# 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 disc. -103 Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: Yes Analytic conductor: $$0.0514036963012$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Projective image $$D_{5}$$ Projective field Galois closure of 5.1.10609.1 Artin image size $$10$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
103.b Odd 1 CM by $$\Q(\sqrt{-103})$$ yes

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(103, [\chi])$$.