LMFDB - Newform orbit 833.2.e.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [833,2,Mod(18,833)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(833, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("833.18");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$833 = 7^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	833.e (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.65153848837$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 2x^{9} + 9x^{8} - 10x^{7} + 44x^{6} - 49x^{5} + 99x^{4} - 20x^{3} + 31x^{2} - 3x + 9$ x^10 - 2x^9 + 9x^8 - 10x^7 + 44x^6 - 49x^5 + 99x^4 - 20x^3 + 31x^2 - 3*x + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 119)
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 basis $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{4} q^{2} + ( - \beta_{8} - \beta_{7} + \beta_{3}) q^{3} + ( - \beta_{8} + 2 \beta_{5} + \beta_{3}) q^{4} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 1) q^{5} + (2 \beta_{9} - \beta_{8} + 2 \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - 4 \beta_{9} + 2 \beta_{8} + \cdots - 10) q^{99}+O(q^{100})$ q - b4 * q^2 + (-b8 - b7 + b3) * q^3 + (-b8 + 2b5 + b3) q^4 + (-b6 - b5 + b4 - b3 + b1 - 1) * q^5 + (2b9 - b8 + 2b4 + 2b2 - 1) q^6 + (b9 - b7 + b4 + b2 - b1 + 1) * q^8 + (b6 - 2b5 + b4 - b3 - b1 - 2) q^9 + (b8 + 4b7 + b6 + b5 - b3 + b2) q^10 + (2b6 + 2b2) * q^11 + (b6 - 3b5 - 2b3 - 3b1 - 3) q^12 + (2b9 + 2b7 + 2b4 + 2b1 - 2) * q^13 + (-3b9 - 3b4 - b2 + 3) * q^15 + (-b4 - b3 - b1) * q^16 - b5 * q^17 + (-5b9 + b8 - 2b7 - 3b6 - 7b5 - b3 - 3b2) q^18 + (2b5 - 2b1 + 2) * q^19 + (-3b9 + b8 - 3b7 - 3b4 - 4b2 - 3b1 + 7) q^20 + (-2b9 - 2b8 - 4b7 - 2b4 - 2b2 - 4b1 + 4) * q^22 + (2b5 - 2b3 + 2) * q^23 + (-5b9 - 3b7 - 2b6 - 4b5 - 2b2) q^24 + (-b8 + b7 + 2b6 + 5b5 + b3 + 2b2) q^25 + (2b6 - 4b5 - 2b1 - 4) q^26 + (-b9 + 2b8 + 2b7 - b4 - b2 + 2b1 + 5) q^27 + (2b9 + 4b7 + 2b4 + 4b1 - 4) * q^29 + (-b6 + 10b5 - 2b4 + 4b3 + 2b1 + 10) * q^30 + (b9 + b8 + b7 + b6 + b5 - b3 + b2) * q^31 + (-2b7 + b5) q^32 + (-2b5 + 4b4 + 2b3 + 4b1 - 2) * q^33 + (-b9 - b4) * q^34 + (-b9 + 4b8 + 7b7 - b4 + 2b2 + 7b1 + 5) * q^36 + (-2b6 - 2b5 + 2b4 - 2b1 - 2) * q^37 + (2b8 - 2b7 - 2b6 - 4b5 - 2b3 - 2b2) * q^38 + (-2b9 + 2b8 + 2b7 - 4b6 + 2b5 - 2b3 - 4b2) q^39 + (-4b6 - b5 - b4 + 2b3 + 4b1 - 1) q^40 + (2b9 - b8 - b7 + 2b4 - b1 - 4) * q^41 + (2b9 - b8 + b7 + 2b4 + 2b2 + b1) q^43 + (-4b6 - 2b5 + 4b4 + 6b1 - 2) * q^44 + (5b9 - 2b8 - 4b7 + 3b6 + b5 + 2b3 + 3b2) * q^45 + (2b7 - 2b6 + 2b5 - 2b2) * q^46 + (-2b5 - 2b4 + 2b3 + 2b1 - 2) * q^47 + (b9 + b7 + b4 + 3b2 + b1 + 4) q^48 + (3b9 - b8 - 6b7 + 3b4 - 2b2 - 6b1 + 7) q^50 + (b3 + b1) * q^51 + (-4b9 - 2b7 - 4b6 - 4b5 - 4b2) q^52 + (-b9 + b8 + b7 + b6 + b5 - b3 + b2) * q^53 + (3b6 + 4b5 - 8b4 + 4b3 + 2b1 + 4) q^54 + (2b8 - 4b7 + 2b2 - 4b1 + 6) * q^55 + (-2b9 - 2b7 - 2b4 - 2b1 + 4) * q^57 + (4b6 + 2b3 - 4b1) q^58 + 4b7 q^59 + (11b9 - 3b8 + 5b6 + 4b5 + 3b3 + 5b2) * q^60 + (4b5 + 2b4 - b3 - b1 + 4) * q^61 + (-2b9 - b8 - 2b7 - 2b4 - 3b2 - 2b1 - 1) q^62 + (b9 - 4b8 + b4 + 2b2 - 4) * q^64 + (2b6 + 4b5 + 2b4 + 2b3 + 4b1 + 4) q^65 + (4b9 + 2b7 + 6b6 - 10b5 + 6b2) q^66 + (b9 + b8 - b7 - b6 + 3b5 - b3 - b2) q^67 + (2b5 + b3 + 2) q^68 + (-2b8 + 2b2 + 6) * q^69 + (-6b7 - 6b1 + 2) * q^71 + (7b6 + 4b5 - 8b4 + 4b3 - 3b1 + 4) q^72 + (-2b9 - b8 - b7 - 4b6 - 4b5 + b3 - 4b2) * q^73 + (-2b9 + 6b8 + 2b7 - 8b5 - 6b3) q^74 + (b6 - 3b5 + 3b4 - 2b3 - 2b1 - 3) * q^75 + (-2b9 + 4b7 - 2b4 + 2b2 + 4b1 - 6) q^76 + (2b9 + 8b8 + 8b7 + 2b4 + 8b1 + 2) q^78 + (-2b5 - 2b4 - 2b3 - 2b1 - 2) * q^79 + (3b9 + b8 + 4b7 + 2b6 + 4b5 - b3 + 2b2) q^80 + (3b9 - 4b8 - 2b7 + 3b6 + 6b5 + 4b3 + 3b2) q^81 + (-2b6 - 9b5 + 6b4 - 3b3 - 9) * q^82 + (2b9 + 2b8 + 4b7 + 2b4 + 4b1) q^83 + (b9 - b8 + b7 + b4 + b2 + b1 - 1) * q^85 + (2b6 - b5 - 2b4 - 3b3 - 6b1 - 1) * q^86 + (2b8 + 4b7 - 4b6 + 6b5 - 2b3 - 4b2) * q^87 + (4b9 - 2b8 + 6b7 + 6b6 - 4b5 + 2b3 + 6b2) q^88 + (-2b6 + 6b5 - 4b4 + 2b3 - 2b1 + 6) q^89 + (4b9 - 12b8 - 4b7 + 4b4 + 