Properties

Label 60.2.j.a
Level $60$
Weight $2$
Character orbit 60.j
Analytic conductor $0.479$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.426337261060096.1
Defining polynomial: \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{10} - \nu^{9} - 4 \nu^{8} + 2 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} - 24 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 24 \nu + 48 \)\()/80\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} - 3 \nu^{7} + 5 \nu^{6} + 24 \nu^{5} + 2 \nu^{4} - 8 \nu^{3} - 24 \nu^{2} - 48 \nu \)\()/160\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{10} + 9 \nu^{9} + 6 \nu^{8} + 2 \nu^{7} - 17 \nu^{6} - 35 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} + 28 \nu^{2} - 56 \nu - 192 \)\()/80\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{11} - \nu^{10} + \nu^{9} + 2 \nu^{8} + 13 \nu^{7} - 5 \nu^{6} - 19 \nu^{5} - 22 \nu^{4} - 22 \nu^{3} + 44 \nu^{2} + 88 \nu \)\()/80\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} + 4 \nu^{10} + 2 \nu^{9} + \nu^{7} + 16 \nu^{6} - 6 \nu^{5} + 4 \nu^{4} + 8 \nu^{3} - 48 \nu^{2} - 160 \nu - 64 \)\()/160\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{8} + 3 \nu^{7} - 4 \nu^{6} - 8 \nu^{5} - 8 \nu^{4} + 12 \nu^{3} + 32 \nu^{2} \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + 38 \nu^{4} + 28 \nu^{3} - 56 \nu^{2} - 192 \nu \)\()/160\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} - 4 \nu^{8} - 5 \nu^{7} + \nu^{6} + 12 \nu^{5} + 8 \nu^{4} + 12 \nu^{3} - 36 \nu^{2} - 16 \nu - 64 \)\()/80\)
\(\beta_{9}\)\(=\)\((\)\( 9 \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 20 \nu^{8} - 31 \nu^{7} + 34 \nu^{6} + 46 \nu^{5} + 16 \nu^{4} - 88 \nu^{3} - 192 \nu^{2} - 160 \nu + 64 \)\()/160\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{11} - 3 \nu^{10} - \nu^{9} + 4 \nu^{8} + 7 \nu^{7} + \nu^{6} - 9 \nu^{5} - 20 \nu^{4} + 4 \nu^{3} + 20 \nu^{2} + 36 \nu + 16 \)\()/40\)
\(\beta_{11}\)\(=\)\((\)\( 11 \nu^{11} + 6 \nu^{10} + 14 \nu^{9} - 12 \nu^{8} - 53 \nu^{7} + 10 \nu^{6} + 14 \nu^{5} + 112 \nu^{4} + 32 \nu^{3} - 224 \nu^{2} - 128 \nu - 320 \)\()/160\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{9} + \beta_{4} - \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} - 2 \beta_{10} + \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} - 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{1} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 4 \beta_{1} + 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{11} - \beta_{9} + 2 \beta_{8} - 2 \beta_{5} - \beta_{4} - \beta_{3} + 8 \beta_{2} - 2 \beta_{1} - 2\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(3 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} - \beta_{4} - 3 \beta_{3} + 6 \beta_{2}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-5 \beta_{11} - 5 \beta_{9} + 10 \beta_{8} - 8 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} + \beta_{3} - 6 \beta_{1} + 10\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(5 