# Properties

 Label 3840.2.a.bp Level 3840 Weight 2 Character orbit 3840.a Self dual yes Analytic conductor 30.663 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + ( -1 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} + ( -1 - \beta_{1} ) q^{7} + q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} -\beta_{2} q^{13} - q^{15} + ( 2 - \beta_{2} ) q^{17} + ( -1 + \beta_{1} ) q^{19} + ( 1 + \beta_{1} ) q^{21} + ( 1 - \beta_{1} ) q^{23} + q^{25} - q^{27} + 2 q^{29} + ( -2 - \beta_{2} ) q^{31} + ( 1 - \beta_{1} + \beta_{2} ) q^{33} + ( -1 - \beta_{1} ) q^{35} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{37} + \beta_{2} q^{39} + ( 4 + 2 \beta_{1} ) q^{41} + ( -2 - 2 \beta_{1} ) q^{43} + q^{45} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( 5 + 2 \beta_{2} ) q^{49} + ( -2 + \beta_{2} ) q^{51} + 2 q^{53} + ( -1 + \beta_{1} - \beta_{2} ) q^{55} + ( 1 - \beta_{1} ) q^{57} + ( -3 - \beta_{1} + \beta_{2} ) q^{59} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -1 - \beta_{1} ) q^{63} -\beta_{2} q^{65} + 4 q^{67} + ( -1 + \beta_{1} ) q^{69} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + 6 q^{73} - q^{75} + ( -4 + 4 \beta_{1} ) q^{77} + ( 6 - \beta_{2} ) q^{79} + q^{81} + ( -6 - 2 \beta_{1} ) q^{83} + ( 2 - \beta_{2} ) q^{85} -2 q^{87} + ( 4 - 2 \beta_{1} ) q^{89} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{91} + ( 2 + \beta_{2} ) q^{93} + ( -1 + \beta_{1} ) q^{95} + ( 6 - 2 \beta_{2} ) q^{97} + ( -1 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 3q^{5} - 2q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 3q^{5} - 2q^{7} + 3q^{9} - 4q^{11} - 3q^{15} + 6q^{17} - 4q^{19} + 2q^{21} + 4q^{23} + 3q^{25} - 3q^{27} + 6q^{29} - 6q^{31} + 4q^{33} - 2q^{35} - 4q^{37} + 10q^{41} - 4q^{43} + 3q^{45} + 4q^{47} + 15q^{49} - 6q^{51} + 6q^{53} - 4q^{55} + 4q^{57} - 8q^{59} + 8q^{61} - 2q^{63} + 12q^{67} - 4q^{69} + 4q^{71} + 18q^{73} - 3q^{75} - 16q^{77} + 18q^{79} + 3q^{81} - 16q^{83} + 6q^{85} - 6q^{87} + 14q^{89} + 16q^{91} + 6q^{93} - 4q^{95} + 18q^{97} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 6$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.34292 0.470683 −1.81361
0 −1.00000 0 1.00000 0 −4.68585 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −0.941367 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 3.62721 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3840.2.a.bp 3
4.b odd 2 1 3840.2.a.br 3
8.b even 2 1 3840.2.a.bq 3
8.d odd 2 1 3840.2.a.bo 3
16.e even 4 2 120.2.k.b 6
16.f odd 4 2 480.2.k.b 6
48.i odd 4 2 360.2.k.f 6
48.k even 4 2 1440.2.k.f 6
80.i odd 4 2 600.2.d.f 6
80.j even 4 2 2400.2.d.f 6
80.k odd 4 2 2400.2.k.c 6
80.q even 4 2 600.2.k.c 6
80.s even 4 2 2400.2.d.e 6
80.t odd 4 2 600.2.d.e 6
