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 discriminant -47
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0234560555938\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.2209.1
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.1.b.a 2
3.b odd 2 1 423.1.d.a 2
4.b odd 2 1 752.1.g.a 2
5.b even 2 1 1175.1.d.c 2
5.c odd 4 2 1175.1.b.b 4
7.b odd 2 1 2303.1.d.c 2
7.c even 3 2 2303.1.f.c 4
7.d odd 6 2 2303.1.f.b 4
8.b even 2 1 3008.1.g.b 2
8.d odd 2 1 3008.1.g.a 2
9.c even 3 2 3807.1.f.b 4
9.d odd 6 2 3807.1.f.a 4
47.b odd 2 1 CM 47.1.b.a 2
47.c even 23 22 2209.1.d.a 44
47.d odd 46 22 2209.1.d.a 44
141.c even 2 1 423.1.d.a 2
188.b even 2 1 752.1.g.a 2
235.b odd 2 1 1175.1.d.c 2
235.e even 4 2 1175.1.b.b 4
329.c even 2 1 2303.1.d.c 2
329.f odd 6 2 2303.1.f.c 4
329.g even 6 2 2303.1.f.b 4
376.e odd 2 1 3008.1.g.b 2
376.h even 2 1 3008.1.g.a 2
423.f odd 6 2 3807.1.f.b 4
423.g even 6 2 3807.1.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.1.b.a 2 1.a even 1 1 trivial
47.1.b.a 2 47.b odd 2 1 CM
423.1.d.a 2 3.b odd 2 1
423.1.d.a 2 141.c even 2 1
752.1.g.a 2 4.b odd 2 1
752.1.g.a 2 188.b even 2 1
1175.1.b.b 4 5.c odd 4 2
1175.1.b.b 4 235.e even 4 2
1175.1.d.c 2 5.b even 2 1
1175.1.d.c 2 235.b odd 2 1
2209.1.d.a 44 47.c even 23 22
2209.1.d.a 44 47.d odd 46 22
2303.1.d.c 2 7.b odd 2 1
2303.1.d.c 2 329.c even 2 1
2303.1.f.b 4 7.d odd 6 2
2303.1.f.b 4 329.g even 6 2
2303.1.f.c 4 7.c even 3 2
2303.1.f.c 4 329.f odd 6 2
3008.1.g.a 2 8.d odd 2 1
3008.1.g.a 2 376.h even 2 1
3008.1.g.b 2 8.b even 2 1
3008.1.g.b 2 376.e odd 2 1
3807.1.f.a 4 9.d odd 6 2
3807.1.f.a 4 423.g even 6 2
3807.1.f.b 4 9.c even 3 2
3807.1.f.b 4 423.f odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(47, [\chi])\).

Hecke characteristic polynomials

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