# Properties

 Label 336.6.q.l Level $336$ Weight $6$ Character orbit 336.q Analytic conductor $53.889$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$53.8889634572$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 564x^{8} + 117814x^{6} + 11067780x^{4} + 427918225x^{2} + 3489248448$$ x^10 + 564*x^8 + 117814*x^6 + 11067780*x^4 + 427918225*x^2 + 3489248448 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}\cdot 3^{3}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 9 \beta_1 q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{4} - 9 \beta_1 + 14) q^{7} + (81 \beta_1 - 81) q^{9}+O(q^{10})$$ q - 9*b1 * q^3 + (-b6 + b1 - 1) * q^5 + (b6 - b5 - b4 - 9*b1 + 14) * q^7 + (81*b1 - 81) * q^9 $$q - 9 \beta_1 q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{4} - 9 \beta_1 + 14) q^{7} + (81 \beta_1 - 81) q^{9} + ( - 3 \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{5} + \beta_{3} - 84 \beta_1) q^{11} + ( - 2 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} + \cdots + 38) q^{13}+ \cdots + (243 \beta_{9} - 162 \beta_{7} + 81 \beta_{6} - 81 \beta_{5} - 81 \beta_{4} + \cdots + 6804) q^{99}+O(q^{100})$$ q - 9*b1 * q^3 + (-b6 + b1 - 1) * q^5 + (b6 - b5 - b4 - 9*b1 + 14) * q^7 + (81*b1 - 81) * q^9 + (-3*b9 - b8 + 2*b7 + b5 + b3 - 84*b1) * q^11 + (-2*b9 - 3*b8 + 2*b7 - b6 + b5 + 3*b4 + 3*b3 - 2*b2 + 38) * q^13 + (9*b6 - 9*b5 + 9) * q^15 + (-2*b9 - 3*b8 - b7 - 2*b4 + 5*b3 - 190*b1) * q^17 + (6*b8 - 8*b7 - 7*b6 + 6*b4 - 8*b3 + 3*b2 + 27*b1 - 27) * q^19 + (-9*b7 + 9*b5 + 9*b4 - 45*b1 - 81) * q^21 + (15*b8 - 8*b7 + 16*b6 + 15*b4 - 8*b3 - 4*b2 + 860*b1 - 860) * q^23 + (-25*b9 - 15*b8 + 10*b7 + 25*b5 - 10*b4 + 25*b3 - 1111*b1) * q^25 + 729 * q^27 + (36*b9 - 4*b8 - 15*b7 + 41*b6 - 41*b5 - 17*b4 - 17*b3 + 36*b2 - 437) * q^29 + (-18*b9 + 19*b8 + 37*b7 - 2*b5 - 39*b4 + 20*b3 - 1623*b1) * q^31 + (9*b8 - 9*b6 + 9*b4 - 27*b2 + 756*b1 - 756) * q^33 + (7*b9 + 63*b8 - 13*b7 - b6 - 63*b5 + 21*b4 - 14*b3 - 35*b2 + 2632*b1 - 3330) * q^35 + (4*b8 - 11*b7 - 41*b6 + 4*b4 - 11*b3 - 24*b2 + 1072*b1 - 1072) * q^37 + (18*b9 + 27*b8 + 9*b7 - 9*b5 - 27*b4 - 342*b1) * q^39 + (-22*b9 - 32*b8 + 50*b7 - 166*b6 + 166*b5 + 4*b4 + 4*b3 - 22*b2 + 2974) * q^41 + (27*b9 + 30*b8 - 23*b7 + 69*b6 - 69*b5 - 34*b4 - 34*b3 + 27*b2 + 4667) * q^43 + (81*b5 - 81*b1) * q^45 + (43*b8 - 46*b7 - 168*b6 + 43*b4 - 46*b3 + 78*b2 - 3456*b1 + 3456) * q^47 + (63*b9 - 28*b8 + 18*b7 + 127*b6 + 23*b5 - 33*b4 + 35*b3 - 7*b2 - 3153*b1 + 3886) * q^49 + (45*b8 - 18*b7 + 45*b4 - 18*b3 - 18*b2 + 1710*b1 - 1710) * q^51 + (44*b9 - 40*b8 - 84*b7 - 195*b5 + 130*b4 - 90*b3 + 2965*b1) * q^53 + (-20*b9 + 64*b8 + 25*b7 + 473*b6 - 473*b5 - 69*b4 - 69*b3 - 20*b2 + 27) * q^55 + (-27*b9 - 72*b8 + 81*b7 + 63*b6 - 63*b5 + 18*b4 + 18*b3 - 27*b2 + 243) * q^57 + (69*b9 + 158*b8 + 89*b7 - 93*b5 + 36*b4 - 194*b3 + 16616*b1) * q^59 + (-126*b8 - 50*b7 + 100*b6 - 126*b4 - 50*b3 + 224*b2 - 2926*b1 + 2926) * q^61 + (81*b7 - 81*b6 + 1134*b1 - 405) * q^63 + (-111*b8 - 14*b7 + 170*b6 - 111*b4 - 14*b3 + 60*b2 + 4696*b1 - 4696) * q^65 + (-61*b9 + 100*b8 + 161*b7 + 255*b5 + 100*b4 - 200*b3 - 7837*b1) * q^67 + (36*b9 - 72*b8 + 99*b7 - 144*b6 + 144*b5 - 63*b4 - 63*b3 + 36*b2 + 7740) * q^69 + (86*b9 + 238*b8 - 97*b7 + 424*b6 - 424*b5 - 227*b4 - 227*b3 + 86*b2 + 5500) * q^71 + (-33*b9 - 31*b8 + 2*b7 - 63*b5 + 2*b4 + 29*b3 - 18085*b1) * q^73 + (225*b8 - 90*b7 - 225*b6 + 225*b4 - 90*b3 - 225*b2 + 9999*b1 - 9999) * q^75 + (-224*b9 + 91*b8 + 82*b7 - 71*b6 + 640*b5 + 157*b4 - 140*b3 - 252*b2 + 4605*b1 + 4161) * q^77 + (64*b8 - 413*b7 + 792*b6 + 64*b4 - 413*b3 + 90*b2 - 8437*b1 + 8437) * q^79 - 6561*b1 * q^81 + (-147*b9 + 124*b8 + 117*b7 + 493*b6 - 493*b5 - 94*b4 - 94*b3 - 147*b2 + 7498) * q^83 + (20*b9 + 214*b8 - 84*b7 + 784*b6 - 784*b5 - 150*b4 - 150*b3 + 20*b2 - 7688) * q^85 + (-324*b9 - 153*b8 + 171*b7 + 369*b5 - 36*b4 + 189*b3 + 3933*b1) * q^87 + (296*b8 - 232*b7 + 498*b6 + 296*b4 - 232*b3 - 296*b2 + 14562*b1 - 14562) * q^89 + (-42*b8 - 106*b7 - 257*b6 + 90*b5 + 20*b4 - 406*b3 + 35*b2 - 1717*b1 - 27939) * q^91 + (180*b8 - 351*b7 + 18*b6 + 180*b4 - 351*b3 - 162*b2 + 14607*b1 - 14607) * q^93 + (-172*b9 - 73*b8 + 99*b7 + 406*b5 + 122*b4 - 49*b3 - 37912*b1) * q^95 + (-223*b9 + 286*b8 - 110*b7 - 87*b6 + 87*b5 + 47*b4 + 47*b3 - 223*b2 + 18362) * q^97 + (243*b9 - 162*b7 + 81*b6 - 81*b5 - 81*b4 - 81*b3 + 243*b2 + 6804) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 45 q^{3} - 6 q^{5} + 97 q^{7} - 405 q^{9}+O(q^{10})$$ 10 * q - 45 * q^3 - 6 * q^5 + 97 * q^7 - 405 * q^9 $$10 q - 45 q^{3} - 6 q^{5} + 97 q^{7} - 405 q^{9} - 424 q^{11} + 374 q^{13} + 108 q^{15} - 952 q^{17} - 139 q^{19} - 1044 q^{21} - 4288 q^{23} - 5605 q^{25} + 7290 q^{27} - 4216 q^{29} - 8131 q^{31} - 3816 q^{33} - 20106 q^{35} - 5425 q^{37} - 1683 q^{39} + 29364 q^{41} + 46862 q^{43} - 486 q^{45} + 17190 q^{47} + 23255 q^{49} - 8568 q^{51} + 15064 q^{53} + 1176 q^{55} + 2502 q^{57} + 83242 q^{59} + 14954 q^{61} + 1539 q^{63} - 23250 q^{65} - 