LMFDB - Newform orbit 4752.2.o.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4752,2,Mod(703,4752)] 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("4752.703"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4752, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4752 = 2^{4} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4752.o (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$37.9449110405$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(i, \sqrt{6}, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 109x^{4} - 156x^{3} - 108x^{2} + 180x + 225 $ x^8 - 4x^7 + 18x^6 - 40x^5 + 109x^4 - 156x^3 - 108x^2 + 180*x + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{7} + \beta_{4} q^{11} + \beta_{3} q^{17} - \beta_{5} q^{19} + ( - \beta_{4} - 2 \beta_1) q^{23} - 5 q^{25} + (\beta_{7} - \beta_{3}) q^{29} + (\beta_{6} - 2) q^{37} + ( - \beta_{7} - \beta_{3}) q^{41}+ \cdots + (\beta_{6} - 8) q^{97}+O(q^{100}) $ q + b2 * q^7 + b4 * q^11 + b3 * q^17 - b5 * q^19 + (-b4 - 2b1) q^23 - 5 * q^25 + (b7 - b3) * q^29 + (b6 - 2) * q^37 + (-b7 - b3) * q^41 + (-b5 - b2) * q^43 + (-2b4 - b1) q^47 + (b6 + 11) * q^49 + (-b4 + 4b1) q^59 + 4b4 q^71 + (-2b7 + b3) q^77 + (b5 - b2) * q^79 + (b6 - 8) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{25} - 16 q^{37} + 88 q^{49} - 64 q^{97}+O(q^{100}) $ 8 * q - 40 * q^25 - 16 * q^37 + 88 * q^49 - 64 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 4x^{7} + 18x^{6} - 40x^{5} + 109x^{4} - 156x^{3} - 108x^{2} + 180x + 225 $ x^8 - 4*x^7 + 18*x^6 - 40*x^5 + 109*x^4 - 156*x^3 - 108*x^2 + 180*x + 225 :

$\beta_{1}$	$=$	$ ( 2\nu^{6} - 6\nu^{5} + 35\nu^{4} - 60\nu^{3} + 203\nu^{2} - 174\nu - 180 ) / 85 $ (2v^6 - 6v^5 + 35v^4 - 60v^3 + 203v^2 - 174v - 180) / 85
$\beta_{2}$	$=$	$ ( -47\nu^{7} + 297\nu^{6} - 1095\nu^{5} + 2525\nu^{4} - 5583\nu^{3} + 3348\nu^{2} + 15495\nu - 59940 ) / 13515 $ (-47v^7 + 297v^6 - 1095v^5 + 2525v^4 - 5583v^3 + 3348v^2 + 15495*v - 59940) / 13515
$\beta_{3}$	$=$	$ ( -5\nu^{7} - 9\nu^{6} - 15\nu^{5} - 205\nu^{4} + 15\nu^{3} - 1836\nu^{2} + 2325\nu + 2250 ) / 795 $ (-5v^7 - 9v^6 - 15v^5 - 205v^4 + 15v^3 - 1836v^2 + 2325*v + 2250) / 795
$\beta_{4}$	$=$	$ ( -88\nu^{7} + 308\nu^{6} - 1536\nu^{5} + 3070\nu^{4} - 9912\nu^{3} + 11952\nu^{2} + 216\nu - 2005 ) / 4505 $ (-88v^7 + 308v^6 - 1536v^5 + 3070v^4 - 9912v^3 + 11952v^2 + 216*v - 2005) / 4505
$\beta_{5}$	$=$	$ ( 94\nu^{7} - 329\nu^{6} + 1395\nu^{5} - 2665\nu^{4} + 7721\nu^{3} - 9081\nu^{2} - 27015\nu + 14940 ) / 4505 $ (94v^7 - 329v^6 + 1395v^5 - 2665v^4 + 7721v^3 - 9081v^2 - 27015*v + 14940) / 4505
$\beta_{6}$	$=$	$ ( 8\nu^{7} - 28\nu^{6} + 130\nu^{5} - 255\nu^{4} + 612\nu^{3} - 677\nu^{2} - 2130\nu + 1170 ) / 265 $ (8v^7 - 28v^6 + 130v^5 - 255v^4 + 612v^3 - 677v^2 - 2130*v + 1170) / 265
$\beta_{7}$	$=$	$ ( -10\nu^{7} + 35\nu^{6} - 189\nu^{5} + 385\nu^{4} - 1295\nu^{3} + 1575\nu^{2} + 39\nu - 270 ) / 265 $ (-10v^7 + 35v^6 - 189v^5 + 385v^4 - 1295v^3 + 1575v^2 + 39*v - 270) / 265

