Properties

Label 4.11.b.a
Level 4
Weight 11
Character orbit 4.b
Analytic conductor 2.541
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.54142901069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.26777625.2
Defining polynomial: \(x^{4} - 2 x^{3} + 59 x^{2} - 58 x + 336\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 + \beta_{1} ) q^{2} + ( -2 \beta_{1} + \beta_{2} ) q^{3} + ( 4 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4} + ( -390 + 64 \beta_{1} + 4 \beta_{3} ) q^{5} + ( 1800 + 8 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} ) q^{6} + ( -460 \beta_{1} - 26 \beta_{2} + 32 \beta_{3} ) q^{7} + ( -9072 - 16 \beta_{1} - 48 \beta_{2} - 52 \beta_{3} ) q^{8} + ( -7191 + 1152 \beta_{1} + 72 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -3 + \beta_{1} ) q^{2} + ( -2 \beta_{1} + \beta_{2} ) q^{3} + ( 4 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{4} + ( -390 + 64 \beta_{1} + 4 \beta_{3} ) q^{5} + ( 1800 + 8 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} ) q^{6} + ( -460 \beta_{1} - 26 \beta_{2} + 32 \beta_{3} ) q^{7} + ( -9072 - 16 \beta_{1} - 48 \beta_{2} - 52 \beta_{3} ) q^{8} + ( -7191 + 1152 \beta_{1} + 72 \beta_{3} ) q^{9} + ( 65810 - 262 \beta_{1} + 256 \beta_{2} - 64 \beta_{3} ) q^{10} + ( 418 \beta_{1} + 303 \beta_{2} - 64 \beta_{3} ) q^{11} + ( -228960 + 1120 \beta_{1} + 160 \beta_{2} + 216 \beta_{3} ) q^{12} + ( 53066 - 7872 \beta_{1} - 492 \beta_{3} ) q^{13} + ( 475440 + 1840 \beta_{1} - 1632 \beta_{2} + 824 \beta_{3} ) q^{14} + ( 14860 \beta_{1} - 2054 \beta_{2} - 672 \beta_{3} ) q^{15} + ( -903104 - 11840 \beta_{1} + 320 \beta_{2} + 688 \beta_{3} ) q^{16} + ( -42846 - 2432 \beta_{1} - 152 \beta_{3} ) q^{17} + ( 1185093 - 4887 \beta_{1} + 4608 \beta_{2} - 1152 \beta_{3} ) q^{18} + ( -38874 \beta_{1} + 8685 \beta_{2} + 1344 \beta_{3} ) q^{19} + ( -777240 + 64536 \beta_{1} - 3096 \beta_{2} - 3322 \beta_{3} ) q^{20} + ( -120960 + 80640 \beta_{1} + 5040 \beta_{3} ) q^{21} + ( -499080 - 1672 \beta_{1} - 752 \beta_{2} - 4660 \beta_{3} ) q^{22} + ( -53956 \beta_{1} - 21918 \beta_{2} + 6112 \beta_{3} ) q^{23} + ( 4475520 - 218752 \beta_{1} + 3200 \beta_{2} - 3360 \beta_{3} ) q^{24} + ( -1339605 - 49920 \beta_{1} - 3120 \beta_{3} ) q^{25} + ( -8109918 + 37322 \beta_{1} - 31488 \beta_{2} + 7872 \beta_{3} ) q^{26} + ( 149724 \beta_{1} + 21906 \beta_{2} - 12096 \beta_{3} ) q^{27} + ( 9024960 + 503360 \beta_{1} + 20416 \beta_{2} + 21008 \beta_{3} ) q^{28} + ( 7511658 - 311744 \beta_{1} - 19484 \beta_{3} ) q^{29} + ( -14664240 - 59440 \beta_{1} + 75872 \beta_{2} + 13896 \beta_{3} ) q^{30} + ( 388624 \beta_{1} + 49400 \beta_{2} - 30464 \beta_{3} ) q^{31} + ( 14514432 - 856320 \beta_{1} - 49920 \beta_{2} + 7360 \beta_{3} ) q^{32} + ( -16384320 + 127872 \beta_{1} + 7992 \beta_{3} ) q^{33} + ( -2327782 - 47710 \beta_{1} - 9728 \beta_{2} + 2432 \beta_{3} ) q^{34} + ( -514360 \beta_{1} - 208740 \beta_{2} + 58240 \beta_{3} ) q^{35} + ( -13991004 + 1162332 \beta_{1} - 56412 \beta_{2} - 59625 \beta_{3} ) q^{36} + ( 33602234 + 525120 \beta_{1} + 32820 \beta_{3} ) q^{37} + ( 37567080 + 155496 \beta_{1} - 224976 \beta_{2} - 82716 \beta_{3} ) q^{38} + ( -1837972 \beta_{1} + 257738 \beta_{2} + 82656 \beta_{3} ) q^{39} + ( -57482080 - 1019808 \beta_{1} + 282912 \beta_{2} - 21192 \beta_{3} ) q^{40} + ( -85045038 + 1336576 \beta_{1} + 83536 \beta_{3} ) q^{41} + ( 81809280 + 40320 \beta_{1} + 322560 \beta_{2} - 80640 \beta_{3} ) q^{42} + ( 2132210 \beta_{1} + 63367 \beta_{2} - 141184 \beta_{3} ) q^{43} + ( -75518880 - 725600 \beta_{1} - 672 \beta_{2} + 12200 \beta_{3} ) q^{44} + ( 151735050 - 909504 \beta_{1} - 56844 \beta_{3} ) q^{45} + ( 60295440 + 215824 \beta_{1} - 40480 \beta_{2} + 360808 \beta_{3} ) q^{46} + ( 1903192 \beta_{1} - 388908 \beta_{2} - 70336 \beta_{3} ) q^{47} + ( -61171200 + 4552192 \beta_{1} - 900608 \beta_{2} + 173952 \beta_{3} ) q^{48} + ( -201230351 - 8117760 \beta_{1} - 507360 \beta_{3} ) q^{49} + ( -46400385 - 1439445 \beta_{1} - 199680 \beta_{2} + 49920 \beta_{3} ) q^{50} + ( -449348 \beta_{1} + 20386 \beta_{2} + 25536 \beta_{3} ) q^{51} + ( 95620904 - 7958312 \beta_{1} + 401192 \beta_{2} + 403510 \beta_{3} ) q^{52} + ( 359392986 + 4582720 \beta_{1} + 286420 \beta_{3} ) q^{53} + ( -157975920 - 598896 \beta_{1} + 423648 \beta_{2} - 456408 \beta_{3} ) q^{54} + ( 4758580 \beta_{1} - 98074 \beta_{2} - 285152 \beta_{3} ) q^{55} + ( 348122880 + 9652480 \beta_{1} + 1850112 \beta_{2} - 789184 \beta_{3} ) q^{56} + ( -652708800 + 14649984 \beta_{1} + 915624 \beta_{3} ) q^{57} + ( -337396414 + 6888170 \beta_{1} - 1246976 \beta_{2} + 311744 \beta_{3} ) q^{58} + ( -18650662 \beta_{1} + 2601747 \beta_{2} + 840448 \beta_{3} ) q^{59} + ( 405771840 - 13482560 \beta_{1} - 844736 \beta_{2} - 1002768 \beta_{3} ) q^{60} + ( 853020842 - 10302144 \beta_{1} - 643884 \beta_{3} ) q^{61} + ( -408252480 - 1554496 \beta_{1} + 1159296 \beta_{2} - 1080224 \beta_{3} ) q^{62} + ( -9179820 \beta_{1} - 3752874 \beta_{2} + 1042848 \beta_{3} ) q^{63} + ( -9092096 + 15424512 \beta_{1} - 3025920 \beta_{2} + 1555200 \beta_{3} ) q^{64} + ( -1038387900 + 6466304 \beta_{1} + 404144 \beta_{3} ) q^{65} + ( 178303680 - 16128576 \beta_{1} + 511488 \beta_{2} - 127872 \beta_{3} ) q^{66} + ( 38524318 \beta_{1} - 2655439 \beta_{2} - 2075840 \beta_{3} ) q^{67} + ( 29304456 - 2221704 \beta_{1} - 113016 \beta_{2} + 183902 \beta_{3} ) q^{68} + ( 1099797120 - 4126464 \beta_{1} - 257904 \beta_{3} ) q^{69} + ( 574744800 + 2057440 \beta_{1} - 387520 \beta_{2} + 3436720 \beta_{3} ) q^{70} + ( 2756436 \beta_{1} + 8102742 \beta_{2} - 1185120 \beta_{3} ) q^{71} + ( -1033126128 - 18353808 \beta_{1} + 5100624 \beta_{2} - 372564 \beta_{3} ) q^{72} + ( -747127534 - 55984512 \beta_{1} - 3499032 \beta_{3} ) q^{73} + ( 429564498 + 