# Properties

 Label 3024.2.q.i Level 3024 Weight 2 Character orbit 3024.q Analytic conductor 24.147 Analytic rank 0 Dimension 10 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 Defining polynomial: $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} - \beta_{6} ) q^{5} + ( 1 - \beta_{1} + \beta_{5} - \beta_{8} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{2} - \beta_{6} ) q^{5} + ( 1 - \beta_{1} + \beta_{5} - \beta_{8} ) q^{7} + ( -\beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{11} + ( \beta_{5} + \beta_{6} + \beta_{9} ) q^{13} + ( -3 + \beta_{1} - 3 \beta_{6} - \beta_{7} ) q^{17} + ( -\beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{23} + ( \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{9} ) q^{25} + ( -1 - \beta_{1} + \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{35} + ( -2 \beta_{5} - 2 \beta_{8} ) q^{37} + ( -\beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{41} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{43} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{47} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{49} + ( 5 - \beta_{1} - 2 \beta_{2} + 5 \beta_{6} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{55} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{59} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{61} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{65} + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{8} - \beta_{9} ) q^{67} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + 2 \beta_{9} ) q^{71} + ( 4 - 3 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{73} + ( -3 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{77} + ( -3 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{5} - \beta_{9} ) q^{79} + ( 1 - 2 \beta_{2} + 4 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{83} + ( -\beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{85} + ( -2 \beta_{5} + 7 \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{89} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{91} + ( 2 - 2 \beta_{3} + \beta_{4} + \beta_{8} ) q^{95} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{6} - 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 4q^{5} + 4q^{7} + O(q^{10})$$ $$10q - 4q^{5} + 4q^{7} + 4q^{11} - 8q^{13} - 12q^{17} - q^{19} + 3q^{23} - q^{25} - 7q^{29} - 6q^{31} + 5q^{35} - 5q^{41} + 7q^{43} - 54q^{47} - 8q^{49} + 21q^{53} - 4q^{55} - 60q^{59} + 28q^{61} - 22q^{65} - 4q^{67} - 6q^{71} + 15q^{73} - 11q^{77} - 8q^{79} + 9q^{83} - 6q^{85} - 28q^{89} + 4q^{91} + 28q^{95} - 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557$$$$)/567$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63$$$$)/63$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180$$$$)/567$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63$$$$)/567$$ $$\beta_{7}$$ $$=$$ $$($$$$-53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368$$$$)/567$$ $$\beta_{8}$$ $$=$$ $$($$$$-82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720$$$$)/567$$ $$\beta_{9}$$ $$=$$ $$($$$$-91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801$$$$)/567$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + 4 \beta_{5} + \beta_{4} - 4 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{7} - 14 \beta_{6} + \beta_{4} + \beta_{2} - 14$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 9 \beta_{6} - 17 \beta_{5} + \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{9} - 10 \beta_{8} - \beta_{5} - 10 \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + \beta_{1} + 70$$ $$\nu^{7}$$ $$=$$ $$11 \beta_{7} + 65 \beta_{6} - 43 \beta_{4} - 19 \beta_{2} + 75 \beta_{1} + 65$$ $$\nu^{8}$$ $$=$$ $$-62 \beta_{9} + 73 \beta_{8} + 118 \beta_{7} + 360 \beta_{6} + 14 \beta_{5} - 118 \beta_{3}$$ $$\nu^{9}$$ $$=$$ $$-135 \beta_{9} + 253 \beta_{8} + 343 \beta_{5} + 253 \beta_{4} - 87 \beta_{3} + 135 \beta_{2} - 343 \beta_{1} - 430$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$ $$1$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2305.1
 0.247934 + 0.429435i 1.19343 + 2.06709i −1.02682 − 1.77851i 0.920620 + 1.59456i −0.335166 − 0.580525i 0.247934 − 0.429435i 1.19343 − 2.06709i −1.02682 + 1.77851i 0.920620 − 1.59456i −0.335166 + 0.580525i
0 0 0 −1.84629 3.19787i 0 −0.926641 + 2.47817i 0 0 0
2305.2 0 0 0 −1.46043 2.52954i 0 0.138560 2.64212i 0 0 0
2305.3 0 0 0 −0.0731228 0.126652i 0 2.33035 + 1.25278i 0 0 0
2305.4 0 0 0 0.667377 + 1.15593i 0 −1.90267 + 1.83844i 0 0 0
2305.5 0 0 0 0.712469 + 1.23403i 0 2.36039 1.19522i 0 0 0
2881.1 0 0 0 −1.84629 + 3.19787i 0 −0.926641 2.47817i 0 0 0
2881.2 0 0 0 −1.46043 + 2.52954i 0 0.138560 + 2.64212i 0 0 0
2881.3 0 0 0 −0.0731228 + 0.126652i 0 2.33035 1.25278i 0 0 0
2881.4 0 0 0 0.667377 1.15593i 0 −1.90267 1.83844i 0 0 0
2881.5 0 0 0 0.712469 1.23403i 0 2.36039 + 1.19522i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2881.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.q.i 10
3.b odd 2 1 1008.2.q.i 10
4.b odd 2 1 189.2.h.b 10
7.c even 3 1 3024.2.t.i 10
9.c even 3 1 3024.2.t.i 10
9.d odd 6 1 1008.2.t.i 10
12.b even 2 1 63.2.h.b yes 10
21.h odd 6 1 1008.2.t.i 10
28.d even 2 1 1323.2.h.f 10
28.f even 6 1 1323.2.f.f 10
28.f even 6 1 1323.2.g.f 10
28.g odd 6 1 189.2.g.b 10
28.g odd 6 1 1323.2.f.e 10
36.f odd 6 1 189.2.g.b 10
36.f odd 6 1 567.2.e.e 10
36.h even 6 1 63.2.g.b 10
36.h even 6 1 567.2.e.f 10
63.h even 3 1 inner 3024.2.q.i 10
63.j odd 6 1 1008.2.q.i 10
84.h odd 2 1 441.2.h.f 10
84.j odd 6 1 441.2.f.f 10
84.j odd 6 1 441.2.g.f 10
84.n even 6 1 63.2.g.b 10
84.n even 6 1 441.2.f.e 10
252.n even 6 1 1323.2.f.f 10
252.o even 6 1 441.2.f.e 10
252.o even 6 1 567.2.e.f 10
252.r odd 6 1 441.2.h.f 10
252.r odd 6 1 3969.2.a.ba 5
252.s odd 6 1 441.2.g.f 10
252.u odd 6 1 189.2.h.b 10
252.u odd 6 1 3969.2.a.bc 5
252.bb even 6 1 63.2.h.b yes 10
252.bb even 6 1 3969.2.a.z 5
252.bi even 6 1 1323.2.g.f 10
252.bj even 6 1 1323.2.h.f 10
252.bj even 6 1 3969.2.a.bb 5
252.bl odd 6 1 567.2.e.e 10
252.bl odd 6 1 1323.2.f.e 10
252.bn odd 6 1 441.2.f.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 36.h even 6 1
63.2.g.b 10 84.n even 6 1
