# Properties

 Label 3822.2.a.bs 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{2}$$. 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 $$q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{8} + q^{9} + \beta q^{10} + ( - \beta - 1) q^{11} + q^{12} + q^{13} + \beta q^{15} + q^{16} + ( - 3 \beta + 1) q^{17} + q^{18} + 3 q^{19} + \beta q^{20} + ( - \beta - 1) q^{22} + (5 \beta - 2) q^{23} + q^{24} - 3 q^{25} + q^{26} + q^{27} + 5 q^{29} + \beta q^{30} + ( - 2 \beta + 4) q^{31} + q^{32} + ( - \beta - 1) q^{33} + ( - 3 \beta + 1) q^{34} + q^{36} + (5 \beta + 2) q^{37} + 3 q^{38} + q^{39} + \beta q^{40} + (\beta + 6) q^{41} + ( - 2 \beta + 2) q^{43} + ( - \beta - 1) q^{44} + \beta q^{45} + (5 \beta - 2) q^{46} + 9 q^{47} + q^{48} - 3 q^{50} + ( - 3 \beta + 1) q^{51} + q^{52} + ( - 8 \beta + 1) q^{53} + q^{54} + ( - \beta - 2) q^{55} + 3 q^{57} + 5 q^{58} + (\beta + 9) q^{59} + \beta q^{60} + (3 \beta - 1) q^{61} + ( - 2 \beta + 4) q^{62} + q^{64} + \beta q^{65} + ( - \beta - 1) q^{66} + (4 \beta - 9) q^{67} + ( - 3 \beta + 1) q^{68} + (5 \beta - 2) q^{69} - q^{71} + q^{72} + (5 \beta + 6) q^{73} + (5 \beta + 2) q^{74} - 3 q^{75} + 3 q^{76} + q^{78} + ( - 8 \beta + 2) q^{79} + \beta q^{80} + q^{81} + (\beta + 6) q^{82} + (4 \beta - 2) q^{83} + (\beta - 6) q^{85} + ( - 2 \beta + 2) q^{86} + 5 q^{87} + ( - \beta - 1) q^{88} + (3 \beta - 2) q^{89} + \beta q^{90} + (5 \beta - 2) q^{92} + ( - 2 \beta + 4) q^{93} + 9 q^{94} + 3 \beta q^{95} + q^{96} + ( - 7 \beta - 2) q^{97} + ( - \beta - 1) q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + b * q^5 + q^6 + q^8 + q^9 + b * q^10 + (-b - 1) * q^11 + q^12 + q^13 + b * q^15 + q^16 + (-3*b + 1) * q^17 + q^18 + 3 * q^19 + b * q^20 + (-b - 1) * q^22 + (5*b - 2) * q^23 + q^24 - 3 * q^25 + q^26 + q^27 + 5 * q^29 + b * q^30 + (-2*b + 4) * q^31 + q^32 + (-b - 1) * q^33 + (-3*b + 1) * q^34 + q^36 + (5*b + 2) * q^37 + 3 * q^38 + q^39 + b * q^40 + (b + 6) * q^41 + (-2*b + 2) * q^43 + (-b - 1) * q^44 + b * q^45 + (5*b - 2) * q^46 + 9 * q^47 + q^48 - 3 * q^50 + (-3*b + 1) * q^51 + q^52 + (-8*b + 1) * q^53 + q^54 + (-b - 2) * q^55 + 3 * q^57 + 5 * q^58 + (b + 9) * q^59 + b * q^60 + (3*b - 1) * q^61 + (-2*b + 4) * q^62 + q^64 + b * q^65 + (-b - 1) * q^66 + (4*b - 9) * q^67 + (-3*b + 1) * q^68 + (5*b - 2) * q^69 - q^71 + q^72 + (5*b + 6) * q^73 + (5*b + 2) * q^74 - 3 * q^75 + 3 * q^76 + q^78 + (-8*b + 2) * q^79 + b * q^80 + q^81 + (b + 6) * q^82 + (4*b - 2) * q^83 + (b - 6) * q^85 + (-2*b + 2) * q^86 + 5 * q^87 + (-b - 1) * q^88 + (3*b - 2) * q^89 + b * q^90 + (5*b - 2) * q^92 + (-2*b + 4) * q^93 + 9 * q^94 + 3*b * q^95 + q^96 + (-7*b - 2) * q^97 + (-b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 