# Properties

 Label 135.4.a.c Level $135$ Weight $4$ Character orbit 135.a Self dual yes Analytic conductor $7.965$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 135.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.96525785077$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{4} + 5 q^{5} - 6 q^{7} - 15 q^{8} + O(q^{10})$$ $$q + q^{2} - 7 q^{4} + 5 q^{5} - 6 q^{7} - 15 q^{8} + 5 q^{10} - 47 q^{11} - 5 q^{13} - 6 q^{14} + 41 q^{16} - 131 q^{17} - 56 q^{19} - 35 q^{20} - 47 q^{22} + 3 q^{23} + 25 q^{25} - 5 q^{26} + 42 q^{28} - 157 q^{29} + 225 q^{31} + 161 q^{32} - 131 q^{34} - 30 q^{35} - 70 q^{37} - 56 q^{38} - 75 q^{40} + 140 q^{41} + 397 q^{43} + 329 q^{44} + 3 q^{46} - 347 q^{47} - 307 q^{49} + 25 q^{50} + 35 q^{52} + 4 q^{53} - 235 q^{55} + 90 q^{56} - 157 q^{58} + 748 q^{59} - 338 q^{61} + 225 q^{62} - 167 q^{64} - 25 q^{65} + 492 q^{67} + 917 q^{68} - 30 q^{70} + 32 q^{71} + 970 q^{73} - 70 q^{74} + 392 q^{76} + 282 q^{77} - 1257 q^{79} + 205 q^{80} + 140 q^{82} - 102 q^{83} - 655 q^{85} + 397 q^{86} + 705 q^{88} - 1488 q^{89} + 30 q^{91} - 21 q^{92} - 347 q^{94} - 280 q^{95} + 974 q^{97} - 307 q^{98} + O(q^{100})$$

## 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
1.00000 0 −7.00000 5.00000 0 −6.00000 −15.0000 0 5.00000
 $$n$$: 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 135.4.a.c yes 1
3.b odd 2 1 135.4.a.b 1
4.b odd 2 1 2160.4.a.p 1
5.b even 2 1 675.4.a.c 1
5.c odd 4 2 675.4.b.e 2
9.c even 3 2 405.4.e.f 2
9.d odd 6 2 405.4.e.h 2
12.b even 2 1 2160.4.a.f 1
15.d odd 2 1 675.4.a.h 1
15.e even 4 2 675.4.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.b 1 3.b odd 2 1
135.4.a.c yes 1 1.a even 1 1 trivial
405.4.e.f 2 9.c even 3 2
405.4.e.h 2 9.d odd 6 2
675.4.a.c 1 5.b even 2 1
675.4.a.h 1 15.d odd 2 1
675.4.b.e 2 5.c odd 4 2
675.4.b.f 2 15.e even 4 2
2160.4.a.f 1 12.b even 2 1
2160.4.a.p 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(135))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$-5 + T$$
$7$ $$6 + T$$
$11$ $$47 + T$$
$13$ $$5 + T$$
$17$ $$131 + T$$
$19$ $$56 + T$$
$23$ $$-3 + T$$
$29$ $$157 + T$$
$31$ $$-225 + T$$
$37$ $$70 + T$$
$41$ $$-140 + T$$
$43$ $$-397 + T$$
$47$ $$347 + T$$
$53$ $$-4 + T$$
$59$ $$-748 + T$$
$61$ $$338 + T$$
$67$ $$-492 + T$$
$71$ $$-32 + T$$
$73$ $$-970 + T$$
$79$ $$1257 + T$$
$83$ $$102 + T$$
$89$ $$1488 + T$$
$97$ $$-974 + T$$