Properties

Label 1568.2.i.t
Level $1568$
Weight $2$
Character orbit 1568.i
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( 3 + 3 \beta_{2} ) q^{9} + \beta_{3} q^{13} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{17} -3 \beta_{2} q^{25} -4 q^{29} + ( 12 + 12 \beta_{2} ) q^{37} -9 \beta_{3} q^{41} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{45} -14 \beta_{2} q^{53} -11 \beta_{1} q^{61} + ( 2 + 2 \beta_{2} ) q^{65} + ( -11 \beta_{1} - 11 \beta_{3} ) q^{73} + 9 \beta_{2} q^{81} -10 q^{85} -3 \beta_{1} q^{89} -5 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9} + O(q^{10}) \) \( 4 q + 6 q^{9} + 6 q^{25} - 16 q^{29} + 24 q^{37} + 28 q^{53} + 4 q^{65} - 18 q^{81} - 40 q^{85} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0 0 0 −0.707107 + 1.22474i 0 0 0 1.50000 2.59808i 0
961.2 0 0 0 0.707107 1.22474i 0 0 0 1.50000 2.59808i 0
1537.1 0 0 0 −0.707107 1.22474i 0 0 0 1.50000 + 2.59808i 0
1537.2 0 0 0 0.707107 + 1.22474i 0 0 0 1.50000 + 2.59808i 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 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.t 4
4.b odd 2 1 CM 1568.2.i.t 4
7.b odd 2 1 inner 1568.2.i.t 4
7.c even 3 1 1568.2.a.o 2
7.c even 3 1 inner 1568.2.i.t 4
7.d odd 6 1 1568.2.a.o 2
7.d odd 6 1 inner 1568.2.i.t 4
28.d even 2 1 inner 1568.2.i.t 4
28.f even 6 1 1568.2.a.o 2
28.f even 6 1 inner 1568.2.i.t 4
28.g odd 6 1 1568.2.a.o 2
28.g odd 6 1 inner 1568.2.i.t 4
56.j odd 6 1 3136.2.a.bi 2
56.k odd 6 1 3136.2.a.bi 2
56.m even 6 1 3136.2.a.bi 2
56.p even 6 1 3136.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.o 2 7.c even 3 1
1568.2.a.o 2 7.d odd 6 1
1568.2.a.o 2 28.f even 6 1
1568.2.a.o 2 28.g odd 6 1
1568.2.i.t 4 1.a even 1 1 trivial
1568.2.i.t 4 4.b odd 2 1 CM
1568.2.i.t 4 7.b odd 2 1 inner
1568.2.i.t 4 7.c even 3 1 inner
1568.2.i.t 4 7.d odd 6 1 inner
1568.2.i.t 4 28.d even 2 1 inner
1568.2.i.t 4 28.f even 6 1 inner
1568.2.i.t 4 28.g odd 6 1 inner
3136.2.a.bi 2 56.j odd 6 1
3136.2.a.bi 2 56.k odd 6 1
3136.2.a.bi 2 56.m even 6 1
3136.2.a.bi 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3} \)
\( T_{5}^{4} + 2 T_{5}^{2} + 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 + 2 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( -2 + T^{2} )^{2} \)
$17$ \( 2500 + 50 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 144 - 12 T + T^{2} )^{2} \)
$41$ \( ( -162 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 196 - 14 T + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( 58564 + 242 T^{2} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 58564 + 242 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 324 + 18 T^{2} + T^{4} \)
$97$ \( ( -50 + T^{2} )^{2} \)
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