↑

Properties

Label 416.3.bb.c
Level $416$
Weight $3$
Character orbit 416.bb
Analytic conductor $11.335$
Analytic rank $0$
Dimension $32$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [416,3,Mod(159,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 416.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,-8,0,0,0,80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3351789974\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{5} + 80 q^{9} - 44 q^{13} + 16 q^{17} - 128 q^{21} + 272 q^{25} - 52 q^{29} - 72 q^{33} + 148 q^{37} + 72 q^{41} - 116 q^{45} + 328 q^{49} - 152 q^{53} - 224 q^{57} + 228 q^{61} - 352 q^{65}+ \cdots - 352 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1 0 −5.02203 + 2.89947i 0 −4.84252 0 11.4418 + 6.60592i 0 12.3138 − 21.3282i 0
159.2 0 −4.36279 + 2.51886i 0 8.41772 0 7.34082 + 4.23822i 0 8.18928 − 14.1842i 0
159.3 0 −4.34136 + 2.50649i 0 −7.18265 0 −5.71633 − 3.30032i 0 8.06494 − 13.9689i 0
159.4 0 −3.45657 + 1.99565i 0 6.36955 0 −5.32276 − 3.07310i 0 3.46526 − 6.00201i 0
159.5 0 −2.08252 + 1.20234i 0 −5.99796 0 −10.7200 − 6.18919i 0 −1.60873 + 2.78641i 0
159.6 0 −2.02906 + 1.17148i 0 −2.43677 0 0.984748 + 0.568544i 0 −1.75527 + 3.04022i 0
159.7 0 −0.576566 + 0.332881i 0 −2.21788 0 1.70346 + 0.983494i 0 −4.27838 + 7.41037i 0
159.8 0 −0.404479 + 0.233526i 0 5.89050 0 −7.23877 − 4.17931i 0 −4.39093 + 7.60532i 0
159.9 0 0.404479 − 0.233526i 0 5.89050 0 7.23877 + 4.17931i 0 −4.39093 + 7.60532i 0
159.10 0 0.576566 − 0.332881i 0 −2.21788 0 −1.70346 − 0.983494i 0 −4.27838 + 7.41037i 0
159.11 0 2.02906 − 1.17148i 0 −2.43677 0 −0.984748 − 0.568544i 0 −1.75527 + 3.04022i 0
159.12 0 2.08252 − 1.20234i 0 −5.99796 0 10.7200 + 6.18919i 0 −1.60873 + 2.78641i 0
159.13 0 3.45657 − 1.99565i 0 6.36955 0 5.32276 + 3.07310i 0 3.46526 − 6.00201i 0
159.14 0 4.34136 − 2.50649i 0 −7.18265 0 5.71633 + 3.30032i 0 8.06494 − 13.9689i 0
159.15 0 4.36279 − 2.51886i 0 8.41772 0 −7.34082 − 4.23822i 0 8.18928 − 14.1842i 0
159.16 0 5.02203 − 2.89947i 0 −4.84252 0 −11.4418 − 6.60592i 0 12.3138 − 21.3282i 0
191.1 0 −5.02203 − 2.89947i 0 −4.84252 0 11.4418 − 6.60592i 0 12.3138 + 21.3282i 0
191.2 0 −4.36279 − 2.51886i 0 8.41772 0 7.34082 − 4.23822i 0 8.18928 + 14.1842i 0
191.3 0 −4.34136 − 2.50649i 0 −7.18265 0 −5.71633 + 3.30032i 0 8.06494 + 13.9689i 0
191.4 0 −3.45657 − 1.99565i 0 6.36955 0 −5.32276 + 3.07310i 0 3.46526 + 6.00201i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.3.bb.c ✓ 32
4.b odd 2 1 inner 416.3.bb.c ✓ 32
13.c even 3 1 inner 416.3.bb.c ✓ 32
52.j odd 6 1 inner 416.3.bb.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.3.bb.c ✓ 32 1.a even 1 1 trivial
416.3.bb.c ✓ 32 4.b odd 2 1 inner
416.3.bb.c ✓ 32 13.c even 3 1 inner
416.3.bb.c ✓ 32 52.j odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 112 T_{3}^{30} + 7634 T_{3}^{28} - 336936 T_{3}^{26} + 10992603 T_{3}^{24} + \cdots + 1099511627776 \) acting on \(S_{3}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display