Properties

Label 45.2.l.a
Level $45$
Weight $2$
Character orbit 45.l
Analytic conductor $0.359$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 45.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + 1720995499366 \nu^{11} - 6827412737484 \nu^{10} + 6402779061545 \nu^{9} - 3349022853273 \nu^{8} - 9000312527194 \nu^{7} + 24987667997140 \nu^{6} - 8831452106135 \nu^{5} + 17087672692002 \nu^{4} + 9515651467064 \nu^{3} - 97367215891956 \nu^{2} + 394928996361 \nu + 33432180594\)\()/ 3707507912227 \)
\(\beta_{3}\)\(=\)\((\)\(94386116202 \nu^{15} - 619740656932 \nu^{14} + 2033008548312 \nu^{13} - 4441986378774 \nu^{12} + 5386888154268 \nu^{11} - 640680728681 \nu^{10} - 7214824090311 \nu^{9} + 16443022873810 \nu^{8} - 16824695358916 \nu^{7} - 2692744559593 \nu^{6} + 17484917305182 \nu^{5} - 49690281510088 \nu^{4} + 74318822560500 \nu^{3} - 6157027329225 \nu^{2} - 993781902738 \nu - 80029143512\)\()/ 3707507912227 \)
\(\beta_{4}\)\(=\)\((\)\(140020985001 \nu^{15} - 843322184609 \nu^{14} + 2535236829090 \nu^{13} - 5074237635644 \nu^{12} + 4819083120574 \nu^{11} + 2468033098841 \nu^{10} - 10010107392923 \nu^{9} + 18125438747586 \nu^{8} - 12670141167362 \nu^{7} - 14375839504346 \nu^{6} + 19963910433530 \nu^{5} - 58749535677614 \nu^{4} + 69548698425511 \nu^{3} + 39778718221026 \nu^{2} + 6229211681286 \nu - 5622211270526\)\()/ 3707507912227 \)
\(\beta_{5}\)\(=\)\((\)\(-140094553942 \nu^{15} + 840493754711 \nu^{14} - 2524530400854 \nu^{13} + 5054110370148 \nu^{12} - 4783342099524 \nu^{11} - 2485960949906 \nu^{10} + 10068880032759 \nu^{9} - 18433708480508 \nu^{8} + 12720523783684 \nu^{7} + 14340027118406 \nu^{6} - 20209428153588 \nu^{5} + 60275329582886 \nu^{4} - 68801672173612 \nu^{3} - 39600205283397 \nu^{2} - 9908294946195 \nu - 1652709999986\)\()/ 3707507912227 \)
\(\beta_{6}\)\(=\)\((\)\(190424165289 \nu^{15} - 1059314239720 \nu^{14} + 2941649532255 \nu^{13} - 5449473018186 \nu^{12} + 3809266070290 \nu^{11} + 5471229679434 \nu^{10} - 11091715879228 \nu^{9} + 18612405885154 \nu^{8} - 7831377865421 \nu^{7} - 23866374631831 \nu^{6} + 15235400765019 \nu^{5} - 70138530799181 \nu^{4} + 64517669865637 \nu^{3} + 87537780396607 \nu^{2} + 51260000406122 \nu + 7166890770427\)\()/ 3707507912227 \)
\(\beta_{7}\)\(=\)\((\)\(-250381984552 \nu^{15} + 1889212496921 \nu^{14} - 6927613311171 \nu^{13} + 16582911879478 \nu^{12} - 24294968776849 \nu^{11} + 12446807908885 \nu^{10} + 21076350180684 \nu^{9} - 61892560799590 \nu^{8} + 81094540846877 \nu^{7} - 23942148678807 \nu^{6} - 64342714660344 \nu^{5} + 171068416087064 \nu^{4} - 307914761564074 \nu^{3} + 166080601083280 \nu^{2} + 37629440983980 \nu + 5660844178894\)\()/ 3707507912227 \)
