# Properties

 Label 3626.2.a.i Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$\Q(\sqrt{21})$$ 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: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} - q^{5} - \beta q^{6} - q^{8} + (\beta + 2) q^{9} +O(q^{10})$$ q - q^2 + b * q^3 + q^4 - q^5 - b * q^6 - q^8 + (b + 2) * q^9 $$q - q^{2} + \beta q^{3} + q^{4} - q^{5} - \beta q^{6} - q^{8} + (\beta + 2) q^{9} + q^{10} + (\beta + 3) q^{11} + \beta q^{12} + ( - 2 \beta + 1) q^{13} - \beta q^{15} + q^{16} + ( - \beta + 2) q^{17} + ( - \beta - 2) q^{18} + (\beta + 3) q^{19} - q^{20} + ( - \beta - 3) q^{22} + ( - 2 \beta - 2) q^{23} - \beta q^{24} - 4 q^{25} + (2 \beta - 1) q^{26} + 5 q^{27} + \beta q^{30} + (2 \beta + 3) q^{31} - q^{32} + (4 \beta + 5) q^{33} + (\beta - 2) q^{34} + (\beta + 2) q^{36} + q^{37} + ( - \beta - 3) q^{38} + ( - \beta - 10) q^{39} + q^{40} + (\beta + 4) q^{41} + (2 \beta + 3) q^{43} + (\beta + 3) q^{44} + ( - \beta - 2) q^{45} + (2 \beta + 2) q^{46} + (\beta + 3) q^{47} + \beta q^{48} + 4 q^{50} + (\beta - 5) q^{51} + ( - 2 \beta + 1) q^{52} + (5 \beta - 4) q^{53} - 5 q^{54} + ( - \beta - 3) q^{55} + (4 \beta + 5) q^{57} + (\beta + 6) q^{59} - \beta q^{60} - 2 q^{61} + ( - 2 \beta - 3) q^{62} + q^{64} + (2 \beta - 1) q^{65} + ( - 4 \beta - 5) q^{66} + (2 \beta + 5) q^{67} + ( - \beta + 2) q^{68} + ( - 4 \beta - 10) q^{69} + (\beta - 11) q^{71} + ( - \beta - 2) q^{72} + (2 \beta + 6) q^{73} - q^{74} - 4 \beta q^{75} + (\beta + 3) q^{76} + (\beta + 10) q^{78} + ( - 4 \beta + 5) q^{79} - q^{80} + (2 \beta - 6) q^{81} + ( - \beta - 4) q^{82} + (2 \beta - 11) q^{83} + (\beta - 2) q^{85} + ( - 2 \beta - 3) q^{86} + ( - \beta - 3) q^{88} + 8 q^{89} + (\beta + 2) q^{90} + ( - 2 \beta - 2) q^{92} + (5 \beta + 10) q^{93} + ( - \beta - 3) q^{94} + ( - \beta - 3) q^{95} - \beta q^{96} + ( - 2 \beta - 5) q^{97} + (6 \beta + 11) q^{99} +O(q^{100})$$ q - q^2 + b * q^3 + q^4 - q^5 - b * q^6 - q^8 + (b + 2) * q^9 + q^10 + (b + 3) * q^11 + b * q^12 + (-2*b + 1) * q^13 - b * q^15 + q^16 + (-b + 2) * q^17 + (-b - 2) * q^18 + (b + 3) * q^19 - q^20 + (-b - 3) * q^22 + (-2*b - 2) * q^23 - b * q^24 - 4 * q^25 + (2*b - 1) * q^26 + 5 * q^27 + b * q^30 + (2*b + 3) * q^31 - q^32 + (4*b + 5) * q^33 + (b - 2) * q^34 + (b + 2) * q^36 + q^37 + (-b - 3) * q^38 + (-b - 10) * q^39 + q^40 + (b + 4) * q^41 + (2*b + 3) * q^43 + (b + 3) * q^44 + (-b - 2) * q^45 + (2*b + 2) * q^46 + (b + 3) * q^47 + b * q^48 + 4 * q^50 + (b - 5) * q^51 + (-2*b + 1) * q^52 + (5*b - 4) * q^53 - 5 * q^54 + (-b - 3) * q^55 + (4*b + 5) * q^57 + (b + 6) * q^59 - b * q^60 - 2 * q^61 + (-2*b - 3) * q^62 + q^64 + (2*b - 1) * q^65 + (-4*b - 5) * q^66 + (2*b + 5) * q^67 + (-b + 2) * q^68 + (-4*b - 10) * q^69 + (b - 11) * q^71 + (-b - 2) * q^72 + (2*b + 6) * q^73 - q^74 - 4*b * q^75 + (b + 3) * q^76 + (b + 10) * q^78 + (-4*b + 5) * q^79 - q^80 + (2*b - 6) * q^81 + (-b - 4) * q^82 + (2*b - 11) * q^83 + (b - 2) * q^85 + (-2*b - 3) * q^86 + (-b - 3) * q^88 + 8 * q^89 + (b + 2) * q^90 + (-2*b - 2) * q^92 + (5*b + 