Properties

Label 26.3.f.b
Level 26
Weight 3
Character orbit 26.f
Analytic conductor 0.708
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 26 = 2 \cdot 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 26.f (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.708448687337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.612074651904.1
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -\beta_{3} - \beta_{4} ) q^{2} \) \( + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{3} \) \( + 2 \beta_{5} q^{4} \) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{5} \) \( + ( -\beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{6} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{7} \) \( + ( -2 + 2 \beta_{4} - 2 \beta_{5} ) q^{8} \) \( + ( -11 + 11 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -\beta_{3} - \beta_{4} ) q^{2} \) \( + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{3} \) \( + 2 \beta_{5} q^{4} \) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{5} \) \( + ( -\beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{6} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{7} \) \( + ( -2 + 2 \beta_{4} - 2 \beta_{5} ) q^{8} \) \( + ( -11 + 11 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{9} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{10} \) \( + ( -\beta_{1} - \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} \) \( + ( 2 - 2 \beta_{3} + 2 \beta_{6} ) q^{12} \) \( + ( 3 + \beta_{2} + \beta_{3} + 6 \beta_{4} - 9 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{13} \) \( + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{14} \) \( + ( 17 - 2 \beta_{1} - \beta_{2} - 15 \beta_{3} - 15 \beta_{4} + 15 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{15} \) \( + 4 \beta_{3} q^{16} \) \( + ( -4 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + 7 \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} \) \( + ( 12 + \beta_{1} - \beta_{2} - 5 \beta_{3} + 13 \beta_{4} - 8 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{18} \) \( + ( 3 + 3 \beta_{1} + \beta_{2} + 9 \beta_{3} - 10 \beta_{4} + 9 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{19} \) \( + ( 2 - 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} ) q^{20} \) \( + ( -3 + \beta_{1} - 19 \beta_{3} + 3 \beta_{4} - 22 \beta_{5} - \beta_{7} ) q^{21} \) \( + ( -10 - \beta_{1} + 10 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{22} \) \( + ( -6 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 13 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{23} \) \( + ( -2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{24} \) \( + ( -14 + 25 \beta_{3} + 11 \beta_{4} - 11 \beta_{5} - 3 \beta_{6} ) q^{25} \) \( + ( 6 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 12 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{26} \) \( + ( 8 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 18 \beta_{4} + 21 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{27} \) \( + ( 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{28} \) \( + ( 1 + 5 \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{29} \) \( + ( -16 + 2 \beta_{1} + 4 \beta_{2} - 16 \beta_{3} - 2 \beta_{4} + 16 \beta_{5} ) q^{30} \) \( + ( 3 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} ) q^{31} \) \( + ( 4 - 4 \beta_{3} - 4 \beta_{5} ) q^{32} \) \( + ( 1 - 5 \beta_{1} - \beta_{2} + 15 \beta_{3} + 12 \beta_{4} - 15 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} ) q^{33} \) \( + ( -11 - 2 \beta_{1} + 6 \beta_{3} + 11 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} ) q^{34} \) \( + ( -18 + \beta_{1} + 18 