# Properties

 Label 225.4.e.b Level $225$ Weight $4$ Character orbit 225.e Analytic conductor $13.275$ 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] = [225,4,Mod(76,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("225.76");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.e (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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 9) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} - 15 \beta_1 + 12) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{2} - 16) q^{8} + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9}+O(q^{10})$$ q + (b3 + b1) * q^2 + (-b3 - b2 - b1 + 1) * q^3 + (3*b3 - 3*b2 + b1 - 4) * q^4 + (-3*b3 + 3*b2 - 15*b1 + 12) * q^6 + (3*b3 + 2*b1) * q^7 + (b2 - 16) * q^8 + (3*b3 - 6*b2 + 21*b1 - 3) * q^9 $$q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} - 15 \beta_1 + 12) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{2} - 16) q^{8} + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9} + (8 \beta_{3} - 37 \beta_1) q^{11} + (2 \beta_{3} + 5 \beta_{2} - 22 \beta_1 + 52) q^{12} + (15 \beta_{3} - 15 \beta_{2} - 2 \beta_1 - 13) q^{13} + (8 \beta_{3} - 8 \beta_{2} + 26 \beta_1 - 34) q^{14} + (9 \beta_{3} - \beta_1) q^{16} + ( - 9 \beta_{2} - 54) q^{17} + (18 \beta_{3} - 27 \beta_{2} - 72) q^{18} + ( - 27 \beta_{2} - 52) q^{19} + ( - 7 \beta_{3} + 8 \beta_{2} - 46 \beta_1 + 34) q^{21} + ( - 21 \beta_{3} + 21 \beta_{2} + 27 \beta_1 - 6) q^{22} + (19 \beta_{3} - 19 \beta_{2} - 26 \beta_1 + 7) q^{23} + (15 \beta_{3} + 18 \beta_{2} + 9 \beta_1 - 24) q^{24} + ( - 28 \beta_{2} - 146) q^{26} + ( - 36 \beta_{3} + 18 \beta_{2} + 18 \beta_1 + 117) q^{27} + ( - 18 \beta_{2} - 92) q^{28} + ( - \beta_{3} + 26 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} + 20 \beta_1 - 23) q^{31} + (9 \beta_{3} - 9 \beta_{2} + 207 \beta_1 - 216) q^{32} + (66 \beta_{3} - 21 \beta_{2} - 165 \beta_1 + 6) q^{33} + ( - 63 \beta_{3} - 117 \beta_1) q^{34} + ( - 15 \beta_{3} - 60 \beta_{2} - 141 \beta_1 - 12) q^{36} + ( - 54 \beta_{2} - 2) q^{37} + ( - 79 \beta_{3} - 241 \beta_1) q^{38} + (17 \beta_{3} + 11 \beta_{2} - 124 \beta_1 + 253) q^{39} + (98 \beta_{3} - 98 \beta_{2} + 17 \beta_1 - 115) q^{41} + ( - 18 \beta_{3} + 60 \beta_{2} - 12 \beta_1 + 162) q^{42} + ( - 6 \beta_{3} + 47 \beta_1) q^{43} + (79 \beta_{2} - 76) q^{44} + ( - 12 \beta_{2} - 138) q^{46} + (91 \beta_{3} + 154 \beta_1) q^{47} + ( - 7 \beta_{3} + 