# Properties

 Label 7623.2.a.ct Level 7623 Weight 2 Character orbit 7623.a Self dual yes Analytic conductor 60.870 Analytic rank 1 Dimension 8 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7623 = 3^{2} \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 7623.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.8699614608$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( -2 - \beta_{2} - \beta_{3} ) q^{5} + q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( -2 - \beta_{2} - \beta_{3} ) q^{5} + q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{10} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{13} -\beta_{1} q^{14} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{19} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{20} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{23} + ( 2 + \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{25} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{26} + ( 1 + \beta_{5} + \beta_{6} ) q^{28} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{29} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( -3 - \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{32} + ( -2 - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( -2 - \beta_{2} - \beta_{3} ) q^{35} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{37} + ( 2 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{38} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{40} + ( 1 + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{41} + ( -2 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{46} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + q^{49} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} - 4 \beta_{7} ) q^{50} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{52} + ( -4 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{56} + ( -2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 3 + 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{62} + ( 1 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( -3 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{65} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( -1 + 5 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{68} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{70} + ( 1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{71} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 - 2 \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{74} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{76} + ( 2 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -3 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{80} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{82} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -1 - 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{6} + \beta_{7} ) q^{86} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{89} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} + ( -7 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{92} + ( -5 + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{94} + ( 2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{95} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10})$$ $$8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - 5 \nu^{5} - 30 \nu^{4} - 30 \nu^{3} + 130 \nu^{2} + 135 \nu - 140$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - 4 \nu^{6} + 35 \nu^{5} + 35 \nu^{4} - 165 \nu^{3} - 60 \nu^{2} + 180 \nu - 45$$$$)/25$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + 6 \nu^{6} - 40 \nu^{5} - 65 \nu^{4} + 160 \nu^{3} + 165 \nu^{2} - 170 \nu - 20$$$$)/25$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{7} + 7 \nu^{6} + 95 \nu^{5} - 55 \nu^{4} - 280 \nu^{3} + 105 \nu^{2} + 210 \nu - 90$$$$)/25$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 7 \nu^{6} - 95 \nu^{5} + 55 \nu^{4} + 280 \nu^{3} - 80 \nu^{2} - 210 \nu + 15$$$$)/25$$ $$\beta_{7}$$ $$=$$ $$($$$$12 \nu^{7} - \nu^{6} - 135 \nu^{5} + 15 \nu^{4} + 440 \nu^{3} - 90 \nu^{2} - 405 \nu + 170$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 6 \beta_{6} + 7 \beta_{5} - \beta_{4} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 7 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{7} + 35 \beta_{6} + 47 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 77$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} + 54 \beta_{4} + 76 \beta_{3} - 42 \beta_{2} + 165 \beta_{1} + 31$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.55194 1.98451 1.11447 0.669744 0.226211 −1.40927 −1.70716 −2.43045
