Properties

Label 1716.1.m.g
Level $1716$
Weight $1$
Character orbit 1716.m
Analytic conductor $0.856$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1716,1,Mod(1715,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.1715");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1716.m (of order \(2\), degree \(1\), 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: \(0.856395561678\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.981552.4

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{3} + \zeta_{8}^{2} q^{4} + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + \zeta_{8}^{3} q^{6} + \zeta_{8}^{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{3} + \zeta_{8}^{2} q^{4} + (\zeta_{8}^{3} - \zeta_{8}) q^{5} + \zeta_{8}^{3} q^{6} + \zeta_{8}^{3} q^{8} - q^{9} + ( - \zeta_{8}^{2} - 1) q^{10} - \zeta_{8} q^{11} - q^{12} + \zeta_{8}^{2} q^{13} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{15} - q^{16} - \zeta_{8} q^{18} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{20} - \zeta_{8}^{2} q^{22} - \zeta_{8} q^{24} + q^{25} + \zeta_{8}^{3} q^{26} - \zeta_{8}^{2} q^{27} + ( - \zeta_{8}^{2} + 1) q^{30} - \zeta_{8} q^{32} - \zeta_{8}^{3} q^{33} - \zeta_{8}^{2} q^{36} - q^{39} + ( - \zeta_{8}^{2} + 1) q^{40} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} - q^{43} - \zeta_{8}^{3} q^{44} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{45} + (\zeta_{8}^{3} + \zeta_{8}) q^{47} - \zeta_{8}^{2} q^{48} + q^{49} + \zeta_{8} q^{50} - q^{52} - \zeta_{8}^{3} q^{54} + (\zeta_{8}^{2} + 1) q^{55} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{59} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{60} + \zeta_{8}^{2} q^{61} - \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{65} + q^{66} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} - \zeta_{8}^{3} q^{72} + \zeta_{8}^{2} q^{75} - \zeta_{8} q^{78} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{80} + q^{81} + (\zeta_{8}^{2} - 1) q^{82} + (\zeta_{8}^{3} - \zeta_{8}) q^{83} - 2 \zeta_{8} q^{86} + q^{88} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{89} + (\zeta_{8}^{2} + 1) q^{90} + (\zeta_{8}^{2} - 1) q^{94} - \zeta_{8}^{3} q^{96} + \zeta_{8} q^{98} + \zeta_{8} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{10} - 4 q^{12} - 4 q^{16} + 4 q^{25} + 4 q^{30} - 4 q^{39} + 4 q^{40} - 8 q^{43} + 4 q^{49} - 4 q^{52} + 4 q^{55} + 4 q^{66} + 4 q^{81} - 4 q^{82} + 4 q^{88} + 4 q^{90} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times\).

\(n\) \(859\) \(925\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1715.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i 1.00000i 1.00000i 1.41421 0.707107 0.707107i 0 0.707107 0.707107i −1.00000 −1.00000 1.00000i
1715.2 −0.707107 + 0.707107i 1.00000i 1.00000i 1.41421 0.707107 + 0.707107i 0 0.707107 + 0.707107i −1.00000 −1.00000 + 1.00000i
1715.3 0.707107 0.707107i 1.00000i 1.00000i −1.41421 −0.707107 0.707107i 0 −0.707107 0.707107i −1.00000 −1.00000 + 1.00000i
1715.4 0.707107 + 0.707107i 1.00000i 1.00000i −1.41421 −0.707107 + 0.707107i 0 −0.707107 + 0.707107i −1.00000 −1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
44.c even 2 1 inner
132.d odd 2 1 inner
572.b even 2 1 inner
1716.m odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.1.m.g 4
3.b odd 2 1 inner 1716.1.m.g 4
4.b odd 2 1 1716.1.m.h yes 4
11.b odd 2 1 1716.1.m.h yes 4
12.b even 2 1 1716.1.m.h yes 4
13.b even 2 1 inner 1716.1.m.g 4
33.d even 2 1 1716.1.m.h yes 4
39.d odd 2 1 CM 1716.1.m.g 4
44.c even 2 1 inner 1716.1.m.g 4
52.b odd 2 1 1716.1.m.h yes 4
132.d odd 2 1 inner 1716.1.m.g 4
143.d odd 2 1 1716.1.m.h yes 4
156.h even 2 1 1716.1.m.h yes 4
429.e even 2 1 1716.1.m.h yes 4
572.b even 2 1 inner 1716.1.m.g 4
1716.m odd 2 1 inner 1716.1.m.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.1.m.g 4 1.a even 1 1 trivial
1716.1.m.g 4 3.b odd 2 1 inner
1716.1.m.g 4 13.b even 2 1 inner
1716.1.m.g 4 39.d odd 2 1 CM
1716.1.m.g 4 44.c even 2 1 inner
1716.1.m.g 4 132.d odd 2 1 inner
1716.1.m.g 4 572.b even 2 1 inner
1716.1.m.g 4 1716.m odd 2 1 inner
1716.1.m.h yes 4 4.b odd 2 1
1716.1.m.h yes 4 11.b odd 2 1
1716.1.m.h yes 4 12.b even 2 1
1716.1.m.h yes 4 33.d even 2 1
1716.1.m.h yes 4 52.b odd 2 1
1716.1.m.h yes 4 143.d odd 2 1
1716.1.m.h yes 4 156.h even 2 1
1716.1.m.h yes 4 429.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1716, [\chi])\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{43} + 2 \) Copy content Toggle raw display
\( T_{73} \) Copy content Toggle raw display

Hecke characteristic polynomials

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