# Properties

 Label 33.6.d.a Level $33$ Weight $6$ Character orbit 33.d Analytic conductor $5.293$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 33.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.29266605383$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 16 - \beta ) q^{3} -32 q^{4} + ( 29 - 58 \beta ) q^{5} + ( 253 - 31 \beta ) q^{9} +O(q^{10})$$ $$q + ( 16 - \beta ) q^{3} -32 q^{4} + ( 29 - 58 \beta ) q^{5} + ( 253 - 31 \beta ) q^{9} + ( 121 - 242 \beta ) q^{11} + ( -512 + 32 \beta ) q^{12} + ( 290 - 899 \beta ) q^{15} + 1024 q^{16} + ( -928 + 1856 \beta ) q^{20} + ( -1501 + 3002 \beta ) q^{23} -6126 q^{25} + ( 3955 - 718 \beta ) q^{27} + 7775 q^{31} + ( 1210 - 3751 \beta ) q^{33} + ( -8096 + 992 \beta ) q^{36} -1267 q^{37} + ( -3872 + 7744 \beta ) q^{44} + ( 1943 - 13775 \beta ) q^{45} + ( 5282 - 10564 \beta ) q^{47} + ( 16384 - 1024 \beta ) q^{48} + 16807 q^{49} + ( 6476 - 12952 \beta ) q^{53} -38599 q^{55} + ( -14281 + 28562 \beta ) q^{59} + ( -9280 + 28768 \beta ) q^{60} -32768 q^{64} + 72917 q^{67} + ( -15010 + 46531 \beta ) q^{69} + ( 16025 - 32050 \beta ) q^{71} + ( -98016 + 6126 \beta ) q^{75} + ( 29696 - 59392 \beta ) q^{80} + ( 61126 - 14725 \beta ) q^{81} + ( -35725 + 71450 \beta ) q^{89} + ( 48032 - 96064 \beta ) q^{92} + ( 124400 - 7775 \beta ) q^{93} -163183 q^{97} + ( 8107 - 57475 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 31q^{3} - 64q^{4} + 475q^{9} + O(q^{10})$$ $$2q + 31q^{3} - 64q^{4} + 475q^{9} - 992q^{12} - 319q^{15} + 2048q^{16} - 12252q^{25} + 7192q^{27} + 15550q^{31} - 1331q^{33} - 15200q^{36} - 2534q^{37} - 9889q^{45} + 31744q^{48} + 33614q^{49} - 77198q^{55} + 10208q^{60} - 65536q^{64} + 145834q^{67} + 16511q^{69} - 189906q^{75} + 107527q^{81} + 241025q^{93} - 326366q^{97} - 41261q^{99} + O(q^{100})$$

## 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 15.5000 1.65831i −32.0000 96.1821i 0 0 0 237.500 51.4077i 0
32.2 0 15.5000 + 1.65831i −32.0000 96.1821i 0 0 0 237.500 + 51.4077i 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 33.6.d.a 2
3.b odd 2 1 inner 33.6.d.a 2
11.b odd 2 1 CM 33.6.d.a 2
33.d even 2 1 inner 33.6.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.d.a 2 1.a even 1 1 trivial
33.6.d.a 2 3.b odd 2 1 inner
33.6.d.a 2 11.b odd 2 1 CM
33.6.d.a 2 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{6}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$243 - 31 T + T^{2}$$
$5$ $$9251 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$161051 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$24783011 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -7775 + T )^{2}$$
$37$ $$( 1267 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$306894764 + T^{2}$$
$53$ $$461324336 + T^{2}$$
$59$ $$2243416571 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$( -72917 + T )^{2}$$
$71$ $$2824806875 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$14039031875 + T^{2}$$
$97$ $$( 163183 + T )^{2}$$