39501 q^{67} + 77184 q^{69} + 56020 q^{71} - 90395 q^{73} - 50445 q^{75} + 63448 q^{77} + 43067 q^{79} - 32805 q^{81} + 75672 q^{83} - 75272 q^{85} + 18972 q^{87} - 72608 q^{89} - 288287 q^{91} - 73179 q^{93} - 190138 q^{95} + 183000 q^{97} + 68688 q^{99}+O(q^{100})$$ 10 * q - 45 * q^3 - 6 * q^5 + 97 * q^7 - 405 * q^9 - 424 * q^11 + 374 * q^13 + 108 * q^15 - 952 * q^17 - 139 * q^19 - 1044 * q^21 - 4288 * q^23 - 5605 * q^25 + 7290 * q^27 - 4216 * q^29 - 8131 * q^31 - 3816 * q^33 - 20106 * q^35 - 5425 * q^37 - 1683 * q^39 + 29364 * q^41 + 46862 * q^43 - 486 * q^45 + 17190 * q^47 + 23255 * q^49 - 8568 * q^51 + 15064 * q^53 + 1176 * q^55 + 2502 * q^57 + 83242 * q^59 + 14954 * q^61 + 1539 * q^63 - 23250 * q^65 - 39501 * q^67 + 77184 * q^69 + 56020 * q^71 - 90395 * q^73 - 50445 * q^75 + 63448 * q^77 + 43067 * q^79 - 32805 * q^81 + 75672 * q^83 - 75272 * q^85 + 18972 * q^87 - 72608 * q^89 - 288287 * q^91 - 73179 * q^93 - 190138 * q^95 + 183000 * q^97 + 68688 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 564x^{8} + 117814x^{6} + 11067780x^{4} + 427918225x^{2} + 3489248448$$ :

 $$\beta_{1}$$ $$=$$ $$( -625\nu^{9} - 281450\nu^{7} - 41638514\nu^{5} - 2184633898\nu^{3} - 19307731905\nu + 221300856 ) / 442601712$$ (-625*v^9 - 281450*v^7 - 41638514*v^5 - 2184633898*v^3 - 19307731905*v + 221300856) / 442601712 $$\beta_{2}$$ $$=$$ $$( - 1520 \nu^{9} - 22127 \nu^{8} - 671941 \nu^{7} - 9881431 \nu^{6} - 97198651 \nu^{5} - 1446793789 \nu^{4} - 4998355379 \nu^{3} + \cdots - 661674519576 ) / 21076272$$ (-1520*v^9 - 22127*v^8 - 671941*v^7 - 9881431*v^6 - 97198651*v^5 - 1446793789*v^4 - 4998355379*v^3 - 75106675469*v^2 - 45518106501*v - 661674519576) / 21076272 $$\beta_{3}$$ $$=$$ $$( 29349 \nu^{9} + 923650 \nu^{8} + 13163751 \nu^{7} + 413303534 \nu^{6} + 1937609757 \nu^{5} + 60656075606 \nu^{4} + 101054602125 \nu^{3} + \cdots + 27626932442592 ) / 221300856$$ (29349*v^9 + 923650*v^8 + 13163751*v^7 + 413303534*v^6 + 1937609757*v^5 + 60656075606*v^4 + 101054602125*v^3 + 3154278485386*v^2 + 884850924762*v + 27626932442592) / 221300856 $$\beta_{4}$$ $$=$$ $$( 92943 \nu^{9} + 1382633 \nu^{8} + 41485257 \nu^{7} + 619097017 \nu^{6} + 6068335779 \nu^{5} + 90929481643 \nu^{4} + 314369414091 \nu^{3} + \cdots + 41458506660144 ) / 442601712$$ (92943*v^9 + 1382633*v^8 + 41485257*v^7 + 619097017*v^6 + 6068335779*v^5 + 90929481643*v^4 + 314369414091*v^3 + 4732201989347*v^2 + 2740916116890*v + 41458506660144) / 442601712 $$\beta_{5}$$ $$=$$ $$( 