# Properties

 Label 256.9.d.h Level $256$ Weight $9$ Character orbit 256.d Analytic conductor $104.289$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# 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: $$\Q(i, \sqrt{39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 19x^{2} + 100$$ x^4 - 19*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 32) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 36) q^{3} + (2 \beta_{3} + 91 \beta_1) q^{5} + (11 \beta_{3} + 236 \beta_1) q^{7} + (72 \beta_{2} + 4719) q^{9}+O(q^{10})$$ q + (b2 + 36) * q^3 + (2*b3 + 91*b1) * q^5 + (11*b3 + 236*b1) * q^7 + (72*b2 + 4719) * q^9 $$q + (\beta_{2} + 36) q^{3} + (2 \beta_{3} + 91 \beta_1) q^{5} + (11 \beta_{3} + 236 \beta_1) q^{7} + (72 \beta_{2} + 4719) q^{9} + (41 \beta_{2} + 10052) q^{11} + (230 \beta_{3} - 1579 \beta_1) q^{13} + (163 \beta_{3} + 23244 \beta_1) q^{15} + ( - 424 \beta_{2} - 97998) q^{17} + (365 \beta_{2} - 166188) q^{19} + (632 \beta_{3} + 118320 \beta_1) q^{21} + (1017 \beta_{3} - 3164 \beta_1) q^{23} + ( - 1456 \beta_{2} + 197757) q^{25} + (750 \beta_{2} + 652536) q^{27} + ( - 898 \beta_{3} - 88187 \beta_1) q^{29} + (3412 \beta_{3} - 339248 \beta_1) q^{31} + (11528 \beta_{2} + 771216) q^{33} + ( - 5892 \beta_{2} - 964496) q^{35} + (1018 \beta_{3} - 555381 \beta_1) q^{37} + (6701 \beta_{3} + 2239476 \beta_1) q^{39} + (38096 \beta_{2} - 738338) q^{41} + ( - 42527 \beta_{2} - 761692) q^{43} + (15990 \beta_{3} + 1867125 \beta_1) q^{45} + (9958 \beta_{3} - 2621800 \beta_1) q^{47} + ( - 20768 \beta_{2} + 709761) q^{49} + ( - 113262 \beta_{2} - 7761144) q^{51} + ( - 6806 \beta_{3} + 562731 \beta_1) q^{53} + (23835 \beta_{3} + 1733420 \beta_1) q^{55} + ( - 153048 \beta_{2} - 2338608) q^{57} + ( - 113435 \beta_{2} + 3544628) q^{59} + ( - 70938 \beta_{3} - 5019883 \beta_1) q^{61} + (68901 \beta_{3} + 9021012 \beta_1) q^{63} + ( - 71088 \beta_{2} - 17795804) q^{65} + (64673 \beta_{2} + 417316) q^{67} + (33448 \beta_{3} + 10039824 \beta_1) q^{69} + (119571 \beta_{3} - 9064564 \beta_1) q^{71} + (104936 \beta_{2} + 1480206) q^{73} + (145341 \beta_{2} - 7417452) q^{75} + (120248 \beta_{3} + 6875056 \beta_1) q^{77} + (71994 \beta_{3} + 10602344 \beta_1) q^{79} + (207144 \beta_{2} + 17937) q^{81} + ( - 189139 \beta_{2} + 39279188) q^{83} + ( - 234580 \beta_{3} - 17384250 \beta_1) q^{85} + ( - 120515 \beta_{3} - 12140364 \beta_1) q^{87} + (451240 \beta_{2} - 33161970) q^{89} + ( - 147644 \beta_{2} - 99547504) q^{91} + ( - 216416 \beta_{3} + 21852480 \beta_1) q^{93} + ( - 299161 \beta_{3} - 7834788 \beta_1) q^{95} + ( - 460680 \beta_{2} + 