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Properties

Label 1224.2.h.c
Level $1224$
Weight $2$
Character orbit 1224.h
Analytic conductor $9.774$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,2,Mod(611,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.611");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.h (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: \(9.77368920740\)
Analytic rank: \(0\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 24 q^{4} - 56 q^{16} - 32 q^{19} + 8 q^{25} - 4 q^{34} - 40 q^{49} - 56 q^{52} + 72 q^{64} - 56 q^{70} + 8 q^{76} - 40 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
611.1 −1.41316 − 0.0545922i 0 1.99404 + 0.154295i 2.17219i 0 2.12610 −2.80947 − 0.326902i 0 0.118585 − 3.06966i
611.2 −1.41316 − 0.0545922i 0 1.99404 + 0.154295i − 2.17219i 0 −2.12610 −2.80947 − 0.326902i 0 −0.118585 + 3.06966i
611.3 −1.41316 + 0.0545922i 0 1.99404 − 0.154295i − 2.17219i 0 2.12610 −2.80947 + 0.326902i 0 0.118585 + 3.06966i
611.4 −1.41316 + 0.0545922i 0 1.99404 − 0.154295i 2.17219i 0 −2.12610 −2.80947 + 0.326902i 0 −0.118585 − 3.06966i
611.5 −1.31153 − 0.529041i 0 1.44023 + 1.38771i 0.806727i 0 −3.59960 −1.15475 − 2.58196i 0 0.426791 − 1.05805i
611.6 −1.31153 − 0.529041i 0 1.44023 + 1.38771i − 0.806727i 0 3.59960 −1.15475 − 2.58196i 0 −0.426791 + 1.05805i
611.7 −1.31153 + 0.529041i 0 1.44023 − 1.38771i − 0.806727i 0 −3.59960 −1.15475 + 2.58196i 0 0.426791 + 1.05805i
611.8 −1.31153 + 0.529041i 0 1.44023 − 1.38771i 0.806727i 0 3.59960 −1.15475 + 2.58196i 0 −0.426791 − 1.05805i
611.9 −1.30554 − 0.543657i 0 1.40887 + 1.41953i 2.39522i 0 −3.95107 −1.06761 − 2.61920i 0 1.30218 − 3.12706i
611.10 −1.30554 − 0.543657i 0 1.40887 + 1.41953i − 2.39522i 0 3.95107 −1.06761 − 2.61920i 0 −1.30218 + 3.12706i
611.11 −1.30554 + 0.543657i 0 1.40887 − 1.41953i − 2.39522i 0 −3.95107 −1.06761 + 2.61920i 0 1.30218 + 3.12706i
611.12 −1.30554 + 0.543657i 0 1.40887 − 1.41953i 2.39522i 0 3.95107 −1.06761 + 2.61920i 0 −1.30218 − 3.12706i
611.13 −1.10492 − 0.882701i 0 0.441679 + 1.95062i 2.78722i 0 −0.404892 1.23380 − 2.54514i 0 2.46028 − 3.07964i
611.14 −1.10492 − 0.882701i 0 0.441679 + 1.95062i − 2.78722i 0 0.404892 1.23380 − 2.54514i 0 −2.46028 + 3.07964i
611.15 −1.10492 + 0.882701i 0 0.441679 − 1.95062i − 2.78722i 0 −0.404892 1.23380 + 2.54514i 0 2.46028 + 3.07964i
611.16 −1.10492 + 0.882701i 0 0.441679 − 1.95062i 2.78722i 0 0.404892 1.23380 + 2.54514i 0 −2.46028 − 3.07964i
611.17 −0.933778 − 1.06210i 0 −0.256118 + 1.98353i 3.62179i 0 −1.18114 2.34587 − 1.58016i 0 3.84671 − 3.38195i
611.18 −0.933778 − 1.06210i 0 −0.256118 + 1.98353i − 3.62179i 0 1.18114 2.34587 − 1.58016i 0 −3.84671 + 3.38195i
611.19 −0.933778 + 1.06210i 0 −0.256118 − 1.98353i − 3.62179i 0 −1.18114 2.34587 + 1.58016i 0 3.84671 + 3.38195i
611.20 −0.933778 + 1.06210i 0 −0.256118 − 1.98353i 3.62179i 0 1.18114 2.34587 + 1.58016i 0 −3.84671 − 3.38195i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 611.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
17.b even 2 1 inner
24.f even 2 1 inner
51.c odd 2 1 inner
136.e odd 2 1 inner
408.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.h.c ✓ 56
3.b odd 2 1 inner 1224.2.h.c ✓ 56
4.b odd 2 1 4896.2.h.c 56
8.b even 2 1 4896.2.h.c 56
8.d odd 2 1 inner 1224.2.h.c ✓ 56
12.b even 2 1 4896.2.h.c 56
17.b even 2 1 inner 1224.2.h.c ✓ 56
24.f even 2 1 inner 1224.2.h.c ✓ 56
24.h odd 2 1 4896.2.h.c 56
51.c odd 2 1 inner 1224.2.h.c ✓ 56
68.d odd 2 1 4896.2.h.c 56
136.e odd 2 1 inner 1224.2.h.c ✓ 56
136.h even 2 1 4896.2.h.c 56
204.h even 2 1 4896.2.h.c 56
408.b odd 2 1 4896.2.h.c 56
408.h even 2 1 inner 1224.2.h.c ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.2.h.c ✓ 56 1.a even 1 1 trivial
1224.2.h.c ✓ 56 3.b odd 2 1 inner
1224.2.h.c ✓ 56 8.d odd 2 1 inner
1224.2.h.c ✓ 56 17.b even 2 1 inner
1224.2.h.c ✓ 56 24.f even 2 1 inner
1224.2.h.c ✓ 56 51.c odd 2 1 inner
1224.2.h.c ✓ 56 136.e odd 2 1 inner
1224.2.h.c ✓ 56 408.h even 2 1 inner
4896.2.h.c 56 4.b odd 2 1
4896.2.h.c 56 8.b even 2 1
4896.2.h.c 56 12.b even 2 1
4896.2.h.c 56 24.h odd 2 1
4896.2.h.c 56 68.d odd 2 1
4896.2.h.c 56 136.h even 2 1
4896.2.h.c 56 204.h even 2 1
4896.2.h.c 56 408.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 34T_{5}^{12} + 432T_{5}^{10} + 2596T_{5}^{8} + 7557T_{5}^{6} + 9510T_{5}^{4} + 3698T_{5}^{2} + 44 \) acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\). Copy content Toggle raw display