LMFDB - Newform orbit 297.2.e.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,2,Mod(100,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, 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("297.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.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:	$2.37155694003$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.508277025.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 $ x^8 - 3x^7 + 5x^6 - 15x^5 + 21x^4 + 3x^3 - 22x^2 + 3*x + 19
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 99)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{6} + \beta_{2}) q^{2} + (3 \beta_{5} + \beta_{3}) q^{4} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{4}) q^{7} + ( - 2 \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{8}+O(q^{10}) $ q + (b6 + b2) * q^2 + (3b5 + b3) q^4 + (b7 - b5 + b3) * q^5 + (-b7 - b4) * q^7 + (-2b6 - b4 - b3 + b1 + 1) q^8	$ q + (\beta_{6} + \beta_{2}) q^{2} + (3 \beta_{5} + \beta_{3}) q^{4} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{4}) q^{7} + ( - 2 \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + (4 \beta_{6} + \beta_{3} - \beta_1 - 5) q^{98}+O(q^{100}) $ q + (b6 + b2) * q^2 + (3b5 + b3) q^4 + (b7 - b5 + b3) * q^5 + (-b7 - b4) * q^7 + (-2b6 - b4 - b3 + b1 + 1) q^8 + (-3b4 + 1) q^10 + (b5 + 1) * q^11 + (b7 + b5 - b3 + b2) * q^13 + (-2b7 - b3) q^14 + (-b7 + 2b6 - 5b5 - b4 + 2b2 - 2b1 - 5) * q^16 + (-b6 + 2b4 + 1) q^17 + (2b4 - b3 + b1 + 2) q^19 + (-4b7 + b6 - 2b5 - 4b4 + b2 - b1 - 2) q^20 + b2 * q^22 + (b7 - 4b5 - b3) q^23 + (b7 - b6 - 3b5 + b4 - b2 + 2b1 - 3) * q^25 + (-b4 + 3b3 - 3b1 - 6) * q^26 + (b6 + 3b4 - b3 + b1 - 1) q^28 + (-b6 - b5 - b2 + b1 - 1) * q^29 + (b3 - b2) * q^31 + (-2b7 + 10b5 + b3 - 3b2) q^32 + (4b7 + b6 - 5b5 + 4b4 + b2 + b1 - 5) q^34 + (2b6 - b4 - b3 + b1 + 2) q^35 + (-2b6 - b4 + 2b3 - 2b1 + 1) q^37 + (5b7 + 3b6 - b5 + 5b4 + 3b2 + b1 - 1) * q^38 + (-3b7 + 4b5 - 2b3 - 3b2) * q^40 + (-3b7 + b5 + b2) q^41 + (b6 + 5b5 + b2 + 5) q^43 + (b3 - b1 - 3) * q^44 + (5b6 - b4 + 2b3 - 2b1 - 1) q^46 + (2b7 - b6 - 2b5 + 2b4 - b2 - 2) q^47 + (-4b5 + b2) q^49 + (4b7 - 7b5 - 2b3 - b2) q^50 + (-3b7 - 7b6 + 5b5 - 3b4 - 7b2 + 5) q^52 + (3b4 - b3 + b1 + 1) q^53 + (-b4 + b3 - b1 + 1) * q^55 + (3b7 + 4b5 + 3b4 + b1 + 4) q^56 + (b7 - 6b5 - 2b3) * q^58 + (b7 - b5 - b3) * q^59 + (b7 - 4b5 + b4 - 4) q^61 + (-b6 - b4 - 2b3 + 2b1 + 6) * q^62 + (-7b6 + b4 - 2b3 + 2b1 + 6) q^64 + (-2b7 - 3b6 + 5b5 - 2b4 - 3b2 - 4b1 + 5) * q^65 + (-3b7 + 4b5 - 2b3 - b2) q^67 + (5b7 + 2b5 + 4b3 - 2b2) * q^68 + (-b7 + 3b6 + 9b5 - b4 + 3b2 + 9) q^70 + (2b6 - 3b4 + 1) * q^71 + (-2b6 - b4 + b3 - b1 + 6) q^73 + (-4b7 - b6 - 8b5 - 4b4 - b2 - b1 - 8) q^74 + (7b7 + 10b5 + 5b3) q^76 - b7 * q^77 + (-2b7 - 2b6 - 2b5 - 2b4 - 2b2 - b1 - 2) q^79 + (-2b3 + 2b1 + 9) * q^80 + (-b6 + 6b4 - 2b3 + 2b1 - 5) q^82 + (2b7 - b6 + 5b5 + 2b4 - b2 + 3b1 + 5) * q^83 + (-4b7 + 4b5 + 3b3 + 4b2) * q^85 + (5b5 + b3 + 5b2) * q^86 + (-b7 - 2b6 + b5 - b4 - 2b2 + b1 + 1) * q^88 + (2b6 + 4b4) * q^89 + (3b4 + 4) q^91 + (-2b7 - 3b6 + 19b5 - 2b4 - 3b2 + 4b1 + 19) * q^92 + (4b7 - 5b5 + b3 - 2b2) q^94 + (b7 + 7b5 + 2b3 + 3b2) q^95 + (-2b7 + 2b6 - 7b5 - 2b4 + 2b2 - 2b1 - 7) * q^97 + (4b6 + b3 - b1 - 5) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 11 q^{4} + 4 q^{5} - q^{7}+O(q^{10}) $ 8 * q + q^2 - 11 * q^4 + 4 * q^5 - q^7	$ 8 q + q^{2} - 11 q^{4} + 4 q^{5} - q^{7} + 2 q^{10} + 4 q^{11} - 7 q^{13} + q^{14} - 17 q^{16} + 10 q^{17} + 18 q^{19} - 10 q^{20} - q^{22} + 14 q^{23} - 14 q^{25} - 44 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{31} - 34 q^{32} - 16 q^{34} + 16 q^{35} + 6 q^{37} + 3 q^{38} - 12 q^{40} - 2 q^{41} + 21 q^{43} - 22 q^{44} + 4 q^{46} - 7 q^{47} + 15 q^{49} + 23 q^{50} + 10 q^{52} + 12 q^{53} + 8 q^{55} + 18 q^{56} + 21 q^{58} + 2 q^{59} - 15 q^{61} + 40 q^{62} + 32 q^{64} + 19 q^{65} - 14 q^{67} - 7 q^{68} + 38 q^{70} + 6 q^{71} + 44 q^{73} - 36 q^{74} - 42 q^{76} + q^{77} - 11 q^{79} + 68 q^{80} - 34 q^{82} + 18 q^{83} - 13 q^{85} - 24 q^{86} + 12 q^{89} + 38 q^{91} + 67 q^{92} + 19 q^{94} - 30 q^{95} - 26 q^{97} - 30 q^{98}+O(q^{100}) $ 8 * q + q^2 - 11 * q^4 + 4 * q^5 - q^7 + 2 * q^10 + 4 * q^11 - 7 * q^13 + q^14 - 17 * q^16 + 10 * q^17 + 18 * q^19 - 10 * q^20 - q^22 + 14 * q^23 - 14 * q^25 - 44 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^31 - 34 * q^32 - 16 * q^34 + 16 * q^35 + 6 * q^37 + 3 * q^38 - 12 * q^40 - 2 * q^41 + 21 * q^43 - 22 * q^44 + 4 * q^46 - 7 * q^47 + 15 * q^49 + 23 * q^50 + 10 * q^52 + 12 * q^53 + 8 * q^55 + 18 * q^56 + 21 * q^58 + 2 * q^59 - 15 * q^61 + 40 * q^62 + 32 * q^64 + 19 * q^65 - 14 * q^67 - 7 * q^68 + 38 * q^70 + 6 * q^71 + 44 * q^73 - 36 * q^74 - 42 * q^76 + q^77 - 11 * q^79 + 68 * q^80 - 34 * q^82 + 18 * q^83 - 13 * q^85 - 24 * q^86 + 12 * q^89 + 38 * q^91 + 67 * q^92 + 19 * q^94 - 30 * q^95 - 26 * q^97 - 30 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 $ x^8 - 3*x^7 + 5*x^6 - 15*x^5 + 21*x^4 + 3*x^3 - 22*x^2 + 3*x + 19 :

