Properties

Label 5.20.a.a
Level 5
Weight 20
Character orbit 5.a
Self dual Yes
Analytic conductor 11.441
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.4408348278\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -335 + \beta_{1} ) q^{2} \) \( + ( -24478 + 18 \beta_{1} - 13 \beta_{2} ) q^{3} \) \( + ( 318162 - 398 \beta_{1} + 184 \beta_{2} ) q^{4} \) \( + 1953125 q^{5} \) \( + ( 17803448 - 46152 \beta_{1} + 11216 \beta_{2} ) q^{6} \) \( + ( -18318318 - 44498 \beta_{1} - 30963 \beta_{2} ) q^{7} \) \( + ( -171462364 + 109668 \beta_{1} - 185104 \beta_{2} ) q^{8} \) \( + ( 469399485 - 1678536 \beta_{1} + 556788 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -335 + \beta_{1} ) q^{2} \) \( + ( -24478 + 18 \beta_{1} - 13 \beta_{2} ) q^{3} \) \( + ( 318162 - 398 \beta_{1} + 184 \beta_{2} ) q^{4} \) \( + 1953125 q^{5} \) \( + ( 17803448 - 46152 \beta_{1} + 11216 \beta_{2} ) q^{6} \) \( + ( -18318318 - 44498 \beta_{1} - 30963 \beta_{2} ) q^{7} \) \( + ( -171462364 + 109668 \beta_{1} - 185104 \beta_{2} ) q^{8} \) \( + ( 469399485 - 1678536 \beta_{1} + 556788 \beta_{2} ) q^{9} \) \( + ( -654296875 + 1953125 \beta_{1} ) q^{10} \) \( + ( -2854948528 + 2097940 \beta_{1} + 1637230 \beta_{2} ) q^{11} \) \( + ( -23777142992 + 28995120 \beta_{1} - 8495552 \beta_{2} ) q^{12} \) \( + ( -28548921090 - 16253608 \beta_{1} + 4569380 \beta_{2} ) q^{13} \) \( + ( -34790122052 - 64436484 \beta_{1} + 10637872 \beta_{2} ) q^{14} \) \( + ( -47808593750 + 35156250 \beta_{1} - 25390625 \beta_{2} ) q^{15} \) \( + ( -79701977272 - 262169144 \beta_{1} + 36253152 \beta_{2} ) q^{16} \) \( + ( 28946200346 - 419322056 \beta_{1} + 205644916 \beta_{2} ) q^{17} \) \( + ( -1231308771243 + 1454872293 \beta_{1} - 647377728 \beta_{2} ) q^{18} \) \( + ( 427611297876 + 823817048 \beta_{1} + 307290116 \beta_{2} ) q^{19} \) \( + ( 621410156250 - 777343750 \beta_{1} + 359375000 \beta_{2} ) q^{20} \) \( + ( 2068504635984 + 227254896 \beta_{1} + 206921832 \beta_{2} ) q^{21} \) \( + ( 2934298505100 - 400295348 \beta_{1} - 609414880 \beta_{2} ) q^{22} \) \( + ( 1362994724638 + 154917186 \beta_{1} - 1481795237 \beta_{2} ) q^{23} \) \( + ( 17490287734368 - 14829867936 \beta_{1} + 4619983488 \beta_{2} ) q^{24} \) \( + 3814697265625 q^{25} \) \( + ( -1060367722330 - 20305323386 \beta_{1} - 5768846912 \beta_{2} ) q^{26} \) \( + ( -35977406241996 + 76963599252 \beta_{1} - 7437917106 \beta_{2} ) q^{27} \) \( + ( -22896992964488 + 9406981624 \beta_{1} - 2090609888 \beta_{2} ) q^{28} \) \( + ( -24413692018026 + 39133027952 \beta_{1} + 27451585384 \beta_{2} ) q^{29} \) \( + ( 34772359375000 - 90140625000 \beta_{1} + 21906250000 \beta_{2} ) q^{30} \) \( + ( -94856262455188 - 42652946780 \beta_{1} - 49909386010 \beta_{2} ) q^{31} \) \( + ( -64972587403120 - 63402957424 \beta_{1} + 26766767040 \beta_{2} ) q^{32} \) \( + ( -18231690238936 - 70610433624 \beta_{1} + 37015537244 \beta_{2} ) q^{33} \) \( + ( -259886153557086 + 380282457154 \beta_{1} - 202187367232 \beta_{2} ) q^{34} \) \( + ( -35777964843750 - 86910156250 \beta_{1} - 