Properties

Label 600.2.m.b
Level 600
Weight 2
Character orbit 600.m
Analytic conductor 4.791
Analytic rank 0
Dimension 8
CM disc. -8
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{2} \) \( -\beta_{3} q^{3} \) \( + 2 q^{4} \) \( + ( -1 + \beta_{5} ) q^{6} \) \( -2 \beta_{2} q^{8} \) \( + ( -\beta_{5} - \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{2} \) \( -\beta_{3} q^{3} \) \( + 2 q^{4} \) \( + ( -1 + \beta_{5} ) q^{6} \) \( -2 \beta_{2} q^{8} \) \( + ( -\beta_{5} - \beta_{6} ) q^{9} \) \( + ( 2 \beta_{5} + \beta_{7} ) q^{11} \) \( -2 \beta_{3} q^{12} \) \( + 4 q^{16} \) \( + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{18} \) \( + ( 2 - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{19} \) \( + ( -3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{22} \) \( + ( -2 + 2 \beta_{5} ) q^{24} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{27} \) \( -4 \beta_{2} q^{32} \) \( + ( \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{33} \) \( + ( 3 + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{34} \) \( + ( -2 \beta_{5} - 2 \beta_{6} ) q^{36} \) \( + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{38} \) \( + ( -1 - 2 \beta_{5} + 2 \beta_{6} ) q^{41} \) \( + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{43} \) \( + ( 4 \beta_{5} + 2 \beta_{7} ) q^{44} \) \( -4 \beta_{3} q^{48} \) \( -7 q^{49} \) \( + ( 4 + 2 \beta_{5} - 3 \beta_{6} ) q^{51} \) \( + ( -3 + 2 \beta_{6} + \beta_{7} ) q^{54} \) \( + ( -3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{57} \) \( + ( 2 - 4 \beta_{6} - 2 \beta_{7} ) q^{59} \) \( + 8 q^{64} \) \( + ( -7 - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{66} \) \( + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{67} \) \( + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{68} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{72} \) \( + ( -\beta_{1} + 6 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} ) q^{73} \) \( + ( 4 - 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{76} \) \( + ( 3 - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{81} \) \( + ( -2 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{82} \) \( + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{83} \) \( + ( -2 + 4 \beta_{6} + 2 \beta_{7} ) q^{86} \) \( + ( -6 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} ) q^{88} \) \( + ( -1 - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{89} \) \( + ( -4 + 4 \beta_{5} ) q^{96} \) \( + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{97} \) \( + 7 \beta_{2} q^{98} \) \( + ( 6 - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 32q^{16} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 16q^{24} \) \(\mathstrut +\mathstrut 32q^{34} \) \(\mathstrut -\mathstrut 8q^{36} \) \(\mathstrut -\mathstrut 56q^{49} \) \(\mathstrut +\mathstrut 20q^{51} \) \(\mathstrut -\mathstrut 16q^{54} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut -\mathstrut 64q^{66} \) \(\mathstrut +\mathstrut 16q^{76} \) \(\mathstrut +\mathstrut 28q^{81} \) \(\mathstrut -\mathstrut 32q^{96} \) \(\mathstrut +\mathstrut 52q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \zeta_{24}^{6} + 3 \zeta_{24}^{2} \)
\(\beta_{2}\)\(=\)\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
\(\beta_{3}\)\(=\)\( -\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} - \zeta_{24} \)
\(\beta_{4}\)\(=\)\( \zeta_{24}^{7} - 3 \zeta_{24}^{6} + 2 \zeta_{24}^{2} + \zeta_{24} \)
\(\beta_{5}\)\(=\)\( -\zeta_{24}^{7} - 2 \zeta_{24}^{4} + \zeta_{24} + 1 \)
\(\beta_{6}\)\(=\)\( -\zeta_{24}^{7} + 2 \zeta_{24}^{5} + \zeta_{24}^{4} + 2 \zeta_{24}^{3} - \zeta_{24} \)
\(\beta_{7}\)\(=\)\( 2 \zeta_{24}^{7} + \zeta_{24}^{5} - 2 \zeta_{24}^{4} + \zeta_{24}^{3} - 3 \zeta_{24} + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{24}\)\(=\)\((\)\(-\)\(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/30\)
\(\zeta_{24}^{2}\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)\()/15\)
\(\zeta_{24}^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut -\mathstrut \) \(1\)\()/10\)
\(\zeta_{24}^{4}\)\(=\)\((\)\(-\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\)\()/15\)
\(\zeta_{24}^{5}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)\()/30\)
\(\zeta_{24}^{6}\)\(=\)\((\)\(-\)\(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/5\)
\(\zeta_{24}^{7}\)\(=\)\((\)\(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/30\)

Character Values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
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
0.258819 + 0.965926i
−1.41421 −0.158919 1.72474i 2.00000 0 0.224745 + 2.43916i 0 −2.82843 −2.94949 + 0.548188i 0
299.2 −1.41421 −0.158919 + 1.72474i 2.00000 0 0.224745 2.43916i 0 −2.82843 −2.94949 0.548188i 0
299.3 −1.41421 1.57313 0.724745i 2.00000 0 −2.22474 + 1.02494i 0 −2.82843 1.94949 2.28024i 0
299.4 −1.41421 1.57313 + 0.724745i 2.00000 0 −2.22474 1.02494i 0 −2.82843 1.94949 + 2.28024i 0
299.5 1.41421 −1.57313 0.724745i 2.00000 0 −2.22474 1.02494i 0 2.82843 1.94949 + 2.28024i 0
299.6 1.41421 −1.57313 + 0.724745i 2.00000 0 −2.22474 + 1.02494i 0 2.82843 1.94949 2.28024i 0
299.7 1.41421 0.158919 1.72474i 2.00000 0 0.224745 2.43916i 0 2.82843 −2.94949 0.548188i 0
299.8 1.41421 0.158919 + 1.72474i 2.00000 0 0.224745 + 2.43916i 0 2.82843 −2.94949 + 0.548188i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.8
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
15.d Odd 1 yes
24.f Even 1 yes
40.e Odd 1 yes
120.m Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\(T_{7} \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 58 T_{11}^{2} \) \(\mathstrut +\mathstrut 625 \)
\(T_{29} \)