# Properties

 Label 7440.2.a.v Level $7440$ Weight $2$ Character orbit 7440.a Self dual yes Analytic conductor $59.409$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7440.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4086991038$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
0 1.00000 0 −1.00000 0 3.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$31$$ $$1$$

## 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 7440.2.a.v 1
4.b odd 2 1 930.2.a.a 1
12.b even 2 1 2790.2.a.y 1
20.d odd 2 1 4650.2.a.bv 1
20.e even 4 2 4650.2.d.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.a 1 4.b odd 2 1
2790.2.a.y 1 12.b even 2 1
4650.2.a.bv 1 20.d odd 2 1
4650.2.d.y 2 20.e even 4 2
7440.2.a.v 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7440))$$:

 $$T_{7} - 3$$ $$T_{11} + 3$$ $$T_{13} + 2$$ $$T_{17} + 4$$ $$T_{19} - 3$$

## Hecke characteristic polynomials

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