LMFDB - Newform orbit 5184.2.d.q

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2593,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("5184.2593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.d (of order $2$, degree $1$, 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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$12$
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^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{8} q^{5} - \beta_{4} q^{7}+O(q^{10}) $ q + b8 * q^5 - b4 * q^7	\( q + \beta_{8} q^{5} - \beta_{4} q^{7} + ( - \beta_{9} - \beta_{7}) q^{11} - \beta_{10} q^{13} - \beta_1 q^{17} + \beta_{7} q^{19} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{23} + 2 q^{25} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{29} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{31} + ( - \beta_{9} - \beta_{7} - \beta_{5}) q^{35} + (\beta_{10} + \beta_{8}) q^{37} + ( - \beta_1 - 3) q^{41} + ( - \beta_{9} + \beta_{7} - \beta_{5}) q^{43} + (2 \beta_{6} + \beta_{4} - \beta_{3}) q^{47} + ( - \beta_{2} + \beta_1 + 4) q^{49} + ( - \beta_{11} + \beta_{10} + 2 \beta_{8}) q^{53} + ( - \beta_{6} + \beta_{4}) q^{55} + (\beta_{9} - 2 \beta_{7} - \beta_{5}) q^{59} + ( - \beta_{11} + \beta_{10} + \beta_{8}) q^{61} + ( - \beta_{2} + \beta_1 - 1) q^{65} + ( - \beta_{9} + \beta_{7} + 2 \beta_{5}) q^{67} + ( - \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{71} + (\beta_{2} - \beta_1) q^{73} + (\beta_{11} + 2 \beta_{10} + 6 \beta_{8}) q^{77} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{79} + (\beta_{9} - 2 \beta_{7} + \beta_{5}) q^{83} + (\beta_{11} - \beta_{10}) q^{85} + ( - \beta_{2} + \beta_1 - 4) q^{89} + (3 \beta_{9} + 3 \beta_{7}) q^{91} - \beta_{3} q^{95} + ( - 3 \beta_1 + 1) q^{97}+O(q^{100}) \) q + b8 * q^5 - b4 * q^7 + (-b9 - b7) * q^11 - b10 * q^13 - b1 * q^17 + b7 * q^19 + (-b6 + b4 - b3) * q^23 + 2 * q^25 + (b11 - b10 + b8) * q^29 + (b6 + b4 - b3) * q^31 + (-b9 - b7 - b5) * q^35 + (b10 + b8) * q^37 + (-b1 - 3) * q^41 + (-b9 + b7 - b5) * q^43 + (2b6 + b4 - b3) q^47 + (-b2 + b1 + 4) * q^49 + (-b11 + b10 + 2b8) q^53 + (-b6 + b4) * q^55 + (b9 - 2b7 - b5) q^59 + (-b11 + b10 + b8) * q^61 + (-b2 + b1 - 1) * q^65 + (-b9 + b7 + 2b5) q^67 + (-b6 - 2b4 - 2b3) * q^71 + (b2 - b1) * q^73 + (b11 + 2b10 + 6b8) * q^77 + (-b6 - b4 - b3) * q^79 + (b9 - 2b7 + b5) q^83 + (b11 - b10) * q^85 + (-b2 + b1 - 4) * q^89 + (3b9 + 3b7) * q^91 - b3 * q^95 + (-3b1 + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 24 q^{25} - 36 q^{41} + 48 q^{49} - 12 q^{65} - 48 q^{89} + 12 q^{97}+O(q^{100}) $ 12 * q + 24 * q^25 - 36 * q^41 + 48 * q^49 - 12 * q^65 - 48 * q^89 + 12 * 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 :

