# Properties

 Label 350.2.q.a Level 350 Weight 2 Character orbit 350.q Analytic conductor 2.795 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.q (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{15}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 - \zeta_{15}^{5}$$ $$-\zeta_{15}^{2} - \zeta_{15}^{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.913545 + 0.406737i 0.669131 − 0.743145i 0.669131 + 0.743145i 0.913545 − 0.406737i −0.978148 + 0.207912i −0.104528 − 0.994522i −0.104528 + 0.994522i −0.978148 − 0.207912i
−0.913545 0.406737i 0.413545 + 0.459289i 0.669131 + 0.743145i 1.11803 + 1.93649i −0.190983 0.587785i −0.500000 2.59808i −0.309017 0.951057i 0.273659 2.60369i −0.233733 2.22382i
81.1 −0.669131 + 0.743145i 0.169131 + 1.60917i −0.104528 0.994522i −1.11803 1.93649i −1.30902 0.951057i −0.500000 2.59808i 0.809017 + 0.587785i 0.373619 0.0794152i 2.18720 + 0.464905i
121.1 −0.669131 0.743145i 0.169131 1.60917i −0.104528 + 0.994522i −1.11803 + 1.93649i −1.30902 + 0.951057i −0.500000 + 2.59808i 0.809017 0.587785i 0.373619 + 0.0794152i 2.18720 0.464905i
191.1 −0.913545 + 0.406737i 0.413545 0.459289i 0.669131 0.743145i 1.11803 1.93649i −0.190983 + 0.587785i −0.500000 + 2.59808i −0.309017 + 0.951057i 0.273659 + 2.60369i −0.233733 + 2.22382i
221.1 0.978148 0.207912i −1.47815 + 0.658114i 0.913545 0.406737i −1.11803 1.93649i −1.30902 + 0.951057i −0.500000 2.59808i 0.809017 0.587785i −0.255585 + 0.283856i −1.49622 1.66172i
261.1 0.104528 + 0.994522i −0.604528 + 0.128496i −0.978148 + 0.207912i 1.11803 1.93649i −0.190983 0.587785i −0.500000 + 2.59808i −0.309017 0.951057i −2.39169 + 1.06485i 2.04275 + 0.909491i
291.1 0.104528 0.994522i −0.604528 0.128496i −0.978148 0.207912i 1.11803 + 1.93649i −0.190983 + 0.587785i −0.500000 2.59808i −0.309017 + 0.951057i −2.39169 1.06485i 2.04275 0.909491i
331.1 0.978148 + 0.207912i −1.47815 0.658114i 0.913545 + 0.406737i −1.11803 + 1.93649i −1.30902 0.951057i −0.500000 + 2.59808i 0.809017 + 0.587785i −0.255585 0.283856i −1.49622 + 1.66172i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.q.a 8
7.c even 3 1 inner 350.2.q.a 8
25.d even 5 1 inner 350.2.q.a 8
175.q even 15 1 inner 350.2.q.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.q.a 8 1.a even 1 1 trivial
350.2.q.a 8 7.c even 3 1 inner
350.2.q.a 8 25.d even 5 1 inner
350.2.q.a 8 175.q even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 3 T_{3}^{7} + 5 T_{3}^{6} + 8 T_{3}^{5} + 9 T_{3}^{4} + 2 T_{3}^{3} + 2 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8}$$
$3$ $$1 + 3 T + 8 T^{2} + 5 T^{3} - 28 T^{5} + 24 T^{6} + 110 T^{7} + 409 T^{8} + 330 T^{9} + 216 T^{10} - 756 T^{11} + 1215 T^{13} + 5832 T^{14} + 6561 T^{15} + 6561 T^{16}$$
$5$ $$( 1 + 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + T + 7 T^{2} )^{4}$$
$11$ $$1 - 8 T + 41 T^{2} - 52 T^{3} - 274 T^{4} + 2156 T^{5} - 2211 T^{6} - 17666 T^{7} + 119427 T^{8} - 194326 T^{9} - 267531 T^{10} + 2869636 T^{11} - 4011634 T^{12} - 8374652 T^{13} + 72634001 T^{14} - 155897368 T^{15} + 214358881 T^{16}$$
$13$ $$( 1 - 10 T + 27 T^{2} + 40 T^{3} - 391 T^{4} + 520 T^{5} + 4563 T^{6} - 21970 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 14 T + 77 T^{2} - 190 T^{3} + 380 T^{4} - 3796 T^{5} + 27351 T^{6} - 109670 T^{7} + 386499 T^{8} - 1864390 T^{9} + 7904439 T^{10} - 18649748 T^{11} + 31737980 T^{12} - 269772830 T^{13} + 1858592813 T^{14} - 5744741422 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 6 T + 19 T^{2} + 126 T^{3} + 756 T^{4} + 4308 T^{5} + 20501 T^{6} + 90468 T^{7} + 412487 T^{8} + 1718892 T^{9} + 7400861 T^{10} + 29548572 T^{11} + 98522676 T^{12} + 311988474 T^{13} + 893871739 T^{14} + 5363230434 T^{15} + 16983563041 T^{16}$$
