# Properties

 Label 475.2.b.e Level $475$ Weight $2$ Character orbit 475.b Analytic conductor $3.793$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (of order $$2$$, degree $$1$$, 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: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2058981376.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1$$ x^8 - 2*x^7 + 2*x^6 + 18*x^4 - 34*x^3 + 32*x^2 - 8*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 95) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + \beta_1 q^{3} + ( - \beta_{3} - 2) q^{4} + ( - \beta_{3} - \beta_{2}) q^{6} + \beta_{7} q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - 2) q^{9}+O(q^{10})$$ q + b6 * q^2 + b1 * q^3 + (-b3 - 2) * q^4 + (-b3 - b2) * q^6 + b7 * q^7 + (-b6 - b5 - b1) * q^8 + (-b4 - b3 - 2) * q^9 $$q + \beta_{6} q^{2} + \beta_1 q^{3} + ( - \beta_{3} - 2) q^{4} + ( - \beta_{3} - \beta_{2}) q^{6} + \beta_{7} q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - 2) q^{9} + ( - \beta_{4} + \beta_{3} + 1) q^{11} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_1) q^{12} + (\beta_{7} + \beta_{5} - \beta_1) q^{13} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{14} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{16} + ( - \beta_{5} + 2 \beta_1) q^{17} + ( - \beta_{7} - 3 \beta_{6} - 4 \beta_{5}) q^{18} - q^{19} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{21} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{22} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_1) q^{23} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 6) q^{24} + ( - 2 \beta_{4} + \beta_{3} + 1) q^{26} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_1) q^{27} + (2 \beta_{6} - \beta_{5} - 2 \beta_1) q^{28} + (\beta_{2} - 1) q^{29} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{31} + (2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_1) q^{32} + (\beta_{7} + 2 \beta_{6} + \beta_{5}) q^{33} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 1) q^{34} + (3 \beta_{4} + 4 \beta_{3} + \beta_{2} + 10) q^{36} + (2 \beta_{6} + \beta_1) q^{37} - \beta_{6} q^{38} + (\beta_{4} - \beta_{3} + 3) q^{39} + ( - 2 \beta_{3} + 4) q^{41} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_1) q^{42} - \beta_{7} q^{43} + (\beta_{4} - \beta_{3} - \beta_{2} - 6) q^{44} + (3 \beta_{4} - \beta_{3} - \beta_{2}) q^{46} + ( - \beta_{7} - 4 \beta_{6} + 2 \beta_{5}) q^{47} + (\beta_{7} + 6 \beta_{6} + 3 \beta_{5} + 3 \beta_1) q^{48} + (4 \beta_{3} - 5) q^{49} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 9) q^{51} + (2 \beta_{6} - 3 \beta_{5} + \beta_1) q^{52} + (2 \beta_{6} - 2 \beta_{5} + \beta_1) q^{53} + (2 \beta_{3} + 3 \beta_{2} + 5) q^{54} + ( - \beta_{4} + 3 \beta_{3} - 3) q^{56} - \beta_1 q^{57} + (\beta_{7} - \beta_{5} + 4 \beta_1) q^{58} + 2 \beta_{4} q^{59} + ( - \beta_{4} + 3 \beta_{3} + 5) q^{61} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - 6 \beta_1) q^{62} + (\beta_{7} + 4 \beta_{5} - 4 \beta_1) q^{63} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{64} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 7) q^{66} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 3 \beta_1) q^{67} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 6 \beta_1) q^{68} + ( - 2 \beta_{4} + \beta_{2} - 7) q^{69} + (2 \beta_{4} - \beta_{2} - 5) q^{71} + (2 \beta_{7} + 9 \beta_{6} + 4 \beta_{5} + 5 \beta_1) q^{72} + (3 \beta_{5} + 2 \beta_1) q^{73} + ( - 3 \beta_{3} - \beta_{2} - 8) q^{74} + (\beta_{3} + 2) q^{76} + ( - 4 \beta_{6} + 6 \beta_{5}) q^{77} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{78} + (2 \beta_{4} + 4) q^{79} + ( - \beta_{4} + 3 \beta_{3} + 4) q^{81} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{82} + ( - \beta_{7} - 2 \beta_1) q^{83} + (2 \beta_{4} - \beta_{2} + 11) q^{84} + (\beta_{4} - \beta_{3} + \beta_{2} - 2) q^{86} + (4 \beta_{6} + 4 \beta_{5} - 2 \beta_1) q^{87} + ( - 2 \beta_{7} - 4 \beta_{6} - \beta_{5} - 2 \beta_1) q^{88} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{89} + (2 \beta_{4} + 4 \beta_{3} - \beta_{2} - 9) q^{91} + ( - 2 \beta_{6} + 5 \beta_{5} - 4 \beta_1) q^{92} + ( - 2 \beta_{7} - 8 \beta_{6} - 4 \beta_{5}) q^{93} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 12) q^{94} + ( - 2 \beta_{4} - 7 \beta_{3} - 13) q^{96} + ( - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - \beta_1) q^{97} + ( - \beta_{6} + 4 \beta_{5} + 4 \beta_1) q^{98} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 1) q^{99}+O(q^{100})$$ q + b6 * q^2 + b1 * q^3 + (-b3 - 2) * q^4 + (-b3 - b2) * q^6 + b7 * q^7 + (-b6 - b5 - b1) * q^8 + (-b4 - b3 - 2) * q^9 + (-b4 + b3 + 1) * q^11 + (-b7 - 2*b6 - 3*b1) * q^12 + (b7 + b5 - b1) * q^13 + (-b4 + b3 - b2 + 2) * q^14 + (b4 + b3 + b2 + 1) * q^16 + (-b5 + 2*b1) * q^17 + (-b7 - 3*b6 - 4*b5) * q^18 - q^19 + (-2*b3 + b2 - 1) * q^21 + (-b7 + 2*b6 - 2*b5 + 2*b1) * q^22 + (-b7 - 2*b5 + 2*b1) * q^23 + (b4 + 2*b3 + 2*b2 + 6) * q^24 + (-2*b4 + b3 + 1) * q^26 + (-b7 - 2*b6 + b5 - 2*b1) * q^27 + (2*b6 - b5 - 2*b1) * q^28 + (b2 - 1) * q^29 + (-2*b3 - b2 + 1) * q^31 + (2*b7 + b6 + b5 + 2*b1) * q^32 + (b7 + 2*b6 + b5) * q^33 + (b4 - b3 - 2*b2 + 1) * q^34 + (3*b4 + 4*b3 + b2 + 10) * q^36 + (2*b6 + b1) * q^37 - b6 * q^38 + (b4 - b3 + 3) * q^39 + (-2*b3 + 4) * q^41 + (b7 - 2*b6 - 3*b5 + 2*b1) * q^42 - b7 * q^43 + (b4 - b3 - b2 - 6) * q^44 + (3*b4 - b3 - b2) * q^46 + (-b7 - 4*b6 + 2*b5) * q^47 + (b7 + 6*b6 + 3*b5 + 3*b1) * q^48 + (4*b3 - 5) * q^49 + (-2*b4 - 2*b3 + b2 - 9) * q^51 + (2*b6 - 3*b5 + b1) * q^52 + (2*b6 - 2*b5 + b1) * q^53 + (2*b3 + 3*b2 + 5) * q^54 + (-b4 + 3*b3 - 3) * q^56 - b1 * q^57 + (b7 - b5 + 4*b1) * q^58 + 2*b4 * q^59 + (-b4 + 3*b3 + 5) * q^61 + (-b7 - 2*b6 - b5 - 6*b1) * q^62 + (b7 + 4*b5 - 4*b1) * q^63 + (-b4 - 2*b2 + 1) * q^64 + (-2*b4 - 2*b3 - b2 - 7) * q^66 + (b7 + 2*b6 + 3*b5 - 3*b1) * q^67 + (-b7 - 2*b6 + 2*b5 - 6*b1) * q^68 + (-2*b4 + b2 - 7) * q^69 + (2*b4 - b2 - 5) * q^71 + (2*b7 + 9*b6 + 4*b5 + 5*b1) * q^72 + (3*b5 + 2*b1) * q^73 + (-3*b3 - b2 - 8) * q^74 + (b3 + 2) * q^76 + (-4*b6 + 6*b5) * q^77 + (b7 + 2*b6 + 2*b5 - 2*b1) * q^78 + (2*b4 + 4) * q^79 + (-b4 + 3*b3 + 4) * q^81 + (2*b6 - 2*b5 - 2*b1) * q^82 + (-b7 - 2*b1) * q^83 + (2*b4 - b2 + 11) * q^84 + (b4 - b3 + b2 - 2) * q^86 + (4*b6 + 4*b5 - 2*b1) * q^87 + (-2*b7 - 4*b6 - b5 - 2*b1) * q^88 + (-2*b4 - 2*b3 - b2 - 1) * q^89 + (2*b4 + 4*b3 - b2 - 9) * q^91 + (-2*b6 + 5*b5 - 4*b1) * q^92 + (-2*b7 - 8*b6 - 4*b5) * q^93 + (-b4 + b3 + b2 + 12) * q^94 + (-2*b4 - 7*b3 - 13) * q^96 + (-2*b7 - 2*b6 - 4*b5 - b1) * q^97 + (-b6 + 4*b5 + 4*b1) * q^98 + (-3*b4 - b3 - 2*b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{4} - 16 q^{9}+O(q^{10})$$ 8 * q - 16 * q^4 - 16 * q^9 $$8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51} + 40 q^{54} - 24 q^{56} + 40 q^{61} + 8 q^{64} - 56 q^{66} - 56 q^{69} - 40 q^{71} - 64 q^{74} + 16 q^{76} + 32 q^{79} + 32 q^{81} + 88 q^{84} - 16 q^{86} - 8 q^{89} - 72 q^{91} + 96 q^{94} - 104 q^{96} + 8 q^{99}+O(q^{100})$$ 8 * q - 16 * q^4 - 16 * q^9 + 8 * q^11 + 16 * q^14 + 8 * q^16 - 8 * q^19 - 8 * q^21 + 48 * q^24 + 8 * q^26 - 8 * q^29 + 8 * q^31 + 8 * q^34 + 80 * q^36 + 24 * q^39 + 32 * q^41 - 48 * q^44 - 40 * q^49 - 72 * q^51 + 40 * q^54 - 24 * q^56 + 40 * q^61 + 8 * q^64 - 56 * q^66 - 56 * q^69 - 40 * q^71 - 64 * q^74 + 16 * q^76 + 32 * q^79 + 32 * q^81 + 88 * q^84 - 16 * q^86 - 8 * q^89 - 72 * q^91 + 96 * q^94 - 104 * q^96 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 120\nu^{7} - 143\nu^{6} + 138\nu^{5} + 30\nu^{4} + 2592\nu^{3} - 2311\nu^{2} + 3834\nu - 552 ) / 1631$$ (120*v^7 - 143*v^6 + 138*v^5 + 30*v^4 + 2592*v^3 - 2311*v^2 + 3834*v - 552) / 1631 $$\beta_{2}$$ $$=$$ $$( -240\nu^{7} + 286\nu^{6} - 276\nu^{5} - 