Properties

Label 585.2.i.f
Level $585$
Weight $2$
Character orbit 585.i
Analytic conductor $4.671$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} - 9 x^{7} + 1221 x^{6} + 133 x^{5} + 1245 x^{4} - 14 x^{3} + 391 x^{2} + 63 x + 81\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{3} + ( \beta_{9} - \beta_{14} ) q^{4} + \beta_{9} q^{5} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{6} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{7} + ( -1 - \beta_{3} - \beta_{10} + \beta_{13} ) q^{8} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{14} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{3} + ( \beta_{9} - \beta_{14} ) q^{4} + \beta_{9} q^{5} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{6} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{7} + ( -1 - \beta_{3} - \beta_{10} + \beta_{13} ) q^{8} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{14} ) q^{9} + \beta_{8} q^{10} + ( 1 - \beta_{1} + \beta_{4} + 2 \beta_{7} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{12} + \beta_{9} q^{13} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{14} -\beta_{7} q^{15} + ( 2 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{14} ) q^{16} + ( 1 + 2 \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{7} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{18} + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{10} + \beta_{14} ) q^{19} + ( -1 + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} ) q^{20} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{21} + ( 3 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{22} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{15} ) q^{23} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{24} + ( -1 - \beta_{9} ) q^{25} + \beta_{8} q^{26} + ( 2 + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{27} + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{28} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{30} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{31} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{32} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} - 4 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{33} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{34} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{14} ) q^{35} + ( 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{36} + ( -2 - \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{10} + \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{15} ) q^{37} + ( -2 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{38} -\beta_{7} q^{39} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{40} + ( 3 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} + 3 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{41} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{42} + ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{43} + ( 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{44} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{13} ) q^{45} + ( 1 + 2 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - 6 \beta_{10} - \beta_{11} + 3 \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{46} + ( 1 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 3 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{47} + ( 3 - 3 \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} - 4 \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{48} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{49} + ( -\beta_{1} - \beta_{8} ) q^{50} + ( -1 