Properties

Label 3600.1.cc.a
Level 3600
Weight 1
Character orbit 3600.cc
Analytic conductor 1.797
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM disc. -20
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3600.cc (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.10497600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{6} q^{3} \) \( + ( -1 - \zeta_{6} ) q^{7} \) \( + \zeta_{6}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{6} q^{3} \) \( + ( -1 - \zeta_{6} ) q^{7} \) \( + \zeta_{6}^{2} q^{9} \) \( + ( \zeta_{6} + \zeta_{6}^{2} ) q^{21} \) \( + ( -1 + \zeta_{6}^{2} ) q^{23} \) \(+ q^{27}\) \( + \zeta_{6}^{2} q^{29} \) \( -\zeta_{6} q^{41} \) \( + ( 1 + \zeta_{6} ) q^{47} \) \( + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{49} \) \( + \zeta_{6}^{2} q^{61} \) \( + ( 1 - \zeta_{6}^{2} ) q^{63} \) \( + ( -1 + \zeta_{6}^{2} ) q^{67} \) \( + ( 1 + \zeta_{6} ) q^{69} \) \( -\zeta_{6} q^{81} \) \( + ( 1 + \zeta_{6} ) q^{83} \) \(+ q^{87}\) \(+ q^{89}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut +\mathstrut 3q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut q^{61} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut -\mathstrut 3q^{67} \) \(\mathstrut +\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut +\mathstrut 3q^{83} \) \(\mathstrut +\mathstrut 2q^{87} \) \(\mathstrut +\mathstrut 2q^{89} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0 0 0 −1.50000 + 0.866025i 0 −0.500000 0.866025i 0
1951.1 0 −0.500000 0.866025i 0 0 0 −1.50000 0.866025i 0 −0.500000 + 0.866025i 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
20.d Odd 1 CM by \(\Q(\sqrt{-5}) \) yes
36.f Odd 1 yes
45.j Even 1 yes

Hecke kernels

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