# Properties

 Label 2205.4.a.ca Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$130.099211563$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.1163891200.1 Defining polynomial: $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ x^6 - 2*x^5 - 23*x^4 + 12*x^3 + 154*x^2 + 152*x + 28 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2\cdot 7$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{4} + 5 q^{5} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 9 \beta_1 + 8) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b5 - b4 - b3 + 2*b2 + 3*b1 + 2) * q^4 + 5 * q^5 + (-2*b5 - 3*b4 - 5*b3 + b2 + 9*b1 + 8) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{4} + 5 q^{5} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 9 \beta_1 + 8) q^{8} + 5 \beta_1 q^{10} + (2 \beta_{5} + 5 \beta_{4} + \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 7) q^{11} + ( - 5 \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 27) q^{13} + ( - 12 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} + \beta_{2} + 27 \beta_1 + 44) q^{16} + (\beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 16 \beta_1 - 1) q^{17} + ( - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 10 \beta_{2} - 6 \beta_1 + 52) q^{19} + ( - 5 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 15 \beta_1 + 10) q^{20} + (22 \beta_{5} + \beta_{4} + 19 \beta_{3} - 17 \beta_{2} - 12 \beta_1 - 48) q^{22} + (6 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} - 3 \beta_1 + 56) q^{23} + 25 q^{25} + ( - 15 \beta_{5} - 5 \beta_{4} - 19 \beta_{3} + 4 \beta_{2} + 58 \beta_1 - 24) q^{26} + (8 \beta_{5} + \beta_{4} - 13 \beta_{3} - 7 \beta_{2} - 39 \beta_1 - 21) q^{29} + (10 \beta_{5} - 17 \beta_{3} + 29 \beta_{2} + 11 \beta_1 + 68) q^{31} + ( - 42 \beta_{5} - 15 \beta_{4} - 31 \beta_{3} + 35 \beta_{2} + 99 \beta_1 + 116) q^{32} + ( - 2 \beta_{5} - 22 \beta_{4} - 11 \beta_{3} + 18 \beta_{2} + 62 \beta_1 + 114) q^{34} + (4 \beta_{5} + 28 \beta_{4} - 9 \beta_{3} - 11 \beta_{2} - 33 \beta_1 - 8) q^{37} + ( - 3 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 16 \beta_1 - 6) q^{38} + ( - 10 \beta_{5} - 15 \beta_{4} - 25 \beta_{3} + 5 \beta_{2} + 45 \beta_1 + 40) q^{40} + ( - 37 \beta_{5} + 15 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} - 11 \beta_1 - 86) q^{41} + (2 \beta_{5} - 34 \beta_{4} - 16 \beta_{3} + 8 \beta_{2} - 74) q^{43} + (\beta_{5} - 10 \beta_{4} + 48 \beta_{3} - 5 \beta_{2} - 122 \beta_1 + 62) q^{44} + (35 \beta_{5} + 20 \beta_{4} + 30 \beta_{3} - 25 \beta_{2} + 42 \beta_1 - 92) q^{46} + (2 \beta_{5} + 9 \beta_{4} - 32 \beta_{3} + 14 \beta_{2} - 50 \beta_1 - 81) q^{47} + 25 \beta_1 q^{50} + ( - 93 \beta_{5} - 32 \beta_{4} - 