\(\beta_{8}\)\(=\)\((\)\(-254444403226 \nu^{15} + 1391318146672 \nu^{14} - 3674576400880 \nu^{13} + 6138067615014 \nu^{12} - 1944897155536 \nu^{11} - 13014144746287 \nu^{10} + 20020382213401 \nu^{9} - 23141068580356 \nu^{8} - 1175929695472 \nu^{7} + 52668080553873 \nu^{6} - 35147821517452 \nu^{5} + 83865626894092 \nu^{4} - 55287519626372 \nu^{3} - 184869896542460 \nu^{2} - 1923868016767 \nu + 146893504700\)\()/ 3707507912227 \)
\(\beta_{9}\)\(=\)\((\)\(196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + 7906540260 \nu^{11} + 2288455604 \nu^{10} - 14534291253 \nu^{9} + 28282811704 \nu^{8} - 22763827488 \nu^{7} - 16547104520 \nu^{6} + 31379154942 \nu^{5} - 90050852224 \nu^{4} + 112985327352 \nu^{3} + 38916218628 \nu^{2} + 6077223668 \nu + 785074438\)\()/ 1973128213 \)
\(\beta_{10}\)\(=\)\((\)\(-677106342206 \nu^{15} + 3890533925017 \nu^{14} - 11114551602021 \nu^{13} + 21019011866538 \nu^{12} - 16017341587322 \nu^{11} - 19694741515785 \nu^{10} + 47355950801010 \nu^{9} - 76380921929866 \nu^{8} + 36855540630045 \nu^{7} + 91030841771699 \nu^{6} - 84926350980924 \nu^{5} + 264557518559700 \nu^{4} - 259338049254711 \nu^{3} - 298633135730430 \nu^{2} - 72960092263182 \nu - 11893459432082\)\()/ 3707507912227 \)
\(\beta_{11}\)\(=\)\((\)\(-785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} - 34368654754 \nu^{11} - 6224799624 \nu^{10} + 58813815140 \nu^{9} - 118164117069 \nu^{8} + 101294734438 \nu^{7} + 57313765188 \nu^{6} - 129597823592 \nu^{5} + 370531312158 \nu^{4} - 484158220100 \nu^{3} - 113116110792 \nu^{2} - 17609140908 \nu - 3343669588\)\()/ 1973128213 \)
\(\beta_{12}\)\(=\)\((\)\(-1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + 118271636061 \nu^{9} - 231117756606 \nu^{8} + 186162436800 \nu^{7} + 134499720676 \nu^{6} - 250255719435 \nu^{5} + 736991238906 \nu^{4} - 924410484114 \nu^{3} - 318408624800 \nu^{2} - 82003950372 \nu - 10233974547\)\()/ 3810388399 \)
\(\beta_{13}\)\(=\)\((\)\(-2029780580195 \nu^{15} + 12360646755715 \nu^{14} - 37626995630777 \nu^{13} + 76333636000087 \nu^{12} - 75496074288529 \nu^{11} - 30557686749160 \nu^{10} + 149878719509936 \nu^{9} - 281602693598733 \nu^{8} + 213102612708737 \nu^{7} + 190394270523687 \nu^{6} - 312115024344538 \nu^{5} + 903298035710624 \nu^{4} - 1096523230221273 \nu^{3} - 494201049779995 \nu^{2} - 88708185746586 \nu - 14351633608601\)\()/ 3707507912227 \)
