# Properties

 Label 3800.1.bd.e Level $3800$ Weight $1$ Character orbit 3800.bd Analytic conductor $1.896$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ 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(1451,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 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("3800.1451");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.bd (of order $$6$$, degree $$2$$, 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: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.72200.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

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

## Character values

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

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1451.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 −1.00000 −1.50000 + 2.59808i 0
3051.1 0.500000 0.866025i 1.00000 1.73205i −0.500000 0.866025i 0 −1.00000 1.73205i 0 −1.00000 −1.50000 2.59808i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
19.c even 3 1 inner
152.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.bd.e yes 2
5.b even 2 1 3800.1.bd.a 2
5.c odd 4 2 3800.1.bn.a 4
8.d odd 2 1 CM 3800.1.bd.e yes 2
19.c even 3 1 inner 3800.1.bd.e yes 2
40.e odd 2 1 3800.1.bd.a 2
40.k even 4 2 3800.1.bn.a 4
95.i even 6 1 3800.1.bd.a 2
95.m odd 12 2 3800.1.bn.a 4
152.k odd 6 1 inner 3800.1.bd.e yes 2
760.bm odd 6 1 3800.1.bd.a 2
760.bw even 12 2 3800.1.bn.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.bd.a 2 5.b even 2 1
3800.1.bd.a 2 40.e odd 2 1
3800.1.bd.a 2 95.i even 6 1
3800.1.bd.a 2 760.bm odd 6 1
3800.1.bd.e yes 2 1.a even 1 1 trivial
3800.1.bd.e yes 2 8.d odd 2 1 CM
3800.1.bd.e yes 2 19.c even 3 1 inner
3800.1.bd.e yes 2 152.k odd 6 1 inner
3800.1.bn.a 4 5.c odd 4 2
3800.1.bn.a 4 40.k even 4 2
3800.1.bn.a 4 95.m odd 12 2
3800.1.bn.a 4 760.bw even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{2} - 2T_{3} + 4$$ T3^2 - 2*T3 + 4 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

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