# Properties

 Label 1232.2.q.o Level $1232$ Weight $2$ Character orbit 1232.q Analytic conductor $9.838$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.939795628203.1 Defining polynomial: $$x^{10} - x^{9} + x^{8} - 9 x^{7} + 10 x^{6} - 26 x^{5} + 87 x^{4} - 48 x^{3} - 65 x^{2} + 30 x + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 616) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{8} ) q^{3} + ( -1 + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{5} + ( 1 + \beta_{1} - \beta_{5} ) q^{7} + ( \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{8} ) q^{3} + ( -1 + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{5} + ( 1 + \beta_{1} - \beta_{5} ) q^{7} + ( \beta_{3} + \beta_{4} ) q^{9} -\beta_{7} q^{11} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{13} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{15} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{17} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{19} + ( -2 - 2 \beta_{1} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{21} + ( \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{25} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{27} + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{29} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{8} ) q^{31} -\beta_{8} q^{33} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 2 \beta_{8} ) q^{35} + ( -2 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{37} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{39} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{41} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{9} ) q^{43} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{45} + ( 3 + \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{7} - \beta_{8} - \beta_{9} ) q^{47} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{49} + ( -4 - 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{51} + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{53} + ( 1 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{57} + ( -2 - 4 \beta_{1} + 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + 2 \beta_{9} ) q^{59} + ( -2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{61} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{8} - 2 \beta_{9} ) q^{63} + ( -1 - \beta_{1} - 4 \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{65} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + 4 \beta_{8} ) q^{67} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{69} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{71} + ( -2 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} ) q^{73} + ( 3 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{75} + ( \beta_{3} - \beta_{7} - \beta_{9} ) q^{77} + ( -2 - 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{79} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{81} + ( 3 + 5 \beta_{2} + \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{9} ) q^{83} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{85} + ( 4 \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{87} + ( -8 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + 9 \beta_{7} + 3 \beta_{8} ) q^{89} + ( 5 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} - \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{91} + ( -1 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{93} + ( -1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 4 \beta_{9} ) q^{95} + ( 3 + \beta_{1} + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{97} + ( -\beta_{4} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - q^{3} - 4 q^{5} + 2 q^{7} + O(q^{10})$$ $$10 q - q^{3} - 4 q^{5} + 2 q^{7} - 5 q^{11} + 2 q^{13} - 14 q^{15} - 9 q^{17} + q^{19} - 12 q^{21} - 8 q^{23} - 5 q^{25} + 8 q^{27} + 18 q^{29} + 3 q^{31} - q^{33} + 15 q^{35} - 2 q^{37} - 