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Properties

Label 56.7.o.a
Level $56$
Weight $7$
Character orbit 56.o
Analytic conductor $12.883$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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] = [56,7,Mod(17,56)] 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("56.17"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 56.o (of order \(6\), degree \(2\), 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: \(12.8830286827\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 564 q^{7} + 2300 q^{9} + 812 q^{11} - 11752 q^{15} - 840 q^{17} - 2268 q^{19} + 68 q^{21} + 1392 q^{23} + 28104 q^{25} + 51464 q^{29} + 139860 q^{31} - 124572 q^{33} - 87564 q^{35} - 22548 q^{37}+ \cdots + 68552 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −36.9900 − 21.3562i 0 165.273 − 95.4202i 0 −120.800 − 321.024i 0 547.675 + 948.600i 0
17.2 0 −30.8184 − 17.7930i 0 −54.4036 + 31.4099i 0 −295.298 − 174.493i 0 268.681 + 465.369i 0
17.3 0 −30.7831 − 17.7726i 0 −131.199 + 75.7478i 0 −2.86620 + 342.988i 0 267.231 + 462.858i 0
17.4 0 −26.2731 − 15.1688i 0 102.414 − 59.1289i 0 322.679 + 116.307i 0 95.6838 + 165.729i 0
17.5 0 −11.6484 − 6.72521i 0 −43.5745 + 25.1577i 0 292.362 − 179.369i 0 −274.043 − 474.657i 0
17.6 0 1.35117 + 0.780100i 0 14.0845 − 8.13166i 0 −123.633 + 319.944i 0 −363.283 − 629.224i 0
17.7 0 7.70960 + 4.45114i 0 −213.648 + 123.350i 0 325.918 − 106.894i 0 −324.875 − 562.700i 0
17.8 0 9.53773 + 5.50661i 0 183.292 − 105.824i 0 46.8195 + 339.790i 0 −303.854 − 526.291i 0
17.9 0 13.7930 + 7.96340i 0 −47.1456 + 27.2195i 0 −316.915 − 131.201i 0 −237.669 − 411.654i 0
17.10 0 22.1092 + 12.7648i 0 122.892 − 70.9516i 0 36.1745 − 341.087i 0 −38.6222 − 66.8956i 0
17.11 0 40.7707 + 23.5390i 0 −60.4339 + 34.8915i 0 −225.250 + 258.673i 0 743.668 + 1288.07i 0
17.12 0 41.2415 + 23.8108i 0 −37.5506 + 21.6798i 0 342.808 + 11.4676i 0 769.408 + 1332.65i 0
33.1 0 −36.9900 + 21.3562i 0 165.273 + 95.4202i 0 −120.800 + 321.024i 0 547.675 − 948.600i 0
33.2 0 −30.8184 + 17.7930i 0 −54.4036 − 31.4099i 0 −295.298 + 174.493i 0 268.681 − 465.369i 0
33.3 0 −30.7831 + 17.7726i 0 −131.199 − 75.7478i 0 −2.86620 − 342.988i 0 267.231 − 462.858i 0
33.4 0 −26.2731 + 15.1688i 0 102.414 + 59.1289i 0 322.679 − 116.307i 0 95.6838 − 165.729i 0
33.5 0 −11.6484 + 6.72521i 0 −43.5745 − 25.1577i 0 292.362 + 179.369i 0 −274.043 + 474.657i 0
33.6 0 1.35117 − 0.780100i 0 14.0845 + 8.13166i 0 −123.633 − 319.944i 0 −363.283 + 629.224i 0
33.7 0 7.70960 − 4.45114i 0 −213.648 − 123.350i 0 325.918 + 106.894i 0 −324.875 + 562.700i 0
33.8 0 9.53773 − 5.50661i 0 183.292 + 105.824i 0 46.8195 − 339.790i 0 −303.854 + 526.291i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.7.o.a ✓ 24
4.b odd 2 1 112.7.s.e 24
7.c even 3 1 392.7.c.c 24
7.d odd 6 1 inner 56.7.o.a ✓ 24
7.d odd 6 1 392.7.c.c 24
28.f even 6 1 112.7.s.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.7.o.a ✓ 24 1.a even 1 1 trivial
56.7.o.a ✓ 24 7.d odd 6 1 inner
112.7.s.e 24 4.b odd 2 1
112.7.s.e 24 28.f even 6 1
392.7.c.c 24 7.c even 3 1
392.7.c.c 24 7.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(56, [\chi])\).