# Properties

 Label 1040.2.dh.a Level $1040$ Weight $2$ Character orbit 1040.dh Analytic conductor $8.304$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.dh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 8 x^{10} + 54 x^{8} - 78 x^{6} + 92 x^{4} - 10 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{10} ) q^{3} + ( \beta_{6} + \beta_{7} ) q^{5} + \beta_{2} q^{7} + ( 1 - \beta_{3} + \beta_{5} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{10} ) q^{3} + ( \beta_{6} + \beta_{7} ) q^{5} + \beta_{2} q^{7} + ( 1 - \beta_{3} + \beta_{5} - \beta_{7} ) q^{9} + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} ) q^{11} + ( 1 - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} ) q^{13} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{15} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{17} + ( -2 - \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{19} + ( 1 - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{21} + ( \beta_{1} - \beta_{10} ) q^{23} + ( -1 - \beta_{2} - \beta_{4} - \beta_{8} - \beta_{11} ) q^{25} + ( \beta_{2} + \beta_{11} ) q^{27} -3 \beta_{5} q^{29} + ( 2 - \beta_{2} + 2 \beta_{4} + \beta_{10} - \beta_{11} ) q^{31} + ( 2 \beta_{1} + \beta_{2} ) q^{33} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - \beta_{9} ) q^{35} + ( -1 + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{39} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{41} + ( 2 \beta_{1} + \beta_{2} ) q^{43} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{5} + \beta_{9} ) q^{45} + ( -2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{47} + ( -1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{49} + ( -1 + \beta_{6} + \beta_{7} + \beta_{8} ) q^{51} + ( \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{55} + ( \beta_{2} + \beta_{11} ) q^{57} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - 2 \beta_{9} ) q^{59} + ( 1 - 2 \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{61} + ( 2 + 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{63} + ( \beta_{1} - \beta_{3} - 5 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{65} + ( -\beta_{1} + \beta_{10} - 2 \beta_{11} ) q^{67} + ( -4 + \beta_{3} - 4 \beta_{5} + \beta_{7} ) q^{69} + ( 2 - \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{71} + ( -2 \beta_{2} + \beta_{6} + \beta_{7} - \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{73} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{75} + ( -2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} ) q^{77} + ( 8 + \beta_{2} - 2 \beta_{4} - \beta_{10} + \beta_{11} ) q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{81} + ( \beta_{2} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{83} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{85} -3 \beta_{1} q^{87} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{89} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{91} + ( 2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{93} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{5} - 3 \beta_{7} + \beta_{9} ) q^{95} + ( -2 