Properties

Label 1008.2.q.c
Level 1008
Weight 2
Character orbit 1008.q
Analytic conductor 8.049
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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 63)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -3 q^{9} + 5 \zeta_{6} q^{11} + 5 \zeta_{6} q^{13} + ( -1 - \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 1 + 4 \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 1 - \zeta_{6} ) q^{29} + ( 10 - 5 \zeta_{6} ) q^{33} + ( -1 + 3 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + ( 10 - 5 \zeta_{6} ) q^{39} + 5 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + 5 q^{55} + ( 2 - \zeta_{6} ) q^{57} -14 q^{61} + ( 9 - 6 \zeta_{6} ) q^{63} + 5 q^{65} -4 q^{67} + ( -3 - 3 \zeta_{6} ) q^{69} + 12 q^{71} + ( -3 + 3 \zeta_{6} ) q^{73} + ( 8 - 4 \zeta_{6} ) q^{75} + ( -10 - 5 \zeta_{6} ) q^{77} -8 q^{79} + 9 q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} + 3 \zeta_{6} q^{85} + ( -1 - \zeta_{6} ) q^{87} + 13 \zeta_{6} q^{89} + ( -10 - 5 \zeta_{6} ) q^{91} + q^{95} + ( 9 - 9 \zeta_{6} ) q^{97} -15 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + q^{5} - 4q^{7} - 6q^{9} + 5q^{11} + 5q^{13} - 3q^{15} - 3q^{17} + q^{19} + 6q^{21} + 3q^{23} + 4q^{25} + q^{29} + 15q^{33} + q^{35} - 3q^{37} + 15q^{39} + 5q^{41} - q^{43} - 3q^{45} + 2q^{49} + 9q^{51} + 9q^{53} + 10q^{55} + 3q^{57} - 28q^{61} + 12q^{63} + 10q^{65} - 8q^{67} - 9q^{69} + 24q^{71} - 3q^{73} + 12q^{75} - 25q^{77} - 16q^{79} + 18q^{81} - 9q^{83} + 3q^{85} - 3q^{87} + 13q^{89} - 25q^{91} + 2q^{95} + 9q^{97} - 15q^{99} + O(q^{100}) \)

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 0.500000 0.866025i 0 −2.00000 + 1.73205i 0 −3.00000 0
625.1 0 1.73205i 0 0.500000 + 0.866025i 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.c 2
3.b odd 2 1 3024.2.q.b 2
4.b odd 2 1 63.2.h.a yes 2
7.c even 3 1 1008.2.t.d 2
9.c even 3 1 1008.2.t.d 2
9.d odd 6 1 3024.2.t.d 2
12.b even 2 1 189.2.h.a 2
21.h odd 6 1 3024.2.t.d 2
28.d even 2 1 441.2.h.a 2
28.f even 6 1 441.2.f.a 2
28.f even 6 1 441.2.g.a 2
28.g odd 6 1 63.2.g.a 2
28.g odd 6 1 441.2.f.b 2
36.f odd 6 1 63.2.g.a 2
36.f odd 6 1 567.2.e.a 2
36.h even 6 1 189.2.g.a 2
36.h even 6 1 567.2.e.b 2
63.h even 3 1 inner 1008.2.q.c 2
63.j odd 6 1 3024.2.q.b 2
84.h odd 2 1 1323.2.h.a 2
84.j odd 6 1 1323.2.f.b 2
84.j odd 6 1 1323.2.g.a 2
84.n even 6 1 189.2.g.a 2
84.n even 6 1 1323.2.f.a 2
252.n even 6 1 441.2.f.a 2
252.o even 6 1 567.2.e.b 2
252.o even 6 1 1323.2.f.a 2
252.r odd 6 1 1323.2.h.a 2
252.r odd 6 1 3969.2.a.a 1
252.s odd 6 1 1323.2.g.a 2
252.u odd 6 1 63.2.h.a yes 2
252.u odd 6 1 3969.2.a.d 1
252.bb even 6 1 189.2.h.a 2
252.bb even 6 1 3969.2.a.c 1
252.bi even 6 1 441.2.g.a 2
252.bj even 6 1 441.2.h.a 2
252.bj even 6 1 3969.2.a.f 1
252.bl odd 6 1 441.2.f.b 2
252.bl odd 6 1 567.2.e.a 2
252.bn odd 6 1 1323.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 28.g odd 6 1
63.2.g.a 2 36.f odd 6 1
63.2.h.a yes 2 4.b odd 2 1
63.2.h.a yes 2 252.u odd 6 1
189.2.g.a 2 36.h even 6 1
189.2.g.a 2 84.n even 6 1
189.2.h.a 2 12.b even 2 1
189.2.h.a 2 252.bb even 6 1
441.2.f.a 2 28.f even 6 1
441.2.f.a 2 252.n even 6 1
441.2.f.b 2 28.g odd 6 1
441.2.f.b 2 252.bl odd 6 1
441.2.g.a 2 28.f even 6 1
441.2.g.a 2 252.bi even 6 1
441.2.h.a 2 28.d even 2 1
441.2.h.a 2 252.bj even 6 1
567.2.e.a 2 36.f odd 6 1
567.2.e.a 2 252.bl odd 6 1
567.2.e.b 2 36.h even 6 1
567.2.e.b 2 252.o even 6 1
1008.2.q.c 2 1.a even 1 1 trivial
1008.2.q.c 2 63.h even 3 1 inner
1008.2.t.d 2 7.c even 3 1
1008.2.t.d 2 9.c even 3 1
1323.2.f.a 2 84.n even 6 1
1323.2.f.a 2 252.o even 6 1
1323.2.f.b 2 84.j odd 6 1
1323.2.f.b 2 252.bn odd 6 1
1323.2.g.a 2 84.j odd 6 1
1323.2.g.a 2 252.s odd 6 1
1323.2.h.a 2 84.h odd 2 1
1323.2.h.a 2 252.r odd 6 1
3024.2.q.b 2 3.b odd 2 1
3024.2.q.b 2 63.j odd 6 1
3024.2.t.d 2 9.d odd 6 1
3024.2.t.d 2 21.h odd 6 1
3969.2.a.a 1 252.r odd 6 1
3969.2.a.c 1 252.bb even 6 1
3969.2.a.d 1 252.u odd 6 1
3969.2.a.f 1 252.bj even 6 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{2} - 5 T_{11} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \)
$43$ \( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \)
show more
show less