# Properties

 Label 21.3.d.a Level 21 Weight 3 Character orbit 21.d Analytic conductor 0.572 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$21 = 3 \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 21.d (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.572208555157$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ q^{2}$$ $$+ \beta q^{3}$$ $$-3 q^{4}$$ $$-4 \beta q^{5}$$ $$+ \beta q^{6}$$ $$+ ( 1 + 4 \beta ) q^{7}$$ $$-7 q^{8}$$ $$-3 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ q^{2}$$ $$+ \beta q^{3}$$ $$-3 q^{4}$$ $$-4 \beta q^{5}$$ $$+ \beta q^{6}$$ $$+ ( 1 + 4 \beta ) q^{7}$$ $$-7 q^{8}$$ $$-3 q^{9}$$ $$-4 \beta q^{10}$$ $$+ 10 q^{11}$$ $$-3 \beta q^{12}$$ $$+ 4 \beta q^{13}$$ $$+ ( 1 + 4 \beta ) q^{14}$$ $$+ 12 q^{15}$$ $$+ 5 q^{16}$$ $$-3 q^{18}$$ $$-12 \beta q^{19}$$ $$+ 12 \beta q^{20}$$ $$+ ( -12 + \beta ) q^{21}$$ $$+ 10 q^{22}$$ $$-14 q^{23}$$ $$-7 \beta q^{24}$$ $$-23 q^{25}$$ $$+ 4 \beta q^{26}$$ $$-3 \beta q^{27}$$ $$+ ( -3 - 12 \beta ) q^{28}$$ $$-38 q^{29}$$ $$+ 12 q^{30}$$ $$+ 16 \beta q^{31}$$ $$+ 33 q^{32}$$ $$+ 10 \beta q^{33}$$ $$+ ( 48 - 4 \beta ) q^{35}$$ $$+ 9 q^{36}$$ $$+ 26 q^{37}$$ $$-12 \beta q^{38}$$ $$-12 q^{39}$$ $$+ 28 \beta q^{40}$$ $$-40 \beta q^{41}$$ $$+ ( -12 + \beta ) q^{42}$$ $$+ 26 q^{43}$$ $$-30 q^{44}$$ $$+ 12 \beta q^{45}$$ $$-14 q^{46}$$ $$+ 16 \beta q^{47}$$ $$+ 5 \beta q^{48}$$ $$+ ( -47 + 8 \beta ) q^{49}$$ $$-23 q^{50}$$ $$-12 \beta q^{52}$$ $$+ 10 q^{53}$$ $$-3 \beta q^{54}$$ $$-40 \beta q^{55}$$ $$+ ( -7 - 28 \beta ) q^{56}$$ $$+ 36 q^{57}$$ $$-38 q^{58}$$ $$+ 44 \beta q^{59}$$ $$-36 q^{60}$$ $$+ 20 \beta q^{61}$$ $$+ 16 \beta q^{62}$$ $$+ ( -3 - 12 \beta ) q^{63}$$ $$+ 13 q^{64}$$ $$+ 48 q^{65}$$ $$+ 10 \beta q^{66}$$ $$+ 74 q^{67}$$ $$-14 \beta q^{69}$$ $$+ ( 48 - 4 \beta ) q^{70}$$ $$-62 q^{71}$$ $$+ 21 q^{72}$$ $$-24 \beta q^{73}$$ $$+ 26 q^{74}$$ $$-23 \beta q^{75}$$ $$+ 36 \beta q^{76}$$ $$+ ( 10 + 40 \beta ) q^{77}$$ $$-12 q^{78}$$ $$-46 q^{79}$$ $$-20 \beta q^{80}$$ $$+ 9 q^{81}$$ $$-40 \beta q^{82}$$ $$+ 52 \beta q^{83}$$ $$+ ( 36 - 3 \beta ) q^{84}$$ $$+ 26 q^{86}$$ $$-38 \beta q^{87}$$ $$-70 q^{88}$$ $$-24 \beta q^{89}$$ $$+ 12 \beta q^{90}$$ $$+ ( -48 + 4 \beta ) q^{91}$$ $$+ 42 q^{92}$$ $$-48 q^{93}$$ $$+ 16 \beta q^{94}$$ $$-144 q^{95}$$ $$+ 33 \beta q^{96}$$ $$+ 32 \beta q^{97}$$ $$+ ( -47 + 8 \beta ) q^{98}$$ $$-30 