# Properties

 Label 24.4.f.b Level $24$ Weight $4$ Character orbit 24.f Analytic conductor $1.416$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,4,Mod(11,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("24.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 24.f (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: $$1.41604584014$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096$$ x^8 - 10*x^6 + 120*x^4 - 640*x^2 + 4096 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} - 2) q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - 3 \beta_{5} - 3 \beta_{4} + \cdots - 6) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b4 - 2) * q^3 + (b2 + 3) * q^4 + (-b7 - b6 + b3 - b1) * q^5 + (b7 + 2*b5 + b4 - b3 - b1 + 1) * q^6 + (-b7 + b6 - 2*b2 - 2) * q^7 + (b7 + b6 + b5 - b4 + 2*b3 + 2*b1 + 1) * q^8 + (-3*b5 - 3*b4 - 3*b3 - 3*b1 - 6) * q^9 $$q + \beta_1 q^{2} + (\beta_{4} - 2) q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - 57 \beta_{5} - 39 \beta_{4} + \cdots + 255) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b4 - 2) * q^3 + (b2 + 3) * q^4 + (-b7 - b6 + b3 - b1) * q^5 + (b7 + 2*b5 + b4 - b3 - b1 + 1) * q^6 + (-b7 + b6 - 2*b2 - 2) * q^7 + (b7 + b6 + b5 - b4 + 2*b3 + 2*b1 + 1) * q^8 + (-3*b5 - 3*b4 - 3*b3 - 3*b1 - 6) * q^9 + (b7 - b6 - 5*b5 - 5*b4 + 2*b2 - 1) * q^10 + (-b5 + b4 + 2*b3 + 2*b1 - 1) * q^11 + (b7 + 3*b6 - b5 + 3*b4 - 4*b3 - 3*b2 + 2*b1) * q^12 + (2*b7 - 2*b6 + 4*b5 + 4*b4 - 4*b2) * q^13 + (-3*b7 - 3*b6 + 5*b5 - 5*b4 - 2*b3 + 5) * q^14 + (b7 + 3*b6 - 4*b5 - 4*b4 + 2*b3 + 6*b2 - 10*b1 + 2) * q^15 + (-2*b7 + 2*b6 + 2*b5 + 2*b4 + 2*b2 - 36) * q^16 + (4*b5 - 4*b4 + 2*b3 + 2*b1 + 4) * q^17 + (-3*b7 - 3*b6 - 3*b5 + 3*b4 + 6*b3 - 6*b2 - 9*b1 + 27) * q^18 + (11*b5 + 11*b4 + 23) * q^19 + (-2*b7 - 2*b6 - 10*b5 + 10*b4 + 12*b3 - 8*b1 - 10) * q^20 + (b7 - 3*b6 - 4*b5 - 4*b4 - 7*b3 + 12*b2 + 23*b1 + 8) * q^21 + (b7 - b6 + 3*b5 + 3*b4 + 4*b2 - 19) * q^22 + (4*b7 + 4*b6 - 12*b3 + 28*b1) * q^23 + (-3*b7 - 3*b6 + 21*b5 + 11*b4 - 6*b3 - 6*b2 + 6*b1 + 35) * q^24 + (-2*b5 - 2*b4 + 37) * q^25 + (2*b7 + 2*b6 - 14*b5 + 14*b4 - 20*b3 + 8*b1 - 14) * q^26 + (-9*b5 - 12*b4 + 18*b3 + 18*b1 - 39) * q^27 + (-2*b7 + 2*b6 - 30*b5 - 30*b4 + 4*b2 + 66) * q^28 + (3*b7 + 3*b6 + 13*b3 - 45*b1) * q^29 + (-b7 + 3*b6 + 19*b5 + b4 + 22*b3 - 12*b2 - 8*b1 - 77) * q^30 + (3*b7 - 3*b6 + 8*b5 + 8*b4 - 10*b2 - 2) * q^31 + (2*b7 + 2*b6 + 18*b5 - 18*b4 + 4*b3 - 36*b1 + 18) * q^32 + (9*b5 + b4 - 9*b3 - 9*b1 - 29) * q^33 + (-4*b7 + 4*b6 - 12*b5 - 12*b4 + 4*b2 - 24) * q^34 + (2*b5 - 2*b4 - 42*b3 - 42*b1 + 2) * q^35 + (-12*b6 - 12*b5 - 12*b3 + 3*b2 + 36*b1 - 63) * q^36 + (-10*b7 + 10*b6 - 4*b5 - 4*b4 - 12*b2 - 16) * q^37 + (11*b7 + 11*b6 + 11*b5 - 11*b4 - 22*b3 + 34*b1 + 11) * q^38 + (-10*b7 - 18*b6 + 4*b5 + 4*b4 + 34*b3 - 74*b1 + 4) * q^39 + (12*b7 - 12*b6 + 20*b5 + 20*b4 + 8*b2 - 140) * q^40 + (-28*b5 + 28*b4 + 32*b3 + 32*b1 - 28) * q^41 + (11*b7 + 9*b6 - 11*b5 + b4 + 28*b3 + 18*b2 - 8*b1 + 169) * q^42 + (-39*b5 - 39*b4 - 19) * q^43 + (8*b7 + 8*b6 - 20*b1) * q^44 + (-9*b7 + 15*b6 - 27*b3 - 24*b2 + 75*b1 - 24) * q^45 + (-4*b7 + 4*b6 + 20*b5 + 20*b4 + 8*b2 + 180) * q^46 + (-24*b7 - 24*b6 - 8*b3 + 72*b1) * q^47 + (8*b7 + 12*b6 - 20*b5 - 40*b4 - 44*b3 + 6*b2 + 52*b1 + 146) * q^48 + (46*b5 + 46*b4 - 119) * q^49 + (-2*b7 - 2*b6 - 2*b5 + 2*b4 + 4*b3 + 35*b1 - 2) * q^50 + (24*b5 + 16*b4 + 6*b3 + 6*b1 + 136) * q^51 + (12*b7 - 12*b6 + 52*b5 + 52*b4 - 16*b2 + 268) * q^52 + (7*b7 + 7*b6 + 25*b3 - 89*b1) * q^53 + (-12*b7 - 9*b6 - 15*b5 + 6*b4 + 21*b3 + 36*b2 - 51*b1 - 192) * q^54 + (6*b7 - 6*b6 - 8*b5 - 8*b4 + 28*b2 + 20) * q^55 + (-28*b7 - 28*b6 - 12*b5 + 12*b4 + 72*b3 + 32*b1 - 12) * q^56 + (-33*b5 + 23*b4 - 33*b3 - 33*b1 + 119) * q^57 + (-3*b7 + 3*b6 + 15*b5 + 15*b4 - 38*b2 - 349) * q^58 + (11*b5 - 11*b4 + 8*b3 + 8*b1 + 11) * q^59 + (-14*b7 + 6*b6 - 10*b5 - 34*b4 - 40*b3 + 12*b2 - 64*b1 - 322) * q^60 + (10*b7 - 10*b6 - 28*b5 - 28*b4 + 76*b2 + 48) * q^61 + (b7 + b6 - 23*b5 + 23*b4 - 42*b3 + 16*b1 - 23) * q^62 + (39*b7 + 33*b6 + 24*b5 + 24*b4 - 12*b3 - 42*b2 - 36*b1 - 18) * q^63 + (-20*b7 + 20*b6 - 44*b5 - 44*b4 - 36*b2 - 208) * q^64 + (76*b5 - 76*b4 + 36*b3 + 36*b1 + 76) * q^65 + (b7 + 9*b6 - 7*b5 - 17*b4 - 10*b3 - 18*b2 - 28*b1 + 91) * q^66 + (9*b5 + 9*b4 - 187) * q^67 + (-12*b7 - 12*b6 + 20*b5 - 20*b4 + 40*b3 - 40*b1 + 20) * q^68 + (28*b7 - 12*b6 + 32*b5 + 32*b4 - 16*b3 - 24*b2 + 32*b1 + 8) * q^69 + (-2*b7 + 2*b6 - 6*b5 - 6*b4 - 84*b2 + 418) * q^70 + (44*b7 + 44*b6 - 52*b3 + 68*b1) * q^71 + (15*b7 - 9*b6 - 57*b5 + 33*b4 + 6*b3 + 36*b2 - 66*b1 + 315) * q^72 + (-148*b5 - 148*b4 + 134) * q^73 + (-26*b7 - 26*b6 + 54*b5 - 54*b4 + 4*b3 - 8*b1 + 54) * q^74 + (6*b5 + 37*b4 + 6*b3 + 6*b1 - 104) * q^75 + (-22*b7 + 22*b6 + 22*b5 + 22*b4 - 10*b2 + 212) * q^76 + (-26*b7 - 26*b6 - 6*b3 + 70*b1) * q^77 + (22*b7 - 6*b6 - 94*b5 - 46*b4 - 16*b3 - 12*b2 + 8*b1 - 478) * q^78 + (-21*b7 + 21*b6 - 56*b5 - 56*b4 + 70*b2 + 14) * q^79 + (40*b7 + 40*b6 - 56*b5 + 56*b4 - 48*b3 - 128*b1 - 56) * q^80 + (144*b5 - 18*b3 - 18*b1 + 9) * q^81 + (28*b7 - 28*b6 + 84*b5 + 84*b4 + 64*b2 - 292) * q^82 + (-21*b5 + 21*b4 + 62*b3 + 62*b1 - 21) * q^83 + (10*b7 + 18*b6 + 74*b5 + 62*b4 + 44*b3 + 152*b1 - 526) * q^84 + (16*b7 - 16*b6 + 32*b5 + 32*b4 - 32*b2) * q^85 + (-39*b7 - 39*b6 - 39*b5 + 39*b4 + 78*b3 - 58*b1 - 39) * q^86 + (-67*b7 - 9*b6 - 20*b5 - 20*b4 + 10*b3 - 18*b2 + 46*b1 - 38) * q^87 + (-8*b7 + 8*b6 + 40*b5 + 40*b4 - 36*b2 - 180) * q^88 + (-112*b5 + 112*b4 - 106*b3 - 106*b1 - 112) * q^89 + (-39*b7 - 33*b6 + 75*b5 - 45*b4 - 24*b3 + 42*b2 + 591) * q^90 + (116*b5 + 116*b4 + 396) * q^91 + (24*b7 + 24*b6 + 56*b5 - 56*b4 - 16*b3 + 192*b1 + 56) * q^92 + (-19*b7 - 39*b6 + 4*b5 + 4*b4 + 61*b3 + 12*b2 - 125*b1 + 16) * q^93 + (24*b7 - 24*b6 - 120*b5 - 120*b4 + 112*b2 + 680) * q^94 + (-12*b7 - 12*b6 + 100*b3 - 276*b1) * q^95 + (-46*b7 - 6*b6 + 10*b5 + 30*b4 + 76*b3 - 12*b2 + 100*b1 + 558) * q^96 + (90*b5 + 90*b4 - 484) * q^97 + (46*b7 + 46*b6 + 46*b5 - 46*b4 - 92*b3 - 73*b1 + 46) * q^98 + (-57*b5 - 39*b4 + 24*b3 + 24*b1 + 255) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{3} + 20 q^{4} - 48 q^{9}+O(q^{10})$$ 8 * q - 12 * q^3 + 20 * q^4 - 48 * q^9 $$8 q - 12 q^{3} + 20 q^{4} - 48 q^{9} - 24 q^{10} + 36 q^{12} - 280 q^{16} + 264 q^{18} + 184 q^{19} - 176 q^{22} + 264 q^{24} + 296 q^{25} - 324 q^{27} + 528 q^{28} - 624 q^{30} - 264 q^{33} - 176 q^{34} - 516 q^{36} - 1248 q^{40} + 1320 q^{42} - 152 q^{43} + 1440 q^{46} + 1080 q^{48} - 952 q^{49} + 1056 q^{51} + 2112 q^{52} - 1584 q^{54} + 1176 q^{57} - 2616 q^{58} - 2640 q^{60} - 1360 q^{64} + 792 q^{66} - 1496 q^{67} + 3696 q^{70} + 2640 q^{72} + 1072 q^{73} - 708 q^{75} + 1912 q^{76} - 3696 q^{78} - 504 q^{81} - 2816 q^{82} - 4224 q^{84} - 1232 q^{88} + 4104 q^{90} + 3168 q^{91} + 4800 q^{94} + 4752 q^{96} - 3872 q^{97} + 2112 q^{99}+O(q^{100})$$ 8 * q - 12 * q^3 + 20 * q^4 - 48 * q^9 - 24 * q^10 + 36 * q^12 - 280 * q^16 + 264 * q^18 + 184 * q^19 - 176 * q^22 + 264 * q^24 + 296 * q^25 - 324 * q^27 + 528 * q^28 - 624 * q^30 - 264 * q^33 - 176 * q^34 - 516 * q^36 - 1248 * q^40 + 1320 * q^42 - 152 * q^43 + 1440 * q^46 + 1080 * q^48 - 952 * q^49 + 1056 * q^51 + 2112 * q^52 - 1584 * q^54 + 1176 * q^57 - 2616 * q^58 - 2640 * q^60 - 1360 * q^64 + 792 * q^66 - 1496 * q^67 + 3696 * q^70 + 2640 * q^72 + 1072 * q^73 - 708 * q^75 + 1912 * q^76 - 3696 * q^78 - 504 * q^81 - 2816 * q^82 - 4224 * q^84 - 1232 * q^88 + 4104 * q^90 + 3168 * q^91 + 4800 * q^94 + 4752 * q^96 - 3872 * q^97 + 2112 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10x^{6} + 120x^{4} - 