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Properties

Label 63.6.p.b
Level $63$
Weight $6$
Character orbit 63.p
Analytic conductor $10.104$
Analytic rank $0$
Dimension $24$
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] = [63,6,Mod(17,63)] 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("63.17"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 304 q^{4} - 436 q^{7} + 1992 q^{10} - 3644 q^{16} + 3804 q^{19} - 5648 q^{22} - 18852 q^{25} - 39172 q^{28} + 38652 q^{31} + 20548 q^{37} + 132060 q^{40} + 2200 q^{43} - 25712 q^{46} - 125676 q^{49}+ \cdots - 1481724 q^{94}+O(q^{100}) \) 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 −9.49029 + 5.47922i 0 44.0437 − 76.2859i −41.3063 − 71.5446i 0 −18.0992 − 128.372i 614.630i 0 784.017 + 452.652i
17.2 −8.14327 + 4.70152i 0 28.2086 − 48.8587i 33.1882 + 57.4837i 0 −116.271 − 57.3420i 229.595i 0 −540.521 − 312.070i
17.3 −6.73742 + 3.88985i 0 14.2619 − 24.7023i −46.8267 − 81.1063i 0 58.7189 + 115.582i − 27.0441i 0 630.983 + 364.298i
17.4 −5.81384 + 3.35662i 0 6.53381 − 11.3169i 43.0174 + 74.5084i 0 95.4202 − 87.7609i − 127.098i 0 −500.193 − 288.786i
17.5 −4.55322 + 2.62880i 0 −2.17881 + 3.77381i −11.2537 − 19.4920i 0 −61.1173 + 114.331i − 191.154i 0 102.481 + 59.1675i
17.6 −1.30242 + 0.751952i 0 −14.8691 + 25.7541i −8.15150 − 14.1188i 0 −67.6518 − 110.590i − 92.8484i 0 21.2333 + 12.2591i
17.7 1.30242 − 0.751952i 0 −14.8691 + 25.7541i 8.15150 + 14.1188i 0 −67.6518 − 110.590i 92.8484i 0 21.2333 + 12.2591i
17.8 4.55322 − 2.62880i 0 −2.17881 + 3.77381i 11.2537 + 19.4920i 0 −61.1173 + 114.331i 191.154i 0 102.481 + 59.1675i
17.9 5.81384 − 3.35662i 0 6.53381 − 11.3169i −43.0174 − 74.5084i 0 95.4202 − 87.7609i 127.098i 0 −500.193 − 288.786i
17.10 6.73742 − 3.88985i 0 14.2619 − 24.7023i 46.8267 + 81.1063i 0 58.7189 + 115.582i 27.0441i 0 630.983 + 364.298i
17.11 8.14327 − 4.70152i 0 28.2086 − 48.8587i −33.1882 − 57.4837i 0 −116.271 − 57.3420i − 229.595i 0 −540.521 − 312.070i
17.12 9.49029 − 5.47922i 0 44.0437 − 76.2859i 41.3063 + 71.5446i 0 −18.0992 − 128.372i − 614.630i 0 784.017 + 452.652i
26.1 −9.49029 − 5.47922i 0 44.0437 + 76.2859i −41.3063 + 71.5446i 0 −18.0992 + 128.372i − 614.630i 0 784.017 − 452.652i
26.2 −8.14327 − 4.70152i 0 28.2086 + 48.8587i 33.1882 − 57.4837i 0 −116.271 + 57.3420i − 229.595i 0 −540.521 + 312.070i
26.3 −6.73742 − 3.88985i 0 14.2619 + 24.7023i −46.8267 + 81.1063i 0 58.7189 − 115.582i 27.0441i 0 630.983 − 364.298i
26.4 −5.81384 − 3.35662i 0 6.53381 + 11.3169i 43.0174 − 74.5084i 0 95.4202 + 87.7609i 127.098i 0 −500.193 + 288.786i
26.5 −4.55322 − 2.62880i 0 −2.17881 − 3.77381i −11.2537 + 19.4920i 0 −61.1173 − 114.331i 191.154i 0 102.481 − 59.1675i
26.6 −1.30242 − 0.751952i 0 −14.8691 − 25.7541i −8.15150 + 14.1188i 0 −67.6518 + 110.590i 92.8484i 0 21.2333 − 12.2591i
26.7 1.30242 + 0.751952i 0 −14.8691 − 25.7541i 8.15150 − 14.1188i 0 −67.6518 + 110.590i − 92.8484i 0 21.2333 − 12.2591i
26.8 4.55322 + 2.62880i 0 −2.17881 − 3.77381i 11.2537 − 19.4920i 0 −61.1173 − 114.331i − 191.154i 0 102.481 − 59.1675i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.12
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
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.p.b ✓ 24
3.b odd 2 1 inner 63.6.p.b ✓ 24
7.c even 3 1 441.6.c.b 24
7.d odd 6 1 inner 63.6.p.b ✓ 24
7.d odd 6 1 441.6.c.b 24
21.g even 6 1 inner 63.6.p.b ✓ 24
21.g even 6 1 441.6.c.b 24
21.h odd 6 1 441.6.c.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.p.b ✓ 24 1.a even 1 1 trivial
63.6.p.b ✓ 24 3.b odd 2 1 inner
63.6.p.b ✓ 24 7.d odd 6 1 inner
63.6.p.b ✓ 24 21.g even 6 1 inner
441.6.c.b 24 7.c even 3 1
441.6.c.b 24 7.d odd 6 1
441.6.c.b 24 21.g even 6 1
441.6.c.b 24 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 344 T_{2}^{22} + 73519 T_{2}^{20} - 9883100 T_{2}^{18} + 975017677 T_{2}^{16} + \cdots + 32\!\cdots\!16 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display