LMFDB - Newform orbit 84.3.l.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [84,3,Mod(67,84)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("84.67"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	84.l (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.28883422063$
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:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 q^{2} + (\zeta_{6} - 2) q^{3} + 4 q^{4} + ( - 5 \zeta_{6} + 5) q^{5} + (2 \zeta_{6} - 4) q^{6} + 7 \zeta_{6} q^{7} + 8 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + (3 \zeta_{6} - 6) q^{11}+ \cdots + (18 \zeta_{6} - 9) q^{99}+O(q^{100}) $ q + 2 * q^2 + (z - 2) * q^3 + 4 * q^4 + (-5z + 5) q^5 + (2z - 4) q^6 + 7z q^7 + 8 * q^8 + (-3z + 3) q^9 + (-10z + 10) q^10 + (3z - 6) q^11 + (4z - 8) q^12 - 16 * q^13 + 14z q^14 + (10z - 5) q^15 + 16 * q^16 - 4z q^17 + (-6z + 6) q^18 + (-16z - 16) q^19 + (-20z + 20) q^20 + (-7z - 7) q^21 + (6z - 12) q^22 + (-12z - 12) q^23 + (8z - 16) q^24 - 32 * q^26 + (6z - 3) q^27 + 28z q^28 + 37 * q^29 + (20z - 10) q^30 + (11z - 22) q^31 + 32 * q^32 + (-9z + 9) q^33 - 8z q^34 + 35 * q^35 + (-12z + 12) q^36 + (68z - 68) q^37 + (-32z - 32) q^38 + (-16z + 32) q^39 + (-40z + 40) q^40 + 40 * q^41 + (-14z - 14) q^42 + (-28z + 14) q^43 + (12z - 24) q^44 - 15z q^45 + (-24z - 24) q^46 + (26z + 26) q^47 + (16z - 32) q^48 + (49z - 49) q^49 + (4z + 4) q^51 - 64 * q^52 - 73z q^53 + (12z - 6) q^54 + (30z - 15) q^55 + 56z q^56 + 48 * q^57 + 74 * q^58 + (-17z + 34) q^59 + (40z - 20) q^60 + (-82z + 82) q^61 + (22z - 44) q^62 + 21 * q^63 + 64 * q^64 + (80z - 80) q^65 + (-18z + 18) q^66 + (-46z + 92) q^67 - 16z q^68 + 36 * q^69 + 70 * q^70 + (-84z + 42) q^71 + (-24z + 24) q^72 + 94z q^73 + (136z - 136) q^74 + (-64z - 64) q^76 + (-21z - 21) q^77 + (-32z + 64) q^78 + (-5z - 5) q^79 + (-80z + 80) q^80 - 9z q^81 + 80 * q^82 + (154z - 77) q^83 + (-28z - 28) q^84 - 20 * q^85 + (-56z + 28) q^86 + (37z - 74) q^87 + (24z - 48) q^88 + (58z - 58) q^89 - 30z q^90 - 112z q^91 + (-48z - 48) q^92 + (-33z + 33) q^93 + (52z + 52) q^94 + (80z - 160) q^95 + (32z - 64) q^96 + 47 * q^97 + (98z - 98) q^98 + (18z - 9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} + 5 q^{5} - 6 q^{6} + 7 q^{7} + 16 q^{8} + 3 q^{9} + 10 q^{10} - 9 q^{11} - 12 q^{12} - 32 q^{13} + 14 q^{14} + 32 q^{16} - 4 q^{17} + 6 q^{18} - 48 q^{19} + 20 q^{20}+ \cdots - 98 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 3 * q^3 + 8 * q^4 + 5 * q^5 - 6 * q^6 + 7 * q^7 + 16 * q^8 + 3 * q^9 + 10 * q^10 - 9 * q^11 - 12 * q^12 - 32 * q^13 + 14 * q^14 + 32 * q^16 - 4 * q^17 + 6 * q^18 - 48 * q^19 + 20 * q^20 - 21 * q^21 - 18 * q^22 - 36 * q^23 - 24 * q^24 - 64 * q^26 + 28 * q^28 + 74 * q^29 - 33 * q^31 + 64 * q^32 + 9 * q^33 - 8 * q^34 + 70 * q^35 + 12 * q^36 - 68 * q^37 - 96 * q^38 + 48 * q^39 + 40 * q^40 + 80 * q^41 - 42 * q^42 - 36 * q^44 - 15 * q^45 - 72 * q^46 + 78 * q^47 - 48 * q^48 - 49 * q^49 + 12 * q^51 - 128 * q^52 - 73 * q^53 + 56 * q^56 + 96 * q^57 + 148 * q^58 + 51 * q^59 + 82 * q^61 - 66 * q^62 + 42 * q^63 + 128 * q^64 - 80 * q^65 + 18 * q^66 + 138 * q^67 - 16 * q^68 + 72 * q^69 + 140 * q^70 + 24 * q^72 + 94 * q^73 - 136 * q^74 - 192 * q^76 - 63 * q^77 + 96 * q^78 - 15 * q^79 + 80 * q^80 - 9 * q^81 + 160 * q^82 - 84 * q^84 - 40 * q^85 - 111 * q^87 - 72 * q^88 - 58 * q^89 - 30 * q^90 - 112 * q^91 - 144 * q^92 + 33 * q^93 + 156 * q^94 - 240 * q^95 - 96 * q^96 + 94 * q^97 - 98 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$-1$	$-1 + \zeta_{6}$