10) * q^93 + (-b - 3) * q^94 + (-b - 3) * q^95 - b * q^96 + (-2*b - 5) * q^97 + (6*b + 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} - 2 q^{8} + 5 q^{9} + 2 q^{10} + 7 q^{11} + q^{12} - q^{15} + 2 q^{16} + 3 q^{17} - 5 q^{18} + 7 q^{19} - 2 q^{20} - 7 q^{22} - 6 q^{23} - q^{24} - 8 q^{25} + 10 q^{27} + q^{30} + 8 q^{31} - 2 q^{32} + 14 q^{33} - 3 q^{34} + 5 q^{36} + 2 q^{37} - 7 q^{38} - 21 q^{39} + 2 q^{40} + 9 q^{41} + 8 q^{43} + 7 q^{44} - 5 q^{45} + 6 q^{46} + 7 q^{47} + q^{48} + 8 q^{50} - 9 q^{51} - 3 q^{53} - 10 q^{54} - 7 q^{55} + 14 q^{57} + 13 q^{59} - q^{60} - 4 q^{61} - 8 q^{62} + 2 q^{64} - 14 q^{66} + 12 q^{67} + 3 q^{68} - 24 q^{69} - 21 q^{71} - 5 q^{72} + 14 q^{73} - 2 q^{74} - 4 q^{75} + 7 q^{76} + 21 q^{78} + 6 q^{79} - 2 q^{80} - 10 q^{81} - 9 q^{82} - 20 q^{83} - 3 q^{85} - 8 q^{86} - 7 q^{88} + 16 q^{89} + 5 q^{90} - 6 q^{92} + 25 q^{93} - 7 q^{94} - 7 q^{95} - q^{96} - 12 q^{97} + 28 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 - 2 * q^8 + 5 * q^9 + 2 * q^10 + 7 * q^11 + q^12 - q^15 + 2 * q^16 + 3 * q^17 - 5 * q^18 + 7 * q^19 - 2 * q^20 - 7 * q^22 - 6 * q^23 - q^24 - 8 * q^25 + 10 * q^27 + q^30 + 8 * q^31 - 2 * q^32 + 14 * q^33 - 3 * q^34 + 5 * q^36 + 2 * q^37 - 7 * q^38 - 21 * q^39 + 2 * q^40 + 9 * q^41 + 8 * q^43 + 7 * q^44 - 5 * q^45 + 6 * q^46 + 7 * q^47 + q^48 + 8 * q^50 - 9 * q^51 - 3 * q^53 - 10 * q^54 - 7 * q^55 + 14 * q^57 + 13 * q^59 - q^60 - 4 * q^61 - 8 * q^62 + 2 * q^64 - 14 * q^66 + 12 * q^67 + 3 * q^68 - 24 * q^69 - 21 * q^71 - 5 * q^72 + 14 * q^73 - 2 * q^74 - 4 * q^75 + 7 * q^76 + 21 * q^78 + 6 * q^79 - 2 * q^80 - 10 * q^81 - 9 * q^82 - 20 * q^83 - 3 * q^85 - 8 * q^86 - 7 * q^88 + 16 * q^89 + 5 * q^90 - 6 * q^92 + 25 * q^93 - 7 * q^94 - 7 * q^95 - q^96 - 12 * q^97 + 28 * 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 −1.79129 1.00000 −1.00000 1.79129 0 −1.00000 0.208712 1.00000
1.2 −1.00000 2.79129 1.00000 −1.00000 −2.79129 0 −1.00000 4.79129 1.00000
 $$n$$: 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.i 2
7.b odd 2 1 3626.2.a.c 2
7.c even 3 2 518.2.e.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.d 4 7.c even 3 2
3626.2.a.c 2 7.b odd 2 1
3626.2.a.i 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} - 5$$ T3^2 - T3 - 5 $$T_{5} + 1$$ T5 + 1 $$T_{11}^{2} - 7T_{11} + 7$$ T11^2 - 7*T11 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - T - 5$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 7T + 7$$
$13$ $$T^{2} - 21$$
$17$ $$T^{2} - 3T - 3$$
$19$ $$T^{2} - 7T + 7$$
$23$ $$T^{2} + 6T - 12$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 8T - 5$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 9T + 15$$
$43$ $$T^{2} - 8T - 5$$
$47$ $$T^{2} - 7T + 7$$
$53$ $$T^{2} + 3T - 129$$
$59$ $$T^{2} - 13T + 37$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} - 12T + 15$$
$71$ $$T^{2} + 21T + 105$$
$73$ $$T^{2} - 14T + 28$$
$79$ $$T^{2} - 6T - 75$$
$83$ $$T^{2} + 20T + 79$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 12T + 15$$
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