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Properties

Label 416.3.j.a
Level $416$
Weight $3$
Character orbit 416.j
Analytic conductor $11.335$
Analytic rank $0$
Dimension $52$
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] = [416,3,Mod(177,416)] 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("416.177"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 416.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3351789974\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 104)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 4 q^{7} - 140 q^{9} - 32 q^{15} + 4 q^{31} + 32 q^{33} - 32 q^{39} + 36 q^{41} + 4 q^{47} - 248 q^{55} - 64 q^{57} + 164 q^{63} + 52 q^{65} - 252 q^{71} - 284 q^{73} + 8 q^{79} + 244 q^{81} + 704 q^{87}+ \cdots + 292 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1 0 − 5.87809i 0 3.41259 − 3.41259i 0 4.31407 − 4.31407i 0 −25.5520 0
177.2 0 − 4.91039i 0 −4.91574 + 4.91574i 0 −2.48876 + 2.48876i 0 −15.1119 0
177.3 0 − 4.42292i 0 0.134151 − 0.134151i 0 −8.22499 + 8.22499i 0 −10.5622 0
177.4 0 − 4.15887i 0 −3.89825 + 3.89825i 0 2.18695 − 2.18695i 0 −8.29620 0
177.5 0 − 4.06957i 0 5.81375 − 5.81375i 0 −7.34046 + 7.34046i 0 −7.56141 0
177.6 0 − 3.93417i 0 −0.100962 + 0.100962i 0 3.41838 − 3.41838i 0 −6.47772 0
177.7 0 − 3.12482i 0 −1.81503 + 1.81503i 0 5.82291 − 5.82291i 0 −0.764530 0
177.8 0 − 1.96436i 0 −5.67132 + 5.67132i 0 −1.23754 + 1.23754i 0 5.14127 0
177.9 0 − 1.85570i 0 3.94501 − 3.94501i 0 9.07028 − 9.07028i 0 5.55638 0
177.10 0 − 1.84380i 0 5.47975 − 5.47975i 0 −0.646575 + 0.646575i 0 5.60039 0
177.11 0 − 1.58909i 0 −0.117449 + 0.117449i 0 −4.75895 + 4.75895i 0 6.47478 0
177.12 0 − 0.910930i 0 4.37050 − 4.37050i 0 4.85989 − 4.85989i 0 8.17021 0
177.13 0 − 0.785543i 0 0.0364474 − 0.0364474i 0 −3.97520 + 3.97520i 0 8.38292 0
177.14 0 0.785543i 0 −0.0364474 + 0.0364474i 0 −3.97520 + 3.97520i 0 8.38292 0
177.15 0 0.910930i 0 −4.37050 + 4.37050i 0 4.85989 − 4.85989i 0 8.17021 0
177.16 0 1.58909i 0 0.117449 − 0.117449i 0 −4.75895 + 4.75895i 0 6.47478 0
177.17 0 1.84380i 0 −5.47975 + 5.47975i 0 −0.646575 + 0.646575i 0 5.60039 0
177.18 0 1.85570i 0 −3.94501 + 3.94501i 0 9.07028 − 9.07028i 0 5.55638 0
177.19 0 1.96436i 0 5.67132 − 5.67132i 0 −1.23754 + 1.23754i 0 5.14127 0
177.20 0 3.12482i 0 1.81503 − 1.81503i 0 5.82291 − 5.82291i 0 −0.764530 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.26
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.d odd 4 1 inner
104.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.3.j.a 52
4.b odd 2 1 104.3.j.a ✓ 52
8.b even 2 1 inner 416.3.j.a 52
8.d odd 2 1 104.3.j.a ✓ 52
13.d odd 4 1 inner 416.3.j.a 52
52.f even 4 1 104.3.j.a ✓ 52
104.j odd 4 1 inner 416.3.j.a 52
104.m even 4 1 104.3.j.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.3.j.a ✓ 52 4.b odd 2 1
104.3.j.a ✓ 52 8.d odd 2 1
104.3.j.a ✓ 52 52.f even 4 1
104.3.j.a ✓ 52 104.m even 4 1
416.3.j.a 52 1.a even 1 1 trivial
416.3.j.a 52 8.b even 2 1 inner
416.3.j.a 52 13.d odd 4 1 inner
416.3.j.a 52 104.j odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(416, [\chi])\).