LMFDB - Newform orbit 603.2.g.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [603,2,Mod(37,603)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(603, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("603.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 603 = 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	603.g (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:	$4.81497924188$
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} + 10x^{8} - 8x^{7} + 49x^{6} - 39x^{5} + 128x^{4} - 14x^{3} + 119x^{2} - 49x + 49 $ x^10 - 2x^9 + 10x^8 - 8x^7 + 49x^6 - 39x^5 + 128x^4 - 14x^3 + 119x^2 - 49*x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 201)
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_1 q^{2} + (\beta_{8} - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{6} - \beta_{5} + \beta_{3} - 1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b8 - b5 - b4 + b3 + b1) * q^4 + b5 * q^5 + (-b9 + b8 - b7 - b5) * q^7 + (b6 - b5 + b3 - 1) * q^8	$ q + \beta_1 q^{2} + (\beta_{8} - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + ( - 5 \beta_{9} + 7 \beta_{8} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b8 - b5 - b4 + b3 + b1) * q^4 + b5 * q^5 + (-b9 + b8 - b7 - b5) * q^7 + (b6 - b5 + b3 - 1) * q^8 + (-b6 + 2b4 + b2 + b1 - 2) q^10 + (-b9 - b7) * q^11 + (b8 - b7 - b6 + b2 + b1) * q^13 + (-b9 + 2b6 + b3 + 3) q^14 + (-b8 + b7 + b6 - b4 - b2 - 2b1 + 1) q^16 + (-b8 + b6 + b4 - b2 + b1 - 1) * q^17 + (b6 - b4 - b2 - b1 + 1) * q^19 + (-b9 + b8 - b7 - b5 - 3b4) q^20 + (-b9 + b6 + 1) * q^22 + (-b4 - 2b1 + 1) q^23 + (b9 + b6 + b3) * q^25 + (-2b9 + 3b8 - 2b7 - 3b5 + 3b3 + 2b2 + 3b1) q^26 + (-3b8 + b7 + b6 + 2b4 - b2 - 2) * q^28 + (-b8 + b5 - 3b4 - 2b3 - 2b2 - 2b1) * q^29 + (-b9 - b7 - 2b4) q^31 + (2b9 - 2b8 + 2b7 + 2b5 + b4 - b3 - b1) * q^32 + (b9 - b8 + b7 + b5 - 5b4 - 2b3 - b2 - 2b1) q^34 + (b9 + 2b8 + b7 - 2b5 - 5b4 - b3 - b2 - b1) q^35 + (b8 - b7 + b6 - 4b4 - b2 + 3b1 + 4) * q^37 + (b9 - 3b8 + b7 + 3b5 + 3b4 - b3 - b1) q^38 + (-b9 + 2b3 - 1) q^40 + (b9 + b8 + b7 - b5 + 2b4 + 3b3 - b2 + 3b1) q^41 + (-b9 + b6 + 3b5 - 3b3 + 1) * q^43 + (-2b8 + b6 - b4 - b2 + 1) q^44 + (-2b8 + 2b5 + 6b4 - b3 - b1) q^46 + (b9 - b8 + b7 + b5 + 6b4 - b3 + b2 - b1) q^47 + (-2b7 - 3b6 + 6b4 + 3b2 + b1 - 6) * q^49 + (-3b8 - b6 + 4b4 + b2 - 2b1 - 4) q^50 + (-2b9 + 3b6 - 5b5 + 6b3 - 1) * q^52 + (-b9 - b6 + 3b5 - b3 + 3) q^53 + (2b9 + 2b8 + 2b7 - 2b5) * q^55 + (-2b8 + 2b5 - b4 - 4b3 - 4b1) * q^56 + (2b9 - b6 + 6b5 - 2b3 + 4) q^58 + (2b9 - b6 + 2b5 - 3b3 - 4) q^59 + (-3b8 + b7 + 2b6 - 2b2) q^61 + (-b9 + b6 + 2b3 + 1) q^62 + (-2b6 - b5 - 1) q^64 + (b8 + b7 - b4 + 2b1 + 1) q^65 + (b8 + b7 + 3b6 - b5 + b4 - b2 - b1) q^67 + (2b9 + 2b5 - b3 + 1) * q^68 + (2b9 + b6 + 3b5 + 5b3 + 6) q^70 + (-2b9 + 4b8 - 2b7 - 4b5 + b4 + 2b3 + 2b1) * q^71 + (b8 + 2b7 - 3b6 - 3b4 + 3b2 + b1 + 3) * q^73 + (b8 - b5 - 6b4 + 7b3 + 2b2 + 7b1) * q^74 + (b9 - 2b6 + b5 - 5b3 - 2) * q^76 + (2b8 - b7 - 2b6 + 8b4 + 2b2 - 8) * q^77 + (b9 - b8 + b7 + b5 - 3b3 - b2 - 3b1) * q^79 + (-b7 + b6 - b4 - b2 - 3b1 + 1) q^80 + (2b9 - b5 + b3 - 8) q^82 + (-b8 - b6 + 3b4 + b2 + b1 - 3) q^83 + (b7 - 2b6 + 5b4 + 2b2 - 5) q^85 + (b8 + 2b7 - 2b6 - 4b4 + 2b2 + 6b1 + 4) q^86 + (-b9 - 2b8 - b7 + 2b5 - 2b4 - 2b3 - 2b1) q^88 + (b9 - b5 + 2b3 - 1) q^89 + (2b9 - 2b6 + b5 + 4b3 - 7) q^91 + (-2b6 + b5 - 5b3 + 1) * q^92 + (-2b6 - b5 - 7b3) * q^94 + (2b8 + b6 - 4b4 - b2 - 3b1 + 4) q^95 + (-3b8 - b7 + 2b4 - 2b1 - 2) q^97 + (-5b9 + 7b8 - 5b7 - 7b5 - b4 - 2b3 + 2b2 - 2b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} - 6 q^{4} - 2 q^{5} - q^{7} - 12 q^{8}+O(q^{10}) $ 10 * q + 2 * q^2 - 6 * q^4 - 2 * q^5 - q^7 - 12 * q^8	$ 10 q + 2 q^{2} - 6 q^{4} - 2 q^{5} - q^{7} - 12 q^{8} - 8 q^{10} - 2 q^{11} + 3 q^{13} + 22 q^{14} - 2 q^{17} + 