Properties

 Label 3240.2.a.p Level $3240$ Weight $2$ Character orbit 3240.a Self dual yes Analytic conductor $25.872$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$25.8715302549$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( 2 + \beta ) q^{7} +O(q^{10})$$ $$q + q^{5} + ( 2 + \beta ) q^{7} + ( -2 - 2 \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} + ( -4 - 2 \beta ) q^{17} -2 q^{19} + ( -2 + \beta ) q^{23} + q^{25} + ( -5 - 2 \beta ) q^{29} -2 q^{31} + ( 2 + \beta ) q^{35} -6 \beta q^{37} + ( -3 + 4 \beta ) q^{41} + ( -8 - 2 \beta ) q^{43} + ( -2 - \beta ) q^{47} + 4 \beta q^{49} -6 q^{53} + ( -2 - 2 \beta ) q^{55} + ( -4 + 6 \beta ) q^{59} + ( -5 + 2 \beta ) q^{61} + ( -2 + 2 \beta ) q^{65} + ( 8 + \beta ) q^{67} + ( -6 + 2 \beta ) q^{71} -4 \beta q^{73} + ( -10 - 6 \beta ) q^{77} + ( 12 - 2 \beta ) q^{79} + ( 8 - 3 \beta ) q^{83} + ( -4 - 2 \beta ) q^{85} + ( 3 - 4 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} -2 q^{95} + ( -2 + 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 4 q^{7} + O(q^{10})$$ $$2 q + 2 q^{5} + 4 q^{7} - 4 q^{11} - 4 q^{13} - 8 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 10 q^{29} - 4 q^{31} + 4 q^{35} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 12 q^{53} - 4 q^{55} - 8 q^{59} - 10 q^{61} - 4 q^{65} + 16 q^{67} - 12 q^{71} - 20 q^{77} + 24 q^{79} + 16 q^{83} - 8 q^{85} + 6 q^{89} + 4 q^{91} - 4 q^{95} - 4 q^{97} + O(q^{100})$$

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
 −1.73205 1.73205
0 0 0 1.00000 0 0.267949 0 0 0
1.2 0 0 0 1.00000 0 3.73205 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

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 3240.2.a.p 2
3.b odd 2 1 3240.2.a.k 2
4.b odd 2 1 6480.2.a.bk 2
9.c even 3 2 1080.2.q.b 4
9.d odd 6 2 360.2.q.b 4
12.b even 2 1 6480.2.a.ba 2
36.f odd 6 2 2160.2.q.h 4
36.h even 6 2 720.2.q.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.b 4 9.d odd 6 2
720.2.q.h 4 36.h even 6 2
1080.2.q.b 4 9.c even 3 2
2160.2.q.h 4 36.f odd 6 2
3240.2.a.k 2 3.b odd 2 1
3240.2.a.p 2 1.a even 1 1 trivial
6480.2.a.ba 2 12.b even 2 1
6480.2.a.bk 2 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3240))$$:

 $$T_{7}^{2} - 4 T_{7} + 1$$ $$T_{11}^{2} + 4 T_{11} - 8$$ $$T_{17}^{2} + 8 T_{17} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$1 - 4 T + T^{2}$$
$11$ $$-8 + 4 T + T^{2}$$
$13$ $$-8 + 4 T + T^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$1 + 4 T + T^{2}$$
$29$ $$13 + 10 T + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$-108 + T^{2}$$
$41$ $$-39 + 6 T + T^{2}$$
$43$ $$52 + 16 T + T^{2}$$
$47$ $$1 + 4 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$-92 + 8 T + T^{2}$$
$61$ $$13 + 10 T + T^{2}$$
$67$ $$61 - 16 T + T^{2}$$
$71$ $$24 + 12 T + T^{2}$$
$73$ $$-48 + T^{2}$$
$79$ $$132 - 24 T + T^{2}$$
$83$ $$37 - 16 T + T^{2}$$
$89$ $$-39 - 6 T + T^{2}$$
$97$ $$-44 + 4 T + T^{2}$$