# Properties

 Label 3822.2.a.bt Level $3822$ Weight $2$ Character orbit 3822.a Self dual yes Analytic conductor $30.519$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + b * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{8} + q^{9} + \beta q^{10} + (\beta + 2) q^{11} + q^{12} - q^{13} + \beta q^{15} + q^{16} - \beta q^{17} + q^{18} + ( - \beta - 2) q^{19} + \beta q^{20} + (\beta + 2) q^{22} + ( - \beta + 2) q^{23} + q^{24} + (\beta + 5) q^{25} - q^{26} + q^{27} - \beta q^{29} + \beta q^{30} + q^{32} + (\beta + 2) q^{33} - \beta q^{34} + q^{36} + ( - \beta + 8) q^{37} + ( - \beta - 2) q^{38} - q^{39} + \beta q^{40} + (2 \beta + 2) q^{41} + (3 \beta - 2) q^{43} + (\beta + 2) q^{44} + \beta q^{45} + ( - \beta + 2) q^{46} + 8 q^{47} + q^{48} + (\beta + 5) q^{50} - \beta q^{51} - q^{52} - 2 q^{53} + q^{54} + (3 \beta + 10) q^{55} + ( - \beta - 2) q^{57} - \beta q^{58} + ( - 4 \beta + 4) q^{59} + \beta q^{60} + ( - \beta - 4) q^{61} + q^{64} - \beta q^{65} + (\beta + 2) q^{66} - 2 \beta q^{67} - \beta q^{68} + ( - \beta + 2) q^{69} - 8 q^{71} + q^{72} + (\beta + 4) q^{73} + ( - \beta + 8) q^{74} + (\beta + 5) q^{75} + ( - \beta - 2) q^{76} - q^{78} + ( - 2 \beta + 4) q^{79} + \beta q^{80} + q^{81} + (2 \beta + 2) q^{82} + ( - 2 \beta + 8) q^{83} + ( - \beta - 10) q^{85} + (3 \beta - 2) q^{86} - \beta q^{87} + (\beta + 2) q^{88} + ( - 4 \beta - 2) q^{89} + \beta q^{90} + ( - \beta + 2) q^{92} + 8 q^{94} + ( - 3 \beta - 10) q^{95} + q^{96} + ( - 4 \beta - 2) q^{97} + (\beta + 2) q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 + b * q^5 + q^6 + q^8 + q^9 + b * q^10 + (b + 2) * q^11 + q^12 - q^13 + b * q^15 + q^16 - b * q^17 + q^18 + (-b - 2) * q^19 + b * q^20 + (b + 2) * q^22 + (-b + 2) * q^23 + q^24 + (b + 5) * q^25 - q^26 + q^27 - b * q^29 + b * q^30 + q^32 + (b + 2) * q^33 - b * q^34 + q^36 + (-b + 8) * q^37 + (-b - 2) * q^38 - q^39 + b * q^40 + (2*b + 2) * q^41 + (3*b - 2) * q^43 + (b + 2) * q^44 + b * q^45 + (-b + 2) * q^46 + 8 * q^47 + q^48 + (b + 5) * q^50 - b * q^51 - q^52 - 2 * q^53 + q^54 + (3*b + 10) * q^55 + (-b - 2) * q^57 - b * q^58 + (-4*b + 4) * q^59 + b * q^60 + (-b - 4) * q^61 + q^64 - b * q^65 + (b + 2) * q^66 - 2*b * q^67 - b * q^68 + (-b + 2) * q^69 - 8 * q^71 + q^72 + (b + 4) * q^73 + (-b + 8) * q^74 + (b + 5) * q^75 + (-b - 2) * q^76 - q^78 + (-2*b + 4) * q^79 + b * q^80 + q^81 + (2*b + 2) * q^82 + (-2*b + 8) * q^83 + (-b - 10) * q^85 + (3*b - 2) * q^86 - b * q^87 + (b + 2) * q^88 + (-4*b - 2) * q^89 + b * q^90 + (-b + 2) * q^92 + 8 * q^94 + (-3*b - 10) * q^95 + q^96 + (-4*b - 2) * q^97 + (b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + q^{10} + 5 q^{11} + 2 q^{12} - 2 q^{13} + q^{15} + 2 q^{16} - q^{17} + 2 q^{18} - 5 q^{19} + q^{20} + 5 q^{22} + 3 