Properties

Label 1008.2.cs.q
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( 2 + 4 \beta_{2} ) q^{13} + ( -6 - 3 \beta_{2} ) q^{17} -\beta_{1} q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{23} + 2 \beta_{2} q^{25} + ( -\beta_{1} - \beta_{3} ) q^{31} + ( \beta_{1} + 2 \beta_{3} ) q^{35} + ( -7 - 7 \beta_{2} ) q^{37} + ( 2 + 4 \beta_{2} ) q^{41} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{43} -3 \beta_{1} q^{47} + 7 q^{49} -3 \beta_{2} q^{53} + 3 \beta_{3} q^{55} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{59} + ( 1 - \beta_{2} ) q^{61} + ( 6 + 6 \beta_{2} ) q^{65} + ( -\beta_{1} + \beta_{3} ) q^{67} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -6 - 3 \beta_{2} ) q^{73} + ( 14 + 7 \beta_{2} ) q^{77} + ( -\beta_{1} - 2 \beta_{3} ) q^{79} -9 q^{85} + ( -1 + \beta_{2} ) q^{89} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} + \beta_{3} ) q^{95} + ( -2 - 4 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} + O(q^{10}) \) \( 4q + 6q^{5} - 18q^{17} - 4q^{25} - 14q^{37} + 28q^{49} + 6q^{53} + 6q^{61} + 12q^{65} - 18q^{73} + 42q^{77} - 36q^{85} - 6q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1 + \beta_{2}\) \(1\) \(1\)

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
1.32288 2.29129i
−1.32288 + 2.29129i
1.32288 + 2.29129i
−1.32288 2.29129i
0 0 0 1.50000 + 0.866025i 0 −2.64575 0 0 0
271.2 0 0 0 1.50000 + 0.866025i 0 2.64575 0 0 0
703.1 0 0 0 1.50000 0.866025i 0 −2.64575 0 0 0
703.2 0 0 0 1.50000 0.866025i 0 2.64575 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 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.q 4
3.b odd 2 1 112.2.p.c 4
4.b odd 2 1 inner 1008.2.cs.q 4
7.c even 3 1 7056.2.b.s 4
7.d odd 6 1 inner 1008.2.cs.q 4
7.d odd 6 1 7056.2.b.s 4
12.b even 2 1 112.2.p.c 4
21.c even 2 1 784.2.p.g 4
21.g even 6 1 112.2.p.c 4
21.g even 6 1 784.2.f.d 4
21.h odd 6 1 784.2.f.d 4
21.h odd 6 1 784.2.p.g 4
24.f even 2 1 448.2.p.c 4
24.h odd 2 1 448.2.p.c 4
28.f even 6 1 inner 1008.2.cs.q 4
28.f even 6 1 7056.2.b.s 4
28.g odd 6 1 7056.2.b.s 4
84.h odd 2 1 784.2.p.g 4
84.j odd 6 1 112.2.p.c 4
84.j odd 6 1 784.2.f.d 4
84.n even 6 1 784.2.f.d 4
84.n even 6 1 784.2.p.g 4
168.s odd 6 1 3136.2.f.f 4
168.v even 6 1 3136.2.f.f 4
168.ba even 6 1 448.2.p.c 4
168.ba even 6 1 3136.2.f.f 4
168.be odd 6 1 448.2.p.c 4
168.be odd 6 1 3136.2.f.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 3.b odd 2 1
112.2.p.c 4 12.b even 2 1
112.2.p.c 4 21.g even 6 1
112.2.p.c 4 84.j odd 6 1
448.2.p.c 4 24.f even 2 1
448.2.p.c 4 24.h odd 2 1
448.2.p.c 4 168.ba even 6 1
448.2.p.c 4 168.be odd 6 1
784.2.f.d 4 21.g even 6 1
784.2.f.d 4 21.h odd 6 1
784.2.f.d 4 84.j odd 6 1
784.2.f.d 4 84.n even 6 1
784.2.p.g 4 21.c even 2 1
784.2.p.g 4 21.h odd 6 1
784.2.p.g 4 84.h odd 2 1
784.2.p.g 4 84.n even 6 1
1008.2.cs.q 4 1.a even 1 1 trivial
1008.2.cs.q 4 4.b odd 2 1 inner
1008.2.cs.q 4 7.d odd 6 1 inner
1008.2.cs.q 4 28.f even 6 1 inner
3136.2.f.f 4 168.s odd 6 1
3136.2.f.f 4 168.v even 6 1
3136.2.f.f 4 168.ba even 6 1
3136.2.f.f 4 168.be odd 6 1
7056.2.b.s 4 7.c even 3 1
7056.2.b.s 4 7.d odd 6 1
7056.2.b.s 4 28.f even 6 1
7056.2.b.s 4 28.g odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 3 \)
\( T_{11}^{4} - 21 T_{11}^{2} + 441 \)
\( T_{13}^{2} + 12 \)
\( T_{17}^{2} + 9 T_{17} + 27 \)
\( T_{19}^{4} + 7 T_{19}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 + T^{2} - 120 T^{4} + 121 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} )^{2} \)
$19$ \( 1 - 31 T^{2} + 600 T^{4} - 11191 T^{6} + 130321 T^{8} \)
$23$ \( 1 + 25 T^{2} + 96 T^{4} + 13225 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 2 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 113 T^{2} + 8280 T^{4} + 507257 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 58 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 137 T^{2} + 12528 T^{4} + 855017 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 + 3 T + 92 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \)
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