# Properties

 Label 48.22.a.i Level $48$ Weight $22$ Character orbit 48.a Self dual yes Analytic conductor $134.149$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 48.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$134.149125258$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\mathbb{Q}[x]/(x^{2} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 797544$$ x^2 - x - 797544 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}\cdot 5$$ Twist minimal: no (minimal twist has level 12) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2880\sqrt{3190177}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 59049 q^{3} + ( - 5 \beta + 14413950) q^{5} + (207 \beta - 254834864) q^{7} + 3486784401 q^{9}+O(q^{10})$$ q + 59049 * q^3 + (-5*b + 14413950) * q^5 + (207*b - 254834864) * q^7 + 3486784401 * q^9 $$q + 59049 q^{3} + ( - 5 \beta + 14413950) q^{5} + (207 \beta - 254834864) q^{7} + 3486784401 q^{9} + (3446 \beta + 38025427644) q^{11} + (33282 \beta - 404663021650) q^{13} + ( - 295245 \beta + 851129333550) q^{15} + ( - 1929874 \beta + 415692948834) q^{17} + (143838 \beta - 14908784326460) q^{19} + (12223143 \beta - 15047743884336) q^{21} + ( - 22074626 \beta - 31340320703064) q^{23} + ( - 144139500 \beta + 392439899119375) q^{25} + 205891132094649 q^{27} + (547532665 \beta - 10\!\cdots\!26) q^{29}+ \cdots + (12015459045846 \beta + 13\!\cdots\!44) q^{99}+O(q^{100})$$ q + 59049 * q^3 + (-5*b + 14413950) * q^5 + (207*b - 254834864) * q^7 + 3486784401 * q^9 + (3446*b + 38025427644) * q^11 + (33282*b - 404663021650) * q^13 + (-295245*b + 851129333550) * q^15 + (-1929874*b + 415692948834) * q^17 + (143838*b - 14908784326460) * q^19 + (12223143*b - 15047743884336) * q^21 + (-22074626*b - 31340320703064) * q^23 + (-144139500*b + 392439899119375) * q^25 + 205891132094649 * q^27 + (547532665*b - 1056895947359226) * q^29 + (289904427*b - 3287869157112824) * q^31 + (203482854*b + 2245363476950556) * q^33 + (4257861970*b - 31059902240560800) * q^35 + (7047881424*b + 17405774448553142) * q^37 + (1965268818*b - 23894946765410850) * q^39 + (1690273114*b + 55814268374687610) * q^41 + (1031839722*b - 125429554629416324) * q^43 + (-17433922005*b + 50258336016793950) * q^45 + (-43854405654*b - 386859448444267296) * q^47 + (-105501633696*b + 640205369284585689) * q^49 + (-113957129826*b + 24546252935698866) * q^51 + (-14778035435*b - 294875840337624162) * q^53 + (-140456666520*b + 92180403994609800) * q^55 + (8493490062*b - 880348805693136540) * q^57 + (643295215144*b + 1432854062031494556) * q^59 + (121916423484*b - 8697352015552368178) * q^61 + (721764371007*b - 888554228626156464) * q^63 + (2503040192150*b - 10236101690657425500) * q^65 + (-1576742198616*b + 7878964908823948324) * q^67 + (-1303484590674*b - 1850614597195226136) * q^69 + (5491929813670*b - 4567238120207698920) * q^71 + (-5603775213456*b + 7181199501557041322) * q^73 + (-8511293335500*b + 23173183603099974375) * q^75 + (6993102580964*b + 9184726361896853184) * q^77 + (-5712837137433*b + 173513604897848152) * q^79 + 12157665459056928801 * q^81 + (5174196765054*b - 160342937059221871164) * q^83 + (-29895572086470*b + 261319936849177290300) * q^85 + (32331256335585*b - 62408648795614936074) * q^87 + (29653481787764*b + 345689834375681029530) * q^89 + (-92246659425198*b + 285419244059466696800) * q^91 + (17118566509923*b - 194145385858355144376) * q^93 + (76617195372400*b - 233924673711385989000) * q^95 + (-12960102665916*b - 587028445708294081342) * q^97 + (12015459045846*b + 132586467950453381244) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9}+O(q^{10})$$ 2 * q + 118098 * q^3 + 28827900 * q^5 - 509669728 * q^7 + 6973568802 * q^9 $$2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9} + 76050855288 q^{11} - 809326043300 q^{13} + 1702258667100 q^{15} + 831385897668 q^{17} - 29817568652920 q^{19} - 30095487768672 q^{21} - 62680641406128 q^{23} + 784879798238750 q^{25} + 411782264189298 q^{27} - 21\!