# Properties

 Label 12.3.c.a Level 12 Weight 3 Character orbit 12.c Self dual Yes Analytic conductor 0.327 Analytic rank 0 Dimension 1 CM disc. -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 12.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: Yes Analytic conductor: $$0.326976317232$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 9q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$q$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut 9q^{9}$$ $$\mathstrut -\mathstrut 22q^{13}$$ $$\mathstrut +\mathstrut 26q^{19}$$ $$\mathstrut -\mathstrut 6q^{21}$$ $$\mathstrut +\mathstrut 25q^{25}$$ $$\mathstrut -\mathstrut 27q^{27}$$ $$\mathstrut -\mathstrut 46q^{31}$$ $$\mathstrut +\mathstrut 26q^{37}$$ $$\mathstrut +\mathstrut 66q^{39}$$ $$\mathstrut -\mathstrut 22q^{43}$$ $$\mathstrut -\mathstrut 45q^{49}$$ $$\mathstrut -\mathstrut 78q^{57}$$ $$\mathstrut +\mathstrut 74q^{61}$$ $$\mathstrut +\mathstrut 18q^{63}$$ $$\mathstrut +\mathstrut 122q^{67}$$ $$\mathstrut -\mathstrut 46q^{73}$$ $$\mathstrut -\mathstrut 75q^{75}$$ $$\mathstrut -\mathstrut 142q^{79}$$ $$\mathstrut +\mathstrut 81q^{81}$$ $$\mathstrut -\mathstrut 44q^{91}$$ $$\mathstrut +\mathstrut 138q^{93}$$ $$\mathstrut +\mathstrut 2q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0
0 −3.00000 0 0 0 2.00000 0 9.00000 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
3.b Odd 1 CM by $$\Q(\sqrt{-3})$$ yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(12, [\chi])$$.