# Properties

 Label 3800.2.a.q 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} -2 \beta q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} -2 \beta q^{7} + ( 3 + 2 \beta ) q^{9} + ( -1 + \beta ) q^{13} + ( -2 + 2 \beta ) q^{17} + q^{19} + ( -10 - 2 \beta ) q^{21} + 2 q^{23} + ( 10 + 2 \beta ) q^{27} + 2 q^{29} + ( -6 - 2 \beta ) q^{31} + ( 7 + \beta ) q^{37} + 4 q^{39} + ( 8 + 2 \beta ) q^{41} + ( 4 + 2 \beta ) q^{43} + ( 8 - 2 \beta ) q^{47} + 13 q^{49} + 8 q^{51} + ( 9 - \beta ) q^{53} + ( 1 + \beta ) q^{57} + ( -6 + 2 \beta ) q^{59} + 2 \beta q^{61} + ( -20 - 6 \beta ) q^{63} + ( -1 - \beta ) q^{67} + ( 2 + 2 \beta ) q^{69} + ( -6 - 2 \beta ) q^{71} + ( 2 - 2 \beta ) q^{73} + ( -2 + 2 \beta ) q^{79} + ( 11 + 6 \beta ) q^{81} + ( -6 + 4 \beta ) q^{83} + ( 2 + 2 \beta ) q^{87} + ( 2 + 4 \beta ) q^{89} + ( -10 + 2 \beta ) q^{91} + ( -16 - 8 \beta ) q^{93} + ( -7 - 5 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 6 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{17} + 2 q^{19} - 20 q^{21} + 4 q^{23} + 20 q^{27} + 4 q^{29} - 12 q^{31} + 14 q^{37} + 8 q^{39} + 16 q^{41} + 8 q^{43} + 16 q^{47} + 26 q^{49} + 16 q^{51} + 18 q^{53} + 2 q^{57} - 12 q^{59} - 40 q^{63} - 2 q^{67} + 4 q^{69} - 12 q^{71} + 4 q^{73} - 4 q^{79} + 22 q^{81} - 12 q^{83} + 4 q^{87} + 4 q^{89} - 20 q^{91} - 32 q^{93} - 14 q^{97} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 −1.23607 0 0 0 4.47214 0 −1.47214 0
1.2 0 3.23607 0 0 0 −4.47214 0 7.47214 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.q 2
4.b odd 2 1 7600.2.a.w 2
5.b even 2 1 3800.2.a.k 2
5.c odd 4 2 760.2.d.b 4
20.d odd 2 1 7600.2.a.be 2
20.e even 4 2 1520.2.d.c 4

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

 $$T_{3}^{2} - 2 T_{3} - 4$$ $$T_{7}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-20 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-4 + 2 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( -2 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$16 + 12 T + T^{2}$$
$37$ $$44 - 14 T + T^{2}$$
$41$ $$44 - 16 T + T^{2}$$
$43$ $$-4 - 8 T + T^{2}$$
$47$ $$44 - 16 T + T^{2}$$
$53$ $$76 - 18 T + T^{2}$$
$59$ $$16 + 12 T + T^{2}$$
$61$ $$-20 + T^{2}$$
$67$ $$-4 + 2 T + T^{2}$$
$71$ $$16 + 12 T + T^{2}$$
$73$ $$-16 - 4 T + T^{2}$$
$79$ $$-16 + 4 T + T^{2}$$
$83$ $$-44 + 12 T + T^{2}$$
$89$ $$-76 - 4 T + T^{2}$$
$97$ $$-76 + 14 T + T^{2}$$