# Properties

 Label 1183.2.c.h Level $1183$ Weight $2$ Character orbit 1183.c Analytic conductor $9.446$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 16 x^{10} + 96 x^{8} + 266 x^{6} + 332 x^{4} + 141 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 16 x^{10} + 96 x^{8} + 266 x^{6} + 332 x^{4} + 141 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{10} + 7 \nu^{8} + 33 \nu^{6} + 218 \nu^{4} + 528 \nu^{2} + 203$$$$)/83$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{10} + 14 \nu^{8} - 17 \nu^{6} - 228 \nu^{4} - 189 \nu^{2} + 157$$$$)/83$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{10} - 14 \nu^{8} + 17 \nu^{6} + 228 \nu^{4} + 272 \nu^{2} + 9$$$$)/83$$ $$\beta_{5}$$ $$=$$ $$($$$$14 \nu^{10} + 181 \nu^{8} + 794 \nu^{6} + 1309 \nu^{4} + 503 \nu^{2} - 146$$$$)/83$$ $$\beta_{6}$$ $$=$$ $$($$$$12 \nu^{11} + 167 \nu^{9} + 811 \nu^{7} + 1620 \nu^{5} + 1273 \nu^{3} + 444 \nu$$$$)/83$$ $$\beta_{7}$$ $$=$$ $$($$$$12 \nu^{10} + 167 \nu^{8} + 811 \nu^{6} + 1620 \nu^{4} + 1190 \nu^{2} + 112$$$$)/83$$ $$\beta_{8}$$ $$=$$ $$($$$$9 \nu^{11} + 146 \nu^{9} + 878 \nu^{7} + 2377 \nu^{5} + 2760 \nu^{3} + 997 \nu$$$$)/83$$ $$\beta_{9}$$ $$=$$ $$($$$$-6 \nu^{11} - 42 \nu^{9} + 134 \nu^{7} + 1597 \nu^{5} + 3555 \nu^{3} + 1936 \nu$$$$)/83$$ $$\beta_{10}$$ $$=$$ $$($$$$8 \nu^{11} + 139 \nu^{9} + 928 \nu^{7} + 2906 \nu^{5} + 4058 \nu^{3} + 1790 \nu$$$$)/83$$ $$\beta_{11}$$ $$=$$ $$($$$$-21 \nu^{11} - 313 \nu^{9} - 1689 \nu^{7} - 3997 \nu^{5} - 3950 \nu^{3} - 1109 \nu$$$$)/83$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{8} + \beta_{6} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - \beta_{5} - 6 \beta_{4} - 5 \beta_{3} + 7$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{11} + \beta_{9} - 9 \beta_{8} - 5 \beta_{6} + 18 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-8 \beta_{7} + 8 \beta_{5} + 33 \beta_{4} + 24 \beta_{3} + 2 \beta_{2} - 29$$ $$\nu^{7}$$ $$=$$ $$41 \beta_{11} + 3 \beta_{10} - 10 \beta_{9} + 57 \beta_{8} + 22 \beta_{6} - 86 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$53 \beta_{7} - 52 \beta_{5} - 175 \beta_{4} - 118 \beta_{3} - 22 \beta_{2} + 133$$ $$\nu^{9}$$ $$=$$ $$-228 \beta_{11} - 39 \beta_{10} + 74 \beta_{9} - 320 \beta_{8} - 96 \beta_{6} + 426 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-325 \beta_{7} + 318 \beta_{5} + 916 \beta_{4} + 596 \beta_{3} + 171 \beta_{2} - 647$$ $$\nu^{11}$$ $$=$$ $$1241 \beta_{11} + 340 \beta_{10} - 489 \beta_{9} + 1710 \beta_{8} + 425 \beta_{6} - 2159 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 1.71083i 0.908891i 1.54570i − 0.0849355i − 2.33192i − 2.10066i 2.10066i 2.33192i 0.0849355i − 1.54570i − 0.908891i − 1.71083i
2.63777i −2.71083 −4.95781 2.08281i 7.15053i 1.00000i 7.80201i 4.34860 5.49396
337.2 2.08281i −0.0911085 −2.33809 2.63777i 0.189762i 1.00000i 0.704173i −2.99170 5.49396
337.3 1.93488i −2.54570 −1.74376 0.312100i 4.92562i 1.00000i 0.495793i 3.48058 −0.603875
337.4 1.90785i −1.08494 −1.63989 1.10591i 2.06989i 1.00000i 0.687029i −1.82292 2.10992
337.5 1.10591i 1.33192 0.776957 1.90785i 1.47298i 1.00000i 3.07107i −1.22600 2.10992
337.6 0.312100i 1.10066 1.90259 1.93488i 0.343514i 1.00000i 1.21800i −1.78856 −0.603875
337.7 0.312100i 1.10066 1.90259 1.93488i 0.343514i 1.00000i 1.21800i −1.78856 −0.603875
337.8 1.10591i 1.33192 0.776957 1.90785i 1.47298i 1.00000i 3.07107i −1.22600 2.10992