29927 \nu^{9} - 75719 \nu^{8} + 13393927 \nu^{7} - 33593455 \nu^{6} + 1966231681 \nu^{5} - 4876676917 \nu^{4} + 102290395157 \nu^{3} + \cdots - 2191308184800 ) / 126457632$$ (29927*v^9 - 75719*v^8 + 13393927*v^7 - 33593455*v^6 + 1966231681*v^5 - 4876676917*v^4 + 102290395157*v^3 - 250707240917*v^2 + 896334262236*v - 2191308184800) / 126457632 $$\beta_{6}$$ $$=$$ $$( 29927 \nu^{9} + 75719 \nu^{8} + 13393927 \nu^{7} + 33593455 \nu^{6} + 1966231681 \nu^{5} + 4876676917 \nu^{4} + 102290395157 \nu^{3} + \cdots + 2191308184800 ) / 126457632$$ (29927*v^9 + 75719*v^8 + 13393927*v^7 + 33593455*v^6 + 1966231681*v^5 + 4876676917*v^4 + 102290395157*v^3 + 250707240917*v^2 + 896334262236*v + 2191308184800) / 126457632 $$\beta_{7}$$ $$=$$ $$( 249 \nu^{9} - 763 \nu^{8} + 111351 \nu^{7} - 340739 \nu^{6} + 16327677 \nu^{5} - 49889441 \nu^{4} + 848076549 \nu^{3} - 2588431825 \nu^{2} + \cdots - 22652476560 ) / 726768$$ (249*v^9 - 763*v^8 + 111351*v^7 - 340739*v^6 + 16327677*v^5 - 49889441*v^4 + 848076549*v^3 - 2588431825*v^2 + 7406597646*v - 22652476560) / 726768 $$\beta_{8}$$ $$=$$ $$( 249 \nu^{9} + 763 \nu^{8} + 111351 \nu^{7} + 340739 \nu^{6} + 16327677 \nu^{5} + 49889441 \nu^{4} + 848076549 \nu^{3} + 2588431825 \nu^{2} + \cdots + 22652476560 ) / 726768$$ (249*v^9 + 763*v^8 + 111351*v^7 + 340739*v^6 + 16327677*v^5 + 49889441*v^4 + 848076549*v^3 + 2588431825*v^2 + 7406597646*v + 22652476560) / 726768 $$\beta_{9}$$ $$=$$ $$( 8741 \nu^{9} - 44254 \nu^{8} + 3901120 \nu^{7} - 19762862 \nu^{6} + 570701284 \nu^{5} - 2893587578 \nu^{4} + 29592575300 \nu^{3} + \cdots - 1318596339816 ) / 21076272$$ (8741*v^9 - 44254*v^8 + 3901120*v^7 - 19762862*v^6 + 570701284*v^5 - 2893587578*v^4 + 29592575300*v^3 - 150171198394*v^2 + 260309438235*v - 1318596339816) / 21076272
 $$\nu$$ $$=$$ $$( \beta_{9} + \beta_{8} + 4\beta_{7} - 4\beta_{6} - 4\beta_{5} - 4\beta_{4} + 4\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 72$$ (b9 + b8 + 4*b7 - 4*b6 - 4*b5 - 4*b4 + 4*b3 - b2 - 2*b1 + 1) / 72 $$\nu^{2}$$ $$=$$ $$( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{2} - 451 ) / 4$$ (-b9 - b8 + 2*b7 - b2 - 451) / 4 $$\nu^{3}$$ $$=$$ $$( - 101 \beta_{9} - 29 \beta_{8} - 656 \beta_{7} + 404 \beta_{6} + 404 \beta_{5} + 728 \beta_{4} - 728 \beta_{3} + 101 \beta_{2} - 25718 \beta _1 + 12859 ) / 72$$ (-101*b9 - 29*b8 - 656*b7 + 404*b6 + 404*b5 + 728*b4 - 728*b3 + 101*b2 - 25718*b1 + 12859) / 72 $$\nu^{4}$$ $$=$$ $$( 75 \beta_{9} + 158 \beta_{8} - 217 \beta_{7} - 