# Properties

 Label 2352.4.a.f Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 6q^{5} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 6q^{5} + 9q^{9} + 30q^{11} + 53q^{13} + 18q^{15} - 84q^{17} + 97q^{19} - 84q^{23} - 89q^{25} - 27q^{27} - 180q^{29} - 179q^{31} - 90q^{33} - 145q^{37} - 159q^{39} + 126q^{41} + 325q^{43} - 54q^{45} + 366q^{47} + 252q^{51} - 768q^{53} - 180q^{55} - 291q^{57} + 264q^{59} + 818q^{61} - 318q^{65} + 523q^{67} + 252q^{69} + 342q^{71} - 43q^{73} + 267q^{75} + 1171q^{79} + 81q^{81} + 810q^{83} + 504q^{85} + 540q^{87} - 600q^{89} + 537q^{93} - 582q^{95} + 386q^{97} + 270q^{99} + O(q^{100})$$

## 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.f 1
4.b odd 2 1 294.4.a.d 1
7.b odd 2 1 2352.4.a.bf 1
7.c even 3 2 336.4.q.f 2
12.b even 2 1 882.4.a.o 1
28.d even 2 1 294.4.a.c 1
28.f even 6 2 294.4.e.i 2
28.g odd 6 2 42.4.e.a 2
84.h odd 2 1 882.4.a.l 1
84.j odd 6 2 882.4.g.g 2
84.n even 6 2 126.4.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 28.g odd 6 2
126.4.g.b 2 84.n even 6 2
294.4.a.c 1 28.d even 2 1
294.4.a.d 1 4.b odd 2 1
294.4.e.i 2 28.f even 6 2
336.4.q.f 2 7.c even 3 2
882.4.a.l 1 84.h odd 2 1
882.4.a.o 1 12.b even 2 1
882.4.g.g 2 84.j odd 6 2
2352.4.a.f 1 1.a even 1 1 trivial
2352.4.a.bf 1 7.b odd 2 1

## Hecke kernels

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

 $$T_{5} + 6$$ $$T_{11} - 30$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$6 + T$$
$7$ $$T$$
$11$ $$-30 + T$$
$13$ $$-53 + T$$
$17$ $$84 + T$$
$19$ $$-97 + T$$
$23$ $$84 + T$$
$29$ $$180 + T$$
$31$ $$179 + T$$
$37$ $$145 + T$$
$41$ $$-126 + T$$
$43$ $$-325 + T$$
$47$ $$-366 + T$$
$53$ $$768 + T$$
$59$ $$-264 + T$$
$61$ $$-818 + T$$
$67$ $$-523 + T$$
$71$ $$-342 + T$$
$73$ $$43 + T$$
$79$ $$-1171 + T$$
$83$ $$-810 + T$$
$89$ $$600 + T$$
$97$ $$-386 + T$$