LMFDB - Newform orbit 513.2.f.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(163,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.163"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.f (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,1,0,-3,5,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 8x^{10} - x^{9} + 41x^{8} - 7x^{7} + 91x^{6} + 9x^{5} + 135x^{4} - 12x^{3} + 45x^{2} + 9x + 9 $ x^12 - x^11 + 8x^10 - x^9 + 41x^8 - 7x^7 + 91x^6 + 9x^5 + 135x^4 - 12x^3 + 45x^2 + 9*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{6} - \beta_1) q^{2} + ( - \beta_{8} + \beta_{3} - \beta_1) q^{4} + ( - \beta_{11} + \beta_{7} + 1) q^{5} + (\beta_{10} + \beta_{2}) q^{7} + (\beta_{10} + \beta_{6} + \beta_{3} - 1) q^{8}+ \cdots + (\beta_{11} + \beta_{10} - 3 \beta_{9} + \cdots - 8) q^{98}+O(q^{100}) $ q + (-b6 - b1) * q^2 + (-b8 + b3 - b1) * q^4 + (-b11 + b7 + 1) * q^5 + (b10 + b2) * q^7 + (b10 + b6 + b3 - 1) * q^8 + (-b9 - b1) * q^10 + (-b3 - b2 - 1) * q^11 + (-b8 + b7 + b5 - b4 + b3 - b1) * q^13 + (-b11 - b10 + b9 + b8 + b7 - b4 - b2 + 1) * q^14 + (b10 + b9 + 2b6 + b4 - b2 + 2b1) * q^16 + (b10 + b9 + b7 + b6 + b4 - b2 + b1 + 1) * q^17 + (b11 - b9 - b8 - b4 + b2 - b1 - 1) * q^19 + (-b11 + b10 + b6 - b5 + b3) * q^20 + (b11 - b7 - b6 - b1 - 1) * q^22 + (b9 + b8 - b4 - b3) * q^23 + (2b8 + 2b7 + 2b5 - 2b3 + b1) * q^25 + (b10 + 4b6 + 2b3 - 2b2) q^26 + (-b8 - b5 + b3 + b1) * q^28 + (2b9 + b8 + b7 - b3 + 2b1) * q^29 + (b11 - b10 - 2b6 + b5 - b3 - b2 - 1) q^31 + (b9 + b8 - b7 - b5 - b4 - b3) * q^32 + (b9 + 2b8 - b7 - b5 + b4 - 2b3) * q^34 + (2b10 - 3b9 + 2b8 + b7 - b6 + 2b4 + 3b2 - b1 + 1) q^35 + (-b11 - b10 - 3b6 - b5 + b3 + b2 + 3) q^37 + (b9 - b7 + 2b6 + b5 - 2b4 + 2b3 - b2 - b1 - 1) q^38 + (b10 - 2b9 + 2b8 + b6 + b4 + 2b2 + b1) q^40 + (-b11 - 2b7 - 2) q^41 + (-2b11 + b9 - b8 - b7 - b6 - b2 - b1 - 1) q^43 + (-b9 + b8 + 4b7 - b3) q^44 + (-b11 - 2b10 - 2b6 - b5 + b3 - b2) * q^46 + (-b9 + 2b8 - 2b7 + b5 + b4 - 2b3 + 2b1) * q^47 + (-3b11 + 4b6 - 3b5 - b2 + 3) q^49 + (-2b10 - 3b6 - b3 - 2b2) q^50 + (2b10 + b9 + 3b8 - 3b7 + 6b6 + 2b4 - b2 + 6b1 - 3) * q^52 + (b9 - 3b8 - 2b7 + b4 + 3b3 - 2b1) * q^53 + (2b11 - 3b10 + 2b9 - 2b8 - b7 + b6 - 3b4 - 2b2 + b1 - 1) * q^55 + (-2b11 - b10 + b6 - 2b5 + b3 - b2 + 5) * q^56 + (-2b11 - 3b10 - 3b6 - 2b5 - 2b3 + 3) q^58 + (b10 - b9 - 4b8 - 2b6 + b4 + b2 - 2b1) q^59 + (b9 + b7 + b5 + b4 - 3b1) q^61 + (b11 - 3b8 + 2b7 - 3b6 - 3b1 + 2) * q^62 + (-b11 + b6 - b5 + b3 - 2b2) q^64 + (b11 + b10 + b5 + 2b2 - 6) q^65 + (2b9 + b8 - 2b7 - 2b5 - b3 + b1) q^67 + (-b11 - b10 - 3b6 - b5 - b3 - 1) q^68 + (2b9 + b8 + 6b7 + 3b5 - b4 - b3 + 2b1) * q^70 + (b11 - 2b10 + b9 + 3b6 - 2b4 - b2 + 3b1) * q^71 + (b11 - b10 - 3b9 - b8 - 2b7 - 3b6 - b4 + 3b2 - 3b1 - 2) q^73 + (-b11 - 2b9 - 4b8 + 6b7 - 4b6 + 2b2 - 4b1 + 6) * q^74 + (2b11 + 2b8 + 5b6 + b5 + b4 + b3 - b2 + 5b1 - 2) * q^76 + (4b11 - b6 + 4b5 - 2b3 + b2 - 6) q^77 + (b10 + b8 - 3b7 - 4b6 + b4 - 4b1 - 3) q^79 + (b9 + b7 - 2b4 + 3b1) * q^80 + (-b9 + 2b1) q^82 + (b11 - 5b10 - 2b6 + b5 + b3 + 1) * q^83 + (2b7 + b5 - 2b4 + 2b1) q^85 + (-2b9 - b8 + b7 - b5 + b3 - 2b1) * q^86 + (3b11 + 3b5 - 3) * q^88 + (b9 + b8 - 5b7 + b5 + 2b4 - b3 + 4b1) q^89 + (-3b9 + b8 - 4b7 - 2b5 + 2b4 - b3 - 2b1) q^91 + (b11 - b9 - 2b8 + 3b7 + b2 + 3) * q^92 + (b11 - b10 - 8b6 + b5 - 3b3 + 1) * q^94 + (b11 - 2b8 - 7b7 + b6 - 2b5 + b4 + 3b3 + b2 - 2b1) q^95 + (-2b11 + b10 + b9 - 2b8 + 2b7 - b6 + b4 - b2 - b1 + 2) q^97 + (b11 + b10 - 3b9 + 4b8 - 8b7 + b6 + b4 + 3b2 + b1 - 8) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} - 3 q^{4} + 5 q^{5} + 2 q^{7} - 12 q^{8} + q^{10} - 4 q^{11} - 5 q^{13} + 2 q^{14} + 3 q^{16} + 10 q^{17} - 9 q^{19} - 2 q^{20} - 4 q^{22} + 3 q^{23} - 5 q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{29}+ \cdots - 59 q^{98}+O(q^{100}) $ 12 * q + q^2 - 3 * q^4 + 5 * q^5 + 2 * q^7 - 12 * q^8 + q^10 - 4 * q^11 - 5 * q^13 + 2 * q^14 + 3 * q^16 + 10 * q^17 - 9 * q^19 - 2 * q^20 - 4 * q^22 + 3 * q^23 - 