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

Downloads

Learn more

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:

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

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