# Properties

 Label 9.24.a.b 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: $$\Q(\sqrt{144169})$$ Defining polynomial: $$x^{2} - x - 36042$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{144169}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -540 - \beta ) q^{2} + ( 12663328 + 1080 \beta ) q^{4} + ( -36534510 + 15040 \beta ) q^{5} + ( -679592200 - 985824 \beta ) q^{7} + ( -24729511680 - 4857920 \beta ) q^{8} +O(q^{10})$$ $$q + ( -540 - \beta ) q^{2} + ( 12663328 + 1080 \beta ) q^{4} + ( -36534510 + 15040 \beta ) q^{5} + ( -679592200 - 985824 \beta ) q^{7} + ( -24729511680 - 4857920 \beta ) q^{8} + ( -292506818040 + 28412910 \beta ) q^{10} + ( -428400984132 - 38671600 \beta ) q^{11} + ( 2188054661030 + 1268350272 \beta ) q^{13} + ( 20833017264864 + 1211937160 \beta ) q^{14} + ( 7978293200896 + 18293091840 \beta ) q^{16} + ( -127014073798770 + 23522231424 \beta ) q^{17} + ( 2130300489980 + 137218594320 \beta ) q^{19} + ( -125434193734080 + 150999182320 \beta ) q^{20} + ( 1034171941088880 + 449283648132 \beta ) q^{22} + ( 4072356539504280 + 106334043808 \beta ) q^{23} + ( -5890137314400425 - 1098958060800 \beta ) q^{25} + ( -27512927329367592 - 2872963807910 \beta ) q^{26} + ( -30709219409854720 - 13217772238272 \beta ) q^{28} + ( -10409216800811670 - 6804021206080 \beta ) q^{29} + ( 68857008588500192 - 14814525283200 \beta ) q^{31} + ( -176632831890800640 + 22894623780864 \beta ) q^{32} + ( -419741827980662664 + 114312068829810 \beta ) q^{34} + ( -282980635625212560 + 25795530098240 \beta ) q^{35} + ( -448860632204483890 + 173410338010176 \beta ) q^{37} + ( -2849854485795480720 - 76228341422780 \beta ) q^{38} + ( -613334262207168000 - 194450128848000 \beta ) q^{40} + ( 1147217738584157478 + 559210547795200 \beta ) q^{41} + ( -875380384309927900 - 240142500532368 \beta ) q^{43} + ( -6292044420016519296 - 952384217947360 \beta ) q^{44} + ( -4406603009025110688 - 4129776923160600 \beta ) q^{46} + ( -7879872108828390480 - 3634099566813376 \beta ) q^{47} + ( -6730990852188100407 + 1339916601945600 \beta ) q^{49} + ( 25995412741892658300 + 6483574667232425 \beta ) q^{50} + ( 56145941891956011200 + 18424634547137616 \beta ) q^{52} + ( 70143626700823398210 - 1360721746009152 \beta ) q^{53} + ( 3576775477530091320 - 5030302844429280 \beta ) q^{55} + ( 116228356027144058880 + 27680350662648320 \beta ) q^{56} + ( 146874743461784344680 + 14083388252094870 \beta ) q^{58} + ( -140436494985670385940 - 31045701436426160 \beta ) q^{59} + ( -90226446258251111818 - 82865402983800000 \beta ) q^{61} + ( 270371737921937051520 - 60857164935572192 \beta ) q^{62} + ( -446845127234676457472 + 10816158495375360 \beta ) q^{64} + ( 316084417404720190380 - 13430213593995520 \beta ) q^{65} + ( 877116581715778812620 - 131620903771013424 \beta ) q^{67} + ( -1081025095071472165440 + 160694532111347472 \beta ) q^{68} + ( -382714328899960626240 + 269051049372162960 \beta ) q^{70} + ( -1527516755097071664312 - 167496041649300000 \beta ) q^{71} + ( -4031704126938803074630 - 117939335115835008 \beta ) q^{73} + ( -3357672141574403878536 + 355219049678988850 \beta ) q^{74} + ( 3103577147256540295040 + 1739944792102275360 \beta ) q^{76} + ( 1082592382178724832800 + 448608889502464768 \beta ) q^{77} + ( 3122458407279819990320 - 1561010165657737920 \beta ) q^{79} + ( 5420268792750897008640 - 548335617017922560 \beta ) q^{80} + ( -12228896445807856225320 - 1449191434393565478 \beta ) q^{82} + ( -3437997041209249488060 - 2448003171672996112 \beta ) q^{83} + ( 11984871543935157451260 - 2769664869115943040 \beta ) q^{85} + ( 5458144406459499621648 + 1005057334597406620 \beta ) q^{86} + ( 14494257334099636853760 + 3037467492718813440 \beta ) q^{88} + ( -3197546543086535002410 + 304549203196268160 \beta ) q^{89} + ( -27445089081352280073008 - 3018997749874317120 \beta ) q^{91} + ( 53953719508595878490880 + 5744687936971695424 \beta ) q^{92} + ( 79700259003267465913536 + 9842285874907613520 \beta ) q^{94} + ( 42766680533350429251000 - 4981174387000684000 \beta ) q^{95} + ( -15573644423127015250270 + 21536924763862843776 \beta ) q^{97} + ( -24182383808187335501820 + 6007435887137476407 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1080q^{2} + 