LMFDB - Newform orbit 693.2.i.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(100,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("693.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.i (of order $3$, degree $2$, 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:	$5.53363286007$
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} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12 $ x^10 + 15x^8 + 72x^6 + 120x^4 + 72x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 231)
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_{3} q^{2} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{4}+ \cdots + (2 \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{8}+O(q^{10}) $ q + b3 * q^2 + (b9 - b7 + b6 - b5 - b4 - 2b2 - 2) q^4 + (b8 + b2 + b1) * q^5 + (b9 - b5 - b4) * q^7 + (2b6 - b5 - b4 + b1) q^8	$ q + \beta_{3} q^{2} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{4}+ \cdots + (\beta_{9} + \beta_{8} - \beta_{7} + \cdots - 2) q^{98}+O(q^{100}) $ q + b3 * q^2 + (b9 - b7 + b6 - b5 - b4 - 2b2 - 2) q^4 + (b8 + b2 + b1) * q^5 + (b9 - b5 - b4) * q^7 + (2b6 - b5 - b4 + b1) q^8 + (-b9 + 3b8 - b7 - b6 - b5 - 2b3 + b1) * q^10 + (b2 + 1) * q^11 + (-b9 - b4 + 2) * q^13 + (-2b8 + b7 - b4 - 2b2 - b1 + 1) * q^14 + (-b8 + b7 + b4 + 3b2 - b1) q^16 + (-b6 - b3) * q^17 + (-b9 + 2b8 + 2b4 - b3 + b1) * q^19 + (-2b5 - 2b4 - b1 + 3) * q^20 - b6 * q^22 + (b8 - b7 - b4 - 3b2 + b1) q^23 + (-2b8 + 2b6 + 2b3 - b2 - 1) q^25 + (-3b8 + 2b7 + 2b4 + 4b3 + 3b2 - 3b1) * q^26 + (-4b8 + 2b7 + b5 + b4 + 3b3 + 4b2 - b1 + 2) * q^28 + (b5 + b4 - 3b1) q^29 + (-b9 + b7 - b6 + b5 + b4 - b2 - 1) * q^31 + (-2b8 - b6 - b3) q^32 + (-b9 - b6 + b5 + 4) * q^34 + (-b9 + b8 + b7 + b4 - 2b3 + 2b1 - 2) * q^35 + (b8 + 2b3 + 4b2 + b1) * q^37 + (b9 + 3b8 - b7 - b6 - b5 - b4 - 2b3 + b2 + 1) * q^38 + (b9 - 5b8 + 3b7 + b4 + 2b3 - 2b2 - 4b1) q^40 + (b6 + b5 + b4 + b1 + 4) * q^41 + (b9 + 2b6 - 3b5 - 2b4 + b1 + 2) q^43 + (-b7 - b4 - 2b2) q^44 + (-2b9 + 4b8 - 2b7 + 5b6 - 2b5 + 3b3 + 2b1) q^46 + (-b7 - b4 + 4b3 + 2b2) * q^47 + (-2b9 + b8 + b7 - 2b6 + b4 + 2b2 + 2b1 + 4) * q^49 + (2b9 + 3b6 + 2b4 + 2b1 - 8) * q^50 + (5b9 - 7b8 + b7 - 3b6 + b5 - 2b4 - 2b3 - 12b2 - 3b1 - 12) q^52 + (-2b9 + 3b8 - 2b7 + 2b6 - 2b5 + 3b2 + 2b1 + 3) q^53 + (b1 - 1) * q^55 + (b9 - b8 - 3b6 + b5 + 2b4 - 2b3 - 5b2 + b1 - 4) * q^56 + (2b9 - 5b8 + b7 - 3b4 + 2b3 + b2 - 3b1) q^58 + (-b9 - b8 + b7 - b6 + b5 + b4 + 3b2 + 3) q^59 + (3b9 - 3b8 + 3b7 - 3b4) * q^61 + (-b6 + b5 + b4 - b1) * q^62 + (b9 + b6 + b5 + 2b4 - 2) q^64 + (-2b9 + 4b8 + 4b4 - 4b3 + 6b2 + 2b1) * q^65 + (3b9 - 2b8 + b7 - b6 + b5 - b4 - 3b2 - 2b1 - 3) * q^67 + (-b9 + b8 - b7 + b4 + 3b3) q^68 + (-2b9 + 3b8 + 2b7 - 4b6 + 2b5 + 4b4 - 4b3 + 9b2 + 7) * q^70 + (-3b9 - b6 + 3b5 - 4) * q^71 + (-4b9 + 2b7 + 2b5 + 3b4 + 2b3 - 2b2 + b1 - 2) * q^73 + (b9 + 3b8 - 3b7 - 2b6 - 3b5 - 2b4 - 5b3 - 8b2 + b1 - 8) q^74 + (b6 - 2b5 - 2b4 - 2b1 + 8) q^76 + (b8 - b7 - b5 - b4 - b3 + b1) * q^77 + (-b9 + 4b8 - 4b7 - 2b4 - 2b3 + 2b2 + 3b1) * q^79 + (2b9 - 5b8 + 2b7 - 2b6 + 2b5 + b2 - 2b1 + 1) * q^80 + (-2b9 + 7b8 - 2b7 + 2b4 + 2b3 + 5b2 + 5b1) q^82 + (-2b9 - 4b6 + 2b5 - 2b1 + 2) * q^83 + (b6 + b5 + b4 + b1) * q^85 + (2b9 - 6b8 + 5b7 + b4 + 2b3 + 2b2 - 4b1) * q^86 + (-b9 + b8 - b7 + 2b6 - b5 + b3 + b1) q^88 + (2b9 + 2b7 - 2b4 + 2b1) * q^89 + (b8 - b7 - 6b6 + b4 - 4b3 - 4b2 - 2b1 - 3) * q^91 + (5b9 + 5b6 - 7b5 - 2b4 - 2b1 - 12) q^92 + (3b9 + b8 - 5b7 + 4b6 - 5b5 - 4b4 - b3 - 16b2 + b1 - 16) * q^94 + (5b9 - 5b8 + b7 + 5b6 + b5 - 2b4 + 6b3 - 4b2 - 3b1 - 4) q^95 + (-b9 + b6 + 5b5 + 4b4 - 2b1 - 4) q^97 + (b9 + b8 - b7 - 4b6 + b4 + b3 - 6b2 - 2b1 - 2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} - 10 q^{4} - 4 q^{5} - q^{7} - 12 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 - 10 * q^4 - 4 * q^5 - q^7 - 12 * q^8	$ 10 q + 2 q^{2} - 10 q^{4} - 4 q^{5} - q^{7} - 12 q^{8} - 2 q^{10} + 5 q^{11} + 10 q^{13} + 10 q^{14} - 16 q^{16} + 2 q^{17} + 3 q^{19} + 16 q^{20} + 4 q^{22} + 16 q^{23} - 7 q^{25} - 10 q^{26} + 4 q^{28} - 5 q^{31} + 4 q^{32} + 40 q^{34} - 26 q^{35} - 15 q^{37} + 6 q^{38} + 6 q^{40} + 44 q^{41} + 6 q^{43} + 10 q^{44} - 16 q^{46} - 2 q^{47} + 31 q^{49} - 68 q^{50} - 40 q^{52} + 6 q^{53} - 8 q^{55} + 12 q^{56} - 12 q^{58} + 16 q^{59} - 12 q^{61} + 8 q^{62} - 8 q^{64} - 28 q^{65} - 7 q^{67} + 10 q^{68} + 32 q^{70} - 48 q^{71} - 17 q^{73} - 36 q^{74} + 60 q^{76} - 2 q^{77} - 7 q^{79} + 16 q^{80} - 8 q^{82} + 24 q^{83} + 4 q^{85} - 18 q^{86} - 6 q^{88} - 6 q^{89} + 11 q^{91} - 136 q^{92} - 82 q^{94} - 18 q^{95} - 28 q^{97} + 38 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 - 10 * q^4 - 4 * q^5 - q^7 - 12 * q^8 - 2 * q^10 + 5 * q^11 + 10 * q^13 + 10 * q^14 - 16 * q^16 + 2 * q^17 + 3 * q^19 + 16 * q^20 + 4 * q^22 + 16 * q^23 - 7 * q^25 - 10 * q^26 + 4 * q^28 - 5 * q^31 + 4 * q^32 + 40 * q^34 - 26 * q^35 - 15 * q^37 + 6 * q^38 + 6 * q^40 + 44 * q^41 + 6 * q^43 + 10 * q^44 - 16 * q^46 - 2 * q^47 + 31 * q^49 - 68 * q^50 - 40 * q^52 + 6 * q^53 - 8 * q^55 + 12 * q^56 - 12 * q^58 + 16 * q^59 - 12 * q^61 + 8 * q^62 - 8 * q^64 - 28 * q^65 - 7 * q^67 + 10 * q^68 + 32 * q^70 - 48 * q^71 - 17 * q^73 - 36 * q^74 + 60 * q^76 - 2 * q^77 - 7 * q^79 + 16 * q^80 - 8 * q^82 + 24 * q^83 + 4 * q^85 - 18 * q^86 - 6 * q^88 - 6 * q^89 + 11 * q^91 - 136 * q^92 - 82 * q^94 - 18 * q^95 - 28 * q^97 + 38 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12 $ x^10 + 15*x^8 + 72*x^6 + 120*x^4 + 72*x^2 + 12 :

