Properties

Label 108.3.g.a
Level $108$
Weight $3$
Character orbit 108.g
Analytic conductor $2.943$
Analytic rank $0$
Dimension $4$
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.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} - 2) q^{5} + ( - 2 \beta_{3} + \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} - 2) q^{5} + ( - 2 \beta_{3} + \beta_1 - 1) q^{7} + (5 \beta_{2} - 2 \beta_1 + 12) q^{11} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{13} + ( - \beta_{3} - 15 \beta_{2} + \beta_1 - 8) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{19} + (\beta_{3} + 17 \beta_{2} - 16) q^{23} + (6 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{25} + (14 \beta_{2} + 7 \beta_1 + 21) q^{29} + (3 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 2) q^{31} + (5 \beta_{3} - 51 \beta_{2} - 5 \beta_1 - 23) q^{35} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 14) q^{37} + (8 \beta_{3} + \beta_{2} + 7) q^{41} + 23 \beta_{2} q^{43} + (12 \beta_{2} - 3 \beta_1 + 27) q^{47} + (\beta_{3} - 26 \beta_{2} - 2 \beta_1 - 25) q^{49} + ( - 8 \beta_{3} + 24 \beta_{2} + 8 \beta_1 + 8) q^{53} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 21) q^{55} + ( - 8 \beta_{3} + 17 \beta_{2} - 25) q^{59} + ( - 6 \beta_{3} - 22 \beta_{2} + 3 \beta_1 - 3) q^{61} + ( - 32 \beta_{2} - 7 \beta_1 - 57) q^{65} + ( - 6 \beta_{3} + 61 \beta_{2} + 12 \beta_1 + 55) q^{67} + (2 \beta_{3} + 30 \beta_{2} - 2 \beta_1 + 16) q^{71} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 20) q^{73} + ( - 17 \beta_{3} - 55 \beta_{2} + 38) q^{77} + ( - 10 \beta_{3} - 44 \beta_{2} + 5 \beta_1 - 5) q^{79} + (12 \beta_{2} - 3 \beta_1 + 27) q^{83} + ( - 6 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 6) q^{85} + (8 \beta_{3} + 120 \beta_{2} - 8 \beta_1 + 64) q^{89} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 77) q^{91} + (4 \beta_{3} - 22 \beta_{2} + 26) q^{95} + ( - 4 \beta_{3} + 97 \beta_{2} + 2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 9 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 9 q^{5} - q^{7} + 36 q^{11} + 5 q^{13} + 2 q^{19} - 99 q^{23} + 13 q^{25} + 63 q^{29} - 7 q^{31} - 64 q^{37} + 18 q^{41} - 46 q^{43} + 81 q^{47} - 51 q^{49} + 90 q^{55} - 126 q^{59} + 41 q^{61} - 171 q^{65} + 116 q^{67} + 86 q^{73} + 279 q^{77} + 83 q^{79} + 81 q^{83} + 18 q^{85} - 302 q^{91} + 144 q^{95} - 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 16\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + \nu^{2} + 8\nu + 12 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 8\beta_{2} + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(55\)
\(\chi(n)\) \(1 + \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.68614 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 1.26217i
0 0 0 −6.55842 3.78651i 0 −4.55842 7.89542i 0 0 0
17.2 0 0 0 2.05842 + 1.18843i 0 4.05842 + 7.02939i 0 0 0
89.1 0 0 0 −6.55842 + 3.78651i 0 −4.55842 + 7.89542i 0 0 0
89.2 0 0 0 2.05842 1.18843i 0 4.05842 7.02939i 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
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.g.a 4
3.b odd 2 1 36.3.g.a 4
