Properties

Label 600.1.bj.b
Level 600
Weight 1
Character orbit 600.bj
Analytic conductor 0.299
Analytic rank 0
Dimension 4
Projective image \(D_{5}\)
CM disc. -24
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 600.bj (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.29943900758\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.225000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{10} q^{2} \) \( -\zeta_{10}^{2} q^{3} \) \( + \zeta_{10}^{2} q^{4} \) \( + \zeta_{10} q^{5} \) \( -\zeta_{10}^{3} q^{6} \) \( + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7} \) \( + \zeta_{10}^{3} q^{8} \) \( + \zeta_{10}^{4} q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{10} q^{2} \) \( -\zeta_{10}^{2} q^{3} \) \( + \zeta_{10}^{2} q^{4} \) \( + \zeta_{10} q^{5} \) \( -\zeta_{10}^{3} q^{6} \) \( + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7} \) \( + \zeta_{10}^{3} q^{8} \) \( + \zeta_{10}^{4} q^{9} \) \( + \zeta_{10}^{2} q^{10} \) \( + ( -1 - \zeta_{10}^{2} ) q^{11} \) \( -\zeta_{10}^{4} q^{12} \) \( + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{14} \) \( -\zeta_{10}^{3} q^{15} \) \( + \zeta_{10}^{4} q^{16} \) \(- q^{18}\) \( + \zeta_{10}^{3} q^{20} \) \( + ( -1 - \zeta_{10}^{4} ) q^{21} \) \( + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{22} \) \(+ q^{24}\) \( + \zeta_{10}^{2} q^{25} \) \( + \zeta_{10} q^{27} \) \( + ( 1 + \zeta_{10}^{4} ) q^{28} \) \( -2 \zeta_{10}^{2} q^{29} \) \( -\zeta_{10}^{4} q^{30} \) \( + ( 1 - \zeta_{10} ) q^{31} \) \(- q^{32}\) \( + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{33} \) \( + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{35} \) \( -\zeta_{10} q^{36} \) \( + \zeta_{10}^{4} q^{40} \) \( + ( 1 - \zeta_{10} ) q^{42} \) \( + ( -\zeta_{10}^{2} - \zeta_{10}^{4} ) q^{44} \) \(- q^{45}\) \( + \zeta_{10} q^{48} \) \( + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{49} \) \( + \zeta_{10}^{3} q^{50} \) \( + ( \zeta_{10} + \zeta_{10}^{3} ) q^{53} \) \( + \zeta_{10}^{2} q^{54} \) \( + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{55} \) \( + ( -1 + \zeta_{10} ) q^{56} \) \( -2 \zeta_{10}^{3} q^{58} \) \( + ( -1 + \zeta_{10}^{3} ) q^{59} \) \(+ q^{60}\) \( + ( \zeta_{10} - \zeta_{10}^{2} ) q^{62} \) \( + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{63} \) \( -\zeta_{10} q^{64} \) \( + ( -1 + \zeta_{10}^{3} ) q^{66} \) \( + ( 1 + \zeta_{10}^{4} ) q^{70} \) \( -\zeta_{10}^{2} q^{72} \) \( -2 \zeta_{10} q^{73} \) \( -\zeta_{10}^{4} q^{75} \) \( + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{77} \) \( + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{79} \) \(- q^{80}\) \( -\zeta_{10}^{3} q^{81} \) \( + ( -1 + \zeta_{10} ) q^{83} \) \( + ( \zeta_{10} - \zeta_{10}^{2} ) q^{84} \) \( + 2 \zeta_{10}^{4} q^{87} \) \( + ( 1 - \zeta_{10}^{3} ) q^{88} \) \( -\zeta_{10} q^{90} \) \( + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{93} \) \( + \zeta_{10}^{2} q^{96} \) \( + ( 1 + \zeta_{10}^{4} ) q^{97} \) \( + ( -1 + \zeta_{10} - \zeta_{10}^{2} ) q^{98} \) \( + ( \zeta_{10} - \zeta_{10}^{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut q^{12} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut -\mathstrut q^{15} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 4q^{18} \) \(\mathstrut +\mathstrut q^{20} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 4q^{24} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut +\mathstrut q^{27} \) \(\mathstrut +\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 4q^{32} \) \(\mathstrut -\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut -\mathstrut q^{36} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut +\mathstrut 3q^{42} \) \(\mathstrut +\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut q^{48} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut +\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut -\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 3q^{56} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut +\mathstrut 2q^{62} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut q^{64} \) \(\mathstrut -\mathstrut 3q^{66} \) \(\mathstrut +\mathstrut 3q^{70} \) \(\mathstrut +\mathstrut q^{72} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut q^{75} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut -\mathstrut 4q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 3q^{83} \) \(\mathstrut +\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 2q^{87} \) \(\mathstrut +\mathstrut 3q^{88} \) \(\mathstrut -\mathstrut q^{90} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut q^{96} \) \(\mathstrut +\mathstrut 3q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.61803 0.809017 0.587785i 0.309017 + 0.951057i −0.809017 0.587785i
341.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.309017 0.951057i 0.618034 −0.309017 + 0.951057i −0.809017 + 0.587785i 0.309017 + 0.951057i
461.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.809017 0.587785i 0.309017 + 0.951057i 0.618034 −0.309017 0.951057i −0.809017 0.587785i 0.309017 0.951057i
581.1 −0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i −0.309017 0.951057i −0.809017 0.587785i −1.61803 0.809017 + 0.587785i 0.309017 0.951057i −0.809017 + 0.587785i
\(n\): 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
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
25.d Even 1 yes
600.bj Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{11}^{4} \) \(\mathstrut +\mathstrut 3 T_{11}^{3} \) \(\mathstrut +\mathstrut 4 T_{11}^{2} \) \(\mathstrut +\mathstrut 2 T_{11} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(600, [\chi])\).