# Properties

 Label 3800.1.o.g Level $3800$ Weight $1$ Character orbit 3800.o Analytic conductor $1.896$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -95 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,1,Mod(1101,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("3800.1101");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.o (of order $$2$$, degree $$1$$, 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: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.66724352000.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{16} q^{2} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{3} + \zeta_{16}^{2} q^{4} + (\zeta_{16}^{6} - \zeta_{16}^{4}) q^{6} - \zeta_{16}^{3} q^{8} + (\zeta_{16}^{6} - \zeta_{16}^{2} + 1) q^{9} +O(q^{10})$$ q - z * q^2 + (-z^5 + z^3) * q^3 + z^2 * q^4 + (z^6 - z^4) * q^6 - z^3 * q^8 + (z^6 - z^2 + 1) * q^9 $$q - \zeta_{16} q^{2} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{3} + \zeta_{16}^{2} q^{4} + (\zeta_{16}^{6} - \zeta_{16}^{4}) q^{6} - \zeta_{16}^{3} q^{8} + (\zeta_{16}^{6} - \zeta_{16}^{2} + 1) q^{9} + ( - \zeta_{16}^{6} - \zeta_{16}^{2}) q^{11} + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{12} + (\zeta_{16}^{7} - \zeta_{16}) q^{13} + \zeta_{16}^{4} q^{16} + ( - \zeta_{16}^{7} + \zeta_{16}^{3} - \zeta_{16}) q^{18} - \zeta_{16}^{4} q^{19} + (\zeta_{16}^{7} + \zeta_{16}^{3}) q^{22} + ( - \zeta_{16}^{6} - 1) q^{24} + (\zeta_{16}^{2} + 1) q^{26} + (\zeta_{16}^{7} - \zeta_{16}^{5} + \cdots - \zeta_{16}) q^{27} + \cdots + ( - \zeta_{16}^{6} + \cdots - \zeta_{16}^{2}) q^{99} +O(q^{100})$$ q - z * q^2 + (-z^5 + z^3) * q^3 + z^2 * q^4 + (z^6 - z^4) * q^6 - z^3 * q^8 + (z^6 - z^2 + 1) * q^9 + (-z^6 - z^2) * q^11 + (-z^7 + z^5) * q^12 + (z^7 - z) * q^13 + z^4 * q^16 + (-z^7 + z^3 - z) * q^18 - z^4 * q^19 + (z^7 + z^3) * q^22 + (-z^6 - 1) * q^24 + (z^2 + 1) * q^26 + (z^7 - z^5 + z^3 - z) * q^27 - z^5 * q^32 + (z^7 - z^5 - z^3 + z) * q^33 + (-z^4 + z^2 - 1) * q^36 + (z^5 - z^3) * q^37 + z^5 * q^38 + (z^6 - z^2) * q^39 + (-z^4 + 1) * q^44 + (z^7 + z) * q^48 - q^49 + (-z^3 - z) * q^52 + (z^5 - z^3) * q^53 + (z^6 - z^4 + z^2 + 1) * q^54 + (-z^7 - z) * q^57 + (z^6 + z^2) * q^61 + z^6 * q^64 + (z^6 + z^4 - z^2 + 1) * q^66 + (z^7 - z) * q^67 + (z^5 - z^3 + z) * q^72 + (-z^6 + z^4) * q^74 - z^6 * q^76 + (-z^7 + z^3) * q^78 + (z^6 - z^2 + 1) * q^81 + (z^5 - z) * q^88 + (-z^2 + 1) * q^96 + (-z^5 - z^3) * q^97 + z * q^98 + (-z^6 + 2*z^4 - z^2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^9 $$8 q + 8 q^{9} - 8 q^{24} + 8 q^{26} - 8 q^{36} + 8 q^{44} - 8 q^{49} + 8 q^{54} + 8 q^{66} + 8 q^{81} + 8 q^{96}+O(q^{100})$$ 8 * q + 8 * q^9 - 8 * q^24 + 8 * q^26 - 8 * q^36 + 8 * q^44 - 8 * q^49 + 8 * q^54 + 8 * q^66 + 8 * q^81 + 8 * q^96

## Character values

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

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1101.1
 0.923880 + 0.382683i 0.923880 − 0.382683i 0.382683 + 0.923880i 0.382683 − 0.923880i −0.382683 + 0.923880i −0.382683 − 0.923880i −0.923880 + 0.382683i −0.923880 − 0.382683i
−0.923880 0.382683i 0.765367 0.707107 + 0.707107i 0 −0.707107 0.292893i 0 −0.382683 0.923880i −0.414214 0
1101.2 −0.923880 + 0.382683i 0.765367 0.707107 0.707107i 0 −0.707107 + 0.292893i 0 −0.382683 + 0.923880i −0.414214 0
1101.3 −0.382683 0.923880i −1.84776 −0.707107 + 0.707107i 0 0.707107 + 1.70711i 0 0.923880 + 0.382683i 2.41421 0
1101.4 −0.382683 + 0.923880i −1.84776 −0.707107 0.707107i 0 0.707107 1.70711i 0 0.923880 0.382683i 2.41421 0
1101.5 0.382683 0.923880i 1.84776 −0.707107 0.707107i 0 0.707107 1.70711i 0 −0.923880 + 0.382683i 2.41421 0
1101.6 0.382683 + 0.923880i 1.84776 −0.707107 + 0.707107i 0 0.707107 + 1.70711i 0 −0.923880 0.382683i 2.41421 0
1101.7 0.923880 0.382683i −0.765367 0.707107 0.707107i 0 −0.707107 + 0.292893i 0 0.382683 0.923880i −0.414214 0
1101.8 0.923880 + 0.382683i −0.765367 0.707107 + 0.707107i 0 −0.707107 0.292893i 0 0.382683 + 0.923880i −0.414214 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1101.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
8.b even 2 1 inner
19.b odd 2 1 inner
40.f even 2 1 inner
152.g odd 2 1 inner
760.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.o.g 8
5.b even 2 1 inner 3800.1.o.g 8
5.c odd 4 2 760.1.b.a 8
8.b even 2 1 inner 3800.1.o.g 8
19.b odd 2 1 inner 3800.1.o.g 8
20.e even 4 2 3040.1.b.a 8
40.f even 2 1 inner 3800.1.o.g 8
40.i odd 4 2 760.1.b.a 8
40.k even 4 2 3040.1.b.a 8
95.d odd 2 1 CM 3800.1.o.g 8
95.g even 4 2 760.1.b.a 8
152.g odd 2 1 inner 3800.1.o.g 8
380.j odd 4 2 3040.1.b.a 8
760.b odd 2 1 inner 3800.1.o.g 8
760.t even 4 2 760.1.b.a 8
760.y odd 4 2 3040.1.b.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.b.a 8 5.c odd 4 2
760.1.b.a 8 40.i odd 4 2
760.1.b.a 8 95.g even 4 2
760.1.b.a 8 760.t even 4 2
3040.1.b.a 8 20.e even 4 2
3040.1.b.a 8 40.k even 4 2
3040.1.b.a 8 380.j odd 4 2
3040.1.b.a 8 760.y odd 4 2
3800.1.o.g 8 1.a even 1 1 trivial
3800.1.o.g 8 5.b even 2 1 inner
3800.1.o.g 8 8.b even 2 1 inner
3800.1.o.g 8 19.b odd 2 1 inner
3800.1.o.g 8 40.f even 2 1 inner
3800.1.o.g 8 95.d odd 2 1 CM
3800.1.o.g 8 152.g odd 2 1 inner
3800.1.o.g 8 760.b odd 2 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

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