# Properties

 Label 3626.2.a.b 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: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + (b + 1) * q^5 + q^6 - q^8 - 2 * q^9 $$q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{8} - 2 q^{9} + ( - \beta - 1) q^{10} + ( - \beta - 1) q^{11} - q^{12} + q^{13} + ( - \beta - 1) q^{15} + q^{16} + (2 \beta - 1) q^{17} + 2 q^{18} + ( - 3 \beta + 1) q^{19} + (\beta + 1) q^{20} + (\beta + 1) q^{22} + ( - 2 \beta + 1) q^{23} + q^{24} + (3 \beta + 1) q^{25} - q^{26} + 5 q^{27} + (\beta + 1) q^{30} + (3 \beta - 5) q^{31} - q^{32} + (\beta + 1) q^{33} + ( - 2 \beta + 1) q^{34} - 2 q^{36} + q^{37} + (3 \beta - 1) q^{38} - q^{39} + ( - \beta - 1) q^{40} + (2 \beta + 5) q^{41} + ( - 3 \beta + 5) q^{43} + ( - \beta - 1) q^{44} + ( - 2 \beta - 2) q^{45} + (2 \beta - 1) q^{46} + ( - \beta - 4) q^{47} - q^{48} + ( - 3 \beta - 1) q^{50} + ( - 2 \beta + 1) q^{51} + q^{52} + ( - 5 \beta + 1) q^{53} - 5 q^{54} + ( - 3 \beta - 6) q^{55} + (3 \beta - 1) q^{57} + ( - 4 \beta + 8) q^{59} + ( - \beta - 1) q^{60} + (3 \beta - 8) q^{61} + ( - 3 \beta + 5) q^{62} + q^{64} + (\beta + 1) q^{65} + ( - \beta - 1) q^{66} - 13 q^{67} + (2 \beta - 1) q^{68} + (2 \beta - 1) q^{69} + ( - 2 \beta + 10) q^{71} + 2 q^{72} + ( - 3 \beta - 8) q^{73} - q^{74} + ( - 3 \beta - 1) q^{75} + ( - 3 \beta + 1) q^{76} + q^{78} + (3 \beta - 4) q^{79} + (\beta + 1) q^{80} + q^{81} + ( - 2 \beta - 5) q^{82} + (\beta - 11) q^{83} + (3 \beta + 9) q^{85} + (3 \beta - 5) q^{86} + (\beta + 1) q^{88} + ( - 3 \beta - 3) q^{89} + (2 \beta + 2) q^{90} + ( - 2 \beta + 1) q^{92} + ( - 3 \beta + 5) q^{93} + (\beta + 4) q^{94} + ( - 5 \beta - 14) q^{95} + q^{96} + (3 \beta - 8) q^{97} + (2 \beta + 2) q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + (b + 1) * q^5 + q^6 - q^8 - 2 * q^9 + (-b - 1) * q^10 + (-b - 1) * q^11 - q^12 + q^13 + (-b - 1) * q^15 + q^16 + (2*b - 1) * q^17 + 2 * q^18 + (-3*b + 1) * q^19 + (b + 1) * q^20 + (b + 1) * q^22 + (-2*b + 1) * q^23 + q^24 + (3*b + 1) * q^25 - q^26 + 5 * q^27 + (b + 1) * q^30 + (3*b - 5) * q^31 - q^32 + (b + 1) * q^33 + (-2*b + 1) * q^34 - 2 * q^36 + q^37 + (3*b - 1) * q^38 - q^39 + (-b - 1) * q^40 + (2*b + 5) * q^41 + (-3*b + 5) * q^43 + (-b - 1) * q^44 + (-2*b - 2) * q^45 + (2*b - 1) * q^46 + (-b - 4) * q^47 - q^48 + (-3*b - 1) * q^50 + (-2*b + 1) * q^51 + q^52 + (-5*b + 1) * q^53 - 5 * q^54 + (-3*b - 6) * q^55 + (3*b - 1) * q^57 + (-4*b + 8) * q^59 + (-b - 1) * q^60 + (3*b - 8) * q^61 + (-3*b + 5) * q^62 + q^64 + (b + 1) * q^65 + (-b - 1) * q^66 - 13 * q^67 + (2*b - 1) * q^68 + (2*b - 1) * q^69 + (-2*b + 10) * q^71 + 2 * q^72 + (-3*b - 8) * q^73 - q^74 + (-3*b - 1) * q^75 + (-3*b + 1) * q^76 + q^78 + (3*b - 4) * q^79 + (b + 1) * q^80 + q^81 + (-2*b - 5) * q^82 + (b - 11) * q^83 + (3*b + 9) * q^85 + (3*b - 5) * q^86 + (b + 1) * q^88 + (-3*b - 3) * q^89 + (2*b + 2) * q^90 + (-2*b + 1) * q^92 + (-3*b + 5) * q^93 + (b + 4) * q^94 + (-5*b - 14) * q^95 + q^96 + (3*b - 8) * q^97 + (2*b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^8 - 4 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{8} - 4 q^{9} - 3 q^{10} - 3 q^{11} - 2 q^{12} + 2 q^{13} - 3 q^{15} + 2 q^{16} + 4 q^{18} - q^{19} + 3 q^{20} + 3 q^{22} + 2 q^{24} + 5 q^{25} - 2 q^{26} + 10 q^{27} + 3 q^{30} - 7 q^{31} - 2 q^{32} + 3 q^{33} - 4 q^{36} + 2 q^{37} + q^{38} - 2 q^{39} - 3 q^{40} + 12 q^{41} + 7 q^{43} - 3 q^{44} - 6 q^{45} - 9 q^{47} - 2 q^{48} - 5 q^{50} + 2 q^{52} - 3 q^{53} - 10 q^{54} - 15 q^{55} + q^{57} + 12 q^{59} - 3 q^{60} - 13 q^{61} + 7 q^{62} + 2 q^{64} + 3 q^{65} - 3 q^{66} - 26 q^{67} + 18 q^{71} + 4 q^{72} - 19 q^{73} - 2 q^{74} - 5 q^{75} - q^{76} + 2 q^{78} - 5 q^{79} + 3 q^{80} + 2 q^{81} - 12 q^{82} - 21 q^{83} + 21 q^{85} - 7 q^{86} + 3 q^{88} - 9 q^{89} + 6 q^{90} + 7 q^{93} + 9 q^{94} - 33 q^{95} + 2 q^{96} - 13 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^8 - 4 * q^9 - 3 * q^10 - 3 * q^11 - 2 * q^12 + 2 * q^13 - 3 * q^15 + 2 * q^16 + 4 * q^18 - q^19 + 3 * q^20 + 3 * q^22 + 2 * q^24 + 5 * q^25 - 2 * q^26 + 10 * q^27 + 3 * q^30 - 7 * q^31 - 2 * q^32 + 3 * q^33 - 4 * q^36 + 2 * q^37 + q^38 - 2 * q^39 - 3 * q^40 + 12 * q^41 + 7 * q^43 - 3 * q^44 - 6 * q^45 - 9 * q^47 - 2 * q^48 - 5 * q^50 + 2 * q^52 - 3 * q^53 - 10 * q^54 - 15 * q^55 + q^57 + 12 * q^59 - 3 * q^60 - 13 * q^61 + 7 * q^62 + 2 * q^64 + 3 * q^65 - 3 * q^66 - 26 * q^67 + 18 * q^71 + 4 * q^72 - 19 * q^73 - 2 * q^74 - 5 * q^75 - q^76 + 2 * q^78 - 5 * q^79 + 3 * q^80 + 2 * q^81 - 12 * q^82 - 21 * q^83 + 21 * q^85 - 7 * q^86 + 3 * q^88 - 9 * q^89 + 6 * q^90 + 7 * q^93 + 9 * q^94 - 33 * q^95 + 2 * q^96 - 13 * q^97 + 6 * 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
 −1.79129 2.79129
−1.00000 −1.00000 1.00000 −0.791288 1.00000 0 −1.00000 −2.00000 0.791288
1.2 −1.00000 −1.00000 1.00000 3.79129 1.00000 0 −1.00000 −2.00000 −3.79129
 $$n$$: 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.b 2
7.b odd 2 1 3626.2.a.j 2
7.d odd 6 2 518.2.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.c 4 7.d odd 6 2
3626.2.a.b 2 1.a even 1 1 trivial
3626.2.a.j 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{3} + 1$$ T3 + 1 $$T_{5}^{2} - 3T_{5} - 3$$ T5^2 - 3*T5 - 3 $$T_{11}^{2} + 3T_{11} - 3$$ T11^2 + 3*T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 3T - 3$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T - 3$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 21$$
$19$ $$T^{2} + T - 47$$
$23$ $$T^{2} - 21$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 7T - 35$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 12T + 15$$
$43$ $$T^{2} - 7T - 35$$
$47$ $$T^{2} + 9T + 15$$
$53$ $$T^{2} + 3T - 129$$
$59$ $$T^{2} - 12T - 48$$
$61$ $$T^{2} + 13T - 5$$
$67$ $$(T + 13)^{2}$$
$71$ $$T^{2} - 18T + 60$$
$73$ $$T^{2} + 19T + 43$$
$79$ $$T^{2} + 5T - 41$$
$83$ $$T^{2} + 21T + 105$$
$89$ $$T^{2} + 9T - 27$$
$97$ $$T^{2} + 13T - 5$$