# Properties

 Label 4008.2.a.m Level $4008$ Weight $2$ Character orbit 4008.a Self dual yes Analytic conductor $32.004$ Analytic rank $0$ Dimension $13$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4008 = 2^{3} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4008.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0040411301$$ Analytic rank: $$0$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{12}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_{1} q^{5} -\beta_{9} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta_{1} q^{5} -\beta_{9} q^{7} + q^{9} + ( 1 + \beta_{5} ) q^{11} + ( 1 + \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( 1 - \beta_{2} ) q^{17} + ( 1 - \beta_{9} + \beta_{10} ) q^{19} -\beta_{9} q^{21} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{23} + ( 3 - \beta_{5} + \beta_{6} ) q^{25} + q^{27} -\beta_{7} q^{29} + ( -1 + \beta_{4} + \beta_{10} ) q^{31} + ( 1 + \beta_{5} ) q^{33} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{35} + ( 1 - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{37} + ( 1 + \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{4} + \beta_{10} - \beta_{12} ) q^{41} + ( -\beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{43} + \beta_{1} q^{45} + ( -1 - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} ) q^{47} + ( \beta_{1} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{49} + ( 1 - \beta_{2} ) q^{51} + ( -1 + \beta_{1} + \beta_{9} + \beta_{10} ) q^{53} + ( -2 \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{55} + ( 1 - \beta_{9} + \beta_{10} ) q^{57} + ( 2 - \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} ) q^{59} + ( 3 + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{61} -\beta_{9} q^{63} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{65} + ( 2 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{67} + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{69} + ( 1 - \beta_{1} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{71} + ( 4 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{12} ) q^{73} + ( 3 - \beta_{5} + \beta_{6} ) q^{75} + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{77} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( 4 - 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{83} + ( 3 + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{85} -\beta_{7} q^{87} + ( 3 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{89} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{91} + ( -1 + \beta_{4} + \beta_{10} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{4} + \beta_{7} + 2 \beta_{10} ) q^{95} + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{97} + ( 1 + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$13q + 13q^{3} + 2q^{5} + q^{7} + 13q^{9} + O(q^{10})$$ $$13q + 13q^{3} + 2q^{5} + q^{7} + 13q^{9} + 11q^{11} + 12q^{13} + 2q^{15} + 15q^{17} + 14q^{19} + q^{21} + 9q^{23} + 37q^{25} + 13q^{27} - 3q^{29} - 17q^{31} + 11q^{33} + 15q^{35} + 16q^{37} + 12q^{39} + 12q^{41} + 20q^{43} + 2q^{45} - 6q^{47} + 26q^{49} + 15q^{51} - 12q^{53} + 7q^{55} + 14q^{57} + 14q^{59} + 24q^{61} + q^{63} + 8q^{65} + 