Properties

Label 600.2.b.d
Level 600
Weight 2
Character orbit 600.b
Analytic conductor 4.791
Analytic rank 0
Dimension 4
CM disc. -8
Inner twists 4

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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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{3} \) \( -2 q^{4} \) \( + ( -1 + \beta_{1} + \beta_{3} ) q^{6} \) \( -2 \beta_{1} q^{8} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{3} \) \( -2 q^{4} \) \( + ( -1 + \beta_{1} + \beta_{3} ) q^{6} \) \( -2 \beta_{1} q^{8} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{11} \) \( + ( -2 - 2 \beta_{2} ) q^{12} \) \( + 4 q^{16} \) \( + ( -1 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{17} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{18} \) \( + ( 1 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{19} \) \( + ( \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{22} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{24} \) \( + ( 5 - \beta_{1} ) q^{27} \) \( + 4 \beta_{1} q^{32} \) \( + ( 4 + 5 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{33} \) \( + ( -6 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{34} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{36} \) \( + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{38} \) \( + ( 1 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41} \) \( -10 q^{43} \) \( + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{44} \) \( + ( 4 + 4 \beta_{2} ) q^{48} \) \( + 7 q^{49} \) \( + ( 1 + 8 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{51} \) \( + ( 2 + 5 \beta_{1} ) q^{54} \) \( + ( 8 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{57} \) \( -10 \beta_{1} q^{59} \) \( -8 q^{64} \) \( + ( -7 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{66} \) \( + ( -5 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} \) \( + ( 2 - 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{68} \) \( + ( -2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{72} \) \( + ( -5 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{73} \) \( + ( -2 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{76} \) \( + ( 6 - \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{81} \) \( + ( 10 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{82} \) \( + ( 3 - 7 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{83} \) \( -10 \beta_{1} q^{86} \) \( + ( -2 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{88} \) \( + ( -3 + 8 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{89} \) \( + ( -4 + 4 \beta_{1} + 4 \beta_{3} ) q^{96} \) \( -10 q^{97} \) \( + 7 \beta_{1} q^{98} \) \( + ( -4 + 7 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 8q^{22} \) \(\mathstrut +\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 20q^{27} \) \(\mathstrut +\mathstrut 22q^{33} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 40q^{43} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut +\mathstrut 28q^{49} \) \(\mathstrut +\mathstrut 10q^{51} \) \(\mathstrut +\mathstrut 8q^{54} \) \(\mathstrut +\mathstrut 34q^{57} \) \(\mathstrut -\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 32q^{66} \) \(\mathstrut -\mathstrut 28q^{67} \) \(\mathstrut -\mathstrut 16q^{72} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut +\mathstrut 32q^{82} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut -\mathstrut 16q^{96} \) \(\mathstrut -\mathstrut 40q^{97} \) \(\mathstrut -\mathstrut 26q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

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

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
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
1.41421i −0.724745 1.57313i −2.00000 0 −2.22474 + 1.02494i 0 2.82843i −1.94949 + 2.28024i 0
251.2 1.41421i 1.72474 + 0.158919i −2.00000 0 0.224745 2.43916i 0 2.82843i 2.94949 + 0.548188i 0
251.3 1.41421i −0.724745 + 1.57313i −2.00000 0 −2.22474 1.02494i 0 2.82843i −1.94949 2.28024i 0
251.4 1.41421i 1.72474 0.158919i −2.00000 0 0.224745 + 2.43916i 0 2.82843i 2.94949 0.548188i 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
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
3.b Odd 1 yes
24.f Even 1 yes

Hecke kernels

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

\(T_{7} \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 58 T_{11}^{2} \) \(\mathstrut +\mathstrut 625 \)
\(T_{23} \)
\(T_{43} \) \(\mathstrut +\mathstrut 10 \)