48 \beta_{3} + 65 \beta_{2} + 196 \beta_1 + 230) q^{52} + ( - 2 \beta_{5} + 60 \beta_{4} + 19 \beta_{3} - 35 \beta_{2} - 73 \beta_1 + 146) q^{53} + (10 \beta_{5} + 25 \beta_{4} + 5 \beta_{3} + 25 \beta_{2} - 25 \beta_1 + 35) q^{55} + (25 \beta_{4} + 15 \beta_{3} - 35 \beta_{2} - 128 \beta_1 - 296) q^{58} + (11 \beta_{5} - 59 \beta_{4} - 47 \beta_{3} + 33 \beta_{2} - 49 \beta_1 - 162) q^{59} + (24 \beta_{5} - 10 \beta_{4} - 4 \beta_{3} + 48 \beta_{2} + 132 \beta_1 + 84) q^{61} + (13 \beta_{5} - 28 \beta_{4} - 6 \beta_{3} + \beta_{2} + 154 \beta_1 - 4) q^{62} + ( - 10 \beta_{5} - 91 \beta_{4} - 63 \beta_{3} + 67 \beta_{2} + 351 \beta_1 + 116) q^{64} + ( - 25 \beta_{4} - 40 \beta_{3} + 20 \beta_{2} + 10 \beta_1 + 135) q^{65} + ( - 12 \beta_{5} + 12 \beta_{4} - 35 \beta_{3} + 31 \beta_{2} - 51 \beta_1 + 330) q^{67} + ( - 78 \beta_{5} - 67 \beta_{4} - 58 \beta_{3} + 15 \beta_{2} + 236 \beta_1 + 420) q^{68} + (14 \beta_{5} + 92 \beta_{3} - 60 \beta_{2} + 116 \beta_1 - 26) q^{71} + ( - 11 \beta_{5} - 85 \beta_{4} + 46 \beta_{3} + 60 \beta_{2} + 64 \beta_1 + 414) q^{73} + (21 \beta_{5} - 4 \beta_{4} + 36 \beta_{3} - 27 \beta_{2} - 132 \beta_1 - 128) q^{74} + (19 \beta_{5} - 43 \beta_{4} - 38 \beta_{3} + 106 \beta_{2} + 76 \beta_1 - 234) q^{76} + (10 \beta_{5} - 55 \beta_{4} + 65 \beta_{3} + 25 \beta_{2} + 67 \beta_1 + 237) q^{79} + ( - 60 \beta_{5} - 15 \beta_{4} - 75 \beta_{3} + 5 \beta_{2} + 135 \beta_1 + 220) q^{80} + (46 \beta_{5} - \beta_{4} + 2 \beta_{3} - 113 \beta_{2} - 96 \beta_1 - 314) q^{82} + (69 \beta_{5} + 41 \beta_{4} + 154 \beta_{3} - 80 \beta_{2} + 32 \beta_1 - 70) q^{83} + (5 \beta_{5} + 10 \beta_{4} - 20 \beta_{3} + 50 \beta_{2} + 80 \beta_1 - 5) q^{85} + ( - 50 \beta_{5} + 18 \beta_{4} - 56 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 172) q^{86} + (22 \beta_{5} + 172 \beta_{4} + 52 \beta_{3} - 144 \beta_{2} - 300 \beta_1 - 840) q^{88} + (33 \beta_{5} + 79 \beta_{4} + 80 \beta_{3} + 62 \beta_{2} - 74 \beta_1 + 378) q^{89} + ( - 60 \beta_{5} + 32 \beta_{4} - 24 \beta_{3} + 86 \beta_{2} - 82 \beta_1 + 412) q^{92} + (23 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} - 92 \beta_{2} - 164 \beta_1 - 536) q^{94} + ( - 15 \beta_{5} + 15 \beta_{4} + 20 \beta_{3} - 50 \beta_{2} - 30 \beta_1 + 260) q^{95} + ( - 77 \beta_{5} - 84 \beta_{4} + 14 \beta_{3} + 72 \beta_{2} + 134 \beta_1 + 63) q^{97}+O(q^{100})$$ q + b1 * q^2 + (-b5 - b4 - b3 + 2*b2 + 3*b1 + 2) * q^4 + 5 * q^5 + (-2*b5 - 3*b4 - 5*b3 + b2 + 9*b1 + 8) * q^8 + 5*b1 * q^10 + (2*b5 + 5*b4 + b3 + 5*b2 - 5*b1 + 7) * q^11 + (-5*b4 - 8*b3 + 4*b2 + 2*b1 + 27) * q^13 + (-12*b5 - 3*b4 - 15*b3 + b2 + 27*b1 + 44) * q^16 + (b5 + 2*b4 - 4*b3 + 10*b2 + 16*b1 - 