Properties

Label 799.1.h.b
Level $799$
Weight $1$
Character orbit 799.h
Analytic conductor $0.399$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -47
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [799,1,Mod(93,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.93");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.h (of order \(8\), degree \(4\), 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: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{40}^{19} + \zeta_{40}^{11}) q^{2} + (\zeta_{40}^{8} - \zeta_{40}^{7}) q^{3} + ( - \zeta_{40}^{18} + \cdots - \zeta_{40}^{2}) q^{4}+ \cdots + (\zeta_{40}^{16} + \cdots + \zeta_{40}^{14}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{40}^{19} + \zeta_{40}^{11}) q^{2} + (\zeta_{40}^{8} - \zeta_{40}^{7}) q^{3} + ( - \zeta_{40}^{18} + \cdots - \zeta_{40}^{2}) q^{4}+ \cdots + ( - \zeta_{40}^{19} + \cdots - \zeta_{40}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71} - 4 q^{79} - 16 q^{83} + 32 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{40}^{15}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−0.453990 0.891007i
−0.987688 + 0.156434i
−0.156434 + 0.987688i
0.891007 + 0.453990i
−0.891007 0.453990i
0.156434 0.987688i
0.987688 0.156434i
0.453990 + 0.891007i
−0.891007 + 0.453990i
0.156434 + 0.987688i
0.987688 + 0.156434i
0.453990 0.891007i
−0.453990 + 0.891007i
−0.987688 0.156434i
−0.156434 0.987688i
0.891007 0.453990i
−1.34500 + 1.34500i −0.652583 + 1.57547i 2.61803i 0 −1.24129 2.99673i 1.84206 0.763007i 2.17625 + 2.17625i −1.34915 1.34915i 0
93.2 −0.831254 + 0.831254i 0.763007 1.84206i 0.381966i 0 0.896969 + 2.16547i 0.431351 0.178671i −0.513743 0.513743i −2.10391 2.10391i 0
93.3 0.831254 0.831254i −0.581990 + 1.40505i 0.381966i 0 0.684170 + 1.65173i −1.57547 + 0.652583i 0.513743 + 0.513743i −0.928339 0.928339i 0
93.4 1.34500 1.34500i 0.178671 0.431351i 2.61803i 0 −0.339853 0.820478i −1.40505 + 0.581990i −2.17625 2.17625i 0.552967 + 0.552967i 0
281.1 −1.34500 + 1.34500i −1.79671 0.744220i 2.61803i 0 3.41754 1.41559i −0.497066 1.20002i 2.17625 + 2.17625i 1.96718 + 1.96718i 0
281.2 −0.831254 + 0.831254i 1.20002 + 0.497066i 0.381966i 0 −1.41071 + 0.584336i 0.399903 + 0.965451i −0.513743 0.513743i 0.485875 + 0.485875i 0
281.3 0.831254 0.831254i −0.144974 0.0600500i 0.381966i 0 −0.170427 + 0.0705930i 0.744220 + 1.79671i 0.513743 + 0.513743i −0.689695 0.689695i 0
281.4 1.34500 1.34500i −0.965451 0.399903i 2.61803i 0 −1.83640 + 0.760661i 0.0600500 + 0.144974i −2.17625 2.17625i 0.0650673 + 0.0650673i 0
563.1 −1.34500 1.34500i −1.79671 + 0.744220i 2.61803i 0 3.41754 + 1.41559i −0.497066 + 1.20002i 2.17625 2.17625i 1.96718 1.96718i 0
563.2 −0.831254 0.831254i 1.20002 0.497066i 0.381966i 0 −1.41071 0.584336i 0.399903 0.965451i −0.513743 + 0.513743i 0.485875 0.485875i 0
563.3 0.831254 + 0.831254i −0.144974 + 0.0600500i 0.381966i 0 −0.170427 0.0705930i 0.744220 1.79671i 0.513743 0.513743i −0.689695 + 0.689695i 0
563.4 1.34500 + 1.34500i −0.965451 + 0.399903i 2.61803i 0 −1.83640 0.760661i 0.0600500 0.144974i −2.17625 + 2.17625i 0.0650673 0.0650673i 0
610.1 −1.34500 1.34500i −0.652583 1.57547i 2.61803i 0 −1.24129 + 2.99673i 1.84206 + 0.763007i 2.17625 2.17625i −1.34915 + 1.34915i 0
610.2 −0.831254 0.831254i 0.763007 + 1.84206i 0.381966i 0 0.896969 2.16547i 0.431351 + 0.178671i −0.513743 + 0.513743i −2.10391 + 2.10391i 0
610.3 0.831254 + 0.831254i −0.581990 1.40505i 0.381966i 0 0.684170 1.65173i −1.57547 0.652583i 0.513743 0.513743i −0.928339 + 0.928339i 0
610.4 1.34500 + 1.34500i 0.178671 + 0.431351i 2.61803i 0 −0.339853 + 0.820478i −1.40505 0.581990i −2.17625 + 2.17625i 0.552967 0.552967i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
17.d even 8 1 inner
799.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.1.h.b 16
17.d even 8 1 inner 799.1.h.b 16
47.b odd 2 1 CM 799.1.h.b 16
799.h odd 8 1 inner 799.1.h.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.1.h.b 16 1.a even 1 1 trivial
799.1.h.b 16 17.d even 8 1 inner
799.1.h.b 16 47.b odd 2 1 CM
799.1.h.b 16 799.h odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 15T_{2}^{4} + 25 \) acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$53$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 7 T^{4} + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} - 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T + 2)^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5 T^{2} + 5)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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