$\beta_{1}$	$=$	$ \nu^{4} + 7\nu^{2} + 5 $ v^4 + 7*v^2 + 5
$\beta_{2}$	$=$	$ ( \nu^{9} + 15\nu^{7} + 70\nu^{5} + 102\nu^{3} + 36\nu - 4 ) / 8 $ (v^9 + 15v^7 + 70v^5 + 102v^3 + 36v - 4) / 8
$\beta_{3}$	$=$	$ ( \nu^{9} + 2\nu^{8} + 13\nu^{7} + 28\nu^{6} + 44\nu^{5} + 118\nu^{4} + 6\nu^{3} + 140\nu^{2} - 36\nu + 40 ) / 8 $ (v^9 + 2v^8 + 13v^7 + 28v^6 + 44v^5 + 118v^4 + 6v^3 + 140v^2 - 36v + 40) / 8
$\beta_{4}$	$=$	$ ( \nu^{9} + \nu^{8} + 14\nu^{7} + 15\nu^{6} + 59\nu^{5} + 70\nu^{4} + 70\nu^{3} + 100\nu^{2} + 18\nu + 32 ) / 4 $ (v^9 + v^8 + 14v^7 + 15v^6 + 59v^5 + 70v^4 + 70v^3 + 100v^2 + 18*v + 32) / 4
$\beta_{5}$	$=$	$ ( -\nu^{9} - \nu^{8} - 14\nu^{7} - 13\nu^{6} - 59\nu^{5} - 48\nu^{4} - 70\nu^{3} - 36\nu^{2} - 18\nu + 4 ) / 4 $ (-v^9 - v^8 - 14v^7 - 13v^6 - 59v^5 - 48v^4 - 70v^3 - 36v^2 - 18*v + 4) / 4
$\beta_{6}$	$=$	$ ( -\nu^{8} - 14\nu^{6} - 59\nu^{4} - 70\nu^{2} - 20 ) / 2 $ (-v^8 - 14v^6 - 59v^4 - 70*v^2 - 20) / 2
$\beta_{7}$	$=$	$ ( \nu^{9} - 2\nu^{8} + 15\nu^{7} - 30\nu^{6} + 74\nu^{5} - 140\nu^{4} + 130\nu^{3} - 204\nu^{2} + 60\nu - 76 ) / 8 $ (v^9 - 2v^8 + 15v^7 - 30v^6 + 74v^5 - 140v^4 + 130v^3 - 204v^2 + 60v - 76) / 8
$\beta_{8}$	$=$	$ ( -3\nu^{9} - 43\nu^{7} - 188\nu^{5} - 4\nu^{4} - 246\nu^{3} - 28\nu^{2} - 96\nu - 20 ) / 8 $ (-3v^9 - 43v^7 - 188v^5 - 4v^4 - 246v^3 - 28v^2 - 96*v - 20) / 8
$\beta_{9}$	$=$	$ ( -\nu^{9} + \nu^{8} - 14\nu^{7} + 15\nu^{6} - 59\nu^{5} + 70\nu^{4} - 70\nu^{3} + 100\nu^{2} - 18\nu + 32 ) / 4 $ (-v^9 + v^8 - 14v^7 + 15v^6 - 59v^5 + 70v^4 - 70v^3 + 100v^2 - 18*v + 32) / 4

