↑

Properties

Label 1998.2.q.b
Level $1998$
Weight $2$
Character orbit 1998.q
Analytic conductor $15.954$
Analytic rank $0$
Dimension $72$
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] = [1998,2,Mod(73,1998)] 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("1998.73"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1998, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1998 = 2 \cdot 3^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1998.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] 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: \(15.9541103239\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 666)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 36 q^{4} + 4 q^{7} + 8 q^{10} + 10 q^{11} - 36 q^{16} + 52 q^{25} + 4 q^{26} + 8 q^{28} + 12 q^{34} - 32 q^{37} - 16 q^{38} + 4 q^{40} - 44 q^{41} + 20 q^{44} + 4 q^{46} - 60 q^{47} - 56 q^{49} + 72 q^{53}+ \cdots - 36 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −3.53299 + 2.03977i 0 0.189560 − 0.328327i − 1.00000i 0 4.07955
73.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −3.56506 + 2.05829i 0 −1.11612 + 1.93317i − 1.00000i 0 4.11657
73.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.61658 + 1.51068i 0 1.88563 − 3.26601i − 1.00000i 0 3.02137
73.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.11882 + 1.22330i 0 −0.0163493 + 0.0283178i − 1.00000i 0 2.44661
73.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.91018 + 1.10284i 0 2.33806 − 4.04964i − 1.00000i 0 2.20569
73.6 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.73582 + 1.00218i 0 −1.84656 + 3.19833i − 1.00000i 0 2.00435
73.7 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.26082 + 0.727936i 0 −0.555043 + 0.961363i − 1.00000i 0 1.45587
73.8 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.893958 − 0.516127i 0 −1.34612 + 2.33155i − 1.00000i 0 −1.03225
73.9 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.0135176 − 0.00780437i 0 0.114323 − 0.198014i − 1.00000i 0 −0.0156087
73.10 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.313745 − 0.181141i 0 1.42679 − 2.47128i − 1.00000i 0 −0.362282
73.11 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.340064 + 0.196336i 0 −2.19539 + 3.80253i − 1.00000i 0 0.392672
73.12 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.622305 + 0.359288i 0 0.930527 − 1.61172i − 1.00000i 0 0.718576
73.13 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.06178 − 0.613020i 0 −0.577625 + 1.00048i − 1.00000i 0 −1.22604
73.14 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.79041 − 1.03369i 0 2.35442 − 4.07797i − 1.00000i 0 −2.06738
73.15 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.09692 − 1.21066i 0 −0.633629 + 1.09748i − 1.00000i 0 −2.42132
73.16 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.09623 − 1.78761i 0 0.750004 − 1.29904i − 1.00000i 0 −3.57521
73.17 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.22658 − 1.86286i 0 −2.33164 + 4.03852i − 1.00000i 0 −3.72573
73.18 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.47746 − 2.00771i 0 1.62916 − 2.82178i − 1.00000i 0 −4.01543
73.19 0.866025 + 0.500000i 0 0.500000 + 0.866025i −3.47746 + 2.00771i 0 1.62916 − 2.82178i 1.00000i 0 −4.01543
73.20 0.866025 + 0.500000i 0 0.500000 + 0.866025i −3.22658 + 1.86286i 0 −2.33164 + 4.03852i 1.00000i 0 −3.72573
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
37.b even 2 1 inner
333.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1998.2.q.b 72
3.b odd 2 1 666.2.q.b ✓ 72
9.c even 3 1 inner 1998.2.q.b 72
9.d odd 6 1 666.2.q.b ✓ 72
37.b even 2 1 inner 1998.2.q.b 72
111.d odd 2 1 666.2.q.b ✓ 72
333.n odd 6 1 666.2.q.b ✓ 72
333.q even 6 1 inner 1998.2.q.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.q.b ✓ 72 3.b odd 2 1
666.2.q.b ✓ 72 9.d odd 6 1
666.2.q.b ✓ 72 111.d odd 2 1
666.2.q.b ✓ 72 333.n odd 6 1
1998.2.q.b 72 1.a even 1 1 trivial
1998.2.q.b 72 9.c even 3 1 inner
1998.2.q.b 72 37.b even 2 1 inner
1998.2.q.b 72 333.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 116 T_{5}^{70} + 7428 T_{5}^{68} - 328096 T_{5}^{66} + 11036840 T_{5}^{64} + \cdots + 34828517376 \) acting on \(S_{2}^{\mathrm{new}}(1998, [\chi])\). Copy content Toggle raw display