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Properties

Label 432.2.k.c
Level $432$
Weight $2$
Character orbit 432.k
Analytic conductor $3.450$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,2,Mod(109,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.k (of order \(4\), degree \(2\), 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: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4} - 4 q^{10} + 16 q^{13} - 20 q^{16} - 16 q^{19} - 12 q^{22} - 12 q^{28} + 32 q^{31} + 28 q^{34} - 8 q^{37} - 36 q^{40} - 64 q^{46} - 16 q^{49} - 36 q^{52} - 32 q^{58} - 16 q^{61} + 16 q^{64} + 48 q^{67} - 24 q^{70} + 16 q^{76} - 48 q^{79} - 16 q^{82} - 16 q^{85} - 60 q^{88} + 96 q^{91} + 84 q^{94} + 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.40946 + 0.115854i 0 1.97316 − 0.326583i −1.78448 + 1.78448i 0 4.77575i −2.74325 + 0.688904i 0 2.30841 − 2.72189i
109.2 −1.37261 − 0.340511i 0 1.76810 + 0.934777i 2.75203 − 2.75203i 0 0.113492i −2.10861 − 1.88514i 0 −4.71455 + 2.84036i
109.3 −1.20402 + 0.741841i 0 0.899344 − 1.78639i −0.431497 + 0.431497i 0 0.145064i 0.242383 + 2.81802i 0 0.199430 − 0.839634i
109.4 −1.15725 − 0.812885i 0 0.678437 + 1.88142i −2.39401 + 2.39401i 0 − 2.42931i 0.744255 − 2.72875i 0 4.71651 − 0.824404i
109.5 −0.936391 − 1.05980i 0 −0.246343 + 1.98477i 1.12951 − 1.12951i 0 1.33035i 2.33413 − 1.59745i 0 −2.25471 − 0.139389i
109.6 −0.680919 + 1.23950i 0 −1.07270 − 1.68799i −2.24693 + 2.24693i 0 − 3.93534i 2.82268 − 0.180219i 0 −1.25508 − 4.31505i
109.7 0.680919 − 1.23950i 0 −1.07270 − 1.68799i 2.24693 − 2.24693i 0 − 3.93534i −2.82268 + 0.180219i 0 −1.25508 − 4.31505i
109.8 0.936391 + 1.05980i 0 −0.246343 + 1.98477i −1.12951 + 1.12951i 0 1.33035i −2.33413 + 1.59745i 0 −2.25471 − 0.139389i
109.9 1.15725 + 0.812885i 0 0.678437 + 1.88142i 2.39401 − 2.39401i 0 − 2.42931i −0.744255 + 2.72875i 0 4.71651 − 0.824404i
109.10 1.20402 − 0.741841i 0 0.899344 − 1.78639i 0.431497 − 0.431497i 0 0.145064i −0.242383 − 2.81802i 0 0.199430 − 0.839634i
109.11 1.37261 + 0.340511i 0 1.76810 + 0.934777i −2.75203 + 2.75203i 0 0.113492i 2.10861 + 1.88514i 0 −4.71455 + 2.84036i
109.12 1.40946 − 0.115854i 0 1.97316 − 0.326583i 1.78448 − 1.78448i 0 4.77575i 2.74325 − 0.688904i 0 2.30841 − 2.72189i
325.1 −1.40946 − 0.115854i 0 1.97316 + 0.326583i −1.78448 − 1.78448i 0 − 4.77575i −2.74325 − 0.688904i 0 2.30841 + 2.72189i
325.2 −1.37261 + 0.340511i 0 1.76810 − 0.934777i 2.75203 + 2.75203i 0 − 0.113492i −2.10861 + 1.88514i 0 −4.71455 − 2.84036i
325.3 −1.20402 − 0.741841i 0 0.899344 + 1.78639i −0.431497 − 0.431497i 0 − 0.145064i 0.242383 − 2.81802i 0 0.199430 + 0.839634i
325.4 −1.15725 + 0.812885i 0 0.678437 − 1.88142i −2.39401 − 2.39401i 0 2.42931i 0.744255 + 2.72875i 0 4.71651 + 0.824404i
325.5 −0.936391 + 1.05980i 0 −0.246343 − 1.98477i 1.12951 + 1.12951i 0 − 1.33035i 2.33413 + 1.59745i 0 −2.25471 + 0.139389i
325.6 −0.680919 − 1.23950i 0 −1.07270 + 1.68799i −2.24693 − 2.24693i 0 3.93534i 2.82268 + 0.180219i 0 −1.25508 + 4.31505i
325.7 0.680919 + 1.23950i 0 −1.07270 + 1.68799i 2.24693 + 2.24693i 0 3.93534i −2.82268 − 0.180219i 0 −1.25508 + 4.31505i
325.8 0.936391 − 1.05980i 0 −0.246343 − 1.98477i −1.12951 − 1.12951i 0 − 1.33035i −2.33413 − 1.59745i 0 −2.25471 + 0.139389i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.k.c ✓ 24
3.b odd 2 1 inner 432.2.k.c ✓ 24
4.b odd 2 1 1728.2.k.c 24
12.b even 2 1 1728.2.k.c 24
16.e even 4 1 inner 432.2.k.c ✓ 24
16.f odd 4 1 1728.2.k.c 24
48.i odd 4 1 inner 432.2.k.c ✓ 24
48.k even 4 1 1728.2.k.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.k.c ✓ 24 1.a even 1 1 trivial
432.2.k.c ✓ 24 3.b odd 2 1 inner
432.2.k.c ✓ 24 16.e even 4 1 inner
432.2.k.c ✓ 24 48.i odd 4 1 inner
1728.2.k.c 24 4.b odd 2 1
1728.2.k.c 24 12.b even 2 1
1728.2.k.c 24 16.f odd 4 1
1728.2.k.c 24 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{24} + 510T_{5}^{20} + 89055T_{5}^{16} + 6358980T_{5}^{12} + 163237055T_{5}^{8} + 834181822T_{5}^{4} + 112550881 \) Copy content Toggle raw display
\( T_{11}^{24} + 1494 T_{11}^{20} + 659871 T_{11}^{16} + 78092340 T_{11}^{12} + 1777212719 T_{11}^{8} + \cdots + 244140625 \) Copy content Toggle raw display