LMFDB - Newform orbit 464.2.y.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [464,2,Mod(33,464)] 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("464.33"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(464, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	464.y (of order $14$, degree $6$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$3.70505865379$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	12.0.7877952219361.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 $ x^12 - 3x^11 + 13x^9 - 18x^8 - 14x^7 + 57x^6 - 28x^5 - 72x^4 + 104x^3 - 96*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 2) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \cdots + 1) q^{5} + ( - 2 \beta_{11} - \beta_{9} + \cdots + 2) q^{7} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 2) q^{9}+ \cdots + ( - \beta_{11} - 3 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) $ q + (-b9 + b8 + b7 + b5 + b1 + 2) * q^3 + (-b11 + b10 - b9 - b6 + b5 - b4 + 2b1 + 1) q^5 + (-2b11 - b9 + b8 - b7 - 2b6 - b5 - 2b3 + b1 + 2) q^7 + (-b11 - b10 - 2b9 + b5 - 2b4 - 3b3 + b2 + 2b1 + 2) * q^9 + (-b11 + b10 - b9 - b8 + b6 + b5 - 2b4 - b3 + b1) q^11 + (b11 + b9 - 2b5 - b3 - 2b2 - 2b1) q^13 + (-b11 + b5 - 2b4 - 2b3 - b2 + b1 + 1) * q^15 + (-b9 + b8 + 2b6 + 2b4 - b3 - b2 - 2b1) q^17 + (b11 + b10 + 2b9 - b8 + b7 - b5 + b4 + b3 - b2 - b1 - 1) q^19 + (-7b11 + b10 + 2b8 - 5b6 + b5 - 4b4 - 3b3 + b2 + 6b1 + 4) * q^21 + (-b11 - 2b10 - b9 - b7 - b3 - 2b1 + 1) * q^23 + (3b11 + b8 + 2b7 + 2b6 + b5 + b4 + 3b3 - b2 - b1 + 1) * q^25 + (-3b11 - b10 - b9 - b7 - b6 - 6b4 - 4b3 + 5b1 + 4) * q^27 + (b10 + b8 + b7 - b6 + b5 + 2b4 - 3b2) * q^29 + (2b11 - 2b10 - 2b9 + b8 - 2b7 + 2b6 + 2b4 - b3 - b2 + 2) * q^31 + (-4b11 - 2b10 - 2b9 - 3b8 - 4b7 + b6 - b5 - 3b4 - 4b3 + b2 + 3b1 - 1) * q^33 + (-b11 + 4b10 + 2b8 + 4b7 - 4b6 + b5 + 2b3 - b2 + 5b1 + 1) * q^35 + (-b11 - 2b10 + b8 - b7 - 2b5 - 3b4 + 2b3 - b2 - b1 + 1) * q^37 + (2b11 + b8 - b6 - b5 + b4 + 4b3 - b2 - 3b1 - 3) q^39 + (2b11 + b9 - b8 + 3b7 + b4 + 2b3 + b2 - 2b1 - 1) * q^41 + (b11 + 3b9 + 3b7 - 4b5 + 2b4 + b3 - 4b1 - 3) q^43 + (b10 + 2b9 - 2b8 + 2b7 - b6 + 2b2 - b1) * q^45 + (-b11 - b10 + b9 - 5b8 + 3b6 + 5b5 - 2b4 + b3 - 3b1) q^47 + (-b11 + 3b10 - 2b9 + 4b8 - 2b6 - 3b5 + 2b4 + b3 - 3b2 + 2b1 + 2) * q^49 + (-6b11 + 2b10 - 2b9 + 2b8 - 2b7 - 2b6 - 2b5 - 2b4 - 4b3 + 2b2 + 4b1) q^51 + (b11 - 2b10 + 2b9 - 2b8 - 2b7 + 2b6 - 2b3 + 2b2 - 2b1 - 3) * q^53 + (-b11 - b10 - b8 - b7 + 5b6 + 2b4 + 4b3 + b2 - 4b1) * q^55 + (5b11 + b10 + b9 + 2b7 + 2b5 + 3b4 + 5b3 - b2 - 4b1 - 3) * q^57 + (6b11 + 2b9 + 2b8 - 4b5 + 6b4 + 4b3 - 2b2 - 6b1 - 8) * q^59 + (b11 - 2b10 - 3b9 - 4b8 - 4b7 + 4b6 + 3b5 - 3b3 + 2b2 - 2b1 - 2) q^61 + (-8b11 - 3b8 - 4b7 - 4b6 + 4b5 - 4b4 - 7b3 + 5b2 + 8b1 + 4) q^63 + (2b11 + 3b10 + b9 - b8 + b7 + b5 + 5b3 + 3b2 + 2b1 - 3) q^65 + (-b11 - 3b10 - 2b9 - 2b8 - 3b6 + 3b5 - b3 + 3b2 + 6b1 + 6) q^67 + (b10 + b9 + 2b8 - b6 - 2b5 + 2b3 + 1) q^69 + (9b11 + b10 + b9 + b8 + 2b7 + 7b6 - b5 + 8b4 + 5b3 - 2b2 - 9b1 - 4) q^71 + (2b9 + 2b7 - 4b6 + 2b5 - b4 - 2b3 + 2b2 + 2b1 + 3) q^73 + (8b11 - b10 - 2b9 + b8 + 3b7 + 8b6 + 3b4 + 3b3 - 2b2 - 6b1 - 2) * q^75 + (2b9 - 5b8 + 3b6 + 3b5 + b4 + 3b2 + b1 - 3) q^77 + (-5b11 + b10 + 4b8 - 2b7 - 5b6 + b5 - 2b4 - 3b3 + 3b2 + 10b1) * q^79 + (3b11 - 4b10 - 2b8 - 4b7 + 6b6 - b5 + 2b4 - 2b3 + b2 - 2b1 - 3) * q^81 + (7b11 + 3b10 + 4b9 + 3b8 + 3b7 - 6b6 - b5 - b4 + b3 + b2 + b1 - 1) * q^83 + (2b11 + 2b8 - 3b7 + 3b6 - 3b5 + 3b4 - 2b2 - 1) q^85 + (6b11 + 4b10 + 3b9 + 3b8 + b7 + b5 + 2b4 + 2b3 - 2b2 - 7b1 - 4) * q^87 + (-b11 + b10 + b9 + 2b8 - 2b7 - 7b6 - 3b5 - 3b4 - 2b3 - 2b2 - 4b1 + 1) * q^89 + (2b11 - 4b10 - 4b9 - 3b8 - 2b7 + 3b6 + b5 + b4 - 4b3 - b2 - b1 + 1) q^91 + (3b11 - 4b9 + 6b8 + 4b7 + b5 - 3b2 + 3b1 + 7) * q^93 + (2b8 - b7 - b6 - 3b4 + 3b3 + b2 + b1 + 1) q^95 + (-b11 - 4b10 - 4b7 - b6 + 3b4 - b3 + 5b1 + 2) * q^97 + (-b11 - 3b10 + 4b9 - 7b8 - 2b6 - b4 + 3b3 + 4b2 + b1 - 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{3} - q^{5} + 11 q^{7} - 3 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} + 7 q^{19} - 7 q^{21} + 5 q^{23} + 13 q^{25} + 7 q^{27} - 15 q^{29} + 21 q^{31} - 17 q^{33} - 19 q^{35} + 7 q^{37} - 21 q^{39}+ \cdots + 14 q^{97}+O(q^{100}) $ 12 * q + 7 * q^3 - q^5 + 11 * q^7 - 3 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 + 7 * q^19 - 7 * q^21 + 5 * q^23 + 13 * q^25 + 7 * q^27 - 15 * q^29 + 21 * q^31 - 17 * q^33 - 19 * q^35 + 7 * q^37 - 21 * q^39 - 7 * q^43 + 16 * q^45 + 7 * q^47 + 13 * q^49 - 20 * q^51 - 10 * q^53 + 35 * q^55 - 14 * q^57 - 44 * q^59 - 7 * q^61 + 13 * q^63 - 6 * q^65 + 37 * q^67 + 21 * q^69 + 21 * q^71 + 14 * q^73 - 7 * q^77 - 49 * q^79 + q^81 - 5 * q^83 + 14 * q^85 - 15 * q^87 + 7 * q^89 + 3 * q^91 + 19 * q^93 + 7 * q^95 + 14 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 $ x^12 - 3*x^11 + 13*x^9 - 18*x^8 - 14*x^7 + 57*x^6 - 28*x^5 - 72*x^4 + 104*x^3 - 96*x + 64 :

