# Properties

 Label 2352.4.a.cr Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.145408.2 Defining polynomial: $$x^{4} - 24 x^{2} + 142$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( 2 + \beta_{2} + 3 \beta_{3} ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( 2 + \beta_{2} + 3 \beta_{3} ) q^{5} + 9 q^{9} + ( 10 + 3 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{11} + ( 12 - 3 \beta_{1} - 23 \beta_{3} ) q^{13} + ( 6 + 3 \beta_{2} + 9 \beta_{3} ) q^{15} + ( -38 - 5 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} ) q^{17} + ( -56 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{19} + ( 2 + 5 \beta_{1} - 13 \beta_{2} - 16 \beta_{3} ) q^{23} + ( -7 + 6 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} ) q^{25} + 27 q^{27} + ( -36 + 3 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} ) q^{29} + ( -100 - \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{31} + ( 30 + 9 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} ) q^{33} + ( -76 + 10 \beta_{1} - 14 \beta_{2} + 16 \beta_{3} ) q^{37} + ( 36 - 9 \beta_{1} - 69 \beta_{3} ) q^{39} + ( -38 - \beta_{1} + 13 \beta_{2} - 23 \beta_{3} ) q^{41} + ( -40 + 6 \beta_{1} - 30 \beta_{2} - 36 \beta_{3} ) q^{43} + ( 18 + 9 \beta_{2} + 27 \beta_{3} ) q^{45} + ( -136 - 5 \beta_{1} - 10 \beta_{2} - 170 \beta_{3} ) q^{47} + ( -114 - 15 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} ) q^{51} + ( -330 - 18 \beta_{1} - 20 \beta_{2} + 128 \beta_{3} ) q^{53} + ( -4 + 7 \beta_{1} + 26 \beta_{2} + 318 \beta_{3} ) q^{55} + ( -168 - 9 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -260 - 23 \beta_{1} + 6 \beta_{2} - 126 \beta_{3} ) q^{59} + ( 224 + 5 \beta_{1} - 6 \beta_{2} + 37 \beta_{3} ) q^{61} + ( -162 - 29 \beta_{1} - 6 \beta_{2} - 298 \beta_{3} ) q^{65} + ( 104 + 6 \beta_{1} - 6 \beta_{2} - 252 \beta_{3} ) q^{67} + ( 6 + 15 \beta_{1} - 39 \beta_{2} - 48 \beta_{3} ) q^{69} + ( -62 - 27 \beta_{1} + 35 \beta_{2} + 28 \beta_{3} ) q^{71} + ( -188 - 26 \beta_{1} - 30 \beta_{2} + 57 \beta_{3} ) q^{73} + ( -21 + 18 \beta_{1} + 12 \beta_{2} + 60 \beta_{3} ) q^{75} + ( -216 + 12 \beta_{1} - 20 \beta_{2} + 260 \beta_{3} ) q^{79} + 81 q^{81} + ( -364 + 10 \beta_{1} - 64 \beta_{2} + 16 \beta_{3} ) q^{83} + ( -402 - 12 \beta_{1} - 74 \beta_{2} - 604 \beta_{3} ) q^{85} + ( -108 + 9 \beta_{1} + 36 \beta_{2} + 54 \beta_{3} ) q^{87} + ( 734 - 5 \beta_{1} + 23 \beta_{2} - 369 \beta_{3} ) q^{89} + ( -300 - 3 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} ) q^{93} + ( 20 - 2 \beta_{1} - 70 \beta_{2} - 444 \beta_{3} ) q^{95} + ( 36 + 24 \beta_{1} + 54 \beta_{2} + 225 \beta_{3} ) q^{97} + ( 90 + 27 \beta_{1} - 9 \beta_{2} + 36 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10})$$ $$4q + 12q^{3} + 8q^{5} + 36q^{9} + 40q^{11} + 48q^{13} + 24q^{15} - 152q^{17} - 224q^{19} + 8q^{23} - 28q^{25} + 108q^{27} - 144q^{29} - 400q^{31} + 120q^{33} - 304q^{37} + 144q^{39} - 152q^{41} - 160q^{43} + 72q^{45} - 544q^{47} - 456q^{51} - 1320q^{53} - 16q^{55} - 672q^{57} - 1040q^{59} + 896q^{61} - 648q^{65} + 416q^{67} + 24q^{69} - 248q^{71} - 752q^{73} - 84q^{75} - 864q^{79} + 324q^{81} - 1456q^{83} - 1608q^{85} - 432q^{87} + 2936q^{89} - 1200q^{93} + 80q^{95} + 144q^{97} + 360q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 24 x^{2} + 142$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} - 24 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 12$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 6 \beta_{1}$$$$)/2$$

