# Properties

 Label 370.2.q.d Level $370$ Weight $2$ Character orbit 370.q Analytic conductor $2.954$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.q (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2x^{11} - 4x^{10} + 6x^{8} + 44x^{7} + 56x^{6} + 32x^{5} + 92x^{4} - 16x^{3} + 36x^{2} - 24x + 4$$ x^12 - 2*x^11 - 4*x^10 + 6*x^8 + 44*x^7 + 56*x^6 + 32*x^5 + 92*x^4 - 16*x^3 + 36*x^2 - 24*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - 1) q^{2} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{3} + \beta_{7} q^{4} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_1) q^{5} + (\beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{11} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{7} + q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + (-b7 - 1) * q^2 + (-b10 + b9 + b8 + b7 + b6 + b5 + 2*b4 - b3 - b1 + 1) * q^3 + b7 * q^4 + (-b10 + b9 + b8 - b5 + b4 - b1) * q^5 + (b10 - b7 - b6 - b5 - b4) * q^6 + (2*b11 + b6 + 2*b5 + 2*b4 - b3 + b2 - b1 + 2) * q^7 + q^8 + (-b11 - b9 - b7 + 2*b6 + b4 - b3 - b2 - b1) * q^9 $$q + ( - \beta_{7} - 1) q^{2} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{3} + \beta_{7} q^{4} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_1) q^{5} + (\beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{11} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{7} + q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{9} + (\beta_{10} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_1) q^{10} + ( - 2 \beta_{8} - 2 \beta_{6} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{12} + ( - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 2) q^{13}+ \cdots + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} - 8 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + \cdots + 7) q^{99}+O(q^{100})$$ q + (-b7 - 1) * q^2 + (-b10 + b9 + b8 + b7 + b6 + b5 + 2*b4 - b3 - b1 + 1) * q^3 + b7 * q^4 + (-b10 + b9 + b8 - b5 + b4 - b1) * q^5 + (b10 - b7 - b6 - b5 - b4) * q^6 + (2*b11 + b6 + 2*b5 + 2*b4 - b3 + b2 - b1 + 2) * q^7 + q^8 + (-b11 - b9 - b7 + 2*b6 + b4 - b3 - b2 - b1) * q^9 + (b10 - b8 - b6 - b4 + b3 + b1) * q^10 + (-2*b8 - 2*b6 + b3 + b1) * q^11 + (-b9 - b8 - b4 + b3 + b1 - 1) * q^12 + (-2*b11 - b10 - b9 + b7 - b6 - b5 - b4 + b3 - b2 - 2) * q^13 + (-b11 - 2*b6 - 2*b5 - b4 + b3 - 2*b2 - b1 - 1) * q^14 + (-2*b11 + b10 - b9 + 3*b8 - b7 + 2*b6 - 3*b5 - 2*b4 - b2 + b1 - 4) * q^15 + (-b7 - 1) * q^16 + (-b11 - b9 + b7 - 2*b6 - b5 - b2 - 2) * q^17 + (-b10 + b9 + 3*b8 - b7 + b6 - b5 + b2 - b1 - 2) * q^18 + (b10 + b9 + b6 - b5 - 2*b3 + b2 - b1) * q^19 + (-b9 + b6 + b5 - b3) * q^20 + (-2*b11 + b10 - 