Properties

Label 561.2.h.b
Level $561$
Weight $2$
Character orbit 561.h
Analytic conductor $4.480$
Analytic rank $0$
Dimension $4$
CM discriminant -51
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [561,2,Mod(560,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.560");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.h (of order \(2\), degree \(1\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 

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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - 2 q^{4} - \beta_{2} q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - 2 q^{4} - \beta_{2} q^{5} + 3 q^{9} + (\beta_{2} - \beta_1) q^{11} - 2 \beta_{2} q^{12} + (2 \beta_{3} - 1) q^{13} - 3 q^{15} + 4 q^{16} + ( - \beta_{2} - 2 \beta_1) q^{17} + ( - 2 \beta_{3} + 1) q^{19} + 2 \beta_{2} q^{20} + 5 \beta_{2} q^{23} - 2 q^{25} + 3 \beta_{2} q^{27} + ( - 2 \beta_{2} - 4 \beta_1) q^{29} + (\beta_{3} + 4) q^{33} - 6 q^{36} + ( - 3 \beta_{2} - 6 \beta_1) q^{39} + ( - \beta_{2} - 2 \beta_1) q^{41} + ( - 2 \beta_{3} + 1) q^{43} + ( - 2 \beta_{2} + 2 \beta_1) q^{44} - 3 \beta_{2} q^{45} + 4 \beta_{2} q^{48} - 7 q^{49} + (2 \beta_{3} - 1) q^{51} + ( - 4 \beta_{3} + 2) q^{52} + ( - \beta_{3} - 4) q^{55} + (3 \beta_{2} + 6 \beta_1) q^{57} + 6 q^{60} - 8 q^{64} + (3 \beta_{2} + 6 \beta_1) q^{65} - 8 q^{67} + (2 \beta_{2} + 4 \beta_1) q^{68} + 15 q^{69} - 2 \beta_{2} q^{71} - 2 \beta_{2} q^{75} + (4 \beta_{3} - 2) q^{76} - 4 \beta_{2} q^{80} + 9 q^{81} + ( - 2 \beta_{3} + 1) q^{85} + (4 \beta_{3} - 2) q^{87} - 10 \beta_{2} q^{92} + ( - 3 \beta_{2} - 6 \beta_1) q^{95} + (3 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 12 q^{9} - 12 q^{15} + 16 q^{16} - 8 q^{25} + 18 q^{33} - 24 q^{36} - 28 q^{49} - 18 q^{55} + 24 q^{60} - 32 q^{64} - 32 q^{67} + 60 q^{69} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{2} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/561\mathbb{Z}\right)^\times\).

\(n\) \(188\) \(409\) \(496\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
560.1
0.866025 + 2.06155i
0.866025 2.06155i
−0.866025 + 2.06155i
−0.866025 2.06155i
0 −1.73205 −2.00000 1.73205 0 0 0 3.00000 0
560.2 0 −1.73205 −2.00000 1.73205 0 0 0 3.00000 0
560.3 0 1.73205 −2.00000 −1.73205 0 0 0 3.00000 0
560.4 0 1.73205 −2.00000 −1.73205 0 0 0 3.00000 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
51.c odd 2 1 CM by \(\Q(\sqrt{-51}) \)
3.b odd 2 1 inner
11.b odd 2 1 inner
17.b even 2 1 inner
33.d even 2 1 inner
187.b odd 2 1 inner
561.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.h.b 4
3.b odd 2 1 inner 561.2.h.b 4
11.b odd 2 1 inner 561.2.h.b 4
17.b even 2 1 inner 561.2.h.b 4
33.d even 2 1 inner 561.2.h.b 4
51.c odd 2 1 CM 561.2.h.b 4
187.b odd 2 1 inner 561.2.h.b 4
561.h even 2 1 inner 561.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.h.b 4 1.a even 1 1 trivial
561.2.h.b 4 3.b odd 2 1 inner
561.2.h.b 4 11.b odd 2 1 inner
561.2.h.b 4 17.b even 2 1 inner
561.2.h.b 4 33.d even 2 1 inner
561.2.h.b 4 51.c odd 2 1 CM
561.2.h.b 4 187.b odd 2 1 inner
561.2.h.b 4 561.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(561, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 5T^{2} + 121 \) Copy content Toggle raw display
$13$ \( (T^{2} + 51)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 17)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 51)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 68)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 17)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 51)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T + 8)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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