# Properties

 Label 1859.1.i.a Level $1859$ Weight $1$ Character orbit 1859.i Analytic conductor $0.928$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -143 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.927761858485$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.20449.1 Artin image: $C_6\times D_5$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{30} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{4} + ( 2 - \beta_{1} + 2 \beta_{3} ) q^{6} -\beta_{1} q^{7} + q^{8} + ( -1 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -1 + \beta_{1} - \beta_{3} ) q^{4} + ( 2 - \beta_{1} + 2 \beta_{3} ) q^{6} -\beta_{1} q^{7} + q^{8} + ( -1 + \beta_{1} - \beta_{3} ) q^{9} -\beta_{3} q^{11} + ( -2 - \beta_{2} ) q^{12} - q^{14} + ( 2 + \beta_{2} ) q^{18} + ( -1 + \beta_{1} - \beta_{3} ) q^{19} + q^{21} + ( 1 - \beta_{1} + \beta_{3} ) q^{22} + ( -\beta_{1} - \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{24} + q^{25} - q^{27} -\beta_{3} q^{28} + ( 1 + \beta_{3} ) q^{32} + ( -1 + \beta_{1} - \beta_{3} ) q^{33} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{36} + ( 2 + \beta_{2} ) q^{38} + ( \beta_{1} + \beta_{2} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{42} + ( -1 - \beta_{2} ) q^{44} + ( -1 - \beta_{3} ) q^{46} + ( \beta_{1} + \beta_{2} ) q^{49} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{50} + ( -1 - \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{54} -\beta_{1} q^{56} + ( -2 - \beta_{2} ) q^{57} -\beta_{3} q^{63} + ( -1 - \beta_{2} ) q^{64} + ( 2 + \beta_{2} ) q^{66} + ( 1 + \beta_{3} ) q^{69} + ( -1 + \beta_{1} - \beta_{3} ) q^{72} + ( 1 + \beta_{2} ) q^{73} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{76} + \beta_{2} q^{77} + ( 1 + \beta_{3} ) q^{82} -\beta_{2} q^{83} + ( -1 + \beta_{1} - \beta_{3} ) q^{84} -\beta_{3} q^{88} + q^{92} + ( 1 + \beta_{2} ) q^{96} + ( 1 + \beta_{3} ) q^{98} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + q^{3} - q^{4} + 3 q^{6} - q^{7} + 4 q^{8} - q^{9} + O(q^{10})$$ $$4 q - q^{2} + q^{3} - q^{4} + 3 q^{6} - q^{7} + 4 q^{8} - q^{9} + 2 q^{11} - 6 q^{12} - 4 q^{14} + 6 q^{18} - q^{19} + 4 q^{21} + q^{22} + q^{23} + q^{24} + 4 q^{25} - 4 q^{27} + 2 q^{28} + 2 q^{32} - q^{33} - 3 q^{36} + 6 q^{38} - q^{41} - q^{42} - 2 q^{44} - 2 q^{46} - q^{49} - q^{50} - 2 q^{53} + q^{54} - q^{56} - 6 q^{57} + 2 q^{63} - 2 q^{64} + 6 q^{66} + 2 q^{69} - q^{72} + 2 q^{73} + q^{75} - 3 q^{76} - 2 q^{77} + 2 q^{82} + 2 q^{83} - q^{84} + 2 q^{88} + 4 q^{92} + 2 q^{96} + 2 q^{98} - 2 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

