# Properties

 Label 136.4.a.b Level $136$ Weight $4$ Character orbit 136.a Self dual yes Analytic conductor $8.024$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.02425976078$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.8396.1 Defining polynomial: $$x^{3} - x^{2} - 20 x + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{3} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{5} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{7} + ( -4 - \beta_{1} - 6 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{3} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{5} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{7} + ( -4 - \beta_{1} - 6 \beta_{2} ) q^{9} + ( -27 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( -15 + 3 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -46 + 4 \beta_{1} + 6 \beta_{2} ) q^{15} -17 q^{17} + ( -72 + 2 \beta_{1} + 6 \beta_{2} ) q^{19} + ( -5 - 5 \beta_{1} + 6 \beta_{2} ) q^{21} + ( -78 - \beta_{1} + \beta_{2} ) q^{23} + ( 49 - 22 \beta_{1} + 8 \beta_{2} ) q^{25} + ( -96 + 4 \beta_{1} ) q^{27} + ( -13 + 15 \beta_{1} + 18 \beta_{2} ) q^{29} + ( -8 - 25 \beta_{1} + 3 \beta_{2} ) q^{31} + ( 39 + 3 \beta_{1} - 34 \beta_{2} ) q^{33} + ( -146 + 20 \beta_{1} - 30 \beta_{2} ) q^{35} + ( 55 + 27 \beta_{1} + 38 \beta_{2} ) q^{37} + ( 44 + 4 \beta_{1} - 28 \beta_{2} ) q^{39} + ( 64 + 14 \beta_{1} - 40 \beta_{2} ) q^{41} + ( -20 - 26 \beta_{1} - 78 \beta_{2} ) q^{43} + ( 239 - 25 \beta_{1} - 26 \beta_{2} ) q^{45} + ( 22 + 30 \beta_{1} + 52 \beta_{2} ) q^{47} + ( 166 + 29 \beta_{1} + 66 \beta_{2} ) q^{49} + ( 17 - 17 \beta_{2} ) q^{51} + ( 142 - 48 \beta_{1} + 12 \beta_{2} ) q^{53} + ( 166 - 40 \beta_{1} + 78 \beta_{2} ) q^{55} + ( 194 - 2 \beta_{1} - 104 \beta_{2} ) q^{57} + ( -8 - 42 \beta_{1} + 86 \beta_{2} ) q^{59} + ( 183 - 25 \beta_{1} + 114 \beta_{2} ) q^{61} + ( 216 + 65 \beta_{1} - 3 \beta_{2} ) q^{63} + ( 148 - 32 \beta_{1} + 68 \beta_{2} ) q^{65} + ( -282 + 102 \beta_{1} + 60 \beta_{2} ) q^{67} + ( 105 - 3 \beta_{1} - 82 \beta_{2} ) q^{69} + ( 136 - 51 \beta_{1} - 59 \beta_{2} ) q^{71} + ( 266 + 56 \beta_{1} - 96 \beta_{2} ) q^{73} + ( 237 - 52 \beta_{1} + 31 \beta_{2} ) q^{75} + ( -285 + 63 \beta_{1} - 14 \beta_{2} ) q^{77} + ( 406 + 23 \beta_{1} - 43 \beta_{2} ) q^{79} + ( 184 + 35 \beta_{1} + 62 \beta_{2} ) q^{81} + ( -424 - 22 \beta_{1} - 102 \beta_{2} ) q^{83} + ( 51 - 17 \beta_{1} + 34 \beta_{2} ) q^{85} + ( 334 + 12 \beta_{1} - 118 \beta_{2} ) q^{87} + ( -827 + 43 \beta_{1} - 18 \beta_{2} ) q^{89} + ( -482 + 14 \beta_{1} - 44 \beta_{2} ) q^{91} + ( 199 - 53 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 56 - 60 \beta_{1} + 188 \beta_{2} ) q^{95} + ( 618 - 152 \beta_{1} - 92 \beta_{2} ) q^{97} + ( -73 - 14 \beta_{1} + 179 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{3} - 8 q^{5} - 2 q^{7} - 5 q^{9} + O(q^{10})$$ $$3 q - 4 q^{3} - 8 q^{5} - 2 q^{7} - 5 q^{9} - 84 q^{11} - 50 q^{13} - 148 q^{15} - 51 q^{17} - 224 q^{19} - 16 q^{21} - 234 q^{23} + 161 q^{25} - 292 q^{27} - 72 q^{29} - 2 q^{31} + 148 q^{33} - 428 q^{35} + 100 q^{37} + 156 q^{39} + 218 q^{41} + 44 q^{43} + 768 q^{45} - 16 q^{47} + 403 q^{49} + 68 q^{51} + 462 q^{53} + 460 q^{55} + 688 q^{57} - 68 q^{59} + 460 q^{61} + 586 q^{63} + 408 q^{65} - 1008 q^{67} + 400 q^{69} + 518 q^{71} + 838 q^{73} + 732 q^{75} - 904 q^{77} + 1238 q^{79} + 455 q^{81} - 1148 q^{83} + 136 q^{85} + 1108 q^{87} - 2506 q^{89} - 1416 q^{91} + 648 q^{93} + 40 q^{95} + 2098 q^{97} - 384 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 20 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 14$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{2} + \beta_{1} + 29$$$$)/2$$

