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Properties

Label 256.5.d.g
Level $256$
Weight $5$
Character orbit 256.d
Analytic conductor $26.463$
Analytic rank $0$
Dimension $8$
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] = [256,5,Mod(127,256)]
 
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, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 256.d (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: \(26.4627105495\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.205520896.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{31} \)
Twist minimal: no (minimal twist has level 128)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2) q^{3} + (\beta_{5} + \beta_1) q^{5} + (2 \beta_{5} - \beta_{4} + 3 \beta_1) q^{7} + (\beta_{6} - 7 \beta_{3} + \beta_{2} + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2) q^{3} + (\beta_{5} + \beta_1) q^{5} + (2 \beta_{5} - \beta_{4} + 3 \beta_1) q^{7} + (\beta_{6} - 7 \beta_{3} + \beta_{2} + 27) q^{9} + ( - \beta_{6} + 2 \beta_{3} + 2 \beta_{2} + 6) q^{11} + (3 \beta_{7} - \beta_{5} + \cdots - 4 \beta_1) q^{13}+ \cdots + (40 \beta_{6} - 169 \beta_{3} + \cdots + 1794) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{3} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{3} + 216 q^{9} + 48 q^{11} + 240 q^{17} - 912 q^{19} - 664 q^{25} - 5536 q^{27} + 992 q^{33} - 9408 q^{35} + 2928 q^{41} - 9104 q^{43} - 5752 q^{49} - 16608 q^{51} - 8608 q^{57} - 21456 q^{59} + 672 q^{65} - 18000 q^{67} - 560 q^{73} - 35024 q^{75} + 14696 q^{81} - 26448 q^{83} + 22992 q^{89} + 15296 q^{91} - 3728 q^{97} + 14352 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 48\nu^{7} - 112\nu^{6} + 208\nu^{5} + 632\nu^{4} - 2096\nu^{3} + 1072\nu^{2} + 1456\nu + 352 ) / 51 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -64\nu^{7} + 172\nu^{6} - 504\nu^{5} + 64\nu^{4} + 664\nu^{3} + 452\nu^{2} + 144\nu - 2464 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 176\nu^{7} - 762\nu^{6} + 2372\nu^{5} - 2896\nu^{4} - 500\nu^{3} + 2514\nu^{2} + 760\nu + 112 ) / 51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 232\nu^{7} - 972\nu^{6} + 2864\nu^{5} - 3088\nu^{4} - 2016\nu^{3} + 2756\nu^{2} + 3048\nu + 704 ) / 51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -292\nu^{7} + 976\nu^{6} - 2580\nu^{5} + 530\nu^{4} + 6812\nu^{3} - 4776\nu^{2} - 5956\nu - 1416 ) / 51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1264\nu^{7} - 5658\nu^{6} + 17604\nu^{5} - 22480\nu^{4} - 2676\nu^{3} + 20466\nu^{2} + 6200\nu + 112 ) / 51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1584\nu^{7} + 6824\nu^{6} - 21280\nu^{5} + 26336\nu^{4} + 1984\nu^{3} - 15928\nu^{2} - 15408\nu - 3456 ) / 51 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - 8\beta_{5} - 6\beta_{4} - 8\beta_{3} + 4\beta_{2} - 18\beta _1 + 128 ) / 256 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 4\beta_{5} - 4\beta_{4} - 14\beta_{3} - 2\beta_{2} - 9\beta _1 - 64 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 11\beta_{6} + 20\beta_{5} - 87\beta_{3} - 18\beta_{2} + 61\beta _1 - 704 ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 64\beta_{5} + 22\beta_{4} + 184\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{7} - 91\beta_{6} + 148\beta_{5} + 44\beta_{4} + 711\beta_{3} + 162\beta_{2} + 429\beta _1 + 6464 ) / 128 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9\beta_{7} - 104\beta_{6} - 364\beta_{5} - 190\beta_{4} + 812\beta_{3} + 182\beta_{2} - 995\beta _1 + 7232 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 167 \beta_{7} + 600 \beta_{6} - 5704 \beta_{5} - 2774 \beta_{4} - 4688 \beta_{3} - 1220 \beta_{2} + \cdots - 50048 ) / 256 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.55050 − 0.642239i
1.55050 + 0.642239i
−0.129640 + 0.312979i
−0.129640 − 0.312979i
−0.550501 − 0.228025i
−0.550501 + 0.228025i
1.12964 + 2.72719i
1.12964 − 2.72719i
0 −16.7135 0 − 10.1891i 0 − 80.4325i 0 198.341 0
127.2 0 −16.7135 0 10.1891i 0 80.4325i 0 198.341 0
127.3 0 −6.29514 0 − 22.7388i 0 − 51.5147i 0 −41.3713 0
127.4 0 −6.29514 0 22.7388i 0 51.5147i 0 −41.3713 0
127.5 0 7.05664 0 − 9.50277i 0 − 6.49140i 0 −31.2038 0
127.6 0 7.05664 0 9.50277i 0 6.49140i 0 −31.2038 0
127.7 0 7.95199 0 − 46.0525i 0 − 57.5735i 0 −17.7658 0
127.8 0 7.95199 0 46.0525i 0 57.5735i 0 −17.7658 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 256.5.d.g 8
4.b odd 2 1 256.5.d.h 8
8.b even 2 1 256.5.d.h 8
8.d odd 2 1 inner 256.5.d.g 8
16.e even 4 1 128.5.c.a ✓ 8
16.e even 4 1 128.5.c.b yes 8
16.f odd 4 1 128.5.c.a ✓ 8
16.f odd 4 1 128.5.c.b yes 8
48.i odd 4 1 1152.5.g.a 8
48.i odd 4 1 1152.5.g.b 8
48.k even 4 1 1152.5.g.a 8
48.k even 4 1 1152.5.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.c.a ✓ 8 16.e even 4 1
128.5.c.a ✓ 8 16.f odd 4 1
128.5.c.b yes 8 16.e even 4 1
128.5.c.b yes 8 16.f odd 4 1
256.5.d.g 8 1.a even 1 1 trivial
256.5.d.g 8 8.d odd 2 1 inner
256.5.d.h 8 4.b odd 2 1
256.5.d.h 8 8.b even 2 1
1152.5.g.a 8 48.i odd 4 1
1152.5.g.a 8 48.k even 4 1
1152.5.g.b 8 48.i odd 4 1
1152.5.g.b 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 8T_{3}^{3} - 184T_{3}^{2} - 288T_{3} + 5904 \) acting on \(S_{5}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 8 T^{3} + \cdots + 5904)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 10280337664 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 2397988519936 \) Copy content Toggle raw display
$11$ \( (T^{4} - 24 T^{3} + \cdots + 25651216)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 71355458150656 \) Copy content Toggle raw display
$17$ \( (T^{4} - 120 T^{3} + \cdots + 1484236816)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 456 T^{3} + \cdots + 4378659088)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 80\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 3022842029552)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4552 T^{3} + \cdots + 462366221584)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 26892080667664)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 180967811812848)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 412684399487248)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 902213637871088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 750969871702288)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 103658484613616)^{2} \) Copy content Toggle raw display
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