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

 $$n$$ $$508$$ $$1354$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
868.1
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
−0.809017 1.40126i 0.809017 + 1.40126i −0.809017 + 1.40126i 0 1.30902 2.26728i 0.309017 0.535233i 1.00000 −0.809017 + 1.40126i 0
868.2 0.309017 + 0.535233i −0.309017 0.535233i 0.309017 0.535233i 0 0.190983 0.330792i −0.809017 + 1.40126i 1.00000 0.309017 0.535233i 0
1330.1 −0.809017 + 1.40126i 0.809017 1.40126i −0.809017 1.40126i 0 1.30902 + 2.26728i 0.309017 + 0.535233i 1.00000 −0.809017 1.40126i 0
1330.2 0.309017 0.535233i −0.309017 + 0.535233i 0.309017 + 0.535233i 0 0.190983 + 0.330792i −0.809017 1.40126i 1.00000 0.309017 + 0.535233i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
143.d odd 2 1 CM by $$\Q(\sqrt{-143})$$
13.c even 3 1 inner
143.i 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.i.a 4
11.b odd 2 1 1859.1.i.b 4
13.b even 2 1 1859.1.i.b 4
13.c even 3 1 143.1.d.b yes 2
13.c even 3 1 inner 1859.1.i.a 4
13.d odd 4 2 1859.1.k.c 8
13.e even 6 1 143.1.d.a 2
13.e even 6 1 1859.1.i.b 4
13.f odd 12 2 1859.1.c.c 4
13.f odd 12 2 1859.1.k.c 8
39.h odd 6 1 1287.1.g.b 2
39.i odd 6 1 1287.1.g.a 2
52.i odd 6 1 2288.1.m.b 2
52.j odd 6 1 2288.1.m.a 2
65.l even 6 1 3575.1.h.f 2
65.n even 6 1 3575.1.h.e 2
65.q odd 12 2 3575.1.c.c 4
65.r odd 12 2 3575.1.c.d 4
143.d odd 2 1 CM 1859.1.i.a 4
143.g even 4 2 1859.1.k.c 8
143.i odd 6 1 143.1.d.b yes 2
143.i odd 6 1 inner 1859.1.i.a 4
143.k odd 6 1 143.1.d.a 2
143.k odd 6 1 1859.1.i.b 4
143.o even 12 2 1859.1.c.c 4
143.o even 12 2 1859.1.k.c 8
143.q even 15 2 1573.1.l.a 4
143.q even 15 2 1573.1.l.c 4
143.t odd 30 2 1573.1.l.b 4
143.t odd 30 2 1573.1.l.d 4
143.u even 30 2 1573.1.l.b 4
143.u even 30 2 1573.1.l.d 4
143.v odd 30 2 1573.1.l.a 4
143.v odd 30 2 1573.1.l.c 4
429.p even 6 1 1287.1.g.b 2
429.t even 6 1 1287.1.g.a 2
572.s even 6 1 2288.1.m.a 2
572.t even 6 1 2288.1.m.b 2
715.x odd 6 1 3575.1.h.f 2
715.bb odd 6 1 3575.1.h.e 2
715.bo even 12 2 3575.1.c.c 4
715.bq even 12 2 3575.1.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 13.e even 6 1
143.1.d.a 2 143.k odd 6 1
143.1.d.b yes 2 13.c even 3 1
143.1.d.b yes 2 143.i odd 6 1
1287.1.g.a 2 39.i odd 6 1
1287.1.g.a 2 429.t even 6 1
1287.1.g.b 2 39.h odd 6 1
1287.1.g.b 2 429.p even 6 1
1573.1.l.a 4 143.q even 15 2
1573.1.l.a 4 143.v odd 30 2
1573.1.l.b 4 143.t odd 30 2
1573.1.l.b 4 143.u even 30 2
1573.1.l.c 4 143.q even 15 2
1573.1.l.c 4 143.v odd 30 2
1573.1.l.d 4 143.t odd 30 2
1573.1.l.d 4 143.u even 30 2
1859.1.c.c 4 13.f odd 12 2
1859.1.c.c 4 143.o even 12 2
1859.1.i.a 4 1.a even 1 1 trivial
1859.1.i.a 4 13.c even 3 1 inner
1859.1.i.a 4 143.d odd 2 1 CM
1859.1.i.a 4 143.i odd 6 1 inner
1859.1.i.b 4 11.b odd 2 1
1859.1.i.b 4 13.b even 2 1
1859.1.i.b 4 13.e even 6 1
1859.1.i.b 4 143.k odd 6 1
1859.1.k.c 8 13.d odd 4 2
1859.1.k.c 8 13.f odd 12 2
1859.1.k.c 8 143.g even 4 2
1859.1.k.c 8 143.o even 12 2
2288.1.m.a 2 52.j odd 6 1
2288.1.m.a 2 572.s even 6 1
2288.1.m.b 2 52.i odd 6 1
2288.1.m.b 2 572.t even 6 1
3575.1.c.c 4 65.q odd 12 2
3575.1.c.c 4 715.bo even 12 2
3575.1.c.d 4 65.r odd 12 2
3575.1.c.d 4 715.bq even 12 2
3575.1.h.e 2 65.n even 6 1
3575.1.h.e 2 715.bb odd 6 1
3575.1.h.f 2 65.l even 6 1
3575.1.h.f 2 715.x odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 2 T_{2}^{2} - T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1859, [\chi])$$.

## Hecke characteristic polynomials

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