Properties

 Label 48.24.a.f Level $48$ Weight $24$ Character orbit 48.a Self dual yes Analytic conductor $160.898$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$24$$ Character orbit: $$[\chi]$$ $$=$$ 48.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$160.897937926$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ Defining polynomial: $$x^{2} - x - 4674852$$ x^2 - x - 4674852 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 3^{3}\cdot 5$$ Twist minimal: no (minimal twist has level 12) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 8640\sqrt{18699409}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 177147 q^{3} + ( - \beta + 36953550) q^{5} + (231 \beta + 765614248) q^{7} + 31381059609 q^{9}+O(q^{10})$$ q - 177147 * q^3 + (-b + 36953550) * q^5 + (231*b + 765614248) * q^7 + 31381059609 * q^9 $$q - 177147 q^{3} + ( - \beta + 36953550) q^{5} + (231 \beta + 765614248) q^{7} + 31381059609 q^{9} + ( - 12194 \beta - 428001419556) q^{11} + ( - 79806 \beta + 770193809510) q^{13} + (177147 \beta - 6546210521850) q^{15} + (3586318 \beta + 93895276928562) q^{17} + (6507462 \beta - 177778332794780) q^{19} + ( - 40920957 \beta - 135626267190456) q^{21} + ( - 158242162 \beta + 569328282916632) q^{23} + ( - 73907100 \beta - 91\!\cdots\!25) q^{25}+ \cdots + ( - 382660640872146 \beta - 13\!\cdots\!04) q^{99}+O(q^{100})$$ q - 177147 * q^3 + (-b + 36953550) * q^5 + (231*b + 765614248) * q^7 + 31381059609 * q^9 + (-12194*b - 428001419556) * q^11 + (-79806*b + 770193809510) * q^13 + (177147*b - 6546210521850) * q^15 + (3586318*b + 93895276928562) * q^17 + (6507462*b - 177778332794780) * q^19 + (-40920957*b - 135626267190456) * q^21 + (-158242162*b + 569328282916632) * q^23 + (-73907100*b - 9159460695389225) * q^25 - 5559060566555523 * q^27 + (-1813934995*b + 24524405097380406) * q^29 + (-3818348613*b - 56243527213601024) * q^31 + (2160130518*b + 75819167470086732) * q^33 + (7770655802*b - 294161521487778000) * q^35 + (-5029372968*b + 297895790147886926) * q^37 + (14137393482*b - 136437522773267970) * q^39 + (107880801674*b + 2070131673912258330) * q^41 + (110015038674*b - 6242488364040010628) * q^43 + (-31381059609*b + 1159641555314161950) * q^45 + (120080180298*b - 20338681881558081168) * q^47 + (353713782576*b + 47704219275392079561) * q^49 + (-635305474746*b - 16633266622063972614) * q^51 + (-1617412037455*b + 67258756863984382686) * q^53 + (-22610169144*b + 1205474227407937800) * q^55 + (-1152777370914*b + 31492898319596892660) * q^57 + (-6064103857624*b - 55536537834271069236) * q^59 + (553782749364*b + 190867113142745619062) * q^61 + (7249024769679*b + 24025786353987709032) * q^63 + (-3719308820810*b + 139862862356325498900) * q^65 + (10829461669992*b + 597771554471933287252) * q^67 + (28032124271814*b - 100854797333832608904) * q^69 + (2509192852790*b + 2656402670591766632520) * q^71 + (37378511392608*b - 1874037692084359547206) * q^73 + (13092421043700*b + 1622570983806115041075) * q^75 + (-108204228057548*b - 4259684230620900163488) * q^77 + (67625614805223*b + 3328880480900848025008) * q^79 + 984770902183611232881 * q^81 + (-471428629377162*b - 2439854517850933486428) * q^83 + (38631904600338*b - 1536389686420231580100) * q^85 + (321333142559265*b - 4344424789785646781682) * q^87 + (-513118829467244*b + 7575089607965991877290) * q^89 + (116814159320922*b - 25144067501213318171920) * q^91 + (676409001747111*b + 9963372115307780598528) * q^93 + (418252155184880*b - 15653328854596511185800) * q^95 + (1385012752665252*b + 45592117826141174725154) * q^97 + (-382660640872146*b - 13431138059823454313604) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 354294 q^{3} + 73907100 q^{5} + 1531228496 q^{7} + 62762119218 q^{9}+O(q^{10})$$ 2 * q - 354294 * q^3 + 73907100 * q^5 + 1531228496 * q^7 + 62762119218 * q^9 $$2 q - 354294 q^{3} + 73907100 q^{5} + 1531228496 q^{7} + 62762119218 q^{9} - 856002839112 q^{11} + 1540387619020 q^{13} - 13092421043700 q^{15} + 187790553857124 q^{17} - 355556665589560 q^{19} - 271252534380912 q^{21} + 11\!