# Properties

 Label 3840.2.d.k Level $3840$ Weight $2$ Character orbit 3840.d Analytic conductor $30.663$ Analytic rank $0$ 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 + 2 i ) q^{5} + 2 i q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 + 2 i ) q^{5} + 2 i q^{7} + q^{9} + 6 i q^{11} + 2 q^{13} + ( -1 - 2 i ) q^{15} + 6 i q^{17} -4 i q^{19} -2 i q^{21} + 8 i q^{23} + ( -3 + 4 i ) q^{25} - q^{27} -8 q^{31} -6 i q^{33} + ( -4 + 2 i ) q^{35} -2 q^{37} -2 q^{39} + 6 q^{41} + 4 q^{43} + ( 1 + 2 i ) q^{45} + 4 i q^{47} + 3 q^{49} -6 i q^{51} + 6 q^{53} + ( -12 + 6 i ) q^{55} + 4 i q^{57} -6 i q^{59} -6 i q^{61} + 2 i q^{63} + ( 2 + 4 i ) q^{65} -8 i q^{69} + 4 q^{71} -12 i q^{73} + ( 3 - 4 i ) q^{75} -12 q^{77} + 8 q^{79} + q^{81} -12 q^{83} + ( -12 + 6 i ) q^{85} + 14 q^{89} + 4 i q^{91} + 8 q^{93} + ( 8 - 4 i ) q^{95} -8 i q^{97} + 6 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} - 2 q^{15} - 6 q^{25} - 2 q^{27} - 16 q^{31} - 8 q^{35} - 4 q^{37} - 4 q^{39} + 12 q^{41} + 8 q^{43} + 2 q^{45} + 6 q^{49} + 12 q^{53} - 24 q^{55} + 4 q^{65} + 8 q^{71} + 6 q^{75} - 24 q^{77} + 16 q^{79} + 2 q^{81} - 24 q^{83} - 24 q^{85} + 28 q^{89} + 16 q^{93} + 16 q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2689.1
 − 1.00000i 1.00000i
0 −1.00000 0 1.00000 2.00000i 0 2.00000i 0 1.00000 0
2689.2 0 −1.00000 0 1.00000 + 2.00000i 0 2.00000i 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
40.f 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.d.k 2
4.b odd 2 1 3840.2.d.bc 2
5.b even 2 1 3840.2.d.u 2
8.b even 2 1 3840.2.d.u 2
8.d odd 2 1 3840.2.d.e 2
16.e even 4 1 480.2.f.a 2
16.e even 4 1 960.2.f.j 2
16.f odd 4 1 480.2.f.b yes 2
16.f odd 4 1 960.2.f.g 2
20.d odd 2 1 3840.2.d.e 2
40.e odd 2 1 3840.2.d.bc 2
40.f even 2 1 inner 3840.2.d.k 2
48.i odd 4 1 1440.2.f.g 2
48.i odd 4 1 2880.2.f.a 2
48.k even 4 1 1440.2.f.e 2
48.k even 4 1 2880.2.f.g 2
80.i odd 4 1 2400.2.a.be 1
80.i odd 4 1 4800.2.a.bu 1
80.j even 4 1 2400.2.a.bf 1
80.j even 4 1 4800.2.a.bt 1
80.k odd 4 1 480.2.f.b yes 2
80.k odd 4 1 960.2.f.g 2
80.q even 4 1 480.2.f.a 2
80.q even 4 1 960.2.f.j 2
80.s even 4 1 2400.2.a.d 1
80.s even 4 1 4800.2.a.z 1
80.t odd 4 1 2400.2.a.c 1
80.t odd 4 1 4800.2.a.ba 1
240.t even 4 1 1440.2.f.e 2
240.t even 4 1 2880.2.f.g 2
240.z odd 4 1 7200.2.a.j 1
240.bb even 4 1 7200.2.a.br 1
240.bd odd 4 1 7200.2.a.bl 1
240.bf even 4 1 7200.2.a.p 1
240.bm odd 4 1 1440.2.f.g 2
240.bm odd 4 1 2880.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.a 2 16.e even 4 1
480.2.f.a 2 80.q even 4 1
480.2.f.b yes 2 16.f odd 4 1
480.2.f.b yes 2 80.k odd 4 1
960.2.f.g 2 16.f odd 4 1
960.2.f.g 2 80.k odd 4 1
960.2.f.j 2 16.e even 4 1
960.2.f.j 2 80.q even 4 1
1440.2.f.e 2 48.k even 4 1
1440.2.f.e 2 240.t even 4 1
1440.2.f.g 2 48.i odd 4 1
1440.2.f.g 2 240.bm odd 4 1
2400.2.a.c 1 80.t odd 4 1
2400.2.a.d 1 80.s even 4 1
2400.2.a.be 1 80.i odd 4 1
2400.2.a.bf 1 80.j even 4 1
2880.2.f.a 2 48.i odd 4 1
2880.2.f.a 2 240.bm odd 4 1
2880.2.f.g 2 48.k even 4 1
2880.2.f.g 2 240.t even 4 1
3840.2.d.e 2 8.d odd 2 1
3840.2.d.e 2 20.d odd 2 1
3840.2.d.k 2 1.a even 1 1 trivial
3840.2.d.k 2 40.f even 2 1 inner
3840.2.d.u 2 5.b even 2 1
3840.2.d.u 2 8.b even 2 1
3840.2.d.bc 2 4.b odd 2 1
3840.2.d.bc 2 40.e odd 2 1
4800.2.a.z 1 80.s even 4 1
4800.2.a.ba 1 80.t odd 4 1
4800.2.a.bt 1 80.j even 4 1
4800.2.a.bu 1 80.i odd 4 1
7200.2.a.j 1 240.z odd 4 1
7200.2.a.p 1 240.bf even 4 1
7200.2.a.bl 1 240.bd odd 4 1
7200.2.a.br 1 240.bb even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 36$$ $$T_{13} - 2$$ $$T_{31} + 8$$ $$T_{37} + 2$$ $$T_{43} - 4$$

## Hecke characteristic polynomials

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