Properties

Label 528.2.d.b
Level $528$
Weight $2$
Character orbit 528.d
Analytic conductor $4.216$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,2,Mod(287,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("528.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.d (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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} - 2 \beta q^{5} + \beta q^{7} + ( - 2 \beta - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{3} - 2 \beta q^{5} + \beta q^{7} + ( - 2 \beta - 1) q^{9} - q^{11} - 6 q^{13} + (2 \beta + 4) q^{15} + \beta q^{17} - 5 \beta q^{19} + ( - \beta - 2) q^{21} - 8 q^{23} - 3 q^{25} + (\beta + 5) q^{27} - 3 \beta q^{29} + 2 \beta q^{31} + ( - \beta + 1) q^{33} + 4 q^{35} + ( - 6 \beta + 6) q^{39} - 7 \beta q^{41} - 3 \beta q^{43} + (2 \beta - 8) q^{45} - 10 q^{47} + 5 q^{49} + ( - \beta - 2) q^{51} - 4 \beta q^{53} + 2 \beta q^{55} + (5 \beta + 10) q^{57} - 4 q^{59} - 2 q^{61} + ( - \beta + 4) q^{63} + 12 \beta q^{65} + 6 \beta q^{67} + ( - 8 \beta + 8) q^{69} + 10 q^{71} + 2 q^{73} + ( - 3 \beta + 3) q^{75} - \beta q^{77} + 9 \beta q^{79} + (4 \beta - 7) q^{81} - 4 q^{83} + 4 q^{85} + (3 \beta + 6) q^{87} + 8 \beta q^{89} - 6 \beta q^{91} + ( - 2 \beta - 4) q^{93} - 20 q^{95} + 8 q^{97} + (2 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{9} - 2 q^{11} - 12 q^{13} + 8 q^{15} - 4 q^{21} - 16 q^{23} - 6 q^{25} + 10 q^{27} + 2 q^{33} + 8 q^{35} + 12 q^{39} - 16 q^{45} - 20 q^{47} + 10 q^{49} - 4 q^{51} + 20 q^{57} - 8 q^{59} - 4 q^{61} + 8 q^{63} + 16 q^{69} + 20 q^{71} + 4 q^{73} + 6 q^{75} - 14 q^{81} - 8 q^{83} + 8 q^{85} + 12 q^{87} - 8 q^{93} - 40 q^{95} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
1.41421i
1.41421i
0 −1.00000 1.41421i 0 2.82843i 0 1.41421i 0 −1.00000 + 2.82843i 0
287.2 0 −1.00000 + 1.41421i 0 2.82843i 0 1.41421i 0 −1.00000 2.82843i 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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.d.b 2
3.b odd 2 1 528.2.d.e yes 2
4.b odd 2 1 528.2.d.e yes 2
8.b even 2 1 2112.2.d.g 2
8.d odd 2 1 2112.2.d.c 2
12.b even 2 1 inner 528.2.d.b 2
24.f even 2 1 2112.2.d.g 2
24.h odd 2 1 2112.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.d.b 2 1.a even 1 1 trivial
528.2.d.b 2 12.b even 2 1 inner
528.2.d.e yes 2 3.b odd 2 1
528.2.d.e yes 2 4.b odd 2 1
2112.2.d.c 2 8.d odd 2 1
2112.2.d.c 2 24.h odd 2 1
2112.2.d.g 2 8.b even 2 1
2112.2.d.g 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 2 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 50 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 8 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 98 \) Copy content Toggle raw display
$43$ \( T^{2} + 18 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 32 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 72 \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 162 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 128 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
show more
show less