# Properties

 Label 33.3.b.b Level $33$ Weight $3$ Character orbit 33.b Analytic conductor $0.899$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{3} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{8} + ( -3 + 3 \beta_{2} - 5 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{3} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{7} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{8} + ( -3 + 3 \beta_{2} - 5 \beta_{3} ) q^{9} -2 q^{10} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{11} + ( 3 + 3 \beta_{1} - 5 \beta_{3} ) q^{12} + ( -4 - 4 \beta_{1} - 4 \beta_{3} ) q^{13} + ( -4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{14} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{15} + ( -3 + 5 \beta_{1} + 5 \beta_{3} ) q^{16} + ( 2 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} ) q^{17} + ( 8 - 5 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{18} + ( -6 + 6 \beta_{1} + 6 \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{20} + ( -8 - 4 \beta_{1} - 2 \beta_{2} + 10 \beta_{3} ) q^{21} + ( 5 - \beta_{1} - \beta_{3} ) q^{22} + ( \beta_{1} - 16 \beta_{2} - \beta_{3} ) q^{23} + ( 8 + 7 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{24} + ( 21 - \beta_{1} - \beta_{3} ) q^{25} + ( -4 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{26} + ( -5 - 16 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} ) q^{27} -16 q^{28} + ( 24 \beta_{1} + 4 \beta_{2} - 24 \beta_{3} ) q^{29} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{30} + ( 14 + 5 \beta_{1} + 5 \beta_{3} ) q^{31} + ( \beta_{1} + 19 \beta_{2} - \beta_{3} ) q^{32} + ( -4 - 5 \beta_{1} - 4 \beta_{2} ) q^{33} + ( 20 - 16 \beta_{1} - 16 \beta_{3} ) q^{34} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{35} + ( -23 - 7 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{36} + ( -26 - 7 \beta_{1} - 7 \beta_{3} ) q^{37} + ( 18 \beta_{1} + 30 \beta_{2} - 18 \beta_{3} ) q^{38} + ( -12 - 12 \beta_{1} + 20 \beta_{3} ) q^{39} + ( -10 - 2 \beta_{1} - 2 \beta_{3} ) q^{40} + ( -4 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -16 + 16 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} ) q^{42} + ( 26 + 4 \beta_{1} + 4 \beta_{3} ) q^{43} + ( \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{44} + ( 4 + 5 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{45} + ( -14 + 16 \beta_{1} + 16 \beta_{3} ) q^{46} + ( 22 \beta_{1} + 14 \beta_{2} - 22 \beta_{3} ) q^{47} + ( 23 + 7 \beta_{1} + 8 \beta_{2} - 25 \beta_{3} ) q^{48} + ( -17 - 4 \beta_{1} - 4 \beta_{3} ) q^{49} + ( -23 \beta_{1} - 25 \beta_{2} + 23 \beta_{3} ) q^{50} + ( 56 - 20 \beta_{1} - 34 \beta_{2} + 30 \beta_{3} ) q^{51} + ( -36 - 4 \beta_{1} - 4 \beta_{3} ) q^{52} + ( -30 \beta_{1} - 38 \beta_{2} + 30 \beta_{3} ) q^{53} + ( -40 + 13 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{54} + ( 6 + \beta_{1} + \beta_{3} ) q^{55} -16 \beta_{2} q^{56} + ( 30 + 6 \beta_{1} + 12 \beta_{2} - 30 \beta_{3} ) q^{57} + ( 52 - 4 \beta_{1} - 4 \beta_{3} ) q^{58} + ( -23 \beta_{1} + 2 \beta_{2} + 23 \beta_{3} ) q^{59} + ( 16 + 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{60} + ( 