# Properties

 Label 405.4.e.v Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.84779568.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64$$ x^6 - x^5 + 13*x^4 - 4*x^3 + 152*x^2 - 96*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - 2 \beta_{3} + 2) q^{2} + (3 \beta_{5} + \beta_{4} - 7 \beta_{3} - \beta_{2} - 3 \beta_1) q^{4} + 5 \beta_{3} q^{5} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2) q^{7} + ( - 5 \beta_{2} - 7 \beta_1 - 29) q^{8}+O(q^{10})$$ q + (b5 - 2*b3 + 2) * q^2 + (3*b5 + b4 - 7*b3 - b2 - 3*b1) * q^4 + 5*b3 * q^5 + (-b5 + 3*b4 - 2*b3 + 2) * q^7 + (-5*b2 - 7*b1 - 29) * q^8 $$q + (\beta_{5} - 2 \beta_{3} + 2) q^{2} + (3 \beta_{5} + \beta_{4} - 7 \beta_{3} - \beta_{2} - 3 \beta_1) q^{4} + 5 \beta_{3} q^{5} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2) q^{7} + ( - 5 \beta_{2} - 7 \beta_1 - 29) q^{8} + (5 \beta_1 + 10) q^{10} + ( - 9 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} - 3) q^{11} + (6 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{13} + (16 \beta_{5} + 5 \beta_{4} + 13 \beta_{3} - 5 \beta_{2} - 16 \beta_1) q^{14} + ( - 37 \beta_{5} - 9 \beta_{4} + 69 \beta_{3} - 69) q^{16} + (5 \beta_{2} - 3 \beta_1 - 51) q^{17} + ( - \beta_{2} - 33 \beta_1 - 28) q^{19} + (15 \beta_{5} + 5 \beta_{4} - 35 \beta_{3} + 35) q^{20} + ( - 37 \beta_{5} - 19 \beta_{4} + 95 \beta_{3} + 19 \beta_{2} + 37 \beta_1) q^{22} + (25 \beta_{5} + 5 \beta_{4} + 85 \beta_{3} - 5 \beta_{2} - 25 \beta_1) q^{23} + (25 \beta_{3} - 25) q^{25} + ( - 10 \beta_{2} - 21 \beta_1 - 72) q^{26} + ( - 2 \beta_{2} - 36 \beta_1 - 124) q^{28} + (\beta_{5} + 25 \beta_{4} - 47 \beta_{3} + 47) q^{29} + ( - 19 \beta_{5} + 17 \beta_{4} + 39 \beta_{3} - 17 \beta_{2} + 19 \beta_1) q^{31} + ( - 95 \beta_{5} - 15 \beta_{4} + 295 \beta_{3} + 15 \beta_{2} + 95 \beta_1) q^{32} + ( - 29 \beta_{5} + 7 \beta_{4} + 145 \beta_{3} - 145) q^{34} + (15 \beta_{2} - 5 \beta_1 + 10) q^{35} + (25 \beta_{2} - 55 \beta_1 - 138) q^{37} + ( - 66 \beta_{5} - 35 \beta_{4} + 417 \beta_{3} - 417) q^{38} + (35 \beta_{5} + 25 \beta_{4} - 145 \beta_{3} - 25 \beta_{2} - 35 \beta_1) q^{40} + (26 \beta_{5} - 10 \beta_{4} + 188 \beta_{3} + 10 \beta_{2} - 26 \beta_1) q^{41} + (17 \beta_{5} + 29 \beta_{4} - 281 \beta_{3} + 