# Properties

 Label 252.2.k.b Level $252$ Weight $2$ Character orbit 252.k Analytic conductor $2.012$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 3 - \zeta_{6} ) q^{7} + 5 q^{13} + ( 1 - \zeta_{6} ) q^{19} + 5 \zeta_{6} q^{25} -11 \zeta_{6} q^{31} + ( -11 + 11 \zeta_{6} ) q^{37} -13 q^{43} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -14 + 14 \zeta_{6} ) q^{61} -5 \zeta_{6} q^{67} -17 \zeta_{6} q^{73} + ( -17 + 17 \zeta_{6} ) q^{79} + ( 15 - 5 \zeta_{6} ) q^{91} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{7} + O(q^{10})$$ $$2q + 5q^{7} + 10q^{13} + q^{19} + 5q^{25} - 11q^{31} - 11q^{37} - 26q^{43} + 11q^{49} - 14q^{61} - 5q^{67} - 17q^{73} - 17q^{79} + 25q^{91} + 28q^{97} + O(q^{100})$$

## 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 0 0 2.50000 0.866025i 0 0 0
109.1 0 0 0 0 0 2.50000 + 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.k.b 2
3.b odd 2 1 CM 252.2.k.b 2
4.b odd 2 1 1008.2.s.i 2
7.b odd 2 1 1764.2.k.f 2
7.c even 3 1 inner 252.2.k.b 2
7.c even 3 1 1764.2.a.f 1
7.d odd 6 1 1764.2.a.d 1
7.d odd 6 1 1764.2.k.f 2
9.c even 3 1 2268.2.i.c 2
9.c even 3 1 2268.2.l.e 2
9.d odd 6 1 2268.2.i.c 2
9.d odd 6 1 2268.2.l.e 2
12.b even 2 1 1008.2.s.i 2
21.c even 2 1 1764.2.k.f 2
21.g even 6 1 1764.2.a.d 1
21.g even 6 1 1764.2.k.f 2
21.h odd 6 1 inner 252.2.k.b 2
21.h odd 6 1 1764.2.a.f 1
28.f even 6 1 7056.2.a.z 1
28.g odd 6 1 1008.2.s.i 2
28.g odd 6 1 7056.2.a.be 1
63.g even 3 1 2268.2.i.c 2
63.h even 3 1 2268.2.l.e 2
63.j odd 6 1 2268.2.l.e 2
63.n odd 6 1 2268.2.i.c 2
84.j odd 6 1 7056.2.a.z 1
84.n even 6 1 1008.2.s.i 2
84.n even 6 1 7056.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.k.b 2 1.a even 1 1 trivial
252.2.k.b 2 3.b odd 2 1 CM
252.2.k.b 2 7.c even 3 1 inner
252.2.k.b 2 21.h odd 6 1 inner
1008.2.s.i 2 4.b odd 2 1
1008.2.s.i 2 12.b even 2 1
1008.2.s.i 2 28.g odd 6 1
1008.2.s.i 2 84.n even 6 1
1764.2.a.d 1 7.d odd 6 1
1764.2.a.d 1 21.g even 6 1
1764.2.a.f 1 7.c even 3 1
1764.2.a.f 1 21.h odd 6 1
1764.2.k.f 2 7.b odd 2 1
1764.2.k.f 2 7.d odd 6 1
1764.2.k.f 2 21.c even 2 1
1764.2.k.f 2 21.g even 6 1
2268.2.i.c 2 9.c even 3 1
2268.2.i.c 2 9.d odd 6 1
2268.2.i.c 2 63.g even 3 1
2268.2.i.c 2 63.n odd 6 1
2268.2.l.e 2 9.c even 3 1
2268.2.l.e 2 9.d odd 6 1
2268.2.l.e 2 63.h even 3 1
2268.2.l.e 2 63.j odd 6 1
7056.2.a.z 1 28.f even 6 1
7056.2.a.z 1 84.j odd 6 1
7056.2.a.be 1 28.g odd 6 1
7056.2.a.be 1 84.n even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 13 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 + 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 14 T + 97 T^{2} )^{2}$$