# Properties

 Label 135.4.a.g Level $135$ Weight $4$ Character orbit 135.a Self dual yes Analytic conductor $7.965$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: $$x^{3} - x^{2} - 23x + 6$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 7) q^{4} - 5 q^{5} + (2 \beta_1 + 14) q^{7} + (\beta_{2} + 8 \beta_1 + 9) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1 + 7) * q^4 - 5 * q^5 + (2*b1 + 14) * q^7 + (b2 + 8*b1 + 9) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 7) q^{4} - 5 q^{5} + (2 \beta_1 + 14) q^{7} + (\beta_{2} + 8 \beta_1 + 9) q^{8} - 5 \beta_1 q^{10} + ( - 4 \beta_{2} + 2 \beta_1 - 12) q^{11} + ( - 4 \beta_{2} - 10 \beta_1 + 14) q^{13} + (2 \beta_{2} + 16 \beta_1 + 30) q^{14} + (17 \beta_1 + 58) q^{16} + (8 \beta_{2} + 2 \beta_1 + 3) q^{17} + ( - 4 \beta_{2} + 14 \beta_1 + 59) q^{19} + ( - 5 \beta_{2} - 5 \beta_1 - 35) q^{20} + (2 \beta_{2} - 42 \beta_1 + 54) q^{22} + ( - 4 \beta_{2} - 32 \beta_1 + 39) q^{23} + 25 q^{25} + ( - 10 \beta_{2} - 28 \beta_1 - 126) q^{26} + (16 \beta_{2} + 46 \beta_1 + 116) q^{28} + (8 \beta_{2} - 42 \beta_1 - 42) q^{29} + (8 \beta_{2} - 30 \beta_1 + 83) q^{31} + (9 \beta_{2} + 11 \beta_1 + 183) q^{32} + (2 \beta_{2} + 69 \beta_1 - 18) q^{34} + ( - 10 \beta_1 - 70) q^{35} + ( - 8 \beta_{2} - 64 \beta_1 + 50) q^{37} + (14 \beta_{2} + 41 \beta_1 + 234) q^{38} + ( - 5 \beta_{2} - 40 \beta_1 - 45) q^{40} + ( - 16 \beta_{2} - 42 \beta_1 + 132) q^{41} + ( - 8 \beta_{2} + 78 \beta_1 - 16) q^{43} + ( - 10 \beta_{2} + 12 \beta_1 - 546) q^{44} + ( - 32 \beta_{2} - 25 \beta_1 - 456) q^{46} + ( - 4 \beta_{2} - 28 \beta_1 + 168) q^{47} + (4 \beta_{2} + 60 \beta_1 - 87) q^{49} + 25 \beta_1 q^{50} + (4 \beta_{2} - 154 \beta_1 - 472) q^{52} + ( - 12 \beta_{2} - 14 \beta_1 - 165) q^{53} + (20 \beta_{2} - 10 \beta_1 + 60) q^{55} + (30 \beta_{2} + 162 \beta_1 + 354) q^{56} + ( - 42 \beta_{2} - 20 \beta_1 - 678) q^{58} + (12 \beta_{2} + 82 \beta_1 - 78) q^{59} + (16 \beta_{2} + 60 \beta_1 + 173) q^{61} + ( - 30 \beta_{2} + 117 \beta_1 - 498) q^{62} + (11 \beta_{2} + 130 \beta_1 - 353) q^{64} + (20 \beta_{2} + 50 \beta_1 - 70) q^{65} + (24 \beta_{2} - 52 \beta_1 + 302) q^{67} + (5 \beta_{2} + 51 \beta_1 + 999) q^{68} + ( - 10 \beta_{2} - 80 \beta_1 - 150) q^{70} + (60 \beta_{2} - 34 \beta_1 + 192) q^{71} + (60 \beta_{2} + 22 \beta_1 + 404) q^{73} + ( - 64 \beta_{2} - 78 \beta_1 - 912) q^{74} + (73 \beta_{2} + 275 \beta_1 + 59) q^{76} + ( - 52 \beta_{2} - 56 \beta_1 - 60) q^{77} + ( - 12 \beta_{2} - 22 \beta_1 + 221) q^{79} + ( - 85 \beta_1 - 290) q^{80} + ( - 42 \beta_{2} - 38 \beta_1 - 534) q^{82} + ( - 60 \beta_{2} - 120 \beta_1 + 489) q^{83} + ( - 40 \beta_{2} - 10 \beta_1 - 15) q^{85} + (78 \beta_{2} - 2 \beta_1 + 1218) q^{86} + ( - 4 \beta_{2} - 278 \beta_1 - 192) q^{88} + (48 \beta_{2} + 66 \beta_1 - 756) q^{89} + ( - 76 \beta_{2} - 196 \beta_1 - 56) q^{91} + (7 \beta_{2} - 481 \beta_1 - 495) q^{92} + ( - 28 \beta_{2} + 108 \beta_1 - 396) q^{94} + (20 \beta_{2} - 70 \beta_1 - 295) q^{95} + (8 \beta_{2} + 64 \beta_1 + 440) q^{97} + (60 \beta_{2} + 5 \beta_1 + 876) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1 + 7) * q^4 - 5 * q^5 + (2*b1 + 14) * q^7 + (b2 + 8*b1 + 9) * q^8 - 5*b1 * q^10 + (-4*b2 + 2*b1 - 12) * q^11 + (-4*b2 - 10*b1 + 14) * q^13 + (2*b2 + 16*b1 + 30) * q^14 + (17*b1 + 58) * q^16 + (8*b2 + 2*b1 + 3) * q^17 + (-4*b2 + 14*b1 + 59) * q^19 + (-5*b2 - 5*b1 - 35) * q^20 + (2*b2 - 42*b1 + 54) * q^22 + (-4*b2 - 