# Properties

 Label 135.2.e.b Level $135$ Weight $2$ Character orbit 135.e Analytic conductor $1.078$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 135.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 Defining polynomial: $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} + \beta_{5} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -1 - \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{4} + \beta_{5} ) q^{2} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -1 - \beta_{2} + \beta_{4} ) q^{8} -\beta_{4} q^{10} + ( \beta_{1} - \beta_{3} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -4 - \beta_{1} - \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{14} + ( -\beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{16} + ( 1 + \beta_{2} ) q^{17} + ( 1 - \beta_{2} ) q^{19} + ( -\beta_{1} + 2 \beta_{3} ) q^{20} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{22} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{23} -\beta_{3} q^{25} + ( 1 + \beta_{2} ) q^{26} + ( 3 + \beta_{2} + 3 \beta_{4} ) q^{28} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{5} ) q^{31} + ( 6 - 6 \beta_{3} + \beta_{5} ) q^{32} + ( -\beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{34} + ( -2 - \beta_{4} ) q^{35} + ( 3 + \beta_{2} + 2 \beta_{4} ) q^{37} + ( \beta_{1} - \beta_{3} ) q^{38} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{40} + ( -4 - \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} ) q^{41} + ( \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 7 + \beta_{2} - 2 \beta_{4} ) q^{44} + ( -2 + \beta_{2} - 4 \beta_{4} ) q^{46} + ( \beta_{1} + 5 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{47} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{49} -\beta_{5} q^{50} + ( \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{52} -2 \beta_{4} q^{53} + ( -1 - \beta_{2} ) q^{55} -3 \beta_{3} q^{56} + ( 6 - 6 \beta_{3} + \beta_{5} ) q^{58} + 2 \beta_{5} q^{59} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -15 - 3 \beta_{2} ) q^{62} + ( -6 - \beta_{2} - 2 \beta_{4} ) q^{64} + ( \beta_{1} + \beta_{3} ) q^{65} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{5} ) q^{67} + ( -7 - \beta_{1} - \beta_{2} + 7 \beta_{3} + 2 \beta_{5} ) q^{68} + ( -\beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{70} + ( 3 - 3 \beta_{2} + 2 \beta_{4} ) q^{71} + ( -4 - 4 \beta_{4} ) q^{73} + ( \beta_{1} - 7 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{74} + ( 3 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{77} + ( -2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{79} + ( -1 + 2 \beta_{4} ) q^{80} + ( -9 - 3 \beta_{2} + 5 \beta_{4} ) q^{82} + ( -6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{83} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{85} + ( 9 + 3 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} - 4 \beta_{5} ) q^{86} + ( -\beta_{1} + 7 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{88} + 3 q^{89} + ( 3 - \beta_{2} ) q^{91} + ( -\beta_{1} + 13 \beta_{3} - \beta_{4} + \beta_{5} ) q^{92} + ( 13 + 4 \beta_{1} + 4 \beta_{2} - 13 \beta_{3} + 4 \beta_{5} ) q^{94} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{95} + ( -4 \beta_{1} + 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{97} + ( 15 + 3 \beta_{2} + 2 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{2} - 5q^{4} + 3q^{5} - 5q^{7} - 6q^{8} + O(q^{10})$$ $$6q + q^{2} - 5q^{4} + 3q^{5} - 5q^{7} - 6q^{8} + 2q^{10} - 2q^{11} - 4q^{13} - 9q^{14} - 5q^{16} + 4q^{17} + 8q^{19} + 5q^{20} + 4q^{22} + 3q^{23} - 3q^{25} + 4q^{26} + 10q^{28} - 7q^{29} - 8q^{31} + 17q^{32} + 4q^{34} - 10q^{35} + 12q^{37} - 2q^{38} - 3q^{40} - 13q^{41} - 10q^{43} + 44q^{44} - 6q^{46} + 13q^{47} + 2q^{49} + q^{50} + 12q^{52} + 4q^{53} - 4q^{55} - 9q^{56} + 17q^{58} - 2q^{59} - q^{61} - 84q^{62} - 30q^{64} + 4q^{65} - 11q^{67} - 22q^{68} + 9q^{70} + 20q^{71} - 16q^{73} - 16q^{74} + 12q^{76} - 2q^{79} - 10q^{80} - 58q^{82} - 15q^{83} + 2q^{85} + 28q^{86} + 24q^{88} + 18q^{89} + 20q^{91} + 39q^{92} + 31q^{94} + 4q^{95} + 18q^{97} + 80q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - \nu^{3} + 9 \nu^{2} - 21 \nu - 9$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} - \nu^{3} - 18 \nu^{2} + 33 \nu - 9$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 36$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} + 7 \nu^{3} + 9 \nu^{2} + 12 \nu + 9$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} + 3 \nu - 72$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} + 7 \beta_{4} - 11 \beta_{3} + \beta_{2} - \beta_{1} + 7$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 20$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{5} - 2 \beta_{4} - 20 \beta_{3} + \beta_{2} - 10 \beta_{1} + 43$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$$.