17 \beta_{2} - 145 \beta_1 + 88) q^{48} + (21 \beta_{3} - 21 \beta_{2} - 267 \beta_1 + 246) q^{49} + (63 \beta_{3} + 36 \beta_{2} + 117 \beta_1 + 18) q^{51} + ( - 54 \beta_{3} - 358 \beta_1) q^{52} + (162 \beta_{2} + 54) q^{53} + (81 \beta_{3} + 54 \beta_{2} - 27 \beta_1 + 324) q^{54} + ( - 46 \beta_{3} - 10 \beta_1) q^{56} + (79 \beta_{3} - 2 \beta_{2} + 241 \beta_1 + 164) q^{57} + (24 \beta_{3} - 24 \beta_{2} + 18 \beta_1 - 42) q^{58} + ( - 136 \beta_{3} + 136 \beta_{2} + 467 \beta_1 - 331) q^{59} + (105 \beta_{3} - 272 \beta_1) q^{61} + ( - 26 \beta_{2} - 70) q^{62} + (57 \beta_{3} - 78 \beta_{2} - 24 \beta_1 - 192) q^{63} + ( - 153 \beta_{2} - 440) q^{64} + ( - 48 \beta_{3} + 33 \beta_{2} + 222 \beta_1 - 330) q^{66} + ( - 66 \beta_{3} + 66 \beta_{2} - 461 \beta_1 + 527) q^{67} + ( - 171 \beta_{3} + 171 \beta_{2} - 261 \beta_1 + 432) q^{68} + (45 \beta_{3} - 33 \beta_{2} - 204 \beta_1 + 297) q^{69} + (144 \beta_{2} + 756) q^{71} + ( - 27 \beta_{3} + 99 \beta_{2} - 333 \beta_1) q^{72} + ( - 243 \beta_{2} + 106) q^{73} + ( - 56 \beta_{3} - 380 \beta_1) q^{74} + ( - 183 \beta_{3} + 183 \beta_{2} - 673 \beta_1 + 856) q^{76} + ( - 71 \beta_{3} + 71 \beta_{2} + 118 \beta_1 - 47) q^{77} + (174 \beta_{3} + 90 \beta_{2} + 342 \beta_1 + 78) q^{78} + ( - 309 \beta_{3} + 556 \beta_1) q^{79} + ( - 135 \beta_{3} - 135 \beta_{2} + 351 \beta_1 - 351) q^{81} + ( - 213 \beta_{2} - 1014) q^{82} + ( - 107 \beta_{3} + 460 \beta_1) q^{83} + (110 \beta_{3} + 56 \beta_{2} + 218 \beta_1 + 52) q^{84} + (35 \beta_{3} - 35 \beta_{2} - \beta_1 - 34) q^{86} + ( - 51 \beta_{3} + 24 \beta_{2} + 42 \beta_1 + 42) q^{87} + ( - 165 \beta_{3} + 693 \beta_1) q^{88} + (72 \beta_{2} - 162) q^{89} + ( - 69 \beta_{2} - 425) q^{91} + (2 \beta_{3} - 430 \beta_1) q^{92} + ( - 17 \beta_{3} + 43 \beta_{2} + 16 \beta_1 + 71) q^{93} + (336 \beta_{3} - 336 \beta_{2} + 882 \beta_1 - 1218) q^{94} + ( - 198 \beta_{3} + 423 \beta_{2} + 342 \beta_1 + 360) q^{96} + (102 \beta_{3} + 317 \beta_1) q^{97} + (225 \beta_{2} + 324) q^{98} + (279 \beta_{3} - 81 \beta_{2} - 1080 \beta_1 + 504) q^{99}+O(q^{100})$$ q + (b3 + b1) * q^2 + (-b3 - b2 - b1 + 1) * q^3 + (3*b3 - 3*b2 + b1 - 4) * q^4 + (-3*b3 + 3*b2 - 15*b1 + 12) * q^6 + (3*b3 + 2*b1) * q^7 + (b2 - 16) * q^8 + (3*b3 - 6*b2 + 21*b1 - 3) * q^9 + (8*b3 - 37*b1) * q^11 + (2*b3 + 5*b2 - 22*b1 + 52) * q^12 + (15*b3 - 15*b2 - 2*b1 - 13) * q^13 + (8*b3 - 8*b2 + 26*b1 - 34) * q^14 + (9*b3 - b1) * q^16 + (-9*b2 - 54) * q^17 + (18*b3 - 27*b2 - 72) * q^18 + (-27*b2 - 52) * q^19 + (-7*b3 + 8*b2 - 46*b1 + 34) * q^21 + (-21*b3 + 21*b2 + 27*b1 - 6) * q^22 + (19*b3 - 19*b2 - 26*b1 + 7) * q^23 + (15*b3 + 18*b2 + 9*b1 - 24) * q^24 + (-28*b2 - 146) * q^26 + (-36*b3 + 18*b2 + 18*b1 + 117) * q^27 + (-18*b2 - 92) * q^28 + (-b3 + 26*b1) * q^29 + (3*b3 - 3*b2 + 20*b1 - 23) * q^31 + (9*b3 - 9*b2 + 207*b1 - 216) * q^32 + (66*b3 - 21*b2 - 165*b1 + 6) * q^33 + (-63*b3 - 117*b1) * q^34 + (-15*b3 - 60*b2 - 141*b1 - 12) * q^36 + (-54*b2 - 2) * q^37 + (-79*b3 - 241*b1) * q^38 + (17*b3 + 11*b2 - 124*b1 + 253) * q^39 + (98*b3 - 98*b2 + 17*b1 - 115) * q^41 + (-18*b3 + 60*b2 - 12*b1 + 162) * q^42 + (-6*b3 + 47*b1) * q^43 + (79*b2 - 76) * q^44 + (-12*b2 - 138) * q^46 + (91*b3 + 154*b1) * q^47 + (-7*b3 + 17*b2 - 145*b1 + 88) * q^48 + (21*b3 - 21*b2 - 267*b1 + 246) * q^49 + (63*b3 + 36*b2 + 117*b1 + 18) * q^51 + (-54*b3 - 358*b1) * q^52 + (162*b2 + 54) * q^53 + (81*b3 + 54*b2 - 27*b1 + 324) * q^54 + (-46*b3 - 10*b1) * q^56 + (79*b3 - 2*b2 + 241*b1 + 164) * q^57 + (24*b3 - 24*b2 + 18*b1 - 42) * q^58 + (-136*b3 + 136*b2 + 467*b1 - 331) * q^59 + (105*b3 - 272*b1) * q^61 + (-26*b2 - 70) * q^62 + (57*b3 - 78*b2 - 24*b1 - 192) * q^63 + (-153*b2 - 440) * q^64 + (-48*b3 + 33*b2 + 222*b1 - 330) * q^66 + (-66*b3 + 66*b2 - 461*b1 + 527) * q^67 + (-171*b3 + 171*b2 - 261*b1 + 432) * q^68 + (45*b3 - 33*b2 - 204*b1 + 297) * q^69 + (144*b2 + 756) * q^71 + (-27*b3 + 99*b2 - 333*b1) * q^72 + (-243*b2 + 106) * q^73 + (-56*b3 - 380*b1) * q^74 + (-183*b3 + 183*b2 - 673*b1 + 856) * q^76 + (-71*b3 + 71*b2 + 118*b1 - 47) * q^77 + (174*b3 + 90*b2 + 342*b1 + 78) * q^78 + (-309*b3 + 556*b1) * q^79 + (-135*b3 - 135*b2 + 351*b1 - 351) * q^81 + (-213*b2 - 1014) * q^82 + (-107*b3 + 460*b1) * q^83 + (110*b3 + 56*b2 + 218*b1 + 52) * q^84 + (35*b3 - 35*b2 - b1 - 34) * q^86 + (-51*b3 + 24*b2 + 42*b1 + 42) * q^87 + (-165*b3 + 693*b1) * q^88 + (72*b2 - 162) * q^89 + (-69*b2 - 425) * q^91 + (2*b3 - 430*b1) * q^92 + (-17*b3 + 43*b2 + 16*b1 + 71) * q^93 + (336*b3 - 336*b2 + 882*b1 - 1218) * q^94 + (-198*b3 + 423*b2 + 342*b1 + 360) * q^96 + (102*b3 + 317*b1) * q^97 + (225*b2 + 324) * q^98 + (279*b3 - 81*b2 - 1080*b1 + 504) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} + 3 q^{3} - 5 q^{4} + 9 q^{6} + 7 q^{7} - 66 q^{8} + 45 q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 + 3 * q^3 - 5 * q^4 + 9 * q^6 + 7 * q^7 - 66 * q^8 + 45 * q^9 $$4 q + 3 q^{2} + 3 q^{3} - 5 q^{4} + 9 q^{6} + 7 q^{7} - 66 q^{8} + 45 q^{9} - 66 q^{11} + 156 q^{12} - 11 q^{13} - 60 q^{14} + 7 q^{16} - 198 q^{17} - 216 q^{18} - 154 q^{19} + 21 q^{21} - 33 q^{22} + 33 q^{23} - 99 q^{24} - 528 q^{26} + 432 q^{27} - 332 q^{28} + 51 q^{29} - 43 q^{31} - 423 q^{32} - 198 q^{33} - 297 q^{34} - 225 q^{36} + 100 q^{37} - 561 q^{38} + 759 q^{39} - 132 q^{41} + 486 q^{42} + 88 q^{43} - 462 q^{44} - 528 q^{46} + 399 q^{47} + 21 q^{48} + 513 q^{49} + 297 q^{51} - 770 q^{52} - 108 q^{53} + 1215 q^{54} - 66 q^{56} + 1221 q^{57} - 60 q^{58} - 798 q^{59} - 439 q^{61} - 228 q^{62} - 603 q^{63} - 1454 q^{64} - 990 q^{66} + 988 q^{67} + 693 q^{68} + 891 q^{69} + 2736 q^{71} - 891 q^{72} + 910 q^{73} - 816 q^{74} + 1529 q^{76} - 165 q^{77} + 990 q^{78} + 803 q^{79} - 567 q^{81} - 3630 q^{82} + 813 q^{83} + 642 q^{84} - 33 q^{86} + 153 q^{87} + 1221 q^{88} - 792 q^{89} - 1562 q^{91} - 858 q^{92} + 213 q^{93} - 2100 q^{94} + 1080 q^{96} + 736 q^{97} + 846 q^{98} + 297 q^{99}+O(q^{100})$$ 4 * q + 3 * q^2 + 3 * q^3 - 5 * q^4 + 9 * q^6 + 7 * q^7 - 66 * q^8 + 45 * q^9 - 66 * q^11 + 156 * q^12 - 11 * q^13 - 60 * q^14 + 7 * q^16 - 198 * q^17 - 216 * q^18 - 154 * q^19 + 21 * q^21 - 33 * q^22 + 33 * q^23 - 99 * q^24 - 528 * q^26 + 432 * q^27 - 332 * q^28 + 51 * q^29 - 43 * q^31 - 423 * q^32 - 198 * q^33 - 297 * q^34 - 225 * q^36 + 100 * q^37 - 561 * q^38 + 759 * q^39 - 132 * q^41 + 486 * q^42 + 88 * q^43 - 462 * q^44 - 528 * q^46 + 399 * q^47 + 21 * q^48 + 513 * q^49 + 297 * q^51 - 770 * q^52 - 108 * q^53 + 1215 * q^54 - 66 * q^56 + 1221 * q^57 - 60 * q^58 - 798 * q^59 - 439 * q^61 - 228 * q^62 - 603 * q^63 - 1454 * q^64 - 990 * q^66 + 988 * q^67 + 693 * q^68 + 891 * q^69 + 2736 * q^71 - 891 * q^72 + 910 * q^73 - 816 * q^74 + 1529 * q^76 - 165 * q^77 + 990 * q^78 + 803 * q^79 - 567 * q^81 - 3630 * q^82 + 813 * q^83 + 642 * q^84 - 33 * q^86 + 153 * q^87 + 1221 * q^88 - 792 * q^89 - 1562 * q^91 - 858 * q^92 + 213 * q^93 - 2100 * q^94 + 1080 * q^96 + 736 * q^97 + 846 * q^98 + 297 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
−0.686141 1.18843i 5.05842 + 1.18843i 3.05842 5.29734i 0 −2.05842 6.82701i −2.55842 4.43132i −19.3723 24.1753 + 12.0232i 0
76.2 2.18614 + 3.78651i −3.55842 3.78651i −5.55842 + 9.62747i 0 6.55842 21.7518i 6.05842 + 10.4935i −13.6277 −1.67527 + 26.9480i 0
151.1 −0.686141 + 1.18843i 5.05842 1.18843i 3.05842 + 5.29734i 0 −2.05842 + 6.82701i −2.55842 + 4.43132i −19.3723 24.1753 12.0232i 0
