Properties

Label 240.2.s.c
Level $240$
Weight $2$
Character orbit 240.s
Analytic conductor $1.916$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + 136 x^{10} + 80 x^{9} - 168 x^{8} - 288 x^{7} + 96 x^{6} + 512 x^{5} + 448 x^{4} - 512 x^{3} - 512 x^{2} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + 136 x^{10} + 80 x^{9} - 168 x^{8} - 288 x^{7} + 96 x^{6} + 512 x^{5} + 448 x^{4} - 512 x^{3} - 512 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{19} + 2 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 16 \nu^{14} - 6 \nu^{13} + 36 \nu^{12} + 42 \nu^{11} - 40 \nu^{10} - 136 \nu^{9} - 80 \nu^{8} + 168 \nu^{7} + 288 \nu^{6} - 96 \nu^{5} - 512 \nu^{4} - 448 \nu^{3} + 512 \nu^{2} + 512 \nu \)\()/256\)
\(\beta_{4}\)\(=\)\((\)\( -18 \nu^{19} - 33 \nu^{18} - 8 \nu^{17} + 104 \nu^{16} + 94 \nu^{15} - 199 \nu^{14} - 616 \nu^{13} - 364 \nu^{12} + 768 \nu^{11} + 1426 \nu^{10} - 152 \nu^{9} - 3332 \nu^{8} - 3808 \nu^{7} + 1024 \nu^{6} + 5536 \nu^{5} + 2976 \nu^{4} - 7040 \nu^{3} - 8448 \nu^{2} - 2816 \nu + 5632 \)\()/512\)
\(\beta_{5}\)\(=\)\((\)\(31 \nu^{19} + 39 \nu^{18} - 30 \nu^{17} - 204 \nu^{16} - 67 \nu^{15} + 497 \nu^{14} + 946 \nu^{13} - 64 \nu^{12} - 2082 \nu^{11} - 2086 \nu^{10} + 2008 \nu^{9} + 6636 \nu^{8} + 3800 \nu^{7} - 7264 \nu^{6} - 11680 \nu^{5} - 736 \nu^{4} + 17728 \nu^{3} + 11776 \nu^{2} - 5120 \nu - 17920\)\()/512\)
\(\beta_{6}\)\(=\)\((\)\(-33 \nu^{19} - 44 \nu^{18} + 32 \nu^{17} + 220 \nu^{16} + 89 \nu^{15} - 508 \nu^{14} - 1012 \nu^{13} + 12 \nu^{12} + 2146 \nu^{11} + 2296 \nu^{10} - 1892 \nu^{9} - 6832 \nu^{8} - 4160 \nu^{7} + 7264 \nu^{6} + 12192 \nu^{5} + 1024 \nu^{4} - 17664 \nu^{3} - 12032 \nu^{2} + 5632 \nu + 18432\)\()/512\)
\(\beta_{7}\)\(=\)\((\)\( -23 \nu^{19} - 30 \nu^{18} + 13 \nu^{17} + 136 \nu^{16} + 59 \nu^{15} - 318 \nu^{14} - 673 \nu^{13} - 96 \nu^{12} + 1254 \nu^{11} + 1480 \nu^{10} - 1030 \nu^{9} - 4336 \nu^{8} - 3220 \nu^{7} + 3712 \nu^{6} + 7152 \nu^{5} + 1472 \nu^{4} - 10656 \nu^{3} - 8320 \nu^{2} + 640 \nu + 9472 \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\( -11 \nu^{19} - 36 \nu^{18} - 44 \nu^{17} + 28 \nu^{16} + 131 \nu^{15} + 12 \nu^{14} - 464 \nu^{13} - 836 \nu^{12} - 266 \nu^{11} + 1096 \nu^{10} + 1356 \nu^{9} - 1184 \nu^{8} - 4816 \nu^{7} - 4448 \nu^{6} + 992 \nu^{5} + 5440 \nu^{4} + 1024 \nu^{3} - 8448 \nu^{2} - 11264 \nu - 5632 \)\()/256\)
\(\beta_{9}\)\(=\)\((\)\(-21 \nu^{19} + 10 \nu^{18} + 104 \nu^{17} + 192 \nu^{16} - 123 \nu^{15} - 594 \nu^{14} - 364 \nu^{13} + 1376 \nu^{12} + 2786 \nu^{11} + 652 \nu^{10} - 4436 \nu^{9} - 5696 \nu^{8} + 3232 \nu^{7} + 15248 \nu^{6} + 11424 \nu^{5} - 7936 \nu^{4} - 21760 \nu^{3} - 128 \nu^{2} + 24064 \nu + 28672\)\()/512\)
