# Properties

 Label 1.24.a.a Level 1 Weight 24 Character orbit 1.a Self dual yes Analytic conductor 3.352 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 1.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.35204037345$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 540 - \beta ) q^{2} + ( 169740 + 48 \beta ) q^{3} + ( 12663328 - 1080 \beta ) q^{4} + ( 36534510 + 15040 \beta ) q^{5} + ( -904836528 - 143820 \beta ) q^{6} + ( -679592200 + 985824 \beta ) q^{7} + ( 24729511680 - 4857920 \beta ) q^{8} + ( -17499697083 + 16295040 \beta ) q^{9} +O(q^{10})$$ $$q + ( 540 - \beta ) q^{2} + ( 169740 + 48 \beta ) q^{3} + ( 12663328 - 1080 \beta ) q^{4} + ( 36534510 + 15040 \beta ) q^{5} + ( -904836528 - 143820 \beta ) q^{6} + ( -679592200 + 985824 \beta ) q^{7} + ( 24729511680 - 4857920 \beta ) q^{8} + ( -17499697083 + 16295040 \beta ) q^{9} + ( -292506818040 - 28412910 \beta ) q^{10} + ( 428400984132 - 38671600 \beta ) q^{11} + ( 1073257476480 + 424520544 \beta ) q^{12} + ( 2188054661030 - 1268350272 \beta ) q^{13} + ( -20833017264864 + 1211937160 \beta ) q^{14} + ( 21188669492520 + 4306546080 \beta ) q^{15} + ( 7978293200896 - 18293091840 \beta ) q^{16} + ( 127014073798770 + 23522231424 \beta ) q^{17} + ( -347740341958260 + 26299018683 \beta ) q^{18} + ( 2130300489980 - 137218594320 \beta ) q^{19} + ( 125434193734080 + 150999182320 \beta ) q^{20} + ( 867015818861472 + 134713340160 \beta ) q^{21} + ( 1034171941088880 - 449283648132 \beta ) q^{22} + ( -4072356539504280 + 106334043808 \beta ) q^{23} + ( -643311157570560 + 362433219840 \beta ) q^{24} + ( -5890137314400425 + 1098958060800 \beta ) q^{25} + ( 27512927329367592 - 2872963807910 \beta ) q^{26} + ( -2712317491358280 - 2592937954080 \beta ) q^{27} + ( -30709219409854720 + 13217772238272 \beta ) q^{28} + ( 10409216800811670 - 6804021206080 \beta ) q^{29} + ( -77963462094322080 - 18863134609320 \beta ) q^{30} + ( 68857008588500192 + 14814525283200 \beta ) q^{31} + ( 176632831890800640 + 22894623780864 \beta ) q^{32} + ( 34180683383000880 + 13999129854336 \beta ) q^{33} + ( -419741827980662664 - 114312068829810 \beta ) q^{34} + ( 282980635625212560 + 25795530098240 \beta ) q^{35} + ( -586958150038787424 + 225249109142760 \beta ) q^{36} + ( -448860632204483890 - 173410338010176 \beta ) q^{37} + ( 2849854485795480720 - 76228341422780 \beta ) q^{38} + ( -892505736832514616 - 110263151439840 \beta ) q^{39} + ( -613334262207168000 + 194450128848000 \beta ) q^{40} + ( -1147217738584157478 + 559210547795200 \beta ) q^{41} + ( -2328505663218698880 - 794270615175072 \beta ) q^{42} + ( -875380384309927900 + 240142500532368 \beta ) q^{43} + ( 6292044420016519296 - 952384217947360 \beta ) q^{44} + ( 4448546345147103270 + 332135857702080 \beta ) q^{45} + ( -4406603009025110688 + 4129776923160600 \beta ) q^{46} + ( 7879872108828390480 - 3634099566813376 \beta ) q^{47} + ( -16874759699788308480 - 2722111335278592 \beta ) q^{48} + ( -6730990852188100407 - 1339916601945600 \beta ) q^{49} + ( -25995412741892658300 + 6483574667232425 \beta ) q^{50} + ( 44999181422539146072 + 10089339104250720 \beta ) q^{51} + ( 56145941891956011200 - 18424634547137616 \beta ) q^{52} + ( -70143626700823398210 - 1360721746009152 \beta ) q^{53} + ( 52365611708519899680 + 1312130996155080 \beta ) q^{54} + ( 3576775477530091320 + 5030302844429280 \beta ) q^{55} + ( -116228356027144058880 + 27680350662648320 \beta ) q^{56} + ( -136376200724313587760 - 23189229776357760 \beta ) q^{57} + ( 146874743461784344680 - 14083388252094870 \beta ) q^{58} + ( 140436494985670385940 - 31045701436426160 \beta ) q^{59} + ( 171761300557468796160 + 31651442506232640 \beta ) q^{60} + ( -90226446258251111818 + 82865402983800000 \beta ) q^{61} + ( -270371737921937051520 - 60857164935572192 \beta ) q^{62} + ( 345387556966967507160 - 28325603459839392 \beta ) q^{63} + ( -446845127234676457472 - 10816158495375360 \beta ) q^{64} + ( -316084417404720190380 - 13430213593995520 \beta ) q^{65} + ( -272169070456825941696 - 26621153261659440 \beta ) q^{66} + ( 877116581715778812620 + 131620903771013424 \beta ) q^{67} + ( 1081025095071472165440 + 160694532111347472 \beta ) q^{68} + ( -585280336086202111776 - 177423973300235520 \beta ) q^{69} + ( -382714328899960626240 - 269051049372162960 \beta ) q^{70} + ( 1527516755097071664312 - 167496041649300000 \beta ) q^{71} + ( -2076147176051519274240 + 487980510459514560 \beta ) q^{72} + ( -4031704126938803074630 + 117939335115835008 \beta ) q^{73} + ( 3357672141574403878536 + 355219049678988850 \beta ) q^{74} + ( 95315544675260442900 - 96189449851028400 \beta ) q^{75} + ( 3103577147256540295040 - 1739944792102275360 \beta ) q^{76} + ( -1082592382178724832800 + 448608889502464768 \beta ) q^{77} + ( 1807146974420404293600 + 832963635055001016 \beta ) q^{78} + ( 3122458407279819990320 + 1561010165657737920 \beta ) q^{79} + ( -5420268792750897008640 - 548335617017922560 \beta ) q^{80} + ( -1396764290464916987799 - 2104383392623854720 \beta ) q^{81} + ( -12228896445807856225320 + 1449191434393565478 \beta ) q^{82} + ( 3437997041209249488060 - 2448003171672996112 \beta ) q^{83} + ( 7958875953595201238016 + 769542128051262720 \beta ) q^{84} + ( 11984871543935157451260 + 2769664869115943040 \beta ) q^{85} + ( -5458144406459499621648 + 1005057334597406620 \beta ) q^{86} + ( -5013320326918837192440 - 655272153081059040 \beta ) q^{87} + ( 14494257334099636853760 - 3037467492718813440 \beta ) q^{88} + ( 3197546543086535002410 + 304549203196268160 \beta ) q^{89} + ( -4493036977163932933080 - 4269192981987980070 \beta ) q^{90} + ( -27445089081352280073008 + 3018997749874317120 \beta ) q^{91} + ( -53953719508595878490880 + 5744687936971695424 \beta ) q^{92} + ( 26450405720678926039680 + 5819753933818377216 \beta ) q^{93} + ( 79700259003267465913536 - 9842285874907613520 \beta ) q^{94} + ( -42766680533350429251000 - 4981174387000684000 \beta ) q^{95} + ( 52796060834792197128192 + 12364509371322286080 \beta ) q^{96} + ( -15573644423127015250270 - 21536924763862843776 \beta ) q^{97} + ( 24182383808187335501820 + 6007435887137476407 \beta ) q^{98} + ( -20579122566156067990956 + 7657552458185248080 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 1080q^{2} + 339480q^{3} + 25326656q^{4} + 73069020q^{5} - 1809673056q^{6} - 1359184400q^{7} + 49459023360q^{8} - 34999394166q^{9} + O(q^{10})$$ $$2q + 1080q^{2} + 339480q^{3} + 25326656q^{4} + 73069020q^{5} - 1809673056q^{6} - 1359184400q^{7} + 49459023360q^{8} - 