# Properties

 Label 135.2.a.b Level $135$ Weight $2$ Character orbit 135.a Self dual yes Analytic conductor $1.078$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 135.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 0 2.00000 1.00000 0 −3.00000 0 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.a.b yes 1
3.b odd 2 1 135.2.a.a 1
4.b odd 2 1 2160.2.a.v 1
5.b even 2 1 675.2.a.a 1
5.c odd 4 2 675.2.b.b 2
7.b odd 2 1 6615.2.a.j 1
8.b even 2 1 8640.2.a.c 1
8.d odd 2 1 8640.2.a.bb 1
9.c even 3 2 405.2.e.b 2
9.d odd 6 2 405.2.e.h 2
12.b even 2 1 2160.2.a.j 1
15.d odd 2 1 675.2.a.i 1
15.e even 4 2 675.2.b.a 2
21.c even 2 1 6615.2.a.a 1
24.f even 2 1 8640.2.a.ce 1
24.h odd 2 1 8640.2.a.bh 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.a 1 3.b odd 2 1
135.2.a.b yes 1 1.a even 1 1 trivial
405.2.e.b 2 9.c even 3 2
405.2.e.h 2 9.d odd 6 2
675.2.a.a 1 5.b even 2 1
675.2.a.i 1 15.d odd 2 1
675.2.b.a 2 15.e even 4 2
675.2.b.b 2 5.c odd 4 2
2160.2.a.j 1 12.b even 2 1
2160.2.a.v 1 4.b odd 2 1
6615.2.a.a 1 21.c even 2 1
6615.2.a.j 1 7.b odd 2 1
8640.2.a.c 1 8.b even 2 1
8640.2.a.bb 1 8.d odd 2 1
8640.2.a.bh 1 24.h odd 2 1
8640.2.a.ce 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(135))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 3$$
$11$ $$T - 2$$
$13$ $$T + 5$$
$17$ $$T - 8$$
$19$ $$T - 1$$
$23$ $$T + 6$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T - 5$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T - 2$$
$59$ $$T - 8$$
$61$ $$T - 7$$
$67$ $$T + 9$$
$71$ $$T + 2$$
$73$ $$T + 5$$
$79$ $$T + 3$$
$83$ $$T + 6$$
$89$ $$T - 12$$
$97$ $$T + 13$$