# Properties

 Label 640.2.o.k Level $640$ Weight $2$ Character orbit 640.o Analytic conductor $5.110$ Analytic rank $0$ Dimension $12$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(63,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("640.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.o (of order $$4$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$5.11042572936$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.125772815663104.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 27x^{8} + 107x^{4} + 1$$ x^12 + 27*x^8 + 107*x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{5}) q^{9}+O(q^{10})$$ q + b2 * q^3 - b3 * q^5 - b1 * q^7 + (-b8 + b7 + b5) * q^9 $$q + \beta_{2} q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{5}) q^{9} + (\beta_{10} - \beta_{6} + \beta_1) q^{11} + (\beta_{8} - 2 \beta_{5} + \beta_{4} + 2) q^{13} + ( - \beta_{11} + \beta_{9} + \cdots - \beta_1) q^{15}+ \cdots + ( - 4 \beta_{11} - 3 \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^3 - b3 * q^5 - b1 * q^7 + (-b8 + b7 + b5) * q^9 + (b10 - b6 + b1) * q^11 + (b8 - 2*b5 + b4 + 2) * q^13 + (-b11 + b9 + b2 - b1) * q^15 + (-b8 - b7 + b5 - b4 - b3 - 1) * q^17 + (b11 - b9 - b6 + b2 - b1) * q^19 + (-b8 + b7) * q^21 + (-2*b11 + b10 + b9 - b6) * q^23 + (b8 + b7 - 2*b5 + 2*b4 + 1) * q^25 + (b10 + b9 - 2*b6) * q^27 + (-b8 - b7 - b4 + b3 - 4) * q^29 + (-2*b11 - b6 - 2*b2 - b1) * q^31 + (b8 - 3*b7 - 2*b5 - b4 + 3*b3 - 2) * q^33 + (-b10 - b6 + b2 + b1) * q^35 + (b8 - 2*b7 - b4 + 2*b3) * q^37 + (-2*b11 - 2*b10 + b6 + 2*b2 - b1) * q^39 + (2*b8 + 2*b7 + b4 - b3) * q^41 + b2 * q^43 + (-b8 + 2*b7 + 6*b5 - b4 - b3 + 2) * q^45 + (b10 - b9 + 2*b2 - b1) * q^47 + (b8 - b7 + b5 - 2*b4 - 2*b3) * q^49 + (-b11 + 2*b10 + b2) * q^51 + (2*b8 + b7 + 2*b5 + 2*b4 + b3 - 2) * q^53 + (b11 + b10 + 2*b9 - 3*b2) * q^55 + (2*b7 + 2*b5 + 2*b3 - 2) * q^57 + (b11 - 3*b9 + b6 + b2 + b1) * q^59 + (4*b5 - b4 - b3) * q^61 + (2*b11 + b10 + b9 + b6) * q^63 + (2*b8 - 2*b7 - b5 + b4 - b3 + 3) * q^65 + (b11 + 2*b10 + 2*b9) * q^67 + (-2*b8 - 2*b7 - b4 + b3 + 4) * q^69 + (4*b11 - 2*b9 - b6 + 4*b2 - b1) * q^71 + (-2*b8 - 3*b5 + 2*b4 - 3) * q^73 + (-3*b10 + b9 + b2 - 2*b1) * q^75 + (-b8 - b7 - 6*b5 + b4 + b3 - 6) * q^77 + (2*b11 - 4*b10 - 2*b2) * q^79 + (-2*b8 - 2*b7 - b4 + b3 - 1) * q^81 + (-b10 + b9 - 3*b2 + 2*b1) * q^83 + (3*b7 - 6*b5 + b4 - 2) * q^85 + (2*b10 - 2*b9 - 4*b2 + 2*b1) * q^87 + (-2*b8 + 2*b7) * q^89 + (-b11 + 2*b6 + b2 - 2*b1) * q^91 + (b8 - b7 - 8*b5 + b4 - b3 + 8) * q^93 + (-b11 + b10 + 2*b9 - b6 + 3*b2 + 3*b1) * q^95 + (-2*b8 - b5 - 2*b4 + 1) * q^97 + (-4*b11 - 3*b9 + b6 - 4*b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 20 q^{13} - 4 q^{17} + 4 q^{25} - 40 q^{29} - 16 q^{33} + 4 q^{37} - 16 q^{41} + 20 q^{45} - 36 q^{53} - 32 q^{57} + 36 q^{65} + 64 q^{69} - 28 q^{73} - 64 q^{77} + 4 q^{81} - 36 q^{85} + 96 q^{93} + 20 q^{97}+O(q^{100})$$ 12 * q + 20 * q^13 - 4 * q^17 + 4 * q^25 - 40 * q^29 - 16 * q^33 + 4 * q^37 - 16 * q^41 + 20 * q^45 - 36 * q^53 - 32 * q^57 + 36 * q^65 + 64 * q^69 - 28 * q^73 - 64 * q^77 + 4 * q^81 - 36 * q^85 + 96 * q^93 + 20 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 27x^{8} + 107x^{4} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{9} + 38\nu^{5} + 373\nu ) / 76$$ (v^9 + 38*v^5 + 373*v) / 76 $$\beta_{2}$$ $$=$$ $$( -7\nu^{9} - 190\nu^{5} - 711\nu ) / 76$$ (-7*v^9 - 190*v^5 - 711*v) / 76 $$\beta_{3}$$ $$=$$ $$( 3\nu^{10} - 3\nu^{8} + 76\nu^{6} - 76\nu^{4} + 245\nu^{2} - 169 ) / 76$$ (3*v^10 - 3*v^8 + 76*v^6 - 76*v^4 + 245*v^2 - 169) / 76 $$\beta_{4}$$ $$=$$ $$( 3\nu^{10} + 3\nu^{8} + 76\nu^{6} + 76\nu^{4} + 245\nu^{2} + 169 ) / 76$$ (3*v^10 + 3*v^8 + 76*v^6 + 76*v^4 + 245*v^2 + 169) / 76 $$\beta_{5}$$ $$=$$ $$( 7\nu^{10} + 190\nu^{6} + 787\nu^{2} ) / 76$$ (7*v^10 + 190*v^6 + 787*v^2) / 76 $$\beta_{6}$$ $$=$$ $$( 35\nu^{11} + 950\nu^{7} + 3859\nu^{3} ) / 76$$ (35*v^11 + 950*v^7 + 3859*v^3) / 76 $$\beta_{7}$$ $$=$$ $$( 17\nu^{10} - 2\nu^{8} + 456\nu^{6} - 38\nu^{4} + 1743\nu^{2} - 24 ) / 76$$ (17*v^10 - 2*v^8 + 456*v^6 - 38*v^4 + 1743*v^2 - 24) / 76 $$\beta_{8}$$ $$=$$ $$( -17\nu^{10} - 2\nu^{8} - 456\nu^{6} - 38\nu^{4} - 1743\nu^{2} - 24 ) / 76$$ (-17*v^10 - 2*v^8 - 456*v^6 - 38*v^4 - 1743*v^2 - 24) / 76 $$\beta_{9}$$ $$=$$ $$( -24\nu^{11} - 3\nu^{9} - 646\nu^{7} - 76\nu^{5} - 2530\nu^{3} - 245\nu ) / 38$$ (-24*v^11 - 3*v^9 - 646*v^7 - 76*v^5 - 2530*v^3 - 245*v) / 38 $$\beta_{10}$$ $$=$$ $$( -24\nu^{11} + 3\nu^{9} - 646\nu^{7} + 76\nu^{5} - 2530\nu^{3} + 245\nu ) / 38$$ (-24*v^11 + 3*v^9 - 646*v^7 + 76*v^5 - 2530*v^3 + 245*v) / 38 $$\beta_{11}$$ $$=$$ $$( -69\nu^{11} - 1862\nu^{7} - 7345\nu^{3} ) / 76$$ (-69*v^11 - 1862*v^7 - 7345*v^3) / 76
