# Properties

 Label 252.2.k.a Level $252$ Weight $2$ Character orbit 252.k Analytic conductor $2.012$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

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

## 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.

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 $$T_{5}^{2} + 2 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.