Properties

Label 1008.4.k.b
Level $1008$
Weight $4$
Character orbit 1008.k
Analytic conductor $59.474$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{111})\)
Defining polynomial: \(x^{4} + 112 x^{2} + 3025\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( 11 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{3} q^{5} + ( 11 + \beta_{2} ) q^{7} -11 \beta_{1} q^{11} + 2 \beta_{2} q^{13} -3 \beta_{3} q^{17} + 6 \beta_{2} q^{19} + 55 \beta_{1} q^{23} + 319 q^{25} + 89 \beta_{1} q^{29} + 16 \beta_{2} q^{31} + ( 222 \beta_{1} + 11 \beta_{3} ) q^{35} -184 q^{37} -5 \beta_{3} q^{41} + 190 q^{43} -2 \beta_{3} q^{47} + ( -101 + 22 \beta_{2} ) q^{49} -253 \beta_{1} q^{53} -22 \beta_{2} q^{55} -4 \beta_{3} q^{59} -44 \beta_{2} q^{61} + 444 \beta_{1} q^{65} -296 q^{67} -233 \beta_{1} q^{71} + 54 \beta_{2} q^{73} + ( -121 \beta_{1} + 11 \beta_{3} ) q^{77} -836 q^{79} + 58 \beta_{3} q^{83} -1332 q^{85} + 33 \beta_{3} q^{89} + ( -444 + 22 \beta_{2} ) q^{91} + 1332 \beta_{1} q^{95} -38 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 44q^{7} + O(q^{10}) \) \( 4q + 44q^{7} + 1276q^{25} - 736q^{37} + 760q^{43} - 404q^{49} - 1184q^{67} - 3344q^{79} - 5328q^{85} - 1776q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 112 x^{2} + 3025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 57 \nu \)\()/55\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 167 \nu \)\()/55\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 112 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 112\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-57 \beta_{2} + 167 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
8.15694i
8.15694i
6.74273i
6.74273i
0 0 0 −21.0713 0 11.0000 14.8997i 0 0 0
881.2 0 0 0 −21.0713 0 11.0000 + 14.8997i 0 0 0
881.3 0 0 0 21.0713 0 11.0000 14.8997i 0 0 0
881.4 0 0 0 21.0713 0 11.0000 + 14.8997i 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.k.b 4
3.b odd 2 1 inner 1008.4.k.b 4
4.b odd 2 1 63.4.c.b 4
7.b odd 2 1 inner 1008.4.k.b 4
12.b even 2 1 63.4.c.b 4
21.c even 2 1 inner 1008.4.k.b 4
28.d even 2 1 63.4.c.b 4
28.f even 6 2 441.4.p.a 8
28.g odd 6 2 441.4.p.a 8
84.h odd 2 1 63.4.c.b 4
84.j odd 6 2 441.4.p.a 8
84.n even 6 2 441.4.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.b 4 4.b odd 2 1
63.4.c.b 4 12.b even 2 1
63.4.c.b 4 28.d even 2 1
63.4.c.b 4 84.h odd 2 1
441.4.p.a 8 28.f even 6 2
441.4.p.a 8 28.g odd 6 2
441.4.p.a 8 84.j odd 6 2
441.4.p.a 8 84.n even 6 2
1008.4.k.b 4 1.a even 1 1 trivial
1008.4.k.b 4 3.b odd 2 1 inner
1008.4.k.b 4 7.b odd 2 1 inner
1008.4.k.b 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 444 \) acting on \(S_{4}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -444 + T^{2} )^{2} \)
$7$ \( ( 343 - 22 T + T^{2} )^{2} \)
$11$ \( ( 242 + T^{2} )^{2} \)
$13$ \( ( 888 + T^{2} )^{2} \)
$17$ \( ( -3996 + T^{2} )^{2} \)
$19$ \( ( 7992 + T^{2} )^{2} \)
$23$ \( ( 6050 + T^{2} )^{2} \)
$29$ \( ( 15842 + T^{2} )^{2} \)
$31$ \( ( 56832 + T^{2} )^{2} \)
$37$ \( ( 184 + T )^{4} \)
$41$ \( ( -11100 + T^{2} )^{2} \)
$43$ \( ( -190 + T )^{4} \)
$47$ \( ( -1776 + T^{2} )^{2} \)
$53$ \( ( 128018 + T^{2} )^{2} \)
$59$ \( ( -7104 + T^{2} )^{2} \)
$61$ \( ( 429792 + T^{2} )^{2} \)
$67$ \( ( 296 + T )^{4} \)
$71$ \( ( 108578 + T^{2} )^{2} \)
$73$ \( ( 647352 + T^{2} )^{2} \)
$79$ \( ( 836 + T )^{4} \)
$83$ \( ( -1493616 + T^{2} )^{2} \)
$89$ \( ( -483516 + T^{2} )^{2} \)
$97$ \( ( 320568 + T^{2} )^{2} \)
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