Properties

Label 772.1.u.a
Level $772$
Weight $1$
Character orbit 772.u
Analytic conductor $0.385$
Analytic rank $0$
Dimension $16$
Projective image $D_{48}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [772,1,Mod(59,772)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(772, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 37]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("772.59");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 772 = 2^{2} \cdot 193 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 772.u (of order \(48\), degree \(16\), 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: \(0.385278189753\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{48}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{48}^{19} q^{2} - \zeta_{48}^{14} q^{4} + ( - \zeta_{48}^{22} - \zeta_{48}^{13}) q^{5} - \zeta_{48}^{9} q^{8} + \zeta_{48}^{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{48}^{19} q^{2} - \zeta_{48}^{14} q^{4} + ( - \zeta_{48}^{22} - \zeta_{48}^{13}) q^{5} - \zeta_{48}^{9} q^{8} + \zeta_{48}^{12} q^{9} + ( - \zeta_{48}^{17} - \zeta_{48}^{8}) q^{10} + (\zeta_{48}^{20} + \zeta_{48}^{19}) q^{13} - \zeta_{48}^{4} q^{16} + (\zeta_{48}^{16} + \zeta_{48}^{13}) q^{17} + \zeta_{48}^{7} q^{18} + ( - \zeta_{48}^{12} - \zeta_{48}^{3}) q^{20} + ( - \zeta_{48}^{20} - \zeta_{48}^{11} - \zeta_{48}^{2}) q^{25} + (\zeta_{48}^{15} + \zeta_{48}^{14}) q^{26} + ( - \zeta_{48}^{5} + \zeta_{48}^{4}) q^{29} + \zeta_{48}^{23} q^{32} + (\zeta_{48}^{11} + \zeta_{48}^{8}) q^{34} + \zeta_{48}^{2} q^{36} + ( - \zeta_{48}^{21} - \zeta_{48}^{10}) q^{37} + (\zeta_{48}^{22} - \zeta_{48}^{7}) q^{40} + ( - \zeta_{48}^{23} + \zeta_{48}^{6}) q^{41} + (\zeta_{48}^{10} + \zeta_{48}) q^{45} - \zeta_{48}^{16} q^{49} + (\zeta_{48}^{21} - \zeta_{48}^{15} - \zeta_{48}^{6}) q^{50} + (\zeta_{48}^{10} + \zeta_{48}^{9}) q^{52} + ( - \zeta_{48}^{6} - \zeta_{48}) q^{53} + ( - \zeta_{48}^{23} - 1) q^{58} + (\zeta_{48}^{22} - \zeta_{48}^{15}) q^{61} + \zeta_{48}^{18} q^{64} + (\zeta_{48}^{18} + \zeta_{48}^{17} + \zeta_{48}^{9} + \zeta_{48}^{8}) q^{65} + (\zeta_{48}^{6} + \zeta_{48}^{3}) q^{68} - \zeta_{48}^{21} q^{72} + (\zeta_{48}^{23} - \zeta_{48}^{18}) q^{73} + ( - \zeta_{48}^{16} - \zeta_{48}^{5}) q^{74} + (\zeta_{48}^{17} - \zeta_{48}^{2}) q^{80} - q^{81} + ( - \zeta_{48}^{18} + \zeta_{48}) q^{82} + (\zeta_{48}^{14} + \zeta_{48}^{11} + \zeta_{48}^{5} + \zeta_{48}^{2}) q^{85} + (\zeta_{48}^{21} - \zeta_{48}^{12}) q^{89} + ( - \zeta_{48}^{20} + \zeta_{48}^{5}) q^{90} + (\zeta_{48}^{7} + \zeta_{48}^{3}) q^{97} - \zeta_{48}^{11} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{10} - 8 q^{17} + 8 q^{34} + 8 q^{49} - 16 q^{58} + 8 q^{65} + 8 q^{74} - 16 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/772\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(387\)
\(\chi(n)\) \(-\zeta_{48}^{11}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−0.991445 + 0.130526i
0.608761 + 0.793353i
−0.130526 0.991445i
0.130526 + 0.991445i
−0.608761 0.793353i
0.991445 0.130526i
0.130526 0.991445i
0.793353 + 0.608761i
0.991445 + 0.130526i
−0.608761 + 0.793353i
−0.793353 + 0.608761i
0.793353 0.608761i
0.608761 0.793353i
−0.991445 0.130526i
−0.793353 0.608761i
−0.130526 + 0.991445i
−0.793353 0.608761i 0 0.258819 + 0.965926i 0.835400 0.732626i 0 0 0.382683 0.923880i 1.00000i −1.10876 + 0.0726721i
