Properties

Label 272.10.a.f
Level $272$
Weight $10$
Character orbit 272.a
Self dual yes
Analytic conductor $140.090$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(140.089747437\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 47 - \beta_{3} ) q^{3} + ( 295 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{5} + ( 2629 + 12 \beta_{1} - 19 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} ) q^{7} + ( 2180 - 54 \beta_{1} - 48 \beta_{2} - 39 \beta_{3} + 25 \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 47 - \beta_{3} ) q^{3} + ( 295 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{5} + ( 2629 + 12 \beta_{1} - 19 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} ) q^{7} + ( 2180 - 54 \beta_{1} - 48 \beta_{2} - 39 \beta_{3} + 25 \beta_{4} ) q^{9} + ( 13658 - 31 \beta_{1} + 30 \beta_{2} + 163 \beta_{3} - 24 \beta_{4} ) q^{11} + ( -31902 + 127 \beta_{1} + 62 \beta_{2} - 645 \beta_{3} + 99 \beta_{4} ) q^{13} + ( 137405 - 471 \beta_{1} - 576 \beta_{2} + 382 \beta_{3} - 28 \beta_{4} ) q^{15} -83521 q^{17} + ( 74186 - 348 \beta_{1} + 682 \beta_{2} - 1774 \beta_{3} + 916 \beta_{4} ) q^{19} + ( 355842 - 387 \beta_{1} - 1062 \beta_{2} - 2933 \beta_{3} - 377 \beta_{4} ) q^{21} + ( -329310 + 221 \beta_{1} - 2617 \beta_{2} + 4775 \beta_{3} - 2450 \beta_{4} ) q^{23} + ( 651223 + 4974 \beta_{1} - 4984 \beta_{2} + 818 \beta_{3} - 2466 \beta_{4} ) q^{25} + ( 599857 - 12825 \beta_{1} - 4380 \beta_{2} - 3048 \beta_{3} + 2576 \beta_{4} ) q^{27} + ( 730621 + 3986 \beta_{1} - 7730 \beta_{2} + 3935 \beta_{3} + 1247 \beta_{4} ) q^{29} + ( 1449686 - 4909 \beta_{1} + 5051 \beta_{2} - 28943 \beta_{3} - 5590 \beta_{4} ) q^{31} + ( -2266941 + 8346 \beta_{1} + 5580 \beta_{2} - 2619 \beta_{3} - 4383 \beta_{4} ) q^{33} + ( 5280999 + 21639 \beta_{1} - 29780 \beta_{2} - 17794 \beta_{3} - 20616 \beta_{4} ) q^{35} + ( -6281293 + 2012 \beta_{1} + 12930 \beta_{2} - 9605 \beta_{3} - 2969 \beta_{4} ) q^{37} + ( 8491683 - 18003 \beta_{1} - 13944 \beta_{2} + 18416 \beta_{3} + 18524 \beta_{4} ) q^{39} + ( -1614442 - 33694 \beta_{1} - 50180 \beta_{2} - 8606 \beta_{3} - 32586 \beta_{4} ) q^{41} + ( 11384145 - 14091 \beta_{1} + 57650 \beta_{2} - 116934 \beta_{3} - 6592 \beta_{4} ) q^{43} + ( 2531451 - 39600 \beta_{1} + 8550 \beta_{2} - 198981 \beta_{3} + 52431 \beta_{4} ) q^{45} + ( 3356661 + 85583 \beta_{1} - 53770 \beta_{2} + 73092 \beta_{3} - 51044 \beta_{4} ) q^{47} + ( -2401070 + 48006 \beta_{1} - 88004 \beta_{2} + 2723 \beta_{3} - 104825 \beta_{4} ) q^{49} + ( -3925487 + 83521 \beta_{3} ) q^{51} + ( -16562618 - 59432 \beta_{1} + 176424 \beta_{2} + 137612 \beta_{3} - 82392 \beta_{4} ) q^{53} + ( -1268447 + 46561 \beta_{1} + 56656 \beta_{2} - 45622 \beta_{3} - 64656 \beta_{4} ) q^{55} + ( 27284960 - 104958 \beta_{1} + 3732 \beta_{2} - 303526 \beta_{3} + 91566 \beta_{4} ) q^{57} + ( 7395563 + 319435 \beta_{1} + 87518 \beta_{2} - 824390 \beta_{3} - 14348 \beta_{4} ) q^{59} + ( -15559405 + 31672 \beta_{1} + 347534 \beta_{2} - 774969 \beta_{3} + 95351 \beta_{4} ) q^{61} + ( 38494982 - 514539 \beta_{1} + 138345 \beta_{2} - 261597 \beta_{3} + 175786 \beta_{4} ) q^{63} + ( -7963902 - 382034 \beta_{1} - 183484 \beta_{2} + 486712 \beta_{3} + 196748 \beta_{4} ) q^{65} + ( 61107664 - 302124 \beta_{1} + 277242 \beta_{2} + 178112 \beta_{3} + 173184 \beta_{4} ) q^{67} + ( -66587132 + 137217 \beta_{1} - 75906 \beta_{2} + 762641 \beta_{3} - 239063 \beta_{4} ) q^{69} + ( 95329113 - 476330 \beta_{1} - 532303 \beta_{2} + 630919 \beta_{3} + 40694 \beta_{4} ) q^{71} + ( -58429074 + 346480 \beta_{1} - 225852 \beta_{2} - 1366700 \beta_{3} + 326100 \beta_{4} ) q^{73} + ( 30564615 + 274086 \beta_{1} - 138456 \beta_{2} - 311699 \beta_{3} - 231044 \beta_{4} ) q^{75} + ( -28932808 + 604647 \beta_{1} - 439734 \beta_{2} + 205723 \beta_{3} - 120901 \beta_{4} ) q^{77} + ( 165338790 + 326109 \beta_{1} - 150393 \beta_{2} - 919765 \beta_{3} + 520718 \beta_{4} ) q^{79} + ( 178091966 - 988092 \beta_{1} + 215304 \beta_{2} - 2287479 \beta_{3} - 131339 \beta_{4} ) q^{81} + ( -39609033 + 1233919 \beta_{1} - 589866 \beta_{2} - 950366 \beta_{3} - 493576 \beta_{4} ) q^{83} + ( -24638695 + 167042 \beta_{1} + 167042 \beta_{2} + 250563 \beta_{3} + 250563 \beta_{4} ) q^{85} + ( -32034693 + 7095 \beta_{1} + 160296 \beta_{2} - 2337078 \beta_{3} - 157404 \beta_{4} ) q^{87} + ( 75436745 + 2195578 \beta_{1} + 863884 \beta_{2} + 803429 \beta_{3} + 403201 \beta_{4} ) q^{89} + ( -38268469 - 1591887 \beta_{1} + 1303718 \beta_{2} - 998004 \beta_{3} + 623796 \beta_{4} ) q^{91} + ( 698817990 - 1562487 \beta_{1} - 1882314 \beta_{2} + 1456171 \beta_{3} + 603235 \beta_{4} ) q^{93} + ( -298979150 - 3155274 \beta_{1} + 429920 \beta_{2} - 231000 \beta_{3} + 1117660 \beta_{4} ) q^{95} + ( 136135752 + 5517586 \beta_{1} - 531448 \beta_{2} - 4776004 \beta_{3} - 767952 \beta_{4} ) q^{97} + ( -405563595 + 1990512 \beta_{1} - 449406 \beta_{2} + 2177757 \beta_{3} + 258108 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + O(q^{10}) \) \( 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + 68036q^{11} - 158862q^{13} + 687324q^{15} - 417605q^{17} + 370992q^{19} + 1783880q^{21} - 1645870q^{23} + 3270239q^{25} + 2998268q^{27} + 3668616q^{29} + 7262362q^{31} - 11334900q^{33} + 26503988q^{35} - 31420708q^{37} + 42449884q^{39} - 7996938q^{41} + 56908268q^{43} + 12799536q^{45} + 16903336q^{47} - 