LMFDB - Newform orbit 1856.1.ca.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1856,1,Mod(193,1856)] 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("1856.193"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1856, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([0, 0, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1856.ca (of order $28$, degree $12$, not 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:	$0.926264663447$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{28})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 928)
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

$q$-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_{28}^{12} + \zeta_{28}^{4}) q^{5} - \zeta_{28}^{3} q^{9} + ( - \zeta_{28}^{13} - \zeta_{28}^{9}) q^{13} + ( - \zeta_{28}^{6} - \zeta_{28}) q^{17} + ( - \zeta_{28}^{10} + \cdots + \zeta_{28}^{2}) q^{25} + \cdots + (\zeta_{28}^{11} - 1) q^{97} +O(q^{100}) $ q + (-z^12 + z^4) * q^5 - z^3 * q^9 + (-z^13 - z^9) * q^13 + (-z^6 - z) * q^17 + (-z^10 + z^8 + z^2) * q^25 - z^11 * q^29 + (z^8 + z^5) * q^37 + (z^5 - z^2) * q^41 + (-z^7 - z) * q^45 + z^10 * q^49 + (z^9 + z^7) * q^53 + (z^11 - z^6) * q^61 + (-z^13 - z^11 - z^7 + z^3) * q^65 + (-z^12 + z^7) * q^73 + z^6 * q^81 + (z^13 - z^10 - z^5 - z^4) * q^85 + (-z^5 + z^4) * q^89 + (z^11 - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{17} - 2 q^{25} - 2 q^{37} - 2 q^{41} + 2 q^{49} - 2 q^{61} + 2 q^{73} + 2 q^{81} - 2 q^{89} - 12 q^{97}+O(q^{100}) $ 12 * q - 2 * q^17 - 2 * q^25 - 2 * q^37 - 2 * q^41 + 2 * q^49 - 2 * q^61 + 2 * q^73 + 2 * q^81 - 2 * q^89 - 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$639$
$\chi(n)$	$-\zeta_{28}^{9}$	$1$	$1$

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} $

193.1

−0.974928	−	0.222521i
−0.781831	+	0.623490i
0.433884	+	0.900969i
0.781831	+	0.623490i
−0.974928	+	0.222521i
0.974928	−	0.222521i
−0.781831	−	0.623490i
−0.433884	−	0.900969i
0.781831	−	0.623490i
0.974928	+	0.222521i
0.433884	−	0.900969i
−0.433884	+	0.900969i

