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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(91,522)] 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("522.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.n (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,4,0,0,-4] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} - 4 q^{7} + 20 q^{13} - 4 q^{16} + 20 q^{22} + 38 q^{25} + 4 q^{28} + 56 q^{31} + 22 q^{34} - 56 q^{37} + 14 q^{40} + 28 q^{43} - 4 q^{49} + 8 q^{52} + 14 q^{55} - 44 q^{58} + 28 q^{61} + 4 q^{64}+ \cdots - 14 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −0.974928 − 0.222521i 0 0.900969 + 0.433884i −0.227953 + 0.998728i 0 2.86761 − 1.38097i −0.781831 − 0.623490i 0 0.444476 − 0.922964i
91.2 −0.974928 − 0.222521i 0 0.900969 + 0.433884i 0.313889 − 1.37524i 0 −2.46664 + 1.18787i −0.781831 − 0.623490i 0 −0.612039 + 1.27091i
91.3 0.974928 + 0.222521i 0 0.900969 + 0.433884i −0.313889 + 1.37524i 0 −2.46664 + 1.18787i 0.781831 + 0.623490i 0 −0.612039 + 1.27091i
91.4 0.974928 + 0.222521i 0 0.900969 + 0.433884i 0.227953 − 0.998728i 0 2.86761 − 1.38097i 0.781831 + 0.623490i 0 0.444476 − 0.922964i
109.1 −0.974928 + 0.222521i 0 0.900969 − 0.433884i −0.227953 − 0.998728i 0 2.86761 + 1.38097i −0.781831 + 0.623490i 0 0.444476 + 0.922964i
109.2 −0.974928 + 0.222521i 0 0.900969 − 0.433884i 0.313889 + 1.37524i 0 −2.46664 − 1.18787i −0.781831 + 0.623490i 0 −0.612039 − 1.27091i
109.3 0.974928 − 0.222521i 0 0.900969 − 0.433884i −0.313889 − 1.37524i 0 −2.46664 − 1.18787i 0.781831 − 0.623490i 0 −0.612039 − 1.27091i
109.4 0.974928 − 0.222521i 0 0.900969 − 0.433884i 0.227953 + 0.998728i 0 2.86761 + 1.38097i 0.781831 − 0.623490i 0 0.444476 + 0.922964i
325.1 −0.781831 − 0.623490i 0 0.222521 + 0.974928i −1.44975 + 1.81792i 0 0.556893 − 2.43991i 0.433884 − 0.900969i 0 2.26691 − 0.517408i
325.2 −0.781831 − 0.623490i 0 0.222521 + 0.974928i −0.0662259 + 0.0830446i 0 −0.834372 + 3.65562i 0.433884 − 0.900969i 0 0.103555 − 0.0236357i
325.3 0.781831 + 0.623490i 0 0.222521 + 0.974928i 0.0662259 − 0.0830446i 0 −0.834372 + 3.65562i −0.433884 + 0.900969i 0 0.103555 − 0.0236357i
325.4 0.781831 + 0.623490i 0 0.222521 + 0.974928i 1.44975 − 1.81792i 0 0.556893 − 2.43991i −0.433884 + 0.900969i 0 2.26691 − 0.517408i
361.1 −0.433884 + 0.900969i 0 −0.623490 − 0.781831i −0.701889 − 0.338012i 0 −0.0349384 + 0.0438113i 0.974928 − 0.222521i 0 0.609077 − 0.485723i
361.2 −0.433884 + 0.900969i 0 −0.623490 − 0.781831i 3.24048 + 1.56053i 0 −1.08855 + 1.36500i 0.974928 − 0.222521i 0 −2.81198 + 2.24248i
361.3 0.433884 − 0.900969i 0 −0.623490 − 0.781831i −3.24048 − 1.56053i 0 −1.08855 + 1.36500i −0.974928 + 0.222521i 0 −2.81198 + 2.24248i
361.4 0.433884 − 0.900969i 0 −0.623490 − 0.781831i 0.701889 + 0.338012i 0 −0.0349384 + 0.0438113i −0.974928 + 0.222521i 0 0.609077 − 0.485723i
415.1 −0.433884 − 0.900969i 0 −0.623490 + 0.781831i −0.701889 + 0.338012i 0 −0.0349384 − 0.0438113i 0.974928 + 0.222521i 0 0.609077 + 0.485723i
415.2 −0.433884 − 0.900969i 0 −0.623490 + 0.781831i 3.24048 − 1.56053i 0 −1.08855 − 1.36500i 0.974928 + 0.222521i 0 −2.81198 − 2.24248i
415.3 0.433884 + 0.900969i 0 −0.623490 + 0.781831i −3.24048 + 1.56053i 0 −1.08855 − 1.36500i −0.974928 − 0.222521i 0 −2.81198 − 2.24248i
415.4 0.433884 + 0.900969i 0 −0.623490 + 0.781831i 0.701889 − 0.338012i 0 −0.0349384 − 0.0438113i −0.974928 − 0.222521i 0 0.609077 + 0.485723i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 91.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.e even 14 1 inner
87.h odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.n.d ✓ 24
3.b odd 2 1 inner 522.2.n.d ✓ 24
29.e even 14 1 inner 522.2.n.d ✓ 24
87.h odd 14 1 inner 522.2.n.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.n.d ✓ 24 1.a even 1 1 trivial
522.2.n.d ✓ 24 3.b odd 2 1 inner
522.2.n.d ✓ 24 29.e even 14 1 inner
522.2.n.d ✓ 24 87.h odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 9 T_{5}^{22} + 101 T_{5}^{20} + 570 T_{5}^{18} + 5780 T_{5}^{16} + 23269 T_{5}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\). Copy content Toggle raw display