60474609375 \beta_{2} ) q^{35} \) \( + ( 1052449653821658 - 1465786253574 \beta_{1} + 369384893592 \beta_{2} ) q^{36} \) \( + ( -143744977087986 + 1379364009648 \beta_{1} + 11520078664 \beta_{2} ) q^{37} \) \( + ( 542016784241564 + 861229207132 \beta_{1} - 35250053696 \beta_{2} ) q^{38} \) \( + ( 240580355688236 + 319523123916 \beta_{1} + 219171817922 \beta_{2} ) q^{39} \) \( + ( -334887429687500 + 214195312500 \beta_{1} - 361531250000 \beta_{2} ) q^{40} \) \( + ( 693182242891602 - 1886964293080 \beta_{1} - 536377941860 \beta_{2} ) q^{41} \) \( + ( -470643788772192 + 2381124072096 \beta_{1} - 83993572992 \beta_{2} ) q^{42} \) \( + ( 365535421859474 - 3394737590086 \beta_{1} + 525514034959 \beta_{2} ) q^{43} \) \( + ( 55536909769944 + 896716834904 \beta_{1} - 561510139232 \beta_{2} ) q^{44} \) \( + ( 916795869140625 - 3278390625000 \beta_{1} + 1087476562500 \beta_{2} ) q^{45} \) \( + ( -747066508537148 - 988001532540 \beta_{1} + 929436266320 \beta_{2} ) q^{46} \) \( + ( -6812102764971382 - 6507187638290 \beta_{1} + 596140161133 \beta_{2} ) q^{47} \) \( + ( -2963998776863552 + 10522349850816 \beta_{1} - 1083529693952 \beta_{2} ) q^{48} \) \( + ( -3628844405523327 + 5694334820840 \beta_{1} - 768318433220 \beta_{2} ) q^{49} \) \( + ( -1277923583984375 + 3814697265625 \beta_{1} ) q^{50} \) \( + ( -18333757436487004 + 25503416244612 \beta_{1} - 5073638027146 \beta_{2} ) q^{51} \) \( + ( -1075601062467348 - 374338838868 \beta_{1} - 2624391681968 \beta_{2} ) q^{52} \) \( + ( 2997775190704582 - 22572443604152 \beta_{1} - 737811714228 \beta_{2} ) q^{53} \) \( + ( 66227354500201776 - 52578022022352 \beta_{1} + 18683555862816 \beta_{2} ) q^{54} \) \( + ( -5576071343750000 + 4097539062500 \beta_{1} + 3197714843750 \beta_{2} ) q^{55} \) \( + ( 32210342438352624 + 6990478893552 \beta_{1} - 2575333204416 \beta_{2} ) q^{56} \) \( + ( -22875339683696872 - 13165370756136 \beta_{1} - 2626870212892 \beta_{2} ) q^{57} \) \( + ( 44231325763815686 + 16494432127718 \beta_{1} - 9490086770304 \beta_{2} ) q^{58} \) \( + ( 1004876734267828 + 10591847238544 \beta_{1} - 23792666129352 \beta_{2} ) q^{59} \) \( + ( -46439732406250000 + 56631093750000 \beta_{1} - 16592875000000 \beta_{2} ) q^{60} \) \( + ( -13586251587741658 + 67547157960800 \beta_{1} + 45709817443600 \beta_{2} ) q^{61} \) \( + ( -12962920151165160 - 171025956703848 \beta_{1} + 22496764486560 \beta_{2} ) q^{62} \) \( + ( -40842546029106534 + 88042476188070 \beta_{1} + 8824065319521 \beta_{2} ) q^{63} \) \( + ( 24544486193169696 + 118765427007264 \beta_{1} - 46947431082112 \beta_{2} ) q^{64} \) \( + ( -55759611503906250 - 31745328125000 \beta_{1} + 8924570312500 \beta_{2} ) q^{65} \) \( + ( -35372187877117024 + 44701315924896 \beta_{1} - 35497766431168 \beta_{2} ) q^{66} \) \( + ( -351763465234425834 - 5238511075346 \beta_{1} + 13790531546061 \beta_{2} ) q^{67} \) \( + ( 294508913141123364 - 383454466488220 \beta_{1} + 85084729673584 \beta_{2} ) q^{68} \) \( + ( 66108470934787440 - 40467462935472 \beta_{1} - 3539193169224 \beta_{2} ) q^{69} \) \( + ( -67949457132812500 - 125852507812500 \beta_{1} + 20777093750000 \beta_{2} ) q^{70} \) \( + ( 98344235061088532 - 414937837689100 \beta_{1} - 86448548028450 \beta_{2} ) q^{71} \) \( + ( -676756840830588108 + 965650234919796 \beta_{1} - 154878311703888 \beta_{2} ) q^{72} \) \( + ( 160488471498859618 + 211121551372376 \beta_{1} - 25653244426332 \beta_{2} ) q^{73} \) \( + ( 1058538305974927806 - 212443185406690 \beta_{1} + 246798769947520 \beta_{2} ) q^{74} \) \( + ( -93376159667968750 + 68664550781250 \beta_{1} - 49591064453125 \beta_{2} ) q^{75} \) \( + ( 213523157291371128 + 146866890744 \beta_{1} + 18789686422048 \beta_{2} ) q^{76} \) \( + ( -330380037685064936 - 110727250876056 \beta_{1} + 158947793404924 \beta_{2} ) q^{77} \) \( + ( 212423867022509648 + 566741871198288 \beta_{1} - 74464210496032 \beta_{2} ) q^{78} \) \( + ( 665876807810103624 + 873237784655752 \beta_{1} - 246712757010516 \beta_{2} ) q^{79} \) \( + ( -155667924359375000 - 512049109375000 \beta_{1} + 70806937500000 \beta_{2} ) q^{80} \) \( + ( 1566030344742328257 - 2079706175201592 \beta_{1} + 414460740148236 \beta_{2} ) q^{81} \) \( + ( -1756214594039786710 - 35416154783158 \beta_{1} - 21083641275840 \beta_{2} ) q^{82} \) \( + ( -500678249285396094 + 716015554178850 \beta_{1} - 903195337524789 \beta_{2} ) q^{83} \) \( + ( 789053010679813440 - 872511465555648 \beta_{1} + 380708288189184 \beta_{2} ) q^{84} \) \( + ( 56535547550781250 - 818988390625000 \beta_{1} + 401650226562500 \beta_{2} ) q^{85} \) \( + ( -2458245518425900064 + 1409716065270112 \beta_{1} - 944144249830896 \beta_{2} ) q^{86} \) \( + ( -841838724630405428 - 920482508460564 \beta_{1} + 342868267193842 \beta_{2} ) q^{87} \) \( + ( -1055152456208811088 - 678272223403344 \beta_{1} + 825902970880832 \beta_{2} ) q^{88} \) \( + ( 347946659763910842 + 1839171741416016 \beta_{1} + 128775009058872 \beta_{2} ) q^{89} \) \( + ( -2404899943833984375 + 2841547447265625 \beta_{1} - 1264409625000000 \beta_{2} ) q^{90} \) \( + ( 359785599161959916 + 2360966545260692 \beta_{1} + 1074995066467214 \beta_{2} ) q^{91} \) \( + ( -932652937690104184 + 702465667184904 \beta_{1} + 29997929306336 \beta_{2} ) q^{92} \) \( + ( 5212554304627096704 - 1976096355274944 \beta_{1} + 1382287429020384 \beta_{2} ) q^{93} \) \( + ( -2307289548058073868 - 5460248489168972 \beta_{1} - 1559775743414224 \beta_{2} ) q^{94} \) \( + ( 835178316164062500 + 1609017671875000 \beta_{1} + 600176007812500 \beta_{2} ) q^{95} \) \( + ( -788439947178468992 + 2436238066520448 \beta_{1} + 172696523516416 \beta_{2} ) q^{96} \) \( + ( 540011182022204218 + 254078334113160 \beta_{1} - 3886383361239892 \beta_{2} ) q^{97} \) \( + ( 5164546238652671465 - 5201530623723847 \beta_{1} + 1514895214432320 \beta_{2} ) q^{98} \) \( + ( 638714469088055280 + 1534986965027268 \beta_{1} - 2477997089288394 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 1006q^{2} \) \(\mathstrut -\mathstrut 73452q^{3} \) \(\mathstrut +\mathstrut 954884q^{4} \) \(\mathstrut +\mathstrut 5859375q^{5} \) \(\mathstrut +\mathstrut 53456496q^{6} \) \(\mathstrut -\mathstrut 54910456q^{7} \) \(\mathstrut -\mathstrut 514496760q^{8} \) \(\mathstrut +\mathstrut 1409876991q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 1006q^{2} \) \(\mathstrut -\mathstrut 73452q^{3} \) \(\mathstrut +\mathstrut 954884q^{4} \) \(\mathstrut +\mathstrut 5859375q^{5} \) \(\mathstrut +\mathstrut 53456496q^{6} \) \(\mathstrut -\mathstrut 54910456q^{7} \) \(\mathstrut -\mathstrut 514496760q^{8} \) \(\mathstrut +\mathstrut 1409876991q^{9} \) \(\mathstrut -\mathstrut 1964843750q^{10} \) \(\mathstrut -\mathstrut 8566943524q^{11} \) \(\mathstrut -\mathstrut 71360424096q^{12} \) \(\mathstrut -\mathstrut 85630509662q^{13} \) \(\mathstrut -\mathstrut 104305929672q^{14} \) \(\mathstrut -\mathstrut 143460937500q^{15} \) \(\mathstrut -\mathstrut 238843762672q^{16} \) \(\mathstrut +\mathstrut 87257923094q^{17} \) \(\mathstrut -\mathstrut 3695381186022q^{18} \) \(\mathstrut +\mathstrut 1282010076580q^{19} \) \(\mathstrut +\mathstrut 1865007812500q^{20} \) \(\mathstrut +\mathstrut 6205286653056q^{21} \) \(\mathstrut +\mathstrut 8803295810648q^{22} \) \(\mathstrut +\mathstrut 4088829256728q^{23} \) \(\mathstrut +\mathstrut 52485693071040q^{24} \) \(\mathstrut +\mathstrut 11444091796875q^{25} \) \(\mathstrut -\mathstrut 3160797843604q^{26} \) \(\mathstrut -\mathstrut 108009182325240q^{27} \) \(\mathstrut -\mathstrut 68700385875088q^{28} \) \(\mathstrut -\mathstrut 73280209082030q^{29} \) \(\mathstrut +\mathstrut 104407218750000q^{30} \) \(\mathstrut -\mathstrut 284526134418784q^{31} \) \(\mathstrut -\mathstrut 194854359251936q^{32} \) \(\mathstrut -\mathstrut 54624460283184q^{33} \) \(\mathstrut -\mathstrut 780038743128412q^{34} \) \(\mathstrut -\mathstrut 107246984375000q^{35} \) \(\mathstrut +\mathstrut 3158814747718548q^{36} \) \(\mathstrut -\mathstrut 432614295273606q^{37} \) \(\mathstrut +\mathstrut 1625189123517560q^{38} \) \(\mathstrut +\mathstrut 721421543940792q^{39} \) \(\mathstrut -\mathstrut 1004876484375000q^{40} \) \(\mathstrut +\mathstrut 2081433692967886q^{41} \) \(\mathstrut -\mathstrut 1414312490388672q^{42} \) \(\mathstrut +\mathstrut 1100001003168508q^{43} \) \(\mathstrut +\mathstrut 165714012474928q^{44} \) \(\mathstrut +\mathstrut 2753665998046875q^{45} \) \(\mathstrut -\mathstrut 2240211524078904q^{46} \) \(\mathstrut -\mathstrut 20429801107275856q^{47} \) \(\mathstrut -\mathstrut 8902518680441472q^{48} \) \(\mathstrut -\mathstrut 10892227551390821q^{49} \) \(\mathstrut -\mathstrut 3837585449218750q^{50} \) \(\mathstrut -\mathstrut 55026775725705624q^{51} \) \(\mathstrut -\mathstrut 3226428848563176q^{52} \) \(\mathstrut +\mathstrut 9015898015717898q^{53} \) \(\mathstrut +\mathstrut 198734641522627680q^{54} \) \(\mathstrut -\mathstrut 16732311570312500q^{55} \) \(\mathstrut +\mathstrut 96624036836164320q^{56} \) \(\mathstrut -\mathstrut 68612853680334480q^{57} \) \(\mathstrut +\mathstrut 132677482859319340q^{58} \) \(\mathstrut +\mathstrut 3004038355564940q^{59} \) \(\mathstrut -\mathstrut 139375828312500000q^{60} \) \(\mathstrut -\mathstrut 