240.t even 4 2 7200.2.k.p 6
240.z odd 4 2 7200.2.d.r 6
240.bb even 4 2 1800.2.d.r 6
240.bd odd 4 2 7200.2.d.q 6
240.bf even 4 2 1800.2.d.q 6
240.bm odd 4 2 1800.2.k.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 16.e even 4 2
360.2.k.f 6 48.i odd 4 2
480.2.k.b 6 16.f odd 4 2
600.2.d.e 6 80.t odd 4 2
600.2.d.f 6 80.i odd 4 2
600.2.k.c 6 80.q even 4 2
1440.2.k.f 6 48.k even 4 2
1800.2.d.q 6 240.bf even 4 2
1800.2.d.r 6 240.bb even 4 2
1800.2.k.p 6 240.bm odd 4 2
2400.2.d.e 6 80.s even 4 2
2400.2.d.f 6 80.j even 4 2
2400.2.k.c 6 80.k odd 4 2
3840.2.a.bo 3 8.d odd 2 1
3840.2.a.bp 3 1.a even 1 1 trivial
3840.2.a.bq 3 8.b even 2 1
3840.2.a.br 3 4.b odd 2 1
7200.2.d.q 6 240.bd odd 4 2
7200.2.d.r 6 240.z odd 4 2
7200.2.k.p 6 240.t even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3840))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 16 T_{7} - 16$$ $$T_{11}^{3} + 4 T_{11}^{2} - 24 T_{11} - 64$$ $$T_{13}^{3} - 28 T_{13} - 16$$ $$T_{17}^{3} - 6 T_{17}^{2} - 16 T_{17} + 32$$ $$T_{19}^{3} + 4 T_{19}^{2} - 12 T_{19} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{3}$$
$5$ $$( 1 - T )^{3}$$
$7$ $$1 + 2 T + 5 T^{2} + 12 T^{3} + 35 T^{4} + 98 T^{5} + 343 T^{6}$$
$11$ $$1 + 4 T + 9 T^{2} + 24 T^{3} + 99 T^{4} + 484 T^{5} + 1331 T^{6}$$
$13$ $$1 + 11 T^{2} - 16 T^{3} + 143 T^{4} + 2197 T^{6}$$
$17$ $$1 - 6 T + 35 T^{2} - 172 T^{3} + 595 T^{4} - 1734 T^{5} + 4913 T^{6}$$
$19$ $$1 + 4 T + 45 T^{2} + 136 T^{3} + 855 T^{4} + 1444 T^{5} + 6859 T^{6}$$
$23$ $$1 - 4 T + 57 T^{2} - 168 T^{3} + 1311 T^{4} - 2116 T^{5} + 12167 T^{6}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{3}$$
$31$ $$1 + 6 T + 77 T^{2} + 308 T^{3} + 2387 T^{4} + 5766 T^{5} + 29791 T^{6}$$
$37$ $$1 + 4 T + 51 T^{2} + 40 T^{3} + 1887 T^{4} + 5476 T^{5} + 50653 T^{6}$$
$41$ $$1 - 10 T + 87 T^{2} - 588 T^{3} + 3567 T^{4} - 16810 T^{5} + 68921 T^{6}$$
$43$ $$1 + 4 T + 65 T^{2} + 216 T^{3} + 2795 T^{4} + 7396 T^{5} + 79507 T^{6}$$
$47$ $$1 - 4 T + 49 T^{2} + 120 T^{3} + 2303 T^{4} - 8836 T^{5} + 103823 T^{6}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{3}$$
$59$ $$1 + 8 T + 169 T^{2} + 912 T^{3} + 9971 T^{4} + 27848 T^{5} + 205379 T^{6}$$
$61$ $$1 - 8 T + 87 T^{2} - 464 T^{3} + 5307 T^{4} - 29768 T^{5} + 226981 T^{6}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{3}$$
$71$ $$1 - 4 T + 101 T^{2} - 632 T^{3} + 7171 T^{4} - 20164 T^{5} + 357911 T^{6}$$
$73$ $$( 1 - 6 T + 73 T^{2} )^{3}$$
$79$ $$1 - 18 T + 317 T^{2} - 2908 T^{3} + 25043 T^{4} - 112338 T^{5} + 493039 T^{6}$$
$83$ $$1 + 16 T + 265 T^{2} + 2400 T^{3} + 21995 T^{4} + 110224 T^{5} + 571787 T^{6}$$
$89$ $$1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6}$$
$97$ $$1 - 18 T + 287 T^{2} - 3164 T^{3} + 27839 T^{4} - 169362 T^{5} + 912673 T^{6}$$