Properties

 Label 354.2.c.b Level 354 Weight 2 Character orbit 354.c Analytic conductor 2.827 Analytic rank 0 Dimension 10 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 354.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.82670423155$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: 10.0.41542366334681088.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} - x^{8} - 2 x^{7} + 2 x^{6} + 14 x^{5} + 6 x^{4} - 18 x^{3} - 27 x^{2} - 81 x + 243$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{9} - 2 \nu^{8} - 23 \nu^{7} - 49 \nu^{6} + 31 \nu^{5} - 47 \nu^{4} + 141 \nu^{3} + 27 \nu^{2} - 1296 \nu - 405$$$$)/648$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{8} - 7 \nu^{7} - 4 \nu^{6} + 13 \nu^{5} - 4 \nu^{4} - 7 \nu^{3} + 12 \nu^{2} + 9 \nu + 27$$$$)/108$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} + 8 \nu^{8} + 17 \nu^{7} - 11 \nu^{6} + 11 \nu^{5} - 49 \nu^{4} + 213 \nu^{3} + 225 \nu^{2} - 54 \nu - 81$$$$)/648$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{9} + 8 \nu^{8} - 19 \nu^{7} + 25 \nu^{6} + 47 \nu^{5} + 23 \nu^{4} - 183 \nu^{3} + 45 \nu^{2} - 594 \nu + 567$$$$)/648$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{9} + 8 \nu^{8} - \nu^{7} + 7 \nu^{6} + 29 \nu^{5} - 13 \nu^{4} + 15 \nu^{3} - 189 \nu^{2} + 243$$$$)/324$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{9} + \nu^{8} + \nu^{7} + 2 \nu^{6} - 2 \nu^{5} - 14 \nu^{4} - 6 \nu^{3} + 18 \nu^{2} + 27 \nu + 81$$$$)/81$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{9} - 4 \nu^{8} - \nu^{7} + 19 \nu^{6} + 17 \nu^{5} - 22 \nu^{4} - 33 \nu^{3} - 45 \nu^{2} + 27 \nu + 324$$$$)/162$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{9} - 2 \nu^{8} + 4 \nu^{7} + 5 \nu^{6} + 4 \nu^{5} - 20 \nu^{4} - 48 \nu^{3} + 81 \nu + 162$$$$)/81$$ $$\beta_{9}$$ $$=$$ $$($$$$8 \nu^{9} + 7 \nu^{8} - 5 \nu^{7} - 22 \nu^{6} - 59 \nu^{5} - 2 \nu^{4} + 159 \nu^{3} + 90 \nu^{2} + 81 \nu - 729$$$$)/324$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} + 1$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{2} - 5 \beta_{1} + 6$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-6 \beta_{9} + 3 \beta_{8} - 12 \beta_{6} + 3 \beta_{5} + \beta_{4} + 15 \beta_{3} + 8 \beta_{2} - 2 \beta_{1} - 12$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{9} - 3 \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 3 \beta_{1} - 9$$ $$\nu^{7}$$ $$=$$ $$($$$$-12 \beta_{9} + 15 \beta_{8} - 18 \beta_{7} - 15 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 27 \beta_{3} - 23 \beta_{2} - \beta_{1} - 6$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-6 \beta_{9} - 15 \beta_{8} - 33 \beta_{7} + 51 \beta_{6} + 36 \beta_{5} + 29 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 16 \beta_{1} - 30$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$30 \beta_{9} + 26 \beta_{8} + 5 \beta_{7} - 45 \beta_{6} + 7 \beta_{5} + 24 \beta_{4} + 16 \beta_{3} - 2 \beta_{1} + 25$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/354\mathbb{Z}\right)^\times$$.

