# Properties

 Label 784.2.x.j Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 4 x^{14} + 4 x^{13} - 13 x^{12} + 32 x^{11} - 4 x^{10} - 34 x^{9} + 121 x^{8} - 68 x^{7} - 16 x^{6} + 256 x^{5} - 208 x^{4} + 128 x^{3} + 256 x^{2} - 256 x + 256$$ x^16 - 2*x^15 + 4*x^14 + 4*x^13 - 13*x^12 + 32*x^11 - 4*x^10 - 34*x^9 + 121*x^8 - 68*x^7 - 16*x^6 + 256*x^5 - 208*x^4 + 128*x^3 + 256*x^2 - 256*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{12} + \beta_{9}) q^{2} + ( - \beta_{13} + \beta_{8}) q^{3} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{2} - 1) q^{4} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}) q^{5} + (\beta_{15} + \beta_{13} + \beta_{9} + \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{6} + (\beta_{15} - \beta_{14} + \beta_{13} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{8} + (\beta_{15} - \beta_{14} - \beta_{7} + \beta_{4} - \beta_{2}) q^{9}+O(q^{10})$$ q + (b12 + b9) * q^2 + (-b13 + b8) * q^3 + (b11 - b10 + b8 - b2 - 1) * q^4 + (b10 - b8 + b7 + b6 + b5 - b4) * q^5 + (b15 + b13 + b9 + b5 - b4 + b3 + 2) * q^6 + (b15 - b14 + b13 + b5 + b4 + b3 + b1 - 1) * q^8 + (b15 - b14 - b7 + b4 - b2) * q^9 $$q + (\beta_{12} + \beta_{9}) q^{2} + ( - \beta_{13} + \beta_{8}) q^{3} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{2} - 1) q^{4} + (\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}) q^{5} + (\beta_{15} + \beta_{13} + \beta_{9} + \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{6} + (\beta_{15} - \beta_{14} + \beta_{13} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{8} + (\beta_{15} - \beta_{14} - \beta_{7} + \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{13} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{3} + 2) q^{10} - 2 \beta_{3} q^{11} + ( - \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{12}+ \cdots + (2 \beta_{14} + 4 \beta_{13} - 4 \beta_{6} + 4 \beta_{5} - 6 \beta_{4} + 6) q^{99}+O(q^{100})$$ q + (b12 + b9) * q^2 + (-b13 + b8) * q^3 + (b11 - b10 + b8 - b2 - 1) * q^4 + (b10 - b8 + b7 + b6 + b5 - b4) * q^5 + (b15 + b13 + b9 + b5 - b4 + b3 + 2) * q^6 + (b15 - b14 + b13 + b5 + b4 + b3 + b1 - 1) * q^8 + (b15 - b14 - b7 + b4 - b2) * q^9 + (-b13 + 2*b10 - b8 + b7 - b3 + 2) * q^10 - 2*b3 * q^11 + (-b15 - b14 + 2*b12 + b11 + b10 + 2*b9 + b7 + b5 - b4 - b2 - b1) * q^12 + (-b13 - 2*b9 + b6 - b5 + b4 + 2*b1 - 1) * q^13 + (b15 + b14 + b13 - b9 + b6 + b5 + b3 + b1 - 2) * q^15 + (-b15 + b14 - b12 + b11 + b10 - b9 + b8 - b7 - b6 + b5 + b4 + b2 - b1) * q^16 + (-b13 - b12 - b11 + 4*b10 - b8 + 4) * q^17 + (-b13 + b11 - b10 + b8 + b7 + b3 + b2 - 1) * q^18 + (2*b14 - b12 - b11 + 2*b10 - b9 - 2*b8 - 2*b7 + 2*b6 - 2*b5 + 2*b4 + 2*b2 + b1) * q^19 + (-2*b13 + b9 - b6 - 2*b5 + 2*b4 + 2) * q^20 + (2*b15 - 2*b13 - 2*b5 + 2*b4 + 2*b3) * q^22 + (b15 - b14 - b12 + b11 - b9 - b8 + b6 - b5 - b2 - b1) * q^23 + (b13 - b12 + 3*b11 - b10 + b8 + b7 - b3 - b2 - 1) * q^24 + (2*b13 - 2*b12 - 2*b11 + 2*b8 - b7 + b3 + b2) * q^25 + (b15 + 2*b14 - 2*b11 - b8 - 3*b7 + b6 + b5 + 3*b4 + 2*b2 + 2*b1) * q^26 + (-2*b15 + b13 - b9 + b6 + b5 - 2*b4 - 