# Properties

 Label 2352.4.a.ch Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.58461.1 Defining polynomial: $$x^{3} - x^{2} - 65 x - 126$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -4 - \beta_{1} ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -4 - \beta_{1} ) q^{5} + 9 q^{9} + ( 6 - \beta_{1} + \beta_{2} ) q^{11} + ( -7 + \beta_{1} - 3 \beta_{2} ) q^{13} + ( 12 + 3 \beta_{1} ) q^{15} + ( -34 + 2 \beta_{1} + \beta_{2} ) q^{17} + ( -69 - 5 \beta_{1} - 8 \beta_{2} ) q^{19} + ( 94 + 2 \beta_{1} - 7 \beta_{2} ) q^{23} + ( 67 + 15 \beta_{1} + 7 \beta_{2} ) q^{25} -27 q^{27} + ( -28 - 11 \beta_{1} - 5 \beta_{2} ) q^{29} + ( 39 - 14 \beta_{1} + \beta_{2} ) q^{31} + ( -18 + 3 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -113 - 13 \beta_{1} - 3 \beta_{2} ) q^{37} + ( 21 - 3 \beta_{1} + 9 \beta_{2} ) q^{39} + ( -166 + 18 \beta_{1} ) q^{41} + ( -13 - 3 \beta_{1} + 20 \beta_{2} ) q^{43} + ( -36 - 9 \beta_{1} ) q^{45} + ( 36 - 18 \beta_{1} + 11 \beta_{2} ) q^{47} + ( 102 - 6 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -134 - 17 \beta_{1} - 4 \beta_{2} ) q^{53} + ( 204 + \beta_{1} + 11 \beta_{2} ) q^{55} + ( 207 + 15 \beta_{1} + 24 \beta_{2} ) q^{57} + ( -94 + 3 \beta_{1} + 16 \beta_{2} ) q^{59} + ( -22 - 32 \beta_{1} - 20 \beta_{2} ) q^{61} + ( -304 + 8 \beta_{1} - 19 \beta_{2} ) q^{65} + ( 25 - 25 \beta_{1} - 6 \beta_{2} ) q^{67} + ( -282 - 6 \beta_{1} + 21 \beta_{2} ) q^{69} + ( -16 - 14 \beta_{1} - 51 \beta_{2} ) q^{71} + ( -47 - 33 \beta_{1} + 39 \beta_{2} ) q^{73} + ( -201 - 45 \beta_{1} - 21 \beta_{2} ) q^{75} + ( 821 + 15 \beta_{2} ) q^{79} + 81 q^{81} + ( 32 - 19 \beta_{1} - 40 \beta_{2} ) q^{83} + ( -164 + 8 \beta_{1} - 10 \beta_{2} ) q^{85} + ( 84 + 33 \beta_{1} + 15 \beta_{2} ) q^{87} + ( -60 - 70 \beta_{1} + 24 \beta_{2} ) q^{89} + ( -117 + 42 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 740 + 156 \beta_{1} + 3 \beta_{2} ) q^{95} + ( -982 - 5 \beta_{1} + 13 \beta_{2} ) q^{97} + ( 54 - 9 \beta_{1} + 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 9q^{3} - 11q^{5} + 27q^{9} + O(q^{10})$$ $$3q - 9q^{3} - 11q^{5} + 27q^{9} + 19q^{11} - 22q^{13} + 33q^{15} - 104q^{17} - 202q^{19} + 280q^{23} + 186q^{25} - 81q^{27} - 73q^{29} + 131q^{31} - 57q^{33} - 326q^{37} + 66q^{39} - 516q^{41} - 36q^{43} - 99q^{45} + 126q^{47} + 312q^{51} - 385q^{53} + 611q^{55} + 606q^{57} - 285q^{59} - 34q^{61} - 920q^{65} + 100q^{67} - 840q^{69} - 34q^{71} - 108q^{73} - 558q^{75} + 2463q^{79} + 243q^{81} + 115q^{83} - 500q^{85} + 219q^{87} - 110q^{89} - 393q^{93} + 2064q^{95} - 2941q^{97} + 171q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 65 x - 126$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + \nu - 45$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{2} + 10 \nu + 84$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{2} + 10 \beta_{1} + 178$$$$)/4$$

## 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
 9.37106 −6.20369 −2.16736
0 −3.00000 0 −21.3959 0 0 0 9.00000 0
1.2 0 −3.00000 0 0.239289 0 0 0 9.00000 0
1.3 0 −3.00000 0 10.1566 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.ch 3
4.b odd 2 1 1176.4.a.y 3
7.b odd 2 1 2352.4.a.cj 3
7.c even 3 2 336.4.q.l 6
28.d even 2 1 1176.4.a.x 3
28.g odd 6 2 168.4.q.e 6
84.n even 6 2 504.4.s.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.e 6 28.g odd 6 2
336.4.q.l 6 7.c even 3 2
504.4.s.g 6 84.n even 6 2
1176.4.a.x 3 28.d even 2 1
1176.4.a.y 3 4.b odd 2 1
2352.4.a.ch 3 1.a even 1 1 trivial
2352.4.a.cj 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{3} + 11 T_{5}^{2} - 220 T_{5} + 52$$ $$T_{11}^{3} - 19 T_{11}^{2} - 624 T_{11} - 3276$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 3 + T )^{3}$$
$5$ $$52 - 220 T + 11 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-3276 - 624 T - 19 T^{2} + T^{3}$$
$13$ $$26976 - 3495 T + 22 T^{2} + T^{3}$$
$17$ $$-4032 + 2560 T + 104 T^{2} + T^{3}$$
$19$ $$-2225968 - 7243 T + 202 T^{2} + T^{3}$$
$23$ $$1532736 + 6976 T - 280 T^{2} + T^{3}$$
$29$ $$970992 - 29024 T + 73 T^{2} + T^{3}$$
$31$ $$4156607 - 47869 T - 131 T^{2} + T^{3}$$
$37$ $$-17796 - 5247 T + 326 T^{2} + T^{3}$$
$41$ $$-15002144 + 4404 T + 516 T^{2} + T^{3}$$
$43$ $$-7204222 - 141111 T + 36 T^{2} + T^{3}$$
$47$ $$-11682152 - 149940 T - 126 T^{2} + T^{3}$$
$53$ $$178128 - 20132 T + 385 T^{2} + T^{3}$$
$59$ $$-1793232 - 50532 T + 285 T^{2} + T^{3}$$
$61$ $$22240152 - 293396 T + 34 T^{2} + T^{3}$$
$67$ $$27246966 - 147039 T - 100 T^{2} + T^{3}$$
$71$ $$-207049704 - 779124 T + 34 T^{2} + T^{3}$$
$73$ $$-409658074 - 978339 T + 108 T^{2} + T^{3}$$
$79$ $$-492780761 + 1949223 T - 2463 T^{2} + T^{3}$$
$83$ $$-111709668 - 486372 T - 115 T^{2} + T^{3}$$
$89$ $$-347278464 - 1727024 T + 110 T^{2} + T^{3}$$
$97$ $$866695284 + 2811496 T + 2941 T^{2} + T^{3}$$
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