# Properties

 Label 21.4.a.b Level $21$ Weight $4$ Character orbit 21.a Self dual yes Analytic conductor $1.239$ 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] = [21,4,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, 4, names="a")

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

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$1.23904011012$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 4 q^{5} - 12 q^{6} - 7 q^{7} + 9 q^{9}+O(q^{10})$$ q + 4 * q^2 - 3 * q^3 + 8 * q^4 - 4 * q^5 - 12 * q^6 - 7 * q^7 + 9 * q^9 $$q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 4 q^{5} - 12 q^{6} - 7 q^{7} + 9 q^{9} - 16 q^{10} + 62 q^{11} - 24 q^{12} - 62 q^{13} - 28 q^{14} + 12 q^{15} - 64 q^{16} + 84 q^{17} + 36 q^{18} + 100 q^{19} - 32 q^{20} + 21 q^{21} + 248 q^{22} - 42 q^{23} - 109 q^{25} - 248 q^{26} - 27 q^{27} - 56 q^{28} - 10 q^{29} + 48 q^{30} - 48 q^{31} - 256 q^{32} - 186 q^{33} + 336 q^{34} + 28 q^{35} + 72 q^{36} - 246 q^{37} + 400 q^{38} + 186 q^{39} - 248 q^{41} + 84 q^{42} + 68 q^{43} + 496 q^{44} - 36 q^{45} - 168 q^{46} + 324 q^{47} + 192 q^{48} + 49 q^{49} - 436 q^{50} - 252 q^{51} - 496 q^{52} + 258 q^{53} - 108 q^{54} - 248 q^{55} - 300 q^{57} - 40 q^{58} + 120 q^{59} + 96 q^{60} + 622 q^{61} - 192 q^{62} - 63 q^{63} - 512 q^{64} + 248 q^{65} - 744 q^{66} + 904 q^{67} + 672 q^{68} + 126 q^{69} + 112 q^{70} - 678 q^{71} - 642 q^{73} - 984 q^{74} + 327 q^{75} + 800 q^{76} - 434 q^{77} + 744 q^{78} + 740 q^{79} + 256 q^{80} + 81 q^{81} - 992 q^{82} + 468 q^{83} + 168 q^{84} - 336 q^{85} + 272 q^{86} + 30 q^{87} + 200 q^{89} - 144 q^{90} + 434 q^{91} - 336 q^{92} + 144 q^{93} + 1296 q^{94} - 400 q^{95} + 768 q^{96} - 1266 q^{97} + 196 q^{98} + 558 q^{99}+O(q^{100})$$ q + 4 * q^2 - 3 * q^3 + 8 * q^4 - 4 * q^5 - 12 * q^6 - 7 * q^7 + 9 * q^9 - 16 * q^10 + 62 * q^11 - 24 * q^12 - 62 * q^13 - 28 * q^14 + 12 * q^15 - 64 * q^16 + 84 * q^17 + 36 * q^18 + 100 * q^19 - 32 * q^20 + 21 * q^21 + 248 * q^22 - 42 * q^23 - 109 * q^25 - 248 * q^26 - 27 * q^27 - 56 * q^28 - 10 * q^29 + 48 * q^30 - 48 * q^31 - 256 * q^32 - 186 * q^33 + 336 * q^34 + 28 * q^35 + 72 * q^36 - 246 * q^37 + 400 * q^38 + 186 * q^39 - 248 * q^41 + 84 * q^42 + 68 * q^43 + 496 * q^44 - 36 * q^45 - 168 * q^46 + 324 * q^47 + 192 * q^48 + 49 * q^49 - 436 * q^50 - 252 * q^51 - 496 * q^52 + 258 * q^53 - 108 * q^54 - 248 * q^55 - 300 * q^57 - 40 * q^58 + 120 * q^59 + 96 * q^60 + 622 * q^61 - 192 * q^62 - 63 * q^63 - 512 * q^64 + 248 * q^65 - 744 * q^66 + 904 * q^67 + 672 * q^68 + 126 * q^69 + 112 * q^70 - 678 * q^71 - 642 * q^73 - 984 * q^74 + 327 * q^75 + 800 * q^76 - 434 * q^77 + 744 * q^78 + 740 * q^79 + 256 * q^80 + 81 * q^81 - 992 * q^82 + 468 * q^83 + 168 * q^84 - 336 * q^85 + 272 * q^86 + 30 * q^87 + 200 * q^89 - 144 * q^90 + 434 * q^91 - 336 * q^92 + 144 * q^93 + 1296 * q^94 - 400 * q^95 + 768 * q^96 - 1266 * q^97 + 196 * q^98 + 558 * 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
4.00000 −3.00000 8.00000 −4.00000 −12.0000 −7.00000 0 9.00000 −16.0000
 $$n$$: 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 21.4.a.b 1
3.b odd 2 1 63.4.a.a 1
4.b odd 2 1 336.4.a.h 1
5.b even 2 1 525.4.a.b 1
5.c odd 4 2 525.4.d.b 2
7.b odd 2 1 147.4.a.g 1
7.c even 3 2 147.4.e.c 2
7.d odd 6 2 147.4.e.b 2
8.b even 2 1 1344.4.a.w 1
8.d odd 2 1 1344.4.a.i 1
12.b even 2 1 1008.4.a.m 1
15.d odd 2 1 1575.4.a.k 1
21.c even 2 1 441.4.a.b 1
21.g even 6 2 441.4.e.n 2
21.h odd 6 2 441.4.e.m 2
28.d even 2 1 2352.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 1.a even 1 1 trivial
63.4.a.a 1 3.b odd 2 1
147.4.a.g 1 7.b odd 2 1
147.4.e.b 2 7.d odd 6 2
147.4.e.c 2 7.c even 3 2
336.4.a.h 1 4.b odd 2 1
441.4.a.b 1 21.c even 2 1
441.4.e.m 2 21.h odd 6 2
441.4.e.n 2 21.g even 6 2
525.4.a.b 1 5.b even 2 1
525.4.d.b 2 5.c odd 4 2
1008.4.a.m 1 12.b even 2 1
1344.4.a.i 1 8.d odd 2 1
1344.4.a.w 1 8.b even 2 1
1575.4.a.k 1 15.d odd 2 1
2352.4.a.l 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(21))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T + 3$$
$5$ $$T + 4$$
$7$ $$T + 7$$
$11$ $$T - 62$$
$13$ $$T + 62$$
$17$ $$T - 84$$
$19$ $$T - 100$$
$23$ $$T + 42$$
$29$ $$T + 10$$
$31$ $$T + 48$$
$37$ $$T + 246$$
$41$ $$T + 248$$
$43$ $$T - 68$$
$47$ $$T - 324$$
$53$ $$T - 258$$
$59$ $$T - 120$$
$61$ $$T - 622$$
$67$ $$T - 904$$
$71$ $$T + 678$$
$73$ $$T + 642$$
$79$ $$T - 740$$
$83$ $$T - 468$$
$89$ $$T - 200$$
$97$ $$T + 1266$$