Properties

Label 136.2.k.b
Level $136$
Weight $2$
Character orbit 136.k
Analytic conductor $1.086$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{3} + ( 2 + 2 i ) q^{5} + ( 2 - 2 i ) q^{7} -i q^{9} +O(q^{10})\) \( q + ( -1 - i ) q^{3} + ( 2 + 2 i ) q^{5} + ( 2 - 2 i ) q^{7} -i q^{9} + ( 1 - i ) q^{11} + 2 q^{13} -4 i q^{15} + ( -4 + i ) q^{17} + 4 i q^{19} -4 q^{21} + ( -4 + 4 i ) q^{23} + 3 i q^{25} + ( -4 + 4 i ) q^{27} + ( 6 + 6 i ) q^{29} + ( -6 - 6 i ) q^{31} -2 q^{33} + 8 q^{35} + ( -8 - 8 i ) q^{37} + ( -2 - 2 i ) q^{39} + ( 1 - i ) q^{41} + 2 i q^{43} + ( 2 - 2 i ) q^{45} -i q^{49} + ( 5 + 3 i ) q^{51} -6 i q^{53} + 4 q^{55} + ( 4 - 4 i ) q^{57} + 14 i q^{59} + ( -4 + 4 i ) q^{61} + ( -2 - 2 i ) q^{63} + ( 4 + 4 i ) q^{65} + 6 q^{67} + 8 q^{69} + ( -9 - 9 i ) q^{73} + ( 3 - 3 i ) q^{75} -4 i q^{77} + ( 4 - 4 i ) q^{79} + 5 q^{81} + 6 i q^{83} + ( -10 - 6 i ) q^{85} -12 i q^{87} + 16 q^{89} + ( 4 - 4 i ) q^{91} + 12 i q^{93} + ( -8 + 8 i ) q^{95} + ( 3 + 3 i ) q^{97} + ( -1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} - 8 q^{17} - 8 q^{21} - 8 q^{23} - 8 q^{27} + 12 q^{29} - 12 q^{31} - 4 q^{33} + 16 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} + 4 q^{45} + 10 q^{51} + 8 q^{55} + 8 q^{57} - 8 q^{61} - 4 q^{63} + 8 q^{65} + 12 q^{67} + 16 q^{69} - 18 q^{73} + 6 q^{75} + 8 q^{79} + 10 q^{81} - 20 q^{85} + 32 q^{89} + 8 q^{91} - 16 q^{95} + 6 q^{97} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
1.00000i
1.00000i
0 −1.00000 1.00000i 0 2.00000 + 2.00000i 0 2.00000 2.00000i 0 1.00000i 0
89.1 0 −1.00000 + 1.00000i 0 2.00000 2.00000i 0 2.00000 + 2.00000i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.k.b 2
3.b odd 2 1 1224.2.w.a 2
4.b odd 2 1 272.2.o.e 2
8.b even 2 1 1088.2.o.m 2
8.d odd 2 1 1088.2.o.c 2
12.b even 2 1 2448.2.be.a 2
17.c even 4 1 inner 136.2.k.b 2
17.d even 8 2 2312.2.a.j 2
17.d even 8 2 2312.2.b.d 2
51.f odd 4 1 1224.2.w.a 2
68.f odd 4 1 272.2.o.e 2
68.g odd 8 2 4624.2.a.o 2
136.i even 4 1 1088.2.o.m 2
136.j odd 4 1 1088.2.o.c 2
204.l even 4 1 2448.2.be.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.k.b 2 1.a even 1 1 trivial
136.2.k.b 2 17.c even 4 1 inner
272.2.o.e 2 4.b odd 2 1
272.2.o.e 2 68.f odd 4 1
1088.2.o.c 2 8.d odd 2 1
1088.2.o.c 2 136.j odd 4 1
1088.2.o.m 2 8.b even 2 1
1088.2.o.m 2 136.i even 4 1
1224.2.w.a 2 3.b odd 2 1
1224.2.w.a 2 51.f odd 4 1
2312.2.a.j 2 17.d even 8 2
2312.2.b.d 2 17.d even 8 2
2448.2.be.a 2 12.b even 2 1
2448.2.be.a 2 204.l even 4 1
4624.2.a.o 2 68.g odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\):

\( T_{3}^{2} + 2 T_{3} + 2 \)
\( T_{5}^{2} - 4 T_{5} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + 2 T + T^{2} \)
$5$ \( 8 - 4 T + T^{2} \)
$7$ \( 8 - 4 T + T^{2} \)
$11$ \( 2 - 2 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 17 + 8 T + T^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( 32 + 8 T + T^{2} \)
$29$ \( 72 - 12 T + T^{2} \)
$31$ \( 72 + 12 T + T^{2} \)
$37$ \( 128 + 16 T + T^{2} \)
$41$ \( 2 - 2 T + T^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 196 + T^{2} \)
$61$ \( 32 + 8 T + T^{2} \)
$67$ \( ( -6 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 162 + 18 T + T^{2} \)
$79$ \( 32 - 8 T + T^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( -16 + T )^{2} \)
$97$ \( 18 - 6 T + T^{2} \)
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