\(\beta_{10}\)\(=\)\((\)\(-39 \nu^{19} - 32 \nu^{18} + 80 \nu^{17} + 284 \nu^{16} - \nu^{15} - 760 \nu^{14} - 1052 \nu^{13} + 780 \nu^{12} + 3326 \nu^{11} + 2208 \nu^{10} - 4156 \nu^{9} - 9008 \nu^{8} - 1664 \nu^{7} + 14656 \nu^{6} + 16608 \nu^{5} - 3840 \nu^{4} - 27648 \nu^{3} - 10752 \nu^{2} + 17920 \nu + 31744\)\()/512\)
\(\beta_{11}\)\(=\)\((\)\( -5 \nu^{19} - 31 \nu^{18} - 54 \nu^{17} - 12 \nu^{16} + 129 \nu^{15} + 119 \nu^{14} - 310 \nu^{13} - 952 \nu^{12} - 746 \nu^{11} + 790 \nu^{10} + 2008 \nu^{9} + 148 \nu^{8} - 4600 \nu^{7} - 6720 \nu^{6} - 1408 \nu^{5} + 6176 \nu^{4} + 5440 \nu^{3} - 6656 \nu^{2} - 14336 \nu - 10752 \)\()/256\)
\(\beta_{12}\)\(=\)\((\)\(-33 \nu^{19} - 16 \nu^{18} + 98 \nu^{17} + 264 \nu^{16} - 47 \nu^{15} - 720 \nu^{14} - 790 \nu^{13} + 1120 \nu^{12} + 3282 \nu^{11} + 1624 \nu^{10} - 4504 \nu^{9} - 7800 \nu^{8} + 760 \nu^{7} + 15952 \nu^{6} + 14848 \nu^{5} - 6336 \nu^{4} - 26176 \nu^{3} - 5248 \nu^{2} + 23296 \nu + 32256\)\()/512\)
\(\beta_{13}\)\(=\)\((\)\( 22 \nu^{19} + 17 \nu^{18} - 44 \nu^{17} - 160 \nu^{16} - 10 \nu^{15} + 407 \nu^{14} + 588 \nu^{13} - 380 \nu^{12} - 1808 \nu^{11} - 1298 \nu^{10} + 2096 \nu^{9} + 4852 \nu^{8} + 1120 \nu^{7} - 7680 \nu^{6} - 8960 \nu^{5} + 1440 \nu^{4} + 14464 \nu^{3} + 5632 \nu^{2} - 9216 \nu - 16896 \)\()/256\)
\(\beta_{14}\)\(=\)\((\)\(16 \nu^{19} - \nu^{18} - 64 \nu^{17} - 136 \nu^{16} + 68 \nu^{15} + 409 \nu^{14} + 312 \nu^{13} - 844 \nu^{12} - 1884 \nu^{11} - 574 \nu^{10} + 2944 \nu^{9} + 4108 \nu^{8} - 1712 \nu^{7} - 10176 \nu^{6} - 8064 \nu^{5} + 5088 \nu^{4} + 15360 \nu^{3} + 1024 \nu^{2} - 15872 \nu - 19968\)\()/256\)
\(\beta_{15}\)\(=\)\((\)\(-31 \nu^{19} + 10 \nu^{18} + 132 \nu^{17} + 260 \nu^{16} - 161 \nu^{15} - 770 \nu^{14} - 480 \nu^{13} + 1804 \nu^{12} + 3622 \nu^{11} + 764 \nu^{10} - 5860 \nu^{9} - 7320 \nu^{8} + 4688 \nu^{7} + 20384 \nu^{6} + 14368 \nu^{5} - 11200 \nu^{4} - 29056 \nu^{3} + 1536 \nu^{2} + 33280 \nu + 39936\)\()/512\)
\(\beta_{16}\)\(=\)\((\)\( 35 \nu^{19} + 62 \nu^{18} + 8 \nu^{17} - 200 \nu^{16} - 163 \nu^{15} + 426 \nu^{14} + 1204 \nu^{13} + 632 \nu^{12} - 1598 \nu^{11} - 2764 \nu^{10} + 588 \nu^{9} + 6816 \nu^{8} + 7392 \nu^{7} - 2480 \nu^{6} - 11168 \nu^{5} - 5440 \nu^{4} + 14208 \nu^{3} + 17536 \nu^{2} + 5632 \nu - 10240 \)\()/256\)
\(\beta_{17}\)\(=\)\((\)\( 31 \nu^{19} + 64 \nu^{18} + 32 \nu^{17} - 164 \nu^{16} - 199 \nu^{15} + 280 \nu^{14} + 1132 \nu^{13} + 956 \nu^{12} - 990 \nu^{11} - 2688 \nu^{10} - 548 \nu^{9} + 5440 \nu^{8} + 8160 \nu^{7} + 928 \nu^{6} - 8736 \nu^{5} - 7680 \nu^{4} + 9216 \nu^{3} + 16896 \nu^{2} + 10752 \nu - 5120 \)\()/256\)
