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Properties

Label 90.12.f.b
Level $90$
Weight $12$
Character orbit 90.f
Analytic conductor $69.151$
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] = [90,12,Mod(17,90)] 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("90.17"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 90.f (of order \(4\), 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: \(69.1508862504\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 47784 q^{7} - 325248 q^{10} - 788856 q^{13} - 25165824 q^{16} + 60462336 q^{22} - 116858016 q^{25} - 48930816 q^{28} + 57472416 q^{31} + 509627616 q^{37} - 786038784 q^{40} + 1088562480 q^{43} - 2939088384 q^{46}+ \cdots - 118418994864 q^{97}+O(q^{100}) \) 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 −22.6274 + 22.6274i 0 − 1024.00i −6464.88 + 2652.06i 0 −59429.2 − 59429.2i 23170.5 + 23170.5i 0 86274.4 − 206293.i
17.2 −22.6274 + 22.6274i 0 − 1024.00i −5092.26 − 4785.08i 0 22542.8 + 22542.8i 23170.5 + 23170.5i 0 223499. − 6950.70i
17.3 −22.6274 + 22.6274i 0 − 1024.00i −2475.04 + 6534.70i 0 45997.8 + 45997.8i 23170.5 + 23170.5i 0 −91859.5 − 203867.i
17.4 −22.6274 + 22.6274i 0 − 1024.00i 2179.32 + 6639.18i 0 −35409.1 − 35409.1i 23170.5 + 23170.5i 0 −199540. − 100915.i
17.5 −22.6274 + 22.6274i 0 − 1024.00i 2582.01 − 6493.18i 0 −13902.0 − 13902.0i 23170.5 + 23170.5i 0 88499.6 + 205348.i
17.6 −22.6274 + 22.6274i 0 − 1024.00i 6827.09 + 1489.60i 0 28253.7 + 28253.7i 23170.5 + 23170.5i 0 −188185. + 120774.i
17.7 22.6274 − 22.6274i 0 − 1024.00i −6827.09 − 1489.60i 0 28253.7 + 28253.7i −23170.5 − 23170.5i 0 −188185. + 120774.i
17.8 22.6274 − 22.6274i 0 − 1024.00i −2582.01 + 6493.18i 0 −13902.0 − 13902.0i −23170.5 − 23170.5i 0 88499.6 + 205348.i
17.9 22.6274 − 22.6274i 0 − 1024.00i −2179.32 − 6639.18i 0 −35409.1 − 35409.1i −23170.5 − 23170.5i 0 −199540. − 100915.i
17.10 22.6274 − 22.6274i 0 − 1024.00i 2475.04 − 6534.70i 0 45997.8 + 45997.8i −23170.5 − 23170.5i 0 −91859.5 − 203867.i
17.11 22.6274 − 22.6274i 0 − 1024.00i 5092.26 + 4785.08i 0 22542.8 + 22542.8i −23170.5 − 23170.5i 0 223499. − 6950.70i
17.12 22.6274 − 22.6274i 0 − 1024.00i 6464.88 − 2652.06i 0 −59429.2 − 59429.2i −23170.5 − 23170.5i 0 86274.4 − 206293.i
53.1 −22.6274 − 22.6274i 0 1024.00i −6464.88 − 2652.06i 0 −59429.2 + 59429.2i 23170.5 − 23170.5i 0 86274.4 + 206293.i
53.2 −22.6274 − 22.6274i 0 1024.00i −5092.26 + 4785.08i 0 22542.8 − 22542.8i 23170.5 − 23170.5i 0 223499. + 6950.70i
53.3 −22.6274 − 22.6274i 0 1024.00i −2475.04 − 6534.70i 0 45997.8 − 45997.8i 23170.5 − 23170.5i 0 −91859.5 + 203867.i
53.4 −22.6274 − 22.6274i 0 1024.00i 2179.32 − 6639.18i 0 −35409.1 + 35409.1i 23170.5 − 23170.5i 0 −199540. + 100915.i
53.5 −22.6274 − 22.6274i 0 1024.00i 2582.01 + 6493.18i 0 −13902.0 + 13902.0i 23170.5 − 23170.5i 0 88499.6 − 205348.i
53.6 −22.6274 − 22.6274i 0 1024.00i 6827.09 − 1489.60i 0 28253.7 − 28253.7i 23170.5 − 23170.5i 0 −188185. − 120774.i
53.7 22.6274 + 22.6274i 0 1024.00i −6827.09 + 1489.60i 0 28253.7 − 28253.7i −23170.5 + 23170.5i 0 −188185. − 120774.i
53.8 22.6274 + 22.6274i 0 1024.00i −2582.01 − 6493.18i 0 −13902.0 + 13902.0i −23170.5 + 23170.5i 0 88499.6 − 205348.i
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
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.12.f.b ✓ 24
3.b odd 2 1 inner 90.12.f.b ✓ 24
5.c odd 4 1 inner 90.12.f.b ✓ 24
15.e even 4 1 inner 90.12.f.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.12.f.b ✓ 24 1.a even 1 1 trivial
90.12.f.b ✓ 24 3.b odd 2 1 inner
90.12.f.b ✓ 24 5.c odd 4 1 inner
90.12.f.b ✓ 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 23892 T_{7}^{11} + 285413832 T_{7}^{10} - 165253563545120 T_{7}^{9} + \cdots + 47\!\cdots\!84 \) acting on \(S_{12}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display