# Properties

 Label 252.4.a.b Level $252$ Weight $4$ Character orbit 252.a Self dual yes Analytic conductor $14.868$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 6q^{5} + 7q^{7} + O(q^{10})$$ $$q - 6q^{5} + 7q^{7} - 36q^{11} + 62q^{13} - 114q^{17} - 76q^{19} + 24q^{23} - 89q^{25} - 54q^{29} - 112q^{31} - 42q^{35} - 178q^{37} - 378q^{41} - 172q^{43} + 192q^{47} + 49q^{49} + 402q^{53} + 216q^{55} - 396q^{59} + 254q^{61} - 372q^{65} - 1012q^{67} - 840q^{71} + 890q^{73} - 252q^{77} + 80q^{79} + 108q^{83} + 684q^{85} + 1638q^{89} + 434q^{91} + 456q^{95} + 1010q^{97} + O(q^{100})$$

## 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
 0
0 0 0 −6.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.a.b 1
3.b odd 2 1 84.4.a.a 1
4.b odd 2 1 1008.4.a.h 1
7.b odd 2 1 1764.4.a.j 1
7.c even 3 2 1764.4.k.l 2
7.d odd 6 2 1764.4.k.f 2
12.b even 2 1 336.4.a.k 1
15.d odd 2 1 2100.4.a.l 1
15.e even 4 2 2100.4.k.j 2
21.c even 2 1 588.4.a.d 1
21.g even 6 2 588.4.i.c 2
21.h odd 6 2 588.4.i.f 2
24.f even 2 1 1344.4.a.d 1
24.h odd 2 1 1344.4.a.q 1
84.h odd 2 1 2352.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 3.b odd 2 1
252.4.a.b 1 1.a even 1 1 trivial
336.4.a.k 1 12.b even 2 1
588.4.a.d 1 21.c even 2 1
588.4.i.c 2 21.g even 6 2
588.4.i.f 2 21.h odd 6 2
1008.4.a.h 1 4.b odd 2 1
1344.4.a.d 1 24.f even 2 1
1344.4.a.q 1 24.h odd 2 1
1764.4.a.j 1 7.b odd 2 1
1764.4.k.f 2 7.d odd 6 2
1764.4.k.l 2 7.c even 3 2
2100.4.a.l 1 15.d odd 2 1
2100.4.k.j 2 15.e even 4 2
2352.4.a.d 1 84.h odd 2 1

## Hecke kernels

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

 $$T_{5} + 6$$ $$T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$6 + T$$
$7$ $$-7 + T$$
$11$ $$36 + T$$
$13$ $$-62 + T$$
$17$ $$114 + T$$
$19$ $$76 + T$$
$23$ $$-24 + T$$
$29$ $$54 + T$$
$31$ $$112 + T$$
$37$ $$178 + T$$
$41$ $$378 + T$$
$43$ $$172 + T$$
$47$ $$-192 + T$$
$53$ $$-402 + T$$
$59$ $$396 + T$$
$61$ $$-254 + T$$
$67$ $$1012 + T$$
$71$ $$840 + T$$
$73$ $$-890 + T$$
$79$ $$-80 + T$$
$83$ $$-108 + T$$
$89$ $$-1638 + T$$
$97$ $$-1010 + T$$