4 + 20 \beta_{1} + 20 \beta_{3} ) q^{61} + ( -4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{62} + ( 40 + 14 \beta_{1} + 22 \beta_{2} - 12 \beta_{3} ) q^{63} + ( 9 + \beta_{1} + \beta_{3} ) q^{64} + ( 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{65} + ( -9 + 3 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{66} + ( -6 + 17 \beta_{1} + 17 \beta_{3} ) q^{67} + ( -44 \beta_{1} - 20 \beta_{2} + 44 \beta_{3} ) q^{68} + ( -68 + 14 \beta_{1} + 31 \beta_{2} - 33 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{70} + ( -3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 16 + 5 \beta_{1} - 17 \beta_{2} - 13 \beta_{3} ) q^{72} + ( -74 - 6 \beta_{1} - 6 \beta_{3} ) q^{73} + ( 12 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} ) q^{74} + ( -25 + 19 \beta_{1} - 22 \beta_{2} + 5 \beta_{3} ) q^{75} + ( 42 - 6 \beta_{1} - 6 \beta_{3} ) q^{76} + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -32 + 20 \beta_{1} + 28 \beta_{2} - 20 \beta_{3} ) q^{78} + ( -32 - 26 \beta_{1} - 26 \beta_{3} ) q^{79} + ( -2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{80} + ( 37 + 35 \beta_{1} + 37 \beta_{2} ) q^{81} + ( 4 - 12 \beta_{1} - 12 \beta_{3} ) q^{82} + ( 30 \beta_{1} + 24 \beta_{2} - 30 \beta_{3} ) q^{83} + ( 16 - 16 \beta_{1} + 16 \beta_{2} ) q^{84} + ( -24 - 14 \beta_{1} - 14 \beta_{3} ) q^{85} + ( -18 \beta_{1} - 10 \beta_{2} + 18 \beta_{3} ) q^{86} + ( -80 - 52 \beta_{1} - 32 \beta_{2} - 16 \beta_{3} ) q^{87} + ( 17 + \beta_{1} + \beta_{3} ) q^{88} + ( -39 \beta_{1} + 26 \beta_{2} + 39 \beta_{3} ) q^{89} + ( 6 - 6 \beta_{2} + 10 \beta_{3} ) q^{90} + 64 q^{91} + ( 50 \beta_{1} + 14 \beta_{2} - 50 \beta_{3} ) q^{92} + ( 6 + 24 \beta_{1} - 9 \beta_{2} - 25 \beta_{3} ) q^{93} + ( 58 - 14 \beta_{1} - 14 \beta_{3} ) q^{94} + 12 \beta_{2} q^{95} + ( 72 - 21 \beta_{1} - 39 \beta_{2} + 37 \beta_{3} ) q^{96} + ( 38 - 33 \beta_{1} - 33 \beta_{3} ) q^{97} + ( 9 \beta_{1} + \beta_{2} - 9 \beta_{3} ) q^{98} + ( -12 + 12 \beta_{2} + 7 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 5q^{3} + 2q^{4} - 2q^{6} + 4q^{7} - 7q^{9} + O(q^{10})$$ $$4q - 5q^{3} + 2q^{4} - 2q^{6} + 4q^{7} - 7q^{9} - 8q^{10} + 14q^{12} - 8q^{13} + 13q^{15} - 22q^{16} + 38q^{18} - 36q^{19} - 38q^{21} + 22q^{22} + 24q^{24} + 86q^{25} - 20q^{27} - 64q^{28} + 10q^{30} + 46q^{31} - 11q^{33} + 112q^{34} - 86q^{36} - 90q^{37} - 56q^{39} - 36q^{40} - 68q^{42} + 96q^{43} + 17q^{45} - 88q^{46} + 110q^{48} - 60q^{49} + 214q^{51} - 136q^{52} - 176q^{54} + 22q^{55} + 144q^{57} + 216q^{58} + 56q^{60} - 24q^{61} + 158q^{63} + 34q^{64} - 44q^{66} - 58q^{67} - 253q^{69} - 8q^{70} + 72q^{72} - 284q^{73} - 124q^{75} + 180q^{76} - 128q^{78} - 76q^{79} + 113q^{81} + 40q^{82} + 80q^{84} - 68q^{85} - 252q^{87} + 66q^{88} + 14q^{90} + 256q^{91} + 25q^{93} + 260q^{94} + 272q^{96} + 218q^{97} - 55q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 4 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu^{2} + 2 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{2} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i −2.68614 1.33591i −2.37228 0.792287i −3.37228 + 6.78073i 6.74456 4.10891i 5.43070 + 7.17687i −2.00000
