Properties

Label 1849.2.a.m
Level $1849$
Weight $2$
Character orbit 1849.a
Self dual yes
Analytic conductor $14.764$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7643393337\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{44})^+\)
Defining polynomial: \(x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 4 \beta - \beta^{3} ) q^{3} + ( -2 + \beta^{2} ) q^{4} + ( 6 \beta - 14 \beta^{3} + 7 \beta^{5} - \beta^{7} ) q^{5} + ( 4 \beta^{2} - \beta^{4} ) q^{6} + ( -6 \beta + 10 \beta^{3} - 6 \beta^{5} + \beta^{7} ) q^{7} + ( -4 \beta + \beta^{3} ) q^{8} + ( -3 + 16 \beta^{2} - 8 \beta^{4} + \beta^{6} ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( 4 \beta - \beta^{3} ) q^{3} + ( -2 + \beta^{2} ) q^{4} + ( 6 \beta - 14 \beta^{3} + 7 \beta^{5} - \beta^{7} ) q^{5} + ( 4 \beta^{2} - \beta^{4} ) q^{6} + ( -6 \beta + 10 \beta^{3} - 6 \beta^{5} + \beta^{7} ) q^{7} + ( -4 \beta + \beta^{3} ) q^{8} + ( -3 + 16 \beta^{2} - 8 \beta^{4} + \beta^{6} ) q^{9} + ( 6 \beta^{2} - 14 \beta^{4} + 7 \beta^{6} - \beta^{8} ) q^{10} + ( 10 - 39 \beta^{2} + 33 \beta^{4} - 10 \beta^{6} + \beta^{8} ) q^{11} + ( -8 \beta + 6 \beta^{3} - \beta^{5} ) q^{12} + ( -19 + 57 \beta^{2} - 53 \beta^{4} + 18 \beta^{6} - 2 \beta^{8} ) q^{13} + ( -6 \beta^{2} + 10 \beta^{4} - 6 \beta^{6} + \beta^{8} ) q^{14} + ( 11 - 31 \beta^{2} + 15 \beta^{4} - 2 \beta^{6} ) q^{15} + ( 4 - 6 \beta^{2} + \beta^{4} ) q^{16} + ( 5 - 51 \beta^{2} + 61 \beta^{4} - 24 \beta^{6} + 3 \beta^{8} ) q^{17} + ( -3 \beta + 16 \beta^{3} - 8 \beta^{5} + \beta^{7} ) q^{18} + ( -4 \beta - 11 \beta^{3} + 19 \beta^{5} - 8 \beta^{7} + \beta^{9} ) q^{19} + ( -12 \beta + 34 \beta^{3} - 28 \beta^{5} + 9 \beta^{7} - \beta^{9} ) q^{20} + ( -11 + 31 \beta^{2} - 31 \beta^{4} + 10 \beta^{6} - \beta^{8} ) q^{21} + ( 10 \beta - 39 \beta^{3} + 33 \beta^{5} - 10 \beta^{7} + \beta^{9} ) q^{22} + ( 18 - 31 \beta^{2} + 14 \beta^{4} - 2 \beta^{6} ) q^{23} + ( -16 \beta^{2} + 8 \beta^{4} - \beta^{6} ) q^{24} + ( -5 + 3 \beta^{2} + 8 \beta^{4} - 6 \beta^{6} + \beta^{8} ) q^{25} + ( -19 \beta + 57 \beta^{3} - 53 \beta^{5} + 18 \beta^{7} - 2 \beta^{9} ) q^{26} + ( -24 \beta + 70 \beta^{3} - 48 \beta^{5} + 12 \beta^{7} - \beta^{9} ) q^{27} + ( 12 \beta - 26 \beta^{3} + 22 \beta^{5} - 8 \beta^{7} + \beta^{9} ) q^{28} + ( -24 \beta + 72 \beta^{3} - 69 \beta^{5} + 25 \beta^{7} - 3 \beta^{9} ) q^{29} + ( 11 \beta - 31 \beta^{3} + 15 \beta^{5} - 2 \beta^{7} ) q^{30} + ( -9 + 59 \beta^{2} - 75 \beta^{4} + 31 \beta^{6} - 4 \beta^{8} ) q^{31} + ( 12 \beta - 8 \beta^{3} + \beta^{5} ) q^{32} + ( 29 \beta - 111 \beta^{3} + 94 \beta^{5} - 29 \beta^{7} + 3 \beta^{9} ) q^{33} + ( 5 \beta - 51 \beta^{3} + 61 \beta^{5} - 24 \beta^{7} + 3 \beta^{9} ) q^{34} + ( -14 \beta^{2} + 23 \beta^{4} - 9 \beta^{6} + \beta^{8} ) q^{35} + ( 6 - 35 \beta^{2} + 32 \beta^{4} - 10 \beta^{6} + \beta^{8} ) q^{36} + ( -12 \beta - 36 \beta^{3} + 50 \beta^{5} - 18 \beta^{7} + 2 \beta^{9} ) q^{37} + ( 11 - 59 \beta^{2} + 66 \beta^{4} - 25 \beta^{6} + 3 \beta^{8} ) q^{38} + ( -54 \beta + 137 \beta^{3} - 115 \beta^{5} + 37 \beta^{7} - 4 \beta^{9} ) q^{39} + ( -11 + 31 \beta^{2} - 15 \beta^{4} + 2 \beta^{6} ) q^{40} + ( -22 + 83 \beta^{2} - 71 \beta^{4} + 21 \beta^{6} - 2 \beta^{8} ) q^{41} + ( -11 \beta + 31 \beta^{3} - 31 \beta^{5} + 10 \beta^{7} - \beta^{9} ) q^{42} + ( -9 + 33 \beta^{2} - 28 \beta^{4} + 9 \beta^{6} - \beta^{8} ) q^{44} + ( 26 \beta - 93 \beta^{3} + 70 \beta^{5} - 20 \beta^{7} + 2 \beta^{9} ) q^{45} + ( 18 \beta - 31 \beta^{3} + 14 \beta^{5} - 2 \beta^{7} ) q^{46} + ( -4 + 22 \beta^{2} - 22 \beta^{4} + 8 \beta^{6} - \beta^{8} ) q^{47} + ( 16 \beta - 28 \beta^{3} + 10 \beta^{5} - \beta^{7} ) q^{48} + ( 4 - 30 \beta^{2} + 23 \beta^{4} - 4 \beta^{6} ) q^{49} + ( -5 \beta + 3 \beta^{3} + 8 \beta^{5} - 6 \beta^{7} + \beta^{9} ) q^{50} + ( -13 \beta - 44 \beta^{3} + 64 \beta^{5} - 25 \beta^{7} + 3 \beta^{9} ) q^{51} + ( 16 - 23 \beta^{2} + 9 \beta^{4} - \beta^{6} ) q^{52} + ( 20 - 74 \beta^{2} + 65 \beta^{4} - 20 \beta^{6} + 2 \beta^{8} ) q^{53} + ( -11 + 31 \beta^{2} - 7 \beta^{4} - 4 \beta^{6} + \beta^{8} ) q^{54} + ( -17 \beta + 77 \beta^{3} - 66 \beta^{5} + 20 \beta^{7} - 2 \beta^{9} ) q^{55} + ( 11 - 31 \beta^{2} + 31 \beta^{4} - 10 \beta^{6} + \beta^{8} ) q^{56} + ( 11 - 82 \beta^{2} + 92 \beta^{4} - 34 \beta^{6} + 4 \beta^{8} ) q^{57} + ( -33 + 141 \beta^{2} - 159 \beta^{4} + 63 \beta^{6} - 8 \beta^{8} ) q^{58} + ( -25 + 121 \beta^{2} - 133 \beta^{4} + 50 \beta^{6} - 6 \beta^{8} ) q^{59} + ( -22 + 73 \beta^{2} - 61 \beta^{4} + 19 \beta^{6} - 2 \beta^{8} ) q^{60} + ( -7 \beta + 62 \beta^{3} - 71 \beta^{5} + 26 \beta^{7} - 3 \beta^{9} ) q^{61} + ( -9 \beta + 59 \beta^{3} - 75 \beta^{5} + 31 \beta^{7} - 4 \beta^{9} ) q^{62} + ( -15 \beta + 50 \beta^{3} - 60 \beta^{5} + 24 \beta^{7} - 3 \beta^{9} ) q^{63} + ( -8 + 24 \beta^{2} - 10 \beta^{4} + \beta^{6} ) q^{64} + ( -15 \beta + 3 \beta^{3} + 16 \beta^{5} - 8 \beta^{7} + \beta^{9} ) q^{65} + ( 33 - 136 \beta^{2} + 120 \beta^{4} - 38 \beta^{6} + 4 \beta^{8} ) q^{66} + ( -26 + 80 \beta^{2} - 54 \beta^{4} + 13 \beta^{6} - \beta^{8} ) q^{67} + ( 23 - 58 \beta^{2} + 58 \beta^{4} - 23 \beta^{6} + 3 \beta^{8} ) q^{68} + ( 72 \beta - 142 \beta^{3} + 87 \beta^{5} - 22 \beta^{7} + 2 \beta^{9} ) q^{69} + ( -14 \beta^{3} + 23 \beta^{5} - 9 \beta^{7} + \beta^{9} ) q^{70} + ( 54 \beta - 129 \beta^{3} + 109 \beta^{5} - 36 \beta^{7} + 4 \beta^{9} ) q^{71} + ( 12 \beta - 67 \beta^{3} + 48 \beta^{5} - 12 \beta^{7} + \beta^{9} ) q^{72} + ( -27 \beta + 71 \beta^{3} - 48 \beta^{5} + 12 \beta^{7} - \beta^{9} ) q^{73} + ( 22 - 122 \beta^{2} + 118 \beta^{4} - 38 \beta^{6} + 4 \beta^{8} ) q^{74} + ( -31 \beta + 72 \beta^{3} - 48 \beta^{5} + 12 \beta^{7} - \beta^{9} ) q^{75} + ( 19 \beta - 37 \beta^{3} + 28 \beta^{5} - 9 \beta^{7} + \beta^{9} ) q^{76} + ( -16 \beta + 59 \beta^{3} - 54 \beta^{5} + 18 \beta^{7} - 2 \beta^{9} ) q^{77} + ( -44 + 166 \beta^{2} - 171 \beta^{4} + 61 \beta^{6} - 7 \beta^{8} ) q^{78} + ( 56 - 204 \beta^{2} + 187 \beta^{4} - 63 \beta^{6} + 7 \beta^{8} ) q^{79} + ( 13 \beta - 37 \beta^{3} + 41 \beta^{5} - 16 \beta^{7} + 2 \beta^{9} ) q^{80} + ( -46 + 142 \beta^{2} - 112 \beta^{4} + 32 \beta^{6} - 3 \beta^{8} ) q^{81} + ( -22 \beta + 83 \beta^{3} - 71 \beta^{5} + 21 \beta^{7} - 2 \beta^{9} ) q^{82} + ( 27 - 100 \beta^{2} + 87 \beta^{4} - 28 \beta^{6} + 3 \beta^{8} ) q^{83} + ( 11 - 18 \beta^{2} + 16 \beta^{4} - 7 \beta^{6} + \beta^{8} ) q^{84} + ( -47 \beta + 141 \beta^{3} - 117 \beta^{5} + 37 \beta^{7} - 4 \beta^{9} ) q^{85} + ( -44 + 157 \beta^{2} - 161 \beta^{4} + 59 \beta^{6} - 7 \beta^{8} ) q^{87} + ( -29 \beta + 111 \beta^{3} - 94 \beta^{5} + 29 \beta^{7} - 3 \beta^{9} ) q^{88} + ( 19 \beta - 76 \beta^{3} + 74 \beta^{5} - 26 \beta^{7} + 3 \beta^{9} ) q^{89} + ( 22 - 84 \beta^{2} + 61 \beta^{4} - 18 \beta^{6} + 2 \beta^{8} ) q^{90} + ( 59 \beta - 169 \beta^{3} + 155 \beta^{5} - 53 \beta^{7} + 6 \beta^{9} ) q^{91} + ( -36 + 80 \beta^{2} - 59 \beta^{4} + 18 \beta^{6} - 2 \beta^{8} ) q^{92} + ( 8 \beta + 25 \beta^{3} - 51 \beta^{5} + 23 \beta^{7} - 3 \beta^{9} ) q^{93} + ( -4 \beta + 22 \beta^{3} - 22 \beta^{5} + 8 \beta^{7} - \beta^{9} ) q^{94} + ( -22 + 75 \beta^{2} - 65 \beta^{4} + 20 \beta^{6} - 2 \beta^{8} ) q^{95} + ( 48 \beta^{2} - 44 \beta^{4} + 12 \beta^{6} - \beta^{8} ) q^{96} + ( -56 + 162 \beta^{2} - 120 \beta^{4} + 33 \beta^{6} - 3 \beta^{8} ) q^{97} + ( 4 \beta - 30 \beta^{3} + 23 \beta^{5} - 4 \beta^{7} ) q^{98} + ( 58 - 240 \beta^{2} + 209 \beta^{4} - 66 \beta^{6} + 7 \beta^{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 2q^{4} + 22q^{6} + 14q^{9} + O(q^{10}) \) \( 10q + 2q^{4} + 22q^{6} + 14q^{9} - 22q^{10} - 10q^{11} - 14q^{13} - 22q^{14} - 22q^{15} - 26q^{16} - 16q^{17} - 44q^{21} - 18q^{23} - 44q^{24} - 6q^{25} - 2q^{31} - 28q^{36} - 22q^{38} + 22q^{40} - 2q^{44} - 18q^{47} + 18q^{49} + 28q^{52} + 2q^{53} + 44q^{56} - 22q^{57} - 22q^{58} + 14q^{59} + 8q^{64} - 22q^{66} + 26q^{67} + 32q^{68} + 44q^{74} - 44q^{78} - 56q^{79} + 2q^{81} - 38q^{83} - 22q^{87} - 22q^{90} - 74q^{92} + 22q^{96} + 34q^{97} - 36q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.97964
−1.81926
−1.51150
−1.08128
−0.563465
0.563465
1.08128
1.51150
1.81926
1.97964
−1.97964 −0.160379 1.91899 3.06092 0.317493 −2.43240 0.160379 −2.97428 −6.05954
1.2 −1.81926 −1.25580 1.30972 −0.160379 2.28463 4.31672 1.25580 −1.42297 0.291772
1.3 −1.51150 −2.59278 0.284630 2.07496 3.91899 3.84858 2.59278 3.72251 −3.13631
1.4 −1.08128 −3.06092 −0.830830 2.59278 3.30972 0.985960 3.06092 6.36926 −2.80353
1.5 −0.563465 −2.07496 −1.68251 −1.25580 1.16917 1.91459 2.07496 1.30548 0.707599
1.6 0.563465 2.07496 −1.68251 1.25580 1.16917 −1.91459 −2.07496 1.30548 0.707599
1.7 1.08128 3.06092 −0.830830 −2.59278 3.30972 −0.985960 −3.06092 6.36926 −2.80353
1.8 1.51150 2.59278 0.284630 −2.07496 3.91899 −3.84858 −2.59278 3.72251 −3.13631
1.9 1.81926 1.25580 1.30972 0.160379 2.28463 −4.31672 −1.25580 −1.42297 0.291772
1.10 1.97964 0.160379 1.91899 −3.06092 0.317493 2.43240 −0.160379 −2.97428 −6.05954
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1849.2.a.m 10
43.b odd 2 1 inner 1849.2.a.m 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1849.2.a.m 10 1.a even 1 1 trivial
