Properties

Label 2016.1.dp.a
Level $2016$
Weight $1$
Character orbit 2016.dp
Analytic conductor $1.006$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2016.dp (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
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: \(D_{8}\)
Projective field: Galois closure of 8.2.59663538192384.15

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{7} - \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{7} - \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{3} - 1) q^{11} - q^{14} - q^{16} + (\zeta_{8} - 1) q^{22} + ( - \zeta_{8}^{2} - 1) q^{23} - \zeta_{8}^{3} q^{25} + \zeta_{8} q^{28} + ( - \zeta_{8} + 1) q^{29} + \zeta_{8} q^{32} + (\zeta_{8}^{2} - \zeta_{8}) q^{37} + (\zeta_{8}^{3} + 1) q^{43} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{44} + (\zeta_{8}^{3} + \zeta_{8}) q^{46} - \zeta_{8}^{2} q^{49} - q^{50} + (\zeta_{8}^{2} + \zeta_{8}) q^{53} - \zeta_{8}^{2} q^{56} + (\zeta_{8}^{2} - \zeta_{8}) q^{58} - \zeta_{8}^{2} q^{64} + ( - \zeta_{8} + 1) q^{67} - \zeta_{8}^{3} q^{71} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{74} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{77} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{79} + ( - \zeta_{8} + 1) q^{86} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{88} + ( - \zeta_{8}^{2} + 1) q^{92} + \zeta_{8}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} - 4 q^{14} - 4 q^{16} - 4 q^{22} - 4 q^{23} + 4 q^{29} + 4 q^{43} - 4 q^{50} + 4 q^{67} + 4 q^{86} + 4 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{8}^{3}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
685.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
1189.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1693.1 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
32.g even 8 1 inner
224.v odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.1.dp.a 4
3.b odd 2 1 2016.1.dp.c yes 4
7.b odd 2 1 CM 2016.1.dp.a 4
21.c even 2 1 2016.1.dp.c yes 4
32.g even 8 1 inner 2016.1.dp.a 4
96.p odd 8 1 2016.1.dp.c yes 4
224.v odd 8 1 inner 2016.1.dp.a 4
672.bo even 8 1 2016.1.dp.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.1.dp.a 4 1.a even 1 1 trivial
2016.1.dp.a 4 7.b odd 2 1 CM
2016.1.dp.a 4 32.g even 8 1 inner
2016.1.dp.a 4 224.v odd 8 1 inner
2016.1.dp.c yes 4 3.b odd 2 1
2016.1.dp.c yes 4 21.c even 2 1
2016.1.dp.c yes 4 96.p odd 8 1
2016.1.dp.c yes 4 672.bo even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 4T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2016, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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