Properties

Label 1008.2.r.k
Level $1008$
Weight $2$
Character orbit 1008.r
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{5} ) q^{3} + ( 2 + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{5} ) q^{3} + ( 2 + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} + ( -1 + 3 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{11} + ( -1 - \beta_{4} ) q^{13} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{15} + ( -5 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{17} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{21} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{23} + ( 2 - \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{25} + ( -2 + 2 \beta_{1} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{27} + ( -\beta_{1} + \beta_{5} ) q^{29} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{31} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{35} + ( -3 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} ) q^{39} + ( 7 - \beta_{2} + 7 \beta_{4} + \beta_{5} ) q^{41} + ( 1 - 5 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{43} + ( -2 - 3 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{45} + ( 3 - 6 \beta_{1} + 3 \beta_{3} ) q^{47} + ( -1 - \beta_{4} ) q^{49} + ( -4 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} ) q^{51} + ( -7 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{53} + ( 1 - \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{55} + ( 1 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{57} + ( -4 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{59} + ( 2 - \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{61} + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{63} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} + ( 2 + 7 \beta_{1} + 6 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} ) q^{67} + ( -7 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{69} + ( -4 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{71} + ( 4 - 5 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} ) q^{73} + ( -8 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{75} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{77} + ( 1 + 4 \beta_{1} + \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{79} + ( 6 + 5 \beta_{1} + \beta_{2} - 2 \beta_{3} + 8 \beta_{4} - \beta_{5} ) q^{81} + ( -1 + 2 \beta_{1} - \beta_{3} + 5 \beta_{4} ) q^{83} + ( -5 - 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} ) q^{85} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{87} + ( 2 - \beta_{1} + 5 \beta_{2} + 7 \beta_{3} - \beta_{4} ) q^{89} + q^{91} + ( 5 - \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{93} + ( 5 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{95} + ( -2 + 7 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{97} + ( 4 + 2 \beta_{1} + 2 \beta_{3} - 8 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{3} + 5q^{5} - 3q^{7} - 4q^{9} + O(q^{10}) \) \( 6q + 4q^{3} + 5q^{5} - 3q^{7} - 4q^{9} - 2q^{11} - 3q^{13} - 11q^{15} - 24q^{17} + 6q^{19} - 2q^{21} - 6q^{25} + 7q^{27} - q^{29} - 3q^{31} + 8q^{33} - 10q^{35} - 6q^{37} - 2q^{39} + 22q^{41} - 3q^{43} + 5q^{45} - 9q^{47} - 3q^{49} - 9q^{51} - 36q^{53} + 12q^{55} + 11q^{57} - 9q^{59} + 6q^{61} + 8q^{63} + 5q^{65} - 39q^{69} - 18q^{71} + 6q^{73} - 31q^{75} - 2q^{77} + 15q^{79} + 32q^{81} - 12q^{83} - 9q^{85} + q^{87} - 4q^{89} + 6q^{91} + 33q^{93} + 16q^{95} - 3q^{97} + 46q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7\)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{4}\)

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 + 0.224437i
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0 −0.349814 1.69636i 0 1.79418 3.10761i 0 −0.500000 0.866025i 0 −2.75526 + 1.18682i 0
337.2 0 0.619562 + 1.61745i 0 −0.590972 + 1.02359i 0 −0.500000 0.866025i 0 −2.23229 + 2.00422i 0
337.3 0 1.73025 + 0.0789082i 0 1.29679 2.24611i 0 −0.500000 0.866025i 0 2.98755 + 0.273062i 0
673.1 0 −0.349814 + 1.69636i 0 1.79418 + 3.10761i 0 −0.500000 + 0.866025i 0 −2.75526 1.18682i 0
673.2 0 0.619562 1.61745i 0 −0.590972 1.02359i 0 −0.500000 + 0.866025i 0 −2.23229 2.00422i 0
673.3 0 1.73025 0.0789082i 0 1.29679 + 2.24611i 0 −0.500000 + 0.866025i 0 2.98755 0.273062i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.3
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.k 6
3.b odd 2 1 3024.2.r.g 6
4.b odd 2 1 63.2.f.b 6
9.c even 3 1 inner 1008.2.r.k 6
9.c even 3 1 9072.2.a.bq 3
9.d odd 6 1 3024.2.r.g 6
9.d odd 6 1 9072.2.a.cd 3
12.b even 2 1 189.2.f.a 6
28.d even 2 1 441.2.f.d 6
28.f even 6 1 441.2.g.d 6
28.f even 6 1 441.2.h.b 6
28.g odd 6 1 441.2.g.e 6
28.g odd 6 1 441.2.h.c 6
36.f odd 6 1 63.2.f.b 6
36.f odd 6 1 567.2.a.d 3
36.h even 6 1 189.2.f.a 6
36.h even 6 1 567.2.a.g 3
84.h odd 2 1 1323.2.f.c 6
84.j odd 6 1 1323.2.g.b 6
84.j odd 6 1 1323.2.h.e 6
84.n even 6 1 1323.2.g.c 6
84.n even 6 1 1323.2.h.d 6
252.n even 6 1 441.2.h.b 6
252.o even 6 1 1323.2.h.d 6
252.r odd 6 1 1323.2.g.b 6
252.s odd 6 1 1323.2.f.c 6
252.s odd 6 1 3969.2.a.p 3
252.u odd 6 1 441.2.g.e 6
252.bb even 6 1 1323.2.g.c 6
252.bi even 6 1 441.2.f.d 6
252.bi even 6 1 3969.2.a.m 3
252.bj even 6 1 441.2.g.d 6
252.bl odd 6 1 441.2.h.c 6
252.bn odd 6 1 1323.2.h.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 4.b odd 2 1
63.2.f.b 6 36.f odd 6 1
189.2.f.a 6 12.b even 2 1
189.2.f.a 6 36.h even 6 1
441.2.f.d 6 28.d even 2 1
441.2.f.d 6 252.bi even 6 1
441.2.g.d 6 28.f even 6 1
441.2.g.d 6 252.bj even 6 1
441.2.g.e 6 28.g odd 6 1
441.2.g.e 6 252.u odd 6 1
441.2.h.b 6 28.f even 6 1
441.2.h.b 6 252.n even 6 1
441.2.h.c 6 28.g odd 6 1
441.2.h.c 6 252.bl odd 6 1
567.2.a.d 3 36.f odd 6 1
567.2.a.g 3 36.h even 6 1
1008.2.r.k 6 1.a even 1 1 trivial
1008.2.r.k 6 9.c even 3 1 inner
1323.2.f.c 6 84.h odd 2 1
1323.2.f.c 6 252.s odd 6 1
1323.2.g.b 6 84.j odd 6 1
1323.2.g.b 6 252.r odd 6 1
1323.2.g.c 6 84.n even 6 1
1323.2.g.c 6 252.bb even 6 1
1323.2.h.d 6 84.n even 6 1
1323.2.h.d 6 252.o even 6 1
1323.2.h.e 6 84.j odd 6 1
1323.2.h.e 6 252.bn odd 6 1
3024.2.r.g 6 3.b odd 2 1
3024.2.r.g 6 9.d odd 6 1
3969.2.a.m 3 252.bi even 6 1
3969.2.a.p 3 252.s odd 6 1
9072.2.a.bq 3 9.c even 3 1
9072.2.a.cd 3 9.d odd 6 1

