# Properties

 Label 21.22.c.a Level $21$ Weight $22$ Character orbit 21.c Analytic conductor $58.690$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 21.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$58.6902423003$$ 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: $$2\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -59049 + 118098 \zeta_{6} ) q^{3} + 2097152 q^{4} + ( -846427447 + 568871874 \zeta_{6} ) q^{7} -10460353203 q^{9} +O(q^{10})$$ $$q + ( -59049 + 118098 \zeta_{6} ) q^{3} + 2097152 q^{4} + ( -846427447 + 568871874 \zeta_{6} ) q^{7} -10460353203 q^{9} + ( -123834728448 + 247669456896 \zeta_{6} ) q^{12} + ( 532697931828 - 1065395863656 \zeta_{6} ) q^{13} + 4398046511104 q^{16} + ( 23048508442770 - 46097016885540 \zeta_{6} ) q^{19} + ( -17201936257749 - 66370073347980 \zeta_{6} ) q^{21} -476837158203125 q^{25} + ( 617673396283947 - 1235346792567894 \zeta_{6} ) q^{27} + ( -1775087013330944 + 1193010788302848 \zeta_{6} ) q^{28} + ( -725516191935270 + 1451032383870540 \zeta_{6} ) q^{31} -21936950640377856 q^{36} + 57776323439003290 q^{37} + 94365840529534716 q^{39} -265258444413820520 q^{43} + ( -259700248434180096 + 519400496868360192 \zeta_{6} ) q^{48} + ( 392824214006665933 - 639402326931579480 \zeta_{6} ) q^{49} + ( 1117148533128953856 - 2234297066257907712 \zeta_{6} ) q^{52} + 4082974125111377190 q^{57} + ( 1438373287720915020 - 2876746575441830040 \zeta_{6} ) q^{61} + ( 8853930056333562741 - 5950600729292512422 \zeta_{6} ) q^{63} + 9223372036854775808 q^{64} + 6944332370266921520 q^{67} + ( 35890654689431889192 - 71781309378863778384 \zeta_{6} ) q^{73} + ( 28156757354736328125 - 56313514709472656250 \zeta_{6} ) q^{75} + ( 48336225577771991040 - 96672451155543982080 \zeta_{6} ) q^{76} + 168068289047083465196 q^{79} + 109418989131512359209 q^{81} + ( -36075075026810830848 - 139188132061862952960 \zeta_{6} ) q^{84} + ( 155183591050483128228 + 598743430163789560560 \zeta_{6} ) q^{91} -128523016852757274690 q^{93} + ( 525465954789943160664 - 1050931909579886321328 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4194304q^{4} - 1123983020q^{7} - 20920706406q^{9} + O(q^{10})$$ $$2q + 4194304q^{4} - 1123983020q^{7} - 20920706406q^{9} + 8796093022208q^{16} - 100773945863478q^{21} - 953674316406250q^{25} - 2357163238359040q^{28} - 43873901280755712q^{36} + 115552646878006580q^{37} + 188731681059069432q^{39} - 530516888827641040q^{43} + 146246101081752386q^{49} + 8165948250222754380q^{57} + 11757259383374613060q^{63} + 18446744073709551616q^{64} + 13888664740533843040q^{67} + 336136578094166930392q^{79} + 218837978263024718418q^{81} - 211338282115484614656q^{84} + 909110612264755817016q^{91} - 257046033705514549380q^{93} + O(q^{100})$$

## 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 102276.i 2.09715e6 0 0 −5.61992e8 4.92657e8i 0 −1.04604e10 0
20.2 0 102276.i 2.09715e6 0 0 −5.61992e8 + 4.92657e8i 0 −1.04604e10 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 21.22.c.a 2
3.b odd 2 1 CM 21.22.c.a 2
7.b odd 2 1 inner 21.22.c.a 2
21.c even 2 1 inner 21.22.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.22.c.a 2 1.a even 1 1 trivial
21.22.c.a 2 3.b odd 2 1 CM
21.22.c.a 2 7.b odd 2 1 inner
21.22.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 $$S_{22}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$10460353203 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$558545864083284007 + 1123983020 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$85\!\cdots\!52$$$$+ T^{2}$$
$17$ $$T^{2}$$
$19$ $$15\!\cdots\!00$$$$+ T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$15\!\cdots\!00$$$$+ T^{2}$$
$37$ $$( -57776323439003290 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( 265258444413820520 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$62\!\cdots\!00$$$$+ T^{2}$$
$67$ $$( -6944332370266921520 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$38\!\cdots\!92$$$$+ T^{2}$$
$79$ $$( -$$$$16\!\cdots\!96$$$$+ T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$82\!\cdots\!88$$$$+ T^{2}$$