# Properties

 Label 3822.2.a.j 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 78) 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 $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} - q^{13} - 2 q^{15} + q^{16} - 2 q^{17} - q^{18} + 8 q^{19} - 2 q^{20} + 4 q^{22} - q^{24} - q^{25} + q^{26} + q^{27} + 6 q^{29} + 2 q^{30} + 4 q^{31} - q^{32} - 4 q^{33} + 2 q^{34} + q^{36} - 2 q^{37} - 8 q^{38} - q^{39} + 2 q^{40} + 10 q^{41} + 4 q^{43} - 4 q^{44} - 2 q^{45} - 8 q^{47} + q^{48} + q^{50} - 2 q^{51} - q^{52} - 10 q^{53} - q^{54} + 8 q^{55} + 8 q^{57} - 6 q^{58} - 4 q^{59} - 2 q^{60} + 2 q^{61} - 4 q^{62} + q^{64} + 2 q^{65} + 4 q^{66} - 16 q^{67} - 2 q^{68} - 8 q^{71} - q^{72} - 2 q^{73} + 2 q^{74} - q^{75} + 8 q^{76} + q^{78} + 8 q^{79} - 2 q^{80} + q^{81} - 10 q^{82} - 12 q^{83} + 4 q^{85} - 4 q^{86} + 6 q^{87} + 4 q^{88} - 14 q^{89} + 2 q^{90} + 4 q^{93} + 8 q^{94} - 16 q^{95} - q^{96} - 10 q^{97} - 4 q^{99}+O(q^{100})$$ 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 + 8 * q^19 - 2 * q^20 + 4 * q^22 - q^24 - q^25 + q^26 + q^27 + 6 * q^29 + 2 * q^30 + 4 * q^31 - q^32 - 4 * q^33 + 2 * q^34 + q^36 - 2 * q^37 - 8 * q^38 - q^39 + 2 * q^40 + 10 * q^41 + 4 * q^43 - 4 * q^44 - 2 * q^45 - 8 * q^47 + q^48 + q^50 - 2 * q^51 - q^52 - 10 * q^53 - q^54 + 8 * q^55 + 8 * q^57 - 6 * q^58 - 4 * q^59 - 2 * q^60 + 2 * q^61 - 4 * q^62 + q^64 + 2 * q^65 + 4 * q^66 - 16 * q^67 - 2 * q^68 - 8 * q^71 - q^72 - 2 * q^73 + 2 * q^74 - q^75 + 8 * q^76 + q^78 + 8 * q^79 - 2 * q^80 + q^81 - 10 * q^82 - 12 * q^83 + 4 * q^85 - 4 * q^86 + 6 * q^87 + 4 * q^88 - 14 * q^89 + 2 * q^90 + 4 * q^93 + 8 * q^94 - 16 * q^95 - q^96 - 10 * 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.j 1
7.b odd 2 1 78.2.a.a 1
21.c even 2 1 234.2.a.c 1
28.d even 2 1 624.2.a.h 1
35.c odd 2 1 1950.2.a.w 1
35.f even 4 2 1950.2.e.i 2
56.e even 2 1 2496.2.a.b 1
56.h odd 2 1 2496.2.a.t 1
63.l odd 6 2 2106.2.e.q 2
63.o even 6 2 2106.2.e.j 2
77.b even 2 1 9438.2.a.t 1
84.h odd 2 1 1872.2.a.c 1
91.b odd 2 1 1014.2.a.d 1
91.i even 4 2 1014.2.b.b 2
91.n odd 6 2 1014.2.e.f 2
91.t odd 6 2 1014.2.e.c 2
91.bc even 12 4 1014.2.i.d 4
105.g even 2 1 5850.2.a.d 1
105.k odd 4 2 5850.2.e.bb 2
168.e odd 2 1 7488.2.a.bk 1
168.i even 2 1 7488.2.a.bz 1
273.g even 2 1 3042.2.a.f 1
273.o odd 4 2 3042.2.b.g 2
364.h even 2 1 8112.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 7.b odd 2 1
234.2.a.c 1 21.c even 2 1
624.2.a.h 1 28.d even 2 1
1014.2.a.d 1 91.b odd 2 1
1014.2.b.b 2 91.i even 4 2
1014.2.e.c 2 91.t odd 6 2
1014.2.e.f 2 91.n odd 6 2
1014.2.i.d 4 91.bc even 12 4
1872.2.a.c 1 84.h odd 2 1
1950.2.a.w 1 35.c odd 2 1
1950.2.e.i 2 35.f even 4 2
2106.2.e.j 2 63.o even 6 2
2106.2.e.q 2 63.l odd 6 2
2496.2.a.b 1 56.e even 2 1
2496.2.a.t 1 56.h odd 2 1
3042.2.a.f 1 273.g even 2 1
3042.2.b.g 2 273.o odd 4 2
3822.2.a.j 1 1.a even 1 1 trivial
5850.2.a.d 1 105.g even 2 1
5850.2.e.bb 2 105.k odd 4 2
7488.2.a.bk 1 168.e odd 2 1
7488.2.a.bz 1 168.i even 2 1
8112.2.a.v 1 364.h even 2 1
9438.2.a.t 1 77.b even 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 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T + 2$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T + 10$$
$59$ $$T + 4$$
$61$ $$T - 2$$
$67$ $$T + 16$$
$71$ $$T + 8$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 12$$
$89$ $$T + 14$$
$97$ $$T + 10$$