# Properties

 Label 103.1.b.a Level $103$ Weight $1$ Character orbit 103.b Self dual yes Analytic conductor $0.051$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -103 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$103$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 103.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0514036963012$$ 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 5.1.10609.1 Artin image: $D_5$ Artin field: Galois closure of 5.1.10609.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{4} -\beta q^{7} - q^{8} + q^{9} + ( -1 + \beta ) q^{13} - q^{14} -\beta q^{17} + ( -1 + \beta ) q^{18} + ( -1 + \beta ) q^{19} + ( -1 + \beta ) q^{23} + q^{25} + ( 2 - \beta ) q^{26} + q^{28} -\beta q^{29} + q^{32} - q^{34} + ( 1 - \beta ) q^{36} + ( 2 - \beta ) q^{38} + ( -1 + \beta ) q^{41} + ( 2 - \beta ) q^{46} + \beta q^{49} + ( -1 + \beta ) q^{50} + ( -2 + \beta ) q^{52} + \beta q^{56} - q^{58} -\beta q^{59} -\beta q^{61} -\beta q^{63} + ( -1 + \beta ) q^{64} + q^{68} - q^{72} + ( -2 + \beta ) q^{76} + ( -1 + \beta ) q^{79} + q^{81} + ( 2 - \beta ) q^{82} -\beta q^{83} - q^{91} + ( -2 + \beta ) q^{92} -\beta q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} - q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - q^{2} + q^{4} - q^{7} - 2 q^{8} + 2 q^{9} - q^{13} - 2 q^{14} - q^{17} - q^{18} - q^{19} - q^{23} + 2 q^{25} + 3 q^{26} + 2 q^{28} - q^{29} + 2 q^{32} - 2 q^{34} + q^{36} + 3 q^{38} - q^{41} + 3 q^{46} + q^{49} - q^{50} - 3 q^{52} + q^{56} - 2 q^{58} - q^{59} - q^{61} - q^{63} - q^{64} + 2 q^{68} - 2 q^{72} - 3 q^{76} - q^{79} + 2 q^{81} + 3 q^{82} - q^{83} - 2 q^{91} - 3 q^{92} - q^{97} + 2 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/103\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
102.1
 −0.618034 1.61803
−1.61803 0 1.61803 0 0 0.618034 −1.00000 1.00000 0
102.2 0.618034 0 −0.618034 0 0 −1.61803 −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
103.b odd 2 1 CM by $$\Q(\sqrt{-103})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.1.b.a 2
3.b odd 2 1 927.1.d.b 2
4.b odd 2 1 1648.1.c.a 2
5.b even 2 1 2575.1.d.d 2
5.c odd 4 2 2575.1.c.b 4
103.b odd 2 1 CM 103.1.b.a 2
309.c even 2 1 927.1.d.b 2
412.d even 2 1 1648.1.c.a 2
515.c odd 2 1 2575.1.d.d 2
515.f even 4 2 2575.1.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 1.a even 1 1 trivial
103.1.b.a 2 103.b odd 2 1 CM
927.1.d.b 2 3.b odd 2 1
927.1.d.b 2 309.c even 2 1
1648.1.c.a 2 4.b odd 2 1
1648.1.c.a 2 412.d even 2 1
2575.1.c.b 4 5.c odd 4 2
2575.1.c.b 4 515.f even 4 2
2575.1.d.d 2 5.b even 2 1
2575.1.d.d 2 515.c odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(103, [\chi])$$.

## Hecke characteristic polynomials

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