LMFDB - Newform orbit 576.2.r.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,2,Mod(97,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("576.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	576.r (of order $6$, 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:	$4.59938315643$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
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} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{5} - \beta_{4}) q^{3} + ( - \beta_{6} - 2) q^{5} + ( - \beta_{11} + \beta_{4} + \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10}) $ q + (b5 - b4) * q^3 + (-b6 - 2) * q^5 + (-b11 + b4 + b1) * q^7 + (b10 - b6 - b3 - 1) * q^9	$ q + (\beta_{5} - \beta_{4}) q^{3} + ( - \beta_{6} - 2) q^{5} + ( - \beta_{11} + \beta_{4} + \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9} + (\beta_{4} - \beta_1) q^{11} + (\beta_{10} - \beta_{7}) q^{13} + ( - \beta_{5} + 2 \beta_{4}) q^{15} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{17}+ \cdots + (\beta_{11} + 3 \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) $ q + (b5 - b4) * q^3 + (-b6 - 2) * q^5 + (-b11 + b4 + b1) * q^7 + (b10 - b6 - b3 - 1) * q^9 + (b4 - b1) * q^11 + (b10 - b7) * q^13 + (-b5 + 2b4) q^15 + (-b10 - b9 + b7 + b3) * q^17 - b8 * q^19 + (-b10 - 4b6 + 2b3 - 2) * q^21 + (-b11 + b8 - b5 + 2b4 - b2 - b1) q^23 + (-2b6 - 2) q^25 + (-b11 - b8 + b5 + b4 - 2b2 + b1) q^27 + (-2b10 - 2b9 + b7 - b6 + b3 + 1) * q^29 + (-b11 + b8 + 3b5 - 2b4 - b2 + b1) * q^31 + (b10 + 4b6 + b3 + 5) q^33 + (2b11 - b5 - b4 - b1) q^35 + (b10 - b9 + 2b6 - b3 + 1) q^37 + (2b11 - b5 + 2b4 - 3b2 - b1) q^39 + (b10 + b9 - 3b6) q^41 + (-b11 + 2b5 - b4 - 2b2 - b1) * q^43 + (-b10 + 2b6 + 2b3 + 1) * q^45 + (-b11 - 2b8 - 4b5 - b4 - b2 - b1) * q^47 + (b10 - 2b9 + b7 + 5b6 - 2b3) q^49 + (b11 + 3b2 + b1) q^51 + (-b10 - b9 - b7 + 4b6 - b3 + 2) q^53 + (-b11 - b5 - b4 + 2b1) q^55 + (-b7 + 1) * q^57 + (b11 - 3b8 + b5 + 2b4 - 3b2) q^59 + (2b10 + 2b9 - b7 - b6 - b3 + 1) * q^61 + (2b11 + 3b8 + 3b4 + 3b2 - b1) * q^63 + (-2b10 + b9 + b7 + b3) q^65 + (-b11 + 2b8 - b5 - 4b4 + 2b2 - b1) q^67 + (b9 + b7 + 2b6 + 3b3 + 7) * q^69 + (-b11 + 2b8 + 2b5 - 4b4 + 4b2 - b1) * q^71 + (-b10 - b9 + 2b7 - b3 - 1) q^73 + 2b4 q^75 + (-2b10 + b9 + b7 - 6b6 - 2b3 - 12) q^77 + (-2b8 + 2b5 + b4 - b2 + b1) * q^79 + (2b9 - b7 - 6b6 + 2b3 - 4) q^81 + (b11 - 2b5 + b4 + 3b2 + b1) * q^83 + (b10 + b9 - 2b7 - 2b3) * q^85 + (3b11 - 3b8 + 2b5 - b4 + 3b2) * q^87 + (b10 + b9 - 2b7 + b3 - 3) q^89 + (-3b11 - 3b5 + 3b4) q^91 + (2b10 + b9 + b7 - 8b6 - b3 - 7) * q^93 + (b8 - b2) * q^95 + (-3b7 - b6 - 3b3 - 1) * q^97 + (b11 + 3b8 - 3b4 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 18 q^{5} - 4 q^{9}+O(q^{10}) $ 12 * q - 18 * q^5 - 4 * q^9	$ 12 q - 18 q^{5} - 4 q^{9} - 6 q^{13} - 6 q^{21} - 12 q^{25} + 18 q^{29} + 30 q^{33} + 18 q^{41} - 6 q^{45} - 24 q^{49} + 8 q^{57} + 18 q^{61} + 6 q^{65} + 66 q^{69} - 90 q^{77} - 20 q^{81} - 48 q^{89} - 30 q^{93} - 6 q^{97}+O(q^{100}) $ 12 * q - 18 * q^5 - 4 * q^9 - 6 * q^13 - 6 * q^21 - 12 * q^25 + 18 * q^29 + 30 * q^33 + 18 * q^41 - 6 * q^45 - 24 * q^49 + 8 * q^57 + 18 * q^61 + 6 * q^65 + 66 * q^69 - 90 * q^77 - 20 * q^81 - 48 * q^89 - 30 * q^93 - 6 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 :

