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Properties

Label 256.8.b.f
Level $256$
Weight $8$
Character orbit 256.b
Analytic conductor $79.971$
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] = [256,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(79.9705665239\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
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.

\(f(q)\) \(=\) \( q + 6 \beta q^{3} - 105 \beta q^{5} + 1016 q^{7} + 2043 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 6 \beta q^{3} - 105 \beta q^{5} + 1016 q^{7} + 2043 q^{9} - 546 \beta q^{11} - 691 \beta q^{13} + 2520 q^{15} + 14706 q^{17} - 19970 \beta q^{19} + 6096 \beta q^{21} + 68712 q^{23} + 34025 q^{25} + 25380 \beta q^{27} + 51285 \beta q^{29} - 227552 q^{31} + 13104 q^{33} - 106680 \beta q^{35} + 80263 \beta q^{37} + 16584 q^{39} - 10842 q^{41} + 315374 \beta q^{43} - 214515 \beta q^{45} - 472656 q^{47} + 208713 q^{49} + 88236 \beta q^{51} - 747009 \beta q^{53} - 229320 q^{55} + 479280 q^{57} - 1320330 \beta q^{59} - 413851 \beta q^{61} + 2075688 q^{63} - 290220 q^{65} - 63002 \beta q^{67} + 412272 \beta q^{69} - 1414728 q^{71} - 980282 q^{73} + 204150 \beta q^{75} - 554736 \beta q^{77} + 3566800 q^{79} + 3858921 q^{81} + 2836446 \beta q^{83} - 1544130 \beta q^{85} - 1230840 q^{87} + 11951190 q^{89} - 702056 \beta q^{91} - 1365312 \beta q^{93} - 8387400 q^{95} + 8682146 q^{97} - 1115478 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2032 q^{7} + 4086 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2032 q^{7} + 4086 q^{9} + 5040 q^{15} + 29412 q^{17} + 137424 q^{23} + 68050 q^{25} - 455104 q^{31} + 26208 q^{33} + 33168 q^{39} - 21684 q^{41} - 945312 q^{47} + 417426 q^{49} - 458640 q^{55} + 958560 q^{57} + 4151376 q^{63} - 580440 q^{65} - 2829456 q^{71} - 1960564 q^{73} + 7133600 q^{79} + 7717842 q^{81} - 2461680 q^{87} + 23902380 q^{89} - 16774800 q^{95} + 17364292 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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 − 12.0000i 0 210.000i 0 1016.00 0 2043.00 0
129.2 0 12.0000i 0 − 210.000i 0 1016.00 0 2043.00 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 256.8.b.f 2
4.b odd 2 1 256.8.b.b 2
8.b even 2 1 inner 256.8.b.f 2
8.d odd 2 1 256.8.b.b 2
16.e even 4 1 16.8.a.b 1
16.e even 4 1 64.8.a.e 1
16.f odd 4 1 2.8.a.a ✓ 1
16.f odd 4 1 64.8.a.c 1
48.i odd 4 1 144.8.a.i 1
48.i odd 4 1 576.8.a.f 1
48.k even 4 1 18.8.a.b 1
48.k even 4 1 576.8.a.g 1
80.i odd 4 1 400.8.c.j 2
80.j even 4 1 50.8.b.c 2
80.k odd 4 1 50.8.a.g 1
80.q even 4 1 400.8.a.l 1
80.s even 4 1 50.8.b.c 2
80.t odd 4 1 400.8.c.j 2
112.j even 4 1 98.8.a.a 1
112.u odd 12 2 98.8.c.d 2
112.v even 12 2 98.8.c.e 2
144.u even 12 2 162.8.c.a 2
144.v odd 12 2 162.8.c.l 2
176.i even 4 1 242.8.a.e 1
208.l even 4 1 338.8.b.d 2
208.o odd 4 1 338.8.a.d 1
208.s even 4 1 338.8.b.d 2
240.t even 4 1 450.8.a.c 1
240.z odd 4 1 450.8.c.g 2
240.bd odd 4 1 450.8.c.g 2
272.k odd 4 1 578.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a ✓ 1 16.f odd 4 1
16.8.a.b 1 16.e even 4 1
18.8.a.b 1 48.k even 4 1
50.8.a.g 1 80.k odd 4 1
50.8.b.c 2 80.j even 4 1
50.8.b.c 2 80.s even 4 1
64.8.a.c 1 16.f odd 4 1
64.8.a.e 1 16.e even 4 1
98.8.a.a 1 112.j even 4 1
98.8.c.d 2 112.u odd 12 2
98.8.c.e 2 112.v even 12 2
144.8.a.i 1 48.i odd 4 1
162.8.c.a 2 144.u even 12 2
162.8.c.l 2 144.v odd 12 2
242.8.a.e 1 176.i even 4 1
256.8.b.b 2 4.b odd 2 1
256.8.b.b 2 8.d odd 2 1
256.8.b.f 2 1.a even 1 1 trivial
256.8.b.f 2 8.b even 2 1 inner
338.8.a.d 1 208.o odd 4 1
338.8.b.d 2 208.l even 4 1
338.8.b.d 2 208.s even 4 1
400.8.a.l 1 80.q even 4 1
400.8.c.j 2 80.i odd 4 1
400.8.c.j 2 80.t odd 4 1
450.8.a.c 1 240.t even 4 1
450.8.c.g 2 240.z odd 4 1
450.8.c.g 2 240.bd odd 4 1
576.8.a.f 1 48.i odd 4 1
576.8.a.g 1 48.k even 4 1
578.8.a.b 1 272.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 144 \) Copy content Toggle raw display
\( T_{7} - 1016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 144 \) Copy content Toggle raw display
$5$ \( T^{2} + 44100 \) Copy content Toggle raw display
$7$ \( (T - 1016)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1192464 \) Copy content Toggle raw display
$13$ \( T^{2} + 1909924 \) Copy content Toggle raw display
$17$ \( (T - 14706)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1595203600 \) Copy content Toggle raw display
$23$ \( (T - 68712)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 10520604900 \) Copy content Toggle raw display
$31$ \( (T + 227552)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 25768596676 \) Copy content Toggle raw display
$41$ \( (T + 10842)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 397843039504 \) Copy content Toggle raw display
$47$ \( (T + 472656)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2232089784324 \) Copy content Toggle raw display
$59$ \( T^{2} + 6973085235600 \) Copy content Toggle raw display
$61$ \( T^{2} + 685090600804 \) Copy content Toggle raw display
$67$ \( T^{2} + 15877008016 \) Copy content Toggle raw display
$71$ \( (T + 1414728)^{2} \) Copy content Toggle raw display
$73$ \( (T + 980282)^{2} \) Copy content Toggle raw display
$79$ \( (T - 3566800)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 32181703643664 \) Copy content Toggle raw display
$89$ \( (T - 11951190)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8682146)^{2} \) Copy content Toggle raw display
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