Properties

Label 136.1.e.a
Level 136
Weight 1
Character orbit 136.e
Self dual yes
Analytic conductor 0.068
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -8, -136, 17
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 136.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{17})\)
Artin image $D_4$
Artin field Galois closure of 4.0.1088.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} + q^{9} + q^{16} - q^{17} - q^{18} - 2q^{19} - q^{25} - q^{32} + q^{34} + q^{36} + 2q^{38} + 2q^{43} - q^{49} + q^{50} + 2q^{59} + q^{64} - 2q^{67} - q^{68} - q^{72} - 2q^{76} + q^{81} + 2q^{83} - 2q^{86} - 2q^{89} + q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0
−1.00000 0 1.00000 0 0 0 −1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
136.e odd 2 1 CM by \(\Q(\sqrt{-34}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.1.e.a 1
3.b odd 2 1 1224.1.n.a 1
4.b odd 2 1 544.1.e.a 1
5.b even 2 1 3400.1.g.a 1
5.c odd 4 2 3400.1.k.a 2
8.b even 2 1 544.1.e.a 1
8.d odd 2 1 CM 136.1.e.a 1
17.b even 2 1 RM 136.1.e.a 1
17.c even 4 2 2312.1.f.a 1
17.d even 8 4 2312.1.j.a 2
17.e odd 16 8 2312.1.p.c 4
24.f even 2 1 1224.1.n.a 1
40.e odd 2 1 3400.1.g.a 1
40.k even 4 2 3400.1.k.a 2
51.c odd 2 1 1224.1.n.a 1
68.d odd 2 1 544.1.e.a 1
85.c even 2 1 3400.1.g.a 1
85.g odd 4 2 3400.1.k.a 2
136.e odd 2 1 CM 136.1.e.a 1
136.h even 2 1 544.1.e.a 1
136.j odd 4 2 2312.1.f.a 1
136.p odd 8 4 2312.1.j.a 2
136.s even 16 8 2312.1.p.c 4
408.h even 2 1 1224.1.n.a 1
680.k odd 2 1 3400.1.g.a 1
680.u even 4 2 3400.1.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.e.a 1 1.a even 1 1 trivial
136.1.e.a 1 8.d odd 2 1 CM
136.1.e.a 1 17.b even 2 1 RM
136.1.e.a 1 136.e odd 2 1 CM
544.1.e.a 1 4.b odd 2 1
544.1.e.a 1 8.b even 2 1
544.1.e.a 1 68.d odd 2 1
544.1.e.a 1 136.h even 2 1
1224.1.n.a 1 3.b odd 2 1
1224.1.n.a 1 24.f even 2 1
1224.1.n.a 1 51.c odd 2 1
1224.1.n.a 1 408.h even 2 1
2312.1.f.a 1 17.c even 4 2
2312.1.f.a 1 136.j odd 4 2
2312.1.j.a 2 17.d even 8 4
2312.1.j.a 2 136.p odd 8 4
2312.1.p.c 4 17.e odd 16 8
2312.1.p.c 4 136.s even 16 8
3400.1.g.a 1 5.b even 2 1
3400.1.g.a 1 40.e odd 2 1
3400.1.g.a 1 85.c even 2 1
3400.1.g.a 1 680.k odd 2 1
3400.1.k.a 2 5.c odd 4 2
3400.1.k.a 2 40.k even 4 2
3400.1.k.a 2 85.g odd 4 2
3400.1.k.a 2 680.u even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( ( 1 - T )( 1 + T ) \)
$5$ \( 1 + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - T )( 1 + T ) \)
$13$ \( ( 1 - T )( 1 + T ) \)
$17$ \( 1 + T \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( 1 + T^{2} \)
$31$ \( 1 + T^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )^{2} \)
$47$ \( ( 1 - T )( 1 + T ) \)
$53$ \( ( 1 - T )( 1 + T ) \)
$59$ \( ( 1 - T )^{2} \)
$61$ \( 1 + T^{2} \)
$67$ \( ( 1 + T )^{2} \)
$71$ \( 1 + T^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( 1 + T^{2} \)
$83$ \( ( 1 - T )^{2} \)
$89$ \( ( 1 + T )^{2} \)
$97$ \( ( 1 - T )( 1 + T ) \)
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