3q^{67} + 9q^{69} + 17q^{71} + 34q^{73} + 37q^{75} + 30q^{77} + 10q^{79} + 13q^{81} + 44q^{83} + 25q^{85} - 3q^{87} + 25q^{89} + 29q^{91} - 17q^{93} - 15q^{95} + 38q^{97} + 11q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + 26911 x^{5} - 72240 x^{4} - 14532 x^{3} + 112850 x^{2} - 72184 x + 12144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-6207348777 \nu^{12} - 12640530894 \nu^{11} + 576861363065 \nu^{10} + 295838273673 \nu^{9} - 16627326024925 \nu^{8} + 5016057310831 \nu^{7} + 201417980697138 \nu^{6} - 196205928405368 \nu^{5} - 1037037612348531 \nu^{4} + 1724063123712248 \nu^{3} + 1650853947740980 \nu^{2} - 4373968597429142 \nu + 1211669325949820$$$$)/ 111451400959900$$ $$\beta_{3}$$ $$=$$ $$($$$$126516435801 \nu^{12} - 351643713690 \nu^{11} - 4532098501229 \nu^{10} + 14015645726951 \nu^{9} + 48247617829913 \nu^{8} - 193085396549923 \nu^{7} - 90263735212158 \nu^{6} + 1012871433621172 \nu^{5} - 714691137288825 \nu^{4} - 993417294727940 \nu^{3} + 598916455112988 \nu^{2} - 3707736754746094 \nu + 3605634089719788$$$$)/ 557257004799500$$ $$\beta_{4}$$ $$=$$ $$($$$$8155241826 \nu^{12} - 24439858116 \nu^{11} - 363301449706 \nu^{10} + 1102737562371 \nu^{9} + 5918945839452 \nu^{8} - 18714943276923 \nu^{7} - 42939671149010 \nu^{6} + 149114772131121 \nu^{5} + 129168526434186 \nu^{4} - 552823809736903 \nu^{3} - 61985566741568 \nu^{2} + 719430531278666 \nu - 261781210799648$$$$)/ 27862850239975$$ $$\beta_{5}$$ $$=$$ $$($$$$42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} - 30214877144177 \nu^{7} - 406570644479846 \nu^{6} + 236986143576626 \nu^{5} + 1772258179746447 \nu^{4} - 967840084997036 \nu^{3} - 2780474603245950 \nu^{2} + 1829362384230014 \nu - 9603799562310$$$$)/ 55725700479950$$ $$\beta_{6}$$ $$=$$ $$($$$$42253210649 \nu^{12} - 42904580362 \nu^{11} - 2192557864275 \nu^{10} + 1857640717639 \nu^{9} + 43420560752365 \nu^{8} - 30214877144177 \nu^{7} - 406570644479846 \nu^{6} + 236986143576626 \nu^{5} + 1772258179746447 \nu^{4} - 967840084997036 \nu^{3} - 2724748902766000 \nu^{2} + 1829362384230014 \nu - 455409403401910$$$$)/ 55725700479950$$ $$\beta_{7}$$ $$=$$ $$($$$$138385219703 \nu^{12} - 105435570025 \nu^{11} - 6791649579897 \nu^{10} + 5771355044103 \nu^{9} + 127066606088584 \nu^{8} - 121899916781369 \nu^{7} - 1131795821929059 \nu^{6} + 1217996115089036 \nu^{5} + 4728856566083805 \nu^{4} - 5762733210647085 \nu^{3} - 6707972583182866 \nu^{2} + 10417867358098218 \nu - 2901404480931816$$$$)/ 139314251199875$$ $$\beta_{8}$$ $$=$$ $$($$$$111023509273 \nu^{12} - 148746957474 \nu^{11} - 5533191476745 \nu^{10} + 7117312024823 \nu^{9} + 104000386651945 \nu^{8} - 129559753690019 \nu^{7} - 912294162144422 \nu^{6} + 1109495790415652 \nu^{5} + 3686070397711199 \nu^{4} - 4408101894281592 \nu^{3} - 5320985228785100 \nu^{2} + 6347787481933558 \nu - 629228580752360$$$$)/ 111451400959900$$ $$\beta_{9}$$ $$=$$ $$($$$$-323942240303 \nu^{12} + 655210507370 \nu^{11} + 16238231918487 \nu^{10} - 30174775561853 \nu^{9} - 309203328646339 \nu^{8} + 524566352898419 \nu^{7} + 2761108370208174 \nu^{6} - 4297098906328266 \nu^{5} - 11231871461142475 \nu^{4} + 16707098673252820 \nu^{3} + 