# Properties

 Label 9680.2.a.cz Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# 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: 6.6.507624192.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 20x^{4} + 100x^{2} - 108$$ x^6 - 20*x^4 + 100*x^2 - 108 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2420) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} - q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} + 3) q^{9}+O(q^{10})$$ q + b4 * q^3 - q^5 + (b2 - b1) * q^7 + (b4 - b3 + 3) * q^9 $$q + \beta_{4} q^{3} - q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} + 3) q^{9} + (\beta_{5} + \beta_{2} + \beta_1) q^{13} - \beta_{4} q^{15} + 2 \beta_1 q^{17} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{19} + ( - 2 \beta_{5} - \beta_{2}) q^{21} + (\beta_{4} + \beta_{3} - 1) q^{23} + q^{25} + (4 \beta_{4} - \beta_{3} + 3) q^{27} - 2 \beta_1 q^{29} + (\beta_{4} - \beta_{3} - 5) q^{31} + ( - \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{4} + 6) q^{37} + (\beta_{5} - 5 \beta_{2} + 3 \beta_1) q^{39} + (\beta_{5} - 2 \beta_{2} + \beta_1) q^{41} + (3 \beta_{2} - \beta_1) q^{43} + ( - \beta_{4} + \beta_{3} - 3) q^{45} + 3 \beta_{4} q^{47} + ( - \beta_{4} - \beta_{3} + 3) q^{49} + (2 \beta_{5} + 2 \beta_{2}) q^{51} + (\beta_{4} + \beta_{3} - 1) q^{53} + ( - \beta_{5} + 7 \beta_{2} - 3 \beta_1) q^{57} + (3 \beta_{4} - \beta_{3} - 3) q^{59} + (\beta_{5} - \beta_1) q^{61} + ( - \beta_{5} + 9 \beta_{2} - 3 \beta_1) q^{63} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{65} + (\beta_{4} - 2 \beta_{3} + 4) q^{67} + ( - 3 \beta_{4} - \beta_{3} + 9) q^{69} + ( - 3 \beta_{4} + \beta_{3} + 3) q^{71} + 4 \beta_{2} q^{73} + \beta_{4} q^{75} + ( - 2 \beta_{5} - 4 \beta_1) q^{79} + (7 \beta_{4} - \beta_{3} + 12) q^{81} + (\beta_{5} - 5 \beta_{2} - \beta_1) q^{83} - 2 \beta_1 q^{85} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{87} + (2 \beta_{4} + 1) q^{89} + ( - 5 \beta_{4} - \beta_{3} - 1) q^{91} + ( - \beta_{4} - \beta_{3} + 3) q^{93} + (\beta_{5} - \beta_{2} - \beta_1) q^{95} + (\beta_{4} + \beta_{3} + 7) q^{97}+O(q^{100})$$ q + b4 * q^3 - q^5 + (b2 - b1) * q^7 + (b4 - b3 + 3) * q^9 + (b5 + b2 + b1) * q^13 - b4 * q^15 + 2*b1 * q^17 + (-b5 + b2 + b1) * q^19 + (-2*b5 - b2) * q^21 + (b4 + b3 - 1) * q^23 + q^25 + (4*b4 - b3 + 3) * q^27 - 2*b1 * q^29 + (b4 - b3 - 5) * q^31 + (-b2 + b1) * q^35 + (-2*b4 + 6) * q^37 + (b5 - 5*b2 + 3*b1) * q^39 + (b5 - 2*b2 + b1) * q^41 + (3*b2 - b1) * q^43 + (-b4 + b3 - 3) * q^45 + 3*b4 * q^47 + (-b4 - b3 + 3) * q^49 + (2*b5 + 2*b2) * q^51 + (b4 + b3 - 1) * q^53 + (-b5 + 7*b2 - 3*b1) * q^57 + (3*b4 - b3 - 3) * q^59 + (b5 - b1) * q^61 + (-b5 + 9*b2 - 3*b1) * q^63 + (-b5 - b2 - b1) * q^65 + (b4 - 2*b3 + 4) * q^67 + (-3*b4 - b3 + 9) * q^69 + (-3*b4 + b3 + 3) * q^71 + 4*b2 * q^73 + b4 * q^75 + (-2*b5 - 4*b1) * q^79 + (7*b4 - b3 + 12) * q^81 + (b5 - 5*b2 - b1) * q^83 - 2*b1 * q^85 + (-2*b5 - 2*b2) * q^87 + (2*b4 + 1) * q^89 + (-5*b4 - b3 - 1) * q^91 + (-b4 - b3 + 3) * q^93 + (b5 - b2 - b1) * q^95 + (b4 + b3 + 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 6 q^{5} + 20 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 - 6 * q^5 + 20 * q^9 $$6 q + 2 q^{3} - 6 q^{5} + 20 q^{9} - 2 q^{15} - 4 q^{23} + 6 q^{25} + 26 q^{27} - 28 q^{31} + 32 q^{37} - 20 q^{45} + 6 q^{47} + 16 q^{49} - 4 q^{53} - 12 q^{59} + 26 q^{67} + 48 q^{69} + 12 q^{71} + 2 q^{75} + 86 q^{81} + 10 q^{89} - 16 q^{91} + 16 q^{93} + 44 q^{97}+O(q^{100})$$ 6 * q + 2 * q^3 - 6 * q^5 + 20 * q^9 - 2 * q^15 - 4 * q^23 + 6 * q^25 + 26 * q^27 - 28 * q^31 + 32 * q^37 - 20 * q^45 + 6 * q^47 + 16 * q^49 - 4 * q^53 - 12 * q^59 + 26 * q^67 + 48 * q^69 + 12 * q^71 + 2 * q^75 + 86 * q^81 + 10 * q^89 - 16 * q^91 + 16 * q^93 + 44 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 20x^{4} + 100x^{2} - 108$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 10\nu ) / 6$$ (v^3 - 10*v) / 6 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 10\nu^{2} ) / 6$$ (v^4 - 10*v^2) / 6 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 16\nu^{2} + 42 ) / 6$$ (v^4 - 16*v^2 + 42) / 6 $$\beta_{5}$$ $$=$$ $$( \nu^{5} - 17\nu^{3} + 52\nu ) / 6$$ (v^5 - 17*v^3 + 52*v) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 7$$ -b4 + b3 + 7 $$\nu^{3}$$ $$=$$ $$6\beta_{2} + 10\beta_1$$ 6*b2 + 10*b1 $$\nu^{4}$$ $$=$$ $$-10\beta_{4} + 16\beta_{3} + 70$$ -10*b4 + 16*b3 + 70 $$\nu^{5}$$ $$=$$ $$6\beta_{5} + 102\beta_{2} + 118\beta_1$$ 6*b5 + 102*b2 + 118*b1

