# Properties

 Label 1805.1.r.a Level $1805$ Weight $1$ Character orbit 1805.r Analytic conductor $0.901$ Analytic rank $0$ Dimension $12$ Projective image $D_{4}$ CM discriminant -19 Inner twists $24$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,1,Mod(28,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([27, 16]))

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

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

chi := DirichletCharacter("1805.28");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1805.r (of order $$36$$, degree $$12$$, 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: $$12$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.2375.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{36}^{7} q^{4} + \zeta_{36}^{2} q^{5} + (\zeta_{36}^{12} + \zeta_{36}^{3}) q^{7} - \zeta_{36} q^{9} +O(q^{10})$$ q - z^7 * q^4 + z^2 * q^5 + (z^12 + z^3) * q^7 - z * q^9 $$q - \zeta_{36}^{7} q^{4} + \zeta_{36}^{2} q^{5} + (\zeta_{36}^{12} + \zeta_{36}^{3}) q^{7} - \zeta_{36} q^{9} + \zeta_{36}^{14} q^{16} + ( - \zeta_{36}^{17} + \zeta_{36}^{8}) q^{17} - \zeta_{36}^{9} q^{20} + ( - \zeta_{36}^{16} + \zeta_{36}^{7}) q^{23} + \zeta_{36}^{4} q^{25} + ( - \zeta_{36}^{10} + \zeta_{36}) q^{28} + (\zeta_{36}^{14} + \zeta_{36}^{5}) q^{35} + \zeta_{36}^{8} q^{36} + ( - \zeta_{36}^{11} - \zeta_{36}^{2}) q^{43} - \zeta_{36}^{3} q^{45} + ( - \zeta_{36}^{10} - \zeta_{36}) q^{47} + \zeta_{36}^{15} q^{49} + ( - \zeta_{36}^{13} - \zeta_{36}^{4}) q^{63} + \zeta_{36}^{3} q^{64} + ( - \zeta_{36}^{15} - \zeta_{36}^{6}) q^{68} + ( - \zeta_{36}^{14} + \zeta_{36}^{5}) q^{73} + \zeta_{36}^{16} q^{80} + \zeta_{36}^{2} q^{81} + (\zeta_{36}^{12} - \zeta_{36}^{3}) q^{83} + (\zeta_{36}^{10} + \zeta_{36}) q^{85} + ( - \zeta_{36}^{14} - \zeta_{36}^{5}) q^{92} +O(q^{100})$$ q - z^7 * q^4 + z^2 * q^5 + (z^12 + z^3) * q^7 - z * q^9 + z^14 * q^16 + (-z^17 + z^8) * q^17 - z^9 * q^20 + (-z^16 + z^7) * q^23 + z^4 * q^25 + (-z^10 + z) * q^28 + (z^14 + z^5) * q^35 + z^8 * q^36 + (-z^11 - z^2) * q^43 - z^3 * q^45 + (-z^10 - z) * q^47 + z^15 * q^49 + (-z^13 - z^4) * q^63 + z^3 * q^64 + (-z^15 - z^6) * q^68 + (-z^14 + z^5) * q^73 + z^16 * q^80 + z^2 * q^81 + (z^12 - z^3) * q^83 + (z^10 + z) * q^85 + (-z^14 - z^5) * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 6 q^{7}+O(q^{10})$$ 12 * q - 6 * q^7 $$12 q - 6 q^{7} - 6 q^{68} - 6 q^{83}+O(q^{100})$$ 12 * q - 6 * q^7 - 6 * q^68 - 6 * q^83