## 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
 0.395276 4.81129 −4.20657
0 −8.11952 0 11.0296 0 5.74786 0 38.9265 0
1.2 0 1.16862 0 1.28534 0 −30.0364 0 −25.6343 0
1.3 0 2.95089 0 −20.3149 0 22.2885 0 −18.2922 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$1$$

## 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 136.4.a.b 3
3.b odd 2 1 1224.4.a.i 3
4.b odd 2 1 272.4.a.j 3
8.b even 2 1 1088.4.a.y 3
8.d odd 2 1 1088.4.a.u 3
12.b even 2 1 2448.4.a.bj 3
17.b even 2 1 2312.4.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.b 3 1.a even 1 1 trivial
272.4.a.j 3 4.b odd 2 1
1088.4.a.u 3 8.d odd 2 1
1088.4.a.y 3 8.b even 2 1
1224.4.a.i 3 3.b odd 2 1
2312.4.a.d 3 17.b even 2 1
2448.4.a.bj 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 4 T_{3}^{2} - 30 T_{3} + 28$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(136))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$28 - 30 T + 4 T^{2} + T^{3}$$
$5$ $$288 - 236 T + 8 T^{2} + T^{3}$$
$7$ $$3848 - 714 T + 2 T^{2} + T^{3}$$
$11$ $$10972 + 2026 T + 84 T^{2} + T^{3}$$
$13$ $$-16048 + 64 T + 50 T^{2} + T^{3}$$
$17$ $$( 17 + T )^{3}$$
$19$ $$322592 + 15336 T + 224 T^{2} + T^{3}$$
$23$ $$463504 + 18118 T + 234 T^{2} + T^{3}$$
$29$ $$-1862624 - 23340 T + 72 T^{2} + T^{3}$$
$31$ $$-1252344 - 52450 T + 2 T^{2} + T^{3}$$
$37$ $$-4014352 - 89196 T - 100 T^{2} + T^{3}$$
$41$ $$7651496 - 66340 T - 218 T^{2} + T^{3}$$
$43$ $$-18647936 - 234152 T - 44 T^{2} + T^{3}$$
$47$ $$-7663872 - 141616 T + 16 T^{2} + T^{3}$$
$53$ $$10502936 - 131316 T - 462 T^{2} + T^{3}$$
$59$ $$-81654208 - 465864 T + 68 T^{2} + T^{3}$$
$61$ $$116249648 - 488892 T - 460 T^{2} + T^{3}$$
$67$ $$-534256128 - 528624 T + 1008 T^{2} + T^{3}$$
$71$ $$93696432 - 192946 T - 518 T^{2} + T^{3}$$
$73$ $$324704504 - 439796 T - 838 T^{2} + T^{3}$$
$79$ $$-7088608 + 385382 T - 1238 T^{2} + T^{3}$$
$83$ $$-158784832 + 71224 T + 1148 T^{2} + T^{3}$$
$89$ $$456757648 + 1918096 T + 2506 T^{2} + T^{3}$$
$97$ $$1961875944 - 468596 T - 2098 T^{2} + T^{3}$$