# Properties

 Label 1280.4.d.ba Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not 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: $$75.5224448073$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.12559936.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 6x^{3} + 49x^{2} - 42x + 18$$ x^6 - 6*x^3 + 49*x^2 - 42*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 640) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_{2}) q^{3} - 5 \beta_{2} q^{5} + (\beta_1 + 7) q^{7} + ( - 2 \beta_1 - 7) q^{9}+O(q^{10})$$ q + (b4 - b2) * q^3 - 5*b2 * q^5 + (b1 + 7) * q^7 + (-2*b1 - 7) * q^9 $$q + (\beta_{4} - \beta_{2}) q^{3} - 5 \beta_{2} q^{5} + (\beta_1 + 7) q^{7} + ( - 2 \beta_1 - 7) q^{9} + (3 \beta_{5} + 3 \beta_{4} + 10 \beta_{2}) q^{11} + (4 \beta_{5} - 2 \beta_{4}) q^{13} + (5 \beta_{3} - 5) q^{15} + ( - 10 \beta_{3} - 4 \beta_1) q^{17} + (5 \beta_{5} - 13 \beta_{4} - 48 \beta_{2}) q^{19} + ( - 2 \beta_{5} + 14 \beta_{4} - 16 \beta_{2}) q^{21} + ( - 8 \beta_{3} - 5 \beta_1 + 53) q^{23} - 25 q^{25} + (4 \beta_{5} + 6 \beta_{4} - 2 \beta_{2}) q^{27} + ( - 18 \beta_{5} + 6 \beta_{4} + 22 \beta_{2}) q^{29} + (16 \beta_{3} + 10 \beta_1 - 50) q^{31} + ( - 34 \beta_{3} - 62) q^{33} + ( - 5 \beta_{5} - 35 \beta_{2}) q^{35} + (4 \beta_{5} + 48 \beta_{4} - 74 \beta_{2}) q^{37} + ( - 26 \beta_{3} + 12 \beta_1 + 102) q^{39} + ( - 12 \beta_{3} - 10 \beta_1 + 116) q^{41} + ( - 34 \beta_{5} + 13 \beta_{4} - 83 \beta_{2}) q^{43} + (10 \beta_{5} + 35 \beta_{2}) q^{45} + ( - 18 \beta_{3} - 19 \beta_1 - 147) q^{47} + ( - 8 \beta_{3} + 6 \beta_1 - 165) q^{49} + ( - 12 \beta_{5} - 38 \beta_{4} - 294 \beta_{2}) q^{51} + ( - 12 \beta_{5} - 10 \beta_{4} - 396 \beta_{2}) q^{53} + (15 \beta_{3} + 15 \beta_1 + 50) q^{55} + (26 \beta_{3} + 36 \beta_1 + 426) q^{57} + (33 \beta_{5} + 11 \beta_{4} + 28 \beta_{2}) q^{59} + (8 \beta_{5} + 64 \beta_{4} + 230 \beta_{2}) q^{61} + (16 \beta_{3} - 5 \beta_1 - 307) q^{63} + ( - 10 \beta_{3} + 20 \beta_1) q^{65} + (14 \beta_{5} + 109 \beta_{4} - 419 \beta_{2}) q^{67} + ( - 6 \beta_{5} + 10 \beta_{4} - 272 \beta_{2}) q^{69} + (14 \beta_{3} - 48 \beta_1 - 14) q^{71} + ( - 150 \beta_{3} - 4 \beta_1 + 232) q^{73} + ( - 25 \beta_{4} + 25 \beta_{2}) q^{75} + (4 \beta_{5} + 18 \beta_{4} + 430 \beta_{2}) q^{77} + ( - 