6 q^{25} + 2 q^{26} + 2 q^{27} + 10 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{33} + 2 q^{34} + 2 q^{36} + 4 q^{37} + 6 q^{38} + 2 q^{39} + 12 q^{41} + 4 q^{43} - 2 q^{44} - 4 q^{46} + 18 q^{47} + 2 q^{48} - 6 q^{50} + 2 q^{51} + 2 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} + 6 q^{57} + 10 q^{58} + 18 q^{59} - 2 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{66} - 18 q^{67} + 2 q^{68} - 4 q^{69} - 2 q^{71} + 2 q^{72} + 12 q^{73} + 4 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 4 q^{79} + 2 q^{81} + 12 q^{82} - 4 q^{83} - 12 q^{85} + 4 q^{86} + 10 q^{87} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 8 q^{93} + 18 q^{94} + 2 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 6 * q^19 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 6 * q^25 + 2 * q^26 + 2 * q^27 + 10 * q^29 + 8 * q^31 + 2 * q^32 - 2 * q^33 + 2 * q^34 + 2 * q^36 + 4 * q^37 + 6 * q^38 + 2 * q^39 + 12 * q^41 + 4 * q^43 - 2 * q^44 - 4 * q^46 + 18 * q^47 + 2 * q^48 - 6 * q^50 + 2 * q^51 + 2 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 + 6 * q^57 + 10 * q^58 + 18 * q^59 - 2 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^66 - 18 * q^67 + 2 * q^68 - 4 * q^69 - 2 * q^71 + 2 * q^72 + 12 * q^73 + 4 * q^74 - 6 * q^75 + 6 * q^76 + 2 * q^78 + 4 * q^79 + 2 * q^81 + 12 * q^82 - 4 * q^83 - 12 * q^85 + 4 * q^86 + 10 * q^87 - 2 * q^88 - 4 * q^89 - 4 * q^92 + 8 * q^93 + 18 * q^94 + 2 * q^96 - 4 * 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 −1.41421 1.00000 0 1.00000 1.00000 −1.41421
1.2 1.00000 1.00000 1.00000 1.41421 1.00000 0 1.00000 1.00000 1.41421
 $$n$$: 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.bs 2
7.b odd 2 1 3822.2.a.bp 2
7.c even 3 2 546.2.i.h 4
21.h odd 6 2 1638.2.j.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.h 4 7.c even 3 2
1638.2.j.n 4 21.h odd 6 2
3822.2.a.bp 2 7.b odd 2 1
3822.2.a.bs 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{11}^{2} + 2T_{11} - 1$$ T11^2 + 2*T11 - 1 $$T_{17}^{2} - 2T_{17} - 17$$ T17^2 - 2*T17 - 17 $$T_{29} - 5$$ T29 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T - 1$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 2T - 17$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} + 4T - 46$$
$29$ $$(T - 5)^{2}$$
$31$ $$T^{2} - 8T + 8$$
$37$ $$T^{2} - 4T - 46$$
$41$ $$T^{2} - 12T + 34$$
$43$ $$T^{2} - 4T - 4$$
$47$ $$(T - 9)^{2}$$
$53$ $$T^{2} - 2T - 127$$
$59$ $$T^{2} - 18T + 79$$
$61$ $$T^{2} + 2T - 17$$
$67$ $$T^{2} + 18T + 49$$
$71$ $$(T + 1)^{2}$$
$73$ $$T^{2} - 12T - 14$$
$79$ $$T^{2} - 4T - 124$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} + 4T - 14$$
$97$ $$T^{2} + 4T - 94$$