Properties

Label 1575.1.bj.a
Level $1575$
Weight $1$
Character orbit 1575.bj
Analytic conductor $0.786$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.bj (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} +O(q^{10})\) \( q -\zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{13} + \zeta_{12}^{4} q^{16} + ( -1 - \zeta_{12}^{2} ) q^{19} + \zeta_{12} q^{28} -\zeta_{12} q^{37} + 2 \zeta_{12}^{3} q^{43} -\zeta_{12}^{4} q^{49} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{52} + ( -1 - \zeta_{12}^{2} ) q^{61} + q^{64} + \zeta_{12}^{5} q^{67} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{73} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{76} + \zeta_{12}^{4} q^{79} + ( 1 - \zeta_{12}^{4} ) q^{91} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{4} + O(q^{10}) \) \( 4q - 2q^{4} - 2q^{16} - 6q^{19} + 2q^{49} - 6q^{61} + 4q^{64} - 2q^{79} + 6q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 −0.500000 + 0.866025i 0 0 −0.866025 0.500000i 0 0 0
199.2 0 0 −0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 0 0 0
649.1 0 0 −0.500000 0.866025i 0 0 −0.866025 + 0.500000i 0 0 0
649.2 0 0 −0.500000 0.866025i 0 0 0.866025 0.500000i 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 \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p 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.bj.a 4
3.b odd 2 1 CM 1575.1.bj.a 4
5.b even 2 1 inner 1575.1.bj.a 4
5.c odd 4 1 1575.1.x.a 2
5.c odd 4 1 1575.1.x.b yes 2
7.d odd 6 1 inner 1575.1.bj.a 4
15.d odd 2 1 inner 1575.1.bj.a 4
15.e even 4 1 1575.1.x.a 2
15.e even 4 1 1575.1.x.b yes 2
21.g even 6 1 inner 1575.1.bj.a 4
35.i odd 6 1 inner 1575.1.bj.a 4
35.k even 12 1 1575.1.x.a 2
35.k even 12 1 1575.1.x.b yes 2
105.p even 6 1 inner 1575.1.bj.a 4
105.w odd 12 1 1575.1.x.a 2
105.w odd 12 1 1575.1.x.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.x.a 2 5.c odd 4 1
1575.1.x.a 2 15.e even 4 1
1575.1.x.a 2 35.k even 12 1
1575.1.x.a 2 105.w odd 12 1
1575.1.x.b yes 2 5.c odd 4 1
1575.1.x.b yes 2 15.e even 4 1
1575.1.x.b yes 2 35.k even 12 1
1575.1.x.b yes 2 105.w odd 12 1
1575.1.bj.a 4 1.a even 1 1 trivial
1575.1.bj.a 4 3.b odd 2 1 CM
1575.1.bj.a 4 5.b even 2 1 inner
1575.1.bj.a 4 7.d odd 6 1 inner
1575.1.bj.a 4 15.d odd 2 1 inner
1575.1.bj.a 4 21.g even 6 1 inner
1575.1.bj.a 4 35.i odd 6 1 inner
1575.1.bj.a 4 105.p even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1575, [\chi])\).

Hecke characteristic polynomials

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