Properties

Label 740.1.bm.a
Level $740$
Weight $1$
Character orbit 740.bm
Analytic conductor $0.369$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,1,Mod(23,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.23");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 740.bm (of order \(12\), degree \(4\), 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: \(0.369308109348\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{5} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{5} q^{2} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{5} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{5} q^{9} - \zeta_{12}^{2} q^{10} - \zeta_{12}^{2} q^{16} + \zeta_{12}^{2} q^{17} + \zeta_{12}^{4} q^{18} - \zeta_{12} q^{20} - q^{25} + (\zeta_{12}^{2} + \zeta_{12}) q^{29} - \zeta_{12} q^{32} + \zeta_{12} q^{34} + \zeta_{12}^{3} q^{36} - \zeta_{12} q^{37} - q^{40} + (\zeta_{12}^{2} + 1) q^{41} + \zeta_{12}^{2} q^{45} + \zeta_{12}^{5} q^{49} + \zeta_{12}^{5} q^{50} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{53} + (\zeta_{12} + 1) q^{58} + ( - \zeta_{12}^{5} - 1) q^{61} - q^{64} + q^{68} + \zeta_{12}^{2} q^{72} + (\zeta_{12}^{3} - 1) q^{73} - q^{74} + \zeta_{12}^{5} q^{80} - \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{82} - \zeta_{12}^{5} q^{85} + (\zeta_{12}^{4} + \zeta_{12}^{3}) q^{89} + \zeta_{12} q^{90} + (\zeta_{12}^{5} - \zeta_{12}) q^{97} + \zeta_{12}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{10} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{25} + 2 q^{29} - 4 q^{40} + 6 q^{41} + 2 q^{45} - 2 q^{53} + 4 q^{58} - 4 q^{61} - 4 q^{64} + 4 q^{68} + 2 q^{72} - 4 q^{73} - 4 q^{74} + 2 q^{81} - 2 q^{89} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(\zeta_{12}\) \(-\zeta_{12}^{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0 1.00000i 0.866025 + 0.500000i −0.500000 + 0.866025i
347.1 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0 1.00000i −0.866025 0.500000i −0.500000 + 0.866025i
547.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.00000i 0 0 1.00000i 0.866025 0.500000i −0.500000 0.866025i
563.1 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 0 1.00000i −0.866025 + 0.500000i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.u even 12 1 inner
740.bm odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.bm.a yes 4
4.b odd 2 1 CM 740.1.bm.a yes 4
5.b even 2 1 3700.1.bu.a 4
5.c odd 4 1 740.1.bj.a 4
5.c odd 4 1 3700.1.br.a 4
20.d odd 2 1 3700.1.bu.a 4
20.e even 4 1 740.1.bj.a 4
20.e even 4 1 3700.1.br.a 4
37.g odd 12 1 740.1.bj.a 4
148.l even 12 1 740.1.bj.a 4
185.p even 12 1 3700.1.bu.a 4
185.q odd 12 1 3700.1.br.a 4
185.u even 12 1 inner 740.1.bm.a yes 4
740.bj odd 12 1 3700.1.bu.a 4
740.bm odd 12 1 inner 740.1.bm.a yes 4
740.bo even 12 1 3700.1.br.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.bj.a 4 5.c odd 4 1
740.1.bj.a 4 20.e even 4 1
740.1.bj.a 4 37.g odd 12 1
740.1.bj.a 4 148.l even 12 1
740.1.bm.a yes 4 1.a even 1 1 trivial
740.1.bm.a yes 4 4.b odd 2 1 CM
740.1.bm.a yes 4 185.u even 12 1 inner
740.1.bm.a yes 4 740.bm odd 12 1 inner
3700.1.br.a 4 5.c odd 4 1
3700.1.br.a 4 20.e even 4 1
3700.1.br.a 4 185.q odd 12 1
3700.1.br.a 4 740.bo even 12 1
3700.1.bu.a 4 5.b even 2 1
3700.1.bu.a 4 20.d odd 2 1
3700.1.bu.a 4 185.p even 12 1
3700.1.bu.a 4 740.bj odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(740, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
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