Properties

Label 528.2.b.a
Level $528$
Weight $2$
Character orbit 528.b
Analytic conductor $4.216$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 1 - 2 \beta ) q^{5} + ( -3 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 1 - 2 \beta ) q^{5} + ( -3 + \beta ) q^{9} + ( 1 - 2 \beta ) q^{11} + ( -6 + \beta ) q^{15} + ( -1 + 2 \beta ) q^{23} -6 q^{25} + ( 3 + 2 \beta ) q^{27} -5 q^{31} + ( -6 + \beta ) q^{33} -7 q^{37} + ( 3 + 5 \beta ) q^{45} + ( 2 - 4 \beta ) q^{47} + 7 q^{49} + ( 4 - 8 \beta ) q^{53} -11 q^{55} + ( -1 + 2 \beta ) q^{59} + 13 q^{67} + ( 6 - \beta ) q^{69} + ( 5 - 10 \beta ) q^{71} + 6 \beta q^{75} + ( 6 - 5 \beta ) q^{81} + ( -5 + 10 \beta ) q^{89} + 5 \beta q^{93} + 17 q^{97} + ( 3 + 5 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{9} + O(q^{10}) \) \( 2 q - q^{3} - 5 q^{9} - 11 q^{15} - 12 q^{25} + 8 q^{27} - 10 q^{31} - 11 q^{33} - 14 q^{37} + 11 q^{45} + 14 q^{49} - 22 q^{55} + 26 q^{67} + 11 q^{69} + 6 q^{75} + 7 q^{81} + 5 q^{93} + 34 q^{97} + 11 q^{99} + O(q^{100}) \)

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.

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 −0.500000 1.65831i 0 3.31662i 0 0 0 −2.50000 + 1.65831i 0
65.2 0 −0.500000 + 1.65831i 0 3.31662i 0 0 0 −2.50000 1.65831i 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 528.2.b.a 2
3.b odd 2 1 inner 528.2.b.a 2
4.b odd 2 1 33.2.d.a 2
8.b even 2 1 2112.2.b.f 2
8.d odd 2 1 2112.2.b.e 2
11.b odd 2 1 CM 528.2.b.a 2
12.b even 2 1 33.2.d.a 2
20.d odd 2 1 825.2.f.a 2
20.e even 4 2 825.2.d.a 4
24.f even 2 1 2112.2.b.e 2
24.h odd 2 1 2112.2.b.f 2
33.d even 2 1 inner 528.2.b.a 2
36.f odd 6 2 891.2.g.a 4
36.h even 6 2 891.2.g.a 4
44.c even 2 1 33.2.d.a 2
44.g even 10 4 363.2.f.c 8
44.h odd 10 4 363.2.f.c 8
60.h even 2 1 825.2.f.a 2
60.l odd 4 2 825.2.d.a 4
88.b odd 2 1 2112.2.b.f 2
88.g even 2 1 2112.2.b.e 2
132.d odd 2 1 33.2.d.a 2
132.n odd 10 4 363.2.f.c 8
132.o even 10 4 363.2.f.c 8
220.g even 2 1 825.2.f.a 2
220.i odd 4 2 825.2.d.a 4
264.m even 2 1 2112.2.b.f 2
264.p odd 2 1 2112.2.b.e 2
396.k even 6 2 891.2.g.a 4
396.o odd 6 2 891.2.g.a 4
660.g odd 2 1 825.2.f.a 2
660.q even 4 2 825.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 4.b odd 2 1
33.2.d.a 2 12.b even 2 1
33.2.d.a 2 44.c even 2 1
33.2.d.a 2 132.d odd 2 1
363.2.f.c 8 44.g even 10 4
363.2.f.c 8 44.h odd 10 4
363.2.f.c 8 132.n odd 10 4
363.2.f.c 8 132.o even 10 4
528.2.b.a 2 1.a even 1 1 trivial
528.2.b.a 2 3.b odd 2 1 inner
528.2.b.a 2 11.b odd 2 1 CM
528.2.b.a 2 33.d even 2 1 inner
825.2.d.a 4 20.e even 4 2
825.2.d.a 4 60.l odd 4 2
825.2.d.a 4 220.i odd 4 2
825.2.d.a 4 660.q even 4 2
825.2.f.a 2 20.d odd 2 1
825.2.f.a 2 60.h even 2 1
825.2.f.a 2 220.g even 2 1
825.2.f.a 2 660.g odd 2 1
891.2.g.a 4 36.f odd 6 2
891.2.g.a 4 36.h even 6 2
891.2.g.a 4 396.k even 6 2
891.2.g.a 4 396.o odd 6 2
2112.2.b.e 2 8.d odd 2 1
2112.2.b.e 2 24.f even 2 1
2112.2.b.e 2 88.g even 2 1
2112.2.b.e 2 264.p odd 2 1
2112.2.b.f 2 8.b even 2 1
2112.2.b.f 2 24.h odd 2 1
2112.2.b.f 2 88.b odd 2 1
2112.2.b.f 2 264.m even 2 1

Hecke kernels

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

\( T_{5}^{2} + 11 \)
\( T_{17} \)
\( T_{29} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T + T^{2} \)
$5$ \( 11 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 11 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 11 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( ( 7 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 44 + T^{2} \)
$53$ \( 176 + T^{2} \)
$59$ \( 11 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( -13 + T )^{2} \)
$71$ \( 275 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 275 + T^{2} \)
$97$ \( ( -17 + T )^{2} \)
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