# Properties

 Label 33.4.d.a Level $33$ Weight $4$ Character orbit 33.d Analytic conductor $1.947$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.94706303019$$ 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{3} -8 q^{4} -4 \beta q^{5} + ( 5 + 8 \beta ) q^{9} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{3} -8 q^{4} -4 \beta q^{5} + ( 5 + 8 \beta ) q^{9} -11 \beta q^{11} + ( 32 + 8 \beta ) q^{12} + ( -44 + 16 \beta ) q^{15} + 64 q^{16} + 32 \beta q^{20} -58 \beta q^{23} -51 q^{25} + ( 68 - 37 \beta ) q^{27} -340 q^{31} + ( -121 + 44 \beta ) q^{33} + ( -40 - 64 \beta ) q^{36} + 434 q^{37} + 88 \beta q^{44} + ( 352 - 20 \beta ) q^{45} + 194 \beta q^{47} + ( -256 - 64 \beta ) q^{48} + 343 q^{49} + 68 \beta q^{53} -484 q^{55} -166 \beta q^{59} + ( 352 - 128 \beta ) q^{60} -512 q^{64} + 416 q^{67} + ( -638 + 232 \beta ) q^{69} -310 \beta q^{71} + ( 204 + 51 \beta ) q^{75} -256 \beta q^{80} + ( -679 + 80 \beta ) q^{81} -40 \beta q^{89} + 464 \beta q^{92} + ( 1360 + 340 \beta ) q^{93} -34 q^{97} + ( 968 - 55 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{3} - 16q^{4} + 10q^{9} + O(q^{10})$$ $$2q - 8q^{3} - 16q^{4} + 10q^{9} + 64q^{12} - 88q^{15} + 128q^{16} - 102q^{25} + 136q^{27} - 680q^{31} - 242q^{33} - 80q^{36} + 868q^{37} + 704q^{45} - 512q^{48} + 686q^{49} - 968q^{55} + 704q^{60} - 1024q^{64} + 832q^{67} - 1276q^{69} + 408q^{75} - 1358q^{81} + 2720q^{93} - 68q^{97} + 1936q^{99} + O(q^{100})$$

## 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 −4.00000 3.31662i −8.00000 13.2665i 0 0 0 5.00000 + 26.5330i 0
32.2 0 −4.00000 + 3.31662i −8.00000 13.2665i 0 0 0 5.00000 26.5330i 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 33.4.d.a 2
3.b odd 2 1 inner 33.4.d.a 2
4.b odd 2 1 528.4.b.a 2
11.b odd 2 1 CM 33.4.d.a 2
12.b even 2 1 528.4.b.a 2
33.d even 2 1 inner 33.4.d.a 2
44.c even 2 1 528.4.b.a 2
132.d odd 2 1 528.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.a 2 1.a even 1 1 trivial
33.4.d.a 2 3.b odd 2 1 inner
33.4.d.a 2 11.b odd 2 1 CM
33.4.d.a 2 33.d even 2 1 inner
528.4.b.a 2 4.b odd 2 1
528.4.b.a 2 12.b even 2 1
528.4.b.a 2 44.c even 2 1
528.4.b.a 2 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 8 T^{2} )^{2}$$
$3$ $$1 + 8 T + 27 T^{2}$$
$5$ $$( 1 - 18 T + 125 T^{2} )( 1 + 18 T + 125 T^{2} )$$
$7$ $$( 1 - 343 T^{2} )^{2}$$
$11$ $$1 + 1331 T^{2}$$
$13$ $$( 1 - 2197 T^{2} )^{2}$$
$17$ $$( 1 + 4913 T^{2} )^{2}$$
$19$ $$( 1 - 6859 T^{2} )^{2}$$
$23$ $$( 1 - 108 T + 12167 T^{2} )( 1 + 108 T + 12167 T^{2} )$$
$29$ $$( 1 + 24389 T^{2} )^{2}$$
$31$ $$( 1 + 340 T + 29791 T^{2} )^{2}$$
$37$ $$( 1 - 434 T + 50653 T^{2} )^{2}$$
$41$ $$( 1 + 68921 T^{2} )^{2}$$
$43$ $$( 1 - 79507 T^{2} )^{2}$$
$47$ $$( 1 - 36 T + 103823 T^{2} )( 1 + 36 T + 103823 T^{2} )$$
$53$ $$( 1 - 738 T + 148877 T^{2} )( 1 + 738 T + 148877 T^{2} )$$
$59$ $$( 1 - 720 T + 205379 T^{2} )( 1 + 720 T + 205379 T^{2} )$$
$61$ $$( 1 - 226981 T^{2} )^{2}$$
$67$ $$( 1 - 416 T + 300763 T^{2} )^{2}$$
$71$ $$( 1 - 612 T + 357911 T^{2} )( 1 + 612 T + 357911 T^{2} )$$
$73$ $$( 1 - 389017 T^{2} )^{2}$$
$79$ $$( 1 - 493039 T^{2} )^{2}$$
$83$ $$( 1 + 571787 T^{2} )^{2}$$
$89$ $$( 1 - 1674 T + 704969 T^{2} )( 1 + 1674 T + 704969 T^{2} )$$
$97$ $$( 1 + 34 T + 912673 T^{2} )^{2}$$