Properties

Label 1008.2.s.e
Level $1008$
Weight $2$
Character orbit 1008.s
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.s (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 56)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( 1 - \zeta_{6} ) q^{11} + 2 q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + 6 q^{29} + ( -1 + \zeta_{6} ) q^{31} + ( 2 + \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + 10 q^{41} + 4 q^{43} -\zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -9 + 9 \zeta_{6} ) q^{53} - q^{55} + ( -3 + 3 \zeta_{6} ) q^{59} -3 \zeta_{6} q^{61} -2 \zeta_{6} q^{65} + ( 11 - 11 \zeta_{6} ) q^{67} + 16 q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} + ( -1 + 3 \zeta_{6} ) q^{77} -11 \zeta_{6} q^{79} -4 q^{83} -3 q^{85} -9 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} + ( 5 - 5 \zeta_{6} ) q^{95} + 6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} - 4q^{7} + O(q^{10}) \) \( 2q - q^{5} - 4q^{7} + q^{11} + 4q^{13} + 3q^{17} + 5q^{19} + 3q^{23} + 4q^{25} + 12q^{29} - q^{31} + 5q^{35} + 5q^{37} + 20q^{41} + 8q^{43} - q^{47} + 2q^{49} - 9q^{53} - 2q^{55} - 3q^{59} - 3q^{61} - 2q^{65} + 11q^{67} + 32q^{71} - 7q^{73} + q^{77} - 11q^{79} - 8q^{83} - 6q^{85} - 9q^{89} - 8q^{91} + 5q^{95} + 12q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 −2.00000 1.73205i 0 0 0
865.1 0 0 0 −0.500000 0.866025i 0 −2.00000 + 1.73205i 0 0 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
7.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.s.e 2
3.b odd 2 1 112.2.i.c 2
4.b odd 2 1 504.2.s.e 2
7.c even 3 1 inner 1008.2.s.e 2
7.c even 3 1 7056.2.a.bi 1
7.d odd 6 1 7056.2.a.s 1
12.b even 2 1 56.2.i.a 2
21.c even 2 1 784.2.i.a 2
21.g even 6 1 784.2.a.j 1
21.g even 6 1 784.2.i.a 2
21.h odd 6 1 112.2.i.c 2
21.h odd 6 1 784.2.a.a 1
24.f even 2 1 448.2.i.f 2
24.h odd 2 1 448.2.i.a 2
28.d even 2 1 3528.2.s.o 2
28.f even 6 1 3528.2.a.k 1
28.f even 6 1 3528.2.s.o 2
28.g odd 6 1 504.2.s.e 2
28.g odd 6 1 3528.2.a.r 1
60.h even 2 1 1400.2.q.g 2
60.l odd 4 2 1400.2.bh.f 4
84.h odd 2 1 392.2.i.f 2
84.j odd 6 1 392.2.a.a 1
84.j odd 6 1 392.2.i.f 2
84.n even 6 1 56.2.i.a 2
84.n even 6 1 392.2.a.f 1
168.s odd 6 1 448.2.i.a 2
168.s odd 6 1 3136.2.a.bc 1
168.v even 6 1 448.2.i.f 2
168.v even 6 1 3136.2.a.b 1
168.ba even 6 1 3136.2.a.a 1
168.be odd 6 1 3136.2.a.bb 1
420.ba even 6 1 1400.2.q.g 2
420.ba even 6 1 9800.2.a.b 1
420.be odd 6 1 9800.2.a.bp 1
420.bp odd 12 2 1400.2.bh.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 12.b even 2 1
56.2.i.a 2 84.n even 6 1
112.2.i.c 2 3.b odd 2 1
112.2.i.c 2 21.h odd 6 1
392.2.a.a 1 84.j odd 6 1
392.2.a.f 1 84.n even 6 1
392.2.i.f 2 84.h odd 2 1
392.2.i.f 2 84.j odd 6 1
448.2.i.a 2 24.h odd 2 1
448.2.i.a 2 168.s odd 6 1
448.2.i.f 2 24.f even 2 1
448.2.i.f 2 168.v even 6 1
504.2.s.e 2 4.b odd 2 1
504.2.s.e 2 28.g odd 6 1
784.2.a.a 1 21.h odd 6 1
784.2.a.j 1 21.g even 6 1
784.2.i.a 2 21.c even 2 1
784.2.i.a 2 21.g even 6 1
1008.2.s.e 2 1.a even 1 1 trivial
1008.2.s.e 2 7.c even 3 1 inner
1400.2.q.g 2 60.h even 2 1
1400.2.q.g 2 420.ba even 6 1
1400.2.bh.f 4 60.l odd 4 2
1400.2.bh.f 4 420.bp odd 12 2
3136.2.a.a 1 168.ba even 6 1
3136.2.a.b 1 168.v even 6 1
3136.2.a.bb 1 168.be odd 6 1
3136.2.a.bc 1 168.s odd 6 1
3528.2.a.k 1 28.f even 6 1
3528.2.a.r 1 28.g odd 6 1
3528.2.s.o 2 28.d even 2 1
3528.2.s.o 2 28.f even 6 1
7056.2.a.s 1 7.d odd 6 1
7056.2.a.bi 1 7.c even 3 1
9800.2.a.b 1 420.ba even 6 1
9800.2.a.bp 1 420.be odd 6 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{13} - 2 \)
\( T_{17}^{2} - 3 T_{17} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \)
$37$ \( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 6 T + 97 T^{2} )^{2} \)
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