Properties

Label 256.4.b.c
Level $256$
Weight $4$
Character orbit 256.b
Analytic conductor $15.104$
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] = [256,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(15.1044889615\)
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.

\(f(q)\) \(=\) \( q + 4 \beta q^{3} + 5 \beta q^{5} - 16 q^{7} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{3} + 5 \beta q^{5} - 16 q^{7} - 37 q^{9} + 20 \beta q^{11} - 25 \beta q^{13} - 80 q^{15} - 30 q^{17} + 20 \beta q^{19} - 64 \beta q^{21} - 48 q^{23} + 25 q^{25} - 40 \beta q^{27} - 17 \beta q^{29} + 320 q^{31} - 320 q^{33} - 80 \beta q^{35} - 155 \beta q^{37} + 400 q^{39} - 410 q^{41} - 76 \beta q^{43} - 185 \beta q^{45} - 416 q^{47} - 87 q^{49} - 120 \beta q^{51} + 205 \beta q^{53} - 400 q^{55} - 320 q^{57} + 100 \beta q^{59} + 15 \beta q^{61} + 592 q^{63} + 500 q^{65} + 388 \beta q^{67} - 192 \beta q^{69} - 400 q^{71} + 630 q^{73} + 100 \beta q^{75} - 320 \beta q^{77} - 1120 q^{79} - 359 q^{81} + 276 \beta q^{83} - 150 \beta q^{85} + 272 q^{87} + 326 q^{89} + 400 \beta q^{91} + 1280 \beta q^{93} - 400 q^{95} - 110 q^{97} - 740 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{7} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{7} - 74 q^{9} - 160 q^{15} - 60 q^{17} - 96 q^{23} + 50 q^{25} + 640 q^{31} - 640 q^{33} + 800 q^{39} - 820 q^{41} - 832 q^{47} - 174 q^{49} - 800 q^{55} - 640 q^{57} + 1184 q^{63} + 1000 q^{65} - 800 q^{71} + 1260 q^{73} - 2240 q^{79} - 718 q^{81} + 544 q^{87} + 652 q^{89} - 800 q^{95} - 220 q^{97}+O(q^{100}) \) 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 10.0000i 0 −16.0000 0 −37.0000 0
129.2 0 8.00000i 0 10.0000i 0 −16.0000 0 −37.0000 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 256.4.b.c 2
4.b odd 2 1 256.4.b.e 2
8.b even 2 1 inner 256.4.b.c 2
8.d odd 2 1 256.4.b.e 2
16.e even 4 1 32.4.a.c yes 1
16.e even 4 1 64.4.a.a 1
16.f odd 4 1 32.4.a.a 1
16.f odd 4 1 64.4.a.e 1
48.i odd 4 1 288.4.a.i 1
48.i odd 4 1 576.4.a.h 1
48.k even 4 1 288.4.a.h 1
48.k even 4 1 576.4.a.g 1
80.i odd 4 1 800.4.c.a 2
80.j even 4 1 800.4.c.b 2
80.k odd 4 1 800.4.a.k 1
80.k odd 4 1 1600.4.a.e 1
80.q even 4 1 800.4.a.a 1
80.q even 4 1 1600.4.a.bw 1
80.s even 4 1 800.4.c.b 2
80.t odd 4 1 800.4.c.a 2
112.j even 4 1 1568.4.a.o 1
112.l odd 4 1 1568.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 16.f odd 4 1
32.4.a.c yes 1 16.e even 4 1
64.4.a.a 1 16.e even 4 1
64.4.a.e 1 16.f odd 4 1
256.4.b.c 2 1.a even 1 1 trivial
256.4.b.c 2 8.b even 2 1 inner
256.4.b.e 2 4.b odd 2 1
256.4.b.e 2 8.d odd 2 1
288.4.a.h 1 48.k even 4 1
288.4.a.i 1 48.i odd 4 1
576.4.a.g 1 48.k even 4 1
576.4.a.h 1 48.i odd 4 1
800.4.a.a 1 80.q even 4 1
800.4.a.k 1 80.k odd 4 1
800.4.c.a 2 80.i odd 4 1
800.4.c.a 2 80.t odd 4 1
800.4.c.b 2 80.j even 4 1
800.4.c.b 2 80.s even 4 1
1568.4.a.c 1 112.l odd 4 1
1568.4.a.o 1 112.j even 4 1
1600.4.a.e 1 80.k odd 4 1
1600.4.a.bw 1 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7} + 16 \) 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
$5$ \( T^{2} + 100 \) Copy content Toggle raw display
$7$ \( (T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1600 \) Copy content Toggle raw display
$13$ \( T^{2} + 2500 \) Copy content Toggle raw display
$17$ \( (T + 30)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1600 \) Copy content Toggle raw display
$23$ \( (T + 48)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1156 \) Copy content Toggle raw display
$31$ \( (T - 320)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 96100 \) Copy content Toggle raw display
$41$ \( (T + 410)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 23104 \) Copy content Toggle raw display
$47$ \( (T + 416)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 168100 \) Copy content Toggle raw display
$59$ \( T^{2} + 40000 \) Copy content Toggle raw display
$61$ \( T^{2} + 900 \) Copy content Toggle raw display
$67$ \( T^{2} + 602176 \) Copy content Toggle raw display
$71$ \( (T + 400)^{2} \) Copy content Toggle raw display
$73$ \( (T - 630)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1120)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 304704 \) Copy content Toggle raw display
$89$ \( (T - 326)^{2} \) Copy content Toggle raw display
$97$ \( (T + 110)^{2} \) Copy content Toggle raw display
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