$\beta_{1}$	$=$	$ ( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + \cdots - 96192 ) / 51972 $ (-275v^11 + 300v^10 - 650v^9 + 1148v^8 + 1747v^7 + 3300v^6 + 5062v^5 - 27262v^4 - 51252v^3 - 44964v^2 - 26082*v - 96192) / 51972
$\beta_{2}$	$=$	$ ( 1969 \nu^{11} - 2148 \nu^{10} + 4654 \nu^{9} - 11338 \nu^{8} - 17186 \nu^{7} - 23628 \nu^{6} + \cdots - 442176 ) / 51972 $ (1969v^11 - 2148v^10 + 4654v^9 - 11338v^8 - 17186v^7 - 23628v^6 - 17534v^5 + 254444v^4 + 441804v^3 + 378072v^2 + 214812*v - 442176) / 51972
$\beta_{3}$	$=$	$ ( - 34 \nu^{11} - 4 \nu^{10} + 17 \nu^{9} - 12 \nu^{8} + 546 \nu^{7} + 896 \nu^{6} - 196 \nu^{5} + \cdots - 2736 ) / 366 $ (-34v^11 - 4v^10 + 17v^9 - 12v^8 + 546v^7 + 896v^6 - 196v^5 - 3468v^4 - 10454v^3 - 13152v^2 - 7260*v - 2736) / 366
$\beta_{4}$	$=$	$ ( - 11251 \nu^{11} - 3656 \nu^{10} + 8584 \nu^{9} - 6576 \nu^{8} + 184040 \nu^{7} + 325576 \nu^{6} + \cdots - 967896 ) / 103944 $ (-11251v^11 - 3656v^10 + 8584v^9 - 6576v^8 + 184040v^7 + 325576v^6 - 51410v^5 - 1183104v^4 - 3657720v^3 - 4638708v^2 - 2561328*v - 967896) / 103944
$\beta_{5}$	$=$	$ ( 5765 \nu^{11} - 3533 \nu^{10} + 2602 \nu^{9} - 2490 \nu^{8} - 88123 \nu^{7} - 86504 \nu^{6} + \cdots + 355320 ) / 51972 $ (5765v^11 - 3533v^10 + 2602v^9 - 2490v^8 - 88123v^7 - 86504v^6 + 51058v^5 + 522216v^4 + 1425948v^3 + 1357440v^2 + 954990*v + 355320) / 51972
$\beta_{6}$	$=$	$ ( 6604 \nu^{11} + 4175 \nu^{10} - 8182 \nu^{9} + 6486 \nu^{8} - 110771 \nu^{7} - 217840 \nu^{6} + \cdots + 631152 ) / 51972 $ (6604v^11 + 4175v^10 - 8182v^9 + 6486v^8 - 110771v^7 - 217840v^6 + 19196v^5 + 732168v^4 + 2347908v^3 + 3011652v^2 + 1663614*v + 631152) / 51972
$\beta_{7}$	$=$	$ ( 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
$\beta_{8}$	$=$	$ ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} + \cdots - 4320 ) / 852 $ (-118v^11 + 90v^10 - 53v^9 + 32v^8 + 1866v^7 + 1416v^6 - 1364v^5 - 10408v^4 - 27758v^3 - 22776v^2 - 12468*v - 4320) / 852
$\beta_{9}$	$=$	$ ( 26035 \nu^{11} - 16590 \nu^{10} + 14290 \nu^{9} - 9780 \nu^{8} - 416670 \nu^{7} - 347068 \nu^{6} + \cdots + 1639800 ) / 103944 $ (26035v^11 - 16590v^10 + 14290v^9 - 9780v^8 - 416670v^7 - 347068v^6 + 155770v^5 + 2327880v^4 + 6525980v^3 + 6240620v^2 + 4402140*v + 1639800) / 103944
$\beta_{10}$	$=$	$ ( - 17920 \nu^{11} + 13314 \nu^{10} - 5985 \nu^{9} + 1146 \nu^{8} + 286651 \nu^{7} + 215040 \nu^{6} + \cdots - 639072 ) / 51972 $ (-17920v^11 + 13314v^10 - 5985v^9 + 1146v^8 + 286651v^7 + 215040v^6 - 211740v^5 - 1622514v^4 - 4157302v^3 - 3407556v^2 - 1863546*v - 639072) / 51972
$\beta_{11}$	$=$	$ ( - 29743 \nu^{11} + 22494 \nu^{10} - 12243 \nu^{9} + 5262 \nu^{8} + 472510 \nu^{7} + 356916 \nu^{6} + \cdots - 1079712 ) / 51972 $ (-29743v^11 + 22494v^10 - 12243v^9 + 5262v^8 + 472510v^7 + 356916v^6 - 347886v^5 - 2656440v^4 - 6971578v^3 - 5717928v^2 - 3128892*v - 1079712) / 51972

