# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{193})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 6 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 6 - \beta ) q^{5} + 9 q^{9} + ( 6 - 7 \beta ) q^{11} + ( 5 - 5 \beta ) q^{13} + ( -18 + 3 \beta ) q^{15} + ( 48 + 4 \beta ) q^{17} + ( -35 + 3 \beta ) q^{19} + ( -48 + 20 \beta ) q^{23} + ( -41 - 11 \beta ) q^{25} -27 q^{27} + ( 132 + 11 \beta ) q^{29} + ( -191 + 20 \beta ) q^{31} + ( -18 + 21 \beta ) q^{33} + ( -25 + 45 \beta ) q^{37} + ( -15 + 15 \beta ) q^{39} + ( -90 + 18 \beta ) q^{41} + ( -359 - 3 \beta ) q^{43} + ( 54 - 9 \beta ) q^{45} + ( 90 + 36 \beta ) q^{47} + ( -144 - 12 \beta ) q^{51} + ( 252 - 9 \beta ) q^{53} + ( 372 - 41 \beta ) q^{55} + ( 105 - 9 \beta ) q^{57} + ( -60 - 53 \beta ) q^{59} + ( -286 + 40 \beta ) q^{61} + ( 270 - 30 \beta ) q^{65} + ( -17 - 77 \beta ) q^{67} + ( 144 - 60 \beta ) q^{69} + ( -822 + 44 \beta ) q^{71} + ( 581 + 53 \beta ) q^{73} + ( 123 + 33 \beta ) q^{75} + ( -761 + 62 \beta ) q^{79} + 81 q^{81} + ( 654 + 101 \beta ) q^{83} + ( 96 - 28 \beta ) q^{85} + ( -396 - 33 \beta ) q^{87} + ( -1008 + 42 \beta ) q^{89} + ( 573 - 60 \beta ) q^{93} + ( -354 + 50 \beta ) q^{95} + ( 266 + 29 \beta ) q^{97} + ( 54 - 63 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 11q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 11q^{5} + 18q^{9} + 5q^{11} + 5q^{13} - 33q^{15} + 100q^{17} - 67q^{19} - 76q^{23} - 93q^{25} - 54q^{27} + 275q^{29} - 362q^{31} - 15q^{33} - 5q^{37} - 15q^{39} - 162q^{41} - 721q^{43} + 99q^{45} + 216q^{47} - 300q^{51} + 495q^{53} + 703q^{55} + 201q^{57} - 173q^{59} - 532q^{61} + 510q^{65} - 111q^{67} + 228q^{69} - 1600q^{71} + 1215q^{73} + 279q^{75} - 1460q^{79} + 162q^{81} + 1409q^{83} + 164q^{85} - 825q^{87} - 1974q^{89} + 1086q^{93} - 658q^{95} + 561q^{97} + 45q^{99} + O(q^{100})$$

## 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
 7.44622 −6.44622
0 −3.00000 0 −1.44622 0 0 0 9.00000 0
1.2 0 −3.00000 0 12.4462 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.bt 2
4.b odd 2 1 588.4.a.i 2
7.b odd 2 1 2352.4.a.bx 2
7.c even 3 2 336.4.q.i 4
12.b even 2 1 1764.4.a.o 2
28.d even 2 1 588.4.a.f 2
28.f even 6 2 588.4.i.j 4
28.g odd 6 2 84.4.i.a 4
84.h odd 2 1 1764.4.a.y 2
84.j odd 6 2 1764.4.k.q 4
84.n even 6 2 252.4.k.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 28.g odd 6 2
252.4.k.f 4 84.n even 6 2
336.4.q.i 4 7.c even 3 2
588.4.a.f 2 28.d even 2 1
588.4.a.i 2 4.b odd 2 1
588.4.i.j 4 28.f even 6 2
1764.4.a.o 2 12.b even 2 1
1764.4.a.y 2 84.h odd 2 1
1764.4.k.q 4 84.j odd 6 2
2352.4.a.bt 2 1.a even 1 1 trivial
2352.4.a.bx 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{2} - 11 T_{5} - 18$$ $$T_{11}^{2} - 5 T_{11} - 2358$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-18 - 11 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-2358 - 5 T + T^{2}$$
$13$ $$-1200 - 5 T + T^{2}$$
$17$ $$1728 - 100 T + T^{2}$$
$19$ $$688 + 67 T + T^{2}$$
$23$ $$-17856 + 76 T + T^{2}$$
$29$ $$13068 - 275 T + T^{2}$$
$31$ $$13461 + 362 T + T^{2}$$
$37$ $$-97700 + 5 T + T^{2}$$
$41$ $$-9072 + 162 T + T^{2}$$
$43$ $$129526 + 721 T + T^{2}$$
$47$ $$-50868 - 216 T + T^{2}$$
$53$ $$57348 - 495 T + T^{2}$$
$59$ $$-128052 + 173 T + T^{2}$$
$61$ $$-6444 + 532 T + T^{2}$$
$67$ $$-282994 + 111 T + T^{2}$$
$71$ $$546588 + 1600 T + T^{2}$$
$73$ $$233522 - 1215 T + T^{2}$$
$79$ $$347427 + 1460 T + T^{2}$$
$83$ $$4122 - 1409 T + T^{2}$$
$89$ $$889056 + 1974 T + T^{2}$$
$97$ $$38102 - 561 T + T^{2}$$
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