# Properties

 Label 1805.1.o.b Level $1805$ Weight $1$ Character orbit 1805.o Analytic conductor $0.901$ Analytic rank $0$ Dimension $12$ Projective image $D_{4}$ CM discriminant -95 Inner twists $24$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,1,Mod(299,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 7]))

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

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

chi := DirichletCharacter("1805.299");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1805.o (of order $$18$$, degree $$6$$, 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: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: 12.0.101559956668416.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 8x^{6} + 64$$ x^12 + 8*x^6 + 64 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: $C_9\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{72} - \cdots)$$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} + \beta_{5}) q^{2} + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{4}) q^{4} - \beta_{2} q^{5} - 2 \beta_{4} q^{6} + \beta_{10} q^{9}+O(q^{10})$$ q + (b11 + b5) * q^2 + b5 * q^3 + (-b10 - b4) * q^4 - b2 * q^5 - 2*b4 * q^6 + b10 * q^9 $$q + (\beta_{11} + \beta_{5}) q^{2} + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{4}) q^{4} - \beta_{2} q^{5} - 2 \beta_{4} q^{6} + \beta_{10} q^{9} + \beta_1 q^{10} + \beta_{3} q^{12} + (\beta_{7} + \beta_1) q^{13} - \beta_{7} q^{15} + (\beta_{8} + \beta_{2}) q^{16} - \beta_{9} q^{18} - q^{20} + \beta_{4} q^{25} + ( - 2 \beta_{6} - 2) q^{26} + 2 \beta_{6} q^{30} + ( - \beta_{7} - \beta_1) q^{32} + \beta_{8} q^{36} + \beta_{9} q^{37} - 2 q^{39} + (\beta_{6} + 1) q^{45} - \beta_1 q^{48} + \beta_{6} q^{49} - \beta_{3} q^{50} - \beta_{11} q^{52} + \beta_{7} q^{53} - \beta_{5} q^{60} + (\beta_{6} + 1) q^{64} + ( - \beta_{9} - \beta_{3}) q^{65} - \beta_1 q^{67} - 2 \beta_{8} q^{74} + \beta_{9} q^{75} + ( - 2 \beta_{11} - 2 \beta_{5}) q^{78} + ( - \beta_{10} - \beta_{4}) q^{80} - \beta_{2} q^{81} + \beta_{11} q^{90} + 2 q^{96} + ( - \beta_{11} - \beta_{5}) q^{97} - \beta_{5} q^{98}+O(q^{100})$$ q + (b11 + b5) * q^2 + b5 * q^3 + (-b10 - b4) * q^4 - b2 * q^5 - 2*b4 * q^6 + b10 * q^9 + b1 * q^10 + b3 * q^12 + (b7 + b1) * q^13 - b7 * q^15 + (b8 + b2) * q^16 - b9 * q^18 - q^20 + b4 * q^25 + (-2*b6 - 2) * q^26 + 2*b6 * q^30 + (-b7 - b1) * q^32 + b8 * q^36 + b9 * q^37 - 2 * q^39 + (b6 + 1) * q^45 - b1 * q^48 + b6 * q^49 - b3 * q^50 - b11 * q^52 + b7 * q^53 - b5 * q^60 + (b6 + 1) * q^64 + (-b9 - b3) * q^65 - b1 * q^67 - 2*b8 * q^74 + b9 * q^75 + (-2*b11 - 2*b5) * q^78 + (-b10 - b4) * q^80 - b2 * q^81 + b11 * q^90 + 2 * q^96 + (-b11 - b5) * q^97 - b5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q - 12 q^{20} - 12 q^{26} - 12 q^{30} - 24 q^{39} + 6 q^{45} - 6 q^{49} + 6 q^{64} + 24 q^{96}+O(q^{100})$$ 12 * q - 12 * q^20 - 12 * q^26 - 12 * q^30 - 24 * q^39 + 6 * q^45 - 6 * q^49 + 6 * q^64 + 24 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 8x^{6} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8 $$\beta_{8}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{9} ) / 16$$ (v^9) / 16 $$\beta_{10}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{11}$$ $$=$$ $$( \nu^{11} ) / 32$$ (v^11) / 32
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3 $$\nu^{4}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{5}$$ $$=$$ $$4\beta_{5}$$ 4*b5 $$\nu^{6}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{7}$$ $$=$$ $$8\beta_{7}$$ 8*b7 $$\nu^{8}$$ $$=$$ $$16\beta_{8}$$ 16*b8 $$\nu^{9}$$ $$=$$ $$16\beta_{9}$$ 16*b9 $$\nu^{10}$$ $$=$$ $$32\beta_{10}$$ 32*b10 $$\nu^{11}$$ $$=$$ $$32\beta_{11}$$ 32*b11

