LMFDB - Newform orbit 882.4.g.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,4,Mod(361,882)] 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("882.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.g (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,-4,-8,0,0,16,0,-16,40,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$52.0396846251$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 294)
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} q^{2} + (4 \zeta_{6} - 4) q^{4} - 8 \zeta_{6} q^{5} + 8 q^{8} + (16 \zeta_{6} - 16) q^{10} + ( - 40 \zeta_{6} + 40) q^{11} - 4 q^{13} - 16 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} + \cdots + 832 q^{97} +O(q^{100}) $ q - 2z q^2 + (4z - 4) q^4 - 8z q^5 + 8 * q^8 + (16z - 16) q^10 + (-40z + 40) q^11 - 4 * q^13 - 16z q^16 + (-84z + 84) q^17 + 148z q^19 + 32 * q^20 - 80 * q^22 + 84z q^23 + (-61z + 61) q^25 + 8z q^26 - 58 * q^29 + (136z - 136) q^31 + (32z - 32) q^32 - 168 * q^34 + 222z q^37 + (-296z + 296) q^38 - 64z q^40 + 420 * q^41 - 164 * q^43 + 160z q^44 + (-168z + 168) q^46 - 488z q^47 - 122 * q^50 + (-16z + 16) q^52 + (-478z + 478) q^53 - 320 * q^55 + 116z q^58 + (548z - 548) q^59 + 692z q^61 + 272 * q^62 + 64 * q^64 + 32z q^65 + (-908z + 908) q^67 + 336z q^68 + 524 * q^71 + (-440z + 440) q^73 + (-444z + 444) q^74 - 592 * q^76 - 1216z q^79 + (128z - 128) q^80 - 840z q^82 - 684 * q^83 - 672 * q^85 + 328z q^86 + (-320z + 320) q^88 - 604z q^89 - 336 * q^92 + (976z - 976) q^94 + (-1184z + 1184) q^95 + 832 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} - 8 q^{5} + 16 q^{8} - 16 q^{10} + 40 q^{11} - 8 q^{13} - 16 q^{16} + 84 q^{17} + 148 q^{19} + 64 q^{20} - 160 q^{22} + 84 q^{23} + 61 q^{25} + 8 q^{26} - 116 q^{29} - 136 q^{31}+ \cdots + 1664 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 - 8 * q^5 + 16 * q^8 - 16 * q^10 + 40 * q^11 - 8 * q^13 - 16 * q^16 + 84 * q^17 + 148 * q^19 + 64 * q^20 - 160 * q^22 + 84 * q^23 + 61 * q^25 + 8 * q^26 - 116 * q^29 - 136 * q^31 - 32 * q^32 - 336 * q^34 + 222 * q^37 + 296 * q^38 - 64 * q^40 + 840 * q^41 - 328 * q^43 + 160 * q^44 + 168 * q^46 - 488 * q^47 - 244 * q^50 + 16 * q^52 + 478 * q^53 - 640 * q^55 + 116 * q^58 - 548 * q^59 + 692 * q^61 + 544 * q^62 + 128 * q^64 + 32 * q^65 + 908 * q^67 + 336 * q^68 + 1048 * q^71 + 440 * q^73 + 444 * q^74 - 1184 * q^76 - 1216 * q^79 - 128 * q^80 - 840 * q^82 - 1368 * q^83 - 1344 * q^85 + 328 * q^86 + 320 * q^88 - 604 * q^89 - 672 * q^92 - 976 * q^94 + 1184 * q^95 + 1664 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-\zeta_{6}$	$1$

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} $

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

−

1.73205i

−2.00000

3.46410i

−4.00000

−

6.92820i

8.00000

−8.00000

13.8564i

667.1

−1.00000

1.73205i

−2.00000

−

3.46410i

−4.00000

6.92820i

8.00000

−8.00000

−

13.8564i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	882.4.g.c		2
3.b	odd	2	1		294.4.e.j		2
7.b	odd	2	1		882.4.g.j		2
7.c	even	3	1		882.4.a.q		1
7.c	even	3	1	inner	882.4.g.c		2
7.d	odd	6	1		882.4.a.j		1
7.d	odd	6	1		882.4.g.j		2
21.c	even	2	1		294.4.e.f		2
21.g	even	6	1		294.4.a.f	yes	1
21.g	even	6	1		294.4.e.f		2
21.h	odd	6	1		294.4.a.b	✓	1
21.h	odd	6	1		294.4.e.j		2
84.j	odd	6	1		2352.4.a.m		1
84.n	even	6	1		2352.4.a.z		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
294.4.a.b	✓	1	21.h	odd	6	1
294.4.a.f	yes	1	21.g	even	6	1
294.4.e.f		2	21.c	even	2	1
294.4.e.f		2	21.g	even	6	1
294.4.e.j		2	3.b	odd	2	1
294.4.e.j		2	21.h	odd	6	1
882.4.a.j		1	7.d	odd	6	1
882.4.a.q		1	7.c	even	3	1
882.4.g.c		2	1.a	even	1	1	trivial
882.4.g.c		2	7.c	even	3	1	inner
882.4.g.j		2	7.b	odd	2	1
882.4.g.j		2	7.d	odd	6	1
2352.4.a.m		1	84.j	odd	6	1
2352.4.a.z		1	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(882, [\chi])$:

$ T_{5}^{2} + 8T_{5} + 64 $

T5^2 + 8*T5 + 64

$ T_{11}^{2} - 40T_{11} + 1600 $

T11^2 - 40*T11 + 1600

$ T_{13} + 4 $

T13 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 8T + 64 $ T^2 + 8*T + 64
$7$	$ T^{2} $ T^2
$11$	$ T^{2} - 40T + 1600 $ T^2 - 40*T + 1600
$13$	$ (T + 4)^{2} $ (T + 4)^2
$17$	$ T^{2} - 84T + 7056 $ T^2 - 84*T + 7056
$19$	$ T^{2} - 148T + 21904 $ T^2 - 148*T + 21904
$23$	$ T^{2} - 84T + 7056 $ T^2 - 84*T + 7056
$29$	$ (T + 58)^{2} $ (T + 58)^2
$31$	$ T^{2} + 136T + 18496 $ T^2 + 136*T + 18496
$37$	$ T^{2} - 222T + 49284 $ T^2 - 222*T + 49284
$41$	$ (T - 420)^{2} $ (T - 420)^2
$43$	$ (T + 164)^{2} $ (T + 164)^2
$47$	$ T^{2} + 488T + 238144 $ T^2 + 488*T + 238144
$53$	$ T^{2} - 478T + 228484 $ T^2 - 478*T + 228484
$59$	$ T^{2} + 548T + 300304 $ T^2 + 548*T + 300304
$61$	$ T^{2} - 692T + 478864 $ T^2 - 692*T + 478864
$67$	$ T^{2} - 908T + 824464 $ T^2 - 908*T + 824464
$71$	$ (T - 524)^{2} $ (T - 524)^2
$73$	$ T^{2} - 440T + 193600 $ T^2 - 440*T + 193600
$79$	$ T^{2} + 1216 T + 1478656 $ T^2 + 1216*T + 1478656
$83$	$ (T + 684)^{2} $ (T + 684)^2
$89$	$ T^{2} + 604T + 364816 $ T^2 + 604*T + 364816
$97$	$ (T - 832)^{2} $ (T - 832)^2
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Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 8 \zeta_{6} q^{5} + 8 q^{8} + (16 \zeta_{6} - 16) q^{10} + ( - 40 \zeta_{6} + 40) q^{11} - 4 q^{13} - 16 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} + \cdots + 832 q^{97} +O(q^{100}) \) q - 2z q^2 + (4z - 4) q^4 - 8z q^5 + 8 * q^8 + (16z - 16) q^10 + (-40z + 40) q^11 - 4 * q^13 - 16z q^16 + (-84z + 84) q^17 + 148z q^19 + 32 * q^20 - 80 * q^22 + 84z q^23 + (-61z + 61) q^25 + 8z q^26 - 58 * q^29 + (136z - 136) q^31 + (32z - 32) q^32 - 168 * q^34 + 222z q^37 + (-296z + 296) q^38 - 64z q^40 + 420 * q^41 - 164 * q^43 + 160z q^44 + (-168z + 168) q^46 - 488z q^47 - 122 * q^50 + (-16z + 16) q^52 + (-478z + 478) q^53 - 320 * q^55 + 116z q^58 + (548z - 548) q^59 + 692z q^61 + 272 * q^62 + 64 * q^64 + 32z q^65 + (-908z + 908) q^67 + 336z q^68 + 524 * q^71 + (-440z + 440) q^73 + (-444z + 444) q^74 - 592 * q^76 - 1216z q^79 + (128z - 128) q^80 - 840z q^82 - 684 * q^83 - 672 * q^85 + 328z q^86 + (-320z + 320) q^88 - 604z q^89 - 336 * q^92 + (976z - 976) q^94 + (-1184z + 1184) q^95 + 832 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} - 8 q^{5} + 16 q^{8} - 16 q^{10} + 40 q^{11} - 8 q^{13} - 16 q^{16} + 84 q^{17} + 148 q^{19} + 64 q^{20} - 160 q^{22} + 84 q^{23} + 61 q^{25} + 8 q^{26} - 116 q^{29} - 136 q^{31}+ \cdots + 1664 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 - 8 * q^5 + 16 * q^8 - 16 * q^10 + 40 * q^11 - 8 * q^13 - 16 * q^16 + 84 * q^17 + 148 * q^19 + 64 * q^20 - 160 * q^22 + 84 * q^23 + 61 * q^25 + 8 * q^26 - 116 * q^29 - 136 * q^31 - 32 * q^32 - 336 * q^34 + 222 * q^37 + 296 * q^38 - 64 * q^40 + 840 * q^41 - 328 * q^43 + 160 * q^44 + 168 * q^46 - 488 * q^47 - 244 * q^50 + 16 * q^52 + 478 * q^53 - 640 * q^55 + 116 * q^58 - 548 * q^59 + 692 * q^61 + 544 * q^62 + 128 * q^64 + 32 * q^65 + 908 * q^67 + 336 * q^68 + 1048 * q^71 + 440 * q^73 + 444 * q^74 - 1184 * q^76 - 1216 * q^79 - 128 * q^80 - 840 * q^82 - 1368 * q^83 - 1344 * q^85 + 328 * q^86 + 320 * q^88 - 604 * q^89 - 672 * q^92 - 976 * q^94 + 1184 * q^95 + 1664 * q^97

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