Properties

 Label 9.24.a.c Level $9$ Weight $24$ Character orbit 9.a Self dual yes Analytic conductor $30.168$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$30.1683633611$$ Analytic rank: $$1$$ 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: no (minimal twist has level 3) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 621) q^{2} + ( - 1242 \beta - 3229358) q^{4} + ( - 62720 \beta + 23404410) q^{5} + (3642624 \beta - 105981952) q^{7} + (10846684 \beta - 1285934508) q^{8}+O(q^{10})$$ q + (-b + 621) * q^2 + (-1242*b - 3229358) * q^4 + (-62720*b + 23404410) * q^5 + (3642624*b - 105981952) * q^7 + (10846684*b - 1285934508) * q^8 $$q + ( - \beta + 621) q^{2} + ( - 1242 \beta - 3229358) q^{4} + ( - 62720 \beta + 23404410) q^{5} + (3642624 \beta - 105981952) q^{7} + (10846684 \beta - 1285934508) q^{8} + ( - 62353530 \beta + 313934895090) q^{10} + ( - 135883264 \beta - 734486183244) q^{11} + (1301405184 \beta + 5245827132374) q^{13} + (2368051456 \beta - 17454277502208) q^{14} + (18440376408 \beta - 25486575338360) q^{16} + (15197836800 \beta + 105444005760414) q^{17} + (142703202816 \beta - 453691224268972) q^{19} + (173477056540 \beta + 296274520879380) q^{20} + (650102676300 \beta + 192537652185252) q^{22} + (114772628992 \beta - 50\!\cdots\!56) q^{23}+ \cdots + ( - 36\!\cdots\!61 \beta + 26\!\cdots\!09) q^{98}+O(q^{100})$$ q + (-b + 621) * q^2 + (-1242*b - 3229358) * q^4 + (-62720*b + 23404410) * q^5 + (3642624*b - 105981952) * q^7 + (10846684*b - 1285934508) * q^8 + (-62353530*b + 313934895090) * q^10 + (-135883264*b - 734486183244) * q^11 + (1301405184*b + 5245827132374) * q^13 + (2368051456*b - 17454277502208) * q^14 + (18440376408*b - 25486575338360) * q^16 + (15197836800*b + 105444005760414) * q^17 + (142703202816*b - 453691224268972) * q^19 + (173477056540*b + 296274520879380) * q^20 + (650102676300*b + 192537652185252) * q^22 + (114772628992*b - 5058461661946056) * q^23 + (-2935849190400*b + 7405252898795575) * q^25 + (-4437654513110*b - 2954740849784802) * q^26 + (-11631707371008*b - 21254217021293056) * q^28 + (10893499728128*b - 9239376772588446) * q^29 + (25314451976448*b - 136396811296372744) * q^31 + (-54050531088144*b - 93067109568453168) * q^32 + (-96006149107614*b - 7067802951794106) * q^34 + (91900653601280*b - 1093084826229411840) * q^35 + (546678772752384*b - 239497518393682) * q^37 + (542309913217708*b - 962951543562314556) * q^38 + (334514051818200*b - 3277601933357892600) * q^40 + (-1766616871075328*b - 2777857480654385706) * q^41 + (-4583930058975744*b - 599161073804660020) * q^43 + (1351047545253560*b + 3177546568147359144) * q^44 + (5129735464550088*b - 3689184346778372904) * q^46 + (667845663375872*b - 13154282836427888640) * q^47 + (-772104803844096*b + 35982116424678135945) * q^49 + (-9228415246033975*b + 18613258168088205675) * q^50 + (-10718020540600380*b - 24656453994293443444) * q^52 + (25226888867419392*b + 20563750949707612314) * q^53 + (42886705790269440*b + 23493336262593844680) * q^55 + (-5833728644316160*b + 188743446110629186560) * q^56 + (16004240103755934*b - 57738961319466798918) * q^58 + (-51173029829187584*b + 153768530954722143348) * q^59 + (-102017420065926144*b + 297480589549251875510) * q^61 + (152117085973746952*b - 205543715599887434856) * q^62 + (-95187359296444320*b + 414018316391103977248) * q^64 + (-298559657240035840*b - 266866207581388222980) * q^65 + (-219887015935518720*b - 815112912563496632044) * q^67 + (-180040711007208588*b - 430621718471470944612) * q^68 + (1150155132115806720*b - 1117503464225417372160) * q^70 + (1000477109551480320*b + 191971299290824636488) * q^71 + (-744202923877896192*b - 1406101175720280846214) * q^73 + (339727015397624146*b - 2609779437678657510378) * q^74 + (102644770902591096*b + 619069442355025463528) * q^76 + (-2661055825190140928*b - 2284958789562030759936) * q^77 + (167160086945199360*b + 