# Properties

 Label 445.2.a.f Level 445 Weight 2 Character orbit 445.a Self dual yes Analytic conductor 3.553 Analytic rank 1 Dimension 7 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$445 = 5 \cdot 89$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.55334288995$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{5} ) q^{3} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} + q^{5} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{6} + ( -2 - \beta_{6} ) q^{7} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 + \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{5} ) q^{3} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} + q^{5} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{6} + ( -2 - \beta_{6} ) q^{7} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 + \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{9} + ( -1 + \beta_{1} ) q^{10} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{12} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{14} + ( -1 - \beta_{5} ) q^{15} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{16} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} + ( 2 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{18} + ( -3 \beta_{1} - \beta_{2} + \beta_{6} ) q^{19} + ( 2 - \beta_{1} + \beta_{3} ) q^{20} + ( 2 - \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{21} + ( 1 - 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{24} + q^{25} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{26} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{27} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{28} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{29} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{30} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{31} + ( -6 - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 5 \beta_{6} ) q^{32} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{33} + ( 5 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{34} + ( -2 - \beta_{6} ) q^{35} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} ) q^{36} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( -6 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{38} + ( -1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{39} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{40} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( 4 + 8 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{42} + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{43} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( 1 + \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{45} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{46} + ( -2 - 3 \beta_{2} + \beta_{4} + \beta_{6} ) q^{47} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{48} + ( 2 + \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{6} ) q^{51} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{52} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{53} + ( -4 - 5 \beta_{1} - \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{54} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{55} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{56} + ( 2 + 7 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{6} ) q^{57} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} ) q^{58} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{59} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{60} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{61} + ( 7 - \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - 5 \beta_{6} ) q^{62} + ( -7 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{63} + ( 7 - \beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 7 \beta_{6} ) q^{64} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( 4 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{5} ) q^{66} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( -12 - \beta_{1} - 8 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} ) q^{68} + ( 3 - 5 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{69} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{70} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 11 + 2 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} ) q^{72} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{73} + ( -5 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{74} + ( -1 - \beta_{5} ) q^{75} + ( 7 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{76} + ( 5 + 2 \beta_{2} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{77} + ( 6 - 4 \beta_{1} - \beta_{4} + 2 \beta_{5} + 6 \beta_{6} ) q^{78} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{79} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{80} + ( 6 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{81} + ( 6 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{82} + ( -8 + 4 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{83} + ( 7 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 7 \beta_{5} + 4 \beta_{6} ) q^{84} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( 8 - 5 \beta_{1} + 2 \beta_{3} ) q^{86} + ( -6 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{87} + ( 4 - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{88} + q^{89} + ( 2 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{90} + ( 5 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - \beta_{6} ) q^{91} + ( 1 + 7 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{92} + ( 4 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{93} + ( 8 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{94} + ( -3 \beta_{1} - \beta_{2} + \beta_{6} ) q^{95} + ( 8 + 6 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{96} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{97} + ( 5 + 6 \beta_{1} + 8 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} ) q^{98} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} - 4 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 4q^{2} - 8q^{3} + 8q^{4} + 7q^{5} - 2q^{6} - 16q^{7} - 12q^{8} + 11q^{9} + O(q^{10})$$ $$7q - 4q^{2} - 8q^{3} + 8q^{4} + 7q^{5} - 2q^{6} - 16q^{7} - 12q^{8} + 11q^{9} - 4q^{10} - 10q^{11} - 11q^{12} - 7q^{13} + 3q^{14} - 8q^{15} + 10q^{16} - 13q^{17} + 4q^{18} - 7q^{19} + 8q^{20} + 16q^{21} + 2q^{22} - 13q^{23} + 4q^{24} + 7q^{25} + q^{26} - 23q^{27} - 21q^{28} - 4q^{29} - 2q^{30} + q^{31} - 13q^{32} - 6q^{33} + 10q^{34} - 16q^{35} + 20q^{36} - 5q^{37} - 40q^{38} - 13q^{39} - 12q^{40} + 5q^{41} + 30q^{42} - 31q^{43} - 21q^{44} + 11q^{45} + 16q^{46} - 14q^{47} - 7q^{48} + 19q^{49} - 4q^{50} - q^{51} - 13q^{53} - 17q^{54} - 10q^{55} - q^{56} + 21q^{57} + 17q^{58} - 14q^{59} - 11q^{60} + 3q^{61} + 26q^{62} - 54q^{63} + 14q^{64} - 7q^{65} + 36q^{66} + q^{67} - 35q^{68} + 31q^{69} + 3q^{70} - 8q^{71} + 53q^{72} + 9q^{73} - 35q^{74} - 8q^{75} + 40q^{76} + 42q^{77} + 46q^{78} + 9q^{79} + 10q^{80} + 35q^{81} + 29q^{82} - 42q^{83} + 55q^{84} - 13q^{85} + 35q^{86} + 6q^{87} + 30q^{88} + 7q^{89} + 4q^{90} + 31q^{91} + 19q^{92} + 24q^{93} + 37q^{94} - 7q^{95} + 44q^{96} - 7q^{97} + 9q^{98} + 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 8 x^{5} + 6 x^{4} + 19 x^{3} - 10 x^{2} - 12 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 4 \nu^{2} - 2 \nu - 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - \nu^{5} - 7 \nu^{4} + 5 \nu^{3} + 13 \nu^{2} - 4 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + \beta_{5} + 8 \beta_{3} + \beta_{2} + 23 \beta_{1} + 29$$

