# Properties

 Label 3840.2.k.r Level $3840$ Weight $2$ Character orbit 3840.k 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.k (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} -i q^{5} - q^{9} +O(q^{10})$$ $$q + i q^{3} -i q^{5} - q^{9} -4 i q^{11} -2 i q^{13} + q^{15} + 2 q^{17} -4 i q^{19} - q^{25} -i q^{27} -2 i q^{29} + 4 q^{33} + 10 i q^{37} + 2 q^{39} -10 q^{41} + 4 i q^{43} + i q^{45} -8 q^{47} -7 q^{49} + 2 i q^{51} + 10 i q^{53} -4 q^{55} + 4 q^{57} -4 i q^{59} -2 i q^{61} -2 q^{65} -12 i q^{67} -8 q^{71} -10 q^{73} -i q^{75} + q^{81} -12 i q^{83} -2 i q^{85} + 2 q^{87} + 6 q^{89} -4 q^{95} + 2 q^{97} + 4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{15} + 4q^{17} - 2q^{25} + 8q^{33} + 4q^{39} - 20q^{41} - 16q^{47} - 14q^{49} - 8q^{55} + 8q^{57} - 4q^{65} - 16q^{71} - 20q^{73} + 2q^{81} + 4q^{87} + 12q^{89} - 8q^{95} + 4q^{97} + O(q^{100})$$

## 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.r 2
4.b odd 2 1 3840.2.k.m 2
8.b even 2 1 inner 3840.2.k.r 2
8.d odd 2 1 3840.2.k.m 2
16.e even 4 1 240.2.a.d 1
16.e even 4 1 960.2.a.a 1
16.f odd 4 1 15.2.a.a 1
16.f odd 4 1 960.2.a.l 1
48.i odd 4 1 720.2.a.c 1
48.i odd 4 1 2880.2.a.bc 1
48.k even 4 1 45.2.a.a 1
48.k even 4 1 2880.2.a.y 1
80.i odd 4 1 1200.2.f.h 2
80.i odd 4 1 4800.2.f.c 2
80.j even 4 1 75.2.b.b 2
80.j even 4 1 4800.2.f.bf 2
80.k odd 4 1 75.2.a.b 1
80.k odd 4 1 4800.2.a.t 1
80.q even 4 1 1200.2.a.e 1
80.q even 4 1 4800.2.a.bz 1
80.s even 4 1 75.2.b.b 2
80.s even 4 1 4800.2.f.bf 2
80.t odd 4 1 1200.2.f.h 2
80.t odd 4 1 4800.2.f.c 2
112.j even 4 1 735.2.a.c 1
112.u odd 12 2 735.2.i.e 2
112.v even 12 2 735.2.i.d 2
144.u even 12 2 405.2.e.c 2
144.v odd 12 2 405.2.e.f 2
176.i even 4 1 1815.2.a.d 1
208.o odd 4 1 2535.2.a.j 1
240.t even 4 1 225.2.a.b 1
240.z odd 4 1 225.2.b.b 2
240.bb even 4 1 3600.2.f.e 2
240.bd odd 4 1 225.2.b.b 2
240.bf even 4 1 3600.2.f.e 2
240.bm odd 4 1 3600.2.a.u 1
272.k odd 4 1 4335.2.a.c 1
304.m even 4 1 5415.2.a.j 1
336.v odd 4 1 2205.2.a.i 1
368.i even 4 1 7935.2.a.d 1
528.s odd 4 1 5445.2.a.c 1
560.be even 4 1 3675.2.a.j 1
624.v even 4 1 7605.2.a.g 1
880.bi even 4 1 9075.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 16.f odd 4 1
45.2.a.a 1 48.k even 4 1
75.2.a.b 1 80.k odd 4 1
75.2.b.b 2 80.j even 4 1
75.2.b.b 2 80.s even 4 1
225.2.a.b 1 240.t even 4 1
225.2.b.b 2 240.z odd 4 1
225.2.b.b 2 240.bd odd 4 1
240.2.a.d 1 16.e even 4 1
405.2.e.c 2 144.u even 12 2
405.2.e.f 2 144.v odd 12 2
720.2.a.c 1 48.i odd 4 1
735.2.a.c 1 112.j even 4 1
735.2.i.d 2 112.v even 12 2
735.2.i.e 2 112.u odd 12 2
960.2.a.a 1 16.e even 4 1
960.2.a.l 1 16.f odd 4 1
1200.2.a.e 1 80.q even 4 1
1200.2.f.h 2 80.i odd 4 1
1200.2.f.h 2 80.t odd 4 1
1815.2.a.d 1 176.i even 4 1
2205.2.a.i 1 336.v odd 4 1
2535.2.a.j 1 208.o odd 4 1
2880.2.a.y 1 48.k even 4 1
2880.2.a.bc 1 48.i odd 4 1
3600.2.a.u 1 240.bm odd 4 1
3600.2.f.e 2 240.bb even 4 1
3600.2.f.e 2 240.bf even 4 1
3675.2.a.j 1 560.be even 4 1
3840.2.k.m 2 4.b odd 2 1
3840.2.k.m 2 8.d odd 2 1
3840.2.k.r 2 1.a even 1 1 trivial
3840.2.k.r 2 8.b even 2 1 inner
4335.2.a.c 1 272.k odd 4 1
4800.2.a.t 1 80.k odd 4 1
4800.2.a.bz 1 80.q even 4 1
4800.2.f.c 2 80.i odd 4 1
4800.2.f.c 2 80.t odd 4 1
4800.2.f.bf 2 80.j even 4 1
4800.2.f.bf 2 80.s even 4 1
5415.2.a.j 1 304.m even 4 1
5445.2.a.c 1 528.s odd 4 1
7605.2.a.g 1 624.v even 4 1
7935.2.a.d 1 368.i even 4 1
9075.2.a.g 1 880.bi even 4 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{2} + 16$$ $$T_{13}^{2} + 4$$ $$T_{17} - 2$$ $$T_{23}$$ $$T_{31}$$ $$T_{47} + 8$$

## Hecke characteristic polynomials

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