# Properties

 Label 825.2.n.k Level $825$ Weight $2$ Character orbit 825.n Analytic conductor $6.588$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ 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 165) 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.

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

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 $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 −0.386111 − 0.280526i 1.69513 + 1.23158i 0.418926 − 1.28932i −0.227943 + 0.701538i 0.418926 + 1.28932i −0.227943 − 0.701538i −0.386111 + 0.280526i 1.69513 − 1.23158i
−1.12474 0.817172i −0.309017 + 0.951057i −0.0207616 0.0638975i 0 1.12474 0.817172i −0.394797 1.21506i −0.888090 + 2.73326i −0.809017 0.587785i 0
301.2 2.24278 + 1.62947i −0.309017 + 0.951057i 1.75683 + 5.40697i 0 −2.24278 + 1.62947i 0.703814 + 2.16612i −3.15700 + 9.71623i −0.809017 0.587785i 0
526.1 −0.758911 + 2.33569i 0.809017 0.587785i −3.26145 2.36959i 0 0.758911 + 2.33569i 2.65911 + 1.93196i 4.03606 2.93237i 0.309017 0.951057i 0
526.2 −0.359123 + 1.10527i 0.809017 0.587785i 0.525387 + 0.381716i 0 0.359123 + 1.10527i −3.46813 2.51974i −2.49097 + 1.80980i 0.309017 0.951057i 0
676.1 −0.758911 2.33569i 0.809017 + 0.587785i −3.26145 + 2.36959i 0 0.758911 2.33569i 2.65911 1.93196i 4.03606 + 2.93237i 0.309017 + 0.951057i 0
676.2 −0.359123 1.10527i 0.809017 + 0.587785i 0.525387 0.381716i 0 0.359123 1.10527i −3.46813 + 2.51974i −2.49097 1.80980i 0.309017 + 0.951057i 0
751.1 −1.12474 + 0.817172i −0.309017 0.951057i −0.0207616 + 0.0638975i 0 1.12474 + 0.817172i −0.394797 + 1.21506i −0.888090 2.73326i −0.809017 + 0.587785i 0
751.2 2.24278 1.62947i −0.309017 0.951057i 1.75683 5.40697i 0 −2.24278 1.62947i 0.703814 2.16612i −3.15700 9.71623i −0.809017 + 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 751.2 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 825.2.n.k 8
5.b even 2 1 165.2.m.a 8
5.c odd 4 2 825.2.bx.h 16
11.c even 5 1 inner 825.2.n.k 8
11.c even 5 1 9075.2.a.cl 4
11.d odd 10 1 9075.2.a.dj 4
15.d odd 2 1 495.2.n.d 8
55.h odd 10 1 1815.2.a.o 4
55.j even 10 1 165.2.m.a 8
55.j even 10 1 1815.2.a.x 4
55.k odd 20 2 825.2.bx.h 16
165.o odd 10 1 495.2.n.d 8
165.o odd 10 1 5445.2.a.be 4
165.r even 10 1 5445.2.a.bv 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 5.b even 2 1
165.2.m.a 8 55.j even 10 1
495.2.n.d 8 15.d odd 2 1
495.2.n.d 8 165.o odd 10 1
825.2.n.k 8 1.a even 1 1 trivial
825.2.n.k 8 11.c even 5 1 inner
825.2.bx.h 16 5.c odd 4 2
825.2.bx.h 16 55.k odd 20 2
1815.2.a.o 4 55.h odd 10 1
1815.2.a.x 4 55.j even 10 1