3 q^{19} - 16 q^{20} + 6 q^{22} + q^{23} - 7 q^{26} - 9 q^{28} - 12 q^{29} - 12 q^{31} + 9 q^{32} - 20 q^{34} - 19 q^{35} + 27 q^{37} + 16 q^{38} - 22 q^{40} + 7 q^{41} + 12 q^{43} + 7 q^{44} + 30 q^{46} + 33 q^{47} - 24 q^{49} - 21 q^{50} - 32 q^{52} + 24 q^{53} + 6 q^{55} + q^{56} + 44 q^{58} - 24 q^{59} + q^{61} - 2 q^{62} - 8 q^{64} + 6 q^{65} + 2 q^{67} + 18 q^{68} + 42 q^{70} + q^{71} + 12 q^{73} - 43 q^{74} + 2 q^{76} - 40 q^{77} + 7 q^{79} + q^{80} - 74 q^{82} - 12 q^{83} - 27 q^{85} + 27 q^{86} - 10 q^{88} - 12 q^{89} - 80 q^{91} + 28 q^{92} + 30 q^{94} + 12 q^{95} - 9 q^{97} - 4 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 - 6 * q^4 - 2 * q^5 - q^7 - 12 * q^8 - 8 * q^10 - 2 * q^11 + 3 * q^13 + 22 * q^14 - 2 * q^17 + 3 * q^19 - 16 * q^20 + 6 * q^22 + q^23 - 7 * q^26 - 9 * q^28 - 12 * q^29 - 12 * q^31 + 9 * q^32 - 20 * q^34 - 19 * q^35 + 27 * q^37 + 16 * q^38 - 22 * q^40 + 7 * q^41 + 12 * q^43 + 7 * q^44 + 30 * q^46 + 33 * q^47 - 24 * q^49 - 21 * q^50 - 32 * q^52 + 24 * q^53 + 6 * q^55 + q^56 + 44 * q^58 - 24 * q^59 + q^61 - 2 * q^62 - 8 * q^64 + 6 * q^65 + 2 * q^67 + 18 * q^68 + 42 * q^70 + q^71 + 12 * q^73 - 43 * q^74 + 2 * q^76 - 40 * q^77 + 7 * q^79 + q^80 - 74 * q^82 - 12 * q^83 - 27 * q^85 + 27 * q^86 - 10 * q^88 - 12 * q^89 - 80 * q^91 + 28 * q^92 + 30 * q^94 + 12 * q^95 - 9 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 49x^{6} - 39x^{5} + 128x^{4} - 14x^{3} + 119x^{2} - 49x + 49 $ x^10 - 2*x^9 + 10*x^8 - 8*x^7 + 49*x^6 - 39*x^5 + 128*x^4 - 14*x^3 + 119*x^2 - 49*x + 49 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 24307 \nu^{9} + 181995 \nu^{8} - 604465 \nu^{7} + 1626651 \nu^{6} - 3301924 \nu^{5} + \cdots + 6437963 ) / 4414074 $ (-24307v^9 + 181995v^8 - 604465v^7 + 1626651v^6 - 3301924v^5 + 7319233v^4 - 12172941v^3 + 14158031v^2 - 13105288*v + 6437963) / 4414074
$\beta_{3}$	$=$	$ ( 32653 \nu^{9} - 100089 \nu^{8} + 396649 \nu^{7} - 539103 \nu^{6} + 1764532 \nu^{5} + \cdots - 2618021 ) / 4414074 $ (32653v^9 - 100089v^8 + 396649v^7 - 539103v^6 + 1764532v^5 - 2509639v^4 + 5937753v^3 - 2828441v^2 + 1453144*v - 2618021) / 4414074
$\beta_{4}$	$=$	$ ( - 53429 \nu^{9} + 74205 \nu^{8} - 434201 \nu^{7} + 30783 \nu^{6} - 2078918 \nu^{5} + \cdots + 1164877 ) / 4414074 $ (-53429v^9 + 74205v^8 - 434201v^7 + 30783v^6 - 2078918v^5 + 319199v^4 - 4329273v^3 - 5189747v^2 - 3529610*v + 1164877) / 4414074
$\beta_{5}$	$=$	$ ( 33718 \nu^{9} - 85104 \nu^{8} + 337264 \nu^{7} - 351819 \nu^{6} + 1500352 \nu^{5} - 2133904 \nu^{4} + \cdots - 7130123 ) / 2207037 $ (33718v^9 - 85104v^8 + 337264v^7 - 351819v^6 + 1500352v^5 - 2133904v^4 + 4154526v^3 - 2404976v^2 + 1235584*v - 7130123) / 2207037
$\beta_{6}$	$=$	$ ( - 31943 \nu^{9} + 110079 \nu^{8} - 436239 \nu^{7} + 663959 \nu^{6} - 1940652 \nu^{5} + \cdots + 1081311 ) / 1471358 $ (-31943v^9 + 110079v^8 - 436239v^7 + 663959v^6 - 1940652v^5 + 2760129v^4 - 5655213v^3 + 3110751v^2 - 1598184*v + 1081311) / 1471358
$\beta_{7}$	$=$	$ ( - 112429 \nu^{9} - 45435 \nu^{8} - 824203 \nu^{7} - 1270707 \nu^{6} - 5458108 \nu^{5} + \cdots - 11599567 ) / 4414074 $ (-112429v^9 - 45435v^8 - 824203v^7 - 1270707v^6 - 5458108v^5 - 6300671v^4 - 13242393v^3 - 19664257v^2 - 25007374*v - 11599567) / 4414074
$\beta_{8}$	$=$	$ ( - 62752 \nu^{9} + 76248 \nu^{8} - 512362 \nu^{7} - 36093 \nu^{6} - 2500291 \nu^{5} + \cdots - 4073797 ) / 2207037 $ (-62752v^9 + 76248v^8 - 512362v^7 - 36093v^6 - 2500291v^5 - 400286v^4 - 5308260v^3 - 6568339v^2 - 6992440*v - 4073797) / 2207037
$\beta_{9}$	$=$	$ ( - 117800 \nu^{9} + 300861 \nu^{8} - 1192301 \nu^{7} + 1489527 \nu^{6} - 5304068 \nu^{5} + \cdots + 12355714 ) / 2207037 $ (-117800v^9 + 300861v^8 - 1192301v^7 + 1489527v^6 - 5304068v^5 + 7543811v^4 - 12249042v^3 + 8502109v^2 - 4368056*v + 12355714) / 2207037

