LMFDB - Newform orbit 2695.1.ck.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2695,1,Mod(109,2695)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2695, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 40, 21]))

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

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

chi := DirichletCharacter("2695.109");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2695 = 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2695.ck (of order $42$, degree $12$, 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:	$1.34498020905$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{42})$
Coefficient field:	$\Q(\zeta_{84})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 $ x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2695\mathbb{Z}\right)^\times$.

$n$	$981$	$1816$	$2157$
$\chi(n)$	$-1$	$\zeta_{84}^{16}$	$-1$

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} $

109.1

−0.930874	−	0.365341i
0.930874	+	0.365341i
0.680173	−	0.733052i
−0.680173	+	0.733052i
−0.294755	−	0.955573i
0.294755	+	0.955573i
−0.563320	−	0.826239i
0.563320	+	0.826239i
−0.563320	+	0.826239i
0.563320	−	0.826239i
−0.930874	+	0.365341i
0.930874	−	0.365341i
−0.997204	+	0.0747301i
0.997204	−	0.0747301i
0.149042	+	0.988831i
−0.149042	−	0.988831i
0.680173	+	0.733052i
−0.680173	−	0.733052i
−0.997204	−	0.0747301i
0.997204	+	0.0747301i

−1.46200

1.35654i

0.222521

−

2.96934i

−0.365341

0.930874i

0.974928

0.222521i

2.45921

3.08375i

0.955573

−

0.294755i

−0.728639

−

1.85654i

109.2

1.46200

−

1.35654i

0.222521

−

2.96934i

−0.365341

0.930874i

−0.974928

−

0.222521i

−2.45921

−

3.08375i

0.955573

−

0.294755i

0.728639

1.85654i

219.1

−0.0222759

0.297251i

0.900969

0.135799i

0.733052

0.680173i

0.433884

−

0.900969i

−0.126766

0.555400i

0.826239

−

0.563320i

−0.218511

0.202749i

219.2

0.0222759

−

0.297251i

0.900969

0.135799i

0.733052

0.680173i

−0.433884

0.900969i

0.126766

−

0.555400i

0.826239

−

0.563320i

0.218511

−

0.202749i

494.1

−1.53825

−

1.04876i

0.900969

2.29563i

−0.955573

0.294755i

−0.433884

0.900969i

0.607374

−

2.66108i

0.0747301

0.997204i

1.77904

0.548760i

494.2

1.53825

1.04876i

0.900969

2.29563i

−0.955573

0.294755i

0.433884

−

0.900969i

−0.607374

2.66108i

0.0747301

0.997204i

−1.77904

−

0.548760i

604.1

−0.496990

−

1.26631i

−0.623490

0.578514i

−0.826239

0.563320i

0.781831

−

0.623490i

−0.183183

−

0.0882162i

−0.988831

0.149042i

1.12397

0.766310i

604.2

0.496990

1.26631i

−0.623490

0.578514i

−0.826239

0.563320i

−0.781831

0.623490i

0.183183

0.0882162i

−0.988831

0.149042i

−1.12397

−

0.766310i

879.1

−0.496990

1.26631i

−0.623490

−

0.578514i

−0.826239

−

0.563320i

0.781831

0.623490i

−0.183183

0.0882162i

−0.988831

−

0.149042i

1.12397

−

0.766310i

879.2

0.496990

−

1.26631i

−0.623490

−

0.578514i

−0.826239

−

0.563320i

−0.781831

−

0.623490i

0.183183

−

0.0882162i

−0.988831

−

0.149042i

−1.12397

0.766310i

989.1

−1.46200

−

1.35654i

0.222521

2.96934i

−0.365341

−

0.930874i

0.974928

−

0.222521i

2.45921

−

3.08375i

0.955573

0.294755i

−0.728639

1.85654i

989.2

1.46200

1.35654i

0.222521

2.96934i

−0.365341

−

0.930874i

−0.974928

0.222521i

−2.45921

3.08375i

0.955573

0.294755i

0.728639

−

1.85654i

1264.1

−0.582926

−

0.0878620i

−0.623490

−

0.192321i

−0.0747301

−

0.997204i

−0.781831

0.623490i

0.877681

0.422669i

0.365341

−

0.930874i

−0.0440542

0.587862i

