# Properties

 Label 3525.2.a.bh Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $0$ Dimension $13$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$0$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} - 923 x^{4} - 107 x^{3} + 293 x^{2} + 12 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 705) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{12}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} - 923 x^{4} - 107 x^{3} + 293 x^{2} + 12 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{9} - 2 \nu^{8} - 13 \nu^{7} + 22 \nu^{6} + 55 \nu^{5} - 72 \nu^{4} - 79 \nu^{3} + 68 \nu^{2} + 32 \nu - 6$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 28 \nu^{9} + 78 \nu^{8} - 142 \nu^{7} - 158 \nu^{6} + 326 \nu^{5} + 99 \nu^{4} - 336 \nu^{3} - 7 \nu^{2} + 134 \nu + 10$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} + 17 \nu^{9} - 12 \nu^{8} - 105 \nu^{7} + 46 \nu^{6} + 281 \nu^{5} - 62 \nu^{4} - 308 \nu^{3} + 13 \nu^{2} + 96 \nu + 4$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{12} + 2 \nu^{11} + 19 \nu^{10} - 36 \nu^{9} - 134 \nu^{8} + 238 \nu^{7} + 430 \nu^{6} - 694 \nu^{5} - 627 \nu^{4} + 824 \nu^{3} + 379 \nu^{2} - 294 \nu - 50$$$$)/8$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{12} + 2 \nu^{11} + 17 \nu^{10} - 32 \nu^{9} - 104 \nu^{8} + 186 \nu^{7} + 268 \nu^{6} - 466 \nu^{5} - 257 \nu^{4} + 440 \nu^{3} + 63 \nu^{2} - 98 \nu - 2$$$$)/4$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{12} - 2 \nu^{11} - 17 \nu^{10} + 32 \nu^{9} + 104 \nu^{8} - 186 \nu^{7} - 268 \nu^{6} + 470 \nu^{5} + 253 \nu^{4} - 468 \nu^{3} - 47 \nu^{2} + 130 \nu - 2$$$$)/4$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{12} - 4 \nu^{11} - 17 \nu^{10} + 68 \nu^{9} + 110 \nu^{8} - 418 \nu^{7} - 326 \nu^{6} + 1106 \nu^{5} + 403 \nu^{4} - 1182 \nu^{3} - 169 \nu^{2} + 390 \nu + 6$$$$)/8$$ $$\beta_{12}$$ $$=$$ $$($$$$\nu^{12} - 2 \nu^{11} - 17 \nu^{10} + 32 \nu^{9} + 106 \nu^{8} - 190 \nu^{7} - 290 \nu^{6} + 506 \nu^{5} + 327 \nu^{4} - 552 \nu^{3} - 113 \nu^{2} + 166 \nu - 2$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + \beta_{9} + \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 28 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$\beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} + 10 \beta_{4} + 10 \beta_{3} + 46 \beta_{2} + 14 \beta_{1} + 97$$ $$\nu^{7}$$ $$=$$ $$\beta_{12} - 2 \beta_{11} + 13 \beta_{10} + 11 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} + 14 \beta_{4} + 53 \beta_{3} + 80 \beta_{2} + 167 \beta_{1} + 111$$ $$\nu^{8}$$ $$=$$ $$15 \beta_{12} - 4 \beta_{11} + 17 \beta_{10} + 15 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 17 \beta_{6} - 2 \beta_{5} + 83 \beta_{4} + 77 \beta_{3} + 302 \beta_{2} + 139 \beta_{1} + 622$$ $$\nu^{9}$$ $$=$$ $$21 \beta_{12} - 34 \beta_{11} + 126 \beta_{10} + 96 \beta_{9} + 17 \beta_{8} + 17 \beta_{7} - 51 \beta_{6} - 15 \beta_{5} + 145 \beta_{4} + 334 \beta_{3} + 597 \beta_{2} + 1036 \beta_{1} + 926$$ $$\nu^{10}$$ $$=$$ $$160 \beta_{12} - 76 \beta_{11} + 202 \beta_{10} + 162 \beta_{9} + 126 \beta_{8} + 38 \beta_{7} - 198 \beta_{6} - 34 \beta_{5} + 660 \beta_{4} + 540 \beta_{3} + 2003 \beta_{2} + 1196 \beta_{1} + 4125$$ $$\nu^{11}$$ $$=$$ $$278 \beta_{12} - 396 \beta_{11} + 1102 \beta_{10} + 786 \beta_{9} + 200 \beta_{8} + 196 \beta_{7} - 592 \beta_{6} - 160 \beta_{5} + 1338 \beta_{4} + 2067 \beta_{3} + 4325 \beta_{2} + 6611 \beta_{1} + 7325$$ $$\nu^{12}$$ $$=$$ $$1498 \beta_{12} - 952 \beta_{11} + 2058 \beta_{10} + 1538 \beta_{9} + 1100 \beta_{8} + 472 \beta_{7} - 1976 \beta_{6} - 396 \beta_{5} + 5185 \beta_{4} + 3611 \beta_{3} + 13441 \beta_{2} + 9557 \beta_{1} + 28020$$

