Properties

Label 79.1.b.a
Level 79
Weight 1
Character orbit 79.b
Self dual yes
Analytic conductor 0.039
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -79
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 79.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0394261359980\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.6241.1
Artin image $D_5$
Artin field Galois closure of 5.1.6241.1

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{5} - q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{5} - q^{8} + q^{9} - q^{10} + ( -1 + \beta ) q^{11} + ( -1 + \beta ) q^{13} + ( -1 + \beta ) q^{18} -\beta q^{19} + q^{20} + ( 2 - \beta ) q^{22} -\beta q^{23} + \beta q^{25} + ( 2 - \beta ) q^{26} + ( -1 + \beta ) q^{31} + q^{32} + ( 1 - \beta ) q^{36} - q^{38} + \beta q^{40} + ( -2 + \beta ) q^{44} -\beta q^{45} - q^{46} + q^{49} + q^{50} + ( -2 + \beta ) q^{52} - q^{55} + ( 2 - \beta ) q^{62} + ( -1 + \beta ) q^{64} - q^{65} -\beta q^{67} - q^{72} -\beta q^{73} + q^{76} + q^{79} + q^{81} + 2 q^{83} + ( 1 - \beta ) q^{88} + ( -1 + \beta ) q^{89} - q^{90} + q^{92} + ( 1 + \beta ) q^{95} -\beta q^{97} + ( -1 + \beta ) q^{98} + ( -1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} - q^{5} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{4} - q^{5} - 2q^{8} + 2q^{9} - 2q^{10} - q^{11} - q^{13} - q^{18} - q^{19} + 2q^{20} + 3q^{22} - q^{23} + q^{25} + 3q^{26} - q^{31} + 2q^{32} + q^{36} - 2q^{38} + q^{40} - 3q^{44} - q^{45} - 2q^{46} + 2q^{49} + 2q^{50} - 3q^{52} - 2q^{55} + 3q^{62} - q^{64} - 2q^{65} - q^{67} - 2q^{72} - q^{73} + 2q^{76} + 2q^{79} + 2q^{81} + 4q^{83} + q^{88} - q^{89} - 2q^{90} + 2q^{92} + 3q^{95} - q^{97} - q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/79\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
78.1
−0.618034
1.61803
−1.61803 0 1.61803 0.618034 0 0 −1.00000 1.00000 −1.00000
78.2 0.618034 0 −0.618034 −1.61803 0 0 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.1.b.a 2
3.b odd 2 1 711.1.d.b 2
4.b odd 2 1 1264.1.e.a 2
5.b even 2 1 1975.1.d.c 2
5.c odd 4 2 1975.1.c.a 4
7.b odd 2 1 3871.1.c.c 2
7.c even 3 2 3871.1.m.c 4
7.d odd 6 2 3871.1.m.b 4
79.b odd 2 1 CM 79.1.b.a 2
237.b even 2 1 711.1.d.b 2
316.d even 2 1 1264.1.e.a 2
395.c odd 2 1 1975.1.d.c 2
395.f even 4 2 1975.1.c.a 4
553.d even 2 1 3871.1.c.c 2
553.l even 6 2 3871.1.m.b 4
553.m odd 6 2 3871.1.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.1.b.a 2 1.a even 1 1 trivial
79.1.b.a 2 79.b odd 2 1 CM
711.1.d.b 2 3.b odd 2 1
711.1.d.b 2 237.b even 2 1
1264.1.e.a 2 4.b odd 2 1
1264.1.e.a 2 316.d even 2 1
1975.1.c.a 4 5.c odd 4 2
1975.1.c.a 4 395.f even 4 2
1975.1.d.c 2 5.b even 2 1
1975.1.d.c 2 395.c odd 2 1
3871.1.c.c 2 7.b odd 2 1
3871.1.c.c 2 553.d even 2 1
3871.1.m.b 4 7.d odd 6 2
3871.1.m.b 4 553.l even 6 2
3871.1.m.c 4 7.c even 3 2
3871.1.m.c 4 553.m odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(79, [\chi])\).

Hecke characteristic polynomials

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