32*b1 + 39) * q^23 + 25 * q^25 + (-10*b2 - 28*b1 - 126) * q^26 + (16*b2 + 46*b1 + 116) * q^28 + (8*b2 - 42*b1 - 42) * q^29 + (8*b2 - 30*b1 + 83) * q^31 + (9*b2 + 11*b1 + 183) * q^32 + (2*b2 + 69*b1 - 18) * q^34 + (-10*b1 - 70) * q^35 + (-8*b2 - 64*b1 + 50) * q^37 + (14*b2 + 41*b1 + 234) * q^38 + (-5*b2 - 40*b1 - 45) * q^40 + (-16*b2 - 42*b1 + 132) * q^41 + (-8*b2 + 78*b1 - 16) * q^43 + (-10*b2 + 12*b1 - 546) * q^44 + (-32*b2 - 25*b1 - 456) * q^46 + (-4*b2 - 28*b1 + 168) * q^47 + (4*b2 + 60*b1 - 87) * q^49 + 25*b1 * q^50 + (4*b2 - 154*b1 - 472) * q^52 + (-12*b2 - 14*b1 - 165) * q^53 + (20*b2 - 10*b1 + 60) * q^55 + (30*b2 + 162*b1 + 354) * q^56 + (-42*b2 - 20*b1 - 678) * q^58 + (12*b2 + 82*b1 - 78) * q^59 + (16*b2 + 60*b1 + 173) * q^61 + (-30*b2 + 117*b1 - 498) * q^62 + (11*b2 + 130*b1 - 353) * q^64 + (20*b2 + 50*b1 - 70) * q^65 + (24*b2 - 52*b1 + 302) * q^67 + (5*b2 + 51*b1 + 999) * q^68 + (-10*b2 - 80*b1 - 150) * q^70 + (60*b2 - 34*b1 + 192) * q^71 + (60*b2 + 22*b1 + 404) * q^73 + (-64*b2 - 78*b1 - 912) * q^74 + (73*b2 + 275*b1 + 59) * q^76 + (-52*b2 - 56*b1 - 60) * q^77 + (-12*b2 - 22*b1 + 221) * q^79 + (-85*b1 - 290) * q^80 + (-42*b2 - 38*b1 - 534) * q^82 + (-60*b2 - 120*b1 + 489) * q^83 + (-40*b2 - 10*b1 - 15) * q^85 + (78*b2 - 2*b1 + 1218) * q^86 + (-4*b2 - 278*b1 - 192) * q^88 + (48*b2 + 66*b1 - 756) * q^89 + (-76*b2 - 196*b1 - 56) * q^91 + (7*b2 - 481*b1 - 495) * q^92 + (-28*b2 + 108*b1 - 396) * q^94 + (20*b2 - 70*b1 - 295) * q^95 + (8*b2 + 64*b1 + 440) * q^97 + (60*b2 + 5*b1 + 876) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 23 q^{4} - 15 q^{5} + 44 q^{7} + 36 q^{8}+O(q^{10})$$ 3 * q + q^2 + 23 * q^4 - 15 * q^5 + 44 * q^7 + 36 * q^8 $$3 q + q^{2} + 23 q^{4} - 15 q^{5} + 44 q^{7} + 36 q^{8} - 5 q^{10} - 38 q^{11} + 28 q^{13} + 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 115 q^{20} + 122 q^{22} + 81 q^{23} + 75 q^{25} - 416 q^{26} + 410 q^{28} - 160 q^{29} + 227 q^{31} + 569 q^{32} + 17 q^{34} - 220 q^{35} + 78 q^{37} + 757 q^{38} - 180 q^{40} + 338 q^{41} + 22 q^{43} - 1636 q^{44} - 1425 q^{46} + 472 q^{47} - 197 q^{49} + 25 q^{50} - 1566 q^{52} - 521 q^{53} + 190 q^{55} + 1254 q^{56} - 2096 q^{58} - 140 q^{59} + 595 q^{61} - 1407 q^{62} - 918 q^{64} - 140 q^{65} + 878 q^{67} + 3053 q^{68} - 540 q^{70} + 602 q^{71} + 1294 q^{73} - 2878 q^{74} + 525 q^{76} - 288 q^{77} + 629 q^{79} - 955 q^{80} - 1682 q^{82} + 1287 q^{83} - 95 q^{85} + 3730 q^{86} - 858 q^{88} - 2154 q^{89} - 440 q^{91} - 1959 q^{92} - 1108 q^{94} - 935 q^{95} + 1392 q^{97} + 2693 q^{98}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 15$$ v^2 - v - 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 15$$ b2 + b1 + 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
 −4.45938 0.258712 5.20067
−4.45938 0 11.8861 −5.00000 0 5.08123 −17.3296 0 22.2969
1.2 0.258712 0 −7.93307 −5.00000 0 14.5174 −4.12208 0 −1.29356
1.3 5.20067 0 19.0470 −5.00000 0 24.4013 57.4517 0 −26.0034
 $$n$$: 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.g yes 3
3.b odd 2 1 135.4.a.f 3
4.b odd 2 1 2160.4.a.be 3
5.b even 2 1 675.4.a.q 3
5.c odd 4 2 675.4.b.k 6
9.c even 3 2 405.4.e.r 6
9.d odd 6 2 405.4.e.t 6
12.b even 2 1 2160.4.a.bm 3
15.d odd 2 1 675.4.a.r 3
15.e even 4 2 675.4.b.l 6

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

## Hecke kernels

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

## Hecke characteristic polynomials

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