# Properties

 Label 252.2.k.a Level $252$ Weight $2$ Character orbit 252.k Analytic conductor $2.012$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,2,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 252.k (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{5} + (3 \zeta_{6} - 1) q^{7}+O(q^{10})$$ q + (2*z - 2) * q^5 + (3*z - 1) * q^7 $$q + (2 \zeta_{6} - 2) q^{5} + (3 \zeta_{6} - 1) q^{7} + 2 \zeta_{6} q^{11} - 3 q^{13} + 8 \zeta_{6} q^{17} + ( - \zeta_{6} + 1) q^{19} + ( - 8 \zeta_{6} + 8) q^{23} + \zeta_{6} q^{25} - 4 q^{29} - 3 \zeta_{6} q^{31} + ( - 2 \zeta_{6} - 4) q^{35} + ( - \zeta_{6} + 1) q^{37} - 6 q^{41} + 11 q^{43} + ( - 6 \zeta_{6} + 6) q^{47} + (3 \zeta_{6} - 8) q^{49} - 12 \zeta_{6} q^{53} - 4 q^{55} + 4 \zeta_{6} q^{59} + ( - 6 \zeta_{6} + 6) q^{61} + ( - 6 \zeta_{6} + 6) q^{65} - 13 \zeta_{6} q^{67} + 10 q^{71} + 11 \zeta_{6} q^{73} + (4 \zeta_{6} - 6) q^{77} + ( - 3 \zeta_{6} + 3) q^{79} - 2 q^{83} - 16 q^{85} + ( - 9 \zeta_{6} + 3) q^{91} + 2 \zeta_{6} q^{95} + 10 q^{97} +O(q^{100})$$ q + (2*z - 2) * q^5 + (3*z - 1) * q^7 + 2*z * q^11 - 3 * q^13 + 8*z * q^17 + (-z + 1) * q^19 + (-8*z + 8) * q^23 + z * q^25 - 4 * q^29 - 3*z * q^31 + (-2*z - 4) * q^35 + (-z + 1) * q^37 - 6 * q^41 + 11 * q^43 + (-6*z + 6) * q^47 + (3*z - 8) * q^49 - 12*z * q^53 - 4 * q^55 + 4*z * q^59 + (-6*z + 6) * q^61 + (-6*z + 6) * q^65 - 13*z * q^67 + 10 * q^71 + 11*z * q^73 + (4*z - 6) * q^77 + (-3*z + 3) * q^79 - 2 * q^83 - 16 * q^85 + (-9*z + 3) * q^91 + 2*z * q^95 + 10 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + q^7 $$2 q - 2 q^{5} + q^{7} + 2 q^{11} - 6 q^{13} + 8 q^{17} + q^{19} + 8 q^{23} + q^{25} - 8 q^{29} - 3 q^{31} - 10 q^{35} + q^{37} - 12 q^{41} + 22 q^{43} + 6 q^{47} - 13 q^{49} - 12 q^{53} - 8 q^{55} + 4 q^{59} + 6 q^{61} + 6 q^{65} - 13 q^{67} + 20 q^{71} + 11 q^{73} - 8 q^{77} + 3 q^{79} - 4 q^{83} - 32 q^{85} - 3 q^{91} + 2 q^{95} + 20 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 + 2 * q^11 - 6 * q^13 + 8 * q^17 + q^19 + 8 * q^23 + q^25 - 8 * q^29 - 3 * q^31 - 10 * q^35 + q^37 - 12 * q^41 + 22 * q^43 + 6 * q^47 - 13 * q^49 - 12 * q^53 - 8 * q^55 + 4 * q^59 + 6 * q^61 + 6 * q^65 - 13 * q^67 + 20 * q^71 + 11 * q^73 - 8 * q^77 + 3 * q^79 - 4 * q^83 - 32 * q^85 - 3 * q^91 + 2 * q^95 + 20 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0.500000 + 2.59808i 0 0 0
109.1 0 0 0 −1.00000 1.73205i 0 0.500000 2.59808i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.k.a 2
3.b odd 2 1 84.2.i.a 2
4.b odd 2 1 1008.2.s.c 2
7.b odd 2 1 1764.2.k.j 2
7.c even 3 1 inner 252.2.k.a 2
