Properties

Label 136.1.j.a
Level 136
Weight 1
Character orbit 136.j
Analytic conductor 0.068
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
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.j (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0678728417181\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.314432.1
Artin image size \(32\)
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.20123648.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -i q^{2} \) \( + ( -1 - i ) q^{3} \) \(- q^{4}\) \( + ( -1 + i ) q^{6} \) \( + i q^{8} \) \( + i q^{9} \) \(+O(q^{10})\) \( q\) \( -i q^{2} \) \( + ( -1 - i ) q^{3} \) \(- q^{4}\) \( + ( -1 + i ) q^{6} \) \( + i q^{8} \) \( + i q^{9} \) \( + ( 1 - i ) q^{11} \) \( + ( 1 + i ) q^{12} \) \(+ q^{16}\) \( -i q^{17} \) \(+ q^{18}\) \( + 2 i q^{19} \) \( + ( -1 - i ) q^{22} \) \( + ( 1 - i ) q^{24} \) \( + i q^{25} \) \( -i q^{32} \) \( -2 q^{33} \) \(- q^{34}\) \( -i q^{36} \) \( + 2 q^{38} \) \( + ( -1 + i ) q^{41} \) \( + ( -1 + i ) q^{44} \) \( + ( -1 - i ) q^{48} \) \( -i q^{49} \) \(+ q^{50}\) \( + ( -1 + i ) q^{51} \) \( + ( 2 - 2 i ) q^{57} \) \(- q^{64}\) \( + 2 i q^{66} \) \( + i q^{68} \) \(- q^{72}\) \( + ( -1 - i ) q^{73} \) \( + ( 1 - i ) q^{75} \) \( -2 i q^{76} \) \(+ q^{81}\) \( + ( 1 + i ) q^{82} \) \( + ( 1 + i ) q^{88} \) \( + ( -1 + i ) q^{96} \) \( + ( 1 + i ) q^{97} \) \(- q^{98}\) \( + ( 1 + i ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 2q^{24} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 2q^{48} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 2q^{72} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut 2q^{88} \) \(\mathstrut -\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
1.00000i
1.00000i
1.00000i −1.00000 1.00000i −1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000i 0
123.1 1.00000i −1.00000 + 1.00000i −1.00000 0 −1.00000 1.00000i 0 1.00000i 1.00000i 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.c Even 1 yes
136.j Odd 1 yes

Hecke kernels

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