23.2 0.792287i 0.186141 + 2.99422i 3.37228 2.52434i 2.37228 0.147477i −4.74456 5.84096i −8.93070 + 1.11469i −2.00000
23.3 0.792287i 0.186141 2.99422i 3.37228 2.52434i 2.37228 + 0.147477i −4.74456 5.84096i −8.93070 1.11469i −2.00000
23.4 2.52434i −2.68614 + 1.33591i −2.37228 0.792287i −3.37228 6.78073i 6.74456 4.10891i 5.43070 7.17687i −2.00000
 $$n$$: 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.b 4
3.b odd 2 1 inner 33.3.b.b 4
4.b odd 2 1 528.3.i.d 4
11.b odd 2 1 363.3.b.h 4
11.c even 5 4 363.3.h.m 16
11.d odd 10 4 363.3.h.l 16
12.b even 2 1 528.3.i.d 4
33.d even 2 1 363.3.b.h 4
33.f even 10 4 363.3.h.l 16
33.h odd 10 4 363.3.h.m 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.b 4 1.a even 1 1 trivial
33.3.b.b 4 3.b odd 2 1 inner
363.3.b.h 4 11.b odd 2 1
363.3.b.h 4 33.d even 2 1
363.3.h.l 16 11.d odd 10 4
363.3.h.l 16 33.f even 10 4
363.3.h.m 16 11.c even 5 4
363.3.h.m 16 33.h odd 10 4
528.3.i.d 4 4.b odd 2 1
528.3.i.d 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7 T_{2}^{2} + 4$$ acting on $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 9 T^{2} + 44 T^{4} - 144 T^{6} + 256 T^{8}$$
$3$ $$1 + 5 T + 16 T^{2} + 45 T^{3} + 81 T^{4}$$
$5$ $$1 - 93 T^{2} + 3404 T^{4} - 58125 T^{6} + 390625 T^{8}$$
$7$ $$( 1 - 2 T + 66 T^{2} - 98 T^{3} + 2401 T^{4} )^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 + 4 T + 210 T^{2} + 676 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$1 + 216 T^{2} + 149006 T^{4} + 18040536 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + 18 T + 506 T^{2} + 6498 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 - 477 T^{2} + 607580 T^{4} - 133484157 T^{6} + 78310985281 T^{8}$$
$29$ $$1 + 188 T^{2} + 206886 T^{4} + 132968828 T^{6} + 500246412961 T^{8}$$
$31$ $$( 1 - 23 T + 1848 T^{2} - 22103 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 + 45 T + 2840 T^{2} + 61605 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 5460 T^{2} + 13000934 T^{4} - 15428655060 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 - 48 T + 4142 T^{2} - 88752 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 6120 T^{2} + 18979214 T^{4} - 29863647720 T^{6} + 23811286661761 T^{8}$$
$53$ $$1 - 3112 T^{2} + 2490798 T^{4} - 24555176872 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 - 9921 T^{2} + 45659744 T^{4} - 120216338481 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 + 12 T + 4178 T^{2} + 44652 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 + 29 T + 6804 T^{2} + 130181 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 19933 T^{2} + 150145500 T^{4} - 506531037373 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 + 142 T + 15402 T^{2} + 756718 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 + 38 T + 7266 T^{2} + 237158 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 22120 T^{2} + 214834542 T^{4} - 1049778060520 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 - 10897 T^{2} + 51258576 T^{4} - 683702200177 T^{6} + 3936588805702081 T^{8}$$
$97$ $$( 1 - 109 T + 12804 T^{2} - 1025581 T^{3} + 88529281 T^{4} )^{2}$$