Properties

Label 136.2.b.a
Level $136$
Weight $2$
Character orbit 136.b
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.b (of order \(2\), degree \(1\), 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: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 2 i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + 2 i q^{7} - q^{9} + 2 i q^{11} -2 q^{13} + ( 1 - 4 i ) q^{17} + 4 q^{19} -4 q^{21} -6 i q^{23} + 5 q^{25} + 4 i q^{27} -8 i q^{29} -6 i q^{31} -4 q^{33} + 8 i q^{37} -4 i q^{39} -12 q^{43} -8 q^{47} + 3 q^{49} + ( 8 + 2 i ) q^{51} + 6 q^{53} + 8 i q^{57} + 4 q^{59} + 8 i q^{61} -2 i q^{63} + 4 q^{67} + 12 q^{69} -6 i q^{71} + 8 i q^{73} + 10 i q^{75} -4 q^{77} + 2 i q^{79} -11 q^{81} -4 q^{83} + 16 q^{87} -14 q^{89} -4 i q^{91} + 12 q^{93} + 8 i q^{97} -2 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} - 4 q^{13} + 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{25} - 8 q^{33} - 24 q^{43} - 16 q^{47} + 6 q^{49} + 16 q^{51} + 12 q^{53} + 8 q^{59} + 8 q^{67} + 24 q^{69} - 8 q^{77} - 22 q^{81} - 8 q^{83} + 32 q^{87} - 28 q^{89} + 24 q^{93} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.00000i
1.00000i
0 2.00000i 0 0 0 2.00000i 0 −1.00000 0
33.2 0 2.00000i 0 0 0 2.00000i 0 −1.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.b.a 2
3.b odd 2 1 1224.2.c.c 2
4.b odd 2 1 272.2.b.b 2
5.b even 2 1 3400.2.c.c 2
5.c odd 4 1 3400.2.o.a 2
5.c odd 4 1 3400.2.o.d 2
8.b even 2 1 1088.2.b.c 2
8.d odd 2 1 1088.2.b.d 2
12.b even 2 1 2448.2.c.e 2
17.b even 2 1 inner 136.2.b.a 2
17.c even 4 1 2312.2.a.b 1
17.c even 4 1 2312.2.a.c 1
51.c odd 2 1 1224.2.c.c 2
68.d odd 2 1 272.2.b.b 2
68.f odd 4 1 4624.2.a.b 1
68.f odd 4 1 4624.2.a.e 1
85.c even 2 1 3400.2.c.c 2
85.g odd 4 1 3400.2.o.a 2
85.g odd 4 1 3400.2.o.d 2
136.e odd 2 1 1088.2.b.d 2
136.h even 2 1 1088.2.b.c 2
204.h even 2 1 2448.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.b.a 2 1.a even 1 1 trivial
136.2.b.a 2 17.b even 2 1 inner
272.2.b.b 2 4.b odd 2 1
272.2.b.b 2 68.d odd 2 1
1088.2.b.c 2 8.b even 2 1
1088.2.b.c 2 136.h even 2 1
1088.2.b.d 2 8.d odd 2 1
1088.2.b.d 2 136.e odd 2 1
1224.2.c.c 2 3.b odd 2 1
1224.2.c.c 2 51.c odd 2 1
2312.2.a.b 1 17.c even 4 1
2312.2.a.c 1 17.c even 4 1
2448.2.c.e 2 12.b even 2 1
2448.2.c.e 2 204.h even 2 1
3400.2.c.c 2 5.b even 2 1
3400.2.c.c 2 85.c even 2 1
3400.2.o.a 2 5.c odd 4 1
3400.2.o.a 2 85.g odd 4 1
3400.2.o.d 2 5.c odd 4 1
3400.2.o.d 2 85.g odd 4 1
4624.2.a.b 1 68.f odd 4 1
4624.2.a.e 1 68.f odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 17 - 2 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( 64 + T^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 12 + T )^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( 36 + T^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( 4 + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 64 + T^{2} \)
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