11784059q^{49} - 19710956q^{51} - 83362982q^{53} - 6363364q^{55} + 136615904q^{57} + 37946604q^{59} - 77685452q^{61} + 191945278q^{63} - 40321288q^{65} + 304503600q^{67} - 333409272q^{69} + 476602922q^{71} - 289980486q^{73} + 153685772q^{75} - 143385648q^{77} + 828240610q^{79} + 891328609q^{81} - 194681148q^{83} - 123611080q^{85} - 158149884q^{87} + 376848106q^{89} - 194543664q^{91} + 3494835920q^{93} - 1498679864q^{95} + 692035246q^{97} - 2027106408q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 8 \nu - 3 \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{4} + 207 \nu^{3} + 6301 \nu^{2} - 192831 \nu - 517832 \)\()/8096\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{4} + 115 \nu^{3} - 13679 \nu^{2} - 91523 \nu + 4588376 \)\()/16192\)
\(\beta_{4}\)\(=\)\((\)\( 59 \nu^{4} + 391 \nu^{3} - 75971 \nu^{2} - 352151 \nu + 12509496 \)\()/16192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 3\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{4} - 28 \beta_{3} + 4 \beta_{2} + 3 \beta_{1} + 5109\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(48 \beta_{4} - 16 \beta_{3} + 272 \beta_{2} + 929 \beta_{1} - 12365\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(7028 \beta_{4} - 35948 \beta_{3} + 3348 \beta_{2} + 3675 \beta_{1} + 4982221\)\()/8\)

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
5.77274
−21.1654
18.8209
33.6330
−35.0613
0 −177.437 0 −1620.18 0 1834.42 0 11801.0 0
1.2 0 3.02373 0 762.851 0 −5573.11 0 −19673.9 0
1.3 0 67.6654 0 2390.67 0 11355.8 0 −15104.4 0
1.4 0 85.9747 0 −1460.58 0 446.232 0 −12291.3 0
1.5 0 256.773 0 1407.25 0 5138.64 0 46249.6 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 272.10.a.f 5
4.b odd 2 1 17.10.a.a 5
12.b even 2 1 153.10.a.c 5
68.d odd 2 1 289.10.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.a 5 4.b odd 2 1
153.10.a.c 5 12.b even 2 1
272.10.a.f 5 1.a even 1 1 trivial
289.10.a.a 5 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 236 T_{3}^{4} - 26850 T_{3}^{3} + 6621804 T_{3}^{2} - 284823432 T_{3} + 801447696 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(272))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 801447696 - 284823432 T + 6621804 T^{2} - 26850 T^{3} - 236 T^{4} + T^{5} \)
$5$ \( -6073215799787520 + 6910711315456 T + 5931144800 T^{2} - 5422732 T^{3} - 1480 T^{4} + T^{5} \)
$7$ \( 266210160692281344 - 769174397227488 T + 392836554656 T^{2} - 7845586 T^{3} - 13202 T^{4} + T^{5} \)
$11$ \( -\)\(18\!\cdots\!36\)\( - 583150347803390760 T + 51654458131068 T^{2} + 161240150 T^{3} - 68036 T^{4} + T^{5} \)
$13$ \( \)\(46\!\cdots\!08\)\( + 15605144350350783296 T - 1715294166968552 T^{2} - 8325650340 T^{3} + 158862 T^{4} + T^{5} \)
