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Properties

Label 936.4.ba.a
Level $936$
Weight $4$
Character orbit 936.ba
Analytic conductor $55.226$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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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] = [936,4,Mod(161,936)] 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("936.161"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 3])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 936.ba (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(55.2257877654\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 56 q^{7} - 72 q^{13} + 408 q^{19} + 544 q^{31} + 512 q^{37} - 1008 q^{55} - 3088 q^{61} - 1184 q^{67} - 760 q^{73} - 2480 q^{79} + 1632 q^{85} + 4208 q^{91} - 3384 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 0 0 −13.4888 − 13.4888i 0 13.4452 + 13.4452i 0 0 0
161.2 0 0 0 −11.1651 − 11.1651i 0 −1.99469 − 1.99469i 0 0 0
161.3 0 0 0 −10.8835 − 10.8835i 0 3.24179 + 3.24179i 0 0 0
161.4 0 0 0 −10.3852 − 10.3852i 0 5.50275 + 5.50275i 0 0 0
161.5 0 0 0 −7.66411 − 7.66411i 0 −14.5038 − 14.5038i 0 0 0
161.6 0 0 0 −6.02636 − 6.02636i 0 −6.72476 − 6.72476i 0 0 0
161.7 0 0 0 −4.20175 − 4.20175i 0 −24.4786 − 24.4786i 0 0 0
161.8 0 0 0 −3.60396 − 3.60396i 0 −2.37597 − 2.37597i 0 0 0
161.9 0 0 0 −2.74653 − 2.74653i 0 17.1547 + 17.1547i 0 0 0
161.10 0 0 0 −2.41835 − 2.41835i 0 24.7335 + 24.7335i 0 0 0
161.11 0 0 0 2.41835 + 2.41835i 0 24.7335 + 24.7335i 0 0 0
161.12 0 0 0 2.74653 + 2.74653i 0 17.1547 + 17.1547i 0 0 0
161.13 0 0 0 3.60396 + 3.60396i 0 −2.37597 − 2.37597i 0 0 0
161.14 0 0 0 4.20175 + 4.20175i 0 −24.4786 − 24.4786i 0 0 0
161.15 0 0 0 6.02636 + 6.02636i 0 −6.72476 − 6.72476i 0 0 0
161.16 0 0 0 7.66411 + 7.66411i 0 −14.5038 − 14.5038i 0 0 0
161.17 0 0 0 10.3852 + 10.3852i 0 5.50275 + 5.50275i 0 0 0
161.18 0 0 0 10.8835 + 10.8835i 0 3.24179 + 3.24179i 0 0 0
161.19 0 0 0 11.1651 + 11.1651i 0 −1.99469 − 1.99469i 0 0 0
161.20 0 0 0 13.4888 + 13.4888i 0 13.4452 + 13.4452i 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.4.ba.a ✓ 40
3.b odd 2 1 inner 936.4.ba.a ✓ 40
13.d odd 4 1 inner 936.4.ba.a ✓ 40
39.f even 4 1 inner 936.4.ba.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.4.ba.a ✓ 40 1.a even 1 1 trivial
936.4.ba.a ✓ 40 3.b odd 2 1 inner
936.4.ba.a ✓ 40 13.d odd 4 1 inner
936.4.ba.a ✓ 40 39.f even 4 1 inner