# Properties

 Label 403.2.bl.a Level 403 Weight 2 Character orbit 403.bl Analytic conductor 3.218 Analytic rank 1 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bl (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/403\mathbb{Z}\right)^\times$$.

 $$n$$ $$249$$ $$313$$ $$\chi(n)$$ $$-1 - \zeta_{15}^{5}$$ $$-\zeta_{15}^{2} - \zeta_{15}^{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.913545 − 0.406737i −0.978148 + 0.207912i 0.913545 + 0.406737i 0.669131 + 0.743145i −0.104528 − 0.994522i 0.669131 − 0.743145i −0.104528 + 0.994522i −0.978148 − 0.207912i
−2.39169 1.06485i 0.233733 2.22382i 3.24803 + 3.60730i −3.00000 −2.92705 + 5.06980i −2.00739 2.22943i −2.30902 7.10642i −1.95630 0.415823i 7.17508 + 3.19455i
35.1 0.373619 + 0.0794152i 1.49622 + 1.66172i −1.69381 0.754131i −3.00000 0.427051 + 0.739674i −2.74064 1.22021i −1.19098 0.865300i −0.209057 + 1.98904i −1.12086 0.238246i
126.1 −2.39169 + 1.06485i 0.233733 + 2.22382i 3.24803 3.60730i −3.00000 −2.92705 5.06980i −2.00739 + 2.22943i −2.30902 + 7.10642i −1.95630 + 0.415823i 7.17508 3.19455i
159.1 −0.255585 + 0.283856i −2.18720 + 0.464905i 0.193806 + 1.84395i −3.00000 0.427051 0.739674i 0.313585 + 2.98357i −1.19098 0.865300i 1.82709 0.813473i 0.766755 0.851568i
250.1 0.273659 2.60369i −2.04275 0.909491i −4.74803 1.00922i −3.00000 −2.92705 + 5.06980i 2.93444 + 0.623735i −2.30902 + 7.10642i 1.33826 + 1.48629i −0.820977 + 7.81108i
256.1 −0.255585 0.283856i −2.18720 0.464905i 0.193806 1.84395i −3.00000 0.427051 + 0.739674i 0.313585 2.98357i −1.19098 + 0.865300i 1.82709 + 0.813473i 0.766755 + 0.851568i
295.1 0.273659 + 2.60369i −2.04275 + 0.909491i −4.74803 + 1.00922i −3.00000 −2.92705 5.06980i 2.93444 0.623735i −2.30902 7.10642i 1.33826 1.48629i −0.820977 7.81108i
380.1 0.373619 0.0794152i 1.49622 1.66172i −1.69381 + 0.754131i −3.00000 0.427051 0.739674i −2.74064 + 1.22021i −1.19098 + 0.865300i −0.209057 1.98904i −1.12086 + 0.238246i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 380.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
31.d even 5 1 inner
403.bl even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bl.a 8
13.c even 3 1 inner 403.2.bl.a 8
31.d even 5 1 inner 403.2.bl.a 8
403.bl even 15 1 inner 403.2.bl.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bl.a 8 1.a even 1 1 trivial
403.2.bl.a 8 13.c even 3 1 inner
403.2.bl.a 8 31.d even 5 1 inner
403.2.bl.a 8 403.bl even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8} )$$
$3$ $$1 + 5 T + 13 T^{2} + 10 T^{3} - 30 T^{4} - 115 T^{5} - 122 T^{6} + 100 T^{7} + 499 T^{8} + 300 T^{9} - 1098 T^{10} - 3105 T^{11} - 2430 T^{12} + 2430 T^{13} + 9477 T^{14} + 10935 T^{15} + 6561 T^{16}$$
$5$ $$( 1 + 3 T + 5 T^{2} )^{8}$$
$7$ $$1 + 3 T + 7 T^{2} + 36 T^{3} + 108 T^{4} + 219 T^{5} + 854 T^{6} + 2628 T^{7} + 5483 T^{8} + 18396 T^{9} + 41846 T^{10} + 75117 T^{11} + 259308 T^{12} + 605052 T^{13} + 823543 T^{14} + 2470629 T^{15} + 5764801 T^{16}$$
$11$ $$1 + 5 T + 21 T^{2} - 30 T^{3} - 310 T^{4} - 1635 T^{5} - 586 T^{6} + 11900 T^{7} + 83499 T^{8} + 130900 T^{9} - 70906 T^{10} - 2176185 T^{11} - 4538710 T^{12} - 4831530 T^{13} + 37202781 T^{14} + 97435855 T^{15} + 214358881 T^{16}$$
$13$ $$1 + 5 T + 12 T^{2} - 5 T^{3} - 181 T^{4} - 65 T^{5} + 2028 T^{6} + 10985 T^{7} + 28561 T^{8}$$
$17$ $$1 + 5 T + 32 T^{2} + 135 T^{3} + 670 T^{4} + 1110 T^{5} + 9032 T^{6} + 10880 T^{7} + 63559 T^{8} + 184960 T^{9} + 2610248 T^{10} + 5453430 T^{11} + 55959070 T^{12} + 191680695 T^{13} + 772402208 T^{14} + 2051693365 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 6 T + 19 T^{2} + 126 T^{3} + 756 T^{4} + 4308 T^{5} + 20501 T^{6} + 90468 T^{7} + 412487 T^{8} + 1718892 T^{9} + 7400861 T^{10} + 29548572 T^{11} + 98522676 T^{12} + 311988474 T^{13} + 893871739 T^{14} + 5363230434 T^{15} + 16983563041 T^{16}$$
