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

Downloads

Learn more

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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -8\zeta_{12}^{3} + 16\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 32\zeta_{12}^{2} - 16 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 16 ) / 32 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

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:

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 \(S_{9}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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