34999394166q^{9} - 585013636080q^{10} + 856801968264q^{11} + 2146514952960q^{12} + 4376109322060q^{13} - 41666034529728q^{14} + 42377338985040q^{15} + 15956586401792q^{16} + 254028147597540q^{17} - 695480683916520q^{18} + 4260600979960q^{19} + 250868387468160q^{20} + 1734031637722944q^{21} + 2068343882177760q^{22} - 8144713079008560q^{23} - 1286622315141120q^{24} - 11780274628800850q^{25} + 55025854658735184q^{26} - 5424634982716560q^{27} - 61418438819709440q^{28} + 20818433601623340q^{29} - 155926924188644160q^{30} + 137714017177000384q^{31} + 353265663781601280q^{32} + 68361366766001760q^{33} - 839483655961325328q^{34} + 565961271250425120q^{35} - 1173916300077574848q^{36} - 897721264408967780q^{37} + 5699708971590961440q^{38} - 1785011473665029232q^{39} - 1226668524414336000q^{40} - 2294435477168314956q^{41} - 4657011326437397760q^{42} - 1750760768619855800q^{43} + 12584088840033038592q^{44} + 8897092690294206540q^{45} - 8813206018050221376q^{46} + 15759744217656780960q^{47} - 33749519399576616960q^{48} - 13461981704376200814q^{49} - 51990825483785316600q^{50} + 89998362845078292144q^{51} + 112291883783912022400q^{52} - 140287253401646796420q^{53} + 104731223417039799360q^{54} + 7153550955060182640q^{55} - 232456712054288117760q^{56} - 272752401448627175520q^{57} + 293749486923568689360q^{58} + 280872989971340771880q^{59} + 343522601114937592320q^{60} - 180452892516502223636q^{61} - 540743475843874103040q^{62} + 690775113933935014320q^{63} - 893690254469352914944q^{64} - 632168834809440380760q^{65} - 544338140913651883392q^{66} + 1754233163431557625240q^{67} + 2162050190142944330880q^{68} - 1170560672172404223552q^{69} - 765428657799921252480q^{70} + 3055033510194143328624q^{71} - 4152294352103038548480q^{72} - 8063408253877606149260q^{73} + 6715344283148807757072q^{74} + 190631089350520885800q^{75} + 6207154294513080590080q^{76} - 2165184764357449665600q^{77} + 3614293948840808587200q^{78} + 6244916814559639980640q^{79} - 10840537585501794017280q^{80} - 2793528580929833975598q^{81} - 24457792891615712450640q^{82} + 6875994082418498976120q^{83} + 15917751907190402476032q^{84} + 23969743087870314902520q^{85} - 10916288812918999243296q^{86} - 10026640653837674384880q^{87} + 28988514668199273707520q^{88} + 6395093086173070004820q^{89} - 8986073954327865866160q^{90} - 54890178162704560146016q^{91} - 107907439017191756981760q^{92} + 52900811441357852079360q^{93} + 159400518006534931827072q^{94} - 85533361066700858502000q^{95} + 105592121669584394256384q^{96} - 31147288846254030500540q^{97} + 48364767616374671003640q^{98} - 41158245132312135981912q^{99} + 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
−4016.35 388445. 7.74247e6 1.05062e8 −1.56013e9 3.81217e9 2.59512e9 5.67462e10 −4.21966e11
1.2 5096.35 −48964.9 1.75842e7 −3.19930e7 −2.49542e8 −5.17135e9 4.68639e10 −9.17456e10 −1.63048e11
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 1.24.a.a 2
3.b odd 2 1 9.24.a.b 2
4.b odd 2 1 16.24.a.b 2
5.b even 2 1 25.24.a.a 2
5.c odd 4 2 25.24.b.a 4
7.b odd 2 1 49.24.a.b 2
8.b even 2 1 64.24.a.d 2
8.d odd 2 1 64.24.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.24.a.a 2 1.a even 1 1 trivial
9.24.a.b 2 3.b odd 2 1
16.24.a.b 2 4.b odd 2 1
