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Properties

Label 3654.2.f.b
Level $3654$
Weight $2$
Character orbit 3654.f
Analytic conductor $29.177$
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] = [3654,2,Mod(755,3654)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3654.755"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3654.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,-40,0,0,-4,0,0,0,0,0,0,0,0,40,0,0,0,0,0,8,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.1773368986\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{4} - 4 q^{7} + 40 q^{16} + 8 q^{22} + 64 q^{25} + 4 q^{28} - 48 q^{37} - 8 q^{43} + 8 q^{46} + 36 q^{49} - 40 q^{58} - 40 q^{64} + 24 q^{67} + 28 q^{70} - 144 q^{79} - 24 q^{85} - 8 q^{88}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
755.1 − 1.00000i 0 −1.00000 4.02309 0 1.07956 + 2.41548i 1.00000i 0 − 4.02309i
755.2 1.00000i 0 −1.00000 4.02309 0 1.07956 − 2.41548i − 1.00000i 0 4.02309i
755.3 − 1.00000i 0 −1.00000 1.81839 0 −0.812712 − 2.51784i 1.00000i 0 − 1.81839i
755.4 1.00000i 0 −1.00000 1.81839 0 −0.812712 + 2.51784i − 1.00000i 0 1.81839i
755.5 − 1.00000i 0 −1.00000 −2.21484 0 2.36568 − 1.18472i 1.00000i 0 2.21484i
755.6 1.00000i 0 −1.00000 −2.21484 0 2.36568 + 1.18472i − 1.00000i 0 − 2.21484i
755.7 − 1.00000i 0 −1.00000 −2.09396 0 −2.34749 − 1.22036i 1.00000i 0 2.09396i
755.8 1.00000i 0 −1.00000 −2.09396 0 −2.34749 + 1.22036i − 1.00000i 0 − 2.09396i
755.9 − 1.00000i 0 −1.00000 0.0793857 0 −2.12738 − 1.57298i 1.00000i 0 − 0.0793857i
755.10 1.00000i 0 −1.00000 0.0793857 0 −2.12738 + 1.57298i − 1.00000i 0 0.0793857i
755.11 − 1.00000i 0 −1.00000 −2.70268 0 −2.02003 + 1.70864i 1.00000i 0 2.70268i
755.12 1.00000i 0 −1.00000 −2.70268 0 −2.02003 − 1.70864i − 1.00000i 0 − 2.70268i
755.13 − 1.00000i 0 −1.00000 −3.89079 0 −2.64517 − 0.0555398i 1.00000i 0 3.89079i
755.14 1.00000i 0 −1.00000 −3.89079 0 −2.64517 + 0.0555398i − 1.00000i 0 − 3.89079i
755.15 − 1.00000i 0 −1.00000 2.84841 0 1.96094 + 1.77615i 1.00000i 0 − 2.84841i
755.16 1.00000i 0 −1.00000 2.84841 0 1.96094 − 1.77615i − 1.00000i 0 2.84841i
755.17 − 1.00000i 0 −1.00000 −2.53682 0 2.41720 + 1.07570i 1.00000i 0 2.53682i
755.18 1.00000i 0 −1.00000 −2.53682 0 2.41720 − 1.07570i − 1.00000i 0 − 2.53682i
755.19 − 1.00000i 0 −1.00000 −0.469078 0 1.12940 + 2.39259i 1.00000i 0 0.469078i
755.20 1.00000i 0 −1.00000 −0.469078 0 1.12940 − 2.39259i − 1.00000i 0 − 0.469078i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 755.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3654.2.f.b ✓ 40
3.b odd 2 1 inner 3654.2.f.b ✓ 40
7.b odd 2 1 inner 3654.2.f.b ✓ 40
21.c even 2 1 inner 3654.2.f.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3654.2.f.b ✓ 40 1.a even 1 1 trivial
3654.2.f.b ✓ 40 3.b odd 2 1 inner
3654.2.f.b ✓ 40 7.b odd 2 1 inner
3654.2.f.b ✓ 40 21.c even 2 1 inner