Properties

Label 600.2.b.g
Level 600
Weight 2
Character orbit 600.b
Analytic conductor 4.791
Analytic rank 0
Dimension 12
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.537291533250985984.1
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{8} q^{2} \) \( -\beta_{5} q^{3} \) \( + ( 1 + \beta_{4} ) q^{4} \) \( + ( -\beta_{4} + \beta_{10} ) q^{6} \) \( + ( 1 - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{7} \) \( + ( -\beta_{2} + \beta_{8} - \beta_{11} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{8} q^{2} \) \( -\beta_{5} q^{3} \) \( + ( 1 + \beta_{4} ) q^{4} \) \( + ( -\beta_{4} + \beta_{10} ) q^{6} \) \( + ( 1 - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{7} \) \( + ( -\beta_{2} + \beta_{8} - \beta_{11} ) q^{8} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} \) \( + ( -\beta_{2} - \beta_{6} - \beta_{11} ) q^{11} \) \( + ( 1 - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{12} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{9} + \beta_{10} ) q^{13} \) \( + ( \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{14} \) \( + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{16} \) \( + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{17} \) \( + ( \beta_{1} + \beta_{3} - 2 \beta_{7} - \beta_{9} - \beta_{11} ) q^{18} \) \( + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{21} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{22} \) \( + ( -\beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{23} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{10} ) q^{24} \) \( + ( -2 \beta_{2} - 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{26} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{11} ) q^{27} \) \( + ( 3 - 3 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{28} \) \( + ( \beta_{6} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{29} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{31} \) \( + ( -2 \beta_{2} - 2 \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{32} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{33} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{34} \) \( + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{36} \) \( + ( 2 \beta_{1} - 2 \beta_{4} ) q^{37} \) \( + ( \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{38} \) \( + ( 1 - \beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{39} \) \( + ( -2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{11} ) q^{41} \) \( + ( 2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{11} ) q^{42} \) \( + ( 3 - 3 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{43} \) \( + ( -2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} ) q^{44} \) \( + ( -3 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{7} + \beta_{9} - \beta_{10} ) q^{46} \) \( + ( -2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{47} \) \( + ( 4 + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{48} \) \( + ( -4 + 3 \beta_{5} - 3 \beta_{7} ) q^{49} \) \( + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} ) q^{51} \) \( + ( -1 - 4 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{52} \) \( + ( 2 \beta_{2} - \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{53} \) \( + ( -4 + \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{54} \) \( + ( -\beta_{2} + 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{56} \) \( + ( -4 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{11} ) q^{57} \) \( + ( -6 + 3 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{7} - \beta_{9} + \beta_{10} ) q^{58} \) \( + ( -2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} ) q^{59} \) \( + ( 3 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} ) q^{61} \) \( + ( 4 \beta_{2} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + 3 \beta_{11} ) q^{62} \) \( + ( 1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{63} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{7} + \beta_{9} - \beta_{10} ) q^{64} \) \( + ( 2 + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{66} \) \( + ( 5 