Properties

Label 9.62.a.a
Level 9
Weight 62
Character orbit 9.a
Self dual yes
Analytic conductor 212.091
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 62 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(212.090564938\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{10}\cdot 5^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -286578000 + \beta_{1} ) q^{2} + ( 1100749418957151232 - 838426626 \beta_{1} + 38 \beta_{2} + \beta_{3} ) q^{4} + ( 131031499005073809450 - 100252987760 \beta_{1} + 5380 \beta_{2} + 120 \beta_{3} ) q^{5} + ( -15834642365846289693295000 + 14513887843120512 \beta_{1} - 5219268334 \beta_{2} - 19628000 \beta_{3} ) q^{7} + ( -2441967096789168519757824000 + 1373003764151712384 \beta_{1} + 833101117568 \beta_{2} - 1190856000 \beta_{3} ) q^{8} +O(q^{10})\) \( q +(-286578000 + \beta_{1}) q^{2} +(1100749418957151232 - 838426626 \beta_{1} + 38 \beta_{2} + \beta_{3}) q^{4} +(\)\(13\!\cdots\!50\)\( - 100252987760 \beta_{1} + 5380 \beta_{2} + 120 \beta_{3}) q^{5} +(-\)\(15\!\cdots\!00\)\( + 14513887843120512 \beta_{1} - 5219268334 \beta_{2} - 19628000 \beta_{3}) q^{7} +(-\)\(24\!\cdots\!00\)\( + 1373003764151712384 \beta_{1} + 833101117568 \beta_{2} - 1190856000 \beta_{3}) q^{8} +(-\)\(37\!\cdots\!00\)\( + \)\(44\!\cdots\!70\)\( \beta_{1} + 100061893877440 \beta_{2} - 141501833440 \beta_{3}) q^{10} +(\)\(10\!\cdots\!88\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} - 14517391394747515 \beta_{2} - 11035029161280 \beta_{3}) q^{11} +(\)\(26\!\cdots\!50\)\( + \)\(39\!\cdots\!92\)\( \beta_{1} - 1387217511377341540 \beta_{2} + 2435574015901000 \beta_{3}) q^{13} +(\)\(52\!\cdots\!76\)\( - \)\(65\!\cdots\!72\)\( \beta_{1} - 16841348233782444064 \beta_{2} + 15743169581051472 \beta_{3}) q^{14} +(\)\(27\!\cdots\!56\)\( - \)\(36\!\cdots\!12\)\( \beta_{1} - \)\(98\!\cdots\!44\)\( \beta_{2} + 603732751100608512 \beta_{3}) q^{16} +(\)\(10\!\cdots\!50\)\( + \)\(38\!\cdots\!72\)\( \beta_{1} - \)\(37\!\cdots\!28\)\( \beta_{2} + 7263830520990282000 \beta_{3}) q^{17} +(-\)\(89\!\cdots\!80\)\( + \)\(48\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!43\)\( \beta_{2} - \)\(10\!\cdots\!36\)\( \beta_{3}) q^{19} +(\)\(12\!\cdots\!00\)\( - \)\(66\!\cdots\!20\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(34\!\cdots\!90\)\( \beta_{3}) q^{20} +(-\)\(43\!\cdots\!00\)\( - \)\(79\!\cdots\!32\)\( \beta_{1} - \)\(11\!\cdots\!40\)\( \beta_{2} - \)\(26\!\cdots\!00\)\( \beta_{3}) q^{22} +(\)\(10\!\cdots\!00\)\( - \)\(24\!\cdots\!20\)\( \beta_{1} + \)\(27\!\cdots\!74\)\( \beta_{2} - \)\(75\!\cdots\!00\)\( \beta_{3}) q^{23} +(-\)\(41\!\cdots\!25\)\( - \)\(79\!\cdots\!00\)\( \beta_{1} - \)\(13\!\cdots\!00\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3}) q^{25} +(\)\(12\!\cdots\!08\)\( + \)\(54\!\cdots\!58\)\( \beta_{1} + \)\(21\!\cdots\!96\)\( \beta_{2} + \)\(11\!\cdots\!92\)\( \beta_{3}) q^{26} +(-\)\(19\!\cdots\!00\)\( + \)\(86\!\cdots\!40\)\( \beta_{1} + \)\(21\!\cdots\!64\)\( \beta_{2} - \)\(48\!\cdots\!00\)\( \beta_{3}) q^{28} +(-\)\(16\!\cdots\!30\)\( - \)\(32\!\cdots\!24\)\( \beta_{1} + \)\(60\!\cdots\!12\)\( \beta_{2} - \)\(93\!