↑

Properties

Label 1792.3.g.f
Level $1792$
Weight $3$
Character orbit 1792.g
Analytic conductor $48.828$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,3,Mod(127,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.g (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: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1997017344.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \) 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 224)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{6} + \beta_{5}) q^{5} - \beta_{5} q^{7} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{6} + \beta_{5}) q^{5} - \beta_{5} q^{7} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{9} + (\beta_{7} - \beta_{2} + 4) q^{11} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4}) q^{13} + ( - 2 \beta_{6} + 6 \beta_{5} + \cdots + \beta_{3}) q^{15}+ \cdots + ( - 5 \beta_{7} - 3 \beta_{2} + \cdots + 68) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 24\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 11\nu^{4} - 16\nu^{2} + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 42\nu^{3} + 48\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 42\nu^{3} + 32\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 51\nu^{3} + 42\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 16\nu^{5} + 77\nu^{3} + 110\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{6} - 41\nu^{4} - 136\nu^{2} - 68 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - 3\beta_{5} + 5\beta_{4} - 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} - 8\beta_{2} - 11\beta _1 + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{6} + 35\beta_{5} - 48\beta_{4} + 22\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11\beta_{7} + 64\beta_{2} + 105\beta _1 - 343 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 75\beta_{6} - 329\beta_{5} + 438\beta_{4} - 176\beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
− 1.92812i
1.92812i
0.277334i
− 0.277334i
1.27733i
− 1.27733i
− 2.92812i
2.92812i
0 −3.85623 0 − 0.490168i 0 − 2.64575i 0 5.87054 0
127.2 0 −3.85623 0 0.490168i 0 2.64575i 0 5.87054 0
127.3 0 −0.554669 0 − 4.57685i 0 − 2.64575i 0 −8.69234 0
127.4 0 −0.554669 0 4.57685i 0 2.64575i 0 −8.69234 0
127.5 0 2.55467 0 − 9.86836i 0 2.64575i 0 −2.47367 0
127.6 0 2.55467 0 9.86836i 0 − 2.64575i 0 −2.47367 0
127.7 0 5.85623 0 − 5.78167i 0 2.64575i 0 25.2955 0
127.8 0 5.85623 0 5.78167i 0 − 2.64575i 0 25.2955 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
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 1792.3.g.f 8
4.b odd 2 1 1792.3.g.d 8
8.b even 2 1 1792.3.g.d 8
8.d odd 2 1 inner 1792.3.g.f 8
16.e even 4 1 224.3.d.b ✓ 8
16.e even 4 1 448.3.d.e 8
16.f odd 4 1 224.3.d.b ✓ 8
16.f odd 4 1 448.3.d.e 8
48.i odd 4 1 2016.3.m.c 8
48.k even 4 1 2016.3.m.c 8
112.j even 4 1 1568.3.d.n 8
112.l odd 4 1 1568.3.d.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.d.b ✓ 8 16.e even 4 1
224.3.d.b ✓ 8 16.f odd 4 1
448.3.d.e 8 16.e even 4 1
448.3.d.e 8 16.f odd 4 1
1568.3.d.n 8 112.j even 4 1
1568.3.d.n 8 112.l odd 4 1
1792.3.g.d 8 4.b odd 2 1
1792.3.g.d 8 8.b even 2 1
1792.3.g.f 8 1.a even 1 1 trivial
1792.3.g.f 8 8.d odd 2 1 inner
2016.3.m.c 8 48.i odd 4 1
2016.3.m.c 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 20T_{3}^{2} + 48T_{3} + 32 \) acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{3} - 20 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 152 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{3} + \cdots - 9728)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 216 T^{6} + \cdots + 2985984 \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{3} + \cdots - 4528)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 44 T^{3} + \cdots - 9056)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 68719476736 \) Copy content Toggle raw display
$29$ \( T^{8} + 1008 T^{6} + \cdots + 186624 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 90943258624 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 34215386769664 \) Copy content Toggle raw display
$41$ \( (T^{4} + 72 T^{3} + \cdots - 139824)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 112 T^{3} + \cdots - 4898304)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40757090320384 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 157564539136 \) Copy content Toggle raw display
$59$ \( (T^{4} - 116 T^{3} + \cdots - 206048)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 9215935922176 \) Copy content Toggle raw display
$67$ \( (T^{4} + 184 T^{3} + \cdots - 4265728)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 28179280429056 \) Copy content Toggle raw display
$73$ \( (T^{4} + 136 T^{3} + \cdots - 3527408)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13710609350656 \) Copy content Toggle raw display
$83$ \( (T^{4} + 212 T^{3} + \cdots - 4523616)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 40 T^{3} + \cdots + 837776)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 264 T^{3} + \cdots + 475344)^{2} \) Copy content Toggle raw display
show more
show less