Properties

Label 600.2.m.d
Level 600
Weight 2
Character orbit 600.m
Analytic conductor 4.791
Analytic rank 0
Dimension 16
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.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{6} q^{3} \) \( + \beta_{2} q^{4} \) \( -\beta_{14} q^{6} \) \( + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{10} ) q^{7} \) \( + \beta_{3} q^{8} \) \( + ( -\beta_{2} + \beta_{8} + \beta_{14} - \beta_{15} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{6} q^{3} \) \( + \beta_{2} q^{4} \) \( -\beta_{14} q^{6} \) \( + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{10} ) q^{7} \) \( + \beta_{3} q^{8} \) \( + ( -\beta_{2} + \beta_{8} + \beta_{14} - \beta_{15} ) q^{9} \) \( + ( -\beta_{2} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{11} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{10} + \beta_{13} ) q^{12} \) \( + ( \beta_{1} - \beta_{5} + \beta_{7} + \beta_{13} ) q^{13} \) \( + ( 1 + 2 \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{12} + \beta_{14} - \beta_{15} ) q^{14} \) \( + ( -1 - \beta_{2} + \beta_{8} + 2 \beta_{12} + \beta_{14} ) q^{16} \) \( + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{10} ) q^{17} \) \( + ( \beta_{1} - \beta_{5} + 2 \beta_{7} - \beta_{10} ) q^{18} \) \( + ( \beta_{2} - 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{19} \) \( + ( -1 - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{21} \) \( + ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{22} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{23} \) \( + ( 1 + \beta_{2} - 3 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{24} \) \( + ( 2 + 2 \beta_{2} - 2 \beta_{8} ) q^{26} \) \( + ( 2 \beta_{1} - \beta_{4} - \beta_{7} - 2 \beta_{13} ) q^{27} \) \( + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{10} + \beta_{13} ) q^{28} \) \( + ( -3 \beta_{2} + \beta_{8} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{29} \) \( + ( -\beta_{2} + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{15} ) q^{31} \) \( + ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} ) q^{32} \) \( + ( 4 \beta_{1} + \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{13} ) q^{33} \) \( + ( 2 \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{15} ) q^{34} \) \( + ( 2 + \beta_{2} - \beta_{9} - 2 \beta_{11} - \beta_{15} ) q^{36} \) \( + ( -\beta_{1} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{13} ) q^{37} \) \( + ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{10} + \beta_{13} ) q^{38} \) \( + ( -2 + 2 \beta_{2} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{39} \) \( + ( -\beta_{2} + \beta_{8} - \beta_{12} - \beta_{15} ) q^{41} \) \( + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{13} ) q^{42} \) \( + ( -\beta_{6} - \beta_{7} ) q^{43} \) \( + ( 4 - 2 \beta_{8} - 2 \beta_{12} + 2 \beta_{15} ) q^{44} \) \( + ( -1 - \beta_{11} + 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{46} \) \( + ( -5 \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{10} + 2 \beta_{13} ) q^{47} \) \( + ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{48} \) \( + ( 3 - \beta_{2} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{49} \) \( + ( 2 - \beta_{2} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{51} \) \( + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{52} \) \( + ( -\beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{53} \) \( + ( -5 + 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{15} ) q^{54} \) \( + ( 4 - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{56} \) \( + ( \beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{7} - 2 \beta_{10} + \beta_{13} ) q^{57} \) \( + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} - 3 \beta_{13} ) q^{58} \) \( + ( \beta_{2} - \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{59} \) \( + ( -\beta_{2} - \beta_{8} - 3 \beta_{12} - \beta_{15} ) q^{61} \) \( + ( 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{62} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} ) q^{63} \) \( + ( -3 - \beta_{2} - \beta_{8} - 2 \beta_{11} - \beta_{14} + 2 \beta_{15} ) q^{64} \) \( + ( -4 + 2 \beta_{2} - \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{66} \) \( + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{10} + 2 \beta_{13} ) q^{67} \) \( + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{13} ) q^{68} \) \( + ( -3 + \beta_{2} - 2 \beta_{8} + 3 \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{69} \) \( + ( -4 - 2 \beta_{2} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{71} \) \( + ( \beta_{3} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{13} ) q^{72} \) \( + ( -6 \beta_{1} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{13} ) q^{73} \) \( + ( -4 - 2 \beta_{2} - 2 \beta_{8} - 2 \beta_{11} + 2 \beta_{14} ) q^{74} \) \( + ( -2 + 2 \beta_{2} - 4 \beta_{8} + 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{76} \) \( + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{13} ) q^{77} \) \( + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{13} ) q^{78} \) \( + ( \beta_{2} - 2 \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{79} \) \( + ( 1 - 3 \beta_{2} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} + 3 \beta_{15} ) q^{81} \) \( + ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{13} ) q^{82} \) \( + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{10} ) q^{83} \) \( + ( -8 + 2 \beta_{2} - 4 \beta_{8} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{84} \) \( + ( -1 + \beta_{11} + \beta_{14} ) q^{86} \) \( + ( 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{13} ) q^{87} \) \( + ( 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{13} ) q^{88} \) \( + ( -2 \beta_{2} + 2 \beta_{8} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{89} \) \( + ( 4 + 2 \beta_{2} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{91} \) \( + ( -2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} + 3 \beta_{10} - \beta_{13} ) q^{92} \) \( + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{13} ) q^{93} \) \( + ( 5 - 4 \beta_{2} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{14} + 4 \beta_{15} ) q^{94} \) \( + ( -5 + \beta_{2} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + \beta_{14} ) q^{96} \) \( + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} ) q^{97} \) \( + ( \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{10} - \beta_{13} ) q^{98} \) \( + ( -2 + \beta_{2} + \beta_{8} + \beta_{9} + 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{14} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 12q^{14} \) \(\mathstrut -\mathstrut 14q^{16} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 8q^{21} \) \(\mathstrut +\mathstrut 22q^{24} \) \(\mathstrut +\mathstrut 32q^{26} \) \(\mathstrut +\mathstrut 26q^{36} \) \(\mathstrut -\mathstrut 32q^{39} \) \(\mathstrut +\mathstrut 60q^{44} \) \(\mathstrut -\mathstrut 16q^{46} \) \(\mathstrut +\mathstrut 32q^{49} \) \(\mathstrut +\mathstrut 40q^{51} \) \(\mathstrut -\mathstrut 82q^{54} \) \(\mathstrut +\mathstrut 60q^{56} \) \(\mathstrut -\mathstrut 50q^{64} \) \(\mathstrut -\mathstrut 68q^{66} \) \(\mathstrut -\mathstrut 40q^{69} \) \(\mathstrut -\mathstrut 48q^{71} \) \(\mathstrut -\mathstrut 64q^{74} \) \(\mathstrut -\mathstrut 24q^{76} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut -\mathstrut 116q^{84} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut +\mathstrut 48q^{91} \) \(\mathstrut +\mathstrut 80q^{94} \) \(\mathstrut -\mathstrut 86q^{96} \) \(\mathstrut -\mathstrut 32q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(x^{14}\mathstrut +\mathstrut \) \(4\) \(x^{12}\mathstrut +\mathstrut \) \(12\) \(x^{10}\mathstrut +\mathstrut \) \(16\) \(x^{8}\mathstrut +\mathstrut \) \(48\) \(x^{6}\mathstrut +\mathstrut \) \(64\) \(x^{4}\mathstrut +\mathstrut \) \(64\) \(x^{2}\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} + 3 \nu^{13} + 10 \nu^{11} + 8 \nu^{9} + 8 \nu^{7} + 64 \nu^{5} + 64 \nu \)\()/192\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{9} + \nu^{7} - 2 \nu^{5} - 4 \nu^{3} + 8 \nu \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} - 3 \nu^{13} + 2 \nu^{11} - 8 \nu^{9} + 4 \nu^{7} + 8 \nu^{5} - 48 \nu^{3} + 32 \nu \)\()/192\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} - 3 \nu^{13} - 10 \nu^{11} - 20 \nu^{9} - 44 \nu^{7} - 40 \nu^{5} - 144 \nu^{3} - 160 \nu \)\()/192\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{10} + \nu^{8} - 2 \nu^{6} - 4 \nu^{4} + 8 \nu^{2} - 16 \)\()/16\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} + 4 \nu^{10} + 32 \nu^{8} + 32 \nu^{6} + 112 \nu^{4} + 192 \nu^{2} + 256 \)\()/192\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{13} - 4 \nu^{11} + 4 \nu^{9} + 28 \nu^{7} + 56 \nu^{5} + 144 \nu^{3} + 224 \nu \)\()/192\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{14} + 3 \nu^{12} + 4 \nu^{10} + 14 \nu^{8} - 4 \nu^{6} + 40 \nu^{4} + 48 \nu^{2} - 32 \)\()/96\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{14} + 3 \nu^{12} + 8 \nu^{10} + 4 \nu^{8} + 40 \nu^{6} + 128 \nu^{4} + 96 \nu^{2} + 320 \)\()/192\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} - \nu^{11} - 2 \nu^{9} - 8 \nu^{7} - 40 \nu^{5} + 32 \nu \)\()/96\)
\(\beta_{14}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 2 \nu^{10} - 10 \nu^{8} - 28 \nu^{6} - 8 \nu^{4} - 48 \nu^{2} - 128 \)\()/96\)
\(\beta_{15}\)\(=\)\((\)\( \nu^{14} + \nu^{12} + 8 \nu^{8} + 16 \nu^{6} + 16 \nu^{4} + 64 \nu^{2} \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{14}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(1\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(3\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{8}\)\(=\)\(-\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(3\) \(\beta_{14}\mathstrut -\mathstrut \) \(4\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(1\)
\(\nu^{9}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{10}\)\(=\)\(-\)\(6\) \(\beta_{15}\mathstrut -\mathstrut \) \(5\) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(6\) \(\beta_{11}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(7\)
\(\nu^{11}\)\(=\)\(7\) \(\beta_{13}\mathstrut -\mathstrut \) \(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{12}\)\(=\)\(6\) \(\beta_{15}\mathstrut -\mathstrut \) \(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(20\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{13}\)\(=\)\(17\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(33\) \(\beta_{6}\mathstrut +\mathstrut \) \(21\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(28\) \(\beta_{1}\)