4.b odd 2 1 432.3.q.b 4
5.b even 2 1 2700.3.p.b 4
5.c odd 4 2 2700.3.u.b 8
8.b even 2 1 1728.3.q.g 4
8.d odd 2 1 1728.3.q.h 4
9.c even 3 1 36.3.g.a 4
9.c even 3 1 324.3.c.b 4
9.d odd 6 1 inner 108.3.g.a 4
9.d odd 6 1 324.3.c.b 4
12.b even 2 1 144.3.q.b 4
15.d odd 2 1 900.3.p.a 4
15.e even 4 2 900.3.u.a 8
24.f even 2 1 576.3.q.g 4
24.h odd 2 1 576.3.q.d 4
36.f odd 6 1 144.3.q.b 4
36.f odd 6 1 1296.3.e.e 4
36.h even 6 1 432.3.q.b 4
36.h even 6 1 1296.3.e.e 4
45.h odd 6 1 2700.3.p.b 4
45.j even 6 1 900.3.p.a 4
45.k odd 12 2 900.3.u.a 8
45.l even 12 2 2700.3.u.b 8
72.j odd 6 1 1728.3.q.g 4
72.l even 6 1 1728.3.q.h 4
72.n even 6 1 576.3.q.d 4
72.p odd 6 1 576.3.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 3.b odd 2 1
36.3.g.a 4 9.c even 3 1
108.3.g.a 4 1.a even 1 1 trivial
108.3.g.a 4 9.d odd 6 1 inner
144.3.q.b 4 12.b even 2 1
144.3.q.b 4 36.f odd 6 1
324.3.c.b 4 9.c even 3 1
324.3.c.b 4 9.d odd 6 1
432.3.q.b 4 4.b odd 2 1
432.3.q.b 4 36.h even 6 1
576.3.q.d 4 24.h odd 2 1
576.3.q.d 4 72.n even 6 1
576.3.q.g 4 24.f even 2 1
576.3.q.g 4 72.p odd 6 1
900.3.p.a 4 15.d odd 2 1
900.3.p.a 4 45.j even 6 1
900.3.u.a 8 15.e even 4 2
900.3.u.a 8 45.k odd 12 2
1296.3.e.e 4 36.f odd 6 1
1296.3.e.e 4 36.h even 6 1
1728.3.q.g 4 8.b even 2 1
1728.3.q.g 4 72.j odd 6 1
1728.3.q.h 4 8.d odd 2 1
1728.3.q.h 4 72.l even 6 1
2700.3.p.b 4 5.b even 2 1
2700.3.p.b 4 45.h odd 6 1
2700.3.u.b 8 5.c odd 4 2
2700.3.u.b 8 45.l even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 9 T^{3} + 9 T^{2} - 162 T + 324 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + 75 T^{2} - 74 T + 5476 \) Copy content Toggle raw display
$11$ \( T^{4} - 36 T^{3} + 441 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( T^{4} + 387 T^{2} + 20736 \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 99 T^{3} + 4059 T^{2} + \cdots + 627264 \) Copy content Toggle raw display
$29$ \( T^{4} - 63 T^{3} + 441 T^{2} + \cdots + 777924 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336 \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T - 932)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} - 1449 T^{2} + \cdots + 2424249 \) Copy content Toggle raw display
$43$ \( (T^{2} + 23 T + 529)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$53$ \( T^{4} + 4032 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121 \) Copy content Toggle raw display
$61$ \( T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504 \) Copy content Toggle raw display
$67$ \( T^{4} - 116 T^{3} + 12765 T^{2} + \cdots + 477481 \) Copy content Toggle raw display
$71$ \( T^{4} + 1548 T^{2} + 331776 \) Copy content Toggle raw display
$73$ \( (T^{2} - 43 T - 206)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 83 T^{3} + 7023 T^{2} + \cdots + 17956 \) Copy content Toggle raw display
$83$ \( T^{4} - 81 T^{3} + 2511 T^{2} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( T^{4} + 24768 T^{2} + \cdots + 84934656 \) Copy content Toggle raw display
$97$ \( T^{4} + 196 T^{3} + \cdots + 86620249 \) Copy content Toggle raw display
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