# Properties

 Label 273.2.c.b Level $273$ Weight $2$ Character orbit 273.c Analytic conductor $2.180$ 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] = [273,2,Mod(64,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("273.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.c (of order $$2$$, degree $$1$$, 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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{5} - \beta_{4} q^{6} - \beta_{3} q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + q^{9}+O(q^{10})$$ q - b4 * q^2 + q^3 + (b2 - b1) * q^4 + (b5 + b4 - b3) * q^5 - b4 * q^6 - b3 * q^7 + (-b5 + b3) * q^8 + q^9 $$q - \beta_{4} q^{2} + q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{5} - \beta_{4} q^{6} - \beta_{3} q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} + q^{9} + (\beta_{2} + \beta_1 + 3) q^{10} + (\beta_{5} - \beta_{4} - 3 \beta_{3}) q^{11} + (\beta_{2} - \beta_1) q^{12} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{13} + \beta_{2} q^{14} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{15} + ( - 2 \beta_1 - 1) q^{16} + 2 \beta_1 q^{17} - \beta_{4} q^{18} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{19} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{20} - \beta_{3} q^{21} + (5 \beta_{2} - \beta_1 - 1) q^{22} + ( - 2 \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{5} + \beta_{3}) q^{24} + ( - 4 \beta_{2} - 2 \beta_1 - 3) q^{25} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{26} + q^{27} + ( - \beta_{5} + \beta_{4}) q^{28} + ( - 4 \beta_{2} - 2) q^{29} + (\beta_{2} + \beta_1 + 3) q^{30} + ( - 4 \beta_{5} + 4 \beta_{3}) q^{31} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{32} + (\beta_{5} - \beta_{4} - 3 \beta_{3}) q^{33} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{34} + ( - \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{2} - \beta_1) q^{36} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{37} + ( - 6 \beta_{2} + 2 \beta_1 + 2) q^{38} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{39} + (3 \beta_{2} + \beta_1 + 5) q^{40} + (3 \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{41} + \beta_{2} q^{42} + (4 \beta_{2} - 4) q^{43} + ( - 3 \beta_{5} + 5 \beta_{4} + 3 \beta_{3}) q^{44} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{45} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3}) q^{46} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3}) q^{47} + ( - 2 \beta_1 - 1) q^{48} - q^{49} + (4 \beta_{5} + \beta_{4} - 10 \beta_{3}) q^{50} + 2 \beta_1 q^{51} + (\beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{52} - 6 q^{53} - \beta_{4} q^{54} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{55} + (\beta_1 + 1) q^{56} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{57} + (4 \beta_{5} - 2 \beta_{4} - 8 \beta_{3}) q^{58} + (\beta_{5} + \beta_{4} - 3 \beta_{3}) q^{59} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{60} + ( - 2 \beta_{2} + 8 \beta_1 + 4) q^{61} + ( - 8 \beta_{2} - 4) q^{62} - \beta_{3} q^{63} + ( - 5 \beta_{2} - \beta_1 + 2) q^{64} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 4) q^{65} + (5 \beta_{2} - \beta_1 - 1) q^{66} + (2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3}) q^{67} + (2 \beta_1 - 4) q^{68} + ( - 2 \beta_{2} + 2 \beta_1) q^{69} + (\beta_{5} + \beta_{4} - 3 \beta_{3}) q^{70} + ( - 5 \beta_{5} - 3 \beta_{4} - \beta_{3}) q^{71} + ( - \beta_{5} + \beta_{3}) q^{72} + (4 \beta_{5} + 2 \beta_{4} - 6 \beta_{3}) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{74} + ( - 4 \beta_{2} - 2 \beta_1 - 3) q^{75} + (2 \beta_{5} - 6 \beta_{4} - 6 \beta_{3}) q^{76} + (\beta_{2} - \beta_1 - 3) q^{77} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{78} + ( - 2 \beta_{2} - 6) q^{79} + ( - \beta_{5} - 5 \beta_{4} + 9 \beta_{3}) q^{80} + q^{81} + ( - \beta_{2} - \beta_1 + 1) q^{82} + (5 \beta_{5} - 7 \beta_{4} + 5 \beta_{3}) q^{83} + ( - \beta_{5} + \beta_{4}) q^{84} + (4 \beta_{4} - 8 \beta_{3}) q^{85} + ( - 4 \beta_{5} + 8 \beta_{4} + 8 \beta_{3}) q^{86} + ( - 4 \beta_{2} - 2) q^{87} + ( - \beta_{2} + 3 \beta_1 + 5) q^{88} + ( - 5 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{89} + (\beta_{2} + \beta_1 + 3) q^{90} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{91} + (4 \beta_{2} - 6) q^{92} + ( - 4 \beta_{5} + 4 \beta_{3}) q^{93} + ( - 3 \beta_{2} + 3 \beta_1 + 5) q^{94} + (4 \beta_{2} + 4) q^{95} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{96} + 2 \beta_{5} q^{97} + \beta_{4} q^{98} + (\beta_{5} - \beta_{4} - 3 \beta_{3}) q^{99}+O(q^{100})$$ q - b4 * q^2 + q^3 + (b2 - b1) * q^4 + (b5 + b4 - b3) * q^5 - b4 * q^6 - b3 * q^7 + (-b5 + b3) * q^8 + q^9 + (b2 + b1 + 3) * q^10 + (b5 - b4 - 3*b3) * q^11 + (b2 - b1) * q^12 + (b5 - 2*b4 + 2*b3 + b2) * q^13 + b2 * q^14 + (b5 + b4 - b3) * q^15 + (-2*b1 - 1) * q^16 + 2*b1 * q^17 - b4 * q^18 + (-2*b5 + 2*b4 + 2*b3) * q^19 + (b5 - b4 + b3) * q^20 - b3 * q^21 + (5*b2 - b1 - 1) * q^22 + (-2*b2 + 2*b1) * q^23 + (-b5 + b3) * q^24 + (-4*b2 - 2*b1 - 3) * q^25 + (-b5 + b4 + 2*b3 + b2 - 2*b1 - 3) * q^26 + q^27 + (-b5 + b4) * q^28 + (-4*b2 - 2) * q^29 + (b2 + b1 + 3) * q^30 + (-4*b5 + 4*b3) * q^31 + (-2*b5 + 3*b4) * q^32 + (b5 - b4 - 3*b3) * q^33 + (-2*b4 + 2*b3) * q^34 + (-b2 - b1 - 1) * q^35 + (b2 - b1) * q^36 + (-2*b5 + 2*b4 - 2*b3) * q^37 + (-6*b2 + 2*b1 + 2) * q^38 + (b5 - 