7 q^{39} + 30 q^{41} - 28 q^{43} - 19 q^{45} + 11 q^{47} - 18 q^{49} - 14 q^{51} + 9 q^{53} + 8 q^{55} + 8 q^{57} + 4 q^{59} + 2 q^{61} + 28 q^{63} - 7 q^{65} - 8 q^{67} + 22 q^{69} - 30 q^{71} - 26 q^{73} + 27 q^{75} + 2 q^{77} - 3 q^{79} + 19 q^{81} + 2 q^{83} + 26 q^{85} + 14 q^{87} - 41 q^{89} + 39 q^{91} - 10 q^{93} - 19 q^{95} + 14 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + x^{8} - 9 x^{7} + 10 x^{6} - 26 x^{5} + 87 x^{4} - 48 x^{3} - 65 x^{2} + 30 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-175931 \nu^{9} + 2122814 \nu^{8} - 2291723 \nu^{7} + 1252650 \nu^{6} - 23214830 \nu^{5} + 19749496 \nu^{4} - 57379965 \nu^{3} + 200646585 \nu^{2} - 28179152 \nu - 166645188$$$$)/41020284$$ $$\beta_{2}$$ $$=$$ $$($$$$-155541 \nu^{9} + 15964 \nu^{8} - 240025 \nu^{7} + 996486 \nu^{6} - 372550 \nu^{5} + 4073080 \nu^{4} - 7882773 \nu^{3} - 1132357 \nu^{2} + 8565432 \nu - 12840162$$$$)/6836714$$ $$\beta_{3}$$ $$=$$ $$($$$$1013393 \nu^{9} - 1615910 \nu^{8} + 4712021 \nu^{7} - 8349810 \nu^{6} + 23040746 \nu^{5} - 53645260 \nu^{4} + 108969915 \nu^{3} - 191378187 \nu^{2} + 118243028 \nu + 92028048$$$$)/41020284$$ $$\beta_{4}$$ $$=$$ $$($$$$-1065671 \nu^{9} - 737758 \nu^{8} - 1671755 \nu^{7} + 6616590 \nu^{6} + 2989378 \nu^{5} + 33742060 \nu^{4} - 34920825 \nu^{3} - 36613935 \nu^{2} + 25557580 \nu - 25274400$$$$)/41020284$$ $$\beta_{5}$$ $$=$$ $$($$$$-276392 \nu^{9} - 372211 \nu^{8} - 612335 \nu^{7} + 1491771 \nu^{6} + 2225113 \nu^{5} + 8490406 \nu^{4} - 3531393 \nu^{3} - 15167460 \nu^{2} + 4530313 \nu - 131250$$$$)/10255071$$ $$\beta_{6}$$ $$=$$ $$($$$$1145905 \nu^{9} - 1178626 \nu^{8} + 950941 \nu^{7} - 9427146 \nu^{6} + 12029062 \nu^{5} - 21697700 \nu^{4} + 99152043 \nu^{3} - 43882995 \nu^{2} - 143152604 \nu + 38723880$$$$)/41020284$$ $$\beta_{7}$$ $$=$$ $$($$$$4793 \nu^{9} - 3458 \nu^{8} + 4601 \nu^{7} - 47178 \nu^{6} + 31478 \nu^{5} - 137080 \nu^{4} + 395715 \nu^{3} - 129855 \nu^{2} - 142600 \nu + 40428$$$$)/144948$$ $$\beta_{8}$$ $$=$$ $$($$$$-673853 \nu^{9} + 685446 \nu^{8} - 904397 \nu^{7} + 6820122 \nu^{6} - 6623314 \nu^{5} + 21055420 \nu^{4} - 65582787 \nu^{3} + 40792183 \nu^{2} + 8194516 \nu + 9226776$$$$)/13673428$$ $$\beta_{9}$$ $$=$$ $$($$$$1505423 \nu^{9} + 725326 \nu^{8} + 2850521 \nu^{7} - 9602418 \nu^{6} + 425318 \nu^{5} - 40881364 \nu^{4} + 69828225 \nu^{3} + 24370503 \nu^{2} - 33065602 \nu - 9415896$$$$)/20510142$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} - 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_{1} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{9} - 6 \beta_{8} - 11 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_{1} + 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} - 8 \beta_{6} - \beta_{5} + 17 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - 7 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{9} + 30 \beta_{8} + 35 \beta_{7} + 12 \beta_{6} + 15 \beta_{5} - 13 \beta_{4} + 13 \beta_{3} + 27 \beta_{2} + 5 \beta_{1} - 5$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$7 \beta_{9} - 50 \beta_{8} - 85 \beta_{7} - 24 \beta_{6} + 33 \beta_{5} - 29 \beta_{4} - 23 \beta_{3} - 51 \beta_{2} - 11 \beta_{1} - 23$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-43 \beta_{9} + 36 \beta_{8} + 3 \beta_{7} + 78 \beta_{6} - 81 \beta_{5} + 87 \beta_{4} + 93 \beta_{3} - 35 \beta_{2} + 85 \beta_{1} + 5$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$53 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} + 82 \beta_{6} + 35 \beta_{5} - 133 \beta_{4} - \beta_{3} + 331 \beta_{2} - 11 \beta_{1} + 435$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$401 \beta_{9} - 104 \beta_{8} + 207 \beta_{7} - 730 \beta_{6} + 275 \beta_{5} + 357 \beta_{4} - 633 \beta_{3} - 467 \beta_{2} - 627 \beta_{1} - 895$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$$.

