374i

160.1

−0.238329

1.35163i

0.973045

0.230616i

−0.830416

−

0.302247i

−0.543613

1.26024i

−0.0798028

0.138223i

0.893633

0.448799i

160.2

−0.0996057

0.564892i

−0.686242

0.727374i

0.630511

0.229487i

−0.342534

−

0.460103i

−0.479241

0.830070i

−0.0581448

−

0.998308i

160.3

0.337935

−

1.91652i

−0.286803

−

0.957990i

−2.61917

−

0.953301i

−1.93293

0.225927i

−1.73909

3.01219i

−0.835488

0.549509i

229.1

−0.238329

−

1.35163i

0.973045

−

0.230616i

−0.830416

0.302247i

−0.543613

−

1.26024i

−0.0798028

−

0.138223i

0.893633

−

0.448799i

229.2

−0.0996057

−

0.564892i

−0.686242

−

0.727374i

0.630511

−

0.229487i

−0.342534

0.460103i

−0.479241

−

0.830070i

−0.0581448

0.998308i

229.3

0.337935

1.91652i

−0.286803

0.957990i

−2.61917

0.953301i

−1.93293

−

0.225927i

−1.73909

−

3.01219i

−0.835488

−

0.549509i

367.1

−1.52173

−

1.27688i

0.597159

0.802123i

0.511583

2.90133i

0.115503

−

1.98312i

1.93293

−

3.34793i

−0.286803

0.957990i

367.2

0.606829

0.509190i

−0.993238

0.116093i

−0.0646810

−

0.366824i

−0.661840

−

0.435299i

0.543613

−

0.941565i

0.973045

−

0.230616i

367.3

0.914900

0.767692i

0.396080

−

0.918216i

0.0740425

0.419916i

1.06728

−

0.536009i

0.342534

−

0.593286i

−0.686242

−

0.727374i

436.1

−1.67948

−

0.611281i

−0.835488

0.549509i

1.68094

1.41048i

1.73909

−

0.412172i

−1.06728

−

1.84858i

0.396080

−

0.918216i

436.2

0.109277

0.0397734i

0.893633

0.448799i

−0.755685

−

0.634095i

0.0798028

0.0845860i

−0.115503

−

0.200058i

0.597159

0.802123i

436.3

1.57020

0.571507i

−0.0581448

−

0.998308i

1.37287

1.15198i

0.479241

−

1.60078i

0.661840

1.14634i

−0.993238

0.116093i

574.1

−1.67948

0.611281i

−0.835488

−

0.549509i

1.68094

−

1.41048i

1.73909

0.412172i

−1.06728

1.84858i

0.396080

0.918216i

574.2

0.109277

−

0.0397734i

0.893633

−

0.448799i

−0.755685

0.634095i

0.0798028

−

0.0845860i

−0.115503

0.200058i

0.597159

−

0.802123i

574.3

1.57020

−

0.571507i

−0.0581448

0.998308i

1.37287

−

1.15198i

0.479241

1.60078i

0.661840

−

1.14634i

−0.993238

−

0.116093i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
27.e	even	9	1	inner
621.k	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	621.1.k.a	✓	18
3.b	odd	2	1		1863.1.k.a		18
23.b	odd	2	1	CM	621.1.k.a	✓	18
27.e	even	9	1	inner	621.1.k.a	✓	18
27.f	odd	18	1		1863.1.k.a		18
69.c	even	2	1		1863.1.k.a		18
621.k	odd	18	1	inner	621.1.k.a	✓	18
621.m	even	18	1		1863.1.k.a		18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
621.1.k.a	✓	18	1.a	even	1	1	trivial
621.1.k.a	✓	18	23.b	odd	2	1	CM
621.1.k.a	✓	18	27.e	even	9	1	inner
621.1.k.a	✓	18	621.k	odd	18	1	inner
1863.1.k.a		18	3.b	odd	2	1
1863.1.k.a		18	27.f	odd	18	1
1863.1.k.a		18	69.c	even	2	1
1863.1.k.a		18	621.m	even	18	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(621, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} - 18 T^{13} + \cdots + 1 $ T^18 - 18T^13 + 30T^12 + 27T^11 - 18T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 441T^4 - 213T^3 + 108T^2 - 18*T + 1
$3$	$ T^{18} + T^{9} + 1 $ T^18 + T^9 + 1
$5$	$ T^{18} $ T^18
$7$	$ T^{18} $ T^18
$11$	$ T^{18} $ T^18
$13$	$ T^{18} - 18 T^{13} + \cdots + 1 $ T^18 - 18T^13 + 30T^12 + 27T^11 - 18T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 441T^4 - 213T^3 + 108T^2 - 18*T + 1
$17$	$ T^{18} $ T^18
$19$	$ T^{18} $ T^18
