# Properties

 Label 33.3.c.a Level $33$ Weight $3$ Character orbit 33.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.39744.5 Defining polynomial: $$x^{4} - 2 x^{3} + 12 x^{2} + 4 x + 22$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{1} q^{3} + ( -5 - 2 \beta_{1} ) q^{4} + ( 1 + 3 \beta_{1} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{6} + \beta_{2} q^{7} + ( -2 \beta_{2} + 3 \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{1} q^{3} + ( -5 - 2 \beta_{1} ) q^{4} + ( 1 + 3 \beta_{1} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{6} + \beta_{2} q^{7} + ( -2 \beta_{2} + 3 \beta_{3} ) q^{8} + 3 q^{9} + ( 3 \beta_{2} - 4 \beta_{3} ) q^{10} + ( 5 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + ( 6 + 5 \beta_{1} ) q^{12} + ( -\beta_{2} + 6 \beta_{3} ) q^{13} + ( 3 - 7 \beta_{1} ) q^{14} + ( -9 - \beta_{1} ) q^{15} + ( 1 + 12 \beta_{1} ) q^{16} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{17} -3 \beta_{3} q^{18} + ( -4 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -23 - 17 \beta_{1} ) q^{20} + ( \beta_{2} + 2 \beta_{3} ) q^{21} + ( -3 - 16 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{22} + ( -23 + 5 \beta_{1} ) q^{23} + ( \beta_{2} - 7 \beta_{3} ) q^{24} + ( 3 + 6 \beta_{1} ) q^{25} + ( 51 + 19 \beta_{1} ) q^{26} -3 \beta_{1} q^{27} + ( -3 \beta_{2} + 4 \beta_{3} ) q^{28} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{29} + ( -\beta_{2} + 10 \beta_{3} ) q^{30} + ( 20 - 18 \beta_{1} ) q^{31} + ( 4 \beta_{2} - \beta_{3} ) q^{32} + ( -3 - 5 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{33} + ( 12 + 18 \beta_{1} ) q^{34} + ( -2 \beta_{2} - 6 \beta_{3} ) q^{35} + ( -15 - 6 \beta_{1} ) q^{36} + ( -16 + 2 \beta_{1} ) q^{37} + ( -30 + 24 \beta_{1} ) q^{38} + ( 5 \beta_{2} - 8 \beta_{3} ) q^{39} + ( -5 \beta_{2} + 24 \beta_{3} ) q^{40} + ( 6 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 21 - 3 \beta_{1} ) q^{42} + ( -12 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -31 - 15 \beta_{1} - 8 \beta_{2} + 15 \beta_{3} ) q^{44} + ( 3 + 9 \beta_{1} ) q^{45} + ( 5 \beta_{2} + 18 \beta_{3} ) q^{46} + ( 25 - 3 \beta_{1} ) q^{47} + ( -36 - \beta_{1} ) q^{48} + ( 25 + 10 \beta_{1} ) q^{49} + ( 6 \beta_{2} - 9 \beta_{3} ) q^{50} -6 \beta_{3} q^{51} + ( 15 \beta_{2} - 46 \beta_{3} ) q^{52} + ( 7 - 11 \beta_{1} ) q^{53} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{54} + ( 14 + 16 \beta_{1} - \beta_{2} - 16 \beta_{3} ) q^{55} + ( 39 + \beta_{1} ) q^{56} + ( -6 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -60 - 44 \beta_{1} ) q^{58} + ( 10 - 42 \beta_{1} ) q^{59} + ( 51 + 23 \beta_{1} ) q^{60} + ( 11 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -18 \beta_{2} - 2 \beta_{3} ) q^{62} + 3 \beta_{2} q^{63} + ( 7 + 18 \beta_{1} ) q^{64} + ( -16 \beta_{2} + 30 \beta_{3} ) q^{65} + ( 48 + 3 