$\beta_{1}$	$=$	$ ( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + \cdots + 96 ) / 128 $ (v^11 + 7v^10 - 10v^9 - 7v^8 + 40v^7 - 14v^6 - 83v^5 + 102v^4 + 68v^3 - 176v^2 + 32v + 96) / 128
$\beta_{2}$	$=$	$ ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} + \cdots - 288 ) / 128 $ (-v^11 + 5v^10 - 10v^9 + 7v^8 + 4v^7 - 58v^6 + 59v^5 + 38v^4 - 196v^3 + 96v^2 + 256v - 288) / 128
$\beta_{3}$	$=$	$ ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} + \cdots - 32 ) / 128 $ (-v^11 + 5v^10 - 10v^9 + 7v^8 + 36v^7 - 58v^6 - 5v^5 + 134v^4 - 100v^3 - 128v^2 + 192v - 32) / 128
$\beta_{4}$	$=$	$ ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + \cdots + 64 ) / 64 $ (v^11 - 3v^10 + 4v^9 + v^8 - 18v^7 + 22v^6 + 17v^5 - 52v^4 + 44v^3 + 40v^2 - 80*v + 64) / 64
$\beta_{5}$	$=$	$ ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + \cdots + 224 ) / 128 $ (-v^11 + 5v^10 - 10v^9 - 25v^8 + 68v^7 + 6v^6 - 165v^5 + 134v^4 + 220v^3 - 288*v^2 + 224) / 128
$\beta_{6}$	$=$	$ ( 3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + \cdots + 96 ) / 128 $ (3v^11 + v^10 - 18v^9 + 11v^8 + 36v^7 - 50v^6 - 49v^5 + 94v^4 + 12v^3 - 128*v^2 + 96) / 128
$\beta_{7}$	$=$	$ ( -\nu^{10} + 3\nu^{9} - 9\nu^{7} + 10\nu^{6} + 6\nu^{5} - 29\nu^{4} + 16\nu^{3} + 20\nu^{2} - 40\nu + 16 ) / 16 $ (-v^10 + 3v^9 - 9v^7 + 10v^6 + 6v^5 - 29v^4 + 16v^3 + 20v^2 - 40v + 16) / 16
$\beta_{8}$	$=$	$ ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 9 \nu^{8} + 36 \nu^{7} - 26 \nu^{6} - 53 \nu^{5} + 86 \nu^{4} + \cdots - 32 ) / 64 $ (-v^11 + 5v^10 - 10v^9 - 9v^8 + 36v^7 - 26v^6 - 53v^5 + 86v^4 + 12v^3 - 96v^2 + 64v - 32) / 64
$\beta_{9}$	$=$	$ ( \nu^{11} - 5\nu^{9} + 5\nu^{8} + 9\nu^{7} - 16\nu^{6} - 5\nu^{5} + 31\nu^{4} - 12\nu^{2} + 8\nu + 16 ) / 32 $ (v^11 - 5v^9 + 5v^8 + 9v^7 - 16v^6 - 5v^5 + 31v^4 - 12v^2 + 8v + 16) / 32
$\beta_{10}$	$=$	$ ( 3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + \cdots - 352 ) / 128 $ (3v^11 - 11v^10 - 14v^9 + 59v^8 - 40v^7 - 122v^6 + 199v^5 + 82v^4 - 420v^3 + 176v^2 + 352*v - 352) / 128
$\beta_{11}$	$=$	$ ( - 15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} + \cdots + 480 ) / 128 $ (-15v^11 + 31v^10 + 30v^9 - 151v^8 + 80v^7 + 290v^6 - 435v^5 - 146v^4 + 740v^3 - 368v^2 - 416*v + 480) / 128