## Character values

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

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$\beta_{4} + \beta_{10}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 1.32893 + 0.483690i −1.32893 − 0.483690i 0.245576 − 1.39273i −0.245576 + 1.39273i 1.08335 + 0.909039i −1.08335 − 0.909039i 1.32893 − 0.483690i −1.32893 + 0.483690i 1.08335 − 0.909039i −1.08335 + 0.909039i 0.245576 + 1.39273i −0.245576 − 1.39273i
−1.32893 + 0.483690i −0.245576 + 1.39273i 0.766044 0.642788i −0.766044 0.642788i −0.347296 1.96962i 0 0 −0.939693 0.342020i 1.32893 + 0.483690i
299.2 1.32893 0.483690i 0.245576 1.39273i 0.766044 0.642788i −0.766044 0.642788i −0.347296 1.96962i 0 0 −0.939693 0.342020i −1.32893 0.483690i
694.1 −0.245576 1.39273i 1.08335 0.909039i −0.939693 + 0.342020i 0.939693 + 0.342020i −1.53209 1.28558i 0 0 0.173648 0.984808i 0.245576 1.39273i
694.2 0.245576 + 1.39273i −1.08335 + 0.909039i −0.939693 + 0.342020i 0.939693 + 0.342020i −1.53209 1.28558i 0 0 0.173648 0.984808i −0.245576 + 1.39273i
849.1 −1.08335 + 0.909039i −1.32893 0.483690i 0.173648 0.984808i −0.173648 0.984808i 1.87939 0.684040i 0 0 0.766044 + 0.642788i 1.08335 + 0.909039i
849.2 1.08335 0.909039i 1.32893 + 0.483690i 0.173648 0.984808i −0.173648 0.984808i 1.87939 0.684040i 0 0 0.766044 + 0.642788i −1.08335 0.909039i
984.1 −1.32893 0.483690i −0.245576 1.39273i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.347296 + 1.96962i 0 0 −0.939693 + 0.342020i 1.32893 0.483690i
984.2 1.32893 + 0.483690i 0.245576 + 1.39273i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.347296 + 1.96962i 0 0 −0.939693 + 0.342020i −1.32893 + 0.483690i
1029.1 −1.08335 0.909039i −1.32893 + 0.483690i 0.173648 + 0.984808i −0.173648 + 0.984808i 1.87939 + 0.684040i 0 0 0.766044 0.642788i 1.08335 0.909039i
1029.2 1.08335 + 0.909039i 1.32893 0.483690i 0.173648 + 0.984808i −0.173648 + 0.984808i 1.87939 + 0.684040i 0 0 0.766044 0.642788i −1.08335 + 0.909039i
1199.1 −0.245576 + 1.39273i 1.08335 + 0.909039i −0.939693 0.342020i 0.939693 0.342020i −1.53209 + 1.28558i 0 0 0.173648 + 0.984808i 0.245576 + 1.39273i
1199.2 0.245576 1.39273i −1.08335 0.909039i −0.939693 0.342020i 0.939693 0.342020i −1.53209 + 1.28558i 0 0 0.173648 + 0.984808i −0.245576 1.39273i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 299.2 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 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner
95.h odd 6 2 inner
95.i even 6 2 inner
