# Properties

 Label 784.2.m.h Level $784$ Weight $2$ Character orbit 784.m Analytic conductor $6.260$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.20138089353117696.1 Defining polynomial: $$x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64$$ x^12 - 3*x^10 - 2*x^9 + 2*x^8 + 4*x^7 + 2*x^6 + 8*x^5 + 8*x^4 - 16*x^3 - 48*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{10} - 3\nu^{8} - 2\nu^{7} + 2\nu^{6} + 4\nu^{5} + 2\nu^{4} + 8\nu^{3} + 8\nu^{2} - 16\nu - 48 ) / 16$$ (v^10 - 3*v^8 - 2*v^7 + 2*v^6 + 4*v^5 + 2*v^4 + 8*v^3 + 8*v^2 - 16*v - 48) / 16 $$\beta_{2}$$ $$=$$ $$( - \nu^{11} + 10 \nu^{10} + 15 \nu^{9} + 12 \nu^{8} - 10 \nu^{7} - 32 \nu^{6} - 34 \nu^{5} - 52 \nu^{4} - 32 \nu^{3} + 144 \nu^{2} + 272 \nu + 96 ) / 128$$ (-v^11 + 10*v^10 + 15*v^9 + 12*v^8 - 10*v^7 - 32*v^6 - 34*v^5 - 52*v^4 - 32*v^3 + 144*v^2 + 272*v + 96) / 128 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - 3\nu^{9} - 2\nu^{8} + 2\nu^{7} + 4\nu^{6} + 2\nu^{5} + 8\nu^{4} + 8\nu^{3} - 16\nu^{2} - 48\nu ) / 32$$ (v^11 - 3*v^9 - 2*v^8 + 2*v^7 + 4*v^6 + 2*v^5 + 8*v^4 + 8*v^3 - 16*v^2 - 48*v) / 32 $$\beta_{4}$$ $$=$$ $$( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + 64 \nu^{3} + 112 \nu^{2} + 48 \nu + 32 ) / 128$$ (5*v^11 + 6*v^10 + 5*v^9 - 4*v^8 - 14*v^7 - 16*v^6 - 22*v^5 - 12*v^4 + 64*v^3 + 112*v^2 + 48*v + 32) / 128 $$\beta_{5}$$ $$=$$ $$( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} - 12 \nu^{4} - 96 \nu^{3} - 144 \nu^{2} - 80 \nu + 160 ) / 64$$ (-3*v^11 - 10*v^10 - 3*v^9 + 12*v^8 + 18*v^7 + 16*v^6 + 26*v^5 - 12*v^4 - 96*v^3 - 144*v^2 - 80*v + 160) / 64 $$\beta_{6}$$ $$=$$ $$( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} - 12 \nu^{4} - 96 \nu^{3} - 144 \nu^{2} + 48 \nu + 160 ) / 64$$ (-3*v^11 - 10*v^10 - 3*v^9 + 12*v^8 + 18*v^7 + 16*v^6 + 26*v^5 - 12*v^4 - 96*v^3 - 144*v^2 + 48*v + 160) / 64 $$\beta_{7}$$ $$=$$ $$( \nu^{11} + \nu^{10} - 3 \nu^{9} - 5 \nu^{8} - 4 \nu^{7} + 2 \nu^{6} + 6 \nu^{5} + 10 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 48 \nu - 64 ) / 16$$ (v^11 + v^10 - 3*v^9 - 5*v^8 - 4*v^7 + 2*v^6 + 6*v^5 + 10*v^4 + 24*v^3 + 16*v^2 - 48*v - 64) / 16 $$\beta_{8}$$ $$=$$ $$( 13 \nu^{11} + 14 \nu^{10} - 19 \nu^{9} - 44 \nu^{8} - 46 \nu^{7} + 26 \nu^{5} + 196 \nu^{4} + 256 \nu^{3} + 112 \nu^{2} - 336 \nu - 480 ) / 128$$ (13*v^11 + 14*v^10 - 19*v^9 - 44*v^8 - 46*v^7 + 26*v^5 + 196*v^4 + 256*v^3 + 112*v^2 - 336*v - 480) / 128 $$\beta_{9}$$ $$=$$ $$( - 7 \nu^{11} - 14 \nu^{10} + 9 \nu^{9} + 32 \nu^{8} + 34 \nu^{7} + 8 \nu^{6} + 2 \nu^{5} - 52 \nu^{4} - 192 \nu^{3} - 144 \nu^{2} + 176 \nu + 352 ) / 64$$ (-7*v^11 - 14*v^10 + 9*v^9 + 32*v^8 + 34*v^7 + 8*v^6 + 2*v^5 - 52*v^4 - 192*v^3 - 144*v^2 + 176*v + 352) / 64 $$\beta_{10}$$ $$=$$ $$( 15 \nu^{11} + 26 \nu^{10} - 17 \nu^{9} - 68 \nu^{8} - 58 \nu^{7} - 32 \nu^{6} + 30 \nu^{5} + 172 \nu^{4} + 384 \nu^{3} + 272 \nu^{2} - 368 \nu - 672 ) / 128$$ (15*v^11 + 26*v^10 - 17*v^9 - 68*v^8 - 58*v^7 - 32*v^6 + 30*v^5 + 172*v^4 + 384*v^3 + 272*v^2 - 368*v - 672) / 128 $$\beta_{11}$$ $$=$$ $$( - 7 \nu^{11} - 8 \nu^{10} + 9 \nu^{9} + 30 \nu^{8} + 22 \nu^{7} + 4 \nu^{6} - 6 \nu^{5} - 72 \nu^{4} - 144 \nu^{3} - 96 \nu^{2} + 208 \nu + 256 ) / 32$$ (-7*v^11 - 8*v^10 + 9*v^9 + 30*v^8 + 22*v^7 + 4*v^6 - 6*v^5 - 72*v^4 - 144*v^3 - 96*v^2 + 208*v + 256) / 32
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} ) / 2$$ (b6 - b5) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{9} + 2\beta_{7} - \beta_{6} - \beta_{5} + 2 ) / 2$$ (2*b9 + 2*b7 - b6 - b5 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{11} + 2\beta_{10} - \beta_{6} - \beta_{5} + 2\beta_{4} - 4\beta_{2} + 2 ) / 2$$ (2*b11 + 2*b10 - b6 - b5 + 2*b4 - 4*b2 + 2) / 2 $$\nu^{4}$$ $$=$$ $$( 2\beta_{9} + 2\beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 2 ) / 2$$ (2*b9 + 2*b8 - b6 - b5 - 2*b4 + 2) / 2 $$\nu^{5}$$ $$=$$ $$( 2\beta_{11} + 4\beta_{10} + 2\beta_{9} + 2\beta_{7} + \beta_{6} + \beta_{5} + 4\beta_{4} + 2\beta _1 + 2 ) / 2$$ (2*b11 + 4*b10 + 2*b9 + 2*b7 + b6 + b5 + 4*b4 + 2*b1 + 2) / 2 $$\nu^{6}$$ $$=$$ $$( 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 6 \beta _1 + 10 ) / 2$$ (2*b11 - 4*b10 + 2*b9 + 4*b8 + 2*b7 - b6 - 5*b5 + 4*b4 + 4*b3 - 4*b2 + 6*b1 + 10) / 2 $$\nu^{7}$$ $$=$$ $$( 2 \beta_{11} + 8 \beta_{10} + 10 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} + 12 \beta_{3} - 4 \beta_{2} + 2 \beta _1 - 2 ) / 2$$ (2*b11 + 8*b10 + 10*b9 - 4*b8 + 2*b7 - 3*b6 - 7*b5 + 12*b3 - 4*b2 + 2*b1 - 2) / 2 $$\nu^{8}$$ $$=$$ $$( 10 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 8 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta _1 - 6 ) / 2$$ (10*b11 + 4*b10 + 2*b9 + 8*b8 + 6*b7 - 9*b6 + 3*b5 + 8*b4 + 4*b3 - 4*b2 - 2*b1 - 6) / 2 $$\nu^{9}$$ $$=$$ $$( 2 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + 20 \beta_{4} - 4 \beta_{3} - 12 \beta_{2} + 14 \beta _1 - 2 ) / 2$$ (2*b11 + 4*b10 + 6*b9 + 4*b8 - 10*b7 - 7*b6 + 5*b5 + 20*b4 - 4*b3 - 12*b2 + 14*b1 - 2) / 2 $$\nu^{10}$$ $$=$$ $$( 6 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} - \beta_{6} + 3 \beta_{5} - 12 \beta_{4} + 28 \beta_{3} + 20 \beta_{2} + 10 \beta _1 + 10 ) / 2$$ (6*b11 + 4*b10 - 6*b9 + 4*b8 - 6*b7 - b6 + 3*b5 - 12*b4 + 28*b3 + 20*b2 + 10*b1 + 10) / 2 $$\nu^{11}$$ $$=$$ $$( - 6 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 5 \beta_{5} + 52 \beta_{4} + 20 \beta_{3} + 12 \beta_{2} + 6 \beta _1 - 58 ) / 2$$ (-6*b11 - 4*b10 + 6*b9 + 4*b8 - 2*b7 + 17*b6 + 5*b5 + 52*b4 + 20*b3 + 12*b2 + 6*b1 - 58) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 −0.605558 − 1.27801i 1.35309 − 0.411286i −1.40471 + 0.163666i 1.37925 + 0.312504i −1.12465 + 0.857418i 0.402577 + 1.35570i −0.605558 + 1.27801i 1.35309 + 0.411286i −1.40471 − 0.163666i 1.37925 − 0.312504i −1.12465 − 0.857418i 0.402577 − 1.35570i
