# Properties

 Label 33.2.e.b Level $33$ Weight $2$ Character orbit 33.e Analytic conductor $0.264$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.263506326670$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i −0.309017 + 0.951057i
0.309017 + 0.224514i −0.309017 + 0.951057i −0.572949 1.76336i −1.30902 + 0.951057i −0.309017 + 0.224514i −0.309017 0.951057i 0.454915 1.40008i −0.809017 0.587785i −0.618034
16.1 −0.809017 2.48990i 0.809017 + 0.587785i −3.92705 + 2.85317i −0.190983 + 0.587785i 0.809017 2.48990i 0.809017 0.587785i 6.04508 + 4.39201i 0.309017 + 0.951057i 1.61803
25.1 0.309017 0.224514i −0.309017 0.951057i −0.572949 + 1.76336i −1.30902 0.951057i −0.309017 0.224514i −0.309017 + 0.951057i 0.454915 + 1.40008i −0.809017 + 0.587785i −0.618034
31.1 −0.809017 + 2.48990i 0.809017 0.587785i −3.92705 2.85317i −0.190983 0.587785i 0.809017 + 2.48990i 0.809017 + 0.587785i 6.04508 4.39201i 0.309017 0.951057i 1.61803
 $$n$$: 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.2.e.b 4
3.b odd 2 1 99.2.f.a 4
4.b odd 2 1 528.2.y.b 4
5.b even 2 1 825.2.n.c 4
5.c odd 4 2 825.2.bx.d 8
9.c even 3 2 891.2.n.c 8
9.d odd 6 2 891.2.n.b 8
11.b odd 2 1 363.2.e.f 4
11.c even 5 1 inner 33.2.e.b 4
11.c even 5 1 363.2.a.d 2
11.c even 5 2 363.2.e.k 4
11.d odd 10 1 363.2.a.i 2
11.d odd 10 2 363.2.e.b 4
11.d odd 10 1 363.2.e.f 4
33.f even 10 1 1089.2.a.l 2
33.h odd 10 1 99.2.f.a 4
33.h odd 10 1 1089.2.a.t 2
44.g even 10 1 5808.2.a.ci 2
44.h odd 10 1 528.2.y.b 4
44.h odd 10 1 5808.2.a.cj 2
55.h odd 10 1 9075.2.a.u 2
55.j even 10 1 825.2.n.c 4
55.j even 10 1 9075.2.a.cb 2
55.k odd 20 2 825.2.bx.d 8
99.m even 15 2 891.2.n.c 8
99.n odd 30 2 891.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 1.a even 1 1 trivial
33.2.e.b 4 11.c even 5 1 inner
99.2.f.a 4 3.b odd 2 1
99.2.f.a 4 33.h odd 10 1
363.2.a.d 2 11.c even 5 1
363.2.a.i 2 11.d odd 10 1
363.2.e.b 4 11.d odd 10 2
363.2.e.f 4 11.b odd 2 1
363.2.e.f 4 11.d odd 10 1
363.2.e.k 4 11.c even 5 2
528.2.y.b 4 4.b odd 2 1
528.2.y.b 4 44.h odd 10 1
825.2.n.c 4 5.b even 2 1
825.2.n.c 4 55.j even 10 1
825.2.bx.d 8 5.c odd 4 2
825.2.bx.d 8 55.k odd 20 2
891.2.n.b 8 9.d odd 6 2
891.2.n.b 8 99.n odd 30 2
891.2.n.c 8 9.c even 3 2
891.2.n.c 8 99.m even 15 2
1089.2.a.l 2 33.f even 10 1
1089.2.a.t 2 33.h odd 10 1
5808.2.a.ci 2 44.g even 10 1
5808.2.a.cj 2 44.h odd 10 1
9075.2.a.u 2 55.h odd 10 1
9075.2.a.cb 2 55.j even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 6 T_{2}^{2} - 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8}$$
$7$ $$1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8}$$
$11$ $$1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4}$$
$13$ $$1 - 7 T + 6 T^{2} + 49 T^{3} - 181 T^{4} + 637 T^{5} + 1014 T^{6} - 15379 T^{7} + 28561 T^{8}$$
$17$ $$1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 3060 T^{5} + 10693 T^{6} - 58956 T^{7} + 83521 T^{8}$$
$19$ $$1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 12 T + 63 T^{2} + 434 T^{3} + 3255 T^{4} + 13454 T^{5} + 60543 T^{6} + 357492 T^{7} + 923521 T^{8}$$
$37$ $$1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 11359 T^{5} - 8214 T^{6} - 455877 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 3 T - 22 T^{2} + 171 T^{3} + 2215 T^{4} + 7011 T^{5} - 36982 T^{6} + 206763 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 41 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 17 T + 67 T^{2} + 335 T^{3} - 4344 T^{4} + 15745 T^{5} + 148003 T^{6} - 1764991 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 8480 T^{5} - 132023 T^{6} - 595508 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 13098 T^{5} + 59177 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 156099 T^{5} + 911645 T^{6} + 4766601 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 103305 T^{5} + 599879 T^{6} - 5368665 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 62050 T^{5} + 335727 T^{6} - 5446238 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 13 T - 14 T^{2} + 421 T^{3} + 1449 T^{4} + 34943 T^{5} - 96446 T^{6} - 7433231 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 12 T + 209 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 74205 T^{5} - 404587 T^{6} - 2738019 T^{7} + 88529281 T^{8}$$