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Properties

Label 27.10.e.a
Level $27$
Weight $10$
Character orbit 27.e
Analytic conductor $13.906$
Analytic rank $0$
Dimension $156$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,10,Mod(4,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.9059675764\)
Analytic rank: \(0\)
Dimension: \(156\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 2382 q^{5} - 6822 q^{6} - 6 q^{7} + 36861 q^{8} + 2538 q^{9} - 3 q^{10} - 121767 q^{11} + 319935 q^{12} - 6 q^{13} - 720417 q^{14} - 706356 q^{15} + 1530 q^{16} + 1002249 q^{17}+ \cdots + 5248107342 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −40.0613 − 14.5811i −125.031 + 63.6419i 1000.08 + 839.170i −306.059 + 1735.74i 5936.87 − 726.489i 764.835 − 641.773i −16914.7 − 29297.2i 11582.4 − 15914.4i 37570.2 − 65073.5i
4.2 −39.3878 − 14.3360i 29.9043 − 137.072i 953.662 + 800.217i 27.4469 − 155.659i −3142.92 + 4970.25i 1724.19 − 1446.77i −15360.3 − 26604.9i −17894.5 − 8198.08i −3312.60 + 5737.59i
4.3 −34.7751 − 12.6571i 37.7394 + 135.125i 656.889 + 551.196i 233.333 − 1323.30i 397.899 − 5176.65i −1098.75 + 921.961i −6393.09 − 11073.2i −16834.5 + 10199.1i −24863.3 + 43064.5i
4.4 −31.5968 − 11.5003i 132.307 + 46.6679i 473.884 + 397.636i −451.726 + 2561.87i −3643.77 − 2996.12i −2730.60 + 2291.24i −1792.39 − 3104.51i 15327.2 + 12349.0i 43735.3 − 75751.7i
4.5 −29.1224 − 10.5997i −114.519 − 81.0463i 343.545 + 288.269i 182.505 − 1035.04i 2475.99 + 3574.12i −6317.76 + 5301.23i 984.496 + 1705.20i 6546.00 + 18562.6i −16286.1 + 28208.3i
4.6 −28.3865 − 10.3319i 136.675 − 31.6686i 306.833 + 257.464i 180.775 − 1025.22i −4206.93 − 513.146i 4261.26 − 3575.62i 1683.47 + 2915.86i 17677.2 − 8656.62i −15724.0 + 27234.8i
4.7 −22.2970 − 8.11544i −117.506 + 76.6512i 39.0810 + 32.7929i 151.895 − 861.438i 3242.08 − 755.482i 4408.97 − 3699.56i 5469.10 + 9472.76i 7932.18 − 18013.9i −10377.8 + 17974.8i
4.8 −20.2233 − 7.36069i −95.1961 − 103.057i −37.4118 − 31.3922i −358.438 + 2032.80i 1166.61 + 2784.86i 8116.71 − 6810.73i 6034.95 + 10452.8i −1558.40 + 19621.2i 22211.6 − 38471.7i
4.9 −15.0855 − 5.49066i 69.4935 − 121.876i −194.791 − 163.449i −120.371 + 682.656i −1717.52 + 1456.99i −4765.46 + 3998.70i 6150.79 + 10653.5i −10024.3 − 16939.1i 5564.08 − 9637.27i
4.10 −12.9570 − 4.71595i −37.0275 + 135.322i −246.572 − 206.898i −222.415 + 1261.38i 1117.94 − 1578.74i −4998.24 + 4194.02i 5748.96 + 9957.50i −16940.9 − 10021.3i 8830.42 − 15294.7i
4.11 −6.47952 − 2.35835i 83.5586 + 112.699i −355.792 − 298.545i −63.8763 + 362.261i −275.637 − 927.294i 6593.75 − 5532.82i 3366.50 + 5830.96i −5718.93 + 18833.9i 1268.23 − 2196.63i
4.12 −4.86289 − 1.76995i 133.111 + 44.3232i −371.700 − 311.893i 375.168 − 2127.68i −568.853 − 451.138i −8670.18 + 7275.14i 2580.30 + 4469.20i 15753.9 + 11799.8i −5590.29 + 9682.67i
4.13 −4.57613 − 1.66557i −24.3571 − 138.166i −374.048 − 313.863i 461.629 − 2618.03i −118.664 + 672.832i 5320.32 − 4464.28i 2435.60 + 4218.58i −18496.5 + 6730.62i −6472.99 + 11211.5i
4.14 3.93000 + 1.43040i −134.904 − 38.5223i −378.816 − 317.864i −245.629 + 1393.03i −475.070 − 344.359i −4348.21 + 3648.58i −2104.72 − 3645.48i 16715.1 + 10393.6i −2957.92 + 5123.27i
4.15 4.82322 + 1.75551i −137.543 + 27.6597i −372.033 − 312.173i 203.603 − 1154.69i −711.954 − 108.048i −440.948 + 369.999i −2560.36 − 4434.68i 18152.9 − 7608.76i 3009.09 − 5211.90i
4.16 8.90982 + 3.24291i 132.389 − 46.4332i −323.346 − 271.320i −128.517 + 728.857i 1330.14 + 15.6152i 4124.79 − 3461.11i −4428.39 − 7670.20i 15370.9 − 12294.5i −3508.68 + 6077.21i
4.17 13.5174 + 4.91993i −10.0180 − 139.938i −233.700 − 196.098i −142.171 + 806.289i 553.068 − 1940.89i −461.968 + 387.637i −5876.77 − 10178.9i −19482.3 + 2803.80i −5888.66 + 10199.5i
4.18 17.4636 + 6.35623i −17.1474 + 139.244i −127.639 − 107.102i 280.318 − 1589.76i −1184.52 + 2322.71i 610.601 − 512.355i −6305.87 − 10922.1i −19094.9 − 4775.35i 15000.2 − 25981.2i
4.19 23.5230 + 8.56168i −83.7573 + 112.551i 87.8153 + 73.6858i −425.267 + 2411.81i −2933.85 + 1930.43i 3548.49 − 2977.54i −4973.56 − 8614.45i −5652.42 − 18853.9i −30652.7 + 53092.0i
4.20 23.7357 + 8.63908i 115.700 + 79.3502i 96.5341 + 81.0017i −237.448 + 1346.63i 2060.71 + 2882.98i −5786.38 + 4855.35i −4874.78 − 8443.36i 7090.08 + 18361.7i −17269.7 + 29911.9i
See next 80 embeddings (of 156 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.26
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 27.10.e.a ✓ 156
3.b odd 2 1 81.10.e.a 156
27.e even 9 1 inner 27.10.e.a ✓ 156
27.f odd 18 1 81.10.e.a 156
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.10.e.a ✓ 156 1.a even 1 1 trivial
27.10.e.a ✓ 156 27.e even 9 1 inner
81.10.e.a 156 3.b odd 2 1
81.10.e.a 156 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(27, [\chi])\).