LMFDB - Newform orbit 4900.2.e.s

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(2549,4900)] 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("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-16,0,8,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0, 0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 700)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + ( - \beta_{5} - \beta_{4} - 2) q^{9} + (\beta_{5} + 1) q^{11} + (3 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 3 \beta_{2}) q^{17} + (\beta_{4} - 1) q^{19} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{23}+ \cdots + ( - 7 \beta_{5} - 2 \beta_{4} - 4) q^{99}+O(q^{100}) $ q + b1 * q^3 + (-b5 - b4 - 2) * q^9 + (b5 + 1) * q^11 + (3b2 - b1) q^13 + (-b3 - 3b2) q^17 + (b4 - 1) * q^19 + (-b3 + b2 + b1) * q^23 + (-b3 - 2b2 - 3b1) * q^27 + (2b5 + b4 - 1) q^29 + (2b5 + b4) q^31 + (b3 + 5b2 + 2b1) * q^33 + (b2 + 3b1) q^37 + (-2b5 + b4 + 5) q^39 + (-3b5 + b4 - 3) q^41 + (b3 + 3b2 + b1) q^43 + (b3 - 2b2 + b1) q^47 + (6b5 - 3) q^51 + (b3 - 6b2 + 3b1) * q^53 + (-3b2 + 2b1) * q^57 + (b5 - 2) * q^59 + (2b5 - b4 - 6) q^61 + (6b2 + 2b1) * q^67 + (b5 - b4 - 8) * q^69 + (2b5 - 7) q^71 + 4b2 q^73 + (-3b5 + b4 + 1) q^79 + (5b5 + 6) q^81 + (-b3 + 11b2 - 4b1) * q^83 + (2b3 + 7b2 + 4b1) q^87 + (-b5 - b4 - 4) * q^89 + (2b3 + 7b2 + 5b1) q^93 + (2b3 - 6b2 + b1) * q^97 + (-7b5 - 2b4 - 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) $ 6 * q - 16 * q^9 + 8 * q^11 - 4 * q^19 + 6 * q^31 + 28 * q^39 - 22 * q^41 - 6 * q^51 - 10 * q^59 - 34 * q^61 - 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 - 28 * q^89 - 42 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9*x^4 + 22*x^2 + 9 :

$\beta_{1}$	$=$	$ \nu^{3} + 5\nu $ v^3 + 5*v
$\beta_{2}$	$=$	$ ( \nu^{5} + 6\nu^{3} + 7\nu ) / 3 $ (v^5 + 6v^3 + 7v) / 3
$\beta_{3}$	$=$	$ ( \nu^{5} + 9\nu^{3} + 13\nu ) / 3 $ (v^5 + 9v^3 + 13v) / 3
$\beta_{4}$	$=$	$ -2\nu^{4} - 9\nu^{2} - 2 $ -2v^4 - 9v^2 - 2
$\beta_{5}$	$=$	$ \nu^{4} + 6\nu^{2} + 6 $ v^4 + 6*v^2 + 6

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

$\nu$	$=$	$ ( -\beta_{3} + \beta_{2} + \beta_1 ) / 3 $ (-b3 + b2 + b1) / 3
$\nu^{2}$	$=$	$ ( 2\beta_{5} + \beta_{4} - 10 ) / 3 $ (2*b5 + b4 - 10) / 3
$\nu^{3}$	$=$	$ ( 5\beta_{3} - 5\beta_{2} - 2\beta_1 ) / 3 $ (5b3 - 5b2 - 2*b1) / 3
$\nu^{4}$	$=$	$ -3\beta_{5} - 2\beta_{4} + 14 $ -3b5 - 2b4 + 14
$\nu^{5}$	$=$	$ ( -23\beta_{3} + 32\beta_{2} + 5\beta_1 ) / 3 $ (-23b3 + 32b2 + 5*b1) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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} $

2549.1

	−	0.713538i
	−	1.91223i
	−	2.19869i
		2.19869i
		1.91223i
		0.713538i

−

3.20440i

−7.26819

2549.2

−

2.56885i

−3.59899

2549.3

−

0.364448i

2.86718

2549.4

0.364448i

2.86718

2549.5

2.56885i

−3.59899

2549.6

3.20440i

−7.26819

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4900.2.e.s		6
5.b	even	2	1	inner	4900.2.e.s		6
5.c	odd	4	1		4900.2.a.bb		3
5.c	odd	4	1		4900.2.a.bd		3
7.b	odd	2	1		4900.2.e.t		6
7.d	odd	6	2		700.2.r.d		12
35.c	odd	2	1		4900.2.e.t		6
35.f	even	4	1		4900.2.a.ba		3
35.f	even	4	1		4900.2.a.bc		3
35.i	odd	6	2		700.2.r.d		12
35.k	even	12	2		700.2.i.d	✓	6
35.k	even	12	2		700.2.i.e	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.i.d	✓	6	35.k	even	12	2
700.2.i.e	yes	6	35.k	even	12	2
700.2.r.d		12	7.d	odd	6	2
700.2.r.d		12	35.i	odd	6	2
4900.2.a.ba		3	35.f	even	4	1
4900.2.a.bb		3	5.c	odd	4	1
4900.2.a.bc		3	35.f	even	4	1
4900.2.a.bd		3	5.c	odd	4	1
4900.2.e.s		6	1.a	even	1	1	trivial
4900.2.e.s		6	5.b	even	2	1	inner
4900.2.e.t		6	7.b	odd	2	1
4900.2.e.t		6	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4900, [\chi])$:

