# Properties

 Label 210.2.m.a Level 210 Weight 2 Character orbit 210.m Analytic conductor 1.677 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.1698758656.6 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + \beta_{3} q^{3} -\beta_{5} q^{4} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{5} -\beta_{5} q^{6} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{7} + \beta_{1} q^{8} -\beta_{5} q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + \beta_{3} q^{3} -\beta_{5} q^{4} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{5} -\beta_{5} q^{6} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{7} + \beta_{1} q^{8} -\beta_{5} q^{9} + ( -\beta_{1} - \beta_{7} ) q^{10} + ( -\beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + \beta_{1} q^{12} + ( -2 + 2 \beta_{3} + 2 \beta_{5} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} ) q^{14} + ( -\beta_{1} - \beta_{7} ) q^{15} - q^{16} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{17} + \beta_{1} q^{18} + ( -2 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{19} + ( 1 + \beta_{6} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} ) q^{21} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{22} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{23} - q^{24} + ( -1 - \beta_{2} - 5 \beta_{3} - \beta_{6} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{26} + \beta_{1} q^{27} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( 1 + \beta_{6} ) q^{30} + ( \beta_{1} + \beta_{3} - 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} -\beta_{3} q^{32} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{34} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{35} - q^{36} + ( -3 - \beta_{1} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{37} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{38} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{39} -\beta_{2} q^{40} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{41} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{42} + ( 2 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{43} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{44} + ( 1 + \beta_{6} ) q^{45} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{46} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{47} -\beta_{3} q^{48} + ( \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{49} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{50} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{51} + ( 2 + 2 \beta_{1} + 2 \beta_{5} ) q^{52} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} - q^{54} + ( 4 + 4 \beta_{1} - 5 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{56} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{57} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{58} + ( 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{59} -\beta_{2} q^{60} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{61} + ( -1 + 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{62} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{63} + \beta_{5} q^{64} + ( -2 - 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{65} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{66} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{67} + ( 1 + \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{68} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{69} + ( 4 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{70} + ( 2 - 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{71} -\beta_{3} q^{72} + ( -4 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{73} + ( -3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{74} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{75} + ( \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{76} + ( -5 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} + \beta_{6} ) q^{77} + ( 2 + 2 \beta_{1} + 2 \beta_{5} ) q^{78} + ( 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{79} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{80} - q^{81} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{82} + ( -2 - \beta_{1} + \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{84} + ( 3 + 5 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{85} + ( 2 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} ) q^{86} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{87} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{88} + ( -6 - 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{89} -\beta_{2} q^{90} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{91} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{92} + ( -1 + 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{93} + ( 2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{94} + ( 4 \beta_{1} - \beta_{2} - \beta_{4} - 6 \beta_{5} ) q^{95} + \beta_{5} q^{96} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{97} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{98} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{7} + O(q^{10})$$ $$8q + 4q^{7} - 4q^{10} + 8q^{11} - 16q^{13} - 8q^{14} - 4q^{15} - 8q^{16} + 12q^{17} - 8q^{19} + 4q^{20} - 8q^{21} + 8q^{22} + 16q^{23} - 8q^{24} - 4q^{25} - 8q^{28} + 4q^{30} + 8q^{33} + 16q^{34} + 8q^{35} - 8q^{36} - 28q^{37} - 4q^{38} - 8q^{42} + 4q^{45} - 8q^{46} + 24q^{47} + 4q^{49} + 16q^{51} + 16q^{52} - 8q^{53} - 8q^{54} + 28q^{55} + 4q^{56} - 4q^{57} - 12q^{58} + 8q^{59} - 4q^{62} - 8q^{63} - 16q^{65} + 12q^{68} - 8q^{69} + 32q^{70} + 8q^{71} - 28q^{73} - 44q^{77} + 16q^{78} - 8q^{81} + 24q^{82} - 16q^{83} + 4q^{84} + 28q^{85} + 8q^{86} - 12q^{87} - 8q^{88} - 64q^{89} - 8q^{91} + 16q^{92} - 4q^{93} + 8q^{94} - 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + 18 \nu^{5} + 28 \nu^{4} + 89 \nu^{3} + 74 \nu^{2} + 104 \nu - 16$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 18 \nu^{5} + 8 \nu^{4} + 105 \nu^{3} + 72 \nu^{2} + 248 \nu + 64$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 18 \nu^{5} - 28 \nu^{4} + 89 \nu^{3} - 74 \nu^{2} + 104 \nu + 16$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 18 \nu^{5} - 8 \nu^{4} + 105 \nu^{3} - 72 \nu^{2} + 248 \nu - 64$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} - 46 \nu^{5} - 179 \nu^{3} - 168 \nu$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 6 \nu^{6} + 10 \nu^{5} - 92 \nu^{4} - 15 \nu^{3} - 358 \nu^{2} - 120 \nu - 336$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 6 \nu^{6} + 10 \nu^{5} + 92 \nu^{4} - 15 \nu^{3} + 358 \nu^{2} - 120 \nu + 336$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{7} - 9 \beta_{6} - 18 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 5 \beta_{2} - 13 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{7} + 9 \beta_{6} - 17 \beta_{4} - 27 \beta_{3} + 17 \beta_{2} + 27 \beta_{1} + 74$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$81 \beta_{7} + 81 \beta_{6} + 178 \beta_{5} + 37 \beta_{4} + 149 \beta_{3} + 37 \beta_{2} + 149 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$89 \beta_{7} - 89 \beta_{6} + 201 \beta_{4} + 235 \beta_{3} - 201 \beta_{2} - 235 \beta_{1} - 650$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-761 \beta_{7} - 761 \beta_{6} - 1810 \beta_{5} - 325 \beta_{4} - 1565 \beta_{3} - 325 \beta_{2} - 1565 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 3.16053i − 2.16053i − 0.692297i 1.69230i − 3.16053i 2.16053i 0.692297i − 1.69230i
−0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i −1.52773 + 1.63280i 1.00000i 2.33991 + 1.23483i 0.707107 + 0.707107i 1.00000i −0.0743018 2.23483i
13.2 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 2.23483 + 0.0743018i 1.00000i 0.781409 2.52773i 0.707107 + 0.707107i 1.00000i −1.63280 + 1.52773i
13.3 0.707107 0.707107i 0.707107 0.707107i 1.00000i −1.19663 1.88893i 1.00000i −2.59604 0.510472i −0.707107 0.707107i 1.00000i −2.18183 0.489528i
13.4 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0.489528 + 2.18183i 1.00000i 1.47472 2.19663i −0.707107 0.707107i 1.00000i 1.88893 + 1.19663i
97.1 −0.707107 0.707107i −0.707107 0.707107i 1.00000i −1.52773 1.63280i 1.00000i 2.33991 1.23483i 0.707107 0.707107i 1.00000i −0.0743018 + 2.23483i
97.2 −0.707107 0.707107i −0.707107 0.707107i 1.00000i 2.23483 0.0743018i 1.00000i 0.781409 + 2.52773i 0.707107 0.707107i 1.00000i −1.63280 1.52773i
97.3 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −1.19663 + 1.88893i 1.00000i −2.59604 + 0.510472i −0.707107 + 0.707107i 1.00000i −2.18183 + 0.489528i
97.4 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0.489528 2.18183i 1.00000i 1.47472 + 2.19663i −0.707107 + 0.707107i 1.00000i 1.88893 1.19663i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.m.a 8
3.b odd 2 1 630.2.p.b 8
