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Properties

Label 168.4.p.a
Level $168$
Weight $4$
Character orbit 168.p
Analytic conductor $9.912$
Analytic rank $0$
Dimension $48$
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] = [168,4,Mod(139,168)] 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("168.139"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 2 q^{2} + 10 q^{4} + 118 q^{8} - 432 q^{9} - 40 q^{11} + 38 q^{14} - 142 q^{16} + 18 q^{18} - 376 q^{22} + 1200 q^{25} - 274 q^{28} + 336 q^{30} + 318 q^{32} - 456 q^{35} - 90 q^{36} + 564 q^{42}+ \cdots + 360 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −2.81425 − 0.282796i − 3.00000i 7.84005 + 1.59172i −4.56033 −0.848389 + 8.44276i 17.6399 − 5.64228i −21.6138 − 6.69665i −9.00000 12.8339 + 1.28964i
139.2 −2.81425 − 0.282796i 3.00000i 7.84005 + 1.59172i 4.56033 0.848389 − 8.44276i −17.6399 − 5.64228i −21.6138 − 6.69665i −9.00000 −12.8339 − 1.28964i
139.3 −2.81425 + 0.282796i − 3.00000i 7.84005 − 1.59172i 4.56033 0.848389 + 8.44276i −17.6399 + 5.64228i −21.6138 + 6.69665i −9.00000 −12.8339 + 1.28964i
139.4 −2.81425 + 0.282796i 3.00000i 7.84005 − 1.59172i −4.56033 −0.848389 − 8.44276i 17.6399 + 5.64228i −21.6138 + 6.69665i −9.00000 12.8339 − 1.28964i
139.5 −2.48994 − 1.34171i − 3.00000i 4.39965 + 6.68155i −20.8990 −4.02512 + 7.46983i −18.5132 − 0.511483i −1.99022 − 22.5397i −9.00000 52.0374 + 28.0403i
139.6 −2.48994 − 1.34171i 3.00000i 4.39965 + 6.68155i 20.8990 4.02512 − 7.46983i 18.5132 − 0.511483i −1.99022 − 22.5397i −9.00000 −52.0374 − 28.0403i
139.7 −2.48994 + 1.34171i − 3.00000i 4.39965 − 6.68155i 20.8990 4.02512 + 7.46983i 18.5132 + 0.511483i −1.99022 + 22.5397i −9.00000 −52.0374 + 28.0403i
139.8 −2.48994 + 1.34171i 3.00000i 4.39965 − 6.68155i −20.8990 −4.02512 − 7.46983i −18.5132 + 0.511483i −1.99022 + 22.5397i −9.00000 52.0374 − 28.0403i
139.9 −2.20291 − 1.77404i − 3.00000i 1.70558 + 7.81607i 4.80791 −5.32211 + 6.60872i 7.97009 + 16.7176i 10.1088 − 20.2438i −9.00000 −10.5914 − 8.52942i
139.10 −2.20291 − 1.77404i 3.00000i 1.70558 + 7.81607i −4.80791 5.32211 − 6.60872i −7.97009 + 16.7176i 10.1088 − 20.2438i −9.00000 10.5914 + 8.52942i
139.11 −2.20291 + 1.77404i − 3.00000i 1.70558 − 7.81607i −4.80791 5.32211 + 6.60872i −7.97009 − 16.7176i 10.1088 + 20.2438i −9.00000 10.5914 − 8.52942i
139.12 −2.20291 + 1.77404i 3.00000i 1.70558 − 7.81607i 4.80791 −5.32211 − 6.60872i 7.97009 − 16.7176i 10.1088 + 20.2438i −9.00000 −10.5914 + 8.52942i
139.13 −1.77579 − 2.20149i − 3.00000i −1.69313 + 7.81878i 1.47916 −6.60448 + 5.32737i 7.43403 − 16.9628i 20.2196 − 10.1571i −9.00000 −2.62669 − 3.25637i
139.14 −1.77579 − 2.20149i 3.00000i −1.69313 + 7.81878i −1.47916 6.60448 − 5.32737i −7.43403 − 16.9628i 20.2196 − 10.1571i −9.00000 2.62669 + 3.25637i
139.15 −1.77579 + 2.20149i − 3.00000i −1.69313 − 7.81878i −1.47916 6.60448 + 5.32737i −7.43403 + 16.9628i 20.2196 + 10.1571i −9.00000 2.62669 − 3.25637i
139.16 −1.77579 + 2.20149i 3.00000i −1.69313 − 7.81878i 1.47916 −6.60448 − 5.32737i 7.43403 + 16.9628i 20.2196 + 10.1571i −9.00000 −2.62669 + 3.25637i
139.17 −1.32895 − 2.49678i − 3.00000i −4.46779 + 6.63618i 18.1258 −7.49033 + 3.98685i −17.7395 + 5.32058i 22.5065 + 2.33591i −9.00000 −24.0883 − 45.2561i
139.18 −1.32895 − 2.49678i 3.00000i −4.46779 + 6.63618i −18.1258 7.49033 − 3.98685i 17.7395 + 5.32058i 22.5065 + 2.33591i −9.00000 24.0883 + 45.2561i
139.19 −1.32895 + 2.49678i − 3.00000i −4.46779 − 6.63618i −18.1258 7.49033 + 3.98685i 17.7395 − 5.32058i 22.5065 − 2.33591i −9.00000 24.0883 − 45.2561i
139.20 −1.32895 + 2.49678i 3.00000i −4.46779 − 6.63618i 18.1258 −7.49033 − 3.98685i −17.7395 − 5.32058i 22.5065 − 2.33591i −9.00000 −24.0883 + 45.2561i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.p.a ✓ 48
4.b odd 2 1 672.4.p.a 48
7.b odd 2 1 inner 168.4.p.a ✓ 48
8.b even 2 1 672.4.p.a 48
8.d odd 2 1 inner 168.4.p.a ✓ 48
28.d even 2 1 672.4.p.a 48
56.e even 2 1 inner 168.4.p.a ✓ 48
56.h odd 2 1 672.4.p.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.p.a ✓ 48 1.a even 1 1 trivial
168.4.p.a ✓ 48 7.b odd 2 1 inner
168.4.p.a ✓ 48 8.d odd 2 1 inner
168.4.p.a ✓ 48 56.e even 2 1 inner
672.4.p.a 48 4.b odd 2 1
672.4.p.a 48 8.b even 2 1
672.4.p.a 48 28.d even 2 1
672.4.p.a 48 56.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(168, [\chi])\).