# Properties

 Label 3822.2.a.by Level $3822$ Weight $2$ Character orbit 3822.a Self dual yes Analytic conductor $30.519$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.10304.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 8$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\beta_2,\beta_3$$ 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_{3} + \beta_{2}) q^{11} + q^{12} - q^{13} + (\beta_1 + 1) q^{15} + q^{16} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{17} - q^{18} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{19} + (\beta_1 + 1) q^{20} + ( - \beta_{3} - \beta_{2}) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} - q^{24} + (2 \beta_{2} + 3 \beta_1) q^{25} + q^{26} + q^{27} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + ( - \beta_1 - 1) q^{30} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{31} - q^{32} + (\beta_{3} + \beta_{2}) q^{33} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{34} + q^{36} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{38} - q^{39} + ( - \beta_1 - 1) q^{40} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 5) q^{41} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{43} + (\beta_{3} + \beta_{2}) q^{44} + (\beta_1 + 1) q^{45} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{46} + (\beta_{3} - 3 \beta_1 + 5) q^{47} + q^{48} + ( - 2 \beta_{2} - 3 \beta_1) q^{50} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{51} - q^{52} + (\beta_{3} - 3 \beta_1 + 3) q^{53} - q^{54} + (3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{58} + ( - \beta_{3} - 5 \beta_{2} - \beta_1 + 5) q^{59} + (\beta_1 + 1) q^{60} + (5 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{61} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{62} + q^{64} + ( - \beta_1 - 1) q^{65} + ( - \beta_{3} - \beta_{2}) q^{66} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{67} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{68} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{69} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{71} - q^{72} + ( - 2 \beta_{3} - 6 \beta_{2} - \beta_1 + 3) q^{73} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{74} + (2 \beta_{2} + 3 \beta_1) q^{75} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{76} + q^{78} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 3) q^{79} + (\beta_1 + 1) q^{80} + q^{81} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 5) q^{82} + (4 \beta_{2} + 2 \beta_1 + 6) q^{83} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 10) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{86} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{87} + ( - \beta_{3} - \beta_{2}) q^{88} + ( - 2 \beta_{3} + \beta_{2} + 8) q^{89} + ( - \beta_1 - 1) q^{90} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{92} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{93} + ( - \beta_{3} + 3 \beta_1 - 5) q^{94} + ( - 2 \beta_{3} + 3 \beta_1 + 5) q^{95} - q^{96} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{97} + (\beta_{3} + \beta_{2}) 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 + (b3 + b2) * q^11 + q^12 - q^13 + (b1 + 1) * q^15 + q^16 + (-b3 - b2 + 2*b1) * q^17 - q^18 + (-2*b3 + 2*b2 + b1 + 1) * q^19 + (b1 + 1) * q^20 + (-b3 - b2) * q^22 + (-2*b3 + 2*b2 - b1 + 1) * q^23 - q^24 + (2*b2 + 3*b1) * q^25 + q^26 + q^27 + (b3 - 2*b2 - 2*b1 + 2) * q^29 + (-b1 - 1) * q^30 + (b3 - 3*b2 + b1 - 1) * q^31 - q^32 + (b3 + b2) * q^33 + (b3 + b2 - 2*b1) * q^34 + q^36 + (2*b3 - b2 - b1 + 1) * q^37 + (2*b3 - 2*b2 - b1 - 1) * q^38 - q^39 + (-b1 - 1) * q^40 + (-b3 + 2*b2 - b1 + 5) * q^41 + (2*b3 - 2*b2 + b1 + 3) * q^43 + (b3 + b2) * q^44 + (b1 + 1) * q^45 + (2*b3 - 2*b2 + b1 - 1) * q^46 + (b3 - 3*b1 + 5) * q^47 + q^48 + (-2*b2 - 3*b1) * q^50 + (-b3 - b2 + 2*b1) * q^51 - q^52 + (b3 - 3*b1 + 3) * q^53 - q^54 + (3*b3 + 3*b2 + 2*b1 - 2) * q^55 + (-2*b3 + 2*b2 + b1 + 1) * q^57 + (-b3 + 2*b2 + 2*b1 - 2) * q^58 + (-b3 - 5*b2 - b1 + 5) * q^59 + (b1 + 1) * q^60 + (5*b3 - b2 + 2*b1 - 4) * q^61 + (-b3 + 3*b2 - b1 + 1) * q^62 + q^64 + (-b1 - 1) * q^65 + (-b3 - b2) * q^66 + (-3*b2 - 2*b1 + 4) * q^67 + (-b3 - b2 + 2*b1) * q^68 + (-2*b3 + 2*b2 - b1 + 1) * q^69 + (-b3 - b2 - b1 - 1) * q^71 - q^72 + (-2*b3 - 6*b2 - b1 + 3) * q^73 + (-2*b3 + b2 + b1 - 1) * q^74 + (2*b2 + 3*b1) * q^75 + (-2*b3 + 2*b2 + b1 + 1) * q^76 + q^78 + (-3*b3 + 5*b2 - 3*b1 - 3) * q^79 + (b1 + 1) * q^80 + q^81 + (b3 - 2*b2 + b1 - 5) * q^82 + (4*b2 + 2*b1 + 6) * q^83 + (-3*b3 + b2 + 2*b1 + 10) * q^85 + (-2*b3 + 2*b2 - b1 - 3) * q^86 + (b3 - 2*b2 - 2*b1 + 2) * q^87 + (-b3 - b2) * q^88 + (-2*b3 + b2 + 8) * q^89 + (-b1 - 1) * q^90 + (-2*b3 + 2*b2 - b1 + 1) * q^92 + (b3 - 3*b2 + b1 - 1) * q^93 + (-b3 + 3*b1 - 5) * q^94 + (-2*b3 + 3*b1 + 5) * q^95 - q^96 + (-2*b3 + 4*b2 + 4*b1 - 2) * q^97 + (b3 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9 $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 6 q^{10} + 2 q^{11} + 4 q^{12} - 4 q^{13} + 6 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 2 q^{19} + 6 q^{20} - 2 q^{22} - 2 q^{23} - 4 q^{24} + 6 q^{25} + 4 q^{26} + 4 q^{27} + 6 q^{29} - 6 q^{30} - 4 q^{32} + 2 q^{33} - 2 q^{34} + 4 q^{36} + 6 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{40} + 16 q^{41} + 18 q^{43} + 2 q^{44} + 6 q^{45} + 2 q^{46} + 16 q^{47} + 4 q^{48} - 6 q^{50} + 2 q^{51} - 4 q^{52} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{57} - 6 q^{58} + 16 q^{59} + 6 q^{60} - 2 q^{61} + 4 q^{64} - 6 q^{65} - 2 q^{66} + 12 q^{67} + 2 q^{68} - 2 q^{69} - 8 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} + 6 q^{75} + 2 q^{76} + 4 q^{78} - 24 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 28 q^{83} + 38 q^{85} - 18 q^{86} + 6 q^{87} - 2 q^{88} + 28 q^{89} - 6 q^{90} - 2 q^{92} - 16 q^{94} + 22 q^{95} - 4 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9 - 6 * q^10 + 2 * q^11 + 4 * q^12 - 4 * q^13 + 6 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 2 * q^19 + 6 * q^20 - 2 * q^22 - 2 * q^23 - 4 * q^24 + 6 * q^25 + 4 * q^26 + 4 * q^27 + 6 * q^29 - 6 * q^30 - 4 * q^32 + 2 * q^33 - 2 * q^34 + 4 * q^36 + 6 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 16 * q^41 + 18 * q^43 + 2 * q^44 + 6 * q^45 + 2 * q^46 + 16 * q^47 + 4 * q^48 - 6 * q^50 + 2 * q^51 - 4 * q^52 + 8 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^57 - 6 * q^58 + 16 * q^59 + 6 * q^60 - 2 * q^61 + 4 * q^64 - 6 * q^65 - 2 * q^66 + 12 * q^67 + 2 * q^68 - 2 * q^69 - 8 * q^71 - 4 * q^72 + 6 * q^73 - 6 * q^74 + 6 * q^75 + 2 * q^76 + 4 * q^78 - 24 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 28 * q^83 + 38 * q^85 - 18 * q^86 + 6 * q^87 - 2 * q^88 + 28 * q^89 - 6 * q^90 - 2 * q^92 - 16 * q^94 + 22 * q^95 - 4 * q^96 - 4 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 7x^{2} + 8x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - \nu - 4 ) / 2$$ (v^2 - v - 4) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2$$ (v^3 - v^2 - 6*v + 2) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2} + \beta _1 + 4$$ 2*b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 2\beta_{2} + 7\beta _1 + 2$$ 2*b3 + 2*b2 + 7*b1 + 2

## 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
 −2.16053 −0.692297 1.69230 3.16053
−1.00000 1.00000 1.00000 −1.16053 −1.00000 0 −1.00000 1.00000 1.16053
1.2 −1.00000 1.00000 1.00000 0.307703 −1.00000 0 −1.00000 1.00000 −0.307703
1.3 −1.00000 1.00000 1.00000 2.69230 −1.00000 0 −1.00000 1.00000 −2.69230
1.4 −1.00000 1.00000 1.00000 4.16053 −1.00000 0 −1.00000 1.00000 −4.16053
 $$n$$: 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.by yes 4
7.b odd 2 1 3822.2.a.bx 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3822.2.a.bx 4 7.b odd 2 1
3822.2.a.by yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3822))$$:

 $$T_{5}^{4} - 6T_{5}^{3} + 5T_{5}^{2} + 12T_{5} - 4$$ T5^4 - 6*T5^3 + 5*T5^2 + 12*T5 - 4 $$T_{11}^{4} - 2T_{11}^{3} - 17T_{11}^{2} + 48T_{11} - 32$$ T11^4 - 2*T11^3 - 17*T11^2 + 48*T11 - 32 $$T_{17}^{4} - 2T_{17}^{3} - 53T_{17}^{2} + 12T_{17} + 316$$ T17^4 - 2*T17^3 - 53*T17^2 + 12*T17 + 316 $$T_{29}^{4} - 6T_{29}^{3} - 41T_{29}^{2} + 108T_{29} + 254$$ T29^4 - 6*T29^3 - 41*T29^2 + 108*T29 + 254

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} - 6 T^{3} + 5 T^{2} + 12 T - 4$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 2 T^{3} - 17 T^{2} + 48 T - 32$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} - 2 T^{3} - 53 T^{2} + 12 T + 316$$
$19$ $$T^{4} - 2 T^{3} - 51 T^{2} + 188 T - 164$$
$23$ $$T^{4} + 2 T^{3} - 47 T^{2} - 56 T + 392$$
$29$ $$T^{4} - 6 T^{3} - 41 T^{2} + 108 T + 254$$
$31$ $$T^{4} - 42 T^{2} + 80 T - 16$$
$37$ $$T^{4} - 6 T^{3} - 35 T^{2} + 132 T - 94$$
$41$ $$T^{4} - 16 T^{3} + 70 T^{2} - 96 T + 16$$
$43$ $$T^{4} - 18 T^{3} + 73 T^{2} + \cdots - 200$$
$47$ $$T^{4} - 16 T^{3} + 6 T^{2} + 456 T + 776$$
$53$ $$T^{4} - 8 T^{3} - 66 T^{2} + \cdots + 1600$$
$59$ $$T^{4} - 16 T^{3} - 42 T^{2} + 1024 T + 16$$
$61$ $$T^{4} + 2 T^{3} - 269 T^{2} + \cdots + 15748$$
$67$ $$T^{4} - 12 T^{3} - 16 T^{2} + \cdots - 1252$$
$71$ $$T^{4} + 8 T^{3} - 2 T^{2} - 24 T - 8$$
$73$ $$T^{4} - 6 T^{3} - 227 T^{2} + \cdots + 5116$$
$79$ $$T^{4} + 24 T^{3} + 14 T^{2} + \cdots - 11912$$
$83$ $$T^{4} - 28 T^{3} + 196 T^{2} + \cdots - 512$$
$89$ $$T^{4} - 28 T^{3} + 256 T^{2} + \cdots + 700$$
$97$ $$T^{4} + 4 T^{3} - 212 T^{2} + \cdots + 5056$$