Properties

Label 256.8.b.k
Level $256$
Weight $8$
Character orbit 256.b
Analytic conductor $79.971$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.50765497344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 101x^{4} + 2512x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{5} + 2 \beta_{3} - 3) q^{7} + ( - 3 \beta_{5} + 7 \beta_{3} + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{5} + 2 \beta_{3} - 3) q^{7} + ( - 3 \beta_{5} + 7 \beta_{3} + 21) q^{9} + ( - 20 \beta_{4} + 32 \beta_{2} - 21 \beta_1) q^{11} + ( - 27 \beta_{4} + 147 \beta_{2} - 85 \beta_1) q^{13} + (3 \beta_{5} - 90 \beta_{3} + 1341) q^{15} + (15 \beta_{5} - 211 \beta_{3} + 104) q^{17} + ( - 68 \beta_{4} - 36 \beta_{2} + 531 \beta_1) q^{19} + ( - 24 \beta_{4} - 1055 \beta_{2} + 392 \beta_1) q^{21} + ( - 51 \beta_{5} + 482 \beta_{3} - 14261) q^{23} + (94 \beta_{5} + 42 \beta_{3} + 13517) q^{25} + ( - 84 \beta_{4} - 1495 \beta_{2} - 1022 \beta_1) q^{27} + (245 \beta_{4} + 7196 \beta_{2} - 2805 \beta_1) q^{29} + ( - 364 \beta_{5} - 20 \beta_{3} + 52424) q^{31} + ( - 63 \beta_{5} - 1549 \beta_{3} - 61662) q^{33} + ( - 360 \beta_{4} + 16650 \beta_{2} - 532 \beta_1) q^{35} + (47 \beta_{4} - 11059 \beta_{2} - 6847 \beta_1) q^{37} + ( - 255 \beta_{5} - 1506 \beta_{3} - 207645) q^{39} + ( - 658 \beta_{5} - 5382 \beta_{3} + 301658) q^{41} + ( - 344 \beta_{4} - 34921 \beta_{2} + 6227 \beta_1) q^{43} + ( - 1107 \beta_{4} + 50445 \beta_{2} - 1917 \beta_1) q^{45} + (1534 \beta_{5} + 2736 \beta_{3} + 284718) q^{47} + ( - 332 \beta_{5} - 932 \beta_{3} - 545711) q^{49} + (2532 \beta_{4} + 114891 \beta_{2} - 9378 \beta_1) q^{51} + (1183 \beta_{4} - 76845 \beta_{2} - 21455 \beta_1) q^{53} + (1967 \beta_{5} - 2814 \beta_{3} - 1236819) q^{55} + (1593 \beta_{5} - 7653 \beta_{3} + 1078674) q^{57} + (3552 \beta_{4} - 92551 \beta_{2} - 849 \beta_1) q^{59} + (3749 \beta_{4} + 175039 \beta_{2} - 13077 \beta_1) q^{61} + ( - 1011 \beta_{5} - 3110 \beta_{3} + 847443) q^{63} + (3014 \beta_{5} - 9086 \beta_{3} - 1595624) q^{65} + ( - 3124 \beta_{4} + 325076 \beta_{2} + 17935 \beta_1) q^{67} + ( - 5784 \beta_{4} - 275773 \beta_{2} + 41320 \beta_1) q^{69} + ( - 729 \beta_{5} + 2966 \beta_{3} - 3312591) q^{71} + (217 \beta_{5} + 22363 \beta_{3} + 1938872) q^{73} + ( - 504 \beta_{4} - 13405 \beta_{2} - 45713 \beta_1) q^{75} + ( - 8040 \beta_{4} + 289925 \beta_{2} + 5144 \beta_1) q^{77} + ( - 2814 \beta_{5} - 21140 \beta_{3} + 328726) q^{79} + ( - 9627 \beta_{5} + 5087 \beta_{3} - 2143389) q^{81} + (6256 \beta_{4} + 617001 \beta_{2} + 18499 \beta_1) q^{83} + (4798 \beta_{4} - 339345 \beta_{2} + 71570 \beta_1) q^{85} + ( - 8415 \beta_{5} + 58514 \beta_{3} - 6040401) q^{87} + ( - 5783 \beta_{5} + 38731 \beta_{3} + 1625288) q^{89} + ( - 12968 \beta_{4} + 330690 \beta_{2} + 30660 \beta_1) q^{91} + (240 \beta_{4} + 78652 \beta_{2} + 74960 \beta_1) q^{93} + (6711 \beta_{5} + 42818 \beta_{3} - 5031307) q^{95} + (18515 \beta_{5} + 69369 \beta_{3} - 1733280) q^{97} + ( - 25152 \beta_{4} + 858271 \beta_{2} + 8417 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{7} + 106 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{7} + 106 q^{9} + 8232 q^{15} + 1076 q^{17} - 86632 q^{23} + 81206 q^{25} + 313856 q^{31} - 367000 q^{33} - 1243368 q^{39} + 1819396 q^{41} + 1705904 q^{47} - 3273066 q^{49} - 7411352 q^{55} + 6490536 q^{57} + 5088856 q^{63} - 9549544 q^{65} - 19882936 q^{71} + 11588940 q^{73} + 2009008 q^{79} - 12889762 q^{81} - 36376264 q^{87} + 9662700 q^{89} - 30260056 q^{95} - 10501388 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 101x^{4} + 2512x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 95\nu^{3} - 1854\nu ) / 44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 95\nu^{3} - 2206\nu ) / 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{4} - 204\nu^{2} + 31 