Properties

Label 91.2.q.a
Level $91$
Weight $2$
Character orbit 91.q
Analytic conductor $0.727$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.58891012706304.1
Defining polynomial: \(x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{10} - 2 x^{9} + 15 x^{8} + 2 x^{7} - 30 x^{6} + 4 x^{5} + 60 x^{4} - 16 x^{3} - 80 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} - 5 \nu^{9} - 2 \nu^{8} + 15 \nu^{7} + 2 \nu^{6} - 30 \nu^{5} + 4 \nu^{4} + 60 \nu^{3} - 16 \nu^{2} - 80 \nu \)\()/32\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64 \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80 \)\()/16\)
\(\beta_{10}\)\(=\)\((\)\( -3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64 \)\()/16\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1\)
\(\nu^{5}\)\(=\)\(-2 \beta_{11} + 4 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} - 2\)
\(\nu^{7}\)\(=\)\(-5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1\)
\(\nu^{8}\)\(=\)\(-3 \beta_{11} + 7 \beta_{8} + \beta_{7} - 5 \beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 10 \beta_{2} + 8 \beta_{1} - 3\)
\(\nu^{9}\)\(=\)\(-7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta_{1} - 8\)
\(\nu^{10}\)\(=\)\(-17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 11\)
\(\nu^{11}\)\(=\)\(18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(1 - \beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
−1.12906 0.851598i
0.759479 1.19298i
1.34408 + 0.439820i
−1.08105 + 0.911778i
1.40744 + 0.138282i
−1.30089 + 0.554694i
−1.12906 + 0.851598i
0.759479 + 1.19298i
1.34408 0.439820i
−1.08105 0.911778i
1.40744 0.138282i
−1.30089 0.554694i
−2.34104 + 1.35160i 0.172975 + 0.299601i 2.65363 4.59623i 3.25812i −0.809880 0.467584i 0.866025 + 0.500000i 8.94020i 1.44016 2.49443i 4.40367 + 7.62739i
36.2 −1.20027 + 0.692976i 1.41289 + 2.44719i −0.0395678 + 0.0685334i 0.518957i −3.39169 1.95819i −0.866025 0.500000i 2.88158i −2.49250 + 4.31714i 0.359625 + 0.622889i
36.3 −0.104235 + 0.0601799i 0.291146 + 0.504280i −0.992757 + 1.71951i 1.68817i −0.0606950 0.0350423i 0.866025 + 0.500000i 0.479696i 1.33047 2.30444i −0.101594 0.175965i
36.4 0.713220 0.411778i −1.33015 2.30388i −0.660878 + 1.14467i 3.16209i −1.89737 1.09545i 0.866025 + 0.500000i 2.73565i −2.03858 + 3.53092i −1.30208 2.25527i
36.5 1.10554 0.638282i 0.583963 + 1.01145i −0.185192 + 0.320762i 1.81487i 1.29118 + 0.745466i −0.866025 0.500000i 3.02595i 0.817975 1.41677i −1.15840 2.00641i
36.6 1.82678 1.05469i −1.13082 1.95864i 1.22476 2.12135i 3.60178i −4.13154 2.38535i −0.866025 0.500000i 0.948212i −1.05753 + 1.83169i 3.79878 + 6.57967i
43.1 −2.34104 1.35160i 0.172975 0.299601i 2.65363 + 4.59623i 3.25812i −0.809880 + 0.467584i 0.866025 0.500000i 8.94020i 1.44016 + 2.49443i 4.40367 7.62739i
43.2 −1.20027 0.692976i 1.41289 2.44719i −0.0395678 0.0685334i 0.518957i −3.39169 + 1.95819i −0.866025 + 0.500000i 2.88158i −2.49250 4.31714i 0.359625 0.622889i
