Properties

Label 5.18.a.b
Level 5
Weight 18
Character orbit 5.a
Self dual Yes
Analytic conductor 9.161
Analytic rank 0
Dimension 3
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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 39 - \beta_{1} ) q^{2} \) \( + ( 5317 + 8 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 5572 - 194 \beta_{1} + 6 \beta_{2} ) q^{4} \) \( + 390625 q^{5} \) \( + ( -810688 - 13124 \beta_{1} - 124 \beta_{2} ) q^{6} \) \( + ( 704579 - 25832 \beta_{1} - 261 \beta_{2} ) q^{7} \) \( + ( 21696960 + 41148 \beta_{1} + 708 \beta_{2} ) q^{8} \) \( + ( 108091313 + 385312 \beta_{1} + 3812 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 39 - \beta_{1} ) q^{2} \) \( + ( 5317 + 8 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 5572 - 194 \beta_{1} + 6 \beta_{2} ) q^{4} \) \( + 390625 q^{5} \) \( + ( -810688 - 13124 \beta_{1} - 124 \beta_{2} ) q^{6} \) \( + ( 704579 - 25832 \beta_{1} - 261 \beta_{2} ) q^{7} \) \( + ( 21696960 + 41148 \beta_{1} + 708 \beta_{2} ) q^{8} \) \( + ( 108091313 + 385312 \beta_{1} + 3812 \beta_{2} ) q^{9} \) \( + ( 15234375 - 390625 \beta_{1} ) q^{10} \) \( + ( 596885622 + 891920 \beta_{1} - 17430 \beta_{2} ) q^{11} \) \( + ( 1037023904 - 1150280 \beta_{1} - 42904 \beta_{2} ) q^{12} \) \( + ( 1803407762 - 4427488 \beta_{1} + 46020 \beta_{2} ) q^{13} \) \( + ( 3501550404 - 2347272 \beta_{1} + 174828 \beta_{2} ) q^{14} \) \( + ( 2076953125 + 3125000 \beta_{1} + 390625 \beta_{2} ) q^{15} \) \( + ( -5399636384 + 3703672 \beta_{1} - 1087128 \beta_{2} ) q^{16} \) \( + ( -9108532266 + 76541216 \beta_{1} - 165948 \beta_{2} ) q^{17} \) \( + ( -47609051573 - 82855117 \beta_{1} - 2601584 \beta_{2} ) q^{18} \) \( + ( -9954572800 + 99735584 \beta_{1} + 6888684 \beta_{2} ) q^{19} \) \( + ( 2176562500 - 75781250 \beta_{1} + 2343750 \beta_{2} ) q^{20} \) \( + ( -74965174008 - 402852288 \beta_{1} - 2772888 \beta_{2} ) q^{21} \) \( + ( -98337289092 - 300948812 \beta_{1} - 4026840 \beta_{2} ) q^{22} \) \( + ( 5688969957 + 785797416 \beta_{1} - 27034611 \beta_{2} ) q^{23} \) \( + ( 299431636800 + 893024112 \beta_{1} + 26415312 \beta_{2} ) q^{24} \) \( + 152587890625 q^{25} \) \( + ( 671484540402 - 2906011342 \beta_{1} + 23067408 \beta_{2} ) q^{26} \) \( + ( 1045863330610 + 5905579088 \beta_{1} + 31049098 \beta_{2} ) q^{27} \) \( + ( 372382772048 - 2061194576 \beta_{1} + 35006496 \beta_{2} ) q^{28} \) \( + ( -843302681850 - 1805173184 \beta_{1} - 175959384 \beta_{2} ) q^{29} \) \( + ( -316675000000 - 5126562500 \beta_{1} - 48437500 \beta_{2} ) q^{30} \) \( + ( -178153880038 - 13226283440 \beta_{1} - 20697990 \beta_{2} ) q^{31} \) \( + ( -3623317258176 + 10415601904 \beta_{1} - 32399280 \beta_{2} ) q^{32} \) \( + ( 581427928124 + 12657852256 \beta_{1} + 791229452 \beta_{2} ) q^{33} \) \( + ( -10708155093426 + 22473752302 \beta_{1} - 446635248 \beta_{2} ) q^{34} \) \( + ( 275226171875 - 