# Properties

 Label 9680.2.a.de Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $1$ Dimension $8$ 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: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100$$ x^8 - x^7 - 21*x^6 + 15*x^5 + 140*x^4 - 60*x^3 - 295*x^2 + 50*x + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{4} + 3) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + (-b4 - 1) * q^7 + (b5 + b4 + 3) * q^9 $$q - \beta_1 q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{4} + 3) q^{9} + ( - \beta_{5} + \beta_1 - 2) q^{13} - \beta_1 q^{15} + ( - \beta_{7} - \beta_{5} - 2 \beta_{2} + \cdots - 2) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots + 2) q^{97}+O(q^{100})$$ q - b1 * q^3 + q^5 + (-b4 - 1) * q^7 + (b5 + b4 + 3) * q^9 + (-b5 + b1 - 2) * q^13 - b1 * q^15 + (-b7 - b5 - 2*b2 + b1 - 2) * q^17 + (b7 + b6 + b5 + b4 + b2 + 1) * q^19 + (2*b7 + b4 - b3 + b2 + 2*b1) * q^21 + (-b3 + b2 - 1) * q^23 + q^25 + (-b7 + b6 - b4 + b2 - 2*b1 - 1) * q^27 + (-b7 - b6 + 2*b3 - b1 - 1) * q^29 + (-b6 + b3 - b2 - 3) * q^31 + (-b4 - 1) * q^35 + (-b7 - b4 + 2*b3 + b2 + 2) * q^37 + (-b7 - b6 - b5 - b4 + b3 - 2*b2 + 3*b1 - 5) * q^39 + (b6 + b5 - 2*b3 - b2 + b1) * q^41 + (b7 - b6 - b5 - b4 - b3 + 2*b2 - 1) * q^43 + (b5 + b4 + 3) * q^45 + (b7 - b6 - b5 + 2*b3 - 2*b1 - 1) * q^47 + (b5 + b4 - 3*b3 + b2 + b1 + 3) * q^49 + (-2*b7 - b6 - b5 + b4 - b3 - 5*b2 + 3*b1 - 5) * q^51 + (-b6 - b5 - b4 - b3 + 2*b1 + 2) * q^53 + (b6 + b5 - 2*b4 + 2*b3 + 6*b2 - 3*b1 + 1) * q^57 + (-b3 - 5*b2 + b1 - 6) * q^59 + (-b6 + b5 - b4 - 3*b2 - 3) * q^61 + (-b7 - b6 - 2*b5 - 4*b4 + 3*b3 - b2 - b1 - 10) * q^63 + (-b5 + b1 - 2) * q^65 + (-b7 + b5 + b4 - b3 + 3*b2 - b1 + 2) * q^67 + (-b7 - b6 + 2*b3 - 6*b2 + b1 - 1) * q^69 + (b7 - b6 + b5 + 2*b4 - b2 + b1 - 3) * q^71 + (b6 + b4 + 2*b2 - b1 - 3) * q^73 - b1 * q^75 + (b7 + b6 - 2*b5 + b1 - 5) * q^79 + (b7 + 3*b4 + b3 + 2*b1 + 5) * q^81 + (2*b6 + b5 - b4 - b3 + 6*b2 + b1) * q^83 + (-b7 - b5 - 2*b2 + b1 - 2) * q^85 + (b7 + 2*b6 + 2*b4 - 3*b3 + 7*b2 + b1 + 6) * q^87 + (2*b7 + 2*b6 + b5 + b4 + 3*b2 - b1 + 5) * q^89 + (-b7 + b6 + b5 + 2*b4 + b3 - b2 - 2*b1) * q^91 + (b7 + b6 - b5 - b4 - 3*b3 + 4*b2 + 3*b1 - 1) * q^93 + (b7 + b6 + b5 + b4 + b2 + 1) * q^95 + (-2*b5 + b4 + 3*b3 - b2 + 3*b1 + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9}+O(q^{10})$$ 8 * q - q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 $$8 q - q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} - 12 q^{13} - q^{15} - 2 q^{17} - 6 q^{19} - 6 q^{21} - 10 q^{23} + 8 q^{25} - 13 q^{27} - 8 q^{29} - 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} + 3 q^{41} - 8 q^{43} + 19 q^{45} - 10 q^{47} + 22 q^{49} - 7 q^{51} + 28 q^{53} - 25 q^{57} - 25 q^{59} - 10 q^{61} - 64 q^{63} - 12 q^{65} + 2 