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Properties

Label 2240.2.bd.b
Level $2240$
Weight $2$
Character orbit 2240.bd
Analytic conductor $17.886$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(561,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.561");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.bd (of order \(4\), degree \(2\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 560)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 4 q^{29} - 12 q^{37} - 36 q^{43} - 52 q^{49} + 8 q^{51} - 4 q^{53} - 24 q^{59} - 16 q^{61} + 68 q^{63} + 40 q^{65} + 12 q^{67} - 72 q^{69} - 4 q^{77} + 16 q^{79} - 116 q^{81} + 16 q^{85} + 8 q^{93} + 32 q^{95} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
561.1 0 −2.37790 + 2.37790i 0 −0.707107 − 0.707107i 0 1.00000i 0 − 8.30879i 0
561.2 0 −2.16124 + 2.16124i 0 0.707107 + 0.707107i 0 1.00000i 0 − 6.34189i 0
561.3 0 −2.07558 + 2.07558i 0 0.707107 + 0.707107i 0 1.00000i 0 − 5.61608i 0
561.4 0 −2.00539 + 2.00539i 0 0.707107 + 0.707107i 0 1.00000i 0 − 5.04319i 0
561.5 0 −1.86516 + 1.86516i 0 −0.707107 − 0.707107i 0 1.00000i 0 − 3.95763i 0
561.6 0 −1.56700 + 1.56700i 0 −0.707107 − 0.707107i 0 1.00000i 0 − 1.91098i 0
561.7 0 −1.51271 + 1.51271i 0 0.707107 + 0.707107i 0 1.00000i 0 − 1.57659i 0
561.8 0 −0.975138 + 0.975138i 0 −0.707107 − 0.707107i 0 1.00000i 0 1.09821i 0
561.9 0 −0.523494 + 0.523494i 0 0.707107 + 0.707107i 0 1.00000i 0 2.45191i 0
561.10 0 −0.401069 + 0.401069i 0 −0.707107 − 0.707107i 0 1.00000i 0 2.67829i 0
561.11 0 −0.363076 + 0.363076i 0 −0.707107 − 0.707107i 0 1.00000i 0 2.73635i 0
561.12 0 −0.359066 + 0.359066i 0 0.707107 + 0.707107i 0 1.00000i 0 2.74214i 0
561.13 0 −0.280441 + 0.280441i 0 0.707107 + 0.707107i 0 1.00000i 0 2.84271i 0
561.14 0 −0.0811559 + 0.0811559i 0 0.707107 + 0.707107i 0 1.00000i 0 2.98683i 0
561.15 0 0.303944 − 0.303944i 0 −0.707107 − 0.707107i 0 1.00000i 0 2.81524i 0
561.16 0 0.367823 − 0.367823i 0 −0.707107 − 0.707107i 0 1.00000i 0 2.72941i 0
561.17 0 0.991372 − 0.991372i 0 0.707107 + 0.707107i 0 1.00000i 0 1.03436i 0
561.18 0 1.01907 − 1.01907i 0 −0.707107 − 0.707107i 0 1.00000i 0 0.922998i 0
561.19 0 1.21993 − 1.21993i 0 0.707107 + 0.707107i 0 1.00000i 0 0.0235503i 0
561.20 0 1.24266 − 1.24266i 0 −0.707107 − 0.707107i 0 1.00000i 0 − 0.0884307i 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 561.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.bd.b 52
4.b odd 2 1 560.2.bd.b ✓ 52
16.e even 4 1 inner 2240.2.bd.b 52
16.f odd 4 1 560.2.bd.b ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.b ✓ 52 4.b odd 2 1
560.2.bd.b ✓ 52 16.f odd 4 1
2240.2.bd.b 52 1.a even 1 1 trivial
2240.2.bd.b 52 16.e even 4 1 inner