# Properties

 Label 546.2.k.e Level $546$ Weight $2$ Character orbit 546.k Analytic conductor $4.360$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} + 15 x^{8} + 14 x^{7} + 110 x^{6} + 36 x^{5} + 233 x^{4} + 164 x^{3} + 345 x^{2} + 76 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 15 x^{8} + 14 x^{7} + 110 x^{6} + 36 x^{5} + 233 x^{4} + 164 x^{3} + 345 x^{2} + 76 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-503 \nu^{9} - 2241 \nu^{8} + 8466 \nu^{7} - 67528 \nu^{6} + 19422 \nu^{5} - 156870 \nu^{4} + 1003571 \nu^{3} - 301041 \nu^{2} - 66732 \nu + 438544$$$$)/2044008$$ $$\beta_{3}$$ $$=$$ $$($$$$-3064 \nu^{9} + 9207 \nu^{8} - 34782 \nu^{7} - 28346 \nu^{6} - 79794 \nu^{5} + 644490 \nu^{4} + 579550 \nu^{3} + 1236807 \nu^{2} + 274164 \nu + 3620228$$$$)/1022004$$ $$\beta_{4}$$ $$=$$ $$($$$$-3977 \nu^{9} - 10833 \nu^{8} - 12699 \nu^{7} - 334006 \nu^{6} - 625302 \nu^{5} - 1723536 \nu^{4} - 478621 \nu^{3} - 3196425 \nu^{2} - 3391749 \nu - 3118196$$$$)/1022004$$ $$\beta_{5}$$ $$=$$ $$($$$$-27409 \nu^{9} + 54315 \nu^{8} - 413376 \nu^{7} - 375260 \nu^{6} - 3082518 \nu^{5} - 967302 \nu^{4} - 6543167 \nu^{3} - 3491505 \nu^{2} - 9757146 \nu - 2149816$$$$)/2044008$$ $$\beta_{6}$$ $$=$$ $$($$$$-14545 \nu^{9} + 29941 \nu^{8} - 216152 \nu^{7} - 200758 \nu^{6} - 1521222 \nu^{5} - 440214 \nu^{4} - 2887781 \nu^{3} - 2349679 \nu^{2} - 3963994 \nu - 1074620$$$$)/340668$$ $$\beta_{7}$$ $$=$$ $$($$$$45928 \nu^{9} - 96426 \nu^{8} + 695481 \nu^{7} + 561428 \nu^{6} + 5037264 \nu^{5} + 744876 \nu^{4} + 10649492 \nu^{3} + 5556402 \nu^{2} + 16896855 \nu + 165664$$$$)/1022004$$ $$\beta_{8}$$ $$=$$ $$($$$$97465 \nu^{9} - 178473 \nu^{8} + 1399728 \nu^{7} + 1662752 \nu^{6} + 10555542 \nu^{5} + 4653846 \nu^{4} + 20509619 \nu^{3} + 17738031 \nu^{2} + 29957226 \nu + 5377120$$$$)/2044008$$ $$\beta_{9}$$ $$=$$ $$($$$$51344 \nu^{9} - 117222 \nu^{8} + 805587 \nu^{7} + 484042 \nu^{6} + 5520312 \nu^{5} + 367938 \nu^{4} + 11604142 \nu^{3} + 5109630 \nu^{2} + 14145267 \nu + 742892$$$$)/1022004$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 4 \beta_{5} - \beta_{4} + 2 \beta_{2} + 2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{8} - 3 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 6$$ $$\nu^{4}$$ $$=$$ $$-16 \beta_{9} - 16 \beta_{8} + 6 \beta_{7} - 18 \beta_{6} - 40 \beta_{5} + 2 \beta_{4} - 37 \beta_{1} - 40$$ $$\nu^{5}$$ $$=$$ $$-65 \beta_{9} + 4 \beta_{8} + 34 \beta_{7} - 4 \beta_{6} - 126 \beta_{5} + 65 \beta_{4} + 34 \beta_{3} - 168 \beta_{2} - 168 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-30 \beta_{9} + 297 \beta_{8} + 267 \beta_{6} + 267 \beta_{4} + 126 \beta_{3} - 657 \beta_{2} + 592$$ $$\nu^{7}$$ $$=$$ $$1080 \beta_{9} + 1080 \beta_{8} - 564 \beta_{7} + 1176 \beta_{6} + 2280 \beta_{5} - 96 \beta_{4} + 2797 \beta_{1} + 2280$$ $$\nu^{8}$$ $$=$$ $$5005 \beta_{9} - 468 \beta_{8} - 2256 \beta_{7} + 468 \beta_{6} + 9784 \beta_{5} - 5005 \beta_{4} - 2256 \beta_{3} + 11402 \beta_{2} + 11402 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$1788 \beta_{9} - 20451 \beta_{8} - 18663 \beta_{6} - 18663 \beta_{4} - 9542 \beta_{3} + 47603 \beta_{2} - 39726$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$-1 - \beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 2.07085 + 3.58682i 0.769836 + 1.33339i −0.114009 − 0.197470i −0.623307 − 1.07960i −1.10337 − 1.91109i 2.07085 − 3.58682i 0.769836 − 1.33339i −0.114009 + 0.197470i −0.623307 + 1.07960i −1.10337 + 1.91109i
0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i −2.07085 + 3.58682i −0.500000 0.866025i 2.11344 + 1.59166i −1.00000 1.00000 −4.14170
373.2 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i −0.769836 + 1.33339i −0.500000 0.866025i −0.131875 2.64246i −1.00000 1.00000 −1.53967
373.3 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0.114009 0.197470i −0.500000 0.866025i 0.848534 2.50599i −1.00000 1.00000 0.228019
373.4 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 0.623307 1.07960i −0.500000 0.866025i −2.27938 + 1.34329i −1.00000 1.00000 1.24661
373.5 0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 1.10337 1.91109i −0.500000 0.866025i 1.44928 + 2.21350i −1.00000 1.00000 2.20674
445.1 0.500000 0.866025i −1.00000 −0.500000 0.866025i −2.07085 3.58682i −0.500000 + 0.866025i 2.11344 1.59166i −1.00000 1.00000 −4.14170
445.2 0.500000 0.866025i −1.00000 −0.500000 0.866025i −0.769836 1.33339i −0.500000 + 0.866025i −0.131875 + 2.64246i −1.00000 1.00000 −1.53967
445.3 0.500000 0.866025i −1.00000 −0.500000 0.866025i 0.114009 + 0.197470i −0.500000 + 0.866025i 0.848534 + 2.50599i −1.00000 1.00000 0.228019
445.4 0.500000 0.866025i −1.00000 −0.500000 0.866025i 0.623307 + 1.07960i −0.500000 + 0.866025i −2.27938 1.34329i −1.00000 1.00000 1.24661
445.5 0.500000 0.866025i −1.00000 −0.500000 0.866025i 1.10337 + 1.91109i −0.500000 + 0.866025i 1.44928 2.21350i −1.00000 1.00000 2.20674
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 445.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.k.e yes 10
3.b odd 2 1 1638.2.p.j 10
7.c even 3 1 546.2.j.e 10
13.c even 3 1 546.2.j.e 10
21.h odd 6 1 1638.2.m.k 10
39.i odd 6 1 1638.2.m.k 10
91.g even 3 1 inner 546.2.k.e yes 10
273.bm odd 6 1 1638.2.p.j 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.e 10 7.c even 3 1
546.2.j.e 10 13.c even 3 1
546.2.k.e yes 10 1.a even 1 1 trivial
546.2.k.e yes 10 91.g even 3 1 inner
1638.2.m.k 10 21.h odd 6 1
1638.2.m.k 10 39.i odd 6 1
1638.2.p.j 10 3.b odd 2 1
1638.2.p.j 10 273.bm odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{5}$$
$3$ $$( 1 + T )^{10}$$
$5$ $$16 - 76 T + 345 T^{2} - 164 T^{3} + 233 T^{4} - 36 T^{5} + 110 T^{6} - 14 T^{7} + 15 T^{8} + 2 T^{9} + T^{10}$$
