# Properties

 Label 361.2.c.h Level $361$ Weight $2$ Character orbit 361.c Analytic conductor $2.883$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.88259951297$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_1 - 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{3} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{4} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{4} + \beta_{3} + 2) q^{8} + (\beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10})$$ q + (b5 + b1 - 1) * q^2 + (b3 - b2 + b1 - 1) * q^3 + (-2*b5 + 2*b4 + 2*b3 - b2 - b1) * q^4 + (b3 - b2 - b1 + 1) * q^5 + (b2 - 2*b1) * q^6 + (-b4 - b3) * q^7 + (-2*b4 + b3 + 2) * q^8 + (b5 - b4 - b3 + 3*b2) * q^9 $$q + (\beta_{5} + \beta_1 - 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{3} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{4} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{4} + \beta_{3} + 2) q^{8} + (\beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{9} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{10} + (\beta_{4} + 2 \beta_{3}) q^{11} + ( - \beta_{4} + 1) q^{12} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{14} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{15} + (3 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{16} + (\beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{17} + (4 \beta_{4} + 1) q^{18} + (3 \beta_{4} + 2 \beta_{3} - 1) q^{20} + (\beta_1 - 1) q^{21} + ( - 3 \beta_1 + 3) q^{22} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{23} + (3 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{24} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{25} + (\beta_{4} - 5) q^{26} + (3 \beta_{4} + 2) q^{27} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{28} + (\beta_{2} - 5 \beta_1) q^{29} + (\beta_{4} - \beta_{3}) q^{30} + ( - \beta_{4} - 3 \beta_{3} + 3) q^{31} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{32} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{33} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{34} + ( - 2 \beta_{5} + \beta_1 - 1) q^{35} + ( - 5 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{36} + ( - 2 \beta_{4} + \beta_{3}) q^{37} + ( - \beta_{4} + \beta_{3} - 4) q^{39} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{40} + (3 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{41} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{42} + ( - 5 \beta_{5} + 3 \beta_{3} - 3 \beta_{2}) q^{43} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{44} + (\beta_{4} + \beta_{3} + 5) q^{45} + (4 \beta_{4} + 2 \beta_{3} - 6) q^{46} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{47} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{48} + (\beta_{3} - 5) q^{49} + (2 \beta_{4} + 2 \beta_{3} - 5) q^{50} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{51} + ( - 3 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 6) q^{52} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{53} + ( - 4 \beta_{5} + 6 \beta_{3} - 6 \beta_{2} - \beta_1 + 1) q^{54} + (3 \beta_{5} - 3 \beta_1 + 