# 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: $$x^{2} - x - 36042$$ 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.

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

$p$ $F_p(T)$
$2$ $$T^{2} - 1080 T - 20468736$$
$3$ $$T^{2} - 339480 T - 19020146544$$
$5$ $$T^{2} - 73069020 T - 33\!\cdots\!00$$
$7$ $$T^{2} + 1359184400 T - 19\!\cdots\!36$$
$11$ $$T^{2} - 856801968264 T + 15\!\cdots\!24$$
$13$ $$T^{2} - 4376109322060 T - 28\!\cdots\!24$$
$17$ $$T^{2} - 254028147597540 T + 46\!\cdots\!64$$
$19$ $$T^{2} - 4260600979960 T - 39\!\cdots\!00$$
$23$ $$T^{2} + \cdots + 16\!\cdots\!96$$
$29$ $$T^{2} + \cdots - 85\!\cdots\!00$$
$31$ $$T^{2} + \cdots + 18\!\cdots\!64$$
$37$ $$T^{2} + \cdots - 42\!\cdots\!36$$
$41$ $$T^{2} + \cdots - 51\!\cdots\!16$$
$43$ $$T^{2} + \cdots - 43\!\cdots\!64$$
$47$ $$T^{2} + \cdots - 21\!\cdots\!36$$
$53$ $$T^{2} + \cdots + 48\!\cdots\!56$$
$59$ $$T^{2} + \cdots - 28\!\cdots\!00$$
$61$ $$T^{2} + \cdots - 13\!\cdots\!76$$
$67$ $$T^{2} + \cdots + 40\!\cdots\!64$$
$71$ $$T^{2} + \cdots + 17\!\cdots\!44$$
$73$ $$T^{2} + \cdots + 15\!\cdots\!96$$
$79$ $$T^{2} + \cdots - 40\!\cdots\!00$$
$83$ $$T^{2} + \cdots - 11\!\cdots\!84$$
$89$ $$T^{2} + \cdots + 82\!\cdots\!00$$
$97$ $$T^{2} + \cdots - 93\!\cdots\!36$$