# Properties

 Label 525.2.t.f Level 525 Weight 2 Character orbit 525.t Analytic conductor 4.192 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.856615824.2 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

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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -1 - \beta_{1} - \beta_{3} - \beta_{7} ) q^{3} + ( -\beta_{2} - \beta_{5} - \beta_{6} ) q^{4} + ( 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{8} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -1 - \beta_{1} - \beta_{3} - \beta_{7} ) q^{3} + ( -\beta_{2} - \beta_{5} - \beta_{6} ) q^{4} + ( 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{6} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} - \beta_{4} + \beta_{5} ) q^{8} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{9} + ( \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{11} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{12} + ( -\beta_{6} + \beta_{7} ) q^{13} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{14} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{16} + ( -4 + 4 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{18} + ( 1 - \beta_{4} + \beta_{7} ) q^{19} + ( -3 - \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{21} + ( 3 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{22} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{24} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{26} + ( -4 + \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{27} + ( 3 + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{28} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{29} + ( 1 - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{31} + ( 4 + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{32} + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{33} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{34} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{36} + ( 2 + \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{37} + ( -2 + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{38} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{39} + ( 1 - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -1 + \beta_{2} - 4 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{42} + ( -3 - \beta_{4} - \beta_{5} ) q^{43} + ( 2 + 6 \beta_{1} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{44} + ( 2 + 8 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{5} + \beta_{6} ) q^{46} + ( 2 - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{47} + ( -4 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{48} + ( 1 + \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{49} + ( 4 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{51} + ( -3 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{52} + ( 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} ) q^{53} + ( 6 + 3 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{54} + ( -\beta_{3} - 3 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{56} + ( -3 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{57} + ( -5 + 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{58} + ( 4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{59} + ( -3 - \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{61} + ( -4 - 4 \beta_{1} - 8 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{62} + ( 8 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{63} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{64} + ( -10 - 6 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{66} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{67} + ( -2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{68} + ( -6 - 6 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} - \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{71} + ( -2 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{72} + ( 1 - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -4 - 5 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{74} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{76} + ( -8 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{77} + ( -3 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{78} + ( -3 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{79} + ( 3 - \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{81} + ( 1 + 6 \beta_{1} + \beta_{2} + 6 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{82} + ( 6 + \beta_{1} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{84} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{86} + ( 8 - 3 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{87} + ( -4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} ) q^{88} + ( -1 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 6 \beta_{7} ) q^{89} + ( 