# Properties

 Label 1456.2.cc.c Level $1456$ Weight $2$ Character orbit 1456.cc Analytic conductor $11.626$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1456 = 2^{4} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1456.cc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.6262185343$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.58891012706304.1 Defining polynomial: $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ x^12 - 5*x^10 - 2*x^9 + 15*x^8 + 2*x^7 - 30*x^6 + 4*x^5 + 60*x^4 - 16*x^3 - 80*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta_{10} + \beta_{2}) q^{3} + ( - \beta_{11} - \beta_{7} - \beta_{5} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{4}) q^{7} + (\beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{9}+O(q^{10})$$ q + (b10 + b2) * q^3 + (-b11 - b7 - b5 - b3) * q^5 + (-b7 - b4) * q^7 + (b10 - 2*b8 - b7 + b6 + b4 + 2*b3 - b2) * q^9 $$q + (\beta_{10} + \beta_{2}) q^{3} + ( - \beta_{11} - \beta_{7} - \beta_{5} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{4}) q^{7} + (\beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{9} + ( - \beta_{11} - \beta_{8} + \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{11} + (2 \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \beta_{3} + \cdots + 1) q^{13}+ \cdots + ( - 4 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + 4 \beta_{8} - \beta_{7} - 2 \beta_{6} + \cdots + 3) q^{99}+O(q^{100})$$ q + (b10 + b2) * q^3 + (-b11 - b7 - b5 - b3) * q^5 + (-b7 - b4) * q^7 + (b10 - 2*b8 - b7 + b6 + b4 + 2*b3 - b2) * q^9 + (-b11 - b8 + b6 + b5 + 2*b3 - b2 + b1) * q^11 + (2*b11 - b10 - b9 - 2*b8 + b7 + 2*b6 + 2*b4 + b3 - b2 + 1) * q^13 + (b10 + b9 + 2*b8 - b6 - 2*b5 + 3*b4 - 2*b3 - b1 - 2) * q^15 + (-b11 + b9 + b6 - b5 + b1 - 1) * q^17 + (-2*b11 + 2*b8 - b6 + 2*b5 + 2*b3 + b1) * q^19 + (-b11 + b8 - b3) * q^21 + (2*b11 + 2*b9 - 4*b8 + 2*b6 - 4*b5 - 2*b3 - 2*b2 - 2*b1) * q^23 + (b11 - b10 + b6 - b5 - b3 - 2*b2 + 2*b1 - 2) * q^25 + (2*b11 - 2*b10 - b8 - b7 + 2*b6 - b5 - 2*b4 - 2*b3 - 1) * q^27 + (2*b11 + b10 - 2*b7 + b6 - b4 + 2*b3 - b1) * q^29 + (2*b10 + 2*b9 - b8 - 3*b7 + 2*b6 - b5 - 2*b2 - 2*b1 - 1) * q^31 + (-b11 - 4*b10 - b9 - b8 - b7 + 3*b6 + b5 - b4 - b3 - 4*b2 + b1 - 1) * q^33 + (b10 + b9 + b6 - b2 - 1) * q^35 + (-b11 + 3*b10 + 2*b9 - b8 + b5 + 3*b4 + 2*b3 - 3*b2 - 4) * q^37 + (b11 - 3*b10 - 2*b9 + b6 + 3*b5 - 4*b4 + b3 + 2*b2 + 4*b1 + 2) * q^39 + (-b11 - b10 - 2*b9 - b8 + b5 - 5*b4 + 2*b3 + b2 + 4) * q^41 + (2*b11 - 3*b10 - b9 - 2*b8 - b7 + b6 + 2*b5 + b4 + 2*b3 + 3*b2 + 4*b1 + 1) * q^43 + (5*b11 + 2*b10 - b9 - 2*b8 + 2*b7 - 2*b6 - 5*b5 + 2*b4 - 2*b3 + 2*b2 - 