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Properties

Label 3024.2.df.e
Level $3024$
Weight $2$
Character orbit 3024.df
Analytic conductor $24.147$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3024,2,Mod(17,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5, 1])) 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("3024.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.df (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{25} - 18 q^{29} - 18 q^{31} + 6 q^{41} + 6 q^{43} + 18 q^{47} - 12 q^{49} + 12 q^{53} + 18 q^{61} + 36 q^{65} + 12 q^{77} - 6 q^{79} + 18 q^{89} - 6 q^{91} - 54 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 0 0 −4.11484 0 2.54819 − 0.711830i 0 0 0
17.2 0 0 0 −4.09884 0 0.146661 + 2.64168i 0 0 0
17.3 0 0 0 −3.53401 0 −1.30083 − 2.30388i 0 0 0
17.4 0 0 0 −2.76937 0 1.21939 + 2.34800i 0 0 0
17.5 0 0 0 −2.59337 0 2.61256 − 0.417759i 0 0 0
17.6 0 0 0 −2.22830 0 −2.51733 − 0.814280i 0 0 0
17.7 0 0 0 −2.20884 0 −2.16520 − 1.52049i 0 0 0
17.8 0 0 0 −2.04899 0 −1.41312 + 2.23676i 0 0 0
17.9 0 0 0 −1.36828 0 −2.64451 − 0.0810554i 0 0 0
17.10 0 0 0 0.0525740 0 −2.44149 + 1.01937i 0 0 0
17.11 0 0 0 0.203178 0 1.27132 + 2.32029i 0 0 0
17.12 0 0 0 0.207028 0 1.37075 − 2.26297i 0 0 0
17.13 0 0 0 0.542075 0 2.62378 − 0.340238i 0 0 0
17.14 0 0 0 0.623597 0 0.996837 − 2.45078i 0 0 0
17.15 0 0 0 1.05582 0 −1.79851 − 1.94045i 0 0 0
17.16 0 0 0 1.07485 0 1.26701 + 2.32265i 0 0 0
17.17 0 0 0 1.28783 0 −1.10056 + 2.40599i 0 0 0
17.18 0 0 0 1.58600 0 1.06431 − 2.42224i 0 0 0
17.19 0 0 0 1.81173 0 −1.69266 + 2.03345i 0 0 0
17.20 0 0 0 2.22094 0 2.45091 + 0.996507i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.df.e 48
3.b odd 2 1 1008.2.df.e 48
4.b odd 2 1 1512.2.cx.a 48
7.d odd 6 1 3024.2.ca.e 48
9.c even 3 1 1008.2.ca.e 48
9.d odd 6 1 3024.2.ca.e 48
12.b even 2 1 504.2.cx.a yes 48
21.g even 6 1 1008.2.ca.e 48
28.f even 6 1 1512.2.bs.a 48
36.f odd 6 1 504.2.bs.a ✓ 48
36.h even 6 1 1512.2.bs.a 48
63.k odd 6 1 1008.2.df.e 48
63.s even 6 1 inner 3024.2.df.e 48
84.j odd 6 1 504.2.bs.a ✓ 48
252.n even 6 1 504.2.cx.a yes 48
252.bn odd 6 1 1512.2.cx.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bs.a ✓ 48 36.f odd 6 1
504.2.bs.a ✓ 48 84.j odd 6 1
504.2.cx.a yes 48 12.b even 2 1
504.2.cx.a yes 48 252.n even 6 1
1008.2.ca.e 48 9.c even 3 1
1008.2.ca.e 48 21.g even 6 1
1008.2.df.e 48 3.b odd 2 1
1008.2.df.e 48 63.k odd 6 1
1512.2.bs.a 48 28.f even 6 1
1512.2.bs.a 48 36.h even 6 1
1512.2.cx.a 48 4.b odd 2 1
1512.2.cx.a 48 252.bn odd 6 1
3024.2.ca.e 48 7.d odd 6 1
3024.2.ca.e 48 9.d odd 6 1
3024.2.df.e 48 1.a even 1 1 trivial
3024.2.df.e 48 63.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 12 T_{5}^{21} + 2175 T_{5}^{20} - 768 T_{5}^{19} - 36005 T_{5}^{18} + \cdots - 6476 \) acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\). Copy content Toggle raw display