Properties

 Label 7920.2.a.cj Level $7920$ Weight $2$ Character orbit 7920.a Self dual yes Analytic conductor $63.242$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7920.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$63.2415184009$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) 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.

 $$f(q)$$ $$=$$ $$q - q^{5} -\beta_{2} q^{7} +O(q^{10})$$ $$q - q^{5} -\beta_{2} q^{7} + q^{11} + ( -1 - \beta_{1} ) q^{13} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( -3 - \beta_{1} + \beta_{2} ) q^{19} -2 \beta_{2} q^{23} + q^{25} + ( 3 - \beta_{1} - \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} ) q^{31} + \beta_{2} q^{35} -2 q^{37} + ( 5 + \beta_{1} + \beta_{2} ) q^{41} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{43} + ( -2 + 2 \beta_{1} ) q^{47} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( 2 - 2 \beta_{2} ) q^{53} - q^{55} + ( 4 + 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{2} ) q^{61} + ( 1 + \beta_{1} ) q^{65} + ( 2 + 2 \beta_{1} ) q^{67} + ( 2 - 2 \beta_{1} ) q^{71} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{73} -\beta_{2} q^{77} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{79} + ( -1 - 3 \beta_{1} ) q^{83} + ( 2 \beta_{1} + \beta_{2} ) q^{85} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -2 + 2 \beta_{1} ) q^{91} + ( 3 + \beta_{1} - \beta_{2} ) q^{95} + ( 8 + 2 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + O(q^{10})$$ $$3 q - 3 q^{5} + 3 q^{11} - 2 q^{13} + 2 q^{17} - 8 q^{19} + 3 q^{25} + 10 q^{29} - 8 q^{31} - 6 q^{37} + 14 q^{41} - 4 q^{43} - 8 q^{47} + 11 q^{49} + 6 q^{53} - 3 q^{55} + 12 q^{59} - 6 q^{61} + 2 q^{65} + 4 q^{67} + 8 q^{71} - 14 q^{73} - 12 q^{79} - 2 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{95} + 22 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 5$$$$)/2$$

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
 −1.48119 2.17009 0.311108
0 0 0 −1.00000 0 −3.35026 0 0 0
1.2 0 0 0 −1.00000 0 −1.07838 0 0 0
1.3 0 0 0 −1.00000 0 4.42864 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

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 7920.2.a.cj 3
3.b odd 2 1 2640.2.a.be 3
4.b odd 2 1 495.2.a.e 3
12.b even 2 1 165.2.a.c 3
20.d odd 2 1 2475.2.a.bb 3
20.e even 4 2 2475.2.c.r 6
44.c even 2 1 5445.2.a.z 3
60.h even 2 1 825.2.a.k 3
60.l odd 4 2 825.2.c.g 6
84.h odd 2 1 8085.2.a.bk 3
132.d odd 2 1 1815.2.a.m 3
660.g odd 2 1 9075.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 12.b even 2 1
495.2.a.e 3 4.b odd 2 1
825.2.a.k 3 60.h even 2 1
825.2.c.g 6 60.l odd 4 2
1815.2.a.m 3 132.d odd 2 1
2475.2.a.bb 3 20.d odd 2 1
2475.2.c.r 6 20.e even 4 2
2640.2.a.be 3 3.b odd 2 1
5445.2.a.z 3 44.c even 2 1
7920.2.a.cj 3 1.a even 1 1 trivial
8085.2.a.bk 3 84.h odd 2 1
9075.2.a.cf 3 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7920))$$:

 $$T_{7}^{3} - 16 T_{7} - 16$$ $$T_{13}^{3} + 2 T_{13}^{2} - 12 T_{13} - 8$$ $$T_{17}^{3} - 2 T_{17}^{2} - 52 T_{17} + 184$$ $$T_{19}^{3} + 8 T_{19}^{2} - 16 T_{19} - 160$$ $$T_{23}^{3} - 64 T_{23} - 128$$ $$T_{29}^{3} - 10 T_{29}^{2} + 12 T_{29} + 40$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-16 - 16 T + T^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-8 - 12 T + 2 T^{2} + T^{3}$$
$17$ $$184 - 52 T - 2 T^{2} + T^{3}$$
$19$ $$-160 - 16 T + 8 T^{2} + T^{3}$$
$23$ $$-128 - 64 T + T^{3}$$
$29$ $$40 + 12 T - 10 T^{2} + T^{3}$$
$31$ $$-128 - 32 T + 8 T^{2} + T^{3}$$
$37$ $$( 2 + T )^{3}$$
$41$ $$-8 + 44 T - 14 T^{2} + T^{3}$$
$43$ $$-400 - 80 T + 4 T^{2} + T^{3}$$
$47$ $$-128 - 32 T + 8 T^{2} + T^{3}$$
$53$ $$-8 - 52 T - 6 T^{2} + T^{3}$$
$59$ $$320 - 16 T - 12 T^{2} + T^{3}$$
$61$ $$-248 - 52 T + 6 T^{2} + T^{3}$$
$67$ $$64 - 48 T - 4 T^{2} + T^{3}$$
$71$ $$128 - 32 T - 8 T^{2} + T^{3}$$
$73$ $$-344 + 4 T + 14 T^{2} + T^{3}$$
$79$ $$-800 - 64 T + 12 T^{2} + T^{3}$$
$83$ $$16 - 120 T + T^{3}$$
$89$ $$200 - 52 T - 10 T^{2} + T^{3}$$
$97$ $$-8 + 108 T - 22 T^{2} + T^{3}$$