Properties

Label 560.2.q.k
Level $560$
Weight $2$
Character orbit 560.q
Analytic conductor $4.472$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(81,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), degree \(2\), 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: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9} + (2 \beta_{2} + 2 \beta_1 + 2) q^{11} + ( - 2 \beta_{3} - 2) q^{13} + (\beta_{3} - 1) q^{15} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + (2 \beta_{3} + 2 \beta_1) q^{19} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + ( - \beta_{3} - 1) q^{27} - q^{29} + ( - 6 \beta_{2} - 6) q^{31} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{33} + (\beta_{3} + 2 \beta_1 + 1) q^{35} + (2 \beta_{2} + 2) q^{39} + ( - 2 \beta_{3} - 5) q^{41} + (\beta_{3} - 5) q^{43} - 2 \beta_1 q^{45} - 2 \beta_{2} q^{47} + (2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{49} + 2 \beta_{2} q^{51} + (4 \beta_{2} + 2 \beta_1 + 4) q^{53} + (2 \beta_{3} - 2) q^{55} + (2 \beta_{3} - 4) q^{57} + ( - 4 \beta_{2} - 6 \beta_1 - 4) q^{59} + ( - 6 \beta_{3} - 3 \beta_{2} - 6 \beta_1) q^{61} + ( - 2 \beta_{3} + 8 \beta_{2} + 4) q^{63} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{65} + (11 \beta_{2} + \beta_1 + 11) q^{67} + (2 \beta_{3} - 3) q^{69} + ( - 6 \beta_{3} + 4) q^{71} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{75} + ( - 4 \beta_{3} + 2 \beta_{2} + 8) q^{77} + (2 \beta_{3} - 12 \beta_{2} + 2 \beta_1) q^{79} + (\beta_{2} + 6 \beta_1 + 1) q^{81} + ( - 9 \beta_{3} - 1) q^{83} + (2 \beta_{3} + 2) q^{85} + ( - \beta_{2} - \beta_1 - 1) q^{87} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{89} + (4 \beta_{3} - 2 \beta_{2} + 6) q^{91} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{93} - 2 \beta_1 q^{95} + (4 \beta_{3} + 6) q^{97} + (4 \beta_{3} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 4 q^{11} - 8 q^{13} - 4 q^{15} - 4 q^{17} + 14 q^{21} - 2 q^{23} - 2 q^{25} - 4 q^{27} - 4 q^{29} - 12 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{39} - 20 q^{41} - 20 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{51} + 8 q^{53} - 8 q^{55} - 16 q^{57} - 8 q^{59} + 6 q^{61} + 4 q^{65} + 22 q^{67} - 12 q^{69} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 28 q^{77} + 24 q^{79} + 2 q^{81} - 4 q^{83} + 8 q^{85} - 2 q^{87} + 6 q^{89} + 28 q^{91} + 12 q^{93} + 24 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 −0.207107 + 0.358719i 0 −0.500000 0.866025i 0 −2.62132 0.358719i 0 1.41421 + 2.44949i 0
81.2 0 1.20711 2.09077i 0 −0.500000 0.866025i 0 1.62132 + 2.09077i 0 −1.41421 2.44949i 0
401.1 0 −0.207107 0.358719i 0 −0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 1.41421 2.44949i 0
401.2 0 1.20711 + 2.09077i 0 −0.500000 + 0.866025i 0 1.62132 2.09077i 0 −1.41421 + 2.44949i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.q.k 4
4.b odd 2 1 35.2.e.a 4
7.c even 3 1 inner 560.2.q.k 4
7.c even 3 1 3920.2.a.bq 2
7.d odd 6 1 3920.2.a.bv 2
12.b even 2 1 315.2.j.e 4
20.d odd 2 1 175.2.e.c 4
20.e even 4 2 175.2.k.a 8
28.d even 2 1 245.2.e.e 4
28.f even 6 1 245.2.a.g 2
28.f even 6 1 245.2.e.e 4
28.g odd 6 1 35.2.e.a 4
28.g odd 6 1 245.2.a.h 2
84.j odd 6 1 2205.2.a.q 2
84.n even 6 1 315.2.j.e 4
84.n even 6 1 2205.2.a.n 2
140.p odd 6 1 175.2.e.c 4
140.p odd 6 1 1225.2.a.k 2
140.s even 6 1 1225.2.a.m 2
140.w even 12 2 175.2.k.a 8
140.w even 12 2 1225.2.b.g 4
140.x odd 12 2 1225.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 4.b odd 2 1
35.2.e.a 4 28.g odd 6 1
175.2.e.c 4 20.d odd 2 1
175.2.e.c 4 140.p odd 6 1
175.2.k.a 8 20.e even 4 2
175.2.k.a 8 140.w even 12 2
245.2.a.g 2 28.f even 6 1
245.2.a.h 2 28.g odd 6 1
245.2.e.e 4 28.d even 2 1
245.2.e.e 4 28.f even 6 1
315.2.j.e 4 12.b even 2 1
315.2.j.e 4 84.n even 6 1
560.2.q.k 4 1.a even 1 1 trivial
560.2.q.k 4 7.c even 3 1 inner
1225.2.a.k 2 140.p odd 6 1
1225.2.a.m 2 140.s even 6 1
1225.2.b.g 4 140.w even 12 2
1225.2.b.h 4 140.x odd 12 2
2205.2.a.n 2 84.n even 6 1
2205.2.a.q 2 84.j odd 6 1
3920.2.a.bq 2 7.c even 3 1
3920.2.a.bv 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} + 20T_{11}^{2} + 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 10 T + 17)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 23)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$67$ \( T^{4} - 22 T^{3} + 365 T^{2} + \cdots + 14161 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} - 24 T^{3} + 440 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 161)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 59 T^{2} + 138 T + 529 \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
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