# Properties

 Label 3822.2.a.bo 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: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + (-b - 1) * q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + ( - \beta - 1) q^{5} - q^{6} + q^{8} + q^{9} + ( - \beta - 1) q^{10} + ( - \beta + 1) q^{11} - q^{12} - q^{13} + (\beta + 1) q^{15} + q^{16} + (3 \beta - 1) q^{17} + q^{18} + (3 \beta - 3) q^{19} + ( - \beta - 1) q^{20} + ( - \beta + 1) q^{22} + ( - \beta - 3) q^{23} - q^{24} + 3 \beta q^{25} - q^{26} - q^{27} + (3 \beta - 1) q^{29} + (\beta + 1) q^{30} + ( - 4 \beta + 4) q^{31} + q^{32} + (\beta - 1) q^{33} + (3 \beta - 1) q^{34} + q^{36} + ( - \beta - 5) q^{37} + (3 \beta - 3) q^{38} + q^{39} + ( - \beta - 1) q^{40} + (2 \beta - 4) q^{41} + (\beta - 9) q^{43} + ( - \beta + 1) q^{44} + ( - \beta - 1) q^{45} + ( - \beta - 3) q^{46} - q^{48} + 3 \beta q^{50} + ( - 3 \beta + 1) q^{51} - q^{52} + (4 \beta + 2) q^{53} - q^{54} + (\beta + 3) q^{55} + ( - 3 \beta + 3) q^{57} + (3 \beta - 1) q^{58} + ( - 4 \beta + 8) q^{59} + (\beta + 1) q^{60} + ( - 3 \beta + 1) q^{61} + ( - 4 \beta + 4) q^{62} + q^{64} + (\beta + 1) q^{65} + (\beta - 1) q^{66} + ( - 2 \beta - 2) q^{67} + (3 \beta - 1) q^{68} + (\beta + 3) q^{69} + 8 q^{71} + q^{72} + (\beta + 1) q^{73} + ( - \beta - 5) q^{74} - 3 \beta q^{75} + (3 \beta - 3) q^{76} + q^{78} + ( - 2 \beta - 6) q^{79} + ( - \beta - 1) q^{80} + q^{81} + (2 \beta - 4) q^{82} + (2 \beta - 14) q^{83} + ( - 5 \beta - 11) q^{85} + (\beta - 9) q^{86} + ( - 3 \beta + 1) q^{87} + ( - \beta + 1) q^{88} - 10 q^{89} + ( - \beta - 1) q^{90} + ( - \beta - 3) q^{92} + (4 \beta - 4) q^{93} + ( - 3 \beta - 9) q^{95} - q^{96} + ( - 8 \beta + 6) q^{97} + ( - \beta + 1) q^{99} +O(q^{100})$$ q + q^2 - q^3 + q^4 + (-b - 1) * q^5 - q^6 + q^8 + q^9 + (-b - 1) * q^10 + (-b + 1) * q^11 - q^12 - q^13 + (b + 1) * q^15 + q^16 + (3*b - 1) * q^17 + q^18 + (3*b - 3) * q^19 + (-b - 1) * q^20 + (-b + 1) * q^22 + (-b - 3) * q^23 - q^24 + 3*b * q^25 - q^26 - q^27 + (3*b - 1) * q^29 + (b + 1) * q^30 + (-4*b + 4) * q^31 + q^32 + (b - 1) * q^33 + (3*b - 1) * q^34 + q^36 + (-b - 5) * q^37 + (3*b - 3) * q^38 + q^39 + (-b - 1) * q^40 + (2*b - 4) * q^41 + (b - 9) * q^43 + (-b + 1) * q^44 + (-b - 1) * q^45 + (-b - 3) * q^46 - q^48 + 3*b * q^50 + (-3*b + 1) * q^51 - q^52 + (4*b + 2) * q^53 - q^54 + (b + 3) * q^55 + (-3*b + 3) * q^57 + (3*b - 1) * q^58 + (-4*b + 8) * q^59 + (b + 1) * q^60 + (-3*b + 1) * q^61 + (-4*b + 4) * q^62 + q^64 + (b + 1) * q^65 + (b - 1) * q^66 + (-2*b - 2) * q^67 + (3*b - 1) * q^68 + (b + 3) * q^69 + 8 * q^71 + q^72 + (b + 1) * q^73 + (-b - 5) * q^74 - 3*b * q^75 + (3*b - 3) * q^76 + q^78 + (-2*b - 6) * q^79 + (-b - 1) * q^80 + q^81 + (2*b - 4) * q^82 + (2*b - 14) * q^83 + (-5*b - 11) * q^85 + (b - 9) * q^86 + (-3*b + 1) * q^87 + (-b + 1) * q^88 - 10 * q^89 + (-b - 1) * q^90 + (-b - 3) * q^92 + (4*b - 4) * q^93 + (-3*b - 9) * q^95 - q^96 + (-8*b + 6) * q^97 + (-b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 2 q^{13} + 3 q^{15} + 2 q^{16} + q^{17} + 2 q^{18} - 3 q^{19} - 3 q^{20} + q^{22} - 7 q^{23} - 2 q^{24} + 3 q^{25} - 2 q^{26} - 2 q^{27} + q^{29} + 3 q^{30} + 4 q^{31} + 2 q^{32} - q^{33} + q^{34} + 2 q^{36} - 11 q^{37} - 3 q^{38} + 2 q^{39} - 3 q^{40} - 6 q^{41} - 17 q^{43} + q^{44} - 3 q^{45} - 7 q^{46} - 2 q^{48} + 3 q^{50} - q^{51} - 2 q^{52} + 8 q^{53} - 2 q^{54} + 7 q^{55} + 3 q^{57} + q^{58} + 12 q^{59} + 3 q^{60} - q^{61} + 4 q^{62} + 2 q^{64} + 3 q^{65} - q^{66} - 6 q^{67} + q^{68} + 7 q^{69} + 16 q^{71} + 2 q^{72} + 3 q^{73} - 11 q^{74} - 3 q^{75} - 3 q^{76} + 2 q^{78} - 14 q^{79} - 3 q^{80} + 2 q^{81} - 6 q^{82} - 26 q^{83} - 27 q^{85} - 17 q^{86} - q^{87} + q^{88} - 20 q^{89} - 3 q^{90} - 7 q^{92} - 4 q^{93} - 21 q^{95} - 2 q^{96} + 4 q^{97} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 2 * q^13 + 3 * q^15 + 2 * q^16 + q^17 + 2 * q^18 - 3 * q^19 - 3 * q^20 + q^22 - 7 * q^23 - 2 * q^24 + 3 * q^25 - 2 * q^26 - 2 * q^27 + q^29 + 3 * q^30 + 4 * q^31 + 2 * q^32 - q^33 + q^34 + 2 * q^36 - 11 * q^37 - 3 * q^38 + 2 * q^39 - 3 * q^40 - 6 * q^41 - 17 * q^43 + q^44 - 3 * q^45 - 7 * q^46 - 2 * q^48 + 3 * q^50 - q^51 - 2 * q^52 + 8 * q^53 - 2 * q^54 + 7 * q^55 + 3 * q^57 + q^58 + 12 * q^59 + 3 * q^60 - q^61 + 4 * q^62 + 2 * q^64 + 3 * q^65 - q^66 - 6 * q^67 + q^68 + 7 * q^69 + 16 * q^71 + 2 * q^72 + 3 * q^73 - 11 * q^74 - 3 * q^75 - 3 * q^76 + 2 * q^78 - 14 * q^79 - 3 * q^80 + 2 * q^81 - 6 * q^82 - 26 * q^83 - 27 * q^85 - 17 * q^86 - q^87 + q^88 - 20 * q^89 - 3 * q^90 - 7 * q^92 - 4 * q^93 - 21 * q^95 - 2 * q^96 + 4 * q^97 + 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
 2.56155 −1.56155
1.00000 −1.00000 1.00000 −3.56155 −1.00000 0 1.00000 1.00000 −3.56155
1.2 1.00000 −1.00000 1.00000 0.561553 −1.00000 0 1.00000 1.00000 0.561553
 $$n$$: 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.bo 2
7.b odd 2 1 546.2.a.j 2
21.c even 2 1 1638.2.a.u 2
28.d even 2 1 4368.2.a.be 2
91.b odd 2 1 7098.2.a.bl 2

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

 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{11}^{2} - T_{11} - 4$$ T11^2 - T11 - 4 $$T_{17}^{2} - T_{17} - 38$$ T17^2 - T17 - 38 $$T_{29}^{2} - T_{29} - 38$$ T29^2 - T29 - 38

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} - T - 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - T - 38$$
$19$ $$T^{2} + 3T - 36$$
$23$ $$T^{2} + 7T + 8$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} - 4T - 64$$
$37$ $$T^{2} + 11T + 26$$
$41$ $$T^{2} + 6T - 8$$
$43$ $$T^{2} + 17T + 68$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$T^{2} - 12T - 32$$
$61$ $$T^{2} + T - 38$$
$67$ $$T^{2} + 6T - 8$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} - 3T - 2$$
$79$ $$T^{2} + 14T + 32$$
$83$ $$T^{2} + 26T + 152$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 4T - 268$$