# Properties

 Label 126.2.l.a Level $126$ Weight $2$ Character orbit 126.l Analytic conductor $1.006$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 126.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561$$ x^16 - 8*x^15 + 23*x^14 - 8*x^13 - 131*x^12 + 380*x^11 - 289*x^10 - 880*x^9 + 2785*x^8 - 2640*x^7 - 2601*x^6 + 10260*x^5 - 10611*x^4 - 1944*x^3 + 16767*x^2 - 17496*x + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} - \beta_{11} q^{3} - q^{4} + ( - \beta_{13} + \beta_{12} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{5} + ( - \beta_{9} - \beta_{4}) q^{6} + (\beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{6} - \beta_{4} + \beta_{2} + 1) q^{7} + \beta_{6} q^{8} + ( - \beta_{15} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q - b6 * q^2 - b11 * q^3 - q^4 + (-b13 + b12 - b9 + b7 + b5 - b4 + b2 + 1) * q^5 + (-b9 - b4) * q^6 + (b15 - b14 + b12 - b11 - b6 - b4 + b2 + 1) * q^7 + b6 * q^8 + (-b15 - b9 + b8 + b7 - b6 - b2 + b1) * q^9 $$q - \beta_{6} q^{2} - \beta_{11} q^{3} - q^{4} + ( - \beta_{13} + \beta_{12} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{5} + ( - \beta_{9} - \beta_{4}) q^{6} + (\beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{6} - \beta_{4} + \beta_{2} + 1) q^{7} + \beta_{6} q^{8} + ( - \beta_{15} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{9} + ( - \beta_{15} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{3} - \beta_1) q^{10} + (\beta_{15} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{11} + \beta_{11} q^{12} + (\beta_{13} + \beta_{11} + \beta_{10} - \beta_{8} + 2 \beta_{6} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{14} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{3} - 1) q^{14} + (\beta_{14} - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} + \cdots - 1) q^{15}+ \cdots + ( - 2 \beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots - 5 \beta_1) q^{99}+O(q^{100})$$ q - b6 * q^2 - b11 * q^3 - q^4 + (-b13 + b12 - b9 + b7 + b5 - b4 + b2 + 1) * q^5 + (-b9 - b4) * q^6 + (b15 - b14 + b12 - b11 - b6 - b4 + b2 + 1) * q^7 + b6 * q^8 + (-b15 - b9 + b8 + b7 - b6 - b2 + b1) * q^9 + (-b15 + b13 + b11 + b10 - b3 - b1) * q^10 + (b15 - b13 - b12 - b11 + b8 - b6 - b5 + b4 - b3 + b2 + b1) * q^11 + b11 * q^12 + (b13 + b11 + b10 - b8 + 2*b6 + b3 - b2 - 2*b1) * q^13 + (-b14 + b10 - b9 + b8 - b6 - b3 - 1) * q^14 + (b14 - 2*b10 + b9 - 2*b8 - b7 + b6 - b5 + 2*b3 - b2 + b1 - 1) * q^15 + q^16 + (-b13 - b12 + b9 - 2*b7 + b2 - 2) * q^17 + (b14 + b11 - b10 + b7 + b6 + b5 - b1) * q^18 + (-b15 + b14 + b13 - b12 + b11 + 2*b9 - b8 + b6 + b5 + b4 - b2 - b1) * q^19 + (b13 - b12 + b9 - b7 - b5 + b4 - b2 - 1) * q^20 + (2*b13 - b12 + b8 - b7 - b6 - b5 + b4 - b3 - b2 + b1 - 2) * q^21 + (-b15 + b14 - b12 + b10 - b9 + b7 - b4 + b3) * q^22 + (b15 - b14 - b13 + b12 - b11 - b10 + b9 - b8 + b5 + b4 + b3 + b1) * q^23 + (b9 + b4) * q^24 + (b15 + b13 + 2*b12 + b11 + 2*b9 + b8 - b7 - b6 + 2*b4 - b3 + b2 - b1) * q^25 + (-b13 + b12 - b10 + b9 - 2*b7 + b4 - b2) * q^26 + (-2*b13 + b12 - b10 - b9 + 2*b7 + b6 + 2*b5 - 2*b4 + b1 + 4) * q^27 + (-b15 + b14 - b12 + b11 + b6 + b4 - b2 - 1) * q^28 + (-b15 + b13 + 2*b11 + b10 + b5 + b4 - 2*b3 - b1) * q^29 + (2*b15 - b14 - b13 + 2*b12 - b11 - b10 + b8 + b7 - b4 + 2*b2 + b1 + 2) * q^30 + (b14 - b12 - b11 + 2*b9 - 2*b8 - 2*b7 + 3*b6 - 2*b5 + b4 + b3 - 1) * q^31 - b6 * q^32 + (-b15 + b14 - 3*b12 - b11 - b8 + b7 + 4*b6 - 3*b5 + 2*b3 - 2*b2 - 2*b1 - 1) * q^33 + (-b15 - b14 + b13 - b11 + b10 - b8 + b3 + 2*b1) * q^34 + (b15 - b11 - b9 + b8 + 2*b7 - b6 - b5 + b4 + 2*b3 + b1 + 2) * q^35 + (b15 + b9 - b8 - b7 + b6 + b2 - b1) * q^36 + (-b14 - 2*b12 + b11 - b9 + b8 + b7 - b5 - 2*b4 - 2*b3) * q^37 + (b15 - b14 - 2*b11 - b10 + b9 - 2*b8 - b7 - b5 + b3) * q^38 + (2*b15 - b14 - b11 + b10 + b8 - 3*b7 - 2*b6 + 2*b5 + 2*b4 + 2*b2 + 2*b1 - 3) * q^39 + (b15 - b13 - b11 - b10 + b3 + b1) * q^40 + (2*b14 - b13 - b12 + b11 - b10 + b8 + b7 + b6 - b5 - b4 + b3 - b2 + b1) * q^41 + (2*b15 + b14 - b13 - b12 - b10 + b7 + b6 + b3 + b1) * q^42 + (-3*b14 - b13 + b12 - b11 - b9 + 3*b8 - b3 + b2) * q^43 + (-b15 + b13 + b12 + b11 - b8 + b6 + b5 - b4 + b3 - b2 - b1) * q^44 + (-3*b15 + b13 + b12 + 3*b11 + 2*b10 - b7 + b6 + 2*b5 - 3*b3 - 2*b1 - 2) * q^45 + (-b13 + b12 - b11 - b9 + b7 - b3 + b2 + 1) * q^46 + (-b15 + b13 + 2*b12 + b10 - 2*b4 - b2 - 1) * q^47 - b11 * q^48 + (b15 + b12 - b10 - 2*b9 - b8 - 3*b6 + b5 + b3 - b2 + 4*b1 - 1) * q^49 + (b15 - b12 - 2*b11 + b10 + b9 + 2*b8 - b7 - b6 + b4 - 2*b3 + b1 - 2) * q^50 + (2*b14 + b13 + b12 + 2*b11 + b9 - 2*b8 - 2*b7 + 2*b6 - 2*b5 + 2*b3 - b2 - 4*b1) * q^51 + (-b13 - b11 - b10 + b8 - 2*b6 - b3 + b2 + 2*b1) * q^52 + (-b15 + 2*b13 - 2*b12 + 3*b11 + b10 - 2*b9 + 2*b7 - b5 - b4 - 3*b3 + b2 - 3*b1 - 2) * q^53 + (-b15 - b14 + b11 - b8 + b7 - 2*b6 - b3 + b2 - 2*b1 + 2) * q^54 + (-2*b15 + 2*b14 - 2*b13 + b12 + b11 - b10 - 2*b9 - b8 + 4*b7 - 3*b6 + 2*b5 - b4 + 2*b3 - b2 + 2) * q^55 + (b14 - b10 + b9 - b8 + b6 + b3 + 1) * q^56 + (-3*b14 + 3*b12 - b11 + 2*b10 - b9 + b8 + b7 - 6*b6 + b5 - b4 - b3 + 2*b2 - b1 + 2) * q^57 + (b14 + b13 - 2*b12 + 2*b9 - b8 - b7 - 2*b5 + 2*b4 - b2 - 1) * q^58 + (-3*b15 + b14 + 3*b13 + 2*b11 + b10 + b9 - b8 + 2*b6 + b5 + b4 - b3 - b2 - 4*b1 + 4) * q^59 + (-b14 + 2*b10 - b9 + 2*b8 + b7 - b6 + b5 - 2*b3 + b2 - b1 + 1) * q^60 + (-b14 + b10 - 3*b9 + b8 - 2*b7 - b6 + 3*b5 - 3*b4 - b3 + b2 - 1) * q^61 + (-b14 + b12 - 2*b11 - b9 + b8 - b6 - b5 - 2*b4 + b3 + 2*b1 + 3) * q^62 + (-b14 - b13 - b12 + b11 + b10 - b9 + 4*b8 + 4*b7 - 2*b6 + b5 - b4 - 4*b3 + 2*b2 - b1 + 4) * q^63 - q^64 + (b14 + b12 + b10 - 4*b9 - b8 + 2*b6 + 4*b5 - 3*b4 + b3 + b2) * q^65 + (-b13 + 2*b12 - 2*b10 - b9 - 2*b7 + 2*b6 + b5 - 2*b4 + 3*b3 - b1 + 2) * q^66 + (b15 + b14 - b13 - 2*b11 - 2*b9 + 3*b8 - b6 - 2*b5 - 2*b4 - b3 + 2*b1 - 2) * q^67 + (b13 + b12 - b9 + 2*b7 - b2 + 2) * q^68 + (-b15 + b13 - 2*b12 + 2*b10 + b9 - b8 - 3*b7 - 4*b6 - b5 + 2*b4 - b3 - 2*b2 + 2*b1 - 6) * q^69 + (2*b14 - b13 + 2*b12 + b11 - b9 + b8 + b7 + 3*b5 - b4 - 2*b1) * q^70 + (-b12 - 2*b11 - 2*b10 - 2*b8 - b6 - b4 - 2*b2) * q^71 + (-b14 - b11 + b10 - b7 - b6 - b5 + b1) * q^72 + (b15 - b14 - b13 - b10 - b8 - 5*b1) * q^73 + (-b14 - 2*b12 + b11 + b9 - b8 + b6 - b5 + 2*b4 + 2*b3 - b1) * q^74 + (b15 - 2*b14 + b13 - 2*b12 - 2*b11 + 2*b10 + b8 - 2*b7 - 3*b6 - 4*b5 - 2*b3 + 2*b2 + 6*b1 + 2) * q^75 + (b15 - b14 - b13 + b12 - b11 - 2*b9 + b8 - b6 - b5 - b4 + b2 + b1) * q^76 + (b13 + b11 - b9 - b7 - 4*b5 - 3*b4 + 2*b3 + b2 + b1 - 2) * q^77 + (-b15 + 2*b14 + 2*b10 - b9 - 2*b8 + 2*b7 + b5 - b2 + 3*b1) * q^78 + (b14 - 4*b12 - b11 - 2*b9 + 2*b8 - 2*b5 + 2*b4 - b3 - 1) * q^79 + (-b13 + b12 - b9 + b7 + b5 - b4 + b2 + 1) * q^80 + (-b15 + b14 + b13 + b12 - b11 - b10 + 2*b9 - 4*b8 + 3*b7 - b6 + 2*b5 + b4 + 2*b3 - 2*b2 - b1) * q^81 + (-b14 - b13 + b12 - b10 + b9 + b7 + b6 + 2*b5 - b4 + b3 + b2 - b1 + 2) * q^82 + (-3*b14 - 2*b13 + 2*b12 - 2*b9 + 3*b8 + 2*b7 + 2*b5 - 2*b4 + 2*b2 + 2) * q^83 + (-2*b13 + b12 - b8 + b7 + b6 + b5 - b4 + b3 + b2 - b1 + 2) * q^84 + (3*b15 + b14 - b13 - 4*b10 + 4*b9 - 2*b7 - 2*b6 - 4*b5 + b3 - b2 - 2*b1) * q^85 + (-b15 + b14 + b13 - b12 + b11 + b10 - b9 + b8 + 2*b5 + 2*b4 - b3) * q^86 + (3*b15 - b14 - 2*b13 + b12 - 2*b11 - b10 + b9 + 2*b7 - 2*b6 - b5 - b4 + 3*b2 + b1 + 1) * q^87 + (b15 - b14 + b12 - b10 + b9 - b7 + b4 - b3) * q^88 + (b15 - b14 + 2*b13 + 3*b12 - 2*b11 + b10 + 3*b9 - 2*b8 + b6 + 3*b4 + b3 + 2*b2 + b1) * q^89 + (-b15 + 3*b13 - 3*b12 + 3*b9 - b8 - 2*b7 + b6 - 3*b5 + 3*b4 - b3 - 2*b2 + b1 - 1) * q^90 + (b15 - b13 + 4*b12 + 2*b9 - 3*b8 + 2*b7 + 5*b6 + 4*b5 + 2*b4 - b3 + 2*b2 + 2) * q^91 + (-b15 + b14 + b13 - b12 + b11 + b10 - b9 + b8 - b5 - b4 - b3 - b1) * q^92 + (-2*b15 + b13 + b12 + 2*b10 + b9 + b8 - 2*b7 + 3*b6 - b5 + 2*b4 + b3 - b2 - 4) * q^93 + (-2*b14 - b10 + 2*b8 + b6 - 2*b3 - b2) * q^94 + (b15 + b14 + b13 - 4*b12 - 2*b11 - 2*b10 + 4*b9 - 3*b8 - 8*b7 + 4*b6 - 4*b5 + b3 - 2*b2 - 4) * q^95 + (-b9 - b4) * q^96 + (4*b15 - 3*b13 + 3*b12 - 2*b11 - 4*b10 + 3*b9 - 2*b7 + 2*b3 + b2 + 8*b1 + 2) * q^97 + (b15 + 2*b14 - 2*b13 + b12 + 2*b11 - b10 + 4*b7 + b6 - b3 + b2 + 1) * q^98 + (-2*b15 - b14 - b13 + 2*b12 + b11 + b10 - 2*b9 + b8 + 7*b6 + 4*b5 + 2*b4 + b3 - b2 - 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4} + 2 q^{7}+O(q^{10})$$ 16 * q - 16 * q^4 + 2 * q^7 $$16 q - 16 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{13} - 6 q^{14} - 18 q^{15} + 16 q^{16} - 18 q^{17} - 12 q^{18} - 12 q^{21} - 6 q^{23} - 8 q^{25} + 12 q^{26} + 36 q^{27} - 2 q^{28} + 6 q^{29} + 30 q^{35} - 2 q^{37} - 12 q^{39} - 6 q^{41} - 2 q^{43} - 12 q^{44} - 30 q^{45} + 6 q^{46} - 36 q^{47} - 8 q^{49} - 12 q^{50} + 6 q^{51} - 6 q^{52} - 36 q^{53} + 18 q^{54} + 6 q^{56} + 6 q^{57} + 6 q^{58} + 60 q^{59} + 18 q^{60} + 36 q^{62} + 36 q^{63} - 16 q^{64} + 24 q^{66} - 28 q^{67} + 18 q^{68} - 42 q^{69} - 18 q^{70} + 12 q^{72} + 18 q^{74} + 60 q^{75} - 42 q^{77} + 32 q^{79} - 36 q^{81} + 12 q^{84} - 12 q^{85} + 24 q^{86} - 24 q^{87} - 24 q^{89} + 18 q^{90} - 12 q^{91} + 6 q^{92} - 42 q^{93} + 6 q^{97} - 24 q^{98} + 18 q^{99}+O(q^{100})$$ 16 * q - 16 * q^4 + 2 * q^7 + 12 * q^11 + 6 * q^13 - 6 * q^14 - 18 * q^15 + 16 * q^16 - 18 * q^17 - 12 * q^18 - 12 * q^21 - 6 * q^23 - 8 * q^25 + 12 * q^26 + 36 * q^27 - 2 * q^28 + 6 * q^29 + 30 * q^35 - 2 * q^37 - 12 * q^39 - 6 * q^41 - 2 * q^43 - 12 * q^44 - 30 * q^45 + 6 * q^46 - 36 * q^47 - 8 * q^49 - 12 * q^50 + 6 * q^51 - 6 * q^52 - 36 * q^53 + 18 * q^54 + 6 * q^56 + 6 * q^57 + 6 * q^58 + 60 * q^59 + 18 * q^60 + 36 * q^62 + 36 * q^63 - 16 * q^64 + 24 * q^66 - 28 * q^67 + 18 * q^68 - 42 * q^69 - 18 * q^70 + 12 * q^72 + 18 * q^74 + 60 * q^75 - 42 * q^77 + 32 * q^79 - 36 * q^81 + 12 * q^84 - 12 * q^85 + 24 * q^86 - 24 * q^87 - 24 * q^89 + 18 * q^90 - 12 * q^91 + 6 * q^92 - 42 * q^93 + 6 * q^97 - 24 * q^98 + 18 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + \cdots + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( - 50 \nu^{15} - 1352 \nu^{14} + 6827 \nu^{13} - 7676 \nu^{12} - 27422 \nu^{11} + 107246 \nu^{10} - 107467 \nu^{9} - 206194 \nu^{8} + 757363 \nu^{7} - 724572 \nu^{6} + \cdots - 2825604 ) / 142155$$ (-50*v^15 - 1352*v^14 + 6827*v^13 - 7676*v^12 - 27422*v^11 + 107246*v^10 - 107467*v^9 - 206194*v^8 + 757363*v^7 - 724572*v^6 - 756198*v^5 + 2730942*v^4 - 2372247*v^3 - 1318518*v^2 + 4444713*v - 2825604) / 142155 $$\beta_{2}$$ $$=$$ $$( 1292 \nu^{15} - 10486 \nu^{14} + 25660 \nu^{13} + 10145 \nu^{12} - 192280 \nu^{11} + 408694 \nu^{10} - 51650 \nu^{9} - 1459361 \nu^{8} + 2979638 \nu^{7} + \cdots - 9270693 ) / 142155$$ (1292*v^15 - 10486*v^14 + 25660*v^13 + 10145*v^12 - 192280*v^11 + 408694*v^10 - 51650*v^9 - 1459361*v^8 + 2979638*v^7 - 