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} + \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{51} + ( -1 + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} ) q^{52} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{53} + ( 4 + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + 4 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{54} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{55} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{56} + ( -4 + 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} + 4 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + \beta_{11} + 3 \beta_{14} + 2 \beta_{15} ) q^{57} + ( 5 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} + 5 \beta_{8} - 3 \beta_{9} - 8 \beta_{10} + \beta_{11} + 5 \beta_{13} + 5 \beta_{14} + 3 \beta_{15} ) q^{58} + ( \beta_{2} - 5 \beta_{3} - 5 \beta_{6} + 5 \beta_{7} - \beta_{9} - \beta_{10} - 5 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{59} + ( \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{60} + ( 3 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{61} + ( -4 + 2 \beta_{1} - \beta_{2} + 6 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{10} + 3 \beta_{11} - \beta_{12} - 5 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{62} + ( -3 - 2 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{63} + ( -2 - \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{64} + ( -1 - \beta_{9} ) q^{65} + ( 5 - \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{8} - \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{66} + ( -2 \beta_{2} + \beta_{3} - \beta_{7} - 2 \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{67} + ( -\beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + 3 \beta_{13} - 3 \beta_{14} ) q^{68} + ( -2 - \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 6 \beta_{6} + \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{69} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{70} + ( -5 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - 3 \beta_{8} + 2 \beta_{13} - 2 \beta_{14} ) q^{71} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{72} + ( -2 + 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + \beta_{6} - 5 \beta_{7} - \beta_{8} + 6 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 7 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{73} + ( 1 + 3 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} - 4 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{74} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{75} + ( -\beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{76} + ( 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 4 \beta_{8} - 3 \beta_{9} + \beta_{11} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{77} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{9} + 2 \beta_{10} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{78} + ( 6 - 5 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{6} + \beta_{7} - 4 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - 4 \beta_{14} ) q^{79} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{80} + ( -2 - 2 \beta_{1} - 6 \beta_{3} - 3 \beta_{4} - 8 \beta_{6} + \beta_{7} - 7 \beta_{8} - \beta_{9} + \beta_{10} - 4 \beta_{11} + \beta_{12} + 4 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{81} + ( -4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 7 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{82} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} + 4 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} + 4 \beta_{14} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{10} + 