Properties

Label 528.4.b.a
Level $528$
Weight $4$
Character orbit 528.b
Analytic conductor $31.153$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,4,Mod(65,528)]
 
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([0, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.b (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: \(31.1530084830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
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 \(\beta = \sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{3} - 4 \beta q^{5} + (8 \beta + 5) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 4) q^{3} - 4 \beta q^{5} + (8 \beta + 5) q^{9} + 11 \beta q^{11} + ( - 16 \beta + 44) q^{15} + 58 \beta q^{23} - 51 q^{25} + (37 \beta - 68) q^{27} + 340 q^{31} + (44 \beta - 121) q^{33} + 434 q^{37} + ( - 20 \beta + 352) q^{45} - 194 \beta q^{47} + 343 q^{49} + 68 \beta q^{53} + 484 q^{55} + 166 \beta q^{59} - 416 q^{67} + (232 \beta - 638) q^{69} + 310 \beta q^{71} + ( - 51 \beta - 204) q^{75} + (80 \beta - 679) q^{81} - 40 \beta q^{89} + (340 \beta + 1360) q^{93} - 34 q^{97} + (55 \beta - 968) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{3} + 10 q^{9} + 88 q^{15} - 102 q^{25} - 136 q^{27} + 680 q^{31} - 242 q^{33} + 868 q^{37} + 704 q^{45} + 686 q^{49} + 968 q^{55} - 832 q^{67} - 1276 q^{69} - 408 q^{75} - 1358 q^{81} + 2720 q^{93} - 68 q^{97} - 1936 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 1.65831i
0.500000 + 1.65831i
0 4.00000 3.31662i 0 13.2665i 0 0 0 5.00000 26.5330i 0
65.2 0 4.00000 + 3.31662i 0 13.2665i 0 0 0 5.00000 + 26.5330i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.4.b.a 2
3.b odd 2 1 inner 528.4.b.a 2
4.b odd 2 1 33.4.d.a 2
11.b odd 2 1 CM 528.4.b.a 2
12.b even 2 1 33.4.d.a 2
33.d even 2 1 inner 528.4.b.a 2
44.c even 2 1 33.4.d.a 2
132.d odd 2 1 33.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.a 2 4.b odd 2 1
33.4.d.a 2 12.b even 2 1
33.4.d.a 2 44.c even 2 1
33.4.d.a 2 132.d odd 2 1
528.4.b.a 2 1.a even 1 1 trivial
528.4.b.a 2 3.b odd 2 1 inner
528.4.b.a 2 11.b odd 2 1 CM
528.4.b.a 2 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{2} + 176 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} + 176 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1331 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 37004 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 340)^{2} \) Copy content Toggle raw display
$37$ \( (T - 434)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 413996 \) Copy content Toggle raw display
$53$ \( T^{2} + 50864 \) Copy content Toggle raw display
$59$ \( T^{2} + 303116 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 416)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 1057100 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 17600 \) Copy content Toggle raw display
$97$ \( (T + 34)^{2} \) Copy content Toggle raw display
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