# Properties

 Label 1520.2.d.e Level $1520$ Weight $2$ Character orbit 1520.d Analytic conductor $12.137$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8}) q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{9}+O(q^{10})$$ q + (z^3 + z^2 + z) * q^3 + (-z^3 - 2*z) * q^5 + (z^3 - 3*z^2 + z) * q^7 + (2*z^3 - 2*z) * q^9 $$q + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{3} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8}) q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{9} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} + (2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 2 \zeta_{8}) q^{13} + ( - 2 \zeta_{8}^{3} - \zeta_{8}^{2} + \cdots + 3) q^{15}+ \cdots + 4 q^{99}+O(q^{100})$$ q + (z^3 + z^2 + z) * q^3 + (-z^3 - 2*z) * q^5 + (z^3 - 3*z^2 + z) * q^7 + (2*z^3 - 2*z) * q^9 + (z^3 - z) * q^11 + (2*z^3 - 3*z^2 + 2*z) * q^13 + (-2*z^3 - z^2 + z + 3) * q^15 + z^2 * q^17 - q^19 + (-2*z^3 + 2*z + 1) * q^21 + (3*z^3 + 5*z^2 + 3*z) * q^23 + (3*z^2 - 4) * q^25 + (z^3 - z^2 + z) * q^27 + (-2*z^3 + 2*z + 3) * q^29 + (-3*z^3 + 3*z - 2) * q^31 + (-z^3 - 2*z^2 - z) * q^33 + (6*z^3 - z^2 - 3*z + 3) * q^35 + (6*z^3 + 6*z) * q^37 + (-z^3 + z - 1) * q^39 + (-3*z^3 + 3*z) * q^41 + (3*z^3 + 6*z^2 + 3*z) * q^43 + (6*z^2 + 2) * q^45 + (-6*z^3 + 6*z - 4) * q^49 + (z^3 - z - 1) * q^51 + (6*z^3 + 3*z^2 + 6*z) * q^53 + (3*z^2 + 1) * q^55 + (-z^3 - z^2 - z) * q^57 + (7*z^3 - 7*z - 3) * q^59 + (3*z^3 - 3*z + 10) * q^61 + (6*z^3 - 4*z^2 + 6*z) * q^63 + (6*z^3 - 2*z^2 - 3*z + 6) * q^65 + (-3*z^3 - 9*z^2 - 3*z) * q^67 + (8*z^3 - 8*z - 11) * q^69 + (z^3 - z + 12) * q^71 + (6*z^3 - 3*z^2 + 6*z) * q^73 + (-z^3 - 4*z^2 - 7*z - 3) * q^75 + (3*z^3 - 2*z^2 + 3*z) * q^77 + (-6*z^3 + 6*z + 2) * q^79 + (6*z^3 - 6*z - 1) * q^81 + (-6*z^3 + 6*z^2 - 6*z) * q^83 + (-2*z^3 + z) * q^85 + (5*z^3 + 7*z^2 + 5*z) * q^87 + (-5*z^3 + 5*z) * q^89 + (-9*z^3 + 9*z - 13) * q^91 + (z^3 + 4*z^2 + z) * q^93 + (z^3 + 2*z) * q^95 + (-4*z^3 - 6*z^2 - 4*z) * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 12 q^{15} - 4 q^{19} + 4 q^{21} - 16 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} - 4 q^{39} + 8 q^{45} - 16 q^{49} - 4 q^{51} + 4 q^{55} - 12 q^{59} + 40 q^{61} + 24 q^{65} - 44 q^{69} + 48 q^{71} - 12 q^{75} + 8 q^{79} - 4 q^{81} - 52 q^{91} + 16 q^{99}+O(q^{100})$$ 4 * q + 12 * q^15 - 4 * q^19 + 4 * q^21 - 16 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 - 4 * q^39 + 8 * q^45 - 16 * q^49 - 4 * q^51 + 4 * q^55 - 12 * q^59 + 40 * q^61 + 24 * q^65 - 44 * q^69 + 48 * q^71 - 12 * q^75 + 8 * q^79 - 4 * q^81 - 52 * q^91 + 16 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
609.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.41421i 0 −0.707107 + 2.12132i 0 1.58579i 0 −2.82843 0
609.2 0 0.414214i 0 0.707107 + 2.12132i 0 4.41421i 0 2.82843 0
609.3 0 0.414214i 0 0.707107 2.12132i 0 4.41421i 0 2.82843 0
609.4 0 2.41421i 0 −0.707107 2.12132i 0 1.58579i 0 −2.82843 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.d.e 4
4.b odd 2 1 190.2.b.a 4
5.b even 2 1 inner 1520.2.d.e 4
5.c odd 4 1 7600.2.a.v 2
5.c odd 4 1 7600.2.a.bg 2
12.b even 2 1 1710.2.d.c 4
20.d odd 2 1 190.2.b.a 4
20.e even 4 1 950.2.a.f 2
20.e even 4 1 950.2.a.g 2
60.h even 2 1 1710.2.d.c 4
60.l odd 4 1 8550.2.a.bn 2
60.l odd 4 1 8550.2.a.cb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 4.b odd 2 1
190.2.b.a 4 20.d odd 2 1
950.2.a.f 2 20.e even 4 1
950.2.a.g 2 20.e even 4 1
1520.2.d.e 4 1.a even 1 1 trivial
1520.2.d.e 4 5.b even 2 1 inner
1710.2.d.c 4 12.b even 2 1
1710.2.d.c 4 60.h even 2 1
7600.2.a.v 2 5.c odd 4 1
7600.2.a.bg 2 5.c odd 4 1
8550.2.a.bn 2 60.l odd 4 1
8550.2.a.cb 2 60.l odd 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 6T_{3}^{2} + 1$$ T3^4 + 6*T3^2 + 1 $$T_{7}^{4} + 22T_{7}^{2} + 49$$ T7^4 + 22*T7^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4} + 8T^{2} + 25$$
$7$ $$T^{4} + 22T^{2} + 49$$
$11$ $$(T^{2} - 2)^{2}$$
$13$ $$T^{4} + 34T^{2} + 1$$
$17$ $$(T^{2} + 1)^{2}$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 86T^{2} + 49$$
$29$ $$(T^{2} - 6 T + 1)^{2}$$
$31$ $$(T^{2} + 4 T - 14)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T^{2} - 18)^{2}$$
$43$ $$T^{4} + 108T^{2} + 324$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 162T^{2} + 3969$$
$59$ $$(T^{2} + 6 T - 89)^{2}$$
$61$ $$(T^{2} - 20 T + 82)^{2}$$
$67$ $$T^{4} + 198T^{2} + 3969$$
$71$ $$(T^{2} - 24 T + 142)^{2}$$
$73$ $$T^{4} + 162T^{2} + 3969$$
$79$ $$(T^{2} - 4 T - 68)^{2}$$
$83$ $$T^{4} + 216T^{2} + 1296$$
$89$ $$(T^{2} - 50)^{2}$$
$97$ $$T^{4} + 136T^{2} + 16$$