Properties

Label 1716.1.m.i
Level $1716$
Weight $1$
Character orbit 1716.m
Analytic conductor $0.856$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -143
Inner twists $4$

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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: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.104051987500032.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{10}^{2} q^{2} - \zeta_{10}^{3} q^{3} + \zeta_{10}^{4} q^{4} - q^{6} + (\zeta_{10}^{4} + \zeta_{10}) q^{7} + \zeta_{10} q^{8} - \zeta_{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10}^{2} q^{2} - \zeta_{10}^{3} q^{3} + \zeta_{10}^{4} q^{4} - q^{6} + (\zeta_{10}^{4} + \zeta_{10}) q^{7} + \zeta_{10} q^{8} - \zeta_{10} q^{9} + q^{11} + \zeta_{10}^{2} q^{12} + q^{13} + ( - \zeta_{10}^{3} + \zeta_{10}) q^{14} - \zeta_{10}^{3} q^{16} + \zeta_{10}^{3} q^{18} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{19} + ( - \zeta_{10}^{4} + \zeta_{10}^{2}) q^{21} - \zeta_{10}^{2} q^{22} + ( - \zeta_{10}^{4} + \zeta_{10}) q^{23} - \zeta_{10}^{4} q^{24} - q^{25} - \zeta_{10}^{2} q^{26} + \zeta_{10}^{4} q^{27} + ( - \zeta_{10}^{3} - 1) q^{28} - q^{32} - \zeta_{10}^{3} q^{33} + q^{36} + ( - \zeta_{10}^{4} + 1) q^{38} - \zeta_{10}^{3} q^{39} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{41} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{42} + \zeta_{10}^{4} q^{44} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{46} - \zeta_{10} q^{48} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{49} + \zeta_{10}^{2} q^{50} + \zeta_{10}^{4} q^{52} + ( - \zeta_{10}^{3} - \zeta_{10}^{2}) q^{53} + \zeta_{10} q^{54} + (\zeta_{10}^{2} - 1) q^{56} + (\zeta_{10} + 1) q^{57} + ( - \zeta_{10}^{2} + 1) q^{63} + \zeta_{10}^{2} q^{64} - q^{66} + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{69} - \zeta_{10}^{2} q^{72} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{73} + \zeta_{10}^{3} q^{75} + ( - \zeta_{10}^{2} - \zeta_{10}) q^{76} + (\zeta_{10}^{4} + \zeta_{10}) q^{77} - q^{78} + \zeta_{10}^{2} q^{81} + (\zeta_{10}^{3} - \zeta_{10}) q^{82} + (\zeta_{10}^{4} - \zeta_{10}) q^{83} + (\zeta_{10}^{3} - \zeta_{10}) q^{84} + \zeta_{10} q^{88} + (\zeta_{10}^{4} + \zeta_{10}) q^{91} + (\zeta_{10}^{3} - 1) q^{92} + \zeta_{10}^{3} q^{96} + ( - \zeta_{10}^{4} + \zeta_{10}^{2} - 1) q^{98} - \zeta_{10} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - q^{4} - 4 q^{6} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{3} - q^{4} - 4 q^{6} + q^{8} - q^{9} + 4 q^{11} - q^{12} + 4 q^{13} - q^{16} + q^{18} + q^{22} + 2 q^{23} + q^{24} - 4 q^{25} + q^{26} - q^{27} - 5 q^{28} - 4 q^{32} - q^{33} + 4 q^{36} + 5 q^{38} - q^{39} - q^{44} - 2 q^{46} - q^{48} - 6 q^{49} - q^{50} - q^{52} + q^{54} - 5 q^{56} + 5 q^{57} + 5 q^{63} - q^{64} - 4 q^{66} + 2 q^{69} + q^{72} + 2 q^{73} + q^{75} - 4 q^{78} - q^{81} - 2 q^{83} + q^{88} - 3 q^{92} + q^{96} - 4 q^{98} - q^{99}+O(q^{100}) \) 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
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
−0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 0 −1.00000 1.17557i 0.809017 + 0.587785i −0.809017 0.587785i 0
1715.2 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 0 −1.00000 1.17557i 0.809017 0.587785i −0.809017 + 0.587785i 0
1715.3 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i 0 −1.00000 1.90211i −0.309017 0.951057i 0.309017 + 0.951057i 0
1715.4 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 0 −1.00000 1.90211i −0.309017 + 0.951057i 0.309017 0.951057i 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
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)
12.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.i yes 4
3.b odd 2 1 1716.1.m.f yes 4
4.b odd 2 1 1716.1.m.f yes 4
11.b odd 2 1 1716.1.m.e 4
12.b even 2 1 inner 1716.1.m.i yes 4
13.b even 2 1 1716.1.m.e 4
33.d even 2 1 1716.1.m.j yes 4
39.d odd 2 1 1716.1.m.j yes 4
44.c even 2 1 1716.1.m.j yes 4
52.b odd 2 1 1716.1.m.j yes 4
132.d odd 2 1 1716.1.m.e 4
143.d odd 2 1 CM 1716.1.m.i yes 4
156.h even 2 1 1716.1.m.e 4
429.e even 2 1 1716.1.m.f yes 4
572.b even 2 1 1716.1.m.f yes 4
1716.m odd 2 1 inner 1716.1.m.i yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.1.m.e 4 11.b odd 2 1
1716.1.m.e 4 13.b even 2 1
1716.1.m.e 4 132.d odd 2 1
1716.1.m.e 4 156.h even 2 1
1716.1.m.f yes 4 3.b odd 2 1
1716.1.m.f yes 4 4.b odd 2 1
1716.1.m.f yes 4 429.e even 2 1
1716.1.m.f yes 4 572.b even 2 1
1716.1.m.i yes 4 1.a even 1 1 trivial
1716.1.m.i yes 4 12.b even 2 1 inner
1716.1.m.i yes 4 143.d odd 2 1 CM
1716.1.m.i yes 4 1716.m odd 2 1 inner
1716.1.m.j yes 4 33.d even 2 1
1716.1.m.j yes 4 39.d odd 2 1
1716.1.m.j yes 4 44.c even 2 1
1716.1.m.j yes 4 52.b odd 2 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{2} + 5 \) Copy content Toggle raw display
\( T_{23}^{2} - T_{23} - 1 \) Copy content Toggle raw display
\( T_{43} \) Copy content Toggle raw display
\( T_{73}^{2} - T_{73} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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