# Properties

 Label 525.2.i.h Level 525 Weight 2 Character orbit 525.i Analytic conductor 4.192 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$68 \nu^{7} - 215 \nu^{6} + 357 \nu^{5} + 646 \nu^{4} - 1444 \nu^{3} + 1156 \nu^{2} + 561 \nu + 5468$$$$)/4243$$ $$\beta_{3}$$ $$=$$ $$($$$$84 \nu^{7} - 16 \nu^{6} + 441 \nu^{5} + 798 \nu^{4} + 3208 \nu^{3} + 1428 \nu^{2} + 693 \nu + 2262$$$$)/4243$$ $$\beta_{4}$$ $$=$$ $$($$$$-754 \nu^{7} + 1760 \nu^{6} - 6080 \nu^{5} + 1323 \nu^{4} - 13440 \nu^{3} + 12640 \nu^{2} - 16828 \nu + 5760$$$$)/12729$$ $$\beta_{5}$$ $$=$$ $$($$$$-815 \nu^{7} + 3388 \nu^{6} - 11704 \nu^{5} + 15594 \nu^{4} - 25872 \nu^{3} + 24332 \nu^{2} - 56579 \nu + 11088$$$$)/12729$$ $$\beta_{6}$$ $$=$$ $$($$$$1052 \nu^{7} - 2827 \nu^{6} + 9766 \nu^{5} - 6978 \nu^{4} + 21588 \nu^{3} - 20303 \nu^{2} + 17165 \nu - 9252$$$$)/12729$$ $$\beta_{7}$$ $$=$$ $$($$$$-556 \nu^{7} + 510 \nu^{6} - 2919 \nu^{5} - 5282 \nu^{4} - 8909 \nu^{3} - 9452 \nu^{2} - 4587 \nu - 13760$$$$)/4243$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 5 \beta_{3} + 2 \beta_{2} - 2$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{6} - 2 \beta_{5} - 9 \beta_{4} - 10 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{7} - 20 \beta_{6} - 8 \beta_{5} - 18 \beta_{4} - 33 \beta_{3} - 20 \beta_{2} - 33 \beta_{1} + 18$$ $$\nu^{6}$$ $$=$$ $$-20 \beta_{7} - 83 \beta_{3} - 61 \beta_{2} + 58$$ $$\nu^{7}$$ $$=$$ $$164 \beta_{6} + 61 \beta_{5} + 146 \beta_{4} + 243 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −0.758290 + 1.31340i −0.276205 + 0.478401i 0.643668 − 1.11487i 1.39083 − 2.40898i −0.758290 − 1.31340i −0.276205 − 0.478401i 0.643668 + 1.11487i 1.39083 + 2.40898i
−1.25829 2.17942i 0.500000 0.866025i −2.16659 + 3.75264i 0 −2.51658 2.29673 + 1.31340i 5.87162 −0.500000 0.866025i 0
151.2 −0.776205 1.34443i 0.500000 0.866025i −0.204988 + 0.355049i 0 −1.55241 −2.60214 + 0.478401i −2.46837 −0.500000 0.866025i 0
151.3 0.143668 + 0.248840i 0.500000 0.866025i 0.958719 1.66055i 0 0.287336 2.39939 1.11487i 1.12562 −0.500000 0.866025i 0
151.4 0.890827 + 1.54296i 0.500000 0.866025i −0.587145 + 1.01696i 0 1.78165 −1.09398 2.40898i 1.47113 −0.500000 0.866025i 0
226.1 −1.25829 + 2.17942i 0.500000 + 0.866025i −2.16659 3.75264i 0 −2.51658 2.29673 1.31340i 5.87162 −0.500000 + 0.866025i 0
226.2 −0.776205 + 1.34443i 0.500000 + 0.866025i −0.204988 0.355049i 0 −1.55241 −2.60214 0.478401i −2.46837 −0.500000 + 0.866025i 0
226.3 0.143668 0.248840i 0.500000 + 0.866025i 0.958719 + 1.66055i 0 0.287336 2.39939 + 1.11487i 1.12562 −0.500000 + 0.866025i 0
226.4 0.890827 1.54296i 0.500000 + 0.866025i −0.587145 1.01696i 0 1.78165 −1.09398 + 2.40898i 1.47113 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 226.4 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.h 8
5.b even 2 1 525.2.i.k 8
5.c odd 4 2 105.2.q.a 16
7.c even 3 1 inner 525.2.i.h 8