## Character values

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

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-\zeta_{36}^{9}$$ $$\zeta_{36}^{16}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 0.984808 + 0.173648i −0.342020 + 0.939693i 0.342020 − 0.939693i 0.984808 − 0.173648i −0.642788 − 0.766044i 0.342020 + 0.939693i 0.642788 − 0.766044i −0.984808 + 0.173648i −0.984808 − 0.173648i 0.642788 + 0.766044i −0.342020 − 0.939693i −0.642788 + 0.766044i
0 0 −0.342020 0.939693i 0.939693 + 0.342020i 0 0.366025 + 1.36603i 0 −0.984808 0.173648i 0
62.1 0 0 −0.642788 0.766044i −0.766044 0.642788i 0 0.366025 1.36603i 0 0.342020 0.939693i 0
423.1 0 0 0.642788 + 0.766044i −0.766044 0.642788i 0 −1.36603 0.366025i 0 −0.342020 + 0.939693i 0
967.1 0 0 −0.342020 + 0.939693i 0.939693 0.342020i 0 0.366025 1.36603i 0 −0.984808 + 0.173648i 0
1137.1 0 0 0.984808 0.173648i −0.173648 + 0.984808i 0 0.366025 1.36603i 0 0.642788 + 0.766044i 0
1182.1 0 0 0.642788 0.766044i −0.766044 + 0.642788i 0 −1.36603 + 0.366025i 0 −0.342020 0.939693i 0
1317.1 0 0 −0.984808 0.173648i −0.173648 0.984808i 0 −1.36603 + 0.366025i 0 −0.642788 + 0.766044i 0
1328.1 0 0 0.342020 0.939693i 0.939693 0.342020i 0 −1.36603 0.366025i 0 0.984808 0.173648i 0
1472.1 0 0 0.342020 + 0.939693i 0.939693 + 0.342020i 0 −1.36603 + 0.366025i 0 0.984808 + 0.173648i 0
1498.1 0 0 −0.984808 + 0.173648i −0.173648 + 0.984808i 0 −1.36603 0.366025i 0 −0.642788 0.766044i 0
1543.1 0 0 −0.642788 + 0.766044i −0.766044 + 0.642788i 0 0.366025 + 1.36603i 0 0.342020 + 0.939693i 0
1678.1 0 0 0.984808 + 0.173648i −0.173648 0.984808i 0 0.366025 + 1.36603i 0 0.642788 0.766044i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.c odd 4 1 inner
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner
95.g even 4 1 inner
95.l even 12 2 inner
95.m odd 12 2 inner
95.q odd 36 3 inner
95.r even 36 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.r.a 12
5.c odd 4 1 inner 1805.1.r.a 12
19.b odd 2 1 CM 1805.1.r.a 12
19.c even 3 2 inner 1805.1.r.a 12
19.d odd 6 2 inner 1805.1.r.a 12
19.e even 9 1 1805.1.f.a 2
19.e even 9 2 1805.1.m.a 4
19.e even 9 3 inner 1805.1.r.a 12
19.f odd 18 1 1805.1.f.a 2
19.f odd 18 2 1805.1.m.a 4
19.f odd 18 3 inner 1805.1.r.a 12
95.g even 4 1 inner 1805.1.r.a 12
95.l even 12 2 inner 1805.1.r.a 12
95.m odd 12 2 inner 1805.1.r.a 12
95.q odd 36 1 1805.1.f.a 2
95.q odd 36 2 1805.1.m.a 4
95.q odd 36 3 inner 1805.1.r.a 12
95.r even 36 1 1805.1.f.a 2
95.r even 36 2 1805.1.m.a 4
95.r even 36 3 inner 1805.1.r.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.1.f.a 2 19.e even 9 1
1805.1.f.a 2 19.f odd 18 1
1805.1.f.a 2 95.q odd 36 1
1805.1.f.a 2 95.r even 36 1
1805.1.m.a 4 19.e even 9 2
1805.1.m.a 4 19.f odd 18 2
1805.1.m.a 4 95.q odd 36 2
1805.1.m.a 4 95.r even 36 2
1805.1.r.a 12 1.a even 1 1 trivial
1805.1.r.a 12 5.c odd 4 1 inner
1805.1.r.a 12 19.b odd 2 1 CM
1805.1.r.a 12 19.c even 3 2 inner
1805.1.r.a 12 19.d odd 6 2 inner
1805.1.r.a 12 19.e even 9 3 inner
1805.1.r.a 12 19.f odd 18 3 inner
1805.1.r.a 12 95.g even 4 1 inner
1805.1.r.a 12 95.l even 12 2 inner
1805.1.r.a 12 95.m odd 12 2 inner
1805.1.r.a 12 95.q odd 36 3 inner
1805.1.r.a 12 95.r even 36 3 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1805, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$(T^{6} - T^{3} + 1)^{2}$$
$7$ $$(T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{3}$$
$11$ $$T^{12}$$
$13$ $$T^{12}$$
$17$ $$T^{12} - 4 T^{9} + \cdots + 64$$
$19$ $$T^{12}$$
$23$ $$T^{12} + 4 T^{9} + \cdots + 64$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$T^{12}$$
$43$ $$T^{12} - 4 T^{9} + \cdots + 64$$
$47$ $$T^{12} - 4 T^{9} + \cdots + 64$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12}$$
$73$ $$T^{12} - 4 T^{9} + \cdots + 64$$
$79$ $$T^{12}$$
$83$ $$(T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{3}$$
$89$ $$T^{12}$$
$97$ $$T^{12}$$