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Properties

Label 1148.2.k.b
Level $1148$
Weight $2$
Character orbit 1148.k
Analytic conductor $9.167$
Analytic rank $0$
Dimension $36$
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] = [1148,2,Mod(337,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.k (of order \(4\), degree \(2\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{11} - 16 q^{17} - 4 q^{19} - 36 q^{23} - 64 q^{25} + 12 q^{27} + 16 q^{29} - 28 q^{31} + 12 q^{35} + 48 q^{37} + 4 q^{41} + 36 q^{45} + 12 q^{47} - 12 q^{51} - 12 q^{53} + 12 q^{55} + 76 q^{57} + 20 q^{59} - 4 q^{65} - 44 q^{67} + 72 q^{69} - 20 q^{71} + 72 q^{75} - 8 q^{79} - 100 q^{81} - 40 q^{83} - 8 q^{85} - 16 q^{89} + 20 q^{93} + 76 q^{95} - 16 q^{97} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1 0 −2.33489 − 2.33489i 0 3.90568i 0 −0.707107 − 0.707107i 0 7.90342i 0
337.2 0 −2.32681 − 2.32681i 0 − 4.24622i 0 0.707107 + 0.707107i 0 7.82813i 0
337.3 0 −1.89621 − 1.89621i 0 − 1.39357i 0 −0.707107 − 0.707107i 0 4.19121i 0
337.4 0 −1.64273 − 1.64273i 0 − 1.39147i 0 −0.707107 − 0.707107i 0 2.39713i 0
337.5 0 −1.38781 − 1.38781i 0 2.11167i 0 0.707107 + 0.707107i 0 0.852014i 0
337.6 0 −0.898855 − 0.898855i 0 3.17533i 0 −0.707107 − 0.707107i 0 − 1.38412i 0
337.7 0 −0.796852 − 0.796852i 0 − 1.53973i 0 0.707107 + 0.707107i 0 − 1.73005i 0
337.8 0 −0.350714 − 0.350714i 0 3.08423i 0 −0.707107 − 0.707107i 0 − 2.75400i 0
337.9 0 −0.197445 − 0.197445i 0 − 3.12585i 0 0.707107 + 0.707107i 0 − 2.92203i 0
337.10 0 0.147004 + 0.147004i 0 2.61619i 0 0.707107 + 0.707107i 0 − 2.95678i 0
337.11 0 0.178034 + 0.178034i 0 − 0.542323i 0 −0.707107 − 0.707107i 0 − 2.93661i 0
337.12 0 0.805830 + 0.805830i 0 − 0.251332i 0 0.707107 + 0.707107i 0 − 1.70128i 0
337.13 0 1.46058 + 1.46058i 0 − 2.07990i 0 −0.707107 − 0.707107i 0 1.26660i 0
337.14 0 1.62776 + 1.62776i 0 3.63967i 0 0.707107 + 0.707107i 0 2.29923i 0
337.15 0 1.65216 + 1.65216i 0 2.95400i 0 −0.707107 − 0.707107i 0 2.45924i 0
337.16 0 1.71130 + 1.71130i 0 − 2.46933i 0 −0.707107 − 0.707107i 0 2.85712i 0
337.17 0 1.86437 + 1.86437i 0 − 2.96253i 0 0.707107 + 0.707107i 0 3.95177i 0
337.18 0 2.38527 + 2.38527i 0 0.515487i 0 0.707107 + 0.707107i 0 8.37900i 0
729.1 0 −2.33489 + 2.33489i 0 − 3.90568i 0 −0.707107 + 0.707107i 0 − 7.90342i 0
729.2 0 −2.32681 + 2.32681i 0 4.24622i 0 0.707107 − 0.707107i 0 − 7.82813i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.k.b ✓ 36
41.c even 4 1 inner 1148.2.k.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.k.b ✓ 36 1.a even 1 1 trivial
1148.2.k.b ✓ 36 41.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 4 T_{3}^{33} + 313 T_{3}^{32} - 56 T_{3}^{31} + 8 T_{3}^{30} - 704 T_{3}^{29} + \cdots + 141376 \) acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\). Copy content Toggle raw display