# Properties

 Label 3626.2.a.l Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $1$ 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] = [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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} + q^{8} + ( - 2 \beta + 1) q^{9}+O(q^{10})$$ q + q^2 + (b - 1) * q^3 + q^4 + (b - 1) * q^6 + q^8 + (-2*b + 1) * q^9 $$q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} + q^{8} + ( - 2 \beta + 1) q^{9} + (2 \beta + 2) q^{11} + (\beta - 1) q^{12} + ( - 2 \beta - 2) q^{13} + q^{16} + ( - \beta - 3) q^{17} + ( - 2 \beta + 1) q^{18} + ( - 2 \beta - 4) q^{19} + (2 \beta + 2) q^{22} + ( - 2 \beta - 2) q^{23} + (\beta - 1) q^{24} - 5 q^{25} + ( - 2 \beta - 2) q^{26} - 4 q^{27} - 4 q^{29} + (\beta + 1) q^{31} + q^{32} + 4 q^{33} + ( - \beta - 3) q^{34} + ( - 2 \beta + 1) q^{36} - q^{37} + ( - 2 \beta - 4) q^{38} - 4 q^{39} + 2 \beta q^{41} - 2 \beta q^{43} + (2 \beta + 2) q^{44} + ( - 2 \beta - 2) q^{46} + ( - 4 \beta + 4) q^{47} + (\beta - 1) q^{48} - 5 q^{50} - 2 \beta q^{51} + ( - 2 \beta - 2) q^{52} + 2 \beta q^{53} - 4 q^{54} + ( - 2 \beta - 2) q^{57} - 4 q^{58} + ( - 4 \beta - 2) q^{59} + (2 \beta + 10) q^{61} + (\beta + 1) q^{62} + q^{64} + 4 q^{66} + (4 \beta + 4) q^{67} + ( - \beta - 3) q^{68} - 4 q^{69} + ( - 4 \beta - 2) q^{71} + ( - 2 \beta + 1) q^{72} + (4 \beta - 6) q^{73} - q^{74} + ( - 5 \beta + 5) q^{75} + ( - 2 \beta - 4) q^{76} - 4 q^{78} + ( - 2 \beta + 2) q^{79} + (2 \beta + 1) q^{81} + 2 \beta q^{82} + (5 \beta - 1) q^{83} - 2 \beta q^{86} + ( - 4 \beta + 4) q^{87} + (2 \beta + 2) q^{88} + (7 \beta + 1) q^{89} + ( - 2 \beta - 2) q^{92} + 2 q^{93} + ( - 4 \beta + 4) q^{94} + (\beta - 1) q^{96} + (7 \beta - 3) q^{97} + ( - 2 \beta - 10) q^{99} +O(q^{100})$$ q + q^2 + (b - 1) * q^3 + q^4 + (b - 1) * q^6 + q^8 + (-2*b + 1) * q^9 + (2*b + 2) * q^11 + (b - 1) * q^12 + (-2*b - 2) * q^13 + q^16 + (-b - 3) * q^17 + (-2*b + 1) * q^18 + (-2*b - 4) * q^19 + (2*b + 2) * q^22 + (-2*b - 2) * q^23 + (b - 1) * q^24 - 5 * q^25 + (-2*b - 2) * q^26 - 4 * q^27 - 4 * q^29 + (b + 1) * q^31 + q^32 + 4 * q^33 + (-b - 3) * q^34 + (-2*b + 1) * q^36 - q^37 + (-2*b - 4) * q^38 - 4 * q^39 + 2*b * q^41 - 2*b * q^43 + (2*b + 2) * q^44 + (-2*b - 2) * q^46 + (-4*b + 4) * q^47 + (b - 1) * q^48 - 5 * q^50 - 2*b * q^51 + (-2*b - 2) * q^52 + 2*b * q^53 - 4 * q^54 + (-2*b - 2) * q^57 - 4 * q^58 + (-4*b - 2) * q^59 + (2*b + 10) * q^61 + (b + 1) * q^62 + q^64 + 4 * q^66 + (4*b + 4) * q^67 + (-b - 3) * q^68 - 4 * q^69 + (-4*b - 2) * q^71 + (-2*b + 1) * q^72 + (4*b - 6) * q^73 - q^74 + (-5*b + 5) * q^75 + (-2*b - 4) * q^76 - 4 * q^78 + (-2*b + 2) * q^79 + (2*b + 1) * q^81 + 2*b * q^82 + (5*b - 1) * q^83 - 2*b * q^86 + (-4*b + 4) * q^87 + (2*b + 2) * q^88 + (7*b + 1) * q^89 + (-2*b - 2) * q^92 + 2 * q^93 + (-4*b + 4) * q^94 + (b - 1) * q^96 + (7*b - 3) * q^97 + (-2*b - 10) * 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 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 4 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 8 q^{19} + 4 q^{22} - 4 q^{23} - 2 q^{24} - 10 q^{25} - 4 q^{26} - 8 q^{27} - 8 q^{29} + 2 q^{31} + 2 q^{32} + 8 q^{33} - 6 q^{34} + 2 q^{36} - 2 q^{37} - 8 q^{38} - 8 q^{39} + 4 q^{44} - 4 q^{46} + 8 q^{47} - 2 q^{48} - 10 q^{50} - 4 q^{52} - 8 q^{54} - 4 q^{57} - 8 q^{58} - 4 q^{59} + 20 q^{61} + 2 q^{62} + 2 q^{64} + 8 q^{66} + 8 q^{67} - 6 q^{68} - 8 q^{69} - 4 q^{71} + 2 q^{72} - 12 q^{73} - 2 q^{74} + 10 q^{75} - 8 q^{76} - 8 q^{78} + 4 q^{79} + 2 q^{81} - 2 q^{83} + 8 q^{87} + 4 q^{88} + 2 q^{89} - 4 q^{92} + 4 q^{93} + 8 q^{94} - 2 q^{96} - 6 q^{97} - 20 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9 + 4 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 6 * q^17 + 2 * q^18 - 8 * q^19 + 4 * q^22 - 4 * q^23 - 2 * q^24 - 10 * q^25 - 4 * q^26 - 8 * q^27 - 8 * q^29 + 2 * q^31 + 2 * q^32 + 8 * q^33 - 6 * q^34 + 2 * q^36 - 2 * q^37 - 8 * q^38 - 8 * q^39 + 4 * q^44 - 4 * q^46 + 8 * q^47 - 2 * q^48 - 10 * q^50 - 4 * q^52 - 8 * q^54 - 4 * q^57 - 8 * q^58 - 4 * q^59 + 20 * q^61 + 2 * q^62 + 2 * q^64 + 8 * q^66 + 8 * q^67 - 6 * q^68 - 8 * q^69 - 4 * q^71 + 2 * q^72 - 12 * q^73 - 2 * q^74 + 10 * q^75 - 8 * q^76 - 8 * q^78 + 4 * q^79 + 2 * q^81 - 2 * q^83 + 8 * q^87 + 4 * q^88 + 2 * q^89 - 4 * q^92 + 4 * q^93 + 8 * q^94 - 2 * q^96 - 6 * q^97 - 20 * 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
 −1.73205 1.73205
1.00000 −2.73205 1.00000 0 −2.73205 0 1.00000 4.46410 0
1.2 1.00000 0.732051 1.00000 0 0.732051 0 1.00000 −2.46410 0
 $$n$$: 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.l 2
7.b odd 2 1 518.2.a.c 2
21.c even 2 1 4662.2.a.r 2
28.d even 2 1 4144.2.a.g 2

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

 $$T_{3}^{2} + 2T_{3} - 2$$ T3^2 + 2*T3 - 2 $$T_{5}$$ T5 $$T_{11}^{2} - 4T_{11} - 8$$ T11^2 - 4*T11 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 2T - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T - 8$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$T^{2} + 6T + 6$$
$19$ $$T^{2} + 8T + 4$$
$23$ $$T^{2} + 4T - 8$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} - 2T - 2$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 12$$
$43$ $$T^{2} - 12$$
$47$ $$T^{2} - 8T - 32$$
$53$ $$T^{2} - 12$$
$59$ $$T^{2} + 4T - 44$$
$61$ $$T^{2} - 20T + 88$$
$67$ $$T^{2} - 8T - 32$$
$71$ $$T^{2} + 4T - 44$$
$73$ $$T^{2} + 12T - 12$$
$79$ $$T^{2} - 4T - 8$$
$83$ $$T^{2} + 2T - 74$$
$89$ $$T^{2} - 2T - 146$$
$97$ $$T^{2} + 6T - 138$$