281) q^{43} + (35 \beta_{2} + 155 \beta_1 + 535) q^{44} + ( - 35 \beta_{2} + 35 \beta_1 - 95) q^{46} + (123 \beta_{5} - 5 \beta_{4} + 9 \beta_{3} - 9) q^{47} + ( - 53 \beta_{5} - 41 \beta_{4} - 161 \beta_{3} + 41 \beta_{2} + 53 \beta_1) q^{49} + ( - 25 \beta_{5} + 50 \beta_{3} + 25 \beta_1) q^{50} + ( - 95 \beta_{5} - 25 \beta_{4} + 315 \beta_{3} - 315) q^{52} + ( - 30 \beta_{2} + 38 \beta_1 + 136) q^{53} + ( - 25 \beta_{2} - 45 \beta_1 - 15) q^{55} + ( - 42 \beta_{5} + 744 \beta_{3} - 744) q^{56} + (173 \beta_{5} + 51 \beta_{4} - 55 \beta_{3} - 51 \beta_{2} - 173 \beta_1) q^{58} + (68 \beta_{5} + 104 \beta_{3} - 68 \beta_1) q^{59} + ( - 43 \beta_{5} - 31 \beta_{4} - 26 \beta_{3} + 26) q^{61} + ( - 15 \beta_{2} - 27 \beta_1 + 321) q^{62} + (53 \beta_{2} + 169 \beta_1 + 1053) q^{64} + (30 \beta_{5} + 10 \beta_{4} - 25 \beta_{3} + 25) q^{65} + (121 \beta_{5} - 3 \beta_{4} - 40 \beta_{3} + 3 \beta_{2} - 121 \beta_1) q^{67} + ( - 115 \beta_{5} - 55 \beta_{4} + 215 \beta_{3} + 55 \beta_{2} + 115 \beta_1) q^{68} + (80 \beta_{5} + 25 \beta_{4} + 65 \beta_{3} - 65) q^{70} + (30 \beta_{2} - 182 \beta_1 - 64) q^{71} + (3 \beta_{2} + 59 \beta_1 - 306) q^{73} + ( - 68 \beta_{5} - 5 \beta_{4} + 931 \beta_{3} - 931) q^{74} + ( - 394 \beta_{5} - 128 \beta_{4} + 1266 \beta_{3} + 128 \beta_{2} + 394 \beta_1) q^{76} + ( - 104 \beta_{5} + 80 \beta_{4} + 658 \beta_{3} - 80 \beta_{2} + 104 \beta_1) q^{77} + (38 \beta_{5} - 54 \beta_{4} + 343 \beta_{3} - 343) q^{79} + ( - 45 \beta_{2} - 185 \beta_1 - 345) q^{80} + ( - 6 \beta_{2} + 212 \beta_1 + 70) q^{82} + ( - 24 \beta_{5} + 60 \beta_{4} - 102 \beta_{3} + 102) q^{83} + (15 \beta_{5} - 25 \beta_{4} - 255 \beta_{3} + 25 \beta_{2} - 15 \beta_1) q^{85} + (443 \beta_{5} + 75 \beta_{4} - 691 \beta_{3} - 75 \beta_{2} - 443 \beta_1) q^{86} + (569 \beta_{5} + 73 \beta_{4} - 1945 \beta_{3} + 1945) q^{88} + ( - 60 \beta_{2} - 360) q^{89} + (23 \beta_{2} - 81 \beta_1 - 230) q^{91} + ( - 35 \beta_{5} + 5 \beta_{4} + 415 \beta_{3} - 415) q^{92} + (89 \beta_{5} + 113 \beta_{4} - 1345 \beta_{3} - 113 \beta_{2} - 89 \beta_1) q^{94} + (165 \beta_{5} + 5 \beta_{4} - 140 \beta_{3} - 5 \beta_{2} - 165 \beta_1) q^{95} + (113 \beta_{5} + \beta_{4} + 202 \beta_{3} - 202) q^{97} + (135 \beta_{2} + 97 \beta_1 + 179) q^{98}+O(q^{100})$$ q + (b5 - 2*b3 + 2) * q^2 + (3*b5 + b4 - 7*b3 - b2 - 3*b1) * q^4 + 5*b3 * q^5 + (-b5 + 3*b4 - 2*b3 + 2) * q^7 + (-5*b2 - 7*b1 - 29) * q^8 + (5*b1 + 10) * q^10 + (-9*b5 - 5*b4 + 3*b3 - 3) * q^11 + (6*b5 + 2*b4 - 5*b3 - 2*b2 - 6*b1) * q^13 + (16*b5 + 5*b4 + 13*b3 - 5*b2 - 16*b1) * q^14 + (-37*b5 - 9*b4 + 69*b3 - 69) * q^16 + (5*b2 - 3*b1 - 51) * q^17 + (-b2 - 33*b1 - 28) * q^19 + (15*b5 + 5*b4 - 35*b3 + 35) * q^20 + (-37*b5 - 19*b4 + 95*b3 + 19*b2 + 37*b1) * q^22 + (25*b5 + 5*b4 + 85*b3 - 5*b2 - 25*b1) * q^23 + (25*b3 - 25) * q^25 + (-10*b2 - 21*b1 - 72) * q^26 + (-2*b2 - 36*b1 - 124) * q^28 + (b5 + 25*b4 - 47*b3 + 47) * q^29 + (-19*b5 + 17*b4 + 39*b3 - 17*b2 + 19*b1) * q^31 + (-95*b5 - 15*b4 + 295*b3 + 15*b2 + 95*b1) * q^32 + (-29*b5 + 7*b4 + 145*b3 - 145) * q^34 + (15*b2 - 5*b1 + 10) * q^35 + (25*b2 - 55*b1 - 138) * q^37 + (-66*b5 - 35*b4 + 417*b3 - 417) * q^38 + (35*b5 + 25*b4 - 145*b3 - 25*b2 - 35*b1) * q^40 + (26*b5 - 10*b4 + 188*b3 + 10*b2 - 26*b1) * q^41 + (17*b5 + 29*b4 - 281*b3 + 281) * q^43 + (35*b2 + 155*b1 + 535) * q^44 + (-35*b2 + 35*b1 - 95) * q^46 + (123*b5 - 5*b4 + 9*b3 - 9) * q^47 + (-53*b5 - 41*b4 - 161*b3 + 41*b2 + 53*b1) * q^49 + (-25*b5 + 50*b3 + 25*b1) * q^50 + (-95*b5 - 25*b4 + 315*b3 - 315) * q^52 + (-30*b2 + 38*b1 + 136) * q^53 + (-25*b2 - 45*b1 - 15) * q^55 + (-42*b5 + 744*b3 - 744) * q^56 + (173*b5 + 51*b4 - 55*b3 - 51*b2 - 173*b1) * q^58 + (68*b5 + 104*b3 - 68*b1) * q^59 + (-43*b5 - 31*b4 - 26*b3 + 26) * q^61 + (-15*b2 - 27*b1 + 321) * q^62 + (53*b2 + 169*b1 + 1053) * q^64 + (30*b5 + 10*b4 - 25*b3 + 25) * q^65 + (121*b5 - 3*b4 - 40*b3 + 3*b2 - 121*b1) * q^67 + (-115*b5 - 55*b4 + 215*b3 + 55*b2 + 115*b1) * q^68 + (80*b5 + 25*b4 + 65*b3 - 65) * q^70 + (30*b2 - 182*b1 - 64) * q^71 + (3*b2 + 59*b1 - 306) * q^73 + (-68*b5 - 5*b4 + 931*b3 - 931) * q^74 + (-394*b5 - 128*b4 + 1266*b3 + 128*b2 + 394*b1) * q^76 + (-104*b5 + 80*b4 + 658*b3 - 80*b2 + 104*b1) * q^77 + (38*b5 - 54*b4 + 343*b3 - 343) * q^79 + (-45*b2 - 185*b1 - 345) * q^80 + (-6*b2 + 212*b1 + 70) * q^82 + (-24*b5 + 60*b4 - 102*b3 + 102) * q^83 + (15*b5 - 25*b4 - 255*b3 + 25*b2 - 15*b1) * q^85 + (443*b5 + 75*b4 - 691*b3 - 75*b2 - 443*b1) * q^86 + (569*b5 + 73*b4 - 1945*b3 + 1945) * q^88 + (-60*b2 - 