Properties

Label 38.10.a.c
Level 38
Weight 10
Character orbit 38.a
Self dual yes
Analytic conductor 19.571
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 4552 x + 85948\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -16 q^{2} + ( 1 + \beta_{1} ) q^{3} + 256 q^{4} + ( 162 + 2 \beta_{1} + \beta_{2} ) q^{5} + ( -16 - 16 \beta_{1} ) q^{6} + ( -4439 - 14 \beta_{1} - 10 \beta_{2} ) q^{7} -4096 q^{8} + ( 7632 - 72 \beta_{1} + 27 \beta_{2} ) q^{9} +O(q^{10})\) \( q -16 q^{2} + ( 1 + \beta_{1} ) q^{3} + 256 q^{4} + ( 162 + 2 \beta_{1} + \beta_{2} ) q^{5} + ( -16 - 16 \beta_{1} ) q^{6} + ( -4439 - 14 \beta_{1} - 10 \beta_{2} ) q^{7} -4096 q^{8} + ( 7632 - 72 \beta_{1} + 27 \beta_{2} ) q^{9} + ( -2592 - 32 \beta_{1} - 16 \beta_{2} ) q^{10} + ( 18656 - 386 \beta_{1} - 43 \beta_{2} ) q^{11} + ( 256 + 256 \beta_{1} ) q^{12} + ( 52727 - 204 \beta_{1} + 67 \beta_{2} ) q^{13} + ( 71024 + 224 \beta_{1} + 160 \beta_{2} ) q^{14} + ( 45220 + 319 \beta_{1} + 129 \beta_{2} ) q^{15} + 65536 q^{16} + ( -209697 + 858 \beta_{1} - 564 \beta_{2} ) q^{17} + ( -122112 + 1152 \beta_{1} - 432 \beta_{2} ) q^{18} -130321 q^{19} + ( 41472 + 512 \beta_{1} + 256 \beta_{2} ) q^{20} + ( -291135 - 6447 \beta_{1} - 1128 \beta_{2} ) q^{21} + ( -298496 + 6176 \beta_{1} + 688 \beta_{2} ) q^{22} + ( -308209 + 5110 \beta_{1} + 3289 \beta_{2} ) q^{23} + ( -4096 - 4096 \beta_{1} ) q^{24} + ( -1601839 + 2040 \beta_{1} + 425 \beta_{2} ) q^{25} + ( -843632 + 3264 \beta_{1} - 1072 \beta_{2} ) q^{26} + ( -2237049 + 1386 \beta_{1} + 81 \beta_{2} ) q^{27} + ( -1136384 - 3584 \beta_{1} - 2560 \beta_{2} ) q^{28} + ( -3279673 - 13893 \beta_{1} - 3628 \beta_{2} ) q^{29} + ( -723520 - 5104 \beta_{1} - 2064 \beta_{2} ) q^{30} + ( 454876 - 29685 \beta_{1} + 8839 \beta_{2} ) q^{31} -1048576 q^{32} + ( -10113038 + 33805 \beta_{1} - 13647 \beta_{2} ) q^{33} + ( 3355152 - 13728 \beta_{1} + 9024 \beta_{2} ) q^{34} + ( -3699190 - 24136 \beta_{1} - 6301 \beta_{2} ) q^{35} + ( 1953792 - 18432 \beta_{1} + 6912 \beta_{2} ) q^{36} + ( -763030 + 3757 \beta_{1} + 14761 \beta_{2} ) q^{37} + 2085136 q^{38} + ( -6160519 + 87920 \beta_{1} - 483 \beta_{2} ) q^{39} + ( -663552 - 8192 \beta_{1} - 4096 \beta_{2} ) q^{40} + ( -4299860 + 13925 \beta_{1} - 15333 \beta_{2} ) q^{41} + ( 4658160 + 103152 \beta_{1} + 18048 \beta_{2} ) q^{42} + ( -7459546 + 106246 \beta_{1} - 9441 \beta_{2} ) q^{43} + ( 4775936 - 98816 \beta_{1} - 11008 \beta_{2} ) q^{44} + ( 4335210 + 21654 \beta_{1} - 1395 \beta_{2} ) q^{45} + ( 4931344 - 81760 \beta_{1} - 52624 \beta_{2} ) q^{46} + ( 19632122 + 15016 \beta_{1} + 44215 \beta_{2} ) q^{47} + ( 65536 + 65536 \beta_{1} ) q^{48} + ( 7431658 + 242228 \beta_{1} + 84492 \beta_{2} ) q^{49} + ( 25629424 - 32640 \beta_{1} - 6800 \beta_{2} ) q^{50} + ( 28623195 - 443223 \beta_{1} - 19134 \beta_{2} ) q^{51} + ( 13498112 - 52224 \beta_{1} + 17152 \beta_{2} ) q^{52} + ( 2923543 - 299391 \beta_{1} - 106356 \beta_{2} ) q^{53} + ( 35792784 - 22176 \beta_{1} - 1296 \beta_{2} ) q^{54} + ( -24471934 - 131576 \beta_{1} - 31053 \beta_{2} ) q^{55} + ( 18182144 + 57344 \beta_{1} + 40960 \beta_{2} ) q^{56} + ( -130321 - 130321 \beta_{1} ) q^{57} + ( 52474768 + 222288 \beta_{1} + 58048 \beta_{2} ) q^{58} + ( 5475433 - 71558 \beta_{1} + 55609 \beta_{2} ) q^{59} + ( 11576320 + 81664 \beta_{1} + 33024 \beta_{2} ) q^{60} + ( -42281260 + 143100 \beta_{1} - 21479 \beta_{2} ) q^{61} + ( -7278016 + 474960 \beta_{1} - 141424 \beta_{2} ) q^{62} + ( -78216696 + 113274 \beta_{1} - 61839 \beta_{2} ) q^{63} + 16777216 q^{64} + ( 15089984 + 113280 \beta_{1} + 27084 \beta_{2} ) q^{65} + ( 161808608 - 540880 \beta_{1} + 218352 \beta_{2} ) q^{66} + ( -96025103 - 461027 \beta_{1} - 162254 \beta_{2} ) q^{67} + ( -53682432 + 219648 \beta_{1} - 144384 \beta_{2} ) q^{68} + ( 107790601 + 315328 \beta_{1} + 384645 \beta_{2} ) q^{69} + ( 59187040 + 386176 \beta_{1} + 100816 \beta_{2} ) q^{70} + ( 26040758 + 583150 \beta_{1} + 105562 \beta_{2} ) q^{71} + ( -31260672 + 294912 \beta_{1} - 110592 \beta_{2} ) q^{72} + ( -185980615 + 492830 \beta_{1} - 464234 \beta_{2} ) q^{73} + ( 12208480 - 60112 \beta_{1} - 236176 \beta_{2} ) q^{74} + ( 50051471 - 1621984 \beta_{1} + 86955 \beta_{2} ) q^{75} -33362176 q^{76} + ( 131337912 + 2609040 \beta_{1} + 350123 \beta_{2} ) q^{77} + ( 98568304 - 1406720 \beta_{1} + 7728 \beta_{2} ) q^{78} + ( -106740674 + 517415 \beta_{1} + 65849 \beta_{2} ) q^{79} + ( 10616832 + 131072 \beta_{1} + 65536 \beta_{2} ) q^{80} + ( -115375671 - 896508 \beta_{1} - 487944 \beta_{2} ) q^{81} + ( 68797760 - 222800 \beta_{1} + 245328 \beta_{2} ) q^{82} + ( 143497154 - 44200 \beta_{1} + 330414 \beta_{2} ) q^{83} + ( -74530560 - 1650432 \beta_{1} - 288768 \beta_{2} ) q^{84} + ( -127809414 - 757656 \beta_{1} - 103821 \beta_{2} ) q^{85} + ( 119352736 - 1699936 \beta_{1} + 151056 \beta_{2} ) q^{86} + ( -348033115 - 3364768 \beta_{1} - 647211 \beta_{2} ) q^{87} + ( -76414976 + 1581056 \beta_{1} + 176128 \beta_{2} ) q^{88} + ( 145896548 + 2108807 \beta_{1} + 726611 \beta_{2} ) q^{89} + ( -69363360 - 346464 \beta_{1} + 22320 \beta_{2} ) q^{90} + ( -336816729 - 28980 \beta_{1} - 463013 \beta_{2} ) q^{91} + ( -78901504 + 1308160 \beta_{1} + 841984 \beta_{2} ) q^{92} + ( -894950444 + 5300098 \beta_{1} - 138570 \beta_{2} ) q^{93} + ( -314113952 - 240256 \beta_{1} - 707440 \beta_{2} ) q^{94} + ( -21112002 - 260642 \beta_{1} - 130321 \beta_{2} ) q^{95} + ( -1048576 - 1048576 \beta_{1} ) q^{96} + ( -128317382 + 2792874 \beta_{1} + 1872892 \beta_{2} ) q^{97} + ( -118906528 - 3875648 \beta_{1} - 1351872 \beta_{2} ) q^{98} + ( 676632474 - 9118206 \beta_{1} + 735579 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 48q^{2} + 3q^{3} + 768q^{4} + 486q^{5} - 48q^{6} - 13317q^{7} - 12288q^{8} + 22896q^{9} + O(q^{10}) \) \( 3q - 48q^{2} + 3q^{3} + 768q^{4} + 486q^{5} - 48q^{6} - 13317q^{7} - 12288q^{8} + 22896q^{9} - 7776q^{10} + 55968q^{11} + 768q^{12} + 158181q^{13} + 213072q^{14} + 135660q^{15} + 196608q^{16} - 629091q^{17} - 366336q^{18} - 390963q^{19} + 124416q^{20} - 873405q^{21} - 895488q^{22} - 924627q^{23} - 12288q^{24} - 4805517q^{25} - 2530896q^{26} - 6711147q^{27} - 3409152q^{28} - 9839019q^{29} - 2170560q^{30} + 1364628q^{31} - 3145728q^{32} - 30339114q^{33} + 10065456q^{34} - 