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{97} + ( -8 - \beta_{2} + 2 \beta_{4} + \beta_{10} - \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 6q^{5} + 6q^{9} + O(q^{10})$$ $$12q - 6q^{5} + 6q^{9} + 4q^{15} - 12q^{19} - 8q^{21} - 2q^{25} + 18q^{29} + 16q^{31} - 10q^{35} + 32q^{39} + 14q^{41} - 29q^{45} + 6q^{49} - 24q^{51} + 26q^{55} + 4q^{59} + 6q^{61} + 23q^{65} - 24q^{69} + 12q^{71} - 2q^{75} + 104q^{79} + 14q^{81} + 21q^{85} + 20q^{89} + 44q^{91} - 20q^{95} - 104q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 8 x^{10} + 54 x^{8} - 78 x^{6} + 92 x^{4} - 10 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$108 \nu^{11} - 729 \nu^{9} + 4768 \nu^{7} - 1242 \nu^{5} + 135 \nu^{3} + 9934 \nu$$$$)/1222$$ $$\beta_{2}$$ $$=$$ $$($$$$108 \nu^{11} - 729 \nu^{9} + 4768 \nu^{7} - 1242 \nu^{5} + 135 \nu^{3} + 12378 \nu$$$$)/1222$$ $$\beta_{3}$$ $$=$$ $$($$$$-92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + 1058 \nu^{5} - 26893 \nu^{4} - 115 \nu^{3} + 28014 \nu^{2} - 5181 \nu - 3045$$$$)/2444$$ $$\beta_{4}$$ $$=$$ $$($$$$135 \nu^{11} - 16 \nu^{10} - 1064 \nu^{9} + 108 \nu^{8} + 7182 \nu^{7} - 729 \nu^{6} - 9801 \nu^{5} + 184 \nu^{4} + 12236 \nu^{3} - 20 \nu^{2} - 108 \nu - 3531$$$$)/1222$$ $$\beta_{5}$$ $$=$$ $$($$$$-135 \nu^{10} + 1064 \nu^{8} - 7182 \nu^{6} + 9801 \nu^{4} - 12236 \nu^{2} + 108$$$$)/1222$$ $$\beta_{6}$$ $$=$$ $$($$$$563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} - 46495 \nu^{5} + 47553 \nu^{4} + 52486 \nu^{3} - 52601 \nu^{2} - 5705 \nu + 524$$$$)/2444$$ $$\beta_{7}$$ $$=$$ $$($$$$92 \nu^{11} + 563 \nu^{10} - 621 \nu^{9} - 4564 \nu^{8} + 4039 \nu^{7} + 30807 \nu^{6} - 1058 \nu^{5} - 46495 \nu^{4} + 115 \nu^{3} + 52486 \nu^{2} + 5181 \nu - 5705$$$$)/2444$$ $$\beta_{8}$$ $$=$$ $$($$$$-655 \nu^{11} - 92 \nu^{10} + 5185 \nu^{9} + 621 \nu^{8} - 34846 \nu^{7} - 4039 \nu^{6} + 47553 \nu^{5} + 1058 \nu^{4} - 52601 \nu^{3} - 115 \nu^{2} + 524 \nu - 5181$$$$)/2444$$ $$\beta_{9}$$ $$=$$ $$($$$$389 \nu^{10} - 3084 \nu^{8} + 20817 \nu^{6} - 29219 \nu^{4} + 35466 \nu^{2} - 1222 \nu - 3855$$$$)/1222$$ $$\beta_{10}$$ $$=$$ $$($$$$1195 \nu^{11} - 9441 \nu^{9} + 63574 \nu^{7} - 86757 \nu^{5} + 100323 \nu^{3} - 956 \nu$$$$)/1222$$ $$\beta_{11}$$ $$=$$ $$($$$$1357 \nu^{11} - 10840 \nu^{9} + 73170 \nu^{7} - 105117 \nu^{5} + 124660 \nu^{3} - 13550 \nu$$$$)/1222$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} - 2 \beta_{9} - 6 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{11} - 3 \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$($$$$-14 \beta_{9} + 2 \beta_{7} - 34 \beta_{5} + 2 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} - 34$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$23 \beta_{11} - 37 \beta_{10} - 16 \beta_{8} + 16 \beta_{6} - 16 \beta_{5} + 16 \beta_{3} + 37 \beta_{1} - 16$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$23 \beta_{11} - 23 \beta_{10} + 8 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} - 46 \beta_{4} + 23 \beta_{2} - 99$$ $$\nu^{7}$$ $$=$$ $$($$$$-108 \beta_{7} - 108 \beta_{5} + 