# Properties

 Label 3800.2.a.g Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 2 q^{7} + q^{9} + O(q^{10})$$ $$q + 2 q^{3} - 2 q^{7} + q^{9} + 4 q^{11} - 8 q^{17} - q^{19} - 4 q^{21} - 6 q^{23} - 4 q^{27} + 2 q^{29} - 8 q^{31} + 8 q^{33} - 6 q^{41} - 10 q^{43} + 6 q^{47} - 3 q^{49} - 16 q^{51} - 2 q^{57} - 4 q^{59} + 6 q^{61} - 2 q^{63} - 2 q^{67} - 12 q^{69} + 16 q^{71} + 16 q^{73} - 8 q^{77} + 8 q^{79} - 11 q^{81} - 10 q^{83} + 4 q^{87} - 10 q^{89} - 16 q^{93} - 4 q^{97} + 4 q^{99} + O(q^{100})$$

## 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
 0
0 2.00000 0 0 0 −2.00000 0 1.00000 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.g 1
4.b odd 2 1 7600.2.a.e 1
5.b even 2 1 3800.2.a.c 1
5.c odd 4 2 760.2.d.a 2
20.d odd 2 1 7600.2.a.q 1
20.e even 4 2 1520.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.a 2 5.c odd 4 2
1520.2.d.a 2 20.e even 4 2
3800.2.a.c 1 5.b even 2 1
3800.2.a.g 1 1.a even 1 1 trivial
7600.2.a.e 1 4.b odd 2 1
7600.2.a.q 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$2 + T$$
$11$ $$-4 + T$$
$13$ $$T$$
$17$ $$8 + T$$
$19$ $$1 + T$$
$23$ $$6 + T$$
$29$ $$-2 + T$$
$31$ $$8 + T$$
$37$ $$T$$
$41$ $$6 + T$$
$43$ $$10 + T$$
$47$ $$-6 + T$$
$53$ $$T$$
$59$ $$4 + T$$
$61$ $$-6 + T$$
$67$ $$2 + T$$
$71$ $$-16 + T$$
$73$ $$-16 + T$$
$79$ $$-8 + T$$
$83$ $$10 + T$$
$89$ $$10 + T$$
$97$ $$4 + T$$