34652474 \beta_{1} + 2100480 \beta_{2} - 525120 \beta_{3} ) q^{74} + ( -8303190 \beta_{1} - 41685 \beta_{2} + 524160 \beta_{3} ) q^{75} + ( -1859493600 + 32091360 \beta_{1} + 2421792 \beta_{2} + 2994168 \beta_{3} ) q^{76} + ( 937050240 + 36476160 \beta_{1} + 2279760 \beta_{3} ) q^{77} + ( 1812874320 + 7351888 \beta_{1} - 9413792 \beta_{2} - 1770360 \beta_{3} ) q^{78} + ( 15125864 \beta_{1} - 2386100 \beta_{2} - 647104 \beta_{3} ) q^{79} + ( 339799680 - 55782016 \beta_{1} - 6342528 \beta_{2} - 2940960 \beta_{3} ) q^{80} + ( -1178945199 + 51456384 \beta_{1} + 3216024 \beta_{3} ) q^{81} + ( 1605076874 - 82371886 \beta_{1} + 5346304 \beta_{2} - 1336576 \beta_{3} ) q^{82} + ( -54031418 \beta_{1} - 17655267 \beta_{2} + 5583872 \beta_{3} ) q^{83} + ( -977840640 + 79833600 \beta_{1} - 2419200 \beta_{2} - 4556160 \beta_{3} ) q^{84} + ( -297699020 - 1793664 \beta_{1} - 112104 \beta_{3} ) q^{85} + ( -2190062280 - 8528840 \beta_{1} + 8021904 \beta_{2} - 3019348 \beta_{3} ) q^{86} + ( -83606996 \beta_{1} + 15617002 \beta_{2} + 3273312 \beta_{3} ) q^{87} + ( 427109760 - 74211712 \beta_{1} - 2897024 \beta_{2} + 735008 \beta_{3} ) q^{88} + ( 1318680498 + 32133760 \beta_{1} + 2008360 \beta_{3} ) q^{89} + ( -1373804190 + 149916042 \beta_{1} - 3638016 \beta_{2} + 909504 \beta_{3} ) q^{90} + ( 60922120 \beta_{1} + 25542524 \beta_{2} - 7000448 \beta_{3} ) q^{91} + ( 5605097280 + 77155520 \beta_{1} + 1187136 \beta_{2} + 350896 \beta_{3} ) q^{92} + ( -1517529600 - 48374784 \beta_{1} - 3023424 \beta_{3} ) q^{93} + ( -1847917920 - 7612768 \beta_{1} + 10724032 \beta_{2} + 3541520 \beta_{3} ) q^{94} + ( 123683100 \beta_{1} - 28849038 \beta_{2} - 4124064 \beta_{3} ) q^{95} + ( 1373644800 - 62777344 \beta_{1} + 25413632 \beta_{2} + 8056320 \beta_{3} ) q^{96} + ( 3585799874 - 47068032 \beta_{1} - 2941752 \beta_{3} ) q^{97} + ( -7595246547 - 217465871 \beta_{1} - 32471040 \beta_{2} + 8117760 \beta_{3} ) q^{98} + ( 85582962 \beta_{1} - 1817145 \beta_{2} - 5121792 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{2} + 16q^{4} - 1560q^{5} + 7200q^{6} - 36288q^{8} - 28764q^{9} + O(q^{10}) \) \( 4q - 12q^{2} + 16q^{4} - 1560q^{5} + 7200q^{6} - 36288q^{8} - 28764q^{9} + 263240q^{10} - 915840q^{12} + 212264q^{13} + 1901760q^{14} - 3612416q^{16} - 171384q^{17} + 4740372q^{18} - 3108960q^{20} - 483840q^{21} - 1996320q^{22} + 17902080q^{24} - 5358420q^{25} - 32439672q^{26} + 36099840q^{28} + 30046632q^{29} - 58656960q^{30} + 58057728q^{32} - 65537280q^{33} - 9311128q^{34} - 55964016q^{36} + 134408936q^{37} + 150268320q^{38} - 229928320q^{40} - 340180152q^{41} + 327237120q^{42} - 302075520q^{44} + 606940200q^{45} + 241181760q^{46} - 244684800q^{48} - 804921404q^{49} - 185601540q^{50} + 382483616q^{52} + 1437571944q^{53} - 631903680q^{54} + 1392491520q^{56} - 2610835200q^{57} - 1349585656q^{58} + 1623087360q^{60} + 3412083368q^{61} - 1633009920q^{62} - 