# Properties

 Label 1805.1.h.b Level $1805$ Weight $1$ Character orbit 1805.h Analytic conductor $0.901$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -95 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,1,Mod(69,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("1805.69");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1805.h (of order $$6$$, degree $$2$$, not 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: $$0.900812347803$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $C_3\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{2} + 1) q^{5} - 2 \beta_{2} q^{6} + \beta_{2} q^{9}+O(q^{10})$$ q - b1 * q^2 + b1 * q^3 + b2 * q^4 + (b2 + 1) * q^5 - 2*b2 * q^6 + b2 * q^9 $$q - \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + (\beta_{2} + 1) q^{5} - 2 \beta_{2} q^{6} + \beta_{2} q^{9} + ( - \beta_{3} - \beta_1) q^{10} + \beta_{3} q^{12} + (\beta_{3} + \beta_1) q^{13} + (\beta_{3} + \beta_1) q^{15} + (\beta_{2} + 1) q^{16} - \beta_{3} q^{18} - q^{20} + \beta_{2} q^{25} + 2 q^{26} + 2 q^{30} + ( - \beta_{3} - \beta_1) q^{32} + ( - \beta_{2} - 1) q^{36} + \beta_{3} q^{37} - 2 q^{39} - q^{45} + (\beta_{3} + \beta_1) q^{48} + q^{49} - \beta_{3} q^{50} - \beta_1 q^{52} + ( - \beta_{3} - \beta_1) q^{53} - \beta_1 q^{60} - q^{64} + \beta_{3} q^{65} + (\beta_{3} + \beta_1) q^{67} + (2 \beta_{2} + 2) q^{74} + \beta_{3} q^{75} + 2 \beta_1 q^{78} + \beta_{2} q^{80} + (\beta_{2} + 1) q^{81} + \beta_1 q^{90} + 2 q^{96} + \beta_1 q^{97} - \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + b1 * q^3 + b2 * q^4 + (b2 + 1) * q^5 - 2*b2 * q^6 + b2 * q^9 + (-b3 - b1) * q^10 + b3 * q^12 + (b3 + b1) * q^13 + (b3 + b1) * q^15 + (b2 + 1) * q^16 - b3 * q^18 - q^20 + b2 * q^25 + 2 * q^26 + 2 * q^30 + (-b3 - b1) * q^32 + (-b2 - 1) * q^36 + b3 * q^37 - 2 * q^39 - q^45 + (b3 + b1) * q^48 + q^49 - b3 * q^50 - b1 * q^52 + (-b3 - b1) * q^53 - b1 * q^60 - q^64 + b3 * q^65 + (b3 + b1) * q^67 + (2*b2 + 2) * q^74 + b3 * q^75 + 2*b1 * q^78 + b2 * q^80 + (b2 + 1) * q^81 + b1 * q^90 + 2 * q^96 + b1 * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 $$4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 2 q^{16} - 4 q^{20} - 2 q^{25} + 8 q^{26} + 8 q^{30} - 2 q^{36} - 8 q^{39} - 4 q^{45} + 4 q^{49} - 4 q^{64} + 4 q^{74} - 2 q^{80} + 2 q^{81} + 8 q^{96}+O(q^{100})$$ 4 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 + 2 * q^16 - 4 * q^20 - 2 * q^25 + 8 * q^26 + 8 * q^30 - 2 * q^36 - 8 * q^39 - 4 * q^45 + 4 * q^49 - 4 * q^64 + 4 * q^74 - 2 * q^80 + 2 * q^81 + 8 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
−0.707107 1.22474i 0.707107 + 1.22474i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 0 −0.500000 + 0.866025i 0.707107 1.22474i
69.2 0.707107 + 1.22474i −0.707107 1.22474i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 0 −0.500000 + 0.866025i −0.707107 + 1.22474i
654.1 −0.707107 + 1.22474i 0.707107 1.22474i −0.500000 0.866025i 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 0.866025i 0.707107 + 1.22474i
654.2 0.707107 1.22474i −0.707107 + 1.22474i −0.500000 0.866025i 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 0.866025i −0.707107 1.22474i
 $$n$$: 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner
19.c even 3 1 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.h.b 4
5.b even 2 1 inner 1805.1.h.b 4
19.b odd 2 1 inner 1805.1.h.b 4
19.c even 3 1 95.1.d.b 2
19.c even 3 1 inner 1805.1.h.b 4
19.d odd 6 1 95.1.d.b 2
19.d odd 6 1 inner 1805.1.h.b 4
19.e even 9 6 1805.1.o.b 12
19.f odd 18 6 1805.1.o.b 12
57.f even 6 1 855.1.g.c 2
57.h odd 6 1 855.1.g.c 2
76.f even 6 1 1520.1.m.b 2
76.g odd 6 1 1520.1.m.b 2
95.d odd 2 1 CM 1805.1.h.b 4
95.h odd 6 1 95.1.d.b 2
95.h odd 6 1 inner 1805.1.h.b 4
95.i even 6 1 95.1.d.b 2
95.i even 6 1 inner 1805.1.h.b 4
95.l even 12 2 475.1.c.b 2
95.m odd 12 2 475.1.c.b 2
95.o odd 18 6 1805.1.o.b 12
95.p even 18 6 1805.1.o.b 12
285.n odd 6 1 855.1.g.c 2
285.q even 6 1 855.1.g.c 2
380.p odd 6 1 1520.1.m.b 2
380.s even 6 1 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 19.c even 3 1
95.1.d.b 2 19.d odd 6 1
95.1.d.b 2 95.h odd 6 1
95.1.d.b 2 95.i even 6 1
475.1.c.b 2 95.l even 12 2
475.1.c.b 2 95.m odd 12 2
855.1.g.c 2 57.f even 6 1
855.1.g.c 2 57.h odd 6 1
855.1.g.c 2 285.n odd 6 1
855.1.g.c 2 285.q even 6 1
1520.1.m.b 2 76.f even 6 1
1520.1.m.b 2 76.g odd 6 1
1520.1.m.b 2 380.p odd 6 1
1520.1.m.b 2 380.s even 6 1
1805.1.h.b 4 1.a even 1 1 trivial
1805.1.h.b 4 5.b even 2 1 inner
1805.1.h.b 4 19.b odd 2 1 inner
1805.1.h.b 4 19.c even 3 1 inner
1805.1.h.b 4 19.d odd 6 1 inner
1805.1.h.b 4 95.d odd 2 1 CM
1805.1.h.b 4 95.h odd 6 1 inner
1805.1.h.b 4 95.i even 6 1 inner
1805.1.o.b 12 19.e even 9 6
1805.1.o.b 12 19.f odd 18 6
1805.1.o.b 12 95.o odd 18 6
1805.1.o.b 12 95.p even 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2T_{2}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1805, [\chi])$$.

## Hecke characteristic polynomials

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