# Properties

 Label 108.2.h.a Level 108 Weight 2 Character orbit 108.h Analytic conductor 0.862 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 108.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.862384341830$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.170772624.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - \nu^{5} - 2 \nu^{3} - 4 \nu^{2} + 4 \nu + 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{5} - \nu^{4} + 2 \nu^{3} + 4 \nu - 4$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 5 \nu^{5} - 2 \nu^{4} + 2 \nu^{3} - 8 \nu^{2} + 20 \nu - 24$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 16 \nu^{2} - 12 \nu + 16$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 5 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} - 12 \nu^{2} + 20 \nu - 24$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} - 5 \nu^{6} + 7 \nu^{5} - 6 \nu^{4} + 10 \nu^{3} - 24 \nu^{2} + 28 \nu - 32$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{5} + \beta_{3} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1} + 5$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 5 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{7} - \beta_{6} - 7 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_{1} - 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$1 + \beta_{7}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 −1.02187 + 0.977642i 0.335728 − 1.37379i 0.774115 + 1.18353i 1.41203 + 0.0786378i −1.02187 − 0.977642i 0.335728 + 1.37379i 0.774115 − 1.18353i 1.41203 − 0.0786378i
−1.02187 + 0.977642i 0 0.0884324 1.99804i 2.18614 1.26217i 0 1.10489 + 0.637910i 1.86301 + 2.12819i 0 −1.00000 + 3.42703i
35.2 0.335728 1.37379i 0 −1.77457 0.922437i 2.18614 1.26217i 0 −1.10489 0.637910i −1.86301 + 2.12819i 0 −1.00000 3.42703i
35.3 0.774115 + 1.18353i 0 −0.801492 + 1.83238i −0.686141 + 0.396143i 0 2.35143 + 1.35760i −2.78912 + 0.469882i 0 −1.00000 0.505408i
35.4 1.41203 + 0.0786378i 0 1.98763 + 0.222077i −0.686141 + 0.396143i 0 −2.35143 1.35760i 2.78912 + 0.469882i 0 −1.00000 + 0.505408i
71.1 −1.02187 0.977642i 0 0.0884324 + 1.99804i 2.18614 + 1.26217i 0 1.10489 0.637910i 1.86301 2.12819i 0 −1.00000 3.42703i
71.2 0.335728 + 1.37379i 0 −1.77457 + 0.922437i 2.18614 + 1.26217i 0 −1.10489 + 0.637910i −1.86301 2.12819i 0 −1.00000 + 3.42703i
71.3 0.774115 1.18353i 0 −0.801492 1.83238i −0.686141 0.396143i 0 2.35143 1.35760i −2.78912 0.469882i 0 −1.00000 + 0.505408i
71.4 1.41203 0.0786378i 0 1.98763 0.222077i −0.686141 0.396143i 0 −2.35143 + 1.35760i 2.78912 0.469882i 0 −1.00000 0.505408i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.2.h.a 8
3.b odd 2 1 36.2.h.a 8
4.b odd 2 1 inner 108.2.h.a 8
8.b even 2 1 1728.2.s.f 8
8.d odd 2 1 1728.2.s.f 8
9.c even 3 1 36.2.h.a 8
9.c even 3 1 324.2.b.b 8
9.d odd 6 1 inner 108.2.h.a 8
9.d odd 6 1 324.2.b.b 8
12.b even 2 1 36.2.h.a 8
15.d odd 2 1 900.2.r.c 8
15.e even 4 2 900.2.o.a 16
24.f even 2 1 576.2.s.f 8
24.h odd 2 1 576.2.s.f 8
36.f odd 6 1 36.2.h.a 8
36.f odd 6 1 324.2.b.b 8
36.h even 6 1 inner 108.2.h.a 8
36.h even 6 1 324.2.b.b 8
45.j even 6 1 900.2.r.c 8
45.k odd 12 2 900.2.o.a 16
60.h even 2 1 900.2.r.c 8
60.l odd 4 2 900.2.o.a 16
72.j odd 6 1 1728.2.s.f 8
72.j odd 6 1 5184.2.c.j 8
72.l even 6 1 1728.2.s.f 8
72.l even 6 1 5184.2.c.j 8
72.n even 6 1 576.2.s.f 8
72.n even 6 1 5184.2.c.j 8
72.p odd 6 1 576.2.s.f 8
72.p odd 6 1 5184.2.c.j 8
180.p odd 6 1 900.2.r.c 8
