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Properties

Label 90.11.k.b
Level $90$
Weight $11$
Character orbit 90.k
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(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 21.8564 − 5.85641i −242.911 + 6.56726i 443.405 − 256.000i −1346.00 + 2820.27i −5270.71 + 1566.12i −3123.77 + 837.012i 8192.00 − 8192.00i 58962.7 − 3190.52i −12902.0 + 69523.7i
7.2 21.8564 − 5.85641i −242.823 + 9.27376i 443.405 − 256.000i −2921.40 − 1109.53i −5252.93 + 1624.76i 71.8804 − 19.2603i 8192.00 − 8192.00i 58877.0 − 4503.76i −70349.1 − 7141.51i
7.3 21.8564 − 5.85641i −233.816 − 66.1741i 443.405 − 256.000i 2804.10 − 1379.37i −5497.92 − 77.0051i 4809.41 − 1288.68i 8192.00 − 8192.00i 50291.0 + 30945.1i 53209.3 − 46570.1i
7.4 21.8564 − 5.85641i −233.185 + 68.3665i 443.405 − 256.000i 2932.88 − 1078.80i −4696.19 + 2859.87i −26794.2 + 7179.49i 8192.00 − 8192.00i 49701.0 − 31884.0i 57784.4 − 40754.9i
7.5 21.8564 − 5.85641i −215.021 − 113.204i 443.405 − 256.000i −962.611 − 2973.05i −5362.55 − 1214.99i 23832.8 − 6385.97i 8192.00 − 8192.00i 33418.6 + 48682.4i −38450.6 − 59342.7i
7.6 21.8564 − 5.85641i −203.957 + 132.101i 443.405 − 256.000i 3034.54 + 746.441i −3684.13 + 4081.70i 29895.5 − 8010.48i 8192.00 − 8192.00i 24147.8 − 53885.7i 70695.7 − 1457.00i
7.7 21.8564 − 5.85641i −197.874 − 141.050i 443.405 − 256.000i 2150.28 + 2267.58i −5150.85 − 1924.01i −3855.01 + 1032.95i 8192.00 − 8192.00i 19259.0 + 55820.0i 60277.3 + 36968.2i
7.8 21.8564 − 5.85641i −161.252 + 181.788i 443.405 − 256.000i 975.463 − 2968.85i −2459.76 + 4917.59i −526.269 + 141.013i 8192.00 − 8192.00i −7044.69 − 58627.3i 3933.30 − 70601.2i
7.9 21.8564 − 5.85641i −156.929 + 185.532i 443.405 − 256.000i −1734.18 − 2599.66i −2343.36 + 4974.11i −12056.6 + 3230.55i 8192.00 − 8192.00i −9795.42 − 58230.9i −53127.6 − 46663.2i
7.10 21.8564 − 5.85641i −123.268 + 209.413i 443.405 − 256.000i −2831.86 + 1321.43i −1467.78 + 5298.93i 29282.6 − 7846.25i 8192.00 − 8192.00i −28659.0 − 51628.0i −54155.5 + 45466.3i
7.11 21.8564 − 5.85641i −105.656 + 218.828i 443.405 − 256.000i 351.923 + 3105.12i −1027.72 + 5401.56i −9471.76 + 2537.95i 8192.00 − 8192.00i −36722.5 − 46241.1i 25876.6 + 65805.8i
7.12 21.8564 − 5.85641i −101.430 − 220.819i 443.405 − 256.000i −3032.36 + 755.269i −3510.10 − 4232.29i −7026.82 + 1882.83i 8192.00 − 8192.00i −38473.0 + 44795.3i −61853.3 + 34266.2i
7.13 21.8564 − 5.85641i −60.1960 − 235.426i 443.405 − 256.000i 2718.24 − 1541.69i −2694.42 − 4793.04i 11257.1 − 3016.33i 8192.00 − 8192.00i −51801.9 + 28343.4i 50382.2 − 49614.9i
7.14 21.8564 − 5.85641i −54.6201 − 236.782i 443.405 − 256.000i 1139.02 − 2910.03i −2580.49 − 4855.32i −2572.53 + 689.308i 8192.00 − 8192.00i −53082.3 + 25866.1i 7852.62 − 70273.3i
7.15 21.8564 − 5.85641i −31.5435 + 240.944i 443.405 − 256.000i −2836.49 + 1311.48i 721.639 + 5450.90i −29964.3 + 8028.92i 8192.00 − 8192.00i −57059.0 − 15200.4i −54314.9 + 45275.8i
7.16 21.8564 − 5.85641i 5.52060 − 242.937i 443.405 − 256.000i 2070.75 + 2340.43i −1302.08 − 5342.07i −29350.0 + 7864.30i 8192.00 − 8192.00i −58988.0 − 2682.32i 58965.7 + 39026.2i
7.17 21.8564 − 5.85641i 51.0847 + 237.570i 443.405 − 256.000i 2874.90 + 1224.99i 2507.83 + 4893.25i 2311.84 − 619.455i 8192.00 − 8192.00i −53829.7 + 24272.3i 70008.9 + 9937.28i
7.18 21.8564 − 5.85641i 64.1341 − 234.384i 443.405 − 256.000i 543.965 + 3077.29i 29.0937 − 5498.39i 20748.0 − 5559.42i 8192.00 − 8192.00i −50822.6 − 30064.0i 29911.0 + 64072.9i
7.19 21.8564 − 5.85641i 81.4250 + 228.952i 443.405 − 256.000i −1692.09 − 2627.25i 3120.49 + 4527.21i 7782.62 − 2085.35i 8192.00 − 8192.00i −45788.9 + 37284.8i −52369.2 − 47512.8i
7.20 21.8564 − 5.85641i 92.2595 − 224.805i 443.405 − 256.000i −2292.39 − 2123.81i 699.913 − 5453.73i 25517.9 − 6837.50i 8192.00 − 8192.00i −42025.4 − 41480.8i −62541.3 − 32993.8i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.c even 3 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.11.k.b ✓ 120
5.c odd 4 1 inner 90.11.k.b ✓ 120
9.c even 3 1 inner 90.11.k.b ✓ 120
45.k odd 12 1 inner 90.11.k.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.11.k.b ✓ 120 1.a even 1 1 trivial
90.11.k.b ✓ 120 5.c odd 4 1 inner
90.11.k.b ✓ 120 9.c even 3 1 inner
90.11.k.b ✓ 120 45.k odd 12 1 inner