Properties

Label 5.18.a.a
Level 5
Weight 18
Character orbit 5.a
Self dual Yes
Analytic conductor 9.161
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(9.16110436723\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{39}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 36\sqrt{39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 340 + \beta ) q^{2} + ( -5490 - 52 \beta ) q^{3} + ( 35072 + 680 \beta ) q^{4} -390625 q^{5} + ( -4494888 - 23170 \beta ) q^{6} + ( -11410350 - 47684 \beta ) q^{7} + ( 1729920 + 135200 \beta ) q^{8} + ( 37670913 + 570960 \beta ) q^{9} +O(q^{10})\) \( q + ( 340 + \beta ) q^{2} + ( -5490 - 52 \beta ) q^{3} + ( 35072 + 680 \beta ) q^{4} -390625 q^{5} + ( -4494888 - 23170 \beta ) q^{6} + ( -11410350 - 47684 \beta ) q^{7} + ( 1729920 + 135200 \beta ) q^{8} + ( 37670913 + 570960 \beta ) q^{9} + ( -132812500 - 390625 \beta ) q^{10} + ( -526677728 - 895000 \beta ) q^{11} + ( -1979781120 - 5556944 \beta ) q^{12} + ( -818712830 + 19322608 \beta ) q^{13} + ( -6289659096 - 27622910 \beta ) q^{14} + ( 2144531250 + 20312500 \beta ) q^{15} + ( 2824764416 - 41431040 \beta ) q^{16} + ( 22642278970 + 49420496 \beta ) q^{17} + ( 41666712660 + 231797313 \beta ) q^{18} + ( 3483245500 - 356859680 \beta ) q^{19} + ( -13700000000 - 265625000 \beta ) q^{20} + ( 187970106492 + 855123360 \beta ) q^{21} + ( -224307307520 - 830977728 \beta ) q^{22} + ( 36099783030 + 83200468 \beta ) q^{23} + ( -364841798400 - 832203840 \beta ) q^{24} + 152587890625 q^{25} + ( 698279536552 + 5750973890 \beta ) q^{26} + ( -998481133980 + 1621830600 \beta ) q^{27} + ( -2039079060480 - 9431411248 \beta ) q^{28} + ( -2094799368250 - 1509301920 \beta ) q^{29} + ( 1755815625000 + 9050781250 \beta ) q^{30} + ( 2306161208412 - 13910715000 \beta ) q^{31} + ( -1360414658560 - 28982723584 \beta ) q^{32} + ( 5243778486720 + 32300791856 \beta ) q^{33} + ( 10196284399624 + 39445247610 \beta ) q^{34} + ( 4457167968750 + 18626562500 \beta ) q^{35} + ( 20945043783936 + 45640929960 \beta ) q^{36} + ( -12281006054590 - 66413262944 \beta ) q^{37} + ( -16852812195920 - 117849045700 \beta ) q^{38} + ( -46290645298404 - 63508050760 \beta ) q^{39} + ( -675750000000 - 52812500000 \beta ) q^{40} + ( -72736365596218 + 4546230000 \beta ) q^{41} + ( 107131191315120 + 478712048892 \beta ) q^{42} + ( -25104519990650 + 225224803788 \beta ) q^{43} + ( -49232719676416 - 389530295040 \beta ) q^{44} + ( -14715200390625 - 223031250000 \beta ) q^{45} + ( 16479210684792 + 64387942150 \beta ) q^{46} + ( 106265963747530 - 459306334564 \beta ) q^{47} + ( 93384748615680 + 80568659968 \beta ) q^{48} + ( 12490693472957 + 1088182258800 \beta ) q^{49} + ( 51879882812500 + 152587890625 \beta ) q^{50} + ( -254197408136148 - 1448717029480 \beta ) q^{51} + ( 635402594777600 + 120957783376 \beta ) q^{52} + ( -239760645565190 + 760411199248 \beta ) q^{53} + ( -257509779706800 - 447058729980 \beta ) q^{54} + ( 205733487500000 + 349609375000 \beta ) q^{55} + ( -345589933651200 - 1625168825280 \beta ) q^{56} + ( 918806996832840 + 1778030877200 \beta ) q^{57} + ( -788517941449480 - 2607962021050 \beta ) q^{58} + ( -1551206649024500 + 3120468771760 \beta ) q^{59} + ( 773352000000000 + 2170681250000 \beta ) q^{60} + ( 231775793933122 + 5321088240000 \beta ) q^{61} + ( 80991631900080 - 2423481891588 \beta ) q^{62} + ( -1805931891361710 - 8311153251492 \beta ) q^{63} + ( -2297691286274048 - 5784091402240 \beta ) q^{64} + ( 319809699218750 - 7547893750000 \beta ) q^{65} + ( 3415495909054464 + 16226047717760 \beta ) q^{66} + ( 1458564342699570 - 11799742792604 \beta ) q^{67} + ( 2492688501916160 + 17130025335312 \beta ) q^{68} + ( -416862600473484 - 2333959286880 \beta ) q^{69} + ( 2456898084375000 + 10790199218750 \beta ) q^{70} + ( 2560588697595092 - 16709267755000 \beta ) q^{71} + ( 3966850688664960 + 6080822560800 \beta ) q^{72} + ( 2379465133435330 + 33511223330768 \beta ) q^{73} + ( -7532334020802136 - 34861515455550 \beta ) q^{74} + ( -837707519531250 - 7934570312500 \beta ) q^{75} + ( -12143074266649600 - 10147175756960 \beta ) q^{76} + ( 8166652599604800 + 