# Properties

 Label 272.2.a.a Level $272$ Weight $2$ Character orbit 272.a Self dual yes Analytic conductor $2.172$ 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] = [272,2,Mod(1,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$272 = 2^{4} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 272.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: $$2.17193093498$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 136) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^9 $$q - 2 q^{3} + q^{9} - 2 q^{11} - 6 q^{13} - q^{17} - 4 q^{19} - 4 q^{23} - 5 q^{25} + 4 q^{27} + 8 q^{31} + 4 q^{33} - 4 q^{37} + 12 q^{39} + 6 q^{41} - 8 q^{43} + 8 q^{47} - 7 q^{49} + 2 q^{51} + 10 q^{53} + 8 q^{57} + 12 q^{61} - 8 q^{67} + 8 q^{69} - 12 q^{71} + 2 q^{73} + 10 q^{75} + 4 q^{79} - 11 q^{81} - 16 q^{83} + 10 q^{89} - 16 q^{93} - 18 q^{97} - 2 q^{99}+O(q^{100})$$ q - 2 * q^3 + q^9 - 2 * q^11 - 6 * q^13 - q^17 - 4 * q^19 - 4 * q^23 - 5 * q^25 + 4 * q^27 + 8 * q^31 + 4 * q^33 - 4 * q^37 + 12 * q^39 + 6 * q^41 - 8 * q^43 + 8 * q^47 - 7 * q^49 + 2 * q^51 + 10 * q^53 + 8 * q^57 + 12 * q^61 - 8 * q^67 + 8 * q^69 - 12 * q^71 + 2 * q^73 + 10 * q^75 + 4 * q^79 - 11 * q^81 - 16 * q^83 + 10 * q^89 - 16 * q^93 - 18 * q^97 - 2 * 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 −2.00000 0 0 0 0 0 1.00000 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
$$2$$ $$1$$
$$17$$ $$1$$

## 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 272.2.a.a 1
3.b odd 2 1 2448.2.a.j 1
4.b odd 2 1 136.2.a.b 1
5.b even 2 1 6800.2.a.w 1
8.b even 2 1 1088.2.a.m 1
8.d odd 2 1 1088.2.a.c 1
12.b even 2 1 1224.2.a.d 1
17.b even 2 1 4624.2.a.f 1
20.d odd 2 1 3400.2.a.b 1
20.e even 4 2 3400.2.e.c 2
24.f even 2 1 9792.2.a.be 1
24.h odd 2 1 9792.2.a.bd 1
28.d even 2 1 6664.2.a.b 1
68.d odd 2 1 2312.2.a.a 1
68.f odd 4 2 2312.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.b 1 4.b odd 2 1
272.2.a.a 1 1.a even 1 1 trivial
1088.2.a.c 1 8.d odd 2 1
1088.2.a.m 1 8.b even 2 1
1224.2.a.d 1 12.b even 2 1
2312.2.a.a 1 68.d odd 2 1
2312.2.b.b 2 68.f odd 4 2
2448.2.a.j 1 3.b odd 2 1
3400.2.a.b 1 20.d odd 2 1
3400.2.e.c 2 20.e even 4 2
4624.2.a.f 1 17.b even 2 1
6664.2.a.b 1 28.d even 2 1
6800.2.a.w 1 5.b even 2 1
9792.2.a.bd 1 24.h odd 2 1
9792.2.a.be 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(272))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 2$$
$13$ $$T + 6$$
$17$ $$T + 1$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T$$
$31$ $$T - 8$$
$37$ $$T + 4$$
$41$ $$T - 6$$
$43$ $$T + 8$$
$47$ $$T - 8$$
$53$ $$T - 10$$
$59$ $$T$$
$61$ $$T - 12$$
$67$ $$T + 8$$
$71$ $$T + 12$$
$73$ $$T - 2$$
$79$ $$T - 4$$
$83$ $$T + 16$$
$89$ $$T - 10$$
$97$ $$T + 18$$