5 * q^25 - 2 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 + 8 * q^32 + 4 * q^34 + 3 * q^35 + 26 * q^37 - 10 * q^38 - 6 * q^40 - 13 * q^41 - 3 * q^43 - 20 * q^44 - 10 * q^46 + 18 * q^47 + 26 * q^49 + 6 * q^50 - 22 * q^52 - q^53 - 6 * q^55 + 48 * q^56 + 28 * q^58 + 11 * q^59 - 13 * q^61 + 22 * q^62 - 72 * q^65 + 9 * q^67 - 10 * q^68 - 30 * q^70 - 6 * q^71 - 15 * q^73 + 43 * q^74 - 33 * q^76 - 58 * q^77 - 13 * q^79 + q^80 + 4 * q^82 - 16 * q^83 - 3 * q^85 - 7 * q^86 - 30 * q^88 + 29 * q^89 + 22 * q^91 + 21 * q^92 + 36 * q^94 + 22 * q^95 + 20 * q^97 - 59 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 8x^{10} - x^{9} + 41x^{8} - 7x^{7} + 91x^{6} + 9x^{5} + 135x^{4} - 12x^{3} + 45x^{2} + 9x + 9 $ x^12 - x^11 + 8*x^10 - x^9 + 41*x^8 - 7*x^7 + 91*x^6 + 9*x^5 + 135*x^4 - 12*x^3 + 45*x^2 + 9*x + 9 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 29682 \nu^{11} - 128524 \nu^{10} - 145321 \nu^{9} - 915818 \nu^{8} - 1810514 \nu^{7} + \cdots + 5891559 ) / 4197747 $ (-29682v^11 - 128524v^10 - 145321v^9 - 915818v^8 - 1810514v^7 - 4108847v^6 - 4175102v^5 - 4498810v^4 - 9585571v^3 - 1920270v^2 - 603642*v + 5891559) / 4197747
$\beta_{3}$	$=$	$ ( - 59825 \nu^{11} - 7498 \nu^{10} - 421642 \nu^{9} - 379953 \nu^{8} - 2547549 \nu^{7} + \cdots + 6303789 ) / 4197747 $ (-59825v^11 - 7498v^10 - 421642v^9 - 379953v^8 - 2547549v^7 - 1775676v^6 - 5361799v^5 - 3221213v^4 - 10953422v^3 - 1632471v^2 - 618153*v + 6303789) / 4197747
$\beta_{4}$	$=$	$ ( - 190049 \nu^{11} + 526361 \nu^{10} - 1996285 \nu^{9} + 2915957 \nu^{8} - 8819761 \nu^{7} + \cdots + 10085472 ) / 12593241 $ (-190049v^11 + 526361v^10 - 1996285v^9 + 2915957v^8 - 8819761v^7 + 14540960v^6 - 25360778v^5 + 29815194v^4 - 34497546v^3 + 41396727v^2 - 56962377*v + 10085472) / 12593241
$\beta_{5}$	$=$	$ ( 284569 \nu^{11} + 249956 \nu^{10} + 840215 \nu^{9} + 4696613 \nu^{8} + 4130102 \nu^{7} + \cdots + 9284058 ) / 12593241 $ (284569v^11 + 249956v^10 + 840215v^9 + 4696613v^8 + 4130102v^7 + 18963080v^6 - 12665762v^5 + 46550697v^4 - 27668184v^3 + 39200583v^2 - 75438468*v + 9284058) / 12593241
$\beta_{6}$	$=$	$ ( 101739 \nu^{11} - 59825 \nu^{10} + 746589 \nu^{9} + 223208 \nu^{8} + 4014554 \nu^{7} + \cdots + 1176054 ) / 4197747 $ (101739v^11 - 59825v^10 + 746589v^9 + 223208v^8 + 4014554v^7 + 754832v^6 + 8237405v^5 + 3791257v^4 + 14304809v^3 + 2130519v^2 + 878556*v + 1176054) / 4197747
$\beta_{7}$	$=$	$ ( 392018 \nu^{11} - 697235 \nu^{10} + 3315619 \nu^{9} - 2631785 \nu^{8} + 15403114 \nu^{7} + \cdots - 11700747 ) / 12593241 $ (392018v^11 - 697235v^10 + 3315619v^9 - 2631785v^8 + 15403114v^7 - 14787788v^6 + 33409142v^5 - 21184053v^4 + 41548659v^3 - 47618643v^2 + 11249253*v - 11700747) / 12593241
$\beta_{8}$	$=$	$ ( 478819 \nu^{11} - 1214995 \nu^{10} + 4391471 \nu^{9} - 5933194 \nu^{8} + 18762566 \nu^{7} + \cdots - 1743174 ) / 12593241 $ (478819v^11 - 1214995v^10 + 4391471v^9 - 5933194v^8 + 18762566v^7 - 31840072v^6 + 42106069v^5 - 53741877v^4 + 40182891v^3 - 89035602v^2 + 7269597*v - 1743174) / 12593241
$\beta_{9}$	$=$	$ ( - 207857 \nu^{11} + 350255 \nu^{10} - 1805715 \nu^{9} + 1596591 \nu^{8} - 8795535 \nu^{7} + \cdots + 6745266 ) / 4197747 $ (-207857v^11 + 350255v^10 - 1805715v^9 + 1596591v^8 - 8795535v^7 + 8745000v^6 - 18838294v^5 + 17529523v^4 - 23790708v^3 + 27494421v^2 - 2844621*v + 6745266) / 4197747
$\beta_{10}$	$=$	$ ( - 448870 \nu^{11} + 306623 \nu^{10} - 3311303 \nu^{9} - 736087 \nu^{8} - 17525221 \nu^{7} + \cdots - 7986312 ) / 4197747 $ (-448870v^11 + 306623v^10 - 3311303v^9 - 736087v^8 - 17525221v^7 - 1998484v^6 - 35825226v^5 - 15735072v^4 - 56372876v^3 - 9020124v^2 - 3774627*v - 7986312) / 4197747
$\beta_{11}$	$=$	$ ( 1963853 \nu^{11} - 1874807 \nu^{10} + 16096396 \nu^{9} - 1527890 \nu^{8} + 83265427 \nu^{7} + \cdots + 19485603 ) / 12593241 $ (1963853v^11 - 1874807v^10 + 16096396v^9 - 1527890v^8 + 83265427v^7 - 8315429v^6 + 191037164v^5 + 30199983v^4 + 278616585v^3 + 5190477v^2 + 94134195*v + 19485603) / 12593241

