# Properties

 Label 3800.1.cv.e Level $3800$ Weight $1$ Character orbit 3800.cv Analytic conductor $1.896$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(251,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(18))

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

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

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

chi := DirichletCharacter("3800.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.cv (of order $$18$$, degree $$6$$, 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: $$1.89644704801$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{9} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{18}^{5} q^{2} + (\zeta_{18}^{4} + 1) q^{3} - \zeta_{18} q^{4} + ( - \zeta_{18}^{5} + 1) q^{6} + \zeta_{18}^{6} q^{8} + (\zeta_{18}^{8} + \zeta_{18}^{4} + 1) q^{9} +O(q^{10})$$ q - z^5 * q^2 + (z^4 + 1) * q^3 - z * q^4 + (-z^5 + 1) * q^6 + z^6 * q^8 + (z^8 + z^4 + 1) * q^9 $$q - \zeta_{18}^{5} q^{2} + (\zeta_{18}^{4} + 1) q^{3} - \zeta_{18} q^{4} + ( - \zeta_{18}^{5} + 1) q^{6} + \zeta_{18}^{6} q^{8} + (\zeta_{18}^{8} + \zeta_{18}^{4} + 1) q^{9} + (\zeta_{18}^{2} - \zeta_{18}) q^{11} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{12} + \zeta_{18}^{2} q^{16} + \zeta_{18}^{5} q^{17} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 1) q^{18} + \zeta_{18}^{4} q^{19} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{22} + (\zeta_{18}^{6} - \zeta_{18}) q^{24} + (\zeta_{18}^{8} + \zeta_{18}^{4} + \cdots + 1) q^{27} + \cdots + (\zeta_{18}^{6} - \zeta_{18}^{5} + \cdots + 1) q^{99} +O(q^{100})$$ q - z^5 * q^2 + (z^4 + 1) * q^3 - z * q^4 + (-z^5 + 1) * q^6 + z^6 * q^8 + (z^8 + z^4 + 1) * q^9 + (z^2 - z) * q^11 + (-z^5 - z) * q^12 + z^2 * q^16 + z^5 * q^17 + (-z^5 + z^4 + 1) * q^18 + z^4 * q^19 + (-z^7 + z^6) * q^22 + (z^6 - z) * q^24 + (z^8 + z^4 - z^3 + 1) * q^27 - z^7 * q^32 + (z^6 - z^5 + z^2 - z) * q^33 + z * q^34 + (-z^5 - z + 1) * q^36 + q^38 + (z^4 + 1) * q^41 - z^8 * q^43 + (-z^3 + z^2) * q^44 + (z^6 + z^2) * q^48 + z^6 * q^49 + (z^5 - 1) * q^51 + (z^8 - z^5 + z^4 + 1) * q^54 + (z^8 + z^4) * q^57 + (-z^7 - z^3) * q^59 - z^3 * q^64 + (-z^7 + z^6 + z^2 - z) * q^66 + (z^6 + z^2) * q^67 - z^6 * q^68 + (z^6 - z^5 - z) * q^72 + (-z^3 - z) * q^73 - z^5 * q^76 + (z^8 - z^7 + z^4 - z^3 + 1) * q^81 + (-z^5 + 1) * q^82 + (z^8 - z^7) * q^83 - z^4 * q^86 + (z^8 - z^7) * q^88 - 2*z^7 * q^89 + (-z^7 + z^2) * q^96 + (z^6 + z^4) * q^97 + z^2 * q^98 + (z^6 - z^5 + z^2 - 2*z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{3} + 6 q^{6} - 3 q^{8} + 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^3 + 6 * q^6 - 3 * q^8 + 6 * q^9 $$6 q + 6 q^{3} + 6 q^{6} - 3 q^{8} + 6 q^{9} + 6 q^{18} - 3 q^{22} - 3 q^{24} + 3 q^{27} - 3 q^{33} + 6 q^{36} + 6 q^{38} + 6 q^{41} - 3 q^{44} - 3 q^{48} - 3 q^{49} - 6 q^{51} + 6 q^{54} - 3 q^{59} - 3 q^{64} - 3 q^{66} - 3 q^{67} + 3 q^{68} - 3 q^{72} - 3 q^{73} + 3 q^{81} + 6 q^{82} - 3 q^{97} + 3 q^{99}+O(q^{100})$$ 6 * q + 6 * q^3 + 6 * q^6 - 3 * q^8 + 6 * q^9 + 6 * q^18 - 3 * q^22 - 3 * q^24 + 3 * q^27 - 3 * q^33 + 6 * q^36 + 6 * q^38 + 6 * q^41 - 3 * q^44 - 3 * q^48 - 3 * q^49 - 6 * q^51 + 6 * q^54 - 3 * q^59 - 3 * q^64 - 3 * q^66 - 3 * q^67 + 3 * q^68 - 3 * q^72 - 3 * q^73 + 3 * q^81 + 6 * q^82 - 3 * q^97 + 3 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$-\zeta_{18}$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.766044 − 0.642788i −0.766044 + 0.642788i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 + 0.342020i 0.939693 − 0.342020i
−0.939693 0.342020i 0.0603074 + 0.342020i 0.766044 + 0.642788i 0 0.0603074 0.342020i 0 −0.500000 0.866025i 0.826352 0.300767i 0
651.1 −0.939693 + 0.342020i 0.0603074 0.342020i 0.766044 0.642788i 0 0.0603074 + 0.342020i 0 −0.500000 + 0.866025i 0.826352 + 0.300767i 0
1051.1 0.766044 0.642788i 1.76604 + 0.642788i 0.173648 0.984808i 0 1.76604 0.642788i 0 −0.500000 0.866025i 1.93969 + 1.62760i 0
1251.1 0.766044 + 0.642788i 1.76604 0.642788i 0.173648 + 0.984808i 0 1.76604 + 0.642788i 0 −0.500000 + 0.866025i 1.93969 1.62760i 0
1651.1 0.173648 0.984808i 1.17365 + 0.984808i −0.939693 0.342020i 0 1.17365 0.984808i 0 −0.500000 + 0.866025i 0.233956 + 1.32683i 0
2251.1 0.173648 + 0.984808i 1.17365 0.984808i −0.939693 + 0.342020i 0 1.17365 + 0.984808i 0 −0.500000 0.866025i 0.233956 1.32683i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
19.e even 9 1 inner
152.u odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.cv.e yes 6
5.b even 2 1 3800.1.cv.a 6
5.c odd 4 2 3800.1.cq.c 12
8.d odd 2 1 CM 3800.1.cv.e yes 6
19.e even 9 1 inner 3800.1.cv.e yes 6
40.e odd 2 1 3800.1.cv.a 6
40.k even 4 2 3800.1.cq.c 12
95.p even 18 1 3800.1.cv.a 6
95.q odd 36 2 3800.1.cq.c 12
152.u odd 18 1 inner 3800.1.cv.e yes 6
760.bz odd 18 1 3800.1.cv.a 6
760.cp even 36 2 3800.1.cq.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.cq.c 12 5.c odd 4 2
3800.1.cq.c 12 40.k even 4 2
3800.1.cq.c 12 95.q odd 36 2
3800.1.cq.c 12 760.cp even 36 2
3800.1.cv.a 6 5.b even 2 1
3800.1.cv.a 6 40.e odd 2 1
3800.1.cv.a 6 95.p even 18 1
3800.1.cv.a 6 760.bz odd 18 1
3800.1.cv.e yes 6 1.a even 1 1 trivial
3800.1.cv.e yes 6 8.d odd 2 1 CM
3800.1.cv.e yes 6 19.e even 9 1 inner
3800.1.cv.e yes 6 152.u odd 18 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 6T_{3}^{5} + 15T_{3}^{4} - 19T_{3}^{3} + 12T_{3}^{2} - 3T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6} - 6 T^{5} + \cdots + 1$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 3 T^{4} + \cdots + 1$$
$13$ $$T^{6}$$
$17$ $$T^{6} - T^{3} + 1$$
$19$ $$T^{6} + T^{3} + 1$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$T^{6} - 6 T^{5} + \cdots + 1$$
$43$ $$T^{6} - T^{3} + 1$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$61$ $$T^{6}$$
$67$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$71$ $$T^{6}$$
$73$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$79$ $$T^{6}$$
$83$ $$T^{6} + 3 T^{4} + \cdots + 1$$
$89$ $$T^{6} + 8T^{3} + 64$$
$97$ $$T^{6} + 3 T^{5} + \cdots + 1$$