Properties

Label 560.5.p.a
Level $560$
Weight $5$
Character orbit 560.p
Self dual yes
Analytic conductor $57.887$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,5,Mod(209,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 560.p (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: \(57.8871793270\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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)
 
\(f(q)\) \(=\) \( q - 17 q^{3} - 25 q^{5} + 49 q^{7} + 208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 17 q^{3} - 25 q^{5} + 49 q^{7} + 208 q^{9} + 73 q^{11} - 23 q^{13} + 425 q^{15} - 263 q^{17} - 833 q^{21} + 625 q^{25} - 2159 q^{27} - 1153 q^{29} - 1241 q^{33} - 1225 q^{35} + 391 q^{39} - 5200 q^{45} - 3457 q^{47} + 2401 q^{49} + 4471 q^{51} - 1825 q^{55} + 10192 q^{63} + 575 q^{65} + 10078 q^{71} + 9502 q^{73} - 10625 q^{75} + 3577 q^{77} - 12167 q^{79} + 19855 q^{81} - 6382 q^{83} + 6575 q^{85} + 19601 q^{87} - 1127 q^{91} - 3383 q^{97} + 15184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0
0 −17.0000 0 −25.0000 0 49.0000 0 208.000 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.p.a 1
4.b odd 2 1 35.5.c.b yes 1
5.b even 2 1 560.5.p.b 1
7.b odd 2 1 560.5.p.b 1
12.b even 2 1 315.5.e.b 1
20.d odd 2 1 35.5.c.a 1
20.e even 4 2 175.5.d.c 2
28.d even 2 1 35.5.c.a 1
35.c odd 2 1 CM 560.5.p.a 1
60.h even 2 1 315.5.e.a 1
84.h odd 2 1 315.5.e.a 1
140.c even 2 1 35.5.c.b yes 1
140.j odd 4 2 175.5.d.c 2
420.o odd 2 1 315.5.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 20.d odd 2 1
35.5.c.a 1 28.d even 2 1
35.5.c.b yes 1 4.b odd 2 1
35.5.c.b yes 1 140.c even 2 1
175.5.d.c 2 20.e even 4 2
175.5.d.c 2 140.j odd 4 2
315.5.e.a 1 60.h even 2 1
315.5.e.a 1 84.h odd 2 1
315.5.e.b 1 12.b even 2 1
315.5.e.b 1 420.o odd 2 1
560.5.p.a 1 1.a even 1 1 trivial
560.5.p.a 1 35.c odd 2 1 CM
560.5.p.b 1 5.b even 2 1
560.5.p.b 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 17 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 17 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 73 \) Copy content Toggle raw display
$13$ \( T + 23 \) Copy content Toggle raw display
$17$ \( T + 263 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 1153 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 3457 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T - 10078 \) Copy content Toggle raw display
$73$ \( T - 9502 \) Copy content Toggle raw display
$79$ \( T + 12167 \) Copy content Toggle raw display
$83$ \( T + 6382 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 3383 \) Copy content Toggle raw display
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