 $$n$$ $$56$$ $$82$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 1.71903 + 0.211943i −1.62241 + 0.606458i 0.403374 − 1.68443i 1.71903 − 0.211943i −1.62241 − 0.606458i 0.403374 + 1.68443i
−1.04307 + 1.80664i 0 −1.17597 2.03684i 0.500000 + 0.866025i 0 −2.04307 + 3.53869i 0.734191 0 −2.08613
46.2 0.285997 0.495361i 0 0.836412 + 1.44871i 0.500000 + 0.866025i 0 −0.714003 + 1.23669i 2.10083 0 0.571993
46.3 1.25707 2.17731i 0 −2.16044 3.74200i 0.500000 + 0.866025i 0 0.257068 0.445256i −5.83502 0 2.51414
91.1 −1.04307 1.80664i 0 −1.17597 + 2.03684i 0.500000 0.866025i 0 −2.04307 3.53869i 0.734191 0 −2.08613
91.2 0.285997 + 0.495361i 0 0.836412 1.44871i 0.500000 0.866025i 0 −0.714003 1.23669i 2.10083 0 0.571993
91.3 1.25707 + 2.17731i 0 −2.16044 + 3.74200i 0.500000 0.866025i 0 0.257068 + 0.445256i −5.83502 0 2.51414
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.e.b 6
3.b odd 2 1 45.2.e.b 6
4.b odd 2 1 2160.2.q.k 6
5.b even 2 1 675.2.e.b 6
5.c odd 4 2 675.2.k.b 12
9.c even 3 1 inner 135.2.e.b 6
9.c even 3 1 405.2.a.i 3
9.d odd 6 1 45.2.e.b 6
9.d odd 6 1 405.2.a.j 3
12.b even 2 1 720.2.q.i 6
15.d odd 2 1 225.2.e.b 6
15.e even 4 2 225.2.k.b 12
36.f odd 6 1 2160.2.q.k 6
36.f odd 6 1 6480.2.a.bs 3
36.h even 6 1 720.2.q.i 6
36.h even 6 1 6480.2.a.bv 3
45.h odd 6 1 225.2.e.b 6
45.h odd 6 1 2025.2.a.n 3
45.j even 6 1 675.2.e.b 6
45.j even 6 1 2025.2.a.o 3
45.k odd 12 2 675.2.k.b 12
45.k odd 12 2 2025.2.b.m 6
45.l even 12 2 225.2.k.b 12
45.l even 12 2 2025.2.b.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 3.b odd 2 1
45.2.e.b 6 9.d odd 6 1
135.2.e.b 6 1.a even 1 1 trivial
135.2.e.b 6 9.c even 3 1 inner
225.2.e.b 6 15.d odd 2 1
225.2.e.b 6 45.h odd 6 1
225.2.k.b 12 15.e even 4 2
225.2.k.b 12 45.l even 12 2
405.2.a.i 3 9.c even 3 1
405.2.a.j 3 9.d odd 6 1
675.2.e.b 6 5.b even 2 1
675.2.e.b 6 45.j even 6 1
675.2.k.b 12 5.c odd 4 2
675.2.k.b 12 45.k odd 12 2
720.2.q.i 6 12.b even 2 1
720.2.q.i 6 36.h even 6 1
2025.2.a.n 3 45.h odd 6 1
