# Properties

 Label 21.6.a.d Level $21$ Weight $6$ Character orbit 21.a Self dual yes Analytic conductor $3.368$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,6,Mod(1,21)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 21.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.36806021607$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 10 q^{2} + 9 q^{3} + 68 q^{4} - 106 q^{5} + 90 q^{6} - 49 q^{7} + 360 q^{8} + 81 q^{9}+O(q^{10})$$ q + 10 * q^2 + 9 * q^3 + 68 * q^4 - 106 * q^5 + 90 * q^6 - 49 * q^7 + 360 * q^8 + 81 * q^9 $$q + 10 q^{2} + 9 q^{3} + 68 q^{4} - 106 q^{5} + 90 q^{6} - 49 q^{7} + 360 q^{8} + 81 q^{9} - 1060 q^{10} + 92 q^{11} + 612 q^{12} + 670 q^{13} - 490 q^{14} - 954 q^{15} + 1424 q^{16} - 222 q^{17} + 810 q^{18} - 908 q^{19} - 7208 q^{20} - 441 q^{21} + 920 q^{22} - 1176 q^{23} + 3240 q^{24} + 8111 q^{25} + 6700 q^{26} + 729 q^{27} - 3332 q^{28} + 1118 q^{29} - 9540 q^{30} + 3696 q^{31} + 2720 q^{32} + 828 q^{33} - 2220 q^{34} + 5194 q^{35} + 5508 q^{36} + 4182 q^{37} - 9080 q^{38} + 6030 q^{39} - 38160 q^{40} - 6662 q^{41} - 4410 q^{42} - 3700 q^{43} + 6256 q^{44} - 8586 q^{45} - 11760 q^{46} - 7056 q^{47} + 12816 q^{48} + 2401 q^{49} + 81110 q^{50} - 1998 q^{51} + 45560 q^{52} - 37578 q^{53} + 7290 q^{54} - 9752 q^{55} - 17640 q^{56} - 8172 q^{57} + 11180 q^{58} + 32700 q^{59} - 64872 q^{60} - 10802 q^{61} + 36960 q^{62} - 3969 q^{63} - 18368 q^{64} - 71020 q^{65} + 8280 q^{66} + 64996 q^{67} - 15096 q^{68} - 10584 q^{69} + 51940 q^{70} - 61320 q^{71} + 29160 q^{72} + 38922 q^{73} + 41820 q^{74} + 72999 q^{75} - 61744 q^{76} - 4508 q^{77} + 60300 q^{78} - 88096 q^{79} - 150944 q^{80} + 6561 q^{81} - 66620 q^{82} + 71892 q^{83} - 29988 q^{84} + 23532 q^{85} - 37000 q^{86} + 10062 q^{87} + 33120 q^{88} + 111818 q^{89} - 85860 q^{90} - 32830 q^{91} - 79968 q^{92} + 33264 q^{93} - 70560 q^{94} + 96248 q^{95} + 24480 q^{96} - 150846 q^{97} + 24010 q^{98} + 7452 q^{99}+O(q^{100})$$ q + 10 * q^2 + 9 * q^3 + 68 * q^4 - 106 * q^5 + 90 * q^6 - 49 * q^7 + 360 * q^8 + 81 * q^9 - 1060 * q^10 + 92 * q^11 + 612 * q^12 + 670 * q^13 - 490 * q^14 - 954 * q^15 + 1424 * q^16 - 222 * q^17 + 810 * q^18 - 908 * q^19 - 7208 * q^20 - 441 * q^21 + 920 * q^22 - 1176 * q^23 + 3240 * q^24 + 8111 * q^25 + 6700 * q^26 + 729 * q^27 - 3332 * q^28 + 1118 * q^29 - 9540 * q^30 + 3696 * q^31 + 2720 * q^32 + 828 * q^33 - 2220 * q^34 + 5194 * q^35 + 5508 * q^36 + 4182 * q^37 - 9080 * q^38 + 6030 * q^39 - 38160 * q^40 - 6662 * q^41 - 4410 * q^42 - 3700 * q^43 + 6256 * q^44 - 8586 * q^45 - 11760 * q^46 - 7056 * q^47 + 12816 * q^48 + 2401 * q^49 + 81110 * q^50 - 1998 * q^51 + 45560 * q^52 - 37578 * q^53 + 7290 * q^54 - 9752 * q^55 - 17640 * q^56 - 8172 * q^57 + 11180 * q^58 + 32700 * q^59 - 64872 * q^60 - 10802 * q^61 + 36960 * q^62 - 3969 * q^63 - 18368 * q^64 - 71020 * q^65 + 8280 * q^66 + 64996 * q^67 - 15096 * q^68 - 10584 * q^69 + 51940 * q^70 - 61320 * q^71 + 29160 * q^72 + 38922 * q^73 + 41820 * q^74 + 72999 * q^75 - 61744 * q^76 - 4508 * q^77 + 60300 * q^78 - 88096 * q^79 - 150944 * q^80 + 6561 * q^81 - 66620 * q^82 + 71892 * q^83 - 29988 * q^84 + 23532 * q^85 - 37000 * q^86 + 10062 * q^87 + 33120 * q^88 + 111818 * q^89 - 85860 * q^90 - 32830 * q^91 - 79968 * q^92 + 33264 * q^93 - 70560 * q^94 + 96248 * q^95 + 24480 * q^96 - 150846 * q^97 + 24010 * q^98 + 7452 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
10.0000 9.00000 68.0000 −106.000 90.0000 −49.0000 360.000 81.0000 −1060.00
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.a.d 1
3.b odd 2 1 63.6.a.a 1
4.b odd 2 1 336.6.a.a 1
5.b even 2 1 525.6.a.a 1
5.c odd 4 2 525.6.d.a 2
7.b odd 2 1 147.6.a.g 1
7.c even 3 2 147.6.e.a 2
7.d odd 6 2 147.6.e.b 2
12.b even 2 1 1008.6.a.bc 1
21.c even 2 1 441.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 1.a even 1 1 trivial
63.6.a.a 1 3.b odd 2 1
147.6.a.g 1 7.b odd 2 1
147.6.e.a 2 7.c even 3 2
147.6.e.b 2 7.d odd 6 2
336.6.a.a 1 4.b odd 2 1
441.6.a.b 1 21.c even 2 1
525.6.a.a 1 5.b even 2 1
525.6.d.a 2 5.c odd 4 2
1008.6.a.bc 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(21))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 10$$
$3$ $$T - 9$$
$5$ $$T + 106$$
$7$ $$T + 49$$
$11$ $$T - 92$$
$13$ $$T - 670$$
$17$ $$T + 222$$
$19$ $$T + 908$$
$23$ $$T + 1176$$
$29$ $$T - 1118$$
$31$ $$T - 3696$$
$37$ $$T - 4182$$
$41$ $$T + 6662$$
$43$ $$T + 3700$$
$47$ $$T + 7056$$
$53$ $$T + 37578$$
$59$ $$T - 32700$$
$61$ $$T + 10802$$
$67$ $$T - 64996$$
$71$ $$T + 61320$$
$73$ $$T - 38922$$
$79$ $$T + 88096$$
$83$ $$T - 71892$$
$89$ $$T - 111818$$
$97$ $$T + 150846$$