# Properties

 Label 9680.2.a.cw Level $9680$ Weight $2$ Character orbit 9680.a Self dual yes Analytic conductor $77.295$ Analytic rank $1$ Dimension $6$ Inner twists $2$

# Learn more

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: $$6$$ Coefficient field: 6.6.27433728.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 9x^{4} + 15x^{2} - 3$$ x^6 - 9*x^4 + 15*x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 605) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 1) q^{3} + q^{5} - \beta_1 q^{7} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (b3 - 1) * q^3 + q^5 - b1 * q^7 + (-2*b3 - b2 + 2) * q^9 $$q + (\beta_{3} - 1) q^{3} + q^{5} - \beta_1 q^{7} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{9} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{13} + (\beta_{3} - 1) q^{15} + (2 \beta_{4} - 2 \beta_1) q^{17} + ( - \beta_{5} + \beta_{4}) q^{19} + ( - \beta_{4} + 2 \beta_1) q^{21} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{23} + q^{25} + (3 \beta_{3} + 3 \beta_{2} - 5) q^{27} + ( - 2 \beta_{5} + 2 \beta_1) q^{29} + ( - 2 \beta_{3} + \beta_{2}) q^{31} - \beta_1 q^{35} + 2 \beta_{3} q^{37} + (\beta_{5} - \beta_{4} - 4 \beta_1) q^{39} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{41} + (2 \beta_{5} - \beta_1) q^{43} + ( - 2 \beta_{3} - \beta_{2} + 2) q^{45} + ( - \beta_{3} - 7) q^{47} + (\beta_{2} - 4) q^{49} + ( - 2 \beta_{5} - 4 \beta_{4} + 6 \beta_1) q^{51} + (4 \beta_{3} + 3 \beta_{2} + 4) q^{53} + ( - \beta_{5} + 3 \beta_{4}) q^{57} + \beta_{2} q^{59} - \beta_{5} q^{61} + (\beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{63} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{65} + ( - \beta_{3} - 5) q^{67} + (2 \beta_{3} + 3 \beta_{2} - 4) q^{69} - \beta_{2} q^{71} + ( - 4 \beta_{4} + 4 \beta_1) q^{73} + (\beta_{3} - 1) q^{75} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{79} + ( - 8 \beta_{3} - 3 \beta_{2} + 5) q^{81} + ( - \beta_{5} - 3 \beta_{4} + 6 \beta_1) q^{83} + (2 \beta_{4} - 2 \beta_1) q^{85} + (10 \beta_{4} - 6 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_{2} - 5) q^{89} + ( - 3 \beta_{2} - 6) q^{91} + (\beta_{2} - 10) q^{93} + ( - \beta_{5} + \beta_{4}) q^{95} + ( - 4 \beta_{3} - \beta_{2} + 4) q^{97}+O(q^{100})$$ q + (b3 - 1) * q^3 + q^5 - b1 * q^7 + (-2*b3 - b2 + 2) * q^9 + (b5 - b4 + 2*b1) * q^13 + (b3 - 1) * q^15 + (2*b4 - 2*b1) * q^17 + (-b5 + b4) * q^19 + (-b4 + 2*b1) * q^21 + (-2*b3 - b2 - 2) * q^23 + q^25 + (3*b3 + 3*b2 - 5) * q^27 + (-2*b5 + 2*b1) * q^29 + (-2*b3 + b2) * q^31 - b1 * q^35 + 2*b3 * q^37 + (b5 - b4 - 4*b1) * q^39 + (b5 - 2*b4 - 2*b1) * q^41 + (2*b5 - b1) * q^43 + (-2*b3 - b2 + 2) * q^45 + (-b3 - 7) * q^47 + (b2 - 4) * q^49 + (-2*b5 - 4*b4 + 6*b1) * q^51 + (4*b3 + 3*b2 + 4) * q^53 + (-b5 + 3*b4) * q^57 + b2 * q^59 - b5 * q^61 + (b5 + 3*b4 - 2*b1) * q^63 + (b5 - b4 + 2*b1) * q^65 + (-b3 - 5) * q^67 + (2*b3 + 3*b2 - 4) * q^69 - b2 * q^71 + (-4*b4 + 4*b1) * q^73 + (b3 - 1) * q^75 + (-2*b5 - 2*b4 - 2*b1) * q^79 + (-8*b3 - 3*b2 + 5) * q^81 + (-b5 - 3*b4 + 6*b1) * q^83 + (2*b4 - 2*b1) * q^85 + (10*b4 - 6*b1) * q^87 + (-2*b3 + 2*b2 - 5) * q^89 + (-3*b2 - 6) * q^91 + (b2 - 10) * q^93 + (-b5 + b4) * q^95 + (-4*b3 - b2 + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} + 6 q^{5} + 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^3 + 6 * q^5 + 12 * q^9 $$6 q - 6 q^{3} + 6 q^{5} + 12 q^{9} - 6 q^{15} - 12 q^{23} + 6 q^{25} - 30 q^{27} + 12 q^{45} - 42 q^{47} - 24 q^{49} + 24 q^{53} - 30 q^{67} - 24 q^{69} - 6 q^{75} + 30 q^{81} - 30 q^{89} - 36 q^{91} - 60 q^{93} + 24 q^{97}+O(q^{100})$$ 6 * q - 6 * q^3 + 6 * q^5 + 12 * q^9 - 6 * q^15 - 12 * q^23 + 6 * q^25 - 30 * q^27 + 12 * q^45 - 42 * q^47 - 24 * q^49 + 24 * q^53 - 30 * q^67 - 24 * q^69 - 6 * q^75 + 30 * q^81 - 30 * q^89 - 36 * q^91 - 60 * q^93 + 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9x^{4} + 15x^{2} - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 8\nu^{2} + 7 ) / 2$$ (v^4 - 8*v^2 + 7) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 8\nu^{3} + 9\nu ) / 2$$ (v^5 - 8*v^3 + 9*v) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 10\nu^{3} - 19\nu ) / 2$$ (-v^5 + 10*v^3 - 19*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 5\beta_1$$ b5 + b4 + 5*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{3} + 8\beta_{2} + 17$$ 2*b3 + 8*b2 + 17 $$\nu^{5}$$ $$=$$ $$8\beta_{5} + 10\beta_{4} + 31\beta_1$$ 8*b5 + 10*b4 + 31*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
 1.37268 −1.37268 2.62383 −2.62383 0.480901 −0.480901
0 −3.26180 0 1.00000 0 −1.37268 0 7.63935 0
1.2 0 −3.26180 0 1.00000 0 1.37268 0 7.63935 0
1.3 0 −1.33988 0 1.00000 0 −2.62383 0 −1.20473 0
1.4 0 −1.33988 0 1.00000 0 2.62383 0 −1.20473 0
1.5 0 1.60168 0 1.00000 0 −0.480901 0 −0.434624 0
1.6 0 1.60168 0 1.00000 0 0.480901 0 −0.434624 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.cw 6
4.b odd 2 1 605.2.a.m 6
11.b odd 2 1 inner 9680.2.a.cw 6
12.b even 2 1 5445.2.a.bx 6
20.d odd 2 1 3025.2.a.bg 6
44.c even 2 1 605.2.a.m 6
44.g even 10 4 605.2.g.q 24
44.h odd 10 4 605.2.g.q 24
132.d odd 2 1 5445.2.a.bx 6
220.g even 2 1 3025.2.a.bg 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.m 6 4.b odd 2 1
605.2.a.m 6 44.c even 2 1
605.2.g.q 24 44.g even 10 4
605.2.g.q 24 44.h odd 10 4
3025.2.a.bg 6 20.d odd 2 1
3025.2.a.bg 6 220.g even 2 1
5445.2.a.bx 6 12.b even 2 1
5445.2.a.bx 6 132.d odd 2 1
9680.2.a.cw 6 1.a even 1 1 trivial
9680.2.a.cw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9680))$$:

 $$T_{3}^{3} + 3T_{3}^{2} - 3T_{3} - 7$$ T3^3 + 3*T3^2 - 3*T3 - 7 $$T_{7}^{6} - 9T_{7}^{4} + 15T_{7}^{2} - 3$$ T7^6 - 9*T7^4 + 15*T7^2 - 3 $$T_{13}^{6} - 72T_{13}^{4} + 1296T_{13}^{2} - 3888$$ T13^6 - 72*T13^4 + 1296*T13^2 - 3888 $$T_{17}^{6} - 48T_{17}^{4} + 384T_{17}^{2} - 768$$ T17^6 - 48*T17^4 + 384*T17^2 - 768

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{3} + 3 T^{2} - 3 T - 7)^{2}$$
$5$ $$(T - 1)^{6}$$
$7$ $$T^{6} - 9 T^{4} + \cdots - 3$$
$11$ $$T^{6}$$
$13$ $$T^{6} - 72 T^{4} + \cdots - 3888$$
$17$ $$T^{6} - 48 T^{4} + \cdots - 768$$
$19$ $$T^{6} - 36 T^{4} + \cdots - 48$$
$23$ $$(T^{3} + 6 T^{2} - 12 T - 84)^{2}$$
$29$ $$T^{6} - 144 T^{4} + \cdots - 6912$$
$31$ $$(T^{3} - 48 T + 124)^{2}$$
$37$ $$(T^{3} - 24 T - 16)^{2}$$
$41$ $$T^{6} - 105 T^{4} + \cdots - 7203$$
$43$ $$T^{6} - 129 T^{4} + \cdots - 43923$$
$47$ $$(T^{3} + 21 T^{2} + \cdots + 303)^{2}$$
$53$ $$(T^{3} - 12 T^{2} + \cdots + 732)^{2}$$
$59$ $$(T^{3} - 12 T - 12)^{2}$$
$61$ $$T^{6} - 33 T^{4} + \cdots - 1083$$
$67$ $$(T^{3} + 15 T^{2} + \cdots + 97)^{2}$$
$71$ $$(T^{3} - 12 T + 12)^{2}$$
$73$ $$T^{6} - 192 T^{4} + \cdots - 49152$$
$79$ $$T^{6} - 276 T^{4} + \cdots - 101568$$
$83$ $$T^{6} - 312 T^{4} + \cdots - 40368$$
$89$ $$(T^{3} + 15 T^{2} + \cdots - 147)^{2}$$
$97$ $$(T^{3} - 12 T^{2} + \cdots + 76)^{2}$$
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