LMFDB - Newform orbit 1725.1.l.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1725,1,Mod(68,1725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 2])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1725.68"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1725.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.860887146792$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{72})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{12} + 1 $ x^24 - x^12 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

$q$-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_{72}^{13} + \zeta_{72}^{5}) q^{2} - \zeta_{72}^{19} q^{3} + (\zeta_{72}^{26} + \cdots + \zeta_{72}^{10}) q^{4} + ( - \zeta_{72}^{32} - \zeta_{72}^{24}) q^{6} + (\zeta_{72}^{31} + \cdots - \zeta_{72}^{3}) q^{8} + \cdots + ( - \zeta_{72}^{31} - \zeta_{72}^{23}) q^{98} +O(q^{100}) $ q + (z^13 + z^5) * q^2 - z^19 * q^3 + (z^26 + z^18 + z^10) * q^4 + (-z^32 - z^24) * q^6 + (z^31 + z^23 + z^15 - z^3) * q^8 - z^2 * q^9 + (-z^29 + z^9 + z) * q^12 + (z^11 + z^7) * q^13 + (z^28 + z^20 - z^16 - z^8 - 1) * q^16 + (-z^15 - z^7) * q^18 - z^27 * q^23 + (-z^34 + z^22 + z^14 + z^6) * q^24 + (z^24 + z^20 + z^16 + z^12) * q^26 + z^21 * q^27 + (-z^22 + z^14) * q^29 + (z^32 - z^4) * q^31 + (z^33 - z^29 + z^25 - z^21 - z^13 - z^5) * q^32 + (-z^28 - z^20 - z^12) * q^36 + (-z^30 - z^26) * q^39 + (-z^28 - z^8) * q^41 + (-z^32 + z^4) * q^46 + (z^29 - z^25) * q^47 + (z^35 + z^27 + z^19 + z^11 + z^3) * q^48 - z^18 * q^49 + (z^33 + z^29 + z^25 + z^21 + z^17 - z) * q^52 + (z^34 + z^26) * q^54 + (-z^35 + z^19) * q^58 + (z^30 - z^6) * q^59 + (-z^17 - 2z^9 - z) q^62 + (-z^34 + z^30 - z^26 - z^18 - z^10 + z^6 - z^2) * q^64 - z^10 * q^69 + (-z^20 - z^16) * q^71 + (-z^33 - z^25 - z^17 + z^5) * q^72 + (-z^31 + z^23) * q^73 + (-z^35 - z^31 + z^7 + z^3) * q^78 + z^4 * q^81 + (-z^33 - z^21 - z^13 + z^5) * q^82 + (-z^33 - z^5) * q^87 + (z^17 + z^9 + z) * q^92 + (z^23 + z^15) * q^93 + (z^34 - z^30 - z^6 + z^2) * q^94 + (z^32 + z^24 + z^16 - z^12 + z^8 - z^4) * q^96 + (-z^31 - z^23) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 12 q^{6} - 24 q^{16} - 12 q^{36} - 24 q^{96}+O(q^{100}) $ 24 * q + 12 * q^6 - 24 * q^16 - 12 * q^36 - 24 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-\zeta_{72}^{18}$	$-1$	$-1$

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} $

68.1

−0.0871557	−	0.996195i
−0.996195	−	0.0871557i
0.819152	−	0.573576i
−0.573576	+	0.819152i
−0.422618	−	0.906308i
−0.906308	−	0.422618i
0.906308	+	0.422618i
0.422618	+	0.906308i
0.573576	−	0.819152i
−0.819152	+	0.573576i
0.996195	+	0.0871557i
0.0871557	+	0.996195i
−0.0871557	+	0.996195i
−0.996195	+	0.0871557i
0.819152	+	0.573576i
−0.573576	−	0.819152i
−0.422618	+	0.906308i
−0.906308	+	0.422618i
0.906308	−	0.422618i
0.422618	−	0.906308i

