Properties

Label 3600.1.cj.a
Level 3600
Weight 1
Character orbit 3600.cj
Analytic conductor 1.797
Analytic rank 0
Dimension 8
Projective image \(D_{10}\)
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.cj (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{10}\)
Projective field Galois closure of 10.0.9492187500000000.9

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{20}^{9} q^{5} \) \(+O(q^{10})\) \( q\) \( + \zeta_{20}^{9} q^{5} \) \( + ( \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{13} \) \( + ( -\zeta_{20} - \zeta_{20}^{7} ) q^{17} \) \( -\zeta_{20}^{8} q^{25} \) \( + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{29} \) \( + ( 1 + \zeta_{20}^{4} ) q^{37} \) \( + ( \zeta_{20} + \zeta_{20}^{3} ) q^{41} \) \(+ q^{49}\) \( + ( \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{53} \) \( + ( -\zeta_{20}^{2} + \zeta_{20}^{4} ) q^{61} \) \( + ( -\zeta_{20}^{5} + \zeta_{20}^{7} ) q^{65} \) \( + ( \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{73} \) \( + ( 1 + \zeta_{20}^{6} ) q^{85} \) \( + ( \zeta_{20} - \zeta_{20}^{5} ) q^{89} \) \( + ( -\zeta_{20}^{4} - \zeta_{20}^{8} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 10q^{85} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-\zeta_{20}^{6}\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.587785 0.809017i
−0.587785 + 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0 0 0 −0.587785 0.809017i 0 0 0 0 0
271.2 0 0 0 0.587785 + 0.809017i 0 0 0 0 0
991.1 0 0 0 −0.951057 0.309017i 0 0 0 0 0
991.2 0 0 0 0.951057 + 0.309017i 0 0 0 0 0
1711.1 0 0 0 −0.951057 + 0.309017i 0 0 0 0 0
1711.2 0 0 0 0.951057 0.309017i 0 0 0 0 0
2431.1 0 0 0 −0.587785 + 0.809017i 0 0 0 0 0
2431.2 0 0 0 0.587785 0.809017i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.2
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.d Even 1 yes
75.j Odd 1 yes
100.j Odd 1 yes
300.n Even 1 yes

Hecke kernels

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