Properties

Label 3600.1.bk.a
Level 3600
Weight 1
Character orbit 3600.bk
Analytic conductor 1.797
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
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.bk (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: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.13500.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(+O(q^{10})\) \( q\) \( + ( -1 - \zeta_{8}^{2} ) q^{13} \) \( -2 \zeta_{8} q^{17} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} \) \( + ( 1 - \zeta_{8}^{2} ) q^{37} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} \) \( -\zeta_{8}^{2} q^{49} \) \( + 2 \zeta_{8}^{3} q^{53} \) \( + ( 1 + \zeta_{8}^{2} ) q^{73} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{89} \) \( + ( 1 - \zeta_{8}^{2} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 0 0 0 0 0 0
143.2 0 0 0 0 0 0 0 0 0
1007.1 0 0 0 0 0 0 0 0 0
1007.2 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
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
3.b Odd 1 yes
5.c Odd 1 yes
12.b Even 1 yes
15.e Even 1 yes
20.e Even 1 yes
60.l Odd 1 yes

Hecke kernels

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