# 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

# Related objects

## Newspace parameters

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

## 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 Defining polynomial: $$x^{8} - 74 x^{6} + 2067 x^{4} - 25778 x^{2} + 121801$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} + 6q^{5} + 6q^{6} - 2q^{7} - 16q^{8} - 42q^{9} + O(q^{10})$$ $$8q - 4q^{2} + 6q^{5} + 6q^{6} - 2q^{7} - 16q^{8} - 42q^{9} - 18q^{10} - 18q^{11} + 36q^{13} + 20q^{14} + 66q^{15} + 16q^{16} - 42q^{17} + 84q^{18} + 46q^{19} + 24q^{20} - 102q^{21} - 42q^{22} - 36q^{23} - 12q^{24} + 40q^{26} + 72q^{27} - 4q^{28} - 6q^{29} - 192q^{30} + 32q^{31} + 16q^{32} + 42q^{33} - 60q^{34} - 78q^{35} - 48q^{36} - 106q^{37} + 12q^{39} - 24q^{40} + 132q^{41} + 102q^{42} - 108q^{43} + 84q^{44} + 240q^{45} + 90q^{46} + 60q^{47} + 258q^{49} + 194q^{50} + 32q^{52} - 132q^{53} - 270q^{54} - 162q^{55} - 12q^{56} - 294q^{57} - 24q^{58} + 18q^{59} - 120q^{60} + 36q^{61} - 12q^{62} - 72q^{63} - 300q^{65} + 108q^{66} - 74q^{67} + 60q^{68} + 258q^{69} + 156q^{70} - 174q^{71} + 132q^{72} + 166q^{73} - 32q^{74} + 6q^{75} - 92q^{76} + 126q^{78} - 96q^{79} + 48q^{80} - 12q^{81} - 252q^{82} - 240q^{83} - 132q^{84} - 24q^{85} + 132q^{86} + 360q^{87} - 12q^{88} + 294q^{89} + 298q^{91} - 216q^{92} + 270q^{93} - 60q^{94} + 714q^{95} - 58q^{97} - 250q^{98} + 252q^{99} + O(q^{100})$$