$ T_{3}^{6} + 17T_{3}^{4} + 70T_{3}^{2} + 9 $

T3^6 + 17*T3^4 + 70*T3^2 + 9

$ T_{11}^{3} - 4T_{11}^{2} - 3T_{11} + 9 $

T11^3 - 4*T11^2 - 3*T11 + 9

$ T_{19}^{3} + 2T_{19}^{2} - 23T_{19} + 21 $

T19^3 + 2*T19^2 - 23*T19 + 21

$ T_{31}^{3} - 3T_{31}^{2} - 42T_{31} - 37 $

T31^3 - 3*T31^2 - 42*T31 - 37

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 17 T^{4} + \cdots + 9 $ T^6 + 17T^4 + 70T^2 + 9
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ (T^{3} - 4 T^{2} - 3 T + 9)^{2} $ (T^3 - 4T^2 - 3T + 9)^2
$13$	$ T^{6} + 38 T^{4} + \cdots + 9 $ T^6 + 38T^4 + 217T^2 + 9
$17$	$ T^{6} + 82 T^{4} + \cdots + 6561 $ T^6 + 82T^4 + 1701T^2 + 6561
$19$	$ (T^{3} + 2 T^{2} - 23 T + 21)^{2} $ (T^3 + 2T^2 - 23T + 21)^2
$23$	$ T^{6} + 81 T^{4} + \cdots + 6561 $ T^6 + 81T^4 + 1782T^2 + 6561
$29$	$ (T^{3} - 45 T - 81)^{2} $ (T^3 - 45*T - 81)^2
$31$	$ (T^{3} - 3 T^{2} - 42 T - 37)^{2} $ (T^3 - 3T^2 - 42T - 37)^2
$37$	$ T^{6} + 162 T^{4} + \cdots + 22201 $ T^6 + 162T^4 + 5757T^2 + 22201
$41$	$ (T^{3} + 11 T^{2} + \cdots - 873)^{2} $ (T^3 + 11T^2 - 78T - 873)^2
$43$	$ T^{6} + 93 T^{4} + \cdots + 5041 $ T^6 + 93T^4 + 1758T^2 + 5041
$47$	$ T^{6} + 58 T^{4} + \cdots + 81 $ T^6 + 58T^4 + 513T^2 + 81
$53$	$ T^{6} + 226 T^{4} + \cdots + 301401 $ T^6 + 226T^4 + 15597T^2 + 301401
$59$	$ (T^{3} + 5 T^{2} - 9)^{2} $ (T^3 + 5*T^2 - 9)^2
$61$	$ (T^{3} + 17 T^{2} + 26 T + 1)^{2} $ (T^3 + 17T^2 + 26T + 1)^2
$67$	$ T^{6} + 200 T^{4} + \cdots + 5184 $ T^6 + 200T^4 + 7120T^2 + 5184
$71$	$ (T^{3} + 19 T^{2} + \cdots + 45)^{2} $ (T^3 + 19T^2 + 87T + 45)^2
$73$	$ (T^{2} + 16)^{3} $ (T^2 + 16)^3
$79$	$ (T^{3} - T^{2} - 118 T - 449)^{2} $ (T^3 - T^2 - 118*T - 449)^2
$83$	$ T^{6} + 526 T^{4} + \cdots + 962361 $ T^6 + 526T^4 + 71577T^2 + 962361
$89$	$ (T^{3} + 14 T^{2} + \cdots - 45)^{2} $ (T^3 + 14T^2 + 39T - 45)^2
$97$	$ T^{6} + 261 T^{4} + \cdots + 25 $ T^6 + 261T^4 + 474T^2 + 25
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Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + ( - \beta_{5} - \beta_{4} - 2) q^{9} + (\beta_{5} + 1) q^{11} + (3 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 3 \beta_{2}) q^{17} + (\beta_{4} - 1) q^{19} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{23}+ \cdots + ( - 7 \beta_{5} - 2 \beta_{4} - 4) q^{99}+O(q^{100}) \) q + b1 * q^3 + (-b5 - b4 - 2) * q^9 + (b5 + 1) * q^11 + (3b2 - b1) q^13 + (-b3 - 3b2) q^17 + (b4 - 1) * q^19 + (-b3 + b2 + b1) * q^23 + (-b3 - 2b2 - 3b1) * q^27 + (2b5 + b4 - 1) q^29 + (2b5 + b4) q^31 + (b3 + 5b2 + 2b1) * q^33 + (b2 + 3b1) q^37 + (-2b5 + b4 + 5) q^39 + (-3b5 + b4 - 3) q^41 + (b3 + 3b2 + b1) q^43 + (b3 - 2b2 + b1) q^47 + (6b5 - 3) q^51 + (b3 - 6b2 + 3b1) * q^53 + (-3b2 + 2b1) * q^57 + (b5 - 2) * q^59 + (2b5 - b4 - 6) q^61 + (6b2 + 2b1) * q^67 + (b5 - b4 - 8) * q^69 + (2b5 - 7) q^71 + 4b2 q^73 + (-3b5 + b4 + 1) q^79 + (5b5 + 6) q^81 + (-b3 + 11b2 - 4b1) * q^83 + (2b3 + 7b2 + 4b1) q^87 + (-b5 - b4 - 4) * q^89 + (2b3 + 7b2 + 5b1) q^93 + (2b3 - 6b2 + b1) * q^97 + (-7b5 - 2b4 - 4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) \) 6 * q - 16 * q^9 + 8 * q^11 - 4 * q^19 + 6 * q^31 + 28 * q^39 - 22 * q^41 - 6 * q^51 - 10 * q^59 - 34 * q^61 - 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 - 28 * q^89 - 42 * q^99

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