Properties

Label 560.2.g.a
Level $560$
Weight $2$
Character orbit 560.g
Analytic conductor $4.472$
Analytic rank $1$
Dimension $2$
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(449,560)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.g (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: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + ( - i - 2) q^{5} - i q^{7} - 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + ( - i - 2) q^{5} - i q^{7} - 6 q^{9} - 3 q^{11} - i q^{13} + ( - 6 i + 3) q^{15} - 5 i q^{17} - 8 q^{19} + 3 q^{21} + 2 i q^{23} + (4 i + 3) q^{25} - 9 i q^{27} + q^{29} + 2 q^{31} - 9 i q^{33} + (2 i - 1) q^{35} + 10 i q^{37} + 3 q^{39} - 6 q^{41} - 4 i q^{43} + (6 i + 12) q^{45} - 11 i q^{47} - q^{49} + 15 q^{51} - 6 i q^{53} + (3 i + 6) q^{55} - 24 i q^{57} - 10 q^{59} + 6 i q^{63} + (2 i - 1) q^{65} + 10 i q^{67} - 6 q^{69} + 10 i q^{73} + (9 i - 12) q^{75} + 3 i q^{77} - 7 q^{79} + 9 q^{81} + 12 i q^{83} + (10 i - 5) q^{85} + 3 i q^{87} - 8 q^{89} - q^{91} + 6 i q^{93} + (8 i + 16) q^{95} + 3 i q^{97} + 18 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 12 q^{9} - 6 q^{11} + 6 q^{15} - 16 q^{19} + 6 q^{21} + 6 q^{25} + 2 q^{29} + 4 q^{31} - 2 q^{35} + 6 q^{39} - 12 q^{41} + 24 q^{45} - 2 q^{49} + 30 q^{51} + 12 q^{55} - 20 q^{59} - 2 q^{65} - 12 q^{69} - 24 q^{75} - 14 q^{79} + 18 q^{81} - 10 q^{85} - 16 q^{89} - 2 q^{91} + 32 q^{95} + 36 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 3.00000i 0 −2.00000 + 1.00000i 0 1.00000i 0 −6.00000 0
449.2 0 3.00000i 0 −2.00000 1.00000i 0 1.00000i 0 −6.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.g.a 2
3.b odd 2 1 5040.2.t.s 2
4.b odd 2 1 140.2.e.a 2
5.b even 2 1 inner 560.2.g.a 2
5.c odd 4 1 2800.2.a.a 1
5.c odd 4 1 2800.2.a.bf 1
8.b even 2 1 2240.2.g.f 2
8.d odd 2 1 2240.2.g.e 2
12.b even 2 1 1260.2.k.c 2
15.d odd 2 1 5040.2.t.s 2
20.d odd 2 1 140.2.e.a 2
20.e even 4 1 700.2.a.a 1
20.e even 4 1 700.2.a.j 1
28.d even 2 1 980.2.e.b 2
28.f even 6 2 980.2.q.c 4
28.g odd 6 2 980.2.q.f 4
40.e odd 2 1 2240.2.g.e 2
40.f even 2 1 2240.2.g.f 2
60.h even 2 1 1260.2.k.c 2
60.l odd 4 1 6300.2.a.c 1
60.l odd 4 1 6300.2.a.t 1
140.c even 2 1 980.2.e.b 2
140.j odd 4 1 4900.2.a.b 1
140.j odd 4 1 4900.2.a.w 1
140.p odd 6 2 980.2.q.f 4
140.s even 6 2 980.2.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 4.b odd 2 1
140.2.e.a 2 20.d odd 2 1
560.2.g.a 2 1.a even 1 1 trivial
560.2.g.a 2 5.b even 2 1 inner
700.2.a.a 1 20.e even 4 1
700.2.a.j 1 20.e even 4 1
980.2.e.b 2 28.d even 2 1
980.2.e.b 2 140.c even 2 1
980.2.q.c 4 28.f even 6 2
980.2.q.c 4 140.s even 6 2
980.2.q.f 4 28.g odd 6 2
980.2.q.f 4 140.p odd 6 2
1260.2.k.c 2 12.b even 2 1
1260.2.k.c 2 60.h even 2 1
2240.2.g.e 2 8.d odd 2 1
2240.2.g.e 2 40.e odd 2 1
2240.2.g.f 2 8.b even 2 1
2240.2.g.f 2 40.f even 2 1
2800.2.a.a 1 5.c odd 4 1
2800.2.a.bf 1 5.c odd 4 1
4900.2.a.b 1 140.j odd 4 1
4900.2.a.w 1 140.j odd 4 1
5040.2.t.s 2 3.b odd 2 1
5040.2.t.s 2 15.d odd 2 1
6300.2.a.c 1 60.l odd 4 1
6300.2.a.t 1 60.l odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 121 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T + 7)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 9 \) Copy content Toggle raw display
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