Properties

 Label 504.2.cj.d Level 504 Weight 2 Character orbit 504.cj Analytic conductor 4.024 Analytic rank 0 Dimension 16 CM no Inner twists 8

Related objects

Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.cj (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{13} ) q^{4} -\beta_{6} q^{5} + ( -\beta_{4} - 2 \beta_{8} + \beta_{13} ) q^{7} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{13} ) q^{4} -\beta_{6} q^{5} + ( -\beta_{4} - 2 \beta_{8} + \beta_{13} ) q^{7} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{12} + \beta_{14} ) q^{8} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{10} + \beta_{11} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{9} ) q^{11} + ( \beta_{2} + 2 \beta_{8} - 3 \beta_{10} - 2 \beta_{13} ) q^{13} + ( \beta_{1} - \beta_{9} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{14} + ( -\beta_{4} - \beta_{5} + \beta_{11} - 2 \beta_{13} ) q^{16} + ( -2 \beta_{1} - \beta_{7} - 2 \beta_{14} ) q^{17} + ( \beta_{5} + \beta_{11} + \beta_{13} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{20} + ( 2 - 2 \beta_{2} + 2 \beta_{10} ) q^{22} + ( \beta_{1} - \beta_{9} - 2 \beta_{12} + \beta_{15} ) q^{23} + ( -1 - \beta_{4} - 2 \beta_{8} ) q^{25} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{12} - 2 \beta_{15} ) q^{26} + ( -2 - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} ) q^{28} + ( 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{15} ) q^{29} + ( 5 + 5 \beta_{4} + \beta_{8} ) q^{31} + ( -2 \beta_{1} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{14} ) q^{32} + ( -5 - 3 \beta_{2} - 4 \beta_{8} + \beta_{10} + 4 \beta_{13} ) q^{34} + ( -\beta_{1} - 2 \beta_{6} + \beta_{9} - 2 \beta_{15} ) q^{35} + ( 5 \beta_{5} - \beta_{11} + 2 \beta_{13} ) q^{37} + ( -\beta_{1} + \beta_{9} + \beta_{14} ) q^{38} + ( 4 \beta_{4} - 2 \beta_{11} - 2 \beta_{13} ) q^{40} + ( -4 \beta_{1} + 4 \beta_{12} + 4 \beta_{14} - 2 \beta_{15} ) q^{41} + ( 5 \beta_{2} + 3 \beta_{8} - \beta_{10} - 3 \beta_{13} ) q^{43} + ( -4 \beta_{3} + 4 \beta_{12} ) q^{44} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + 4 \beta_{8} - \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{46} + ( 3 \beta_{1} - 3 \beta_{9} - 6 \beta_{12} + 3 \beta_{15} ) q^{47} + ( 5 + 5 \beta_{4} - 2 \beta_{8} + 4 \beta_{13} ) q^{49} + ( \beta_{1} + \beta_{3} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{50} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + 4 \beta_{8} + 4 \beta_{10} + 4 \beta_{11} + 3 \beta_{13} ) q^{52} + ( 2 \beta_{1} + 3 \beta_{7} - 2 \beta_{9} ) q^{53} + ( -4 \beta_{8} + 4 \beta_{13} ) q^{55} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{6} - 2 \beta_{9} + \beta_{12} - 2 \beta_{15} ) q^{56} + ( -4 \beta_{11} - 4 \beta_{13} ) q^{58} + ( -\beta_{1} + 4 \beta_{7} + \beta_{9} ) q^{59} + ( -4 \beta_{5} - 2 \beta_{13} ) q^{61} + ( 4 \beta_{1} - 5 \beta_{3} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{62} + ( -2 - 2 \beta_{2} - 4 \beta_{8} + 4 \beta_{10} + 4 \beta_{13} ) q^{64} + ( -4 \beta_{1} + 4 \beta_{3} + \beta_{6} + 4 \beta_{9} + 