LMFDB - Newform orbit 4704.2.c.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(2353,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4704.2353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.c (of order $2$, degree $1$, not 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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.84396412309504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 39 x^{8} - 28 x^{7} + 13 x^{6} + 10 x^{5} + 6 x^{4} + \cdots + 2 $ x^12 - 6x^11 + 18x^10 - 32x^9 + 39x^8 - 28x^7 + 13x^6 + 10x^5 + 6x^4 + 14x^3 + 17x^2 + 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 1176)
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_{7} q^{3} + ( - \beta_{10} + \beta_{7}) q^{5} - q^{9}+O(q^{10}) $ q - b7 * q^3 + (-b10 + b7) * q^5 - q^9	\( q - \beta_{7} q^{3} + ( - \beta_{10} + \beta_{7}) q^{5} - q^{9} - \beta_{11} q^{11} + (\beta_{10} + \beta_{8} - 2 \beta_{7}) q^{13} + (\beta_1 + 1) q^{15} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{17} + (\beta_{10} - \beta_{8}) q^{19} + (\beta_{4} + 1) q^{23} + ( - \beta_{4} - \beta_1 - 1) q^{25} + \beta_{7} q^{27} + (\beta_{11} - \beta_{6}) q^{29} + ( - \beta_{5} - \beta_{2}) q^{31} - \beta_{5} q^{33} + (\beta_{11} - \beta_{6}) q^{37} + ( - \beta_{4} - \beta_1 - 2) q^{39} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{41} + (\beta_{9} - \beta_{6}) q^{43} + (\beta_{10} - \beta_{7}) q^{45} + 2 \beta_{3} q^{47} + (\beta_{11} - \beta_{9} - \beta_{6}) q^{51} + ( - \beta_{11} + 2 \beta_{9} + \beta_{6}) q^{53} + (3 \beta_{5} - \beta_{2}) q^{55} + (\beta_{4} - \beta_1) q^{57} + ( - 4 \beta_{10} - 2 \beta_{8} + 2 \beta_{7}) q^{59} + ( - 3 \beta_{10} - \beta_{8} - 4 \beta_{7}) q^{61} + (2 \beta_{4} + 4 \beta_1 + 6) q^{65} + (\beta_{9} + \beta_{6}) q^{67} + (\beta_{8} - \beta_{7}) q^{69} + (\beta_{4} - 2 \beta_1 - 1) q^{71} + 2 \beta_{3} q^{73} + ( - \beta_{10} - \beta_{8} + \beta_{7}) q^{75} + (2 \beta_1 + 2) q^{79} + q^{81} + ( - 2 \beta_{8} + 6 \beta_{7}) q^{83} + ( - 3 \beta_{11} + 2 \beta_{9} + \beta_{6}) q^{85} + (\beta_{5} - \beta_{2}) q^{87} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2}) q^{89} + (\beta_{11} + \beta_{6}) q^{93} + ( - 2 \beta_1 + 6) q^{95} - 2 \beta_{3} q^{97} + \beta_{11} q^{99}+O(q^{100}) \) q - b7 * q^3 + (-b10 + b7) * q^5 - q^9 - b11 * q^11 + (b10 + b8 - 2b7) q^13 + (b1 + 1) * q^15 + (-b5 + b3 + b2) * q^17 + (b10 - b8) * q^19 + (b4 + 1) * q^23 + (-b4 - b1 - 1) * q^25 + b7 * q^27 + (b11 - b6) * q^29 + (-b5 - b2) * q^31 - b5 * q^33 + (b11 - b6) * q^37 + (-b4 - b1 - 2) * q^39 + (-b5 - b3 + b2) * q^41 + (b9 - b6) * q^43 + (b10 - b7) * q^45 + 2b3 q^47 + (b11 - b9 - b6) * q^51 + (-b11 + 2b9 + b6) q^53 + (3b5 - b2) q^55 + (b4 - b1) * q^57 + (-4b10 - 2b8 + 2b7) q^59 + (-3b10 - b8 - 4b7) * q^61 + (2b4 + 4b1 + 6) * q^65 + (b9 + b6) * q^67 + (b8 - b7) * q^69 + (b4 - 2b1 - 1) q^71 + 2b3 q^73 + (-b10 - b8 + b7) * q^75 + (2b1 + 2) q^79 + q^81 + (-2b8 + 6b7) * q^83 + (-3b11 + 2b9 + b6) * q^85 + (b5 - b2) * q^87 + (-3b5 + b3 - b2) q^89 + (b11 + b6) * q^93 + (-2b1 + 6) q^95 - 2b3 q^97 + b11 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 8 q^{15} + 8 q^{23} - 4 q^{25} - 16 q^{39} + 48 q^{65} - 8 q^{71} + 16 q^{79} + 12 q^{81} + 80 q^{95}+O(q^{100}) $ 12 * q - 12 * q^9 + 8 * q^15 + 8 * q^23 - 4 * q^25 - 16 * q^39 + 48 * q^65 - 8 * q^71 + 16 * q^79 + 12 * q^81 + 80 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 39 x^{8} - 28 x^{7} + 13 x^{6} + 10 x^{5} + 6 x^{4} + \cdots + 2 $ x^12 - 6*x^11 + 18*x^10 - 32*x^9 + 39*x^8 - 28*x^7 + 13*x^6 + 10*x^5 + 6*x^4 + 14*x^3 + 17*x^2 + 10*x + 2 :

$\beta_{1}$	$=$	$ ( 5 \nu^{11} - 30 \nu^{10} + 91 \nu^{9} - 166 \nu^{8} + 210 \nu^{7} - 154 \nu^{6} + 59 \nu^{5} + \cdots + 48 ) / 16 $ (5v^11 - 30v^10 + 91v^9 - 166v^8 + 210v^7 - 154v^6 + 59v^5 + 80v^4 - 3v^3 + 102v^2 + 78*v + 48) / 16
$\beta_{2}$	$=$	$ ( - 5 \nu^{11} + 32 \nu^{10} - 107 \nu^{9} + 224 \nu^{8} - 326 \nu^{7} + 286 \nu^{6} - 119 \nu^{5} + \cdots - 12 ) / 16 $ (-5v^11 + 32v^10 - 107v^9 + 224v^8 - 326v^7 + 286v^6 - 119v^5 - 114v^4 + 71v^3 - 120v^2 - 114*v - 12) / 16
$\beta_{3}$	$=$	$ ( - 16 \nu^{11} + 103 \nu^{10} - 330 \nu^{9} + 633 \nu^{8} - 818 \nu^{7} + 630 \nu^{6} - 238 \nu^{5} + \cdots - 70 ) / 16 $ (-16v^11 + 103v^10 - 330v^9 + 633v^8 - 818v^7 + 630v^6 - 238v^5 - 279v^4 + 128v^3 - 241v^2 - 222*v - 70) / 16
$\beta_{4}$	$=$	$ ( 19 \nu^{11} - 118 \nu^{10} + 361 \nu^{9} - 650 \nu^{8} + 782 \nu^{7} - 542 \nu^{6} + 189 \nu^{5} + \cdots + 64 ) / 16 $ (19v^11 - 118v^10 + 361v^9 - 650v^8 + 782v^7 - 542v^6 + 189v^5 + 276v^4 - 41v^3 + 274v^2 + 218*v + 64) / 16