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

$\nu$	$=$	$ ( \beta_{7} - \beta_{5} - 3\beta_{4} + 3 ) / 6 $ (b7 - b5 - 3*b4 + 3) / 6
$\nu^{2}$	$=$	$ ( 2\beta_{7} - 2\beta_{5} - 3\beta_{4} - 6\beta_{3} - 6\beta_{2} - 6\beta _1 - 15 ) / 6 $ (2b7 - 2b5 - 3b4 - 6b3 - 6b2 - 6b1 - 15) / 6
$\nu^{3}$	$=$	$ ( -\beta_{7} - 3\beta_{6} + 3\beta_{5} + 2\beta_{4} - 3\beta_{3} - 3\beta_{2} - 3\beta _1 - 8 ) / 2 $ (-b7 - 3b6 + 3b5 + 2b4 - 3b3 - 3b2 - 3b1 - 8) / 2
$\nu^{4}$	$=$	$ ( -20\beta_{7} - 18\beta_{6} + 20\beta_{5} + 15\beta_{4} + 60\beta_{3} - 12\beta_{2} + 90\beta _1 - 33 ) / 6 $ (-20b7 - 18b6 + 20b5 + 15b4 + 60b3 - 12b2 + 90*b1 - 33) / 6
$\nu^{5}$	$=$	$ ( -94\beta_{7} + 60\beta_{6} - 86\beta_{5} + 132\beta_{4} + 165\beta_{3} - 15\beta_{2} + 240\beta _1 - 42 ) / 6 $ (-94b7 + 60b6 - 86b5 + 132b4 + 165b3 - 15b2 + 240*b1 - 42) / 6
$\nu^{6}$	$=$	$ ( -46\beta_{7} + 75\beta_{6} - 74\beta_{5} + 119\beta_{4} - 72\beta_{3} + 168\beta_{2} - 87\beta _1 + 685 ) / 2 $ (-46b7 + 75b6 - 74b5 + 119b4 - 72b3 + 168b2 - 87*b1 + 685) / 2
$\nu^{7}$	$=$	$ ( 1072 \beta_{7} + 126 \beta_{6} + 134 \beta_{5} - 1929 \beta_{4} - 1344 \beta_{3} + 1806 \beta_{2} + \cdots + 7311 ) / 6 $ (1072b7 + 126b6 + 134b5 - 1929b4 - 1344b3 + 1806b2 - 1764*b1 + 7311) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4752\mathbb{Z}\right)^\times$.

$n$	$353$	$1189$	$1729$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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} $

703.1

−0.724745	+	0.433568i
−0.724745	−	0.433568i
1.72474	+	0.433568i
1.72474	−	0.433568i
−0.724745	+	2.88306i
−0.724745	−	2.88306i
1.72474	+	2.88306i
1.72474	−	2.88306i

