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Properties

Label 4012.2.b.a
Level $4012$
Weight $2$
Character orbit 4012.b
Analytic conductor $32.036$
Analytic rank $0$
Dimension $40$
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] = [4012,2,Mod(237,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4012.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1 0 − 3.27953i 0 4.21610i 0 0.547734i 0 −7.75529 0
237.2 0 − 3.07288i 0 − 1.34148i 0 0.0167420i 0 −6.44259 0
237.3 0 − 2.97688i 0 − 2.24434i 0 − 1.80545i 0 −5.86181 0
237.4 0 − 2.95725i 0 2.24490i 0 1.49294i 0 −5.74532 0
237.5 0 − 2.44713i 0 − 1.66009i 0 3.65944i 0 −2.98842 0
237.6 0 − 2.32869i 0 − 4.13581i 0 − 2.86231i 0 −2.42278 0
237.7 0 − 2.23721i 0 0.538464i 0 − 0.0679637i 0 −2.00513 0
237.8 0 − 2.21522i 0 2.69018i 0 − 2.88510i 0 −1.90721 0
237.9 0 − 2.18094i 0 − 0.351694i 0 3.15514i 0 −1.75651 0
237.10 0 − 1.86007i 0 2.89135i 0 5.10759i 0 −0.459864 0
237.11 0 − 1.67902i 0 − 0.714826i 0 1.81204i 0 0.180890 0
237.12 0 − 1.51601i 0 − 1.74989i 0 − 0.991299i 0 0.701704 0
237.13 0 − 1.46162i 0 3.03678i 0 − 1.90612i 0 0.863667 0
237.14 0 − 1.18944i 0 − 0.609716i 0 − 4.31103i 0 1.58523 0
237.15 0 − 1.02370i 0 3.54148i 0 0.166941i 0 1.95203 0
237.16 0 − 0.622620i 0 0.657734i 0 − 2.30157i 0 2.61234 0
237.17 0 − 0.510359i 0 − 1.04821i 0 1.04964i 0 2.73953 0
237.18 0 − 0.409767i 0 0.188112i 0 − 3.66007i 0 2.83209 0
237.19 0 − 0.346659i 0 − 2.94947i 0 − 4.18117i 0 2.87983 0
237.20 0 − 0.0489925i 0 − 3.24065i 0 0.683113i 0 2.99760 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4012.2.b.a ✓ 40
17.b even 2 1 inner 4012.2.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.b.a ✓ 40 1.a even 1 1 trivial
4012.2.b.a ✓ 40 17.b even 2 1 inner