Properties

Label 108.2.b.a
Level 108
Weight 2
Character orbit 108.b
Analytic conductor 0.862
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -1 - 2 \beta_{2} ) q^{7} + ( -\beta_{1} + 2 \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( -1 - 2 \beta_{2} ) q^{7} + ( -\beta_{1} + 2 \beta_{3} ) q^{8} + ( -2 + \beta_{2} ) q^{10} + ( \beta_{1} - \beta_{3} ) q^{11} + 2 q^{13} + ( \beta_{1} - 4 \beta_{3} ) q^{14} + ( -4 - \beta_{2} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{20} + ( 2 + \beta_{2} ) q^{22} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{23} + 2 \beta_{1} q^{26} + ( 8 + \beta_{2} ) q^{28} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( 1 + 2 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{32} + ( 4 - 2 \beta_{2} ) q^{34} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{35} -4 q^{37} + ( -4 - 3 \beta_{2} ) q^{40} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{41} + ( 2 + 4 \beta_{2} ) q^{43} + ( \beta_{1} + 2 \beta_{3} ) q^{44} + ( -8 - 4 \beta_{2} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{47} -8 q^{49} + 2 \beta_{2} q^{52} + ( \beta_{1} + \beta_{3} ) q^{53} + ( 1 + 2 \beta_{2} ) q^{55} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{56} + ( 4 - 2 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{59} -4 q^{61} + ( -\beta_{1} + 4 \beta_{3} ) q^{62} + ( 4 - 3 \beta_{2} ) q^{64} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{65} + ( -2 - 4 \beta_{2} ) q^{67} + ( 6 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 10 + 5 \beta_{2} ) q^{70} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{71} + 5 q^{73} -4 \beta_{1} q^{74} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{77} + ( 2 + 4 \beta_{2} ) q^{79} + ( -\beta_{1} - 6 \beta_{3} ) q^{80} + ( -8 + 4 \beta_{2} ) q^{82} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{83} + 10 q^{85} + ( -2 \beta_{1} + 8 \beta_{3} ) q^{86} + ( -4 + \beta_{2} ) q^{88} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -2 - 4 \beta_{2} ) q^{91} + ( -4 \beta_{1} - 8 \beta_{3} ) q^{92} + ( -4 - 2 \beta_{2} ) q^{94} + 11 q^{97} -8 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{4} + O(q^{10}) \) \( 4q - 2q^{4} - 10q^{10} + 8q^{13} - 14q^{16} + 6q^{22} + 30q^{28} + 20q^{34} - 16q^{37} - 10q^{40} - 24q^{46} - 32q^{49} - 4q^{52} + 20q^{58} - 16q^{61} + 22q^{64} + 30q^{70} + 20q^{73} - 40q^{82} + 40q^{85} - 18q^{88} - 12q^{94} + 44q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.866025 1.11803i
−0.866025 + 1.11803i
0.866025 1.11803i
0.866025 + 1.11803i
−0.866025 1.11803i 0 −0.500000 + 1.93649i 2.23607i 0 3.87298i 2.59808 1.11803i 0 −2.50000 + 1.93649i
107.2 −0.866025 + 1.11803i 0 −0.500000 1.93649i 2.23607i 0 3.87298i 2.59808 + 1.11803i 0 −2.50000 1.93649i
107.3 0.866025 1.11803i 0 −0.500000 1.93649i 2.23607i 0 3.87298i −2.59808 1.11803i 0 −2.50000 1.93649i
107.4 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 2.23607i 0 3.87298i −2.59808 + 1.11803i 0 −2.50000 + 1.93649i
\(n\): 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
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.2.b.a 4
3.b odd 2 1 inner 108.2.b.a 4
4.b odd 2 1 inner 108.2.b.a 4
8.b even 2 1 1728.2.c.c 4
8.d odd 2 1 1728.2.c.c 4
9.c even 3 2 324.2.h.d 8
9.d odd 6 2 324.2.h.d 8
12.b even 2 1 inner 108.2.b.a 4
24.f even 2 1 1728.2.c.c 4
24.h odd 2 1 1728.2.c.c 4
36.f odd 6 2 324.2.h.d 8
36.h even 6 2 324.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.a 4 1.a even 1 1 trivial
108.2.b.a 4 3.b odd 2 1 inner
108.2.b.a 4 4.b odd 2 1 inner
108.2.b.a 4 12.b even 2 1 inner
324.2.h.d 8 9.c even 3 2
324.2.h.d 8 9.d odd 6 2
324.2.h.d 8 36.f odd 6 2
324.2.h.d 8 36.h even 6 2
1728.2.c.c 4 8.b even 2 1
1728.2.c.c 4 8.d odd 2 1
1728.2.c.c 4 24.f even 2 1
1728.2.c.c 4 24.h odd 2 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 19 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 - 2 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 47 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 2 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 26 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 101 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 4 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 74 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 5 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 19 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 158 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 11 T + 97 T^{2} )^{4} \)
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