Properties

Label 600.2.b.i
Level 600
Weight 2
Character orbit 600.b
Analytic conductor 4.791
Analytic rank 0
Dimension 16
CM No
Inner twists 8

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(24\) \(x^{14}\mathstrut +\mathstrut \) \(192\) \(x^{12}\mathstrut +\mathstrut \) \(672\) \(x^{10}\mathstrut +\mathstrut \) \(1092\) \(x^{8}\mathstrut +\mathstrut \) \(880\) \(x^{6}\mathstrut +\mathstrut \) \(352\) \(x^{4}\mathstrut +\mathstrut \) \(64\) \(x^{2}\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{13} + 69 \nu^{11} + 506 \nu^{9} + 1488 \nu^{7} + 1638 \nu^{5} + 594 \nu^{3} + 44 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{14} - 24 \nu^{12} - 191 \nu^{10} - 651 \nu^{8} - 962 \nu^{6} - 584 \nu^{4} - 142 \nu^{2} - 22 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{14} - 164 \nu^{12} - 1250 \nu^{10} - 3984 \nu^{8} - 5334 \nu^{6} - 3024 \nu^{4} - 596 \nu^{2} - 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{15} + 4 \nu^{14} + 113 \nu^{13} + 90 \nu^{12} + 799 \nu^{11} + 630 \nu^{10} + 2176 \nu^{9} + 1674 \nu^{8} + 1922 \nu^{7} + 1352 \nu^{6} + 218 \nu^{5} + 20 \nu^{4} - 282 \nu^{3} - 212 \nu^{2} - 56 \nu - 28 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( 15 \nu^{14} + 3 \nu^{13} + 352 \nu^{12} + 69 \nu^{11} + 2692 \nu^{10} + 506 \nu^{9} + 8638 \nu^{8} + 1488 \nu^{7} + 11726 \nu^{6} + 1638 \nu^{5} + 6808 \nu^{4} + 594 \nu^{3} + 1496 \nu^{2} + 28 \nu + 76 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -15 \nu^{14} + 3 \nu^{13} - 352 \nu^{12} + 69 \nu^{11} - 2692 \nu^{10} + 506 \nu^{9} - 8638 \nu^{8} + 1488 \nu^{7} - 11726 \nu^{6} + 1638 \nu^{5} - 6808 \nu^{4} + 594 \nu^{3} - 1496 \nu^{2} + 28 \nu - 76 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{15} - 20 \nu^{14} - 107 \nu^{13} - 474 \nu^{12} - 657 \nu^{11} - 3698 \nu^{10} - 1074 \nu^{9} - 12334 \nu^{8} + 1686 \nu^{7} - 18152 \nu^{6} + 4770 \nu^{5} - 12164 \nu^{4} + 3014 \nu^{3} - 3428 \nu^{2} + 484 \nu - 300 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( -11 \nu^{15} - 14 \nu^{14} - 260 \nu^{13} - 330 \nu^{12} - 2018 \nu^{11} - 2548 \nu^{10} - 6672 \nu^{9} - 8350 \nu^{8} - 9702 \nu^{7} - 11972 \nu^{6} - 6544 \nu^{5} - 8004 \nu^{4} - 2004 \nu^{3} - 2440 \nu^{2} - 272 \nu - 236 \)\()/16\)
\(\beta_{9}\)\(=\)\((\)\( 19 \nu^{15} - \nu^{14} + 451 \nu^{13} - 20 \nu^{12} + 3529 \nu^{11} - 100 \nu^{10} + 11832 \nu^{9} + 4 \nu^{8} + 17582 \nu^{7} + 918 \nu^{6} + 11966 \nu^{5} + 1440 \nu^{4} + 3498 \nu^{3} + 664 \nu^{2} + 344 \nu + 72 \)\()/16\)
\(\beta_{10}\)\(=\)\((\)\( 19 \nu^{15} + \nu^{14} + 451 \nu^{13} + 20 \nu^{12} + 3529 \nu^{11} + 100 \nu^{10} + 11832 \nu^{9} - 4 \nu^{8} + 17582 \nu^{7} - 918 \nu^{6} + 11966 \nu^{5} - 1440 \nu^{4} + 3498 \nu^{3} - 664 \nu^{2} + 344 \nu - 72 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( 11 \nu^{15} - 28 \nu^{14} + 260 \nu^{13} - 658 \nu^{12} + 2018 \nu^{11} - 5048 \nu^{10} + 6672 \nu^{9} - 16318 \nu^{8} + 9702 \nu^{7} - 22640 \nu^{6} + 6544 \nu^{5} - 14052 \nu^{4} + 2004 \nu^{3} - 3632 \nu^{2} + 272 \nu - 268 \)\()/16\)