11097570q^{35} + 5861376q^{36} - 2289090q^{37} + 6255408q^{38} - 18481557q^{39} - 1990656q^{40} - 12899580q^{41} + 13974480q^{42} - 22378638q^{43} + 14327808q^{44} + 13005630q^{45} + 14794032q^{46} + 58896366q^{47} + 196608q^{48} + 22294974q^{49} + 76888272q^{50} + 85869585q^{51} + 40494336q^{52} + 8770629q^{53} + 107378352q^{54} - 73415802q^{55} + 54546432q^{56} - 390963q^{57} + 157424304q^{58} + 16426299q^{59} + 34728960q^{60} - 126843780q^{61} - 21834048q^{62} - 234650088q^{63} + 50331648q^{64} + 45269952q^{65} + 485425824q^{66} - 288075309q^{67} - 161047296q^{68} + 323371803q^{69} + 177561120q^{70} + 78122274q^{71} - 93782016q^{72} - 557941845q^{73} + 36625440q^{74} + 150154413q^{75} - 100086528q^{76} + 394013736q^{77} + 295704912q^{78} - 320222022q^{79} + 31850496q^{80} - 346127013q^{81} + 206393280q^{82} + 430491462q^{83} - 223591680q^{84} - 383428242q^{85} + 358058208q^{86} - 1044099345q^{87} - 229244928q^{88} + 437689644q^{89} - 208090080q^{90} - 1010450187q^{91} - 236704512q^{92} - 2684851332q^{93} - 942341856q^{94} - 63336006q^{95} - 3145728q^{96} - 384952146q^{97} - 356719584q^{98} + 2029897422q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4552 x + 85948\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 24 \nu - 3043 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\(3 \beta_{2} - 8 \beta_{1} + 3035\)

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
−74.9878
20.7493
55.2385
−16.0000 −224.963 256.000 −29.7721 3599.41 −3877.06 −4096.00 30925.5 476.353
1.2 −16.0000 62.2478 256.000 −420.333 −995.965 1751.81 −4096.00 −15808.2 6725.32
1.3 −16.0000 165.716 256.000 936.105 −2651.45 −11191.8 −4096.00 7778.66 −14977.7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(1\)

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 38.10.a.c 3
3.b odd 2 1 342.10.a.e 3
4.b odd 2 1 304.10.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.c 3 1.a even 1 1 trivial
304.10.a.c 3 4.b odd 2 1
342.10.a.e 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 3 T_{3}^{2} - 40968 T_{3} + 2320596 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 16 T )^{3} \)
$3$ \( 1 - 3 T + 18081 T^{2} + 2202498 T^{3} + 355888323 T^{4} - 1162261467 T^{5} + 7625597484987 T^{6} \)
$5$ \( 1 - 486 T + 5450544 T^{2} - 1910152084 T^{3} + 10645593750000 T^{4} - 1853942871093750 T^{5} + 7450580596923828125 T^{6} \)
$7$ \( 1 + 13317 T + 138054168 T^{2} + 998764858749 T^{3} + 5570983640183976 T^{4} + 21685583883373449333 T^{5} + \)\(65\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 - 55968 T + 1785343494 T^{2} - 110373793234486 T^{3} + 4209746569319172354 T^{4} - \)\(31\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 - 158181 T + 36345696879 T^{2} - 3315103030674770 T^{3} + \)\(38\!\cdots\!67\)\( T^{4} - \)\(17\!\cdots\!49\)\( T^{5} + \)\(11\!\cdots\!17\)\( T^{6} \)
$17$ \( 1 + 629091 T + 322401568650 T^{2} + 106985872063124811 T^{3} + \)\(38\!\cdots\!50\)\( T^{4} + \)\(88\!\cdots\!19\)\( T^{5} + \)\(16\!\cdots\!73\)\( T^{6} \)
$19$ \( ( 1 + 130321 T )^{3} \)
$23$ \( 1 + 924627 T + 978574003653 T^{2} + 526819541184348842 T^{3} + \)\(17\!\cdots\!39\)\( T^{4} + \)\(29\!\cdots\!63\)\( T^{5} + \)\(58\!\cdots\!47\)\( T^{6} \)