131.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i −1.05217 0.357164i 0 0 0.382683 0.923880i 1.00000i 0.491445 0.996552i
147.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i 0.0255190 0.389345i 0 0 0.923880 + 0.382683i 1.00000i 0.293353 + 0.257264i
239.1 0.608761 0.793353i 0 −0.258819 0.965926i −1.95737 0.128293i 0 0 −0.923880 0.382683i 1.00000i −1.29335 + 1.47479i
255.1 0.130526 0.991445i 0 −0.965926 0.258819i 0.534534 1.57469i 0 0 −0.382683 + 0.923880i 1.00000i −1.49144 0.735499i
327.1 0.793353 + 0.608761i 0 0.258819 + 0.965926i 1.09645 + 1.25026i 0 0 −0.382683 + 0.923880i 1.00000i 0.108761 + 1.65938i
407.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i −1.95737 + 0.128293i 0 0 −0.923880 + 0.382683i 1.00000i −1.29335 1.47479i
531.1 −0.991445 + 0.130526i 0 0.965926 0.258819i 0.867580 1.75928i 0 0 −0.923880 + 0.382683i 1.00000i −0.630526 + 1.85747i
543.1 0.793353 0.608761i 0 0.258819 0.965926i 1.09645 1.25026i 0 0 −0.382683 0.923880i 1.00000i 0.108761 1.65938i
551.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i 0.534534 + 1.57469i 0 0 −0.382683 0.923880i 1.00000i −1.49144 + 0.735499i
575.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i −0.349942 + 0.172572i 0 0 0.923880 + 0.382683i 1.00000i −0.369474 + 0.125419i
583.1 −0.991445 0.130526i 0 0.965926 + 0.258819i 0.867580 + 1.75928i 0 0 −0.923880 0.382683i 1.00000i −0.630526 1.85747i
607.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i −1.05217 + 0.357164i 0 0 0.382683 + 0.923880i 1.00000i 0.491445 + 0.996552i
615.1 −0.793353 + 0.608761i 0 0.258819 0.965926i 0.835400 + 0.732626i 0 0 0.382683 + 0.923880i 1.00000i −1.10876 0.0726721i
627.1 0.991445 0.130526i 0 0.965926 0.258819i −0.349942 0.172572i 0 0 0.923880 0.382683i 1.00000i −0.369474 0.125419i
751.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i 0.0255190 + 0.389345i 0 0 0.923880 0.382683i 1.00000i 0.293353 0.257264i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
193.k even 48 1 inner
772.u odd 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 772.1.u.a 16
4.b odd 2 1 CM 772.1.u.a 16
193.k even 48 1 inner 772.1.u.a 16
772.u odd 48 1 inner 772.1.u.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
772.1.u.a 16 1.a even 1 1 trivial
772.1.u.a 16 4.b odd 2 1 CM
772.1.u.a 16 193.k even 48 1 inner
772.1.u.a 16 772.u odd 48 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(772, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 16 T^{13} - 2 T^{12} + 88 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{14} + 10 T^{12} - 16 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + 36 T^{14} + 112 T^{13} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 4 T^{14} + 10 T^{12} + 16 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 2 T^{12} - 32 T^{11} - 20 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} + 4 T^{12} + 16 T^{11} - 20 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 8 T^{13} + 4 T^{12} + 52 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 2 T^{12} - 32 T^{11} - 20 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 8 T^{13} + 4 T^{12} + 52 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{8} + 4 T^{6} + 6 T^{4} + 8 T^{3} + 4 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 24 T^{12} + 191 T^{8} - 24 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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