Properties

Label 1575.1.cn.a
Level 1575
Weight 1
Character orbit 1575.cn
Analytic conductor 0.786
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM discriminant -35
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.cn (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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.843908625.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{9} q^{3} + \zeta_{24}^{2} q^{4} -\zeta_{24}^{7} q^{7} -\zeta_{24}^{6} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{9} q^{3} + \zeta_{24}^{2} q^{4} -\zeta_{24}^{7} q^{7} -\zeta_{24}^{6} q^{9} + ( -1 + \zeta_{24}^{8} ) q^{11} -\zeta_{24}^{11} q^{12} + \zeta_{24}^{5} q^{13} + \zeta_{24}^{4} q^{16} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{17} -\zeta_{24}^{4} q^{21} -\zeta_{24}^{3} q^{27} -\zeta_{24}^{9} q^{28} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{33} -\zeta_{24}^{8} q^{36} + \zeta_{24}^{2} q^{39} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{44} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{47} + \zeta_{24} q^{48} -\zeta_{24}^{2} q^{49} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{51} + \zeta_{24}^{7} q^{52} -\zeta_{24} q^{63} + \zeta_{24}^{6} q^{64} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{68} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{71} + \zeta_{24}^{9} q^{73} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{77} + \zeta_{24}^{10} q^{79} - q^{81} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{83} -\zeta_{24}^{6} q^{84} + q^{91} -\zeta_{24}^{7} q^{97} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 12q^{11} + 4q^{16} - 4q^{21} + 4q^{36} - 8q^{81} + 8q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-1\) \(-\zeta_{24}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0 −0.707107 + 0.707107i 0.866025 + 0.500000i 0 0 −0.258819 + 0.965926i 0 1.00000i 0
293.2 0 0.707107 0.707107i 0.866025 + 0.500000i 0 0 0.258819 0.965926i 0 1.00000i 0
482.1 0 −0.707107 0.707107i −0.866025 0.500000i 0 0 0.965926 + 0.258819i 0 1.00000i 0
482.2 0 0.707107 + 0.707107i −0.866025 0.500000i 0 0 −0.965926 0.258819i 0 1.00000i 0
1343.1 0 −0.707107 + 0.707107i −0.866025 + 0.500000i 0 0 0.965926 0.258819i 0 1.00000i 0
1343.2 0 0.707107 0.707107i −0.866025 + 0.500000i 0 0 −0.965926 + 0.258819i 0 1.00000i 0
1532.1 0 −0.707107 0.707107i 0.866025 0.500000i 0 0 −0.258819 0.965926i 0 1.00000i 0
1532.2 0 0.707107 + 0.707107i 0.866025 0.500000i 0 0 0.258819 + 0.965926i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1532.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
9.d odd 6 1 inner
35.f even 4 2 inner
45.h odd 6 1 inner
45.l even 12 2 inner
63.o even 6 1 inner
315.z even 6 1 inner
315.cf odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.cn.a 8
5.b even 2 1 inner 1575.1.cn.a 8
5.c odd 4 2 inner 1575.1.cn.a 8
7.b odd 2 1 inner 1575.1.cn.a 8
9.d odd 6 1 inner 1575.1.cn.a 8
35.c odd 2 1 CM 1575.1.cn.a 8
35.f even 4 2 inner 1575.1.cn.a 8
45.h odd 6 1 inner 1575.1.cn.a 8
45.l even 12 2 inner 1575.1.cn.a 8
63.o even 6 1 inner 1575.1.cn.a 8
315.z even 6 1 inner 1575.1.cn.a 8
315.cf odd 12 2 inner 1575.1.cn.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.cn.a 8 1.a even 1 1 trivial
1575.1.cn.a 8 5.b even 2 1 inner
1575.1.cn.a 8 5.c odd 4 2 inner
1575.1.cn.a 8 7.b odd 2 1 inner
1575.1.cn.a 8 9.d odd 6 1 inner
1575.1.cn.a 8 35.c odd 2 1 CM
1575.1.cn.a 8 35.f even 4 2 inner
1575.1.cn.a 8 45.h odd 6 1 inner
1575.1.cn.a 8 45.l even 12 2 inner
1575.1.cn.a 8 63.o even 6 1 inner
1575.1.cn.a 8 315.z even 6 1 inner
1575.1.cn.a 8 315.cf odd 12 2 inner

Hecke kernels

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

Hecke characteristic polynomials

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