−1.27801 0.605558i −1.39123 + 1.39123i 1.26660 + 1.54781i −2.16478 2.16478i 2.62048 0.935533i 0 −0.681431 2.74511i 0.871066i 1.45570 + 4.07750i
197.2 −0.411286 + 1.35309i −2.21570 + 2.21570i −1.66169 1.11301i 0.393125 + 0.393125i −2.08675 3.90932i 0 2.18943 1.79064i 6.81864i −0.693620 + 0.370246i
197.3 0.163666 1.40471i 2.05500 2.05500i −1.94643 0.459808i −2.72766 2.72766i −2.55034 3.22301i 0 −0.964462 + 2.65891i 5.44602i −4.27801 + 3.38515i
197.4 0.312504 + 1.37925i 0.599978 0.599978i −1.80468 + 0.862045i −0.974969 0.974969i 1.01502 + 0.640026i 0 −1.75295 2.21972i 2.28005i 1.04005 1.64941i
197.5 0.857418 1.12465i −0.416854 + 0.416854i −0.529667 1.92859i 1.13169 + 1.13169i 0.111396 + 0.826233i 0 −2.62313 1.05792i 2.65247i 2.24309 0.302422i
197.6 1.35570 + 0.402577i −0.631188 + 0.631188i 1.67586 + 1.09155i 2.34259 + 2.34259i −1.10981 + 0.601602i 0 1.83254 + 2.15448i 2.20320i 2.23279 + 4.11894i
589.1 −1.27801 + 0.605558i −1.39123 1.39123i 1.26660 1.54781i −2.16478 + 2.16478i 2.62048 + 0.935533i 0 −0.681431 + 2.74511i 0.871066i 1.45570 4.07750i
589.2 −0.411286 1.35309i −2.21570 2.21570i −1.66169 + 1.11301i 0.393125 0.393125i −2.08675 + 3.90932i 0 2.18943 + 1.79064i 6.81864i −0.693620 0.370246i
589.3 0.163666 + 1.40471i 2.05500 + 2.05500i −1.94643 + 0.459808i −2.72766 + 2.72766i −2.55034 + 3.22301i 0 −0.964462 2.65891i 5.44602i −4.27801 3.38515i
589.4 0.312504 1.37925i 0.599978 + 0.599978i −1.80468 0.862045i −0.974969 + 0.974969i 1.01502 0.640026i 0 −1.75295 + 2.21972i 2.28005i 1.04005 + 1.64941i
589.5 0.857418 + 1.12465i −0.416854 0.416854i −0.529667 + 1.92859i 1.13169 1.13169i 0.111396 0.826233i 0 −2.62313 + 1.05792i 2.65247i 2.24309 + 0.302422i
589.6 1.35570 0.402577i −0.631188 0.631188i 1.67586 1.09155i 2.34259 2.34259i −1.10981 0.601602i 0 1.83254 2.15448i 2.20320i 2.23279 4.11894i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 589.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.h 12
7.b odd 2 1 112.2.m.d 12
7.c even 3 2 784.2.x.m 24
7.d odd 6 2 784.2.x.l 24
16.e even 4 1 inner 784.2.m.h 12
28.d even 2 1 448.2.m.d 12
56.e even 2 1 896.2.m.h 12
56.h odd 2 1 896.2.m.g 12
112.j even 4 1 448.2.m.d 12
112.j even 4 1 896.2.m.h 12
112.l odd 4 1 112.2.m.d 12
112.l odd 4 1 896.2.m.g 12
112.w even 12 2 784.2.x.m 24
112.x odd 12 2 784.2.x.l 24
224.v odd 8 2 7168.2.a.bj 12
