# Properties

 Label 1520.2.a.g Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$12.1372611072$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^5 + q^7 - 2 * q^9 $$q + q^{3} - q^{5} + q^{7} - 2 q^{9} - 3 q^{13} - q^{15} - 7 q^{17} + q^{19} + q^{21} + 5 q^{23} + q^{25} - 5 q^{27} - 5 q^{29} - 10 q^{31} - q^{35} + 2 q^{37} - 3 q^{39} + 2 q^{41} - 6 q^{43} + 2 q^{45} - 6 q^{49} - 7 q^{51} + 9 q^{53} + q^{57} + 7 q^{59} - 4 q^{61} - 2 q^{63} + 3 q^{65} - 7 q^{67} + 5 q^{69} - 9 q^{73} + q^{75} + 10 q^{79} + q^{81} + 2 q^{83} + 7 q^{85} - 5 q^{87} - 10 q^{89} - 3 q^{91} - 10 q^{93} - q^{95} - 18 q^{97}+O(q^{100})$$ q + q^3 - q^5 + q^7 - 2 * q^9 - 3 * q^13 - q^15 - 7 * q^17 + q^19 + q^21 + 5 * q^23 + q^25 - 5 * q^27 - 5 * q^29 - 10 * q^31 - q^35 + 2 * q^37 - 3 * q^39 + 2 * q^41 - 6 * q^43 + 2 * q^45 - 6 * q^49 - 7 * q^51 + 9 * q^53 + q^57 + 7 * q^59 - 4 * q^61 - 2 * q^63 + 3 * q^65 - 7 * q^67 + 5 * q^69 - 9 * q^73 + q^75 + 10 * q^79 + q^81 + 2 * q^83 + 7 * q^85 - 5 * q^87 - 10 * q^89 - 3 * q^91 - 10 * q^93 - q^95 - 18 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 1.00000 0 −2.00000 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
$$2$$ $$-1$$
$$5$$ $$+1$$
$$19$$ $$-1$$

## 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 1520.2.a.g 1
4.b odd 2 1 190.2.a.a 1
5.b even 2 1 7600.2.a.g 1
8.b even 2 1 6080.2.a.i 1
8.d odd 2 1 6080.2.a.r 1
12.b even 2 1 1710.2.a.r 1
20.d odd 2 1 950.2.a.e 1
20.e even 4 2 950.2.b.d 2
28.d even 2 1 9310.2.a.i 1
60.h even 2 1 8550.2.a.l 1
76.d even 2 1 3610.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.a 1 4.b odd 2 1
950.2.a.e 1 20.d odd 2 1
950.2.b.d 2 20.e even 4 2
1520.2.a.g 1 1.a even 1 1 trivial
1710.2.a.r 1 12.b even 2 1
3610.2.a.h 1 76.d even 2 1
6080.2.a.i 1 8.b even 2 1
6080.2.a.r 1 8.d odd 2 1
7600.2.a.g 1 5.b even 2 1
8550.2.a.l 1 60.h even 2 1
9310.2.a.i 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 3$$
$17$ $$T + 7$$
$19$ $$T - 1$$
$23$ $$T - 5$$
$29$ $$T + 5$$
$31$ $$T + 10$$
$37$ $$T - 2$$
$41$ $$T - 2$$
$43$ $$T + 6$$
$47$ $$T$$
$53$ $$T - 9$$
$59$ $$T - 7$$
$61$ $$T + 4$$
$67$ $$T + 7$$
$71$ $$T$$
$73$ $$T + 9$$
$79$ $$T - 10$$
$83$ $$T - 2$$
$89$ $$T + 10$$
$97$ $$T + 18$$