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{8} - \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1 $ b8 - b5 - 3*b4 + b3 + b1
$\nu^{3}$	$=$	$ \beta_{6} - \beta_{5} + 5\beta_{3} - 1 $ b6 - b5 + 5*b3 - 1
$\nu^{4}$	$=$	$ -7\beta_{8} + \beta_{7} + \beta_{6} + 13\beta_{4} - \beta_{2} - 8\beta _1 - 13 $ -7b8 + b7 + b6 + 13b4 - b2 - 8*b1 - 13
$\nu^{5}$	$=$	$ 2\beta_{9} - 10\beta_{8} + 2\beta_{7} + 10\beta_{5} + 9\beta_{4} - 29\beta_{3} - 8\beta_{2} - 29\beta_1 $ 2b9 - 10b8 + 2b7 + 10b5 + 9b4 - 29b3 - 8b2 - 29b1
$\nu^{6}$	$=$	$ 10\beta_{9} - 12\beta_{6} + 45\beta_{5} - 56\beta_{3} + 65 $ 10b9 - 12b6 + 45b5 - 56b3 + 65
$\nu^{7}$	$=$	$ 80\beta_{8} - 22\beta_{7} - 55\beta_{6} - 68\beta_{4} + 55\beta_{2} + 178\beta _1 + 68 $ 80b8 - 22b7 - 55b6 - 68b4 + 55b2 + 178b1 + 68
$\nu^{8}$	$=$	$ -77\beta_{9} + 288\beta_{8} - 77\beta_{7} - 288\beta_{5} - 352\beta_{4} + 381\beta_{3} + 102\beta_{2} + 381\beta_1 $ -77b9 + 288b8 - 77b7 - 288b5 - 352b4 + 381b3 + 102b2 + 381b1
$\nu^{9}$	$=$	$ -179\beta_{9} + 365\beta_{6} - 585\beta_{5} + 1123\beta_{3} - 490 $ -179b9 + 365b6 - 585b5 + 1123b3 - 490