14675109344372636 \nu^{2} - 25352491816897518 \nu + 6802024064231936$$$$)/ 278628502399750$$ $$\beta_{10}$$ $$=$$ $$($$$$208422301596 \nu^{12} + 101272023955 \nu^{11} - 9810853937919 \nu^{10} - 3015289235129 \nu^{9} + 173349897433318 \nu^{8} + 10485608743617 \nu^{7} - 1435122721494978 \nu^{6} + 348943217873482 \nu^{5} + 5522030332256130 \nu^{4} - 3276596851987855 \nu^{3} - 7596810398586632 \nu^{2} + 7530986189108326 \nu - 1313465392184157$$$$)/ 139314251199875$$ $$\beta_{11}$$ $$=$$ $$($$$$889692540773 \nu^{12} - 1710596413190 \nu^{11} - 43690158347557 \nu^{10} + 79678278406623 \nu^{9} + 807810238699029 \nu^{8} - 1397243777337779 \nu^{7} - 6930554296860074 \nu^{6} + 11473935236033836 \nu^{5} + 26745397139049795 \nu^{4} - 44471075457654780 \nu^{3} - 31856351450553896 \nu^{2} + 67210663867287138 \nu - 20950363454455596$$$$)/ 557257004799500$$ $$\beta_{12}$$ $$=$$ $$($$$$224267911141 \nu^{12} + 63390661414 \nu^{11} - 11028950583521 \nu^{10} - 1496522255749 \nu^{9} + 205661421513157 \nu^{8} - 13254732421323 \nu^{7} - 1812887391077730 \nu^{6} + 558179494985656 \nu^{5} + 7475418034130291 \nu^{4} - 4406202874244528 \nu^{3} - 10980877487070868 \nu^{2} + 10312125439960266 \nu - 2219186123392628$$$$)/ 111451400959900$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + 8$$ $$\nu^{3}$$ $$=$$ $$-\beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 11 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{12} - \beta_{11} + 5 \beta_{10} - \beta_{9} + 2 \beta_{8} + 19 \beta_{6} - 17 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 91$$ $$\nu^{5}$$ $$=$$ $$-30 \beta_{12} - 18 \beta_{11} + 29 \beta_{10} + 7 \beta_{9} + 20 \beta_{8} + 3 \beta_{7} + 46 \beta_{6} - 16 \beta_{5} + 43 \beta_{4} - 21 \beta_{3} - 7 \beta_{2} + 143 \beta_{1} - 7$$ $$\nu^{6}$$ $$=$$ $$-137 \beta_{12} - 28 \beta_{11} + 144 \beta_{10} - 42 \beta_{9} + 41 \beta_{8} - 12 \beta_{7} + 335 \beta_{6} - 280 \beta_{5} - 100 \beta_{4} + 40 \beta_{3} - 102 \beta_{2} + 49 \beta_{1} + 1184$$ $$\nu^{7}$$ $$=$$ $$-665 \beta_{12} - 281 \beta_{11} + 632 \beta_{10} + 213 \beta_{9} + 350 \beta_{8} + 97 \beta_{7} + 905 \beta_{6} - 247 \beta_{5} + 753 \beta_{4} - 381 \beta_{3} - 199 \beta_{2} + 2061 \beta_{1} + 6$$ $$\nu^{8}$$ $$=$$ $$-2931 \beta_{12} - 676 \beta_{11} + 3142 \beta_{10} - 1097 \beta_{9} + 716 \beta_{8} - 320 \beta_{7} + 5857 \beta_{6} - 4654 \beta_{5} - 1956 \beta_{4} + 668 \beta_{3} - 2055 \beta_{2} + 1017 \beta_{1} + 16711$$ $$\nu^{9}$$ $$=$$ $$-13266 \beta_{12} - 4390 \beta_{11} + 12482 \beta_{10} + 4644 \beta_{9} + 6123 \beta_{8} + 2314 \beta_{7} + 16957 \beta_{6} - 4186 \beta_{5} + 12395 \beta_{4} - 6687 \beta_{3} - 4266 \beta_{2} + 31753 \beta_{1} + 1836$$ $$\nu^{10}$$ $$=$$ $$-57319 \beta_{12} - 14661 \beta_{11} + 61925 \beta_{10} - 23592 \beta_{9} + 12118 \beta_{8} - 6102 \beta_{7} + 102280 \beta_{6} - 78127 \beta_{5} - 34966 \beta_{4} + 10790 \beta_{3} - 38346 \beta_{2} + 20530 \beta_{1} + 249558$$ $$\nu^{11}$$ $$=$$ $$-251329 \beta_{12} - 70391 \beta_{11} + 235453 \beta_{10} + 89347 \beta_{9} + 107880 \beta_{8} + 48175 \beta_{7} + 310864 \beta_{6} - 76539 \beta_{5} + 199760 \beta_{4} - 116067 \beta_{3} - 83231 \beta_{2} + 510061 \beta_{1} + 60252$$ $$\nu^{12}$$ $$=$$ $$-1073038 \beta_{12} - 294937 \beta_{11} + 1162932 \beta_{10} - 459487 \beta_{9} + 205019 \beta_{8} - 103543 \beta_{7} + 1786032 \beta_{6} - 1320651 \beta_{5} - 597828 \beta_{4} + 173012 \beta_{3} - 691732 \beta_{2} + 410169 \beta_{1} + 3880740$$

