Properties

Label 54.3.d.a
Level $54$
Weight $3$
Character orbit 54.d
Analytic conductor $1.471$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 54.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47139342755\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{2} + 6) q^{5} + (6 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{2} + 6) q^{5} + (6 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8} + ( - 3 \beta_{3} + 6 \beta_1) q^{10} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{11} + ( - 6 \beta_{3} - 5 \beta_{2} - 6 \beta_1) q^{13} + ( - \beta_{3} + 6 \beta_{2} + \beta_1 - 12) q^{14} + (4 \beta_{2} - 4) q^{16} + ( - 6 \beta_{3} + 12 \beta_{2} - 6) q^{17} + ( - 6 \beta_{3} + 12 \beta_1 - 10) q^{19} + (6 \beta_{2} + 6) q^{20} + ( - 3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{22} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 6) q^{23} + ( - 2 \beta_{2} + 2) q^{25} + ( - 5 \beta_{3} - 24 \beta_{2} + 12) q^{26} + (6 \beta_{3} - 12 \beta_1 + 2) q^{28} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{29} + (9 \beta_{3} + 19 \beta_{2} + 9 \beta_1) q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (12 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 12) q^{34} + (27 \beta_{3} - 6 \beta_{2} + 3) q^{35} + (6 \beta_{3} - 12 \beta_1 + 32) q^{37} + (12 \beta_{2} - 10 \beta_1 + 12) q^{38} + (6 \beta_{3} + 6 \beta_1) q^{40} + ( - 18 \beta_{3} - 21 \beta_{2} + 18 \beta_1 + 42) q^{41} + ( - 18 \beta_{3} + 23 \beta_{2} + 9 \beta_1 - 23) q^{43} + ( - 6 \beta_{3} - 12 \beta_{2} + 6) q^{44} + (3 \beta_{3} - 6 \beta_1 - 6) q^{46} + ( - 9 \beta_{2} + 21 \beta_1 - 9) q^{47} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{50} + ( - 24 \beta_{3} - 10 \beta_{2} + 12 \beta_1 + 10) q^{52} + ( - 30 \beta_{3} + 60 \beta_{2} - 30) q^{53} + (9 \beta_{3} - 18 \beta_1 - 27) q^{55} + ( - 12 \beta_{2} + 2 \beta_1 - 12) q^{56} + ( - 3 \beta_{3} + 12 \beta_{2} - 3 \beta_1) q^{58} + (39 \beta_{3} + 21 \beta_{2} - 39 \beta_1 - 42) q^{59} + ( - 36 \beta_{3} - 31 \beta_{2} + 18 \beta_1 + 31) q^{61} + (19 \beta_{3} + 36 \beta_{2} - 18) q^{62} - 8 q^{64} + ( - 15 \beta_{2} - 54 \beta_1 - 15) q^{65} + (9 \beta_{3} - 53 \beta_{2} + 9 \beta_1) q^{67} + ( - 12 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 24) q^{68} + ( - 6 \beta_{3} + 54 \beta_{2} + 3 \beta_1 - 54) q^{70} + ( - 24 \beta_{3} - 60 \beta_{2} + 30) q^{71} + (18 \beta_{3} - 36 \beta_1 - 52) q^{73} + ( - 12 \beta_{2} + 32 \beta_1 - 12) q^{74} + (12 \beta_{3} - 20 \beta_{2} + 12 \beta_1) q^{76} + ( - 24 \beta_{3} - 15 \beta_{2} + 24 \beta_1 + 30) q^{77} + (30 \beta_{3} - 7 \beta_{2} - 15 \beta_1 + 7) q^{79} + (24 \beta_{2} - 12) q^{80} + ( - 21 \beta_{3} + 42 \beta_1 + 36) q^{82} + (63 \beta_{2} - 15 \beta_1 + 63) q^{83} + ( - 18 \beta_{3} + 54 \beta_{2} - 18 \beta_1) q^{85} + (23 \beta_{3} - 18 \beta_{2} - 23 \beta_1 + 36) q^{86} + ( - 12 \beta_{3} - 12 \beta_{2} + 6 \beta_1 + 12) q^{88} + (66 \beta_{3} - 60 \beta_{2} + 30) q^{89} + ( - 9 \beta_{3} + 18 \beta_1 + 103) q^{91} + ( - 6 \beta_{2} - 6 \beta_1 - 6) q^{92} + ( - 9 \beta_{3} + 42 \beta_{2} - 9 \beta_1) q^{94} + ( - 54 \beta_{3} + 30 \beta_{2} + 54 \beta_1 - 60) q^{95} + (84 \beta_{3} - 7 \beta_{2} - 42 \beta_1 + 7) q^{97} + ( - 6 \beta_{3} + 24 \beta_{2} - 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 18 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 18 q^{5} + 2 q^{7} - 18 q^{11} - 10 q^{13} - 36 q^{14} - 8 q^{16} - 40 q^{19} + 36 q^{20} - 12 q^{22} - 18 q^{23} + 4 q^{25} + 8 q^{28} - 18 q^{29} + 38 q^{31} + 24 q^{34} + 128 q^{37} + 72 q^{38} + 126 q^{41} - 46 q^{43} - 24 q^{46} - 54 q^{47} - 12 q^{49} + 20 q^{52} - 108 q^{55} - 72 q^{56} + 24 q^{58} - 126 q^{59} + 62 q^{61} - 32 q^{64} - 90 q^{65} - 106 q^{67} - 72 q^{68} - 108 q^{70} - 208 q^{73} - 72 q^{74} - 40 q^{76} + 90 q^{77} + 14 q^{79} + 144 q^{82} + 378 q^{83} + 108 q^{85} + 108 q^{86} + 24 q^{88} + 412 q^{91} - 36 q^{92} + 84 q^{94} - 180 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(\beta_{2}\)