\cdots\!64 q^{23}+ \cdots - 26\!\cdots\!08 q^{99}+O(q^{100})$$ 2 * q - 354294 * q^3 + 73907100 * q^5 + 1531228496 * q^7 + 62762119218 * q^9 - 856002839112 * q^11 + 1540387619020 * q^13 - 13092421043700 * q^15 + 187790553857124 * q^17 - 355556665589560 * q^19 - 271252534380912 * q^21 + 1138656565833264 * q^23 - 18318921390778450 * q^25 - 11118121133111046 * q^27 + 49048810194760812 * q^29 - 112487054427202048 * q^31 + 151638334940173464 * q^33 - 588323042975556000 * q^35 + 595791580295773852 * q^37 - 272875045546535940 * q^39 + 4140263347824516660 * q^41 - 12484976728080021256 * q^43 + 2319283110628323900 * q^45 - 40677363763116162336 * q^47 + 95408438550784159122 * q^49 - 33266533244127945228 * q^51 + 134517513727968765372 * q^53 + 2410948454815875600 * q^55 + 62985796639193785320 * q^57 - 111073075668542138472 * q^59 + 381734226285491238124 * q^61 + 48051572707975418064 * q^63 + 279725724712650997800 * q^65 + 1195543108943866574504 * q^67 - 201709594667665217808 * q^69 + 5312805341183533265040 * q^71 - 3748075384168719094412 * q^73 + 3245141967612230082150 * q^75 - 8519368461241800326976 * q^77 + 6657760961801696050016 * q^79 + 1969541804367222465762 * q^81 - 4879709035701866972856 * q^83 - 3072779372840463160200 * q^85 - 8688849579571293563364 * q^87 + 15150179215931983754580 * q^89 - 50288135002426636343840 * q^91 + 19926744230615561197056 * q^93 - 31306657709193022371600 * q^95 + 91184235652282349450308 * q^97 - 26862276119646908627208 * 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
 2162.64 −2161.64
0 −177147. 0 −408241. 0 9.39619e9 0 3.13811e10 0
1.2 0 −177147. 0 7.43153e7 0 −7.86496e9 0 3.13811e10 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

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 48.24.a.f 2
4.b odd 2 1 12.24.a.b 2
12.b even 2 1 36.24.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.24.a.b 2 4.b odd 2 1
36.24.a.b 2 12.b even 2 1
48.24.a.f 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 73907100T_{5} - 30338544483900$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(48))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 177147)^{2}$$
$5$ $$T^{2} - 73907100 T - 30338544483900$$
$7$ $$T^{2} - 1531228496 T - 73\!\cdots\!96$$
$11$ $$T^{2} + 856002839112 T - 24\!\cdots\!64$$
$13$ $$T^{2} - 1540387619020 T - 82\!\cdots\!00$$
$17$ $$T^{2} - 187790553857124 T - 91\!\cdots\!56$$
$19$ $$T^{2} + 355556665589560 T - 27\!\cdots\!00$$
$23$ $$T^{2} + \cdots - 34\!\cdots\!76$$
$29$ $$T^{2} + \cdots - 39\!\cdots\!64$$
$31$ $$T^{2} + \cdots - 17\!\cdots\!24$$
$37$ $$T^{2} + \cdots + 53\!\cdots\!76$$
$41$ $$T^{2} + \cdots - 11\!\cdots\!00$$
$43$ $$T^{2} + \cdots + 22\!\cdots\!84$$
$47$ $$T^{2} + \cdots + 39\!\cdots\!24$$
$53$ $$T^{2} + \cdots + 87\!\cdots\!96$$
$59$ $$T^{2} + \cdots - 48\!\cdots\!04$$
$61$ $$T^{2} + \cdots + 36\!\cdots\!44$$
$67$ $$T^{2} + \cdots + 19\!\cdots\!04$$
$71$ $$T^{2} + \cdots + 70\!\cdots\!00$$
$73$ $$T^{2} + \cdots + 15\!\cdots\!36$$
$79$ $$T^{2} + \cdots + 46\!\cdots\!64$$
$83$ $$T^{2} + \cdots - 30\!\cdots\!16$$
$89$ $$T^{2} + \cdots - 31\!\cdots\!00$$
$97$ $$T^{2} + \cdots - 59\!\cdots\!84$$