# Properties

 Label 1859.1.k.b Level $1859$ Weight $1$ Character orbit 1859.k Analytic conductor $0.928$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1859.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.927761858485$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.6424482779.1 Artin image: $C_6\times D_7$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{42} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} -\beta_{5} q^{4} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} -\beta_{5} q^{4} + \beta_{2} q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{9} + ( 1 - \beta_{5} ) q^{11} + \beta_{3} q^{12} + ( -1 + \beta_{1} + \beta_{5} ) q^{15} + ( -1 + \beta_{5} ) q^{16} + ( \beta_{1} - \beta_{2} ) q^{20} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{23} + ( \beta_{2} - \beta_{3} ) q^{25} + ( -1 + \beta_{2} ) q^{27} + ( 1 - \beta_{2} + \beta_{3} ) q^{31} + ( \beta_{3} + \beta_{4} ) q^{33} + ( -1 + \beta_{1} + \beta_{5} ) q^{36} -\beta_{4} q^{37} - q^{44} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{45} + ( 1 - \beta_{2} + \beta_{3} ) q^{47} + ( -\beta_{3} - \beta_{4} ) q^{48} + ( -1 + \beta_{5} ) q^{49} -\beta_{2} q^{53} + \beta_{1} q^{55} + ( \beta_{3} + \beta_{4} ) q^{59} + ( 1 - \beta_{2} ) q^{60} + q^{64} + ( 2 - 2 \beta_{5} ) q^{67} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{69} + ( \beta_{3} + \beta_{4} ) q^{71} + ( 1 - \beta_{5} ) q^{75} -\beta_{1} q^{80} -\beta_{4} q^{81} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{89} + ( -1 + \beta_{2} - \beta_{3} ) q^{92} + ( -1 + \beta_{4} + \beta_{5} ) q^{93} + ( \beta_{1} - \beta_{2} ) q^{97} + ( -1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{3} - 3 q^{4} + 2 q^{5} - 2 q^{9} + O(q^{10})$$ $$6 q + q^{3} - 3 q^{4} + 2 q^{5} - 2 q^{9} + 3 q^{11} - 2 q^{12} - 2 q^{15} - 3 q^{16} - q^{20} + q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{31} - q^{33} - 2 q^{36} - q^{37} - 6 q^{44} + 4 q^{45} + 2 q^{47} + q^{48} - 3 q^{49} - 2 q^{53} + q^{55} - q^{59} + 4 q^{60} + 6 q^{64} + 6 q^{67} + 2 q^{69} - q^{71} + 3 q^{75} - q^{80} - q^{81} - q^{89} - 2 q^{92} - 2 q^{93} - q^{97} - 4 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times$$.

 $$n$$ $$508$$ $$1354$$ $$\chi(n)$$ $$-1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1374.1
 0.222521 − 0.385418i 0.900969 − 1.56052i −0.623490 + 1.07992i 0.222521 + 0.385418i 0.900969 + 1.56052i −0.623490 − 1.07992i
0 −0.623490 + 1.07992i −0.500000 0.866025i 0.445042 0 0 0 −0.277479 0.480608i 0
1374.2 0 0.222521 0.385418i −0.500000 0.866025i 1.80194 0 0 0 0.400969 + 0.694498i 0
1374.3 0 0.900969 1.56052i −0.500000 0.866025i −1.24698 0 0 0 −1.12349 1.94594i 0
1836.1 0 −0.623490 1.07992i −0.500000 + 0.866025i 0.445042 0 0 0 −0.277479 + 0.480608i 0
1836.2 0 0.222521 + 0.385418i −0.500000 + 0.866025i 1.80194 0 0 0 0.400969 0.694498i 0
1836.3 0 0.900969 + 1.56052i −0.500000 + 0.866025i −1.24698 0 0 0 −1.12349 + 1.94594i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1836.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
13.c even 3 1 inner
143.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.1.k.b 6
11.b odd 2 1 CM 1859.1.k.b 6
13.b even 2 1 1859.1.k.a 6
13.c even 3 1 1859.1.c.b yes 3
13.c even 3 1 inner 1859.1.k.b 6
13.d odd 4 2 1859.1.i.c 12
13.e even 6 1 1859.1.c.a 3
13.e even 6 1 1859.1.k.a 6
13.f odd 12 2 1859.1.d.a 6
13.f odd 12 2 1859.1.i.c 12
143.d odd 2 1 1859.1.k.a 6
143.g even 4 2 1859.1.i.c 12
143.i odd 6 1 1859.1.c.a 3
143.i odd 6 1 1859.1.k.a 6
143.k odd 6 1 1859.1.c.b yes 3
143.k odd 6 1 inner 1859.1.k.b 6
143.o even 12 2 1859.1.d.a 6
143.o even 12 2 1859.1.i.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.1.c.a 3 13.e even 6 1
1859.1.c.a 3 143.i odd 6 1
1859.1.c.b yes 3 13.c even 3 1
1859.1.c.b yes 3 143.k odd 6 1
1859.1.d.a 6 13.f odd 12 2
1859.1.d.a 6 143.o even 12 2
1859.1.i.c 12 13.d odd 4 2
1859.1.i.c 12 13.f odd 12 2
1859.1.i.c 12 143.g even 4 2
1859.1.i.c 12 143.o even 12 2
1859.1.k.a 6 13.b even 2 1
1859.1.k.a 6 13.e even 6 1
1859.1.k.a 6 143.d odd 2 1
1859.1.k.a 6 143.i odd 6 1
1859.1.k.b 6 1.a even 1 1 trivial
1859.1.k.b 6 11.b odd 2 1 CM
1859.1.k.b 6 13.c even 3 1 inner
1859.1.k.b 6 143.k odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1859, [\chi])$$:

 $$T_{2}$$ $$T_{5}^{3} - T_{5}^{2} - 2 T_{5} + 1$$

## Hecke characteristic polynomials

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