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 disc. -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 \(20\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
3.b Odd 1 yes
12.b Even 1 yes
25.f Odd 1 yes
75.l Even 1 yes
100.l Even 1 yes
300.u Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(3600, [\chi])\).