5445.2.a.be 4 165.o odd 10 1
5445.2.a.bv 4 165.r even 10 1
9075.2.a.cl 4 11.c even 5 1
9075.2.a.dj 4 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{8} + 3 T_{2}^{6} - 5 T_{2}^{5} + 24 T_{2}^{4} + 85 T_{2}^{3} + 177 T_{2}^{2} + 165 T_{2} + 121$$ $$T_{13}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$121 + 165 T + 177 T^{2} + 85 T^{3} + 24 T^{4} - 5 T^{5} + 3 T^{6} + T^{8}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$1681 + 164 T + 1027 T^{2} - 253 T^{3} + 180 T^{4} - 7 T^{5} - 3 T^{6} + T^{7} + T^{8}$$
$11$ $$14641 + 3993 T + 968 T^{2} + 11 T^{3} - 85 T^{4} + T^{5} + 8 T^{6} + 3 T^{7} + T^{8}$$
$13$ $$1 + 3 T + 11 T^{2} + 39 T^{3} + 94 T^{4} + 87 T^{5} + 39 T^{6} + 6 T^{7} + T^{8}$$
$17$ $$( 625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$19$ $$961 - 3813 T + 38671 T^{2} - 2379 T^{3} + 874 T^{4} + 123 T^{5} + 9 T^{6} - 6 T^{7} + T^{8}$$
$23$ $$( -1 + 5 T - T^{2} - 5 T^{3} + T^{4} )^{2}$$
$29$ $$290521 + 18865 T + 34153 T^{2} - 5285 T^{3} + 844 T^{4} + 95 T^{5} + 17 T^{6} + T^{8}$$
$31$ $$19321 + 4309 T + 2829 T^{2} + 253 T^{3} + 174 T^{4} + T^{5} + 11 T^{6} - 3 T^{7} + T^{8}$$
$37$ $$185761 - 227999 T + 136645 T^{2} - 49391 T^{3} + 12234 T^{4} - 2111 T^{5} + 255 T^{6} - 19 T^{7} + T^{8}$$
$41$ $$4289041 + 2091710 T + 708673 T^{2} + 150785 T^{3} + 23634 T^{4} + 2855 T^{5} + 327 T^{6} + 25 T^{7} + T^{8}$$
$43$ $$( 1861 + 63 T - 92 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$47$ $$121 + 990 T + 3837 T^{2} + 6815 T^{3} + 10194 T^{4} + 665 T^{5} + 123 T^{6} + 15 T^{7} + T^{8}$$
$53$ $$1615441 + 557969 T + 343174 T^{2} + 35863 T^{3} + 7329 T^{4} - 379 T^{5} - 14 T^{6} + 7 T^{7} + T^{8}$$
$59$ $$5285401 - 3253085 T + 1296507 T^{2} - 331435 T^{3} + 58754 T^{4} - 7285 T^{5} + 633 T^{6} - 35 T^{7} + T^{8}$$
$61$ $$3575881 + 1760521 T + 459192 T^{2} + 41503 T^{3} + 8805 T^{4} - 1043 T^{5} + 242 T^{6} - 21 T^{7} + T^{8}$$
$67$ $$( -3379 + 1768 T - 136 T^{2} - 13 T^{3} + T^{4} )^{2}$$
$71$ $$5527201 - 2879975 T + 1218443 T^{2} - 269975 T^{3} + 37544 T^{4} - 3625 T^{5} + 347 T^{6} - 25 T^{7} + T^{8}$$
$73$ $$84621601 + 18664771 T + 3462030 T^{2} + 293209 T^{3} + 19089 T^{4} - 141 T^{5} + 30 T^{6} + T^{7} + T^{8}$$
$79$ $$1437601 - 1402830 T + 669978 T^{2} - 183930 T^{3} + 37084 T^{4} - 4980 T^{5} + 487 T^{6} - 30 T^{7} + T^{8}$$
$83$ $$149352841 + 29978113 T + 2466101 T^{2} - 85701 T^{3} + 18074 T^{4} + 297 T^{5} + 359 T^{6} + 11 T^{7} + T^{8}$$
$89$ $$( -271 + 132 T + 45 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$97$ $$625 + 1875 T + 13000 T^{2} + 3375 T^{3} + 2675 T^{4} - 225 T^{5} - 20 T^{6} + 5 T^{7} + T^{8}$$