6 \beta_{12} - 4 \beta_{15} ) q^{65} + ( \beta_{2} - \beta_{5} + 3 \beta_{8} - 5 \beta_{10} - 5 \beta_{11} - 6 \beta_{13} ) q^{67} + ( 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} - 4 \beta_{9} + 4 \beta_{15} ) q^{68} + ( -1 + \beta_{2} + 3 \beta_{10} + 4 \beta_{11} + 4 \beta_{13} ) q^{70} + ( 10 \beta_{1} - 8 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{71} + ( 3 + 3 \beta_{4} ) q^{73} + ( \beta_{1} + 6 \beta_{7} + 2 \beta_{9} + 2 \beta_{14} ) q^{74} + ( 2 + 2 \beta_{8} + \beta_{10} - 2 \beta_{13} ) q^{76} + ( 4 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{9} + 4 \beta_{15} ) q^{77} + ( 3 \beta_{4} - 3 \beta_{13} ) q^{79} + ( 6 \beta_{1} + 2 \beta_{7} - 2 \beta_{14} ) q^{80} + ( 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{11} - 4 \beta_{13} ) q^{82} + ( 4 \beta_{6} - 4 \beta_{7} - 3 \beta_{15} ) q^{83} + ( 4 \beta_{2} + 2 \beta_{8} - 2 \beta_{13} ) q^{85} + ( -3 \beta_{1} + 5 \beta_{3} + 4 \beta_{6} + 3 \beta_{9} - 3 \beta_{12} - 3 \beta_{15} ) q^{86} + ( -4 - 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} - 8 \beta_{8} + 4 \beta_{13} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{9} + 4 \beta_{12} - 2 \beta_{15} ) q^{89} + ( -8 \beta_{2} + 5 \beta_{5} - 2 \beta_{8} - 4 \beta_{10} + \beta_{11} + 5 \beta_{13} ) q^{91} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{92} + ( -3 + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 12 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} - 6 \beta_{13} ) q^{94} + ( \beta_{1} + \beta_{9} - 2 \beta_{14} ) q^{95} + ( -2 - 2 \beta_{8} + 2 \beta_{13} ) q^{97} + ( 7 \beta_{1} - 5 \beta_{3} - 4 \beta_{9} - 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{4} + 8q^{7} + O(q^{10})$$ $$16q - 8q^{4} + 8q^{7} - 8q^{10} + 8q^{16} + 32q^{22} - 8q^{25} - 16q^{28} + 40q^{31} - 80q^{34} - 32q^{40} - 8q^{46} + 40q^{49} + 8q^{52} - 32q^{64} - 16q^{70} + 24q^{73} + 32q^{76} - 24q^{79} - 16q^{82} - 32q^{88} - 24q^{94} - 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 4 x^{14} + 6 x^{12} + 8 x^{10} + 20 x^{8} + 32 x^{6} + 96 x^{4} + 256 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{15} - 44 \nu^{13} - 6 \nu^{11} - 24 \nu^{9} - 148 \nu^{7} - 64 \nu^{5} - 384 \nu^{3} - 2048 \nu$$$$)/896$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{14} - 44 \nu^{12} - 6 \nu^{10} - 24 \nu^{8} - 148 \nu^{6} - 64 \nu^{4} - 384 \nu^{2} - 2496$$$$)/448$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} + 4 \nu^{13} + 6 \nu^{11} + 8 \nu^{9} + 20 \nu^{7} + 32 \nu^{5} + 96 \nu^{3} + 256 \nu$$$$)/128$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{14} + 3 \nu^{12} + 2 \nu^{10} - 6 \nu^{8} + 12 \nu^{6} + 12 \nu^{4} - 96 \nu^{2} + 160$$$$)/224$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{14} - 3 \nu^{12} - 2 \nu^{10} + 6 \nu^{8} - 12 \nu^{6} - 12 \nu^{4} + 320 \nu^{2} - 160$$$$)/224$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{15} + 30 \nu^{13} + 6 \nu^{11} + 52 \nu^{9} + 148 \nu^{7} + 8 \nu^{5} + 608 \nu^{3} + 2048 \nu$$$$)/448$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{15} + 