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

$\nu$	$=$	$ -\beta_{8} + \beta_{5} + \beta_{2} $ -b8 + b5 + b2
$\nu^{2}$	$=$	$ \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + 1 $ b9 - b6 - b4 - b3 + 1
$\nu^{3}$	$=$	$ \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1 $ b9 + b7 - b6 + b5 + b4 + b3 + b2 - 1
$\nu^{4}$	$=$	$ \beta_{10} + \beta_{9} + 2\beta_{8} - 3\beta_{6} - \beta_{5} - \beta_{3} - 2\beta_{2} + 2\beta _1 + 1 $ b10 + b9 + 2b8 - 3b6 - b5 - b3 - 2b2 + 2b1 + 1
$\nu^{5}$	$=$	$ -\beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 2\beta _1 - 2 $ -b10 + b8 - b7 - b5 + 5b4 + 4b3 - 2*b1 - 2
$\nu^{6}$	$=$	$ 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 4 $ 2b11 + 2b10 + 2b9 - 3b8 - 3b7 + 2b6 - 2b5 + 2b4 + 2b3 - 2b2 - 2*b1 - 4
$\nu^{7}$	$=$	$ -5\beta_{10} + \beta_{9} - 6\beta_{8} - 5\beta_{7} + 5\beta_{6} + 5\beta_{3} + \beta_{2} - 10\beta _1 - 5 $ -5b10 + b9 - 6b8 - 5b7 + 5b6 + 5b3 + b2 - 10b1 - 5
$\nu^{8}$	$=$	$ 4 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots - 10 $ 4b11 + 4b10 + 10b9 - 11b8 + 4b7 + 4b6 + b5 - 6b4 + 8b3 + b2 - 4*b1 - 10
$\nu^{9}$	$=$	$ - 6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 11 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} + \cdots + 5 $ -6b11 - 10b10 - 3b9 - 4b8 - 11b6 + 4b5 - 15b4 - 5b3 + 6*b1 + 5
$\nu^{10}$	$=$	$ 2\beta_{11} + 9\beta_{9} + 9\beta_{7} - 7\beta_{6} - 9\beta_{5} + \beta_{4} + 9\beta_{3} + 5\beta_{2} + 18\beta _1 - 1 $ 2b11 + 9b9 + 9b7 - 7b6 - 9b5 + b4 + 9b3 + 5b2 + 18b1 - 1
$\nu^{11}$	$=$	$ - 18 \beta_{11} - 29 \beta_{10} - 29 \beta_{9} - 8 \beta_{8} - 28 \beta_{7} - 7 \beta_{6} + \beta_{5} + \cdots + 11 $ -18b11 - 29b10 - 29b9 - 8b8 - 28b7 - 7b6 + b5 - 39b3 + 8b2 + 36*b1 + 11

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$1$	$\beta_{11}$

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} $

33.1

0.639551	+	1.26134i
1.38491	−	0.286410i
0.911180	−	1.08155i
−1.25719	+	0.647667i
−1.41140	−	0.0891373i
1.23295	−	0.692694i
0.639551	−	1.26134i
1.38491	+	0.286410i
0.911180	+	1.08155i
−1.25719	−	0.647667i
−1.41140	+	0.0891373i
1.23295	+	0.692694i

−0.855966

−

1.77743i

0.610610

−

2.67526i

4.03077

−

1.94112i

−0.556117

0.697349i

33.2

1.23248

2.55926i

0.0128801

−

0.0564316i

−1.40728

0.677709i

−3.16036

3.96297i

129.1

0.343489

0.273923i

2.32488

1.11960i

−0.0468435

0.0587399i

−0.624612

−

2.73660i

129.2

0.879032

0.701005i

−2.54740

−

1.22676i

1.82432

−

2.28763i

−0.386273

−

1.69237i

209.1

−0.960118

−

0.219141i

−1.18424

1.48499i

0.339509

−

1.48749i

−1.82910

−

0.880850i

209.2

2.86109

0.653024i

0.283269

−

0.355208i

0.759522

−

3.32768i

5.05647

2.43507i

225.1