2 \beta_{11} + \beta_{12} - 3 \beta_{13} ) q^{84} + ( -\beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{85} + ( 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} + 5 \beta_{8} - 5 \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{86} + ( -2 - 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} - 2 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} + 3 \beta_{12} + 6 \beta_{13} + 4 \beta_{15} ) q^{87} + ( 2 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{88} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} + 4 \beta_{8} - 3 \beta_{10} + \beta_{11} + 2 \beta_{14} ) q^{89} + ( 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{90} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{14} ) q^{91} + ( -2 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{92} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{93} + ( \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} - 6 \beta_{9} - 2 \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{94} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{95} + ( -2 - \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - \beta_{7} + 7 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{96} + ( 1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{12} + 2 \beta_{14} ) q^{97} + ( 7 + 4 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - 8 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{98} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} - 2 q^{3} - 5 q^{4} - 8 q^{5} - 13 q^{6} + 6 q^{7} - 12 q^{8} + 4 q^{9} + O(q^{10}) \) \( 16 q + q^{2} - 2 q^{3} - 5 q^{4} - 8 q^{5} - 13 q^{6} + 6 q^{7} - 12 q^{8} + 4 q^{9} - 2 q^{10} + 9 q^{11} - 16 q^{12} - 8 q^{13} - 3 q^{14} + q^{15} + 13 q^{16} + 12 q^{17} + 23 q^{18} - 22 q^{19} - 5 q^{20} + 15 q^{21} + 4 q^{22} + 3 q^{23} - 12 q^{24} - 8 q^{25} - 2 q^{26} + 22 q^{27} - 26 q^{28} + 8 q^{29} + 8 q^{30} + 18 q^{31} + 3 q^{32} - 26 q^{33} + 9 q^{34} - 12 q^{35} + 5 q^{36} - 36 q^{37} - 8 q^{38} + q^{39} + 6 q^{40} - 17 q^{41} - 45 q^{42} + 17 q^{43} + 10 q^{44} + q^{45} + 6 q^{46} + 11 q^{47} + 35 q^{48} + 16 q^{49} + q^{50} - 16 q^{51} - 5 q^{52} + 20 q^{53} + 44 q^{54} - 18 q^{55} - q^{56} - 25 q^{57} + 10 q^{58} + 7 q^{59} + 5 q^{60} + 21 q^{61} - 58 q^{62} - 30 q^{63} - 20 q^{64} - 8 q^{65} + 68 q^{66} + 13 q^{67} - 16 q^{68} - 13 q^{69} - 3 q^{70} - 68 q^{71} + 36 q^{72} - 32 q^{73} - 4 q^{74} + q^{75} + 2 q^{76} + 18 q^{77} + 8 q^{78} + 37 q^{79} - 26 q^{80} - 32 q^{81} + 2 q^{82} + 3 q^{83} + 27 q^{84} - 6 q^{85} - 2 q^{86} - 20 q^{87} + 19 q^{88} + 28 q^{89} - 16 q^{90} - 12 q^{91} - 14 q^{92} + 19 q^{93} + 44 q^{94} + 11 q^{95} - 35 q^{96} + 17 q^{97} + 90 q^{98} - 44 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} - 9 x^{7} + 1221 x^{6} + 133 x^{5} + 1245 x^{4} - 14 x^{3} + 391 x^{2} + 63 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-482086147342 \nu^{15} - 82999536459994 \nu^{14} + 51742196981879 \nu^{13} - 899262551344468 \nu^{12} + 29888665074896 \nu^{11} - 6008483088362072 \nu^{10} - 655850999572883 \nu^{9} - 22854640199651823 \nu^{8} - 6835814771962126 \nu^{7} - 61053441921082007 \nu^{6} - 16621010179418725 \nu^{5} - 88736525406721413 \nu^{4} - 21804108527677176 \nu^{3} - 85814814008979952 \nu^{2} - 8427487924594911 \nu - 15267025981692153\)\()/ 9139478381478873 \)
\(\beta_{3}\)\(=\)\((\)\(-3469225716849 \nu^{15} + 7812059681008 \nu^{14} - 8290755936386 \nu^{13} - 2697108934580 \nu^{12} + 189377660393089 \nu^{11} + 16556478346708 \nu^{10} + 1498667977954772 \nu^{9} + 543181503433067 \nu^{8} + 6528933517925721 \nu^{7} + 5162807733352603 \nu^{6} + 13241752875048308 \nu^{5} + 18517847455679833 \nu^{4} + 17630686105303827 \nu^{3} + 28007951530950219 \nu^{2} + 82246147650769 \nu + 12833642664201015\)\()/ 9139478381478873 \)