1.52446

0.347948i

0.781831

0.623490i

321.1

−0.678448

0.541044i

−0.433884

−

0.900969i

385.1

−0.846011

−

1.75676i

0.974928

0.222521i

449.1

−0.678448

−

0.541044i

0.433884

−

0.900969i

577.1

1.52446

−

0.347948i

0.781831

−

0.623490i

641.1

1.52446

−

0.347948i

−0.781831

0.623490i

769.1

−0.678448

−

0.541044i

−0.433884

0.900969i

833.1

−0.846011

−

1.75676i

−0.974928

−

0.222521i

897.1

−0.678448

0.541044i

0.433884

0.900969i

1025.1

1.52446

0.347948i

−0.781831

−

0.623490i

1345.1

−0.846011

1.75676i

0.974928

−

0.222521i

1729.1

−0.846011

1.75676i

−0.974928

0.222521i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
29.f	odd	28	1	inner
116.l	even	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1856.1.ca.a		12
4.b	odd	2	1	CM	1856.1.ca.a		12
8.b	even	2	1		928.1.bs.a	✓	12
8.d	odd	2	1		928.1.bs.a	✓	12
29.f	odd	28	1	inner	1856.1.ca.a		12
116.l	even	28	1	inner	1856.1.ca.a		12
232.u	odd	28	1		928.1.bs.a	✓	12
232.v	even	28	1		928.1.bs.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
928.1.bs.a	✓	12	8.b	even	2	1
928.1.bs.a	✓	12	8.d	odd	2	1
928.1.bs.a	✓	12	232.u	odd	28	1
928.1.bs.a	✓	12	232.v	even	28	1
1856.1.ca.a		12	1.a	even	1	1	trivial
1856.1.ca.a		12	4.b	odd	2	1	CM
1856.1.ca.a		12	29.f	odd	28	1	inner
1856.1.ca.a		12	116.l	even	28	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1856, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ (T^{6} - 7 T^{3} + 7 T + 7)^{2} $ (T^6 - 7T^3 + 7T + 7)^2
$7$	$ T^{12} $ T^12
$11$	$ T^{12} $ T^12
$13$	$ T^{12} - 4 T^{10} + \cdots + 1 $ T^12 - 4T^10 + 16T^8 - 29T^6 + 18T^4 + 5*T^2 + 1
$17$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 2T^10 + 17T^8 + 34T^7 + 34T^6 - 2T^5 + 26T^4 + 42T^3 + 32T^2 + 8T + 1
$19$	$ T^{12} $ T^12
$23$	$ T^{12} $ T^12
$29$	$ T^{12} - T^{10} + \cdots + 1 $ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$31$	$ T^{12} $ T^12
$37$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 2T^10 - 4T^8 - 22T^7 - T^6 + 40T^5 + 82T^4 + 84T^3 + 39T^2 + 8*T + 1
$41$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 2T^10 + 17T^8 + 34T^7 + 34T^6 - 2T^5 + 26T^4 + 42T^3 + 32T^2 + 8T + 1
$43$	$ T^{12} $ T^12
$47$	$ T^{12} $ T^12
$53$	$ T^{12} + 7 T^{10} + \cdots + 49 $ T^12 + 7T^10 + 21T^8 + 35T^6 + 49T^4 - 49*T^2 + 49
$59$	$ T^{12} $ T^12
$61$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 2T^10 + 14T^9 + 24T^8 + 20T^7 + 55T^6 + 68T^5 + 26T^4 + 14T^3 + 11T^2 - 6*T + 1
$67$	$ T^{12} $ T^12
$71$	$ T^{12} $ T^12
$73$	$ T^{12} - 2 T^{11} + \cdots + 1 $ T^12 - 2T^11 + 9T^10 - 14T^9 + 31T^8 - 34T^7 + 41T^6 - 12T^5 - 23T^4 + 28T^3 + 11T^2 - 8*T + 1
$79$	$ T^{12} $ T^12
$83$	$ T^{12} $ T^12
$89$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 + 2T^10 + 14T^9 + 24T^8 + 20T^7 + 55T^6 + 68T^5 + 26T^4 + 14T^3 + 11T^2 - 6*T + 1
$97$	$ T^{12} + 12 T^{11} + \cdots + 1 $ T^12 + 12T^11 + 65T^10 + 210T^9 + 451T^8 + 680T^7 + 741T^6 + 590T^5 + 341T^4 + 140T^3 + 39T^2 + 6*T + 1
show more
show less