$23$ $$( 1 - 20 T + 217 T^{2} - 1600 T^{3} + 8769 T^{4} - 36800 T^{5} + 114793 T^{6} - 243340 T^{7} + 279841 T^{8} )( 1 + 13 T + 46 T^{2} - 61 T^{3} - 891 T^{4} - 1403 T^{5} + 24334 T^{6} + 158171 T^{7} + 279841 T^{8} )$$
$29$ $$( 1 - 11 T + 32 T^{2} - 113 T^{3} + 1115 T^{4} - 3277 T^{5} + 26912 T^{6} - 268279 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 - 6 T + 31 T^{2} + 318 T^{3} - 2898 T^{4} + 16284 T^{5} - 1873 T^{6} - 480582 T^{7} + 3789491 T^{8} - 14898042 T^{9} - 1799953 T^{10} + 485116644 T^{11} - 2676363858 T^{12} + 9104070018 T^{13} + 27512614111 T^{14} - 165075684666 T^{15} + 852891037441 T^{16}$$
$37$ $$1 + T + 32 T^{2} - 245 T^{3} - 590 T^{4} - 2246 T^{5} - 32624 T^{6} + 408180 T^{7} - 1426941 T^{8} + 15102660 T^{9} - 44662256 T^{10} - 113766638 T^{11} - 1105754990 T^{12} - 16989269465 T^{13} + 82103245088 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 + 8 T + 73 T^{2} + 556 T^{3} + 2205 T^{4} + 22796 T^{5} + 122713 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 81 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$1 - 2 T - 13 T^{2} - 294 T^{3} - 212 T^{4} + 11364 T^{5} + 91289 T^{6} - 391562 T^{7} - 4902677 T^{8} - 18403414 T^{9} + 201657401 T^{10} + 1179844572 T^{11} - 1034492372 T^{12} - 67427432058 T^{13} - 140129799277 T^{14} - 1013246240926 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 7 T + 83 T^{2} - 180 T^{3} + 1240 T^{4} + 23277 T^{5} - 22906 T^{6} + 667850 T^{7} + 3038809 T^{8} + 35396050 T^{9} - 64342954 T^{10} + 3465409929 T^{11} + 9784196440 T^{12} - 75275188740 T^{13} + 1839641973707 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 15 T + 94 T^{2} - 825 T^{3} - 17940 T^{4} - 160230 T^{5} - 233584 T^{6} + 8132460 T^{7} + 106709249 T^{8} + 479815140 T^{9} - 813105904 T^{10} - 32907877170 T^{11} - 217385456340 T^{12} - 589812546675 T^{13} + 3964970162254 T^{14} + 37329772272285 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 11 T + 131 T^{2} - 1292 T^{3} + 13072 T^{4} - 79991 T^{5} + 834742 T^{6} - 4920222 T^{7} + 38414121 T^{8} - 300133542 T^{9} + 3106074982 T^{10} - 18156437171 T^{11} + 180992833552 T^{12} - 1091218420892 T^{13} + 6749169041291 T^{14} - 34570171196231 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + T + 67 T^{2} + 200 T^{3} + 200 T^{4} - 8711 T^{5} - 282874 T^{6} - 1742200 T^{7} - 21893321 T^{8} - 116727400 T^{9} - 1269821386 T^{10} - 2619946493 T^{11} + 4030224200 T^{12} + 270025021400 T^{13} + 6060711605323 T^{14} + 6060711605323 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 - 13 T + 218 T^{2} - 1831 T^{3} + 21125 T^{4} - 130001 T^{5} + 1098938 T^{6} - 4652843 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 4 T + 43 T^{2} - 1052 T^{3} + 2828 T^{4} + 28448 T^{5} + 223961 T^{6} + 1434004 T^{7} - 44342897 T^{8} + 104682292 T^{9} + 1193488169 T^{10} + 11066755616 T^{11} + 80310225548 T^{12} - 2180871315836 T^{13} + 6507371730427 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 20 T + 239 T^{2} + 20 T^{3} - 25040 T^{4} - 374440 T^{5} - 1011119 T^{6} + 19593600 T^{7} + 350987679 T^{8} + 1547894400 T^{9} - 6310393679 T^{10} - 184613523160 T^{11} - 975310028240 T^{12} + 61541127980 T^{13} + 58097901869519 T^{14} + 384078179723180 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 13 T - 4 T^{2} + 1151 T^{3} - 11791 T^{4} + 95533 T^{5} - 27556 T^{6} - 7433231 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 - T - 136 T^{2} - 891 T^{3} + 7966 T^{4} + 107562 T^{5} + 814376 T^{6} - 6768808 T^{7} - 97658633 T^{8} - 602423912 T^{9} + 6450672296 T^{10} + 75827875578 T^{11} + 499804691806 T^{12} - 4975396969059 T^{13} - 67589455570696 T^{14} - 44231334895529 T^{15} + 3936588805702081 T^{16}$$
$97$ $$( 1 - 4 T - T^{2} + 682 T^{3} + 829 T^{4} + 66154 T^{5} - 9409 T^{6} - 3650692 T^{7} + 88529281 T^{8} )^{2}$$