 $$\nu$$ $$=$$ $$( \beta_{10} - \beta_{9} + 2\beta_{2} + 2\beta_1 ) / 4$$ (b10 - b9 + 2*b2 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{8} - \beta_{7} + 4\beta_{5} + \beta_{4} + \beta_{3} ) / 2$$ (b8 - b7 + 4*b5 + b4 + b3) / 2 $$\nu^{3}$$ $$=$$ $$( 10\beta_{11} - 5\beta_{10} - 5\beta_{9} + 6\beta_{6} ) / 4$$ (10*b11 - 5*b10 - 5*b9 + 6*b6) / 4 $$\nu^{4}$$ $$=$$ $$3\beta_{8} + 3\beta_{7} + 2\beta_{4} - 2\beta_{3} - 7$$ 3*b8 + 3*b7 + 2*b4 - 2*b3 - 7 $$\nu^{5}$$ $$=$$ $$( -25\beta_{10} + 25\beta_{9} - 46\beta_{2} - 22\beta_1 ) / 4$$ (-25*b10 + 25*b9 - 46*b2 - 22*b1) / 4 $$\nu^{6}$$ $$=$$ $$( -17\beta_{8} + 17\beta_{7} - 56\beta_{5} - 31\beta_{4} - 31\beta_{3} ) / 2$$ (-17*b8 + 17*b7 - 56*b5 - 31*b4 - 31*b3) / 2 $$\nu^{7}$$ $$=$$ $$( -214\beta_{11} + 121\beta_{10} + 121\beta_{9} - 90\beta_{6} ) / 4$$ (-214*b11 + 121*b10 + 121*b9 - 90*b6) / 4 $$\nu^{8}$$ $$=$$ $$-76\beta_{8} - 76\beta_{7} - 38\beta_{4} + 38\beta_{3} + 121$$ -76*b8 - 76*b7 - 38*b4 + 38*b3 + 121 $$\nu^{9}$$ $$=$$ $$( 577\beta_{10} - 577\beta_{9} + 1002\beta_{2} + 394\beta_1 ) / 4$$ (577*b10 - 577*b9 + 1002*b2 + 394*b1) / 4 $$\nu^{10}$$ $$=$$ $$( 349\beta_{8} - 349\beta_{7} + 1092\beta_{5} + 729\beta_{4} + 729\beta_{3} ) / 2$$ (349*b8 - 349*b7 + 1092*b5 + 729*b4 + 729*b3) / 2 $$\nu^{11}$$ $$=$$ $$( 4706\beta_{11} - 2733\beta_{10} - 2733\beta_{9} + 1790\beta_{6} ) / 4$$ (4706*b11 - 2733*b10 - 2733*b9 + 1790*b6) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-\beta_{5}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 0.219986 + 0.219986i −1.53448 − 1.53448i −1.04736 − 1.04736i 1.04736 + 1.04736i 1.53448 + 1.53448i −0.219986 − 0.219986i 0.219986 − 0.219986i −1.53448 + 1.53448i −1.04736 + 1.04736i 1.04736 − 1.04736i 1.53448 − 1.53448i −0.219986 + 0.219986i
0 −2.05288 2.05288i 0 2.21432 0.311108i 0 −1.07864 1.07864i 0 5.42864i 0
63.2 0 −1.20864 1.20864i 0 −0.539189 2.17009i 0 0.446112 + 0.446112i 0 0.0783777i 0
63.3 0 −0.569973 0.569973i 0 −1.67513 + 1.48119i 0 2.93897 + 2.93897i 0 2.35026i 0
63.4 0 0.569973 + 0.569973i 0 −1.67513 + 1.48119i 0 −2.93897 2.93897i 0 2.35026i 0
63.5 0 1.20864 + 1.20864i 0 −0.539189 2.17009i 0 −0.446112 0.446112i 0 0.0783777i 0
63.6 0 2.05288 + 2.05288i 0 2.21432 0.311108i 0 1.07864 + 1.07864i 0 5.42864i 0
447.1 0 −2.05288 + 2.05288i 0 2.21432 + 0.311108i 0 −1.07864 + 1.07864i 0 5.42864i 0
447.2 0 −1.20864 + 1.20864i 0 −0.539189 + 2.17009i 0 0.446112 0.446112i 0 0.0783777i 0
447.3 0 −0.569973 + 0.569973i 0 −1.67513 1.48119i 0 2.93897 2.93897i 0 2.35026i 0
447.4 0 0.569973 0.569973i 0 −1.67513 1.48119i 0 −2.93897 + 2.93897i 0 2.35026i 0
447.5 0 1.20864 1.20864i 0 −0.539189 + 2.17009i 0 −0.446112 + 0.446112i 0 0.0783777i 0
447.6 0 2.05288 2.05288i 0 2.21432 + 0.311108i 0 1.07864 1.07864i 0 5.42864i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
40.i odd 4 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.o.k yes 12