## 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
 0.498937 0.885013 −1.07810 −1.49803 1.89340 −1.96388 2.26266
−2.75106 −0.459905 5.56834 1.00000 1.26523 0.587818 −9.81674 −2.78849 −2.75106
1.2 −2.21675 −2.76669 2.91399 1.00000 6.13307 −3.75132 −2.02609 4.65459 −2.21675
1.3 −1.83770 2.10226 1.37716 1.00000 −3.86333 −4.72699 1.14460 1.41950 −1.83770
1.4 −0.755898 −2.82660 −1.42862 1.00000 2.13662 0.0498231 2.59169 4.98967 −0.755898
1.5 0.584976 0.172659 −1.65780 1.00000 0.101002 −2.74591 −2.13973 −2.97019 0.584976
1.6 0.856822 −0.931146 −1.26586 1.00000 −0.797826 −0.580377 −2.79826 −2.13297 0.856822
1.7 2.11962 −3.29058 2.49279 1.00000 −6.97477 −4.83304 1.04452 7.82788 2.11962
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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 445.2.a.f 7
3.b odd 2 1 4005.2.a.o 7
4.b odd 2 1 7120.2.a.bj 7
5.b even 2 1 2225.2.a.k 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.a.f 7 1.a even 1 1 trivial
2225.2.a.k 7 5.b even 2 1
4005.2.a.o 7 3.b odd 2 1
7120.2.a.bj 7 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$89$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} + 4 T_{2}^{6} - 3 T_{2}^{5} - 24 T_{2}^{4} - 8 T_{2}^{3} + 29 T_{2}^{2} + 6 T_{2} - 9$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(445))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 11 T^{2} + 24 T^{3} + 46 T^{4} + 77 T^{5} + 118 T^{6} + 171 T^{7} + 236 T^{8} + 308 T^{9} + 368 T^{10} + 384 T^{11} + 352 T^{12} + 256 T^{13} + 128 T^{14}$$
$3$ $$1 + 8 T + 37 T^{2} + 125 T^{3} + 340 T^{4} + 780 T^{5} + 1576 T^{6} + 2866 T^{7} + 4728 T^{8} + 7020 T^{9} + 9180 T^{10} + 10125 T^{11} + 8991 T^{12} + 5832 T^{13} + 2187 T^{14}$$
$5$ $$( 1 - T )^{7}$$
$7$ $$1 + 16 T + 143 T^{2} + 908 T^{3} + 4508 T^{4} + 18272 T^{5} + 61958 T^{6} + 177804 T^{7} + 433706 T^{8} + 895328 T^{9} + 1546244 T^{10} + 2180108 T^{11} + 2403401 T^{12} + 1882384 T^{13} + 823543 T^{14}$$
$11$ $$1 + 10 T + 91 T^{2} + 511 T^{3} + 2672 T^{4} + 10626 T^{5} + 41830 T^{6} + 136602 T^{7} + 460130 T^{8} + 1285746 T^{9} + 3556432 T^{10} + 7481551 T^{11} + 14655641 T^{12} + 17715610 T^{13} + 19487171 T^{14}$$
$13$ $$1 + 7 T + 47 T^{2} + 168 T^{3} + 622 T^{4} + 2167 T^{5} + 9118 T^{6} + 36132 T^{7} + 118534 T^{8} + 366223 T^{9} + 1366534 T^{10} + 4798248 T^{11} + 17450771 T^{12} + 33787663 T^{13} + 62748517 T^{14}$$
$17$ $$1 + 13 T + 108 T^{2} + 591 T^{3} + 3017 T^{4} + 14845 T^{5} + 78122 T^{6} + 344702 T^{7} + 1328074 T^{8} + 4290205 T^{9} + 14822521 T^{10} + 49360911 T^{11} + 153344556 T^{12} + 313788397 T^{13} + 410338673 T^{14}$$
$19$ $$1 + 7 T + 81 T^{2} + 366 T^{3} + 2784 T^{4} + 9257 T^{5} + 61132 T^{6} + 176376 T^{7} + 1161508 T^{8} + 3341777 T^{9} + 19095456 T^{10} + 47697486 T^{11} + 200564019 T^{12} + 329321167 T^{13} + 893871739 T^{14}$$
$23$ $$1 + 13 T + 144 T^{2} + 1105 T^{3} + 7943 T^{4} + 47855 T^{5} + 274740 T^{6} + 1359354 T^{7} + 6319020 T^{8} + 25315295 T^{9} + 96642481 T^{10} + 309224305 T^{11} + 926833392 T^{12} + 1924466557 T^{13} + 3404825447 T^{14}$$
$29$ $$1 + 4 T + 113 T^{2} + 625 T^{3} + 6740 T^{4} + 40902 T^{5} + 266814 T^{6} + 1538922 T^{7} + 7737606 T^{8} + 34398582 T^{9} + 164381860 T^{10} + 442050625 T^{11} + 2317759837 T^{12} + 2379293284 T^{13} + 17249876309 T^{14}$$