180.x even 12 2 900.2.o.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a 8 3.b odd 2 1
36.2.h.a 8 9.c even 3 1
36.2.h.a 8 12.b even 2 1
36.2.h.a 8 36.f odd 6 1
108.2.h.a 8 1.a even 1 1 trivial
108.2.h.a 8 4.b odd 2 1 inner
108.2.h.a 8 9.d odd 6 1 inner
108.2.h.a 8 36.h even 6 1 inner
324.2.b.b 8 9.c even 3 1
324.2.b.b 8 9.d odd 6 1
324.2.b.b 8 36.f odd 6 1
324.2.b.b 8 36.h even 6 1
576.2.s.f 8 24.f even 2 1
576.2.s.f 8 24.h odd 2 1
576.2.s.f 8 72.n even 6 1
576.2.s.f 8 72.p odd 6 1
900.2.o.a 16 15.e even 4 2
900.2.o.a 16 45.k odd 12 2
900.2.o.a 16 60.l odd 4 2
900.2.o.a 16 180.x even 12 2
900.2.r.c 8 15.d odd 2 1
900.2.r.c 8 45.j even 6 1
900.2.r.c 8 60.h even 2 1
900.2.r.c 8 180.p odd 6 1
1728.2.s.f 8 8.b even 2 1
1728.2.s.f 8 8.d odd 2 1
1728.2.s.f 8 72.j odd 6 1
1728.2.s.f 8 72.l even 6 1
5184.2.c.j 8 72.j odd 6 1
5184.2.c.j 8 72.l even 6 1
5184.2.c.j 8 72.n even 6 1
5184.2.c.j 8 72.p odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(108, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 5 T^{2} - 6 T^{3} + 6 T^{4} - 12 T^{5} + 20 T^{6} - 24 T^{7} + 16 T^{8}$$
$3$ 1
$5$ $$( 1 - 3 T + 11 T^{2} - 24 T^{3} + 54 T^{4} - 120 T^{5} + 275 T^{6} - 375 T^{7} + 625 T^{8} )^{2}$$
$7$ $$1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 76342 T^{10} + 434581 T^{12} + 2235331 T^{14} + 5764801 T^{16}$$
$11$ $$1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 863456 T^{10} + 8184319 T^{12} - 56689952 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + T - 17 T^{2} - 8 T^{3} + 142 T^{4} - 104 T^{5} - 2873 T^{6} + 2197 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 61 T^{2} + 1500 T^{4} - 17629 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 49 T^{2} + 1248 T^{4} - 17689 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 60040442 T^{10} + 950619877 T^{12} - 11398763453 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 + 3 T + 59 T^{2} + 168 T^{3} + 2382 T^{4} + 4872 T^{5} + 49619 T^{6} + 73167 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 12790910 T^{10} + 1242135745 T^{12} + 48812702455 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 + 2 T + 42 T^{2} + 74 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 12 T + 131 T^{2} + 996 T^{3} + 7176 T^{4} + 40836 T^{5} + 220211 T^{6} + 827052 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 117270976 T^{10} - 2027348993 T^{12} + 404567235136 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 428735974 T^{10} + 10017985093 T^{12} - 571298412437 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 - 136 T^{2} + 9054 T^{4} - 382024 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 625549624 T^{10} - 7476411737 T^{12} - 2362109883896 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 488 T^{5} - 420473 T^{6} + 226981 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 4573034080 T^{10} + 206649745855 T^{12} + 14473341147040 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 + 140 T^{2} + 10230 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - T + 138 T^{2} - 73 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 2367024070 T^{10} - 99517456955 T^{12} + 27955057384915 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 18577166738 T^{10} + 1084944676381 T^{12} - 72253822514549 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 184 T^{2} + 21006 T^{4} - 1457464 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 25414 T^{5} - 555131 T^{6} - 1825346 T^{7} + 88529281 T^{8} )^{2}$$