# Properties

 Label 592.2.bc.d Level $592$ Weight $2$ Character orbit 592.bc Analytic conductor $4.727$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 592.bc (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.72714379966$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625$$ x^12 - 24*x^10 + 264*x^8 - 1687*x^6 + 6600*x^4 - 15000*x^2 + 15625 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{9} + \beta_1) q^{3} + (\beta_{5} + \beta_1 - 1) q^{5} + (\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{7} + (\beta_{11} - 2 \beta_{6} - \beta_{3}) q^{9}+O(q^{10})$$ q + (b9 + b1) * q^3 + (b5 + b1 - 1) * q^5 + (b10 + b8 + b7 - b6 + b5 - b4 - b2 - b1) * q^7 + (b11 - 2*b6 - b3) * q^9 $$q + (\beta_{9} + \beta_1) q^{3} + (\beta_{5} + \beta_1 - 1) q^{5} + (\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{7} + (\beta_{11} - 2 \beta_{6} - \beta_{3}) q^{9} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{4} - \beta_1) q^{11} + (\beta_{11} + \beta_{10} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{13} + (\beta_{11} - \beta_{9} + \beta_{6} + \beta_{3} - \beta_1) q^{15} + (\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{17} + ( - \beta_{10} + 3 \beta_{7} - \beta_{5} - 4 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{19} + (\beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta_1) q^{21} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 4 \beta_1 + 4) q^{23} + ( - \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_1) q^{25} + (\beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{7} - 4 \beta_{6} + 4 \beta_{5}) q^{27} + ( - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_1) q^{29} + (\beta_{9} - \beta_{8} - 4 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{31}+ \cdots + ( - \beta_{11} + 3 \beta_{9} + \beta_{8} + 4 \beta_{7} + 7 \beta_{6} - 5 \beta_{5} + \cdots + 7 \beta_1) q^{99}+O(q^{100})$$ q + (b9 + b1) * q^3 + (b5 + b1 - 1) * q^5 + (b10 + b8 + b7 - b6 + b5 - b4 - b2 - b1) * q^7 + (b11 - 2*b6 - b3) * q^9 + (b11 - b9 - b8 + b4 - b1) * q^11 + (b11 + b10 + b6 - b5 + b2 - b1) * q^13 + (b11 - b9 + b6 + b3 - b1) * q^15 + (b9 + b7 - b6 + b5 - b2) * q^17 + (-b10 + 3*b7 - b5 - 4*b4 - b3 + b2 + b1) * q^19 + (b10 - b9 + b8 + b7 + b5 - 5*b4 + b3 - b2 - 6*b1) * q^21 + (b10 + b9 + b8 + b7 - b6 + b5 - b4 - b2 - 4*b1 + 4) * q^23 + (-b7 - 3*b6 + 3*b5 + b4 - b1) * q^25 + (b11 - b10 - b9 - 5*b7 - 4*b6 + 4*b5) * q^27 + (-b10 + b9 - b7 + b6 - b5 - b4 + 2*b1) * q^29 + (b9 - b8 - 4*b7 + 4*b6 - 2*b5 + b4 - 2*b3 + 2*b2 + b1 - 4) * q^31 + (-b11 - b10 - b8 - b7 + 5*b5 + b4 + b3 - b2 + b1 + 