Properties

Label 256.9.d.b
Level $256$
Weight $9$
Character orbit 256.d
Analytic conductor $104.289$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(104.288924176\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 32)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 36) q^{3} + ( - 2 \beta_{3} + 91 \beta_1) q^{5} + (11 \beta_{3} - 236 \beta_1) q^{7} + ( - 72 \beta_{2} + 4719) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 36) q^{3} + ( - 2 \beta_{3} + 91 \beta_1) q^{5} + (11 \beta_{3} - 236 \beta_1) q^{7} + ( - 72 \beta_{2} + 4719) q^{9} + (41 \beta_{2} - 10052) q^{11} + ( - 230 \beta_{3} - 1579 \beta_1) q^{13} + (163 \beta_{3} - 23244 \beta_1) q^{15} + (424 \beta_{2} - 97998) q^{17} + (365 \beta_{2} + 166188) q^{19} + ( - 632 \beta_{3} + 118320 \beta_1) q^{21} + (1017 \beta_{3} + 3164 \beta_1) q^{23} + (1456 \beta_{2} + 197757) q^{25} + (750 \beta_{2} - 652536) q^{27} + (898 \beta_{3} - 88187 \beta_1) q^{29} + (3412 \beta_{3} + 339248 \beta_1) q^{31} + ( - 11528 \beta_{2} + 771216) q^{33} + ( - 5892 \beta_{2} + 964496) q^{35} + ( - 1018 \beta_{3} - 555381 \beta_1) q^{37} + (6701 \beta_{3} - 2239476 \beta_1) q^{39} + ( - 38096 \beta_{2} - 738338) q^{41} + ( - 42527 \beta_{2} + 761692) q^{43} + ( - 15990 \beta_{3} + 1867125 \beta_1) q^{45} + (9958 \beta_{3} + 2621800 \beta_1) q^{47} + (20768 \beta_{2} + 709761) q^{49} + ( - 113262 \beta_{2} + 7761144) q^{51} + (6806 \beta_{3} + 562731 \beta_1) q^{53} + (23835 \beta_{3} - 1733420 \beta_1) q^{55} + (153048 \beta_{2} - 2338608) q^{57} + ( - 113435 \beta_{2} - 3544628) q^{59} + (70938 \beta_{3} - 5019883 \beta_1) q^{61} + (68901 \beta_{3} - 9021012 \beta_1) q^{63} + (71088 \beta_{2} - 17795804) q^{65} + (64673 \beta_{2} - 417316) q^{67} + ( - 33448 \beta_{3} + 10039824 \beta_1) q^{69} + (119571 \beta_{3} + 9064564 \beta_1) q^{71} + ( - 104936 \beta_{2} + 1480206) q^{73} + (145341 \beta_{2} + 7417452) q^{75} + ( - 120248 \beta_{3} + 6875056 \beta_1) q^{77} + (71994 \beta_{3} - 10602344 \beta_1) q^{79} + ( - 207144 \beta_{2} + 17937) q^{81} + ( - 189139 \beta_{2} - 39279188) q^{83} + (234580 \beta_{3} - 17384250 \beta_1) q^{85} + ( - 120515 \beta_{3} + 12140364 \beta_1) q^{87} + ( - 451240 \beta_{2} - 33161970) q^{89} + ( - 147644 \beta_{2} + 99547504) q^{91} + (216416 \beta_{3} + 21852480 \beta_1) q^{93} + ( - 299161 \beta_{3} + 7834788 \beta_1) q^{95} + (460680 \beta_{2} + 14111218) q^{97} + (917223 \beta_{2} - 76908156) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 144 q^{3} + 18876 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 144 q^{3} + 18876 q^{9} - 40208 q^{11} - 391992 q^{17} + 664752 q^{19} + 791028 q^{25} - 2610144 q^{27} + 3084864 q^{33} + 3857984 q^{35} - 2953352 q^{41} + 3046768 q^{43} + 2839044 q^{49} + 31044576 q^{51} - 9354432 q^{57} - 14178512 q^{59} - 71183216 q^{65} - 1669264 q^{67} + 5920824 q^{73} + 29669808 q^{75} + 71748 q^{81} - 157116752 q^{83} - 132647880 q^{89} + 398190016 q^{91} + 56444872 q^{97} - 307632624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 19x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 9\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{3} + 232\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 64\nu^{2} - 608 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 608 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{2} + 232\beta_1 ) / 32 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−3.12250 0.500000i
−3.12250 + 0.500000i
3.12250 + 0.500000i
3.12250 0.500000i
0 −135.920 0 581.680i 0 2670.24i 0 11913.2 0
127.2 0 −135.920 0 581.680i 0 2670.24i 0 11913.2 0
127.3 0 63.9200 0 217.680i 0 1726.24i 0 −2475.24 0
127.4 0 63.9200 0 217.680i 0 1726.24i 0 −2475.24 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.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.d.b 4
4.b odd 2 1 256.9.d.h 4
8.b even 2 1 256.9.d.h 4
8.d odd 2 1 inner 256.9.d.b 4
16.e even 4 1 32.9.c.a 4
16.e even 4 1 64.9.c.f 4
16.f odd 4 1 32.9.c.a 4
16.f odd 4 1 64.9.c.f 4
48.i odd 4 1 288.9.g.b 4
48.k even 4 1 288.9.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.9.c.a 4 16.e even 4 1
32.9.c.a 4 16.f odd 4 1
64.9.c.f 4 16.e even 4 1
64.9.c.f 4 16.f odd 4 1
256.9.d.b 4 1.a even 1 1 trivial
256.9.d.b 4 8.d odd 2 1 inner
256.9.d.h 4 4.b odd 2 1
256.9.d.h 4 8.b even 2 1
288.9.g.b 4 48.i odd 4 1
288.9.g.b 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 72T_{3} - 8688 \) acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 72 T - 8688)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 16032624400 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 21247232118784 \) Copy content Toggle raw display
$11$ \( (T^{2} + 20104 T + 84259600)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( (T^{2} + 195996 T + 7808724420)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 332376 T + 26288332944)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 13944688274300)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots - 17476345855472)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots - 115904724604016)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots - 41584895095280)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 107748446132028)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{2} + \cdots + 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 933201241117500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
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