\(\beta_{4}\)\(=\)\((\)\(6246753964615 \nu^{15} - 59652178722282 \nu^{14} + 157853267854272 \nu^{13} - 554550721812852 \nu^{12} + 939963055735038 \nu^{11} - 3319089373556393 \nu^{10} + 5043192392654460 \nu^{9} - 11225648306363066 \nu^{8} + 13576539103281376 \nu^{7} - 26605135287110714 \nu^{6} + 36230288966505302 \nu^{5} - 33863989763810635 \nu^{4} + 36951118425472407 \nu^{3} - 29817520384644188 \nu^{2} + 47347580732583266 \nu + 188168453461782\)\()/ 9139478381478873 \)
\(\beta_{5}\)\(=\)\((\)\(-10150138176959 \nu^{15} + 69644913861 \nu^{14} - 83292291525414 \nu^{13} - 118768769872326 \nu^{12} - 568340771522781 \nu^{11} - 634932225031991 \nu^{10} - 2850156002058894 \nu^{9} - 2287548272381948 \nu^{8} - 10896063663200009 \nu^{7} - 3020771374915802 \nu^{6} - 28231418467650946 \nu^{5} + 346318987982825 \nu^{4} - 43280949748322271 \nu^{3} + 1711574395907992 \nu^{2} - 26296054994954299 \nu + 10022852458122111\)\()/ 9139478381478873 \)
\(\beta_{6}\)\(=\)\((\)\(16744116205687 \nu^{15} - 45341940014915 \nu^{14} + 176286820995922 \nu^{13} - 394886776051364 \nu^{12} + 1026293635905532 \nu^{11} - 2740194677046448 \nu^{10} + 2842579273044968 \nu^{9} - 9946106370147720 \nu^{8} + 3410762945368948 \nu^{7} - 29202452252119336 \nu^{6} - 7293677405066531 \nu^{5} - 42465191302472856 \nu^{4} - 21257885787172131 \nu^{3} - 45244610063730803 \nu^{2} - 14181352848452430 \nu + 3465152003612358\)\()/ 9139478381478873 \)
\(\beta_{7}\)\(=\)\((\)\(26595513341993 \nu^{15} + 73952194191366 \nu^{14} + 112052967333795 \nu^{13} + 1032516115451622 \nu^{12} + 899822933969919 \nu^{11} + 6370987760378243 \nu^{10} + 2174607198860307 \nu^{9} + 24965334872155991 \nu^{8} + 8193067138457009 \nu^{7} + 58933039323116513 \nu^{6} + 3455848364026450 \nu^{5} + 86744344986601537 \nu^{4} + 8345422433200593 \nu^{3} + 66520663704511511 \nu^{2} - 15105298015423202 \nu + 7952721863900868\)\()/ 9139478381478873 \)
\(\beta_{8}\)\(=\)\((\)\(10058702437115 \nu^{15} - 6203109654385 \nu^{14} + 103630645608798 \nu^{13} + 1974780746302 \nu^{12} + 701826108448870 \nu^{11} + 80541385038920 \nu^{10} + 2653504647339595 \nu^{9} + 914336412587371 \nu^{8} + 7278933995715497 \nu^{7} + 2179877825131728 \nu^{6} + 10645673810520968 \nu^{5} + 4368639176864571 \nu^{4} + 12190149830375740 \nu^{3} + 2099036328412212 \nu^{2} + 993128556611358 \nu + 845713476880083\)\()/ 3046492793826291 \)
\(\beta_{9}\)\(=\)\((\)\(-31322721365929 \nu^{15} + 61498828677274 \nu^{14} - 363159263988374 \nu^{13} + 436182822290110 \nu^{12} - 2311957038839840 \nu^{11} + 2669287309933332 \nu^{10} - 8810642319636721 \nu^{9} + 8085804827482501 \nu^{8} - 21814004313126223 \nu^{7} + 22118706479439852 \nu^{6} - 31705409312404125 \nu^{5} + 27771099489894347 \nu^{4} - 25890870569987892 \nu^{3} + 37008967590250226 \nu^{2} - 5950075068841603 \nu + 1006054223780547\)\()/ 9139478381478873 \)
\(\beta_{10}\)\(=\)\((\)\(35353445168842 \nu^{15} - 42906025379669 \nu^{14} + 360549986338030 \nu^{13} - 261408818119538 \nu^{12} + 2346977874531172 \nu^{11} - 1738189450969768 \nu^{10} + 7939379548056260 \nu^{9} - 5381830987437618 \nu^{8} + 18436346491318150 \nu^{7} - 17754813181134811 \nu^{6} + 15550848768311545 \nu^{5} - 28360469660381838 \nu^{4} + 8592989216359623 \nu^{3} - 39267507171071219 \nu^{2} - 11838012848643870 \nu - 18971878017965067\)\()/ 9139478381478873 \)