4) * q^33 + (b9 - 2*b8 + b6 + 2*b4 + b3 + b1) * q^35 + (2*b11 - b10 - b9 - b8 + 4*b6 - 5*b5 + b4 - b3 + b2) * q^37 + (-b9 - b8 - 5*b6 + 4*b5 + 5*b4 - b3 - 5*b1 + 4) * q^39 + (b11 - 2*b10 - 2*b8 + 2*b7 + 3*b6 - 2*b5 + 2*b4 + 2*b3 - 2*b2 - b1 - 1) * q^41 + (-b11 - b10 + b9 - b8 - b7 + 2*b5 + b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^43 + (b10 - b9 + 2*b7 + b6 + b5 + b3 - 2*b1) * q^45 + (b11 + b10 + b9 + 2*b8 + b7 + 5*b6 - b5 + 2*b4 + b2) * q^47 + (-2*b11 + b9 - b8 - 4*b7 + 2*b6 - 3*b5 + 5*b4 - b3 + 2*b2 - 3*b1) * q^49 + (-5*b6 + b5 - 4*b4 - b2) * q^51 + (b10 + b8 + b7 + 3*b5 - b4 - b1) * q^53 + (-b10 - b9 - b7 - b5 - b3 + b2) * q^55 + (b11 + 3*b10 - b7 - b6 + 4*b4 - b3 + b1 - 4) * q^57 + (-b11 + b9 - b6 - 4*b5 - b3 + b1) * q^59 + (-b11 + b10 + 2*b8 - 3*b7 - 5*b6 - 2*b4 - 2*b3 + b2 + 3*b1 - 3) * q^61 + (-2*b11 + 3*b10 - b9 + 2*b8 + 7*b7 + 6*b6 - 6*b4 + 2*b3 - 8*b1) * q^63 + (b11 - b10 - b9 - b5 - b4 - b3 + b2 - b1 + 1) * q^65 + (b11 - b10 - b9 - 2*b8 - b7 + 5*b6 - 2*b5 + 2*b4 - b3 + 2*b2 + 4*b1) * q^67 + (-3*b11 + b10 + 3*b9 + b8 + b7 - 5*b6 + b5 - 5*b4 - b2 - 2*b1) * q^69 + (-2*b9 - 4*b5 + 2*b1) * q^71 + (-2*b11 - 2*b10 + b9 - b8 - 2*b7 + 2*b5 - 3*b4 - 2*b3 + 2*b2 + 3*b1 + 3) * q^73 + (-b11 - b10 - b7 + 3*b5 + 3*b3 - 3*b2 + b1) * q^75 + (b9 + 4*b7 - 4*b6 - 4*b4 + b1 + 4) * q^77 + (b11 + b10 - b9 + b8 + b7 - b6 - 3*b5 + 3*b4 - b2 - 6*b1 + 4) * q^79 + (b11 - b9 + 2*b8 + 6*b5 + b4 + 2*b3 - 2*b2 + 3*b1 - 4) * q^81 + (-b11 + b9 + 4*b6 + b5 + b3 - b2 + b1) * q^83 + (b11 - b9 - b8 - b7 + b6 + 2*b4 + 2*b3) * q^85 + (2*b11 + b10 + 2*b8 + b7 - 5*b6 - b5 - 2*b4 - 2*b3 + b2 + 3*b1 - 4) * q^87 + (3*b11 + 2*b10 - 2*b9 + 2*b8 + b7 + 3*b6 - 5*b5 - b4 + 2*b3 + b2 - 4*b1 + 1) * q^89 + (-b10 - b7 - 4*b6 - 4*b4 + b1 + 4) * q^91 + (-3*b10 - 3*b9 + b8 - 11*b7 - 3*b6 - 4*b5 + 7*b4 - 3*b3 + 4*b2 + 4) * q^93 + (-b11 + 2*b10 + 2*b8 + 2*b7 - b6 + b5 + 2*b4 - b2 + 2*b1) * q^95 + (-b11 - b8 + 3*b7 + b6 + 2*b4 - 3*b2 + 8*b1 - 8) * q^97 + (-b11 + 3*b9 + b8 + 4*b7 + 7*b6 - 5*b5 - 5*b4 + 2*b3 + b2 + 7*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 3 q^{3} - 6 q^{5} - 6 q^{7} - 3 q^{9}+O(q^{10})$$ 12 * q + 3 * q^3 - 6 * q^5 - 6 * q^7 - 3 * q^9 $$12 q + 3 q^{3} - 6 q^{5} - 6 q^{7} - 3 q^{9} - 3 q^{11} - 6 q^{13} - 6 q^{15} - 3 q^{17} + 3 q^{19} - 33 q^{21} + 21 q^{23} - 6 q^{25} - 3 q^{27} + 6 q^{29} - 42 q^{31} + 57 q^{33} + 9 q^{35} - 3 q^{37} + 24 q^{39} - 21 q^{41} - 36 q^{43} - 6 q^{45} - 9 q^{47} - 12 q^{49} - 6 q^{53} - 36 q^{57} + 6 q^{59} - 18 q^{61} - 36 q^{63} + 3 q^{65} + 27 q^{67} - 12 q^{69} + 18 q^{71} + 54 q^{73} + 6 q^{75} + 