2*b4 + 2*b3 + b2) * q^39 + (3*b2 + b1 + 5) * q^40 + (3*b5 - b4 + 5*b3) * q^41 + b2 * q^42 + (4*b2 - 4) * q^43 + (-3*b5 + 5*b4 + 3*b3) * q^44 + (b5 + b4 - b3) * q^45 + (2*b5 - 4*b4 - 2*b3) * q^46 + (-b5 + 3*b4 - b3) * q^47 + (-2*b1 - 1) * q^48 - q^49 + (4*b5 + b4 - 10*b3) * q^50 + 2*b1 * q^51 + (b5 + 2*b4 + 4*b3 - 2*b2 + b1 + 1) * q^52 - 6 * q^53 - b4 * q^54 + (-4*b2 - 2*b1 - 4) * q^55 + (b1 + 1) * q^56 + (-2*b5 + 2*b4 + 2*b3) * q^57 + (4*b5 - 2*b4 - 8*b3) * q^58 + (b5 + b4 - 3*b3) * q^59 + (b5 - b4 + b3) * q^60 + (-2*b2 + 8*b1 + 4) * q^61 + (-8*b2 - 4) * q^62 - b3 * q^63 + (-5*b2 - b1 + 2) * q^64 + (b5 + b4 - 3*b3 + 2*b2 + 4*b1 + 4) * q^65 + (5*b2 - b1 - 1) * q^66 + (2*b5 - 2*b4 + 6*b3) * q^67 + (2*b1 - 4) * q^68 + (-2*b2 + 2*b1) * q^69 + (b5 + b4 - 3*b3) * q^70 + (-5*b5 - 3*b4 - b3) * q^71 + (-b5 + b3) * q^72 + (4*b5 + 2*b4 - 6*b3) * q^73 + (-2*b2 + 2*b1 + 2) * q^74 + (-4*b2 - 2*b1 - 3) * q^75 + (2*b5 - 6*b4 - 6*b3) * q^76 + (b2 - b1 - 3) * q^77 + (-b5 + b4 + 2*b3 + b2 - 2*b1 - 3) * q^78 + (-2*b2 - 6) * q^79 + (-b5 - 5*b4 + 9*b3) * q^80 + q^81 + (-b2 - b1 + 1) * q^82 + (5*b5 - 7*b4 + 5*b3) * q^83 + (-b5 + b4) * q^84 + (4*b4 - 8*b3) * q^85 + (-4*b5 + 8*b4 + 8*b3) * q^86 + (-4*b2 - 2) * q^87 + (-b2 + 3*b1 + 5) * q^88 + (-5*b5 + 3*b4 - 3*b3) * q^89 + (b2 + b1 + 3) * q^90 + (b4 + 2*b2 - b1 + 2) * q^91 + (4*b2 - 6) * q^92 + (-4*b5 + 4*b3) * q^93 + (-3*b2 + 3*b1 + 5) * q^94 + (4*b2 + 4) * q^95 + (-2*b5 + 3*b4) * q^96 + 2*b5 * q^97 + b4 * q^98 + (b5 - b4 - 3*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^3 - 2 * q^4 + 6 * q^9 $$6 q + 6 q^{3} - 2 q^{4} + 6 q^{9} + 16 q^{10} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 6 q^{16} - 16 q^{22} + 4 q^{23} - 10 q^{25} - 20 q^{26} + 6 q^{27} - 4 q^{29} + 16 q^{30} - 4 q^{35} - 2 q^{36} + 24 q^{38} - 2 q^{39} + 24 q^{40} - 2 q^{42} - 32 q^{43} - 6 q^{48} - 6 q^{49} + 10 q^{52} - 36 q^{53} - 16 q^{55} + 6 q^{56} + 28 q^{61} - 8 q^{62} + 22 q^{64} + 20 q^{65} - 16 q^{66} - 24 q^{68} + 4 q^{69} + 16 q^{74} - 10 q^{75} - 20 q^{77} - 20 q^{78} - 32 q^{79} + 6 q^{81} + 8 q^{82} - 4 q^{87} + 32 q^{88} + 16 q^{90} + 8 q^{91} - 44 q^{92} + 36 q^{94} + 16 q^{95}+O(q^{100})$$ 6 * q + 6 * q^3 - 2 * q^4 + 6 * q^9 + 16 * q^10 - 2 * q^12 - 2 * q^13 - 2 * q^14 - 6 * q^16 - 16 * q^22 + 4 * q^23 - 10 * q^25 - 20 * q^26 + 6 * q^27 - 4 * q^29 + 16 * q^30 - 4 * q^35 - 2 * q^36 + 24 * q^38 - 2 * q^39 + 24 * q^40 - 2 * q^42 - 32 * q^43 - 6 * q^48 - 6 * q^49 + 10 * q^52 - 36 * q^53 - 16 * q^55 + 6 * q^56 + 28 * q^61 - 8 * q^62 + 22 * q^64 + 20 * q^65 - 16 * q^66 - 24 * q^68 + 4 * q^69 + 16 * q^74 - 10 * q^75 - 20 * q^77 - 20 * q^78 - 32 * q^79 + 6 * q^81 + 8 * q^82 - 4 * q^87 + 32 * q^88 + 16 * q^90 + 8 * q^91 - 44 * q^92 + 36 * q^94 + 16 * q^95