1138791*v^6 - 5534208*v^5 + 10987326*v^4 - 5516586*v^3 - 9628389*v^2 + 17565255*v - 9270693) / 142155 $$\beta_{3}$$ $$=$$ $$( 2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + 878875 \nu^{10} - 166676 \nu^{9} - 3023495 \nu^{8} + 6386423 \nu^{7} + \cdots - 20783061 ) / 142155$$ (2846*v^15 - 22369*v^14 + 55246*v^13 + 17972*v^12 - 402586*v^11 + 878875*v^10 - 166676*v^9 - 3023495*v^8 + 6386423*v^7 - 2756469*v^6 - 11333493*v^5 + 23486949*v^4 - 12667509*v^3 - 19623951*v^2 + 37934244*v - 20783061) / 142155 $$\beta_{4}$$ $$=$$ $$( - 2782 \nu^{15} + 15918 \nu^{14} - 26947 \nu^{13} - 42629 \nu^{12} + 270897 \nu^{11} - 425335 \nu^{10} - 220488 \nu^{9} + 2001225 \nu^{8} - 3082271 \nu^{7} + \cdots + 7405182 ) / 47385$$ (-2782*v^15 + 15918*v^14 - 26947*v^13 - 42629*v^12 + 270897*v^11 - 425335*v^10 - 220488*v^9 + 2001225*v^8 - 3082271*v^7 - 70562*v^6 + 7458951*v^5 - 11186748*v^4 + 2636253*v^3 + 12673422*v^2 - 17180343*v + 7405182) / 47385 $$\beta_{5}$$ $$=$$ $$( - 3648 \nu^{15} + 23120 \nu^{14} - 45376 \nu^{13} - 47012 \nu^{12} + 401996 \nu^{11} - 719704 \nu^{10} - 150629 \nu^{9} + 2988436 \nu^{8} - 5221301 \nu^{7} + \cdots + 14281839 ) / 47385$$ (-3648*v^15 + 23120*v^14 - 45376*v^13 - 47012*v^12 + 401996*v^11 - 719704*v^10 - 150629*v^9 + 2988436*v^8 - 5221301*v^7 + 915140*v^6 + 11160666*v^5 - 19038915*v^4 + 7000290*v^3 + 19081575*v^2 - 29765799*v + 14281839) / 47385 $$\beta_{6}$$ $$=$$ $$( - 16948 \nu^{15} + 107120 \nu^{14} - 210206 \nu^{13} - 216202 \nu^{12} + 1856546 \nu^{11} - 3329594 \nu^{10} - 672599 \nu^{9} + 13772146 \nu^{8} + \cdots + 66620394 ) / 142155$$ (-16948*v^15 + 107120*v^14 - 210206*v^13 - 216202*v^12 + 1856546*v^11 - 3329594*v^10 - 672599*v^9 + 13772146*v^8 - 24135496*v^7 + 4376130*v^6 + 51333696*v^5 - 87961680*v^4 + 32792850*v^3 + 87596640*v^2 - 137624994*v + 66620394) / 142155 $$\beta_{7}$$ $$=$$ $$( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} - 771188 \nu^{10} - 200900 \nu^{9} + 3269857 \nu^{8} - 5585140 \nu^{7} + \cdots + 14935023 ) / 28431$$ (-4120*v^15 + 25571*v^14 - 48788*v^13 - 55006*v^12 + 441224*v^11 - 771188*v^10 - 200900*v^9 + 3269857*v^8 - 5585140*v^7 + 796053*v^6 + 12187116*v^5 - 20322468*v^4 + 7028856*v^3 + 20790351*v^2 - 31653180*v + 14935023) / 28431 $$\beta_{8}$$ $$=$$ $$( 24706 \nu^{15} - 155855 \nu^{14} + 305147 \nu^{13} + 316579 \nu^{12} - 2700647 \nu^{11} + 4831208 \nu^{10} + 1007453 \nu^{9} - 20043787 \nu^{8} + 35017972 \nu^{7} + \cdots - 96324228 ) / 142155$$ (24706*v^15 - 155855*v^14 + 305147*v^13 + 316579*v^12 - 2700647*v^11 + 4831208*v^10 + 1007453*v^9 - 20043787*v^8 + 35017972*v^7 - 6181845*v^6 - 74736702*v^5 + 127624545*v^4 - 47130660*v^3 - 127644255*v^2 + 199674558*v - 96324228) / 142155 $$\beta_{9}$$ $$=$$ $$( 2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + 93322 \nu^{9} - 1605386 \nu^{8} + 2760692 \nu^{7} - 423483 \nu^{6} + \cdots - 7453296 ) / 10935$$ (2015*v^15 - 12538*v^14 + 24088*v^13 + 26576*v^12 - 216643*v^11 + 381184*v^10 + 93322*v^9 - 1605386*v^8 + 2760692*v^7 - 423483*v^6 - 5980032*v^5 + 10047753*v^4 - 3553308*v^3 - 10193607*v^2 + 15657462*v - 7453296) / 10935 $$\beta_{10}$$ $$=$$ $$( - 26777 \nu^{15} + 172141 \nu^{14} - 345205 \nu^{13} - 329150 \nu^{12} + 2992915 \nu^{11} - 5473354 \nu^{10} - 873115 \nu^{9} + 22225271 \nu^{8} + \cdots + 112005018 ) / 142155$$ (-26777*v^15 + 172141*v^14 - 345205*v^13 - 329150*v^12 + 2992915*v^11 - 5473354*v^10 - 873115*v^9 + 22225271*v^8 - 39691043*v^7 + 8309796*v^6 + 82849923*v^5 - 144806481*v^4 + 56755161*v^3 + 141385419*v^2 - 227291265*v + 