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{10} + \beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_1 ) / 12 $ (b11 - b10 + b6 + 3b5 + 2b4 + 3*b1) / 12
$\nu^{2}$	$=$	$ ( 2\beta_{9} - 7\beta_{7} + 2\beta_{5} + 6\beta_{4} - 9\beta_{3} ) / 12 $ (2b9 - 7b7 + 2b5 + 6b4 - 9*b3) / 12
$\nu^{3}$	$=$	$ ( -4\beta_{11} + 4\beta_{10} + 6\beta_{8} + 4\beta_{6} + 8\beta_{4} - 3\beta_{3} ) / 6 $ (-4b11 + 4b10 + 6b8 + 4b6 + 8b4 - 3b3) / 6
$\nu^{4}$	$=$	$ ( -9\beta_{11} + 3\beta_{10} + 30\beta_{8} + 2\beta_{2} + 7\beta _1 + 32 ) / 6 $ (-9b11 + 3b10 + 30b8 + 2b2 + 7*b1 + 32) / 6
$\nu^{5}$	$=$	$ ( - 19 \beta_{11} + 16 \beta_{10} + 6 \beta_{9} + 39 \beta_{8} - 57 \beta_{7} - 16 \beta_{6} + \cdots + 120 ) / 12 $ (-19b11 + 16b10 + 6b9 + 39b8 - 57b7 - 16b6 + 54b5 - 38b4 + 21b3 + 3b2 + 54*b1 + 120) / 12
$\nu^{6}$	$=$	$ ( 16\beta_{9} - 65\beta_{7} + 40\beta_{5} ) / 3 $ (16b9 - 65b7 + 40*b5) / 3
$\nu^{7}$	$=$	$ ( - 47 \beta_{11} + 35 \beta_{10} + 24 \beta_{9} + 108 \beta_{8} - 156 \beta_{7} + 35 \beta_{6} + \cdots - 336 ) / 6 $ (-47b11 + 35b10 + 24b9 + 108b8 - 156b7 + 35b6 + 129b5 + 94b4 - 60b3 - 12b2 - 129*b1 - 336) / 6
$\nu^{8}$	$=$	$ ( -249\beta_{11} + 144\beta_{10} + 669\beta_{8} - 35\beta_{2} - 214\beta _1 - 704 ) / 6 $ (-249b11 + 144b10 + 669b8 - 35b2 - 214*b1 - 704) / 6
$\nu^{9}$	$=$	$ ( -236\beta_{11} + 164\beta_{10} + 570\beta_{8} - 164\beta_{6} - 472\beta_{4} + 321\beta_{3} ) / 3 $ (-236b11 + 164b10 + 570b8 - 164b6 - 472b4 + 321b3) / 3
$\nu^{10}$	$=$	$ ( 164\beta_{9} - 790\beta_{7} - 393\beta_{6} + 557\beta_{5} - 1278\beta_{4} + 954\beta_{3} ) / 3 $ (164b9 - 790b7 - 393b6 + 557b5 - 1278b4 + 954b3) / 3
$\nu^{11}$	$=$	$ ( 1193 \beta_{11} - 800 \beta_{10} + 786 \beta_{9} - 2949 \beta_{8} - 4227 \beta_{7} - 800 \beta_{6} + \cdots - 9240 ) / 6 $ (1193b11 - 800b10 + 786b9 - 2949b8 - 4227b7 - 800b6 + 3186b5 - 2386b4 + 1671b3 - 393b2 - 3186*b1 - 9240) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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} $