95.o odd 18 3 inner
95.p even 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.o.b 12
5.b even 2 1 inner 1805.1.o.b 12
19.b odd 2 1 inner 1805.1.o.b 12
19.c even 3 2 inner 1805.1.o.b 12
19.d odd 6 2 inner 1805.1.o.b 12
19.e even 9 1 95.1.d.b 2
19.e even 9 2 1805.1.h.b 4
19.e even 9 3 inner 1805.1.o.b 12
19.f odd 18 1 95.1.d.b 2
19.f odd 18 2 1805.1.h.b 4
19.f odd 18 3 inner 1805.1.o.b 12
57.j even 18 1 855.1.g.c 2
57.l odd 18 1 855.1.g.c 2
76.k even 18 1 1520.1.m.b 2
76.l odd 18 1 1520.1.m.b 2
95.d odd 2 1 CM 1805.1.o.b 12
95.h odd 6 2 inner 1805.1.o.b 12
95.i even 6 2 inner 1805.1.o.b 12
95.o odd 18 1 95.1.d.b 2
95.o odd 18 2 1805.1.h.b 4
95.o odd 18 3 inner 1805.1.o.b 12
95.p even 18 1 95.1.d.b 2
95.p even 18 2 1805.1.h.b 4
95.p even 18 3 inner 1805.1.o.b 12
95.q odd 36 2 475.1.c.b 2
95.r even 36 2 475.1.c.b 2
285.bd odd 18 1 855.1.g.c 2
285.bf even 18 1 855.1.g.c 2
380.ba odd 18 1 1520.1.m.b 2
380.bb even 18 1 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 19.e even 9 1
95.1.d.b 2 19.f odd 18 1
95.1.d.b 2 95.o odd 18 1
95.1.d.b 2 95.p even 18 1
475.1.c.b 2 95.q odd 36 2
475.1.c.b 2 95.r even 36 2
855.1.g.c 2 57.j even 18 1
855.1.g.c 2 57.l odd 18 1
855.1.g.c 2 285.bd odd 18 1
855.1.g.c 2 285.bf even 18 1
1520.1.m.b 2 76.k even 18 1
1520.1.m.b 2 76.l odd 18 1
1520.1.m.b 2 380.ba odd 18 1
1520.1.m.b 2 380.bb even 18 1
1805.1.h.b 4 19.e even 9 2
1805.1.h.b 4 19.f odd 18 2
1805.1.h.b 4 95.o odd 18 2
1805.1.h.b 4 95.p even 18 2
1805.1.o.b 12 1.a even 1 1 trivial
1805.1.o.b 12 5.b even 2 1 inner
1805.1.o.b 12 19.b odd 2 1 inner
1805.1.o.b 12 19.c even 3 2 inner
1805.1.o.b 12 19.d odd 6 2 inner
1805.1.o.b 12 19.e even 9 3 inner
1805.1.o.b 12 19.f odd 18 3 inner
1805.1.o.b 12 95.d odd 2 1 CM
1805.1.o.b 12 95.h odd 6 2 inner
1805.1.o.b 12 95.i even 6 2 inner
1805.1.o.b 12 95.o odd 18 3 inner
1805.1.o.b 12 95.p even 18 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 8T_{2}^{6} + 64$$ acting on $$S_{1}^{\mathrm{new}}(1805, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 8T^{6} + 64$$
$3$ $$T^{12} + 8T^{6} + 64$$
$5$ $$(T^{6} - T^{3} + 1)^{2}$$
$7$ $$T^{12}$$
$11$ $$T^{12}$$
$13$ $$T^{12} + 8T^{6} + 64$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$(T^{2} - 2)^{6}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12} + 8T^{6} + 64$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$T^{12} + 8T^{6} + 64$$
$71$ $$T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12} + 8T^{6} + 64$$