51 q^{77} + 12 q^{79} - 36 q^{81} + 6 q^{83} + 3 q^{85} - 39 q^{87} - 15 q^{89} + 51 q^{91} + 45 q^{93} + 15 q^{95} - 42 q^{97} + 33 q^{99}+O(q^{100})$$ 12 * q + 3 * q^3 - 6 * q^5 - 6 * q^7 - 3 * q^9 - 3 * q^11 - 6 * q^13 - 6 * q^15 - 3 * q^17 + 3 * q^19 - 33 * q^21 + 21 * q^23 - 6 * q^25 - 3 * q^27 + 6 * q^29 - 42 * q^31 + 57 * q^33 + 9 * q^35 - 3 * q^37 + 24 * q^39 - 21 * q^41 - 36 * q^43 - 6 * q^45 - 9 * q^47 - 12 * q^49 - 6 * q^53 - 36 * q^57 + 6 * q^59 - 18 * q^61 - 36 * q^63 + 3 * q^65 + 27 * q^67 - 12 * q^69 + 18 * q^71 + 54 * q^73 + 6 * q^75 + 51 * q^77 + 12 * q^79 - 36 * q^81 + 6 * q^83 + 3 * q^85 - 39 * q^87 - 15 * q^89 + 51 * q^91 + 45 * q^93 + 15 * q^95 - 42 * q^97 + 33 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625$$ :

 $$\beta_{1}$$ $$=$$ $$( -12\nu^{11} + 263\nu^{9} - 2568\nu^{7} + 13644\nu^{5} - 40150\nu^{3} + 52500\nu + 3125 ) / 6250$$ (-12*v^11 + 263*v^9 - 2568*v^7 + 13644*v^5 - 40150*v^3 + 52500*v + 3125) / 6250 $$\beta_{2}$$ $$=$$ $$( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} + 125\nu - 7500 ) / 250$$ (v^10 - 24*v^8 + 264*v^6 - 1562*v^4 + 5100*v^2 + 125*v - 7500) / 250 $$\beta_{3}$$ $$=$$ $$( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} - 125\nu - 7500 ) / 250$$ (v^10 - 24*v^8 + 264*v^6 - 1562*v^4 + 5100*v^2 - 125*v - 7500) / 250 $$\beta_{4}$$ $$=$$ $$( \nu^{11} - 45 \nu^{10} - 24 \nu^{9} + 955 \nu^{8} + 264 \nu^{7} - 8880 \nu^{6} - 1687 \nu^{5} + 46040 \nu^{4} + 6600 \nu^{3} - 133000 \nu^{2} - 11875 \nu + 175000 ) / 6250$$ (v^11 - 45*v^10 - 24*v^9 + 955*v^8 + 264*v^7 - 8880*v^6 - 1687*v^5 + 46040*v^4 + 6600*v^3 - 133000*v^2 - 11875*v + 175000) / 6250 $$\beta_{5}$$ $$=$$ $$( \nu^{11} + 45 \nu^{10} - 24 \nu^{9} - 955 \nu^{8} + 264 \nu^{7} + 8880 \nu^{6} - 1687 \nu^{5} - 46040 \nu^{4} + 6600 \nu^{3} + 133000 \nu^{2} - 11875 \nu - 175000 ) / 6250$$ (v^11 + 45*v^10 - 24*v^9 - 955*v^8 + 264*v^7 + 8880*v^6 - 1687*v^5 - 46040*v^4 + 6600*v^3 + 133000*v^2 - 11875*v - 175000) / 6250 $$\beta_{6}$$ $$=$$ $$( 12 \nu^{11} + 40 \nu^{10} - 188 \nu^{9} - 835 \nu^{8} + 1393 \nu^{7} + 7560 \nu^{6} - 5719 \nu^{5} - 37605 \nu^{4} + 13000 \nu^{3} + 103125 \nu^{2} - 11875 \nu - 125000 ) / 6250$$ (12*v^11 + 40*v^10 - 188*v^9 - 835*v^8 + 1393*v^7 + 7560*v^6 - 5719*v^5 - 37605*v^4 + 13000*v^3 + 103125*v^2 - 11875*v - 125000) / 6250 $$\beta_{7}$$ $$=$$ $$( 12 \nu^{11} - 40 \nu^{10} - 188 \nu^{9} + 835 \nu^{8} + 1393 \nu^{7} - 7560 \nu^{6} - 5719 \nu^{5} + 37605 \nu^{4} + 13000 \nu^{3} - 103125 \nu^{2} - 11875 \nu + 125000 ) / 6250$$ (12*v^11 - 40*v^10 - 188*v^9 + 835*v^8 + 1393*v^7 - 7560*v^6 - 