Properties

Label 600.2.m.a
Level 600
Weight 2
Character orbit 600.m
Analytic conductor 4.791
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \zeta_{8}^{2} \)
\(\beta_{2}\)\(=\)\( \zeta_{8}^{3} + \zeta_{8} \)
\(\beta_{3}\)\(=\)\( -\zeta_{8}^{3} + \zeta_{8} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{8}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\zeta_{8}^{2}\)\(=\)\(\beta_{1}\)
\(\zeta_{8}^{3}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−1.41421 −1.41421 1.00000i 2.00000 0 2.00000 + 1.41421i 0 −2.82843 1.00000 + 2.82843i 0
299.2 −1.41421 −1.41421 + 1.00000i 2.00000 0 2.00000 1.41421i 0 −2.82843 1.00000 2.82843i 0
299.3 1.41421 1.41421 1.00000i 2.00000 0 2.00000 1.41421i 0 2.82843 1.00000 2.82843i 0
299.4 1.41421 1.41421 + 1.00000i 2.00000 0 2.00000 + 1.41421i 0 2.82843 1.00000 + 2.82843i 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
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 \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\(T_{7} \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 8 \)
\(T_{29} \)