2*b3 - b1 - 2) * q^27 + (-2*b14 + 2*b13 + 2*b9 - 2*b6 + 2*b5 + b4 - 2*b1 - 1) * q^29 + (-b12 - 2*b11 + 4*b10 - b9 - b8 + b6 + 2*b5 + 2*b1) * q^30 + (b13 + b12 + 3*b11 + 3*b8) * q^31 + (2*b13 - 2*b12 - 2*b11 + 4*b10 + 4) * q^32 + (-2*b15 - 2*b14 - 2*b11 + 4*b10 - 2*b8 + 2*b6 - 2*b2 + 2*b1) * q^33 + (-b15 - b14 - b13 + 3*b9 - b6 - b5 - b4 - b3 - b1 + 3) * q^34 + (b14 + 2*b13 - b6 + 2*b5 - b1 + 3) * q^36 + (4*b15 + 2*b12 - 2*b11 + 3*b10 + 2*b9 - 2*b8 + 3*b7 + 2*b6 + 2*b5 - 3*b4 + 2*b1) * q^37 + (-2*b13 - 3*b11 - b10 + b8 - 2*b3 - b2 - 1) * q^38 + (-3*b13 + 3*b12 + b11 - b8 - 3*b3 - 3*b2) * q^39 + (-b15 - b14 + b12 + b11 - 5*b10 + b9 + 3*b8 - b7 - 3*b6 - b5 + b4 - b2 - b1) * q^40 + (-4*b15 + 4*b14 - 3*b13 - 3*b9 + b6 - 3*b5 - 4*b3 - b1) * q^41 + (-4*b15 - 2*b9 - 4*b4 - 4*b3 - 2*b1 - 4) * q^43 + (-2*b15 - 4*b10 + 2*b8 + 2*b7 - 2*b6 + 2*b5 - 2*b4) * q^44 + (-b13 - b10 + b8 + b7 - 1) * q^45 + (-2*b13 + b12 - 2*b10 - b8 + 4*b7 + 2*b2 - 2) * q^46 + (-b12 + 3*b11 - 4*b10 - b9 + 3*b8 - 3*b6 + b5 - 3*b1) * q^47 + (2*b15 - 2*b14 - 2*b6 + 6*b4 + 2*b3 - 2) * q^48 + (b15 - b14 + 3*b13 + 2*b9 - b6 + 3*b5 - 7*b4 + b3 - 3*b1 - 1) * q^50 + (2*b14 - 2*b12 - 2*b11 + 2*b10 - 2*b9 + 2*b8 - 2*b7 - 2*b6 + 2*b5 + 2*b4 + 2*b2 + 2*b1) * q^51 + (-2*b13 + b12 - 2*b11 + 4*b10 + 3*b8 - 2*b7 - 2*b2 + 4) * q^52 + (-2*b12 + 2*b11 + b10 + b7 + 1) * q^53 + (3*b15 + b14 - b12 - b11 + 3*b10 - b9 - 3*b8 + b7 + 3*b6 - b5 - b4 + b2 + b1) * q^54 + (2*b6 - 2*b1) * q^55 + (-b15 + b14 - 2*b13 - 2*b9 - 2*b6 - 2*b5 - 2*b4 - b3 + 2*b1) * q^57 + (-2*b15 - 3*b12 + 4*b11 - 3*b9 + 3*b8 + 2*b7 - 3*b6 - 2*b5 - 2*b4 - 4*b1) * q^58 + (-3*b13 - 2*b10 - 3*b8 - 2*b7 - 2*b3 - 2) * q^59 + (-b13 - 3*b12 - b11 + 5*b10 - b8 - 3*b7 - b3 + b2 + 5) * q^60 + (-b12 - b11 - 3*b10 - b9 + 3*b7 - 3*b4 + b1) * q^61 + (3*b15 + b14 + 3*b13 + 3*b9 + b6 + 3*b5 + 3*b4 + 3*b3 + b1 - 3) * q^62 + (-2*b14 + 4*b9 - 2*b6 - 4*b4 - 2*b1 - 2) * q^64 + (-b15 - b14 + 2*b12 + 2*b11 - 8*b10 + 2*b9 + 2*b8 - 2*b6 - 2*b5 - b2 - 2*b1) * q^65 + (-2*b12 + 2*b11 - 2*b10 - 4*b7 - 4*b3 + 2*b2 - 2) * q^66 + (-4*b13 + 4*b12 + 4*b11 - 2*b10 + 4*b8 + 2*b7 - 4*b2 - 2) * q^67 + (-2*b14 + 2*b12 + 4*b11 - 6*b10 + 2*b9 + 2*b8 - 2*b6 - 2*b5 - 2*b2 - 4*b1) * q^68 + (-4*b15 + b13 + b9 + b6 + b5 - 6*b4 - 4*b3 + b1 - 6) * q^69 + (-2*b15 + 2*b14 - 6*b13 - 6*b9 - 6*b5 + 10*b4 - 2*b3) * q^71 + (-b15 - b14 + 2*b12 - b11 + 5*b10 + 2*b9 + b7 - b5 - b4 - b2 + b1) * q^72 + (-2*b13 + 2*b12 - 2*b11 + 2*b8 + 6*b7 - 2*b3 - 2*b2) * q^73 + (-6*b13 - b12 + 2*b11 + 2*b10 - b8 + 2*b7 + 2*b3 - 2*b2 + 2) * q^74 + (-2*b15 - 4*b12 + 4*b11 - 6*b10 - 4*b9 + 5*b8 - 6*b7 - 5*b6 - 5*b5 + 6*b4 - 4*b1) * q^75 + (3*b15 - b14 - b13 - b5 - 5*b4 + 3*b3 + b1 + 3) * q^76 + (2*b15 - 4*b13 - b9 + 3*b6 - 4*b5 + 6*b4 + 2*b3 + 6*b1) * q^78 + (2*b15 + 2*b14 + 4*b12 + 2*b11 - 6*b10 + 4*b9 + 2*b8 - 2*b6 - 4*b5 + 2*b2 - 2*b1) * q^79 + (4*b13 + 2*b12 + 2*b11 - 4*b10 + 2*b8 - 2*b7 + 2*b3 - 4) * q^80 + (2*b13 + 2*b12 - 2*b11 - 5*b10 - 2*b8 - b3 + b2 - 5) * q^81 + (5*b15 - b14 + b12 - 7*b11 + b10 + b9 - 3*b8 - b7 + 3*b6 - 3*b5 + b4 - b2 + 7*b1) * q^82 + (b13 - b6 + b5) * q^83 + (2*b15 + 6*b13 + 6*b6 + 6*b5 - 8*b4 + 2*b3 - 8) * q^85 + (4*b15 + 2*b14 - 4*b12 - 2*b11 + 2*b10 - 4*b9 - 6*b8 + 6*b6 - 4*b5 + 2*b2 + 2*b1) * q^86 + (7*b13 - 7*b12 + b11 - b8 + 2*b3 + 2*b2) * q^87 + (4*b13 + 2*b12 + 4*b10 - 2*b8 - 4*b7 + 4) * q^88 + (4*b15 - 4*b14 + 6*b12 + 6*b9 + 6*b7 + 6*b5 - 6*b4 - 4*b2) * q^89 + (b15 + b13 - b6 + b5 - b4 + b3 + 2) * q^90 + (-b15 + 3*b14 - b13 - 3*b9 - 3*b6 - b5 + b4 - b3 - b1 + 5) * q^92 + (-6*b14 + 4*b12 + 4*b11 + 4*b9 + 4*b8 - 4*b6 + 4*b5 - 6*b2 - 4*b1) * q^93 + (3*b13 + b12 - b11 + 3*b10 - b8 + 3*b7 + 3*b3 + b2 + 3) * q^94 + (3*b13 + 3*b12 + b11 - 4*b10 + b8 - b3 + b2 - 4) * q^95 + (-4*b15 + 2*b14 - 4*b12 + 2*b11 - 2*b10 - 4*b9 + 6*b8 - 6*b6 + 2*b2 - 2*b1) * q^96 + (-2*b15 - 2*b14 + 3*b13 - 3*b9 - 3*b6 + 3*b5 - 2*b3 - 3*b1 + 4) * q^97 + (2*b14 + 4*b13 - 4*b6 + 4*b5 - 6*b4 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 32 q^{6} - 8 q^{8}+O(q^{10})$$ 16 * q - 2 * q^2 - 4 * q^4 - 4 * q^5 + 32 * q^6 - 8 * q^8 $$16 q - 2 q^{2} - 4 q^{4} - 4 q^{5} + 32 q^{6} - 8 q^{8} + 12 q^{10} - 12 q^{12} - 16 q^{15} - 8 q^{16} + 24 q^{17} - 6 q^{18} - 12 q^{19} + 16 q^{20} - 8 q^{22} + 8 q^{26} - 24 q^{27} - 32 q^{29} - 20 q^{30} + 16 q^{31} + 28 q^{32} - 24 q^{33} + 24 q^{34} + 48 q^{36} - 16 q^{37} - 16 q^{38} + 28 q^{40} - 64 q^{43} + 32 q^{44} - 8 q^{45} - 20 q^{46} + 24 q^{47} - 40 q^{48} - 28 q^{50} - 8 q^{51} + 32 q^{52} + 8 q^{53} - 16 q^{54} - 12 q^{58} - 28 q^{59} + 28 q^{60} + 28 q^{61} - 40 q^{62} - 64 q^{64} + 48 q^{65} - 16 q^{66} + 28 q^{68} - 88 q^{69} - 44 q^{72} + 4 q^{74} + 28 q^{75} + 48 q^{76} + 24 q^{78} + 24 q^{79} - 12 q^{80} - 40 q^{81} + 4 q^{82} - 80 q^{85} + 40 q^{88} + 32 q^{90} + 72 q^{92} - 16 q^{93} + 28 q^{94} - 16 q^{95} + 8 q^{96} + 64 q^{97} + 96 q^{99}+O(q^{100})$$ 16 * q - 2 * q^2 - 4 * q^4 - 4 * q^5 + 32 * q^6 - 8 * q^8 + 12 * q^10 - 12 * q^12 - 16 * q^15 - 8 * q^16 + 24 * q^17 - 6 * q^18 - 12 * q^19 + 16 * q^20 - 8 * q^22 + 8 * q^26 - 24 * q^27 - 32 * q^29 - 20 * q^30 + 16 * q^31 + 28 * q^32 - 24 * q^33 + 24 * q^34 + 48 * q^36 - 16 * q^37 - 16 * q^38 + 28 * q^40 - 64 * q^43 + 32 * q^44 - 8 * q^45 - 20 * q^46 + 24 * q^47 - 40 * q^48 - 28 * q^50 - 8 * q^51 + 32 * q^52 + 8 * q^53 - 16 * q^54 - 12 * q^58 - 28 * q^59 + 28 * q^60 + 28 * q^61 - 40 * q^62 - 64 * q^64 + 48 * q^65 - 16 * q^66 + 28 * q^68 - 88 * q^69 - 44 * q^72 + 4 * q^74 + 28 * q^75 + 48 * q^76 + 24 * q^78 + 24 * q^79 - 12 * q^80 - 40 * q^81 + 4 * q^82 - 80 * q^85 + 40 * q^88 + 32 * q^90 + 72 * q^92 - 16 * q^93 + 28 * q^94 - 16 * q^95 + 8 * q^96 + 64 * q^97 + 96 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 4 x^{14} + 4 x^{13} - 13 x^{12} + 32 x^{11} - 4 x^{10} - 34 x^{9} + 121 x^{8} - 68 x^{7} - 16 x^{6} + 256 x^{5} - 208 x^{4} + 128 x^{3} + 256 x^{2} - 256 x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( 46 \nu^{15} + 465 \nu^{14} - 180 \nu^{13} + 992 \nu^{12} + 3150 \nu^{11} - 1905 \nu^{10} + 6738 \nu^{9} + 9600 \nu^{8} - 7980 \nu^{7} + 25377 \nu^{6} + 18090 \nu^{5} - 12840 \nu^{4} + \cdots + 70080 ) / 3008$$ (46*v^15 + 