# Properties

 Label 1520.2.a.d 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 $$q - q^{3} + q^{5} + q^{7} - 2 q^{9} - q^{13} - q^{15} - 3 q^{17} - q^{19} - q^{21} - 3 q^{23} + q^{25} + 5 q^{27} - 3 q^{29} - 2 q^{31} + q^{35} - 10 q^{37} + q^{39} + 6 q^{41} - 2 q^{43} - 2 q^{45} - 6 q^{49} + 3 q^{51} + 3 q^{53} + q^{57} - 3 q^{59} + 8 q^{61} - 2 q^{63} - q^{65} + 7 q^{67} + 3 q^{69} - 12 q^{71} - 13 q^{73} - q^{75} - 14 q^{79} + q^{81} - 6 q^{83} - 3 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} + 2 q^{93} - q^{95} - 10 q^{97}+O(q^{100})$$ q - q^3 + q^5 + q^7 - 2 * q^9 - q^13 - q^15 - 3 * q^17 - q^19 - q^21 - 3 * q^23 + q^25 + 5 * q^27 - 3 * q^29 - 2 * q^31 + q^35 - 10 * q^37 + q^39 + 6 * q^41 - 2 * q^43 - 2 * q^45 - 6 * q^49 + 3 * q^51 + 3 * q^53 + q^57 - 3 * q^59 + 8 * q^61 - 2 * q^63 - q^65 + 7 * q^67 + 3 * q^69 - 12 * q^71 - 13 * q^73 - q^75 - 14 * q^79 + q^81 - 6 * q^83 - 3 * q^85 + 3 * q^87 + 6 * q^89 - q^91 + 2 * q^93 - q^95 - 10 * 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.d 1
4.b odd 2 1 190.2.a.c 1
5.b even 2 1 7600.2.a.m 1
8.b even 2 1 6080.2.a.p 1
8.d odd 2 1 6080.2.a.h 1
12.b even 2 1 1710.2.a.d 1
20.d odd 2 1 950.2.a.a 1
20.e even 4 2 950.2.b.e 2
28.d even 2 1 9310.2.a.o 1
60.h even 2 1 8550.2.a.bd 1
76.d even 2 1 3610.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 4.b odd 2 1
950.2.a.a 1 20.d odd 2 1
950.2.b.e 2 20.e even 4 2
1520.2.a.d 1 1.a even 1 1 trivial
1710.2.a.d 1 12.b even 2 1
3610.2.a.b 1 76.d even 2 1
6080.2.a.h 1 8.d odd 2 1
6080.2.a.p 1 8.b even 2 1
7600.2.a.m 1 5.b even 2 1
8550.2.a.bd 1 60.h even 2 1
9310.2.a.o 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

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