# Properties

 Label 5.24.a.a Level $5$ Weight $24$ Character orbit 5.a Self dual yes Analytic conductor $16.760$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 5.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.7602018673$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - x^{2} - 215756x + 18660756$$ x^3 - x^2 - 215756*x + 18660756 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}\cdot 3^{3}\cdot 5$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 222) q^{2} + ( - \beta_{2} + 64 \beta_1 - 46476) q^{3} + (12 \beta_{2} - 334 \beta_1 - 3161172) q^{4} + 48828125 q^{5} + ( - 232 \beta_{2} - 247376 \beta_1 + 319966272) q^{6} + (24969 \beta_{2} + 134848 \beta_1 - 810894448) q^{7} + (7992 \beta_{2} - 9380284 \beta_1 - 4280140520) q^{8} + ( - 122796 \beta_{2} - 19484928 \beta_1 + 564361317) q^{9}+O(q^{10})$$ q + (b1 + 222) * q^2 + (-b2 + 64*b1 - 46476) * q^3 + (12*b2 - 334*b1 - 3161172) * q^4 + 48828125 * q^5 + (-232*b2 - 247376*b1 + 319966272) * q^6 + (24969*b2 + 134848*b1 - 810894448) * q^7 + (7992*b2 - 9380284*b1 - 4280140520) * q^8 + (-122796*b2 - 19484928*b1 + 564361317) * q^9 $$q + (\beta_1 + 222) q^{2} + ( - \beta_{2} + 64 \beta_1 - 46476) q^{3} + (12 \beta_{2} - 334 \beta_1 - 3161172) q^{4} + 48828125 q^{5} + ( - 232 \beta_{2} - 247376 \beta_1 + 319966272) q^{6} + (24969 \beta_{2} + 134848 \beta_1 - 810894448) q^{7} + (7992 \beta_{2} - 9380284 \beta_1 - 4280140520) q^{8} + ( - 122796 \beta_{2} - 19484928 \beta_1 + 564361317) q^{9} + (48828125 \beta_1 + 10839843750) q^{10} + ( - 3176250 \beta_{2} - 369424000 \beta_1 - 55095343468) q^{11} + (5188096 \beta_{2} - 117716896 \beta_1 - 820308397248) q^{12} + (7666020 \beta_{2} - 1088653568 \beta_1 - 1184990244666) q^{13} + (26587176 \beta_{2} + 3241905268 \beta_1 + 546154822136) q^{14} + ( - 48828125 \beta_{2} + 3125000000 \beta_1 - 2269335937500) q^{15} + ( - 205234704 \beta_{2} + 5058297928 \beta_1 - 22995961492304) q^{16} + (392947092 \beta_{2} + 52011875072 \beta_1 - 138701923089838) q^{17} + ( - 356615136 \beta_{2} + \cdots - 100907890084746) q^{18}+ \cdots + (11\!\cdots\!78 \beta_{2} + \cdots + 65\!\cdots\!44) q^{99}+O(q^{100})$$ q + (b1 + 222) * q^2 + (-b2 + 64*b1 - 46476) * q^3 + (12*b2 - 334*b1 - 3161172) * q^4 + 48828125 * q^5 + (-232*b2 - 247376*b1 + 319966272) * q^6 + (24969*b2 + 134848*b1 - 810894448) * q^7 + (7992*b2 - 9380284*b1 - 4280140520) * q^8 + (-122796*b2 - 19484928*b1 + 564361317) * q^9 + (48828125*b1 + 10839843750) * q^10 + (-3176250*b2 - 369424000*b1 - 55095343468) * q^11 + (5188096*b2 - 117716896*b1 - 820308397248) * q^12 + (7666020*b2 - 1088653568*b1 - 1184990244666) * q^13 + (26587176*b2 + 3241905268*b1 + 546154822136) * q^14 + (-48828125*b2 + 3125000000*b1 - 2269335937500) * q^15 + (-205234704*b2 + 5058297928*b1 - 22995961492304) * q^16 + (392947092*b2 + 52011875072*b1 - 138701923089838) * q^17 + (-356615136*b2 - 8902162251*b1 - 100907890084746) * q^18 + (-76236972*b2 - 238564061696*b1 - 325234681022820) * q^19 + (585937500*b2 - 16308593750*b1 - 154354101562500) * q^20 + (5020118376*b2 - 92936965632*b1 - 1700794779330528) * q^21 + (-7609338000*b2 - 374780544468*b1 - 1928715152127896) * q^22 + (-6575183169*b2 + 735154573120*b1 - 1659888970777456) * q^23 + (5721650304*b2 + 2177957767872*b1 - 3469936902975360) * q^24 + 2384185791015625 * q^25 + (-5397822816*b2 + 687616901462*b1 - 5891712530262508) * q^26 + (73066331718*b2 - 863743844736*b1 + 6682392692667720) * q^27 + (-143965113936*b2 + 2007754069160*b1 + 23740318991405296) * q^28 + (221499175272*b2 - 18079820219904*b1 + 31842287955802070) * q^29 + (-11328125000*b2 - 12078906250000*b1 + 15623353125000000) * q^30 + (-283700003250*b2 - 3265191408000*b1 + 67999176240265392) * q^31 + (-211576884000*b2 + 18950569937936*b1 + 56762545019997792) * q^32 + (-476827806532*b2 + 93842739874048*b1 + 107359740698802768) * q^33 + (1017089592864*b2 - 102660084168798*b1 + 238972797974167036) * q^34 + (1219189453125*b2 + 6584375000000*b1 - 39594455468750000) * q^35 + (566646424956*b2 + 8538947204058*b1 - 73631125415009124) * q^36 + (-1507517641272*b2 + 330639588451840*b1 - 161010059061744818) * q^37 + (-2939005712352*b2 - 205196253982996*b1 - 1307608288853841880) * q^38 + (2488613064986*b2 + 207164803262848*b1 - 851916686263830216) * q^39 + (390234375000*b2 - 458021679687500*b1 - 208991236328125000) * q^40 + (-10025979826500*b2 - 749931865824000*b1 - 1452708290578190278) * q^41 + (3904874788416*b2 - 819215936992320*b1 - 853206767473850496) * q^42 + (9151901654523*b2 + 1786831882165056*b1 - 1774717895615178836) * q^43 + (14537611626384*b2 + 120670631580312*b1 - 1915177746959035504) * q^44 + (-5995898437500*b2 - 951412500000000*b1 + 27556704931640625) * q^45 + (2246671708440*b2 - 3155617894198580*b1 + 3430897137054501512) * q^46 + (7451007190041*b2 + 10139748041326784*b1 + 4086087513107791592) * q^47 + (-11663760091904*b2 - 2747520184735360*b1 + 17395111531946559744) * q^48 + (-138103575543780*b2 + 7620615462216960*b1 + 17955316993492090793) * q^49 + (2384185791015625*b1 + 529289245605468750) * q^50 + (204445258357262*b2 - 22573133605491584*b1 - 4435098279752311128) * q^51 + (-61453656698616*b2 + 1965914025628108*b1 + 12187009698169586632) * q^52 + (139294979645004*b2 - 18891364110000896*b1 + 23880327980920028574) * q^53 + (62701405581168*b2 + 19241667964633824*b1 - 2907433362802101120) * q^54 + (-155090332031250*b2 - 18038281250000000*b1 - 2690202317773437500) * q^55 + (-342901462397088*b2 - 28370801512878384*b1 + 10924405953696503520) * q^56 + (314764328577476*b2 + 41607125960904448*b1 - 58234239606558378960) * q^57 + (4541332633152*b2 + 78512025657334646*b1 - 86303481071016040620) * q^58 + (-335630712239976*b2 + 16358437670616832*b1 - 140281138812701502060) * q^59 + (253325000000000*b2 - 5747895312500000*b1 - 40054120959375000000) * q^60 + (-247406271654000*b2 - 23901724378752000*b1 - 193908445941406671818) * q^61 + (-322882300146000*b2 + 22914472925836392*b1 - 2128955618811896976) * q^62 + (197615718837117*b2 - 98394267612174912*b1 - 233792654256433557936) * q^63 + (1737463435107264*b2 - 31183094486042848*b1 + 303397825906030900928) * q^64 + (374317382812500*b2 - 53156912500000000*b1 - 57860851790332031250) * q^65 + (649285071956576*b2 - 23644088335812032*b1 + 509232843106519048704) * q^66 + (-2014946101493343*b2 - 61496051260486208*b1 - 466556181056373557788) * q^67 + (-3511110536689512*b2 + 27885756581943972*b1 + 686115389111705039576) * q^68 + (543751136891448*b2 - 297019279288859136*b1 + 789484302319037853024) * q^69 + (1298201953125000*b2 + 158296155664062500*b1 + 26667715924609375000) * q^70 + (-1048180970768250*b2 + 688917794172944000*b1 - 136383891668990165688) * q^71 + (3660618374175384*b2 + 89973689803597332*b1 + 874979979035949306360) * q^72 + (-96940008304764*b2 + 603928614175727360*b1 + 1084986169325580653594) * q^73 + (2460157420150080*b2 - 594062456625489810*b1 + 1674672734010233268836) * q^74 + (-2384185791015625*b2 + 152587890625000000*b1 - 110807418823242187500) * q^75 + (-4761838686932976*b2 + 321836556473079832*b1 + 1372154640464188765840) * q^76 + (6203367841640508*b2 - 2122221854836100864*b1 - 5893059062455251022336) * q^77 + (4974590704140176*b2 - 555692759426748128*b1 + 886387067860784899968) * q^78 + (-3130025058096708*b2 + 1292856678944563456*b1 - 3387849515951436370480) * q^79 + (-10021225781250000*b2 + 246987203515625000*b1 - 1122849682241406250000) * q^80 + (17207972183289708*b2 + 2471115870557708544*b1 - 5866601567725440911799) * q^81 + (-19025162216388000*b2 - 2693201054177720278*b1 - 4216969311223019965716) * q^82 + (4363465832004087*b2 - 4314776175415261632*b1 - 4788450003009155216796) * q^83 + (-48037521625352448*b2 + 1027427347411979136*b1 + 9840228953862946012416) * q^84 + (19186869726562500*b2 + 2539642337500000000*b1 - 6772554838370996093750) * q^85 + (30593884240503672*b2 - 1255240648179825704*b1 + 8868729560709178691952) * q^86 + (5690359170864874*b2 + 6716151304014292352*b1 - 23268488447952989996040) * q^87 + (79817412826851744*b2 + 3564916259078228912*b1 + 16395166676969421283360) * q^88 + (-7512492359115144*b2 - 4061331204010132992*b1 + 18421036999708339733610) * q^89 + (-17412848437500000*b2 - 434675891162109375*b1 - 4927143070544238281250) * q^90 + (-89553395043850458*b2 - 1416229703775388544*b1 + 13853866870730972967392) * q^91 + (19535891110995792*b2 - 610092066929637928*b1 - 1651940752589585570288) * q^92 + (-115807423274239392*b2 + 5271578843797721088*b1 + 16020062575411057233408) * q^93 + (129127983685962408*b2 - 319841693241082356*b1 + 53420596644223211745736) * q^94 + (-3722508398437500*b2 - 11648635825000000000*b1 - 15880599659317382812500) * q^95 + (-92630683822065152*b2 - 1275507363926983936*b1 + 18729540519848122286592) * q^96 + (24921572003036556*b2 - 3252233106127280896*b1 - 77174992601432766599358) * q^97 + (-46656189997176480*b2 - 9112475898096073447*b1 + 43292415602379291430446) * q^98 + (110315676980611278*b2 + 17212119094957970304*b1 + 