# Properties

 Label 3240.2.q.d 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 360) 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.

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

## 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.d 2
3.b odd 2 1 3240.2.q.n 2
9.c even 3 1 360.2.a.d yes 1
9.c even 3 1 inner 3240.2.q.d 2
9.d odd 6 1 360.2.a.c 1
9.d odd 6 1 3240.2.q.n 2
36.f odd 6 1 720.2.a.i 1
36.h even 6 1 720.2.a.a 1
45.h odd 6 1 1800.2.a.i 1
45.j even 6 1 1800.2.a.f 1
45.k odd 12 2 1800.2.f.d 2
45.l even 12 2 1800.2.f.h 2
72.j odd 6 1 2880.2.a.bd 1
72.l even 6 1 2880.2.a.w 1
72.n even 6 1 2880.2.a.n 1
72.p odd 6 1 2880.2.a.e 1
180.n even 6 1 3600.2.a.bd 1
180.p odd 6 1 3600.2.a.bh 1
180.v odd 12 2 3600.2.f.g 2
180.x even 12 2 3600.2.f.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.a.c 1 9.d odd 6 1
360.2.a.d yes 1 9.c even 3 1
720.2.a.a 1 36.h even 6 1
720.2.a.i 1 36.f odd 6 1
1800.2.a.f 1 45.j even 6 1
1800.2.a.i 1 45.h odd 6 1
1800.2.f.d 2 45.k odd 12 2
1800.2.f.h 2 45.l even 12 2
2880.2.a.e 1 72.p odd 6 1
2880.2.a.n 1 72.n even 6 1
2880.2.a.w 1 72.l even 6 1
2880.2.a.bd 1 72.j odd 6 1
3240.2.q.d 2 1.a even 1 1 trivial
3240.2.q.d 2 9.c even 3 1 inner
3240.2.q.n 2 3.b odd 2 1
3240.2.q.n 2 9.d odd 6 1
3600.2.a.bd 1 180.n even 6 1
3600.2.a.bh 1 180.p odd 6 1
3600.2.f.g 2 180.v odd 12 2
3600.2.f.q 2 180.x even 12 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$16 + 4 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$100 + 10 T + T^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 - 8 T + T^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$196 + 14 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$144 - 12 T + T^{2}$$
$83$ $$16 - 4 T + T^{2}$$
$89$ $$( -12 + T )^{2}$$
$97$ $$196 - 14 T + T^{2}$$