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Properties

Label 1560.4.g.e
Level $1560$
Weight $4$
Character orbit 1560.g
Analytic conductor $92.043$
Analytic rank $0$
Dimension $24$
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] = [1560,4,Mod(961,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.961");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.g (of order \(2\), degree \(1\), 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: \(92.0429796090\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 72 q^{3} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 72 q^{3} + 216 q^{9} + 98 q^{13} + 54 q^{17} - 210 q^{23} - 600 q^{25} + 648 q^{27} + 80 q^{29} - 90 q^{35} + 294 q^{39} - 2140 q^{43} - 2014 q^{49} + 162 q^{51} - 366 q^{53} + 150 q^{55} + 298 q^{61} + 70 q^{65} - 630 q^{69} - 1800 q^{75} - 1474 q^{77} + 2246 q^{79} + 1944 q^{81} + 240 q^{87} - 190 q^{91} + 120 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1 0 3.00000 0 − 5.00000i 0 − 33.6012i 0 9.00000 0
961.2 0 3.00000 0 − 5.00000i 0 − 32.0135i 0 9.00000 0
961.3 0 3.00000 0 − 5.00000i 0 − 14.0088i 0 9.00000 0
961.4 0 3.00000 0 − 5.00000i 0 − 12.0126i 0 9.00000 0
961.5 0 3.00000 0 − 5.00000i 0 − 10.4785i 0 9.00000 0
961.6 0 3.00000 0 − 5.00000i 0 − 8.41107i 0 9.00000 0
961.7 0 3.00000 0 − 5.00000i 0 − 1.70962i 0 9.00000 0
961.8 0 3.00000 0 − 5.00000i 0 9.37093i 0 9.00000 0
961.9 0 3.00000 0 − 5.00000i 0 13.6833i 0 9.00000 0
961.10 0 3.00000 0 − 5.00000i 0 24.2550i 0 9.00000 0
961.11 0 3.00000 0 − 5.00000i 0 24.9748i 0 9.00000 0
961.12 0 3.00000 0 − 5.00000i 0 30.9513i 0 9.00000 0
961.13 0 3.00000 0 5.00000i 0 − 30.9513i 0 9.00000 0
961.14 0 3.00000 0 5.00000i 0 − 24.9748i 0 9.00000 0
961.15 0 3.00000 0 5.00000i 0 − 24.2550i 0 9.00000 0
961.16 0 3.00000 0 5.00000i 0 − 13.6833i 0 9.00000 0
961.17 0 3.00000 0 5.00000i 0 − 9.37093i 0 9.00000 0
961.18 0 3.00000 0 5.00000i 0 1.70962i 0 9.00000 0
961.19 0 3.00000 0 5.00000i 0 8.41107i 0 9.00000 0
961.20 0 3.00000 0 5.00000i 0 10.4785i 0 9.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.4.g.e ✓ 24
13.b even 2 1 inner 1560.4.g.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.g.e ✓ 24 1.a even 1 1 trivial
1560.4.g.e ✓ 24 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 5123 T_{7}^{22} + 11073995 T_{7}^{20} + 13201017377 T_{7}^{18} + 9547536543976 T_{7}^{16} + \cdots + 42\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(1560, [\chi])\). Copy content Toggle raw display