1) * q^45 + (2*b11 + 2*b10 - 6*b9 - 3*b8 - 3*b7 + 2*b6 - b5 + 2*b3 - 2*b2 - 2*b1 + 3) * q^47 + b9 * q^49 + (-b11 - b10 + 2*b7 + b6 + b5 + 4*b4 + b3 - 2*b2 + 2*b1 - 4) * q^51 + (2*b11 + 2*b10 - 2*b8 - 2*b6 - 2*b3 + 2*b2 - 2*b1 - 3) * q^53 + (5*b11 + 3*b10 - b9 - 4*b8 + 2*b7 - 4*b5 + b4 + b3 + 3*b2) * q^55 + (-2*b9 - 2*b8 + b7 - 2*b5 - 3*b2 - 3*b1 + 1) * q^57 + (3*b11 - 2*b10 - b9 - 3*b8 + b7 - 3*b5 + b4 - 3*b3 - 2*b2 + 2*b1 - 1) * q^59 + (2*b11 - 2*b9 + 3*b8 + b7 + b6 + 2*b5 - b4 - 3*b3 + b1 + 2) * q^61 + (-b11 + 2*b10 + b9 + b8 - 2*b6 - b5 - 2*b1 - 2) * q^63 + (3*b11 - 2*b9 - 2*b8 - 2*b7 - 4*b6 - b5 - b4 - b3 + 6*b2 + b1 + 6) * q^65 + (3*b11 - b10 - b9 + b8 + 2*b6 - b5 + 5*b4 - 4*b3 - b2 + 2*b1 + 2) * q^67 + (2*b11 - 4*b10 + 4*b8 + 4*b6 + 2*b5 - 4*b3 + 4*b2 + 8*b1) * q^69 + (-2*b10 + 2*b9 - 2*b8 + 2*b6 - 2*b3 - 2*b2 + 2) * q^71 + (b11 + b10 - 4*b9 + 5*b7 + b6 + b5 + b3 + b2 + b1 + 2) * q^73 + (-3*b11 - 3*b10 - 7*b9 + 4*b8 + 6*b7 - 2*b6 + 4*b5 + 3*b4 + b3 - b2 + 2*b1) * q^75 + (2*b10 + b8 - 2*b6 - b5 + b2 - b1) * q^77 + (-2*b11 + 4*b10 + 2*b8 - 4*b6 + 2*b3 + 4) * q^79 + (b11 + 3*b10 - b8 - 6*b7 - b5 - 3*b4 + 3*b2) * q^81 + (-2*b11 - 2*b9 + 3*b8 + 5*b7 + b5 - 2*b3 - 2*b2 - 2*b1 + 1) * q^83 + (4*b11 + b10 - 4*b9 + 3*b7 - b6 - 4*b5 + 3*b4 + b2 - 4) * q^85 + (b11 - 2*b10 - b9 - b8 + 3*b7 + 3*b6 + b5 - 3*b4 + b3 + 2*b2 + 5*b1 + 1) * q^87 + (-3*b10 - 2*b8 + 2*b5 + 6*b4 + 2*b3 + 3*b2) * q^89 + (b11 - b9 - b8 - b6 - 2*b5 - b4 - b3 - 2*b1 - 1) * q^91 + (b11 - 4*b10 + 2*b9 + 3*b8 + 5*b6 - 3*b5 + 2*b4 - 4*b3 - b2 + 5*b1 - 4) * q^93 + (b10 - 5*b9 - 4*b8 + b7 + b6 - b4 + 4*b3 - b2 + 5) * q^95 + (b11 - 4*b10 + b9 - 3*b8 + 5*b7 + 5*b6 - b5 + 5*b4 - 3*b3 - 4*b2 - b1 + 1) * q^97 + (-4*b11 - 2*b10 - 6*b9 + 4*b8 - b7 - 2*b6 - 4*b3 - b2 - b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{9}+O(q^{10})$$ 12 * q - 4 * q^9 $$12 q - 4 q^{9} - 6 q^{11} + 4 q^{13} - 6 q^{15} - 4 q^{17} + 12 q^{23} - 20 q^{25} - 12 q^{27} + 8 q^{29} - 30 q^{33} - 6 q^{35} - 42 q^{37} + 4 q^{39} + 30 q^{41} - 2 q^{43} + 6 q^{49} - 52 q^{51} - 44 q^{53} + 6 q^{55} - 18 q^{59} + 14 q^{61} - 12 q^{63} + 60 q^{65} + 24 q^{67} + 4 q^{69} + 24 q^{71} - 46 q^{75} + 8 q^{77} + 56 q^{79} + 2 q^{81} - 48 q^{85} + 2 q^{87} - 12 q^{89} - 14 q^{91} - 18 q^{93} + 22 q^{95} + 6 q^{97}+O(q^{100})$$ 12 * q - 4 * q^9 - 6 * q^11 + 4 * q^13 - 6 * q^15 - 4 * q^17 + 12 * q^23 - 20 * q^25 - 12 * q^27 + 8 * q^29 - 30 * q^33 - 6 * q^35 - 42 * q^37 + 4 * q^39 + 30 * q^41 - 2 * q^43 + 6 * q^49 - 52 * q^51 - 44 * q^53 + 6 * q^55 - 18 * q^59 + 14 * q^61 - 12 * q^63 + 60 * q^65 + 24 * q^67 + 4 * q^69 + 24 * q^71 - 46 * q^75 + 8 * q^77 + 56 * q^79 + 2 * q^81 - 48 * q^85 + 2 * q^87 - 12 * q^89 - 14 * q^91 - 18 * q^93 + 22 * q^95 + 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32$$ (v^11 - 5*v^9 - 2*v^8 + 15*v^7 + 2*v^6 - 30*v^5 + 4*v^4 + 60*v^3 - 16*v^2 - 80*v) / 32 $$\beta_{3}$$ $$=$$ $$( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96 ) / 32$$ (-3*v^11 + 4*v^10 + 7*v^9 - 6*v^8 - 13*v^7 + 30*v^6 - 6*v^5 - 28*v^4 - 4*v^3 + 16*v^2 - 48*v + 96) / 32 $$\beta_{4}$$ $$=$$ $$( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64 ) / 32$$ (3*v^11 + 2*v^10 - 11*v^9 - 8*v^8 + 21*v^7 + 4*v^6 - 42*v^5 + 84*v^3 + 24*v^2 - 64*v - 64) / 32 $$\beta_{5}$$ $$=$$ $$( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64 ) / 16$$ (v^11 + 3*v^10 - 3*v^9 - 13*v^8 + 7*v^7 + 23*v^6 - 26*v^5 - 38*v^4 + 60*v^3 + 68*v^2 - 56*v - 64) / 16 $$\beta_{6}$$ $$=$$ $$( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64 ) / 32$$ (-5*v^11 + 4*v^10 + 13*v^9 - 10*v^8 - 23*v^7 + 42*v^6 + 10*v^5 - 68*v^4 - 20*v^3 + 48*v^2 - 32*v + 64) / 32 $$\beta_{7}$$ $$=$$ $$( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32$$ (-5*v^11 + 2*v^10 + 17*v^9 - 8*v^8 - 39*v^7 + 44*v^6 + 50*v^5 - 88*v^4 - 68*v^3 + 120*v^2 + 16*v - 32) / 32 $$\beta_{8}$$ $$=$$ $$( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32 ) / 8$$ (-v^11 + v^10 + 5*v^9 - 3*v^8 - 13*v^7 + 13*v^6 + 20*v^5 - 34*v^4 - 28*v^3 + 52*v^2 + 24*v - 32) / 8 $$\beta_{9}$$ $$=$$ $$( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80 ) / 16$$ (-v^11 + 3*v^10 + 5*v^9 - 13*v^8 - 13*v^7 + 35*v^6 + 12*v^5 - 70*v^4 + 8*v^3 + 108*v^2 - 16*v - 80) / 16 $$\beta_{10}$$ $$=$$ $$( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64 ) / 16$$ (-3*v^11 + 15*v^9 - 2*v^8 - 37*v^7 + 18*v^6 + 66*v^5 - 68*v^4 - 92*v^3 + 96*v^2 + 96*v - 64) / 16 $$\beta_{11}$$ $$=$$ $$( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160 ) / 16$$ (-v^11 + 4*v^10 + 9*v^9 - 18*v^8 - 27*v^7 + 50*v^6 + 34*v^5 - 116*v^4 - 20*v^3 + 192*v^2 + 16*v - 160) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1$$ -b11 + b8 + b7 - b6 + b5 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1$$ -b11 + b10 + b9 - b6 + b4 + b3 + b2 + b1 $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1$$ -b11 + b9 + b7 - 2*b6 + b5 - b4 + b3 - b2 - 1 $$\nu^{5}$$ $$=$$ $$-2\beta_{11} + 4\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta _1 + 1$$ -2*b11 + 4*b9 + b8 - b7 - b6 - b5 - b1 + 1 $$\nu^{6}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta _1 - 2$$ 2*b11 - 2*b10 + b9 - b8 - b6 - 2*b5 - 3*b4 + b3 + b2 + 4*b1 - 2 $$\nu^{7}$$ $$=$$ $$- 5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1$$ -5*b11 - 