Properties

Label 1008.2.r.f
Level $1008$
Weight $2$
Character orbit 1008.r
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{3} ) q^{3} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{9} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} -2 \beta_{2} q^{13} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{15} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} -5 q^{19} + ( 1 - \beta_{1} + \beta_{3} ) q^{21} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{23} + ( -7 + 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{25} + ( 5 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{29} + 2 \beta_{2} q^{31} + ( -1 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{33} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + 2 q^{37} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( -1 + 2 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{41} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{43} + ( 1 - 4 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{45} -\beta_{2} q^{49} + ( 5 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + 6 q^{55} + ( -5 \beta_{2} - 5 \beta_{3} ) q^{57} + ( 3 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( 4 + 3 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} ) q^{61} + ( -2 \beta_{2} + \beta_{3} ) q^{63} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 3 - 6 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{67} + ( 7 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{69} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -13 + 4 \beta_{1} + 15 \beta_{2} - 7 \beta_{3} ) q^{75} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( 2 + 3 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{79} + ( -\beta_{2} + 5 \beta_{3} ) q^{81} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{83} -6 \beta_{2} q^{85} + ( 8 - 2 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{87} + ( 8 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} -2 q^{91} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -5 + 10 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{95} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -8 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + O(q^{10}) \) \( 4q + q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + 3q^{11} - 4q^{13} - 15q^{15} - 6q^{17} - 20q^{19} + 2q^{21} - 9q^{23} - 11q^{25} + 16q^{27} + 6q^{29} + 4q^{31} + 3q^{33} - 6q^{35} + 8q^{37} + 2q^{39} - 15q^{41} + q^{43} - 15q^{45} - 2q^{49} + 15q^{51} - 12q^{53} + 24q^{55} - 5q^{57} - 3q^{59} + 11q^{61} - 5q^{63} - 6q^{65} + 13q^{67} + 21q^{69} - 6q^{71} + 14q^{73} - 11q^{75} - 3q^{77} + 7q^{79} - 7q^{81} + 12q^{83} - 12q^{85} + 6q^{87} + 36q^{89} - 8q^{91} - 2q^{93} + 15q^{95} - q^{97} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 −1.18614 1.26217i 0 0.686141 1.18843i 0 0.500000 + 0.866025i 0 −0.186141 + 2.99422i 0
337.2 0 1.68614 + 0.396143i 0 −2.18614 + 3.78651i 0 0.500000 + 0.866025i 0 2.68614 + 1.33591i 0
673.1 0 −1.18614 + 1.26217i 0 0.686141 + 1.18843i 0 0.500000 0.866025i 0 −0.186141 2.99422i 0
673.2 0 1.68614 0.396143i 0 −2.18614 3.78651i 0 0.500000 0.866025i 0 2.68614 1.33591i 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.f 4
3.b odd 2 1 3024.2.r.f 4
4.b odd 2 1 126.2.f.d 4
9.c even 3 1 inner 1008.2.r.f 4
9.c even 3 1 9072.2.a.bm 2
9.d odd 6 1 3024.2.r.f 4
9.d odd 6 1 9072.2.a.bb 2
12.b even 2 1 378.2.f.c 4
28.d even 2 1 882.2.f.k 4
28.f even 6 1 882.2.e.k 4
28.f even 6 1 882.2.h.n 4
28.g odd 6 1 882.2.e.l 4
28.g odd 6 1 882.2.h.m 4
36.f odd 6 1 126.2.f.d 4
36.f odd 6 1 1134.2.a.k 2
36.h even 6 1 378.2.f.c 4
36.h even 6 1 1134.2.a.n 2
84.h odd 2 1 2646.2.f.j 4
84.j odd 6 1 2646.2.e.m 4
84.j odd 6 1 2646.2.h.l 4
84.n even 6 1 2646.2.e.n 4
84.n even 6 1 2646.2.h.k 4
252.n even 6 1 882.2.e.k 4
252.o even 6 1 2646.2.e.n 4
252.r odd 6 1 2646.2.h.l 4
252.s odd 6 1 2646.2.f.j 4
252.s odd 6 1 7938.2.a.bs 2
252.u odd 6 1 882.2.h.m 4
252.bb even 6 1 2646.2.h.k 4
252.bi even 6 1 882.2.f.k 4
252.bi even 6 1 7938.2.a.bh 2
252.bj even 6 1 882.2.h.n 4
252.bl odd 6 1 882.2.e.l 4
252.bn odd 6 1 2646.2.e.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 4.b odd 2 1
126.2.f.d 4 36.f odd 6 1
378.2.f.c 4 12.b even 2 1
378.2.f.c 4 36.h even 6 1
882.2.e.k 4 28.f even 6 1
882.2.e.k 4 252.n even 6 1
882.2.e.l 4 28.g odd 6 1
882.2.e.l 4 252.bl odd 6 1
882.2.f.k 4 28.d even 2 1
882.2.f.k 4 252.bi even 6 1
882.2.h.m 4 28.g odd 6 1
882.2.h.m 4 252.u odd 6 1
882.2.h.n 4 28.f even 6 1
882.2.h.n 4 252.bj even 6 1
1008.2.r.f 4 1.a even 1 1 trivial
1008.2.r.f 4 9.c even 3 1 inner
1134.2.a.k 2 36.f odd 6 1
1134.2.a.n 2 36.h even 6 1
2646.2.e.m 4 84.j odd 6 1
2646.2.e.m 4 252.bn odd 6 1
2646.2.e.n 4 84.n even 6 1
2646.2.e.n 4 252.o even 6 1
2646.2.f.j 4 84.h odd 2 1
2646.2.f.j 4 252.s odd 6 1
2646.2.h.k 4 84.n even 6 1
2646.2.h.k 4 252.bb even 6 1
2646.2.h.l 4 84.j odd 6 1
2646.2.h.l 4 252.r odd 6 1
3024.2.r.f 4 3.b odd 2 1
3024.2.r.f 4 9.d odd 6 1
7938.2.a.bh 2 252.bi even 6 1
7938.2.a.bs 2 252.s odd 6 1
9072.2.a.bb 2 9.d odd 6 1
9072.2.a.bm 2 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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} + 3 T_{5}^{3} + 15 T_{5}^{2} - 18 T_{5} + 36 \)
\( T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 3 T - 2 T^{2} - T^{3} + T^{4} \)
$5$ \( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( ( -6 + 3 T + T^{2} )^{2} \)
$19$ \( ( 5 + T )^{4} \)
$23$ \( 144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4} \)
$29$ \( 576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( ( 4 - 2 T + T^{2} )^{2} \)
$37$ \( ( -2 + T )^{4} \)
$41$ \( 2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4} \)
$43$ \( 5476 + 74 T + 75 T^{2} - T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -24 + 6 T + T^{2} )^{2} \)
$59$ \( 5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4} \)
$61$ \( 1936 + 484 T + 165 T^{2} - 11 T^{3} + T^{4} \)
$67$ \( 1024 + 416 T + 201 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( ( -72 + 3 T + T^{2} )^{2} \)
$73$ \( ( -62 - 7 T + T^{2} )^{2} \)
$79$ \( 3844 + 434 T + 111 T^{2} - 7 T^{3} + T^{4} \)
$83$ \( 9216 + 1152 T + 240 T^{2} - 12 T^{3} + T^{4} \)
$89$ \( ( 48 - 18 T + T^{2} )^{2} \)
$97$ \( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} \)
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