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Properties

Label 256.3.f.b
Level $256$
Weight $3$
Character orbit 256.f
Analytic conductor $6.975$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,3,Mod(63,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.63");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 256.f (of order \(4\), degree \(2\), 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: \(6.97549476762\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.8540717056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 31x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{6} - 2 \beta_1 - 2) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} - \beta_{6} + 11 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{6} - 2 \beta_1 - 2) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} - \beta_{6} + 11 \beta_1) q^{9} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{11} + (\beta_{7} - 4 \beta_1 + 4) q^{13} + (\beta_{4} + 9 \beta_{3} - \beta_{2}) q^{15} + ( - \beta_{7} - \beta_{6} + 8) q^{17} + (5 \beta_{5} + 5 \beta_{3} - 3 \beta_{2}) q^{19} + (4 \beta_{6} - 22 \beta_1 - 22) q^{21} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}) q^{23} + ( - 4 \beta_{7} + 4 \beta_{6} + 21 \beta_1) q^{25} + ( - 9 \beta_{5} - 4 \beta_{4} + 9 \beta_{3}) q^{27} + ( - 5 \beta_{7} - 14 \beta_1 + 14) q^{29} + ( - 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2}) q^{31} + (5 \beta_{7} + 5 \beta_{6} + 8) q^{33} + ( - 6 \beta_{5} - 6 \beta_{3} - 2 \beta_{2}) q^{35} + ( - 3 \beta_{6} - 16 \beta_1 - 16) q^{37} + ( - 9 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}) q^{39} + (2 \beta_{7} - 2 \beta_{6} + 16 \beta_1) q^{41} + (8 \beta_{5} + \beta_{4} - 8 \beta_{3}) q^{43} + (7 \beta_{7} + 16 \beta_1 - 16) q^{45} + 6 \beta_{3} q^{47} + ( - 6 \beta_{7} - 6 \beta_{6} + 7) q^{49} + (9 \beta_{5} + 9 \beta_{3} + 10 \beta_{2}) q^{51} + ( - 3 \beta_{6} + 36 \beta_1 + 36) q^{53} + (27 \beta_{5} - 13 \beta_{4} - 13 \beta_{2}) q^{55} + (7 \beta_{7} - 7 \beta_{6} - 40 \beta_1) q^{57} + ( - 6 \beta_{5} + 13 \beta_{4} + 6 \beta_{3}) q^{59} + (\beta_{7} + 8 \beta_1 - 8) q^{61} + (17 \beta_{4} - 27 \beta_{3} - 17 \beta_{2}) q^{63} + ( - 6 \beta_{7} - 6 \beta_{6} - 54) q^{65} + ( - 9 \beta_{5} - 9 \beta_{3} + \beta_{2}) q^{67} + (54 \beta_1 + 54) q^{69} + ( - 27 \beta_{5} - 5 \beta_{4} - 5 \beta_{2}) q^{71} + ( - \beta_{7} + \beta_{6} - 74 \beta_1) q^{73} + (36 \beta_{5} - 13 \beta_{4} - 36 \beta_{3}) q^{75} + ( - 8 \beta_{7} - 14 \beta_1 + 14) q^{77} + ( - 12 \beta_{4} - 8 \beta_{3} + 12 \beta_{2}) q^{79} + (13 \beta_{7} + 13 \beta_{6} - 17) q^{81} + ( - 12 \beta_{5} - 12 \beta_{3} - 3 \beta_{2}) q^{83} + ( - 4 \beta_{6} + 22 \beta_1 + 22) q^{85} + (45 \beta_{5} + 19 \beta_{4} + 19 \beta_{2}) q^{87} + ( - 7 \beta_{7} + 7 \beta_{6} - 38 \beta_1) q^{89} + (4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3}) q^{91} + (12 \beta_{7} + 84 \beta_1 - 84) q^{93} + ( - 23 \beta_{4} - 57 \beta_{3} + 23 \beta_{2}) q^{95} + ( - \beta_{7} - \beta_{6} + 56) q^{97} + ( - 18 \beta_{5} - 18 \beta_{3} - 11 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{5} + 32 q^{13} + 64 q^{17} - 176 q^{21} + 112 q^{29} + 64 q^{33} - 128 q^{37} - 128 q^{45} + 56 q^{49} + 288 q^{53} - 64 q^{61} - 432 q^{65} + 432 q^{69} + 112 q^{77} - 136 q^{81} + 176 q^{85} - 672 q^{93} + 448 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 31x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 56\nu^{2} ) / 225 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{7} + 25\nu^{5} - \nu^{3} + 2525\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} - 50\nu^{5} - 2\nu^{3} - 550\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{7} + 25\nu^{5} - 559\nu^{3} + 275\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{7} + 50\nu^{5} - 2\nu^{3} + 550\nu ) / 1125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} + 50\nu^{4} - 54\nu^{2} + 775 ) / 225 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{6} - 50\nu^{4} - 54\nu^{2} - 775 ) / 225 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 18\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{5} - 8\beta_{4} - 9\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} - 62 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 56\beta_{5} - 45\beta_{3} - 22\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{7} - 14\beta_{6} - 27\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 280\beta_{5} - 2\beta_{4} + 279\beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(5\) \(255\)
