# Properties

 Label 136.4.a.c Level $136$ Weight $4$ Character orbit 136.a Self dual yes Analytic conductor $8.024$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1556.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - \beta_{2} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 4 - \beta_{1} + \beta_{2} ) q^{7} + ( 22 + 3 \beta_{1} - 6 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 3 - \beta_{2} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 4 - \beta_{1} + \beta_{2} ) q^{7} + ( 22 + 3 \beta_{1} - 6 \beta_{2} ) q^{9} + ( 23 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( 17 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( 20 - 2 \beta_{1} + 6 \beta_{2} ) q^{15} + 17 q^{17} + ( 38 + 10 \beta_{2} ) q^{19} + ( -11 - 5 \beta_{1} + 6 \beta_{2} ) q^{21} + ( 18 - 11 \beta_{1} + 17 \beta_{2} ) q^{23} + ( -25 - 8 \beta_{1} + 8 \beta_{2} ) q^{25} + ( 174 + 24 \beta_{1} - 34 \beta_{2} ) q^{27} + ( -25 + 5 \beta_{1} + 16 \beta_{2} ) q^{29} + ( 104 + 5 \beta_{1} - 17 \beta_{2} ) q^{31} + ( -5 + \beta_{1} - 34 \beta_{2} ) q^{33} + ( 86 - 12 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -185 - 15 \beta_{1} + 28 \beta_{2} ) q^{37} + ( 114 + 8 \beta_{1} - 30 \beta_{2} ) q^{39} + ( 8 + 30 \beta_{1} - 24 \beta_{2} ) q^{41} + ( 14 - 12 \beta_{1} + 22 \beta_{2} ) q^{43} + ( -173 + 5 \beta_{1} + 12 \beta_{2} ) q^{45} + ( 16 - 20 \beta_{1} - 12 \beta_{2} ) q^{47} + ( -222 - 13 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 51 - 17 \beta_{2} ) q^{51} + ( -196 + 30 \beta_{1} + 40 \beta_{2} ) q^{53} + ( -192 - 10 \beta_{1} - 22 \beta_{2} ) q^{55} + ( -286 - 30 \beta_{1} - 8 \beta_{2} ) q^{57} + ( 26 + 54 \beta_{2} ) q^{59} + ( -197 + 45 \beta_{1} - 68 \beta_{2} ) q^{61} + ( -296 - \beta_{1} + 37 \beta_{2} ) q^{63} + ( -48 - 8 \beta_{1} + 4 \beta_{2} ) q^{65} + ( -112 + 12 \beta_{1} - 8 \beta_{2} ) q^{67} + ( -439 - 73 \beta_{1} + 110 \beta_{2} ) q^{69} + ( -102 + 63 \beta_{1} - 25 \beta_{2} ) q^{71} + ( -206 - 52 \beta_{1} + 92 \beta_{2} ) q^{73} + ( -259 - 40 \beta_{1} + 105 \beta_{2} ) q^{75} + ( -49 + \beta_{1} + 18 \beta_{2} ) q^{77} + ( -278 + 81 \beta_{1} + 85 \beta_{2} ) q^{79} + ( 880 + 69 \beta_{1} - 282 \beta_{2} ) q^{81} + ( 322 - 20 \beta_{1} - 70 \beta_{2} ) q^{83} + ( 17 - 17 \beta_{1} ) q^{85} + ( -800 - 38 \beta_{1} + 38 \beta_{2} ) q^{87} + ( 601 + 37 \beta_{1} - 50 \beta_{2} ) q^{89} + ( -60 - 10 \beta_{1} + 22 \beta_{2} ) q^{91} + ( 907 + 61 \beta_{1} - 190 \beta_{2} ) q^{93} + ( -132 - 48 \beta_{1} - 60 \beta_{2} ) q^{95} + ( 258 + 40 \beta_{1} + 24 \beta_{2} ) q^{97} + ( 707 + 50 \beta_{1} - 131 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + O(q^{10})$$ $$3 q + 8 q^{3} + 