$\beta_{1}$	$=$	$ ( - 778 \nu^{11} + 5496 \nu^{10} - 7293 \nu^{9} + 5400 \nu^{8} + 10120 \nu^{7} - 68622 \nu^{6} + \cdots + 73584 ) / 51972 $ (-778v^11 + 5496v^10 - 7293v^9 + 5400v^8 + 10120v^7 - 68622v^6 - 13980v^5 + 12828v^4 + 200318v^3 + 699840v^2 + 180024*v + 73584) / 51972
$\beta_{2}$	$=$	$ ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + \cdots + 15984 ) / 51972 $ (962v^11 - 2392v^10 + 2887v^9 - 2220v^8 - 13990v^7 + 14442v^6 + 11380v^5 + 57708v^4 + 98118v^3 - 132080v^2 + 49284*v + 15984) / 51972
$\beta_{3}$	$=$	$ ( 1360 \nu^{11} - 864 \nu^{10} - 400 \nu^{9} + 2902 \nu^{8} - 25539 \nu^{7} - 16320 \nu^{6} + \cdots + 219540 ) / 51972 $ (1360v^11 - 864v^10 - 400v^9 + 2902v^8 - 25539v^7 - 16320v^6 + 27692v^5 + 92050v^4 + 330320v^3 + 266532v^2 + 143634*v + 219540) / 51972
$\beta_{4}$	$=$	$ ( 5984 \nu^{11} - 7522 \nu^{10} + 8393 \nu^{9} - 6600 \nu^{8} - 90816 \nu^{7} - 28498 \nu^{6} + \cdots + 272448 ) / 103944 $ (5984v^11 - 7522v^10 + 8393v^9 - 6600v^8 - 90816v^7 - 28498v^6 + 55616v^5 + 473748v^4 + 1167562v^3 + 792408v^2 + 746328*v + 272448) / 103944
$\beta_{5}$	$=$	$ ( - 6823 \nu^{11} - 186 \nu^{10} + 2391 \nu^{9} - 2376 \nu^{8} + 113464 \nu^{7} + 159834 \nu^{6} + \cdots - 548280 ) / 103944 $ (-6823v^11 - 186v^10 + 2391v^9 - 2376v^8 + 113464v^7 + 159834v^6 - 23754v^5 - 683700v^4 - 2089522v^3 - 2446620v^2 - 1454952*v - 548280) / 103944
$\beta_{6}$	$=$	$ ( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + \cdots + 3468 ) / 1704 $ (118v^11 - 90v^10 + 53v^9 - 32v^8 - 1866v^7 - 1416v^6 + 1364v^5 + 10408v^4 + 27758v^3 + 22776v^2 + 12468*v + 3468) / 1704
$\beta_{7}$	$=$	$ ( 4689 \nu^{11} - 3876 \nu^{10} + 3854 \nu^{9} - 5534 \nu^{8} - 68264 \nu^{7} - 56268 \nu^{6} + \cdots + 48876 ) / 51972 $ (4689v^11 - 3876v^10 + 3854v^9 - 5534v^8 - 68264v^7 - 56268v^6 + 37850v^5 + 438544v^4 + 1102444v^3 + 911136v^2 + 502080*v + 48876) / 51972
$\beta_{8}$	$=$	$ ( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} + \cdots - 105120 ) / 12993 $ (-1688v^11 + 1054v^10 - 840v^9 + 684v^8 + 26378v^7 + 24587v^6 - 12648v^5 - 151968v^4 - 420176v^3 - 400856v^2 - 282372*v - 105120) / 12993
$\beta_{9}$	$=$	$ ( - 4502 \nu^{11} + 3246 \nu^{10} - 927 \nu^{9} - 1080 \nu^{8} + 73378 \nu^{7} + 54024 \nu^{6} + \cdots - 228796 ) / 34648 $ (-4502v^11 + 3246v^10 - 927v^9 - 1080v^8 + 73378v^7 + 54024v^6 - 56276v^5 - 403744v^4 - 1035370v^3 - 847128v^2 - 462516*v - 228796) / 34648
$\beta_{10}$	$=$	$ ( 25054 \nu^{11} - 18618 \nu^{10} + 8389 \nu^{9} + 272 \nu^{8} - 404246 \nu^{7} - 300648 \nu^{6} + \cdots + 1030836 ) / 103944 $ (25054v^11 - 18618v^10 + 8389v^9 + 272v^8 - 404246v^7 - 300648v^6 + 310036v^5 + 2217896v^4 + 5869134v^3 + 4806792v^2 + 2626812*v + 1030836) / 103944
$\beta_{11}$	$=$	$ ( - 13366 \nu^{11} + 8843 \nu^{10} - 7504 \nu^{9} + 5514 \nu^{8} + 211707 \nu^{7} + 177716 \nu^{6} + \cdots - 830016 ) / 51972 $ (-13366v^11 + 8843v^10 - 7504v^9 + 5514v^8 + 211707v^7 + 177716v^6 - 88792v^5 - 1193088v^4 - 3315152v^3 - 3104220v^2 - 2229918*v - 830016) / 51972

