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 discriminant -131
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0653775166549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.17161.1
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
131.b odd 2 1 CM by \(\Q(\sqrt{-131}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 131.1.b.a 2
3.b odd 2 1 1179.1.c.a 2
4.b odd 2 1 2096.1.h.b 2
5.b even 2 1 3275.1.c.d 2
5.c odd 4 2 3275.1.d.a 4
131.b odd 2 1 CM 131.1.b.a 2
393.d even 2 1 1179.1.c.a 2
524.b even 2 1 2096.1.h.b 2
655.d odd 2 1 3275.1.c.d 2
655.e even 4 2 3275.1.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
131.1.b.a 2 1.a even 1 1 trivial
131.1.b.a 2 131.b odd 2 1 CM
1179.1.c.a 2 3.b odd 2 1
1179.1.c.a 2 393.d even 2 1
2096.1.h.b 2 4.b odd 2 1
2096.1.h.b 2 524.b even 2 1
3275.1.c.d 2 5.b even 2 1
3275.1.c.d 2 655.d odd 2 1
3275.1.d.a 4 5.c odd 4 2
3275.1.d.a 4 655.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(131, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$11$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$13$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$17$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$19$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$23$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$29$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$41$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$43$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{4} \)
$59$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$61$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$67$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$79$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( ( 1 - T )^{4} \)
$97$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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