Properties

Label 600.2.f.c
Level 600
Weight 2
Character orbit 600.f
Analytic conductor 4.791
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + i q^{3} \) \( -4 i q^{7} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + i q^{3} \) \( -4 i q^{7} \) \(- q^{9}\) \( -6 i q^{13} \) \( + 2 i q^{17} \) \( -4 q^{19} \) \( + 4 q^{21} \) \( -8 i q^{23} \) \( -i q^{27} \) \( + 6 q^{29} \) \( + 6 i q^{37} \) \( + 6 q^{39} \) \( + 10 q^{41} \) \( -4 i q^{43} \) \( -8 i q^{47} \) \( -9 q^{49} \) \( -2 q^{51} \) \( + 10 i q^{53} \) \( -4 i q^{57} \) \( + 6 q^{61} \) \( + 4 i q^{63} \) \( + 4 i q^{67} \) \( + 8 q^{69} \) \( -14 i q^{73} \) \( -16 q^{79} \) \(+ q^{81}\) \( + 12 i q^{83} \) \( + 6 i q^{87} \) \( -2 q^{89} \) \( -24 q^{91} \) \( -2 i q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut +\mathstrut 20q^{41} \) \(\mathstrut -\mathstrut 18q^{49} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 12q^{61} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut -\mathstrut 32q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut -\mathstrut 48q^{91} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 4.00000i 0 −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
5.b Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\(T_{7}^{2} \) \(\mathstrut +\mathstrut 16 \)
\(T_{11} \)