36368384q^{64} - 4153551600q^{65} + 713214720q^{66} + 117217824q^{68} + 4399188480q^{69} + 2298979200q^{70} - 4132504512q^{72} - 2988510136q^{73} + 1718257992q^{74} - 7437974400q^{76} + 3748200960q^{77} + 7251497280q^{78} + 1359198720q^{80} - 4715780796q^{81} + 6420307496q^{82} - 3911362560q^{84} - 1190796080q^{85} - 8760249120q^{86} + 1708439040q^{88} + 5274721992q^{89} - 5495216760q^{90} + 22420389120q^{92} - 6070118400q^{93} - 7391671680q^{94} + 5494579200q^{96} + 14343199496q^{97} - 30380986188q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 59 x^{2} - 58 x + 336\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} + 4 \nu^{2} + 78 \nu + 161 \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{3} + 8 \nu^{2} + 492 \nu + 154 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( -32 \nu^{3} + 160 \nu^{2} - 1472 \nu + 3920 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1} + 24\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{3} + 2 \beta_{2} + 44 \beta_{1} - 2736\)\()/96\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} - 41 \beta_{2} + 202 \beta_{1} - 2064\)\()/48\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 + 7.15697i
0.500000 7.15697i
0.500000 2.50555i
0.500000 + 2.50555i
−25.4722 19.3692i 343.535i 273.666 + 986.754i −3266.44 6654.00 8750.58i 10902.2i 12141.8 30435.5i −58967.0 83203.5 + 63268.4i
3.2 −25.4722 + 19.3692i 343.535i 273.666 986.754i −3266.44 6654.00 + 8750.58i 10902.2i 12141.8 + 30435.5i −58967.0 83203.5 63268.4i
3.3 19.4722 25.3936i 120.267i −265.666 988.937i 2486.44 −3054.00 2341.85i 29129.9i −30285.8 12510.6i 44585.0 48416.5 63139.6i
3.4 19.4722 + 25.3936i 120.267i −265.666 + 988.937i 2486.44 −3054.00 + 2341.85i 29129.9i −30285.8 + 12510.6i 44585.0 48416.5 + 63139.6i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.11.b.a 4
3.b odd 2 1 36.11.d.c 4
4.b odd 2 1 inner 4.11.b.a 4
5.b even 2 1 100.11.b.d 4
5.c odd 4 2 100.11.d.a 8
8.b even 2 1 64.11.c.d 4
8.d odd 2 1 64.11.c.d 4
12.b even 2 1 36.11.d.c 4
16.e even 4 2 256.11.d.f 8
16.f odd 4 2 256.11.d.f 8
20.d odd 2 1 100.11.b.d 4
20.e even 4 2 100.11.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.11.b.a 4 1.a even 1 1 trivial
4.11.b.a 4 4.b odd 2 1 inner
36.11.d.c 4 3.b odd 2 1
36.11.d.c 4 12.b even 2 1
64.11.c.d 4 8.b even 2 1
64.11.c.d 4 8.d odd 2 1
100.11.b.d 4 5.b even 2 1
100.11.b.d 4 20.d odd 2 1
100.11.d.a 8 5.c odd 4 2
100.11.d.a 8 20.e even 4 2
256.11.d.f 8 16.e even 4 2
256.11.d.f 8 16.f odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 12 T + 64 T^{2} + 12288 T^{3} + 1048576 T^{4} \)
$3$ \( 1 - 103716 T^{2} + 6982070886 T^{4} - 361635330934116 T^{6} + 12157665459056928801 T^{8} \)
$5$ \( ( 1 + 780 T + 11409430 T^{2} + 7617187500 T^{3} + 95367431640625 T^{4} )^{2} \)
$7$ \( 1 - 162489796 T^{2} + 33071250662702406 T^{4} - \)\(12\!\cdots\!96\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - 91776568804 T^{2} + \)\(34\!\cdots\!26\)\( T^{4} - \)\(61\!\cdots\!04\)\( T^{6} + \)\(45\!\cdots\!01\)\( T^{8} \)
