Properties

Label 128.3.f.a
Level $128$
Weight $3$
Character orbit 128.f
Analytic conductor $3.488$
Analytic rank $0$
Dimension $6$
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] = [128,3,Mod(31,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), 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: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{4} + \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{4} + \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{9} + ( - \beta_{5} - 2 \beta_{3} - 4 \beta_1 - 4) q^{11} + (3 \beta_{5} - \beta_{3} + \beta_1 + 1) q^{13} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 8 \beta_1) q^{15} + (3 \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{17} + ( - \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 4) q^{19} + ( - 4 \beta_{4} - 2 \beta_1 + 2) q^{21} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 8) q^{23} + ( - 4 \beta_{5} + 4 \beta_{4} - \beta_1) q^{25} + (2 \beta_{5} + 2 \beta_{3} + 12 \beta_1 + 12) q^{27} + ( - 7 \beta_{5} + \beta_{3} + \beta_1 + 1) q^{29} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 24 \beta_1) q^{31} + ( - 7 \beta_{5} - 7 \beta_{4} - \beta_{3} - \beta_{2} - 6) q^{33} + ( - 2 \beta_{4} + 4 \beta_{2} + 16 \beta_1 - 16) q^{35} + (7 \beta_{4} - 5 \beta_{2} + 7 \beta_1 - 7) q^{37} + ( - 4 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 32) q^{39} + (6 \beta_{5} - 6 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 8 \beta_1) q^{41} + (\beta_{5} + 8 \beta_{3} - 16 \beta_1 - 16) q^{43} + (\beta_{5} + 5 \beta_{3} - 9 \beta_1 - 9) q^{45} + (10 \beta_{5} - 10 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 24 \beta_1) q^{47} + ( - 2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} - 13) q^{49} + (6 \beta_{2} - 28 \beta_1 + 28) q^{51} + ( - 5 \beta_{4} - \beta_{2} + 15 \beta_1 - 15) q^{53} + ( - 8 \beta_{5} - 8 \beta_{4} - \beta_{3} - \beta_{2} - 48) q^{55} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 6 \beta_1) q^{57} + ( - 3 \beta_{5} - 4 \beta_{3} + 32 \beta_1 + 32) q^{59} + ( - \beta_{5} - 5 \beta_{3} - 7 \beta_1 - 7) q^{61} + ( - 6 \beta_{5} + 6 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 40 \beta_1) q^{63} + (6 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 10) q^{65} + ( - 5 \beta_{4} - 14 \beta_{2} + 44 \beta_1 - 44) q^{67} + ( - 8 \beta_{4} + 4 \beta_{2} - 18 \beta_1 + 18) q^{69} + (18 \beta_{5} + 18 \beta_{4} + \beta_{3} + \beta_{2} + 56) q^{71} + ( - 13 \beta_{5} + 13 \beta_{4} - 11 \beta_{3} + 11 \beta_{2} - 24 \beta_1) q^{73} + (7 \beta_{5} - 8 \beta_{3} - 40 \beta_1 - 40) q^{75} + ( - 4 \beta_{3} + 34 \beta_1 + 34) q^{77} + ( - 20 \beta_{5} + 20 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 64 \beta_1) q^{79} + (5 \beta_{5} + 5 \beta_{4} + 11 \beta_{3} + 11 \beta_{2} + 25) q^{81} + (9 \beta_{4} - 56 \beta_1 + 56) q^{83} + (16 \beta_{4} + 4 \beta_{2} - 42 \beta_1 + 42) q^{85} + (10 \beta_{5} + 10 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} - 72) q^{87} + (13 \beta_{5} - 13 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - 8 \beta_1) q^{89} + ( - 14 \beta_{5} - 8 \beta_{3} + 24 \beta_1 + 24) q^{91} + (28 \beta_{5} + 4 \beta_{3} + 16 \beta_1 + 16) q^{93} + ( - 7 \beta_{3} + 7 \beta_{2} + 32 \beta_1) q^{95} + (15 \beta_{5} + 15 \beta_{4} - 11 \beta_{3} - 11 \beta_{2} + 2) q^{97} + (13 \beta_{4} + 4 \beta_{2} + 32 \beta_1 - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 2 q^{5} + 4 q^{7} - 18 q^{11} + 2 q^{13} - 4 q^{17} + 30 q^{19} + 20 q^{21} - 60 q^{23} + 64 q^{27} + 18 q^{29} - 4 q^{33} - 100 q^{35} - 46 q^{37} + 196 q^{39} - 114 q^{43} - 66 q^{45} - 46 q^{49} + 156 q^{51} - 78 q^{53} - 252 q^{55} + 206 q^{59} - 30 q^{61} + 12 q^{65} - 226 q^{67} + 116 q^{69} + 260 q^{71} - 238 q^{75} + 212 q^{77} + 86 q^{81} + 318 q^{83} + 212 q^{85} - 444 q^{87} + 188 q^{91} + 32 q^{93} - 4 q^{97} - 226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 4\nu^{4} - 5\nu^{3} + 8\nu^{2} - 14\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 4\nu^{4} - 9\nu^{3} + 8\nu^{2} + 2\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - 4\nu^{4} + 9\nu^{3} - 8\nu^{2} + 14\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{5} + 4\nu^{4} - 7\nu^{3} + 16\nu^{2} - 10\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} + \beta_{4} - \beta_{3} - 4\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + \beta_{2} - 10\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{5} + 3\beta_{4} + \beta_{3} + 4\beta_{2} + 4\beta _1 + 10 ) / 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.264658 1.38923i
