Properties

Label 504.3.n.a
Level $504$
Weight $3$
Character orbit 504.n
Analytic conductor $13.733$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,3,Mod(197,504)] 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("504.197"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.n (of order \(2\), degree \(1\), 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: \(13.7330053238\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{4} - 56 q^{10} + 140 q^{16} + 132 q^{22} + 240 q^{25} - 28 q^{28} + 256 q^{31} - 96 q^{34} + 216 q^{40} + 28 q^{46} + 336 q^{49} + 48 q^{52} - 512 q^{55} - 196 q^{58} + 208 q^{64} + 168 q^{70}+ \cdots - 24 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.99917 − 0.0576227i 0 3.99336 + 0.230395i 7.03286 0 −2.64575 −7.97013 − 0.690708i 0 −14.0599 − 0.405253i
197.2 −1.99917 + 0.0576227i 0 3.99336 − 0.230395i 7.03286 0 −2.64575 −7.97013 + 0.690708i 0 −14.0599 + 0.405253i
197.3 −1.98329 − 0.258031i 0 3.86684 + 1.02350i 3.92423 0 2.64575 −7.40495 − 3.02765i 0 −7.78287 − 1.01257i
197.4 −1.98329 + 0.258031i 0 3.86684 − 1.02350i 3.92423 0 2.64575 −7.40495 + 3.02765i 0 −7.78287 + 1.01257i
197.5 −1.96504 − 0.372294i 0 3.72279 + 1.46315i −6.52341 0 2.64575 −6.77073 − 4.26112i 0 12.8188 + 2.42863i
197.6 −1.96504 + 0.372294i 0 3.72279 − 1.46315i −6.52341 0 2.64575 −6.77073 + 4.26112i 0 12.8188 − 2.42863i
197.7 −1.84205 − 0.778997i 0 2.78633 + 2.86991i −6.09763 0 −2.64575 −2.89692 − 7.45707i 0 11.2322 + 4.75003i
197.8 −1.84205 + 0.778997i 0 2.78633 − 2.86991i −6.09763 0 −2.64575 −2.89692 + 7.45707i 0 11.2322 − 4.75003i
197.9 −1.67199 − 1.09747i 0 1.59111 + 3.66993i 0.953944 0 −2.64575 1.36733 − 7.88228i 0 −1.59499 − 1.04693i
197.10 −1.67199 + 1.09747i 0 1.59111 − 3.66993i 0.953944 0 −2.64575 1.36733 + 7.88228i 0 −1.59499 + 1.04693i
197.11 −1.24838 − 1.56254i 0 −0.883085 + 3.90130i 5.36369 0 2.64575 7.19838 − 3.49046i 0 −6.69594 − 8.38101i
197.12 −1.24838 + 1.56254i 0 −0.883085 − 3.90130i 5.36369 0 2.64575 7.19838 + 3.49046i 0 −6.69594 + 8.38101i
197.13 −1.19681 − 1.60238i 0 −1.13527 + 3.83551i −0.617081 0 2.64575 7.50468 − 2.77124i 0 0.738531 + 0.988801i
197.14 −1.19681 + 1.60238i 0 −1.13527 − 3.83551i −0.617081 0 2.64575 7.50468 + 2.77124i 0 0.738531 − 0.988801i
197.15 −1.19442 − 1.60417i 0 −1.14670 + 3.83211i 9.31660 0 −2.64575 7.51699 − 2.73767i 0 −11.1280 − 14.9454i
197.16 −1.19442 + 1.60417i 0 −1.14670 − 3.83211i 9.31660 0 −2.64575 7.51699 + 2.73767i 0 −11.1280 + 14.9454i
197.17 −0.616098 − 1.90274i 0 −3.24085 + 2.34455i −1.92658 0 −2.64575 6.45775 + 4.72202i 0 1.18696 + 3.66578i
197.18 −0.616098 + 1.90274i 0 −3.24085 − 2.34455i −1.92658 0 −2.64575 6.45775 − 4.72202i 0 1.18696 − 3.66578i
197.19 −0.412084 − 1.95709i 0 −3.66037 + 1.61297i 1.39175 0 −2.64575 4.66510 + 6.49899i 0 −0.573518 − 2.72377i
197.20 −0.412084 + 1.95709i 0 −3.66037 − 1.61297i 1.39175 0 −2.64575 4.66510 − 6.49899i 0 −0.573518 + 2.72377i
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 197.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.3.n.a ✓ 48
3.b odd 2 1 inner 504.3.n.a ✓ 48
4.b odd 2 1 2016.3.n.a 48
8.b even 2 1 inner 504.3.n.a ✓ 48
8.d odd 2 1 2016.3.n.a 48
12.b even 2 1 2016.3.n.a 48
24.f even 2 1 2016.3.n.a 48
24.h odd 2 1 inner 504.3.n.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.n.a ✓ 48 1.a even 1 1 trivial
504.3.n.a ✓ 48 3.b odd 2 1 inner
504.3.n.a ✓ 48 8.b even 2 1 inner
504.3.n.a ✓ 48 24.h odd 2 1 inner
2016.3.n.a 48 4.b odd 2 1
2016.3.n.a 48 8.d odd 2 1
2016.3.n.a 48 12.b even 2 1
2016.3.n.a 48 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(504, [\chi])\).