LMFDB - Newform orbit 792.2.q.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [792,2,Mod(265,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("792.265"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 792 = 2^{3} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	792.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,1,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.32415184009$
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} + 7x^{10} - 2x^{9} + 39x^{8} - 9x^{7} + 67x^{6} - 18x^{5} + 88x^{4} - 16x^{3} + 24x^{2} + 4x + 4 $ x^12 + 7x^10 - 2x^9 + 39x^8 - 9x^7 + 67x^6 - 18x^5 + 88x^4 - 16x^3 + 24x^2 + 4x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 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_{10} q^{3} + (\beta_{8} - \beta_{6} - \beta_{4}) q^{5} + (\beta_{6} - \beta_{2} - 1) q^{7} + ( - \beta_{11} + \beta_{6} - \beta_{4}) q^{9} + (\beta_{6} - 1) q^{11} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{4}) q^{13}+ \cdots + (\beta_{11} + \beta_{8} + \cdots - \beta_{5}) q^{99}+O(q^{100}) $ q - b10 * q^3 + (b8 - b6 - b4) * q^5 + (b6 - b2 - 1) * q^7 + (-b11 + b6 - b4) * q^9 + (b6 - 1) * q^11 + (-b11 + b10 + b8 + b7 - b4) * q^13 + (-b11 + b10 + b9 + 2b7 + b6 - b3 + 2b1 - 1) * q^15 + (b10 - b3 + b1 + 1) * q^17 + (-b9 + b4 - b2 + 1) * q^19 + (-b11 + b10 - b9 + b8 + b7 + b5 + b3 - b2 + 1) * q^21 + (-b11 + b10 + b8 - b6 - b4 - b3) * q^23 + (b10 + b7 - b3) * q^25 + (-b11 + b9 + b8 + 2b7 + b5 - b4 + 2b2 + b1 - 1) * q^27 + (b10 + b8 + b7 + 2b6 + 2b5 + b3 - 2) * q^29 + (-b11 + b10 + b9 + b8 + b7 - b4) * q^31 + b3 * q^33 + (b10 + b9 - b7 + b4 + b2 + 2) * q^35 + (-2b11 - b10 + b7 + 2b6 + 2b5 + 2b3 - 2b1) q^37 + (b10 + b9 + b8 + 2b7 - 2b6 - b3 + b2 + 3b1 + 2) q^39 + (b11 - 2b9 - 2b7 - b6 - b3 - b1) * q^41 + (b10 + b7 + b6 - b3 + b2 - 1) * q^43 + (b10 + b8 + 3b6 + 2b5 - b4 + b3) * q^45 + (-3b10 - 3b7 + 2b6 - b5 - b2 - 2b1 - 2) * q^47 + (b11 + b10 + 2b8 + b7 - b6 - 2b4 + 2b3 - 2b1) * q^49 + (-b10 - b8 + 3b6 + b5 + b4 - 3) q^51 + (-2b11 + 3b10 + b9 + b7 + 2b6 + 2b5 - b4 - 2b3 + b2 + 2b1 + 1) * q^53 + (b4 + 1) * q^55 + (-2b9 - b7 - b6 - b5 + b4 - b3 - b2 + b1 + 2) q^57 + (-2b11 + 3b10 - b9 - b8 - b7 + b6 + b4 - 3b3 - b1) q^59 + (b6 + b5 - 2b3 - 3b1 - 1) * q^61 + (b11 + b10 - 2b9 + b8 + 2b7 - 2b6 - 2b5 + b4 - 2b3 - b2 + 4b1 + 2) * q^63 + (-b10 - b7 + 3b6 + 2b5 + 2b3 - b1 - 3) q^65 + (b11 + b10 - 3b9 - b7 - 2b1) * q^67 + (-b11 + 2b10 + b9 + 2b7 - b6 + b5 - 2b3 + b2 + 3b1 - 1) * q^69 + (-b11 - b10 + b9 - 2b7 + b6 + b5 + 4b3 + b2 - 4b1 + 2) q^71 + (-4b10 + 2b9 + 2b7 - b4 + 2b3 + 2b2 - 2b1 - 2) * q^73 + (-b8 + b5 + b4 + 3) * q^75 + (-b9 - b6) * q^77 + (-b8 - b6 - 3b5 - 2b3 + 2b2 + b1 + 1) q^79 + (b11 + 2b9 - b8 + b7 + 3b6 - b5 - 2b4 + b2 + 2b1 - 2) * q^81 + (-3b10 - b8 - 3b7 + 2b6 - 3b5 - 2) * q^83 + (b11 + b10 - b9 + b8 - b4 + b3 - 2b1) q^85 + (2b11 - 2b9 - b8 + 2b7 + b6 - 2b5 + b4 + 2b3 - 3b2 + 2b1 - 1) q^87 + (-b10 - b9 - 3b7 - 2b4 + 4b3 - b2 - 4b1 + 4) * q^89 + (3b11 + b10 + b9 - 3b6 - 3b5 + 2b4 - 4b3 + b2 + 4b1 + 3) * q^91 + (b10 + b9 + 2b8 + 2b7 - 3b6 + b5 - b4 + 2b1 + 2) * q^93 + (b9 - b7 - 4b6 - b3) q^95 + (2b10 - 2b8 + 2b7 - b6 - 4b3 - 2b1 + 1) q^97 + (b11 + b8 - b6 - b5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{3} - 4 q^{5} - 5 q^{7} + 5 q^{9} - 6 q^{11} - 3 q^{13} - 7 q^{15} + 14 q^{17} + 10 q^{19} + 9 q^{21} - 8 q^{23} + 2 q^{25} - 11 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{33} + 16 q^{35} + 6 q^{37}+ \cdots - 7 q^{99}+O(q^{100}) $ 12 * q + q^3 - 4 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 3 * q^13 - 7 * q^15 + 14 * q^17 + 10 * q^19 + 9 * q^21 - 8 * q^23 + 2 * q^25 - 11 * q^27 - 8 * q^29 - 4 * q^31 - 2 * q^33 + 16 * q^35 + 6 * q^37 + 14 * q^39 - 5 * q^43 + 25 * q^45 - 17 * q^47 - 3 * q^49 - 15 * q^51 + 28 * q^53 + 8 * q^55 + 15 * q^57 - 4 * q^59 - q^61 + 15 * q^63 - 15 * q^65 + 4 * q^67 - 14 * q^69 + 14 * q^71 - 24 * q^73 + 38 * q^75 - 5 * q^77 - q^79 + 5 * q^81 - 22 * q^83 + 3 * q^85 - q^87 + 44 * q^89 + 22 * q^91 + 12 * q^93 - 24 * q^95 + 16 * q^97 - 7 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + 7x^{10} - 2x^{9} + 39x^{8} - 9x^{7} + 67x^{6} - 18x^{5} + 88x^{4} - 16x^{3} + 24x^{2} + 4x + 4 $ x^12 + 7*x^10 - 2*x^9 + 39*x^8 - 9*x^7 + 67*x^6 - 18*x^5 + 88*x^4 - 16*x^3 + 24*x^2 + 4*x + 4 :

