# Properties

 Label 8670.2.a.be Level $8670$ Weight $2$ Character orbit 8670.a Self dual yes Analytic conductor $69.230$ Analytic rank $0$ 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] = [8670,2,Mod(1,8670)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8670.a (trivial)

## Newform invariants

comment: select newform

sage: f = N # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$69.2302985525$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 510) 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 $$\beta = 2\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + \beta q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + b * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + \beta q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + (\beta + 2) q^{13} - \beta q^{14} - q^{15} + q^{16} - q^{18} + 4 q^{19} - q^{20} + \beta q^{21} + 4 q^{23} - q^{24} + q^{25} + ( - \beta - 2) q^{26} + q^{27} + \beta q^{28} - 6 q^{29} + q^{30} - 4 q^{31} - q^{32} - \beta q^{35} + q^{36} - 6 q^{37} - 4 q^{38} + (\beta + 2) q^{39} + q^{40} + (\beta - 2) q^{41} - \beta q^{42} + (\beta + 4) q^{43} - q^{45} - 4 q^{46} + 2 \beta q^{47} + q^{48} + 17 q^{49} - q^{50} + (\beta + 2) q^{52} + ( - 2 \beta + 2) q^{53} - q^{54} - \beta q^{56} + 4 q^{57} + 6 q^{58} + \beta q^{59} - q^{60} + ( - 2 \beta - 2) q^{61} + 4 q^{62} + \beta q^{63} + q^{64} + ( - \beta - 2) q^{65} + (\beta - 4) q^{67} + 4 q^{69} + \beta q^{70} + (\beta + 4) q^{71} - q^{72} + (\beta + 6) q^{73} + 6 q^{74} + q^{75} + 4 q^{76} + ( - \beta - 2) q^{78} + (2 \beta - 4) q^{79} - q^{80} + q^{81} + ( - \beta + 2) q^{82} + (2 \beta + 4) q^{83} + \beta q^{84} + ( - \beta - 4) q^{86} - 6 q^{87} + ( - 2 \beta + 2) q^{89} + q^{90} + (2 \beta + 24) q^{91} + 4 q^{92} - 4 q^{93} - 2 \beta q^{94} - 4 q^{95} - q^{96} + (3 \beta - 2) q^{97} - 17 q^{98} +O(q^{100})$$ q - q^2 + q^3 + q^4 - q^5 - q^6 + b * q^7 - q^8 + q^9 + q^10 + q^12 + (b + 2) * q^13 - b * q^14 - q^15 + q^16 - q^18 + 4 * q^19 - q^20 + b * q^21 + 4 * q^23 - q^24 + q^25 + (-b - 2) * q^26 + q^27 + b * q^28 - 6 * q^29 + q^30 - 4 * q^31 - q^32 - b * q^35 + q^36 - 6 * q^37 - 4 * q^38 + (b + 2) * q^39 + q^40 + (b - 2) * q^41 - b * q^42 + (b + 4) * q^43 - q^45 - 4 * q^46 + 2*b * q^47 + q^48 + 17 * q^49 - q^50 + (b + 2) * q^52 + (-2*b + 2) * q^53 - q^54 - b * q^56 + 4 * q^57 + 6 * q^58 + b * q^59 - q^60 + (-2*b - 2) * q^61 + 4 * q^62 + b * q^63 + q^64 + (-b - 2) * q^65 + (b - 4) * q^67 + 4 * q^69 + b * q^70 + (b + 4) * q^71 - q^72 + (b + 6) * q^73 + 6 * q^74 + q^75 + 4 * q^76 + (-b - 2) * q^78 + (2*b - 4) * q^79 - q^80 + q^81 + (-b + 2) * q^82 + (2*b + 4) * q^83 + b * q^84 + (-b - 4) * q^86 - 6 * q^87 + (-2*b + 2) * q^89 + q^90 + (2*b + 24) * q^91 + 4 * q^92 - 4 * q^93 - 2*b * q^94 - 4 * q^95 - q^96 + (3*b - 2) * q^97 - 17 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 4 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{19} - 2 q^{20} + 8 q^{23} - 2 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 12 q^{29} + 2 q^{30} - 8 q^{31} - 2 q^{32} + 2 q^{36} - 12 q^{37} - 8 q^{38} + 4 q^{39} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 2 q^{45} - 8 q^{46} + 2 q^{48} + 34 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} - 2 q^{54} + 8 q^{57} + 12 q^{58} - 2 q^{60} - 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{69} + 8 q^{71} - 2 q^{72} + 12 q^{73} + 12 q^{74} + 2 q^{75} + 8 q^{76} - 4 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 8 q^{86} - 12 q^{87} + 4 q^{89} + 2 q^{90} + 48 q^{91} + 8 q^{92} - 8 q^{93} - 8 q^{95} - 2 q^{96} - 4 q^{97} - 34 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 4 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^18 + 8 * q^19 - 2 * q^20 + 8 * q^23 - 2 * q^24 + 2 * q^25 - 4 * q^26 + 2 * q^27 - 12 * q^29 + 2 * q^30 - 8 * q^31 - 2 * q^32 + 2 * q^36 - 12 * q^37 - 8 * q^38 + 4 * q^39 + 2 * q^40 - 4 * q^41 + 8 * q^43 - 2 * q^45 - 8 * q^46 + 2 * q^48 + 34 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 - 2 * q^54 + 8 * q^57 + 12 * q^58 - 2 * q^60 - 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^69 + 8 * q^71 - 2 * q^72 + 12 * q^73 + 12 * q^74 + 2 * q^75 + 8 * q^76 - 4 * q^78 - 8 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 - 8 * q^86 - 12 * q^87 + 4 * q^89 + 2 * q^90 + 48 * q^91 + 8 * q^92 - 8 * q^93 - 8 * q^95 - 2 * q^96 - 4 * q^97 - 34 * q^98

## 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
 −2.44949 2.44949
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −4.89898 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 4.89898 −1.00000 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$17$$ $$1$$

## 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 8670.2.a.be 2
17.b even 2 1 510.2.a.h 2
51.c odd 2 1 1530.2.a.s 2
68.d odd 2 1 4080.2.a.bq 2
85.c even 2 1 2550.2.a.bl 2
85.g odd 4 2 2550.2.d.u 4
255.h odd 2 1 7650.2.a.cu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.h 2 17.b even 2 1
1530.2.a.s 2 51.c odd 2 1
2550.2.a.bl 2 85.c even 2 1
2550.2.d.u 4 85.g odd 4 2
4080.2.a.bq 2 68.d odd 2 1
7650.2.a.cu 2 255.h odd 2 1
8670.2.a.be 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8670))$$:

 $$T_{7}^{2} - 24$$ T7^2 - 24 $$T_{11}$$ T11 $$T_{13}^{2} - 4T_{13} - 20$$ T13^2 - 4*T13 - 20 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 24$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T - 20$$
$17$ $$T^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 4T - 20$$
$43$ $$T^{2} - 8T - 8$$
$47$ $$T^{2} - 96$$
$53$ $$T^{2} - 4T - 92$$
$59$ $$T^{2} - 24$$
$61$ $$T^{2} + 4T - 92$$
$67$ $$T^{2} + 8T - 8$$
$71$ $$T^{2} - 8T - 8$$
$73$ $$T^{2} - 12T + 12$$
$79$ $$T^{2} + 8T - 80$$
$83$ $$T^{2} - 8T - 80$$
$89$ $$T^{2} - 4T - 92$$
$97$ $$T^{2} + 4T - 212$$