\cdots\!76\)\( \beta_{3}) q^{29} +(\)\(81\!\cdots\!32\)\( - \)\(32\!\cdots\!40\)\( \beta_{1} + \)\(53\!\cdots\!20\)\( \beta_{2} - \)\(92\!\cdots\!60\)\( \beta_{3}) q^{31} +(-\)\(73\!\cdots\!00\)\( + \)\(27\!\cdots\!36\)\( \beta_{1} - \)\(16\!\cdots\!20\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3}) q^{32} +(\)\(99\!\cdots\!04\)\( + \)\(22\!\cdots\!58\)\( \beta_{1} + \)\(60\!\cdots\!96\)\( \beta_{2} - \)\(38\!\cdots\!08\)\( \beta_{3}) q^{34} +(-\)\(23\!\cdots\!00\)\( + \)\(10\!\cdots\!80\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2} - \)\(57\!\cdots\!60\)\( \beta_{3}) q^{35} +(-\)\(17\!\cdots\!50\)\( - \)\(22\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!68\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{37} +(\)\(16\!\cdots\!00\)\( - \)\(61\!\cdots\!24\)\( \beta_{1} - \)\(92\!\cdots\!48\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3}) q^{38} +(-\)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( \beta_{1} + \)\(33\!\cdots\!00\)\( \beta_{2} - \)\(62\!\cdots\!00\)\( \beta_{3}) q^{40} +(-\)\(40\!\cdots\!42\)\( + \)\(27\!\cdots\!40\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2} + \)\(34\!\cdots\!60\)\( \beta_{3}) q^{41} +(\)\(18\!\cdots\!00\)\( + \)\(70\!\cdots\!16\)\( \beta_{1} + \)\(16\!\cdots\!63\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3}) q^{43} +(-\)\(38\!\cdots\!84\)\( - \)\(67\!\cdots\!28\)\( \beta_{1} + \)\(96\!\cdots\!64\)\( \beta_{2} + \)\(13\!\cdots\!28\)\( \beta_{3}) q^{44} +(-\)\(37\!\cdots\!28\)\( - \)\(52\!\cdots\!20\)\( \beta_{1} - \)\(62\!\cdots\!40\)\( \beta_{2} + \)\(62\!\cdots\!20\)\( \beta_{3}) q^{46} +(\)\(54\!\cdots\!00\)\( + \)\(53\!\cdots\!44\)\( \beta_{1} - \)\(10\!\cdots\!24\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3}) q^{47} +(\)\(38\!\cdots\!57\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} - \)\(56\!\cdots\!40\)\( \beta_{2} + \)\(74\!\cdots\!20\)\( \beta_{3}) q^{49} +(\)\(93\!\cdots\!00\)\( - \)\(40\!\cdots\!25\)\( \beta_{1} + \)\(31\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{50} +(\)\(85\!\cdots\!00\)\( + \)\(29\!\cdots\!92\)\( \beta_{1} + \)\(46\!\cdots\!36\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3}) q^{52} +(-\)\(21\!\cdots\!50\)\( + \)\(18\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!16\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3}) q^{53} +(-\)\(47\!\cdots\!00\)\( - \)\(80\!\cdots\!80\)\( \beta_{1} + \)\(12\!\cdots\!90\)\( \beta_{2} + \)\(16\!\cdots\!60\)\( \beta_{3}) q^{55} +(\)\(22\!\cdots\!80\)\( - \)\(19\!\cdots\!36\)\( \beta_{1} + \)\(24\!\cdots\!68\)\( \beta_{2} + \)\(97\!\cdots\!36\)\( \beta_{3}) q^{56} +(-\)\(10\!\cdots\!00\)\( - \)\(17\!\cdots\!34\)\( \beta_{1} - \)\(76\!\cdots\!68\)\( \beta_{2} + \)\(82\!\cdots\!00\)\( \beta_{3}) q^{58} +(\)\(53\!\cdots\!40\)\( + \)\(31\!\cdots\!12\)\( \beta_{1} + \)\(74\!\cdots\!69\)\( \beta_{2} + \)\(38\!\cdots\!88\)\( \beta_{3}) q^{59} +(\)\(10\!\cdots\!62\)\( + \)\(63\!\cdots\!00\)\( \beta_{1} + \)\(27\!\cdots\!00\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3}) q^{61} +(-\)\(12\!\cdots\!00\)\( - \)\(94\!\cdots\!08\)\( \beta_{1} - \)\(80\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{62} +(\)\(49\!