−1.32893

−

1.32893i

−0.996195

0.0871557i

2.53209i

1.43969

1.20805i

2.03603

−

2.03603i

0.984808

−

0.173648i

68.2

−1.32893

−

1.32893i

−0.0871557

0.996195i

2.53209i

1.43969

−

1.20805i

2.03603

−

2.03603i

−0.984808

−

0.173648i

68.3

−1.08335

−

1.08335i

−0.573576

−

0.819152i

1.34730i

−0.266044

1.50881i

0.376244

−

0.376244i

−0.342020

0.939693i

68.4

−1.08335

−

1.08335i

0.819152

0.573576i

1.34730i

−0.266044

−

1.50881i

0.376244

−

0.376244i

0.342020

0.939693i

68.5

−0.245576

−

0.245576i

−0.906308

0.422618i

−

0.879385i

0.326352

0.118782i

−0.461531

0.461531i

0.642788

−

0.766044i

68.6

−0.245576

−

0.245576i

−0.422618

0.906308i

−

0.879385i

0.326352

−

0.118782i

−0.461531

0.461531i

−0.642788

−

0.766044i

68.7

0.245576

0.245576i

0.422618

−

0.906308i

−

0.879385i

0.326352

−

0.118782i

0.461531

−

0.461531i

−0.642788

−

0.766044i

68.8

0.245576

0.245576i

0.906308

−

0.422618i

−

0.879385i

0.326352

0.118782i

0.461531

−

0.461531i

0.642788

−

0.766044i

68.9

1.08335

1.08335i

−0.819152

−

0.573576i

1.34730i

−0.266044

−

1.50881i

−0.376244

0.376244i

0.342020

0.939693i

68.10

1.08335

1.08335i

0.573576

0.819152i

1.34730i

−0.266044

1.50881i

−0.376244

0.376244i

−0.342020

0.939693i

68.11

1.32893

1.32893i

0.0871557

−

0.996195i

2.53209i

1.43969

−

1.20805i

−2.03603

2.03603i

−0.984808

−

0.173648i

68.12

1.32893

1.32893i

0.996195

−

0.0871557i

2.53209i

1.43969

1.20805i

−2.03603

2.03603i

0.984808

−

0.173648i

482.1

−1.32893

1.32893i

−0.996195

−

0.0871557i

−

2.53209i

1.43969

−

1.20805i

2.03603

2.03603i

0.984808

0.173648i

482.2

−1.32893

1.32893i

−0.0871557

−

0.996195i

−

2.53209i

1.43969

1.20805i

2.03603

2.03603i

−0.984808

0.173648i

482.3

−1.08335

1.08335i

−0.573576

0.819152i

−

1.34730i

−0.266044

−

1.50881i

0.376244

0.376244i

−0.342020

−

0.939693i

482.4

−1.08335

1.08335i

0.819152

−

0.573576i

−

1.34730i

−0.266044

1.50881i

0.376244

0.376244i

0.342020

−

0.939693i

482.5

−0.245576

0.245576i

−0.906308

−

0.422618i

0.879385i

0.326352

−

0.118782i

−0.461531

−

0.461531i

0.642788

0.766044i

482.6

−0.245576

0.245576i

−0.422618

−

0.906308i

0.879385i

0.326352

0.118782i

−0.461531

−

0.461531i

−0.642788

0.766044i

482.7

0.245576

−

0.245576i

0.422618

0.906308i

0.879385i

0.326352

0.118782i

0.461531

0.461531i

−0.642788

0.766044i

482.8

0.245576

−

0.245576i

0.906308

0.422618i

0.879385i

0.326352

−

0.118782i

0.461531

0.461531i

0.642788

0.766044i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner
69.c	even	2	1	inner
115.c	odd	2	1	inner
115.e	even	4	2	inner
345.h	even	2	1	inner
345.l	odd	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1725.1.l.c	✓	24
3.b	odd	2	1	inner	1725.1.l.c	✓	24
5.b	even	2	1	inner	1725.1.l.c	✓	24
5.c	odd	4	2	inner	1725.1.l.c	✓	24
15.d	odd	2	1	inner	1725.1.l.c	✓	24
15.e	even	4	2	inner	1725.1.l.c	✓	24
23.b	odd	2	1	CM	1725.1.l.c	✓	24
69.c	even	2	1	inner	1725.1.l.c	✓	24
115.c	odd	2	1	inner	1725.1.l.c	✓	24
115.e	even	4	2	inner	1725.1.l.c	✓	24
345.h	even	2	1	inner	1725.1.l.c	✓	24
345.l	odd	4	2	inner	1725.1.l.c	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1725.1.l.c	✓	24	1.a	even	1	1	trivial
1725.1.l.c	✓	24	3.b	odd	2	1	inner
1725.1.l.c	✓	24	5.b	even	2	1	inner