\(\beta_{18}\)\(=\)\((\)\(21 \nu^{19} + 4 \nu^{18} - 80 \nu^{17} - 184 \nu^{16} + 59 \nu^{15} + 532 \nu^{14} + 484 \nu^{13} - 976 \nu^{12} - 2450 \nu^{11} - 1000 \nu^{10} + 3556 \nu^{9} + 5544 \nu^{8} - 1360 \nu^{7} - 12464 \nu^{6} - 10944 \nu^{5} + 5504 \nu^{4} + 19200 \nu^{3} + 2816 \nu^{2} - 18944 \nu - 24064\)\()/256\)
\(\beta_{19}\)\(=\)\((\)\(-180 \nu^{19} - 217 \nu^{18} + 214 \nu^{17} + 1228 \nu^{16} + 376 \nu^{15} - 2983 \nu^{14} - 5454 \nu^{13} + 816 \nu^{12} + 12724 \nu^{11} + 12266 \nu^{10} - 12364 \nu^{9} - 39068 \nu^{8} - 20280 \nu^{7} + 45936 \nu^{6} + 70208 \nu^{5} + 2144 \nu^{4} - 105024 \nu^{3} - 65152 \nu^{2} + 39424 \nu + 110080\)\()/512\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{18} + \beta_{15} + \beta_{14} - \beta_{12} - \beta_{9} - \beta_{5} - \beta_{4} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{19} - \beta_{16} - \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{5}\)\(=\)\(-\beta_{19} - \beta_{18} - \beta_{17} - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(\beta_{19} + \beta_{17} + 2 \beta_{16} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} - 5 \beta_{9} + 2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - \beta_{1} - 2\)
\(\nu^{7}\)\(=\)\(-2 \beta_{19} - \beta_{17} - 4 \beta_{16} + \beta_{15} - \beta_{14} - 2 \beta_{13} + 4 \beta_{12} + 3 \beta_{11} + \beta_{10} - 2 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 3 \beta_{1} - 5\)
\(\nu^{8}\)\(=\)\(-2 \beta_{19} - \beta_{18} - 4 \beta_{17} + 2 \beta_{16} - 3 \beta_{15} - 4 \beta_{14} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 5 \beta_{9} - 2 \beta_{7} + \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 3 \beta_{1} - 1\)
\(\nu^{9}\)\(=\)\(7 \beta_{19} + 5 \beta_{18} + 3 \beta_{17} + 7 \beta_{16} - 2 \beta_{15} + 2 \beta_{13} - 5 \beta_{12} - 3 \beta_{11} + 5 \beta_{10} - 4 \beta_{9} + \beta_{8} + 5 \beta_{7} - \beta_{6} + 13 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 7 \beta_{2} + \beta_{1} + 2\)
\(\nu^{10}\)\(=\)\(\beta_{19} + 2 \beta_{18} - 5 \beta_{17} - 8 \beta_{16} - 10 \beta_{14} + 5 \beta_{13} + 8 \beta_{12} - \beta_{11} - 13 \beta_{10} + 5 \beta_{9} - 4 \beta_{7} - 5 \beta_{6} + 15 \beta_{5} - 6 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 5 \beta_{1} + 2\)
\(\nu^{11}\)\(=\)\(10 \beta_{19} + 2 \beta_{18} + 3 \beta_{17} + 8 \beta_{16} + 7 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} - 6 \beta_{12} - \beta_{11} - 15 \beta_{10} + 10 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} - 13 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} + \beta_{2} + \beta_{1} + 13\)
