# Properties

 Label 2352.4.a.ce Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{137})$$ Defining polynomial: $$x^{2} - x - 34$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1176) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{137}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( 9 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( 9 - \beta ) q^{5} + 9 q^{9} + ( -29 - 3 \beta ) q^{11} + ( 24 + 4 \beta ) q^{13} + ( 27 - 3 \beta ) q^{15} + ( -3 - \beta ) q^{17} + ( -42 - 10 \beta ) q^{19} + ( -3 + 15 \beta ) q^{23} + ( 93 - 18 \beta ) q^{25} + 27 q^{27} + ( -52 + 6 \beta ) q^{29} + ( -126 - 2 \beta ) q^{31} + ( -87 - 9 \beta ) q^{33} + ( 112 - 6 \beta ) q^{37} + ( 72 + 12 \beta ) q^{39} + ( 219 - 23 \beta ) q^{41} + 388 q^{43} + ( 81 - 9 \beta ) q^{45} + ( 120 + 32 \beta ) q^{47} + ( -9 - 3 \beta ) q^{51} + ( 540 - 6 \beta ) q^{53} + ( 150 + 2 \beta ) q^{55} + ( -126 - 30 \beta ) q^{57} + ( -156 - 16 \beta ) q^{59} + ( -330 + 22 \beta ) q^{61} + ( -332 + 12 \beta ) q^{65} + ( 198 - 42 \beta ) q^{67} + ( -9 + 45 \beta ) q^{69} + ( -33 - 51 \beta ) q^{71} + ( 486 + 18 \beta ) q^{73} + ( 279 - 54 \beta ) q^{75} + ( 430 + 18 \beta ) q^{79} + 81 q^{81} + ( -1260 + 16 \beta ) q^{83} + ( 110 - 6 \beta ) q^{85} + ( -156 + 18 \beta ) q^{87} + ( 843 - 15 \beta ) q^{89} + ( -378 - 6 \beta ) q^{93} + ( 992 - 48 \beta ) q^{95} + ( 1050 + 22 \beta ) q^{97} + ( -261 - 27 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 18q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 18q^{5} + 18q^{9} - 58q^{11} + 48q^{13} + 54q^{15} - 6q^{17} - 84q^{19} - 6q^{23} + 186q^{25} + 54q^{27} - 104q^{29} - 252q^{31} - 174q^{33} + 224q^{37} + 144q^{39} + 438q^{41} + 776q^{43} + 162q^{45} + 240q^{47} - 18q^{51} + 1080q^{53} + 300q^{55} - 252q^{57} - 312q^{59} - 660q^{61} - 664q^{65} + 396q^{67} - 18q^{69} - 66q^{71} + 972q^{73} + 558q^{75} + 860q^{79} + 162q^{81} - 2520q^{83} + 220q^{85} - 312q^{87} + 1686q^{89} - 756q^{93} + 1984q^{95} + 2100q^{97} - 522q^{99} + O(q^{100})$$

## 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
 6.35235 −5.35235
0 3.00000 0 −2.70470 0 0 0 9.00000 0
1.2 0 3.00000 0 20.7047 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.ce 2
4.b odd 2 1 1176.4.a.s 2
7.b odd 2 1 2352.4.a.bm 2
28.d even 2 1 1176.4.a.t yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.s 2 4.b odd 2 1
1176.4.a.t yes 2 28.d even 2 1
2352.4.a.bm 2 7.b odd 2 1
2352.4.a.ce 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} - 18 T_{5} - 56$$ $$T_{11}^{2} + 58 T_{11} - 392$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-56 - 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-392 + 58 T + T^{2}$$
$13$ $$-1616 - 48 T + T^{2}$$
$17$ $$-128 + 6 T + T^{2}$$
$19$ $$-11936 + 84 T + T^{2}$$
$23$ $$-30816 + 6 T + T^{2}$$
$29$ $$-2228 + 104 T + T^{2}$$
$31$ $$15328 + 252 T + T^{2}$$
$37$ $$7612 - 224 T + T^{2}$$
$41$ $$-24512 - 438 T + T^{2}$$
$43$ $$( -388 + T )^{2}$$
$47$ $$-125888 - 240 T + T^{2}$$
$53$ $$286668 - 1080 T + T^{2}$$
$59$ $$-10736 + 312 T + T^{2}$$
$61$ $$42592 + 660 T + T^{2}$$
$67$ $$-202464 - 396 T + T^{2}$$
$71$ $$-355248 + 66 T + T^{2}$$
$73$ $$191808 - 972 T + T^{2}$$
$79$ $$140512 - 860 T + T^{2}$$
$83$ $$1552528 + 2520 T + T^{2}$$
$89$ $$679824 - 1686 T + T^{2}$$
$97$ $$1036192 - 2100 T + T^{2}$$