3*b10 + 5*b9 + 7*b8 - b7 - 2*b6 + 3*b5 - 7*b4 - 3*b3 + b2 + 1 $$\nu^{8}$$ $$=$$ $$-3\beta_{11} + 7\beta_{8} + \beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 8\beta _1 - 3$$ -3*b11 + 7*b8 + b7 - 5*b6 - b5 + 2*b4 + 4*b3 + 10*b2 + 8*b1 - 3 $$\nu^{9}$$ $$=$$ $$- 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta _1 - 8$$ -7*b11 + 3*b10 + 3*b9 + 6*b8 - 4*b7 - 11*b6 + 12*b5 - 17*b4 + 7*b3 + 7*b2 + b1 - 8 $$\nu^{10}$$ $$=$$ $$- 17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta _1 - 11$$ -17*b11 + 4*b10 + 13*b9 + 18*b8 - 9*b7 - 8*b6 + 3*b5 + 7*b4 + 9*b3 + 3*b2 - 6*b1 - 11 $$\nu^{11}$$ $$=$$ $$18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta _1 - 7$$ 18*b11 + 4*b10 - 6*b9 - 13*b8 - 21*b7 - 11*b6 - b5 - 26*b4 + 22*b3 + 14*b2 + 3*b1 - 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$$.

 $$n$$ $$561$$ $$911$$ $$1093$$ $$1249$$ $$\chi(n)$$ $$1 - \beta_{9}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
225.1
 0.759479 + 1.19298i 1.40744 − 0.138282i 1.34408 − 0.439820i −1.12906 + 0.851598i −1.30089 − 0.554694i −1.08105 − 0.911778i 0.759479 − 1.19298i 1.40744 + 0.138282i 1.34408 + 0.439820i −1.12906 − 0.851598i −1.30089 + 0.554694i −1.08105 + 0.911778i
0 −1.41289 + 2.44719i 0 0.518957i 0 0.866025 0.500000i 0 −2.49250 4.31714i 0
225.2 0 −0.583963 + 1.01145i 0 1.81487i 0 0.866025 0.500000i 0 0.817975 + 1.41677i 0
225.3 0 −0.291146 + 0.504280i 0 1.68817i 0 −0.866025 + 0.500000i 0 1.33047 + 2.30444i 0
225.4 0 −0.172975 + 0.299601i 0 3.25812i 0 −0.866025 + 0.500000i 0 1.44016 + 2.49443i 0
225.5 0 1.13082 1.95864i 0 3.60178i 0 0.866025 0.500000i 0 −1.05753 1.83169i 0
225.6 0 1.33015 2.30388i 0 3.16209i 0 −0.866025 + 0.500000i 0 −2.03858 3.53092i 0
673.1 0 −1.41289 2.44719i 0 0.518957i 0 0.866025 + 0.500000i 0 −2.49250 + 4.31714i 0
673.2 0 −0.583963 1.01145i 0 1.81487i 0 0.866025 + 0.500000i 0 0.817975 1.41677i 0
673.3 0 −0.291146 0.504280i 0 1.68817i 0 −0.866025 0.500000i 0 1.33047 2.30444i 0
673.4 0 −0.172975 0.299601i 0 3.25812i 0 −0.866025 0.500000i 0 1.44016 2.49443i 0
673.5 0 1.13082 + 1.95864i 0 3.60178i 0 0.866025 + 0.500000i 0 −1.05753 + 1.83169i 0
673.6 0 1.33015 + 2.30388i 0 3.16209i 0 −0.866025 0.500000i 0 −2.03858 + 3.53092i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 673.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.cc.c 12
4.b odd 2 1 91.2.q.a 12
12.b even 2 1 819.2.ct.a 12
13.e even 6 1 inner 1456.2.cc.c 12
28.d even 2 1 637.2.q.h 12
28.f even 6 1 637.2.k.g 12
28.f even 6 1 637.2.u.i 12
28.g odd 6 1 637.2.k.h 12
28.g odd 6 1 637.2.u.h 12
52.i odd 6 1 91.2.q.a 12
52.i odd 6 1 1183.2.c.i 12
52.j odd 6 1 1183.2.c.i 12
