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Properties

Label 432.2.u.e
Level $432$
Weight $2$
Character orbit 432.u
Analytic conductor $3.450$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,2,Mod(49,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.u (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: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{7} + 6 q^{9} - 6 q^{11} + 12 q^{13} + 3 q^{15} + 6 q^{17} - 9 q^{19} - 18 q^{21} - 24 q^{23} - 24 q^{25} - 9 q^{29} + 27 q^{31} + 21 q^{33} + 18 q^{35} + 15 q^{37} + 15 q^{39} - 6 q^{41} - 39 q^{43} - 69 q^{45} + 36 q^{47} + 3 q^{49} + 36 q^{51} - 18 q^{53} + 54 q^{55} + 27 q^{57} + 30 q^{59} + 12 q^{61} - 18 q^{63} - 18 q^{65} - 54 q^{67} - 57 q^{69} + 36 q^{73} + 51 q^{75} - 24 q^{77} + 45 q^{79} + 18 q^{81} - 33 q^{83} - 57 q^{85} - 90 q^{87} + 9 q^{89} - 39 q^{91} + 42 q^{93} - 87 q^{95} + 57 q^{97} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.73099 + 0.0606946i 0 −0.00848388 + 0.00308788i 0 0.356397 + 2.02123i 0 2.99263 − 0.210123i 0
49.2 0 −0.119504 − 1.72792i 0 0.307563 − 0.111944i 0 −0.551939 − 3.13020i 0 −2.97144 + 0.412989i 0
49.3 0 0.278117 + 1.70958i 0 2.42978 − 0.884366i 0 0.245784 + 1.39391i 0 −2.84530 + 0.950925i 0
49.4 0 1.57237 − 0.726388i 0 −1.78916 + 0.651202i 0 0.623407 + 3.53552i 0 1.94472 − 2.28431i 0
97.1 0 −1.73099 − 0.0606946i 0 −0.00848388 − 0.00308788i 0 0.356397 − 2.02123i 0 2.99263 + 0.210123i 0
97.2 0 −0.119504 + 1.72792i 0 0.307563 + 0.111944i 0 −0.551939 + 3.13020i 0 −2.97144 − 0.412989i 0
97.3 0 0.278117 − 1.70958i 0 2.42978 + 0.884366i 0 0.245784 − 1.39391i 0 −2.84530 − 0.950925i 0
97.4 0 1.57237 + 0.726388i 0 −1.78916 − 0.651202i 0 0.623407 − 3.53552i 0 1.94472 + 2.28431i 0
193.1 0 −1.56067 − 0.751197i 0 −0.0770674 + 0.437071i 0 −0.935232 + 0.784753i 0 1.87141 + 2.34475i 0
193.2 0 −0.887369 + 1.48747i 0 0.444259 − 2.51952i 0 0.612199 − 0.513696i 0 −1.42515 − 2.63988i 0
193.3 0 0.747543 + 1.56243i 0 −0.738874 + 4.19036i 0 2.50342 − 2.10062i 0 −1.88236 + 2.33596i 0
193.4 0 1.70050 − 0.329088i 0 0.198034 − 1.12311i 0 −0.914338 + 0.767221i 0 2.78340 − 1.11923i 0
241.1 0 −1.56946 + 0.732664i 0 0.407020 − 0.341530i 0 −0.507439 + 0.184693i 0 1.92641 − 2.29977i 0
241.2 0 −1.05517 − 1.37354i 0 −1.80911 + 1.51802i 0 3.12406 − 1.13706i 0 −0.773215 + 2.89864i 0
241.3 0 0.962247 + 1.44017i 0 2.75433 − 2.31115i 0 1.28512 − 0.467744i 0 −1.14816 + 2.77159i 0
241.4 0 1.66239 + 0.486282i 0 −2.11828 + 1.77745i 0 −4.34143 + 1.58015i 0 2.52706 + 1.61678i 0
337.1 0 −1.56946 − 0.732664i 0 0.407020 + 0.341530i 0 −0.507439 − 0.184693i 0 1.92641 + 2.29977i 0
337.2 0 −1.05517 + 1.37354i 0 −1.80911 − 1.51802i 0 3.12406 + 1.13706i 0 −0.773215 − 2.89864i 0
337.3 0 0.962247 − 1.44017i 0 2.75433 + 2.31115i 0 1.28512 + 0.467744i 0 −1.14816 − 2.77159i 0
337.4 0 1.66239 − 0.486282i 0 −2.11828 − 1.77745i 0 −4.34143 − 1.58015i 0 2.52706 − 1.61678i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
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 432.2.u.e 24
4.b odd 2 1 216.2.q.a ✓ 24
12.b even 2 1 648.2.q.a 24
27.e even 9 1 inner 432.2.u.e 24
108.j odd 18 1 216.2.q.a ✓ 24
108.j odd 18 1 5832.2.a.h 12
108.l even 18 1 648.2.q.a 24
108.l even 18 1 5832.2.a.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.q.a ✓ 24 4.b odd 2 1
216.2.q.a ✓ 24 108.j odd 18 1
432.2.u.e 24 1.a even 1 1 trivial
432.2.u.e 24 27.e even 9 1 inner
648.2.q.a 24 12.b even 2 1
648.2.q.a 24 108.l even 18 1
5832.2.a.h 12 108.j odd 18 1
5832.2.a.i 12 108.l even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 12 T_{5}^{22} - 19 T_{5}^{21} + 45 T_{5}^{20} + 309 T_{5}^{19} + 1702 T_{5}^{18} - 1152 T_{5}^{17} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display