# Properties

 Label 1014.2.e.k 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: $$\Z[a_1, \ldots, a_{5}]$$ 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} + ( - \beta_{5} - \beta_{4} - 3 \beta_1 + 1) 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} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{17} + q^{18} + (4 \beta_{4} + 4 \beta_{3}) q^{19} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{20} + (2 \beta_{3} - \beta_{2} + 2) q^{21} + (\beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{22} + (6 \beta_{5} + 2 \beta_{4} - 6) q^{23} + (\beta_{5} - 1) q^{24} + ( - 4 \beta_{3} + 7 \beta_{2} - 3) q^{25} + q^{27} + (2 \beta_{5} + 2 \beta_{4} + \beta_1 - 2) q^{28} + (7 \beta_{5} + 3 \beta_{4} + 5 \beta_1 - 7) q^{29} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{30} + ( - 5 \beta_{3} + \beta_{2} - 5) q^{31} - \beta_{5} q^{32} + (\beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{33} + ( - 4 \beta_{3} + 2 \beta_{2} + 2) q^{34} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{35} + (\beta_{5} - 1) q^{36} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_1 + 4) q^{37} - 4 \beta_{3} q^{38} + (\beta_{3} - 2 \beta_{2}) q^{40} + ( - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 6) q^{41} + (2 \beta_{5} + 2 \beta_{4} + \beta_1 - 2) q^{42} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{43} + ( - \beta_{3} + 3 \beta_{2} - 1) q^{44} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{45} + ( - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{46} + 4 \beta_{2} q^{47} - \beta_{5} q^{48} + (2 \beta_{5} + 7 \beta_{4} + 3 \beta_1 - 2) q^{49} + ( - 3 \beta_{5} - 4 \beta_{4} - 7 \beta_1 + 3) q^{50} + ( - 4 \beta_{3} + 2 \beta_{2} + 2) q^{51} + (3 \beta_{3} + 6 \beta_{2} + 4) q^{53} + (\beta_{5} - 1) q^{54} + ( - 3 \beta_{5} + 5 \beta_{4} + 8 \beta_1 + 3) q^{55} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{56} - 4 \beta_{3} q^{57} + ( - 7 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{58} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{59} + (\beta_{3} - 2 \beta_{2}) q^{60} + (4 \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{61} + ( - 5 \beta_{5} - 5 \beta_{4} - \beta_1 + 5) q^{62} + (2 \beta_{5} + 2 \beta_{4} + \beta_1 - 2) q^{63} + q^{64} + ( - \beta_{3} + 3 \beta_{2} - 1) q^{66} + (2 \beta_{5} + 10 \beta_{4} + 2 \beta_1 - 2) q^{67} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_1 - 2) q^{68} + ( - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{69} + (\beta_{2} + 1) q^{70} + ( - 4 \beta_{5} - 6 \beta_{4} - 6 \beta_{3}) q^{71} - \beta_{5} q^{72} + ( - \beta_{3} + 4 \beta_{2}) q^{73} + (4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{74} + ( - 3 \beta_{5} - 4 \beta_{4} - 7 \beta_1 + 3) q^{75} - 4 \beta_{4} q^{76} + ( - \beta_{3} - \beta_{2} - 2) q^{77} + ( - 6 \beta_{3} - 5 \beta_{2} - 2) q^{79} + (\beta_{4} + 2 \beta_1) q^{80} + (\beta_{5} - 1) q^{81} + (6 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{82} + ( - 