Properties

Label 588.5.c.a
Level 588
Weight 5
Character orbit 588.c
Self dual yes
Analytic conductor 60.782
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7815382933\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 9q^{3} + 81q^{9} + O(q^{10}) \) \( q - 9q^{3} + 81q^{9} - 146q^{13} + 46q^{19} + 625q^{25} - 729q^{27} - 194q^{31} - 2062q^{37} + 1314q^{39} - 3214q^{43} - 414q^{57} + 1966q^{61} + 5906q^{67} + 8542q^{73} - 5625q^{75} + 7682q^{79} + 6561q^{81} + 1746q^{93} + 18814q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
0
0 −9.00000 0 0 0 0 0 81.0000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.5.c.a 1
3.b odd 2 1 CM 588.5.c.a 1
7.b odd 2 1 12.5.c.a 1
21.c even 2 1 12.5.c.a 1
28.d even 2 1 48.5.e.a 1
35.c odd 2 1 300.5.g.b 1
35.f even 4 2 300.5.b.a 2
56.e even 2 1 192.5.e.b 1
56.h odd 2 1 192.5.e.a 1
63.l odd 6 2 324.5.g.b 2
63.o even 6 2 324.5.g.b 2
84.h odd 2 1 48.5.e.a 1
105.g even 2 1 300.5.g.b 1
105.k odd 4 2 300.5.b.a 2
168.e odd 2 1 192.5.e.b 1
168.i even 2 1 192.5.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.c.a 1 7.b odd 2 1
12.5.c.a 1 21.c even 2 1
48.5.e.a 1 28.d even 2 1
48.5.e.a 1 84.h odd 2 1
192.5.e.a 1 56.h odd 2 1
192.5.e.a 1 168.i even 2 1
192.5.e.b 1 56.e even 2 1
192.5.e.b 1 168.e odd 2 1
300.5.b.a 2 35.f even 4 2
300.5.b.a 2 105.k odd 4 2
300.5.g.b 1 35.c odd 2 1
300.5.g.b 1 105.g even 2 1
324.5.g.b 2 63.l odd 6 2
324.5.g.b 2 63.o even 6 2
588.5.c.a 1 1.a even 1 1 trivial
588.5.c.a 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(588, [\chi])\):

\( T_{5} \)
\( T_{13} + 146 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 9 T \)
$5$ \( ( 1 - 25 T )( 1 + 25 T ) \)
$7$ 1
$11$ \( ( 1 - 121 T )( 1 + 121 T ) \)
$13$ \( 1 + 146 T + 28561 T^{2} \)
$17$ \( ( 1 - 289 T )( 1 + 289 T ) \)
$19$ \( 1 - 46 T + 130321 T^{2} \)
$23$ \( ( 1 - 529 T )( 1 + 529 T ) \)
$29$ \( ( 1 - 841 T )( 1 + 841 T ) \)
$31$ \( 1 + 194 T + 923521 T^{2} \)
$37$ \( 1 + 2062 T + 1874161 T^{2} \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( 1 + 3214 T + 3418801 T^{2} \)
$47$ \( ( 1 - 2209 T )( 1 + 2209 T ) \)
$53$ \( ( 1 - 2809 T )( 1 + 2809 T ) \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( 1 - 1966 T + 13845841 T^{2} \)
$67$ \( 1 - 5906 T + 20151121 T^{2} \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( 1 - 8542 T + 28398241 T^{2} \)
$79$ \( 1 - 7682 T + 38950081 T^{2} \)
$83$ \( ( 1 - 6889 T )( 1 + 6889 T ) \)
$89$ \( ( 1 - 7921 T )( 1 + 7921 T ) \)
$97$ \( 1 - 18814 T + 88529281 T^{2} \)
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