$\beta_{1}$	$=$	$ ( -43\nu^{7} - 317\nu^{6} - 270\nu^{5} + 36\nu^{4} + 2226\nu^{3} + 5037\nu^{2} - 3419\nu - 2806 ) / 2799 $ (-43v^7 - 317v^6 - 270v^5 + 36v^4 + 2226v^3 + 5037v^2 - 3419*v - 2806) / 2799
$\beta_{2}$	$=$	$ ( 193\nu^{7} + 620\nu^{6} - 1023\nu^{5} + 1878\nu^{4} - 10143\nu^{3} + 12564\nu^{2} + 2696\nu - 10991 ) / 5598 $ (193v^7 + 620v^6 - 1023v^5 + 1878v^4 - 10143v^3 + 12564v^2 + 2696*v - 10991) / 5598
$\beta_{3}$	$=$	$ ( 130\nu^{7} - 235\nu^{6} - 312\nu^{5} - 1389\nu^{4} - 828\nu^{3} + 6795\nu^{2} + 2048\nu - 3125 ) / 2799 $ (130v^7 - 235v^6 - 312v^5 - 1389v^4 - 828v^3 + 6795v^2 + 2048*v - 3125) / 2799
$\beta_{4}$	$=$	$ ( 145\nu^{7} - 298\nu^{6} + 585\nu^{5} - 1944\nu^{4} + 1086\nu^{3} + 1371\nu^{2} - 2596\nu + 3799 ) / 2799 $ (145v^7 - 298v^6 + 585v^5 - 1944v^4 + 1086v^3 + 1371v^2 - 2596*v + 3799) / 2799
$\beta_{5}$	$=$	$ ( -145\nu^{7} + 298\nu^{6} - 585\nu^{5} + 1944\nu^{4} - 2019\nu^{3} - 438\nu^{2} + 730\nu - 67 ) / 1866 $ (-145v^7 + 298v^6 - 585v^5 + 1944v^4 - 2019v^3 - 438v^2 + 730*v - 67) / 1866
$\beta_{6}$	$=$	$ ( 242\nu^{7} - 581\nu^{6} + 912\nu^{5} - 3045\nu^{4} + 3138\nu^{3} + 1812\nu^{2} - 6752\nu + 929 ) / 2799 $ (242v^7 - 581v^6 + 912v^5 - 3045v^4 + 3138v^3 + 1812v^2 - 6752*v + 929) / 2799
$\beta_{7}$	$=$	$ ( -254\nu^{7} + 818\nu^{6} - 1443\nu^{5} + 4422\nu^{4} - 6162\nu^{3} + 1221\nu^{2} + 1697\nu - 2363 ) / 2799 $ (-254v^7 + 818v^6 - 1443v^5 + 4422v^4 - 6162v^3 + 1221v^2 + 1697*v - 2363) / 2799

