Properties

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

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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 4)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - 305 \beta_1 q^{5} - 7 \beta_{3} q^{7} + 3423 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - 305 \beta_1 q^{5} - 7 \beta_{3} q^{7} + 3423 q^{9} - 185 \beta_{2} q^{11} - 2735 \beta_1 q^{13} + 305 \beta_{3} q^{15} + 73090 q^{17} + 195 \beta_{2} q^{19} + 69888 \beta_1 q^{21} + 1187 \beta_{3} q^{23} + 18525 q^{25} + 3138 \beta_{2} q^{27} - 64111 \beta_1 q^{29} - 340 \beta_{3} q^{31} + 1847040 q^{33} - 8540 \beta_{2} q^{35} + 1736015 \beta_1 q^{37} + 2735 \beta_{3} q^{39} - 2146882 q^{41} - 59329 \beta_{2} q^{43} - 1044015 \beta_1 q^{45} + 38162 \beta_{3} q^{47} + 3807937 q^{49} - 73090 \beta_{2} q^{51} - 412145 \beta_1 q^{53} + 56425 \beta_{3} q^{55} - 1946880 q^{57} - 37285 \beta_{2} q^{59} - 7373039 \beta_1 q^{61} - 23961 \beta_{3} q^{63} - 3336700 q^{65} - 152689 \beta_{2} q^{67} - 11851008 \beta_1 q^{69} + 5985 \beta_{3} q^{71} + 5725630 q^{73} - 18525 \beta_{2} q^{75} + 12929280 \beta_1 q^{77} + 179710 \beta_{3} q^{79} - 53788095 q^{81} + 520019 \beta_{2} q^{83} - 22292450 \beta_1 q^{85} + 64111 \beta_{3} q^{87} + 83324222 q^{89} - 76580 \beta_{2} q^{91} + 3394560 \beta_1 q^{93} - 59475 \beta_{3} q^{95} + 120619010 q^{97} - 633255 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13692 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 13692 q^{9} + 292360 q^{17} + 74100 q^{25} + 7388160 q^{33} - 8587528 q^{41} + 15231748 q^{49} - 7787520 q^{57} - 13346800 q^{65} + 22902520 q^{73} - 215152380 q^{81} + 333296888 q^{89} + 482476040 q^{97}+O(q^{100}) \) 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 −99.9200 0 610.000i 0 1398.88i 0 3423.00 0
127.2 0 −99.9200 0 610.000i 0 1398.88i 0 3423.00 0
127.3 0 99.9200 0 610.000i 0 1398.88i 0 3423.00 0
127.4 0 99.9200 0 610.000i 0 1398.88i 0 3423.00 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.e 4
4.b odd 2 1 inner 256.9.d.e 4
8.b even 2 1 inner 256.9.d.e 4
8.d odd 2 1 inner 256.9.d.e 4
16.e even 4 1 4.9.b.b 2
16.e even 4 1 64.9.c.b 2
16.f odd 4 1 4.9.b.b 2
16.f odd 4 1 64.9.c.b 2
48.i odd 4 1 36.9.d.b 2
48.k even 4 1 36.9.d.b 2
80.i odd 4 1 100.9.d.b 4
80.j even 4 1 100.9.d.b 4
80.k odd 4 1 100.9.b.c 2
80.q even 4 1 100.9.b.c 2
80.s even 4 1 100.9.d.b 4
80.t odd 4 1 100.9.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 16.e even 4 1
4.9.b.b 2 16.f odd 4 1
36.9.d.b 2 48.i odd 4 1
36.9.d.b 2 48.k even 4 1
64.9.c.b 2 16.e even 4 1
64.9.c.b 2 16.f odd 4 1
100.9.b.c 2 80.k odd 4 1
100.9.b.c 2 80.q even 4 1
100.9.d.b 4 80.i odd 4 1
100.9.d.b 4 80.j even 4 1
100.9.d.b 4 80.s even 4 1
100.9.d.b 4 80.t odd 4 1
256.9.d.e 4 1.a even 1 1 trivial
256.9.d.e 4 4.b odd 2 1 inner
256.9.d.e 4 8.b even 2 1 inner
256.9.d.e 4 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 9984 \) 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
$3$ \( (T^{2} - 9984)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 372100)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1956864)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 341702400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 29920900)^{2} \) Copy content Toggle raw display
$17$ \( (T - 73090)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 379641600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 56268585984)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16440881284)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4616601600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12054992320900)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2146882)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 35142983526144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 58160324112384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 679454004100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 13879469510400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 217446816382084)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 232766284318464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1430516505600)^{2} \) Copy content Toggle raw display
$73$ \( (T - 5725630)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T - 83324222)^{4} \) Copy content Toggle raw display
$97$ \( (T - 120619010)^{4} \) Copy content Toggle raw display
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