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{7}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 −0.574432 + 0.269593i 1.13198 + 0.475002i 1.60172 − 0.387640i −0.0561490 + 2.08099i −1.60312 − 1.57192i −0.574432 − 0.269593i 1.13198 − 0.475002i 1.60172 + 0.387640i −0.0561490 − 2.08099i −1.60312 + 1.57192i
0 −1.29867 2.24937i 0 −1.07443 + 1.86097i 0 −0.726377 2.54409i 0 −1.87311 + 3.24431i 0
177.2 0 −0.746498 1.29297i 0 0.631979 1.09462i 0 2.25927 + 1.37685i 0 0.385481 0.667672i 0
177.3 0 −0.142113 0.246147i 0 1.10172 1.90824i 0 −1.07405 + 2.41794i 0 1.45961 2.52811i 0
177.4 0 0.666828 + 1.15498i 0 −0.556149 + 0.963278i 0 2.01243 1.71759i 0 0.610680 1.05773i 0
177.5 0 1.02046 + 1.76748i 0 −2.10312 + 3.64271i 0 −1.47127 + 2.19895i 0 −0.582662 + 1.00920i 0
529.1 0 −1.29867 + 2.24937i 0 −1.07443 1.86097i 0 −0.726377 + 2.54409i 0 −1.87311 3.24431i 0
529.2 0 −0.746498 + 1.29297i 0 0.631979 + 1.09462i 0 2.25927 1.37685i 0 0.385481 + 0.667672i 0
529.3 0 −0.142113 + 0.246147i 0 1.10172 + 1.90824i 0 −1.07405 2.41794i 0 1.45961 + 2.52811i 0
529.4 0 0.666828 1.15498i 0 −0.556149 0.963278i 0 2.01243 + 1.71759i 0 0.610680 + 1.05773i 0
529.5 0 1.02046 1.76748i 0 −2.10312 3.64271i 0 −1.47127 2.19895i 0 −0.582662 1.00920i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.5 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.o 10
4.b odd 2 1 616.2.q.f 10
7.c even 3 1 inner 1232.2.q.o 10
7.c even 3 1 8624.2.a.dc 5
7.d odd 6 1 8624.2.a.db 5
28.f even 6 1 4312.2.a.bg 5
28.g odd 6 1 616.2.q.f 10
28.g odd 6 1 4312.2.a.bf 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.f 10 4.b odd 2 1
616.2.q.f 10 28.g odd 6 1
1232.2.q.o 10 1.a even 1 1 trivial
1232.2.q.o 10 7.c even 3 1 inner
4312.2.a.bf 5 28.g odd 6 1
4312.2.a.bg 5 28.f even 6 1
8624.2.a.db 5 7.d odd 6 1
8624.2.a.dc 5 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{10} + \cdots$$ $$T_{13}^{5} - T_{13}^{4} - 39 T_{13}^{3} + 10 T_{13}^{2} + 12 T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$9 + 30 T + 112 T^{2} + 2 T^{3} + 83 T^{4} + 11 T^{5} + 43 T^{6} + T^{7} + 8 T^{8} + T^{9} + T^{10}$$
$5$ $$784 + 280 T + 800 T^{2} + 142 T^{3} + 583 T^{4} + 123 T^{5} + 139 T^{6} + 22 T^{7} + 23 T^{8} + 4 T^{9} + T^{10}$$
$7$ $$16807 - 4802 T + 3773 T^{2} - 2205 T^{3} + 1099 T^{4} - 267 T^{5} + 157 T^{6} - 45 T^{7} + 11 T^{8} - 2 T^{9} + T^{10}$$
$11$ $$( 1 + T + T^{2} )^{5}$$
