Properties

Label 1008.3.cg.b
Level $1008$
Weight $3$
Character orbit 1008.cg
Analytic conductor $27.466$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + ( - 3 \zeta_{6} + 3) q^{11} + (8 \zeta_{6} - 4) q^{13} + (10 \zeta_{6} + 10) q^{17} + (6 \zeta_{6} - 12) q^{19} - 36 \zeta_{6} q^{23} + (22 \zeta_{6} - 22) q^{25} + 51 q^{29} + (7 \zeta_{6} + 7) q^{31} + ( - 2 \zeta_{6} + 13) q^{35} - 22 \zeta_{6} q^{37} + ( - 28 \zeta_{6} + 14) q^{41} - 10 q^{43} + ( - 52 \zeta_{6} + 104) q^{47} + (39 \zeta_{6} + 16) q^{49} + ( - 51 \zeta_{6} + 51) q^{53} + (6 \zeta_{6} - 3) q^{55} + (43 \zeta_{6} + 43) q^{59} + ( - 40 \zeta_{6} + 80) q^{61} - 12 \zeta_{6} q^{65} + ( - 68 \zeta_{6} + 68) q^{67} + ( - 12 \zeta_{6} - 12) q^{73} + (15 \zeta_{6} - 24) q^{77} + 125 \zeta_{6} q^{79} + (178 \zeta_{6} - 89) q^{83} - 30 q^{85} + (42 \zeta_{6} - 84) q^{89} + ( - 52 \zeta_{6} + 44) q^{91} + ( - 18 \zeta_{6} + 18) q^{95} + ( - 170 \zeta_{6} + 85) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - 13 q^{7} + 3 q^{11} + 30 q^{17} - 18 q^{19} - 36 q^{23} - 22 q^{25} + 102 q^{29} + 21 q^{31} + 24 q^{35} - 22 q^{37} - 20 q^{43} + 156 q^{47} + 71 q^{49} + 51 q^{53} + 129 q^{59} + 120 q^{61} - 12 q^{65} + 68 q^{67} - 36 q^{73} - 33 q^{77} + 125 q^{79} - 60 q^{85} - 126 q^{89} + 36 q^{91} + 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −1.50000 0.866025i 0 −6.50000 + 2.59808i 0 0 0
577.1 0 0 0 −1.50000 + 0.866025i 0 −6.50000 2.59808i 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.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.b 2
3.b odd 2 1 336.3.bh.b 2
4.b odd 2 1 252.3.z.b 2
7.d odd 6 1 inner 1008.3.cg.b 2
12.b even 2 1 84.3.m.a 2
21.g even 6 1 336.3.bh.b 2
21.g even 6 1 2352.3.f.c 2
21.h odd 6 1 2352.3.f.c 2
28.d even 2 1 1764.3.z.e 2
28.f even 6 1 252.3.z.b 2
28.f even 6 1 1764.3.d.c 2
28.g odd 6 1 1764.3.d.c 2
28.g odd 6 1 1764.3.z.e 2
60.h even 2 1 2100.3.bd.b 2
60.l odd 4 2 2100.3.be.c 4
84.h odd 2 1 588.3.m.a 2
84.j odd 6 1 84.3.m.a 2
84.j odd 6 1 588.3.d.a 2
84.n even 6 1 588.3.d.a 2
84.n even 6 1 588.3.m.a 2
420.be odd 6 1 2100.3.bd.b 2
420.br even 12 2 2100.3.be.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.a 2 12.b even 2 1
84.3.m.a 2 84.j odd 6 1
252.3.z.b 2 4.b odd 2 1
252.3.z.b 2 28.f even 6 1
336.3.bh.b 2 3.b odd 2 1
336.3.bh.b 2 21.g even 6 1
588.3.d.a 2 84.j odd 6 1
588.3.d.a 2 84.n even 6 1
588.3.m.a 2 84.h odd 2 1
588.3.m.a 2 84.n even 6 1
1008.3.cg.b 2 1.a even 1 1 trivial
1008.3.cg.b 2 7.d odd 6 1 inner
1764.3.d.c 2 28.f even 6 1
1764.3.d.c 2 28.g odd 6 1
1764.3.z.e 2 28.d even 2 1
1764.3.z.e 2 28.g odd 6 1
2100.3.bd.b 2 60.h even 2 1
2100.3.bd.b 2 420.be odd 6 1
2100.3.be.c 4 60.l odd 4 2
2100.3.be.c 4 420.br even 12 2
2352.3.f.c 2 21.g even 6 1
2352.3.f.c 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} - 30T + 300 \) Copy content Toggle raw display
$19$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$23$ \( T^{2} + 36T + 1296 \) Copy content Toggle raw display
$29$ \( (T - 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 21T + 147 \) Copy content Toggle raw display
$37$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$41$ \( T^{2} + 588 \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 156T + 8112 \) Copy content Toggle raw display
$53$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$59$ \( T^{2} - 129T + 5547 \) Copy content Toggle raw display
$61$ \( T^{2} - 120T + 4800 \) Copy content Toggle raw display
$67$ \( T^{2} - 68T + 4624 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36T + 432 \) Copy content Toggle raw display
$79$ \( T^{2} - 125T + 15625 \) Copy content Toggle raw display
$83$ \( T^{2} + 23763 \) Copy content Toggle raw display
$89$ \( T^{2} + 126T + 5292 \) Copy content Toggle raw display
$97$ \( T^{2} + 21675 \) Copy content Toggle raw display
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