# Properties

 Label 3800.2.a.j Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.3431527681$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + ( 1 - 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + ( 1 - 2 \beta ) q^{9} + 2 q^{11} + ( -1 - \beta ) q^{13} + ( -4 + 2 \beta ) q^{17} + q^{19} + ( -2 - 2 \beta ) q^{23} -4 q^{27} + 2 \beta q^{29} + 4 q^{31} + ( -2 + 2 \beta ) q^{33} + ( 1 + 5 \beta ) q^{37} -2 q^{39} + ( 4 + 2 \beta ) q^{41} + ( -4 - 4 \beta ) q^{43} + ( -4 - 4 \beta ) q^{47} -7 q^{49} + ( 10 - 6 \beta ) q^{51} + ( 3 - 5 \beta ) q^{53} + ( -1 + \beta ) q^{57} + ( -2 - 2 \beta ) q^{59} + ( -10 - 2 \beta ) q^{61} + ( -1 - 3 \beta ) q^{67} -4 q^{69} + ( 4 - 4 \beta ) q^{71} -6 \beta q^{73} + ( 2 + 6 \beta ) q^{79} + ( 1 + 2 \beta ) q^{81} + ( 2 - 2 \beta ) q^{83} + ( 6 - 2 \beta ) q^{87} + 2 \beta q^{89} + ( -4 + 4 \beta ) q^{93} + ( 7 - 5 \beta ) q^{97} + ( 2 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 2 q^{19} - 4 q^{23} - 8 q^{27} + 8 q^{31} - 4 q^{33} + 2 q^{37} - 4 q^{39} + 8 q^{41} - 8 q^{43} - 8 q^{47} - 14 q^{49} + 20 q^{51} + 6 q^{53} - 2 q^{57} - 4 q^{59} - 20 q^{61} - 2 q^{67} - 8 q^{69} + 8 q^{71} + 4 q^{79} + 2 q^{81} + 4 q^{83} + 12 q^{87} - 8 q^{93} + 14 q^{97} + 4 q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −2.73205 0 0 0 0 0 4.46410 0
1.2 0 0.732051 0 0 0 0 0 −2.46410 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 3800.2.a.j 2
4.b odd 2 1 7600.2.a.bd 2
5.b even 2 1 760.2.a.g 2
5.c odd 4 2 3800.2.d.h 4
15.d odd 2 1 6840.2.a.y 2
20.d odd 2 1 1520.2.a.k 2
40.e odd 2 1 6080.2.a.bk 2
40.f even 2 1 6080.2.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.g 2 5.b even 2 1
1520.2.a.k 2 20.d odd 2 1
3800.2.a.j 2 1.a even 1 1 trivial
3800.2.d.h 4 5.c odd 4 2
6080.2.a.ba 2 40.f even 2 1
6080.2.a.bk 2 40.e odd 2 1
6840.2.a.y 2 15.d odd 2 1
7600.2.a.bd 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

 $$T_{3}^{2} + 2 T_{3} - 2$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-2 + 2 T + T^{2}$$
$17$ $$4 + 8 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-8 + 4 T + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$-74 - 2 T + T^{2}$$
$41$ $$4 - 8 T + T^{2}$$
$43$ $$-32 + 8 T + T^{2}$$
$47$ $$-32 + 8 T + T^{2}$$
$53$ $$-66 - 6 T + T^{2}$$
$59$ $$-8 + 4 T + T^{2}$$
$61$ $$88 + 20 T + T^{2}$$
$67$ $$-26 + 2 T + T^{2}$$
$71$ $$-32 - 8 T + T^{2}$$
$73$ $$-108 + T^{2}$$
$79$ $$-104 - 4 T + T^{2}$$
$83$ $$-8 - 4 T + T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$-26 - 14 T + T^{2}$$