\(\beta_{11}\)\(=\)\((\)\(47477396037032 \nu^{15} - 55102847496393 \nu^{14} + 436243278002799 \nu^{13} - 112029436571388 \nu^{12} + 2518026227560113 \nu^{11} - 864033044630458 \nu^{10} + 8674336787121810 \nu^{9} - 1319639334852235 \nu^{8} + 19687039264476329 \nu^{7} - 11054480958405496 \nu^{6} + 24635102803156015 \nu^{5} - 19746766936098476 \nu^{4} + 10463030136046236 \nu^{3} - 33052810323229990 \nu^{2} - 10729277200479152 \nu - 18277046259484959\)\()/ 9139478381478873 \)
\(\beta_{12}\)\(=\)\((\)\(67174987872150 \nu^{15} - 20896415688563 \nu^{14} + 620697587330554 \nu^{13} + 489098601157807 \nu^{12} + 3837661637756542 \nu^{11} + 4030094128429255 \nu^{10} + 13772024529051905 \nu^{9} + 22166122749372422 \nu^{8} + 36094915770719934 \nu^{7} + 59485031448387424 \nu^{6} + 53253791588998406 \nu^{5} + 109067758838777617 \nu^{4} + 52649488556200071 \nu^{3} + 91314609178178160 \nu^{2} + 5618911102245064 \nu + 27855550205743248\)\()/ 9139478381478873 \)
\(\beta_{13}\)\(=\)\((\)\(-88820209793387 \nu^{15} + 39343350153959 \nu^{14} - 891308516903932 \nu^{13} - 287803296009742 \nu^{12} - 5885557766462179 \nu^{11} - 2688129593090100 \nu^{10} - 22404008242064108 \nu^{9} - 15810686435053003 \nu^{8} - 62381927939342093 \nu^{7} - 38750539349362944 \nu^{6} - 98955484082891763 \nu^{5} - 62266292327076857 \nu^{4} - 110918644261366557 \nu^{3} - 36447991581067544 \nu^{2} - 23673309380329397 \nu - 7147318694846175\)\()/ 9139478381478873 \)
\(\beta_{14}\)\(=\)\((\)\(-31322721365929 \nu^{15} + 61498828677274 \nu^{14} - 363159263988374 \nu^{13} + 436182822290110 \nu^{12} - 2311957038839840 \nu^{11} + 2669287309933332 \nu^{10} - 8810642319636721 \nu^{9} + 8085804827482501 \nu^{8} - 21814004313126223 \nu^{7} + 22118706479439852 \nu^{6} - 31705409312404125 \nu^{5} + 27771099489894347 \nu^{4} - 25890870569987892 \nu^{3} + 33962474796423935 \nu^{2} - 5950075068841603 \nu + 1006054223780547\)\()/ 3046492793826291 \)
\(\beta_{15}\)\(=\)\((\)\(241239132189497 \nu^{15} - 269626419785329 \nu^{14} + 2707842518071070 \nu^{13} - 1384847430059254 \nu^{12} + 18188750599418894 \nu^{11} - 7008331821372974 \nu^{10} + 70587911675040145 \nu^{9} - 12238829708683665 \nu^{8} + 187862907036005150 \nu^{7} - 30755186597641787 \nu^{6} + 280378078725409877 \nu^{5} - 17879932341420474 \nu^{4} + 262801827503435319 \nu^{3} - 43331342630160394 \nu^{2} + 47651869820136828 \nu - 1957620602448711\)\()/ 9139478381478873 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{14} + 3 \beta_{9}\)
\(\nu^{3}\)\(=\)\(\beta_{13} - \beta_{10} + 4 \beta_{8} - \beta_{3} - 1\)
\(\nu^{4}\)\(=\)\(5 \beta_{14} - 4 \beta_{10} - 12 \beta_{9} + 5 \beta_{8} - \beta_{7} + 6 \beta_{6} - \beta_{5} - \beta_{2} - \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(-\beta_{15} + 8 \beta_{14} - 7 \beta_{13} + \beta_{11} + 8 \beta_{10} - 9 \beta_{9} - 11 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} + 6 \beta_{4} + 8 \beta_{3} - \beta_{2} - 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{15} + 9 \beta_{14} - 13 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 28 \beta_{10} - 25 \beta_{8} - 2 \beta_{7} - 14 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} + 21 \beta_{3} + 6 \beta_{2} + 9 \beta_{1} + 54\)
\(\nu^{7}\)\(=\)\(-32 \beta_{14} - 11 \beta_{13} - 10 \beta_{12} + 11 \beta_{11} + \beta_{10} + 63 \beta_{9} - 32 \beta_{8} + 11 \beta_{7} - 52 \beta_{6} + 23 \beta_{5} - 12 \beta_{4} + 10 \beta_{3} + 20 \beta_{2} + 65 \beta_{1} + 63\)
\(\nu^{8}\)\(=\)\(23 \beta_{15} - 170 \beta_{14} + 86 \beta_{13} - 51 \beta_{11} - 110 \beta_{10} + 263 \beta_{9} + 23 \beta_{8} + 75 \beta_{7} - 125 \beta_{6} - 63 \beta_{4} - 138 \beta_{3} + 36 \beta_{2} + 23 \beta_{1}\)
