Properties

 Label 825.2.n.g 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.

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

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

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$1$$

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
 0.418926 + 1.28932i −0.227943 − 0.701538i 1.69513 + 1.23158i −0.386111 − 0.280526i 1.69513 − 1.23158i −0.386111 + 0.280526i 0.418926 − 1.28932i −0.227943 + 0.701538i
−1.90578 1.38463i −0.309017 + 0.951057i 1.09676 + 3.37549i 0 1.90578 1.38463i −0.0598032 0.184055i 1.12773 3.47080i −0.809017 0.587785i 0
301.2 −0.212253 0.154211i −0.309017 + 0.951057i −0.596764 1.83665i 0 0.212253 0.154211i 0.986854 + 3.03722i −0.318714 + 0.980901i −0.809017 0.587785i 0
526.1 −0.338464 + 1.04169i 0.809017 0.587785i 0.647481 + 0.470423i 0 0.338464 + 1.04169i −0.570387 0.414410i −2.48141 + 1.80285i 0.309017 0.951057i 0
526.2 0.456498 1.40496i 0.809017 0.587785i −0.147481 0.107152i 0 −0.456498 1.40496i −1.85666 1.34895i 2.17239 1.57833i 0.309017 0.951057i 0
676.1 −0.338464 1.04169i 0.809017 + 0.587785i 0.647481 0.470423i 0 0.338464 1.04169i −0.570387 + 0.414410i −2.48141 1.80285i 0.309017 + 0.951057i 0
676.2 0.456498 + 1.40496i 0.809017 + 0.587785i −0.147481 + 0.107152i 0 −0.456498 + 1.40496i −1.85666 + 1.34895i 2.17239 + 1.57833i 0.309017 + 0.951057i 0
751.1 −1.90578 + 1.38463i −0.309017 0.951057i 1.09676 3.37549i 0 1.90578 + 1.38463i −0.0598032 + 0.184055i 1.12773 + 3.47080i −0.809017 + 0.587785i 0
751.2 −0.212253 + 0.154211i −0.309017 0.951057i −0.596764 + 1.83665i 0 0.212253 + 0.154211i 0.986854 3.03722i −0.318714 0.980901i −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.g 8
5.b even 2 1 165.2.m.d 8
5.c odd 4 2 825.2.bx.f 16
11.c even 5 1 inner 825.2.n.g 8
11.c even 5 1 9075.2.a.di 4
11.d odd 10 1 9075.2.a.cm 4
15.d odd 2 1 495.2.n.a 8
55.h odd 10 1 1815.2.a.w 4
55.j even 10 1 165.2.m.d 8
55.j even 10 1 1815.2.a.p 4
55.k odd 20 2 825.2.bx.f 16
165.o odd 10 1 495.2.n.a 8
165.o odd 10 1 5445.2.a.bt 4
165.r even 10 1 5445.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.d 8 5.b even 2 1
165.2.m.d 8 55.j even 10 1
495.2.n.a 8 15.d odd 2 1
495.2.n.a 8 165.o odd 10 1
825.2.n.g 8 1.a even 1 1 trivial
825.2.n.g 8 11.c even 5 1 inner
825.2.bx.f 16 5.c odd 4 2
825.2.bx.f 16 55.k odd 20 2
1815.2.a.p 4 55.j even 10 1
1815.2.a.w 4 55.h odd 10 1
5445.2.a.bf 4 165.r even 10 1
5445.2.a.bt 4 165.o odd 10 1
9075.2.a.cm 4 11.d odd 10 1
9075.2.a.di 4 11.c even 5 1