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/603\mathbb{Z}\right)^\times$.

$n$	$136$	$470$
$\chi(n)$	$-\beta_{4}$	$1$

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} $

37.1

−1.03354	−	1.79014i
−0.527336	−	0.913372i
0.330147	+	0.571831i
0.947050	+	1.64034i
1.28367	+	2.22339i
−1.03354	+	1.79014i
−0.527336	+	0.913372i
0.330147	−	0.571831i
0.947050	−	1.64034i
1.28367	−	2.22339i

−1.03354

−

1.79014i

−1.13639

1.96829i

3.33986

−2.29539

3.97573i

0.563865

−3.45186

−

5.97880i

37.2

−0.527336

−

0.913372i

0.443834

−

0.768743i

−0.832996

1.13029

−

1.95772i

−3.04554

0.439269

0.760836i

37.3

0.330147

0.571831i

0.782006

−

1.35447i

−3.22431

−1.02143

1.76916i

2.35330

−1.06450

−

1.84376i

37.4

0.947050

1.64034i

−0.793808

1.37492i

−1.30648

2.53710

−

4.39438i

0.781096

−1.23731

−

2.14308i

37.5

1.28367

2.22339i

−2.29564

3.97616i

1.02393

−0.850574

1.47324i

−6.65272

1.31439

2.27659i

163.1