# Properties

 Label 2940.2.a.d Level $2940$ Weight $2$ Character orbit 2940.a Self dual yes Analytic conductor $23.476$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2940.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.4760181943$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - q^{15} - 4 q^{17} - q^{19} + 4 q^{23} + q^{25} - q^{27} - 5 q^{31} + 2 q^{33} - 5 q^{37} - q^{39} + 2 q^{41} - 9 q^{43} + q^{45} - 2 q^{47} + 4 q^{51} + 12 q^{53} - 2 q^{55} + q^{57} - 8 q^{59} - 14 q^{61} + q^{65} + 9 q^{67} - 4 q^{69} + 2 q^{71} + q^{73} - q^{75} - 3 q^{79} + q^{81} - 18 q^{83} - 4 q^{85} - 4 q^{89} + 5 q^{93} - q^{95} + 10 q^{97} - 2 q^{99} + O(q^{100})$$

## 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 1.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 2940.2.a.d 1
3.b odd 2 1 8820.2.a.j 1
7.b odd 2 1 2940.2.a.h 1
7.c even 3 2 420.2.q.a 2
7.d odd 6 2 2940.2.q.h 2
21.c even 2 1 8820.2.a.y 1
21.h odd 6 2 1260.2.s.d 2
28.g odd 6 2 1680.2.bg.a 2
35.j even 6 2 2100.2.q.a 2
35.l odd 12 4 2100.2.bc.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 7.c even 3 2
1260.2.s.d 2 21.h odd 6 2
1680.2.bg.a 2 28.g odd 6 2
2100.2.q.a 2 35.j even 6 2
2100.2.bc.c 4 35.l odd 12 4
2940.2.a.d 1 1.a even 1 1 trivial
2940.2.a.h 1 7.b odd 2 1
2940.2.q.h 2 7.d odd 6 2
8820.2.a.j 1 3.b odd 2 1
8820.2.a.y 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2940))$$:

 $$T_{11} + 2$$ $$T_{13} - 1$$ $$T_{17} + 4$$ $$T_{31} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$2 + T$$
$13$ $$-1 + T$$
$17$ $$4 + T$$
$19$ $$1 + T$$
$23$ $$-4 + T$$
$29$ $$T$$
$31$ $$5 + T$$
$37$ $$5 + T$$
$41$ $$-2 + T$$
$43$ $$9 + T$$
$47$ $$2 + T$$
$53$ $$-12 + T$$
$59$ $$8 + T$$
$61$ $$14 + T$$
$67$ $$-9 + T$$
$71$ $$-2 + T$$
$73$ $$-1 + T$$
$79$ $$3 + T$$
$83$ $$18 + T$$
$89$ $$4 + T$$
$97$ $$-10 + T$$