2593.1

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

−

1.73205i

−4.35461

2593.2

−

1.73205i

−3.61328

2593.3

−

1.73205i

−0.990721

2593.4

−

1.73205i

0.990721

2593.5

−

1.73205i

3.61328

2593.6

−

1.73205i

4.35461

2593.7

1.73205i

−4.35461

2593.8

1.73205i

−3.61328

2593.9

1.73205i

−0.990721

2593.10

1.73205i

0.990721

2593.11

1.73205i

3.61328

2593.12

1.73205i

4.35461

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner

Twists

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

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(5184, [\chi])$:

$ T_{5}^{2} + 3 $

$ T_{7}^{6} - 33T_{7}^{4} + 279T_{7}^{2} - 243 $

$ T_{17}^{3} - 24T_{17} + 36 $

$ T_{23}^{6} - 117T_{23}^{4} + 4347T_{23}^{2} - 49923 $

$ T_{41}^{3} + 9T_{41}^{2} + 3T_{41} - 9 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ (T^{2} + 3)^{6} $ (T^2 + 3)^6
$7$	$ (T^{6} - 33 T^{4} + \cdots - 243)^{2} $ (T^6 - 33T^4 + 279T^2 - 243)^2
$11$	$ (T^{6} + 39 T^{4} + \cdots + 81)^{2} $ (T^6 + 39T^4 + 171T^2 + 81)^2
$13$	$ (T^{6} + 57 T^{4} + \cdots + 243)^{2} $ (T^6 + 57T^4 + 675T^2 + 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^{6} - 117 T^{4} + \cdots - 49923)^{2} $ (T^6 - 117T^4 + 4347T^2 - 49923)^2
$29$	$ (T^{6} + 153 T^{4} + \cdots + 93987)^{2} $ (T^6 + 153T^4 + 7155T^2 + 93987)^2
$31$	$ (T^{6} - 117 T^{4} + \cdots - 9747)^{2} $ (T^6 - 117T^4 + 2619T^2 - 9747)^2
$37$	$ (T^{6} + 60 T^{4} + \cdots + 3888)^{2} $ (T^6 + 60T^4 + 1008T^2 + 3888)^2
$41$	$ (T^{3} + 9 T^{2} + 3 T - 9)^{4} $ (T^3 + 9T^2 + 3T - 9)^4
$43$	$ (T^{6} + 123 T^{4} + \cdots + 6889)^{2} $ (T^6 + 123T^4 + 1935T^2 + 6889)^2
$47$	$ (T^{6} - 297 T^{4} + \cdots - 717363)^{2} $ (T^6 - 297T^4 + 27135T^2 - 717363)^2
$53$	$ (T^{6} + 180 T^{4} + \cdots + 432)^{2} $ (T^6 + 180T^4 + 1728T^2 + 432)^2
$59$	$ (T^{6} + 147 T^{4} + \cdots + 729)^{2} $ (T^6 + 147T^4 + 711T^2 + 729)^2
$61$	$ (T^{6} + 153 T^{4} + \cdots + 4563)^{2} $ (T^6 + 153T^4 + 3267T^2 + 4563)^2
$67$	$ (T^{6} + 231 T^{4} + \cdots + 18769)^{2} $ (T^6 + 231T^4 + 8091T^2 + 18769)^2
$71$	$ (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^{6} - 93 T^{4} + \cdots - 2187)^{2} $ (T^6 - 93T^4 + 891T^2 - 2187)^2
$83$	$ (T^{6} + 171 T^{4} + \cdots + 6561)^{2} $ (T^6 + 171T^4 + 3159T^2 + 6561)^2
$89$	$ (T^{3} + 12 T^{2} + \cdots - 108)^{4} $ (T^3 + 12T^2 - 36T - 108)^4
$97$	$ (T^{3} - 3 T^{2} + \cdots + 1187)^{4} $ (T^3 - 3T^2 - 213T + 1187)^4
