# Properties

 Label 3822.2.a.bu 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + (\beta + 2) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + (b + 2) * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + (\beta + 2) q^{5} + q^{6} + q^{8} + q^{9} + (\beta + 2) q^{10} + (\beta + 1) q^{11} + q^{12} - q^{13} + (\beta + 2) q^{15} + q^{16} + ( - \beta + 1) q^{17} + q^{18} + (2 \beta + 5) q^{19} + (\beta + 2) q^{20} + (\beta + 1) q^{22} - \beta q^{23} + q^{24} + (4 \beta + 1) q^{25} - q^{26} + q^{27} + (2 \beta + 1) q^{29} + (\beta + 2) q^{30} - 6 \beta q^{31} + q^{32} + (\beta + 1) q^{33} + ( - \beta + 1) q^{34} + q^{36} - \beta q^{37} + (2 \beta + 5) q^{38} - q^{39} + (\beta + 2) q^{40} - 7 \beta q^{41} + ( - 6 \beta - 2) q^{43} + (\beta + 1) q^{44} + (\beta + 2) q^{45} - \beta q^{46} - q^{47} + q^{48} + (4 \beta + 1) q^{50} + ( - \beta + 1) q^{51} - q^{52} + ( - 6 \beta + 1) q^{53} + q^{54} + (3 \beta + 4) q^{55} + (2 \beta + 5) q^{57} + (2 \beta + 1) q^{58} + (5 \beta + 5) q^{59} + (\beta + 2) q^{60} + ( - \beta + 3) q^{61} - 6 \beta q^{62} + q^{64} + ( - \beta - 2) q^{65} + (\beta + 1) q^{66} + ( - 2 \beta - 1) q^{67} + ( - \beta + 1) q^{68} - \beta q^{69} - 5 q^{71} + q^{72} + \beta q^{73} - \beta q^{74} + (4 \beta + 1) q^{75} + (2 \beta + 5) q^{76} - q^{78} + (4 \beta - 6) q^{79} + (\beta + 2) q^{80} + q^{81} - 7 \beta q^{82} + (4 \beta - 2) q^{83} - \beta q^{85} + ( - 6 \beta - 2) q^{86} + (2 \beta + 1) q^{87} + (\beta + 1) q^{88} + ( - \beta - 4) q^{89} + (\beta + 2) q^{90} - \beta q^{92} - 6 \beta q^{93} - q^{94} + (9 \beta + 14) q^{95} + q^{96} + (5 \beta + 8) q^{97} + (\beta + 1) q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 + (b + 2) * q^5 + q^6 + q^8 + q^9 + (b + 2) * q^10 + (b + 1) * q^11 + q^12 - q^13 + (b + 2) * q^15 + q^16 + (-b + 1) * q^17 + q^18 + (2*b + 5) * q^19 + (b + 2) * q^20 + (b + 1) * q^22 - b * q^23 + q^24 + (4*b + 1) * q^25 - q^26 + q^27 + (2*b + 1) * q^29 + (b + 2) * q^30 - 6*b * q^31 + q^32 + (b + 1) * q^33 + (-b + 1) * q^34 + q^36 - b * q^37 + (2*b + 5) * q^38 - q^39 + (b + 2) * q^40 - 7*b * q^41 + (-6*b - 2) * q^43 + (b + 1) * q^44 + (b + 2) * q^45 - b * q^46 - q^47 + q^48 + (4*b + 1) * q^50 + (-b + 1) * q^51 - q^52 + (-6*b + 1) * q^53 + q^54 + (3*b + 4) * q^55 + (2*b + 5) * q^57 + (2*b + 1) * q^58 + (5*b + 5) * q^59 + (b + 2) * q^60 + (-b + 3) * q^61 - 6*b * q^62 + q^64 + (-b - 2) * q^65 + (b + 1) * q^66 + (-2*b - 1) * q^67 + (-b + 1) * q^68 - b * q^69 - 5 * q^71 + q^72 + b * q^73 - b * q^74 + (4*b + 1) * q^75 + (2*b + 5) * q^76 - q^78 + (4*b - 6) * q^79 + (b + 2) * q^80 + q^81 - 7*b * q^82 + (4*b - 2) * q^83 - b * q^85 + (-6*b - 2) * q^86 + (2*b + 1) * q^87 + (b + 1) * q^88 + (-b - 4) * q^89 + (b + 2) * q^90 - b * q^92 - 6*b * q^93 - q^94 + (9*b + 14) * q^95 + q^96 + (5*b + 8) * q^97 + (b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27} + 2 q^{29} + 4 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 10 q^{38} - 2 q^{39} + 4 q^{40} - 4 q^{43} + 2 q^{44} + 4 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 8 q^{55} + 10 q^{57} + 2 q^{58} + 10 q^{59} + 4 q^{60} + 6 q^{61} + 2 q^{64} - 4 q^{65} + 2 q^{66} - 2 q^{67} + 2 q^{68} - 10 q^{71} + 2 q^{72} + 2 q^{75} + 10 q^{76} - 2 q^{78} - 12 q^{79} + 4 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{86} + 2 q^{87} + 2 q^{88} - 8 q^{89} + 4 q^{90} - 2 q^{94} + 28 q^{95} + 2 q^{96} + 16 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 4 * q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 10 * q^19 + 4 * q^20 + 2 * q^22 + 2 * q^24 + 2 * q^25 - 2 * q^26 + 2 * q^27 + 2 * q^29 + 4 * q^30 + 2 * q^32 + 2 * q^33 + 2 * q^34 + 2 * q^36 + 10 * q^38 - 2 * q^39 + 4 * q^40 - 4 * q^43 + 2 * q^44 + 4 * q^45 - 2 * q^47 + 2 * q^48 + 2 * q^50 + 2 * q^51 - 2 * q^52 + 2 * q^53 + 2 * q^54 + 8 * q^55 + 10 * q^57 + 2 * q^58 + 10 * q^59 + 4 * q^60 + 6 * q^61 + 2 * q^64 - 4 * q^65 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 10 * q^71 + 2 * q^72 + 2 * q^75 + 10 * q^76 - 2 * q^78 - 12 * q^79 + 4 * q^80 + 2 * q^81 - 4 * q^83 - 4 * q^86 + 2 * q^87 + 2 * q^88 - 8 * q^89 + 4 * q^90 - 2 * q^94 + 28 * q^95 + 2 * q^96 + 16 * 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 0.585786 1.00000 0 1.00000 1.00000 0.585786
1.2 1.00000 1.00000 1.00000 3.41421 1.00000 0 1.00000 1.00000 3.41421
 $$n$$: 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.bu 2
7.b odd 2 1 3822.2.a.bn 2
7.d odd 6 2 546.2.i.i 4
21.g even 6 2 1638.2.j.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.i 4 7.d odd 6 2
1638.2.j.m 4 21.g even 6 2
3822.2.a.bn 2 7.b odd 2 1
3822.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{11}^{2} - 2T_{11} - 1$$ T11^2 - 2*T11 - 1 $$T_{17}^{2} - 2T_{17} - 1$$ T17^2 - 2*T17 - 1 $$T_{29}^{2} - 2T_{29} - 7$$ T29^2 - 2*T29 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 2T - 1$$
$19$ $$T^{2} - 10T + 17$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 2T - 7$$
$31$ $$T^{2} - 72$$
$37$ $$T^{2} - 2$$
$41$ $$T^{2} - 98$$
$43$ $$T^{2} + 4T - 68$$
$47$ $$(T + 1)^{2}$$
$53$ $$T^{2} - 2T - 71$$
$59$ $$T^{2} - 10T - 25$$
$61$ $$T^{2} - 6T + 7$$
$67$ $$T^{2} + 2T - 7$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} - 2$$
$79$ $$T^{2} + 12T + 4$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} + 8T + 14$$
$97$ $$T^{2} - 16T + 14$$