# Properties

 Label 3520.1.cg.b Level $3520$ Weight $1$ Character orbit 3520.cg Analytic conductor $1.757$ Analytic rank $0$ Dimension $8$ Projective image $D_{10}$ CM discriminant -40 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,1,Mod(1249,3520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 5, 5, 9]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3520.1249");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3520.cg (of order $$10$$, degree $$4$$, 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: $$1.75670884447$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.241453843558400000.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{20}^{8} q^{5} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{7} + \zeta_{20}^{2} q^{9} +O(q^{10})$$ q - z^8 * q^5 + (z^5 + z^3) * q^7 + z^2 * q^9 $$q - \zeta_{20}^{8} q^{5} + (\zeta_{20}^{5} + \zeta_{20}^{3}) q^{7} + \zeta_{20}^{2} q^{9} + \zeta_{20}^{9} q^{11} + ( - \zeta_{20}^{4} + 1) q^{13} + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{19} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{23} - \zeta_{20}^{6} q^{25} + (\zeta_{20}^{3} + \zeta_{20}) q^{35} + (\zeta_{20}^{6} + \zeta_{20}^{2}) q^{37} + ( - \zeta_{20}^{2} - 1) q^{41} + q^{45} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{47} + (\zeta_{20}^{8} + \zeta_{20}^{6} - 1) q^{49} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{53} + \zeta_{20}^{7} q^{55} + (\zeta_{20}^{7} - \zeta_{20}) q^{59} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{63} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{65} + ( - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{77} + \zeta_{20}^{4} q^{81} + (\zeta_{20}^{6} - \zeta_{20}^{4}) q^{89} + ( - \zeta_{20}^{9} - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{3}) q^{91} + ( - \zeta_{20}^{7} - \zeta_{20}) q^{95} - \zeta_{20} q^{99} +O(q^{100})$$ q - z^8 * q^5 + (z^5 + z^3) * q^7 + z^2 * q^9 + z^9 * q^11 + (-z^4 + 1) * q^13 + (-z^9 - z^3) * q^19 + (-z^9 - z) * q^23 - z^6 * q^25 + (z^3 + z) * q^35 + (z^6 + z^2) * q^37 + (-z^2 - 1) * q^41 + q^45 + (-z^9 + z^3) * q^47 + (z^8 + z^6 - 1) * q^49 + (-z^8 + z^6) * q^53 + z^7 * q^55 + (z^7 - z) * q^59 + (z^7 + z^5) * q^63 + (-z^8 - z^2) * q^65 + (-z^4 - z^2) * q^77 + z^4 * q^81 + (z^6 - z^4) * q^89 + (-z^9 - z^7 + z^5 + z^3) * q^91 + (-z^7 - z) * q^95 - z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 8 * q + 2 * q^5 + 2 * q^9 $$8 q + 2 q^{5} + 2 q^{9} + 10 q^{13} - 2 q^{25} + 4 q^{37} - 10 q^{41} + 8 q^{45} - 8 q^{49} + 4 q^{53} - 2 q^{81} + 4 q^{89}+O(q^{100})$$ 8 * q + 2 * q^5 + 2 * q^9 + 10 * q^13 - 2 * q^25 + 4 * q^37 - 10 * q^41 + 8 * q^45 - 8 * q^49 + 4 * q^53 - 2 * q^81 + 4 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$1541$$ $$2751$$ $$2817$$ $$\chi(n)$$ $$-\zeta_{20}^{4}$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 0.587785 − 0.809017i −0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.951057 + 0.309017i 0.951057 − 0.309017i 0.587785 + 0.809017i −0.587785 − 0.809017i
0 0 0 −0.309017 + 0.951057i 0 −0.951057 + 0.690983i 0 −0.309017 0.951057i 0
1249.2 0 0 0 −0.309017 + 0.951057i 0 0.951057 0.690983i 0 −0.309017 0.951057i 0
1569.1 0 0 0 0.809017 0.587785i 0 −0.587785 1.80902i 0 0.809017 + 0.587785i 0
1569.2 0 0 0 0.809017 0.587785i 0 0.587785 + 1.80902i 0 0.809017 + 0.587785i 0
1889.1 0 0 0 0.809017 + 0.587785i 0 −0.587785 + 1.80902i 0 0.809017 0.587785i 0
1889.2 0 0 0 0.809017 + 0.587785i 0 0.587785 1.80902i 0 0.809017 0.587785i 0
3489.1 0 0 0 −0.309017 0.951057i 0 −0.951057 0.690983i 0 −0.309017 + 0.951057i 0
3489.2 0 0 0 −0.309017 0.951057i 0 0.951057 + 0.690983i 0 −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
4.b odd 2 1 inner
11.d odd 10 1 inner
40.f even 2 1 inner
44.g even 10 1 inner
440.ba odd 10 1 inner
440.bm even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.1.cg.b yes 8
4.b odd 2 1 inner 3520.1.cg.b yes 8
5.b even 2 1 3520.1.cg.a 8
8.b even 2 1 3520.1.cg.a 8
8.d odd 2 1 3520.1.cg.a 8
11.d odd 10 1 inner 3520.1.cg.b yes 8
20.d odd 2 1 3520.1.cg.a 8
40.e odd 2 1 CM 3520.1.cg.b yes 8
40.f even 2 1 inner 3520.1.cg.b yes 8
44.g even 10 1 inner 3520.1.cg.b yes 8
55.h odd 10 1 3520.1.cg.a 8
88.k even 10 1 3520.1.cg.a 8
88.p odd 10 1 3520.1.cg.a 8
220.o even 10 1 3520.1.cg.a 8
440.ba odd 10 1 inner 3520.1.cg.b yes 8
440.bm even 10 1 inner 3520.1.cg.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3520.1.cg.a 8 5.b even 2 1
3520.1.cg.a 8 8.b even 2 1
3520.1.cg.a 8 8.d odd 2 1
3520.1.cg.a 8 20.d odd 2 1
3520.1.cg.a 8 55.h odd 10 1
3520.1.cg.a 8 88.k even 10 1
3520.1.cg.a 8 88.p odd 10 1
3520.1.cg.a 8 220.o even 10 1
3520.1.cg.b yes 8 1.a even 1 1 trivial
3520.1.cg.b yes 8 4.b odd 2 1 inner
3520.1.cg.b yes 8 11.d odd 10 1 inner
3520.1.cg.b yes 8 40.e odd 2 1 CM
3520.1.cg.b yes 8 40.f even 2 1 inner
3520.1.cg.b yes 8 44.g even 10 1 inner
3520.1.cg.b yes 8 440.ba odd 10 1 inner
3520.1.cg.b yes 8 440.bm even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} - 5T_{13}^{3} + 10T_{13}^{2} - 10T_{13} + 5$$ acting on $$S_{1}^{\mathrm{new}}(3520, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$7$ $$T^{8} + 5 T^{6} + 10 T^{4} + 25$$
$11$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$13$ $$(T^{4} - 5 T^{3} + 10 T^{2} - 10 T + 5)^{2}$$
$17$ $$T^{8}$$
$19$ $$T^{8} + 10 T^{4} + 25 T^{2} + 25$$
$23$ $$(T^{4} + 3 T^{2} + 1)^{2}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2}$$
$41$ $$(T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2}$$
$43$ $$T^{8}$$
$47$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$53$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2}$$
$59$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$(T^{2} - T - 1)^{4}$$
$97$ $$T^{8}$$