Properties

Label 74.3.g.b
Level $74$
Weight $3$
Character orbit 74.g
Analytic conductor $2.016$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01635395627\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 82 x^{10} + 2505 x^{8} + 34456 x^{6} + 196096 x^{4} + 262464 x^{2} + 69696\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{4} - \beta_{7} ) q^{2} -\beta_{3} q^{3} + 2 \beta_{6} q^{4} + ( 1 + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{3} + \beta_{5} ) q^{6} + ( -1 - \beta_{4} - \beta_{7} - \beta_{11} ) q^{7} + ( -2 - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( \beta_{4} - \beta_{7} ) q^{2} -\beta_{3} q^{3} + 2 \beta_{6} q^{4} + ( 1 + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{2} + \beta_{3} + \beta_{5} ) q^{6} + ( -1 - \beta_{4} - \beta_{7} - \beta_{11} ) q^{7} + ( -2 - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{9} + ( -3 + \beta_{6} + \beta_{7} ) q^{10} + ( -1 - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{11} -2 \beta_{8} q^{12} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{5} + 6 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{13} + ( 2 - \beta_{1} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{14} + ( 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{8} ) q^{15} -4 \beta_{4} q^{16} + ( 6 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{17} + ( 4 + 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{18} + ( -3 - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{19} + ( -2 - 4 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{20} + ( 6 - \beta_{1} - 5 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 5 \beta_{7} - \beta_{8} + \beta_{11} ) q^{21} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{22} + ( -6 + \beta_{1} + 3 \beta_{3} + \beta_{4} - 7 \beta_{6} + 7 \beta_{7} + 3 \beta_{8} + \beta_{10} + \beta_{11} ) q^{23} + ( 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{8} ) q^{24} + ( -6 - 3 \beta_{4} - 19 \beta_{7} ) q^{25} + ( -10 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{26} + ( -9 - 3 \beta_{2} - 3 \beta_{3} - 20 \beta_{4} - \beta_{5} - 7 \beta_{6} + 7 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{27} + ( -2 - 2 \beta_{7} - 2 \beta_{9} ) q^{28} + ( 2 \beta_{1} + 5 \beta_{2} + \beta_{3} - 4 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{29} + ( 3 \beta_{3} - \beta_{5} - \beta_{8} ) q^{30} + ( 2 - 3 \beta_{2} + 11 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{31} + ( 4 + 4 \beta_{4} - 4 \beta_{6} ) q^{32} + ( -2 - 4 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} + 32 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} ) q^{33} + ( 1 - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 6 \beta_{5} + 5 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} - 2 \beta_{11} ) q^{34} + ( -2 \beta_{1} - \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{35} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{36} + ( -9 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{37} + ( 6 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{38} + ( 18 - 2 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} + \beta_{4} + 6 \beta_{5} + 17 \beta_{6} - 2 \beta_{7} - 12 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{39} + ( 2 - 2 \beta_{4} - 6 \beta_{6} ) q^{40} + ( 3 - 2 \beta_{1} + 6 \beta_{3} - 