60\nu^{4} - 5184\nu^{3} + 4622\nu^{2} - 1144\nu - 527 ) / 1631$$ (-240*v^7 + 286*v^6 - 276*v^5 - 60*v^4 - 5184*v^3 + 4622*v^2 - 1144*v - 527) / 1631 $$\beta_{3}$$ $$=$$ $$( 388\nu^{7} - 408\nu^{6} + 120\nu^{5} + 97\nu^{4} + 7076\nu^{3} - 6548\nu^{2} + 1632\nu - 480 ) / 1631$$ (388*v^7 - 408*v^6 + 120*v^5 + 97*v^4 + 7076*v^3 - 6548*v^2 + 1632*v - 480) / 1631 $$\beta_{4}$$ $$=$$ $$( -582\nu^{7} + 612\nu^{6} - 180\nu^{5} - 961\nu^{4} - 10614\nu^{3} + 9822\nu^{2} - 2448\nu - 7435 ) / 1631$$ (-582*v^7 + 612*v^6 - 180*v^5 - 961*v^4 - 10614*v^3 + 9822*v^2 - 2448*v - 7435) / 1631 $$\beta_{5}$$ $$=$$ $$( 1104\nu^{7} - 1968\nu^{6} + 1922\nu^{5} + 276\nu^{4} + 19932\nu^{3} - 32352\nu^{2} + 30706\nu - 4426 ) / 1631$$ (1104*v^7 - 1968*v^6 + 1922*v^5 + 276*v^4 + 19932*v^3 - 32352*v^2 + 30706*v - 4426) / 1631 $$\beta_{6}$$ $$=$$ $$( -1104\nu^{7} + 1968\nu^{6} - 1922\nu^{5} - 276\nu^{4} - 19932\nu^{3} + 33983\nu^{2} - 30706\nu + 4426 ) / 1631$$ (-1104*v^7 + 1968*v^6 - 1922*v^5 - 276*v^4 - 19932*v^3 + 33983*v^2 - 30706*v + 4426) / 1631 $$\beta_{7}$$ $$=$$ $$( 2168\nu^{7} - 4432\nu^{6} + 3798\nu^{5} + 542\nu^{4} + 39000\nu^{3} - 76438\nu^{2} + 60134\nu - 8668 ) / 1631$$ (2168*v^7 - 4432*v^6 + 3798*v^5 + 542*v^4 + 39000*v^3 - 76438*v^2 + 60134*v - 8668) / 1631
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 1 ) / 4$$ (b2 + 2*b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5}$$ b6 + b5 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{3} - 4\beta_{2} + 8\beta _1 - 2 ) / 4$$ (b7 + 2*b6 - b5 - 2*b3 - 4*b2 + 8*b1 - 2) / 4 $$\nu^{4}$$ $$=$$ $$-2\beta_{4} - 3\beta_{3} - 10$$ -2*b4 - 3*b3 - 10 $$\nu^{5}$$ $$=$$ $$( -6\beta_{7} - 12\beta_{6} + 4\beta_{5} - 12\beta_{3} - 17\beta_{2} - 34\beta _1 - 9 ) / 4$$ (-6*b7 - 12*b6 + 4*b5 - 12*b3 - 17*b2 - 34*b1 - 9) / 4 $$\nu^{6}$$ $$=$$ $$-3\beta_{7} - 23\beta_{6} - 17\beta_{5} - \beta_1$$ -3*b7 - 23*b6 - 17*b5 - b1 $$\nu^{7}$$ $$=$$ $$( -29\beta_{7} - 62\beta_{6} + 13\beta_{5} + 2\beta_{4} + 60\beta_{3} + 74\beta_{2} - 148\beta _1 + 50 ) / 4$$ (-29*b7 - 62*b6 + 13*b5 + 2*b4 + 60*b3 + 74*b2 - 148*b1 + 50) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 1.52153 − 1.52153i −1.43917 + 1.43917i 0.148421 + 0.148421i 0.769222 + 0.769222i 0.769222 − 0.769222i 0.148421 − 0.148421i −1.43917 − 1.43917i 1.52153 + 1.52153i
2.63010i 3.04306i −4.91744 0 −8.00355 0.574672i 7.67316i −6.26020 0
324.2 2.14243i 2.87834i −2.59002 0 6.16666 3.10482i 1.26409i −5.28487 0
324.3 1.95594i 0.296842i −1.82571 0 0.580605 3.56331i 0.340899i 2.91188 0
324.4 0.816594i 1.53844i 1.33317 0 1.25628 5.03316i 2.72185i 0.633188 0