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{8} - 2\beta_{7} + \beta_{6} + \beta _1 - 2 $ b8 - 2*b7 + b6 + b1 - 2
$\nu^{3}$	$=$	$ \beta_{10} + 5\beta_{6} + \beta_{3} - 1 $ b10 + 5*b6 + b3 - 1
$\nu^{4}$	$=$	$ -\beta_{9} - 6\beta_{8} + 8\beta_{7} - \beta_{4} + 6\beta_{3} - 8\beta_1 $ -b9 - 6b8 + 8b7 - b4 + 6b3 - 8b1
$\nu^{5}$	$=$	$ -\beta_{11} - 7\beta_{10} - \beta_{9} - 9\beta_{8} + 9\beta_{7} - 28\beta_{6} - 7\beta_{4} + \beta_{2} - 28\beta _1 + 9 $ -b11 - 7b10 - b9 - 9b8 + 9b7 - 28b6 - 7b4 + b2 - 28b1 + 9
$\nu^{6}$	$=$	$ -\beta_{11} - 10\beta_{10} - 55\beta_{6} - \beta_{5} - 35\beta_{3} + 8\beta_{2} + 40 $ -b11 - 10b10 - 55b6 - b5 - 35b3 + 8b2 + 40
$\nu^{7}$	$=$	$ 11\beta_{9} + 65\beta_{8} - 65\beta_{7} - 8\beta_{5} + 43\beta_{4} - 65\beta_{3} + 165\beta_1 $ 11b9 + 65b8 - 65b7 - 8b5 + 43b4 - 65b3 + 165*b1
$\nu^{8}$	$=$	$ 11 \beta_{11} + 76 \beta_{10} + 51 \beta_{9} + 208 \beta_{8} - 222 \beta_{7} + 360 \beta_{6} + \cdots - 222 $ 11b11 + 76b10 + 51b9 + 208b8 - 222b7 + 360b6 + 76b4 - 51b2 + 360*b1 - 222
$\nu^{9}$	$=$	$ 51\beta_{11} + 259\beta_{10} + 998\beta_{6} + 51\beta_{5} + 436\beta_{3} - 87\beta_{2} - 436 $ 51b11 + 259b10 + 998b6 + 51b5 + 436b3 - 87b2 - 436
$\nu^{10}$	$=$	$ -310\beta_{9} - 1257\beta_{8} + 1301\beta_{7} + 87\beta_{5} - 523\beta_{4} + 1257\beta_{3} - 2306\beta_1 $ -310b9 - 1257b8 + 1301b7 + 87b5 - 523b4 + 1257b3 - 2306*b1
$\nu^{11}$	$=$	$ - 310 \beta_{11} - 1567 \beta_{10} - 610 \beta_{9} - 2829 \beta_{8} + 2832 \beta_{7} - 6121 \beta_{6} + \cdots + 2832 $ -310b11 - 1567b10 - 610b9 - 2829b8 + 2832b7 - 6121b6 - 1567b4 + 610b2 - 6121*b1 + 2832

