Properties

Label 2007.1.d.a
Level 2007
Weight 1
Character orbit 2007.d
Self dual yes
Analytic conductor 1.002
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -3, -223, 669
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 2007 = 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2007.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.00162348035\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{-223})\)
Artin image $D_4$
Artin field Galois closure of 4.0.6021.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{4} - 2q^{7} + O(q^{10}) \) \( q - q^{4} - 2q^{7} + q^{16} + 2q^{19} + q^{25} + 2q^{28} - 2q^{31} + 2q^{37} + 2q^{43} + 3q^{49} - q^{64} - 2q^{73} - 2q^{76} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

\(n\) \(226\) \(893\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1783.1
0
0 0 −1.00000 0 0 −2.00000 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
223.b odd 2 1 CM by \(\Q(\sqrt{-223}) \)
669.c even 2 1 RM by \(\Q(\sqrt{669}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.d.a 1
3.b odd 2 1 CM 2007.1.d.a 1
223.b odd 2 1 CM 2007.1.d.a 1
669.c even 2 1 RM 2007.1.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.d.a 1 1.a even 1 1 trivial
2007.1.d.a 1 3.b odd 2 1 CM
2007.1.d.a 1 223.b odd 2 1 CM
2007.1.d.a 1 669.c even 2 1 RM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\).

Hecke characteristic polynomials

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