Properties

Label 600.2.b.c
Level 600
Weight 2
Character orbit 600.b
Analytic conductor 4.791
Analytic rank 0
Dimension 4
CM disc. -15
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{2} \) \( + ( \beta_{1} - \beta_{2} ) q^{3} \) \( + \beta_{3} q^{4} \) \( + ( -2 + \beta_{3} ) q^{6} \) \( + ( 2 \beta_{1} - \beta_{2} ) q^{8} \) \( -3 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{2} \) \( + ( \beta_{1} - \beta_{2} ) q^{3} \) \( + \beta_{3} q^{4} \) \( + ( -2 + \beta_{3} ) q^{6} \) \( + ( 2 \beta_{1} - \beta_{2} ) q^{8} \) \( -3 q^{9} \) \( + ( 2 \beta_{1} + \beta_{2} ) q^{12} \) \( + ( -4 + \beta_{3} ) q^{16} \) \( + ( 4 \beta_{1} - 4 \beta_{2} ) q^{17} \) \( + 3 \beta_{2} q^{18} \) \( -4 q^{19} \) \( + ( -4 \beta_{1} - 4 \beta_{2} ) q^{23} \) \( + ( -4 - \beta_{3} ) q^{24} \) \( + ( -3 \beta_{1} + 3 \beta_{2} ) q^{27} \) \( + ( 2 - 4 \beta_{3} ) q^{31} \) \( + ( 2 \beta_{1} + 3 \beta_{2} ) q^{32} \) \( + ( -8 + 4 \beta_{3} ) q^{34} \) \( -3 \beta_{3} q^{36} \) \( + 4 \beta_{2} q^{38} \) \( + ( 8 + 4 \beta_{3} ) q^{46} \) \( + ( -4 \beta_{1} - 4 \beta_{2} ) q^{47} \) \( + ( -2 \beta_{1} + 5 \beta_{2} ) q^{48} \) \( + 7 q^{49} \) \( -12 q^{51} \) \( + ( -2 \beta_{1} - 2 \beta_{2} ) q^{53} \) \( + ( 6 - 3 \beta_{3} ) q^{54} \) \( + ( -4 \beta_{1} + 4 \beta_{2} ) q^{57} \) \( + ( -4 + 8 \beta_{3} ) q^{61} \) \( + ( -8 \beta_{1} + 2 \beta_{2} ) q^{62} \) \( + ( -4 - 3 \beta_{3} ) q^{64} \) \( + ( 8 \beta_{1} + 4 \beta_{2} ) q^{68} \) \( + ( -4 + 8 \beta_{3} ) q^{69} \) \( + ( -6 \beta_{1} + 3 \beta_{2} ) q^{72} \) \( -4 \beta_{3} q^{76} \) \( + ( 2 - 4 \beta_{3} ) q^{79} \) \( + 9 q^{81} \) \( + ( -2 \beta_{1} + 2 \beta_{2} ) q^{83} \) \( + ( 8 \beta_{1} - 12 \beta_{2} ) q^{92} \) \( + ( -6 \beta_{1} - 6 \beta_{2} ) q^{93} \) \( + ( 8 + 4 \beta_{3} ) q^{94} \) \( + ( 4 - 5 \beta_{3} ) q^{96} \) \( -7 \beta_{2} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 14q^{16} \) \(\mathstrut -\mathstrut 16q^{19} \) \(\mathstrut -\mathstrut 18q^{24} \) \(\mathstrut -\mathstrut 24q^{34} \) \(\mathstrut -\mathstrut 6q^{36} \) \(\mathstrut +\mathstrut 40q^{46} \) \(\mathstrut +\mathstrut 28q^{49} \) \(\mathstrut -\mathstrut 48q^{51} \) \(\mathstrut +\mathstrut 18q^{54} \) \(\mathstrut -\mathstrut 22q^{64} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 36q^{81} \) \(\mathstrut +\mathstrut 40q^{94} \) \(\mathstrut +\mathstrut 6q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(2\) \(x^{2}\mathstrut +\mathstrut \) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} + \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} + 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.309017 + 0.535233i
−0.309017 0.535233i
0.809017 1.40126i
0.809017 + 1.40126i
−1.11803 0.866025i 1.73205i 0.500000 + 1.93649i 0 −1.50000 + 1.93649i 0 1.11803 2.59808i −3.00000 0
251.2 −1.11803 + 0.866025i 1.73205i 0.500000 1.93649i 0 −1.50000 1.93649i 0 1.11803 + 2.59808i −3.00000 0
251.3 1.11803 0.866025i 1.73205i 0.500000 1.93649i 0 −1.50000 1.93649i 0 −1.11803 2.59808i −3.00000 0
251.4 1.11803 + 0.866025i 1.73205i 0.500000 + 1.93649i 0 −1.50000 + 1.93649i 0 −1.11803 + 2.59808i −3.00000 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
3.b Odd 1 yes
5.b Even 1 yes
8.d Odd 1 no
24.f Even 1 no
40.e Odd 1 no
120.m Even 1 no

Hecke kernels

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

\(T_{7} \)
\(T_{11} \)
\(T_{23}^{2} \) \(\mathstrut -\mathstrut 80 \)
\(T_{43} \)