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Properties

Label 3640.1.ij.b
Level $3640$
Weight $1$
Character orbit 3640.ij
Analytic conductor $1.817$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3640,1,Mod(59,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 6, 2, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3640.59");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3640.ij (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: \(1.81659664598\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{9} q^{2} - \zeta_{24}^{6} q^{4} + \zeta_{24}^{7} q^{5} - \zeta_{24}^{9} q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{9} q^{2} - \zeta_{24}^{6} q^{4} + \zeta_{24}^{7} q^{5} - \zeta_{24}^{9} q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{8} q^{9} + \zeta_{24}^{4} q^{10} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{11} - \zeta_{24}^{9} q^{13} - \zeta_{24}^{6} q^{14} - q^{16} - \zeta_{24}^{5} q^{18} + (\zeta_{24}^{10} - 1) q^{19} + \zeta_{24} q^{20} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{22} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{23} - \zeta_{24}^{2} q^{25} - \zeta_{24}^{6} q^{26} - \zeta_{24}^{3} q^{28} + \zeta_{24}^{9} q^{32} + \zeta_{24}^{4} q^{35} - \zeta_{24}^{2} q^{36} - \zeta_{24}^{3} q^{37} + (\zeta_{24}^{9} + \zeta_{24}^{7}) q^{38} - \zeta_{24}^{10} q^{40} + ( - \zeta_{24}^{6} + \zeta_{24}^{4}) q^{41} + (\zeta_{24}^{10} - \zeta_{24}^{4}) q^{44} + \zeta_{24}^{3} q^{45} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{46} + ( - \zeta_{24}^{11} + \zeta_{24}^{3}) q^{47} - \zeta_{24}^{6} q^{49} + \zeta_{24}^{11} q^{50} - \zeta_{24}^{3} q^{52} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{53} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{55} - q^{56} + (\zeta_{24}^{6} - 1) q^{59} - \zeta_{24}^{5} q^{63} + \zeta_{24}^{6} q^{64} + \zeta_{24}^{4} q^{65} + \zeta_{24} q^{70} + \zeta_{24}^{11} q^{72} - q^{74} + (\zeta_{24}^{6} + \zeta_{24}^{4}) q^{76} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{77} - \zeta_{24}^{7} q^{80} - \zeta_{24}^{4} q^{81} + ( - \zeta_{24}^{3} + \zeta_{24}) q^{82} + (\zeta_{24}^{7} - \zeta_{24}) q^{88} + ( - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{89} + q^{90} - \zeta_{24}^{6} q^{91} + ( - \zeta_{24}^{11} + \zeta_{24}) q^{92} + ( - \zeta_{24}^{8} + 1) q^{94} + ( - \zeta_{24}^{7} - \zeta_{24}^{5}) q^{95} - \zeta_{24}^{3} q^{98} + ( - \zeta_{24}^{6} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} + 4 q^{10} - 4 q^{11} - 8 q^{16} - 8 q^{19} + 4 q^{35} + 4 q^{41} - 4 q^{44} + 4 q^{46} - 8 q^{56} - 8 q^{59} + 4 q^{65} - 8 q^{74} + 4 q^{76} - 4 q^{81} - 4 q^{89} + 8 q^{90} + 12 q^{94} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3640\mathbb{Z}\right)^\times\).

\(n\) \(521\) \(561\) \(911\) \(1457\) \(1821\)
\(\chi(n)\) \(\zeta_{24}^{4}\) \(\zeta_{24}^{10}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.707107 − 0.707107i 0 1.00000i −0.965926 − 0.258819i 0 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
59.2 0.707107 + 0.707107i 0 1.00000i 0.965926 + 0.258819i 0 0.707107 + 0.707107i −0.707107 + 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
1419.1 −0.707107 + 0.707107i 0 − 1.00000i −0.965926 + 0.258819i 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
1419.2 0.707107 − 0.707107i 0 − 1.00000i 0.965926 − 0.258819i 0 0.707107 − 0.707107i −0.707107 − 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
2819.1 −0.707107 − 0.707107i 0 1.00000i 0.258819 + 0.965926i 0 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
2819.2 0.707107 + 0.707107i 0 1.00000i −0.258819 − 0.965926i 0 0.707107 + 0.707107i −0.707107 + 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
3139.1 −0.707107 + 0.707107i 0 − 1.00000i 0.258819 − 0.965926i 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
3139.2 0.707107 − 0.707107i 0 − 1.00000i −0.258819 + 0.965926i 0 0.707107 − 0.707107i −0.707107 − 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
8.d odd 2 1 inner
91.ba even 12 1 inner
455.cl even 12 1 inner
728.el odd 12 1 inner
3640.ij odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3640.1.ij.b ✓ 8
5.b even 2 1 inner 3640.1.ij.b ✓ 8
7.d odd 6 1 3640.1.pj.b yes 8
8.d odd 2 1 inner 3640.1.ij.b ✓ 8
13.f odd 12 1 3640.1.pj.b yes 8
35.i odd 6 1 3640.1.pj.b yes 8
40.e odd 2 1 CM 3640.1.ij.b ✓ 8
56.m even 6 1 3640.1.pj.b yes 8
65.s odd 12 1 3640.1.pj.b yes 8
91.ba even 12 1 inner 3640.1.ij.b ✓ 8
104.u even 12 1 3640.1.pj.b yes 8
280.ba even 6 1 3640.1.pj.b yes 8
455.cl even 12 1 inner 3640.1.ij.b ✓ 8
520.cz even 12 1 3640.1.pj.b yes 8
728.el odd 12 1 inner 3640.1.ij.b ✓ 8
3640.ij odd 12 1 inner 3640.1.ij.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.1.ij.b ✓ 8 1.a even 1 1 trivial
3640.1.ij.b ✓ 8 5.b even 2 1 inner
3640.1.ij.b ✓ 8 8.d odd 2 1 inner
3640.1.ij.b ✓ 8 40.e odd 2 1 CM
3640.1.ij.b ✓ 8 91.ba even 12 1 inner
3640.1.ij.b ✓ 8 455.cl even 12 1 inner
3640.1.ij.b ✓ 8 728.el odd 12 1 inner
3640.1.ij.b ✓ 8 3640.ij odd 12 1 inner
3640.1.pj.b yes 8 7.d odd 6 1
3640.1.pj.b yes 8 13.f odd 12 1
3640.1.pj.b yes 8 35.i odd 6 1
3640.1.pj.b yes 8 56.m even 6 1
3640.1.pj.b yes 8 65.s odd 12 1
3640.1.pj.b yes 8 104.u even 12 1
3640.1.pj.b yes 8 280.ba even 6 1
3640.1.pj.b yes 8 520.cz even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 2T_{11}^{3} + 2T_{11}^{2} + 4T_{11} + 4 \) acting on \(S_{1}^{\mathrm{new}}(3640, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 2)^{4} \) 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
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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