# Properties

 Label 21.26.g.a Level $21$ Weight $26$ Character orbit 21.g Analytic conductor $83.159$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$26$$ Character orbit: $$[\chi]$$ $$=$$ 21.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$83.1593237900$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -531441 - 531441 \zeta_{6} ) q^{3} + ( -33554432 + 33554432 \zeta_{6} ) q^{4} + ( -35728842238 - 1722717363 \zeta_{6} ) q^{7} + 847288609443 \zeta_{6} q^{9} +O(q^{10})$$ $$q +(-531441 - 531441 \zeta_{6}) q^{3} +(-33554432 + 33554432 \zeta_{6}) q^{4} +(-35728842238 - 1722717363 \zeta_{6}) q^{7} +847288609443 \zeta_{6} q^{9} +(35664401793024 - 17832200896512 \zeta_{6}) q^{12} +(21081468058559 - 42162936117118 \zeta_{6}) q^{13} -1125899906842624 \zeta_{6} q^{16} +(5642266816761114 - 2821133408380557 \zeta_{6}) q^{19} +(18072249009694875 + 20818816924025124 \zeta_{6}) q^{21} +(298023223876953125 - 298023223876953125 \zeta_{6}) q^{25} +(450283905890997363 - 900567811781994726 \zeta_{6}) q^{27} +(1256665809925701632 - 1198861007313698816 \zeta_{6}) q^{28} +(-1420701470158304875 - 1420701470158304875 \zeta_{6}) q^{31} -28430288029929701376 q^{36} -70792710091786388161 \zeta_{6} q^{37} +(-33610669399525960557 + 33610669399525960557 \zeta_{6}) q^{39} +$$$$18\!\cdots\!75$$$$q^{43} +(-$$$$59\!\cdots\!84$$$$+$$$$11\!\cdots\!68$$$$\zeta_{6}) q^{48} +($$$$12\!\cdots\!75$$$$+$$$$12\!\cdots\!57$$$$\zeta_{6}) q^{49} +($$$$70\!\cdots\!88$$$$+$$$$70\!\cdots\!88$$$$\zeta_{6}) q^{52} -$$$$44\!\cdots\!11$$$$q^{57} +(-$$$$42\!\cdots\!52$$$$+$$$$21\!\cdots\!76$$$$\zeta_{6}) q^{61} +($$$$14\!\cdots\!09$$$$-$$$$31\!\cdots\!43$$$$\zeta_{6}) q^{63} +$$$$37\!\cdots\!68$$$$q^{64} +($$$$55\!\cdots\!89$$$$-$$$$55\!\cdots\!89$$$$\zeta_{6}) q^{67} +($$$$97\!\cdots\!91$$$$+$$$$97\!\cdots\!91$$$$\zeta_{6}) q^{73} +(-$$$$31\!\cdots\!50$$$$+$$$$15\!\cdots\!25$$$$\zeta_{6}) q^{75} +(-$$$$94\!\cdots\!24$$$$+$$$$18\!\cdots\!48$$$$\zeta_{6}) q^{76} +$$$$10\!\cdots\!93$$$$\zeta_{6} q^{79} +(-$$$$71\!\cdots\!49$$$$+$$$$71\!\cdots\!49$$$$\zeta_{6}) q^{81} +(-$$$$13\!\cdots\!68$$$$+$$$$60\!\cdots\!00$$$$\zeta_{6}) q^{84} +(-$$$$82\!\cdots\!76$$$$+$$$$15\!\cdots\!01$$$$\zeta_{6}) q^{91} +$$$$22\!\cdots\!25$$$$\zeta_{6} q^{93} +(-$$$$48\!\cdots\!88$$$$+$$$$96\!\cdots\!76$$$$\zeta_{6}) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1594323q^{3} - 33554432q^{4} - 73180401839q^{7} + 847288609443q^{9} + O(q^{10})$$ $$2q - 1594323q^{3} - 33554432q^{4} - 73180401839q^{7} + 847288609443q^{9} + 53496602689536q^{12} - 1125899906842624q^{16} + 8463400225141671q^{19} + 56963314943414874q^{21} + 298023223876953125q^{25} + 1314470612537704448q^{28} - 4262104410474914625q^{31} - 56860576059859402752q^{36} - 70792710091786388161q^{37} - 33610669399525960557q^{39} +$$$$36\!\cdots\!50$$$$q^{43} +$$$$26\!\cdots\!07$$$$q^{49} +$$$$21\!\cdots\!64$$$$q^{52} -$$$$89\!\cdots\!22$$$$q^{57} -$$$$63\!\cdots\!28$$$$q^{61} -$$$$28\!\cdots\!25$$$$q^{63} +$$$$75\!\cdots\!36$$$$q^{64} +$$$$55\!\cdots\!89$$$$q^{67} +$$$$29\!\cdots\!73$$$$q^{73} -$$$$47\!\cdots\!75$$$$q^{75} +$$$$10\!\cdots\!93$$$$q^{79} -$$$$71\!\cdots\!49$$$$q^{81} -$$$$20\!\cdots\!36$$$$q^{84} -$$$$10\!\cdots\!51$$$$q^{91} +$$$$22\!\cdots\!25$$$$q^{93} + O(q^{100})$$

## 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 −797161. + 460241.i −1.67772e7 2.90590e7i 0 0 −3.65902e10 + 1.49192e9i 0 4.23644e11 7.33773e11i 0
17.1 0 −797161. 460241.i −1.67772e7 + 2.90590e7i 0 0 −3.65902e10 1.49192e9i 0 4.23644e11 + 7.33773e11i 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.26.g.a 2
3.b odd 2 1 CM 21.26.g.a 2
7.d odd 6 1 inner 21.26.g.a 2
21.g even 6 1 inner 21.26.g.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{26}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$847288609443 + 1594323 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$13\!\cdots\!07$$$$+ 73180401839 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13\!\cdots\!43$$$$+ T^{2}$$
$17$ $$T^{2}$$
$19$ $$23\!\cdots\!47$$$$- 8463400225141671 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$60\!\cdots\!75$$$$+ 4262104410474914625 T + T^{2}$$
$37$ $$50\!\cdots\!21$$$$+ 70792710091786388161 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -$$$$18\!\cdots\!75$$$$+ T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$13\!\cdots\!28$$$$+$$$$63\!\cdots\!28$$$$T + T^{2}$$
$67$ $$31\!\cdots\!21$$$$-$$$$55\!\cdots\!89$$$$T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$28\!\cdots\!43$$$$-$$$$29\!\cdots\!73$$$$T + T^{2}$$
$79$ $$10\!\cdots\!49$$$$-$$$$10\!\cdots\!93$$$$T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$69\!\cdots\!32$$$$+ T^{2}$$