Hecke kernels

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

\( T_{5}^{6} - 5 T_{5}^{5} + 23 T_{5}^{4} - 32 T_{5}^{3} + 59 T_{5}^{2} + 22 T_{5} + 121 \)
\( T_{11}^{6} + 2 T_{11}^{5} + 23 T_{11}^{4} + 56 T_{11}^{3} + 455 T_{11}^{2} + 893 T_{11} + 2209 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 27 - 36 T + 30 T^{2} - 21 T^{3} + 10 T^{4} - 4 T^{5} + T^{6} \)
$5$ \( 121 + 22 T + 59 T^{2} - 32 T^{3} + 23 T^{4} - 5 T^{5} + T^{6} \)
$7$ \( ( 1 + T + T^{2} )^{3} \)
$11$ \( 2209 + 893 T + 455 T^{2} + 56 T^{3} + 23 T^{4} + 2 T^{5} + T^{6} \)
$13$ \( ( 1 + T + T^{2} )^{3} \)
$17$ \( ( 27 + 39 T + 12 T^{2} + T^{3} )^{2} \)
$19$ \( ( 7 - 6 T - 3 T^{2} + T^{3} )^{2} \)
$23$ \( 81 + 297 T + 1089 T^{2} + 18 T^{3} + 33 T^{4} + T^{6} \)
$29$ \( 1 + 4 T + 17 T^{2} - 2 T^{3} + 5 T^{4} + T^{5} + T^{6} \)
$31$ \( 729 - 648 T + 495 T^{2} - 126 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} \)
$37$ \( ( 81 - 54 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( 124609 - 54715 T + 16259 T^{2} - 2704 T^{3} + 329 T^{4} - 22 T^{5} + T^{6} \)
$43$ \( 14641 - 7986 T + 3993 T^{2} - 440 T^{3} + 75 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( 35721 + 10206 T + 4617 T^{2} - 108 T^{3} + 135 T^{4} + 9 T^{5} + T^{6} \)
$53$ \( ( 9 + 75 T + 18 T^{2} + T^{3} )^{2} \)
$59$ \( 3969 + 378 T + 603 T^{2} + 72 T^{3} + 87 T^{4} + 9 T^{5} + T^{6} \)
$61$ \( 4489 - 1407 T + 843 T^{2} - 8 T^{3} + 57 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 466489 - 141381 T + 42849 T^{2} - 1366 T^{3} + 207 T^{4} + T^{6} \)
$71$ \( ( -81 - 6 T + 9 T^{2} + T^{3} )^{2} \)
$73$ \( ( -243 - 168 T - 3 T^{2} + T^{3} )^{2} \)
$79$ \( 591361 - 36912 T + 13839 T^{2} - 818 T^{3} + 273 T^{4} - 15 T^{5} + T^{6} \)
$83$ \( 729 + 1053 T + 1197 T^{2} + 414 T^{3} + 105 T^{4} + 12 T^{5} + T^{6} \)
$89$ \( ( 379 - 151 T + 2 T^{2} + T^{3} )^{2} \)
$97$ \( 363609 + 68742 T + 14805 T^{2} + 864 T^{3} + 123 T^{4} + 3 T^{5} + T^{6} \)
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