34 \beta_{6} + 34 \beta_{5} - 16 \beta_{4} - 16 \beta_{3} + 75 \beta_{2} + 32979 ) / 2$$ (75*b9 + 158*b8 - 217*b7 - 34*b6 + 34*b5 - 16*b4 - 16*b3 + 75*b2 + 32979) / 2 $$\nu^{5}$$ $$=$$ $$( 14737 \beta_{9} - 16367 \beta_{8} + 108412 \beta_{7} - 37348 \beta_{6} - 37348 \beta_{5} - 139516 \beta_{4} + 139516 \beta_{3} - 14737 \beta_{2} + 7250590 \beta _1 - 3625295 ) / 72$$ (14737*b9 - 16367*b8 + 108412*b7 - 37348*b6 - 37348*b5 - 139516*b4 + 139516*b3 - 14737*b2 + 7250590*b1 - 3625295) / 72 $$\nu^{6}$$ $$=$$ $$( - 22521 \beta_{9} - 77665 \beta_{8} + 89430 \beta_{7} + 21712 \beta_{6} - 21712 \beta_{5} + 10756 \beta_{4} + 10756 \beta_{3} - 22521 \beta_{2} - 10614395 ) / 4$$ (-22521*b9 - 77665*b8 + 89430*b7 + 21712*b6 - 21712*b5 + 10756*b4 + 10756*b3 - 22521*b2 - 10614395) / 4 $$\nu^{7}$$ $$=$$ $$( - 2418317 \beta_{9} + 5917627 \beta_{8} - 19082264 \beta_{7} + 2430356 \beta_{6} + 2430356 \beta_{5} + 27418208 \beta_{4} - 27418208 \beta_{3} + 2418317 \beta_{2} + \cdots + 855464563 ) / 72$$ (-2418317*b9 + 5917627*b8 - 19082264*b7 + 2430356*b6 + 2430356*b5 + 27418208*b4 - 27418208*b3 + 2418317*b2 - 1710929126*b1 + 855464563) / 72 $$\nu^{8}$$ $$=$$ $$( 1820969 \beta_{9} + 8707940 \beta_{8} - 9173387 \beta_{7} - 2624930 \beta_{6} + 2624930 \beta_{5} - 1355522 \beta_{4} - 1355522 \beta_{3} + 1820969 \beta_{2} + 919333837 ) / 2$$ (1820969*b9 + 8707940*b8 - 9173387*b7 - 2624930*b6 + 2624930*b5 - 1355522*b4 - 1355522*b3 + 1820969*b2 + 919333837) / 2 $$\nu^{9}$$ $$=$$ $$( 429358129 \beta_{9} - 1503955055 \beta_{8} + 3539964052 \beta_{7} + 105168572 \beta_{6} + 105168572 \beta_{5} - 5473277236 \beta_{4} + 5473277236 \beta_{3} + \cdots - 188662697615 ) / 72$$ (429358129*b9 - 1503955055*b8 + 3539964052*b7 + 105168572*b6 + 105168572*b5 - 5473277236*b4 + 5473277236*b3 - 429358129*b2 + 377325395230*b1 - 188662697615) / 72

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 10.6668i 10.7938i 3.29825i − 14.3017i 10.8764i 10.6668i − 10.7938i − 3.29825i 14.3017i − 10.8764i
0 −4.50000 + 7.79423i 0 −50.8171 88.0179i 0 119.572 + 50.0952i 0 −40.5000 70.1481i 0
193.2 0 −4.50000 + 7.79423i 0 −19.4585 33.7032i 0 −106.179 74.3848i 0 −40.5000 70.1481i 0
193.3 0 −4.50000 + 7.79423i 0 8.04054 + 13.9266i 0 77.3451 + 104.042i 0 −40.5000 70.1481i 0
193.4 0 −4.50000 + 7.79423i 0 13.3782 + 23.1717i 0 66.5893 111.233i 0 −40.5000 70.1481i 0
193.5 0 −4.50000 + 7.79423i 0 45.8569 + 79.4265i 0 −108.828 + 70.4522i 0 −40.5000 70.1481i 0
