# Properties

 Label 450.3.i.b Level $450$ Weight $3$ Character orbit 450.i Analytic conductor $12.262$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 450.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2616118962$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{6} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} - 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9}+O(q^{10})$$ q + (-b3 + b1) * q^2 + (b3 - 2*b2 - 2*b1 + 1) * q^3 + (-2*b2 + 2) * q^4 + (-b3 + 2*b2 - b1 - 4) * q^6 + (3*b3 - b2 + 3*b1) * q^7 - 2*b3 * q^8 + (6*b3 + 3) * q^9 $$q + ( - \beta_{3} + \beta_1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{6} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} - 2 \beta_{3} q^{8} + (6 \beta_{3} + 3) q^{9} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 6) q^{11} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{12} + ( - 12 \beta_{3} - 5 \beta_{2} + 6 \beta_1 + 5) q^{13} + (6 \beta_{2} - \beta_1 + 6) q^{14} - 4 \beta_{2} q^{16} + (6 \beta_{3} - 12 \beta_{2} + 6) q^{17} + ( - 3 \beta_{3} + 12 \beta_{2} + 3 \beta_1) q^{18} + ( - 6 \beta_{3} + 12 \beta_1 - 10) q^{19} + ( - 8 \beta_{3} - 17 \beta_{2} + 10 \beta_1 - 2) q^{21} + ( - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 6) q^{22} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{23} + (2 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 4) q^{24} + ( - 5 \beta_{3} - 24 \beta_{2} + 12) q^{26} + ( - 3 \beta_{3} - 30 \beta_{2} + 6 \beta_1 + 15) q^{27} + ( - 6 \beta_{3} + 12 \beta_1 - 2) q^{28} + (6 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 6) q^{29} + ( - 18 \beta_{3} - 19 \beta_{2} + 9 \beta_1 + 19) q^{31} - 4 \beta_1 q^{32} + (6 \beta_{3} - 3 \beta_{2} - 12 \beta_1 - 12) q^{33} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{34} + ( - 6 \beta_{2} + 12 \beta_1 + 6) q^{36} + ( - 6 \beta_{3} + 12 \beta_1 - 32) q^{37} + (10 \beta_{3} - 12 \beta_{2} - 10 \beta_1 + 24) q^{38} + (10 \beta_{3} + 31 \beta_{2} - 23 \beta_1 - 41) q^{39} + ( - 21 \beta_{2} - 18 \beta_1 - 21) q^{41} + (2 \beta_{3} - 16 \beta_{2} - 19 \beta_1 + 20) q^{42} + ( - 9 \beta_{3} + 23 \beta_{2} - 9 \beta_1) q^{43} + ( - 6 \beta_{3} - 12 \beta_{2} + 6) q^{44} + (3 \beta_{3} - 6 \beta_1 - 6) q^{46} + ( - 21 \beta_{3} + 9 \beta_{2} + 21 \beta_1 - 18) q^{47} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 8) q^{48} + ( - 12 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 6) q^{49} + (12 \beta_{3} - 24 \beta_{2} + 