# Properties

 Label 1136.1.h.a Level $1136$ Weight $1$ Character orbit 1136.h Self dual yes Analytic conductor $0.567$ Analytic rank $0$ Dimension $3$ Projective image $D_{7}$ CM discriminant -71 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1136 = 2^{4} \cdot 71$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1136.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.566937854351$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 71) Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.357911.1 Artin image: $D_{14}$ Artin field: Galois closure of 14.0.2098795051761664.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( 1 + \beta_{2} ) q^{9} + ( 1 - \beta_{1} ) q^{15} -\beta_{2} q^{19} + ( 1 - \beta_{1} ) q^{25} + ( 1 + \beta_{2} ) q^{27} -\beta_{1} q^{29} + \beta_{2} q^{37} + ( 1 - \beta_{1} + \beta_{2} ) q^{43} - q^{45} + q^{49} + ( -1 - \beta_{2} ) q^{57} - q^{71} + ( -1 + \beta_{1} - \beta_{2} ) q^{73} + ( -2 + \beta_{1} - \beta_{2} ) q^{75} + ( 1 - \beta_{1} + \beta_{2} ) q^{79} + \beta_{1} q^{81} -\beta_{2} q^{83} + ( -2 - \beta_{2} ) q^{87} -\beta_{1} q^{89} + ( \beta_{1} - \beta_{2} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - q^{5} + 2 q^{9} + O(q^{10})$$ $$3 q + q^{3} - q^{5} + 2 q^{9} + 2 q^{15} + q^{19} + 2 q^{25} + 2 q^{27} - q^{29} - q^{37} + q^{43} - 3 q^{45} + 3 q^{49} - 2 q^{57} - 3 q^{71} - q^{73} - 4 q^{75} + q^{79} + q^{81} + q^{83} - 5 q^{87} - q^{89} + 2 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1136\mathbb{Z}\right)^\times$$.

 $$n$$ $$143$$ $$433$$ $$853$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
993.1
 −1.24698 0.445042 1.80194
0 −1.24698 0 −1.80194 0 0 0 0.554958 0
993.2 0 0.445042 0 1.24698 0 0 0 −0.801938 0
993.3 0 1.80194 0 −0.445042 0 0 0 2.24698 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
71.b odd 2 1 CM by $$\Q(\sqrt{-71})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1136.1.h.a 3
4.b odd 2 1 71.1.b.a 3
12.b even 2 1 639.1.d.a 3
20.d odd 2 1 1775.1.d.b 3
20.e even 4 2 1775.1.c.a 6
28.d even 2 1 3479.1.d.e 3
28.f even 6 2 3479.1.g.d 6
28.g odd 6 2 3479.1.g.e 6
71.b odd 2 1 CM 1136.1.h.a 3
284.c even 2 1 71.1.b.a 3
852.d odd 2 1 639.1.d.a 3
1420.g even 2 1 1775.1.d.b 3
1420.l odd 4 2 1775.1.c.a 6
1988.g odd 2 1 3479.1.d.e 3
1988.l odd 6 2 3479.1.g.d 6
1988.n even 6 2 3479.1.g.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 4.b odd 2 1
71.1.b.a 3 284.c even 2 1
639.1.d.a 3 12.b even 2 1
639.1.d.a 3 852.d odd 2 1
1136.1.h.a 3 1.a even 1 1 trivial
1136.1.h.a 3 71.b odd 2 1 CM
1775.1.c.a 6 20.e even 4 2
1775.1.c.a 6 1420.l odd 4 2
1775.1.d.b 3 20.d odd 2 1
1775.1.d.b 3 1420.g even 2 1
3479.1.d.e 3 28.d even 2 1
3479.1.d.e 3 1988.g odd 2 1
3479.1.g.d 6 28.f even 6 2
3479.1.g.d 6 1988.l odd 6 2
3479.1.g.e 6 28.g odd 6 2
3479.1.g.e 6 1988.n even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1136, [\chi])$$.

## Hecke characteristic polynomials

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