10090625000 \beta_{1} - 101953125 \beta_{2} ) q^{35} \) \( + ( -4992591100364 + 7799424422 \beta_{1} + 195204622 \beta_{2} ) q^{36} \) \( + ( 10586395166894 - 3949553088 \beta_{1} - 388153272 \beta_{2} ) q^{37} \) \( + ( -13431274105860 - 36908335828 \beta_{1} - 1121953488 \beta_{2} ) q^{38} \) \( + ( 14322889450246 - 34232512912 \beta_{1} + 952457038 \beta_{2} ) q^{39} \) \( + ( 8475375000000 + 16073437500 \beta_{1} + 276562500 \beta_{2} ) q^{40} \) \( + ( 28919253090582 - 75896980640 \beta_{1} - 173297940 \beta_{2} ) q^{41} \) \( + ( 51336461764608 + 37609387104 \beta_{1} + 2627853216 \beta_{2} ) q^{42} \) \( + ( 29790253726697 + 108422499944 \beta_{1} - 4292294403 \beta_{2} ) q^{43} \) \( + ( -41658461319216 - 28784693528 \beta_{1} + 4396317672 \beta_{2} ) q^{44} \) \( + ( 42223169140625 + 150512500000 \beta_{1} + 1489062500 \beta_{2} ) q^{45} \) \( + ( -107658803587908 + 360691755240 \beta_{1} - 2660154060 \beta_{2} ) q^{46} \) \( + ( 115903925728899 - 478747843240 \beta_{1} - 7786889709 \beta_{2} ) q^{47} \) \( + ( -243252665565568 - 249222726944 \beta_{1} - 1742195296 \beta_{2} ) q^{48} \) \( + ( -129102040290843 + 19282959200 \beta_{1} + 4570435500 \beta_{2} ) q^{49} \) \( + ( 5950927734375 - 152587890625 \beta_{1} ) q^{50} \) \( + ( -3832314220198 + 916493947024 \beta_{1} + 3833833874 \beta_{2} ) q^{51} \) \( + ( 183932306647544 - 750287431452 \beta_{1} + 9651011604 \beta_{2} ) q^{52} \) \( + ( 14522038786242 - 418616374432 \beta_{1} + 29766479436 \beta_{2} ) q^{53} \) \( + ( -755236880329600 - 411399761576 \beta_{1} - 37793205976 \beta_{2} ) q^{54} \) \( + ( 233158446093750 + 348406250000 \beta_{1} - 6808593750 \beta_{2} ) q^{55} \) \( + ( -163714427937600 - 700910065056 \beta_{1} - 13208381856 \beta_{2} ) q^{56} \) \( + ( 1431969402141220 + 2881235405216 \beta_{1} - 13047985964 \beta_{2} ) q^{57} \) \( + ( 199957959636210 + 2155405385378 \beta_{1} + 24203952288 \beta_{2} ) q^{58} \) \( + ( -711496097017500 - 1226719451008 \beta_{1} - 31707355608 \beta_{2} ) q^{59} \) \( + ( 405087462500000 - 449328125000 \beta_{1} - 16759375000 \beta_{2} ) q^{60} \) \( + ( 940270880260622 - 2084755504000 \beta_{1} + 93053706000 \beta_{2} ) q^{61} \) \( + ( 1778924509336968 - 1684665337632 \beta_{1} + 80930747880 \beta_{2} ) q^{62} \) \( + ( -1456545333867753 - 3328923314760 \beta_{1} - 99135472017 \beta_{2} ) q^{63} \) \( + ( -842994592907648 + 5045404143072 \beta_{1} + 82460775072 \beta_{2} ) q^{64} \) \( + ( 704456157031250 - 1729487500000 \beta_{1} + 17976562500 \beta_{2} ) q^{65} \) \( + ( -1637896838087936 - 5777713680688 \beta_{1} - 136080551888 \beta_{2} ) q^{66} \) \( + ( 125300623996259 + 1507377473656 \beta_{1} - 221032358433 \beta_{2} ) q^{67} \) \( + ( -2288573435839992 + 8199885525340 \beta_{1} - 79147098708 \beta_{2} ) q^{68} \) \( + ( -4598767997198184 + 4242566288064 \beta_{1} + 209714171064 \beta_{2} ) q^{69} \) \( + ( 