# Properties

 Label 3240.2.q.ba Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8715302549$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + \beta_{3} q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + 2 \beta_{2} q^{17} + q^{19} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} -\beta_{1} q^{25} + \beta_{3} q^{29} + ( -1 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{31} -\beta_{2} q^{35} + 6 q^{37} + ( -7 + 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} -2 \beta_{3} q^{43} + ( 7 \beta_{1} - \beta_{3} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{49} -3 \beta_{2} q^{53} + ( -1 - \beta_{2} ) q^{55} + ( -5 + 5 \beta_{1} ) q^{59} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{61} + ( \beta_{1} - \beta_{3} ) q^{65} + ( -6 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -1 + \beta_{2} ) q^{71} + ( 8 - 2 \beta_{2} ) q^{73} + ( -8 + 8 \beta_{1} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{79} + 10 \beta_{1} q^{83} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -1 + \beta_{2} ) q^{89} + ( -8 + \beta_{2} ) q^{91} + ( -1 + \beta_{1} ) q^{95} -2 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{5} - q^{7} + O(q^{10})$$ $$4q - 2q^{5} - q^{7} + q^{11} + q^{13} - 4q^{17} + 4q^{19} - 5q^{23} - 2q^{25} + q^{29} - 5q^{31} + 2q^{35} + 24q^{37} - 12q^{41} - 2q^{43} + 13q^{47} - 3q^{49} + 6q^{53} - 2q^{55} - 10q^{59} - 14q^{61} + q^{65} - 10q^{67} - 6q^{71} + 36q^{73} - 16q^{77} - 6q^{79} + 20q^{83} + 2q^{85} - 6q^{89} - 34q^{91} - 2q^{95} - 2q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$$.

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1081.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0 0 0 −0.500000 + 0.866025i 0 −1.68614 2.92048i 0 0 0
1081.2 0 0 0 −0.500000 + 0.866025i 0 1.18614 + 2.05446i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −1.68614 + 2.92048i 0 0 0
2161.2 0 0 0 −0.500000 0.866025i 0 1.18614 2.05446i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.2.q.ba 4
3.b odd 2 1 3240.2.q.bd 4
9.c even 3 1 3240.2.a.m yes 2
9.c even 3 1 inner 3240.2.q.ba 4
9.d odd 6 1 3240.2.a.i 2
9.d odd 6 1 3240.2.q.bd 4
36.f odd 6 1 6480.2.a.bo 2
36.h even 6 1 6480.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.i 2 9.d odd 6 1
3240.2.a.m yes 2 9.c even 3 1
3240.2.q.ba 4 1.a even 1 1 trivial
3240.2.q.ba 4 9.c even 3 1 inner
3240.2.q.bd 4 3.b odd 2 1
3240.2.q.bd 4 9.d odd 6 1
6480.2.a.bd 2 36.h even 6 1
6480.2.a.bo 2 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3240, [\chi])$$:

 $$T_{7}^{4} + T_{7}^{3} + 9 T_{7}^{2} - 8 T_{7} + 64$$ $$T_{11}^{4} - T_{11}^{3} + 9 T_{11}^{2} + 8 T_{11} + 64$$ $$T_{17}^{2} + 2 T_{17} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$64 - 8 T + 9 T^{2} + T^{3} + T^{4}$$
$11$ $$64 + 8 T + 9 T^{2} - T^{3} + T^{4}$$
$13$ $$64 + 8 T + 9 T^{2} - T^{3} + T^{4}$$
$17$ $$( -32 + 2 T + T^{2} )^{2}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$4 - 10 T + 27 T^{2} + 5 T^{3} + T^{4}$$
$29$ $$64 + 8 T + 9 T^{2} - T^{3} + T^{4}$$
$31$ $$4624 - 340 T + 93 T^{2} + 5 T^{3} + T^{4}$$
$37$ $$( -6 + T )^{4}$$
$41$ $$9 + 36 T + 141 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4}$$
$47$ $$1156 - 442 T + 135 T^{2} - 13 T^{3} + T^{4}$$
$53$ $$( -72 - 3 T + T^{2} )^{2}$$
$59$ $$( 25 + 5 T + T^{2} )^{2}$$
$61$ $$256 + 224 T + 180 T^{2} + 14 T^{3} + T^{4}$$
$67$ $$64 - 80 T + 108 T^{2} + 10 T^{3} + T^{4}$$
$71$ $$( -6 + 3 T + T^{2} )^{2}$$
$73$ $$( 48 - 18 T + T^{2} )^{2}$$
$79$ $$576 - 144 T + 60 T^{2} + 6 T^{3} + T^{4}$$
$83$ $$( 100 - 10 T + T^{2} )^{2}$$
$89$ $$( -6 + 3 T + T^{2} )^{2}$$
$97$ $$1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4}$$