1) * q^17 + (-3*b5 + 3*b4 + 4*b3 - 10*b2 - 6*b1 + 52) * q^19 + (-5*b5 - 5*b4 - 5*b3 + 10*b2 + 15*b1 + 10) * q^20 + (22*b5 + b4 + 19*b3 - 17*b2 - 12*b1 - 48) * q^22 + (6*b5 - 8*b4 + 9*b3 + 11*b2 - 3*b1 + 56) * q^23 + 25 * q^25 + (-15*b5 - 5*b4 - 19*b3 + 4*b2 + 58*b1 - 24) * q^26 + (8*b5 + b4 - 13*b3 - 7*b2 - 39*b1 - 21) * q^29 + (10*b5 - 17*b3 + 29*b2 + 11*b1 + 68) * q^31 + (-42*b5 - 15*b4 - 31*b3 + 35*b2 + 99*b1 + 116) * q^32 + (-2*b5 - 22*b4 - 11*b3 + 18*b2 + 62*b1 + 114) * q^34 + (4*b5 + 28*b4 - 9*b3 - 11*b2 - 33*b1 - 8) * q^37 + (-3*b5 + 7*b4 + 4*b3 - 2*b2 + 16*b1 - 6) * q^38 + (-10*b5 - 15*b4 - 25*b3 + 5*b2 + 45*b1 + 40) * q^40 + (-37*b5 + 15*b4 + 3*b3 + 7*b2 - 11*b1 - 86) * q^41 + (2*b5 - 34*b4 - 16*b3 + 8*b2 - 74) * q^43 + (b5 - 10*b4 + 48*b3 - 5*b2 - 122*b1 + 62) * q^44 + (35*b5 + 20*b4 + 30*b3 - 25*b2 + 42*b1 - 92) * q^46 + (2*b5 + 9*b4 - 32*b3 + 14*b2 - 50*b1 - 81) * q^47 + 25*b1 * q^50 + (-93*b5 - 32*b4 - 48*b3 + 65*b2 + 196*b1 + 230) * q^52 + (-2*b5 + 60*b4 + 19*b3 - 35*b2 - 73*b1 + 146) * q^53 + (10*b5 + 25*b4 + 5*b3 + 25*b2 - 25*b1 + 35) * q^55 + (25*b4 + 15*b3 - 35*b2 - 128*b1 - 296) * q^58 + (11*b5 - 59*b4 - 47*b3 + 33*b2 - 49*b1 - 162) * q^59 + (24*b5 - 10*b4 - 4*b3 + 48*b2 + 132*b1 + 84) * q^61 + (13*b5 - 28*b4 - 6*b3 + b2 + 154*b1 - 4) * q^62 + (-10*b5 - 91*b4 - 63*b3 + 67*b2 + 351*b1 + 116) * q^64 + (-25*b4 - 40*b3 + 20*b2 + 10*b1 + 135) * q^65 + (-12*b5 + 12*b4 - 35*b3 + 31*b2 - 51*b1 + 330) * q^67 + (-78*b5 - 67*b4 - 58*b3 + 15*b2 + 236*b1 + 420) * q^68 + (14*b5 + 92*b3 - 60*b2 + 116*b1 - 26) * q^71 + (-11*b5 - 85*b4 + 46*b3 + 60*b2 + 64*b1 + 414) * q^73 + (21*b5 - 4*b4 + 36*b3 - 27*b2 - 132*b1 - 128) * q^74 + (19*b5 - 43*b4 - 38*b3 + 106*b2 + 76*b1 - 234) * q^76 + (10*b5 - 55*b4 + 65*b3 + 25*b2 + 67*b1 + 237) * q^79 + (-60*b5 - 15*b4 - 75*b3 + 5*b2 + 135*b1 + 220) * q^80 + (46*b5 - b4 + 2*b3 - 113*b2 - 96*b1 - 314) * q^82 + (69*b5 + 41*b4 + 154*b3 - 80*b2 + 32*b1 - 70) * q^83 + (5*b5 + 10*b4 - 20*b3 + 50*b2 + 80*b1 - 5) * q^85 + (-50*b5 + 18*b4 - 56*b3 + 4*b2 - 2*b1 - 172) * q^86 + (22*b5 + 172*b4 + 52*b3 - 144*b2 - 300*b1 - 840) * q^88 + (33*b5 + 79*b4 + 80*b3 + 62*b2 - 74*b1 + 378) * q^89 + (-60*b5 + 32*b4 - 24*b3 + 86*b2 - 82*b1 + 412) * q^92 + (23*b5 + 9*b4 + 11*b3 - 92*b2 - 164*b1 - 536) * q^94 + (-15*b5 + 15*b4 + 20*b3 - 50*b2 - 30*b1 + 260) * q^95 + (-77*b5 - 84*b4 + 14*b3 + 72*b2 + 134*b1 + 63) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8}+O(q^{10})$$ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 $$6 q + 2 q^{2} + 14 q^{4} + 30 q^{5} + 66 q^{8} + 10 q^{10} + 16 q^{11} + 168 q^{13} + 298 q^{16} + 4 q^{17} + 308 q^{19} + 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} - 56 q^{26} - 