 $$n$$ $$61$$ $$119$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
353.1
 1.65530 − 0.509894i 1.65530 + 0.509894i 1.43134 − 0.975334i 1.43134 + 0.975334i −0.0738304 − 1.73048i −0.0738304 + 1.73048i −0.869925 − 1.49774i −0.869925 + 1.49774i −1.64288 − 0.548592i −1.64288 + 0.548592i
1.00000 −1.65530 0.509894i 1.00000 2.90260i −1.65530 0.509894i −3.06916 1.00000 2.48002 + 1.68805i 2.90260i
353.2 1.00000 −1.65530 + 0.509894i 1.00000 2.90260i −1.65530 + 0.509894i −3.06916 1.00000 2.48002 1.68805i 2.90260i
353.3 1.00000 −1.43134 0.975334i 1.00000 0.0999120i −1.43134 0.975334i 2.48626 1.00000 1.09745 + 2.79206i 0.0999120i
353.4 1.00000 −1.43134 + 0.975334i 1.00000 0.0999120i −1.43134 + 0.975334i 2.48626 1.00000 1.09745 2.79206i 0.0999120i
353.5 1.00000 0.0738304 1.73048i 1.00000 2.30520i 0.0738304 1.73048i 3.25240 1.00000 −2.98910 0.255524i 2.30520i
353.6 1.00000 0.0738304 + 1.73048i 1.00000 2.30520i 0.0738304 + 1.73048i 3.25240 1.00000 −2.98910 + 0.255524i 2.30520i
353.7 1.00000 0.869925 1.49774i 1.00000 1.66014i 0.869925 1.49774i −3.57946 1.00000 −1.48646 2.60585i 1.66014i
353.8 1.00000 0.869925 + 1.49774i 1.00000 1.66014i 0.869925 + 1.49774i −3.57946 1.00000 −1.48646 + 2.60585i 1.66014i
353.9 1.00000 1.64288 0.548592i 1.00000 2.54852i 1.64288 0.548592i −0.0900537 1.00000 2.39809 1.80254i 2.54852i
353.10 1.00000 1.64288 + 0.548592i 1.00000 2.54852i 1.64288 + 0.548592i −0.0900537 1.00000 2.39809 + 1.80254i 2.54852i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
177.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.2.c.b yes 10
3.b odd 2 1 354.2.c.a 10
59.b odd 2 1 354.2.c.a 10
177.d even 2 1 inner 354.2.c.b yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.c.a 10 3.b odd 2 1
354.2.c.a 10 59.b odd 2 1
354.2.c.b yes 10 1.a even 1 1 trivial
354.2.c.b yes 10 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{5} + 2 T_{11}^{4} - 25 T_{11}^{3} - 54 T_{11}^{2} + 54 T_{11} + 108$$ acting on $$S_{2}^{\mathrm{new}}(354, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{10}$$
$3$ $$1 + T - T^{2} + 2 T^{3} + 2 T^{4} - 14 T^{5} + 6 T^{6} + 18 T^{7} - 27 T^{8} + 81 T^{9} + 243 T^{10}$$
$5$ $$1 - 27 T^{2} + 395 T^{4} - 3938 T^{6} + 29068 T^{8} - 165022 T^{10} + 726700 T^{12} - 2461250 T^{14} + 6171875 T^{16} - 10546875 T^{18} + 9765625 T^{20}$$
$7$ $$( 1 + T + 16 T^{2} + 17 T^{3} + 179 T^{4} + 148 T^{5} + 1253 T^{6} + 833 T^{7} + 5488 T^{8} + 2401 T^{9} + 16807 T^{10} )^{2}$$
$11$ $$( 1 + 2 T + 30 T^{2} + 34 T^{3} + 439 T^{4} + 372 T^{5} + 4829 T^{6} + 4114 T^{7} + 39930 T^{8} + 29282 T^{9} + 161051 T^{10} )^{2}$$
$13$ $$1 - 52 T^{2} + 1144 T^{4} - 10254 T^{6} - 57459 T^{8} + 2166640 T^{10} - 9710571 T^{12} - 292864494 T^{14} + 5521869496 T^{16} - 42417997492 T^{18} + 137858491849 T^{20}$$
$17$ $$1 - 108 T^{2} + 5426 T^{4} - 171278 T^{6} + 3938173 T^{8} - 73013740 T^{10} + 1138131997 T^{12} - 14305309838 T^{14} + 130970449394 T^{16} - 753381803628 T^{18} + 2015993900449 T^{20}$$
$19$ $$( 1 + 3 T + 77 T^{2} + 178 T^{3} + 2608 T^{4} + 4670 T^{5} + 49552 T^{6} + 64258 T^{7} + 528143 T^{8} + 390963 T^{9} + 2476099 T^{10} )^{2}$$
$23$ $$( 1 - 4 T + 29 T^{2} - 188 T^{3} + 1084 T^{4} - 2688 T^{5} + 24932 T^{6} - 99452 T^{7} + 352843 T^{8} - 1119364 T^{9} + 6436343 T^{10} )^{2}$$