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

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −0.854638 − 0.854638i 0.403032 − 0.403032i 1.45161 + 1.45161i 1.45161 − 1.45161i 0.403032 + 0.403032i −0.854638 + 0.854638i
2.17009i 1.00000 −2.70928 0.630898i 2.17009i 1.00000i 1.53919i 1.00000 1.36910
64.2 1.48119i 1.00000 −0.193937 4.15633i 1.48119i 1.00000i 2.67513i 1.00000 6.15633
64.3 0.311108i 1.00000 1.90321 1.52543i 0.311108i 1.00000i 1.21432i 1.00000 0.474572
64.4 0.311108i 1.00000 1.90321 1.52543i 0.311108i 1.00000i 1.21432i 1.00000 0.474572
64.5 1.48119i 1.00000 −0.193937 4.15633i 1.48119i 1.00000i 2.67513i 1.00000 6.15633
64.6 2.17009i 1.00000 −2.70928 0.630898i 2.17009i 1.00000i 1.53919i 1.00000 1.36910
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.c.b 6
3.b odd 2 1 819.2.c.c 6
4.b odd 2 1 4368.2.h.o 6
7.b odd 2 1 1911.2.c.h 6
13.b even 2 1 inner 273.2.c.b 6
13.d odd 4 1 3549.2.a.k 3
13.d odd 4 1 3549.2.a.q 3
39.d odd 2 1 819.2.c.c 6
52.b odd 2 1 4368.2.h.o 6
91.b odd 2 1 1911.2.c.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 1.a even 1 1 trivial
273.2.c.b 6 13.b even 2 1 inner
819.2.c.c 6 3.b odd 2 1
819.2.c.c 6 39.d odd 2 1
1911.2.c.h 6 7.b odd 2 1
1911.2.c.h 6 91.b odd 2 1
3549.2.a.k 3 13.d odd 4 1
3549.2.a.q 3 13.d odd 4 1
4368.2.h.o 6 4.b odd 2 1
4368.2.h.o 6 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 7 T^{4} + 11 T^{2} + 1$$
$3$ $$(T - 1)^{6}$$
$5$ $$T^{6} + 20 T^{4} + 48 T^{2} + 16$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400$$
$13$ $$T^{6} + 2 T^{5} + 27 T^{4} + \cdots + 2197$$
$17$ $$(T^{3} - 16 T + 16)^{2}$$
$19$ $$T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024$$
$23$ $$(T^{3} - 2 T^{2} - 20 T + 8)^{2}$$
$29$ $$(T^{3} + 2 T^{2} - 52 T - 40)^{2}$$
$31$ $$T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400$$
$37$ $$T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024$$
$41$ $$T^{6} + 132 T^{4} + 464 T^{2} + \cdots + 400$$
$43$ $$(T^{3} + 16 T^{2} + 32 T - 128)^{2}$$
$47$ $$T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704$$
$53$ $$(T + 6)^{6}$$
$59$ $$T^{6} + 40 T^{4} + 80 T^{2} + 16$$
$61$ $$(T^{3} - 14 T^{2} - 172 T + 2392)^{2}$$
$67$ $$T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 1024$$
$71$ $$T^{6} + 332 T^{4} + 30784 T^{2} + \cdots + 547600$$
$73$ $$T^{6} + 272 T^{4} + 6720 T^{2} + \cdots + 43264$$
$79$ $$(T^{3} + 16 T^{2} + 72 T + 80)^{2}$$
$83$ $$T^{6} + 408 T^{4} + 49616 T^{2} + \cdots + 1567504$$
$89$ $$T^{6} + 212 T^{4} + 6832 T^{2} + \cdots + 5776$$
$97$ $$T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256$$