40826301921185774q^{61} \) \(\mathstrut -\mathstrut 38717734496791632q^{62} \) \(\mathstrut -\mathstrut 122615680563507672q^{63} \) \(\mathstrut +\mathstrut 73514693152501824q^{64} \) \(\mathstrut -\mathstrut 167247089183593750q^{65} \) \(\mathstrut -\mathstrut 106161264947275968q^{66} \) \(\mathstrut -\mathstrut 1055285157192202156q^{67} \) \(\mathstrut +\mathstrut 883910193889858312q^{68} \) \(\mathstrut +\mathstrut 198365880267297792q^{69} \) \(\mathstrut -\mathstrut 203722518890625000q^{70} \) \(\mathstrut +\mathstrut 295447643020954696q^{71} \) \(\mathstrut -\mathstrut 2031236172726684120q^{72} \) \(\mathstrut +\mathstrut 481254292945206478q^{73} \) \(\mathstrut +\mathstrut 3175827361110190108q^{74} \) \(\mathstrut -\mathstrut 280197143554687500q^{75} \) \(\mathstrut +\mathstrut 640569325007222640q^{76} \) \(\mathstrut -\mathstrut 991029385804318752q^{77} \) \(\mathstrut +\mathstrut 636704859196330656q^{78} \) \(\mathstrut +\mathstrut 1996757185645655120q^{79} \) \(\mathstrut -\mathstrut 466491723968750000q^{80} \) \(\mathstrut +\mathstrut 4700170740402186363q^{81} \) \(\mathstrut -\mathstrut 5268608365964576972q^{82} \) \(\mathstrut -\mathstrut 1502750763410367132q^{83} \) \(\mathstrut +\mathstrut 2368031543504995968q^{84} \) \(\mathstrut +\mathstrut 170425631042968750q^{85} \) \(\mathstrut -\mathstrut 7376146271342970304q^{86} \) \(\mathstrut -\mathstrut 2524595691382755720q^{87} \) \(\mathstrut -\mathstrut 3164779096403029920q^{88} \) \(\mathstrut +\mathstrut 1042000807550316510q^{89} \) \(\mathstrut -\mathstrut 7217541378949218750q^{90} \) \(\mathstrut +\mathstrut 1076995830940619056q^{91} \) \(\mathstrut -\mathstrut 2798661278737497456q^{92} \) \(\mathstrut +\mathstrut 15639639010236565056q^{93} \) \(\mathstrut -\mathstrut 6916408395685052632q^{94} \) \(\mathstrut +\mathstrut 2503925930820312500q^{95} \) \(\mathstrut -\mathstrut 2367756079601927424q^{96} \) \(\mathstrut +\mathstrut 1619779467732499494q^{97} \) \(\mathstrut +\mathstrut 15498840246581738242q^{98} \) \(\mathstrut +\mathstrut 1914608420299138572q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(22777\) \(x\mathstrut -\mathstrut \) \(646704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{2} + 104 \nu - 30381 \)\()/35\)
\(\beta_{2}\)\(=\)\((\)\( -6 \nu^{2} + 738 \nu + 91108 \)\()/35\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(52\) \(\beta_{2}\mathstrut +\mathstrut \) \(369\) \(\beta_{1}\mathstrut +\mathstrut \) \(455663\)\()/30\)

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
−29.5226
−133.978
163.500
−1240.95 −64590.2 1.01566e6 1.95313e6 8.01531e7 −3.47039e7 −6.09772e8 3.00964e9 −2.42373e9
1.2 −575.417 14082.6 −193183. 1.95313e6 −8.10336e6 9.45293e7 4.12845e8 −9.63942e8 −1.12386e9
1.3 810.365 −22944.3 132403. 1.95313e6 −1.85933e7 −1.14736e8 −3.17570e8 −6.35819e8 1.58274e9
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut +\mathstrut 1006 T_{2}^{2} \) \(\mathstrut -\mathstrut 757856 T_{2} \) \(\mathstrut -\mathstrut 578651136 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\).