\(\beta_{14}\)\(=\)\((\)\(2802573231023 \nu^{15} - 17485025240258 \nu^{14} + 54572756591878 \nu^{13} - 113605741903926 \nu^{12} + 121395144569897 \nu^{11} + 23632381842971 \nu^{10} - 209882210310499 \nu^{9} + 419879212252284 \nu^{8} - 356956659982399 \nu^{7} - 208763256058877 \nu^{6} + 460873742162788 \nu^{5} - 1317355191257852 \nu^{4} + 1711060513528000 \nu^{3} + 423279276478650 \nu^{2} + 69635354161975 \nu + 12411058785072\)\()/ 3707507912227 \)
\(\beta_{15}\)\(=\)\((\)\(-2852772870096 \nu^{15} + 17521813816267 \nu^{14} - 53872302373806 \nu^{13} + 110572117714578 \nu^{12} - 113406120362540 \nu^{11} - 33753433646508 \nu^{10} + 208555923938660 \nu^{9} - 406505628692916 \nu^{8} + 326089380120540 \nu^{7} + 239007446234759 \nu^{6} - 439893314072684 \nu^{5} + 1297608293014434 \nu^{4} - 1622375494927184 \nu^{3} - 574674020284387 \nu^{2} - 143994055202790 \nu - 17972736434118\)\()/ 3707507912227 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{3} - 2 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 4 \beta_{3} - \beta_{2} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{15} + 5 \beta_{14} + 2 \beta_{13} + \beta_{12} + 8 \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 8\)
\(\nu^{5}\)\(=\)\(-16 \beta_{15} + 8 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 3 \beta_{9} - 8 \beta_{8} - 8 \beta_{6} - 8 \beta_{5} + 8 \beta_{3} + 7 \beta_{2} + 11 \beta_{1} + 14\)
\(\nu^{6}\)\(=\)\(-35 \beta_{15} - 16 \beta_{14} + 8 \beta_{13} + 30 \beta_{12} - 8 \beta_{11} + 16 \beta_{10} - \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 16 \beta_{6} - 57 \beta_{5} - 8 \beta_{4} + 34 \beta_{3} + 16 \beta_{2} + 25 \beta_{1} + 16\)
\(\nu^{7}\)\(=\)\(\beta_{15} - \beta_{14} - 10 \beta_{12} - 10 \beta_{11} + 51 \beta_{10} - \beta_{8} + 51 \beta_{7} - 98 \beta_{5} + \beta_{4} + 97 \beta_{3} + 10 \beta_{2} + 10\)
\(\nu^{8}\)\(=\)\(50 \beta_{15} + 148 \beta_{14} + 50 \beta_{13} - 50 \beta_{12} + 145 \beta_{11} + 50 \beta_{10} - 10 \beta_{9} + 100 \beta_{7} + 50 \beta_{6} + 10 \beta_{5} + 50 \beta_{4} + 128 \beta_{3} - 138 \beta_{1} + 50\)
\(\nu^{9}\)\(=\)\(-288 \beta_{15} + 278 \beta_{14} + 298 \beta_{13} + 228 \beta_{12} + 456 \beta_{11} - 55 \beta_{9} - 278 \beta_{8} - 10 \beta_{4} - 10 \beta_{3} + 158 \beta_{2} - 278 \beta_{1} + 228\)
\(\nu^{10}\)\(=\)\(-1385 \beta_{15} - 288 \beta_{14} + 576 \beta_{13} + 1032 \beta_{12} + 288 \beta_{11} + 288 \beta_{10} - 140 \beta_{9} - 1097 \beta_{8} - 288 \beta_{7} - 288 \beta_{6} - 1315 \beta_{5} - 576 \beta_{4} - 218 \beta_{3} + 744 \beta_{2} - 70 \beta_{1} + 288\)
