# Properties

 Label 256.9.d.f Level $256$ Weight $9$ Character orbit 256.d Analytic conductor $104.289$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(127,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 256.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: $$104.288924176$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 17x^{2} + 81$$ x^4 - 17*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{10}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 16) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - 255 \beta_1 q^{5} - 9 \beta_{3} q^{7} + 13599 q^{9}+O(q^{10})$$ q - b2 * q^3 - 255*b1 * q^5 - 9*b3 * q^7 + 13599 * q^9 $$q - \beta_{2} q^{3} - 255 \beta_1 q^{5} - 9 \beta_{3} q^{7} + 13599 q^{9} + 135 \beta_{2} q^{11} + 13855 \beta_1 q^{13} + 255 \beta_{3} q^{15} + 50370 q^{17} - 765 \beta_{2} q^{19} + 181440 \beta_1 q^{21} + 621 \beta_{3} q^{23} + 130525 q^{25} - 7038 \beta_{2} q^{27} - 27489 \beta_1 q^{29} - 4140 \beta_{3} q^{31} - 2721600 q^{33} - 9180 \beta_{2} q^{35} + 396865 \beta_1 q^{37} - 13855 \beta_{3} q^{39} + 75582 q^{41} + 3519 \beta_{2} q^{43} - 3467745 \beta_1 q^{45} - 10098 \beta_{3} q^{47} - 767039 q^{49} - 50370 \beta_{2} q^{51} + 5583105 \beta_1 q^{53} - 34425 \beta_{3} q^{55} + 15422400 q^{57} - 153765 \beta_{2} q^{59} + 11913311 \beta_1 q^{61} - 122391 \beta_{3} q^{63} + 14132100 q^{65} - 52785 \beta_{2} q^{67} - 12519360 \beta_1 q^{69} - 35505 \beta_{3} q^{71} - 6516610 q^{73} - 130525 \beta_{2} q^{75} - 24494400 \beta_1 q^{77} + 171810 \beta_{3} q^{79} + 52663041 q^{81} - 517293 \beta_{2} q^{83} - 12844350 \beta_1 q^{85} + 27489 \beta_{3} q^{87} - 86795778 q^{89} + 498780 \beta_{2} q^{91} + 83462400 \beta_1 q^{93} + 195075 \beta_{3} q^{95} - 46670270 q^{97} + 1835865 \beta_{2} q^{99}+O(q^{100})$$ q - b2 * q^3 - 255*b1 * q^5 - 9*b3 * q^7 + 13599 * q^9 + 135*b2 * q^11 + 13855*b1 * q^13 + 255*b3 * q^15 + 50370 * q^17 - 765*b2 * q^19 + 181440*b1 * q^21 + 621*b3 * q^23 + 130525 * q^25 - 7038*b2 * q^27 - 27489*b1 * q^29 - 4140*b3 * q^31 - 2721600 * q^33 - 9180*b2 * q^35 + 396865*b1 * q^37 - 13855*b3 * q^39 + 75582 * q^41 + 3519*b2 * q^43 - 3467745*b1 * q^45 - 10098*b3 * q^47 - 767039 * q^49 - 50370*b2 * q^51 + 5583105*b1 * q^53 - 34425*b3 * q^55 + 15422400 * q^57 - 153765*b2 * q^59 + 11913311*b1 * q^61 - 122391*b3 * q^63 + 14132100 * q^65 - 52785*b2 * q^67 - 12519360*b1 * q^69 - 35505*b3 * q^71 - 6516610 * q^73 - 130525*b2 * q^75 - 24494400*b1 * q^77 + 171810*b3 * q^79 + 52663041 * q^81 - 517293*b2 * q^83 - 12844350*b1 * q^85 + 27489*b3 * q^87 - 86795778 * q^89 + 498780*b2 * q^91 + 83462400*b1 * q^93 + 195075*b3 * q^95 - 46670270 * q^97 + 1835865*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 54396 q^{9}+O(q^{10})$$ 4 * q + 54396 * q^9 $$4 q + 54396 q^{9} + 201480 q^{17} + 522100 q^{25} - 10886400 q^{33} + 302328 q^{41} - 3068156 q^{49} + 61689600 q^{57} + 56528400 q^{65} - 26066440 q^{73} + 210652164 q^{81} - 347183112 q^{89} - 186681080 q^{97}+O(q^{100})$$ 