1264.2

0.582926

0.0878620i

−0.623490

−

0.192321i

−0.0747301

−

0.997204i

0.781831

−

0.623490i

−0.877681

−

0.422669i

0.365341

−

0.930874i

0.0440542

−

0.587862i

1374.1

−1.07659

−

0.332083i

0.222521

0.151712i

0.988831

−

0.149042i

0.974928

0.222521i

0.513267

0.643616i

−0.733052

−

0.680173i

−1.11406

−

0.167917i

1374.2

1.07659

0.332083i

0.222521

0.151712i

0.988831

−

0.149042i

−0.974928

−

0.222521i

−0.513267

−

0.643616i

−0.733052

−

0.680173i

1.11406

0.167917i

1649.1

−0.0222759

−

0.297251i

0.900969

−

0.135799i

0.733052

−

0.680173i

0.433884

0.900969i

−0.126766

−

0.555400i

0.826239

0.563320i

−0.218511

−

0.202749i

1649.2

0.0222759

0.297251i

0.900969

−

0.135799i

0.733052

−

0.680173i

−0.433884

−

0.900969i

0.126766

0.555400i

0.826239

0.563320i

0.218511

0.202749i

1759.1

−0.582926

0.0878620i

−0.623490

0.192321i

−0.0747301

0.997204i

−0.781831

−

0.623490i

0.877681

−

0.422669i

0.365341

0.930874i

−0.0440542

−

0.587862i

1759.2

0.582926

−

0.0878620i

−0.623490

0.192321i

−0.0747301

0.997204i

0.781831

0.623490i

−0.877681

0.422669i

0.365341

0.930874i

0.0440542

0.587862i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
55.d	odd	2	1	CM by $\Q(\sqrt{-55}) $
5.b	even	2	1	inner
11.b	odd	2	1	inner
49.g	even	21	1	inner
245.t	even	42	1	inner
539.x	odd	42	1	inner
2695.ck	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2695.1.ck.c	✓	24
5.b	even	2	1	inner	2695.1.ck.c	✓	24
11.b	odd	2	1	inner	2695.1.ck.c	✓	24
49.g	even	21	1	inner	2695.1.ck.c	✓	24
55.d	odd	2	1	CM	2695.1.ck.c	✓	24
245.t	even	42	1	inner	2695.1.ck.c	✓	24
539.x	odd	42	1	inner	2695.1.ck.c	✓	24
2695.ck	odd	42	1	inner	2695.1.ck.c	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2695.1.ck.c	✓	24	1.a	even	1	1	trivial
2695.1.ck.c	✓	24	5.b	even	2	1	inner
2695.1.ck.c	✓	24	11.b	odd	2	1	inner
2695.1.ck.c	✓	24	49.g	even	21	1	inner
2695.1.ck.c	✓	24	55.d	odd	2	1	CM
2695.1.ck.c	✓	24	245.t	even	42	1	inner
2695.1.ck.c	✓	24	539.x	odd	42	1	inner
2695.1.ck.c	✓	24	2695.ck	odd	42	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{24} - 3 T_{2}^{22} + 28 T_{2}^{20} - 43 T_{2}^{18} + 171 T_{2}^{16} - 28 T_{2}^{14} + 148 T_{2}^{12} + \cdots + 1 $ T2^24 - 3*T2^22 + 28*T2^20 - 43*T2^18 + 171*T2^16 - 28*T2^14 + 148*T2^12 - 875*T2^10 + 1440*T2^8 - 519*T2^6 + 16*T2^2 + 1 acting on $S_{1}^{\mathrm{new}}(2695, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 + 28T^20 - 43T^18 + 171T^16 - 28T^14 + 148T^12 - 875T^10 + 1440T^8 - 519T^6 + 16T^2 + 1
$3$	$ T^{24} $ T^24
$5$	$ (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} $ (T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1)^2
$7$	$ (T^{12} - T^{10} + T^{8} + \cdots + 1)^{2} $ (T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$11$	$ (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} $ (T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1)^2
$13$	$ T^{24} - T^{22} + \cdots + 1 $ T^24 - T^22 + 13T^20 + 10T^18 + 132T^16 + 408T^14 + 301T^12 - 108T^10 + 927T^8 + 668T^6 + 159T^4 + 10T^2 + 1
$17$	$ T^{24} - 14 T^{20} + \cdots + 2401 $ T^24 - 14T^20 - 28T^18 + 147T^16 + 98T^14 - 392T^12 + 3430T^10 + 3087T^8 - 6174T^6 + 7203T^4 - 4802T^2 + 2401
$19$	$ T^{24} $ T^24
$23$	$ T^{24} $ T^24
$29$	$ T^{24} $ T^24
$31$	$ (T^{2} + T + 1)^{12} $ (T^2 + T + 1)^12
$37$	$ T^{24} $ T^24
$41$	$ T^{24} $ T^24
$43$	$ T^{24} + 6 T^{22} + \cdots + 1 $ T^24 + 6T^22 + 13T^20 + 38T^18 + 258T^16 + 716T^14 + 1519T^12 + 1978T^10 + 1284T^8 + 234T^6 + 96T^4 - 11*T^2 + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} $ T^24