\(\beta_{12}\)\(=\)\((\)\( -11 \nu^{15} - 261 \nu^{13} - 2042 \nu^{11} - 6862 \nu^{9} - 10334 \nu^{7} - 7410 \nu^{5} - 2412 \nu^{3} - 252 \nu \)\()/8\)
\(\beta_{13}\)\(=\)\((\)\( -11 \nu^{15} - 12 \nu^{14} - 258 \nu^{13} - 282 \nu^{12} - 1971 \nu^{11} - 2164 \nu^{10} - 6311 \nu^{9} - 7004 \nu^{8} - 8530 \nu^{7} - 9752 \nu^{6} - 4916 \nu^{5} - 6092 \nu^{4} - 1046 \nu^{3} - 1608 \nu^{2} - 38 \nu - 136 \)\()/8\)
\(\beta_{14}\)\(=\)\((\)\( -11 \nu^{15} + 12 \nu^{14} - 258 \nu^{13} + 282 \nu^{12} - 1971 \nu^{11} + 2164 \nu^{10} - 6311 \nu^{9} + 7004 \nu^{8} - 8530 \nu^{7} + 9752 \nu^{6} - 4916 \nu^{5} + 6092 \nu^{4} - 1046 \nu^{3} + 1608 \nu^{2} - 38 \nu + 136 \)\()/8\)
\(\beta_{15}\)\(=\)\((\)\( 38 \nu^{15} + 896 \nu^{13} + 6921 \nu^{11} + 22674 \nu^{9} + 32332 \nu^{7} + 20960 \nu^{5} + 5842 \nu^{3} + 540 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(3\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(9\) \(\beta_{14}\mathstrut +\mathstrut \) \(27\) \(\beta_{13}\mathstrut -\mathstrut \) \(18\) \(\beta_{12}\mathstrut -\mathstrut \) \(12\) \(\beta_{11}\mathstrut +\mathstrut \) \(20\) \(\beta_{10}\mathstrut -\mathstrut \) \(20\) \(\beta_{9}\mathstrut -\mathstrut \) \(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(48\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(50\) \(\beta_{15}\mathstrut +\mathstrut \) \(20\) \(\beta_{14}\mathstrut -\mathstrut \) \(9\) \(\beta_{13}\mathstrut +\mathstrut \) \(55\) \(\beta_{12}\mathstrut -\mathstrut \) \(40\) \(\beta_{11}\mathstrut -\mathstrut \) \(31\) \(\beta_{10}\mathstrut -\mathstrut \) \(31\) \(\beta_{9}\mathstrut +\mathstrut \) \(40\) \(\beta_{8}\mathstrut +\mathstrut \) \(29\) \(\beta_{7}\mathstrut -\mathstrut \) \(95\) \(\beta_{6}\mathstrut -\mathstrut \) \(95\) \(\beta_{5}\mathstrut -\mathstrut \) \(29\) \(\beta_{4}\mathstrut +\mathstrut \) \(40\) \(\beta_{3}\mathstrut +\mathstrut \) \(83\) \(\beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(47\) \(\beta_{14}\mathstrut -\mathstrut \) \(171\) \(\beta_{13}\mathstrut +\mathstrut \) \(124\) \(\beta_{12}\mathstrut +\mathstrut \) \(72\) \(\beta_{11}\mathstrut -\mathstrut \) \(140\) \(\beta_{10}\mathstrut +\mathstrut \) \(140\) \(\beta_{9}\mathstrut +\mathstrut \) \(72\) \(\beta_{8}\mathstrut +\mathstrut \) \(124\) \(\beta_{7}\mathstrut +\mathstrut \) \(37\) \(\beta_{6}\mathstrut -\mathstrut \) \(37\) \(\beta_{5}\mathstrut +\mathstrut \) \(124\) \(\beta_{4}\mathstrut -\mathstrut \) \(67\) \(\beta_{3}\mathstrut -\mathstrut \) \(57\) \(\beta_{2}\mathstrut -\mathstrut \) \(252\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(672\) \(\beta_{15}\mathstrut -\mathstrut \) \(279\) \(\beta_{14}\mathstrut +\mathstrut \) \(93\) \(\beta_{13}\mathstrut -\mathstrut \) \(756\) \(\beta_{12}\mathstrut +\mathstrut \) \(497\) \(\beta_{11}\mathstrut +\mathstrut \) \(413\) \(\beta_{10}\mathstrut +\mathstrut \) \(413\) \(\beta_{9}\mathstrut -\mathstrut \) \(497\) \(\beta_{8}\mathstrut -\mathstrut \) \(372\) \(\beta_{7}\mathstrut +\mathstrut \) \(1091\) \(\beta_{6}\mathstrut +\mathstrut \) \(1091\) \(\beta_{5}\mathstrut +\mathstrut \) \(372\) \(\beta_{4}\mathstrut -\mathstrut \) \(497\) \(\beta_{3}\mathstrut -\mathstrut \) \(956\) \(\beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\(-\)\(540\) \(\beta_{14}\mathstrut +\mathstrut \) \(2120\) \(\beta_{13}\mathstrut -\mathstrut \) \(1580\) \(\beta_{12}\mathstrut -\mathstrut \) \(876\) \(\beta_{11}\mathstrut +\mathstrut \) \(1792\) \(\beta_{10}\mathstrut -\mathstrut \) \(1792\) \(\beta_{9}\mathstrut -\mathstrut \) \(876\) \(\beta_{8}\mathstrut -\mathstrut \) \(1580\) \(\beta_{7}\mathstrut -\mathstrut \) \(396\) \(\beta_{6}\mathstrut +\mathstrut \) \(396\) \(\beta_{5}\mathstrut -\mathstrut \) \(1580\) \(\beta_{4}\mathstrut +\mathstrut \) \(784\) \(\beta_{3}\mathstrut +\mathstrut \) \(680\) \(\beta_{2}\mathstrut +\mathstrut \) \(2910\)
\(\nu^{9}\)\(=\)\(4248\) \(\beta_{15}\mathstrut +\mathstrut \) \(1782\) \(\beta_{14}\mathstrut -\mathstrut \) \(534\) \(\beta_{13}\mathstrut +\mathstrut \) \(4812\) \(\beta_{12}\mathstrut -\mathstrut \) \(3054\) \(\beta_{11}\mathstrut -\mathstrut \) \(2610\) \(\beta_{10}\mathstrut -\mathstrut \) \(2610\) \(\beta_{9}\mathstrut +\mathstrut \) \(3054\) \(\beta_{8}\mathstrut +\mathstrut \) \(2316\) \(\beta_{7}\mathstrut -\mathstrut \) \(6517\) \(\beta_{6}\mathstrut -\mathstrut \) \(6517\) \(\beta_{5}\mathstrut -\mathstrut \) \(2316\) \(\beta_{4}\mathstrut +\mathstrut \) \(3054\) \(\beta_{3}\mathstrut +\mathstrut \) \(5723\) \(\beta_{1}\)
\(\nu^{10}\)\(=\)\(6459\) \(\beta_{14}\mathstrut -\mathstrut \) \(26078\) \(\beta_{13}\mathstrut +\mathstrut \) \(19619\) \(\beta_{12}\mathstrut +\mathstrut \) \(10707\) \(\beta_{11}\mathstrut -\mathstrut \) \(22287\) \(\beta_{10}\mathstrut +\mathstrut \) \(22287\) \(\beta_{9}\mathstrut +\mathstrut \) \(10707\) \(\beta_{8}\mathstrut +\mathstrut \) \(19619\) \(\beta_{7}\mathstrut +\mathstrut \) \(4587\) \(\beta_{6}\mathstrut -\mathstrut \) \(4587\) \(\beta_{5}\mathstrut +\mathstrut \) \(19619\) \(\beta_{4}\mathstrut -\mathstrut \) \(9423\) \(\beta_{3}\mathstrut -\mathstrut \) \(8243\) \(\beta_{2}\mathstrut -\mathstrut \) \(34858\)
\(\nu^{11}\)\(=\)\(-\)\(52613\) \(\beta_{15}\mathstrut -\mathstrut \) \(22139\) \(\beta_{14}\mathstrut +\mathstrut \) \(6403\) \(\beta_{13}\mathstrut -\mathstrut \) \(59741\) \(\beta_{12}\mathstrut +\mathstrut \) \(37433\) \(\beta_{11}\mathstrut +\mathstrut \) \(32346\) \(\beta_{10}\mathstrut +\mathstrut \) \(32346\) \(\beta_{9}\mathstrut -\mathstrut \) \(37433\) \(\beta_{8}\mathstrut -\mathstrut \) \(28542\) \(\beta_{7}\mathstrut +\mathstrut \) \(79079\) \(\beta_{6}\mathstrut +\mathstrut \) \(79079\) \(\beta_{5}\mathstrut +\mathstrut \) \(28542\) \(\beta_{4}\mathstrut -\mathstrut \) \(37433\) \(\beta_{3}\mathstrut -\mathstrut \) \(69508\) \(\beta_{1}\)
