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Properties

Label 256.2.g.b
Level $256$
Weight $2$
Character orbit 256.g
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
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] = [256,2,Mod(33,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.g (of order \(8\), degree \(4\), 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: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{2} - \zeta_{8}) q^{3} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + \cdots + 1) q^{5}+ \cdots + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{2} - \zeta_{8}) q^{3} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + \cdots + 1) q^{5}+ \cdots + ( - 5 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 4 q^{7} - 4 q^{9} + 8 q^{11} - 4 q^{13} + 8 q^{19} + 4 q^{21} + 12 q^{23} + 4 q^{25} - 12 q^{27} + 4 q^{29} - 16 q^{31} + 8 q^{33} - 4 q^{35} - 4 q^{37} + 4 q^{39} - 12 q^{41} - 16 q^{43} - 8 q^{45} - 8 q^{51} - 4 q^{53} - 20 q^{55} - 20 q^{57} + 16 q^{59} - 4 q^{61} - 8 q^{63} - 8 q^{65} + 8 q^{67} + 4 q^{69} - 12 q^{71} + 28 q^{73} + 4 q^{75} + 4 q^{77} - 16 q^{83} - 8 q^{85} - 4 q^{87} + 12 q^{89} - 4 q^{91} + 64 q^{95} - 40 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(\zeta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.707107 − 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 −0.707107 + 1.70711i 0 3.12132 − 1.29289i 0 1.00000 − 1.00000i 0 −0.292893 − 0.292893i 0
97.1 0 0.707107 − 0.292893i 0 −1.12132 + 2.70711i 0 1.00000 + 1.00000i 0 −1.70711 + 1.70711i 0
161.1 0 0.707107 + 0.292893i 0 −1.12132 − 2.70711i 0 1.00000 − 1.00000i 0 −1.70711 − 1.70711i 0
225.1 0 −0.707107 − 1.70711i 0 3.12132 + 1.29289i 0 1.00000 + 1.00000i 0 −0.292893 + 0.292893i 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
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.2.g.b 4
4.b odd 2 1 256.2.g.a 4
8.b even 2 1 32.2.g.a ✓ 4
8.d odd 2 1 128.2.g.a 4
16.e even 4 1 512.2.g.a 4
16.e even 4 1 512.2.g.d 4
16.f odd 4 1 512.2.g.b 4
16.f odd 4 1 512.2.g.c 4
24.f even 2 1 1152.2.v.a 4
24.h odd 2 1 288.2.v.a 4
32.g even 8 1 32.2.g.a ✓ 4
32.g even 8 1 inner 256.2.g.b 4
32.g even 8 1 512.2.g.a 4
32.g even 8 1 512.2.g.d 4
32.h odd 8 1 128.2.g.a 4
32.h odd 8 1 256.2.g.a 4
32.h odd 8 1 512.2.g.b 4
32.h odd 8 1 512.2.g.c 4
40.f even 2 1 800.2.y.a 4
40.i odd 4 1 800.2.ba.a 4
40.i odd 4 1 800.2.ba.b 4
64.i even 16 2 4096.2.a.e 4
64.j odd 16 2 4096.2.a.f 4
96.o even 8 1 1152.2.v.a 4
96.p odd 8 1 288.2.v.a 4
160.v odd 8 1 800.2.ba.b 4
160.z even 8 1 800.2.y.a 4
160.bb odd 8 1 800.2.ba.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a ✓ 4 8.b even 2 1
32.2.g.a ✓ 4 32.g even 8 1
128.2.g.a 4 8.d odd 2 1
128.2.g.a 4 32.h odd 8 1
256.2.g.a 4 4.b odd 2 1
256.2.g.a 4 32.h odd 8 1
256.2.g.b 4 1.a even 1 1 trivial
256.2.g.b 4 32.g even 8 1 inner
288.2.v.a 4 24.h odd 2 1
288.2.v.a 4 96.p odd 8 1
512.2.g.a 4 16.e even 4 1
512.2.g.a 4 32.g even 8 1
512.2.g.b 4 16.f odd 4 1
512.2.g.b 4 32.h odd 8 1
512.2.g.c 4 16.f odd 4 1
512.2.g.c 4 32.h odd 8 1
512.2.g.d 4 16.e even 4 1
512.2.g.d 4 32.g even 8 1
800.2.y.a 4 40.f even 2 1
800.2.y.a 4 160.z even 8 1
800.2.ba.a 4 40.i odd 4 1
800.2.ba.a 4 160.bb odd 8 1
800.2.ba.b 4 40.i odd 4 1
800.2.ba.b 4 160.v odd 8 1
1152.2.v.a 4 24.f even 2 1
1152.2.v.a 4 96.o even 8 1
4096.2.a.e 4 64.i even 16 2
4096.2.a.f 4 64.j odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2T_{3}^{2} - 4T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 98 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + \cdots + 578 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 98 \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$43$ \( T^{4} + 16 T^{3} + \cdots + 1922 \) Copy content Toggle raw display
$47$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 98 \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + \cdots + 1058 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + \cdots + 578 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$73$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} + \cdots + 1058 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$97$ \( (T^{2} + 20 T + 28)^{2} \) Copy content Toggle raw display
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