\(\nu^{11}\)\(=\)\(-1533 \beta_{15} - 1603 \beta_{14} + 817 \beta_{12} - 1245 \beta_{11} + 1673 \beta_{10} - 1603 \beta_{8} - 4331 \beta_{5} - 1533 \beta_{4} + 1245 \beta_{2} - 817\)
\(\nu^{12}\)\(=\)\(3206 \beta_{15} - 1603 \beta_{13} - 3206 \beta_{12} - 1603 \beta_{11} + 3206 \beta_{10} + 428 \beta_{9} + 1603 \beta_{7} + 3206 \beta_{6} - 3545 \beta_{5} - 1195 \beta_{4} - 428 \beta_{3} - 3545 \beta_{1} - 3923\)
\(\nu^{13}\)\(=\)\(8782 \beta_{15} + 8782 \beta_{14} - 6751 \beta_{12} + 6751 \beta_{11} + 428 \beta_{8} + 9210 \beta_{6} + 9210 \beta_{5} + 428 \beta_{4} - 8782 \beta_{3} - 2459 \beta_{2} - 14219 \beta_{1} - 6751\)
\(\nu^{14}\)\(=\)\(-8782 \beta_{15} + 8782 \beta_{14} + 8782 \beta_{13} + 8782 \beta_{12} + 17564 \beta_{11} - 8782 \beta_{10} - 2459 \beta_{9} - 15075 \beta_{8} - 17564 \beta_{7} + 8782 \beta_{6} + 15105 \beta_{5} - 8782 \beta_{4} - 36503 \beta_{3} + 3376 \beta_{2} - 21398 \beta_{1} - 8782\)
\(\nu^{15}\)\(=\)\(-47744 \beta_{15} - 45285 \beta_{14} + 36503 \beta_{12} - 22803 \beta_{11} - 45285 \beta_{8} - 50203 \beta_{7} - 45285 \beta_{5} - 47744 \beta_{4} - 78841 \beta_{3} + 22803 \beta_{2} - 36503\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1 + \beta_{11}\) \(-\beta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.601150 2.24352i
0.430324 1.60599i
−0.0499037 + 0.186243i
−0.347596 + 1.29724i
0.601150 + 2.24352i
0.430324 + 1.60599i
−0.0499037 0.186243i
−0.347596 1.29724i
2.24352 0.601150i
1.60599 0.430324i
−0.186243 + 0.0499037i
−1.29724 + 0.347596i
2.24352 + 0.601150i
1.60599 + 0.430324i
−0.186243 0.0499037i
−1.29724 0.347596i
−0.601150 2.24352i −1.72336 0.173261i −2.93996 + 1.69739i 1.70912 1.44185i 0.647285 + 3.97056i 0.751454 0.201351i 2.29074 + 2.29074i 2.93996 + 0.597183i −4.26225 2.96768i
2.2 −0.430324 1.60599i 1.35314 + 1.08121i −0.661975 + 0.382191i −2.23073 + 0.154373i 1.15412 2.63840i −1.73749 + 0.465559i −1.45267 1.45267i 0.661975 + 2.92605i 1.20786 + 3.51610i
2.3 0.0499037 + 0.186243i −0.806271 1.53295i 1.69985 0.981412i −0.250705 + 2.22197i 0.245265 0.226662i −2.35868 + 0.632007i 0.540289 + 0.540289i −1.69985 + 2.47194i −0.426337 + 0.0641924i
2.4 0.347596 + 1.29724i −1.18953 + 1.25897i 0.170031 0.0981673i −1.59371 1.56847i −2.04667 1.10550i 1.97869 0.530190i 2.08575 + 2.08575i −0.170031 2.99518i 1.48073 2.61262i
23.1 −0.601150 + 2.24352i −1.72336 + 0.173261i −2.93996 1.69739i 1.70912 + 1.44185i 0.647285 3.97056i 0.751454 + 0.201351i 2.29074 2.29074i 2.93996 0.597183i −4.26225 + 2.96768i