25326656q^{4} - 73069020q^{5} - 1359184400q^{7} - 49459023360q^{8} + O(q^{10})$$ $$2q - 1080q^{2} + 25326656q^{4} - 73069020q^{5} - 1359184400q^{7} - 49459023360q^{8} - 585013636080q^{10} - 856801968264q^{11} + 4376109322060q^{13} + 41666034529728q^{14} + 15956586401792q^{16} - 254028147597540q^{17} + 4260600979960q^{19} - 250868387468160q^{20} + 2068343882177760q^{22} + 8144713079008560q^{23} - 11780274628800850q^{25} - 55025854658735184q^{26} - 61418438819709440q^{28} - 20818433601623340q^{29} + 137714017177000384q^{31} - 353265663781601280q^{32} - 839483655961325328q^{34} - 565961271250425120q^{35} - 897721264408967780q^{37} - 5699708971590961440q^{38} - 1226668524414336000q^{40} + 2294435477168314956q^{41} - 1750760768619855800q^{43} - 12584088840033038592q^{44} - 8813206018050221376q^{46} - 15759744217656780960q^{47} - 13461981704376200814q^{49} + 51990825483785316600q^{50} + 112291883783912022400q^{52} + 140287253401646796420q^{53} + 7153550955060182640q^{55} + 232456712054288117760q^{56} + 293749486923568689360q^{58} - 280872989971340771880q^{59} - 180452892516502223636q^{61} + 540743475843874103040q^{62} - 893690254469352914944q^{64} + 632168834809440380760q^{65} + 1754233163431557625240q^{67} - 2162050190142944330880q^{68} - 765428657799921252480q^{70} - 3055033510194143328624q^{71} - 8063408253877606149260q^{73} - 6715344283148807757072q^{74} + 6207154294513080590080q^{76} + 2165184764357449665600q^{77} + 6244916814559639980640q^{79} + 10840537585501794017280q^{80} - 24457792891615712450640q^{82} - 6875994082418498976120q^{83} + 23969743087870314902520q^{85} + 10916288812918999243296q^{86} + 28988514668199273707520q^{88} - 6395093086173070004820q^{89} - 54890178162704560146016q^{91} + 107907439017191756981760q^{92} + 159400518006534931827072q^{94} + 85533361066700858502000q^{95} - 31147288846254030500540q^{97} - 48364767616374671003640q^{98} + O(q^{100})$$

## 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
 190.348 −189.348
−5096.35 0 1.75842e7 3.19930e7 0 −5.17135e9 −4.68639e10 0 −1.63048e11
1.2 4016.35 0 7.74247e6 −1.05062e8 0 3.81217e9 −2.59512e9 0 −4.21966e11
 $$n$$: 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.b 2
3.b odd 2 1 1.24.a.a 2
12.b even 2 1 16.24.a.b 2
15.d odd 2 1 25.24.a.a 2
15.e even 4 2 25.24.b.a 4
21.c even 2 1 49.24.a.b 2
24.f even 2 1 64.24.a.g 2
24.h odd 2 1 64.24.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.24.a.a 2 3.b odd 2 1
9.24.a.b 2 1.a even 1 1 trivial
16.24.a.b 2 12.b even 2 1
25.24.a.a 2 15.d odd 2 1
25.24.b.a 4 15.e even 4 2
49.24.a.b 2 21.c even 2 1
64.24.a.d 2 24.h odd 2 1
64.24.a.g 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1080 T_{2} - 20468736$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-20468736 + 1080 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-3361250798797500 + 73069020 T + T^{2}$$
$7$ $$-19714065371291135936 + 1359184400 T + T^{2}$$
$11$ $$15\!\cdots\!24$$$$+ 856801968264 T + T^{2}$$
$13$ $$-$$$$28\!\cdots\!24$$$$- 4376109322060 T + T^{2}$$
$17$ $$46\!\cdots\!64$$$$+ 254028147597540 T + T^{2}$$
$19$ $$-$$$$39\!\cdots\!00$$$$- 4260600979960 T + T^{2}$$
$23$ $$16\!\cdots\!96$$$$- 8144713079008560 T + T^{2}$$
$29$ $$-$$$$85\!\cdots\!00$$$$+ 20818433601623340 T + T^{2}$$
$31$ $$18\!\cdots\!64$$$$- 137714017177000384 T + T^{2}$$
$37$ $$-$$$$42\!\cdots\!36$$$$+ 897721264408967780 T + T^{2}$$
$41$ $$-$$$$51\!\cdots\!16$$$$- 2294435477168314956 T + T^{2}$$
$43$ $$-$$$$43\!\cdots\!64$$$$+ 1750760768619855800 T + T^{2}$$
$47$ $$-$$$$21\!\cdots\!36$$$$+ 15759744217656780960 T + T^{2}$$
$53$ $$48\!\cdots\!56$$$$-$$$$14\!\cdots\!20$$$$T + T^{2}$$
$59$ $$-$$$$28\!\cdots\!00$$$$+$$$$28\!\cdots\!80$$$$T + T^{2}$$
$61$ $$-$$$$13\!\cdots\!76$$$$+$$$$18\!\cdots\!36$$$$T + T^{2}$$
$67$ $$40\!\cdots\!64$$$$-$$$$17\!\cdots\!40$$$$T + T^{2}$$
$71$ $$17\!\cdots\!44$$$$+$$$$30\!\cdots\!24$$$$T + T^{2}$$
$73$ $$15\!\cdots\!96$$$$+$$$$80\!\cdots\!60$$$$T + T^{2}$$
$79$ $$-$$$$40\!\cdots\!00$$$$-$$$$62\!\cdots\!40$$$$T + T^{2}$$
$83$ $$-$$$$11\!\cdots\!84$$$$+$$$$68\!\cdots\!20$$$$T + T^{2}$$
$89$ $$82\!\cdots\!00$$$$+$$$$63\!\cdots\!20$$$$T + T^{2}$$
$97$ $$-$$$$93\!\cdots\!36$$$$+$$$$31\!\cdots\!40$$$$T + T^{2}$$