# Properties

 Label 1520.2.a.o 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{3} + q^{5} + ( - 2 \beta + 2) q^{7} + (4 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 2) * q^3 + q^5 + (-2*b + 2) * q^7 + (4*b + 3) * q^9 $$q + (\beta + 2) q^{3} + q^{5} + ( - 2 \beta + 2) q^{7} + (4 \beta + 3) q^{9} + 2 q^{11} + ( - 3 \beta - 2) q^{13} + (\beta + 2) q^{15} + (2 \beta - 2) q^{17} + q^{19} - 2 \beta q^{21} + 6 q^{23} + q^{25} + (8 \beta + 8) q^{27} + ( - 6 \beta + 2) q^{29} + (2 \beta + 4) q^{31} + (2 \beta + 4) q^{33} + ( - 2 \beta + 2) q^{35} + (3 \beta - 6) q^{37} + ( - 8 \beta - 10) q^{39} + (4 \beta - 2) q^{41} + ( - 2 \beta - 2) q^{43} + (4 \beta + 3) q^{45} + (2 \beta + 2) q^{47} + ( - 8 \beta + 5) q^{49} + 2 \beta q^{51} + (5 \beta + 2) q^{53} + 2 q^{55} + (\beta + 2) q^{57} + ( - 4 \beta - 8) q^{59} + ( - 4 \beta - 8) q^{61} + (2 \beta - 10) q^{63} + ( - 3 \beta - 2) q^{65} + (\beta + 2) q^{67} + (6 \beta + 12) q^{69} + (2 \beta - 8) q^{71} + ( - 6 \beta + 6) q^{73} + (\beta + 2) q^{75} + ( - 4 \beta + 4) q^{77} + ( - 4 \beta + 4) q^{79} + (12 \beta + 23) q^{81} + (8 \beta + 2) q^{83} + (2 \beta - 2) q^{85} + ( - 10 \beta - 8) q^{87} + ( - 6 \beta + 2) q^{89} + ( - 2 \beta + 8) q^{91} + (8 \beta + 12) q^{93} + q^{95} + ( - 3 \beta - 6) q^{97} + (8 \beta + 6) q^{99} +O(q^{100})$$ q + (b + 2) * q^3 + q^5 + (-2*b + 2) * q^7 + (4*b + 3) * q^9 + 2 * q^11 + (-3*b - 2) * q^13 + (b + 2) * q^15 + (2*b - 2) * q^17 + q^19 - 2*b * q^21 + 6 * q^23 + q^25 + (8*b + 8) * q^27 + (-6*b + 2) * q^29 + (2*b + 4) * q^31 + (2*b + 4) * q^33 + (-2*b + 2) * q^35 + (3*b - 6) * q^37 + (-8*b - 10) * q^39 + (4*b - 2) * q^41 + (-2*b - 2) * q^43 + (4*b + 3) * q^45 + (2*b + 2) * q^47 + (-8*b + 5) * q^49 + 2*b * q^51 + (5*b + 2) * q^53 + 2 * q^55 + (b + 2) * q^57 + (-4*b - 8) * q^59 + (-4*b - 8) * q^61 + (2*b - 10) * q^63 + (-3*b - 2) * q^65 + (b + 2) * q^67 + (6*b + 12) * q^69 + (2*b - 8) * q^71 + (-6*b + 6) * q^73 + (b + 2) * q^75 + (-4*b + 4) * q^77 + (-4*b + 4) * q^79 + (12*b + 23) * q^81 + (8*b + 2) * q^83 + (2*b - 2) * q^85 + (-10*b - 8) * q^87 + (-6*b + 2) * q^89 + (-2*b + 8) * q^91 + (8*b + 12) * q^93 + q^95 + (-3*b - 6) * q^97 + (8*b + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9 $$2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} - 4 q^{17} + 2 q^{19} + 12 q^{23} + 2 q^{25} + 16 q^{27} + 4 q^{29} + 8 q^{31} + 8 q^{33} + 4 q^{35} - 12 q^{37} - 20 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{45} + 4 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{55} + 4 q^{57} - 16 q^{59} - 16 q^{61} - 20 q^{63} - 4 q^{65} + 4 q^{67} + 24 q^{69} - 16 q^{71} + 12 q^{73} + 4 q^{75} + 8 q^{77} + 8 q^{79} + 46 q^{81} + 4 q^{83} - 4 q^{85} - 16 q^{87} + 4 q^{89} + 16 q^{91} + 24 q^{93} + 2 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 - 4 * q^17 + 2 * q^19 + 12 * q^23 + 2 * q^25 + 16 * q^27 + 4 * q^29 + 8 * q^31 + 8 * q^33 + 4 * q^35 - 12 * q^37 - 20 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^45 + 4 * q^47 + 10 * q^49 + 4 * q^53 + 4 * q^55 + 4 * q^57 - 16 * q^59 - 16 * q^61 - 20 * q^63 - 4 * q^65 + 4 * q^67 + 24 * q^69 - 16 * q^71 + 12 * q^73 + 4 * q^75 + 8 * q^77 + 8 * q^79 + 46 * q^81 + 4 * q^83 - 4 * q^85 - 16 * q^87 + 4 * q^89 + 16 * q^91 + 24 * 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
 −1.41421 1.41421
0 0.585786 0 1.00000 0 4.82843 0 −2.65685 0
1.2 0 3.41421 0 1.00000 0 −0.828427 0 8.65685 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.o 2
4.b odd 2 1 380.2.a.c 2
5.b even 2 1 7600.2.a.u 2
8.b even 2 1 6080.2.a.y 2
8.d odd 2 1 6080.2.a.bl 2
12.b even 2 1 3420.2.a.g 2
20.d odd 2 1 1900.2.a.e 2
20.e even 4 2 1900.2.c.d 4
76.d even 2 1 7220.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.c 2 4.b odd 2 1
1520.2.a.o 2 1.a even 1 1 trivial
1900.2.a.e 2 20.d odd 2 1
1900.2.c.d 4 20.e even 4 2
3420.2.a.g 2 12.b even 2 1
6080.2.a.y 2 8.b even 2 1
6080.2.a.bl 2 8.d odd 2 1
7220.2.a.m 2 76.d even 2 1
7600.2.a.u 2 5.b even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 4T + 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 4T - 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 4T - 14$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$(T - 1)^{2}$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 4T - 68$$
$31$ $$T^{2} - 8T + 8$$
$37$ $$T^{2} + 12T + 18$$
$41$ $$T^{2} + 4T - 28$$
$43$ $$T^{2} + 4T - 4$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} - 4T - 46$$
$59$ $$T^{2} + 16T + 32$$
$61$ $$T^{2} + 16T + 32$$
$67$ $$T^{2} - 4T + 2$$
$71$ $$T^{2} + 16T + 56$$
$73$ $$T^{2} - 12T - 36$$
$79$ $$T^{2} - 8T - 16$$
$83$ $$T^{2} - 4T - 124$$
$89$ $$T^{2} - 4T - 68$$
$97$ $$T^{2} + 12T + 18$$