Properties

Label 672.2.p.a
Level 672
Weight 2
Character orbit 672.p
Analytic conductor 5.366
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + x^{14} - 4 x^{12} - 4 x^{10} + 16 x^{8} - 16 x^{6} - 64 x^{4} + 64 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{2} q^{5} + \beta_{7} q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + \beta_{2} q^{5} + \beta_{7} q^{7} - q^{9} + \beta_{3} q^{11} + ( \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{13} -\beta_{1} q^{15} -\beta_{15} q^{17} + ( 2 \beta_{4} + \beta_{9} - \beta_{13} ) q^{19} + \beta_{6} q^{21} + ( \beta_{5} + \beta_{6} + \beta_{8} ) q^{23} + ( 1 + \beta_{3} + \beta_{11} ) q^{25} -\beta_{4} q^{27} + ( -\beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{29} + ( \beta_{2} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{31} -\beta_{9} q^{33} + ( -2 - \beta_{9} - \beta_{11} - \beta_{15} ) q^{35} + ( \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} ) q^{37} + ( -\beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{39} + ( 4 \beta_{4} - 2 \beta_{13} + \beta_{15} ) q^{41} + ( \beta_{3} + \beta_{14} ) q^{43} -\beta_{2} q^{45} + ( -2 \beta_{8} - 2 \beta_{12} ) q^{47} + ( -1 + 2 \beta_{4} + \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{49} -\beta_{14} q^{51} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{53} + ( 3 \beta_{2} - 2 \beta_{6} + \beta_{8} - \beta_{12} ) q^{55} + ( -2 + \beta_{3} - \beta_{11} ) q^{57} + 2 \beta_{13} q^{59} + ( \beta_{2} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{12} ) q^{61} -\beta_{7} q^{63} -2 \beta_{14} q^{65} + ( 4 - \beta_{3} + 2 \beta_{11} - \beta_{14} ) q^{67} + ( -\beta_{6} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{69} + ( -\beta_{5} + \beta_{6} + \beta_{8} ) q^{71} -2 \beta_{15} q^{73} + ( \beta_{4} - \beta_{9} - \beta_{13} ) q^{75} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{12} ) q^{77} + ( 2 \beta_{1} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{79} + q^{81} + ( -2 \beta_{9} - 2 \beta_{13} + 2 \beta_{15} ) q^{83} + ( \beta_{1} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} ) q^{85} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} ) q^{87} + ( -4 \beta_{4} + 2 \beta_{13} + \beta_{15} ) q^{89} + ( 4 - \beta_{3} - 2 \beta_{4} + \beta_{9} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{91} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{93} + ( -2 \beta_{6} - 2 \beta_{8} ) q^{95} + ( -4 \beta_{4} + 2 \beta_{9} - 2 \beta_{13} + 2 \beta_{15} ) q^{97} -\beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{9} + O(q^{10}) \) \( 16q - 16q^{9} + 8q^{11} + 16q^{25} - 24q^{35} + 8q^{43} - 8q^{49} - 16q^{57} + 40q^{67} + 16q^{81} + 56q^{91} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + x^{14} - 4 x^{12} - 4 x^{10} + 16 x^{8} - 16 x^{6} - 64 x^{4} + 64 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} + \nu^{12} + 16 \nu^{6} - 16 \nu^{4} \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} + \nu^{9} + 2 \nu^{7} + 32 \nu \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{14} - \nu^{12} + 16 \nu^{6} + 48 \nu^{4} \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} + 2 \nu^{11} - 8 \nu^{9} - 8 \nu^{7} + 32 \nu^{5} - 64 \nu \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{14} - \nu^{12} + 8 \nu^{10} + 8 \nu^{8} - 16 \nu^{6} + 16 \nu^{4} - 128 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{13} - \nu^{11} + 4 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 