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} ) q^{66} -34 q^{67} + ( 10 \beta_{2} - 22 \beta_{3} ) q^{68} + ( -15 + 23 \beta_{1} ) q^{69} + ( -60 + 2 \beta_{1} ) q^{70} + ( -71 + \beta_{1} ) q^{71} + ( -6 \beta_{2} + 9 \beta_{3} ) q^{72} + ( 10 \beta_{2} - 4 \beta_{3} ) q^{73} + ( 2 \beta_{2} + 14 \beta_{3} ) q^{74} + ( -18 - 3 \beta_{1} ) q^{75} + ( 8 \beta_{2} - 2 \beta_{3} ) q^{76} + ( -45 + 13 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -57 - 51 \beta_{1} ) q^{78} + ( -5 \beta_{2} + 24 \beta_{3} ) q^{79} + ( 109 + 15 \beta_{1} ) q^{80} + 9 q^{81} + ( 54 - 34 \beta_{1} ) q^{82} + ( 6 \beta_{2} - 16 \beta_{3} ) q^{83} + ( \beta_{2} - 10 \beta_{3} ) q^{84} + ( -2 \beta_{2} + 20 \beta_{3} ) q^{85} + ( -54 + 80 \beta_{1} ) q^{86} + ( -4 \beta_{2} + 16 \beta_{3} ) q^{87} + ( 99 + 22 \beta_{1} - 11 \beta_{2} + 22 \beta_{3} ) q^{88} + ( 76 + 18 \beta_{1} ) q^{89} + ( 9 \beta_{2} - 12 \beta_{3} ) q^{90} + ( 6 + 32 \beta_{1} ) q^{91} + ( 85 + 21 \beta_{1} ) q^{92} + ( 54 - 20 \beta_{1} ) q^{93} + ( -3 \beta_{2} - 22 \beta_{3} ) q^{94} + ( 14 \beta_{2} + 16 \beta_{3} ) q^{95} + ( 3 \beta_{2} + 9 \beta_{3} ) q^{96} + ( -94 - 42 \beta_{1} ) q^{97} + ( 10 \beta_{2} - 35 \beta_{3} ) q^{98} + ( 15 + 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 20q^{4} + 4q^{5} + 12q^{9} + O(q^{10})$$ $$4q - 20q^{4} + 4q^{5} + 12q^{9} + 20q^{11} + 24q^{12} + 12q^{14} - 36q^{15} + 4q^{16} - 92q^{20} - 12q^{22} - 92q^{23} + 12q^{25} + 204q^{26} + 80q^{31} - 12q^{33} + 48q^{34} - 60q^{36} - 64q^{37} - 120q^{38} + 84q^{42} - 124q^{44} + 12q^{45} + 100q^{47} - 144q^{48} + 100q^{49} + 28q^{53} + 56q^{55} + 156q^{56} - 240q^{58} + 40q^{59} + 204q^{60} + 28q^{64} + 192q^{66} - 136q^{67} - 60q^{69} - 240q^{70} - 284q^{71} - 72q^{75} - 180q^{77} - 228q^{78} + 436q^{80} + 36q^{81} + 216q^{82} - 216q^{86} + 396q^{88} + 304q^{89} + 24q^{91} + 340q^{92} + 216q^{93} - 376q^{97} + 60q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 12 x^{2} + 4 x + 22$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 2 \nu + 23$$$$)/13$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 28 \nu + 10$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{3} + 10 \nu^{2} - 32 \nu + 9$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 2 \beta_{1} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} - 2 \beta_{2} + 12 \beta_{1} - 19$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −0.366025 − 1.29224i 1.36603 + 3.21405i 1.36603 − 3.21405i −0.366025 + 1.29224i
3.53045i −1.73205 −8.46410 6.19615 6.11492i 2.58447i 15.7603i 3.00000 21.8752i
10.2 2.35285i 1.73205 −1.53590 −4.19615 4.07525i 6.42810i 5.79766i 3.00000 9.87291i
10.3 2.35285i 1.73205 −1.53590 −4.19615 4.07525i 6.42810i 5.79766i 3.00000 9.87291i
10.4 3.53045i −1.73205 −8.46410 6.19615 6.11492i 2.58447i 15.7603i 3.00000 21.8752i
 $$n$$: 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.c.a 4