−2.55194 0 4.51241 −3.42898 0 1.00000 −6.41153 0 8.75055
1.2 −1.98451 0 1.93830 0.0269243 0 1.00000 0.122446 0 −0.0534317
1.3 −1.11447 0 −0.757964 −3.45608 0 1.00000 3.07366 0 3.85168
1.4 −0.669744 0 −1.55144 −2.14378 0 1.00000 2.37856 0 1.43578
1.5 −0.226211 0 −1.94883 2.49552 0 1.00000 0.893270 0 −0.564516
1.6 1.40927 0 −0.0139645 1.83139 0 1.00000 −2.83822 0 2.58091
1.7 1.70716 0 0.914391 −4.06637 0 1.00000 −1.85331 0 −6.94194
1.8 2.43045 0 3.90710 −1.25863 0 1.00000 4.63512 0 −3.05904
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.ct 8
3.b odd 2 1 847.2.a.p 8
11.b odd 2 1 7623.2.a.cw 8
11.c even 5 2 693.2.m.i 16
21.c even 2 1 5929.2.a.bt 8
33.d even 2 1 847.2.a.o 8
33.f even 10 2 847.2.f.v 16
33.f even 10 2 847.2.f.x 16
33.h odd 10 2 77.2.f.b 16
33.h odd 10 2 847.2.f.w 16
231.h odd 2 1 5929.2.a.bs 8
231.u even 10 2 539.2.f.e 16
231.z odd 30 4 539.2.q.g 32
231.bc even 30 4 539.2.q.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 33.h odd 10 2
539.2.f.e 16 231.u even 10 2
539.2.q.f 32 231.bc even 30 4
539.2.q.g 32 231.z odd 30 4
693.2.m.i 16 11.c even 5 2
847.2.a.o 8 33.d even 2 1
847.2.a.p 8 3.b odd 2 1
847.2.f.v 16 33.f even 10 2
847.2.f.w 16 33.h odd 10 2
847.2.f.x 16 33.f even 10 2
5929.2.a.bs 8 231.h odd 2 1
5929.2.a.bt 8 21.c even 2 1
7623.2.a.ct 8 1.a even 1 1 trivial
7623.2.a.cw 8 11.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{8} + \cdots$$ $$T_{5}^{8} + \cdots$$ $$T_{13}^{8} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 5 T^{2} + 4 T^{3} + 15 T^{4} + 14 T^{5} + 38 T^{6} + 30 T^{7} + 75 T^{8} + 60 T^{9} + 152 T^{10} + 112 T^{11} + 240 T^{12} + 128 T^{13} + 320 T^{14} + 128 T^{15} + 256 T^{16}$$
$3$ 
$5$ $$1 + 10 T + 62 T^{2} + 283 T^{3} + 1075 T^{4} + 3513 T^{5} + 10230 T^{6} + 26646 T^{7} + 62784 T^{8} + 133230 T^{9} + 255750 T^{10} + 439125 T^{11} + 671875 T^{12} + 884375 T^{13} + 968750 T^{14} + 781250 T^{15} + 390625 T^{16}$$
$7$ $$( 1 - T )^{8}$$
$11$ 
$13$ $$1 + 6 T + 68 T^{2} + 285 T^{3} + 2119 T^{4} + 7443 T^{5} + 44016 T^{6} + 130942 T^{7} + 656120 T^{8} + 1702246 T^{9} + 7438704 T^{10} + 16352271 T^{11} + 60520759 T^{12} + 105818505 T^{13} + 328223012 T^{14} + 376491102 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 5 T + 99 T^{2} - 384 T^{3} + 4383 T^{4} - 13616 T^{5} + 119813 T^{6} - 309659 T^{7} + 2344384 T^{8} - 5264203 T^{9} + 34625957 T^{10} - 66895408 T^{11} + 366072543 T^{12} - 545225088 T^{13} + 2389619331 T^{14} - 2051693365 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 13 T + 167 T^{2} + 1424 T^{3} + 11403 T^{4} + 72606 T^{5} + 434133 T^{6} + 2182431 T^{7} + 10302272 T^{8} + 41466189 T^{9} + 156722013 T^{10} + 498004554 T^{11} + 1486050363 T^{12} + 3525964976 T^{13} + 7856662127 T^{14} + 11620332607 T^{15} + 16983563041 T^{16}$$
$23$ $$1 + 16 T + 224 T^{2} + 2234 T^{3} + 19567 T^{4} + 141712 T^{5} + 926670 T^{6} + 5230592 T^{7} + 26816927 T^{8} + 120303616 T^{9} + 490208430 T^{10} + 1724209904 T^{11} + 5475648847 T^{12} + 14378790262 T^{13} + 33160039136 T^{14} + 54477207152 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 9 T + 187 T^{2} + 1382 T^{3} + 16028 T^{4} + 97473 T^{5} + 828470 T^{6} + 4208558 T^{7} + 28798593 T^{8} + 122048182 T^{9} + 696743270 T^{10} + 2377268997 T^{11} + 11336299868 T^{12} + 28346407918 T^{13} + 111231961027 T^{14} + 155248886781 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 9 T + 142 T^{2} - 963 T^{3} + 9327 T^{4} - 55589 T^{5} + 