$23$	$ (T^{6} + T^{3} + 1)^{3} $ (T^6 + T^3 + 1)^3
$29$	$ T^{18} + 9 T^{13} + \cdots + 1 $ T^18 + 9T^13 + 30T^12 - 54T^11 + 9T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 468T^4 - 213T^3 + 27T^2 + 9*T + 1
$31$	$ T^{18} + 9 T^{13} + \cdots + 1 $ T^18 + 9T^13 + 30T^12 + 27T^11 + 9T^10 + 2T^9 + 81T^8 + 540T^7 + 1386T^6 + 1782T^5 + 1278T^4 + 516T^3 + 108T^2 + 9*T + 1
$37$	$ T^{18} $ T^18
$41$	$ T^{18} + 9 T^{13} + \cdots + 1 $ T^18 + 9T^13 + 30T^12 + 27T^11 + 9T^10 + 2T^9 + 81T^8 + 540T^7 + 1386T^6 + 1782T^5 + 1278T^4 + 516T^3 + 108T^2 + 9*T + 1
$43$	$ T^{18} $ T^18
$47$	$ T^{18} - 18 T^{13} + \cdots + 1 $ T^18 - 18T^13 + 30T^12 + 27T^11 - 18T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 441T^4 - 213T^3 + 108T^2 - 18*T + 1
$53$	$ T^{18} $ T^18
$59$	$ (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{3} $ (T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1)^3
$61$	$ T^{18} $ T^18
$67$	$ T^{18} $ T^18
$71$	$ T^{18} + 9 T^{16} + \cdots + 1 $ T^18 + 9T^16 + 54T^14 + 183T^12 + 450T^10 + 2T^9 + 648T^8 + 9T^7 + 657T^6 - 27T^5 + 270T^4 - 60T^3 + 81T^2 - 9*T + 1
$73$	$ T^{18} + 9 T^{16} + \cdots + 1 $ T^18 + 9T^16 + 54T^14 + 183T^12 + 450T^10 + 2T^9 + 648T^8 + 9T^7 + 657T^6 - 27T^5 + 270T^4 - 60T^3 + 81T^2 - 9*T + 1
$79$	$ T^{18} $ T^18
$83$	$ T^{18} $ T^18
$89$	$ T^{18} $ T^18
$97$	$ T^{18} $ T^18
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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:	\( N \)	\(=\)	\( 621 = 3^{3} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	621.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{54}^{20} - \zeta_{54}) q^{2} - \zeta_{54}^{5} q^{3} + ( - \zeta_{54}^{21} + \cdots + \zeta_{54}^{2}) q^{4} + ( - \zeta_{54}^{25} + \zeta_{54}^{6}) q^{6} + (\zeta_{54}^{22} + \cdots - \zeta_{54}^{3}) q^{8} + \cdots + ( - \zeta_{54}^{13} - \zeta_{54}^{5}) q^{98} +O(q^{100}) \) q + (z^20 - z) * q^2 - z^5 * q^3 + (-z^21 - z^13 + z^2) * q^4 + (-z^25 + z^6) * q^6 + (z^22 + z^14 + z^6 - z^3) * q^8 + z^10 * q^9 + (z^26 + z^18 - z^7) * q^12 + (-z^17 + z^16) * q^13 + (z^26 - z^23 - z^15 - z^7 + z^4) * q^16 + (-z^11 - z^3) * q^18 - z^21 * q^23 + (-z^19 - z^11 + z^8 + 1) * q^24 + z^24 * q^25 + (z^18 - z^17 + z^10 - z^9) * q^26 - z^15 * q^27 + (-z^13 + z^8) * q^29 + (-z^23 - z^19) * q^31 + (z^24 - z^19 + z^16 + z^8 - z^5 + 1) * q^32 + (-z^23 + z^12 + z^4) * q^36 + (z^22 - z^21) * q^39 + (z^4 + z^2) * q^41 + (z^22 + z^14) * q^46 + (-z^25 + z^14) * q^47 + (z^20 + z^12 - z^9 + z^4 - z) * q^48 + z^12 * q^49 + (-z^25 - z^17) * q^50 + (-z^19 + z^18 - z^11 + z^10 - z^3 + z^2) * q^52 + (z^16 + z^8) * q^54 + (z^14 - z^9 + z^6 - z) * q^58 + (-z^15 - z^9) * q^59 + (z^24 + z^20 + z^16 + z^12) * q^62 + (-z^25 + z^20 - z^17 + z^12 - z^9 + z^6 - z) * q^64 + z^26 * q^69 + (-z^11 - z^7) * q^71 + (z^24 + z^16 - z^13 - z^5) * q^72 + (-z^25 - z^11) * q^73 + z^2 * q^75 + (-z^23 + z^22 - z^15 + z^14) * q^78 + z^20 * q^81 + (z^24 + z^22 - z^5 - z^3) * q^82 + (z^18 - z^13) * q^87 + (-z^23 - z^15 - z^7) * q^92 + (z^24 - z) * q^93 + (z^26 + z^18 - z^15 - z^7) * q^94 + (z^24 - z^21 - z^13 + z^10 - z^5 + z^2) * q^96 + (-z^13 - z^5) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 9 q^{12} + 18 q^{24} - 18 q^{26} + 18 q^{32} - 9 q^{48} - 9 q^{52} - 9 q^{58} - 9 q^{59} - 9 q^{64} - 9 q^{87} - 9 q^{94}+O(q^{100}) \) 18 * q - 9 * q^12 + 18 * q^24 - 18 * q^26 + 18 * q^32 - 9 * q^48 - 9 * q^52 - 9 * q^58 - 9 * q^59 - 9 * q^64 - 9 * q^87 - 9 * q^94