\(\nu^{9}\)\(=\)\(76 \beta_{15} - 148 \beta_{14} + 357 \beta_{13} + 76 \beta_{12} - 173 \beta_{11} - 368 \beta_{10} + 370 \beta_{8} + 87 \beta_{7} - 14 \beta_{6} - 184 \beta_{5} - 173 \beta_{4} - 443 \beta_{3} - 36 \beta_{2} - 148 \beta_{1} - 403\)
\(\nu^{10}\)\(=\)\(514 \beta_{14} + 198 \beta_{13} + 184 \beta_{12} - 198 \beta_{11} - 222 \beta_{10} - 1358 \beta_{9} + 514 \beta_{8} - 307 \beta_{7} + 921 \beta_{6} - 519 \beta_{5} - 101 \beta_{4} - 184 \beta_{3} - 407 \beta_{2} - 590 \beta_{1} - 1358\)
\(\nu^{11}\)\(=\)\(-519 \beta_{15} + 1886 \beta_{14} - 1638 \beta_{13} + 590 \beta_{11} + 2241 \beta_{10} - 2473 \beta_{9} - 966 \beta_{8} - 1193 \beta_{7} + 2016 \beta_{6} + 1119 \beta_{4} + 2312 \beta_{3} - 815 \beta_{2} - 966 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-1277 \beta_{15} + 2535 \beta_{14} - 5171 \beta_{13} - 1277 \beta_{12} + 3222 \beta_{11} + 6063 \beta_{10} - 3825 \beta_{8} - 1405 \beta_{7} - 205 \beta_{6} + 3350 \beta_{5} + 3222 \beta_{4} + 6988 \beta_{3} + 443 \beta_{2} + 2535 \beta_{1} + 7316\)
\(\nu^{13}\)\(=\)\(-4940 \beta_{14} - 3909 \beta_{13} - 3350 \beta_{12} + 3909 \beta_{11} + 526 \beta_{10} + 14858 \beta_{9} - 4940 \beta_{8} + 3797 \beta_{7} - 11113 \beta_{6} + 8265 \beta_{5} + 1200 \beta_{4} + 3350 \beta_{3} + 6173 \beta_{2} + 11097 \beta_{1} + 14858\)
\(\nu^{14}\)\(=\)\(8265 \beta_{15} - 28774 \beta_{14} + 23287 \beta_{13} - 10699 \beta_{11} - 32526 \beta_{10} + 40602 \beta_{9} + 8068 \beta_{8} + 19938 \beta_{7} - 29518 \beta_{6} - 15022 \beta_{4} - 34960 \beta_{3} + 13707 \beta_{2} + 8068 \beta_{1}\)
\(\nu^{15}\)\(=\)\(20912 \beta_{15} - 37783 \beta_{14} + 83331 \beta_{13} + 20912 \beta_{12} - 47996 \beta_{11} - 86483 \beta_{10} + 53709 \beta_{8} + 24406 \beta_{7} - 7061 \beta_{6} - 51490 \beta_{5} - 47996 \beta_{4} - 106921 \beta_{3} - 3164 \beta_{2} - 37783 \beta_{1} - 88279\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(\beta_{9}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1
−0.947115 + 1.64045i
−0.921972 + 1.59690i
−0.624867 + 1.08230i
−0.237622 + 0.411573i
0.316605 0.548376i
0.715744 1.23970i
0.987387 1.71020i
1.21184 2.09897i
−0.947115 1.64045i
−0.921972 1.59690i
−0.624867 1.08230i
−0.237622 0.411573i
0.316605 + 0.548376i
0.715744 + 1.23970i
0.987387 + 1.71020i
1.21184 + 2.09897i
−0.947115 + 1.64045i 1.31847 1.12323i −0.794055 1.37534i −0.500000 0.866025i 0.593868 + 3.22671i −0.837925 + 1.45133i −0.780216 0.476703 2.96188i 1.89423
196.2 −0.921972 + 1.59690i −0.495399 + 1.65969i −0.700064 1.21255i −0.500000 0.866025i −2.19362 2.32129i 2.38510 4.13111i −1.10613 −2.50916 1.64442i 1.84394
196.3 −0.624867 + 1.08230i 1.28189 + 1.16480i 0.219082 + 0.379462i −0.500000 0.866025i −2.06168 + 0.659540i 0.148724 0.257597i −3.04706 0.286466 + 2.98629i 1.24973
196.4 −0.237622 + 0.411573i −1.28679 1.15939i 0.887072 + 1.53645i −0.500000 0.866025i 0.782942 0.254110i −1.24774 + 2.16115i −1.79364 0.311632 + 2.98377i 0.475244
196.5 0.316605 0.548376i −1.15302 + 1.29250i 0.799523 + 1.38481i −0.500000 0.866025i 0.343725 + 1.04150i 0.896323 1.55248i 2.27895 −0.341107 2.98054i −0.633210
196.6 0.715744 1.23970i −1.70726 0.291973i −0.0245786 0.0425715i −0.500000 0.866025i −1.58392 + 1.90753i 0.625355 1.08315i 2.79261 2.82950 + 0.996950i −1.43149
196.7 0.987387 1.71020i 1.73142 + 0.0465761i −0.949865 1.64522i −0.500000 0.866025i 1.78924 2.91510i 0.124489 0.215622i 0.198009 2.99566 + 0.161286i −1.97477
196.8 1.21184 2.09897i −0.689312 1.58898i −1.93711 3.35518i −0.500000 0.866025i −4.17055 0.478743i 0.905675 1.56867i −4.54253 −2.04970 + 2.19060i −2.42368
