# Properties

 Label 1444.1.l.b Level $1444$ Weight $1$ Character orbit 1444.l Analytic conductor $0.721$ Analytic rank $0$ Dimension $12$ Projective image $D_{5}$ CM discriminant -4 Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,1,Mod(99,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 2]))

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

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

chi := DirichletCharacter("1444.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.l (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: $$0.720649878242$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: 12.0.6053445140625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1$$ x^12 - 4*x^9 + 17*x^6 + 4*x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.2085136.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{9} + \beta_{3}) q^{2} + (\beta_{11} - \beta_{5}) q^{4} + (\beta_{9} - \beta_{7} + \beta_1) q^{5} + \beta_{8} q^{8} + \beta_{5} q^{9}+O(q^{10})$$ q + (-b9 + b3) * q^2 + (b11 - b5) * q^4 + (b9 - b7 + b1) * q^5 + b8 * q^8 + b5 * q^9 $$q + ( - \beta_{9} + \beta_{3}) q^{2} + (\beta_{11} - \beta_{5}) q^{4} + (\beta_{9} - \beta_{7} + \beta_1) q^{5} + \beta_{8} q^{8} + \beta_{5} q^{9} + (\beta_{5} - \beta_{4}) q^{10} + ( - \beta_{10} - \beta_{4}) q^{13} + \beta_{3} q^{16} - \beta_{7} q^{17} - q^{18} + ( - \beta_{2} - 1) q^{20} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{25} + (\beta_{6} - \beta_{2}) q^{26} + \beta_{4} q^{29} + \beta_{11} q^{32} + \beta_{10} q^{34} + (\beta_{9} - \beta_{3}) q^{36} + (\beta_{2} + 1) q^{37} + (\beta_{9} - \beta_{7} - \beta_{3}) q^{40} + (\beta_{3} + \beta_1) q^{41} + ( - \beta_{8} - \beta_{6} + \beta_{2} + 1) q^{45} - \beta_{8} q^{49} + ( - \beta_{8} - \beta_{6} + \beta_{2} + 1) q^{50} + ( - \beta_{7} + \beta_1) q^{52} + (\beta_{11} - \beta_{10} - \beta_{5}) q^{53} + \beta_{2} q^{58} - \beta_{10} q^{61} + (\beta_{8} - 1) q^{64} - \beta_{8} q^{65} - \beta_{6} q^{68} + ( - \beta_{11} + \beta_{5}) q^{72} + ( - \beta_{3} - \beta_1) q^{73} + ( - \beta_{9} + \beta_{7} + \beta_{3}) q^{74} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{80} - \beta_{9} q^{81} + (\beta_{11} - \beta_{10} - \beta_{4}) q^{82} - \beta_{5} q^{85} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{89} + ( - \beta_{9} + \beta_{7} - \beta_1) q^{90} + \beta_{7} q^{97} - \beta_{3} q^{98}+O(q^{100})$$ q + (-b9 + b3) * q^2 + (b11 - b5) * q^4 + (b9 - b7 + b1) * q^5 + b8 * q^8 + b5 * q^9 + (b5 - b4) * q^10 + (-b10 - b4) * q^13 + b3 * q^16 - b7 * q^17 - q^18 + (-b2 - 1) * q^20 + (-b11 + b10 + b4) * q^25 + (b6 - b2) * q^26 + b4 * q^29 + b11 * q^32 + b10 * q^34 + (b9 - b3) * q^36 + (b2 + 1) * q^37 + (b9 - b7 - b3) * q^40 + (b3 + b1) * q^41 + (-b8 - b6 + b2 + 1) * q^45 - b8 * q^49 + (-b8 - b6 + b2 + 1) * q^50 + (-b7 + b1) * q^52 + (b11 - b10 - b5) * q^53 + b2 * q^58 - b10 * q^61 + (b8 - 1) * q^64 - b8 * q^65 - b6 * q^68 + (-b11 + b5) * q^72 + (-b3 - b1) * q^73 + (-b9 + b7 + b3) * q^74 + (-b11 + b10 + b5) * q^80 - b9 * q^81 + (b11 - b10 - b4) * q^82 - b5 * q^85 + (-b11 + b10 + b4) * q^89 + (-b9 + b7 - b1) * q^90 + b7 * q^97 - b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{8}+O(q^{10})$$ 12 * q + 6 * q^8 $$12 q + 6 q^{8} - 12 q^{18} - 6 q^{20} + 3 q^{26} + 6 q^{37} + 3 q^{45} - 6 q^{49} + 3 q^{50} - 6 q^{58} - 6 q^{64} - 6 q^{65} + 3 q^{68}+O(q^{100})$$ 12 * q + 6 * q^8 - 12 * q^18 - 6 * q^20 + 3 * q^26 + 6 * q^37 + 3 * q^45 - 6 * q^49 + 3 * q^50 - 6 * q^58 - 6 * q^64 - 6 * q^65 + 3 * q^68

