Properties

Label 112.10.p.b
Level $112$
Weight $10$
Character orbit 112.p
Analytic conductor $57.684$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,10,Mod(31,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.31");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.p (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.6840136504\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 1704 q^{5} - 80428 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57} - 593611308 q^{61} + 160281180 q^{65} - 1676922516 q^{73} + 1645751688 q^{77} - 528698056 q^{81} + 1370082456 q^{85} - 941603484 q^{89} - 1432348316 q^{93}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −128.906 + 223.272i 0 125.209 72.2894i 0 5420.95 + 3311.64i 0 −23392.1 40516.3i 0
31.2 0 −104.726 + 181.391i 0 1126.36 650.304i 0 −5633.01 2936.47i 0 −12093.7 20946.9i 0
31.3 0 −84.0026 + 145.497i 0 −2357.52 + 1361.12i 0 −2187.92 5963.77i 0 −4271.38 7398.25i 0
31.4 0 −48.7996 + 84.5234i 0 −62.4476 + 36.0541i 0 6259.38 + 1083.41i 0 5078.69 + 8796.55i 0
31.5 0 −41.2793 + 71.4979i 0 −557.845 + 322.072i 0 −3078.62 + 5556.59i 0 6433.53 + 11143.2i 0
31.6 0 −29.1853 + 50.5504i 0 2152.25 1242.60i 0 2216.90 5953.06i 0 8137.94 + 14095.3i 0
31.7 0 29.1853 50.5504i 0 2152.25 1242.60i 0 −2216.90 + 5953.06i 0 8137.94 + 14095.3i 0
31.8 0 41.2793 71.4979i 0 −557.845 + 322.072i 0 3078.62 5556.59i 0 6433.53 + 11143.2i 0
31.9 0 48.7996 84.5234i 0 −62.4476 + 36.0541i 0 −6259.38 1083.41i 0 5078.69 + 8796.55i 0
31.10 0 84.0026 145.497i 0 −2357.52 + 1361.12i 0 2187.92 + 5963.77i 0 −4271.38 7398.25i 0
31.11 0 104.726 181.391i 0 1126.36 650.304i 0 5633.01 + 2936.47i 0 −12093.7 20946.9i 0
31.12 0 128.906 223.272i 0 125.209 72.2894i 0 −5420.95 3311.64i 0 −23392.1 40516.3i 0
47.1 0 −128.906 223.272i 0 125.209 + 72.2894i 0 5420.95 3311.64i 0 −23392.1 + 40516.3i 0
47.2 0 −104.726 181.391i 0 1126.36 + 650.304i 0 −5633.01 + 2936.47i 0 −12093.7 + 20946.9i 0
47.3 0 −84.0026 145.497i 0 −2357.52 1361.12i 0 −2187.92 + 5963.77i 0 −4271.38 + 7398.25i 0
47.4 0 −48.7996 84.5234i 0 −62.4476 36.0541i 0 6259.38 1083.41i 0 5078.69 8796.55i 0
47.5 0 −41.2793 71.4979i 0 −557.845 322.072i 0 −3078.62 5556.59i 0 6433.53 11143.2i 0
47.6 0 −29.1853 50.5504i 0 2152.25 + 1242.60i 0 2216.90 + 5953.06i 0 8137.94 14095.3i 0
47.7 0 29.1853 + 50.5504i 0 2152.25 + 1242.60i 0 −2216.90 5953.06i 0 8137.94 14095.3i 0
47.8 0 41.2793 + 71.4979i 0 −557.845 322.072i 0 3078.62 + 5556.59i 0 6433.53 11143.2i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.10.p.b 24
4.b odd 2 1 inner 112.10.p.b 24
7.d odd 6 1 inner 112.10.p.b 24
28.f even 6 1 inner 112.10.p.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.10.p.b 24 1.a even 1 1 trivial
112.10.p.b 24 4.b odd 2 1 inner
112.10.p.b 24 7.d odd 6 1 inner
112.10.p.b 24 28.f even 6 1 inner