$23$ $$1 + 11 T + 93 T^{2} + 798 T^{3} + 5738 T^{4} + 37203 T^{5} + 214886 T^{6} + 1165784 T^{7} + 5972343 T^{8} + 26813032 T^{9} + 113674694 T^{10} + 452648901 T^{11} + 1605727658 T^{12} + 5136201714 T^{13} + 13767337677 T^{14} + 37453079917 T^{15} + 78310985281 T^{16}$$
$29$ $$1 + 7 T + 54 T^{2} + 411 T^{3} + 2762 T^{4} + 11112 T^{5} + 80708 T^{6} + 338866 T^{7} + 1602651 T^{8} + 9827114 T^{9} + 67875428 T^{10} + 271010568 T^{11} + 1953510122 T^{12} + 8430082239 T^{13} + 32120459334 T^{14} + 120749134163 T^{15} + 500246412961 T^{16}$$
$31$ $$( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 5 T - 24 T^{2} + 125 T^{3} + 107 T^{4} + 4625 T^{5} - 32856 T^{6} - 253265 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$1 - 17 T + 146 T^{2} - 153 T^{3} - 7514 T^{4} + 82824 T^{5} - 245596 T^{6} - 1697654 T^{7} + 23405647 T^{8} - 69603814 T^{9} - 412846876 T^{10} + 5708312904 T^{11} - 21232768154 T^{12} - 17725998753 T^{13} + 693515219186 T^{14} - 3310822655977 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 12 T + 123 T^{2} + 340 T^{3} + 240 T^{4} - 20312 T^{5} + 86989 T^{6} + 1467420 T^{7} + 19111999 T^{8} + 63099060 T^{9} + 160842661 T^{10} - 1614946184 T^{11} + 820512240 T^{12} + 49982870620 T^{13} + 777527655027 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16}$$
$47$ $$( 1 + 14 T + 29 T^{2} - 352 T^{3} - 2851 T^{4} - 16544 T^{5} + 64061 T^{6} + 1453522 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 8480 T^{5} - 132023 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 - 4 T - 21 T^{2} - 144 T^{3} + 836 T^{4} + 4008 T^{5} + 237701 T^{6} - 1329022 T^{7} - 10035333 T^{8} - 78412298 T^{9} + 827437181 T^{10} + 823159032 T^{11} + 10130113796 T^{12} - 102949099056 T^{13} - 885791206461 T^{14} - 9954605939276 T^{15} + 146830437604321 T^{16}$$
$61$ $$( 1 + 3 T - 14 T^{2} - 297 T^{3} - 3741 T^{4} - 18117 T^{5} - 52094 T^{6} + 680943 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 20 T + 171 T^{2} + 1900 T^{3} + 21152 T^{4} + 127300 T^{5} + 767619 T^{6} + 6015260 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$1 + 19 T + 291 T^{2} + 4008 T^{3} + 50192 T^{4} + 565539 T^{5} + 5694542 T^{6} + 53394988 T^{7} + 475063971 T^{8} + 3791044148 T^{9} + 28706186222 T^{10} + 202412629029 T^{11} + 1275463092752 T^{12} + 7231351238808 T^{13} + 37277182621011 T^{14} + 172807283009429 T^{15} + 645753531245761 T^{16}$$
$73$ $$( 1 + 22 T + 171 T^{2} + 1076 T^{3} + 10229 T^{4} + 78548 T^{5} + 911259 T^{6} + 8558374 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 27 T + 470 T^{2} + 6117 T^{3} + 61789 T^{4} + 483243 T^{5} + 2933270 T^{6} + 13312053 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 23 T + 166 T^{2} + 709 T^{3} + 6369 T^{4} + 58847 T^{5} + 1143574 T^{6} + 13151101 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 - 9 T + 134 T^{2} + 51 T^{3} - 1674 T^{4} + 122868 T^{5} - 284444 T^{6} + 806238 T^{7} + 66134807 T^{8} + 71755182 T^{9} - 2253080924 T^{10} + 86618131092 T^{11} - 105030511434 T^{12} + 284787031899 T^{13} + 66595492988774 T^{14} - 398082014059761 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 + 3 T + 102 T^{2} - 89 T^{3} - 282 T^{4} - 55676 T^{5} - 856316 T^{6} - 2476602 T^{7} - 69420377 T^{8} - 240230394 T^{9} - 8057077244 T^{10} - 50813981948 T^{11} - 24965257242 T^{12} - 764273282873 T^{13} + 84963144502758 T^{14} + 242394853434339 T^{15} + 7837433594376961 T^{16}$$