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Properties

Label 2240.1.dl.b
Level $2240$
Weight $1$
Character orbit 2240.dl
Analytic conductor $1.118$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,1,Mod(417,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 3, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.417");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2240.dl (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: \(1.11790562830\)
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: \(S_{4}\)
Projective field: Galois closure of 4.2.49000.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{3} - \zeta_{24}^{7} q^{5} - \zeta_{24}^{9} q^{7} + \zeta_{24}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{3} - \zeta_{24}^{7} q^{5} - \zeta_{24}^{9} q^{7} + \zeta_{24}^{10} q^{9} - \zeta_{24}^{2} q^{11} + \zeta_{24}^{9} q^{13} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{15} - \zeta_{24}^{4} q^{19} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{21} + \zeta_{24}^{7} q^{23} - \zeta_{24}^{2} q^{25} + ( - \zeta_{24}^{6} - 1) q^{27} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{29} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{31} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{33} - \zeta_{24}^{4} q^{35} + \zeta_{24}^{7} q^{37} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{39} - q^{41} + \zeta_{24}^{5} q^{45} + \zeta_{24} q^{47} - \zeta_{24}^{6} q^{49} + \zeta_{24}^{5} q^{53} + \zeta_{24}^{9} q^{55} + (\zeta_{24}^{6} - 1) q^{57} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{61} + \zeta_{24}^{7} q^{63} + \zeta_{24}^{4} q^{65} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{67} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{69} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{71} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{75} + \zeta_{24}^{11} q^{77} + \zeta_{24}^{8} q^{81} + (\zeta_{24}^{6} - 1) q^{83} - \zeta_{24}^{5} q^{87} + \zeta_{24}^{6} q^{91} + ( - \zeta_{24}^{7} - 2 \zeta_{24}) q^{93} + \zeta_{24}^{11} q^{95} + ( - \zeta_{24}^{6} + 1) q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{19} + 4 q^{33} - 4 q^{35} - 8 q^{41} - 8 q^{57} + 4 q^{65} + 4 q^{67} + 4 q^{75} - 4 q^{81} - 8 q^{83} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(1\) \(-1\) \(-\zeta_{24}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
−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 1.36603 − 0.366025i 0 −0.965926 − 0.258819i 0 0.707107 + 0.707107i 0 0.866025 − 0.500000i 0
417.2 0 1.36603 − 0.366025i 0 0.965926 + 0.258819i 0 −0.707107 − 0.707107i 0 0.866025 − 0.500000i 0
737.1 0 −0.366025 + 1.36603i 0 −0.258819 − 0.965926i 0 −0.707107 − 0.707107i 0 −0.866025 − 0.500000i 0
737.2 0 −0.366025 + 1.36603i 0 0.258819 + 0.965926i 0 0.707107 + 0.707107i 0 −0.866025 − 0.500000i 0
1313.1 0 −0.366025 − 1.36603i 0 −0.258819 + 0.965926i 0 −0.707107 + 0.707107i 0 −0.866025 + 0.500000i 0
1313.2 0 −0.366025 − 1.36603i 0 0.258819 − 0.965926i 0 0.707107 − 0.707107i 0 −0.866025 + 0.500000i 0
1633.1 0 1.36603 + 0.366025i 0 −0.965926 + 0.258819i 0 0.707107 − 0.707107i 0 0.866025 + 0.500000i 0
1633.2 0 1.36603 + 0.366025i 0 0.965926 − 0.258819i 0 −0.707107 + 0.707107i 0 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
20.e even 4 1 inner
40.i odd 4 1 inner
56.k odd 6 1 inner
140.w even 12 1 inner
280.bt odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.1.dl.b yes 8
4.b odd 2 1 2240.1.dl.a ✓ 8
5.c odd 4 1 2240.1.dl.a ✓ 8
7.c even 3 1 inner 2240.1.dl.b yes 8
8.b even 2 1 2240.1.dl.a ✓ 8
8.d odd 2 1 inner 2240.1.dl.b yes 8
20.e even 4 1 inner 2240.1.dl.b yes 8
28.g odd 6 1 2240.1.dl.a ✓ 8
35.l odd 12 1 2240.1.dl.a ✓ 8
40.i odd 4 1 inner 2240.1.dl.b yes 8
40.k even 4 1 2240.1.dl.a ✓ 8
56.k odd 6 1 inner 2240.1.dl.b yes 8
56.p even 6 1 2240.1.dl.a ✓ 8
140.w even 12 1 inner 2240.1.dl.b yes 8
280.br even 12 1 2240.1.dl.a ✓ 8
280.bt odd 12 1 inner 2240.1.dl.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.1.dl.a ✓ 8 4.b odd 2 1
2240.1.dl.a ✓ 8 5.c odd 4 1
2240.1.dl.a ✓ 8 8.b even 2 1
2240.1.dl.a ✓ 8 28.g odd 6 1
2240.1.dl.a ✓ 8 35.l odd 12 1
2240.1.dl.a ✓ 8 40.k even 4 1
2240.1.dl.a ✓ 8 56.p even 6 1
2240.1.dl.a ✓ 8 280.br even 12 1
2240.1.dl.b yes 8 1.a even 1 1 trivial
2240.1.dl.b yes 8 7.c even 3 1 inner
2240.1.dl.b yes 8 8.d odd 2 1 inner
2240.1.dl.b yes 8 20.e even 4 1 inner
2240.1.dl.b yes 8 40.i odd 4 1 inner
2240.1.dl.b yes 8 56.k odd 6 1 inner
2240.1.dl.b yes 8 140.w even 12 1 inner
2240.1.dl.b yes 8 280.bt odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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