391.1 −0.947115 1.64045i 1.31847 + 1.12323i −0.794055 + 1.37534i −0.500000 + 0.866025i 0.593868 3.22671i −0.837925 1.45133i −0.780216 0.476703 + 2.96188i 1.89423
391.2 −0.921972 1.59690i −0.495399 1.65969i −0.700064 + 1.21255i −0.500000 + 0.866025i −2.19362 + 2.32129i 2.38510 + 4.13111i −1.10613 −2.50916 + 1.64442i 1.84394
391.3 −0.624867 1.08230i 1.28189 1.16480i 0.219082 0.379462i −0.500000 + 0.866025i −2.06168 0.659540i 0.148724 + 0.257597i −3.04706 0.286466 2.98629i 1.24973
391.4 −0.237622 0.411573i −1.28679 + 1.15939i 0.887072 1.53645i −0.500000 + 0.866025i 0.782942 + 0.254110i −1.24774 2.16115i −1.79364 0.311632 2.98377i 0.475244
391.5 0.316605 + 0.548376i −1.15302 1.29250i 0.799523 1.38481i −0.500000 + 0.866025i 0.343725 1.04150i 0.896323 + 1.55248i 2.27895 −0.341107 + 2.98054i −0.633210
391.6 0.715744 + 1.23970i −1.70726 + 0.291973i −0.0245786 + 0.0425715i −0.500000 + 0.866025i −1.58392 1.90753i 0.625355 + 1.08315i 2.79261 2.82950 0.996950i −1.43149
391.7 0.987387 + 1.71020i 1.73142 0.0465761i −0.949865 + 1.64522i −0.500000 + 0.866025i 1.78924 + 2.91510i 0.124489 + 0.215622i 0.198009 2.99566 0.161286i −1.97477
391.8 1.21184 + 2.09897i −0.689312 + 1.58898i −1.93711 + 3.35518i −0.500000 + 0.866025i −4.17055 + 0.478743i 0.905675 + 1.56867i −4.54253 −2.04970 2.19060i −2.42368
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.i.f 16
3.b odd 2 1 1755.2.i.e 16
9.c even 3 1 inner 585.2.i.f 16
9.c even 3 1 5265.2.a.bb 8
9.d odd 6 1 1755.2.i.e 16
9.d odd 6 1 5265.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.f 16 1.a even 1 1 trivial
585.2.i.f 16 9.c even 3 1 inner
1755.2.i.e 16 3.b odd 2 1
1755.2.i.e 16 9.d odd 6 1
5265.2.a.bb 8 9.c even 3 1
5265.2.a.be 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\(T_{2}^{16} - \cdots\)
\(T_{7}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 81 + 63 T + 391 T^{2} - 14 T^{3} + 1245 T^{4} + 133 T^{5} + 1221 T^{6} - 9 T^{7} + 784 T^{8} - 4 T^{9} + 289 T^{10} - 18 T^{11} + 74 T^{12} - 4 T^{13} + 11 T^{14} - T^{15} + T^{16} \)
$3$ \( 6561 + 4374 T - 2430 T^{3} - 648 T^{4} + 810 T^{5} + 513 T^{6} - 189 T^{7} - 324 T^{8} - 63 T^{9} + 57 T^{10} + 30 T^{11} - 8 T^{12} - 10 T^{13} + 2 T^{15} + T^{16} \)
$5$ \( ( 1 + T + T^{2} )^{8} \)
$7$ \( 36 - 306 T + 1827 T^{2} - 5679 T^{3} + 13110 T^{4} - 14373 T^{5} + 14472 T^{6} - 9159 T^{7} + 7015 T^{8} - 3575 T^{9} + 2210 T^{10} - 763 T^{11} + 353 T^{12} - 88 T^{13} + 38 T^{14} - 6 T^{15} + T^{16} \)
$11$ \( 6561 + 40257 T + 221413 T^{2} + 238538 T^{3} + 364508 T^{4} + 33764 T^{5} + 231355 T^{6} - 10536 T^{7} + 103840 T^{8} - 32957 T^{9} + 20707 T^{10} - 4077 T^{11} + 1607 T^{12} - 298 T^{13} + 83 T^{14} - 9 T^{15} + T^{16} \)
$13$ \( ( 1 + T + T^{2} )^{8} \)
$17$ \( ( 822 + 1441 T - 1684 T^{2} - 1014 T^{3} + 466 T^{4} + 159 T^{5} - 36 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$19$ \( ( 51733 + 79819 T + 41174 T^{2} + 5521 T^{3} - 1956 T^{4} - 648 T^{5} - 21 T^{6} + 11 T^{7} + T^{8} )^{2} \)
$23$ \( 898560576 + 2093493864 T + 3777336745 T^{2} + 2807165779 T^{3} + 1756897499 T^{4} + 432939472 T^{5} + 185228593 T^{6} + 33477786 T^{7} + 13317103 T^{8} + 1454120 T^{9} + 437740 T^{10} + 16236 T^{11} + 9599 T^{12} + 124 T^{13} + 125 T^{14} - 3 T^{15} + T^{16} \)
$29$ \( 310831895529 + 111814582788 T + 123110211069 T^{2} + 2962146186 T^{3} + 23970707298 T^{4} + 1979793117 T^{5} + 1598604138 T^{6} + 25449315 T^{7} + 54222865 T^{8} - 325280 T^{9} + 1314307 T^{10} - 42107 T^{11} + 19936 T^{12} - 692 T^{13} + 205 T^{14} - 8 T^{15} + T^{16} \)
