# Properties

 Label 1520.2.a.t Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + 2) q^{9}+O(q^{10})$$ q + b2 * q^3 - q^5 + (b1 - 1) * q^7 + (-b3 + 2) * q^9 $$q + \beta_{2} q^{3} - q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + 2) q^{9} + ( - \beta_1 - 1) q^{11} + (\beta_{2} + \beta_1 + 1) q^{13} - \beta_{2} q^{15} + (2 \beta_{2} + 2) q^{17} - q^{19} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{21} + (2 \beta_{2} + \beta_1 + 3) q^{23} + q^{25} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{27} + (2 \beta_{2} + 2) q^{29} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{31} + ( - \beta_{3} + \beta_1 + 2) q^{33} + ( - \beta_1 + 1) q^{35} + (\beta_{3} + \beta_{2} - 1) q^{37} + ( - \beta_1 + 3) q^{39} + (\beta_{3} - \beta_1 + 4) q^{41} + (\beta_1 - 1) q^{43} + (\beta_{3} - 2) q^{45} + (2 \beta_{3} - \beta_1 + 3) q^{47} + (2 \beta_{3} - 2 \beta_1 + 5) q^{49} + ( - 2 \beta_{3} + 2 \beta_{2} + 10) q^{51} + ( - \beta_{3} - \beta_{2} - 3) q^{53} + (\beta_1 + 1) q^{55} - \beta_{2} q^{57} + ( - \beta_{3} - \beta_1) q^{59} + ( - \beta_{3} + 2 \beta_1 + 5) q^{61} + ( - 4 \beta_{2} - \beta_1 - 7) q^{63} + ( - \beta_{2} - \beta_1 - 1) q^{65} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{67} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 8) q^{69} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + 6) q^{73} + \beta_{2} q^{75} + ( - 2 \beta_{3} - 10) q^{77} + ( - \beta_{3} - \beta_1 + 4) q^{79} + ( - \beta_{3} + 2 \beta_1 + 4) q^{81} + ( - 2 \beta_{2} + \beta_1 - 1) q^{83} + ( - 2 \beta_{2} - 2) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2} + 10) q^{87} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{89} + (3 \beta_{3} - 2 \beta_{2} - \beta_1 + 8) q^{91} + (4 \beta_{3} - 2 \beta_1 - 10) q^{93} + q^{95} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{97} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 3) q^{99}+O(q^{100})$$ q + b2 * q^3 - q^5 + (b1 - 1) * q^7 + (-b3 + 2) * q^9 + (-b1 - 1) * q^11 + (b2 + b1 + 1) * q^13 - b2 * q^15 + (2*b2 + 2) * q^17 - q^19 + (b3 - 2*b2 - b1 - 2) * q^21 + (2*b2 + b1 + 3) * q^23 + q^25 + (b3 + 2*b2 - b1 + 2) * q^27 + (2*b2 + 2) * q^29 + (-b3 - 2*b2 + b1 - 2) * q^31 + (-b3 + b1 + 2) * q^33 + (-b1 + 1) * q^35 + (b3 + b2 - 1) * q^37 + (-b1 + 3) * q^39 + (b3 - b1 + 4) * q^41 + (b1 - 1) * q^43 + (b3 - 2) * q^45 + (2*b3 - b1 + 3) * q^47 + (2*b3 - 2*b1 + 5) * q^49 + (-2*b3 + 2*b2 + 10) * q^51 + (-b3 - b2 - 3) * q^53 + (b1 + 1) * q^55 - b2 * q^57 + (-b3 - b1) * q^59 + (-b3 + 2*b1 + 5) * q^61 + (-4*b2 - b1 - 7) * q^63 + (-b2 - b1 - 1) * q^65 + (-b3 + 3*b2 + b1 + 6) * q^67 + (-b3 + 2*b2 - b1 + 8) * q^69 + (b3 - 2*b2 + b1 + 4) * q^71 + (-2*b2 + 6) * q^73 + b2 * q^75 + (-2*b3 - 10) * q^77 + (-b3 - b1 + 4) * q^79 + (-b3 + 2*b1 + 4) * q^81 + (-2*b2 + b1 - 1) * q^83 + (-2*b2 - 2) * q^85 + (-2*b3 + 2*b2 + 10) * q^87 + (-2*b3 - 2*b2) * q^89 + (3*b3 - 2*b2 - b1 + 8) * q^91 + (4*b3 - 2*b1 - 10) * q^93 + q^95 + (-b3 - b2 + 2*b1 + 7) * q^97 + (2*b3 + 4*b2 + b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 $$4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 - 6 * q^37 + 12 * q^39 + 16 * q^41 - 4 * q^43 - 8 * q^45 + 12 * q^47 + 20 * q^49 + 36 * q^51 - 10 * q^53 + 4 * q^55 + 2 * q^57 + 20 * q^61 - 20 * q^63 - 2 * q^65 + 18 * q^67 + 28 * q^69 + 20 * q^71 + 28 * q^73 - 2 * q^75 - 40 * q^77 + 16 * q^79 + 16 * q^81 - 4 * q^85 + 36 * q^87 + 4 * q^89 + 36 * q^91 - 40 * q^93 + 4 * q^95 + 30 * q^97 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 2$$ v^3 - 2*v^2 - 3*v + 2 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 2\nu - 5$$ 2*v^2 - 2*v - 5
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta _1 + 6 ) / 2$$ (b3 + b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 11 ) / 2$$ (2*b3 + 2*b2 + 5*b1 + 11) / 2

