# Properties

 Label 21.39.h.a Level $21$ Weight $39$ Character orbit 21.h Analytic conductor $192.073$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$39$$ Character orbit: $$[\chi]$$ $$=$$ 21.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$192.073284970$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 1162261467 \zeta_{6} q^{3} -274877906944 \zeta_{6} q^{4} + ( -1387488121601405 - 10641641639802267 \zeta_{6} ) q^{7} + ( -1350851717672992089 + 1350851717672992089 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q +1162261467 \zeta_{6} q^{3} -274877906944 \zeta_{6} q^{4} +(-1387488121601405 - 10641641639802267 \zeta_{6}) q^{7} +(-1350851717672992089 + 1350851717672992089 \zeta_{6}) q^{9} +($$$$31\!\cdots\!48$$$$-$$$$31\!\cdots\!48$$$$\zeta_{6}) q^{12} -$$$$28\!\cdots\!13$$$$q^{13} +(-$$$$75\!\cdots\!36$$$$+$$$$75\!\cdots\!36$$$$\zeta_{6}) q^{16} +(-$$$$25\!\cdots\!91$$$$+$$$$25\!\cdots\!91$$$$\zeta_{6}) q^{19} +($$$$12\!\cdots\!89$$$$-$$$$13\!\cdots\!24$$$$\zeta_{6}) q^{21} -$$$$36\!\cdots\!25$$$$\zeta_{6} q^{25} -$$$$15\!\cdots\!63$$$$q^{27} +(-$$$$29\!\cdots\!48$$$$+$$$$33\!\cdots\!68$$$$\zeta_{6}) q^{28} +$$$$33\!\cdots\!33$$$$\zeta_{6} q^{31} +$$$$37\!\cdots\!16$$$$q^{36} +(-$$$$12\!\cdots\!63$$$$+$$$$12\!\cdots\!63$$$$\zeta_{6}) q^{37} -$$$$33\!\cdots\!71$$$$\zeta_{6} q^{39} +$$$$14\!\cdots\!11$$$$q^{43} -$$$$87\!\cdots\!12$$$$q^{48} +(-$$$$11\!\cdots\!64$$$$+$$$$14\!\cdots\!59$$$$\zeta_{6}) q^{49} +$$$$78\!\cdots\!72$$$$\zeta_{6} q^{52} -$$$$29\!\cdots\!97$$$$q^{57} +(-$$$$30\!\cdots\!74$$$$+$$$$30\!\cdots\!74$$$$\zeta_{6}) q^{61} +($$$$16\!\cdots\!08$$$$-$$$$18\!\cdots\!45$$$$\zeta_{6}) q^{63} +$$$$20\!\cdots\!84$$$$q^{64} -$$$$95\!\cdots\!23$$$$\zeta_{6} q^{67} -$$$$47\!\cdots\!67$$$$\zeta_{6} q^{73} +($$$$42\!\cdots\!75$$$$-$$$$42\!\cdots\!75$$$$\zeta_{6}) q^{75} +$$$$70\!\cdots\!04$$$$q^{76} +($$$$22\!\cdots\!09$$$$-$$$$22\!\cdots\!09$$$$\zeta_{6}) q^{79} -$$$$18\!\cdots\!21$$$$\zeta_{6} q^{81} +(-$$$$38\!\cdots\!56$$$$+$$$$44\!\cdots\!40$$$$\zeta_{6}) q^{84} +($$$$39\!\cdots\!65$$$$+$$$$30\!\cdots\!71$$$$\zeta_{6}) q^{91} +(-$$$$38\!\cdots\!11$$$$+$$$$38\!\cdots\!11$$$$\zeta_{6}) q^{93} -$$$$21\!\cdots\!62$$$$q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 1162261467q^{3} - 274877906944q^{4} - 13416617883005077q^{7} - 1350851717672992089q^{9} + O(q^{10})$$ $$2q + 1162261467q^{3} - 274877906944q^{4} - 13416617883005077q^{7} - 1350851717672992089q^{9} +$$$$31\!\cdots\!48$$$$q^{12} -$$$$57\!\cdots\!26$$$$q^{13} -$$$$75\!\cdots\!36$$$$q^{16} -$$$$25\!\cdots\!91$$$$q^{19} +$$$$10\!\cdots\!54$$$$q^{21} -$$$$36\!\cdots\!25$$$$q^{25} -$$$$31\!\cdots\!26$$$$q^{27} -$$$$25\!\cdots\!28$$$$q^{28} +$$$$33\!\cdots\!33$$$$q^{31} +$$$$74\!\cdots\!32$$$$q^{36} -$$$$12\!\cdots\!63$$$$q^{37} -$$$$33\!\cdots\!71$$$$q^{39} +$$$$28\!\cdots\!22$$$$q^{43} -$$$$17\!\cdots\!24$$$$q^{48} -$$$$79\!\cdots\!69$$$$q^{49} +$$$$78\!\cdots\!72$$$$q^{52} -$$$$59\!\cdots\!94$$$$q^{57} -$$$$30\!\cdots\!74$$$$q^{61} +$$$$30\!\cdots\!71$$$$q^{63} +$$$$41\!\cdots\!68$$$$q^{64} -$$$$95\!\cdots\!23$$$$q^{67} -$$$$47\!\cdots\!67$$$$q^{73} +$$$$42\!\cdots\!75$$$$q^{75} +$$$$14\!\cdots\!08$$$$q^{76} +$$$$22\!\cdots\!09$$$$q^{79} -$$$$18\!\cdots\!21$$$$q^{81} -$$$$72\!\cdots\!72$$$$q^{84} +$$$$38\!\cdots\!01$$$$q^{91} -$$$$38\!\cdots\!11$$$$q^{93} -$$$$43\!\cdots\!24$$$$q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 5.81131e8 + 1.00655e9i −1.37439e11 2.38051e11i 0 0 −6.70831e15 9.21593e15i 0 −6.75426e17 + 1.16987e18i 0
11.1 0 5.81131e8 1.00655e9i −1.37439e11 + 2.38051e11i 0 0 −6.70831e15 + 9.21593e15i 0 −6.75426e17 1.16987e18i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.39.h.a 2
3.b odd 2 1 CM 21.39.h.a 2
7.c even 3 1 inner 21.39.h.a 2
21.h odd 6 1 inner 21.39.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.39.h.a 2 1.a even 1 1 trivial
21.39.h.a 2 3.b odd 2 1 CM
21.39.h.a 2 7.c even 3 1 inner
21.39.h.a 2 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{39}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1350851717672992089 - 1162261467 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$12\!\cdots\!49$$$$+ 13416617883005077 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$($$$$28\!\cdots\!13$$$$+ T )^{2}$$
$17$ $$T^{2}$$
$19$ $$65\!\cdots\!81$$$$+$$$$25\!\cdots\!91$$$$T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$11\!\cdots\!89$$$$-$$$$33\!\cdots\!33$$$$T + T^{2}$$
$37$ $$15\!\cdots\!69$$$$+$$$$12\!\cdots\!63$$$$T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -$$$$14\!\cdots\!11$$$$+ T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$92\!\cdots\!76$$$$+$$$$30\!\cdots\!74$$$$T + T^{2}$$
$67$ $$91\!\cdots\!29$$$$+$$$$95\!\cdots\!23$$$$T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$22\!\cdots\!89$$$$+$$$$47\!\cdots\!67$$$$T + T^{2}$$
$79$ $$48\!\cdots\!81$$$$-$$$$22\!\cdots\!09$$$$T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$($$$$21\!\cdots\!62$$$$+ T )^{2}$$