# Properties

 Label 3800.2.a.s Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## 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
 −1.53209 −0.347296 1.87939
0 −1.53209 0 0 0 −2.22668 0 −0.652704 0
1.2 0 −0.347296 0 0 0 3.41147 0 −2.87939 0
1.3 0 1.87939 0 0 0 −1.18479 0 0.532089 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.s 3
4.b odd 2 1 7600.2.a.br 3
5.b even 2 1 3800.2.a.t yes 3
5.c odd 4 2 3800.2.d.o 6
20.d odd 2 1 7600.2.a.bs 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.s 3 1.a even 1 1 trivial
3800.2.a.t yes 3 5.b even 2 1
3800.2.d.o 6 5.c odd 4 2
7600.2.a.br 3 4.b odd 2 1
7600.2.a.bs 3 20.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{3} - 3 T_{3} - 1$$ $$T_{7}^{3} - 9 T_{7} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-1 - 3 T + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-9 - 9 T + T^{3}$$
$11$ $$-17 - 6 T + 3 T^{2} + T^{3}$$
$13$ $$17 - 18 T + 3 T^{2} + T^{3}$$
$17$ $$-1 + 9 T - 6 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-89 - 27 T + 6 T^{2} + T^{3}$$
$29$ $$51 - 36 T + 3 T^{2} + T^{3}$$
$31$ $$89 + 66 T + 15 T^{2} + T^{3}$$
$37$ $$37 - 30 T - 9 T^{2} + T^{3}$$
$41$ $$-109 - 12 T + 9 T^{2} + T^{3}$$
$43$ $$-19 - 9 T + 3 T^{2} + T^{3}$$
$47$ $$17 - 18 T + 3 T^{2} + T^{3}$$
$53$ $$-111 - 45 T + 3 T^{2} + T^{3}$$
$59$ $$296 - 72 T - 6 T^{2} + T^{3}$$
$61$ $$-233 - 66 T + 9 T^{2} + T^{3}$$
$67$ $$17 - 6 T - 3 T^{2} + T^{3}$$
$71$ $$-181 - 21 T + 9 T^{2} + T^{3}$$
$73$ $$9 + 18 T + 9 T^{2} + T^{3}$$
$79$ $$-71 + 21 T + 12 T^{2} + T^{3}$$
$83$ $$-289 - 102 T + 9 T^{2} + T^{3}$$
$89$ $$51 - 9 T - 6 T^{2} + T^{3}$$
$97$ $$-107 + 90 T - 21 T^{2} + T^{3}$$