# Properties

 Label 3626.2.a.bd Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.24635632.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 10x^{4} + 13x^{3} + 10x^{2} - 13x - 1$$ x^6 - x^5 - 10*x^4 + 13*x^3 + 10*x^2 - 13*x - 1 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{2} + 1) q^{6} - q^{8} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+O(q^{10})$$ q - q^2 + (-b2 - 1) * q^3 + q^4 + (b5 + b3) * q^5 + (b2 + 1) * q^6 - q^8 + (-b5 + 2*b4 + b3 + 2*b2 + b1 + 1) * q^9 $$q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{2} + 1) q^{6} - q^{8} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + (\beta_{4} - 3 \beta_{2} + 3 \beta_1 - 8) q^{99}+O(q^{100})$$ q - q^2 + (-b2 - 1) * q^3 + q^4 + (b5 + b3) * q^5 + (b2 + 1) * q^6 - q^8 + (-b5 + 2*b4 + b3 + 2*b2 + b1 + 1) * q^9 + (-b5 - b3) * q^10 + (-b4 - b3 - b2 - 2*b1) * q^11 + (-b2 - 1) * q^12 + (b4 - b3 - b2 + b1 - 2) * q^13 + (-b5 + b4 - 2*b3 + 2*b2 - b1 + 1) * q^15 + q^16 + (b2 + 1) * q^17 + (b5 - 2*b4 - b3 - 2*b2 - b1 - 1) * q^18 + (b5 - b4 + b3 + b1 - 2) * q^19 + (b5 + b3) * q^20 + (b4 + b3 + b2 + 2*b1) * q^22 + (-b5 - b4 - 1) * q^23 + (b2 + 1) * q^24 + (-2*b4 + b2 - b1 + 5) * q^25 + (-b4 + b3 + b2 - b1 + 2) * q^26 + (b5 - 3*b4 + b3 - 4*b2 - 2) * q^27 + (-2*b4 - 2*b2 + 2*b1 - 2) * q^29 + (b5 - b4 + 2*b3 - 2*b2 + b1 - 1) * q^30 + (-2*b4 - b3 - b2 - b1 - 4) * q^31 - q^32 + (b4 + 2*b3 + 2*b2 + 2*b1 + 3) * q^33 + (-b2 - 1) * q^34 + (-b5 + 2*b4 + b3 + 2*b2 + b1 + 1) * q^36 - q^37 + (-b5 + b4 - b3 - b1 + 2) * q^38 + (b4 + b3 + 2*b2 + b1 + 4) * q^39 + (-b5 - b3) * q^40 + (-b5 + b4 - b3 + b2 + 2*b1 + 2) * q^41 + (-b4 + 2*b2 - 2*b1 - 1) * q^43 + (-b4 - b3 - b2 - 2*b1) * q^44 + (b5 - 6*b4 - b3 - 6*b2 + b1 - 7) * q^45 + (b5 + b4 + 1) * q^46 + (-3*b3 + 2*b2 - 3*b1 + 3) * q^47 + (-b2 - 1) * q^48 + (2*b4 - b2 + b1 - 5) * q^50 + (b5 - 2*b4 - b3 - 2*b2 - b1 - 4) * q^51 + (b4 - b3 - b2 + b1 - 2) * q^52 + (-b5 + 3*b4 + 2*b2 + 5*b1 + 3) * q^53 + (-b5 + 3*b4 - b3 + 4*b2 + 2) * q^54 + (b4 - 2*b3 + b2 - 6*b1) * q^55 + (-b5 + b4 - 4*b3 + 5*b2 - 3*b1 + 1) * q^57 + (2*b4 + 2*b2 - 2*b1 + 2) * q^58 + (-b5 + 4*b4 - b3 + 4*b2 - b1 - 3) * q^59 + (-b5 + b4 - 2*b3 + 2*b2 - b1 + 1) * q^60 + (-b5 + b4 + 3*b2 - b1 - 2) * q^61 + (2*b4 + b3 + b2 + b1 + 4) * q^62 + q^64 + (-2*b5 + 3*b4 - 4*b3 + 2*b1 - 3) * q^65 + (-b4 - 2*b3 - 2*b2 - 2*b1 - 3) * q^66 + (-b5 - b4 + 3*b3 + 2*b1 + 4) * q^67 + (b2 + 1) * q^68 + b3 * q^69 + (-b5 - 3*b4 - b3 - 3*b2 - 3) * q^71 + (b5 - 2*b4 - b3 - 2*b2 - b1 - 1) * q^72 + (3*b4 + 4*b3 + 2*b2) * q^73 + q^74 + (b5 - 2*b4 - 2*b3 - 4*b2 - 2*b1 - 9) * q^75 + (b5 - b4 + b3 + b1 - 2) * q^76 + (-b4 - b3 - 2*b2 - b1 - 4) * q^78 + (-b5 + b4 - 3*b3 - 3*b2 + b1 - 3) * q^79 + (b5 + b3) * q^80 + (-2*b5 + 3*b4 - 4*b3 + 5*b2 - 3*b1 + 9) * q^81 + (b5 - b4 + b3 - b2 - 2*b1 - 2) * q^82 + (-b5 - b4 - b2 + 3*b1 + 1) * q^83 + (b5 - b4 + 2*b3 - 2*b2 + b1 - 1) * q^85 + (b4 - 2*b2 + 2*b1 + 1) * q^86 + (-2*b5 + 4*b4 - 2*b3 + 6*b2 - 2*b1 + 4) * q^87 + (b4 + b3 + b2 + 2*b1) * q^88 + (-b5 - 2*b2 + 4*b1 - 1) * q^89 + (-b5 + 6*b4 + b3 + 6*b2 - b1 + 7) * q^90 + (-b5 - b4 - 1) * q^92 + (b4 + 7*b2 + 5) * q^93 + (3*b3 - 2*b2 + 3*b1 - 3) * q^94 + (-2*b5 + 2*b4 + 3*b3 + 4*b2 + 5*b1 + 9) * q^95 + (b2 + 1) * q^96 + (-2*b5 + b4 - b3 - 2*b2 - 5) * q^97 + (b4 - 3*b2 + 3*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} - 3 q^{3} + 6 q^{4} + 3 q^{6} - 6 q^{8} + 5 q^{9}+O(q^{10})$$ 6 * q - 6 * q^2 - 3 * q^3 + 6 * q^4 + 3 * q^6 - 6 * q^8 + 5 * q^9 $$6 q - 6 q^{2} - 3 q^{3} + 6 q^{4} + 3 q^{6} - 6 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} - 6 q^{13} + q^{15} + 6 q^{16} + 3 q^{17} - 5 q^{18} - 13 q^{19} + q^{22} - 8 q^{23} + 3 q^{24} + 22 q^{25} + 6 q^{26} - 6 q^{27} - 8 q^{29} - q^{30} - 26 q^{31} - 6 q^{32} + 16 q^{33} - 3 q^{34} + 5 q^{36} - 6 q^{37} + 13 q^{38} + 21 q^{39} + 13 q^{41} - 16 q^{43} - q^{44} - 35 q^{45} + 8 q^{46} + 9 q^{47} - 3 q^{48} - 22 q^{50} - 23 q^{51} - 6 q^{52} + 23 q^{53} + 6 q^{54} - 7 q^{55} - 10 q^{57} + 8 q^{58} - 23 q^{59} + q^{60} - 20 q^{61} + 26 q^{62} + 6 q^{64} - 10 q^{65} - 16 q^{66} + 24 q^{67} + 3 q^{68} - 15 q^{71} - 5 q^{72} + 6 q^{74} - 48 q^{75} - 13 q^{76} - 21 q^{78} - 6 q^{79} + 42 q^{81} - 13 q^{82} + 10 q^{83} - q^{85} + 16 q^{86} + 12 q^{87} + q^{88} + 4 q^{89} + 35 q^{90} - 8 q^{92} + 11 q^{93} - 9 q^{94} + 51 q^{95} + 3 q^{96} - 22 q^{97} - 34 q^{99}+O(q^{100})$$ 6 * q - 6 * q^2 - 3 * q^3 + 6 * q^4 + 3 * q^6 - 6 * q^8 + 5 * q^9 - q^11 - 3 * q^12 - 6 * q^13 + q^15 + 6 * q^16 + 3 * q^17 - 5 * q^18 - 13 * q^19 + q^22 - 8 * q^23 + 3 * q^24 + 22 * q^25 + 6 * q^26 - 6 * q^27 - 8 * q^29 - q^30 - 26 * q^31 - 6 * q^32 + 16 * q^33 - 3 * q^34 + 5 * q^36 - 6 * q^37 + 13 * q^38 + 21 * q^39 + 13 * q^41 - 16 * q^43 - q^44 - 35 * q^45 + 8 * q^46 + 9 * q^47 - 3 * q^48 - 22 * q^50 - 23 * q^51 - 6 * q^52 + 23 * q^53 + 6 * q^54 - 7 * q^55 - 10 * q^57 + 8 * q^58 - 23 * q^59 + q^60 - 20 * q^61 + 26 * q^62 + 6 * q^64 - 10 * q^65 - 16 * q^66 + 24 * q^67 + 3 * q^68 - 15 * q^71 - 5 * q^72 + 6 * q^74 - 48 * q^75 - 13 * q^76 - 21 * q^78 - 6 * q^79 + 42 * q^81 - 13 * q^82 + 10 * q^83 - q^85 + 16 * q^86 + 12 * q^87 + q^88 + 4 * q^89 + 35 * q^90 - 8 * q^92 + 11 * q^93 - 9 * q^94 + 51 * q^95 + 3 * q^96 - 22 * q^97 - 34 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 10x^{4} + 13x^{3} + 10x^{2} - 13x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 9\nu^{2} + 4\nu + 5 ) / 2$$ (v^4 - 9*v^2 + 4*v + 5) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 9\nu^{3} + 4\nu^{2} + 5\nu - 2 ) / 2$$ (v^5 - 9*v^3 + 4*v^2 + 5*v - 2) / 2 $$\beta_{4}$$ $$=$$ $$( -3\nu^{5} - \nu^{4} + 29\nu^{3} - \nu^{2} - 35\nu - 1 ) / 2$$ (-3*v^5 - v^4 + 29*v^3 - v^2 - 35*v - 1) / 2 $$\beta_{5}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} + 38\nu^{3} - 3\nu^{2} - 38\nu - 7 ) / 2$$ (-4*v^5 - v^4 + 38*v^3 - 3*v^2 - 38*v - 7) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 4$$ b5 - b4 + b3 - b1 + 4 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 9\beta _1 - 3$$ -b5 + 2*b4 + 2*b3 + b2 + 9*b1 - 3 $$\nu^{4}$$ $$=$$ $$9\beta_{5} - 9\beta_{4} + 9\beta_{3} + 2\beta_{2} - 13\beta _1 + 31$$ 9*b5 - 9*b4 + 9*b3 + 2*b2 - 13*b1 + 31 $$\nu^{5}$$ $$=$$ $$-13\beta_{5} + 22\beta_{4} + 16\beta_{3} + 9\beta_{2} + 80\beta _1 - 41$$ -13*b5 + 22*b4 + 16*b3 + 9*b2 + 80*b1 - 41

## 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
 −0.0732141 2.65756 1.19066 −3.10233 1.35453 −1.02721
−1.00000 −3.32946 1.00000 −3.29499 3.32946 0 −1.00000 8.08534 3.29499
1.2 −1.00000 −1.97359 1.00000 3.55591 1.97359 0 −1.00000 0.895047 −3.55591
1.3 −1.00000 −0.506692 1.00000 −3.55591 0.506692 0 −1.00000 −2.74326 3.55591
1.4 −1.00000 −0.300349 1.00000 3.29499 0.300349 0 −1.00000 −2.90979 −3.29499
1.5 −1.00000 0.364177 1.00000 1.58067 −0.364177 0 −1.00000 −2.86737 −1.58067
1.6 −1.00000 2.74591 1.00000 −1.58067 −2.74591 0 −1.00000 4.54005 1.58067
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.bd 6
7.b odd 2 1 3626.2.a.bf 6
7.c even 3 2 518.2.e.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.h 12 7.c even 3 2
3626.2.a.bd 6 1.a even 1 1 trivial
3626.2.a.bf 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{6} + 3T_{3}^{5} - 7T_{3}^{4} - 22T_{3}^{3} - 7T_{3}^{2} + 3T_{3} + 1$$ T3^6 + 3*T3^5 - 7*T3^4 - 22*T3^3 - 7*T3^2 + 3*T3 + 1 $$T_{5}^{6} - 26T_{5}^{4} + 196T_{5}^{2} - 343$$ T5^6 - 26*T5^4 + 196*T5^2 - 343 $$T_{11}^{6} + T_{11}^{5} - 32T_{11}^{4} - 11T_{11}^{3} + 110T_{11}^{2} + 31T_{11} - 47$$ T11^6 + T11^5 - 32*T11^4 - 11*T11^3 + 110*T11^2 + 31*T11 - 47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{6}$$
$3$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$5$ $$T^{6} - 26 T^{4} + \cdots - 343$$
$7$ $$T^{6}$$
$11$ $$T^{6} + T^{5} + \cdots - 47$$
$13$ $$T^{6} + 6 T^{5} + \cdots + 67$$
$17$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$19$ $$T^{6} + 13 T^{5} + \cdots - 2267$$
$23$ $$T^{6} + 8 T^{5} + \cdots - 28$$
$29$ $$(T^{3} + 4 T^{2} - 44 T - 64)^{2}$$
$31$ $$T^{6} + 26 T^{5} + \cdots - 4679$$
$37$ $$(T + 1)^{6}$$
$41$ $$T^{6} - 13 T^{5} + \cdots - 593$$
$43$ $$(T^{3} + 8 T^{2} - 14 T - 59)^{2}$$
$47$ $$T^{6} - 9 T^{5} + \cdots + 99397$$
$53$ $$T^{6} - 23 T^{5} + \cdots - 104713$$
$59$ $$T^{6} + 23 T^{5} + \cdots - 61904$$
$61$ $$T^{6} + 20 T^{5} + \cdots - 27292$$
$67$ $$T^{6} - 24 T^{5} + \cdots + 23969$$
$71$ $$T^{6} + 15 T^{5} + \cdots + 30272$$
$73$ $$T^{6} - 201 T^{4} + \cdots + 24668$$
$79$ $$T^{6} + 6 T^{5} + \cdots - 81961$$
$83$ $$T^{6} - 10 T^{5} + \cdots - 227$$
$89$ $$T^{6} - 4 T^{5} + \cdots + 38448$$
$97$ $$T^{6} + 22 T^{5} + \cdots - 17533$$