# Properties

 Label 21.4.e.a Level $21$ Weight $4$ Character orbit 21.e Analytic conductor $1.239$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 21.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.23904011012$$ 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{2} -3 \zeta_{6} q^{3} -\zeta_{6} q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} -9 q^{6} + ( -14 + 21 \zeta_{6} ) q^{7} + 21 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{2} -3 \zeta_{6} q^{3} -\zeta_{6} q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} -9 q^{6} + ( -14 + 21 \zeta_{6} ) q^{7} + 21 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} -9 \zeta_{6} q^{10} + 15 \zeta_{6} q^{11} + ( -3 + 3 \zeta_{6} ) q^{12} -64 q^{13} + ( 21 + 42 \zeta_{6} ) q^{14} -9 q^{15} + ( 71 - 71 \zeta_{6} ) q^{16} -84 \zeta_{6} q^{17} + 27 \zeta_{6} q^{18} + ( 16 - 16 \zeta_{6} ) q^{19} -3 q^{20} + ( 63 - 21 \zeta_{6} ) q^{21} + 45 q^{22} + ( 84 - 84 \zeta_{6} ) q^{23} -63 \zeta_{6} q^{24} + 116 \zeta_{6} q^{25} + ( -192 + 192 \zeta_{6} ) q^{26} + 27 q^{27} + ( 21 - 7 \zeta_{6} ) q^{28} -297 q^{29} + ( -27 + 27 \zeta_{6} ) q^{30} + 253 \zeta_{6} q^{31} -45 \zeta_{6} q^{32} + ( 45 - 45 \zeta_{6} ) q^{33} -252 q^{34} + ( 21 + 42 \zeta_{6} ) q^{35} + 9 q^{36} + ( 316 - 316 \zeta_{6} ) q^{37} -48 \zeta_{6} q^{38} + 192 \zeta_{6} q^{39} + ( 63 - 63 \zeta_{6} ) q^{40} + 360 q^{41} + ( 126 - 189 \zeta_{6} ) q^{42} + 26 q^{43} + ( 15 - 15 \zeta_{6} ) q^{44} + 27 \zeta_{6} q^{45} -252 \zeta_{6} q^{46} + ( 30 - 30 \zeta_{6} ) q^{47} -213 q^{48} + ( -245 - 147 \zeta_{6} ) q^{49} + 348 q^{50} + ( -252 + 252 \zeta_{6} ) q^{51} + 64 \zeta_{6} q^{52} -363 \zeta_{6} q^{53} + ( 81 - 81 \zeta_{6} ) q^{54} + 45 q^{55} + ( -294 + 441 \zeta_{6} ) q^{56} -48 q^{57} + ( -891 + 891 \zeta_{6} ) q^{58} + 15 \zeta_{6} q^{59} + 9 \zeta_{6} q^{60} + ( 118 - 118 \zeta_{6} ) q^{61} + 759 q^{62} + ( -63 - 126 \zeta_{6} ) q^{63} + 433 q^{64} + ( -192 + 192 \zeta_{6} ) q^{65} -135 \zeta_{6} q^{66} + 370 \zeta_{6} q^{67} + ( -84 + 84 \zeta_{6} ) q^{68} -252 q^{69} + ( 189 - 63 \zeta_{6} ) q^{70} -342 q^{71} + ( -189 + 189 \zeta_{6} ) q^{72} -362 \zeta_{6} q^{73} -948 \zeta_{6} q^{74} + ( 348 - 348 \zeta_{6} ) q^{75} -16 q^{76} + ( -315 + 105 \zeta_{6} ) q^{77} + 576 q^{78} + ( -467 + 467 \zeta_{6} ) q^{79} -213 \zeta_{6} q^{80} -81 \zeta_{6} q^{81} + ( 1080 - 1080 \zeta_{6} ) q^{82} + 477 q^{83} + ( -21 - 42 \zeta_{6} ) q^{84} -252 q^{85} + ( 78 - 78 \zeta_{6} ) q^{86} + 891 \zeta_{6} q^{87} + 315 \zeta_{6} q^{88} + ( -906 + 906 \zeta_{6} ) q^{89} + 81 q^{90} + ( 896 - 1344 \zeta_{6} ) q^{91} -84 q^{92} + ( 759 - 759 \zeta_{6} ) q^{93} -90 \zeta_{6} q^{94} -48 \zeta_{6} q^{95} + ( -135 + 135 \zeta_{6} ) q^{96} + 503 q^{97} + ( -1176 + 735 \zeta_{6} ) q^{98} -135 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} - 3q^{3} - q^{4} + 3q^{5} - 18q^{6} - 7q^{7} + 42q^{8} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{2} - 3q^{3} - q^{4} + 3q^{5} - 18q^{6} - 7q^{7} + 42q^{8} - 9q^{9} - 9q^{10} + 15q^{11} - 3q^{12} - 128q^{13} + 