Properties

Label 256.6.b.h
Level $256$
Weight $6$
Character orbit 256.b
Analytic conductor $41.058$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,6,Mod(129,256)]
 
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([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.b (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: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta q^{3} + 7 \beta q^{5} + 208 q^{7} + 179 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{3} + 7 \beta q^{5} + 208 q^{7} + 179 q^{9} - 268 \beta q^{11} - 347 \beta q^{13} - 112 q^{15} - 1278 q^{17} - 556 \beta q^{19} + 832 \beta q^{21} - 3216 q^{23} + 2929 q^{25} + 1688 \beta q^{27} - 1459 \beta q^{29} - 2624 q^{31} + 4288 q^{33} + 1456 \beta q^{35} - 4729 \beta q^{37} + 5552 q^{39} - 170 q^{41} - 9964 \beta q^{43} + 1253 \beta q^{45} + 32 q^{47} + 26457 q^{49} - 5112 \beta q^{51} - 11089 \beta q^{53} + 7504 q^{55} + 8896 q^{57} + 20740 \beta q^{59} - 7731 \beta q^{61} + 37232 q^{63} + 9716 q^{65} + 10372 \beta q^{67} - 12864 \beta q^{69} - 28592 q^{71} + 53670 q^{73} + 11716 \beta q^{75} - 55744 \beta q^{77} - 69152 q^{79} + 16489 q^{81} + 18900 \beta q^{83} - 8946 \beta q^{85} + 23344 q^{87} + 126806 q^{89} - 72176 \beta q^{91} - 10496 \beta q^{93} + 15568 q^{95} + 62290 q^{97} - 47972 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 416 q^{7} + 358 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 416 q^{7} + 358 q^{9} - 224 q^{15} - 2556 q^{17} - 6432 q^{23} + 5858 q^{25} - 5248 q^{31} + 8576 q^{33} + 11104 q^{39} - 340 q^{41} + 64 q^{47} + 52914 q^{49} + 15008 q^{55} + 17792 q^{57} + 74464 q^{63} + 19432 q^{65} - 57184 q^{71} + 107340 q^{73} - 138304 q^{79} + 32978 q^{81} + 46688 q^{87} + 253612 q^{89} + 31136 q^{95} + 124580 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 8.00000i 0 14.0000i 0 208.000 0 179.000 0
129.2 0 8.00000i 0 14.0000i 0 208.000 0 179.000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.6.b.h 2
4.b odd 2 1 256.6.b.b 2
8.b even 2 1 inner 256.6.b.h 2
8.d odd 2 1 256.6.b.b 2
16.e even 4 1 32.6.a.a 1
16.e even 4 1 64.6.a.e 1
16.f odd 4 1 32.6.a.c yes 1
16.f odd 4 1 64.6.a.c 1
48.i odd 4 1 288.6.a.d 1
48.i odd 4 1 576.6.a.u 1
48.k even 4 1 288.6.a.e 1
48.k even 4 1 576.6.a.v 1
80.i odd 4 1 800.6.c.a 2
80.j even 4 1 800.6.c.b 2
80.k odd 4 1 800.6.a.a 1
80.q even 4 1 800.6.a.e 1
80.s even 4 1 800.6.c.b 2
80.t odd 4 1 800.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.a 1 16.e even 4 1
32.6.a.c yes 1 16.f odd 4 1
64.6.a.c 1 16.f odd 4 1
64.6.a.e 1 16.e even 4 1
256.6.b.b 2 4.b odd 2 1
256.6.b.b 2 8.d odd 2 1
256.6.b.h 2 1.a even 1 1 trivial
256.6.b.h 2 8.b even 2 1 inner
288.6.a.d 1 48.i odd 4 1
288.6.a.e 1 48.k even 4 1
576.6.a.u 1 48.i odd 4 1
576.6.a.v 1 48.k even 4 1
800.6.a.a 1 80.k odd 4 1
800.6.a.e 1 80.q even 4 1
800.6.c.a 2 80.i odd 4 1
800.6.c.a 2 80.t odd 4 1
800.6.c.b 2 80.j even 4 1
800.6.c.b 2 80.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7} - 208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} + 196 \) Copy content Toggle raw display
$7$ \( (T - 208)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 287296 \) Copy content Toggle raw display
$13$ \( T^{2} + 481636 \) Copy content Toggle raw display
$17$ \( (T + 1278)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1236544 \) Copy content Toggle raw display
$23$ \( (T + 3216)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8514724 \) Copy content Toggle raw display
$31$ \( (T + 2624)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 89453764 \) Copy content Toggle raw display
$41$ \( (T + 170)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 397125184 \) Copy content Toggle raw display
$47$ \( (T - 32)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 491863684 \) Copy content Toggle raw display
$59$ \( T^{2} + 1720590400 \) Copy content Toggle raw display
$61$ \( T^{2} + 239073444 \) Copy content Toggle raw display
$67$ \( T^{2} + 430313536 \) Copy content Toggle raw display
$71$ \( (T + 28592)^{2} \) Copy content Toggle raw display
$73$ \( (T - 53670)^{2} \) Copy content Toggle raw display
$79$ \( (T + 69152)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1428840000 \) Copy content Toggle raw display
$89$ \( (T - 126806)^{2} \) Copy content Toggle raw display
$97$ \( (T - 62290)^{2} \) Copy content Toggle raw display
show more
show less