Properties

Label 108.5.c.a
Level 108
Weight 5
Character orbit 108.c
Self dual yes
Analytic conductor 11.164
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 23q^{7} + O(q^{10}) \) \( q + 23q^{7} + 191q^{13} + 647q^{19} + 625q^{25} + 194q^{31} + 2591q^{37} - 3214q^{43} - 1872q^{49} - 5233q^{61} - 8809q^{67} + 9791q^{73} - 12361q^{79} + 4393q^{91} + 9743q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0
0 0 0 0 0 23.0000 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
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 108.5.c.a 1
3.b odd 2 1 CM 108.5.c.a 1
4.b odd 2 1 432.5.e.b 1
9.c even 3 2 324.5.g.a 2
9.d odd 6 2 324.5.g.a 2
12.b even 2 1 432.5.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.c.a 1 1.a even 1 1 trivial
108.5.c.a 1 3.b odd 2 1 CM
324.5.g.a 2 9.c even 3 2
324.5.g.a 2 9.d odd 6 2
432.5.e.b 1 4.b odd 2 1
432.5.e.b 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{5}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 25 T )( 1 + 25 T ) \)
$7$ \( 1 - 23 T + 2401 T^{2} \)
$11$ \( ( 1 - 121 T )( 1 + 121 T ) \)
$13$ \( 1 - 191 T + 28561 T^{2} \)
$17$ \( ( 1 - 289 T )( 1 + 289 T ) \)
$19$ \( 1 - 647 T + 130321 T^{2} \)
$23$ \( ( 1 - 529 T )( 1 + 529 T ) \)
$29$ \( ( 1 - 841 T )( 1 + 841 T ) \)
$31$ \( 1 - 194 T + 923521 T^{2} \)
$37$ \( 1 - 2591 T + 1874161 T^{2} \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( 1 + 3214 T + 3418801 T^{2} \)
$47$ \( ( 1 - 2209 T )( 1 + 2209 T ) \)
$53$ \( ( 1 - 2809 T )( 1 + 2809 T ) \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( 1 + 5233 T + 13845841 T^{2} \)
$67$ \( 1 + 8809 T + 20151121 T^{2} \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( 1 - 9791 T + 28398241 T^{2} \)
$79$ \( 1 + 12361 T + 38950081 T^{2} \)
$83$ \( ( 1 - 6889 T )( 1 + 6889 T ) \)
$89$ \( ( 1 - 7921 T )( 1 + 7921 T ) \)
$97$ \( 1 - 9743 T + 88529281 T^{2} \)
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