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.3.f.b 8
3.b odd 2 1 234.3.bb.f 8
4.b odd 2 1 208.3.bd.f 8
13.b even 2 1 338.3.f.i 8
13.c even 3 1 338.3.d.g 8
13.c even 3 1 338.3.f.h 8
13.d odd 4 1 338.3.f.h 8
13.d odd 4 1 338.3.f.j 8
13.e even 6 1 338.3.d.f 8
13.e even 6 1 338.3.f.j 8
13.f odd 12 1 inner 26.3.f.b 8
13.f odd 12 1 338.3.d.f 8
13.f odd 12 1 338.3.d.g 8
13.f odd 12 1 338.3.f.i 8
39.k even 12 1 234.3.bb.f 8
52.l even 12 1 208.3.bd.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.b 8 1.a even 1 1 trivial
26.3.f.b 8 13.f odd 12 1 inner
208.3.bd.f 8 4.b odd 2 1
208.3.bd.f 8 52.l even 12 1
234.3.bb.f 8 3.b odd 2 1
234.3.bb.f 8 39.k even 12 1
338.3.d.f 8 13.e even 6 1
338.3.d.f 8 13.f odd 12 1
338.3.d.g 8 13.c even 3 1
338.3.d.g 8 13.f odd 12 1
338.3.f.h 8 13.c even 3 1
338.3.f.h 8 13.d odd 4 1
338.3.f.i 8 13.b even 2 1
338.3.f.i 8 13.f odd 12 1
338.3.f.j 8 13.d odd 4 1
338.3.f.j 8 13.e even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2}$$
$3$ $$1 + 3 T^{2} - 24 T^{3} - 87 T^{4} + 72 T^{5} - 54 T^{6} + 1800 T^{7} + 630 T^{8} + 16200 T^{9} - 4374 T^{10} + 52488 T^{11} - 570807 T^{12} - 1417176 T^{13} + 1594323 T^{14} + 43046721 T^{16}$$
$5$ $$1 - 6 T + 18 T^{2} - 240 T^{3} + 445 T^{4} + 2988 T^{5} + 2862 T^{6} + 36666 T^{7} - 716556 T^{8} + 916650 T^{9} + 1788750 T^{10} + 46687500 T^{11} + 173828125 T^{12} - 2343750000 T^{13} + 4394531250 T^{14} - 36621093750 T^{15} + 152587890625 T^{16}$$
$7$ $$1 + 2 T - 127 T^{2} - 434 T^{3} + 8255 T^{4} + 22156 T^{5} - 385248 T^{6} - 453708 T^{7} + 19028482 T^{8} - 22231692 T^{9} - 924980448 T^{10} + 2606631244 T^{11} + 47588432255 T^{12} - 122594258066 T^{13} - 1757843474527 T^{14} + 1356446145698 T^{15} + 33232930569601 T^{16}$$
$11$ $$1 + 18 T + 105 T^{2} - 3114 T^{3} - 59393 T^{4} - 565908 T^{5} + 897792 T^{6} + 64810956 T^{7} + 1103342370 T^{8} + 7842125676 T^{9} + 13144572672 T^{10} - 1002540542388 T^{11} - 12731417019233 T^{12} - 80769140207514 T^{13} + 329534979555705 T^{14} + 6835497004498338 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 - 36 T + 589 T^{2} - 7824 T^{3} + 106236 T^{4} - 1322256 T^{5} + 16822429 T^{6} - 173765124 T^{7} + 815730721 T^{8}$$
$17$ $$1 + 42 T + 1726 T^{2} + 47796 T^{3} + 1303465 T^{4} + 30227796 T^{5} + 646060522 T^{6} + 12441458526 T^{7} + 218725843324 T^{8} + 3595581514014 T^{9} + 53959620857962 T^{10} + 729625511667924 T^{11} + 9092655672833065 T^{12} + 96356444465860404 T^{13} + 1005605981458567486 T^{14} + 7071868715494839018 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 - 46 T + 977 T^{2} - 19730 T^{3} + 483311 T^{4} - 8783612 T^{5} + 118611072 T^{6} - 1680839364 T^{7} + 28608116338 T^{8} - 606783010404 T^{9} + 15457513514112 T^{10} - 413232764902172 T^{11} + 8208342836908751 T^{12} - 120965937266413730 T^{13} + 2162408675927639297 T^{14} - 36754307546012669566 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$1 + 36 T + 2131 T^{2} + 61164 T^{3} + 2152429 T^{4} + 44226000 T^{5} + 1303450510 T^{6} + 22452294672 T^{7} + 648751140166 T^{8} + 11877263881488 T^{9} + 364758894168910 T^{10} + 6547035226914000 T^{11} + 168558835737397549 T^{12} + 2533811131871627436 T^{13} + 46700064664635304051 T^{14} +$$$$41\!\cdots\!24$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$1 + 6 T - 2398 T^{2} - 14772 T^{3} + 2997253 T^{4} + 14035500 T^{5} - 3325399546 T^{6} - 4535559294 T^{7} + 3230740374436 T^{8} - 3814405366254 T^{9} - 2351991916294426 T^{10} + 8348642721895500 T^{11} + 1499365061986596133 T^{12} - 6214687250310569172 T^{13} -$$$$84\!\cdots\!18$$$$T^{14} +$$$$17\!\cdots\!86$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16}$$
$31$ $$1 - 32 T + 512 T^{2} - 24592 T^{3} + 1881968 T^{4} - 48317872 T^{5} + 884987520 T^{6} - 50977297056 T^{7} + 2940963189598 T^{8} - 48989182470816 T^{9} + 817304559457920 T^{10} - 42882289258086832 T^{11} + 1605113639950763888 T^{12} - 20156298833431858192 T^{13} +$$$$40\!\cdots\!32$$$$T^{14} -$$$$24\!\cdots\!72$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 106 T + 8342 T^{2} + 393068 T^{3} + 15188585 T^{4} + 276977564 T^{5} - 1726711374 T^{6} - 651456022674 T^{7} - 28349233687076 T^{8} - 891843295040706 T^{9} - 3236135115407214 T^{10} + 710648650655287676 T^{11} + 53349592746632691785 T^{12} +$$$$18\!\cdots\!32$$$$T^{13} +$$$$54\!\cdots\!02$$$$T^{14} +$$$$95\!\cdots\!34$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 - 132 T + 10686 T^{2} - 553992 T^{3} + 19988317 T^{4} - 213687144 T^{5} - 26569998546 T^{6} + 2518966024308 T^{7} - 124070043071700 T^{8} + 4234381886861748 T^{9} - 75080465661343506 T^{10} - 1015036208961577704 T^{11} +$$$$15\!