1849.2.a.m 10 43.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 11 T_{2}^{8} + 44 T_{2}^{6} - 77 T_{2}^{4} + 55 T_{2}^{2} - 11 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1849))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 9 T^{2} + 48 T^{4} + 179 T^{6} + 511 T^{8} + 1145 T^{10} + 2044 T^{12} + 2864 T^{14} + 3072 T^{16} + 2304 T^{18} + 1024 T^{20} \)
$3$ \( 1 + 8 T^{2} + 42 T^{4} + 182 T^{6} + 653 T^{8} + 2089 T^{10} + 5877 T^{12} + 14742 T^{14} + 30618 T^{16} + 52488 T^{18} + 59049 T^{20} \)
$5$ \( 1 + 28 T^{2} + 410 T^{4} + 4066 T^{6} + 29885 T^{8} + 169289 T^{10} + 747125 T^{12} + 2541250 T^{14} + 6406250 T^{16} + 10937500 T^{18} + 9765625 T^{20} \)
$7$ \( 1 + 26 T^{2} + 401 T^{4} + 4541 T^{6} + 42232 T^{8} + 324259 T^{10} + 2069368 T^{12} + 10902941 T^{14} + 47177249 T^{16} + 149884826 T^{18} + 282475249 T^{20} \)
$11$ \( ( 1 + 5 T + 54 T^{2} + 208 T^{3} + 1182 T^{4} + 3367 T^{5} + 13002 T^{6} + 25168 T^{7} + 71874 T^{8} + 73205 T^{9} + 161051 T^{10} )^{2} \)
$13$ \( ( 1 + 7 T + 56 T^{2} + 292 T^{3} + 1458 T^{4} + 5203 T^{5} + 18954 T^{6} + 49348 T^{7} + 123032 T^{8} + 199927 T^{9} + 371293 T^{10} )^{2} \)
$17$ \( ( 1 + 8 T + 60 T^{2} + 273 T^{3} + 1594 T^{4} + 6285 T^{5} + 27098 T^{6} + 78897 T^{7} + 294780 T^{8} + 668168 T^{9} + 1419857 T^{10} )^{2} \)
$19$ \( 1 + 146 T^{2} + 10261 T^{4} + 453469 T^{6} + 13907084 T^{8} + 308703099 T^{10} + 5020457324 T^{12} + 59096533549 T^{14} + 482737784941 T^{16} + 2479600203986 T^{18} + 6131066257801 T^{20} \)
$23$ \( ( 1 + 9 T + 55 T^{2} + 27 T^{3} - 1431 T^{4} - 10897 T^{5} - 32913 T^{6} + 14283 T^{7} + 669185 T^{8} + 2518569 T^{9} + 6436343 T^{10} )^{2} \)
$29$ \( 1 + 92 T^{2} + 4944 T^{4} + 212793 T^{6} + 8043434 T^{8} + 260511681 T^{10} + 6764527994 T^{12} + 150504445833 T^{14} + 2940806499024 T^{16} + 46022669992412 T^{18} + 420707233300201 T^{20} \)
$31$ \( ( 1 + T + 74 T^{2} - 11 T^{3} + 3015 T^{4} - 2141 T^{5} + 93465 T^{6} - 10571 T^{7} + 2204534 T^{8} + 923521 T^{9} + 28629151 T^{10} )^{2} \)
$37$ \( 1 - 70 T^{2} + 5461 T^{4} - 259752 T^{6} + 13400962 T^{8} - 494954340 T^{10} + 18345916978 T^{12} - 486817068072 T^{14} + 14011431919549 T^{16} - 245873561774470 T^{18} + 4808584372417849 T^{20} \)