14111218) q^{97} + (917223 \beta_{2} + 76908156) q^{99}+O(q^{100})$$ q + (b2 + 36) * q^3 + (2*b3 + 91*b1) * q^5 + (11*b3 + 236*b1) * q^7 + (72*b2 + 4719) * q^9 + (41*b2 + 10052) * q^11 + (230*b3 - 1579*b1) * q^13 + (163*b3 + 23244*b1) * q^15 + (-424*b2 - 97998) * q^17 + (365*b2 - 166188) * q^19 + (632*b3 + 118320*b1) * q^21 + (1017*b3 - 3164*b1) * q^23 + (-1456*b2 + 197757) * q^25 + (750*b2 + 652536) * q^27 + (-898*b3 - 88187*b1) * q^29 + (3412*b3 - 339248*b1) * q^31 + (11528*b2 + 771216) * q^33 + (-5892*b2 - 964496) * q^35 + (1018*b3 - 555381*b1) * q^37 + (6701*b3 + 2239476*b1) * q^39 + (38096*b2 - 738338) * q^41 + (-42527*b2 - 761692) * q^43 + (15990*b3 + 1867125*b1) * q^45 + (9958*b3 - 2621800*b1) * q^47 + (-20768*b2 + 709761) * q^49 + (-113262*b2 - 7761144) * q^51 + (-6806*b3 + 562731*b1) * q^53 + (23835*b3 + 1733420*b1) * q^55 + (-153048*b2 - 2338608) * q^57 + (-113435*b2 + 3544628) * q^59 + (-70938*b3 - 5019883*b1) * q^61 + (68901*b3 + 9021012*b1) * q^63 + (-71088*b2 - 17795804) * q^65 + (64673*b2 + 417316) * q^67 + (33448*b3 + 10039824*b1) * q^69 + (119571*b3 - 9064564*b1) * q^71 + (104936*b2 + 1480206) * q^73 + (145341*b2 - 7417452) * q^75 + (120248*b3 + 6875056*b1) * q^77 + (71994*b3 + 10602344*b1) * q^79 + (207144*b2 + 17937) * q^81 + (-189139*b2 + 39279188) * q^83 + (-234580*b3 - 17384250*b1) * q^85 + (-120515*b3 - 12140364*b1) * q^87 + (451240*b2 - 33161970) * q^89 + (-147644*b2 - 99547504) * q^91 + (-216416*b3 + 21852480*b1) * q^93 + (-299161*b3 - 7834788*b1) * q^95 + (-460680*b2 + 14111218) * q^97 + (917223*b2 + 76908156) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 144 q^{3} + 18876 q^{9}+O(q^{10})$$ 4 * q + 144 * q^3 + 18876 * q^9 $$4 q + 144 q^{3} + 18876 q^{9} + 40208 q^{11} - 391992 q^{17} - 664752 q^{19} + 791028 q^{25} + 2610144 q^{27} + 3084864 q^{33} - 3857984 q^{35} - 2953352 q^{41} - 3046768 q^{43} + 2839044 q^{49} - 31044576 q^{51} - 9354432 q^{57} + 14178512 q^{59} - 71183216 q^{65} + 1669264 q^{67} + 5920824 q^{73} - 29669808 q^{75} + 71748 q^{81} + 157116752 q^{83} - 132647880 q^{89} - 398190016 q^{91} + 56444872 q^{97} + 307632624 q^{99}+O(q^{100})$$ 4 * q + 144 * q^3 + 18876 * q^9 + 40208 * q^11 - 391992 * q^17 - 664752 * q^19 + 791028 * q^25 + 2610144 * q^27 + 3084864 * q^33 - 3857984 * q^35 - 2953352 * q^41 - 3046768 * q^43 + 2839044 * q^49 - 31044576 * q^51 - 9354432 * q^57 + 14178512 * q^59 - 71183216 * q^65 + 1669264 * q^67 + 5920824 * q^73 - 29669808 * q^75 + 71748 * q^81 + 157116752 * q^83 - 132647880 * q^89 - 398190016 * q^91 + 56444872 * q^97 + 307632624 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 19x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 9\nu ) / 5$$ (v^3 - 9*v) / 5 $$\beta_{2}$$ $$=$$ $$( -8\nu^{3} + 232\nu ) / 5$$ (-8*v^3 + 232*v) / 5 $$\beta_{3}$$ $$=$$ $$64\nu^{2} - 608$$ 64*v^2 - 608