3 \beta_{4} - 14 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{41} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} + 9 \beta_{6} - 5 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{42} + ( 14 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + 13 \beta_{6} - 13 \beta_{7} - 8 \beta_{8} + \beta_{9} + \beta_{10} ) q^{43} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{11} ) q^{44} + ( 6 - \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 7 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{45} + ( -1 - \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 13 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{46} + ( -11 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{9} + \beta_{10} ) q^{47} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{48} + ( -\beta_{1} - 7 \beta_{2} - 14 \beta_{3} + 28 \beta_{4} + 4 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - \beta_{11} ) q^{49} + ( 22 + 16 \beta_{4} + 16 \beta_{6} + 6 \beta_{7} ) q^{50} + ( -26 + 3 \beta_{1} + 9 \beta_{2} + 10 \beta_{3} + 20 \beta_{4} - 9 \beta_{5} + 43 \beta_{6} - 23 \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{51} + ( -2 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{52} + ( 2 - \beta_{1} + 5 \beta_{2} + 10 \beta_{3} - 13 \beta_{4} - 4 \beta_{6} + \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{53} + ( 19 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 20 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{54} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{55} + ( 2 + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} ) q^{56} + ( -6 - 2 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} - 17 \beta_{4} - 2 \beta_{5} - 7 \beta_{6} + 16 \beta_{7} + 11 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{57} + ( 9 + 3 \beta_{1} - 6 \beta_{2} + 4 \beta_{4} + 8 \beta_{5} - 5 \beta_{7} - 4 \beta_{8} + \beta_{9} - 3 \beta_{11} ) q^{58} + ( -3 - \beta_{1} - 12 \beta_{2} - 6 \beta_{3} + 16 \beta_{4} + 6 \beta_{5} + \beta_{6} + 14 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{59} + ( -4 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{8} ) q^{60} + ( 35 + 10 \beta_{2} + 8 \beta_{3} + 2 \beta_{5} - 36 \beta_{7} + 8 \beta_{8} - \beta_{10} - \beta_{11} ) q^{61} + ( -21 - \beta_{1} + 3 \beta_{2} - 11 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{11} ) q^{62} + ( -66 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{5} - 7 \beta_{6} - 7 \beta_{7} - \beta_{9} + \beta_{10} ) q^{63} + ( 8 \beta_{6} - 8 \beta_{7} ) q^{64} + ( -11 + 2 \beta_{1} - 3 \beta_{2} - 7 \beta_{4} - 6 \beta_{5} + 16 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{11} ) q^{65} + ( -12 - 2 \beta_{1} + 4 \beta_{2} - 32 \beta_{4} - 4 \beta_{5} - 22 \beta_{6} - 14 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{66} + ( 25 + 2 \beta_{1} - 25 \beta_{4} + 3 \beta_{5} + 11 \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{10} + \beta_{11} ) q^{67} + ( 6 - 2 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{68} + ( -1 + 2 \beta_{1} + 6 \beta_{2} + 16 \beta_{3} + 39 \beta_{4} - 10 \beta_{5} - 39 \beta_{6} + 43 \beta_{7} + 16 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{69} + ( 1 + 3 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{70} + ( 14 - 10 \beta_{2} - 5 \beta_{3} + 16 \beta_{4} + 9 \beta_{5} + 21 \beta_{6} - 38 \beta_{7} - 9 \beta_{8} - 2 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{71} + ( 2 + 2 \beta_{1} + 4 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} + 10 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{72} + ( -22 + 2 \beta_{1} - 48 \beta_{4} + 8 \beta_{5} + 30 \beta_{6} - 26 \beta_{7} - 16 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{73} + ( -16 - 2 \beta_{1} - 6 \beta_{3} - 23 \beta_{4} + 8 \beta_{5} + 8 \beta_{7} - 4 \beta_{8} - 3 \beta_{10} + \beta_{11} ) q^{74} + ( -3 \beta_{2} + 3 \beta_{3} + 19 \beta_{5} ) q^{75} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{76} + ( 8 - 8 \beta_{4} - 12 \beta_{5} - 86 \beta_{6} - 12 \beta_{8} + 2 \beta_{10} ) q^{77} + ( -17 + 2 \beta_{1} + 18 \beta_{3} + 17 \beta_{4} - 6 \beta_{5} - 15 \beta_{6} + \beta_{7} - 6 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{78} + ( 26 - 5 \beta_{2} - 19 \beta_{3} + 30 \beta_{4} - 14 \beta_{5} + 30 \beta_{6} - 4 \beta_{7} + 19 \beta_{8} ) q^{79} + ( 8 + 4 \beta_{4} + 4 \beta_{6} - 8 \beta_{7} ) q^{80} + ( 3 - 8 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} - 10 \beta_{6} + 26 \beta_{7} + 8 \beta_{8} + 6 \beta_{11} ) q^{81} + ( 17 + 2 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + 11 \beta_{6} - 15 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{82} + ( 1 + 7 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} - 2 \beta_{6} + 8 \beta_{7} - 15 \beta_{8} + \beta_{9} + 2 \beta_{10} + 7 \beta_{11} ) q^{83} + ( -10 - 2 \beta_{2} + 2 \beta_{3} + 10 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{84} + ( 11 - \beta_{1} - 12 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} - 6 \beta_{5} + 9 \beta_{6} - 11 \beta_{7} + 12 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} ) q^{85} + ( -2 \beta_{1} + 8 \beta_{2} + 16 \beta_{3} + 26 \beta_{4} - 2 \beta_{7} - 2 \beta_{11} ) q^{86} + ( 64 - 5 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} - 14 \beta_{4} - 11 \beta_{5} - 14 \beta_{6} + 69 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{87} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{88} + ( 4 - \beta_{1} + 10 \beta_{2} + 2 \beta_{3} - 42 \beta_{4} + 2 \beta_{5} + 7 \beta_{6} + 45 \beta_{7} - 10 \beta_{8} - 3 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} ) q^{89} + ( -1 + 3 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 12 \beta_{4} + 8 \beta_{6} - \beta_{7} - 6 \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{90} + ( 65 - \beta_{1} - 15 \beta_{2} - 4 \beta_{4} - 67 \beta_{6} - 6 \beta_{7} - 15 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{91} + ( 14 + 2 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 14 \beta_{4} - 14 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{92} + ( -41 + 2 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - 41 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{93} + ( -9 + \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - 13 \beta_{4} + 8 \beta_{5} - 6 \beta_{6} + 16 \beta_{7} - 7 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{94} + ( 3 - \beta_{1} + 6 \beta_{2} + 3 \beta_{4} + 6 \beta_{5} - 8 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + \beta_{11} ) q^{95} + ( 4 \beta_{2} + 4 \beta_{8} ) q^{96} + ( -42 - 4 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} + 3 \beta_{5} - 27 \beta_{6} + 38 \beta_{7} + 6 \beta_{8} - \beta_{9} - 4 \beta_{10} - 3 \beta_{11} ) q^{97} + ( -30 + 14 \beta_{2} + 4 \beta_{3} - 26 \beta_{4} + 10 \beta_{5} + 26 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + \beta_{10} + \beta_{11} ) q^{98} + ( -27 + 2 \beta_{1} - 4 \beta_{2} - 14 \beta_{4} - 30 \beta_{5} - 37 \beta_{7} + 15 \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{2} + 4q^{6} - 8q^{7} - 24q^{8} + 28q^{9} + O(q^{10}) \) \( 12q - 6q^{2} + 4q^{6} - 8q^{7} - 24q^{8} + 28q^{9} - 36q^{10} - 4q^{12} + 16q^{13} + 16q^{14} + 24q^{16} + 40q^{17} + 28q^{18} - 26q^{19} + 66q^{21} + 4q^{22} - 80q^{23} - 4q^{24} - 54q^{25} - 124q^{26} - 12q^{28} + 16q^{29} - 6q^{30} - 32q^{31} + 24q^{32} - 20q^{33} - 10q^{34} + 12q^{35} - 148q^{37} + 92q^{38} + 216q^{39} + 36q^{40} + 66q^{41} - 46q^{42} + 152q^{43} - 16q^{44} + 84q^{45} + 80q^{46} - 112q^{47} - 160q^{49} + 168q^{50} - 446q^{51} + 32q^{52} + 74q^{53} + 230q^{54} + 28q^{56} + 50q^{57} + 84q^{58} - 114q^{59} - 12q^{60} + 448q^{61} - 204q^{62} - 784q^{63} - 138q^{65} + 40q^{66} + 468q^{67} + 20q^{68} - 278q^{69} + 18q^{70} + 116q^{71} - 56q^{72} - 2q^{74} + 76q^{75} - 52q^{76} + 60q^{77} - 366q^{78} + 114q^{79} + 72q^{80} + 14q^{81} + 128q^{82} - 20q^{83} - 80q^{84} - 152q^{86} + 770q^{87} + 32q^{88} + 340q^{89} - 84q^{90} + 792q^{91} + 68q^{92} - 498q^{93} + 20q^{94} + 60q^{95} + 8q^{96} - 356q^{97} - 160q^{98} - 348q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 82 x^{10} + 2505 x^{8} + 34456 x^{6} + 196096 x^{4} + 262464 x^{2} + 69696\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 11 \nu^{10} + 617 \nu^{8} + 10275 \nu^{6} + 46025 \nu^{4} + 86420 \nu^{2} + 1280592 \)\()/113880\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{10} + 62 \nu^{8} + 1265 \nu^{6} + 9260 \nu^{4} + 15160 \nu^{2} + 2080 \nu + 2112 \)\()/4160\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{10} - 62 \nu^{8} - 1265 \nu^{6} - 9260 \nu^{4} - 15160 \nu^{2} + 2080 \nu - 2112 \)\()/4160\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} + 49 \nu^{9} + 459 \nu^{7} - 7289 \nu^{5} - 109484 \nu^{3} - 237816 \nu - 68640 \)\()/137280\)
\(\beta_{5}\)\(=\)\((\)\( -35 \nu^{10} - 2222 \nu^{8} - 47187 \nu^{6} - 368768 \nu^{4} - 685976 \nu^{2} - 252384 \)\()/91104\)
\(\beta_{6}\)\(=\)\((\)\(-1195 \nu^{11} - 1716 \nu^{10} - 86440 \nu^{9} - 130416 \nu^{8} - 2260215 \nu^{7} - 3516084 \nu^{6} - 25603210 \nu^{5} - 39533208 \nu^{4} - 112641280 \nu^{3} - 160933344 \nu^{2} - 87272400 \nu - 104113152\)\()/60128640\)
\(\beta_{7}\)\(=\)\((\)\(1195 \nu^{11} - 1716 \nu^{10} + 86440 \nu^{9} - 130416 \nu^{8} + 2260215 \nu^{7} - 3516084 \nu^{6} + 25603210 \nu^{5} - 39533208 \nu^{4} + 112641280 \nu^{3} - 160933344 \nu^{2} + 87272400 \nu - 104113152\)\()/60128640\)
\(\beta_{8}\)\(=\)\((\)\( -26 \nu^{11} - 175 \nu^{10} - 1976 \nu^{9} - 11110 \nu^{8} - 53274 \nu^{7} - 235935 \nu^{6} - 598988 \nu^{5} - 1843840 \nu^{4} - 2438384 \nu^{3} - 3429880 \nu^{2} - 1577472 \nu - 1261920 \)\()/911040\)
\(\beta_{9}\)\(=\)\((\)\(-2269 \nu^{11} - 8184 \nu^{10} - 260044 \nu^{9} - 390720 \nu^{8} - 10295001 \nu^{7} - 2315016 \nu^{6} - 171615082 \nu^{5} + 107128032 \nu^{4} - 1077031216 \nu^{3} + 1285864800 \nu^{2} - 917077008 \nu + 954410688\)\()/60128640\)
