# Properties

 Label 1520.2.a.n Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} + \beta q^{7} + (\beta + 1) q^{9}+O(q^{10})$$ q + b * q^3 + q^5 + b * q^7 + (b + 1) * q^9 $$q + \beta q^{3} + q^{5} + \beta q^{7} + (\beta + 1) q^{9} - 4 q^{11} + (3 \beta - 2) q^{13} + \beta q^{15} + ( - \beta + 6) q^{17} + q^{19} + (\beta + 4) q^{21} - 3 \beta q^{23} + q^{25} + ( - \beta + 4) q^{27} + ( - 3 \beta + 2) q^{29} + 2 \beta q^{31} - 4 \beta q^{33} + \beta q^{35} - 6 q^{37} + (\beta + 12) q^{39} + (4 \beta + 2) q^{41} + ( - 2 \beta + 8) q^{43} + (\beta + 1) q^{45} + ( - 4 \beta + 4) q^{47} + (\beta - 3) q^{49} + (5 \beta - 4) q^{51} + ( - \beta - 2) q^{53} - 4 q^{55} + \beta q^{57} - \beta q^{59} + (2 \beta + 6) q^{61} + (2 \beta + 4) q^{63} + (3 \beta - 2) q^{65} + \beta q^{67} + ( - 3 \beta - 12) q^{69} - 4 \beta q^{71} + ( - 3 \beta + 6) q^{73} + \beta q^{75} - 4 \beta q^{77} + 2 \beta q^{79} - 7 q^{81} + (2 \beta - 8) q^{83} + ( - \beta + 6) q^{85} + ( - \beta - 12) q^{87} + 2 q^{89} + (\beta + 12) q^{91} + (2 \beta + 8) q^{93} + q^{95} + 6 q^{97} + ( - 4 \beta - 4) q^{99} +O(q^{100})$$ q + b * q^3 + q^5 + b * q^7 + (b + 1) * q^9 - 4 * q^11 + (3*b - 2) * q^13 + b * q^15 + (-b + 6) * q^17 + q^19 + (b + 4) * q^21 - 3*b * q^23 + q^25 + (-b + 4) * q^27 + (-3*b + 2) * q^29 + 2*b * q^31 - 4*b * q^33 + b * q^35 - 6 * q^37 + (b + 12) * q^39 + (4*b + 2) * q^41 + (-2*b + 8) * q^43 + (b + 1) * q^45 + (-4*b + 4) * q^47 + (b - 3) * q^49 + (5*b - 4) * q^51 + (-b - 2) * q^53 - 4 * q^55 + b * q^57 - b * q^59 + (2*b + 6) * q^61 + (2*b + 4) * q^63 + (3*b - 2) * q^65 + b * q^67 + (-3*b - 12) * q^69 - 4*b * q^71 + (-3*b + 6) * q^73 + b * q^75 - 4*b * q^77 + 2*b * q^79 - 7 * q^81 + (2*b - 8) * q^83 + (-b + 6) * q^85 + (-b - 12) * q^87 + 2 * q^89 + (b + 12) * q^91 + (2*b + 8) * q^93 + q^95 + 6 * q^97 + (-4*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 $$2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 - 8 * q^11 - q^13 + q^15 + 11 * q^17 + 2 * q^19 + 9 * q^21 - 3 * q^23 + 2 * q^25 + 7 * q^27 + q^29 + 2 * q^31 - 4 * q^33 + q^35 - 12 * q^37 + 25 * q^39 + 8 * q^41 + 14 * q^43 + 3 * q^45 + 4 * q^47 - 5 * q^49 - 3 * q^51 - 5 * q^53 - 8 * q^55 + q^57 - q^59 + 14 * q^61 + 10 * q^63 - q^65 + q^67 - 27 * q^69 - 4 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 2 * q^79 - 14 * q^81 - 14 * q^83 + 11 * q^85 - 25 * q^87 + 4 * q^89 + 25 * q^91 + 18 * q^93 + 2 * q^95 + 12 * q^97 - 12 * q^99

## 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
 −1.56155 2.56155
0 −1.56155 0 1.00000 0 −1.56155 0 −0.561553 0
1.2 0 2.56155 0 1.00000 0 2.56155 0 3.56155 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.n 2
4.b odd 2 1 190.2.a.d 2
5.b even 2 1 7600.2.a.y 2
8.b even 2 1 6080.2.a.bb 2
8.d odd 2 1 6080.2.a.bh 2
12.b even 2 1 1710.2.a.w 2
20.d odd 2 1 950.2.a.h 2
20.e even 4 2 950.2.b.f 4
28.d even 2 1 9310.2.a.bc 2
60.h even 2 1 8550.2.a.br 2
76.d even 2 1 3610.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 4.b odd 2 1
950.2.a.h 2 20.d odd 2 1
950.2.b.f 4 20.e even 4 2
1520.2.a.n 2 1.a even 1 1 trivial
1710.2.a.w 2 12.b even 2 1
3610.2.a.t 2 76.d even 2 1
6080.2.a.bb 2 8.b even 2 1
6080.2.a.bh 2 8.d odd 2 1
7600.2.a.y 2 5.b even 2 1
8550.2.a.br 2 60.h even 2 1
9310.2.a.bc 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{7}^{2} - T_{7} - 4$$ T7^2 - T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - T - 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + T - 38$$
$17$ $$T^{2} - 11T + 26$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 3T - 36$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - 2T - 16$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 8T - 52$$
$43$ $$T^{2} - 14T + 32$$
$47$ $$T^{2} - 4T - 64$$
$53$ $$T^{2} + 5T + 2$$
$59$ $$T^{2} + T - 4$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} - T - 4$$
$71$ $$T^{2} + 4T - 64$$
$73$ $$T^{2} - 9T - 18$$
$79$ $$T^{2} - 2T - 16$$
$83$ $$T^{2} + 14T + 32$$
$89$ $$(T - 2)^{2}$$
$97$ $$(T - 6)^{2}$$