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{9} + 21 ) / 34$$ (v^9 + 21) / 34 $$\beta_{3}$$ $$=$$ $$( \nu^{10} + 55\nu ) / 34$$ (v^10 + 55*v) / 34 $$\beta_{4}$$ $$=$$ $$( -\nu^{11} - 55\nu^{2} ) / 34$$ (-v^11 - 55*v^2) / 34 $$\beta_{5}$$ $$=$$ $$( \nu^{11} + 89\nu^{2} ) / 34$$ (v^11 + 89*v^2) / 34 $$\beta_{6}$$ $$=$$ $$( -4\nu^{9} + 17\nu^{6} - 85\nu^{3} + 1 ) / 34$$ (-4*v^9 + 17*v^6 - 85*v^3 + 1) / 34 $$\beta_{7}$$ $$=$$ $$( 4\nu^{10} - 17\nu^{7} + 68\nu^{4} + 16\nu ) / 17$$ (4*v^10 - 17*v^7 + 68*v^4 + 16*v) / 17 $$\beta_{8}$$ $$=$$ $$( -4\nu^{9} + 17\nu^{6} - 68\nu^{3} + 1 ) / 17$$ (-4*v^9 + 17*v^6 - 68*v^3 + 1) / 17 $$\beta_{9}$$ $$=$$ $$( -12\nu^{10} + 51\nu^{7} - 221\nu^{4} + 3\nu ) / 34$$ (-12*v^10 + 51*v^7 - 221*v^4 + 3*v) / 34 $$\beta_{10}$$ $$=$$ $$( -12\nu^{11} + 51\nu^{8} - 221\nu^{5} + 3\nu^{2} ) / 34$$ (-12*v^11 + 51*v^8 - 221*v^5 + 3*v^2) / 34 $$\beta_{11}$$ $$=$$ $$( 21\nu^{11} - 85\nu^{8} + 357\nu^{5} + 84\nu^{2} ) / 34$$ (21*v^11 - 85*v^8 + 357*v^5 + 84*v^2) / 34
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4}$$ b5 + b4 $$\nu^{3}$$ $$=$$ $$\beta_{8} - 2\beta_{6}$$ b8 - 2*b6 $$\nu^{4}$$ $$=$$ $$-2\beta_{9} - 3\beta_{7} + 3\beta_1$$ -2*b9 - 3*b7 + 3*b1 $$\nu^{5}$$ $$=$$ $$-3\beta_{11} - 5\beta_{10} + 3\beta_{5}$$ -3*b11 - 5*b10 + 3*b5 $$\nu^{6}$$ $$=$$ $$5\beta_{8} - 8\beta_{6} + 8\beta_{2} - 5$$ 5*b8 - 8*b6 + 8*b2 - 5 $$\nu^{7}$$ $$=$$ $$-8\beta_{9} - 13\beta_{7} + 8\beta_{3}$$ -8*b9 - 13*b7 + 8*b3 $$\nu^{8}$$ $$=$$ $$-13\beta_{11} - 21\beta_{10} - 21\beta_{4}$$ -13*b11 - 21*b10 - 21*b4 $$\nu^{9}$$ $$=$$ $$34\beta_{2} - 21$$ 34*b2 - 21 $$\nu^{10}$$ $$=$$ $$34\beta_{3} - 55\beta_1$$ 34*b3 - 55*b1 $$\nu^{11}$$ $$=$$ $$-55\beta_{5} - 89\beta_{4}$$ -55*b5 - 89*b4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$$.