12 \beta_1 - 6) q^{51} + ( - 12 \beta_{3} - 10 \beta_{2} - 12 \beta_1) q^{52} + (30 \beta_{3} - 60 \beta_{2} + 30) q^{53} + ( - 15 \beta_{3} - 6 \beta_{2} - 15 \beta_1 + 12) q^{54} + (2 \beta_{3} - 12 \beta_{2} - 2 \beta_1 + 24) q^{56} + ( - 28 \beta_{3} + 20 \beta_{2} + 20 \beta_1 - 46) q^{57} + ( - 6 \beta_{3} + 12 \beta_{2} + 3 \beta_1 - 12) q^{58} + (21 \beta_{2} + 39 \beta_1 + 21) q^{59} + (18 \beta_{3} + 31 \beta_{2} + 18 \beta_1) q^{61} + ( - 19 \beta_{3} - 36 \beta_{2} + 18) q^{62} + (3 \beta_{3} + 33 \beta_{2} + 15 \beta_1 - 72) q^{63} - 8 q^{64} + (12 \beta_{3} + 12 \beta_{2} - 15 \beta_1 - 24) q^{66} + (18 \beta_{3} - 53 \beta_{2} - 9 \beta_1 + 53) q^{67} + ( - 12 \beta_{2} + 12 \beta_1 - 12) q^{68} + (6 \beta_{3} + 15 \beta_{2} + 6 \beta_1 - 3) q^{69} + ( - 24 \beta_{3} - 60 \beta_{2} + 30) q^{71} + ( - 6 \beta_{3} + 24) q^{72} + ( - 18 \beta_{3} + 36 \beta_1 + 52) q^{73} + (32 \beta_{3} - 12 \beta_{2} - 32 \beta_1 + 24) q^{74} + ( - 24 \beta_{3} + 20 \beta_{2} + 12 \beta_1 - 20) q^{76} + (15 \beta_{2} + 24 \beta_1 + 15) q^{77} + (41 \beta_{3} + 20 \beta_{2} - 10 \beta_1 - 46) q^{78} + ( - 15 \beta_{3} + 7 \beta_{2} - 15 \beta_1) q^{79} + (36 \beta_{3} - 63) q^{81} + (21 \beta_{3} - 42 \beta_1 - 36) q^{82} + (15 \beta_{3} - 63 \beta_{2} - 15 \beta_1 + 126) q^{83} + ( - 20 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 38) q^{84} + ( - 18 \beta_{2} + 23 \beta_1 - 18) q^{86} + (15 \beta_{3} - 21 \beta_{2} - 3 \beta_1 + 24) q^{87} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{88} + (66 \beta_{3} - 60 \beta_{2} + 30) q^{89} + ( - 9 \beta_{3} + 18 \beta_1 + 103) q^{91} + (6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 - 12) q^{92} + (38 \beta_{3} + 35 \beta_{2} - 46 \beta_1 - 73) q^{93} + (18 \beta_{3} - 42 \beta_{2} - 9 \beta_1 + 42) q^{94} + (8 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 8) q^{96} + (42 \beta_{3} - 7 \beta_{2} + 42 \beta_1) q^{97} + (6 \beta_{3} - 24 \beta_{2} + 12) q^{98} + (9 \beta_{3} + 27 \beta_{2} + 27 \beta_1 + 18) q^{99}+O(q^{100})$$ q + (-b3 + b1) * q^2 + (b3 - 2*b2 - 2*b1 + 1) * q^3 + (-2*b2 + 2) * q^4 + (-b3 + 2*b2 - b1 - 4) * q^6 + (3*b3 - b2 + 3*b1) * q^7 - 2*b3 * q^8 + (6*b3 + 3) * q^9 + (-3*b3 - 3*b2 + 3*b1 + 6) * q^11 + (4*b3 - 2*b2 - 2*b1 - 2) * q^12 + (-12*b3 - 5*b2 + 6*b1 + 5) * q^13 + (6*b2 - b1 + 6) * q^14 - 4*b2 * q^16 + (6*b3 - 12*b2 + 6) * q^17 + (-3*b3 + 12*b2 + 3*b1) * q^18 + (-6*b3 + 12*b1 - 10) * q^19 + (-8*b3 - 17*b2 + 