Properties

Label 2007.1.d.c
Level 2007
Weight 1
Character orbit 2007.d
Self dual yes
Analytic conductor 1.002
Analytic rank 0
Dimension 6
Projective image \(D_{14}\)
CM discriminant -223
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: \(6\)
Coefficient field: \(\Q(\zeta_{28})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{14}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{6} - 7 x^{4} + 14 x^{2} - 7\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -1 + \beta^{2} ) q^{4} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{7} + ( 2 \beta - \beta^{3} ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -1 + \beta^{2} ) q^{4} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{7} + ( 2 \beta - \beta^{3} ) q^{8} + ( 2 \beta - 4 \beta^{3} + \beta^{5} ) q^{14} + ( 1 - 3 \beta^{2} + \beta^{4} ) q^{16} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{17} + ( 2 - \beta^{2} ) q^{19} + q^{25} + ( -5 + 8 \beta^{2} - 2 \beta^{4} ) q^{28} + \beta q^{29} + ( 5 - 5 \beta^{2} + \beta^{4} ) q^{31} + ( -3 \beta + 4 \beta^{3} - \beta^{5} ) q^{32} + ( 7 - 9 \beta^{2} + 2 \beta^{4} ) q^{34} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{37} + ( -2 \beta + \beta^{3} ) q^{38} + ( -3 \beta + \beta^{3} ) q^{41} + ( -5 + 5 \beta^{2} - \beta^{4} ) q^{43} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{47} + ( -4 + 5 \beta^{2} - \beta^{4} ) q^{49} -\beta q^{50} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{53} + ( 3 \beta - 4 \beta^{3} + \beta^{5} ) q^{56} -\beta^{2} q^{58} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{62} + ( 6 - 8 \beta^{2} + 2 \beta^{4} ) q^{64} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{68} + ( 5 - 5 \beta^{2} + \beta^{4} ) q^{73} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{74} + ( -2 + 3 \beta^{2} - \beta^{4} ) q^{76} + ( 3 \beta^{2} - \beta^{4} ) q^{82} + \beta q^{83} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{86} + ( 3 \beta - \beta^{3} ) q^{89} + ( -7 + 9 \beta^{2} - 2 \beta^{4} ) q^{94} + ( 4 \beta - 5 \beta^{3} + \beta^{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 8q^{4} + 2q^{7} + O(q^{10}) \) \( 6q + 8q^{4} + 2q^{7} + 6q^{16} - 2q^{19} + 6q^{25} - 2q^{28} + 2q^{31} - 2q^{37} - 2q^{43} + 4q^{49} - 14q^{58} + 8q^{64} + 2q^{73} - 12q^{76} + O(q^{100}) \)

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
1.94986
1.56366
0.867767
−0.867767
−1.56366
−1.94986
−1.94986 0 2.80194 0 0 −1.24698 −3.51352 0 0
1783.2 −1.56366 0 1.44504 0 0 1.80194 −0.695895 0 0
1783.3 −0.867767 0 −0.246980 0 0 0.445042 1.08209 0 0
1783.4 0.867767 0 −0.246980 0 0 0.445042 −1.08209 0 0
1783.5 1.56366 0 1.44504 0 0 1.80194 0.695895 0 0
1783.6 1.94986 0 2.80194 0 0 −1.24698 3.51352 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1783.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
223.b odd 2 1 CM by \(\Q(\sqrt{-223}) \)
3.b odd 2 1 inner
669.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.d.c 6
3.b odd 2 1 inner 2007.1.d.c 6
223.b odd 2 1 CM 2007.1.d.c 6
669.c even 2 1 inner 2007.1.d.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.d.c 6 1.a even 1 1 trivial
2007.1.d.c 6 3.b odd 2 1 inner
2007.1.d.c 6 223.b odd 2 1 CM
2007.1.d.c 6 669.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 7 T_{2}^{4} + 14 T_{2}^{2} - 7 \) acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\).

Hecke Characteristic Polynomials

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