\cdots\!52\)\( - \)\(57\!\cdots\!12\)\( \beta_{1} + \)\(12\!\cdots\!56\)\( \beta_{2} + \)\(25\!\cdots\!12\)\( \beta_{3}) q^{64} +(\)\(10\!\cdots\!00\)\( + \)\(36\!\cdots\!40\)\( \beta_{1} + \)\(55\!\cdots\!80\)\( \beta_{2} + \)\(25\!\cdots\!20\)\( \beta_{3}) q^{65} +(-\)\(37\!\cdots\!00\)\( + \)\(19\!\cdots\!08\)\( \beta_{1} + \)\(65\!\cdots\!13\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3}) q^{67} +(\)\(49\!\cdots\!00\)\( - \)\(18\!\cdots\!72\)\( \beta_{1} + \)\(68\!\cdots\!12\)\( \beta_{2} + \)\(15\!\cdots\!50\)\( \beta_{3}) q^{68} +(\)\(41\!\cdots\!00\)\( - \)\(41\!\cdots\!60\)\( \beta_{1} - \)\(43\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!20\)\( \beta_{3}) q^{70} +(-\)\(66\!\cdots\!72\)\( + \)\(92\!\cdots\!00\)\( \beta_{1} - \)\(79\!\cdots\!50\)\( \beta_{2} - \)\(40\!\cdots\!00\)\( \beta_{3}) q^{71} +(\)\(10\!\cdots\!50\)\( + \)\(79\!\cdots\!00\)\( \beta_{1} + \)\(19\!\cdots\!36\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{73} +(-\)\(68\!\cdots\!64\)\( - \)\(10\!\cdots\!70\)\( \beta_{1} - \)\(35\!\cdots\!40\)\( \beta_{2} - \)\(45\!\cdots\!80\)\( \beta_{3}) q^{74} +(-\)\(22\!\cdots\!60\)\( + \)\(14\!\cdots\!32\)\( \beta_{1} + \)\(64\!\cdots\!84\)\( \beta_{2} - \)\(59\!\cdots\!32\)\( \beta_{3}) q^{76} +(\)\(15\!\cdots\!00\)\( + \)\(14\!\cdots\!76\)\( \beta_{1} - \)\(43\!\cdots\!52\)\( \beta_{2} - \)\(34\!\cdots\!00\)\( \beta_{3}) q^{77} +(-\)\(54\!\cdots\!20\)\( - \)\(32\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3}) q^{79} +(\)\(20\!\cdots\!00\)\( - \)\(22\!\cdots\!60\)\( \beta_{1} - \)\(23\!\cdots\!20\)\( \beta_{2} + \)\(83\!\cdots\!20\)\( \beta_{3}) q^{80} +(\)\(10\!\cdots\!00\)\( + \)\(18\!\cdots\!98\)\( \beta_{1} + \)\(32\!\cdots\!80\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3}) q^{82} +(\)\(48\!\cdots\!00\)\( - \)\(14\!\cdots\!08\)\( \beta_{1} - \)\(72\!\cdots\!37\)\( \beta_{2} + \)\(66\!\cdots\!00\)\( \beta_{3}) q^{83} +(\)\(59\!\cdots\!00\)\( - \)\(22\!\cdots\!80\)\( \beta_{1} + \)\(82\!\cdots\!40\)\( \beta_{2} + \)\(18\!\cdots\!60\)\( \beta_{3}) q^{85} +(\)\(17\!\cdots\!68\)\( + \)\(99\!\cdots\!04\)\( \beta_{1} - \)\(10\!\cdots\!52\)\( \beta_{2} + \)\(29\!\cdots\!96\)\( \beta_{3}) q^{86} +(-\)\(11\!\cdots\!00\)\( + \)\(47\!\cdots\!92\)\( \beta_{1} + \)\(36\!\cdots\!84\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3}) q^{88} +(\)\(15\!\cdots\!10\)\( - \)\(39\!\cdots\!72\)\( \beta_{1} + \)\(22\!\cdots\!36\)\( \beta_{2} - \)\(35\!\cdots\!28\)\( \beta_{3}) q^{89} +(-\)\(11\!\cdots\!88\)\( - \)\(72\!\cdots\!84\)\( \beta_{1} - \)\(10\!\cdots\!08\)\( \beta_{2} - \)\(80\!\cdots\!16\)\( \beta_{3}) q^{91} +(-\)\(40\!\cdots\!00\)\( + \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(16\!\cdots\!88\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3}) q^{92} +(\)\(16\!\cdots\!44\)\( + \)\(17\!\cdots\!96\)\( \beta_{1} - \)\(13\!\cdots\!48\)\( \beta_{2} + \)\(41\!\cdots\!04\)\( \beta_{3}) q^{94} +(-\)\(27\!\cdots\!00\)\( + \)\(17\!\cdots\!00\)\( \beta_{1} + \)\(78\!\cdots\!50\)\( \beta_{2} - \)\(70\!\cdots\!00\)\( \beta_{3}) q^{95} +(-\)\(20\!\cdots\!50\)\( + \)\(47\!