# Properties

 Label 3648.2.a.bk 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: $$\Q(\sqrt{33})$$ 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 228) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -1 - \beta ) q^{5} + ( 1 - \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -1 - \beta ) q^{5} + ( 1 - \beta ) q^{7} + q^{9} + ( 1 + \beta ) q^{11} -2 q^{13} + ( 1 + \beta ) q^{15} + ( -1 - \beta ) q^{17} - q^{19} + ( -1 + \beta ) q^{21} + ( -4 + 2 \beta ) q^{23} + ( 4 + 3 \beta ) q^{25} - q^{27} + ( 2 + 2 \beta ) q^{29} + ( -2 + 2 \beta ) q^{31} + ( -1 - \beta ) q^{33} + ( 7 + \beta ) q^{35} + 2 \beta q^{37} + 2 q^{39} + ( -1 + \beta ) q^{43} + ( -1 - \beta ) q^{45} + ( -11 + \beta ) q^{47} + ( 2 - \beta ) q^{49} + ( 1 + \beta ) q^{51} + ( -2 - 2 \beta ) q^{53} + ( -9 - 3 \beta ) q^{55} + q^{57} + ( 5 + \beta ) q^{61} + ( 1 - \beta ) q^{63} + ( 2 + 2 \beta ) q^{65} -4 \beta q^{67} + ( 4 - 2 \beta ) q^{69} -12 q^{71} + ( -3 + \beta ) q^{73} + ( -4 - 3 \beta ) q^{75} + ( -7 - \beta ) q^{77} + 8 q^{79} + q^{81} + ( -4 + 2 \beta ) q^{83} + ( 9 + 3 \beta ) q^{85} + ( -2 - 2 \beta ) q^{87} + ( 10 - 2 \beta ) q^{89} + ( -2 + 2 \beta ) q^{91} + ( 2 - 2 \beta ) q^{93} + ( 1 + \beta ) q^{95} + 14 q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 3q^{5} + q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 3q^{5} + q^{7} + 2q^{9} + 3q^{11} - 4q^{13} + 3q^{15} - 3q^{17} - 2q^{19} - q^{21} - 6q^{23} + 11q^{25} - 2q^{27} + 6q^{29} - 2q^{31} - 3q^{33} + 15q^{35} + 2q^{37} + 4q^{39} - q^{43} - 3q^{45} - 21q^{47} + 3q^{49} + 3q^{51} - 6q^{53} - 21q^{55} + 2q^{57} + 11q^{61} + q^{63} + 6q^{65} - 4q^{67} + 6q^{69} - 24q^{71} - 5q^{73} - 11q^{75} - 15q^{77} + 16q^{79} + 2q^{81} - 6q^{83} + 21q^{85} - 6q^{87} + 18q^{89} - 2q^{91} + 2q^{93} + 3q^{95} + 28q^{97} + 3q^{99} + O(q^{100})$$

## 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
 3.37228 −2.37228
0 −1.00000 0 −4.37228 0 −2.37228 0 1.00000 0
1.2 0 −1.00000 0 1.37228 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.bk 2
4.b odd 2 1 3648.2.a.bq 2
8.b even 2 1 228.2.a.c 2
8.d odd 2 1 912.2.a.n 2
24.f even 2 1 2736.2.a.y 2
24.h odd 2 1 684.2.a.d 2
40.f even 2 1 5700.2.a.t 2
40.i odd 4 2 5700.2.f.m 4
152.g odd 2 1 4332.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.a.c 2 8.b even 2 1
684.2.a.d 2 24.h odd 2 1
912.2.a.n 2 8.d odd 2 1
2736.2.a.y 2 24.f even 2 1
3648.2.a.bk 2 1.a even 1 1 trivial
3648.2.a.bq 2 4.b odd 2 1
4332.2.a.i 2 152.g odd 2 1
5700.2.a.t 2 40.f even 2 1
5700.2.f.m 4 40.i odd 4 2

## Hecke kernels

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

 $$T_{5}^{2} + 3 T_{5} - 6$$ $$T_{7}^{2} - T_{7} - 8$$ $$T_{11}^{2} - 3 T_{11} - 6$$ $$T_{23}^{2} + 6 T_{23} - 24$$ $$T_{31}^{2} + 2 T_{31} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-6 + 3 T + T^{2}$$
$7$ $$-8 - T + T^{2}$$
$11$ $$-6 - 3 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-6 + 3 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-24 + 6 T + T^{2}$$
$29$ $$-24 - 6 T + T^{2}$$
$31$ $$-32 + 2 T + T^{2}$$
$37$ $$-32 - 2 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$-8 + T + T^{2}$$
$47$ $$102 + 21 T + T^{2}$$
$53$ $$-24 + 6 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$22 - 11 T + T^{2}$$
$67$ $$-128 + 4 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$-2 + 5 T + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-24 + 6 T + T^{2}$$
$89$ $$48 - 18 T + T^{2}$$
$97$ $$( -14 + T )^{2}$$