q^{23} + 2 q^{24} + 11 q^{25} - 2 q^{26} + 2 q^{27} - q^{29} + q^{30} + 2 q^{32} + 5 q^{33} - q^{34} + 2 q^{36} + 15 q^{37} - 5 q^{38} - 2 q^{39} + q^{40} + 6 q^{41} - q^{43} + 5 q^{44} + q^{45} + 3 q^{46} + 16 q^{47} + 2 q^{48} + 11 q^{50} - q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 23 q^{55} - 5 q^{57} - q^{58} + 4 q^{59} + q^{60} - 9 q^{61} + 2 q^{64} - q^{65} + 5 q^{66} - 2 q^{67} - q^{68} + 3 q^{69} - 16 q^{71} + 2 q^{72} + 9 q^{73} + 15 q^{74} + 11 q^{75} - 5 q^{76} - 2 q^{78} + 6 q^{79} + q^{80} + 2 q^{81} + 6 q^{82} + 14 q^{83} - 21 q^{85} - q^{86} - q^{87} + 5 q^{88} - 8 q^{89} + q^{90} + 3 q^{92} + 16 q^{94} - 23 q^{95} + 2 q^{96} - 8 q^{97} + 5 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + q^10 + 5 * q^11 + 2 * q^12 - 2 * q^13 + q^15 + 2 * q^16 - q^17 + 2 * q^18 - 5 * q^19 + q^20 + 5 * q^22 + 3 * q^23 + 2 * q^24 + 11 * q^25 - 2 * q^26 + 2 * q^27 - q^29 + q^30 + 2 * q^32 + 5 * q^33 - q^34 + 2 * q^36 + 15 * q^37 - 5 * q^38 - 2 * q^39 + q^40 + 6 * q^41 - q^43 + 5 * q^44 + q^45 + 3 * q^46 + 16 * q^47 + 2 * q^48 + 11 * q^50 - q^51 - 2 * q^52 - 4 * q^53 + 2 * q^54 + 23 * q^55 - 5 * q^57 - q^58 + 4 * q^59 + q^60 - 9 * q^61 + 2 * q^64 - q^65 + 5 * q^66 - 2 * q^67 - q^68 + 3 * q^69 - 16 * q^71 + 2 * q^72 + 9 * q^73 + 15 * q^74 + 11 * q^75 - 5 * q^76 - 2 * q^78 + 6 * q^79 + q^80 + 2 * q^81 + 6 * q^82 + 14 * q^83 - 21 * q^85 - q^86 - q^87 + 5 * q^88 - 8 * q^89 + q^90 + 3 * q^92 + 16 * q^94 - 23 * q^95 + 2 * q^96 - 8 * 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
 −2.70156 3.70156
1.00000 1.00000 1.00000 −2.70156 1.00000 0 1.00000 1.00000 −2.70156
1.2 1.00000 1.00000 1.00000 3.70156 1.00000 0 1.00000 1.00000 3.70156
 $$n$$: 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.bt 2
7.b odd 2 1 546.2.a.i 2
21.c even 2 1 1638.2.a.w 2
28.d even 2 1 4368.2.a.bg 2
91.b odd 2 1 7098.2.a.bh 2

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

 $$T_{5}^{2} - T_{5} - 10$$ T5^2 - T5 - 10 $$T_{11}^{2} - 5T_{11} - 4$$ T11^2 - 5*T11 - 4 $$T_{17}^{2} + T_{17} - 10$$ T17^2 + T17 - 10 $$T_{29}^{2} + T_{29} - 10$$ T29^2 + T29 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - T - 10$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T - 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + T - 10$$
$19$ $$T^{2} + 5T - 4$$
$23$ $$T^{2} - 3T - 8$$
$29$ $$T^{2} + T - 10$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 15T + 46$$
$41$ $$T^{2} - 6T - 32$$
$43$ $$T^{2} + T - 92$$
$47$ $$(T - 8)^{2}$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 4T - 160$$
$61$ $$T^{2} + 9T + 10$$
$67$ $$T^{2} + 2T - 40$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 9T + 10$$
$79$ $$T^{2} - 6T - 32$$
$83$ $$T^{2} - 14T + 8$$
$89$ $$T^{2} + 8T - 148$$
$97$ $$T^{2} + 8T - 148$$