# Properties

 Label 275.2.h.a Level $275$ Weight $2$ Character orbit 275.h Analytic conductor $2.196$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.h (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.19588605559$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$\beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −0.227943 + 0.701538i 0.418926 − 1.28932i −0.386111 − 0.280526i 1.69513 + 1.23158i −0.227943 − 0.701538i 0.418926 + 1.28932i −0.386111 + 0.280526i 1.69513 − 1.23158i
−0.596764 0.433574i 0.868820 2.67395i −0.449894 1.38463i 0 −1.67784 + 1.21902i −0.318714 0.980901i −0.787747 + 2.42443i −3.96813 2.88301i 0
26.2 1.09676 + 0.796845i −0.177837 + 0.547326i −0.0501062 0.154211i 0 −0.631180 + 0.458579i 1.12773 + 3.47080i 0.905781 2.78771i 2.15911 + 1.56869i 0
126.1 −0.147481 0.453901i 0.261370 + 0.189896i 1.43376 1.04169i 0 0.0476470 0.146642i 2.17239 1.57833i −1.45650 1.05821i −0.894797 2.75390i 0
126.2 0.647481 + 1.99274i 1.54765 + 1.12443i −1.93376 + 1.40496i 0 −1.23863 + 3.81211i −2.48141 + 1.80285i −0.661536 0.480634i 0.203814 + 0.627276i 0
201.1 −0.596764 + 0.433574i 0.868820 + 2.67395i −0.449894 + 1.38463i 0 −1.67784 1.21902i −0.318714 + 0.980901i −0.787747 2.42443i −3.96813 + 2.88301i 0
201.2 1.09676 0.796845i −0.177837 0.547326i −0.0501062 + 0.154211i 0 −0.631180 0.458579i 1.12773 3.47080i 0.905781 + 2.78771i 2.15911 1.56869i 0
251.1 −0.147481 + 0.453901i 0.261370 0.189896i 1.43376 + 1.04169i 0 0.0476470 + 0.146642i 2.17239 + 1.57833i −1.45650 + 1.05821i −0.894797 + 2.75390i 0
251.2 0.647481 1.99274i 1.54765 1.12443i −1.93376 1.40496i 0 −1.23863 3.81211i −2.48141 1.80285i −0.661536 + 0.480634i 0.203814 0.627276i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.h.a 8
5.b even 2 1 55.2.g.b 8
5.c odd 4 2 275.2.z.a 16
11.c even 5 1 inner 275.2.h.a 8
11.c even 5 1 3025.2.a.bd 4
11.d odd 10 1 3025.2.a.w 4
15.d odd 2 1 495.2.n.e 8
20.d odd 2 1 880.2.bo.h 8
55.d odd 2 1 605.2.g.k 8
55.h odd 10 1 605.2.a.k 4
55.h odd 10 2 605.2.g.e 8
55.h odd 10 1 605.2.g.k 8
55.j even 10 1 55.2.g.b 8
55.j even 10 1 605.2.a.j 4
55.j even 10 2 605.2.g.m 8
55.k odd 20 2 275.2.z.a 16
165.o odd 10 1 495.2.n.e 8
165.o odd 10 1 5445.2.a.bp 4
165.r even 10 1 5445.2.a.bi 4
220.n odd 10 1 880.2.bo.h 8
220.n odd 10 1 9680.2.a.cn 4
220.o even 10 1 9680.2.a.cm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 5.b even 2 1
55.2.g.b 8 55.j even 10 1
275.2.h.a 8 1.a even 1 1 trivial
275.2.h.a 8 11.c even 5 1 inner
275.2.z.a 16 5.c odd 4 2
275.2.z.a 16 55.k odd 20 2
495.2.n.e 8 15.d odd 2 1
495.2.n.e 8 165.o odd 10 1
605.2.a.j 4 55.j even 10 1
605.2.a.k 4 55.h odd 10 1
605.2.g.e 8 55.h odd 10 2
605.2.g.k 8 55.d odd 2 1
605.2.g.k 8 55.h odd 10 1