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

$\nu$	$=$	$ ( -2\beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{7} + 4\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_1 ) / 6 $ (-2b11 + b10 + b9 - 2b7 + 4b5 - 2b4 - 2*b3 + b1) / 6
$\nu^{2}$	$=$	$ ( -2\beta_{11} + 4\beta_{5} - 2\beta_{4} - 9\beta_{2} - 2\beta_1 ) / 6 $ (-2b11 + 4b5 - 2b4 - 9b2 - 2*b1) / 6
$\nu^{3}$	$=$	$ ( - 4 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} - 12 \beta_{6} + 8 \beta_{5} + \cdots - 6 ) / 6 $ (-4b11 + 4b10 + 4b9 - 3b8 + 4b7 - 12b6 + 8b5 - 16b4 + 4b3 - 6b2 - 4*b1 - 6) / 6
$\nu^{4}$	$=$	$ ( 11\beta_{10} + 5\beta_{9} + 2\beta_{7} - 30\beta_{6} - 4\beta_{3} ) / 3 $ (11b10 + 5b9 + 2b7 - 30b6 - 4*b3) / 3
$\nu^{5}$	$=$	$ ( - 22 \beta_{11} + 38 \beta_{10} + 35 \beta_{9} + 42 \beta_{8} - 16 \beta_{7} - 39 \beta_{6} + 32 \beta_{5} + \cdots + 39 ) / 6 $ (-22b11 + 38b10 + 35b9 + 42b8 - 16b7 - 39b6 + 32b5 + 38b4 - 22b3 + 21b2 + 38*b1 + 39) / 6
$\nu^{6}$	$=$	$ ( -56\beta_{11} + 81\beta_{8} + 64\beta_{5} + 16\beta_{4} + 40\beta_1 ) / 3 $ (-56b11 + 81b8 + 64b5 + 16b4 + 40*b1) / 3
$\nu^{7}$	$=$	$ ( - 94 \beta_{11} - 35 \beta_{10} - 47 \beta_{9} + 60 \beta_{8} + 82 \beta_{7} - 108 \beta_{6} + \cdots - 216 ) / 3 $ (-94b11 - 35b10 - 47b9 + 60b8 + 82b7 - 108b6 + 164b5 - 70b4 + 94b3 - 60b2 + 35*b1 - 216) / 3
$\nu^{8}$	$=$	$ ( 70\beta_{10} - 35\beta_{9} + 214\beta_{7} - 669\beta_{6} + 214\beta_{3} - 669 ) / 3 $ (70b10 - 35b9 + 214b7 - 669b6 + 214*b3 - 669) / 3
$\nu^{9}$	$=$	$ ( 164 \beta_{11} + 308 \beta_{10} + 164 \beta_{9} + 321 \beta_{8} + 236 \beta_{7} - 1140 \beta_{6} + \cdots - 570 ) / 3 $ (164b11 + 308b10 + 164b9 + 321b8 + 236b7 - 1140b6 - 472b5 + 800b4 + 164b3 + 642b2 + 308*b1 - 570) / 3
$\nu^{10}$	$=$	$ ( -328\beta_{11} + 1908\beta_{8} - 328\beta_{5} + 2228\beta_{4} + 1908\beta_{2} + 1442\beta_1 ) / 3 $ (-328b11 + 1908b8 - 328b5 + 2228b4 + 1908b2 + 1442b1) / 3
$\nu^{11}$	$=$	$ ( - 1586 \beta_{11} - 2386 \beta_{10} - 1993 \beta_{9} + 3342 \beta_{8} + 800 \beta_{7} + 2949 \beta_{6} + \cdots - 2949 ) / 3 $ (-1586b11 - 2386b10 - 1993b9 + 3342b8 + 800b7 + 2949b6 + 1600b5 + 2386b4 + 1586b3 + 1671b2 + 2386*b1 - 2949) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$\beta_{6}$	$1$	$-1$

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} $

97.1

0.583700	−	2.17840i
−1.50511	−	0.403293i
−0.673288	−	0.180407i
−0.180407	+	0.673288i
−0.403293	+	1.50511i
2.17840	+	0.583700i
0.583700	+	2.17840i
−1.50511	+	0.403293i
−0.673288	+	0.180407i
−0.180407	−	0.673288i
−0.403293	−	1.50511i
2.17840	−	0.583700i