3b2 - 4b1 - 20) * q^90 + (2b9 + 2b7 + 2b4 + 2b1 + 4) * q^92 + (-3b6 + 3b5 + 3b4 + 2b1 + 3) * q^93 + (2b9 - 4b8 + 4b6 + 10b5 + 4b3 + 4b2) * q^94 + (2b9 - 4b7 - 4b6 - 6b5 - 4b2) q^95 + (-4b5 + 2b4 - 3b3 - b1 - 4) q^96 + (-b9 - 3b8 + 3b7 - b4 - b2 + 3b1 - 3) q^97 + (-4b9 + 2b8 - 4b4 - 4b2 - 10) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 2 q^{2} - 2 q^{3} - 10 q^{4} + 2 q^{6} + 12 q^{8} - 11 q^{9} + 4 q^{10} + 2 q^{11} - 22 q^{12} - 4 q^{13} + 16 q^{15} - 4 q^{16} + 5 q^{17} + 18 q^{18} + 6 q^{19} + 38 q^{20} + 12 q^{22} + 10 q^{23}+ \cdots - 124 q^{99}+O(q^{100})$ 10 * q - 2 * q^2 - 2 * q^3 - 10 * q^4 + 2 * q^6 + 12 * q^8 - 11 * q^9 + 4 * q^10 + 2 * q^11 - 22 * q^12 - 4 * q^13 + 16 * q^15 - 4 * q^16 + 5 * q^17 + 18 * q^18 + 6 * q^19 + 38 * q^20 + 12 * q^22 + 10 * q^23 + 2 * q^24 - 21 * q^25 - 26 * q^26 + 52 * q^27 - 16 * q^29 + 51 * q^30 - 9 * q^32 + 6 * q^33 - 4 * q^34 + 78 * q^36 - 8 * q^37 + 14 * q^38 - 14 * q^39 + 5 * q^40 - 36 * q^41 + 16 * q^43 + 14 * q^44 - 8 * q^46 - 10 * q^47 + 54 * q^48 + 54 * q^50 + 2 * q^51 + 4 * q^52 - 4 * q^53 + 5 * q^54 + 48 * q^55 + 24 * q^57 - 12 * q^58 + 8 * q^59 + 7 * q^60 + 22 * q^61 - 32 * q^62 - 32 * q^64 + 30 * q^65 + 68 * q^66 - 16 * q^67 + 10 * q^68 + 64 * q^69 - 4 * q^71 - 9 * q^72 + 10 * q^73 + 40 * q^74 - 14 * q^75 - 48 * q^76 + 60 * q^78 - 18 * q^79 - 4 * q^80 - 25 * q^81 - 31 * q^82 + 24 * q^83 - 23 * q^86 - 26 * q^87 + 46 * q^88 + 20 * q^89 - 194 * q^90 + 56 * q^92 + 28 * q^93 - 42 * q^94 + 22 * q^95 - 18 * q^96 - 24 * q^97 - 124 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{10} - 2x^{9} + 9x^{8} - 10x^{7} + 44x^{6} - 49x^{5} + 99x^{4} - 20x^{3} + 31x^{2} - 3x + 9$ x^10 - 2*x^9 + 9*x^8 - 10*x^7 + 44*x^6 - 49*x^5 + 99*x^4 - 20*x^3 + 31*x^2 - 3*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 4212 \nu^{9} + 3675 \nu^{8} - 15575 \nu^{7} - 15678 \nu^{6} - 74025 \nu^{5} + 88200 \nu^{4} + \cdots + 2078070 ) / 773879$ (-4212v^9 + 3675v^8 - 15575v^7 - 15678v^6 - 74025v^5 + 88200v^4 + 119004v^3 + 26600v^2 - 5775*v + 2078070) / 773879
$\beta_{3}$	$=$	$( - 4749 \nu^{9} + 22333 \nu^{8} - 57798 \nu^{7} + 111303 \nu^{6} - 118188 \nu^{5} + 535992 \nu^{4} + \cdots + 37908 ) / 773879$ (-4749v^9 + 22333v^8 - 57798v^7 + 111303v^6 - 118188v^5 + 535992v^4 - 57640v^3 + 124797v^2 + 2839313*v + 37908) / 773879
$\beta_{4}$	$=$	$( - 19451 \nu^{9} + 168550 \nu^{8} - 235263 \nu^{7} + 1002431 \nu^{6} - 631225 \nu^{5} + \cdots + 1479609 ) / 2321637$ (-19451v^9 + 168550v^8 - 235263v^7 + 1002431v^6 - 631225v^5 + 4819079v^4 - 1048908v^3 + 4610308v^2 + 7695034*v + 1479609) / 2321637
$\beta_{5}$	$=$	$( 28988 \nu^{9} + 23213 \nu^{8} + 85878 \nu^{7} + 451846 \nu^{6} + 416857 \nu^{5} + 2104870 \nu^{4} + \cdots + 188058 ) / 2321637$ (28988v^9 + 23213v^8 + 85878v^7 + 451846v^6 + 416857v^5 + 2104870v^4 - 1330524v^3 + 7722551v^2 - 368140*v + 188058) / 2321637
$\beta_{6}$	$=$	$( - 24776 \nu^{9} - 26888 \nu^{8} - 70303 \nu^{7} - 436168 \nu^{6} - 342832 \nu^{5} - 2193070 \nu^{4} + \cdots - 2266128 ) / 773879$ (-24776v^9 - 26888v^8 - 70303v^7 - 436168v^6 - 342832v^5 - 2193070v^4 + 1211520v^3 - 6975272v^2 + 373915*v - 2266128) / 773879
$\beta_{7}$	$=$	$( - 27063 \nu^{9} + 58338 \nu^{8} - 247242 \nu^{7} + 286205 \nu^{6} - 1175094 \nu^{5} + 1400112 \nu^{4} + \cdots + 86964 ) / 773879$ (-27063v^9 + 58338v^8 - 247242v^7 + 286205v^6 - 1175094v^5 + 1400112v^4 - 2767437v^3 + 422256v^2 - 865553*v + 86964) / 773879
$\beta_{8}$	$=$	$( - 108252 \nu^{9} + 233352 \nu^{8} - 988968 \nu^{7} + 1144820 \nu^{6} - 4700376 \nu^{5} + \cdots + 347856 ) / 773879$ (-108252v^9 + 233352v^8 - 988968v^7 + 1144820v^6 - 4700376v^5 + 5600448v^4 - 10295869v^3 + 1689024v^2 - 366696*v + 347856) / 773879
$\beta_{9}$	$=$	$( - 329980 \nu^{9} + 541460 \nu^{8} - 2773827 \nu^{7} + 2469163 \nu^{6} - 13670405 \nu^{5} + \cdots + 681951 ) / 2321637$ (-329980v^9 + 541460v^8 - 2773827v^7 + 2469163v^6 - 13670405v^5 + 12221161v^4 - 27964188v^3 + 528812v^2 - 8810764*v + 681951) / 2321637

