# Properties

 Label 33.3.b.a Level 33 Weight 3 Character orbit 33.b Analytic conductor 0.899 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + 3 q^{3} -7 q^{4} + 2 \beta q^{5} -3 \beta q^{6} -8 q^{7} + 3 \beta q^{8} + 9 q^{9} +O(q^{10})$$ $$q -\beta q^{2} + 3 q^{3} -7 q^{4} + 2 \beta q^{5} -3 \beta q^{6} -8 q^{7} + 3 \beta q^{8} + 9 q^{9} + 22 q^{10} -\beta q^{11} -21 q^{12} + 4 q^{13} + 8 \beta q^{14} + 6 \beta q^{15} + 5 q^{16} -4 \beta q^{17} -9 \beta q^{18} -6 q^{19} -14 \beta q^{20} -24 q^{21} -11 q^{22} -2 \beta q^{23} + 9 \beta q^{24} -19 q^{25} -4 \beta q^{26} + 27 q^{27} + 56 q^{28} + 12 \beta q^{29} + 66 q^{30} -26 q^{31} + 7 \beta q^{32} -3 \beta q^{33} -44 q^{34} -16 \beta q^{35} -63 q^{36} + 30 q^{37} + 6 \beta q^{38} + 12 q^{39} -66 q^{40} -4 \beta q^{41} + 24 \beta q^{42} + 42 q^{43} + 7 \beta q^{44} + 18 \beta q^{45} -22 q^{46} -26 \beta q^{47} + 15 q^{48} + 15 q^{49} + 19 \beta q^{50} -12 \beta q^{51} -28 q^{52} + 18 \beta q^{53} -27 \beta q^{54} + 22 q^{55} -24 \beta q^{56} -18 q^{57} + 132 q^{58} -20 \beta q^{59} -42 \beta q^{60} + 12 q^{61} + 26 \beta q^{62} -72 q^{63} + 97 q^{64} + 8 \beta q^{65} -33 q^{66} + 2 q^{67} + 28 \beta q^{68} -6 \beta q^{69} -176 q^{70} + 18 \beta q^{71} + 27 \beta q^{72} -74 q^{73} -30 \beta q^{74} -57 q^{75} + 42 q^{76} + 8 \beta q^{77} -12 \beta q^{78} -40 q^{79} + 10 \beta q^{80} + 81 q^{81} -44 q^{82} + 12 \beta q^{83} + 168 q^{84} + 88 q^{85} -42 \beta q^{86} + 36 \beta q^{87} + 33 q^{88} + 36 \beta q^{89} + 198 q^{90} -32 q^{91} + 14 \beta q^{92} -78 q^{93} -286 q^{94} -12 \beta q^{95} + 21 \beta q^{96} + 62 q^{97} -15 \beta q^{98} -9 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 14q^{4} - 16q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 14q^{4} - 16q^{7} + 18q^{9} + 44q^{10} - 42q^{12} + 8q^{13} + 10q^{16} - 12q^{19} - 48q^{21} - 22q^{22} - 38q^{25} + 54q^{27} + 112q^{28} + 132q^{30} - 52q^{31} - 88q^{34} - 126q^{36} + 60q^{37} + 24q^{39} - 132q^{40} + 84q^{43} - 44q^{46} + 30q^{48} + 30q^{49} - 56q^{52} + 44q^{55} - 36q^{57} + 264q^{58} + 24q^{61} - 144q^{63} + 194q^{64} - 66q^{66} + 4q^{67} - 352q^{70} - 148q^{73} - 114q^{75} + 84q^{76} - 80q^{79} + 162q^{81} - 88q^{82} + 336q^{84} + 176q^{85} + 66q^{88} + 396q^{90} - 64q^{91} - 156q^{93} - 572q^{94} + 124q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 + 1.65831i 0.5 − 1.65831i
3.31662i 3.00000 −7.00000 6.63325i 9.94987i −8.00000 9.94987i 9.00000 22.0000
23.2 3.31662i 3.00000 −7.00000 6.63325i 9.94987i −8.00000 9.94987i 9.00000 22.0000
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.b.a 2
3.b odd 2 1 inner 33.3.b.a 2
4.b odd 2 1 528.3.i.a 2
11.b odd 2 1 363.3.b.d 2
11.c even 5 4 363.3.h.e 8
11.d odd 10 4 363.3.h.d 8
12.b even 2 1 528.3.i.a 2
33.d even 2 1 363.3.b.d 2
33.f even 10 4 363.3.h.d 8
33.h odd 10 4 363.3.h.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.a 2 1.a even 1 1 trivial
33.3.b.a 2 3.b odd 2 1 inner
363.3.b.d 2 11.b odd 2 1
363.3.b.d 2 33.d even 2 1
363.3.h.d 8 11.d odd 10 4
363.3.h.d 8 33.f even 10 4
363.3.h.e 8 11.c even 5 4
363.3.h.e 8 33.h odd 10 4
528.3.i.a 2 4.b odd 2 1
528.3.i.a 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 11$$ acting on $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + 16 T^{4}$$
$3$ $$( 1 - 3 T )^{2}$$
$5$ $$1 - 6 T^{2} + 625 T^{4}$$
$7$ $$( 1 + 8 T + 49 T^{2} )^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$( 1 - 4 T + 169 T^{2} )^{2}$$
$17$ $$1 - 402 T^{2} + 83521 T^{4}$$
$19$ $$( 1 + 6 T + 361 T^{2} )^{2}$$
$23$ $$1 - 1014 T^{2} + 279841 T^{4}$$
$29$ $$1 - 98 T^{2} + 707281 T^{4}$$
$31$ $$( 1 + 26 T + 961 T^{2} )^{2}$$
$37$ $$( 1 - 30 T + 1369 T^{2} )^{2}$$
$41$ $$1 - 3186 T^{2} + 2825761 T^{4}$$
$43$ $$( 1 - 42 T + 1849 T^{2} )^{2}$$
$47$ $$1 + 3018 T^{2} + 4879681 T^{4}$$
$53$ $$1 - 2054 T^{2} + 7890481 T^{4}$$
$59$ $$1 - 2562 T^{2} + 12117361 T^{4}$$
$61$ $$( 1 - 12 T + 3721 T^{2} )^{2}$$
$67$ $$( 1 - 2 T + 4489 T^{2} )^{2}$$
$71$ $$1 - 6518 T^{2} + 25411681 T^{4}$$
$73$ $$( 1 + 74 T + 5329 T^{2} )^{2}$$
$79$ $$( 1 + 40 T + 6241 T^{2} )^{2}$$
$83$ $$1 - 12194 T^{2} + 47458321 T^{4}$$
$89$ $$1 - 1586 T^{2} + 62742241 T^{4}$$
$97$ $$( 1 - 62 T + 9409 T^{2} )^{2}$$