−1.59470

−

0.675970i

−1.50000

0.866025i

1.80664

−

3.12920i

2.08613

2.15594i

97.2

−1.10182

−

1.33641i

−1.50000

0.866025i

0.495361

−

0.857990i

−0.571993

2.94497i

97.3

−0.492881

1.66044i

−1.50000

0.866025i

2.17731

−

3.77121i

−2.51414

−

1.63680i

97.4

0.492881

−

1.66044i

−1.50000

0.866025i

−2.17731

3.77121i

−2.51414

−

1.63680i

97.5

1.10182

1.33641i

−1.50000

0.866025i

−0.495361

0.857990i

−0.571993

2.94497i

97.6

1.59470

0.675970i

−1.50000

0.866025i

−1.80664

3.12920i

2.08613

2.15594i

481.1

−1.59470

0.675970i

−1.50000

−

0.866025i

1.80664

3.12920i

2.08613

−

2.15594i

481.2

−1.10182

1.33641i

−1.50000

−

0.866025i

0.495361

0.857990i

−0.571993

−

2.94497i

481.3

−0.492881

−

1.66044i

−1.50000

−

0.866025i

2.17731

3.77121i

−2.51414

1.63680i

481.4

0.492881

1.66044i

−1.50000

−

0.866025i

−2.17731

−

3.77121i

−2.51414

1.63680i

481.5

1.10182

−

1.33641i

−1.50000

−

0.866025i

−0.495361

−

0.857990i

−0.571993

−

2.94497i

481.6

1.59470

−

0.675970i

−1.50000

−

0.866025i

−1.80664

−

3.12920i

2.08613

−

2.15594i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
72.n	even	6	1	inner
72.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.2.r.e	✓	12
3.b	odd	2	1		1728.2.r.f		12
4.b	odd	2	1	inner	576.2.r.e	✓	12
8.b	even	2	1		576.2.r.f	yes	12
8.d	odd	2	1		576.2.r.f	yes	12
9.c	even	3	1		576.2.r.f	yes	12
9.c	even	3	1		5184.2.d.q		12
9.d	odd	6	1		1728.2.r.e		12
9.d	odd	6	1		5184.2.d.r		12
12.b	even	2	1		1728.2.r.f		12
24.f	even	2	1		1728.2.r.e		12
24.h	odd	2	1		1728.2.r.e		12
36.f	odd	6	1		576.2.r.f	yes	12
36.f	odd	6	1		5184.2.d.q		12
36.h	even	6	1		1728.2.r.e		12
36.h	even	6	1		5184.2.d.r		12
72.j	odd	6	1		1728.2.r.f		12
72.j	odd	6	1		5184.2.d.r		12
72.l	even	6	1		1728.2.r.f		12
72.l	even	6	1		5184.2.d.r		12
72.n	even	6	1	inner	576.2.r.e	✓	12
72.n	even	6	1		5184.2.d.q		12
72.p	odd	6	1	inner	576.2.r.e	✓	12
72.p	odd	6	1		5184.2.d.q		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
576.2.r.e	✓	12	1.a	even	1	1	trivial
576.2.r.e	✓	12	4.b	odd	2	1	inner
576.2.r.e	✓	12	72.n	even	6	1	inner
576.2.r.e	✓	12	72.p	odd	6	1	inner
576.2.r.f	yes	12	8.b	even	2	1
576.2.r.f	yes	12	8.d	odd	2	1
576.2.r.f	yes	12	9.c	even	3	1
576.2.r.f	yes	12	36.f	odd	6	1
1728.2.r.e		12	9.d	odd	6	1
1728.2.r.e		12	24.f	even	2	1
1728.2.r.e		12	24.h	odd	2	1
1728.2.r.e		12	36.h	even	6	1
1728.2.r.f		12	3.b	odd	2	1
1728.2.r.f		12	12.b	even	2	1
1728.2.r.f		12	72.j	odd	6	1
1728.2.r.f		12	72.l	even	6	1
5184.2.d.q		12	9.c	even	3	1
5184.2.d.q		12	36.f	odd	6	1
5184.2.d.q		12	72.n	even	6	1
5184.2.d.q		12	72.p	odd	6	1
5184.2.d.r		12	9.d	odd	6	1
5184.2.d.r		12	36.h	even	6	1
5184.2.d.r		12	72.j	odd	6	1
5184.2.d.r		12	72.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} + 3T_{5} + 3 $ T5^2 + 3*T5 + 3 acting on $S_{2}^{\mathrm{new}}(576, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 2 T^{10} + \cdots + 729 $ T^12 + 2T^10 + 7T^8 + 12T^6 + 63T^4 + 162*T^2 + 729
$5$	$ (T^{2} + 3 T + 3)^{6} $ (T^2 + 3*T + 3)^6
$7$	$ T^{12} + 33 T^{10} + \cdots + 59049 $ T^12 + 33T^10 + 810T^8 + 8721T^6 + 69822T^4 + 67797*T^2 + 59049
$11$	$ T^{12} - 39 T^{10} + \cdots + 6561 $ T^12 - 39T^10 + 1350T^8 - 6507T^6 + 26082T^4 - 13851*T^2 + 6561
$13$	$ (T^{6} + 3 T^{5} + \cdots + 243)^{2} $ (T^6 + 3T^5 - 24T^4 - 81T^3 + 756T^2 - 729*T + 243)^2
$17$	$ (T^{3} - 24 T + 36)^{4} $ (T^3 - 24*T + 36)^4
$19$	$ (T^{2} + 4)^{6} $ (T^2 + 4)^6
$23$	$ T^{12} + \cdots + 2492305929 $ T^12 + 117T^10 + 9342T^8 + 408753T^6 + 13055418T^4 + 217015281*T^2 + 2492305929
$29$	$ (T^{6} - 9 T^{5} + \cdots + 93987)^{2} $ (T^6 - 9T^5 - 36T^4 + 567T^3 + 2376T^2 - 33453*T + 93987)^2
$31$	$ T^{12} + 117 T^{10} + \cdots + 95004009 $ T^12 + 117T^10 + 11070T^8 + 286929T^6 + 5718762T^4 + 25527393*T^2 + 95004009
$37$	$ (T^{6} + 60 T^{4} + \cdots + 3888)^{2} $ (T^6 + 60T^4 + 1008T^2 + 3888)^2
$41$	$ (T^{6} - 9 T^{5} + 78 T^{4} + \cdots + 81)^{2} $ (T^6 - 9T^5 + 78T^4 - 45T^3 + 90T^2 + 27*T + 81)^2
$43$	$ T^{12} - 123 T^{10} + \cdots + 47458321 $ T^12 - 123T^10 + 13194T^8 - 224227T^6 + 2896878T^4 - 13330215*T^2 + 47458321