\(\nu^{12}\)\(=\)\(-\)\(78431\) \(\beta_{14}\mathstrut +\mathstrut \) \(319865\) \(\beta_{13}\mathstrut -\mathstrut \) \(241434\) \(\beta_{12}\mathstrut -\mathstrut \) \(131044\) \(\beta_{11}\mathstrut +\mathstrut \) \(274428\) \(\beta_{10}\mathstrut -\mathstrut \) \(274428\) \(\beta_{9}\mathstrut -\mathstrut \) \(131044\) \(\beta_{8}\mathstrut -\mathstrut \) \(241434\) \(\beta_{7}\mathstrut -\mathstrut \) \(54976\) \(\beta_{6}\mathstrut +\mathstrut \) \(54976\) \(\beta_{5}\mathstrut -\mathstrut \) \(241434\) \(\beta_{4}\mathstrut +\mathstrut \) \(114548\) \(\beta_{3}\mathstrut +\mathstrut \) \(100538\) \(\beta_{2}\mathstrut +\mathstrut \) \(423384\)
\(\nu^{13}\)\(=\)\(646906\) \(\beta_{15}\mathstrut +\mathstrut \) \(272464\) \(\beta_{14}\mathstrut -\mathstrut \) \(77907\) \(\beta_{13}\mathstrut +\mathstrut \) \(735189\) \(\beta_{12}\mathstrut -\mathstrut \) \(458484\) \(\beta_{11}\mathstrut -\mathstrut \) \(397897\) \(\beta_{10}\mathstrut -\mathstrut \) \(397897\) \(\beta_{9}\mathstrut +\mathstrut \) \(458484\) \(\beta_{8}\mathstrut +\mathstrut \) \(350371\) \(\beta_{7}\mathstrut -\mathstrut \) \(965133\) \(\beta_{6}\mathstrut -\mathstrut \) \(965133\) \(\beta_{5}\mathstrut -\mathstrut \) \(350371\) \(\beta_{4}\mathstrut +\mathstrut \) \(458484\) \(\beta_{3}\mathstrut +\mathstrut \) \(848621\) \(\beta_{1}\)
\(\nu^{14}\)\(=\)\(957558\) \(\beta_{14}\mathstrut -\mathstrut \) \(3919222\) \(\beta_{13}\mathstrut +\mathstrut \) \(2961664\) \(\beta_{12}\mathstrut +\mathstrut \) \(1604464\) \(\beta_{11}\mathstrut -\mathstrut \) \(3367136\) \(\beta_{10}\mathstrut +\mathstrut \) \(3367136\) \(\beta_{9}\mathstrut +\mathstrut \) \(1604464\) \(\beta_{8}\mathstrut +\mathstrut \) \(2961664\) \(\beta_{7}\mathstrut +\mathstrut \) \(667774\) \(\beta_{6}\mathstrut -\mathstrut \) \(667774\) \(\beta_{5}\mathstrut +\mathstrut \) \(2961664\) \(\beta_{4}\mathstrut -\mathstrut \) \(1398722\) \(\beta_{3}\mathstrut -\mathstrut \) \(1229198\) \(\beta_{2}\mathstrut -\mathstrut \) \(5168936\)
\(\nu^{15}\)\(=\)\(-\)\(7933264\) \(\beta_{15}\mathstrut -\mathstrut \) \(3342241\) \(\beta_{14}\mathstrut +\mathstrut \) \(952247\) \(\beta_{13}\mathstrut -\mathstrut \) \(9018824\) \(\beta_{12}\mathstrut +\mathstrut \) \(5614479\) \(\beta_{11}\mathstrut +\mathstrut \) \(4880807\) \(\beta_{10}\mathstrut +\mathstrut \) \(4880807\) \(\beta_{9}\mathstrut -\mathstrut \) \(5614479\) \(\beta_{8}\mathstrut -\mathstrut \) \(4294488\) \(\beta_{7}\mathstrut +\mathstrut \) \(11804009\) \(\beta_{6}\mathstrut +\mathstrut \) \(11804009\) \(\beta_{5}\mathstrut +\mathstrut \) \(4294488\) \(\beta_{4}\mathstrut -\mathstrut \) \(5614479\) \(\beta_{3}\mathstrut -\mathstrut \) \(10380392\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.528036i
