Properties

 Label 7800.2.a.bk Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2700.1 Defining polynomial: $$x^{3} - 15 x - 20$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{1} q^{7} + q^{9} -\beta_{1} q^{11} - q^{13} + ( -2 + \beta_{2} ) q^{17} + ( 1 - \beta_{2} ) q^{19} -\beta_{1} q^{21} + 2 \beta_{2} q^{23} - q^{27} + q^{29} -\beta_{1} q^{31} + \beta_{1} q^{33} + ( -3 - \beta_{2} ) q^{37} + q^{39} + ( 1 + \beta_{2} ) q^{41} + 2 q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{47} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{49} + ( 2 - \beta_{2} ) q^{51} + ( -1 - 2 \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{57} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 - 3 \beta_{2} ) q^{61} + \beta_{1} q^{63} + ( 1 + \beta_{1} + \beta_{2} ) q^{67} -2 \beta_{2} q^{69} + ( 3 + \beta_{2} ) q^{71} + ( 4 + 2 \beta_{1} ) q^{73} + ( -10 - 2 \beta_{1} - \beta_{2} ) q^{77} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{83} - q^{87} + ( 2 + 2 \beta_{2} ) q^{89} -\beta_{1} q^{91} + \beta_{1} q^{93} + ( -4 - 2 \beta_{2} ) q^{97} -\beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15 x - 20$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 10$$

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
 −2.80560 −1.61323 4.41883
0 −1.00000 0 0 0 −2.80560 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.61323 0 1.00000 0
1.3 0 −1.00000 0 0 0 4.41883 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$13$$ $$1$$

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 7800.2.a.bk 3
5.b even 2 1 7800.2.a.bq yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bk 3 1.a even 1 1 trivial
7800.2.a.bq yes 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7800))$$:

 $$T_{7}^{3} - 15 T_{7} - 20$$ $$T_{11}^{3} - 15 T_{11} + 20$$ $$T_{17}^{3} + 6 T_{17}^{2} - 3 T_{17} - 12$$ $$T_{19}^{3} - 3 T_{19}^{2} - 12 T_{19} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$-20 - 15 T + T^{3}$$
$11$ $$20 - 15 T + T^{3}$$
$13$ $$( 1 + T )^{3}$$
$17$ $$-12 - 3 T + 6 T^{2} + T^{3}$$
$19$ $$4 - 12 T - 3 T^{2} + T^{3}$$
$23$ $$80 - 60 T + T^{3}$$
$29$ $$( -1 + T )^{3}$$
$31$ $$20 - 15 T + T^{3}$$
$37$ $$-28 + 12 T + 9 T^{2} + T^{3}$$
$41$ $$24 - 12 T - 3 T^{2} + T^{3}$$
$43$ $$( -2 + T )^{3}$$
$47$ $$31 - 27 T + 3 T^{2} + T^{3}$$
$53$ $$-139 - 57 T + 3 T^{2} + T^{3}$$
$59$ $$-58 - 63 T - 6 T^{2} + T^{3}$$
$61$ $$-532 - 123 T + 6 T^{2} + T^{3}$$
$67$ $$49 - 27 T - 3 T^{2} + T^{3}$$
$71$ $$28 + 12 T - 9 T^{2} + T^{3}$$
$73$ $$16 - 12 T - 12 T^{2} + T^{3}$$
$79$ $$100 - 15 T^{2} + T^{3}$$
$83$ $$342 - 63 T - 6 T^{2} + T^{3}$$
$89$ $$192 - 48 T - 6 T^{2} + T^{3}$$
$97$ $$-256 - 12 T + 12 T^{2} + T^{3}$$