# Properties

 Label 21.38.g.a Level $21$ Weight $38$ Character orbit 21.g Analytic conductor $182.099$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$38$$ Character orbit: $$[\chi]$$ $$=$$ 21.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$182.099480062$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -387420489 - 387420489 \zeta_{6} ) q^{3} + ( -137438953472 + 137438953472 \zeta_{6} ) q^{4} + ( 4758916148715458 - 3635106806344107 \zeta_{6} ) q^{7} + 450283905890997363 \zeta_{6} q^{9} +O(q^{10})$$ $$q +(-387420489 - 387420489 \zeta_{6}) q^{3} +(-137438953472 + 137438953472 \zeta_{6}) q^{4} +(4758916148715458 - 3635106806344107 \zeta_{6}) q^{7} +450283905890997363 \zeta_{6} q^{9} +($$$$10\!\cdots\!16$$$$- 53246666561770487808 \zeta_{6}) q^{12} +(-$$$$20\!\cdots\!89$$$$+$$$$40\!\cdots\!78$$$$\zeta_{6}) q^{13} -$$$$18\!\cdots\!84$$$$\zeta_{6} q^{16} +($$$$10\!\cdots\!74$$$$-$$$$52\!\cdots\!37$$$$\zeta_{6}) q^{19} +(-$$$$32\!\cdots\!85$$$$+$$$$97\!\cdots\!84$$$$\zeta_{6}) q^{21} +($$$$72\!\cdots\!25$$$$-$$$$72\!\cdots\!25$$$$\zeta_{6}) q^{25} +($$$$17\!\cdots\!07$$$$-$$$$34\!\cdots\!14$$$$\zeta_{6}) q^{27} +(-$$$$15\!\cdots\!72$$$$+$$$$65\!\cdots\!76$$$$\zeta_{6}) q^{28} +($$$$44\!\cdots\!85$$$$+$$$$44\!\cdots\!85$$$$\zeta_{6}) q^{31} -$$$$61\!\cdots\!36$$$$q^{36} -$$$$19\!\cdots\!09$$$$\zeta_{6} q^{37} +($$$$23\!\cdots\!63$$$$-$$$$23\!\cdots\!63$$$$\zeta_{6}) q^{39} -$$$$31\!\cdots\!65$$$$q^{43} +(-$$$$73\!\cdots\!76$$$$+$$$$14\!\cdots\!52$$$$\zeta_{6}) q^{48} +($$$$94\!\cdots\!15$$$$-$$$$21\!\cdots\!63$$$$\zeta_{6}) q^{49} +(-$$$$27\!\cdots\!08$$$$-$$$$27\!\cdots\!08$$$$\zeta_{6}) q^{52} -$$$$60\!\cdots\!79$$$$q^{57} +(-$$$$23\!\cdots\!32$$$$+$$$$11\!\cdots\!16$$$$\zeta_{6}) q^{61} +($$$$16\!\cdots\!41$$$$+$$$$50\!\cdots\!13$$$$\zeta_{6}) q^{63} +$$$$25\!\cdots\!48$$$$q^{64} +(-$$$$10\!\cdots\!39$$$$+$$$$10\!\cdots\!39$$$$\zeta_{6}) q^{67} +($$$$34\!\cdots\!59$$$$+$$$$34\!\cdots\!59$$$$\zeta_{6}) q^{73} +(-$$$$56\!\cdots\!50$$$$+$$$$28\!\cdots\!25$$$$\zeta_{6}) q^{75} +(-$$$$71\!\cdots\!64$$$$+$$$$14\!\cdots\!28$$$$\zeta_{6}) q^{76} -$$$$22\!\cdots\!67$$$$\zeta_{6} q^{79} +(-$$$$20\!\cdots\!69$$$$+$$$$20\!\cdots\!69$$$$\zeta_{6}) q^{81} +($$$$31\!\cdots\!72$$$$-$$$$44\!\cdots\!20$$$$\zeta_{6}) q^{84} +($$$$51\!\cdots\!84$$$$+$$$$11\!\cdots\!01$$$$\zeta_{6}) q^{91} -$$$$52\!\cdots\!95$$$$\zeta_{6} q^{93} +($$$$36\!\cdots\!28$$$$-$$$$73\!