3) q^{55} + ( - 5 \beta_{4} - 3 \beta_{3} + 1) q^{56} + ( - 4 \beta_{4} - 5 \beta_{3} + 6) q^{58} + ( - 2 \beta_{5} + 7 \beta_1 - 7) q^{59} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{60} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{61} + (2 \beta_{5} + \beta_{3} - \beta_{2} + 7 \beta_1 - 7) q^{62} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1) q^{63} + (3 \beta_{4} + 4) q^{64} + ( - 5 \beta_{4} - 5 \beta_{3} - 4) q^{65} + ( - 3 \beta_{2} + 3 \beta_1) q^{66} + ( - 6 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{67} + ( - 7 \beta_{4} - 3 \beta_{3} + 5) q^{68} + (2 \beta_{3} - 4) q^{69} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{70} + ( - 2 \beta_{5} + 10 \beta_1 - 10) q^{71} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 13 \beta_1) q^{72} + 4 \beta_{5} q^{73} + (5 \beta_{5} - 5 \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{74} + ( - \beta_{4} + 3 \beta_{3} - 5) q^{75} + ( - \beta_{4} - 2 \beta_{3} - 3) q^{77} + ( - \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 4) q^{78} + ( - 7 \beta_{5} + 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{79} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} + 4 \beta_1) q^{80} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{81} + ( - 11 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} - 7 \beta_{2} - 6 \beta_1) q^{82} + ( - 6 \beta_{4} + 3 \beta_{3}) q^{83} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{84} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{85} + (8 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} + 8 \beta_{2} + 7 \beta_1) q^{86} + (\beta_{4} - 6 \beta_{3} + 7) q^{87} + (6 \beta_{4} + 3 \beta_{3} + 3) q^{88} + (\beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 5 \beta_1) q^{89} + (4 \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{90} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 5 \beta_1) q^{91} + ( - 8 \beta_{5} + 6 \beta_{3} - 6 \beta_{2} - 8 \beta_1 + 8) q^{92} + ( - 2 \beta_{5} + 7 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 8) q^{93} + ( - 2 \beta_{4} + 2 \beta_{3} - 3) q^{94} + 3 q^{96} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 5) q^{97} + ( - 4 \beta_{5} - \beta_{3} + \beta_{2} - 6 \beta_1 + 6) q^{98} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100})$$ q + (b5 + b1 - 1) * q^2 + (b3 - b2 + b1 - 1) * q^3 + (-2*b5 + 2*b4 + 2*b3 - b2 - b1) * q^4 + (b3 - b2 - b1 + 1) * q^5 + (b2 - 2*b1) * q^6 + (-b4 - b3) * q^7 + (-2*b4 + b3 + 2) * q^8 + (b5 - b4 - b3 + 3*b2) * q^9 + (2*b5 - 2*b4 - 2*b3 + b2) * q^10 + (b4 + 2*b3) * q^11 + (-b4 + 1) * q^12 + (2*b5 - 2*b4 - 2*b3 - b2) * q^13 + (b5 - b3 + b2 + 2*b1 - 2) * q^14 + (b5 - b4 - b3 + b2 - b1) * q^15 + (3*b5 - 3*b3 + 3*b2 + b1 - 1) * q^16 + (b5 - b3 + b2 + 2*b1 - 2) * q^17 + (4*b4 + 1) * q^18 + (3*b4 + 2*b3 - 1) * q^20 + (b1 - 1) * q^21 + (-3*b1 + 3) * q^22 + (2*b5 - 2*b4 - 2*b3 + 2*b1) * q^23 + (3*b5 - 4*b3 + 4*b2 - 2*b1 + 2) * q^24 + (b5 - b4 - b3 - b2 + 2*b1) * q^25 + (b4 - 5) * q^26 + (3*b4 + 2) * q^27 + (-2*b5 + 2*b4 + 2*b3 - 2*b2 - 3*b1) * q^28 + (b2 - 5*b1) * q^29 + (b4 - b3) * q^30 + (-b4 - 3*b3 + 3) * q^31 + (-3*b5 + 3*b4 + 3*b3) * q^32 + (b5 - 2*b3 + 2*b2 - 3*b1 + 3) * q^33 + (-4*b5 + 4*b4 + 4*b3 - 2*b2 - 3*b1) * q^34 + (-2*b5 + b1 - 1) * q^35 + (-5*b5 + 2*b3 - 2*b2 - 3*b1 + 3) * q^36 + (-2*b4 + b3) * q^37 + (-b4 + b3 - 4) * q^39 + (-b5 + 2*b3 - 2*b2 - 6*b1 + 6) * q^40 + (3*b5 - 4*b3 + 4*b2 + 4*b1 - 4) * q^41 + (-b5 + b4 + b3 - b1) * q^42 + (-5*b5 + 3*b3 - 3*b2) * q^43 + (b5 - b4 - b3 + 2*b2 + 3*b1) * q^44 + (b4 + b3 + 5) * q^45 + (4*b4 + 2*b3 - 6) * q^46 + (-b5 + b4 + b3 - 3*b2 + 2*b1) * q^47 + (-3*b5 + 3*b4 + 3*b3 - 5*b2 + 2*b1) * q^48 + (b3 - 5) * q^49 + (2*b4 + 2*b3 - 5) * q^50 + (-b5 + b4 + b3 - b1) * q^51 + (-3*b5 + 4*b3 - 4*b2 - 6*b1 + 6) * q^52 + (3*b5 - 3*b4 - 3*b3 + b2 - 2*b1) * q^53 + (-4*b5 + 6*b3 - 6*b2 - b1 + 1) * q^54 + (3*b5 - 3*b1 + 3) * q^55 + (-5*b4 - 3*b3 + 1) * q^56 + (-4*b4 - 5*b3 + 6) * q^58 + (-2*b5 + 7*b1 - 7) * q^59 + (-b5 + b3 - b2 - 2*b1 + 2) * q^60 + (4*b5 - 4*b4 - 4*b3 + 4*b2 - 3*b1) * q^61 + (2*b5 + b3 - b2 + 7*b1 - 7) * q^62 + (3*b5 - 3*b4 - 3*b3 + b2 - b1) * q^63 + (3*b4 + 4) * q^64 + (-5*b4 - 5*b3 - 4) * q^65 + (-3*b2 + 3*b1) * q^66 + (-6*b5 + 6*b4 + 6*b3 - 2*b2 + 6*b1) * q^67 + (-7*b4 - 3*b3 + 5) * q^68 + (2*b3 - 4) * q^69 + (b5 - b4 - b3 + 2*b2 + 3*b1) * q^70 + (-2*b5 + 10*b1 - 10) * q^71 + (2*b5 - 2*b4 - 2*b3 - b2 + 13*b1) * q^72 + 4*b5 * q^73 + (5*b5 - 5*b3 + 5*b2 + b1 - 1) * q^74 + (-b4 + 3*b3 - 5) * q^75 + (-b4 - 2*b3 - 3) * q^77 + (-b5 - 3*b3 + 3*b2 - 4*b1 + 4) * q^78 + (-7*b5 + 6*b3 - 6*b2 + 3*b1 - 3) * q^79 + (3*b5 - 3*b4 - 3*b3 + b2 + 4*b1) * q^80 + (-b3 + b2 + 5*b1 - 5) * q^81 + (-11*b5 + 11*b4 + 11*b3 - 7*b2 - 6*b1) * q^82 + (-6*b4 + 3*b3) * q^83 + (-2*b4 - b3 + 1) * q^84 + (b5 - b4 - b3 + 2*b2 + 3*b1) * q^85 + (8*b5 - 8*b4 - 8*b3 + 8*b2 + 7*b1) * q^86 + (b4 - 6*b3 + 7) * q^87 + (6*b4 + 3*b3 + 3) * q^88 + (b5 - b4 - b3 + 3*b2 - 5*b1) * q^89 + (4*b5 + b3 - b2 + 3*b1 - 3) * q^90 + (-b5 + b4 + b3 + 2*b2 + 5*b1) * q^91 + (-8*b5 + 6*b3 - 6*b2 - 8*b1 + 8) * q^92 + (-2*b5 + 7*b3 - 7*b2 + 8*b1 - 8) * q^93 + (-2*b4 + 2*b3 - 3) * q^94 + 3 * q^96 + (-2*b5 - 2*b3 + 2*b2 - 5*b1 + 5) * q^97 + (-4*b5 - b3 + b2 - 6*b1 + 6) * q^98 + (-5*b5 + 5*b4 + 5*b3 - 4*b2 + 6*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10})$$ 6 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 + 3 * q^5 - 6 * q^6 + 12 * q^8 $$6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} + 3 q^{5} - 6 q^{6} + 12 q^{8} + 6 q^{12} - 6 q^{14} - 3 q^{15} - 3 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{20} - 3 q^{21} + 9 q^{22} + 6 q^{23} + 6 q^{24} + 6 q^{25} - 30 q^{26} + 12 q^{27} - 9 q^{28} - 15 q^{29} + 18 q^{31} + 9 q^{33} - 9 q^{34} - 3 q^{35} + 9 q^{36} - 24 q^{39} + 18 q^{40} - 12 q^{41} - 3 q^{42} + 9 q^{44} + 30 q^{45} - 36 q^{46} + 6 q^{47} + 6 q^{48} - 30 q^{49} - 30 q^{50} - 3 q^{51} + 18 q^{52} - 6 q^{53} + 3 q^{54} + 9 q^{55} + 6 q^{56} + 36 q^{58} - 21 q^{59} + 6 q^{60} - 9 q^{61} - 21 q^{62} - 3 q^{63} + 24 q^{64} - 24 q^{65} + 9 q^{66} + 18 q^{67} + 30 q^{68} - 24 q^{69} + 9 q^{70} - 30 q^{71} + 39 