12 \nu^{13} - 6 \nu^{11} - 24 \nu^{9} + 76 \nu^{7} - 64 \nu^{5} - 384 \nu^{3} + 1088 \nu$$$$)/448$$ $$\beta_{8}$$ $$=$$ $$($$$$11 \nu^{14} + 24 \nu^{12} + 2 \nu^{10} + 64 \nu^{8} + 124 \nu^{6} + 208 \nu^{4} + 800 \nu^{2} + 1280$$$$)/448$$ $$\beta_{9}$$ $$=$$ $$($$$$15 \nu^{15} + 20 \nu^{13} + 18 \nu^{11} + 72 \nu^{9} - 4 \nu^{7} + 192 \nu^{5} + 1152 \nu^{3} + 1664 \nu$$$$)/896$$ $$\beta_{10}$$ $$=$$ $$($$$$15 \nu^{14} + 20 \nu^{12} + 18 \nu^{10} + 72 \nu^{8} - 4 \nu^{6} + 192 \nu^{4} + 1152 \nu^{2} + 768$$$$)/448$$ $$\beta_{11}$$ $$=$$ $$($$$$-5 \nu^{14} - 9 \nu^{12} - 6 \nu^{10} - 38 \nu^{8} - 36 \nu^{6} - 36 \nu^{4} - 384 \nu^{2} - 480$$$$)/112$$ $$\beta_{12}$$ $$=$$ $$($$$$-11 \nu^{15} - 24 \nu^{13} - 2 \nu^{11} - 64 \nu^{9} - 124 \nu^{7} - 208 \nu^{5} - 800 \nu^{3} - 1280 \nu$$$$)/448$$ $$\beta_{13}$$ $$=$$ $$($$$$-6 \nu^{14} - 15 \nu^{12} - 10 \nu^{10} - 26 \nu^{8} - 60 \nu^{6} - 60 \nu^{4} - 472 \nu^{2} - 800$$$$)/112$$ $$\beta_{14}$$ $$=$$ $$($$$$65 \nu^{15} + 124 \nu^{13} + 78 \nu^{11} + 312 \nu^{9} + 580 \nu^{7} + 832 \nu^{5} + 4992 \nu^{3} + 6912 \nu$$$$)/896$$ $$\beta_{15}$$ $$=$$ $$($$$$17 \nu^{15} + 39 \nu^{13} + 26 \nu^{11} + 90 \nu^{9} + 156 \nu^{7} + 156 \nu^{5} + 1216 \nu^{3} + 2080 \nu$$$$)/224$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{7} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{14} + \beta_{12} - \beta_{7} + \beta_{6} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} + 2 \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} + 1$$ $$\nu^{5}$$ $$=$$ $$-\beta_{15} - 2 \beta_{12} + \beta_{9} - \beta_{6} + 2 \beta_{3} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{13} - 4 \beta_{10} + 2 \beta_{8} + 2 \beta_{2} - 2$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{14} - 6 \beta_{9} + 2 \beta_{7} + 4 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$4 \beta_{13} - 6 \beta_{11} - 2 \beta_{5} + 2 \beta_{4}$$ $$\nu^{9}$$ $$=$$ $$6 \beta_{15} - 12 \beta_{14} - 12 \beta_{12} - 6 \beta_{7} + 6 \beta_{6} + 4 \beta_{3} + 8 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-12 \beta_{13} - 12 \beta_{8} - 12 \beta_{5} + 4 \beta_{4} + 12 \beta_{2} + 4$$ $$\nu^{11}$$ $$=$$ $$4 \beta_{15} + 12 \beta_{12} - 4 \beta_{9} - 4 \beta_{6} + 16 \beta_{3} + 4 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$8 \beta_{13} + 12 \beta_{10} - 8 \beta_{8} - 20 \beta_{2} - 52$$ $$\nu^{13}$$ $$=$$ $$-8 \beta_{14} - 4 \beta_{9} - 28 \beta_{7} - 60 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-16 \beta_{13} + 16 \beta_{11} - 40 \beta_{5} - 104 \beta_{4}$$ $$\nu^{15}$$ $$=$$ $$56 \beta_{15} - 8 \beta_{14} - 8 \beta_{12} + 88 \beta_{7} - 88 \beta_{6} - 64 \beta_{3} + 72 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −1.33546 − 0.465333i −1.01214 − 0.987711i −0.349313 − 1.37039i −0.264742 + 1.38921i 0.264742 − 1.38921i 0.349313 + 1.37039i 1.01214 + 0.987711i 1.33546 + 0.465333i −1.33546 + 0.465333i −1.01214 + 0.987711i −0.349313 + 1.37039i −0.264742 − 1.38921i 0.264742 + 1.38921i 0.349313 − 1.37039i 1.01214 − 0.987711i 1.33546 − 0.465333i
