# Properties

 Label 3626.2.a.v 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: 3.3.733.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 8$$ x^3 - x^2 - 7*x + 8 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5

## 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
 2.51820 1.17819 −2.69639
−1.00000 −2.51820 1.00000 −2.34132 2.51820 0 −1.00000 3.34132 2.34132
1.2 −1.00000 −1.17819 1.00000 2.61186 1.17819 0 −1.00000 −1.61186 −2.61186
1.3 −1.00000 2.69639 1.00000 −3.27053 −2.69639 0 −1.00000 4.27053 3.27053
 $$n$$: 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.v 3
7.b odd 2 1 518.2.a.e 3
21.c even 2 1 4662.2.a.bb 3
28.d even 2 1 4144.2.a.l 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.a.e 3 7.b odd 2 1
3626.2.a.v 3 1.a even 1 1 trivial
4144.2.a.l 3 28.d even 2 1
4662.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{3} + T_{3}^{2} - 7T_{3} - 8$$ T3^3 + T3^2 - 7*T3 - 8 $$T_{5}^{3} + 3T_{5}^{2} - 7T_{5} - 20$$ T5^3 + 3*T5^2 - 7*T5 - 20 $$T_{11}^{3} + 3T_{11}^{2} - 7T_{11} - 20$$ T11^3 + 3*T11^2 - 7*T11 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} + T^{2} - 7T - 8$$
$5$ $$T^{3} + 3 T^{2} + \cdots - 20$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 3 T^{2} + \cdots - 20$$
$13$ $$T^{3} + 11 T^{2} + \cdots + 28$$
$17$ $$T^{3} + 2 T^{2} + \cdots - 64$$
$19$ $$T^{3} - 40T + 88$$
$23$ $$T^{3} - 3 T^{2} + \cdots + 260$$
$29$ $$T^{3} - 5T^{2} + T + 2$$
$31$ $$T^{3} + 15 T^{2} + \cdots + 86$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3} - 13 T^{2} + \cdots + 154$$
$43$ $$T^{3} - 10 T^{2} + \cdots + 16$$
$47$ $$T^{3} + 10 T^{2} + \cdots - 16$$
$53$ $$T^{3} + 2 T^{2} + \cdots + 32$$
$59$ $$T^{3} + 2 T^{2} + \cdots + 32$$
$61$ $$T^{3} + T^{2} + \cdots + 464$$
$67$ $$T^{3} - 17 T^{2} + \cdots + 404$$
$71$ $$T^{3} + 12 T^{2} + \cdots - 1280$$
$73$ $$T^{3} + 23 T^{2} + \cdots + 298$$
$79$ $$T^{3} - 27 T^{2} + \cdots + 44$$
$83$ $$T^{3} - 8 T^{2} + \cdots + 896$$
$89$ $$T^{3} - 14 T^{2} + \cdots + 704$$
$97$ $$T^{3} - 20 T^{2} + \cdots - 80$$