# Properties

 Label 3626.2.a.z Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.2624.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ x^4 - 2*x^3 - 3*x^2 + 2*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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + (\beta_1 - 2) q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{6} + q^{8} + 2 \beta_1 q^{9}+O(q^{10})$$ q + q^2 + (-b2 - b1 + 1) * q^3 + q^4 + (b1 - 2) * q^5 + (-b2 - b1 + 1) * q^6 + q^8 + 2*b1 * q^9 $$q + q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + (\beta_1 - 2) q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{6} + q^{8} + 2 \beta_1 q^{9} + (\beta_1 - 2) q^{10} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{12} + ( - \beta_{3} - 3) q^{13} + ( - \beta_{3} + \beta_{2} - 2) q^{15} + q^{16} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{17} + 2 \beta_1 q^{18} + (2 \beta_{3} + \beta_{2} - 2) q^{19} + (\beta_1 - 2) q^{20} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{22} + (3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{23} + ( - \beta_{2} - \beta_1 + 1) q^{24} + (\beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{3} - 3) q^{26} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{27} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{29} + ( - \beta_{3} + \beta_{2} - 2) q^{30} + (3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{31} + q^{32} + (4 \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{33} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{34} + 2 \beta_1 q^{36} + q^{37} + (2 \beta_{3} + \beta_{2} - 2) q^{38} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 3) q^{39} + (\beta_1 - 2) q^{40} + (2 \beta_{2} + 2 \beta_1 - 3) q^{41} + ( - 4 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{43} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{44} + (2 \beta_{2} + 2) q^{45} + (3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{46} + ( - 3 \beta_{3} + 2 \beta_{2} + \cdots - 7) q^{47}+ \cdots + (4 \beta_{3} + 2 \beta_1 - 6) q^{99}+O(q^{100})$$ q + q^2 + (-b2 - b1 + 1) * q^3 + q^4 + (b1 - 2) * q^5 + (-b2 - b1 + 1) * q^6 + q^8 + 2*b1 * q^9 + (b1 - 2) * q^10 + (b3 + 2*b2 - b1) * q^11 + (-b2 - b1 + 1) * q^12 + (-b3 - 3) * q^13 + (-b3 + b2 - 2) * q^15 + q^16 + (-2*b3 + b2 + b1 - 1) * q^17 + 2*b1 * q^18 + (2*b3 + b2 - 2) * q^19 + (b1 - 2) * q^20 + (b3 + 2*b2 - b1) * q^22 + (3*b3 + b2 - b1 - 1) * q^23 + (-b2 - b1 + 1) * q^24 + (b2 - 2*b1) * q^25 + (-b3 - 3) * q^26 + (-2*b3 + b2 - b1 - 3) * q^27 + (-2*b3 - 2*b2) * q^29 + (-b3 + b2 - 2) * q^30 + (3*b3 - 3*b2 - 4*b1 + 2) * q^31 + q^32 + (4*b3 + b2 - 2*b1 - 4) * q^33 + (-2*b3 + b2 + b1 - 1) * q^34 + 2*b1 * q^36 + q^37 + (2*b3 + b2 - 2) * q^38 + (-b3 + 3*b2 + 5*b1 - 3) * q^39 + (b1 - 2) * q^40 + (2*b2 + 2*b1 - 3) * q^41 + (-4*b3 + 3*b2 + 4*b1) * q^43 + (b3 + 2*b2 - b1) * q^44 + (2*b2 + 2) * q^45 + (3*b3 + b2 - b1 - 1) * q^46 + (-3*b3 + 2*b2 + 3*b1 - 7) * q^47 + (-b2 - b1 + 1) * q^48 + (b2 - 2*b1) * q^50 + (-2*b3 + 2*b1 - 3) * q^51 + (-b3 - 3) * q^52 + (b3 - 3*b2) * q^53 + (-2*b3 + b2 - b1 - 3) * q^54 + (-4*b2 + 3*b1 - 3) * q^55 + (3*b3 + 2*b2 - 3*b1 - 4) * q^57 + (-2*b3 - 2*b2) * q^58 + (2*b3 - 4*b2 - 2*b1 + 2) * q^59 + (-b3 + b2 - 2) * q^60 + (-3*b3 + 2*b2 + 3*b1 - 7) * q^61 + (3*b3 - 3*b2 - 4*b1 + 2) * q^62 + q^64 + (2*b3 - b2 - 4*b1 + 6) * q^65 + (4*b3 + b2 - 2*b1 - 4) * q^66 + (-2*b3 - b2 + 9*b1 - 5) * q^67 + (-2*b3 + b2 + b1 - 1) * q^68 + (5*b3 + 2*b2 - 4*b1 - 3) * q^69 + (-4*b3 - 4*b2 - 2) * q^71 + 2*b1 * q^72 + (-2*b3 + 3*b2 - 2*b1 - 3) * q^73 + q^74 + (3*b3 + 2*b2 + 3*b1 - 2) * q^75 + (2*b3 + b2 - 2) * q^76 + (-b3 + 3*b2 + 5*b1 - 3) * q^78 + (-4*b3 - b1 + 1) * q^79 + (b1 - 2) * q^80 + (4*b2 + 2*b1 - 5) * q^81 + (2*b2 + 2*b1 - 3) * q^82 + (-9*b3 + 5*b1) * q^83 + (5*b3 - 3*b2 - 2*b1 + 2) * q^85 + (-4*b3 + 3*b2 + 4*b1) * q^86 + (-4*b3 + 6*b1 + 4) * q^87 + (b3 + 2*b2 - b1) * q^88 + (-b3 + 7*b1 - 2) * q^89 + (2*b2 + 2) * q^90 + (3*b3 + b2 - b1 - 1) * q^92 + (4*b3 + 2*b2 + 3*b1 + 8) * q^93 + (-3*b3 + 2*b2 + 3*b1 - 7) * q^94 + (-3*b3 + b1 + 3) * q^95 + (-b2 - b1 + 1) * q^96 + (7*b3 + b2 - 2*b1 - 3) * q^97 + (4*b3 + 2*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} - 6 q^{5} + 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^5 + 4 * q^8 + 4 * q^9 $$4 q + 4 q^{2} + 4 q^{4} - 6 q^{5} + 4 q^{8} + 4 q^{9} - 6 q^{10} + 2 q^{11} - 12 q^{13} - 6 q^{15} + 4 q^{16} + 4 q^{18} - 6 q^{19} - 6 q^{20} + 2 q^{22} - 4 q^{23} - 2 q^{25} - 12 q^{26} - 12 q^{27} - 4 q^{29} - 6 q^{30} - 6 q^{31} + 4 q^{32} - 18 q^{33} + 4 q^{36} + 4 q^{37} - 6 q^{38} + 4 q^{39} - 6 q^{40} - 4 q^{41} + 14 q^{43} + 2 q^{44} + 12 q^{45} - 4 q^{46} - 18 q^{47} - 2 q^{50} - 8 q^{51} - 12 q^{52} - 6 q^{53} - 12 q^{54} - 14 q^{55} - 18 q^{57} - 4 q^{58} - 4 q^{59} - 6 q^{60} - 18 q^{61} - 6 q^{62} + 4 q^{64} + 14 q^{65} - 18 q^{66} - 4 q^{67} - 16 q^{69} - 16 q^{71} + 4 q^{72} - 10 q^{73} + 4 q^{74} + 2 q^{75} - 6 q^{76} + 4 q^{78} + 2 q^{79} - 6 q^{80} - 8 q^{81} - 4 q^{82} + 10 q^{83} - 2 q^{85} + 14 q^{86} + 28 q^{87} + 2 q^{88} + 6 q^{89} + 12 q^{90} - 4 q^{92} + 42 q^{93} - 18 q^{94} + 14 q^{95} - 14 q^{97} - 20 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^5 + 4 * q^8 + 4 * q^9 - 6 * q^10 + 2 * q^11 - 12 * q^13 - 6 * q^15 + 4 * q^16 + 4 * q^18 - 6 * q^19 - 6 * q^20 + 2 * q^22 - 4 * q^23 - 2 * q^25 - 12 * q^26 - 12 * q^27 - 4 * q^29 - 6 * q^30 - 6 * q^31 + 4 * q^32 - 18 * q^33 + 4 * q^36 + 4 * q^37 - 6 * q^38 + 4 * q^39 - 6 * q^40 - 4 * q^41 + 14 * q^43 + 2 * q^44 + 12 * q^45 - 4 * q^46 - 18 * q^47 - 2 * q^50 - 8 * q^51 - 12 * q^52 - 6 * q^53 - 12 * q^54 - 14 * q^55 - 18 * q^57 - 4 * q^58 - 4 * q^59 - 6 * q^60 - 18 * q^61 - 6 * q^62 + 4 * q^64 + 14 * q^65 - 18 * q^66 - 4 * q^67 - 16 * q^69 - 16 * q^71 + 4 * q^72 - 10 * q^73 + 4 * q^74 + 2 * q^75 - 6 * q^76 + 4 * q^78 + 2 * q^79 - 6 * q^80 - 8 * q^81 - 4 * q^82 + 10 * q^83 - 2 * q^85 + 14 * q^86 + 28 * q^87 + 2 * q^88 + 6 * q^89 + 12 * q^90 - 4 * q^92 + 42 * q^93 - 18 * q^94 + 14 * q^95 - 14 * q^97 - 20 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 1$$ v^2 - 2*v - 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 2\nu + 1$$ v^3 - 2*v^2 - 2*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 1$$ b2 + 2*b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 6\beta _1 + 1$$ b3 + 2*b2 + 6*b1 + 1

## 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
 2.77462 −1.22833 −0.360409 0.814115
1.00000 −2.92391 1.00000 0.774623 −2.92391 0 1.00000 5.54925 0.774623
1.2 1.00000 −0.737118 1.00000 −3.22833 −0.737118 0 1.00000 −2.45666 −3.22833
1.3 1.00000 1.50970 1.00000 −2.36041 1.50970 0 1.00000 −0.720819 −2.36041
1.4 1.00000 2.15133 1.00000 −1.18589 2.15133 0 1.00000 1.62823 −1.18589
 $$n$$: 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.z 4
7.b odd 2 1 3626.2.a.ba 4
7.c even 3 2 518.2.e.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.g 8 7.c even 3 2
3626.2.a.z 4 1.a even 1 1 trivial
3626.2.a.ba 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{4} - 8T_{3}^{2} + 4T_{3} + 7$$ T3^4 - 8*T3^2 + 4*T3 + 7 $$T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} - 2T_{5} - 7$$ T5^4 + 6*T5^3 + 9*T5^2 - 2*T5 - 7 $$T_{11}^{4} - 2T_{11}^{3} - 35T_{11}^{2} + 86T_{11} - 49$$ T11^4 - 2*T11^3 - 35*T11^2 + 86*T11 - 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4} - 8 T^{2} + \cdots + 7$$
$5$ $$T^{4} + 6 T^{3} + \cdots - 7$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 2 T^{3} + \cdots - 49$$
$13$ $$(T^{2} + 6 T + 7)^{2}$$
$17$ $$T^{4} - 16 T^{2} + \cdots - 1$$
$19$ $$T^{4} + 6 T^{3} + \cdots + 17$$
$23$ $$T^{4} + 4 T^{3} + \cdots + 47$$
$29$ $$T^{4} + 4 T^{3} + \cdots - 272$$
$31$ $$T^{4} + 6 T^{3} + \cdots + 167$$
$37$ $$(T - 1)^{4}$$
$41$ $$T^{4} + 4 T^{3} + \cdots + 49$$
$43$ $$T^{4} - 14 T^{3} + \cdots - 1951$$
$47$ $$T^{4} + 18 T^{3} + \cdots + 17$$
$53$ $$T^{4} + 6 T^{3} + \cdots + 175$$
$59$ $$T^{4} + 4 T^{3} + \cdots + 1808$$
$61$ $$T^{4} + 18 T^{3} + \cdots + 17$$
$67$ $$T^{4} + 4 T^{3} + \cdots + 20143$$
$71$ $$T^{4} + 16 T^{3} + \cdots - 8176$$
$73$ $$T^{4} + 10 T^{3} + \cdots - 3791$$
$79$ $$T^{4} - 2 T^{3} + \cdots + 1471$$
$83$ $$T^{4} - 10 T^{3} + \cdots - 1841$$
$89$ $$T^{4} - 6 T^{3} + \cdots + 4463$$
$97$ $$T^{4} + 14 T^{3} + \cdots + 2303$$