Properties

Label 777.1.dp.a
Level $777$
Weight $1$
Character orbit 777.dp
Analytic conductor $0.388$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [777,1,Mod(5,777)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(777, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 30, 23]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("777.5");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 777 = 3 \cdot 7 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 777.dp (of order \(36\), degree \(12\), 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: \(0.387773514816\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{36}^{14} q^{3} - \zeta_{36}^{17} q^{4} - \zeta_{36} q^{7} - \zeta_{36}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36}^{14} q^{3} - \zeta_{36}^{17} q^{4} - \zeta_{36} q^{7} - \zeta_{36}^{10} q^{9} - \zeta_{36}^{13} q^{12} + (\zeta_{36}^{10} - \zeta_{36}^{3}) q^{13} - \zeta_{36}^{16} q^{16} + ( - \zeta_{36}^{8} + \zeta_{36}^{5}) q^{19} + \zeta_{36}^{15} q^{21} + \zeta_{36}^{7} q^{25} - \zeta_{36}^{6} q^{27} - q^{28} + (\zeta_{36}^{13} + \zeta_{36}^{8}) q^{31} - \zeta_{36}^{9} q^{36} + \zeta_{36}^{12} q^{37} + (\zeta_{36}^{17} + \zeta_{36}^{6}) q^{39} + (\zeta_{36}^{16} - \zeta_{36}^{11}) q^{43} - \zeta_{36}^{12} q^{48} + \zeta_{36}^{2} q^{49} + (\zeta_{36}^{9} - \zeta_{36}^{2}) q^{52} + ( - \zeta_{36}^{4} + \zeta_{36}) q^{57} + (\zeta_{36}^{14} - \zeta_{36}^{5}) q^{61} + \zeta_{36}^{11} q^{63} - \zeta_{36}^{15} q^{64} + ( - \zeta_{36}^{15} - \zeta_{36}^{7}) q^{67} + (\zeta_{36}^{11} + \zeta_{36}) q^{73} + \zeta_{36}^{3} q^{75} + ( - \zeta_{36}^{7} + \zeta_{36}^{4}) q^{76} + ( - \zeta_{36}^{4} + \zeta_{36}^{3}) q^{79} - \zeta_{36}^{2} q^{81} + \zeta_{36}^{14} q^{84} + ( - \zeta_{36}^{11} + \zeta_{36}^{4}) q^{91} + (\zeta_{36}^{9} + \zeta_{36}^{4}) q^{93} + (\zeta_{36}^{16} - \zeta_{36}^{5}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{27} - 12 q^{28} - 6 q^{37} + 6 q^{39} + 6 q^{48}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/777\mathbb{Z}\right)^\times\).

\(n\) \(260\) \(556\) \(631\)
\(\chi(n)\) \(-1\) \(-\zeta_{36}^{12}\) \(\zeta_{36}^{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.342020 0.939693i
−0.984808 + 0.173648i
0.342020 0.939693i
−0.642788 + 0.766044i
−0.342020 + 0.939693i
−0.984808 0.173648i
0.984808 0.173648i
0.984808 + 0.173648i
0.642788 0.766044i
0.642788 + 0.766044i
−0.642788 0.766044i
0.342020 + 0.939693i
0 0.173648 + 0.984808i −0.342020 + 0.939693i 0 0 0.342020 + 0.939693i 0 −0.939693 + 0.342020i 0
17.1 0 0.766044 + 0.642788i −0.984808 0.173648i 0 0 0.984808 0.173648i 0 0.173648 + 0.984808i 0
59.1 0 0.173648 0.984808i 0.342020 + 0.939693i 0 0 −0.342020 + 0.939693i 0 −0.939693 0.342020i 0
278.1 0 −0.939693 0.342020i −0.642788 0.766044i 0 0 0.642788 0.766044i 0 0.766044 + 0.642788i 0
311.1 0 0.173648 0.984808i −0.342020 0.939693i 0 0 0.342020 0.939693i 0 −0.939693 0.342020i 0
320.1 0 0.766044 0.642788i −0.984808 + 0.173648i 0 0 0.984808 + 0.173648i 0 0.173648 0.984808i 0
353.1 0 0.766044 + 0.642788i 0.984808 + 0.173648i 0 0 −0.984808 + 0.173648i 0 0.173648 + 0.984808i 0
383.1 0 0.766044 0.642788i 0.984808 0.173648i 0 0 −0.984808 0.173648i 0 0.173648 0.984808i 0
425.1 0 −0.939693 0.342020i 0.642788 + 0.766044i 0 0 −0.642788 + 0.766044i 0 0.766044 + 0.642788i 0
479.1 0 −0.939693 + 0.342020i 0.642788 0.766044i 0 0 −0.642788 0.766044i 0 0.766044 0.642788i 0
668.1 0 −0.939693 + 0.342020i −0.642788 + 0.766044i 0 0 0.642788 + 0.766044i 0 0.766044 0.642788i 0
698.1 0 0.173648 + 0.984808i 0.342020 0.939693i 0 0 −0.342020 0.939693i 0 −0.939693 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
259.bt even 36 1 inner
777.dp odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 777.1.dp.a yes 12
3.b odd 2 1 CM 777.1.dp.a yes 12
7.d odd 6 1 777.1.dg.a 12
21.g even 6 1 777.1.dg.a 12
37.i odd 36 1 777.1.dg.a 12
111.q even 36 1 777.1.dg.a 12
259.bt even 36 1 inner 777.1.dp.a yes 12
777.dp odd 36 1 inner 777.1.dp.a yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
777.1.dg.a 12 7.d odd 6 1
777.1.dg.a 12 21.g even 6 1
777.1.dg.a 12 37.i odd 36 1
777.1.dg.a 12 111.q even 36 1
777.1.dp.a yes 12 1.a even 1 1 trivial
777.1.dp.a yes 12 3.b odd 2 1 CM
777.1.dp.a yes 12 259.bt even 36 1 inner
777.1.dp.a yes 12 777.dp odd 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(777, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{10} - 2 T^{9} + 6 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{9} + 53 T^{6} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 2 T^{9} - 9 T^{8} + 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{9} + 18 T^{8} + 6 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + 4 T^{9} + 8 T^{6} + 32 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{10} - 6 T^{8} + 8 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 6 T^{10} + 27 T^{8} + 48 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$79$ \( T^{12} - 3 T^{10} - 2 T^{9} + 6 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + 2 T^{9} - 9 T^{8} - 12 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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