52.l even 12 1 1183.2.a.m 6
52.l even 12 1 1183.2.a.p 6
156.r even 6 1 819.2.ct.a 12
364.s odd 6 1 637.2.k.h 12
364.w even 6 1 637.2.u.i 12
364.bc even 6 1 637.2.q.h 12
364.bk odd 6 1 637.2.u.h 12
364.bp even 6 1 637.2.k.g 12
364.bv odd 12 1 8281.2.a.by 6
364.bv odd 12 1 8281.2.a.ch 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 4.b odd 2 1
91.2.q.a 12 52.i odd 6 1
637.2.k.g 12 28.f even 6 1
637.2.k.g 12 364.bp even 6 1
637.2.k.h 12 28.g odd 6 1
637.2.k.h 12 364.s odd 6 1
637.2.q.h 12 28.d even 2 1
637.2.q.h 12 364.bc even 6 1
637.2.u.h 12 28.g odd 6 1
637.2.u.h 12 364.bk odd 6 1
637.2.u.i 12 28.f even 6 1
637.2.u.i 12 364.w even 6 1
819.2.ct.a 12 12.b even 2 1
819.2.ct.a 12 156.r even 6 1
1183.2.a.m 6 52.l even 12 1
1183.2.a.p 6 52.l even 12 1
1183.2.c.i 12 52.i odd 6 1
1183.2.c.i 12 52.j odd 6 1
1456.2.cc.c 12 1.a even 1 1 trivial
1456.2.cc.c 12 13.e even 6 1 inner
8281.2.a.by 6 364.bv odd 12 1
8281.2.a.ch 6 364.bv odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 11 T_{3}^{10} + 4 T_{3}^{9} + 96 T_{3}^{8} + 42 T_{3}^{7} + 287 T_{3}^{6} + 390 T_{3}^{5} + 709 T_{3}^{4} + 516 T_{3}^{3} + 300 T_{3}^{2} + 80 T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(1456, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 11 T^{10} + 4 T^{9} + 96 T^{8} + \cdots + 16$$
$5$ $$T^{12} + 40 T^{10} + 600 T^{8} + \cdots + 3481$$
$7$ $$(T^{4} - T^{2} + 1)^{3}$$
$11$ $$T^{12} + 6 T^{11} - 7 T^{10} - 114 T^{9} + \cdots + 256$$
$13$ $$T^{12} - 4 T^{11} + 21 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} + 4 T^{11} + 37 T^{10} + \cdots + 241081$$
$19$ $$T^{12} - 29 T^{10} + 748 T^{8} + \cdots + 55696$$
$23$ $$T^{12} - 12 T^{11} + 164 T^{10} + \cdots + 38539264$$
$29$ $$T^{12} - 8 T^{11} + 108 T^{10} + \cdots + 10042561$$
$31$ $$T^{12} + 136 T^{10} + 5854 T^{8} + \cdots + 913936$$
$37$ $$T^{12} + 42 T^{11} + \cdots + 1755945216$$
$41$ $$T^{12} - 30 T^{11} + \cdots + 884705536$$
$43$ $$T^{12} + 2 T^{11} + 113 T^{10} + \cdots + 2408704$$
$47$ $$T^{12} + 272 T^{10} + 21782 T^{8} + \cdots + 9461776$$
$53$ $$(T^{6} + 22 T^{5} + 91 T^{4} - 700 T^{3} + \cdots - 2339)^{2}$$
$59$ $$T^{12} + 18 T^{11} + \cdots + 4571923456$$
$61$ $$T^{12} - 14 T^{11} + 283 T^{10} + \cdots + 5607424$$
$67$ $$T^{12} - 24 T^{11} + \cdots + 613651984$$
$71$ $$T^{12} - 24 T^{11} + 212 T^{10} + \cdots + 46895104$$
$73$ $$T^{12} + 334 T^{10} + \cdots + 1386221824$$
$79$ $$(T^{6} - 28 T^{5} + 212 T^{4} - 192 T^{3} + \cdots - 512)^{2}$$
$83$ $$T^{12} + 304 T^{10} + \cdots + 141324544$$
$89$ $$T^{12} + 12 T^{11} + \cdots + 1834580224$$
$97$ $$T^{12} - 6 T^{11} - 173 T^{10} + \cdots + 53465344$$