$47$	$ T^{12} + \cdots + 514609673769 $ T^12 + 297T^10 + 61074T^8 + 6624369T^6 + 523251414T^4 + 19465645005*T^2 + 514609673769
$53$	$ (T^{6} + 180 T^{4} + \cdots + 432)^{2} $ (T^6 + 180T^4 + 1728T^2 + 432)^2
$59$	$ T^{12} - 147 T^{10} + \cdots + 531441 $ T^12 - 147T^10 + 20898T^8 - 103059T^6 + 398358T^4 - 518319*T^2 + 531441
$61$	$ (T^{6} - 9 T^{5} + \cdots + 4563)^{2} $ (T^6 - 9T^5 - 36T^4 + 567T^3 + 4320T^2 + 7371*T + 4563)^2
$67$	$ T^{12} + \cdots + 352275361 $ T^12 - 231T^10 + 45270T^8 - 1831483T^6 + 61128642T^4 - 151859979*T^2 + 352275361
$71$	$ (T^{6} - 288 T^{4} + \cdots - 124848)^{2} $ (T^6 - 288T^4 + 19872T^2 - 124848)^2
$73$	$ (T^{3} - 84 T - 164)^{4} $ (T^3 - 84*T - 164)^4
$79$	$ T^{12} + 93 T^{10} + \cdots + 4782969 $ T^12 + 93T^10 + 7758T^8 + 78489T^6 + 590490T^4 + 1948617*T^2 + 4782969
$83$	$ T^{12} - 171 T^{10} + \cdots + 43046721 $ T^12 - 171T^10 + 26082T^8 - 527067T^6 + 8857350T^4 - 20726199*T^2 + 43046721
$89$	$ (T^{3} + 12 T^{2} + \cdots - 108)^{4} $ (T^3 + 12T^2 - 36T - 108)^4
$97$	$ (T^{6} + 3 T^{5} + \cdots + 1408969)^{2} $ (T^6 + 3T^5 + 222T^4 + 1735T^3 + 48930T^2 + 252831*T + 1408969)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} - \beta_{4}) q^{3} + ( - \beta_{6} - 2) q^{5} + ( - \beta_{11} + \beta_{4} + \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10}) \) q + (b5 - b4) * q^3 + (-b6 - 2) * q^5 + (-b11 + b4 + b1) * q^7 + (b10 - b6 - b3 - 1) * q^9	\( q + (\beta_{5} - \beta_{4}) q^{3} + ( - \beta_{6} - 2) q^{5} + ( - \beta_{11} + \beta_{4} + \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9} + (\beta_{4} - \beta_1) q^{11} + (\beta_{10} - \beta_{7}) q^{13} + ( - \beta_{5} + 2 \beta_{4}) q^{15} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{17}+ \cdots + (\beta_{11} + 3 \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) \) q + (b5 - b4) * q^3 + (-b6 - 2) * q^5 + (-b11 + b4 + b1) * q^7 + (b10 - b6 - b3 - 1) * q^9 + (b4 - b1) * q^11 + (b10 - b7) * q^13 + (-b5 + 2b4) q^15 + (-b10 - b9 + b7 + b3) * q^17 - b8 * q^19 + (-b10 - 4b6 + 2b3 - 2) * q^21 + (-b11 + b8 - b5 + 2b4 - b2 - b1) q^23 + (-2b6 - 2) q^25 + (-b11 - b8 + b5 + b4 - 2b2 + b1) q^27 + (-2b10 - 2b9 + b7 - b6 + b3 + 1) * q^29 + (-b11 + b8 + 3b5 - 2b4 - b2 + b1) * q^31 + (b10 + 4b6 + b3 + 5) q^33 + (2b11 - b5 - b4 - b1) q^35 + (b10 - b9 + 2b6 - b3 + 1) q^37 + (2b11 - b5 + 2b4 - 3b2 - b1) q^39 + (b10 + b9 - 3b6) q^41 + (-b11 + 2b5 - b4 - 2b2 - b1) * q^43 + (-b10 + 2b6 + 2b3 + 1) * q^45 + (-b11 - 2b8 - 4b5 - b4 - b2 - b1) * q^47 + (b10 - 2b9 + b7 + 5b6 - 2b3) q^49 + (b11 + 3b2 + b1) q^51 + (-b10 - b9 - b7 + 4b6 - b3 + 2) q^53 + (-b11 - b5 - b4 + 2b1) q^55 + (-b7 + 1) * q^57 + (b11 - 3b8 + b5 + 2b4 - 3b2) q^59 + (2b10 + 2b9 - b7 - b6 - b3 + 1) * q^61 + (2b11 + 3b8 + 3b4 + 3b2 - b1) * q^63 + (-2b10 + b9 + b7 + b3) q^65 + (-b11 + 2b8 - b5 - 4b4 + 2b2 - b1) q^67 + (b9 + b7 + 2b6 + 3b3 + 7) * q^69 + (-b11 + 2b8 + 2b5 - 4b4 + 4b2 - b1) * q^71 + (-b10 - b9 + 2b7 - b3 - 1) q^73 + 2b4 q^75 + (-2b10 + b9 + b7 - 6b6 - 2b3 - 12) q^77 + (-2b8 + 2b5 + b4 - b2 + b1) * q^79 + (2b9 - b7 - 6b6 + 2b3 - 4) q^81 + (b11 - 2b5 + b4 + 3b2 + b1) * q^83 + (b10 + b9 - 2b7 - 2b3) * q^85 + (3b11 - 3b8 + 2b5 - b4 + 3b2) * q^87 + (b10 + b9 - 2b7 + b3 - 3) q^89 + (-3b11 - 3b5 + 3b4) q^91 + (2b10 + b9 + b7 - 8b6 - b3 - 7) * q^93 + (b8 - b2) * q^95 + (-3b7 - b6 - 3b3 - 1) * q^97 + (b11 + 3b8 - 3b4 + b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 18 q^{5} - 4 q^{9}+O(q^{10}) \) 12 * q - 18 * q^5 - 4 * q^9	\( 12 q - 18 q^{5} - 4 q^{9} - 6 q^{13} - 6 q^{21} - 12 q^{25} + 18 q^{29} + 30 q^{33} + 18 q^{41} - 6 q^{45} - 24 q^{49} + 8 q^{57} + 18 q^{61} + 6 q^{65} + 66 q^{69} - 90 q^{77} - 20 q^{81} - 48 q^{89} - 30 q^{93} - 6 q^{97}+O(q^{100}) \) 12 * q - 18 * q^5 - 4 * q^9 - 6 * q^13 - 6 * q^21 - 12 * q^25 + 18 * q^29 + 30 * q^33 + 18 * q^41 - 6 * q^45 - 24 * q^49 + 8 * q^57 + 18 * q^61 + 6 * q^65 + 66 * q^69 - 90 * q^77 - 20 * q^81 - 48 * q^89 - 30 * q^93 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	576.2.r.e