4.b odd 2 1 inner 640.2.o.k yes 12
5.c odd 4 1 640.2.o.j 12
8.b even 2 1 640.2.o.j 12
8.d odd 2 1 640.2.o.j 12
16.e even 4 1 1280.2.n.r 12
16.e even 4 1 1280.2.n.s 12
16.f odd 4 1 1280.2.n.r 12
16.f odd 4 1 1280.2.n.s 12
20.e even 4 1 640.2.o.j 12
40.i odd 4 1 inner 640.2.o.k yes 12
40.k even 4 1 inner 640.2.o.k yes 12
80.i odd 4 1 1280.2.n.s 12
80.j even 4 1 1280.2.n.r 12
80.s even 4 1 1280.2.n.s 12
80.t odd 4 1 1280.2.n.r 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.o.j 12 5.c odd 4 1
640.2.o.j 12 8.b even 2 1
640.2.o.j 12 8.d odd 2 1
640.2.o.j 12 20.e even 4 1
640.2.o.k yes 12 1.a even 1 1 trivial
640.2.o.k yes 12 4.b odd 2 1 inner
640.2.o.k yes 12 40.i odd 4 1 inner
640.2.o.k yes 12 40.k even 4 1 inner
1280.2.n.r 12 16.e even 4 1
1280.2.n.r 12 16.f odd 4 1
1280.2.n.r 12 80.j even 4 1
1280.2.n.r 12 80.t odd 4 1
1280.2.n.s 12 16.e even 4 1
1280.2.n.s 12 16.f odd 4 1
1280.2.n.s 12 80.i odd 4 1
1280.2.n.s 12 80.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(640, [\chi])$$:

 $$T_{3}^{12} + 80T_{3}^{8} + 640T_{3}^{4} + 256$$ T3^12 + 80*T3^8 + 640*T3^4 + 256 $$T_{7}^{12} + 304T_{7}^{8} + 1664T_{7}^{4} + 256$$ T7^12 + 304*T7^8 + 1664*T7^4 + 256 $$T_{13}^{6} - 10T_{13}^{5} + 50T_{13}^{4} - 80T_{13}^{3} + 36T_{13}^{2} + 120T_{13} + 200$$ T13^6 - 10*T13^5 + 50*T13^4 - 80*T13^3 + 36*T13^2 + 120*T13 + 200

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 80 T^{8} + \cdots + 256$$
$5$ $$(T^{6} - T^{4} - 16 T^{3} + \cdots + 125)^{2}$$
$7$ $$T^{12} + 304 T^{8} + \cdots + 256$$
$11$ $$(T^{6} - 48 T^{4} + \cdots - 3200)^{2}$$
$13$ $$(T^{6} - 10 T^{5} + \cdots + 200)^{2}$$
$17$ $$(T^{6} + 2 T^{5} + \cdots + 200)^{2}$$
$19$ $$(T^{6} + 88 T^{4} + \cdots + 12800)^{2}$$
$23$ $$T^{12} + 4208 T^{8} + \cdots + 256$$
$29$ $$(T^{3} + 10 T^{2} + \cdots - 40)^{4}$$
$31$ $$(T^{6} + 136 T^{4} + \cdots + 80000)^{2}$$
$37$ $$(T^{6} - 2 T^{5} + \cdots + 57800)^{2}$$
$41$ $$(T^{3} + 4 T^{2} - 48 T + 80)^{4}$$
$43$ $$T^{12} + 80 T^{8} + \cdots + 256$$
$47$ $$T^{12} + \cdots + 479785216$$
$53$ $$(T^{6} + 18 T^{5} + \cdots + 200)^{2}$$
$59$ $$(T^{6} + 184 T^{4} + \cdots + 12800)^{2}$$
$61$ $$(T^{6} + 60 T^{4} + \cdots + 64)^{2}$$
$67$ $$T^{12} + \cdots + 133090713856$$
$71$ $$(T^{6} + 424 T^{4} + \cdots + 1692800)^{2}$$
$73$ $$(T^{6} + 14 T^{5} + \cdots + 200)^{2}$$
$79$ $$(T^{6} - 352 T^{4} + \cdots - 204800)^{2}$$
$83$ $$T^{12} + \cdots + 133090713856$$
$89$ $$(T^{6} + 128 T^{4} + \cdots + 16384)^{2}$$
$97$ $$(T^{6} - 10 T^{5} + \cdots + 5000)^{2}$$