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Properties

Label 38.4.e.b
Level $38$
Weight $4$
Character orbit 38.e
Analytic conductor $2.242$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.24207258022\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 165 x^{16} - 56 x^{15} + 18435 x^{14} - 11748 x^{13} + 1092662 x^{12} - 1833567 x^{11} + 46842546 x^{10} - 93643115 x^{9} + 1273086000 x^{8} + \cdots + 3892796082289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{8} q^{2} + (\beta_{9} + \beta_{6} + \beta_{4} + \beta_{3}) q^{3} + 4 \beta_{7} q^{4} + ( - \beta_{16} + \beta_{5}) q^{5} + (2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{4} - 2 \beta_{3} - 2) q^{6} + ( - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - 5 \beta_{9} - \beta_{8} + 4 \beta_{7} + \cdots - 4) q^{7}+ \cdots + (\beta_{13} + \beta_{12} - 3 \beta_{11} + 12 \beta_{9} - 6 \beta_{8} - 12 \beta_{7} + 2 \beta_{6} + \cdots + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{8} q^{2} + (\beta_{9} + \beta_{6} + \beta_{4} + \beta_{3}) q^{3} + 4 \beta_{7} q^{4} + ( - \beta_{16} + \beta_{5}) q^{5} + (2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{4} - 2 \beta_{3} - 2) q^{6} + ( - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - 5 \beta_{9} - \beta_{8} + 4 \beta_{7} + \cdots - 4) q^{7}+ \cdots + (13 \beta_{17} + 24 \beta_{16} + 7 \beta_{15} - 17 \beta_{14} + 92 \beta_{13} + \cdots + 35) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 6 q^{3} - 12 q^{6} - 33 q^{7} + 72 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 6 q^{3} - 12 q^{6} - 33 q^{7} + 72 q^{8} + 42 q^{9} - 75 q^{11} + 36 q^{12} + 99 q^{13} + 162 q^{14} - 183 q^{15} - 111 q^{17} - 408 q^{18} - 372 q^{19} + 24 q^{20} - 207 q^{21} - 180 q^{22} + 198 q^{23} - 48 q^{24} + 534 q^{25} + 180 q^{26} + 678 q^{27} + 216 q^{28} + 669 q^{29} - 42 q^{31} + 315 q^{33} - 48 q^{34} - 1995 q^{35} + 168 q^{36} - 1056 q^{37} - 180 q^{38} + 1812 q^{39} - 210 q^{41} - 342 q^{42} - 399 q^{43} + 360 q^{44} + 1494 q^{45} + 672 q^{46} + 1149 q^{47} - 192 q^{48} - 858 q^{49} + 1068 q^{50} + 2646 q^{51} - 468 q^{52} - 633 q^{53} - 2898 q^{54} - 3483 q^{55} - 528 q^{56} - 2814 q^{57} + 636 q^{58} + 51 q^{59} - 84 q^{60} - 4104 q^{61} - 1326 q^{62} + 1215 q^{63} - 576 q^{64} + 1755 q^{65} + 2340 q^{66} - 675 q^{67} - 948 q^{68} + 3693 q^{69} + 3990 q^{70} + 2964 q^{71} + 672 q^{72} - 2004 q^{73} - 486 q^{74} - 4446 q^{75} - 408 q^{76} + 5820 q^{77} + 4992 q^{78} + 543 q^{79} + 1722 q^{81} + 420 q^{82} + 381 q^{83} + 1092 q^{84} + 1266 q^{85} - 3396 q^{86} - 4506 q^{87} + 600 q^{88} + 4386 q^{89} - 2148 q^{90} - 1356 q^{91} - 2628 q^{92} - 8604 q^{93} - 3264 q^{94} + 921 q^{95} + 576 q^{96} + 7599 q^{97} - 954 q^{98} - 5055 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 165 x^{16} - 56 x^{15} + 18435 x^{14} - 11748 x^{13} + 1092662 x^{12} - 1833567 x^{11} + 46842546 x^{10} - 93643115 x^{9} + 1273086000 x^{8} + \cdots + 3892796082289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 64\!\cdots\!63 \nu^{17} + \cdots - 68\!\cdots\!38 ) / 31\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!97 \nu^{17} + \cdots - 84\!\cdots\!58 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!06 \nu^{17} + \cdots + 97\!\cdots\!92 ) / 32\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!39 \nu^{17} + \cdots - 52\!\cdots\!50 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 50\!\cdots\!79 \nu^{17} + \cdots + 31\!\cdots\!66 ) / 40\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55\!\cdots\!10 \nu^{17} + \cdots + 10\!\cdots\!79 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83\!\cdots\!42 \nu^{17} + \cdots - 14\!