1725.1.l.c	✓	24	5.c	odd	4	2	inner
1725.1.l.c	✓	24	15.d	odd	2	1	inner
1725.1.l.c	✓	24	15.e	even	4	2	inner
1725.1.l.c	✓	24	23.b	odd	2	1	CM
1725.1.l.c	✓	24	69.c	even	2	1	inner
1725.1.l.c	✓	24	115.c	odd	2	1	inner
1725.1.l.c	✓	24	115.e	even	4	2	inner
1725.1.l.c	✓	24	345.h	even	2	1	inner
1725.1.l.c	✓	24	345.l	odd	4	2	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{12} + 18T_{2}^{8} + 69T_{2}^{4} + 1 $ T2^12 + 18*T2^8 + 69*T2^4 + 1 acting on $S_{1}^{\mathrm{new}}(1725, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{12} + 18 T^{8} + \cdots + 1)^{2} $ (T^12 + 18T^8 + 69T^4 + 1)^2
$3$	$ T^{24} - T^{12} + 1 $ T^24 - T^12 + 1
$5$	$ T^{24} $ T^24
$7$	$ T^{24} $ T^24
$11$	$ T^{24} $ T^24
$13$	$ (T^{12} + 18 T^{8} + \cdots + 9)^{2} $ (T^12 + 18T^8 + 45T^4 + 9)^2
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ (T^{4} + 1)^{6} $ (T^4 + 1)^6
$29$	$ (T^{6} - 6 T^{4} + 9 T^{2} - 3)^{4} $ (T^6 - 6T^4 + 9T^2 - 3)^4
$31$	$ (T^{3} - 3 T + 1)^{8} $ (T^3 - 3*T + 1)^8
$37$	$ T^{24} $ T^24
$41$	$ (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{4} $ (T^6 + 6T^4 + 9T^2 + 3)^4
$43$	$ T^{24} $ T^24
$47$	$ (T^{12} + 18 T^{8} + \cdots + 1)^{2} $ (T^12 + 18T^8 + 69T^4 + 1)^2
$53$	$ T^{24} $ T^24
$59$	$ (T^{2} - 3)^{12} $ (T^2 - 3)^12
$61$	$ T^{24} $ T^24
$67$	$ T^{24} $ T^24
$71$	$ (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{4} $ (T^6 + 6T^4 + 9T^2 + 3)^4
$73$	$ (T^{12} + 18 T^{8} + \cdots + 9)^{2} $ (T^12 + 18T^8 + 45T^4 + 9)^2
$79$	$ T^{24} $ T^24
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ 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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1725.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{72}^{13} + \zeta_{72}^{5}) q^{2} - \zeta_{72}^{19} q^{3} + (\zeta_{72}^{26} + \cdots + \zeta_{72}^{10}) q^{4} + ( - \zeta_{72}^{32} - \zeta_{72}^{24}) q^{6} + (\zeta_{72}^{31} + \cdots - \zeta_{72}^{3}) q^{8} + \cdots + ( - \zeta_{72}^{31} - \zeta_{72}^{23}) q^{98} +O(q^{100}) \) q + (z^13 + z^5) * q^2 - z^19 * q^3 + (z^26 + z^18 + z^10) * q^4 + (-z^32 - z^24) * q^6 + (z^31 + z^23 + z^15 - z^3) * q^8 - z^2 * q^9 + (-z^29 + z^9 + z) * q^12 + (z^11 + z^7) * q^13 + (z^28 + z^20 - z^16 - z^8 - 1) * q^16 + (-z^15 - z^7) * q^18 - z^27 * q^23 + (-z^34 + z^22 + z^14 + z^6) * q^24 + (z^24 + z^20 + z^16 + z^12) * q^26 + z^21 * q^27 + (-z^22 + z^14) * q^29 + (z^32 - z^4) * q^31 + (z^33 - z^29 + z^25 - z^21 - z^13 - z^5) * q^32 + (-z^28 - z^20 - z^12) * q^36 + (-z^30 - z^26) * q^39 + (-z^28 - z^8) * q^41 + (-z^32 + z^4) * q^46 + (z^29 - z^25) * q^47 + (z^35 + z^27 + z^19 + z^11 + z^3) * q^48 - z^18 * q^49 + (z^33 + z^29 + z^25 + z^21 + z^17 - z) * q^52 + (z^34 + z^26) * q^54 + (-z^35 + z^19) * q^58 + (z^30 - z^6) * q^59 + (-z^17 - 2z^9 - z) q^62 + (-z^34 + z^30 - z^26 - z^18 - z^10 + z^6 - z^2) * q^64 - z^10 * q^69 + (-z^20 - z^16) * q^71 + (-z^33 - z^25 - z^17 + z^5) * q^72 + (-z^31 + z^23) * q^73 + (-z^35 - z^31 + z^7 + z^3) * q^78 + z^4 * q^81 + (-z^33 - z^21 - z^13 + z^5) * q^82 + (-z^33 - z^5) * q^87 + (z^17 + z^9 + z) * q^92 + (z^23 + z^15) * q^93 + (z^34 - z^30 - z^6 + z^2) * q^94 + (z^32 + z^24 + z^16 - z^12 + z^8 - z^4) * q^96 + (-z^31 - z^23) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 12 q^{6} - 24 q^{16} - 12 q^{36} - 24 q^{96}+O(q^{100}) \) 24 * q + 12 * q^6 - 24 * q^16 - 12 * q^36 - 24 * q^96

