Properties

Label 1568.2.i.u
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: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 224)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} + 3 \beta_{2} q^{11} - \beta_{3} q^{15} + 5 \beta_1 q^{17} + ( - \beta_{3} + \beta_{2}) q^{19} + ( - \beta_{3} + \beta_{2}) q^{23} + 4 \beta_1 q^{25} - 3 \beta_{3} q^{27} + 8 q^{29} + 5 \beta_{2} q^{31} + ( - 9 \beta_1 + 9) q^{33} + ( - 5 \beta_1 + 5) q^{37} - 4 q^{41} - 4 \beta_{3} q^{43} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{47} + (5 \beta_{3} - 5 \beta_{2}) q^{51} + \beta_1 q^{53} + 3 \beta_{3} q^{55} + 3 q^{57} - \beta_{2} q^{59} + ( - 11 \beta_1 + 11) q^{61} - 7 \beta_{2} q^{67} + 3 q^{69} + 8 \beta_{3} q^{71} + 15 \beta_1 q^{73} + (4 \beta_{3} - 4 \beta_{2}) q^{75} + ( - \beta_{3} + \beta_{2}) q^{79} + 9 \beta_1 q^{81} + 4 \beta_{3} q^{83} + 5 q^{85} - 8 \beta_{2} q^{87} + ( - 7 \beta_1 + 7) q^{89} + ( - 15 \beta_1 + 15) q^{93} + \beta_{2} q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 10 q^{17} + 8 q^{25} + 32 q^{29} + 18 q^{33} + 10 q^{37} - 16 q^{41} + 2 q^{53} + 12 q^{57} + 22 q^{61} + 12 q^{69} + 30 q^{73} + 18 q^{81} + 20 q^{85} + 14 q^{89} + 30 q^{93} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 0.866025i 0 0 0 0 0
961.2 0 0.866025 + 1.50000i 0 0.500000 0.866025i 0 0 0 0 0
1537.1 0 −0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0 0 0 0
1537.2 0 0.866025 1.50000i 0 0.500000 + 0.866025i 0 0 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.c even 3 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.u 4
4.b odd 2 1 inner 1568.2.i.u 4
7.b odd 2 1 224.2.i.b 4
7.c even 3 1 1568.2.a.n 2
7.c even 3 1 inner 1568.2.i.u 4
7.d odd 6 1 224.2.i.b 4
7.d odd 6 1 1568.2.a.s 2
21.c even 2 1 2016.2.s.r 4
21.g even 6 1 2016.2.s.r 4
28.d even 2 1 224.2.i.b 4
28.f even 6 1 224.2.i.b 4
28.f even 6 1 1568.2.a.s 2
28.g odd 6 1 1568.2.a.n 2
28.g odd 6 1 inner 1568.2.i.u 4
56.e even 2 1 448.2.i.i 4
56.h odd 2 1 448.2.i.i 4
56.j odd 6 1 448.2.i.i 4
56.j odd 6 1 3136.2.a.bh 2
56.k odd 6 1 3136.2.a.bu 2
56.m even 6 1 448.2.i.i 4
56.m even 6 1 3136.2.a.bh 2
56.p even 6 1 3136.2.a.bu 2
84.h odd 2 1 2016.2.s.r 4
84.j odd 6 1 2016.2.s.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.b 4 7.b odd 2 1
224.2.i.b 4 7.d odd 6 1
224.2.i.b 4 28.d even 2 1
224.2.i.b 4 28.f even 6 1
448.2.i.i 4 56.e even 2 1
448.2.i.i 4 56.h odd 2 1
448.2.i.i 4 56.j odd 6 1
448.2.i.i 4 56.m even 6 1
1568.2.a.n 2 7.c even 3 1
1568.2.a.n 2 28.g odd 6 1
1568.2.a.s 2 7.d odd 6 1
1568.2.a.s 2 28.f even 6 1
1568.2.i.u 4 1.a even 1 1 trivial
1568.2.i.u 4 4.b odd 2 1 inner
1568.2.i.u 4 7.c even 3 1 inner
1568.2.i.u 4 28.g odd 6 1 inner
2016.2.s.r 4 21.c even 2 1
2016.2.s.r 4 21.g even 6 1
2016.2.s.r 4 84.h odd 2 1
2016.2.s.r 4 84.j odd 6 1
3136.2.a.bh 2 56.j odd 6 1
3136.2.a.bh 2 56.m even 6 1
3136.2.a.bu 2 56.k odd 6 1
3136.2.a.bu 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 \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 27T_{11}^{2} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T - 8)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$37$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T + 4)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$71$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 225)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$97$ \( (T + 12)^{4} \) Copy content Toggle raw display
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