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Properties

Label 1120.2.cb.b
Level $1120$
Weight $2$
Character orbit 1120.cb
Analytic conductor $8.943$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1120,2,Mod(81,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.cb (of order \(6\), 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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 8 q^{7} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 8 q^{7} + 38 q^{9} - 12 q^{17} + 2 q^{23} + 30 q^{25} - 36 q^{41} + 30 q^{47} + 12 q^{49} + 44 q^{55} - 32 q^{57} + 74 q^{63} + 2 q^{65} + 8 q^{71} - 32 q^{79} - 22 q^{81} - 48 q^{87} + 4 q^{89} - 2 q^{95} - 96 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −2.81779 − 1.62685i 0 −0.866025 + 0.500000i 0 0.452341 − 2.60680i 0 3.79330 + 6.57019i 0
81.2 0 −2.73909 − 1.58141i 0 0.866025 − 0.500000i 0 2.49644 − 0.876244i 0 3.50174 + 6.06518i 0
81.3 0 −2.56394 − 1.48029i 0 0.866025 − 0.500000i 0 −2.64452 − 0.0805794i 0 2.88251 + 4.99266i 0
81.4 0 −2.20766 − 1.27459i 0 0.866025 − 0.500000i 0 0.879470 + 2.49530i 0 1.74917 + 3.02964i 0
81.5 0 −2.17556 − 1.25606i 0 −0.866025 + 0.500000i 0 0.986215 + 2.45507i 0 1.65537 + 2.86718i 0
81.6 0 −1.96523 − 1.13463i 0 −0.866025 + 0.500000i 0 2.64045 + 0.167479i 0 1.07475 + 1.86152i 0
81.7 0 −1.92077 − 1.10896i 0 −0.866025 + 0.500000i 0 −2.33530 + 1.24353i 0 0.959573 + 1.66203i 0
81.8 0 −1.69968 − 0.981313i 0 −0.866025 + 0.500000i 0 −1.68530 − 2.03955i 0 0.425950 + 0.737767i 0
81.9 0 −1.50534 − 0.869106i 0 0.866025 − 0.500000i 0 0.830882 − 2.51190i 0 0.0106917 + 0.0185186i 0
81.10 0 −1.12126 − 0.647359i 0 −0.866025 + 0.500000i 0 −2.45429 − 0.988163i 0 −0.661853 − 1.14636i 0
81.11 0 −0.992927 − 0.573266i 0 0.866025 − 0.500000i 0 −2.64418 − 0.0910290i 0 −0.842731 − 1.45965i 0
81.12 0 −0.970453 − 0.560291i 0 0.866025 − 0.500000i 0 1.15341 − 2.38110i 0 −0.872147 − 1.51060i 0
81.13 0 −0.501254 − 0.289399i 0 0.866025 − 0.500000i 0 −1.67292 + 2.04972i 0 −1.33250 − 2.30795i 0
81.14 0 −0.433972 − 0.250554i 0 0.866025 − 0.500000i 0 −0.366713 + 2.62021i 0 −1.37445 − 2.38061i 0
81.15 0 −0.214333 − 0.123745i 0 −0.866025 + 0.500000i 0 2.36404 − 1.18800i 0 −1.46937 − 2.54503i 0
81.16 0 0.214333 + 0.123745i 0 0.866025 − 0.500000i 0 2.36404 − 1.18800i 0 −1.46937 − 2.54503i 0
81.17 0 0.433972 + 0.250554i 0 −0.866025 + 0.500000i 0 −0.366713 + 2.62021i 0 −1.37445 − 2.38061i 0
81.18 0 0.501254 + 0.289399i 0 −0.866025 + 0.500000i 0 −1.67292 + 2.04972i 0 −1.33250 − 2.30795i 0
81.19 0 0.970453 + 0.560291i 0 −0.866025 + 0.500000i 0 1.15341 − 2.38110i 0 −0.872147 − 1.51060i 0
81.20 0 0.992927 + 0.573266i 0 −0.866025 + 0.500000i 0 −2.64418 − 0.0910290i 0 −0.842731 − 1.45965i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.cb.b 60
4.b odd 2 1 280.2.bl.b ✓ 60
7.c even 3 1 inner 1120.2.cb.b 60
8.b even 2 1 inner 1120.2.cb.b 60
8.d odd 2 1 280.2.bl.b ✓ 60
28.g odd 6 1 280.2.bl.b ✓ 60
56.k odd 6 1 280.2.bl.b ✓ 60
56.p even 6 1 inner 1120.2.cb.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bl.b ✓ 60 4.b odd 2 1
280.2.bl.b ✓ 60 8.d odd 2 1
280.2.bl.b ✓ 60 28.g odd 6 1
280.2.bl.b ✓ 60 56.k odd 6 1
1120.2.cb.b 60 1.a even 1 1 trivial
1120.2.cb.b 60 7.c even 3 1 inner
1120.2.cb.b 60 8.b even 2 1 inner
1120.2.cb.b 60 56.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - 64 T_{3}^{58} + 2274 T_{3}^{56} - 55664 T_{3}^{54} + 1036397 T_{3}^{52} + \cdots + 25600000000 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\). Copy content Toggle raw display