$13$ $$( 1 + 12 T + 10 T^{2} - 39 T^{3} - T^{4} + T^{5} )^{2}$$
$17$ $$71824 - 38592 T + 54772 T^{2} + 17216 T^{3} + 14005 T^{4} + 2606 T^{5} + 1291 T^{6} + 272 T^{7} + 79 T^{8} + 9 T^{9} + T^{10}$$
$19$ $$29584 + 29240 T + 28728 T^{2} + 9114 T^{3} + 4593 T^{4} + 486 T^{5} + 507 T^{6} + 24 T^{7} + 27 T^{8} - T^{9} + T^{10}$$
$23$ $$753424 + 845432 T + 663104 T^{2} + 287462 T^{3} + 96679 T^{4} + 20967 T^{5} + 3967 T^{6} + 506 T^{7} + 83 T^{8} + 8 T^{9} + T^{10}$$
$29$ $$( 401 - 1084 T + 474 T^{2} - 35 T^{3} - 9 T^{4} + T^{5} )^{2}$$
$31$ $$256 + 1088 T + 4672 T^{2} + 628 T^{3} + 1825 T^{4} + 502 T^{5} + 599 T^{6} + 84 T^{7} + 35 T^{8} - 3 T^{9} + T^{10}$$
$37$ $$9216 + 98304 T + 1074688 T^{2} - 257024 T^{3} + 188480 T^{4} + 26464 T^{5} + 12064 T^{6} + 320 T^{7} + 116 T^{8} + 2 T^{9} + T^{10}$$
$41$ $$( -516 - 2114 T + 645 T^{2} + 8 T^{3} - 15 T^{4} + T^{5} )^{2}$$
$43$ $$( -5488 - 6668 T - 2099 T^{2} - 109 T^{3} + 14 T^{4} + T^{5} )^{2}$$
$47$ $$1296 + 2016 T + 5764 T^{2} - 2792 T^{3} + 5941 T^{4} + 46 T^{5} + 1183 T^{6} - 344 T^{7} + 103 T^{8} - 11 T^{9} + T^{10}$$
$53$ $$6091024 + 9778216 T + 13542880 T^{2} + 4209098 T^{3} + 1386565 T^{4} - 58912 T^{5} + 26999 T^{6} - 378 T^{7} + 233 T^{8} - 9 T^{9} + T^{10}$$
$59$ $$206008609 + 81855159 T + 33629390 T^{2} + 4727949 T^{3} + 1089881 T^{4} + 73837 T^{5} + 26389 T^{6} + 874 T^{7} + 196 T^{8} - 4 T^{9} + T^{10}$$
$61$ $$1577536 + 447136 T + 875312 T^{2} - 501056 T^{3} + 311764 T^{4} - 71220 T^{5} + 14773 T^{6} - 962 T^{7} + 119 T^{8} - 2 T^{9} + T^{10}$$
$67$ $$17106496 + 10538528 T + 8808464 T^{2} - 624496 T^{3} + 527668 T^{4} + 17688 T^{5} + 11341 T^{6} + 344 T^{7} + 161 T^{8} + 8 T^{9} + T^{10}$$
$71$ $$( -5348 + 4350 T - 703 T^{2} - 88 T^{3} + 15 T^{4} + T^{5} )^{2}$$
$73$ $$116553616 + 57974520 T + 24702032 T^{2} + 5489838 T^{3} + 1281215 T^{4} + 207547 T^{5} + 40609 T^{6} + 4900 T^{7} + 517 T^{8} + 26 T^{9} + T^{10}$$
$79$ $$46063369 - 32021066 T + 21200752 T^{2} - 2677090 T^{3} + 719371 T^{4} - 12787 T^{5} + 16199 T^{6} - 117 T^{7} + 152 T^{8} + 3 T^{9} + T^{10}$$
$83$ $$( -37116 + 13972 T - 23 T^{2} - 292 T^{3} - T^{4} + T^{5} )^{2}$$
$89$ $$147573904 + 138147056 T + 82516140 T^{2} + 29433084 T^{3} + 7615317 T^{4} + 1360620 T^{5} + 181119 T^{6} + 16566 T^{7} + 1089 T^{8} + 41 T^{9} + T^{10}$$
$97$ $$( -709 + 70 T + 670 T^{2} - 133 T^{3} - 7 T^{4} + T^{5} )^{2}$$