224.x even 8 2 7168.2.a.bi 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d 12 7.b odd 2 1
112.2.m.d 12 112.l odd 4 1
448.2.m.d 12 28.d even 2 1
448.2.m.d 12 112.j even 4 1
784.2.m.h 12 1.a even 1 1 trivial
784.2.m.h 12 16.e even 4 1 inner
784.2.x.l 24 7.d odd 6 2
784.2.x.l 24 112.x odd 12 2
784.2.x.m 24 7.c even 3 2
784.2.x.m 24 112.w even 12 2
896.2.m.g 12 56.h odd 2 1
896.2.m.g 12 112.l odd 4 1
896.2.m.h 12 56.e even 2 1
896.2.m.h 12 112.j even 4 1
7168.2.a.bi 12 224.x even 8 2
7168.2.a.bj 12 224.v odd 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{12} + 4 T_{3}^{11} + 8 T_{3}^{10} + 4 T_{3}^{9} + 76 T_{3}^{8} + 288 T_{3}^{7} + 552 T_{3}^{6} + 376 T_{3}^{5} + 164 T_{3}^{4} + 144 T_{3}^{3} + 288 T_{3}^{2} + 192 T_{3} + 64$$ T3^12 + 4*T3^11 + 8*T3^10 + 4*T3^9 + 76*T3^8 + 288*T3^7 + 552*T3^6 + 376*T3^5 + 164*T3^4 + 144*T3^3 + 288*T3^2 + 192*T3 + 64 $$T_{5}^{12} + 4 T_{5}^{11} + 8 T_{5}^{10} - 12 T_{5}^{9} + 140 T_{5}^{8} + 504 T_{5}^{7} + 968 T_{5}^{6} - 1288 T_{5}^{5} + 1828 T_{5}^{4} + 3072 T_{5}^{3} + 4608 T_{5}^{2} - 4608 T_{5} + 2304$$ T5^12 + 4*T5^11 + 8*T5^10 - 12*T5^9 + 140*T5^8 + 504*T5^7 + 968*T5^6 - 1288*T5^5 + 1828*T5^4 + 3072*T5^3 + 4608*T5^2 - 4608*T5 + 2304

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 2 T^{11} + 5 T^{10} - 6 T^{9} + \cdots + 64$$
$3$ $$T^{12} + 4 T^{11} + 8 T^{10} + 4 T^{9} + \cdots + 64$$
$5$ $$T^{12} + 4 T^{11} + 8 T^{10} - 12 T^{9} + \cdots + 2304$$
$7$ $$T^{12}$$
$11$ $$T^{12} + 32 T^{9} + 740 T^{8} + \cdots + 541696$$
$13$ $$T^{12} - 20 T^{9} + 1724 T^{8} + \cdots + 3211264$$
$17$ $$(T^{6} - 4 T^{5} - 44 T^{4} + 200 T^{3} + \cdots - 96)^{2}$$
$19$ $$T^{12} + 76 T^{9} + 2444 T^{8} + \cdots + 2849344$$
$23$ $$T^{12} + 136 T^{10} + \cdots + 10137856$$
$29$ $$T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 8620096$$
$31$ $$(T^{6} - 4 T^{5} - 16 T^{4} + 72 T^{3} + \cdots + 64)^{2}$$
$37$ $$T^{12} + 20 T^{11} + 200 T^{10} + \cdots + 5053504$$
$41$ $$T^{12} + 120 T^{10} + 4704 T^{8} + \cdots + 25600$$
$43$ $$T^{12} - 16 T^{11} + 128 T^{10} + \cdots + 23040000$$
$47$ $$(T^{6} + 8 T^{5} - 104 T^{4} - 1032 T^{3} + \cdots + 19776)^{2}$$
$53$ $$T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 78400$$
$59$ $$T^{12} - 16 T^{11} + \cdots + 119596096$$
$61$ $$T^{12} - 20 T^{11} + 200 T^{10} + \cdots + 256$$
$67$ $$T^{12} - 24 T^{11} + 288 T^{10} + \cdots + 3686400$$
$71$ $$T^{12} + 432 T^{10} + 60960 T^{8} + \cdots + 6553600$$
$73$ $$T^{12} + 616 T^{10} + \cdots + 3050573824$$
$79$ $$(T^{6} - 12 T^{5} - 44 T^{4} + 576 T^{3} + \cdots - 2240)^{2}$$
$83$ $$T^{12} - 20 T^{11} + \cdots + 320093429824$$
$89$ $$T^{12} + 552 T^{10} + \cdots + 35476475904$$
$97$ $$(T^{6} + 24 T^{5} - 60 T^{4} - 2568 T^{3} + \cdots - 39008)^{2}$$