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Properties

Label 4012.2.b.b
Level $4012$
Weight $2$
Character orbit 4012.b
Analytic conductor $32.036$
Analytic rank $0$
Dimension $46$
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: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 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 − 3.24132i 0 0.276658i 0 − 5.07344i 0 −7.50616 0
237.2 0 − 3.20235i 0 − 1.49915i 0 2.89750i 0 −7.25507 0
237.3 0 − 3.18403i 0 − 1.45906i 0 − 1.23253i 0 −7.13805 0
237.4 0 − 3.13247i 0 − 3.91603i 0 2.98385i 0 −6.81236 0
237.5 0 − 2.81557i 0 1.77357i 0 − 1.01302i 0 −4.92746 0
237.6 0 − 2.74502i 0 1.20235i 0 − 3.93843i 0 −4.53516 0
237.7 0 − 2.57223i 0 2.31913i 0 3.71324i 0 −3.61638 0
237.8 0 − 2.34087i 0 − 2.91532i 0 − 3.43927i 0 −2.47969 0
237.9 0 − 2.27623i 0 2.55864i 0 − 1.00518i 0 −2.18122 0
237.10 0 − 1.82354i 0 − 0.284793i 0 − 0.645277i 0 −0.325298 0
237.11 0 − 1.79340i 0 1.19915i 0 − 1.20458i 0 −0.216279 0
237.12 0 − 1.77972i 0 1.91545i 0 2.45686i 0 −0.167415 0
237.13 0 − 1.58213i 0 − 3.58424i 0 − 1.89303i 0 0.496852 0
237.14 0 − 1.50352i 0 4.31120i 0 − 4.93216i 0 0.739426 0
237.15 0 − 1.45665i 0 − 3.69601i 0 3.33013i 0 0.878183 0
237.16 0 − 1.43806i 0 2.66437i 0 3.93282i 0 0.931993 0
237.17 0 − 1.38906i 0 − 2.35139i 0 − 0.244801i 0 1.07052 0
237.18 0 − 1.13278i 0 − 2.09684i 0 2.71039i 0 1.71682 0
237.19 0 − 0.593296i 0 3.87533i 0 2.92807i 0 2.64800 0
237.20 0 − 0.405081i 0 1.20499i 0 − 0.122460i 0 2.83591 0
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.46
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.b ✓ 46
17.b even 2 1 inner 4012.2.b.b ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.b.b ✓ 46 1.a even 1 1 trivial
4012.2.b.b ✓ 46 17.b even 2 1 inner