640x^{2} + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 10\nu^{5} + 120\nu^{3} - 384\nu ) / 256$$ (v^7 - 10*v^5 + 120*v^3 - 384*v) / 256 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 4\nu^{5} + 10\nu^{4} + 8\nu^{3} - 56\nu^{2} - 160\nu + 384 ) / 128$$ (-v^6 - 4*v^5 + 10*v^4 + 8*v^3 - 56*v^2 - 160*v + 384) / 128 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 10\nu^{4} - 8\nu^{3} - 56\nu^{2} + 160\nu + 256 ) / 128$$ (-v^6 + 4*v^5 + 10*v^4 - 8*v^3 - 56*v^2 + 160*v + 256) / 128 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} + 2\nu^{5} + 44\nu^{4} + 24\nu^{3} - 16\nu^{2} - 192\nu + 2048 ) / 256$$ (-v^7 + 2*v^6 + 2*v^5 + 44*v^4 + 24*v^3 - 16*v^2 - 192*v + 2048) / 256 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} - 2\nu^{6} + 2\nu^{5} - 44\nu^{4} + 24\nu^{3} + 16\nu^{2} - 192\nu - 2048 ) / 256$$ (-v^7 - 2*v^6 + 2*v^5 - 44*v^4 + 24*v^3 + 16*v^2 - 192*v - 2048) / 256
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta _1 + 1$$ b7 + b6 + b5 - b4 + 2*b3 + 2*b1 + 1 $$\nu^{4}$$ $$=$$ $$-2\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{2} - 36$$ -2*b7 + 2*b6 + 2*b5 + 2*b4 + 2*b2 - 36 $$\nu^{5}$$ $$=$$ $$2\beta_{7} + 2\beta_{6} + 18\beta_{5} - 18\beta_{4} + 4\beta_{3} - 36\beta _1 + 18$$ 2*b7 + 2*b6 + 18*b5 - 18*b4 + 4*b3 - 36*b1 + 18 $$\nu^{6}$$ $$=$$ $$-20\beta_{7} + 20\beta_{6} - 44\beta_{5} - 44\beta_{4} - 36\beta_{2} - 208$$ -20*b7 + 20*b6 - 44*b5 - 44*b4 - 36*b2 - 208 $$\nu^{7}$$ $$=$$ $$-100\beta_{7} - 100\beta_{6} + 60\beta_{5} - 60\beta_{4} + 56\beta_{3} - 216\beta _1 + 60$$ -100*b7 - 100*b6 + 60*b5 - 60*b4 + 56*b3 - 216*b1 + 60

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −2.58576 − 1.14624i −2.58576 + 1.14624i −1.95291 − 2.04601i −1.95291 + 2.04601i 1.95291 − 2.04601i 1.95291 + 2.04601i 2.58576 − 1.14624i 2.58576 + 1.14624i
−2.58576 1.14624i 1.37228 + 5.01167i 5.37228 + 5.92778i 12.2683 2.19618 14.5319i 14.0624i −7.09677 21.4857i −23.2337 + 13.7548i −31.7228 14.0624i
11.2 −2.58576 + 1.14624i 1.37228 5.01167i 5.37228 5.92778i 12.2683 2.19618 + 14.5319i 14.0624i −7.09677 + 21.4857i −23.2337 13.7548i −31.7228 + 14.0624i
11.3 −1.95291 2.04601i −4.37228 2.80770i −0.372281 + 7.99133i −13.1715 2.79411 + 14.4289i 26.9490i 17.0773 14.8447i 11.2337 + 24.5521i 25.7228 + 26.9490i
11.4 −1.95291 + 2.04601i −4.37228 + 2.80770i −0.372281 7.99133i −13.1715 2.79411 14.4289i 26.9490i 17.0773 + 14.8447i 11.2337 24.5521i 25.7228 26.9490i
