# Properties

 Label 3840.2.k.k Level $3840$ Weight $2$ Character orbit 3840.k Analytic conductor $30.663$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{5} - q^{9} +O(q^{10})$$ $$q + i q^{3} + i q^{5} - q^{9} -4 i q^{11} + 2 i q^{13} - q^{15} -2 q^{17} + 8 i q^{19} -4 q^{23} - q^{25} -i q^{27} -6 i q^{29} + 4 q^{33} -2 i q^{37} -2 q^{39} + 6 q^{41} -4 i q^{43} -i q^{45} -12 q^{47} -7 q^{49} -2 i q^{51} + 6 i q^{53} + 4 q^{55} -8 q^{57} -12 i q^{59} + 14 i q^{61} -2 q^{65} -12 i q^{67} -4 i q^{69} -2 q^{73} -i q^{75} -8 q^{79} + q^{81} -4 i q^{83} -2 i q^{85} + 6 q^{87} -2 q^{89} -8 q^{95} -14 q^{97} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 2q^{15} - 4q^{17} - 8q^{23} - 2q^{25} + 8q^{33} - 4q^{39} + 12q^{41} - 24q^{47} - 14q^{49} + 8q^{55} - 16q^{57} - 4q^{65} - 4q^{73} - 16q^{79} + 2q^{81} + 12q^{87} - 4q^{89} - 16q^{95} - 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$1537$$ $$2561$$ $$2821$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1921.1
 − 1.00000i 1.00000i
0 1.00000i 0 1.00000i 0 0 0 −1.00000 0
1921.2 0 1.00000i 0 1.00000i 0 0 0 −1.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.2.k.k 2
4.b odd 2 1 3840.2.k.p 2
8.b even 2 1 inner 3840.2.k.k 2
8.d odd 2 1 3840.2.k.p 2
16.e even 4 1 480.2.a.e yes 1
16.e even 4 1 960.2.a.f 1
16.f odd 4 1 480.2.a.b 1
16.f odd 4 1 960.2.a.o 1
48.i odd 4 1 1440.2.a.j 1
48.i odd 4 1 2880.2.a.j 1
48.k even 4 1 1440.2.a.k 1
48.k even 4 1 2880.2.a.i 1
80.i odd 4 1 2400.2.f.n 2
80.i odd 4 1 4800.2.f.j 2
80.j even 4 1 2400.2.f.e 2
80.j even 4 1 4800.2.f.ba 2
80.k odd 4 1 2400.2.a.y 1
80.k odd 4 1 4800.2.a.u 1
80.q even 4 1 2400.2.a.j 1
80.q even 4 1 4800.2.a.ca 1
80.s even 4 1 2400.2.f.e 2
80.s even 4 1 4800.2.f.ba 2
80.t odd 4 1 2400.2.f.n 2
80.t odd 4 1 4800.2.f.j 2
240.t even 4 1 7200.2.a.bg 1
240.z odd 4 1 7200.2.f.bb 2
240.bb even 4 1 7200.2.f.b 2
240.bd odd 4 1 7200.2.f.bb 2
240.bf even 4 1 7200.2.f.b 2
240.bm odd 4 1 7200.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.b 1 16.f odd 4 1
480.2.a.e yes 1 16.e even 4 1
960.2.a.f 1 16.e even 4 1
960.2.a.o 1 16.f odd 4 1
1440.2.a.j 1 48.i odd 4 1
1440.2.a.k 1 48.k even 4 1
2400.2.a.j 1 80.q even 4 1
2400.2.a.y 1 80.k odd 4 1
2400.2.f.e 2 80.j even 4 1
2400.2.f.e 2 80.s even 4 1
2400.2.f.n 2 80.i odd 4 1
2400.2.f.n 2 80.t odd 4 1
2880.2.a.i 1 48.k even 4 1
2880.2.a.j 1 48.i odd 4 1
3840.2.k.k 2 1.a even 1 1 trivial
3840.2.k.k 2 8.b even 2 1 inner
3840.2.k.p 2 4.b odd 2 1
3840.2.k.p 2 8.d odd 2 1
4800.2.a.u 1 80.k odd 4 1
4800.2.a.ca 1 80.q even 4 1
4800.2.f.j 2 80.i odd 4 1
4800.2.f.j 2 80.t odd 4 1
4800.2.f.ba 2 80.j even 4 1
4800.2.f.ba 2 80.s even 4 1
7200.2.a.u 1 240.bm odd 4 1
7200.2.a.bg 1 240.t even 4 1
7200.2.f.b 2 240.bb even 4 1
7200.2.f.b 2 240.bf even 4 1
7200.2.f.bb 2 240.z odd 4 1
7200.2.f.bb 2 240.bd odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3840, [\chi])$$:

 $$T_{7}$$ $$T_{11}^{2} + 16$$ $$T_{13}^{2} + 4$$ $$T_{17} + 2$$ $$T_{23} + 4$$ $$T_{31}$$ $$T_{47} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$64 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$( 12 + T )^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$144 + T^{2}$$
$61$ $$196 + T^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$