Properties

Label 1008.2.q.a
Level 1008
Weight 2
Character orbit 1008.q
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.q (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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -3 + 3 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -3 + 3 \zeta_{6} ) q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} -3 \zeta_{6} q^{11} + \zeta_{6} q^{13} + ( 3 + 3 \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} -7 \zeta_{6} q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( -9 + 9 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -3 + 3 \zeta_{6} ) q^{29} -8 q^{31} + ( -6 + 3 \zeta_{6} ) q^{33} + ( -9 + 3 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( 2 - \zeta_{6} ) q^{39} -3 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( 9 - 9 \zeta_{6} ) q^{45} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} + ( -3 + 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( -14 + 7 \zeta_{6} ) q^{57} + 2 q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} -3 q^{65} + 4 q^{67} + ( 9 + 9 \zeta_{6} ) q^{69} -12 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + ( -8 + 4 \zeta_{6} ) q^{75} + ( 6 - 9 \zeta_{6} ) q^{77} + 16 q^{79} + 9 q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} + ( 3 + 3 \zeta_{6} ) q^{87} -3 \zeta_{6} q^{89} + ( -2 + 3 \zeta_{6} ) q^{91} + ( -8 + 16 \zeta_{6} ) q^{93} + 21 q^{95} + ( 1 - \zeta_{6} ) q^{97} + 9 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{5} + 4q^{7} - 6q^{9} - 3q^{11} + q^{13} + 9q^{15} - 3q^{17} - 7q^{19} + 6q^{21} - 9q^{23} - 4q^{25} - 3q^{29} - 16q^{31} - 9q^{33} - 15q^{35} + q^{37} + 3q^{39} - 3q^{41} - q^{43} + 9q^{45} + 2q^{49} + 9q^{51} - 3q^{53} + 18q^{55} - 21q^{57} + 4q^{61} - 12q^{63} - 6q^{65} + 8q^{67} + 27q^{69} - 24q^{71} - 11q^{73} - 12q^{75} + 3q^{77} + 32q^{79} + 18q^{81} - 9q^{83} - 9q^{85} + 9q^{87} - 3q^{89} - q^{91} + 42q^{95} + q^{97} + 9q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 2.00000 + 1.73205i 0 −3.00000 0
625.1 0 1.73205i 0 −1.50000 2.59808i 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
63.h 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.q.a 2
3.b odd 2 1 3024.2.q.f 2
4.b odd 2 1 126.2.e.a 2
7.c even 3 1 1008.2.t.f 2
9.c even 3 1 1008.2.t.f 2
9.d odd 6 1 3024.2.t.a 2
12.b even 2 1 378.2.e.b 2
21.h odd 6 1 3024.2.t.a 2
28.d even 2 1 882.2.e.c 2
28.f even 6 1 882.2.f.g 2
28.f even 6 1 882.2.h.i 2
28.g odd 6 1 126.2.h.b yes 2
28.g odd 6 1 882.2.f.i 2
36.f odd 6 1 126.2.h.b yes 2
36.f odd 6 1 1134.2.g.e 2
36.h even 6 1 378.2.h.a 2
36.h even 6 1 1134.2.g.c 2
63.h even 3 1 inner 1008.2.q.a 2
63.j odd 6 1 3024.2.q.f 2
84.h odd 2 1 2646.2.e.g 2
84.j odd 6 1 2646.2.f.a 2
84.j odd 6 1 2646.2.h.d 2
84.n even 6 1 378.2.h.a 2
84.n even 6 1 2646.2.f.d 2
252.n even 6 1 882.2.f.g 2
252.o even 6 1 1134.2.g.c 2
252.o even 6 1 2646.2.f.d 2
252.r odd 6 1 2646.2.e.g 2
252.r odd 6 1 7938.2.a.be 1
252.s odd 6 1 2646.2.h.d 2
252.u odd 6 1 126.2.e.a 2
252.u odd 6 1 7938.2.a.m 1
252.bb even 6 1 378.2.e.b 2
252.bb even 6 1 7938.2.a.t 1
252.bi even 6 1 882.2.h.i 2
252.bj even 6 1 882.2.e.c 2
252.bj even 6 1 7938.2.a.b 1
252.bl odd 6 1 882.2.f.i 2
252.bl odd 6 1 1134.2.g.e 2
252.bn odd 6 1 2646.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 4.b odd 2 1
126.2.e.a 2 252.u odd 6 1
126.2.h.b yes 2 28.g odd 6 1
126.2.h.b yes 2 36.f odd 6 1
378.2.e.b 2 12.b even 2 1
378.2.e.b 2 252.bb even 6 1
378.2.h.a 2 36.h even 6 1
378.2.h.a 2 84.n even 6 1
882.2.e.c 2 28.d even 2 1
882.2.e.c 2 252.bj even 6 1
882.2.f.g 2 28.f even 6 1
882.2.f.g 2 252.n even 6 1
882.2.f.i 2 28.g odd 6 1
882.2.f.i 2 252.bl odd 6 1
882.2.h.i 2 28.f even 6 1
882.2.h.i 2 252.bi even 6 1
1008.2.q.a 2 1.a even 1 1 trivial
1008.2.q.a 2 63.h even 3 1 inner
1008.2.t.f 2 7.c even 3 1
1008.2.t.f 2 9.c even 3 1
1134.2.g.c 2 36.h even 6 1
1134.2.g.c 2 252.o even 6 1
1134.2.g.e 2 36.f odd 6 1
1134.2.g.e 2 252.bl odd 6 1
2646.2.e.g 2 84.h odd 2 1
2646.2.e.g 2 252.r odd 6 1
2646.2.f.a 2 84.j odd 6 1
2646.2.f.a 2 252.bn odd 6 1
2646.2.f.d 2 84.n even 6 1
2646.2.f.d 2 252.o even 6 1
2646.2.h.d 2 84.j odd 6 1
2646.2.h.d 2 252.s odd 6 1
3024.2.q.f 2 3.b odd 2 1
3024.2.q.f 2 63.j odd 6 1
3024.2.t.a 2 9.d odd 6 1
3024.2.t.a 2 21.h odd 6 1
7938.2.a.b 1 252.bj even 6 1
7938.2.a.m 1 252.u odd 6 1
7938.2.a.t 1 252.bb even 6 1
7938.2.a.be 1 252.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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 16 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \)
$97$ \( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \)
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