Properties

Label 1575.1.ci.a
Level 1575
Weight 1
Character orbit 1575.ci
Analytic conductor 0.786
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM discriminant -3
Inner twists 16

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.ci (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.67528125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{2} q^{4} -\zeta_{24}^{11} q^{7} +O(q^{10})\) \( q -\zeta_{24}^{2} q^{4} -\zeta_{24}^{11} q^{7} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{13} + \zeta_{24}^{4} q^{16} -\zeta_{24}^{10} q^{19} -\zeta_{24} q^{28} -2 \zeta_{24}^{8} q^{31} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{37} -\zeta_{24}^{10} q^{49} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{52} + \zeta_{24}^{4} q^{61} -\zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{67} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{73} - q^{76} + \zeta_{24}^{10} q^{79} + ( -1 - \zeta_{24}^{4} ) q^{91} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{16} + 8q^{31} + 4q^{61} - 8q^{76} - 12q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-\zeta_{24}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0 0 0.866025 0.500000i 0 0 −0.258819 + 0.965926i 0 0 0
793.2 0 0 0.866025 0.500000i 0 0 0.258819 0.965926i 0 0 0
982.1 0 0 −0.866025 + 0.500000i 0 0 −0.965926 0.258819i 0 0 0
982.2 0 0 −0.866025 + 0.500000i 0 0 0.965926 + 0.258819i 0 0 0
1243.1 0 0 −0.866025 0.500000i 0 0 −0.965926 + 0.258819i 0 0 0
1243.2 0 0 −0.866025 0.500000i 0 0 0.965926 0.258819i 0 0 0
1432.1 0 0 0.866025 + 0.500000i 0 0 −0.258819 0.965926i 0 0 0
1432.2 0 0 0.866025 + 0.500000i 0 0 0.258819 + 0.965926i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1432.2
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
5.c odd 4 2 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.h odd 6 1 inner
35.j even 6 1 inner
35.l odd 12 2 inner
105.o odd 6 1 inner
105.x even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.ci.a 8
3.b odd 2 1 CM 1575.1.ci.a 8
5.b even 2 1 inner 1575.1.ci.a 8
5.c odd 4 2 inner 1575.1.ci.a 8
7.c even 3 1 inner 1575.1.ci.a 8
15.d odd 2 1 inner 1575.1.ci.a 8
15.e even 4 2 inner 1575.1.ci.a 8
21.h odd 6 1 inner 1575.1.ci.a 8
35.j even 6 1 inner 1575.1.ci.a 8
35.l odd 12 2 inner 1575.1.ci.a 8
105.o odd 6 1 inner 1575.1.ci.a 8
105.x even 12 2 inner 1575.1.ci.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.ci.a 8 1.a even 1 1 trivial
1575.1.ci.a 8 3.b odd 2 1 CM
1575.1.ci.a 8 5.b even 2 1 inner
1575.1.ci.a 8 5.c odd 4 2 inner
1575.1.ci.a 8 7.c even 3 1 inner
1575.1.ci.a 8 15.d odd 2 1 inner
1575.1.ci.a 8 15.e even 4 2 inner
1575.1.ci.a 8 21.h odd 6 1 inner
1575.1.ci.a 8 35.j even 6 1 inner
1575.1.ci.a 8 35.l odd 12 2 inner
1575.1.ci.a 8 105.o odd 6 1 inner
1575.1.ci.a 8 105.x even 12 2 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 - T^{4} + T^{8} \)
$11$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$13$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$17$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$19$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$23$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$29$ \( ( 1 - T )^{8}( 1 + T )^{8} \)
$31$ \( ( 1 - T + T^{2} )^{8} \)
$37$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$41$ \( ( 1 + T^{2} )^{8} \)
$43$ \( ( 1 + T^{4} )^{4} \)
$47$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$53$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$59$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$61$ \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
$67$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$71$ \( ( 1 + T^{2} )^{8} \)
$73$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$79$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$83$ \( ( 1 + T^{4} )^{4} \)
$89$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$97$ \( ( 1 - T^{4} + T^{8} )^{2} \)
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