# Properties

 Label 3800.2.a.m 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( -2 + 2 \beta ) q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( -2 + 2 \beta ) q^{7} - q^{9} + ( 2 - 2 \beta ) q^{11} + ( -2 - \beta ) q^{13} + 2 \beta q^{17} - q^{19} + ( 4 - 2 \beta ) q^{21} + ( 2 - 4 \beta ) q^{23} -4 \beta q^{27} + ( -2 - 4 \beta ) q^{29} + ( 4 - 2 \beta ) q^{31} + ( -4 + 2 \beta ) q^{33} + ( 2 + \beta ) q^{37} + ( -2 - 2 \beta ) q^{39} + ( -2 - 2 \beta ) q^{41} + ( -6 + 2 \beta ) q^{43} + ( -2 - 2 \beta ) q^{47} + ( 5 - 8 \beta ) q^{49} + 4 q^{51} + ( -2 + 7 \beta ) q^{53} -\beta q^{57} + ( -4 + 2 \beta ) q^{59} + 4 \beta q^{61} + ( 2 - 2 \beta ) q^{63} -7 \beta q^{67} + ( -8 + 2 \beta ) q^{69} + 6 \beta q^{71} + ( -4 - 2 \beta ) q^{73} + ( -12 + 8 \beta ) q^{77} -6 \beta q^{79} -5 q^{81} + ( 10 - 4 \beta ) q^{83} + ( -8 - 2 \beta ) q^{87} + ( -2 + 8 \beta ) q^{89} -2 \beta q^{91} + ( -4 + 4 \beta ) q^{93} + ( -6 - 5 \beta ) q^{97} + ( -2 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} - 2 q^{9} + O(q^{10})$$ $$2 q - 4 q^{7} - 2 q^{9} + 4 q^{11} - 4 q^{13} - 2 q^{19} + 8 q^{21} + 4 q^{23} - 4 q^{29} + 8 q^{31} - 8 q^{33} + 4 q^{37} - 4 q^{39} - 4 q^{41} - 12 q^{43} - 4 q^{47} + 10 q^{49} + 8 q^{51} - 4 q^{53} - 8 q^{59} + 4 q^{63} - 16 q^{69} - 8 q^{73} - 24 q^{77} - 10 q^{81} + 20 q^{83} - 16 q^{87} - 4 q^{89} - 8 q^{93} - 12 q^{97} - 4 q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −1.41421 0 0 0 −4.82843 0 −1.00000 0
1.2 0 1.41421 0 0 0 0.828427 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.m 2
4.b odd 2 1 7600.2.a.bb 2
5.b even 2 1 3800.2.a.o 2
5.c odd 4 2 760.2.d.d 4
20.d odd 2 1 7600.2.a.z 2
20.e even 4 2 1520.2.d.g 4

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

 $$T_{3}^{2} - 2$$ $$T_{7}^{2} + 4 T_{7} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-4 + 4 T + T^{2}$$
$11$ $$-4 - 4 T + T^{2}$$
$13$ $$2 + 4 T + T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-28 - 4 T + T^{2}$$
$29$ $$-28 + 4 T + T^{2}$$
$31$ $$8 - 8 T + T^{2}$$
$37$ $$2 - 4 T + T^{2}$$
$41$ $$-4 + 4 T + T^{2}$$
$43$ $$28 + 12 T + T^{2}$$
$47$ $$-4 + 4 T + T^{2}$$
$53$ $$-94 + 4 T + T^{2}$$
$59$ $$8 + 8 T + T^{2}$$
$61$ $$-32 + T^{2}$$
$67$ $$-98 + T^{2}$$
$71$ $$-72 + T^{2}$$
$73$ $$8 + 8 T + T^{2}$$
$79$ $$-72 + T^{2}$$
$83$ $$68 - 20 T + T^{2}$$
$89$ $$-124 + 4 T + T^{2}$$
$97$ $$-14 + 12 T + T^{2}$$