\(\beta_{10}\)\(=\)\((\)\(-2269 \nu^{11} + 8184 \nu^{10} - 260044 \nu^{9} + 390720 \nu^{8} - 10295001 \nu^{7} + 2315016 \nu^{6} - 171615082 \nu^{5} - 107128032 \nu^{4} - 1077031216 \nu^{3} - 1285864800 \nu^{2} - 917077008 \nu - 1014539328\)\()/60128640\)
\(\beta_{11}\)\(=\)\((\)\(5849 \nu^{11} - 1188 \nu^{10} + 400286 \nu^{9} - 32472 \nu^{8} + 9894465 \nu^{7} + 803484 \nu^{6} + 110211188 \nu^{5} + 27382608 \nu^{4} + 568949912 \nu^{3} + 138118464 \nu^{2} + 1103072640 \nu - 233963136\)\()/60128640\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{3} - \beta_{2} + \beta_{1} - 13\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{9} + 2 \beta_{8} + 7 \beta_{7} - 7 \beta_{6} - \beta_{5} - 20 \beta_{4} - 21 \beta_{3} - 21 \beta_{2} - 9\)
\(\nu^{4}\)\(=\)\(-10 \beta_{7} - 10 \beta_{6} + 35 \beta_{5} - 31 \beta_{3} + 31 \beta_{2} - 21 \beta_{1} + 267\)
\(\nu^{5}\)\(=\)\(20 \beta_{11} - 35 \beta_{10} - 35 \beta_{9} - 110 \beta_{8} - 309 \beta_{7} + 329 \beta_{6} + 55 \beta_{5} + 680 \beta_{4} + 479 \beta_{3} + 479 \beta_{2} + 10 \beta_{1} + 305\)
\(\nu^{6}\)\(=\)\(-20 \beta_{10} + 20 \beta_{9} + 550 \beta_{7} + 550 \beta_{6} - 1083 \beta_{5} + 879 \beta_{3} - 879 \beta_{2} + 449 \beta_{1} - 5907\)
\(\nu^{7}\)\(=\)\(-860 \beta_{11} + 1023 \beta_{10} + 1023 \beta_{9} + 4286 \beta_{8} + 10345 \beta_{7} - 11205 \beta_{6} - 2143 \beta_{5} - 19920 \beta_{4} - 11383 \beta_{3} - 11383 \beta_{2} - 430 \beta_{1} - 8937\)
\(\nu^{8}\)\(=\)\(1120 \beta_{10} - 1120 \beta_{9} - 22210 \beta_{7} - 22210 \beta_{6} + 31819 \beta_{5} - 24183 \beta_{3} + 24183 \beta_{2} - 10093 \beta_{1} + 136851\)
\(\nu^{9}\)\(=\)\(28180 \beta_{11} - 28459 \beta_{10} - 28459 \beta_{9} - 145198 \beta_{8} - 314829 \beta_{7} + 343009 \beta_{6} + 72599 \beta_{5} + 554760 \beta_{4} + 279351 \beta_{3} + 279351 \beta_{2} + 14090 \beta_{1} + 248921\)
\(\nu^{10}\)\(=\)\(-44140 \beta_{10} + 44140 \beta_{9} + 773870 \beta_{7} + 773870 \beta_{6} - 911723 \beta_{5} + 657231 \beta_{3} - 657231 \beta_{2} + 237081 \beta_{1} - 3289859\)
\(\nu^{11}\)\(=\)\(-840300 \beta_{11} + 779303 \beta_{10} + 779303 \beta_{9} + 4564606 \beta_{8} + 9192353 \beta_{7} - 10032653 \beta_{6} - 2282303 \beta_{5} - 15135840 \beta_{4} - 7033319 \beta_{3} - 7033319 \beta_{2} - 420150 \beta_{1} - 6788617\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
3.64005i
0.594165i
5.23421i
3.64005i
0.594165i
5.23421i
4.36026i
1.17841i
4.53867i
4.36026i
1.17841i
4.53867i
0.366025 + 1.36603i −3.15238 + 1.82002i −1.73205 + 1.00000i −0.866025 + 3.23205i −3.64005 3.64005i −5.59738 9.69495i −2.00000 2.00000i 2.12498 3.68058i −4.73205
23.2 0.366025 + 1.36603i −0.514562 + 0.297082i −1.73205 + 1.00000i −0.866025 + 3.23205i −0.594165 0.594165i 5.01184 + 8.68076i −2.00000 2.00000i −4.32348 + 7.48849i −4.73205
23.3 0.366025 + 1.36603i 4.53296 2.61711i −1.73205 + 1.00000i −0.866025 + 3.23205i 5.23421 + 5.23421i −0.548432 0.949912i −2.00000 2.00000i 9.19850 15.9323i −4.73205
29.1 0.366025 1.36603i −3.15238 1.82002i −1.73205 1.00000i −0.866025 3.23205i −3.64005 + 3.64005i −5.59738 + 9.69495i −2.00000 + 2.00000i 2.12498 + 3.68058i −4.73205
29.2 0.366025 1.36603i −0.514562 0.297082i −1.73205 1.00000i −0.866025 3.23205i −0.594165 + 0.594165i 5.01184 8.68076i −2.00000 + 2.00000i −4.32348 7.48849i −4.73205
29.3 0.366025 1.36603i 4.53296 + 2.61711i −1.73205 1.00000i −0.866025 3.23205i 5.23421 5.23421i −0.548432 + 0.949912i −2.00000 + 2.00000i 9.19850 + 15.9323i −4.73205
