# Properties

 Label 1014.2.e.l Level $1014$ Weight $2$ Character orbit 1014.e Analytic conductor $8.097$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$$ (-v^5 + 3*v^4 - 9*v^3 + 5*v^2 - 2*v + 6) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$$ (-3*v^5 + 9*v^4 - 14*v^3 + 15*v^2 - 6*v + 18) / 13 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$$ (-4*v^5 - v^4 - 10*v^3 - 6*v^2 - 34*v - 2) / 13 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$$ (-6*v^5 + 5*v^4 - 15*v^3 - 9*v^2 - 25*v + 10) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ -b5 + b4 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_{2}$$ b3 - 3*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$$ 2*b5 - 3*b4 - 4*b1 - 2 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$$ b5 - 4*b4 - 4*b3 + 9*b2 - 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 −0.623490 + 1.07992i 0.222521 − 0.385418i 0.900969 − 1.56052i −0.623490 − 1.07992i 0.222521 + 0.385418i 0.900969 + 1.56052i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −4.29590 0.500000 + 0.866025i −2.17845 3.77318i 1.00000 −0.500000 0.866025i 2.14795 3.72036i
529.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 2.13706 0.500000 + 0.866025i 0.0244587 + 0.0423637i 1.00000 −0.500000 0.866025i −1.06853 + 1.85075i
529.3 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 3.15883 0.500000 + 0.866025i −2.34601 4.06341i 1.00000 −0.500000 0.866025i −1.57942 + 2.73563i
991.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −4.29590 0.500000 0.866025i −2.17845 + 3.77318i 1.00000 −0.500000 + 0.866025i 2.14795 + 3.72036i
991.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 2.13706 0.500000 0.866025i 0.0244587 0.0423637i 1.00000 −0.500000 + 0.866025i −1.06853 1.85075i
991.3 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 3.15883 0.500000 0.866025i −2.34601 + 4.06341i 1.00000 −0.500000 + 0.866025i −1.57942 2.73563i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.e.l 6
13.b even 2 1 1014.2.e.n 6
13.c even 3 1 1014.2.a.n yes 3
13.c even 3 1 inner 1014.2.e.l 6
13.d odd 4 2 1014.2.i.h 12
13.e even 6 1 1014.2.a.l 3
13.e even 6 1 1014.2.e.n 6
13.f odd 12 2 1014.2.b.f 6
13.f odd 12 2 1014.2.i.h 12
39.h odd 6 1 3042.2.a.bh 3
39.i odd 6 1 3042.2.a.ba 3
39.k even 12 2 3042.2.b.o 6
52.i odd 6 1 8112.2.a.cj 3
52.j odd 6 1 8112.2.a.cm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l 3 13.e even 6 1
1014.2.a.n yes 3 13.c even 3 1
1014.2.b.f 6 13.f odd 12 2
1014.2.e.l 6 1.a even 1 1 trivial
1014.2.e.l 6 13.c even 3 1 inner
1014.2.e.n 6 13.b even 2 1
1014.2.e.n 6 13.e even 6 1
1014.2.i.h 12 13.d odd 4 2
1014.2.i.h 12 13.f odd 12 2
3042.2.a.ba 3 39.i odd 6 1
3042.2.a.bh 3 39.h odd 6 1
3042.2.b.o 6 39.k even 12 2
8112.2.a.cj 3 52.i odd 6 1
8112.2.a.cm 3 52.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5}^{3} - T_{5}^{2} - 16T_{5} + 29$$ T5^3 - T5^2 - 16*T5 + 29 $$T_{7}^{6} + 9T_{7}^{5} + 61T_{7}^{4} + 182T_{7}^{3} + 409T_{7}^{2} - 20T_{7} + 1$$ T7^6 + 9*T7^5 + 61*T7^4 + 182*T7^3 + 409*T7^2 - 20*T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$(T^{3} - T^{2} - 16 T + 29)^{2}$$
$7$ $$T^{6} + 9 T^{5} + 61 T^{4} + 182 T^{3} + \cdots + 1$$
$11$ $$T^{6} + 5 T^{5} + 33 T^{4} - 42 T^{3} + \cdots + 1$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 8 T^{5} + 52 T^{4} - 112 T^{3} + \cdots + 64$$
$19$ $$T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 4096$$
$23$ $$T^{6} + 28 T^{4} + 112 T^{3} + \cdots + 3136$$
$29$ $$T^{6} + 11 T^{5} + 97 T^{4} + \cdots + 841$$
$31$ $$(T^{3} - 5 T^{2} - 50 T + 125)^{2}$$
$37$ $$T^{6} - 8 T^{5} + 108 T^{4} + 336 T^{3} + \cdots + 64$$
$41$ $$T^{6} + 2 T^{5} + 68 T^{4} + \cdots + 53824$$
$43$ $$T^{6} + 12 T^{5} + 124 T^{4} + \cdots + 10816$$
$47$ $$(T^{3} - 4 T^{2} - 32 T + 64)^{2}$$
$53$ $$(T^{3} - 5 T^{2} - 36 T - 43)^{2}$$
$59$ $$T^{6} - 5 T^{5} + 61 T^{4} + \cdots + 27889$$
$61$ $$T^{6} + 22 T^{5} + 332 T^{4} + \cdots + 107584$$
$67$ $$T^{6} + 6 T^{5} + 52 T^{4} + \cdots + 10816$$
$71$ $$T^{6} - 18 T^{5} + 244 T^{4} + \cdots + 64$$
$73$ $$(T^{3} + 13 T^{2} - 30 T - 13)^{2}$$
$79$ $$(T^{3} - 31 T^{2} + 276 T - 533)^{2}$$
$83$ $$(T^{3} - 13 T^{2} - 128 T + 1567)^{2}$$
$89$ $$T^{6} - 14 T^{5} + 252 T^{4} + \cdots + 3136$$
$97$ $$T^{6} - 23 T^{5} + 439 T^{4} + \cdots + 9409$$