35326364031952 \beta ) q^{77} + ( -18948770319070800 - 67883382556804 \beta ) q^{78} + ( -1304667048579000 + 37974652152880 \beta ) q^{79} + ( -1103423600000000 + 16184000000000 \beta ) q^{80} + ( -3645804323641419 - 30716698493520 \beta ) q^{81} + ( -24500579653594120 - 71190647396218 \beta ) q^{82} + ( 25319165760930390 + 13258309308828 \beta ) q^{83} + ( 35983009048218624 + 157810558896480 \beta ) q^{84} + ( -8844640222656250 - 19304881250000 \beta ) q^{85} + ( 2848225685839672 + 51471913297270 \beta ) q^{86} + ( 15467328656405460 + 117215634689800 \beta ) q^{87} + ( -7027136511221760 - 72755107225600 \beta ) q^{88} + ( 30596432088320250 - 109608527628960 \beta ) q^{89} + ( -16276059632812500 - 90545825390625 \beta ) q^{90} + ( -37228392360299868 - 181438217607080 \beta ) q^{91} + ( 4125685019550720 + 27465859274096 \beta ) q^{92} + ( 23900540271738120 - 43550557487424 \beta ) q^{93} + ( 12915248299957384 - 49898190004230 \beta ) q^{94} + ( -1360642773437500 + 139398312500000 \beta ) q^{95} + ( 83643621078638592 + 229856714721280 \beta ) q^{96} + ( -128762398757527870 + 82347302612336 \beta ) q^{97} + ( 59247919869592580 + 382472661464957 \beta ) q^{98} + ( -45668879875325664 - 334427382713880 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 680q^{2} - 10980q^{3} + 70144q^{4} - 781250q^{5} - 8989776q^{6} - 22820700q^{7} + 3459840q^{8} + 75341826q^{9} + O(q^{10}) \) \( 2q + 680q^{2} - 10980q^{3} + 70144q^{4} - 781250q^{5} - 8989776q^{6} - 22820700q^{7} + 3459840q^{8} + 75341826q^{9} - 265625000q^{10} - 1053355456q^{11} - 3959562240q^{12} - 1637425660q^{13} - 12579318192q^{14} + 4289062500q^{15} + 5649528832q^{16} + 45284557940q^{17} + 83333425320q^{18} + 6966491000q^{19} - 27400000000q^{20} + 375940212984q^{21} - 448614615040q^{22} + 72199566060q^{23} - 729683596800q^{24} + 305175781250q^{25} + 1396559073104q^{26} - 1996962267960q^{27} - 4078158120960q^{28} - 4189598736500q^{29} + 3511631250000q^{30} + 4612322416824q^{31} - 2720829317120q^{32} + 10487556973440q^{33} + 20392568799248q^{34} + 8914335937500q^{35} + 41890087567872q^{36} - 24562012109180q^{37} - 33705624391840q^{38} - 92581290596808q^{39} - 1351500000000q^{40} - 145472731192436q^{41} + 214262382630240q^{42} - 50209039981300q^{43} - 98465439352832q^{44} - 29430400781250q^{45} + 32958421369584q^{46} + 212531927495060q^{47} + 186769497231360q^{48} + 24981386945914q^{49} + 103759765625000q^{50} - 508394816272296q^{51} + 1270805189555200q^{52} - 479521291130380q^{53} - 515019559413600q^{54} + 411466975000000q^{55} - 691179867302400q^{56} + 1837613993665680q^{57} - 1577035882898960q^{58} - 3102413298049000q^{59} + 1546704000000000q^{60} + 463551587866244q^{61} + 161983263800160q^{62} - 3611863782723420q^{63} - 4595382572548096q^{64} + 639619398437500q^{65} + 6830991818108928q^{66} + 2917128685399140q^{67} + 4985377003832320q^{68} - 833725200946968q^{69} + 4913796168750000q^{70} + 5121177395190184q^{71} + 7933701377329920q^{72} + 4758930266870660q^{73} - 15064668041604272q^{74} - 1675415039062500q^{75} - 24286148533299200q^{76} + 16333305199209600q^{77} - 37897540638141600q^{78} - 2609334097158000q^{79} - 2206847200000000q^{80} - 7291608647282838q^{81} - 49001159307188240q^{82} + 50638331521860780q^{83} + 71966018096437248q^{84} - 17689280445312500q^{85} + 5696451371679344q^{86} + 30934657312810920q^{87} - 14054273022443520q^{88} + 61192864176640500q^{89} - 32552119265625000q^{90} - 74456784720599736q^{91} + 8251370039101440q^{92} + 47801080543476240q^{93} + 25830496599914768q^{94} - 2721285546875000q^{95} + 167287242157277184q^{96} - 257524797515055740q^{97} + 118495839739185160q^{98} - 91337759750651328q^{99} + 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
−6.24500
6.24500
115.180 6200.64 −117806. −390625. 714190. −690037. −2.86657e7 −9.06923e7 −4.49922e7
1.2 564.820 −17180.6 187950. −390625. −9.70397e6 −2.21307e7 3.21256e7 1.66034e8 −2.20633e8
\(n\): 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
\(5\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} - 680 T_{2} + 65056 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(5))\).