4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{91} + ( 8 - 3 \beta_{1} - 16 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} ) q^{92} + ( -5 + 2 \beta_{1} + 4 \beta_{3} - \beta_{4} - 4 \beta_{6} + 3 \beta_{7} ) q^{93} + ( -4 - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{94} + ( -2 - 5 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{96} + ( -1 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{97} + ( -6 - 8 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{98} + ( -1 + 3 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 3q^{2} - 2q^{3} + 3q^{4} - 5q^{6} - 2q^{7} + 4q^{9} + O(q^{10})$$ $$8q - 3q^{2} - 2q^{3} + 3q^{4} - 5q^{6} - 2q^{7} + 4q^{9} + 18q^{12} + 12q^{14} + q^{16} - 12q^{17} - 26q^{18} + 9q^{19} - 22q^{21} + 40q^{22} + 27q^{23} - 7q^{24} + 6q^{26} + 4q^{27} - 3q^{28} - 21q^{31} + 21q^{32} - 4q^{33} + 9q^{36} - 7q^{37} - 12q^{38} + 15q^{39} + 30q^{41} + 5q^{42} - 16q^{43} - 7q^{46} - 6q^{47} - 25q^{48} - 4q^{49} + 12q^{51} - 30q^{52} + 24q^{53} + 7q^{54} + 21q^{56} - 6q^{57} + 13q^{58} + 12q^{59} + 15q^{61} + 24q^{62} + 44q^{63} + 38q^{64} - 16q^{66} - 4q^{67} + 13q^{69} - 13q^{72} - 15q^{73} - 54q^{74} - 36q^{77} + 6q^{78} - 29q^{79} + 28q^{81} - 27q^{82} + 30q^{83} - 51q^{84} - 9q^{86} + 29q^{87} + 2q^{88} + 3q^{89} - 3q^{91} - 45q^{93} - 24q^{94} - 42q^{96} - 39q^{98} - 34q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 7 \nu^{3} + 10 \nu + 2$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 7 \nu^{4} + 10 \nu^{2} - 2 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 9 \nu^{4} + 2 \nu^{3} + 22 \nu^{2} + 10 \nu + 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + 9 \nu^{4} - 2 \nu^{3} + 22 \nu^{2} - 10 \nu + 8$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} - 10 \nu^{5} - 9 \nu^{4} - 29 \nu^{3} - 20 \nu^{2} - 20 \nu - 6$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 29 \nu^{3} - 20 \nu^{2} + 20 \nu - 6$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-6 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{5} - 7 \beta_{4} + 4 \beta_{2} + 25 \beta_{1} - 2$$ $$\nu^{6}$$ $$=$$ $$32 \beta_{7} + 32 \beta_{6} + 25 \beta_{5} + 25 \beta_{4} + 18 \beta_{3} + 9 \beta_{1} - 4$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{6} - 41 \beta_{5} + 41 \beta_{4} - 40 \beta_{2} - 125 \beta_{1} + 20$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\beta_{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 − 2.33086i − 1.07834i − 0.385731i 2.06288i 2.33086i 1.07834i 0.385731i − 2.06288i
−2.01859 1.16543i 1.67602 0.437000i 1.71646 + 2.97300i 0 −3.89248 1.07116i −1.11699 + 2.39840i 3.33995i 2.61806 1.46484i 0
26.2 −0.933868 0.539169i −1.73096 + 0.0613278i −0.418594 0.725026i 0 1.64956 + 0.876010i 2.47720 0.929227i 3.05945i 2.99248 0.212312i 0
26.3 −0.334053 0.192865i 0.139098 + 1.72646i −0.925606 1.60320i 0 0.286507 0.603555i −2.36975 + 1.17656i 1.48553i −2.96130 + 0.480295i 0
26.4 1.78651 + 1.03144i −1.08415 1.35078i 1.12774 + 1.95330i 0 −0.543588 3.53142i 0.00953166 2.64573i 0.527019i −0.649237 + 2.92891i 0
101.1 −2.01859 + 1.16543i 1.67602 + 0.437000i 1.71646 2.97300i 0 −3.89248 + 1.07116i −1.11699 2.39840i 3.33995i 2.61806 + 1.46484i 0
101.2 −0.933868 + 0.539169i −1.73096 0.0613278i −0.418594 + 0.725026i 0 1.64956 0.876010i 2.47720 + 0.929227i 3.05945i 2.99248 + 0.212312i 0
101.3 −0.334053 + 0.192865i 0.139098 1.72646i −0.925606 + 1.60320i 0 0.286507 + 0.603555i −2.36975 1.17656i 1.48553i −2.96130 0.480295i 0
101.4 1.78651 1.03144i −1.08415 + 1.35078i 1.12774 1.95330i 0 −0.543588 + 3.53142i 0.00953166 + 2.64573i 0.527019i −0.649237 2.92891i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g 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.t.f 8
3.b odd 2 1 525.2.t.g 8
5.b even 2 1 105.2.s.d yes 8
5.c odd 4 2 525.2.q.e 16
7.d odd 6 1 525.2.t.g 8
15.d odd 2 1 105.2.s.c 8
15.e even 4 2 525.2.q.f 16
21.g even 6 1 inner 525.2.t.f 8
35.c odd 2 1 735.2.s.l 8
35.i odd 6 1 105.2.s.c 8
35.i odd 6 1 735.2.b.d 8
35.j even 6 1 735.2.b.c 8
35.j even 6 1 735.2.s.k 8
35.k even 12 2 525.2.q.f 16
105.g even 2 1 735.2.s.k 8
105.o odd 6 1 735.2.b.d 8
105.o odd 6 1 735.2.s.l 8
105.p even 6 1 105.2.s.d yes 8
105.p even 6 1 735.2.b.c 8
105.w odd 12 2 525.2.q.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c 8 15.d odd 2 1
105.2.s.c 8 35.i odd 6 1
105.2.s.d yes 8 5.b even 2 1
105.2.s.d yes 8 105.p even 6 1
525.2.q.e 16 5.c odd 4 2
525.2.q.e 16 105.w odd 12 2
525.2.q.f 16 15.e even 4 2
525.2.q.f 16 35.k even 12 2
525.2.t.f 8 1.a even 1 1 trivial
525.2.t.f 8 21.g even 6 1 inner
525.2.t.g 8 3.b odd 2 1
525.2.t.g 8 7.d odd 6 1
735.2.b.c 8 35.j even 6 1
735.2.b.c 8 105.p even 6 1
735.2.b.d 8 35.i odd 6 1
735.2.b.d 8 105.o odd 6 1
735.2.s.k 8 35.j even 6 1
735.2.s.k 8 105.g even 2 1
735.2.s.l 8 35.c odd 2 1
735.2.s.l 8 105.o odd 6 1