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
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i 4.50000 + 2.59808i 0 4.17423 + 7.22999i 2.82843i 0 −7.34847
17.2 1.22474 0.707107i 0 1.00000 1.73205i 4.50000 + 2.59808i 0 −3.17423 5.49794i 2.82843i 0 7.34847
35.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 4.50000 2.59808i 0 4.17423 7.22999i 2.82843i 0 −7.34847
35.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 4.50000 2.59808i 0 −3.17423 + 5.49794i 2.82843i 0 7.34847
\(n\): 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 54.3.d.a 4
3.b odd 2 1 18.3.d.a 4
4.b odd 2 1 432.3.q.d 4
5.b even 2 1 1350.3.i.b 4
5.c odd 4 2 1350.3.k.a 8
8.b even 2 1 1728.3.q.d 4
8.d odd 2 1 1728.3.q.c 4
9.c even 3 1 18.3.d.a 4
9.c even 3 1 162.3.b.a 4
9.d odd 6 1 inner 54.3.d.a 4
9.d odd 6 1 162.3.b.a 4
12.b even 2 1 144.3.q.c 4
15.d odd 2 1 450.3.i.b 4
15.e even 4 2 450.3.k.a 8
24.f even 2 1 576.3.q.e 4
24.h odd 2 1 576.3.q.f 4
36.f odd 6 1 144.3.q.c 4
36.f odd 6 1 1296.3.e.g 4
36.h even 6 1 432.3.q.d 4
36.h even 6 1 1296.3.e.g 4
45.h odd 6 1 1350.3.i.b 4
45.j even 6 1 450.3.i.b 4
45.k odd 12 2 450.3.k.a 8
45.l even 12 2 1350.3.k.a 8
72.j odd 6 1 1728.3.q.d 4
72.l even 6 1 1728.3.q.c 4
72.n even 6 1 576.3.q.f 4
72.p odd 6 1 576.3.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 3.b odd 2 1
18.3.d.a 4 9.c even 3 1
54.3.d.a 4 1.a even 1 1 trivial
54.3.d.a 4 9.d odd 6 1 inner
144.3.q.c 4 12.b even 2 1
144.3.q.c 4 36.f odd 6 1
162.3.b.a 4 9.c even 3 1
162.3.b.a 4 9.d odd 6 1
432.3.q.d 4 4.b odd 2 1
432.3.q.d 4 36.h even 6 1
450.3.i.b 4 15.d odd 2 1
450.3.i.b 4 45.j even 6 1
450.3.k.a 8 15.e even 4 2
450.3.k.a 8 45.k odd 12 2
576.3.q.e 4 24.f even 2 1
576.3.q.e 4 72.p odd 6 1
576.3.q.f 4 24.h odd 2 1
576.3.q.f 4 72.n even 6 1
1296.3.e.g 4 36.f odd 6 1
1296.3.e.g 4 36.h even 6 1
1350.3.i.b 4 5.b even 2 1
1350.3.i.b 4 45.h odd 6 1
1350.3.k.a 8 5.c odd 4 2
1350.3.k.a 8 45.l even 12 2
1728.3.q.c 4 8.d odd 2 1
1728.3.q.c 4 72.l even 6 1
1728.3.q.d 4 8.b even 2 1
1728.3.q.d 4 72.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(54, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 57 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 291 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 18 T^{3} + 63 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( (T^{2} - 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 126 T^{3} + 5967 T^{2} + \cdots + 455625 \) Copy content Toggle raw display
$43$ \( T^{4} + 46 T^{3} + 2073 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} + 54 T^{3} + 333 T^{2} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 126 T^{3} + 3573 T^{2} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289 \) Copy content Toggle raw display
$67$ \( T^{4} + 106 T^{3} + 8913 T^{2} + \cdots + 5396329 \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + \cdots + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} + 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 14 T^{3} + 1497 T^{2} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{4} - 378 T^{3} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + \cdots + 36144144 \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + \cdots + 110986225 \) Copy content Toggle raw display
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