Properties

Label 3600.1.bh.a
Level 3600
Weight 1
Character orbit 3600.bh
Analytic conductor 1.797
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM disc. -3, -15, 5
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: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{5})\)
Artin image size \(16\)
Artin image $OD_{16}$
Artin field Galois closure of 8.4.14580000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+O(q^{10})\) \( q\) \( -2 i q^{19} \) \( + 2 q^{31} \) \( -i q^{49} \) \( + 2 q^{61} \) \( + 2 i q^{79} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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
1.00000i
1.00000i
0 0 0 0 0 0 0 0 0
3457.1 0 0 0 0 0 0 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 RM by \(\Q(\sqrt{5}) \) yes
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
5.c Odd 2 yes
15.e Even 2 yes

Hecke kernels

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