# Properties

 Label 210.2.t.e Level 210 Weight 2 Character orbit 210.t Analytic conductor 1.677 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 14 \nu^{4} + 7 \nu^{2} - 36$$$$)/63$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 144$$$$)/63$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} - 7 \nu^{5} - 35 \nu^{3} + 180 \nu$$$$)/189$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{5} + 7 \nu^{3} - 36 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} + 7 \nu^{2} + 162$$$$)/63$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - 2 \beta_{4} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + 3 \beta_{3} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 4 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-5 \beta_{6} - 7 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{7} + 7 \beta_{2} + 22$$ $$\nu^{7}$$ $$=$$ $$-21 \beta_{5} - 21 \beta_{3} + 29 \beta_{1}$$

## Character values

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

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 −1.01575 + 1.40294i −1.72286 + 0.178197i 1.72286 − 0.178197i 1.01575 − 1.40294i −1.01575 − 1.40294i −1.72286 − 0.178197i 1.72286 + 0.178197i 1.01575 + 1.40294i
−0.500000 0.866025i −1.72286 0.178197i −0.500000 + 0.866025i 2.20711 + 0.358719i 0.707107 + 1.58114i −1.41421 + 2.23607i 1.00000 2.93649 + 0.614017i −0.792893 2.09077i
59.2 −0.500000 0.866025i −1.01575 1.40294i −0.500000 + 0.866025i 0.792893 2.09077i −0.707107 + 1.58114i 1.41421 2.23607i 1.00000 −0.936492 + 2.85008i −2.20711 + 0.358719i
59.3 −0.500000 0.866025i 1.01575 + 1.40294i −0.500000 + 0.866025i 2.20711 + 0.358719i 0.707107 1.58114i −1.41421 2.23607i 1.00000 −0.936492 + 2.85008i −0.792893 2.09077i
59.4 −0.500000 0.866025i 1.72286 + 0.178197i −0.500000 + 0.866025i 0.792893 2.09077i −0.707107 1.58114i 1.41421 + 2.23607i 1.00000 2.93649 + 0.614017i −2.20711 + 0.358719i
89.1 −0.500000 + 0.866025i −1.72286 + 0.178197i −0.500000 0.866025i 2.20711 0.358719i 0.707107 1.58114i −1.41421 2.23607i 1.00000 2.93649 0.614017i −0.792893 + 2.09077i
89.2 −0.500000 + 0.866025i −1.01575 + 1.40294i −0.500000 0.866025i 0.792893 + 2.09077i −0.707107 1.58114i 1.41421 + 2.23607i 1.00000 −0.936492 2.85008i −2.20711 0.358719i
89.3 −0.500000 + 0.866025i 1.01575 1.40294i −0.500000 0.866025i 2.20711 0.358719i 0.707107 + 1.58114i −1.41421 + 2.23607i 1.00000 −0.936492 2.85008i −0.792893 + 2.09077i
89.4 −0.500000 + 0.866025i 1.72286 0.178197i −0.500000 0.866025i 0.792893 + 2.09077i −0.707107 + 1.58114i 1.41421 2.23607i 1.00000 2.93649 0.614017i −2.20711 0.358719i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.t.e 8
3.b odd 2 1 210.2.t.f yes 8
5.b even 2 1 210.2.t.f yes 8
5.c odd 4 2 1050.2.s.i 16
7.c even 3 1 1470.2.d.f 8
7.d odd 6 1 inner 210.2.t.e 8
7.d odd 6 1 1470.2.d.f 8
15.d odd 2 1 inner 210.2.t.e 8
15.e even 4 2 1050.2.s.i 16
21.g even 6 1 210.2.t.f yes 8
21.g even 6 1 1470.2.d.e 8
21.h odd 6 1 1470.2.d.e 8
35.i odd 6 1 210.2.t.f yes 8
35.i odd 6 1 1470.2.d.e 8
35.j even 6 1 1470.2.d.e 8
35.k even 12 2 1050.2.s.i 16
105.o odd 6 1 1470.2.d.f 8
105.p even 6 1 inner 210.2.t.e 8
105.p even 6 1 1470.2.d.f 8
105.w odd 12 2 1050.2.s.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.e 8 1.a even 1 1 trivial
210.2.t.e 8 7.d odd 6 1 inner
210.2.t.e 8 15.d odd 2 1 inner
210.2.t.e 8 105.p even 6 1 inner
210.2.t.f yes 8 3.b odd 2 1
210.2.t.f yes 8 5.b even 2 1
210.2.t.f yes 8 21.g even 6 1
210.2.t.f yes 8 35.i odd 6 1
1050.2.s.i 16 5.c odd 4 2
1050.2.s.i 16 15.e even 4 2
1050.2.s.i 16 35.k even 12 2
1050.2.s.i 16 105.w odd 12 2
1470.2.d.e 8 21.g even 6 1
1470.2.d.e 8 21.h odd 6 1
1470.2.d.e 8 35.i odd 6 1
1470.2.d.e 8 35.j even 6 1
1470.2.d.f 8 7.c even 3 1
1470.2.d.f 8 7.d odd 6 1
1470.2.d.f 8 105.o odd 6 1
1470.2.d.f 8 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$:

 $$T_{11}^{8} - 22 T_{11}^{6} + 483 T_{11}^{4} - 22 T_{11}^{2} + 1$$ $$T_{13}^{4} - 46 T_{13}^{2} + 49$$ $$T_{23}^{4} - 2 T_{23}^{3} + 33 T_{23}^{2} + 58 T_{23} + 841$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{4}$$
$3$ $$1 - 4 T^{2} + 7 T^{4} - 36 T^{6} + 81 T^{8}$$
$5$ $$( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} )^{2}$$
$7$ $$( 1 + 6 T^{2} + 49 T^{4} )^{2}$$
$11$ $$1 + 22 T^{2} + 241 T^{4} + 22 T^{6} - 14156 T^{8} + 2662 T^{10} + 3528481 T^{12} + 38974342 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + 6 T^{2} - 133 T^{4} + 1014 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 24 T^{2} + 287 T^{4} + 6936 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 6 T + 43 T^{2} - 186 T^{3} + 828 T^{4} - 3534 T^{5} + 15523 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 2 T - 13 T^{2} + 58 T^{3} - 332 T^{4} + 1334 T^{5} - 6877 T^{6} - 24334 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 64 T^{2} + 2226 T^{4} - 53824 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 12 T + 112 T^{2} + 768 T^{3} + 4623 T^{4} + 23808 T^{5} + 107632 T^{6} + 357492 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 + 90 T^{2} + 3817 T^{4} + 139050 T^{6} + 5381028 T^{8} + 190359450 T^{10} + 7153672537 T^{12} + 230915376810 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 118 T^{2} + 6363 T^{4} + 198358 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 120 T^{2} + 6818 T^{4} - 221880 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 6 T + 19 T^{2} + 42 T^{3} - 1596 T^{4} + 1974 T^{5} + 41971 T^{6} + 622938 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 5 T - 28 T^{2} + 265 T^{3} + 2809 T^{4} )^{4}$$
$59$ $$1 - 100 T^{2} + 2458 T^{4} - 58000 T^{6} + 9954403 T^{8} - 201898000 T^{10} + 29784473338 T^{12} - 4218053364100 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 + 36 T^{2} - 326 T^{4} - 264816 T^{6} - 23337981 T^{8} - 1188759024 T^{10} - 6569265446 T^{12} + 3256501758084 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 232 T^{2} + 23058 T^{4} - 1169512 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 136 T^{2} + 7534 T^{4} - 41344 T^{6} - 6797981 T^{8} - 220322176 T^{10} + 213952347694 T^{12} - 20581454775304 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 12 T - 20 T^{2} + 72 T^{3} + 9279 T^{4} + 5688 T^{5} - 124820 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 36 T^{2} - 3178 T^{4} - 248004 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$1 - 220 T^{2} + 22378 T^{4} - 2239600 T^{6} + 231025843 T^{8} - 17739871600 T^{10} + 1404045869098 T^{12} - 109335884011420 T^{14} + 3936588805702081 T^{16}$$
$97$ $$( 1 + 232 T^{2} + 27954 T^{4} + 2182888 T^{6} + 88529281 T^{8} )^{2}$$