# Properties

 Label 3600.1.da.a Level $3600$ Weight $1$ Character orbit 3600.da Analytic conductor $1.797$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -20 Inner twists $16$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(1343,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 0, 2, 9]))

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

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

chi := DirichletCharacter("3600.1343");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.da (of order $$12$$, 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.157464000.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{7} q^{3} + \zeta_{24}^{5} q^{7} - \zeta_{24}^{2} q^{9} +O(q^{10})$$ q - z^7 * q^3 + z^5 * q^7 - z^2 * q^9 $$q - \zeta_{24}^{7} q^{3} + \zeta_{24}^{5} q^{7} - \zeta_{24}^{2} q^{9} + q^{21} + ( - \zeta_{24}^{11} + \zeta_{24}^{3}) q^{23} + \zeta_{24}^{9} q^{27} + ( - \zeta_{24}^{10} - \zeta_{24}^{6}) q^{29} + ( - \zeta_{24}^{8} + 1) q^{41} + 2 \zeta_{24}^{11} q^{43} + ( - \zeta_{24}^{9} + \zeta_{24}) q^{47} - \zeta_{24}^{8} q^{61} - \zeta_{24}^{7} q^{63} + \zeta_{24} q^{67} + ( - \zeta_{24}^{10} - \zeta_{24}^{6}) q^{69} + \zeta_{24}^{4} q^{81} + (\zeta_{24}^{7} + \zeta_{24}^{3}) q^{83} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{87} + (\zeta_{24}^{10} - \zeta_{24}^{2}) q^{89} +O(q^{100})$$ q - z^7 * q^3 + z^5 * q^7 - z^2 * q^9 + q^21 + (-z^11 + z^3) * q^23 + z^9 * q^27 + (-z^10 - z^6) * q^29 + (-z^8 + 1) * q^41 + 2*z^11 * q^43 + (-z^9 + z) * q^47 - z^8 * q^61 - z^7 * q^63 + z * q^67 + (-z^10 - z^6) * q^69 + z^4 * q^81 + (z^7 + z^3) * q^83 + (-z^5 - z) * q^87 + (z^10 - z^2) * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 8 q^{21} + 12 q^{41} + 4 q^{61} + 4 q^{81}+O(q^{100})$$ 8 * q + 8 * q^21 + 12 * q^41 + 4 * q^61 + 4 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$\zeta_{24}^{6}$$ $$1$$ $$\zeta_{24}^{4}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1343.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
0 −0.965926 0.258819i 0 0 0 −0.965926 + 0.258819i 0 0.866025 + 0.500000i 0
1343.2 0 0.965926 + 0.258819i 0 0 0 0.965926 0.258819i 0 0.866025 + 0.500000i 0
2207.1 0 −0.258819 + 0.965926i 0 0 0 −0.258819 0.965926i 0 −0.866025 0.500000i 0
2207.2 0 0.258819 0.965926i 0 0 0 0.258819 + 0.965926i 0 −0.866025 0.500000i 0
2543.1 0 −0.258819 0.965926i 0 0 0 −0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0
2543.2 0 0.258819 + 0.965926i 0 0 0 0.258819 0.965926i 0 −0.866025 + 0.500000i 0
3407.1 0 −0.965926 + 0.258819i 0 0 0 −0.965926 0.258819i 0 0.866025 0.500000i 0
3407.2 0 0.965926 0.258819i 0 0 0 0.965926 + 0.258819i 0 0.866025 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1343.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
9.d odd 6 1 inner
20.e even 4 2 inner
36.h even 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner
180.n even 6 1 inner
180.v odd 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.da.a 8
4.b odd 2 1 inner 3600.1.da.a 8
5.b even 2 1 inner 3600.1.da.a 8
5.c odd 4 2 inner 3600.1.da.a 8
9.d odd 6 1 inner 3600.1.da.a 8
20.d odd 2 1 CM 3600.1.da.a 8
20.e even 4 2 inner 3600.1.da.a 8
36.h even 6 1 inner 3600.1.da.a 8
45.h odd 6 1 inner 3600.1.da.a 8
45.l even 12 2 inner 3600.1.da.a 8
180.n even 6 1 inner 3600.1.da.a 8
180.v odd 12 2 inner 3600.1.da.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.da.a 8 1.a even 1 1 trivial
3600.1.da.a 8 4.b odd 2 1 inner
3600.1.da.a 8 5.b even 2 1 inner
3600.1.da.a 8 5.c odd 4 2 inner
3600.1.da.a 8 9.d odd 6 1 inner
3600.1.da.a 8 20.d odd 2 1 CM
3600.1.da.a 8 20.e even 4 2 inner
3600.1.da.a 8 36.h even 6 1 inner
3600.1.da.a 8 45.h odd 6 1 inner
3600.1.da.a 8 45.l even 12 2 inner
3600.1.da.a 8 180.n even 6 1 inner
3600.1.da.a 8 180.v odd 12 2 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{4} + 1$$
$5$ $$T^{8}$$
$7$ $$T^{8} - T^{4} + 1$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8} - 9T^{4} + 81$$
$29$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{2} - 3 T + 3)^{4}$$
$43$ $$T^{8} - 16T^{4} + 256$$
$47$ $$T^{8} - 9T^{4} + 81$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$(T^{2} - T + 1)^{4}$$
$67$ $$T^{8} - T^{4} + 1$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8} - 9T^{4} + 81$$
$89$ $$(T^{2} - 3)^{4}$$
$97$ $$T^{8}$$
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