# Properties

 Label 3648.2.a.bt 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: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{5} -\beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta q^{5} -\beta q^{7} + q^{9} + ( -4 + \beta ) q^{11} + 4 q^{13} + \beta q^{15} + ( -2 - \beta ) q^{17} - q^{19} -\beta q^{21} + ( 2 - 2 \beta ) q^{23} + ( 3 + \beta ) q^{25} + q^{27} + ( 2 + 2 \beta ) q^{29} + 6 q^{31} + ( -4 + \beta ) q^{33} + ( -8 - \beta ) q^{35} + 4 q^{37} + 4 q^{39} + ( 2 + 2 \beta ) q^{41} + ( 4 + \beta ) q^{43} + \beta q^{45} + ( -2 + 3 \beta ) q^{47} + ( 1 + \beta ) q^{49} + ( -2 - \beta ) q^{51} + 10 q^{53} + ( 8 - 3 \beta ) q^{55} - q^{57} + ( 4 - 2 \beta ) q^{59} + ( -6 - \beta ) q^{61} -\beta q^{63} + 4 \beta q^{65} -2 \beta q^{67} + ( 2 - 2 \beta ) q^{69} + ( 8 + 2 \beta ) q^{71} + ( 6 - \beta ) q^{73} + ( 3 + \beta ) q^{75} + ( -8 + 3 \beta ) q^{77} + 2 q^{79} + q^{81} + ( -8 - 3 \beta ) q^{85} + ( 2 + 2 \beta ) q^{87} + ( 6 - 4 \beta ) q^{89} -4 \beta q^{91} + 6 q^{93} -\beta q^{95} + ( 14 - 2 \beta ) q^{97} + ( -4 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + q^{5} - q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + q^{5} - q^{7} + 2q^{9} - 7q^{11} + 8q^{13} + q^{15} - 5q^{17} - 2q^{19} - q^{21} + 2q^{23} + 7q^{25} + 2q^{27} + 6q^{29} + 12q^{31} - 7q^{33} - 17q^{35} + 8q^{37} + 8q^{39} + 6q^{41} + 9q^{43} + q^{45} - q^{47} + 3q^{49} - 5q^{51} + 20q^{53} + 13q^{55} - 2q^{57} + 6q^{59} - 13q^{61} - q^{63} + 4q^{65} - 2q^{67} + 2q^{69} + 18q^{71} + 11q^{73} + 7q^{75} - 13q^{77} + 4q^{79} + 2q^{81} - 19q^{85} + 6q^{87} + 8q^{89} - 4q^{91} + 12q^{93} - q^{95} + 26q^{97} - 7q^{99} + O(q^{100})$$

## 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
 −2.37228 3.37228
0 1.00000 0 −2.37228 0 2.37228 0 1.00000 0
1.2 0 1.00000 0 3.37228 0 −3.37228 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.bt 2
4.b odd 2 1 3648.2.a.bm 2
8.b even 2 1 1824.2.a.o 2
8.d odd 2 1 1824.2.a.r yes 2
24.f even 2 1 5472.2.a.be 2
24.h odd 2 1 5472.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.o 2 8.b even 2 1
1824.2.a.r yes 2 8.d odd 2 1
3648.2.a.bm 2 4.b odd 2 1
3648.2.a.bt 2 1.a even 1 1 trivial
5472.2.a.bd 2 24.h odd 2 1
5472.2.a.be 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3648))$$:

 $$T_{5}^{2} - T_{5} - 8$$ $$T_{7}^{2} + T_{7} - 8$$ $$T_{11}^{2} + 7 T_{11} + 4$$ $$T_{23}^{2} - 2 T_{23} - 32$$ $$T_{31} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-8 - T + T^{2}$$
$7$ $$-8 + T + T^{2}$$
$11$ $$4 + 7 T + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$-2 + 5 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-32 - 2 T + T^{2}$$
$29$ $$-24 - 6 T + T^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$-24 - 6 T + T^{2}$$
$43$ $$12 - 9 T + T^{2}$$
$47$ $$-74 + T + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$-24 - 6 T + T^{2}$$
$61$ $$34 + 13 T + T^{2}$$
$67$ $$-32 + 2 T + T^{2}$$
$71$ $$48 - 18 T + T^{2}$$
$73$ $$22 - 11 T + T^{2}$$
$79$ $$( -2 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$-116 - 8 T + T^{2}$$
$97$ $$136 - 26 T + T^{2}$$