 $$\nu$$ $$=$$ $$( \beta_{2} + 8\beta_1 ) / 32$$ (b2 + 8*b1) / 32 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 608 ) / 64$$ (b3 + 608) / 64 $$\nu^{3}$$ $$=$$ $$( 9\beta_{2} + 232\beta_1 ) / 32$$ (9*b2 + 232*b1) / 32

## 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
 −3.12250 + 0.500000i −3.12250 − 0.500000i 3.12250 − 0.500000i 3.12250 + 0.500000i
0 −63.9200 0 217.680i 0 1726.24i 0 −2475.24 0
127.2 0 −63.9200 0 217.680i 0 1726.24i 0 −2475.24 0
127.3 0 135.920 0 581.680i 0 2670.24i 0 11913.2 0
127.4 0 135.920 0 581.680i 0 2670.24i 0 11913.2 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
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.h 4
4.b odd 2 1 256.9.d.b 4
8.b even 2 1 256.9.d.b 4
8.d odd 2 1 inner 256.9.d.h 4
16.e even 4 1 32.9.c.a 4
16.e even 4 1 64.9.c.f 4
16.f odd 4 1 32.9.c.a 4
16.f odd 4 1 64.9.c.f 4
48.i odd 4 1 288.9.g.b 4
48.k even 4 1 288.9.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.9.c.a 4 16.e even 4 1
32.9.c.a 4 16.f odd 4 1
64.9.c.f 4 16.e even 4 1
64.9.c.f 4 16.f odd 4 1
256.9.d.b 4 4.b odd 2 1
256.9.d.b 4 8.b even 2 1
256.9.d.h 4 1.a even 1 1 trivial
256.9.d.h 4 8.d odd 2 1 inner
288.9.g.b 4 48.i odd 4 1
288.9.g.b 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 72T_{3} - 8688$$ acting on $$S_{9}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 72 T - 8688)^{2}$$
$5$ $$T^{4} + \cdots + 16032624400$$
$7$ $$T^{4} + \cdots + 21247232118784$$
$11$ $$(T^{2} - 20104 T + 84259600)^{2}$$
$13$ $$T^{4} + \cdots + 44\!\cdots\!96$$
$17$ $$(T^{2} + 195996 T + 7808724420)^{2}$$
$19$ $$(T^{2} + 332376 T + 26288332944)^{2}$$
$23$ $$T^{4} + \cdots + 17\!\cdots\!00$$
$29$ $$T^{4} + \cdots + 12\!\cdots\!24$$
$31$ $$T^{4} + \cdots + 20\!\cdots\!24$$
$37$ $$T^{4} + \cdots + 14\!\cdots\!00$$
$41$ $$(T^{2} + \cdots - 13944688274300)^{2}$$
$43$ $$(T^{2} + \cdots - 17476345855472)^{2}$$
$47$ $$T^{4} + \cdots + 55\!\cdots\!16$$
$53$ $$T^{4} + \cdots + 34\!\cdots\!04$$
$59$ $$(T^{2} + \cdots - 115904724604016)^{2}$$
$61$ $$T^{4} + \cdots + 10\!\cdots\!84$$
$67$ $$(T^{2} + \cdots - 41584895095280)^{2}$$
$71$ $$T^{4} + \cdots + 58\!\cdots\!64$$
$73$ $$(T^{2} + \cdots - 107748446132028)^{2}$$
$79$ $$T^{4} + \cdots + 58\!\cdots\!04$$
$83$ $$(T^{2} + \cdots + 11\!\cdots\!80)^{2}$$
$89$ $$(T^{2} + \cdots - 933201241117500)^{2}$$
$97$ $$(T^{2} + \cdots - 19\!\cdots\!76)^{2}$$