$41$ \( ( 1 + 128 T^{2} - 66 T^{3} + 8549 T^{4} - 3223 T^{5} + 350509 T^{6} - 110946 T^{7} + 8821888 T^{8} + 115856201 T^{10} )^{2} \)
$43$ 1
$47$ \( ( 1 + 9 T + 219 T^{2} + 1639 T^{3} + 19871 T^{4} + 114327 T^{5} + 933937 T^{6} + 3620551 T^{7} + 22737237 T^{8} + 43917129 T^{9} + 229345007 T^{10} )^{2} \)
$53$ \( ( 1 - T + 217 T^{2} - 165 T^{3} + 20879 T^{4} - 12533 T^{5} + 1106587 T^{6} - 463485 T^{7} + 32306309 T^{8} - 7890481 T^{9} + 418195493 T^{10} )^{2} \)
$59$ \( ( 1 - 7 T + 198 T^{2} - 1030 T^{3} + 16660 T^{4} - 72387 T^{5} + 982940 T^{6} - 3585430 T^{7} + 40665042 T^{8} - 84821527 T^{9} + 714924299 T^{10} )^{2} \)
$61$ \( 1 + 379 T^{2} + 71239 T^{4} + 8701202 T^{6} + 774928669 T^{8} + 53283270877 T^{10} + 2883509577349 T^{12} + 120475459400882 T^{14} + 3670259949103279 T^{16} + 72657071625969499 T^{18} + 713342911662882601 T^{20} \)
$67$ \( ( 1 - 13 T + 308 T^{2} - 3053 T^{3} + 39181 T^{4} - 293685 T^{5} + 2625127 T^{6} - 13704917 T^{7} + 92635004 T^{8} - 261964573 T^{9} + 1350125107 T^{10} )^{2} \)
$71$ \( 1 + 336 T^{2} + 58644 T^{4} + 7325325 T^{6} + 719409326 T^{8} + 56796003109 T^{10} + 3626542412366 T^{12} + 186148822121325 T^{14} + 7512313050263124 T^{16} + 216973186498575696 T^{18} + 3255243551009881201 T^{20} \)
$73$ \( 1 + 609 T^{2} + 173420 T^{4} + 30452178 T^{6} + 3655326744 T^{8} + 313602113089 T^{10} + 19479236218776 T^{12} + 864788289818898 T^{14} + 26244381523038380 T^{16} + 491134195963495329 T^{18} + 4297625829703557649 T^{20} \)
$79$ \( ( 1 + 28 T + 449 T^{2} + 5527 T^{3} + 64884 T^{4} + 640171 T^{5} + 5125836 T^{6} + 34494007 T^{7} + 221374511 T^{8} + 1090602268 T^{9} + 3077056399 T^{10} )^{2} \)
$83$ \( ( 1 + 19 T + 500 T^{2} + 6136 T^{3} + 88868 T^{4} + 757387 T^{5} + 7376044 T^{6} + 42270904 T^{7} + 285893500 T^{8} + 901708099 T^{9} + 3939040643 T^{10} )^{2} \)
$89$ \( 1 + 769 T^{2} + 274572 T^{4} + 59995506 T^{6} + 8889881144 T^{8} + 933750066017 T^{10} + 70416748541624 T^{12} + 3764252496368946 T^{14} + 136457147021743692 T^{16} + 3027236791584900289 T^{18} + 31181719929966183601 T^{20} \)
$97$ \( ( 1 - 17 T + 264 T^{2} - 2439 T^{3} + 23811 T^{4} - 156801 T^{5} + 2309667 T^{6} - 22948551 T^{7} + 240945672 T^{8} - 1504997777 T^{9} + 8587340257 T^{10} )^{2} \)
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