\cdots\!57$$$$T^{12} -$$$$74\!\cdots\!92$$$$T^{13} +$$$$24\!\cdots\!66$$$$T^{14} -$$$$50\!\cdots\!52$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$1 + 108 T + 10879 T^{2} + 755028 T^{3} + 48271669 T^{4} + 2548436904 T^{5} + 130030600102 T^{6} + 5877290089008 T^{7} + 263654368009582 T^{8} + 10867109374575792 T^{9} + 444548745659317702 T^{10} + 16109594877653560296 T^{11} +$$$$56\!\cdots\!69$$$$T^{12} +$$$$16\!\cdots\!72$$$$T^{13} +$$$$43\!\cdots\!79$$$$T^{14} +$$$$79\!\cdots\!92$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 60 T + 1800 T^{2} - 154932 T^{3} + 21951088 T^{4} - 867532140 T^{5} + 24541932312 T^{6} - 2003115348324 T^{7} + 163148147612766 T^{8} - 4424881804447716 T^{9} + 119756800806152472 T^{10} - 9351315741888174060 T^{11} +$$$$52\!\cdots\!68$$$$T^{12} -$$$$81\!\cdots\!68$$$$T^{13} +$$$$20\!\cdots\!00$$$$T^{14} -$$$$15\!\cdots\!40$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 + 66 T + 5917 T^{2} + 184818 T^{3} + 12226368 T^{4} + 519153762 T^{5} + 46687976077 T^{6} + 1462847834514 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 - 18 T + 5565 T^{2} - 318294 T^{3} + 23254759 T^{4} - 1693659348 T^{5} + 98133474288 T^{6} - 8885846915100 T^{7} + 327983542866810 T^{8} - 30931633111463100 T^{9} + 1189118734131913968 T^{10} - 71439455104708126068 T^{11} +$$$$34\!\cdots\!39$$$$T^{12} -$$$$16\!\cdots\!94$$$$T^{13} +$$$$99\!\cdots\!65$$$$T^{14} -$$$$11\!\cdots\!98$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 36 T - 11878 T^{2} + 285336 T^{3} + 88972381 T^{4} - 1311266160 T^{5} - 475974647662 T^{6} + 1883345164380 T^{7} + 2030181383859340 T^{8} + 7007927356657980 T^{9} - 6590269291559073742 T^{10} - 67556923450110923760 T^{11} +$$$$17\!\cdots\!61$$$$T^{12} +$$$$20\!\cdots\!36$$$$T^{13} -$$$$31\!\cdots\!38$$$$T^{14} -$$$$35\!\cdots\!76$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$1 + 74 T + 15917 T^{2} + 695782 T^{3} + 97615703 T^{4} + 493359916 T^{5} + 227553513504 T^{6} - 21342931471668 T^{7} + 178591314167674 T^{8} - 95808419376317652 T^{9} + 4585458384594237984 T^{10} + 44628539828393737804 T^{11} +$$$$39\!\cdots\!23$$$$T^{12} +$$$$12\!\cdots\!18$$$$T^{13} +$$$$13\!\cdots\!37$$$$T^{14} +$$$$27\!\cdots\!46$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$1 + 174 T + 14793 T^{2} + 561450 T^{3} - 2330369 T^{4} - 584834196 T^{5} + 83898109968 T^{6} + 14951669162460 T^{7} + 1250020614182514 T^{8} + 75371364247960860 T^{9} + 2131992007009736208 T^{10} - 74917426554309762516 T^{11} -$$$$15\!\cdots\!09$$$$T^{12} +$$$$18\!\cdots\!50$$$$T^{13} +$$$$24\!\cdots\!13$$$$T^{14} +$$$$14\!\cdots\!94$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$1 - 166 T + 13778 T^{2} - 1402664 T^{3} + 172017437 T^{4} - 14047512068 T^{5} + 945563904750 T^{6} - 83152659919950 T^{7} + 7211040518499604 T^{8} - 443120524713413550 T^{9} + 26852351647991544750 T^{10} -$$$$21\!\cdots\!52$$$$T^{11} +$$$$13\!\cdots\!97$$$$T^{12} -$$$$60\!\cdots\!36$$$$T^{13} +$$$$31\!\cdots\!38$$$$T^{14} -$$$$20\!\cdots\!94$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$( 1 + 48 T + 10084 T^{2} + 724368 T^{3} + 50280774 T^{4} + 4520780688 T^{5} + 392772616804 T^{6} + 11668197865008 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 + 240 T + 28800 T^{2} + 2896512 T^{3} + 298194640 T^{4} + 28074339648 T^{5} + 2344726766592 T^{6} + 177350340390480 T^{7} + 13560551099314782 T^{8} + 1221766494950016720 T^{9} +$$$$11\!\cdots\!32$$$$T^{10} +$$$$91\!\cdots\!12$$$$T^{11} +$$$$67\!\cdots\!40$$$$T^{12} +$$$$44\!\cdots\!88$$$$T^{13} +$$$$30\!\cdots\!00$$$$T^{14} +$$$$17\!\cdots\!60$$$$T^{15} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 294 T + 63735 T^{2} - 10176306 T^{3} + 1386303445 T^{4} - 163006886868 T^{5} + 17292939386730 T^{6} - 1689166620371832 T^{7} + 154185854226666234 T^{8} - 13379888799965281272 T^{9} +$$$$10\!\cdots\!30$$$$T^{10} -$$$$81\!\cdots\!48$$$$T^{11} +$$$$54\!\cdots\!45$$$$T^{12} -$$$$31\!\cdots\!06$$$$T^{13} +$$$$15\!\cdots\!35$$$$T^{14} -$$$$57\!\cdots\!54$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 + 58 T + 14363 T^{2} + 998 T^{3} - 4472611 T^{4} - 1301881588 T^{5} - 229071912798 T^{6} + 178692170625312 T^{7} + 7600464823934194 T^{8} + 1681314633413560608 T^{9} - 20279571737301638238 T^{10} -$$$$10\!\cdots\!52$$$$T^{11} -$$$$35\!\cdots\!71$$$$T^{12} +$$$$73\!\cdots\!02$$$$T^{13} +$$$$99\!\cdots\!83$$$$T^{14} +$$$$37\!\cdots\!02$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$