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{35} \) \( + ( -8 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 20 \beta_{4} - 2 \beta_{5} ) q^{36} \) \( + ( -12 + 6 \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 18 \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{37} \) \( + ( 7 + \beta_{1} - \beta_{2} - 19 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{38} \) \( + ( 10 + 2 \beta_{1} - 5 \beta_{2} - 16 \beta_{3} + 29 \beta_{4} - 43 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{39} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{40} \) \( + ( 17 + 8 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} + 25 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{41} \) \( + ( -1 - \beta_{2} + 25 \beta_{3} - 19 \beta_{4} + 39 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{42} \) \( + ( -6 + \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - \beta_{4} - 16 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{43} \) \( + ( 12 + 2 \beta_{2} + 10 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{44} \) \( + ( 28 - 8 \beta_{1} - 7 \beta_{2} - \beta_{3} - 28 \beta_{4} - \beta_{5} - \beta_{6} + 8 \beta_{7} ) q^{45} \) \( + ( 18 - \beta_{1} + 2 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + 10 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{46} \) \( + ( 11 - \beta_{1} + \beta_{2} - 9 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{47} \) \( + ( 4 \beta_{1} + 4 \beta_{4} ) q^{48} \) \( + ( 43 - 4 \beta_{1} - 2 \beta_{2} - 21 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{49} \) \( + ( 25 - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 25 \beta_{5} + 3 \beta_{7} ) q^{50} \) \( + ( -10 - 3 \beta_{1} + 3 \beta_{2} + 26 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} + 6 \beta_{6} ) q^{51} \) \( + ( 12 + 2 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} ) q^{52} \) \( + ( -14 - 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 26 \beta_{4} + 24 \beta_{5} + 5 \beta_{6} - 10 \beta_{7} ) q^{53} \) \( + ( -37 - 4 \beta_{1} - 5 \beta_{2} + 13 \beta_{3} + 13 \beta_{4} - 41 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} ) q^{54} \) \( + ( -5 - 3 \beta_{2} - 43 \beta_{3} - 5 \beta_{4} + 13 \beta_{5} - 10 \beta_{6} + 5 \beta_{7} ) q^{55} \) \( + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{56} \) \( + ( -31 + 7 \beta_{1} - 15 \beta_{3} - 31 \beta_{4} + 46 \beta_{5} + 7 \beta_{7} ) q^{57} \) \( + ( -1 + 6 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{58} \) \( + ( -21 + 7 \beta_{1} + \beta_{2} + 37 \beta_{3} + 22 \beta_{4} - 37 \beta_{5} - 6 \beta_{6} + 7 \beta_{7} ) q^{59} \) \( + ( -30 + 2 \beta_{1} - 2 \beta_{2} + 34 \beta_{3} + 32 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{60} \) \( + ( 7 - 2 \beta_{1} - 7 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{61} \) \( + ( -2 - 2 \beta_{3} - 8 \beta_{4} + 6 \beta_{7} ) q^{62} \) \( + ( -30 - 10 \beta_{1} + 4 \beta_{2} + 34 \beta_{3} - 34 \beta_{4} + 20 \beta_{5} - 10 \beta_{6} - 4 \beta_{7} ) q^{63} \) \( + ( -8 \beta_{4} + 8 \beta_{5} ) q^{64} \) \( + ( -27 - 4 \beta_{1} + 9 \beta_{2} - 16 \beta_{3} - 7 \beta_{4} + 6 \beta_{5} + 8 \beta_{6} + 6 \beta_{7} ) q^{65} \) \( + ( 13 + 9 \beta_{1} + 9 \beta_{2} - \beta_{3} - 16 \beta_{4} - 7 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{66} \) \( + ( -8 - 11 \beta_{1} + \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 19 \beta_{5} + 11 \beta_{6} + \beta_{7} ) q^{67} \) \( + ( 2 + 2 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} - 14 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{68} \) \( + ( 14 + 2 \beta_{1} + 4 \beta_{2} + 