Properties

Label 256.9.d.i
Level $256$
Weight $9$
Character orbit 256.d
Analytic conductor $104.289$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 316 x^{14} - 2552 x^{13} + 15056 x^{12} + 854200 x^{11} + 10172664 x^{10} + \cdots + 50894124966450 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{151} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 14) q^{3} - \beta_{5} q^{5} + (\beta_{12} - \beta_{5}) q^{7} + (\beta_{6} + 9 \beta_1 + 2187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 14) q^{3} - \beta_{5} q^{5} + (\beta_{12} - \beta_{5}) q^{7} + (\beta_{6} + 9 \beta_1 + 2187) q^{9} + (\beta_{10} + \beta_{6} + 16 \beta_1 + 2442) q^{11} + (\beta_{14} - 4 \beta_{12} + \cdots - 30 \beta_{2}) q^{13}+ \cdots + (1341 \beta_{11} - 807 \beta_{10} + \cdots + 63677070) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 224 q^{3} + 34992 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 224 q^{3} + 34992 q^{9} + 39072 q^{11} + 77280 q^{17} + 285216 q^{19} - 1604144 q^{25} + 2241856 q^{27} + 1652672 q^{33} - 6029952 q^{35} - 4938528 q^{41} - 3693536 q^{43} - 25310960 q^{49} + 40422336 q^{51} + 10158272 q^{57} + 19174560 q^{59} - 37587648 q^{65} + 57931680 q^{67} - 74289760 q^{73} + 137112224 q^{75} + 90834128 q^{81} - 2013792 q^{83} - 32417376 q^{89} + 415140992 q^{91} + 131455712 q^{97} + 1018833120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 316 x^{14} - 2552 x^{13} + 15056 x^{12} + 854200 x^{11} + 10172664 x^{10} + \cdots + 50894124966450 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 46\!\cdots\!32 \nu^{15} + \cdots + 44\!\cdots\!00 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\!\cdots\!32 \nu^{15} + \cdots - 18\!\cdots\!08 ) / 41\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\!\cdots\!68 \nu^{15} + \cdots + 29\!\cdots\!00 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!08 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\!\cdots\!92 \nu^{15} + \cdots - 90\!\cdots\!50 ) / 22\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!84 \nu^{15} + \cdots - 39\!\cdots\!00 ) / 26\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!52 \nu^{15} + \cdots + 52\!\cdots\!00 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!32 \nu^{15} + \cdots + 11\!\cdots\!80 ) / 22\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!12 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 69\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!76 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!24 \nu^{15} + \cdots + 70\!\cdots\!00 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!96 \nu^{15} + \cdots + 68\!\cdots\!50 ) / 63\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 53\!\cdots\!56 \nu^{15} + \cdots + 16\!\cdots\!50 ) / 86\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!36 \nu^{15} + \cdots + 40\!\cdots\!38 ) / 45\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 39\!\cdots\!04 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 56 \beta_{15} - 8 \beta_{14} + 69 \beta_{13} - 616 \beta_{12} + 160 \beta_{11} + \cdots - 14046 \beta_1 ) / 524288 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 344 \beta_{15} + 232 \beta_{14} + 569 \beta_{13} - 3896 \beta_{12} + 1056 \beta_{11} + \cdots + 10354688 ) / 262144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2736 \beta_{15} + 784 \beta_{14} + 69537 \beta_{13} - 99312 \beta_{12} + 21344 \beta_{11} + \cdots + 250871808 ) / 524288 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5124 \beta_{15} + 29692 \beta_{14} + 78552 \beta_{13} - 700820 \beta_{12} + 24240 \beta_{11} + \cdots + 571342848 ) / 65536 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 136452 \beta_{15} - 300420 \beta_{14} + 15717372 \beta_{13} - 30716564 \beta_{12} + \cdots - 3913154560 ) / 262144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 20759224 \beta_{15} + 45031816 \beta_{14} + 95580913 \beta_{13} - 705910808 \beta_{12} + \cdots - 113623957504 ) / 262144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 31184688 \beta_{15} + 21385456 \beta_{14} + 2032043835 \beta_{13} - 4685742256 \beta_{12} + \cdots - 112267624448 ) / 131072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 327186840 \beta_{15} + 966803240 \beta_{14} + 4339255797 \beta_{13} - 23893373560 \beta_{12} + \cdots - 28165747867648 ) / 32768 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15590945244 \beta_{15} + 19809760580 \beta_{14} + 356728005921 \beta_{13} - 923193675660 \beta_{12} + \cdots - 12\!