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Properties

Label 54.8.e.a
Level $54$
Weight $8$
Character orbit 54.e
Analytic conductor $16.869$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,8,Mod(7,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 54.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: \(16.8687913761\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 213 q^{5} - 72 q^{6} + 1677 q^{7} - 15360 q^{8} - 3024 q^{9} + 3000 q^{10} - 15744 q^{11} + 2880 q^{12} + 9924 q^{13} + 13416 q^{14} + 2322 q^{15} + 29478 q^{17} + 49680 q^{18} + 61731 q^{19} - 27264 q^{20}+ \cdots + 36391797 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −7.51754 + 2.73616i −46.7582 − 0.818523i 49.0268 − 41.1384i 7.81200 + 44.3040i 353.746 − 121.785i −714.875 − 599.852i −256.000 + 443.405i 2185.66 + 76.5453i −179.950 − 311.683i
7.2 −7.51754 + 2.73616i −42.7347 + 18.9932i 49.0268 − 41.1384i −81.8382 − 464.127i 269.292 − 259.711i 742.983 + 623.437i −256.000 + 443.405i 1465.52 − 1623.34i 1885.15 + 3265.17i
7.3 −7.51754 + 2.73616i −28.0663 − 37.4070i 49.0268 − 41.1384i 4.03154 + 22.8640i 313.341 + 204.415i 250.482 + 210.179i −256.000 + 443.405i −611.569 + 2099.75i −92.8669 − 160.850i
7.4 −7.51754 + 2.73616i −22.4250 + 41.0380i 49.0268 − 41.1384i 54.0576 + 306.576i 56.2941 − 369.863i −1071.67 − 899.235i −256.000 + 443.405i −1181.24 − 1840.55i −1245.22 − 2156.79i
7.5 −7.51754 + 2.73616i −11.2841 + 45.3836i 49.0268 − 41.1384i −4.70644 − 26.6915i −39.3482 − 372.048i 243.254 + 204.114i −256.000 + 443.405i −1932.34 − 1024.22i 108.413 + 187.777i
7.6 −7.51754 + 2.73616i 14.5144 − 44.4560i 49.0268 − 41.1384i −80.0788 − 454.149i 12.5263 + 373.913i −1075.14 − 902.152i −256.000 + 443.405i −1765.67 − 1290.50i 1844.62 + 3194.98i
7.7 −7.51754 + 2.73616i 20.9112 − 41.8297i 49.0268 − 41.1384i 71.6637 + 406.425i −42.7476 + 371.673i −194.379 − 163.103i −256.000 + 443.405i −1312.45 − 1749.41i −1650.78 − 2859.23i
7.8 −7.51754 + 2.73616i 28.8790 + 36.7832i 49.0268 − 41.1384i 67.5637 + 383.173i −317.744 − 197.501i 619.597 + 519.903i −256.000 + 443.405i −519.005 + 2124.52i −1556.34 − 2695.65i
7.9 −7.51754 + 2.73616i 32.1805 + 33.9325i 49.0268 − 41.1384i −56.1070 − 318.199i −334.763 − 167.038i −287.281 − 241.057i −256.000 + 443.405i −115.825 + 2183.93i 1292.43 + 2238.55i
7.10 −7.51754 + 2.73616i 44.9983 − 12.7340i 49.0268 − 41.1384i −22.9934 − 130.402i −303.434 + 218.851i 812.804 + 682.023i −256.000 + 443.405i 1862.69 − 1146.01i 529.655 + 917.389i
13.1 1.38919 + 7.87846i −46.5298 + 4.68829i −60.1403 + 21.8893i 269.395 + 226.049i −101.575 − 360.070i 989.779 + 360.250i −256.000 − 443.405i 2143.04 − 436.290i −1406.68 + 2436.44i
13.2 1.38919 + 7.87846i −43.7763 − 16.4511i −60.1403 + 21.8893i −250.391 − 210.103i 68.7958 − 367.743i −714.560 − 260.079i −256.000 − 443.405i 1645.72 + 1440.33i 1307.45 − 2264.57i
13.3 1.38919 + 7.87846i −25.6010 + 39.1355i −60.1403 + 21.8893i 192.012 + 161.117i −343.892 − 147.330i −1238.25 − 450.687i −256.000 − 443.405i −876.175 − 2003.82i −1002.62 + 1736.58i
13.4 1.38919 + 7.87846i −20.8780 − 41.8463i −60.1403 + 21.8893i 134.896 + 113.191i 300.681 − 222.619i 168.898 + 61.4738i −256.000 − 443.405i −1315.22 + 1747.33i −704.377 + 1220.02i
13.5 1.38919 + 7.87846i −14.0270 + 44.6122i −60.1403 + 21.8893i −224.372 − 188.270i −370.961 − 48.5362i 356.182 + 129.640i −256.000 − 443.405i −1793.49 − 1251.55i 1171.59 − 2029.25i
13.6 1.38919 + 7.87846i 14.6775 − 44.4024i −60.1403 + 21.8893i −379.082 − 318.088i 370.212 + 53.9532i 1350.21 + 491.436i −256.000 − 443.405i −1756.14 − 1303.43i 1979.43 − 3428.47i
13.7 1.38919 + 7.87846i 28.7664 + 36.8713i −60.1403 + 21.8893i 32.6732 + 27.4161i −250.528 + 277.856i 1319.07 + 480.101i −256.000 − 443.405i −531.991 + 2121.31i −170.607 + 295.501i
13.8 1.38919 + 7.87846i 36.2871 − 29.5000i −60.1403 + 21.8893i 329.484 + 276.470i 282.824 + 244.906i 438.081 + 159.448i −256.000 − 443.405i 446.505 − 2140.93i −1720.44 + 2979.90i
13.9 1.38919 + 7.87846i 37.6126 + 27.7902i −60.1403 + 21.8893i 100.967 + 84.7217i −166.693 + 334.935i −905.523 − 329.583i −256.000 − 443.405i 642.413 + 2090.52i −527.214 + 913.162i
13.10 1.38919 + 7.87846i 46.4505 − 5.41765i −60.1403 + 21.8893i −199.121 − 167.082i 107.211 + 358.432i −314.460 − 114.454i −256.000 − 443.405i 2128.30 − 503.305i 1039.74 − 1800.88i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
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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 54.8.e.a ✓ 60
27.e even 9 1 inner 54.8.e.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.e.a ✓ 60 1.a even 1 1 trivial
54.8.e.a ✓ 60 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 213 T_{5}^{59} + 194193 T_{5}^{58} - 84858978 T_{5}^{57} + 22693137750 T_{5}^{56} + \cdots + 10\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display