\(\nu^{12}\)\(=\)\(-\beta_{18} - 8 \beta_{17} - 4 \beta_{16} - 3 \beta_{15} + 4 \beta_{14} + 10 \beta_{13} + 13 \beta_{12} - 7 \beta_{11} + 13 \beta_{10} + 7 \beta_{9} + 2 \beta_{8} + 6 \beta_{7} - 13 \beta_{6} + 37 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 11\)
\(\nu^{13}\)\(=\)\(21 \beta_{19} + 7 \beta_{18} + 21 \beta_{17} + 7 \beta_{16} + 6 \beta_{15} + 26 \beta_{13} - 19 \beta_{12} - 13 \beta_{11} - 37 \beta_{10} + 12 \beta_{9} + 9 \beta_{8} + 15 \beta_{7} + 5 \beta_{6} + 19 \beta_{5} + 42 \beta_{4} - 4 \beta_{3} - \beta_{2} + 11 \beta_{1} + 30\)
\(\nu^{14}\)\(=\)\(-\beta_{19} + 6 \beta_{18} + 5 \beta_{17} - 16 \beta_{16} + 28 \beta_{15} + 38 \beta_{14} + 15 \beta_{13} + 8 \beta_{12} - 27 \beta_{11} - 19 \beta_{10} + 11 \beta_{9} + 12 \beta_{8} - 8 \beta_{7} + 21 \beta_{6} - 27 \beta_{5} - 14 \beta_{4} - 14 \beta_{3} + 38 \beta_{2} + 9 \beta_{1} - 54\)
\(\nu^{15}\)\(=\)\(-18 \beta_{19} - 22 \beta_{18} + 9 \beta_{17} - 4 \beta_{16} - 15 \beta_{15} + 15 \beta_{14} - 34 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} + 27 \beta_{10} + 10 \beta_{9} + 22 \beta_{8} - \beta_{7} - 13 \beta_{6} + 5 \beta_{5} + 51 \beta_{4} - 14 \beta_{3} + 19 \beta_{2} - 53 \beta_{1} + 35\)
\(\nu^{16}\)\(=\)\(-28 \beta_{19} - 35 \beta_{18} + 24 \beta_{17} + 7 \beta_{15} + 28 \beta_{14} - 6 \beta_{13} - 33 \beta_{12} - 9 \beta_{11} - 5 \beta_{10} - 39 \beta_{9} - 38 \beta_{8} + 34 \beta_{7} + 69 \beta_{6} - 77 \beta_{5} + 127 \beta_{4} - 6 \beta_{3} - 99 \beta_{2} + 53 \beta_{1} - 23\)
\(\nu^{17}\)\(=\)\(-37 \beta_{19} + 121 \beta_{18} + 59 \beta_{17} - 19 \beta_{16} - 86 \beta_{15} - 32 \beta_{14} - 82 \beta_{13} + 91 \beta_{12} + 5 \beta_{11} + 77 \beta_{10} + 36 \beta_{9} - 13 \beta_{8} + 9 \beta_{7} + 155 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} - 76 \beta_{3} + 73 \beta_{2} + 5 \beta_{1} - 134\)
\(\nu^{18}\)\(=\)\(-55 \beta_{19} - 118 \beta_{18} - 85 \beta_{17} + 32 \beta_{16} - 44 \beta_{15} - 30 \beta_{14} - 151 \beta_{13} - 200 \beta_{12} - 45 \beta_{11} + 3 \beta_{10} + 45 \beta_{9} + 76 \beta_{8} - 72 \beta_{7} + 43 \beta_{6} - 77 \beta_{5} - 50 \beta_{4} + 198 \beta_{3} - 126 \beta_{2} - 97 \beta_{1} + 86\)
\(\nu^{19}\)\(=\)\(-22 \beta_{19} + 30 \beta_{18} + 111 \beta_{17} + 20 \beta_{16} - 105 \beta_{15} - 39 \beta_{14} - 150 \beta_{13} + 134 \beta_{12} + 27 \beta_{11} + 77 \beta_{10} - 458 \beta_{9} - 214 \beta_{8} + 97 \beta_{7} + 5 \beta_{6} - 277 \beta_{5} + 101 \beta_{4} - 2 \beta_{3} - 227 \beta_{2} + 141 \beta_{1} + 101\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−1.38431 + 0.289262i
−1.13207 + 0.847599i
−1.04932 0.948122i
−0.720859 1.21670i
−0.491956 + 1.32589i
−0.0861743 + 1.41159i
1.15787 + 0.811989i
1.18701 0.768775i
1.19834 + 0.750988i
1.32147 0.503713i
−1.38431 0.289262i
