Properties

Label 1805.2.b.j
Level $1805$
Weight $2$
Character orbit 1805.b
Analytic conductor $14.413$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 22x^{14} + 190x^{12} + 820x^{10} + 1862x^{8} + 2154x^{6} + 1163x^{4} + 256x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{14} - \beta_{12} + \beta_{5} - \beta_{4} - 1) q^{6} + \beta_{15} q^{7} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_1) q^{8} + ( - \beta_{10} + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{14} - \beta_{12} + \beta_{5} - \beta_{4} - 1) q^{6} + \beta_{15} q^{7} + (\beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_1) q^{8} + ( - \beta_{10} + \beta_{7}) q^{9} + ( - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{10} + ( - \beta_{13} + \beta_{12} - \beta_{7} + \beta_{2} - 2) q^{11} + (\beta_{14} - \beta_{13} + \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + \beta_1) q^{12} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_1) q^{13} + (\beta_{10} - \beta_{7}) q^{14} + (\beta_{13} - \beta_{11} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{15} + (\beta_{13} - \beta_{12} - \beta_{10} + 3 \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + 2) q^{16} + ( - \beta_{15} + \beta_{11} + \beta_{9} - 2 \beta_{8} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{17} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} - \beta_{6}) q^{18} + ( - \beta_{13} - \beta_{11} - \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{20} + (\beta_{14} + \beta_{13} + \beta_{7} - \beta_{5} + \beta_{4} + 2) q^{21} + (\beta_{11} + \beta_{9} - \beta_{8} - 2 \beta_{4} - 2 \beta_{3}) q^{22} + (\beta_{11} - \beta_{9} - 2 \beta_{8} - \beta_{4} - \beta_{3} + \beta_1) q^{23} + (\beta_{10} - \beta_{7} - \beta_{4} + \beta_{3}) q^{24} + ( - \beta_{15} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{25} + (2 \beta_{14} + 2 \beta_{12} - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{26} + (\beta_{14} - \beta_{13} + \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{27} + (\beta_{15} - \beta_{14} + \beta_{13} + \beta_{9} + \beta_{8} + \beta_{6}) q^{28} + ( - \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3}) q^{29} + ( - \beta_{14} - \beta_{10} - \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{4} + \cdots - 3 \beta_1) q^{30}+ \cdots + (\beta_{14} + \beta_{12} + 3 \beta_{10} - 5 \beta_{7} + \beta_{4} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{4} + 4 q^{5} - 10 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{4} + 4 q^{5} - 10 q^{6} - 6 q^{9} + 16 q^{10} - 22 q^{11} + 6 q^{14} - 10 q^{15} + 8 q^{16} - 14 q^{20} + 20 q^{21} + 14 q^{24} + 4 q^{25} - 16 q^{26} + 2 q^{29} - 12 q^{30} - 16 q^{31} - 8 q^{34} - 10 q^{35} + 18 q^{36} + 36 q^{39} - 38 q^{40} - 26 q^{41} + 64 q^{44} - 2 q^{45} - 2 q^{46} + 20 q^{49} + 48 q^{50} + 38 q^{51} + 12 q^{54} - 10 q^{55} - 6 q^{56} + 10 q^{59} + 10 q^{60} - 30 q^{61} + 16 q^{64} + 36 q^{65} + 4 q^{66} + 68 q^{69} + 2 q^{70} + 20 q^{71} + 40 q^{74} + 32 q^{75} + 12 q^{79} + 40 q^{80} - 48 q^{81} - 2 q^{84} - 2 q^{85} + 20 q^{86} - 30 q^{90} + 86 q^{91} - 38 q^{94} + 22 q^{96} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 22x^{14} + 190x^{12} + 820x^{10} + 1862x^{8} + 2154x^{6} + 1163x^{4} + 256x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{15} - 6 \nu^{14} - 