# Properties

 Label 9680.2.a.db 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: 6.6.22733568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2$$ x^6 - 8*x^4 - 2*x^3 + 16*x^2 + 8*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4840) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + (\beta_{4} + \beta_{3}) q^{7} + (\beta_{3} - \beta_1) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + (b4 + b3) * q^7 + (b3 - b1) * q^9 $$q - \beta_1 q^{3} + q^{5} + (\beta_{4} + \beta_{3}) q^{7} + (\beta_{3} - \beta_1) q^{9} + (\beta_{5} + \beta_{4}) q^{13} - \beta_1 q^{15} + ( - 2 \beta_1 - 2) q^{17} + (\beta_{5} + \beta_{4} + 2) q^{19} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{23} + q^{25} + (\beta_{3} - 2 \beta_{2} + 2) q^{27} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_1 - 2) q^{29} + ( - 2 \beta_{5} - \beta_{3} - \beta_1) q^{31} + (\beta_{4} + \beta_{3}) q^{35} + ( - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{37} + ( - \beta_{5} - \beta_{4} + 2) q^{39} + ( - \beta_{5} + \beta_{4} + \beta_{2} - 6) q^{41} + (\beta_{4} - \beta_{3} + \cdots + 2 \beta_1) q^{43}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{97}+O(q^{100})$$ q - b1 * q^3 + q^5 + (b4 + b3) * q^7 + (b3 - b1) * q^9 + (b5 + b4) * q^13 - b1 * q^15 + (-2*b1 - 2) * q^17 + (b5 + b4 + 2) * q^19 + (-2*b5 - b2 - 2*b1) * q^21 + (-2*b4 - b3 - b1 + 2) * q^23 + q^25 + (b3 - 2*b2 + 2) * q^27 + (2*b5 + 2*b4 + 2*b1 - 2) * q^29 + (-2*b5 - b3 - b1) * q^31 + (b4 + b3) * q^35 + (-2*b5 - 2*b2 - 2*b1) * q^37 + (-b5 - b4 + 2) * q^39 + (-b5 + b4 + b2 - 6) * q^41 + (b4 - b3 - 2*b2 + 2*b1) * q^43 + (b3 - b1) * q^45 + (2*b5 + 2*b2 + b1 + 2) * q^47 + (-2*b5 - b3 + 2*b2 - b1 + 2) * q^49 + (2*b3 + 6) * q^51 + (2*b5 + 2*b4 + 3*b3 - b1) * q^53 + (-b5 - b4 - 2*b1 + 2) * q^57 + (2*b5 - 2*b4 + b3 + 2*b2 - b1) * q^59 + (3*b5 - b4 + b2 - 2*b1 - 2) * q^61 + (-b5 - b4 + 2*b2 - 2*b1 + 4) * q^63 + (b5 + b4) * q^65 + (2*b4 - 2*b3 + 2*b2 - b1) * q^67 + (4*b5 + b3 - b1 + 2) * q^69 + (2*b5 + 2*b4 + 3*b3 - 2*b2 + b1 + 4) * q^71 + (4*b3 - 4) * q^73 - b1 * q^75 + (2*b5 + 2*b3) * q^79 + (2*b5 - b3 - 2*b2 - b1 - 1) * q^81 + (5*b5 + b4 + 2*b2 + 2*b1 + 4) * q^83 + (-2*b1 - 2) * q^85 + (-2*b5 - 2*b4 - 2*b3 + 4*b1 - 2) * q^87 + (2*b5 - 2*b3 - 2*b2 - 4*b1 + 3) * q^89 + (-2*b5 - b3 - 4*b2 + b1 + 4) * q^91 + (-2*b5 + 2*b4 + b3 + 4*b2 + b1 + 2) * q^93 + (b5 + b4 + 2) * q^95 + (-2*b5 - 2*b4 - b3 + 4*b2 - b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 $$6 q + 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} + 2 q^{15} - 8 q^{17} + 12 q^{19} + 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} - 16 q^{29} + 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} - 32 q^{41} - 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} + 40 q^{51} + 8 q^{53} + 16 q^{57} - 4 q^{59} - 16 q^{61} + 28 q^{63} + 2 q^{67} + 8 q^{69} + 28 q^{71} - 16 q^{73} + 2 q^{75} - 10 q^{81} + 12 q^{83} - 8 q^{85} - 24 q^{87} + 18 q^{89} + 24 q^{91} + 20 q^{93} + 12 q^{95}+O(q^{100})$$ 6 * q + 2 * q^3 + 6 * q^5 + 4 * q^7 + 4 * q^9 + 2 * q^15 - 8 * q^17 + 12 * q^19 + 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 - 16 * q^29 + 4 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^39 - 32 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 + 40 * q^51 + 8 * q^53 + 16 * q^57 - 4 * q^59 - 16 * q^61 + 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 - 16 * q^73 + 2 * q^75 - 10 * q^81 + 12 * q^83 - 8 * q^85 - 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 + 12 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 3$$ v^4 - 5*v^2 + 3 $$\beta_{4}$$ $$=$$ $$-\nu^{5} + 7\nu^{3} - 10\nu$$ -v^5 + 7*v^3 - 10*v $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 7\nu^{3} + 4\nu^{2} + 12\nu$$ v^5 - v^4 - 7*v^3 + 4*v^2 + 12*v
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 ) / 2$$ (b5 + b4 + b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 3$$ b1 + 3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1$$ 2*b5 + 2*b4 + 2*b3 + b2 + 2*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5\beta _1 + 12$$ b3 + 5*b1 + 12 $$\nu^{5}$$ $$=$$ $$9\beta_{5} + 8\beta_{4} + 9\beta_{3} + 7\beta_{2} + 9\beta _1 + 7$$ 9*b5 + 8*b4 + 9*b3 + 7*b2 + 9*b1 + 7

## 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
 2.27997 −2.08589 1.90131 −1.45825 −0.821728 0.184585
0 −2.19828 0 1.00000 0 2.58485 0 1.83244 0
1.2 0 −1.35095 0 1.00000 0 −3.00679 0 −1.17493 0
1.3 0 −0.614975 0 1.00000 0 2.24598 0 −2.62181 0
1.4 0 0.873518 0 1.00000 0 −3.64048 0 −2.23697 0
1.5 0 2.32476 0 1.00000 0 4.78768 0 2.40452 0
1.6 0 2.96593 0 1.00000 0 1.02876 0 5.79673 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.db 6
4.b odd 2 1 4840.2.a.bc 6
11.b odd 2 1 9680.2.a.da 6
44.c even 2 1 4840.2.a.bd yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.bc 6 4.b odd 2 1
4840.2.a.bd yes 6 44.c even 2 1
9680.2.a.da 6 11.b odd 2 1
9680.2.a.db 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{6} - 2T_{3}^{5} - 9T_{3}^{4} + 12T_{3}^{3} + 23T_{3}^{2} - 10T_{3} - 11$$ T3^6 - 2*T3^5 - 9*T3^4 + 12*T3^3 + 23*T3^2 - 10*T3 - 11 $$T_{7}^{6} - 4T_{7}^{5} - 21T_{7}^{4} + 84T_{7}^{3} + 71T_{7}^{2} - 440T_{7} + 313$$ T7^6 - 4*T7^5 - 21*T7^4 + 84*T7^3 + 71*T7^2 - 440*T7 + 313 $$T_{13}^{6} - 32T_{13}^{4} + 16T_{13}^{3} + 64T_{13}^{2} - 64T_{13} + 16$$ T13^6 - 32*T13^4 + 16*T13^3 + 64*T13^2 - 64*T13 + 16 $$T_{17}^{6} + 8T_{17}^{5} - 16T_{17}^{4} - 192T_{17}^{3} + 1024T_{17} + 256$$ T17^6 + 8*T17^5 - 16*T17^4 - 192*T17^3 + 1024*T17 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 2 T^{5} + \cdots - 11$$
$5$ $$(T - 1)^{6}$$
$7$ $$T^{6} - 4 T^{5} + \cdots + 313$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 32 T^{4} + \cdots + 16$$
$17$ $$T^{6} + 8 T^{5} + \cdots + 256$$
$19$ $$T^{6} - 12 T^{5} + \cdots - 176$$
$23$ $$T^{6} - 8 T^{5} + \cdots - 9776$$
$29$ $$T^{6} + 16 T^{5} + \cdots - 5888$$
$31$ $$T^{6} - 4 T^{5} + \cdots - 368$$
$37$ $$T^{6} - 8 T^{5} + \cdots + 3328$$
$41$ $$T^{6} + 32 T^{5} + \cdots - 70187$$
$43$ $$T^{6} + 4 T^{5} + \cdots + 9097$$
$47$ $$T^{6} - 6 T^{5} + \cdots + 277$$
$53$ $$T^{6} - 8 T^{5} + \cdots + 89296$$
$59$ $$T^{6} + 4 T^{5} + \cdots - 221168$$
$61$ $$T^{6} + 16 T^{5} + \cdots + 252013$$
$67$ $$T^{6} - 2 T^{5} + \cdots - 1315739$$
$71$ $$T^{6} - 28 T^{5} + \cdots - 10736$$
$73$ $$T^{6} + 16 T^{5} + \cdots + 212992$$
$79$ $$T^{6} - 132 T^{4} + \cdots + 19008$$
$83$ $$T^{6} - 12 T^{5} + \cdots + 927952$$
$89$ $$T^{6} - 18 T^{5} + \cdots + 75673$$
$97$ $$T^{6} - 240 T^{4} + \cdots + 103248$$