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Properties

Label 256.9.c.n
Level $256$
Weight $9$
Character orbit 256.c
Analytic conductor $104.289$
Analytic rank $0$
Dimension $12$
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] = [256,9,Mod(255,256)]
 
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([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.255");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 256.c (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: \(104.288924176\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 155x^{10} + 13530x^{8} - 838070x^{6} + 36386725x^{4} - 1689017631x^{2} + 57111440400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{100} \)
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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + ( - 3 \beta_{2} - 2724) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + ( - 3 \beta_{2} - 2724) q^{9} + ( - \beta_{9} + 16 \beta_{7} + 36 \beta_{6}) q^{11} + (\beta_{10} + 10 \beta_{8}) q^{13} + \beta_{5} q^{15} + (\beta_{3} + 17 \beta_{2} - 33507) q^{17} + ( - 39 \beta_{9} + \cdots - 140 \beta_{6}) q^{19}+ \cdots + (9384 \beta_{9} + \cdots - 261243 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 32676 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 32676 q^{9} - 402152 q^{17} + 895860 q^{25} + 3473712 q^{33} - 7634200 q^{41} - 25310964 q^{49} - 15047472 q^{57} + 54120960 q^{65} + 110969832 q^{73} - 158290884 q^{81} - 284346904 q^{89} + 250387224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 155x^{10} + 13530x^{8} - 838070x^{6} + 36386725x^{4} - 1689017631x^{2} + 57111440400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1143659 \nu^{10} - 140873804 \nu^{8} + 5378497326 \nu^{6} - 542334198044 \nu^{4} + \cdots - 815482060624560 ) / 1537321820160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57421 \nu^{10} + 10916844 \nu^{8} - 1001972606 \nu^{6} + 53551641404 \nu^{4} + \cdots + 55026150186800 ) / 36602900480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34549 \nu^{10} - 3540556 \nu^{8} + 199520494 \nu^{6} - 14298242076 \nu^{4} + \cdots - 25059831768240 ) / 2287681280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57389627 \nu^{10} + 1060654708 \nu^{8} - 1942517203122 \nu^{6} + 203301910045348 \nu^{4} + \cdots + 42\!\cdots\!20 ) / 1537321820160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17088703 \nu^{10} + 1591452900 \nu^{8} - 127345951626 \nu^{6} + 5293327288148 \nu^{4} + \cdots + 13\!\cdots\!00 ) / 102488121344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 130921547 \nu^{11} - 9277857268 \nu^{9} + 1113660695762 \nu^{7} - 48919014874788 \nu^{5} + \cdots - 12\!\cdots\!60 \nu ) / 81\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3930953 \nu^{11} - 295624516 \nu^{9} + 16862220174 \nu^{7} - 386804032996 \nu^{5} + \cdots - 24\!\cdots\!04 \nu ) / 35877848494320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41392187 \nu^{11} + 4666156660 \nu^{9} - 295996703922 \nu^{7} + \cdots + 17\!\cdots\!56 \nu ) / 349894446268416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2867369963 \nu^{11} + 372964657012 \nu^{9} - 33056712635538 \nu^{7} + \cdots + 22\!\cdots\!20 \nu ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3471594961 \nu^{11} + 320892945116 \nu^{9} - 28733860743894 \nu^{7} + \cdots + 10\!\cdots\!12 \nu ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20843270929 \nu^{11} + 5708977640924 \nu^{9} - 486580437091926 \nu^{7} + \cdots + 53\!