$59$	$ (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} $ (T^12 + T^11 - T^9 + 6T^8 - 21T^7 - 20T^6 + 69T^4 - 29T^3 + 49T^2 - 13*T + 1)^2
$61$	$ T^{24} $ T^24
$67$	$ T^{24} $ T^24
$71$	$ (T^{12} + 5 T^{11} + \cdots + 1)^{2} $ (T^12 + 5T^11 + 17T^10 + 38T^9 + 68T^8 + 92T^7 + 105T^6 + 92T^5 + 61T^4 + 10T^3 + 45T^2 + 12*T + 1)^2
$73$	$ T^{24} - 3 T^{22} + \cdots + 1 $ T^24 - 3T^22 - 8T^18 + 220T^16 - 679T^14 + 1233T^12 - 1015T^10 - 163T^8 + 461T^6 + 119T^4 + 2T^2 + 1
$79$	$ T^{24} $ T^24
$83$	$ (T^{12} + 3 T^{10} + \cdots + 729)^{2} $ (T^12 + 3T^10 + 9T^8 + 27T^6 + 81T^4 + 243*T^2 + 729)^2
$89$	$ (T^{12} - 8 T^{11} + \cdots + 1)^{2} $ (T^12 - 8T^11 + 35T^10 - 104T^9 + 230T^8 - 392T^7 + 519T^6 - 518T^5 + 349T^4 - 118T^3 + 6T + 1)^2
$97$	$ T^{24} $ T^24
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2695.ck (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{84}^{23} + \zeta_{84}^{15}) q^{2} + ( - \zeta_{84}^{38} + \cdots - \zeta_{84}^{4}) q^{4}+ \cdots + \zeta_{84}^{16} q^{9}+O(q^{10}) \) q + (-z^23 + z^15) * q^2 + (-z^38 + z^30 - z^4) * q^4 + z^22 * q^5 + z^9 * q^7 + (z^27 - z^19 + z^11 - z^3) * q^8 + z^16 * q^9	\( q + ( - \zeta_{84}^{23} + \zeta_{84}^{15}) q^{2} + ( - \zeta_{84}^{38} + \cdots - \zeta_{84}^{4}) q^{4}+ \cdots - q^{99}+O(q^{100}) \) q + (-z^23 + z^15) * q^2 + (-z^38 + z^30 - z^4) * q^4 + z^22 * q^5 + z^9 * q^7 + (z^27 - z^19 + z^11 - z^3) * q^8 + z^16 * q^9 + (z^37 + z^3) * q^10 + z^26 * q^11 + (-z^17 - z^7) * q^13 + (-z^32 + z^24) * q^14 + (-z^34 + z^26 - z^18 + z^8 - 1) * q^16 + (z^17 - z^5) * q^17 + (-z^39 + z^31) * q^18 + (-z^26 + z^18 - z^10) * q^20 + (z^41 + z^7) * q^22 - z^2 * q^25 + (z^40 - z^32 + z^30 - z^22) * q^26 + (z^39 - z^13 + z^5) * q^28 - z^14 * q^31 + (z^41 - z^33 - z^31 + z^23 - z^15 + z^7) * q^32 + (-z^40 + z^32 + z^28 - z^20) * q^34 + z^31 * q^35 + (-z^20 + z^12 - z^4) * q^36 + (-z^41 + z^33 - z^25 - z^7) * q^40 + (z^29 - z^25) * q^43 + (-z^30 + z^22 - z^14) * q^44 + z^38 * q^45 + z^18 * q^49 + (z^25 - z^17) * q^50 + (-z^37 + z^21 - z^13 + z^11 + z^5 - z^3) * q^52 - z^6 * q^55 + (z^36 - z^28 + z^20 - z^12) * q^56 + (z^36 + z^4) * q^59 + (z^37 - z^29) * q^62 + z^25 * q^63 + (z^38 - z^30 + z^22 - z^14 - z^12 + z^6 + z^4) * q^64 + (-z^39 - z^29) * q^65 + (-z^35 - z^21 + z^13 + z^9 - z^5 - z) * q^68 + (z^12 - z^4) * q^70 + (z^28 + z^20) * q^71 + (-z^35 + z^27 - z^19 - z) * q^72 + (-z^31 + z^15) * q^73 + z^35 * q^77 + (-z^40 + z^30 - z^22 + z^14 - z^6) * q^80 + z^32 * q^81 + (-z^37 - z^23) * q^83 + (z^39 - z^27) * q^85 + (-z^40 + z^10 - z^6 - z^2) * q^86 + (z^37 - z^29 - z^11 + z^3) * q^88 + (z^18 + z^14) * q^89 + (z^19 - z^11) * q^90 + (-z^26 - z^16) * q^91 + (-z^41 + z^33) * q^98 - q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 24 * q + 4 * q^4 - 2 * q^5 + 2 * q^9	\( 24 q + 4 q^{4} - 2 q^{5} + 2 q^{9} - 2 q^{11} - 6 q^{14} - 26 q^{16} + 8 q^{20} + 2 q^{25} + 6 q^{26} - 12 q^{31} - 14 q^{34} - 8 q^{36} - 18 q^{44} - 2 q^{45} + 4 q^{49} - 4 q^{55} + 14 q^{56} - 2 q^{59} - 10 q^{64} - 6 q^{70} - 10 q^{71} + 12 q^{80} + 2 q^{81} - 6 q^{86} + 16 q^{89} - 24 q^{99}+O(q^{100}) \) 24 * q + 4 * q^4 - 2 * q^5 + 2 * q^9 - 2 * q^11 - 6 * q^14 - 26 * q^16 + 8 * q^20 + 2 * q^25 + 6 * q^26 - 12 * q^31 - 14 * q^34 - 8 * q^36 - 18 * q^44 - 2 * q^45 + 4 * q^49 - 4 * q^55 + 14 * q^56 - 2 * q^59 - 10 * q^64 - 6 * q^70 - 10 * q^71 + 12 * q^80 + 2 * q^81 - 6 * q^86 + 16 * q^89 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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	2695.1.ck.c