43.3 −0.104235 0.0601799i 0.291146 0.504280i −0.992757 1.71951i 1.68817i −0.0606950 + 0.0350423i 0.866025 0.500000i 0.479696i 1.33047 + 2.30444i −0.101594 + 0.175965i
43.4 0.713220 + 0.411778i −1.33015 + 2.30388i −0.660878 1.14467i 3.16209i −1.89737 + 1.09545i 0.866025 0.500000i 2.73565i −2.03858 3.53092i −1.30208 + 2.25527i
43.5 1.10554 + 0.638282i 0.583963 1.01145i −0.185192 0.320762i 1.81487i 1.29118 0.745466i −0.866025 + 0.500000i 3.02595i 0.817975 + 1.41677i −1.15840 + 2.00641i
43.6 1.82678 + 1.05469i −1.13082 + 1.95864i 1.22476 + 2.12135i 3.60178i −4.13154 + 2.38535i −0.866025 + 0.500000i 0.948212i −1.05753 1.83169i 3.79878 6.57967i
\(n\): 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
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.q.a 12
3.b odd 2 1 819.2.ct.a 12
4.b odd 2 1 1456.2.cc.c 12
7.b odd 2 1 637.2.q.h 12
7.c even 3 1 637.2.k.h 12
7.c even 3 1 637.2.u.h 12
7.d odd 6 1 637.2.k.g 12
7.d odd 6 1 637.2.u.i 12
13.c even 3 1 1183.2.c.i 12
13.e even 6 1 inner 91.2.q.a 12
13.e even 6 1 1183.2.c.i 12
13.f odd 12 1 1183.2.a.m 6
13.f odd 12 1 1183.2.a.p 6
39.h odd 6 1 819.2.ct.a 12
52.i odd 6 1 1456.2.cc.c 12
91.k even 6 1 637.2.u.h 12
91.l odd 6 1 637.2.u.i 12
91.p odd 6 1 637.2.k.g 12
91.t odd 6 1 637.2.q.h 12
91.u even 6 1 637.2.k.h 12
91.bc even 12 1 8281.2.a.by 6
91.bc even 12 1 8281.2.a.ch 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 1.a even 1 1 trivial
91.2.q.a 12 13.e even 6 1 inner
637.2.k.g 12 7.d odd 6 1
637.2.k.g 12 91.p odd 6 1
637.2.k.h 12 7.c even 3 1
637.2.k.h 12 91.u even 6 1
637.2.q.h 12 7.b odd 2 1
637.2.q.h 12 91.t odd 6 1
637.2.u.h 12 7.c even 3 1
637.2.u.h 12 91.k even 6 1
637.2.u.i 12 7.d odd 6 1
637.2.u.i 12 91.l odd 6 1
819.2.ct.a 12 3.b odd 2 1
819.2.ct.a 12 39.h odd 6 1
1183.2.a.m 6 13.f odd 12 1
1183.2.a.p 6 13.f odd 12 1
1183.2.c.i 12 13.c even 3 1
1183.2.c.i 12 13.e even 6 1
1456.2.cc.c 12 4.b odd 2 1
1456.2.cc.c 12 52.i odd 6 1
8281.2.a.by 6 91.bc even 12 1
8281.2.a.ch 6 91.bc even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 12 T + 36 T^{2} - 144 T^{3} + 112 T^{4} + 72 T^{5} - 82 T^{6} - 36 T^{7} + 52 T^{8} - 8 T^{10} + T^{12} \)
$3$ \( 16 - 80 T + 300 T^{2} - 516 T^{3} + 709 T^{4} - 390 T^{5} + 287 T^{6} - 42 T^{7} + 96 T^{8} - 4 T^{9} + 11 T^{10} + T^{12} \)
$5$ \( 3481 + 16148 T^{2} + 13040 T^{4} + 4146 T^{6} + 600 T^{8} + 40 T^{10} + T^{12} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$11$ \( 256 + 768 T - 80 T^{2} - 2544 T^{3} + 2025 T^{4} + 1494 T^{5} - 771 T^{6} - 522 T^{7} + 248 T^{8} + 114 T^{9} - 7 T^{10} - 6 T^{11} + T^{12} \)
$13$ \( 4826809 - 1485172 T + 599781 T^{2} - 70304 T^{3} - 23998 T^{4} + 12012 T^{5} - 6587 T^{6} + 924 T^{7} - 142 T^{8} - 32 T^{9} + 21 T^{10} - 4 T^{11} + T^{12} \)
$17$ \( 241081 - 109984 T + 132173 T^{2} - 21512 T^{3} + 31018 T^{4} - 2576 T^{5} + 5229 T^{6} + 148 T^{7} + 514 T^{8} + 36 T^{9} + 37 T^{10} + 4 T^{11} + T^{12} \)