2 \beta_{3} - 9 \beta_{2}) q^{83} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{84} + (2 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} + 14 \beta_{2} - 14 \beta_1) q^{85} + ( - 2 \beta_{3} - 4 \beta_{2} - 2) q^{86} + ( - 7 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{87} + ( - \beta_{5} - \beta_{4} - 3 \beta_1 + 1) q^{88} + ( - 4 \beta_{5} - 2 \beta_{4} + 4) q^{89} + (\beta_{3} - 2 \beta_{2}) q^{90} + (2 \beta_{3} + 6) q^{92} + ( - 5 \beta_{5} - 5 \beta_{4} - \beta_1 + 5) q^{93} - 4 \beta_1 q^{94} + (4 \beta_{2} - 4 \beta_1) q^{95} + q^{96} + (3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{97} + ( - 2 \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{98} + ( - \beta_{3} + 3 \beta_{2} - 1) 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 + (-b5 - b4 - 3*b1 + 1) * q^11 + q^12 + (2*b3 - b2 + 2) * q^14 + (b4 + 2*b1) * q^15 + (b5 - 1) * q^16 + (-2*b5 + 4*b4 + 4*b3 - 2*b2 + 2*b1) * q^17 + q^18 + (4*b4 + 4*b3) * q^19 + (-b4 - b3 + 2*b2 - 2*b1) * q^20 + (2*b3 - b2 + 2) * q^21 + (b5 + b4 + b3 - 3*b2 + 3*b1) * q^22 + (6*b5 + 2*b4 - 6) * q^23 + (b5 - 1) * q^24 + (-4*b3 + 7*b2 - 3) * q^25 + q^27 + (2*b5 + 2*b4 + b1 - 2) * q^28 + (7*b5 + 3*b4 + 5*b1 - 7) * q^29 + (-b4 - b3 + 2*b2 - 2*b1) * q^30 + (-5*b3 + b2 - 5) * q^31 - b5 * q^32 + (b5 + b4 + b3 - 3*b2 + 3*b1) * q^33 + (-4*b3 + 2*b2 + 2) * q^34 + (-b5 - b2 + b1) * q^35 + (b5 - 1) * q^36 + (-4*b5 + 2*b4 - 2*b1 + 4) * q^37 - 4*b3 * q^38 + (b3 - 2*b2) * q^40 + (-6*b5 - 2*b4 - 2*b1 + 6) * q^41 + (2*b5 + 2*b4 + b1 - 2) * q^42 + (2*b5 + 2*b4 + 2*b3 + 4*b2 - 4*b1) * q^43 + (-b3 + 3*b2 - 1) * q^44 + (-b4 - b3 + 2*b2 - 2*b1) * q^45 + (-6*b5 - 2*b4 - 2*b3) * q^46 + 4*b2 * q^47 - b5 * q^48 + (2*b5 + 7*b4 + 3*b1 - 2) * q^49 + (-3*b5 - 4*b4 - 7*b1 + 3) * q^50 + (-4*b3 + 2*b2 + 2) * q^51 + (3*b3 + 6*b2 + 4) * q^53 + (b5 - 1) * q^54 + (-3*b5 + 5*b4 + 8*b1 + 3) * q^55 + (-2*b5 - 2*b4 - 2*b3 + b2 - b1) * q^56 - 4*b3 * q^57 + (-7*b5 - 3*b4 - 3*b3 + 5*b2 - 5*b1) * q^58 + (4*b5 + 2*b4 + 2*b3 - b2 + b1) * q^59 + (b3 - 2*b2) * q^60 + (4*b5 - 2*b2 + 2*b1) * q^61 + (-5*b5 - 5*b4 - b1 + 5) * q^62 + (2*b5 + 2*b4 + b1 - 2) * q^63 + q^64 + (-b3 + 3*b2 - 1) * q^66 + (2*b5 + 10*b4 + 2*b1 - 2) * q^67 + (2*b5 - 4*b4 - 2*b1 - 2) * q^68 + (-6*b5 - 2*b4 - 2*b3) * q^69 + (b2 + 1) * q^70 + (-4*b5 - 6*b4 - 6*b3) * q^71 - b5 * q^72 + (-b3 + 4*b2) * q^73 + (4*b5 - 2*b4 - 2*b3 - 2*b2 + 2*b1) * q^74 + (-3*b5 - 4*b4 - 7*b1 + 3) * q^75 - 4*b4 * q^76 + (-b3 - b2 - 2) * q^77 + (-6*b3 - 5*b2 - 2) * q^79 + (b4 + 2*b1) * q^80 + (b5 - 1) * q^81 + (6*b5 + 2*b4 + 2*b3 - 2*b2 + 2*b1) * q^82 + (-2*b3 - 9*b2) * q^83 + (-2*b5 - 2*b4 - 2*b3 + b2 - b1) * q^84 + (2*b5 - 6*b4 - 6*b3 + 14*b2 - 14*b1) * q^85 + (-2*b3 - 4*b2 - 2) * q^86 + (-7*b5 - 3*b4 - 3*b3 + 5*b2 - 5*b1) * q^87 + (-b5 - b4 - 3*b1 + 1) * q^88 + (-4*b5 - 2*b4 + 4) * q^89 + (b3 - 2*b2) * q^90 + (2*b3 + 6) * q^92 + (-5*b5 - 5*b4 - b1 + 5) * q^93 - 4*b1 * q^94 + (4*b2 - 