2*b9 - 3*b7 + 4*b6 - b5 - b3 - b2 + 1) * q^21 + (2*b6 + b5 + b4 - b3 - b1) * q^22 + (-b11 + b10 - b9 + b8 - 2*b6 - 2*b5 - 2*b4 + b3 - b2 + b1 - 2) * q^23 + (-b10 + b9 + b8 + b7 + b6 + b5 + 2*b4 - b3 - b1 + 1) * q^24 + (-b11 - b9 - 2*b8 + b7 + b6 - 2*b5 - b4 + b3 + b2 + b1 - 2) * q^25 + (b11 - b10 + 2*b9 + 2*b6 + 2*b5 + 2*b4 - 2*b3 + 2*b2 + 2) * q^26 + (b11 - 2*b10 + 4*b9 + 2*b8 - b7 + 3*b6 + 2*b5 + 4*b4 - 3*b3 - 2*b1 + 3) * q^27 + (-b11 + b6 - b4 + b2 + 2*b1 - 1) * q^28 + (b11 - b10 + b8 - 2*b7 + b6 - b3 + 2*b2 - b1) * q^29 + (b11 - b10 + 2*b7 - b6 + b5 + b4 + b3 + 2*b2 + 2) * q^30 + (-b11 - b10 + 2*b9 - b7 + b6 + b4 - b3 - 1) * q^31 + b7 * q^32 + (-3*b8 + 3*b7 - 2*b6 + 2*b5 + 2*b2 + 3*b1 + 4) * q^33 + (-b10 + b9 - b8 + b7 + b6 + b4 + b2 - b1 + 2) * q^34 + (-2*b11 + b10 - 4*b9 + 2*b8 - 3*b7 - 4*b5 - 3*b4 + 3*b3 - 3*b2 - b1 - 5) * q^35 + (b11 + b10 - 3*b8 + 2*b7 - 3*b6 + b5 - b4 + b3 + 2*b1 + 2) * q^36 + (b11 + 5*b7 - b6 + b5 + 2*b4 - b3 + b2 + 5) * q^37 + (b11 + b10 - 2*b9 - b5 - 3*b4 + 3*b3 + 2*b1 + 1) * q^38 + (b11 + 2*b9 - 5*b8 + 5*b7 - 6*b6 + 3*b3 - b2 + 2*b1 + 1) * q^39 + (-b10 + b9 + b8 - b5 + b4 - b1) * q^40 + (2*b11 - 2*b10 - 3*b8 + 2*b6 + 2*b5 - 2*b3 + 2*b2 + 2) * q^41 + (b11 - 2*b10 + b9 + 5*b8 - 3*b7 + 2*b6 + b4 + 2*b2 - 2*b1 - 5) * q^42 + (-b11 + b10 - 3*b9 - b8 - 2*b6 + b5 + b4 + 2*b3 - 3*b2 - 3*b1 + 3) * q^43 + (2*b8 - b5 - b4) * q^44 + (2*b9 + b8 - 2*b7 - 5*b6 - 3*b5 - b4 - 2*b2 - b1 - 7) * q^45 + (-b10 - 2*b8 + b7 + b5 + b4 + b2 + 1) * q^46 + (-b10 + 4*b8 - 4*b7 + 2*b5 - b4 - 2*b1 - 4) * q^47 + (b10 - b7 - b6 - b5 - b4) * q^48 + (-3*b11 + 5*b10 - 3*b9 - 3*b7 + 4*b6 - 3*b4 - 2*b3 + 2*b2 + 2*b1) * q^49 + (2*b11 - b10 + b9 + 2*b8 + b7 + 3*b6 + 3*b5 + 2*b4 + b2 - b1 + 4) * q^50 + (-2*b8 - 2*b6 - b5 - 4*b4 + 2*b3 + 4*b1 - 2) * q^51 + (b11 + 2*b10 - b9 - b7 - b6 - b5 - b4 + b3 - b2) * q^52 + (-4*b11 - 2*b10 + b9 + 3*b8 + 4*b7 - 6*b5 - 2*b4 + 3*b3 - 4*b2 - b1 - 3) * q^53 + (-b11 + 4*b10 - 2*b9 + b8 - 2*b7 - 3*b6 - 4*b5 - 4*b4 + 2*b3 - b2 + 2*b1 - 4) * q^54 + (b11 + b9 - b8 - 3*b7 + 3*b6 + 2*b5 + 3*b4 - b3 + b2 - 2*b1 + 1) * q^55 + (2*b11 + b6 + 2*b5 + 2*b4 - b3 + b2 - b1 + 2) * q^56 + (b11 - b9 + 8*b8 - 6*b7 + 5*b6 - b4 - b3 + b2 - b1 - 5) * q^57 + (b11 + b9 - 2*b8 + b7 - 2*b6 + 2*b3 - b2 + b1) * q^58 + (2*b11 + b7 + 3*b6 + 2*b5 + 2*b4 - b3 + 2*b2 + 3) * q^59 + (b11 + b9 - 3*b8 - b7 - b6 + 2*b5 + b4 - b3 - b2 - b1 + 2) * q^60 + (-4*b8 + 4*b7 - 6*b6 + b5 + 3*b3 - 2*b2 + b1 + 4) * q^61 + (b11 + 2*b10 - b9 + b8 + b6 - b4 + b2 + b1) * q^62 + (8*b11 - 2*b10 + 4*b9 - 2*b7 + 2*b6 + 7*b5 + 9*b4 - 6*b3 - 7*b1 + 8) * q^63 + q^64 + (-2*b11 - b10 + 4*b9 - 2*b8 + 3*b7 - b6 + 2*b5 + 4*b4 - 4*b3 - b2 - b1 + 1) * q^65 + (2*b11 - b8 - 4*b7 + 3*b6 + 2*b5 + 3*b4 - 2*b3 - 3*b1 + 1) * q^66 + (2*b10 + 2*b9 - 2*b8 + 2*b7 - 7*b6 + b5 + 4*b3 - 3*b2 + b1 + 4) * q^67 + (b11 + b10 + b8 - 2*b7 + b6 + b5 - b4 + b1) * q^68 + (-b10 + b9 + 2*b8 + 2*b7 - 4*b6 - 3*b5 - 4*b4 + 3*b3 + 2*b1 - 4) * q^69 + (-b11 - 4*b10 + 3*b9 + b8 + 3*b7 + 2*b5 + 2*b4 - b3 + 2*b2 - 1) * q^70 + (5*b10 - 5*b9 - 2*b8 - 2*b7 + b6 + b5 - 6*b4 + 5*b2 + 5*b1) * q^71 + (-b11 - b9 - b7 + 2*b6 + b4 - b3 - b2 - b1) * q^72 + (b11 + 5*b10 + 3*b8 - 4*b7 + b6 - b5 - 2*b4 - b3 + 2*b2 + 3*b1 - 2) * q^73 + (-2*b8 - 4*b7 - b6 - b5 + b3 - b2 - 2*b1 + 1) * q^74 + (3*b11 + 2*b10 - 8*b8 + 3*b7 - 9*b6 + 2*b5 - 2*b4 + b3 + 6*b1 + 1) * q^75 + (-b11 - 2*b10 + b9 - b6 + 2*b5 + 3*b4 - b3 - b2 - b1 - 1) * q^76 + (-2*b10 - 2*b9 - 4*b8 + 4*b7 - 3*b6 + 2*b5 + 3*b3 - b2 + 2) * q^77 + (-2*b11 + 2*b10 - 2*b9 + 4*b6 + b5 - 2*b3 - b2 + 3) * q^78 + (-b10 - b9 + 5*b8 - 5*b7 + 6*b6 + 2*b3 - 2*b2 - 2*b1 - 8) * q^79 + (b10 - b8 - b6 - b4 + b3 + b1) * q^80 + (-2*b11 - 3*b10 + b9 + 5*b8 + 7*b6 - b5 + 4*b4 - 2*b3 - 3*b2 - 5*b1 - 2) * q^81 + (2*b9 + 3*b8 + b6 - 2*b5 + 2*b4 + 2*b3 - 2*b2) * q^82 + (4*b11 - 4*b10 + 2*b9 + b7 + 5*b6 + 4*b5 - 2*b3 + 4*b2 - 2*b1 + 5) * q^83 + (b11 + b10 + b9 - 5*b8 + 6*b7 - 6*b6 + b5 - b4 + b3 - b2 + 2*b1 + 4) * q^84 + (-2*b11 + b10 - b9 - 4*b8 - 2*b7 - 3*b6 + 2*b5 + 2*b4 - 2*b3 - b2 + b1 - 4) * q^85 + (-2*b11 - 3*b10 + 2*b9 + 2*b8 - 4*b7 + 2*b6 - b4 - 4*b3 + b2 - b1 - 6) * q^86 + (-4*b11 + 4*b10 + 3*b8 + b7 - 4*b6 - 6*b5 - 4*b4 + 2*b3 - 4*b2 + 2*b1 - 2) * q^87 + (-2*b8 - 2*b6 + b3 + b1) * q^88 + (b11 - 2*b9 - 4*b8 + 2*b7 - 3*b6 - 4*b4 + 2*b3 - b2 + 6*b1 - 1) * q^89 + (-2*b11 + 2*b10 - 2*b9 - 4*b8 + 7*b7 - b6 - 2*b5 - 3*b4 + 3*b3 + 4*b1 + 3) * q^90 + (4*b11 - 4*b8 + 8*b7 - 12*b6 - b4 + 4*b3 - 4*b2 + 2*b1 + 8) * q^91 + (b11 + b9 + b8 - b7 + 2*b6 + b5 + b4 - b3 - b1 + 1) * q^92 + (b11 + b10 + b8 - 2*b7 + 3*b6 + 2*b5 + b4 - 3*b3 + b2 - 2*b1 + 1) * q^93 + (b9 - 4*b8 + 4*b7 - 4*b6 - b4 - 2*b3 + 3*b1) * q^94 + (3*b11 + 2*b10 - 5*b9 + 2*b7 + b6 + 2*b5 - 3*b4 - b3 + 5*b2 + b1 + 1) * q^95 + (-b9 - b8 - b4 + b3 + b1 - 1) * q^96 + (-2*b9 + 3*b8 + 10*b7 + 5*b6 + b5 - b4 + 2*b2 + b1 + 5) * q^97 + (5*b11 - 3*b10 - 2*b9 + 2*b8 - 3*b7 + 3*b6 + 3*b5 - b3 + 3*b2 - 2*b1 - 1) * q^98 + (-b11 - 2*b10 + b9 - 8*b8 + 8*b7 - 3*b6 + 4*b5 + 3*b4 + b3 + b2 + 3*b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 6 q^{2} - 6 q^{4} + 2 q^{5} + 6 q^{6} - 4 q^{7} + 12 q^{8} + 18 q^{9}+O(q^{10})$$ 12 * q - 6 * q^2 - 6 * q^4 + 2 * q^5 + 6 * q^6 - 4 * q^7 + 12 * q^8 + 18 * q^9 $$12 q - 6 