$19$ \( 55696 - 138768 T + 137196 T^{2} - 54684 T^{3} - 4499 T^{4} + 9486 T^{5} + 299 T^{6} - 2370 T^{7} + 748 T^{8} - 29 T^{10} + T^{12} \)
$23$ \( 38539264 + 9138176 T + 12198912 T^{2} + 5170176 T^{3} + 3630592 T^{4} + 1115904 T^{5} + 332096 T^{6} + 52416 T^{7} + 9312 T^{8} + 976 T^{9} + 164 T^{10} + 12 T^{11} + T^{12} \)
$29$ \( 10042561 - 1064784 T + 4587524 T^{2} - 3112876 T^{3} + 2323356 T^{4} - 854112 T^{5} + 261878 T^{6} - 47832 T^{7} + 7876 T^{8} - 780 T^{9} + 108 T^{10} - 8 T^{11} + T^{12} \)
$31$ \( 913936 + 1285560 T^{2} + 568225 T^{4} + 96896 T^{6} + 5854 T^{8} + 136 T^{10} + T^{12} \)
$37$ \( 1755945216 - 633588480 T - 260200512 T^{2} + 121383360 T^{3} + 65323584 T^{4} + 8644320 T^{5} - 396036 T^{6} - 159840 T^{7} + 8109 T^{8} + 5670 T^{9} + 723 T^{10} + 42 T^{11} + T^{12} \)
$41$ \( 884705536 - 421175040 T - 8238656 T^{2} + 35739840 T^{3} - 872384 T^{4} - 2742240 T^{5} + 452252 T^{6} + 35760 T^{7} - 11651 T^{8} - 450 T^{9} + 315 T^{10} - 30 T^{11} + T^{12} \)
$43$ \( 2408704 + 5860352 T + 10500784 T^{2} + 8862336 T^{3} + 5690569 T^{4} + 1037954 T^{5} + 282645 T^{6} + 3650 T^{7} + 9640 T^{8} + 38 T^{9} + 113 T^{10} - 2 T^{11} + T^{12} \)
$47$ \( 9461776 + 47561752 T^{2} + 10113609 T^{4} + 722232 T^{6} + 21782 T^{8} + 272 T^{10} + T^{12} \)
$53$ \( ( -2339 + 7302 T - 3353 T^{2} - 700 T^{3} + 91 T^{4} + 22 T^{5} + T^{6} )^{2} \)
$59$ \( 4571923456 - 3919023360 T + 1190986848 T^{2} - 61031880 T^{3} - 34227911 T^{4} + 3548430 T^{5} + 1035110 T^{6} - 185580 T^{7} - 2237 T^{8} + 1980 T^{9} - 2 T^{10} - 18 T^{11} + T^{12} \)
$61$ \( 5607424 - 3788800 T + 7030784 T^{2} - 3685376 T^{3} + 6036160 T^{4} - 2984960 T^{5} + 1823136 T^{6} - 177656 T^{7} + 29281 T^{8} - 1614 T^{9} + 283 T^{10} - 14 T^{11} + T^{12} \)
$67$ \( 613651984 - 1248211536 T + 790406444 T^{2} + 113725716 T^{3} - 37845559 T^{4} - 4849356 T^{5} + 1364502 T^{6} + 185988 T^{7} - 11949 T^{8} - 2352 T^{9} + 94 T^{10} + 24 T^{11} + T^{12} \)
$71$ \( 46895104 + 26296320 T - 3083264 T^{2} - 4485120 T^{3} + 367104 T^{4} + 620544 T^{5} + 82752 T^{6} - 17280 T^{7} - 3808 T^{8} + 480 T^{9} + 212 T^{10} + 24 T^{11} + T^{12} \)
$73$ \( 1386221824 + 513361280 T^{2} + 55965104 T^{4} + 2238456 T^{6} + 40473 T^{8} + 334 T^{10} + T^{12} \)
$79$ \( ( -512 + 1664 T - 1584 T^{2} + 192 T^{3} + 212 T^{4} + 28 T^{5} + T^{6} )^{2} \)
$83$ \( 141324544 + 454322976 T^{2} + 48190849 T^{4} + 1905008 T^{6} + 35086 T^{8} + 304 T^{10} + T^{12} \)
$89$ \( 1834580224 + 1271596416 T - 422744080 T^{2} - 496650552 T^{3} + 291555473 T^{4} - 41341974 T^{5} - 2811261 T^{6} + 792798 T^{7} + 66288 T^{8} - 3660 T^{9} - 257 T^{10} + 12 T^{11} + T^{12} \)
$97$ \( 53465344 + 190755456 T + 211235504 T^{2} - 55750056 T^{3} - 14421439 T^{4} + 4289964 T^{5} + 938031 T^{6} - 349092 T^{7} + 28032 T^{8} + 1110 T^{9} - 173 T^{10} - 6 T^{11} + T^{12} \)
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