q^{67} + 18 q^{69} - 25 q^{71} - 38 q^{73} - q^{75} - 38 q^{79} + 32 q^{81} - 28 q^{83} - 2 q^{85} + 15 q^{87} + 12 q^{89} - 8 q^{91} - 15 q^{93} - 6 q^{95} + 21 q^{97}+O(q^{100})$$ 8 * q - q^3 + 8 * q^5 - 6 * q^7 + 19 * q^9 - 12 * q^13 - q^15 - 2 * q^17 - 6 * q^19 - 6 * q^21 - 10 * q^23 + 8 * q^25 - 13 * q^27 - 8 * q^29 - 19 * q^31 - 6 * q^35 + 12 * q^37 - 21 * q^39 + 3 * q^41 - 8 * q^43 + 19 * q^45 - 10 * q^47 + 22 * q^49 - 7 * q^51 + 28 * q^53 - 25 * q^57 - 25 * q^59 - 10 * q^61 - 64 * q^63 - 12 * q^65 + 2 * q^67 + 18 * q^69 - 25 * q^71 - 38 * q^73 - q^75 - 38 * q^79 + 32 * q^81 - 28 * q^83 - 2 * q^85 + 15 * q^87 + 12 * q^89 - 8 * q^91 - 15 * q^93 - 6 * q^95 + 21 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 3\nu^{6} - 23\nu^{5} - 35\nu^{4} + 140\nu^{3} + 80\nu^{2} - 185\nu - 60 ) / 70$$ (v^7 + 3*v^6 - 23*v^5 - 35*v^4 + 140*v^3 + 80*v^2 - 185*v - 60) / 70 $$\beta_{3}$$ $$=$$ $$( -2\nu^{7} + \nu^{6} + 25\nu^{5} - 70\nu^{3} - 55\nu^{2} + 55\nu + 50 ) / 35$$ (-2*v^7 + v^6 + 25*v^5 - 70*v^3 - 55*v^2 + 55*v + 50) / 35 $$\beta_{4}$$ $$=$$ $$( 3\nu^{7} - 5\nu^{6} - 41\nu^{5} + 77\nu^{4} + 140\nu^{3} - 320\nu^{2} - 135\nu + 100 ) / 70$$ (3*v^7 - 5*v^6 - 41*v^5 + 77*v^4 + 140*v^3 - 320*v^2 - 135*v + 100) / 70 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 5\nu^{6} + 41\nu^{5} - 77\nu^{4} - 140\nu^{3} + 390\nu^{2} + 135\nu - 520 ) / 70$$ (-3*v^7 + 5*v^6 + 41*v^5 - 77*v^4 - 140*v^3 + 390*v^2 + 135*v - 520) / 70 $$\beta_{6}$$ $$=$$ $$( -3\nu^{7} + 5\nu^{6} + 55\nu^{5} - 49\nu^{4} - 350\nu^{3} + 40\nu^{2} + 765\nu + 110 ) / 70$$ (-3*v^7 + 5*v^6 + 55*v^5 - 49*v^4 - 350*v^3 + 40*v^2 + 765*v + 110) / 70 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 13\nu^{6} + 73\nu^{5} - 161\nu^{4} - 280\nu^{3} + 440\nu^{2} + 155\nu - 120 ) / 70$$ (-5*v^7 + 13*v^6 + 73*v^5 - 161*v^4 - 280*v^3 + 440*v^2 + 155*v - 120) / 70
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + 6$$ b5 + b4 + 6 $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 8\beta _1 + 1$$ b7 - b6 + b4 - b2 + 8*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{7} + 9\beta_{5} + 12\beta_{4} + \beta_{3} + 2\beta _1 + 50$$ b7 + 9*b5 + 12*b4 + b3 + 2*b1 + 50 $$\nu^{5}$$ $$=$$ $$13\beta_{7} - 10\beta_{6} + 2\beta_{5} + 16\beta_{4} - 2\beta_{3} - 15\beta_{2} + 71\beta _1 + 20$$ 13*b7 - 10*b6 + 2*b5 + 16*b4 - 2*b3 - 15*b2 + 71*b1 + 20 $$\nu^{6}$$ $$=$$ $$19\beta_{7} + 81\beta_{5} + 123\beta_{4} + 9\beta_{3} + 5\beta_{2} + 38\beta _1 + 450$$ 19*b7 + 81*b5 + 123*b4 + 9*b3 + 5*b2 + 38*b1 + 450 $$\nu^{7}$$ $$=$$ $$137\beta_{7} - 90\beta_{6} + 38\beta_{5} + 199\beta_{4} - 38\beta_{3} - 150\beta_{2} + 654\beta _1 + 300$$ 137*b7 - 90*b6 + 38*b5 + 199*b4 - 38*b3 - 150*b2 + 654*b1 + 300