$29$ $$1 - 155 T^{2} + 12907 T^{4} - 729682 T^{6} + 30825980 T^{8} - 1008918398 T^{10} + 25924649180 T^{12} - 516090214642 T^{14} + 7677384604147 T^{16} - 77538194008955 T^{18} + 420707233300201 T^{20}$$
$31$ $$1 - 66 T^{2} + 2027 T^{4} - 68420 T^{6} + 2683060 T^{8} - 85000756 T^{10} + 2578420660 T^{12} - 63187306820 T^{14} + 1798969961387 T^{16} - 56290808471106 T^{18} + 819628286980801 T^{20}$$
$37$ $$1 - 188 T^{2} + 14592 T^{4} - 563614 T^{6} + 9281381 T^{8} - 55855104 T^{10} + 12706210589 T^{12} - 1056303377854 T^{14} + 37439079760128 T^{16} - 660346137337148 T^{18} + 4808584372417849 T^{20}$$
$41$ $$1 - 209 T^{2} + 23428 T^{4} - 1817599 T^{6} + 106670111 T^{8} - 4914350696 T^{10} + 179312456591 T^{12} - 5136100367839 T^{14} + 111285442158148 T^{16} - 1668849372886289 T^{18} + 13422659310152401 T^{20}$$
$43$ $$1 - 124 T^{2} + 8986 T^{4} - 558234 T^{6} + 29482965 T^{8} - 1340267732 T^{10} + 54514002285 T^{12} - 1908490957434 T^{14} + 56803768358314 T^{16} - 1449336834422524 T^{18} + 21611482313284249 T^{20}$$
$47$ $$( 1 + 149 T^{2} - 36 T^{3} + 11260 T^{4} - 2520 T^{5} + 529220 T^{6} - 79524 T^{7} + 15469627 T^{8} + 229345007 T^{10} )^{2}$$
$53$ $$1 - 355 T^{2} + 60783 T^{4} - 6738486 T^{6} + 540355392 T^{8} - 32753382654 T^{10} + 1517858296128 T^{12} - 53169895751766 T^{14} + 1347216362504007 T^{16} - 22102190096033155 T^{18} + 174887470365513049 T^{20}$$
$59$ $$1 + 20 T + 71 T^{2} - 896 T^{3} - 6134 T^{4} - 9960 T^{5} - 361906 T^{6} - 3118976 T^{7} + 14581909 T^{8} + 242347220 T^{9} + 714924299 T^{10}$$
$61$ $$1 - 226 T^{2} + 28531 T^{4} - 2771220 T^{6} + 216253284 T^{8} - 14141970452 T^{10} + 804678469764 T^{12} - 38369871496020 T^{14} + 1469927800893691 T^{16} - 43325852737385506 T^{18} + 713342911662882601 T^{20}$$
$67$ $$1 - 226 T^{2} + 24469 T^{4} - 2099544 T^{6} + 180779250 T^{8} - 13661508812 T^{10} + 811518053250 T^{12} - 42308165188824 T^{14} + 2213426153293261 T^{16} - 91771295127800866 T^{18} + 1822837804551761449 T^{20}$$
$71$ $$1 - 404 T^{2} + 85792 T^{4} - 12232858 T^{6} + 1288692677 T^{8} - 103955631344 T^{10} + 6496299784757 T^{12} - 310857485214298 T^{14} + 10989979558150432 T^{16} - 260884426623287444 T^{18} + 3255243551009881201 T^{20}$$
$73$ $$1 - 234 T^{2} + 40625 T^{4} - 4874480 T^{6} + 488701726 T^{8} - 38421788236 T^{10} + 2604291497854 T^{12} - 138426657789680 T^{14} + 6147952942990625 T^{16} - 188711661503214954 T^{18} + 4297625829703557649 T^{20}$$
$79$ $$( 1 - 3 T + 192 T^{2} - 299 T^{3} + 24163 T^{4} - 44052 T^{5} + 1908877 T^{6} - 1866059 T^{7} + 94663488 T^{8} - 116850243 T^{9} + 3077056399 T^{10} )^{2}$$
$83$ $$( 1 + 6 T + 260 T^{2} + 1794 T^{3} + 33211 T^{4} + 211464 T^{5} + 2756513 T^{6} + 12358866 T^{7} + 148664620 T^{8} + 284749926 T^{9} + 3939040643 T^{10} )^{2}$$
$89$ $$( 1 - 8 T + 201 T^{2} - 220 T^{3} + 15682 T^{4} + 31200 T^{5} + 1395698 T^{6} - 1742620 T^{7} + 141698769 T^{8} - 501937928 T^{9} + 5584059449 T^{10} )^{2}$$
$97$ $$1 - 682 T^{2} + 227521 T^{4} - 48709872 T^{6} + 7388315838 T^{8} - 828434696780 T^{10} + 69516663719742 T^{12} - 4312249945762032 T^{14} + 189518623533451009 T^{16} - 5345129711365087402 T^{18} + 73742412689492826049 T^{20}$$