360) * q^89 + (23*b2 - 81*b1 - 230) * q^91 + (-35*b5 + 5*b4 + 415*b3 - 415) * q^92 + (89*b5 + 113*b4 - 1345*b3 - 113*b2 - 89*b1) * q^94 + (165*b5 + 5*b4 - 140*b3 - 5*b2 - 165*b1) * q^95 + (113*b5 + b4 + 202*b3 - 202) * q^97 + (135*b2 + 97*b1 + 179) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 5 q^{2} - 17 q^{4} + 15 q^{5} + 4 q^{7} - 150 q^{8}+O(q^{10})$$ 6 * q + 5 * q^2 - 17 * q^4 + 15 * q^5 + 4 * q^7 - 150 * q^8 $$6 q + 5 q^{2} - 17 q^{4} + 15 q^{5} + 4 q^{7} - 150 q^{8} + 50 q^{10} + 5 q^{11} - 7 q^{13} + 60 q^{14} - 161 q^{16} - 310 q^{17} - 100 q^{19} + 85 q^{20} + 229 q^{22} + 285 q^{23} - 75 q^{25} - 370 q^{26} - 668 q^{28} + 115 q^{29} + 115 q^{31} + 775 q^{32} - 413 q^{34} + 40 q^{35} - 768 q^{37} - 1150 q^{38} - 375 q^{40} + 580 q^{41} + 797 q^{43} + 2830 q^{44} - 570 q^{46} - 145 q^{47} - 577 q^{49} + 125 q^{50} - 825 q^{52} + 800 q^{53} + 50 q^{55} - 2190 q^{56} + 59 q^{58} + 380 q^{59} + 152 q^{61} + 2010 q^{62} + 5874 q^{64} + 35 q^{65} - 2 q^{67} + 475 q^{68} - 300 q^{70} - 80 q^{71} - 1960 q^{73} - 2720 q^{74} + 3276 q^{76} + 1950 q^{77} - 1013 q^{79} - 1610 q^{80} + 8 q^{82} + 270 q^{83} - 775 q^{85} - 1555 q^{86} + 5193 q^{88} - 2040 q^{89} - 1264 q^{91} - 1215 q^{92} - 3833 q^{94} - 250 q^{95} - 720 q^{97} + 610 q^{98}+O(q^{100})$$ 6 * q + 5 * q^2 - 17 * q^4 + 15 * q^5 + 4 * q^7 - 150 * q^8 + 50 * q^10 + 5 * q^11 - 7 * q^13 + 60 * q^14 - 161 * q^16 - 310 * q^17 - 100 * q^19 + 85 * q^20 + 229 * q^22 + 285 * q^23 - 75 * q^25 - 370 * q^26 - 668 * q^28 + 115 * q^29 + 115 * q^31 + 775 * q^32 - 413 * q^34 + 40 * q^35 - 768 * q^37 - 1150 * q^38 - 375 * q^40 + 580 * q^41 + 797 * q^43 + 2830 * q^44 - 570 * q^46 - 145 * q^47 - 577 * q^49 + 125 * q^50 - 825 * q^52 + 800 * q^53 + 50 * q^55 - 2190 * q^56 + 59 * q^58 + 380 * q^59 + 152 * q^61 + 2010 * q^62 + 5874 * q^64 + 35 * q^65 - 2 * q^67 + 475 * q^68 - 300 * q^70 - 80 * q^71 - 1960 * q^73 - 2720 * q^74 + 3276 * q^76 + 1950 * q^77 - 1013 * q^79 - 1610 * q^80 + 8 * q^82 + 270 * q^83 - 775 * q^85 - 1555 * q^86 + 5193 * q^88 - 2040 * q^89 - 1264 * q^91 - 1215 * q^92 - 3833 * q^94 - 250 * q^95 - 720 * q^97 + 610 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 13x^{4} - 4x^{3} + 152x^{2} - 96x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( -11\nu^{5} + 