Properties

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

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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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{2} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{2} ) q^{3} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{5} + ( 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -3 + \beta_{3} ) q^{13} + 4 q^{15} -2 \beta_{2} q^{17} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + 4 \beta_{1} q^{23} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{25} + ( 10 - 2 \beta_{3} ) q^{27} -2 \beta_{3} q^{29} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} + 8 \beta_{1} q^{33} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 8 + 8 \beta_{1} + 4 \beta_{2} ) q^{39} + ( 4 + 2 \beta_{3} ) q^{41} + ( 2 + 2 \beta_{3} ) q^{43} + ( -7 - 7 \beta_{1} - \beta_{2} ) q^{45} + ( -6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 10 + 10 \beta_{1} ) q^{53} + ( 12 + 4 \beta_{3} ) q^{55} + ( -6 + 2 \beta_{3} ) q^{57} + ( -7 - 7 \beta_{1} + \beta_{2} ) q^{59} + ( -9 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -4 - 4 \beta_{1} ) q^{67} + ( 4 - 4 \beta_{3} ) q^{69} + ( 4 - 4 \beta_{3} ) q^{71} + ( 6 + 6 \beta_{1} + 4 \beta_{2} ) q^{73} + ( -9 \beta_{1} - \beta_{2} - \beta_{3} ) q^{75} + ( 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{79} + ( -11 - 11 \beta_{1} - 6 \beta_{2} ) q^{81} + ( 7 + \beta_{3} ) q^{83} + ( 10 + 2 \beta_{3} ) q^{85} + ( -10 - 10 \beta_{1} - 2 \beta_{2} ) q^{87} + 6 \beta_{1} q^{89} + ( -12 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{93} + ( 4 + 4 \beta_{1} ) q^{95} + ( -8 - 2 \beta_{3} ) q^{97} + ( 14 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 2q^{5} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 2q^{5} - 6q^{9} + 4q^{11} - 12q^{13} + 16q^{15} + 2q^{19} - 8q^{23} - 2q^{25} + 40q^{27} + 4q^{31} - 16q^{33} + 16q^{39} + 16q^{41} + 8q^{43} - 14q^{45} + 12q^{47} - 20q^{51} + 20q^{53} + 48q^{55} - 24q^{57} - 14q^{59} + 18q^{61} + 4q^{65} - 8q^{67} + 16q^{69} + 16q^{71} + 12q^{73} + 18q^{75} - 8q^{79} - 22q^{81} + 28q^{83} + 40q^{85} - 20q^{87} - 12q^{89} + 24q^{93} + 8q^{95} - 32q^{97} + 56q^{99} + O(q^{100}) \)

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

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

Character values

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

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

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
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
0 −1.61803 2.80252i 0 −0.618034 + 1.07047i 0 0 0 −3.73607 + 6.47106i 0
961.2 0 0.618034 + 1.07047i 0 1.61803 2.80252i 0 0 0 0.736068 1.27491i 0
1537.1 0 −1.61803 + 2.80252i 0 −0.618034 1.07047i 0 0 0 −3.73607 6.47106i 0
1537.2 0 0.618034 1.07047i 0 1.61803 + 2.80252i 0 0 0 0.736068 + 1.27491i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.n 4
4.b odd 2 1 1568.2.i.w 4
7.b odd 2 1 1568.2.i.v 4
7.c even 3 1 1568.2.a.v 2
7.c even 3 1 inner 1568.2.i.n 4
7.d odd 6 1 224.2.a.c 2
7.d odd 6 1 1568.2.i.v 4
21.g even 6 1 2016.2.a.r 2
28.d even 2 1 1568.2.i.m 4
28.f even 6 1 224.2.a.d yes 2
28.f even 6 1 1568.2.i.m 4
28.g odd 6 1 1568.2.a.k 2
28.g odd 6 1 1568.2.i.w 4
35.i odd 6 1 5600.2.a.bk 2
56.j odd 6 1 448.2.a.j 2
56.k odd 6 1 3136.2.a.by 2
56.m even 6 1 448.2.a.i 2
56.p even 6 1 3136.2.a.bf 2
84.j odd 6 1 2016.2.a.o 2
112.v even 12 2 1792.2.b.m 4
112.x odd 12 2 1792.2.b.k 4
140.s even 6 1 5600.2.a.z 2
168.ba even 6 1 4032.2.a.bw 2
168.be odd 6 1 4032.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 7.d odd 6 1
224.2.a.d yes 2 28.f even 6 1
448.2.a.i 2 56.m even 6 1
448.2.a.j 2 56.j odd 6 1
1568.2.a.k 2 28.g odd 6 1
1568.2.a.v 2 7.c even 3 1
1568.2.i.m 4 28.d even 2 1
1568.2.i.m 4 28.f even 6 1
1568.2.i.n 4 1.a even 1 1 trivial
1568.2.i.n 4 7.c even 3 1 inner
1568.2.i.v 4 7.b odd 2 1
1568.2.i.v 4 7.d odd 6 1
1568.2.i.w 4 4.b odd 2 1
1568.2.i.w 4 28.g odd 6 1
1792.2.b.k 4 112.x odd 12 2
1792.2.b.m 4 112.v even 12 2
2016.2.a.o 2 84.j odd 6 1
2016.2.a.r 2 21.g even 6 1
3136.2.a.bf 2 56.p even 6 1
3136.2.a.by 2 56.k odd 6 1
4032.2.a.bv 2 168.be odd 6 1
4032.2.a.bw 2 168.ba even 6 1
5600.2.a.z 2 140.s even 6 1
5600.2.a.bk 2 35.i odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 2 T_{3}^{3} + 8 T_{3}^{2} - 8 T_{3} + 16 \)
\( T_{5}^{4} - 2 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 16 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 32 T_{11}^{2} + 64 T_{11} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 - 8 T + 8 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 256 + 64 T + 32 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( ( 4 + 6 T + T^{2} )^{2} \)
$17$ \( 400 + 20 T^{2} + T^{4} \)
$19$ \( 16 + 8 T + 8 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( ( 16 + 4 T + T^{2} )^{2} \)
$29$ \( ( -20 + T^{2} )^{2} \)
$31$ \( 256 + 64 T + 32 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( 400 + 20 T^{2} + T^{4} \)
$41$ \( ( -4 - 8 T + T^{2} )^{2} \)
$43$ \( ( -16 - 4 T + T^{2} )^{2} \)
$47$ \( 256 - 192 T + 128 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( ( 100 - 10 T + T^{2} )^{2} \)
$59$ \( 1936 + 616 T + 152 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( 5776 - 1368 T + 248 T^{2} - 18 T^{3} + T^{4} \)
$67$ \( ( 16 + 4 T + T^{2} )^{2} \)
$71$ \( ( -64 - 8 T + T^{2} )^{2} \)
$73$ \( 1936 + 528 T + 188 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( 4096 - 512 T + 128 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( ( 44 - 14 T + T^{2} )^{2} \)
$89$ \( ( 36 + 6 T + T^{2} )^{2} \)
$97$ \( ( 44 + 16 T + T^{2} )^{2} \)
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