7.c even 3 1 1764.2.a.h 1
7.d odd 6 1 1764.2.a.c 1
7.d odd 6 1 1764.2.k.j 2
9.c even 3 1 2268.2.i.b 2
9.c even 3 1 2268.2.l.g 2
9.d odd 6 1 2268.2.i.g 2
9.d odd 6 1 2268.2.l.b 2
12.b even 2 1 336.2.q.c 2
15.d odd 2 1 2100.2.q.b 2
15.e even 4 2 2100.2.bc.a 4
21.c even 2 1 588.2.i.b 2
21.g even 6 1 588.2.a.f 1
21.g even 6 1 588.2.i.b 2
21.h odd 6 1 84.2.i.a 2
21.h odd 6 1 588.2.a.a 1
24.f even 2 1 1344.2.q.n 2
24.h odd 2 1 1344.2.q.b 2
28.f even 6 1 7056.2.a.o 1
28.g odd 6 1 1008.2.s.c 2
28.g odd 6 1 7056.2.a.bs 1
63.g even 3 1 2268.2.i.b 2
63.h even 3 1 2268.2.l.g 2
63.j odd 6 1 2268.2.l.b 2
63.n odd 6 1 2268.2.i.g 2
84.h odd 2 1 2352.2.q.q 2
84.j odd 6 1 2352.2.a.k 1
84.j odd 6 1 2352.2.q.q 2
84.n even 6 1 336.2.q.c 2
84.n even 6 1 2352.2.a.o 1
105.o odd 6 1 2100.2.q.b 2
105.x even 12 2 2100.2.bc.a 4
168.s odd 6 1 1344.2.q.b 2
168.s odd 6 1 9408.2.a.cx 1
168.v even 6 1 1344.2.q.n 2
168.v even 6 1 9408.2.a.bi 1
168.ba even 6 1 9408.2.a.i 1
168.be odd 6 1 9408.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 3.b odd 2 1
84.2.i.a 2 21.h odd 6 1
252.2.k.a 2 1.a even 1 1 trivial
252.2.k.a 2 7.c even 3 1 inner
336.2.q.c 2 12.b even 2 1
336.2.q.c 2 84.n even 6 1
588.2.a.a 1 21.h odd 6 1
588.2.a.f 1 21.g even 6 1
588.2.i.b 2 21.c even 2 1
588.2.i.b 2 21.g even 6 1
1008.2.s.c 2 4.b odd 2 1
1008.2.s.c 2 28.g odd 6 1
1344.2.q.b 2 24.h odd 2 1
1344.2.q.b 2 168.s odd 6 1
1344.2.q.n 2 24.f even 2 1
1344.2.q.n 2 168.v even 6 1
1764.2.a.c 1 7.d odd 6 1
1764.2.a.h 1 7.c even 3 1
1764.2.k.j 2 7.b odd 2 1
1764.2.k.j 2 7.d odd 6 1
2100.2.q.b 2 15.d odd 2 1
2100.2.q.b 2 105.o odd 6 1
2100.2.bc.a 4 15.e even 4 2
2100.2.bc.a 4 105.x even 12 2
2268.2.i.b 2 9.c even 3 1
2268.2.i.b 2 63.g even 3 1
2268.2.i.g 2 9.d odd 6 1
2268.2.i.g 2 63.n odd 6 1
2268.2.l.b 2 9.d odd 6 1
2268.2.l.b 2 63.j odd 6 1
2268.2.l.g 2 9.c even 3 1
2268.2.l.g 2 63.h even 3 1
2352.2.a.k 1 84.j odd 6 1
2352.2.a.o 1 84.n even 6 1
2352.2.q.q 2 84.h odd 2 1
2352.2.q.q 2 84.j odd 6 1
7056.2.a.o 1 28.f even 6 1
7056.2.a.bs 1 28.g odd 6 1
9408.2.a.i 1 168.ba even 6 1
9408.2.a.bi 1 168.v even 6 1
9408.2.a.bx 1 168.be odd 6 1
9408.2.a.cx 1 168.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} - 8T + 64$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} + 3T + 9$$
$37$ $$T^{2} - T + 1$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 11)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} - 6T + 36$$
$67$ $$T^{2} + 13T + 169$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$(T + 2)^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 10)^{2}$$