# Properties

 Label 127.1.b.a Level $127$ Weight $1$ Character orbit 127.b Self dual yes Analytic conductor $0.063$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -127 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$127$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 127.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0633812566044$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.16129.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.16129.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} - q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} - q^{8} + q^{9} -\beta q^{11} -\beta q^{13} + ( -1 + \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( -1 + \beta ) q^{19} - q^{22} + q^{25} - q^{26} + ( -1 + \beta ) q^{31} + q^{32} + ( 2 - \beta ) q^{34} + ( 1 - \beta ) q^{36} + ( -1 + \beta ) q^{37} + ( 2 - \beta ) q^{38} -\beta q^{41} + q^{44} -\beta q^{47} + q^{49} + ( -1 + \beta ) q^{50} + q^{52} + ( -1 + \beta ) q^{61} + ( 2 - \beta ) q^{62} + ( -1 + \beta ) q^{64} + ( -2 + \beta ) q^{68} + ( -1 + \beta ) q^{71} - q^{72} -\beta q^{73} + ( 2 - \beta ) q^{74} + ( -2 + \beta ) q^{76} -\beta q^{79} + q^{81} - q^{82} + \beta q^{88} - q^{94} + ( -1 + \beta ) q^{98} -\beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{4} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{4} - 2q^{8} + 2q^{9} - q^{11} - q^{13} - q^{17} - q^{18} - q^{19} - 2q^{22} + 2q^{25} - 2q^{26} - q^{31} + 2q^{32} + 3q^{34} + q^{36} - q^{37} + 3q^{38} - q^{41} + 2q^{44} - q^{47} + 2q^{49} - q^{50} + 2q^{52} - q^{61} + 3q^{62} - q^{64} - 3q^{68} - q^{71} - 2q^{72} - q^{73} + 3q^{74} - 3q^{76} - q^{79} + 2q^{81} - 2q^{82} + q^{88} - 2q^{94} - q^{98} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/127\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 −0.618034 1.61803
−1.61803 0 1.61803 0 0 0 −1.00000 1.00000 0
126.2 0.618034 0 −0.618034 0 0 0 −1.00000 1.00000 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
127.b odd 2 1 CM by $$\Q(\sqrt{-127})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 127.1.b.a 2
3.b odd 2 1 1143.1.d.b 2
4.b odd 2 1 2032.1.b.a 2
5.b even 2 1 3175.1.d.d 2
5.c odd 4 2 3175.1.c.b 4
127.b odd 2 1 CM 127.1.b.a 2
381.c even 2 1 1143.1.d.b 2
508.d even 2 1 2032.1.b.a 2
635.c odd 2 1 3175.1.d.d 2
635.f even 4 2 3175.1.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.1.b.a 2 1.a even 1 1 trivial
127.1.b.a 2 127.b odd 2 1 CM
1143.1.d.b 2 3.b odd 2 1
1143.1.d.b 2 381.c even 2 1
2032.1.b.a 2 4.b odd 2 1
2032.1.b.a 2 508.d even 2 1
3175.1.c.b 4 5.c odd 4 2
3175.1.c.b 4 635.f even 4 2
3175.1.d.d 2 5.b even 2 1
3175.1.d.d 2 635.c odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(127, [\chi])$$.

## Hecke characteristic polynomials

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