\cdots\!51 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!57 \nu^{17} + \cdots - 71\!\cdots\!30 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!22 \nu^{17} + \cdots - 21\!\cdots\!81 ) / 40\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\!\cdots\!05 \nu^{17} + \cdots - 42\!\cdots\!11 ) / 40\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!35 \nu^{17} + \cdots + 11\!\cdots\!41 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!14 \nu^{17} + \cdots + 39\!\cdots\!97 ) / 40\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!39 \nu^{17} + \cdots - 20\!\cdots\!55 ) / 17\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!44 \nu^{17} + \cdots + 56\!\cdots\!27 ) / 17\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 55\!\cdots\!46 \nu^{17} + \cdots - 77\!\cdots\!15 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 55\!\cdots\!90 \nu^{17} + \cdots - 49\!\cdots\!95 ) / 20\!\cdots\!72 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - 3\beta_{11} - 5\beta_{9} + 5\beta_{7} - 3\beta_{6} + 36\beta_{4} + 5\beta_{3} - \beta_{2} - \beta _1 - 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + 2 \beta_{13} - \beta_{12} + 6 \beta_{11} - 8 \beta_{10} + 24 \beta_{9} - 88 \beta_{8} + 88 \beta_{7} - 8 \beta_{6} + 112 \beta_{3} + 57 \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{17} + 7 \beta_{16} + 56 \beta_{14} + 296 \beta_{13} + 7 \beta_{12} + 296 \beta_{11} - 274 \beta_{10} + 622 \beta_{9} - 622 \beta_{8} - 222 \beta_{7} - 22 \beta_{6} - 2 \beta_{5} - 2078 \beta_{4} + 222 \beta_{3} + 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 40 \beta_{17} - 40 \beta_{16} + 6 \beta_{15} + 879 \beta_{13} + 156 \beta_{12} + 268 \beta_{11} + 879 \beta_{10} + 9012 \beta_{9} - 2569 \beta_{8} - 11581 \beta_{7} + 268 \beta_{6} + 156 \beta_{5} - 1595 \beta_{4} - 9012 \beta_{3} + \cdots + 1595 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 893 \beta_{17} - 400 \beta_{16} - 3312 \beta_{15} - 3312 \beta_{14} - 23021 \beta_{13} - 493 \beta_{12} - 2780 \beta_{11} + 25801 \beta_{10} - 31030 \beta_{9} + 35155 \beta_{8} - 35155 \beta_{7} + \cdots + 145459 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14918 \beta_{17} - 14572 \beta_{16} - 1818 \beta_{14} - 128486 \beta_{13} - 14572 \beta_{12} - 128486 \beta_{11} + 37444 \beta_{10} - 1038778 \beta_{9} + 1038778 \beta_{8} + 233570 \beta_{7} + \cdots + 295792 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 58696 \beta_{17} - 58696 \beta_{16} + 224404 \beta_{15} - 292430 \beta_{13} - 34846 \beta_{12} - 1918802 \beta_{11} - 292430 \beta_{10} - 3361386 \beta_{9} + 3261130 \beta_{8} + \cdots - 11160210 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1409632 \beta_{17} + 1293906 \beta_{16} + 386589 \beta_{15} + 386589 \beta_{14} + 4671837 \beta_{13} + 115726 \beta_{12} + 8449912 \beta_{11} - 13121749 \beta_{10} + 20972780 \beta_{9} + \cdots - 40208848 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3551481 \beta_{17} + 9258444 \beta_{16} + 16810526 \beta_{14} + 190372536 \beta_{13} + 9258444 \beta_{12} + 190372536 \beta_{11} - 160533576 \beta_{10} + 640135440 \beta_{9} + \cdots - 74540953 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 17339398 \beta_{17} + 17339398 \beta_{16} - 50766222 \beta_{15} + 750873190 \beta_{13} + 109791635 \beta_{12} + 529267958 \beta_{11} + 750873190 \beta_{10} + \cdots + 4346923604 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 890919814 \beta_{17} - 375717320 \beta_{16} - 1339981704 \beta_{15} - 1339981704 \beta_{14} - 13528021390 \beta_{13} - 515202494 \beta_{12} - 3012820279 \beta_{11} + \cdots + 75363031909 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9302450500 \beta_{17} - 11254259423 \beta_{16} - 5647093956 \beta_{14} - 121694400067 \beta_{13} - 11254259423 \beta_{12} - 121694400067 \beta_{11} + \cdots + 159584620144 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 45203148824 \beta_{17} - 45203148824 \beta_{16} + 110912225908 \beta_{15} - 300460695808 \beta_{13} - 38993259452 \beta_{12} - 1149648891760 \beta_{11} + \cdots - 6433442474541 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 994183242116 \beta_{17} + 792214993472 \beta_{16} + 580600861908 \beta_{15} + 580600861908 \beta_{14} + 5697403289404 \beta_{13} + 201968248644 \beta_{12} + \cdots - 42530549970820 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 3930296819656 \beta_{17} + 7858333786556 \beta_{16} + 9402098067641 \beta_{14} + 128074633720903 \beta_{13} + 7858333786556 \beta_{12} + \cdots - 83123860453267 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 20122050426723 \beta_{17} + 20122050426723 \beta_{16} - 57162326555028 \beta_{15} + 493750285485482 \beta_{13} + 67957658485045 \beta_{12} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−4.21239 + 7.29608i
1.11938 − 1.93882i
2.15332 − 3.72966i
−3.03492 − 5.25663i
−1.54550 − 2.67688i
4.75406 + 8.23428i
−3.03492 + 5.25663i
−1.54550 + 2.67688i
4.75406 − 8.23428i
−4.21239 − 7.29608i
1.11938 + 1.93882i
2.15332 + 3.72966i
2.89395 + 5.01246i
1.58432 + 2.74412i
−3.71222 − 6.42975i
2.89395 − 5.01246i
1.58432 − 2.74412i
−3.71222 + 6.42975i
1.87939 − 0.684040i −1.31024 + 7.43077i 3.06418 − 2.57115i 3.45478 + 2.89891i 2.62049 + 14.8615i 4.21085 + 7.29340i 4.00000 − 6.92820i −28.1279 − 10.2377i 8.47584 + 3.08495i
5.2 1.87939 − 0.684040i 0.541460 − 3.07077i 3.06418 − 2.57115i 8.27240 + 6.94137i −1.08292 − 6.14154i −13.7717 − 23.8533i 4.00000 − 6.92820i 16.2353 + 5.90915i 20.2952 + 7.38685i
5.3 1.87939 − 0.684040i 0.900544 − 5.10724i 3.06418 − 2.57115i −10.9611 − 9.19749i −1.80109 − 10.2145i 16.0725 + 27.8383i 4.00000 − 6.92820i 0.0987659 + 0.0359478i −26.8917 − 9.78776i
9.1 −0.347296 − 1.96962i −5.68185 + 4.76764i −3.75877 + 1.36808i 18.4168 + 6.70318i 11.3637 + 9.53528i −15.6039 + 27.0267i 4.00000 + 6.92820i 4.86455 − 27.5882i 6.80659 − 38.6021i
9.2 −0.347296 − 1.96962i −3.39993 + 2.85288i −3.75877 + 1.36808i −20.2092 − 7.35554i 6.79986 + 5.70576i 2.98693 − 5.17352i 4.00000 + 6.92820i −1.26790 + 7.19064i −7.46901 + 42.3588i
9.3 −0.347296 − 1.96962i 6.25156 − 5.24568i −3.75877 + 1.36808i 0.852642 + 0.310336i −12.5031 − 10.4914i −4.68031 + 8.10654i 4.00000 + 6.92820i 6.87632 − 38.9975i 0.315124 − 1.78716i
17.1 −0.347296 + 1.96962i −5.68185 − 4.76764i −3.75877 − 1.36808i 18.4168 − 6.70318i 11.3637 − 9.53528i −15.6039 − 27.0267i 4.00000 − 6.92820i 4.86455 + 27.5882i 6.80659 + 38.6021i
17.2 −0.347296 + 1.96962i −3.39993 − 2.85288i −3.75877 − 1.36808i −20.2092 + 7.35554i 6.79986 − 5.70576i 2.98693 + 5.17352i 4.00000 − 6.92820i −1.26790 − 7.19064i −7.46901 − 42.3588i
17.3 −0.347296 + 1.96962i 6.25156 + 5.24568i −3.75877 − 1.36808i 0.852642 − 0.310336i −12.5031 + 10.4914i −4.68031 − 8.10654i 4.00000 − 6.92820i 6.87632 + 38.9975i 0.315124 + 1.78716i
23.1 1.87939 + 0.684040i −1.31024 − 7.43077i 3.06418 + 2.57115i 3.45478 − 2.89891i 2.62049 − 14.8615i 4.21085 − 7.29340i 4.00000 + 6.92820i −28.1279 + 10.2377i 8.47584 − 3.08495i
23.2 1.87939 + 0.684040i 0.541460 + 3.07077i 3.06418 + 2.57115i 8.27240 − 6.94137i −1.08292 + 6.14154i −13.7717 + 23.8533i 4.00000 + 6.92820i 16.2353 − 5.90915i 20.2952 − 7.38685i
23.3 1.87939 + 0.684040i 0.900544 + 5.10724i 3.06418 + 2.57115i −10.9611 + 9.19749i −1.80109 + 10.2145i 16.0725 − 27.8383i 4.00000 + 6.92820i 0.0987659 − 0.0359478i −26.8917 + 9.78776i