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$-1$$ $$-\beta_{5} + \beta_{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0.107320 + 0.608645i −0.280969 − 1.59345i −0.580762 + 0.211380i 1.52045 − 0.553400i 0.107320 − 0.608645i −0.280969 + 1.59345i −0.580762 − 0.211380i 1.52045 + 0.553400i −1.23949 − 1.04005i 0.473442 + 0.397265i −1.23949 + 1.04005i 0.473442 − 0.397265i
0.939693 + 0.342020i 0 0.766044 + 0.642788i −1.23949 + 1.04005i 0 0 0.500000 + 0.866025i −0.939693 + 0.342020i −1.52045 + 0.553400i
99.2 0.939693 + 0.342020i 0 0.766044 + 0.642788i 0.473442 0.397265i 0 0 0.500000 + 0.866025i −0.939693 + 0.342020i 0.580762 0.211380i
415.1 −0.766044 0.642788i 0 0.173648 + 0.984808i −0.280969 + 1.59345i 0 0 0.500000 0.866025i 0.766044 0.642788i 1.23949 1.04005i
415.2 −0.766044 0.642788i 0 0.173648 + 0.984808i 0.107320 0.608645i 0 0 0.500000 0.866025i 0.766044 0.642788i −0.473442 + 0.397265i
423.1 0.939693 0.342020i 0 0.766044 0.642788i −1.23949 1.04005i 0 0 0.500000 0.866025i −0.939693 0.342020i −1.52045 0.553400i
423.2 0.939693 0.342020i 0 0.766044 0.642788i 0.473442 + 0.397265i 0 0 0.500000 0.866025i −0.939693 0.342020i 0.580762 + 0.211380i
595.1 −0.766044 + 0.642788i 0 0.173648 0.984808i −0.280969 1.59345i 0 0 0.500000 + 0.866025i 0.766044 + 0.642788i 1.23949 + 1.04005i
595.2 −0.766044 + 0.642788i 0 0.173648 0.984808i 0.107320 + 0.608645i 0 0 0.500000 + 0.866025i 0.766044 + 0.642788i −0.473442 0.397265i
967.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i −0.580762 + 0.211380i 0 0 0.500000 0.866025i 0.173648 + 0.984808i −0.107320 0.608645i
967.2 −0.173648 + 0.984808i 0 −0.939693 0.342020i 1.52045 0.553400i 0 0 0.500000 0.866025i 0.173648 + 0.984808i 0.280969 + 1.59345i
1111.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −0.580762 0.211380i 0 0 0.500000 + 0.866025i 0.173648 0.984808i −0.107320 + 0.608645i
1111.2 −0.173648 0.984808i 0 −0.939693 + 0.342020i 1.52045 + 0.553400i 0 0 0.500000 + 0.866025i 0.173648 0.984808i 0.280969 1.59345i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
19.c even 3 2 inner
19.e even 9 3 inner
76.g odd 6 2 inner
76.l odd 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.l.b 12
4.b odd 2 1 CM 1444.1.l.b 12
19.b odd 2 1 1444.1.l.a 12
19.c even 3 2 inner 1444.1.l.b 12
19.d odd 6 2 1444.1.l.a 12
19.e even 9 1 1444.1.b.a 2
19.e even 9 2 1444.1.g.b 4
19.e even 9 3 inner 1444.1.l.b 12
19.f odd 18 1 1444.1.b.b yes 2
19.f odd 18 2 1444.1.g.a 4
19.f odd 18 3 1444.1.l.a 12
76.d even 2 1 1444.1.l.a 12
76.f even 6 2 1444.1.l.a 12
76.g odd 6 2 inner 1444.1.l.b 12
76.k even 18 1 1444.1.b.b yes 2
76.k even 18 2 1444.1.g.a 4
76.k even 18 3 1444.1.l.a 12
76.l odd 18 1 1444.1.b.a 2
76.l odd 18 2 1444.1.g.b 4
76.l odd 18 3 inner 1444.1.l.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.b.a 2 19.e even 9 1
1444.1.b.a 2 76.l odd 18 1
1444.1.b.b yes 2 19.f odd 18 1
1444.1.b.b yes 2 76.k even 18 1
1444.1.g.a 4 19.f odd 18 2
1444.1.g.a 4 76.k even 18 2
1444.1.g.b 4 19.e even 9 2
1444.1.g.b 4 76.l odd 18 2
1444.1.l.a 12 19.b odd 2 1
1444.1.l.a 12 19.d odd 6 2
1444.1.l.a 12 19.f odd 18 3
1444.1.l.a 12 76.d even 2 1
1444.1.l.a 12 76.f even 6 2
1444.1.l.a 12 76.k even 18 3
1444.1.l.b 12 1.a even 1 1 trivial
1444.1.l.b 12 4.b odd 2 1 CM
1444.1.l.b 12 19.c even 3 2 inner
1444.1.l.b 12 19.e even 9 3 inner
1444.1.l.b 12 76.g odd 6 2 inner
1444.1.l.b 12 76.l odd 18 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{12} + 4T_{13}^{9} + 17T_{13}^{6} - 4T_{13}^{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

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