337.9 1.90785i −1.08494 −1.63989 1.10591i 2.06989i 1.00000i 0.687029i −1.82292 2.10992
337.10 1.93488i −2.54570 −1.74376 0.312100i 4.92562i 1.00000i 0.495793i 3.48058 −0.603875
337.11 2.08281i −0.0911085 −2.33809 2.63777i 0.189762i 1.00000i 0.704173i −2.99170 5.49396
337.12 2.63777i −2.71083 −4.95781 2.08281i 7.15053i 1.00000i 7.80201i 4.34860 5.49396
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.12 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.h 12
13.b even 2 1 inner 1183.2.c.h 12
13.d odd 4 1 1183.2.a.n 6
13.d odd 4 1 1183.2.a.o yes 6
91.i even 4 1 8281.2.a.cb 6
91.i even 4 1 8281.2.a.cg 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.n 6 13.d odd 4 1
1183.2.a.o yes 6 13.d odd 4 1
1183.2.c.h 12 1.a even 1 1 trivial
1183.2.c.h 12 13.b even 2 1 inner
8281.2.a.cb 6 91.i even 4 1
8281.2.a.cg 6 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 20 T_{2}^{10} + 152 T_{2}^{8} + 547 T_{2}^{6} + 924 T_{2}^{4} + 588 T_{2}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 + 588 T^{2} + 924 T^{4} + 547 T^{6} + 152 T^{8} + 20 T^{10} + T^{12}$$
$3$ $$( 1 + 11 T - T^{2} - 14 T^{3} - T^{4} + 4 T^{5} + T^{6} )^{2}$$
$5$ $$49 + 588 T^{2} + 924 T^{4} + 547 T^{6} + 152 T^{8} + 20 T^{10} + T^{12}$$
$7$ $$( 1 + T^{2} )^{6}$$
$11$ $$790321 + 1499400 T^{2} + 446712 T^{4} + 55399 T^{6} + 3352 T^{8} + 96 T^{10} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$( -7351 + 5190 T - 319 T^{2} - 585 T^{3} + 190 T^{4} - 23 T^{5} + T^{6} )^{2}$$
$19$ $$337561 + 388325 T^{2} + 153538 T^{4} + 26412 T^{6} + 2085 T^{8} + 75 T^{10} + T^{12}$$
$23$ $$( -587 + 2649 T - 1198 T^{2} - 21 T^{3} + 97 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$29$ $$( 3569 + 6707 T - 996 T^{2} - 598 T^{3} + 9 T^{4} + 15 T^{5} + T^{6} )^{2}$$
$31$ $$460660369 + 893966229 T^{2} + 75982338 T^{4} + 2525488 T^{6} + 40869 T^{8} + 323 T^{10} + T^{12}$$
$37$ $$49 + 41013 T^{2} + 48034 T^{4} + 19356 T^{6} + 3065 T^{8} + 159 T^{10} + T^{12}$$
$41$ $$253009 + 1553078 T^{2} + 1152303 T^{4} + 241780 T^{6} + 17055 T^{8} + 246 T^{10} + T^{12}$$
$43$ $$( 181 + 19 T - 1107 T^{2} + 374 T^{3} + 53 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$47$ $$914034289 + 313905368 T^{2} + 36090852 T^{4} + 1647143 T^{6} + 33136 T^{8} + 300 T^{10} + T^{12}$$
$53$ $$( 24193 + 14872 T - 7043 T^{2} - 1218 T^{3} + 111 T^{4} + 25 T^{5} + T^{6} )^{2}$$
$59$ $$8632453921 + 5904453009 T^{2} + 429711240 T^{4} + 11015495 T^{6} + 119101 T^{8} + 570 T^{10} + T^{12}$$
$61$ $$( -12979 - 12037 T + 2271 T^{2} + 1088 T^{3} - 55 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$67$ $$715402009 + 572173833 T^{2} + 91460684 T^{4} + 4443594 T^{6} + 73708 T^{8} + 468 T^{10} + T^{12}$$
$71$ $$317982082201 + 27037393999 T^{2} + 899160661 T^{4} + 14938540 T^{6} + 131050 T^{8} + 577 T^{10} + T^{12}$$
$73$ $$2058164689 + 489609470 T^{2} + 44754577 T^{4} + 1965111 T^{6} + 41844 T^{8} + 377 T^{10} + T^{12}$$
$79$ $$( -10277 + 5533 T + 7248 T^{2} + 101 T^{3} - 167 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$83$ $$1697687209 + 6882262579 T^{2} + 471787913 T^{4} + 11538652 T^{6} + 124982 T^{8} + 593 T^{10} + T^{12}$$
$89$ $$49398174049 + 12313908201 T^{2} + 600526976 T^{4} + 12350122 T^{6} + 123700 T^{8} + 584 T^{10} + T^{12}$$
$97$ $$1929932761 + 1419409616 T^{2} + 241137916 T^{4} + 9287054 T^{6} + 124792 T^{8} + 617 T^{10} + T^{12}$$