# Properties

 Label 9680.2.a.dd Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $0$ 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] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.25903625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5$$ x^6 - 3*x^5 - 7*x^4 + 17*x^3 + 16*x^2 - 20*x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{3} - \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (-b5 + b3 - b2 + 1) * q^7 + (b3 - b2 + b1) * q^9 $$q + \beta_1 q^{3} - q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{3} - \beta_{2} + \beta_1) q^{9} + (\beta_{5} - 2 \beta_{2} - 1) q^{13} - \beta_1 q^{15} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{3} - 8 \beta_{2} + \cdots - 8) q^{97}+O(q^{100})$$ q + b1 * q^3 - q^5 + (-b5 + b3 - b2 + 1) * q^7 + (b3 - b2 + b1) * q^9 + (b5 - 2*b2 - 1) * q^13 - b1 * q^15 + (-b5 - b4 + b3 - b2 - b1 + 1) * q^17 + (2*b5 - b4 + b1) * q^19 + (b3 + 2*b2 + 3*b1 + 2) * q^21 + (-b5 + 2*b3 - 3*b2 - b1 + 1) * q^23 + q^25 + (b5 + b4 + 2*b3 - 2*b2 + 3) * q^27 + (-2*b5 + b4 + b3 - b1 + 3) * q^29 + (b5 - 2*b4 + 2*b2 + 2*b1 - 1) * q^31 + (b5 - b3 + b2 - 1) * q^35 + (b5 - b4 - b2 - b1) * q^37 + (3*b5 + b4 - 3*b2 + b1 - 2) * q^39 + (-b5 + 2*b3 + b1 - 1) * q^41 + (-b5 + b4 - b2 - 2*b1 - 2) * q^43 + (-b3 + b2 - b1) * q^45 + (-b5 - b4 + b3 - 4*b2 - 3*b1) * q^47 + (-3*b5 + b3 - 2*b2 + b1 + 2) * q^49 + (-b5 - b3 + 3*b2 + 2*b1 - 2) * q^51 + (-b5 - 2*b4 - b3 + 2*b2 + 3*b1 - 5) * q^53 + (b5 + 2*b4 - 7*b2 + b1 - 2) * q^57 + (-b5 + 5*b2 - 2*b1 + 4) * q^59 + (2*b4 - b3 - b2 + 2*b1) * q^61 + (b5 + b4 + b3 - b2 + 4*b1 + 6) * q^63 + (-b5 + 2*b2 + 1) * q^65 + (-3*b5 - b4 - 2*b3 - 3*b2 - 2*b1 + 3) * q^67 + (2*b5 + b4 + b3 + 2*b2 + 5*b1 - 1) * q^69 + (-b5 + b4 - b3 + b2 - 2*b1 + 4) * q^71 + (b5 - 4*b4 - b3 + 4*b2 + 2*b1 + 1) * q^73 + b1 * q^75 + (-2*b5 - b4 - b3 + 4*b2 + b1 + 5) * q^79 + (4*b5 + 3*b4 - 2*b2 + 4*b1 - 1) * q^81 + (-3*b5 - 2*b4 - b3 - b2 + b1 - 3) * q^83 + (b5 + b4 - b3 + b2 + b1 - 1) * q^85 + (-b5 - b4 + b3 + 6*b2 + 3*b1 + 2) * q^87 + (-2*b5 - 2*b4 + 3*b3 + 2*b2 + 2*b1 - 4) * q^89 + (4*b5 + 3*b4 + b3 - 5*b2 - 2*b1 - 3) * q^91 + (-3*b5 + b4 - 5*b2 - b1 + 2) * q^93 + (-2*b5 + b4 - b1) * q^95 + (-2*b5 + b3 - 8*b2 - 2*b1 - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10})$$ 6 * q + 3 * q^3 - 6 * q^5 + 7 * q^7 + 5 * q^9 $$6 q + 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} + q^{13} - 3 q^{15} + 6 q^{17} + 7 q^{19} + 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} + 10 q^{29} - q^{31} - 7 q^{35} + 3 q^{37} + q^{39} - 6 q^{41} - 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} - 15 q^{51} - 23 q^{53} + 9 q^{57} + 2 q^{59} + 6 q^{61} + 49 q^{63} - q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} + 10 q^{73} + 3 q^{75} + 22 q^{79} + 10 q^{81} - 10 q^{83} - 6 q^{85} + 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} - 7 q^{95} - 33 q^{97}+O(q^{100})$$ 6 * q + 3 * q^3 - 6 * q^5 + 7 * q^7 + 5 * q^9 + q^13 - 3 * q^15 + 6 * q^17 + 7 * q^19 + 14 * q^21 + 9 * q^23 + 6 * q^25 + 21 * q^27 + 10 * q^29 - q^31 - 7 * q^35 + 3 * q^37 + q^39 - 6 * q^41 - 18 * q^43 - 5 * q^45 + 3 * q^47 + 17 * q^49 - 15 * q^51 - 23 * q^53 + 9 * q^57 + 2 * q^59 + 6 * q^61 + 49 * q^63 - q^65 + 22 * q^67 + 2 * q^69 + 13 * q^71 + 10 * q^73 + 3 * q^75 + 22 * q^79 + 10 * q^81 - 10 * q^83 - 6 * q^85 + 3 * q^87 - 25 * q^89 - 12 * q^91 + 19 * q^93 - 7 * q^95 - 33 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{5} - 5\nu^{4} + 2\nu^{3} + 16\nu^{2} - 13\nu - 3$$ v^5 - 5*v^4 + 2*v^3 + 16*v^2 - 13*v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{5} - 5\nu^{4} + 2\nu^{3} + 17\nu^{2} - 14\nu - 6$$ v^5 - 5*v^4 + 2*v^3 + 17*v^2 - 14*v - 6 $$\beta_{4}$$ $$=$$ $$-2\nu^{5} + 9\nu^{4} - 31\nu^{2} + 14\nu + 8$$ -2*v^5 + 9*v^4 - 31*v^2 + 14*v + 8 $$\beta_{5}$$ $$=$$ $$2\nu^{5} - 9\nu^{4} + \nu^{3} + 29\nu^{2} - 18\nu - 5$$ 2*v^5 - 9*v^4 + v^3 + 29*v^2 - 18*v - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2} + \beta _1 + 3$$ b3 - b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 2\beta_{3} - 2\beta_{2} + 6\beta _1 + 3$$ b5 + b4 + 2*b3 - 2*b2 + 6*b1 + 3 $$\nu^{4}$$ $$=$$ $$4\beta_{5} + 3\beta_{4} + 9\beta_{3} - 11\beta_{2} + 13\beta _1 + 17$$ 4*b5 + 3*b4 + 9*b3 - 11*b2 + 13*b1 + 17 $$\nu^{5}$$ $$=$$ $$18\beta_{5} + 13\beta_{4} + 25\beta_{3} - 34\beta_{2} + 50\beta _1 + 34$$ 18*b5 + 13*b4 + 25*b3 - 34*b2 + 50*b1 + 34