\cdots\!68 \nu ) / 17\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 10\beta_{10} - 64\beta_{9} + 51\beta_{8} - 12\beta_{7} - 832\beta_{6} ) / 65536 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -37\beta_{5} - 7\beta_{4} - 144\beta_{3} + 1792\beta_{2} + 1942\beta _1 + 3386624 ) / 131072 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{11} + 110\beta_{10} - 3776\beta_{9} + 33329\beta_{8} + 27708\beta_{7} - 114624\beta_{6} ) / 65536 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3763\beta_{5} + 575\beta_{4} - 31216\beta_{3} + 53504\beta_{2} + 68026\beta _1 - 66282752 ) / 131072 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4211 \beta_{11} + 29058 \beta_{10} + 29888 \beta_{9} + 1711895 \beta_{8} + \cdots + 454080 \beta_{6} ) / 65536 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -151141\beta_{5} + 825\beta_{4} - 2551440\beta_{3} - 6609152\beta_{2} - 4024554\beta _1 - 1167979264 ) / 131072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 233509 \beta_{11} + 2400910 \beta_{10} + 2721088 \beta_{9} + 62516385 \beta_{8} + \cdots + 1017824320 \beta_{6} ) / 65536 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3171523 \beta_{5} - 4298609 \beta_{4} - 76748016 \beta_{3} + 13508864 \beta_{2} + \cdots + 374562949888 ) / 131072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12736893 \beta_{11} + 39489762 \beta_{10} + 281096384 \beta_{9} + 3421918695 \beta_{8} + \cdots + 89174079936 \beta_{6} ) / 65536 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 646660821 \beta_{5} - 138669719 \beta_{4} + 4041790576 \beta_{3} + 44121478912 \beta_{2} + \cdots + 79559030545664 ) / 131072 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2104851051 \beta_{11} + 1126969390 \beta_{10} - 3458012864 \beta_{9} + \cdots + 2604305537088 \beta_{6} ) / 65536 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
7.17456 − 4.53916i
−7.17456 − 4.53916i
−7.99733 + 0.793226i
7.99733 + 0.793226i
4.16154 + 5.83239i
−4.16154 + 5.83239i
4.16154 − 5.83239i
−4.16154 − 5.83239i
−7.99733 − 0.793226i
7.99733 − 0.793226i
7.17456 + 4.53916i
−7.17456 + 4.53916i
0 − 117.615i 0 −306.178 0 − 4321.70i 0 −7272.39 0
255.2 0 − 117.615i 0 306.178 0 4321.70i 0 −7272.39 0
255.3 0 − 117.102i 0 −816.841 0 1333.52i 0 −7151.84 0
255.4 0 − 117.102i 0 816.841 0 − 1333.52i 0 −7151.84 0
255.5 0 − 17.4864i 0 −796.785 0 1779.55i 0 6255.23 0
255.6 0 − 17.4864i 0 796.785 0 − 1779.55i 0 6255.23 0
255.7 0 17.4864i 0 −796.785 0 − 1779.55i 0 6255.23 0
255.8 0 17.4864i 0 796.785 0 1779.55i 0 6255.23 0
255.9 0 117.102i 0 −816.841 0 − 1333.52i 0 −7151.84 0
255.10 0 117.102i 0 816.841 0 1333.52i 0 −7151.84 0
255.11 0 117.615i 0 −306.178 0 4321.70i 0 −7272.39 0
255.12 0 117.615i 0 306.178 0 − 4321.70i 0 −7272.39 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.9.c.n 12
4.b odd 2 1 inner 256.9.c.n 12
8.b even 2 1 inner 256.9.c.n 12
8.d odd 2 1 inner 256.9.c.n 12
16.e even 4 1 8.9.d.b ✓ 6
16.e even 4 1 32.9.d.b 6
16.f odd 4 1 8.9.d.b ✓ 6
16.f odd 4 1 32.9.d.b 6
48.i odd 4 1 72.9.b.b 6
48.i odd 4 1 288.9.b.b 6
48.k even 4 1 72.9.b.b 6
48.k even 4 1 288.9.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.9.d.b ✓ 6 16.e even 4 1
8.9.d.b ✓ 6 16.f odd 4 1
32.9.d.b 6 16.e even 4 1
32.9.d.b 6 16.f odd 4 1
72.9.b.b 6 48.i odd 4 1
72.9.b.b 6 48.k even 4 1
256.9.c.n 12 1.a even 1 1 trivial
256.9.c.n 12 4.b odd 2 1 inner
256.9.c.n 12 8.b even 2 1 inner
256.9.c.n 12 8.d odd 2 1 inner
288.9.b.b 6 48.i odd 4 1
288.9.b.b 6 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{6} + 27852T_{3}^{4} + 198117936T_{3}^{2} + 58003905600 \) Copy content Toggle raw display
\( T_{5}^{6} - 1395840T_{5}^{4} + 545665781760T_{5}^{2} - 39710399791104000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 27852 T^{4} + \cdots + 58003905600)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 24\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 54\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots - 13136750437640)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 26\!\cdots\!28)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 35\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 66\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 49\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 13\!\cdots\!20)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 46\!\cdots\!72)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 47\!\cdots\!60)^{4} \) Copy content Toggle raw display
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