176 q^{29} + 392 q^{31} + 770 q^{32} + 812 q^{34} - 140 q^{37} - 20 q^{38} + 330 q^{40} - 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} - 628 q^{47} + 50 q^{50} + 1520 q^{52} + 676 q^{53} + 80 q^{55} - 2012 q^{58} - 996 q^{59} + 740 q^{61} + 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} + 2940 q^{68} + 224 q^{71} + 2640 q^{73} - 928 q^{74} - 1340 q^{76} + 1636 q^{79} + 1490 q^{80} - 1756 q^{82} - 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} + 1904 q^{89} + 1952 q^{92} - 3332 q^{94} + 1540 q^{95} + 516 q^{97}+O(q^{100})$$ 6 * q + 2 * q^2 + 14 * q^4 + 30 * q^5 + 66 * q^8 + 10 * q^10 + 16 * q^11 + 168 * q^13 + 298 * q^16 + 4 * q^17 + 308 * q^19 + 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 - 56 * q^26 - 176 * q^29 + 392 * q^31 + 770 * q^32 + 812 * q^34 - 140 * q^37 - 20 * q^38 + 330 * q^40 - 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 - 628 * q^47 + 50 * q^50 + 1520 * q^52 + 676 * q^53 + 80 * q^55 - 2012 * q^58 - 996 * q^59 + 740 * q^61 + 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 + 2940 * q^68 + 224 * q^71 + 2640 * q^73 - 928 * q^74 - 1340 * q^76 + 1636 * q^79 + 1490 * q^80 - 1756 * q^82 - 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 + 1904 * q^89 + 1952 * q^92 - 3332 * q^94 + 1540 * q^95 + 516 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26$$ (-v^5 + 23*v^3 + 8*v^2 - 86*v - 64) / 26 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26$$ (v^5 - 23*v^3 + 18*v^2 + 60*v - 144) / 26 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26$$ (-5*v^5 + 26*v^4 + 89*v^3 - 298*v^2 - 638*v - 164) / 26 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26$$ (-v^5 + 49*v^3 + 8*v^2 - 476*v - 428) / 26 $$\beta_{5}$$ $$=$$ $$( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26$$ (9*v^5 - 26*v^4 - 155*v^3 + 240*v^2 + 800*v + 264) / 26
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7$$ (b5 - b4 + b3 + b2 + 6*b1) / 7 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7$$ (b5 - b4 + b3 + 8*b2 + 13*b1 + 56) / 7 $$\nu^{3}$$ $$=$$ $$( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7$$ (15*b5 - 8*b4 + 15*b3 + 15*b2 + 83*b1 + 98) / 7 $$\nu^{4}$$ $$=$$ $$( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7$$ (36*b5 - 29*b4 + 43*b3 + 127*b2 + 265*b1 + 784) / 7 $$\nu^{5}$$ $$=$$ $$( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7$$ (267*b5 - 106*b4 + 267*b3 + 323*b2 + 1315*b1 + 2254) / 7

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.05323 −0.241849 −2.05886 −1.04490 4.29508 4.10376
−4.46745 0 11.9581 5.00000 0 0 −17.6824 0 −22.3372
1.2 −1.65606 0 −5.25746 5.00000 0 0 21.9552 0 −8.28031
1.3 −0.644648 0 −7.58443 5.00000 0 0 10.0465 0 −3.22324