Level	$576$
Weight	$2$
Character orbit	576.r
Analytic conductor	$4.599$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 778 \nu^{11} + 5496 \nu^{10} - 7293 \nu^{9} + 5400 \nu^{8} + 10120 \nu^{7} - 68622 \nu^{6} + \cdots + 73584 ) / 51972 \) (-778v^11 + 5496v^10 - 7293v^9 + 5400v^8 + 10120v^7 - 68622v^6 - 13980v^5 + 12828v^4 + 200318v^3 + 699840v^2 + 180024*v + 73584) / 51972
\(\beta_{2}\)	\(=\)	\( ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + \cdots + 15984 ) / 51972 \) (962v^11 - 2392v^10 + 2887v^9 - 2220v^8 - 13990v^7 + 14442v^6 + 11380v^5 + 57708v^4 + 98118v^3 - 132080v^2 + 49284*v + 15984) / 51972
\(\beta_{3}\)	\(=\)	\( ( 1360 \nu^{11} - 864 \nu^{10} - 400 \nu^{9} + 2902 \nu^{8} - 25539 \nu^{7} - 16320 \nu^{6} + \cdots + 219540 ) / 51972 \) (1360v^11 - 864v^10 - 400v^9 + 2902v^8 - 25539v^7 - 16320v^6 + 27692v^5 + 92050v^4 + 330320v^3 + 266532v^2 + 143634*v + 219540) / 51972
\(\beta_{4}\)	\(=\)	\( ( 5984 \nu^{11} - 7522 \nu^{10} + 8393 \nu^{9} - 6600 \nu^{8} - 90816 \nu^{7} - 28498 \nu^{6} + \cdots + 272448 ) / 103944 \) (5984v^11 - 7522v^10 + 8393v^9 - 6600v^8 - 90816v^7 - 28498v^6 + 55616v^5 + 473748v^4 + 1167562v^3 + 792408v^2 + 746328*v + 272448) / 103944
\(\beta_{5}\)	\(=\)	\( ( - 6823 \nu^{11} - 186 \nu^{10} + 2391 \nu^{9} - 2376 \nu^{8} + 113464 \nu^{7} + 159834 \nu^{6} + \cdots - 548280 ) / 103944 \) (-6823v^11 - 186v^10 + 2391v^9 - 2376v^8 + 113464v^7 + 159834v^6 - 23754v^5 - 683700v^4 - 2089522v^3 - 2446620v^2 - 1454952*v - 548280) / 103944
\(\beta_{6}\)	\(=\)	\( ( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + \cdots + 3468 ) / 1704 \) (118v^11 - 90v^10 + 53v^9 - 32v^8 - 1866v^7 - 1416v^6 + 1364v^5 + 10408v^4 + 27758v^3 + 22776v^2 + 12468*v + 3468) / 1704
\(\beta_{7}\)	\(=\)	\( ( 4689 \nu^{11} - 3876 \nu^{10} + 3854 \nu^{9} - 5534 \nu^{8} - 68264 \nu^{7} - 56268 \nu^{6} + \cdots + 48876 ) / 51972 \) (4689v^11 - 3876v^10 + 3854v^9 - 5534v^8 - 68264v^7 - 56268v^6 + 37850v^5 + 438544v^4 + 1102444v^3 + 911136v^2 + 502080*v + 48876) / 51972
\(\beta_{8}\)	\(=\)	\( ( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} + \cdots - 105120 ) / 12993 \) (-1688v^11 + 1054v^10 - 840v^9 + 684v^8 + 26378v^7 + 24587v^6 - 12648v^5 - 151968v^4 - 420176v^3 - 400856v^2 - 282372*v - 105120) / 12993
\(\beta_{9}\)	\(=\)	\( ( - 4502 \nu^{11} + 3246 \nu^{10} - 927 \nu^{9} - 1080 \nu^{8} + 73378 \nu^{7} + 54024 \nu^{6} + \cdots - 228796 ) / 34648 \) (-4502v^11 + 3246v^10 - 927v^9 - 1080v^8 + 73378v^7 + 54024v^6 - 56276v^5 - 403744v^4 - 1035370v^3 - 847128v^2 - 462516*v - 228796) / 34648
\(\beta_{10}\)	\(=\)	\( ( 25054 \nu^{11} - 18618 \nu^{10} + 8389 \nu^{9} + 272 \nu^{8} - 404246 \nu^{7} - 300648 \nu^{6} + \cdots + 1030836 ) / 103944 \) (25054v^11 - 18618v^10 + 8389v^9 + 272v^8 - 404246v^7 - 300648v^6 + 310036v^5 + 2217896v^4 + 5869134v^3 + 4806792v^2 + 2626812*v + 1030836) / 103944
\(\beta_{11}\)	\(=\)	\( ( - 13366 \nu^{11} + 8843 \nu^{10} - 7504 \nu^{9} + 5514 \nu^{8} + 211707 \nu^{7} + 177716 \nu^{6} + \cdots - 830016 ) / 51972 \) (-13366v^11 + 8843v^10 - 7504v^9 + 5514v^8 + 211707v^7 + 177716v^6 - 88792v^5 - 1193088v^4 - 3315152v^3 - 3104220v^2 - 2229918*v - 830016) / 51972