4*b1) * q^95 + q^96 + (3*b4 + 3*b3 - 4*b2 + 4*b1) * q^97 + (-2*b5 - 7*b4 - 7*b3 + 3*b2 - 3*b1) * q^98 + (-b3 + 3*b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 6 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 6 * q^5 - 3 * q^6 - 3 * q^7 + 6 * q^8 - 3 * q^9 $$6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 6 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} - 3 q^{9} + 3 q^{10} - q^{11} + 6 q^{12} + 6 q^{14} + 3 q^{15} - 3 q^{16} - 12 q^{17} + 6 q^{18} - 4 q^{19} + 3 q^{20} + 6 q^{21} - q^{22} - 16 q^{23} - 3 q^{24} + 4 q^{25} + 6 q^{27} - 3 q^{28} - 13 q^{29} + 3 q^{30} - 18 q^{31} - 3 q^{32} - q^{33} + 24 q^{34} - 4 q^{35} - 3 q^{36} + 12 q^{37} + 8 q^{38} - 6 q^{40} + 14 q^{41} - 3 q^{42} + 8 q^{43} + 2 q^{44} + 3 q^{45} - 16 q^{46} + 8 q^{47} - 3 q^{48} + 4 q^{49} - 2 q^{50} + 24 q^{51} + 30 q^{53} - 3 q^{54} + 22 q^{55} - 3 q^{56} + 8 q^{57} - 13 q^{58} + 9 q^{59} - 6 q^{60} + 10 q^{61} + 9 q^{62} - 3 q^{63} + 6 q^{64} + 2 q^{66} + 6 q^{67} - 12 q^{68} - 16 q^{69} + 8 q^{70} - 6 q^{71} - 3 q^{72} + 10 q^{73} + 12 q^{74} - 2 q^{75} - 4 q^{76} - 12 q^{77} - 10 q^{79} + 3 q^{80} - 3 q^{81} + 14 q^{82} - 14 q^{83} - 3 q^{84} + 26 q^{85} - 16 q^{86} - 13 q^{87} - q^{88} + 10 q^{89} - 6 q^{90} + 32 q^{92} + 9 q^{93} - 4 q^{94} + 4 q^{95} + 6 q^{96} - 7 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100})$$ 6 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 6 * q^5 - 3 * q^6 - 3 * q^7 + 6 * q^8 - 3 * q^9 + 3 * q^10 - q^11 + 6 * q^12 + 6 * q^14 + 3 * q^15 - 3 * q^16 - 12 * q^17 + 6 * q^18 - 4 * q^19 + 3 * q^20 + 6 * q^21 - q^22 - 16 * q^23 - 3 * q^24 + 4 * q^25 + 6 * q^27 - 3 * q^28 - 13 * q^29 + 3 * q^30 - 18 * q^31 - 3 * q^32 - q^33 + 24 * q^34 - 4 * q^35 - 3 * q^36 + 12 * q^37 + 8 * q^38 - 6 * q^40 + 14 * q^41 - 3 * q^42 + 8 * q^43 + 2 * q^44 + 3 * q^45 - 16 * q^46 + 8 * q^47 - 3 * q^48 + 4 * q^49 - 2 * q^50 + 24 * q^51 + 30 * q^53 - 3 * q^54 + 22 * q^55 - 3 * q^56 + 8 * q^57 - 13 * q^58 + 9 * q^59 - 6 * q^60 + 10 * q^61 + 9 * q^62 - 3 * q^63 + 6 * q^64 + 2 * q^66 + 6 * q^67 - 12 * q^68 - 16 * q^69 + 8 * q^70 - 6 * q^71 - 3 * q^72 + 10 * q^73 + 12 * q^74 - 2 * q^75 - 4 * q^76 - 12 * q^77 - 10 * q^79 + 3 * q^80 - 3 * q^81 + 14 * q^82 - 14 * q^83 - 3 * q^84 + 26 * q^85 - 16 * q^86 - 13 * q^87 - q^88 + 10 * q^89 - 6 * q^90 + 32 * q^92 + 9 * q^93 - 4 * q^94 + 4 * q^95 + 6 * q^96 - 7 * q^97 + 4 * q^98 + 2 * 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
 0.900969 − 1.56052i 0.222521 − 0.385418i −0.623490 + 1.07992i 0.900969 + 1.56052i 0.222521 + 0.385418i −0.623490 − 1.07992i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −4.04892 −0.500000 0.866025i 0.346011 + 0.599308i 1.00000 −0.500000 0.866025i 2.02446 3.50647i
529.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.356896 −0.500000 0.866025i −2.02446 3.50647i 1.00000 −0.500000 0.866025i −0.178448 + 0.309081i
529.3 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0.692021 −0.500000 0.866025i 0.178448 + 0.309081i 1.00000 −0.500000 0.866025i −0.346011 + 0.599308i
