# Properties

 Label 3626.2.a.w Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $0$ 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] = [3626,2,Mod(1,3626)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3626, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3626.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3626 = 2 \cdot 7^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3626.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: $$28.9537557729$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 518) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_{2} q^{3} + q^{4} + (\beta_1 + 1) q^{5} + \beta_{2} q^{6} - q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q - q^2 - b2 * q^3 + q^4 + (b1 + 1) * q^5 + b2 * q^6 - q^8 + (-2*b2 + b1) * q^9 $$q - q^{2} - \beta_{2} q^{3} + q^{4} + (\beta_1 + 1) q^{5} + \beta_{2} q^{6} - q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9} + ( - \beta_1 - 1) q^{10} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{11} - \beta_{2} q^{12} + (\beta_{2} + 4) q^{13} + ( - \beta_{2} - \beta_1 - 1) q^{15} + q^{16} - 2 \beta_{2} q^{17} + (2 \beta_{2} - \beta_1) q^{18} + (2 \beta_{2} + 4) q^{19} + (\beta_1 + 1) q^{20} + (2 \beta_{2} - 3 \beta_1 - 1) q^{22} + ( - \beta_{2} - 4 \beta_1 + 2) q^{23} + \beta_{2} q^{24} + (\beta_{2} + 2 \beta_1 - 1) q^{25} + ( - \beta_{2} - 4) q^{26} + ( - \beta_{2} + \beta_1 + 5) q^{27} + (\beta_{2} + 2 \beta_1 - 2) q^{29} + (\beta_{2} + \beta_1 + 1) q^{30} + ( - 2 \beta_{2} + 5 \beta_1 + 1) q^{31} - q^{32} + ( - 5 \beta_{2} - \beta_1 + 3) q^{33} + 2 \beta_{2} q^{34} + ( - 2 \beta_{2} + \beta_1) q^{36} - q^{37} + ( - 2 \beta_{2} - 4) q^{38} + ( - 2 \beta_{2} - \beta_1 - 3) q^{39} + ( - \beta_1 - 1) q^{40} - \beta_{2} q^{41} + (2 \beta_{2} + 4 \beta_1 - 4) q^{43} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{44} + ( - \beta_{2} - \beta_1 + 1) q^{45} + (\beta_{2} + 4 \beta_1 - 2) q^{46} + (2 \beta_{2} - 4) q^{47} - \beta_{2} q^{48} + ( - \beta_{2} - 2 \beta_1 + 1) q^{50} + ( - 4 \beta_{2} + 2 \beta_1 + 6) q^{51} + (\beta_{2} + 4) q^{52} + (2 \beta_{2} - 2 \beta_1 - 4) q^{53} + (\beta_{2} - \beta_1 - 5) q^{54} + (\beta_{2} + 2 \beta_1 + 8) q^{55} + ( - 2 \beta_1 - 6) q^{57} + ( - \beta_{2} - 2 \beta_1 + 2) q^{58} + ( - 4 \beta_{2} - 4) q^{59} + ( - \beta_{2} - \beta_1 - 1) q^{60} + ( - \beta_1 + 7) q^{61} + (2 \beta_{2} - 5 \beta_1 - 1) q^{62} + q^{64} + (\beta_{2} + 5 \beta_1 + 5) q^{65} + (5 \beta_{2} + \beta_1 - 3) q^{66} + (2 \beta_{2} - 5 \beta_1 - 7) q^{67} - 2 \beta_{2} q^{68} + ( - 4 \beta_{2} + 5 \beta_1 + 7) q^{69} + (4 \beta_1 + 4) q^{71} + (2 \beta_{2} - \beta_1) q^{72} + (4 \beta_{2} - \beta_1 + 3) q^{73} + q^{74} + (3 \beta_{2} - 3 \beta_1 - 5) q^{75} + (2 \beta_{2} + 4) q^{76} + (2 \beta_{2} + \beta_1 + 3) q^{78} + (3 \beta_{2} + 2) q^{79} + (\beta_1 + 1) q^{80} + ( - \beta_{2} - 3 \beta_1 + 2) q^{81} + \beta_{2} q^{82} + (6 \beta_{2} - 2 \beta_1 + 4) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{85} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{86} + (4 \beta_{2} - 