LMFDB - Newform orbit 9248.2.a.u

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9248,2,Mod(1,9248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,2,0,2,0,3,0,10,0,-6,0,-8,0,0,0,-4,0,0,0,6,0,13,0,8,0, 10,0,10,0,-4,0,-16,0,10,0,8,0,-6,0,8,0,10,0,4,0,-1,0,0,0,-6,0,16,0,-24, 0,0,0,18,0,-18,0,4,0,8,0,8,0,30,0,-14,0,-34,0,8,0,-10,0,-1,0,-8,0,0,0, 16,0,10,0,-24,0,-16,0,16,0,10,0,-2,0,18,0,0,0,-8,0,10,0,10,0,16,0,2,0, 56,0,26,0,0,0,11,0,20,0,36,0,28,0,8,0,10,0,0,0,40,0,10,0,14,0,-16,0,-32, 0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.8456517893$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 basis $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_1 + 3) q^{11} + (\beta_{2} - \beta_1 - 2) q^{13} + ( - 2 \beta_1 - 2) q^{15}+ \cdots + (2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) $ q + (-b1 + 1) * q^3 + (b2 + b1) * q^5 + (-b2 + 1) * q^7 + (b2 - b1 + 1) * q^9 + (b1 + 3) * q^11 + (b2 - b1 - 2) * q^13 + (-2b1 - 2) q^15 + (2b1 - 2) q^19 + (-b2 + b1) * q^21 + (3b2 + 1) q^23 + (4b1 + 3) q^25 + (2b2 + 2) q^27 + (-b2 - b1 + 4) * q^29 + (-b2 + 2b1 + 3) q^31 + (-b2 - 3b1) q^33 + (2b2 - 6) q^35 + (-b2 - b1 + 4) * q^37 + (2b2 + 2) q^39 + (2b2 - 2b1 - 2) * q^41 + (2b2 + 2) q^43 + (-b2 - b1 + 4) * q^45 + (2b2 + 2b1) * q^47 + (-3b2 - b1 + 1) q^49 + (-2b2 + 2b1 - 2) * q^53 + (4b2 + 6b1 + 2) * q^55 + (-2b2 + 2b1 - 8) * q^57 + (-2b2 - 4b1 + 2) * q^59 + (-b2 - 5b1 + 8) q^61 + (b2 + 2b1 - 7) q^63 + (-4b2 - 4b1 + 4) * q^65 + (-4b1 + 4) q^67 + (3b2 - 7b1 + 4) * q^69 + (-b2 - 2b1 + 11) q^71 + (4b1 - 6) q^73 + (-4b2 - 3b1 - 9) * q^75 + (-3b2 - b1 + 4) q^77 + (3b2 - 4b1 - 3) * q^79 + (-b2 - 3b1 + 1) q^81 + (-2b2 - 2) q^83 + (-2b1 + 6) q^87 + (3b2 + b1 + 2) q^89 + (4b2 + 2b1 - 10) * q^91 + (-3b2 - b1 - 4) q^93 + (4b1 + 4) q^95 + (4b1 + 2) q^97 + (2b2 - b1 - 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + 10 q^{11} - 6 q^{13} - 8 q^{15} - 4 q^{19} + 6 q^{23} + 13 q^{25} + 8 q^{27} + 10 q^{29} + 10 q^{31} - 4 q^{33} - 16 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 + 10 * q^11 - 6 * q^13 - 8 * q^15 - 4 * q^19 + 6 * q^23 + 13 * q^25 + 8 * q^27 + 10 * q^29 + 10 * q^31 - 4 * q^33 - 16 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 8 * q^43 + 10 * q^45 + 4 * q^47 - q^49 - 6 * q^53 + 16 * q^55 - 24 * q^57 + 18 * q^61 - 18 * q^63 + 4 * q^65 + 8 * q^67 + 8 * q^69 + 30 * q^71 - 14 * q^73 - 34 * q^75 + 8 * q^77 - 10 * q^79 - q^81 - 8 * q^83 + 16 * q^87 + 10 * q^89 - 24 * q^91 - 16 * q^93 + 16 * q^95 + 10 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1 :