45.1 −1.36603 0.366025i −3.77609 2.18013i 1.73205 + 1.00000i 0.866025 0.232051i 4.36026 + 4.36026i −5.10188 + 8.83671i −2.00000 2.00000i 5.00591 + 8.67050i −1.26795
45.2 −1.36603 0.366025i −1.02054 0.589207i 1.73205 + 1.00000i 0.866025 0.232051i 1.17841 + 1.17841i 4.87434 8.44261i −2.00000 2.00000i −3.80567 6.59162i −1.26795
45.3 −1.36603 0.366025i 3.93060 + 2.26933i 1.73205 + 1.00000i 0.866025 0.232051i −4.53867 4.53867i −2.63849 + 4.57000i −2.00000 2.00000i 5.79976 + 10.0455i −1.26795
51.1 −1.36603 + 0.366025i −3.77609 + 2.18013i 1.73205 1.00000i 0.866025 + 0.232051i 4.36026 4.36026i −5.10188 8.83671i −2.00000 + 2.00000i 5.00591 8.67050i −1.26795
51.2 −1.36603 + 0.366025i −1.02054 + 0.589207i 1.73205 1.00000i 0.866025 + 0.232051i 1.17841 1.17841i 4.87434 + 8.44261i −2.00000 + 2.00000i −3.80567 + 6.59162i −1.26795
51.3 −1.36603 + 0.366025i 3.93060 2.26933i 1.73205 1.00000i 0.866025 + 0.232051i −4.53867 + 4.53867i −2.63849 4.57000i −2.00000 + 2.00000i 5.79976 10.0455i −1.26795
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.3.g.b 12
37.g odd 12 1 inner 74.3.g.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.g.b 12 1.a even 1 1 trivial
74.3.g.b 12 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} )^{3} \)
$3$ \( 69696 + 316800 T + 588768 T^{2} + 494400 T^{3} + 156568 T^{4} - 24720 T^{5} - 16220 T^{6} + 1260 T^{7} + 1269 T^{8} - 41 T^{10} + T^{12} \)
$5$ \( ( 9 - 18 T + 9 T^{2} + T^{4} )^{3} \)
$7$ \( 4174193664 + 4533931008 T + 4351468800 T^{2} + 816942336 T^{3} + 196855648 T^{4} + 14542016 T^{5} + 3559864 T^{6} + 221504 T^{7} + 41185 T^{8} + 1448 T^{9} + 259 T^{10} + 8 T^{11} + T^{12} \)
$11$ \( 196795155456 + 39578998272 T^{2} + 1704382528 T^{4} + 30090304 T^{6} + 242448 T^{8} + 832 T^{10} + T^{12} \)
$13$ \( 57672420423696 - 6052909861440 T - 614415814992 T^{2} + 87861972672 T^{3} - 2535877044 T^{4} - 115607736 T^{5} + 54748116 T^{6} - 4636764 T^{7} + 264397 T^{8} - 19082 T^{9} + 701 T^{10} - 16 T^{11} + T^{12} \)
$17$ \( 70631132318289 - 16105619994210 T + 8110291048695 T^{2} - 745114262652 T^{3} + 111980344870 T^{4} - 13408884710 T^{5} + 555267743 T^{6} + 4725100 T^{7} - 958070 T^{8} + 32894 T^{9} - 85 T^{10} - 40 T^{11} + T^{12} \)
$19$ \( 6088062760000 + 864182176000 T + 25794058400 T^{2} + 16727566400 T^{3} + 427987704 T^{4} - 356915008 T^{5} - 5793836 T^{6} - 392228 T^{7} + 148845 T^{8} + 13588 T^{9} + 437 T^{10} + 26 T^{11} + T^{12} \)
$23$ \( 1879443581184 + 289145166336 T + 22241935872 T^{2} - 570306048 T^{3} + 24843043264 T^{4} + 3692787904 T^{5} + 274207232 T^{6} + 7195552 T^{7} + 368356 T^{8} + 41888 T^{9} + 3200 T^{10} + 80 T^{11} + T^{12} \)
$29$ \( 597244517304513156 - 31118458079328984 T + 810688424236488 T^{2} + 261846634454412 T^{3} + 30083553056233 T^{4} + 301717146796 T^{5} + 844661384 T^{6} - 37623356 T^{7} + 10401298 T^{8} + 56108 T^{9} + 128 T^{10} - 16 T^{11} + T^{12} \)
$31$ \( 2864518849440000 + 837987850944000 T + 122572703347200 T^{2} + 5912328084480 T^{3} + 186865753536 T^{4} + 11035043136 T^{5} + 1333698048 T^{6} + 68174496 T^{7} + 1831588 T^{8} + 13408 T^{9} + 512 T^{10} + 32 T^{11} + T^{12} \)
$37$ \( 6582952005840035281 + 711670487117841652 T + 36178538375386300 T^{2} + 1025561897299844 T^{3} + 9047999509360 T^{4} - 515731623980 T^{5} - 23808226978 T^{6} - 376721420 T^{7} + 4827760 T^{8} + 399716 T^{9} + 10300 T^{10} + 148 T^{11} + T^{12} \)
