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Properties

Label 128.5.c.b
Level $128$
Weight $5$
Character orbit 128.c
Analytic conductor $13.231$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [128,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.205520896.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{39} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 6) q^{5} + (\beta_{4} + \beta_1) q^{7} + ( - \beta_{3} - \beta_{2} - 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} + 6) q^{5} + (\beta_{4} + \beta_1) q^{7} + ( - \beta_{3} - \beta_{2} - 27) q^{9} + (\beta_{6} + \beta_{4} - 2 \beta_1) q^{11} + (\beta_{7} + \beta_{3} + 30) q^{13} + ( - 2 \beta_{6} - \beta_{5} + 11 \beta_1) q^{15} + (2 \beta_{7} + \beta_{3} - 3 \beta_{2} + 30) q^{17} + (3 \beta_{6} + 2 \beta_{5} + \cdots - 10 \beta_1) q^{19}+ \cdots + (28 \beta_{6} - 10 \beta_{5} + \cdots - 201 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 48 q^{5} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 48 q^{5} - 216 q^{9} + 240 q^{13} + 240 q^{17} - 1216 q^{21} + 664 q^{25} + 432 q^{29} + 992 q^{33} + 2800 q^{37} - 2928 q^{41} - 4880 q^{45} - 5752 q^{49} + 1776 q^{53} + 8608 q^{57} + 12656 q^{61} + 672 q^{65} - 4416 q^{69} + 560 q^{73} - 31296 q^{77} + 14696 q^{81} - 14432 q^{85} - 22992 q^{89} + 56320 q^{93} - 3728 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 152\nu^{7} - 638\nu^{6} + 1996\nu^{5} - 2362\nu^{4} - 268\nu^{3} + 1570\nu^{2} + 1528\nu + 344 ) / 51 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 144\nu^{7} - 676\nu^{6} + 2120\nu^{5} - 2864\nu^{4} - 168\nu^{3} + 2740\nu^{2} + 832\nu - 1120 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -176\nu^{7} + 524\nu^{6} - 1624\nu^{5} + 720\nu^{4} + 1656\nu^{3} + 580\nu^{2} + 192\nu - 2016 ) / 51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -824\nu^{7} + 3226\nu^{6} - 9396\nu^{5} + 8406\nu^{4} + 9620\nu^{3} - 10662\nu^{2} - 12120\nu - 2824 ) / 51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1288\nu^{7} - 3742\nu^{6} + 8732\nu^{5} + 6430\nu^{4} - 42620\nu^{3} + 24674\nu^{2} + 32360\nu + 7768 ) / 51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1488\nu^{7} - 7008\nu^{6} + 22768\nu^{5} - 33720\nu^{4} + 11184\nu^{3} + 11200\nu^{2} + 8144\nu + 1664 ) / 51 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4672\nu^{7} + 20784\nu^{6} - 64672\nu^{5} + 81920\nu^{4} + 10528\nu^{3} - 73968\nu^{2} - 22400\nu - 896 ) / 51 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{7} - 8\beta_{6} - 5\beta_{5} - 13\beta_{4} + 6\beta_{3} + 26\beta_{2} + 94\beta _1 + 512 ) / 1024 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 6\beta_{6} - 3\beta_{5} - 5\beta_{4} + 5\beta_{3} - 21\beta_{2} + 52\beta _1 - 256 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 7\beta_{6} + 6\beta_{5} + 19\beta_{4} + 13\beta_{3} - 85\beta_{2} - 13\beta _1 - 1408 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 60\beta_{6} + 43\beta_{5} + 111\beta_{4} - 350\beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27\beta_{7} + 67\beta_{6} + 49\beta_{5} + 130\beta_{4} - 101\beta_{3} + 749\beta_{2} - 363\beta _1 + 12928 ) / 256 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 61\beta_{7} - 390\beta_{6} - 254\beta_{5} - 592\beta_{4} - 233\beta_{3} + 1689\beta_{2} + 2766\beta _1 + 28928 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 394 \beta_{7} - 5960 \beta_{6} - 3949 \beta_{5} - 9397 \beta_{4} + 1218 \beta_{3} - 10978 \beta_{2} + \cdots - 200192 ) / 1024 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.55050 − 0.642239i
1.12964 − 2.72719i
−0.550501 − 0.228025i
−0.129640 − 0.312979i
−0.129640 + 0.312979i
−0.550501 + 0.228025i
1.12964 + 2.72719i
1.55050 + 0.642239i
0 − 16.7135i 0 10.1891 0 − 80.4325i 0 −198.341 0
127.2 0 − 7.95199i 0 46.0525 0 57.5735i 0 17.7658 0
127.3 0 − 7.05664i 0 −9.50277 0 − 6.49140i 0 31.2038 0
127.4 0 − 6.29514i 0 −22.7388 0 51.5147i 0 41.3713 0
127.5 0 6.29514i 0 −22.7388 0 − 51.5147i 0 41.3713 0
127.6 0 7.05664i 0 −9.50277 0 6.49140i 0 31.2038 0
127.7 0 7.95199i 0 46.0525 0 − 57.5735i 0 17.7658 0
127.8 0 16.7135i 0 10.1891 0 80.4325i 0 −198.341 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
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 128.5.c.b yes 8
3.b odd 2 1 1152.5.g.a 8
4.b odd 2 1 inner 128.5.c.b yes 8
8.b even 2 1 128.5.c.a ✓ 8
8.d odd 2 1 128.5.c.a ✓ 8
12.b even 2 1 1152.5.g.a 8
16.e even 4 1 256.5.d.g 8
16.e even 4 1 256.5.d.h 8
16.f odd 4 1 256.5.d.g 8
16.f odd 4 1 256.5.d.h 8
24.f even 2 1 1152.5.g.b 8
24.h odd 2 1 1152.5.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.c.a ✓ 8 8.b even 2 1
128.5.c.a ✓ 8 8.d odd 2 1
128.5.c.b yes 8 1.a even 1 1 trivial
128.5.c.b yes 8 4.b odd 2 1 inner
256.5.d.g 8 16.e even 4 1
256.5.d.g 8 16.f odd 4 1
256.5.d.h 8 16.e even 4 1
256.5.d.h 8 16.f odd 4 1
1152.5.g.a 8 3.b odd 2 1
1152.5.g.a 8 12.b even 2 1
1152.5.g.b 8 24.f even 2 1
1152.5.g.b 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 24T_{5}^{3} - 1128T_{5}^{2} + 2976T_{5} + 101392 \) acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 432 T^{6} + \cdots + 34857216 \) Copy content Toggle raw display
$5$ \( (T^{4} - 24 T^{3} + \cdots + 101392)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 2397988519936 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 657984882278656 \) Copy content Toggle raw display
$13$ \( (T^{4} - 120 T^{3} + \cdots - 8447216)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 120 T^{3} + \cdots + 1484236816)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{4} - 216 T^{3} + \cdots + 191421236752)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 80\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{4} - 1400 T^{3} + \cdots - 564755606768)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 3022842029552)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 3952010710256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 61339635871472)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 412684399487248)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 750969871702288)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 103658484613616)^{2} \) Copy content Toggle raw display
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