# Properties

 Label 2352.4.a.cj Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$3$$ 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
 −2.16736 −6.20369 9.37106
0 3.00000 0 −10.1566 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 21.3959 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.cj 3
4.b odd 2 1 1176.4.a.x 3
7.b odd 2 1 2352.4.a.ch 3
7.d odd 6 2 336.4.q.l 6
28.d even 2 1 1176.4.a.y 3
28.f even 6 2 168.4.q.e 6
84.j odd 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.f even 6 2
336.4.q.l 6 7.d odd 6 2
504.4.s.g 6 84.j odd 6 2
1176.4.a.x 3 4.b odd 2 1
1176.4.a.y 3 28.d even 2 1
2352.4.a.ch 3 7.b odd 2 1
2352.4.a.cj 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

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

## Hecke characteristic polynomials

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