# Properties

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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - q^{11} + q^{12} - q^{13} - q^{15} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} - q^{20} - q^{22} - 6 q^{23} + q^{24} - 4 q^{25} - q^{26} + q^{27} + 3 q^{29} - q^{30} - 11 q^{31} + q^{32} - q^{33} - 6 q^{34} + q^{36} + 4 q^{37} - 4 q^{38} - q^{39} - q^{40} + 12 q^{41} - 8 q^{43} - q^{44} - q^{45} - 6 q^{46} - 8 q^{47} + q^{48} - 4 q^{50} - 6 q^{51} - q^{52} - 5 q^{53} + q^{54} + q^{55} - 4 q^{57} + 3 q^{58} - 5 q^{59} - q^{60} + 12 q^{61} - 11 q^{62} + q^{64} + q^{65} - q^{66} + 16 q^{67} - 6 q^{68} - 6 q^{69} + 6 q^{71} + q^{72} - 10 q^{73} + 4 q^{74} - 4 q^{75} - 4 q^{76} - q^{78} + 7 q^{79} - q^{80} + q^{81} + 12 q^{82} - 17 q^{83} + 6 q^{85} - 8 q^{86} + 3 q^{87} - q^{88} - 12 q^{89} - q^{90} - 6 q^{92} - 11 q^{93} - 8 q^{94} + 4 q^{95} + q^{96} + 13 q^{97} - q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 + q^9 - q^10 - q^11 + q^12 - q^13 - q^15 + q^16 - 6 * q^17 + q^18 - 4 * q^19 - q^20 - q^22 - 6 * q^23 + q^24 - 4 * q^25 - q^26 + q^27 + 3 * q^29 - q^30 - 11 * q^31 + q^32 - q^33 - 6 * q^34 + q^36 + 4 * q^37 - 4 * q^38 - q^39 - q^40 + 12 * q^41 - 8 * q^43 - q^44 - q^45 - 6 * q^46 - 8 * q^47 + q^48 - 4 * q^50 - 6 * q^51 - q^52 - 5 * q^53 + q^54 + q^55 - 4 * q^57 + 3 * q^58 - 5 * q^59 - q^60 + 12 * q^61 - 11 * q^62 + q^64 + q^65 - q^66 + 16 * q^67 - 6 * q^68 - 6 * q^69 + 6 * q^71 + q^72 - 10 * q^73 + 4 * q^74 - 4 * q^75 - 4 * q^76 - q^78 + 7 * q^79 - q^80 + q^81 + 12 * q^82 - 17 * q^83 + 6 * q^85 - 8 * q^86 + 3 * q^87 - q^88 - 12 * q^89 - q^90 - 6 * q^92 - 11 * q^93 - 8 * q^94 + 4 * q^95 + q^96 + 13 * q^97 - 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 −1.00000 1.00000 0 1.00000 1.00000 −1.00000
 $$n$$: 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.be 1
7.b odd 2 1 3822.2.a.v 1
7.c even 3 2 546.2.i.b 2
21.h odd 6 2 1638.2.j.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.b 2 7.c even 3 2
1638.2.j.h 2 21.h odd 6 2
3822.2.a.v 1 7.b odd 2 1
3822.2.a.be 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5} + 1$$ T5 + 1 $$T_{11} + 1$$ T11 + 1 $$T_{17} + 6$$ T17 + 6 $$T_{29} - 3$$ T29 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 1$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T + 6$$
$29$ $$T - 3$$
$31$ $$T + 11$$
$37$ $$T - 4$$
$41$ $$T - 12$$
$43$ $$T + 8$$
$47$ $$T + 8$$
$53$ $$T + 5$$
$59$ $$T + 5$$
$61$ $$T - 12$$
$67$ $$T - 16$$
$71$ $$T - 6$$
$73$ $$T + 10$$
$79$ $$T - 7$$
$83$ $$T + 17$$
$89$ $$T + 12$$
$97$ $$T - 13$$