−1.13207 0.847599i
−1.04932 + 0.948122i
−0.720859 + 1.21670i
−0.491956 1.32589i
−0.0861743 1.41159i
1.15787 0.811989i
1.18701 + 0.768775i
1.19834 0.750988i
1.32147 + 0.503713i
−1.38431 + 0.289262i −0.707107 0.707107i 1.83266 0.800859i −0.707107 + 0.707107i 1.18340 + 0.774320i 2.60796i −2.30531 + 1.63876i 1.00000i 0.774320 1.18340i
61.2 −1.13207 + 0.847599i 0.707107 + 0.707107i 0.563151 1.91908i 0.707107 0.707107i −1.39984 0.201149i 4.27253i 0.989085 + 2.64985i 1.00000i −0.201149 + 1.39984i
61.3 −1.04932 0.948122i −0.707107 0.707107i 0.202128 + 1.98976i −0.707107 + 0.707107i 0.0715547 + 1.41240i 0.740019i 1.67444 2.27953i 1.00000i 1.41240 0.0715547i
61.4 −0.720859 1.21670i 0.707107 + 0.707107i −0.960724 + 1.75414i 0.707107 0.707107i 0.350613 1.37006i 0.0588949i 2.82681 0.0955746i 1.00000i −1.37006 0.350613i
61.5 −0.491956 + 1.32589i 0.707107 + 0.707107i −1.51596 1.30456i 0.707107 0.707107i −1.28541 + 0.589679i 3.46600i 2.47548 1.36821i 1.00000i 0.589679 + 1.28541i
61.6 −0.0861743 + 1.41159i −0.707107 0.707107i −1.98515 0.243285i −0.707107 + 0.707107i 1.05908 0.937207i 2.76462i 0.514486 2.78124i 1.00000i −0.937207 1.05908i
61.7 1.15787 + 0.811989i 0.707107 + 0.707107i 0.681349 + 1.88036i 0.707107 0.707107i 0.244579 + 1.39290i 2.18060i −0.737916 + 2.73047i 1.00000i 1.39290 0.244579i
61.8 1.18701 0.768775i 0.707107 + 0.707107i 0.817970 1.82508i 0.707107 0.707107i 1.38295 + 0.295735i 4.92824i −0.432142 2.79522i 1.00000i 0.295735 1.38295i
61.9 1.19834 + 0.750988i −0.707107 0.707107i 0.872033 + 1.79988i −0.707107 + 0.707107i −0.316325 1.37838i 3.79862i −0.306697 + 2.81175i 1.00000i −1.37838 + 0.316325i
61.10 1.32147 0.503713i −0.707107 0.707107i 1.49255 1.33128i −0.707107 + 0.707107i −1.29060 0.578239i 2.69529i 1.30176 2.51106i 1.00000i −0.578239 + 1.29060i
181.1 −1.38431 0.289262i −0.707107 + 0.707107i 1.83266 + 0.800859i −0.707107 0.707107i 1.18340 0.774320i 2.60796i −2.30531 1.63876i 1.00000i 0.774320 + 1.18340i
181.2 −1.13207 0.847599i 0.707107 0.707107i 0.563151 + 1.91908i 0.707107 + 0.707107i −1.39984 + 0.201149i 4.27253i 0.989085 2.64985i 1.00000i −0.201149 1.39984i
181.3 −1.04932 + 0.948122i −0.707107 + 0.707107i 0.202128 1.98976i −0.707107 0.707107i 0.0715547 1.41240i 0.740019i 1.67444 + 2.27953i 1.00000i 1.41240 + 0.0715547i
181.4 −0.720859 + 1.21670i 0.707107 0.707107i −0.960724 1.75414i 0.707107 + 0.707107i 0.350613 + 1.37006i 0.0588949i 2.82681 + 0.0955746i 1.00000i −1.37006 + 0.350613i
181.5 −0.491956 1.32589i 0.707107 0.707107i −1.51596 + 1.30456i 0.707107 + 0.707107i −1.28541 0.589679i 3.46600i 2.47548 + 1.36821i 1.00000i 0.589679 1.28541i
