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Properties

Label 3240.1.bv.c
Level $3240$
Weight $1$
Character orbit 3240.bv
Analytic conductor $1.617$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -24
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,1,Mod(757,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.757");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3240.bv (of order \(12\), degree \(4\), not 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: \(1.61697064093\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.9000.2

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} - \zeta_{24}^{5} q^{5} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{7} - \zeta_{24}^{9} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} - \zeta_{24}^{5} q^{5} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{7} - \zeta_{24}^{9} q^{8} + q^{10} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{11} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{14} + \zeta_{24}^{4} q^{16} + \zeta_{24}^{7} q^{20} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{22} + \zeta_{24}^{10} q^{25} + ( - \zeta_{24}^{6} - 1) q^{28} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{29} + \zeta_{24}^{11} q^{32} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{35} - \zeta_{24}^{2} q^{40} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{44} + \zeta_{24}^{2} q^{49} - \zeta_{24}^{5} q^{50} + ( - \zeta_{24}^{6} - 1) q^{55} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{56} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{58} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{59} - \zeta_{24}^{6} q^{64} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{70} + (\zeta_{24}^{6} + 1) q^{73} + ( - 2 \zeta_{24}^{11} - \zeta_{24}^{5}) q^{77} - \zeta_{24}^{9} q^{80} - \zeta_{24} q^{83} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{88} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{97} + \zeta_{24}^{9} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 8 q^{10} + 4 q^{16} - 4 q^{22} - 8 q^{28} - 8 q^{55} + 4 q^{58} + 4 q^{70} + 8 q^{73} - 4 q^{88} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-1\) \(1\) \(\zeta_{24}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i 0 0.866025 + 0.500000i −0.965926 + 0.258819i 0 −0.366025 + 1.36603i −0.707107 − 0.707107i 0 1.00000
757.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.965926 − 0.258819i 0 −0.366025 + 1.36603i 0.707107 + 0.707107i 0 1.00000
1837.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.258819 + 0.965926i 0 1.36603 − 0.366025i 0.707107 + 0.707107i 0 1.00000
1837.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.258819 − 0.965926i 0 1.36603 − 0.366025i −0.707107 − 0.707107i 0 1.00000
2053.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.258819 − 0.965926i 0 1.36603 + 0.366025i 0.707107 − 0.707107i 0 1.00000
2053.2 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.258819 + 0.965926i 0 1.36603 + 0.366025i −0.707107 + 0.707107i 0 1.00000
3133.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −0.965926 − 0.258819i 0 −0.366025 − 1.36603i −0.707107 + 0.707107i 0 1.00000
3133.2 0.965926 − 0.258819i 0 0.866025 − 0.500000i 0.965926 + 0.258819i 0 −0.366025 − 1.36603i 0.707107 − 0.707107i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
40.i odd 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner
72.j odd 6 1 inner
72.n even 6 1 inner
120.w even 4 1 inner
360.br even 12 1 inner
360.bu odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.1.bv.c 8
3.b odd 2 1 inner 3240.1.bv.c 8
5.c odd 4 1 inner 3240.1.bv.c 8
8.b even 2 1 inner 3240.1.bv.c 8
9.c even 3 1 360.1.u.a ✓ 4
9.c even 3 1 inner 3240.1.bv.c 8
9.d odd 6 1 360.1.u.a ✓ 4
9.d odd 6 1 inner 3240.1.bv.c 8
15.e even 4 1 inner 3240.1.bv.c 8
24.h odd 2 1 CM 3240.1.bv.c 8
36.f odd 6 1 1440.1.y.a 4
36.h even 6 1 1440.1.y.a 4
40.i odd 4 1 inner 3240.1.bv.c 8
45.h odd 6 1 1800.1.u.b 4
45.j even 6 1 1800.1.u.b 4
45.k odd 12 1 360.1.u.a ✓ 4
45.k odd 12 1 1800.1.u.b 4
45.k odd 12 1 inner 3240.1.bv.c 8
45.l even 12 1 360.1.u.a ✓ 4
45.l even 12 1 1800.1.u.b 4
45.l even 12 1 inner 3240.1.bv.c 8
72.j odd 6 1 360.1.u.a ✓ 4
72.j odd 6 1 inner 3240.1.bv.c 8
72.l even 6 1 1440.1.y.a 4
72.n even 6 1 360.1.u.a ✓ 4
72.n even 6 1 inner 3240.1.bv.c 8
72.p odd 6 1 1440.1.y.a 4
120.w even 4 1 inner 3240.1.bv.c 8
180.v odd 12 1 1440.1.y.a 4
180.x even 12 1 1440.1.y.a 4
360.bh odd 6 1 1800.1.u.b 4
360.bk even 6 1 1800.1.u.b 4
360.bo even 12 1 1440.1.y.a 4
360.br even 12 1 360.1.u.a ✓ 4
360.br even 12 1 1800.1.u.b 4
360.br even 12 1 inner 3240.1.bv.c 8
360.bt odd 12 1 1440.1.y.a 4
360.bu odd 12 1 360.1.u.a ✓ 4
360.bu odd 12 1 1800.1.u.b 4
360.bu odd 12 1 inner 3240.1.bv.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.u.a ✓ 4 9.c even 3 1
360.1.u.a ✓ 4 9.d odd 6 1
360.1.u.a ✓ 4 45.k odd 12 1
360.1.u.a ✓ 4 45.l even 12 1
360.1.u.a ✓ 4 72.j odd 6 1
360.1.u.a ✓ 4 72.n even 6 1
360.1.u.a ✓ 4 360.br even 12 1
360.1.u.a ✓ 4 360.bu odd 12 1
1440.1.y.a 4 36.f odd 6 1
1440.1.y.a 4 36.h even 6 1
1440.1.y.a 4 72.l even 6 1
1440.1.y.a 4 72.p odd 6 1
1440.1.y.a 4 180.v odd 12 1
1440.1.y.a 4 180.x even 12 1
1440.1.y.a 4 360.bo even 12 1
1440.1.y.a 4 360.bt odd 12 1
1800.1.u.b 4 45.h odd 6 1
1800.1.u.b 4 45.j even 6 1
1800.1.u.b 4 45.k odd 12 1
1800.1.u.b 4 45.l even 12 1
1800.1.u.b 4 360.bh odd 6 1
1800.1.u.b 4 360.bk even 6 1
1800.1.u.b 4 360.br even 12 1
1800.1.u.b 4 360.bu odd 12 1
3240.1.bv.c 8 1.a even 1 1 trivial
3240.1.bv.c 8 3.b odd 2 1 inner
3240.1.bv.c 8 5.c odd 4 1 inner
3240.1.bv.c 8 8.b even 2 1 inner
3240.1.bv.c 8 9.c even 3 1 inner
3240.1.bv.c 8 9.d odd 6 1 inner
3240.1.bv.c 8 15.e even 4 1 inner
3240.1.bv.c 8 24.h odd 2 1 CM
3240.1.bv.c 8 40.i odd 4 1 inner
3240.1.bv.c 8 45.k odd 12 1 inner
3240.1.bv.c 8 45.l even 12 1 inner
3240.1.bv.c 8 72.j odd 6 1 inner
3240.1.bv.c 8 72.n even 6 1 inner
3240.1.bv.c 8 120.w even 4 1 inner
3240.1.bv.c 8 360.br even 12 1 inner
3240.1.bv.c 8 360.bu odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} - 4T_{7} + 4 \) acting on \(S_{1}^{\mathrm{new}}(3240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
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