↑

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 $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,4,Mod(323,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.323");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.b (of order \(2\), degree \(1\), 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: \(19.1166188419\)
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.

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

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \( T_{5}^{12} + 876T_{5}^{10} + 265998T_{5}^{8} + 30811636T_{5}^{6} + 875662665T_{5}^{4} + 5418919608T_{5}^{2} + 7678666384 \) acting on \(S_{4}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display