# Properties

 Label 21.4.a.a Level $21$ Weight $4$ Character orbit 21.a Self dual yes Analytic conductor $1.239$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} - 3 q^{3} + q^{4} - 18 q^{5} + 9 q^{6} + 7 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10})$$ q - 3 * q^2 - 3 * q^3 + q^4 - 18 * q^5 + 9 * q^6 + 7 * q^7 + 21 * q^8 + 9 * q^9 $$q - 3 q^{2} - 3 q^{3} + q^{4} - 18 q^{5} + 9 q^{6} + 7 q^{7} + 21 q^{8} + 9 q^{9} + 54 q^{10} - 36 q^{11} - 3 q^{12} - 34 q^{13} - 21 q^{14} + 54 q^{15} - 71 q^{16} + 42 q^{17} - 27 q^{18} - 124 q^{19} - 18 q^{20} - 21 q^{21} + 108 q^{22} - 63 q^{24} + 199 q^{25} + 102 q^{26} - 27 q^{27} + 7 q^{28} + 102 q^{29} - 162 q^{30} - 160 q^{31} + 45 q^{32} + 108 q^{33} - 126 q^{34} - 126 q^{35} + 9 q^{36} + 398 q^{37} + 372 q^{38} + 102 q^{39} - 378 q^{40} - 318 q^{41} + 63 q^{42} - 268 q^{43} - 36 q^{44} - 162 q^{45} + 240 q^{47} + 213 q^{48} + 49 q^{49} - 597 q^{50} - 126 q^{51} - 34 q^{52} - 498 q^{53} + 81 q^{54} + 648 q^{55} + 147 q^{56} + 372 q^{57} - 306 q^{58} - 132 q^{59} + 54 q^{60} + 398 q^{61} + 480 q^{62} + 63 q^{63} + 433 q^{64} + 612 q^{65} - 324 q^{66} + 92 q^{67} + 42 q^{68} + 378 q^{70} - 720 q^{71} + 189 q^{72} - 502 q^{73} - 1194 q^{74} - 597 q^{75} - 124 q^{76} - 252 q^{77} - 306 q^{78} - 1024 q^{79} + 1278 q^{80} + 81 q^{81} + 954 q^{82} - 204 q^{83} - 21 q^{84} - 756 q^{85} + 804 q^{86} - 306 q^{87} - 756 q^{88} + 354 q^{89} + 486 q^{90} - 238 q^{91} + 480 q^{93} - 720 q^{94} + 2232 q^{95} - 135 q^{96} - 286 q^{97} - 147 q^{98} - 324 q^{99}+O(q^{100})$$ q - 3 * q^2 - 3 * q^3 + q^4 - 18 * q^5 + 9 * q^6 + 7 * q^7 + 21 * q^8 + 9 * q^9 + 54 * q^10 - 36 * q^11 - 3 * q^12 - 34 * q^13 - 21 * q^14 + 54 * q^15 - 71 * q^16 + 42 * q^17 - 27 * q^18 - 124 * q^19 - 18 * q^20 - 21 * q^21 + 108 * q^22 - 63 * q^24 + 199 * q^25 + 102 * q^26 - 27 * q^27 + 7 * q^28 + 102 * q^29 - 162 * q^30 - 160 * q^31 + 45 * q^32 + 108 * q^33 - 126 * q^34 - 126 * q^35 + 9 * q^36 + 398 * q^37 + 372 * q^38 + 102 * q^39 - 378 * q^40 - 318 * q^41 + 63 * q^42 - 268 * q^43 - 36 * q^44 - 162 * q^45 + 240 * q^47 + 213 * q^48 + 49 * q^49 - 597 * q^50 - 126 * q^51 - 34 * q^52 - 498 * q^53 + 81 * q^54 + 648 * q^55 + 147 * q^56 + 372 * q^57 - 306 * q^58 - 132 * q^59 + 54 * q^60 + 398 * q^61 + 480 * q^62 + 63 * q^63 + 433 * q^64 + 612 * q^65 - 324 * q^66 + 92 * q^67 + 42 * q^68 + 378 * q^70 - 720 * q^71 + 189 * q^72 - 502 * q^73 - 1194 * q^74 - 597 * q^75 - 124 * q^76 - 252 * q^77 - 306 * q^78 - 1024 * q^79 + 1278 * q^80 + 81 * q^81 + 954 * q^82 - 204 * q^83 - 21 * q^84 - 756 * q^85 + 804 * q^86 - 306 * q^87 - 756 * q^88 + 354 * q^89 + 486 * q^90 - 238 * q^91 + 480 * q^93 - 720 * q^94 + 2232 * q^95 - 135 * q^96 - 286 * q^97 - 147 * q^98 - 324 * 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
−3.00000 −3.00000 1.00000 −18.0000 9.00000 7.00000 21.0000 9.00000 54.0000
 $$n$$: 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.a 1
3.b odd 2 1 63.4.a.c 1
4.b odd 2 1 336.4.a.f 1
5.b even 2 1 525.4.a.g 1
5.c odd 4 2 525.4.d.c 2
7.b odd 2 1 147.4.a.c 1
7.c even 3 2 147.4.e.i 2
7.d odd 6 2 147.4.e.g 2
8.b even 2 1 1344.4.a.ba 1
8.d odd 2 1 1344.4.a.n 1
12.b even 2 1 1008.4.a.v 1
15.d odd 2 1 1575.4.a.b 1
21.c even 2 1 441.4.a.j 1
21.g even 6 2 441.4.e.d 2
21.h odd 6 2 441.4.e.b 2
28.d even 2 1 2352.4.a.r 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(21))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T + 3$$
$5$ $$T + 18$$
$7$ $$T - 7$$
$11$ $$T + 36$$
$13$ $$T + 34$$
$17$ $$T - 42$$
$19$ $$T + 124$$
$23$ $$T$$
$29$ $$T - 102$$
$31$ $$T + 160$$
$37$ $$T - 398$$
$41$ $$T + 318$$
$43$ $$T + 268$$
$47$ $$T - 240$$
$53$ $$T + 498$$
$59$ $$T + 132$$
$61$ $$T - 398$$
$67$ $$T - 92$$
$71$ $$T + 720$$
$73$ $$T + 502$$
$79$ $$T + 1024$$
$83$ $$T + 204$$
$89$ $$T - 354$$
$97$ $$T + 286$$