10*b1 - 2) * q^21 + (-6*b3 - 6*b2 + 3*b1 + 6) * q^22 + (-3*b2 - 3*b1 - 3) * q^23 + (2*b3 + 8*b2 - 4*b1 - 4) * q^24 + (-5*b3 - 24*b2 + 12) * q^26 + (-3*b3 - 30*b2 + 6*b1 + 15) * q^27 + (-6*b3 + 12*b1 - 2) * q^28 + (6*b3 - 3*b2 - 6*b1 + 6) * q^29 + (-18*b3 - 19*b2 + 9*b1 + 19) * q^31 - 4*b1 * q^32 + (6*b3 - 3*b2 - 12*b1 - 12) * q^33 + (-6*b3 + 12*b2 - 6*b1) * q^34 + (-6*b2 + 12*b1 + 6) * q^36 + (-6*b3 + 12*b1 - 32) * q^37 + (10*b3 - 12*b2 - 10*b1 + 24) * q^38 + (10*b3 + 31*b2 - 23*b1 - 41) * q^39 + (-21*b2 - 18*b1 - 21) * q^41 + (2*b3 - 16*b2 - 19*b1 + 20) * q^42 + (-9*b3 + 23*b2 - 9*b1) * q^43 + (-6*b3 - 12*b2 + 6) * q^44 + (3*b3 - 6*b1 - 6) * q^46 + (-21*b3 + 9*b2 + 21*b1 - 18) * q^47 + (4*b3 + 4*b2 + 4*b1 - 8) * q^48 + (-12*b3 + 6*b2 + 6*b1 - 6) * q^49 + (12*b3 - 24*b2 + 12*b1 - 6) * q^51 + (-12*b3 - 10*b2 - 12*b1) * q^52 + (30*b3 - 60*b2 + 30) * q^53 + (-15*b3 - 6*b2 - 15*b1 + 12) * q^54 + (2*b3 - 12*b2 - 2*b1 + 24) * q^56 + (-28*b3 + 20*b2 + 20*b1 - 46) * q^57 + (-6*b3 + 12*b2 + 3*b1 - 12) * q^58 + (21*b2 + 39*b1 + 21) * q^59 + (18*b3 + 31*b2 + 18*b1) * q^61 + (-19*b3 - 36*b2 + 18) * q^62 + (3*b3 + 33*b2 + 15*b1 - 72) * q^63 - 8 * q^64 + (12*b3 + 12*b2 - 15*b1 - 24) * q^66 + (18*b3 - 53*b2 - 9*b1 + 53) * q^67 + (-12*b2 + 12*b1 - 12) * q^68 + (6*b3 + 15*b2 + 6*b1 - 3) * q^69 + (-24*b3 - 60*b2 + 30) * q^71 + (-6*b3 + 24) * q^72 + (-18*b3 + 36*b1 + 52) * q^73 + (32*b3 - 12*b2 - 32*b1 + 24) * q^74 + (-24*b3 + 20*b2 + 12*b1 - 20) * q^76 + (15*b2 + 24*b1 + 15) * q^77 + (41*b3 + 20*b2 - 10*b1 - 46) * q^78 + (-15*b3 + 7*b2 - 15*b1) * q^79 + (36*b3 - 63) * q^81 + (21*b3 - 42*b1 - 36) * q^82 + (15*b3 - 63*b2 - 15*b1 + 126) * q^83 + (-20*b3 + 4*b2 + 4*b1 - 38) * q^84 + (-18*b2 + 23*b1 - 18) * q^86 + (15*b3 - 21*b2 - 3*b1 + 24) * q^87 + (-6*b3 - 12*b2 - 6*b1) * q^88 + (66*b3 - 60*b2 + 30) * q^89 + (-9*b3 + 18*b1 + 103) * q^91 + (6*b3 + 6*b2 - 6*b1 - 12) * q^92 + (38*b3 + 35*b2 - 46*b1 - 73) * q^93 + (18*b3 - 42*b2 - 9*b1 + 42) * q^94 + (8*b3 + 8*b2 - 4*b1 + 8) * q^96 + (42*b3 - 7*b2 + 42*b1) * q^97 + (6*b3 - 24*b2 + 12) * q^98 + (9*b3 + 27*b2 + 27*b1 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 12 q^{6} - 2 q^{7} + 12 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 12 * q^6 - 2 * q^7 + 12 * q^9 $$4 q + 4 q^{4} - 12 q^{6} - 2 q^{7} + 12 q^{9} + 18 q^{11} - 12 q^{12} + 10 q^{13} + 36 q^{14} - 8 q^{16} + 24 q^{18} - 40 q^{19} - 42 q^{21} + 12 q^{22} - 18 q^{23} - 8 q^{28} + 18 q^{29} + 38 q^{31} - 54 