# 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

## Newspace parameters

 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

 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 Defining polynomial: $$x^{12} - 4 x^{9} + 17 x^{6} + 4 x^{3} + 1$$ 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

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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} + 21$$$$)/34$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{10} + 55 \nu$$$$)/34$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} - 55 \nu^{2}$$$$)/34$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{11} + 89 \nu^{2}$$$$)/34$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{9} + 17 \nu^{6} - 85 \nu^{3} + 1$$$$)/34$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{10} - 17 \nu^{7} + 68 \nu^{4} + 16 \nu$$$$)/17$$ $$\beta_{8}$$ $$=$$ $$($$$$-4 \nu^{9} + 17 \nu^{6} - 68 \nu^{3} + 1$$$$)/17$$ $$\beta_{9}$$ $$=$$ $$($$$$-12 \nu^{10} + 51 \nu^{7} - 221 \nu^{4} + 3 \nu$$$$)/34$$ $$\beta_{10}$$ $$=$$ $$($$$$-12 \nu^{11} + 51 \nu^{8} - 221 \nu^{5} + 3 \nu^{2}$$$$)/34$$ $$\beta_{11}$$ $$=$$ $$($$$$21 \nu^{11} - 85 \nu^{8} + 357 \nu^{5} + 84 \nu^{2}$$$$)/34$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} - 2 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{9} - 3 \beta_{7} + 3 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{11} - 5 \beta_{10} + 3 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$5 \beta_{8} - 8 \beta_{6} + 8 \beta_{2} - 5$$ $$\nu^{7}$$ $$=$$ $$-8 \beta_{9} - 13 \beta_{7} + 8 \beta_{3}$$ $$\nu^{8}$$ $$=$$ $$-13 \beta_{11} - 21 \beta_{10} - 21 \beta_{4}$$ $$\nu^{9}$$ $$=$$ $$34 \beta_{2} - 21$$ $$\nu^{10}$$ $$=$$ $$34 \beta_{3} - 55 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-55 \beta_{5} - 89 \beta_{4}$$

## 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.

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 1111.2 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 $$T_{13}^{12} + 4 T_{13}^{9} + 17 T_{13}^{6} - 4 T_{13}^{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

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