# Properties

 Label 8470.2.a.bq 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: $$1$$ Twist minimal: no (minimal twist has level 770) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} + q^{5} - \beta q^{6} - q^{7} - q^{8} + (\beta - 2) q^{9} +O(q^{10})$$ q - q^2 + b * q^3 + q^4 + q^5 - b * q^6 - q^7 - q^8 + (b - 2) * q^9 $$q - q^{2} + \beta q^{3} + q^{4} + q^{5} - \beta q^{6} - q^{7} - q^{8} + (\beta - 2) q^{9} - q^{10} + \beta q^{12} + ( - \beta - 4) q^{13} + q^{14} + \beta q^{15} + q^{16} + ( - \beta - 3) q^{17} + ( - \beta + 2) q^{18} + (2 \beta + 2) q^{19} + q^{20} - \beta q^{21} + (4 \beta + 2) q^{23} - \beta q^{24} + q^{25} + (\beta + 4) q^{26} + ( - 4 \beta + 1) q^{27} - q^{28} + (3 \beta + 1) q^{29} - \beta q^{30} + (2 \beta + 2) q^{31} - q^{32} + (\beta + 3) q^{34} - q^{35} + (\beta - 2) q^{36} + ( - 6 \beta + 6) q^{37} + ( - 2 \beta - 2) q^{38} + ( - 5 \beta - 1) q^{39} - q^{40} + ( - 8 \beta + 4) q^{41} + \beta q^{42} - 2 \beta q^{43} + (\beta - 2) q^{45} + ( - 4 \beta - 2) q^{46} + ( - 5 \beta - 5) q^{47} + \beta q^{48} + q^{49} - q^{50} + ( - 4 \beta - 1) q^{51} + ( - \beta - 4) q^{52} + 4 \beta q^{53} + (4 \beta - 1) q^{54} + q^{56} + (4 \beta + 2) q^{57} + ( - 3 \beta - 1) q^{58} + ( - 2 \beta + 8) q^{59} + \beta q^{60} + (6 \beta + 2) q^{61} + ( - 2 \beta - 2) q^{62} + ( - \beta + 2) q^{63} + q^{64} + ( - \beta - 4) q^{65} + ( - 2 \beta + 2) q^{67} + ( - \beta - 3) q^{68} + (6 \beta + 4) q^{69} + q^{70} + ( - 7 \beta + 4) q^{71} + ( - \beta + 2) q^{72} + ( - 7 \beta + 7) q^{73} + (6 \beta - 6) q^{74} + \beta q^{75} + (2 \beta + 2) q^{76} + (5 \beta + 1) q^{78} + (3 \beta - 4) q^{79} + q^{80} + ( - 6 \beta + 2) q^{81} + (8 \beta - 4) q^{82} + ( - \beta - 12) q^{83} - \beta q^{84} + ( - \beta - 3) q^{85} + 2 \beta q^{86} + (4 \beta + 3) q^{87} + ( - 6 \beta + 10) q^{89} + ( - \beta + 2) q^{90} + (\beta + 4) q^{91} + (4 \beta + 2) q^{92} + (4 \beta + 2) q^{93} + (5 \beta + 5) q^{94} + (2 \beta + 2) q^{95} - \beta q^{96} + (\beta - 5) q^{97} - q^{98} +O(q^{100})$$ q - q^2 + b * q^3 + q^4 + q^5 - b * q^6 - q^7 - q^8 + (b - 2) * q^9 - q^10 + b * q^12 + (-b - 4) * q^13 + q^14 + b * q^15 + q^16 + (-b - 3) * q^17 + (-b + 2) * q^18 + (2*b + 2) * q^19 + q^20 - b * q^21 + (4*b + 2) * q^23 - b * q^24 + q^25 + (b + 4) * q^26 + (-4*b + 1) * q^27 - q^28 + (3*b + 1) * q^29 - b * q^30 + (2*b + 2) * q^31 - q^32 + (b + 3) * q^34 - q^35 + (b - 2) * q^36 + (-6*b + 6) * q^37 + (-2*b - 2) * q^38 + (-5*b - 1) * q^39 - q^40 + (-8*b + 4) * q^41 + b * q^42 - 2*b * q^43 + (b - 2) * q^45 + (-4*b - 2) * q^46 + (-5*b - 5) * q^47 + b * q^48 + q^49 - q^50 + (-4*b - 1) * q^51 + (-b - 4) * q^52 + 4*b * q^53 + (4*b - 1) * q^54 + q^56 + (4*b + 2) * q^57 + (-3*b - 1) * q^58 + (-2*b + 8) * q^59 + b * q^60 + (6*b + 2) * q^61 + (-2*b - 2) * q^62 + (-b + 2) * q^63 + q^64 + (-b - 4) * q^65 + (-2*b + 2) * q^67 + (-b - 3) * q^68 + (6*b + 4) * q^69 + q^70 + (-7*b + 4) * q^71 + (-b + 2) * q^72 + (-7*b + 7) * q^73 + (6*b - 6) * q^74 + b * q^75 + (2*b + 2) * q^76 + (5*b + 1) * q^78 + (3*b - 4) * q^79 + q^80 + (-6*b + 2) * q^81 + (8*b - 4) * q^82 + (-b - 12) * q^83 - b * q^84 + (-b - 3) * q^85 + 2*b * q^86 + (4*b + 3) * q^87 + (-6*b + 10) * q^89 + (-b + 2) * q^90 + (b + 4) * q^91 + (4*b + 2) * q^92 + (4*b + 2) * q^93 + (5*b + 5) * q^94 + (2*b + 2) * q^95 - b * q^96 + (b - 5) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} - 2 q^{10} + q^{12} - 9 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - 7 q^{17} + 3 q^{18} + 6 q^{19} + 2 q^{20} - q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 9 q^{26} - 2 q^{27} - 2 q^{28} + 5 q^{29} - q^{30} + 6 q^{31} - 2 q^{32} + 7 q^{34} - 2 q^{35} - 3 q^{36} + 6 q^{37} - 6 q^{38} - 7 q^{39} - 2 q^{40} + q^{42} - 2 q^{43} - 3 q^{45} - 8 q^{46} - 15 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} - 6 q^{51} - 9 q^{52} + 4 q^{53} + 2 q^{54} + 2 q^{56} + 8 q^{57} - 5 q^{58} + 14 q^{59} + q^{60} + 10 q^{61} - 6 q^{62} + 3 q^{63} + 2 q^{64} - 9 q^{65} + 2 q^{67} - 7 q^{68} + 14 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} + 7 q^{73} - 6 q^{74} + q^{75} + 6 q^{76} + 7 q^{78} - 5 q^{79} + 2 q^{80} - 2 q^{81} - 25 q^{83} - q^{84} - 7 q^{85} + 2 q^{86} + 10 q^{87} + 14 q^{89} + 3 q^{90} + 9 q^{91} + 8 q^{92} + 8 q^{93} + 15 q^{94} + 6 q^{95} - q^{96} - 9 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 + 2 * q^5 - q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9 - 2 * q^10 + q^12 - 9 * q^13 + 2 * q^14 + q^15 + 2 * q^16 - 7 * q^17 + 3 * q^18 + 6 * q^19 + 2 * q^20 - q^21 + 8 * q^23 - q^24 + 2 * q^25 + 9 * q^26 - 2 * q^27 - 2 * q^28 + 5 * q^29 - q^30 + 6 * q^31 - 2 * q^32 + 7 * q^34 - 2 * q^35 - 3 * q^36 + 6 * q^37 - 6 * q^38 - 7 * q^39 - 2 * q^40 + q^42 - 2 * q^43 - 3 * q^45 - 8 * q^46 - 15 * q^47 + q^48 + 2 * q^49 - 2 * q^50 - 6 * q^51 - 9 * q^52 + 4 * q^53 + 2 * q^54 + 2 * q^56 + 8 * q^57 - 5 * q^58 + 14 * q^59 + q^60 + 10 * q^61 - 6 * q^62 + 3 * q^63 + 2 * q^64 - 9 * q^65 + 2 * q^67 - 7 * q^68 + 14 * q^69 + 2 * q^70 + q^71 + 3 * q^72 + 7 * q^73 - 6 * q^74 + q^75 + 6 * q^76 + 7 * q^78 - 5 * q^79 + 2 * q^80 - 2 * q^81 - 25 * q^83 - q^84 - 7 * q^85 + 2 * q^86 + 10 * q^87 + 14 * q^89 + 3 * q^90 + 9 * q^91 + 8 * q^92 + 8 * q^93 + 15 * q^94 + 6 * q^95 - q^96 - 9 * 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
 −0.618034 1.61803
−1.00000 −0.618034 1.00000 1.00000 0.618034 −1.00000 −1.00000 −2.61803 −1.00000
1.2 −1.00000 1.61803 1.00000 1.00000 −1.61803 −1.00000 −1.00000 −0.381966 −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.bq 2
11.b odd 2 1 8470.2.a.cc 2
11.d odd 10 2 770.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.a 4 11.d odd 10 2
8470.2.a.bq 2 1.a even 1 1 trivial
8470.2.a.cc 2 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}^{2} - T_{3} - 1$$ T3^2 - T3 - 1 $$T_{13}^{2} + 9T_{13} + 19$$ T13^2 + 9*T13 + 19 $$T_{17}^{2} + 7T_{17} + 11$$ T17^2 + 7*T17 + 11 $$T_{19}^{2} - 6T_{19} + 4$$ T19^2 - 6*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 9T + 19$$
$17$ $$T^{2} + 7T + 11$$
$19$ $$T^{2} - 6T + 4$$
$23$ $$T^{2} - 8T - 4$$
$29$ $$T^{2} - 5T - 5$$
$31$ $$T^{2} - 6T + 4$$
$37$ $$T^{2} - 6T - 36$$
$41$ $$T^{2} - 80$$
$43$ $$T^{2} + 2T - 4$$
$47$ $$T^{2} + 15T + 25$$
$53$ $$T^{2} - 4T - 16$$
$59$ $$T^{2} - 14T + 44$$
$61$ $$T^{2} - 10T - 20$$
$67$ $$T^{2} - 2T - 4$$
$71$ $$T^{2} - T - 61$$
$73$ $$T^{2} - 7T - 49$$
$79$ $$T^{2} + 5T - 5$$
$83$ $$T^{2} + 25T + 155$$
$89$ $$T^{2} - 14T + 4$$
$97$ $$T^{2} + 9T + 19$$