# Properties

 Label 48.28.a.a Level $48$ Weight $28$ Character orbit 48.a Self dual yes Analytic conductor $221.691$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,28,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, 28, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ 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: $$221.690675922$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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)

 $$f(q)$$ $$=$$ $$q - 1594323 q^{3} + 1220703150 q^{5} - 96889207016 q^{7} + 2541865828329 q^{9}+O(q^{10})$$ q - 1594323 * q^3 + 1220703150 * q^5 - 96889207016 * q^7 + 2541865828329 * q^9 $$q - 1594323 q^{3} + 1220703150 q^{5} - 96889207016 q^{7} + 2541865828329 q^{9} - 34495064342052 q^{11} + 300892562137622 q^{13} - 19\!\cdots\!50 q^{15}+ \cdots - 87\!\cdots\!08 q^{99}+O(q^{100})$$ q - 1594323 * q^3 + 1220703150 * q^5 - 96889207016 * q^7 + 2541865828329 * q^9 - 34495064342052 * q^11 + 300892562137622 * q^13 - 1946195108217450 * q^15 + 11406510312331986 * q^17 - 62694436994411420 * q^19 + 154472691197370168 * q^21 + 894750379460289528 * q^23 - 5960464416503905625 * q^25 - 4052555153018976267 * q^27 + 104977877616797619030 * q^29 - 242604665669598327632 * q^31 + 54996274467013370796 * q^33 - 118272960205433300400 * q^35 + 1615296776273185563326 * q^37 - 479719932344939919906 * q^39 - 3845227056141271336998 * q^41 + 457279560936215164348 * q^43 + 3102863623518569536350 * q^45 + 48040279177582596731664 * q^47 - 56324843927344976515287 * q^49 - 18185661740688068915478 * q^51 - 277436637507624408709218 * q^53 - 42108233701795553863800 * q^55 + 99955182872240998368660 * q^57 - 257780373920421815739540 * q^59 + 143521812506920157597702 * q^61 - 246279364447864798356264 * q^63 + 367300498412965908909300 * q^65 + 5989037992702629692139124 * q^67 - 1426521109232267181149544 * q^69 + 5167007853435876621826728 * q^71 - 12558020215909057855312678 * q^73 + 9502905509913756327766875 * q^75 + 3342199430067316062236832 * q^77 + 60783355718993851022367520 * q^79 + 6461081889226673298932241 * q^81 - 3971159179443692816062812 * q^83 + 13923963068771139155955900 * q^85 - 167368644775645630364766690 * q^87 + 312460635134078620023752010 * q^89 - 29153241742526701439955952 * q^91 + 386790198384351014505233136 * q^93 - 76531296726554552789973000 * q^95 - 548671865528572742983955614 * q^97 - 87681825297072158367591108 * 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
 0
0 −1.59432e6 0 1.22070e9 0 −9.68892e10 0 2.54187e12 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 48.28.a.a 1
4.b odd 2 1 6.28.a.c 1
12.b even 2 1 18.28.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.28.a.c 1 4.b odd 2 1
18.28.a.b 1 12.b even 2 1
48.28.a.a 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1220703150$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1594323$$
$5$ $$T - 1220703150$$
$7$ $$T + 96889207016$$
$11$ $$T + 34495064342052$$
$13$ $$T - 300892562137622$$
$17$ $$T - 11\!\cdots\!86$$
$19$ $$T + 62\!\cdots\!20$$
$23$ $$T - 89\!\cdots\!28$$
$29$ $$T - 10\!\cdots\!30$$
$31$ $$T + 24\!\cdots\!32$$
$37$ $$T - 16\!\cdots\!26$$
$41$ $$T + 38\!\cdots\!98$$
$43$ $$T - 45\!\cdots\!48$$
$47$ $$T - 48\!\cdots\!64$$
$53$ $$T + 27\!\cdots\!18$$
$59$ $$T + 25\!\cdots\!40$$
$61$ $$T - 14\!\cdots\!02$$
$67$ $$T - 59\!\cdots\!24$$
$71$ $$T - 51\!\cdots\!28$$
$73$ $$T + 12\!\cdots\!78$$
$79$ $$T - 60\!\cdots\!20$$
$83$ $$T + 39\!\cdots\!12$$
$89$ $$T - 31\!\cdots\!10$$
$97$ $$T + 54\!\cdots\!14$$