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{6} + 3\beta_{5} + \beta_{2}$ b6 + 3*b5 + b2
$\nu^{3}$	$=$	$\beta_{8} - 4\beta_{7} - 4\beta_1$ b8 - 4b7 - 4b1
$\nu^{4}$	$=$	$-5\beta_{6} - 13\beta_{5} - \beta_{4} + \beta_{3} - 13$ -5b6 - 13b5 - b4 + b3 - 13
$\nu^{5}$	$=$	$2\beta_{9} - 7\beta_{8} + 19\beta_{7} + \beta_{5} + 7\beta_{3}$ 2b9 - 7b8 + 19b7 + b5 + 7b3
$\nu^{6}$	$=$	$9\beta_{9} - 9\beta_{8} + \beta_{7} + 9\beta_{4} - 24\beta_{2} + \beta _1 + 60$ 9b9 - 9b8 + b7 + 9b4 - 24b2 + b1 + 60
$\nu^{7}$	$=$	$\beta_{6} - 6\beta_{5} + 18\beta_{4} - 42\beta_{3} + 93\beta _1 - 6$ b6 - 6b5 + 18b4 - 42b3 + 93b1 - 6
$\nu^{8}$	$=$	$-60\beta_{9} + 61\beta_{8} - 13\beta_{7} + 117\beta_{6} + 285\beta_{5} - 61\beta_{3} + 117\beta_{2}$ -60b9 + 61b8 - 13b7 + 117b6 + 285b5 - 61b3 + 117*b2
$\nu^{9}$	$=$	$-121\beta_{9} + 238\beta_{8} - 462\beta_{7} - 121\beta_{4} + 14\beta_{2} - 462\beta _1 + 20$ -121b9 + 238b8 - 462b7 - 121b4 + 14b2 - 462b1 + 20