605.2.g.m 8 55.j even 10 2
880.2.bo.h 8 20.d odd 2 1
880.2.bo.h 8 220.n odd 10 1
3025.2.a.w 4 11.d odd 10 1
3025.2.a.bd 4 11.c even 5 1
5445.2.a.bi 4 165.r even 10 1
5445.2.a.bp 4 165.o odd 10 1
9680.2.a.cm 4 220.o even 10 1
9680.2.a.cn 4 220.n odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} + 2 T^{3} - T^{4} - 2 T^{5} + 5 T^{6} - 2 T^{7} + T^{8}$$
$3$ $$1 - 5 T + 12 T^{2} - 15 T^{3} + 39 T^{4} - 35 T^{5} + 18 T^{6} - 5 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$961 + 341 T + 777 T^{2} - 197 T^{3} + 30 T^{4} + 17 T^{5} + 7 T^{6} - T^{7} + T^{8}$$
$11$ $$14641 - 3993 T + 2178 T^{2} + 99 T^{3} + 75 T^{4} + 9 T^{5} + 18 T^{6} - 3 T^{7} + T^{8}$$
$13$ $$19321 + 3336 T + 1439 T^{2} - 128 T^{3} + 129 T^{4} + 4 T^{5} + 21 T^{6} - 2 T^{7} + T^{8}$$
$17$ $$361 - 931 T + 2544 T^{2} - 2537 T^{3} + 1379 T^{4} - 449 T^{5} + 96 T^{6} - 13 T^{7} + T^{8}$$
$19$ $$625 + 3750 T + 67125 T^{2} - 33875 T^{3} + 7950 T^{4} - 1025 T^{5} + 135 T^{6} - 15 T^{7} + T^{8}$$
$23$ $$( -11 - 10 T + 4 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$29$ $$203401 + 18942 T - 2329 T^{2} - 429 T^{3} + 1384 T^{4} + 363 T^{5} + 99 T^{6} + 9 T^{7} + T^{8}$$
$31$ $$390625 - 156250 T + 78125 T^{2} - 6250 T^{3} - 625 T^{4} + 250 T^{5} + 125 T^{6} + 10 T^{7} + T^{8}$$
$37$ $$1324801 - 740093 T + 124041 T^{2} + 46831 T^{3} + 19344 T^{4} + 3133 T^{5} + 359 T^{6} + 24 T^{7} + T^{8}$$
$41$ $$101761 - 120263 T + 69513 T^{2} - 23171 T^{3} + 5430 T^{4} - 831 T^{5} + 93 T^{6} - 8 T^{7} + T^{8}$$
$43$ $$( 211 - 289 T + 121 T^{2} - 19 T^{3} + T^{4} )^{2}$$
$47$ $$28561 + 3887 T^{2} + 1170 T^{3} + 259 T^{4} - 90 T^{5} + 23 T^{6} + T^{8}$$
$53$ $$885481 - 313353 T + 81665 T^{2} - 13753 T^{3} + 5824 T^{4} + 133 T^{5} + 85 T^{6} + 13 T^{7} + T^{8}$$
$59$ $$687241 + 395433 T + 157155 T^{2} + 54243 T^{3} + 18994 T^{4} + 3177 T^{5} + 385 T^{6} + 27 T^{7} + T^{8}$$
$61$ $$28561 + 26364 T + 17745 T^{2} + 6396 T^{3} + 1504 T^{4} + 96 T^{5} + 10 T^{6} - 6 T^{7} + T^{8}$$
$67$ $$( -4079 + 1014 T + 22 T^{2} - 19 T^{3} + T^{4} )^{2}$$
$71$ $$17161 + 9825 T + 11477 T^{2} + 10445 T^{3} + 5634 T^{4} + 1325 T^{5} + 213 T^{6} + 20 T^{7} + T^{8}$$
$73$ $$121 - 319 T + 584 T^{2} - 633 T^{3} + 479 T^{4} - 201 T^{5} + 56 T^{6} + 13 T^{7} + T^{8}$$
$79$ $$45954841 - 17171207 T + 6346418 T^{2} - 1002329 T^{3} + 104405 T^{4} - 8839 T^{5} + 698 T^{6} - 37 T^{7} + T^{8}$$
$83$ $$2886601 - 2101663 T + 461353 T^{2} + 174129 T^{3} + 42080 T^{4} + 5379 T^{5} + 493 T^{6} + 27 T^{7} + T^{8}$$
$89$ $$( 1861 - 472 T - 102 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$97$ $$9066121 - 1951128 T - 356419 T^{2} + 209856 T^{3} + 96589 T^{4} + 7968 T^{5} + 749 T^{6} + 24 T^{7} + T^{8}$$