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{5} - \beta_{4} q^{7}+O(q^{10}) \) q + b8 * q^5 - b4 * q^7	\( q + \beta_{8} q^{5} - \beta_{4} q^{7} + ( - \beta_{9} - \beta_{7}) q^{11} - \beta_{10} q^{13} - \beta_1 q^{17} + \beta_{7} q^{19} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{23} + 2 q^{25} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{29} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{31} + ( - \beta_{9} - \beta_{7} - \beta_{5}) q^{35} + (\beta_{10} + \beta_{8}) q^{37} + ( - \beta_1 - 3) q^{41} + ( - \beta_{9} + \beta_{7} - \beta_{5}) q^{43} + (2 \beta_{6} + \beta_{4} - \beta_{3}) q^{47} + ( - \beta_{2} + \beta_1 + 4) q^{49} + ( - \beta_{11} + \beta_{10} + 2 \beta_{8}) q^{53} + ( - \beta_{6} + \beta_{4}) q^{55} + (\beta_{9} - 2 \beta_{7} - \beta_{5}) q^{59} + ( - \beta_{11} + \beta_{10} + \beta_{8}) q^{61} + ( - \beta_{2} + \beta_1 - 1) q^{65} + ( - \beta_{9} + \beta_{7} + 2 \beta_{5}) q^{67} + ( - \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{71} + (\beta_{2} - \beta_1) q^{73} + (\beta_{11} + 2 \beta_{10} + 6 \beta_{8}) q^{77} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{79} + (\beta_{9} - 2 \beta_{7} + \beta_{5}) q^{83} + (\beta_{11} - \beta_{10}) q^{85} + ( - \beta_{2} + \beta_1 - 4) q^{89} + (3 \beta_{9} + 3 \beta_{7}) q^{91} - \beta_{3} q^{95} + ( - 3 \beta_1 + 1) q^{97}+O(q^{100}) \) q + b8 * q^5 - b4 * q^7 + (-b9 - b7) * q^11 - b10 * q^13 - b1 * q^17 + b7 * q^19 + (-b6 + b4 - b3) * q^23 + 2 * q^25 + (b11 - b10 + b8) * q^29 + (b6 + b4 - b3) * q^31 + (-b9 - b7 - b5) * q^35 + (b10 + b8) * q^37 + (-b1 - 3) * q^41 + (-b9 + b7 - b5) * q^43 + (2b6 + b4 - b3) q^47 + (-b2 + b1 + 4) * q^49 + (-b11 + b10 + 2b8) q^53 + (-b6 + b4) * q^55 + (b9 - 2b7 - b5) q^59 + (-b11 + b10 + b8) * q^61 + (-b2 + b1 - 1) * q^65 + (-b9 + b7 + 2b5) q^67 + (-b6 - 2b4 - 2b3) * q^71 + (b2 - b1) * q^73 + (b11 + 2b10 + 6b8) * q^77 + (-b6 - b4 - b3) * q^79 + (b9 - 2b7 + b5) q^83 + (b11 - b10) * q^85 + (-b2 + b1 - 4) * q^89 + (3b9 + 3b7) * q^91 - b3 * q^95 + (-3b1 + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 24 q^{25} - 36 q^{41} + 48 q^{49} - 12 q^{65} - 48 q^{89} + 12 q^{97}+O(q^{100}) \) 12 * q + 24 * q^25 - 36 * q^41 + 48 * q^49 - 12 * q^65 - 48 * q^89 + 12 * q^97

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Newspace parameters

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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	5184.2.d.q

