Properties

Label 4.13.b.a
Level 4
Weight 13
Character orbit 4.b
Self dual Yes
Analytic conductor 3.656
Analytic rank 0
Dimension 1
CM disc. -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(3.65597526911\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 64q^{2} \) \(\mathstrut +\mathstrut 4096q^{4} \) \(\mathstrut +\mathstrut 23506q^{5} \) \(\mathstrut -\mathstrut 262144q^{8} \) \(\mathstrut +\mathstrut 531441q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 64q^{2} \) \(\mathstrut +\mathstrut 4096q^{4} \) \(\mathstrut +\mathstrut 23506q^{5} \) \(\mathstrut -\mathstrut 262144q^{8} \) \(\mathstrut +\mathstrut 531441q^{9} \) \(\mathstrut -\mathstrut 1504384q^{10} \) \(\mathstrut +\mathstrut 6911282q^{13} \) \(\mathstrut +\mathstrut 16777216q^{16} \) \(\mathstrut -\mathstrut 47295038q^{17} \) \(\mathstrut -\mathstrut 34012224q^{18} \) \(\mathstrut +\mathstrut 96280576q^{20} \) \(\mathstrut +\mathstrut 308391411q^{25} \) \(\mathstrut -\mathstrut 442322048q^{26} \) \(\mathstrut -\mathstrut 173439758q^{29} \) \(\mathstrut -\mathstrut 1073741824q^{32} \) \(\mathstrut +\mathstrut 3026882432q^{34} \) \(\mathstrut +\mathstrut 2176782336q^{36} \) \(\mathstrut -\mathstrut 2050092718q^{37} \) \(\mathstrut -\mathstrut 6161956864q^{40} \) \(\mathstrut -\mathstrut 2285065118q^{41} \) \(\mathstrut +\mathstrut 12492052146q^{45} \) \(\mathstrut +\mathstrut 13841287201q^{49} \) \(\mathstrut -\mathstrut 19737050304q^{50} \) \(\mathstrut +\mathstrut 28308611072q^{52} \) \(\mathstrut -\mathstrut 43462597358q^{53} \) \(\mathstrut +\mathstrut 11100144512q^{58} \) \(\mathstrut -\mathstrut 47844884878q^{61} \) \(\mathstrut +\mathstrut 68719476736q^{64} \) \(\mathstrut +\mathstrut 162456594692q^{65} \) \(\mathstrut -\mathstrut 193720475648q^{68} \) \(\mathstrut -\mathstrut 139314069504q^{72} \) \(\mathstrut -\mathstrut 119852347678q^{73} \) \(\mathstrut +\mathstrut 131205933952q^{74} \) \(\mathstrut +\mathstrut 394365239296q^{80} \) \(\mathstrut +\mathstrut 282429536481q^{81} \) \(\mathstrut +\mathstrut 146244167552q^{82} \) \(\mathstrut -\mathstrut 1111717163228q^{85} \) \(\mathstrut +\mathstrut 907573615522q^{89} \) \(\mathstrut -\mathstrut 799491337344q^{90} \) \(\mathstrut +\mathstrut 502341690242q^{97} \) \(\mathstrut -\mathstrut 885842380864q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
−64.0000 0 4096.00 23506.0 0 0 −262144. 531441. −1.50438e6
\(n\): 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
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{13}^{\mathrm{new}}(4, [\chi])\).