181.6 −0.0861743 1.41159i −0.707107 + 0.707107i −1.98515 + 0.243285i −0.707107 0.707107i 1.05908 + 0.937207i 2.76462i 0.514486 + 2.78124i 1.00000i −0.937207 + 1.05908i
181.7 1.15787 0.811989i 0.707107 0.707107i 0.681349 1.88036i 0.707107 + 0.707107i 0.244579 1.39290i 2.18060i −0.737916 2.73047i 1.00000i 1.39290 + 0.244579i
181.8 1.18701 + 0.768775i 0.707107 0.707107i 0.817970 + 1.82508i 0.707107 + 0.707107i 1.38295 0.295735i 4.92824i −0.432142 + 2.79522i 1.00000i 0.295735 + 1.38295i
181.9 1.19834 0.750988i −0.707107 + 0.707107i 0.872033 1.79988i −0.707107 0.707107i −0.316325 + 1.37838i 3.79862i −0.306697 2.81175i 1.00000i −1.37838 0.316325i
181.10 1.32147 + 0.503713i −0.707107 + 0.707107i 1.49255 + 1.33128i −0.707107 0.707107i −1.29060 + 0.578239i 2.69529i 1.30176 + 2.51106i 1.00000i −0.578239 1.29060i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.s.c 20
3.b odd 2 1 720.2.t.d 20
4.b odd 2 1 960.2.s.c 20
8.b even 2 1 1920.2.s.e 20
8.d odd 2 1 1920.2.s.f 20
12.b even 2 1 2880.2.t.d 20
16.e even 4 1 inner 240.2.s.c 20
16.e even 4 1 1920.2.s.e 20
16.f odd 4 1 960.2.s.c 20
16.f odd 4 1 1920.2.s.f 20
48.i odd 4 1 720.2.t.d 20
48.k even 4 1 2880.2.t.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.c 20 1.a even 1 1 trivial
240.2.s.c 20 16.e even 4 1 inner
720.2.t.d 20 3.b odd 2 1
720.2.t.d 20 48.i odd 4 1
960.2.s.c 20 4.b odd 2 1
960.2.s.c 20 16.f odd 4 1
1920.2.s.e 20 8.b even 2 1
1920.2.s.e 20 16.e even 4 1
1920.2.s.f 20 8.d odd 2 1
1920.2.s.f 20 16.f odd 4 1
2880.2.t.d 20 12.b even 2 1
2880.2.t.d 20 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1024 - 512 T^{2} - 512 T^{3} + 448 T^{4} + 512 T^{5} + 96 T^{6} - 288 T^{7} - 168 T^{8} + 80 T^{9} + 136 T^{10} + 40 T^{11} - 42 T^{12} - 36 T^{13} + 6 T^{14} + 16 T^{15} + 7 T^{16} - 4 T^{17} - 2 T^{18} + T^{20} \)
$3$ \( ( 1 + T^{4} )^{5} \)
$5$ \( ( 1 + T^{4} )^{5} \)
$7$ \( 262144 + 76283904 T^{2} + 204587008 T^{4} + 145248256 T^{6} + 49590528 T^{8} + 9697664 T^{10} + 1161104 T^{12} + 86368 T^{14} + 3880 T^{16} + 96 T^{18} + T^{20} \)
$11$ \( 1048576 + 115343360 T + 6343884800 T^{2} - 15595208704 T^{3} + 18952028160 T^{4} - 11886264320 T^{5} + 4560683008 T^{6} - 1115578368 T^{7} + 251479040 T^{8} - 84426752 T^{9} + 30799872 T^{10} - 7170048 T^{11} + 1093696 T^{12} - 173952 T^{13} + 57984 T^{14} - 13696 T^{15} + 1808 T^{16} - 112 T^{17} + 32 T^{18} - 8 T^{19} + T^{20} \)
$13$ \( 167981940736 - 262517686272 T + 205127811072 T^{2} - 90035355648 T^{3} + 25928249344 T^{4} - 7034568704 T^{5} + 3460349952 T^{6} - 1511043072 T^{7} + 413145856 T^{8} - 60063232 T^{9} + 12435968 T^{10} - 5323392 T^{11} + 1636752 T^{12} - 175680 T^{13} + 10368 T^{14} - 4192 T^{15} + 2088 T^{16} - 144 T^{17} + T^{20} \)
