# Properties

 Label 6272.2.a.g Level $6272$ Weight $2$ Character orbit 6272.a Self dual yes Analytic conductor $50.082$ 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] = [6272,2,Mod(1,6272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6272, 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("6272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6272 = 2^{7} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6272.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: $$50.0821721477$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 2 q^{5} + q^{9}+O(q^{10})$$ q + 2 * q^3 - 2 * q^5 + q^9 $$q + 2 q^{3} - 2 q^{5} + q^{9} + 2 q^{11} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 2 q^{19} - 4 q^{23} - q^{25} - 4 q^{27} - 6 q^{29} + 4 q^{33} + 10 q^{37} - 4 q^{39} + 6 q^{41} - 6 q^{43} - 2 q^{45} - 8 q^{47} + 4 q^{51} - 6 q^{53} - 4 q^{55} + 4 q^{57} + 14 q^{59} - 2 q^{61} + 4 q^{65} - 10 q^{67} - 8 q^{69} - 12 q^{71} - 14 q^{73} - 2 q^{75} + 8 q^{79} - 11 q^{81} - 6 q^{83} - 4 q^{85} - 12 q^{87} + 2 q^{89} - 4 q^{95} + 2 q^{97} + 2 q^{99}+O(q^{100})$$ q + 2 * q^3 - 2 * q^5 + q^9 + 2 * q^11 - 2 * q^13 - 4 * q^15 + 2 * q^17 + 2 * q^19 - 4 * q^23 - q^25 - 4 * q^27 - 6 * q^29 + 4 * q^33 + 10 * q^37 - 4 * q^39 + 6 * q^41 - 6 * q^43 - 2 * q^45 - 8 * q^47 + 4 * q^51 - 6 * q^53 - 4 * q^55 + 4 * q^57 + 14 * q^59 - 2 * q^61 + 4 * q^65 - 10 * q^67 - 8 * q^69 - 12 * q^71 - 14 * q^73 - 2 * q^75 + 8 * q^79 - 11 * q^81 - 6 * q^83 - 4 * q^85 - 12 * q^87 + 2 * q^89 - 4 * q^95 + 2 * 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 −2.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$

## 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 6272.2.a.g 1
4.b odd 2 1 6272.2.a.a 1
7.b odd 2 1 128.2.a.b yes 1
8.b even 2 1 6272.2.a.b 1
8.d odd 2 1 6272.2.a.h 1
21.c even 2 1 1152.2.a.h 1
28.d even 2 1 128.2.a.d yes 1
35.c odd 2 1 3200.2.a.u 1
35.f even 4 2 3200.2.c.k 2
56.e even 2 1 128.2.a.a 1
56.h odd 2 1 128.2.a.c yes 1
84.h odd 2 1 1152.2.a.c 1
112.j even 4 2 256.2.b.c 2
112.l odd 4 2 256.2.b.a 2
140.c even 2 1 3200.2.a.h 1
140.j odd 4 2 3200.2.c.e 2
168.e odd 2 1 1152.2.a.m 1
168.i even 2 1 1152.2.a.r 1
224.v odd 8 4 1024.2.e.m 4
224.x even 8 4 1024.2.e.i 4
280.c odd 2 1 3200.2.a.e 1
280.n even 2 1 3200.2.a.x 1
280.s even 4 2 3200.2.c.f 2
280.y odd 4 2 3200.2.c.l 2
336.v odd 4 2 2304.2.d.r 2
336.y even 4 2 2304.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 56.e even 2 1
128.2.a.b yes 1 7.b odd 2 1
128.2.a.c yes 1 56.h odd 2 1
128.2.a.d yes 1 28.d even 2 1
256.2.b.a 2 112.l odd 4 2
256.2.b.c 2 112.j even 4 2
1024.2.e.i 4 224.x even 8 4
1024.2.e.m 4 224.v odd 8 4
1152.2.a.c 1 84.h odd 2 1
1152.2.a.h 1 21.c even 2 1
1152.2.a.m 1 168.e odd 2 1
1152.2.a.r 1 168.i even 2 1
2304.2.d.b 2 336.y even 4 2
2304.2.d.r 2 336.v odd 4 2
3200.2.a.e 1 280.c odd 2 1
3200.2.a.h 1 140.c even 2 1
3200.2.a.u 1 35.c odd 2 1
3200.2.a.x 1 280.n even 2 1
3200.2.c.e 2 140.j odd 4 2
3200.2.c.f 2 280.s even 4 2
3200.2.c.k 2 35.f even 4 2
3200.2.c.l 2 280.y odd 4 2
6272.2.a.a 1 4.b odd 2 1
6272.2.a.b 1 8.b even 2 1
6272.2.a.g 1 1.a even 1 1 trivial
6272.2.a.h 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6272))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{5} + 2$$ T5 + 2 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 2$$ T13 + 2 $$T_{23} + 4$$ T23 + 4 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T + 2$$
$17$ $$T - 2$$
$19$ $$T - 2$$
$23$ $$T + 4$$
$29$ $$T + 6$$
$31$ $$T$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T + 6$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T - 14$$
$61$ $$T + 2$$
$67$ $$T + 10$$
$71$ $$T + 12$$
$73$ $$T + 14$$
$79$ $$T - 8$$
$83$ $$T + 6$$
$89$ $$T - 2$$
$97$ $$T - 2$$