Properties

Label 1620.1.w.c
Level $1620$
Weight $1$
Character orbit 1620.w
Analytic conductor $0.808$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1620.w (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.808485320465\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} -\zeta_{24}^{5} q^{5} -\zeta_{24}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} -\zeta_{24}^{5} q^{5} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{10} q^{10} + ( \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{13} -\zeta_{24}^{8} q^{16} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{3} q^{20} + \zeta_{24}^{10} q^{25} + ( \zeta_{24} + \zeta_{24}^{11} ) q^{26} + ( -\zeta_{24}^{7} - \zeta_{24}^{9} ) q^{29} + \zeta_{24} q^{32} + ( 1 + \zeta_{24}^{4} ) q^{34} + ( \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{37} + \zeta_{24}^{8} q^{40} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{41} -\zeta_{24}^{10} q^{49} -\zeta_{24}^{3} q^{50} + ( -\zeta_{24}^{4} + \zeta_{24}^{6} ) q^{52} + ( 1 + \zeta_{24}^{2} ) q^{58} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{61} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24} - \zeta_{24}^{11} ) q^{65} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{68} + ( \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{73} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{74} -\zeta_{24} q^{80} + ( -1 - \zeta_{24}^{6} ) q^{82} + ( -1 - \zeta_{24}^{4} ) q^{85} + ( -\zeta_{24}^{5} + \zeta_{24}^{7} ) q^{89} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{97} + \zeta_{24}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{13} + 4q^{16} + 12q^{34} + 4q^{37} - 4q^{40} - 4q^{52} + 8q^{58} - 4q^{73} - 8q^{82} - 12q^{85} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{6}\) \(\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i 0 −0.866025 + 0.500000i 0.258819 + 0.965926i 0 0 0.707107 + 0.707107i 0 0.866025 0.500000i
107.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.258819 0.965926i 0 0 −0.707107 0.707107i 0 0.866025 0.500000i
863.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0.258819 0.965926i 0 0 0.707107 0.707107i 0 0.866025 + 0.500000i
863.2 0.258819 0.965926i 0 −0.866025 0.500000i −0.258819 + 0.965926i 0 0 −0.707107 + 0.707107i 0 0.866025 + 0.500000i
1187.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0.965926 + 0.258819i 0 0 −0.707107 0.707107i 0 −0.866025 0.500000i
1187.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.965926 0.258819i 0 0 0.707107 + 0.707107i 0 −0.866025 0.500000i
1403.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.965926 0.258819i 0 0 −0.707107 + 0.707107i 0 −0.866025 + 0.500000i
1403.2 0.965926 0.258819i 0 0.866025 0.500000i −0.965926 + 0.258819i 0 0 0.707107 0.707107i 0 −0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1403.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner
180.v odd 12 1 inner
180.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.w.c 8
3.b odd 2 1 inner 1620.1.w.c 8
4.b odd 2 1 CM 1620.1.w.c 8
5.c odd 4 1 1620.1.w.a 8
9.c even 3 1 1620.1.m.a 8
9.c even 3 1 1620.1.w.a 8
9.d odd 6 1 1620.1.m.a 8
9.d odd 6 1 1620.1.w.a 8
12.b even 2 1 inner 1620.1.w.c 8
15.e even 4 1 1620.1.w.a 8
20.e even 4 1 1620.1.w.a 8
36.f odd 6 1 1620.1.m.a 8
36.f odd 6 1 1620.1.w.a 8
36.h even 6 1 1620.1.m.a 8
36.h even 6 1 1620.1.w.a 8
45.k odd 12 1 1620.1.m.a 8
45.k odd 12 1 inner 1620.1.w.c 8
45.l even 12 1 1620.1.m.a 8
45.l even 12 1 inner 1620.1.w.c 8
60.l odd 4 1 1620.1.w.a 8
180.v odd 12 1 1620.1.m.a 8
180.v odd 12 1 inner 1620.1.w.c 8
180.x even 12 1 1620.1.m.a 8
180.x even 12 1 inner 1620.1.w.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.1.m.a 8 9.c even 3 1
1620.1.m.a 8 9.d odd 6 1
1620.1.m.a 8 36.f odd 6 1
1620.1.m.a 8 36.h even 6 1
1620.1.m.a 8 45.k odd 12 1
1620.1.m.a 8 45.l even 12 1
1620.1.m.a 8 180.v odd 12 1
1620.1.m.a 8 180.x even 12 1
1620.1.w.a 8 5.c odd 4 1
1620.1.w.a 8 9.c even 3 1
1620.1.w.a 8 9.d odd 6 1
1620.1.w.a 8 15.e even 4 1
1620.1.w.a 8 20.e even 4 1
1620.1.w.a 8 36.f odd 6 1
1620.1.w.a 8 36.h even 6 1
1620.1.w.a 8 60.l odd 4 1
1620.1.w.c 8 1.a even 1 1 trivial
1620.1.w.c 8 3.b odd 2 1 inner
1620.1.w.c 8 4.b odd 2 1 CM
1620.1.w.c 8 12.b even 2 1 inner
1620.1.w.c 8 45.k odd 12 1 inner
1620.1.w.c 8 45.l even 12 1 inner
1620.1.w.c 8 180.v odd 12 1 inner
1620.1.w.c 8 180.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} - 2 T_{13}^{3} + 5 T_{13}^{2} - 4 T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1620, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 1 - T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 1 - 4 T + 5 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$17$ \( ( 9 + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( 1 + 4 T^{2} + 15 T^{4} + 4 T^{6} + T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 1 + 2 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$41$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 1 - 2 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 1 - 4 T^{2} + T^{4} )^{2} \)
$97$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
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