$31$ \( 18344089 - 197497696 T + 1931598515 T^{2} - 2068738808 T^{3} + 1934245988 T^{4} - 493051303 T^{5} + 212803033 T^{6} - 49224396 T^{7} + 15570025 T^{8} - 2958261 T^{9} + 659606 T^{10} - 103903 T^{11} + 18456 T^{12} - 2284 T^{13} + 269 T^{14} - 18 T^{15} + T^{16} \)
$37$ \( ( 236259 - 512577 T + 205125 T^{2} + 78171 T^{3} - 4781 T^{4} - 2411 T^{5} - 61 T^{6} + 18 T^{7} + T^{8} )^{2} \)
$41$ \( 3109623696 + 2321511084 T + 3973849209 T^{2} + 101147526 T^{3} + 2122532586 T^{4} + 469001556 T^{5} + 319112871 T^{6} + 41727141 T^{7} + 23413458 T^{8} + 3594165 T^{9} + 976969 T^{10} + 108871 T^{11} + 21110 T^{12} + 2175 T^{13} + 288 T^{14} + 17 T^{15} + T^{16} \)
$43$ \( 16841810176 + 2971481072 T + 13443213857 T^{2} - 7464680412 T^{3} + 8915095398 T^{4} - 2333698976 T^{5} + 837878257 T^{6} - 153471491 T^{7} + 41662896 T^{8} - 6550925 T^{9} + 1304699 T^{10} - 157147 T^{11} + 24498 T^{12} - 2445 T^{13} + 286 T^{14} - 17 T^{15} + T^{16} \)
$47$ \( 10850436 - 86194098 T + 657032407 T^{2} - 286136817 T^{3} + 341417302 T^{4} - 40005193 T^{5} + 97081173 T^{6} - 10298712 T^{7} + 13302295 T^{8} + 314467 T^{9} + 953329 T^{10} - 48720 T^{11} + 17263 T^{12} - 783 T^{13} + 218 T^{14} - 11 T^{15} + T^{16} \)
$53$ \( ( -405738 - 80877 T + 292659 T^{2} - 82800 T^{3} - 276 T^{4} + 2246 T^{5} - 166 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$59$ \( 1642567767129 + 2130729238413 T + 2912243416245 T^{2} + 396288063666 T^{3} + 356445367992 T^{4} - 70772266404 T^{5} + 51291807639 T^{6} - 6962482878 T^{7} + 1406953854 T^{8} - 83819331 T^{9} + 14599609 T^{10} - 653691 T^{11} + 107929 T^{12} - 2432 T^{13} + 397 T^{14} - 7 T^{15} + T^{16} \)
$61$ \( 28213508866321 - 23574381791750 T + 13217688198413 T^{2} - 4755955760690 T^{3} + 1357378138549 T^{4} - 293787725350 T^{5} + 55275885616 T^{6} - 8594403759 T^{7} + 1266702505 T^{8} - 154924263 T^{9} + 17696226 T^{10} - 1493620 T^{11} + 121854 T^{12} - 7152 T^{13} + 509 T^{14} - 21 T^{15} + T^{16} \)
$67$ \( 9767162560516 + 920463078150 T + 1528089679349 T^{2} + 240415203090 T^{3} + 182712582083 T^{4} + 24299642850 T^{5} + 8949794795 T^{6} + 616687027 T^{7} + 240786024 T^{8} + 10319917 T^{9} + 4344165 T^{10} - 1297 T^{11} + 44223 T^{12} - 1057 T^{13} + 360 T^{14} - 13 T^{15} + T^{16} \)
$71$ \( ( 1374696 + 972195 T - 22074 T^{2} - 150522 T^{3} - 34941 T^{4} - 1525 T^{5} + 297 T^{6} + 34 T^{7} + T^{8} )^{2} \)
$73$ \( ( 373998 - 1081671 T + 187068 T^{2} + 168582 T^{3} + 6328 T^{4} - 3387 T^{5} - 196 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$79$ \( 5500298896 - 20140643316 T + 53560501025 T^{2} - 57982784440 T^{3} + 43758913215 T^{4} - 20759109136 T^{5} + 7447377404 T^{6} - 1937720694 T^{7} + 410539489 T^{8} - 68215809 T^{9} + 10395334 T^{10} - 1334875 T^{11} + 162755 T^{12} - 13994 T^{13} + 953 T^{14} - 37 T^{15} + T^{16} \)
$83$ \( 568679695164036 - 69779272737738 T + 59987214981865 T^{2} + 333757326428 T^{3} + 3507063290555 T^{4} + 101065345842 T^{5} + 130390975414 T^{6} + 9341987214 T^{7} + 3006771373 T^{8} + 120786641 T^{9} + 25189324 T^{10} + 490941 T^{11} + 148377 T^{12} + 1318 T^{13} + 469 T^{14} - 3 T^{15} + T^{16} \)
$89$ \( ( 1052154 - 957871 T + 259278 T^{2} + 9537 T^{3} - 16719 T^{4} + 2789 T^{5} - 96 T^{6} - 14 T^{7} + T^{8} )^{2} \)
$97$ \( 71550365121 - 153594591201 T + 231125682528 T^{2} - 175391191269 T^{3} + 98924379975 T^{4} - 32706919125 T^{5} + 8937932481 T^{6} - 1587640047 T^{7} + 311378731 T^{8} - 43066902 T^{9} + 7270341 T^{10} - 634370 T^{11} + 68597 T^{12} - 3990 T^{13} + 397 T^{14} - 17 T^{15} + T^{16} \)
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