q^{33} + 24 q^{34} + 12 q^{36} - 128 q^{37} + 72 q^{38} - 102 q^{39} - 126 q^{41} + 48 q^{42} + 46 q^{43} - 24 q^{46} - 54 q^{47} - 24 q^{48} - 12 q^{49} - 72 q^{51} - 20 q^{52} + 36 q^{54} + 72 q^{56} - 144 q^{57} - 24 q^{58} + 126 q^{59} + 62 q^{61} - 222 q^{63} - 32 q^{64} - 72 q^{66} + 106 q^{67} - 72 q^{68} + 18 q^{69} + 96 q^{72} + 208 q^{73} + 72 q^{74} - 40 q^{76} + 90 q^{77} - 144 q^{78} + 14 q^{79} - 252 q^{81} - 144 q^{82} + 378 q^{83} - 144 q^{84} - 108 q^{86} + 54 q^{87} - 24 q^{88} + 412 q^{91} - 36 q^{92} - 222 q^{93} + 84 q^{94} + 48 q^{96} - 14 q^{97} + 126 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 - 12 * q^6 - 2 * q^7 + 12 * q^9 + 18 * q^11 - 12 * q^12 + 10 * q^13 + 36 * q^14 - 8 * q^16 + 24 * q^18 - 40 * q^19 - 42 * q^21 + 12 * q^22 - 18 * q^23 - 8 * q^28 + 18 * q^29 + 38 * q^31 - 54 * q^33 + 24 * q^34 + 12 * q^36 - 128 * q^37 + 72 * q^38 - 102 * q^39 - 126 * q^41 + 48 * q^42 + 46 * q^43 - 24 * q^46 - 54 * q^47 - 24 * q^48 - 12 * q^49 - 72 * q^51 - 20 * q^52 + 36 * q^54 + 72 * q^56 - 144 * q^57 - 24 * q^58 + 126 * q^59 + 62 * q^61 - 222 * q^63 - 32 * q^64 - 72 * q^66 + 106 * q^67 - 72 * q^68 + 18 * q^69 + 96 * q^72 + 208 * q^73 + 72 * q^74 - 40 * q^76 + 90 * q^77 - 144 * q^78 + 14 * q^79 - 252 * q^81 - 144 * q^82 + 378 * q^83 - 144 * q^84 - 108 * q^86 + 54 * q^87 - 24 * q^88 + 412 * q^91 - 36 * q^92 - 222 * q^93 + 84 * q^94 + 48 * q^96 - 14 * q^97 + 126 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 0.707107i 2.44949 + 1.73205i 1.00000 + 1.73205i 0 −1.77526 3.85337i −4.17423 + 7.22999i 2.82843i 3.00000 + 8.48528i 0
101.2 1.22474 + 0.707107i −2.44949 + 1.73205i 1.00000 + 1.73205i 0 −4.22474 + 0.389270i 3.17423 5.49794i 2.82843i 3.00000 8.48528i 0
401.1 −1.22474 + 0.707107i 2.44949 1.73205i 1.00000 1.73205i 0 −1.77526 + 3.85337i −4.17423 7.22999i 2.82843i 3.00000 8.48528i 0
401.2 1.22474 0.707107i −2.44949 1.73205i 1.00000 1.73205i 0 −4.22474 0.389270i 3.17423 + 5.49794i 2.82843i 3.00000 + 8.48528i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.i.b 4
3.b odd 2 1 1350.3.i.b 4
5.b even 2 1 18.3.d.a 4
5.c odd 4 2 450.3.k.a 8
9.c even 3 1 1350.3.i.b 4
9.d odd 6 1 inner 450.3.i.b 4
15.d odd 2 1 54.3.d.a 4
15.e even 4 2 1350.3.k.a 8
20.d odd 2 1 144.3.q.c 4
40.e odd 2 1 576.3.q.e 4
40.f even 2 1 576.3.q.f 4
45.h odd 6 1 18.3.d.a 4
45.h odd 6 1 162.3.b.a 4
45.j even 6 1 54.3.d.a 4
45.j even 6 1 162.3.b.a 4
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
60.h even 2 1 432.3.q.d 4
