Properties

Label 1568.2.i.r
Level $1568$
Weight $2$
Character orbit 1568.i
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{9} + (2 \beta_{2} + 2) q^{11} - 2 \beta_{3} q^{13} - 3 \beta_1 q^{17} + (3 \beta_{3} + 3 \beta_1) q^{19} - 8 \beta_{2} q^{23} + (5 \beta_{2} + 5) q^{25} - 4 \beta_{3} q^{27} + 6 q^{29} + 6 \beta_1 q^{31} + (2 \beta_{3} + 2 \beta_1) q^{33} - 2 \beta_{2} q^{37} + (4 \beta_{2} + 4) q^{39} + 3 \beta_{3} q^{41} - 6 q^{43} + (2 \beta_{3} + 2 \beta_1) q^{47} - 6 \beta_{2} q^{51} + ( - 6 \beta_{2} - 6) q^{53} - 6 q^{57} + 9 \beta_1 q^{59} + ( - 4 \beta_{3} - 4 \beta_1) q^{61} + (12 \beta_{2} + 12) q^{67} - 8 \beta_{3} q^{69} + 4 q^{71} + \beta_1 q^{73} + (5 \beta_{3} + 5 \beta_1) q^{75} - 12 \beta_{2} q^{79} + (5 \beta_{2} + 5) q^{81} - 7 \beta_{3} q^{83} + 6 \beta_1 q^{87} + (3 \beta_{3} + 3 \beta_1) q^{89} + 12 \beta_{2} q^{93} + 13 \beta_{3} q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 4 q^{11} + 16 q^{23} + 10 q^{25} + 24 q^{29} + 4 q^{37} + 8 q^{39} - 24 q^{43} + 12 q^{51} - 12 q^{53} - 24 q^{57} + 24 q^{67} + 16 q^{71} + 24 q^{79} + 10 q^{81} - 24 q^{93} + 8 q^{99}+O(q^{100}) \) 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

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

Character values

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

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

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 −0.707107 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
961.2 0 0.707107 + 1.22474i 0 0 0 0 0 0.500000 0.866025i 0
1537.1 0 −0.707107 + 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 0
1537.2 0 0.707107 1.22474i 0 0 0 0 0 0.500000 + 0.866025i 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d 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.r 4
4.b odd 2 1 1568.2.i.q 4
7.b odd 2 1 inner 1568.2.i.r 4
7.c even 3 1 1568.2.a.q 2
7.c even 3 1 inner 1568.2.i.r 4
7.d odd 6 1 1568.2.a.q 2
7.d odd 6 1 inner 1568.2.i.r 4
28.d even 2 1 1568.2.i.q 4
28.f even 6 1 1568.2.a.r yes 2
28.f even 6 1 1568.2.i.q 4
28.g odd 6 1 1568.2.a.r yes 2
28.g odd 6 1 1568.2.i.q 4
56.j odd 6 1 3136.2.a.bo 2
56.k odd 6 1 3136.2.a.bl 2
56.m even 6 1 3136.2.a.bl 2
56.p even 6 1 3136.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.q 2 7.c even 3 1
1568.2.a.q 2 7.d odd 6 1
1568.2.a.r yes 2 28.f even 6 1
1568.2.a.r yes 2 28.g odd 6 1
1568.2.i.q 4 4.b odd 2 1
1568.2.i.q 4 28.d even 2 1
1568.2.i.q 4 28.f even 6 1
1568.2.i.q 4 28.g odd 6 1
1568.2.i.r 4 1.a even 1 1 trivial
1568.2.i.r 4 7.b odd 2 1 inner
1568.2.i.r 4 7.c even 3 1 inner
1568.2.i.r 4 7.d odd 6 1 inner
3136.2.a.bl 2 56.k odd 6 1
3136.2.a.bl 2 56.m even 6 1
3136.2.a.bo 2 56.j odd 6 1
3136.2.a.bo 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 \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$19$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 162 T^{2} + 26244 \) Copy content Toggle raw display
$61$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$71$ \( (T - 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$97$ \( (T^{2} - 338)^{2} \) Copy content Toggle raw display
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