Properties

Label 1008.2.r.c
Level 1008
Weight 2
Character orbit 1008.r
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.r (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 504)
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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} -3 q^{9} + ( -6 + 6 \zeta_{6} ) q^{11} -6 \zeta_{6} q^{13} + ( -2 + \zeta_{6} ) q^{15} -2 q^{17} -7 q^{19} + ( 1 + \zeta_{6} ) q^{21} -\zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -2 + 2 \zeta_{6} ) q^{29} + 10 \zeta_{6} q^{31} + ( -6 - 6 \zeta_{6} ) q^{33} + q^{35} -6 q^{37} + ( 12 - 6 \zeta_{6} ) q^{39} + 8 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{45} + ( 8 - 8 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + ( 2 - 4 \zeta_{6} ) q^{51} + 2 q^{53} -6 q^{55} + ( 7 - 14 \zeta_{6} ) q^{57} + ( -7 + 7 \zeta_{6} ) q^{61} + ( -3 + 3 \zeta_{6} ) q^{63} + ( 6 - 6 \zeta_{6} ) q^{65} -12 \zeta_{6} q^{67} + ( 2 - \zeta_{6} ) q^{69} -15 q^{71} -2 q^{73} + ( 4 + 4 \zeta_{6} ) q^{75} + 6 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + 9 q^{81} + ( 12 - 12 \zeta_{6} ) q^{83} -2 \zeta_{6} q^{85} + ( -2 - 2 \zeta_{6} ) q^{87} + 4 q^{89} -6 q^{91} + ( -20 + 10 \zeta_{6} ) q^{93} -7 \zeta_{6} q^{95} + ( 2 - 2 \zeta_{6} ) q^{97} + ( 18 - 18 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + q^{5} + q^{7} - 6q^{9} - 6q^{11} - 6q^{13} - 3q^{15} - 4q^{17} - 14q^{19} + 3q^{21} - q^{23} + 4q^{25} - 2q^{29} + 10q^{31} - 18q^{33} + 2q^{35} - 12q^{37} + 18q^{39} + 8q^{41} - 10q^{43} - 3q^{45} + 8q^{47} - q^{49} + 4q^{53} - 12q^{55} - 7q^{61} - 3q^{63} + 6q^{65} - 12q^{67} + 3q^{69} - 30q^{71} - 4q^{73} + 12q^{75} + 6q^{77} + q^{79} + 18q^{81} + 12q^{83} - 2q^{85} - 6q^{87} + 8q^{89} - 12q^{91} - 30q^{93} - 7q^{95} + 2q^{97} + 18q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −3.00000 0
673.1 0 1.73205i 0 0.500000 + 0.866025i 0 0.500000 0.866025i 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
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.c 2
3.b odd 2 1 3024.2.r.b 2
4.b odd 2 1 504.2.r.a 2
9.c even 3 1 inner 1008.2.r.c 2
9.c even 3 1 9072.2.a.i 1
9.d odd 6 1 3024.2.r.b 2
9.d odd 6 1 9072.2.a.n 1
12.b even 2 1 1512.2.r.a 2
36.f odd 6 1 504.2.r.a 2
36.f odd 6 1 4536.2.a.d 1
36.h even 6 1 1512.2.r.a 2
36.h even 6 1 4536.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.a 2 4.b odd 2 1
504.2.r.a 2 36.f odd 6 1
1008.2.r.c 2 1.a even 1 1 trivial
1008.2.r.c 2 9.c even 3 1 inner
1512.2.r.a 2 12.b even 2 1
1512.2.r.a 2 36.h even 6 1
3024.2.r.b 2 3.b odd 2 1
3024.2.r.b 2 9.d odd 6 1
4536.2.a.d 1 36.f odd 6 1
4536.2.a.g 1 36.h even 6 1
9072.2.a.i 1 9.c even 3 1
9072.2.a.n 1 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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{2} + 6 T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \)
$29$ \( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \)
$31$ \( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 8 T + 23 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 15 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 4 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \)
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