7.c even 3 1 3675.2.a.bz 4
7.d odd 6 1 3675.2.a.cb 4
15.e even 4 2 315.2.bf.b 16
20.e even 4 2 1680.2.di.d 16
35.f even 4 2 735.2.q.g 16
35.i odd 6 1 3675.2.a.bn 4
35.j even 6 1 525.2.i.k 8
35.j even 6 1 3675.2.a.bp 4
35.k even 12 2 735.2.d.e 8
35.k even 12 2 735.2.q.g 16
35.l odd 12 2 105.2.q.a 16
35.l odd 12 2 735.2.d.d 8
105.w odd 12 2 2205.2.d.o 8
105.x even 12 2 315.2.bf.b 16
105.x even 12 2 2205.2.d.s 8
140.w even 12 2 1680.2.di.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a 16 5.c odd 4 2
105.2.q.a 16 35.l odd 12 2
315.2.bf.b 16 15.e even 4 2
315.2.bf.b 16 105.x even 12 2
525.2.i.h 8 1.a even 1 1 trivial
525.2.i.h 8 7.c even 3 1 inner
525.2.i.k 8 5.b even 2 1
525.2.i.k 8 35.j even 6 1
735.2.d.d 8 35.l odd 12 2
735.2.d.e 8 35.k even 12 2
735.2.q.g 16 35.f even 4 2
735.2.q.g 16 35.k even 12 2
1680.2.di.d 16 20.e even 4 2
1680.2.di.d 16 140.w even 12 2
2205.2.d.o 8 105.w odd 12 2
2205.2.d.s 8 105.x even 12 2
3675.2.a.bn 4 35.i odd 6 1
3675.2.a.bp 4 35.j even 6 1
3675.2.a.bz 4 7.c even 3 1
3675.2.a.cb 4 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - 4 T^{3} - 6 T^{4} - 4 T^{5} + 4 T^{6} + 12 T^{7} + 12 T^{8} + 24 T^{9} + 16 T^{10} - 32 T^{11} - 96 T^{12} - 128 T^{13} + 256 T^{15} + 256 T^{16}$$
$3$ $$( 1 - T + T^{2} )^{4}$$
$5$ 
$7$ $$1 - 2 T - 8 T^{2} + 14 T^{3} + 41 T^{4} + 98 T^{5} - 392 T^{6} - 686 T^{7} + 2401 T^{8}$$
$11$ $$1 - 26 T^{2} - 28 T^{3} + 316 T^{4} + 518 T^{5} - 2872 T^{6} - 2968 T^{7} + 31007 T^{8} - 32648 T^{9} - 347512 T^{10} + 689458 T^{11} + 4626556 T^{12} - 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 - 2 T + 24 T^{2} - 42 T^{3} + 413 T^{4} - 546 T^{5} + 4056 T^{6} - 4394 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 + 2 T - 36 T^{2} - 80 T^{3} + 524 T^{4} + 896 T^{5} - 9840 T^{6} - 2574 T^{7} + 223639 T^{8} - 43758 T^{9} - 2843760 T^{10} + 4402048 T^{11} + 43765004 T^{12} - 113588560 T^{13} - 868952484 T^{14} + 820677346 T^{15} + 6975757441 T^{16}$$
$19$ $$1 - 12 T + 30 T^{2} + 80 T^{3} + 689 T^{4} - 9436 T^{5} + 28846 T^{6} - 59944 T^{7} + 291332 T^{8} - 1138936 T^{9} + 10413406 T^{10} - 64721524 T^{11} + 89791169 T^{12} + 198087920 T^{13} + 1411376430 T^{14} - 10726460868 T^{15} + 16983563041 T^{16}$$
$23$ $$1 + 10 T + 40 T^{2} + 392 T^{3} + 2364 T^{4} + 4712 T^{5} + 38616 T^{6} + 172414 T^{7} + 38807 T^{8} + 3965522 T^{9} + 20427864 T^{10} + 57330904 T^{11} + 661544124 T^{12} + 2523046456 T^{13} + 5921435560 T^{14} + 34048254470 T^{15} + 78310985281 T^{16}$$
$29$ $$( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 9628 T^{5} + 65598 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 1872896 T^{9} - 151543934 T^{10} - 276222152 T^{11} + 3033766485 T^{12} + 10077461152 T^{13} - 47925198774 T^{14} - 220100912888 T^{15} + 852891037441 T^{16}$$