84q^{14} - 18q^{15} + 71q^{16} - 84q^{17} + 27q^{18} + 16q^{19} - 6q^{20} + 105q^{21} + 90q^{22} + 84q^{23} - 63q^{24} + 116q^{25} - 192q^{26} + 54q^{27} + 35q^{28} - 594q^{29} - 27q^{30} + 253q^{31} - 45q^{32} + 45q^{33} - 504q^{34} + 84q^{35} + 18q^{36} + 316q^{37} - 48q^{38} + 192q^{39} + 63q^{40} + 720q^{41} + 63q^{42} + 52q^{43} + 15q^{44} + 27q^{45} - 252q^{46} + 30q^{47} - 426q^{48} - 637q^{49} + 696q^{50} - 252q^{51} + 64q^{52} - 363q^{53} + 81q^{54} + 90q^{55} - 147q^{56} - 96q^{57} - 891q^{58} + 15q^{59} + 9q^{60} + 118q^{61} + 1518q^{62} - 252q^{63} + 866q^{64} - 192q^{65} - 135q^{66} + 370q^{67} - 84q^{68} - 504q^{69} + 315q^{70} - 684q^{71} - 189q^{72} - 362q^{73} - 948q^{74} + 348q^{75} - 32q^{76} - 525q^{77} + 1152q^{78} - 467q^{79} - 213q^{80} - 81q^{81} + 1080q^{82} + 954q^{83} - 84q^{84} - 504q^{85} + 78q^{86} + 891q^{87} + 315q^{88} - 906q^{89} + 162q^{90} + 448q^{91} - 168q^{92} + 759q^{93} - 90q^{94} - 48q^{95} - 135q^{96} + 1006q^{97} - 1617q^{98} - 270q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 + 0.866025i 1.50000 + 2.59808i −9.00000 −3.50000 18.1865i 21.0000 −4.50000 7.79423i −4.50000 + 7.79423i
16.1 1.50000 2.59808i −1.50000 2.59808i −0.500000 0.866025i 1.50000 2.59808i −9.00000 −3.50000 + 18.1865i 21.0000 −4.50000 + 7.79423i −4.50000 7.79423i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.4.e.a 2
3.b odd 2 1 63.4.e.a 2
4.b odd 2 1 336.4.q.e 2
7.b odd 2 1 147.4.e.h 2
7.c even 3 1 inner 21.4.e.a 2
7.c even 3 1 147.4.a.b 1
7.d odd 6 1 147.4.a.a 1
7.d odd 6 1 147.4.e.h 2
21.c even 2 1 441.4.e.c 2
21.g even 6 1 441.4.a.k 1
21.g even 6 1 441.4.e.c 2
21.h odd 6 1 63.4.e.a 2
21.h odd 6 1 441.4.a.l 1
28.f even 6 1 2352.4.a.bd 1
28.g odd 6 1 336.4.q.e 2
28.g odd 6 1 2352.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 1.a even 1 1 trivial
21.4.e.a 2 7.c even 3 1 inner
63.4.e.a 2 3.b odd 2 1
63.4.e.a 2 21.h odd 6 1
147.4.a.a 1 7.d odd 6 1
147.4.a.b 1 7.c even 3 1
147.4.e.h 2 7.b odd 2 1
147.4.e.h 2 7.d odd 6 1
336.4.q.e 2 4.b odd 2 1
336.4.q.e 2 28.g odd 6 1
441.4.a.k 1 21.g even 6 1
441.4.a.l 1 21.h odd 6 1
441.4.e.c 2 21.c even 2 1
441.4.e.c 2 21.g even 6 1
2352.4.a.i 1 28.g odd 6 1
2352.4.a.bd 1 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3 T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 - 3 T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$343 + 7 T + T^{2}$$
$11$ $$225 - 15 T + T^{2}$$
$13$ $$( 64 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$256 - 16 T + T^{2}$$
$23$ $$7056 - 84 T + T^{2}$$
$29$ $$( 297 + T )^{2}$$
$31$ $$64009 - 253 T + T^{2}$$
$37$ $$99856 - 316 T + T^{2}$$
$41$ $$( -360 + T )^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$900 - 30 T + T^{2}$$
$53$ $$131769 + 363 T + T^{2}$$
$59$ $$225 - 15 T + T^{2}$$
$61$ $$13924 - 118 T + T^{2}$$
$67$ $$136900 - 370 T + T^{2}$$
$71$ $$( 342 + T )^{2}$$
$73$ $$131044 + 362 T + T^{2}$$
$79$ $$218089 + 467 T + T^{2}$$
$83$ $$( -477 + T )^{2}$$
$89$ $$820836 + 906 T + T^{2}$$
$97$ $$( -503 + T )^{2}$$
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