# Properties

 Label 3822.2.a.bw Level $3822$ Weight $2$ Character orbit 3822.a Self dual yes Analytic conductor $30.519$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.2700.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 15x - 20$$ x^3 - 15*x - 20 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + (\beta_1 - 1) q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + (b1 - 1) * q^5 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + (\beta_1 - 1) q^{5} - q^{6} - q^{8} + q^{9} + ( - \beta_1 + 1) q^{10} + (\beta_{2} - \beta_1 - 1) q^{11} + q^{12} - q^{13} + (\beta_1 - 1) q^{15} + q^{16} + ( - 2 \beta_{2} - \beta_1 - 2) q^{17} - q^{18} + ( - \beta_{2} - 2) q^{19} + (\beta_1 - 1) q^{20} + ( - \beta_{2} + \beta_1 + 1) q^{22} + (\beta_{2} - \beta_1) q^{23} - q^{24} + (\beta_{2} + 6) q^{25} + q^{26} + q^{27} - 3 q^{29} + ( - \beta_1 + 1) q^{30} + (\beta_{2} + 5) q^{31} - q^{32} + (\beta_{2} - \beta_1 - 1) q^{33} + (2 \beta_{2} + \beta_1 + 2) q^{34} + q^{36} + ( - \beta_{2} - \beta_1 + 2) q^{37} + (\beta_{2} + 2) q^{38} - q^{39} + ( - \beta_1 + 1) q^{40} + (\beta_{2} + \beta_1 - 6) q^{41} + ( - 2 \beta_1 - 4) q^{43} + (\beta_{2} - \beta_1 - 1) q^{44} + (\beta_1 - 1) q^{45} + ( - \beta_{2} + \beta_1) q^{46} + ( - \beta_{2} + 2 \beta_1 + 2) q^{47} + q^{48} + ( - \beta_{2} - 6) q^{50} + ( - 2 \beta_{2} - \beta_1 - 2) q^{51} - q^{52} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{53} - q^{54} + ( - 4 \beta_{2} - \beta_1 - 9) q^{55} + ( - \beta_{2} - 2) q^{57} + 3 q^{58} + (\beta_{2} + \beta_1 + 1) q^{59} + (\beta_1 - 1) q^{60} + (2 \beta_{2} - \beta_1) q^{61} + ( - \beta_{2} - 5) q^{62} + q^{64} + ( - \beta_1 + 1) q^{65} + ( - \beta_{2} + \beta_1 + 1) q^{66} + ( - 3 \beta_{2} + 2) q^{67} + ( - 2 \beta_{2} - \beta_1 - 2) q^{68} + (\beta_{2} - \beta_1) q^{69} + ( - \beta_{2} - 12) q^{71} - q^{72} + (\beta_{2} - \beta_1 - 8) q^{73} + (\beta_{2} + \beta_1 - 2) q^{74} + (\beta_{2} + 6) q^{75} + ( - \beta_{2} - 2) q^{76} + q^{78} + (\beta_{2} + 2 \beta_1 + 5) q^{79} + (\beta_1 - 1) q^{80} + q^{81} + ( - \beta_{2} - \beta_1 + 6) q^{82} + (3 \beta_{2} - 3) q^{83} + (5 \beta_{2} - 5 \beta_1 - 8) q^{85} + (2 \beta_1 + 4) q^{86} - 3 q^{87} + ( - \beta_{2} + \beta_1 + 1) q^{88} + ( - \beta_{2} - \beta_1 - 10) q^{89} + ( - \beta_1 + 1) q^{90} + (\beta_{2} - \beta_1) q^{92} + (\beta_{2} + 5) q^{93} + (\beta_{2} - 2 \beta_1 - 2) q^{94} + (3 \beta_{2} - 3 \beta_1 + 2) q^{95} - q^{96} + (2 \beta_{2} + \beta_1 - 1) q^{97} + (\beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + (b1 - 1) * q^5 - q^6 - q^8 + q^9 + (-b1 + 1) * q^10 + (b2 - b1 - 1) * q^11 + q^12 - q^13 + (b1 - 1) * q^15 + q^16 + (-2*b2 - b1 - 2) * q^17 - q^18 + (-b2 - 2) * q^19 + (b1 - 1) * q^20 + (-b2 + b1 + 1) * q^22 + (b2 - b1) * q^23 - q^24 + (b2 + 6) * q^25 + q^26 + q^27 - 3 * q^29 + (-b1 + 1) * q^30 + (b2 + 5) * q^31 - q^32 + (b2 - b1 - 1) * q^33 + (2*b2 + b1 + 2) * q^34 + q^36 + (-b2 - b1 + 2) * q^37 + (b2 + 2) * q^38 - q^39 + (-b1 + 1) * q^40 + (b2 + b1 - 6) * q^41 + (-2*b1 - 4) * q^43 + (b2 - b1 - 1) * q^44 + (b1 - 1) * q^45 + (-b2 + b1) * q^46 + (-b2 + 2*b1 + 2) * q^47 + q^48 + (-b2 - 6) * q^50 + (-2*b2 - b1 - 2) * q^51 - q^52 + (-2*b2 - 2*b1 + 1) * q^53 - q^54 + (-4*b2 - b1 - 9) * q^55 + (-b2 - 2) * q^57 + 3 * q^58 + (b2 + b1 + 1) * q^59 + (b1 - 1) * q^60 + (2*b2 - b1) * q^61 + (-b2 - 5) * q^62 + q^64 + (-b1 + 1) * q^65 + (-b2 + b1 + 1) * q^66 + (-3*b2 + 2) * q^67 + (-2*b2 - b1 - 2) * q^68 + (b2 - b1) * q^69 + (-b2 - 12) * q^71 - q^72 + (b2 - b1 - 8) * q^73 + (b2 + b1 - 2) * q^74 + (b2 + 6) * q^75 + (-b2 - 2) * q^76 + q^78 + (b2 + 2*b1 + 5) * q^79 + (b1 - 1) * q^80 + q^81 + (-b2 - b1 + 6) * q^82 + (3*b2 - 3) * q^83 + (5*b2 - 5*b1 - 8) * q^85 + (2*b1 + 4) * q^86 - 3 * q^87 + (-b2 + b1 + 1) * q^88 + (-b2 - b1 - 10) * q^89 + (-b1 + 1) * q^90 + (b2 - b1) * q^92 + (b2 + 5) * q^93 + (b2 - 2*b1 - 2) * q^94 + (3*b2 - 3*b1 + 2) * q^95 - q^96 + (2*b2 + b1 - 1) * q^97 + (b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 $$3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 3 q^{11} + 3 q^{12} - 3 q^{13} - 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} - 6 q^{19} - 3 q^{20} + 3 q^{22} - 3 q^{24} + 18 q^{25} + 3 q^{26} + 3 q^{27} - 9 q^{29} + 3 q^{30} + 15 q^{31} - 3 q^{32} - 3 q^{33} + 6 q^{34} + 3 q^{36} + 6 q^{37} + 6 q^{38} - 3 q^{39} + 3 q^{40} - 18 q^{41} - 12 q^{43} - 3 q^{44} - 3 q^{45} + 6 q^{47} + 3 q^{48} - 18 q^{50} - 6 q^{51} - 3 q^{52} + 3 q^{53} - 3 q^{54} - 27 q^{55} - 6 q^{57} + 9 q^{58} + 3 q^{59} - 3 q^{60} - 15 q^{62} + 3 q^{64} + 3 q^{65} + 3 q^{66} + 6 q^{67} - 6 q^{68} - 36 q^{71} - 3 q^{72} - 24 q^{73} - 6 q^{74} + 18 q^{75} - 6 q^{76} + 3 q^{78} + 15 q^{79} - 3 q^{80} + 3 q^{81} + 18 q^{82} - 9 q^{83} - 24 q^{85} + 12 q^{86} - 9 q^{87} + 3 q^{88} - 30 q^{89} + 3 q^{90} + 15 q^{93} - 6 q^{94} + 6 q^{95} - 3 q^{96} - 3 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 3 * q^10 - 3 * q^11 + 3 * q^12 - 3 * q^13 - 3 * q^15 + 3 * q^16 - 6 * q^17 - 3 * q^18 - 6 * q^19 - 3 * q^20 + 3 * q^22 - 3 * q^24 + 18 * q^25 + 3 * q^26 + 3 * q^27 - 9 * q^29 + 3 * q^30 + 15 * q^31 - 3 * q^32 - 3 * q^33 + 6 * q^34 + 3 * q^36 + 6 * q^37 + 6 * q^38 - 3 * q^39 + 3 * q^40 - 18 * q^41 - 12 * q^43 - 3 * q^44 - 3 * q^45 + 6 * q^47 + 3 * q^48 - 18 * q^50 - 6 * q^51 - 3 * q^52 + 3 * q^53 - 3 * q^54 - 27 * q^55 - 6 * q^57 + 9 * q^58 + 3 * q^59 - 3 * q^60 - 15 * q^62 + 3 * q^64 + 3 * q^65 + 3 * q^66 + 6 * q^67 - 6 * q^68 - 36 * q^71 - 3 * q^72 - 24 * q^73 - 6 * q^74 + 18 * q^75 - 6 * q^76 + 3 * q^78 + 15 * q^79 - 3 * q^80 + 3 * q^81 + 18 * q^82 - 9 * q^83 - 24 * q^85 + 12 * q^86 - 9 * q^87 + 3 * q^88 - 30 * q^89 + 3 * q^90 + 15 * q^93 - 6 * q^94 + 6 * q^95 - 3 * q^96 - 3 * q^97 - 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15x - 20$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 10$$ b2 + 2*b1 + 10

## 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
 −2.80560 −1.61323 4.41883
−1.00000 1.00000 1.00000 −3.80560 −1.00000 0 −1.00000 1.00000 3.80560
1.2 −1.00000 1.00000 1.00000 −2.61323 −1.00000 0 −1.00000 1.00000 2.61323
1.3 −1.00000 1.00000 1.00000 3.41883 −1.00000 0 −1.00000 1.00000 −3.41883
 $$n$$: 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.bw 3
7.b odd 2 1 3822.2.a.bv 3
7.d odd 6 2 546.2.i.k 6
21.g even 6 2 1638.2.j.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.k 6 7.d odd 6 2
1638.2.j.q 6 21.g even 6 2
3822.2.a.bv 3 7.b odd 2 1
3822.2.a.bw 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5}^{3} + 3T_{5}^{2} - 12T_{5} - 34$$ T5^3 + 3*T5^2 - 12*T5 - 34 $$T_{11}^{3} + 3T_{11}^{2} - 27T_{11} - 89$$ T11^3 + 3*T11^2 - 27*T11 - 89 $$T_{17}^{3} + 6T_{17}^{2} - 63T_{17} - 382$$ T17^3 + 6*T17^2 - 63*T17 - 382 $$T_{29} + 3$$ T29 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} + 3 T^{2} + \cdots - 34$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 3 T^{2} + \cdots - 89$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} + 6 T^{2} + \cdots - 382$$
$19$ $$T^{3} + 6 T^{2} + \cdots - 32$$
$23$ $$T^{3} - 30T - 60$$
$29$ $$(T + 3)^{3}$$
$31$ $$T^{3} - 15 T^{2} + \cdots - 40$$
$37$ $$T^{3} - 6 T^{2} + \cdots + 32$$
$41$ $$T^{3} + 18 T^{2} + \cdots + 56$$
$43$ $$T^{3} + 12 T^{2} + \cdots - 16$$
$47$ $$T^{3} - 6 T^{2} + \cdots + 212$$
$53$ $$T^{3} - 3 T^{2} + \cdots - 41$$
$59$ $$T^{3} - 3 T^{2} + \cdots + 49$$
$61$ $$T^{3} - 75T - 200$$
$67$ $$T^{3} - 6 T^{2} + \cdots - 8$$
$71$ $$T^{3} + 36 T^{2} + \cdots + 1538$$
$73$ $$T^{3} + 24 T^{2} + \cdots + 212$$
$79$ $$T^{3} - 15T^{2} + 100$$
$83$ $$T^{3} + 9 T^{2} + \cdots - 108$$
$89$ $$T^{3} + 30 T^{2} + \cdots + 680$$
$97$ $$T^{3} + 3 T^{2} + \cdots + 166$$