Properties

Label 600.2.k.e
Level 600
Weight 2
Character orbit 600.k
Analytic conductor 4.791
Analytic rank 0
Dimension 8
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.k (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.214798336.3
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{3} q^{2} \) \( -\beta_{2} q^{3} \) \( + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{4} \) \( + \beta_{4} q^{6} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{7} \) \( + ( -1 - \beta_{2} + \beta_{6} - \beta_{7} ) q^{8} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( -\beta_{3} q^{2} \) \( -\beta_{2} q^{3} \) \( + ( -\beta_{4} - \beta_{5} + \beta_{6} ) q^{4} \) \( + \beta_{4} q^{6} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{7} \) \( + ( -1 - \beta_{2} + \beta_{6} - \beta_{7} ) q^{8} \) \(- q^{9}\) \( + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} \) \( + ( -\beta_{1} - \beta_{2} ) q^{12} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{14} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{16} \) \( + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{17} \) \( + \beta_{3} q^{18} \) \( + ( -1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{19} \) \( + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{21} \) \( + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{22} \) \( + ( -1 + 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{23} \) \( + ( -1 + \beta_{2} + \beta_{6} + \beta_{7} ) q^{24} \) \( + ( 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{7} ) q^{26} \) \( + \beta_{2} q^{27} \) \( + ( 2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{28} \) \( + ( -\beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{29} \) \( + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{31} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{7} ) q^{32} \) \( + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{33} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{34} \) \( + ( \beta_{4} + \beta_{5} - \beta_{6} ) q^{36} \) \( + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{37} \) \( + ( -6 + \beta_{1} + 3 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} ) q^{38} \) \( + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{39} \) \( + ( \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{41} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{42} \) \( + ( -1 - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{43} \) \( + ( -2 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{44} \) \( + ( 2 + 3 \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{46} \) \( + ( -4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{47} \) \( + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{48} \) \( + ( 4 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{49} \) \( + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{51} \) \( + ( 4 - 4 \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{52} \) \( + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - 3 \beta_{7} ) q^{53} \) \( -\beta_{4} q^{54} \) \( + ( 3 + 5 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{56} \) \( + ( -1 - \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{57} \) \( + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{58} \) \( + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{59} \) \( + ( 1 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{61} \) \( + ( -4 + 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{62} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} ) q^{63} \) \( + ( -4 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{64} \) \( + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{66} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{67} \) \( + ( 4 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{68} \) \( + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{69} \) \( + ( -5 + 2 \beta_{1} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{71} \) \( + ( 1 + \beta_{2} - \beta_{6} + \beta_{7} ) q^{72} \) \( + ( 2 - 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{73} \) \( + ( 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{74} \) \( + ( -2 - 3 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{76} \) \( + ( 4 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{77} \) \( + ( 2 - \beta_{4} - 2 \beta_{5} ) q^{78} \) \( + ( -2 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{79} \) \(+ q^{81}\) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{82} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{83} \) \( + ( 4 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{84} \) \( + ( -2 - \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{86} \) \( + ( -3 + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{87} \) \( + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{88} \) \( + ( 2 \beta_{1} - 6 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{89} \) \( + ( -3 - 5 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} \) \( + ( 2 - 4 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{92} \) \( + ( -1 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{93} \) \( + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} ) q^{94} \) \( + ( 2 + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{96} \) \( + ( 1 - 2 \beta_{1} + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{97} \) \( + ( 4 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{98} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 12q^{22} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut 28q^{32} \) \(\mathstrut +\mathstrut 12q^{34} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 30q^{38} \) \(\mathstrut +\mathstrut 6q^{42} \) \(\mathstrut -\mathstrut 12q^{44} \) \(\mathstrut +\mathstrut 20q^{46} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut +\mathstrut 20q^{52} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 8q^{56} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 30q^{62} \) \(\mathstrut +\mathstrut 8q^{63} \) \(\mathstrut -\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 20q^{66} \) \(\mathstrut +\mathstrut 28q^{68} \) \(\mathstrut -\mathstrut 40q^{71} \) \(\mathstrut +\mathstrut 4q^{72} \) \(\mathstrut +\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 8q^{74} \) \(\mathstrut -\mathstrut 20q^{76} \) \(\mathstrut +\mathstrut 22q^{78} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut 24q^{82} \) \(\mathstrut +\mathstrut 24q^{84} \) \(\mathstrut -\mathstrut 18q^{86} \) \(\mathstrut -\mathstrut 24q^{87} \) \(\mathstrut +\mathstrut 8q^{88} \) \(\mathstrut +\mathstrut 36q^{92} \) \(\mathstrut -\mathstrut 4q^{94} \) \(\mathstrut +\mathstrut 12q^{96} \) \(\mathstrut +\mathstrut 8q^{97} \) \(\mathstrut +\mathstrut 48q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut +\mathstrut \) \(9\) \(x^{4}\mathstrut -\mathstrut \) \(4\) \(x^{3}\mathstrut -\mathstrut \) \(16\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} - 4 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{7} - 4 \nu^{6} - 4 \nu^{5} - 18 \nu^{4} + 25 \nu^{3} + 10 \nu^{2} + 24 \nu - 64 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} - 2 \nu^{6} - 2 \nu^{5} - 10 \nu^{4} + 15 \nu^{3} + 8 \nu^{2} + 10 \nu - 36 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} - 12 \nu^{2} - 8 \nu + 36 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 12 \nu^{4} - 18 \nu^{3} - 9 \nu^{2} - 10 \nu + 44 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{7} - 4 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 31 \nu^{3} + 18 \nu^{2} + 26 \nu - 80 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(7\) \(\beta_{7}\mathstrut -\mathstrut \) \(5\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
1.23291 + 0.692769i
1.23291 0.692769i
−1.08003 + 0.912978i
−1.08003 0.912978i
−0.565036 + 1.29643i
−0.565036 1.29643i
1.41216 + 0.0762223i
1.41216 0.0762223i
−1.40101 0.192769i 1.00000i 1.92568 + 0.540143i 0 −0.192769 + 1.40101i 0.0802864 −2.59378 1.12796i −1.00000 0
301.2 −1.40101 + 0.192769i 1.00000i 1.92568 0.540143i 0 −0.192769 1.40101i 0.0802864 −2.59378 + 1.12796i −1.00000 0
301.3 −0.0591148 1.41298i 1.00000i −1.99301 + 0.167056i 0 1.41298 0.0591148i −1.33411 0.353863 + 2.80620i −1.00000 0
301.4 −0.0591148 + 1.41298i 1.00000i −1.99301 0.167056i 0 1.41298 + 0.0591148i −1.33411 0.353863 2.80620i −1.00000 0
301.5 1.16863 0.796431i 1.00000i 0.731395 1.86147i 0 −0.796431 1.16863i −4.72294 −0.627801 2.75787i −1.00000 0
301.6 1.16863 + 0.796431i 1.00000i 0.731395 + 1.86147i 0 −0.796431 + 1.16863i −4.72294 −0.627801 + 2.75787i −1.00000 0
301.7 1.29150 0.576222i 1.00000i 1.33594 1.48838i 0 0.576222 + 1.29150i 1.97676 0.867721 2.69204i −1.00000 0
301.8 1.29150 + 0.576222i 1.00000i 1.33594 + 1.48838i 0 0.576222 1.29150i 1.97676 0.867721 + 2.69204i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{4} \) \(\mathstrut +\mathstrut 4 T_{7}^{3} \) \(\mathstrut -\mathstrut 6 T_{7}^{2} \) \(\mathstrut -\mathstrut 12 T_{7} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).