Properties

Label 108.3.d.a
Level $108$
Weight $3$
Character orbit 108.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta - 2) q^{4} + 7 q^{5} + 5 \beta q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta - 2) q^{4} + 7 q^{5} + 5 \beta q^{7} + 8 q^{8} + ( - 7 \beta - 7) q^{10} - 5 \beta q^{11} + 20 q^{13} + ( - 5 \beta + 15) q^{14} + ( - 8 \beta - 8) q^{16} - 8 q^{17} - 6 \beta q^{19} + (14 \beta - 14) q^{20} + (5 \beta - 15) q^{22} + 2 \beta q^{23} + 24 q^{25} + ( - 20 \beta - 20) q^{26} + ( - 10 \beta - 30) q^{28} + 10 q^{29} + 31 \beta q^{31} + (16 \beta - 16) q^{32} + (8 \beta + 8) q^{34} + 35 \beta q^{35} - 10 q^{37} + (6 \beta - 18) q^{38} + 56 q^{40} - 50 q^{41} - 10 \beta q^{43} + (10 \beta + 30) q^{44} + ( - 2 \beta + 6) q^{46} - 50 \beta q^{47} - 26 q^{49} + ( - 24 \beta - 24) q^{50} + (40 \beta - 40) q^{52} - 47 q^{53} - 35 \beta q^{55} + 40 \beta q^{56} + ( - 10 \beta - 10) q^{58} + 20 \beta q^{59} - 64 q^{61} + ( - 31 \beta + 93) q^{62} + 64 q^{64} + 140 q^{65} - 50 \beta q^{67} + ( - 16 \beta + 16) q^{68} + ( - 35 \beta + 105) q^{70} - 55 q^{73} + (10 \beta + 10) q^{74} + (12 \beta + 36) q^{76} + 75 q^{77} - 4 \beta q^{79} + ( - 56 \beta - 56) q^{80} + (50 \beta + 50) q^{82} - 17 \beta q^{83} - 56 q^{85} + (10 \beta - 30) q^{86} - 40 \beta q^{88} + 10 q^{89} + 100 \beta q^{91} + ( - 4 \beta - 12) q^{92} + (50 \beta - 150) q^{94} - 42 \beta q^{95} - 25 q^{97} + (26 \beta + 26) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 14 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 14 q^{5} + 16 q^{8} - 14 q^{10} + 40 q^{13} + 30 q^{14} - 16 q^{16} - 16 q^{17} - 28 q^{20} - 30 q^{22} + 48 q^{25} - 40 q^{26} - 60 q^{28} + 20 q^{29} - 32 q^{32} + 16 q^{34} - 20 q^{37} - 36 q^{38} + 112 q^{40} - 100 q^{41} + 60 q^{44} + 12 q^{46} - 52 q^{49} - 48 q^{50} - 80 q^{52} - 94 q^{53} - 20 q^{58} - 128 q^{61} + 186 q^{62} + 128 q^{64} + 280 q^{65} + 32 q^{68} + 210 q^{70} - 110 q^{73} + 20 q^{74} + 72 q^{76} + 150 q^{77} - 112 q^{80} + 100 q^{82} - 112 q^{85} - 60 q^{86} + 20 q^{89} - 24 q^{92} - 300 q^{94} - 50 q^{97} + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 7.00000 0 8.66025i 8.00000 0 −7.00000 12.1244i
55.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.00000 0 8.66025i 8.00000 0 −7.00000 + 12.1244i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.d.a 2
3.b odd 2 1 108.3.d.b yes 2
4.b odd 2 1 inner 108.3.d.a 2
8.b even 2 1 1728.3.g.a 2
8.d odd 2 1 1728.3.g.a 2
9.c even 3 1 324.3.f.c 2
9.c even 3 1 324.3.f.i 2
9.d odd 6 1 324.3.f.b 2
9.d odd 6 1 324.3.f.h 2
12.b even 2 1 108.3.d.b yes 2
24.f even 2 1 1728.3.g.f 2
24.h odd 2 1 1728.3.g.f 2
36.f odd 6 1 324.3.f.c 2
36.f odd 6 1 324.3.f.i 2
36.h even 6 1 324.3.f.b 2
36.h even 6 1 324.3.f.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 1.a even 1 1 trivial
108.3.d.a 2 4.b odd 2 1 inner
108.3.d.b yes 2 3.b odd 2 1
108.3.d.b yes 2 12.b even 2 1
324.3.f.b 2 9.d odd 6 1
324.3.f.b 2 36.h even 6 1
324.3.f.c 2 9.c even 3 1
324.3.f.c 2 36.f odd 6 1
324.3.f.h 2 9.d odd 6 1
324.3.f.h 2 36.h even 6 1
324.3.f.i 2 9.c even 3 1
324.3.f.i 2 36.f odd 6 1
1728.3.g.a 2 8.b even 2 1
1728.3.g.a 2 8.d odd 2 1
1728.3.g.f 2 24.f even 2 1
1728.3.g.f 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 7 \) acting on \(S_{3}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 75 \) Copy content Toggle raw display
$11$ \( T^{2} + 75 \) Copy content Toggle raw display
$13$ \( (T - 20)^{2} \) Copy content Toggle raw display
$17$ \( (T + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 108 \) Copy content Toggle raw display
$23$ \( T^{2} + 12 \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2883 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( (T + 50)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 300 \) Copy content Toggle raw display
$47$ \( T^{2} + 7500 \) Copy content Toggle raw display
$53$ \( (T + 47)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1200 \) Copy content Toggle raw display
$61$ \( (T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 7500 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 55)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 48 \) Copy content Toggle raw display
$83$ \( T^{2} + 867 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( (T + 25)^{2} \) Copy content Toggle raw display
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