Properties

Label 324.4.b.c
Level 324
Weight 4
Character orbit 324.b
Analytic conductor 19.117
Analytic rank 0
Dimension 24
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1166188419\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{4} + 96q^{10} + 432q^{13} + 144q^{16} + 384q^{22} - 504q^{25} - 672q^{28} + 1320q^{34} + 1248q^{37} - 1272q^{40} + 960q^{46} - 696q^{49} - 264q^{52} - 1032q^{58} + 528q^{61} + 960q^{64} - 1128q^{70} - 4776q^{73} + 1200q^{76} - 4104q^{82} - 1440q^{85} - 3912q^{88} + 2376q^{94} - 1176q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −2.82820 0.0360939i 0 7.99739 + 0.204161i 16.9163i 0 3.56763i −22.6108 0.866066i 0 0.610574 47.8425i
323.2 −2.82820 + 0.0360939i 0 7.99739 0.204161i 16.9163i 0 3.56763i −22.6108 + 0.866066i 0 0.610574 + 47.8425i
323.3 −2.75517 0.639551i 0 7.18195 + 3.52415i 5.44403i 0 24.1745i −17.5336 14.3029i 0 3.48173 14.9992i
323.4 −2.75517 + 0.639551i 0 7.18195 3.52415i 5.44403i 0 24.1745i −17.5336 + 14.3029i 0 3.48173 + 14.9992i
323.5 −2.36704 1.54827i 0 3.20572 + 7.32962i 1.43006i 0 27.5763i 3.76017 22.3128i 0 2.21411 3.38499i
323.6 −2.36704 + 1.54827i 0 3.20572 7.32962i 1.43006i 0 27.5763i 3.76017 + 22.3128i 0 2.21411 + 3.38499i
323.7 −2.04896 1.94981i 0 0.396477 + 7.99017i 16.5019i 0 22.2418i 14.7670 17.1446i 0 −32.1756 + 33.8118i
323.8 −2.04896 + 1.94981i 0 0.396477 7.99017i 16.5019i 0 22.2418i 14.7670 + 17.1446i 0 −32.1756 33.8118i
323.9 −1.16011 2.57956i 0 −5.30829 + 5.98515i 16.7343i 0 19.3037i 21.5973 + 6.74964i 0 43.1673 19.4137i
323.10 −1.16011 + 2.57956i 0 −5.30829 5.98515i 16.7343i 0 19.3037i 21.5973 6.74964i 0 43.1673 + 19.4137i
323.11 −0.513200 2.78148i 0 −7.47325 + 2.85491i 2.40947i 0 2.66000i 11.7761 + 19.3216i 0 6.70190 1.23654i
323.12 −0.513200 + 2.78148i 0 −7.47325 2.85491i 2.40947i 0 2.66000i 11.7761 19.3216i 0 6.70190 + 1.23654i
323.13 0.513200 2.78148i 0 −7.47325 2.85491i 2.40947i 0 2.66000i −11.7761 + 19.3216i 0 6.70190 + 1.23654i
323.14 0.513200 + 2.78148i 0 −7.47325 + 2.85491i 2.40947i 0 2.66000i −11.7761 19.3216i 0 6.70190 1.23654i
323.15 1.16011 2.57956i 0 −5.30829 5.98515i 16.7343i 0 19.3037i −21.5973 + 6.74964i 0 43.1673 + 19.4137i
323.16 1.16011 + 2.57956i 0 −5.30829 + 5.98515i 16.7343i 0 19.3037i −21.5973 6.74964i 0 43.1673 19.4137i
323.17 2.04896 1.94981i 0 0.396477 7.99017i 16.5019i 0 22.2418i −14.7670 17.1446i 0 −32.1756 33.8118i
323.18 2.04896 + 1.94981i 0 0.396477 + 7.99017i 16.5019i 0 22.2418i −14.7670 + 17.1446i 0 −32.1756 + 33.8118i
323.19 2.36704 1.54827i 0 3.20572 7.32962i 1.43006i 0 27.5763i −3.76017 22.3128i 0 2.21411 + 3.38499i
323.20 2.36704 + 1.54827i 0 3.20572 + 7.32962i 1.43006i 0 27.5763i −3.76017 + 22.3128i 0 2.21411 3.38499i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.b.c 24
3.b odd 2 1 inner 324.4.b.c 24
4.b odd 2 1 inner 324.4.b.c 24
9.c even 3 1 36.4.h.b 24
9.c even 3 1 108.4.h.b 24
9.d odd 6 1 36.4.h.b 24
9.d odd 6 1 108.4.h.b 24
12.b even 2 1 inner 324.4.b.c 24
36.f odd 6 1 36.4.h.b 24
36.f odd 6 1 108.4.h.b 24
36.h even 6 1 36.4.h.b 24
36.h even 6 1 108.4.h.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.b 24 9.c even 3 1
36.4.h.b 24 9.d odd 6 1
36.4.h.b 24 36.f odd 6 1
36.4.h.b 24 36.h even 6 1
108.4.h.b 24 9.c even 3 1
108.4.h.b 24 9.d odd 6 1
108.4.h.b 24 36.f odd 6 1
108.4.h.b 24 36.h even 6 1
324.4.b.c 24 1.a even 1 1 trivial
324.4.b.c 24 3.b odd 2 1 inner
324.4.b.c 24 4.b odd 2 1 inner
324.4.b.c 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 876 T_{5}^{10} + 265998 T_{5}^{8} + 30811636 T_{5}^{6} + 875662665 T_{5}^{4} + 5418919608 T_{5}^{2} + 7678666384 \) acting on \(S_{4}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database