112005018) / 142155 $$\beta_{11}$$ $$=$$ $$( 32006 \nu^{15} - 201190 \nu^{14} + 390892 \nu^{13} + 415394 \nu^{12} - 3479542 \nu^{11} + 6180118 \nu^{10} + 1385833 \nu^{9} - 25793657 \nu^{8} + \cdots - 121376313 ) / 142155$$ (32006*v^15 - 201190*v^14 + 390892*v^13 + 415394*v^12 - 3479542*v^11 + 6180118*v^10 + 1385833*v^9 - 25793657*v^8 + 44738537*v^7 - 7406040*v^6 - 96117777*v^5 + 162796500*v^4 - 58760640*v^3 - 164045655*v^2 + 253970478*v - 121376313) / 142155 $$\beta_{12}$$ $$=$$ $$( - 10990 \nu^{15} + 68842 \nu^{14} - 133062 \nu^{13} - 143724 \nu^{12} + 1190062 \nu^{11} - 2105781 \nu^{10} - 489283 \nu^{9} + 8822519 \nu^{8} - 15253998 \nu^{7} + \cdots + 41433444 ) / 47385$$ (-10990*v^15 + 68842*v^14 - 133062*v^13 - 143724*v^12 + 1190062*v^11 - 2105781*v^10 - 489283*v^9 + 8822519*v^8 - 15253998*v^7 + 2459777*v^6 + 32873688*v^5 - 55540512*v^4 + 19941417*v^3 + 56061558*v^2 - 86668623*v + 41433444) / 47385 $$\beta_{13}$$ $$=$$ $$( - 35897 \nu^{15} + 219337 \nu^{14} - 409846 \nu^{13} - 492632 \nu^{12} + 3774151 \nu^{11} - 6478657 \nu^{10} - 1950109 \nu^{9} + 27930713 \nu^{8} + \cdots + 122931270 ) / 142155$$ (-35897*v^15 + 219337*v^14 - 409846*v^13 - 492632*v^12 + 3774151*v^11 - 6478657*v^10 - 1950109*v^9 + 27930713*v^8 - 46902542*v^7 + 5479452*v^6 + 103995882*v^5 - 170423757*v^4 + 56059857*v^3 + 177093783*v^2 - 264354354*v + 122931270) / 142155 $$\beta_{14}$$ $$=$$ $$( - 14807 \nu^{15} + 92828 \nu^{14} - 179757 \nu^{13} - 193044 \nu^{12} + 1605257 \nu^{11} - 2844540 \nu^{10} - 652043 \nu^{9} + 11900965 \nu^{8} - 20604021 \nu^{7} + \cdots + 55960227 ) / 47385$$ (-14807*v^15 + 92828*v^14 - 179757*v^13 - 193044*v^12 + 1605257*v^11 - 2844540*v^10 - 652043*v^9 + 11900965*v^8 - 20604021*v^7 + 3361183*v^6 + 44347701*v^5 - 75011778*v^4 + 26997138*v^3 + 75667932*v^2 - 117064278*v + 55960227) / 47385 $$\beta_{15}$$ $$=$$ $$( - 68699 \nu^{15} + 433198 \nu^{14} - 844756 \nu^{13} - 889547 \nu^{12} + 7506256 \nu^{11} - 13372516 \nu^{10} - 2912329 \nu^{9} + 55693424 \nu^{8} + \cdots + 264515463 ) / 142155$$ (-68699*v^15 + 433198*v^14 - 844756*v^13 - 889547*v^12 + 7506256*v^11 - 13372516*v^10 - 2912329*v^9 + 55693424*v^8 - 96903605*v^7 + 16506963*v^6 + 207637920*v^5 - 352968678*v^4 + 128769183*v^3 + 354359367*v^2 - 551274174*v + 264515463) / 142155
 $$\nu$$ $$=$$ $$( -\beta_{15} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$$ (-b15 - b9 - b8 + b7 + b6 - 2*b4 + 2*b3 - 2*b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( - \beta_{15} + 4 \beta_{14} + \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta _1 + 3 ) / 3$$ (-b15 + 4*b14 + b13 - 2*b12 + 2*b11 - b10 - b9 - b8 - b6 - b5 - 2*b4 - b3 - 2*b2 + 2*b1 + 3) / 3 $$\nu^{3}$$ $$=$$ $$( - 3 \beta_{15} - \beta_{14} + 5 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{3} - 6 \beta_{2} - \beta _1 - 4 ) / 3$$ (-3*b15 - b14 + 5*b13 + 2*b12 + b11 + b10 - 3*b8 - 2*b7 - b6 - 2*b5 - 3*b3 - 6*b2 - b1 - 4) / 3 $$\nu^{4}$$ $$=$$ $$( - \beta_{15} + 2 \beta_{14} + 11 \beta_{13} - 4 \beta_{12} + 7 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} - 4 \beta_{8} - \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + 7 \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 15 \beta _1 + 7 ) / 3$$ (-b15 + 2*b14 + 11*b13 - 4*b12 + 7*b11 + 4*b10 + 8*b9 - 4*b8 - b7 - 6*b6 - 2*b5 + 7*b4 - 10*b3 - 2*b2 - 15*b1 + 7) / 3 $$\nu^{5}$$ $$=$$ $$( - 11 \beta_{15} - 6 \beta_{14} + 9 \beta_{13} + 12 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \beta_{8} + 23 \beta_{7} + 17 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} - 