202 \nu^{13} - 164 \nu^{12} - 1342 \nu^{11} - 1768 \nu^{10} - 3844 \nu^{9} - 9564 \nu^{8} - 3934 \nu^{7} - 27252 \nu^{6} + 1846 \nu^{5} - 38832 \nu^{4} + \cdots - 3016 ) / 592 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{15} + 6 \nu^{14} - 202 \nu^{13} + 164 \nu^{12} - 1342 \nu^{11} + 1768 \nu^{10} - 3844 \nu^{9} + 9564 \nu^{8} - 3934 \nu^{7} + 27252 \nu^{6} + 1846 \nu^{5} + 38832 \nu^{4} + \cdots + 3016 ) / 592 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{15} - 26 \nu^{14} - 202 \nu^{13} - 612 \nu^{12} - 1342 \nu^{11} - 5688 \nu^{10} - 3844 \nu^{9} - 26348 \nu^{8} - 3934 \nu^{7} - 62740 \nu^{6} + 1846 \nu^{5} - 70592 \nu^{4} + \cdots - 2808 ) / 592 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{15} - 306 \nu^{13} - 2918 \nu^{11} - 14580 \nu^{9} - 40990 \nu^{7} - 63490 \nu^{5} - 46879 \nu^{3} - 10284 \nu ) / 592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\nu^{14} + 650\nu^{12} + 5262\nu^{10} + 20776\nu^{8} + 41494\nu^{6} + 39090\nu^{4} + 14393\nu^{2} + 1276 ) / 296 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\nu^{15} + 366\nu^{13} + 3110\nu^{11} + 13408\nu^{9} + 31334\nu^{7} + 38762\nu^{5} + 22015\nu^{3} + 3760\nu ) / 296 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31 \nu^{15} - 650 \nu^{13} - 5262 \nu^{11} - 20776 \nu^{9} - 41494 \nu^{7} - 39090 \nu^{5} - 14393 \nu^{3} - 1572 \nu ) / 296 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 61\nu^{14} + 1322\nu^{12} + 11142\nu^{10} + 46100\nu^{8} + 96798\nu^{6} + 95610\nu^{4} + 36519\nu^{2} + 3332 ) / 296 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} + 22\nu^{13} + 190\nu^{11} + 820\nu^{9} + 1862\nu^{7} + 2154\nu^{5} + 1163\nu^{3} + 248\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 39 \nu^{15} + 17 \nu^{14} - 844 \nu^{13} + 292 \nu^{12} - 7126 \nu^{11} + 1704 \nu^{10} - 29754 \nu^{9} + 3492 \nu^{8} - 63992 \nu^{7} - 856 \nu^{6} - 66446 \nu^{5} - 9856 \nu^{4} + \cdots - 2012 ) / 296 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 39 \nu^{15} - 25 \nu^{14} - 844 \nu^{13} - 560 \nu^{12} - 7126 \nu^{11} - 4900 \nu^{10} - 29754 \nu^{9} - 21128 \nu^{8} - 63992 \nu^{7} - 46284 \nu^{6} - 66446 \nu^{5} - 47100 \nu^{4} + \cdots - 924 ) / 296 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39 \nu^{15} - 25 \nu^{14} + 844 \nu^{13} - 560 \nu^{12} + 7126 \nu^{11} - 4900 \nu^{10} + 29754 \nu^{9} - 21128 \nu^{8} + 63992 \nu^{7} - 46284 \nu^{6} + 66446 \nu^{5} - 47100 \nu^{4} + \cdots - 924 ) / 296 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 257 \nu^{15} - 5594 \nu^{13} - 47486 \nu^{11} - 198980 \nu^{9} - 427886 \nu^{7} - 442970 \nu^{5} - 185851 \nu^{3} - 21540 \nu ) / 592 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{10} + 3\beta_{7} - \beta_{5} + \beta_{3} - 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} - 7 \beta_{11} - 9 \beta_{9} + 8 \beta_{8} + \beta_{6} + 9 \beta_{4} + 9 \beta_{3} + 12 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - 9\beta_{13} + 10\beta_{12} + 12\beta_{10} - 32\beta_{7} + 11\beta_{5} - 11\beta_{3} + 46\beta_{2} - 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{15} - 14 \beta_{14} + 14 \beta_{13} + 46 \beta_{11} + 66 \beta_{9} - 52 \beta_{8} - 8 \beta_{6} - 66 \beta_{4} - 66 \beta_{3} - 59 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{14} + 66 \beta_{13} - 