Properties

Label 324.5.k.a
Level 324
Weight 5
Character orbit 324.k
Analytic conductor 33.492
Analytic rank 0
Dimension 72
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.k (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72q - 9q^{5} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q - 9q^{5} - 18q^{11} + 1278q^{23} + 441q^{25} - 1854q^{29} - 1665q^{31} + 2673q^{35} + 5472q^{41} + 1260q^{43} - 5103q^{47} - 5904q^{49} + 10944q^{59} + 8352q^{61} - 8757q^{65} + 378q^{67} + 19764q^{71} + 6111q^{73} + 5679q^{77} - 5652q^{79} + 20061q^{83} + 26100q^{85} - 15633q^{89} - 6039q^{91} - 48024q^{95} - 37530q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 0 0 −47.5857 8.39064i 0 −52.2512 + 43.8440i 0 0 0
17.2 0 0 0 −33.5251 5.91137i 0 16.7096 14.0210i 0 0 0
17.3 0 0 0 −26.8634 4.73675i 0 65.2588 54.7587i 0 0 0
17.4 0 0 0 −12.7195 2.24280i 0 −50.9819 + 42.7789i 0 0 0
17.5 0 0 0 −8.34491 1.47143i 0 44.7323 37.5349i 0 0 0
17.6 0 0 0 −8.21900 1.44923i 0 −41.3160 + 34.6682i 0 0 0
17.7 0 0 0 −5.61224 0.989589i 0 26.5803 22.3035i 0 0 0
17.8 0 0 0 15.6508 + 2.75966i 0 −42.2367 + 35.4408i 0 0 0
17.9 0 0 0 16.9805 + 2.99411i 0 1.82205 1.52888i 0 0 0
17.10 0 0 0 32.0223 + 5.64639i 0 −0.626400 + 0.525612i 0 0 0
17.11 0 0 0 34.9786 + 6.16768i 0 32.9047 27.6104i 0 0 0
17.12 0 0 0 39.4395 + 6.95424i 0 −0.595664 + 0.499822i 0 0 0
89.1 0 0 0 −29.8374 35.5588i 0 −70.9709 + 25.8313i 0 0 0
89.2 0 0 0 −23.3523 27.8302i 0 44.9114 16.3464i 0 0 0
89.3 0 0 0 −18.1168 21.5908i 0 22.9581 8.35608i 0 0 0
89.4 0 0 0 −11.0592 13.1798i 0 7.77764 2.83083i 0 0 0
89.5 0 0 0 −6.32074 7.53276i 0 66.1128 24.0631i 0 0 0
89.6 0 0 0 −1.79646 2.14093i 0 −63.3056 + 23.0413i 0 0 0
89.7 0 0 0 −0.0201223 0.0239808i 0 −60.3934 + 21.9814i 0 0 0
89.8 0 0 0 8.22970 + 9.80778i 0 2.75000 1.00092i 0 0 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
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 324.5.k.a 72
3.b odd 2 1 108.5.k.a 72
27.e even 9 1 108.5.k.a 72
27.f odd 18 1 inner 324.5.k.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.k.a 72 3.b odd 2 1
108.5.k.a 72 27.e even 9 1
324.5.k.a 72 1.a even 1 1 trivial
324.5.k.a 72 27.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database