# Properties

 Label 21.2.g.a Level 21 Weight 2 Character orbit 21.g Analytic conductor 0.168 Analytic rank 0 Dimension 2 CM disc. -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$21 = 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 21.g (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.167685844245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 - \zeta_{6} ) q^{3}$$ $$+ ( -2 + 2 \zeta_{6} ) q^{4}$$ $$+ ( 2 - 3 \zeta_{6} ) q^{7}$$ $$+ 3 \zeta_{6} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 - \zeta_{6} ) q^{3}$$ $$+ ( -2 + 2 \zeta_{6} ) q^{4}$$ $$+ ( 2 - 3 \zeta_{6} ) q^{7}$$ $$+ 3 \zeta_{6} q^{9}$$ $$+ ( 4 - 2 \zeta_{6} ) q^{12}$$ $$+ ( -1 + 2 \zeta_{6} ) q^{13}$$ $$-4 \zeta_{6} q^{16}$$ $$+ ( -6 + 3 \zeta_{6} ) q^{19}$$ $$+ ( -5 + 4 \zeta_{6} ) q^{21}$$ $$+ ( 5 - 5 \zeta_{6} ) q^{25}$$ $$+ ( 3 - 6 \zeta_{6} ) q^{27}$$ $$+ ( 2 + 4 \zeta_{6} ) q^{28}$$ $$+ ( 5 + 5 \zeta_{6} ) q^{31}$$ $$-6 q^{36}$$ $$-\zeta_{6} q^{37}$$ $$+ ( 3 - 3 \zeta_{6} ) q^{39}$$ $$-5 q^{43}$$ $$+ ( -4 + 8 \zeta_{6} ) q^{48}$$ $$+ ( -5 - 3 \zeta_{6} ) q^{49}$$ $$+ ( -2 - 2 \zeta_{6} ) q^{52}$$ $$+ 9 q^{57}$$ $$+ ( 8 - 4 \zeta_{6} ) q^{61}$$ $$+ ( 9 - 3 \zeta_{6} ) q^{63}$$ $$+ 8 q^{64}$$ $$+ ( -11 + 11 \zeta_{6} ) q^{67}$$ $$+ ( -9 - 9 \zeta_{6} ) q^{73}$$ $$+ ( -10 + 5 \zeta_{6} ) q^{75}$$ $$+ ( 6 - 12 \zeta_{6} ) q^{76}$$ $$+ 13 \zeta_{6} q^{79}$$ $$+ ( -9 + 9 \zeta_{6} ) q^{81}$$ $$+ ( 2 - 10 \zeta_{6} ) q^{84}$$ $$+ ( 4 + \zeta_{6} ) q^{91}$$ $$-15 \zeta_{6} q^{93}$$ $$+ ( -8 + 16 \zeta_{6} ) q^{97}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 3q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 3q^{3}$$ $$\mathstrut -\mathstrut 2q^{4}$$ $$\mathstrut +\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 3q^{9}$$ $$\mathstrut +\mathstrut 6q^{12}$$ $$\mathstrut -\mathstrut 4q^{16}$$ $$\mathstrut -\mathstrut 9q^{19}$$ $$\mathstrut -\mathstrut 6q^{21}$$ $$\mathstrut +\mathstrut 5q^{25}$$ $$\mathstrut +\mathstrut 8q^{28}$$ $$\mathstrut +\mathstrut 15q^{31}$$ $$\mathstrut -\mathstrut 12q^{36}$$ $$\mathstrut -\mathstrut q^{37}$$ $$\mathstrut +\mathstrut 3q^{39}$$ $$\mathstrut -\mathstrut 10q^{43}$$ $$\mathstrut -\mathstrut 13q^{49}$$ $$\mathstrut -\mathstrut 6q^{52}$$ $$\mathstrut +\mathstrut 18q^{57}$$ $$\mathstrut +\mathstrut 12q^{61}$$ $$\mathstrut +\mathstrut 15q^{63}$$ $$\mathstrut +\mathstrut 16q^{64}$$ $$\mathstrut -\mathstrut 11q^{67}$$ $$\mathstrut -\mathstrut 27q^{73}$$ $$\mathstrut -\mathstrut 15q^{75}$$ $$\mathstrut +\mathstrut 13q^{79}$$ $$\mathstrut -\mathstrut 9q^{81}$$ $$\mathstrut -\mathstrut 6q^{84}$$ $$\mathstrut +\mathstrut 9q^{91}$$ $$\mathstrut -\mathstrut 15q^{93}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i −1.00000 1.73205i 0 0 0.500000 + 2.59808i 0 1.50000 2.59808i 0
17.1 0 −1.50000 0.866025i −1.00000 + 1.73205i 0 0 0.500000 2.59808i 0 1.50000 + 2.59808i 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
7.d Odd 1 yes
21.g Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(21, [\chi])$$.