8 \beta_{3} + 12 \beta_1 - 684) q^{79} + ( - 32 \beta_{3} - 58 \beta_1 - 353) q^{81} + ( - 24 \beta_{5} - 39 \beta_{4} - 633 \beta_{2}) q^{83} + (20 \beta_{5} + 50 \beta_{4}) q^{85} + (98 \beta_{3} - 48 \beta_1 - 338) q^{87} + ( - 120 \beta_{3} - 28 \beta_1 + 646) q^{89} + ( - 2 \beta_{5} - 60 \beta_{4} + 534 \beta_{2}) q^{91} + (12 \beta_{5} + 36 \beta_{4} + 488 \beta_{2}) q^{93} + ( - 65 \beta_{3} + 25 \beta_1 - 240) q^{95} + ( - 74 \beta_{3} + 88 \beta_1 - 252) q^{97} + (13 \beta_{5} - 15 \beta_{4} - 790 \beta_{2}) q^{99}+O(q^{100})$$ q + (b4 - b2) * q^3 - 5*b2 * q^5 + (b1 + 7) * q^7 + (-2*b1 - 7) * q^9 + (3*b5 + 3*b4 + 10*b2) * q^11 + (4*b5 - 2*b4) * q^13 + (5*b3 - 5) * q^15 + (-10*b3 - 4*b1) * q^17 + (5*b5 - 13*b4 - 48*b2) * q^19 + (-2*b5 + 14*b4 - 16*b2) * q^21 + (-8*b3 - 5*b1 + 53) * q^23 - 25 * q^25 + (4*b5 + 6*b4 - 2*b2) * q^27 + (-18*b5 + 6*b4 + 22*b2) * q^29 + (16*b3 + 10*b1 - 50) * q^31 + (-34*b3 - 62) * q^33 + (-5*b5 - 35*b2) * q^35 + (4*b5 + 48*b4 - 74*b2) * q^37 + (-26*b3 + 12*b1 + 102) * q^39 + (-12*b3 - 10*b1 + 116) * q^41 + (-34*b5 + 13*b4 - 83*b2) * q^43 + (10*b5 + 35*b2) * q^45 + (-18*b3 - 19*b1 - 147) * q^47 + (-8*b3 + 6*b1 - 165) * q^49 + (-12*b5 - 38*b4 - 294*b2) * q^51 + (-12*b5 - 10*b4 - 396*b2) * q^53 + (15*b3 + 15*b1 + 50) * q^55 + (26*b3 + 36*b1 + 426) * q^57 + (33*b5 + 11*b4 + 28*b2) * q^59 + (8*b5 + 64*b4 + 230*b2) * q^61 + (16*b3 - 5*b1 - 307) * q^63 + (-10*b3 + 20*b1) * q^65 + (14*b5 + 109*b4 - 419*b2) * q^67 + (-6*b5 + 10*b4 - 272*b2) * q^69 + (14*b3 - 48*b1 - 14) * q^71 + (-150*b3 - 4*b1 + 232) * q^73 + (-25*b4 + 25*b2) * q^75 + (4*b5 + 18*b4 + 430*b2) * q^77 + (-8*b3 + 12*b1 - 684) * q^79 + (-32*b3 - 58*b1 - 353) * q^81 + (-24*b5 - 39*b4 - 633*b2) * q^83 + (20*b5 + 50*b4) * q^85 + (98*b3 - 48*b1 - 338) * q^87 + (-120*b3 - 28*b1 + 646) * q^89 + (-2*b5 - 60*b4 + 534*b2) * q^91 + (12*b5 + 36*b4 + 488*b2) * q^93 + (-65*b3 + 25*b1 - 240) * q^95 + (-74*b3 + 88*b1 - 252) * q^97 + (13*b5 - 15*b4 - 790*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 40 q^{7} - 38 q^{9}+O(q^{10})$$ 6 * q + 40 * q^7 - 38 * q^9 $$6 q + 40 q^{7} - 38 q^{9} - 20 q^{15} - 12 q^{17} + 312 q^{23} - 150 q^{25} - 288 q^{31} - 440 q^{33} + 536 q^{39} + 692 q^{41} - 880 q^{47} - 1018 q^{49} + 300 q^{55} + 2536 q^{57} - 1800 