16 \beta_{2} - 31 \beta _1 + 10 ) / 3$$ (-11*b15 - 6*b14 + 9*b13 + 12*b11 + 6*b10 - 2*b9 + b8 + 23*b7 + 17*b6 + 3*b5 + 8*b4 - 5*b3 - 16*b2 - 31*b1 + 10) / 3 $$\nu^{6}$$ $$=$$ $$( - 5 \beta_{15} + 26 \beta_{14} + 2 \beta_{13} - 31 \beta_{12} + 28 \beta_{11} + 13 \beta_{10} - 14 \beta_{9} + 4 \beta_{8} + 39 \beta_{7} - 20 \beta_{6} - 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 10 \beta_{2} - 20 \beta _1 + 15 ) / 3$$ (-5*b15 + 26*b14 + 2*b13 - 31*b12 + 28*b11 + 13*b10 - 14*b9 + 4*b8 + 39*b7 - 20*b6 - 5*b5 - b4 - 2*b3 - 10*b2 - 20*b1 + 15) / 3 $$\nu^{7}$$ $$=$$ $$( - 15 \beta_{15} + 16 \beta_{14} - 20 \beta_{13} + 16 \beta_{12} + 29 \beta_{11} - 13 \beta_{10} - 60 \beta_{9} + 6 \beta_{8} + 26 \beta_{7} - 5 \beta_{6} - 10 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 30 \beta_{2} - 14 \beta _1 - 11 ) / 3$$ (-15*b15 + 16*b14 - 20*b13 + 16*b12 + 29*b11 - 13*b10 - 60*b9 + 6*b8 + 26*b7 - 5*b6 - 10*b5 - 3*b4 - 3*b3 - 30*b2 - 14*b1 - 11) / 3 $$\nu^{8}$$ $$=$$ $$( 43 \beta_{15} + 28 \beta_{14} - 8 \beta_{13} + 10 \beta_{12} + 44 \beta_{11} - 16 \beta_{10} + 25 \beta_{9} - 92 \beta_{8} - 65 \beta_{7} - 114 \beta_{6} - 52 \beta_{5} - 7 \beta_{4} + 31 \beta_{3} + 26 \beta_{2} - 27 \beta _1 - 1 ) / 3$$ (43*b15 + 28*b14 - 8*b13 + 10*b12 + 44*b11 - 16*b10 + 25*b9 - 92*b8 - 65*b7 - 114*b6 - 52*b5 - 7*b4 + 31*b3 + 26*b2 - 27*b1 - 1) / 3 $$\nu^{9}$$ $$=$$ $$( 50 \beta_{15} - 12 \beta_{14} - 18 \beta_{13} + 102 \beta_{12} + 102 \beta_{11} - 144 \beta_{10} + 80 \beta_{9} - 106 \beta_{8} - 104 \beta_{7} + 229 \beta_{6} - 84 \beta_{5} + 34 \beta_{4} + 44 \beta_{3} - 8 \beta_{2} - 194 \beta _1 + 134 ) / 3$$ (50*b15 - 12*b14 - 18*b13 + 102*b12 + 102*b11 - 144*b10 + 80*b9 - 106*b8 - 104*b7 + 229*b6 - 84*b5 + 34*b4 + 44*b3 - 8*b2 - 194*b1 + 134) / 3 $$\nu^{10}$$ $$=$$ $$( 140 \beta_{15} + 139 \beta_{14} + 31 \beta_{13} - 224 \beta_{12} + 95 \beta_{11} - 46 \beta_{10} + 320 \beta_{9} - 220 \beta_{8} - 117 \beta_{7} + 251 \beta_{6} - 337 \beta_{5} + 10 \beta_{4} + 224 \beta_{3} - 80 \beta_{2} - 109 \beta_1 ) / 3$$ (140*b15 + 139*b14 + 31*b13 - 224*b12 + 95*b11 - 46*b10 + 320*b9 - 220*b8 - 117*b7 + 251*b6 - 337*b5 + 10*b4 + 224*b3 - 80*b2 - 109*b1) / 3 $$\nu^{11}$$ $$=$$ $$( 60 \beta_{15} + 503 \beta_{14} + 23 \beta_{13} - 313 \beta_{12} + 133 \beta_{11} - 218 \beta_{10} + 114 \beta_{9} + 171 \beta_{8} - 482 \beta_{7} + 779 \beta_{6} - 524 \beta_{5} + 270 \beta_{4} - 12 \beta_{3} - 399 \beta_{2} + \cdots - 193 ) / 3$$ (60*b15 + 503*b14 + 23*b13 - 313*b12 + 133*b11 - 218*b10 + 114*b9 + 171*b8 - 482*b7 + 779*b6 - 524*b5 + 270*b4 - 12*b3 - 399*b2 - 166*b1 - 193) / 3 $$\nu^{12}$$ $$=$$ $$( - 94 \beta_{15} + 722 \beta_{14} + 191 \beta_{13} - 343 \beta_{12} - 182 \beta_{11} + 238 \beta_{10} + 569 \beta_{9} - 454 \beta_{8} - 1069 \beta_{7} - 192 \beta_{6} - 674 \beta_{5} + 382 \beta_{4} + 398 \beta_{3} + \cdots - 1112 ) / 3$$ (-94*b15 + 722*b14 + 191*b13 - 343*b12 - 182*b11 + 238*b10 + 569*b9 - 454*b8 - 1069*b7 - 192*b6 - 674*b5 + 382*b4 + 398*b3 - 719*b2 - 84*b1 - 1112) / 3 $$\nu^{13}$$ $$=$$ $$( - 425 \beta_{15} + 1353 \beta_{14} + 255 \beta_{13} + 1086 \beta_{12} + 1068 \beta_{11} - 588 \beta_{10} + 691 \beta_{9} - 122 \beta_{8} - 1675 \beta_{7} + 662 \beta_{6} + 498 \beta_{5} + 707 \beta_{4} - 206 \beta_{3} + \cdots + 727 ) / 3$$ (-425*b15 + 1353*b14 + 255*b13 + 1086*b12 + 1068*b11 - 588*b10 + 691*b9 - 122*b8 - 1675*b7 + 662*b6 + 498*b5 + 707*b4 - 206*b3 - 454*b2 - 1759*b1 + 727) / 3 $$\nu^{14}$$ $$=$$ $$( - 953 \beta_{15} + 446 \beta_{14} + 35 \beta_{13} + 