# Properties

 Label 930.2.j.c Level $930$ Weight $2$ Character orbit 930.j Analytic conductor $7.426$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.1698758656.6 Defining polynomial: $$x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64$$ x^8 + 18*x^6 + 97*x^4 + 176*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64$$ (v^7 + 2*v^6 + 18*v^5 + 28*v^4 + 89*v^3 + 74*v^2 + 104*v - 16) / 64 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64$$ (v^7 - 2*v^6 + 18*v^5 - 28*v^4 + 89*v^3 - 74*v^2 + 104*v + 16) / 64 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - 2\nu^{6} - 10\nu^{5} - 20\nu^{4} + 15\nu^{3} - 2\nu^{2} + 120\nu + 80 ) / 64$$ (-v^7 - 2*v^6 - 10*v^5 - 20*v^4 + 15*v^3 - 2*v^2 + 120*v + 80) / 64 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 2\nu^{6} - 10\nu^{5} - 20\nu^{4} + 15\nu^{3} - 2\nu^{2} + 184\nu + 80 ) / 64$$ (-v^7 - 2*v^6 - 10*v^5 - 20*v^4 + 15*v^3 - 2*v^2 + 184*v + 80) / 64 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + 10\nu^{5} - 20\nu^{4} - 15\nu^{3} - 2\nu^{2} - 120\nu + 80 ) / 64$$ (v^7 - 2*v^6 + 10*v^5 - 20*v^4 - 15*v^3 - 2*v^2 - 120*v + 80) / 64 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 14\nu^{4} + 45\nu^{2} + 32 ) / 8$$ (v^6 + 14*v^4 + 45*v^2 + 32) / 8 $$\beta_{7}$$ $$=$$ $$( 3\nu^{7} + 46\nu^{5} + 179\nu^{3} + 168\nu ) / 64$$ (3*v^7 + 46*v^5 + 179*v^3 + 168*v) / 64
 $$\nu$$ $$=$$ $$\beta_{4} - \beta_{3}$$ b4 - b3 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{2} - 2\beta _1 - 5$$ b6 + 2*b2 - 2*b1 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{7} - 2\beta_{5} - 5\beta_{4} + 7\beta_{3} - 4\beta_{2} - 4\beta_1$$ 4*b7 - 2*b5 - 5*b4 + 7*b3 - 4*b2 - 4*b1 $$\nu^{4}$$ $$=$$ $$-9\beta_{6} + 4\beta_{5} + 4\beta_{3} - 22\beta_{2} + 22\beta _1 + 37$$ -9*b6 + 4*b5 + 4*b3 - 22*b2 + 22*b1 + 37 $$\nu^{5}$$ $$=$$ $$-52\beta_{7} + 22\beta_{5} + 37\beta_{4} - 59\beta_{3} + 56\beta_{2} + 56\beta_1$$ -52*b7 + 22*b5 + 37*b4 - 59*b3 + 56*b2 + 56*b1 $$\nu^{6}$$ $$=$$ $$89\beta_{6} - 56\beta_{5} - 56\beta_{3} + 218\beta_{2} - 218\beta _1 - 325$$ 89*b6 - 56*b5 - 56*b3 + 218*b2 - 218*b1 - 325 $$\nu^{7}$$ $$=$$ $$580\beta_{7} - 218\beta_{5} - 325\beta_{4} + 543\beta_{3} - 620\beta_{2} - 620\beta_1$$ 580*b7 - 218*b5 - 325*b4 + 543*b3 - 620*b2 - 620*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$\beta_{7}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
497.1
 2.16053i − 3.16053i − 1.69230i 0.692297i − 2.16053i 3.16053i 1.69230i − 0.692297i
−0.707107 + 0.707107i −1.70711 0.292893i 1.00000i −2.23483 + 0.0743018i 1.41421 1.00000i 0.218591 + 0.218591i 0.707107 + 0.707107i 2.82843 + 1.00000i 1.52773 1.63280i
497.2 −0.707107 + 0.707107i −1.70711 0.292893i 1.00000i 1.52773 + 1.63280i 1.41421 1.00000i −1.33991 1.33991i 0.707107 + 0.707107i 2.82843 + 1.00000i −2.23483 0.0743018i
497.3 0.707107 0.707107i −0.292893 1.70711i 1.00000i −0.489528 + 2.18183i −1.41421 1.00000i −0.474719 0.474719i −0.707107 0.707107i −2.82843 + 1.00000i 1.19663 + 1.88893i
497.4 0.707107 0.707107i −0.292893 1.70711i 1.00000i 1.19663 1.88893i −1.41421 1.00000i 3.59604 + 3.59604i −0.707107 0.707107i −2.82843 + 1.00000i −0.489528 2.18183i