$17$ \( ( -20032 + 151040 T - 145472 T^{2} - 106144 T^{3} + 25976 T^{4} + 15824 T^{5} - 676 T^{6} - 792 T^{7} - 34 T^{8} + 12 T^{9} + T^{10} )^{2} \)
$19$ \( 19716653056 + 59527397376 T + 89860866048 T^{2} + 68702330880 T^{3} + 29049219072 T^{4} + 7396435968 T^{5} + 9631844352 T^{6} + 7963704832 T^{7} + 3224447552 T^{8} - 73408640 T^{9} + 28386432 T^{10} + 20915520 T^{11} + 7892400 T^{12} - 209856 T^{13} + 29888 T^{14} + 17152 T^{15} + 5164 T^{16} - 72 T^{17} + 8 T^{18} + 4 T^{19} + T^{20} \)
$23$ \( 217558810624 + 1211058880512 T^{2} + 1734629007360 T^{4} + 505220718592 T^{6} + 64474080320 T^{8} + 4370425280 T^{10} + 169125872 T^{12} + 3828512 T^{14} + 49980 T^{16} + 348 T^{18} + T^{20} \)
$29$ \( 19723262623744 - 2510316109824 T + 159752650752 T^{2} - 1868828770304 T^{3} + 2276071636992 T^{4} - 779917459456 T^{5} + 169368092672 T^{6} - 67031334912 T^{7} + 46757634048 T^{8} - 18135728128 T^{9} + 4155932672 T^{10} - 541822976 T^{11} + 52284928 T^{12} - 7973888 T^{13} + 1863680 T^{14} - 231168 T^{15} + 15632 T^{16} - 576 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$31$ \( ( -28698368 - 4962560 T + 6148800 T^{2} + 621056 T^{3} - 468112 T^{4} - 19536 T^{5} + 14932 T^{6} + 168 T^{7} - 204 T^{8} + T^{10} )^{2} \)
$37$ \( 18939904 + 171573248 T + 777125888 T^{2} + 1848475648 T^{3} + 2665852928 T^{4} + 2241855488 T^{5} + 1128316928 T^{6} + 422612992 T^{7} + 451001088 T^{8} + 394871296 T^{9} + 184033792 T^{10} + 12409216 T^{11} - 2713200 T^{12} - 581952 T^{13} + 678528 T^{14} - 81440 T^{15} + 4904 T^{16} - 80 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$41$ \( 3311118843904 + 11279143534592 T^{2} + 4549133533184 T^{4} + 806185730048 T^{6} + 78819631104 T^{8} + 4616884224 T^{10} + 166691328 T^{12} + 3697152 T^{14} + 48592 T^{16} + 344 T^{18} + T^{20} \)
$43$ \( 3288334336 + 31474057216 T + 150625845248 T^{2} + 354645180416 T^{3} + 430934327296 T^{4} + 72320286720 T^{5} + 76928778240 T^{6} + 118577430528 T^{7} + 83002048512 T^{8} + 32131072000 T^{9} + 7850893312 T^{10} + 1219307520 T^{11} + 121272576 T^{12} + 8994816 T^{13} + 1246208 T^{14} + 208640 T^{15} + 20960 T^{16} + 704 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$47$ \( ( -12544 + 145152 T + 283712 T^{2} + 35072 T^{3} - 97448 T^{4} - 11520 T^{5} + 7980 T^{6} + 160 T^{7} - 166 T^{8} + T^{10} )^{2} \)
$53$ \( 4398046511104 + 37383395344384 T + 158879430213632 T^{2} + 217797791580160 T^{3} + 164036679303168 T^{4} + 64499805585408 T^{5} + 15258273972224 T^{6} + 2110865276928 T^{7} + 243227688960 T^{8} + 46965587968 T^{9} + 11259117568 T^{10} + 1463160832 T^{11} + 113316096 T^{12} + 8978432 T^{13} + 2199552 T^{14} + 279040 T^{15} + 18720 T^{16} + 512 T^{17} + 128 T^{18} + 16 T^{19} + T^{20} \)