8 \nu^{6} + 16 \nu^{5} + 32 \nu^{4} - 48 \nu^{3} + 32 \nu^{2} - 32 \nu - 128 \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} + \nu^{13} - 4 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 4 \nu^{9} - 8 \nu^{8} + 8 \nu^{7} - 16 \nu^{6} - 64 \nu^{5} - 32 \nu^{3} + 128 \nu^{2} + 128 \nu - 128 \)\()/128\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{13} + \nu^{11} + 4 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} - 8 \nu^{6} - 16 \nu^{5} + 32 \nu^{4} + 48 \nu^{3} + 32 \nu^{2} + 32 \nu - 128 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{15} + \nu^{13} + 32 \nu^{3} \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} - \nu^{13} - 4 \nu^{12} + 4 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 8 \nu^{8} - 8 \nu^{7} - 16 \nu^{6} + 64 \nu^{5} + 32 \nu^{3} + 128 \nu^{2} - 128 \nu - 128 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 4 \nu^{10} + 16 \nu^{8} - 48 \nu^{4} + 128 \nu^{2} + 128 \)\()/64\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{15} - 7 \nu^{11} - 4 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + 24 \nu^{7} + 8 \nu^{6} - 32 \nu^{5} - 32 \nu^{4} - 16 \nu^{3} - 32 \nu^{2} + 96 \nu + 128 \)\()/64\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{13} + 4 \nu^{11} - 8 \nu^{9} + 8 \nu^{7} + 48 \nu^{5} - 64 \nu^{3} - 64 \nu \)\()/64\)
\(\beta_{14}\)\(=\)\((\)\( -3 \nu^{14} + \nu^{12} + 12 \nu^{10} - 8 \nu^{8} - 32 \nu^{6} + 80 \nu^{4} + 128 \nu^{2} - 256 \)\()/64\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} + 4 \nu^{11} - 2 \nu^{9} - 4 \nu^{7} + 32 \nu^{5} - 128 \nu \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{13} - \beta_{12} - \beta_{6} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14} + \beta_{11} - \beta_{5} + \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} - \beta_{13} + \beta_{12} + 2 \beta_{9} + 2 \beta_{8} - \beta_{6} + 4 \beta_{4} + \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{14} + \beta_{11} - 2 \beta_{10} + 4 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 2 \beta_{3} - 3 \beta_{1} + 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{15} + \beta_{13} - \beta_{12} + 4 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} - 4 \beta_{4} + 3 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{14} - \beta_{11} + 2 \beta_{10} - 4 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + \beta_{5} + 6 \beta_{3} + 11 \beta_{1} - 4\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{15} + 7 \beta_{13} + 9 \beta_{12} + 4 \beta_{10} - 2 \beta_{9} + 10 \beta_{8} - 4 \beta_{7} - \beta_{6} - 12 \beta_{4} - 3 \beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-\beta_{14} + 9 \beta_{11} - 10 \beta_{10} + 4 \beta_{8} - 10 \beta_{7} + 4 \beta_{6} + 7 \beta_{5} + 2 \beta_{3} - 3 \beta_{1} - 12\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(15 \beta_{15} - 7 \beta_{13} - 9 \beta_{12} - 4 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} + 4 \beta_{7} + \beta_{6} - 52 \beta_{4} + 19 \beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(\beta_{14} - 9 \beta_{11} + 10 \beta_{10} - 4 \beta_{8} + 10 \beta_{7} - 4 \beta_{6} + 25 \beta_{5} - 2 \beta_{3} + 35 \beta_{1} + 76\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-15 \beta_{15} + 39 \beta_{13} - 23 \beta_{12} + 4 \beta_{10} - 2 \beta_{9} + 10 \beta_{8} - 4 \beta_{7} - 33 \beta_{6} - 76 \beta_{4} - 19 \beta_{2}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(31 \beta_{14} + 9 \beta_{11} - 42 \beta_{10} + 4 \beta_{8} - 42 \beta_{7} + 4 \beta_{6} - 25 \beta_{5} - 30 \beta_{3} + 29 \beta_{1} - 12\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(47 \beta_{15} - 7 \beta_{13} - 9 \beta_{12} - 68 \beta_{10} + 66 \beta_{9} - 10 \beta_{8} + 68 \beta_{7} + \beta_{6} + 76 \beta_{4} + 51 \beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-63 \beta_{14} + 23 \beta_{11} - 22 \beta_{10} + 124 \beta_{8} - 22 \beta_{7} + 124 \beta_{6} - 7 \beta_{5} - 34 \beta_{3} + 3 \beta_{1} + 140\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-79 \beta_{15} + 39 \beta_{13} - 23 \beta_{12} + 68 \beta_{10} + 126 \beta_{9} - 54 \beta_{8} - 68 \beta_{7} + 31 \beta_{6} - 204 \beta_{4} - 83 \beta_{2}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.40199 + 0.185533i
−0.310478 + 1.37971i
0.474920 1.33209i
1.20933 0.733159i
−1.20933 0.733159i
−0.474920 1.33209i
0.310478 + 1.37971i
1.40199 + 0.185533i
−1.40199 0.185533i
−0.310478 1.37971i
0.474920 + 1.33209i
1.20933 + 0.733159i
−1.20933 + 0.733159i
−0.474920 + 1.33209i
0.310478 1.37971i
1.40199 0.185533i
0 1.00000i 0 −3.84444 0 1.62140 + 2.09071i 0 −1.00000 0
559.2 0 1.00000i 0 −2.33443 0 0.490487 2.59989i 0 −1.00000 0
559.3 0 1.00000i 0 −1.58069 0 −2.37995 + 1.15578i 0 −1.00000 0
559.4 0 1.00000i 0 −1.12786 0 2.11337 1.59175i 0 −1.00000 0
559.5 0 1.00000i 0 1.12786 0 −2.11337 + 1.59175i 0 −1.00000 0
559.6 0 1.00000i 0 1.58069 0 2.37995 1.15578i 0 −1.00000 0
559.7 0 1.00000i 0 2.33443 0 −0.490487 + 2.59989i 0 −1.00000 0
559.8 0 1.00000i 0 3.84444 0 −1.62140 2.09071i 0 −1.00000 0
559.9 0 1.00000i 0 −3.84444 0 1.62140 2.09071i 0 −1.00000 0
559.10 0 1.00000i 0 −2.33443 0 0.490487 + 2.59989i 0 −1.00000 0
559.11 0 1.00000i 0 −1.58069 0 −2.37995 1.15578i 0 −1.00000 0
559.12 0 1.00000i 0 −1.12786 0 2.11337 + 1.59175i 0 −1.00000 0
559.13 0 1.00000i 0 1.12786 0 −2.11337 1.59175i 0 −1.00000 0
559.14 0 1.00000i 0 1.58069 0 2.37995 + 1.15578i 0 −1.00000 0
559.15 0 1.00000i 0 2.33443 0 −0.490487 2.59989i 0 −1.00000 0
559.16 0 1.00000i 0 3.84444 0 −1.62140 + 2.09071i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.p.a 16
3.b odd 2 1 2016.2.p.g 16
4.b odd 2 1 168.2.p.a 16
7.b odd 2 1 inner 672.2.p.a 16
8.b even 2 1 168.2.p.a 16
8.d odd 2 1 inner 672.2.p.a 16
12.b even 2 1 504.2.p.g 16
21.c even 2 1 2016.2.p.g 16
24.f even 2 1 2016.2.p.g 16
24.h odd 2 1 504.2.p.g 16
28.d even 2 1 168.2.p.a 16
56.e even 2 1 inner 672.2.p.a 16
56.h odd 2 1 168.2.p.a 16
84.h odd 2 1 504.2.p.g 16
168.e odd 2 1 2016.2.p.g 16
168.i even 2 1 504.2.p.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.p.a 16 4.b odd 2 1
168.2.p.a 16 8.b even 2 1
168.2.p.a 16 28.d even 2 1
168.2.p.a 16 56.h odd 2 1
504.2.p.g 16 12.b even 2 1
504.2.p.g 16 24.h odd 2 1
504.2.p.g 16 84.h odd 2 1
504.2.p.g 16 168.i even 2 1
672.2.p.a 16 1.a even 1 1 trivial
672.2.p.a 16 7.b odd 2 1 inner
672.2.p.a 16 8.d odd 2 1 inner
672.2.p.a 16 56.e even 2 1 inner
2016.2.p.g 16 3.b odd 2 1
2016.2.p.g 16 21.c even 2 1
2016.2.p.g 16 24.f even 2 1
2016.2.p.g 16 168.e odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T^{2} )^{8} \)
$5$ \( ( 1 + 16 T^{2} + 140 T^{4} + 832 T^{6} + 4326 T^{8} + 20800 T^{10} + 87500 T^{12} + 250000 T^{14} + 390625 T^{16} )^{2} \)
$7$ \( 1 + 4 T^{2} + 68 T^{4} + 572 T^{6} + 3382 T^{8} + 28028 T^{10} + 163268 T^{12} + 470596 T^{14} + 5764801 T^{16} \)
