Properties

Label 128.9.c.a
Level $128$
Weight $9$
Character orbit 128.c
Analytic conductor $52.144$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,9,Mod(127,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 128.c (of order \(2\), degree \(1\), 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: \(52.1444620878\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{183} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_1 - 42) q^{5} + ( - \beta_{6} - 5 \beta_{2}) q^{7} + (\beta_{3} - 2187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_1 - 42) q^{5} + ( - \beta_{6} - 5 \beta_{2}) q^{7} + (\beta_{3} - 2187) q^{9} + (\beta_{4} + 12 \beta_{2}) q^{11} + (\beta_{5} - 5 \beta_1 - 3570) q^{13} + ( - \beta_{13} - \beta_{10} + \cdots + 39 \beta_{2}) q^{15}+ \cdots + ( - 124 \beta_{15} + \cdots - 279987 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 672 q^{5} - 34992 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 672 q^{5} - 34992 q^{9} - 57120 q^{13} + 77280 q^{17} - 683392 q^{21} + 1604144 q^{25} - 2667936 q^{29} + 1652672 q^{33} + 6053600 q^{37} + 4938528 q^{41} + 1028320 q^{45} - 25310960 q^{49} - 3112224 q^{53} - 10158272 q^{57} - 30994976 q^{61} - 37587648 q^{65} + 152334720 q^{69} + 74289760 q^{73} + 30451584 q^{77} + 90834128 q^{81} - 342254272 q^{85} + 32417376 q^{89} - 122685440 q^{93} + 131455712 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 28\!\cdots\!44 \nu^{15} + \cdots + 37\!\cdots\!60 ) / 69\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!96 \nu^{15} + \cdots - 45\!\cdots\!50 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36\!\cdots\!04 \nu^{15} + \cdots + 47\!\cdots\!00 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 35\!\cdots\!12 \nu^{15} + \cdots - 11\!\cdots\!50 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!08 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!24 \nu^{15} + \cdots - 47\!\cdots\!50 ) / 37\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!08 \nu^{15} + \cdots - 26\!\cdots\!00 ) / 38\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 54\!\cdots\!96 \nu^{15} + \cdots - 66\!\cdots\!00 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56\!\cdots\!96 \nu^{15} + \cdots + 54\!\cdots\!00 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 71\!\cdots\!72 \nu^{15} + \cdots - 26\!\cdots\!00 ) / 28\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 99\!\cdots\!16 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 26\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 79\!\cdots\!12 \nu^{15} + \cdots - 25\!\cdots\!00 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24\!\cdots\!32 \nu^{15} + \cdots - 80\!\cdots\!00 ) / 22\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!64 \nu^{15} + \cdots - 50\!\cdots\!00 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 31\!\cdots\!96 \nu^{15} + \cdots - 12\!\cdots\!20 ) / 22\!\cdots\!45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 16 \beta_{15} - 176 \beta_{14} - 176 \beta_{13} - 72 \beta_{12} - 4 \beta_{11} + \cdots + 54040 \beta_1 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 208 \beta_{15} + 240 \beta_{14} + 240 \beta_{13} - 80 \beta_{12} - 1772 \beta_{11} + \cdots + 165675008 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29216 \beta_{15} - 20960 \beta_{14} + 56352 \beta_{13} - 46220 \beta_{12} + 1606 \beta_{11} + \cdots + 4013948928 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16216 \beta_{15} - 68984 \beta_{14} - 129656 \beta_{13} - 82402 \beta_{12} - 39946 \beta_{11} + \cdots + 9141485568 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3390792 \beta_{15} - 6107736 \beta_{14} + 8886952 \beta_{13} - 13377458 \beta_{12} + \cdots - 62610472960 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18035888 \beta_{15} - 59583376 \beta_{14} - 85544848 \beta_{13} - 58317056 \beta_{12} + \cdots - 1817983320064 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 442269056 \beta_{15} - 852941888 \beta_{14} + 926268608 \beta_{13} - 1716553568 \beta_{12} + \cdots - 1796281991168 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 239721808 \beta_{15} - 2623958128 \beta_{14} - 2217005680 \beta_{13} - 3883746144 \beta_{12} + \cdots - 450651965882368 ) / 524288 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 44720185004 \beta_{15} - 74234795300 \beta_{14} + 73121670876 \beta_{13} - 139952385447 \beta_{12} + \cdots - 96\!