# Properties

 Label 471.2.b.b Level $471$ Weight $2$ Character orbit 471.b Analytic conductor $3.761$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 471.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76095393520$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 24 x^{12} + 224 x^{10} + 1027 x^{8} + 2399 x^{6} + 2652 x^{4} + 1094 x^{2} + 147$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 24 x^{12} + 224 x^{10} + 1027 x^{8} + 2399 x^{6} + 2652 x^{4} + 1094 x^{2} + 147$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 5 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$25 \nu^{13} + 593 \nu^{11} + 5418 \nu^{9} + 23876 \nu^{7} + 51582 \nu^{5} + 47750 \nu^{3} + 10522 \nu$$$$)/133$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{12} + 52 \nu^{10} + 533 \nu^{8} + 2702 \nu^{6} + 6858 \nu^{4} + 7468 \nu^{2} + 1791$$$$)/19$$ $$\beta_{6}$$ $$=$$ $$($$$$39 \nu^{13} + 957 \nu^{11} + 9149 \nu^{9} + 42790 \nu^{7} + 99721 \nu^{5} + 101223 \nu^{3} + 25187 \nu$$$$)/266$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{12} + 78 \nu^{10} + 790 \nu^{8} + 3901 \nu^{6} + 9508 \nu^{4} + 9853 \nu^{2} + 2259$$$$)/19$$ $$\beta_{8}$$ $$=$$ $$($$$$7 \nu^{13} + 163 \nu^{11} + 1457 \nu^{9} + 6265 \nu^{7} + 13230 \nu^{5} + 12192 \nu^{3} + 2953 \nu$$$$)/19$$ $$\beta_{9}$$ $$=$$ $$($$$$7 \nu^{12} + 163 \nu^{10} + 1457 \nu^{8} + 6265 \nu^{6} + 13211 \nu^{4} + 12021 \nu^{2} + 2687$$$$)/19$$ $$\beta_{10}$$ $$=$$ $$($$$$-7 \nu^{12} - 163 \nu^{10} - 1457 \nu^{8} - 6265 \nu^{6} - 13230 \nu^{4} - 12173 \nu^{2} - 2858$$$$)/19$$ $$\beta_{11}$$ $$=$$ $$($$$$21 \nu^{12} + 489 \nu^{10} + 4371 \nu^{8} + 18776 \nu^{6} + 39443 \nu^{4} + 35645 \nu^{2} + 7985$$$$)/38$$ $$\beta_{12}$$ $$=$$ $$($$$$137 \nu^{13} + 3239 \nu^{11} + 29547 \nu^{9} + 130500 \nu^{7} + 284675 \nu^{5} + 269517 \nu^{3} + 62539 \nu$$$$)/266$$ $$\beta_{13}$$ $$=$$ $$($$$$-93 \nu^{13} - 2190 \nu^{11} - 19873 \nu^{9} - 87111 \nu^{7} - 187778 \nu^{5} - 174172 \nu^{3} - 38280 \nu$$$$)/133$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{10} - \beta_{9} - 8 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{12} + \beta_{8} + \beta_{6} - 9 \beta_{3} + 30 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{11} + 10 \beta_{10} + 13 \beta_{9} + 58 \beta_{2} - 88$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{13} + 15 \beta_{12} - 12 \beta_{8} - 13 \beta_{6} + 68 \beta_{3} - 195 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$32 \beta_{11} - 78 \beta_{10} - 126 \beta_{9} - 2 \beta_{7} + 3 \beta_{5} - 414 \beta_{2} + 559$$ $$\nu^{9}$$ $$=$$ $$-34 \beta_{13} - 157 \beta_{12} + 109 \beta_{8} + 127 \beta_{6} - 9 \beta_{4} - 490 \beta_{3} + 1322 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-352 \beta_{11} + 565 \beta_{10} + 1092 \beta_{9} + 43 \beta_{7} - 61 \beta_{5} + 2958 \beta_{2} - 3714$$ $$\nu^{11}$$ $$=$$ $$395 \beta_{13} + 1426 \beta_{12} - 899 \beta_{8} - 1110 \beta_{6} + 190 \beta_{4} + 3480 \beta_{3} - 9181 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$3326 \beta_{11} - 3984 \beta_{10} - 8947 \beta_{9} - 585 \beta_{7} + 796 \beta_{5} - 21237 \beta_{2} + 25350$$ $$\nu^{13}$$ $$=$$ $$-3911 \beta_{13} - 12062 \beta_{12} + 7099 \beta_{8} + 9158 \beta_{6} - 2551 \beta_{4} - 24636 \beta_{3} + 64733 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$$.