439146 T^{6} - 2365595 T^{7} + 15862880 T^{8} - 73333445 T^{9} + 422019306 T^{10} - 1656051899 T^{11} + 8613680367 T^{12} - 27569872413 T^{13} + 126025522702 T^{14} - 247613526999 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 7 T + 231 T^{2} - 1552 T^{3} + 24962 T^{4} - 154999 T^{5} + 1656240 T^{6} - 9060906 T^{7} + 73868707 T^{8} - 335253522 T^{9} + 2267392560 T^{10} - 7851164347 T^{11} + 46782806882 T^{12} - 107621821264 T^{13} + 592682800479 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 10 T + 261 T^{2} - 2176 T^{3} + 30891 T^{4} - 221954 T^{5} + 2245611 T^{6} - 13854084 T^{7} + 110613464 T^{8} - 568017444 T^{9} + 3774872091 T^{10} - 15297291634 T^{11} + 87290583051 T^{12} - 252103093376 T^{13} + 1239777206901 T^{14} - 1947542738810 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 + 16 T + 320 T^{2} + 3319 T^{3} + 41705 T^{4} + 344729 T^{5} + 3361468 T^{6} + 23054530 T^{7} + 186100580 T^{8} + 1083562910 T^{9} + 7425482812 T^{10} + 35790798967 T^{11} + 203507096105 T^{12} + 761196078233 T^{13} + 3449348905280 T^{14} + 8105969927408 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 37 T + 857 T^{2} + 14496 T^{3} + 199396 T^{4} + 2305797 T^{5} + 23105040 T^{6} + 202528658 T^{7} + 1568879259 T^{8} + 10734018874 T^{9} + 64902057360 T^{10} + 343280139969 T^{11} + 1573330349476 T^{12} + 6062161866528 T^{13} + 18994857487553 T^{14} + 43464312173969 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + T + 332 T^{2} + 463 T^{3} + 53293 T^{4} + 80577 T^{5} + 5426300 T^{6} + 7692267 T^{7} + 381400788 T^{8} + 453843753 T^{9} + 18888950300 T^{10} + 16548823683 T^{11} + 645770519773 T^{12} + 331009950437 T^{13} + 14003937168812 T^{14} + 2488651484819 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 19 T + 492 T^{2} - 7043 T^{3} + 104067 T^{4} - 1170549 T^{5} + 12549376 T^{6} - 113336285 T^{7} + 952230800 T^{8} - 6913513385 T^{9} + 46696228096 T^{10} - 265692382569 T^{11} + 1440895135347 T^{12} - 5948491747943 T^{13} + 25348024185612 T^{14} - 59712113884399 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 + 13 T + 398 T^{2} + 3457 T^{3} + 57783 T^{4} + 323737 T^{5} + 4210848 T^{6} + 14543752 T^{7} + 249426533 T^{8} + 1032606392 T^{9} + 21226884768 T^{10} + 115869033407 T^{11} + 1468363163223 T^{12} + 6237220866407 T^{13} + 50983913000558 T^{14} + 118236562059083 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 25 T + 616 T^{2} + 9963 T^{3} + 152093 T^{4} + 1861173 T^{5} + 21394652 T^{6} + 209466223 T^{7} + 1925408604 T^{8} + 15291034279 T^{9} + 114012100508 T^{10} + 724027936941 T^{11} + 4319173668413 T^{12} + 20654012281059 T^{13} + 93221883394024 T^{14} + 276184962977425 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 337 T^{2} + 1005 T^{3} + 57248 T^{4} + 275700 T^{5} + 6743044 T^{6} + 36745800 T^{7} + 601257265 T^{8} + 2902918200 T^{9} + 42083337604 T^{10} + 135930852300 T^{11} + 2229814237088 T^{12} + 3092441680995 T^{13} + 81920472510577 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 - 25 T + 786 T^{2} - 13983 T^{3} + 250863 T^{4} - 3398873 T^{5} + 43717402 T^{6} - 464690723 T^{7} + 4604658344 T^{8} - 38569330009 T^{9} + 301169182378 T^{10} - 1943431396051 T^{11} + 11905536781023 T^{12} - 55079605311069 T^{13} + 256975133468034 T^{14} - 678401274740675 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 + 37 T + 1032 T^{2} + 20901 T^{3} + 357823 T^{4} + 5152639 T^{5} + 65606528 T^{6} + 731972999 T^{7} + 7339853952 T^{8} + 65145596911 T^{9} + 519669308288 T^{10} + 3632450763191 T^{11} + 22450616901343 T^{12} + 116712426543549 T^{13} + 512884692271752 T^{14} + 1636559391134573 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 - 15 T + 554 T^{2} - 7041 T^{3} + 141913 T^{4} - 1604409 T^{5} + 23035698 T^{6} - 230491951 T^{7} + 2630434004 T^{8} - 22357719247 T^{9} + 216742882482 T^{10} - 1464300775257 T^{11} + 12563455854553 T^{12} - 60463462749537 T^{13} + 461466490730666 T^{14} - 1211974267171695 T^{15} + 7837433594376961 T^{16}$$