Properties

 Label 3626.2.a.k 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: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + ( - 2 \beta + 1) q^{5} + ( - \beta - 1) q^{6} + q^{8} + (3 \beta - 1) q^{9}+O(q^{10})$$ q + q^2 + (-b - 1) * q^3 + q^4 + (-2*b + 1) * q^5 + (-b - 1) * q^6 + q^8 + (3*b - 1) * q^9 $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + ( - 2 \beta + 1) q^{5} + ( - \beta - 1) q^{6} + q^{8} + (3 \beta - 1) q^{9} + ( - 2 \beta + 1) q^{10} + (\beta - 2) q^{11} + ( - \beta - 1) q^{12} + q^{13} + (3 \beta + 1) q^{15} + q^{16} + (\beta - 5) q^{17} + (3 \beta - 1) q^{18} + (3 \beta - 4) q^{19} + ( - 2 \beta + 1) q^{20} + (\beta - 2) q^{22} + ( - 2 \beta + 4) q^{23} + ( - \beta - 1) q^{24} + q^{26} + ( - 2 \beta + 1) q^{27} + (8 \beta - 4) q^{29} + (3 \beta + 1) q^{30} + (6 \beta - 5) q^{31} + q^{32} + q^{33} + (\beta - 5) q^{34} + (3 \beta - 1) q^{36} + q^{37} + (3 \beta - 4) q^{38} + ( - \beta - 1) q^{39} + ( - 2 \beta + 1) q^{40} + ( - 5 \beta + 1) q^{41} + ( - 6 \beta - 3) q^{43} + (\beta - 2) q^{44} + ( - \beta - 7) q^{45} + ( - 2 \beta + 4) q^{46} + (7 \beta - 2) q^{47} + ( - \beta - 1) q^{48} + (3 \beta + 4) q^{51} + q^{52} + (5 \beta - 1) q^{53} + ( - 2 \beta + 1) q^{54} + (3 \beta - 4) q^{55} + ( - 2 \beta + 1) q^{57} + (8 \beta - 4) q^{58} + ( - \beta - 1) q^{59} + (3 \beta + 1) q^{60} - 2 q^{61} + (6 \beta - 5) q^{62} + q^{64} + ( - 2 \beta + 1) q^{65} + q^{66} + 3 q^{67} + (\beta - 5) q^{68} - 2 q^{69} + ( - 5 \beta + 4) q^{71} + (3 \beta - 1) q^{72} + (6 \beta - 4) q^{73} + q^{74} + (3 \beta - 4) q^{76} + ( - \beta - 1) q^{78} + ( - 12 \beta + 7) q^{79} + ( - 2 \beta + 1) q^{80} + ( - 6 \beta + 4) q^{81} + ( - 5 \beta + 1) q^{82} + (4 \beta - 5) q^{83} + (9 \beta - 7) q^{85} + ( - 6 \beta - 3) q^{86} + ( - 12 \beta - 4) q^{87} + (\beta - 2) q^{88} - 12 q^{89} + ( - \beta - 7) q^{90} + ( - 2 \beta + 4) q^{92} + ( - 7 \beta - 1) q^{93} + (7 \beta - 2) q^{94} + (5 \beta - 10) q^{95} + ( - \beta - 1) q^{96} - 3 q^{97} + ( - 4 \beta + 5) q^{99}+O(q^{100})$$ q + q^2 + (-b - 1) * q^3 + q^4 + (-2*b + 1) * q^5 + (-b - 1) * q^6 + q^8 + (3*b - 1) * q^9 + (-2*b + 1) * q^10 + (b - 2) * q^11 + (-b - 1) * q^12 + q^13 + (3*b + 1) * q^15 + q^16 + (b - 5) * q^17 + (3*b - 1) * q^18 + (3*b - 4) * q^19 + (-2*b + 1) * q^20 + (b - 2) * q^22 + (-2*b + 4) * q^23 + (-b - 1) * q^24 + q^26 + (-2*b + 1) * q^27 + (8*b - 4) * q^29 + (3*b + 1) * q^30 + (6*b - 5) * q^31 + q^32 + q^33 + (b - 5) * q^34 + (3*b - 1) * q^36 + q^37 + (3*b - 4) * q^38 + (-b - 1) * q^39 + (-2*b + 1) * q^40 + (-5*b + 1) * q^41 + (-6*b - 3) * q^43 + (b - 2) * q^44 + (-b - 7) * q^45 + (-2*b + 4) * q^46 + (7*b - 2) * q^47 + (-b - 1) * q^48 + (3*b + 4) * q^51 + q^52 + (5*b - 1) * q^53 + (-2*b + 1) * q^54 + (3*b - 4) * q^55 + (-2*b + 1) * q^57 + (8*b - 4) * q^58 + (-b - 1) * q^59 + (3*b + 1) * q^60 - 2 * q^61 + (6*b - 5) * q^62 + q^64 + (-2*b + 1) * q^65 + q^66 + 3 * q^67 + (b - 5) * q^68 - 2 * q^69 + (-5*b + 4) * q^71 + (3*b - 1) * q^72 + (6*b - 4) * q^73 + q^74 + (3*b - 4) * q^76 + (-b - 1) * q^78 + (-12*b + 7) * q^79 + (-2*b + 1) * q^80 + (-6*b + 4) * q^81 + (-5*b + 1) * q^82 + (4*b - 5) * q^83 + (9*b - 7) * q^85 + (-6*b - 3) * q^86 + (-12*b - 4) * q^87 + (b - 2) * q^88 - 12 * q^89 + (-b - 7) * q^90 + (-2*b + 4) * q^92 + (-7*b - 1) * q^93 + (7*b - 2) * q^94 + (5*b - 10) * q^95 + (-b - 1) * q^96 - 3 * q^97 + (-4*b + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{8} + q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 + 2 * q^8 + q^9 $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} + 2 q^{8} + q^{9} - 3 q^{11} - 3 q^{12} + 2 q^{13} + 5 q^{15} + 2 q^{16} - 9 q^{17} + q^{18} - 5 q^{19} - 3 q^{22} + 6 q^{23} - 3 q^{24} + 2 q^{26} + 5 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{33} - 9 q^{34} + q^{36} + 2 q^{37} - 5 q^{38} - 3 q^{39} - 3 q^{41} - 12 q^{43} - 3 q^{44} - 15 q^{45} + 6 q^{46} + 3 q^{47} - 3 q^{48} + 11 q^{51} + 2 q^{52} + 3 q^{53} - 5 q^{55} - 3 q^{59} + 5 q^{60} - 4 q^{61} - 4 q^{62} + 2 q^{64} + 2 q^{66} + 6 q^{67} - 9 q^{68} - 4 q^{69} + 3 q^{71} + q^{72} - 2 q^{73} + 2 q^{74} - 5 q^{76} - 3 q^{78} + 2 q^{79} + 2 q^{81} - 3 q^{82} - 6 q^{83} - 5 q^{85} - 12 q^{86} - 20 q^{87} - 3 q^{88} - 24 q^{89} - 15 q^{90} + 6 q^{92} - 9 q^{93} + 3 q^{94} - 15 q^{95} - 3 q^{96} - 6 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 + 2 * q^8 + q^9 - 3 * q^11 - 3 * q^12 + 2 * q^13 + 5 * q^15 + 2 * q^16 - 9 * q^17 + q^18 - 5 * q^19 - 3 * q^22 + 6 * q^23 - 3 * q^24 + 2 * q^26 + 5 * q^30 - 4 * q^31 + 2 * q^32 + 2 * q^33 - 9 * q^34 + q^36 + 2 * q^37 - 5 * q^38 - 3 * q^39 - 3 * q^41 - 12 * q^43 - 3 * q^44 - 15 * q^45 + 6 * q^46 + 3 * q^47 - 3 * q^48 + 11 * q^51 + 2 * q^52 + 3 * q^53 - 5 * q^55 - 3 * q^59 + 5 * q^60 - 4 * q^61 - 4 * q^62 + 2 * q^64 + 2 * q^66 + 6 * q^67 - 9 * q^68 - 4 * q^69 + 3 * q^71 + q^72 - 2 * q^73 + 2 * q^74 - 5 * q^76 - 3 * q^78 + 2 * q^79 + 2 * q^81 - 3 * q^82 - 6 * q^83 - 5 * q^85 - 12 * q^86 - 20 * q^87 - 3 * q^88 - 24 * q^89 - 15 * q^90 + 6 * q^92 - 9 * q^93 + 3 * q^94 - 15 * q^95 - 3 * q^96 - 6 * 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
 1.61803 −0.618034
1.00000 −2.61803 1.00000 −2.23607 −2.61803 0 1.00000 3.85410 −2.23607
1.2 1.00000 −0.381966 1.00000 2.23607 −0.381966 0 1.00000 −2.85410 2.23607
 $$n$$: 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.k 2
7.b odd 2 1 3626.2.a.t 2
7.c even 3 2 518.2.e.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.b 4 7.c even 3 2
3626.2.a.k 2 1.a even 1 1 trivial
3626.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$T^{2} - 5$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 1$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 9T + 19$$
$19$ $$T^{2} + 5T - 5$$
$23$ $$T^{2} - 6T + 4$$
$29$ $$T^{2} - 80$$
$31$ $$T^{2} + 4T - 41$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 3T - 29$$
$43$ $$T^{2} + 12T - 9$$
$47$ $$T^{2} - 3T - 59$$
$53$ $$T^{2} - 3T - 29$$
$59$ $$T^{2} + 3T + 1$$
$61$ $$(T + 2)^{2}$$
$67$ $$(T - 3)^{2}$$
$71$ $$T^{2} - 3T - 29$$
$73$ $$T^{2} + 2T - 44$$
$79$ $$T^{2} - 2T - 179$$
$83$ $$T^{2} + 6T - 11$$
$89$ $$(T + 12)^{2}$$
$97$ $$(T + 3)^{2}$$