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{67} \) \( + ( -2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{68} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{69} \) \( + ( \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{71} \) \( + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{72} \) \( + ( 4 \beta_{1} + 4 \beta_{4} + \beta_{5} - \beta_{7} ) q^{73} \) \( + ( 4 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{11} ) q^{74} \) \( + ( -5 - \beta_{1} + \beta_{3} + 3 \beta_{5} - 3 \beta_{7} - \beta_{9} + \beta_{10} ) q^{76} \) \( + ( \beta_{6} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{77} \) \( + ( -4 + 3 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{78} \) \( + ( 4 - 4 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 4 \beta_{9} - 4 \beta_{10} ) q^{79} \) \( + ( -2 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{81} \) \( + ( -6 + \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{82} \) \( + ( -6 \beta_{2} + 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{11} ) q^{83} \) \( + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{84} \) \( + ( 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{86} \) \( + ( 2 - 4 \beta_{1} - \beta_{2} + 4 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{87} \) \( + ( -2 - 4 \beta_{1} - 2 \beta_{5} + 2 \beta_{7} + \beta_{9} - \beta_{10} ) q^{88} \) \( + ( -2 \beta_{2} + 3 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} ) q^{89} \) \( + ( -1 + 5 \beta_{1} + 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{91} \) \( + ( 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{92} \) \( + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} + \beta_{10} - \beta_{11} ) q^{93} \) \( + ( 4 + 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{94} \) \( + ( -2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{96} \) \( + ( 3 + 4 \beta_{1} + 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{97} \) \( + ( -4 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{98} \) \( + ( 5 + \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 10q^{4} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 3q^{12} \) \(\mathstrut -\mathstrut 6q^{16} \) \(\mathstrut +\mathstrut 5q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 2q^{22} \) \(\mathstrut +\mathstrut 5q^{24} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 20q^{28} \) \(\mathstrut -\mathstrut 18q^{33} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 19q^{36} \) \(\mathstrut +\mathstrut 14q^{42} \) \(\mathstrut +\mathstrut 40q^{43} \) \(\mathstrut -\mathstrut 16q^{46} \) \(\mathstrut +\mathstrut 27q^{48} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 30q^{51} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 30q^{54} \) \(\mathstrut -\mathstrut 42q^{57} \) \(\mathstrut -\mathstrut 52q^{58} \) \(\mathstrut +\mathstrut 10q^{64} \) \(\mathstrut +\mathstrut 7q^{66} \) \(\mathstrut +\mathstrut 60q^{67} \) \(\mathstrut +\mathstrut 39q^{72} \) \(\mathstrut -\mathstrut 12q^{73} \) \(\mathstrut -\mathstrut 38q^{76} \) \(\mathstrut -\mathstrut 54q^{78} \) \(\mathstrut -\mathstrut 10q^{81} \) \(\mathstrut -\mathstrut 58q^{82} \) \(\mathstrut -\mathstrut 34q^{84} \) \(\mathstrut -\mathstrut 34q^{88} \) \(\mathstrut -\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut 28q^{94} \) \(\mathstrut -\mathstrut 31q^{96} \) \(\mathstrut +\mathstrut 32q^{97} \) \(\mathstrut +\mathstrut 58q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(5\) \(x^{10}\mathstrut +\mathstrut \) \(14\) \(x^{8}\mathstrut -\mathstrut \) \(30\) \(x^{6}\mathstrut +\mathstrut \) \(56\) \(x^{4}\mathstrut -\mathstrut \) \(80\) \(x^{2}\mathstrut +\mathstrut \) \(64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{9} + \nu^{7} - 2 \nu^{5} + 6 \nu^{3} \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{10} - 5 \nu^{8} + 14 \nu^{6} - 14 \nu^{4} + 40 \nu^{2} - 48 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} + 5 \nu^{8} - 14 \nu^{6} + 30 \nu^{4} - 56 \nu^{2} + 64 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{11} + 6 \nu^{10} + \nu^{9} - 22 \nu^{8} - 10 \nu^{7} + 44 \nu^{6} + 22 \nu^{5} - 100 \nu^{4} - 