## 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
 3.21195 3.08163 1.88088 0.713352 −0.569801 −1.64111 −2.77190 −2.90500
0 −3.21195 0 1.00000 0 −4.90937 0 7.31662 0
1.2 0 −3.08163 0 1.00000 0 0.0385298 0 6.49642 0
1.3 0 −1.88088 0 1.00000 0 3.67750 0 0.537727 0
1.4 0 −0.713352 0 1.00000 0 0.376172 0 −2.49113 0
1.5 0 0.569801 0 1.00000 0 −1.82111 0 −2.67533 0
1.6 0 1.64111 0 1.00000 0 3.37642 0 −0.306743 0
1.7 0 2.77190 0 1.00000 0 −4.55506 0 4.68342 0
1.8 0 2.90500 0 1.00000 0 −2.18309 0 5.43902 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.de 8
4.b odd 2 1 4840.2.a.bh 8
11.b odd 2 1 9680.2.a.df 8
11.d odd 10 2 880.2.bo.k 16
44.c even 2 1 4840.2.a.bg 8
44.g even 10 2 440.2.y.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.d 16 44.g even 10 2
880.2.bo.k 16 11.d odd 10 2
4840.2.a.bg 8 44.c even 2 1
4840.2.a.bh 8 4.b odd 2 1
9680.2.a.de 8 1.a even 1 1 trivial
9680.2.a.df 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{8} + T_{3}^{7} - 21T_{3}^{6} - 15T_{3}^{5} + 140T_{3}^{4} + 60T_{3}^{3} - 295T_{3}^{2} - 50T_{3} + 100$$ T3^8 + T3^7 - 21*T3^6 - 15*T3^5 + 140*T3^4 + 60*T3^3 - 295*T3^2 - 50*T3 + 100 $$T_{7}^{8} + 6T_{7}^{7} - 21T_{7}^{6} - 151T_{7}^{5} + 55T_{7}^{4} + 954T_{7}^{3} + 709T_{7}^{2} - 444T_{7} + 16$$ T7^8 + 6*T7^7 - 21*T7^6 - 151*T7^5 + 55*T7^4 + 954*T7^3 + 709*T7^2 - 444*T7 + 16 $$T_{13}^{8} + 12T_{13}^{7} + 17T_{13}^{6} - 261T_{13}^{5} - 871T_{13}^{4} + 1006T_{13}^{3} + 6127T_{13}^{2} + 2508T_{13} - 6064$$ T13^8 + 12*T13^7 + 17*T13^6 - 261*T13^5 - 871*T13^4 + 1006*T13^3 + 6127*T13^2 + 2508*T13 - 6064 $$T_{17}^{8} + 2T_{17}^{7} - 96T_{17}^{6} - 95T_{17}^{5} + 2549T_{17}^{4} - 171T_{17}^{3} - 12545T_{17}^{2} - 5794T_{17} + 3124$$ T17^8 + 2*T17^7 - 96*T17^6 - 95*T17^5 + 2549*T17^4 - 171*T17^3 - 12545*T17^2 - 5794*T17 + 3124

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + T^{7} + \cdots + 100$$
$5$ $$(T - 1)^{8}$$
$7$ $$T^{8} + 6 T^{7} + \cdots + 16$$
$11$ $$T^{8}$$
$13$ $$T^{8} + 12 T^{7} + \cdots - 6064$$
$17$ $$T^{8} + 2 T^{7} + \cdots + 3124$$
$19$ $$T^{8} + 6 T^{7} + \cdots + 74855$$
$23$ $$T^{8} + 10 T^{7} + \cdots + 4$$
$29$ $$T^{8} + 8 T^{7} + \cdots + 69520$$
$31$ $$T^{8} + 19 T^{7} + \cdots - 2480$$
$37$ $$T^{8} - 12 T^{7} + \cdots - 1020420$$
$41$ $$T^{8} - 3 T^{7} + \cdots - 126755$$
$43$ $$T^{8} + 8 T^{7} + \cdots - 76780$$
$47$ $$T^{8} + 10 T^{7} + \cdots + 886724$$
$53$ $$T^{8} - 28 T^{7} + \cdots + 4820$$
$59$ $$T^{8} + 25 T^{7} + \cdots + 11749$$
$61$ $$T^{8} + 10 T^{7} + \cdots - 1596416$$
$67$ $$T^{8} - 2 T^{7} + \cdots + 2034124$$
$71$ $$T^{8} + 25 T^{7} + \cdots + 490000$$
$73$ $$T^{8} + 38 T^{7} + \cdots - 53420$$
$79$ $$T^{8} + 38 T^{7} + \cdots + 12393280$$
$83$ $$T^{8} + 28 T^{7} + \cdots + 7869644$$
$89$ $$T^{8} - 12 T^{7} + \cdots + 2898841$$
$97$ $$T^{8} - 21 T^{7} + \cdots + 105621104$$