Properties

Label 1008.2.r.b
Level 1008
Weight 2
Character orbit 1008.r
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.r (of order \(3\), degree \(2\), not 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: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + 6 \zeta_{6} q^{13} + ( -2 + 4 \zeta_{6} ) q^{15} + 7 q^{17} - q^{19} + ( -2 + \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 4 - 4 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} + ( 6 - 3 \zeta_{6} ) q^{33} -2 q^{35} -6 q^{37} + ( 6 - 12 \zeta_{6} ) q^{39} -7 \zeta_{6} q^{41} + ( 11 - 11 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{45} + ( 8 - 8 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + ( -7 - 7 \zeta_{6} ) q^{51} -4 q^{53} + 6 q^{55} + ( 1 + \zeta_{6} ) q^{57} + 9 \zeta_{6} q^{59} + ( -4 + 4 \zeta_{6} ) q^{61} + 3 q^{63} + ( 12 - 12 \zeta_{6} ) q^{65} + 9 \zeta_{6} q^{67} + ( -4 + 8 \zeta_{6} ) q^{69} + 13 q^{73} + ( -2 + \zeta_{6} ) q^{75} + 3 \zeta_{6} q^{77} + ( 10 - 10 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -14 \zeta_{6} q^{85} + ( -8 + 4 \zeta_{6} ) q^{87} + 10 q^{89} + 6 q^{91} + ( 10 - 20 \zeta_{6} ) q^{93} + 2 \zeta_{6} q^{95} + ( -7 + 7 \zeta_{6} ) q^{97} -9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 2q^{5} + q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 2q^{5} + q^{7} + 3q^{9} - 3q^{11} + 6q^{13} + 14q^{17} - 2q^{19} - 3q^{21} - 4q^{23} + q^{25} + 4q^{29} + 10q^{31} + 9q^{33} - 4q^{35} - 12q^{37} - 7q^{41} + 11q^{43} + 6q^{45} + 8q^{47} - q^{49} - 21q^{51} - 8q^{53} + 12q^{55} + 3q^{57} + 9q^{59} - 4q^{61} + 6q^{63} + 12q^{65} + 9q^{67} + 26q^{73} - 3q^{75} + 3q^{77} + 10q^{79} - 9q^{81} - 14q^{85} - 12q^{87} + 20q^{89} + 12q^{91} + 2q^{95} - 7q^{97} - 18q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 + 0.866025i 0 −1.00000 + 1.73205i 0 0.500000 + 0.866025i 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −1.00000 1.73205i 0 0.500000 0.866025i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.b 2
3.b odd 2 1 3024.2.r.d 2
4.b odd 2 1 504.2.r.b 2
9.c even 3 1 inner 1008.2.r.b 2
9.c even 3 1 9072.2.a.s 1
9.d odd 6 1 3024.2.r.d 2
9.d odd 6 1 9072.2.a.d 1
12.b even 2 1 1512.2.r.b 2
36.f odd 6 1 504.2.r.b 2
36.f odd 6 1 4536.2.a.h 1
36.h even 6 1 1512.2.r.b 2
36.h even 6 1 4536.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.b 2 4.b odd 2 1
504.2.r.b 2 36.f odd 6 1
1008.2.r.b 2 1.a even 1 1 trivial
1008.2.r.b 2 9.c even 3 1 inner
1512.2.r.b 2 12.b even 2 1
1512.2.r.b 2 36.h even 6 1
3024.2.r.d 2 3.b odd 2 1
3024.2.r.d 2 9.d odd 6 1
4536.2.a.c 1 36.h even 6 1
4536.2.a.h 1 36.f odd 6 1
9072.2.a.d 1 9.d odd 6 1
9072.2.a.s 1 9.c even 3 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 7 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 11 T + 78 T^{2} - 473 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 13 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 83 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} \)
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