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Properties

Label 90.11.j.a
Level $90$
Weight $11$
Character orbit 90.j
Analytic conductor $57.182$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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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(29,90)] 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("90.29"); 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, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.j (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: \(120\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 30720 q^{4} + 9918 q^{5} + 12160 q^{6} - 172900 q^{9} - 653220 q^{11} - 175680 q^{14} + 1781914 q^{15} - 15728640 q^{16} - 5078016 q^{20} - 18127640 q^{21} - 3112960 q^{24} - 9921306 q^{25} - 10065420 q^{29}+ \cdots - 43749358340 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −11.3137 + 19.5959i −242.369 − 17.5048i −256.000 − 443.405i −125.593 + 3122.48i 3085.11 − 4551.39i 9255.10 + 5343.44i 11585.2 58436.2 + 8485.24i −59766.8 − 37787.9i
29.2 −11.3137 + 19.5959i −239.901 − 38.6843i −256.000 − 443.405i −2664.37 − 1633.02i 3472.23 − 4263.42i −181.476 − 104.775i 11585.2 56056.0 + 18560.8i 62144.4 − 33735.3i
29.3 −11.3137 + 19.5959i −232.077 − 72.0370i −256.000 − 443.405i 3089.59 − 469.107i 4037.28 − 3732.75i −25178.8 − 14537.0i 11585.2 48670.3 + 33436.2i −25762.1 + 65850.7i
29.4 −11.3137 + 19.5959i −221.671 + 99.5537i −256.000 − 443.405i −3061.09 − 628.750i 557.075 − 5470.17i 16087.1 + 9287.87i 11585.2 39227.1 − 44136.4i 46953.3 − 52871.5i
29.5 −11.3137 + 19.5959i −217.418 − 108.528i −256.000 − 443.405i 3042.13 − 714.871i 4586.51 − 3032.65i 15642.1 + 9030.96i 11585.2 35492.3 + 47191.9i −20409.3 + 67701.3i
29.6 −11.3137 + 19.5959i −213.769 + 115.550i −256.000 − 443.405i −51.6544 − 3124.57i 154.200 − 5496.30i −13596.7 − 7850.04i 11585.2 32345.2 − 49402.2i 61813.3 + 34338.3i
29.7 −11.3137 + 19.5959i −199.236 − 139.118i −256.000 − 443.405i −2955.06 + 1016.48i 4980.25 − 2330.28i −23229.0 − 13411.3i 11585.2 20341.2 + 55434.8i 13513.9 − 69407.3i
29.8 −11.3137 + 19.5959i −179.081 + 164.253i −256.000 − 443.405i 2513.23 + 1857.23i −1192.61 − 5367.57i 16219.2 + 9364.16i 11585.2 5091.07 − 58829.1i −64828.0 + 28236.9i
29.9 −11.3137 + 19.5959i −146.334 + 193.998i −256.000 − 443.405i 2975.63 + 954.586i −2145.99 − 5062.39i −6562.41 − 3788.81i 11585.2 −16221.5 − 56777.2i −52371.4 + 47510.4i
29.10 −11.3137 + 19.5959i −141.377 − 197.640i −256.000 − 443.405i 407.746 − 3098.28i 5472.43 − 534.363i 9861.77 + 5693.69i 11585.2 −19074.3 + 55883.4i 56100.6 + 43043.2i
29.11 −11.3137 + 19.5959i −105.385 − 218.959i −256.000 − 443.405i 847.202 + 3007.97i 5483.00 + 412.131i −8445.46 − 4875.99i 11585.2 −36837.2 + 46149.8i −68528.9 − 17429.6i
29.12 −11.3137 + 19.5959i −65.2972 − 234.063i −256.000 − 443.405i −2746.79 + 1490.23i 5325.42 + 1368.56i 26709.9 + 15421.0i 11585.2 −50521.6 + 30567.2i 1873.85 − 70685.8i
29.13 −11.3137 + 19.5959i −58.3912 + 235.880i −256.000 − 443.405i 768.307 − 3029.08i −3961.67 − 3812.91i 26853.7 + 15504.0i 11585.2 −52229.9 − 27546.6i 50665.2 + 49325.8i
29.14 −11.3137 + 19.5959i −11.8731 + 242.710i −256.000 − 443.405i −2773.01 − 1440.85i −4621.79 − 2978.61i −6061.44 − 3499.57i 11585.2 −58767.1 − 5763.45i 59607.7 − 38038.4i
29.15 −11.3137 + 19.5959i −7.74189 + 242.877i −256.000 − 443.405i −2708.71 + 1558.36i −4671.80 − 2899.55i 5471.01 + 3158.69i 11585.2 −58929.1 − 3760.65i 108.079 − 70710.6i
29.16 −11.3137 + 19.5959i 1.44736 − 242.996i −256.000 − 443.405i −1281.09 − 2850.34i 4745.35 + 2777.54i −13939.3 − 8047.83i 11585.2 −59044.8 − 703.403i 70348.9 + 7143.84i
29.17 −11.3137 + 19.5959i 13.6495 + 242.616i −256.000 − 443.405i 2554.16 − 1800.52i −4908.72 − 2477.42i −7776.56 − 4489.80i 11585.2 −58676.4 + 6623.21i 6385.83 + 70421.7i
29.18 −11.3137 + 19.5959i 43.5187 − 239.071i −256.000 − 443.405i 2997.40 − 883.859i 4192.46 + 3557.57i −5973.28 − 3448.67i 11585.2 −55261.2 − 20808.2i −16591.7 + 68736.6i
29.19 −11.3137 + 19.5959i 91.4469 − 225.137i −256.000 − 443.405i −2624.82 + 1695.86i 3377.15 + 4339.12i −17804.6 − 10279.5i 11585.2 −42323.9 − 41176.1i −3535.47 − 70622.2i
29.20 −11.3137 + 19.5959i 104.977 + 219.155i −256.000 − 443.405i 1840.61 + 2525.43i −5482.22 − 422.333i −14963.8 − 8639.33i 11585.2 −37008.7 + 46012.4i −70312.2 + 7496.48i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 29.60
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h 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.j.a ✓ 120
5.b even 2 1 inner 90.11.j.a ✓ 120
9.d odd 6 1 inner 90.11.j.a ✓ 120
45.h odd 6 1 inner 90.11.j.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.11.j.a ✓ 120 1.a even 1 1 trivial
90.11.j.a ✓ 120 5.b even 2 1 inner
90.11.j.a ✓ 120 9.d odd 6 1 inner
90.11.j.a ✓ 120 45.h odd 6 1 inner

Hecke kernels

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