↑

Properties

Label 990.2.t.a
Level $990$
Weight $2$
Character orbit 990.t
Analytic conductor $7.905$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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] = [990,2,Mod(131,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("990.131"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{2} - 2 q^{3} - 24 q^{4} - 2 q^{6} + 48 q^{8} + 10 q^{9} - 6 q^{11} + 4 q^{12} + 24 q^{13} - 4 q^{15} - 24 q^{16} - 12 q^{17} - 8 q^{18} - 16 q^{21} + 12 q^{22} + 36 q^{23} - 2 q^{24} + 24 q^{25}+ \cdots - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −0.500000 − 0.866025i −1.72761 + 0.123910i −0.500000 + 0.866025i 0.866025 + 0.500000i 0.971116 + 1.43420i 4.54797 − 2.62577i 1.00000 2.96929 − 0.428137i − 1.00000i
131.2 −0.500000 − 0.866025i −1.69057 + 0.376790i −0.500000 + 0.866025i −0.866025 − 0.500000i 1.17159 + 1.27568i −0.803327 + 0.463801i 1.00000 2.71606 − 1.27398i 1.00000i
131.3 −0.500000 − 0.866025i −1.68610 − 0.396321i −0.500000 + 0.866025i 0.866025 + 0.500000i 0.499825 + 1.65837i 0.0653962 − 0.0377565i 1.00000 2.68586 + 1.33647i − 1.00000i
131.4 −0.500000 − 0.866025i −1.62493 − 0.599680i −0.500000 + 0.866025i −0.866025 − 0.500000i 0.293125 + 1.70707i −3.18423 + 1.83842i 1.00000 2.28077 + 1.94887i 1.00000i
131.5 −0.500000 − 0.866025i −1.60551 + 0.649887i −0.500000 + 0.866025i −0.866025 − 0.500000i 1.36557 + 1.06546i 3.00605 − 1.73554i 1.00000 2.15529 − 2.08679i 1.00000i
131.6 −0.500000 − 0.866025i −1.28087 + 1.16592i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.65015 + 0.526309i −0.700188 + 0.404254i 1.00000 0.281268 − 2.98679i − 1.00000i
131.7 −0.500000 − 0.866025i −1.21393 − 1.23547i −0.500000 + 0.866025i 0.866025 + 0.500000i −0.462983 + 1.66903i −1.10421 + 0.637517i 1.00000 −0.0527607 + 2.99954i − 1.00000i
131.8 −0.500000 − 0.866025i −1.19434 + 1.25442i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.68352 + 0.407116i −1.93804 + 1.11893i 1.00000 −0.147126 − 2.99639i − 1.00000i
131.9 −0.500000 − 0.866025i −1.02646 − 1.39512i −0.500000 + 0.866025i −0.866025 − 0.500000i −0.694980 + 1.58651i −1.11114 + 0.641516i 1.00000 −0.892741 + 2.86409i 1.00000i
131.10 −0.500000 − 0.866025i −0.805671 + 1.53326i −0.500000 + 0.866025i −0.866025 − 0.500000i 1.73068 − 0.0689002i −1.00951 + 0.582842i 1.00000 −1.70179 − 2.47061i 1.00000i
131.11 −0.500000 − 0.866025i −0.234765 − 1.71607i −0.500000 + 0.866025i −0.866025 − 0.500000i −1.36878 + 1.06135i 3.20594 − 1.85095i 1.00000 −2.88977 + 0.805744i 1.00000i
131.12 −0.500000 − 0.866025i −0.152357 − 1.72534i −0.500000 + 0.866025i −0.866025 − 0.500000i −1.41801 + 0.994614i 0.0947420 − 0.0546993i 1.00000 −2.95357 + 0.525735i 1.00000i
131.13 −0.500000 − 0.866025i 0.0978520 − 1.72928i −0.500000 + 0.866025i 0.866025 + 0.500000i −1.54653 + 0.779900i 1.01491 − 0.585956i 1.00000 −2.98085 − 0.338428i − 1.00000i
131.14 −0.500000 − 0.866025i 0.216395 + 1.71848i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.38005 − 1.04664i 3.44565 − 1.98935i 1.00000 −2.90635 + 0.743740i − 1.00000i
131.15 −0.500000 − 0.866025i 0.713195 + 1.57840i −0.500000 + 0.866025i 0.866025 + 0.500000i 1.01034 − 1.40685i −3.86969 + 2.23417i 1.00000 −1.98271 + 2.25142i − 1.00000i
131.16 −0.500000 − 0.866025i 0.761656 − 1.55560i −0.500000 + 0.866025i 0.866025 + 0.500000i −1.72801 + 0.118185i −2.83014 + 1.63398i 1.00000 −1.83976 − 2.36966i − 1.00000i
131.17 −0.500000 − 0.866025i 0.795729 + 1.53845i −0.500000 + 0.866025i −0.866025 − 0.500000i 0.934468 − 1.45834i 1.95421 − 1.12826i 1.00000 −1.73363 + 2.44837i 1.00000i
131.18 −0.500000 − 0.866025i 1.07742 − 1.35616i −0.500000 + 0.866025i −0.866025 − 0.500000i −1.71318 − 0.254992i 2.99719 − 1.73043i 1.00000 −0.678339 − 2.92230i 1.00000i
131.19 −0.500000 − 0.866025i 1.50274 + 0.861268i −0.500000 + 0.866025i −0.866025 − 0.500000i −0.00548788 − 1.73204i −1.26719 + 0.731611i 1.00000 1.51643 + 2.58852i 1.00000i
131.20 −0.500000 − 0.866025i 1.51330 − 0.842574i −0.500000 + 0.866025i 0.866025 + 0.500000i −1.48634 − 0.889267i 2.62958 − 1.51819i 1.00000 1.58014 − 2.55013i − 1.00000i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.t.a ✓ 48
3.b odd 2 1 2970.2.t.b 48
9.c even 3 1 2970.2.t.a 48
9.d odd 6 1 990.2.t.b yes 48
11.b odd 2 1 990.2.t.b yes 48
33.d even 2 1 2970.2.t.a 48
99.g even 6 1 inner 990.2.t.a ✓ 48
99.h odd 6 1 2970.2.t.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.t.a ✓ 48 1.a even 1 1 trivial
990.2.t.a ✓ 48 99.g even 6 1 inner
990.2.t.b yes 48 9.d odd 6 1
990.2.t.b yes 48 11.b odd 2 1
2970.2.t.a 48 9.c even 3 1
2970.2.t.a 48 33.d even 2 1
2970.2.t.b 48 3.b odd 2 1
2970.2.t.b 48 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} - 96 T_{7}^{46} + 5316 T_{7}^{44} + 870 T_{7}^{43} - 198414 T_{7}^{42} - 65058 T_{7}^{41} + \cdots + 406829660224 \) acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\). Copy content Toggle raw display