Properties

Label 600.1.q.a
Level 600
Weight 1
Character orbit 600.q
Analytic conductor 0.299
Analytic rank 0
Dimension 4
Projective image \(D_{2}\)
CM/RM disc. -8, -15, 120
Inner twists 16

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.29943900758\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{-15})\)
Artin image size \(32\)
Artin image $OD_{16}:C_2$
Artin field Galois closure of 16.0.164025000000000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{8}^{3} q^{2} \) \( -\zeta_{8} q^{3} \) \( -\zeta_{8}^{2} q^{4} \) \(+ q^{6}\) \( + \zeta_{8} q^{8} \) \( + \zeta_{8}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{8}^{3} q^{2} \) \( -\zeta_{8} q^{3} \) \( -\zeta_{8}^{2} q^{4} \) \(+ q^{6}\) \( + \zeta_{8} q^{8} \) \( + \zeta_{8}^{2} q^{9} \) \( + \zeta_{8}^{3} q^{12} \) \(- q^{16}\) \( -2 \zeta_{8}^{3} q^{17} \) \( -\zeta_{8} q^{18} \) \( -2 \zeta_{8}^{2} q^{19} \) \( -\zeta_{8}^{2} q^{24} \) \( -\zeta_{8}^{3} q^{27} \) \( -\zeta_{8}^{3} q^{32} \) \( + 2 \zeta_{8}^{2} q^{34} \) \(+ q^{36}\) \( + 2 \zeta_{8} q^{38} \) \( + \zeta_{8} q^{48} \) \( + \zeta_{8}^{2} q^{49} \) \( -2 q^{51} \) \( + \zeta_{8}^{2} q^{54} \) \( + 2 \zeta_{8}^{3} q^{57} \) \( + \zeta_{8}^{2} q^{64} \) \( -2 \zeta_{8} q^{68} \) \( + \zeta_{8}^{3} q^{72} \) \( -2 q^{76} \) \(- q^{81}\) \( + 2 \zeta_{8} q^{83} \) \(- q^{96}\) \( -\zeta_{8} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut +\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut -\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 4q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 0 0.707107 + 0.707107i 1.00000i 0
107.2 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 0 −0.707107 0.707107i 1.00000i 0
443.1 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 0 0.707107 0.707107i 1.00000i 0
443.2 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 0 −0.707107 + 0.707107i 1.00000i 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
15.d Odd 1 CM by \(\Q(\sqrt{-15}) \) yes
120.m Even 1 RM by \(\Q(\sqrt{30}) \) yes
3.b Odd 1 yes
5.b Even 1 yes
5.c Odd 2 yes
15.e Even 2 yes
24.f Even 1 yes
40.e Odd 1 yes
40.k Even 2 yes
120.q Odd 2 yes