# Properties

 Label 252.4.a.f Level $252$ Weight $4$ Character orbit 252.a Self dual yes Analytic conductor $14.868$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 7 q^{7} +O(q^{10})$$ $$q + \beta q^{5} + 7 q^{7} + \beta q^{11} + 26 q^{13} -5 \beta q^{17} + 68 q^{19} + 3 \beta q^{23} + 127 q^{25} -16 \beta q^{29} + 212 q^{31} + 7 \beta q^{35} + 218 q^{37} + 25 \beta q^{41} + 260 q^{43} + 26 \beta q^{47} + 49 q^{49} -30 \beta q^{53} + 252 q^{55} -18 \beta q^{59} -322 q^{61} + 26 \beta q^{65} + 356 q^{67} -71 \beta q^{71} -226 q^{73} + 7 \beta q^{77} + 440 q^{79} -16 \beta q^{83} -1260 q^{85} + 13 \beta q^{89} + 182 q^{91} + 68 \beta q^{95} -1330 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{7} + O(q^{10})$$ $$2q + 14q^{7} + 52q^{13} + 136q^{19} + 254q^{25} + 424q^{31} + 436q^{37} + 520q^{43} + 98q^{49} + 504q^{55} - 644q^{61} + 712q^{67} - 452q^{73} + 880q^{79} - 2520q^{85} + 364q^{91} - 2660q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
0 0 0 −15.8745 0 7.00000 0 0 0
1.2 0 0 0 15.8745 0 7.00000 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 252.4.a.f 2
3.b odd 2 1 inner 252.4.a.f 2
4.b odd 2 1 1008.4.a.bc 2
7.b odd 2 1 1764.4.a.s 2
7.c even 3 2 1764.4.k.v 4
7.d odd 6 2 1764.4.k.u 4
12.b even 2 1 1008.4.a.bc 2
21.c even 2 1 1764.4.a.s 2
21.g even 6 2 1764.4.k.u 4
21.h odd 6 2 1764.4.k.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.a.f 2 1.a even 1 1 trivial
252.4.a.f 2 3.b odd 2 1 inner
1008.4.a.bc 2 4.b odd 2 1
1008.4.a.bc 2 12.b even 2 1
1764.4.a.s 2 7.b odd 2 1
1764.4.a.s 2 21.c even 2 1
1764.4.k.u 4 7.d odd 6 2
1764.4.k.u 4 21.g even 6 2
1764.4.k.v 4 7.c even 3 2
1764.4.k.v 4 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(252))$$:

 $$T_{5}^{2} - 252$$ $$T_{11}^{2} - 252$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-252 + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-252 + T^{2}$$
$13$ $$( -26 + T )^{2}$$
$17$ $$-6300 + T^{2}$$
$19$ $$( -68 + T )^{2}$$
$23$ $$-2268 + T^{2}$$
$29$ $$-64512 + T^{2}$$
$31$ $$( -212 + T )^{2}$$
$37$ $$( -218 + T )^{2}$$
$41$ $$-157500 + T^{2}$$
$43$ $$( -260 + T )^{2}$$
$47$ $$-170352 + T^{2}$$
$53$ $$-226800 + T^{2}$$
$59$ $$-81648 + T^{2}$$
$61$ $$( 322 + T )^{2}$$
$67$ $$( -356 + T )^{2}$$
$71$ $$-1270332 + T^{2}$$
$73$ $$( 226 + T )^{2}$$
$79$ $$( -440 + T )^{2}$$
$83$ $$-64512 + T^{2}$$
$89$ $$-42588 + T^{2}$$
$97$ $$( 1330 + T )^{2}$$