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Properties

Label 588.2.t.b
Level $588$
Weight $2$
Character orbit 588.t
Analytic conductor $4.695$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,2,Mod(41,588)] 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("588.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.t (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96] 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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 16 q^{9} - 6 q^{15} - 20 q^{25} - 21 q^{27} - 38 q^{37} - 82 q^{39} + 4 q^{43} - 91 q^{45} + 28 q^{49} - 6 q^{51} + 154 q^{55} - 29 q^{57} + 168 q^{61} + 84 q^{63} + 4 q^{67} + 28 q^{69} - 84 q^{75}+ \cdots + 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −1.72777 − 0.121731i 0 2.50195 − 1.20487i 0 −2.23394 − 1.41757i 0 2.97036 + 0.420645i 0
41.2 0 −1.69980 + 0.332698i 0 −1.95786 + 0.942857i 0 1.61724 − 2.09393i 0 2.77862 − 1.13104i 0
41.3 0 −1.53519 + 0.801984i 0 0.0974025 − 0.0469066i 0 −0.153754 + 2.64128i 0 1.71364 − 2.46240i 0
41.4 0 −1.47234 − 0.912255i 0 −3.41268 + 1.64346i 0 −2.56878 + 0.633554i 0 1.33558 + 2.68630i 0
41.5 0 −1.46785 − 0.919466i 0 2.07533 − 0.999426i 0 2.61464 − 0.404536i 0 1.30916 + 2.69928i 0
41.6 0 −0.622565 + 1.61630i 0 2.95718 − 1.42410i 0 2.57412 + 0.611477i 0 −2.22483 − 2.01250i 0
41.7 0 −0.443612 + 1.67428i 0 −1.58576 + 0.763660i 0 −1.04034 − 2.43263i 0 −2.60642 − 1.48546i 0
41.8 0 0.196321 − 1.72089i 0 −2.07533 + 0.999426i 0 2.61464 − 0.404536i 0 −2.92292 − 0.675694i 0
41.9 0 0.204761 − 1.71990i 0 3.41268 − 1.64346i 0 −2.56878 + 0.633554i 0 −2.91615 − 0.704338i 0
41.10 0 0.290780 + 1.70747i 0 2.29274 − 1.10412i 0 −2.33365 + 1.24663i 0 −2.83089 + 0.992995i 0
41.11 0 0.982073 − 1.42672i 0 −2.50195 + 1.20487i 0 −2.23394 − 1.41757i 0 −1.07107 − 2.80229i 0
41.12 0 1.15365 + 1.29193i 0 −2.29274 + 1.10412i 0 −2.33365 + 1.24663i 0 −0.338166 + 2.98088i 0
41.13 0 1.31992 − 1.12152i 0 1.95786 − 0.942857i 0 1.61724 − 2.09393i 0 0.484379 − 2.96064i 0
41.14 0 1.58419 − 0.700234i 0 −0.0974025 + 0.0469066i 0 −0.153754 + 2.64128i 0 2.01934 − 2.21861i 0
41.15 0 1.58559 + 0.697065i 0 1.58576 − 0.763660i 0 −1.04034 − 2.43263i 0 2.02820 + 2.21052i 0
41.16 0 1.65183 + 0.521003i 0 −2.95718 + 1.42410i 0 2.57412 + 0.611477i 0 2.45711 + 1.72122i 0
125.1 0 −1.65771 + 0.502000i 0 0.784282 − 3.43617i 0 −0.657650 + 2.56271i 0 2.49599 − 1.66434i 0
125.2 0 −1.55575 + 0.761336i 0 −0.124821 + 0.546875i 0 2.28593 + 1.33210i 0 1.84074 − 2.36890i 0
125.3 0 −1.28593 + 1.16034i 0 −0.272909 + 1.19569i 0 0.474570 − 2.60284i 0 0.307211 − 2.98423i 0
125.4 0 −1.27573 − 1.17154i 0 −0.784282 + 3.43617i 0 −0.657650 + 2.56271i 0 0.254994 + 2.98914i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
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
49.f odd 14 1 inner
147.k even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.t.b ✓ 96
3.b odd 2 1 inner 588.2.t.b ✓ 96
49.f odd 14 1 inner 588.2.t.b ✓ 96
147.k even 14 1 inner 588.2.t.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.t.b ✓ 96 1.a even 1 1 trivial
588.2.t.b ✓ 96 3.b odd 2 1 inner
588.2.t.b ✓ 96 49.f odd 14 1 inner
588.2.t.b ✓ 96 147.k even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{96} + 50 T_{5}^{94} + 1559 T_{5}^{92} + 37246 T_{5}^{90} + 748720 T_{5}^{88} + \cdots + 68\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display