# Properties

 Label 3822.2.a.br Level $3822$ Weight $2$ Character orbit 3822.a Self dual yes Analytic conductor $30.519$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + ( - \beta - 5) q^{11} + q^{12} - q^{13} - q^{15} + q^{16} + (3 \beta - 1) q^{17} + q^{18} + ( - 4 \beta + 1) q^{19} - q^{20} + ( - \beta - 5) q^{22} + (2 \beta - 5) q^{23} + q^{24} - 4 q^{25} - q^{26} + q^{27} + (6 \beta - 1) q^{29} - q^{30} + \beta q^{31} + q^{32} + ( - \beta - 5) q^{33} + (3 \beta - 1) q^{34} + q^{36} + ( - \beta - 7) q^{37} + ( - 4 \beta + 1) q^{38} - q^{39} - q^{40} + (2 \beta - 8) q^{41} + ( - 4 \beta - 3) q^{43} + ( - \beta - 5) q^{44} - q^{45} + (2 \beta - 5) q^{46} + 6 q^{47} + q^{48} - 4 q^{50} + (3 \beta - 1) q^{51} - q^{52} - 4 \beta q^{53} + q^{54} + (\beta + 5) q^{55} + ( - 4 \beta + 1) q^{57} + (6 \beta - 1) q^{58} + ( - 5 \beta - 4) q^{59} - q^{60} + (\beta + 9) q^{61} + \beta q^{62} + q^{64} + q^{65} + ( - \beta - 5) q^{66} - 11 \beta q^{67} + (3 \beta - 1) q^{68} + (2 \beta - 5) q^{69} + (3 \beta - 10) q^{71} + q^{72} + 11 q^{73} + ( - \beta - 7) q^{74} - 4 q^{75} + ( - 4 \beta + 1) q^{76} - q^{78} + (3 \beta - 2) q^{79} - q^{80} + q^{81} + (2 \beta - 8) q^{82} + ( - 4 \beta + 2) q^{83} + ( - 3 \beta + 1) q^{85} + ( - 4 \beta - 3) q^{86} + (6 \beta - 1) q^{87} + ( - \beta - 5) q^{88} + ( - 3 \beta - 10) q^{89} - q^{90} + (2 \beta - 5) q^{92} + \beta q^{93} + 6 q^{94} + (4 \beta - 1) q^{95} + q^{96} + (8 \beta - 4) q^{97} + ( - \beta - 5) q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 + q^9 - q^10 + (-b - 5) * q^11 + q^12 - q^13 - q^15 + q^16 + (3*b - 1) * q^17 + q^18 + (-4*b + 1) * q^19 - q^20 + (-b - 5) * q^22 + (2*b - 5) * q^23 + q^24 - 4 * q^25 - q^26 + q^27 + (6*b - 1) * q^29 - q^30 + b * q^31 + q^32 + (-b - 5) * q^33 + (3*b - 1) * q^34 + q^36 + (-b - 7) * q^37 + (-4*b + 1) * q^38 - q^39 - q^40 + (2*b - 8) * q^41 + (-4*b - 3) * q^43 + (-b - 5) * q^44 - q^45 + (2*b - 5) * q^46 + 6 * q^47 + q^48 - 4 * q^50 + (3*b - 1) * q^51 - q^52 - 4*b * q^53 + q^54 + (b + 5) * q^55 + (-4*b + 1) * q^57 + (6*b - 1) * q^58 + (-5*b - 4) * q^59 - q^60 + (b + 9) * q^61 + b * q^62 + q^64 + q^65 + (-b - 5) * q^66 - 11*b * q^67 + (3*b - 1) * q^68 + (2*b - 5) * q^69 + (3*b - 10) * q^71 + q^72 + 11 * q^73 + (-b - 7) * q^74 - 4 * q^75 + (-4*b + 1) * q^76 - q^78 + (3*b - 2) * q^79 - q^80 + q^81 + (2*b - 8) * q^82 + (-4*b + 2) * q^83 + (-3*b + 1) * q^85 + (-4*b - 3) * q^86 + (6*b - 1) * q^87 + (-b - 5) * q^88 + (-3*b - 10) * q^89 - q^90 + (2*b - 5) * q^92 + b * q^93 + 6 * q^94 + (4*b - 1) * q^95 + q^96 + (8*b - 4) * q^97 + (-b - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 10 q^{11} + 2 q^{12} - 2 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} - 