−0.671462 + 1.24464i
1.40680 + 0.144584i
0.264658 + 1.38923i
−0.671462 1.24464i
1.40680 0.144584i
0 −3.24914 3.24914i 0 0.0586332 + 0.0586332i 0 −4.61555 0 12.1138i 0
31.2 0 0.146365 + 0.146365i 0 −3.68585 3.68585i 0 9.66442 0 8.95715i 0
31.3 0 2.10278 + 2.10278i 0 4.62721 + 4.62721i 0 −3.04888 0 0.156674i 0
95.1 0 −3.24914 + 3.24914i 0 0.0586332 0.0586332i 0 −4.61555 0 12.1138i 0
95.2 0 0.146365 0.146365i 0 −3.68585 + 3.68585i 0 9.66442 0 8.95715i 0
95.3 0 2.10278 2.10278i 0 4.62721 4.62721i 0 −3.04888 0 0.156674i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.3.f.a 6
3.b odd 2 1 1152.3.m.b 6
4.b odd 2 1 128.3.f.b 6
8.b even 2 1 64.3.f.a 6
8.d odd 2 1 16.3.f.a 6
12.b even 2 1 1152.3.m.a 6
16.e even 4 1 16.3.f.a 6
16.e even 4 1 128.3.f.b 6
16.f odd 4 1 64.3.f.a 6
16.f odd 4 1 inner 128.3.f.a 6
24.f even 2 1 144.3.m.a 6
24.h odd 2 1 576.3.m.a 6
32.g even 8 2 1024.3.c.j 12
32.g even 8 2 1024.3.d.k 12
32.h odd 8 2 1024.3.c.j 12
32.h odd 8 2 1024.3.d.k 12
40.e odd 2 1 400.3.r.c 6
40.k even 4 1 400.3.k.c 6
40.k even 4 1 400.3.k.d 6
48.i odd 4 1 144.3.m.a 6
48.i odd 4 1 1152.3.m.a 6
48.k even 4 1 576.3.m.a 6
48.k even 4 1 1152.3.m.b 6
80.i odd 4 1 400.3.k.d 6
80.q even 4 1 400.3.r.c 6
80.t odd 4 1 400.3.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a 6 8.d odd 2 1
16.3.f.a 6 16.e even 4 1
64.3.f.a 6 8.b even 2 1
64.3.f.a 6 16.f odd 4 1
128.3.f.a 6 1.a even 1 1 trivial
128.3.f.a 6 16.f odd 4 1 inner
128.3.f.b 6 4.b odd 2 1
128.3.f.b 6 16.e even 4 1
144.3.m.a 6 24.f even 2 1
144.3.m.a 6 48.i odd 4 1
400.3.k.c 6 40.k even 4 1
400.3.k.c 6 80.t odd 4 1
400.3.k.d 6 40.k even 4 1
400.3.k.d 6 80.i odd 4 1
400.3.r.c 6 40.e odd 2 1
400.3.r.c 6 80.q even 4 1
576.3.m.a 6 24.h odd 2 1
576.3.m.a 6 48.k even 4 1
1024.3.c.j 12 32.g even 8 2
1024.3.c.j 12 32.h odd 8 2
1024.3.d.k 12 32.g even 8 2
1024.3.d.k 12 32.h odd 8 2
1152.3.m.a 6 12.b even 2 1
1152.3.m.a 6 48.i odd 4 1
1152.3.m.b 6 3.b odd 2 1
1152.3.m.b 6 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 2T_{3}^{5} + 2T_{3}^{4} - 32T_{3}^{3} + 196T_{3}^{2} - 56T_{3} + 8 \) acting on \(S_{3}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 2 T^{4} + 64 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 60 T - 136)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 587528 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 1286408 \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 260 T - 1544)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 30 T^{5} + 450 T^{4} + \cdots + 13448 \) Copy content Toggle raw display
$23$ \( (T^{3} + 30 T^{2} + 164 T - 968)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 19046792 \) Copy content Toggle raw display
$31$ \( T^{6} + 1920 T^{4} + \cdots + 16777216 \) Copy content Toggle raw display
$37$ \( T^{6} + 46 T^{5} + 1058 T^{4} + \cdots + 42632 \) Copy content Toggle raw display
$41$ \( T^{6} + 4992 T^{4} + \cdots + 67108864 \) Copy content Toggle raw display
$43$ \( T^{6} + 114 T^{5} + 6498 T^{4} + \cdots + 42632 \) Copy content Toggle raw display
$47$ \( T^{6} + 8576 T^{4} + \cdots + 6056574976 \) Copy content Toggle raw display
$53$ \( T^{6} + 78 T^{5} + 3042 T^{4} + \cdots + 783752 \) Copy content Toggle raw display
$59$ \( T^{6} - 206 T^{5} + \cdots + 8410007432 \) Copy content Toggle raw display
$61$ \( T^{6} + 30 T^{5} + \cdots + 151449608 \) Copy content Toggle raw display
$67$ \( T^{6} + 226 T^{5} + \cdots + 87233303432 \) Copy content Toggle raw display
$71$ \( (T^{3} - 130 T^{2} - 3548 T + 391864)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 18848 T^{4} + \cdots + 7310934016 \) Copy content Toggle raw display
$79$ \( T^{6} + 37376 T^{4} + \cdots + 1550483193856 \) Copy content Toggle raw display
$83$ \( T^{6} - 318 T^{5} + \cdots + 105636303368 \) Copy content Toggle raw display
$89$ \( T^{6} + 16288 T^{4} + \cdots + 25681985536 \) Copy content Toggle raw display
$97$ \( (T^{3} + 2 T^{2} - 17540 T + 519928)^{2} \) Copy content Toggle raw display
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