# Properties

 Label 3.24.a.b Level $3$ Weight $24$ Character orbit 3.a Self dual yes Analytic conductor $10.056$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 3.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.0561211204$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{530401})$$ Defining polynomial: $$x^{2} - x - 132600$$ x^2 - x - 132600 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{530401}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 621) q^{2} - 177147 q^{3} + (1242 \beta - 3229358) q^{4} + ( - 62720 \beta - 23404410) q^{5} + (177147 \beta + 110008287) q^{6} + ( - 3642624 \beta - 105981952) q^{7} + (10846684 \beta + 1285934508) q^{8} + 31381059609 q^{9}+O(q^{10})$$ q + (-b - 621) * q^2 - 177147 * q^3 + (1242*b - 3229358) * q^4 + (-62720*b - 23404410) * q^5 + (177147*b + 110008287) * q^6 + (-3642624*b - 105981952) * q^7 + (10846684*b + 1285934508) * q^8 + 31381059609 * q^9 $$q + ( - \beta - 621) q^{2} - 177147 q^{3} + (1242 \beta - 3229358) q^{4} + ( - 62720 \beta - 23404410) q^{5} + (177147 \beta + 110008287) q^{6} + ( - 3642624 \beta - 105981952) q^{7} + (10846684 \beta + 1285934508) q^{8} + 31381059609 q^{9} + (62353530 \beta + 313934895090) q^{10} + ( - 135883264 \beta + 734486183244) q^{11} + ( - 220016574 \beta + 572071081626) q^{12} + ( - 1301405184 \beta + 5245827132374) q^{13} + (2368051456 \beta + 17454277502208) q^{14} + (11110659840 \beta + 4146021018270) q^{15} + ( - 18440376408 \beta - 25486575338360) q^{16} + (15197836800 \beta - 105444005760414) q^{17} + ( - 31381059609 \beta - 19487638017189) q^{18} + ( - 142703202816 \beta - 453691224268972) q^{19} + (173477056540 \beta - 296274520879380) q^{20} + (645279913728 \beta + 18774384850944) q^{21} + ( - 650102676300 \beta + 192537652185252) q^{22} + (114772628992 \beta + 50\!\cdots\!56) q^{23}+ \cdots + ( - 42\!\cdots\!76 \beta + 23\!\cdots\!96) q^{99}+O(q^{100})$$ q + (-b - 621) * q^2 - 177147 * q^3 + (1242*b - 3229358) * q^4 + (-62720*b - 23404410) * q^5 + (177147*b + 110008287) * q^6 + (-3642624*b - 105981952) * q^7 + (10846684*b + 1285934508) * q^8 + 31381059609 * q^9 + (62353530*b + 313934895090) * q^10 + (-135883264*b + 734486183244) * q^11 + (-220016574*b + 572071081626) * q^12 + (-1301405184*b + 5245827132374) * q^13 + (2368051456*b + 17454277502208) * q^14 + (11110659840*b + 4146021018270) * q^15 + (-18440376408*b - 25486575338360) * q^16 + (15197836800*b - 105444005760414) * q^17 + (-31381059609*b - 19487638017189) * q^18 + (-142703202816*b - 453691224268972) * q^19 + (173477056540*b - 296274520879380) * q^20 + (645279913728*b + 18774384850944) * q^21 + (-650102676300*b + 192537652185252) * q^22 + (114772628992*b + 5058461661946056) * q^23 + (-1921457530548*b - 227799440288676) * q^24 + (2935849190400*b + 7405252898795575) * q^25 + (-4437654513110*b + 2954740849784802) * q^26 - 5559060566555523 * q^27 + (11631707371008*b - 21254217021293056) * q^28 + (10893499728128*b + 9239376772588446) * q^29 + (-11045740778910*b - 55612624860508230) * q^30 + (-25314451976448*b - 136396811296372744) * q^31 + (-54050531088144*b + 93067109568453168) * q^32 + (24071312567808*b - 130112023903124868) * q^33 + (96006149107614*b - 7067802951794106) * q^34 + (91900653601280*b + 1093084826229411840) * q^35 + (38975276034378*b - 101340675896801022) * q^36 + (-546678772752384*b - 239497518393682) * q^37 + (542309913217708*b + 962951543562314556) * q^38 + (230540024130048*b - 929282539018656978) * q^39 + (-334514051818200*b - 3277601933357892600) * q^40 + (-1766616871075328*b + 2777857480654385706) * q^41 + (-419493211276032*b - 3091972896683640576) * q^42 + (4583930058975744*b - 599161073804660020) * q^43 + (1351047545253560*b - 3177546568147359144) * q^44 + (-1968220058676480*b - 734455185323475690) * q^45 + (-5129735464550088*b - 3689184346778372904) * q^46 + (667845663375872*b + 13154282836427888640) * q^47 + (3266657359547976*b + 4514870361464458920) * q^48 + (772104803844096*b + 35982116424678135945) * q^49 + (-9228415246033975*b - 18613258168088205675) * q^50 + (-2692251195609600*b + 18679089288440058858) * q^51 + (10718020540600380*b - 24656453994293443444) * q^52 + (25226888867419392*b - 20563750949707612314) * q^53 + (5559060566555523*b + 3452176611830979783) * q^54 + (-42886705790269440*b + 23493336262593844680) * q^55 + (-5833728644316160*b - 188743446110629186560) * q^56 + (25279444269245952*b + 80370039305575582884) * q^57 + (-16004240103755934*b - 57738961319466798918) * q^58 + (-51173029829187584*b - 153768530954722143348) * q^59 + (-30730940134891380*b + 52484142550219528860) * q^60 + (102017420065926144*b + 297480589549251875510) * q^61 + (152117085973746952*b + 205543715599887434856) * q^62 + (-114309400877174016*b - 3325825953190176768) * q^63 + (95187359296444320*b + 414018316391103977248) * q^64 + (-298559657240035840*b + 266866207581388222980) * q^65 + (115163738798516100*b - 34107467471660836044) * q^66 + (219887015935518720*b - 815112912563496632044) * q^67 + (-180040711007208588*b + 430621718471470944612) * q^68 + (-20331626908045824*b - 896091308028757982232) * q^69 + (-1150155132115806720*b - 1117503464225417372160) * q^70 + (1000477109551480320*b - 191971299290824636488) * q^71 + (340380437163986556*b + 40353987448818087372) * q^72 + (744202923877896192*b - 1406101175720280846214) * q^73 + (339727015397624146*b + 2609779437678657510378) * q^74 + (-520076876531788800*b - 1311818335262939724525) * q^75 + (-102644770902591096*b + 619069442355025463528) * q^76 + (-2661055825190140928*b + 2284958789562030759936) * q^77 + (786117184033897170*b - 523423477316828319894) * q^78 + (-167160086945199360*b + 10534694287656642440) * q^79 + (2030104135229098480*b + 6117560905046011291440) * q^80 + 984770902183611232881 * q^81 + (-1680788403716607018*b + 6708088699830651895326) * q^82 + (1186031845274398208*b - 7408631040162201348012) * q^83 + (-2060522065651954176*b + 3765120782671000991232) * q^84 + (6257751637712878080*b - 2082389091920491438260) * q^85 + (-2247459492819277004*b - 21509810758064448467676) * q^86 + (-1929750796338690816*b - 1636727877133725443562) * q^87 + (7792002553776488784*b - 6091239192053213678832) * q^88 + (-7489945387427920896*b - 21724902630442228278582) * q^89 + (1956719841761569770*b + 9851609656164451419810) * q^90 + (-18970650350493470208*b + 22073472513261952208896) * q^91 + (5911967476520654416*b - 15655117104549130329072) * q^92 + (4484379224271833856*b + 24162285930718542481368) * q^93 + (-13569014993384305152*b - 11356843710723751807488) * q^94 + (31795397853168742400*b + 53343822301422237214200) * q^95 + (9574889430671445168*b - 16486559258722773351696) * q^96 + (19434897144083543040*b + 53568286522621084225346) * q^97 + (-36461593507865319561*b - 26030620740298533684309) * q^98 + (-4264160807449483776*b + 23048954698366860991596) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 1242 q^{2} - 354294 q^{3} - 6458716 q^{4} - 46808820 q^{5} + 220016574 q^{6} - 211963904 q^{7} + 2571869016 q^{8} + 62762119218 q^{9}+O(q^{10})$$ 2 * q - 1242 * q^2 - 354294 * q^3 - 6458716 * q^4 - 46808820 * q^5 + 220016574 * q^6 - 211963904 * q^7 + 2571869016 * q^8 + 62762119218 * q^9 $$2 q - 1242 q^{2} - 354294 q^{3} - 6458716 q^{4} - 46808820 q^{5} + 220016574 q^{6} - 211963904 q^{7} + 2571869016 q^{8} + 62762119218 q^{9} + 627869790180 q^{10} + 1468972366488 q^{11} + 1144142163252 q^{12} + 10491654264748 q^{13} + 34908555004416 q^{14} + 8292042036540 q^{15} - 50973150676720 q^{16} - 210888011520828 q^{17} - 38975276034378 q^{18} - 907382448537944 q^{19} - 592549041758760 q^{20} + 37548769701888 q^{21} + 385075304370504 q^{22} + 10\!\cdots\!12 q^{23}+ \cdots + 46\!\cdots\!92 q^{99}+O(q^{100})$$ 2 * q - 1242 * q^2 - 354294 * q^3 - 6458716 * q^4 - 46808820 * q^5 + 220016574 * q^6 - 211963904 * q^7 + 2571869016 * q^8 + 62762119218 * q^9 + 627869790180 * q^10 + 1468972366488 * q^11 + 1144142163252 * q^12 + 10491654264748 * q^13 + 34908555004416 * q^14 + 8292042036540 * q^15 - 50973150676720 * q^16 - 210888011520828 * q^17 - 38975276034378 * q^18 - 907382448537944 * q^19 - 592549041758760 * q^20 + 37548769701888 * q^21 + 385075304370504 * q^22 + 10116923323892112 * q^23 - 455598880577352 * q^24 + 14810505797591150 * q^25 + 5909481699569604 * q^26 - 11118121133111046 * q^27 - 42508434042586112 * q^28 + 18478753545176892 * q^29 - 111225249721016460 * q^30 - 272793622592745488 * q^31 + 186134219136906336 * q^32 - 260224047806249736 * q^33 - 14135605903588212 * q^34 + 2186169652458823680 * q^35 - 202681351793602044 * q^36 - 478995036787364 * q^37 + 1925903087124629112 * q^38 - 1858565078037313956 * q^39 - 6555203866715785200 * q^40 + 5555714961308771412 * q^41 - 6183945793367281152 * q^42 - 1198322147609320040 * q^43 - 6355093136294718288 * q^44 - 1468910370646951380 * q^45 - 7378368693556745808 * q^46 + 26308565672855777280 * q^47 + 9029740722928917840 * q^48 + 71964232849356271890 * q^49 - 37226516336176411350 * q^50 + 37358178576880117716 * q^51 - 49312907988586886888 * q^52 - 41127501899415224628 * q^53 + 6904353223661959566 * q^54 + 46986672525187689360 * q^55 - 377486892221258373120 * q^56 + 160740078611151165768 * q^57 - 115477922638933597836 * q^58 - 307537061909444286696 * q^59 + 104968285100439057720 * q^60 + 594961179098503751020 * q^61 + 411087431199774869712 * q^62 - 6651651906380353536 * q^63 + 828036632782207954496 * q^64 + 533732415162776445960 * q^65 - 68214934943321672088 * q^66 - 1630225825126993264088 * q^67 + 861243436942941889224 * q^68 - 1792182616057515964464 * q^69 - 2235006928450834744320 * q^70 - 383942598581649272976 * q^71 + 80707974897636174744 * q^72 - 2812202351440561692428 * q^73 + 5219558875357315020756 * q^74 - 2623636670525879449050 * q^75 + 1238138884710050927056 * q^76 + 4569917579124061519872 * q^77 - 1046846954633656639788 * q^78 + 21069388575313284880 * q^79 + 12235121810092022582880 * q^80 + 1969541804367222465762 * q^81 + 13416177399661303790652 * q^82 - 14817262080324402696024 * q^83 + 7530241565342001982464 * q^84 - 4164778183840982876520 * q^85 - 43019621516128896935352 * q^86 - 3273455754267450887124 * q^87 - 12182478384106427357664 * q^88 - 43449805260884456557164 * q^89 + 19703219312328902839620 * q^90 + 44146945026523904417792 * q^91 - 31310234209098260658144 * q^92 + 48324571861437084962736 * q^93 - 22713687421447503614976 * q^94 + 106687644602844474428400 * q^95 - 32973118517445546703392 * q^96 + 107136573045242168450692 * q^97 - 52061241480597067368618 * q^98 + 46097909396733721983192 * q^99

## 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
 364.643 −363.643
−2805.86 −177147. −515763. −1.60439e8 4.97050e8 −8.06460e9 2.49844e10 3.13811e10 4.50169e11
1.2 1563.86 −177147. −5.94295e6 1.13630e8 −2.77033e8 7.85264e9 −2.24125e10 3.13811e10 1.77701e11
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$

## 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 3.24.a.b 2
3.b odd 2 1 9.24.a.c 2
4.b odd 2 1 48.24.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.b 2 1.a even 1 1 trivial
9.24.a.c 2 3.b odd 2 1
48.24.a.g 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1242T_{2} - 4387968$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1242 T - 4387968$$
$3$ $$(T + 177147)^{2}$$
$5$ $$T^{2} + 46808820 T - 18\!\cdots\!00$$
$7$ $$T^{2} + 211963904 T - 63\!\cdots\!80$$
$11$ $$T^{2} - 1468972366488 T + 45\!\cdots\!72$$
$13$ $$T^{2} - 10491654264748 T + 19\!\cdots\!72$$
$17$ $$T^{2} + 210888011520828 T + 10\!\cdots\!96$$
$19$ $$T^{2} + 907382448537944 T + 10\!\cdots\!80$$
$23$ $$T^{2} + \cdots + 25\!\cdots\!60$$
$29$ $$T^{2} + \cdots - 48\!\cdots\!40$$
$31$ $$T^{2} + \cdots + 15\!\cdots\!00$$
$37$ $$T^{2} + 478995036787364 T - 14\!\cdots\!80$$
$41$ $$T^{2} + \cdots - 71\!\cdots\!20$$
$43$ $$T^{2} + \cdots - 99\!\cdots\!24$$
$47$ $$T^{2} + \cdots + 17\!\cdots\!44$$
$53$ $$T^{2} + \cdots - 26\!\cdots\!80$$
$59$ $$T^{2} + \cdots + 11\!\cdots\!00$$
$61$ $$T^{2} + \cdots + 38\!\cdots\!76$$
$67$ $$T^{2} + \cdots + 43\!\cdots\!36$$
$71$ $$T^{2} + \cdots - 47\!\cdots\!56$$
$73$ $$T^{2} + \cdots - 66\!\cdots\!80$$
$79$ $$T^{2} + \cdots - 13\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 48\!\cdots\!68$$
$89$ $$T^{2} + \cdots + 20\!\cdots\!80$$
$97$ $$T^{2} + \cdots + 10\!\cdots\!16$$