−1.33546 + 0.465333i 0 1.56693 1.24287i 0.937379 + 0.541196i 0 −1.62132 2.09077i −1.51423 + 2.38896i 0 −1.50367 0.286555i
37.2 −1.01214 + 0.987711i 0 0.0488544 1.99940i 2.26303 + 1.30656i 0 2.62132 + 0.358719i 1.92538 + 2.07193i 0 −3.58101 + 0.912798i
37.3 −0.349313 + 1.37039i 0 −1.75596 0.957392i −2.26303 1.30656i 0 2.62132 + 0.358719i 1.92538 2.07193i 0 2.58101 2.64485i
37.4 −0.264742 1.38921i 0 −1.85982 + 0.735566i 0.937379 + 0.541196i 0 −1.62132 2.09077i 1.51423 + 2.38896i 0 0.503673 1.44550i
37.5 0.264742 + 1.38921i 0 −1.85982 + 0.735566i −0.937379 0.541196i 0 −1.62132 2.09077i −1.51423 2.38896i 0 0.503673 1.44550i
37.6 0.349313 1.37039i 0 −1.75596 0.957392i 2.26303 + 1.30656i 0 2.62132 + 0.358719i −1.92538 + 2.07193i 0 2.58101 2.64485i
37.7 1.01214 0.987711i 0 0.0488544 1.99940i −2.26303 1.30656i 0 2.62132 + 0.358719i −1.92538 2.07193i 0 −3.58101 + 0.912798i
37.8 1.33546 0.465333i 0 1.56693 1.24287i −0.937379 0.541196i 0 −1.62132 2.09077i 1.51423 2.38896i 0 −1.50367 0.286555i
109.1 −1.33546 0.465333i 0 1.56693 + 1.24287i 0.937379 0.541196i 0 −1.62132 + 2.09077i −1.51423 2.38896i 0 −1.50367 + 0.286555i
109.2 −1.01214 0.987711i 0 0.0488544 + 1.99940i 2.26303 1.30656i 0 2.62132 0.358719i 1.92538 2.07193i 0 −3.58101 0.912798i
109.3 −0.349313 1.37039i 0 −1.75596 + 0.957392i −2.26303 + 1.30656i 0 2.62132 0.358719i 1.92538 + 2.07193i 0 2.58101 + 2.64485i
109.4 −0.264742 + 1.38921i 0 −1.85982 0.735566i 0.937379 0.541196i 0 −1.62132 + 2.09077i 1.51423 2.38896i 0 0.503673 + 1.44550i
109.5 0.264742 1.38921i 0 −1.85982 0.735566i −0.937379 + 0.541196i 0 −1.62132 + 2.09077i −1.51423 + 2.38896i 0 0.503673 + 1.44550i
109.6 0.349313 + 1.37039i 0 −1.75596 + 0.957392i 2.26303 1.30656i 0 2.62132 0.358719i −1.92538 2.07193i 0 2.58101 + 2.64485i
109.7 1.01214 + 0.987711i 0 0.0488544 + 1.99940i −2.26303 + 1.30656i 0 2.62132 0.358719i −1.92538 + 2.07193i 0 −3.58101 0.912798i
109.8 1.33546 + 0.465333i 0 1.56693 + 1.24287i −0.937379 + 0.541196i 0 −1.62132 + 2.09077i 1.51423 + 2.38896i 0 −1.50367 + 0.286555i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.d 16
3.b odd 2 1 inner 504.2.cj.d 16
4.b odd 2 1 2016.2.cr.d 16
7.c even 3 1 inner 504.2.cj.d 16
8.b even 2 1 inner 504.2.cj.d 16
8.d odd 2 1 2016.2.cr.d 16
12.b even 2 1 2016.2.cr.d 16
21.h odd 6 1 inner 504.2.cj.d 16
24.f even 2 1 2016.2.cr.d 16
24.h odd 2 1 inner 504.2.cj.d 16
28.g odd 6 1 2016.2.cr.d 16
56.k odd 6 1 2016.2.cr.d 16
56.p even 6 1 inner 504.2.cj.d 16
84.n even 6 1 2016.2.cr.d 16
168.s odd 6 1 inner 504.2.cj.d 16
168.v even 6 1 2016.2.cr.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.d 16 1.a even 1 1 trivial
504.2.cj.d 16 3.b odd 2 1 inner
504.2.cj.d 16 7.c even 3 1 inner
504.2.cj.d 16 8.b even 2 1 inner
504.2.cj.d 16 21.h odd 6 1 inner
504.2.cj.d 16 24.h odd 2 1 inner
504.2.cj.d 16 56.p even 6 1 inner
504.2.cj.d 16 168.s odd 6 1 inner
2016.2.cr.d 16 4.b odd 2 1
2016.2.cr.d 16 8.d odd 2 1
2016.2.cr.d 16 12.b even 2 1
2016.2.cr.d 16 24.f even 2 1
2016.2.cr.d 16 28.g odd 6 1
2016.2.cr.d 16 56.k odd 6 1
2016.2.cr.d 16 84.n even 6 1
