# Properties

 Label 33.8.d.a Level 33 Weight 8 Character orbit 33.d Analytic conductor 10.309 Analytic rank 0 Dimension 2 CM discriminant -11 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 33.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3087058410$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -35 - 13 \beta ) q^{3} -128 q^{4} + ( 71 - 142 \beta ) q^{5} + ( 718 + 1079 \beta ) q^{9} +O(q^{10})$$ $$q + ( -35 - 13 \beta ) q^{3} -128 q^{4} + ( 71 - 142 \beta ) q^{5} + ( 718 + 1079 \beta ) q^{9} + ( -1331 + 2662 \beta ) q^{11} + ( 4480 + 1664 \beta ) q^{12} + ( -8023 + 5893 \beta ) q^{15} + 16384 q^{16} + ( -9088 + 18176 \beta ) q^{20} + ( -21853 + 43706 \beta ) q^{23} + 22674 q^{25} + ( 16951 - 61126 \beta ) q^{27} + 39065 q^{31} + ( 150403 - 110473 \beta ) q^{33} + ( -91904 - 138112 \beta ) q^{36} -562471 q^{37} + ( 170368 - 340736 \beta ) q^{44} + ( 510632 - 178565 \beta ) q^{45} + ( -164446 + 328892 \beta ) q^{47} + ( -573440 - 212992 \beta ) q^{48} + 823543 q^{49} + ( -644332 + 1288664 \beta ) q^{53} + 1039511 q^{55} + ( -950221 + 1900442 \beta ) q^{59} + ( 1026944 - 754304 \beta ) q^{60} -2097152 q^{64} + 684671 q^{67} + ( 2469389 - 1813799 \beta ) q^{69} + ( -568615 + 1137230 \beta ) q^{71} + ( -793590 - 294762 \beta ) q^{75} + ( 1163264 - 2326528 \beta ) q^{80} + ( -2977199 + 2713685 \beta ) q^{81} + ( 3782165 - 7564330 \beta ) q^{89} + ( 2797184 - 5594368 \beta ) q^{92} + ( -1367275 - 507845 \beta ) q^{93} -15182479 q^{97} + ( -9572552 + 3347465 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 83q^{3} - 256q^{4} + 2515q^{9} + O(q^{10})$$ $$2q - 83q^{3} - 256q^{4} + 2515q^{9} + 10624q^{12} - 10153q^{15} + 32768q^{16} + 45348q^{25} - 27224q^{27} + 78130q^{31} + 190333q^{33} - 321920q^{36} - 1124942q^{37} + 842699q^{45} - 1359872q^{48} + 1647086q^{49} + 2079022q^{55} + 1299584q^{60} - 4194304q^{64} + 1369342q^{67} + 3124979q^{69} - 1881942q^{75} - 3240713q^{81} - 3242395q^{93} - 30364958q^{97} - 15797639q^{99} + O(q^{100})$$

## 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 −41.5000 21.5581i −128.000 235.480i 0 0 0 1257.50 + 1789.32i 0
32.2 0 −41.5000 + 21.5581i −128.000 235.480i 0 0 0 1257.50 1789.32i 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 33.8.d.a 2
3.b odd 2 1 inner 33.8.d.a 2
11.b odd 2 1 CM 33.8.d.a 2
33.d even 2 1 inner 33.8.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.d.a 2 1.a even 1 1 trivial
33.8.d.a 2 3.b odd 2 1 inner
33.8.d.a 2 11.b odd 2 1 CM
33.8.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 $$S_{8}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 128 T^{2} )^{2}$$
$3$ $$1 + 83 T + 2187 T^{2}$$
$5$ $$( 1 - 507 T + 78125 T^{2} )( 1 + 507 T + 78125 T^{2} )$$
$7$ $$( 1 - 823543 T^{2} )^{2}$$
$11$ $$1 + 19487171 T^{2}$$
$13$ $$( 1 - 62748517 T^{2} )^{2}$$
$17$ $$( 1 + 410338673 T^{2} )^{2}$$
$19$ $$( 1 - 893871739 T^{2} )^{2}$$
$23$ $$( 1 - 91467 T + 3404825447 T^{2} )( 1 + 91467 T + 3404825447 T^{2} )$$
$29$ $$( 1 + 17249876309 T^{2} )^{2}$$
$31$ $$( 1 - 39065 T + 27512614111 T^{2} )^{2}$$
$37$ $$( 1 + 562471 T + 94931877133 T^{2} )^{2}$$
$41$ $$( 1 + 194754273881 T^{2} )^{2}$$
$43$ $$( 1 - 271818611107 T^{2} )^{2}$$
$47$ $$( 1 - 1314924 T + 506623120463 T^{2} )( 1 + 1314924 T + 506623120463 T^{2} )$$
$53$ $$( 1 - 363378 T + 1174711139837 T^{2} )( 1 + 363378 T + 1174711139837 T^{2} )$$
$59$ $$( 1 - 149955 T + 2488651484819 T^{2} )( 1 + 149955 T + 2488651484819 T^{2} )$$
$61$ $$( 1 - 3142742836021 T^{2} )^{2}$$
$67$ $$( 1 - 684671 T + 6060711605323 T^{2} )^{2}$$
$71$ $$( 1 - 5729217 T + 9095120158391 T^{2} )( 1 + 5729217 T + 9095120158391 T^{2} )$$
$73$ $$( 1 - 11047398519097 T^{2} )^{2}$$
$79$ $$( 1 - 19203908986159 T^{2} )^{2}$$
$83$ $$( 1 + 27136050989627 T^{2} )^{2}$$
$89$ $$( 1 - 4424121 T + 44231334895529 T^{2} )( 1 + 4424121 T + 44231334895529 T^{2} )$$
$97$ $$( 1 + 15182479 T + 80798284478113 T^{2} )^{2}$$