Properties

Label 1008.2.bf.b
Level $1008$
Weight $2$
Character orbit 1008.bf
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.bf (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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} + ( 1 + \zeta_{6} ) q^{13} + ( 1 + \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( 2 - 4 \zeta_{6} ) q^{23} + 5 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} -3 \zeta_{6} q^{29} -\zeta_{6} q^{31} + 6 q^{33} -7 \zeta_{6} q^{37} + ( -3 + 3 \zeta_{6} ) q^{39} + ( 1 + \zeta_{6} ) q^{41} + ( -2 + \zeta_{6} ) q^{43} + ( 9 - 9 \zeta_{6} ) q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + ( 10 - 5 \zeta_{6} ) q^{57} + 15 \zeta_{6} q^{59} + ( 1 + \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + ( -18 + 9 \zeta_{6} ) q^{67} + 6 q^{69} + ( 6 - 12 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + ( -10 + 8 \zeta_{6} ) q^{77} + ( 1 + \zeta_{6} ) q^{79} + 9 q^{81} -9 \zeta_{6} q^{83} + ( 6 - 3 \zeta_{6} ) q^{87} + ( 2 - \zeta_{6} ) q^{89} + ( 1 - 5 \zeta_{6} ) q^{91} + ( 2 - \zeta_{6} ) q^{93} + ( 2 - \zeta_{6} ) q^{97} + ( -6 + 12 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{7} - 6 q^{9} + 3 q^{13} + 3 q^{17} - 5 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{29} - q^{31} + 12 q^{33} - 7 q^{37} - 3 q^{39} + 3 q^{41} - 3 q^{43} + 9 q^{47} + 2 q^{49} - 3 q^{51} + 9 q^{53} + 15 q^{57} + 15 q^{59} + 3 q^{61} + 12 q^{63} - 27 q^{67} + 12 q^{69} + 3 q^{73} - 12 q^{77} + 3 q^{79} + 18 q^{81} - 9 q^{83} + 9 q^{87} + 3 q^{89} - 3 q^{91} + 3 q^{93} + 3 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
943.1 0 1.73205i 0 0 0 −2.00000 + 1.73205i 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.n 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.bf.b 2
3.b odd 2 1 3024.2.bf.b 2
4.b odd 2 1 1008.2.bf.c yes 2
7.d odd 6 1 1008.2.cz.d yes 2
9.c even 3 1 1008.2.cz.a yes 2
9.d odd 6 1 3024.2.cz.a 2
12.b even 2 1 3024.2.bf.c 2
21.g even 6 1 3024.2.cz.b 2
28.f even 6 1 1008.2.cz.a yes 2
36.f odd 6 1 1008.2.cz.d yes 2
36.h even 6 1 3024.2.cz.b 2
63.k odd 6 1 1008.2.bf.c yes 2
63.s even 6 1 3024.2.bf.c 2
84.j odd 6 1 3024.2.cz.a 2
252.n even 6 1 inner 1008.2.bf.b 2
252.bn odd 6 1 3024.2.bf.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.b 2 1.a even 1 1 trivial
1008.2.bf.b 2 252.n even 6 1 inner
1008.2.bf.c yes 2 4.b odd 2 1
1008.2.bf.c yes 2 63.k odd 6 1
1008.2.cz.a yes 2 9.c even 3 1
1008.2.cz.a yes 2 28.f even 6 1
1008.2.cz.d yes 2 7.d odd 6 1
1008.2.cz.d yes 2 36.f odd 6 1
3024.2.bf.b 2 3.b odd 2 1
3024.2.bf.b 2 252.bn odd 6 1
3024.2.bf.c 2 12.b even 2 1
3024.2.bf.c 2 63.s even 6 1
3024.2.cz.a 2 9.d odd 6 1
3024.2.cz.a 2 84.j odd 6 1
3024.2.cz.b 2 21.g even 6 1
3024.2.cz.b 2 36.h even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{19}^{2} + 5 T_{19} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 3 - 3 T + T^{2} \)
$17$ \( 3 - 3 T + T^{2} \)
$19$ \( 25 + 5 T + T^{2} \)
$23$ \( 12 + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( 1 + T + T^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( 3 - 3 T + T^{2} \)
$43$ \( 3 + 3 T + T^{2} \)
$47$ \( 81 - 9 T + T^{2} \)
$53$ \( 81 - 9 T + T^{2} \)
$59$ \( 225 - 15 T + T^{2} \)
$61$ \( 3 - 3 T + T^{2} \)
$67$ \( 243 + 27 T + T^{2} \)
$71$ \( 108 + T^{2} \)
$73$ \( 3 - 3 T + T^{2} \)
$79$ \( 3 - 3 T + T^{2} \)
$83$ \( 81 + 9 T + T^{2} \)
$89$ \( 3 - 3 T + T^{2} \)
$97$ \( 3 - 3 T + T^{2} \)
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