14 \beta_{3} - 2 \beta_{4} - 26 \beta_{5} - 15 \beta_{6} + 15 \beta_{7} ) q^{69} \) \( + ( 24 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 22 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{70} \) \( + ( -31 + 5 \beta_{1} + 11 \beta_{2} + 15 \beta_{3} + 10 \beta_{4} + 15 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} ) q^{71} \) \( + ( 26 + 2 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} + 6 \beta_{4} + 18 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{72} \) \( + ( 36 - 2 \beta_{1} - 3 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{73} \) \( + ( -11 - 4 \beta_{1} + 11 \beta_{3} + 30 \beta_{4} - 17 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{74} \) \( + ( 1 + 28 \beta_{1} + 14 \beta_{2} + 5 \beta_{3} + 68 \beta_{4} + 14 \beta_{5} - 11 \beta_{7} ) q^{75} \) \( + ( -20 - 2 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} + 18 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{76} \) \( + ( 2 + 7 \beta_{1} - 7 \beta_{2} - 28 \beta_{4} + 35 \beta_{5} + 4 \beta_{6} ) q^{77} \) \( + ( -8 - \beta_{1} - 4 \beta_{2} + 28 \beta_{3} - 53 \beta_{4} + 36 \beta_{5} - 13 \beta_{6} + 13 \beta_{7} ) q^{78} \) \( + ( -12 - 16 \beta_{1} - 16 \beta_{2} - 24 \beta_{4} - 40 \beta_{5} ) q^{79} \) \( + ( 8 - 4 \beta_{1} + 4 \beta_{5} + 4 \beta_{6} ) q^{80} \) \( + ( 12 - 12 \beta_{2} + 3 \beta_{3} + 36 \beta_{4} - 60 \beta_{5} + 24 \beta_{6} - 12 \beta_{7} ) q^{81} \) \( + ( -23 - 8 \beta_{1} - 16 \beta_{2} - 23 \beta_{3} + 8 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{82} \) \( + ( -47 - 3 \beta_{1} + 5 \beta_{2} + 43 \beta_{3} - 52 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} - 8 \beta_{7} ) q^{83} \) \( + ( 6 - 2 \beta_{1} - 2 \beta_{2} - 46 \beta_{3} + 40 \beta_{4} - 46 \beta_{5} + 2 \beta_{7} ) q^{84} \) \( + ( 11 + 2 \beta_{1} - 11 \beta_{2} - 42 \beta_{3} - 18 \beta_{4} + 42 \beta_{5} - 13 \beta_{6} + 2 \beta_{7} ) q^{85} \) \( + ( 5 + 7 \beta_{1} + 5 \beta_{2} + 19 \beta_{3} - 10 \beta_{4} + 29 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{86} \) \( + ( 92 - 3 \beta_{1} - 92 \beta_{3} - 19 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{87} \) \( + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 22 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{88} \) \( + ( 66 - \beta_{1} - 7 \beta_{2} - 63 \beta_{3} + 63 \beta_{4} - 67 \beta_{5} - \beta_{6} + 7 \beta_{7} ) q^{89} \) \( + ( 36 - 7 \beta_{1} + 7 \beta_{2} - 63 \beta_{3} - 29 \beta_{4} + 22 \beta_{5} + 9 \beta_{6} ) q^{90} \) \( + ( 31 - \beta_{1} - 7 \beta_{2} + 17 \beta_{3} + 38 \beta_{4} + 39 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{91} \) \( + ( -26 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{92} \) \( + ( 7 + 2 \beta_{1} - 3 \beta_{2} + 53 \beta_{3} + 53 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{93} \) \( + ( 2 - 14 \beta_{3} - 8 \beta_{4} + 16 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{94} \) \( + ( 66 - \beta_{1} - 2 \beta_{2} + 66 \beta_{3} + \beta_{4} - 32 \beta_{5} + 13 \beta_{6} - 13 \beta_{7} ) q^{95} \) \( + ( -4 - 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{96} \) \( + ( -22 - 3 \beta_{1} + 7 \beta_{2} + 18 \beta_{3} - 6 \beta_{4} + 18 \beta_{5} - 10 \beta_{6} + 3 \beta_{7} ) q^{97} \) \( + ( -19 + 3 \beta_{1} + 4 \beta_{2} - 25 \beta_{3} - 45 \beta_{4} + 25 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{98} \) \( + ( 10 + 2 \beta_{1} + 4 \beta_{2} + 38 \beta_{3} - 14 \beta_{4} + 52 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 42q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 42q^{9} \) \(\mathstrut -\mathstrut 