\cdots\!56\)\( \beta_{1} - \)\(33\!\cdots\!76\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3}) q^{97} +(-\)\(42\!\cdots\!00\)\( + \)\(25\!\cdots\!37\)\( \beta_{1} + \)\(54\!\cdots\!60\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 1146312000q^{2} + 4402997675828604928q^{4} + 524125996020295237800q^{5} - 63338569463385158773180000q^{7} - 9767868387156674079031296000q^{8} + O(q^{10}) \) \( 4q - 1146312000q^{2} + 4402997675828604928q^{4} + \)\(52\!\cdots\!00\)\(q^{5} - \)\(63\!\cdots\!00\)\(q^{7} - \)\(97\!\cdots\!00\)\(q^{8} - \)\(14\!\cdots\!00\)\(q^{10} + \)\(41\!\cdots\!52\)\(q^{11} + \)\(10\!\cdots\!00\)\(q^{13} + \)\(21\!\cdots\!04\)\(q^{14} + \)\(10\!\cdots\!24\)\(q^{16} + \)\(40\!\cdots\!00\)\(q^{17} - \)\(35\!\cdots\!20\)\(q^{19} + \)\(50\!\cdots\!00\)\(q^{20} - \)\(17\!\cdots\!00\)\(q^{22} + \)\(41\!\cdots\!00\)\(q^{23} - \)\(16\!\cdots\!00\)\(q^{25} + \)\(49\!\cdots\!32\)\(q^{26} - \)\(79\!\cdots\!00\)\(q^{28} - \)\(66\!\cdots\!20\)\(q^{29} + \)\(32\!\cdots\!28\)\(q^{31} - \)\(29\!\cdots\!00\)\(q^{32} + \)\(39\!\cdots\!16\)\(q^{34} - \)\(94\!\cdots\!00\)\(q^{35} - \)\(71\!\cdots\!00\)\(q^{37} + \)\(65\!\cdots\!00\)\(q^{38} - \)\(69\!\cdots\!00\)\(q^{40} - \)\(16\!\cdots\!68\)\(q^{41} + \)\(75\!\cdots\!00\)\(q^{43} - \)\(15\!\cdots\!36\)\(q^{44} - \)\(15\!\cdots\!12\)\(q^{46} + \)\(21\!\cdots\!00\)\(q^{47} + \)\(15\!\cdots\!28\)\(q^{49} + \)\(37\!\cdots\!00\)\(q^{50} + \)\(34\!\cdots\!00\)\(q^{52} - \)\(84\!\cdots\!00\)\(q^{53} - \)\(18\!\cdots\!00\)\(q^{55} + \)\(89\!\cdots\!20\)\(q^{56} - \)\(42\!\cdots\!00\)\(q^{58} + \)\(21\!\cdots\!60\)\(q^{59} + \)\(42\!\cdots\!48\)\(q^{61} - \)\(51\!\cdots\!00\)\(q^{62} + \)\(19\!\cdots\!08\)\(q^{64} + \)\(40\!\cdots\!00\)\(q^{65} - \)\(15\!\cdots\!00\)\(q^{67} + \)\(19\!\cdots\!00\)\(q^{68} + \)\(16\!\cdots\!00\)\(q^{70} - \)\(26\!\cdots\!88\)\(q^{71} + \)\(43\!\cdots\!00\)\(q^{73} - \)\(27\!\cdots\!56\)\(q^{74} - \)\(91\!\cdots\!40\)\(q^{76} + \)\(62\!\cdots\!00\)\(q^{77} - \)\(21\!\cdots\!80\)\(q^{79} + \)\(81\!\cdots\!00\)\(q^{80} + \)\(41\!\cdots\!00\)\(q^{82} + \)\(19\!\cdots\!00\)\(q^{83} + \)\(23\!\cdots\!00\)\(q^{85} + \)\(71\!\cdots\!72\)\(q^{86} - \)\(45\!\cdots\!00\)\(q^{88} + \)\(60\!\cdots\!40\)\(q^{89} - \)\(45\!\cdots\!52\)\(q^{91} - \)\(16\!\cdots\!00\)\(q^{92} + \)\(64\!\cdots\!76\)\(q^{94} - \)\(11\!\cdots\!00\)\(q^{95} - \)\(80\!\cdots\!00\)\(q^{97} - \)\(17\!\cdots\!00\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 192 \nu - 48 \)
\(\beta_{2}\)\(=\)\((\)\( 108 \nu^{3} + 186256044 \nu^{2} - 15885966350153172 \nu - 3340459070129947691688 \)\()/13402613\)
\(\beta_{3}\)\(=\)\((\)\( -4104 \nu^{3} + 486996195960 \nu^{2} + 1286287826053639416 \nu - 44429586960648335446356720 \)\()/13402613\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 48\)\()/192\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 38 \beta_{2} - 265270530 \beta_{1} + 3324465478086847488\)\()/36864\)
\(\nu^{3}\)\(=\)\((\)\(-5173779 \beta_{3} + 13527672110 \beta_{2} + 86097604964916454 \beta_{1} - 13779415522130239634116608\)\()/110592\)

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
−1.33086e7
−3.61406e6
4.76281e6
