# Properties

 Label 136.2.a.c Level $136$ Weight $2$ Character orbit 136.a Self dual yes Analytic conductor $1.086$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.08596546749$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + 2 q^{5} + ( 1 + \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + 2 q^{5} + ( 1 + \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} + ( 1 + \beta ) q^{11} -2 \beta q^{13} + ( -2 - 2 \beta ) q^{15} + q^{17} + ( -2 + 2 \beta ) q^{19} + ( -6 - 2 \beta ) q^{21} + ( 1 + \beta ) q^{23} - q^{25} + ( -10 - 2 \beta ) q^{27} + 2 q^{29} + ( -1 - \beta ) q^{31} + ( -6 - 2 \beta ) q^{33} + ( 2 + 2 \beta ) q^{35} + ( -2 + 4 \beta ) q^{37} + ( 10 + 2 \beta ) q^{39} + 2 q^{41} + ( -6 - 2 \beta ) q^{43} + ( 6 + 4 \beta ) q^{45} + ( 4 - 4 \beta ) q^{47} + ( -1 + 2 \beta ) q^{49} + ( -1 - \beta ) q^{51} -2 q^{53} + ( 2 + 2 \beta ) q^{55} -8 q^{57} + ( 10 - 2 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} + ( 13 + 5 \beta ) q^{63} -4 \beta q^{65} -12 q^{67} + ( -6 - 2 \beta ) q^{69} + ( 7 - \beta ) q^{71} + ( 6 - 4 \beta ) q^{73} + ( 1 + \beta ) q^{75} + ( 6 + 2 \beta ) q^{77} + ( 5 - 3 \beta ) q^{79} + ( 11 + 6 \beta ) q^{81} + ( 6 + 2 \beta ) q^{83} + 2 q^{85} + ( -2 - 2 \beta ) q^{87} + ( -12 - 2 \beta ) q^{89} + ( -10 - 2 \beta ) q^{91} + ( 6 + 2 \beta ) q^{93} + ( -4 + 4 \beta ) q^{95} + 2 q^{97} + ( 13 + 5 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 6 q^{9} + 2 q^{11} - 4 q^{15} + 2 q^{17} - 4 q^{19} - 12 q^{21} + 2 q^{23} - 2 q^{25} - 20 q^{27} + 4 q^{29} - 2 q^{31} - 12 q^{33} + 4 q^{35} - 4 q^{37} + 20 q^{39} + 4 q^{41} - 12 q^{43} + 12 q^{45} + 8 q^{47} - 2 q^{49} - 2 q^{51} - 4 q^{53} + 4 q^{55} - 16 q^{57} + 20 q^{59} - 4 q^{61} + 26 q^{63} - 24 q^{67} - 12 q^{69} + 14 q^{71} + 12 q^{73} + 2 q^{75} + 12 q^{77} + 10 q^{79} + 22 q^{81} + 12 q^{83} + 4 q^{85} - 4 q^{87} - 24 q^{89} - 20 q^{91} + 12 q^{93} - 8 q^{95} + 4 q^{97} + 26 q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −3.23607 0 2.00000 0 3.23607 0 7.47214 0
1.2 0 1.23607 0 2.00000 0 −1.23607 0 −1.47214 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 136.2.a.c 2
3.b odd 2 1 1224.2.a.i 2
4.b odd 2 1 272.2.a.f 2
5.b even 2 1 3400.2.a.i 2
5.c odd 4 2 3400.2.e.f 4
7.b odd 2 1 6664.2.a.i 2
8.b even 2 1 1088.2.a.s 2
8.d odd 2 1 1088.2.a.o 2
12.b even 2 1 2448.2.a.u 2
17.b even 2 1 2312.2.a.m 2
17.c even 4 2 2312.2.b.g 4
20.d odd 2 1 6800.2.a.bd 2
24.f even 2 1 9792.2.a.da 2
24.h odd 2 1 9792.2.a.db 2
68.d odd 2 1 4624.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.c 2 1.a even 1 1 trivial
272.2.a.f 2 4.b odd 2 1
1088.2.a.o 2 8.d odd 2 1
1088.2.a.s 2 8.b even 2 1
1224.2.a.i 2 3.b odd 2 1
2312.2.a.m 2 17.b even 2 1
2312.2.b.g 4 17.c even 4 2
2448.2.a.u 2 12.b even 2 1
3400.2.a.i 2 5.b even 2 1
3400.2.e.f 4 5.c odd 4 2
4624.2.a.h 2 68.d odd 2 1
6664.2.a.i 2 7.b odd 2 1
6800.2.a.bd 2 20.d odd 2 1
9792.2.a.da 2 24.f even 2 1
9792.2.a.db 2 24.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 + 2 T + T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$-4 - 2 T + T^{2}$$
$11$ $$-4 - 2 T + T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$-16 + 4 T + T^{2}$$
$23$ $$-4 - 2 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-4 + 2 T + T^{2}$$
$37$ $$-76 + 4 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + 12 T + T^{2}$$
$47$ $$-64 - 8 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$80 - 20 T + T^{2}$$
$61$ $$-76 + 4 T + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$44 - 14 T + T^{2}$$
$73$ $$-44 - 12 T + T^{2}$$
$79$ $$-20 - 10 T + T^{2}$$
$83$ $$16 - 12 T + T^{2}$$
$89$ $$124 + 24 T + T^{2}$$
$97$ $$( -2 + T )^{2}$$