1367793126562500 - 916903125000 \beta_{1} + 68292187500 \beta_{2} ) q^{70} \) \( + ( -3011082276541158 - 3500766662000 \beta_{1} + 40355399250 \beta_{2} ) q^{71} \) \( + ( 5003905741164480 + 15295471565964 \beta_{1} + 279362720244 \beta_{2} ) q^{72} \) \( + ( -1971925206744418 - 9176510514784 \beta_{1} + 34167857364 \beta_{2} ) q^{73} \) \( + ( 922117223551914 - 7686953243750 \beta_{1} + 53196967200 \beta_{2} ) q^{74} \) \( + ( 811309814453125 + 1220703125000 \beta_{1} + 152587890625 \beta_{2} ) q^{75} \) \( + ( 5697503239139600 + 4788252792408 \beta_{1} - 596195109192 \beta_{2} ) q^{76} \) \( + ( -1790525580502212 - 9409181734944 \beta_{1} - 322452784932 \beta_{2} ) q^{77} \) \( + ( 5244133509540224 - 28245807774392 \beta_{1} + 133008342584 \beta_{2} ) q^{78} \) \( + ( 3059202702265700 - 6498016256864 \beta_{1} - 708980432364 \beta_{2} ) q^{79} \) \( + ( -2109232962500000 + 1446746875000 \beta_{1} - 424659375000 \beta_{2} ) q^{80} \) \( + ( 3849279886662581 + 45260492955616 \beta_{1} + 1428973504316 \beta_{2} ) q^{81} \) \( + ( 11372372426293398 - 39115458626602 \beta_{1} + 468552527280 \beta_{2} ) q^{82} \) \( + ( 5351679788862477 + 63469144591560 \beta_{1} + 630895655673 \beta_{2} ) q^{83} \) \( + ( 6911442769185024 - 16478539715904 \beta_{1} - 61925191104 \beta_{2} ) q^{84} \) \( + ( -3558020416406250 + 29898912500000 \beta_{1} - 64823437500 \beta_{2} ) q^{85} \) \( + ( -13758680528255928 + 25847621228564 \beta_{1} - 324320625036 \beta_{2} ) q^{86} \) \( + ( -41657160262368170 - 72401859897616 \beta_{1} - 643276001186 \beta_{2} ) q^{87} \) \( + ( 15430732768053120 + 36869310530256 \beta_{1} + 366393990576 \beta_{2} ) q^{88} \) \( + ( -10416093326009550 + 22733407766208 \beta_{1} - 1648351156392 \beta_{2} ) q^{89} \) \( + ( -18597285770703125 - 32365280078125 \beta_{1} - 1016243750000 \beta_{2} ) q^{90} \) \( + ( 14308769253611482 - 65229426688816 \beta_{1} + 352794374634 \beta_{2} ) q^{91} \) \( + ( -53849521528884816 + 84636400520976 \beta_{1} + 1581501710112 \beta_{2} ) q^{92} \) \( + ( -18568774806513696 - 184003564011264 \beta_{1} - 2275897426848 \beta_{2} ) q^{93} \) \( + ( 68720045595489084 - 119661850233776 \beta_{1} + 3464290677324 \beta_{2} ) q^{94} \) \( + ( -3888505000000000 + 38959212500000 \beta_{1} + 2690892187500 \beta_{2} ) q^{95} \) \( + ( -15167876499415808 + 103334327324096 \beta_{1} - 1834564570304 \beta_{2} ) q^{96} \) \( + ( 9434581491045374 + 95289079350240 \beta_{1} - 823011989604 \beta_{2} ) q^{97} \) \( + ( -7352919650002977 + 90742168998343 \beta_{1} - 463050853200 \beta_{2} ) q^{98} \) \( + ( 97787876678156686 + 245706427455824 \beta_{1} + 2673007356274 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 118q^{2} \) \(\mathstrut +\mathstrut 15944q^{3} \) \(\mathstrut +\mathstrut 16916q^{4} \) \(\mathstrut +\mathstrut 1171875q^{5} \) \(\mathstrut -\mathstrut 2419064q^{6} \) \(\mathstrut +\mathstrut 