# Properties

 Label 1859.1.c.c Level $1859$ Weight $1$ Character orbit 1859.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.927761858485$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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: $C_4\times D_5$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-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.

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

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

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

## Character values

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

 $$n$$ $$508$$ $$1354$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
846.1
 1.61803i 0.618034i − 0.618034i − 1.61803i
1.61803i −1.61803 −1.61803 0 2.61803i 0.618034i 1.00000i 1.61803 0
846.2 0.618034i 0.618034 0.618034 0 0.381966i 1.61803i 1.00000i −0.618034 0
846.3 0.618034i 0.618034 0.618034 0 0.381966i 1.61803i 1.00000i −0.618034 0
846.4 1.61803i −1.61803 −1.61803 0 2.61803i 0.618034i 1.00000i 1.61803 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 $$\Q(\sqrt{-143})$$
11.b odd 2 1 inner
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.1.c.c 4
11.b odd 2 1 inner 1859.1.c.c 4
13.b even 2 1 inner 1859.1.c.c 4
13.c even 3 2 1859.1.k.c 8
13.d odd 4 1 143.1.d.a 2
13.d odd 4 1 143.1.d.b yes 2
13.e even 6 2 1859.1.k.c 8
13.f odd 12 2 1859.1.i.a 4
13.f odd 12 2 1859.1.i.b 4
39.f even 4 1 1287.1.g.a 2
39.f even 4 1 1287.1.g.b 2
52.f even 4 1 2288.1.m.a 2
52.f even 4 1 2288.1.m.b 2
65.f even 4 1 3575.1.c.c 4
65.f even 4 1 3575.1.c.d 4
65.g odd 4 1 3575.1.h.e 2
65.g odd 4 1 3575.1.h.f 2
65.k even 4 1 3575.1.c.c 4
65.k even 4 1 3575.1.c.d 4
143.d odd 2 1 CM 1859.1.c.c 4
143.g even 4 1 143.1.d.a 2
143.g even 4 1 143.1.d.b yes 2
143.i odd 6 2 1859.1.k.c 8
143.k odd 6 2 1859.1.k.c 8
143.o even 12 2 1859.1.i.a 4
143.o even 12 2 1859.1.i.b 4
143.r odd 20 2 1573.1.l.a 4
143.r odd 20 2 1573.1.l.b 4
143.r odd 20 2 1573.1.l.c 4
143.r odd 20 2 1573.1.l.d 4
143.s even 20 2 1573.1.l.a 4
143.s even 20 2 1573.1.l.b 4
143.s even 20 2 1573.1.l.c 4
143.s even 20 2 1573.1.l.d 4
429.l odd 4 1 1287.1.g.a 2
429.l odd 4 1 1287.1.g.b 2
572.k odd 4 1 2288.1.m.a 2
572.k odd 4 1 2288.1.m.b 2
715.k odd 4 1 3575.1.c.c 4
715.k odd 4 1 3575.1.c.d 4
715.l even 4 1 3575.1.h.e 2
715.l even 4 1 3575.1.h.f 2
715.u odd 4 1 3575.1.c.c 4
715.u odd 4 1 3575.1.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 13.d odd 4 1
143.1.d.a 2 143.g even 4 1
143.1.d.b yes 2 13.d odd 4 1
143.1.d.b yes 2 143.g even 4 1
1287.1.g.a 2 39.f even 4 1
1287.1.g.a 2 429.l odd 4 1
1287.1.g.b 2 39.f even 4 1
1287.1.g.b 2 429.l odd 4 1
1573.1.l.a 4 143.r odd 20 2
1573.1.l.a 4 143.s even 20 2
1573.1.l.b 4 143.r odd 20 2
1573.1.l.b 4 143.s even 20 2
1573.1.l.c 4 143.r odd 20 2
1573.1.l.c 4 143.s even 20 2
1573.1.l.d 4 143.r odd 20 2
1573.1.l.d 4 143.s even 20 2
1859.1.c.c 4 1.a even 1 1 trivial
1859.1.c.c 4 11.b odd 2 1 inner
1859.1.c.c 4 13.b even 2 1 inner
1859.1.c.c 4 143.d odd 2 1 CM
1859.1.i.a 4 13.f odd 12 2
1859.1.i.a 4 143.o even 12 2
1859.1.i.b 4 13.f odd 12 2
1859.1.i.b 4 143.o even 12 2
1859.1.k.c 8 13.c even 3 2
1859.1.k.c 8 13.e even 6 2
1859.1.k.c 8 143.i odd 6 2
1859.1.k.c 8 143.k odd 6 2
2288.1.m.a 2 52.f even 4 1
2288.1.m.a 2 572.k odd 4 1
2288.1.m.b 2 52.f even 4 1
2288.1.m.b 2 572.k odd 4 1
3575.1.c.c 4 65.f even 4 1
3575.1.c.c 4 65.k even 4 1
3575.1.c.c 4 715.k odd 4 1
3575.1.c.c 4 715.u odd 4 1
3575.1.c.d 4 65.f even 4 1
3575.1.c.d 4 65.k even 4 1
3575.1.c.d 4 715.k odd 4 1
3575.1.c.d 4 715.u odd 4 1
3575.1.h.e 2 65.g odd 4 1
3575.1.h.e 2 715.l even 4 1
3575.1.h.f 2 65.g odd 4 1
3575.1.h.f 2 715.l even 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 3 T_{2}^{2} + 1$$ $$T_{5}$$

## Hecke characteristic polynomials

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