# Properties

 Label 3800.2.a.p Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $0$ 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: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{7} + 2 \beta q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{7} + 2 \beta q^{9} + \beta q^{11} + ( -1 + 2 \beta ) q^{13} + q^{17} - q^{19} + ( -3 - 2 \beta ) q^{21} + ( 1 + 3 \beta ) q^{23} + ( 1 - \beta ) q^{27} + ( 1 + 2 \beta ) q^{29} + ( 2 + \beta ) q^{31} + ( 2 + \beta ) q^{33} + ( 8 - 2 \beta ) q^{37} + ( 3 + \beta ) q^{39} -5 \beta q^{41} + ( -2 + 3 \beta ) q^{43} + 8 q^{47} + ( -4 + 2 \beta ) q^{49} + ( 1 + \beta ) q^{51} + ( 1 - 2 \beta ) q^{53} + ( -1 - \beta ) q^{57} + ( 13 + \beta ) q^{59} + ( 2 + \beta ) q^{61} + ( -4 - 2 \beta ) q^{63} + ( 1 - 5 \beta ) q^{67} + ( 7 + 4 \beta ) q^{69} + ( -4 + 5 \beta ) q^{71} + ( 11 + 2 \beta ) q^{73} + ( -2 - \beta ) q^{77} + ( 2 - 2 \beta ) q^{79} + ( -1 - 6 \beta ) q^{81} + ( 6 - 6 \beta ) q^{83} + ( 5 + 3 \beta ) q^{87} + ( -8 + 3 \beta ) q^{89} + ( -3 - \beta ) q^{91} + ( 4 + 3 \beta ) q^{93} + ( 2 + 4 \beta ) q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{7} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{7} - 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} + 16 q^{37} + 6 q^{39} - 4 q^{43} + 16 q^{47} - 8 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{57} + 26 q^{59} + 4 q^{61} - 8 q^{63} + 2 q^{67} + 14 q^{69} - 8 q^{71} + 22 q^{73} - 4 q^{77} + 4 q^{79} - 2 q^{81} + 12 q^{83} + 10 q^{87} - 16 q^{89} - 6 q^{91} + 8 q^{93} + 4 q^{97} + 8 q^{99} + O(q^{100})$$

## 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 −0.414214 0 0 0 0.414214 0 −2.82843 0
1.2 0 2.41421 0 0 0 −2.41421 0 2.82843 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.p 2
4.b odd 2 1 7600.2.a.x 2
5.b even 2 1 3800.2.a.l 2
5.c odd 4 2 760.2.d.c 4
20.d odd 2 1 7600.2.a.bc 2
20.e even 4 2 1520.2.d.d 4

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

 $$T_{3}^{2} - 2 T_{3} - 1$$ $$T_{7}^{2} + 2 T_{7} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + 2 T + T^{2}$$
$11$ $$-2 + T^{2}$$
$13$ $$-7 + 2 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-17 - 2 T + T^{2}$$
$29$ $$-7 - 2 T + T^{2}$$
$31$ $$2 - 4 T + T^{2}$$
$37$ $$56 - 16 T + T^{2}$$
$41$ $$-50 + T^{2}$$
$43$ $$-14 + 4 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$-7 - 2 T + T^{2}$$
$59$ $$167 - 26 T + T^{2}$$
$61$ $$2 - 4 T + T^{2}$$
$67$ $$-49 - 2 T + T^{2}$$
$71$ $$-34 + 8 T + T^{2}$$
$73$ $$113 - 22 T + T^{2}$$
$79$ $$-4 - 4 T + T^{2}$$
$83$ $$-36 - 12 T + T^{2}$$
$89$ $$46 + 16 T + T^{2}$$
$97$ $$-28 - 4 T + T^{2}$$