# Properties

 Label 8470.2.a.bv Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ 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] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.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: $$67.6332905120$$ 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: $$2$$ Twist minimal: yes 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + q^{5} + ( - \beta - 1) q^{6} - q^{7} + q^{8} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + q^2 + (-b - 1) * q^3 + q^4 + q^5 + (-b - 1) * q^6 - q^7 + q^8 + (2*b + 3) * q^9 $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + q^{5} + ( - \beta - 1) q^{6} - q^{7} + q^{8} + (2 \beta + 3) q^{9} + q^{10} + ( - \beta - 1) q^{12} + 2 \beta q^{13} - q^{14} + ( - \beta - 1) q^{15} + q^{16} + 2 q^{17} + (2 \beta + 3) q^{18} + ( - \beta - 3) q^{19} + q^{20} + (\beta + 1) q^{21} + ( - \beta - 5) q^{23} + ( - \beta - 1) q^{24} + q^{25} + 2 \beta q^{26} + ( - 2 \beta - 10) q^{27} - q^{28} + ( - \beta + 1) q^{29} + ( - \beta - 1) q^{30} + 4 \beta q^{31} + q^{32} + 2 q^{34} - q^{35} + (2 \beta + 3) q^{36} + ( - \beta - 1) q^{37} + ( - \beta - 3) q^{38} + ( - 2 \beta - 10) q^{39} + q^{40} + ( - 3 \beta - 1) q^{41} + (\beta + 1) q^{42} - 8 q^{43} + (2 \beta + 3) q^{45} + ( - \beta - 5) q^{46} + (2 \beta - 4) q^{47} + ( - \beta - 1) q^{48} + q^{49} + q^{50} + ( - 2 \beta - 2) q^{51} + 2 \beta q^{52} + (\beta - 3) q^{53} + ( - 2 \beta - 10) q^{54} - q^{56} + (4 \beta + 8) q^{57} + ( - \beta + 1) q^{58} - 8 q^{59} + ( - \beta - 1) q^{60} + (2 \beta + 10) q^{61} + 4 \beta q^{62} + ( - 2 \beta - 3) q^{63} + q^{64} + 2 \beta q^{65} - 6 q^{67} + 2 q^{68} + (6 \beta + 10) q^{69} - q^{70} - 8 q^{71} + (2 \beta + 3) q^{72} + ( - 2 \beta - 4) q^{73} + ( - \beta - 1) q^{74} + ( - \beta - 1) q^{75} + ( - \beta - 3) q^{76} + ( - 2 \beta - 10) q^{78} + (\beta + 3) q^{79} + q^{80} + (6 \beta + 11) q^{81} + ( - 3 \beta - 1) q^{82} + 8 q^{83} + (\beta + 1) q^{84} + 2 q^{85} - 8 q^{86} + 4 q^{87} + ( - 4 \beta - 6) q^{89} + (2 \beta + 3) q^{90} - 2 \beta q^{91} + ( - \beta - 5) q^{92} + ( - 4 \beta - 20) q^{93} + (2 \beta - 4) q^{94} + ( - \beta - 3) q^{95} + ( - \beta - 1) q^{96} + (3 \beta - 5) q^{97} + q^{98} +O(q^{100})$$ q + q^2 + (-b - 1) * q^3 + q^4 + q^5 + (-b - 1) * q^6 - q^7 + q^8 + (2*b + 3) * q^9 + q^10 + (-b - 1) * q^12 + 2*b * q^13 - q^14 + (-b - 1) * q^15 + q^16 + 2 * q^17 + (2*b + 3) * q^18 + (-b - 3) * q^19 + q^20 + (b + 1) * q^21 + (-b - 5) * q^23 + (-b - 1) * q^24 + q^25 + 2*b * q^26 + (-2*b - 10) * q^27 - q^28 + (-b + 1) * q^29 + (-b - 1) * q^30 + 4*b * q^31 + q^32 + 2 * q^34 - q^35 + (2*b + 3) * q^36 + (-b - 1) * q^37 + (-b - 3) * q^38 + (-2*b - 10) * q^39 + q^40 + (-3*b - 1) * q^41 + (b + 1) * q^42 - 8 * q^43 + (2*b + 3) * q^45 + (-b - 5) * q^46 + (2*b - 4) * q^47 + (-b - 1) * q^48 + q^49 + q^50 + (-2*b - 2) * q^51 + 2*b * q^52 + (b - 3) * q^53 + (-2*b - 10) * q^54 - q^56 + (4*b + 8) * q^57 + (-b + 1) * q^58 - 8 * q^59 + (-b - 1) * q^60 + (2*b + 10) * q^61 + 4*b * q^62 + (-2*b - 3) * q^63 + q^64 + 2*b * q^65 - 6 * q^67 + 2 * q^68 + (6*b + 10) * q^69 - q^70 - 8 * q^71 + (2*b + 3) * q^72 + (-2*b - 4) * q^73 + (-b - 1) * q^74 + (-b - 1) * q^75 + (-b - 3) * q^76 + (-2*b - 10) * q^78 + (b + 3) * q^79 + q^80 + (6*b + 