63.2.h.b yes 10 12.b even 2 1
63.2.h.b yes 10 252.bb even 6 1
189.2.g.b 10 28.g odd 6 1
189.2.g.b 10 36.f odd 6 1
189.2.h.b 10 4.b odd 2 1
189.2.h.b 10 252.u odd 6 1
441.2.f.e 10 84.n even 6 1
441.2.f.e 10 252.o even 6 1
441.2.f.f 10 84.j odd 6 1
441.2.f.f 10 252.bn odd 6 1
441.2.g.f 10 84.j odd 6 1
441.2.g.f 10 252.s odd 6 1
441.2.h.f 10 84.h odd 2 1
441.2.h.f 10 252.r odd 6 1
567.2.e.e 10 36.f odd 6 1
567.2.e.e 10 252.bl odd 6 1
567.2.e.f 10 36.h even 6 1
567.2.e.f 10 252.o even 6 1
1008.2.q.i 10 3.b odd 2 1
1008.2.q.i 10 63.j odd 6 1
1008.2.t.i 10 9.d odd 6 1
1008.2.t.i 10 21.h odd 6 1
1323.2.f.e 10 28.g odd 6 1
1323.2.f.e 10 252.bl odd 6 1
1323.2.f.f 10 28.f even 6 1
1323.2.f.f 10 252.n even 6 1
1323.2.g.f 10 28.f even 6 1
1323.2.g.f 10 252.bi even 6 1
1323.2.h.f 10 28.d even 2 1
1323.2.h.f 10 252.bj even 6 1
3024.2.q.i 10 1.a even 1 1 trivial
3024.2.q.i 10 63.h even 3 1 inner
3024.2.t.i 10 7.c even 3 1
3024.2.t.i 10 9.c even 3 1
3969.2.a.z 5 252.bb even 6 1
3969.2.a.ba 5 252.r odd 6 1
3969.2.a.bb 5 252.bj even 6 1
3969.2.a.bc 5 252.u odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}^{10} + \cdots$$ $$T_{11}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 4 T - 4 T^{2} - 44 T^{3} - 41 T^{4} + 119 T^{5} + 222 T^{6} + 456 T^{7} + 1623 T^{8} - 2021 T^{9} - 16541 T^{10} - 10105 T^{11} + 40575 T^{12} + 57000 T^{13} + 138750 T^{14} + 371875 T^{15} - 640625 T^{16} - 3437500 T^{17} - 1562500 T^{18} + 7812500 T^{19} + 9765625 T^{20}$$
$7$ $$1 - 4 T + 12 T^{2} - 47 T^{3} + 146 T^{4} - 309 T^{5} + 1022 T^{6} - 2303 T^{7} + 4116 T^{8} - 9604 T^{9} + 16807 T^{10}$$
$11$ $$1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 1493063 T^{11} + 11971740 T^{12} + 36595845 T^{13} - 122764785 T^{14} - 401339092 T^{15} + 1075337527 T^{16} + 2611280914 T^{17} - 6645125311 T^{18} - 9431790764 T^{19} + 25937424601 T^{20}$$
$13$ $$1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20}$$
$17$ $$1 + 12 T + 14 T^{2} - 192 T^{3} + 1185 T^{4} + 11847 T^{5} - 6180 T^{6} - 65736 T^{7} + 1002861 T^{8} + 2436261 T^{9} - 7749777 T^{10} + 41416437 T^{11} + 289826829 T^{12} - 322960968 T^{13} - 516159780 T^{14} + 16821045879 T^{15} + 28603019265 T^{16} - 78785025216 T^{17} + 97660604174 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20}$$
$19$ $$1 + T - 53 T^{2} - 190 T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 13710704 T^{11} + 24535004 T^{12} - 796398490 T^{13} - 1708638631 T^{14} + 17350025693 T^{15} + 59371901822 T^{16} - 169835630410 T^{17} - 900128841173 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20}$$
$23$ $$1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 75250227 T^{11} + 265749498 T^{12} - 1642873509 T^{13} + 6098295072 T^{14} - 32999130561 T^{15} + 888215334 T^{16} + 1001018681418 T^{17} - 3367372367083 T^{18} - 5403457984389 T^{19} + 41426511213649 T^{20}$$
$29$ $$1 + 7 T - 76 T^{2} - 419 T^{3} + 4561 T^{4} + 15146 T^{5} - 199563 T^{6} - 341373 T^{7} + 6918636 T^{8} + 2570041 T^{9} - 219913241 T^{10} + 74531189 T^{11} + 5818572876 T^{12} - 8325746097 T^{13} - 141147118203 T^{14} + 310661862754 T^{15} + 2712989167081 T^{16} - 7227698173471 T^{17} - 38018727385036 T^{18} + 101550021831083 T^{19} + 420707233300201 T^{20}$$
