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Properties

Label 112.10.p.a
Level $112$
Weight $10$
Character orbit 112.p
Analytic conductor $57.684$
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] = [112,10,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [24,0,-162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −122.918 + 212.900i 0 −1692.79 + 977.330i 0 −3573.64 + 5251.92i 0 −20376.0 − 35292.2i 0
31.2 0 −116.983 + 202.620i 0 2050.66 − 1183.95i 0 4080.46 + 4868.62i 0 −17528.4 − 30360.1i 0
31.3 0 −80.9515 + 140.212i 0 −916.446 + 529.111i 0 4644.00 − 4334.38i 0 −3264.80 − 5654.80i 0
31.4 0 −79.5164 + 137.726i 0 85.2371 − 49.2117i 0 −6144.84 − 1610.75i 0 −2804.21 − 4857.03i 0
31.5 0 −54.9814 + 95.2305i 0 1115.20 − 643.863i 0 1141.43 − 6249.06i 0 3795.60 + 6574.17i 0
31.6 0 −5.53558 + 9.58790i 0 654.896 − 378.104i 0 680.140 + 6315.93i 0 9780.21 + 16939.8i 0
31.7 0 12.6907 − 21.9809i 0 −1449.89 + 837.095i 0 5644.15 + 2914.99i 0 9519.39 + 16488.1i 0
31.8 0 22.7708 − 39.4402i 0 −1500.67 + 866.412i 0 −6349.75 − 185.284i 0 8804.48 + 15249.8i 0
31.9 0 44.5044 − 77.0839i 0 1678.45 − 969.052i 0 −6146.89 − 1602.93i 0 5880.22 + 10184.8i 0
31.10 0 85.3530 − 147.836i 0 −1415.83 + 817.429i 0 −796.914 − 6302.26i 0 −4728.77 − 8190.47i 0
31.11 0 86.8868 − 150.492i 0 1206.43 − 696.532i 0 5663.78 − 2876.67i 0 −5257.14 − 9105.63i 0
31.12 0 127.679 − 221.147i 0 −241.245 + 139.283i 0 −2213.93 + 5954.17i 0 −22762.6 − 39426.0i 0
47.1 0 −122.918 − 212.900i 0 −1692.79 − 977.330i 0 −3573.64 − 5251.92i 0 −20376.0 + 35292.2i 0
47.2 0 −116.983 − 202.620i 0 2050.66 + 1183.95i 0 4080.46 − 4868.62i 0 −17528.4 + 30360.1i 0
47.3 0 −80.9515 − 140.212i 0 −916.446 − 529.111i 0 4644.00 + 4334.38i 0 −3264.80 + 5654.80i 0
47.4 0 −79.5164 − 137.726i 0 85.2371 + 49.2117i 0 −6144.84 + 1610.75i 0 −2804.21 + 4857.03i 0
47.5 0 −54.9814 − 95.2305i 0 1115.20 + 643.863i 0 1141.43 + 6249.06i 0 3795.60 − 6574.17i 0
47.6 0 −5.53558 − 9.58790i 0 654.896 + 378.104i 0 680.140 − 6315.93i 0 9780.21 − 16939.8i 0
47.7 0 12.6907 + 21.9809i 0 −1449.89 − 837.095i 0 5644.15 − 2914.99i 0 9519.39 − 16488.1i 0
47.8 0 22.7708 + 39.4402i 0 −1500.67 − 866.412i 0 −6349.75 + 185.284i 0 8804.48 − 15249.8i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
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Inner twists

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