Properties

Label 320.3.h.a
Level $320$
Weight $3$
Character orbit 320.h
Self dual yes
Analytic conductor $8.719$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,3,Mod(319,320)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.h (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9} - 20 q^{15} + 16 q^{21} + 44 q^{23} + 25 q^{25} + 8 q^{27} + 22 q^{29} - 20 q^{35} + 62 q^{41} + 76 q^{43} + 35 q^{45} - 4 q^{47} - 33 q^{49} + 58 q^{61} - 28 q^{63} - 116 q^{67} - 176 q^{69} - 100 q^{75} - 95 q^{81} + 76 q^{83} - 88 q^{87} - 142 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
0
0 −4.00000 0 5.00000 0 −4.00000 0 7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.h.a 1
4.b odd 2 1 320.3.h.b 1
5.b even 2 1 320.3.h.b 1
5.c odd 4 2 1600.3.b.f 2
8.b even 2 1 20.3.d.a 1
8.d odd 2 1 20.3.d.b yes 1
16.e even 4 2 1280.3.e.c 2
16.f odd 4 2 1280.3.e.b 2
20.d odd 2 1 CM 320.3.h.a 1
20.e even 4 2 1600.3.b.f 2
24.f even 2 1 180.3.f.a 1
24.h odd 2 1 180.3.f.b 1
40.e odd 2 1 20.3.d.a 1
40.f even 2 1 20.3.d.b yes 1
40.i odd 4 2 100.3.b.c 2
40.k even 4 2 100.3.b.c 2
80.k odd 4 2 1280.3.e.c 2
80.q even 4 2 1280.3.e.b 2
120.i odd 2 1 180.3.f.a 1
120.m even 2 1 180.3.f.b 1
120.q odd 4 2 900.3.c.h 2
120.w even 4 2 900.3.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 8.b even 2 1
20.3.d.a 1 40.e odd 2 1
20.3.d.b yes 1 8.d odd 2 1
20.3.d.b yes 1 40.f even 2 1
100.3.b.c 2 40.i odd 4 2
100.3.b.c 2 40.k even 4 2
180.3.f.a 1 24.f even 2 1
180.3.f.a 1 120.i odd 2 1
180.3.f.b 1 24.h odd 2 1
180.3.f.b 1 120.m even 2 1
320.3.h.a 1 1.a even 1 1 trivial
320.3.h.a 1 20.d odd 2 1 CM
320.3.h.b 1 4.b odd 2 1
320.3.h.b 1 5.b even 2 1
900.3.c.h 2 120.q odd 4 2
900.3.c.h 2 120.w even 4 2
1280.3.e.b 2 16.f odd 4 2
1280.3.e.b 2 80.q even 4 2
1280.3.e.c 2 16.e even 4 2
1280.3.e.c 2 80.k odd 4 2
1600.3.b.f 2 5.c odd 4 2
1600.3.b.f 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 4 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 4 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 44 \) Copy content Toggle raw display
$29$ \( T - 22 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 62 \) Copy content Toggle raw display
$43$ \( T - 76 \) Copy content Toggle raw display
$47$ \( T + 4 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 58 \) Copy content Toggle raw display
$67$ \( T + 116 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 76 \) Copy content Toggle raw display
$89$ \( T + 142 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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