# Properties

 Label 48.26.a.c Level $48$ Weight $26$ Character orbit 48.a Self dual yes Analytic conductor $190.078$ 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] = [48,26,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, 26, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$26$$ 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: $$190.078454377$$ 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)

 $$f(q)$$ $$=$$ $$q + 531441 q^{3} - 292754850 q^{5} - 3580644032 q^{7} + 282429536481 q^{9}+O(q^{10})$$ q + 531441 * q^3 - 292754850 * q^5 - 3580644032 * q^7 + 282429536481 * q^9 $$q + 531441 q^{3} - 292754850 q^{5} - 3580644032 q^{7} + 282429536481 q^{9} - 15111573238212 q^{11} + 1221071681246 q^{13} - 155581930238850 q^{15} + 25\!\cdots\!82 q^{17}+ \cdots - 42\!\cdots\!72 q^{99}+O(q^{100})$$ q + 531441 * q^3 - 292754850 * q^5 - 3580644032 * q^7 + 282429536481 * q^9 - 15111573238212 * q^11 + 1221071681246 * q^13 - 155581930238850 * q^15 + 2518250853863682 * q^17 + 7992693407413060 * q^19 - 1902901045010112 * q^21 + 99645642629247624 * q^23 - 212317821678430625 * q^25 + 150094635296999121 * q^27 - 2080672742244316890 * q^29 + 4937672075835729208 * q^31 - 8030909593288623492 * q^33 + 1048250906491555200 * q^35 + 19829154107621718182 * q^37 + 648927555353055486 * q^39 + 224696060863159376442 * q^41 + 72221008334482349884 * q^43 - 82682616588064682850 * q^45 - 189872435947262116992 * q^47 - 1328247607980067683783 * q^49 + 1338301752028169025762 * q^51 - 2645676034335389555874 * q^53 + 4423986356616768328200 * q^55 + 4247644977129004019460 * q^57 + 16454608826354674865340 * q^59 - 35546954389065591688738 * q^61 - 1011279634261218931392 * q^63 - 357474656882420543100 * q^65 - 106703750402023286661692 * q^67 + 52955779964529986546184 * q^69 - 73672004836753334994312 * q^71 - 262402855870448192600374 * q^73 - 112834395470606849780625 * q^75 + 54109164529534712150784 * q^77 + 1002642123108883497568840 * q^79 + 79766443076872509863361 * q^81 - 1558588706101147601425596 * q^83 - 737230150985234144357700 * q^85 - 1105754802811062012338490 * q^87 + 2181670205644666928498490 * q^89 - 4372223028097696223872 * q^91 + 2624081385654215766028728 * q^93 - 2339899759583199268341000 * q^95 - 440165308375605500117758 * q^97 - 4267954625166899357211972 * 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 531441. 0 −2.92755e8 0 −3.58064e9 0 2.82430e11 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 48.26.a.c 1
4.b odd 2 1 6.26.a.a 1
12.b even 2 1 18.26.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.26.a.a 1 4.b odd 2 1
18.26.a.d 1 12.b even 2 1
48.26.a.c 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 292754850$$ acting on $$S_{26}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 531441$$
$5$ $$T + 292754850$$
$7$ $$T + 3580644032$$
$11$ $$T + 15111573238212$$
$13$ $$T - 1221071681246$$
$17$ $$T - 2518250853863682$$
$19$ $$T - 7992693407413060$$
$23$ $$T - 99\!\cdots\!24$$
$29$ $$T + 20\!\cdots\!90$$
$31$ $$T - 49\!\cdots\!08$$
$37$ $$T - 19\!\cdots\!82$$
$41$ $$T - 22\!\cdots\!42$$
$43$ $$T - 72\!\cdots\!84$$
$47$ $$T + 18\!\cdots\!92$$
$53$ $$T + 26\!\cdots\!74$$
$59$ $$T - 16\!\cdots\!40$$
$61$ $$T + 35\!\cdots\!38$$
$67$ $$T + 10\!\cdots\!92$$
$71$ $$T + 73\!\cdots\!12$$
$73$ $$T + 26\!\cdots\!74$$
$79$ $$T - 10\!\cdots\!40$$
$83$ $$T + 15\!\cdots\!96$$
$89$ $$T - 21\!\cdots\!90$$
$97$ $$T + 44\!\cdots\!58$$