# Properties

 Label 252.6.a.c Level $252$ Weight $6$ Character orbit 252.a Self dual yes Analytic conductor $40.417$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 34 q^{5} - 49 q^{7}+O(q^{10})$$ q + 34 * q^5 - 49 * q^7 $$q + 34 q^{5} - 49 q^{7} + 332 q^{11} - 1026 q^{13} - 922 q^{17} + 452 q^{19} + 3776 q^{23} - 1969 q^{25} - 1166 q^{29} - 9792 q^{31} - 1666 q^{35} + 2390 q^{37} + 7230 q^{41} + 4652 q^{43} - 24672 q^{47} + 2401 q^{49} - 1110 q^{53} + 11288 q^{55} - 46892 q^{59} - 9762 q^{61} - 34884 q^{65} - 26252 q^{67} - 65440 q^{71} - 5606 q^{73} - 16268 q^{77} - 9840 q^{79} - 61108 q^{83} - 31348 q^{85} + 62958 q^{89} + 50274 q^{91} + 15368 q^{95} - 37838 q^{97}+O(q^{100})$$ q + 34 * q^5 - 49 * q^7 + 332 * q^11 - 1026 * q^13 - 922 * q^17 + 452 * q^19 + 3776 * q^23 - 1969 * q^25 - 1166 * q^29 - 9792 * q^31 - 1666 * q^35 + 2390 * q^37 + 7230 * q^41 + 4652 * q^43 - 24672 * q^47 + 2401 * q^49 - 1110 * q^53 + 11288 * q^55 - 46892 * q^59 - 9762 * q^61 - 34884 * q^65 - 26252 * q^67 - 65440 * q^71 - 5606 * q^73 - 16268 * q^77 - 9840 * q^79 - 61108 * q^83 - 31348 * q^85 + 62958 * q^89 + 50274 * q^91 + 15368 * q^95 - 37838 * 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 34.0000 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.c 1
3.b odd 2 1 84.6.a.b 1
4.b odd 2 1 1008.6.a.u 1
12.b even 2 1 336.6.a.e 1
21.c even 2 1 588.6.a.b 1
21.g even 6 2 588.6.i.e 2
21.h odd 6 2 588.6.i.c 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 34$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(252))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 34$$
$7$ $$T + 49$$
$11$ $$T - 332$$
$13$ $$T + 1026$$
$17$ $$T + 922$$
$19$ $$T - 452$$
$23$ $$T - 3776$$
$29$ $$T + 1166$$
$31$ $$T + 9792$$
$37$ $$T - 2390$$
$41$ $$T - 7230$$
$43$ $$T - 4652$$
$47$ $$T + 24672$$
$53$ $$T + 1110$$
$59$ $$T + 46892$$
$61$ $$T + 9762$$
$67$ $$T + 26252$$
$71$ $$T + 65440$$
$73$ $$T + 5606$$
$79$ $$T + 9840$$
$83$ $$T + 61108$$
$89$ $$T - 62958$$
$97$ $$T + 37838$$