q^{63} - 60 q^{65} + 40 q^{71} + 1100 q^{73} - 4144 q^{79} - 2066 q^{81} - 1736 q^{87} + 3692 q^{89} - 1620 q^{95} - 1836 q^{97}+O(q^{100})$$ 6 * q + 40 * q^7 - 38 * q^9 - 20 * q^15 - 12 * q^17 + 312 * q^23 - 150 * q^25 - 288 * q^31 - 440 * q^33 + 536 * q^39 + 692 * q^41 - 880 * q^47 - 1018 * q^49 + 300 * q^55 + 2536 * q^57 - 1800 * q^63 - 60 * q^65 + 40 * q^71 + 1100 * q^73 - 4144 * q^79 - 2066 * q^81 - 1736 * q^87 + 3692 * q^89 - 1620 * q^95 - 1836 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 6x^{3} + 49x^{2} - 42x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{5} + 46\nu^{4} - 28\nu^{3} + 12\nu^{2} + 1671 ) / 167$$ (-4*v^5 + 46*v^4 - 28*v^3 + 12*v^2 + 1671) / 167 $$\beta_{2}$$ $$=$$ $$( 49\nu^{5} + 21\nu^{4} + 9\nu^{3} - 147\nu^{2} + 2338\nu - 1056 ) / 1002$$ (49*v^5 + 21*v^4 + 9*v^3 - 147*v^2 + 2338*v - 1056) / 1002 $$\beta_{3}$$ $$=$$ $$( -10\nu^{5} - 52\nu^{4} - 70\nu^{3} + 30\nu^{2} - 1083 ) / 167$$ (-10*v^5 - 52*v^4 - 70*v^3 + 30*v^2 - 1083) / 167 $$\beta_{4}$$ $$=$$ $$( -133\nu^{5} - 57\nu^{4} + 71\nu^{3} + 1067\nu^{2} - 5678\nu + 2580 ) / 334$$ (-133*v^5 - 57*v^4 + 71*v^3 + 1067*v^2 - 5678*v + 2580) / 334 $$\beta_{5}$$ $$=$$ $$( 161\nu^{5} + 69\nu^{4} + 125\nu^{3} - 1151\nu^{2} + 8350\nu - 3756 ) / 334$$ (161*v^5 + 69*v^4 + 125*v^3 - 1151*v^2 + 8350*v - 3756) / 334
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 ) / 8$$ (b5 + b4 + b3 + b1) / 8 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + \beta_{4} + 18\beta_{2} ) / 4$$ (-b5 + b4 + 18*b2) / 4 $$\nu^{3}$$ $$=$$ $$( 7\beta_{5} + 7\beta_{4} - 7\beta_{3} - 24\beta_{2} - 7\beta _1 + 24 ) / 8$$ (7*b5 + 7*b4 - 7*b3 - 24*b2 - 7*b1 + 24) / 8 $$\nu^{4}$$ $$=$$ $$( -2\beta_{3} + 5\beta _1 - 63 ) / 2$$ (-2*b3 + 5*b1 - 63) / 2 $$\nu^{5}$$ $$=$$ $$( -55\beta_{5} - 43\beta_{4} - 43\beta_{3} + 276\beta_{2} - 55\beta _1 + 276 ) / 8$$ (-55*b5 - 43*b4 - 43*b3 + 276*b2 - 55*b1 + 276) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 0.455589 + 0.455589i 1.60096 − 1.60096i −2.05655 − 2.05655i −2.05655 + 2.05655i 1.60096 + 1.60096i 0.455589 − 0.455589i
0 7.34740i 0 5.00000i 0 16.9921 0 −26.9842 0
641.2 0 6.65606i 0 5.00000i 0 12.1516 0 −17.3032 0
641.3 0 1.30867i 0 5.00000i 0 −9.14370 0 25.2874 0
641.4 0 1.30867i 0 5.00000i 0 −9.14370 0 25.2874 0