$\beta_{1}$	$=$	$ \nu^{2} - 2 $ v^2 - 2
$\beta_{2}$	$=$	$ -\nu^{2} + 2\nu + 2 $ -v^2 + 2*v + 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{2} + \beta_1 ) / 2 $ (b2 + b1) / 2
$\nu^{2}$	$=$	$ \beta _1 + 2 $ b1 + 2

Display $\beta_i$ in terms of $\nu^j$

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} $

1.1

2.17009
−1.48119
0.311108

−1.70928

4.34017

−0.630898

−0.0783777

1.2

0.806063

−2.96239

4.15633

−2.35026

1.3

2.90321

0.622216

−1.52543

5.42864

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$17$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9248.2.a.u		3
4.b	odd	2	1		9248.2.a.t		3
17.b	even	2	1		544.2.a.i	✓	3
51.c	odd	2	1		4896.2.a.be		3
68.d	odd	2	1		544.2.a.j	yes	3
136.e	odd	2	1		1088.2.a.u		3
136.h	even	2	1		1088.2.a.v		3
204.h	even	2	1		4896.2.a.bf		3
408.b	odd	2	1		9792.2.a.dc		3
408.h	even	2	1		9792.2.a.dd		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
544.2.a.i	✓	3	17.b	even	2	1
544.2.a.j	yes	3	68.d	odd	2	1
1088.2.a.u		3	136.e	odd	2	1
1088.2.a.v		3	136.h	even	2	1
4896.2.a.be		3	51.c	odd	2	1
4896.2.a.bf		3	204.h	even	2	1
9248.2.a.t		3	4.b	odd	2	1
9248.2.a.u		3	1.a	even	1	1	trivial
9792.2.a.dc		3	408.b	odd	2	1
9792.2.a.dd		3	408.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9248))$:

$ T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4 $

T3^3 - 2*T3^2 - 4*T3 + 4

$ T_{5}^{3} - 2T_{5}^{2} - 12T_{5} + 8 $

T5^3 - 2*T5^2 - 12*T5 + 8

$ T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4 $

T7^3 - 2*T7^2 - 8*T7 - 4

$ T_{19}^{3} + 4T_{19}^{2} - 16T_{19} - 32 $

T19^3 + 4*T19^2 - 16*T19 - 32

$ T_{43}^{3} - 8T_{43}^{2} - 16T_{43} + 160 $

T43^3 - 8*T43^2 - 16*T43 + 160

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{3} $ T^3
$3$	$ T^{3} - 2 T^{2} + \cdots + 4 $ T^3 - 2T^2 - 4T + 4
$5$	$ T^{3} - 2 T^{2} + \cdots + 8 $ T^3 - 2T^2 - 12T + 8
$7$	$ T^{3} - 2 T^{2} + \cdots - 4 $ T^3 - 2T^2 - 8T - 4
$11$	$ T^{3} - 10 T^{2} + \cdots - 20 $ T^3 - 10T^2 + 28T - 20
$13$	$ T^{3} + 6 T^{2} + \cdots - 40 $ T^3 + 6T^2 - 4T - 40
$17$	$ T^{3} $ T^3
$19$	$ T^{3} + 4 T^{2} + \cdots - 32 $ T^3 + 4T^2 - 16T - 32
$23$	$ T^{3} - 6 T^{2} + \cdots + 428 $ T^3 - 6T^2 - 72T + 428
$29$	$ T^{3} - 10 T^{2} + \cdots + 8 $ T^3 - 10T^2 + 20T + 8
$31$	$ T^{3} - 10T^{2} + 148 $ T^3 - 10*T^2 + 148
$37$	$ T^{3} - 10 T^{2} + \cdots + 8 $ T^3 - 10T^2 + 20T + 8
$41$	$ T^{3} + 6 T^{2} + \cdots - 248 $ T^3 + 6T^2 - 52T - 248
$43$	$ T^{3} - 8 T^{2} + \cdots + 160 $ T^3 - 8T^2 - 16T + 160
$47$	$ T^{3} - 4 T^{2} + \cdots + 64 $ T^3 - 4T^2 - 48T + 64
$53$	$ T^{3} + 6 T^{2} + \cdots + 8 $ T^3 + 6T^2 - 52T + 8
$59$	$ T^{3} - 112T + 416 $ T^3 - 112*T + 416
$61$	$ T^{3} - 18 T^{2} + \cdots + 1096 $ T^3 - 18T^2 - 28T + 1096
$67$	$ T^{3} - 8 T^{2} + \cdots + 256 $ T^3 - 8T^2 - 64T + 256
$71$	$ T^{3} - 30 T^{2} + \cdots - 668 $ T^3 - 30T^2 + 272T - 668
$73$	$ T^{3} + 14 T^{2} + \cdots - 344 $ T^3 + 14T^2 - 20T - 344
$79$	$ T^{3} + 10 T^{2} + \cdots - 1444 $ T^3 + 10T^2 - 152T - 1444
$83$	$ T^{3} + 8 T^{2} + \cdots - 160 $ T^3 + 8T^2 - 16T - 160
$89$	$ T^{3} - 10 T^{2} + \cdots + 536 $ T^3 - 10T^2 - 52T + 536
$97$	$ T^{3} - 10 T^{2} + \cdots + 200 $ T^3 - 10T^2 - 52T + 200
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Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_1 + 3) q^{11} + (\beta_{2} - \beta_1 - 2) q^{13} + ( - 2 \beta_1 - 2) q^{15}+ \cdots + (2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) q + (-b1 + 1) * q^3 + (b2 + b1) * q^5 + (-b2 + 1) * q^7 + (b2 - b1 + 1) * q^9 + (b1 + 3) * q^11 + (b2 - b1 - 2) * q^13 + (-2b1 - 2) q^15 + (2b1 - 2) q^19 + (-b2 + b1) * q^21 + (3b2 + 1) q^23 + (4b1 + 3) q^25 + (2b2 + 2) q^27 + (-b2 - b1 + 4) * q^29 + (-b2 + 2b1 + 3) q^31 + (-b2 - 3b1) q^33 + (2b2 - 6) q^35 + (-b2 - b1 + 4) * q^37 + (2b2 + 2) q^39 + (2b2 - 2b1 - 2) * q^41 + (2b2 + 2) q^43 + (-b2 - b1 + 4) * q^45 + (2b2 + 2b1) * q^47 + (-3b2 - b1 + 1) q^49 + (-2b2 + 2b1 - 2) * q^53 + (4b2 + 6b1 + 2) * q^55 + (-2b2 + 2b1 - 8) * q^57 + (-2b2 - 4b1 + 2) * q^59 + (-b2 - 5b1 + 8) q^61 + (b2 + 2b1 - 7) q^63 + (-4b2 - 4b1 + 4) * q^65 + (-4b1 + 4) q^67 + (3b2 - 7b1 + 4) * q^69 + (-b2 - 2b1 + 11) q^71 + (4b1 - 6) q^73 + (-4b2 - 3b1 - 9) * q^75 + (-3b2 - b1 + 4) q^77 + (3b2 - 4b1 - 3) * q^79 + (-b2 - 3b1 + 1) q^81 + (-2b2 - 2) q^83 + (-2b1 + 6) q^87 + (3b2 + b1 + 2) q^89 + (4b2 + 2b1 - 10) * q^91 + (-3b2 - b1 - 4) q^93 + (4b1 + 4) q^95 + (4b1 + 2) q^97 + (2b2 - b1 - 1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + 10 q^{11} - 6 q^{13} - 8 q^{15} - 4 q^{19} + 6 q^{23} + 13 q^{25} + 8 q^{27} + 10 q^{29} + 10 q^{31} - 4 q^{33} - 16 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 + 10 * q^11 - 6 * q^13 - 8 * q^15 - 4 * q^19 + 6 * q^23 + 13 * q^25 + 8 * q^27 + 10 * q^29 + 10 * q^31 - 4 * q^33 - 16 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 8 * q^43 + 10 * q^45 + 4 * q^47 - q^49 - 6 * q^53 + 16 * q^55 - 24 * q^57 + 18 * q^61 - 18 * q^63 + 4 * q^65 + 8 * q^67 + 8 * q^69 + 30 * q^71 - 14 * q^73 - 34 * q^75 + 8 * q^77 - 10 * q^79 - q^81 - 8 * q^83 + 16 * q^87 + 10 * q^89 - 24 * q^91 - 16 * q^93 + 16 * q^95 + 10 * q^97 - 2 * q^99

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