Properties

Label 1008.3.cg.f
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 4) q^{5} + 7 \zeta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 4) q^{5} + 7 \zeta_{6} q^{7} + (10 \zeta_{6} - 10) q^{11} + (14 \zeta_{6} - 7) q^{13} + (4 \zeta_{6} + 4) q^{17} + (19 \zeta_{6} - 38) q^{19} - 40 \zeta_{6} q^{23} + (13 \zeta_{6} - 13) q^{25} - 16 q^{29} + ( - 3 \zeta_{6} - 3) q^{31} + (14 \zeta_{6} + 14) q^{35} - 5 \zeta_{6} q^{37} + ( - 28 \zeta_{6} + 14) q^{41} + 19 q^{43} + (30 \zeta_{6} - 60) q^{47} + (49 \zeta_{6} - 49) q^{49} + (32 \zeta_{6} - 32) q^{53} + (40 \zeta_{6} - 20) q^{55} + (24 \zeta_{6} + 24) q^{59} + ( - 12 \zeta_{6} + 24) q^{61} + 42 \zeta_{6} q^{65} + ( - 59 \zeta_{6} + 59) q^{67} - 26 q^{71} + ( - 11 \zeta_{6} - 11) q^{73} - 70 q^{77} + 47 \zeta_{6} q^{79} + ( - 28 \zeta_{6} + 14) q^{83} + 24 q^{85} + (68 \zeta_{6} - 136) q^{89} + (49 \zeta_{6} - 98) q^{91} + (114 \zeta_{6} - 114) q^{95} + ( - 56 \zeta_{6} + 28) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} + 7 q^{7} - 10 q^{11} + 12 q^{17} - 57 q^{19} - 40 q^{23} - 13 q^{25} - 32 q^{29} - 9 q^{31} + 42 q^{35} - 5 q^{37} + 38 q^{43} - 90 q^{47} - 49 q^{49} - 32 q^{53} + 72 q^{59} + 36 q^{61} + 42 q^{65} + 59 q^{67} - 52 q^{71} - 33 q^{73} - 140 q^{77} + 47 q^{79} + 48 q^{85} - 204 q^{89} - 147 q^{91} - 114 q^{95}+O(q^{100}) \) 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 3.00000 + 1.73205i 0 3.50000 6.06218i 0 0 0
577.1 0 0 0 3.00000 1.73205i 0 3.50000 + 6.06218i 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.f 2
3.b odd 2 1 336.3.bh.c 2
4.b odd 2 1 63.3.m.a 2
7.d odd 6 1 inner 1008.3.cg.f 2
12.b even 2 1 21.3.f.c 2
21.g even 6 1 336.3.bh.c 2
21.g even 6 1 2352.3.f.b 2
21.h odd 6 1 2352.3.f.b 2
28.d even 2 1 441.3.m.b 2
28.f even 6 1 63.3.m.a 2
28.f even 6 1 441.3.d.d 2
28.g odd 6 1 441.3.d.d 2
28.g odd 6 1 441.3.m.b 2
60.h even 2 1 525.3.o.b 2
60.l odd 4 2 525.3.s.d 4
84.h odd 2 1 147.3.f.e 2
84.j odd 6 1 21.3.f.c 2
84.j odd 6 1 147.3.d.a 2
84.n even 6 1 147.3.d.a 2
84.n even 6 1 147.3.f.e 2
420.be odd 6 1 525.3.o.b 2
420.br even 12 2 525.3.s.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.c 2 12.b even 2 1
21.3.f.c 2 84.j odd 6 1
63.3.m.a 2 4.b odd 2 1
63.3.m.a 2 28.f even 6 1
147.3.d.a 2 84.j odd 6 1
147.3.d.a 2 84.n even 6 1
147.3.f.e 2 84.h odd 2 1
147.3.f.e 2 84.n even 6 1
336.3.bh.c 2 3.b odd 2 1
336.3.bh.c 2 21.g even 6 1
441.3.d.d 2 28.f even 6 1
441.3.d.d 2 28.g odd 6 1
441.3.m.b 2 28.d even 2 1
441.3.m.b 2 28.g odd 6 1
525.3.o.b 2 60.h even 2 1
525.3.o.b 2 420.be odd 6 1
525.3.s.d 4 60.l odd 4 2
525.3.s.d 4 420.br even 12 2
1008.3.cg.f 2 1.a even 1 1 trivial
1008.3.cg.f 2 7.d odd 6 1 inner
2352.3.f.b 2 21.g even 6 1
2352.3.f.b 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 \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{2} - 6T_{5} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} + 10T_{11} + 100 \) Copy content Toggle raw display
\( T_{13}^{2} + 147 \) 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
$5$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$13$ \( T^{2} + 147 \) Copy content Toggle raw display
$17$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$19$ \( T^{2} + 57T + 1083 \) Copy content Toggle raw display
$23$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$29$ \( (T + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$37$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$41$ \( T^{2} + 588 \) Copy content Toggle raw display
$43$ \( (T - 19)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 90T + 2700 \) Copy content Toggle raw display
$53$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$59$ \( T^{2} - 72T + 1728 \) Copy content Toggle raw display
$61$ \( T^{2} - 36T + 432 \) Copy content Toggle raw display
$67$ \( T^{2} - 59T + 3481 \) Copy content Toggle raw display
$71$ \( (T + 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 33T + 363 \) Copy content Toggle raw display
$79$ \( T^{2} - 47T + 2209 \) Copy content Toggle raw display
$83$ \( T^{2} + 588 \) Copy content Toggle raw display
$89$ \( T^{2} + 204T + 13872 \) Copy content Toggle raw display
$97$ \( T^{2} + 2352 \) Copy content Toggle raw display
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