Properties

Label 927.1.d.b
Level $927$
Weight $1$
Character orbit 927.d
Self dual yes
Analytic conductor $0.463$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -103
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 927.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.462633266711\)
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: no (minimal twist has level 103)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.10609.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.0.27349864083.1

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{7} + q^{8} +O(q^{10})\) \( q + ( 1 - \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{7} + q^{8} + ( -1 + \beta ) q^{13} + q^{14} + \beta q^{17} + ( -1 + \beta ) q^{19} + ( 1 - \beta ) q^{23} + q^{25} + ( -2 + \beta ) q^{26} + q^{28} + \beta q^{29} - q^{32} - q^{34} + ( -2 + \beta ) q^{38} + ( 1 - \beta ) q^{41} + ( 2 - \beta ) q^{46} + \beta q^{49} + ( 1 - \beta ) q^{50} + ( -2 + \beta ) q^{52} -\beta q^{56} - q^{58} + \beta q^{59} -\beta q^{61} + ( -1 + \beta ) q^{64} - q^{68} + ( -2 + \beta ) q^{76} + ( -1 + \beta ) q^{79} + ( 2 - \beta ) q^{82} + \beta q^{83} - q^{91} + ( 2 - \beta ) q^{92} -\beta q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + q^{2} + q^{4} - q^{7} + 2q^{8} - q^{13} + 2q^{14} + q^{17} - q^{19} + q^{23} + 2q^{25} - 3q^{26} + 2q^{28} + q^{29} - 2q^{32} - 2q^{34} - 3q^{38} + q^{41} + 3q^{46} + q^{49} + q^{50} - 3q^{52} - q^{56} - 2q^{58} + q^{59} - q^{61} - q^{64} - 2q^{68} - 3q^{76} - q^{79} + 3q^{82} + q^{83} - 2q^{91} + 3q^{92} - q^{97} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\) \(722\) \(829\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
514.1
1.61803
−0.618034
−0.618034 0 −0.618034 0 0 −1.61803 1.00000 0 0
514.2 1.61803 0 1.61803 0 0 0.618034 1.00000 0 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
103.b odd 2 1 CM by \(\Q(\sqrt{-103}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.1.d.b 2
3.b odd 2 1 103.1.b.a 2
12.b even 2 1 1648.1.c.a 2
15.d odd 2 1 2575.1.d.d 2
15.e even 4 2 2575.1.c.b 4
103.b odd 2 1 CM 927.1.d.b 2
309.c even 2 1 103.1.b.a 2
1236.f odd 2 1 1648.1.c.a 2
1545.g even 2 1 2575.1.d.d 2
1545.m odd 4 2 2575.1.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 3.b odd 2 1
103.1.b.a 2 309.c even 2 1
927.1.d.b 2 1.a even 1 1 trivial
927.1.d.b 2 103.b odd 2 1 CM
1648.1.c.a 2 12.b even 2 1
1648.1.c.a 2 1236.f odd 2 1
2575.1.c.b 4 15.e even 4 2
2575.1.c.b 4 1545.m odd 4 2
2575.1.d.d 2 15.d odd 2 1
2575.1.d.d 2 1545.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(927, [\chi])\).

Hecke characteristic polynomials

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