Properties

 Label 7800.2.a.bg Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.788.1 Defining polynomial: $$x^{3} - x^{2} - 7 x - 3$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} -\beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -\beta_{1} q^{7} + q^{9} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} + q^{13} + q^{17} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{19} + \beta_{1} q^{21} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{23} - q^{27} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} ) q^{33} + ( 5 - \beta_{2} ) q^{37} - q^{39} + ( 1 - 3 \beta_{2} ) q^{41} + ( -1 - \beta_{2} ) q^{43} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{49} - q^{51} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{57} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{59} + ( -3 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} -\beta_{1} q^{63} + ( 1 - \beta_{1} + \beta_{2} ) q^{67} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{69} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( -5 + 2 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -5 - \beta_{2} ) q^{79} + q^{81} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} -\beta_{1} q^{91} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{93} + ( 8 + 2 \beta_{1} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} + q^{21} - 3 q^{27} - q^{29} - 7 q^{31} + 3 q^{33} + 14 q^{37} - 3 q^{39} - 4 q^{43} - q^{47} - 6 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{57} - 7 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 10 q^{71} - 10 q^{73} + 19 q^{77} - 16 q^{79} + 3 q^{81} + 11 q^{83} + q^{87} - 8 q^{89} - q^{91} + 7 q^{93} + 26 q^{97} - 3 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

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
 3.35386 −0.476452 −1.87740
0 −1.00000 0 0 0 −3.35386 0 1.00000 0
1.2 0 −1.00000 0 0 0 0.476452 0 1.00000 0
1.3 0 −1.00000 0 0 0 1.87740 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.bg 3
5.b even 2 1 7800.2.a.br yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bg 3 1.a even 1 1 trivial
7800.2.a.br 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7800))$$:

 $$T_{7}^{3} + T_{7}^{2} - 7 T_{7} + 3$$ $$T_{11}^{3} + 3 T_{11}^{2} - 17 T_{11} - 53$$ $$T_{17} - 1$$ $$T_{19}^{3} + 6 T_{19}^{2} - 20 T_{19} - 100$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$3 - 7 T + T^{2} + T^{3}$$
$11$ $$-53 - 17 T + 3 T^{2} + T^{3}$$
$13$ $$( -1 + T )^{3}$$
$17$ $$( -1 + T )^{3}$$
$19$ $$-100 - 20 T + 6 T^{2} + T^{3}$$
$23$ $$-44 - 32 T + T^{3}$$
$29$ $$-93 - 53 T + T^{2} + T^{3}$$
$31$ $$-425 - 65 T + 7 T^{2} + T^{3}$$
$37$ $$-60 + 56 T - 14 T^{2} + T^{3}$$
$41$ $$-52 - 84 T + T^{3}$$
$43$ $$-12 - 4 T + 4 T^{2} + T^{3}$$
$47$ $$-89 - 83 T + T^{2} + T^{3}$$
$53$ $$781 - 125 T - 5 T^{2} + T^{3}$$
$59$ $$9 - 35 T + 7 T^{2} + T^{3}$$
$61$ $$-9 - 45 T + 5 T^{2} + T^{3}$$
$67$ $$-15 - 17 T - 3 T^{2} + T^{3}$$
$71$ $$-180 - 12 T + 10 T^{2} + T^{3}$$
$73$ $$52 - 60 T + 10 T^{2} + T^{3}$$
$79$ $$100 + 76 T + 16 T^{2} + T^{3}$$
$83$ $$255 - 11 T - 11 T^{2} + T^{3}$$
$89$ $$-48 - 32 T + 8 T^{2} + T^{3}$$
$97$ $$-440 + 196 T - 26 T^{2} + T^{3}$$