Properties

Label 3600.1.da.a
Level 3600
Weight 1
Character orbit 3600.da
Analytic conductor 1.797
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM disc. -20
Inner twists 16

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Newspace parameters

Level: \( N \) = \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3600.da (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.7873200.3

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{24}^{7} q^{3} \) \( + \zeta_{24}^{5} q^{7} \) \( -\zeta_{24}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{24}^{7} q^{3} \) \( + \zeta_{24}^{5} q^{7} \) \( -\zeta_{24}^{2} q^{9} \) \(+ q^{21}\) \( + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{23} \) \( + \zeta_{24}^{9} q^{27} \) \( + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{29} \) \( + ( 1 - \zeta_{24}^{8} ) q^{41} \) \( + 2 \zeta_{24}^{11} q^{43} \) \( + ( \zeta_{24} - \zeta_{24}^{9} ) q^{47} \) \( -\zeta_{24}^{8} q^{61} \) \( -\zeta_{24}^{7} q^{63} \) \( + \zeta_{24} q^{67} \) \( + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{69} \) \( + \zeta_{24}^{4} q^{81} \) \( + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{83} \) \( + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{87} \) \( + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{89} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(\zeta_{24}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0 −0.965926 0.258819i 0 0 0 −0.965926 + 0.258819i 0 0.866025 + 0.500000i 0
1343.2 0 0.965926 + 0.258819i 0 0 0 0.965926 0.258819i 0 0.866025 + 0.500000i 0
2207.1 0 −0.258819 + 0.965926i 0 0 0 −0.258819 0.965926i 0 −0.866025 0.500000i 0
2207.2 0 0.258819 0.965926i 0 0 0 0.258819 + 0.965926i 0 −0.866025 0.500000i 0
2543.1 0 −0.258819 0.965926i 0 0 0 −0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0
2543.2 0 0.258819 + 0.965926i 0 0 0 0.258819 0.965926i 0 −0.866025 + 0.500000i 0
3407.1 0 −0.965926 + 0.258819i 0 0 0 −0.965926 0.258819i 0 0.866025 0.500000i 0
3407.2 0 0.965926 0.258819i 0 0 0 0.965926 + 0.258819i 0 0.866025 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3407.2
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
4.b Odd 1 yes
5.b Even 1 yes
5.c Odd 2 yes
9.d Odd 1 yes
20.e Even 2 yes
36.h Even 1 yes
45.h Odd 1 yes
45.l Even 2 yes
180.n Even 1 yes
180.v Odd 2 yes

Hecke kernels

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