25.1 −1.53209 + 1.28558i −3.05945 − 1.11355i 0.694593 − 3.93923i −1.91957 − 10.8864i 6.11891 − 2.22710i 9.80910 − 16.9899i 4.00000 + 6.92820i −12.5629 − 10.5416i 16.9363 + 14.2112i
25.2 −1.53209 + 1.28558i −0.598158 − 0.217712i 0.694593 − 3.93923i 3.08282 + 17.4836i 1.19632 − 0.435424i −11.6036 + 20.0980i 4.00000 + 6.92820i −20.3728 − 17.0948i −27.1996 − 22.8232i
25.3 −1.53209 + 1.28558i 9.35607 + 3.40533i 0.694593 − 3.93923i −0.989599 − 5.61230i −18.7121 + 6.81067i −3.91986 + 6.78940i 4.00000 + 6.92820i 55.2566 + 46.3658i 8.73118 + 7.32633i
35.1 −1.53209 − 1.28558i −3.05945 + 1.11355i 0.694593 + 3.93923i −1.91957 + 10.8864i 6.11891 + 2.22710i 9.80910 + 16.9899i 4.00000 − 6.92820i −12.5629 + 10.5416i 16.9363 − 14.2112i
35.2 −1.53209 − 1.28558i −0.598158 + 0.217712i 0.694593 + 3.93923i 3.08282 − 17.4836i 1.19632 + 0.435424i −11.6036 − 20.0980i 4.00000 − 6.92820i −20.3728 + 17.0948i −27.1996 + 22.8232i
35.3 −1.53209 − 1.28558i 9.35607 − 3.40533i 0.694593 + 3.93923i −0.989599 + 5.61230i −18.7121 − 6.81067i −3.91986 − 6.78940i 4.00000 − 6.92820i 55.2566 − 46.3658i 8.73118 − 7.32633i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.4.e.b ✓ 18
19.e even 9 1 inner 38.4.e.b ✓ 18
19.e even 9 1 722.4.a.t 9
19.f odd 18 1 722.4.a.u 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.b ✓ 18 1.a even 1 1 trivial
38.4.e.b ✓ 18 19.e even 9 1 inner
722.4.a.t 9 19.e even 9 1
722.4.a.u 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 6 T_{3}^{17} - 3 T_{3}^{16} - 298 T_{3}^{15} + 3495 T_{3}^{14} + 10455 T_{3}^{13} + 207596 T_{3}^{12} - 770469 T_{3}^{11} + 3933852 T_{3}^{10} + 76842539 T_{3}^{9} + 766535988 T_{3}^{8} + \cdots + 457499373769 \) acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 8 T^{3} + 64)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} - 6 T^{17} + \cdots + 457499373769 \) Copy content Toggle raw display
$5$ \( T^{18} - 267 T^{16} + \cdots + 88\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{17} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{18} + 75 T^{17} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{18} - 99 T^{17} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{18} + 111 T^{17} + \cdots + 35\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{18} + 372 T^{17} + \cdots + 33\!\cdots\!39 \) Copy content Toggle raw display
$23$ \( T^{18} - 198 T^{17} + \cdots + 85\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{18} - 669 T^{17} + \cdots + 86\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{18} + 42 T^{17} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{9} + 528 T^{8} + \cdots - 64\!\cdots\!08)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + 210 T^{17} + \cdots + 96\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{18} + 399 T^{17} + \cdots + 33\!\cdots\!29 \) Copy content Toggle raw display
$47$ \( T^{18} - 1149 T^{17} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{18} + 633 T^{17} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{18} - 51 T^{17} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{18} + 4104 T^{17} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{18} + 675 T^{17} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{18} - 2964 T^{17} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{18} + 2004 T^{17} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{18} - 543 T^{17} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{18} - 381 T^{17} + \cdots + 40\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{18} - 4386 T^{17} + \cdots + 53\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( T^{18} - 7599 T^{17} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
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