q^{2} - 6 q^{4} + 2 q^{5} + 6 q^{6} - 4 q^{7} + 12 q^{8} + 18 q^{9} - 4 q^{10} - 6 q^{12} + 8 q^{14} - 10 q^{15} - 6 q^{16} - 12 q^{17} - 18 q^{18} - 8 q^{19} + 2 q^{20} + 54 q^{21} - 6 q^{22} - 4 q^{23} - 14 q^{25} + 18 q^{27} - 4 q^{28} + 4 q^{29} - 4 q^{30} - 2 q^{31} - 6 q^{32} + 20 q^{33} + 12 q^{34} + 6 q^{35} + 14 q^{37} + 16 q^{38} - 30 q^{39} + 2 q^{40} + 12 q^{41} - 54 q^{42} + 44 q^{43} + 6 q^{44} - 60 q^{45} + 2 q^{46} - 28 q^{47} + 6 q^{48} + 24 q^{49} + 10 q^{50} - 8 q^{51} + 8 q^{53} - 18 q^{54} - 4 q^{56} - 28 q^{57} - 14 q^{58} + 14 q^{60} + 24 q^{61} - 14 q^{62} + 8 q^{63} + 12 q^{64} - 10 q^{65} + 8 q^{66} + 18 q^{67} - 40 q^{69} - 30 q^{70} + 18 q^{72} - 28 q^{73} + 38 q^{74} - 30 q^{75} - 8 q^{76} + 10 q^{77} + 48 q^{78} - 56 q^{79} - 4 q^{80} + 4 q^{81} + 14 q^{83} - 28 q^{85} - 22 q^{86} + 18 q^{87} - 2 q^{89} + 18 q^{90} + 38 q^{91} + 2 q^{92} - 16 q^{94} - 28 q^{95} - 6 q^{96} - 24 q^{98} + 28 q^{99}+O(q^{100})$$ 12 * q - 6 * q^2 - 6 * q^4 + 2 * q^5 + 6 * q^6 - 4 * q^7 + 12 * q^8 + 18 * q^9 - 4 * q^10 - 6 * q^12 + 8 * q^14 - 10 * q^15 - 6 * q^16 - 12 * q^17 - 18 * q^18 - 8 * q^19 + 2 * q^20 + 54 * q^21 - 6 * q^22 - 4 * q^23 - 14 * q^25 + 18 * q^27 - 4 * q^28 + 4 * q^29 - 4 * q^30 - 2 * q^31 - 6 * q^32 + 20 * q^33 + 12 * q^34 + 6 * q^35 + 14 * q^37 + 16 * q^38 - 30 * q^39 + 2 * q^40 + 12 * q^41 - 54 * q^42 + 44 * q^43 + 6 * q^44 - 60 * q^45 + 2 * q^46 - 28 * q^47 + 6 * q^48 + 24 * q^49 + 10 * q^50 - 8 * q^51 + 8 * q^53 - 18 * q^54 - 4 * q^56 - 28 * q^57 - 14 * q^58 + 14 * q^60 + 24 * q^61 - 14 * q^62 + 8 * q^63 + 12 * q^64 - 10 * q^65 + 8 * q^66 + 18 * q^67 - 40 * q^69 - 30 * q^70 + 18 * q^72 - 28 * q^73 + 38 * q^74 - 30 * q^75 - 8 * q^76 + 10 * q^77 + 48 * q^78 - 56 * q^79 - 4 * q^80 + 4 * q^81 + 14 * q^83 - 28 * q^85 - 22 * q^86 + 18 * q^87 - 2 * q^89 + 18 * q^90 + 38 * q^91 + 2 * q^92 - 16 * q^94 - 28 * q^95 - 6 * q^96 - 24 * q^98 + 28 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2x^{11} - 4x^{10} + 6x^{8} + 44x^{7} + 56x^{6} + 32x^{5} + 92x^{4} - 16x^{3} + 36x^{2} - 24x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 48023 \nu^{11} - 173427 \nu^{10} + 812758 \nu^{9} + 795518 \nu^{8} - 312576 \nu^{7} - 3181332 \nu^{6} - 14801278 \nu^{5} - 14727942 \nu^{4} + \cdots - 16072116 ) / 8800836$$ (-48023*v^11 - 173427*v^10 + 812758*v^9 + 795518*v^8 - 312576*v^7 - 3181332*v^6 - 14801278*v^5 - 14727942*v^4 - 11185716*v^3 - 21629056*v^2 + 9352612*v - 16072116) / 8800836 $$\beta_{3}$$ $$=$$ $$( 2660983 \nu^{11} - 7621528 \nu^{10} - 6209960 \nu^{9} + 9898720 \nu^{8} + 18275158 \nu^{7} + 95878200 \nu^{6} + 40728518 \nu^{5} - 