1.21599e7
−2.84183e9 0 5.77017e18 6.89620e20 0 −9.80127e25 −9.84504e27 0 −1.95979e30
1.2 −9.80478e8 0 −1.34451e18 −1.59503e20 0 1.63507e25 3.57909e27 0 1.56389e29
1.3 6.27881e8 0 −1.91161e18 −2.33985e20 0 6.25792e25 −2.64806e27 0 −1.46915e29
1.4 2.04812e9 0 1.88894e18 2.27994e20 0 −4.42559e25 −8.53860e26 0 4.66958e29
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.62.a.a 4
3.b odd 2 1 1.62.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.62.a.a 4 3.b odd 2 1
9.62.a.a 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1146312000 T_{2}^{3} - \)\(61\!\cdots\!68\)\( T_{2}^{2} - \)\(25\!\cdots\!00\)\( T_{2} + \)\(35\!\cdots\!56\)\( \) acting on \(S_{62}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 1146312000 T + 3067202781185085440 T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!08\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!16\)\( T^{8} \)
$3$ 1
$5$ \( 1 - \)\(52\!\cdots\!00\)\( T + \)\(17\!\cdots\!00\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!00\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + \)\(63\!\cdots\!00\)\( T + \)\(83\!\cdots\!00\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!98\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(41\!\cdots\!52\)\( T + \)\(70\!\cdots\!08\)\( T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(81\!\cdots\!44\)\( T^{5} + \)\(78\!\cdots\!68\)\( T^{6} - \)\(15\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \)
$13$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(17\!\cdots\!60\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!40\)\( T^{6} - \)\(76\!\cdots\!00\)\( T^{7} + \)\(63\!\cdots\!61\)\( T^{8} \)
$17$ \( 1 - \)\(40\!\cdots\!00\)\( T + \)\(40\!\cdots\!20\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!78\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!80\)\( T^{6} - \)\(60\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} \)
$19$ \( 1 + \)\(35\!\cdots\!20\)\( T + \)\(16\!\cdots\!76\)\( T^{2} + \)\(29\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!66\)\( T^{4} + \)\(29\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!36\)\( T^{6} + \)\(36\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} \)
$23$ \( 1 - \)\(41\!\cdots\!00\)\( T + \)\(43\!\cdots\!40\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!58\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(58\!\cdots\!60\)\( T^{6} - \)\(64\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \)
$29$ \( 1 + \)\(66\!\cdots\!20\)\( T + \)\(43\!\cdots\!16\)\( T^{2} - \)\(46\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!46\)\( T^{4} - \)\(74\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(27\!\cdots\!80\)\( T^{7} + \)\(66\!\cdots\!81\)\( T^{8} \)
$31$ \( 1 - \)\(32\!\cdots\!28\)\( T + \)\(28\!\cdots\!68\)\( T^{2} - \)\(75\!\cdots\!76\)\( T^{3} + \)\(38\!\cdots\!70\)\( T^{4} - \)\(70\!\cdots\!56\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} - \)\(27\!\cdots\!48\)\( T^{7} + \)\(78\!\cdots\!21\)\( T^{8} \)
$37$ \( 1 + \)\(71\!\cdots\!00\)\( T + \)\(98\!\cdots\!60\)\( T^{2} + \)\(68\!\cdots\!00\)\( T^{3} + \)\(69\!\cdots\!38\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} + \)\(67\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!61\)\( T^{8} \)