3.b odd 2 1 99.3.c.b 4
4.b odd 2 1 528.3.j.c 4
5.b even 2 1 825.3.b.a 4
5.c odd 4 2 825.3.h.a 8
8.b even 2 1 2112.3.j.a 4
8.d odd 2 1 2112.3.j.d 4
11.b odd 2 1 inner 33.3.c.a 4
11.c even 5 4 363.3.g.e 16
11.d odd 10 4 363.3.g.e 16
12.b even 2 1 1584.3.j.f 4
33.d even 2 1 99.3.c.b 4
44.c even 2 1 528.3.j.c 4
55.d odd 2 1 825.3.b.a 4
55.e even 4 2 825.3.h.a 8
88.b odd 2 1 2112.3.j.a 4
88.g even 2 1 2112.3.j.d 4
132.d odd 2 1 1584.3.j.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a 4 1.a even 1 1 trivial
33.3.c.a 4 11.b odd 2 1 inner
99.3.c.b 4 3.b odd 2 1
99.3.c.b 4 33.d even 2 1
363.3.g.e 16 11.c even 5 4
363.3.g.e 16 11.d odd 10 4
528.3.j.c 4 4.b odd 2 1
528.3.j.c 4 44.c even 2 1
825.3.b.a 4 5.b even 2 1
825.3.b.a 4 55.d odd 2 1
825.3.h.a 8 5.c odd 4 2
825.3.h.a 8 55.e even 4 2
1584.3.j.f 4 12.b even 2 1
1584.3.j.f 4 132.d odd 2 1
2112.3.j.a 4 8.b even 2 1
2112.3.j.a 4 88.b odd 2 1
2112.3.j.d 4 8.d odd 2 1
2112.3.j.d 4 88.g even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 21 T^{4} + 32 T^{6} + 256 T^{8}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ $$( 1 - 2 T + 24 T^{2} - 50 T^{3} + 625 T^{4} )^{2}$$
$7$ $$1 - 148 T^{2} + 9978 T^{4} - 355348 T^{6} + 5764801 T^{8}$$
$11$ $$1 - 20 T + 330 T^{2} - 2420 T^{3} + 14641 T^{4}$$
$13$ $$1 - 52 T^{2} - 6150 T^{4} - 1485172 T^{6} + 815730721 T^{8}$$
$17$ $$1 - 940 T^{2} + 386214 T^{4} - 78509740 T^{6} + 6975757441 T^{8}$$
$19$ $$1 - 508 T^{2} + 116070 T^{4} - 66203068 T^{6} + 16983563041 T^{8}$$
$23$ $$( 1 + 46 T + 1512 T^{2} + 24334 T^{3} + 279841 T^{4} )^{2}$$
$29$ $$1 - 1828 T^{2} + 1730790 T^{4} - 1292909668 T^{6} + 500246412961 T^{8}$$
$31$ $$( 1 - 40 T + 1350 T^{2} - 38440 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 + 32 T + 2982 T^{2} + 43808 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 4420 T^{2} + 9212934 T^{4} - 12489863620 T^{6} + 7984925229121 T^{8}$$
$43$ $$1 - 124 T^{2} - 2536026 T^{4} - 423931324 T^{6} + 11688200277601 T^{8}$$
$47$ $$( 1 - 50 T + 5016 T^{2} - 110450 T^{3} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 - 14 T + 5304 T^{2} - 39326 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$( 1 - 20 T + 1770 T^{2} - 69620 T^{3} + 12117361 T^{4} )^{2}$$
$61$ $$1 - 8740 T^{2} + 39948282 T^{4} - 121012650340 T^{6} + 191707312997281 T^{8}$$
$67$ $$( 1 + 34 T + 4489 T^{2} )^{4}$$
$71$ $$( 1 + 142 T + 15120 T^{2} + 715822 T^{3} + 25411681 T^{4} )^{2}$$
$73$ $$1 - 16708 T^{2} + 126086406 T^{4} - 474477810628 T^{6} + 806460091894081 T^{8}$$
$79$ $$1 - 14836 T^{2} + 112926714 T^{4} - 577863401716 T^{6} + 1517108809906561 T^{8}$$
$83$ $$1 - 22372 T^{2} + 213329190 T^{4} - 1061737557412 T^{6} + 2252292232139041 T^{8}$$
$89$ $$( 1 - 152 T + 20646 T^{2} - 1203992 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 + 188 T + 22362 T^{2} + 1768892 T^{3} + 88529281 T^{4} )^{2}$$