$7$ $$16807 - 9604 T + 6174 T^{2} - 1274 T^{3} + 434 T^{4} + 3 T^{5} + 62 T^{6} - 26 T^{7} + 18 T^{8} - 4 T^{9} + T^{10}$$
$11$ $$( 30 - 21 T - 65 T^{2} - 12 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$13$ $$371293 + 114244 T + 52728 T^{2} + 7436 T^{3} + 1118 T^{4} + 15 T^{5} + 86 T^{6} + 44 T^{7} + 24 T^{8} + 4 T^{9} + T^{10}$$
$17$ $$784 + 476 T + 2277 T^{2} - 2327 T^{3} + 4589 T^{4} - 1584 T^{5} + 701 T^{6} - 62 T^{7} + 36 T^{8} - 4 T^{9} + T^{10}$$
$19$ $$( -111 + 1101 T - 109 T^{2} - 69 T^{3} + 3 T^{4} + T^{5} )^{2}$$
$23$ $$129600 + 197640 T + 494721 T^{2} - 346653 T^{3} + 246681 T^{4} - 45612 T^{5} + 8955 T^{6} - 642 T^{7} + 108 T^{8} - 6 T^{9} + T^{10}$$
$29$ $$236196 - 196830 T + 177147 T^{2} - 47385 T^{3} + 25029 T^{4} - 2106 T^{5} + 3195 T^{6} - 54 T^{7} + 60 T^{8} + T^{10}$$
$31$ $$1254400 + 1647520 T + 1604961 T^{2} + 675789 T^{3} + 221955 T^{4} + 41274 T^{5} + 7137 T^{6} + 738 T^{7} + 126 T^{8} + 10 T^{9} + T^{10}$$
$37$ $$319225 + 207920 T + 145029 T^{2} + 36684 T^{3} + 14838 T^{4} + 1947 T^{5} + 1059 T^{6} + 72 T^{7} + 39 T^{8} - T^{9} + T^{10}$$
$41$ $$770884 + 194038 T + 160347 T^{2} + 28125 T^{3} + 19689 T^{4} + 3174 T^{5} + 1311 T^{6} + 126 T^{7} + 48 T^{8} + 4 T^{9} + T^{10}$$
$43$ $$308025 - 213120 T + 277881 T^{2} - 22980 T^{3} + 92728 T^{4} - 22221 T^{5} + 10725 T^{6} - 164 T^{7} + 111 T^{8} - 3 T^{9} + T^{10}$$
$47$ $$271854144 + 17164008 T + 19550241 T^{2} + 614784 T^{3} + 1063294 T^{4} + 45738 T^{5} + 18675 T^{6} + 1430 T^{7} + 279 T^{8} + 15 T^{9} + T^{10}$$
$53$ $$256 + 66704 T + 17364273 T^{2} + 4244362 T^{3} + 1078286 T^{4} + 131550 T^{5} + 21575 T^{6} + 2206 T^{7} + 279 T^{8} + 17 T^{9} + T^{10}$$
$59$ $$1607824 - 2367356 T + 2927769 T^{2} - 1244992 T^{3} + 502853 T^{4} + 79680 T^{5} + 25142 T^{6} + 1214 T^{7} + 171 T^{8} - 2 T^{9} + T^{10}$$
$61$ $$( 80 - 1016 T - 400 T^{2} - 8 T^{3} + 11 T^{4} + T^{5} )^{2}$$
$67$ $$( -20 + 719 T + 199 T^{2} - 137 T^{3} - T^{4} + T^{5} )^{2}$$
$71$ $$202500 + 2544750 T + 32555925 T^{2} - 7241610 T^{3} + 1686319 T^{4} - 192492 T^{5} + 28812 T^{6} - 2726 T^{7} + 315 T^{8} - 18 T^{9} + T^{10}$$
$73$ $$37941975369 - 3240866106 T + 887285502 T^{2} - 50704044 T^{3} + 11876944 T^{4} - 622851 T^{5} + 90666 T^{6} - 3100 T^{7} + 408 T^{8} - 12 T^{9} + T^{10}$$
$79$ $$817216 - 709640 T + 580065 T^{2} - 159768 T^{3} + 60951 T^{4} - 4344 T^{5} + 4416 T^{6} - 204 T^{7} + 87 T^{8} + 4 T^{9} + T^{10}$$
$83$ $$( 486 + 405 T - 27 T^{2} - 60 T^{3} + T^{5} )^{2}$$
$89$ $$205435562500 - 22923118750 T + 4266583125 T^{2} - 229948250 T^{3} + 34506950 T^{4} - 1494480 T^{5} + 191111 T^{6} - 4292 T^{7} + 513 T^{8} - 7 T^{9} + T^{10}$$
$97$ $$1734489 + 1457919 T + 1797027 T^{2} + 159624 T^{3} + 449455 T^{4} + 93495 T^{5} + 60546 T^{6} - 590 T^{7} + 279 T^{8} + 6 T^{9} + T^{10}$$