$\beta_{1}$	$=$	$ ( - 227 \nu^{11} + 27283 \nu^{10} - 6521 \nu^{9} + 181033 \nu^{8} - 106013 \nu^{7} + 993670 \nu^{6} + \cdots - 6600 ) / 240564 $ (-227v^11 + 27283v^10 - 6521v^9 + 181033v^8 - 106013v^7 + 993670v^6 - 468778v^5 + 1443341v^4 - 923458v^3 + 1603488v^2 - 742314*v - 6600) / 240564
$\beta_{2}$	$=$	$ ( 6647 \nu^{11} + 42545 \nu^{10} - 15881 \nu^{9} + 277191 \nu^{8} - 264513 \nu^{7} + 1665582 \nu^{6} + \cdots + 740800 ) / 721692 $ (6647v^11 + 42545v^10 - 15881v^9 + 277191v^8 - 264513v^7 + 1665582v^6 - 2265832v^5 + 2968745v^4 - 3776456v^3 + 3877124v^2 - 2646058*v + 740800) / 721692
$\beta_{3}$	$=$	$ ( 3664 \nu^{11} - 19652 \nu^{10} + 1841 \nu^{9} - 132954 \nu^{8} + 26763 \nu^{7} - 657714 \nu^{6} + \cdots + 380300 ) / 360846 $ (3664v^11 - 19652v^10 + 1841v^9 - 132954v^8 + 26763v^7 - 657714v^6 - 429749v^5 - 680639v^4 - 503041v^3 - 466670v^2 - 931250*v + 380300) / 360846
$\beta_{4}$	$=$	$ ( 4540 \nu^{11} - 4391 \nu^{10} + 30185 \nu^{9} - 32247 \nu^{8} + 175701 \nu^{7} - 167199 \nu^{6} + \cdots + 653222 ) / 360846 $ (4540v^11 - 4391v^10 + 30185v^9 - 32247v^8 + 175701v^7 - 167199v^6 + 274222v^5 - 79328v^4 + 366719v^3 + 5440v^2 + 11500*v + 653222) / 360846
$\beta_{5}$	$=$	$ ( - 5314 \nu^{11} + 19879 \nu^{10} - 40674 \nu^{9} + 142775 \nu^{8} - 272146 \nu^{7} + 778577 \nu^{6} + \cdots + 114850 ) / 120282 $ (-5314v^11 + 19879v^10 - 40674v^9 + 142775v^8 - 272146v^7 + 778577v^6 - 674471v^5 + 1179117v^4 - 1085500v^3 + 1416528v^2 - 952402*v + 114850) / 120282
$\beta_{6}$	$=$	$ ( 39985 \nu^{11} - 34235 \nu^{10} + 270815 \nu^{9} - 310833 \nu^{8} + 1567515 \nu^{7} - 1630536 \nu^{6} + \cdots + 49112 ) / 721692 $ (39985v^11 - 34235v^10 + 270815v^9 - 310833v^8 + 1567515v^7 - 1630536v^6 + 2635708v^5 - 2679077v^4 + 3586466v^3 - 3493784v^2 + 773962*v + 49112) / 721692
$\beta_{7}$	$=$	$ ( - 14427 \nu^{11} - 27867 \nu^{10} - 106231 \nu^{9} - 170621 \nu^{8} - 532631 \nu^{7} + \cdots - 439952 ) / 240564 $ (-14427v^11 - 27867v^10 - 106231v^9 - 170621v^8 - 532631v^7 - 961076v^6 - 840050v^5 - 1648883v^4 - 766968v^3 - 1948864v^2 + 75878*v - 439952) / 240564
$\beta_{8}$	$=$	$ ( - 11815 \nu^{11} + 9948 \nu^{10} - 80210 \nu^{9} + 92862 \nu^{8} - 463938 \nu^{7} + 487779 \nu^{6} + \cdots + 201370 ) / 120282 $ (-11815v^11 + 9948v^10 - 80210v^9 + 92862v^8 - 463938v^7 + 487779v^6 - 787162v^5 + 866583v^4 - 1073249v^3 + 1046126v^2 - 254154*v + 201370) / 120282
$\beta_{9}$	$=$	$ ( 41411 \nu^{11} - 5685 \nu^{10} + 305421 \nu^{9} - 119847 \nu^{8} + 1711377 \nu^{7} - 605628 \nu^{6} + \cdots + 156984 ) / 240564 $ (41411v^11 - 5685v^10 + 305421v^9 - 119847v^8 + 1711377v^7 - 605628v^6 + 3260036v^5 - 1114611v^4 + 3869790v^3 - 1066300v^2 + 1021494*v + 156984) / 240564
$\beta_{10}$	$=$	$ ( 52023 \nu^{11} - 13317 \nu^{10} + 347805 \nu^{9} - 203471 \nu^{8} + 1937689 \nu^{7} - 982652 \nu^{6} + \cdots + 70592 ) / 240564 $ (52023v^11 - 13317v^10 + 347805v^9 - 203471v^8 + 1937689v^7 - 982652v^6 + 2970940v^5 - 1819103v^4 + 3746686v^3 - 1599004v^2 + 289238*v + 70592) / 240564
$\beta_{11}$	$=$	$ ( - 86060 \nu^{11} - 17441 \nu^{10} - 607687 \nu^{9} + 33177 \nu^{8} - 3372969 \nu^{7} + \cdots - 971548 ) / 360846 $ (-86060v^11 - 17441v^10 - 607687v^9 + 33177v^8 - 3372969v^7 - 7725v^6 - 5884328v^5 - 139940v^4 - 7960921v^3 - 791960v^2 - 2578064*v - 971548) / 360846

