# Properties

 Label 3800.2.a.y Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( -2 + 2 \beta_{1} ) q^{11} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( 4 + \beta_{2} ) q^{17} - q^{19} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{21} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{23} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{27} -3 \beta_{2} q^{29} + ( -4 + 2 \beta_{2} ) q^{31} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -4 - \beta_{1} + \beta_{2} ) q^{37} + ( -4 \beta_{1} + \beta_{2} ) q^{39} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{49} + ( -2 + 4 \beta_{1} - \beta_{2} ) q^{51} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{1} q^{57} + ( -2 + \beta_{2} ) q^{59} + ( 8 - 4 \beta_{1} ) q^{61} + ( 4 + 2 \beta_{1} ) q^{63} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 10 + 6 \beta_{1} + 3 \beta_{2} ) q^{69} + ( -4 + 4 \beta_{1} ) q^{71} + ( 4 + 4 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -8 + 4 \beta_{1} - 4 \beta_{2} ) q^{77} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{79} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( 8 - 2 \beta_{2} ) q^{83} + ( 6 + 3 \beta_{2} ) q^{87} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -14 + 2 \beta_{1} - 3 \beta_{2} ) q^{91} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 4 - 7 \beta_{1} - \beta_{2} ) q^{97} + ( 10 + 6 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 5 q^{7} + 4 q^{9} + O(q^{10})$$ $$3 q + q^{3} + 5 q^{7} + 4 q^{9} - 4 q^{11} - 5 q^{13} + 11 q^{17} - 3 q^{19} - 3 q^{21} + 9 q^{23} + 19 q^{27} + 3 q^{29} - 14 q^{31} + 24 q^{33} - 14 q^{37} - 5 q^{39} - 10 q^{41} + 10 q^{43} + 4 q^{49} - q^{51} + 7 q^{53} - q^{57} - 7 q^{59} + 20 q^{61} + 14 q^{63} + q^{67} + 33 q^{69} - 8 q^{71} + 13 q^{73} - 16 q^{77} + 14 q^{79} + 15 q^{81} + 26 q^{83} + 15 q^{87} + 18 q^{89} - 37 q^{91} - 14 q^{93} + 6 q^{97} + 36 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.76156 −0.363328 3.12489
0 −1.76156 0 0 0 4.62620 0 0.103084 0
1.2 0 −0.363328 0 0 0 −1.14134 0 −2.86799 0
1.3 0 3.12489 0 0 0 1.51514 0 6.76491 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.y 3
4.b odd 2 1 7600.2.a.bo 3
5.b even 2 1 760.2.a.h 3
5.c odd 4 2 3800.2.d.k 6
15.d odd 2 1 6840.2.a.bj 3
20.d odd 2 1 1520.2.a.r 3
40.e odd 2 1 6080.2.a.bs 3
40.f even 2 1 6080.2.a.bw 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 5.b even 2 1
1520.2.a.r 3 20.d odd 2 1
3800.2.a.y 3 1.a even 1 1 trivial
3800.2.d.k 6 5.c odd 4 2
6080.2.a.bs 3 40.e odd 2 1
6080.2.a.bw 3 40.f even 2 1
6840.2.a.bj 3 15.d odd 2 1
7600.2.a.bo 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

 $$T_{3}^{3} - T_{3}^{2} - 6 T_{3} - 2$$ $$T_{7}^{3} - 5 T_{7}^{2} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-2 - 6 T - T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$8 - 5 T^{2} + T^{3}$$
$11$ $$-64 - 20 T + 4 T^{2} + T^{3}$$
$13$ $$-106 - 22 T + 5 T^{2} + T^{3}$$
$17$ $$-20 + 32 T - 11 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$160 - 16 T - 9 T^{2} + T^{3}$$
$29$ $$108 - 72 T - 3 T^{2} + T^{3}$$
$31$ $$-64 + 32 T + 14 T^{2} + T^{3}$$
$37$ $$-20 + 46 T + 14 T^{2} + T^{3}$$
$41$ $$-472 - 44 T + 10 T^{2} + T^{3}$$
$43$ $$848 - 88 T - 10 T^{2} + T^{3}$$
$47$ $$-64 - 40 T + T^{3}$$
$53$ $$-2 - 14 T - 7 T^{2} + T^{3}$$
$59$ $$-8 + 8 T + 7 T^{2} + T^{3}$$
$61$ $$640 + 32 T - 20 T^{2} + T^{3}$$
$67$ $$-530 - 134 T - T^{2} + T^{3}$$
$71$ $$-512 - 80 T + 8 T^{2} + T^{3}$$
$73$ $$500 - 64 T - 13 T^{2} + T^{3}$$
$79$ $$-16 - 56 T - 14 T^{2} + T^{3}$$
$83$ $$-352 + 192 T - 26 T^{2} + T^{3}$$
$89$ $$-40 + 68 T - 18 T^{2} + T^{3}$$
$97$ $$2308 - 274 T - 6 T^{2} + T^{3}$$