\(\nu\)	\(=\)	\( ( -2\beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{7} + 4\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_1 ) / 6 \) (-2b11 + b10 + b9 - 2b7 + 4b5 - 2b4 - 2*b3 + b1) / 6
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{11} + 4\beta_{5} - 2\beta_{4} - 9\beta_{2} - 2\beta_1 ) / 6 \) (-2b11 + 4b5 - 2b4 - 9b2 - 2*b1) / 6
\(\nu^{3}\)	\(=\)	\( ( - 4 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} - 12 \beta_{6} + 8 \beta_{5} + \cdots - 6 ) / 6 \) (-4b11 + 4b10 + 4b9 - 3b8 + 4b7 - 12b6 + 8b5 - 16b4 + 4b3 - 6b2 - 4*b1 - 6) / 6
\(\nu^{4}\)	\(=\)	\( ( 11\beta_{10} + 5\beta_{9} + 2\beta_{7} - 30\beta_{6} - 4\beta_{3} ) / 3 \) (11b10 + 5b9 + 2b7 - 30b6 - 4*b3) / 3
\(\nu^{5}\)	\(=\)	\( ( - 22 \beta_{11} + 38 \beta_{10} + 35 \beta_{9} + 42 \beta_{8} - 16 \beta_{7} - 39 \beta_{6} + 32 \beta_{5} + \cdots + 39 ) / 6 \) (-22b11 + 38b10 + 35b9 + 42b8 - 16b7 - 39b6 + 32b5 + 38b4 - 22b3 + 21b2 + 38*b1 + 39) / 6
\(\nu^{6}\)	\(=\)	\( ( -56\beta_{11} + 81\beta_{8} + 64\beta_{5} + 16\beta_{4} + 40\beta_1 ) / 3 \) (-56b11 + 81b8 + 64b5 + 16b4 + 40*b1) / 3
\(\nu^{7}\)	\(=\)	\( ( - 94 \beta_{11} - 35 \beta_{10} - 47 \beta_{9} + 60 \beta_{8} + 82 \beta_{7} - 108 \beta_{6} + \cdots - 216 ) / 3 \) (-94b11 - 35b10 - 47b9 + 60b8 + 82b7 - 108b6 + 164b5 - 70b4 + 94b3 - 60b2 + 35*b1 - 216) / 3
\(\nu^{8}\)	\(=\)	\( ( 70\beta_{10} - 35\beta_{9} + 214\beta_{7} - 669\beta_{6} + 214\beta_{3} - 669 ) / 3 \) (70b10 - 35b9 + 214b7 - 669b6 + 214*b3 - 669) / 3
\(\nu^{9}\)	\(=\)	\( ( 164 \beta_{11} + 308 \beta_{10} + 164 \beta_{9} + 321 \beta_{8} + 236 \beta_{7} - 1140 \beta_{6} + \cdots - 570 ) / 3 \) (164b11 + 308b10 + 164b9 + 321b8 + 236b7 - 1140b6 - 472b5 + 800b4 + 164b3 + 642b2 + 308*b1 - 570) / 3
\(\nu^{10}\)	\(=\)	\( ( -328\beta_{11} + 1908\beta_{8} - 328\beta_{5} + 2228\beta_{4} + 1908\beta_{2} + 1442\beta_1 ) / 3 \) (-328b11 + 1908b8 - 328b5 + 2228b4 + 1908b2 + 1442b1) / 3
\(\nu^{11}\)	\(=\)	\( ( - 1586 \beta_{11} - 2386 \beta_{10} - 1993 \beta_{9} + 3342 \beta_{8} + 800 \beta_{7} + 2949 \beta_{6} + \cdots - 2949 ) / 3 \) (-1586b11 - 2386b10 - 1993b9 + 3342b8 + 800b7 + 2949b6 + 1600b5 + 2386b4 + 1586b3 + 1671b2 + 2386*b1 - 2949) / 3

