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Properties

Label 990.2.bh.d
Level $990$
Weight $2$
Character orbit 990.bh
Analytic conductor $7.905$
Analytic rank $0$
Dimension $96$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(73,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 15, 14])) 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("990.73"); 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.bh (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 40 q^{7} + 24 q^{16} + 4 q^{22} + 8 q^{25} + 20 q^{28} - 16 q^{31} + 64 q^{37} - 40 q^{46} - 40 q^{52} - 36 q^{55} - 12 q^{58} - 80 q^{61} - 48 q^{67} - 52 q^{70} + 20 q^{73} + 48 q^{82} - 160 q^{85}+ \cdots + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i −2.20829 − 0.351353i 0 −0.320682 + 0.629375i 0.453990 + 0.891007i 0 −0.00157399 + 2.23607i
73.2 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i −1.30314 + 1.81709i 0 −0.794006 + 1.55833i 0.453990 + 0.891007i 0 1.99858 + 1.00284i
73.3 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i −1.24280 − 1.85888i 0 0.999017 − 1.96068i 0.453990 + 0.891007i 0 −1.64158 + 1.51830i
73.4 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i 0.235568 + 2.22362i 0 −0.307021 + 0.602562i 0.453990 + 0.891007i 0 2.15940 − 0.580519i
73.5 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i 0.405031 − 2.19908i 0 −2.37763 + 4.66637i 0.453990 + 0.891007i 0 −2.23537 − 0.0560326i
73.6 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i 2.23494 + 0.0710376i 0 1.11262 − 2.18365i 0.453990 + 0.891007i 0 −0.279459 − 2.21854i
73.7 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −2.23494 − 0.0710376i 0 1.11262 − 2.18365i −0.453990 − 0.891007i 0 −0.279459 − 2.21854i
73.8 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −0.405031 + 2.19908i 0 −2.37763 + 4.66637i −0.453990 − 0.891007i 0 −2.23537 − 0.0560326i
73.9 0.156434 + 0.987688i 0 −0.951057 + 0.309017i −0.235568 − 2.22362i 0 −0.307021 + 0.602562i −0.453990 − 0.891007i 0 2.15940 − 0.580519i
73.10 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 1.24280 + 1.85888i 0 0.999017 − 1.96068i −0.453990 − 0.891007i 0 −1.64158 + 1.51830i
73.11 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 1.30314 − 1.81709i 0 −0.794006 + 1.55833i −0.453990 − 0.891007i 0 1.99858 + 1.00284i
73.12 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 2.20829 + 0.351353i 0 −0.320682 + 0.629375i −0.453990 − 0.891007i 0 −0.00157399 + 2.23607i
127.1 −0.891007 − 0.453990i 0 0.587785 + 0.809017i −2.20499 − 0.371538i 0 −0.564879 + 3.56651i −0.156434 − 0.987688i 0 1.79598 + 1.33209i
127.2 −0.891007 − 0.453990i 0 0.587785 + 0.809017i −1.88505 + 1.20273i 0 0.433371 − 2.73620i −0.156434 − 0.987688i 0 2.22563 − 0.215847i
127.3 −0.891007 − 0.453990i 0 0.587785 + 0.809017i −0.169731 − 2.22962i 0 0.743917 − 4.69691i −0.156434 − 0.987688i 0 −0.860994 + 2.06366i
127.4 −0.891007 − 0.453990i 0 0.587785 + 0.809017i 0.320687 − 2.21295i 0 −0.171071 + 1.08010i −0.156434 − 0.987688i 0 −1.29039 + 1.82617i
127.5 −0.891007 − 0.453990i 0 0.587785 + 0.809017i 1.29590 + 1.82226i 0 0.522655 − 3.29991i −0.156434 − 0.987688i 0 −0.327367 − 2.21197i
127.6 −0.891007 − 0.453990i 0 0.587785 + 0.809017i 2.10948 + 0.741671i 0 −0.0230381 + 0.145457i −0.156434 − 0.987688i 0 −1.54285 − 1.61852i
127.7 0.891007 + 0.453990i 0 0.587785 + 0.809017i −2.10948 − 0.741671i 0 −0.0230381 + 0.145457i 0.156434 + 0.987688i 0 −1.54285 − 1.61852i
127.8 0.891007 + 0.453990i 0 0.587785 + 0.809017i −1.29590 − 1.82226i 0 0.522655 − 3.29991i 0.156434 + 0.987688i 0 −0.327367 − 2.21197i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.12
Significant digits:
Format:

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
11.d odd 10 1 inner
15.e even 4 1 inner
33.f even 10 1 inner
55.l even 20 1 inner
165.u odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.bh.d ✓ 96
3.b odd 2 1 inner 990.2.bh.d ✓ 96
5.c odd 4 1 inner 990.2.bh.d ✓ 96
11.d odd 10 1 inner 990.2.bh.d ✓ 96
15.e even 4 1 inner 990.2.bh.d ✓ 96
33.f even 10 1 inner 990.2.bh.d ✓ 96
55.l even 20 1 inner 990.2.bh.d ✓ 96
165.u odd 20 1 inner 990.2.bh.d ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.bh.d ✓ 96 1.a even 1 1 trivial
990.2.bh.d ✓ 96 3.b odd 2 1 inner
990.2.bh.d ✓ 96 5.c odd 4 1 inner
990.2.bh.d ✓ 96 11.d odd 10 1 inner
990.2.bh.d ✓ 96 15.e even 4 1 inner
990.2.bh.d ✓ 96 33.f even 10 1 inner
990.2.bh.d ✓ 96 55.l even 20 1 inner
990.2.bh.d ✓ 96 165.u odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{48} + 20 T_{7}^{47} + 200 T_{7}^{46} + 1370 T_{7}^{45} + 6953 T_{7}^{44} + 25060 T_{7}^{43} + \cdots + 133090713856 \) acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\). Copy content Toggle raw display