Properties

Label 108.3.c.b
Level 108
Weight 3
Character orbit 108.c
Analytic conductor 2.943
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: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 108.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 9 i q^{5} -7 q^{7} +O(q^{10})\) \( q + 9 i q^{5} -7 q^{7} + 9 i q^{11} + 14 q^{13} + 18 i q^{17} + 8 q^{19} -36 i q^{23} -56 q^{25} -18 i q^{29} + 35 q^{31} -63 i q^{35} + 44 q^{37} + 36 i q^{41} -22 q^{43} -54 i q^{47} + 9 i q^{53} -81 q^{55} + 18 i q^{59} + 20 q^{61} + 126 i q^{65} + 14 q^{67} + 126 i q^{71} + 89 q^{73} -63 i q^{77} + 110 q^{79} + 27 i q^{83} -162 q^{85} + 18 i q^{89} -98 q^{91} + 72 i q^{95} + 11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 14q^{7} + O(q^{10}) \) \( 2q - 14q^{7} + 28q^{13} + 16q^{19} - 112q^{25} + 70q^{31} + 88q^{37} - 44q^{43} - 162q^{55} + 40q^{61} + 28q^{67} + 178q^{73} + 220q^{79} - 324q^{85} - 196q^{91} + 22q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 0 0 9.00000i 0 −7.00000 0 0 0
53.2 0 0 0 9.00000i 0 −7.00000 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 108.3.c.b 2
3.b odd 2 1 inner 108.3.c.b 2
4.b odd 2 1 432.3.e.g 2
5.b even 2 1 2700.3.g.m 2
5.c odd 4 1 2700.3.b.a 2
5.c odd 4 1 2700.3.b.f 2
8.b even 2 1 1728.3.e.e 2
8.d odd 2 1 1728.3.e.n 2
9.c even 3 2 324.3.g.c 4
9.d odd 6 2 324.3.g.c 4
12.b even 2 1 432.3.e.g 2
15.d odd 2 1 2700.3.g.m 2
15.e even 4 1 2700.3.b.a 2
15.e even 4 1 2700.3.b.f 2
24.f even 2 1 1728.3.e.n 2
24.h odd 2 1 1728.3.e.e 2
36.f odd 6 2 1296.3.q.d 4
36.h even 6 2 1296.3.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.b 2 1.a even 1 1 trivial
108.3.c.b 2 3.b odd 2 1 inner
324.3.g.c 4 9.c even 3 2
324.3.g.c 4 9.d odd 6 2
432.3.e.g 2 4.b odd 2 1
432.3.e.g 2 12.b even 2 1
1296.3.q.d 4 36.f odd 6 2
1296.3.q.d 4 36.h even 6 2
1728.3.e.e 2 8.b even 2 1
1728.3.e.e 2 24.h odd 2 1
1728.3.e.n 2 8.d odd 2 1
1728.3.e.n 2 24.f even 2 1
2700.3.b.a 2 5.c odd 4 1
2700.3.b.a 2 15.e even 4 1
2700.3.b.f 2 5.c odd 4 1
2700.3.b.f 2 15.e even 4 1
2700.3.g.m 2 5.b even 2 1
2700.3.g.m 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(108, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 31 T^{2} + 625 T^{4} \)
$7$ \( ( 1 + 7 T + 49 T^{2} )^{2} \)
$11$ \( 1 - 161 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 - 14 T + 169 T^{2} )^{2} \)
$17$ \( 1 - 254 T^{2} + 83521 T^{4} \)
$19$ \( ( 1 - 8 T + 361 T^{2} )^{2} \)
$23$ \( 1 + 238 T^{2} + 279841 T^{4} \)
$29$ \( 1 - 1358 T^{2} + 707281 T^{4} \)
$31$ \( ( 1 - 35 T + 961 T^{2} )^{2} \)
$37$ \( ( 1 - 44 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 2066 T^{2} + 2825761 T^{4} \)
$43$ \( ( 1 + 22 T + 1849 T^{2} )^{2} \)
$47$ \( 1 - 1502 T^{2} + 4879681 T^{4} \)
$53$ \( 1 - 5537 T^{2} + 7890481 T^{4} \)
$59$ \( 1 - 6638 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 - 20 T + 3721 T^{2} )^{2} \)
$67$ \( ( 1 - 14 T + 4489 T^{2} )^{2} \)
$71$ \( 1 + 5794 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 - 89 T + 5329 T^{2} )^{2} \)
$79$ \( ( 1 - 110 T + 6241 T^{2} )^{2} \)
$83$ \( 1 - 13049 T^{2} + 47458321 T^{4} \)
$89$ \( 1 - 15518 T^{2} + 62742241 T^{4} \)
$97$ \( ( 1 - 11 T + 9409 T^{2} )^{2} \)
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