2 q^{5} + 12 q^{7} + 63 q^{9} + 72 q^{11} + 50 q^{13} + 64 q^{15} + 51 q^{17} + 124 q^{19} - 32 q^{21} + 60 q^{23} - 75 q^{25} + 512 q^{27} - 54 q^{29} + 300 q^{31} - 48 q^{33} + 248 q^{35} - 542 q^{37} + 320 q^{39} + 30 q^{41} + 52 q^{43} - 502 q^{45} + 16 q^{47} - 677 q^{49} + 136 q^{51} - 518 q^{53} - 608 q^{55} - 896 q^{57} + 132 q^{59} - 614 q^{61} - 852 q^{63} - 148 q^{65} - 332 q^{67} - 1280 q^{69} - 268 q^{71} - 578 q^{73} - 712 q^{75} - 128 q^{77} - 668 q^{79} + 2427 q^{81} + 876 q^{83} + 34 q^{85} - 2400 q^{87} + 1790 q^{89} - 168 q^{91} + 2592 q^{93} - 504 q^{95} + 838 q^{97} + 2040 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} + 2 \nu - 13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} - \beta_{1} + 25$$$$)/4$$

## 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
 2.80979 −3.08060 1.27082
0 −5.40937 0 −9.23914 0 2.17022 0 2.26124 0
1.2 0 3.18096 0 14.3224 0 17.1415 0 −16.8815 0
1.3 0 10.2284 0 −3.08327 0 −7.31168 0 77.6202 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.c 3
3.b odd 2 1 1224.4.a.f 3
4.b odd 2 1 272.4.a.g 3
8.b even 2 1 1088.4.a.s 3
8.d odd 2 1 1088.4.a.z 3
12.b even 2 1 2448.4.a.bf 3
17.b even 2 1 2312.4.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.c 3 1.a even 1 1 trivial
272.4.a.g 3 4.b odd 2 1
1088.4.a.s 3 8.b even 2 1
1088.4.a.z 3 8.d odd 2 1
1224.4.a.f 3 3.b odd 2 1
2312.4.a.c 3 17.b even 2 1
2448.4.a.bf 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 8 T_{3}^{2} - 40 T_{3} + 176$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(136))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$176 - 40 T - 8 T^{2} + T^{3}$$
$5$ $$-408 - 148 T - 2 T^{2} + T^{3}$$
$7$ $$272 - 104 T - 12 T^{2} + T^{3}$$
$11$ $$4752 + 952 T - 72 T^{2} + T^{3}$$
$13$ $$-1496 + 556 T - 50 T^{2} + T^{3}$$
$17$ $$( -17 + T )^{3}$$
$19$ $$151488 - 1008 T - 124 T^{2} + T^{3}$$
$23$ $$1168976 - 23624 T - 60 T^{2} + T^{3}$$
$29$ $$-1826616 - 23156 T + 54 T^{2} + T^{3}$$
$31$ $$-122448 + 13528 T - 300 T^{2} + T^{3}$$
$37$ $$-451992 + 40876 T + 542 T^{2} + T^{3}$$
$41$ $$13345816 - 127188 T - 30 T^{2} + T^{3}$$
$43$ $$2509632 - 34800 T - 52 T^{2} + T^{3}$$
$47$ $$1737728 - 82560 T - 16 T^{2} + T^{3}$$
$53$ $$-97851384 - 213492 T + 518 T^{2} + T^{3}$$
$59$ $$2838592 - 173040 T - 132 T^{2} + T^{3}$$
$61$ $$-115549176 - 280820 T + 614 T^{2} + T^{3}$$
$67$ $$78912 + 16944 T + 332 T^{2} + T^{3}$$
$71$ $$104748848 - 514696 T + 268 T^{2} + T^{3}$$
$73$ $$17940632 - 530900 T + 578 T^{2} + T^{3}$$
$79$ $$-976074672 - 1678088 T + 668 T^{2} + T^{3}$$
$83$ $$211499712 - 186608 T - 876 T^{2} + T^{3}$$
$89$ $$-72949736 + 818796 T - 1790 T^{2} + T^{3}$$
$97$ $$60169976 - 96500 T - 838 T^{2} + T^{3}$$