# Properties

 Label 48.22.a.a Level $48$ Weight $22$ Character orbit 48.a Self dual yes Analytic conductor $134.149$ Analytic rank $0$ 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] = [48,22,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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 59049 q^{3} - 23245050 q^{5} + 1322977768 q^{7} + 3486784401 q^{9}+O(q^{10})$$ q - 59049 * q^3 - 23245050 * q^5 + 1322977768 * q^7 + 3486784401 * q^9 $$q - 59049 q^{3} - 23245050 q^{5} + 1322977768 q^{7} + 3486784401 q^{9} + 109174443828 q^{11} + 468325115966 q^{13} + 1372596957450 q^{15} + 2654798072562 q^{17} + 43712786306860 q^{19} - 78120514222632 q^{21} + 216861233964744 q^{23} + 63495191299375 q^{25} - 205891132094649 q^{27} + 25\!\cdots\!10 q^{29}+ \cdots + 38\!\cdots\!28 q^{99}+O(q^{100})$$ q - 59049 * q^3 - 23245050 * q^5 + 1322977768 * q^7 + 3486784401 * q^9 + 109174443828 * q^11 + 468325115966 * q^13 + 1372596957450 * q^15 + 2654798072562 * q^17 + 43712786306860 * q^19 - 78120514222632 * q^21 + 216861233964744 * q^23 + 63495191299375 * q^25 - 205891132094649 * q^27 + 2535247265345310 * q^29 - 5132915444930672 * q^31 - 6446641733599572 * q^33 - 30752684366048400 * q^35 - 8126962096433578 * q^37 - 27654129772676334 * q^39 - 28546174551317718 * q^41 - 60426656396902316 * q^43 - 81050477740465050 * q^45 + 316578527337771888 * q^47 + 1191724310538977817 * q^49 - 156763171386713538 * q^51 + 237962956198086486 * q^53 - 2537765405504051400 * q^55 - 2581196318633776140 * q^57 + 1932323017293179940 * q^59 - 5540333075504971378 * q^61 + 4612938244332196968 * q^63 - 10886240736885468300 * q^65 - 19977690098489923172 * q^67 - 12805439004384168456 * q^69 + 43257338948428460568 * q^71 + 1683404149334166506 * q^73 - 3749327551036794375 * q^75 + 144435362018208815904 * q^77 - 159081535530354452960 * q^79 + 12157665459056928801 * q^81 + 265853420659068579564 * q^83 - 61710913936607318100 * q^85 - 149703815771375210190 * q^87 - 17496064778396968710 * q^89 + 619583716619039843888 * q^91 + 303093524107711250928 * q^93 - 1016105903342276043000 * q^95 - 361656712539028124638 * q^97 + 380667747727321127028 * 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 −59049.0 0 −2.32450e7 0 1.32298e9 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.a 1
4.b odd 2 1 6.22.a.c 1
12.b even 2 1 18.22.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.22.a.c 1 4.b odd 2 1
18.22.a.c 1 12.b even 2 1
48.22.a.a 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 23245050$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 59049$$
$5$ $$T + 23245050$$
$7$ $$T - 1322977768$$
$11$ $$T - 109174443828$$
$13$ $$T - 468325115966$$
$17$ $$T - 2654798072562$$
$19$ $$T - 43712786306860$$
$23$ $$T - 216861233964744$$
$29$ $$T - 2535247265345310$$
$31$ $$T + 5132915444930672$$
$37$ $$T + 8126962096433578$$
$41$ $$T + 28\!\cdots\!18$$
$43$ $$T + 60\!\cdots\!16$$
$47$ $$T - 31\!\cdots\!88$$
$53$ $$T - 23\!\cdots\!86$$
$59$ $$T - 19\!\cdots\!40$$
$61$ $$T + 55\!\cdots\!78$$
$67$ $$T + 19\!\cdots\!72$$
$71$ $$T - 43\!\cdots\!68$$
$73$ $$T - 16\!\cdots\!06$$
$79$ $$T + 15\!\cdots\!60$$
$83$ $$T - 26\!\cdots\!64$$
$89$ $$T + 17\!\cdots\!10$$
$97$ $$T + 36\!\cdots\!38$$