324.5 0.816594i 1.53844i 1.33317 0 1.25628 5.03316i 2.72185i 0.633188 0
324.6 1.95594i 0.296842i −1.82571 0 0.580605 3.56331i 0.340899i 2.91188 0
324.7 2.14243i 2.87834i −2.59002 0 6.16666 3.10482i 1.26409i −5.28487 0
324.8 2.63010i 3.04306i −4.91744 0 −8.00355 0.574672i 7.67316i −6.26020 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.b.e 8
5.b even 2 1 inner 475.2.b.e 8
5.c odd 4 1 95.2.a.b 4
5.c odd 4 1 475.2.a.i 4
15.e even 4 1 855.2.a.m 4
15.e even 4 1 4275.2.a.bo 4
20.e even 4 1 1520.2.a.t 4
20.e even 4 1 7600.2.a.cf 4
35.f even 4 1 4655.2.a.y 4
40.i odd 4 1 6080.2.a.cc 4
40.k even 4 1 6080.2.a.ch 4
95.g even 4 1 1805.2.a.p 4
95.g even 4 1 9025.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 5.c odd 4 1
475.2.a.i 4 5.c odd 4 1
475.2.b.e 8 1.a even 1 1 trivial
475.2.b.e 8 5.b even 2 1 inner
855.2.a.m 4 15.e even 4 1
1520.2.a.t 4 20.e even 4 1
1805.2.a.p 4 95.g even 4 1
4275.2.a.bo 4 15.e even 4 1
4655.2.a.y 4 35.f even 4 1
6080.2.a.cc 4 40.i odd 4 1
6080.2.a.ch 4 40.k even 4 1
7600.2.a.cf 4 20.e even 4 1
9025.2.a.bf 4 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 16T_{2}^{6} + 86T_{2}^{4} + 172T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 16 T^{6} + 86 T^{4} + 172 T^{2} + \cdots + 81$$
$3$ $$T^{8} + 20 T^{6} + 120 T^{4} + \cdots + 16$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 48 T^{6} + 704 T^{4} + \cdots + 1024$$
$11$ $$(T^{4} - 4 T^{3} - 16 T^{2} + 32 T + 48)^{2}$$
$13$ $$T^{8} + 52 T^{6} + 744 T^{4} + \cdots + 400$$
$17$ $$T^{8} + 80 T^{6} + 1248 T^{4} + \cdots + 2304$$
$19$ $$(T + 1)^{8}$$
$23$ $$T^{8} + 112 T^{6} + 3968 T^{4} + \cdots + 82944$$
$29$ $$(T^{4} + 4 T^{3} - 32 T^{2} - 16 T + 48)^{2}$$
$31$ $$(T^{4} - 4 T^{3} - 80 T^{2} + 512 T - 640)^{2}$$
$37$ $$T^{8} + 84 T^{6} + 1064 T^{4} + \cdots + 16$$
$41$ $$(T^{4} - 16 T^{3} + 56 T^{2} + 32 T - 240)^{2}$$
$43$ $$T^{8} + 48 T^{6} + 704 T^{4} + \cdots + 1024$$
$47$ $$T^{8} + 272 T^{6} + 21952 T^{4} + \cdots + 1115136$$
$53$ $$T^{8} + 100 T^{6} + 2984 T^{4} + \cdots + 121104$$
$59$ $$(T^{4} - 64 T^{2} + 224 T - 192)^{2}$$
$61$ $$(T^{4} - 20 T^{3} + 56 T^{2} + 688 T - 2656)^{2}$$
$67$ $$T^{8} + 308 T^{6} + 15480 T^{4} + \cdots + 1157776$$
$71$ $$(T^{4} + 20 T^{3} + 32 T^{2} - 1024 T - 4224)^{2}$$
$73$ $$T^{8} + 272 T^{6} + 21984 T^{4} + \cdots + 30976$$
$79$ $$(T^{4} - 16 T^{3} + 32 T^{2} + 480 T - 1856)^{2}$$
$83$ $$T^{8} + 144 T^{6} + 6144 T^{4} + \cdots + 230400$$
$89$ $$(T^{4} + 4 T^{3} - 144 T^{2} + 176 T + 240)^{2}$$
$97$ $$T^{8} + 452 T^{6} + 46920 T^{4} + \cdots + 1926544$$