# Properties

 Label 3648.2.a.r Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 3q^{5} + 5q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + 3q^{5} + 5q^{7} + q^{9} + q^{11} - 2q^{13} - 3q^{15} - q^{17} - q^{19} - 5q^{21} + 4q^{23} + 4q^{25} - q^{27} + 2q^{29} + 6q^{31} - q^{33} + 15q^{35} + 2q^{39} - q^{43} + 3q^{45} + 9q^{47} + 18q^{49} + q^{51} - 10q^{53} + 3q^{55} + q^{57} - 8q^{59} + q^{61} + 5q^{63} - 6q^{65} + 8q^{67} - 4q^{69} + 12q^{71} - 11q^{73} - 4q^{75} + 5q^{77} - 16q^{79} + q^{81} + 12q^{83} - 3q^{85} - 2q^{87} - 6q^{89} - 10q^{91} - 6q^{93} - 3q^{95} - 10q^{97} + q^{99} + O(q^{100})$$

## 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
 0
0 −1.00000 0 3.00000 0 5.00000 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.r 1
4.b odd 2 1 3648.2.a.bh 1
8.b even 2 1 912.2.a.g 1
8.d odd 2 1 57.2.a.a 1
24.f even 2 1 171.2.a.d 1
24.h odd 2 1 2736.2.a.v 1
40.e odd 2 1 1425.2.a.j 1
40.k even 4 2 1425.2.c.b 2
56.e even 2 1 2793.2.a.b 1
88.g even 2 1 6897.2.a.f 1
104.h odd 2 1 9633.2.a.o 1
120.m even 2 1 4275.2.a.b 1
152.b even 2 1 1083.2.a.e 1
168.e odd 2 1 8379.2.a.p 1
456.l odd 2 1 3249.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.a 1 8.d odd 2 1
171.2.a.d 1 24.f even 2 1
912.2.a.g 1 8.b even 2 1
1083.2.a.e 1 152.b even 2 1
1425.2.a.j 1 40.e odd 2 1
1425.2.c.b 2 40.k even 4 2
2736.2.a.v 1 24.h odd 2 1
2793.2.a.b 1 56.e even 2 1
3249.2.a.b 1 456.l odd 2 1
3648.2.a.r 1 1.a even 1 1 trivial
3648.2.a.bh 1 4.b odd 2 1
4275.2.a.b 1 120.m even 2 1
6897.2.a.f 1 88.g even 2 1
8379.2.a.p 1 168.e odd 2 1
9633.2.a.o 1 104.h odd 2 1

## Hecke kernels

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

 $$T_{5} - 3$$ $$T_{7} - 5$$ $$T_{11} - 1$$ $$T_{23} - 4$$ $$T_{31} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-3 + T$$
$7$ $$-5 + T$$
$11$ $$-1 + T$$
$13$ $$2 + T$$
$17$ $$1 + T$$
$19$ $$1 + T$$
$23$ $$-4 + T$$
$29$ $$-2 + T$$
$31$ $$-6 + T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$1 + T$$
$47$ $$-9 + T$$
$53$ $$10 + T$$
$59$ $$8 + T$$
$61$ $$-1 + T$$
$67$ $$-8 + T$$
$71$ $$-12 + T$$
$73$ $$11 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$6 + T$$
$97$ $$10 + T$$