$31$ $$1 - T + 77 T^{2} - 34 T^{3} + 3570 T^{4} - 2367 T^{5} + 134894 T^{6} - 138904 T^{7} + 4181714 T^{8} - 2274687 T^{9} + 106353870 T^{10} - 31399714 T^{11} + 2204444627 T^{12} - 887503681 T^{13} + 27512614111 T^{14}$$
$37$ $$1 + 5 T + 168 T^{2} + 803 T^{3} + 14161 T^{4} + 59989 T^{5} + 771494 T^{6} + 2748646 T^{7} + 28545278 T^{8} + 82124941 T^{9} + 717297133 T^{10} + 1504951283 T^{11} + 11649784776 T^{12} + 12828632045 T^{13} + 94931877133 T^{14}$$
$41$ $$1 - 5 T + 92 T^{2} - 673 T^{3} + 7237 T^{4} - 46413 T^{5} + 387726 T^{6} - 2290138 T^{7} + 15896766 T^{8} - 78020253 T^{9} + 498781277 T^{10} - 1901737153 T^{11} + 10658770492 T^{12} - 23750521205 T^{13} + 194754273881 T^{14}$$
$43$ $$1 + 31 T + 687 T^{2} + 10436 T^{3} + 129854 T^{4} + 1291529 T^{5} + 10962460 T^{6} + 77409324 T^{7} + 471385780 T^{8} + 2388037121 T^{9} + 10324301978 T^{10} + 35678607236 T^{11} + 100994800341 T^{12} + 195962254519 T^{13} + 271818611107 T^{14}$$
$47$ $$1 + 14 T + 271 T^{2} + 2621 T^{3} + 31508 T^{4} + 240678 T^{5} + 2207362 T^{6} + 13822254 T^{7} + 103746014 T^{8} + 531657702 T^{9} + 3271255084 T^{10} + 12789643901 T^{11} + 62152496897 T^{12} + 150909014606 T^{13} + 506623120463 T^{14}$$
$53$ $$1 + 13 T + 269 T^{2} + 2568 T^{3} + 34060 T^{4} + 265121 T^{5} + 2673394 T^{6} + 17174660 T^{7} + 141689882 T^{8} + 744724889 T^{9} + 5070750620 T^{10} + 20262755208 T^{11} + 112494587617 T^{12} + 288136694677 T^{13} + 1174711139837 T^{14}$$
$59$ $$1 + 14 T + 253 T^{2} + 2446 T^{3} + 31472 T^{4} + 260310 T^{5} + 2563464 T^{6} + 17624256 T^{7} + 151244376 T^{8} + 906139110 T^{9} + 6463687888 T^{10} + 29639065006 T^{11} + 180875847647 T^{12} + 590527470974 T^{13} + 2488651484819 T^{14}$$
$61$ $$1 - 3 T + 222 T^{2} - 759 T^{3} + 29403 T^{4} - 83651 T^{5} + 2524030 T^{6} - 6527094 T^{7} + 153965830 T^{8} - 311265371 T^{9} + 6673922343 T^{10} - 10508993319 T^{11} + 187500378822 T^{12} - 154561123083 T^{13} + 3142742836021 T^{14}$$
$67$ $$1 - T + 328 T^{2} - 279 T^{3} + 53091 T^{4} - 39287 T^{5} + 5339384 T^{6} - 3265170 T^{7} + 357738728 T^{8} - 176359343 T^{9} + 15967808433 T^{10} - 5622162759 T^{11} + 442841035096 T^{12} - 90458382169 T^{13} + 6060711605323 T^{14}$$
$71$ $$1 + 8 T + 187 T^{2} + 1328 T^{3} + 22776 T^{4} + 168048 T^{5} + 2047314 T^{6} + 13028752 T^{7} + 145359294 T^{8} + 847129968 T^{9} + 8151780936 T^{10} + 33746712368 T^{11} + 337390888637 T^{12} + 1024802271368 T^{13} + 9095120158391 T^{14}$$
$73$ $$1 - 9 T + 286 T^{2} - 1613 T^{3} + 37567 T^{4} - 166989 T^{5} + 3689378 T^{6} - 14466938 T^{7} + 269324594 T^{8} - 889884381 T^{9} + 14614201639 T^{10} - 45806362733 T^{11} + 592898475598 T^{12} - 1362008036601 T^{13} + 11047398519097 T^{14}$$
$79$ $$1 - 9 T + 430 T^{2} - 2867 T^{3} + 81347 T^{4} - 413191 T^{5} + 9305818 T^{6} - 38366618 T^{7} + 735159622 T^{8} - 2578725031 T^{9} + 40107243533 T^{10} - 111669882227 T^{11} + 1323134251570 T^{12} - 2187787099689 T^{13} + 19203908986159 T^{14}$$
$83$ $$1 + 42 T + 1008 T^{2} + 16390 T^{3} + 206255 T^{4} + 2129142 T^{5} + 19880600 T^{6} + 179268272 T^{7} + 1650089800 T^{8} + 14667659238 T^{9} + 117933927685 T^{10} + 777841881190 T^{11} + 3970552968144 T^{12} + 13731495681498 T^{13} + 27136050989627 T^{14}$$
$89$ $$( 1 - T )^{7}$$
$97$ $$1 + 7 T + 609 T^{2} + 3724 T^{3} + 164856 T^{4} + 856227 T^{5} + 25710150 T^{6} + 108804964 T^{7} + 2493884550 T^{8} + 8056239843 T^{9} + 150459620088 T^{10} + 329683042444 T^{11} + 5229690216513 T^{12} + 5830804034503 T^{13} + 80798284478113 T^{14}$$