4 * q + 54396 * q^9 + 201480 * q^17 + 522100 * q^25 - 10886400 * q^33 + 302328 * q^41 - 3068156 * q^49 + 61689600 * q^57 + 56528400 * q^65 - 26066440 * q^73 + 210652164 * q^81 - 347183112 * q^89 - 186681080 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 17x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} - 16\nu ) / 9$$ (2*v^3 - 16*v) / 9 $$\beta_{2}$$ $$=$$ $$( -8\nu^{3} + 208\nu ) / 3$$ (-8*v^3 + 208*v) / 3 $$\beta_{3}$$ $$=$$ $$96\nu^{2} - 816$$ 96*v^2 - 816
 $$\nu$$ $$=$$ $$( \beta_{2} + 12\beta_1 ) / 48$$ (b2 + 12*b1) / 48 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 816 ) / 96$$ (b3 + 816) / 96 $$\nu^{3}$$ $$=$$ $$( \beta_{2} + 39\beta_1 ) / 6$$ (b2 + 39*b1) / 6

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 2.95804 + 0.500000i 2.95804 − 0.500000i −2.95804 + 0.500000i −2.95804 − 0.500000i
0 −141.986 0 510.000i 0 2555.75i 0 13599.0 0
127.2 0 −141.986 0 510.000i 0 2555.75i 0 13599.0 0
127.3 0 141.986 0 510.000i 0 2555.75i 0 13599.0 0
127.4 0 141.986 0 510.000i 0 2555.75i 0 13599.0 0
 $$n$$: e.g. 2-40 or 990-1000 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
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.9.d.f 4
4.b odd 2 1 inner 256.9.d.f 4
8.b even 2 1 inner 256.9.d.f 4
8.d odd 2 1 inner 256.9.d.f 4
16.e even 4 1 16.9.c.a 2
16.e even 4 1 64.9.c.d 2
16.f odd 4 1 16.9.c.a 2
16.f odd 4 1 64.9.c.d 2
48.i odd 4 1 144.9.g.g 2
48.k even 4 1 144.9.g.g 2
80.i odd 4 1 400.9.h.b 4
80.j even 4 1 400.9.h.b 4
80.k odd 4 1 400.9.b.c 2
80.q even 4 1 400.9.b.c 2
80.s even 4 1 400.9.h.b 4
80.t odd 4 1 400.9.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.a 2 16.e even 4 1
16.9.c.a 2 16.f odd 4 1
64.9.c.d 2 16.e even 4 1
64.9.c.d 2 16.f odd 4 1
144.9.g.g 2 48.i odd 4 1
144.9.g.g 2 48.k even 4 1
256.9.d.f 4 1.a even 1 1 trivial
256.9.d.f 4 4.b odd 2 1 inner
256.9.d.f 4 8.b even 2 1 inner
256.9.d.f 4 8.d odd 2 1 inner
400.9.b.c 2 80.k odd 4 1
400.9.b.c 2 80.q even 4 1
400.9.h.b 4 80.i odd 4 1
400.9.h.b 4 80.j even 4 1
400.9.h.b 4 80.s even 4 1
400.9.h.b 4 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 20160$$ acting on $$S_{9}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 20160)^{2}$$
$5$ $$(T^{2} + 260100)^{2}$$
$7$ $$(T^{2} + 6531840)^{2}$$
$11$ $$(T^{2} - 367416000)^{2}$$
$13$ $$(T^{2} + 767844100)^{2}$$
$17$ $$(T - 50370)^{4}$$
$19$ $$(T^{2} - 11798136000)^{2}$$
$23$ $$(T^{2} + 31098090240)^{2}$$
$29$ $$(T^{2} + 3022580484)^{2}$$
$31$ $$(T^{2} + 1382137344000)^{2}$$
$37$ $$(T^{2} + 630007312900)^{2}$$
$41$ $$(T - 75582)^{4}$$
$43$ $$(T^{2} - 249648557760)^{2}$$
$47$ $$(T^{2} + 8222828866560)^{2}$$
$53$ $$(T^{2} + 124684245764100)^{2}$$
$59$ $$(T^{2} - 476656492536000)^{2}$$
$61$ $$(T^{2} + 567707915930884)^{2}$$
$67$ $$(T^{2} - 56170925496000)^{2}$$
$71$ $$(T^{2} + 101655189216000)^{2}$$
$73$ $$(T + 6516610)^{4}$$
$79$ $$(T^{2} + 23\!\cdots\!00)^{2}$$
$83$ $$(T^{2} - 53\!\cdots\!40)^{2}$$
$89$ $$(T + 86795778)^{4}$$
$97$ $$(T + 46670270)^{4}$$