Properties

Label 1008.2.cs.p
Level $1008$
Weight $2$
Character orbit 1008.cs
Analytic conductor $8.049$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( 3 + \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( 3 + \beta_{1} ) q^{7} + ( -\beta_{2} + \beta_{3} ) q^{11} + ( 2 + 4 \beta_{1} ) q^{13} -4 \beta_{1} q^{19} + 2 \beta_{2} q^{23} + ( 10 + 10 \beta_{1} ) q^{25} + ( 2 \beta_{2} - \beta_{3} ) q^{29} + ( -1 - \beta_{1} ) q^{31} + ( -2 \beta_{2} - \beta_{3} ) q^{35} + 4 \beta_{1} q^{37} + 2 \beta_{3} q^{41} + ( -4 - 8 \beta_{1} ) q^{43} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 8 + 5 \beta_{1} ) q^{49} + ( \beta_{2} + \beta_{3} ) q^{53} + 15 q^{55} + ( \beta_{2} + \beta_{3} ) q^{59} + ( 12 + 6 \beta_{1} ) q^{61} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( -4 + 4 \beta_{1} ) q^{67} + 2 \beta_{3} q^{71} + ( -4 + 4 \beta_{1} ) q^{73} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{77} + ( 14 + 7 \beta_{1} ) q^{79} + ( -2 \beta_{2} + \beta_{3} ) q^{83} -2 \beta_{2} q^{89} + ( 2 + 10 \beta_{1} ) q^{91} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( -3 - 6 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{7} + O(q^{10}) \) \( 4q + 10q^{7} + 8q^{19} + 20q^{25} - 2q^{31} - 8q^{37} + 22q^{49} + 60q^{55} + 36q^{61} - 24q^{67} - 24q^{73} + 42q^{79} - 12q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 2 \nu^{2} + 6 \nu - 5 \)\()/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 2 \nu^{2} + 6 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 3 \beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - \beta_{2} + 9 \beta_{1}\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 2 \beta_{2} - 6\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
0 0 0 −3.35410 1.93649i 0 2.50000 + 0.866025i 0 0 0
271.2 0 0 0 3.35410 + 1.93649i 0 2.50000 + 0.866025i 0 0 0
703.1 0 0 0 −3.35410 + 1.93649i 0 2.50000 0.866025i 0 0 0
703.2 0 0 0 3.35410 1.93649i 0 2.50000 0.866025i 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
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cs.p yes 4
3.b odd 2 1 inner 1008.2.cs.p yes 4
4.b odd 2 1 1008.2.cs.o 4
7.c even 3 1 7056.2.b.o 4
7.d odd 6 1 1008.2.cs.o 4
7.d odd 6 1 7056.2.b.v 4
12.b even 2 1 1008.2.cs.o 4
21.g even 6 1 1008.2.cs.o 4
21.g even 6 1 7056.2.b.v 4
21.h odd 6 1 7056.2.b.o 4
28.f even 6 1 inner 1008.2.cs.p yes 4
28.f even 6 1 7056.2.b.o 4
28.g odd 6 1 7056.2.b.v 4
84.j odd 6 1 inner 1008.2.cs.p yes 4
84.j odd 6 1 7056.2.b.o 4
84.n even 6 1 7056.2.b.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cs.o 4 4.b odd 2 1
1008.2.cs.o 4 7.d odd 6 1
1008.2.cs.o 4 12.b even 2 1
1008.2.cs.o 4 21.g even 6 1
1008.2.cs.p yes 4 1.a even 1 1 trivial
1008.2.cs.p yes 4 3.b odd 2 1 inner
1008.2.cs.p yes 4 28.f even 6 1 inner
1008.2.cs.p yes 4 84.j odd 6 1 inner
7056.2.b.o 4 7.c even 3 1
7056.2.b.o 4 21.h odd 6 1
7056.2.b.o 4 28.f even 6 1
7056.2.b.o 4 84.j odd 6 1
7056.2.b.v 4 7.d odd 6 1
7056.2.b.v 4 21.g even 6 1
7056.2.b.v 4 28.g odd 6 1
7056.2.b.v 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} - 15 T_{5}^{2} + 225 \)
\( T_{11}^{4} - 15 T_{11}^{2} + 225 \)
\( T_{13}^{2} + 12 \)
\( T_{17} \)
\( T_{19}^{2} - 4 T_{19} + 16 \)