Properties

 Label 39.1.d.a Level 39 Weight 1 Character orbit 39.d Self dual Yes Analytic conductor 0.019 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM disc. -3, -39, 13 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$39 = 3 \cdot 13$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 39.d (of order $$2$$ and degree $$1$$)

Newform invariants

 Self dual: Yes Analytic conductor: $$0.0194635354927$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image size $$8$$ Artin image $D_4$ Artin field Galois closure of 4.0.117.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q$$ $$\mathstrut -\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$q$$ $$\mathstrut -\mathstrut q^{3}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut +\mathstrut q^{9}$$ $$\mathstrut +\mathstrut q^{12}$$ $$\mathstrut -\mathstrut q^{13}$$ $$\mathstrut +\mathstrut q^{16}$$ $$\mathstrut -\mathstrut q^{25}$$ $$\mathstrut -\mathstrut q^{27}$$ $$\mathstrut -\mathstrut q^{36}$$ $$\mathstrut +\mathstrut q^{39}$$ $$\mathstrut +\mathstrut 2q^{43}$$ $$\mathstrut -\mathstrut q^{48}$$ $$\mathstrut +\mathstrut q^{49}$$ $$\mathstrut +\mathstrut q^{52}$$ $$\mathstrut -\mathstrut 2q^{61}$$ $$\mathstrut -\mathstrut q^{64}$$ $$\mathstrut +\mathstrut q^{75}$$ $$\mathstrut -\mathstrut 2q^{79}$$ $$\mathstrut +\mathstrut q^{81}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1
 0
0 −1.00000 −1.00000 0 0 0 0 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. Type Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by $$\Q(\sqrt{-3})$$ yes
13.b Even 1 RM by $$\Q(\sqrt{13})$$ yes
39.d Odd 1 CM by $$\Q(\sqrt{-39})$$ yes

Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(39, \chi)$$.