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 disc. -79
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(0.039426135998\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.6241.1
Artin image size \(10\)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
79.b Odd 1 CM by \(\Q(\sqrt{-79}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(79, [\chi])\).