q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 6q^{4}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 14q^{8}$$ $$\mathstrut -\mathstrut 6q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 6q^{4}$$ $$\mathstrut +\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 14q^{8}$$ $$\mathstrut -\mathstrut 6q^{9}$$ $$\mathstrut +\mathstrut 20q^{11}$$ $$\mathstrut +\mathstrut 2q^{14}$$ $$\mathstrut +\mathstrut 24q^{15}$$ $$\mathstrut +\mathstrut 10q^{16}$$ $$\mathstrut -\mathstrut 6q^{18}$$ $$\mathstrut -\mathstrut 24q^{21}$$ $$\mathstrut +\mathstrut 20q^{22}$$ $$\mathstrut -\mathstrut 28q^{23}$$ $$\mathstrut -\mathstrut 46q^{25}$$ $$\mathstrut -\mathstrut 6q^{28}$$ $$\mathstrut -\mathstrut 76q^{29}$$ $$\mathstrut +\mathstrut 24q^{30}$$ $$\mathstrut +\mathstrut 66q^{32}$$ $$\mathstrut +\mathstrut 96q^{35}$$ $$\mathstrut +\mathstrut 18q^{36}$$ $$\mathstrut +\mathstrut 52q^{37}$$ $$\mathstrut -\mathstrut 24q^{39}$$ $$\mathstrut -\mathstrut 24q^{42}$$ $$\mathstrut +\mathstrut 52q^{43}$$ $$\mathstrut -\mathstrut 60q^{44}$$ $$\mathstrut -\mathstrut 28q^{46}$$ $$\mathstrut -\mathstrut 94q^{49}$$ $$\mathstrut -\mathstrut 46q^{50}$$ $$\mathstrut +\mathstrut 20q^{53}$$ $$\mathstrut -\mathstrut 14q^{56}$$ $$\mathstrut +\mathstrut 72q^{57}$$ $$\mathstrut -\mathstrut 76q^{58}$$ $$\mathstrut -\mathstrut 72q^{60}$$ $$\mathstrut -\mathstrut 6q^{63}$$ $$\mathstrut +\mathstrut 26q^{64}$$ $$\mathstrut +\mathstrut 96q^{65}$$ $$\mathstrut +\mathstrut 148q^{67}$$ $$\mathstrut +\mathstrut 96q^{70}$$ $$\mathstrut -\mathstrut 124q^{71}$$ $$\mathstrut +\mathstrut 42q^{72}$$ $$\mathstrut +\mathstrut 52q^{74}$$ $$\mathstrut +\mathstrut 20q^{77}$$ $$\mathstrut -\mathstrut 24q^{78}$$ $$\mathstrut -\mathstrut 92q^{79}$$ $$\mathstrut +\mathstrut 18q^{81}$$ $$\mathstrut +\mathstrut 72q^{84}$$ $$\mathstrut +\mathstrut 52q^{86}$$ $$\mathstrut -\mathstrut 140q^{88}$$ $$\mathstrut -\mathstrut 96q^{91}$$ $$\mathstrut +\mathstrut 84q^{92}$$ $$\mathstrut -\mathstrut 96q^{93}$$ $$\mathstrut -\mathstrut 288q^{95}$$ $$\mathstrut -\mathstrut 94q^{98}$$ $$\mathstrut -\mathstrut 60q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 6.92820i −7.00000 −3.00000 6.92820i
13.2 1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 + 6.92820i −7.00000 −3.00000 6.92820i
 $$n$$: 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
7.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(21, [\chi])$$.