$17$ \( ( 83521 + T )^{5} \)
$19$ \( \)\(23\!\cdots\!16\)\( - \)\(17\!\cdots\!04\)\( T + 176324676443571104 T^{2} - 413150145000 T^{3} - 370992 T^{4} + T^{5} \)
$23$ \( -\)\(28\!\cdots\!00\)\( - \)\(13\!\cdots\!96\)\( T - 4397519995817681960 T^{2} - 2266461594322 T^{3} + 1645870 T^{4} + T^{5} \)
$29$ \( \)\(14\!\cdots\!00\)\( - \)\(47\!\cdots\!52\)\( T + 43135444415107067232 T^{2} - 8671259026316 T^{3} - 3668616 T^{4} + T^{5} \)
$31$ \( \)\(63\!\cdots\!72\)\( - \)\(15\!\cdots\!16\)\( T + \)\(61\!\cdots\!12\)\( T^{2} - 55088066425306 T^{3} - 7262362 T^{4} + T^{5} \)
$37$ \( \)\(12\!\cdots\!48\)\( + \)\(31\!\cdots\!60\)\( T + \)\(16\!\cdots\!40\)\( T^{2} + 352949672725220 T^{3} + 31420708 T^{4} + T^{5} \)
$41$ \( \)\(84\!\cdots\!52\)\( + \)\(16\!\cdots\!32\)\( T - \)\(34\!\cdots\!32\)\( T^{2} - 944595954515528 T^{3} + 7996938 T^{4} + T^{5} \)
$43$ \( \)\(35\!\cdots\!88\)\( - \)\(76\!\cdots\!72\)\( T + \)\(42\!\cdots\!64\)\( T^{2} + 104989881324440 T^{3} - 56908268 T^{4} + T^{5} \)
$47$ \( \)\(53\!\cdots\!68\)\( + \)\(13\!\cdots\!68\)\( T - \)\(13\!\cdots\!84\)\( T^{2} - 2082901585389616 T^{3} - 16903336 T^{4} + T^{5} \)
$53$ \( \)\(46\!\cdots\!44\)\( - \)\(15\!\cdots\!36\)\( T - \)\(92\!\cdots\!16\)\( T^{2} - 9379424557396600 T^{3} + 83362982 T^{4} + T^{5} \)
$59$ \( -\)\(43\!\cdots\!28\)\( + \)\(40\!\cdots\!72\)\( T + \)\(23\!\cdots\!92\)\( T^{2} - 31008104213596424 T^{3} - 37946604 T^{4} + T^{5} \)
$61$ \( -\)\(81\!\cdots\!20\)\( + \)\(24\!\cdots\!92\)\( T - \)\(36\!\cdots\!32\)\( T^{2} - 33463527209865196 T^{3} + 77685452 T^{4} + T^{5} \)
$67$ \( \)\(17\!\cdots\!00\)\( - \)\(58\!\cdots\!40\)\( T + \)\(55\!\cdots\!76\)\( T^{2} + 7020713481287504 T^{3} - 304503600 T^{4} + T^{5} \)
$71$ \( -\)\(27\!\cdots\!64\)\( - \)\(34\!\cdots\!28\)\( T + \)\(14\!\cdots\!84\)\( T^{2} + 9603673423875238 T^{3} - 476602922 T^{4} + T^{5} \)
$73$ \( \)\(39\!\cdots\!92\)\( + \)\(18\!\cdots\!28\)\( T - \)\(24\!\cdots\!16\)\( T^{2} - 108975991091496536 T^{3} + 289980486 T^{4} + T^{5} \)
$79$ \( -\)\(88\!\cdots\!72\)\( - \)\(60\!\cdots\!44\)\( T + \)\(47\!\cdots\!40\)\( T^{2} + 104201235865665870 T^{3} - 828240610 T^{4} + T^{5} \)
$83$ \( \)\(48\!\cdots\!16\)\( - \)\(32\!\cdots\!40\)\( T - \)\(93\!\cdots\!08\)\( T^{2} - 285056400431908168 T^{3} + 194681148 T^{4} + T^{5} \)
$89$ \( \)\(26\!\cdots\!96\)\( + \)\(21\!\cdots\!28\)\( T + \)\(27\!\cdots\!24\)\( T^{2} - 834395900612571220 T^{3} - 376848106 T^{4} + T^{5} \)
$97$ \( \)\(10\!\cdots\!28\)\( + \)\(30\!\cdots\!16\)\( T + \)\(49\!\cdots\!44\)\( T^{2} - 3215802378794508088 T^{3} - 692035246 T^{4} + T^{5} \)
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