80 \beta_{12} - 106 \beta_{10} + 266 \beta_{7} - 88 \beta_{5} + 88 \beta_{3} - 303 \beta_{2} + 605 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 106 \beta_{15} + 134 \beta_{14} - 134 \beta_{13} - 303 \beta_{11} - 461 \beta_{9} + 317 \beta_{8} + 42 \beta_{6} + 457 \beta_{4} + 457 \beta_{3} + 333 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 140 \beta_{14} - 457 \beta_{13} + 597 \beta_{12} + 835 \beta_{10} - 2025 \beta_{7} + 627 \beta_{5} + 6 \beta_{4} - 633 \beta_{3} + 2011 \beta_{2} - 3916 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 835 \beta_{15} - 1109 \beta_{14} + 1109 \beta_{13} + 2011 \beta_{11} + 3171 \beta_{9} - 1882 \beta_{8} - 151 \beta_{6} - 3101 \beta_{4} - 3101 \beta_{3} - 2050 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1219 \beta_{14} + 3101 \beta_{13} - 4320 \beta_{12} - 6224 \beta_{10} + 14784 \beta_{7} - 4251 \beta_{5} - 110 \beta_{4} + 4361 \beta_{3} - 13434 \beta_{2} + 25758 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6224 \beta_{15} + 8552 \beta_{14} - 8552 \beta_{13} - 13434 \beta_{11} - 21680 \beta_{9} + 11056 \beta_{8} + 60 \beta_{6} + 20896 \beta_{4} + 20896 \beta_{3} + 13283 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9840 \beta_{14} - 20896 \beta_{13} + 30736 \beta_{12} + 45008 \beta_{10} - 105472 \beta_{7} + 28220 \beta_{5} + 1288 \beta_{4} - 29508 \beta_{3} + 90189 \beta_{2} - 171171 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 45008 \beta_{15} - 63400 \beta_{14} + 63400 \beta_{13} + 90189 \beta_{11} + 147821 \beta_{9} - 64837 \beta_{8} + 5672 \beta_{6} - 140593 \beta_{4} - 140593 \beta_{3} - 88535 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1084.1
2.61137i
2.31447i
1.93600i
1.85244i
1.24938i
0.805332i
0.578047i
0.317290i
0.317290i
0.578047i
0.805332i
1.24938i
1.85244i
1.93600i
2.31447i
2.61137i
2.61137i 0.146779i −4.81924 0.948930 + 2.02473i 0.383293 1.14057i 7.36208i 2.97846 5.28732 2.47801i
1084.2 2.31447i 2.90369i −3.35679 1.84346 + 1.26556i −6.72051 2.34671i 3.14026i −5.43140 2.92911 4.26665i
1084.3 1.93600i 2.41423i −1.74811 −0.825597 + 2.07807i 4.67396 1.46100i 0.487655i −2.82851 4.02316 + 1.59836i
1084.4 1.85244i 1.45440i −1.43152 −0.323421 2.21255i −2.69418 0.477604i 1.05306i 0.884731 −4.09862 + 0.599118i
1084.5 1.24938i 1.51315i 0.439042 −1.95031 1.09375i −1.89050 0.568589i 3.04730i 0.710386 −1.36651 + 2.43669i
1084.6 0.805332i 1.95838i 1.35144 2.22523 + 0.219874i 1.57714 1.03713i 2.69902i −0.835236 0.177072 1.79205i
1084.7 0.578047i 0.551888i 1.66586 1.93245 + 1.12501i 0.319017 4.66297i 2.11904i 2.69542 0.650310 1.11705i
1084.8 0.317290i 2.04300i 1.89933 −1.85074 + 1.25489i −0.648223 3.69961i 1.23722i −1.17385 0.398165 + 0.587221i
1084.9 0.317290i 2.04300i 1.89933 −1.85074 1.25489i −0.648223 3.69961i 1.23722i −1.17385 0.398165 0.587221i
1084.10 0.578047i 0.551888i 1.66586 1.93245 1.12501i 0.319017 4.66297i 2.11904i 2.69542 0.650310 + 1.11705i
1084.11 0.805332i 1.95838i 1.35144 2.22523 0.219874i 1.57714 1.03713i 2.69902i −0.835236 0.177072 + 1.79205i
1084.12 1.24938i 1.51315i 0.439042 −1.95031 + 1.09375i −1.89050 0.568589i 3.04730i 0.710386 −1.36651 2.43669i
1084.13 1.85244i 1.45440i −1.43152 −0.323421 + 2.21255i −2.69418 0.477604i 1.05306i 0.884731 −4.09862 0.599118i
1084.14 1.93600i 2.41423i −1.74811 −0.825597 2.07807i 4.67396 1.46100i 0.487655i −2.82851 4.02316 1.59836i
