# Properties

 Label 21.5.f.a Level $21$ Weight $5$ Character orbit 21.f Analytic conductor $2.171$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17076922476$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( 3 + 3 \zeta_{6} ) q^{3} + ( 12 - 12 \zeta_{6} ) q^{4} + ( 12 - 6 \zeta_{6} ) q^{5} + ( 6 - 12 \zeta_{6} ) q^{6} + ( 56 - 35 \zeta_{6} ) q^{7} -56 q^{8} + 27 \zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( 3 + 3 \zeta_{6} ) q^{3} + ( 12 - 12 \zeta_{6} ) q^{4} + ( 12 - 6 \zeta_{6} ) q^{5} + ( 6 - 12 \zeta_{6} ) q^{6} + ( 56 - 35 \zeta_{6} ) q^{7} -56 q^{8} + 27 \zeta_{6} q^{9} + ( -12 - 12 \zeta_{6} ) q^{10} + ( -194 + 194 \zeta_{6} ) q^{11} + ( 72 - 36 \zeta_{6} ) q^{12} + ( -95 + 190 \zeta_{6} ) q^{13} + ( -70 - 42 \zeta_{6} ) q^{14} + 54 q^{15} -80 \zeta_{6} q^{16} + ( -140 - 140 \zeta_{6} ) q^{17} + ( 54 - 54 \zeta_{6} ) q^{18} + ( 302 - 151 \zeta_{6} ) q^{19} + ( 72 - 144 \zeta_{6} ) q^{20} + ( 273 - 42 \zeta_{6} ) q^{21} + 388 q^{22} + 112 \zeta_{6} q^{23} + ( -168 - 168 \zeta_{6} ) q^{24} + ( -517 + 517 \zeta_{6} ) q^{25} + ( 380 - 190 \zeta_{6} ) q^{26} + ( -81 + 162 \zeta_{6} ) q^{27} + ( 252 - 672 \zeta_{6} ) q^{28} + 1040 q^{29} -108 \zeta_{6} q^{30} + ( -673 - 673 \zeta_{6} ) q^{31} + ( -1056 + 1056 \zeta_{6} ) q^{32} + ( -1164 + 582 \zeta_{6} ) q^{33} + ( -280 + 560 \zeta_{6} ) q^{34} + ( 462 - 546 \zeta_{6} ) q^{35} + 324 q^{36} + 1075 \zeta_{6} q^{37} + ( -302 - 302 \zeta_{6} ) q^{38} + ( -855 + 855 \zeta_{6} ) q^{39} + ( -672 + 336 \zeta_{6} ) q^{40} + ( 754 - 1508 \zeta_{6} ) q^{41} + ( -84 - 462 \zeta_{6} ) q^{42} -1087 q^{43} + 2328 \zeta_{6} q^{44} + ( 162 + 162 \zeta_{6} ) q^{45} + ( 224 - 224 \zeta_{6} ) q^{46} + ( 2500 - 1250 \zeta_{6} ) q^{47} + ( 240 - 480 \zeta_{6} ) q^{48} + ( 1911 - 2695 \zeta_{6} ) q^{49} + 1034 q^{50} -1260 \zeta_{6} q^{51} + ( 1140 + 1140 \zeta_{6} ) q^{52} + ( 2200 - 2200 \zeta_{6} ) q^{53} + ( 324 - 162 \zeta_{6} ) q^{54} + ( -1164 + 2328 \zeta_{6} ) q^{55} + ( -3136 + 1960 \zeta_{6} ) q^{56} + 1359 q^{57} -2080 \zeta_{6} q^{58} + ( -3088 - 3088 \zeta_{6} ) q^{59} + ( 648 - 648 \zeta_{6} ) q^{60} + ( 808 - 404 \zeta_{6} ) q^{61} + ( -1346 + 2692 \zeta_{6} ) q^{62} + ( 945 + 567 \zeta_{6} ) q^{63} + 832 q^{64} + 1710 \zeta_{6} q^{65} + ( 1164 + 1164 \zeta_{6} ) q^{66} + ( -2375 + 2375 \zeta_{6} ) q^{67} + ( -3360 + 1680 \zeta_{6} ) q^{68} + ( -336 + 672 \zeta_{6} ) q^{69} + ( -1092 + 168 \zeta_{6} ) q^{70} -8938 q^{71} -1512 \zeta_{6} q^{72} + ( 5269 + 5269 \zeta_{6} ) q^{73} + ( 2150 - 2150 \zeta_{6} ) q^{74} + ( -3102 + 1551 \zeta_{6} ) q^{75} + ( 1812 - 3624 \zeta_{6} ) q^{76} + ( -4074 + 10864 \zeta_{6} ) q^{77} + 1710 q^{78} -8147 \zeta_{6} q^{79} + ( -480 - 480 \zeta_{6} ) q^{80} + ( -729 + 729 \zeta_{6} ) q^{81} + ( -3016 + 1508 \zeta_{6} ) q^{82} + ( 3854 - 7708 \zeta_{6} ) q^{83} + ( 2772 - 3276 \zeta_{6} ) q^{84} -2520 q^{85} + 2174 \zeta_{6} q^{86} + ( 3120 + 3120 \zeta_{6} ) q^{87} + ( 10864 - 10864 \zeta_{6} ) q^{88} + ( 15752 - 7876 \zeta_{6} ) q^{89} + ( 324 - 648 \zeta_{6} ) q^{90} + ( 1330 + 7315 \zeta_{6} ) q^{91} + 1344 q^{92} -6057 \zeta_{6} q^{93} + ( -2500 - 2500 \zeta_{6} ) q^{94} + ( 2718 - 2718 \zeta_{6} ) q^{95} + ( -6336 + 3168 \zeta_{6} ) q^{96} + ( -2020 + 4040 \zeta_{6} ) q^{97} + ( -5390 + 1568 \zeta_{6} ) q^{98} -5238 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 9q^{3} + 12q^{4} + 18q^{5} + 77q^{7} - 112q^{8} + 27q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 9q^{3} + 12q^{4} + 18q^{5} + 77q^{7} - 112q^{8} + 27q^{9} - 36q^{10} - 194q^{11} + 108q^{12} - 182q^{14} + 108q^{15} - 80q^{16} - 420q^{17} + 54q^{18} + 453q^{19} + 504q^{21} + 776q^{22} + 112q^{23} - 504q^{24} - 517q^{25} + 570q^{26} - 168q^{28} + 2080q^{29} - 108q^{30} - 2019q^{31} - 1056q^{32} - 1746q^{33} + 378q^{35} + 648q^{36} + 1075q^{37} - 906q^{38} - 855q^{39} - 1008q^{40} - 630q^{42} - 2174q^{43} + 2328q^{44} + 486q^{45} + 224q^{46} + 3750q^{47} + 1127q^{49} + 2068q^{50} - 1260q^{51} + 3420q^{52} + 2200q^{53} + 486q^{54} - 4312q^{56} + 2718q^{57} - 2080q^{58} - 9264q^{59} + 648q^{60} + 1212q^{61} + 2457q^{63} + 1664q^{64} + 1710q^{65} + 3492q^{66} - 2375q^{67} - 5040q^{68} - 2016q^{70} - 17876q^{71} - 1512q^{72} + 15807q^{73} + 2150q^{74} - 4653q^{75} + 2716q^{77} + 3420q^{78} - 8147q^{79} - 1440q^{80} - 729q^{81} - 4524q^{82} + 2268q^{84} - 5040q^{85} + 2174q^{86} + 9360q^{87} + 10864q^{88} + 23628q^{89} + 9975q^{91} + 2688q^{92} - 6057q^{93} - 7500q^{94} + 2718q^{95} - 9504q^{96} - 9212q^{98} - 10476q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 4.50000 + 2.59808i 6.00000 10.3923i 9.00000 5.19615i 10.3923i 38.5000 30.3109i −56.0000 13.5000 + 23.3827i −18.0000 10.3923i
19.1 −1.00000 + 1.73205i 4.50000 2.59808i 6.00000 + 10.3923i 9.00000 + 5.19615i 10.3923i 38.5000 + 30.3109i −56.0000 13.5000 23.3827i −18.0000 + 10.3923i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.5.f.a 2
3.b odd 2 1 63.5.m.c 2
4.b odd 2 1 336.5.bh.b 2
7.b odd 2 1 147.5.f.a 2
7.c even 3 1 147.5.d.b 2
7.c even 3 1 147.5.f.a 2
7.d odd 6 1 inner 21.5.f.a 2
7.d odd 6 1 147.5.d.b 2
21.g even 6 1 63.5.m.c 2
21.g even 6 1 441.5.d.a 2
21.h odd 6 1 441.5.d.a 2
28.f even 6 1 336.5.bh.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.f.a 2 1.a even 1 1 trivial
21.5.f.a 2 7.d odd 6 1 inner
63.5.m.c 2 3.b odd 2 1
63.5.m.c 2 21.g even 6 1
147.5.d.b 2 7.c even 3 1
147.5.d.b 2 7.d odd 6 1
147.5.f.a 2 7.b odd 2 1
147.5.f.a 2 7.c even 3 1
336.5.bh.b 2 4.b odd 2 1
336.5.bh.b 2 28.f even 6 1
441.5.d.a 2 21.g even 6 1
441.5.d.a 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} + 4$$ acting on $$S_{5}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$27 - 9 T + T^{2}$$
$5$ $$108 - 18 T + T^{2}$$
$7$ $$2401 - 77 T + T^{2}$$
$11$ $$37636 + 194 T + T^{2}$$
$13$ $$27075 + T^{2}$$
$17$ $$58800 + 420 T + T^{2}$$
$19$ $$68403 - 453 T + T^{2}$$
$23$ $$12544 - 112 T + T^{2}$$
$29$ $$( -1040 + T )^{2}$$
$31$ $$1358787 + 2019 T + T^{2}$$
$37$ $$1155625 - 1075 T + T^{2}$$
$41$ $$1705548 + T^{2}$$
$43$ $$( 1087 + T )^{2}$$
$47$ $$4687500 - 3750 T + T^{2}$$
$53$ $$4840000 - 2200 T + T^{2}$$
$59$ $$28607232 + 9264 T + T^{2}$$
$61$ $$489648 - 1212 T + T^{2}$$
$67$ $$5640625 + 2375 T + T^{2}$$
$71$ $$( 8938 + T )^{2}$$
$73$ $$83287083 - 15807 T + T^{2}$$
$79$ $$66373609 + 8147 T + T^{2}$$
$83$ $$44559948 + T^{2}$$
$89$ $$186094128 - 23628 T + T^{2}$$
$97$ $$12241200 + T^{2}$$