$11$ \( ( 1 - 2 T + 20 T^{2} - 6 T^{3} + 182 T^{4} - 66 T^{5} + 2420 T^{6} - 2662 T^{7} + 14641 T^{8} )^{4} \)
$13$ \( ( 1 + 36 T^{2} + 692 T^{4} + 12604 T^{6} + 195894 T^{8} + 2130076 T^{10} + 19764212 T^{12} + 173765124 T^{14} + 815730721 T^{16} )^{2} \)
$17$ \( ( 1 - 72 T^{2} + 2652 T^{4} - 66024 T^{6} + 1253414 T^{8} - 19080936 T^{10} + 221497692 T^{12} - 1737904968 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 56 T^{2} + 1532 T^{4} - 36808 T^{6} + 804262 T^{8} - 13287688 T^{10} + 199651772 T^{12} - 2634569336 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 - 88 T^{2} + 3548 T^{4} - 96952 T^{6} + 2287782 T^{8} - 51287608 T^{10} + 992875868 T^{12} - 13027158232 T^{14} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 - 164 T^{2} + 12980 T^{4} - 650812 T^{6} + 22500790 T^{8} - 547332892 T^{10} + 9180507380 T^{12} - 97551024644 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 + 68 T^{2} + 5252 T^{4} + 204988 T^{6} + 8264182 T^{8} + 196993468 T^{10} + 4850332292 T^{12} + 60350250308 T^{14} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 - 232 T^{2} + 25148 T^{4} - 1670040 T^{6} + 74486502 T^{8} - 2286284760 T^{10} + 47131400828 T^{12} - 595248526888 T^{14} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 - 120 T^{2} + 10204 T^{4} - 627512 T^{6} + 28679654 T^{8} - 1054847672 T^{10} + 28834065244 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 - 2 T + 120 T^{2} - 250 T^{3} + 6782 T^{4} - 10750 T^{5} + 221880 T^{6} - 159014 T^{7} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 + 120 T^{2} + 10396 T^{4} + 591944 T^{6} + 31424966 T^{8} + 1307604296 T^{10} + 50729163676 T^{12} + 1293505839480 T^{14} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 - 260 T^{2} + 33908 T^{4} - 2877468 T^{6} + 176791542 T^{8} - 8082807612 T^{10} + 267550429748 T^{12} - 5762733893540 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 - 264 T^{2} + 34076 T^{4} - 3023352 T^{6} + 203740198 T^{8} - 10524288312 T^{10} + 412911193436 T^{12} - 11135660881224 T^{14} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 + 324 T^{2} + 46708 T^{4} + 4167580 T^{6} + 280236854 T^{8} + 15507565180 T^{10} + 646711541428 T^{12} + 16692601292964 T^{14} + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 - 10 T + 208 T^{2} - 1122 T^{3} + 16622 T^{4} - 75174 T^{5} + 933712 T^{6} - 3007630 T^{7} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 - 456 T^{2} + 94300 T^{4} - 11817416 T^{6} + 1002933542 T^{8} - 59571594056 T^{10} + 2396321518300 T^{12} - 58413729467976 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 - 328 T^{2} + 54492 T^{4} - 6043256 T^{6} + 500200454 T^{8} - 32204511224 T^{10} + 1547476948572 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 444 T^{2} + 95492 T^{4} - 12990660 T^{6} + 1221053302 T^{8} - 81074709060 T^{10} + 3719421134852 T^{12} - 107930830251324 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( ( 1 - 120 T^{2} + 20284 T^{4} - 1531784 T^{6} + 164981030 T^{8} - 10552459976 T^{10} + 962644583164 T^{12} - 39232844804280 T^{14} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 - 312 T^{2} + 51484 T^{4} - 5696504 T^{6} + 530357222 T^{8} - 45122008184 T^{10} + 3230221535644 T^{12} - 155058162779832 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 232 T^{2} + 17244 T^{4} + 409768 T^{6} - 157308730 T^{8} + 3855507112 T^{10} + 1526598921564 T^{12} - 193249505143528 T^{14} + 7837433594376961 T^{16} )^{2} \)
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