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Properties

Label 1148.3.b.a
Level $1148$
Weight $3$
Character orbit 1148.b
Analytic conductor $31.281$
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] = [1148,3,Mod(657,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.657");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1148.b (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: \(31.2807343486\)
Analytic rank: \(0\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 20 q^{7} - 140 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 20 q^{7} - 140 q^{9} + 8 q^{11} - 84 q^{15} - 32 q^{21} + 80 q^{23} - 240 q^{25} + 100 q^{29} + 46 q^{35} - 76 q^{37} - 4 q^{39} - 60 q^{43} + 52 q^{49} - 356 q^{51} - 176 q^{53} + 140 q^{57} + 334 q^{63} + 116 q^{65} + 208 q^{67} + 40 q^{71} + 120 q^{77} - 216 q^{79} + 716 q^{81} + 456 q^{85} - 208 q^{91} + 272 q^{93} - 44 q^{95} - 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
657.1 0 − 5.72284i 0 − 4.30537i 0 2.46143 + 6.55297i 0 −23.7509 0
657.2 0 − 5.67537i 0 − 3.85005i 0 −6.95623 − 0.781539i 0 −23.2098 0
657.3 0 − 5.59243i 0 7.47255i 0 0.199096 − 6.99717i 0 −22.2753 0
657.4 0 − 5.16657i 0 2.35704i 0 −5.78890 + 3.93556i 0 −17.6935 0
657.5 0 − 4.73917i 0 − 6.40629i 0 −5.88287 − 3.79365i 0 −13.4597 0
657.6 0 − 4.52392i 0 0.243446i 0 6.99917 − 0.107926i 0 −11.4658 0
657.7 0 − 4.42350i 0 − 9.43649i 0 5.67259 − 4.10143i 0 −10.5673 0
657.8 0 − 4.28628i 0 − 1.08374i 0 3.85649 − 5.84187i 0 −9.37221 0
657.9 0 − 4.04643i 0 3.94881i 0 −2.52185 − 6.52995i 0 −7.37359 0
657.10 0 − 3.91391i 0 8.74883i 0 −2.13771 + 6.66560i 0 −6.31871 0
657.11 0 − 3.56705i 0 − 7.33948i 0 2.27942 + 6.61848i 0 −3.72382 0
657.12 0 − 3.16590i 0 3.15339i 0 −6.52511 + 2.53435i 0 −1.02293 0
657.13 0 − 3.11584i 0 − 2.04176i 0 −4.98785 + 4.91135i 0 −0.708477 0
657.14 0 − 2.59222i 0 − 0.777043i 0 6.62458 − 2.26163i 0 2.28039 0
657.15 0 − 2.46358i 0 − 5.88014i 0 −3.16439 − 6.24393i 0 2.93078 0
657.16 0 − 2.20737i 0 − 2.07426i 0 −6.26099 + 3.13049i 0 4.12750 0
657.17 0 − 1.89517i 0 8.25389i 0 5.86664 − 3.81871i 0 5.40834 0
657.18 0 − 1.88842i 0 7.30559i 0 −6.45348 − 2.71156i 0 5.43387 0
657.19 0 − 1.71095i 0 7.16281i 0 6.87975 + 1.29194i 0 6.07265 0
657.20 0 − 1.49702i 0 − 4.77871i 0 0.894412 + 6.94262i 0 6.75894 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 657.52
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.3.b.a ✓ 52
7.b odd 2 1 inner 1148.3.b.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.3.b.a ✓ 52 1.a even 1 1 trivial
1148.3.b.a ✓ 52 7.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1148, [\chi])\).