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Properties

Label 1716.1.bt.a
Level $1716$
Weight $1$
Character orbit 1716.bt
Analytic conductor $0.856$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
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(623,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.623");
 
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.bt (of order \(10\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{15} q^{2} + \zeta_{40}^{14} q^{3} - \zeta_{40}^{10} q^{4} + ( - \zeta_{40}^{13} - \zeta_{40}^{11}) q^{5} - \zeta_{40}^{9} q^{6} + \zeta_{40}^{5} q^{8} - \zeta_{40}^{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{40}^{15} q^{2} + \zeta_{40}^{14} q^{3} - \zeta_{40}^{10} q^{4} + ( - \zeta_{40}^{13} - \zeta_{40}^{11}) q^{5} - \zeta_{40}^{9} q^{6} + \zeta_{40}^{5} q^{8} - \zeta_{40}^{8} q^{9} + (\zeta_{40}^{8} + \zeta_{40}^{6}) q^{10} - \zeta_{40}^{17} q^{11} + \zeta_{40}^{4} q^{12} + \zeta_{40}^{18} q^{13} + (\zeta_{40}^{7} + \zeta_{40}^{5}) q^{15} - q^{16} + \zeta_{40}^{3} q^{18} + ( - \zeta_{40}^{3} - \zeta_{40}) q^{20} + \zeta_{40}^{12} q^{22} + \zeta_{40}^{19} q^{24} + ( - \zeta_{40}^{6} + \cdots - \zeta_{40}^{2}) q^{25} + \cdots - \zeta_{40}^{5} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{9} - 4 q^{10} + 4 q^{12} - 16 q^{16} + 4 q^{22} - 4 q^{25} - 16 q^{30} - 4 q^{39} + 4 q^{40} - 8 q^{43} - 4 q^{49} - 4 q^{52} + 4 q^{55} + 20 q^{75} - 4 q^{81} + 4 q^{82} + 4 q^{90} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(859\) \(925\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{40}^{16}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
623.1
−0.891007 − 0.453990i
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.891007 + 0.453990i
0.987688 − 0.156434i
0.156434 + 0.987688i
−0.156434 − 0.987688i
−0.987688 + 0.156434i
0.156434 − 0.987688i
0.987688 + 0.156434i
−0.987688 − 0.156434i
−0.156434 + 0.987688i
0.453990 + 0.891007i
−0.891007 + 0.453990i
0.891007 − 0.453990i
−0.453990 − 0.891007i
−0.707107 − 0.707107i 0.951057 + 0.309017i 1.00000i 1.44168 − 1.04744i −0.453990 − 0.891007i 0 0.707107 − 0.707107i 0.809017 + 0.587785i −1.76007 − 0.278768i
623.2 −0.707107 + 0.707107i −0.951057 − 0.309017i − 1.00000i −0.734572 + 0.533698i 0.891007 − 0.453990i 0 0.707107 + 0.707107i 0.809017 + 0.587785i 0.142040 − 0.896802i
623.3 0.707107 − 0.707107i −0.951057 − 0.309017i − 1.00000i 0.734572 − 0.533698i −0.891007 + 0.453990i 0 −0.707107 − 0.707107i 0.809017 + 0.587785i 0.142040 − 0.896802i
623.4 0.707107 + 0.707107i 0.951057 + 0.309017i 1.00000i −1.44168 + 1.04744i 0.453990 + 0.891007i 0 −0.707107 + 0.707107i 0.809017 + 0.587785i −1.76007 − 0.278768i
1091.1 −0.707107 − 0.707107i −0.587785 − 0.809017i 1.00000i 0.610425 + 1.87869i −0.156434 + 0.987688i 0 0.707107 − 0.707107i −0.309017 + 0.951057i 0.896802 − 1.76007i
1091.2 −0.707107 + 0.707107i 0.587785 + 0.809017i − 1.00000i 0.0966818 + 0.297556i −0.987688 − 0.156434i 0 0.707107 + 0.707107i −0.309017 + 0.951057i −0.278768 − 0.142040i
1091.3 0.707107 − 0.707107i 0.587785 + 0.809017i − 1.00000i −0.0966818 − 0.297556i 0.987688 + 0.156434i 0 −0.707107 − 0.707107i −0.309017 + 0.951057i −0.278768 − 0.142040i
