Properties

Label 131.1.b.a
Level 131
Weight 1
Character orbit 131.b
Self dual Yes
Analytic conductor 0.065
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM disc. -131
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 131 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 131.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0653775166549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.17161.1
Artin image size \(10\)
Artin image $D_5$
Artin field Galois closure of 5.1.17161.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta ) q^{3} \) \(+ q^{4}\) \( -\beta q^{5} \) \( -\beta q^{7} \) \( + ( 1 - \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta ) q^{3} \) \(+ q^{4}\) \( -\beta q^{5} \) \( -\beta q^{7} \) \( + ( 1 - \beta ) q^{9} \) \( + ( -1 + \beta ) q^{11} \) \( + ( -1 + \beta ) q^{12} \) \( + ( -1 + \beta ) q^{13} \) \(- q^{15}\) \(+ q^{16}\) \( -\beta q^{20} \) \(- q^{21}\) \( + \beta q^{25} \) \(- q^{27}\) \( -\beta q^{28} \) \( + ( 2 - \beta ) q^{33} \) \( + ( 1 + \beta ) q^{35} \) \( + ( 1 - \beta ) q^{36} \) \( + ( 2 - \beta ) q^{39} \) \( + ( -1 + \beta ) q^{41} \) \( -\beta q^{43} \) \( + ( -1 + \beta ) q^{44} \) \(+ q^{45}\) \( + ( -1 + \beta ) q^{48} \) \( + \beta q^{49} \) \( + ( -1 + \beta ) q^{52} \) \( + 2 q^{53} \) \(- q^{55}\) \( -\beta q^{59} \) \(- q^{60}\) \( -\beta q^{61} \) \(+ q^{63}\) \(+ q^{64}\) \(- q^{65}\) \(+ q^{75}\) \(- q^{77}\) \( -\beta q^{80} \) \(- q^{84}\) \( + 2 q^{89} \) \(- q^{91}\) \( + ( -2 + \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut q^{28} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 3q^{35} \) \(\mathstrut +\mathstrut q^{36} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut q^{43} \) \(\mathstrut -\mathstrut q^{44} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut q^{59} \) \(\mathstrut -\mathstrut 2q^{60} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 2q^{77} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/131\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
130.1
−0.618034
1.61803
0 −1.61803 1.00000 0.618034 0 0.618034 0 1.61803 0
130.2 0 0.618034 1.00000 −1.61803 0 −1.61803 0 −0.618034 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
131.b Odd 1 CM by \(\Q(\sqrt{-131}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(131, [\chi])\).