2231 \beta_{12} + 1294 \beta_{11} + 82 \beta_{10} + 1990 \beta_{9} - 1151 \beta_{8} + 1755 \beta_{7} + 91 \beta_{6} + 688 \beta_{5} - 970 \beta_{4} + 1378 \beta_{3} + \cdots + 1896 ) / 3$$ (-953*b15 + 446*b14 + 35*b13 + 2231*b12 + 1294*b11 + 82*b10 + 1990*b9 - 1151*b8 + 1755*b7 + 91*b6 + 688*b5 - 970*b4 + 1378*b3 - 484*b2 - 1505*b1 + 1896) / 3 $$\nu^{15}$$ $$=$$ $$( 1563 \beta_{15} + 1963 \beta_{14} - 725 \beta_{13} + 2308 \beta_{12} + 3086 \beta_{11} - 1306 \beta_{10} + 714 \beta_{9} + 4854 \beta_{8} + 2906 \beta_{7} - 1550 \beta_{6} + 32 \beta_{5} - 1380 \beta_{4} - 2391 \beta_{3} + \cdots + 6694 ) / 3$$ (1563*b15 + 1963*b14 - 725*b13 + 2308*b12 + 3086*b11 - 1306*b10 + 714*b9 + 4854*b8 + 2906*b7 - 1550*b6 + 32*b5 - 1380*b4 - 2391*b3 + 1551*b2 + 1375*b1 + 6694) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$-\beta_{7}$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 1.73109 − 0.0577511i 0.320287 + 1.70218i −1.68301 + 0.409224i 0.765614 − 1.55365i 1.71298 + 0.256290i 1.27866 − 1.16834i −1.70672 − 0.295146i 1.58110 + 0.707199i 1.71298 − 0.256290i 1.27866 + 1.16834i −1.70672 + 0.295146i 1.58110 − 0.707199i 1.73109 + 0.0577511i 0.320287 − 1.70218i −1.68301 − 0.409224i 0.765614 + 1.55365i
1.00000i −0.890915 + 1.48535i −1.00000 1.14095 + 1.97618i 1.48535 + 0.890915i 1.42337 + 2.23025i 1.00000i −1.41254 2.64665i 1.97618 1.14095i
5.2 1.00000i −0.290993 1.70743i −1.00000 −0.0338034 0.0585493i −1.70743 + 0.290993i 1.19767 2.35915i 1.00000i −2.83065 + 0.993700i −0.0585493 + 0.0338034i
5.3 1.00000i 1.38631 + 1.03834i −1.00000 0.714925 + 1.23829i 1.03834 1.38631i 0.327442 2.62541i 1.00000i 0.843698 + 2.87892i 1.23829 0.714925i
5.4 1.00000i 1.52765 0.816261i −1.00000 −1.82207 3.15592i −0.816261 1.52765i −1.58246 + 2.12034i 1.00000i 1.66744 2.49392i −3.15592 + 1.82207i
5.5 1.00000i −1.56012 0.752355i −1.00000 −1.80966 3.13442i 0.752355 1.56012i −2.41308 1.08492i 1.00000i 1.86792 + 2.34752i 3.13442 1.80966i
5.6 1.00000i −1.08509 1.35003i −1.00000 1.77612 + 3.07634i 1.35003 1.08509i 2.63804 0.201867i 1.00000i −0.645160 + 2.92981i −3.07634 + 1.77612i
5.7 1.00000i −0.734581 + 1.56856i −1.00000 0.483662 + 0.837727i −1.56856 0.734581i −2.16249 + 1.52435i 1.00000i −1.92078 2.30447i −0.837727 + 0.483662i
5.8 1.00000i 1.64774 + 0.533822i −1.00000 −0.450129 0.779646i −0.533822 + 1.64774i 1.57151 + 2.12847i 1.00000i 2.43007 + 1.75919i 0.779646 0.450129i
101.1 1.00000i −1.56012 + 0.752355i −1.00000 −1.80966 + 3.13442i 0.752355 + 1.56012i −2.41308 + 1.08492i 1.00000i 1.86792 2.34752i 3.13442 + 1.80966i
101.2 1.00000i −1.08509 + 1.35003i −1.00000 1.77612 3.07634i 1.35003 + 1.08509i 2.63804 + 0.201867i 1.00000i −0.645160 2.92981i −3.07634 1.77612i
101.3 1.00000i −0.734581 1.56856i −1.00000 0.483662 0.837727i −1.56856 + 0.734581i −2.16249 1.52435i 1.00000i −1.92078 + 2.30447i −0.837727 0.483662i
101.4 1.00000i 1.64774 0.533822i −1.00000 −0.450129 + 0.779646i −0.533822 1.64774i 1.57151 2.12847i 1.00000i 2.43007 1.75919i 0.779646 + 0.450129i
101.5 1.00000i −0.890915 1.48535i −1.00000 1.14095 1.97618i 1.48535 0.890915i 1.42337 2.23025i 1.00000i −1.41254 + 2.64665i 1.97618 + 1.14095i
101.6 1.00000i −0.290993 + 1.70743i −1.00000 −0.0338034 + 0.0585493i −1.70743 0.290993i 1.19767 + 2.35915i 1.00000i −2.83065 0.993700i −0.0585493 0.0338034i
101.7 1.00000i 1.38631 1.03834i −1.00000 0.714925 1.23829i 1.03834 + 1.38631i 0.327442 + 2.62541i 1.00000i 0.843698 2.87892i 1.23829 + 0.714925i