Introduction

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Varieties

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	621.1.k.a

Level	$621$
Weight	$1$
Character orbit	621.k
Analytic conductor	$0.310$
Analytic rank	$0$
Dimension	$18$
Projective image	$D_{27}$
CM discriminant	-23
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.309919372845\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{54})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - x^{9} + 1 \) x^18 - x^9 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{27}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{27} + \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 22.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} - 18 T^{13} + \cdots + 1 \) T^18 - 18T^13 + 30T^12 + 27T^11 - 18T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 441T^4 - 213T^3 + 108T^2 - 18*T + 1
$3$	\( T^{18} + T^{9} + 1 \) T^18 + T^9 + 1
$5$	\( T^{18} \) T^18
$7$	\( T^{18} \) T^18
$11$	\( T^{18} \) T^18
$13$	\( T^{18} - 18 T^{13} + \cdots + 1 \) T^18 - 18T^13 + 30T^12 + 27T^11 - 18T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 441T^4 - 213T^3 + 108T^2 - 18*T + 1
$17$	\( T^{18} \) T^18
$19$	\( T^{18} \) T^18
$23$	\( (T^{6} + T^{3} + 1)^{3} \) (T^6 + T^3 + 1)^3
$29$	\( T^{18} + 9 T^{13} + \cdots + 1 \) T^18 + 9T^13 + 30T^12 - 54T^11 + 9T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 468T^4 - 213T^3 + 27T^2 + 9*T + 1
$31$	\( T^{18} + 9 T^{13} + \cdots + 1 \) T^18 + 9T^13 + 30T^12 + 27T^11 + 9T^10 + 2T^9 + 81T^8 + 540T^7 + 1386T^6 + 1782T^5 + 1278T^4 + 516T^3 + 108T^2 + 9*T + 1
$37$	\( T^{18} \) T^18
$41$	\( T^{18} + 9 T^{13} + \cdots + 1 \) T^18 + 9T^13 + 30T^12 + 27T^11 + 9T^10 + 2T^9 + 81T^8 + 540T^7 + 1386T^6 + 1782T^5 + 1278T^4 + 516T^3 + 108T^2 + 9*T + 1
$43$	\( T^{18} \) T^18
$47$	\( T^{18} - 18 T^{13} + \cdots + 1 \) T^18 - 18T^13 + 30T^12 + 27T^11 - 18T^10 + 2T^9 + 81T^8 - 270T^7 + 657T^6 - 648T^5 + 441T^4 - 213T^3 + 108T^2 - 18*T + 1
$53$	\( T^{18} \) T^18
$59$	\( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{3} \) (T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1)^3
$61$	\( T^{18} \) T^18
$67$	\( T^{18} \) T^18
$71$	\( T^{18} + 9 T^{16} + \cdots + 1 \) T^18 + 9T^16 + 54T^14 + 183T^12 + 450T^10 + 2T^9 + 648T^8 + 9T^7 + 657T^6 - 27T^5 + 270T^4 - 60T^3 + 81T^2 - 9*T + 1
$73$	\( T^{18} + 9 T^{16} + \cdots + 1 \) T^18 + 9T^16 + 54T^14 + 183T^12 + 450T^10 + 2T^9 + 648T^8 + 9T^7 + 657T^6 - 27T^5 + 270T^4 - 60T^3 + 81T^2 - 9*T + 1
$79$	\( T^{18} \) T^18
$83$	\( T^{18} \) T^18
$89$	\( T^{18} \) T^18
$97$	\( T^{18} \) T^18
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\(n\)	\(28\)	\(461\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{54}^{21}\)