# Properties

 Label 252.6.a.b Level $252$ Weight $6$ Character orbit 252.a Self dual yes Analytic conductor $40.417$ 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] = [252,6,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 252.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: $$40.4167225929$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{5} + 49 q^{7}+O(q^{10})$$ q - 6 * q^5 + 49 * q^7 $$q - 6 q^{5} + 49 q^{7} + 108 q^{11} - 346 q^{13} + 1398 q^{17} - 1012 q^{19} + 1536 q^{23} - 3089 q^{25} + 3762 q^{29} - 736 q^{31} - 294 q^{35} + 2054 q^{37} + 15534 q^{41} + 11036 q^{43} - 4560 q^{47} + 2401 q^{49} + 7962 q^{53} - 648 q^{55} + 7020 q^{59} + 26870 q^{61} + 2076 q^{65} + 52148 q^{67} + 2544 q^{71} - 9766 q^{73} + 5292 q^{77} + 68672 q^{79} + 61668 q^{83} - 8388 q^{85} + 41454 q^{89} - 16954 q^{91} + 6072 q^{95} - 111262 q^{97}+O(q^{100})$$ q - 6 * q^5 + 49 * q^7 + 108 * q^11 - 346 * q^13 + 1398 * q^17 - 1012 * q^19 + 1536 * q^23 - 3089 * q^25 + 3762 * q^29 - 736 * q^31 - 294 * q^35 + 2054 * q^37 + 15534 * q^41 + 11036 * q^43 - 4560 * q^47 + 2401 * q^49 + 7962 * q^53 - 648 * q^55 + 7020 * q^59 + 26870 * q^61 + 2076 * q^65 + 52148 * q^67 + 2544 * q^71 - 9766 * q^73 + 5292 * q^77 + 68672 * q^79 + 61668 * q^83 - 8388 * q^85 + 41454 * q^89 - 16954 * q^91 + 6072 * q^95 - 111262 * q^97

## 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 0 0 −6.00000 0 49.0000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 252.6.a.b 1
3.b odd 2 1 84.6.a.a 1
4.b odd 2 1 1008.6.a.o 1
12.b even 2 1 336.6.a.n 1
21.c even 2 1 588.6.a.e 1
21.g even 6 2 588.6.i.b 2
21.h odd 6 2 588.6.i.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.a 1 3.b odd 2 1
252.6.a.b 1 1.a even 1 1 trivial
336.6.a.n 1 12.b even 2 1
588.6.a.e 1 21.c even 2 1
588.6.i.b 2 21.g even 6 2
588.6.i.f 2 21.h odd 6 2
1008.6.a.o 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 6$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(252))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T - 49$$
$11$ $$T - 108$$
$13$ $$T + 346$$
$17$ $$T - 1398$$
$19$ $$T + 1012$$
$23$ $$T - 1536$$
$29$ $$T - 3762$$
$31$ $$T + 736$$
$37$ $$T - 2054$$
$41$ $$T - 15534$$
$43$ $$T - 11036$$
$47$ $$T + 4560$$
$53$ $$T - 7962$$
$59$ $$T - 7020$$
$61$ $$T - 26870$$
$67$ $$T - 52148$$
$71$ $$T - 2544$$
$73$ $$T + 9766$$
$79$ $$T - 68672$$
$83$ $$T - 61668$$
$89$ $$T - 41454$$
$97$ $$T + 111262$$