# Properties

 Label 2352.4.a.d Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $2$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$2$$ 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

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 6q^{5} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 6q^{5} + 9q^{9} - 36q^{11} - 62q^{13} + 18q^{15} - 114q^{17} - 76q^{19} + 24q^{23} - 89q^{25} - 27q^{27} + 54q^{29} - 112q^{31} + 108q^{33} - 178q^{37} + 186q^{39} - 378q^{41} + 172q^{43} - 54q^{45} - 192q^{47} + 342q^{51} - 402q^{53} + 216q^{55} + 228q^{57} + 396q^{59} - 254q^{61} + 372q^{65} + 1012q^{67} - 72q^{69} - 840q^{71} - 890q^{73} + 267q^{75} - 80q^{79} + 81q^{81} - 108q^{83} + 684q^{85} - 162q^{87} + 1638q^{89} + 336q^{93} + 456q^{95} - 1010q^{97} - 324q^{99} + O(q^{100})$$

## 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 −3.00000 0 −6.00000 0 0 0 9.00000 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 2352.4.a.d 1
4.b odd 2 1 588.4.a.d 1
7.b odd 2 1 336.4.a.k 1
12.b even 2 1 1764.4.a.j 1
21.c even 2 1 1008.4.a.h 1
28.d even 2 1 84.4.a.a 1
28.f even 6 2 588.4.i.f 2
28.g odd 6 2 588.4.i.c 2
56.e even 2 1 1344.4.a.q 1
56.h odd 2 1 1344.4.a.d 1
84.h odd 2 1 252.4.a.b 1
84.j odd 6 2 1764.4.k.l 2
84.n even 6 2 1764.4.k.f 2
140.c even 2 1 2100.4.a.l 1
140.j odd 4 2 2100.4.k.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 28.d even 2 1
252.4.a.b 1 84.h odd 2 1
336.4.a.k 1 7.b odd 2 1
588.4.a.d 1 4.b odd 2 1
588.4.i.c 2 28.g odd 6 2
588.4.i.f 2 28.f even 6 2
1008.4.a.h 1 21.c even 2 1
1344.4.a.d 1 56.h odd 2 1
1344.4.a.q 1 56.e even 2 1
1764.4.a.j 1 12.b even 2 1
1764.4.k.f 2 84.n even 6 2
1764.4.k.l 2 84.j odd 6 2
2100.4.a.l 1 140.c even 2 1
2100.4.k.j 2 140.j odd 4 2
2352.4.a.d 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

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

## Hecke characteristic polynomials

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