Level	$2695$
Weight	$1$
Character orbit	2695.ck
Analytic conductor	$1.345$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{42}$
CM discriminant	-55
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.34498020905\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(2\) over \(\Q(\zeta_{42})\)
Coefficient field:	\(\Q(\zeta_{84})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \) x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{42}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

\(n\)	\(981\)	\(1816\)	\(2157\)
\(\chi(n)\)	\(-1\)	\(\zeta_{84}^{16}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 109.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 + 28T^20 - 43T^18 + 171T^16 - 28T^14 + 148T^12 - 875T^10 + 1440T^8 - 519T^6 + 16T^2 + 1
$3$	\( T^{24} \) T^24
$5$	\( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \) (T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1)^2
$7$	\( (T^{12} - T^{10} + T^{8} + \cdots + 1)^{2} \) (T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$11$	\( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \) (T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1)^2
$13$	\( T^{24} - T^{22} + \cdots + 1 \) T^24 - T^22 + 13T^20 + 10T^18 + 132T^16 + 408T^14 + 301T^12 - 108T^10 + 927T^8 + 668T^6 + 159T^4 + 10T^2 + 1
$17$	\( T^{24} - 14 T^{20} + \cdots + 2401 \) T^24 - 14T^20 - 28T^18 + 147T^16 + 98T^14 - 392T^12 + 3430T^10 + 3087T^8 - 6174T^6 + 7203T^4 - 4802T^2 + 2401
$19$	\( T^{24} \) T^24
$23$	\( T^{24} \) T^24
$29$	\( T^{24} \) T^24
$31$	\( (T^{2} + T + 1)^{12} \) (T^2 + T + 1)^12
$37$	\( T^{24} \) T^24
$41$	\( T^{24} \) T^24
$43$	\( T^{24} + 6 T^{22} + \cdots + 1 \) T^24 + 6T^22 + 13T^20 + 38T^18 + 258T^16 + 716T^14 + 1519T^12 + 1978T^10 + 1284T^8 + 234T^6 + 96T^4 - 11*T^2 + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} \) T^24
$59$	\( (T^{12} + T^{11} - T^{9} + \cdots + 1)^{2} \) (T^12 + T^11 - T^9 + 6T^8 - 21T^7 - 20T^6 + 69T^4 - 29T^3 + 49T^2 - 13*T + 1)^2
$61$	\( T^{24} \) T^24
$67$	\( T^{24} \) T^24
$71$	\( (T^{12} + 5 T^{11} + \cdots + 1)^{2} \) (T^12 + 5T^11 + 17T^10 + 38T^9 + 68T^8 + 92T^7 + 105T^6 + 92T^5 + 61T^4 + 10T^3 + 45T^2 + 12*T + 1)^2
$73$	\( T^{24} - 3 T^{22} + \cdots + 1 \) T^24 - 3T^22 - 8T^18 + 220T^16 - 679T^14 + 1233T^12 - 1015T^10 - 163T^8 + 461T^6 + 119T^4 + 2T^2 + 1
$79$	\( T^{24} \) T^24
$83$	\( (T^{12} + 3 T^{10} + \cdots + 729)^{2} \) (T^12 + 3T^10 + 9T^8 + 27T^6 + 81T^4 + 243*T^2 + 729)^2
$89$	\( (T^{12} - 8 T^{11} + \cdots + 1)^{2} \) (T^12 - 8T^11 + 35T^10 - 104T^9 + 230T^8 - 392T^7 + 519T^6 - 518T^5 + 349T^4 - 118T^3 + 6T + 1)^2
$97$	\( T^{24} \) T^24
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