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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + 7 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + 7 \beta_1 q^{7} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{11} + (2 \beta_1 + 1) q^{13} + ( - \beta_{3} + \beta_{2}) q^{17} + ( - 17 \beta_1 - 34) q^{19} + ( - \beta_{3} + 2 \beta_{2}) q^{23} + ( - \beta_1 - 1) q^{25} + ( - 8 \beta_{3} + 4 \beta_{2}) q^{29} + ( - 7 \beta_1 + 7) q^{31} + ( - 7 \beta_{3} + 7 \beta_{2}) q^{35} - 47 \beta_1 q^{37} - 14 \beta_{2} q^{41} - 31 q^{43} + 17 \beta_{3} q^{47} + ( - 49 \beta_1 - 49) q^{49} + (9 \beta_{3} + 9 \beta_{2}) q^{53} + ( - 96 \beta_1 - 48) q^{55} + (17 \beta_{3} - 17 \beta_{2}) q^{59} + ( - 48 \beta_1 - 96) q^{61} + ( - \beta_{3} + 2 \beta_{2}) q^{65} + ( - 31 \beta_1 - 31) q^{67} + ( - 14 \beta_{3} + 7 \beta_{2}) q^{71} + (47 \beta_1 - 47) q^{73} + (28 \beta_{3} - 14 \beta_{2}) q^{77} - 41 \beta_1 q^{79} - \beta_{2} q^{83} - 24 q^{85} + 12 \beta_{3} q^{89} + ( - 7 \beta_1 - 14) q^{91} + ( - 17 \beta_{3} - 17 \beta_{2}) q^{95} + ( - 48 \beta_1 - 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{7} - 102 q^{19} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 124 q^{43} - 98 q^{49} - 288 q^{61} - 62 q^{67} - 282 q^{73} + 82 q^{79} - 96 q^{85} - 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} - 24T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{11}^{4} + 288T_{11}^{2} + 82944 \) Copy content Toggle raw display
\( T_{13}^{2} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

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