−5.28676

−5.28676

−2.83727

−2.83727

2.83727

2.83727

5.28676

5.28676

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
132.d	odd	2	1	CM by $\Q(\sqrt{-33}) $
3.b	odd	2	1	inner
4.b	odd	2	1	inner
11.b	odd	2	1	inner
12.b	even	2	1	inner
33.d	even	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4752.2.o.c	✓	8
3.b	odd	2	1	inner	4752.2.o.c	✓	8
4.b	odd	2	1	inner	4752.2.o.c	✓	8
11.b	odd	2	1	inner	4752.2.o.c	✓	8
12.b	even	2	1	inner	4752.2.o.c	✓	8
33.d	even	2	1	inner	4752.2.o.c	✓	8
44.c	even	2	1	inner	4752.2.o.c	✓	8
132.d	odd	2	1	CM	4752.2.o.c	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4752.2.o.c	✓	8	1.a	even	1	1	trivial
4752.2.o.c	✓	8	3.b	odd	2	1	inner
4752.2.o.c	✓	8	4.b	odd	2	1	inner
4752.2.o.c	✓	8	11.b	odd	2	1	inner
4752.2.o.c	✓	8	12.b	even	2	1	inner
4752.2.o.c	✓	8	33.d	even	2	1	inner
4752.2.o.c	✓	8	44.c	even	2	1	inner
4752.2.o.c	✓	8	132.d	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4752, [\chi])$:

$ T_{5} $

T5

$ T_{7}^{4} - 36T_{7}^{2} + 225 $

T7^4 - 36*T7^2 + 225

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ (T^{4} - 36 T^{2} + 225)^{2} $ (T^4 - 36*T^2 + 225)^2
$11$	$ (T^{2} + 11)^{4} $ (T^2 + 11)^4
$13$	$ T^{8} $ T^8
$17$	$ (T^{4} + 36 T^{2} + 225)^{2} $ (T^4 + 36*T^2 + 225)^2
$19$	$ (T^{2} - 54)^{4} $ (T^2 - 54)^4
$23$	$ (T^{4} + 94 T^{2} + 625)^{2} $ (T^4 + 94*T^2 + 625)^2
$29$	$ (T^{4} + 108 T^{2} + 441)^{2} $ (T^4 + 108*T^2 + 441)^2
$31$	$ T^{8} $ T^8
$37$	$ (T^{2} + 4 T - 95)^{4} $ (T^2 + 4*T - 95)^4
$41$	$ (T^{4} + 180 T^{2} + 3249)^{2} $ (T^4 + 180*T^2 + 3249)^2
$43$	$ (T^{4} - 108 T^{2} + 441)^{2} $ (T^4 - 108*T^2 + 441)^2
$47$	$ (T^{4} + 106 T^{2} + 1225)^{2} $ (T^4 + 106*T^2 + 1225)^2
$53$	$ T^{8} $ T^8
$59$	$ (T^{4} + 310 T^{2} + 17689)^{2} $ (T^4 + 310*T^2 + 17689)^2
$61$	$ T^{8} $ T^8
$67$	$ T^{8} $ T^8
$71$	$ (T^{2} + 176)^{4} $ (T^2 + 176)^4
$73$	$ T^{8} $ T^8
$79$	$ (T^{4} - 180 T^{2} + 3249)^{2} $ (T^4 - 180*T^2 + 3249)^2
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ (T^{2} + 16 T - 35)^{4} $ (T^2 + 16*T - 35)^4
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Level:	\( N \)	\(=\)	\( 4752 = 2^{4} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4752.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{7} + \beta_{4} q^{11} + \beta_{3} q^{17} - \beta_{5} q^{19} + ( - \beta_{4} - 2 \beta_1) q^{23} - 5 q^{25} + (\beta_{7} - \beta_{3}) q^{29} + (\beta_{6} - 2) q^{37} + ( - \beta_{7} - \beta_{3}) q^{41}+ \cdots + (\beta_{6} - 8) q^{97}+O(q^{100}) \) q + b2 * q^7 + b4 * q^11 + b3 * q^17 - b5 * q^19 + (-b4 - 2b1) q^23 - 5 * q^25 + (b7 - b3) * q^29 + (b6 - 2) * q^37 + (-b7 - b3) * q^41 + (-b5 - b2) * q^43 + (-2b4 - b1) q^47 + (b6 + 11) * q^49 + (-b4 + 4b1) q^59 + 4b4 q^71 + (-2b7 + b3) q^77 + (b5 - b2) * q^79 + (b6 - 8) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{25} - 16 q^{37} + 88 q^{49} - 64 q^{97}+O(q^{100}) \) 8 * q - 40 * q^25 - 16 * q^37 + 88 * q^49 - 64 * q^97

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