Level	$5184$
Weight	$2$
Character orbit	5184.d
Analytic conductor	$41.394$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(41.3944484078\)
Analytic rank:	\(0\)
Dimension:	\(12\)
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:	\( 2^{12}\cdot 3^{4} \)
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + \cdots - 96192 ) / 51972 \) (-275v^11 + 300v^10 - 650v^9 + 1148v^8 + 1747v^7 + 3300v^6 + 5062v^5 - 27262v^4 - 51252v^3 - 44964v^2 - 26082*v - 96192) / 51972
\(\beta_{2}\)	\(=\)	\( ( 1969 \nu^{11} - 2148 \nu^{10} + 4654 \nu^{9} - 11338 \nu^{8} - 17186 \nu^{7} - 23628 \nu^{6} + \cdots - 442176 ) / 51972 \) (1969v^11 - 2148v^10 + 4654v^9 - 11338v^8 - 17186v^7 - 23628v^6 - 17534v^5 + 254444v^4 + 441804v^3 + 378072v^2 + 214812*v - 442176) / 51972
\(\beta_{3}\)	\(=\)	\( ( - 34 \nu^{11} - 4 \nu^{10} + 17 \nu^{9} - 12 \nu^{8} + 546 \nu^{7} + 896 \nu^{6} - 196 \nu^{5} + \cdots - 2736 ) / 366 \) (-34v^11 - 4v^10 + 17v^9 - 12v^8 + 546v^7 + 896v^6 - 196v^5 - 3468v^4 - 10454v^3 - 13152v^2 - 7260*v - 2736) / 366
\(\beta_{4}\)	\(=\)	\( ( - 11251 \nu^{11} - 3656 \nu^{10} + 8584 \nu^{9} - 6576 \nu^{8} + 184040 \nu^{7} + 325576 \nu^{6} + \cdots - 967896 ) / 103944 \) (-11251v^11 - 3656v^10 + 8584v^9 - 6576v^8 + 184040v^7 + 325576v^6 - 51410v^5 - 1183104v^4 - 3657720v^3 - 4638708v^2 - 2561328*v - 967896) / 103944
\(\beta_{5}\)	\(=\)	\( ( 5765 \nu^{11} - 3533 \nu^{10} + 2602 \nu^{9} - 2490 \nu^{8} - 88123 \nu^{7} - 86504 \nu^{6} + \cdots + 355320 ) / 51972 \) (5765v^11 - 3533v^10 + 2602v^9 - 2490v^8 - 88123v^7 - 86504v^6 + 51058v^5 + 522216v^4 + 1425948v^3 + 1357440v^2 + 954990*v + 355320) / 51972
\(\beta_{6}\)	\(=\)	\( ( 6604 \nu^{11} + 4175 \nu^{10} - 8182 \nu^{9} + 6486 \nu^{8} - 110771 \nu^{7} - 217840 \nu^{6} + \cdots + 631152 ) / 51972 \) (6604v^11 + 4175v^10 - 8182v^9 + 6486v^8 - 110771v^7 - 217840v^6 + 19196v^5 + 732168v^4 + 2347908v^3 + 3011652v^2 + 1663614*v + 631152) / 51972
\(\beta_{7}\)	\(=\)	\( ( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + \cdots + 105120 ) / 12993 \) (1688v^11 - 1054v^10 + 840v^9 - 684v^8 - 26378v^7 - 24587v^6 + 12648v^5 + 151968v^4 + 420176v^3 + 400856v^2 + 282372*v + 105120) / 12993
\(\beta_{8}\)	\(=\)	\( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} + \cdots - 4320 ) / 852 \) (-118v^11 + 90v^10 - 53v^9 + 32v^8 + 1866v^7 + 1416v^6 - 1364v^5 - 10408v^4 - 27758v^3 - 22776v^2 - 12468*v - 4320) / 852
\(\beta_{9}\)	\(=\)	\( ( 26035 \nu^{11} - 16590 \nu^{10} + 14290 \nu^{9} - 9780 \nu^{8} - 416670 \nu^{7} - 347068 \nu^{6} + \cdots + 1639800 ) / 103944 \) (26035v^11 - 16590v^10 + 14290v^9 - 9780v^8 - 416670v^7 - 347068v^6 + 155770v^5 + 2327880v^4 + 6525980v^3 + 6240620v^2 + 4402140*v + 1639800) / 103944
\(\beta_{10}\)	\(=\)	\( ( - 17920 \nu^{11} + 13314 \nu^{10} - 5985 \nu^{9} + 1146 \nu^{8} + 286651 \nu^{7} + 215040 \nu^{6} + \cdots - 639072 ) / 51972 \) (-17920v^11 + 13314v^10 - 5985v^9 + 1146v^8 + 286651v^7 + 215040v^6 - 211740v^5 - 1622514v^4 - 4157302v^3 - 3407556v^2 - 1863546*v - 639072) / 51972
\(\beta_{11}\)	\(=\)	\( ( - 29743 \nu^{11} + 22494 \nu^{10} - 12243 \nu^{9} + 5262 \nu^{8} + 472510 \nu^{7} + 356916 \nu^{6} + \cdots - 1079712 ) / 51972 \) (-29743v^11 + 22494v^10 - 12243v^9 + 5262v^8 + 472510v^7 + 356916v^6 - 347886v^5 - 2656440v^4 - 6971578v^3 - 5717928v^2 - 3128892*v - 1079712) / 51972

