# Properties

 Label 135.4.a.e Level $135$ Weight $4$ Character orbit 135.a Self dual yes Analytic conductor $7.965$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.1772.1 Defining polynomial: $$x^{3} - x^{2} - 12 x + 8$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{1} ) q^{2} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{4} -5 q^{5} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{7} + ( -29 + 7 \beta_{1} - 5 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -2 + \beta_{1} ) q^{2} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{4} -5 q^{5} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{7} + ( -29 + 7 \beta_{1} - 5 \beta_{2} ) q^{8} + ( 10 - 5 \beta_{1} ) q^{10} + ( 3 - 9 \beta_{1} + 5 \beta_{2} ) q^{11} + ( 5 - 6 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -13 - 16 \beta_{1} + 5 \beta_{2} ) q^{14} + ( 69 - 37 \beta_{1} + 9 \beta_{2} ) q^{16} + ( -51 + 3 \beta_{1} + 5 \beta_{2} ) q^{17} + ( -28 + 33 \beta_{1} - \beta_{2} ) q^{19} + ( -35 + 15 \beta_{1} - 5 \beta_{2} ) q^{20} + ( -95 + 37 \beta_{1} - 19 \beta_{2} ) q^{22} + ( -85 - 25 \beta_{1} + 5 \beta_{2} ) q^{23} + 25 q^{25} + ( -72 + 21 \beta_{1} - 10 \beta_{2} ) q^{26} + ( -124 + 36 \beta_{1} - 2 \beta_{2} ) q^{28} + ( -47 + \beta_{1} - 25 \beta_{2} ) q^{29} + ( -39 + 19 \beta_{1} + 17 \beta_{2} ) q^{31} + ( -295 + 95 \beta_{1} - 15 \beta_{2} ) q^{32} + ( 145 - 29 \beta_{1} - 7 \beta_{2} ) q^{34} + ( 10 + 5 \beta_{1} + 15 \beta_{2} ) q^{35} + ( -138 + 55 \beta_{1} + 25 \beta_{2} ) q^{37} + ( 417 - 66 \beta_{1} + 35 \beta_{2} ) q^{38} + ( 145 - 35 \beta_{1} + 25 \beta_{2} ) q^{40} + ( -188 - 26 \beta_{1} - 10 \beta_{2} ) q^{41} + ( -281 + 17 \beta_{1} - 29 \beta_{2} ) q^{43} + ( 535 - 155 \beta_{1} + 35 \beta_{2} ) q^{44} + ( -95 - 35 \beta_{1} - 35 \beta_{2} ) q^{46} + ( 9 + 123 \beta_{1} + 5 \beta_{2} ) q^{47} + ( 161 + 53 \beta_{1} - 41 \beta_{2} ) q^{49} + ( -50 + 25 \beta_{1} ) q^{50} + ( 315 - 95 \beta_{1} + 25 \beta_{2} ) q^{52} + ( 136 - 38 \beta_{1} - 30 \beta_{2} ) q^{53} + ( -15 + 45 \beta_{1} - 25 \beta_{2} ) q^{55} + ( 744 - 42 \beta_{1} ) q^{56} + ( 55 - 173 \beta_{1} + 51 \beta_{2} ) q^{58} + ( -104 - 68 \beta_{1} ) q^{59} + ( -26 - 43 \beta_{1} + 31 \beta_{2} ) q^{61} + ( 321 + 27 \beta_{1} - 15 \beta_{2} ) q^{62} + ( 1053 - 169 \beta_{1} + 53 \beta_{2} ) q^{64} + ( -25 + 30 \beta_{1} - 10 \beta_{2} ) q^{65} + ( 40 - 121 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -215 + 115 \beta_{1} - 55 \beta_{2} ) q^{68} + ( 65 + 80 \beta_{1} - 25 \beta_{2} ) q^{70} + ( -64 + 182 \beta_{1} + 30 \beta_{2} ) q^{71} + ( -306 - 59 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 931 - 68 \beta_{1} + 5 \beta_{2} ) q^{74} + ( -1266 + 394 \beta_{1} - 128 \beta_{2} ) q^{76} + ( -658 + 104 \beta_{1} + 80 \beta_{2} ) q^{77} + ( 343 + 38 \beta_{1} + 54 \beta_{2} ) q^{79} + ( -345 + 185 \beta_{1} - 45 \beta_{2} ) q^{80} + ( 70 - 212 \beta_{1} - 6 \beta_{2} ) q^{82} + ( -102 - 24 \beta_{1} - 60 \beta_{2} ) q^{83} + ( 255 - 15 \beta_{1} - 25 \beta_{2} ) q^{85} + ( 691 - 443 \beta_{1} + 75 \beta_{2} ) q^{86} + ( -1945 + 569 \beta_{1} - 73 \beta_{2} ) q^{88} + ( -360 - 60 \beta_{2} ) q^{89} + ( -230 + 81 \beta_{1} + 23 \beta_{2} ) q^{91} + ( 415 - 35 \beta_{1} - 5 \beta_{2} ) q^{92} + ( 1345 - 89 \beta_{1} + 113 \beta_{2} ) q^{94} + ( 140 - 165 \beta_{1} + 5 \beta_{2} ) q^{95} + ( 202 + 113 \beta_{1} - \beta_{2} ) q^{97} + ( 179 - 97 \beta_{1} + 135 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8} + O(q^{10})$$ $$3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8} + 25 q^{10} - 5 q^{11} + 7 q^{13} - 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} - 85 q^{20} - 229 q^{22} - 285 q^{23} + 75 q^{25} - 185 q^{26} - 334 q^{28} - 115 q^{29} - 115 q^{31} - 775 q^{32} + 413 q^{34} + 20 q^{35} - 384 q^{37} + 1150 q^{38} + 375 q^{40} - 580 q^{41} - 797 q^{43} + 1415 q^{44} - 285 q^{46} + 145 q^{47} + 577 q^{49} - 125 q^{50} + 825 q^{52} + 400 q^{53} + 25 q^{55} + 2190 q^{56} - 59 q^{58} - 380 q^{59} - 152 q^{61} + 1005 q^{62} + 2937 q^{64} - 35 q^{65} + 2 q^{67} - 475 q^{68} + 300 q^{70} - 40 q^{71} - 980 q^{73} + 2720 q^{74} - 3276 q^{76} - 1950 q^{77} + 1013 q^{79} - 805 q^{80} + 4 q^{82} - 270 q^{83} + 775 q^{85} + 1555 q^{86} - 5193 q^{88} - 1020 q^{89} - 632 q^{91} + 1215 q^{92} + 3833 q^{94} + 250 q^{95} + 720 q^{97} + 305 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 12 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + \nu - 8$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{2} + 5 \nu + 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{2} + 5 \beta_{1} + 23$$$$)/3$$

## 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
 0.654334 −3.32803 3.67370
−5.45876 0 21.7980 −5.00000 0 −11.8065 −75.3201 0 27.2938
1.2 −2.12612 0 −3.47962 −5.00000 0 30.7000 24.4070 0 10.6306
1.3 2.58488 0 −1.31841 −5.00000 0 −22.8935 −24.0869 0 −12.9244
 $$n$$: 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.e 3
3.b odd 2 1 135.4.a.h yes 3
4.b odd 2 1 2160.4.a.bi 3
5.b even 2 1 675.4.a.s 3
5.c odd 4 2 675.4.b.m 6
9.c even 3 2 405.4.e.v 6
9.d odd 6 2 405.4.e.q 6
12.b even 2 1 2160.4.a.bq 3
15.d odd 2 1 675.4.a.p 3
15.e even 4 2 675.4.b.n 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 5 T_{2}^{2} - 8 T_{2} - 30$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(135))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-30 - 8 T + 5 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$( 5 + T )^{3}$$
$7$ $$-8298 - 795 T + 4 T^{2} + T^{3}$$
$11$ $$-31260 - 2888 T + 5 T^{2} + T^{3}$$
$13$ $$-6425 - 769 T - 7 T^{2} + T^{3}$$
$17$ $$41760 + 5608 T + 155 T^{2} + T^{3}$$
$19$ $$-368012 - 16663 T + 50 T^{2} + T^{3}$$
$23$ $$-553500 + 16200 T + 285 T^{2} + T^{3}$$
$29$ $$-6440340 - 47408 T + 115 T^{2} + T^{3}$$
$31$ $$-938304 - 29232 T + 115 T^{2} + T^{3}$$
$37$ $$-22667198 - 67923 T + 384 T^{2} + T^{3}$$
$41$ $$3917280 + 89812 T + 580 T^{2} + T^{3}$$
$43$ $$-5357936 + 142520 T + 797 T^{2} + T^{3}$$
$47$ $$-14388240 - 249152 T - 145 T^{2} + T^{3}$$
$53$ $$12658320 - 58172 T - 400 T^{2} + T^{3}$$
$59$ $$-5205120 - 27392 T + 380 T^{2} + T^{3}$$
$61$ $$-5069066 - 87475 T + 152 T^{2} + T^{3}$$
$67$ $$20769300 - 243999 T - 2 T^{2} + T^{3}$$
$71$ $$-216071280 - 677372 T + 40 T^{2} + T^{3}$$
$73$ $$16447954 + 264533 T + 980 T^{2} + T^{3}$$
$79$ $$90596925 + 52215 T - 1013 T^{2} + T^{3}$$
$83$ $$-84539160 - 301428 T + 270 T^{2} + T^{3}$$
$89$ $$-125064000 + 46800 T + 1020 T^{2} + T^{3}$$
$97$ $$27430558 - 34563 T - 720 T^{2} + T^{3}$$