Properties

Label 3600.1.j.b
Level 3600
Weight 1
Character orbit 3600.j
Analytic conductor 1.797
Analytic rank 0
Dimension 4
Projective image \(D_{6}\)
CM disc. -3
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.j (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.1440000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{7} \) \(+O(q^{10})\) \( q\) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{7} \) \( -\zeta_{12}^{3} q^{13} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{19} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{31} \) \( + 2 \zeta_{12}^{3} q^{37} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{43} \) \( + ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{49} \) \(- q^{61}\) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{67} \) \( + 2 \zeta_{12}^{3} q^{73} \) \( + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{91} \) \( + \zeta_{12}^{3} q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 0 0 −1.73205 0 0 0
1999.2 0 0 0 0 0 −1.73205 0 0 0
1999.3 0 0 0 0 0 1.73205 0 0 0
1999.4 0 0 0 0 0 1.73205 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
4.b Odd 1 yes
5.b Even 1 yes
12.b Even 1 yes
15.d Odd 1 yes
20.d Odd 1 yes
60.h Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{2} \) \(\mathstrut -\mathstrut 3 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).