32 \nu^{3} + 160 \nu^{2} + 16 \nu - 160 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{9} - 14 \nu^{7} + 30 \nu^{5} - 24 \nu^{3} + 16 \nu \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} - 6 \nu^{10} + \nu^{9} + 22 \nu^{8} - 10 \nu^{7} - 44 \nu^{6} + 22 \nu^{5} + 100 \nu^{4} - 32 \nu^{3} - 160 \nu^{2} + 16 \nu + 160 \)\()/64\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{9} - 14 \nu^{7} + 30 \nu^{5} - 56 \nu^{3} + 80 \nu \)\()/32\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{11} + 2 \nu^{10} + 13 \nu^{9} - 18 \nu^{8} - 26 \nu^{7} + 36 \nu^{6} + 62 \nu^{5} - 76 \nu^{4} - 80 \nu^{3} + 160 \nu^{2} + 80 \nu - 224 \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( -5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 18 \nu^{8} - 26 \nu^{7} - 36 \nu^{6} + 62 \nu^{5} + 76 \nu^{4} - 80 \nu^{3} - 160 \nu^{2} + 80 \nu + 224 \)\()/64\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{9} + 8 \nu^{7} - 14 \nu^{5} + 20 \nu^{3} - 24 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\)
\(\nu^{6}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\)
\(\nu^{7}\)\(=\)\(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\)
\(\nu^{8}\)\(=\)\(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(6\)
\(\nu^{9}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\)
\(\nu^{10}\)\(=\)\(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut -\mathstrut \) \(22\)
\(\nu^{11}\)\(=\)\(4\) \(\beta_{11}\mathstrut -\mathstrut \) \(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.39298 + 0.244153i
−1.39298 0.244153i
−1.26128 + 0.639662i
−1.26128 0.639662i
−0.847808 + 1.13191i
−0.847808 1.13191i
0.847808 + 1.13191i
0.847808 1.13191i
1.26128 + 0.639662i
1.26128 0.639662i
1.39298 + 0.244153i
1.39298 0.244153i
−1.39298 0.244153i 1.31310 + 1.12950i 1.88078 + 0.680200i 0 −1.55335 1.89397i 4.34495i −2.45381 1.40670i 0.448458 + 2.96629i 0
251.2 −1.39298 + 0.244153i 1.31310 1.12950i 1.88078 0.680200i 0 −1.55335 + 1.89397i 4.34495i −2.45381 + 1.40670i 0.448458 2.96629i 0
251.3 −1.26128 0.639662i −1.57067 0.730070i 1.18166 + 1.61359i 0 1.51406 + 1.92552i 1.25539i −0.458259 2.79106i 1.93400 + 2.29339i 0
251.4 −1.26128 + 0.639662i −1.57067 + 0.730070i 1.18166 1.61359i 0 1.51406 1.92552i 1.25539i −0.458259 + 2.79106i 1.93400 2.29339i 0
251.5 −0.847808 1.13191i −0.242431 + 1.71500i −0.562443 + 1.91929i 0 2.14676 1.17958i 3.08957i 2.64930 0.990551i −2.88245 0.831539i 0
251.6 −0.847808 + 1.13191i −0.242431 1.71500i −0.562443 1.91929i 0 2.14676 + 1.17958i 3.08957i 2.64930 + 0.990551i −2.88245 + 0.831539i 0
251.7 0.847808 1.13191i −0.242431 + 1.71500i −0.562443 1.91929i 0 1.73569 + 1.72840i 3.08957i −2.64930 0.990551i −2.88245 0.831539i 0
251.8 0.847808 + 1.13191i −0.242431 1.71500i −0.562443 + 1.91929i 0 1.73569 1.72840i 3.08957i −2.64930 + 0.990551i −2.88245 + 0.831539i 0
251.9 1.26128 0.639662i −1.57067 0.730070i 1.18166 1.61359i 0 −2.44805 + 0.0838735i 1.25539i 0.458259 2.79106i 1.93400 + 2.29339i 0
251.10 1.26128 + 0.639662i −1.57067 + 0.730070i 1.18166 + 1.61359i 0 −2.44805 0.0838735i 1.25539i 0.458259 + 2.79106i 1.93400 2.29339i 0
251.11 1.39298 0.244153i 1.31310 + 1.12950i 1.88078 0.680200i 0 2.10489 + 1.25277i 4.34495i 2.45381 1.40670i 0.448458 + 2.96629i 0
251.12 1.39298 + 0.244153i 1.31310 1.12950i 1.88078 + 0.680200i 0 2.10489 1.25277i 4.34495i 2.45381 + 1.40670i 0.448458 2.96629i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.12
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
8.d Odd 1 no
24.f Even 1 no

Hecke kernels

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

\(T_{7}^{6} \) \(\mathstrut +\mathstrut 30 T_{7}^{4} \) \(\mathstrut +\mathstrut 225 T_{7}^{2} \) \(\mathstrut +\mathstrut 284 \)
\(T_{11}^{6} \) \(\mathstrut +\mathstrut 19 T_{11}^{4} \) \(\mathstrut +\mathstrut 112 T_{11}^{2} \) \(\mathstrut +\mathstrut 200 \)
\(T_{23}^{6} \) \(\mathstrut -\mathstrut 104 T_{23}^{4} \) \(\mathstrut +\mathstrut 2136 T_{23}^{2} \) \(\mathstrut -\mathstrut 9088 \)
\(T_{43}^{3} \) \(\mathstrut -\mathstrut 10 T_{43}^{2} \) \(\mathstrut -\mathstrut 29 T_{43} \) \(\mathstrut +\mathstrut 148 \)