151.2 2.18614 3.78651i −3.55842 + 3.78651i −5.55842 9.62747i 0 6.55842 + 21.7518i 6.05842 10.4935i −13.6277 −1.67527 26.9480i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.e.b 4
5.b even 2 1 9.4.c.a 4
5.c odd 4 2 225.4.k.b 8
9.c even 3 1 inner 225.4.e.b 4
9.c even 3 1 2025.4.a.g 2
9.d odd 6 1 2025.4.a.n 2
15.d odd 2 1 27.4.c.a 4
20.d odd 2 1 144.4.i.c 4
45.h odd 6 1 27.4.c.a 4
45.h odd 6 1 81.4.a.a 2
45.j even 6 1 9.4.c.a 4
45.j even 6 1 81.4.a.d 2
45.k odd 12 2 225.4.k.b 8
60.h even 2 1 432.4.i.c 4
180.n even 6 1 432.4.i.c 4
180.n even 6 1 1296.4.a.i 2
180.p odd 6 1 144.4.i.c 4
180.p odd 6 1 1296.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 5.b even 2 1
9.4.c.a 4 45.j even 6 1
27.4.c.a 4 15.d odd 2 1
27.4.c.a 4 45.h odd 6 1
81.4.a.a 2 45.h odd 6 1
81.4.a.d 2 45.j even 6 1
144.4.i.c 4 20.d odd 2 1
144.4.i.c 4 180.p odd 6 1
225.4.e.b 4 1.a even 1 1 trivial
225.4.e.b 4 9.c even 3 1 inner
225.4.k.b 8 5.c odd 4 2
225.4.k.b 8 45.k odd 12 2
432.4.i.c 4 60.h even 2 1
432.4.i.c 4 180.n even 6 1
1296.4.a.i 2 180.n even 6 1
1296.4.a.u 2 180.p odd 6 1
2025.4.a.g 2 9.c even 3 1
2025.4.a.n 2 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3T_{2}^{3} + 15T_{2}^{2} + 18T_{2} + 36$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36$$
$3$ $$T^{4} - 3 T^{3} - 18 T^{2} - 81 T + 729$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 7 T^{3} + 111 T^{2} + \cdots + 3844$$
$11$ $$T^{4} + 66 T^{3} + 3795 T^{2} + \cdots + 314721$$
$13$ $$T^{4} + 11 T^{3} + 1947 T^{2} + \cdots + 3334276$$
$17$ $$(T^{2} + 99 T + 1782)^{2}$$
$19$ $$(T^{2} + 77 T - 4532)^{2}$$
$23$ $$T^{4} - 33 T^{3} + 3795 T^{2} + \cdots + 7322436$$
$29$ $$T^{4} - 51 T^{3} + 1959 T^{2} + \cdots + 412164$$
$31$ $$T^{4} + 43 T^{3} + 1461 T^{2} + \cdots + 150544$$
$37$ $$(T^{2} - 50 T - 23432)^{2}$$
$41$ $$T^{4} + 132 T^{3} + \cdots + 5606565129$$
$43$ $$T^{4} - 88 T^{3} + 6105 T^{2} + \cdots + 2686321$$
$47$ $$T^{4} - 399 T^{3} + \cdots + 813276324$$
$53$ $$(T^{2} + 54 T - 215784)^{2}$$
$59$ $$T^{4} + 798 T^{3} + \cdots + 43678881$$
$61$ $$T^{4} + 439 T^{3} + \cdots + 1829786176$$
$67$ $$T^{4} - 988 T^{3} + \cdots + 43305193801$$
$71$ $$(T^{2} - 1368 T + 296784)^{2}$$
$73$ $$(T^{2} - 455 T - 435398)^{2}$$
$79$ $$T^{4} - 803 T^{3} + \cdots + 392522298256$$
$83$ $$T^{4} - 813 T^{3} + \cdots + 5010940944$$
$89$ $$(T^{2} + 396 T - 3564)^{2}$$
$97$ $$T^{4} - 736 T^{3} + \cdots + 2459267281$$