65241913027317883852644) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 666 q^{2} - 139428 q^{3} - 9483516 q^{4} + 146484375 q^{5} + 959898816 q^{6} - 2432683344 q^{7} - 12840421560 q^{8} + 1693083951 q^{9}+O(q^{10})$$ 3 * q + 666 * q^2 - 139428 * q^3 - 9483516 * q^4 + 146484375 * q^5 + 959898816 * q^6 - 2432683344 * q^7 - 12840421560 * q^8 + 1693083951 * q^9 $$3 q + 666 q^{2} - 139428 q^{3} - 9483516 q^{4} + 146484375 q^{5} + 959898816 q^{6} - 2432683344 q^{7} - 12840421560 q^{8} + 1693083951 q^{9} + 32519531250 q^{10} - 165286030404 q^{11} - 2460925191744 q^{12} - 3554970733998 q^{13} + 1638464466408 q^{14} - 6808007812500 q^{15} - 68987884476912 q^{16} - 416105769269514 q^{17} - 302723670254238 q^{18} - 975704043068460 q^{19} - 463062304687500 q^{20} - 51\!\cdots\!84 q^{21}+ \cdots + 19\!\cdots\!32 q^{99}+O(q^{100})$$ 3 * q + 666 * q^2 - 139428 * q^3 - 9483516 * q^4 + 146484375 * q^5 + 959898816 * q^6 - 2432683344 * q^7 - 12840421560 * q^8 + 1693083951 * q^9 + 32519531250 * q^10 - 165286030404 * q^11 - 2460925191744 * q^12 - 3554970733998 * q^13 + 1638464466408 * q^14 - 6808007812500 * q^15 - 68987884476912 * q^16 - 416105769269514 * q^17 - 302723670254238 * q^18 - 975704043068460 * q^19 - 463062304687500 * q^20 - 5102384337991584 * q^21 - 5786145456383688 * q^22 - 4979666912332368 * q^23 - 10409810708926080 * q^24 + 7152557373046875 * q^25 - 17675137590787524 * q^26 + 20047178078003160 * q^27 + 71220956974215888 * q^28 + 95526863867406210 * q^29 + 46870059375000000 * q^30 + 203997528720796176 * q^31 + 170287635059993376 * q^32 + 322079222096408304 * q^33 + 716918393922501108 * q^34 - 118783366406250000 * q^35 - 220893376245027372 * q^36 - 483030177185234454 * q^37 - 3922824866561525640 * q^38 - 2555750058791490648 * q^39 - 626973708984375000 * q^40 - 4358124871734570834 * q^41 - 2559620302421551488 * q^42 - 5324153686845536508 * q^43 - 5745533240877106512 * q^44 + 82670114794921875 * q^45 + 10292691411163504536 * q^46 + 12258262539323374776 * q^47 + 52185334595839679232 * q^48 + 53865950980476272379 * q^49 + 1587867736816406250 * q^50 - 13305294839256933384 * q^51 + 36561029094508759896 * q^52 + 71640983942760085722 * q^53 - 8722300088406303360 * q^54 - 8070606953320312500 * q^55 + 32773217861089510560 * q^56 - 174702718819675136880 * q^57 - 258910443213048121860 * q^58 - 420843416438104506180 * q^59 - 120162362878125000000 * q^60 - 581725337824220015454 * q^61 - 6386866856435690928 * q^62 - 701377962769300673808 * q^63 + 910193477718092702784 * q^64 - 173582555370996093750 * q^65 + 1527698529319557146112 * q^66 - 1399668543169120673364 * q^67 + 2058346167335115118728 * q^68 + 2368452906957113559072 * q^69 + 80003147773828125000 * q^70 - 409151675006970497064 * q^71 + 2624939937107847919080 * q^72 + 3254958507976741960782 * q^73 + 5024018202030699806508 * q^74 - 332422256469726562500 * q^75 + 4116463921392566297520 * q^76 - 17679177187365753067008 * q^77 + 2659161203582354699904 * q^78 - 10163548547854309111440 * q^79 - 