Properties

Label 1008.2.cz.f
Level $1008$
Weight $2$
Character orbit 1008.cz
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - 2 \beta_{1} ) q^{3} + ( 2 + \beta_{1} + \beta_{3} ) q^{5} + ( 1 + \beta_{2} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 - 2 \beta_{1} ) q^{3} + ( 2 + \beta_{1} + \beta_{3} ) q^{5} + ( 1 + \beta_{2} ) q^{7} -3 q^{9} + ( -1 + \beta_{1} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{15} + ( 2 + \beta_{1} + 2 \beta_{3} ) q^{17} + ( 1 + \beta_{1} ) q^{19} + ( -1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{21} + ( -6 - 3 \beta_{1} - \beta_{3} ) q^{23} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 3 + 6 \beta_{1} ) q^{27} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + 4 q^{31} + ( 3 + 3 \beta_{1} ) q^{33} + ( 2 + 7 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( 3 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{39} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{43} + ( -6 - 3 \beta_{1} - 3 \beta_{3} ) q^{45} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{47} + ( -5 + 2 \beta_{2} ) q^{49} + ( -3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{55} + ( 1 - \beta_{1} ) q^{57} + 6 q^{59} + ( -4 - 8 \beta_{1} ) q^{61} + ( -3 - 3 \beta_{2} ) q^{63} + 3 q^{65} + ( 9 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( -2 - 4 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -10 - 5 \beta_{1} ) q^{73} + ( 4 - 4 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( -4 - 8 \beta_{1} - 2 \beta_{2} ) q^{79} + 9 q^{81} + ( 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 15 + 15 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{85} + ( 6 + 3 \beta_{1} + 3 \beta_{3} ) q^{87} + ( -1 + \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{89} + ( 5 + 7 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -4 - 8 \beta_{1} ) q^{93} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{97} + ( 3 - 3 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} + 4q^{7} - 12q^{9} + O(q^{10}) \) \( 4q + 6q^{5} + 4q^{7} - 12q^{9} - 6q^{11} - 6q^{13} + 6q^{15} + 6q^{17} + 2q^{19} - 18q^{23} + 8q^{25} - 6q^{29} + 16q^{31} + 6q^{33} - 6q^{35} - 2q^{37} + 6q^{39} - 18q^{41} + 6q^{43} - 18q^{45} - 20q^{49} + 6q^{51} + 6q^{53} - 12q^{55} + 6q^{57} + 24q^{59} - 12q^{63} + 12q^{65} - 18q^{69} - 30q^{73} + 24q^{75} - 6q^{77} + 36q^{81} - 6q^{83} + 30q^{85} + 18q^{87} - 6q^{89} + 6q^{91} - 6q^{97} + 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{3} + 2 \beta_{2}\)\()/3\)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1 + \beta_{1}\) \(1\) \(-1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 1.73205i 0 −0.621320 0.358719i 0 1.00000 2.44949i 0 −3.00000 0
367.2 0 1.73205i 0 3.62132 + 2.09077i 0 1.00000 + 2.44949i 0 −3.00000 0
607.1 0 1.73205i 0 −0.621320 + 0.358719i 0 1.00000 + 2.44949i 0 −3.00000 0
607.2 0 1.73205i 0 3.62132 2.09077i 0 1.00000 2.44949i 0 −3.00000 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.f yes 4
3.b odd 2 1 3024.2.cz.f 4
4.b odd 2 1 1008.2.cz.e yes 4
7.d odd 6 1 1008.2.bf.e 4
9.c even 3 1 1008.2.bf.f yes 4
9.d odd 6 1 3024.2.bf.e 4
12.b even 2 1 3024.2.cz.e 4
21.g even 6 1 3024.2.bf.f 4
28.f even 6 1 1008.2.bf.f yes 4
36.f odd 6 1 1008.2.bf.e 4
36.h even 6 1 3024.2.bf.f 4
63.i even 6 1 3024.2.cz.e 4
63.t odd 6 1 1008.2.cz.e yes 4
84.j odd 6 1 3024.2.bf.e 4
252.r odd 6 1 3024.2.cz.f 4
252.bj even 6 1 inner 1008.2.cz.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.e 4 7.d odd 6 1
1008.2.bf.e 4 36.f odd 6 1
1008.2.bf.f yes 4 9.c even 3 1
1008.2.bf.f yes 4 28.f even 6 1
1008.2.cz.e yes 4 4.b odd 2 1
1008.2.cz.e yes 4 63.t odd 6 1
1008.2.cz.f yes 4 1.a even 1 1 trivial
1008.2.cz.f yes 4 252.bj even 6 1 inner
3024.2.bf.e 4 9.d odd 6 1
3024.2.bf.e 4 84.j odd 6 1
3024.2.bf.f 4 21.g even 6 1
3024.2.bf.f 4 36.h even 6 1
3024.2.cz.e 4 12.b even 2 1
3024.2.cz.e 4 63.i even 6 1
3024.2.cz.f 4 3.b odd 2 1
3024.2.cz.f 4 252.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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} - 6 T_{5}^{3} + 9 T_{5}^{2} + 18 T_{5} + 9 \)
\( T_{11}^{2} + 3 T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( 9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( ( 7 - 2 T + T^{2} )^{2} \)
$11$ \( ( 3 + 3 T + T^{2} )^{2} \)
$13$ \( 9 - 18 T + 9 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 441 + 126 T - 9 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( ( 1 - T + T^{2} )^{2} \)
$23$ \( 441 + 378 T + 129 T^{2} + 18 T^{3} + T^{4} \)
$29$ \( 81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( 289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 9 + 54 T + 111 T^{2} + 18 T^{3} + T^{4} \)
$43$ \( 441 + 126 T - 9 T^{2} - 6 T^{3} + T^{4} \)
$47$ \( ( -72 + T^{2} )^{2} \)
$53$ \( 81 + 54 T + 45 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( ( -6 + T )^{4} \)
$61$ \( ( 48 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( 7056 + 216 T^{2} + T^{4} \)
$73$ \( ( 75 + 15 T + T^{2} )^{2} \)
$79$ \( 576 + 144 T^{2} + T^{4} \)
$83$ \( 3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4} \)
$89$ \( 45369 - 1278 T - 201 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( 441 - 126 T - 9 T^{2} + 6 T^{3} + T^{4} \)
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