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

$\nu$	$=$	$ ( -2\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2 ) / 3 $ (-2*b7 - b6 + b5 - b4 + b2 + 2) / 3
$\nu^{2}$	$=$	$ ( 2\beta_{7} - 3\beta_{6} - 4\beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta _1 - 2 ) / 3 $ (2b7 - 3b6 - 4b5 + 4b4 - b3 + 2*b1 - 2) / 3
$\nu^{3}$	$=$	$ ( 6\beta_{7} - \beta_{6} - 12\beta_{5} - 3\beta_{4} - \beta_{3} - 2\beta_{2} + 2\beta _1 + 6 ) / 3 $ (6b7 - b6 - 12b5 - 3b4 - b3 - 2b2 + 2*b1 + 6) / 3
$\nu^{4}$	$=$	$ ( -17\beta_{7} + 6\beta_{6} + 16\beta_{5} - 19\beta_{4} - 5\beta_{3} + 12\beta_{2} + \beta _1 + 29 ) / 3 $ (-17b7 + 6b6 + 16b5 - 19b4 - 5b3 + 12b2 + b1 + 29) / 3
$\nu^{5}$	$=$	$ ( -13\beta_{7} - 22\beta_{6} + 5\beta_{5} + 31\beta_{4} - 17\beta_{3} + 13\beta_{2} + 10\beta _1 - 29 ) / 3 $ (-13b7 - 22b6 + 5b5 + 31b4 - 17b3 + 13b2 + 10*b1 - 29) / 3
$\nu^{6}$	$=$	$ 37\beta_{7} - 13\beta_{6} - 54\beta_{5} + 19\beta_{4} - 3\beta_{3} - 14\beta_{2} + 5\beta _1 - 18 $ 37b7 - 13b6 - 54b5 + 19b4 - 3b3 - 14b2 + 5*b1 - 18
$\nu^{7}$	$=$	$ ( -47\beta_{7} + 107\beta_{6} + 7\beta_{5} - 238\beta_{4} + 55\beta_{2} - 30\beta _1 + 326 ) / 3 $ (-47b7 + 107b6 + 7b5 - 238b4 + 55b2 - 30b1 + 326) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$\beta_{5}$	$1$

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} $

100.1

1.86526	+	0.199842i
0.947217	+	0.807294i
−0.577806	−	2.22188i
−0.734668	+	0.348716i
1.86526	−	0.199842i
0.947217	−	0.807294i
−0.577806	+	2.22188i
−0.734668	−	0.348716i

−1.36526

2.36469i

−2.72785

−

4.72478i

−0.468293

−

0.811107i

0.259560

−

0.449571i

9.43585

2.55736

100.2

−0.447217

0.774602i

0.599994

1.03922i

1.87447

3.24667i

−0.725528

1.25665i

−2.86218

−3.35317

100.3

1.07781

−

1.86682i

−1.32333

−

2.29208i

1.81197

3.13842i

1.13530

−

1.96640i

−1.39396

7.81179

100.4

1.23467

−

2.13851i

−2.04881

−

3.54864i

−1.21814

−

2.10988i

−1.16933

2.02534i

−5.17972

−6.01598

199.1