$13$ \( ( 1 - 106132 T + 153356848374 T^{2} - 14631197456918068 T^{3} + \)\(19\!\cdots\!01\)\( T^{4} )^{2} \)
$17$ \( ( 1 + 85692 T + 4021876040134 T^{2} + 172754549317275708 T^{3} + \)\(40\!\cdots\!01\)\( T^{4} )^{2} \)
$19$ \( 1 - 9955343221924 T^{2} + \)\(46\!\cdots\!66\)\( T^{4} - \)\(37\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 - 93909691628356 T^{2} + \)\(51\!\cdots\!06\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{6} + \)\(29\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 - 15023316 T + 701527143006646 T^{2} - \)\(63\!\cdots\!16\)\( T^{3} + \)\(17\!\cdots\!01\)\( T^{4} )^{2} \)
$31$ \( 1 - 2332612996197124 T^{2} + \)\(24\!\cdots\!66\)\( T^{4} - \)\(15\!\cdots\!24\)\( T^{6} + \)\(45\!\cdots\!01\)\( T^{8} \)
$37$ \( ( 1 - 67204468 T + 10189261825538454 T^{2} - \)\(32\!\cdots\!32\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} )^{2} \)
$41$ \( ( 1 + 170090076 T + 30469377593098726 T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(18\!\cdots\!01\)\( T^{4} )^{2} \)
$43$ \( 1 - 67155685993274596 T^{2} + \)\(20\!\cdots\!06\)\( T^{4} - \)\(31\!\cdots\!96\)\( T^{6} + \)\(21\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 178889386570200196 T^{2} + \)\(13\!\cdots\!26\)\( T^{4} - \)\(49\!\cdots\!96\)\( T^{6} + \)\(76\!\cdots\!01\)\( T^{8} \)
$53$ \( ( 1 - 718785972 T + 436515587468254294 T^{2} - \)\(12\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!01\)\( T^{4} )^{2} \)
$59$ \( 1 + 93873365134043036 T^{2} - \)\(34\!\cdots\!54\)\( T^{4} + \)\(24\!\cdots\!36\)\( T^{6} + \)\(68\!\cdots\!01\)\( T^{8} \)
$61$ \( ( 1 - 1706041684 T + 1939939354798747446 T^{2} - \)\(12\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!01\)\( T^{4} )^{2} \)
$67$ \( 1 - 586380343444775716 T^{2} + \)\(45\!\cdots\!86\)\( T^{4} - \)\(19\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!01\)\( T^{8} \)
$71$ \( 1 - 5068560656654470084 T^{2} + \)\(27\!\cdots\!86\)\( T^{4} - \)\(53\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!01\)\( T^{8} \)
$73$ \( ( 1 + 1494255068 T + 2822234732034185574 T^{2} + \)\(64\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!01\)\( T^{4} )^{2} \)
$79$ \( 1 - 36325713459635253124 T^{2} + \)\(50\!\cdots\!66\)\( T^{4} - \)\(32\!\cdots\!24\)\( T^{6} + \)\(80\!\cdots\!01\)\( T^{8} \)
$83$ \( 1 - 10916459259555656356 T^{2} + \)\(15\!\cdots\!06\)\( T^{4} - \)\(26\!\cdots\!56\)\( T^{6} + \)\(57\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 - 2637360996 T + 62016549481627943206 T^{2} - \)\(82\!\cdots\!96\)\( T^{3} + \)\(97\!\cdots\!01\)\( T^{4} )^{2} \)
$97$ \( ( 1 - 7171599748 T + \)\(15\!\cdots\!94\)\( T^{2} - \)\(52\!\cdots\!52\)\( T^{3} + \)\(54\!\cdots\!01\)\( T^{4} )^{2} \)
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