# Properties

 Label 256.9.d.e 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: $$\Q(i, \sqrt{39})$$ 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 4) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - 305 \beta_1 q^{5} - 7 \beta_{3} q^{7} + 3423 q^{9}+O(q^{10})$$ q - b2 * q^3 - 305*b1 * q^5 - 7*b3 * q^7 + 3423 * q^9 $$q - \beta_{2} q^{3} - 305 \beta_1 q^{5} - 7 \beta_{3} q^{7} + 3423 q^{9} - 185 \beta_{2} q^{11} - 2735 \beta_1 q^{13} + 305 \beta_{3} q^{15} + 73090 q^{17} + 195 \beta_{2} q^{19} + 69888 \beta_1 q^{21} + 1187 \beta_{3} q^{23} + 18525 q^{25} + 3138 \beta_{2} q^{27} - 64111 \beta_1 q^{29} - 340 \beta_{3} q^{31} + 1847040 q^{33} - 8540 \beta_{2} q^{35} + 1736015 \beta_1 q^{37} + 2735 \beta_{3} q^{39} - 2146882 q^{41} - 59329 \beta_{2} q^{43} - 1044015 \beta_1 q^{45} + 38162 \beta_{3} q^{47} + 3807937 q^{49} - 73090 \beta_{2} q^{51} - 412145 \beta_1 q^{53} + 56425 \beta_{3} q^{55} - 1946880 q^{57} - 37285 \beta_{2} q^{59} - 7373039 \beta_1 q^{61} - 23961 \beta_{3} q^{63} - 3336700 q^{65} - 152689 \beta_{2} q^{67} - 11851008 \beta_1 q^{69} + 5985 \beta_{3} q^{71} + 5725630 q^{73} - 18525 \beta_{2} q^{75} + 12929280 \beta_1 q^{77} + 179710 \beta_{3} q^{79} - 53788095 q^{81} + 520019 \beta_{2} q^{83} - 22292450 \beta_1 q^{85} + 64111 \beta_{3} q^{87} + 83324222 q^{89} - 76580 \beta_{2} q^{91} + 3394560 \beta_1 q^{93} - 59475 \beta_{3} q^{95} + 120619010 q^{97} - 633255 \beta_{2} q^{99}+O(q^{100})$$ q - b2 * q^3 - 305*b1 * q^5 - 7*b3 * q^7 + 3423 * q^9 - 185*b2 * q^11 - 2735*b1 * q^13 + 305*b3 * q^15 + 73090 * q^17 + 195*b2 * q^19 + 69888*b1 * q^21 + 1187*b3 * q^23 + 18525 * q^25 + 3138*b2 * q^27 - 64111*b1 * q^29 - 340*b3 * q^31 + 1847040 * q^33 - 8540*b2 * q^35 + 1736015*b1 * q^37 + 2735*b3 * q^39 - 2146882 * q^41 - 59329*b2 * q^43 - 1044015*b1 * q^45 + 38162*b3 * q^47 + 3807937 * q^49 - 73090*b2 * q^51 - 412145*b1 * q^53 + 56425*b3 * q^55 - 1946880 * q^57 - 37285*b2 * q^59 - 7373039*b1 * q^61 - 23961*b3 * q^63 - 3336700 * q^65 - 152689*b2 * q^67 - 11851008*b1 * q^69 + 5985*b3 * q^71 + 5725630 * q^73 - 18525*b2 * q^75 + 12929280*b1 * q^77 + 179710*b3 * q^79 - 53788095 * q^81 + 520019*b2 * q^83 - 22292450*b1 * q^85 + 64111*b3 * q^87 + 83324222 * q^89 - 76580*b2 * q^91 + 3394560*b1 * q^93 - 59475*b3 * q^95 + 120619010 * q^97 - 633255*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 13692 q^{9}+O(q^{10})$$ 4 * q + 13692 * q^9 $$4 q + 13692 q^{9} + 292360 q^{17} + 74100 q^{25} + 7388160 q^{33} - 8587528 q^{41} + 15231748 q^{49} - 7787520 q^{57} - 13346800 q^{65} + 22902520 q^{73} - 215152380 q^{81} + 333296888 q^{89} + 482476040 q^{97}+O(q^{100})$$ 4 * q + 13692 * q^9 + 292360 * q^17 + 74100 * q^25 + 7388160 * q^33 - 8587528 * q^41 + 15231748 * q^49 - 7787520 * q^57 - 13346800 * q^65 + 22902520 * q^73 - 215152380 * q^81 + 333296888 * q^89 + 482476040 * q^97

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 −99.9200 0 610.000i 0 1398.88i 0 3423.00 0
127.2 0 −99.9200 0 610.000i 0 1398.88i 0 3423.00 0
127.3 0 99.9200 0 610.000i 0 1398.88i 0 3423.00 0
127.4 0 99.9200 0 610.000i 0 1398.88i 0 3423.00 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.e 4
4.b odd 2 1 inner 256.9.d.e 4
8.b even 2 1 inner 256.9.d.e 4
8.d odd 2 1 inner 256.9.d.e 4
16.e even 4 1 4.9.b.b 2
16.e even 4 1 64.9.c.b 2
16.f odd 4 1 4.9.b.b 2
16.f odd 4 1 64.9.c.b 2
48.i odd 4 1 36.9.d.b 2
48.k even 4 1 36.9.d.b 2
80.i odd 4 1 100.9.d.b 4
80.j even 4 1 100.9.d.b 4
80.k odd 4 1 100.9.b.c 2
80.q even 4 1 100.9.b.c 2
80.s even 4 1 100.9.d.b 4
80.t odd 4 1 100.9.d.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 9984$$ acting on $$S_{9}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 9984)^{2}$$
$5$ $$(T^{2} + 372100)^{2}$$
$7$ $$(T^{2} + 1956864)^{2}$$
$11$ $$(T^{2} - 341702400)^{2}$$
$13$ $$(T^{2} + 29920900)^{2}$$
$17$ $$(T - 73090)^{4}$$
$19$ $$(T^{2} - 379641600)^{2}$$
$23$ $$(T^{2} + 56268585984)^{2}$$
$29$ $$(T^{2} + 16440881284)^{2}$$
$31$ $$(T^{2} + 4616601600)^{2}$$
$37$ $$(T^{2} + 12054992320900)^{2}$$
$41$ $$(T + 2146882)^{4}$$
$43$ $$(T^{2} - 35142983526144)^{2}$$
$47$ $$(T^{2} + 58160324112384)^{2}$$
$53$ $$(T^{2} + 679454004100)^{2}$$
$59$ $$(T^{2} - 13879469510400)^{2}$$
$61$ $$(T^{2} + 217446816382084)^{2}$$
$67$ $$(T^{2} - 232766284318464)^{2}$$
$71$ $$(T^{2} + 1430516505600)^{2}$$
$73$ $$(T - 5725630)^{4}$$
$79$ $$(T^{2} + 12\!\cdots\!00)^{2}$$
$83$ $$(T^{2} - 26\!\cdots\!24)^{2}$$
$89$ $$(T - 83324222)^{4}$$
$97$ $$(T - 120619010)^{4}$$