Properties

Label 3.20.a.b
Level 3
Weight 20
Character orbit 3.a
Self dual Yes
Analytic conductor 6.865
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.86450089669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{87481}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{87481}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 351 - \beta ) q^{2} \) \( -19683 q^{3} \) \( + ( 386242 - 702 \beta ) q^{4} \) \( + ( 3008070 - 4160 \beta ) q^{5} \) \( + ( -6908733 + 19683 \beta ) q^{6} \) \( + ( 56946032 + 63936 \beta ) q^{7} \) \( + ( 504250812 - 108356 \beta ) q^{8} \) \( + 387420489 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 351 - \beta ) q^{2} \) \( -19683 q^{3} \) \( + ( 386242 - 702 \beta ) q^{4} \) \( + ( 3008070 - 4160 \beta ) q^{5} \) \( + ( -6908733 + 19683 \beta ) q^{6} \) \( + ( 56946032 + 63936 \beta ) q^{7} \) \( + ( 504250812 - 108356 \beta ) q^{8} \) \( + 387420489 q^{9} \) \( + ( 4331121210 - 4468230 \beta ) q^{10} \) \( + ( -3325035636 + 10995584 \beta ) q^{11} \) \( + ( -7602401286 + 13817466 \beta ) q^{12} \) \( + ( -22036178074 - 39982464 \beta ) q^{13} \) \( + ( -30350609712 - 34504496 \beta ) q^{14} \) \( + ( -59207841810 + 81881280 \beta ) q^{15} \) \( + ( 59801810440 - 174233592 \beta ) q^{16} \) \( + ( 168140735874 + 582591360 \beta ) q^{17} \) \( + ( 135984591639 - 387420489 \beta ) q^{18} \) \( + ( 301059462548 + 1038738816 \beta ) q^{19} \) \( + ( 3461095598220 - 3718431860 \beta ) q^{20} \) \( + ( -1120868747856 - 1258452288 \beta ) q^{21} \) \( + ( -9824229663372 + 7184485620 \beta ) q^{22} \) \( + ( 1184126082984 + 4325606272 \beta ) q^{23} \) \( + ( -9925168732596 + 2132771148 \beta ) q^{24} \) \( + ( 3600199539175 - 25027142400 \beta ) q^{25} \) \( + ( 23744654894682 + 8002333210 \beta ) q^{26} \) \( -7625597484987 q^{27} \) \( + ( -13342794902944 - 15281345952 \beta ) q^{28} \) \( + ( 140488625985246 + 14942987072 \beta ) q^{29} \) \( + ( -85249458776430 + 87948171090 \beta ) q^{30} \) \( + ( 20805074626856 + 20829313728 \beta ) q^{31} \) \( + ( -106203054501648 - 64148050704 \beta ) q^{32} \) \( + ( 65446676423388 - 216426079872 \beta ) q^{33} \) \( + ( -399673674585666 + 36348831486 \beta ) q^{34} \) \( + ( -38111204008800 - 44571529600 \beta ) q^{35} \) \( + ( 149638064512338 - 271969183278 \beta ) q^{36} \) \( + ( 318997081994942 + 1169925037056 \beta ) q^{37} \) \( + ( -712157321908116 + 63537861868 \beta ) q^{38} \) \( + ( 433738093030542 + 786974838912 \beta ) q^{39} \) \( + ( 1871718915928680 - 2423625810840 \beta ) q^{40} \) \( + ( 575019215140266 - 1296527879552 \beta ) q^{41} \) \( + ( 597391050961296 + 679151994768 \beta ) q^{42} \) \( + ( 1410259476769100 - 1209933369216 \beta ) q^{43} \) \( + ( -7361582207025384 + 6581131371800 \beta ) q^{44} \) \( + ( 1165387950346230 - 1611669234240 \beta ) q^{45} \) \( + ( -2990047005400104 + 334161718488 \beta ) q^{46} \) \( + ( -2801545659750240 - 4731612028288 \beta ) q^{47} \) \( + ( -1177079034890520 + 3429439791336 \beta ) q^{48} \) \( + ( -4937591615096535 + 7281803003904 \beta ) q^{49} \) \( + ( 20968265036900025 - 12384726521575 \beta ) q^{50} \) \( + ( -3309514104207942 - 11467145738880 \beta ) q^{51} \) \( + ( 13587208594198604 + 26490147660 \beta ) q^{52} \) \( + ( 13768934428823814 + 3629613529152 \beta ) q^{53} \) \( + ( -2676584717230437 + 7625597484987 \beta ) q^{54} \) \( + ( -46015651310948280 + 46907634608640 \beta ) q^{55} \) \( + ( 23260586280793920 + 26069335672640 \beta ) q^{56} \) \( + ( -5925753401332284 - 20445496115328 \beta ) q^{57} \) \( + ( 37546460652410658 - 135243637522974 \beta ) q^{58} \) \( + ( -27576737210917908 + 572349981184 \beta ) q^{59} \) \( + ( -68124744659764260 + 73189894300380 \beta ) q^{60} \) \( + ( -43596284533102810 - 8516769041664 \beta ) q^{61} \) \( + ( -9096941554126056 - 13493985508328 \beta ) q^{62} \) \( + ( 22062059564049648 + 24770116384704 \beta ) q^{63} \) \( + ( -18125023109315552 + 175035670187040 \beta ) q^{64} \) \( + ( 64667743959351780 - 28599549696640 \beta ) q^{65} \) \( + ( 193370312464151076 - 141412230458460 \beta ) q^{66} \) \( + ( 249649855438939124 - 47573122936320 \beta ) q^{67} \) \( + ( -257058119054517372 + 106986455485572 \beta ) q^{68} \) \( + ( -23307153691374072 - 85140908251776 \beta ) q^{69} \) \( + ( 21715425221349600 + 22466597119200 \beta ) q^{70} \) \( + ( 42457524218485272 - 312831875103360 \beta ) q^{71} \) \( + ( 195357096163687068 - 41979334506084 \beta ) q^{72} \) \( + ( -120716855580864166 + 219879620692992 \beta ) q^{73} \) \( + ( -809147933720038782 + 91646606011714 \beta ) q^{74} \) \( + ( -70862727529581525 + 492609243859200 \beta ) q^{75} \) \( + ( -457834284736785112 + 189860815060776 \beta ) q^{76} \) \( + ( 364155455101978944 + 413565399899392 \beta ) q^{77} \) \( + ( -467366042292025806 - 157509924572430 \beta ) q^{78} \) \( + ( 565795733016761720 - 1425076089053760 \beta ) q^{79} \) \( + ( 750553336514245680 - 772882372517840 \beta ) q^{80} \) \( + 150094635296999121 q^{81} \) \( + ( 1222625743394029974 - 1030100500863018 \beta ) q^{82} \) \( + ( 803871638001791892 + 2762772599881088 \beta ) q^{83} \) \( + ( 262626232074646752 + 300782732373216 \beta ) q^{84} \) \( + ( -1402375759809647220 + 1053010131039360 \beta ) q^{85} \) \( + ( 1447616705997418164 - 1834946089363916 \beta ) q^{86} \) \( + ( -2765237625267597018 - 294122814538176 \beta ) q^{87} \) \( + ( -2614705214743852848 + 5904819721788624 \beta ) q^{88} \) \( + ( 2161398813184788138 - 4053972006476544 \beta ) q^{89} \) \( + ( 1677965097096471690 - 1731083851564470 \beta ) q^{90} \) \( + ( -3267536840656172384 - 3685747755722112 \beta ) q^{91} \) \( + ( -1933424806346390448 + 839474307455056 \beta ) q^{92} \) \( + ( -409506283880406648 - 409983382108224 \beta ) q^{93} \) \( + ( 2741992840047628512 + 1140749837821152 \beta ) q^{94} \) \( + ( -2496561506465087880 + 1872191706045440 \beta ) q^{95} \) \( + ( 2090394721755937584 + 1262626082006832 \beta ) q^{96} \) \( + ( 3951100617649729154 - 2571793123864320 \beta ) q^{97} \) \( + ( -7466269334159616201 + 7493504469466839 \beta ) q^{98} \) \( + ( -1288186932041546004 + 4259914530120576 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 702q^{2} \) \(\mathstrut -\mathstrut 39366q^{3} \) \(\mathstrut +\mathstrut 772484q^{4} \) \(\mathstrut +\mathstrut 6016140q^{5} \) \(\mathstrut -\mathstrut 13817466q^{6} \) \(\mathstrut +\mathstrut 113892064q^{7} \) \(\mathstrut +\mathstrut 1008501624q^{8} \) \(\mathstrut +\mathstrut 774840978q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 702q^{2} \) \(\mathstrut -\mathstrut 39366q^{3} \) \(\mathstrut +\mathstrut 772484q^{4} \) \(\mathstrut +\mathstrut 6016140q^{5} \) \(\mathstrut -\mathstrut 13817466q^{6} \) \(\mathstrut +\mathstrut 113892064q^{7} \) \(\mathstrut +\mathstrut 1008501624q^{8} \) \(\mathstrut +\mathstrut 774840978q^{9} \) \(\mathstrut +\mathstrut 8662242420q^{10} \) \(\mathstrut -\mathstrut 6650071272q^{11} \) \(\mathstrut -\mathstrut 15204802572q^{12} \) \(\mathstrut -\mathstrut 44072356148q^{13} \) \(\mathstrut -\mathstrut 60701219424q^{14} \) \(\mathstrut -\mathstrut 118415683620q^{15} \) \(\mathstrut +\mathstrut 119603620880q^{16} \) \(\mathstrut +\mathstrut 336281471748q^{17} \) \(\mathstrut +\mathstrut 271969183278q^{18} \) \(\mathstrut +\mathstrut 602118925096q^{19} \) \(\mathstrut +\mathstrut 6922191196440q^{20} \) \(\mathstrut -\mathstrut 2241737495712q^{21} \) \(\mathstrut -\mathstrut 19648459326744q^{22} \) \(\mathstrut +\mathstrut 2368252165968q^{23} \) \(\mathstrut -\mathstrut 19850337465192q^{24} \) \(\mathstrut +\mathstrut 7200399078350q^{25} \) \(\mathstrut +\mathstrut 47489309789364q^{26} \) \(\mathstrut -\mathstrut 15251194969974q^{27} \) \(\mathstrut -\mathstrut 26685589805888q^{28} \) \(\mathstrut +\mathstrut 280977251970492q^{29} \) \(\mathstrut -\mathstrut 170498917552860q^{30} \) \(\mathstrut +\mathstrut 41610149253712q^{31} \) \(\mathstrut -\mathstrut 212406109003296q^{32} \) \(\mathstrut +\mathstrut 130893352846776q^{33} \) \(\mathstrut -\mathstrut 799347349171332q^{34} \) \(\mathstrut -\mathstrut 76222408017600q^{35} \) \(\mathstrut +\mathstrut 299276129024676q^{36} \) \(\mathstrut +\mathstrut 637994163989884q^{37} \) \(\mathstrut -\mathstrut 1424314643816232q^{38} \) \(\mathstrut +\mathstrut 867476186061084q^{39} \) \(\mathstrut +\mathstrut 3743437831857360q^{40} \) \(\mathstrut +\mathstrut 1150038430280532q^{41} \) \(\mathstrut +\mathstrut 1194782101922592q^{42} \) \(\mathstrut +\mathstrut 2820518953538200q^{43} \) \(\mathstrut -\mathstrut 14723164414050768q^{44} \) \(\mathstrut +\mathstrut 2330775900692460q^{45} \) \(\mathstrut -\mathstrut 5980094010800208q^{46} \) \(\mathstrut -\mathstrut 5603091319500480q^{47} \) \(\mathstrut -\mathstrut 2354158069781040q^{48} \) \(\mathstrut -\mathstrut 9875183230193070q^{49} \) \(\mathstrut +\mathstrut 41936530073800050q^{50} \) \(\mathstrut -\mathstrut 