# 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 $4$

# Related objects

## 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
 0.5 + 1.65831i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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