Properties

Label 600.2.r.a
Level 600
Weight 2
Character orbit 600.r
Analytic conductor 4.791
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} \) \( + ( -2 + 2 \zeta_{8}^{2} ) q^{7} \) \( + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} \) \( + ( -2 + 2 \zeta_{8}^{2} ) q^{7} \) \( + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} \) \( + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} \) \( -4 \zeta_{8} q^{17} \) \( + 4 \zeta_{8}^{2} q^{19} \) \( + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} \) \( + 6 \zeta_{8}^{3} q^{23} \) \( + ( -1 + 5 \zeta_{8} + \zeta_{8}^{2} ) q^{27} \) \( + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} \) \( + 8 q^{31} \) \( + ( 4 + 4 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{33} \) \( + ( 8 - 8 \zeta_{8}^{2} ) q^{37} \) \( + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} \) \( + ( -2 - 2 \zeta_{8}^{2} ) q^{43} \) \( + 2 \zeta_{8} q^{47} \) \( -\zeta_{8}^{2} q^{49} \) \( + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51} \) \( + 8 \zeta_{8}^{3} q^{53} \) \( + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{57} \) \( + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{59} \) \( -6 q^{61} \) \( + ( -2 - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{63} \) \( + ( -6 + 6 \zeta_{8}^{2} ) q^{67} \) \( + ( 6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{69} \) \( + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{71} \) \( + ( 8 + 8 \zeta_{8}^{2} ) q^{73} \) \( -16 \zeta_{8} q^{77} \) \( + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} \) \( + 14 \zeta_{8}^{3} q^{83} \) \( + ( -4 + 8 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{87} \) \( + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} \) \( + ( -8 - 8 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{93} \) \( + ( -8 + 8 \zeta_{8}^{2} ) q^{97} \) \( + ( -4 \zeta_{8} - 16 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 16q^{21} \) \(\mathstrut -\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 32q^{31} \) \(\mathstrut +\mathstrut 16q^{33} \) \(\mathstrut +\mathstrut 32q^{37} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 24q^{61} \) \(\mathstrut -\mathstrut 8q^{63} \) \(\mathstrut -\mathstrut 24q^{67} \) \(\mathstrut +\mathstrut 32q^{73} \) \(\mathstrut +\mathstrut 28q^{81} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 32q^{93} \) \(\mathstrut -\mathstrut 32q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 −1.70711 + 0.292893i 0 0 0 −2.00000 2.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 −2.00000 2.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 −2.00000 + 2.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 −2.00000 + 2.00000i 0 −2.82843 + 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
3.b Odd 1 yes
5.c Odd 1 yes
15.e Even 1 yes

Hecke kernels

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

\(T_{7}^{2} \) \(\mathstrut +\mathstrut 4 T_{7} \) \(\mathstrut +\mathstrut 8 \)
\(T_{17}^{4} \) \(\mathstrut +\mathstrut 256 \)