Properties

Label 1008.3.cg.i
Level $1008$
Weight $3$
Character orbit 1008.cg
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + 7 \beta_{1} q^{7} +O(q^{10})\) \( q + \beta_{3} q^{5} + 7 \beta_{1} q^{7} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 1 + 2 \beta_{1} ) q^{13} + ( \beta_{2} - \beta_{3} ) q^{17} + ( -34 - 17 \beta_{1} ) q^{19} + ( 2 \beta_{2} - \beta_{3} ) q^{23} + ( -1 - \beta_{1} ) q^{25} + ( 4 \beta_{2} - 8 \beta_{3} ) q^{29} + ( 7 - 7 \beta_{1} ) q^{31} + ( 7 \beta_{2} - 7 \beta_{3} ) q^{35} -47 \beta_{1} q^{37} -14 \beta_{2} q^{41} -31 q^{43} + 17 \beta_{3} q^{47} + ( -49 - 49 \beta_{1} ) q^{49} + ( 9 \beta_{2} + 9 \beta_{3} ) q^{53} + ( -48 - 96 \beta_{1} ) q^{55} + ( -17 \beta_{2} + 17 \beta_{3} ) q^{59} + ( -96 - 48 \beta_{1} ) q^{61} + ( 2 \beta_{2} - \beta_{3} ) q^{65} + ( -31 - 31 \beta_{1} ) q^{67} + ( 7 \beta_{2} - 14 \beta_{3} ) q^{71} + ( -47 + 47 \beta_{1} ) q^{73} + ( -14 \beta_{2} + 28 \beta_{3} ) q^{77} -41 \beta_{1} q^{79} -\beta_{2} q^{83} -24 q^{85} + 12 \beta_{3} q^{89} + ( -14 - 7 \beta_{1} ) q^{91} + ( -17 \beta_{2} - 17 \beta_{3} ) q^{95} + ( -24 - 48 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{7} + O(q^{10}) \) \( 4 q - 14 q^{7} - 102 q^{19} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 124 q^{43} - 98 q^{49} - 288 q^{61} - 62 q^{67} - 282 q^{73} + 82 q^{79} - 96 q^{85} - 42 q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/6\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + \beta_{2}\)\()/3\)

Character values

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

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

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 0 0 −4.24264 2.44949i 0 −3.50000 + 6.06218i 0 0 0
145.2 0 0 0 4.24264 + 2.44949i 0 −3.50000 + 6.06218i 0 0 0
577.1 0 0 0 −4.24264 + 2.44949i 0 −3.50000 6.06218i 0 0 0
577.2 0 0 0 4.24264 2.44949i 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 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.i 4
3.b odd 2 1 inner 1008.3.cg.i 4
4.b odd 2 1 126.3.n.b 4
7.d odd 6 1 inner 1008.3.cg.i 4
12.b even 2 1 126.3.n.b 4
21.g even 6 1 inner 1008.3.cg.i 4
28.d even 2 1 882.3.n.c 4
28.f even 6 1 126.3.n.b 4
28.f even 6 1 882.3.c.c 4
28.g odd 6 1 882.3.c.c 4
28.g odd 6 1 882.3.n.c 4
84.h odd 2 1 882.3.n.c 4
84.j odd 6 1 126.3.n.b 4
84.j odd 6 1 882.3.c.c 4
84.n even 6 1 882.3.c.c 4
84.n even 6 1 882.3.n.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.n.b 4 4.b odd 2 1
126.3.n.b 4 12.b even 2 1
126.3.n.b 4 28.f even 6 1
126.3.n.b 4 84.j odd 6 1
882.3.c.c 4 28.f even 6 1
882.3.c.c 4 28.g odd 6 1
882.3.c.c 4 84.j odd 6 1
882.3.c.c 4 84.n even 6 1
882.3.n.c 4 28.d even 2 1
882.3.n.c 4 28.g odd 6 1
882.3.n.c 4 84.h odd 2 1
882.3.n.c 4 84.n even 6 1
1008.3.cg.i 4 1.a even 1 1 trivial
1008.3.cg.i 4 3.b odd 2 1 inner
1008.3.cg.i 4 7.d odd 6 1 inner
1008.3.cg.i 4 21.g even 6 1 inner

Hecke kernels

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

\( T_{5}^{4} - 24 T_{5}^{2} + 576 \)
\( T_{11}^{4} + 288 T_{11}^{2} + 82944 \)
\( T_{13}^{2} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 576 - 24 T^{2} + T^{4} \)
$7$ \( ( 49 + 7 T + T^{2} )^{2} \)
$11$ \( 82944 + 288 T^{2} + T^{4} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( 576 - 24 T^{2} + T^{4} \)
$19$ \( ( 867 + 51 T + T^{2} )^{2} \)
$23$ \( 5184 + 72 T^{2} + T^{4} \)
$29$ \( ( -1152 + T^{2} )^{2} \)
$31$ \( ( 147 - 21 T + T^{2} )^{2} \)
$37$ \( ( 2209 - 47 T + T^{2} )^{2} \)
$41$ \( ( 4704 + T^{2} )^{2} \)
$43$ \( ( 31 + T )^{4} \)
$47$ \( 48108096 - 6936 T^{2} + T^{4} \)
$53$ \( 34012224 + 5832 T^{2} + T^{4} \)
$59$ \( 48108096 - 6936 T^{2} + T^{4} \)
$61$ \( ( 6912 + 144 T + T^{2} )^{2} \)
$67$ \( ( 961 + 31 T + T^{2} )^{2} \)
$71$ \( ( -3528 + T^{2} )^{2} \)
$73$ \( ( 6627 + 141 T + T^{2} )^{2} \)
$79$ \( ( 1681 - 41 T + T^{2} )^{2} \)
$83$ \( ( 24 + T^{2} )^{2} \)
$89$ \( 11943936 - 3456 T^{2} + T^{4} \)
$97$ \( ( 1728 + T^{2} )^{2} \)
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