Properties

Label 600.1.z.a
Level 600
Weight 1
Character orbit 600.z
Analytic conductor 0.299
Analytic rank 0
Dimension 8
Projective image \(D_{10}\)
CM disc. -24
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.29943900758\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{10}\)
Projective field Galois closure of 10.0.759375000000000000.15

$q$-expansion

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

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

Character Values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{20}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.951057 + 0.309017i
−0.951057 0.309017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.951057 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0.951057 + 0.309017i 0.809017 0.587785i 1.17557i −0.587785 0.809017i −0.309017 0.951057i −0.809017 0.587785i
29.2 0.951057 + 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i −0.951057 0.309017i 0.809017 0.587785i 1.17557i 0.587785 + 0.809017i −0.309017 0.951057i −0.809017 0.587785i
269.1 −0.951057 + 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0.951057 0.309017i 0.809017 + 0.587785i 1.17557i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.809017 + 0.587785i
269.2 0.951057 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i −0.951057 + 0.309017i 0.809017 + 0.587785i 1.17557i 0.587785 0.809017i −0.309017 + 0.951057i −0.809017 + 0.587785i
389.1 −0.587785 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0.587785 + 0.809017i −0.309017 0.951057i 1.90211i 0.951057 0.309017i 0.809017 + 0.587785i 0.309017 0.951057i
389.2 0.587785 + 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i −0.587785 0.809017i −0.309017 0.951057i 1.90211i −0.951057 + 0.309017i 0.809017 + 0.587785i 0.309017 0.951057i
509.1 −0.587785 + 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0.587785 0.809017i −0.309017 + 0.951057i 1.90211i 0.951057 + 0.309017i 0.809017 0.587785i 0.309017 + 0.951057i
509.2 0.587785 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.90211i −0.951057 0.309017i 0.809017 0.587785i 0.309017 + 0.951057i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.2
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
3.b Odd 1 yes
8.b Even 1 yes
25.e Even 1 yes
75.h Odd 1 yes
200.o Even 1 yes
600.z Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(600, [\chi])\).