\cdots\!08 ) / 131072 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 255446564392 \beta_{15} + 523831075160 \beta_{14} + 2827666906711 \beta_{13} + \cdots - 43\!\cdots\!76 ) / 131072 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 250167803316 \beta_{15} + 2054490579724 \beta_{14} + 21943998929547 \beta_{13} + \cdots - 48\!\cdots\!24 ) / 131072 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 27436225645536 \beta_{15} - 74339428358752 \beta_{14} - 136669104426927 \beta_{13} + \cdots - 14\!\cdots\!60 ) / 131072 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14\!\cdots\!12 \beta_{15} + \cdots - 60\!\cdots\!56 ) / 524288 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 64\!\cdots\!96 \beta_{15} + \cdots - 59\!\cdots\!64 ) / 262144 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 44\!\cdots\!28 \beta_{15} + \cdots - 32\!\cdots\!08 ) / 131072 \) 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
5.93298 2.42889i
5.93298 + 2.42889i
1.49256 + 0.108386i
1.49256 0.108386i
−14.5863 + 4.36773i
−14.5863 4.36773i
16.6303 + 4.63467i
16.6303 4.63467i
−3.42845 12.5625i
−3.42845 + 12.5625i
3.43006 0.984106i
3.43006 + 0.984106i
−3.53658 + 6.06559i
−3.53658 6.06559i
−5.93458 + 6.32105i
−5.93458 6.32105i
0 −135.082 0 93.0802i 0 2522.45i 0 11686.2 0
127.2 0 −135.082 0 93.0802i 0 2522.45i 0 11686.2 0
127.3 0 −115.263 0 1176.43i 0 4245.70i 0 6724.51 0
127.4 0 −115.263 0 1176.43i 0 4245.70i 0 6724.51 0
127.5 0 −63.7444 0 944.632i 0 1875.82i 0 −2497.66 0
127.6 0 −63.7444 0 944.632i 0 1875.82i 0 −2497.66 0
127.7 0 −47.6247 0 1.66205i 0 4641.59i 0 −4292.89 0
127.8 0 −47.6247 0 1.66205i 0 4641.59i 0 −4292.89 0
127.9 0 −25.5081 0 679.831i 0 2310.00i 0 −5910.34 0
127.10 0 −25.5081 0 679.831i 0 2310.00i 0 −5910.34 0
127.11 0 36.1265 0 710.671i 0 51.7339i 0 −5255.88 0
127.12 0 36.1265 0 710.671i 0 51.7339i 0 −5255.88 0
127.13 0 91.4359 0 122.296i 0 544.420i 0 1799.53 0
127.14 0 91.4359 0 122.296i 0 544.420i 0 1799.53 0
127.15 0 147.660 0 812.351i 0 1920.17i 0 15242.5 0
127.16 0 147.660 0 812.351i 0 1920.17i 0 15242.5 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.16
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.i 16
4.b odd 2 1 256.9.d.j 16
8.b even 2 1 256.9.d.j 16
8.d odd 2 1 inner 256.9.d.i 16
16.e even 4 1 128.9.c.a 16
16.e even 4 1 128.9.c.b yes 16
16.f odd 4 1 128.9.c.a 16
16.f odd 4 1 128.9.c.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.9.c.a 16 16.e even 4 1
128.9.c.a 16 16.f odd 4 1
128.9.c.b yes 16 16.e even 4 1
128.9.c.b yes 16 16.f odd 4 1
256.9.d.i 16 1.a even 1 1 trivial
256.9.d.i 16 8.d odd 2 1 inner
256.9.d.j 16 4.b odd 2 1
256.9.d.j 16 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 112 T_{3}^{7} - 28720 T_{3}^{6} - 3568704 T_{3}^{5} + 142691424 T_{3}^{4} + \cdots - 588091959795456 \) acting on \(S_{9}^{\mathrm{new}}(256, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots - 588091959795456)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 37\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 74\!\cdots\!84)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 30\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 34\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 33\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 89\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 28\!\cdots\!04)^{2} \) Copy content Toggle raw display
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