## 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
 −1.71280 −1.68797 −0.220878 1.03795 2.29082 3.29288
0 −1.71280 0 −1.00000 0 3.70505 0 −0.0663151 0
1.2 0 −1.68797 0 −1.00000 0 −0.193967 0 −0.150741 0
1.3 0 −0.220878 0 −1.00000 0 −2.08772 0 −2.95121 0
1.4 0 1.03795 0 −1.00000 0 −2.60210 0 −1.92266 0
1.5 0 2.29082 0 −1.00000 0 4.66366 0 2.24785 0
1.6 0 3.29288 0 −1.00000 0 3.51508 0 7.84307 0
 $$n$$: 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$11$$ $$-1$$

## 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 9680.2.a.dd 6
4.b odd 2 1 4840.2.a.ba 6
11.b odd 2 1 9680.2.a.dc 6
11.d odd 10 2 880.2.bo.i 12
44.c even 2 1 4840.2.a.bb 6
44.g even 10 2 440.2.y.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.c 12 44.g even 10 2
880.2.bo.i 12 11.d odd 10 2
4840.2.a.ba 6 4.b odd 2 1
4840.2.a.bb 6 44.c even 2 1
9680.2.a.dc 6 11.b odd 2 1
9680.2.a.dd 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{6} - 3T_{3}^{5} - 7T_{3}^{4} + 17T_{3}^{3} + 16T_{3}^{2} - 20T_{3} - 5$$ T3^6 - 3*T3^5 - 7*T3^4 + 17*T3^3 + 16*T3^2 - 20*T3 - 5 $$T_{7}^{6} - 7T_{7}^{5} - 5T_{7}^{4} + 93T_{7}^{3} - 13T_{7}^{2} - 336T_{7} - 64$$ T7^6 - 7*T7^5 - 5*T7^4 + 93*T7^3 - 13*T7^2 - 336*T7 - 64 $$T_{13}^{6} - T_{13}^{5} - 33T_{13}^{4} + 71T_{13}^{3} + 91T_{13}^{2} - 220T_{13} + 80$$ T13^6 - T13^5 - 33*T13^4 + 71*T13^3 + 91*T13^2 - 220*T13 + 80 $$T_{17}^{6} - 6T_{17}^{5} - 30T_{17}^{4} + 197T_{17}^{3} + 79T_{17}^{2} - 825T_{17} - 605$$ T17^6 - 6*T17^5 - 30*T17^4 + 197*T17^3 + 79*T17^2 - 825*T17 - 605

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 3 T^{5} + \cdots - 5$$
$5$ $$(T + 1)^{6}$$
$7$ $$T^{6} - 7 T^{5} + \cdots - 64$$
$11$ $$T^{6}$$
$13$ $$T^{6} - T^{5} + \cdots + 80$$
$17$ $$T^{6} - 6 T^{5} + \cdots - 605$$
$19$ $$T^{6} - 7 T^{5} + \cdots - 9791$$
$23$ $$T^{6} - 9 T^{5} + \cdots + 5956$$
$29$ $$T^{6} - 10 T^{5} + \cdots + 1100$$
$31$ $$T^{6} + T^{5} + \cdots + 3596$$
$37$ $$T^{6} - 3 T^{5} + \cdots - 1900$$
$41$ $$T^{6} + 6 T^{5} + \cdots + 3775$$
$43$ $$T^{6} + 18 T^{5} + \cdots + 3751$$
$47$ $$T^{6} - 3 T^{5} + \cdots - 9284$$
$53$ $$T^{6} + 23 T^{5} + \cdots + 199804$$
$59$ $$T^{6} - 2 T^{5} + \cdots + 11759$$
$61$ $$T^{6} - 6 T^{5} + \cdots + 6620$$
$67$ $$T^{6} - 22 T^{5} + \cdots + 126475$$
$71$ $$T^{6} - 13 T^{5} + \cdots + 620$$
$73$ $$T^{6} - 10 T^{5} + \cdots - 832031$$
$79$ $$T^{6} - 22 T^{5} + \cdots + 6724$$
$83$ $$T^{6} + 10 T^{5} + \cdots - 1375$$
$89$ $$T^{6} + 25 T^{5} + \cdots - 22429$$
$97$ $$T^{6} + 33 T^{5} + \cdots - 164695$$