465*v^14 - 180*v^13 + 992*v^12 + 3150*v^11 - 1905*v^10 + 6738*v^9 + 9600*v^8 - 7980*v^7 + 25377*v^6 + 18090*v^5 - 12840*v^4 + 60728*v^3 + 17760*v^2 - 5280*v + 70080) / 3008 $$\beta_{2}$$ $$=$$ $$( - 99 \nu^{15} + 1526 \nu^{14} - 900 \nu^{13} + 1012 \nu^{12} + 10439 \nu^{11} - 9008 \nu^{10} + 12164 \nu^{9} + 33430 \nu^{8} - 36939 \nu^{7} + 56620 \nu^{6} + 63096 \nu^{5} + \cdots + 166912 ) / 6016$$ (-99*v^15 + 1526*v^14 - 900*v^13 + 1012*v^12 + 10439*v^11 - 9008*v^10 + 12164*v^9 + 33430*v^8 - 36939*v^7 + 56620*v^6 + 63096*v^5 - 72848*v^4 + 146096*v^3 + 54208*v^2 - 64000*v + 166912) / 6016 $$\beta_{3}$$ $$=$$ $$( 117 \nu^{15} - 26 \nu^{14} + 4 \nu^{13} + 684 \nu^{12} - 1057 \nu^{11} + 528 \nu^{10} + 1724 \nu^{9} - 4130 \nu^{8} + 4141 \nu^{7} + 1244 \nu^{6} - 9520 \nu^{5} + 11064 \nu^{4} - 8176 \nu^{3} + \cdots - 20608 ) / 6016$$ (117*v^15 - 26*v^14 + 4*v^13 + 684*v^12 - 1057*v^11 + 528*v^10 + 1724*v^9 - 4130*v^8 + 4141*v^7 + 1244*v^6 - 9520*v^5 + 11064*v^4 - 8176*v^3 - 11424*v^2 + 11648*v - 20608) / 6016 $$\beta_{4}$$ $$=$$ $$( - 44 \nu^{15} - 79 \nu^{14} + 23 \nu^{13} - 438 \nu^{12} - 520 \nu^{11} + 263 \nu^{10} - 2213 \nu^{9} - 1540 \nu^{8} + 1098 \nu^{7} - 7041 \nu^{6} - 2899 \nu^{5} + 1484 \nu^{4} + \cdots - 16224 ) / 1504$$ (-44*v^15 - 79*v^14 + 23*v^13 - 438*v^12 - 520*v^11 + 263*v^10 - 2213*v^9 - 1540*v^8 + 1098*v^7 - 7041*v^6 - 2899*v^5 + 1484*v^4 - 15052*v^3 - 2896*v^2 + 48*v - 16224) / 1504 $$\beta_{5}$$ $$=$$ $$( 243 \nu^{15} - 336 \nu^{14} + 688 \nu^{13} + 1276 \nu^{12} - 2687 \nu^{11} + 4994 \nu^{10} + 2308 \nu^{9} - 9590 \nu^{8} + 19519 \nu^{7} - 1574 \nu^{6} - 17256 \nu^{5} + 42936 \nu^{4} + \cdots - 31232 ) / 6016$$ (243*v^15 - 336*v^14 + 688*v^13 + 1276*v^12 - 2687*v^11 + 4994*v^10 + 2308*v^9 - 9590*v^8 + 19519*v^7 - 1574*v^6 - 17256*v^5 + 42936*v^4 - 18080*v^3 - 14240*v^2 + 45248*v - 31232) / 6016 $$\beta_{6}$$ $$=$$ $$( - 70 \nu^{15} + 26 \nu^{14} - 51 \nu^{13} - 402 \nu^{12} + 258 \nu^{11} - 434 \nu^{10} - 1301 \nu^{9} + 1028 \nu^{8} - 1838 \nu^{7} - 2748 \nu^{6} + 1953 \nu^{5} - 4484 \nu^{4} - 3480 \nu^{3} + \cdots - 1952 ) / 1504$$ (-70*v^15 + 26*v^14 - 51*v^13 - 402*v^12 + 258*v^11 - 434*v^10 - 1301*v^9 + 1028*v^8 - 1838*v^7 - 2748*v^6 + 1953*v^5 - 4484*v^4 - 3480*v^3 + 1648*v^2 - 4880*v - 1952) / 1504 $$\beta_{7}$$ $$=$$ $$( 429 \nu^{15} - 1474 \nu^{14} + 1080 \nu^{13} + 1380 \nu^{12} - 11145 \nu^{11} + 10584 \nu^{10} - 3768 \nu^{9} - 38706 \nu^{8} + 45765 \nu^{7} - 38052 \nu^{6} - 74700 \nu^{5} + \cdots - 167808 ) / 6016$$ (429*v^15 - 1474*v^14 + 1080*v^13 + 1380*v^12 - 11145*v^11 + 10584*v^10 - 3768*v^9 - 38706*v^8 + 45765*v^7 - 38052*v^6 - 74700*v^5 + 98472*v^4 - 125232*v^3 - 70464*v^2 + 94848*v - 167808) / 6016 $$\beta_{8}$$ $$=$$ $$( 119 \nu^{15} - 392 \nu^{14} + 317 \nu^{13} + 392 \nu^{12} - 2939 \nu^{11} + 3022 \nu^{10} - 989 \nu^{9} - 10076 \nu^{8} + 12863 \nu^{7} - 10312 \nu^{6} - 18863 \nu^{5} + 27626 \nu^{4} + \cdots - 44960 ) / 1504$$ (119*v^15 - 392*v^14 + 317*v^13 + 392*v^12 - 2939*v^11 + 3022*v^10 - 989*v^9 - 10076*v^8 + 12863*v^7 - 10312*v^6 - 18863*v^5 + 27626*v^4 - 33752*v^3 - 17616*v^2 + 26720*v - 44960) / 1504 $$\beta_{9}$$ $$=$$ $$( - 262 \nu^{15} + 53 \nu^{14} - 254 \nu^{13} - 1980 \nu^{12} + 826 \nu^{11} - 2085 \nu^{10} - 7108 \nu^{9} + 3896 \nu^{8} - 8848 \nu^{7} - 16719 \nu^{6} + 7432 \nu^{5} - 22944 \nu^{4} + \cdots - 17920 ) / 3008$$ (-262*v^15 + 53*v^14 - 254*v^13 - 1980*v^12 + 826*v^11 - 2085*v^10 - 7108*v^9 + 3896*v^8 - 8848*v^7 - 16719*v^6 + 7432*v^5 - 22944*v^4 - 24520*v^3 + 5760*v^2 - 26752*v - 17920) / 3008 $$\beta_{10}$$ $$=$$ $$( - 293 \nu^{15} + 274 \nu^{14} - 476 \nu^{13} - 1872 \nu^{12} + 2361 \nu^{11} - 3800 \nu^{10} - 5124 \nu^{9} + 9062 \nu^{8} - 15917 \nu^{7} - 6660 \nu^{6} + 16912 \nu^{5} + \cdots + 15872 ) / 3008$$ (-293*v^15 + 274*v^14 - 476*v^13 - 1872*v^12 + 2361*v^11 - 3800*v^10 - 5124*v^9 + 9062*v^8 - 15917*v^7 - 6660*v^6 + 16912*v^5 - 38028*v^4 + 2112*v^3 + 13376*v^2 - 41536*v + 15872) / 3008 $$\beta_{11}$$ $$=$$ $$( - 607 \nu^{15} + 2182 \nu^{14} - 1536 \nu^{13} - 1900 \nu^{12} + 16211 \nu^{11} - 15128 \nu^{10} + 5760 \nu^{9} + 55318 \nu^{8} - 65511 \nu^{7} + 55660 \nu^{6} + 107180 \nu^{5} + \cdots + 243072 ) / 6016$$ (-607*v^15 + 2182*v^14 - 1536*v^13 - 1900*v^12 + 16211*v^11 - 15128*v^10 + 5760*v^9 + 55318*v^8 - 65511*v^7 + 55660*v^6 + 107180*v^5 - 140776*v^4 + 181968*v^3 + 101920*v^2 - 135296*v + 243072) / 6016 $$\beta_{12}$$ $$=$$ $$( - 619 \nu^{15} + 806 \nu^{14} - 1252 \nu^{13} - 3532 \nu^{12} + 6447 \nu^{11} - 10352 \nu^{10} - 7196 \nu^{9} + 23878 \nu^{8} - 42643 \nu^{7} + 1292 \nu^{6} + 43576 \nu^{5} + \cdots + 79360 ) / 6016$$ (-619*v^15 + 806*v^14 - 1252*v^13 - 3532*v^12 + 6447*v^11 - 10352*v^10 - 7196*v^9 + 23878*v^8 - 42643*v^7 + 1292*v^6 + 43576*v^5 - 97456*v^4 + 41392*v^3 + 32288*v^2 - 102400*v + 79360) / 6016 $$\beta_{13}$$ $$=$$ $$( - 767 \nu^{15} + 630 \nu^{14} - 1384 \nu^{13} - 4860 \nu^{12} + 5843 \nu^{11} - 10856 \nu^{10} - 12952 \nu^{9} + 22646 \nu^{8} - 44359 \nu^{7} - 15508 \nu^{6} + 42084 \nu^{5} + \cdots + 49536 ) / 6016$$ (-767*v^15 + 630*v^14 - 1384*v^13 - 4860*v^12 + 5843*v^11 - 10856*v^10 - 12952*v^9 + 22646*v^8 - 44359*v^7 - 15508*v^6 + 42084*v^5 - 104616*v^4 + 8144*v^3 + 34784*v^2 - 113792*v + 49536) / 6016 $$\beta_{14}$$ $$=$$ $$( - 108 \nu^{15} - 164 \nu^{14} - 29 \nu^{13} - 1058 \nu^{12} - 926 \nu^{11} - 68 \nu^{10} - 4791 \nu^{9} - 2276 \nu^{8} - 330 \nu^{7} - 14330 \nu^{6} - 4253 \nu^{5} - 3980 \nu^{4} + \cdots - 26560 ) / 752$$ (-108*v^15 - 164*v^14 - 29*v^13 - 1058*v^12 - 926*v^11 - 68*v^10 - 4791*v^9 - 2276*v^8 - 330*v^7 - 14330*v^6 - 4253*v^5 - 3980*v^4 - 28238*v^3 - 4784*v^2 - 8496*v - 26560) / 752 $$\beta_{15}$$ $$=$$ $$( 1311 \nu^{15} - 1764 \nu^{14} + 2296 \nu^{13} + 7404 \nu^{12} - 14283 \nu^{11} + 19662 \nu^{10} + 15924 \nu^{9} - 52110 \nu^{8} + 81219 \nu^{7} + 2382 \nu^{6} - 97080 \nu^{5} + \cdots - 145920 ) / 6016$$ (1311*v^15 - 1764*v^14 + 2296*v^13 + 7404*v^12 - 14283*v^11 + 19662*v^10 + 15924*v^9 - 52110*v^8 + 81219*v^7 + 2382*v^6 - 97080*v^5 + 186216*v^4 - 75744*v^3 - 82656*v^2 + 188672*v - 145920) / 6016
