# Properties

 Label 21.20.g.a Level $21$ Weight $20$ Character orbit 21.g Analytic conductor $48.052$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 21.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.0515062768$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 19683 + 19683 \zeta_{6} ) q^{3} + ( -524288 + 524288 \zeta_{6} ) q^{4} + ( 103363346 - 109870239 \zeta_{6} ) q^{7} + 1162261467 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 19683 + 19683 \zeta_{6} ) q^{3} + ( -524288 + 524288 \zeta_{6} ) q^{4} + ( 103363346 - 109870239 \zeta_{6} ) q^{7} + 1162261467 \zeta_{6} q^{9} + ( -20639121408 + 10319560704 \zeta_{6} ) q^{12} + ( 43912988783 - 87825977566 \zeta_{6} ) q^{13} -274877906944 \zeta_{6} q^{16} + ( 1365151658466 - 682575829233 \zeta_{6} ) q^{19} + ( 4197076653555 - 2290651089156 \zeta_{6} ) q^{21} + ( 19073486328125 - 19073486328125 \zeta_{6} ) q^{25} + ( -22876792454961 + 45753584909922 \zeta_{6} ) q^{27} + ( 3411485917184 + 54192161947648 \zeta_{6} ) q^{28} + ( -159714263521255 - 159714263521255 \zeta_{6} ) q^{31} -609359740010496 q^{36} + 1581011344413647 \zeta_{6} q^{37} + ( 2593018074647367 - 2593018074647367 \zeta_{6} ) q^{39} + 6012529514786335 q^{43} + ( 5410421842378752 - 10820843684757504 \zeta_{6} ) q^{48} + ( -1387488121601405 - 10641641639802267 \zeta_{6} ) q^{49} + ( 23023053063061504 + 23023053063061504 \zeta_{6} ) q^{52} + 40305420140379417 q^{57} + ( 134845233947509688 - 67422616973754844 \zeta_{6} ) q^{61} + ( 127697945159780613 - 7562711003792031 \zeta_{6} ) q^{63} + 144115188075855872 q^{64} + ( 441174519368848177 - 441174519368848177 \zeta_{6} ) q^{67} + ( -95532016723177113 - 95532016723177113 \zeta_{6} ) q^{73} + ( 750846862792968750 - 375423431396484375 \zeta_{6} ) q^{75} + ( -357866316356911104 + 715732632713822208 \zeta_{6} ) q^{76} -262389124703851223 \zeta_{6} q^{79} + ( -1350851717672992089 + 1350851717672992089 \zeta_{6} ) q^{81} + ( -999516046307622912 + 2200476924539043840 \zeta_{6} ) q^{84} + ( -5110467692113710356 - 4253256334150166699 \zeta_{6} ) q^{91} -9430967546666586495 \zeta_{6} q^{93} + ( -6681938461866185624 + 13363876923732371248 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 59049q^{3} - 524288q^{4} + 96856453q^{7} + 1162261467q^{9} + O(q^{10})$$ $$2q + 59049q^{3} - 524288q^{4} + 96856453q^{7} + 1162261467q^{9} - 30958682112q^{12} - 274877906944q^{16} + 2047727487699q^{19} + 6103502217954q^{21} + 19073486328125q^{25} + 61015133782016q^{28} - 479142790563765q^{31} - 1218719480020992q^{36} + 1581011344413647q^{37} + 2593018074647367q^{39} + 12025059029572670q^{43} - 13416617883005077q^{49} + 69069159189184512q^{52} + 80610840280758834q^{57} + 202267850921264532q^{61} + 247833179315769195q^{63} + 288230376151711744q^{64} + 441174519368848177q^{67} - 286596050169531339q^{73} + 1126270294189453125q^{75} - 262389124703851223q^{79} - 1350851717672992089q^{81} + 201444831923798016q^{84} - 14474191718377587411q^{91} - 9430967546666586495q^{93} + O(q^{100})$$

## 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 29524.5 17046.0i −262144. 454047.i 0 0 4.84282e7 + 9.51504e7i 0 5.81131e8 1.00655e9i 0
17.1 0 29524.5 + 17046.0i −262144. + 454047.i 0 0 4.84282e7 9.51504e7i 0 5.81131e8 + 1.00655e9i 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 21.20.g.a 2
3.b odd 2 1 CM 21.20.g.a 2
7.d odd 6 1 inner 21.20.g.a 2
21.g even 6 1 inner 21.20.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.20.g.a 2 1.a even 1 1 trivial
21.20.g.a 2 3.b odd 2 1 CM
21.20.g.a 2 7.d odd 6 1 inner
21.20.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 $$S_{20}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1162261467 - 59049 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$11398895185373143 - 96856453 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$57\!\cdots\!67$$$$+ T^{2}$$
$17$ $$T^{2}$$
$19$ $$13\!\cdots\!67$$$$- 2047727487699 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$76\!\cdots\!75$$$$+ 479142790563765 T + T^{2}$$
$37$ $$24\!\cdots\!09$$$$- 1581011344413647 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -6012529514786335 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$13\!\cdots\!08$$$$- 202267850921264532 T + T^{2}$$
$67$ $$19\!\cdots\!29$$$$- 441174519368848177 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$27\!\cdots\!07$$$$+ 286596050169531339 T + T^{2}$$
$79$ $$68\!\cdots\!29$$$$+ 262389124703851223 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$13\!\cdots\!28$$$$+ T^{2}$$