# Properties

 Label 3822.2.a.bf 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 5 q^{11} + q^{12} + q^{13} + q^{16} - 7 q^{17} + q^{18} - 7 q^{19} - 5 q^{22} + 2 q^{23} + q^{24} - 5 q^{25} + q^{26} + q^{27} - 9 q^{29} + q^{32} - 5 q^{33} - 7 q^{34} + q^{36} + 4 q^{37} - 7 q^{38} + q^{39} - 4 q^{41} + 2 q^{43} - 5 q^{44} + 2 q^{46} + 3 q^{47} + q^{48} - 5 q^{50} - 7 q^{51} + q^{52} + q^{53} + q^{54} - 7 q^{57} - 9 q^{58} - 7 q^{59} - 13 q^{61} + q^{64} - 5 q^{66} + 3 q^{67} - 7 q^{68} + 2 q^{69} + 9 q^{71} + q^{72} + 10 q^{73} + 4 q^{74} - 5 q^{75} - 7 q^{76} + q^{78} + 14 q^{79} + q^{81} - 4 q^{82} - 16 q^{83} + 2 q^{86} - 9 q^{87} - 5 q^{88} + 12 q^{89} + 2 q^{92} + 3 q^{94} + q^{96} - 6 q^{97} - 5 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + q^8 + q^9 - 5 * q^11 + q^12 + q^13 + q^16 - 7 * q^17 + q^18 - 7 * q^19 - 5 * q^22 + 2 * q^23 + q^24 - 5 * q^25 + q^26 + q^27 - 9 * q^29 + q^32 - 5 * q^33 - 7 * q^34 + q^36 + 4 * q^37 - 7 * q^38 + q^39 - 4 * q^41 + 2 * q^43 - 5 * q^44 + 2 * q^46 + 3 * q^47 + q^48 - 5 * q^50 - 7 * q^51 + q^52 + q^53 + q^54 - 7 * q^57 - 9 * q^58 - 7 * q^59 - 13 * q^61 + q^64 - 5 * q^66 + 3 * q^67 - 7 * q^68 + 2 * q^69 + 9 * q^71 + q^72 + 10 * q^73 + 4 * q^74 - 5 * q^75 - 7 * q^76 + q^78 + 14 * q^79 + q^81 - 4 * q^82 - 16 * q^83 + 2 * q^86 - 9 * q^87 - 5 * q^88 + 12 * q^89 + 2 * q^92 + 3 * q^94 + q^96 - 6 * q^97 - 5 * 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 0 1.00000 0 1.00000 1.00000 0
 $$n$$: 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.bf 1
7.b odd 2 1 3822.2.a.u 1
7.d odd 6 2 546.2.i.d 2
21.g even 6 2 1638.2.j.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.d 2 7.d odd 6 2
1638.2.j.i 2 21.g even 6 2
3822.2.a.u 1 7.b odd 2 1
3822.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5}$$ T5 $$T_{11} + 5$$ T11 + 5 $$T_{17} + 7$$ T17 + 7 $$T_{29} + 9$$ T29 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 5$$
$13$ $$T - 1$$
$17$ $$T + 7$$
$19$ $$T + 7$$
$23$ $$T - 2$$
$29$ $$T + 9$$
$31$ $$T$$
$37$ $$T - 4$$
$41$ $$T + 4$$
$43$ $$T - 2$$
$47$ $$T - 3$$
$53$ $$T - 1$$
$59$ $$T + 7$$
$61$ $$T + 13$$
$67$ $$T - 3$$
$71$ $$T - 9$$
$73$ $$T - 10$$
$79$ $$T - 14$$
$83$ $$T + 16$$
$89$ $$T - 12$$
$97$ $$T + 6$$