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Properties

Label 777.1.n.d
Level $777$
Weight $1$
Character orbit 777.n
Analytic conductor $0.388$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [777,1,Mod(524,777)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("777.524"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(777, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 777 = 3 \cdot 7 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 777.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.387773514816\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.7445991.2

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + q^{3} + i q^{4} + i q^{7} + q^{9} + i q^{12} + ( - i - 1) q^{13} - q^{16} + ( - i - 1) q^{19} + i q^{21} + i q^{25} + q^{27} - q^{28} + ( - i + 1) q^{31} + i q^{36} + q^{37} + ( - i - 1) q^{39}+ \cdots + (i + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9} - 2 q^{13} - 2 q^{16} - 2 q^{19} + 2 q^{27} - 2 q^{28} + 2 q^{31} + 2 q^{37} - 2 q^{39} + 2 q^{43} - 2 q^{48} - 2 q^{49} + 2 q^{52} - 2 q^{57} - 2 q^{61} + 2 q^{76} - 2 q^{79} + 2 q^{81}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(260\) \(556\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(i\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
− 1.00000i
1.00000i
0 1.00000 − 1.00000i 0 0 − 1.00000i 0 1.00000 0
734.1 0 1.00000 1.00000i 0 0 1.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.j even 4 1 inner
777.n odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 777.1.n.d yes 2
3.b odd 2 1 CM 777.1.n.d yes 2
7.b odd 2 1 777.1.n.a ✓ 2
21.c even 2 1 777.1.n.a ✓ 2
37.d odd 4 1 777.1.n.a ✓ 2
111.g even 4 1 777.1.n.a ✓ 2
259.j even 4 1 inner 777.1.n.d yes 2
777.n odd 4 1 inner 777.1.n.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
777.1.n.a ✓ 2 7.b odd 2 1
777.1.n.a ✓ 2 21.c even 2 1
777.1.n.a ✓ 2 37.d odd 4 1
777.1.n.a ✓ 2 111.g even 4 1
777.1.n.d yes 2 1.a even 1 1 trivial
777.1.n.d yes 2 3.b odd 2 1 CM
777.1.n.d yes 2 259.j even 4 1 inner
777.1.n.d yes 2 777.n odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(777, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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