# Properties

 Label 770.2.n.e Level $770$ Weight $2$ Character orbit 770.n Analytic conductor $6.148$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.682515625.5 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} + 2 x^{5} + 19 x^{4} + 28 x^{3} + 100 x^{2} + 88 x + 121$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} + 2 x^{5} + 19 x^{4} + 28 x^{3} + 100 x^{2} + 88 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + 319181 \nu + 440220$$$$)/1168519$$ $$\beta_{3}$$ $$=$$ $$($$$$5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + 501961 \nu + 528473$$$$)/1168519$$ $$\beta_{4}$$ $$=$$ $$($$$$7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + 18601 \nu - 701074$$$$)/1168519$$ $$\beta_{5}$$ $$=$$ $$($$$$8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + 2179908 \nu + 2809884$$$$)/1168519$$ $$\beta_{6}$$ $$=$$ $$($$$$-11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} - 1085964 \nu - 2305776$$$$)/1168519$$ $$\beta_{7}$$ $$=$$ $$($$$$-13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} - 742824 \nu + 665808$$$$)/1168519$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3} - 5 \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 6 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 10 \beta_{2} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{7} + 10 \beta_{6} + 13 \beta_{5} + 13 \beta_{4} + 14 \beta_{3} - 13 \beta_{2} - 10 \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$43 \beta_{7} + 25 \beta_{5} + 49 \beta_{4} + 18 \beta_{2} - 25 \beta_{1} - 62$$ $$\nu^{6}$$ $$=$$ $$97 \beta_{7} - 92 \beta_{6} + 92 \beta_{4} - 97 \beta_{3} + 221 \beta_{2} - 44 \beta_{1} - 221$$ $$\nu^{7}$$ $$=$$ $$-449 \beta_{6} - 260 \beta_{5} - 412 \beta_{3} + 896 \beta_{2} - 412$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/770\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 2.51217 + 1.82520i −1.20316 − 0.874145i 2.51217 − 1.82520i −1.20316 + 0.874145i −0.390899 − 1.20306i 0.581882 + 1.79085i −0.390899 + 1.20306i 0.581882 − 1.79085i
−0.309017 0.951057i −1.55261 1.12804i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.593044 + 1.82520i 0.809017 0.587785i 0.809017 + 0.587785i 0.211078 + 0.649631i 1.00000
71.2 −0.309017 0.951057i 0.743592 + 0.540251i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.284027 0.874145i 0.809017 0.587785i 0.809017 + 0.587785i −0.665993 2.04972i 1.00000
141.1 −0.309017 + 0.951057i −1.55261 + 1.12804i −0.809017 0.587785i −0.309017 0.951057i −0.593044 1.82520i 0.809017 + 0.587785i 0.809017 0.587785i 0.211078 0.649631i 1.00000
141.2 −0.309017 + 0.951057i 0.743592 0.540251i −0.809017 0.587785i −0.309017 0.951057i 0.284027 + 0.874145i 0.809017 + 0.587785i 0.809017 0.587785i −0.665993 + 2.04972i 1.00000
421.1 0.809017 0.587785i −0.632489 1.94660i 0.309017 0.951057i 0.809017 + 0.587785i −1.65588 1.20306i −0.309017 + 0.951057i −0.309017 0.951057i −0.962157 + 0.699048i 1.00000
