Properties

 Label 256.9.d.d 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: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - 129 \beta_1 q^{5} - 119 \beta_{3} q^{7} - 6369 q^{9}+O(q^{10})$$ q - b2 * q^3 - 129*b1 * q^5 - 119*b3 * q^7 - 6369 * q^9 $$q - \beta_{2} q^{3} - 129 \beta_1 q^{5} - 119 \beta_{3} q^{7} - 6369 q^{9} + 1671 \beta_{2} q^{11} + 9569 \beta_1 q^{13} + 129 \beta_{3} q^{15} - 58686 q^{17} + 11011 \beta_{2} q^{19} + 22848 \beta_1 q^{21} + 9747 \beta_{3} q^{23} + 324061 q^{25} + 12930 \beta_{2} q^{27} + 421089 \beta_1 q^{29} + 37932 \beta_{3} q^{31} - 320832 q^{33} - 61404 \beta_{2} q^{35} - 1274305 \beta_1 q^{37} - 9569 \beta_{3} q^{39} + 4324158 q^{41} - 147009 \beta_{2} q^{43} + 821601 \beta_1 q^{45} - 261006 \beta_{3} q^{47} - 5110847 q^{49} + 58686 \beta_{2} q^{51} - 596097 \beta_1 q^{53} - 215559 \beta_{3} q^{55} - 2114112 q^{57} + 24411 \beta_{2} q^{59} + 4207393 \beta_1 q^{61} + 757911 \beta_{3} q^{63} + 4937604 q^{65} - 1257521 \beta_{2} q^{67} - 1871424 \beta_1 q^{69} + 1114161 \beta_{3} q^{71} - 12735874 q^{73} - 324061 \beta_{2} q^{75} - 38179008 \beta_1 q^{77} - 226850 \beta_{3} q^{79} + 39304449 q^{81} + 5995347 \beta_{2} q^{83} + 7570494 \beta_1 q^{85} - 421089 \beta_{3} q^{87} + 16802814 q^{89} + 4554844 \beta_{2} q^{91} - 7282944 \beta_1 q^{93} - 1420419 \beta_{3} q^{95} + 120994882 q^{97} - 10642599 \beta_{2} q^{99}+O(q^{100})$$ q - b2 * q^3 - 129*b1 * q^5 - 119*b3 * q^7 - 6369 * q^9 + 1671*b2 * q^11 + 9569*b1 * q^13 + 129*b3 * q^15 - 58686 * q^17 + 11011*b2 * q^19 + 22848*b1 * q^21 + 9747*b3 * q^23 + 324061 * q^25 + 12930*b2 * q^27 + 421089*b1 * q^29 + 37932*b3 * q^31 - 320832 * q^33 - 61404*b2 * q^35 - 1274305*b1 * q^37 - 9569*b3 * q^39 + 4324158 * q^41 - 147009*b2 * q^43 + 821601*b1 * q^45 - 261006*b3 * q^47 - 5110847 * q^49 + 58686*b2 * q^51 - 596097*b1 * q^53 - 215559*b3 * q^55 - 2114112 * q^57 + 24411*b2 * q^59 + 4207393*b1 * q^61 + 757911*b3 * q^63 + 4937604 * q^65 - 1257521*b2 * q^67 - 1871424*b1 * q^69 + 1114161*b3 * q^71 - 12735874 * q^73 - 324061*b2 * q^75 - 38179008*b1 * q^77 - 226850*b3 * q^79 + 39304449 * q^81 + 5995347*b2 * q^83 + 7570494*b1 * q^85 - 421089*b3 * q^87 + 16802814 * q^89 + 4554844*b2 * q^91 - 7282944*b1 * q^93 - 1420419*b3 * q^95 + 120994882 * q^97 - 10642599*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 25476 q^{9}+O(q^{10})$$ 4 * q - 25476 * q^9 $$4 q - 25476 q^{9} - 234744 q^{17} + 1296244 q^{25} - 1283328 q^{33} + 17296632 q^{41} - 20443388 q^{49} - 8456448 q^{57} + 19750416 q^{65} - 50943496 q^{73} + 157217796 q^{81} + 67211256 q^{89} + 483979528 q^{97}+O(q^{100})$$ 4 * q - 25476 * q^9 - 234744 * q^17 + 1296244 * q^25 - 1283328 * q^33 + 17296632 * q^41 - 20443388 * q^49 - 8456448 * q^57 + 19750416 * q^65 - 50943496 * q^73 + 157217796 * q^81 + 67211256 * q^89 + 483979528 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{2}$$ $$=$$ $$-8\zeta_{12}^{3} + 16\zeta_{12}$$ -8*v^3 + 16*v $$\beta_{3}$$ $$=$$ $$32\zeta_{12}^{2} - 16$$ 32*v^2 - 16
 $$\zeta_{12}$$ $$=$$ $$( \beta_{2} + 4\beta_1 ) / 16$$ (b2 + 4*b1) / 16 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{3} + 16 ) / 32$$ (b3 + 16) / 32 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2

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
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 −13.8564 0 258.000i 0 3297.82i 0 −6369.00 0
127.2 0 −13.8564 0 258.000i 0 3297.82i 0 −6369.00 0
127.3 0 13.8564 0 258.000i 0 3297.82i 0 −6369.00 0
127.4 0 13.8564 0 258.000i 0 3297.82i 0 −6369.00 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.d 4
4.b odd 2 1 inner 256.9.d.d 4
8.b even 2 1 inner 256.9.d.d 4
8.d odd 2 1 inner 256.9.d.d 4
16.e even 4 1 16.9.c.b 2
16.e even 4 1 64.9.c.c 2
16.f odd 4 1 16.9.c.b 2
16.f odd 4 1 64.9.c.c 2
48.i odd 4 1 144.9.g.d 2
48.k even 4 1 144.9.g.d 2
80.i odd 4 1 400.9.h.a 4
80.j even 4 1 400.9.h.a 4
80.k odd 4 1 400.9.b.e 2
80.q even 4 1 400.9.b.e 2
80.s even 4 1 400.9.h.a 4
80.t odd 4 1 400.9.h.a 4

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 192$$ acting on $$S_{9}^{\mathrm{new}}(256, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 192)^{2}$$
$5$ $$(T^{2} + 66564)^{2}$$
$7$ $$(T^{2} + 10875648)^{2}$$
$11$ $$(T^{2} - 536110272)^{2}$$
$13$ $$(T^{2} + 366263044)^{2}$$
$17$ $$(T + 58686)^{4}$$
$19$ $$(T^{2} - 23278487232)^{2}$$
$23$ $$(T^{2} + 72963078912)^{2}$$
$29$ $$(T^{2} + 709263783684)^{2}$$
$31$ $$(T^{2} + 1105026527232)^{2}$$
$37$ $$(T^{2} + 6495412932100)^{2}$$
$41$ $$(T - 4324158)^{4}$$
$43$ $$(T^{2} - 4149436047552)^{2}$$
$47$ $$(T^{2} + 52319333403648)^{2}$$
$53$ $$(T^{2} + 1421326533636)^{2}$$
$59$ $$(T^{2} - 114412208832)^{2}$$
$61$ $$(T^{2} + 70808623425796)^{2}$$
$67$ $$(T^{2} - 303620940564672)^{2}$$
$71$ $$(T^{2} + 953360435651328)^{2}$$
$73$ $$(T + 12735874)^{4}$$
$79$ $$(T^{2} + 39521988480000)^{2}$$
$83$ $$(T^{2} - 69\!\cdots\!28)^{2}$$
$89$ $$(T - 16802814)^{4}$$
$97$ $$(T - 120994882)^{4}$$