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/513\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$\beta_{7}$

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} $

163.1

−0.963952	−	1.66961i
−0.690206	−	1.19547i
−0.216506	−	0.375000i
0.328657	+	0.569250i
0.792283	+	1.37227i
1.24972	+	2.16459i
−0.963952	+	1.66961i
−0.690206	+	1.19547i
−0.216506	+	0.375000i
0.328657	−	0.569250i
0.792283	−	1.37227i
1.24972	−	2.16459i

−0.963952

1.66961i

−0.858406

−

1.48680i

0.100478

−

0.174033i

3.71133

−0.545959

0.193712

0.335518i

163.2

−0.690206

1.19547i

0.0472324

0.0818090i

1.59642

−

2.76507i

−5.01270

−2.89122

2.20371

3.81694i

163.3

−0.216506

0.375000i

0.906250

1.56967i

−1.01591

1.75962i

−1.15054

−1.65086

−0.439904

−

0.761936i

163.4

0.328657

−

0.569250i

0.783970

1.35788i

2.06470

−

3.57617i

3.83431

2.34525

−1.35716

−

2.35066i

163.5

0.792283

−

1.37227i

−0.255424

−

0.442408i

−0.780905

1.35257i

−0.188849

2.35966

1.23740

2.14323i

163.6

1.24972

−

2.16459i

−2.12362

−

3.67822i

0.535221

−

0.927030i

−0.193560

−5.61687

−1.33776

−

2.31706i

406.1