10534694287656642440) * q^79 + (2030104135229098480*b - 6117560905046011291440) * q^80 + (1680788403716607018*b + 6708088699830651895326) * q^82 + (1186031845274398208*b + 7408631040162201348012) * q^83 + (-6257751637712878080*b - 2082389091920491438260) * q^85 + (-2247459492819277004*b + 21509810758064448467676) * q^86 + (-7792002553776488784*b - 6091239192053213678832) * q^88 + (-7489945387427920896*b + 21724902630442228278582) * q^89 + (18970650350493470208*b + 22073472513261952208896) * q^91 + (5911967476520654416*b + 15655117104549130329072) * q^92 + (13569014993384305152*b - 11356843710723751807488) * q^94 + (31795397853168742400*b - 53343822301422237214200) * q^95 + (-19434897144083543040*b + 53568286522621084225346) * q^97 + (-36461593507865319561*b + 26030620740298533684309) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 1242 q^{2} - 6458716 q^{4} + 46808820 q^{5} - 211963904 q^{7} - 2571869016 q^{8}+O(q^{10})$$ 2 * q + 1242 * q^2 - 6458716 * q^4 + 46808820 * q^5 - 211963904 * q^7 - 2571869016 * q^8 $$2 q + 1242 q^{2} - 6458716 q^{4} + 46808820 q^{5} - 211963904 q^{7} - 2571869016 q^{8} + 627869790180 q^{10} - 1468972366488 q^{11} + 10491654264748 q^{13} - 34908555004416 q^{14} - 50973150676720 q^{16} + 210888011520828 q^{17} - 907382448537944 q^{19} + 592549041758760 q^{20} + 385075304370504 q^{22} - 10\!\cdots\!12 q^{23}+ \cdots + 52\!\cdots\!18 q^{98}+O(q^{100})$$ 2 * q + 1242 * q^2 - 6458716 * q^4 + 46808820 * q^5 - 211963904 * q^7 - 2571869016 * q^8 + 627869790180 * q^10 - 1468972366488 * q^11 + 10491654264748 * q^13 - 34908555004416 * q^14 - 50973150676720 * q^16 + 210888011520828 * q^17 - 907382448537944 * q^19 + 592549041758760 * q^20 + 385075304370504 * q^22 - 10116923323892112 * q^23 + 14810505797591150 * q^25 - 5909481699569604 * q^26 - 42508434042586112 * q^28 - 18478753545176892 * q^29 - 272793622592745488 * q^31 - 186134219136906336 * q^32 - 14135605903588212 * q^34 - 2186169652458823680 * q^35 - 478995036787364 * q^37 - 1925903087124629112 * q^38 - 6555203866715785200 * q^40 - 5555714961308771412 * q^41 - 1198322147609320040 * q^43 + 6355093136294718288 * q^44 - 7378368693556745808 * q^46 - 26308565672855777280 * q^47 + 71964232849356271890 * q^49 + 37226516336176411350 * q^50 - 49312907988586886888 * q^52 + 41127501899415224628 * q^53 + 46986672525187689360 * q^55 + 377486892221258373120 * q^56 - 115477922638933597836 * q^58 + 307537061909444286696 * q^59 + 594961179098503751020 * q^61 - 411087431199774869712 * q^62 + 828036632782207954496 * q^64 - 533732415162776445960 * q^65 - 1630225825126993264088 * q^67 - 861243436942941889224 * q^68 - 2235006928450834744320 * q^70 + 383942598581649272976 * q^71 - 2812202351440561692428 * q^73 - 5219558875357315020756 * q^74 + 1238138884710050927056 * q^76 - 4569917579124061519872 * q^77 + 21069388575313284880 * q^79 - 12235121810092022582880 * q^80 + 13416177399661303790652 * q^82 + 14817262080324402696024 * q^83 - 4164778183840982876520 * q^85 + 43019621516128896935352 * q^86 - 12182478384106427357664 * q^88 + 43449805260884456557164 * q^89 + 44146945026523904417792 * q^91 + 31310234209098260658144 * q^92 - 22713687421447503614976 * q^94 - 106687644602844474428400 * q^95 + 107136573045242168450692 * q^97 + 52061241480597067368618 * q^98

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
−1563.86 0 −5.94295e6 −1.13630e8 0 7.85264e9 2.24125e10 0 1.77701e11
1.2 2805.86 0 −515763. 1.60439e8 0 −8.06460e9 −2.49844e10 0 4.50169e11
 $$n$$: 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 9.24.a.c 2
3.b odd 2 1 3.24.a.b 2
12.b even 2 1 48.24.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.b 2 3.b odd 2 1
9.24.a.c 2 1.a even 1 1 trivial
48.24.a.g 2 12.b even 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 $$S_{24}^{\mathrm{new}}(\Gamma_0(9))$$.

Hecke characteristic polynomials

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