2139308q^{7} \) \(\mathstrut +\mathstrut 65050440q^{8} \) \(\mathstrut +\mathstrut 323892439q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 118q^{2} \) \(\mathstrut +\mathstrut 15944q^{3} \) \(\mathstrut +\mathstrut 16916q^{4} \) \(\mathstrut +\mathstrut 1171875q^{5} \) \(\mathstrut -\mathstrut 2419064q^{6} \) \(\mathstrut +\mathstrut 2139308q^{7} \) \(\mathstrut +\mathstrut 65050440q^{8} \) \(\mathstrut +\mathstrut 323892439q^{9} \) \(\mathstrut +\mathstrut 46093750q^{10} \) \(\mathstrut +\mathstrut 1789747516q^{11} \) \(\mathstrut +\mathstrut 3112179088q^{12} \) \(\mathstrut +\mathstrut 5414696794q^{13} \) \(\mathstrut +\mathstrut 10507173312q^{14} \) \(\mathstrut +\mathstrut 6228125000q^{15} \) \(\mathstrut -\mathstrut 16203699952q^{16} \) \(\mathstrut -\mathstrut 27402303962q^{17} \) \(\mathstrut -\mathstrut 142746901186q^{18} \) \(\mathstrut -\mathstrut 29956565300q^{19} \) \(\mathstrut +\mathstrut 6607812500q^{20} \) \(\mathstrut -\mathstrut 224495442624q^{21} \) \(\mathstrut -\mathstrut 294714945304q^{22} \) \(\mathstrut +\mathstrut 16254077844q^{23} \) \(\mathstrut +\mathstrut 897428301600q^{24} \) \(\mathstrut +\mathstrut 457763671875q^{25} \) \(\mathstrut +\mathstrut 2017382699956q^{26} \) \(\mathstrut +\mathstrut 3131715461840q^{27} \) \(\mathstrut +\mathstrut 1119244517216q^{28} \) \(\mathstrut -\mathstrut 2528278831750q^{29} \) \(\mathstrut -\mathstrut 944946875000q^{30} \) \(\mathstrut -\mathstrut 521256054664q^{31} \) \(\mathstrut -\mathstrut 10880399775712q^{32} \) \(\mathstrut +\mathstrut 1732417161568q^{33} \) \(\mathstrut -\mathstrut 32147385667828q^{34} \) \(\mathstrut +\mathstrut 835667187500q^{35} \) \(\mathstrut -\mathstrut 14985377520892q^{36} \) \(\mathstrut +\mathstrut 31762746900498q^{37} \) \(\mathstrut -\mathstrut 40258035935240q^{38} \) \(\mathstrut +\mathstrut 43003853320688q^{39} \) \(\mathstrut +\mathstrut 25410328125000q^{40} \) \(\mathstrut +\mathstrut 86833482954446q^{41} \) \(\mathstrut +\mathstrut 153974403759936q^{42} \) \(\mathstrut +\mathstrut 89258046385744q^{43} \) \(\mathstrut -\mathstrut 124942202946448q^{44} \) \(\mathstrut +\mathstrut 126520483984375q^{45} \) \(\mathstrut -\mathstrut 323339762673024q^{46} \) \(\mathstrut +\mathstrut 348182738140228q^{47} \) \(\mathstrut -\mathstrut 729510516165056q^{48} \) \(\mathstrut -\mathstrut 387320833396229q^{49} \) \(\mathstrut +\mathstrut 18005371093750q^{50} \) \(\mathstrut -\mathstrut 12409602773744q^{51} \) \(\mathstrut +\mathstrut 552556858385688q^{52} \) \(\mathstrut +\mathstrut 44014499212594q^{53} \) \(\mathstrut -\mathstrut 2265337034433200q^{54} \) \(\mathstrut +\mathstrut 699120123437500q^{55} \) \(\mathstrut -\mathstrut 490455582129600q^{56} \) \(\mathstrut +\mathstrut 4293013923032480q^{57} \) \(\mathstrut +\mathstrut 597742677475540q^{58} \) \(\mathstrut -\mathstrut 2133293278957100q^{59} \) \(\mathstrut +\mathstrut 