# Properties

 Label 272.4.a.d Level $272$ Weight $4$ Character orbit 272.a Self dual yes Analytic conductor $16.049$ 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] = [272,4,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, 4, names="a")

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

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$272 = 2^{4} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$16.0485195216$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) 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)

 $$f(q)$$ $$=$$ $$q + 8 q^{3} + 6 q^{5} + 28 q^{7} + 37 q^{9}+O(q^{10})$$ q + 8 * q^3 + 6 * q^5 + 28 * q^7 + 37 * q^9 $$q + 8 q^{3} + 6 q^{5} + 28 q^{7} + 37 q^{9} + 24 q^{11} - 58 q^{13} + 48 q^{15} + 17 q^{17} - 116 q^{19} + 224 q^{21} + 60 q^{23} - 89 q^{25} + 80 q^{27} + 30 q^{29} + 172 q^{31} + 192 q^{33} + 168 q^{35} - 58 q^{37} - 464 q^{39} - 342 q^{41} + 148 q^{43} + 222 q^{45} - 288 q^{47} + 441 q^{49} + 136 q^{51} + 318 q^{53} + 144 q^{55} - 928 q^{57} - 252 q^{59} + 110 q^{61} + 1036 q^{63} - 348 q^{65} + 484 q^{67} + 480 q^{69} + 708 q^{71} + 362 q^{73} - 712 q^{75} + 672 q^{77} + 484 q^{79} - 359 q^{81} - 756 q^{83} + 102 q^{85} + 240 q^{87} - 774 q^{89} - 1624 q^{91} + 1376 q^{93} - 696 q^{95} - 382 q^{97} + 888 q^{99}+O(q^{100})$$ q + 8 * q^3 + 6 * q^5 + 28 * q^7 + 37 * q^9 + 24 * q^11 - 58 * q^13 + 48 * q^15 + 17 * q^17 - 116 * q^19 + 224 * q^21 + 60 * q^23 - 89 * q^25 + 80 * q^27 + 30 * q^29 + 172 * q^31 + 192 * q^33 + 168 * q^35 - 58 * q^37 - 464 * q^39 - 342 * q^41 + 148 * q^43 + 222 * q^45 - 288 * q^47 + 441 * q^49 + 136 * q^51 + 318 * q^53 + 144 * q^55 - 928 * q^57 - 252 * q^59 + 110 * q^61 + 1036 * q^63 - 348 * q^65 + 484 * q^67 + 480 * q^69 + 708 * q^71 + 362 * q^73 - 712 * q^75 + 672 * q^77 + 484 * q^79 - 359 * q^81 - 756 * q^83 + 102 * q^85 + 240 * q^87 - 774 * q^89 - 1624 * q^91 + 1376 * q^93 - 696 * q^95 - 382 * q^97 + 888 * 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 8.00000 0 6.00000 0 28.0000 0 37.0000 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 272.4.a.d 1
3.b odd 2 1 2448.4.a.f 1
4.b odd 2 1 17.4.a.a 1
8.b even 2 1 1088.4.a.a 1
8.d odd 2 1 1088.4.a.l 1
12.b even 2 1 153.4.a.d 1
20.d odd 2 1 425.4.a.d 1
20.e even 4 2 425.4.b.c 2
28.d even 2 1 833.4.a.a 1
44.c even 2 1 2057.4.a.d 1
68.d odd 2 1 289.4.a.a 1
68.f odd 4 2 289.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 4.b odd 2 1
153.4.a.d 1 12.b even 2 1
272.4.a.d 1 1.a even 1 1 trivial
289.4.a.a 1 68.d odd 2 1
289.4.b.a 2 68.f odd 4 2
425.4.a.d 1 20.d odd 2 1
425.4.b.c 2 20.e even 4 2
833.4.a.a 1 28.d even 2 1
1088.4.a.a 1 8.b even 2 1
1088.4.a.l 1 8.d odd 2 1
2057.4.a.d 1 44.c even 2 1
2448.4.a.f 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(272))$$:

 $$T_{3} - 8$$ T3 - 8 $$T_{5} - 6$$ T5 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T - 6$$
$7$ $$T - 28$$
$11$ $$T - 24$$
$13$ $$T + 58$$
$17$ $$T - 17$$
$19$ $$T + 116$$
$23$ $$T - 60$$
$29$ $$T - 30$$
$31$ $$T - 172$$
$37$ $$T + 58$$
$41$ $$T + 342$$
$43$ $$T - 148$$
$47$ $$T + 288$$
$53$ $$T - 318$$
$59$ $$T + 252$$
$61$ $$T - 110$$
$67$ $$T - 484$$
$71$ $$T - 708$$
$73$ $$T - 362$$
$79$ $$T - 484$$
$83$ $$T + 756$$
$89$ $$T + 774$$
$97$ $$T + 382$$