LMFDB - Newform orbit 546.4.l.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(211,546)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("546.211"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.l (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,-3,-4,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.2150428631$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (2 \zeta_{6} - 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + 7 q^{5} - 6 \zeta_{6} q^{6} + 7 \zeta_{6} q^{7} + 8 q^{8} - 9 \zeta_{6} q^{9} + (14 \zeta_{6} - 14) q^{10} + ( - 16 \zeta_{6} + 16) q^{11} + \cdots - 144 q^{99} +O(q^{100}) $ q + (2z - 2) q^2 + (3z - 3) q^3 - 4z q^4 + 7 * q^5 - 6z q^6 + 7z q^7 + 8 * q^8 - 9z q^9 + (14z - 14) q^10 + (-16z + 16) q^11 + 12 * q^12 + (13z + 39) q^13 - 14 * q^14 + (21z - 21) q^15 + (16z - 16) q^16 + 33z q^17 + 18 * q^18 - 144z q^19 - 28z q^20 - 21 * q^21 + 32z q^22 + (-44z + 44) q^23 + (24z - 24) q^24 - 76 * q^25 + (78z - 104) q^26 + 27 * q^27 + (-28z + 28) q^28 + (-185z + 185) q^29 - 42z q^30 + 184 * q^31 - 32z q^32 + 48z q^33 - 66 * q^34 + 49z q^35 + (36z - 36) q^36 + (-225z + 225) q^37 + 288 * q^38 + (117z - 156) q^39 + 56 * q^40 + (-165z + 165) q^41 + (-42z + 42) q^42 + 20z q^43 - 64 * q^44 - 63z q^45 + 88z q^46 - 88 * q^47 - 48z q^48 + (49z - 49) q^49 + (-152z + 152) q^50 - 99 * q^51 + (-208z + 52) q^52 + 111 * q^53 + (54z - 54) q^54 + (-112z + 112) q^55 + 56z q^56 + 432 * q^57 + 370z q^58 - 220z q^59 + 84 * q^60 + 269z q^61 + (368z - 368) q^62 + (-63z + 63) q^63 + 64 * q^64 + (91z + 273) q^65 - 96 * q^66 + (700z - 700) q^67 + (-132z + 132) q^68 + 132z q^69 - 98 * q^70 - 1016z q^71 - 72z q^72 + 947 * q^73 + 450z q^74 + (-228z + 228) q^75 + (576z - 576) q^76 + 112 * q^77 + (-312z + 78) q^78 + 1244 * q^79 + (112z - 112) q^80 + (81z - 81) q^81 + 330z q^82 + 1140 * q^83 + 84z q^84 + 231z q^85 - 40 * q^86 + 555z q^87 + (-128z + 128) q^88 + (858z - 858) q^89 + 126 * q^90 + (364z - 91) q^91 - 176 * q^92 + (552z - 552) q^93 + (-176z + 176) q^94 - 1008z q^95 + 96 * q^96 + 862z q^97 - 98z q^98 - 144 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} + 14 q^{5} - 6 q^{6} + 7 q^{7} + 16 q^{8} - 9 q^{9} - 14 q^{10} + 16 q^{11} + 24 q^{12} + 91 q^{13} - 28 q^{14} - 21 q^{15} - 16 q^{16} + 33 q^{17} + 36 q^{18} - 144 q^{19}+ \cdots - 288 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 3 * q^3 - 4 * q^4 + 14 * q^5 - 6 * q^6 + 7 * q^7 + 16 * q^8 - 9 * q^9 - 14 * q^10 + 16 * q^11 + 24 * q^12 + 91 * q^13 - 28 * q^14 - 21 * q^15 - 16 * q^16 + 33 * q^17 + 36 * q^18 - 144 * q^19 - 28 * q^20 - 42 * q^21 + 32 * q^22 + 44 * q^23 - 24 * q^24 - 152 * q^25 - 130 * q^26 + 54 * q^27 + 28 * q^28 + 185 * q^29 - 42 * q^30 + 368 * q^31 - 32 * q^32 + 48 * q^33 - 132 * q^34 + 49 * q^35 - 36 * q^36 + 225 * q^37 + 576 * q^38 - 195 * q^39 + 112 * q^40 + 165 * q^41 + 42 * q^42 + 20 * q^43 - 128 * q^44 - 63 * q^45 + 88 * q^46 - 176 * q^47 - 48 * q^48 - 49 * q^49 + 152 * q^50 - 198 * q^51 - 104 * q^52 + 222 * q^53 - 54 * q^54 + 112 * q^55 + 56 * q^56 + 864 * q^57 + 370 * q^58 - 220 * q^59 + 168 * q^60 + 269 * q^61 - 368 * q^62 + 63 * q^63 + 128 * q^64 + 637 * q^65 - 192 * q^66 - 700 * q^67 + 132 * q^68 + 132 * q^69 - 196 * q^70 - 1016 * q^71 - 72 * q^72 + 1894 * q^73 + 450 * q^74 + 228 * q^75 - 576 * q^76 + 224 * q^77 - 156 * q^78 + 2488 * q^79 - 112 * q^80 - 81 * q^81 + 330 * q^82 + 2280 * q^83 + 84 * q^84 + 231 * q^85 - 80 * q^86 + 555 * q^87 + 128 * q^88 - 858 * q^89 + 252 * q^90 + 182 * q^91 - 352 * q^92 - 552 * q^93 + 176 * q^94 - 1008 * q^95 + 192 * q^96 + 862 * q^97 - 98 * q^98 - 288 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$.