\cdots\!56$$$$\zeta_{6}) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1162261467q^{3} - 137438953472q^{4} + 5882725491086809q^{7} + 450283905890997363q^{9} + O(q^{10})$$ $$2q - 1162261467q^{3} - 137438953472q^{4} + 5882725491086809q^{7} + 450283905890997363q^{9} +$$$$15\!\cdots\!24$$$$q^{12} -$$$$18\!\cdots\!84$$$$q^{16} +$$$$15\!\cdots\!11$$$$q^{19} -$$$$55\!\cdots\!86$$$$q^{21} +$$$$72\!\cdots\!25$$$$q^{25} +$$$$34\!\cdots\!32$$$$q^{28} +$$$$13\!\cdots\!55$$$$q^{31} -$$$$12\!\cdots\!72$$$$q^{36} -$$$$19\!\cdots\!09$$$$q^{37} +$$$$23\!\cdots\!63$$$$q^{39} -$$$$62\!\cdots\!30$$$$q^{43} -$$$$25\!\cdots\!33$$$$q^{49} -$$$$83\!\cdots\!24$$$$q^{52} -$$$$12\!\cdots\!58$$$$q^{57} -$$$$35\!\cdots\!48$$$$q^{61} +$$$$37\!\cdots\!95$$$$q^{63} +$$$$51\!\cdots\!96$$$$q^{64} -$$$$10\!\cdots\!39$$$$q^{67} +$$$$10\!\cdots\!77$$$$q^{73} -$$$$84\!\cdots\!75$$$$q^{75} -$$$$22\!\cdots\!67$$$$q^{79} -$$$$20\!\cdots\!69$$$$q^{81} +$$$$17\!\cdots\!24$$$$q^{84} +$$$$22\!\cdots\!69$$$$q^{91} -$$$$52\!\cdots\!95$$$$q^{93} + O(q^{100})$$

## 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 −5.81131e8 + 3.35516e8i −6.87195e10 1.19026e11i 0 0 2.94136e15 + 3.14809e15i 0 2.25142e17 3.89957e17i 0
17.1 0 −5.81131e8 3.35516e8i −6.87195e10 + 1.19026e11i 0 0 2.94136e15 3.14809e15i 0 2.25142e17 + 3.89957e17i 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 21.38.g.a 2
3.b odd 2 1 CM 21.38.g.a 2
7.d odd 6 1 inner 21.38.g.a 2
21.g even 6 1 inner 21.38.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.38.g.a 2 1.a even 1 1 trivial
21.38.g.a 2 3.b odd 2 1 CM
21.38.g.a 2 7.d odd 6 1 inner
21.38.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 $$S_{38}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$450283905890997363 + 1162261467 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$18\!\cdots\!07$$$$- 5882725491086809 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$12\!\cdots\!63$$$$+ T^{2}$$
$17$ $$T^{2}$$
$19$ $$82\!\cdots\!07$$$$-$$$$15\!\cdots\!11$$$$T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$60\!\cdots\!75$$$$-$$$$13\!\cdots\!55$$$$T + T^{2}$$
$37$ $$39\!\cdots\!81$$$$+$$$$19\!\cdots\!09$$$$T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$($$$$31\!\cdots\!65$$$$+ T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$42\!\cdots\!68$$$$+$$$$35\!\cdots\!48$$$$T + T^{2}$$
$67$ $$10\!\cdots\!21$$$$+$$$$10\!\cdots\!39$$$$T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$34\!\cdots\!43$$$$-$$$$10\!\cdots\!77$$$$T + T^{2}$$
$79$ $$50\!\cdots\!89$$$$+$$$$22\!\cdots\!67$$$$T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$40\!\cdots\!52$$$$+ T^{2}$$
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