Properties

Label 528.3.i.a
Level $528$
Weight $3$
Character orbit 528.i
Analytic conductor $14.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.3869579582\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q - 3 q^{3} - 2 \beta q^{5} + 8 q^{7} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 2 \beta q^{5} + 8 q^{7} + 9 q^{9} - \beta q^{11} + 4 q^{13} + 6 \beta q^{15} + 4 \beta q^{17} + 6 q^{19} - 24 q^{21} - 2 \beta q^{23} - 19 q^{25} - 27 q^{27} - 12 \beta q^{29} + 26 q^{31} + 3 \beta q^{33} - 16 \beta q^{35} + 30 q^{37} - 12 q^{39} + 4 \beta q^{41} - 42 q^{43} - 18 \beta q^{45} - 26 \beta q^{47} + 15 q^{49} - 12 \beta q^{51} - 18 \beta q^{53} - 22 q^{55} - 18 q^{57} - 20 \beta q^{59} + 12 q^{61} + 72 q^{63} - 8 \beta q^{65} - 2 q^{67} + 6 \beta q^{69} + 18 \beta q^{71} - 74 q^{73} + 57 q^{75} - 8 \beta q^{77} + 40 q^{79} + 81 q^{81} + 12 \beta q^{83} + 88 q^{85} + 36 \beta q^{87} - 36 \beta q^{89} + 32 q^{91} - 78 q^{93} - 12 \beta q^{95} + 62 q^{97} - 9 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 16 q^{7} + 18 q^{9} + 8 q^{13} + 12 q^{19} - 48 q^{21} - 38 q^{25} - 54 q^{27} + 52 q^{31} + 60 q^{37} - 24 q^{39} - 84 q^{43} + 30 q^{49} - 44 q^{55} - 36 q^{57} + 24 q^{61} + 144 q^{63} - 4 q^{67} - 148 q^{73} + 114 q^{75} + 80 q^{79} + 162 q^{81} + 176 q^{85} + 64 q^{91} - 156 q^{93} + 124 q^{97}+O(q^{100}) \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
0.500000 + 1.65831i
0.500000 1.65831i
0 −3.00000 0 6.63325i 0 8.00000 0 9.00000 0
353.2 0 −3.00000 0 6.63325i 0 8.00000 0 9.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.3.i.a 2
3.b odd 2 1 inner 528.3.i.a 2
4.b odd 2 1 33.3.b.a 2
12.b even 2 1 33.3.b.a 2
44.c even 2 1 363.3.b.d 2
44.g even 10 4 363.3.h.d 8
44.h odd 10 4 363.3.h.e 8
132.d odd 2 1 363.3.b.d 2
132.n odd 10 4 363.3.h.d 8
132.o even 10 4 363.3.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.a 2 4.b odd 2 1
33.3.b.a 2 12.b even 2 1
363.3.b.d 2 44.c even 2 1
363.3.b.d 2 132.d odd 2 1
363.3.h.d 8 44.g even 10 4
363.3.h.d 8 132.n odd 10 4
363.3.h.e 8 44.h odd 10 4
363.3.h.e 8 132.o even 10 4
528.3.i.a 2 1.a even 1 1 trivial
528.3.i.a 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 44 \) acting on \(S_{3}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 44 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 11 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 176 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 44 \) Copy content Toggle raw display
$29$ \( T^{2} + 1584 \) Copy content Toggle raw display
$31$ \( (T - 26)^{2} \) Copy content Toggle raw display
$37$ \( (T - 30)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 176 \) Copy content Toggle raw display
$43$ \( (T + 42)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7436 \) Copy content Toggle raw display
$53$ \( T^{2} + 3564 \) Copy content Toggle raw display
$59$ \( T^{2} + 4400 \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3564 \) Copy content Toggle raw display
$73$ \( (T + 74)^{2} \) Copy content Toggle raw display
$79$ \( (T - 40)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1584 \) Copy content Toggle raw display
$89$ \( T^{2} + 14256 \) Copy content Toggle raw display
$97$ \( (T - 62)^{2} \) Copy content Toggle raw display
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