## 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
 −4.02595 −3.60065 −3.05600 −2.71099 −2.04519 0.284022 0.649069 1.25667 1.68837 2.79706 3.18209 3.38131 4.20017
0 1.00000 0 −4.02595 0 0.910115 0 1.00000 0
1.2 0 1.00000 0 −3.60065 0 −4.85776 0 1.00000 0
1.3 0 1.00000 0 −3.05600 0 2.59989 0 1.00000 0
1.4 0 1.00000 0 −2.71099 0 −0.775439 0 1.00000 0
1.5 0 1.00000 0 −2.04519 0 1.81275 0 1.00000 0
1.6 0 1.00000 0 0.284022 0 −3.90646 0 1.00000 0
1.7 0 1.00000 0 0.649069 0 4.19646 0 1.00000 0
1.8 0 1.00000 0 1.25667 0 −4.55521 0 1.00000 0
1.9 0 1.00000 0 1.68837 0 0.722089 0 1.00000 0
1.10 0 1.00000 0 2.79706 0 4.56518 0 1.00000 0
1.11 0 1.00000 0 3.18209 0 −1.39408 0 1.00000 0
1.12 0 1.00000 0 3.38131 0 2.18014 0 1.00000 0
1.13 0 1.00000 0 4.20017 0 −0.497670 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$167$$ $$1$$

## 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 4008.2.a.m 13
4.b odd 2 1 8016.2.a.bg 13

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.m 13 1.a even 1 1 trivial
8016.2.a.bg 13 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4008))$$:

 $$T_{5}^{13} - \cdots$$ $$T_{7}^{13} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{13}$$
$3$ $$( -1 + T )^{13}$$
$5$ $$12144 - 72184 T + 112850 T^{2} - 14532 T^{3} - 72240 T^{4} + 26911 T^{5} + 16968 T^{6} - 7582 T^{7} - 1879 T^{8} + 901 T^{9} + 99 T^{10} - 49 T^{11} - 2 T^{12} + T^{13}$$
$7$ $$-6016 - 2176 T + 30480 T^{2} - 1584 T^{3} - 44740 T^{4} + 13575 T^{5} + 19342 T^{6} - 8413 T^{7} - 2082 T^{8} + 1147 T^{9} + 81 T^{10} - 58 T^{11} - T^{12} + T^{13}$$
$11$ $$123008 + 111808 T - 789008 T^{2} + 523856 T^{3} + 280568 T^{4} - 329672 T^{5} + 28920 T^{6} + 47078 T^{7} - 12423 T^{8} - 1261 T^{9} + 704 T^{10} - 29 T^{11} - 11 T^{12} + T^{13}$$
$13$ $$1786432 + 10814368 T + 12821008 T^{2} + 26648 T^{3} - 4217920 T^{4} - 373800 T^{5} + 561750 T^{6} + 26020 T^{7} - 36717 T^{8} + 442 T^{9} + 1104 T^{10} - 61 T^{11} - 12 T^{12} + T^{13}$$
$17$ $$-23680624 - 108002816 T + 26237008 T^{2} + 35174692 T^{3} - 8057516 T^{4} - 4143086 T^{5} + 1050316 T^{6} + 202612 T^{7} - 63971 T^{8} - 2773 T^{9} + 1714 T^{10} - 54 T^{11} - 15 T^{12} + T^{13}$$
$19$ $$180736 + 336128 T - 1282752 T^{2} + 518768 T^{3} + 680048 T^{4} - 468804 T^{5} - 50652 T^{6} + 96660 T^{7} - 15685 T^{8} - 3680 T^{9} + 1124 T^{10} - 25 T^{11} - 14 T^{12} + T^{13}$$
$23$ $$368197632 - 647776256 T + 134828800 T^{2} + 172605600 T^{3} - 69395312 T^{4} - 6849140 T^{5} + 5775312 T^{6} - 203746 T^{7} - 172701 T^{8} + 13617 T^{9} + 2108 T^{10} - 209 T^{11} - 9 T^{12} + T^{13}$$
$29$ $$-134409216 + 100284160 T + 37467104 T^{2} - 35403376 T^{3} - 2612552 T^{4} + 4563532 T^{5} - 36576 T^{6} - 282724 T^{7} + 9211 T^{8} + 9171 T^{9} - 300 T^{10} - 151 T^{11} + 3 T^{12} + T^{13}$$