q^{72} - 3 q^{74} - 30 q^{75} - 18 q^{77} + 12 q^{78} - 9 q^{79} + 12 q^{80} - 15 q^{81} - 18 q^{82} + 6 q^{84} + 9 q^{85} + 21 q^{86} + 42 q^{87} + 18 q^{88} - 15 q^{89} - 9 q^{90} + 15 q^{91} + 24 q^{92} - 24 q^{93} - 18 q^{94} + 18 q^{96} + 15 q^{97} + 18 q^{98} + 18 q^{99}+O(q^{100})$$ 6 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 + 3 * q^5 - 6 * q^6 + 12 * q^8 + 6 * q^12 - 6 * q^14 - 3 * q^15 - 3 * q^16 - 6 * q^17 + 6 * q^18 - 6 * q^20 - 3 * q^21 + 9 * q^22 + 6 * q^23 + 6 * q^24 + 6 * q^25 - 30 * q^26 + 12 * q^27 - 9 * q^28 - 15 * q^29 + 18 * q^31 + 9 * q^33 - 9 * q^34 - 3 * q^35 + 9 * q^36 - 24 * q^39 + 18 * q^40 - 12 * q^41 - 3 * q^42 + 9 * q^44 + 30 * q^45 - 36 * q^46 + 6 * q^47 + 6 * q^48 - 30 * q^49 - 30 * q^50 - 3 * q^51 + 18 * q^52 - 6 * q^53 + 3 * q^54 + 9 * q^55 + 6 * q^56 + 36 * q^58 - 21 * q^59 + 6 * q^60 - 9 * q^61 - 21 * q^62 - 3 * q^63 + 24 * q^64 - 24 * q^65 + 9 * q^66 + 18 * q^67 + 30 * q^68 - 24 * q^69 + 9 * q^70 - 30 * q^71 + 39 * q^72 - 3 * q^74 - 30 * q^75 - 18 * q^77 + 12 * q^78 - 9 * q^79 + 12 * q^80 - 15 * q^81 - 18 * q^82 + 6 * q^84 + 9 * q^85 + 21 * q^86 + 42 * q^87 + 18 * q^88 - 15 * q^89 - 9 * q^90 + 15 * q^91 + 24 * q^92 - 24 * q^93 - 18 * q^94 + 18 * q^96 + 15 * q^97 + 18 * q^98 + 18 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ -v^4 + v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ -v^5 + v^4 $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$ -v^5 - v^4 + v
 $$\zeta_{18}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$$ (b5 + b4 + 2*b2) / 3 $$\zeta_{18}^{2}$$ $$=$$ $$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$$ (-2*b5 + b4 + 3*b3 - b2) / 3 $$\zeta_{18}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{18}^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$$ (-b5 + 2*b4 + b2) / 3 $$\zeta_{18}^{5}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$$ (-b5 - b4 + b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 − 0.342020i
−1.26604 + 2.19285i −0.326352 + 0.565258i −2.20574 3.82045i 0.673648 1.16679i −0.826352 1.43128i 1.53209 6.10607 1.28699 + 2.22913i 1.70574 + 2.95442i
68.2 −0.673648 + 1.16679i −1.43969 + 2.49362i 0.0923963 + 0.160035i −0.439693 + 0.761570i −1.93969 3.35965i 0.347296 −2.94356 −2.64543 4.58202i −0.592396 1.02606i
68.3 0.439693 0.761570i 0.266044 0.460802i 0.613341 + 1.06234i 1.26604 2.19285i −0.233956 0.405223i −1.87939 2.83750 1.35844 + 2.35289i −1.11334 1.92836i
292.1 −1.26604 2.19285i −0.326352 0.565258i −2.20574 + 3.82045i 0.673648 + 1.16679i −0.826352 + 1.43128i 1.53209 6.10607 1.28699 2.22913i 1.70574 2.95442i
292.2 −0.673648 1.16679i −1.43969 2.49362i 0.0923963 0.160035i −0.439693 0.761570i −1.93969 + 3.35965i 0.347296 −2.94356 −2.64543 + 4.58202i −0.592396 + 1.02606i
292.3 0.439693 + 0.761570i 0.266044 + 0.460802i 0.613341 1.06234i 1.26604 + 2.19285i −0.233956 + 0.405223i −1.87939 2.83750 1.35844 2.35289i −1.11334 + 1.92836i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 292.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.c.h 6
19.b odd 2 1 361.2.c.i 6
