LMFDB - Newform orbit 252.3.y.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,3,Mod(163,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.163"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	252.y (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:	$6.86650266188$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
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} + 4 q^{4} + (5 \zeta_{6} - 5) q^{5} + 7 \zeta_{6} q^{7} - 8 q^{8} + ( - 10 \zeta_{6} + 10) q^{10} + ( - 3 \zeta_{6} + 6) q^{11} - 16 q^{13} - 14 \zeta_{6} q^{14} + 16 q^{16} + 4 \zeta_{6} q^{17} + \cdots + ( - 98 \zeta_{6} + 98) q^{98} +O(q^{100}) $ q - 2 * q^2 + 4 * q^4 + (5z - 5) q^5 + 7z q^7 - 8 * q^8 + (-10z + 10) q^10 + (-3z + 6) q^11 - 16 * q^13 - 14z q^14 + 16 * q^16 + 4z q^17 + (-16z - 16) q^19 + (20z - 20) q^20 + (6z - 12) q^22 + (12z + 12) q^23 + 32 * q^26 + 28z q^28 - 37 * q^29 + (11z - 22) q^31 - 32 * q^32 - 8z q^34 - 35 * q^35 + (68z - 68) q^37 + (32z + 32) q^38 + (-40z + 40) q^40 - 40 * q^41 + (-28z + 14) q^43 + (-12z + 24) q^44 + (-24z - 24) q^46 + (-26z - 26) q^47 + (49z - 49) q^49 - 64 * q^52 + 73z q^53 + (30z - 15) q^55 - 56z q^56 + 74 * q^58 + (17z - 34) q^59 + (-82z + 82) q^61 + (-22z + 44) q^62 + 64 * q^64 + (-80z + 80) q^65 + (-46z + 92) q^67 + 16z q^68 + 70 * q^70 + (84z - 42) q^71 + 94z q^73 + (-136z + 136) q^74 + (-64z - 64) q^76 + (21z + 21) q^77 + (-5z - 5) q^79 + (80z - 80) q^80 + 80 * q^82 + (-154z + 77) q^83 - 20 * q^85 + (56z - 28) q^86 + (24z - 48) q^88 + (-58z + 58) q^89 - 112z q^91 + (48z + 48) q^92 + (52z + 52) q^94 + (-80z + 160) q^95 + 47 * q^97 + (-98z + 98) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 5 q^{5} + 7 q^{7} - 16 q^{8} + 10 q^{10} + 9 q^{11} - 32 q^{13} - 14 q^{14} + 32 q^{16} + 4 q^{17} - 48 q^{19} - 20 q^{20} - 18 q^{22} + 36 q^{23} + 64 q^{26} + 28 q^{28} - 74 q^{29}+ \cdots + 98 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 5 * q^5 + 7 * q^7 - 16 * q^8 + 10 * q^10 + 9 * q^11 - 32 * q^13 - 14 * q^14 + 32 * q^16 + 4 * q^17 - 48 * q^19 - 20 * q^20 - 18 * q^22 + 36 * q^23 + 64 * q^26 + 28 * q^28 - 74 * q^29 - 33 * q^31 - 64 * q^32 - 8 * q^34 - 70 * q^35 - 68 * q^37 + 96 * q^38 + 40 * q^40 - 80 * q^41 + 36 * q^44 - 72 * q^46 - 78 * q^47 - 49 * q^49 - 128 * q^52 + 73 * q^53 - 56 * q^56 + 148 * q^58 - 51 * q^59 + 82 * q^61 + 66 * q^62 + 128 * q^64 + 80 * q^65 + 138 * q^67 + 16 * q^68 + 140 * q^70 + 94 * q^73 + 136 * q^74 - 192 * q^76 + 63 * q^77 - 15 * q^79 - 80 * q^80 + 160 * q^82 - 40 * q^85 - 72 * q^88 + 58 * q^89 - 112 * q^91 + 144 * q^92 + 156 * q^94 + 240 * q^95 + 94 * q^97 + 98 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1 + \zeta_{6}$	$-1$

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.

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} $

163.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.00000

4.00000

−2.50000

4.33013i

3.50000

6.06218i

−8.00000

5.00000

−

8.66025i

235.1

−2.00000

4.00000

−2.50000

−

4.33013i

3.50000

−

6.06218i

−8.00000

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	252.3.y.a		2
3.b	odd	2	1		84.3.l.b	yes	2
4.b	odd	2	1		252.3.y.b		2
7.c	even	3	1		252.3.y.b		2
12.b	even	2	1		84.3.l.a	✓	2
21.g	even	6	1		588.3.g.c		2
21.h	odd	6	1		84.3.l.a	✓	2
21.h	odd	6	1		588.3.g.a		2
28.g	odd	6	1	inner	252.3.y.a		2
84.j	odd	6	1		588.3.g.c		2
84.n	even	6	1		84.3.l.b	yes	2
84.n	even	6	1		588.3.g.a		2

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(252, [\chi])$:

$ 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} $ T^2
$5$	$ 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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	252.y (of order \(6\), degree \(2\), minimal)

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

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