## 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
 2.74237 2.59761 2.57687 1.67335 1.32490 0.747203 0.0655745 −0.105426 −0.697808 −1.22883 −2.12231 −2.24873 −2.32477
−2.74237 −1.00000 5.52059 0 2.74237 −1.18262 −9.65475 1.00000 0
1.2 −2.59761 −1.00000 4.74755 0 2.59761 4.83808 −7.13706 1.00000 0
1.3 −2.57687 −1.00000 4.64024 0 2.57687 −1.51741 −6.80354 1.00000 0
1.4 −1.67335 −1.00000 0.800100 0 1.67335 2.75180 2.00785 1.00000 0
1.5 −1.32490 −1.00000 −0.244653 0 1.32490 −1.72082 2.97393 1.00000 0
1.6 −0.747203 −1.00000 −1.44169 0 0.747203 −0.654660 2.57164 1.00000 0
1.7 −0.0655745 −1.00000 −1.99570 0 0.0655745 −1.54896 0.262016 1.00000 0
1.8 0.105426 −1.00000 −1.98889 0 −0.105426 3.91364 −0.420531 1.00000 0
1.9 0.697808 −1.00000 −1.51306 0 −0.697808 −3.46613 −2.45144 1.00000 0
1.10 1.22883 −1.00000 −0.489976 0 −1.22883 1.54483 −3.05976 1.00000 0
1.11 2.12231 −1.00000 2.50418 0 −2.12231 −4.63433 1.07002 1.00000 0
1.12 2.24873 −1.00000 3.05678 0 −2.24873 1.38491 2.37640 1.00000 0
1.13 2.32477 −1.00000 3.40453 0 −2.32477 4.29166 3.26521 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$47$$ $$1$$

## 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 3525.2.a.bh 13
5.b even 2 1 3525.2.a.bi 13
5.c odd 4 2 705.2.c.c 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.c 26 5.c odd 4 2
3525.2.a.bh 13 1.a even 1 1 trivial
3525.2.a.bi 13 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{13} + \cdots$$ $$T_{7}^{13} - \cdots$$ $$T_{11}^{13} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 12 T - 293 T^{2} - 107 T^{3} + 923 T^{4} + 309 T^{5} - 852 T^{6} - 288 T^{7} + 316 T^{8} + 106 T^{9} - 51 T^{10} - 17 T^{11} + 3 T^{12} + T^{13}$$
$3$ $$( 1 + T )^{13}$$
$5$ $$T^{13}$$
$7$ $$24064 + 56192 T - 1792 T^{2} - 72464 T^{3} - 22104 T^{4} + 33677 T^{5} + 12580 T^{6} - 7272 T^{7} - 2388 T^{8} + 842 T^{9} + 172 T^{10} - 48 T^{11} - 4 T^{12} + T^{13}$$
$11$ $$-2451456 - 6953472 T - 505856 T^{2} + 4668608 T^{3} - 98752 T^{4} - 1014460 T^{5} + 104072 T^{6} + 94552 T^{7} - 15976 T^{8} - 3548 T^{9} + 892 T^{10} + 17 T^{11} - 16 T^{12} + T^{13}$$
$13$ $$892224 - 2715936 T + 954080 T^{2} + 1698016 T^{3} - 840412 T^{4} - 295298 T^{5} + 190888 T^{6} + 12789 T^{7} - 17462 T^{8} + 771 T^{9} + 662 T^{10} - 65 T^{11} - 8 T^{12} + T^{13}$$
$17$ $$60736 - 121504 T - 895200 T^{2} - 130688 T^{3} + 1161588 T^{4} - 164530 T^{5} - 260040 T^{6} + 21977 T^{7} + 23986 T^{8} - 205 T^{9} - 954 T^{10} - 49 T^{11} + 12 T^{12} + T^{13}$$
$19$ $$-899200 - 2726880 T - 616000 T^{2} + 3390552 T^{3} + 1681560 T^{4} - 742456 T^{5} - 338742 T^{6} + 99255 T^{7} + 22906 T^{8} - 7923 T^{9} - 144 T^{10} + 245 T^{11} - 28 T^{12} + T^{13}$$
$23$ $$60416 + 1701760 T + 69056 T^{2} - 2467504 T^{3} + 154772 T^{4} + 962277 T^{5} - 125830 T^{6} - 119328 T^{7} + 15422 T^{8} + 5618 T^{9} - 546 T^{10} - 120 T^{11} + 6 T^{12} + T^{13}$$