10 q^{22} - 10 q^{23} + 2 q^{24} - 8 q^{25} - 2 q^{26} + 2 q^{27} - 2 q^{29} - 2 q^{30} + 2 q^{32} - 10 q^{33} - 2 q^{34} + 2 q^{36} - 14 q^{37} + 2 q^{38} - 2 q^{39} - 2 q^{40} - 16 q^{41} - 6 q^{43} - 10 q^{44} - 2 q^{45} - 10 q^{46} + 12 q^{47} + 2 q^{48} - 8 q^{50} - 2 q^{51} - 2 q^{52} + 2 q^{54} + 10 q^{55} + 2 q^{57} - 2 q^{58} - 8 q^{59} - 2 q^{60} + 18 q^{61} + 2 q^{64} + 2 q^{65} - 10 q^{66} - 2 q^{68} - 10 q^{69} - 20 q^{71} + 2 q^{72} + 22 q^{73} - 14 q^{74} - 8 q^{75} + 2 q^{76} - 2 q^{78} - 4 q^{79} - 2 q^{80} + 2 q^{81} - 16 q^{82} + 4 q^{83} + 2 q^{85} - 6 q^{86} - 2 q^{87} - 10 q^{88} - 20 q^{89} - 2 q^{90} - 10 q^{92} + 12 q^{94} - 2 q^{95} + 2 q^{96} - 8 q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 10 * q^11 + 2 * q^12 - 2 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 2 * q^18 + 2 * q^19 - 2 * q^20 - 10 * q^22 - 10 * q^23 + 2 * q^24 - 8 * q^25 - 2 * q^26 + 2 * q^27 - 2 * q^29 - 2 * q^30 + 2 * q^32 - 10 * q^33 - 2 * q^34 + 2 * q^36 - 14 * q^37 + 2 * q^38 - 2 * q^39 - 2 * q^40 - 16 * q^41 - 6 * q^43 - 10 * q^44 - 2 * q^45 - 10 * q^46 + 12 * q^47 + 2 * q^48 - 8 * q^50 - 2 * q^51 - 2 * q^52 + 2 * q^54 + 10 * q^55 + 2 * q^57 - 2 * q^58 - 8 * q^59 - 2 * q^60 + 18 * q^61 + 2 * q^64 + 2 * q^65 - 10 * q^66 - 2 * q^68 - 10 * q^69 - 20 * q^71 + 2 * q^72 + 22 * q^73 - 14 * q^74 - 8 * q^75 + 2 * q^76 - 2 * q^78 - 4 * q^79 - 2 * q^80 + 2 * q^81 - 16 * q^82 + 4 * q^83 + 2 * q^85 - 6 * q^86 - 2 * q^87 - 10 * q^88 - 20 * q^89 - 2 * q^90 - 10 * q^92 + 12 * q^94 - 2 * q^95 + 2 * q^96 - 8 * q^97 - 10 * 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
 1.41421 −1.41421
1.00000 1.00000 1.00000 −1.00000 1.00000 0 1.00000 1.00000 −1.00000
1.2 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.br yes 2
7.b odd 2 1 3822.2.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3822.2.a.bq 2 7.b odd 2 1
3822.2.a.br yes 2 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 $$T_{11}^{2} + 10T_{11} + 23$$ T11^2 + 10*T11 + 23 $$T_{17}^{2} + 2T_{17} - 17$$ T17^2 + 2*T17 - 17 $$T_{29}^{2} + 2T_{29} - 71$$ T29^2 + 2*T29 - 71

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 10T + 23$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 2T - 17$$
$19$ $$T^{2} - 2T - 31$$
$23$ $$T^{2} + 10T + 17$$
$29$ $$T^{2} + 2T - 71$$
$31$ $$T^{2} - 2$$
$37$ $$T^{2} + 14T + 47$$
$41$ $$T^{2} + 16T + 56$$
$43$ $$T^{2} + 6T - 23$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} - 32$$
$59$ $$T^{2} + 8T - 34$$
$61$ $$T^{2} - 18T + 79$$
$67$ $$T^{2} - 242$$
$71$ $$T^{2} + 20T + 82$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} + 4T - 14$$
$83$ $$T^{2} - 4T - 28$$
$89$ $$T^{2} + 20T + 82$$
$97$ $$T^{2} + 8T - 112$$