Properties

Label 1008.2.cz.d
Level $1008$
Weight $2$
Character orbit 1008.cz
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cz (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - \zeta_{6} ) q^{3} + ( 2 - 3 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 2 - \zeta_{6} ) q^{3} + ( 2 - 3 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 4 - 2 \zeta_{6} ) q^{11} + ( -2 + \zeta_{6} ) q^{13} + ( 1 + \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( 1 - 5 \zeta_{6} ) q^{21} + ( -2 - 2 \zeta_{6} ) q^{23} -5 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -3 + 3 \zeta_{6} ) q^{29} - q^{31} + ( 6 - 6 \zeta_{6} ) q^{33} -7 \zeta_{6} q^{37} + ( -3 + 3 \zeta_{6} ) q^{39} + ( -2 + \zeta_{6} ) q^{41} + ( -1 - \zeta_{6} ) q^{43} + 9 q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 3 q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + ( 5 + 5 \zeta_{6} ) q^{57} + 15 q^{59} + ( 1 - 2 \zeta_{6} ) q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} + ( -9 + 18 \zeta_{6} ) q^{67} -6 q^{69} + ( -6 + 12 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} + ( -5 - 5 \zeta_{6} ) q^{75} + ( 2 - 10 \zeta_{6} ) q^{77} + ( -1 + 2 \zeta_{6} ) q^{79} -9 \zeta_{6} q^{81} + ( 9 - 9 \zeta_{6} ) q^{83} + ( -3 + 6 \zeta_{6} ) q^{87} + ( 2 - \zeta_{6} ) q^{89} + ( -1 + 5 \zeta_{6} ) q^{91} + ( -2 + \zeta_{6} ) q^{93} + ( -1 - \zeta_{6} ) q^{97} + ( 6 - 12 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + q^{7} + 3q^{9} + 6q^{11} - 3q^{13} + 3q^{17} + 5q^{19} - 3q^{21} - 6q^{23} - 5q^{25} - 3q^{29} - 2q^{31} + 6q^{33} - 7q^{37} - 3q^{39} - 3q^{41} - 3q^{43} + 18q^{47} - 13q^{49} + 6q^{51} + 9q^{53} + 15q^{57} + 30q^{59} - 12q^{63} - 12q^{69} + 3q^{73} - 15q^{75} - 6q^{77} - 9q^{81} + 9q^{83} + 3q^{89} + 3q^{91} - 3q^{93} - 3q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 0.866025i 0 0 0 0.500000 2.59808i 0 1.50000 2.59808i 0
607.1 0 1.50000 + 0.866025i 0 0 0 0.500000 + 2.59808i 0 1.50000 + 2.59808i 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
252.bj even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cz.d yes 2
3.b odd 2 1 3024.2.cz.b 2
4.b odd 2 1 1008.2.cz.a yes 2
7.d odd 6 1 1008.2.bf.b 2
9.c even 3 1 1008.2.bf.c yes 2
9.d odd 6 1 3024.2.bf.c 2
12.b even 2 1 3024.2.cz.a 2
21.g even 6 1 3024.2.bf.b 2
28.f even 6 1 1008.2.bf.c yes 2
36.f odd 6 1 1008.2.bf.b 2
36.h even 6 1 3024.2.bf.b 2
63.i even 6 1 3024.2.cz.a 2
63.t odd 6 1 1008.2.cz.a yes 2
84.j odd 6 1 3024.2.bf.c 2
252.r odd 6 1 3024.2.cz.b 2
252.bj even 6 1 inner 1008.2.cz.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.b 2 7.d odd 6 1
1008.2.bf.b 2 36.f odd 6 1
1008.2.bf.c yes 2 9.c even 3 1
1008.2.bf.c yes 2 28.f even 6 1
1008.2.cz.a yes 2 4.b odd 2 1
1008.2.cz.a yes 2 63.t odd 6 1
1008.2.cz.d yes 2 1.a even 1 1 trivial
1008.2.cz.d yes 2 252.bj even 6 1 inner
3024.2.bf.b 2 21.g even 6 1
3024.2.bf.b 2 36.h even 6 1
3024.2.bf.c 2 9.d odd 6 1
3024.2.bf.c 2 84.j odd 6 1
3024.2.cz.a 2 12.b even 2 1
3024.2.cz.a 2 63.i even 6 1
3024.2.cz.b 2 3.b odd 2 1
3024.2.cz.b 2 252.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} - 6 T_{11} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 12 - 6 T + T^{2} \)
$13$ \( 3 + 3 T + T^{2} \)
$17$ \( 3 - 3 T + T^{2} \)
$19$ \( 25 - 5 T + T^{2} \)
$23$ \( 12 + 6 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( 3 + 3 T + T^{2} \)
$43$ \( 3 + 3 T + T^{2} \)
$47$ \( ( -9 + T )^{2} \)
$53$ \( 81 - 9 T + T^{2} \)
$59$ \( ( -15 + T )^{2} \)
$61$ \( 3 + T^{2} \)
$67$ \( 243 + T^{2} \)
$71$ \( 108 + T^{2} \)
$73$ \( 3 - 3 T + T^{2} \)
$79$ \( 3 + T^{2} \)
$83$ \( 81 - 9 T + T^{2} \)
$89$ \( 3 - 3 T + T^{2} \)
$97$ \( 3 + 3 T + T^{2} \)
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