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} $

67.1

0.500000	−	0.866025i
0.500000	+	0.866025i

2.00000

−1.50000

−

0.866025i

4.00000

2.50000

4.33013i

−3.00000

−

1.73205i

3.50000

−

6.06218i

8.00000

1.50000

2.59808i

5.00000

8.66025i

79.1

2.00000

−1.50000

0.866025i

4.00000

2.50000

−

4.33013i

−3.00000

1.73205i

3.50000

6.06218i

8.00000

1.50000

−

2.59808i

5.00000

−

8.66025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	84.3.l.b	yes	2
3.b	odd	2	1		252.3.y.a		2
4.b	odd	2	1		84.3.l.a	✓	2
7.c	even	3	1		84.3.l.a	✓	2
7.c	even	3	1		588.3.g.a		2
7.d	odd	6	1		588.3.g.c		2
12.b	even	2	1		252.3.y.b		2
21.h	odd	6	1		252.3.y.b		2
28.f	even	6	1		588.3.g.c		2
28.g	odd	6	1	inner	84.3.l.b	yes	2
28.g	odd	6	1		588.3.g.a		2
84.n	even	6	1		252.3.y.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.3.l.a	✓	2	4.b	odd	2	1
84.3.l.a	✓	2	7.c	even	3	1
84.3.l.b	yes	2	1.a	even	1	1	trivial
84.3.l.b	yes	2	28.g	odd	6	1	inner
252.3.y.a		2	3.b	odd	2	1
252.3.y.a		2	84.n	even	6	1
252.3.y.b		2	12.b	even	2	1
252.3.y.b		2	21.h	odd	6	1
588.3.g.a		2	7.c	even	3	1
588.3.g.a		2	28.g	odd	6	1
588.3.g.c		2	7.d	odd	6	1
588.3.g.c		2	28.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(84, [\chi])$:

$ T_{5}^{2} - 5T_{5} + 25 $

T5^2 - 5*T5 + 25

$ T_{11}^{2} + 9T_{11} + 27 $

T11^2 + 9*T11 + 27

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 2)^{2} $ (T - 2)^2
$3$	$ T^{2} + 3T + 3 $ T^2 + 3*T + 3
$5$	$ T^{2} - 5T + 25 $ T^2 - 5*T + 25
$7$	$ T^{2} - 7T + 49 $ T^2 - 7*T + 49
$11$	$ T^{2} + 9T + 27 $ T^2 + 9*T + 27
$13$	$ (T + 16)^{2} $ (T + 16)^2
$17$	$ T^{2} + 4T + 16 $ T^2 + 4*T + 16
$19$	$ T^{2} + 48T + 768 $ T^2 + 48*T + 768
$23$	$ T^{2} + 36T + 432 $ T^2 + 36*T + 432
$29$	$ (T - 37)^{2} $ (T - 37)^2
$31$	$ T^{2} + 33T + 363 $ T^2 + 33*T + 363
$37$	$ T^{2} + 68T + 4624 $ T^2 + 68*T + 4624
$41$	$ (T - 40)^{2} $ (T - 40)^2
$43$	$ T^{2} + 588 $ T^2 + 588
$47$	$ T^{2} - 78T + 2028 $ T^2 - 78*T + 2028
$53$	$ T^{2} + 73T + 5329 $ T^2 + 73*T + 5329
$59$	$ T^{2} - 51T + 867 $ T^2 - 51*T + 867
$61$	$ T^{2} - 82T + 6724 $ T^2 - 82*T + 6724
$67$	$ T^{2} - 138T + 6348 $ T^2 - 138*T + 6348
$71$	$ T^{2} + 5292 $ T^2 + 5292
$73$	$ T^{2} - 94T + 8836 $ T^2 - 94*T + 8836
$79$	$ T^{2} + 15T + 75 $ T^2 + 15*T + 75
$83$	$ T^{2} + 17787 $ T^2 + 17787
$89$	$ T^{2} + 58T + 3364 $ T^2 + 58*T + 3364
$97$	$ (T - 47)^{2} $ (T - 47)^2
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Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	84.l (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + 2 q^{2} + (\zeta_{6} - 2) q^{3} + 4 q^{4} + ( - 5 \zeta_{6} + 5) q^{5} + (2 \zeta_{6} - 4) q^{6} + 7 \zeta_{6} q^{7} + 8 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + (3 \zeta_{6} - 6) q^{11}+ \cdots + (18 \zeta_{6} - 9) q^{99}+O(q^{100}) \) q + 2 * q^2 + (z - 2) * q^3 + 4 * q^4 + (-5z + 5) q^5 + (2z - 4) q^6 + 7z q^7 + 8 * q^8 + (-3z + 3) q^9 + (-10z + 10) q^10 + (3z - 6) q^11 + (4z - 8) q^12 - 16 * q^13 + 14z q^14 + (10z - 5) q^15 + 16 * q^16 - 4z q^17 + (-6z + 6) q^18 + (-16z - 16) q^19 + (-20z + 20) q^20 + (-7z - 7) q^21 + (6z - 12) q^22 + (-12z - 12) q^23 + (8z - 16) q^24 - 32 * q^26 + (6z - 3) q^27 + 28z q^28 + 37 * q^29 + (20z - 10) q^30 + (11z - 22) q^31 + 32 * q^32 + (-9z + 9) q^33 - 8z q^34 + 35 * q^35 + (-12z + 12) q^36 + (68z - 68) q^37 + (-32z - 32) q^38 + (-16z + 32) q^39 + (-40z + 40) q^40 + 40 * q^41 + (-14z - 14) q^42 + (-28z + 14) q^43 + (12z - 24) q^44 - 15z q^45 + (-24z - 24) q^46 + (26z + 26) q^47 + (16z - 32) q^48 + (49z - 49) q^49 + (4z + 4) q^51 - 64 * q^52 - 73z q^53 + (12z - 6) q^54 + (30z - 15) q^55 + 56z q^56 + 48 * q^57 + 74 * q^58 + (-17z + 34) q^59 + (40z - 20) q^60 + (-82z + 82) q^61 + (22z - 44) q^62 + 21 * q^63 + 64 * q^64 + (80z - 80) q^65 + (-18z + 18) q^66 + (-46z + 92) q^67 - 16z q^68 + 36 * q^69 + 70 * q^70 + (-84z + 42) q^71 + (-24z + 24) q^72 + 94z q^73 + (136z - 136) q^74 + (-64z - 64) q^76 + (-21z - 21) q^77 + (-32z + 64) q^78 + (-5z - 5) q^79 + (-80z + 80) q^80 - 9z q^81 + 80 * q^82 + (154z - 77) q^83 + (-28z - 28) q^84 - 20 * q^85 + (-56z + 28) q^86 + (37z - 74) q^87 + (24z - 48) q^88 + (58z - 58) q^89 - 30z q^90 - 112z q^91 + (-48z - 48) q^92 + (-33z + 33) q^93 + (52z + 52) q^94 + (80z - 160) q^95 + (32z - 64) q^96 + 47 * q^97 + (98z - 98) q^98 + (18z - 9) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} + 5 q^{5} - 6 q^{6} + 7 q^{7} + 16 q^{8} + 3 q^{9} + 10 q^{10} - 9 q^{11} - 12 q^{12} - 32 q^{13} + 14 q^{14} + 32 q^{16} - 4 q^{17} + 6 q^{18} - 48 q^{19} + 20 q^{20}+ \cdots - 98 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 3 * q^3 + 8 * q^4 + 5 * q^5 - 6 * q^6 + 7 * q^7 + 16 * q^8 + 3 * q^9 + 10 * q^10 - 9 * q^11 - 12 * q^12 - 32 * q^13 + 14 * q^14 + 32 * q^16 - 4 * q^17 + 6 * q^18 - 48 * q^19 + 20 * q^20 - 21 * q^21 - 18 * q^22 - 36 * q^23 - 24 * q^24 - 64 * q^26 + 28 * q^28 + 74 * q^29 - 33 * q^31 + 64 * q^32 + 9 * q^33 - 8 * q^34 + 70 * q^35 + 12 * q^36 - 68 * q^37 - 96 * q^38 + 48 * q^39 + 40 * q^40 + 80 * q^41 - 42 * q^42 - 36 * q^44 - 15 * q^45 - 72 * q^46 + 78 * q^47 - 48 * q^48 - 49 * q^49 + 12 * q^51 - 128 * q^52 - 73 * q^53 + 56 * q^56 + 96 * q^57 + 148 * q^58 + 51 * q^59 + 82 * q^61 - 66 * q^62 + 42 * q^63 + 128 * q^64 - 80 * q^65 + 18 * q^66 + 138 * q^67 - 16 * q^68 + 72 * q^69 + 140 * q^70 + 24 * q^72 + 94 * q^73 - 136 * q^74 - 192 * q^76 - 63 * q^77 + 96 * q^78 - 15 * q^79 + 80 * q^80 - 9 * q^81 + 160 * q^82 - 84 * q^84 - 40 * q^85 - 111 * q^87 - 72 * q^88 - 58 * q^89 - 30 * q^90 - 112 * q^91 - 144 * q^92 + 33 * q^93 + 156 * q^94 - 240 * q^95 - 96 * q^96 + 94 * q^97 - 98 * q^98

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