Properties

Label 600.1.q.b
Level 600
Weight 1
Character orbit 600.q
Analytic conductor 0.299
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM disc. -8
Inner twists 16

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Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 600.q (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.29943900758\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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 6.0.5400000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{24}^{3} q^{2} \) \( -\zeta_{24} q^{3} \) \( + \zeta_{24}^{6} q^{4} \) \( -\zeta_{24}^{4} q^{6} \) \( + \zeta_{24}^{9} q^{8} \) \( + \zeta_{24}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{24}^{3} q^{2} \) \( -\zeta_{24} q^{3} \) \( + \zeta_{24}^{6} q^{4} \) \( -\zeta_{24}^{4} q^{6} \) \( + \zeta_{24}^{9} q^{8} \) \( + \zeta_{24}^{2} q^{9} \) \( + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{11} \) \( -\zeta_{24}^{7} q^{12} \) \(- q^{16}\) \( + \zeta_{24}^{3} q^{17} \) \( + \zeta_{24}^{5} q^{18} \) \( -\zeta_{24}^{6} q^{19} \) \( + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{22} \) \( -\zeta_{24}^{10} q^{24} \) \( -\zeta_{24}^{3} q^{27} \) \( -\zeta_{24}^{3} q^{32} \) \( + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{33} \) \( + \zeta_{24}^{6} q^{34} \) \( + \zeta_{24}^{8} q^{36} \) \( -\zeta_{24}^{9} q^{38} \) \( + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{41} \) \( + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{44} \) \( + \zeta_{24} q^{48} \) \( -\zeta_{24}^{6} q^{49} \) \( -\zeta_{24}^{4} q^{51} \) \( -\zeta_{24}^{6} q^{54} \) \( + \zeta_{24}^{7} q^{57} \) \( -\zeta_{24}^{6} q^{64} \) \( + ( 1 - \zeta_{24}^{8} ) q^{66} \) \( + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{67} \) \( + \zeta_{24}^{9} q^{68} \) \( + \zeta_{24}^{11} q^{72} \) \( + ( \zeta_{24} + \zeta_{24}^{5} ) q^{73} \) \(+ q^{76}\) \( + \zeta_{24}^{4} q^{81} \) \( + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{82} \) \( -\zeta_{24}^{9} q^{83} \) \( + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{88} \) \( + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{89} \) \( + \zeta_{24}^{4} q^{96} \) \( -\zeta_{24}^{9} q^{98} \) \( + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 12q^{66} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut +\mathstrut 4q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 0 −0.500000 0.866025i 0 0.707107 + 0.707107i −0.866025 0.500000i 0
107.2 −0.707107 + 0.707107i 0.965926 0.258819i 1.00000i 0 −0.500000 + 0.866025i 0 0.707107 + 0.707107i 0.866025 0.500000i 0
107.3 0.707107 0.707107i −0.965926 + 0.258819i 1.00000i 0 −0.500000 + 0.866025i 0 −0.707107 0.707107i 0.866025 0.500000i 0
107.4 0.707107 0.707107i 0.258819 0.965926i 1.00000i 0 −0.500000 0.866025i 0 −0.707107 0.707107i −0.866025 0.500000i 0
443.1 −0.707107 0.707107i −0.258819 0.965926i 1.00000i 0 −0.500000 + 0.866025i 0 0.707107 0.707107i −0.866025 + 0.500000i 0
443.2 −0.707107 0.707107i 0.965926 + 0.258819i 1.00000i 0 −0.500000 0.866025i 0 0.707107 0.707107i 0.866025 + 0.500000i 0
443.3 0.707107 + 0.707107i −0.965926 0.258819i 1.00000i 0 −0.500000 0.866025i 0 −0.707107 + 0.707107i 0.866025 + 0.500000i 0
443.4 0.707107 + 0.707107i 0.258819 + 0.965926i 1.00000i 0 −0.500000 + 0.866025i 0 −0.707107 + 0.707107i −0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
3.b Odd 1 yes
5.b Even 1 yes
5.c Odd 2 yes
15.d Odd 1 yes
15.e Even 2 yes
24.f Even 1 yes
40.e Odd 1 yes
40.k Even 2 yes
120.m Even 1 yes
120.q Odd 2 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{11}^{2} \) \(\mathstrut +\mathstrut 3 \) acting on \(S_{1}^{\mathrm{new}}(600, [\chi])\).