$59$ \( 16482977321058304 - 15913855322423296 T + 7682191945367552 T^{2} + 225773976551424 T^{3} - 19284448182272 T^{4} - 8752961814528 T^{5} + 18984867758080 T^{6} + 1690337058816 T^{7} + 60533785600 T^{8} + 4783368192 T^{9} + 14857752576 T^{10} + 1869032960 T^{11} + 117256256 T^{12} + 208256 T^{13} + 2543744 T^{14} + 332800 T^{15} + 21776 T^{16} - 112 T^{17} + 128 T^{18} + 16 T^{19} + T^{20} \)
$61$ \( 74350019584 - 90727790592 T + 55356622848 T^{2} + 405824488448 T^{3} + 635588871424 T^{4} + 400694880256 T^{5} + 145373775872 T^{6} + 32054490112 T^{7} + 8596122240 T^{8} + 3542995712 T^{9} + 1229804288 T^{10} + 246492544 T^{11} + 31363232 T^{12} + 3663488 T^{13} + 926848 T^{14} + 186688 T^{15} + 21588 T^{16} + 840 T^{17} + 8 T^{18} + 4 T^{19} + T^{20} \)
$67$ \( 210453397504 + 2284922601472 T + 12403865550848 T^{2} + 27655294418944 T^{3} + 28263770488832 T^{4} - 13407377948672 T^{5} + 5667880435712 T^{6} + 2867528204288 T^{7} + 724321959936 T^{8} - 392953856 T^{9} + 1266810880 T^{10} + 1069056000 T^{11} + 402550784 T^{12} - 10567680 T^{13} + 430080 T^{14} + 164864 T^{15} + 42176 T^{16} - 960 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$71$ \( 84783728164864 + 286334329552896 T^{2} + 119444957298688 T^{4} + 19751533281280 T^{6} + 1633300643840 T^{8} + 72970174464 T^{10} + 1802900480 T^{12} + 24765952 T^{14} + 184320 T^{16} + 688 T^{18} + T^{20} \)
$73$ \( 1224592353918976 + 9210388927217664 T^{2} + 5934864196173824 T^{4} + 620748281806848 T^{6} + 26910405677056 T^{8} + 619035629568 T^{10} + 8344750848 T^{12} + 68179456 T^{14} + 332272 T^{16} + 888 T^{18} + T^{20} \)
$79$ \( ( 46268416 - 77168640 T - 50255872 T^{2} + 3687424 T^{3} + 3393408 T^{4} - 266432 T^{5} - 70364 T^{6} + 9352 T^{7} - 104 T^{8} - 28 T^{9} + T^{10} )^{2} \)
$83$ \( 761801736963751936 + 875555940458299392 T + 503148107752538112 T^{2} + 158089032079769600 T^{3} + 31122539714969600 T^{4} + 4222033294524416 T^{5} + 700248334794752 T^{6} + 152230226034688 T^{7} + 27414010302464 T^{8} + 3259538489344 T^{9} + 322663907328 T^{10} + 40555229184 T^{11} + 6124317440 T^{12} + 684328960 T^{13} + 51499008 T^{14} + 2781696 T^{15} + 181424 T^{16} + 15936 T^{17} + 1152 T^{18} + 48 T^{19} + T^{20} \)
$89$ \( 1518596177800462336 + 469438536991899648 T^{2} + 52248775078248448 T^{4} + 2876984841338880 T^{6} + 88667808735232 T^{8} + 1624662126592 T^{10} + 18159960576 T^{12} + 123870720 T^{14} + 500432 T^{16} + 1096 T^{18} + T^{20} \)
$97$ \( ( -37943296 + 87896064 T - 52876288 T^{2} + 5846528 T^{3} + 2172480 T^{4} - 392640 T^{5} - 22480 T^{6} + 6080 T^{7} - 44 T^{8} - 28 T^{9} + T^{10} )^{2} \)
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