# Properties

 Label 3120.2.a.a Level $3120$ Weight $2$ Character orbit 3120.a Self dual yes Analytic conductor $24.913$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.9133254306$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - 4q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - q^{5} - 4q^{7} + q^{9} - q^{13} + q^{15} - 2q^{17} - 4q^{19} + 4q^{21} - 8q^{23} + q^{25} - q^{27} + 2q^{29} + 8q^{31} + 4q^{35} + 2q^{37} + q^{39} - 6q^{41} - 12q^{43} - q^{45} + 9q^{49} + 2q^{51} + 10q^{53} + 4q^{57} - 10q^{61} - 4q^{63} + q^{65} + 4q^{67} + 8q^{69} + 16q^{71} - 6q^{73} - q^{75} + 8q^{79} + q^{81} + 4q^{83} + 2q^{85} - 2q^{87} - 14q^{89} + 4q^{91} - 8q^{93} + 4q^{95} - 6q^{97} + O(q^{100})$$

## 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 −4.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 3120.2.a.a 1
3.b odd 2 1 9360.2.a.bc 1
4.b odd 2 1 390.2.a.c 1
12.b even 2 1 1170.2.a.n 1
20.d odd 2 1 1950.2.a.n 1
20.e even 4 2 1950.2.e.e 2
52.b odd 2 1 5070.2.a.u 1
52.f even 4 2 5070.2.b.i 2
60.h even 2 1 5850.2.a.c 1
60.l odd 4 2 5850.2.e.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.c 1 4.b odd 2 1
1170.2.a.n 1 12.b even 2 1
1950.2.a.n 1 20.d odd 2 1
1950.2.e.e 2 20.e even 4 2
3120.2.a.a 1 1.a even 1 1 trivial
5070.2.a.u 1 52.b odd 2 1
5070.2.b.i 2 52.f even 4 2
5850.2.a.c 1 60.h even 2 1
5850.2.e.m 2 60.l odd 4 2
9360.2.a.bc 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3120))$$:

 $$T_{7} + 4$$ $$T_{11}$$ $$T_{17} + 2$$ $$T_{19} + 4$$ $$T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$4 + T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$8 + T$$
$29$ $$-2 + T$$
$31$ $$-8 + T$$
$37$ $$-2 + T$$
$41$ $$6 + T$$
$43$ $$12 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$-4 + T$$
$71$ $$-16 + T$$
$73$ $$6 + T$$
$79$ $$-8 + T$$
$83$ $$-4 + T$$
$89$ $$14 + T$$
$97$ $$6 + T$$