Properties

 Label 3120.2.a.o Level $3120$ Weight $2$ Character orbit 3120.a Self dual yes Analytic conductor $24.913$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - 2q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{5} - 2q^{7} + q^{9} - 4q^{11} - q^{13} - q^{15} + 8q^{17} + 6q^{19} - 2q^{21} - 6q^{23} + q^{25} + q^{27} - 4q^{29} - 4q^{33} + 2q^{35} - 2q^{37} - q^{39} - 2q^{41} + 4q^{43} - q^{45} - 3q^{49} + 8q^{51} - 10q^{53} + 4q^{55} + 6q^{57} - 4q^{59} - 10q^{61} - 2q^{63} + q^{65} - 12q^{67} - 6q^{69} + 8q^{71} - 8q^{73} + q^{75} + 8q^{77} - 8q^{79} + q^{81} - 12q^{83} - 8q^{85} - 4q^{87} - 14q^{89} + 2q^{91} - 6q^{95} - 16q^{97} - 4q^{99} + O(q^{100})$$

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 −2.00000 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.o 1
3.b odd 2 1 9360.2.a.bh 1
4.b odd 2 1 390.2.a.e 1
12.b even 2 1 1170.2.a.e 1
20.d odd 2 1 1950.2.a.h 1
20.e even 4 2 1950.2.e.f 2
52.b odd 2 1 5070.2.a.e 1
52.f even 4 2 5070.2.b.e 2
60.h even 2 1 5850.2.a.bi 1
60.l odd 4 2 5850.2.e.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.e 1 4.b odd 2 1
1170.2.a.e 1 12.b even 2 1
1950.2.a.h 1 20.d odd 2 1
1950.2.e.f 2 20.e even 4 2
3120.2.a.o 1 1.a even 1 1 trivial
5070.2.a.e 1 52.b odd 2 1
5070.2.b.e 2 52.f even 4 2
5850.2.a.bi 1 60.h even 2 1
5850.2.e.i 2 60.l odd 4 2
9360.2.a.bh 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3120))$$:

 $$T_{7} + 2$$ $$T_{11} + 4$$ $$T_{17} - 8$$ $$T_{19} - 6$$ $$T_{31}$$

Hecke characteristic polynomials

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