# Properties

 Label 3648.2.a.bl Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ Analytic rank $1$ 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: $$1$$ 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 456) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} -\beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -\beta q^{5} + \beta q^{7} + q^{9} + ( -4 + \beta ) q^{11} + 2 \beta q^{13} + \beta q^{15} + ( 2 - 3 \beta ) q^{17} - q^{19} -\beta q^{21} + ( 6 - 2 \beta ) q^{23} + ( -1 + \beta ) q^{25} - q^{27} + ( 2 - 4 \beta ) q^{29} -2 q^{31} + ( 4 - \beta ) q^{33} + ( -4 - \beta ) q^{35} + 8 q^{37} -2 \beta q^{39} + ( -2 + 2 \beta ) q^{41} -\beta q^{43} -\beta q^{45} + ( -2 + 3 \beta ) q^{47} + ( -3 + \beta ) q^{49} + ( -2 + 3 \beta ) q^{51} + ( 2 + 4 \beta ) q^{53} + ( -4 + 3 \beta ) q^{55} + q^{57} -12 q^{59} + ( -2 + 3 \beta ) q^{61} + \beta q^{63} + ( -8 - 2 \beta ) q^{65} + 4 \beta q^{67} + ( -6 + 2 \beta ) q^{69} + ( -6 + 7 \beta ) q^{73} + ( 1 - \beta ) q^{75} + ( 4 - 3 \beta ) q^{77} + ( 2 - 6 \beta ) q^{79} + q^{81} -4 q^{83} + ( 12 + \beta ) q^{85} + ( -2 + 4 \beta ) q^{87} -6 q^{89} + ( 8 + 2 \beta ) q^{91} + 2 q^{93} + \beta q^{95} + ( -2 - 4 \beta ) q^{97} + ( -4 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} + q^{15} + q^{17} - 2 q^{19} - q^{21} + 10 q^{23} - q^{25} - 2 q^{27} - 4 q^{31} + 7 q^{33} - 9 q^{35} + 16 q^{37} - 2 q^{39} - 2 q^{41} - q^{43} - q^{45} - q^{47} - 5 q^{49} - q^{51} + 8 q^{53} - 5 q^{55} + 2 q^{57} - 24 q^{59} - q^{61} + q^{63} - 18 q^{65} + 4 q^{67} - 10 q^{69} - 5 q^{73} + q^{75} + 5 q^{77} - 2 q^{79} + 2 q^{81} - 8 q^{83} + 25 q^{85} - 12 q^{89} + 18 q^{91} + 4 q^{93} + q^{95} - 8 q^{97} - 7 q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
0 −1.00000 0 −2.56155 0 2.56155 0 1.00000 0
1.2 0 −1.00000 0 1.56155 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.bl 2
4.b odd 2 1 3648.2.a.br 2
8.b even 2 1 456.2.a.f 2
8.d odd 2 1 912.2.a.m 2
24.f even 2 1 2736.2.a.z 2
24.h odd 2 1 1368.2.a.k 2
152.g odd 2 1 8664.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.f 2 8.b even 2 1
912.2.a.m 2 8.d odd 2 1
1368.2.a.k 2 24.h odd 2 1
2736.2.a.z 2 24.f even 2 1
3648.2.a.bl 2 1.a even 1 1 trivial
3648.2.a.br 2 4.b odd 2 1
8664.2.a.r 2 152.g odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} - 4$$ $$T_{7}^{2} - T_{7} - 4$$ $$T_{11}^{2} + 7 T_{11} + 8$$ $$T_{23}^{2} - 10 T_{23} + 8$$ $$T_{31} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-4 + T + T^{2}$$
$7$ $$-4 - T + T^{2}$$
$11$ $$8 + 7 T + T^{2}$$
$13$ $$-16 - 2 T + T^{2}$$
$17$ $$-38 - T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$8 - 10 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$( -8 + T )^{2}$$
$41$ $$-16 + 2 T + T^{2}$$
$43$ $$-4 + T + T^{2}$$
$47$ $$-38 + T + T^{2}$$
$53$ $$-52 - 8 T + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$-38 + T + T^{2}$$
$67$ $$-64 - 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$-202 + 5 T + T^{2}$$
$79$ $$-152 + 2 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$-52 + 8 T + T^{2}$$