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 $1$

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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: \(x^{2} - x - 132600\)
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.

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

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:

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 \( T_{2}^{2} - 1242 T_{2} - 4387968 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4387968 - 1242 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -18230649038977500 - 46808820 T + T^{2} \)
$7$ \( -63328399416459591680 + 211963904 T + T^{2} \)
$11$ \( \)\(45\!\cdots\!72\)\( + 1468972366488 T + T^{2} \)
$13$ \( \)\(19\!\cdots\!72\)\( - 10491654264748 T + T^{2} \)
$17$ \( \)\(10\!\cdots\!96\)\( - 210888011520828 T + T^{2} \)
$19$ \( \)\(10\!\cdots\!80\)\( + 907382448537944 T + T^{2} \)
$23$ \( \)\(25\!\cdots\!60\)\( + 10116923323892112 T + T^{2} \)
$29$ \( -\)\(48\!\cdots\!40\)\( + 18478753545176892 T + T^{2} \)
$31$ \( \)\(15\!\cdots\!00\)\( + 272793622592745488 T + T^{2} \)
$37$ \( -\)\(14\!\cdots\!80\)\( + 478995036787364 T + T^{2} \)
$41$ \( -\)\(71\!\cdots\!20\)\( + 5555714961308771412 T + T^{2} \)
$43$ \( -\)\(99\!\cdots\!24\)\( + 1198322147609320040 T + T^{2} \)
$47$ \( \)\(17\!\cdots\!44\)\( + 26308565672855777280 T + T^{2} \)
$53$ \( -\)\(26\!\cdots\!80\)\( - 41127501899415224628 T + T^{2} \)
$59$ \( \)\(11\!\cdots\!00\)\( - \)\(30\!\cdots\!96\)\( T + T^{2} \)
$61$ \( \)\(38\!\cdots\!76\)\( - \)\(59\!\cdots\!20\)\( T + T^{2} \)
$67$ \( \)\(43\!\cdots\!36\)\( + \)\(16\!\cdots\!88\)\( T + T^{2} \)
$71$ \( -\)\(47\!\cdots\!56\)\( - \)\(38\!\cdots\!76\)\( T + T^{2} \)
$73$ \( -\)\(66\!\cdots\!80\)\( + \)\(28\!\cdots\!28\)\( T + T^{2} \)
$79$ \( -\)\(13\!\cdots\!00\)\( - 21069388575313284880 T + T^{2} \)
$83$ \( \)\(48\!\cdots\!68\)\( - \)\(14\!\cdots\!24\)\( T + T^{2} \)
$89$ \( \)\(20\!\cdots\!80\)\( - \)\(43\!\cdots\!64\)\( T + T^{2} \)
$97$ \( \)\(10\!\cdots\!16\)\( - \)\(10\!\cdots\!92\)\( T + T^{2} \)
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