# Properties

 Label 3840.2.d.a 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + (i - 2) q^{5} + 4 i q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (i - 2) * q^5 + 4*i * q^7 + q^9 $$q - q^{3} + (i - 2) q^{5} + 4 i q^{7} + q^{9} - 4 q^{13} + ( - i + 2) q^{15} - 8 i q^{19} - 4 i q^{21} + 4 i q^{23} + ( - 4 i + 3) q^{25} - q^{27} - 6 i q^{29} - 8 q^{31} + ( - 8 i - 4) q^{35} + 4 q^{37} + 4 q^{39} - 6 q^{41} + 4 q^{43} + (i - 2) q^{45} - 4 i q^{47} - 9 q^{49} - 12 q^{53} + 8 i q^{57} + 6 i q^{61} + 4 i q^{63} + ( - 4 i + 8) q^{65} + 12 q^{67} - 4 i q^{69} + 16 q^{71} + (4 i - 3) q^{75} + 8 q^{79} + q^{81} + 12 q^{83} + 6 i q^{87} - 10 q^{89} - 16 i q^{91} + 8 q^{93} + (16 i + 8) q^{95} + 8 i q^{97} +O(q^{100})$$ q - q^3 + (i - 2) * q^5 + 4*i * q^7 + q^9 - 4 * q^13 + (-i + 2) * q^15 - 8*i * q^19 - 4*i * q^21 + 4*i * q^23 + (-4*i + 3) * q^25 - q^27 - 6*i * q^29 - 8 * q^31 + (-8*i - 4) * q^35 + 4 * q^37 + 4 * q^39 - 6 * q^41 + 4 * q^43 + (i - 2) * q^45 - 4*i * q^47 - 9 * q^49 - 12 * q^53 + 8*i * q^57 + 6*i * q^61 + 4*i * q^63 + (-4*i + 8) * q^65 + 12 * q^67 - 4*i * q^69 + 16 * q^71 + (4*i - 3) * q^75 + 8 * q^79 + q^81 + 12 * q^83 + 6*i * q^87 - 10 * q^89 - 16*i * q^91 + 8 * q^93 + (16*i + 8) * q^95 + 8*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 $$2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 8 q^{13} + 4 q^{15} + 6 q^{25} - 2 q^{27} - 16 q^{31} - 8 q^{35} + 8 q^{37} + 8 q^{39} - 12 q^{41} + 8 q^{43} - 4 q^{45} - 18 q^{49} - 24 q^{53} + 16 q^{65} + 24 q^{67} + 32 q^{71} - 6 q^{75} + 16 q^{79} + 2 q^{81} + 24 q^{83} - 20 q^{89} + 16 q^{93} + 16 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 8 * q^13 + 4 * q^15 + 6 * q^25 - 2 * q^27 - 16 * q^31 - 8 * q^35 + 8 * q^37 + 8 * q^39 - 12 * q^41 + 8 * q^43 - 4 * q^45 - 18 * q^49 - 24 * q^53 + 16 * q^65 + 24 * q^67 + 32 * q^71 - 6 * q^75 + 16 * q^79 + 2 * q^81 + 24 * q^83 - 20 * q^89 + 16 * q^93 + 16 * q^95

## 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.a 2
4.b odd 2 1 3840.2.d.q 2
5.b even 2 1 3840.2.d.bf 2
8.b even 2 1 3840.2.d.bf 2
8.d odd 2 1 3840.2.d.p 2
16.e even 4 1 480.2.f.d yes 2
16.e even 4 1 960.2.f.e 2
16.f odd 4 1 480.2.f.c 2
16.f odd 4 1 960.2.f.d 2
20.d odd 2 1 3840.2.d.p 2
40.e odd 2 1 3840.2.d.q 2
40.f even 2 1 inner 3840.2.d.a 2
48.i odd 4 1 1440.2.f.d 2
48.i odd 4 1 2880.2.f.m 2
48.k even 4 1 1440.2.f.b 2
48.k even 4 1 2880.2.f.o 2
80.i odd 4 1 2400.2.a.t 1
80.i odd 4 1 4800.2.a.cr 1
80.j even 4 1 2400.2.a.s 1
80.j even 4 1 4800.2.a.cp 1
80.k odd 4 1 480.2.f.c 2
80.k odd 4 1 960.2.f.d 2
80.q even 4 1 480.2.f.d yes 2
80.q even 4 1 960.2.f.e 2
80.s even 4 1 2400.2.a.p 1
80.s even 4 1 4800.2.a.e 1
80.t odd 4 1 2400.2.a.o 1
80.t odd 4 1 4800.2.a.c 1
240.t even 4 1 1440.2.f.b 2
240.t even 4 1 2880.2.f.o 2
240.z odd 4 1 7200.2.a.ca 1
240.bb even 4 1 7200.2.a.c 1
240.bd odd 4 1 7200.2.a.a 1
240.bf even 4 1 7200.2.a.by 1
240.bm odd 4 1 1440.2.f.d 2
240.bm odd 4 1 2880.2.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.c 2 16.f odd 4 1
480.2.f.c 2 80.k odd 4 1
480.2.f.d yes 2 16.e even 4 1
480.2.f.d yes 2 80.q even 4 1
960.2.f.d 2 16.f odd 4 1
960.2.f.d 2 80.k odd 4 1
960.2.f.e 2 16.e even 4 1
960.2.f.e 2 80.q even 4 1
1440.2.f.b 2 48.k even 4 1
1440.2.f.b 2 240.t even 4 1
1440.2.f.d 2 48.i odd 4 1
1440.2.f.d 2 240.bm odd 4 1
2400.2.a.o 1 80.t odd 4 1
2400.2.a.p 1 80.s even 4 1
2400.2.a.s 1 80.j even 4 1
2400.2.a.t 1 80.i odd 4 1
2880.2.f.m 2 48.i odd 4 1
2880.2.f.m 2 240.bm odd 4 1
2880.2.f.o 2 48.k even 4 1
2880.2.f.o 2 240.t even 4 1
3840.2.d.a 2 1.a even 1 1 trivial
3840.2.d.a 2 40.f even 2 1 inner
3840.2.d.p 2 8.d odd 2 1
3840.2.d.p 2 20.d odd 2 1
3840.2.d.q 2 4.b odd 2 1
3840.2.d.q 2 40.e odd 2 1
3840.2.d.bf 2 5.b even 2 1
3840.2.d.bf 2 8.b even 2 1
4800.2.a.c 1 80.t odd 4 1
4800.2.a.e 1 80.s even 4 1
4800.2.a.cp 1 80.j even 4 1
4800.2.a.cr 1 80.i odd 4 1
7200.2.a.a 1 240.bd odd 4 1
7200.2.a.c 1 240.bb even 4 1
7200.2.a.by 1 240.bf even 4 1
7200.2.a.ca 1 240.z odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}$$ T11 $$T_{13} + 4$$ T13 + 4 $$T_{31} + 8$$ T31 + 8 $$T_{37} - 4$$ T37 - 4 $$T_{43} - 4$$ T43 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 36$$
$31$ $$(T + 8)^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 16$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 36$$
$67$ $$(T - 12)^{2}$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 64$$