40308836 \nu^{4} + \cdots - 77578192 ) / 132012540$$ (2660983*v^11 - 7621528*v^10 - 6209960*v^9 + 9898720*v^8 + 18275158*v^7 + 95878200*v^6 + 40728518*v^5 - 40308836*v^4 + 180194580*v^3 - 172657228*v^2 + 188165900*v - 77578192) / 132012540 $$\beta_{4}$$ $$=$$ $$( 1499 \nu^{11} - 1684 \nu^{10} - 8260 \nu^{9} - 6690 \nu^{8} + 8734 \nu^{7} + 77220 \nu^{6} + 146974 \nu^{5} + 127492 \nu^{4} + 163380 \nu^{3} + 66376 \nu^{2} + 66060 \nu - 22456 ) / 40260$$ (1499*v^11 - 1684*v^10 - 8260*v^9 - 6690*v^8 + 8734*v^7 + 77220*v^6 + 146974*v^5 + 127492*v^4 + 163380*v^3 + 66376*v^2 + 66060*v - 22456) / 40260 $$\beta_{5}$$ $$=$$ $$( - 5362957 \nu^{11} + 9940372 \nu^{10} + 23030390 \nu^{9} + 1380680 \nu^{8} - 29137702 \nu^{7} - 231395340 \nu^{6} - 335137622 \nu^{5} + \cdots + 32870848 ) / 132012540$$ (-5362957*v^11 + 9940372*v^10 + 23030390*v^9 + 1380680*v^8 - 29137702*v^7 - 231395340*v^6 - 335137622*v^5 - 231091216*v^4 - 633675960*v^3 - 69045068*v^2 - 161253080*v + 32870848) / 132012540 $$\beta_{6}$$ $$=$$ $$( - 8217712 \nu^{11} + 11072467 \nu^{10} + 42811220 \nu^{9} + 23030390 \nu^{8} - 47925592 \nu^{7} - 390717030 \nu^{6} - 691587212 \nu^{5} + \cdots + 35972008 ) / 132012540$$ (-8217712*v^11 + 11072467*v^10 + 42811220*v^9 + 23030390*v^8 - 47925592*v^7 - 390717030*v^6 - 691587212*v^5 - 598104406*v^4 - 987120720*v^3 - 502192568*v^2 - 364882700*v + 35972008) / 132012540 $$\beta_{7}$$ $$=$$ $$( - 5614 \nu^{11} + 9729 \nu^{10} + 24140 \nu^{9} + 8260 \nu^{8} - 26994 \nu^{7} - 255750 \nu^{6} - 391604 \nu^{5} - 326622 \nu^{4} - 643980 \nu^{3} - 73556 \nu^{2} + \cdots + 68676 ) / 40260$$ (-5614*v^11 + 9729*v^10 + 24140*v^9 + 8260*v^8 - 26994*v^7 - 255750*v^6 - 391604*v^5 - 326622*v^4 - 643980*v^3 - 73556*v^2 - 268480*v + 68676) / 40260 $$\beta_{8}$$ $$=$$ $$( - 19394548 \nu^{11} + 36128113 \nu^{10} + 85199720 \nu^{9} + 6209960 \nu^{8} - 126266008 \nu^{7} - 871635270 \nu^{6} - 1181972888 \nu^{5} + \cdots + 277303252 ) / 132012540$$ (-19394548*v^11 + 36128113*v^10 + 85199720*v^9 + 6209960*v^8 - 126266008*v^7 - 871635270*v^6 - 1181972888*v^5 - 661354054*v^4 - 1743989580*v^3 + 130118188*v^2 - 525546500*v + 277303252) / 132012540 $$\beta_{9}$$ $$=$$ $$( 12537719 \nu^{11} - 22791104 \nu^{10} - 56749885 \nu^{9} - 5276635 \nu^{8} + 85217429 \nu^{7} + 561759990 \nu^{6} + 785090434 \nu^{5} + \cdots - 249596696 ) / 66006270$$ (12537719*v^11 - 22791104*v^10 - 56749885*v^9 - 5276635*v^8 + 85217429*v^7 + 561759990*v^6 + 785090434*v^5 + 453290102*v^4 + 1111025700*v^3 - 35355044*v^2 + 227562640*v - 249596696) / 66006270 $$\beta_{10}$$ $$=$$ $$( - 15731201 \nu^{11} + 24513101 \nu^{10} + 