683.1 −0.707107 0.707107i −1.70711 + 0.292893i 1.00000i −2.23483 0.0743018i 1.41421 + 1.00000i 0.218591 0.218591i 0.707107 0.707107i 2.82843 1.00000i 1.52773 + 1.63280i
683.2 −0.707107 0.707107i −1.70711 + 0.292893i 1.00000i 1.52773 1.63280i 1.41421 + 1.00000i −1.33991 + 1.33991i 0.707107 0.707107i 2.82843 1.00000i −2.23483 + 0.0743018i
683.3 0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i −0.489528 2.18183i −1.41421 + 1.00000i −0.474719 + 0.474719i −0.707107 + 0.707107i −2.82843 1.00000i 1.19663 1.88893i
683.4 0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i 1.19663 + 1.88893i −1.41421 + 1.00000i 3.59604 3.59604i −0.707107 + 0.707107i −2.82843 1.00000i −0.489528 + 2.18183i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 683.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.c 8
3.b odd 2 1 930.2.j.f yes 8
5.c odd 4 1 930.2.j.f yes 8
15.e even 4 1 inner 930.2.j.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.c 8 1.a even 1 1 trivial
930.2.j.c 8 15.e even 4 1 inner
930.2.j.f yes 8 3.b odd 2 1
930.2.j.f yes 8 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} + 48T_{7}^{5} + 117T_{7}^{4} + 52T_{7}^{3} + 8T_{7}^{2} - 8T_{7} + 4$$ T7^8 - 4*T7^7 + 8*T7^6 + 48*T7^5 + 117*T7^4 + 52*T7^3 + 8*T7^2 - 8*T7 + 4 $$T_{11}^{8} + 22T_{11}^{6} + 97T_{11}^{4} + 144T_{11}^{2} + 64$$ T11^8 + 22*T11^6 + 97*T11^4 + 144*T11^2 + 64 $$T_{17}^{8} - 4T_{17}^{7} + 8T_{17}^{6} + 168T_{17}^{5} + 3540T_{17}^{4} - 7424T_{17}^{3} + 15488T_{17}^{2} + 339328T_{17} + 3717184$$ T17^8 - 4*T17^7 + 8*T17^6 + 168*T17^5 + 3540*T17^4 - 7424*T17^3 + 15488*T17^2 + 339328*T17 + 3717184

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{2}$$
$3$ $$(T^{4} + 4 T^{3} + 8 T^{2} + 12 T + 9)^{2}$$
$5$ $$T^{8} + 2 T^{6} + 16 T^{5} + 2 T^{4} + \cdots + 625$$
$7$ $$T^{8} - 4 T^{7} + 8 T^{6} + 48 T^{5} + \cdots + 4$$
$11$ $$T^{8} + 22 T^{6} + 97 T^{4} + 144 T^{2} + \cdots + 64$$
$13$ $$T^{8} + 4 T^{7} + 8 T^{6} + 24 T^{5} + \cdots + 3136$$
$17$ $$T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 3717184$$
$19$ $$T^{8} + 118 T^{6} + 4417 T^{4} + \cdots + 107584$$
$23$ $$T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 37636$$
$29$ $$(T^{4} + 4 T^{3} - 62 T^{2} + 8 T + 248)^{2}$$
$31$ $$(T - 1)^{8}$$
$37$ $$(T^{2} + 4 T + 8)^{4}$$
$41$ $$T^{8} + 196 T^{6} + 11140 T^{4} + \cdots + 1048576$$
$43$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 85264$$
$47$ $$T^{8} - 28 T^{7} + 392 T^{6} + \cdots + 16384$$
$53$ $$T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 15376$$
$59$ $$(T^{4} - 12 T^{3} - 30 T^{2} + 512 T - 1024)^{2}$$
$61$ $$(T^{4} - 28 T^{3} + 186 T^{2} + 720 T - 7792)^{2}$$
$67$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 2408704$$
$71$ $$T^{8} + 286 T^{6} + 22561 T^{4} + \cdots + 583696$$
$73$ $$T^{8} + 36 T^{7} + 648 T^{6} + \cdots + 38416$$
$79$ $$T^{8} + 278 T^{6} + 20289 T^{4} + \cdots + 602176$$
$83$ $$T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 18800896$$
$89$ $$(T^{4} - 18 T^{3} + 113 T^{2} - 288 T + 248)^{2}$$
$97$ $$T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 4129024$$