# Properties

 Label 48.24.a.g Level $48$ Weight $24$ Character orbit 48.a Self dual yes Analytic conductor $160.898$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{530401})$$ Defining polynomial: $$x^{2} - x - 132600$$ x^2 - x - 132600 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 768\sqrt{530401}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 177147 q^{3} + ( - 245 \beta - 23404410) q^{5} + (14229 \beta + 105981952) q^{7} + 31381059609 q^{9}+O(q^{10})$$ q + 177147 * q^3 + (-245*b - 23404410) * q^5 + (14229*b + 105981952) * q^7 + 31381059609 * q^9 $$q + 177147 q^{3} + ( - 245 \beta - 23404410) q^{5} + (14229 \beta + 105981952) q^{7} + 31381059609 q^{9} + (530794 \beta - 734486183244) q^{11} + ( - 5083614 \beta + 5245827132374) q^{13} + ( - 43401015 \beta - 4146021018270) q^{15} + (59366550 \beta - 105444005760414) q^{17} + (557434386 \beta + 453691224268972) q^{19} + (2520624663 \beta + 18774384850944) q^{21} + ( - 448330582 \beta - 50\!\cdots\!56) q^{23}+ \cdots + (16\!\cdots\!46 \beta - 23\!\cdots\!96) q^{99}+O(q^{100})$$ q + 177147 * q^3 + (-245*b - 23404410) * q^5 + (14229*b + 105981952) * q^7 + 31381059609 * q^9 + (530794*b - 734486183244) * q^11 + (-5083614*b + 5245827132374) * q^13 + (-43401015*b - 4146021018270) * q^15 + (59366550*b - 105444005760414) * q^17 + (557434386*b + 453691224268972) * q^19 + (2520624663*b + 18774384850944) * q^21 + (-448330582*b - 5058461661946056) * q^23 + (11468160900*b + 7405252898795575) * q^25 + 5559060566555523 * q^27 + (42552733313*b + 9239376772588446) * q^29 + (98884578033*b + 136396811296372744) * q^31 + (94028564718*b - 130112023903124868) * q^33 + (-358986928130*b - 1093084826229411840) * q^35 + (-2135463956064*b - 239497518393682) * q^37 + (-900546969258*b + 929282539018656978) * q^39 + (-6900847152638*b + 2777857480654385706) * q^41 + (-17905976792874*b + 599161073804660020) * q^43 + (-7688359604205*b - 734455185323475690) * q^45 + (-2608772122562*b - 13154282836427888640) * q^47 + (3016034390016*b + 35982116424678135945) * q^49 + (10516606232850*b - 18679089288440058858) * q^51 + (98542534638357*b - 20563750949707612314) * q^53 + (167526194493240*b - 23493336262593844680) * q^55 + (98747829176742*b + 80370039305575582884) * q^57 + (199894647770264*b + 153768530954722143348) * q^59 + (398505547132524*b + 297480589549251875510) * q^61 + (446521097176461*b + 3325825953190176768) * q^63 + (-1166248661093890*b + 266866207581388222980) * q^65 + (-858933655998120*b + 815112912563496632044) * q^67 + (-79420417609554*b - 896091308028757982232) * q^69 + (-3908113709185470*b + 191971299290824636488) * q^71 + (2907042671398032*b - 1406101175720280846214) * q^73 + (2031550298952300*b + 1311818335262939724525) * q^75 + (-10394749317148988*b + 2284958789562030759936) * q^77 + (652969089629685*b - 10534694287656642440) * q^79 + 984770902183611232881 * q^81 + (-4632936895603118*b + 7408631040162201348012) * q^83 + (24444342334815930*b - 2082389091920491438260) * q^85 + (7538089048198011*b + 1636727877133725443562) * q^87 + (-29257599169640316*b - 21724902630442228278582) * q^89 + (74104102931615118*b - 22073472513261952208896) * q^91 + (17517106344811851*b + 24162285930718542481368) * q^93 + (-124200772863940400*b - 53343822301422237214200) * q^95 + (75917566969076340*b + 53568286522621084225346) * q^97 + (16656878154099546*b - 23048954698366860991596) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 354294 q^{3} - 46808820 q^{5} + 211963904 q^{7} + 62762119218 q^{9}+O(q^{10})$$ 2 * q + 354294 * q^3 - 46808820 * q^5 + 211963904 * q^7 + 62762119218 * q^9 $$2 q + 354294 q^{3} - 46808820 q^{5} + 211963904 q^{7} + 62762119218 q^{9} - 1468972366488 q^{11} + 10491654264748 q^{13} - 8292042036540 q^{15} - 210888011520828 q^{17} + 907382448537944 q^{19} + 37548769701888 q^{21} - 10\!