# Properties

 Label 252.4.k.a Level $252$ Weight $4$ Character orbit 252.k Analytic conductor $14.868$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -19 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -19 + \zeta_{6} ) q^{7} + 89 q^{13} + ( 163 - 163 \zeta_{6} ) q^{19} + 125 \zeta_{6} q^{25} + 19 \zeta_{6} q^{31} + ( 433 - 433 \zeta_{6} ) q^{37} + 449 q^{43} + ( 360 - 37 \zeta_{6} ) q^{49} + ( -182 + 182 \zeta_{6} ) q^{61} -1007 \zeta_{6} q^{67} + 919 \zeta_{6} q^{73} + ( -503 + 503 \zeta_{6} ) q^{79} + ( -1691 + 89 \zeta_{6} ) q^{91} -1330 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 37q^{7} + O(q^{10})$$ $$2q - 37q^{7} + 178q^{13} + 163q^{19} + 125q^{25} + 19q^{31} + 433q^{37} + 898q^{43} + 683q^{49} - 182q^{61} - 1007q^{67} + 919q^{73} - 503q^{79} - 3293q^{91} - 2660q^{97} + O(q^{100})$$

## 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 −18.5000 + 0.866025i 0 0 0
109.1 0 0 0 0 0 −18.5000 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 252.4.k.a 2
3.b odd 2 1 CM 252.4.k.a 2
7.b odd 2 1 1764.4.k.h 2
7.c even 3 1 inner 252.4.k.a 2
7.c even 3 1 1764.4.a.h 1
7.d odd 6 1 1764.4.a.e 1
7.d odd 6 1 1764.4.k.h 2
21.c even 2 1 1764.4.k.h 2
21.g even 6 1 1764.4.a.e 1
21.g even 6 1 1764.4.k.h 2
21.h odd 6 1 inner 252.4.k.a 2
21.h odd 6 1 1764.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.k.a 2 1.a even 1 1 trivial
252.4.k.a 2 3.b odd 2 1 CM
252.4.k.a 2 7.c even 3 1 inner
252.4.k.a 2 21.h odd 6 1 inner
1764.4.a.e 1 7.d odd 6 1
1764.4.a.e 1 21.g even 6 1
1764.4.a.h 1 7.c even 3 1
1764.4.a.h 1 21.h odd 6 1
1764.4.k.h 2 7.b odd 2 1
1764.4.k.h 2 7.d odd 6 1
1764.4.k.h 2 21.c even 2 1
1764.4.k.h 2 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}$$ $$T_{13} - 89$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 + 37 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -89 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$26569 - 163 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$361 - 19 T + T^{2}$$
$37$ $$187489 - 433 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -449 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$33124 + 182 T + T^{2}$$
$67$ $$1014049 + 1007 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$844561 - 919 T + T^{2}$$
$79$ $$253009 + 503 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 1330 + T )^{2}$$