73188085 \nu^{9} + 35555635 \nu^{8} - 80700356 \nu^{7} - 738220065 \nu^{6} - 1202248126 \nu^{5} + \cdots + 164311814 ) / 66006270$$ (-15731201*v^11 + 24513101*v^10 + 73188085*v^9 + 35555635*v^8 - 80700356*v^7 - 738220065*v^6 - 1202248126*v^5 - 1043296028*v^4 - 1868935770*v^3 - 480892054*v^2 - 847123570*v + 164311814) / 66006270 $$\beta_{11}$$ $$=$$ $$( - 16622963 \nu^{11} + 31633798 \nu^{10} + 68509300 \nu^{9} + 8136630 \nu^{8} - 92738998 \nu^{7} - 739730970 \nu^{6} - 1013258878 \nu^{5} + \cdots + 249278272 ) / 44004180$$ (-16622963*v^11 + 31633798*v^10 + 68509300*v^9 + 8136630*v^8 - 92738998*v^7 - 739730970*v^6 - 1013258878*v^5 - 674881324*v^4 - 1668477300*v^3 + 52920608*v^2 - 671227140*v + 249278272) / 44004180
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{7} - 2\beta_{6} - \beta_{3} + \beta_{2} + \beta _1 + 1$$ b10 - b7 - 2*b6 - b3 + b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{8} - 2\beta_{7} - \beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 2$$ b11 + b10 - b8 - 2*b7 - b6 - 3*b5 + b4 - b3 + 2*b2 + 2*b1 + 2 $$\nu^{4}$$ $$=$$ $$6 \beta_{11} + 7 \beta_{10} - \beta_{9} - 8 \beta_{8} - 16 \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 7 \beta_{2} + 7 \beta _1 + 6$$ 6*b11 + 7*b10 - b9 - 8*b8 - 16*b7 - 9*b6 - 3*b5 + 2*b4 + 7*b2 + 7*b1 + 6 $$\nu^{5}$$ $$=$$ $$13 \beta_{11} + 18 \beta_{10} - 9 \beta_{9} - 24 \beta_{8} - 34 \beta_{7} - 21 \beta_{6} - 16 \beta_{5} + 16 \beta_{4} + 8 \beta_{3} + 13 \beta_{2} + 9 \beta _1 + 3$$ 13*b11 + 18*b10 - 9*b9 - 24*b8 - 34*b7 - 21*b6 - 16*b5 + 16*b4 + 8*b3 + 13*b2 + 9*b1 + 3 $$\nu^{6}$$ $$=$$ $$54 \beta_{11} + 54 \beta_{10} - 42 \beta_{9} - 112 \beta_{8} - 124 \beta_{7} - 70 \beta_{6} - 34 \beta_{5} + 34 \beta_{4} + 36 \beta_{3} + 42 \beta_{2} + 2 \beta _1 - 8$$ 54*b11 + 54*b10 - 42*b9 - 112*b8 - 124*b7 - 70*b6 - 34*b5 + 34*b4 + 36*b3 + 42*b2 + 2*b1 - 8 $$\nu^{7}$$ $$=$$ $$152 \beta_{11} + 124 \beta_{10} - 124 \beta_{9} - 310 \beta_{8} - 310 \beta_{7} - 148 \beta_{6} - 112 \beta_{5} + 124 \beta_{4} + 188 \beta_{3} + 76 \beta_{2} - 62 \beta _1 - 72$$ 152*b11 + 124*b10 - 124*b9 - 310*b8 - 310*b7 - 148*b6 - 112*b5 + 124*b4 + 188*b3 + 76*b2 - 62*b1 - 72 $$\nu^{8}$$ $$=$$ $$436 \beta_{11} + 326 \beta_{10} - 436 \beta_{9} - 996 \beta_{8} - 874 \beta_{7} - 388 \beta_{6} - 224 \beta_{5} + 310 \beta_{4} + 636 \beta_{3} + 110 \beta_{2} - 334 \beta _1 - 438$$ 436*b11 + 326*b10 - 436*b9 - 996*b8 - 874*b7 - 388*b6 - 224*b5 + 310*b4 + 636*b3 + 110*b2 - 334*b1 - 438 $$\nu^{9}$$ $$=$$ $$1080 \beta_{11} + 636 \beta_{10} - 1272 \beta_{9} - 2662 \beta_{8} - 1940 \beta_{7} - 722 \beta_{6} - 498 \beta_{5} + 