108 \beta_{3} - 145 \beta_{2} + 237 \beta_{1} - 108$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$299 \beta_{11} - 299 \beta_{10} + 598 \beta_{9} + 108 \beta_{8} + 108 \beta_{6} + 1378 \beta_{5} - 598 \beta_{4} - 108 \beta_{3} + 598 \beta_{2} - 299 \beta_{1} + 108$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$-467 \beta_{11} + 766 \beta_{10} + 353 \beta_{8} - 353 \beta_{7} - 353 \beta_{6} - 467 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$($$$$3878 \beta_{9} - 706 \beta_{7} + 8918 \beta_{5} - 706 \beta_{3} + 1939 \beta_{2} - 1939 \beta_{1} + 8918$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-6045 \beta_{11} + 9923 \beta_{10} + 4584 \beta_{8} - 4584 \beta_{6} + 4584 \beta_{5} - 4584 \beta_{3} - 9923 \beta_{1} + 4584$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 - \beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.286513 + 0.165418i −2.20467 − 1.27287i −1.02826 − 0.593667i 1.02826 + 0.593667i 2.20467 + 1.27287i −0.286513 − 0.165418i 0.286513 − 0.165418i −2.20467 + 1.27287i −1.02826 + 0.593667i 1.02826 − 0.593667i 2.20467 − 1.27287i −0.286513 + 0.165418i
0 −2.33117 + 1.34590i 0 −2.12291 + 0.702335i 0 2.90420 + 1.67674i 0 2.12291 3.67698i 0
289.2 0 −1.86449 + 1.07646i 0 −0.817544 2.08125i 0 −2.54486 1.46928i 0 0.817544 1.41603i 0
289.3 0 −0.298874 + 0.172555i 0 1.44045 + 1.71029i 0 −1.75765 1.01478i 0 −1.44045 + 2.49493i 0
289.4 0 0.298874 0.172555i 0 1.44045 1.71029i 0 1.75765 + 1.01478i 0 −1.44045 + 2.49493i 0
289.5 0 1.86449 1.07646i 0 −0.817544 + 2.08125i 0 2.54486 + 1.46928i 0 0.817544 1.41603i 0
289.6 0 2.33117 1.34590i 0 −2.12291 0.702335i 0 −2.90420 1.67674i 0 2.12291 3.67698i 0
529.1 0 −2.33117 1.34590i 0 −2.12291 0.702335i 0 2.90420 1.67674i 0 2.12291 + 3.67698i 0
529.2 0 −1.86449 1.07646i 0 −0.817544 + 2.08125i 0 −2.54486 + 1.46928i 0 0.817544 + 1.41603i 0
529.3 0 −0.298874 0.172555i 0 1.44045 1.71029i 0 −1.75765 + 1.01478i 0 −1.44045 2.49493i 0
529.4 0 0.298874 + 0.172555i 0 1.44045 + 1.71029i 0 1.75765 1.01478i 0 −1.44045 2.49493i 0
529.5 0 1.86449 + 1.07646i 0 −0.817544 2.08125i 0 2.54486 1.46928i 0 0.817544 + 1.41603i 0
529.6 0 2.33117 + 1.34590i 0 −2.12291 + 0.702335i 0 −2.90420 + 1.67674i 0 2.12291 + 3.67698i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.dh.a 12
4.b odd 2 1 65.2.n.a 12
5.b even 2 1 inner 1040.2.dh.a 12
12.b even 2 1 585.2.bs.a 12
13.c even 3 1 inner 1040.2.dh.a 12
20.d odd 2 1 65.2.n.a 12
20.e even 4 2 325.2.e.e 12
52.b odd 2 1 845.2.n.e 12
52.f even 4 2 845.2.l.f 24
52.i odd 6 1 845.2.b.e 6
52.i odd 6 1 845.2.n.e 12
52.j odd 6 1 65.2.n.a 12
52.j odd 6 1 845.2.b.d 6
52.l even 12 2 845.2.d.d 12
52.l even 12 2 845.2.l.f 24
60.h even 2 1 585.2.bs.a 12
65.n even 6 1 inner 1040.2.dh.a 12
156.p even 6 1 585.2.bs.a 12
260.g odd 2 1 845.2.n.e 12
260.u even 4 2 845.2.l.f 24
260.v odd 6 1 65.2.n.a 12
260.v odd 6 1 845.2.b.d 6
260.w odd 6 1 845.2.b.e 6
260.w odd 6 1 845.2.n.e 12
260.bc even 12 2 845.2.d.d 12
260.bc even 12 2 845.2.l.f 24
260.bg even 12 2 4225.2.a.bq 6
260.bj even 12 2 325.2.e.e 12