991.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −4.04892 −0.500000 + 0.866025i 0.346011 0.599308i 1.00000 −0.500000 + 0.866025i 2.02446 + 3.50647i
991.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.356896 −0.500000 + 0.866025i −2.02446 + 3.50647i 1.00000 −0.500000 + 0.866025i −0.178448 0.309081i
991.3 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0.692021 −0.500000 + 0.866025i 0.178448 0.309081i 1.00000 −0.500000 + 0.866025i −0.346011 0.599308i
 $$n$$: 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.k 6
13.b even 2 1 1014.2.e.m 6
13.c even 3 1 1014.2.a.o yes 3
13.c even 3 1 inner 1014.2.e.k 6
13.d odd 4 2 1014.2.i.g 12
13.e even 6 1 1014.2.a.m 3
13.e even 6 1 1014.2.e.m 6
13.f odd 12 2 1014.2.b.g 6
13.f odd 12 2 1014.2.i.g 12
39.h odd 6 1 3042.2.a.be 3
39.i odd 6 1 3042.2.a.bd 3
39.k even 12 2 3042.2.b.r 6
52.i odd 6 1 8112.2.a.ce 3
52.j odd 6 1 8112.2.a.bz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 13.e even 6 1
1014.2.a.o yes 3 13.c even 3 1
1014.2.b.g 6 13.f odd 12 2
1014.2.e.k 6 1.a even 1 1 trivial
1014.2.e.k 6 13.c even 3 1 inner
1014.2.e.m 6 13.b even 2 1
1014.2.e.m 6 13.e even 6 1
1014.2.i.g 12 13.d odd 4 2
1014.2.i.g 12 13.f odd 12 2
3042.2.a.bd 3 39.i odd 6 1
3042.2.a.be 3 39.h odd 6 1
3042.2.b.r 6 39.k even 12 2
8112.2.a.bz 3 52.j odd 6 1
8112.2.a.ce 3 52.i odd 6 1

## Hecke kernels

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

 $$T_{5}^{3} + 3T_{5}^{2} - 4T_{5} + 1$$ T5^3 + 3*T5^2 - 4*T5 + 1 $$T_{7}^{6} + 3T_{7}^{5} + 13T_{7}^{4} - 14T_{7}^{3} + 13T_{7}^{2} - 4T_{7} + 1$$ T7^6 + 3*T7^5 + 13*T7^4 - 14*T7^3 + 13*T7^2 - 4*T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$(T^{2} + T + 1)^{3}$$
$5$ $$(T^{3} + 3 T^{2} - 4 T + 1)^{2}$$
$7$ $$T^{6} + 3 T^{5} + 13 T^{4} - 14 T^{3} + \cdots + 1$$
$11$ $$T^{6} + T^{5} + 17 T^{4} - 42 T^{3} + \cdots + 169$$
$13$ $$T^{6}$$
$17$ $$T^{6} + 12 T^{5} + 124 T^{4} + \cdots + 10816$$
$19$ $$T^{6} + 4 T^{5} + 48 T^{4} + \cdots + 4096$$
$23$ $$T^{6} + 16 T^{5} + 180 T^{4} + \cdots + 10816$$
$29$ $$T^{6} + 13 T^{5} + 157 T^{4} + \cdots + 49729$$
$31$ $$(T^{3} + 9 T^{2} - 22 T - 29)^{2}$$
$37$ $$T^{6} - 12 T^{5} + 124 T^{4} + \cdots + 64$$
$41$ $$T^{6} - 14 T^{5} + 140 T^{4} + \cdots + 3136$$
$43$ $$T^{6} - 8 T^{5} + 108 T^{4} + \cdots + 118336$$
$47$ $$(T^{3} - 4 T^{2} - 32 T + 64)^{2}$$
$53$ $$(T^{3} - 15 T^{2} - 72 T + 1247)^{2}$$
$59$ $$T^{6} - 9 T^{5} + 61 T^{4} - 154 T^{3} + \cdots + 169$$
$61$ $$T^{6} - 10 T^{5} + 76 T^{4} - 224 T^{3} + \cdots + 64$$
$67$ $$T^{6} - 6 T^{5} + 220 T^{4} + \cdots + 1236544$$
$71$ $$T^{6} + 6 T^{5} + 108 T^{4} + \cdots + 10816$$
$73$ $$(T^{3} - 5 T^{2} - 22 T + 13)^{2}$$
$79$ $$(T^{3} + 5 T^{2} - 204 T - 1469)^{2}$$
$83$ $$(T^{3} + 7 T^{2} - 224 T - 1477)^{2}$$
$89$ $$T^{6} - 10 T^{5} + 76 T^{4} - 224 T^{3} + \cdots + 64$$
$97$ $$T^{6} + 7 T^{5} + 63 T^{4} - 84 T^{3} + \cdots + 49$$