# Properties

 Label 135.4.a.f Level $135$ Weight $4$ Character orbit 135.a Self dual yes Analytic conductor $7.965$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.5637.1 Defining polynomial: $$x^{3} - x^{2} - 23 x + 6$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + 5 q^{5} + ( 14 + 2 \beta_{1} ) q^{7} + ( -9 - 8 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + 5 q^{5} + ( 14 + 2 \beta_{1} ) q^{7} + ( -9 - 8 \beta_{1} - \beta_{2} ) q^{8} -5 \beta_{1} q^{10} + ( 12 - 2 \beta_{1} + 4 \beta_{2} ) q^{11} + ( 14 - 10 \beta_{1} - 4 \beta_{2} ) q^{13} + ( -30 - 16 \beta_{1} - 2 \beta_{2} ) q^{14} + ( 58 + 17 \beta_{1} ) q^{16} + ( -3 - 2 \beta_{1} - 8 \beta_{2} ) q^{17} + ( 59 + 14 \beta_{1} - 4 \beta_{2} ) q^{19} + ( 35 + 5 \beta_{1} + 5 \beta_{2} ) q^{20} + ( 54 - 42 \beta_{1} + 2 \beta_{2} ) q^{22} + ( -39 + 32 \beta_{1} + 4 \beta_{2} ) q^{23} + 25 q^{25} + ( 126 + 28 \beta_{1} + 10 \beta_{2} ) q^{26} + ( 116 + 46 \beta_{1} + 16 \beta_{2} ) q^{28} + ( 42 + 42 \beta_{1} - 8 \beta_{2} ) q^{29} + ( 83 - 30 \beta_{1} + 8 \beta_{2} ) q^{31} + ( -183 - 11 \beta_{1} - 9 \beta_{2} ) q^{32} + ( -18 + 69 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 70 + 10 \beta_{1} ) q^{35} + ( 50 - 64 \beta_{1} - 8 \beta_{2} ) q^{37} + ( -234 - 41 \beta_{1} - 14 \beta_{2} ) q^{38} + ( -45 - 40 \beta_{1} - 5 \beta_{2} ) q^{40} + ( -132 + 42 \beta_{1} + 16 \beta_{2} ) q^{41} + ( -16 + 78 \beta_{1} - 8 \beta_{2} ) q^{43} + ( 546 - 12 \beta_{1} + 10 \beta_{2} ) q^{44} + ( -456 - 25 \beta_{1} - 32 \beta_{2} ) q^{46} + ( -168 + 28 \beta_{1} + 4 \beta_{2} ) q^{47} + ( -87 + 60 \beta_{1} + 4 \beta_{2} ) q^{49} -25 \beta_{1} q^{50} + ( -472 - 154 \beta_{1} + 4 \beta_{2} ) q^{52} + ( 165 + 14 \beta_{1} + 12 \beta_{2} ) q^{53} + ( 60 - 10 \beta_{1} + 20 \beta_{2} ) q^{55} + ( -354 - 162 \beta_{1} - 30 \beta_{2} ) q^{56} + ( -678 - 20 \beta_{1} - 42 \beta_{2} ) q^{58} + ( 78 - 82 \beta_{1} - 12 \beta_{2} ) q^{59} + ( 173 + 60 \beta_{1} + 16 \beta_{2} ) q^{61} + ( 498 - 117 \beta_{1} + 30 \beta_{2} ) q^{62} + ( -353 + 130 \beta_{1} + 11 \beta_{2} ) q^{64} + ( 70 - 50 \beta_{1} - 20 \beta_{2} ) q^{65} + ( 302 - 52 \beta_{1} + 24 \beta_{2} ) q^{67} + ( -999 - 51 \beta_{1} - 5 \beta_{2} ) q^{68} + ( -150 - 80 \beta_{1} - 10 \beta_{2} ) q^{70} + ( -192 + 34 \beta_{1} - 60 \beta_{2} ) q^{71} + ( 404 + 22 \beta_{1} + 60 \beta_{2} ) q^{73} + ( 912 + 78 \beta_{1} + 64 \beta_{2} ) q^{74} + ( 59 + 275 \beta_{1} + 73 \beta_{2} ) q^{76} + ( 60 + 56 \beta_{1} + 52 \beta_{2} ) q^{77} + ( 221 - 22 \beta_{1} - 12 \beta_{2} ) q^{79} + ( 290 + 85 \beta_{1} ) q^{80} + ( -534 - 38 \beta_{1} - 42 \beta_{2} ) q^{82} + ( -489 + 120 \beta_{1} + 60 \beta_{2} ) q^{83} + ( -15 - 10 \beta_{1} - 40 \beta_{2} ) q^{85} + ( -1218 + 2 \beta_{1} - 78 \beta_{2} ) q^{86} + ( -192 - 278 \beta_{1} - 4 \beta_{2} ) q^{88} + ( 756 - 66 \beta_{1} - 48 \beta_{2} ) q^{89} + ( -56 - 196 \beta_{1} - 76 \beta_{2} ) q^{91} + ( 495 + 481 \beta_{1} - 7 \beta_{2} ) q^{92} + ( -396 + 108 \beta_{1} - 28 \beta_{2} ) q^{94} + ( 295 + 70 \beta_{1} - 20 \beta_{2} ) q^{95} + ( 440 + 64 \beta_{1} + 8 \beta_{2} ) q^{97} + ( -876 - 5 \beta_{1} - 60 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} + O(q^{10})$$ $$3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 23 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 15$$

## 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
 5.20067 0.258712 −4.45938
−5.20067 0 19.0470 5.00000 0 24.4013 −57.4517 0 −26.0034
1.2 −0.258712 0 −7.93307 5.00000 0 14.5174 4.12208 0 −1.29356
1.3 4.45938 0 11.8861 5.00000 0 5.08123 17.3296 0 22.2969
 $$n$$: 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.f 3
3.b odd 2 1 135.4.a.g yes 3
4.b odd 2 1 2160.4.a.bm 3
5.b even 2 1 675.4.a.r 3
5.c odd 4 2 675.4.b.l 6
9.c even 3 2 405.4.e.t 6
9.d odd 6 2 405.4.e.r 6
12.b even 2 1 2160.4.a.be 3
15.d odd 2 1 675.4.a.q 3
15.e even 4 2 675.4.b.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 1.a even 1 1 trivial
135.4.a.g yes 3 3.b odd 2 1
405.4.e.r 6 9.d odd 6 2
405.4.e.t 6 9.c even 3 2
675.4.a.q 3 15.d odd 2 1
675.4.a.r 3 5.b even 2 1
675.4.b.k 6 15.e even 4 2
675.4.b.l 6 5.c odd 4 2
2160.4.a.be 3 12.b even 2 1
2160.4.a.bm 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 23 T_{2} - 6$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(135))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 - 23 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$( -5 + T )^{3}$$
$7$ $$-1800 + 552 T - 44 T^{2} + T^{3}$$
$11$ $$83280 - 2612 T - 38 T^{2} + T^{3}$$
$13$ $$100120 - 4576 T - 28 T^{2} + T^{3}$$
$17$ $$-553887 - 11477 T + 19 T^{2} + T^{3}$$
$19$ $$525871 + 3587 T - 187 T^{2} + T^{3}$$
$23$ $$-2043981 - 23301 T + 81 T^{2} + T^{3}$$
$29$ $$7892760 - 47768 T - 160 T^{2} + T^{3}$$
$31$ $$-246321 - 17973 T - 227 T^{2} + T^{3}$$
$37$ $$13637080 - 99924 T - 78 T^{2} + T^{3}$$
$41$ $$-12116640 - 42812 T + 338 T^{2} + T^{3}$$
$43$ $$18464560 - 159916 T - 22 T^{2} + T^{3}$$
$47$ $$-283200 + 54208 T + 472 T^{2} + T^{3}$$
$53$ $$939789 + 61387 T - 521 T^{2} + T^{3}$$
$59$ $$34131480 - 166448 T - 140 T^{2} + T^{3}$$
$61$ $$1782607 - 2749 T - 595 T^{2} + T^{3}$$
$67$ $$11295000 + 75948 T - 878 T^{2} + T^{3}$$
$71$ $$-280550880 - 583652 T + 602 T^{2} + T^{3}$$
$73$ $$404091280 - 95908 T - 1294 T^{2} + T^{3}$$
$79$ $$-2010303 + 97059 T - 629 T^{2} + T^{3}$$
$83$ $$-346404411 - 365877 T + 1287 T^{2} + T^{3}$$
$89$ $$-74325600 + 1057572 T - 2154 T^{2} + T^{3}$$
$97$ $$-63595520 + 543936 T - 1392 T^{2} + T^{3}$$