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1856.ca (of order \(28\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{28}^{12} + \zeta_{28}^{4}) q^{5} - \zeta_{28}^{3} q^{9} + ( - \zeta_{28}^{13} - \zeta_{28}^{9}) q^{13} + ( - \zeta_{28}^{6} - \zeta_{28}) q^{17} + ( - \zeta_{28}^{10} + \cdots + \zeta_{28}^{2}) q^{25} + \cdots + (\zeta_{28}^{11} - 1) q^{97} +O(q^{100}) \) q + (-z^12 + z^4) * q^5 - z^3 * q^9 + (-z^13 - z^9) * q^13 + (-z^6 - z) * q^17 + (-z^10 + z^8 + z^2) * q^25 - z^11 * q^29 + (z^8 + z^5) * q^37 + (z^5 - z^2) * q^41 + (-z^7 - z) * q^45 + z^10 * q^49 + (z^9 + z^7) * q^53 + (z^11 - z^6) * q^61 + (-z^13 - z^11 - z^7 + z^3) * q^65 + (-z^12 + z^7) * q^73 + z^6 * q^81 + (z^13 - z^10 - z^5 - z^4) * q^85 + (-z^5 + z^4) * q^89 + (z^11 - 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{17} - 2 q^{25} - 2 q^{37} - 2 q^{41} + 2 q^{49} - 2 q^{61} + 2 q^{73} + 2 q^{81} - 2 q^{89} - 12 q^{97}+O(q^{100}) \) 12 * q - 2 * q^17 - 2 * q^25 - 2 * q^37 - 2 * q^41 + 2 * q^49 - 2 * q^61 + 2 * q^73 + 2 * q^81 - 2 * q^89 - 12 * q^97

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	1856.1.ca.a

Level	$1856$
Weight	$1$
Character orbit	1856.ca
Analytic conductor	$0.926$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{28}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{28})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 928)
Projective image:	\(D_{28}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

\(n\)	\(321\)	\(581\)	\(639\)
\(\chi(n)\)	\(-\zeta_{28}^{9}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 193.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
29.f	odd	28	1	inner
116.l	even	28	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( (T^{6} - 7 T^{3} + 7 T + 7)^{2} \) (T^6 - 7T^3 + 7T + 7)^2
$7$	\( T^{12} \) T^12
$11$	\( T^{12} \) T^12
$13$	\( T^{12} - 4 T^{10} + \cdots + 1 \) T^12 - 4T^10 + 16T^8 - 29T^6 + 18T^4 + 5*T^2 + 1
$17$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 2T^10 + 17T^8 + 34T^7 + 34T^6 - 2T^5 + 26T^4 + 42T^3 + 32T^2 + 8T + 1
$19$	\( T^{12} \) T^12
$23$	\( T^{12} \) T^12
$29$	\( T^{12} - T^{10} + \cdots + 1 \) T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$31$	\( T^{12} \) T^12
$37$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 2T^10 - 4T^8 - 22T^7 - T^6 + 40T^5 + 82T^4 + 84T^3 + 39T^2 + 8*T + 1
$41$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 2T^10 + 17T^8 + 34T^7 + 34T^6 - 2T^5 + 26T^4 + 42T^3 + 32T^2 + 8T + 1
$43$	\( T^{12} \) T^12
$47$	\( T^{12} \) T^12
$53$	\( T^{12} + 7 T^{10} + \cdots + 49 \) T^12 + 7T^10 + 21T^8 + 35T^6 + 49T^4 - 49*T^2 + 49
$59$	\( T^{12} \) T^12
$61$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 2T^10 + 14T^9 + 24T^8 + 20T^7 + 55T^6 + 68T^5 + 26T^4 + 14T^3 + 11T^2 - 6*T + 1
$67$	\( T^{12} \) T^12
$71$	\( T^{12} \) T^12
$73$	\( T^{12} - 2 T^{11} + \cdots + 1 \) T^12 - 2T^11 + 9T^10 - 14T^9 + 31T^8 - 34T^7 + 41T^6 - 12T^5 - 23T^4 + 28T^3 + 11T^2 - 8*T + 1
$79$	\( T^{12} \) T^12
$83$	\( T^{12} \) T^12
$89$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 + 2T^10 + 14T^9 + 24T^8 + 20T^7 + 55T^6 + 68T^5 + 26T^4 + 14T^3 + 11T^2 - 6*T + 1
$97$	\( T^{12} + 12 T^{11} + \cdots + 1 \) T^12 + 12T^11 + 65T^10 + 210T^9 + 451T^8 + 680T^7 + 741T^6 + 590T^5 + 341T^4 + 140T^3 + 39T^2 + 6*T + 1
show more
show less