$41$ \( 77494672219065201 + 5824176294317922 T - 366978726493119 T^{2} - 38546346552534 T^{3} + 2056487144830 T^{4} + 164842933098 T^{5} - 872315051 T^{6} - 209291082 T^{7} + 1329054 T^{8} + 152790 T^{9} - 863 T^{10} - 66 T^{11} + T^{12} \)
$43$ \( 8420345554088116224 - 649697087040159744 T + 25064666419986432 T^{2} - 263319683936256 T^{3} + 14475508950528 T^{4} - 993467275776 T^{5} + 37682256384 T^{6} - 379011648 T^{7} + 7473412 T^{8} - 364432 T^{9} + 11552 T^{10} - 152 T^{11} + T^{12} \)
$47$ \( ( 104298336 + 13184736 T - 254552 T^{2} - 77008 T^{3} - 1512 T^{4} + 56 T^{5} + T^{6} )^{2} \)
$53$ \( 9764779418305007616 + 706941591836553216 T + 69173333389185024 T^{2} + 505343883242496 T^{3} + 84266200451008 T^{4} - 178497741824 T^{5} + 87108441664 T^{6} - 701018264 T^{7} + 36689929 T^{8} - 239138 T^{9} + 10063 T^{10} - 74 T^{11} + T^{12} \)
$59$ \( 13512273275568301056 - 5029970094622006272 T + 249960644111188224 T^{2} + 51239896389531648 T^{3} + 2210417231619168 T^{4} + 49936523628672 T^{5} + 945330345000 T^{6} + 15767741016 T^{7} + 193479993 T^{8} + 2133036 T^{9} + 20421 T^{10} + 114 T^{11} + T^{12} \)
$61$ \( \)\(58\!\cdots\!41\)\( - 81567920021930340462 T + 5989765671058710231 T^{2} - 302509847542052556 T^{3} + 10679457283452486 T^{4} - 269815989581082 T^{5} + 5651694957375 T^{6} - 99727346628 T^{7} + 1352362666 T^{8} - 13432574 T^{9} + 96011 T^{10} - 448 T^{11} + T^{12} \)
$67$ \( 45233799848981941824 - 2330058940305009408 T - 100097276798281824 T^{2} + 7217039509034112 T^{3} + 364814178511320 T^{4} - 49235114687184 T^{5} + 2276221105836 T^{6} - 61644858516 T^{7} + 1091977477 T^{8} - 12966876 T^{9} + 100715 T^{10} - 468 T^{11} + T^{12} \)
$71$ \( \)\(12\!\cdots\!24\)\( + 12787454835952422912 T + 1322057715805655040 T^{2} + 23924052810286080 T^{3} + 1153126623529408 T^{4} - 17479771154816 T^{5} + 1019693722048 T^{6} - 10045471688 T^{7} + 193526065 T^{8} - 1210244 T^{9} + 21703 T^{10} - 116 T^{11} + T^{12} \)
$73$ \( \)\(26\!\cdots\!96\)\( + 6047844614002980864 T^{2} + 5243375966838528 T^{4} + 2126857971968 T^{6} + 409594608 T^{8} + 34104 T^{10} + T^{12} \)
$79$ \( \)\(45\!\cdots\!00\)\( - \)\(44\!\cdots\!00\)\( T + 893741891888546400 T^{2} + 576474692482543040 T^{3} + 7446092846739576 T^{4} + 79611390836352 T^{5} + 916104906572 T^{6} - 32121618732 T^{7} + 355885653 T^{8} - 4433444 T^{9} + 27573 T^{10} - 114 T^{11} + T^{12} \)
$83$ \( 48721437177679895616 - 20841493590084091392 T + 8219141632655397408 T^{2} - 299406629437043136 T^{3} + 10425242305995352 T^{4} - 104337967621456 T^{5} + 2022428293468 T^{6} - 9196620556 T^{7} + 274602733 T^{8} - 617044 T^{9} + 19807 T^{10} + 20 T^{11} + T^{12} \)
$89$ \( \)\(82\!\cdots\!25\)\( - \)\(11\!\cdots\!50\)\( T + 7638176238016234275 T^{2} - 357469998794079960 T^{3} + 11864160179380246 T^{4} - 281310627194390 T^{5} + 5734410874787 T^{6} - 98611302680 T^{7} + 1148081674 T^{8} - 8571730 T^{9} + 56567 T^{10} - 340 T^{11} + T^{12} \)
$97$ \( \)\(33\!\cdots\!56\)\( + \)\(29\!\cdots\!60\)\( T + \)\(12\!\cdots\!00\)\( T^{2} + 3504970437885855504 T^{3} + 65810342884153113 T^{4} + 875494202378220 T^{5} + 8627047593768 T^{6} + 71718690828 T^{7} + 670040194 T^{8} + 7149028 T^{9} + 63368 T^{10} + 356 T^{11} + T^{12} \)
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