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Properties

Label 504.3.g.c
Level $504$
Weight $3$
Character orbit 504.g
Analytic conductor $13.733$
Analytic rank $0$
Dimension $24$
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(379,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.379"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7330053238\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{4} + 12 q^{10} + 66 q^{16} - 64 q^{19} - 144 q^{22} - 168 q^{25} - 14 q^{28} + 12 q^{34} + 196 q^{40} + 224 q^{43} - 84 q^{46} - 168 q^{49} - 364 q^{52} + 348 q^{58} + 214 q^{64} - 32 q^{67}+ \cdots - 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1 −1.99926 − 0.0543236i 0 3.99410 + 0.217214i 9.01528i 0 − 2.64575i −7.97345 − 0.651241i 0 0.489742 − 18.0239i
379.2 −1.99926 + 0.0543236i 0 3.99410 − 0.217214i − 9.01528i 0 2.64575i −7.97345 + 0.651241i 0 0.489742 + 18.0239i
379.3 −1.90930 − 0.595458i 0 3.29086 + 2.27382i 2.67381i 0 2.64575i −4.92928 − 6.30097i 0 1.59214 − 5.10512i
379.4 −1.90930 + 0.595458i 0 3.29086 − 2.27382i − 2.67381i 0 − 2.64575i −4.92928 + 6.30097i 0 1.59214 + 5.10512i
379.5 −1.60008 − 1.19989i 0 1.12051 + 3.83985i − 1.24577i 0 − 2.64575i 2.81450 − 7.48856i 0 −1.49479 + 1.99333i
379.6 −1.60008 + 1.19989i 0 1.12051 − 3.83985i 1.24577i 0 2.64575i 2.81450 + 7.48856i 0 −1.49479 − 1.99333i
379.7 −1.02753 − 1.71586i 0 −1.88836 + 3.52620i 6.98819i 0 2.64575i 7.99082 − 0.383129i 0 11.9908 − 7.18059i
379.8 −1.02753 + 1.71586i 0 −1.88836 − 3.52620i − 6.98819i 0 − 2.64575i 7.99082 + 0.383129i 0 11.9908 + 7.18059i
379.9 −0.550248 − 1.92282i 0 −3.39445 + 2.11605i 2.10762i 0 − 2.64575i 5.93658 + 5.36256i 0 4.05258 − 1.15972i
379.10 −0.550248 + 1.92282i 0 −3.39445 − 2.11605i − 2.10762i 0 2.64575i 5.93658 − 5.36256i 0 4.05258 + 1.15972i
379.11 −0.434362 − 1.95226i 0 −3.62266 + 1.69598i − 6.98187i 0 2.64575i 4.88454 + 6.33572i 0 −13.6304 + 3.03266i
379.12 −0.434362 + 1.95226i 0 −3.62266 − 1.69598i 6.98187i 0 − 2.64575i 4.88454 − 6.33572i 0 −13.6304 − 3.03266i
379.13 0.434362 − 1.95226i 0 −3.62266 − 1.69598i − 6.98187i 0 − 2.64575i −4.88454 + 6.33572i 0 −13.6304 − 3.03266i
379.14 0.434362 + 1.95226i 0 −3.62266 + 1.69598i 6.98187i 0 2.64575i −4.88454 − 6.33572i 0 −13.6304 + 3.03266i
379.15 0.550248 − 1.92282i 0 −3.39445 − 2.11605i 2.10762i 0 2.64575i −5.93658 + 5.36256i 0 4.05258 + 1.15972i
379.16 0.550248 + 1.92282i 0 −3.39445 + 2.11605i − 2.10762i 0 − 2.64575i −5.93658 − 5.36256i 0 4.05258 − 1.15972i
379.17 1.02753 − 1.71586i 0 −1.88836 − 3.52620i 6.98819i 0 − 2.64575i −7.99082 − 0.383129i 0 11.9908 + 7.18059i
379.18 1.02753 + 1.71586i 0 −1.88836 + 3.52620i − 6.98819i 0 2.64575i −7.99082 + 0.383129i 0 11.9908 − 7.18059i
379.19 1.60008 − 1.19989i 0 1.12051 − 3.83985i − 1.24577i 0 2.64575i −2.81450 − 7.48856i 0 −1.49479 − 1.99333i
379.20 1.60008 + 1.19989i 0 1.12051 + 3.83985i 1.24577i 0 − 2.64575i −2.81450 + 7.48856i 0 −1.49479 + 1.99333i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.24
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.d odd 2 1 inner
24.f even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 192T_{5}^{10} + 12712T_{5}^{8} + 337952T_{5}^{6} + 3064720T_{5}^{4} + 10133120T_{5}^{2} + 9535744 \) acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display