 $$\nu$$ $$=$$ $$( - \beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 ) / 2$$ (-b15 - b14 + b12 + b11 - b10 + b9 + b8 + b7 - b6 + b5 - b4 - b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_{2} - 1 ) / 2$$ (b13 - b12 + b11 - b10 - b8 + b7 + b3 - b2 - 1) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{15} + \beta_{14} - 3\beta_{13} - \beta_{9} - \beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 3 ) / 2$$ (-b15 + b14 - 3*b13 - b9 - b6 - 3*b5 + b4 - b3 + b1 - 3) / 2 $$\nu^{4}$$ $$=$$ $$( 3 \beta_{15} + \beta_{14} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - 5 \beta_{7} + \beta_{6} - 3 \beta_{5} + 5 \beta_{4} + \beta_{2} + 3 \beta_1 ) / 2$$ (3*b15 + b14 + b12 - 3*b11 - b10 + b9 - b8 - 5*b7 + b6 - 3*b5 + 5*b4 + b2 + 3*b1) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{13} + 3\beta_{12} + 3\beta_{11} - 3\beta_{10} + 7\beta_{8} - \beta_{7} - \beta_{3} + \beta_{2} - 3 ) / 2$$ (b13 + 3*b12 + 3*b11 - 3*b10 + 7*b8 - b7 - b3 + b2 - 3) / 2 $$\nu^{6}$$ $$=$$ $$( \beta_{15} - 3\beta_{14} + 9\beta_{13} - 3\beta_{9} - \beta_{6} + 9\beta_{5} + 5\beta_{4} + \beta_{3} - 3\beta _1 - 1 ) / 2$$ (b15 - 3*b14 + 9*b13 - 3*b9 - b6 + 9*b5 + 5*b4 + b3 - 3*b1 - 1) / 2 $$\nu^{7}$$ $$=$$ $$( - 9 \beta_{15} + 5 \beta_{14} - 13 \beta_{12} - \beta_{11} + 3 \beta_{10} - 13 \beta_{9} + \beta_{8} + 3 \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{4} + 5 \beta_{2} + \beta_1 ) / 2$$ (-9*b15 + 5*b14 - 13*b12 - b11 + 3*b10 - 13*b9 + b8 + 3*b7 - b6 + b5 - 3*b4 + 5*b2 + b1) / 2 $$\nu^{8}$$ $$=$$ $$( -5\beta_{13} - 3\beta_{12} - 15\beta_{11} + 7\beta_{10} - 9\beta_{8} - 17\beta_{7} + 5\beta_{3} - 3\beta_{2} + 7 ) / 2$$ (-5*b13 - 3*b12 - 15*b11 + 7*b10 - 9*b8 - 17*b7 + 5*b3 - 3*b2 + 7) / 2 $$\nu^{9}$$ $$=$$ $$( 11 \beta_{15} + 7 \beta_{14} + \beta_{13} + 21 \beta_{9} + 19 \beta_{6} + \beta_{5} - 33 \beta_{4} + 11 \beta_{3} + 15 \beta _1 - 7 ) / 2$$ (11*b15 + 7*b14 + b13 + 21*b9 + 19*b6 + b5 - 33*b4 + 11*b3 + 15*b1 - 7) / 2 $$\nu^{10}$$ $$=$$ $$( 21 \beta_{15} - 19 \beta_{14} + 21 \beta_{12} + 11 \beta_{11} + 37 \beta_{10} + 21 \beta_{9} - 3 \beta_{8} + 11 \beta_{7} + 3 \beta_{6} + 13 \beta_{5} - 11 \beta_{4} - 19 \beta_{2} - 11 \beta_1 ) / 2$$ (21*b15 - 19*b14 + 21*b12 + 11*b11 + 37*b10 + 21*b9 - 3*b8 + 11*b7 + 3*b6 + 13*b5 - 11*b4 - 19*b2 - 11*b1) / 2 $$\nu^{11}$$ $$=$$ $$( 11 \beta_{13} - 25 \beta_{12} - \beta_{11} + 19 \beta_{10} - 15 \beta_{8} + 39 \beta_{7} - 31 \beta_{3} + 25 \beta_{2} + 19 ) / 2$$ (11*b13 - 25*b12 - b11 + 19*b10 - 15*b8 + 39*b7 - 31*b3 + 25*b2 + 19) / 2 $$\nu^{12}$$ $$=$$ $$( - 43 \beta_{15} - 17 \beta_{14} - 53 \beta_{13} - 49 \beta_{9} - 25 \beta_{6} - 53 \beta_{5} + 43 \beta_{4} - 43 \beta_{3} - 67 \beta _1 + 71 ) / 2$$ (-43*b15 - 17*b14 - 53*b13 - 49*b9 - 25*b6 - 53*b5 + 43*b4 - 43*b3 - 67*b1 + 71) / 2 $$\nu^{13}$$ $$=$$ $$( - 17 \beta_{15} + 23 \beta_{14} + 37 \beta_{12} + 13 \beta_{11} - 157 \beta_{10} + 37 \beta_{9} - 23 \beta_{8} + 25 \beta_{7} + 23 \beta_{6} + 25 \beta_{5} - 25 \beta_{4} + 23 \beta_{2} - 13 \beta_1 ) / 2$$ (-17*b15 + 23*b14 + 37*b12 + 13*b11 - 157*b10 + 37*b9 - 23*b8 + 25*b7 + 23*b6 + 