Properties

Label 927.1.d.c
Level $927$
Weight $1$
Character orbit 927.d
Self dual yes
Analytic conductor $0.463$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -103
Inner twists $4$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.27349864083.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} -\beta_{2} q^{13} + ( \beta_{1} + \beta_{3} ) q^{14} + ( 2 + 2 \beta_{2} ) q^{16} + \beta_{3} q^{17} + \beta_{2} q^{19} + \beta_{1} q^{23} + q^{25} + \beta_{3} q^{26} + ( -3 - 2 \beta_{2} ) q^{28} -\beta_{3} q^{29} + ( -\beta_{1} - \beta_{3} ) q^{32} + ( -1 - 2 \beta_{2} ) q^{34} -\beta_{3} q^{38} + \beta_{1} q^{41} + ( -3 - \beta_{2} ) q^{46} + ( 1 + \beta_{2} ) q^{49} -\beta_{1} q^{50} + ( -1 - \beta_{2} ) q^{52} + ( 2 \beta_{1} + \beta_{3} ) q^{56} + ( 1 + 2 \beta_{2} ) q^{58} -\beta_{3} q^{59} + ( 1 + \beta_{2} ) q^{61} + ( 2 + \beta_{2} ) q^{64} + ( \beta_{1} + \beta_{3} ) q^{68} + ( 1 + \beta_{2} ) q^{76} -\beta_{2} q^{79} + ( -3 - \beta_{2} ) q^{82} + \beta_{3} q^{83} + q^{91} + ( 2 \beta_{1} + \beta_{3} ) q^{92} + ( 1 + \beta_{2} ) q^{97} + ( -\beta_{1} - \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} - 2q^{7} + O(q^{10}) \) \( 4q + 6q^{4} - 2q^{7} + 2q^{13} + 4q^{16} - 2q^{19} + 4q^{25} - 8q^{28} - 10q^{46} + 2q^{49} - 2q^{52} + 2q^{61} + 6q^{64} + 2q^{76} + 2q^{79} - 10q^{82} + 4q^{91} + 2q^{97} + O(q^{100}) \)

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
1.90211
1.17557
−1.17557
−1.90211
−1.90211 0 2.61803 0 0 −1.61803 −3.07768 0 0
514.2 −1.17557 0 0.381966 0 0 0.618034 0.726543 0 0
514.3 1.17557 0 0.381966 0 0 0.618034 −0.726543 0 0
514.4 1.90211 0 2.61803 0 0 −1.61803 3.07768 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 \(\Q(\sqrt{-103}) \)
3.b odd 2 1 inner
309.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.1.d.c 4
3.b odd 2 1 inner 927.1.d.c 4
103.b odd 2 1 CM 927.1.d.c 4
309.c even 2 1 inner 927.1.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.1.d.c 4 1.a even 1 1 trivial
927.1.d.c 4 3.b odd 2 1 inner
927.1.d.c 4 103.b odd 2 1 CM
927.1.d.c 4 309.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5 T_{2}^{2} + 5 \) acting on \(S_{1}^{\mathrm{new}}(927, [\chi])\).

Hecke characteristic polynomials

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