Properties

Label 47.1.b.a
Level 47
Weight 1
Character orbit 47.b
Self dual Yes
Analytic conductor 0.023
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM disc. -47
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 47 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 47.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0234560555938\)
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 5.1.2209.1
Artin image size \(10\)
Artin image $D_5$
Artin field Galois closure of 5.1.2209.1

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta ) q^{2} \) \( -\beta q^{3} \) \( + ( 1 - \beta ) q^{4} \) \(- q^{6}\) \( + ( -1 + \beta ) q^{7} \) \(- q^{8}\) \( + \beta q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta ) q^{2} \) \( -\beta q^{3} \) \( + ( 1 - \beta ) q^{4} \) \(- q^{6}\) \( + ( -1 + \beta ) q^{7} \) \(- q^{8}\) \( + \beta q^{9} \) \(+ q^{12}\) \( + ( 2 - \beta ) q^{14} \) \( -\beta q^{17} \) \(+ q^{18}\) \(- q^{21}\) \( + \beta q^{24} \) \(+ q^{25}\) \(- q^{27}\) \( + ( -2 + \beta ) q^{28} \) \(+ q^{32}\) \(- q^{34}\) \(- q^{36}\) \( -\beta q^{37} \) \( + ( 1 - \beta ) q^{42} \) \(+ q^{47}\) \( + ( 1 - \beta ) q^{49} \) \( + ( -1 + \beta ) q^{50} \) \( + ( 1 + \beta ) q^{51} \) \( + ( -1 + \beta ) q^{53} \) \( + ( 1 - \beta ) q^{54} \) \( + ( 1 - \beta ) q^{56} \) \( + ( -1 + \beta ) q^{59} \) \( + ( -1 + \beta ) q^{61} \) \(+ q^{63}\) \( + ( -1 + \beta ) q^{64} \) \(+ q^{68}\) \( -\beta q^{71} \) \( -\beta q^{72} \) \(- q^{74}\) \( -\beta q^{75} \) \( -\beta q^{79} \) \( + 2 q^{83} \) \( + ( -1 + \beta ) q^{84} \) \( + ( -1 + \beta ) q^{89} \) \( + ( -1 + \beta ) q^{94} \) \( -\beta q^{96} \) \( + ( -1 + \beta ) q^{97} \) \( + ( -2 + \beta ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 3q^{14} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut q^{24} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut +\mathstrut q^{42} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut q^{49} \) \(\mathstrut -\mathstrut q^{50} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut +\mathstrut q^{54} \) \(\mathstrut +\mathstrut q^{56} \) \(\mathstrut -\mathstrut q^{59} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut +\mathstrut 2q^{68} \) \(\mathstrut -\mathstrut q^{71} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut -\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut q^{84} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut -\mathstrut q^{94} \) \(\mathstrut -\mathstrut q^{96} \) \(\mathstrut -\mathstrut q^{97} \) \(\mathstrut -\mathstrut 3q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/47\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−0.618034
1.61803
−1.61803 0.618034 1.61803 0 −1.00000 −1.61803 −1.00000 −0.618034 0
46.2 0.618034 −1.61803 −0.618034 0 −1.00000 0.618034 −1.00000 1.61803 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
47.b Odd 1 CM by \(\Q(\sqrt{-47}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(47, [\chi])\).