\cdots\!52 q^{29}+ \cdots + 26\!\cdots\!88 q^{99}+O(q^{100})$$ 2 * q + 118098 * q^3 + 28827900 * q^5 - 509669728 * q^7 + 6973568802 * q^9 + 76050855288 * q^11 - 809326043300 * q^13 + 1702258667100 * q^15 + 831385897668 * q^17 - 29817568652920 * q^19 - 30095487768672 * q^21 - 62680641406128 * q^23 + 784879798238750 * q^25 + 411782264189298 * q^27 - 2113791894718452 * q^29 - 6575738314225648 * q^31 + 4490726953901112 * q^33 - 62119804481121600 * q^35 + 34811548897106284 * q^37 - 47789893530821700 * q^39 + 111628536749375220 * q^41 - 250859109258832648 * q^43 + 100516672033587900 * q^45 - 773718896888534592 * q^47 + 1280410738569171378 * q^49 + 49092505871397732 * q^51 - 589751680675248324 * q^53 + 184360807989219600 * q^55 - 1760697611386273080 * q^57 + 2865708124062989112 * q^59 - 17394704031104736356 * q^61 - 1777108457252312928 * q^63 - 20472203381314851000 * q^65 + 15757929817647896648 * q^67 - 3701229194390452272 * q^69 - 9134476240415397840 * q^71 + 14362399003114082644 * q^73 + 46346367206199948750 * q^75 + 18369452723793706368 * q^77 + 347027209795696304 * q^79 + 24315330918113857602 * q^81 - 320685874118443742328 * q^83 + 522639873698354580600 * q^85 - 124817297591229872148 * q^87 + 691379668751362059060 * q^89 + 570838488118933393600 * q^91 - 388290771716710288752 * q^93 - 467849347422771978000 * q^95 - 1174056891416588162684 * q^97 + 265172935900906762488 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 893.553 −892.553
0 59049.0 0 −1.13060e7 0 8.09970e8 0 3.48678e9 0
1.2 0 59049.0 0 4.01339e7 0 −1.31964e9 0 3.48678e9 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 48.22.a.i 2
4.b odd 2 1 12.22.a.b 2
12.b even 2 1 36.22.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.22.a.b 2 4.b odd 2 1
36.22.a.b 2 12.b even 2 1
48.22.a.i 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 28827900T_{5} - 453753148117500$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 59049)^{2}$$
$5$ $$T^{2} + \cdots - 453753148117500$$
$7$ $$T^{2} + 509669728 T - 10\!\cdots\!04$$
$11$ $$T^{2} - 76050855288 T + 11\!\cdots\!36$$
$13$ $$T^{2} + 809326043300 T + 13\!\cdots\!00$$
$17$ $$T^{2} - 831385897668 T - 98\!\cdots\!44$$
$19$ $$T^{2} + 29817568652920 T + 22\!\cdots\!00$$
$23$ $$T^{2} + 62680641406128 T - 11\!\cdots\!04$$
$29$ $$T^{2} + \cdots - 68\!\cdots\!24$$
$31$ $$T^{2} + \cdots + 85\!\cdots\!76$$
$37$ $$T^{2} + \cdots - 10\!\cdots\!36$$
$41$ $$T^{2} + \cdots + 30\!\cdots\!00$$
$43$ $$T^{2} + \cdots + 15\!\cdots\!76$$
$47$ $$T^{2} + \cdots + 98\!\cdots\!16$$
$53$ $$T^{2} + \cdots + 81\!\cdots\!44$$
$59$ $$T^{2} + \cdots - 88\!\cdots\!64$$
$61$ $$T^{2} + \cdots + 75\!\cdots\!84$$
$67$ $$T^{2} + \cdots - 37\!\cdots\!24$$
$71$ $$T^{2} + \cdots - 77\!\cdots\!00$$
$73$ $$T^{2} + \cdots - 77\!\cdots\!16$$
$79$ $$T^{2} + \cdots - 86\!\cdots\!96$$
$83$ $$T^{2} + \cdots + 25\!\cdots\!96$$
$89$ $$T^{2} + \cdots + 96\!\cdots\!00$$
$97$ $$T^{2} + \cdots + 34\!\cdots\!64$$