$31$ $$5259264 - 12827648 T - 13335328 T^{2} + 9475888 T^{3} + 7480684 T^{4} - 1439287 T^{5} - 1147353 T^{6} + 89324 T^{7} + 69262 T^{8} - 1249 T^{9} - 1822 T^{10} - 49 T^{11} + 17 T^{12} + T^{13}$$
$37$ $$108176 - 2308328 T + 8294524 T^{2} + 25716434 T^{3} - 6942088 T^{4} - 6227789 T^{5} + 1891898 T^{6} + 273566 T^{7} - 115291 T^{8} - 757 T^{9} + 2393 T^{10} - 97 T^{11} - 16 T^{12} + T^{13}$$
$41$ $$-262115744 + 569390256 T - 378281688 T^{2} - 3774804 T^{3} + 110453400 T^{4} - 47194370 T^{5} + 5431570 T^{6} + 990928 T^{7} - 275897 T^{8} + 5366 T^{9} + 3355 T^{10} - 210 T^{11} - 12 T^{12} + T^{13}$$
$43$ $$-1580774304 + 2840099648 T - 1843845176 T^{2} + 404327408 T^{3} + 79156500 T^{4} - 50571518 T^{5} + 4280880 T^{6} + 1458990 T^{7} - 263847 T^{8} - 9488 T^{9} + 4235 T^{10} - 112 T^{11} - 20 T^{12} + T^{13}$$
$47$ $$232066544 + 3130521520 T - 2099949156 T^{2} - 540010980 T^{3} + 353281876 T^{4} + 57028409 T^{5} - 16041051 T^{6} - 2414838 T^{7} + 290635 T^{8} + 44031 T^{9} - 2234 T^{10} - 351 T^{11} + 6 T^{12} + T^{13}$$
$53$ $$56844352 - 4859184 T - 47347322 T^{2} + 1927542 T^{3} + 13412066 T^{4} - 179681 T^{5} - 1634463 T^{6} - 16770 T^{7} + 88175 T^{8} + 3701 T^{9} - 2000 T^{10} - 143 T^{11} + 12 T^{12} + T^{13}$$
$59$ $$-6069184 + 42051200 T - 49135560 T^{2} - 49656186 T^{3} + 7247752 T^{4} + 11115267 T^{5} + 572647 T^{6} - 814630 T^{7} - 101321 T^{8} + 17287 T^{9} + 2194 T^{10} - 187 T^{11} - 14 T^{12} + T^{13}$$
$61$ $$-2717696 + 8985088 T + 82373376 T^{2} + 50435280 T^{3} - 32886736 T^{4} - 16313828 T^{5} + 5344744 T^{6} + 1102310 T^{7} - 272093 T^{8} - 17428 T^{9} + 5292 T^{10} - 87 T^{11} - 24 T^{12} + T^{13}$$
$67$ $$-7323689064 + 10066789960 T + 4964675718 T^{2} - 3775853066 T^{3} - 382484242 T^{4} + 282075619 T^{5} + 12938871 T^{6} - 7879585 T^{7} - 208758 T^{8} + 96039 T^{9} + 1364 T^{10} - 516 T^{11} - 3 T^{12} + T^{13}$$
$71$ $$618043392 + 1029017088 T + 234785088 T^{2} - 283462720 T^{3} - 99833008 T^{4} + 25356252 T^{5} + 10289372 T^{6} - 893250 T^{7} - 371681 T^{8} + 20095 T^{9} + 4879 T^{10} - 274 T^{11} - 17 T^{12} + T^{13}$$
$73$ $$3166561152 - 2181329856 T - 1037604928 T^{2} + 611693056 T^{3} + 87384920 T^{4} - 60020252 T^{5} - 1236548 T^{6} + 2551202 T^{7} - 112599 T^{8} - 44794 T^{9} + 3936 T^{10} + 211 T^{11} - 34 T^{12} + T^{13}$$
$79$ $$-341278848 + 2546496 T + 220632272 T^{2} - 8898804 T^{3} - 50401364 T^{4} + 3432254 T^{5} + 5062938 T^{6} - 469880 T^{7} - 213583 T^{8} + 24220 T^{9} + 2641 T^{10} - 300 T^{11} - 10 T^{12} + T^{13}$$
$83$ $$43837825568 - 15139875424 T - 13546867568 T^{2} + 3967456566 T^{3} + 1363225396 T^{4} - 366183789 T^{5} - 46996629 T^{6} + 14255724 T^{7} + 142797 T^{8} - 209419 T^{9} + 12536 T^{10} + 291 T^{11} - 44 T^{12} + T^{13}$$
$89$ $$259651296 - 1511624112 T + 3078781120 T^{2} - 2704068552 T^{3} + 1005599062 T^{4} - 122058759 T^{5} - 19292841 T^{6} + 6684011 T^{7} - 396280 T^{8} - 58263 T^{9} + 8168 T^{10} - 118 T^{11} - 25 T^{12} + T^{13}$$
$97$ $$-428877315008 - 626043658240 T + 212159782104 T^{2} + 112893177956 T^{3} - 13785003646 T^{4} - 3204623641 T^{5} + 350291802 T^{6} + 32850667 T^{7} - 4048341 T^{8} - 107857 T^{9} + 20959 T^{10} - 170 T^{11} - 38 T^{12} + T^{13}$$