LMFDB - Newform orbit 2320.1.fg.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,1,Mod(47,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 0, 7, 11])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.47"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$2320 = 2^{4} \cdot 5 \cdot 29$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2320.fg (of order $28$ , degree $12$ , 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:	$1.15783082931$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + \zeta_{28}^{9} q^{5} + \zeta_{28}^{12} q^{9} + ( - \zeta_{28}^{10} + \zeta_{28}) q^{13} + (\zeta_{28}^{11} + \zeta_{28}^{3}) q^{17} - \zeta_{28}^{4} q^{25} + \zeta_{28}^{9} q^{29} + (\zeta_{28}^{13} + \zeta_{28}^{11}) q^{37} + \cdots + (\zeta_{28}^{2} - 1) q^{97} +O(q^{100})$ q + z^9 * q^5 + z^12 * q^9 + (-z^10 + z) * q^13 + (z^11 + z^3) * q^17 - z^4 * q^25 + z^9 * q^29 + (z^13 + z^11) * q^37 + (z^6 + z) * q^41 - z^7 * q^45 + z^5 * q^49 + (z^8 + z^7) * q^53 + (-z^3 - z^2) * q^61 + (z^10 + z^5) * q^65 + (z^6 + 1) * q^73 - z^10 * q^81 + (z^12 - z^6) * q^85 + (-z^13 - z^2) * q^89 + (z^2 - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 2 q^{9} - 2 q^{13} + 2 q^{25} + 2 q^{41} - 2 q^{53} - 2 q^{61} + 2 q^{65} + 14 q^{73} - 2 q^{81} - 4 q^{85} - 2 q^{89} - 10 q^{97}+O(q^{100})$ 12 * q - 2 * q^9 - 2 * q^13 + 2 * q^25 + 2 * q^41 - 2 * q^53 - 2 * q^61 + 2 * q^65 + 14 * q^73 - 2 * q^81 - 4 * q^85 - 2 * q^89 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$ .

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$\zeta_{28}$	$1$	$-\zeta_{28}^{7}$	$-1$

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}

47.1

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

−0.974928

−

0.222521i

−0.222521

−

0.974928i

287.1

0.781831

0.623490i

0.623490

0.781831i

543.1

−0.974928

0.222521i

−0.222521

0.974928i

607.1

0.433884

−

0.900969i

−0.900969

0.433884i

623.1

0.433884

0.900969i

−0.900969

−

0.433884i

1407.1

−0.433884

−

0.900969i

−0.900969

−

0.433884i

1423.1

−0.433884

0.900969i

−0.900969

0.433884i

1487.1

0.974928

−

0.222521i

−0.222521

0.974928i

1743.1

−0.781831

−

0.623490i

0.623490

0.781831i

1983.1

0.974928

0.222521i

−0.222521

−

0.974928i

2127.1

−0.781831

0.623490i

0.623490

−

0.781831i

2223.1

0.781831

−

0.623490i

0.623490

−

0.781831i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
145.t	even	28	1	inner
580.bm	odd	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2320.1.fg.a	yes	12
4.b	odd	2	1	CM	2320.1.fg.a	yes	12
5.c	odd	4	1		2320.1.dr.a	✓	12
20.e	even	4	1		2320.1.dr.a	✓	12
29.f	odd	28	1		2320.1.dr.a	✓	12
116.l	even	28	1		2320.1.dr.a	✓	12
145.t	even	28	1	inner	2320.1.fg.a	yes	12
580.bm	odd	28	1	inner	2320.1.fg.a	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2320.1.dr.a	✓	12	5.c	odd	4	1
2320.1.dr.a	✓	12	20.e	even	4	1
2320.1.dr.a	✓	12	29.f	odd	28	1
2320.1.dr.a	✓	12	116.l	even	28	1
2320.1.fg.a	yes	12	1.a	even	1	1	trivial
2320.1.fg.a	yes	12	4.b	odd	2	1	CM
2320.1.fg.a	yes	12	145.t	even	28	1	inner
2320.1.fg.a	yes	12	580.bm	odd	28	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2320, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12}$ T^12
$3$	$T^{12}$ T^12
$5$	$T^{12} - T^{10} + \cdots + 1$ T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$7$	$T^{12}$ T^12
$11$	$T^{12}$ T^12
$13$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 2T^10 - 4T^8 - 22T^7 - T^6 + 40T^5 + 82T^4 + 84T^3 + 39T^2 + 8*T + 1
$17$	$(T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2}$ (T^6 + 5T^4 + 6T^2 + 1)^2
$19$	$T^{12}$ 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
$37$	$T^{12} + 35 T^{6} + \cdots + 49$ T^12 + 35T^6 + 98T^4 + 49*T^2 + 49
$41$	$T^{12} - 2 T^{11} + \cdots + 1$ 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
$53$	$T^{12} + 2 T^{11} + \cdots + 1$ 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
$59$	$T^{12}$ T^12
$61$	$T^{12} + 2 T^{11} + \cdots + 1$ T^12 + 2T^11 + 2T^10 - 4T^8 - 22T^7 - T^6 + 40T^5 + 82T^4 + 84T^3 + 39T^2 + 8*T + 1
$67$	$T^{12}$ T^12
$71$	$T^{12}$ T^12
$73$	$(T^{6} - 7 T^{5} + 21 T^{4} + \cdots + 7)^{2}$ (T^6 - 7T^5 + 21T^4 - 35T^3 + 35T^2 - 21*T + 7)^2
$79$	$T^{12}$ 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
$97$	$(T^{6} + 5 T^{5} + 11 T^{4} + \cdots + 1)^{2}$ (T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1)^2
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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	2320.1.fg.a

Level	$2320$
Weight	$1$
Character orbit	2320.fg
Analytic conductor	$1.158$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{28}$
CM discriminant	-4
Inner twists	$4$