874 \beta_{4} + 2076 \beta_{3} - 1510 \beta _1 - 1582$$ 1080*b11 + 636*b10 - 1272*b9 - 2662*b8 - 1940*b7 - 722*b6 - 498*b5 + 874*b4 + 2076*b3 - 1510*b1 - 1582 $$\nu^{10}$$ $$=$$ $$2644 \beta_{11} + 942 \beta_{10} - 3586 \beta_{9} - 7104 \beta_{8} - 4084 \beta_{7} - 942 \beta_{6} - 722 \beta_{5} + 1940 \beta_{4} + 6248 \beta_{3} - 942 \beta_{2} - 5306 \beta _1 - 5524$$ 2644*b11 + 942*b10 - 3586*b9 - 7104*b8 - 4084*b7 - 942*b6 - 722*b5 + 1940*b4 + 6248*b3 - 942*b2 - 5306*b1 - 5524 $$\nu^{11}$$ $$=$$ $$5306 \beta_{11} - 9130 \beta_{9} - 16376 \beta_{8} - 6028 \beta_{7} + 722 \beta_{6} + 4084 \beta_{4} + 17716 \beta_{3} - 5306 \beta_{2} - 17298 \beta _1 - 17098$$ 5306*b11 - 9130*b9 - 16376*b8 - 6028*b7 + 722*b6 + 4084*b4 + 17716*b3 - 5306*b2 - 17298*b1 - 17098

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\beta_{8}$$ $$-\beta_{6} - \beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.270517 + 0.0724848i 2.78638 + 0.746609i −1.69087 − 0.453068i 0.270517 − 0.0724848i 2.78638 − 0.746609i −1.69087 + 0.453068i −0.208144 − 0.776804i −0.431120 − 1.60896i 0.273238 + 1.01974i −0.208144 + 0.776804i −0.431120 + 1.60896i 0.273238 − 1.01974i
−0.500000 + 0.866025i −0.176529 + 0.658815i −0.500000 0.866025i 0.242559 + 2.22287i −0.482286 0.482286i 1.09930 4.10264i 1.00000 2.19520 + 1.26740i −2.04634 0.901374i
97.2 −0.500000 + 0.866025i 0.434319 1.62090i −0.500000 0.866025i 1.34690 + 1.78490i 1.18658 + 1.18658i −0.632846 + 2.36181i 1.00000 0.159392 + 0.0920251i −2.21922 + 0.274000i
97.3 −0.500000 + 0.866025i 0.608236 2.26997i −0.500000 0.866025i 0.642592 2.14175i 1.66173 + 1.66173i 0.265598 0.991227i 1.00000 −2.18472 1.26135i 1.53351 + 1.62737i
103.1 −0.500000 0.866025i −0.176529 0.658815i −0.500000 + 0.866025i 0.242559 2.22287i −0.482286 + 0.482286i 1.09930 + 4.10264i 1.00000 2.19520 1.26740i −2.04634 + 0.901374i
103.2 −0.500000 0.866025i 0.434319 + 1.62090i −0.500000 + 0.866025i 1.34690 1.78490i 1.18658 1.18658i −0.632846 2.36181i 1.00000 0.159392 0.0920251i −2.21922 0.274000i
103.3 −0.500000 0.866025i 0.608236 + 2.26997i −0.500000 + 0.866025i 0.642592 + 2.14175i 1.66173 1.66173i 0.265598 + 0.991227i 1.00000 −2.18472 + 1.26135i 1.53351 1.62737i
267.1 −0.500000 0.866025i −2.78144 + 0.745286i −0.500000 + 0.866025i −2.02520 + 0.947920i 2.03616 + 2.03616i −4.90683 + 1.31478i 1.00000 4.58291 2.64594i 1.83352 + 1.27992i
267.2 −0.500000 0.866025i −1.08428 + 0.290531i −0.500000 + 0.866025i 1.94546 + 1.10235i 0.793745 + 0.793745i −1.13508 + 0.304145i 1.00000 −1.50683 + 0.869970i −0.0180717 2.23599i
