# Properties

 Label 252.6.a.g Level $252$ Weight $6$ Character orbit 252.a Self dual yes Analytic conductor $40.417$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{91})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 91$$ x^2 - 91 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{91}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 49 q^{7}+O(q^{10})$$ q + b * q^5 + 49 * q^7 $$q + \beta q^{5} + 49 q^{7} - 9 \beta q^{11} - 670 q^{13} - 17 \beta q^{17} + 284 q^{19} - 31 \beta q^{23} + 151 q^{25} + 120 \beta q^{29} + 1532 q^{31} + 49 \beta q^{35} - 15118 q^{37} + 93 \beta q^{41} - 10996 q^{43} - 338 \beta q^{47} + 2401 q^{49} + 410 \beta q^{53} - 29484 q^{55} - 414 \beta q^{59} - 14602 q^{61} - 670 \beta q^{65} - 36628 q^{67} + 1187 \beta q^{71} - 54802 q^{73} - 441 \beta q^{77} - 31768 q^{79} - 1296 \beta q^{83} - 55692 q^{85} - 15 \beta q^{89} - 32830 q^{91} + 284 \beta q^{95} + 14126 q^{97} +O(q^{100})$$ q + b * q^5 + 49 * q^7 - 9*b * q^11 - 670 * q^13 - 17*b * q^17 + 284 * q^19 - 31*b * q^23 + 151 * q^25 + 120*b * q^29 + 1532 * q^31 + 49*b * q^35 - 15118 * q^37 + 93*b * q^41 - 10996 * q^43 - 338*b * q^47 + 2401 * q^49 + 410*b * q^53 - 29484 * q^55 - 414*b * q^59 - 14602 * q^61 - 670*b * q^65 - 36628 * q^67 + 1187*b * q^71 - 54802 * q^73 - 441*b * q^77 - 31768 * q^79 - 1296*b * q^83 - 55692 * q^85 - 15*b * q^89 - 32830 * q^91 + 284*b * q^95 + 14126 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 98 q^{7}+O(q^{10})$$ 2 * q + 98 * q^7 $$2 q + 98 q^{7} - 1340 q^{13} + 568 q^{19} + 302 q^{25} + 3064 q^{31} - 30236 q^{37} - 21992 q^{43} + 4802 q^{49} - 58968 q^{55} - 29204 q^{61} - 73256 q^{67} - 109604 q^{73} - 63536 q^{79} - 111384 q^{85} - 65660 q^{91} + 28252 q^{97}+O(q^{100})$$ 2 * q + 98 * q^7 - 1340 * q^13 + 568 * q^19 + 302 * q^25 + 3064 * q^31 - 30236 * q^37 - 21992 * q^43 + 4802 * q^49 - 58968 * q^55 - 29204 * q^61 - 73256 * q^67 - 109604 * q^73 - 63536 * q^79 - 111384 * q^85 - 65660 * q^91 + 28252 * 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
 −9.53939 9.53939
0 0 0 −57.2364 0 49.0000 0 0 0
1.2 0 0 0 57.2364 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.6.a.g 2
3.b odd 2 1 inner 252.6.a.g 2
4.b odd 2 1 1008.6.a.bk 2
12.b even 2 1 1008.6.a.bk 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.6.a.g 2 1.a even 1 1 trivial
252.6.a.g 2 3.b odd 2 1 inner
1008.6.a.bk 2 4.b odd 2 1
1008.6.a.bk 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3276$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(252))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3276$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} - 265356$$
$13$ $$(T + 670)^{2}$$
$17$ $$T^{2} - 946764$$
$19$ $$(T - 284)^{2}$$
$23$ $$T^{2} - 3148236$$
$29$ $$T^{2} - 47174400$$
$31$ $$(T - 1532)^{2}$$
$37$ $$(T + 15118)^{2}$$
$41$ $$T^{2} - 28334124$$
$43$ $$(T + 10996)^{2}$$
$47$ $$T^{2} - 374263344$$
$53$ $$T^{2} - 550695600$$
$59$ $$T^{2} - 561493296$$
$61$ $$(T + 14602)^{2}$$
$67$ $$(T + 36628)^{2}$$
$71$ $$T^{2} - 4615782444$$
$73$ $$(T + 54802)^{2}$$
$79$ $$(T + 31768)^{2}$$
$83$ $$T^{2} - 5502422016$$
$89$ $$T^{2} - 737100$$
$97$ $$(T - 14126)^{2}$$