18q^{10} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 36q^{13} \) \(\mathstrut +\mathstrut 20q^{14} \) \(\mathstrut +\mathstrut 66q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 42q^{17} \) \(\mathstrut +\mathstrut 84q^{18} \) \(\mathstrut +\mathstrut 46q^{19} \) \(\mathstrut +\mathstrut 24q^{20} \) \(\mathstrut -\mathstrut 102q^{21} \) \(\mathstrut -\mathstrut 42q^{22} \) \(\mathstrut -\mathstrut 36q^{23} \) \(\mathstrut -\mathstrut 12q^{24} \) \(\mathstrut +\mathstrut 40q^{26} \) \(\mathstrut +\mathstrut 72q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 192q^{30} \) \(\mathstrut +\mathstrut 32q^{31} \) \(\mathstrut +\mathstrut 16q^{32} \) \(\mathstrut +\mathstrut 42q^{33} \) \(\mathstrut -\mathstrut 60q^{34} \) \(\mathstrut -\mathstrut 78q^{35} \) \(\mathstrut -\mathstrut 48q^{36} \) \(\mathstrut -\mathstrut 106q^{37} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut -\mathstrut 24q^{40} \) \(\mathstrut +\mathstrut 132q^{41} \) \(\mathstrut +\mathstrut 102q^{42} \) \(\mathstrut -\mathstrut 108q^{43} \) \(\mathstrut +\mathstrut 84q^{44} \) \(\mathstrut +\mathstrut 240q^{45} \) \(\mathstrut +\mathstrut 90q^{46} \) \(\mathstrut +\mathstrut 60q^{47} \) \(\mathstrut +\mathstrut 258q^{49} \) \(\mathstrut +\mathstrut 194q^{50} \) \(\mathstrut +\mathstrut 32q^{52} \) \(\mathstrut -\mathstrut 132q^{53} \) \(\mathstrut -\mathstrut 270q^{54} \) \(\mathstrut -\mathstrut 162q^{55} \) \(\mathstrut -\mathstrut 12q^{56} \) \(\mathstrut -\mathstrut 294q^{57} \) \(\mathstrut -\mathstrut 24q^{58} \) \(\mathstrut +\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 120q^{60} \) \(\mathstrut +\mathstrut 36q^{61} \) \(\mathstrut -\mathstrut 12q^{62} \) \(\mathstrut -\mathstrut 72q^{63} \) \(\mathstrut -\mathstrut 300q^{65} \) \(\mathstrut +\mathstrut 108q^{66} \) \(\mathstrut -\mathstrut 74q^{67} \) \(\mathstrut +\mathstrut 60q^{68} \) \(\mathstrut +\mathstrut 258q^{69} \) \(\mathstrut +\mathstrut 156q^{70} \) \(\mathstrut -\mathstrut 174q^{71} \) \(\mathstrut +\mathstrut 132q^{72} \) \(\mathstrut +\mathstrut 166q^{73} \) \(\mathstrut -\mathstrut 32q^{74} \) \(\mathstrut +\mathstrut 6q^{75} \) \(\mathstrut -\mathstrut 92q^{76} \) \(\mathstrut +\mathstrut 126q^{78} \) \(\mathstrut -\mathstrut 96q^{79} \) \(\mathstrut +\mathstrut 48q^{80} \) \(\mathstrut -\mathstrut 12q^{81} \) \(\mathstrut -\mathstrut 252q^{82} \) \(\mathstrut -\mathstrut 240q^{83} \) \(\mathstrut -\mathstrut 132q^{84} \) \(\mathstrut -\mathstrut 24q^{85} \) \(\mathstrut +\mathstrut 132q^{86} \) \(\mathstrut +\mathstrut 360q^{87} \) \(\mathstrut -\mathstrut 12q^{88} \) \(\mathstrut +\mathstrut 294q^{89} \) \(\mathstrut +\mathstrut 298q^{91} \) \(\mathstrut -\mathstrut 216q^{92} \) \(\mathstrut +\mathstrut 270q^{93} \) \(\mathstrut -\mathstrut 60q^{94} \) \(\mathstrut +\mathstrut 714q^{95} \) \(\mathstrut -\mathstrut 58q^{97} \) \(\mathstrut -\mathstrut 250q^{98} \) \(\mathstrut +\mathstrut 252q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(74\) \(x^{6}\mathstrut +\mathstrut \) \(2067\) \(x^{4}\mathstrut -\mathstrut \) \(25778\) \(x^{2}\mathstrut +\mathstrut \) \(121801\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{4} - 37 \nu^{2} + 4 \nu + 349 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 37 \nu^{2} + 4 \nu - 349 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 74 \nu^{5} + 1718 \nu^{3} - 12865 \nu + 1396 \)\()/2792\)
\(\beta_{4}\)\(=\)\((\)\( 40 \nu^{7} - 349 \nu^{6} - 2262 \nu^{5} + 19195 \nu^{4} + 44290 \nu^{3} - 345859 \nu^{2} - 296126 \nu + 2024898 \)\()/86552\)