6619028208415884q^{51} \) \(\mathstrut +\mathstrut 27174417188397208q^{52} \) \(\mathstrut +\mathstrut 27537868857647628q^{53} \) \(\mathstrut -\mathstrut 5353169434460874q^{54} \) \(\mathstrut -\mathstrut 92031302621896560q^{55} \) \(\mathstrut +\mathstrut 46521172561587840q^{56} \) \(\mathstrut -\mathstrut 11851506802664568q^{57} \) \(\mathstrut +\mathstrut 75092921304821316q^{58} \) \(\mathstrut -\mathstrut 55153474421835816q^{59} \) \(\mathstrut -\mathstrut 136249489319528520q^{60} \) \(\mathstrut -\mathstrut 87192569066205620q^{61} \) \(\mathstrut -\mathstrut 18193883108252112q^{62} \) \(\mathstrut +\mathstrut 44124119128099296q^{63} \) \(\mathstrut -\mathstrut 36250046218631104q^{64} \) \(\mathstrut +\mathstrut 129335487918703560q^{65} \) \(\mathstrut +\mathstrut 386740624928302152q^{66} \) \(\mathstrut +\mathstrut 499299710877878248q^{67} \) \(\mathstrut -\mathstrut 514116238109034744q^{68} \) \(\mathstrut -\mathstrut 46614307382748144q^{69} \) \(\mathstrut +\mathstrut 43430850442699200q^{70} \) \(\mathstrut +\mathstrut 84915048436970544q^{71} \) \(\mathstrut +\mathstrut 390714192327374136q^{72} \) \(\mathstrut -\mathstrut 241433711161728332q^{73} \) \(\mathstrut -\mathstrut 1618295867440077564q^{74} \) \(\mathstrut -\mathstrut 141725455059163050q^{75} \) \(\mathstrut -\mathstrut 915668569473570224q^{76} \) \(\mathstrut +\mathstrut 728310910203957888q^{77} \) \(\mathstrut -\mathstrut 934732084584051612q^{78} \) \(\mathstrut +\mathstrut 1131591466033523440q^{79} \) \(\mathstrut +\mathstrut 1501106673028491360q^{80} \) \(\mathstrut +\mathstrut 300189270593998242q^{81} \) \(\mathstrut +\mathstrut 2445251486788059948q^{82} \) \(\mathstrut +\mathstrut 1607743276003583784q^{83} \) \(\mathstrut +\mathstrut 525252464149293504q^{84} \) \(\mathstrut -\mathstrut 2804751519619294440q^{85} \) \(\mathstrut +\mathstrut 2895233411994836328q^{86} \) \(\mathstrut -\mathstrut 5530475250535194036q^{87} \) \(\mathstrut -\mathstrut 5229410429487705696q^{88} \) \(\mathstrut +\mathstrut 4322797626369576276q^{89} \) \(\mathstrut +\mathstrut 3355930194192943380q^{90} \) \(\mathstrut -\mathstrut 6535073681312344768q^{91} \) \(\mathstrut -\mathstrut 3866849612692780896q^{92} \) \(\mathstrut -\mathstrut 819012567760813296q^{93} \) \(\mathstrut +\mathstrut 5483985680095257024q^{94} \) \(\mathstrut -\mathstrut 4993123012930175760q^{95} \) \(\mathstrut +\mathstrut 4180789443511875168q^{96} \) \(\mathstrut +\mathstrut 7902201235299458308q^{97} \) \(\mathstrut -\mathstrut 14932538668319232402q^{98} \) \(\mathstrut -\mathstrut 2576373864083092008q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
148.386
−147.386
−536.316 −19683.0 −236654. −683163. 1.05563e7 1.13677e8 4.08105e8 3.87420e8 3.66391e8
1.2 1238.32 −19683.0 1.00914e6 6.69930e6 −2.43738e7 214621. 6.00397e8 3.87420e8 8.29585e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 702 T_{2} \) \(\mathstrut -\mathstrut 664128 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(3))\).