# Properties

 Label 3240.2.q.c Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{11} -\zeta_{6} q^{13} - q^{17} + 4 q^{19} -\zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( 5 - 5 \zeta_{6} ) q^{29} -\zeta_{6} q^{31} + 2 q^{35} + 6 q^{37} + ( -7 + 7 \zeta_{6} ) q^{43} + ( -7 + 7 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} -12 q^{53} + q^{55} + 4 \zeta_{6} q^{59} + ( -10 + 10 \zeta_{6} ) q^{61} + ( -1 + \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 12 q^{71} + 6 q^{73} -2 \zeta_{6} q^{77} + ( -15 + 15 \zeta_{6} ) q^{79} + ( -2 + 2 \zeta_{6} ) q^{83} + \zeta_{6} q^{85} -12 q^{89} + 2 q^{91} -4 \zeta_{6} q^{95} + ( -10 + 10 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} - 2q^{7} + O(q^{10})$$ $$2q - q^{5} - 2q^{7} - q^{11} - q^{13} - 2q^{17} + 8q^{19} - q^{23} - q^{25} + 5q^{29} - q^{31} + 4q^{35} + 12q^{37} - 7q^{43} - 7q^{47} + 3q^{49} - 24q^{53} + 2q^{55} + 4q^{59} - 10q^{61} - q^{65} + 4q^{67} + 24q^{71} + 12q^{73} - 2q^{77} - 15q^{79} - 2q^{83} + q^{85} - 24q^{89} + 4q^{91} - 4q^{95} - 10q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 −1.00000 1.73205i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −1.00000 + 1.73205i 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.c 2
3.b odd 2 1 3240.2.q.o 2
9.c even 3 1 1080.2.a.k yes 1
9.c even 3 1 inner 3240.2.q.c 2
9.d odd 6 1 1080.2.a.f 1
9.d odd 6 1 3240.2.q.o 2
36.f odd 6 1 2160.2.a.n 1
36.h even 6 1 2160.2.a.d 1
45.h odd 6 1 5400.2.a.m 1
45.j even 6 1 5400.2.a.n 1
45.k odd 12 2 5400.2.f.p 2
45.l even 12 2 5400.2.f.m 2
72.j odd 6 1 8640.2.a.cb 1
72.l even 6 1 8640.2.a.bk 1
72.n even 6 1 8640.2.a.v 1
72.p odd 6 1 8640.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.f 1 9.d odd 6 1
1080.2.a.k yes 1 9.c even 3 1
2160.2.a.d 1 36.h even 6 1
2160.2.a.n 1 36.f odd 6 1
3240.2.q.c 2 1.a even 1 1 trivial
3240.2.q.c 2 9.c even 3 1 inner
3240.2.q.o 2 3.b odd 2 1
3240.2.q.o 2 9.d odd 6 1
5400.2.a.m 1 45.h odd 6 1
5400.2.a.n 1 45.j even 6 1
5400.2.f.m 2 45.l even 12 2
5400.2.f.p 2 45.k odd 12 2
8640.2.a.i 1 72.p odd 6 1
8640.2.a.v 1 72.n even 6 1
8640.2.a.bk 1 72.l even 6 1
8640.2.a.cb 1 72.j odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$1 + T + T^{2}$$
$29$ $$25 - 5 T + T^{2}$$
$31$ $$1 + T + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$49 + 7 T + T^{2}$$
$47$ $$49 + 7 T + T^{2}$$
$53$ $$( 12 + T )^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$100 + 10 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$225 + 15 T + T^{2}$$
$83$ $$4 + 2 T + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$100 + 10 T + T^{2}$$