Properties

Label 1575.1.n.d
Level 1575
Weight 1
Character orbit 1575.n
Analytic conductor 0.786
Analytic rank 0
Dimension 8
Projective image \(D_{12}\)
CM discriminant -7
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.n (of order \(4\), degree \(2\), minimal)

Newform invariants

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

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{24}^{8} - \zeta_{24}^{10} ) q^{2} + ( -\zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{4} + \zeta_{24}^{3} q^{7} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{24}^{8} - \zeta_{24}^{10} ) q^{2} + ( -\zeta_{24}^{4} - \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{4} + \zeta_{24}^{3} q^{7} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{11} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{14} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{16} + ( -\zeta_{24}^{7} + \zeta_{24}^{11} ) q^{22} + ( \zeta_{24}^{8} - \zeta_{24}^{10} ) q^{23} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} - \zeta_{24}^{11} ) q^{28} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{29} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{32} + \zeta_{24}^{3} q^{37} + \zeta_{24}^{9} q^{43} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{44} + ( \zeta_{24}^{4} - \zeta_{24}^{8} ) q^{46} + \zeta_{24}^{6} q^{49} + ( 1 + \zeta_{24}^{6} ) q^{53} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{56} + ( \zeta_{24}^{7} + 2 \zeta_{24}^{9} + \zeta_{24}^{11} ) q^{58} + ( \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{64} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{67} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{71} + ( \zeta_{24} - \zeta_{24}^{11} ) q^{74} + ( \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{77} -\zeta_{24}^{6} q^{79} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{86} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{88} + ( 1 - \zeta_{24}^{6} ) q^{92} + ( \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 12q^{8} + O(q^{10}) \) \( 8q + 4q^{2} - 12q^{8} - 16q^{16} - 4q^{23} - 8q^{32} + 8q^{46} + 8q^{53} - 4q^{77} + 8q^{92} + 4q^{98} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
818.1
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.366025 0.366025i 0 0.732051i 0 0 −0.707107 + 0.707107i −0.633975 + 0.633975i 0 0
818.2 −0.366025 0.366025i 0 0.732051i 0 0 0.707107 0.707107i −0.633975 + 0.633975i 0 0
818.3 1.36603 + 1.36603i 0 2.73205i 0 0 −0.707107 + 0.707107i −2.36603 + 2.36603i 0 0
818.4 1.36603 + 1.36603i 0 2.73205i 0 0 0.707107 0.707107i −2.36603 + 2.36603i 0 0
1007.1 −0.366025 + 0.366025i 0 0.732051i 0 0 −0.707107 0.707107i −0.633975 0.633975i 0 0
1007.2 −0.366025 + 0.366025i 0 0.732051i 0 0 0.707107 + 0.707107i −0.633975 0.633975i 0 0
1007.3 1.36603 1.36603i 0 2.73205i 0 0 −0.707107 0.707107i −2.36603 2.36603i 0 0
1007.4 1.36603 1.36603i 0 2.73205i 0 0 0.707107 + 0.707107i −2.36603 2.36603i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1007.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.c odd 4 1 inner
15.d odd 2 1 inner
15.e even 4 1 inner
35.f even 4 1 inner
105.g even 2 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.n.d yes 8
3.b odd 2 1 1575.1.n.c 8
5.b even 2 1 1575.1.n.c 8
5.c odd 4 1 1575.1.n.c 8
5.c odd 4 1 inner 1575.1.n.d yes 8
7.b odd 2 1 CM 1575.1.n.d yes 8
15.d odd 2 1 inner 1575.1.n.d yes 8
15.e even 4 1 1575.1.n.c 8
15.e even 4 1 inner 1575.1.n.d yes 8
21.c even 2 1 1575.1.n.c 8
35.c odd 2 1 1575.1.n.c 8
35.f even 4 1 1575.1.n.c 8
35.f even 4 1 inner 1575.1.n.d yes 8
105.g even 2 1 inner 1575.1.n.d yes 8
105.k odd 4 1 1575.1.n.c 8
105.k odd 4 1 inner 1575.1.n.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.n.c 8 3.b odd 2 1
1575.1.n.c 8 5.b even 2 1
1575.1.n.c 8 5.c odd 4 1
1575.1.n.c 8 15.e even 4 1
1575.1.n.c 8 21.c even 2 1
1575.1.n.c 8 35.c odd 2 1
1575.1.n.c 8 35.f even 4 1
1575.1.n.c 8 105.k odd 4 1
1575.1.n.d yes 8 1.a even 1 1 trivial
1575.1.n.d yes 8 5.c odd 4 1 inner
1575.1.n.d yes 8 7.b odd 2 1 CM
1575.1.n.d yes 8 15.d odd 2 1 inner
1575.1.n.d yes 8 15.e even 4 1 inner
1575.1.n.d yes 8 35.f even 4 1 inner
1575.1.n.d yes 8 105.g even 2 1 inner
1575.1.n.d yes 8 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\).

Hecke characteristic polynomials

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