## 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
 1.28734 −1.51658 2.78165 −0.552409
0 −3.04306 0 −1.00000 0 0.574672 0 6.26020 0
1.2 0 −1.53844 0 −1.00000 0 −5.03316 0 −0.633188 0
1.3 0 −0.296842 0 −1.00000 0 3.56331 0 −2.91188 0
1.4 0 2.87834 0 −1.00000 0 −3.10482 0 5.28487 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.t 4
4.b odd 2 1 95.2.a.b 4
5.b even 2 1 7600.2.a.cf 4
8.b even 2 1 6080.2.a.ch 4
8.d odd 2 1 6080.2.a.cc 4
12.b even 2 1 855.2.a.m 4
20.d odd 2 1 475.2.a.i 4
20.e even 4 2 475.2.b.e 8
28.d even 2 1 4655.2.a.y 4
60.h even 2 1 4275.2.a.bo 4
76.d even 2 1 1805.2.a.p 4
380.d even 2 1 9025.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 4.b odd 2 1
475.2.a.i 4 20.d odd 2 1
475.2.b.e 8 20.e even 4 2
855.2.a.m 4 12.b even 2 1
1520.2.a.t 4 1.a even 1 1 trivial
1805.2.a.p 4 76.d even 2 1
4275.2.a.bo 4 60.h even 2 1
4655.2.a.y 4 28.d even 2 1
6080.2.a.cc 4 8.d odd 2 1
6080.2.a.ch 4 8.b even 2 1
7600.2.a.cf 4 5.b even 2 1
9025.2.a.bf 4 380.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

 $$T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 16T_{3} - 4$$ T3^4 + 2*T3^3 - 8*T3^2 - 16*T3 - 4 $$T_{7}^{4} + 4T_{7}^{3} - 16T_{7}^{2} - 48T_{7} + 32$$ T7^4 + 4*T7^3 - 16*T7^2 - 48*T7 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} - 8 T^{2} - 16 T - 4$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32$$
$11$ $$T^{4} + 4 T^{3} - 16 T^{2} - 32 T + 48$$
$13$ $$T^{4} - 2 T^{3} - 24 T^{2} + 32 T + 20$$
$17$ $$T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} - 8 T^{3} - 24 T^{2} + 176 T + 288$$
$29$ $$T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48$$
$31$ $$T^{4} + 4 T^{3} - 80 T^{2} - 512 T - 640$$
$37$ $$T^{4} + 6 T^{3} - 24 T^{2} - 40 T + 4$$
$41$ $$T^{4} - 16 T^{3} + 56 T^{2} + \cdots - 240$$
$43$ $$T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32$$
$47$ $$T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 1056$$
$53$ $$T^{4} + 10 T^{3} - 184 T - 348$$
$59$ $$T^{4} - 64 T^{2} + 224 T - 192$$
$61$ $$T^{4} - 20 T^{3} + 56 T^{2} + \cdots - 2656$$
$67$ $$T^{4} - 18 T^{3} + 8 T^{2} + \cdots - 1076$$
$71$ $$T^{4} - 20 T^{3} + 32 T^{2} + \cdots - 4224$$
$73$ $$T^{4} - 28 T^{3} + 256 T^{2} + \cdots + 176$$
$79$ $$T^{4} - 16 T^{3} + 32 T^{2} + \cdots - 1856$$
$83$ $$T^{4} - 72 T^{2} + 112 T + 480$$
$89$ $$T^{4} - 4 T^{3} - 144 T^{2} + \cdots + 240$$
$97$ $$T^{4} - 30 T^{3} + 224 T^{2} + \cdots - 1388$$