3 \beta_1 - 5) q^{87} + (2 \beta_{2} - 3 \beta_1 - 1) q^{88} + (6 \beta_1 - 6) q^{89} + (\beta_{2} + \beta_1 - 1) q^{90} + ( - \beta_{2} - 4 \beta_1 + 2) q^{92} + ( - 5 \beta_{2} - 3 \beta_1 + 1) q^{93} + ( - 2 \beta_{2} + 4) q^{94} + (2 \beta_{2} + 6 \beta_1 + 6) q^{95} + \beta_{2} q^{96} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{97} + ( - 7 \beta_{2} - 3 \beta_1 + 13) q^{99}+O(q^{100})$$ q - q^2 - b2 * q^3 + q^4 + (b1 + 1) * q^5 + b2 * q^6 - q^8 + (-2*b2 + b1) * q^9 + (-b1 - 1) * q^10 + (-2*b2 + 3*b1 + 1) * q^11 - b2 * q^12 + (b2 + 4) * q^13 + (-b2 - b1 - 1) * q^15 + q^16 - 2*b2 * q^17 + (2*b2 - b1) * q^18 + (2*b2 + 4) * q^19 + (b1 + 1) * q^20 + (2*b2 - 3*b1 - 1) * q^22 + (-b2 - 4*b1 + 2) * q^23 + b2 * q^24 + (b2 + 2*b1 - 1) * q^25 + (-b2 - 4) * q^26 + (-b2 + b1 + 5) * q^27 + (b2 + 2*b1 - 2) * q^29 + (b2 + b1 + 1) * q^30 + (-2*b2 + 5*b1 + 1) * q^31 - q^32 + (-5*b2 - b1 + 3) * q^33 + 2*b2 * q^34 + (-2*b2 + b1) * q^36 - q^37 + (-2*b2 - 4) * q^38 + (-2*b2 - b1 - 3) * q^39 + (-b1 - 1) * q^40 - b2 * q^41 + (2*b2 + 4*b1 - 4) * q^43 + (-2*b2 + 3*b1 + 1) * q^44 + (-b2 - b1 + 1) * q^45 + (b2 + 4*b1 - 2) * q^46 + (2*b2 - 4) * q^47 - b2 * q^48 + (-b2 - 2*b1 + 1) * q^50 + (-4*b2 + 2*b1 + 6) * q^51 + (b2 + 4) * q^52 + (2*b2 - 2*b1 - 4) * q^53 + (b2 - b1 - 5) * q^54 + (b2 + 2*b1 + 8) * q^55 + (-2*b1 - 6) * q^57 + (-b2 - 2*b1 + 2) * q^58 + (-4*b2 - 4) * q^59 + (-b2 - b1 - 1) * q^60 + (-b1 + 7) * q^61 + (2*b2 - 5*b1 - 1) * q^62 + q^64 + (b2 + 5*b1 + 5) * q^65 + (5*b2 + b1 - 3) * q^66 + (2*b2 - 5*b1 - 7) * q^67 - 2*b2 * q^68 + (-4*b2 + 5*b1 + 7) * q^69 + (4*b1 + 4) * q^71 + (2*b2 - b1) * q^72 + (4*b2 - b1 + 3) * q^73 + q^74 + (3*b2 - 3*b1 - 5) * q^75 + (2*b2 + 4) * q^76 + (2*b2 + b1 + 3) * q^78 + (3*b2 + 2) * q^79 + (b1 + 1) * q^80 + (-b2 - 3*b1 + 2) * q^81 + b2 * q^82 + (6*b2 - 2*b1 + 4) * q^83 + (-2*b2 - 2*b1 - 2) * q^85 + (-2*b2 - 4*b1 + 4) * q^86 + (4*b2 - 3*b1 - 5) * q^87 + (2*b2 - 3*b1 - 1) * q^88 + (6*b1 - 6) * q^89 + (b2 + b1 - 1) * q^90 + (-b2 - 4*b1 + 2) * q^92 + (-5*b2 - 3*b1 + 1) * q^93 + (-2*b2 + 4) * q^94 + (2*b2 + 6*b1 + 6) * q^95 + b2 * q^96 + (-2*b2 - 6*b1 + 2) * q^97 + (-7*b2 - 3*b1 + 13) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + q^{3} + 3 q^{4} + 3 q^{5} - q^{6} - 3 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 + q^3 + 3 * q^4 + 3 * q^5 - q^6 - 3 * q^8 + 2 * q^9 $$3 q - 3 q^{2} + q^{3} + 3 q^{4} + 3 q^{5} - q^{6} - 3 q^{8} + 2 q^{9} - 3 q^{10} + 5 q^{11} + q^{12} + 11 q^{13} - 2 q^{15} + 3 q^{16} + 2 q^{17} - 2 q^{18} + 10 q^{19} + 3 q^{20} - 5 q^{22} + 7 q^{23} - q^{24} - 4 q^{25} - 11 q^{26} + 16 q^{27} - 7 q^{29} + 2 q^{30} + 5 q^{31} - 3 q^{32} + 14 q^{33} - 2 q^{34} + 2 q^{36} - 3 q^{37} - 10 q^{38} - 7 q^{39} - 3 q^{40} + q^{41} - 14 q^{43} + 5 q^{44} + 4 q^{45} - 7 q^{46} - 14 q^{47} + q^{48} + 4 q^{50} + 22 q^{51} + 11 q^{52} - 14 q^{53} - 16 q^{54} + 23 q^{55} - 18 q^{57} + 7 q^{58} - 8 q^{59} - 2 q^{60} + 21 q^{61} - 5 q^{62} + 3 q^{64} + 14 q^{65} - 14 q^{66} - 23 q^{67} + 2 q^{68} + 25 q^{69} + 12 q^{71} - 2 q^{72} + 5 q^{73} + 3 q^{74} - 18 