2025.2.a.o 3 45.j even 6 1
2025.2.b.l 6 45.l even 12 2
2025.2.b.m 6 45.k odd 12 2
2160.2.q.k 6 4.b odd 2 1
2160.2.q.k 6 36.f odd 6 1
6480.2.a.bs 3 36.f odd 6 1
6480.2.a.bv 3 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - T_{2}^{5} + 6 T_{2}^{4} - T_{2}^{3} + 28 T_{2}^{2} - 15 T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(135, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 15 T + 28 T^{2} - T^{3} + 6 T^{4} - T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 1 - T + T^{2} )^{3}$$
$7$ $$9 - 9 T + 24 T^{2} + 21 T^{3} + 22 T^{4} + 5 T^{5} + T^{6}$$
$11$ $$144 + 96 T + 88 T^{2} + 8 T^{3} + 12 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$16 + 16 T + 32 T^{2} - 8 T^{3} + 20 T^{4} + 4 T^{5} + T^{6}$$
$17$ $$( 12 - 8 T - 2 T^{2} + T^{3} )^{2}$$
$19$ $$( 4 - 4 T - 4 T^{2} + T^{3} )^{2}$$
$23$ $$13689 - 3861 T + 1440 T^{2} - 135 T^{3} + 42 T^{4} - 3 T^{5} + T^{6}$$
$29$ $$2601 + 1479 T + 1198 T^{2} - 101 T^{3} + 78 T^{4} + 7 T^{5} + T^{6}$$
$31$ $$219024 + 28080 T + 7344 T^{2} + 456 T^{3} + 124 T^{4} + 8 T^{5} + T^{6}$$
$37$ $$( 4 - 12 T - 6 T^{2} + T^{3} )^{2}$$
$41$ $$9 + 57 T + 322 T^{2} + 241 T^{3} + 150 T^{4} + 13 T^{5} + T^{6}$$
$43$ $$16 + 16 T + 56 T^{2} - 32 T^{3} + 104 T^{4} + 10 T^{5} + T^{6}$$
$47$ $$136161 - 4059 T + 4918 T^{2} - 595 T^{3} + 180 T^{4} - 13 T^{5} + T^{6}$$
$53$ $$( 24 - 20 T - 2 T^{2} + T^{3} )^{2}$$
$59$ $$576 + 480 T + 448 T^{2} + 8 T^{3} + 24 T^{4} + 2 T^{5} + T^{6}$$
$61$ $$5041 - 2627 T + 1298 T^{2} - 179 T^{3} + 38 T^{4} + T^{5} + T^{6}$$
$67$ $$257049 + 19773 T + 7098 T^{2} + 585 T^{3} + 160 T^{4} + 11 T^{5} + T^{6}$$
$71$ $$( 708 - 92 T - 10 T^{2} + T^{3} )^{2}$$
$73$ $$( -128 - 64 T + 8 T^{2} + T^{3} )^{2}$$
$79$ $$576 - 2016 T + 7008 T^{2} - 216 T^{3} + 88 T^{4} + 2 T^{5} + T^{6}$$
$83$ $$6561 - 2187 T + 1944 T^{2} + 567 T^{3} + 198 T^{4} + 15 T^{5} + T^{6}$$
$89$ $$( -3 + T )^{6}$$
$97$ $$1700416 - 46944 T + 24768 T^{2} - 1960 T^{3} + 360 T^{4} - 18 T^{5} + T^{6}$$