# Properties

 Label 33.2.d.a Level $33$ Weight $2$ Character orbit 33.d Analytic conductor $0.264$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 33.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.263506326670$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -2 q^{4} + ( 1 - 2 \beta ) q^{5} + ( -3 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} -2 q^{4} + ( 1 - 2 \beta ) q^{5} + ( -3 + \beta ) q^{9} + ( -1 + 2 \beta ) q^{11} -2 \beta q^{12} + ( 6 - \beta ) q^{15} + 4 q^{16} + ( -2 + 4 \beta ) q^{20} + ( 1 - 2 \beta ) q^{23} -6 q^{25} + ( -3 - 2 \beta ) q^{27} + 5 q^{31} + ( -6 + \beta ) q^{33} + ( 6 - 2 \beta ) q^{36} -7 q^{37} + ( 2 - 4 \beta ) q^{44} + ( 3 + 5 \beta ) q^{45} + ( -2 + 4 \beta ) q^{47} + 4 \beta q^{48} + 7 q^{49} + ( 4 - 8 \beta ) q^{53} + 11 q^{55} + ( 1 - 2 \beta ) q^{59} + ( -12 + 2 \beta ) q^{60} -8 q^{64} -13 q^{67} + ( 6 - \beta ) q^{69} + ( -5 + 10 \beta ) q^{71} -6 \beta q^{75} + ( 4 - 8 \beta ) q^{80} + ( 6 - 5 \beta ) q^{81} + ( -5 + 10 \beta ) q^{89} + ( -2 + 4 \beta ) q^{92} + 5 \beta q^{93} + 17 q^{97} + ( -3 - 5 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 4q^{4} - 5q^{9} + O(q^{10})$$ $$2q + q^{3} - 4q^{4} - 5q^{9} - 2q^{12} + 11q^{15} + 8q^{16} - 12q^{25} - 8q^{27} + 10q^{31} - 11q^{33} + 10q^{36} - 14q^{37} + 11q^{45} + 4q^{48} + 14q^{49} + 22q^{55} - 22q^{60} - 16q^{64} - 26q^{67} + 11q^{69} - 6q^{75} + 7q^{81} + 5q^{93} + 34q^{97} - 11q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 0.5 − 1.65831i 0.5 + 1.65831i
0 0.500000 1.65831i −2.00000 3.31662i 0 0 0 −2.50000 1.65831i 0
32.2 0 0.500000 + 1.65831i −2.00000 3.31662i 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 $$\Q(\sqrt{-11})$$
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 33.2.d.a 2
3.b odd 2 1 inner 33.2.d.a 2
4.b odd 2 1 528.2.b.a 2
5.b even 2 1 825.2.f.a 2
5.c odd 4 2 825.2.d.a 4
8.b even 2 1 2112.2.b.e 2
8.d odd 2 1 2112.2.b.f 2
9.c even 3 2 891.2.g.a 4
9.d odd 6 2 891.2.g.a 4
11.b odd 2 1 CM 33.2.d.a 2
11.c even 5 4 363.2.f.c 8
11.d odd 10 4 363.2.f.c 8
12.b even 2 1 528.2.b.a 2
15.d odd 2 1 825.2.f.a 2
15.e even 4 2 825.2.d.a 4
24.f even 2 1 2112.2.b.f 2
24.h odd 2 1 2112.2.b.e 2
33.d even 2 1 inner 33.2.d.a 2
33.f even 10 4 363.2.f.c 8
33.h odd 10 4 363.2.f.c 8
44.c even 2 1 528.2.b.a 2
55.d odd 2 1 825.2.f.a 2
55.e even 4 2 825.2.d.a 4
88.b odd 2 1 2112.2.b.e 2
88.g even 2 1 2112.2.b.f 2
99.g even 6 2 891.2.g.a 4
99.h odd 6 2 891.2.g.a 4
132.d odd 2 1 528.2.b.a 2
165.d even 2 1 825.2.f.a 2
165.l odd 4 2 825.2.d.a 4
264.m even 2 1 2112.2.b.e 2
264.p odd 2 1 2112.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 1.a even 1 1 trivial
33.2.d.a 2 3.b odd 2 1 inner
33.2.d.a 2 11.b odd 2 1 CM
33.2.d.a 2 33.d even 2 1 inner
363.2.f.c 8 11.c even 5 4
363.2.f.c 8 11.d odd 10 4
363.2.f.c 8 33.f even 10 4
363.2.f.c 8 33.h odd 10 4
528.2.b.a 2 4.b odd 2 1
528.2.b.a 2 12.b even 2 1
528.2.b.a 2 44.c even 2 1
528.2.b.a 2 132.d odd 2 1
825.2.d.a 4 5.c odd 4 2
825.2.d.a 4 15.e even 4 2
825.2.d.a 4 55.e even 4 2
825.2.d.a 4 165.l odd 4 2
825.2.f.a 2 5.b even 2 1
825.2.f.a 2 15.d odd 2 1
825.2.f.a 2 55.d odd 2 1
825.2.f.a 2 165.d even 2 1
891.2.g.a 4 9.c even 3 2
891.2.g.a 4 9.d odd 6 2
891.2.g.a 4 99.g even 6 2
891.2.g.a 4 99.h odd 6 2
2112.2.b.e 2 8.b even 2 1
2112.2.b.e 2 24.h odd 2 1
2112.2.b.e 2 88.b odd 2 1
2112.2.b.e 2 264.m even 2 1
2112.2.b.f 2 8.d odd 2 1
2112.2.b.f 2 24.f even 2 1
2112.2.b.f 2 88.g even 2 1
2112.2.b.f 2 264.p odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ $$1 - T + 3 T^{2}$$
$5$ $$( 1 - 3 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{2}$$
$23$ $$( 1 - 9 T + 23 T^{2} )( 1 + 9 T + 23 T^{2} )$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 5 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 7 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 43 T^{2} )^{2}$$
$47$ $$( 1 - 12 T + 47 T^{2} )( 1 + 12 T + 47 T^{2} )$$
$53$ $$( 1 - 6 T + 53 T^{2} )( 1 + 6 T + 53 T^{2} )$$
$59$ $$( 1 - 15 T + 59 T^{2} )( 1 + 15 T + 59 T^{2} )$$
$61$ $$( 1 - 61 T^{2} )^{2}$$
$67$ $$( 1 + 13 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 3 T + 71 T^{2} )( 1 + 3 T + 71 T^{2} )$$
$73$ $$( 1 - 73 T^{2} )^{2}$$
$79$ $$( 1 - 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 - 9 T + 89 T^{2} )( 1 + 9 T + 89 T^{2} )$$
$97$ $$( 1 - 17 T + 97 T^{2} )^{2}$$