25*b5 - 25*b4 + 23*b2 - 13*b1) / 2 $$\nu^{14}$$ $$=$$ $$( - 57 \beta_{13} + 69 \beta_{12} + 123 \beta_{11} - 15 \beta_{10} + 33 \beta_{8} + 67 \beta_{7} + 111 \beta_{3} - 75 \beta_{2} - 15 ) / 2$$ (-57*b13 + 69*b12 + 123*b11 - 15*b10 + 33*b8 + 67*b7 + 111*b3 - 75*b2 - 15) / 2 $$\nu^{15}$$ $$=$$ $$( 49 \beta_{15} + 11 \beta_{14} + 79 \beta_{13} + 109 \beta_{9} - 167 \beta_{6} + 79 \beta_{5} + 35 \beta_{4} + 49 \beta_{3} + 103 \beta _1 - 85 ) / 2$$ (49*b15 + 11*b14 + 79*b13 + 109*b9 - 167*b6 + 79*b5 + 35*b4 + 49*b3 + 103*b1 - 85) / 2

## Character values

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

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$\beta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 −1.33068 − 0.478848i 0.640069 + 1.26108i 0.840224 − 1.13755i 1.21641 + 0.721349i −1.40526 − 0.158880i 0.772089 − 1.18485i 0.0165007 − 1.41412i 0.250645 + 1.39183i −1.40526 + 0.158880i 0.772089 + 1.18485i 0.0165007 + 1.41412i 0.250645 − 1.39183i −1.33068 + 0.478848i 0.640069 − 1.26108i 0.840224 + 1.13755i 1.21641 − 0.721349i
−1.19412 + 0.757684i −0.538823 2.01092i 0.851831 1.80953i −0.129523 + 0.483385i 2.16706 + 1.99301i 0 0.353863 + 2.80620i −1.15537 + 0.667056i −0.211588 0.675356i
165.2 −1.14477 0.830359i 0.261809 + 0.977085i 0.621007 + 1.90114i −0.317608 + 1.18533i 0.511620 1.33594i 0 0.867721 2.69204i 1.71192 0.988380i 1.34784 1.09320i
165.3 0.105414 1.41028i 0.719263 + 2.68432i −1.97778 0.297327i 0.229791 0.857592i 3.86147 0.731395i 0 −0.627801 + 2.75787i −4.09018 + 2.36147i −1.18522 0.414472i
165.4 0.867450 + 1.11693i −0.442248 1.65049i −0.495063 + 1.93776i 0.949390 3.54317i 1.45986 1.92568i 0 −2.59378 + 1.12796i 0.0695300 0.0401432i 4.78102 2.01312i
373.1 −1.27404 + 0.613848i −2.68432 0.719263i 1.24638 1.56414i −0.857592 + 0.229791i 3.86147 0.731395i 0 −0.627801 + 2.75787i 4.09018 + 2.36147i 0.951553 0.819195i
373.2 −0.146726 + 1.40658i −0.977085 0.261809i −1.95694 0.412764i 1.18533 0.317608i 0.511620 1.33594i 0 0.867721 2.69204i −1.71192 0.988380i 0.272823 + 1.71386i
373.3 0.533564 1.30970i 1.65049 + 0.442248i −1.43062 1.39762i −3.54317 + 0.949390i 1.45986 1.92568i 0 −2.59378 + 1.12796i −0.0695300 0.0401432i −0.647096 + 5.14705i
373.4 1.25323 + 0.655294i 2.01092 + 0.538823i 1.14118 + 1.64247i 0.483385 0.129523i 2.16706 + 1.99301i 0 0.353863 + 2.80620i 1.15537 + 0.667056i 0.690669 + 0.154437i
557.1 −1.27404 0.613848i −2.68432 + 0.719263i 1.24638 + 1.56414i −0.857592 0.229791i 3.86147 + 0.731395i 0 −0.627801 2.75787i 4.09018 2.36147i 0.951553 + 0.819195i
557.2 −0.146726 1.40658i −0.977085 + 0.261809i −1.95694 + 0.412764i 1.18533 + 0.317608i 0.511620 + 1.33594i 0 0.867721 + 2.69204i −1.71192 + 0.988380i 0.272823 1.71386i
557.3 0.533564 + 1.30970i 1.65049 0.442248i −1.43062 + 1.39762i −3.54317 0.949390i 1.45986 + 1.92568i 0 −2.59378 1.12796i −0.0695300 + 0.0401432i −0.647096 5.14705i
557.4 1.25323 0.655294i 2.01092 0.538823i 1.14118 1.64247i 0.483385 + 0.129523i 2.16706 1.99301i 0 0.353863 2.80620i 1.15537 0.667056i 0.690669 0.154437i