3.49930i
0.528036i
3.49930i
1.05636i
0.724535i
1.05636i
0.724535i
0.357857i
2.13875i
0.357857i
2.13875i
0.886177i
2.08509i
0.886177i
2.08509i
−1.30656 0.541196i −1.13705 + 1.30656i 1.41421 + 1.41421i 0 2.19274 1.09174i 2.27411i −1.08239 2.61313i −0.414214 2.97127i 0
251.2 −1.30656 0.541196i 1.13705 + 1.30656i 1.41421 + 1.41421i 0 −0.778527 2.32248i 2.27411i −1.08239 2.61313i −0.414214 + 2.97127i 0
251.3 −1.30656 + 0.541196i −1.13705 1.30656i 1.41421 1.41421i 0 2.19274 + 1.09174i 2.27411i −1.08239 + 2.61313i −0.414214 + 2.97127i 0
251.4 −1.30656 + 0.541196i 1.13705 1.30656i 1.41421 1.41421i 0 −0.778527 + 2.32248i 2.27411i −1.08239 + 2.61313i −0.414214 2.97127i 0
251.5 −0.541196 1.30656i −1.64533 0.541196i −1.41421 + 1.41421i 0 0.183339 + 2.44262i 3.29066i 2.61313 + 1.08239i 2.41421 + 1.78089i 0
251.6 −0.541196 1.30656i 1.64533 0.541196i −1.41421 + 1.41421i 0 −1.59755 1.85683i 3.29066i 2.61313 + 1.08239i 2.41421 1.78089i 0
251.7 −0.541196 + 1.30656i −1.64533 + 0.541196i −1.41421 1.41421i 0 0.183339 2.44262i 3.29066i 2.61313 1.08239i 2.41421 1.78089i 0
251.8 −0.541196 + 1.30656i 1.64533 + 0.541196i −1.41421 1.41421i 0 −1.59755 + 1.85683i 3.29066i 2.61313 1.08239i 2.41421 + 1.78089i 0
251.9 0.541196 1.30656i −1.64533 0.541196i −1.41421 1.41421i 0 −1.59755 + 1.85683i 3.29066i −2.61313 + 1.08239i 2.41421 + 1.78089i 0
251.10 0.541196 1.30656i 1.64533 0.541196i −1.41421 1.41421i 0 0.183339 2.44262i 3.29066i −2.61313 + 1.08239i 2.41421 1.78089i 0
251.11 0.541196 + 1.30656i −1.64533 + 0.541196i −1.41421 + 1.41421i 0 −1.59755 1.85683i 3.29066i −2.61313 1.08239i 2.41421 1.78089i 0
251.12 0.541196 + 1.30656i 1.64533 + 0.541196i −1.41421 + 1.41421i 0 0.183339 + 2.44262i 3.29066i −2.61313 1.08239i 2.41421 + 1.78089i 0
251.13 1.30656 0.541196i −1.13705 + 1.30656i 1.41421 1.41421i 0 −0.778527 + 2.32248i 2.27411i 1.08239 2.61313i −0.414214 2.97127i 0
251.14 1.30656 0.541196i 1.13705 + 1.30656i 1.41421 1.41421i 0 2.19274 + 1.09174i 2.27411i 1.08239 2.61313i −0.414214 + 2.97127i 0
251.15 1.30656 + 0.541196i −1.13705 1.30656i 1.41421 + 1.41421i 0 −0.778527 2.32248i 2.27411i 1.08239 + 2.61313i −0.414214 + 2.97127i 0
251.16 1.30656 + 0.541196i 1.13705 1.30656i 1.41421 + 1.41421i 0 2.19274 1.09174i 2.27411i 1.08239 + 2.61313i −0.414214 2.97127i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
5.b Even 1 yes
8.d Odd 1 no
15.d Odd 1 yes
24.f Even 1 no
40.e Odd 1 no
120.m Even 1 no

Hecke kernels

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

\(T_{7}^{4} \) \(\mathstrut +\mathstrut 16 T_{7}^{2} \) \(\mathstrut +\mathstrut 56 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 24 T_{11}^{2} \) \(\mathstrut +\mathstrut 112 \)
\(T_{23}^{4} \) \(\mathstrut -\mathstrut 8 T_{23}^{2} \) \(\mathstrut +\mathstrut 8 \)
\(T_{43}^{4} \) \(\mathstrut -\mathstrut 112 T_{43}^{2} \) \(\mathstrut +\mathstrut 2744 \)