Properties

 Label 3822.2.a.h 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + 2 * q^5 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} - q^{13} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + 2 q^{20} + 4 q^{22} - 4 q^{23} + q^{24} - q^{25} + q^{26} - q^{27} - 2 q^{29} + 2 q^{30} - q^{32} + 4 q^{33} - 2 q^{34} + q^{36} - 2 q^{37} - 4 q^{38} + q^{39} - 2 q^{40} - 2 q^{41} + 4 q^{43} - 4 q^{44} + 2 q^{45} + 4 q^{46} + 12 q^{47} - q^{48} + q^{50} - 2 q^{51} - q^{52} + 6 q^{53} + q^{54} - 8 q^{55} - 4 q^{57} + 2 q^{58} - 2 q^{60} + 10 q^{61} + q^{64} - 2 q^{65} - 4 q^{66} + 4 q^{67} + 2 q^{68} + 4 q^{69} - 8 q^{71} - q^{72} + 6 q^{73} + 2 q^{74} + q^{75} + 4 q^{76} - q^{78} + 8 q^{79} + 2 q^{80} + q^{81} + 2 q^{82} - 8 q^{83} + 4 q^{85} - 4 q^{86} + 2 q^{87} + 4 q^{88} + 6 q^{89} - 2 q^{90} - 4 q^{92} - 12 q^{94} + 8 q^{95} + q^{96} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + 2 * q^5 + q^6 - q^8 + q^9 - 2 * q^10 - 4 * q^11 - q^12 - q^13 - 2 * q^15 + q^16 + 2 * q^17 - q^18 + 4 * q^19 + 2 * q^20 + 4 * q^22 - 4 * q^23 + q^24 - q^25 + q^26 - q^27 - 2 * q^29 + 2 * q^30 - q^32 + 4 * q^33 - 2 * q^34 + q^36 - 2 * q^37 - 4 * q^38 + q^39 - 2 * q^40 - 2 * q^41 + 4 * q^43 - 4 * q^44 + 2 * q^45 + 4 * q^46 + 12 * q^47 - q^48 + q^50 - 2 * q^51 - q^52 + 6 * q^53 + q^54 - 8 * q^55 - 4 * q^57 + 2 * q^58 - 2 * q^60 + 10 * q^61 + q^64 - 2 * q^65 - 4 * q^66 + 4 * q^67 + 2 * q^68 + 4 * q^69 - 8 * q^71 - q^72 + 6 * q^73 + 2 * q^74 + q^75 + 4 * q^76 - q^78 + 8 * q^79 + 2 * q^80 + q^81 + 2 * q^82 - 8 * q^83 + 4 * q^85 - 4 * q^86 + 2 * q^87 + 4 * q^88 + 6 * q^89 - 2 * q^90 - 4 * q^92 - 12 * q^94 + 8 * q^95 + q^96 - 2 * q^97 - 4 * 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 2.00000 1.00000 0 −1.00000 1.00000 −2.00000
 $$n$$: 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.h 1
7.b odd 2 1 546.2.a.b 1
21.c even 2 1 1638.2.a.s 1
28.d even 2 1 4368.2.a.d 1
91.b odd 2 1 7098.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.b 1 7.b odd 2 1
1638.2.a.s 1 21.c even 2 1
3822.2.a.h 1 1.a even 1 1 trivial
4368.2.a.d 1 28.d even 2 1
7098.2.a.be 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{11} + 4$$ T11 + 4 $$T_{17} - 2$$ T17 - 2 $$T_{29} + 2$$ T29 + 2

Hecke characteristic polynomials

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