Properties

Label 256.4.b.a
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 8)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} - \beta q^{5} - 24 q^{7} + 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{3} - \beta q^{5} - 24 q^{7} + 11 q^{9} - 22 \beta q^{11} - 11 \beta q^{13} + 8 q^{15} + 50 q^{17} - 22 \beta q^{19} - 48 \beta q^{21} + 56 q^{23} + 121 q^{25} + 76 \beta q^{27} - 99 \beta q^{29} - 160 q^{31} + 176 q^{33} + 24 \beta q^{35} - 81 \beta q^{37} + 88 q^{39} + 198 q^{41} + 26 \beta q^{43} - 11 \beta q^{45} + 528 q^{47} + 233 q^{49} + 100 \beta q^{51} - 121 \beta q^{53} - 88 q^{55} + 176 q^{57} - 334 \beta q^{59} - 275 \beta q^{61} - 264 q^{63} - 44 q^{65} - 94 \beta q^{67} + 112 \beta q^{69} - 728 q^{71} - 154 q^{73} + 242 \beta q^{75} + 528 \beta q^{77} - 656 q^{79} - 311 q^{81} - 118 \beta q^{83} - 50 \beta q^{85} + 792 q^{87} - 714 q^{89} + 264 \beta q^{91} - 320 \beta q^{93} - 88 q^{95} - 478 q^{97} - 242 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 48 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 48 q^{7} + 22 q^{9} + 16 q^{15} + 100 q^{17} + 112 q^{23} + 242 q^{25} - 320 q^{31} + 352 q^{33} + 176 q^{39} + 396 q^{41} + 1056 q^{47} + 466 q^{49} - 176 q^{55} + 352 q^{57} - 528 q^{63} - 88 q^{65} - 1456 q^{71} - 308 q^{73} - 1312 q^{79} - 622 q^{81} + 1584 q^{87} - 1428 q^{89} - 176 q^{95} - 956 q^{97}+O(q^{100}) \) 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 4.00000i 0 2.00000i 0 −24.0000 0 11.0000 0
129.2 0 4.00000i 0 2.00000i 0 −24.0000 0 11.0000 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.a 2
4.b odd 2 1 256.4.b.g 2
8.b even 2 1 inner 256.4.b.a 2
8.d odd 2 1 256.4.b.g 2
16.e even 4 1 8.4.a.a 1
16.e even 4 1 64.4.a.d 1
16.f odd 4 1 16.4.a.a 1
16.f odd 4 1 64.4.a.b 1
48.i odd 4 1 72.4.a.c 1
48.i odd 4 1 576.4.a.k 1
48.k even 4 1 144.4.a.e 1
48.k even 4 1 576.4.a.j 1
80.i odd 4 1 200.4.c.e 2
80.j even 4 1 400.4.c.i 2
80.k odd 4 1 400.4.a.g 1
80.k odd 4 1 1600.4.a.bm 1
80.q even 4 1 200.4.a.g 1
80.q even 4 1 1600.4.a.o 1
80.s even 4 1 400.4.c.i 2
80.t odd 4 1 200.4.c.e 2
112.j even 4 1 784.4.a.e 1
112.l odd 4 1 392.4.a.e 1
112.w even 12 2 392.4.i.g 2
112.x odd 12 2 392.4.i.b 2
144.w odd 12 2 648.4.i.e 2
144.x even 12 2 648.4.i.h 2
176.i even 4 1 1936.4.a.l 1
176.l odd 4 1 968.4.a.a 1
208.p even 4 1 1352.4.a.a 1
240.bb even 4 1 1800.4.f.u 2
240.bf even 4 1 1800.4.f.u 2
240.bm odd 4 1 1800.4.a.d 1
272.r even 4 1 2312.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 16.e even 4 1
16.4.a.a 1 16.f odd 4 1
64.4.a.b 1 16.f odd 4 1
64.4.a.d 1 16.e even 4 1
72.4.a.c 1 48.i odd 4 1
144.4.a.e 1 48.k even 4 1
200.4.a.g 1 80.q even 4 1
200.4.c.e 2 80.i odd 4 1
200.4.c.e 2 80.t odd 4 1
256.4.b.a 2 1.a even 1 1 trivial
256.4.b.a 2 8.b even 2 1 inner
256.4.b.g 2 4.b odd 2 1
256.4.b.g 2 8.d odd 2 1
392.4.a.e 1 112.l odd 4 1
392.4.i.b 2 112.x odd 12 2
392.4.i.g 2 112.w even 12 2
400.4.a.g 1 80.k odd 4 1
400.4.c.i 2 80.j even 4 1
400.4.c.i 2 80.s even 4 1
576.4.a.j 1 48.k even 4 1
576.4.a.k 1 48.i odd 4 1
648.4.i.e 2 144.w odd 12 2
648.4.i.h 2 144.x even 12 2
784.4.a.e 1 112.j even 4 1
968.4.a.a 1 176.l odd 4 1
1352.4.a.a 1 208.p even 4 1
1600.4.a.o 1 80.q even 4 1
1600.4.a.bm 1 80.k odd 4 1
1800.4.a.d 1 240.bm odd 4 1
1800.4.f.u 2 240.bb even 4 1
1800.4.f.u 2 240.bf even 4 1
1936.4.a.l 1 176.i even 4 1
2312.4.a.a 1 272.r even 4 1

Hecke kernels

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

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 24)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1936 \) Copy content Toggle raw display
$13$ \( T^{2} + 484 \) Copy content Toggle raw display
$17$ \( (T - 50)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1936 \) Copy content Toggle raw display
$23$ \( (T - 56)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 39204 \) Copy content Toggle raw display
$31$ \( (T + 160)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 26244 \) Copy content Toggle raw display
$41$ \( (T - 198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2704 \) Copy content Toggle raw display
$47$ \( (T - 528)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 58564 \) Copy content Toggle raw display
$59$ \( T^{2} + 446224 \) Copy content Toggle raw display
$61$ \( T^{2} + 302500 \) Copy content Toggle raw display
$67$ \( T^{2} + 35344 \) Copy content Toggle raw display
$71$ \( (T + 728)^{2} \) Copy content Toggle raw display
$73$ \( (T + 154)^{2} \) Copy content Toggle raw display
$79$ \( (T + 656)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 55696 \) Copy content Toggle raw display
$89$ \( (T + 714)^{2} \) Copy content Toggle raw display
$97$ \( (T + 478)^{2} \) Copy content Toggle raw display
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