19.c even 3 1 361.2.a.h 3
19.c even 3 1 inner 361.2.c.h 6
19.d odd 6 1 361.2.a.g 3
19.d odd 6 1 361.2.c.i 6
19.e even 9 2 361.2.e.a 6
19.e even 9 2 361.2.e.b 6
19.e even 9 2 361.2.e.h 6
19.f odd 18 2 19.2.e.a 6
19.f odd 18 2 361.2.e.f 6
19.f odd 18 2 361.2.e.g 6
57.f even 6 1 3249.2.a.z 3
57.h odd 6 1 3249.2.a.s 3
57.j even 18 2 171.2.u.c 6
76.f even 6 1 5776.2.a.br 3
76.g odd 6 1 5776.2.a.bi 3
76.k even 18 2 304.2.u.b 6
95.h odd 6 1 9025.2.a.bd 3
95.i even 6 1 9025.2.a.x 3
95.o odd 18 2 475.2.l.a 6
95.r even 36 4 475.2.u.a 12
133.ba even 18 2 931.2.w.a 6
133.bb even 18 2 931.2.v.a 6
133.bd odd 18 2 931.2.v.b 6
133.be odd 18 2 931.2.x.a 6
133.bf even 18 2 931.2.x.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.f odd 18 2
171.2.u.c 6 57.j even 18 2
304.2.u.b 6 76.k even 18 2
361.2.a.g 3 19.d odd 6 1
361.2.a.h 3 19.c even 3 1
361.2.c.h 6 1.a even 1 1 trivial
361.2.c.h 6 19.c even 3 1 inner
361.2.c.i 6 19.b odd 2 1
361.2.c.i 6 19.d odd 6 1
361.2.e.a 6 19.e even 9 2
361.2.e.b 6 19.e even 9 2
361.2.e.f 6 19.f odd 18 2
361.2.e.g 6 19.f odd 18 2
361.2.e.h 6 19.e even 9 2
475.2.l.a 6 95.o odd 18 2
475.2.u.a 12 95.r even 36 4
931.2.v.a 6 133.bb even 18 2
931.2.v.b 6 133.bd odd 18 2
931.2.w.a 6 133.ba even 18 2
931.2.x.a 6 133.be odd 18 2
931.2.x.b 6 133.bf even 18 2
3249.2.a.s 3 57.h odd 6 1
3249.2.a.z 3 57.f even 6 1
5776.2.a.bi 3 76.g odd 6 1
5776.2.a.br 3 76.f even 6 1
9025.2.a.x 3 95.i even 6 1
9025.2.a.bd 3 95.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(361, [\chi])$$:

 $$T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 6T_{2}^{3} + 9T_{2}^{2} + 9$$ T2^6 + 3*T2^5 + 9*T2^4 + 6*T2^3 + 9*T2^2 + 9 $$T_{3}^{6} + 3T_{3}^{5} + 9T_{3}^{4} + 2T_{3}^{3} + 3T_{3}^{2} + 1$$ T3^6 + 3*T3^5 + 9*T3^4 + 2*T3^3 + 3*T3^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9$$
$3$ $$T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1$$
$5$ $$T^{6} - 3 T^{5} + 9 T^{4} - 6 T^{3} + \cdots + 9$$
$7$ $$(T^{3} - 3 T + 1)^{2}$$
$11$ $$(T^{3} - 9 T - 9)^{2}$$
$13$ $$T^{6} + 21 T^{4} - 74 T^{3} + \cdots + 1369$$
$17$ $$T^{6} + 6 T^{5} + 27 T^{4} + 48 T^{3} + \cdots + 9$$
$19$ $$T^{6}$$
$23$ $$T^{6} - 6 T^{5} + 36 T^{4} - 48 T^{3} + \cdots + 576$$
$29$ $$T^{6} + 15 T^{5} + 153 T^{4} + \cdots + 12321$$
$31$ $$(T^{3} - 9 T^{2} + 6 T + 53)^{2}$$
$37$ $$(T^{3} - 21 T + 17)^{2}$$
$41$ $$T^{6} + 12 T^{5} + 135 T^{4} + \cdots + 12321$$
$43$ $$T^{6} + 57 T^{4} + 326 T^{3} + \cdots + 26569$$
$47$ $$T^{6} - 6 T^{5} + 45 T^{4} + 60 T^{3} + \cdots + 9$$
$53$ $$T^{6} + 6 T^{5} + 45 T^{4} + \cdots + 2601$$
$59$ $$T^{6} + 21 T^{5} + 306 T^{4} + \cdots + 71289$$
$61$ $$T^{6} + 9 T^{5} + 102 T^{4} + \cdots + 32761$$
$67$ $$T^{6} - 18 T^{5} + 300 T^{4} + \cdots + 179776$$
$71$ $$T^{6} + 30 T^{5} + 612 T^{4} + \cdots + 788544$$
$73$ $$T^{6} + 48 T^{4} + 128 T^{3} + \cdots + 4096$$
$79$ $$T^{6} + 9 T^{5} + 183 T^{4} + \cdots + 654481$$
$83$ $$(T^{3} - 189 T + 459)^{2}$$
$89$ $$T^{6} + 15 T^{5} + 171 T^{4} + \cdots + 3249$$
$97$ $$T^{6} - 15 T^{5} + 186 T^{4} + \cdots + 16129$$
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