## 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
 3.25358 −3.66254 −3.25358 3.66254
0 3.00000 0 −11.4452 0 0 0 9.00000 0
1.2 0 3.00000 0 −4.11659 0 0 0 9.00000 0
1.3 0 3.00000 0 6.95987 0 0 0 9.00000 0
1.4 0 3.00000 0 16.6019 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.cr 4
4.b odd 2 1 1176.4.a.bb 4
7.b odd 2 1 2352.4.a.ck 4
28.d even 2 1 1176.4.a.bc yes 4

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

 $$T_{5}^{4} - 8 T_{5}^{3} - 204 T_{5}^{2} + 688 T_{5} + 5444$$ $$T_{11}^{4} - 40 T_{11}^{3} - 2920 T_{11}^{2} + 79968 T_{11} + 2039184$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T )^{4}$$
$5$ $$5444 + 688 T - 204 T^{2} - 8 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$2039184 + 79968 T - 2920 T^{2} - 40 T^{3} + T^{4}$$
$13$ $$-692156 + 153312 T - 4708 T^{2} - 48 T^{3} + T^{4}$$
$17$ $$-8992508 - 917008 T - 3820 T^{2} + 152 T^{3} + T^{4}$$
$19$ $$-1381824 + 254976 T + 14960 T^{2} + 224 T^{3} + T^{4}$$
$23$ $$112149392 - 1217568 T - 38888 T^{2} - 8 T^{3} + T^{4}$$
$29$ $$37205568 - 3494016 T - 26928 T^{2} + 144 T^{3} + T^{4}$$
$31$ $$38706752 + 2505088 T + 52176 T^{2} + 400 T^{3} + T^{4}$$
$37$ $$-163223296 - 5559552 T - 33440 T^{2} + 304 T^{3} + T^{4}$$
$41$ $$165653252 - 2580880 T - 25452 T^{2} + 152 T^{3} + T^{4}$$
$43$ $$5188983808 - 21880832 T - 170688 T^{2} + 160 T^{3} + T^{4}$$
$47$ $$1123620416 - 15397376 T - 36624 T^{2} + 544 T^{3} + T^{4}$$
$53$ $$-51605020912 - 126878816 T + 363608 T^{2} + 1320 T^{3} + T^{4}$$
$59$ $$-13864209856 - 85236352 T + 140880 T^{2} + 1040 T^{3} + T^{4}$$
$61$ $$1143730116 - 34465920 T + 280988 T^{2} - 896 T^{3} + T^{4}$$
$67$ $$12240034816 + 39976960 T - 207552 T^{2} - 416 T^{3} + T^{4}$$
$71$ $$259707024 - 19978272 T - 434728 T^{2} + 248 T^{3} + T^{4}$$
$73$ $$-31874858428 - 236159968 T - 283236 T^{2} + 752 T^{3} + T^{4}$$
$79$ $$4954883072 - 42533888 T - 107200 T^{2} + 864 T^{3} + T^{4}$$
$83$ $$55227560192 - 367160576 T + 10080 T^{2} + 1456 T^{3} + T^{4}$$
$89$ $$26681619204 - 681366576 T + 2584084 T^{2} - 2936 T^{3} + T^{4}$$
$97$ $$-32096683836 - 429875424 T - 1058724 T^{2} - 144 T^{3} + T^{4}$$