11.5 1.95291 2.04601i −4.37228 2.80770i −0.372281 7.99133i 13.1715 −14.2832 + 3.46254i 26.9490i −17.0773 14.8447i 11.2337 + 24.5521i 25.7228 26.9490i
11.6 1.95291 + 2.04601i −4.37228 + 2.80770i −0.372281 + 7.99133i 13.1715 −14.2832 3.46254i 26.9490i −17.0773 + 14.8447i 11.2337 24.5521i 25.7228 + 26.9490i
11.7 2.58576 1.14624i 1.37228 + 5.01167i 5.37228 5.92778i −12.2683 9.29295 + 11.3860i 14.0624i 7.09677 21.4857i −23.2337 + 13.7548i −31.7228 + 14.0624i
11.8 2.58576 + 1.14624i 1.37228 5.01167i 5.37228 + 5.92778i −12.2683 9.29295 11.3860i 14.0624i 7.09677 + 21.4857i −23.2337 13.7548i −31.7228 14.0624i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.4.f.b 8
3.b odd 2 1 inner 24.4.f.b 8
4.b odd 2 1 96.4.f.b 8
8.b even 2 1 96.4.f.b 8
8.d odd 2 1 inner 24.4.f.b 8
12.b even 2 1 96.4.f.b 8
16.e even 4 2 768.4.c.v 16
16.f odd 4 2 768.4.c.v 16
24.f even 2 1 inner 24.4.f.b 8
24.h odd 2 1 96.4.f.b 8
48.i odd 4 2 768.4.c.v 16
48.k even 4 2 768.4.c.v 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.b 8 1.a even 1 1 trivial
24.4.f.b 8 3.b odd 2 1 inner
24.4.f.b 8 8.d odd 2 1 inner
24.4.f.b 8 24.f even 2 1 inner
96.4.f.b 8 4.b odd 2 1
96.4.f.b 8 8.b even 2 1
96.4.f.b 8 12.b even 2 1
96.4.f.b 8 24.h odd 2 1
768.4.c.v 16 16.e even 4 2
768.4.c.v 16 16.f odd 4 2
768.4.c.v 16 48.i odd 4 2
768.4.c.v 16 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 324T_{5}^{2} + 26112$$ acting on $$S_{4}^{\mathrm{new}}(24, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 10 T^{6} + \cdots + 4096$$
$3$ $$(T^{4} + 6 T^{3} + \cdots + 729)^{2}$$
$5$ $$(T^{4} - 324 T^{2} + 26112)^{2}$$
$7$ $$(T^{4} + 924 T^{2} + 143616)^{2}$$
$11$ $$(T^{4} + 484 T^{2} + 352)^{2}$$
$13$ $$(T^{4} + 5808 T^{2} + 574464)^{2}$$
$17$ $$(T^{4} + 2464 T^{2} + 90112)^{2}$$
$19$ $$(T^{2} - 46 T - 3464)^{4}$$
$23$ $$(T^{4} - 17472 T^{2} + 1671168)^{2}$$
$29$ $$(T^{4} - 43620 T^{2} + 448108032)^{2}$$
$31$ $$(T^{4} + 20988 T^{2} + 41505024)^{2}$$
$37$ $$(T^{4} + 77616 T^{2} + 1494180864)^{2}$$
$41$ $$(T^{4} + 193600 T^{2} + 3151126528)^{2}$$
$43$ $$(T^{2} + 38 T - 49832)^{4}$$
$47$ $$(T^{4} - 321792 T^{2} + 427819008)^{2}$$
$53$ $$(T^{4} - 177156 T^{2} + 7033554432)^{2}$$
$59$ $$(T^{4} + 21604 T^{2} + 296032)^{2}$$
$61$ $$(T^{4} + 550704 T^{2} + 18820015104)^{2}$$
$67$ $$(T^{2} + 374 T + 32296)^{4}$$
$71$ $$(T^{4} - 654912 T^{2} + 103614087168)^{2}$$
$73$ $$(T^{2} - 268 T - 704876)^{4}$$
$79$ $$(T^{4} + 1028412 T^{2} + 99653562624)^{2}$$
$83$ $$(T^{4} + 396484 T^{2} + 2128431712)^{2}$$
$89$ $$(T^{4} + 2644576 T^{2} + 147293673472)^{2}$$
$97$ $$(T^{2} + 968 T - 33044)^{4}$$