Hecke kernels

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

 $$T_{2}^{8} + \cdots$$ $$T_{13}^{8} - \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T + 21 T^{2} + 21 T^{3} + 24 T^{4} + 13 T^{5} + 9 T^{6} + 4 T^{7} + T^{8}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$1 + 6 T + 39 T^{2} + 87 T^{3} + 94 T^{4} + 39 T^{5} + 11 T^{6} + 3 T^{7} + T^{8}$$
$11$ $$14641 - 3993 T - 2662 T^{2} + 209 T^{3} + 335 T^{4} + 19 T^{5} - 22 T^{6} - 3 T^{7} + T^{8}$$
$13$ $$121 - 77 T - 19 T^{2} + 39 T^{3} + 74 T^{4} - 33 T^{5} + 29 T^{6} - 4 T^{7} + T^{8}$$
$17$ $$1 + 20 T + 163 T^{2} + 290 T^{3} + 559 T^{4} + 250 T^{5} + 47 T^{6} + T^{8}$$
$19$ $$1 + 7 T + 195 T^{2} - 403 T^{3} + 354 T^{4} - 137 T^{5} + 25 T^{6} - 2 T^{7} + T^{8}$$
$23$ $$( 449 + 129 T - 55 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$29$ $$249001 + 237025 T + 84767 T^{2} - 10335 T^{3} + 11144 T^{4} - 1125 T^{5} + 113 T^{6} - 10 T^{7} + T^{8}$$
$31$ $$1560001 - 1287719 T + 498345 T^{2} - 97231 T^{3} + 17274 T^{4} - 2071 T^{5} + 235 T^{6} - 19 T^{7} + T^{8}$$
$37$ $$12538681 + 322231 T + 356977 T^{2} - 14947 T^{3} + 4080 T^{4} + 157 T^{5} + 37 T^{6} - T^{7} + T^{8}$$
$41$ $$1681 - 12956 T + 40225 T^{2} - 26019 T^{3} + 6434 T^{4} + 441 T^{5} + 205 T^{6} + 9 T^{7} + T^{8}$$
$43$ $$( 275 - 375 T - 110 T^{2} + T^{4} )^{2}$$
$47$ $$961 + 4526 T + 9525 T^{2} + 7039 T^{3} + 6104 T^{4} - 1751 T^{5} + 255 T^{6} - 19 T^{7} + T^{8}$$
$53$ $$410881 - 631385 T + 326418 T^{2} + 121925 T^{3} + 31229 T^{4} + 4415 T^{5} + 432 T^{6} + 25 T^{7} + T^{8}$$
$59$ $$346921 - 325717 T + 228195 T^{2} - 86697 T^{3} + 18894 T^{4} - 2053 T^{5} + 165 T^{6} - 13 T^{7} + T^{8}$$
$61$ $$1324801 + 1545793 T + 639460 T^{2} - 110177 T^{3} + 22389 T^{4} - 2123 T^{5} + 210 T^{6} - 13 T^{7} + T^{8}$$
$67$ $$( 619 - 112 T - 100 T^{2} + T^{3} + T^{4} )^{2}$$
$71$ $$78961 - 75589 T + 176745 T^{2} - 4771 T^{3} - 3276 T^{4} + 189 T^{5} + 225 T^{6} + 11 T^{7} + T^{8}$$
$73$ $$866761 + 188993 T + 83398 T^{2} + 2471 T^{3} + 555 T^{4} + 31 T^{5} + 48 T^{6} - 7 T^{7} + T^{8}$$
$79$ $$28561 + 57122 T + 47658 T^{2} + 8554 T^{3} + 9780 T^{4} + 2044 T^{5} + 283 T^{6} + 22 T^{7} + T^{8}$$
$83$ $$1681 - 11931 T + 34155 T^{2} - 19059 T^{3} + 10144 T^{4} - 2109 T^{5} + 275 T^{6} - 21 T^{7} + T^{8}$$
$89$ $$( 209 - 310 T - 89 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$97$ $$1343281 + 763781 T + 326730 T^{2} + 84259 T^{3} + 17109 T^{4} + 2629 T^{5} + 430 T^{6} + 31 T^{7} + T^{8}$$