Properties

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

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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 128)
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.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} - 47 \beta q^{5} + 244 q^{7} + 207 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{3} - 47 \beta q^{5} + 244 q^{7} + 207 q^{9} + 179 \beta q^{11} - 385 \beta q^{13} + 564 q^{15} + 670 q^{17} + 515 \beta q^{19} + 732 \beta q^{21} + 2828 q^{23} - 5711 q^{25} + 1350 \beta q^{27} - 381 \beta q^{29} - 4992 q^{31} - 2148 q^{33} - 11468 \beta q^{35} + 1781 \beta q^{37} + 4620 q^{39} - 858 q^{41} - 6393 \beta q^{43} - 9729 \beta q^{45} - 3560 q^{47} + 42729 q^{49} + 2010 \beta q^{51} + 4557 \beta q^{53} + 33652 q^{55} - 6180 q^{57} + 4123 \beta q^{59} + 2207 \beta q^{61} + 50508 q^{63} - 72380 q^{65} - 14993 \beta q^{67} + 8484 \beta q^{69} + 49572 q^{71} + 24370 q^{73} - 17133 \beta q^{75} + 43676 \beta q^{77} + 65176 q^{79} + 34101 q^{81} - 19689 \beta q^{83} - 31490 \beta q^{85} + 4572 q^{87} - 11134 q^{89} - 93940 \beta q^{91} - 14976 \beta q^{93} + 96820 q^{95} + 478 q^{97} + 37053 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 488 q^{7} + 414 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 488 q^{7} + 414 q^{9} + 1128 q^{15} + 1340 q^{17} + 5656 q^{23} - 11422 q^{25} - 9984 q^{31} - 4296 q^{33} + 9240 q^{39} - 1716 q^{41} - 7120 q^{47} + 85458 q^{49} + 67304 q^{55} - 12360 q^{57} + 101016 q^{63} - 144760 q^{65} + 99144 q^{71} + 48740 q^{73} + 130352 q^{79} + 68202 q^{81} + 9144 q^{87} - 22268 q^{89} + 193640 q^{95} + 956 q^{97}+O(q^{100}) \) 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 6.00000i 0 94.0000i 0 244.000 0 207.000 0
129.2 0 6.00000i 0 94.0000i 0 244.000 0 207.000 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.i 2
4.b odd 2 1 256.6.b.a 2
8.b even 2 1 inner 256.6.b.i 2
8.d odd 2 1 256.6.b.a 2
16.e even 4 1 128.6.a.a 1
16.e even 4 1 128.6.a.d yes 1
16.f odd 4 1 128.6.a.b yes 1
16.f odd 4 1 128.6.a.c yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.6.a.a 1 16.e even 4 1
128.6.a.b yes 1 16.f odd 4 1
128.6.a.c yes 1 16.f odd 4 1
128.6.a.d yes 1 16.e even 4 1
256.6.b.a 2 4.b odd 2 1
256.6.b.a 2 8.d odd 2 1
256.6.b.i 2 1.a even 1 1 trivial
256.6.b.i 2 8.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{7} - 244 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 36 \) Copy content Toggle raw display
$5$ \( T^{2} + 8836 \) Copy content Toggle raw display
$7$ \( (T - 244)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 128164 \) Copy content Toggle raw display
$13$ \( T^{2} + 592900 \) Copy content Toggle raw display
$17$ \( (T - 670)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1060900 \) Copy content Toggle raw display
$23$ \( (T - 2828)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 580644 \) Copy content Toggle raw display
$31$ \( (T + 4992)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12687844 \) Copy content Toggle raw display
$41$ \( (T + 858)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 163481796 \) Copy content Toggle raw display
$47$ \( (T + 3560)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 83064996 \) Copy content Toggle raw display
$59$ \( T^{2} + 67996516 \) Copy content Toggle raw display
$61$ \( T^{2} + 19483396 \) Copy content Toggle raw display
$67$ \( T^{2} + 899160196 \) Copy content Toggle raw display
$71$ \( (T - 49572)^{2} \) Copy content Toggle raw display
$73$ \( (T - 24370)^{2} \) Copy content Toggle raw display
$79$ \( (T - 65176)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1550626884 \) Copy content Toggle raw display
$89$ \( (T + 11134)^{2} \) Copy content Toggle raw display
$97$ \( (T - 478)^{2} \) Copy content Toggle raw display
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