# Properties

 Label 525.2.r.a Level $525$ Weight $2$ Character orbit 525.r 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(424,525)]

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, 3, 4]))

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

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

chi := DirichletCharacter("525.424");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.r (of order $$6$$, 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: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
424.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−2.36603 + 1.36603i −0.866025 0.500000i 2.73205 4.73205i 0 2.73205 2.50000 + 0.866025i 9.46410i 0.500000 + 0.866025i 0
424.2 −0.633975 + 0.366025i 0.866025 + 0.500000i −0.732051 + 1.26795i 0 −0.732051 2.50000 + 0.866025i 2.53590i 0.500000 + 0.866025i 0
499.1 −2.36603 1.36603i −0.866025 + 0.500000i 2.73205 + 4.73205i 0 2.73205 2.50000 0.866025i 9.46410i 0.500000 0.866025i 0
499.2 −0.633975 0.366025i 0.866025 0.500000i −0.732051 1.26795i 0 −0.732051 2.50000 0.866025i 2.53590i 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
35.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{3} + 14T_{2}^{2} + 12T_{2} + 4$$ T2^4 + 6*T2^3 + 14*T2^2 + 12*T2 + 4 $$T_{11}^{4} - 2T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 4$$ T11^4 - 2*T11^3 + 6*T11^2 + 4*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 5 T + 7)^{2}$$
$11$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$13$ $$T^{4} + 38T^{2} + 169$$
$17$ $$T^{4} - 6 T^{3} - 10 T^{2} + 132 T + 484$$
$19$ $$T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121$$
$23$ $$T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36$$
$29$ $$(T^{2} + 2 T - 26)^{2}$$
$31$ $$T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9$$
$37$ $$T^{4} + 18 T^{3} + 131 T^{2} + \cdots + 529$$
$41$ $$(T^{2} - 2 T - 2)^{2}$$
$43$ $$T^{4} + 62T^{2} + 529$$
$47$ $$T^{4} - 4T^{2} + 16$$
$53$ $$T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816$$
$59$ $$T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4$$
$61$ $$(T^{2} + 4 T + 16)^{2}$$
$67$ $$T^{4} - 30 T^{3} + 339 T^{2} + \cdots + 1521$$
$71$ $$(T^{2} - 2 T - 26)^{2}$$
$73$ $$T^{4} - 30 T^{3} + 359 T^{2} + \cdots + 3481$$
$79$ $$T^{4} - 6 T^{3} + 135 T^{2} + \cdots + 9801$$
$83$ $$T^{4} + 312 T^{2} + 19044$$
$89$ $$T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044$$
$97$ $$T^{4} + 224T^{2} + 256$$