1091.4 0.707107 + 0.707107i −0.587785 − 0.809017i 1.00000i −0.610425 − 1.87869i 0.156434 − 0.987688i 0 −0.707107 + 0.707107i −0.309017 + 0.951057i 0.896802 − 1.76007i
1403.1 −0.707107 − 0.707107i 0.587785 − 0.809017i 1.00000i 0.0966818 − 0.297556i −0.987688 + 0.156434i 0 0.707107 − 0.707107i −0.309017 − 0.951057i −0.278768 + 0.142040i
1403.2 −0.707107 + 0.707107i −0.587785 + 0.809017i − 1.00000i 0.610425 − 1.87869i −0.156434 − 0.987688i 0 0.707107 + 0.707107i −0.309017 − 0.951057i 0.896802 + 1.76007i
1403.3 0.707107 − 0.707107i −0.587785 + 0.809017i − 1.00000i −0.610425 + 1.87869i 0.156434 + 0.987688i 0 −0.707107 − 0.707107i −0.309017 − 0.951057i 0.896802 + 1.76007i
1403.4 0.707107 + 0.707107i 0.587785 − 0.809017i 1.00000i −0.0966818 + 0.297556i 0.987688 − 0.156434i 0 −0.707107 + 0.707107i −0.309017 − 0.951057i −0.278768 + 0.142040i
1559.1 −0.707107 − 0.707107i −0.951057 + 0.309017i 1.00000i −0.734572 − 0.533698i 0.891007 + 0.453990i 0 0.707107 − 0.707107i 0.809017 − 0.587785i 0.142040 + 0.896802i
1559.2 −0.707107 + 0.707107i 0.951057 − 0.309017i − 1.00000i 1.44168 + 1.04744i −0.453990 + 0.891007i 0 0.707107 + 0.707107i 0.809017 − 0.587785i −1.76007 + 0.278768i
1559.3 0.707107 − 0.707107i 0.951057 − 0.309017i − 1.00000i −1.44168 − 1.04744i 0.453990 − 0.891007i 0 −0.707107 − 0.707107i 0.809017 − 0.587785i −1.76007 + 0.278768i
1559.4 0.707107 + 0.707107i −0.951057 + 0.309017i 1.00000i 0.734572 + 0.533698i −0.891007 − 0.453990i 0 −0.707107 + 0.707107i 0.809017 − 0.587785i 0.142040 + 0.896802i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.4
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.g even 10 1 inner
132.n odd 10 1 inner
572.bb even 10 1 inner
1716.bt odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.1.bt.a ✓ 16
3.b odd 2 1 inner 1716.1.bt.a ✓ 16
4.b odd 2 1 1716.1.bt.b yes 16
11.d odd 10 1 1716.1.bt.b yes 16
12.b even 2 1 1716.1.bt.b yes 16
13.b even 2 1 inner 1716.1.bt.a ✓ 16
33.f even 10 1 1716.1.bt.b yes 16
39.d odd 2 1 CM 1716.1.bt.a ✓ 16
44.g even 10 1 inner 1716.1.bt.a ✓ 16
52.b odd 2 1 1716.1.bt.b yes 16
132.n odd 10 1 inner 1716.1.bt.a ✓ 16
143.l odd 10 1 1716.1.bt.b yes 16
156.h even 2 1 1716.1.bt.b yes 16
429.y even 10 1 1716.1.bt.b yes 16
572.bb even 10 1 inner 1716.1.bt.a ✓ 16
1716.bt odd 10 1 inner 1716.1.bt.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.1.bt.a ✓ 16 1.a even 1 1 trivial
1716.1.bt.a ✓ 16 3.b odd 2 1 inner
1716.1.bt.a ✓ 16 13.b even 2 1 inner
1716.1.bt.a ✓ 16 39.d odd 2 1 CM
1716.1.bt.a ✓ 16 44.g even 10 1 inner
1716.1.bt.a ✓ 16 132.n odd 10 1 inner
1716.1.bt.a ✓ 16 572.bb even 10 1 inner
1716.1.bt.a ✓ 16 1716.bt odd 10 1 inner
1716.1.bt.b yes 16 4.b odd 2 1
1716.1.bt.b yes 16 11.d odd 10 1
1716.1.bt.b yes 16 12.b even 2 1
1716.1.bt.b yes 16 33.f even 10 1
1716.1.bt.b yes 16 52.b odd 2 1
1716.1.bt.b yes 16 143.l odd 10 1
1716.1.bt.b yes 16 156.h even 2 1
1716.1.bt.b yes 16 429.y even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{43}^{2} + T_{43} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1716, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 1)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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