Properties

Label 136.1.p.a
Level 136
Weight 1
Character orbit 136.p
Analytic conductor 0.068
Analytic rank 0
Dimension 4
Projective image \(D_{8}\)
CM disc. -8
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 136.p (of order \(8\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{8}\)
Projective field Galois closure of 8.0.1680747204608.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( + \zeta_{8}^{3} q^{2} \) \( + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} \) \( -\zeta_{8}^{2} q^{4} \) \( + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{6} \) \( + \zeta_{8} q^{8} \) \( + ( -1 + \zeta_{8} - \zeta_{8}^{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{8}^{3} q^{2} \) \( + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} \) \( -\zeta_{8}^{2} q^{4} \) \( + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{6} \) \( + \zeta_{8} q^{8} \) \( + ( -1 + \zeta_{8} - \zeta_{8}^{2} ) q^{9} \) \( + ( -1 + \zeta_{8}^{3} ) q^{11} \) \( + ( 1 - \zeta_{8} ) q^{12} \) \(- q^{16}\) \( -\zeta_{8}^{3} q^{17} \) \( + ( -1 + \zeta_{8} - \zeta_{8}^{3} ) q^{18} \) \( + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} \) \( + ( 1 + \zeta_{8}^{3} ) q^{24} \) \( -\zeta_{8} q^{25} \) \( + ( 1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} \) \( -\zeta_{8}^{3} q^{32} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{33} \) \( + \zeta_{8}^{2} q^{34} \) \( + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{36} \) \( + ( \zeta_{8} + \zeta_{8}^{2} ) q^{41} \) \( + ( 1 + \zeta_{8}^{2} ) q^{43} \) \( + ( \zeta_{8} + \zeta_{8}^{2} ) q^{44} \) \( + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{48} \) \( + \zeta_{8}^{3} q^{49} \) \(+ q^{50}\) \( + ( \zeta_{8} - \zeta_{8}^{2} ) q^{51} \) \( + ( 1 + \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{54} \) \( + ( -1 - \zeta_{8}^{2} ) q^{59} \) \( + \zeta_{8}^{2} q^{64} \) \( + ( 1 - \zeta_{8}^{2} ) q^{66} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{67} \) \( -\zeta_{8} q^{68} \) \( + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{72} \) \( + ( \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{73} \) \( + ( -1 - \zeta_{8}^{3} ) q^{75} \) \( + ( -\zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{81} \) \( + ( -1 - \zeta_{8} ) q^{82} \) \( + ( 1 - \zeta_{8}^{2} ) q^{83} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{86} \) \( + ( -1 - \zeta_{8} ) q^{88} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{89} \) \( + ( \zeta_{8} - \zeta_{8}^{2} ) q^{96} \) \( + ( -1 + \zeta_{8} ) q^{97} \) \( -\zeta_{8}^{2} q^{98} \) \( + ( \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 4q^{18} \) \(\mathstrut +\mathstrut 4q^{24} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut +\mathstrut 4q^{43} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 4q^{54} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 4q^{66} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0.707107 0.292893i 1.00000i 0 −0.707107 0.292893i 0 0.707107 0.707107i −0.292893 + 0.292893i 0
43.1 −0.707107 + 0.707107i 0.707107 + 0.292893i 1.00000i 0 −0.707107 + 0.292893i 0 0.707107 + 0.707107i −0.292893 0.292893i 0
59.1 0.707107 0.707107i −0.707107 + 1.70711i 1.00000i 0 0.707107 + 1.70711i 0 −0.707107 0.707107i −1.70711 1.70711i 0
83.1 0.707107 + 0.707107i −0.707107 1.70711i 1.00000i 0 0.707107 1.70711i 0 −0.707107 + 0.707107i −1.70711 + 1.70711i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
17.d Even 1 yes
136.p Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(136, [\chi])\).