3368549046724218750000 * q^80 - 17599804703176322735397 * q^81 - 12650907933669059897148 * q^82 - 14365350009027465650388 * q^83 + 29520686861588838037248 * q^84 - 20317664515112988281250 * q^85 + 26606188682127536075856 * q^86 - 69805465343858969988120 * q^87 + 49185500030908263850080 * q^88 + 55263110999125019200830 * q^89 - 14781429211632714843750 * q^90 + 41561600612192918902176 * q^91 - 4955822257768756710864 * q^92 + 48060187726233171700224 * q^93 + 160261789932669635237208 * q^94 - 47641798977952148437500 * q^95 + 56188621559544366859776 * q^96 - 231524977804298299798074 * q^97 + 129877246807137874291338 * q^98 + 195725739081953651557932 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 215756x + 18660756$$ :

 $$\beta_{1}$$ $$=$$ $$6\nu - 2$$ 6*v - 2 $$\beta_{2}$$ $$=$$ $$3\nu^{2} + 387\nu - 431642$$ 3*v^2 + 387*v - 431642
 $$\nu$$ $$=$$ $$( \beta _1 + 2 ) / 6$$ (b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} - 129\beta _1 + 863026 ) / 6$$ (2*b2 - 129*b1 + 863026) / 6

## 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
 −502.392 89.8102 413.582
−2794.35 −370647. −580214. 4.88281e7 1.03572e9 2.05641e9 2.50620e10 4.32361e10 −1.36443e11
1.2 758.861 360571. −7.81274e6 4.88281e7 2.73623e8 −1.00441e10 −1.22946e10 3.58682e10 3.70538e10
1.3 2701.49 −129352. −1.09056e6 4.88281e7 −3.49442e8 5.55505e9 −2.56079e10 −7.74113e10 1.31909e11
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$

## 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 5.24.a.a 3
3.b odd 2 1 45.24.a.a 3
5.b even 2 1 25.24.a.b 3
5.c odd 4 2 25.24.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.24.a.a 3 1.a even 1 1 trivial
25.24.a.b 3 5.b even 2 1
25.24.b.b 6 5.c odd 4 2
45.24.a.a 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 666T_{2}^{2} - 7619376T_{2} + 5728572416$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 666 T^{2} + \cdots + 5728572416$$
$3$ $$T^{3} + 139428 T^{2} + \cdots - 17\!\cdots\!72$$
$5$ $$(T - 48828125)^{3}$$
$7$ $$T^{3} + 2432683344 T^{2} + \cdots + 11\!\cdots\!16$$
$11$ $$T^{3} + 165286030404 T^{2} + \cdots + 10\!\cdots\!32$$
$13$ $$T^{3} + 3554970733998 T^{2} + \cdots - 29\!\cdots\!32$$
$17$ $$T^{3} + 416105769269514 T^{2} + \cdots - 53\!\cdots\!84$$
$19$ $$T^{3} + 975704043068460 T^{2} + \cdots - 15\!\cdots\!00$$
$23$ $$T^{3} + \cdots - 80\!\cdots\!92$$
$29$ $$T^{3} + \cdots + 28\!\cdots\!00$$
$31$ $$T^{3} + \cdots + 60\!\cdots\!12$$
$37$ $$T^{3} + \cdots + 23\!\cdots\!16$$
$41$ $$T^{3} + \cdots - 54\!\cdots\!48$$
$43$ $$T^{3} + \cdots - 12\!\cdots\!12$$
$47$ $$T^{3} + \cdots + 53\!\cdots\!16$$
$53$ $$T^{3} + \cdots + 40\!\cdots\!28$$
$59$ $$T^{3} + \cdots + 27\!\cdots\!00$$
$61$ $$T^{3} + \cdots + 55\!\cdots\!32$$
$67$ $$T^{3} + \cdots - 15\!\cdots\!84$$
$71$ $$T^{3} + \cdots + 19\!\cdots\!72$$
$73$ $$T^{3} + \cdots + 27\!\cdots\!08$$
$79$ $$T^{3} + \cdots + 11\!\cdots\!00$$
$83$ $$T^{3} + \cdots - 11\!\cdots\!52$$
$89$ $$T^{3} + \cdots - 36\!\cdots\!00$$
$97$ $$T^{3} + \cdots + 44\!\cdots\!16$$