Properties

Label 3600.1.bb.a
Level 3600
Weight 1
Character orbit 3600.bb
Analytic conductor 1.797
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -15
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3600.bb (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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.153600.2

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -i q^{2} \) \(- q^{4}\) \( + i q^{8} \) \(+O(q^{10})\) \( q\) \( -i q^{2} \) \(- q^{4}\) \( + i q^{8} \) \(+ q^{16}\) \( + ( -1 + i ) q^{17} \) \( + ( -1 + i ) q^{19} \) \( + ( -1 + i ) q^{23} \) \( -i q^{32} \) \( + ( 1 + i ) q^{34} \) \( + ( 1 + i ) q^{38} \) \( + ( 1 + i ) q^{46} \) \( + ( -1 + i ) q^{47} \) \( -i q^{49} \) \( -2 q^{53} \) \( + ( 1 + i ) q^{61} \) \(- q^{64}\) \( + ( 1 - i ) q^{68} \) \( + ( 1 - i ) q^{76} \) \( + 2 i q^{79} \) \( -2 i q^{83} \) \( + ( 1 - i ) q^{92} \) \( + ( 1 + i ) q^{94} \) \(- q^{98}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 2q^{68} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut +\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut 2q^{94} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 0 0
1693.1 1.00000i 0 −1.00000 0 0 0 1.00000i 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
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
80.i Odd 1 yes
240.bf Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{17}^{2} \) \(\mathstrut +\mathstrut 2 T_{17} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).