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

$\nu$	$=$	$ ( 2\beta_{9} - 2\beta_{8} + 2\beta_{7} + \beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 3 $ (2b9 - 2b8 + 2b7 + b5 - b4 + 2b3 - 2*b2 - b1 - 1) / 3
$\nu^{2}$	$=$	$ -\beta_{9} - \beta_{6} + \beta_{5} - 3 $ -b9 - b6 + b5 - 3
$\nu^{3}$	$=$	$ -3\beta_{9} + 2\beta_{8} - 4\beta_{7} - 2\beta_{5} + \beta_{4} - 4\beta_{3} + 2\beta_{2} + \beta _1 + 1 $ -3b9 + 2b8 - 4b7 - 2b5 + b4 - 4b3 + 2b2 + b1 + 1
$\nu^{4}$	$=$	$ 7\beta_{9} + 7\beta_{6} - 7\beta_{5} + \beta _1 + 16 $ 7b9 + 7b6 - 7*b5 + b1 + 16
$\nu^{5}$	$=$	$ 17 \beta_{9} - 10 \beta_{8} + 26 \beta_{7} - \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 24 \beta_{3} + \cdots - 6 $ 17b9 - 10b8 + 26b7 - b6 + 13b5 - 4b4 + 24b3 - 12b2 - 5b1 - 6
$\nu^{6}$	$=$	$ -45\beta_{9} - 45\beta_{6} + 47\beta_{5} + 2\beta_{4} - 11\beta _1 - 98 $ -45b9 - 45b6 + 47b5 + 2b4 - 11*b1 - 98
$\nu^{7}$	$=$	$ - 101 \beta_{9} + 58 \beta_{8} - 170 \beta_{7} + 11 \beta_{6} - 85 \beta_{5} + 16 \beta_{4} - 148 \beta_{3} + \cdots + 44 $ -101b9 + 58b8 - 170b7 + 11b6 - 85b5 + 16b4 - 148b3 + 88b2 + 29*b1 + 44
$\nu^{8}$	$=$	$ 287\beta_{9} + 285\beta_{6} - 315\beta_{5} - 28\beta_{4} + 95\beta _1 + 618 $ 287b9 + 285b6 - 315b5 - 28b4 + 95*b1 + 618
$\nu^{9}$	$=$	$ 607 \beta_{9} - 350 \beta_{8} + 1114 \beta_{7} - 95 \beta_{6} + 557 \beta_{5} - 50 \beta_{4} + \cdots - 326 $ 607b9 - 350b8 + 1114b7 - 95b6 + 557b5 - 50b4 + 924b3 - 652b2 - 175*b1 - 326

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$442$
$\chi(n)$	$1$	$\beta_{2}$	$1$

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} $

100.1

	−	0.886226i
		2.42024i
	−	2.57330i
	−	1.21103i
		0.518255i
		0.886226i
	−	2.42024i
		2.57330i
		1.21103i
	−	0.518255i

−1.24646

2.15892i

−2.10730

−

3.64995i

−0.440463

0.762904i

−2.14580

−

1.54775i

5.52081

−1.09804

−

1.90185i

100.2

−0.534421

0.925645i

0.428788

0.742682i

−1.34592

2.33120i

−0.855706

2.50355i

−3.05430

−1.43858

−

2.49169i

100.3

0.307468

−

0.532550i

0.810927

1.40457i

0.747986

−

1.29555i

2.59895

−

0.495442i

2.22721

−0.459963

−

0.796680i

100.4

1.17616

−

2.03717i

−1.76671

−

3.06003i

−2.05761

3.56389i

2.51585

0.818848i

−3.60708

4.84017

8.38342i

100.5

1.29725

−

2.24690i

−2.36571

−

4.09752i

1.09601

−

1.89835i

−2.61329

−

0.413178i

−7.08664

−2.84359

−

4.92525i

298.1