Properties

Label 165.2.d.a
Level $165$
Weight $2$
Character orbit 165.d
Analytic conductor $1.318$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.31753163335\)
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: yes
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.

\(f(q)\) \(=\) \( q -\beta q^{3} + 2 q^{4} + ( 1 + \beta ) q^{5} + ( -3 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + 2 q^{4} + ( 1 + \beta ) q^{5} + ( -3 + \beta ) q^{9} + ( 1 - 2 \beta ) q^{11} -2 \beta q^{12} + ( 3 - 2 \beta ) q^{15} + 4 q^{16} + ( 2 + 2 \beta ) q^{20} -9 q^{23} + ( -2 + 3 \beta ) q^{25} + ( 3 + 2 \beta ) q^{27} -5 q^{31} + ( -6 + \beta ) q^{33} + ( -6 + 2 \beta ) q^{36} + ( -3 + 6 \beta ) q^{37} + ( 2 - 4 \beta ) q^{44} + ( -6 - \beta ) q^{45} + 12 q^{47} -4 \beta q^{48} -7 q^{49} -6 q^{53} + ( 7 - 3 \beta ) q^{55} + ( 1 - 2 \beta ) q^{59} + ( 6 - 4 \beta ) q^{60} + 8 q^{64} + ( 3 - 6 \beta ) q^{67} + 9 \beta q^{69} + ( -5 + 10 \beta ) q^{71} + ( 9 - \beta ) q^{75} + ( 4 + 4 \beta ) q^{80} + ( 6 - 5 \beta ) q^{81} + ( 5 - 10 \beta ) q^{89} -18 q^{92} + 5 \beta q^{93} + ( 3 - 6 \beta ) q^{97} + ( 3 + 5 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 4q^{4} + 3q^{5} - 5q^{9} + O(q^{10}) \) \( 2q - q^{3} + 4q^{4} + 3q^{5} - 5q^{9} - 2q^{12} + 4q^{15} + 8q^{16} + 6q^{20} - 18q^{23} - q^{25} + 8q^{27} - 10q^{31} - 11q^{33} - 10q^{36} - 13q^{45} + 24q^{47} - 4q^{48} - 14q^{49} - 12q^{53} + 11q^{55} + 8q^{60} + 16q^{64} + 9q^{69} + 17q^{75} + 12q^{80} + 7q^{81} - 36q^{92} + 5q^{93} + 11q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1
0.500000 + 1.65831i
0.500000 1.65831i
0 −0.500000 1.65831i 2.00000 1.50000 + 1.65831i 0 0 0 −2.50000 + 1.65831i 0
164.2 0 −0.500000 + 1.65831i 2.00000 1.50000 1.65831i 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 \(\Q(\sqrt{-11}) \)
15.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.d.a 2
3.b odd 2 1 165.2.d.b yes 2
5.b even 2 1 165.2.d.b yes 2
5.c odd 4 2 825.2.f.b 4
11.b odd 2 1 CM 165.2.d.a 2
15.d odd 2 1 inner 165.2.d.a 2
15.e even 4 2 825.2.f.b 4
33.d even 2 1 165.2.d.b yes 2
55.d odd 2 1 165.2.d.b yes 2
55.e even 4 2 825.2.f.b 4
165.d even 2 1 inner 165.2.d.a 2
165.l odd 4 2 825.2.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.a 2 1.a even 1 1 trivial
165.2.d.a 2 11.b odd 2 1 CM
165.2.d.a 2 15.d odd 2 1 inner
165.2.d.a 2 165.d even 2 1 inner
165.2.d.b yes 2 3.b odd 2 1
165.2.d.b yes 2 5.b even 2 1
165.2.d.b yes 2 33.d even 2 1
165.2.d.b yes 2 55.d odd 2 1
825.2.f.b 4 5.c odd 4 2
825.2.f.b 4 15.e even 4 2
825.2.f.b 4 55.e even 4 2
825.2.f.b 4 165.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\):

\( T_{2} \)
\( T_{23} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T + T^{2} \)
$5$ \( 5 - 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 11 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( ( 9 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 5 + T )^{2} \)
$37$ \( 99 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( ( -12 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 11 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 99 + T^{2} \)
$71$ \( 275 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 275 + T^{2} \)
$97$ \( 99 + T^{2} \)
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