\cdots\!12 q^{23}+ \cdots - 46\!\cdots\!92 q^{99}+O(q^{100})$$ 2 * q + 354294 * q^3 - 46808820 * q^5 + 211963904 * q^7 + 62762119218 * q^9 - 1468972366488 * q^11 + 10491654264748 * q^13 - 8292042036540 * q^15 - 210888011520828 * q^17 + 907382448537944 * q^19 + 37548769701888 * q^21 - 10116923323892112 * q^23 + 14810505797591150 * q^25 + 11118121133111046 * q^27 + 18478753545176892 * q^29 + 272793622592745488 * q^31 - 260224047806249736 * q^33 - 2186169652458823680 * q^35 - 478995036787364 * q^37 + 1858565078037313956 * q^39 + 5555714961308771412 * q^41 + 1198322147609320040 * q^43 - 1468910370646951380 * q^45 - 26308565672855777280 * q^47 + 71964232849356271890 * q^49 - 37358178576880117716 * q^51 - 41127501899415224628 * q^53 - 46986672525187689360 * q^55 + 160740078611151165768 * q^57 + 307537061909444286696 * q^59 + 594961179098503751020 * q^61 + 6651651906380353536 * q^63 + 533732415162776445960 * q^65 + 1630225825126993264088 * q^67 - 1792182616057515964464 * q^69 + 383942598581649272976 * q^71 - 2812202351440561692428 * q^73 + 2623636670525879449050 * q^75 + 4569917579124061519872 * q^77 - 21069388575313284880 * q^79 + 1969541804367222465762 * q^81 + 14817262080324402696024 * q^83 - 4164778183840982876520 * q^85 + 3273455754267450887124 * q^87 - 43449805260884456557164 * q^89 - 44146945026523904417792 * q^91 + 48324571861437084962736 * q^93 - 106687644602844474428400 * q^95 + 107136573045242168450692 * q^97 - 46097909396733721983192 * 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
 364.643 −363.643
0 177147. 0 −1.60439e8 0 8.06460e9 0 3.13811e10 0
1.2 0 177147. 0 1.13630e8 0 −7.85264e9 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.g 2
4.b odd 2 1 3.24.a.b 2
12.b even 2 1 9.24.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.b 2 4.b odd 2 1
9.24.a.c 2 12.b even 2 1
48.24.a.g 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 46808820T_{5} - 18230649038977500$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 177147)^{2}$$
$5$ $$T^{2} + 46808820 T - 18\!\cdots\!00$$
$7$ $$T^{2} - 211963904 T - 63\!\cdots\!80$$
$11$ $$T^{2} + 1468972366488 T + 45\!\cdots\!72$$
$13$ $$T^{2} - 10491654264748 T + 19\!\cdots\!72$$
$17$ $$T^{2} + 210888011520828 T + 10\!\cdots\!96$$
$19$ $$T^{2} - 907382448537944 T + 10\!\cdots\!80$$
$23$ $$T^{2} + \cdots + 25\!\cdots\!60$$
$29$ $$T^{2} + \cdots - 48\!\cdots\!40$$
$31$ $$T^{2} + \cdots + 15\!\cdots\!00$$
$37$ $$T^{2} + 478995036787364 T - 14\!\cdots\!80$$
$41$ $$T^{2} + \cdots - 71\!\cdots\!20$$
$43$ $$T^{2} + \cdots - 99\!\cdots\!24$$
$47$ $$T^{2} + \cdots + 17\!\cdots\!44$$
$53$ $$T^{2} + \cdots - 26\!\cdots\!80$$
$59$ $$T^{2} + \cdots + 11\!\cdots\!00$$
$61$ $$T^{2} + \cdots + 38\!\cdots\!76$$
$67$ $$T^{2} + \cdots + 43\!\cdots\!36$$
$71$ $$T^{2} + \cdots - 47\!\cdots\!56$$
$73$ $$T^{2} + \cdots - 66\!\cdots\!80$$
$79$ $$T^{2} + \cdots - 13\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 48\!\cdots\!68$$
$89$ $$T^{2} + \cdots + 20\!\cdots\!80$$
$97$ $$T^{2} + \cdots + 10\!\cdots\!16$$