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Properties

Label 320.4.o.c
Level $320$
Weight $4$
Character orbit 320.o
Analytic conductor $18.881$
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] = [320,4,Mod(223,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.o (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: \(18.8806112018\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{7} + 88 q^{15} - 72 q^{17} + 296 q^{23} - 600 q^{25} + 672 q^{33} + 112 q^{39} + 448 q^{41} + 2776 q^{47} - 160 q^{55} - 2272 q^{57} + 3192 q^{63} - 1720 q^{65} + 584 q^{73} - 5248 q^{79} + 6600 q^{81} + 5328 q^{87} - 8096 q^{95} - 4504 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 0 −5.74980 − 5.74980i 0 −7.97890 − 7.83180i 0 −2.03807 − 2.03807i 0 39.1204i 0
223.2 0 −5.34367 − 5.34367i 0 −4.10965 + 10.3976i 0 −22.7795 − 22.7795i 0 30.1095i 0
223.3 0 −4.83541 − 4.83541i 0 0.463770 − 11.1707i 0 5.05184 + 5.05184i 0 19.7625i 0
223.4 0 −2.79085 − 2.79085i 0 7.62030 + 8.18113i 0 24.4209 + 24.4209i 0 − 11.4223i 0
223.5 0 −2.16177 − 2.16177i 0 11.1305 − 1.05448i 0 0.200550 + 0.200550i 0 − 17.6535i 0
223.6 0 −1.59426 − 1.59426i 0 −6.10538 + 9.36613i 0 −6.85573 − 6.85573i 0 − 21.9167i 0
223.7 0 1.59426 + 1.59426i 0 6.10538 − 9.36613i 0 −6.85573 − 6.85573i 0 − 21.9167i 0
223.8 0 2.16177 + 2.16177i 0 −11.1305 + 1.05448i 0 0.200550 + 0.200550i 0 − 17.6535i 0
223.9 0 2.79085 + 2.79085i 0 −7.62030 − 8.18113i 0 24.4209 + 24.4209i 0 − 11.4223i 0
223.10 0 4.83541 + 4.83541i 0 −0.463770 + 11.1707i 0 5.05184 + 5.05184i 0 19.7625i 0
223.11 0 5.34367 + 5.34367i 0 4.10965 − 10.3976i 0 −22.7795 − 22.7795i 0 30.1095i 0
223.12 0 5.74980 + 5.74980i 0 7.97890 + 7.83180i 0 −2.03807 − 2.03807i 0 39.1204i 0
287.1 0 −5.74980 + 5.74980i 0 −7.97890 + 7.83180i 0 −2.03807 + 2.03807i 0 − 39.1204i 0
287.2 0 −5.34367 + 5.34367i 0 −4.10965 − 10.3976i 0 −22.7795 + 22.7795i 0 − 30.1095i 0
287.3 0 −4.83541 + 4.83541i 0 0.463770 + 11.1707i 0 5.05184 − 5.05184i 0 − 19.7625i 0
287.4 0 −2.79085 + 2.79085i 0 7.62030 − 8.18113i 0 24.4209 − 24.4209i 0 11.4223i 0
287.5 0 −2.16177 + 2.16177i 0 11.1305 + 1.05448i 0 0.200550 − 0.200550i 0 17.6535i 0
287.6 0 −1.59426 + 1.59426i 0 −6.10538 − 9.36613i 0 −6.85573 + 6.85573i 0 21.9167i 0
287.7 0 1.59426 − 1.59426i 0 6.10538 + 9.36613i 0 −6.85573 + 6.85573i 0 21.9167i 0
287.8 0 2.16177 − 2.16177i 0 −11.1305 − 1.05448i 0 0.200550 − 0.200550i 0 17.6535i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
20.e even 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.o.c ✓ 24
4.b odd 2 1 320.4.o.d yes 24
5.c odd 4 1 320.4.o.d yes 24
8.b even 2 1 inner 320.4.o.c ✓ 24
8.d odd 2 1 320.4.o.d yes 24
20.e even 4 1 inner 320.4.o.c ✓ 24
40.i odd 4 1 320.4.o.d yes 24
40.k even 4 1 inner 320.4.o.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.4.o.c ✓ 24 1.a even 1 1 trivial
320.4.o.c ✓ 24 8.b even 2 1 inner
320.4.o.c ✓ 24 20.e even 4 1 inner
320.4.o.c ✓ 24 40.k even 4 1 inner
320.4.o.d yes 24 4.b odd 2 1
320.4.o.d yes 24 5.c odd 4 1
320.4.o.d yes 24 8.d odd 2 1
320.4.o.d yes 24 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{24} + 10176 T_{3}^{20} + 34475520 T_{3}^{16} + 42487360640 T_{3}^{12} + 12021454417920 T_{3}^{8} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
\( T_{7}^{12} + 4 T_{7}^{11} + 8 T_{7}^{10} + 3400 T_{7}^{9} + 1269184 T_{7}^{8} + 9113936 T_{7}^{7} + \cdots + 3969000000 \) Copy content Toggle raw display