# Properties

 Label 3840.2.d.j 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 30) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 - 2 i ) q^{5} + 2 i q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 - 2 i ) q^{5} + 2 i q^{7} + q^{9} -2 i q^{11} -6 q^{13} + ( -1 + 2 i ) q^{15} + 2 i q^{17} -2 i q^{21} + 4 i q^{23} + ( -3 - 4 i ) q^{25} - q^{27} + 8 q^{31} + 2 i q^{33} + ( 4 + 2 i ) q^{35} -2 q^{37} + 6 q^{39} -2 q^{41} + 4 q^{43} + ( 1 - 2 i ) q^{45} + 8 i q^{47} + 3 q^{49} -2 i q^{51} + 6 q^{53} + ( -4 - 2 i ) q^{55} + 10 i q^{59} -2 i q^{61} + 2 i q^{63} + ( -6 + 12 i ) q^{65} + 8 q^{67} -4 i q^{69} + 12 q^{71} -4 i q^{73} + ( 3 + 4 i ) q^{75} + 4 q^{77} + q^{81} -4 q^{83} + ( 4 + 2 i ) q^{85} -10 q^{89} -12 i q^{91} -8 q^{93} -8 i q^{97} -2 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} + 2q^{9} - 12q^{13} - 2q^{15} - 6q^{25} - 2q^{27} + 16q^{31} + 8q^{35} - 4q^{37} + 12q^{39} - 4q^{41} + 8q^{43} + 2q^{45} + 6q^{49} + 12q^{53} - 8q^{55} - 12q^{65} + 16q^{67} + 24q^{71} + 6q^{75} + 8q^{77} + 2q^{81} - 8q^{83} + 8q^{85} - 20q^{89} - 16q^{93} + O(q^{100})$$

## 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 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.j 2
4.b odd 2 1 3840.2.d.y 2
5.b even 2 1 3840.2.d.x 2
8.b even 2 1 3840.2.d.x 2
8.d odd 2 1 3840.2.d.g 2
16.e even 4 1 240.2.f.a 2
16.e even 4 1 960.2.f.i 2
16.f odd 4 1 30.2.c.a 2
16.f odd 4 1 960.2.f.h 2
20.d odd 2 1 3840.2.d.g 2
40.e odd 2 1 3840.2.d.y 2
40.f even 2 1 inner 3840.2.d.j 2
48.i odd 4 1 720.2.f.f 2
48.i odd 4 1 2880.2.f.c 2
48.k even 4 1 90.2.c.a 2
48.k even 4 1 2880.2.f.e 2
80.i odd 4 1 1200.2.a.m 1
80.i odd 4 1 4800.2.a.cj 1
80.j even 4 1 150.2.a.c 1
80.j even 4 1 4800.2.a.cg 1
80.k odd 4 1 30.2.c.a 2
80.k odd 4 1 960.2.f.h 2
80.q even 4 1 240.2.f.a 2
80.q even 4 1 960.2.f.i 2
80.s even 4 1 150.2.a.a 1
80.s even 4 1 4800.2.a.l 1
80.t odd 4 1 1200.2.a.g 1
80.t odd 4 1 4800.2.a.m 1
112.j even 4 1 1470.2.g.g 2
112.u odd 12 2 1470.2.n.h 4
112.v even 12 2 1470.2.n.a 4
144.u even 12 2 810.2.i.b 4
144.v odd 12 2 810.2.i.e 4
240.t even 4 1 90.2.c.a 2
240.t even 4 1 2880.2.f.e 2
240.z odd 4 1 450.2.a.f 1
240.bb even 4 1 3600.2.a.o 1
240.bd odd 4 1 450.2.a.b 1
240.bf even 4 1 3600.2.a.bg 1
240.bm odd 4 1 720.2.f.f 2
240.bm odd 4 1 2880.2.f.c 2
560.u odd 4 1 7350.2.a.bg 1
560.be even 4 1 1470.2.g.g 2
560.bm odd 4 1 7350.2.a.cc 1
560.co even 12 2 1470.2.n.a 4
560.cs odd 12 2 1470.2.n.h 4
720.cz odd 12 2 810.2.i.e 4
720.da even 12 2 810.2.i.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 16.f odd 4 1
30.2.c.a 2 80.k odd 4 1
90.2.c.a 2 48.k even 4 1
90.2.c.a 2 240.t even 4 1
150.2.a.a 1 80.s even 4 1
150.2.a.c 1 80.j even 4 1
240.2.f.a 2 16.e even 4 1
240.2.f.a 2 80.q even 4 1
450.2.a.b 1 240.bd odd 4 1
450.2.a.f 1 240.z odd 4 1
720.2.f.f 2 48.i odd 4 1
720.2.f.f 2 240.bm odd 4 1
810.2.i.b 4 144.u even 12 2
810.2.i.b 4 720.da even 12 2
810.2.i.e 4 144.v odd 12 2
810.2.i.e 4 720.cz odd 12 2
960.2.f.h 2 16.f odd 4 1
960.2.f.h 2 80.k odd 4 1
960.2.f.i 2 16.e even 4 1
960.2.f.i 2 80.q even 4 1
1200.2.a.g 1 80.t odd 4 1
1200.2.a.m 1 80.i odd 4 1
1470.2.g.g 2 112.j even 4 1
1470.2.g.g 2 560.be even 4 1
1470.2.n.a 4 112.v even 12 2
1470.2.n.a 4 560.co even 12 2
1470.2.n.h 4 112.u odd 12 2
1470.2.n.h 4 560.cs odd 12 2
2880.2.f.c 2 48.i odd 4 1
2880.2.f.c 2 240.bm odd 4 1
2880.2.f.e 2 48.k even 4 1
2880.2.f.e 2 240.t even 4 1
3600.2.a.o 1 240.bb even 4 1
3600.2.a.bg 1 240.bf even 4 1
3840.2.d.g 2 8.d odd 2 1
3840.2.d.g 2 20.d odd 2 1
3840.2.d.j 2 1.a even 1 1 trivial
3840.2.d.j 2 40.f even 2 1 inner
3840.2.d.x 2 5.b even 2 1
3840.2.d.x 2 8.b even 2 1
3840.2.d.y 2 4.b odd 2 1
3840.2.d.y 2 40.e odd 2 1
4800.2.a.l 1 80.s even 4 1
4800.2.a.m 1 80.t odd 4 1
4800.2.a.cg 1 80.j even 4 1
4800.2.a.cj 1 80.i odd 4 1
7350.2.a.bg 1 560.u odd 4 1
7350.2.a.cc 1 560.bm odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11}^{2} + 4$$ $$T_{13} + 6$$ $$T_{31} - 8$$ $$T_{37} + 2$$ $$T_{43} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$1 - 18 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 6 T + 13 T^{2} )^{2}$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 19 T^{2} )^{2}$$
$23$ $$1 - 30 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}$$
$59$ $$1 - 18 T^{2} + 3481 T^{4}$$
$61$ $$1 - 118 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 130 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$( 1 + 4 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} )$$