120.i odd 2 1 1728.3.q.d 4
120.m even 2 1 1728.3.q.c 4
180.n even 6 1 144.3.q.c 4
180.n even 6 1 1296.3.e.g 4
180.p odd 6 1 432.3.q.d 4
180.p odd 6 1 1296.3.e.g 4
360.z odd 6 1 1728.3.q.c 4
360.bd even 6 1 576.3.q.e 4
360.bh odd 6 1 576.3.q.f 4
360.bk even 6 1 1728.3.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 5.b even 2 1
18.3.d.a 4 45.h odd 6 1
54.3.d.a 4 15.d odd 2 1
54.3.d.a 4 45.j even 6 1
144.3.q.c 4 20.d odd 2 1
144.3.q.c 4 180.n even 6 1
162.3.b.a 4 45.h odd 6 1
162.3.b.a 4 45.j even 6 1
432.3.q.d 4 60.h even 2 1
432.3.q.d 4 180.p odd 6 1
450.3.i.b 4 1.a even 1 1 trivial
450.3.i.b 4 9.d odd 6 1 inner
450.3.k.a 8 5.c odd 4 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 40.e odd 2 1
576.3.q.e 4 360.bd even 6 1
576.3.q.f 4 40.f even 2 1
576.3.q.f 4 360.bh odd 6 1
1296.3.e.g 4 180.n even 6 1
1296.3.e.g 4 180.p odd 6 1
1350.3.i.b 4 3.b odd 2 1
1350.3.i.b 4 9.c even 3 1
1350.3.k.a 8 15.e even 4 2
1350.3.k.a 8 45.k odd 12 2
1728.3.q.c 4 120.m even 2 1
1728.3.q.c 4 360.z odd 6 1
1728.3.q.d 4 120.i odd 2 1
1728.3.q.d 4 360.bk even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 2T_{7}^{3} + 57T_{7}^{2} - 106T_{7} + 2809$$ acting on $$S_{3}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4} - 6T^{2} + 81$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2 T^{3} + 57 T^{2} + \cdots + 2809$$
$11$ $$T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81$$
$13$ $$T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481$$
$17$ $$T^{4} + 360T^{2} + 1296$$
$19$ $$(T^{2} + 20 T - 116)^{2}$$
$23$ $$T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81$$
$29$ $$T^{4} - 18 T^{3} + 63 T^{2} + \cdots + 2025$$
$31$ $$T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625$$
$37$ $$(T^{2} + 64 T + 808)^{2}$$
$41$ $$T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625$$
$43$ $$T^{4} - 46 T^{3} + 2073 T^{2} + \cdots + 1849$$
$47$ $$T^{4} + 54 T^{3} + 333 T^{2} + \cdots + 408321$$
$53$ $$T^{4} + 9000 T^{2} + 810000$$
$59$ $$T^{4} - 126 T^{3} + 3573 T^{2} + \cdots + 2954961$$
$61$ $$T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289$$
$67$ $$T^{4} - 106 T^{3} + 8913 T^{2} + \cdots + 5396329$$
$71$ $$T^{4} + 7704 T^{2} + \cdots + 2396304$$
$73$ $$(T^{2} - 104 T + 760)^{2}$$
$79$ $$T^{4} - 14 T^{3} + 1497 T^{2} + \cdots + 1692601$$
$83$ $$T^{4} - 378 T^{3} + \cdots + 131262849$$
$89$ $$T^{4} + 22824 T^{2} + \cdots + 36144144$$
$97$ $$T^{4} + 14 T^{3} + \cdots + 110986225$$