$29$ \( 1 + 9839019 T + 64313088760239 T^{2} + \)\(29\!\cdots\!62\)\( T^{3} + \)\(93\!\cdots\!91\)\( T^{4} + \)\(20\!\cdots\!59\)\( T^{5} + \)\(30\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 - 1364628 T + 6528423117837 T^{2} - \)\(10\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(95\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 + 2289090 T + 309609164810043 T^{2} + \)\(75\!\cdots\!08\)\( T^{3} + \)\(40\!\cdots\!11\)\( T^{4} + \)\(38\!\cdots\!10\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \)
$41$ \( 1 + 12899580 T + 933941039944047 T^{2} + \)\(77\!\cdots\!84\)\( T^{3} + \)\(30\!\cdots\!67\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{5} + \)\(35\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + 22378638 T + 1149456564786162 T^{2} + \)\(23\!\cdots\!16\)\( T^{3} + \)\(57\!\cdots\!66\)\( T^{4} + \)\(56\!\cdots\!62\)\( T^{5} + \)\(12\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - 58896366 T + 3778439737379502 T^{2} - \)\(11\!\cdots\!64\)\( T^{3} + \)\(42\!\cdots\!34\)\( T^{4} - \)\(73\!\cdots\!74\)\( T^{5} + \)\(14\!\cdots\!63\)\( T^{6} \)
$53$ \( 1 - 8770629 T + 2855843385985791 T^{2} + \)\(18\!\cdots\!78\)\( T^{3} + \)\(94\!\cdots\!03\)\( T^{4} - \)\(95\!\cdots\!81\)\( T^{5} + \)\(35\!\cdots\!37\)\( T^{6} \)
$59$ \( 1 - 16426299 T + 24576394162259001 T^{2} - \)\(25\!\cdots\!90\)\( T^{3} + \)\(21\!\cdots\!39\)\( T^{4} - \)\(12\!\cdots\!79\)\( T^{5} + \)\(65\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + 126843780 T + 39342501567258564 T^{2} + \)\(30\!\cdots\!18\)\( T^{3} + \)\(46\!\cdots\!24\)\( T^{4} + \)\(17\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + 288075309 T + 92688504651310905 T^{2} + \)\(15\!\cdots\!54\)\( T^{3} + \)\(25\!\cdots\!35\)\( T^{4} + \)\(21\!\cdots\!81\)\( T^{5} + \)\(20\!\cdots\!23\)\( T^{6} \)
$71$ \( 1 - 78122274 T + 123167736240843729 T^{2} - \)\(73\!\cdots\!48\)\( T^{3} + \)\(56\!\cdots\!99\)\( T^{4} - \)\(16\!\cdots\!14\)\( T^{5} + \)\(96\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + 557941845 T + 181729649630455470 T^{2} + \)\(43\!\cdots\!53\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} + \)\(19\!\cdots\!05\)\( T^{5} + \)\(20\!\cdots\!97\)\( T^{6} \)
$79$ \( 1 + 320222022 T + 382092522680784861 T^{2} + \)\(76\!\cdots\!36\)\( T^{3} + \)\(45\!\cdots\!59\)\( T^{4} + \)\(45\!\cdots\!42\)\( T^{5} + \)\(17\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 - 430491462 T + 580489855508328153 T^{2} - \)\(15\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!59\)\( T^{4} - \)\(15\!\cdots\!58\)\( T^{5} + \)\(65\!\cdots\!27\)\( T^{6} \)
$89$ \( 1 - 437689644 T + 775511286211782651 T^{2} - \)\(33\!\cdots\!72\)\( T^{3} + \)\(27\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!64\)\( T^{5} + \)\(43\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 + 384952146 T + 823894073491956771 T^{2} + \)\(18\!\cdots\!72\)\( T^{3} + \)\(62\!\cdots\!07\)\( T^{4} + \)\(22\!\cdots\!94\)\( T^{5} + \)\(43\!\cdots\!13\)\( T^{6} \)
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