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	1725.1.l.c

Level	$1725$
Weight	$1$
Character orbit	1725.l
Analytic conductor	$0.861$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{18}$
CM discriminant	-23
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(0.860887146792\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{72})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{12} + 1 \) x^24 - x^12 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{18}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

\(n\)	\(277\)	\(1151\)	\(1201\)
\(\chi(n)\)	\(-\zeta_{72}^{18}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 68.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{12} + 18 T^{8} + \cdots + 1)^{2} \) (T^12 + 18T^8 + 69T^4 + 1)^2
$3$	\( T^{24} - T^{12} + 1 \) T^24 - T^12 + 1
$5$	\( T^{24} \) T^24
$7$	\( T^{24} \) T^24
$11$	\( T^{24} \) T^24
$13$	\( (T^{12} + 18 T^{8} + \cdots + 9)^{2} \) (T^12 + 18T^8 + 45T^4 + 9)^2
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( (T^{4} + 1)^{6} \) (T^4 + 1)^6
$29$	\( (T^{6} - 6 T^{4} + 9 T^{2} - 3)^{4} \) (T^6 - 6T^4 + 9T^2 - 3)^4
$31$	\( (T^{3} - 3 T + 1)^{8} \) (T^3 - 3*T + 1)^8
$37$	\( T^{24} \) T^24
$41$	\( (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{4} \) (T^6 + 6T^4 + 9T^2 + 3)^4
$43$	\( T^{24} \) T^24
$47$	\( (T^{12} + 18 T^{8} + \cdots + 1)^{2} \) (T^12 + 18T^8 + 69T^4 + 1)^2
$53$	\( T^{24} \) T^24
$59$	\( (T^{2} - 3)^{12} \) (T^2 - 3)^12
$61$	\( T^{24} \) T^24
$67$	\( T^{24} \) T^24
$71$	\( (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{4} \) (T^6 + 6T^4 + 9T^2 + 3)^4
$73$	\( (T^{12} + 18 T^{8} + \cdots + 9)^{2} \) (T^12 + 18T^8 + 45T^4 + 9)^2
$79$	\( T^{24} \) T^24
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} \) T^24
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