# Properties

 Label 525.2.i.f Level $525$ Weight $2$ Character orbit 525.i Analytic conductor $4.192$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(151,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("525.151");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.i (of order $$3$$, degree $$2$$, 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: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.36603 2.36603i −0.500000 + 0.866025i −2.73205 + 4.73205i 0 2.73205 −0.866025 + 2.50000i 9.46410 −0.500000 0.866025i 0
151.2 0.366025 + 0.633975i −0.500000 + 0.866025i 0.732051 1.26795i 0 −0.732051 0.866025 2.50000i 2.53590 −0.500000 0.866025i 0
226.1 −1.36603 + 2.36603i −0.500000 0.866025i −2.73205 4.73205i 0 2.73205 −0.866025 2.50000i 9.46410 −0.500000 + 0.866025i 0
226.2 0.366025 0.633975i −0.500000 0.866025i 0.732051 + 1.26795i 0 −0.732051 0.866025 + 2.50000i 2.53590 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.f 4
5.b even 2 1 105.2.i.d 4
5.c odd 4 1 525.2.r.a 4
5.c odd 4 1 525.2.r.f 4
7.c even 3 1 inner 525.2.i.f 4
7.c even 3 1 3675.2.a.bg 2
7.d odd 6 1 3675.2.a.be 2
15.d odd 2 1 315.2.j.c 4
20.d odd 2 1 1680.2.bg.o 4
35.c odd 2 1 735.2.i.l 4
35.i odd 6 1 735.2.a.h 2
35.i odd 6 1 735.2.i.l 4
35.j even 6 1 105.2.i.d 4
35.j even 6 1 735.2.a.g 2
35.l odd 12 1 525.2.r.a 4
35.l odd 12 1 525.2.r.f 4
105.o odd 6 1 315.2.j.c 4
105.o odd 6 1 2205.2.a.z 2
105.p even 6 1 2205.2.a.ba 2
140.p odd 6 1 1680.2.bg.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d 4 5.b even 2 1
105.2.i.d 4 35.j even 6 1
315.2.j.c 4 15.d odd 2 1
315.2.j.c 4 105.o odd 6 1
525.2.i.f 4 1.a even 1 1 trivial
525.2.i.f 4 7.c even 3 1 inner
525.2.r.a 4 5.c odd 4 1
525.2.r.a 4 35.l odd 12 1
525.2.r.f 4 5.c odd 4 1
525.2.r.f 4 35.l odd 12 1
735.2.a.g 2 35.j even 6 1
735.2.a.h 2 35.i odd 6 1
735.2.i.l 4 35.c odd 2 1
735.2.i.l 4 35.i odd 6 1
1680.2.bg.o 4 20.d odd 2 1
1680.2.bg.o 4 140.p odd 6 1
2205.2.a.z 2 105.o odd 6 1
2205.2.a.ba 2 105.p even 6 1
3675.2.a.be 2 7.d odd 6 1
3675.2.a.bg 2 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2T_{2}^{3} + 6T_{2}^{2} - 4T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11T^{2} + 49$$
$11$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$13$ $$(T^{2} + 8 T + 13)^{2}$$
$17$ $$T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484$$
$19$ $$T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121$$
$23$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$29$ $$(T^{2} - 2 T - 26)^{2}$$
$31$ $$T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9$$
$37$ $$T^{4} - 4 T^{3} + 39 T^{2} + 92 T + 529$$
$41$ $$(T^{2} - 2 T - 2)^{2}$$
$43$ $$(T^{2} - 4 T - 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} - 4 T^{3} + 120 T^{2} + \cdots + 10816$$
$59$ $$T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4$$
$61$ $$(T^{2} + 4 T + 16)^{2}$$
$67$ $$T^{4} + 12 T^{3} + 183 T^{2} + \cdots + 1521$$
$71$ $$(T^{2} - 2 T - 26)^{2}$$
$73$ $$T^{4} - 8 T^{3} + 123 T^{2} + \cdots + 3481$$
$79$ $$T^{4} + 6 T^{3} + 135 T^{2} + \cdots + 9801$$
$83$ $$(T^{2} + 6 T - 138)^{2}$$
$89$ $$T^{4} + 6 T^{3} + 174 T^{2} + \cdots + 19044$$
$97$ $$(T^{2} + 16 T + 16)^{2}$$