\(\nu^{14}\)\(=\)\(42\) \(\beta_{15}\mathstrut +\mathstrut \) \(27\) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(6\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(57\)
\(\nu^{15}\)\(=\)\(-\)\(57\) \(\beta_{13}\mathstrut -\mathstrut \) \(39\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(55\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\) \(\beta_{5}\mathstrut -\mathstrut \) \(19\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(60\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
−1.29041 0.578647i
−1.29041 + 0.578647i
−1.15595 0.814732i
−1.15595 + 0.814732i
−0.842022 1.13622i
−0.842022 + 1.13622i
−0.199044 1.40014i
−0.199044 + 1.40014i
0.199044 1.40014i
0.199044 + 1.40014i
0.842022 1.13622i
0.842022 + 1.13622i
1.15595 0.814732i
1.15595 + 0.814732i
1.29041 0.578647i
1.29041 + 0.578647i
−1.29041 0.578647i 1.56044 0.751690i 1.33034 + 1.49339i 0 −2.44857 + 0.0670494i −4.28591 −0.852541 2.69688i 1.86993 2.34593i 0
299.2 −1.29041 + 0.578647i 1.56044 + 0.751690i 1.33034 1.49339i 0 −2.44857 0.0670494i −4.28591 −0.852541 + 2.69688i 1.86993 + 2.34593i 0
299.3 −1.15595 0.814732i −0.887900 + 1.48716i 0.672424 + 1.88357i 0 2.23800 0.995672i −0.797253 0.757320 2.72515i −1.42327 2.64089i 0
299.4 −1.15595 + 0.814732i −0.887900 1.48716i 0.672424 1.88357i 0 2.23800 + 0.995672i −0.797253 0.757320 + 2.72515i −1.42327 + 2.64089i 0
299.5 −0.842022 1.13622i 0.218455 + 1.71822i −0.581998 + 1.91345i 0 1.76833 1.69499i 3.64426 2.66415 0.949886i −2.90455 + 0.750707i 0
299.6 −0.842022 + 1.13622i 0.218455 1.71822i −0.581998 1.91345i 0 1.76833 + 1.69499i 3.64426 2.66415 + 0.949886i −2.90455 0.750707i 0
299.7 −0.199044 1.40014i 1.65195 0.520627i −1.92076 + 0.557378i 0 −1.05776 2.20933i 1.92736 1.16272 + 2.57839i 2.45790 1.72010i 0
299.8 −0.199044 + 1.40014i 1.65195 + 0.520627i −1.92076 0.557378i 0 −1.05776 + 2.20933i 1.92736 1.16272 2.57839i 2.45790 + 1.72010i 0
299.9 0.199044 1.40014i −1.65195 0.520627i −1.92076 0.557378i 0 −1.05776 + 2.20933i −1.92736 −1.16272 + 2.57839i 2.45790 + 1.72010i 0
299.10 0.199044 + 1.40014i −1.65195 + 0.520627i −1.92076 + 0.557378i 0 −1.05776 2.20933i −1.92736 −1.16272 2.57839i 2.45790 1.72010i 0
299.11 0.842022 1.13622i −0.218455 + 1.71822i −0.581998 1.91345i 0 1.76833 + 1.69499i −3.64426 −2.66415 0.949886i −2.90455 0.750707i 0
299.12 0.842022 + 1.13622i −0.218455 1.71822i −0.581998 + 1.91345i 0 1.76833 1.69499i −3.64426 −2.66415 + 0.949886i −2.90455 + 0.750707i 0
299.13 1.15595 0.814732i 0.887900 + 1.48716i 0.672424 1.88357i 0 2.23800 + 0.995672i 0.797253 −0.757320 2.72515i −1.42327 + 2.64089i 0
299.14 1.15595 + 0.814732i 0.887900 1.48716i 0.672424 + 1.88357i 0 2.23800 0.995672i 0.797253 −0.757320 + 2.72515i −1.42327 2.64089i 0
299.15 1.29041 0.578647i −1.56044 0.751690i 1.33034 1.49339i 0 −2.44857 0.0670494i 4.28591 0.852541 2.69688i 1.86993 + 2.34593i 0
299.16 1.29041 + 0.578647i −1.56044 + 0.751690i 1.33034 + 1.49339i 0 −2.44857 + 0.0670494i 4.28591 0.852541 + 2.69688i 1.86993 2.34593i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes
24.f Even 1 yes
120.m Even 1 yes

Hecke kernels

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

\(T_{7}^{8} \) \(\mathstrut -\mathstrut 36 T_{7}^{6} \) \(\mathstrut +\mathstrut 384 T_{7}^{4} \) \(\mathstrut -\mathstrut 1136 T_{7}^{2} \) \(\mathstrut +\mathstrut 576 \)
\(T_{11}^{8} \) \(\mathstrut +\mathstrut 48 T_{11}^{6} \) \(\mathstrut +\mathstrut 672 T_{11}^{4} \) \(\mathstrut +\mathstrut 2560 T_{11}^{2} \) \(\mathstrut +\mathstrut 256 \)
\(T_{29}^{4} \) \(\mathstrut -\mathstrut 64 T_{29}^{2} \) \(\mathstrut +\mathstrut 112 T_{29} \) \(\mathstrut -\mathstrut 48 \)