# Properties

 Label 2352.4.a.cg 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: 3.3.57516.1 Defining polynomial: $$x^{3} - x^{2} - 24 x + 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -4 - \beta_{2} ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -4 - \beta_{2} ) q^{5} + 9 q^{9} + ( -11 + \beta_{1} + 2 \beta_{2} ) q^{11} + ( -20 - \beta_{1} + 2 \beta_{2} ) q^{13} + ( 12 + 3 \beta_{2} ) q^{15} + ( -17 - \beta_{1} - 3 \beta_{2} ) q^{17} + ( -67 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( -73 + 7 \beta_{1} - 3 \beta_{2} ) q^{23} + ( 46 - 7 \beta_{1} + 8 \beta_{2} ) q^{25} -27 q^{27} + ( 21 - 5 \beta_{1} + 10 \beta_{2} ) q^{29} + ( -34 + 5 \beta_{1} - 7 \beta_{2} ) q^{31} + ( 33 - 3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 86 + 5 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 60 + 3 \beta_{1} - 6 \beta_{2} ) q^{39} + ( -78 + 4 \beta_{1} + 10 \beta_{2} ) q^{41} + ( -125 + 18 \beta_{1} - 15 \beta_{2} ) q^{43} + ( -36 - 9 \beta_{2} ) q^{45} + ( -71 - 25 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 51 + 3 \beta_{1} + 9 \beta_{2} ) q^{51} + ( 134 - 20 \beta_{1} + 9 \beta_{2} ) q^{53} + ( -339 + 11 \beta_{1} + 14 \beta_{2} ) q^{55} + ( 201 + 6 \beta_{1} - 3 \beta_{2} ) q^{57} + ( 368 + 10 \beta_{1} - 39 \beta_{2} ) q^{59} + ( 10 + 20 \beta_{1} - 40 \beta_{2} ) q^{61} + ( -157 + 17 \beta_{1} + \beta_{2} ) q^{65} + ( 199 - 16 \beta_{1} - 31 \beta_{2} ) q^{67} + ( 219 - 21 \beta_{1} + 9 \beta_{2} ) q^{69} + ( -99 - 33 \beta_{1} + 21 \beta_{2} ) q^{71} + ( -332 + 31 \beta_{1} - 8 \beta_{2} ) q^{73} + ( -138 + 21 \beta_{1} - 24 \beta_{2} ) q^{75} + ( -302 - 3 \beta_{1} - 45 \beta_{2} ) q^{79} + 81 q^{81} + ( 162 + 6 \beta_{1} - 33 \beta_{2} ) q^{83} + ( 606 - 18 \beta_{1} + 18 \beta_{2} ) q^{85} + ( -63 + 15 \beta_{1} - 30 \beta_{2} ) q^{87} + ( -580 - 48 \beta_{1} + 26 \beta_{2} ) q^{89} + ( 102 - 15 \beta_{1} + 21 \beta_{2} ) q^{93} + ( 259 + 13 \beta_{1} + 41 \beta_{2} ) q^{95} + ( -23 + \beta_{1} - 50 \beta_{2} ) q^{97} + ( -99 + 9 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 9q^{3} - 11q^{5} + 27q^{9} + O(q^{10})$$ $$3q - 9q^{3} - 11q^{5} + 27q^{9} - 35q^{11} - 62q^{13} + 33q^{15} - 48q^{17} - 202q^{19} - 216q^{23} + 130q^{25} - 81q^{27} + 53q^{29} - 95q^{31} + 105q^{33} + 262q^{37} + 186q^{39} - 244q^{41} - 360q^{43} - 99q^{45} - 210q^{47} + 144q^{51} + 393q^{53} - 1031q^{55} + 606q^{57} + 1143q^{59} + 70q^{61} - 472q^{65} + 628q^{67} + 648q^{69} - 318q^{71} - 988q^{73} - 390q^{75} - 861q^{79} + 243q^{81} + 519q^{83} + 1800q^{85} - 159q^{87} - 1766q^{89} + 285q^{93} + 736q^{95} - 19q^{97} - 315q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 24 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + 2 \nu - 17$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 33$$$$)/2$$

## 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
 −4.55637 5.30829 0.248072
0 −3.00000 0 −17.8732 0 0 0 9.00000 0
1.2 0 −3.00000 0 −5.56140 0 0 0 9.00000 0
1.3 0 −3.00000 0 12.4346 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.cg 3
4.b odd 2 1 147.4.a.m 3
7.b odd 2 1 2352.4.a.ci 3
7.d odd 6 2 336.4.q.k 6
12.b even 2 1 441.4.a.t 3
28.d even 2 1 147.4.a.l 3
28.f even 6 2 21.4.e.b 6
28.g odd 6 2 147.4.e.n 6
84.h odd 2 1 441.4.a.s 3
84.j odd 6 2 63.4.e.c 6
84.n even 6 2 441.4.e.w 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 28.f even 6 2
63.4.e.c 6 84.j odd 6 2
147.4.a.l 3 28.d even 2 1
147.4.a.m 3 4.b odd 2 1
147.4.e.n 6 28.g odd 6 2
336.4.q.k 6 7.d odd 6 2
441.4.a.s 3 84.h odd 2 1
441.4.a.t 3 12.b even 2 1
441.4.e.w 6 84.n even 6 2
2352.4.a.cg 3 1.a even 1 1 trivial
2352.4.a.ci 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

 $$T_{5}^{3} + 11 T_{5}^{2} - 192 T_{5} - 1236$$ $$T_{11}^{3} + 35 T_{11}^{2} - 1368 T_{11} + 9564$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 3 + T )^{3}$$
$5$ $$-1236 - 192 T + 11 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$9564 - 1368 T + 35 T^{2} + T^{3}$$
$13$ $$-18452 + 425 T + 62 T^{2} + T^{3}$$
$17$ $$-112896 - 2400 T + 48 T^{2} + T^{3}$$
$19$ $$233804 + 12281 T + 202 T^{2} + T^{3}$$
$23$ $$-1580544 - 672 T + 216 T^{2} + T^{3}$$
$29$ $$-824976 - 20472 T - 53 T^{2} + T^{3}$$
$31$ $$-11823 - 10001 T + 95 T^{2} + T^{3}$$
$37$ $$-49152 + 14089 T - 262 T^{2} + T^{3}$$
$41$ $$300384 - 18780 T + 244 T^{2} + T^{3}$$
$43$ $$-18269746 - 72363 T + 360 T^{2} + T^{3}$$
$47$ $$5119128 - 246516 T + 210 T^{2} + T^{3}$$
$53$ $$33169392 - 80736 T - 393 T^{2} + T^{3}$$
$59$ $$100468944 + 133104 T - 1143 T^{2} + T^{3}$$
$61$ $$84631000 - 340900 T - 70 T^{2} + T^{3}$$
$67$ $$-27993002 - 304963 T - 628 T^{2} + T^{3}$$
$71$ $$28535976 - 330804 T + 318 T^{2} + T^{3}$$
$73$ $$-143207118 - 4355 T + 988 T^{2} + T^{3}$$
$79$ $$-193956337 - 257901 T + 861 T^{2} + T^{3}$$
$83$ $$47916036 - 131616 T - 519 T^{2} + T^{3}$$
$89$ $$-13004544 + 277920 T + 1766 T^{2} + T^{3}$$
$97$ $$-44776452 - 569600 T + 19 T^{2} + T^{3}$$