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Properties

Label 504.3.g.d
Level $504$
Weight $3$
Character orbit 504.g
Analytic conductor $13.733$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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 = [24,2] 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: no (minimal twist has level 168)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 10 q^{4} - 10 q^{8} + 12 q^{10} - 32 q^{11} + 14 q^{14} + 66 q^{16} - 16 q^{17} - 64 q^{19} - 20 q^{20} + 12 q^{22} - 72 q^{25} - 100 q^{26} - 14 q^{28} - 98 q^{32} - 108 q^{34} + 72 q^{38}+ \cdots - 14 q^{98}+O(q^{100}) \) 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 −1.97309 − 0.326994i 0 3.78615 + 1.29038i − 4.09875i 0 2.64575i −7.04846 − 3.78407i 0 −1.34027 + 8.08719i
379.2 −1.97309 + 0.326994i 0 3.78615 − 1.29038i 4.09875i 0 − 2.64575i −7.04846 + 3.78407i 0 −1.34027 − 8.08719i
379.3 −1.94205 − 0.477965i 0 3.54310 + 1.85646i − 6.16432i 0 − 2.64575i −5.99354 − 5.29882i 0 −2.94633 + 11.9714i
379.4 −1.94205 + 0.477965i 0 3.54310 − 1.85646i 6.16432i 0 2.64575i −5.99354 + 5.29882i 0 −2.94633 − 11.9714i
379.5 −1.81394 − 0.842398i 0 2.58073 + 3.05611i 5.94177i 0 − 2.64575i −2.10682 − 7.71760i 0 5.00533 − 10.7780i
379.6 −1.81394 + 0.842398i 0 2.58073 − 3.05611i − 5.94177i 0 2.64575i −2.10682 + 7.71760i 0 5.00533 + 10.7780i
379.7 −1.18386 − 1.61198i 0 −1.19696 + 3.81671i − 2.78060i 0 2.64575i 7.56949 − 2.58898i 0 −4.48227 + 3.29183i
379.8 −1.18386 + 1.61198i 0 −1.19696 − 3.81671i 2.78060i 0 − 2.64575i 7.56949 + 2.58898i 0 −4.48227 − 3.29183i
379.9 −0.434385 − 1.95226i 0 −3.62262 + 1.69606i − 3.78720i 0 2.64575i 4.88476 + 6.33554i 0 −7.39360 + 1.64510i
379.10 −0.434385 + 1.95226i 0 −3.62262 − 1.69606i 3.78720i 0 − 2.64575i 4.88476 − 6.33554i 0 −7.39360 − 1.64510i
379.11 0.0411377 − 1.99958i 0 −3.99662 − 0.164516i 3.19600i 0 − 2.64575i −0.493374 + 7.98477i 0 6.39065 + 0.131476i
379.12 0.0411377 + 1.99958i 0 −3.99662 + 0.164516i − 3.19600i 0 2.64575i −0.493374 − 7.98477i 0 6.39065 − 0.131476i
379.13 0.121960 − 1.99628i 0 −3.97025 − 0.486933i 9.26683i 0 2.64575i −1.45627 + 7.86634i 0 18.4992 + 1.13018i
379.14 0.121960 + 1.99628i 0 −3.97025 + 0.486933i − 9.26683i 0 − 2.64575i −1.45627 − 7.86634i 0 18.4992 − 1.13018i
379.15 1.05960 − 1.69625i 0 −1.75451 − 3.59468i 2.47223i 0 2.64575i −7.95653 − 0.832818i 0 4.19351 + 2.61956i
379.16 1.05960 + 1.69625i 0 −1.75451 + 3.59468i − 2.47223i 0 − 2.64575i −7.95653 + 0.832818i 0 4.19351 − 2.61956i
379.17 1.59973 − 1.20036i 0 1.11825 − 3.84051i − 0.769463i 0 − 2.64575i −2.82111 − 7.48608i 0 −0.923636 − 1.23093i
379.18 1.59973 + 1.20036i 0 1.11825 + 3.84051i 0.769463i 0 2.64575i −2.82111 + 7.48608i 0 −0.923636 + 1.23093i
379.19 1.61092 − 1.18531i 0 1.19010 − 3.81886i − 8.04930i 0 − 2.64575i −2.60937 − 7.56248i 0 −9.54089 − 12.9667i
379.20 1.61092 + 1.18531i 0 1.19010 + 3.81886i 8.04930i 0 2.64575i −2.60937 + 7.56248i 0 −9.54089 + 12.9667i
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
8.d 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.g.d 24
3.b odd 2 1 168.3.g.a ✓ 24
4.b odd 2 1 2016.3.g.d 24
8.b even 2 1 2016.3.g.d 24
8.d odd 2 1 inner 504.3.g.d 24
12.b even 2 1 672.3.g.a 24
24.f even 2 1 168.3.g.a ✓ 24
24.h odd 2 1 672.3.g.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.g.a ✓ 24 3.b odd 2 1
168.3.g.a ✓ 24 24.f even 2 1
504.3.g.d 24 1.a even 1 1 trivial
504.3.g.d 24 8.d odd 2 1 inner
672.3.g.a 24 12.b even 2 1
672.3.g.a 24 24.h odd 2 1
2016.3.g.d 24 4.b odd 2 1
2016.3.g.d 24 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 336 T_{5}^{22} + 47408 T_{5}^{20} + 3680896 T_{5}^{18} + 173353056 T_{5}^{16} + \cdots + 9028053827584 \) acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display