# Properties

 Label 3648.2.a.bo Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.1294266574$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1824) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 + \beta ) q^{5} + ( -1 + \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 + \beta ) q^{5} + ( -1 + \beta ) q^{7} + q^{9} + ( 1 + \beta ) q^{11} + ( 4 - 2 \beta ) q^{13} + ( -1 - \beta ) q^{15} + ( 1 + \beta ) q^{17} - q^{19} + ( 1 - \beta ) q^{21} -2 \beta q^{23} + 3 \beta q^{25} - q^{27} + 4 q^{29} + ( -2 + 2 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( 3 + \beta ) q^{35} + ( 4 + 2 \beta ) q^{37} + ( -4 + 2 \beta ) q^{39} -4 q^{41} + ( 1 + 3 \beta ) q^{43} + ( 1 + \beta ) q^{45} + ( -7 + \beta ) q^{47} + ( -2 - \beta ) q^{49} + ( -1 - \beta ) q^{51} + ( 2 + 2 \beta ) q^{53} + ( 5 + 3 \beta ) q^{55} + q^{57} + ( 2 - 2 \beta ) q^{59} + ( 9 + \beta ) q^{61} + ( -1 + \beta ) q^{63} -4 q^{65} + ( -10 - 2 \beta ) q^{67} + 2 \beta q^{69} + ( 2 - 6 \beta ) q^{71} + ( 1 + \beta ) q^{73} -3 \beta q^{75} + ( 3 + \beta ) q^{77} + ( 2 - 2 \beta ) q^{79} + q^{81} + ( 8 - 6 \beta ) q^{83} + ( 5 + 3 \beta ) q^{85} -4 q^{87} + ( 2 + 2 \beta ) q^{89} + ( -12 + 4 \beta ) q^{91} + ( 2 - 2 \beta ) q^{93} + ( -1 - \beta ) q^{95} + ( -4 - 6 \beta ) q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 3q^{5} - q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 3q^{5} - q^{7} + 2q^{9} + 3q^{11} + 6q^{13} - 3q^{15} + 3q^{17} - 2q^{19} + q^{21} - 2q^{23} + 3q^{25} - 2q^{27} + 8q^{29} - 2q^{31} - 3q^{33} + 7q^{35} + 10q^{37} - 6q^{39} - 8q^{41} + 5q^{43} + 3q^{45} - 13q^{47} - 5q^{49} - 3q^{51} + 6q^{53} + 13q^{55} + 2q^{57} + 2q^{59} + 19q^{61} - q^{63} - 8q^{65} - 22q^{67} + 2q^{69} - 2q^{71} + 3q^{73} - 3q^{75} + 7q^{77} + 2q^{79} + 2q^{81} + 10q^{83} + 13q^{85} - 8q^{87} + 6q^{89} - 20q^{91} + 2q^{93} - 3q^{95} - 14q^{97} + 3q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
0 −1.00000 0 −0.561553 0 −2.56155 0 1.00000 0
1.2 0 −1.00000 0 3.56155 0 1.56155 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$19$$ $$1$$

## 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 3648.2.a.bo 2
4.b odd 2 1 3648.2.a.bv 2
8.b even 2 1 1824.2.a.p yes 2
8.d odd 2 1 1824.2.a.n 2
24.f even 2 1 5472.2.a.bk 2
24.h odd 2 1 5472.2.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.n 2 8.d odd 2 1
1824.2.a.p yes 2 8.b even 2 1
3648.2.a.bo 2 1.a even 1 1 trivial
3648.2.a.bv 2 4.b odd 2 1
5472.2.a.bj 2 24.h odd 2 1
5472.2.a.bk 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3648))$$:

 $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{7}^{2} + T_{7} - 4$$ $$T_{11}^{2} - 3 T_{11} - 2$$ $$T_{23}^{2} + 2 T_{23} - 16$$ $$T_{31}^{2} + 2 T_{31} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 - 3 T + T^{2}$$
$7$ $$-4 + T + T^{2}$$
$11$ $$-2 - 3 T + T^{2}$$
$13$ $$-8 - 6 T + T^{2}$$
$17$ $$-2 - 3 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-16 + 2 T + T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$-16 + 2 T + T^{2}$$
$37$ $$8 - 10 T + T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$-32 - 5 T + T^{2}$$
$47$ $$38 + 13 T + T^{2}$$
$53$ $$-8 - 6 T + T^{2}$$
$59$ $$-16 - 2 T + T^{2}$$
$61$ $$86 - 19 T + T^{2}$$
$67$ $$104 + 22 T + T^{2}$$
$71$ $$-152 + 2 T + T^{2}$$
$73$ $$-2 - 3 T + T^{2}$$
$79$ $$-16 - 2 T + T^{2}$$
$83$ $$-128 - 10 T + T^{2}$$
$89$ $$-8 - 6 T + T^{2}$$
$97$ $$-104 + 14 T + T^{2}$$