Properties

Label 1575.1.x.b
Level $1575$
Weight $1$
Character orbit 1575.x
Analytic conductor $0.786$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.283618125.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{4} + \zeta_{6} q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{4} + \zeta_{6} q^{7} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{13} + \zeta_{6}^{2} q^{16} + ( 1 + \zeta_{6} ) q^{19} + \zeta_{6}^{2} q^{28} + \zeta_{6}^{2} q^{37} -2 q^{43} + \zeta_{6}^{2} q^{49} + ( 1 - \zeta_{6}^{2} ) q^{52} + ( -1 - \zeta_{6} ) q^{61} - q^{64} + \zeta_{6} q^{67} + ( 1 - \zeta_{6}^{2} ) q^{73} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{76} -\zeta_{6}^{2} q^{79} + ( 1 - \zeta_{6}^{2} ) q^{91} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{4} + q^{7} + O(q^{10}) \) \( 2q + q^{4} + q^{7} - q^{16} + 3q^{19} - q^{28} - q^{37} - 4q^{43} - q^{49} + 3q^{52} - 3q^{61} - 2q^{64} + q^{67} + 3q^{73} + q^{79} + 3q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-\zeta_{6}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 0
901.1 0 0 0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.x.b yes 2
3.b odd 2 1 CM 1575.1.x.b yes 2
5.b even 2 1 1575.1.x.a 2
5.c odd 4 2 1575.1.bj.a 4
7.d odd 6 1 inner 1575.1.x.b yes 2
15.d odd 2 1 1575.1.x.a 2
15.e even 4 2 1575.1.bj.a 4
21.g even 6 1 inner 1575.1.x.b yes 2
35.i odd 6 1 1575.1.x.a 2
35.k even 12 2 1575.1.bj.a 4
105.p even 6 1 1575.1.x.a 2
105.w odd 12 2 1575.1.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.x.a 2 5.b even 2 1
1575.1.x.a 2 15.d odd 2 1
1575.1.x.a 2 35.i odd 6 1
1575.1.x.a 2 105.p even 6 1
1575.1.x.b yes 2 1.a even 1 1 trivial
1575.1.x.b yes 2 3.b odd 2 1 CM
1575.1.x.b yes 2 7.d odd 6 1 inner
1575.1.x.b yes 2 21.g even 6 1 inner
1575.1.bj.a 4 5.c odd 4 2
1575.1.bj.a 4 15.e even 4 2
1575.1.bj.a 4 35.k even 12 2
1575.1.bj.a 4 105.w odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{2} + T_{37} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\).

Hecke characteristic polynomials

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