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Properties

Label 27.11.f.a
Level $27$
Weight $11$
Character orbit 27.f
Analytic conductor $17.155$
Analytic rank $0$
Dimension $174$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,11,Mod(2,27)]
 
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([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.2");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), 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: \(17.1546458222\)
Analytic rank: \(0\)
Dimension: \(174\)
Relative dimension: \(29\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 174 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 4965 q^{5} + 18234 q^{6} - 6 q^{7} - 9 q^{8} + 119124 q^{9} - 3 q^{10} - 978 q^{11} - 1015107 q^{12} - 6 q^{13} + 2134569 q^{14} - 2548539 q^{15} - 3078 q^{16} - 9 q^{17}+ \cdots - 72892391589 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −56.8307 − 10.0208i −240.581 + 34.2027i 2167.07 + 788.750i 3512.61 + 4186.16i 14015.1 + 467.049i −10298.5 + 3748.35i −64076.9 − 36994.8i 56709.4 − 16457.0i −157675. − 273102.i
2.2 −56.1659 − 9.90357i −197.148 − 142.062i 2094.29 + 762.257i −3678.77 − 4384.19i 9666.10 + 9931.50i 4723.20 − 1719.10i −59501.5 − 34353.2i 18685.9 + 56014.5i 163203. + 282675.i
2.3 −54.5581 − 9.62006i 241.646 + 25.6213i 1921.79 + 699.476i −1536.17 − 1830.74i −12937.2 − 3722.49i −13051.0 + 4750.19i −48991.4 − 28285.2i 57736.1 + 12382.5i 66198.7 + 114660.i
2.4 −54.1567 − 9.54929i 57.4013 + 236.123i 1879.52 + 684.088i 613.604 + 731.264i −853.859 − 13335.8i 30273.5 − 11018.7i −46488.3 − 26840.0i −52459.2 + 27107.5i −26247.7 − 45462.4i
2.5 −50.9348 − 8.98117i 58.0426 − 235.966i 1551.44 + 564.679i 1305.93 + 1556.34i −5075.64 + 11497.6i 599.564 − 218.223i −28084.6 − 16214.7i −52311.1 − 27392.2i −52539.3 − 91000.7i
2.6 −41.9909 − 7.40413i −31.5484 + 240.943i 746.171 + 271.584i −460.776 − 549.132i 3108.72 − 9883.84i −25768.4 + 9378.91i 8490.87 + 4902.20i −57058.4 − 15202.7i 15282.6 + 26470.2i
2.7 −34.9718 − 6.16647i −206.178 + 128.606i 222.756 + 81.0766i −1482.02 − 1766.21i 8003.47 − 3226.20i 4406.20 − 1603.73i 24201.5 + 13972.8i 25969.9 − 53031.6i 40937.8 + 70906.3i
2.8 −29.9307 − 5.27758i 216.698 − 109.959i −94.2540 − 34.3057i 378.278 + 450.814i −7066.23 + 2147.50i 11074.9 − 4030.93i 29592.3 + 17085.1i 34867.1 − 47655.7i −8942.89 − 15489.5i
2.9 −27.8667 − 4.91365i −146.293 − 194.029i −209.837 − 76.3742i 287.607 + 342.757i 3123.30 + 6125.79i 15362.6 − 5591.54i 30565.9 + 17647.2i −16245.9 + 56770.2i −6330.48 − 10964.7i
2.10 −26.8887 − 4.74121i 199.948 + 138.094i −261.720 − 95.2582i 3602.84 + 4293.69i −4721.61 − 4661.16i −9989.71 + 3635.96i 30798.7 + 17781.7i 20909.3 + 55223.1i −76518.4 − 132534.i
2.11 −19.7178 − 3.47679i −201.076 − 136.446i −585.540 − 213.119i 1387.93 + 1654.07i 3490.39 + 3389.51i −22901.7 + 8335.55i 28560.3 + 16489.3i 21814.1 + 54871.9i −21616.2 − 37440.3i
2.12 −18.0069 − 3.17510i 88.7734 − 226.204i −648.078 − 235.881i −3624.02 − 4318.93i −2316.76 + 3791.37i −24038.2 + 8749.19i 27136.0 + 15667.0i −43287.6 − 40161.8i 51544.2 + 89277.3i
2.13 −15.0843 − 2.65976i 178.198 + 165.210i −741.784 − 269.987i −2989.41 − 3562.64i −2248.57 − 2966.04i 13386.2 − 4872.18i 24054.4 + 13887.8i 4460.33 + 58880.3i 35617.3 + 61691.0i
2.14 −6.94178 − 1.22402i −197.177 + 142.021i −915.555 − 333.235i 3117.17 + 3714.90i 1542.60 − 744.531i 24236.9 − 8821.52i 12198.7 + 7042.92i 18708.9 − 56006.8i −17091.6 − 29603.5i
2.15 2.92749 + 0.516196i −14.4961 + 242.567i −953.942 − 347.206i 237.873 + 283.486i −167.649 + 702.630i −6094.69 + 2218.29i −5249.61 − 3030.86i −58628.7 − 7032.56i 550.037 + 952.692i
2.16 9.76434 + 1.72172i 232.294 − 71.3331i −869.867 − 316.606i 730.470 + 870.540i 2391.01 − 296.575i 5090.69 − 1852.86i −16741.3 − 9665.57i 48872.2 − 33140.5i 5633.73 + 9757.91i
2.17 9.99045 + 1.76159i 58.8240 − 235.773i −865.539 − 315.031i 2619.44 + 3121.73i 1003.01 − 2251.85i −15129.3 + 5506.60i −17088.5 − 9866.04i −52128.5 − 27738.2i 20670.2 + 35801.9i
2.18 12.7838 + 2.25414i −11.4870 − 242.728i −803.900 − 292.596i −1623.88 − 1935.26i 400.294 − 3128.89i 27986.1 − 10186.1i −21129.1 − 12198.9i −58785.1 + 5576.46i −16397.0 − 28400.5i
2.19 12.8751 + 2.27023i −242.999 + 0.792606i −801.631 − 291.770i −2001.83 − 2385.69i −3130.43 − 541.458i −5414.20 + 1970.61i −21252.6 − 12270.2i 59047.7 − 385.204i −20357.7 − 35260.6i
2.20 24.9445 + 4.39838i 228.011 + 84.0242i −359.365 − 130.798i −928.191 − 1106.17i 5318.04 + 3098.82i −25337.3 + 9222.03i −30851.1 − 17811.9i 44928.9 + 38316.9i −18287.8 − 31675.5i
See next 80 embeddings (of 174 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.11.f.a ✓ 174
3.b odd 2 1 81.11.f.a 174
27.e even 9 1 81.11.f.a 174
27.f odd 18 1 inner 27.11.f.a ✓ 174
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.11.f.a ✓ 174 1.a even 1 1 trivial
27.11.f.a ✓ 174 27.f odd 18 1 inner
81.11.f.a 174 3.b odd 2 1
81.11.f.a 174 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(27, [\chi])\).