5719*v^5 + 37605*v^4 + 13000*v^3 - 103125*v^2 - 11875*v + 125000) / 6250 $$\beta_{8}$$ $$=$$ $$( \nu^{11} - 20 \nu^{10} - 24 \nu^{9} + 355 \nu^{8} + 264 \nu^{7} - 2905 \nu^{6} - 1687 \nu^{5} + 13240 \nu^{4} + 5975 \nu^{3} - 33000 \nu^{2} - 10000 \nu + 34375 ) / 1250$$ (v^11 - 20*v^10 - 24*v^9 + 355*v^8 + 264*v^7 - 2905*v^6 - 1687*v^5 + 13240*v^4 + 5975*v^3 - 33000*v^2 - 10000*v + 34375) / 1250 $$\beta_{9}$$ $$=$$ $$( - \nu^{11} - 20 \nu^{10} + 24 \nu^{9} + 355 \nu^{8} - 264 \nu^{7} - 2905 \nu^{6} + 1687 \nu^{5} + 13240 \nu^{4} - 5975 \nu^{3} - 33000 \nu^{2} + 10000 \nu + 34375 ) / 1250$$ (-v^11 - 20*v^10 + 24*v^9 + 355*v^8 - 264*v^7 - 2905*v^6 + 1687*v^5 + 13240*v^4 - 5975*v^3 - 33000*v^2 + 10000*v + 34375) / 1250 $$\beta_{10}$$ $$=$$ $$( 28 \nu^{11} + 100 \nu^{10} - 572 \nu^{9} - 1775 \nu^{8} + 4992 \nu^{7} + 14525 \nu^{6} - 23961 \nu^{5} - 66200 \nu^{4} + 62975 \nu^{3} + 168125 \nu^{2} - 72500 \nu - 184375 ) / 6250$$ (28*v^11 + 100*v^10 - 572*v^9 - 1775*v^8 + 4992*v^7 + 14525*v^6 - 23961*v^5 - 66200*v^4 + 62975*v^3 + 168125*v^2 - 72500*v - 184375) / 6250 $$\beta_{11}$$ $$=$$ $$( 28 \nu^{11} - 100 \nu^{10} - 572 \nu^{9} + 1775 \nu^{8} + 4992 \nu^{7} - 14525 \nu^{6} - 23961 \nu^{5} + 66200 \nu^{4} + 62975 \nu^{3} - 168125 \nu^{2} - 72500 \nu + 184375 ) / 6250$$ (28*v^11 - 100*v^10 - 572*v^9 + 1775*v^8 + 4992*v^7 - 14525*v^6 - 23961*v^5 + 66200*v^4 + 62975*v^3 - 168125*v^2 - 72500*v + 184375) / 6250
 $$\nu$$ $$=$$ $$-\beta_{3} + \beta_{2}$$ -b3 + b2 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 4$$ -b11 + b10 + b9 + b8 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} + 5\beta_{5} + 5\beta_{4} - 3\beta_{3} + 3\beta_{2}$$ b9 - b8 + 5*b5 + 5*b4 - 3*b3 + 3*b2 $$\nu^{4}$$ $$=$$ $$- 7 \beta_{11} + 7 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 8$$ -7*b11 + 7*b10 + 7*b9 + 7*b8 - 5*b7 + 5*b6 - 5*b5 + 5*b4 + b3 + b2 + 8 $$\nu^{5}$$ $$=$$ $$-\beta_{11} - \beta_{10} + 11\beta_{9} - 11\beta_{8} + 35\beta_{5} + 35\beta_{4} - \beta_{3} + \beta_{2} - 8\beta _1 + 4$$ -b11 - b10 + 11*b9 - 11*b8 + 35*b5 + 35*b4 - b3 + b2 - 8*b1 + 4 $$\nu^{6}$$ $$=$$ $$- 26 \beta_{11} + 26 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} - 50 \beta_{7} + 50 \beta_{6} - 55 \beta_{5} + 55 \beta_{4} + 15 \beta_{3} + 15 \beta_{2} - 21$$ -26*b11 + 26*b10 + 25*b9 + 25*b8 - 50*b7 + 50*b6 - 55*b5 + 55*b4 + 15*b3 + 15*b2 - 21 $$\nu^{7}$$ $$=$$ $$- 19 \beta_{11} - 19 \beta_{10} + 65 \beta_{9} - 65 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 125 \beta_{5} + 125 \beta_{4} + 46 \beta_{3} - 46 \beta_{2} - 112 \beta _1 + 56$$ -19*b11 - 19*b10 + 65*b9 - 65*b8 + 5*b7 + 5*b6 + 125*b5 + 125*b4 + 46*b3 - 46*b2 - 112*b1 + 56 $$\nu^{8}$$ $$=$$ $$- 38 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} + 14 \beta_{8} - 230 \beta_{7} + 230 \beta_{6} - 325 \beta_{5} + 325 \beta_{4} + 121 \beta_{3} + 121 \beta_{2} - 268$$ -38*b11 + 38*b10 + 14*b9 + 14*b8 - 230*b7 + 230*b6 - 325*b5 + 325*b4 + 121*b3 + 121*b2 - 268 $$\nu^{9}$$ $$=$$ $$- 192 \beta_{11} - 192 \beta_{10} + 218 \beta_{9} - 218 \beta_{8} + 120 \beta_{7} + 120 \beta_{6} + 70 \beta_{5} + 70 \beta_{4} + 282 \beta_{3} - 282 \beta_{2} - 826 \beta _1 + 413$$ -192*b11 - 192*b10 + 218*b9 - 218*b8 + 120*b7 + 120*b6 + 70*b5 + 70*b4 + 282*b3 - 282*b2 - 826*b1 + 413 $$\nu^{10}$$ $$=$$ $$118 \beta_{11} - 118 \beta_{10} - 430 \beta_{9} - 430 \beta_{8} - 130 \beta_{7} + 130 \beta_{6} - 1090 \beta_{5} + 1090 \beta_{4} + 631 \beta_{3} + 631 \beta_{2} - 1292$$ 118*b11 - 118*b10 - 430*b9 - 430*b8 - 130*b7 + 130*b6 - 1090*b5 + 1090*b4 + 631*b3 + 631*b2 - 1292 $$\nu^{11}$$ $$=$$ $$- 1279 \beta_{11} - 1279 \beta_{10} + 29 \beta_{9} - 29 \beta_{8} + 1560 \beta_{7} + 1560 \beta_{6} - 2150 \beta_{5} - 2150 \beta_{4} + 862 \beta_{3} - 862 \beta_{2} - 3752 \beta _1 + 1876$$ -1279*b11 - 1279*b10 + 29*b9 - 29*b8 + 1560*b7 + 1560*b6 - 2150*b5 - 2150*b4 + 862*b3 - 862*b2 - 3752*b1 + 1876

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$$.

 $$n$$ $$113$$ $$149$$ $$223$$ $$\chi(n)$$ $$\beta_{7}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 −2.14169 + 0.642788i 2.14169 + 0.642788i −2.20976 − 0.342020i 2.20976 − 0.342020i 2.00752 − 0.984808i −2.00752 − 0.984808i −2.20976 + 0.342020i 2.20976 + 0.342020i −2.14169 − 0.642788i 2.14169 − 0.642788i 2.00752 + 0.984808i −2.00752 + 0.984808i
0 −0.504922 2.86356i 0 0.266044 0.223238i 0 −0.773586 + 0.649116i 0 −5.12593 + 1.86569i 0
33.2 0 0.238878 + 1.35474i 0 0.266044 0.223238i 0 0.365982 0.307095i 0 1.04081 0.378824i 0
49.1 0 −0.972925 0.816381i 0 −1.43969 + 0.524005i 0 1.82850 0.665520i 0 −0.240839 1.36587i 0
49.2 0 2.41262 + 2.02443i 0 −1.43969 + 0.524005i 0 −4.53424 + 1.65033i 0 1.20148 + 6.81391i 0
81.1 0 −1.72328 0.627223i 0 −0.326352 1.85083i 0 −0.598489 3.39420i 0 0.278152 + 0.233397i 0
81.2 0 2.04963 + 0.746005i 0 −0.326352 1.85083i 0 0.711830 + 4.03699i 0 1.34633 + 1.12971i 0
145.1 0 −0.972925 + 0.816381i 0 −1.43969 0.524005i 0 1.82850 + 0.665520i 0 −0.240839 + 1.36587i 0
145.2 0 2.41262 2.02443i 0 −1.43969 0.524005i 0 −4.53424 1.65033i 0 1.20148 6.81391i 0
305.1 0 −0.504922 + 2.86356i 0 0.266044 + 0.223238i 0 −0.773586 0.649116i 0 −5.12593 1.86569i 0
