Properties

 Label 2368.2.a.s Level $2368$ Weight $2$ Character orbit 2368.a Self dual yes Analytic conductor $18.909$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2368 = 2^{6} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2368.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$18.9085751986$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + \beta q^{5} + ( 2 - 2 \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + \beta q^{5} + ( 2 - 2 \beta ) q^{7} + ( 1 + 3 \beta ) q^{9} + \beta q^{11} + ( 1 - \beta ) q^{13} + ( -3 - 2 \beta ) q^{15} -6 q^{17} -2 q^{19} + ( 4 + 2 \beta ) q^{21} + ( -3 + 3 \beta ) q^{23} + ( -2 + \beta ) q^{25} + ( -7 - 4 \beta ) q^{27} + ( -3 + 3 \beta ) q^{29} + ( 2 - \beta ) q^{31} + ( -3 - 2 \beta ) q^{33} -6 q^{35} - q^{37} + ( 2 + \beta ) q^{39} + ( 3 + 3 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + ( 9 + 4 \beta ) q^{45} + 2 \beta q^{47} + ( 9 - 4 \beta ) q^{49} + ( 6 + 6 \beta ) q^{51} + 6 q^{53} + ( 3 + \beta ) q^{55} + ( 2 + 2 \beta ) q^{57} + ( -6 - 2 \beta ) q^{59} + ( 4 - 5 \beta ) q^{61} + ( -16 - 2 \beta ) q^{63} -3 q^{65} + ( -8 + 5 \beta ) q^{67} + ( -6 - 3 \beta ) q^{69} + 6 q^{71} + ( -10 - \beta ) q^{73} - q^{75} -6 q^{77} + ( -7 + 7 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( -12 + 4 \beta ) q^{83} -6 \beta q^{85} + ( -6 - 3 \beta ) q^{87} -4 \beta q^{89} + ( 8 - 2 \beta ) q^{91} + q^{93} -2 \beta q^{95} + ( 2 - 8 \beta ) q^{97} + ( 9 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9} + O(q^{10})$$ $$2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9} + q^{11} + q^{13} - 8 q^{15} - 12 q^{17} - 4 q^{19} + 10 q^{21} - 3 q^{23} - 3 q^{25} - 18 q^{27} - 3 q^{29} + 3 q^{31} - 8 q^{33} - 12 q^{35} - 2 q^{37} + 5 q^{39} + 9 q^{41} + 6 q^{43} + 22 q^{45} + 2 q^{47} + 14 q^{49} + 18 q^{51} + 12 q^{53} + 7 q^{55} + 6 q^{57} - 14 q^{59} + 3 q^{61} - 34 q^{63} - 6 q^{65} - 11 q^{67} - 15 q^{69} + 12 q^{71} - 21 q^{73} - 2 q^{75} - 12 q^{77} - 7 q^{79} + 38 q^{81} - 20 q^{83} - 6 q^{85} - 15 q^{87} - 4 q^{89} + 14 q^{91} + 2 q^{93} - 2 q^{95} - 4 q^{97} + 22 q^{99} + O(q^{100})$$

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
 2.30278 −1.30278
0 −3.30278 0 2.30278 0 −2.60555 0 7.90833 0
1.2 0 0.302776 0 −1.30278 0 4.60555 0 −2.90833 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
$$2$$ $$1$$
$$37$$ $$1$$

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 2368.2.a.s 2
4.b odd 2 1 2368.2.a.ba 2
8.b even 2 1 74.2.a.a 2
8.d odd 2 1 592.2.a.f 2
24.f even 2 1 5328.2.a.bf 2
24.h odd 2 1 666.2.a.j 2
40.f even 2 1 1850.2.a.u 2
40.i odd 4 2 1850.2.b.i 4
56.h odd 2 1 3626.2.a.a 2
88.b odd 2 1 8954.2.a.p 2
296.e even 2 1 2738.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 8.b even 2 1
592.2.a.f 2 8.d odd 2 1
666.2.a.j 2 24.h odd 2 1
1850.2.a.u 2 40.f even 2 1
1850.2.b.i 4 40.i odd 4 2
2368.2.a.s 2 1.a even 1 1 trivial
2368.2.a.ba 2 4.b odd 2 1
2738.2.a.l 2 296.e even 2 1
3626.2.a.a 2 56.h odd 2 1
5328.2.a.bf 2 24.f even 2 1
8954.2.a.p 2 88.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2368))$$:

 $$T_{3}^{2} + 3 T_{3} - 1$$ $$T_{5}^{2} - T_{5} - 3$$ $$T_{7}^{2} - 2 T_{7} - 12$$ $$T_{11}^{2} - T_{11} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 3 T + T^{2}$$
$5$ $$-3 - T + T^{2}$$
$7$ $$-12 - 2 T + T^{2}$$
$11$ $$-3 - T + T^{2}$$
$13$ $$-3 - T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-27 + 3 T + T^{2}$$
$29$ $$-27 + 3 T + T^{2}$$
$31$ $$-1 - 3 T + T^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$-9 - 9 T + T^{2}$$
$43$ $$-4 - 6 T + T^{2}$$
$47$ $$-12 - 2 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$36 + 14 T + T^{2}$$
$61$ $$-79 - 3 T + T^{2}$$
$67$ $$-51 + 11 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$107 + 21 T + T^{2}$$
$79$ $$-147 + 7 T + T^{2}$$
$83$ $$48 + 20 T + T^{2}$$
$89$ $$-48 + 4 T + T^{2}$$
$97$ $$-204 + 4 T + T^{2}$$