\(\chi(n)\) \(\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
−1.18755 − 1.89466i
−1.89466 − 1.18755i
1.89466 + 1.18755i
1.18755 + 1.89466i
−1.18755 + 1.89466i
−1.89466 + 1.18755i
1.89466 − 1.18755i
1.18755 − 1.89466i
0 −3.78931 − 3.78931i 0 2.35890 + 2.35890i 0 10.4071 0 19.7178i 0
63.2 0 −2.37510 − 2.37510i 0 −6.35890 − 6.35890i 0 1.92177 0 2.28220i 0
63.3 0 2.37510 + 2.37510i 0 −6.35890 − 6.35890i 0 −1.92177 0 2.28220i 0
63.4 0 3.78931 + 3.78931i 0 2.35890 + 2.35890i 0 −10.4071 0 19.7178i 0
191.1 0 −3.78931 + 3.78931i 0 2.35890 − 2.35890i 0 10.4071 0 − 19.7178i 0
191.2 0 −2.37510 + 2.37510i 0 −6.35890 + 6.35890i 0 1.92177 0 − 2.28220i 0
191.3 0 2.37510 − 2.37510i 0 −6.35890 + 6.35890i 0 −1.92177 0 − 2.28220i 0
191.4 0 3.78931 − 3.78931i 0 2.35890 − 2.35890i 0 −10.4071 0 − 19.7178i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.3.f.b ✓ 8
4.b odd 2 1 inner 256.3.f.b ✓ 8
8.b even 2 1 256.3.f.c yes 8
8.d odd 2 1 256.3.f.c yes 8
16.e even 4 1 inner 256.3.f.b ✓ 8
16.e even 4 1 256.3.f.c yes 8
16.f odd 4 1 inner 256.3.f.b ✓ 8
16.f odd 4 1 256.3.f.c yes 8
32.g even 8 2 1024.3.c.f 8
32.g even 8 1 1024.3.d.f 4
32.g even 8 1 1024.3.d.j 4
32.h odd 8 2 1024.3.c.f 8
32.h odd 8 1 1024.3.d.f 4
32.h odd 8 1 1024.3.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.3.f.b ✓ 8 1.a even 1 1 trivial
256.3.f.b ✓ 8 4.b odd 2 1 inner
256.3.f.b ✓ 8 16.e even 4 1 inner
256.3.f.b ✓ 8 16.f odd 4 1 inner
256.3.f.c yes 8 8.b even 2 1
256.3.f.c yes 8 8.d odd 2 1
256.3.f.c yes 8 16.e even 4 1
256.3.f.c yes 8 16.f odd 4 1
1024.3.c.f 8 32.g even 8 2
1024.3.c.f 8 32.h odd 8 2
1024.3.d.f 4 32.g even 8 1
1024.3.d.f 4 32.h odd 8 1
1024.3.d.j 4 32.g even 8 1
1024.3.d.j 4 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{8} + 952T_{3}^{4} + 104976 \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{3} + 32T_{5}^{2} - 240T_{5} + 900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 952 T^{4} + 104976 \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{3} + \cdots + 900)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 112 T^{2} + 400)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 57592 T^{4} + 108243216 \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{3} + \cdots + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 16 T - 12)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 818552 T^{4} + 193877776 \) Copy content Toggle raw display
$23$ \( (T^{4} - 720 T^{2} + 104976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 56 T^{3} + \cdots + 311364)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1472 T^{2} + 230400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 64 T^{3} + \cdots + 28900)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1120 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1310796010000 \) Copy content Toggle raw display
$47$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 144 T^{3} + \cdots + 5062500)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 91166213610000 \) Copy content Toggle raw display
$61$ \( (T^{4} + 32 T^{3} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 2115384078096 \) Copy content Toggle raw display
$71$ \( (T^{4} - 15824 T^{2} + 36144144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11104 T^{2} + 29160000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 11008 T^{2} + 29593600)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 34867844010000 \) Copy content Toggle raw display
$89$ \( (T^{4} + 10336 T^{2} + 5198400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 112 T + 3060)^{4} \) Copy content Toggle raw display
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