641.5 0 6.65606i 0 5.00000i 0 12.1516 0 −17.3032 0
641.6 0 7.34740i 0 5.00000i 0 16.9921 0 −26.9842 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 641.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.4.d.ba 6
4.b odd 2 1 1280.4.d.z 6
8.b even 2 1 inner 1280.4.d.ba 6
8.d odd 2 1 1280.4.d.z 6
16.e even 4 1 640.4.a.n yes 3
16.e even 4 1 640.4.a.o yes 3
16.f odd 4 1 640.4.a.m 3
16.f odd 4 1 640.4.a.p yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.m 3 16.f odd 4 1
640.4.a.n yes 3 16.e even 4 1
640.4.a.o yes 3 16.e even 4 1
640.4.a.p yes 3 16.f odd 4 1
1280.4.d.z 6 4.b odd 2 1
1280.4.d.z 6 8.d odd 2 1
1280.4.d.ba 6 1.a even 1 1 trivial
1280.4.d.ba 6 8.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{6} + 100T_{3}^{4} + 2560T_{3}^{2} + 4096$$ T3^6 + 100*T3^4 + 2560*T3^2 + 4096 $$T_{7}^{3} - 20T_{7}^{2} - 60T_{7} + 1888$$ T7^3 - 20*T7^2 - 60*T7 + 1888 $$T_{11}^{6} + 4332T_{11}^{4} + 5338416T_{11}^{2} + 1591690816$$ T11^6 + 4332*T11^4 + 5338416*T11^2 + 1591690816

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 100 T^{4} + 2560 T^{2} + \cdots + 4096$$
$5$ $$(T^{2} + 25)^{3}$$
$7$ $$(T^{3} - 20 T^{2} - 60 T + 1888)^{2}$$
$11$ $$T^{6} + 4332 T^{4} + \cdots + 1591690816$$
$13$ $$T^{6} + 6892 T^{4} + \cdots + 317980224$$
$17$ $$(T^{3} + 6 T^{2} - 7268 T + 154536)^{2}$$
$19$ $$T^{6} + 37516 T^{4} + \cdots + 1645637414976$$
$23$ $$(T^{3} - 156 T^{2} + 868 T + 249248)^{2}$$
$29$ $$T^{6} + 135564 T^{4} + \cdots + 2278578174016$$
$31$ $$(T^{3} + 144 T^{2} - 22064 T - 1385600)^{2}$$
$37$ $$T^{6} + \cdots + 266301927937600$$
$41$ $$(T^{3} - 346 T^{2} + 15708 T + 710312)^{2}$$
$43$ $$T^{6} + 493764 T^{4} + \cdots + 13\!\cdots\!00$$
$47$ $$(T^{3} + 440 T^{2} - 14860 T - 14079984)^{2}$$
$53$ $$T^{6} + 529932 T^{4} + \cdots + 23\!\cdots\!36$$
$59$ $$T^{6} + 420748 T^{4} + \cdots + 17\!\cdots\!64$$
$61$ $$T^{6} + 595276 T^{4} + \cdots + 35335796249664$$
$67$ $$T^{6} + 1641156 T^{4} + \cdots + 10\!\cdots\!00$$
$71$ $$(T^{3} - 20 T^{2} - 467520 T - 67907072)^{2}$$
$73$ $$(T^{3} - 550 T^{2} - 1001060 T + 367588568)^{2}$$
$79$ $$(T^{3} + 2072 T^{2} + 1398272 T + 307052544)^{2}$$
$83$ $$T^{6} + 1558980 T^{4} + \cdots + 23\!\cdots\!64$$
$89$ $$(T^{3} - 1846 T^{2} + 336652 T + 533827256)^{2}$$
$97$ $$(T^{3} + 918 T^{2} - 1607972 T - 660976152)^{2}$$