$31$ $$( 1 + 3 T + 134 T^{2} + 308 T^{3} + 7750 T^{4} + 13615 T^{5} + 240250 T^{6} + 295988 T^{7} + 3991994 T^{8} + 2770563 T^{9} + 28629151 T^{10} )^{2}$$
$37$ $$1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20}$$
$41$ $$1 + 5 T - 136 T^{2} - 733 T^{3} + 10507 T^{4} + 54412 T^{5} - 554055 T^{6} - 2345451 T^{7} + 23706084 T^{8} + 41392439 T^{9} - 952045937 T^{10} + 1697089999 T^{11} + 39849927204 T^{12} - 161650828371 T^{13} - 1565627010855 T^{14} + 6303967608812 T^{15} + 49909345260187 T^{16} - 142754882754773 T^{17} - 1085949831160456 T^{18} + 1636909671969805 T^{19} + 13422659310152401 T^{20}$$
$43$ $$1 - 7 T - 77 T^{2} + 66 T^{3} + 7014 T^{4} + 3843 T^{5} - 95427 T^{6} - 1632678 T^{7} - 3708600 T^{8} + 15416324 T^{9} + 670279801 T^{10} + 662901932 T^{11} - 6857201400 T^{12} - 129809329746 T^{13} - 326245923027 T^{14} + 564953446449 T^{15} + 44338040425686 T^{16} + 17940028333062 T^{17} - 899991421375277 T^{18} - 3518148283557901 T^{19} + 21611482313284249 T^{20}$$
$47$ $$( 1 + 27 T + 448 T^{2} + 5169 T^{3} + 48091 T^{4} + 359985 T^{5} + 2260277 T^{6} + 11418321 T^{7} + 46512704 T^{8} + 131751387 T^{9} + 229345007 T^{10} )^{2}$$
$53$ $$1 - 21 T + 41 T^{2} + 924 T^{3} + 12966 T^{4} - 177027 T^{5} - 601755 T^{6} + 3783942 T^{7} + 110973258 T^{8} - 340111866 T^{9} - 4044436041 T^{10} - 18025928898 T^{11} + 311723881722 T^{12} + 563341933134 T^{13} - 4748136394155 T^{14} - 74031893539311 T^{15} + 287383106398614 T^{16} + 1085433093209388 T^{17} + 2552647306865801 T^{18} - 69295035427844793 T^{19} + 174887470365513049 T^{20}$$
$59$ $$( 1 + 30 T + 601 T^{2} + 8193 T^{3} + 88864 T^{4} + 752289 T^{5} + 5242976 T^{6} + 28519833 T^{7} + 123432779 T^{8} + 363520830 T^{9} + 714924299 T^{10} )^{2}$$
$61$ $$( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 2648498 T^{6} - 12684889 T^{7} + 76946559 T^{8} - 193841774 T^{9} + 844596301 T^{10} )^{2}$$
$67$ $$( 1 + 2 T + 132 T^{2} + 196 T^{3} + 10871 T^{4} + 15429 T^{5} + 728357 T^{6} + 879844 T^{7} + 39700716 T^{8} + 40302242 T^{9} + 1350125107 T^{10} )^{2}$$
$71$ $$( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2}$$
$73$ $$1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20}$$
$79$ $$( 1 + 4 T + 300 T^{2} + 1488 T^{3} + 39873 T^{4} + 184983 T^{5} + 3149967 T^{6} + 9286608 T^{7} + 147911700 T^{8} + 155800324 T^{9} + 3077056399 T^{10} )^{2}$$
$83$ $$1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20}$$
$89$ $$1 + 28 T + 104 T^{2} - 1736 T^{3} + 31273 T^{4} + 611939 T^{5} - 1780638 T^{6} - 18973932 T^{7} + 740914101 T^{8} + 3271180573 T^{9} - 40614588329 T^{10} + 291135070997 T^{11} + 5868780594021 T^{12} - 13376033868108 T^{13} - 111721218529758 T^{14} + 3417103755161611 T^{15} + 15542095912223353 T^{16} - 76785597378638344 T^{17} + 409405235793016424 T^{18} + 9809979303809585852 T^{19} + 31181719929966183601 T^{20}$$
$97$ $$1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20}$$