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Properties

Label 90.11.h.a
Level $90$
Weight $11$
Character orbit 90.h
Analytic conductor $57.182$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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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] = [90,11,Mod(11,90)] 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("90.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 44 q^{3} + 20480 q^{4} - 13568 q^{6} - 24476 q^{7} - 103396 q^{9} + 1944 q^{11} + 45056 q^{12} + 561100 q^{13} - 175680 q^{14} - 687500 q^{15} - 10485760 q^{16} - 3888896 q^{18} + 5932480 q^{19}+ \cdots - 78067269744 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −19.5959 − 11.3137i −242.976 + 3.39723i 256.000 + 443.405i −1210.31 + 698.771i 4799.78 + 2682.39i 8841.03 − 15313.1i − 11585.2i 59025.9 − 1650.89i 31622.8
11.2 −19.5959 − 11.3137i −242.497 − 15.6231i 256.000 + 443.405i 1210.31 − 698.771i 4575.20 + 3049.69i −2506.88 + 4342.04i − 11585.2i 58560.8 + 7577.13i −31622.8
11.3 −19.5959 − 11.3137i −208.723 + 124.434i 256.000 + 443.405i −1210.31 + 698.771i 5497.92 − 76.9600i 4668.95 − 8086.86i − 11585.2i 28081.6 − 51944.3i 31622.8
11.4 −19.5959 − 11.3137i −192.466 − 148.344i 256.000 + 443.405i −1210.31 + 698.771i 2093.22 + 5084.44i −12296.7 + 21298.5i − 11585.2i 15037.0 + 57102.3i 31622.8
11.5 −19.5959 − 11.3137i −164.534 + 178.823i 256.000 + 443.405i 1210.31 − 698.771i 5247.34 − 1642.72i −3815.60 + 6608.81i − 11585.2i −4906.41 − 58844.8i −31622.8
11.6 −19.5959 − 11.3137i −156.662 − 185.758i 256.000 + 443.405i 1210.31 − 698.771i 968.329 + 5412.53i −10611.8 + 18380.2i − 11585.2i −9962.94 + 58202.4i −31622.8
11.7 −19.5959 − 11.3137i −126.394 − 207.542i 256.000 + 443.405i −1210.31 + 698.771i 128.746 + 5496.95i 6493.94 − 11247.8i − 11585.2i −27098.0 + 52464.1i 31622.8
11.8 −19.5959 − 11.3137i −113.066 + 215.093i 256.000 + 443.405i 1210.31 − 698.771i 4649.14 − 2935.74i 12618.1 − 21855.1i − 11585.2i −33481.0 − 48639.6i −31622.8
11.9 −19.5959 − 11.3137i −57.9788 − 235.982i 256.000 + 443.405i 1210.31 − 698.771i −1533.68 + 5280.24i 4212.63 − 7296.48i − 11585.2i −52325.9 + 27363.9i −31622.8
11.10 −19.5959 − 11.3137i −27.3656 + 241.454i 256.000 + 443.405i −1210.31 + 698.771i 3268.00 − 4421.91i −3628.74 + 6285.17i − 11585.2i −57551.2 − 13215.1i 31622.8
11.11 −19.5959 − 11.3137i 60.9172 − 235.241i 256.000 + 443.405i −1210.31 + 698.771i −3855.17 + 3920.55i −7265.28 + 12583.8i − 11585.2i −51627.2 − 28660.4i 31622.8
11.12 −19.5959 − 11.3137i 73.1613 + 231.725i 256.000 + 443.405i −1210.31 + 698.771i 1188.00 − 5368.59i 9389.73 − 16263.5i − 11585.2i −48343.8 + 33906.6i 31622.8
11.13 −19.5959 − 11.3137i 105.689 + 218.812i 256.000 + 443.405i 1210.31 − 698.771i 404.512 − 5483.56i −8869.13 + 15361.8i − 11585.2i −36708.8 + 46252.0i −31622.8
11.14 −19.5959 − 11.3137i 157.398 − 185.135i 256.000 + 443.405i 1210.31 − 698.771i −5178.91 + 1847.14i 9074.69 − 15717.8i − 11585.2i −9501.01 − 58279.6i −31622.8
11.15 −19.5959 − 11.3137i 169.897 − 173.735i 256.000 + 443.405i 1210.31 − 698.771i −5294.88 + 1482.33i −13757.7 + 23829.0i − 11585.2i −1318.83 − 59034.3i −31622.8
11.16 −19.5959 − 11.3137i 213.450 + 116.138i 256.000 + 443.405i −1210.31 + 698.771i −2868.79 − 4690.75i −12940.6 + 22413.8i − 11585.2i 32072.8 + 49579.4i 31622.8
11.17 −19.5959 − 11.3137i 214.506 + 114.176i 256.000 + 443.405i 1210.31 − 698.771i −2911.68 − 4664.24i 15545.1 − 26924.9i − 11585.2i 32976.6 + 48982.9i −31622.8
11.18 −19.5959 − 11.3137i 216.480 − 110.387i 256.000 + 443.405i −1210.31 + 698.771i −5491.02 − 286.069i 7828.11 − 13558.7i − 11585.2i 34678.6 − 47793.1i 31622.8
11.19 −19.5959 − 11.3137i 235.959 − 58.0713i 256.000 + 443.405i −1210.31 + 698.771i −5280.84 − 1531.61i −2866.21 + 4964.43i − 11585.2i 52304.5 − 27404.9i 31622.8
11.20 −19.5959 − 11.3137i 239.500 + 41.0931i 256.000 + 443.405i 1210.31 − 698.771i −4228.31 − 3514.89i −5111.91 + 8854.09i − 11585.2i 55671.7 + 19683.6i −31622.8
See all 80 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.40
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.11.h.a ✓ 80
3.b odd 2 1 270.11.h.a 80
9.c even 3 1 270.11.h.a 80
9.d odd 6 1 inner 90.11.h.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.11.h.a ✓ 80 1.a even 1 1 trivial
90.11.h.a ✓ 80 9.d odd 6 1 inner
270.11.h.a 80 3.b odd 2 1
270.11.h.a 80 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(90, [\chi])\).