## 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
 2.36936 −2.36936 3.59084 −3.59084 1.22147 −1.22147
0 −2.71774 0 −1.00000 0 −4.10141 0 4.38612 0
1.2 0 −2.71774 0 −1.00000 0 4.10141 0 4.38612 0
1.3 0 0.325397 0 −1.00000 0 −1.85879 0 −2.89412 0
1.4 0 0.325397 0 −1.00000 0 1.85879 0 −2.89412 0
1.5 0 3.39234 0 −1.00000 0 −2.95353 0 8.50800 0
1.6 0 3.39234 0 −1.00000 0 2.95353 0 8.50800 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.cz 6
4.b odd 2 1 2420.2.a.o 6
11.b odd 2 1 inner 9680.2.a.cz 6
44.c even 2 1 2420.2.a.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2420.2.a.o 6 4.b odd 2 1
2420.2.a.o 6 44.c even 2 1
9680.2.a.cz 6 1.a even 1 1 trivial
9680.2.a.cz 6 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{3} - T_{3}^{2} - 9T_{3} + 3$$ T3^3 - T3^2 - 9*T3 + 3 $$T_{7}^{6} - 29T_{7}^{4} + 235T_{7}^{2} - 507$$ T7^6 - 29*T7^4 + 235*T7^2 - 507 $$T_{13}^{6} - 68T_{13}^{4} + 1504T_{13}^{2} - 10800$$ T13^6 - 68*T13^4 + 1504*T13^2 - 10800 $$T_{17}^{6} - 80T_{17}^{4} + 1600T_{17}^{2} - 6912$$ T17^6 - 80*T17^4 + 1600*T17^2 - 6912

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{3} - T^{2} - 9 T + 3)^{2}$$
$5$ $$(T + 1)^{6}$$
$7$ $$T^{6} - 29 T^{4} + \cdots - 507$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 68 T^{4} + \cdots - 10800$$
$17$ $$T^{6} - 80 T^{4} + \cdots - 6912$$
$19$ $$T^{6} - 104 T^{4} + \cdots - 40368$$
$23$ $$(T^{3} + 2 T^{2} - 44 T + 12)^{2}$$
$29$ $$T^{6} - 80 T^{4} + \cdots - 6912$$
$31$ $$(T^{3} + 14 T^{2} + \cdots - 20)^{2}$$
$37$ $$(T^{3} - 16 T^{2} + \cdots + 48)^{2}$$
$41$ $$T^{6} - 113 T^{4} + \cdots - 27$$
$43$ $$T^{6} - 101 T^{4} + \cdots - 6075$$
$47$ $$(T^{3} - 3 T^{2} - 81 T + 81)^{2}$$
$53$ $$(T^{3} + 2 T^{2} - 44 T + 12)^{2}$$
$59$ $$(T^{3} + 6 T^{2} + \cdots - 540)^{2}$$
$61$ $$T^{6} - 89 T^{4} + \cdots - 18723$$
$67$ $$(T^{3} - 13 T^{2} + \cdots + 895)^{2}$$
$71$ $$(T^{3} - 6 T^{2} + \cdots + 540)^{2}$$
$73$ $$(T^{2} - 48)^{3}$$
$79$ $$T^{6} - 452 T^{4} + \cdots - 192$$
$83$ $$T^{6} - 344 T^{4} + \cdots - 73008$$
$89$ $$(T^{3} - 5 T^{2} - 29 T + 57)^{2}$$
$97$ $$(T^{3} - 22 T^{2} + \cdots - 20)^{2}$$