# Properties

 Label 1859.1.k.a 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: $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ 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: $C_3\times D_7$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{21} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} - \beta_{5} q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + b4 * q^3 - b5 * q^4 - b2 * q^5 + (-b5 + b2 - b1) * q^9 $$q + \beta_{4} q^{3} - \beta_{5} q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{9} + (\beta_{5} - 1) q^{11} + \beta_{3} q^{12} + ( - \beta_{5} - \beta_1 + 1) q^{15} + (\beta_{5} - 1) q^{16} + (\beta_{2} - \beta_1) q^{20} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{23} + ( - \beta_{3} + \beta_{2}) q^{25} + (\beta_{2} - 1) q^{27} + ( - \beta_{3} + \beta_{2} - 1) q^{31} + ( - \beta_{4} - \beta_{3}) q^{33} + (\beta_{5} + \beta_1 - 1) q^{36} + \beta_{4} q^{37} + q^{44} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{45} + ( - \beta_{3} + \beta_{2} - 1) q^{47} + ( - \beta_{4} - \beta_{3}) q^{48} + (\beta_{5} - 1) q^{49} - \beta_{2} q^{53} + \beta_1 q^{55} + ( - \beta_{4} - \beta_{3}) q^{59} + (\beta_{2} - 1) q^{60} + q^{64} + (2 \beta_{5} - 2) q^{67} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{69} + ( - \beta_{4} - \beta_{3}) q^{71} + ( - \beta_{5} + 1) q^{75} + \beta_1 q^{80} - \beta_{4} q^{81} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{89} + ( - \beta_{3} + \beta_{2} - 1) q^{92} + ( - \beta_{5} - \beta_{4} + 1) q^{93} + (\beta_{2} - \beta_1) q^{97} + ( - \beta_{2} + 1) q^{99}+O(q^{100})$$ q + b4 * q^3 - b5 * q^4 - b2 * q^5 + (-b5 + b2 - b1) * q^9 + (b5 - 1) * q^11 + b3 * q^12 + (-b5 - b1 + 1) * q^15 + (b5 - 1) * q^16 + (b2 - b1) * q^20 + (-b5 - b4 - b1 + 1) * q^23 + (-b3 + b2) * q^25 + (b2 - 1) * q^27 + (-b3 + b2 - 1) * q^31 + (-b4 - b3) * q^33 + (b5 + b1 - 1) * q^36 + b4 * q^37 + q^44 + (-b5 + b4 + b3) * q^45 + (-b3 + b2 - 1) * q^47 + (-b4 - b3) * q^48 + (b5 - 1) * q^49 - b2 * q^53 + b1 * q^55 + (-b4 - b3) * q^59 + (b2 - 1) * q^60 + q^64 + (2*b5 - 2) * q^67 + (b5 + b4 + b3) * q^69 + (-b4 - b3) * q^71 + (-b5 + 1) * q^75 + b1 * q^80 - b4 * q^81 + (-b5 - b4 - b1 + 1) * q^89 + (-b3 + b2 - 1) * q^92 + (-b5 - b4 + 1) * q^93 + (b2 - b1) * q^97 + (-b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{3} - 3 q^{4} - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 6 * q + q^3 - 3 * q^4 - 2 * q^5 - 2 * q^9 $$6 q + q^{3} - 3 q^{4} - 2 q^{5} - 2 q^{9} - 3 q^{11} - 2 q^{12} + 2 q^{15} - 3 q^{16} + q^{20} + q^{23} + 4 q^{25} - 4 q^{27} - 2 q^{31} + q^{33} - 2 q^{36} + q^{37} + 6 q^{44} - 4 q^{45} - 2 q^{47} + q^{48} - 3 q^{49} - 2 q^{53} + q^{55} + q^{59} - 4 q^{60} + 6 q^{64} - 6 q^{67} + 2 q^{69} + q^{71} + 3 q^{75} + q^{80} - q^{81} + q^{89} - 2 q^{92} + 2 q^{93} + q^{97} + 4 q^{99}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$$ (-v^5 + 3*v^4 - 9*v^3 + 5*v^2 - 2*v + 6) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$$ (-3*v^5 + 9*v^4 - 14*v^3 + 15*v^2 - 6*v + 18) / 13 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$$ (-4*v^5 - v^4 - 10*v^3 - 6*v^2 - 34*v - 2) / 13 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$$ (-6*v^5 + 5*v^4 - 15*v^3 - 9*v^2 - 25*v + 10) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ -b5 + b4 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_{2}$$ b3 - 3*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$$ 2*b5 - 3*b4 - 4*b1 - 2 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$$ b5 - 4*b4 - 4*b3 + 9*b2 - 9*b1

## 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.a 6
11.b odd 2 1 CM 1859.1.k.a 6
13.b even 2 1 1859.1.k.b 6
13.c even 3 1 1859.1.c.a 3
13.c even 3 1 inner 1859.1.k.a 6
13.d odd 4 2 1859.1.i.c 12
13.e even 6 1 1859.1.c.b yes 3
13.e even 6 1 1859.1.k.b 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.b 6
143.g even 4 2 1859.1.i.c 12
143.i odd 6 1 1859.1.c.b yes 3
143.i odd 6 1 1859.1.k.b 6
143.k odd 6 1 1859.1.c.a 3
143.k odd 6 1 inner 1859.1.k.a 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.c even 3 1
1859.1.c.a 3 143.k odd 6 1
1859.1.c.b yes 3 13.e even 6 1
1859.1.c.b yes 3 143.i 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 1.a even 1 1 trivial
1859.1.k.a 6 11.b odd 2 1 CM
1859.1.k.a 6 13.c even 3 1 inner
1859.1.k.a 6 143.k odd 6 1 inner
1859.1.k.b 6 13.b even 2 1
1859.1.k.b 6 13.e even 6 1
1859.1.k.b 6 143.d odd 2 1
1859.1.k.b 6 143.i odd 6 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}^{3} + T_{5}^{2} - 2T_{5} - 1$$ T5^3 + T5^2 - 2*T5 - 1

## Hecke characteristic polynomials

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