11) * q^81 + (-3*b - 1) * q^82 + 8 * q^83 + (b + 1) * q^84 + 2 * q^85 - 8 * q^86 + 4 * q^87 + (-4*b - 6) * q^89 + (2*b + 3) * q^90 - 2*b * q^91 + (-b - 5) * q^92 + (-4*b - 20) * q^93 + (2*b - 4) * q^94 + (-b - 3) * q^95 + (-b - 1) * q^96 + (3*b - 5) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 6 q^{9} + 2 q^{10} - 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} + 6 q^{18} - 6 q^{19} + 2 q^{20} + 2 q^{21} - 10 q^{23} - 2 q^{24} + 2 q^{25} - 20 q^{27} - 2 q^{28} + 2 q^{29} - 2 q^{30} + 2 q^{32} + 4 q^{34} - 2 q^{35} + 6 q^{36} - 2 q^{37} - 6 q^{38} - 20 q^{39} + 2 q^{40} - 2 q^{41} + 2 q^{42} - 16 q^{43} + 6 q^{45} - 10 q^{46} - 8 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} - 6 q^{53} - 20 q^{54} - 2 q^{56} + 16 q^{57} + 2 q^{58} - 16 q^{59} - 2 q^{60} + 20 q^{61} - 6 q^{63} + 2 q^{64} - 12 q^{67} + 4 q^{68} + 20 q^{69} - 2 q^{70} - 16 q^{71} + 6 q^{72} - 8 q^{73} - 2 q^{74} - 2 q^{75} - 6 q^{76} - 20 q^{78} + 6 q^{79} + 2 q^{80} + 22 q^{81} - 2 q^{82} + 16 q^{83} + 2 q^{84} + 4 q^{85} - 16 q^{86} + 8 q^{87} - 12 q^{89} + 6 q^{90} - 10 q^{92} - 40 q^{93} - 8 q^{94} - 6 q^{95} - 2 q^{96} - 10 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 6 * q^9 + 2 * q^10 - 2 * q^12 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 4 * q^17 + 6 * q^18 - 6 * q^19 + 2 * q^20 + 2 * q^21 - 10 * q^23 - 2 * q^24 + 2 * q^25 - 20 * q^27 - 2 * q^28 + 2 * q^29 - 2 * q^30 + 2 * q^32 + 4 * q^34 - 2 * q^35 + 6 * q^36 - 2 * q^37 - 6 * q^38 - 20 * q^39 + 2 * q^40 - 2 * q^41 + 2 * q^42 - 16 * q^43 + 6 * q^45 - 10 * q^46 - 8 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 - 6 * q^53 - 20 * q^54 - 2 * q^56 + 16 * q^57 + 2 * q^58 - 16 * q^59 - 2 * q^60 + 20 * q^61 - 6 * q^63 + 2 * q^64 - 12 * q^67 + 4 * q^68 + 20 * q^69 - 2 * q^70 - 16 * q^71 + 6 * q^72 - 8 * q^73 - 2 * q^74 - 2 * q^75 - 6 * q^76 - 20 * q^78 + 6 * q^79 + 2 * q^80 + 22 * q^81 - 2 * q^82 + 16 * q^83 + 2 * q^84 + 4 * q^85 - 16 * q^86 + 8 * q^87 - 12 * q^89 + 6 * q^90 - 10 * q^92 - 40 * q^93 - 8 * q^94 - 6 * q^95 - 2 * q^96 - 10 * q^97 + 2 * 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
 1.61803 −0.618034
1.00000 −3.23607 1.00000 1.00000 −3.23607 −1.00000 1.00000 7.47214 1.00000
1.2 1.00000 1.23607 1.00000 1.00000 1.23607 −1.00000 1.00000 −1.47214 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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 8470.2.a.bv yes 2
11.b odd 2 1 8470.2.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.bj 2 11.b odd 2 1
8470.2.a.bv yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{2} + 2T_{3} - 4$$ T3^2 + 2*T3 - 4 $$T_{13}^{2} - 20$$ T13^2 - 20 $$T_{17} - 2$$ T17 - 2 $$T_{19}^{2} + 6T_{19} + 4$$ T19^2 + 6*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 20$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 6T + 4$$
$23$ $$T^{2} + 10T + 20$$
$29$ $$T^{2} - 2T - 4$$
$31$ $$T^{2} - 80$$
$37$ $$T^{2} + 2T - 4$$
$41$ $$T^{2} + 2T - 44$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 8T - 4$$
$53$ $$T^{2} + 6T + 4$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} - 20T + 80$$
$67$ $$(T + 6)^{2}$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 8T - 4$$
$79$ $$T^{2} - 6T + 4$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} + 12T - 44$$
$97$ $$T^{2} + 10T - 20$$