Properties

 Label 2352.4.a.by Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{113})$$ Defining polynomial: $$x^{2} - x - 28$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{113}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( -3 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( -3 - \beta ) q^{5} + 9 q^{9} + ( -1 - 3 \beta ) q^{11} -4 \beta q^{13} + ( -9 - 3 \beta ) q^{15} + ( -15 - \beta ) q^{17} + ( 6 - 2 \beta ) q^{19} + ( -39 - 9 \beta ) q^{23} + ( -3 + 6 \beta ) q^{25} + 27 q^{27} + ( 28 - 18 \beta ) q^{29} + ( 66 - 10 \beta ) q^{31} + ( -3 - 9 \beta ) q^{33} + ( 160 + 18 \beta ) q^{37} -12 \beta q^{39} + ( -9 + 25 \beta ) q^{41} -188 q^{43} + ( -27 - 9 \beta ) q^{45} + ( -120 - 16 \beta ) q^{47} + ( -45 - 3 \beta ) q^{51} + ( -36 - 30 \beta ) q^{53} + ( 342 + 10 \beta ) q^{55} + ( 18 - 6 \beta ) q^{57} + ( 276 + 32 \beta ) q^{59} + ( 246 + 14 \beta ) q^{61} + ( 452 + 12 \beta ) q^{65} + ( 270 - 42 \beta ) q^{67} + ( -117 - 27 \beta ) q^{69} + ( -429 - 27 \beta ) q^{71} + ( -450 + 18 \beta ) q^{73} + ( -9 + 18 \beta ) q^{75} + ( 310 + 66 \beta ) q^{79} + 81 q^{81} + ( 612 - 32 \beta ) q^{83} + ( 158 + 18 \beta ) q^{85} + ( 84 - 54 \beta ) q^{87} + ( 711 + 33 \beta ) q^{89} + ( 198 - 30 \beta ) q^{93} + 208 q^{95} + ( -558 - 58 \beta ) q^{97} + ( -9 - 27 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 6q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 6q^{5} + 18q^{9} - 2q^{11} - 18q^{15} - 30q^{17} + 12q^{19} - 78q^{23} - 6q^{25} + 54q^{27} + 56q^{29} + 132q^{31} - 6q^{33} + 320q^{37} - 18q^{41} - 376q^{43} - 54q^{45} - 240q^{47} - 90q^{51} - 72q^{53} + 684q^{55} + 36q^{57} + 552q^{59} + 492q^{61} + 904q^{65} + 540q^{67} - 234q^{69} - 858q^{71} - 900q^{73} - 18q^{75} + 620q^{79} + 162q^{81} + 1224q^{83} + 316q^{85} + 168q^{87} + 1422q^{89} + 396q^{93} + 416q^{95} - 1116q^{97} - 18q^{99} + O(q^{100})$$

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
 5.81507 −4.81507
0 3.00000 0 −13.6301 0 0 0 9.00000 0
1.2 0 3.00000 0 7.63015 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.by 2
4.b odd 2 1 1176.4.a.q 2
7.b odd 2 1 2352.4.a.bs 2
28.d even 2 1 1176.4.a.v yes 2

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

 $$T_{5}^{2} + 6 T_{5} - 104$$ $$T_{11}^{2} + 2 T_{11} - 1016$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-104 + 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1016 + 2 T + T^{2}$$
$13$ $$-1808 + T^{2}$$
$17$ $$112 + 30 T + T^{2}$$
$19$ $$-416 - 12 T + T^{2}$$
$23$ $$-7632 + 78 T + T^{2}$$
$29$ $$-35828 - 56 T + T^{2}$$
$31$ $$-6944 - 132 T + T^{2}$$
$37$ $$-11012 - 320 T + T^{2}$$
$41$ $$-70544 + 18 T + T^{2}$$
$43$ $$( 188 + T )^{2}$$
$47$ $$-14528 + 240 T + T^{2}$$
$53$ $$-100404 + 72 T + T^{2}$$
$59$ $$-39536 - 552 T + T^{2}$$
$61$ $$38368 - 492 T + T^{2}$$
$67$ $$-126432 - 540 T + T^{2}$$
$71$ $$101664 + 858 T + T^{2}$$
$73$ $$165888 + 900 T + T^{2}$$
$79$ $$-396128 - 620 T + T^{2}$$
$83$ $$258832 - 1224 T + T^{2}$$
$89$ $$382464 - 1422 T + T^{2}$$
$97$ $$-68768 + 1116 T + T^{2}$$