 $$n$$ $$158$$ $$319$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
313.1
 − 2.72109i − 2.49123i − 2.14537i − 1.66261i − 1.53977i − 0.580612i − 0.560879i 0.560879i 0.580612i 1.53977i 1.66261i 2.14537i 2.49123i 2.72109i
2.72109i 1.00000 −5.40431 1.84852i 2.72109i 5.01715i 9.26343i 1.00000 5.02998
313.2 2.49123i 1.00000 −4.20622 1.87721i 2.49123i 2.20605i 5.49621i 1.00000 −4.67656
313.3 2.14537i 1.00000 −2.60262 4.18337i 2.14537i 4.37069i 1.29283i 1.00000 8.97487
313.4 1.66261i 1.00000 −0.764270 1.85555i 1.66261i 0.948177i 2.05454i 1.00000 −3.08505
313.5 1.53977i 1.00000 −0.370884 3.16196i 1.53977i 0.921702i 2.50846i 1.00000 −4.86868
313.6 0.580612i 1.00000 1.66289 2.44616i 0.580612i 1.30605i 2.12672i 1.00000 1.42027
313.7 0.560879i 1.00000 1.68542 0.365796i 0.560879i 4.51711i 2.06707i 1.00000 0.205167
313.8 0.560879i 1.00000 1.68542 0.365796i 0.560879i 4.51711i 2.06707i 1.00000 0.205167
313.9 0.580612i 1.00000 1.66289 2.44616i 0.580612i 1.30605i 2.12672i 1.00000 1.42027
313.10 1.53977i 1.00000 −0.370884 3.16196i 1.53977i 0.921702i 2.50846i 1.00000 −4.86868
313.11 1.66261i 1.00000 −0.764270 1.85555i 1.66261i 0.948177i 2.05454i 1.00000 −3.08505
313.12 2.14537i 1.00000 −2.60262 4.18337i 2.14537i 4.37069i 1.29283i 1.00000 8.97487
313.13 2.49123i 1.00000 −4.20622 1.87721i 2.49123i 2.20605i 5.49621i 1.00000 −4.67656
313.14 2.72109i 1.00000 −5.40431 1.84852i 2.72109i 5.01715i 9.26343i 1.00000 5.02998
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.b.b 14
3.b odd 2 1 1413.2.b.e 14
157.b even 2 1 inner 471.2.b.b 14
471.d odd 2 1 1413.2.b.e 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.b.b 14 1.a even 1 1 trivial
471.2.b.b 14 157.b even 2 1 inner
1413.2.b.e 14 3.b odd 2 1
1413.2.b.e 14 471.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{14} + 24 T_{2}^{12} + 224 T_{2}^{10} + 1027 T_{2}^{8} + 2399 T_{2}^{6} + 2652 T_{2}^{4} + 1094 T_{2}^{2} + 147$$ acting on $$S_{2}^{\mathrm{new}}(471, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$147 + 1094 T^{2} + 2652 T^{4} + 2399 T^{6} + 1027 T^{8} + 224 T^{10} + 24 T^{12} + T^{14}$$
$3$ $$( -1 + T )^{14}$$
$5$ $$5808 + 50324 T^{2} + 54976 T^{4} + 25240 T^{6} + 5914 T^{8} + 729 T^{10} + 44 T^{12} + T^{14}$$
$7$ $$62208 + 200448 T^{2} + 229152 T^{4} + 111368 T^{6} + 22679 T^{8} + 1943 T^{10} + 73 T^{12} + T^{14}$$
