# Properties

 Label 3822.2.a.a Level $3822$ Weight $2$ Character orbit 3822.a Self dual yes Analytic conductor $30.519$ 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] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.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: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 - 3 * q^5 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} - q^{8} + q^{9} + 3 q^{10} + 3 q^{11} - q^{12} - q^{13} + 3 q^{15} + q^{16} + 3 q^{17} - q^{18} + 7 q^{19} - 3 q^{20} - 3 q^{22} + 9 q^{23} + q^{24} + 4 q^{25} + q^{26} - q^{27} - 9 q^{29} - 3 q^{30} + 4 q^{31} - q^{32} - 3 q^{33} - 3 q^{34} + q^{36} - 7 q^{37} - 7 q^{38} + q^{39} + 3 q^{40} - 12 q^{41} - q^{43} + 3 q^{44} - 3 q^{45} - 9 q^{46} - q^{48} - 4 q^{50} - 3 q^{51} - q^{52} - 6 q^{53} + q^{54} - 9 q^{55} - 7 q^{57} + 9 q^{58} - 12 q^{59} + 3 q^{60} + q^{61} - 4 q^{62} + q^{64} + 3 q^{65} + 3 q^{66} + 14 q^{67} + 3 q^{68} - 9 q^{69} + 12 q^{71} - q^{72} + 7 q^{73} + 7 q^{74} - 4 q^{75} + 7 q^{76} - q^{78} - 10 q^{79} - 3 q^{80} + q^{81} + 12 q^{82} + 6 q^{83} - 9 q^{85} + q^{86} + 9 q^{87} - 3 q^{88} + 6 q^{89} + 3 q^{90} + 9 q^{92} - 4 q^{93} - 21 q^{95} + q^{96} + 10 q^{97} + 3 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 - 3 * q^5 + q^6 - q^8 + q^9 + 3 * q^10 + 3 * q^11 - q^12 - q^13 + 3 * q^15 + q^16 + 3 * q^17 - q^18 + 7 * q^19 - 3 * q^20 - 3 * q^22 + 9 * q^23 + q^24 + 4 * q^25 + q^26 - q^27 - 9 * q^29 - 3 * q^30 + 4 * q^31 - q^32 - 3 * q^33 - 3 * q^34 + q^36 - 7 * q^37 - 7 * q^38 + q^39 + 3 * q^40 - 12 * q^41 - q^43 + 3 * q^44 - 3 * q^45 - 9 * q^46 - q^48 - 4 * q^50 - 3 * q^51 - q^52 - 6 * q^53 + q^54 - 9 * q^55 - 7 * q^57 + 9 * q^58 - 12 * q^59 + 3 * q^60 + q^61 - 4 * q^62 + q^64 + 3 * q^65 + 3 * q^66 + 14 * q^67 + 3 * q^68 - 9 * q^69 + 12 * q^71 - q^72 + 7 * q^73 + 7 * q^74 - 4 * q^75 + 7 * q^76 - q^78 - 10 * q^79 - 3 * q^80 + q^81 + 12 * q^82 + 6 * q^83 - 9 * q^85 + q^86 + 9 * q^87 - 3 * q^88 + 6 * q^89 + 3 * q^90 + 9 * q^92 - 4 * q^93 - 21 * q^95 + q^96 + 10 * q^97 + 3 * 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
−1.00000 −1.00000 1.00000 −3.00000 1.00000 0 −1.00000 1.00000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 3822.2.a.a 1
7.b odd 2 1 546.2.a.d 1
21.c even 2 1 1638.2.a.l 1
28.d even 2 1 4368.2.a.l 1
91.b odd 2 1 7098.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.d 1 7.b odd 2 1
1638.2.a.l 1 21.c even 2 1
3822.2.a.a 1 1.a even 1 1 trivial
4368.2.a.l 1 28.d even 2 1
7098.2.a.w 1 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{11} - 3$$ T11 - 3 $$T_{17} - 3$$ T17 - 3 $$T_{29} + 9$$ T29 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 3$$
$7$ $$T$$
$11$ $$T - 3$$
$13$ $$T + 1$$
$17$ $$T - 3$$
$19$ $$T - 7$$
$23$ $$T - 9$$
$29$ $$T + 9$$
$31$ $$T - 4$$
$37$ $$T + 7$$
$41$ $$T + 12$$
$43$ $$T + 1$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T - 1$$
$67$ $$T - 14$$
$71$ $$T - 12$$
$73$ $$T - 7$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T - 6$$
$97$ $$T - 10$$