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

$\nu$	$=$	$ ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 $ (-b3 + b2 - b1) / 2
$\nu^{2}$	$=$	$ -\beta_{8} - 2\beta_{6} + \beta_{4} $ -b8 - 2*b6 + b4
$\nu^{3}$	$=$	$ \beta_{11} + 2\beta_{10} - 2\beta_{9} - 3\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{2} + 1 $ b11 + 2b10 - 2b9 - 3b7 - b6 - b5 - 2b2 + 1
$\nu^{4}$	$=$	$ \beta_{10} + 6\beta_{8} + \beta_{7} + 8\beta_{6} - \beta_{3} - 8 $ b10 + 6b8 + b7 + 8b6 - b3 - 8
$\nu^{5}$	$=$	$ -6\beta_{11} - 10\beta_{10} + 9\beta_{9} - \beta_{8} + 16\beta_{7} - \beta_{6} + \beta_{4} + 10\beta_{3} + 16\beta_1 $ -6b11 - 10b10 + 9b9 - b8 + 16b7 - b6 + b4 + 10b3 + 16b1
$\nu^{6}$	$=$	$ \beta_{11} + \beta_{10} - \beta_{9} - 9 \beta_{7} - \beta_{6} - \beta_{5} - 32 \beta_{4} + 7 \beta_{3} + \cdots + 39 $ b11 + b10 - b9 - 9b7 - b6 - b5 - 32b4 + 7b3 - b2 - 7b1 + 39
$\nu^{7}$	$=$	$ \beta_{10} + 11\beta_{8} + \beta_{7} + 43\beta_{6} + 32\beta_{5} - 52\beta_{3} + 44\beta_{2} - 83\beta _1 - 43 $ b10 + 11b8 + b7 + 43b6 + 32b5 - 52b3 + 44b2 - 83b1 - 43
$\nu^{8}$	$=$	$ - 11 \beta_{11} - 52 \beta_{10} + 12 \beta_{9} - 167 \beta_{8} + 24 \beta_{7} - 191 \beta_{6} + \cdots + 63 \beta_1 $ -11b11 - 52b10 + 12b9 - 167b8 + 24b7 - 191b6 + 167b4 + 13b3 + 63*b1
$\nu^{9}$	$=$	$ 167 \beta_{11} + 262 \beta_{10} - 223 \beta_{9} - 441 \beta_{7} - 167 \beta_{6} - 167 \beta_{5} + \cdots + 256 $ 167b11 + 262b10 - 223b9 - 441b7 - 167b6 - 167b5 - 87b4 + 12b3 - 223b2 - 12b1 + 256
$\nu^{10}$	$=$	$ 206 \beta_{10} + 870 \beta_{8} + 206 \beta_{7} + 1069 \beta_{6} + 87 \beta_{5} - 318 \beta_{3} + \cdots - 1069 $ 206b10 + 870b8 + 206b7 + 1069b6 + 87b5 - 318b3 + 100b2 - 199b1 - 1069
$\nu^{11}$	$=$	$ - 870 \beta_{11} - 1454 \beta_{10} + 1149 \beta_{9} - 604 \beta_{8} + 2225 \beta_{7} + \cdots + 2324 \beta_1 $ -870b11 - 1454b10 + 1149b9 - 604b8 + 2225b7 - 630b6 + 604b4 + 1355b3 + 2324*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/792\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$	$397$
$\chi(n)$	$1$	$1$	$-\beta_{6}$	$1$

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} $

265.1

−1.17557	−	2.03615i
−0.189597	−	0.328392i
0.613168	+	1.06204i
0.322589	+	0.558741i
−0.654157	−	1.13303i
1.08357	+	1.87680i
−1.17557	+	2.03615i
−0.189597	+	0.328392i
0.613168	−	1.06204i
0.322589	−	0.558741i
−0.654157	+	1.13303i
1.08357	−	1.87680i

−1.72146

−

0.191265i

1.26394

−

2.18921i

0.129686

0.224623i

2.92684

0.658510i

265.2

−1.40604

1.01146i

−1.42811

2.47355i

−1.52684

−

2.64457i

0.953888

−

2.84431i

265.3

−0.215092

−

1.71864i

−0.748050

1.29566i

−1.94143

−

3.36265i

−2.90747

0.739331i

265.4

0.893881

1.48357i

−1.29187

2.23759i

−0.251298

−

0.435260i

−1.40195

2.65227i

265.5

1.23257

−

1.21686i

−0.644157

1.11571i

2.04089

3.53492i

0.0384795

−

2.99975i

265.6

1.71613

−

0.234283i

0.848245

−

1.46920i

−0.951006

−

1.64719i

2.89022

−

0.804122i

529.1