$29$ $$17120 + 42920 T - 14008 T^{2} - 109156 T^{3} - 63390 T^{4} + 41837 T^{5} + 38906 T^{6} - 3988 T^{7} - 7382 T^{8} + 122 T^{9} + 542 T^{10} - 24 T^{11} - 12 T^{12} + T^{13}$$
$31$ $$17920 + 121792 T + 57152 T^{2} - 537984 T^{3} + 85872 T^{4} + 429192 T^{5} - 245120 T^{6} + 852 T^{7} + 29504 T^{8} - 6752 T^{9} - 116 T^{10} + 212 T^{11} - 26 T^{12} + T^{13}$$
$37$ $$-389395456 - 990185728 T + 853411968 T^{2} - 7905600 T^{3} - 108466992 T^{4} + 12371284 T^{5} + 5264848 T^{6} - 814904 T^{7} - 117876 T^{8} + 21416 T^{9} + 1184 T^{10} - 247 T^{11} - 4 T^{12} + T^{13}$$
$41$ $$58496 - 1073632 T + 2372256 T^{2} - 675784 T^{3} - 2924832 T^{4} + 3805552 T^{5} - 1983698 T^{6} + 438337 T^{7} + 1264 T^{8} - 16709 T^{9} + 1986 T^{10} + 79 T^{11} - 24 T^{12} + T^{13}$$
$43$ $$-308908544 + 208028928 T + 198561792 T^{2} - 77406336 T^{3} - 36547072 T^{4} + 10249248 T^{5} + 2571040 T^{6} - 596176 T^{7} - 81008 T^{8} + 16416 T^{9} + 1152 T^{10} - 210 T^{11} - 6 T^{12} + T^{13}$$
$47$ $$( 1 + T )^{13}$$
$53$ $$-25854912 - 65768736 T - 35673568 T^{2} + 19537120 T^{3} + 18044580 T^{4} + 1196574 T^{5} - 1568552 T^{6} - 227755 T^{7} + 55868 T^{8} + 9733 T^{9} - 922 T^{10} - 167 T^{11} + 6 T^{12} + T^{13}$$
$59$ $$-178880 + 1839840 T - 2652128 T^{2} - 1780928 T^{3} + 3530484 T^{4} + 111654 T^{5} - 934736 T^{6} + 111057 T^{7} + 73712 T^{8} - 16669 T^{9} - 282 T^{10} + 359 T^{11} - 34 T^{12} + T^{13}$$
$61$ $$27648112896 + 287734272 T - 12942005728 T^{2} + 1856748176 T^{3} + 1544438816 T^{4} - 479031528 T^{5} + 17277482 T^{6} + 9035681 T^{7} - 1001276 T^{8} - 33321 T^{9} + 9098 T^{10} - 193 T^{11} - 24 T^{12} + T^{13}$$
$67$ $$2592738688 + 1533792704 T - 10175277696 T^{2} + 5629868416 T^{3} - 485480512 T^{4} - 263977096 T^{5} + 45824024 T^{6} + 3697388 T^{7} - 1015112 T^{8} - 4212 T^{9} + 8652 T^{10} - 232 T^{11} - 24 T^{12} + T^{13}$$
$71$ $$-9039415936 + 17507763008 T - 12575574688 T^{2} + 3718046672 T^{3} - 98906840 T^{4} - 177853748 T^{5} + 27120946 T^{6} + 2167695 T^{7} - 659258 T^{8} + 4137 T^{9} + 6108 T^{10} - 219 T^{11} - 20 T^{12} + T^{13}$$
$73$ $$594804992 + 1141024640 T - 122391168 T^{2} - 805047424 T^{3} - 55790448 T^{4} + 102344184 T^{5} + 7850224 T^{6} - 4364972 T^{7} - 240232 T^{8} + 69584 T^{9} + 2256 T^{10} - 456 T^{11} - 6 T^{12} + T^{13}$$
$79$ $$953661440 - 71421440 T - 715824896 T^{2} - 82899008 T^{3} + 157579776 T^{4} + 42185316 T^{5} - 5191360 T^{6} - 2290088 T^{7} + 3736 T^{8} + 42488 T^{9} + 960 T^{10} - 335 T^{11} - 6 T^{12} + T^{13}$$
$83$ $$-35253854208 - 39110252544 T - 11405691904 T^{2} + 1446557184 T^{3} + 1075276672 T^{4} + 40694432 T^{5} - 36190272 T^{6} - 2842688 T^{7} + 589856 T^{8} + 51736 T^{9} - 4696 T^{10} - 378 T^{11} + 14 T^{12} + T^{13}$$
$89$ $$-461035624960 + 4082174720 T + 182296593024 T^{2} + 27432461376 T^{3} - 10146710240 T^{4} - 1063681584 T^{5} + 240465304 T^{6} + 13592796 T^{7} - 2832024 T^{8} - 52568 T^{9} + 16228 T^{10} - 142 T^{11} - 36 T^{12} + T^{13}$$
$97$ $$27544367104 - 25975038976 T - 16186004992 T^{2} + 7723190016 T^{3} + 457742144 T^{4} - 439306800 T^{5} + 16610432 T^{6} + 9210320 T^{7} - 776120 T^{8} - 68240 T^{9} + 9252 T^{10} + T^{11} - 32 T^{12} + T^{13}$$