1084.15 2.31447i 2.90369i −3.35679 1.84346 1.26556i −6.72051 2.34671i 3.14026i −5.43140 2.92911 + 4.26665i
1084.16 2.61137i 0.146779i −4.81924 0.948930 2.02473i 0.383293 1.14057i 7.36208i 2.97846 5.28732 + 2.47801i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1084.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.j yes 16
5.b even 2 1 inner 1805.2.b.j yes 16
5.c odd 4 2 9025.2.a.ck 16
19.b odd 2 1 1805.2.b.i 16
95.d odd 2 1 1805.2.b.i 16
95.g even 4 2 9025.2.a.cl 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.i 16 19.b odd 2 1
1805.2.b.i 16 95.d odd 2 1
1805.2.b.j yes 16 1.a even 1 1 trivial
1805.2.b.j yes 16 5.b even 2 1 inner
9025.2.a.ck 16 5.c odd 4 2
9025.2.a.cl 16 95.g even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1805, [\chi])\):

\( T_{2}^{16} + 22T_{2}^{14} + 190T_{2}^{12} + 820T_{2}^{10} + 1862T_{2}^{8} + 2154T_{2}^{6} + 1163T_{2}^{4} + 256T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{29}^{8} - T_{29}^{7} - 116T_{29}^{6} + 381T_{29}^{5} + 2570T_{29}^{4} - 14309T_{29}^{3} + 16029T_{29}^{2} + 16054T_{29} - 27484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 22 T^{14} + 190 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} + 27 T^{14} + 291 T^{12} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + 6 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 46 T^{14} + 709 T^{12} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( (T^{8} + 11 T^{7} + 14 T^{6} - 189 T^{5} + \cdots - 3716)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 389036176 \) Copy content Toggle raw display
$17$ \( T^{16} + 135 T^{14} + \cdots + 10137856 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} + 169 T^{14} + 9247 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{8} - T^{7} - 116 T^{6} + 381 T^{5} + \cdots - 27484)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 8 T^{7} - 156 T^{6} + \cdots + 1577684)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 262 T^{14} + \cdots + 7255632400 \) Copy content Toggle raw display
$41$ \( (T^{8} + 13 T^{7} - 112 T^{6} + \cdots + 54305)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 371 T^{14} + \cdots + 2834253058576 \) Copy content Toggle raw display
$47$ \( T^{16} + 471 T^{14} + \cdots + 33594857521 \) Copy content Toggle raw display
$53$ \( T^{16} + 367 T^{14} + 44688 T^{12} + \cdots + 400 \) Copy content Toggle raw display
$59$ \( (T^{8} - 5 T^{7} - 264 T^{6} + \cdots - 3274820)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 15 T^{7} - 91 T^{6} - 1706 T^{5} + \cdots - 7156)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 438 T^{14} + \cdots + 3421867329241 \) Copy content Toggle raw display
$71$ \( (T^{8} - 10 T^{7} - 217 T^{6} + \cdots + 606524)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 484 T^{14} + \cdots + 25761065687296 \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} - 433 T^{6} + \cdots + 9098224)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 404 T^{14} + \cdots + 178917376 \) Copy content Toggle raw display
$89$ \( (T^{8} - 261 T^{6} + 344 T^{5} + \cdots - 726211)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 938 T^{14} + \cdots + 14388396585616 \) Copy content Toggle raw display
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