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Properties

Label 27.12.e.a
Level $27$
Weight $12$
Character orbit 27.e
Analytic conductor $20.745$
Analytic rank $0$
Dimension $192$
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] = [27,12,Mod(4,27)] 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("27.4"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), degree \(6\), 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: \(20.7452658751\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2931 q^{5} + 4410 q^{6} - 6 q^{7} - 294915 q^{8} + 96228 q^{9} - 3 q^{10} - 1594293 q^{11} + 323823 q^{12} - 6 q^{13} + 5263143 q^{14} + 3836529 q^{15} + 6138 q^{16}+ \cdots - 581308635231 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −84.4378 − 30.7329i 356.904 + 223.084i 4616.38 + 3873.60i 25.9620 − 147.238i −23280.2 − 29805.4i 50087.8 − 42028.6i −178737. − 309581.i 77614.4 + 159239.i −6717.22 + 11634.6i
4.2 −76.1538 − 27.7177i −338.466 + 250.176i 3462.27 + 2905.19i 1617.62 − 9173.99i 32709.8 − 9670.35i −29844.0 + 25042.1i −100154. − 173472.i 51971.0 − 169352.i −377470. + 653798.i
4.3 −71.9272 − 26.1793i 244.045 − 342.913i 2919.30 + 2449.58i −1677.00 + 9510.73i −26530.7 + 18275.8i −35545.3 + 29826.0i −67468.1 − 116858.i −58031.2 − 167372.i 369606. − 640177.i
4.4 −71.3644 − 25.9745i −358.855 − 219.931i 2849.35 + 2390.89i −900.499 + 5106.99i 19896.9 + 25016.4i −5535.58 + 4644.91i −63472.8 − 109938.i 80407.5 + 157847.i 196915. − 341067.i
4.5 −64.5257 − 23.4854i −92.3316 − 410.636i 2043.14 + 1714.40i 1145.62 − 6497.13i −3686.20 + 28665.0i 50780.0 − 42609.5i −21256.9 − 36818.0i −160097. + 75829.4i −226510. + 392327.i
4.6 −63.6523 − 23.1675i 339.140 − 249.262i 1946.02 + 1632.90i 2126.74 − 12061.4i −27361.8 + 8009.05i −32409.0 + 27194.4i −16675.1 − 28882.2i 52884.2 − 169069.i −414804. + 718462.i
4.7 −58.6349 − 21.3414i 176.782 + 381.962i 1413.74 + 1186.27i −404.877 + 2296.17i −2213.98 − 26169.1i −56857.8 + 47709.4i 6317.45 + 10942.1i −114643. + 135048.i 72743.5 − 125995.i
4.8 −57.1473 − 20.7999i −177.998 + 381.397i 1264.32 + 1060.89i −1067.28 + 6052.87i 18105.1 − 18093.5i 31997.4 − 26849.0i 12088.4 + 20937.7i −113780. − 135776.i 186892. − 323706.i
4.9 −45.2381 − 16.4653i 419.279 + 36.7703i 206.522 + 173.293i −251.754 + 1427.77i −18362.0 − 8566.99i 11983.1 − 10055.1i 42807.5 + 74144.7i 174443. + 30834.0i 34897.6 − 60444.4i
4.10 −35.0843 − 12.7696i −420.886 + 1.52023i −501.016 − 420.403i 1084.25 − 6149.07i 14785.9 + 5321.22i 14859.0 − 12468.2i 50441.4 + 87367.0i 177142. − 1279.69i −116561. + 201890.i
4.11 −29.6810 − 10.8030i 163.426 + 387.865i −804.602 − 675.141i 2010.04 − 11399.5i −660.534 − 13277.7i 26564.4 − 22290.2i 48931.8 + 84752.3i −123731. + 126774.i −182809. + 316634.i
4.12 −29.6174 − 10.7799i 231.317 − 351.624i −807.872 − 677.885i −1554.64 + 8816.79i −10641.5 + 7920.64i 51229.3 − 42986.5i 48894.2 + 84687.3i −70132.0 − 162673.i 141088. − 244372.i
4.13 −26.2525 − 9.55513i −115.768 − 404.654i −970.966 − 814.737i 256.464 − 1454.48i −827.328 + 11729.4i −34352.6 + 28825.3i 46313.2 + 80216.8i −150343. + 93691.7i −20630.5 + 35733.1i
4.14 −24.4318 − 8.89246i −419.811 + 30.0951i −1051.02 − 881.911i −2059.92 + 11682.4i 10524.4 + 2997.87i −22736.8 + 19078.4i 44459.8 + 77006.7i 175336. − 25268.5i 154213. − 267104.i
4.15 −3.15277 − 1.14751i 419.946 + 28.1467i −1560.24 − 1309.19i −145.055 + 822.646i −1291.69 − 570.634i −28559.8 + 23964.5i 6852.37 + 11868.7i 175563. + 23640.2i 1401.32 − 2427.16i
4.16 −2.61958 − 0.953448i −238.888 + 346.525i −1562.91 − 1311.43i 728.437 − 4131.17i 956.178 − 679.982i −46162.6 + 38735.0i 5698.37 + 9869.86i −63012.4 − 165561.i −5847.05 + 10127.4i
4.17 2.52138 + 0.917709i 221.074 + 358.153i −1563.34 − 1311.80i −1902.92 + 10792.0i 228.733 + 1105.92i 3029.77 − 2542.28i −5485.54 − 9501.23i −79399.5 + 158357.i −14701.9 + 25464.5i
4.18 10.1300 + 3.68703i 279.148 − 314.997i −1479.84 − 1241.73i 1299.27 − 7368.50i 3989.18 − 2161.70i 12402.9 − 10407.2i −21451.4 − 37154.8i −21299.6 − 175862.i 40329.5 − 69852.7i
4.19 15.4824 + 5.63514i −381.743 − 177.256i −1360.91 − 1141.94i 480.358 − 2724.25i −4911.44 − 4895.51i 55813.1 − 46832.8i −31506.6 − 54571.1i 114308. + 135332.i 22788.6 − 39471.0i
4.20 16.1971 + 5.89528i −183.592 + 378.736i −1341.27 − 1125.46i −217.464 + 1233.30i −5206.42 + 5052.11i 49804.8 − 41791.2i −32740.1 − 56707.5i −109735. − 139066.i −10792.9 + 18693.9i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.12.e.a ✓ 192
3.b odd 2 1 81.12.e.a 192
27.e even 9 1 inner 27.12.e.a ✓ 192
27.f odd 18 1 81.12.e.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.12.e.a ✓ 192 1.a even 1 1 trivial
27.12.e.a ✓ 192 27.e even 9 1 inner
81.12.e.a 192 3.b odd 2 1
81.12.e.a 192 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(27, [\chi])\).