Properties

Label 3600.1.bh.b
Level 3600
Weight 1
Character orbit 3600.bh
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.bh (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.450000.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{7} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{7} \) \( + \beta_{3} q^{13} \) \( + \beta_{2} q^{19} \) \(- q^{31}\) \( + \beta_{3} q^{43} \) \( + 2 \beta_{2} q^{49} \) \(- q^{61}\) \( + \beta_{1} q^{67} \) \( + 2 \beta_{2} q^{79} \) \( + 3 q^{91} \) \( -\beta_{1} q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 12q^{91} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{3}\)

Character Values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2593.1
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
0 0 0 0 0 −1.22474 + 1.22474i 0 0 0
2593.2 0 0 0 0 0 1.22474 1.22474i 0 0 0
3457.1 0 0 0 0 0 −1.22474 1.22474i 0 0 0
3457.2 0 0 0 0 0 1.22474 + 1.22474i 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
5.b Even 1 yes
5.c Odd 2 yes
15.d Odd 1 yes
15.e Even 2 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{4} \) \(\mathstrut +\mathstrut 9 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).