Properties

Label 256.9.d.f
Level $256$
Weight $9$
Character orbit 256.d
Analytic conductor $104.289$
Analytic rank $0$
Dimension $4$
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(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: \(\Q(i, \sqrt{35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - 255 \beta_1 q^{5} - 9 \beta_{3} q^{7} + 13599 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - 255 \beta_1 q^{5} - 9 \beta_{3} q^{7} + 13599 q^{9} + 135 \beta_{2} q^{11} + 13855 \beta_1 q^{13} + 255 \beta_{3} q^{15} + 50370 q^{17} - 765 \beta_{2} q^{19} + 181440 \beta_1 q^{21} + 621 \beta_{3} q^{23} + 130525 q^{25} - 7038 \beta_{2} q^{27} - 27489 \beta_1 q^{29} - 4140 \beta_{3} q^{31} - 2721600 q^{33} - 9180 \beta_{2} q^{35} + 396865 \beta_1 q^{37} - 13855 \beta_{3} q^{39} + 75582 q^{41} + 3519 \beta_{2} q^{43} - 3467745 \beta_1 q^{45} - 10098 \beta_{3} q^{47} - 767039 q^{49} - 50370 \beta_{2} q^{51} + 5583105 \beta_1 q^{53} - 34425 \beta_{3} q^{55} + 15422400 q^{57} - 153765 \beta_{2} q^{59} + 11913311 \beta_1 q^{61} - 122391 \beta_{3} q^{63} + 14132100 q^{65} - 52785 \beta_{2} q^{67} - 12519360 \beta_1 q^{69} - 35505 \beta_{3} q^{71} - 6516610 q^{73} - 130525 \beta_{2} q^{75} - 24494400 \beta_1 q^{77} + 171810 \beta_{3} q^{79} + 52663041 q^{81} - 517293 \beta_{2} q^{83} - 12844350 \beta_1 q^{85} + 27489 \beta_{3} q^{87} - 86795778 q^{89} + 498780 \beta_{2} q^{91} + 83462400 \beta_1 q^{93} + 195075 \beta_{3} q^{95} - 46670270 q^{97} + 1835865 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 54396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 54396 q^{9} + 201480 q^{17} + 522100 q^{25} - 10886400 q^{33} + 302328 q^{41} - 3068156 q^{49} + 61689600 q^{57} + 56528400 q^{65} - 26066440 q^{73} + 210652164 q^{81} - 347183112 q^{89} - 186681080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 16\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{3} + 208\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 96\nu^{2} - 816 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 12\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 816 ) / 96 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 39\beta_1 ) / 6 \) 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
2.95804 + 0.500000i
2.95804 0.500000i
−2.95804 + 0.500000i
−2.95804 0.500000i
0 −141.986 0 510.000i 0 2555.75i 0 13599.0 0
127.2 0 −141.986 0 510.000i 0 2555.75i 0 13599.0 0
127.3 0 141.986 0 510.000i 0 2555.75i 0 13599.0 0
127.4 0 141.986 0 510.000i 0 2555.75i 0 13599.0 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
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.d.f 4
4.b odd 2 1 inner 256.9.d.f 4
8.b even 2 1 inner 256.9.d.f 4
8.d odd 2 1 inner 256.9.d.f 4
16.e even 4 1 16.9.c.a 2
16.e even 4 1 64.9.c.d 2
16.f odd 4 1 16.9.c.a 2
16.f odd 4 1 64.9.c.d 2
48.i odd 4 1 144.9.g.g 2
48.k even 4 1 144.9.g.g 2
80.i odd 4 1 400.9.h.b 4
80.j even 4 1 400.9.h.b 4
80.k odd 4 1 400.9.b.c 2
80.q even 4 1 400.9.b.c 2
80.s even 4 1 400.9.h.b 4
80.t odd 4 1 400.9.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.a 2 16.e even 4 1
16.9.c.a 2 16.f odd 4 1
64.9.c.d 2 16.e even 4 1
64.9.c.d 2 16.f odd 4 1
144.9.g.g 2 48.i odd 4 1
144.9.g.g 2 48.k even 4 1
256.9.d.f 4 1.a even 1 1 trivial
256.9.d.f 4 4.b odd 2 1 inner
256.9.d.f 4 8.b even 2 1 inner
256.9.d.f 4 8.d odd 2 1 inner
400.9.b.c 2 80.k odd 4 1
400.9.b.c 2 80.q even 4 1
400.9.h.b 4 80.i odd 4 1
400.9.h.b 4 80.j even 4 1
400.9.h.b 4 80.s even 4 1
400.9.h.b 4 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20160 \) 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
$3$ \( (T^{2} - 20160)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 260100)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6531840)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 367416000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 767844100)^{2} \) Copy content Toggle raw display
$17$ \( (T - 50370)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 11798136000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 31098090240)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3022580484)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1382137344000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 630007312900)^{2} \) Copy content Toggle raw display
$41$ \( (T - 75582)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 249648557760)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8222828866560)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 124684245764100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 476656492536000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 567707915930884)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 56170925496000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 101655189216000)^{2} \) Copy content Toggle raw display
$73$ \( (T + 6516610)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 53\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T + 86795778)^{4} \) Copy content Toggle raw display
$97$ \( (T + 46670270)^{4} \) Copy content Toggle raw display
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