Properties

Label 320.6.a.o
Level $320$
Weight $6$
Character orbit 320.a
Self dual yes
Analytic conductor $51.323$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,6,Mod(1,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

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: yes
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 24 q^{3} - 25 q^{5} + 172 q^{7} + 333 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 24 q^{3} - 25 q^{5} + 172 q^{7} + 333 q^{9} + 132 q^{11} + 946 q^{13} - 600 q^{15} - 222 q^{17} + 500 q^{19} + 4128 q^{21} - 3564 q^{23} + 625 q^{25} + 2160 q^{27} - 2190 q^{29} - 2312 q^{31} + 3168 q^{33} - 4300 q^{35} + 11242 q^{37} + 22704 q^{39} + 1242 q^{41} + 20624 q^{43} - 8325 q^{45} - 6588 q^{47} + 12777 q^{49} - 5328 q^{51} + 21066 q^{53} - 3300 q^{55} + 12000 q^{57} + 7980 q^{59} - 16622 q^{61} + 57276 q^{63} - 23650 q^{65} + 1808 q^{67} - 85536 q^{69} + 24528 q^{71} + 20474 q^{73} + 15000 q^{75} + 22704 q^{77} + 46240 q^{79} - 29079 q^{81} - 51576 q^{83} + 5550 q^{85} - 52560 q^{87} - 110310 q^{89} + 162712 q^{91} - 55488 q^{93} - 12500 q^{95} - 78382 q^{97} + 43956 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 24.0000 0 −25.0000 0 172.000 0 333.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.a.o 1
4.b odd 2 1 320.6.a.b 1
8.b even 2 1 80.6.a.a 1
8.d odd 2 1 10.6.a.b 1
24.f even 2 1 90.6.a.d 1
24.h odd 2 1 720.6.a.j 1
40.e odd 2 1 50.6.a.d 1
40.f even 2 1 400.6.a.n 1
40.i odd 4 2 400.6.c.b 2
40.k even 4 2 50.6.b.a 2
56.e even 2 1 490.6.a.a 1
120.m even 2 1 450.6.a.l 1
120.q odd 4 2 450.6.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 8.d odd 2 1
50.6.a.d 1 40.e odd 2 1
50.6.b.a 2 40.k even 4 2
80.6.a.a 1 8.b even 2 1
90.6.a.d 1 24.f even 2 1
320.6.a.b 1 4.b odd 2 1
320.6.a.o 1 1.a even 1 1 trivial
400.6.a.n 1 40.f even 2 1
400.6.c.b 2 40.i odd 4 2
450.6.a.l 1 120.m even 2 1
450.6.c.h 2 120.q odd 4 2
490.6.a.a 1 56.e even 2 1
720.6.a.j 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 24 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T - 172 \) Copy content Toggle raw display
$11$ \( T - 132 \) Copy content Toggle raw display
$13$ \( T - 946 \) Copy content Toggle raw display
$17$ \( T + 222 \) Copy content Toggle raw display
$19$ \( T - 500 \) Copy content Toggle raw display
$23$ \( T + 3564 \) Copy content Toggle raw display
$29$ \( T + 2190 \) Copy content Toggle raw display
$31$ \( T + 2312 \) Copy content Toggle raw display
$37$ \( T - 11242 \) Copy content Toggle raw display
$41$ \( T - 1242 \) Copy content Toggle raw display
$43$ \( T - 20624 \) Copy content Toggle raw display
$47$ \( T + 6588 \) Copy content Toggle raw display
$53$ \( T - 21066 \) Copy content Toggle raw display
$59$ \( T - 7980 \) Copy content Toggle raw display
$61$ \( T + 16622 \) Copy content Toggle raw display
$67$ \( T - 1808 \) Copy content Toggle raw display
$71$ \( T - 24528 \) Copy content Toggle raw display
$73$ \( T - 20474 \) Copy content Toggle raw display
$79$ \( T - 46240 \) Copy content Toggle raw display
$83$ \( T + 51576 \) Copy content Toggle raw display
$89$ \( T + 110310 \) Copy content Toggle raw display
$97$ \( T + 78382 \) Copy content Toggle raw display
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