q^{75} + 10 q^{76} + 7 q^{78} + 3 q^{79} + 3 q^{80} + 7 q^{81} - q^{82} + 6 q^{83} - 4 q^{85} + 14 q^{86} - 19 q^{87} - 5 q^{88} - 18 q^{89} - 4 q^{90} + 7 q^{92} + 8 q^{93} + 14 q^{94} + 16 q^{95} - q^{96} + 8 q^{97} + 46 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 + q^3 + 3 * q^4 + 3 * q^5 - q^6 - 3 * q^8 + 2 * q^9 - 3 * q^10 + 5 * q^11 + q^12 + 11 * q^13 - 2 * q^15 + 3 * q^16 + 2 * q^17 - 2 * q^18 + 10 * q^19 + 3 * q^20 - 5 * q^22 + 7 * q^23 - q^24 - 4 * q^25 - 11 * q^26 + 16 * q^27 - 7 * q^29 + 2 * q^30 + 5 * q^31 - 3 * q^32 + 14 * q^33 - 2 * q^34 + 2 * q^36 - 3 * q^37 - 10 * q^38 - 7 * q^39 - 3 * q^40 + q^41 - 14 * q^43 + 5 * q^44 + 4 * q^45 - 7 * q^46 - 14 * q^47 + q^48 + 4 * q^50 + 22 * q^51 + 11 * q^52 - 14 * q^53 - 16 * q^54 + 23 * q^55 - 18 * q^57 + 7 * q^58 - 8 * q^59 - 2 * q^60 + 21 * q^61 - 5 * q^62 + 3 * q^64 + 14 * q^65 - 14 * q^66 - 23 * q^67 + 2 * q^68 + 25 * q^69 + 12 * q^71 - 2 * q^72 + 5 * q^73 + 3 * q^74 - 18 * q^75 + 10 * q^76 + 7 * q^78 + 3 * q^79 + 3 * q^80 + 7 * q^81 - q^82 + 6 * q^83 - 4 * q^85 + 14 * q^86 - 19 * q^87 - 5 * q^88 - 18 * q^89 - 4 * q^90 + 7 * q^92 + 8 * q^93 + 14 * q^94 + 16 * q^95 - q^96 + 8 * q^97 + 46 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 2.11491 −1.86081 −0.254102
−1.00000 −1.47283 1.00000 3.11491 1.47283 0 −1.00000 −0.830760 −3.11491
1.2 −1.00000 −0.462598 1.00000 −0.860806 0.462598 0 −1.00000 −2.78600 0.860806
1.3 −1.00000 2.93543 1.00000 0.745898 −2.93543 0 −1.00000 5.61676 −0.745898
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$37$$ $$1$$

## 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 3626.2.a.w 3
7.b odd 2 1 518.2.a.d 3
21.c even 2 1 4662.2.a.be 3
28.d even 2 1 4144.2.a.n 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.a.d 3 7.b odd 2 1
3626.2.a.w 3 1.a even 1 1 trivial
4144.2.a.n 3 28.d even 2 1
4662.2.a.be 3 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{3} - T_{3}^{2} - 5T_{3} - 2$$ T3^3 - T3^2 - 5*T3 - 2 $$T_{5}^{3} - 3T_{5}^{2} - T_{5} + 2$$ T5^3 - 3*T5^2 - T5 + 2 $$T_{11}^{3} - 5T_{11}^{2} - 31T_{11} + 148$$ T11^3 - 5*T11^2 - 31*T11 + 148

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} - T^{2} - 5T - 2$$
$5$ $$T^{3} - 3T^{2} - T + 2$$
$7$ $$T^{3}$$
$11$ $$T^{3} - 5 T^{2} + \cdots + 148$$
$13$ $$T^{3} - 11 T^{2} + \cdots - 26$$
$17$ $$T^{3} - 2 T^{2} + \cdots - 16$$
$19$ $$T^{3} - 10 T^{2} + \cdots + 64$$
$23$ $$T^{3} - 7 T^{2} + \cdots + 424$$
$29$ $$T^{3} + 7 T^{2} + \cdots - 106$$
$31$ $$T^{3} - 5 T^{2} + \cdots + 446$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3} - T^{2} - 5T - 2$$
$43$ $$T^{3} + 14 T^{2} + \cdots - 848$$
$47$ $$T^{3} + 14 T^{2} + \cdots + 32$$
$53$ $$T^{3} + 14 T^{2} + \cdots - 32$$
$59$ $$T^{3} + 8 T^{2} + \cdots - 448$$
$61$ $$T^{3} - 21 T^{2} + \cdots - 314$$
$67$ $$T^{3} + 23 T^{2} + \cdots - 548$$
$71$ $$T^{3} - 12 T^{2} + \cdots + 128$$
$73$ $$T^{3} - 5 T^{2} + \cdots + 386$$
$79$ $$T^{3} - 3 T^{2} + \cdots + 148$$
$83$ $$T^{3} - 6 T^{2} + \cdots + 1184$$
$89$ $$T^{3} + 18 T^{2} + \cdots - 864$$
$97$ $$T^{3} - 8 T^{2} + \cdots + 1568$$