# Properties

 Label 1520.2.d.b Level $1520$ Weight $2$ Character orbit 1520.d Analytic conductor $12.137$ Analytic rank $0$ Dimension $2$ CM no 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{5} - \beta q^{7} + 3 q^{9} +O(q^{10})$$ q + (-b - 1) * q^5 - b * q^7 + 3 * q^9 $$q + ( - \beta - 1) q^{5} - \beta q^{7} + 3 q^{9} + 4 q^{11} + \beta q^{13} + 2 \beta q^{17} + q^{19} - 3 \beta q^{23} + (2 \beta - 3) q^{25} + 6 q^{29} + 4 q^{31} + (\beta - 4) q^{35} - 5 \beta q^{37} - 10 q^{41} + \beta q^{43} + ( - 3 \beta - 3) q^{45} + 3 \beta q^{47} + 3 q^{49} - 5 \beta q^{53} + ( - 4 \beta - 4) q^{55} + 2 q^{61} - 3 \beta q^{63} + ( - \beta + 4) q^{65} - 4 \beta q^{67} - 4 q^{71} - 2 \beta q^{73} - 4 \beta q^{77} + 4 q^{79} + 9 q^{81} - 9 \beta q^{83} + ( - 2 \beta + 8) q^{85} + 2 q^{89} + 4 q^{91} + ( - \beta - 1) q^{95} + 3 \beta q^{97} + 12 q^{99} +O(q^{100})$$ q + (-b - 1) * q^5 - b * q^7 + 3 * q^9 + 4 * q^11 + b * q^13 + 2*b * q^17 + q^19 - 3*b * q^23 + (2*b - 3) * q^25 + 6 * q^29 + 4 * q^31 + (b - 4) * q^35 - 5*b * q^37 - 10 * q^41 + b * q^43 + (-3*b - 3) * q^45 + 3*b * q^47 + 3 * q^49 - 5*b * q^53 + (-4*b - 4) * q^55 + 2 * q^61 - 3*b * q^63 + (-b + 4) * q^65 - 4*b * q^67 - 4 * q^71 - 2*b * q^73 - 4*b * q^77 + 4 * q^79 + 9 * q^81 - 9*b * q^83 + (-2*b + 8) * q^85 + 2 * q^89 + 4 * q^91 + (-b - 1) * q^95 + 3*b * q^97 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 6 * q^9 $$2 q - 2 q^{5} + 6 q^{9} + 8 q^{11} + 2 q^{19} - 6 q^{25} + 12 q^{29} + 8 q^{31} - 8 q^{35} - 20 q^{41} - 6 q^{45} + 6 q^{49} - 8 q^{55} + 4 q^{61} + 8 q^{65} - 8 q^{71} + 8 q^{79} + 18 q^{81} + 16 q^{85} + 4 q^{89} + 8 q^{91} - 2 q^{95} + 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 + 6 * q^9 + 8 * q^11 + 2 * q^19 - 6 * q^25 + 12 * q^29 + 8 * q^31 - 8 * q^35 - 20 * q^41 - 6 * q^45 + 6 * q^49 - 8 * q^55 + 4 * q^61 + 8 * q^65 - 8 * q^71 + 8 * q^79 + 18 * q^81 + 16 * q^85 + 4 * q^89 + 8 * q^91 - 2 * q^95 + 24 * 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
 1.00000i − 1.00000i
0 0 0 −1.00000 2.00000i 0 2.00000i 0 3.00000 0
609.2 0 0 0 −1.00000 + 2.00000i 0 2.00000i 0 3.00000 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.b 2
4.b odd 2 1 95.2.b.a 2
5.b even 2 1 inner 1520.2.d.b 2
5.c odd 4 1 7600.2.a.i 1
5.c odd 4 1 7600.2.a.l 1
12.b even 2 1 855.2.c.b 2
20.d odd 2 1 95.2.b.a 2
20.e even 4 1 475.2.a.a 1
20.e even 4 1 475.2.a.c 1
60.h even 2 1 855.2.c.b 2
60.l odd 4 1 4275.2.a.e 1
60.l odd 4 1 4275.2.a.p 1
76.d even 2 1 1805.2.b.c 2
380.d even 2 1 1805.2.b.c 2
380.j odd 4 1 9025.2.a.c 1
380.j odd 4 1 9025.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 4.b odd 2 1
95.2.b.a 2 20.d odd 2 1
475.2.a.a 1 20.e even 4 1
475.2.a.c 1 20.e even 4 1
855.2.c.b 2 12.b even 2 1
855.2.c.b 2 60.h even 2 1
1520.2.d.b 2 1.a even 1 1 trivial
1520.2.d.b 2 5.b even 2 1 inner
1805.2.b.c 2 76.d even 2 1
1805.2.b.c 2 380.d even 2 1
4275.2.a.e 1 60.l odd 4 1
4275.2.a.p 1 60.l odd 4 1
7600.2.a.i 1 5.c odd 4 1
7600.2.a.l 1 5.c odd 4 1
9025.2.a.c 1 380.j odd 4 1
9025.2.a.h 1 380.j odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 324$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 36$$