Properties

Label 1568.3.d.a
Level 1568
Weight 3
Character orbit 1568.d
Analytic conductor 42.725
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 i q^{3} -9 q^{5} -16 q^{9} +O(q^{10})\) \( q + 5 i q^{3} -9 q^{5} -16 q^{9} -3 i q^{11} -16 q^{13} -45 i q^{15} + 7 q^{17} -11 i q^{19} -19 i q^{23} + 56 q^{25} -35 i q^{27} -32 q^{29} + 11 i q^{31} + 15 q^{33} - q^{37} -80 i q^{39} + 40 q^{41} -40 i q^{43} + 144 q^{45} + 85 i q^{47} + 35 i q^{51} + 7 q^{53} + 27 i q^{55} + 55 q^{57} + 53 i q^{59} -79 q^{61} + 144 q^{65} + 11 i q^{67} + 95 q^{69} -48 i q^{71} -143 q^{73} + 280 i q^{75} -35 i q^{79} + 31 q^{81} + 8 i q^{83} -63 q^{85} -160 i q^{87} + 97 q^{89} -55 q^{93} + 99 i q^{95} + 88 q^{97} + 48 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{5} - 32q^{9} + O(q^{10}) \) \( 2q - 18q^{5} - 32q^{9} - 32q^{13} + 14q^{17} + 112q^{25} - 64q^{29} + 30q^{33} - 2q^{37} + 80q^{41} + 288q^{45} + 14q^{53} + 110q^{57} - 158q^{61} + 288q^{65} + 190q^{69} - 286q^{73} + 62q^{81} - 126q^{85} + 194q^{89} - 110q^{93} + 176q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.00000i
1.00000i
0 5.00000i 0 −9.00000 0 0 0 −16.0000 0
1471.2 0 5.00000i 0 −9.00000 0 0 0 −16.0000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.d.a 2
4.b odd 2 1 inner 1568.3.d.a 2
7.b odd 2 1 1568.3.d.e 2
7.d odd 6 2 224.3.r.a 4
28.d even 2 1 1568.3.d.e 2
28.f even 6 2 224.3.r.a 4
56.j odd 6 2 448.3.r.c 4
56.m even 6 2 448.3.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.a 4 7.d odd 6 2
224.3.r.a 4 28.f even 6 2
448.3.r.c 4 56.j odd 6 2
448.3.r.c 4 56.m even 6 2
1568.3.d.a 2 1.a even 1 1 trivial
1568.3.d.a 2 4.b odd 2 1 inner
1568.3.d.e 2 7.b odd 2 1
1568.3.d.e 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{2} + 25 \)
\( T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 7 T^{2} + 81 T^{4} \)
$5$ \( ( 1 + 9 T + 25 T^{2} )^{2} \)
$7$ 1
$11$ \( 1 - 233 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 + 16 T + 169 T^{2} )^{2} \)
$17$ \( ( 1 - 7 T + 289 T^{2} )^{2} \)
$19$ \( 1 - 601 T^{2} + 130321 T^{4} \)
$23$ \( 1 - 697 T^{2} + 279841 T^{4} \)
$29$ \( ( 1 + 32 T + 841 T^{2} )^{2} \)
$31$ \( 1 - 1801 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 + T + 1369 T^{2} )^{2} \)
$41$ \( ( 1 - 40 T + 1681 T^{2} )^{2} \)
$43$ \( 1 - 2098 T^{2} + 3418801 T^{4} \)
$47$ \( 1 + 2807 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 - 7 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 4153 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 79 T + 3721 T^{2} )^{2} \)
$67$ \( 1 - 8857 T^{2} + 20151121 T^{4} \)
$71$ \( 1 - 7778 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 + 143 T + 5329 T^{2} )^{2} \)
$79$ \( 1 - 11257 T^{2} + 38950081 T^{4} \)
$83$ \( 1 - 13714 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 97 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 - 88 T + 9409 T^{2} )^{2} \)
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