143\nu^{4} + 21\nu^{3} + 1672\nu^{2} - 1056\nu + 13728 ) / 3760$$ (-11*v^5 + 143*v^4 + 21*v^3 + 1672*v^2 - 1056*v + 13728) / 3760 $$\beta_{2}$$ $$=$$ $$( -17\nu^{5} + 221\nu^{4} - 993\nu^{3} + 2584\nu^{2} - 1632\nu + 17456 ) / 3760$$ (-17*v^5 + 221*v^4 - 993*v^3 + 2584*v^2 - 1632*v + 17456) / 3760 $$\beta_{3}$$ $$=$$ $$( 39\nu^{5} - 37\nu^{4} + 481\nu^{3} + 182\nu^{2} + 5624\nu + 208 ) / 3760$$ (39*v^5 - 37*v^4 + 481*v^3 + 182*v^2 + 5624*v + 208) / 3760 $$\beta_{4}$$ $$=$$ $$( -61\nu^{5} + 88\nu^{4} - 1144\nu^{3} + 1047\nu^{2} - 13376\nu + 8448 ) / 1880$$ (-61*v^5 + 88*v^4 - 1144*v^3 + 1047*v^2 - 13376*v + 8448) / 1880 $$\beta_{5}$$ $$=$$ $$( -31\nu^{5} + 27\nu^{4} - 351\nu^{3} + 200\nu^{2} - 4104\nu + 2592 ) / 752$$ (-31*v^5 + 27*v^4 - 351*v^3 + 200*v^2 - 4104*v + 2592) / 752
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 3$$ (b5 - b4 + b3 + b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( 5\beta_{5} + \beta_{4} + 23\beta_{3} - 23 ) / 3$$ (5*b5 + b4 + 23*b3 - 23) / 3 $$\nu^{3}$$ $$=$$ $$( -11\beta_{2} + 17\beta _1 - 11 ) / 3$$ (-11*b2 + 17*b1 - 11) / 3 $$\nu^{4}$$ $$=$$ $$-23\beta_{5} - 3\beta_{4} - 93\beta_{3} + 3\beta_{2} + 23\beta_1$$ -23*b5 - 3*b4 - 93*b3 + 3*b2 + 23*b1 $$\nu^{5}$$ $$=$$ $$( -233\beta_{5} + 131\beta_{4} - 227\beta_{3} + 227 ) / 3$$ (-233*b5 + 131*b4 - 227*b3 + 227) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 1.83685 + 3.18152i −1.66402 − 2.88216i 0.327167 + 0.566669i 1.83685 − 3.18152i −1.66402 + 2.88216i 0.327167 − 0.566669i
−1.29244 + 2.23857i 0 0.659207 + 1.14178i 2.50000 + 4.33013i 0 11.4468 19.8264i −24.0869 0 −12.9244
136.2 1.06306 1.84127i 0 1.73981 + 3.01344i 2.50000 + 4.33013i 0 −15.3500 + 26.5870i 24.4070 0 10.6306
136.3 2.72938 4.72742i 0 −10.8990 18.8776i 2.50000 + 4.33013i 0 5.90326 10.2247i −75.3201 0 27.2938
271.1 −1.29244 2.23857i 0 0.659207 1.14178i 2.50000 4.33013i 0 11.4468 + 19.8264i −24.0869 0 −12.9244
271.2 1.06306 + 1.84127i 0 1.73981 3.01344i 2.50000 4.33013i 0 −15.3500 26.5870i 24.4070 0 10.6306
271.3 2.72938 + 4.72742i 0 −10.8990 + 18.8776i 2.50000 4.33013i 0 5.90326 + 10.2247i −75.3201 0 27.2938
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.v 6
3.b odd 2 1 405.4.e.q 6
9.c even 3 1 135.4.a.e 3
