# Properties

 Label 21.28.c.a Level $21$ Weight $28$ Character orbit 21.c Analytic conductor $96.990$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 21.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$96.9896707160$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1594323 - 3188646 \zeta_{6} ) q^{3} + 134217728 q^{4} + ( 173476059101 + 120968265618 \zeta_{6} ) q^{7} -7625597484987 q^{9} +O(q^{10})$$ $$q +(1594323 - 3188646 \zeta_{6}) q^{3} +134217728 q^{4} +(173476059101 + 120968265618 \zeta_{6}) q^{7} -7625597484987 q^{9} +(213986410758144 - 427972821516288 \zeta_{6}) q^{12} +(-331524223037316 + 663048446074632 \zeta_{6}) q^{13} +18014398509481984 q^{16} +(-211512059159899590 + 423024118319799180 \zeta_{6}) q^{19} +(662301847263856851 - 746016230093053860 \zeta_{6}) q^{21} -7450580596923828125 q^{25} +(-12157665459056928801 + 24315330918113857602 \zeta_{6}) q^{27} +(23283562514929942528 + 16236085771348475904 \zeta_{6}) q^{28} +(-$$$$13\!\cdots\!90$$$$+$$$$27\!\cdots\!80$$$$\zeta_{6}) q^{31} -$$$$10\!\cdots\!36$$$$q^{36} -$$$$23\!\cdots\!90$$$$q^{37} +$$$$15\!\cdots\!04$$$$q^{39} -$$$$20\!\cdots\!20$$$$q^{43} +($$$$28\!\cdots\!32$$$$-$$$$57\!\cdots\!64$$$$\zeta_{6}) q^{48} +($$$$15\!\cdots\!77$$$$+$$$$56\!\cdots\!60$$$$\zeta_{6}) q^{49} +(-$$$$44\!\cdots\!48$$$$+$$$$88\!\cdots\!96$$$$\zeta_{6}) q^{52} +$$$$10\!\cdots\!10$$$$q^{57} +($$$$23\!\cdots\!40$$$$-$$$$46\!\cdots\!80$$$$\zeta_{6}) q^{61} +(-$$$$13\!\cdots\!87$$$$-$$$$92\!\cdots\!66$$$$\zeta_{6}) q^{63} +$$$$24\!\cdots\!52$$$$q^{64} -$$$$77\!\cdots\!20$$$$q^{67} +($$$$64\!\cdots\!36$$$$-$$$$12\!\cdots\!72$$$$\zeta_{6}) q^{73} +(-$$$$11\!\cdots\!75$$$$+$$$$23\!\cdots\!50$$$$\zeta_{6}) q^{75} +(-$$$$28\!\cdots\!20$$$$+$$$$56\!\cdots\!40$$$$\zeta_{6}) q^{76} -$$$$12\!\cdots\!56$$$$q^{79} +$$$$58\!\cdots\!69$$$$q^{81} +($$$$88\!\cdots\!28$$$$-$$$$10\!\cdots\!80$$$$\zeta_{6}) q^{84} +(-$$$$13\!\cdots\!92$$$$+$$$$15\!\cdots\!20$$$$\zeta_{6}) q^{91} +$$$$66\!\cdots\!10$$$$q^{93} +(-$$$$61\!\cdots\!72$$$$+$$$$12\!\cdots\!44$$$$\zeta_{6}) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 268435456q^{4} + 467920383820q^{7} - 15251194969974q^{9} + O(q^{10})$$ $$2q + 268435456q^{4} + 467920383820q^{7} - 15251194969974q^{9} + 36028797018963968q^{16} + 578587464434659842q^{21} - 14901161193847656250q^{25} + 62803210801208360960q^{28} -$$$$20\!\cdots\!72$$$$q^{36} -$$$$47\!\cdots\!80$$$$q^{37} +$$$$31\!\cdots\!08$$$$q^{39} -$$$$40\!\cdots\!40$$$$q^{43} +$$$$87\!\cdots\!14$$$$q^{49} +$$$$20\!\cdots\!20$$$$q^{57} -$$$$35\!\cdots\!40$$$$q^{63} +$$$$48\!\cdots\!04$$$$q^{64} -$$$$15\!\cdots\!40$$$$q^{67} -$$$$25\!\cdots\!12$$$$q^{79} +$$$$11\!\cdots\!38$$$$q^{81} +$$$$77\!\cdots\!76$$$$q^{84} -$$$$12\!\cdots\!64$$$$q^{91} +$$$$13\!\cdots\!20$$$$q^{93} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 2.76145e6i 1.34218e8 0 0 2.33960e11 + 1.04762e11i 0 −7.62560e12 0
20.2 0 2.76145e6i 1.34218e8 0 0 2.33960e11 1.04762e11i 0 −7.62560e12 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.28.c.a 2
3.b odd 2 1 CM 21.28.c.a 2
7.b odd 2 1 inner 21.28.c.a 2
21.c even 2 1 inner 21.28.c.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{28}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$7625597484987 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$65\!\cdots\!43$$$$- 467920383820 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$32\!\cdots\!68$$$$+ T^{2}$$
$17$ $$T^{2}$$
$19$ $$13\!\cdots\!00$$$$+ T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$57\!\cdots\!00$$$$+ T^{2}$$
$37$ $$($$$$23\!\cdots\!90$$$$+ T )^{2}$$
$41$ $$T^{2}$$
$43$ $$($$$$20\!\cdots\!20$$$$+ T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$16\!\cdots\!00$$$$+ T^{2}$$
$67$ $$($$$$77\!\cdots\!20$$$$+ T )^{2}$$
$71$ $$T^{2}$$
$73$ $$12\!\cdots\!88$$$$+ T^{2}$$
$79$ $$($$$$12\!\cdots\!56$$$$+ T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$11\!\cdots\!52$$$$+ T^{2}$$