765.1 −1.19412 0.757684i −0.538823 + 2.01092i 0.851831 + 1.80953i −0.129523 0.483385i 2.16706 1.99301i 0 0.353863 2.80620i −1.15537 0.667056i −0.211588 + 0.675356i
765.2 −1.14477 + 0.830359i 0.261809 0.977085i 0.621007 1.90114i −0.317608 1.18533i 0.511620 + 1.33594i 0 0.867721 + 2.69204i 1.71192 + 0.988380i 1.34784 + 1.09320i
765.3 0.105414 + 1.41028i 0.719263 2.68432i −1.97778 + 0.297327i 0.229791 + 0.857592i 3.86147 + 0.731395i 0 −0.627801 2.75787i −4.09018 2.36147i −1.18522 + 0.414472i
765.4 0.867450 1.11693i −0.442248 + 1.65049i −0.495063 1.93776i 0.949390 + 3.54317i 1.45986 + 1.92568i 0 −2.59378 1.12796i 0.0695300 + 0.0401432i 4.78102 + 2.01312i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 765.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{16} + 8 T_{3}^{13} - 44 T_{3}^{12} - 32 T_{3}^{11} + 32 T_{3}^{10} - 136 T_{3}^{9} + 1708 T_{3}^{8} - 1344 T_{3}^{7} + 832 T_{3}^{6} - 4176 T_{3}^{5} - 3056 T_{3}^{4} + 6720 T_{3}^{3} + 800 T_{3}^{2} + 4000 T_{3} + 10000$$ T3^16 + 8*T3^13 - 44*T3^12 - 32*T3^11 + 32*T3^10 - 136*T3^9 + 1708*T3^8 - 1344*T3^7 + 832*T3^6 - 4176*T3^5 - 3056*T3^4 + 6720*T3^3 + 800*T3^2 + 4000*T3 + 10000 $$T_{5}^{16} + 4 T_{5}^{15} + 8 T_{5}^{14} + 56 T_{5}^{13} + 100 T_{5}^{12} - 184 T_{5}^{11} + 32 T_{5}^{10} - 248 T_{5}^{9} + 108 T_{5}^{8} + 240 T_{5}^{7} - 64 T_{5}^{6} + 208 T_{5}^{5} + 80 T_{5}^{4} - 128 T_{5}^{3} + 32 T_{5}^{2} - 32 T_{5} + 16$$ T5^16 + 4*T5^15 + 8*T5^14 + 56*T5^13 + 100*T5^12 - 184*T5^11 + 32*T5^10 - 248*T5^9 + 108*T5^8 + 240*T5^7 - 64*T5^6 + 208*T5^5 + 80*T5^4 - 128*T5^3 + 32*T5^2 - 32*T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 2 T^{15} + 4 T^{14} + 8 T^{13} + \cdots + 256$$
$3$ $$T^{16} + 8 T^{13} - 44 T^{12} + \cdots + 10000$$
$5$ $$T^{16} + 4 T^{15} + 8 T^{14} + 56 T^{13} + \cdots + 16$$
$7$ $$T^{16}$$
$11$ $$T^{16} + 64 T^{13} - 544 T^{12} + \cdots + 65536$$
$13$ $$(T^{8} + 4 T^{5} + 444 T^{4} + 112 T^{3} + \cdots + 28900)^{2}$$
$17$ $$(T^{8} - 12 T^{7} + 104 T^{6} - 400 T^{5} + \cdots + 64)^{2}$$
$19$ $$T^{16} + 12 T^{15} + \cdots + 1536953616$$
$23$ $$T^{16} - 80 T^{14} + \cdots + 136048896$$
$29$ $$(T^{8} + 16 T^{7} + 128 T^{6} + 224 T^{5} + \cdots + 144)^{2}$$
$31$ $$(T^{8} - 8 T^{7} + 96 T^{6} - 112 T^{5} + \cdots + 238144)^{2}$$
$37$ $$T^{16} + 16 T^{15} + \cdots + 189747360000$$
$41$ $$(T^{8} + 288 T^{6} + 27344 T^{4} + \cdots + 5198400)^{2}$$
$43$ $$(T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 10863616)^{2}$$
$47$ $$(T^{8} - 12 T^{7} + 152 T^{6} + \cdots + 141376)^{2}$$
$53$ $$T^{16} - 8 T^{15} + 32 T^{14} + \cdots + 21381376$$
$59$ $$T^{16} + 28 T^{15} + \cdots + 95489560559376$$
$61$ $$T^{16} - 28 T^{15} + \cdots + 26115852816$$
$67$ $$T^{16} - 2048 T^{13} + \cdots + 245635219456$$
$71$ $$(T^{8} + 496 T^{6} + 85280 T^{4} + \cdots + 145926400)^{2}$$
$73$ $$T^{16} - 272 T^{14} + \cdots + 959512576$$
$79$ $$(T^{8} - 12 T^{7} + 304 T^{6} + \cdots + 4665600)^{2}$$
$83$ $$(T^{8} + 4 T^{5} + 44 T^{4} + 32 T^{3} + \cdots + 100)^{2}$$
$89$ $$T^{16} - 432 T^{14} + \cdots + 1665379926016$$
$97$ $$(T^{4} - 16 T^{3} - 24 T^{2} + 648 T + 712)^{4}$$