# Properties

 Label 252.4.a.c 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 28) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 6 q^{5} + 7 q^{7}+O(q^{10})$$ q - 6 * q^5 + 7 * q^7 $$q - 6 q^{5} + 7 q^{7} + 12 q^{11} - 82 q^{13} + 30 q^{17} + 68 q^{19} - 216 q^{23} - 89 q^{25} - 246 q^{29} - 112 q^{31} - 42 q^{35} + 110 q^{37} + 246 q^{41} - 172 q^{43} - 192 q^{47} + 49 q^{49} - 558 q^{53} - 72 q^{55} - 540 q^{59} + 110 q^{61} + 492 q^{65} + 140 q^{67} + 840 q^{71} - 550 q^{73} + 84 q^{77} - 208 q^{79} - 516 q^{83} - 180 q^{85} + 1398 q^{89} - 574 q^{91} - 408 q^{95} + 1586 q^{97}+O(q^{100})$$ q - 6 * q^5 + 7 * q^7 + 12 * q^11 - 82 * q^13 + 30 * q^17 + 68 * q^19 - 216 * q^23 - 89 * q^25 - 246 * q^29 - 112 * q^31 - 42 * q^35 + 110 * q^37 + 246 * q^41 - 172 * q^43 - 192 * q^47 + 49 * q^49 - 558 * q^53 - 72 * q^55 - 540 * q^59 + 110 * q^61 + 492 * q^65 + 140 * q^67 + 840 * q^71 - 550 * q^73 + 84 * q^77 - 208 * q^79 - 516 * q^83 - 180 * q^85 + 1398 * q^89 - 574 * q^91 - 408 * q^95 + 1586 * q^97

## 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.c 1
3.b odd 2 1 28.4.a.b 1
4.b odd 2 1 1008.4.a.f 1
7.b odd 2 1 1764.4.a.k 1
7.c even 3 2 1764.4.k.k 2
7.d odd 6 2 1764.4.k.e 2
12.b even 2 1 112.4.a.c 1
15.d odd 2 1 700.4.a.e 1
15.e even 4 2 700.4.e.f 2
21.c even 2 1 196.4.a.b 1
21.g even 6 2 196.4.e.d 2
21.h odd 6 2 196.4.e.c 2
24.f even 2 1 448.4.a.m 1
24.h odd 2 1 448.4.a.d 1
84.h odd 2 1 784.4.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 3.b odd 2 1
112.4.a.c 1 12.b even 2 1
196.4.a.b 1 21.c even 2 1
196.4.e.c 2 21.h odd 6 2
196.4.e.d 2 21.g even 6 2
252.4.a.c 1 1.a even 1 1 trivial
448.4.a.d 1 24.h odd 2 1
448.4.a.m 1 24.f even 2 1
700.4.a.e 1 15.d odd 2 1
700.4.e.f 2 15.e even 4 2
784.4.a.n 1 84.h odd 2 1
1008.4.a.f 1 4.b odd 2 1
1764.4.a.k 1 7.b odd 2 1
1764.4.k.e 2 7.d odd 6 2
1764.4.k.k 2 7.c even 3 2

## Hecke kernels

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

 $$T_{5} + 6$$ T5 + 6 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T - 7$$
$11$ $$T - 12$$
$13$ $$T + 82$$
$17$ $$T - 30$$
$19$ $$T - 68$$
$23$ $$T + 216$$
$29$ $$T + 246$$
$31$ $$T + 112$$
$37$ $$T - 110$$
$41$ $$T - 246$$
$43$ $$T + 172$$
$47$ $$T + 192$$
$53$ $$T + 558$$
$59$ $$T + 540$$
$61$ $$T - 110$$
$67$ $$T - 140$$
$71$ $$T - 840$$
$73$ $$T + 550$$
$79$ $$T + 208$$
$83$ $$T + 516$$
$89$ $$T - 1398$$
$97$ $$T - 1586$$