9.c even 3 1 inner 405.4.e.v 6
9.d odd 6 1 135.4.a.h yes 3
9.d odd 6 1 405.4.e.q 6
36.f odd 6 1 2160.4.a.bi 3
36.h even 6 1 2160.4.a.bq 3
45.h odd 6 1 675.4.a.p 3
45.j even 6 1 675.4.a.s 3
45.k odd 12 2 675.4.b.m 6
45.l even 12 2 675.4.b.n 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 9.c even 3 1
135.4.a.h yes 3 9.d odd 6 1
405.4.e.q 6 3.b odd 2 1
405.4.e.q 6 9.d odd 6 1
405.4.e.v 6 1.a even 1 1 trivial
405.4.e.v 6 9.c even 3 1 inner
675.4.a.p 3 45.h odd 6 1
675.4.a.s 3 45.j even 6 1
675.4.b.m 6 45.k odd 12 2
675.4.b.n 6 45.l even 12 2
2160.4.a.bi 3 36.f odd 6 1
2160.4.a.bq 3 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{6} - 5T_{2}^{5} + 33T_{2}^{4} - 20T_{2}^{3} + 214T_{2}^{2} - 240T_{2} + 900$$ T2^6 - 5*T2^5 + 33*T2^4 - 20*T2^3 + 214*T2^2 - 240*T2 + 900 $$T_{7}^{6} - 4T_{7}^{5} + 811T_{7}^{4} - 13416T_{7}^{3} + 665217T_{7}^{2} - 6596910T_{7} + 68856804$$ T7^6 - 4*T7^5 + 811*T7^4 - 13416*T7^3 + 665217*T7^2 - 6596910*T7 + 68856804

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 5 T^{5} + 33 T^{4} - 20 T^{3} + \cdots + 900$$
$3$ $$T^{6}$$
$5$ $$(T^{2} - 5 T + 25)^{3}$$
$7$ $$T^{6} - 4 T^{5} + 811 T^{4} + \cdots + 68856804$$
$11$ $$T^{6} - 5 T^{5} + 2913 T^{4} + \cdots + 977187600$$
$13$ $$T^{6} + 7 T^{5} + 818 T^{4} + \cdots + 41280625$$
$17$ $$(T^{3} + 155 T^{2} + 5608 T + 41760)^{2}$$
$19$ $$(T^{3} + 50 T^{2} - 16663 T - 368012)^{2}$$
$23$ $$T^{6} - 285 T^{5} + \cdots + 306362250000$$
$29$ $$T^{6} - 115 T^{5} + \cdots + 41477979315600$$
$31$ $$T^{6} - 115 T^{5} + \cdots + 880414396416$$
$37$ $$(T^{3} + 384 T^{2} - 67923 T - 22667198)^{2}$$
$41$ $$T^{6} - 580 T^{5} + \cdots + 15345082598400$$
$43$ $$T^{6} - 797 T^{5} + \cdots + 28707478180096$$
$47$ $$T^{6} + \cdots + 207021450297600$$
$53$ $$(T^{3} - 400 T^{2} - 58172 T + 12658320)^{2}$$
$59$ $$T^{6} - 380 T^{5} + \cdots + 27093274214400$$
$61$ $$T^{6} - 152 T^{5} + \cdots + 25695430112356$$
$67$ $$T^{6} + \cdots + 431363822490000$$
$71$ $$(T^{3} + 40 T^{2} - 677372 T - 216071280)^{2}$$
$73$ $$(T^{3} + 980 T^{2} + 264533 T + 16447954)^{2}$$
$79$ $$T^{6} + 1013 T^{5} + \cdots + 82\!\cdots\!25$$
$83$ $$T^{6} - 270 T^{5} + \cdots + 71\!\cdots\!00$$
$89$ $$(T^{3} + 1020 T^{2} + 46800 T - 125064000)^{2}$$
$97$ $$T^{6} + \cdots + 752435512191364$$