$37$ $$1 + 24 T + 224 T^{2} + 1676 T^{3} + 18423 T^{4} + 154738 T^{5} + 897620 T^{6} + 6581218 T^{7} + 49111756 T^{8} + 243505066 T^{9} + 1228841780 T^{10} + 7837943914 T^{11} + 34527668103 T^{12} + 116220471932 T^{13} + 574722715616 T^{14} + 2278365051192 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 14186 T^{5} + 191634 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 8 T + 140 T^{2} - 878 T^{3} + 8293 T^{4} - 37754 T^{5} + 258860 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 10 T - 104 T^{2} + 652 T^{3} + 13518 T^{4} - 45086 T^{5} - 905784 T^{6} + 563558 T^{7} + 53075567 T^{8} + 26487226 T^{9} - 2000876856 T^{10} - 4680963778 T^{11} + 65963527758 T^{12} + 149532944564 T^{13} - 1121038394216 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 20 T + 68 T^{2} - 136 T^{3} + 12794 T^{4} + 117580 T^{5} - 98000 T^{6} + 2920764 T^{7} + 73537971 T^{8} + 154800492 T^{9} - 275282000 T^{10} + 17504957660 T^{11} + 100950813914 T^{12} - 56874587048 T^{13} + 1507176556772 T^{14} + 23494222796740 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 2 T - 226 T^{2} - 236 T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 24029402 T^{9} - 10310513140 T^{10} + 4070611780 T^{11} + 387949429776 T^{12} - 168722134564 T^{13} - 9532800602866 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 48723872 T^{9} - 921438672 T^{10} + 3040637476 T^{11} + 55355672318 T^{12} + 317568209176 T^{13} - 4121629948880 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 6 T - 160 T^{2} - 148 T^{3} + 18063 T^{4} - 22466 T^{5} - 1278868 T^{6} + 541096 T^{7} + 68029204 T^{8} + 36253432 T^{9} - 5740838452 T^{10} - 6756941558 T^{11} + 363989698623 T^{12} - 199818515836 T^{13} - 14473341147040 T^{14} + 36364269631938 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 117008 T^{5} + 977954 T^{6} + 5010754 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 + 12 T - 84 T^{2} - 932 T^{3} + 8735 T^{4} + 42250 T^{5} - 655952 T^{6} - 2845850 T^{7} + 12256412 T^{8} - 207747050 T^{9} - 3495568208 T^{10} + 16435968250 T^{11} + 248058635135 T^{12} - 1932102724676 T^{13} - 12712075008276 T^{14} + 132568782229164 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 8 T - 98 T^{2} + 1128 T^{3} - 3647 T^{4} + 31028 T^{5} - 546546 T^{6} - 5486636 T^{7} + 152161556 T^{8} - 433444244 T^{9} - 3410993586 T^{10} + 15298014092 T^{11} - 142050945407 T^{12} + 3470919618072 T^{13} - 23822570641058 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 + 6 T + 276 T^{2} + 1256 T^{3} + 32400 T^{4} + 104248 T^{5} + 1901364 T^{6} + 3430722 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 2927032 T^{7} - 142835833 T^{8} - 260505848 T^{9} + 8329596864 T^{10} + 62946682010 T^{11} + 287610432744 T^{12} - 5606395686796 T^{13} - 48704166514178 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16}$$
$97$ $$( 1 + 2 T + 380 T^{2} + 566 T^{3} + 54898 T^{4} + 54902 T^{5} + 3575420 T^{6} + 1825346 T^{7} + 88529281 T^{8} )^{2}$$