# Properties

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3822.2.a.bj 2 7.b odd 2 1
3822.2.a.bl 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 $$T_{11}^{2} - 2T_{11} - 1$$ T11^2 - 2*T11 - 1 $$T_{17}^{2} + 2T_{17} - 1$$ T17^2 + 2*T17 - 1 $$T_{29}^{2} + 6T_{29} - 23$$ T29^2 + 6*T29 - 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 1$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 2T - 1$$
$19$ $$T^{2} + 2T - 7$$
$23$ $$T^{2} + 6T - 23$$
$29$ $$T^{2} + 6T - 23$$
$31$ $$T^{2} - 18$$
$37$ $$T^{2} - 2T - 17$$
$41$ $$T^{2} + 8T + 8$$
$43$ $$(T + 7)^{2}$$
$47$ $$T^{2} + 4T - 28$$
$53$ $$T^{2} + 16T + 32$$
$59$ $$T^{2} - 98$$
$61$ $$T^{2} - 14T + 47$$
$67$ $$T^{2} - 2$$
$71$ $$T^{2} + 4T - 46$$
$73$ $$T^{2} - 18T + 73$$
$79$ $$T^{2} - 12T + 18$$
$83$ $$T^{2} + 12T + 4$$
$89$ $$T^{2} + 4T + 2$$
$97$ $$T^{2} - 8T - 112$$