1.4 0.369315 0 −7.86361 5.00000 0 0 −5.85867 0 1.84657
1.5 2.88087 0 0.299392 5.00000 0 0 −22.1844 0 14.4043
1.6 5.51797 0 22.4480 5.00000 0 0 79.7239 0 27.5899
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.4.a.ca 6
3.b odd 2 1 245.4.a.p yes 6
7.b odd 2 1 2205.4.a.bz 6
15.d odd 2 1 1225.4.a.bi 6
21.c even 2 1 245.4.a.o 6
21.g even 6 2 245.4.e.q 12
21.h odd 6 2 245.4.e.p 12
105.g even 2 1 1225.4.a.bj 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o 6 21.c even 2 1
245.4.a.p yes 6 3.b odd 2 1
245.4.e.p 12 21.h odd 6 2
245.4.e.q 12 21.g even 6 2
1225.4.a.bi 6 15.d odd 2 1
1225.4.a.bj 6 105.g even 2 1
2205.4.a.bz 6 7.b odd 2 1
2205.4.a.ca 6 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2}^{6} - 2T_{2}^{5} - 29T_{2}^{4} + 28T_{2}^{3} + 134T_{2}^{2} + 24T_{2} - 28$$ T2^6 - 2*T2^5 - 29*T2^4 + 28*T2^3 + 134*T2^2 + 24*T2 - 28 $$T_{11}^{6} - 16T_{11}^{5} - 7174T_{11}^{4} + 121780T_{11}^{3} + 14188145T_{11}^{2} - 145959100T_{11} - 9225436100$$ T11^6 - 16*T11^5 - 7174*T11^4 + 121780*T11^3 + 14188145*T11^2 - 145959100*T11 - 9225436100 $$T_{13}^{6} - 168T_{13}^{5} + 5774T_{13}^{4} + 335452T_{13}^{3} - 18677143T_{13}^{2} - 142488452T_{13} + 12513937372$$ T13^6 - 168*T13^5 + 5774*T13^4 + 335452*T13^3 - 18677143*T13^2 - 142488452*T13 + 12513937372 $$T_{17}^{6} - 4T_{17}^{5} - 15392T_{17}^{4} + 258596T_{17}^{3} + 56343501T_{17}^{2} - 1617286496T_{17} - 3807091762$$ T17^6 - 4*T17^5 - 15392*T17^4 + 258596*T17^3 + 56343501*T17^2 - 1617286496*T17 - 3807091762

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} - 29 T^{4} + 28 T^{3} + \cdots - 28$$
$3$ $$T^{6}$$
$5$ $$(T - 5)^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 16 T^{5} + \cdots - 9225436100$$
$13$ $$T^{6} - 168 T^{5} + \cdots + 12513937372$$
$17$ $$T^{6} - 4 T^{5} + \cdots - 3807091762$$
$19$ $$T^{6} - 308 T^{5} + \cdots + 1617325344$$
$23$ $$T^{6} - 336 T^{5} + \cdots + 298897696$$
$29$ $$T^{6} + 176 T^{5} + \cdots - 544215793700$$
$31$ $$T^{6} - 392 T^{5} + \cdots + 16539893268192$$
$37$ $$T^{6} + 140 T^{5} + \cdots + 271258136464$$
$41$ $$T^{6} + \cdots + 144691772208184$$
$43$ $$T^{6} + 388 T^{5} + \cdots + 440374360000$$
$47$ $$T^{6} + \cdots + 251564448569400$$
$53$ $$T^{6} + \cdots + 590408333736048$$
$59$ $$T^{6} + 996 T^{5} + \cdots + 14\!\cdots\!04$$
$61$ $$T^{6} + \cdots - 842000334839552$$
$67$ $$T^{6} - 1768 T^{5} + \cdots + 885207397312$$
$71$ $$T^{6} - 224 T^{5} + \cdots + 12\!\cdots\!88$$
$73$ $$T^{6} - 2640 T^{5} + \cdots - 85\!\cdots\!92$$
$79$ $$T^{6} - 1636 T^{5} + \cdots - 12\!\cdots\!68$$
$83$ $$T^{6} + 140 T^{5} + \cdots + 22\!\cdots\!00$$
$89$ $$T^{6} - 1904 T^{5} + \cdots + 45\!\cdots\!76$$
$97$ $$T^{6} + \cdots - 510868966648482$$