# 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 $2$

# Related objects

## 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: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{2} + 6) q^{5} + (6 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 2*b2 * q^4 + (-3*b2 + 6) * q^5 + (6*b3 - b2 - 3*b1 + 1) * q^7 + 2*b3 * q^8 $$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})$$ q + b1 * q^2 + 2*b2 * q^4 + (-3*b2 + 6) * q^5 + (6*b3 - b2 - 3*b1 + 1) * q^7 + 2*b3 * q^8 + (-3*b3 + 6*b1) * q^10 + (-3*b2 - 3*b1 - 3) * q^11 + (-6*b3 - 5*b2 - 6*b1) * q^13 + (-b3 + 6*b2 + b1 - 12) * q^14 + (4*b2 - 4) * q^16 + (-6*b3 + 12*b2 - 6) * q^17 + (-6*b3 + 12*b1 - 10) * q^19 + (6*b2 + 6) * q^20 + (-3*b3 - 6*b2 - 3*b1) * q^22 + (3*b3 + 3*b2 - 3*b1 - 6) * q^23 + (-2*b2 + 2) * q^25 + (-5*b3 - 24*b2 + 12) * q^26 + (6*b3 - 12*b1 + 2) * q^28 + (-3*b2 + 6*b1 - 3) * q^29 + (9*b3 + 19*b2 + 9*b1) * q^31 + (4*b3 - 4*b1) * q^32 + (12*b3 - 12*b2 - 6*b1 + 12) * q^34 + (27*b3 - 6*b2 + 3) * q^35 + (6*b3 - 12*b1 + 32) * q^37 + (12*b2 - 10*b1 + 12) * q^38 + (6*b3 + 6*b1) * q^40 + (-18*b3 - 21*b2 + 18*b1 + 42) * q^41 + (-18*b3 + 23*b2 + 9*b1 - 23) * q^43 + (-6*b3 - 12*b2 + 6) * q^44 + (3*b3 - 6*b1 - 6) * q^46 + (-9*b2 + 21*b1 - 9) * q^47 + (6*b3 - 6*b2 + 6*b1) * q^49 + (-2*b3 + 2*b1) * q^50 + (-24*b3 - 10*b2 + 12*b1 + 10) * q^52 + (-30*b3 + 60*b2 - 30) * q^53 + (9*b3 - 18*b1 - 27) * q^55 + (-12*b2 + 2*b1 - 12) * q^56 + (-3*b3 + 12*b2 - 3*b1) * q^58 + (39*b3 + 21*b2 - 39*b1 - 42) * q^59 + (-36*b3 - 31*b2 + 18*b1 + 31) * q^61 + (19*b3 + 36*b2 - 18) * q^62 - 8 * q^64 + (-15*b2 - 54*b1 - 15) * q^65 + (9*b3 - 53*b2 + 9*b1) * q^67 + (-12*b3 + 12*b2 + 12*b1 - 24) * q^68 + (-6*b3 + 54*b2 + 3*b1 - 54) * q^70 + (-24*b3 - 60*b2 + 30) * q^71 + (18*b3 - 36*b1 - 52) * q^73 + (-12*b2 + 32*b1 - 12) * q^74 + (12*b3 - 20*b2 + 12*b1) * q^76 + (-24*b3 - 15*b2 + 24*b1 + 30) * q^77 + (30*b3 - 7*b2 - 15*b1 + 7) * q^79 + (24*b2 - 12) * q^80 + (-21*b3 + 42*b1 + 36) * q^82 + (63*b2 - 15*b1 + 63) * q^83 + (-18*b3 + 54*b2 - 18*b1) * q^85 + (23*b3 - 18*b2 - 23*b1 + 36) * q^86 + (-12*b3 - 12*b2 + 6*b1 + 12) * q^88 + (66*b3 - 60*b2 + 30) * q^89 + (-9*b3 + 18*b1 + 103) * q^91 + (-6*b2 - 6*b1 - 6) * q^92 + (-9*b3 + 42*b2 - 9*b1) * q^94 + (-54*b3 + 30*b2 + 54*b1 - 60) * q^95 + (84*b3 - 7*b2 - 42*b1 + 7) * q^97 + (-6*b3 + 24*b2 - 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 18 q^{5} + 2 q^{7}+O(q^{10})$$ 4 * q + 4 * q^4 + 18 * q^5 + 2 * q^7 $$4 q + 4 q^{4} + 18 q^{5} + 2 q^{7} - 18 q^{11} - 10 q^{13} - 36 q^{14} - 8 q^{16} - 40 q^{19} + 36 q^{20} - 12 q^{22} - 18 q^{23} + 4 q^{25} + 8 q^{28} - 18 q^{29} + 38 q^{31} + 24 q^{34} + 128 q^{37} + 72 q^{38} + 126 q^{41} - 46 q^{43} - 24 q^{46} - 54 q^{47} - 12 q^{49} + 20 q^{52} - 108 q^{55} - 72 q^{56} + 24 q^{58} - 126 q^{59} + 62 q^{61} - 32 q^{64} - 90 q^{65} - 106 q^{67} - 72 q^{68} - 108 q^{70} - 208 q^{73} - 72 q^{74} - 40 q^{76} + 90 q^{77} + 14 q^{79} + 144 q^{82} + 378 q^{83} + 108 q^{85} + 108 q^{86} + 24 q^{88} + 412 q^{91} - 36 q^{92} + 84 q^{94} - 180 q^{95} + 14 q^{97}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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