421.2 0.809017 0.587785i 0.941506 + 2.89766i 0.309017 0.951057i 0.809017 + 0.587785i 2.46489 + 1.79085i −0.309017 + 0.951057i −0.309017 0.951057i −5.08293 + 3.69296i 1.00000
631.1 0.809017 + 0.587785i −0.632489 + 1.94660i 0.309017 + 0.951057i 0.809017 0.587785i −1.65588 + 1.20306i −0.309017 0.951057i −0.309017 + 0.951057i −0.962157 0.699048i 1.00000
631.2 0.809017 + 0.587785i 0.941506 2.89766i 0.309017 + 0.951057i 0.809017 0.587785i 2.46489 1.79085i −0.309017 0.951057i −0.309017 + 0.951057i −5.08293 3.69296i 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.e 8
11.c even 5 1 inner 770.2.n.e 8
11.c even 5 1 8470.2.a.cq 4
11.d odd 10 1 8470.2.a.ct 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.e 8 1.a even 1 1 trivial
770.2.n.e 8 11.c even 5 1 inner
8470.2.a.cq 4 11.c even 5 1
8470.2.a.ct 4 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$3$ $$121 - 99 T + 20 T^{2} + 29 T^{3} + 49 T^{4} + 19 T^{5} + 10 T^{6} + T^{7} + T^{8}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$11$ $$14641 - 10648 T + 2783 T^{2} - 726 T^{3} + 275 T^{4} - 66 T^{5} + 23 T^{6} - 8 T^{7} + T^{8}$$
$13$ $$16 - 72 T + 140 T^{2} - 82 T^{3} + 199 T^{4} - 113 T^{5} + 40 T^{6} - 8 T^{7} + T^{8}$$
$17$ $$361 - 361 T + 190 T^{2} - 19 T^{3} + 289 T^{4} + 151 T^{5} + 50 T^{6} + 9 T^{7} + T^{8}$$
$19$ $$55696 - 26904 T + 12832 T^{2} - 2172 T^{3} + 505 T^{4} + 12 T^{5} + 42 T^{6} + 9 T^{7} + T^{8}$$
$23$ $$( 2 + T )^{8}$$
$29$ $$1936 - 2640 T + 2248 T^{2} - 1650 T^{3} + 1339 T^{4} - 465 T^{5} + 72 T^{6} + T^{8}$$
$31$ $$( 1936 - 704 T + 136 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$37$ $$4096 + 7168 T + 37376 T^{2} - 4736 T^{3} + 2384 T^{4} + 152 T^{5} - 16 T^{6} - 4 T^{7} + T^{8}$$
$41$ $$1048576 + 262144 T + 67584 T^{2} + 14048 T^{3} + 5609 T^{4} - 439 T^{5} + 66 T^{6} - 8 T^{7} + T^{8}$$
$43$ $$( 1076 - 298 T - 67 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$47$ $$112444816 - 106040 T + 1140232 T^{2} - 53460 T^{3} + 11189 T^{4} + 555 T^{5} + 118 T^{6} + T^{8}$$
$53$ $$4096 + 5120 T + 3328 T^{2} + 320 T^{3} + 144 T^{4} - 320 T^{5} + 192 T^{6} - 10 T^{7} + T^{8}$$
$59$ $$7862416 - 7043648 T + 2908936 T^{2} - 640934 T^{3} + 93609 T^{4} - 9392 T^{5} + 684 T^{6} - 31 T^{7} + T^{8}$$
$61$ $$952576 - 679296 T + 615104 T^{2} - 154112 T^{3} + 23824 T^{4} - 2944 T^{5} + 376 T^{6} - 28 T^{7} + T^{8}$$
$67$ $$( -284 - 198 T + 19 T^{2} + 13 T^{3} + T^{4} )^{2}$$
$71$ $$1860496 - 373736 T + 342032 T^{2} - 149528 T^{3} + 37505 T^{4} - 5587 T^{5} + 572 T^{6} - 34 T^{7} + T^{8}$$
$73$ $$755161 - 818598 T + 345161 T^{2} + 10476 T^{3} + 19834 T^{4} + 858 T^{5} + 264 T^{6} + 24 T^{7} + T^{8}$$
$79$ $$1936 - 6600 T + 8988 T^{2} - 1890 T^{3} + 859 T^{4} - 255 T^{5} + 52 T^{6} + T^{8}$$
$83$ $$6241 - 16037 T + 82446 T^{2} - 31481 T^{3} + 3519 T^{4} + 907 T^{5} + 224 T^{6} + 21 T^{7} + T^{8}$$
$89$ $$( 1084 - 646 T - 105 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$97$ $$( 21025 - 4350 T + 460 T^{2} - 25 T^{3} + T^{4} )^{2}$$
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