2016.2.cr.d 16 168.v even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 8 T_{5}^{6} + 56 T_{5}^{4} - 64 T_{5}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 6 T^{4} + 8 T^{6} + 20 T^{8} + 32 T^{10} + 96 T^{12} + 256 T^{14} + 256 T^{16}$$
$3$ 
$5$ $$( 1 + 12 T^{2} + 66 T^{4} + 336 T^{6} + 1859 T^{8} + 8400 T^{10} + 41250 T^{12} + 187500 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{4}$$
$11$ $$( 1 + 12 T^{2} - 6 T^{4} - 1104 T^{6} - 9565 T^{8} - 133584 T^{10} - 87846 T^{12} + 21258732 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 + 10 T^{2} + 235 T^{4} + 1690 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 - 12 T^{2} + 178 T^{4} + 7344 T^{6} - 123981 T^{8} + 2122416 T^{10} + 14866738 T^{12} - 289650828 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 + 66 T^{2} + 2553 T^{4} + 71346 T^{6} + 1547972 T^{8} + 25755906 T^{10} + 332709513 T^{12} + 3105028146 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 68 T^{2} + 2418 T^{4} - 78064 T^{6} + 2140499 T^{8} - 41295856 T^{10} + 676655538 T^{12} - 10066440452 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 52 T^{2} + 1846 T^{4} - 43732 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 10 T + 15 T^{2} - 230 T^{3} + 3164 T^{4} - 7130 T^{5} + 14415 T^{6} - 297910 T^{7} + 923521 T^{8} )^{4}$$
$37$ $$( 1 + 6 T^{2} - 2583 T^{4} - 714 T^{6} + 4935716 T^{8} - 977466 T^{10} - 4840957863 T^{12} + 15394358454 T^{14} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 68 T^{2} + 4390 T^{4} + 114308 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 50 T^{2} + 1435 T^{4} - 92450 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 28 T^{2} - 3182 T^{4} - 12656 T^{6} + 9117619 T^{8} - 27957104 T^{10} - 15527144942 T^{12} + 301818029212 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 + 108 T^{2} + 5442 T^{4} + 65232 T^{6} - 1941373 T^{8} + 183236688 T^{10} + 42939997602 T^{12} + 2393751001932 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 100 T^{2} + 930 T^{4} + 210800 T^{6} + 35337539 T^{8} + 733794800 T^{10} + 11269145730 T^{12} + 4218053364100 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 + 164 T^{2} + 13242 T^{4} + 1018768 T^{6} + 72505859 T^{8} + 3790835728 T^{10} + 183346626522 T^{12} + 8449341395204 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 98 T^{2} + 537 T^{4} + 8722 T^{6} + 18947012 T^{8} + 39153058 T^{10} + 10821151977 T^{12} + 8864921452562 T^{14} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 + 36 T^{2} + 1694 T^{4} + 181476 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} )^{8}$$
$79$ $$( 1 + 6 T - 113 T^{2} - 54 T^{3} + 13116 T^{4} - 4266 T^{5} - 705233 T^{6} + 2958234 T^{7} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 132 T^{2} + 15822 T^{4} - 909348 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 260 T^{2} + 34986 T^{4} - 4360720 T^{6} + 465471155 T^{8} - 34541263120 T^{10} + 2195100043626 T^{12} - 129215135649860 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 + 4 T + 190 T^{2} + 388 T^{3} + 9409 T^{4} )^{8}$$