\cdots\!64 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 514901073232 \beta_{15} - 1433963137776 \beta_{14} - 266861580528 \beta_{13} + \cdots - 70\!\cdots\!16 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 591799370404 \beta_{15} - 6155507868268 \beta_{14} - 1970625620460 \beta_{13} + \cdots - 38\!\cdots\!92 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 7804521807616 \beta_{15} + 109172825778560 \beta_{14} + 224165501621120 \beta_{13} + \cdots - 22\!\cdots\!60 ) / 2097152 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 16\!\cdots\!48 \beta_{15} + \cdots - 96\!\cdots\!96 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10\!\cdots\!64 \beta_{15} + \cdots - 95\!\cdots\!24 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 24\!\cdots\!44 \beta_{15} + \cdots - 52\!\cdots\!28 ) / 2097152 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
−5.93458 + 6.32105i
5.93298 2.42889i
1.49256 + 0.108386i
−3.53658 6.06559i
−14.5863 4.36773i
16.6303 4.63467i
3.43006 + 0.984106i
−3.42845 + 12.5625i
−3.42845 12.5625i
3.43006 0.984106i
16.6303 + 4.63467i
−14.5863 + 4.36773i
−3.53658 + 6.06559i
1.49256 0.108386i
5.93298 + 2.42889i
−5.93458 6.32105i
0 147.660i 0 −812.351 0 1920.17i 0 −15242.5 0
127.2 0 135.082i 0 93.0802 0 2522.45i 0 −11686.2 0
127.3 0 115.263i 0 1176.43 0 4245.70i 0 −6724.51 0
127.4 0 91.4359i 0 122.296 0 544.420i 0 −1799.53 0
127.5 0 63.7444i 0 −944.632 0 1875.82i 0 2497.66 0
127.6 0 47.6247i 0 −1.66205 0 4641.59i 0 4292.89 0
127.7 0 36.1265i 0 710.671 0 51.7339i 0 5255.88 0
127.8 0 25.5081i 0 −679.831 0 2310.00i 0 5910.34 0
127.9 0 25.5081i 0 −679.831 0 2310.00i 0 5910.34 0
127.10 0 36.1265i 0 710.671 0 51.7339i 0 5255.88 0
127.11 0 47.6247i 0 −1.66205 0 4641.59i 0 4292.89 0
127.12 0 63.7444i 0 −944.632 0 1875.82i 0 2497.66 0
127.13 0 91.4359i 0 122.296 0 544.420i 0 −1799.53 0
127.14 0 115.263i 0 1176.43 0 4245.70i 0 −6724.51 0
127.15 0 135.082i 0 93.0802 0 2522.45i 0 −11686.2 0
127.16 0 147.660i 0 −812.351 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.9.c.a 16
4.b odd 2 1 inner 128.9.c.a 16
8.b even 2 1 128.9.c.b yes 16
8.d odd 2 1 128.9.c.b yes 16
16.e even 4 1 256.9.d.i 16
16.e even 4 1 256.9.d.j 16
16.f odd 4 1 256.9.d.i 16
16.f odd 4 1 256.9.d.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.9.c.a 16 1.a even 1 1 trivial
128.9.c.a 16 4.b odd 2 1 inner
128.9.c.b yes 16 8.b even 2 1
128.9.c.b yes 16 8.d odd 2 1
256.9.d.i 16 16.e even 4 1
256.9.d.i 16 16.f odd 4 1
256.9.d.j 16 16.e even 4 1
256.9.d.j 16 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 336 T_{5}^{7} - 1907088 T_{5}^{6} - 752265024 T_{5}^{5} + 880188962912 T_{5}^{4} + \cdots + 82\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 50\!\cdots\!96)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 33\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 21\!\cdots\!72)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 37\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 28\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 61\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 94\!\cdots\!04 \) 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
$83$ \( T^{16} + \cdots + 11\!\cdots\!84 \) 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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