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Properties

Label 595.2.x.b
Level $595$
Weight $2$
Character orbit 595.x
Analytic conductor $4.751$
Analytic rank $0$
Dimension $92$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [595,2,Mod(16,595)] 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("595.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(595, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 595 = 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 595.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.75109892027\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 52 q^{4} + 40 q^{9} + 24 q^{13} - 4 q^{15} - 48 q^{16} + 12 q^{17} - 12 q^{18} - 8 q^{19} - 56 q^{21} + 46 q^{25} - 44 q^{26} - 20 q^{33} - 72 q^{34} + 6 q^{35} - 176 q^{36} + 8 q^{38} + 20 q^{42}+ \cdots - 152 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.39384 + 2.41421i −1.84849 + 1.06723i −2.88560 − 4.99800i 0.866025 + 0.500000i − 5.95018i 2.49361 + 0.884254i 10.5129 0.777940 − 1.34743i −2.41421 + 1.39384i
16.2 −1.39384 + 2.41421i 1.84849 − 1.06723i −2.88560 − 4.99800i −0.866025 − 0.500000i 5.95018i −2.49361 − 0.884254i 10.5129 0.777940 − 1.34743i 2.41421 − 1.39384i
16.3 −1.27624 + 2.21052i −2.06818 + 1.19407i −2.25759 − 3.91027i −0.866025 − 0.500000i − 6.09567i −2.39657 + 1.12092i 6.41999 1.35159 − 2.34102i 2.21052 − 1.27624i
16.4 −1.27624 + 2.21052i 2.06818 − 1.19407i −2.25759 − 3.91027i 0.866025 + 0.500000i 6.09567i 2.39657 − 1.12092i 6.41999 1.35159 − 2.34102i −2.21052 + 1.27624i
16.5 −1.15797 + 2.00566i −1.02661 + 0.592714i −1.68177 − 2.91291i 0.866025 + 0.500000i − 2.74537i −2.40091 − 1.11159i 3.15787 −0.797381 + 1.38110i −2.00566 + 1.15797i
16.6 −1.15797 + 2.00566i 1.02661 − 0.592714i −1.68177 − 2.91291i −0.866025 − 0.500000i 2.74537i 2.40091 + 1.11159i 3.15787 −0.797381 + 1.38110i 2.00566 − 1.15797i
16.7 −1.00968 + 1.74882i −2.89186 + 1.66962i −1.03890 − 1.79943i −0.866025 − 0.500000i − 6.74312i 2.60766 − 0.447341i 0.157121 4.07525 − 7.05854i 1.74882 − 1.00968i
16.8 −1.00968 + 1.74882i 2.89186 − 1.66962i −1.03890 − 1.79943i 0.866025 + 0.500000i 6.74312i −2.60766 + 0.447341i 0.157121 4.07525 − 7.05854i −1.74882 + 1.00968i
16.9 −1.00254 + 1.73645i −0.439248 + 0.253600i −1.01016 − 1.74965i 0.866025 + 0.500000i − 1.01697i 0.111277 + 2.64341i 0.0407467 −1.37137 + 2.37529i −1.73645 + 1.00254i
16.10 −1.00254 + 1.73645i 0.439248 − 0.253600i −1.01016 − 1.74965i −0.866025 − 0.500000i 1.01697i −0.111277 − 2.64341i 0.0407467 −1.37137 + 2.37529i 1.73645 − 1.00254i
16.11 −0.837274 + 1.45020i −1.69643 + 0.979434i −0.402056 − 0.696381i 0.866025 + 0.500000i − 3.28022i 1.19780 − 2.35908i −2.00257 0.418584 − 0.725008i −1.45020 + 0.837274i
16.12 −0.837274 + 1.45020i 1.69643 − 0.979434i −0.402056 − 0.696381i −0.866025 − 0.500000i 3.28022i −1.19780 + 2.35908i −2.00257 0.418584 − 0.725008i 1.45020 − 0.837274i
16.13 −0.754544 + 1.30691i −1.14054 + 0.658489i −0.138672 − 0.240187i −0.866025 − 0.500000i − 1.98743i 2.59066 + 0.537082i −2.59964 −0.632785 + 1.09602i 1.30691 − 0.754544i
16.14 −0.754544 + 1.30691i 1.14054 − 0.658489i −0.138672 − 0.240187i 0.866025 + 0.500000i 1.98743i −2.59066 − 0.537082i −2.59964 −0.632785 + 1.09602i −1.30691 + 0.754544i
16.15 −0.689672 + 1.19455i −0.402406 + 0.232329i 0.0487037 + 0.0843574i −0.866025 − 0.500000i − 0.640925i −2.48784 − 0.900371i −2.89305 −1.39205 + 2.41109i 1.19455 − 0.689672i
16.16 −0.689672 + 1.19455i 0.402406 − 0.232329i 0.0487037 + 0.0843574i 0.866025 + 0.500000i 0.640925i 2.48784 + 0.900371i −2.89305 −1.39205 + 2.41109i −1.19455 + 0.689672i
16.17 −0.420367 + 0.728097i −1.87558 + 1.08286i 0.646583 + 1.11991i −0.866025 − 0.500000i − 1.82080i −0.576334 + 2.58222i −2.76868 0.845190 − 1.46391i 0.728097 − 0.420367i
16.18 −0.420367 + 0.728097i 1.87558 − 1.08286i 0.646583 + 1.11991i 0.866025 + 0.500000i 1.82080i 0.576334 − 2.58222i −2.76868 0.845190 − 1.46391i −0.728097 + 0.420367i
16.19 −0.349678 + 0.605661i −1.89728 + 1.09540i 0.755450 + 1.30848i 0.866025 + 0.500000i − 1.53215i 0.894806 − 2.48984i −2.45537 0.899791 − 1.55848i −0.605661 + 0.349678i
16.20 −0.349678 + 0.605661i 1.89728 − 1.09540i 0.755450 + 1.30848i −0.866025 − 0.500000i 1.53215i −0.894806 + 2.48984i −2.45537 0.899791 − 1.55848i 0.605661 − 0.349678i
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.46
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.b even 2 1 inner
119.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 595.2.x.b ✓ 92
7.c even 3 1 inner 595.2.x.b ✓ 92
17.b even 2 1 inner 595.2.x.b ✓ 92
119.j even 6 1 inner 595.2.x.b ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
595.2.x.b ✓ 92 1.a even 1 1 trivial
595.2.x.b ✓ 92 7.c even 3 1 inner
595.2.x.b ✓ 92 17.b even 2 1 inner
595.2.x.b ✓ 92 119.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} + 36 T_{2}^{44} + 739 T_{2}^{42} + 10338 T_{2}^{40} + 6 T_{2}^{39} + 109093 T_{2}^{38} + \cdots + 86436 \) acting on \(S_{2}^{\mathrm{new}}(595, [\chi])\). Copy content Toggle raw display