Properties

Label 3600.1.dx.a
Level 3600
Weight 1
Character orbit 3600.dx
Analytic conductor 1.797
Analytic rank 0
Dimension 16
Projective image \(D_{20}\)
CM discriminant -4
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.dx (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{20}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{40}^{7} q^{5} +O(q^{10})\) \( q -\zeta_{40}^{7} q^{5} + ( -\zeta_{40}^{4} + \zeta_{40}^{18} ) q^{13} + ( \zeta_{40} - \zeta_{40}^{13} ) q^{17} + \zeta_{40}^{14} q^{25} + ( -\zeta_{40}^{7} - \zeta_{40}^{9} ) q^{29} + ( \zeta_{40}^{10} + \zeta_{40}^{12} ) q^{37} + ( -\zeta_{40}^{3} - \zeta_{40}^{9} ) q^{41} -\zeta_{40}^{10} q^{49} + ( -\zeta_{40}^{11} + \zeta_{40}^{15} ) q^{53} + ( -\zeta_{40}^{2} - \zeta_{40}^{6} ) q^{61} + ( \zeta_{40}^{5} + \zeta_{40}^{11} ) q^{65} + ( -\zeta_{40}^{2} - \zeta_{40}^{16} ) q^{73} + ( -1 - \zeta_{40}^{8} ) q^{85} + ( -\zeta_{40}^{13} + \zeta_{40}^{15} ) q^{89} + ( \zeta_{40}^{12} - \zeta_{40}^{14} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 4q^{13} + 4q^{37} + 4q^{73} - 12q^{85} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-\zeta_{40}^{18}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.156434 + 0.987688i
0.156434 0.987688i
0.987688 + 0.156434i
−0.987688 0.156434i
−0.891007 + 0.453990i
0.891007 0.453990i
0.987688 0.156434i
−0.987688 + 0.156434i
0.453990 0.891007i
−0.453990 + 0.891007i
−0.891007 0.453990i
0.891007 + 0.453990i
−0.156434 0.987688i
0.156434 + 0.987688i
0.453990 + 0.891007i
−0.453990 0.891007i
0 0 0 −0.891007 + 0.453990i 0 0 0 0 0
287.2 0 0 0 0.891007 0.453990i 0 0 0 0 0
863.1 0 0 0 −0.453990 0.891007i 0 0 0 0 0
863.2 0 0 0 0.453990 + 0.891007i 0 0 0 0 0
1583.1 0 0 0 −0.987688 + 0.156434i 0 0 0 0 0
1583.2 0 0 0 0.987688 0.156434i 0 0 0 0 0
1727.1 0 0 0 −0.453990 + 0.891007i 0 0 0 0 0
1727.2 0 0 0 0.453990 0.891007i 0 0 0 0 0
2303.1 0 0 0 −0.156434 + 0.987688i 0 0 0 0 0
2303.2 0 0 0 0.156434 0.987688i 0 0 0 0 0
2447.1 0 0 0 −0.987688 0.156434i 0 0 0 0 0
2447.2 0 0 0 0.987688 + 0.156434i 0 0 0 0 0
3023.1 0 0 0 −0.891007 0.453990i 0 0 0 0 0
3023.2 0 0 0 0.891007 + 0.453990i 0 0 0 0 0
3167.1 0 0 0 −0.156434 0.987688i 0 0 0 0 0
3167.2 0 0 0 0.156434 + 0.987688i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3167.2
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
25.f odd 20 1 inner
75.l even 20 1 inner
100.l even 20 1 inner
300.u odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.dx.a 16
3.b odd 2 1 inner 3600.1.dx.a 16
4.b odd 2 1 CM 3600.1.dx.a 16
12.b even 2 1 inner 3600.1.dx.a 16
25.f odd 20 1 inner 3600.1.dx.a 16
75.l even 20 1 inner 3600.1.dx.a 16
100.l even 20 1 inner 3600.1.dx.a 16
300.u odd 20 1 inner 3600.1.dx.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.dx.a 16 1.a even 1 1 trivial
3600.1.dx.a 16 3.b odd 2 1 inner
3600.1.dx.a 16 4.b odd 2 1 CM
3600.1.dx.a 16 12.b even 2 1 inner
3600.1.dx.a 16 25.f odd 20 1 inner
3600.1.dx.a 16 75.l even 20 1 inner
3600.1.dx.a 16 100.l even 20 1 inner
3600.1.dx.a 16 300.u odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$7$ \( ( 1 + T^{4} )^{8} \)
$11$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$17$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$19$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$23$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$29$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$37$ \( ( 1 + T^{2} )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$41$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$43$ \( ( 1 + T^{4} )^{8} \)
$47$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$53$ \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$61$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$67$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$71$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$73$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$79$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$83$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$89$ \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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