101.8 1.00000i 1.52765 + 0.816261i −1.00000 −1.82207 + 3.15592i −0.816261 + 1.52765i −1.58246 2.12034i 1.00000i 1.66744 + 2.49392i −3.15592 1.82207i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.l.a 16
3.b odd 2 1 378.2.l.a 16
4.b odd 2 1 1008.2.ca.c 16
7.b odd 2 1 882.2.l.b 16
7.c even 3 1 882.2.m.b 16
7.c even 3 1 882.2.t.a 16
7.d odd 6 1 126.2.t.a yes 16
7.d odd 6 1 882.2.m.a 16
9.c even 3 1 378.2.t.a 16
9.c even 3 1 1134.2.k.a 16
9.d odd 6 1 126.2.t.a yes 16
9.d odd 6 1 1134.2.k.b 16
12.b even 2 1 3024.2.ca.c 16
21.c even 2 1 2646.2.l.a 16
21.g even 6 1 378.2.t.a 16
21.g even 6 1 2646.2.m.a 16
21.h odd 6 1 2646.2.m.b 16
21.h odd 6 1 2646.2.t.b 16
28.f even 6 1 1008.2.df.c 16
36.f odd 6 1 3024.2.df.c 16
36.h even 6 1 1008.2.df.c 16
63.g even 3 1 2646.2.m.a 16
63.h even 3 1 2646.2.l.a 16
63.i even 6 1 inner 126.2.l.a 16
63.j odd 6 1 882.2.l.b 16
63.k odd 6 1 1134.2.k.b 16
63.k odd 6 1 2646.2.m.b 16
63.l odd 6 1 2646.2.t.b 16
63.n odd 6 1 882.2.m.a 16
63.o even 6 1 882.2.t.a 16
63.s even 6 1 882.2.m.b 16
63.s even 6 1 1134.2.k.a 16
63.t odd 6 1 378.2.l.a 16
84.j odd 6 1 3024.2.df.c 16
252.r odd 6 1 1008.2.ca.c 16
252.bj even 6 1 3024.2.ca.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a 16 1.a even 1 1 trivial
126.2.l.a 16 63.i even 6 1 inner
126.2.t.a yes 16 7.d odd 6 1
126.2.t.a yes 16 9.d odd 6 1
378.2.l.a 16 3.b odd 2 1
378.2.l.a 16 63.t odd 6 1
378.2.t.a 16 9.c even 3 1
378.2.t.a 16 21.g even 6 1
882.2.l.b 16 7.b odd 2 1
882.2.l.b 16 63.j odd 6 1
882.2.m.a 16 7.d odd 6 1
882.2.m.a 16 63.n odd 6 1
882.2.m.b 16 7.c even 3 1
882.2.m.b 16 63.s even 6 1
882.2.t.a 16 7.c even 3 1
882.2.t.a 16 63.o even 6 1
1008.2.ca.c 16 4.b odd 2 1
1008.2.ca.c 16 252.r odd 6 1
1008.2.df.c 16 28.f even 6 1
1008.2.df.c 16 36.h even 6 1
1134.2.k.a 16 9.c even 3 1
1134.2.k.a 16 63.s even 6 1
1134.2.k.b 16 9.d odd 6 1
1134.2.k.b 16 63.k odd 6 1
2646.2.l.a 16 21.c even 2 1
2646.2.l.a 16 63.h even 3 1
2646.2.m.a 16 21.g even 6 1
2646.2.m.a 16 63.g even 3 1
2646.2.m.b 16 21.h odd 6 1
2646.2.m.b 16 63.k odd 6 1
2646.2.t.b 16 21.h odd 6 1
2646.2.t.b 16 63.l odd 6 1
3024.2.ca.c 16 12.b even 2 1
3024.2.ca.c 16 252.bj even 6 1
3024.2.df.c 16 36.f odd 6 1
3024.2.df.c 16 84.j odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$T^{16} - 12 T^{13} + 9 T^{12} + \cdots + 6561$$
$5$ $$T^{16} + 24 T^{14} - 24 T^{13} + 423 T^{12} + \cdots + 81$$
$7$ $$T^{16} - 2 T^{15} + 6 T^{14} + \cdots + 5764801$$
$11$ $$T^{16} - 12 T^{15} + 18 T^{14} + \cdots + 61732449$$
$13$ $$T^{16} - 6 T^{15} - 57 T^{14} + \cdots + 390971529$$
$17$ $$T^{16} + 18 T^{15} + 231 T^{14} + \cdots + 56070144$$
$19$ $$T^{16} - 72 T^{14} + 4167 T^{12} + \cdots + 9199089$$
$23$ $$T^{16} + 6 T^{15} - 54 T^{14} + \cdots + 187388721$$
$29$ $$T^{16} - 6 T^{15} - 36 T^{14} + \cdots + 1108809$$
$31$ $$T^{16} + 204 T^{14} + \cdots + 65610000$$
$37$ $$T^{16} + 2 T^{15} + \cdots + 32746159681$$
$41$ $$T^{16} + 6 T^{15} + 105 T^{14} - 210 T^{13} + \cdots + 81$$
$43$ $$T^{16} + 2 T^{15} + \cdots + 2999643361$$
$47$ $$(T^{8} + 18 T^{7} + 3 T^{6} - 1650 T^{5} + \cdots + 766944)^{2}$$
$53$ $$T^{16} + 36 T^{15} + \cdots + 36759242529$$
$59$ $$(T^{8} - 30 T^{7} + 228 T^{6} + \cdots + 465300)^{2}$$
$61$ $$T^{16} + 504 T^{14} + \cdots + 547560000$$
$67$ $$(T^{8} + 14 T^{7} - 101 T^{6} + \cdots + 51028)^{2}$$
$71$ $$T^{16} + 486 T^{14} + \cdots + 65610000$$
$73$ $$T^{16} - 150 T^{14} + \cdots + 71115489$$
$79$ $$(T^{8} - 16 T^{7} - 149 T^{6} + \cdots - 985100)^{2}$$
$83$ $$T^{16} + 177 T^{14} + \cdots + 953512641$$
$89$ $$T^{16} + \cdots + 131145120363321$$
$97$ $$T^{16} - 6 T^{15} + \cdots + 9120206721024$$