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 2 \beta_{7} + 10 \beta_{6} + 5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 20 \beta_{1} + 4\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(9 \beta_{11} - 7 \beta_{9} - 10 \beta_{8} + 2 \beta_{5} + 9 \beta_{4} + \beta_{3} + 40 \beta_{2} + 10 \beta_{1} + 10\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(13 \beta_{11} - 14 \beta_{10} - 13 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 14 \beta_{6} + 26 \beta_{5} + 25 \beta_{4} - 21 \beta_{3} - 6 \beta_{2}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-11 \beta_{11} + 29 \beta_{9} + 54 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} - 18 \beta_{5} + 11 \beta_{4} + 15 \beta_{3} - 26 \beta_{1} + 54\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(\beta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.41127 + 0.0912546i
1.19252 + 0.760198i
−0.0912546 1.41127i
−0.394157 + 1.35818i
−0.760198 1.19252i
−1.35818 + 0.394157i
1.41127 0.0912546i
1.19252 0.760198i
−0.0912546 + 1.41127i
−0.394157 1.35818i
−0.760198 + 1.19252i
−1.35818 0.394157i
−1.41127 + 0.0912546i −0.707107 0.707107i 1.98335 0.257569i 1.32001 1.80487i 1.06244 + 0.933389i 1.86678 1.86678i −2.77552 + 0.544488i 1.00000i −1.69819 + 2.66761i
7.2 −1.19252 + 0.760198i 0.707107 + 0.707107i 0.844199 1.81310i 0.432320 + 2.19388i −1.38078 0.305697i −0.611393 + 0.611393i 0.371591 + 2.80391i 1.00000i −2.18333 2.28759i
7.3 0.0912546 1.41127i 0.707107 + 0.707107i −1.98335 0.257569i 1.32001 1.80487i 1.06244 0.933389i −1.86678 + 1.86678i −0.544488 + 2.77552i 1.00000i −2.42670 2.02759i
7.4 0.394157 + 1.35818i 0.707107 + 0.707107i −1.68928 + 1.07067i −1.75233 1.38900i −0.681664 + 1.23909i 2.47817 2.47817i −2.12000 1.87233i 1.00000i 1.19582 2.92746i
7.5 0.760198 1.19252i −0.707107 0.707107i −0.844199 1.81310i 0.432320 + 2.19388i −1.38078 + 0.305697i 0.611393 0.611393i −2.80391 0.371591i 1.00000i 2.94489 + 1.15223i
7.6 1.35818 + 0.394157i −0.707107 0.707107i 1.68928 + 1.07067i −1.75233 1.38900i −0.681664 1.23909i −2.47817 + 2.47817i 1.87233 + 2.12000i 1.00000i −1.83249 2.57720i
43.1 −1.41127 0.0912546i −0.707107 + 0.707107i 1.98335 + 0.257569i 1.32001 + 1.80487i 1.06244 0.933389i 1.86678 + 1.86678i −2.77552 0.544488i 1.00000i −1.69819 2.66761i
43.2 −1.19252 0.760198i 0.707107 0.707107i 0.844199 + 1.81310i 0.432320 2.19388i −1.38078 + 0.305697i −0.611393 0.611393i 0.371591 2.80391i 1.00000i −2.18333 + 2.28759i
43.3 0.0912546 + 1.41127i 0.707107 0.707107i −1.98335 + 0.257569i 1.32001 + 1.80487i 1.06244 + 0.933389i −1.86678 1.86678i −0.544488 2.77552i 1.00000i −2.42670 + 2.02759i
43.4 0.394157 1.35818i 0.707107 0.707107i −1.68928 1.07067i −1.75233 + 1.38900i −0.681664 1.23909i 2.47817 + 2.47817i −2.12000 + 1.87233i 1.00000i 1.19582 + 2.92746i
43.5 0.760198 + 1.19252i −0.707107 + 0.707107i −0.844199 + 1.81310i 0.432320 2.19388i −1.38078 0.305697i 0.611393 + 0.611393i −2.80391 + 0.371591i 1.00000i 2.94489 1.15223i
