# Properties

 Label 3626.2.a.q Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ 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: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{5} + \beta q^{6} + q^{8} - q^{9}+O(q^{10})$$ q + q^2 + b * q^3 + q^4 + b * q^5 + b * q^6 + q^8 - q^9 $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{5} + \beta q^{6} + q^{8} - q^{9} + \beta q^{10} + 4 q^{11} + \beta q^{12} + 3 \beta q^{13} + 2 q^{15} + q^{16} - 2 \beta q^{17} - q^{18} + 2 \beta q^{19} + \beta q^{20} + 4 q^{22} + 8 q^{23} + \beta q^{24} - 3 q^{25} + 3 \beta q^{26} - 4 \beta q^{27} - 4 q^{29} + 2 q^{30} + 3 \beta q^{31} + q^{32} + 4 \beta q^{33} - 2 \beta q^{34} - q^{36} - q^{37} + 2 \beta q^{38} + 6 q^{39} + \beta q^{40} + \beta q^{41} - 4 q^{43} + 4 q^{44} - \beta q^{45} + 8 q^{46} - 8 \beta q^{47} + \beta q^{48} - 3 q^{50} - 4 q^{51} + 3 \beta q^{52} - 2 q^{53} - 4 \beta q^{54} + 4 \beta q^{55} + 4 q^{57} - 4 q^{58} - 8 \beta q^{59} + 2 q^{60} + \beta q^{61} + 3 \beta q^{62} + q^{64} + 6 q^{65} + 4 \beta q^{66} + 12 q^{67} - 2 \beta q^{68} + 8 \beta q^{69} - q^{72} + \beta q^{73} - q^{74} - 3 \beta q^{75} + 2 \beta q^{76} + 6 q^{78} + 14 q^{79} + \beta q^{80} - 5 q^{81} + \beta q^{82} + 11 \beta q^{83} - 4 q^{85} - 4 q^{86} - 4 \beta q^{87} + 4 q^{88} - 4 \beta q^{89} - \beta q^{90} + 8 q^{92} + 6 q^{93} - 8 \beta q^{94} + 4 q^{95} + \beta q^{96} - 4 \beta q^{97} - 4 q^{99} +O(q^{100})$$ q + q^2 + b * q^3 + q^4 + b * q^5 + b * q^6 + q^8 - q^9 + b * q^10 + 4 * q^11 + b * q^12 + 3*b * q^13 + 2 * q^15 + q^16 - 2*b * q^17 - q^18 + 2*b * q^19 + b * q^20 + 4 * q^22 + 8 * q^23 + b * q^24 - 3 * q^25 + 3*b * q^26 - 4*b * q^27 - 4 * q^29 + 2 * q^30 + 3*b * q^31 + q^32 + 4*b * q^33 - 2*b * q^34 - q^36 - q^37 + 2*b * q^38 + 6 * q^39 + b * q^40 + b * q^41 - 4 * q^43 + 4 * q^44 - b * q^45 + 8 * q^46 - 8*b * q^47 + b * q^48 - 3 * q^50 - 4 * q^51 + 3*b * q^52 - 2 * q^53 - 4*b * q^54 + 4*b * q^55 + 4 * q^57 - 4 * q^58 - 8*b * q^59 + 2 * q^60 + b * q^61 + 3*b * q^62 + q^64 + 6 * q^65 + 4*b * q^66 + 12 * q^67 - 2*b * q^68 + 8*b * q^69 - q^72 + b * q^73 - q^74 - 3*b * q^75 + 2*b * q^76 + 6 * q^78 + 14 * q^79 + b * q^80 - 5 * q^81 + b * q^82 + 11*b * q^83 - 4 * q^85 - 4 * q^86 - 4*b * q^87 + 4 * q^88 - 4*b * q^89 - b * q^90 + 8 * q^92 + 6 * q^93 - 8*b * q^94 + 4 * q^95 + b * q^96 - 4*b * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 8 q^{11} + 4 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{22} + 16 q^{23} - 6 q^{25} - 8 q^{29} + 4 q^{30} + 2 q^{32} - 2 q^{36} - 2 q^{37} + 12 q^{39} - 8 q^{43} + 8 q^{44} + 16 q^{46} - 6 q^{50} - 8 q^{51} - 4 q^{53} + 8 q^{57} - 8 q^{58} + 4 q^{60} + 2 q^{64} + 12 q^{65} + 24 q^{67} - 2 q^{72} - 2 q^{74} + 12 q^{78} + 28 q^{79} - 10 q^{81} - 8 q^{85} - 8 q^{86} + 8 q^{88} + 16 q^{92} + 12 q^{93} + 8 q^{95} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 8 * q^11 + 4 * q^15 + 2 * q^16 - 2 * q^18 + 8 * q^22 + 16 * q^23 - 6 * q^25 - 8 * q^29 + 4 * q^30 + 2 * q^32 - 2 * q^36 - 2 * q^37 + 12 * q^39 - 8 * q^43 + 8 * q^44 + 16 * q^46 - 6 * q^50 - 8 * q^51 - 4 * q^53 + 8 * q^57 - 8 * q^58 + 4 * q^60 + 2 * q^64 + 12 * q^65 + 24 * q^67 - 2 * q^72 - 2 * q^74 + 12 * q^78 + 28 * q^79 - 10 * q^81 - 8 * q^85 - 8 * q^86 + 8 * q^88 + 16 * q^92 + 12 * q^93 + 8 * q^95 - 8 * 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 −1.41421 1.00000 −1.41421 −1.41421 0 1.00000 −1.00000 −1.41421
1.2 1.00000 1.41421 1.00000 1.41421 1.41421 0 1.00000 −1.00000 1.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.q 2
7.b odd 2 1 inner 3626.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.q 2 1.a even 1 1 trivial
3626.2.a.q 2 7.b odd 2 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - 18$$
$17$ $$T^{2} - 8$$
$19$ $$T^{2} - 8$$
$23$ $$(T - 8)^{2}$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} - 18$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 2$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 128$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 128$$
$61$ $$T^{2} - 2$$
$67$ $$(T - 12)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} - 242$$
$89$ $$T^{2} - 32$$
$97$ $$T^{2} - 32$$