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 73\nu^{5} + 7639\nu^{3} + 197118\nu ) / 66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -20\nu^{4} - 1724\nu^{2} - 23517 ) / 33 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -3\beta_{2} + 4\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 5\beta_{3} - 2152 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12\beta_{4} + 1053\beta_{2} - 820\beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 153\beta_{5} - 431\beta_{3} + 110248 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1140\beta_{4} - 77787\beta_{2} + 42604\beta_1 ) / 128 \) Copy content Toggle raw display

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
7.53231i
6.65206i
0.119748i
0.119748i
6.65206i
7.53231i
0 62.2585i 0 203.009i 0 −534.535 0 −1689.12 0
129.2 0 51.2165i 0 145.477i 0 −192.521 0 −436.129 0
129.3 0 2.95798i 0 362.486i 0 715.055 0 2178.25 0
129.4 0 2.95798i 0 362.486i 0 715.055 0 2178.25 0
129.5 0 51.2165i 0 145.477i 0 −192.521 0 −436.129 0
129.6 0 62.2585i 0 203.009i 0 −534.535 0 −1689.12 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.6
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.k 6
4.b odd 2 1 256.8.b.l 6
8.b even 2 1 inner 256.8.b.k 6
8.d odd 2 1 256.8.b.l 6
16.e even 4 1 128.8.a.b yes 3
16.e even 4 1 128.8.a.c yes 3
16.f odd 4 1 128.8.a.a 3
16.f odd 4 1 128.8.a.d yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.8.a.a 3 16.f odd 4 1
128.8.a.b yes 3 16.e even 4 1
128.8.a.c yes 3 16.e even 4 1
128.8.a.d yes 3 16.f odd 4 1
256.8.b.k 6 1.a even 1 1 trivial
256.8.b.k 6 8.b even 2 1 inner
256.8.b.l 6 4.b odd 2 1
256.8.b.l 6 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{6} + 6508T_{3}^{4} + 10224432T_{3}^{2} + 88962624 \) Copy content Toggle raw display
\( T_{7}^{3} + 12T_{7}^{2} - 416976T_{7} - 73585600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 6508 T^{4} + \cdots + 88962624 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 114603876302400 \) Copy content Toggle raw display
$7$ \( (T^{3} + 12 T^{2} - 416976 T - 73585600)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 81591084 T^{4} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{6} + 199327532 T^{4} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( (T^{3} - 538 T^{2} + \cdots - 5603612104824)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 2547517676 T^{4} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{3} + 43316 T^{2} + \cdots + 36954823141312)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 65975706892 T^{4} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{3} - 156928 T^{2} + \cdots - 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 338211325452 T^{4} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{3} - 909698 T^{2} + \cdots + 31\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 462753523084 T^{4} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{3} - 852952 T^{2} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 4538710858188 T^{4} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + 4089116028396 T^{4} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{6} + 9057050008236 T^{4} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{6} + 24990280042380 T^{4} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{3} + 9941468 T^{2} + \cdots + 35\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 5794470 T^{2} + \cdots + 48\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 1004504 T^{2} + \cdots + 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 84914470467660 T^{4} + \cdots + 98\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{3} - 4831350 T^{2} + \cdots + 95\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 5250694 T^{2} + \cdots - 83\!\cdots\!64)^{2} \) Copy content Toggle raw display
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