23.2 −0.430324 + 1.60599i 1.35314 1.08121i −0.661975 0.382191i −2.23073 0.154373i 1.15412 + 2.63840i −1.73749 0.465559i −1.45267 + 1.45267i 0.661975 2.92605i 1.20786 3.51610i
23.3 0.0499037 0.186243i −0.806271 + 1.53295i 1.69985 + 0.981412i −0.250705 2.22197i 0.245265 + 0.226662i −2.35868 0.632007i 0.540289 0.540289i −1.69985 2.47194i −0.426337 0.0641924i
23.4 0.347596 1.29724i −1.18953 1.25897i 0.170031 + 0.0981673i −1.59371 + 1.56847i −2.04667 + 1.10550i 1.97869 + 0.530190i 2.08575 2.08575i −0.170031 + 2.99518i 1.48073 + 2.61262i
32.1 −2.24352 0.601150i 0.173261 + 1.72336i 2.93996 + 1.69739i 2.10323 + 0.759216i 0.647285 3.97056i −0.201351 + 0.751454i −2.29074 2.29074i −2.93996 + 0.597183i −4.26225 2.96768i
32.2 −1.60599 0.430324i −1.08121 1.35314i 0.661975 + 0.382191i −1.24906 1.85468i 1.15412 + 2.63840i 0.465559 1.73749i 1.45267 + 1.45267i −0.661975 + 2.92605i 1.20786 + 3.51610i
32.3 0.186243 + 0.0499037i 1.53295 + 0.806271i −1.69985 0.981412i −2.04963 + 0.893868i 0.245265 + 0.226662i 0.632007 2.35868i −0.540289 0.540289i 1.69985 + 2.47194i −0.426337 + 0.0641924i
32.4 1.29724 + 0.347596i −1.25897 + 1.18953i −0.170031 0.0981673i 0.561484 2.16443i −2.04667 + 1.10550i −0.530190 + 1.97869i −2.08575 2.08575i 0.170031 2.99518i 1.48073 2.61262i
38.1 −2.24352 + 0.601150i 0.173261 1.72336i 2.93996 1.69739i 2.10323 0.759216i 0.647285 + 3.97056i −0.201351 0.751454i −2.29074 + 2.29074i −2.93996 0.597183i −4.26225 + 2.96768i
38.2 −1.60599 + 0.430324i −1.08121 + 1.35314i 0.661975 0.382191i −1.24906 + 1.85468i 1.15412 2.63840i 0.465559 + 1.73749i 1.45267 1.45267i −0.661975 2.92605i 1.20786 3.51610i
38.3 0.186243 0.0499037i 1.53295 0.806271i −1.69985 + 0.981412i −2.04963 0.893868i 0.245265 0.226662i 0.632007 + 2.35868i −0.540289 + 0.540289i 1.69985 2.47194i −0.426337 0.0641924i
38.4 1.29724 0.347596i −1.25897 1.18953i −0.170031 + 0.0981673i 0.561484 + 2.16443i −2.04667 1.10550i −0.530190 1.97869i −2.08575 + 2.08575i 0.170031 + 2.99518i 1.48073 + 2.61262i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.l.a 16
3.b odd 2 1 135.2.m.a 16
4.b odd 2 1 720.2.cu.c 16
5.b even 2 1 225.2.p.b 16
5.c odd 4 1 inner 45.2.l.a 16
5.c odd 4 1 225.2.p.b 16
9.c even 3 1 135.2.m.a 16
9.c even 3 1 405.2.f.a 16
9.d odd 6 1 inner 45.2.l.a 16
9.d odd 6 1 405.2.f.a 16
15.d odd 2 1 675.2.q.a 16
15.e even 4 1 135.2.m.a 16
15.e even 4 1 675.2.q.a 16
20.e even 4 1 720.2.cu.c 16
36.h even 6 1 720.2.cu.c 16
45.h odd 6 1 225.2.p.b 16
45.j even 6 1 675.2.q.a 16
45.k odd 12 1 135.2.m.a 16
45.k odd 12 1 405.2.f.a 16
45.k odd 12 1 675.2.q.a 16
45.l even 12 1 inner 45.2.l.a 16
45.l even 12 1 225.2.p.b 16
