Properties

Label 179.1.b.a
Level $179$
Weight $1$
Character orbit 179.b
Self dual yes
Analytic conductor $0.089$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -179
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 179 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 179.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0893326372613\)
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 5.1.32041.1
Artin image: $D_5$
Artin field: Galois closure of 5.1.32041.1

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} + ( 1 - \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + q^{4} -\beta q^{5} + ( 1 - \beta ) q^{9} + ( -1 + \beta ) q^{12} -\beta q^{13} - q^{15} + q^{16} + ( -1 + \beta ) q^{17} + ( -1 + \beta ) q^{19} -\beta q^{20} + \beta q^{25} - q^{27} + ( -1 + \beta ) q^{29} -\beta q^{31} + ( 1 - \beta ) q^{36} - q^{39} -\beta q^{43} + q^{45} + 2 q^{47} + ( -1 + \beta ) q^{48} + q^{49} + ( 2 - \beta ) q^{51} -\beta q^{52} + ( 2 - \beta ) q^{57} + ( -1 + \beta ) q^{59} - q^{60} + ( -1 + \beta ) q^{61} + q^{64} + ( 1 + \beta ) q^{65} + ( -1 + \beta ) q^{67} + ( -1 + \beta ) q^{68} + q^{75} + ( -1 + \beta ) q^{76} -\beta q^{80} -\beta q^{83} - q^{85} + ( 2 - \beta ) q^{87} -\beta q^{89} - q^{93} - q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{4} - q^{5} + q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{4} - q^{5} + q^{9} - q^{12} - q^{13} - 2q^{15} + 2q^{16} - q^{17} - q^{19} - q^{20} + q^{25} - 2q^{27} - q^{29} - q^{31} + q^{36} - 2q^{39} - q^{43} + 2q^{45} + 4q^{47} - q^{48} + 2q^{49} + 3q^{51} - q^{52} + 3q^{57} - q^{59} - 2q^{60} - q^{61} + 2q^{64} + 3q^{65} - q^{67} - q^{68} + 2q^{75} - q^{76} - q^{80} - q^{83} - 2q^{85} + 3q^{87} - q^{89} - 2q^{93} - 2q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/179\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
178.1
−0.618034
1.61803
0 −1.61803 1.00000 0.618034 0 0 0 1.61803 0
178.2 0 0.618034 1.00000 −1.61803 0 0 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
179.b odd 2 1 CM by \(\Q(\sqrt{-179}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 179.1.b.a 2
3.b odd 2 1 1611.1.c.a 2
4.b odd 2 1 2864.1.h.b 2
179.b odd 2 1 CM 179.1.b.a 2
537.d even 2 1 1611.1.c.a 2
716.c even 2 1 2864.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
179.1.b.a 2 1.a even 1 1 trivial
179.1.b.a 2 179.b odd 2 1 CM
1611.1.c.a 2 3.b odd 2 1
1611.1.c.a 2 537.d even 2 1
2864.1.h.b 2 4.b odd 2 1
2864.1.h.b 2 716.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(179, [\chi])\).

Hecke characteristic polynomials

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