267.3 −0.500000 0.866025i 2.99969 0.803766i −0.500000 + 0.866025i −1.15231 1.91629i −2.19593 2.19593i 3.30986 0.886874i 1.00000 5.75405 3.32210i −1.08340 + 1.95608i
273.1 −0.500000 + 0.866025i −2.78144 0.745286i −0.500000 0.866025i −2.02520 0.947920i 2.03616 2.03616i −4.90683 1.31478i 1.00000 4.58291 + 2.64594i 1.83352 1.27992i
273.2 −0.500000 + 0.866025i −1.08428 0.290531i −0.500000 0.866025i 1.94546 1.10235i 0.793745 0.793745i −1.13508 0.304145i 1.00000 −1.50683 0.869970i −0.0180717 + 2.23599i
273.3 −0.500000 + 0.866025i 2.99969 + 0.803766i −0.500000 0.866025i −1.15231 + 1.91629i −2.19593 + 2.19593i 3.30986 + 0.886874i 1.00000 5.75405 + 3.32210i −1.08340 1.95608i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 273.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.d 12
5.c odd 4 1 370.2.r.d yes 12
37.g odd 12 1 370.2.r.d yes 12
185.p even 12 1 inner 370.2.q.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.d 12 1.a even 1 1 trivial
370.2.q.d 12 185.p even 12 1 inner
370.2.r.d yes 12 5.c odd 4 1
370.2.r.d yes 12 37.g odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 9 T_{3}^{10} + 6 T_{3}^{9} - 10 T_{3}^{8} - 96 T_{3}^{7} + 351 T_{3}^{6} + 522 T_{3}^{5} + 886 T_{3}^{4} + 2340 T_{3}^{3} + 2511 T_{3}^{2} + 1458 T_{3} + 729$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{6}$$
$3$ $$T^{12} - 9 T^{10} + 6 T^{9} - 10 T^{8} + \cdots + 729$$
$5$ $$T^{12} - 2 T^{11} + 9 T^{10} + \cdots + 15625$$
$7$ $$T^{12} + 4 T^{11} - 4 T^{10} + \cdots + 47524$$
$11$ $$T^{12} + 36 T^{10} + 428 T^{8} + \cdots + 324$$
$13$ $$T^{12} + 39 T^{10} + 92 T^{9} + \cdots + 97969$$
$17$ $$T^{12} + 12 T^{11} + 38 T^{10} + \cdots + 47524$$
$19$ $$T^{12} + 8 T^{11} + 56 T^{10} + \cdots + 30976$$
$23$ $$(T^{6} + 2 T^{5} - 26 T^{4} + 2 T^{3} + \cdots - 74)^{2}$$
$29$ $$T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 238144$$
$31$ $$T^{12} + 2 T^{11} + 2 T^{10} - 170 T^{9} + \cdots + 529$$
$37$ $$T^{12} - 14 T^{11} + \cdots + 2565726409$$
$41$ $$T^{12} - 12 T^{11} - 23 T^{10} + \cdots + 1390041$$
$43$ $$(T^{6} - 22 T^{5} - T^{4} + 2744 T^{3} + \cdots + 366553)^{2}$$
$47$ $$T^{12} + 28 T^{11} + \cdots + 16333351204$$
$53$ $$T^{12} - 8 T^{11} + \cdots + 24894212841$$
$59$ $$T^{12} + 18 T^{10} - 140 T^{9} + \cdots + 1747684$$
$61$ $$T^{12} - 24 T^{11} + \cdots + 898081024$$
$67$ $$T^{12} - 18 T^{11} + \cdots + 47924215056$$
$71$ $$T^{12} + 304 T^{10} + \cdots + 165408143616$$
$73$ $$T^{12} + 28 T^{11} + \cdots + 573985764$$
$79$ $$T^{12} + 56 T^{11} + \cdots + 4859205264$$
$83$ $$T^{12} - 14 T^{11} + \cdots + 6754209856$$
$89$ $$T^{12} + 2 T^{11} + \cdots + 2304384016$$
$97$ $$T^{12} + 640 T^{10} + \cdots + 234403159104$$