## Hecke kernels

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

 $$T_{2}^{8} + \cdots$$ $$T_{13}^{8} + 21 T_{13}^{6} + 123 T_{13}^{4} + 135 T_{13}^{2} + 36$$ $$T_{37}^{8} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16}$$
$3$ $$1 + 2 T - 4 T^{3} - 11 T^{4} - 12 T^{5} + 54 T^{7} + 81 T^{8}$$
$5$ 
$7$ $$1 + 2 T + 4 T^{2} - 10 T^{3} - 41 T^{4} - 70 T^{5} + 196 T^{6} + 686 T^{7} + 2401 T^{8}$$
$11$ $$1 + 16 T^{2} - 2 T^{4} - 30 T^{5} + 268 T^{6} - 1548 T^{7} + 21079 T^{8} - 17028 T^{9} + 32428 T^{10} - 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16}$$
$17$ $$1 + 12 T + 34 T^{2} + 12 T^{3} + 1078 T^{4} + 6882 T^{5} + 8740 T^{6} + 70272 T^{7} + 637627 T^{8} + 1194624 T^{9} + 2525860 T^{10} + 33811266 T^{11} + 90035638 T^{12} + 17038284 T^{13} + 820677346 T^{14} + 4924064076 T^{15} + 6975757441 T^{16}$$
$19$ $$1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 13379724 T^{9} + 53787556 T^{10} - 180007596 T^{11} + 615506083 T^{12} - 1626797043 T^{13} + 4704588100 T^{14} - 8044845651 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 27 T + 391 T^{2} - 3996 T^{3} + 31651 T^{4} - 205875 T^{5} + 1157938 T^{6} - 5917779 T^{7} + 28782226 T^{8} - 136108917 T^{9} + 612549202 T^{10} - 2504881125 T^{11} + 8857247491 T^{12} - 25719626628 T^{13} + 57882032599 T^{14} - 91930287069 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16}$$
$31$ $$1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16}$$
$37$ $$1 + 7 T - 24 T^{2} - 493 T^{3} - 973 T^{4} + 16188 T^{5} + 118336 T^{6} - 258098 T^{7} - 5756772 T^{8} - 9549626 T^{9} + 162001984 T^{10} + 819970764 T^{11} - 1823558653 T^{12} - 34186570801 T^{13} - 61577433816 T^{14} + 664523139931 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 73431 T^{5} + 366458 T^{6} - 1033815 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 184 T^{2} + 1022 T^{3} + 12127 T^{4} + 43946 T^{5} + 340216 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 + 6 T - 116 T^{2} - 252 T^{3} + 10126 T^{4} + 1986 T^{5} - 595736 T^{6} - 157218 T^{7} + 25623007 T^{8} - 7389246 T^{9} - 1315980824 T^{10} + 206192478 T^{11} + 49411649806 T^{12} - 57794941764 T^{13} - 1250388978164 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 24 T + 340 T^{2} - 3552 T^{3} + 29050 T^{4} - 180120 T^{5} + 750160 T^{6} - 1659096 T^{7} + 1273315 T^{8} - 87932088 T^{9} + 2107199440 T^{10} - 26815725240 T^{11} + 229218473050 T^{12} - 1485430391136 T^{13} + 7535882783860 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 4 T - 234 T^{2} - 412 T^{3} + 35255 T^{4} + 28434 T^{5} - 3551522 T^{6} - 717722 T^{7} + 271900824 T^{8} - 48087374 T^{9} - 15942782258 T^{10} + 8551895142 T^{11} + 710427770855 T^{12} - 556251544084 T^{13} - 21167261427546 T^{14} + 24242846421292 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 + 15 T + 280 T^{2} + 3075 T^{3} + 38779 T^{4} + 422928 T^{5} + 4017052 T^{6} + 39506334 T^{7} + 310273396 T^{8} + 2883962382 T^{9} + 21406870108 T^{10} + 164526181776 T^{11} + 1101255387739 T^{12} + 6374695148475 T^{13} + 42373583360920 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 15 T + 380 T^{2} - 3759 T^{3} + 49260 T^{4} - 311997 T^{5} + 2617820 T^{6} - 8576805 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16}$$