# Properties

 Label 3840.2.k.s Level $3840$ Weight $2$ Character orbit 3840.k Analytic conductor $30.663$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + i q^{5} - q^{9} +O(q^{10})$$ q - i * q^3 + i * q^5 - q^9 $$q - i q^{3} + i q^{5} - q^{9} - 2 i q^{13} + q^{15} + 6 q^{17} - 4 i q^{19} + 8 q^{23} - q^{25} + i q^{27} + 2 i q^{29} - 4 q^{31} + 10 i q^{37} - 2 q^{39} - 2 q^{41} + 4 i q^{43} - i q^{45} - 8 q^{47} - 7 q^{49} - 6 i q^{51} - 2 i q^{53} - 4 q^{57} - 8 i q^{59} + 2 i q^{61} + 2 q^{65} - 12 i q^{67} - 8 i q^{69} + 8 q^{71} + 14 q^{73} + i q^{75} + 12 q^{79} + q^{81} - 4 i q^{83} + 6 i q^{85} + 2 q^{87} + 14 q^{89} + 4 i q^{93} + 4 q^{95} + 2 q^{97} +O(q^{100})$$ q - i * q^3 + i * q^5 - q^9 - 2*i * q^13 + q^15 + 6 * q^17 - 4*i * q^19 + 8 * q^23 - q^25 + i * q^27 + 2*i * q^29 - 4 * q^31 + 10*i * q^37 - 2 * q^39 - 2 * q^41 + 4*i * q^43 - i * q^45 - 8 * q^47 - 7 * q^49 - 6*i * q^51 - 2*i * q^53 - 4 * q^57 - 8*i * q^59 + 2*i * q^61 + 2 * q^65 - 12*i * q^67 - 8*i * q^69 + 8 * q^71 + 14 * q^73 + i * q^75 + 12 * q^79 + q^81 - 4*i * q^83 + 6*i * q^85 + 2 * q^87 + 14 * q^89 + 4*i * q^93 + 4 * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{15} + 12 q^{17} + 16 q^{23} - 2 q^{25} - 8 q^{31} - 4 q^{39} - 4 q^{41} - 16 q^{47} - 14 q^{49} - 8 q^{57} + 4 q^{65} + 16 q^{71} + 28 q^{73} + 24 q^{79} + 2 q^{81} + 4 q^{87} + 28 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^15 + 12 * q^17 + 16 * q^23 - 2 * q^25 - 8 * q^31 - 4 * q^39 - 4 * q^41 - 16 * q^47 - 14 * q^49 - 8 * q^57 + 4 * q^65 + 16 * q^71 + 28 * q^73 + 24 * q^79 + 2 * q^81 + 4 * q^87 + 28 * q^89 + 8 * q^95 + 4 * q^97

## 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.s 2
4.b odd 2 1 3840.2.k.n 2
8.b even 2 1 inner 3840.2.k.s 2
8.d odd 2 1 3840.2.k.n 2
16.e even 4 1 480.2.a.g yes 1
16.e even 4 1 960.2.a.b 1
16.f odd 4 1 480.2.a.d 1
16.f odd 4 1 960.2.a.k 1
48.i odd 4 1 1440.2.a.d 1
48.i odd 4 1 2880.2.a.z 1
48.k even 4 1 1440.2.a.c 1
48.k even 4 1 2880.2.a.ba 1
80.i odd 4 1 2400.2.f.g 2
80.i odd 4 1 4800.2.f.v 2
80.j even 4 1 2400.2.f.l 2
80.j even 4 1 4800.2.f.o 2
80.k odd 4 1 2400.2.a.z 1
80.k odd 4 1 4800.2.a.s 1
80.q even 4 1 2400.2.a.i 1
80.q even 4 1 4800.2.a.cb 1
80.s even 4 1 2400.2.f.l 2
80.s even 4 1 4800.2.f.o 2
80.t odd 4 1 2400.2.f.g 2
80.t odd 4 1 4800.2.f.v 2
240.t even 4 1 7200.2.a.z 1
240.z odd 4 1 7200.2.f.s 2
240.bb even 4 1 7200.2.f.k 2
240.bd odd 4 1 7200.2.f.s 2
240.bf even 4 1 7200.2.f.k 2
240.bm odd 4 1 7200.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.d 1 16.f odd 4 1
480.2.a.g yes 1 16.e even 4 1
960.2.a.b 1 16.e even 4 1
960.2.a.k 1 16.f odd 4 1
1440.2.a.c 1 48.k even 4 1
1440.2.a.d 1 48.i odd 4 1
2400.2.a.i 1 80.q even 4 1
2400.2.a.z 1 80.k odd 4 1
2400.2.f.g 2 80.i odd 4 1
2400.2.f.g 2 80.t odd 4 1
2400.2.f.l 2 80.j even 4 1
2400.2.f.l 2 80.s even 4 1
2880.2.a.z 1 48.i odd 4 1
2880.2.a.ba 1 48.k even 4 1
3840.2.k.n 2 4.b odd 2 1
3840.2.k.n 2 8.d odd 2 1
3840.2.k.s 2 1.a even 1 1 trivial
3840.2.k.s 2 8.b even 2 1 inner
4800.2.a.s 1 80.k odd 4 1
4800.2.a.cb 1 80.q even 4 1
4800.2.f.o 2 80.j even 4 1
4800.2.f.o 2 80.s even 4 1
4800.2.f.v 2 80.i odd 4 1
4800.2.f.v 2 80.t odd 4 1
7200.2.a.z 1 240.t even 4 1
7200.2.a.ba 1 240.bm odd 4 1
7200.2.f.k 2 240.bb even 4 1
7200.2.f.k 2 240.bf even 4 1
7200.2.f.s 2 240.z odd 4 1
7200.2.f.s 2 240.bd odd 4 1

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17} - 6$$ T17 - 6 $$T_{23} - 8$$ T23 - 8 $$T_{31} + 4$$ T31 + 4 $$T_{47} + 8$$ T47 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 8)^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2} + 64$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T - 14)^{2}$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 14)^{2}$$
$97$ $$(T - 2)^{2}$$
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