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/833\mathbb{Z}\right)^\times$ .

$n$	$52$	$785$
$\chi(n)$	$-1 - \beta_{5}$	$1$

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}

18.1

0.304720	+	0.527790i
−1.08840	−	1.88516i
1.16091	+	2.01076i
−0.272099	−	0.471289i
0.894862	+	1.54995i
0.304720	−	0.527790i
−1.08840	+	1.88516i
1.16091	−	2.01076i
−0.272099	+	0.471289i
0.894862	−	1.54995i

−1.24613

−

2.15837i

−1.41042

2.44292i

−2.10570

3.64718i

1.25944

2.18142i

7.03030

5.51141

−2.47858

−

4.29302i

3.13887

−

5.43669i

18.2

−1.18400

−

2.05075i

0.284689

−

0.493096i

−1.80371

3.12411i

−2.07732

−

3.59803i

−1.34829

3.80636

1.33790

2.31732i

−4.91910

8.52013i

18.3

−0.438917

−

0.760227i

0.453789

−

0.785986i

0.614704

−

1.06470i

1.51909

2.63114i

−0.796703

−2.83488

1.08815

1.88473i

1.33351

−

2.30971i

18.4

0.704339

1.21995i

1.27991

−

2.21687i

0.00781334

−

0.0135331i

0.883300

1.52992i

3.60597

2.83937

−1.77635

−

3.07673i

−1.24428

2.15516i

18.5

1.16471

2.01734i

−1.60797

2.78508i

−1.71311

2.96719i

−1.58451

−

2.74445i

−7.49128

−3.32226

−3.67113

−

6.35858i

3.69100

−

6.39300i

324.1