1215694956250000q^{60} \) \(\mathstrut +\mathstrut 2822990449991866q^{61} \) \(\mathstrut +\mathstrut 5338539124096416q^{62} \) \(\mathstrut -\mathstrut 4366406213760516q^{63} \) \(\mathstrut -\mathstrut 2533946722090944q^{64} \) \(\mathstrut +\mathstrut 2115115935156250q^{65} \) \(\mathstrut -\mathstrut 4908048881135008q^{66} \) \(\mathstrut +\mathstrut 374173462156688q^{67} \) \(\mathstrut -\mathstrut 6873999340144024q^{68} \) \(\mathstrut -\mathstrut 13800336843711552q^{69} \) \(\mathstrut +\mathstrut 4104364575000000q^{70} \) \(\mathstrut -\mathstrut 9029705707562224q^{71} \) \(\mathstrut +\mathstrut 14996701114647720q^{72} \) \(\mathstrut -\mathstrut 5906564941861106q^{73} \) \(\mathstrut +\mathstrut 2774091820866692q^{74} \) \(\mathstrut +\mathstrut 2432861328125000q^{75} \) \(\mathstrut +\mathstrut 17087125269517200q^{76} \) \(\mathstrut -\mathstrut 5362490012556624q^{77} \) \(\mathstrut +\mathstrut 15760779344737648q^{78} \) \(\mathstrut +\mathstrut 9183397142621600q^{79} \) \(\mathstrut -\mathstrut 6329570293750000q^{80} \) \(\mathstrut +\mathstrut 11504008140536443q^{81} \) \(\mathstrut +\mathstrut 34156701290034076q^{82} \) \(\mathstrut +\mathstrut 15992201117651544q^{83} \) \(\mathstrut +\mathstrut 20750744922079872q^{84} \) \(\mathstrut -\mathstrut 10704024985156250q^{85} \) \(\mathstrut -\mathstrut 41302213526621384q^{86} \) \(\mathstrut -\mathstrut 124899722203208080q^{87} \) \(\mathstrut +\mathstrut 46255695387619680q^{88} \) \(\mathstrut -\mathstrut 31272661736951250q^{89} \) \(\mathstrut -\mathstrut 55760508275781250q^{90} \) \(\mathstrut +\mathstrut 42991889981897896q^{91} \) \(\mathstrut -\mathstrut 161631619485465312q^{92} \) \(\mathstrut -\mathstrut 55524596752956672q^{93} \) \(\mathstrut +\mathstrut 206283262927378352q^{94} \) \(\mathstrut -\mathstrut 11701783320312500q^{95} \) \(\mathstrut -\mathstrut 45608798390141824q^{96} \) \(\mathstrut +\mathstrut 28207632381796278q^{97} \) \(\mathstrut -\mathstrut 22149964169860474q^{98} \) \(\mathstrut +\mathstrut 293120596614370508q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(50686\) \(x\mathstrut +\mathstrut \) \(2014936\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} + 114 \nu - 67619 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(57\) \(\beta_{1}\mathstrut +\mathstrut \) \(67562\)\()/2\)

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
202.308
41.0886
−242.397
−364.616 20979.7 1872.98 390625. −7.64953e6 −1.29668e7 4.71081e7 3.11007e8 −1.42428e8
1.2 −42.1772 −13886.4 −129293. 390625. 585688. 3.78919e6 1.09815e7 6.36910e7 −1.64755e7
1.3 524.793 8850.68 144336. 390625. 4.64478e6 1.13170e7 6.96092e6 −5.08056e7 2.04997e8
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut -\mathstrut 118 T_{2}^{2} \) \(\mathstrut -\mathstrut 198104 T_{2} \) \(\mathstrut -\mathstrut 8070528 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(5))\).