45.l even 12 1 405.2.f.a 16
180.v odd 12 1 720.2.cu.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.l.a 16 1.a even 1 1 trivial
45.2.l.a 16 5.c odd 4 1 inner
45.2.l.a 16 9.d odd 6 1 inner
45.2.l.a 16 45.l even 12 1 inner
135.2.m.a 16 3.b odd 2 1
135.2.m.a 16 9.c even 3 1
135.2.m.a 16 15.e even 4 1
135.2.m.a 16 45.k odd 12 1
225.2.p.b 16 5.b even 2 1
225.2.p.b 16 5.c odd 4 1
225.2.p.b 16 45.h odd 6 1
225.2.p.b 16 45.l even 12 1
405.2.f.a 16 9.c even 3 1
405.2.f.a 16 9.d odd 6 1
405.2.f.a 16 45.k odd 12 1
405.2.f.a 16 45.l even 12 1
675.2.q.a 16 15.d odd 2 1
675.2.q.a 16 15.e even 4 1
675.2.q.a 16 45.j even 6 1
675.2.q.a 16 45.k odd 12 1
720.2.cu.c 16 4.b odd 2 1
720.2.cu.c 16 20.e even 4 1
720.2.cu.c 16 36.h even 6 1
720.2.cu.c 16 180.v odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 502 T^{4} + 432 T^{5} + 144 T^{6} + 102 T^{7} - 93 T^{8} - 132 T^{9} - 72 T^{10} - 18 T^{11} + 34 T^{12} + 36 T^{13} + 18 T^{14} + 6 T^{15} + T^{16} \)
$3$ \( 6561 + 13122 T + 13122 T^{2} + 7290 T^{3} + 2430 T^{4} + 972 T^{5} + 1134 T^{6} + 1188 T^{7} + 819 T^{8} + 396 T^{9} + 126 T^{10} + 36 T^{11} + 30 T^{12} + 30 T^{13} + 18 T^{14} + 6 T^{15} + T^{16} \)
$5$ \( 390625 + 468750 T + 250000 T^{2} + 75000 T^{3} + 13750 T^{4} + 3750 T^{5} + 3400 T^{6} + 3150 T^{7} + 1831 T^{8} + 630 T^{9} + 136 T^{10} + 30 T^{11} + 22 T^{12} + 24 T^{13} + 16 T^{14} + 6 T^{15} + T^{16} \)
$7$ \( 2401 - 2744 T + 1568 T^{2} - 5124 T^{3} + 1850 T^{4} + 3356 T^{5} + 424 T^{6} + 990 T^{7} + 283 T^{8} - 216 T^{9} - 32 T^{10} - 62 T^{11} - 10 T^{12} + 12 T^{13} + 2 T^{14} + 2 T^{15} + T^{16} \)
$11$ \( ( 4 + 36 T + 128 T^{2} + 180 T^{3} + 102 T^{4} - 10 T^{6} + T^{8} )^{2} \)
$13$ \( 4477456 + 389344 T + 16928 T^{2} + 797088 T^{3} - 989488 T^{4} - 289648 T^{5} + 49504 T^{6} - 85560 T^{7} + 223084 T^{8} - 24216 T^{9} + 2128 T^{10} - 2228 T^{11} - 376 T^{12} + 108 T^{13} + 2 T^{14} + 2 T^{15} + T^{16} \)
$17$ \( 16 + 832 T^{4} + 6504 T^{8} + 964 T^{12} + T^{16} \)
$19$ \( ( 324 + 1836 T^{2} + 864 T^{4} + 60 T^{6} + T^{8} )^{2} \)
$23$ \( 62742241 - 143813676 T + 164820168 T^{2} - 125930016 T^{3} + 65763382 T^{4} - 21955764 T^{5} + 3946176 T^{6} + 340374 T^{7} - 477093 T^{8} + 128484 T^{9} - 19008 T^{10} - 6 T^{11} + 3034 T^{12} - 972 T^{13} + 162 T^{14} - 18 T^{15} + T^{16} \)
$29$ \( 981506241 + 542492964 T^{2} + 231985242 T^{4} + 32243184 T^{6} + 3205683 T^{8} + 147312 T^{10} + 4890 T^{12} + 84 T^{14} + T^{16} \)