25.24.a.a 2 5.b even 2 1
25.24.b.a 4 5.c odd 4 2
49.24.a.b 2 7.b odd 2 1
64.24.a.d 2 8.b even 2 1
64.24.a.g 2 8.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{24}^{\mathrm{new}}(\Gamma_0(1))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1080 T - 3691520 T^{2} - 9059696640 T^{3} + 70368744177664 T^{4}$$
$3$ $$1 - 339480 T + 169266211110 T^{2} - 31959726348189960 T^{3} +$$$$88\!\cdots\!29$$$$T^{4}$$
$5$ $$1 - 73069020 T + 20480607111358750 T^{2} -$$$$87\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!25$$$$T^{4}$$
$7$ $$1 + 1359184400 T + 35023429308870696750 T^{2} +$$$$37\!\cdots\!00$$$$T^{3} +$$$$74\!\cdots\!49$$$$T^{4}$$
$11$ $$1 - 856801968264 T +$$$$19\!\cdots\!86$$$$T^{2} -$$$$76\!\cdots\!84$$$$T^{3} +$$$$80\!\cdots\!61$$$$T^{4}$$
$13$ $$1 - 4376109322060 T +$$$$54\!\cdots\!70$$$$T^{2} -$$$$18\!\cdots\!20$$$$T^{3} +$$$$17\!\cdots\!09$$$$T^{4}$$
$17$ $$1 - 254028147597540 T +$$$$44\!\cdots\!90$$$$T^{2} -$$$$50\!\cdots\!20$$$$T^{3} +$$$$39\!\cdots\!69$$$$T^{4}$$
$19$ $$1 - 4260600979960 T +$$$$12\!\cdots\!18$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} +$$$$66\!\cdots\!81$$$$T^{4}$$
$23$ $$1 + 8144713079008560 T +$$$$58\!\cdots\!30$$$$T^{2} +$$$$17\!\cdots\!20$$$$T^{3} +$$$$43\!\cdots\!89$$$$T^{4}$$
$29$ $$1 - 20818433601623340 T +$$$$77\!\cdots\!78$$$$T^{2} -$$$$89\!\cdots\!60$$$$T^{3} +$$$$18\!\cdots\!21$$$$T^{4}$$
$31$ $$1 - 137714017177000384 T +$$$$40\!\cdots\!46$$$$T^{2} -$$$$27\!\cdots\!44$$$$T^{3} +$$$$40\!\cdots\!81$$$$T^{4}$$
$37$ $$1 + 897721264408967780 T +$$$$19\!\cdots\!70$$$$T^{2} +$$$$10\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!09$$$$T^{4}$$
$41$ $$1 + 2294435477168314956 T +$$$$19\!\cdots\!26$$$$T^{2} +$$$$28\!\cdots\!76$$$$T^{3} +$$$$15\!\cdots\!41$$$$T^{4}$$
$43$ $$1 + 1750760768619855800 T +$$$$73\!\cdots\!50$$$$T^{2} +$$$$65\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 15759744217656780960 T +$$$$36\!\cdots\!10$$$$T^{2} -$$$$45\!\cdots\!80$$$$T^{3} +$$$$82\!\cdots\!29$$$$T^{4}$$
$53$ $$1 +$$$$14\!\cdots\!20$$$$T +$$$$13\!\cdots\!10$$$$T^{2} +$$$$63\!\cdots\!40$$$$T^{3} +$$$$20\!\cdots\!29$$$$T^{4}$$
$59$ $$1 -$$$$28\!\cdots\!80$$$$T +$$$$10\!\cdots\!58$$$$T^{2} -$$$$15\!\cdots\!20$$$$T^{3} +$$$$28\!\cdots\!41$$$$T^{4}$$
$61$ $$1 +$$$$18\!\cdots\!36$$$$T +$$$$96\!\cdots\!86$$$$T^{2} +$$$$20\!\cdots\!16$$$$T^{3} +$$$$13\!\cdots\!61$$$$T^{4}$$
$67$ $$1 -$$$$17\!\cdots\!40$$$$T +$$$$24\!\cdots\!90$$$$T^{2} -$$$$17\!\cdots\!20$$$$T^{3} +$$$$99\!\cdots\!69$$$$T^{4}$$
$71$ $$1 -$$$$30\!\cdots\!24$$$$T +$$$$93\!\cdots\!66$$$$T^{2} -$$$$11\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!21$$$$T^{4}$$
$73$ $$1 +$$$$80\!\cdots\!60$$$$T +$$$$30\!\cdots\!30$$$$T^{2} +$$$$57\!\cdots\!20$$$$T^{3} +$$$$51\!\cdots\!89$$$$T^{4}$$
$79$ $$1 -$$$$62\!\cdots\!40$$$$T +$$$$47\!\cdots\!78$$$$T^{2} -$$$$27\!\cdots\!60$$$$T^{3} +$$$$19\!\cdots\!21$$$$T^{4}$$
$83$ $$1 -$$$$68\!\cdots\!20$$$$T +$$$$16\!\cdots\!90$$$$T^{2} -$$$$94\!\cdots\!40$$$$T^{3} +$$$$18\!\cdots\!69$$$$T^{4}$$
$89$ $$1 -$$$$63\!\cdots\!20$$$$T +$$$$13\!\cdots\!38$$$$T^{2} -$$$$43\!\cdots\!80$$$$T^{3} +$$$$46\!\cdots\!61$$$$T^{4}$$
$97$ $$1 +$$$$31\!\cdots\!40$$$$T +$$$$53\!\cdots\!10$$$$T^{2} +$$$$15\!\cdots\!20$$$$T^{3} +$$$$24\!\cdots\!29$$$$T^{4}$$