Properties

Label 600.1.n.b
Level 600
Weight 1
Character orbit 600.n
Self dual Yes
Analytic conductor 0.299
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM disc. -15, -24, 40
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.n (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.29943900758\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-6}, \sqrt{10})\)
Artin image size \(8\)
Artin image $D_4$
Artin field Galois closure of 4.0.9000.2

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut +\mathstrut q^{18} \) \(\mathstrut -\mathstrut q^{24} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut q^{32} \) \(\mathstrut +\mathstrut q^{36} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut -\mathstrut q^{54} \) \(\mathstrut -\mathstrut 2q^{62} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut +\mathstrut q^{72} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut q^{96} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0
1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000 1.00000 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
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
40.f Even 1 RM by \(\Q(\sqrt{10}) \) yes