305.2 0 0.238878 1.35474i 0 0.266044 + 0.223238i 0 0.365982 + 0.307095i 0 1.04081 + 0.378824i 0
497.1 0 −1.72328 + 0.627223i 0 −0.326352 + 1.85083i 0 −0.598489 + 3.39420i 0 0.278152 0.233397i 0
497.2 0 2.04963 0.746005i 0 −0.326352 + 1.85083i 0 0.711830 4.03699i 0 1.34633 1.12971i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 497.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.bc.d 12
4.b odd 2 1 74.2.f.b 12
12.b even 2 1 666.2.x.g 12
37.f even 9 1 inner 592.2.bc.d 12
148.o odd 18 1 2738.2.a.q 6
148.p odd 18 1 74.2.f.b 12
148.p odd 18 1 2738.2.a.t 6
444.z even 18 1 666.2.x.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.b 12 4.b odd 2 1
74.2.f.b 12 148.p odd 18 1
592.2.bc.d 12 1.a even 1 1 trivial
592.2.bc.d 12 37.f even 9 1 inner
666.2.x.g 12 12.b even 2 1
666.2.x.g 12 444.z even 18 1
2738.2.a.q 6 148.o odd 18 1
2738.2.a.t 6 148.p odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 3 T_{3}^{11} + 6 T_{3}^{10} - 8 T_{3}^{9} + 24 T_{3}^{8} + 126 T_{3}^{7} - 151 T_{3}^{6} - 504 T_{3}^{5} + 384 T_{3}^{4} + 512 T_{3}^{3} + 1536 T_{3}^{2} + 3072 T_{3} + 4096$$ acting on $$S_{2}^{\mathrm{new}}(592, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 3 T^{11} + 6 T^{10} - 8 T^{9} + \cdots + 4096$$
$5$ $$(T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + 3 T^{2} + \cdots + 1)^{2}$$
$7$ $$T^{12} + 6 T^{11} + 24 T^{10} + \cdots + 4096$$
$11$ $$T^{12} + 3 T^{11} + 33 T^{10} + \cdots + 4096$$
$13$ $$T^{12} + 6 T^{11} - 24 T^{10} + \cdots + 516961$$
$17$ $$T^{12} + 3 T^{11} + 18 T^{10} + \cdots + 26569$$
$19$ $$T^{12} - 3 T^{11} - 9 T^{10} - 228 T^{9} + \cdots + 4096$$
$23$ $$T^{12} - 21 T^{11} + 297 T^{10} + \cdots + 4096$$
$29$ $$T^{12} - 6 T^{11} + 60 T^{10} + \cdots + 1369$$
$31$ $$(T^{6} + 21 T^{5} + 24 T^{4} + \cdots + 130112)^{2}$$
$37$ $$T^{12} + 3 T^{11} + \cdots + 2565726409$$
$41$ $$T^{12} + 21 T^{11} + \cdots + 466905664$$
$43$ $$(T^{6} + 18 T^{5} + 51 T^{4} - 639 T^{3} + \cdots + 8704)^{2}$$
$47$ $$T^{12} + 9 T^{11} + \cdots + 6890328064$$
$53$ $$T^{12} + 6 T^{11} + 6 T^{10} + 109 T^{9} + \cdots + 289$$
$59$ $$T^{12} - 6 T^{11} + 12 T^{10} + \cdots + 1183744$$
$61$ $$T^{12} + 18 T^{11} + \cdots + 2801373184$$
$67$ $$T^{12} - 27 T^{11} + 444 T^{10} + \cdots + 262144$$
$71$ $$T^{12} - 18 T^{11} + 192 T^{10} + \cdots + 262144$$
$73$ $$(T^{6} - 27 T^{5} + 108 T^{4} + \cdots + 216289)^{2}$$
$79$ $$T^{12} - 12 T^{11} - 54 T^{10} + \cdots + 95883264$$
$83$ $$T^{12} - 6 T^{11} - 24 T^{10} + \cdots + 1183744$$
$89$ $$T^{12} + 15 T^{11} + \cdots + 48144697561$$
$97$ $$T^{12} + 42 T^{11} + 1215 T^{10} + \cdots + 7529536$$