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(\beta_{6}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 97.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 2 T^{10} + \cdots + 729 \) T^12 + 2T^10 + 7T^8 + 12T^6 + 63T^4 + 162*T^2 + 729
$5$	\( (T^{2} + 3 T + 3)^{6} \) (T^2 + 3*T + 3)^6
$7$	\( T^{12} + 33 T^{10} + \cdots + 59049 \) T^12 + 33T^10 + 810T^8 + 8721T^6 + 69822T^4 + 67797*T^2 + 59049
$11$	\( T^{12} - 39 T^{10} + \cdots + 6561 \) T^12 - 39T^10 + 1350T^8 - 6507T^6 + 26082T^4 - 13851*T^2 + 6561
$13$	\( (T^{6} + 3 T^{5} + \cdots + 243)^{2} \) (T^6 + 3T^5 - 24T^4 - 81T^3 + 756T^2 - 729*T + 243)^2
$17$	\( (T^{3} - 24 T + 36)^{4} \) (T^3 - 24*T + 36)^4
$19$	\( (T^{2} + 4)^{6} \) (T^2 + 4)^6
$23$	\( T^{12} + \cdots + 2492305929 \) T^12 + 117T^10 + 9342T^8 + 408753T^6 + 13055418T^4 + 217015281*T^2 + 2492305929
$29$	\( (T^{6} - 9 T^{5} + \cdots + 93987)^{2} \) (T^6 - 9T^5 - 36T^4 + 567T^3 + 2376T^2 - 33453*T + 93987)^2
$31$	\( T^{12} + 117 T^{10} + \cdots + 95004009 \) T^12 + 117T^10 + 11070T^8 + 286929T^6 + 5718762T^4 + 25527393*T^2 + 95004009
$37$	\( (T^{6} + 60 T^{4} + \cdots + 3888)^{2} \) (T^6 + 60T^4 + 1008T^2 + 3888)^2
$41$	\( (T^{6} - 9 T^{5} + 78 T^{4} + \cdots + 81)^{2} \) (T^6 - 9T^5 + 78T^4 - 45T^3 + 90T^2 + 27*T + 81)^2
$43$	\( T^{12} - 123 T^{10} + \cdots + 47458321 \) T^12 - 123T^10 + 13194T^8 - 224227T^6 + 2896878T^4 - 13330215*T^2 + 47458321
$47$	\( T^{12} + \cdots + 514609673769 \) T^12 + 297T^10 + 61074T^8 + 6624369T^6 + 523251414T^4 + 19465645005*T^2 + 514609673769
$53$	\( (T^{6} + 180 T^{4} + \cdots + 432)^{2} \) (T^6 + 180T^4 + 1728T^2 + 432)^2
$59$	\( T^{12} - 147 T^{10} + \cdots + 531441 \) T^12 - 147T^10 + 20898T^8 - 103059T^6 + 398358T^4 - 518319*T^2 + 531441
$61$	\( (T^{6} - 9 T^{5} + \cdots + 4563)^{2} \) (T^6 - 9T^5 - 36T^4 + 567T^3 + 4320T^2 + 7371*T + 4563)^2
$67$	\( T^{12} + \cdots + 352275361 \) T^12 - 231T^10 + 45270T^8 - 1831483T^6 + 61128642T^4 - 151859979*T^2 + 352275361
$71$	\( (T^{6} - 288 T^{4} + \cdots - 124848)^{2} \) (T^6 - 288T^4 + 19872T^2 - 124848)^2
$73$	\( (T^{3} - 84 T - 164)^{4} \) (T^3 - 84*T - 164)^4
$79$	\( T^{12} + 93 T^{10} + \cdots + 4782969 \) T^12 + 93T^10 + 7758T^8 + 78489T^6 + 590490T^4 + 1948617*T^2 + 4782969
$83$	\( T^{12} - 171 T^{10} + \cdots + 43046721 \) T^12 - 171T^10 + 26082T^8 - 527067T^6 + 8857350T^4 - 20726199*T^2 + 43046721
$89$	\( (T^{3} + 12 T^{2} + \cdots - 108)^{4} \) (T^3 + 12T^2 - 36T - 108)^4
$97$	\( (T^{6} + 3 T^{5} + \cdots + 1408969)^{2} \) (T^6 + 3T^5 + 222T^4 + 1735T^3 + 48930T^2 + 252831*T + 1408969)^2
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