\(\beta_{5}\)\(=\)\((\)\( -40 \nu^{7} - 349 \nu^{6} + 2262 \nu^{5} + 19195 \nu^{4} - 44290 \nu^{3} - 345859 \nu^{2} + 296126 \nu + 2024898 \)\()/86552\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 55 \nu^{4} + 1053 \nu^{2} - 6980 \)\()/62\)
\(\beta_{7}\)\(=\)\((\)\( -318 \nu^{7} + 698 \nu^{6} + 16901 \nu^{5} - 38390 \nu^{4} - 292601 \nu^{3} + 734994 \nu^{2} + 1626083 \nu - 4828764 \)\()/86552\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\)
\(\nu^{3}\)\(=\)\(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(37\) \(\beta_{6}\mathstrut +\mathstrut \) \(74\) \(\beta_{5}\mathstrut +\mathstrut \) \(74\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(354\)
\(\nu^{5}\)\(=\)\(140\) \(\beta_{7}\mathstrut -\mathstrut \) \(70\) \(\beta_{6}\mathstrut -\mathstrut \) \(727\) \(\beta_{5}\mathstrut +\mathstrut \) \(727\) \(\beta_{4}\mathstrut -\mathstrut \) \(440\) \(\beta_{3}\mathstrut +\mathstrut \) \(317\) \(\beta_{2}\mathstrut +\mathstrut \) \(317\) \(\beta_{1}\mathstrut +\mathstrut \) \(150\)
\(\nu^{6}\)\(=\)\(1044\) \(\beta_{6}\mathstrut +\mathstrut \) \(1964\) \(\beta_{5}\mathstrut +\mathstrut \) \(1964\) \(\beta_{4}\mathstrut -\mathstrut \) \(220\) \(\beta_{2}\mathstrut +\mathstrut \) \(220\) \(\beta_{1}\mathstrut +\mathstrut \) \(6443\)
\(\nu^{7}\)\(=\)\(3488\) \(\beta_{7}\mathstrut -\mathstrut \) \(1744\) \(\beta_{6}\mathstrut -\mathstrut \) \(21156\) \(\beta_{5}\mathstrut +\mathstrut \) \(21156\) \(\beta_{4}\mathstrut -\mathstrut \) \(16024\) \(\beta_{3}\mathstrut +\mathstrut \) \(5399\) \(\beta_{2}\mathstrut +\mathstrut \) \(5399\) \(\beta_{1}\mathstrut +\mathstrut \) \(6268\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(\beta_{5}\)

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
4.71318 + 0.500000i
−4.71318 + 0.500000i
3.90972 + 0.500000i
−3.90972 + 0.500000i
4.71318 0.500000i
−4.71318 0.500000i
3.90972 0.500000i
−3.90972 0.500000i
−1.36603 + 0.366025i −2.78960 + 4.83174i 1.73205 1.00000i 0.323893 + 0.323893i 2.04213 7.62134i 7.67890 + 2.05755i −2.00000 + 2.00000i −11.0638 19.1630i −0.560999 0.323893i
7.2 −1.36603 + 0.366025i 1.92358 3.33174i 1.73205 1.00000i 3.77418 + 3.77418i −1.40816 + 5.25532i −9.91095 2.65563i −2.00000 + 2.00000i −2.90031 5.02349i −6.53708 3.77418i
11.1 0.366025 1.36603i −1.52185 2.63592i −1.73205 1.00000i 4.79174 + 4.79174i −4.15776 + 1.11407i 1.13983 + 4.25390i −2.00000 + 2.00000i −0.132034 + 0.228689i 8.29953 4.79174i
11.2 0.366025 1.36603i 2.38787 + 4.13592i −1.73205 1.00000i −5.88981 5.88981i 6.52379 1.74804i 0.0922225 + 0.344179i −2.00000 + 2.00000i −6.90386 + 11.9578i −10.2015 + 5.88981i
15.1 −1.36603 0.366025i −2.78960 4.83174i 1.73205 + 1.00000i 0.323893 0.323893i 2.04213 + 7.62134i 7.67890 2.05755i −2.00000 2.00000i −11.0638 + 19.1630i −0.560999 + 0.323893i
15.2 −1.36603 0.366025i 1.92358 + 3.33174i 1.73205 + 1.00000i 3.77418 3.77418i −1.40816 5.25532i −9.91095 + 2.65563i −2.00000 2.00000i −2.90031 + 5.02349i −6.53708 + 3.77418i
19.1 0.366025 + 1.36603i −1.52185 + 2.63592i −1.73205 + 1.00000i 4.79174 4.79174i −4.15776 1.11407i 1.13983 4.25390i −2.00000 2.00000i −0.132034 0.228689i 8.29953 + 4.79174i
19.2 0.366025 + 1.36603i 2.38787 4.13592i −1.73205 + 1.00000i −5.88981 + 5.88981i 6.52379 + 1.74804i 0.0922225 0.344179i −2.00000 2.00000i −6.90386 11.9578i −10.2015 5.88981i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.f Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} \) \(\mathstrut +\mathstrut 39 T_{3}^{6} \) \(\mathstrut -\mathstrut 24 T_{3}^{5} \) \(\mathstrut +\mathstrut 1209 T_{3}^{4} \) \(\mathstrut -\mathstrut 468 T_{3}^{3} \) \(\mathstrut +\mathstrut 12312 T_{3}^{2} \) \(\mathstrut +\mathstrut 3744 T_{3} \) \(\mathstrut +\mathstrut 97344 \) acting on \(S_{3}^{\mathrm{new}}(26, [\chi])\).