4.b odd 2 1 1680.2.cz.b 8
5.b even 2 1 1050.2.m.b 8
5.c odd 4 1 210.2.m.b yes 8
5.c odd 4 1 1050.2.m.a 8
7.b odd 2 1 210.2.m.b yes 8
15.e even 4 1 630.2.p.c 8
20.e even 4 1 1680.2.cz.a 8
21.c even 2 1 630.2.p.c 8
28.d even 2 1 1680.2.cz.a 8
35.c odd 2 1 1050.2.m.a 8
35.f even 4 1 inner 210.2.m.a 8
35.f even 4 1 1050.2.m.b 8
105.k odd 4 1 630.2.p.b 8
140.j odd 4 1 1680.2.cz.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.m.a 8 1.a even 1 1 trivial
210.2.m.a 8 35.f even 4 1 inner
210.2.m.b yes 8 5.c odd 4 1
210.2.m.b yes 8 7.b odd 2 1
630.2.p.b 8 3.b odd 2 1
630.2.p.b 8 105.k odd 4 1
630.2.p.c 8 15.e even 4 1
630.2.p.c 8 21.c even 2 1
1050.2.m.a 8 5.c odd 4 1
1050.2.m.a 8 35.c odd 2 1
1050.2.m.b 8 5.b even 2 1
1050.2.m.b 8 35.f even 4 1
1680.2.cz.a 8 20.e even 4 1
1680.2.cz.a 8 28.d even 2 1
1680.2.cz.b 8 4.b odd 2 1
1680.2.cz.b 8 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} + 8 T_{13}^{3} + 32 T_{13}^{2} + 32 T_{13} + 16$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8}$$
$7$ $$1 - 4 T + 6 T^{2} + 28 T^{3} - 126 T^{4} + 196 T^{5} + 294 T^{6} - 1372 T^{7} + 2401 T^{8}$$
$11$ $$( 1 - 4 T + 12 T^{2} + 4 T^{3} + 18 T^{4} + 44 T^{5} + 1452 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 12 T + 72 T^{2} - 388 T^{3} + 1696 T^{4} - 4732 T^{5} + 9944 T^{6} - 2420 T^{7} - 98754 T^{8} - 41140 T^{9} + 2873816 T^{10} - 23248316 T^{11} + 141651616 T^{12} - 550904516 T^{13} + 1737904968 T^{14} - 4924064076 T^{15} + 6975757441 T^{16}$$
$19$ $$( 1 + 4 T + 62 T^{2} + 212 T^{3} + 1666 T^{4} + 4028 T^{5} + 22382 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 - 16 T + 128 T^{2} - 656 T^{3} + 3300 T^{4} - 21520 T^{5} + 137088 T^{6} - 691344 T^{7} + 3238918 T^{8} - 15900912 T^{9} + 72519552 T^{10} - 261833840 T^{11} + 923475300 T^{12} - 4222241008 T^{13} + 18948593792 T^{14} - 54477207152 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16}$$
$31$ $$1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16}$$
$37$ $$1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 - 32 T^{3} - 6588 T^{4} - 1632 T^{5} + 512 T^{6} + 117440 T^{7} + 17663206 T^{8} + 5049920 T^{9} + 946688 T^{10} - 129755424 T^{11} - 22523060988 T^{12} - 4704270176 T^{13} + 11688200277601 T^{16}$$
$47$ $$1 - 24 T + 288 T^{2} - 2728 T^{3} + 26308 T^{4} - 243208 T^{5} + 1981280 T^{6} - 14819768 T^{7} + 104930310 T^{8} - 696529096 T^{9} + 4376647520 T^{10} - 25250584184 T^{11} + 128374647748 T^{12} - 625653179096 T^{13} + 3104414014752 T^{14} - 12158954891112 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 8 T + 32 T^{2} + 264 T^{3} + 5828 T^{4} + 46136 T^{5} + 217440 T^{6} + 2204664 T^{7} + 21970150 T^{8} + 116847192 T^{9} + 610788960 T^{10} + 6868589272 T^{11} + 45985723268 T^{12} + 110403610152 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16}$$
$59$ $$( 1 - 4 T + 70 T^{2} - 116 T^{3} + 802 T^{4} - 6844 T^{5} + 243670 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 64 T^{3} + 9348 T^{4} + 10560 T^{5} + 2048 T^{6} + 428416 T^{7} + 51599014 T^{8} + 28703872 T^{9} + 9193472 T^{10} + 3176057280 T^{11} + 188372679108 T^{12} - 86408006848 T^{13} + 406067677556641 T^{16}$$
$71$ $$( 1 - 4 T + 158 T^{2} + 220 T^{3} + 10018 T^{4} + 15620 T^{5} + 796478 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 + 28 T + 392 T^{2} + 4756 T^{3} + 46736 T^{4} + 302508 T^{5} + 1459480 T^{6} + 3207460 T^{7} - 21483554 T^{8} + 234144580 T^{9} + 7777568920 T^{10} + 117680754636 T^{11} + 1327220191376 T^{12} + 9859528496308 T^{13} + 59323016705288 T^{14} + 309327158534716 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 + 16 T + 128 T^{2} + 976 T^{3} + 16964 T^{4} + 235664 T^{5} + 2075520 T^{6} + 16972368 T^{7} + 135669670 T^{8} + 1408706544 T^{9} + 14298257280 T^{10} + 134749611568 T^{11} + 805082957444 T^{12} + 3844503667568 T^{13} + 41848367791232 T^{14} + 434176815834032 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 32 T + 628 T^{2} + 8320 T^{3} + 88630 T^{4} + 740480 T^{5} + 4974388 T^{6} + 22559008 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 28 T + 392 T^{2} + 4900 T^{3} + 61488 T^{4} + 704956 T^{5} + 7640472 T^{6} + 89011524 T^{7} + 981394654 T^{8} + 8634117828 T^{9} + 71889201048 T^{10} + 643394307388 T^{11} + 5443488430128 T^{12} + 42077967259300 T^{13} + 326525025932168 T^{14} + 2262351965387164 T^{15} + 7837433594376961 T^{16}$$