43.6 1.35818 0.394157i −0.707107 + 0.707107i 1.68928 1.07067i −1.75233 + 1.38900i −0.681664 + 1.23909i −2.47817 2.47817i 1.87233 2.12000i 1.00000i −1.83249 + 2.57720i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.j.a 12
3.b odd 2 1 180.2.k.e 12
4.b odd 2 1 inner 60.2.j.a 12
5.b even 2 1 300.2.j.d 12
5.c odd 4 1 inner 60.2.j.a 12
5.c odd 4 1 300.2.j.d 12
8.b even 2 1 960.2.w.g 12
8.d odd 2 1 960.2.w.g 12
12.b even 2 1 180.2.k.e 12
15.d odd 2 1 900.2.k.n 12
15.e even 4 1 180.2.k.e 12
15.e even 4 1 900.2.k.n 12
20.d odd 2 1 300.2.j.d 12
20.e even 4 1 inner 60.2.j.a 12
20.e even 4 1 300.2.j.d 12
40.i odd 4 1 960.2.w.g 12
40.k even 4 1 960.2.w.g 12
60.h even 2 1 900.2.k.n 12
60.l odd 4 1 180.2.k.e 12
60.l odd 4 1 900.2.k.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a 12 1.a even 1 1 trivial
60.2.j.a 12 4.b odd 2 1 inner
60.2.j.a 12 5.c odd 4 1 inner
60.2.j.a 12 20.e even 4 1 inner
180.2.k.e 12 3.b odd 2 1
180.2.k.e 12 12.b even 2 1
180.2.k.e 12 15.e even 4 1
180.2.k.e 12 60.l odd 4 1
300.2.j.d 12 5.b even 2 1
300.2.j.d 12 5.c odd 4 1
300.2.j.d 12 20.d odd 2 1
300.2.j.d 12 20.e even 4 1
900.2.k.n 12 15.d odd 2 1
900.2.k.n 12 15.e even 4 1
900.2.k.n 12 60.h even 2 1
900.2.k.n 12 60.l odd 4 1
960.2.w.g 12 8.b even 2 1
960.2.w.g 12 8.d odd 2 1
960.2.w.g 12 40.i odd 4 1
960.2.w.g 12 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 32 T^{3} - 12 T^{4} - 8 T^{5} + 8 T^{6} - 4 T^{7} - 3 T^{8} + 4 T^{9} + T^{12} \)
$3$ \( ( 1 + T^{4} )^{3} \)
$5$ \( ( 125 + 25 T^{2} + 8 T^{3} + 5 T^{4} + T^{6} )^{2} \)
$7$ \( 4096 + 7440 T^{4} + 200 T^{8} + T^{12} \)
$11$ \( ( 128 + 260 T^{2} + 36 T^{4} + T^{6} )^{2} \)
$13$ \( ( 32 - 96 T + 144 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$17$ \( ( 800 - 160 T + 16 T^{2} + 80 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$19$ \( ( -512 + 400 T^{2} - 40 T^{4} + T^{6} )^{2} \)
$23$ \( 65536 + 37120 T^{4} + 4640 T^{8} + T^{12} \)
$29$ \( ( 64 + 100 T^{2} + 20 T^{4} + T^{6} )^{2} \)
$31$ \( ( 32768 + 3648 T^{2} + 112 T^{4} + T^{6} )^{2} \)
$37$ \( ( 32 + 96 T + 144 T^{2} + 32 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$41$ \( ( 64 - 20 T - 4 T^{2} + T^{3} )^{4} \)
$43$ \( 15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12} \)
$47$ \( 40960000 + 901376 T^{4} + 4896 T^{8} + T^{12} \)
$53$ \( ( 128 - 256 T + 256 T^{2} + 16 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$59$ \( ( -512 + 1860 T^{2} - 100 T^{4} + T^{6} )^{2} \)
$61$ \( ( 176 - 100 T + 8 T^{2} + T^{3} )^{4} \)
$67$ \( 15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12} \)
$71$ \( ( 204800 + 15616 T^{2} + 256 T^{4} + T^{6} )^{2} \)
$73$ \( ( 55112 + 4648 T + 196 T^{2} - 640 T^{3} + 242 T^{4} - 22 T^{5} + T^{6} )^{2} \)
$79$ \( ( -2048 + 12864 T^{2} - 304 T^{4} + T^{6} )^{2} \)
$83$ \( 65536 + 10264832 T^{4} + 16672 T^{8} + T^{12} \)
$89$ \( ( 1024 + 1040 T^{2} + 72 T^{4} + T^{6} )^{2} \)
$97$ \( ( 35912 - 47704 T + 31684 T^{2} - 2048 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2} \)
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