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Properties

Label 81.8.e.a
Level $81$
Weight $8$
Character orbit 81.e
Analytic conductor $25.303$
Analytic rank $0$
Dimension $120$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,8,Mod(10,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.10");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.e (of order \(9\), degree \(6\), not 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: \(25.3031870642\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 27)
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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{2} - 6 q^{4} + 219 q^{5} - 6 q^{7} + 4611 q^{8} - 3 q^{10} - 9399 q^{11} - 6 q^{13} - 16647 q^{14} + 378 q^{16} + 58959 q^{17} - 3 q^{19} - 240243 q^{20} + 105762 q^{22} + 144084 q^{23} - 107997 q^{25}+ \cdots + 88493274 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.78531 − 21.4676i 0 −326.247 + 118.744i −5.44265 − 4.56692i 0 −1251.57 − 455.534i 2388.97 + 4137.82i 0 −77.4386 + 134.128i
10.2 −3.37304 − 19.1294i 0 −234.277 + 85.2699i 84.6713 + 71.0477i 0 549.429 + 199.976i 1178.22 + 2040.74i 0 1073.50 − 1859.36i
10.3 −3.12991 − 17.7506i 0 −185.007 + 67.3369i −339.472 − 284.851i 0 −71.6195 − 26.0674i 620.762 + 1075.19i 0 −3993.75 + 6917.39i
10.4 −2.62261 − 14.8736i 0 −94.0647 + 34.2368i 415.153 + 348.355i 0 −977.702 − 355.854i −210.675 − 364.901i 0 4092.50 − 7088.41i
10.5 −2.40701 − 13.6508i 0 −60.2708 + 21.9368i −33.9324 − 28.4726i 0 674.611 + 245.538i −442.603 − 766.611i 0 −307.000 + 531.739i
10.6 −2.28109 − 12.9367i 0 −41.8741 + 15.2409i 86.4596 + 72.5482i 0 1303.70 + 474.509i −548.035 − 949.224i 0 741.312 − 1283.99i
10.7 −1.81920 − 10.3172i 0 17.1451 − 6.24032i −372.643 − 312.684i 0 −1217.91 − 443.285i −766.062 − 1326.86i 0 −2548.12 + 4413.47i
10.8 −0.995547 − 5.64603i 0 89.3941 − 32.5368i 164.412 + 137.958i 0 −753.757 − 274.345i −639.620 − 1107.85i 0 615.234 − 1065.62i
10.9 −0.678939 − 3.85046i 0 105.916 − 38.5501i 178.863 + 150.084i 0 −703.275 − 255.971i −470.577 − 815.063i 0 456.455 − 790.603i
10.10 −0.649023 − 3.68079i 0 107.154 − 39.0007i −80.9975 − 67.9649i 0 791.204 + 287.975i −452.304 − 783.413i 0 −197.596 + 342.246i
10.11 0.0387020 + 0.219490i 0 120.234 − 43.7616i −199.491 − 167.393i 0 −867.130 − 315.610i 28.5226 + 49.4026i 0 29.0204 − 50.2647i
10.12 0.898071 + 5.09321i 0 95.1464 − 34.6305i −331.543 − 278.197i 0 981.429 + 357.211i 592.822 + 1026.80i 0 1119.17 − 1938.46i
10.13 1.10962 + 6.29299i 0 81.9102 − 29.8129i 292.309 + 245.276i 0 1306.69 + 475.598i 687.466 + 1190.73i 0 −1219.17 + 2111.66i
10.14 1.12676 + 6.39019i 0 80.7157 − 29.3781i 233.744 + 196.135i 0 100.767 + 36.6762i 693.960 + 1201.97i 0 −989.964 + 1714.67i
10.15 1.90051 + 10.7783i 0 7.72077 − 2.81013i −235.564 − 197.662i 0 277.012 + 100.824i 745.415 + 1291.10i 0 1682.77 − 2914.64i
10.16 2.26179 + 12.8272i 0 −39.1414 + 14.2463i 4.19097 + 3.51664i 0 −1350.63 − 491.588i 562.337 + 973.996i 0 −35.6297 + 61.7124i
10.17 2.63245 + 14.9293i 0 −95.6749 + 34.8228i 128.706 + 107.997i 0 −237.509 − 86.4462i 198.477 + 343.772i 0 −1273.52 + 2205.80i
10.18 3.35424 + 19.0228i 0 −230.336 + 83.8354i −74.1505 − 62.2197i 0 −569.519 − 207.288i −1131.15 − 1959.20i 0 934.875 − 1619.25i
10.19 3.52519 + 19.9924i 0 −266.987 + 97.1752i −333.918 − 280.190i 0 999.181 + 363.672i −1584.69 − 2744.76i 0 4424.54 − 7663.53i
10.20 3.56548 + 20.2208i 0 −275.889 + 100.415i 330.525 + 277.343i 0 1014.82 + 369.366i −1700.06 − 2944.59i 0 −4429.63 + 7672.35i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.20
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 81.8.e.a 120
3.b odd 2 1 27.8.e.a ✓ 120
27.e even 9 1 inner 81.8.e.a 120
27.f odd 18 1 27.8.e.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.8.e.a ✓ 120 3.b odd 2 1
27.8.e.a ✓ 120 27.f odd 18 1
81.8.e.a 120 1.a even 1 1 trivial
81.8.e.a 120 27.e even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(81, [\chi])\).