260.bj even 12 2 4225.2.a.br 6
780.br even 6 1 585.2.bs.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 4.b odd 2 1
65.2.n.a 12 20.d odd 2 1
65.2.n.a 12 52.j odd 6 1
65.2.n.a 12 260.v odd 6 1
325.2.e.e 12 20.e even 4 2
325.2.e.e 12 260.bj even 12 2
585.2.bs.a 12 12.b even 2 1
585.2.bs.a 12 60.h even 2 1
585.2.bs.a 12 156.p even 6 1
585.2.bs.a 12 780.br even 6 1
845.2.b.d 6 52.j odd 6 1
845.2.b.d 6 260.v odd 6 1
845.2.b.e 6 52.i odd 6 1
845.2.b.e 6 260.w odd 6 1
845.2.d.d 12 52.l even 12 2
845.2.d.d 12 260.bc even 12 2
845.2.l.f 24 52.f even 4 2
845.2.l.f 24 52.l even 12 2
845.2.l.f 24 260.u even 4 2
845.2.l.f 24 260.bc even 12 2
845.2.n.e 12 52.b odd 2 1
845.2.n.e 12 52.i odd 6 1
845.2.n.e 12 260.g odd 2 1
845.2.n.e 12 260.w odd 6 1
1040.2.dh.a 12 1.a even 1 1 trivial
1040.2.dh.a 12 5.b even 2 1 inner
1040.2.dh.a 12 13.c even 3 1 inner
1040.2.dh.a 12 65.n even 6 1 inner
4225.2.a.bq 6 260.bg even 12 2
4225.2.a.br 6 260.bj even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 12 T_{3}^{10} + 109 T_{3}^{8} - 412 T_{3}^{6} + 1177 T_{3}^{4} - 140 T_{3}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$16 - 140 T^{2} + 1177 T^{4} - 412 T^{6} + 109 T^{8} - 12 T^{10} + T^{12}$$
$5$ $$( 125 + 75 T + 25 T^{2} + 10 T^{3} + 5 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$7$ $$160000 - 71600 T^{2} + 22441 T^{4} - 3496 T^{6} + 397 T^{8} - 24 T^{10} + T^{12}$$
$11$ $$( 64 + 104 T + 169 T^{2} + 16 T^{3} + 13 T^{4} + T^{6} )^{2}$$
$13$ $$4826809 - 428415 T^{2} + 6591 T^{4} - 322 T^{6} + 39 T^{8} - 15 T^{10} + T^{12}$$
$17$ $$28561 - 27547 T^{2} + 20654 T^{4} - 5367 T^{6} + 1062 T^{8} - 35 T^{10} + T^{12}$$
$19$ $$( 100 + 10 T + 61 T^{2} + 14 T^{3} + 37 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$23$ $$16 - 140 T^{2} + 1177 T^{4} - 412 T^{6} + 109 T^{8} - 12 T^{10} + T^{12}$$
$29$ $$( 9 - 3 T + T^{2} )^{6}$$
$31$ $$( -40 - 40 T - 4 T^{2} + T^{3} )^{4}$$
$37$ $$28561 - 27547 T^{2} + 20654 T^{4} - 5367 T^{6} + 1062 T^{8} - 35 T^{10} + T^{12}$$
$41$ $$( 25 + 145 T + 806 T^{2} + 213 T^{3} + 78 T^{4} - 7 T^{5} + T^{6} )^{2}$$
$43$ $$65536 - 72448 T^{2} + 59609 T^{4} - 22128 T^{6} + 6117 T^{8} - 80 T^{10} + T^{12}$$
$47$ $$( 270400 + 14640 T^{2} + 236 T^{4} + T^{6} )^{2}$$
$53$ $$( 400 + 1040 T^{2} + 171 T^{4} + T^{6} )^{2}$$
$59$ $$( 18496 - 7480 T + 3297 T^{2} - 162 T^{3} + 59 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$61$ $$( 13225 - 5635 T + 2746 T^{2} - 83 T^{3} + 58 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$67$ $$406586896 - 52486892 T^{2} + 4759209 T^{4} - 219972 T^{6} + 7397 T^{8} - 100 T^{10} + T^{12}$$
$71$ $$( 676 - 26 T + 157 T^{2} - 46 T^{3} + 37 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$73$ $$( 250000 + 13900 T^{2} + 215 T^{4} + T^{6} )^{2}$$
$79$ $$( -160 + 180 T - 26 T^{2} + T^{3} )^{4}$$
$83$ $$( 640000 + 23600 T^{2} + 276 T^{4} + T^{6} )^{2}$$
$89$ $$( 2515396 - 249002 T + 40509 T^{2} - 1602 T^{3} + 257 T^{4} - 10 T^{5} + T^{6} )^{2}$$
$97$ $$41740124416 - 2934418352 T^{2} + 149090649 T^{4} - 3613032 T^{6} + 64037 T^{8} - 280 T^{10} + T^{12}$$