289.1 0 −4.50000 7.79423i 0 −50.8171 + 88.0179i 0 119.572 50.0952i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 7.79423i 0 −19.4585 + 33.7032i 0 −106.179 + 74.3848i 0 −40.5000 + 70.1481i 0
289.3 0 −4.50000 7.79423i 0 8.04054 13.9266i 0 77.3451 104.042i 0 −40.5000 + 70.1481i 0
289.4 0 −4.50000 7.79423i 0 13.3782 23.1717i 0 66.5893 + 111.233i 0 −40.5000 + 70.1481i 0
289.5 0 −4.50000 7.79423i 0 45.8569 79.4265i 0 −108.828 70.4522i 0 −40.5000 + 70.1481i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.6.q.l 10
4.b odd 2 1 168.6.q.c 10
7.c even 3 1 inner 336.6.q.l 10
12.b even 2 1 504.6.s.c 10
28.g odd 6 1 168.6.q.c 10
84.n even 6 1 504.6.s.c 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.q.c 10 4.b odd 2 1
168.6.q.c 10 28.g odd 6 1
336.6.q.l 10 1.a even 1 1 trivial
336.6.q.l 10 7.c even 3 1 inner
504.6.s.c 10 12.b even 2 1
504.6.s.c 10 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + 6 T_{5}^{9} + 10633 T_{5}^{8} - 145618 T_{5}^{7} + 100355305 T_{5}^{6} - 731091322 T_{5}^{5} + 126550850680 T_{5}^{4} - 2828330981128 T_{5}^{3} + \cdots + 24\!\cdots\!76$$ acting on $$S_{6}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$(T^{2} + 9 T + 81)^{5}$$
$5$ $$T^{10} + 6 T^{9} + \cdots + 24\!\cdots\!76$$
$7$ $$T^{10} - 97 T^{9} + \cdots + 13\!\cdots\!07$$
$11$ $$T^{10} + 424 T^{9} + \cdots + 68\!\cdots\!36$$
$13$ $$(T^{5} - 187 T^{4} + \cdots + 735027021216)^{2}$$
$17$ $$T^{10} + 952 T^{9} + \cdots + 15\!\cdots\!24$$
$19$ $$T^{10} + 139 T^{9} + \cdots + 13\!\cdots\!84$$
$23$ $$T^{10} + 4288 T^{9} + \cdots + 26\!\cdots\!76$$
$29$ $$(T^{5} + 2108 T^{4} + \cdots + 12\!\cdots\!76)^{2}$$
$31$ $$T^{10} + 8131 T^{9} + \cdots + 68\!\cdots\!25$$
$37$ $$T^{10} + 5425 T^{9} + \cdots + 12\!\cdots\!64$$
$41$ $$(T^{5} - 14682 T^{4} + \cdots + 25\!\cdots\!44)^{2}$$
$43$ $$(T^{5} - 23431 T^{4} + \cdots + 33\!\cdots\!72)^{2}$$
$47$ $$T^{10} - 17190 T^{9} + \cdots + 13\!\cdots\!24$$
$53$ $$T^{10} - 15064 T^{9} + \cdots + 40\!\cdots\!04$$
$59$ $$T^{10} - 83242 T^{9} + \cdots + 22\!\cdots\!24$$
$61$ $$T^{10} - 14954 T^{9} + \cdots + 36\!\cdots\!36$$
$67$ $$T^{10} + 39501 T^{9} + \cdots + 50\!\cdots\!84$$
$71$ $$(T^{5} - 28010 T^{4} + \cdots - 29\!\cdots\!68)^{2}$$
$73$ $$T^{10} + 90395 T^{9} + \cdots + 13\!\cdots\!16$$
$79$ $$T^{10} - 43067 T^{9} + \cdots + 49\!\cdots\!89$$
$83$ $$(T^{5} - 37836 T^{4} + \cdots - 10\!\cdots\!96)^{2}$$
$89$ $$T^{10} + 72608 T^{9} + \cdots + 64\!\cdots\!96$$
$97$ $$(T^{5} - 91500 T^{4} + \cdots - 14\!\cdots\!48)^{2}$$