Properties

Label 108.5.c.b
Level 108
Weight 5
Character orbit 108.c
Analytic conductor 11.164
Analytic rank 0
Dimension 2
CM no
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: no
Analytic conductor: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 18\sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -31 q^{7} +O(q^{10})\) \( q + \beta q^{5} -31 q^{7} -5 \beta q^{11} -241 q^{13} + 5 \beta q^{17} -271 q^{19} + 5 \beta q^{23} -1319 q^{25} + 10 \beta q^{29} -778 q^{31} -31 \beta q^{35} + 1079 q^{37} -50 \beta q^{41} -298 q^{43} + 75 \beta q^{47} -1440 q^{49} + 70 \beta q^{53} + 9720 q^{55} + 65 \beta q^{59} -2641 q^{61} -241 \beta q^{65} + 5609 q^{67} -100 \beta q^{71} + 7199 q^{73} + 155 \beta q^{77} + 329 q^{79} + 30 \beta q^{83} -9720 q^{85} + 185 \beta q^{89} + 7471 q^{91} -271 \beta q^{95} -15961 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 62q^{7} + O(q^{10}) \) \( 2q - 62q^{7} - 482q^{13} - 542q^{19} - 2638q^{25} - 1556q^{31} + 2158q^{37} - 596q^{43} - 2880q^{49} + 19440q^{55} - 5282q^{61} + 11218q^{67} + 14398q^{73} + 658q^{79} - 19440q^{85} + 14942q^{91} - 31922q^{97} + O(q^{100}) \)

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
2.44949i
2.44949i
0 0 0 44.0908i 0 −31.0000 0 0 0
53.2 0 0 0 44.0908i 0 −31.0000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.c.b 2
3.b odd 2 1 inner 108.5.c.b 2
4.b odd 2 1 432.5.e.f 2
9.c even 3 2 324.5.g.e 4
9.d odd 6 2 324.5.g.e 4
12.b even 2 1 432.5.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.c.b 2 1.a even 1 1 trivial
108.5.c.b 2 3.b odd 2 1 inner
324.5.g.e 4 9.c even 3 2
324.5.g.e 4 9.d odd 6 2
432.5.e.f 2 4.b odd 2 1
432.5.e.f 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1944 \) acting on \(S_{5}^{\mathrm{new}}(108, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 694 T^{2} + 390625 T^{4} \)
$7$ \( ( 1 + 31 T + 2401 T^{2} )^{2} \)
$11$ \( 1 + 19318 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 + 241 T + 28561 T^{2} )^{2} \)
$17$ \( 1 - 118442 T^{2} + 6975757441 T^{4} \)
$19$ \( ( 1 + 271 T + 130321 T^{2} )^{2} \)
$23$ \( 1 - 511082 T^{2} + 78310985281 T^{4} \)
$29$ \( 1 - 1220162 T^{2} + 500246412961 T^{4} \)
$31$ \( ( 1 + 778 T + 923521 T^{2} )^{2} \)
$37$ \( ( 1 - 1079 T + 1874161 T^{2} )^{2} \)
$41$ \( 1 - 791522 T^{2} + 7984925229121 T^{4} \)
$43$ \( ( 1 + 298 T + 3418801 T^{2} )^{2} \)
$47$ \( 1 + 1175638 T^{2} + 23811286661761 T^{4} \)
$53$ \( 1 - 6255362 T^{2} + 62259690411361 T^{4} \)
$59$ \( 1 - 16021322 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 + 2641 T + 13845841 T^{2} )^{2} \)
$67$ \( ( 1 - 5609 T + 20151121 T^{2} )^{2} \)
$71$ \( 1 - 31383362 T^{2} + 645753531245761 T^{4} \)
$73$ \( ( 1 - 7199 T + 28398241 T^{2} )^{2} \)
$79$ \( ( 1 - 329 T + 38950081 T^{2} )^{2} \)
$83$ \( 1 - 93167042 T^{2} + 2252292232139041 T^{4} \)
$89$ \( 1 - 58951082 T^{2} + 3936588805702081 T^{4} \)
$97$ \( ( 1 + 15961 T + 88529281 T^{2} )^{2} \)
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