$31$ \( ( 676 + 3692 T + 19072 T^{2} + 6068 T^{3} + 2074 T^{4} + 200 T^{5} + 46 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$37$ \( ( 4 + 56 T + 392 T^{2} + 1340 T^{3} + 2308 T^{4} + 124 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$41$ \( ( 32761 - 33666 T + 7550 T^{2} + 4092 T^{3} - 441 T^{4} - 264 T^{5} + 26 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$43$ \( 11316496 - 3902240 T + 672800 T^{2} - 2680992 T^{3} - 1085200 T^{4} + 1045808 T^{5} + 21472 T^{6} + 183048 T^{7} + 213388 T^{8} - 15096 T^{9} - 464 T^{10} - 1532 T^{11} - 400 T^{12} + 60 T^{13} + 2 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( 33243864241 - 15415187634 T + 3574013058 T^{2} - 552423564 T^{3} - 255703838 T^{4} + 83612562 T^{5} - 6690816 T^{6} + 3723372 T^{7} + 1113267 T^{8} - 350922 T^{9} + 34848 T^{10} - 11508 T^{11} - 1826 T^{12} + 288 T^{13} + 72 T^{14} + 12 T^{15} + T^{16} \)
$53$ \( 409600000000 + 132589551616 T^{4} + 104540160 T^{8} + 20032 T^{12} + T^{16} \)
$59$ \( 592240896 + 1944933120 T^{2} + 6232429440 T^{4} + 501282432 T^{6} + 28916784 T^{8} + 756000 T^{10} + 14376 T^{12} + 144 T^{14} + T^{16} \)
$61$ \( ( 11449 + 29318 T + 64162 T^{2} + 27092 T^{3} + 9415 T^{4} + 956 T^{5} + 118 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$67$ \( 539415333601 - 686070844370 T + 436299428450 T^{2} - 300113128236 T^{3} + 171721343210 T^{4} - 54722745634 T^{5} + 14192591128 T^{6} - 3975597636 T^{7} + 599752603 T^{8} - 12674142 T^{9} + 502984 T^{10} + 31204 T^{11} - 27850 T^{12} + 900 T^{13} + 8 T^{14} - 4 T^{15} + T^{16} \)
$71$ \( ( 128164 + 43580 T^{2} + 3972 T^{4} + 116 T^{6} + T^{8} )^{2} \)
$73$ \( ( 270400 + 29120 T + 1568 T^{2} - 9472 T^{3} + 16384 T^{4} - 584 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$79$ \( 1737217064964096 - 95814503755776 T^{2} + 3389951831040 T^{4} - 73485107712 T^{6} + 1169408448 T^{8} - 12312000 T^{10} + 92928 T^{12} - 372 T^{14} + T^{16} \)
$83$ \( 13841287201 + 71183762748 T + 183043961352 T^{2} + 313789648032 T^{3} + 342486827410 T^{4} + 204350665848 T^{5} + 78625297296 T^{6} + 21779149266 T^{7} + 4574200323 T^{8} + 745444644 T^{9} + 95661720 T^{10} + 9726150 T^{11} + 777886 T^{12} + 47916 T^{13} + 2178 T^{14} + 66 T^{15} + T^{16} \)
$89$ \( ( 3969 - 49356 T^{2} + 8262 T^{4} - 300 T^{6} + T^{8} )^{2} \)
$97$ \( 6146560000 - 10712576000 T + 9335244800 T^{2} - 7398881280 T^{3} + 5125458944 T^{4} - 2379530752 T^{5} + 815938048 T^{6} - 245889792 T^{7} + 63795904 T^{8} - 13214784 T^{9} + 2334592 T^{10} - 369728 T^{11} + 46976 T^{12} - 4560 T^{13} + 392 T^{14} - 28 T^{15} + T^{16} \)
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