Properties

 Label 21.21.h.a Level $21$ Weight $21$ Character orbit 21.h Analytic conductor $53.238$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$53.2378906716$$ 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.

 $$f(q)$$ $$=$$ $$q -59049 \zeta_{6} q^{3} -1048576 \zeta_{6} q^{4} + ( -122884025 - 200219799 \zeta_{6} ) q^{7} + ( -3486784401 + 3486784401 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -59049 \zeta_{6} q^{3} -1048576 \zeta_{6} q^{4} + ( -122884025 - 200219799 \zeta_{6} ) q^{7} + ( -3486784401 + 3486784401 \zeta_{6} ) q^{9} + ( -61917364224 + 61917364224 \zeta_{6} ) q^{12} + 38170922351 q^{13} + ( -1099511627776 + 1099511627776 \zeta_{6} ) q^{16} + ( -12005816399399 + 12005816399399 \zeta_{6} ) q^{19} + ( -11822778911151 + 19078957703376 \zeta_{6} ) q^{21} -95367431640625 \zeta_{6} q^{25} + 205891132094649 q^{27} + ( -209945675956224 + 338798915354624 \zeta_{6} ) q^{28} -845125195444727 \zeta_{6} q^{31} + 3656158440062976 q^{36} + ( 721096304301649 - 721096304301649 \zeta_{6} ) q^{37} -2253954793904199 \zeta_{6} q^{39} + 1343935055601623 q^{43} + 64925062108545024 q^{48} + ( -24987484311399776 + 89295597483222351 \zeta_{6} ) q^{49} -40025113075122176 \zeta_{6} q^{52} + 708931452568111551 q^{57} + ( 1387788230779990126 - 1387788230779990126 \zeta_{6} ) q^{61} + ( 1126593373426649424 - 428470101502094025 \zeta_{6} ) q^{63} + 1152921504606846976 q^{64} + 2055850947949627849 \zeta_{6} q^{67} + 4762721323358202049 \zeta_{6} q^{73} + ( -5631351470947265625 + 5631351470947265625 \zeta_{6} ) q^{75} + 12589010936816205824 q^{76} + ( 14522848453745208601 - 14522848453745208601 \zeta_{6} ) q^{79} -12157665459056928801 \zeta_{6} q^{81} + ( 20005737152775192576 - 7608654933236121600 \zeta_{6} ) q^{84} + ( -4690596576453342775 - 7642574400761827449 \zeta_{6} ) q^{91} + ( -49903797665815684623 + 49903797665815684623 \zeta_{6} ) q^{93} -146701748198870395774 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 59049q^{3} - 1048576q^{4} - 445987849q^{7} - 3486784401q^{9} + O(q^{10})$$ $$2q - 59049q^{3} - 1048576q^{4} - 445987849q^{7} - 3486784401q^{9} - 61917364224q^{12} + 76341844702q^{13} - 1099511627776q^{16} - 12005816399399q^{19} - 4566600118926q^{21} - 95367431640625q^{25} + 411782264189298q^{27} - 81092436557824q^{28} - 845125195444727q^{31} + 7312316880125952q^{36} + 721096304301649q^{37} - 2253954793904199q^{39} + 2687870111203246q^{43} + 129850124217090048q^{48} + 39320628860422799q^{49} - 40025113075122176q^{52} + 1417862905136223102q^{57} + 1387788230779990126q^{61} + 1824716645351204823q^{63} + 2305843009213693952q^{64} + 2055850947949627849q^{67} + 4762721323358202049q^{73} - 5631351470947265625q^{75} + 25178021873632411648q^{76} + 14522848453745208601q^{79} - 12157665459056928801q^{81} + 32402819372314263552q^{84} - 17023767553668512999q^{91} - 49903797665815684623q^{93} - 293403496397740791548q^{97} + O(q^{100})$$

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 −29524.5 51137.9i −524288. 908093.i 0 0 −2.22994e8 1.73395e8i 0 −1.74339e9 + 3.01964e9i 0
11.1 0 −29524.5 + 51137.9i −524288. + 908093.i 0 0 −2.22994e8 + 1.73395e8i 0 −1.74339e9 3.01964e9i 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 21.21.h.a 2
3.b odd 2 1 CM 21.21.h.a 2
7.c even 3 1 inner 21.21.h.a 2
21.h odd 6 1 inner 21.21.h.a 2

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3486784401 + 59049 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$79792266297612001 + 445987849 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -38170922351 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$14\!\cdots\!01$$$$+ 12005816399399 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$71\!\cdots\!29$$$$+ 845125195444727 T + T^{2}$$
$37$ $$51\!\cdots\!01$$$$- 721096304301649 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -1343935055601623 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$19\!\cdots\!76$$$$- 1387788230779990126 T + T^{2}$$
$67$ $$42\!\cdots\!01$$$$- 2055850947949627849 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$22\!\cdots\!01$$$$- 4762721323358202049 T + T^{2}$$
$79$ $$21\!\cdots\!01$$$$- 14522848453745208601 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$($$$$14\!\cdots\!74$$$$+ T )^{2}$$