# Properties

 Label 3840.2.d.be 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: $$1$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 2 + i ) q^{5} + 4 i q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 2 + i ) q^{5} + 4 i q^{7} + q^{9} + 4 i q^{11} + ( 2 + i ) q^{15} + 4 i q^{17} + 4 i q^{21} -4 i q^{23} + ( 3 + 4 i ) q^{25} + q^{27} -6 i q^{29} -4 q^{31} + 4 i q^{33} + ( -4 + 8 i ) q^{35} + 8 q^{37} + 10 q^{41} -4 q^{43} + ( 2 + i ) q^{45} + 4 i q^{47} -9 q^{49} + 4 i q^{51} -12 q^{53} + ( -4 + 8 i ) q^{55} + 4 i q^{59} -2 i q^{61} + 4 i q^{63} + 4 q^{67} -4 i q^{69} -8 i q^{73} + ( 3 + 4 i ) q^{75} -16 q^{77} -12 q^{79} + q^{81} + 4 q^{83} + ( -4 + 8 i ) q^{85} -6 i q^{87} -10 q^{89} -4 q^{93} + 8 i q^{97} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + 4q^{15} + 6q^{25} + 2q^{27} - 8q^{31} - 8q^{35} + 16q^{37} + 20q^{41} - 8q^{43} + 4q^{45} - 18q^{49} - 24q^{53} - 8q^{55} + 8q^{67} + 6q^{75} - 32q^{77} - 24q^{79} + 2q^{81} + 8q^{83} - 8q^{85} - 20q^{89} - 8q^{93} + O(q^{100})$$

## 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 2.00000 1.00000i 0 4.00000i 0 1.00000 0
2689.2 0 1.00000 0 2.00000 + 1.00000i 0 4.00000i 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.be 2
4.b odd 2 1 3840.2.d.o 2
5.b even 2 1 3840.2.d.b 2
8.b even 2 1 3840.2.d.b 2
8.d odd 2 1 3840.2.d.r 2
16.e even 4 1 240.2.f.b 2
16.e even 4 1 960.2.f.c 2
16.f odd 4 1 60.2.d.a 2
16.f odd 4 1 960.2.f.f 2
20.d odd 2 1 3840.2.d.r 2
40.e odd 2 1 3840.2.d.o 2
40.f even 2 1 inner 3840.2.d.be 2
48.i odd 4 1 720.2.f.c 2
48.i odd 4 1 2880.2.f.p 2
48.k even 4 1 180.2.d.a 2
48.k even 4 1 2880.2.f.l 2
80.i odd 4 1 1200.2.a.a 1
80.i odd 4 1 4800.2.a.bf 1
80.j even 4 1 300.2.a.a 1
80.j even 4 1 4800.2.a.bj 1
80.k odd 4 1 60.2.d.a 2
80.k odd 4 1 960.2.f.f 2
80.q even 4 1 240.2.f.b 2
80.q even 4 1 960.2.f.c 2
80.s even 4 1 300.2.a.d 1
80.s even 4 1 4800.2.a.bn 1
80.t odd 4 1 1200.2.a.s 1
80.t odd 4 1 4800.2.a.bk 1
112.j even 4 1 2940.2.k.c 2
112.u odd 12 2 2940.2.bb.d 4
112.v even 12 2 2940.2.bb.e 4
144.u even 12 2 1620.2.r.d 4
144.v odd 12 2 1620.2.r.c 4
240.t even 4 1 180.2.d.a 2
240.t even 4 1 2880.2.f.l 2
240.z odd 4 1 900.2.a.h 1
240.bb even 4 1 3600.2.a.d 1
240.bd odd 4 1 900.2.a.a 1
240.bf even 4 1 3600.2.a.bm 1
240.bm odd 4 1 720.2.f.c 2
240.bm odd 4 1 2880.2.f.p 2
560.be even 4 1 2940.2.k.c 2
560.co even 12 2 2940.2.bb.e 4
560.cs odd 12 2 2940.2.bb.d 4
720.cz odd 12 2 1620.2.r.c 4
720.da even 12 2 1620.2.r.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 16.f odd 4 1
60.2.d.a 2 80.k odd 4 1
180.2.d.a 2 48.k even 4 1
180.2.d.a 2 240.t even 4 1
240.2.f.b 2 16.e even 4 1
240.2.f.b 2 80.q even 4 1
300.2.a.a 1 80.j even 4 1
300.2.a.d 1 80.s even 4 1
720.2.f.c 2 48.i odd 4 1
720.2.f.c 2 240.bm odd 4 1
900.2.a.a 1 240.bd odd 4 1
900.2.a.h 1 240.z odd 4 1
960.2.f.c 2 16.e even 4 1
960.2.f.c 2 80.q even 4 1
960.2.f.f 2 16.f odd 4 1
960.2.f.f 2 80.k odd 4 1
1200.2.a.a 1 80.i odd 4 1
1200.2.a.s 1 80.t odd 4 1
1620.2.r.c 4 144.v odd 12 2
1620.2.r.c 4 720.cz odd 12 2
1620.2.r.d 4 144.u even 12 2
1620.2.r.d 4 720.da even 12 2
2880.2.f.l 2 48.k even 4 1
2880.2.f.l 2 240.t even 4 1
2880.2.f.p 2 48.i odd 4 1
2880.2.f.p 2 240.bm odd 4 1
2940.2.k.c 2 112.j even 4 1
2940.2.k.c 2 560.be even 4 1
2940.2.bb.d 4 112.u odd 12 2
2940.2.bb.d 4 560.cs odd 12 2
2940.2.bb.e 4 112.v even 12 2
2940.2.bb.e 4 560.co even 12 2
3600.2.a.d 1 240.bb even 4 1
3600.2.a.bm 1 240.bf even 4 1
3840.2.d.b 2 5.b even 2 1
3840.2.d.b 2 8.b even 2 1
3840.2.d.o 2 4.b odd 2 1
3840.2.d.o 2 40.e odd 2 1
3840.2.d.r 2 8.d odd 2 1
3840.2.d.r 2 20.d odd 2 1
3840.2.d.be 2 1.a even 1 1 trivial
3840.2.d.be 2 40.f even 2 1 inner
4800.2.a.bf 1 80.i odd 4 1
4800.2.a.bj 1 80.j even 4 1
4800.2.a.bk 1 80.t odd 4 1
4800.2.a.bn 1 80.s even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ $$T_{11}^{2} + 16$$ $$T_{13}$$ $$T_{31} + 4$$ $$T_{37} - 8$$ $$T_{43} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$36 + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$( -8 + T )^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$( 12 + T )^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$4 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$64 + T^{2}$$