$41$ \( 1 + \)\(16\!\cdots\!68\)\( T + \)\(80\!\cdots\!48\)\( T^{2} + \)\(85\!\cdots\!16\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!88\)\( T^{6} + \)\(22\!\cdots\!28\)\( T^{7} + \)\(33\!\cdots\!61\)\( T^{8} \)
$43$ \( 1 - \)\(75\!\cdots\!00\)\( T + \)\(13\!\cdots\!00\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!98\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(63\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(21\!\cdots\!00\)\( T + \)\(36\!\cdots\!80\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!18\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(98\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 + \)\(84\!\cdots\!00\)\( T + \)\(55\!\cdots\!80\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(89\!\cdots\!18\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!81\)\( T^{8} \)
$59$ \( 1 - \)\(21\!\cdots\!60\)\( T + \)\(37\!\cdots\!36\)\( T^{2} - \)\(50\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!86\)\( T^{4} - \)\(53\!\cdots\!80\)\( T^{5} + \)\(41\!\cdots\!16\)\( T^{6} - \)\(24\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$61$ \( 1 - \)\(42\!\cdots\!48\)\( T + \)\(31\!\cdots\!08\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!70\)\( T^{4} - \)\(81\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!68\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(41\!\cdots\!41\)\( T^{8} \)
$67$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(14\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!78\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{5} + \)\(89\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!21\)\( T^{8} \)
$71$ \( 1 + \)\(26\!\cdots\!88\)\( T + \)\(15\!\cdots\!88\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(64\!\cdots\!70\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!81\)\( T^{8} \)
$73$ \( 1 - \)\(43\!\cdots\!00\)\( T + \)\(13\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!60\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} \)
$79$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(11\!\cdots\!16\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} + \)\(38\!\cdots\!56\)\( T^{6} + \)\(40\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!81\)\( T^{8} \)
$83$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(16\!\cdots\!20\)\( T^{2} - \)\(86\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!78\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} - \)\(29\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!21\)\( T^{8} \)
$89$ \( 1 - \)\(60\!\cdots\!40\)\( T + \)\(42\!\cdots\!56\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!26\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!76\)\( T^{6} - \)\(33\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} \)
$97$ \( 1 + \)\(80\!\cdots\!00\)\( T + \)\(80\!\cdots\!80\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!18\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{7} + \)\(59\!\cdots\!81\)\( T^{8} \)
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