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Properties

Label 3936.2.j.h
Level $3936$
Weight $2$
Character orbit 3936.j
Analytic conductor $31.429$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3936,2,Mod(3361,3936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3936.3361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3936 = 2^{5} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3936.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,0,-4,0,0,0,-22,0,0,0,0,0,0,0,0,0,0,0,4,0,-8,0,30,0,0, 0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.4291182356\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{5} - 22 q^{9} + 4 q^{21} - 8 q^{23} + 30 q^{25} - 16 q^{31} - 4 q^{33} + 8 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 4 q^{45} - 30 q^{49} + 20 q^{51} - 4 q^{57} + 12 q^{59} + 16 q^{61} - 4 q^{73}+ \cdots + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3361.1 0 − 1.00000i 0 −3.38755 0 − 4.17077i 0 −1.00000 0
3361.2 0 1.00000i 0 −3.38755 0 4.17077i 0 −1.00000 0
3361.3 0 − 1.00000i 0 2.18458 0 4.41839i 0 −1.00000 0
3361.4 0 1.00000i 0 2.18458 0 − 4.41839i 0 −1.00000 0
3361.5 0 − 1.00000i 0 3.45502 0 2.50089i 0 −1.00000 0
3361.6 0 1.00000i 0 3.45502 0 − 2.50089i 0 −1.00000 0
3361.7 0 − 1.00000i 0 3.19623 0 − 1.38035i 0 −1.00000 0
3361.8 0 1.00000i 0 3.19623 0 1.38035i 0 −1.00000 0
3361.9 0 − 1.00000i 0 0.765367 0 − 2.53688i 0 −1.00000 0
3361.10 0 1.00000i 0 0.765367 0 2.53688i 0 −1.00000 0
3361.11 0 − 1.00000i 0 −0.559456 0 − 0.446077i 0 −1.00000 0
3361.12 0 1.00000i 0 −0.559456 0 0.446077i 0 −1.00000 0
3361.13 0 − 1.00000i 0 0.275338 0 1.41851i 0 −1.00000 0
3361.14 0 1.00000i 0 0.275338 0 − 1.41851i 0 −1.00000 0
3361.15 0 − 1.00000i 0 −2.14706 0 1.11744i 0 −1.00000 0
3361.16 0 1.00000i 0 −2.14706 0 − 1.11744i 0 −1.00000 0
3361.17 0 − 1.00000i 0 −4.19411 0 0.996315i 0 −1.00000 0
3361.18 0 1.00000i 0 −4.19411 0 − 0.996315i 0 −1.00000 0
3361.19 0 − 1.00000i 0 −2.68663 0 4.28573i 0 −1.00000 0
3361.20 0 1.00000i 0 −2.68663 0 − 4.28573i 0 −1.00000 0
See all 22 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3361.22
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3936.2.j.h ✓ 22
4.b odd 2 1 3936.2.j.i yes 22
41.b even 2 1 inner 3936.2.j.h ✓ 22
164.d odd 2 1 3936.2.j.i yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3936.2.j.h ✓ 22 1.a even 1 1 trivial
3936.2.j.h ✓ 22 41.b even 2 1 inner
3936.2.j.i yes 22 4.b odd 2 1
3936.2.j.i yes 22 164.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3936, [\chi])\):

\( T_{5}^{11} + 2 T_{5}^{10} - 33 T_{5}^{9} - 50 T_{5}^{8} + 384 T_{5}^{7} + 380 T_{5}^{6} - 1864 T_{5}^{5} + \cdots + 256 \) Copy content Toggle raw display
\( T_{23}^{11} + 4 T_{23}^{10} - 124 T_{23}^{9} - 652 T_{23}^{8} + 3896 T_{23}^{7} + 29304 T_{23}^{6} + \cdots + 1441792 \) Copy content Toggle raw display
\( T_{31}^{11} + 8 T_{31}^{10} - 104 T_{31}^{9} - 780 T_{31}^{8} + 4042 T_{31}^{7} + 27340 T_{31}^{6} + \cdots - 2236352 \) Copy content Toggle raw display