$n$	$157$	$365$	$379$
$\chi(n)$	$1$	$1$	$-\zeta_{6}$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

211.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

−1.50000

−

2.59808i

−2.00000

3.46410i

7.00000

−3.00000

5.19615i

3.50000

−

6.06218i

8.00000

−4.50000

7.79423i

−7.00000

−

12.1244i

295.1

−1.00000

1.73205i

−1.50000

2.59808i

−2.00000

−

3.46410i

7.00000

−3.00000

−

5.19615i

3.50000

6.06218i

8.00000

−4.50000

−

7.79423i

−7.00000

12.1244i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	546.4.l.a	✓	2
13.c	even	3	1	inner	546.4.l.a	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.4.l.a	✓	2	1.a	even	1	1	trivial
546.4.l.a	✓	2	13.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 7 $ T5 - 7 acting on $S_{4}^{\mathrm{new}}(546, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$3$	$ T^{2} + 3T + 9 $ T^2 + 3*T + 9
$5$	$ (T - 7)^{2} $ (T - 7)^2
$7$	$ T^{2} - 7T + 49 $ T^2 - 7*T + 49
$11$	$ T^{2} - 16T + 256 $ T^2 - 16*T + 256
$13$	$ T^{2} - 91T + 2197 $ T^2 - 91*T + 2197
$17$	$ T^{2} - 33T + 1089 $ T^2 - 33*T + 1089
$19$	$ T^{2} + 144T + 20736 $ T^2 + 144*T + 20736
$23$	$ T^{2} - 44T + 1936 $ T^2 - 44*T + 1936
$29$	$ T^{2} - 185T + 34225 $ T^2 - 185*T + 34225
$31$	$ (T - 184)^{2} $ (T - 184)^2
$37$	$ T^{2} - 225T + 50625 $ T^2 - 225*T + 50625
$41$	$ T^{2} - 165T + 27225 $ T^2 - 165*T + 27225
$43$	$ T^{2} - 20T + 400 $ T^2 - 20*T + 400
$47$	$ (T + 88)^{2} $ (T + 88)^2
$53$	$ (T - 111)^{2} $ (T - 111)^2
$59$	$ T^{2} + 220T + 48400 $ T^2 + 220*T + 48400
$61$	$ T^{2} - 269T + 72361 $ T^2 - 269*T + 72361
$67$	$ T^{2} + 700T + 490000 $ T^2 + 700*T + 490000
$71$	$ T^{2} + 1016 T + 1032256 $ T^2 + 1016*T + 1032256
$73$	$ (T - 947)^{2} $ (T - 947)^2
$79$	$ (T - 1244)^{2} $ (T - 1244)^2
$83$	$ (T - 1140)^{2} $ (T - 1140)^2
$89$	$ T^{2} + 858T + 736164 $ T^2 + 858*T + 736164
$97$	$ T^{2} - 862T + 743044 $ T^2 - 862*T + 743044
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Elliptic curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.l (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + 7 q^{5} - 6 \zeta_{6} q^{6} + 7 \zeta_{6} q^{7} + 8 q^{8} - 9 \zeta_{6} q^{9} + (14 \zeta_{6} - 14) q^{10} + ( - 16 \zeta_{6} + 16) q^{11} + \cdots - 144 q^{99} +O(q^{100}) \) q + (2z - 2) q^2 + (3z - 3) q^3 - 4z q^4 + 7 * q^5 - 6z q^6 + 7z q^7 + 8 * q^8 - 9z q^9 + (14z - 14) q^10 + (-16z + 16) q^11 + 12 * q^12 + (13z + 39) q^13 - 14 * q^14 + (21z - 21) q^15 + (16z - 16) q^16 + 33z q^17 + 18 * q^18 - 144z q^19 - 28z q^20 - 21 * q^21 + 32z q^22 + (-44z + 44) q^23 + (24z - 24) q^24 - 76 * q^25 + (78z - 104) q^26 + 27 * q^27 + (-28z + 28) q^28 + (-185z + 185) q^29 - 42z q^30 + 184 * q^31 - 32z q^32 + 48z q^33 - 66 * q^34 + 49z q^35 + (36z - 36) q^36 + (-225z + 225) q^37 + 288 * q^38 + (117z - 156) q^39 + 56 * q^40 + (-165z + 165) q^41 + (-42z + 42) q^42 + 20z q^43 - 64 * q^44 - 63z q^45 + 88z q^46 - 88 * q^47 - 48z q^48 + (49z - 49) q^49 + (-152z + 152) q^50 - 99 * q^51 + (-208z + 52) q^52 + 111 * q^53 + (54z - 54) q^54 + (-112z + 112) q^55 + 56z q^56 + 432 * q^57 + 370z q^58 - 220z q^59 + 84 * q^60 + 269z q^61 + (368z - 368) q^62 + (-63z + 63) q^63 + 64 * q^64 + (91z + 273) q^65 - 96 * q^66 + (700z - 700) q^67 + (-132z + 132) q^68 + 132z q^69 - 98 * q^70 - 1016z q^71 - 72z q^72 + 947 * q^73 + 450z q^74 + (-228z + 228) q^75 + (576z - 576) q^76 + 112 * q^77 + (-312z + 78) q^78 + 1244 * q^79 + (112z - 112) q^80 + (81z - 81) q^81 + 330z q^82 + 1140 * q^83 + 84z q^84 + 231z q^85 - 40 * q^86 + 555z q^87 + (-128z + 128) q^88 + (858z - 858) q^89 + 126 * q^90 + (364z - 91) q^91 - 176 * q^92 + (552z - 552) q^93 + (-176z + 176) q^94 - 1008z q^95 + 96 * q^96 + 862z q^97 - 98z q^98 - 144 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} + 14 q^{5} - 6 q^{6} + 7 q^{7} + 16 q^{8} - 9 q^{9} - 14 q^{10} + 16 q^{11} + 24 q^{12} + 91 q^{13} - 28 q^{14} - 21 q^{15} - 16 q^{16} + 33 q^{17} + 36 q^{18} - 144 q^{19}+ \cdots - 288 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 3 * q^3 - 4 * q^4 + 14 * q^5 - 6 * q^6 + 7 * q^7 + 16 * q^8 - 9 * q^9 - 14 * q^10 + 16 * q^11 + 24 * q^12 + 91 * q^13 - 28 * q^14 - 21 * q^15 - 16 * q^16 + 33 * q^17 + 36 * q^18 - 144 * q^19 - 28 * q^20 - 42 * q^21 + 32 * q^22 + 44 * q^23 - 24 * q^24 - 152 * q^25 - 130 * q^26 + 54 * q^27 + 28 * q^28 + 185 * q^29 - 42 * q^30 + 368 * q^31 - 32 * q^32 + 48 * q^33 - 132 * q^34 + 49 * q^35 - 36 * q^36 + 225 * q^37 + 576 * q^38 - 195 * q^39 + 112 * q^40 + 165 * q^41 + 42 * q^42 + 20 * q^43 - 128 * q^44 - 63 * q^45 + 88 * q^46 - 176 * q^47 - 48 * q^48 - 49 * q^49 + 152 * q^50 - 198 * q^51 - 104 * q^52 + 222 * q^53 - 54 * q^54 + 112 * q^55 + 56 * q^56 + 864 * q^57 + 370 * q^58 - 220 * q^59 + 168 * q^60 + 269 * q^61 - 368 * q^62 + 63 * q^63 + 128 * q^64 + 637 * q^65 - 192 * q^66 - 700 * q^67 + 132 * q^68 + 132 * q^69 - 196 * q^70 - 1016 * q^71 - 72 * q^72 + 1894 * q^73 + 450 * q^74 + 228 * q^75 - 576 * q^76 + 224 * q^77 - 156 * q^78 + 2488 * q^79 - 112 * q^80 - 81 * q^81 + 330 * q^82 + 2280 * q^83 + 84 * q^84 + 231 * q^85 - 80 * q^86 + 555 * q^87 + 128 * q^88 - 858 * q^89 + 252 * q^90 + 182 * q^91 - 352 * q^92 - 552 * q^93 + 176 * q^94 - 1008 * q^95 + 192 * q^96 + 862 * q^97 - 98 * q^98 - 288 * q^99

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