# Properties

 Label 49.24.a.b Level $49$ Weight $24$ Character orbit 49.a Self dual yes Analytic conductor $164.250$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$164.249978299$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 540 - \beta ) q^{2} + ( -169740 - 48 \beta ) q^{3} + ( 12663328 - 1080 \beta ) q^{4} + ( -36534510 - 15040 \beta ) q^{5} + ( 904836528 + 143820 \beta ) q^{6} + ( 24729511680 - 4857920 \beta ) q^{8} + ( -17499697083 + 16295040 \beta ) q^{9} +O(q^{10})$$ $$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} + ( 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} + ( 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} + ( 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} + ( 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} + ( -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} + ( -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} + ( -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} + ( -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} + ( -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} + ( 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} + ( 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} + ( 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} + ( -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} + ( -20579122566156067990956 + 7657552458185248080 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 1080q^{2} - 339480q^{3} + 25326656q^{4} - 73069020q^{5} + 1809673056q^{6} + 49459023360q^{8} - 34999394166q^{9} + O(q^{10})$$ $$2q + 1080q^{2} - 339480q^{3} + 25326656q^{4} - 73069020q^{5} + 1809673056q^{6} + 49459023360q^{8} - 34999394166q^{9} + 585013636080q^{10} + 856801968264q^{11} - 2146514952960q^{12} - 4376109322060q^{13} + 42377338985040q^{15} + 15956586401792q^{16} - 254028147597540q^{17} - 695480683916520q^{18} - 4260600979960q^{19} - 250868387468160q^{20} + 2068343882177760q^{22} - 8144713079008560q^{23} + 1286622315141120q^{24} - 11780274628800850q^{25} - 55025854658735184q^{26} + 5424634982716560q^{27} + 20818433601623340q^{29} - 155926924188644160q^{30} - 137714017177000384q^{31} + 353265663781601280q^{32} - 68361366766001760q^{33} + 839483655961325328q^{34} - 1173916300077574848q^{36} - 897721264408967780q^{37} - 5699708971590961440q^{38} - 1785011473665029232q^{39} + 1226668524414336000q^{40} + 2294435477168314956q^{41} - 1750760768619855800q^{43} + 12584088840033038592q^{44} - 8897092690294206540q^{45} - 8813206018050221376q^{46} - 15759744217656780960q^{47} + 33749519399576616960q^{48} - 51990825483785316600q^{50} + 89998362845078292144q^{51} - 112291883783912022400q^{52} - 140287253401646796420q^{53} - 104731223417039799360q^{54} - 7153550955060182640q^{55} - 272752401448627175520q^{57} + 293749486923568689360q^{58} - 280872989971340771880q^{59} + 343522601114937592320q^{60} + 180452892516502223636q^{61} + 540743475843874103040q^{62} - 893690254469352914944q^{64} - 632168834809440380760q^{65} + 544338140913651883392q^{66} + 1754233163431557625240q^{67} - 2162050190142944330880q^{68} + 1170560672172404223552q^{69} + 3055033510194143328624q^{71} - 4152294352103038548480q^{72} + 8063408253877606149260q^{73} + 6715344283148807757072q^{74} - 190631089350520885800q^{75} - 6207154294513080590080q^{76} + 3614293948840808587200q^{78} + 6244916814559639980640q^{79} + 10840537585501794017280q^{80} - 2793528580929833975598q^{81} + 24457792891615712450640q^{82} - 6875994082418498976120q^{83} + 23969743087870314902520q^{85} - 10916288812918999243296q^{86} + 10026640653837674384880q^{87} + 28988514668199273707520q^{88} - 6395093086173070004820q^{89} + 8986073954327865866160q^{90} - 107907439017191756981760q^{92} + 52900811441357852079360q^{93} - 159400518006534931827072q^{94} - 85533361066700858502000q^{95} - 105592121669584394256384q^{96} + 31147288846254030500540q^{97} - 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 0 2.59512e9 5.67462e10 4.21966e11
1.2 5096.35 48964.9 1.75842e7 3.19930e7 2.49542e8 0 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

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

## 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 49.24.a.b 2
7.b odd 2 1 1.24.a.a 2
21.c even 2 1 9.24.a.b 2
28.d even 2 1 16.24.a.b 2
35.c odd 2 1 25.24.a.a 2
35.f even 4 2 25.24.b.a 4
56.e even 2 1 64.24.a.g 2
56.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 7.b odd 2 1
9.24.a.b 2 21.c even 2 1
16.24.a.b 2 28.d even 2 1
25.24.a.a 2 35.c odd 2 1
25.24.b.a 4 35.f even 4 2
49.24.a.b 2 1.a even 1 1 trivial
64.24.a.d 2 56.h odd 2 1
64.24.a.g 2 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2}^{2} - 1080 T_{2} - 20468736$$ $$T_{3}^{2} + 339480 T_{3} - 19020146544$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-20468736 - 1080 T + T^{2}$$
$3$ $$-19020146544 + 339480 T + T^{2}$$
$5$ $$-3361250798797500 + 73069020 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$15\!\cdots\!24$$$$- 856801968264 T + T^{2}$$
$13$ $$-$$$$28\!\cdots\!24$$$$+ 4376109322060 T + T^{2}$$
$17$ $$46\!\cdots\!64$$$$+ 254028147597540 T + T^{2}$$
$19$ $$-$$$$39\!\cdots\!00$$$$+ 4260600979960 T + T^{2}$$
$23$ $$16\!\cdots\!96$$$$+ 8144713079008560 T + T^{2}$$
$29$ $$-$$$$85\!\cdots\!00$$$$- 20818433601623340 T + T^{2}$$
$31$ $$18\!\cdots\!64$$$$+ 137714017177000384 T + T^{2}$$
$37$ $$-$$$$42\!\cdots\!36$$$$+ 897721264408967780 T + T^{2}$$
$41$ $$-$$$$51\!\cdots\!16$$$$- 2294435477168314956 T + T^{2}$$
$43$ $$-$$$$43\!\cdots\!64$$$$+ 1750760768619855800 T + T^{2}$$
$47$ $$-$$$$21\!\cdots\!36$$$$+ 15759744217656780960 T + T^{2}$$
$53$ $$48\!\cdots\!56$$$$+$$$$14\!\cdots\!20$$$$T + T^{2}$$
$59$ $$-$$$$28\!\cdots\!00$$$$+$$$$28\!\cdots\!80$$$$T + T^{2}$$
$61$ $$-$$$$13\!\cdots\!76$$$$-$$$$18\!\cdots\!36$$$$T + T^{2}$$
$67$ $$40\!\cdots\!64$$$$-$$$$17\!\cdots\!40$$$$T + T^{2}$$
$71$ $$17\!\cdots\!44$$$$-$$$$30\!\cdots\!24$$$$T + T^{2}$$
$73$ $$15\!\cdots\!96$$$$-$$$$80\!\cdots\!60$$$$T + T^{2}$$
$79$ $$-$$$$40\!\cdots\!00$$$$-$$$$62\!\cdots\!40$$$$T + T^{2}$$
$83$ $$-$$$$11\!\cdots\!84$$$$+$$$$68\!\cdots\!20$$$$T + T^{2}$$
$89$ $$82\!\cdots\!00$$$$+$$$$63\!\cdots\!20$$$$T + T^{2}$$
$97$ $$-$$$$93\!\cdots\!36$$$$-$$$$31\!\cdots\!40$$$$T + T^{2}$$