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{10} + \beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_1 ) / 12 \) (b11 - b10 + b6 + 3b5 + 2b4 + 3*b1) / 12
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{9} - 7\beta_{7} + 2\beta_{5} + 6\beta_{4} - 9\beta_{3} ) / 12 \) (2b9 - 7b7 + 2b5 + 6b4 - 9*b3) / 12
\(\nu^{3}\)	\(=\)	\( ( -4\beta_{11} + 4\beta_{10} + 6\beta_{8} + 4\beta_{6} + 8\beta_{4} - 3\beta_{3} ) / 6 \) (-4b11 + 4b10 + 6b8 + 4b6 + 8b4 - 3b3) / 6
\(\nu^{4}\)	\(=\)	\( ( -9\beta_{11} + 3\beta_{10} + 30\beta_{8} + 2\beta_{2} + 7\beta _1 + 32 ) / 6 \) (-9b11 + 3b10 + 30b8 + 2b2 + 7*b1 + 32) / 6
\(\nu^{5}\)	\(=\)	\( ( - 19 \beta_{11} + 16 \beta_{10} + 6 \beta_{9} + 39 \beta_{8} - 57 \beta_{7} - 16 \beta_{6} + \cdots + 120 ) / 12 \) (-19b11 + 16b10 + 6b9 + 39b8 - 57b7 - 16b6 + 54b5 - 38b4 + 21b3 + 3b2 + 54*b1 + 120) / 12
\(\nu^{6}\)	\(=\)	\( ( 16\beta_{9} - 65\beta_{7} + 40\beta_{5} ) / 3 \) (16b9 - 65b7 + 40*b5) / 3
\(\nu^{7}\)	\(=\)	\( ( - 47 \beta_{11} + 35 \beta_{10} + 24 \beta_{9} + 108 \beta_{8} - 156 \beta_{7} + 35 \beta_{6} + \cdots - 336 ) / 6 \) (-47b11 + 35b10 + 24b9 + 108b8 - 156b7 + 35b6 + 129b5 + 94b4 - 60b3 - 12b2 - 129*b1 - 336) / 6
\(\nu^{8}\)	\(=\)	\( ( -249\beta_{11} + 144\beta_{10} + 669\beta_{8} - 35\beta_{2} - 214\beta _1 - 704 ) / 6 \) (-249b11 + 144b10 + 669b8 - 35b2 - 214*b1 - 704) / 6
\(\nu^{9}\)	\(=\)	\( ( -236\beta_{11} + 164\beta_{10} + 570\beta_{8} - 164\beta_{6} - 472\beta_{4} + 321\beta_{3} ) / 3 \) (-236b11 + 164b10 + 570b8 - 164b6 - 472b4 + 321b3) / 3
\(\nu^{10}\)	\(=\)	\( ( 164\beta_{9} - 790\beta_{7} - 393\beta_{6} + 557\beta_{5} - 1278\beta_{4} + 954\beta_{3} ) / 3 \) (164b9 - 790b7 - 393b6 + 557b5 - 1278b4 + 954b3) / 3
\(\nu^{11}\)	\(=\)	\( ( 1193 \beta_{11} - 800 \beta_{10} + 786 \beta_{9} - 2949 \beta_{8} - 4227 \beta_{7} - 800 \beta_{6} + \cdots - 9240 ) / 6 \) (1193b11 - 800b10 + 786b9 - 2949b8 - 4227b7 - 800b6 + 3186b5 - 2386b4 + 1671b3 - 393b2 - 3186*b1 - 9240) / 6