$11$ $$( -1152 - 768 T + 480 T^{2} + 304 T^{3} - 54 T^{4} - 35 T^{5} + T^{6} + T^{7} )^{2}$$
$13$ $$( 2464 - 144 T - 1864 T^{2} + 708 T^{3} + 75 T^{4} - 52 T^{5} + T^{7} )^{2}$$
$17$ $$( -1296 - 11616 T - 6144 T^{2} + 956 T^{3} + 517 T^{4} - 57 T^{5} - 9 T^{6} + T^{7} )^{2}$$
$19$ $$( -96 - 480 T - 112 T^{2} + 748 T^{3} - 160 T^{4} - 48 T^{5} + 6 T^{6} + T^{7} )^{2}$$
$23$ $$4747692 + 6248732 T^{2} + 2710752 T^{4} + 534730 T^{6} + 54893 T^{8} + 3049 T^{10} + 87 T^{12} + T^{14}$$
$29$ $$181398528 + 151296660 T^{2} + 43095564 T^{4} + 5231952 T^{6} + 317322 T^{8} + 10111 T^{10} + 161 T^{12} + T^{14}$$
$31$ $$( -144 - 204 T + 476 T^{2} + 100 T^{3} - 118 T^{4} - 19 T^{5} + 7 T^{6} + T^{7} )^{2}$$
$37$ $$( -1296 + 360 T + 1652 T^{2} + 132 T^{3} - 320 T^{4} - 44 T^{5} + 7 T^{6} + T^{7} )^{2}$$
$41$ $$4026589488 + 5556312788 T^{2} + 1174401744 T^{4} + 84511976 T^{6} + 2773570 T^{8} + 45089 T^{10} + 348 T^{12} + T^{14}$$
$43$ $$110592 + 42034176 T^{2} + 36605184 T^{4} + 7804928 T^{6} + 591664 T^{8} + 18684 T^{10} + 243 T^{12} + T^{14}$$
$47$ $$( 27648 - 11904 T - 11136 T^{2} + 1184 T^{3} + 692 T^{4} - 61 T^{5} - 11 T^{6} + T^{7} )^{2}$$
$53$ $$1063932672 + 1560405632 T^{2} + 476411232 T^{4} + 42983700 T^{6} + 1712168 T^{8} + 33244 T^{10} + 302 T^{12} + T^{14}$$
$59$ $$20635140288 + 6483645344 T^{2} + 802784108 T^{4} + 50482484 T^{6} + 1738844 T^{8} + 32574 T^{10} + 301 T^{12} + T^{14}$$
$61$ $$73638346752 + 93524742144 T^{2} + 10796630016 T^{4} + 471472256 T^{6} + 9709760 T^{8} + 101233 T^{10} + 514 T^{12} + T^{14}$$
$67$ $$( 36352 + 25136 T - 14736 T^{2} - 2816 T^{3} + 1124 T^{4} + 41 T^{5} - 21 T^{6} + T^{7} )^{2}$$
$71$ $$( -28224 + 68352 T - 14016 T^{2} - 6680 T^{3} + 1806 T^{4} - 23 T^{5} - 19 T^{6} + T^{7} )^{2}$$
$73$ $$429632335872 + 108303547392 T^{2} + 10254127104 T^{4} + 461404640 T^{6} + 10357572 T^{8} + 112141 T^{10} + 555 T^{12} + T^{14}$$
$79$ $$19554905088 + 9116709120 T^{2} + 1365311616 T^{4} + 89608064 T^{6} + 2822244 T^{8} + 44729 T^{10} + 342 T^{12} + T^{14}$$
$83$ $$30950832 + 1250144132 T^{2} + 1801194764 T^{4} + 202310032 T^{6} + 7111462 T^{8} + 96883 T^{10} + 533 T^{12} + T^{14}$$
$89$ $$( -240048 + 707424 T + 326016 T^{2} - 12884 T^{3} - 6953 T^{4} - 188 T^{5} + 24 T^{6} + T^{7} )^{2}$$
$97$ $$134547222528 + 207880178688 T^{2} + 38737383936 T^{4} + 1875857120 T^{6} + 33924516 T^{8} + 256673 T^{10} + 846 T^{12} + T^{14}$$