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2593.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( (T^{2} + 3)^{6} \) (T^2 + 3)^6
$7$	\( (T^{6} - 33 T^{4} + \cdots - 243)^{2} \) (T^6 - 33T^4 + 279T^2 - 243)^2
$11$	\( (T^{6} + 39 T^{4} + \cdots + 81)^{2} \) (T^6 + 39T^4 + 171T^2 + 81)^2
$13$	\( (T^{6} + 57 T^{4} + \cdots + 243)^{2} \) (T^6 + 57T^4 + 675T^2 + 243)^2
$17$	\( (T^{3} - 24 T + 36)^{4} \) (T^3 - 24*T + 36)^4
$19$	\( (T^{2} + 4)^{6} \) (T^2 + 4)^6
$23$	\( (T^{6} - 117 T^{4} + \cdots - 49923)^{2} \) (T^6 - 117T^4 + 4347T^2 - 49923)^2
$29$	\( (T^{6} + 153 T^{4} + \cdots + 93987)^{2} \) (T^6 + 153T^4 + 7155T^2 + 93987)^2
$31$	\( (T^{6} - 117 T^{4} + \cdots - 9747)^{2} \) (T^6 - 117T^4 + 2619T^2 - 9747)^2
$37$	\( (T^{6} + 60 T^{4} + \cdots + 3888)^{2} \) (T^6 + 60T^4 + 1008T^2 + 3888)^2
$41$	\( (T^{3} + 9 T^{2} + 3 T - 9)^{4} \) (T^3 + 9T^2 + 3T - 9)^4
$43$	\( (T^{6} + 123 T^{4} + \cdots + 6889)^{2} \) (T^6 + 123T^4 + 1935T^2 + 6889)^2
$47$	\( (T^{6} - 297 T^{4} + \cdots - 717363)^{2} \) (T^6 - 297T^4 + 27135T^2 - 717363)^2
$53$	\( (T^{6} + 180 T^{4} + \cdots + 432)^{2} \) (T^6 + 180T^4 + 1728T^2 + 432)^2
$59$	\( (T^{6} + 147 T^{4} + \cdots + 729)^{2} \) (T^6 + 147T^4 + 711T^2 + 729)^2
$61$	\( (T^{6} + 153 T^{4} + \cdots + 4563)^{2} \) (T^6 + 153T^4 + 3267T^2 + 4563)^2
$67$	\( (T^{6} + 231 T^{4} + \cdots + 18769)^{2} \) (T^6 + 231T^4 + 8091T^2 + 18769)^2
$71$	\( (T^{6} - 288 T^{4} + \cdots - 124848)^{2} \) (T^6 - 288T^4 + 19872T^2 - 124848)^2
$73$	\( (T^{3} - 84 T - 164)^{4} \) (T^3 - 84*T - 164)^4
$79$	\( (T^{6} - 93 T^{4} + \cdots - 2187)^{2} \) (T^6 - 93T^4 + 891T^2 - 2187)^2
$83$	\( (T^{6} + 171 T^{4} + \cdots + 6561)^{2} \) (T^6 + 171T^4 + 3159T^2 + 6561)^2
$89$	\( (T^{3} + 12 T^{2} + \cdots - 108)^{4} \) (T^3 + 12T^2 - 36T - 108)^4
$97$	\( (T^{3} - 3 T^{2} + \cdots + 1187)^{4} \) (T^3 - 3T^2 - 213T + 1187)^4
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\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)