$\beta_{5}$	$=$	$ ( - 21 \nu^{11} + 130 \nu^{10} - 403 \nu^{9} + 746 \nu^{8} - 938 \nu^{7} + 690 \nu^{6} - 259 \nu^{5} + \cdots - 152 ) / 16 $ (-21v^11 + 130v^10 - 403v^9 + 746v^8 - 938v^7 + 690v^6 - 259v^5 - 340v^4 + 67v^3 - 362v^2 - 294*v - 152) / 16
$\beta_{6}$	$=$	$ ( 24 \nu^{11} - 151 \nu^{10} + 478 \nu^{9} - 917 \nu^{8} + 1226 \nu^{7} - 1062 \nu^{6} + 670 \nu^{5} + \cdots + 158 ) / 16 $ (24v^11 - 151v^10 + 478v^9 - 917v^8 + 1226v^7 - 1062v^6 + 670v^5 - 9v^4 + 204v^3 + 261v^2 + 414*v + 158) / 16
$\beta_{7}$	$=$	$ ( 85 \nu^{11} - 545 \nu^{10} + 1753 \nu^{9} - 3431 \nu^{8} + 4692 \nu^{7} - 4248 \nu^{6} + 2793 \nu^{5} + \cdots + 422 ) / 32 $ (85v^11 - 545v^10 + 1753v^9 - 3431v^8 + 4692v^7 - 4248v^6 + 2793v^5 - 285v^4 + 657v^3 + 887v^2 + 1092*v + 422) / 32
$\beta_{8}$	$=$	$ ( - 129 \nu^{11} + 835 \nu^{10} - 2721 \nu^{9} + 5437 \nu^{8} - 7656 \nu^{7} + 7284 \nu^{6} - 5089 \nu^{5} + \cdots - 570 ) / 32 $ (-129v^11 + 835v^10 - 2721v^9 + 5437v^8 - 7656v^7 + 7284v^6 - 5089v^5 + 931v^4 - 965v^3 - 1549v^2 - 1504*v - 570) / 32
$\beta_{9}$	$=$	$ ( - 76 \nu^{11} + 497 \nu^{10} - 1630 \nu^{9} + 3267 \nu^{8} - 4574 \nu^{7} + 4290 \nu^{6} - 2910 \nu^{5} + \cdots - 322 ) / 16 $ (-76v^11 + 497v^10 - 1630v^9 + 3267v^8 - 4574v^7 + 4290v^6 - 2910v^5 + 487v^4 - 536v^3 - 819v^2 - 826*v - 322) / 16
$\beta_{10}$	$=$	$ ( - 181 \nu^{11} + 1159 \nu^{10} - 3725 \nu^{9} + 7297 \nu^{8} - 10024 \nu^{7} + 9172 \nu^{6} + \cdots - 866 ) / 32 $ (-181v^11 + 1159v^10 - 3725v^9 + 7297v^8 - 10024v^7 + 9172v^6 - 6133v^5 + 727v^4 - 1409v^3 - 1953v^2 - 2256*v - 866) / 32
$\beta_{11}$	$=$	$ ( - 262 \nu^{11} + 1683 \nu^{10} - 5428 \nu^{9} + 10677 \nu^{8} - 14718 \nu^{7} + 13522 \nu^{6} + \cdots - 1238 ) / 16 $ (-262v^11 + 1683v^10 - 5428v^9 + 10677v^8 - 14718v^7 + 13522v^6 - 9064v^5 + 1153v^4 - 2006v^3 - 2829v^2 - 3218*v - 1238) / 16

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

$\nu$	$=$	$ ( \beta_{11} - \beta_{10} - \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 4 ) / 8 $ (b11 - b10 - b8 + 2*b7 + b6 + b5 + b4 - b2 - b1 + 4) / 8
$\nu^{2}$	$=$	$ ( 3 \beta_{11} - 5 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - \beta_1 ) / 8 $ (3b11 - 5b10 - 2b9 - b8 + 2b7 + b6 - b5 + b4 + 2*b3 - b2 - b1) / 8
$\nu^{3}$	$=$	$ ( 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots - 5 ) / 4 $ (2b11 - 6b10 - 2b9 + b8 - 3b7 + b6 - 2b5 - b4 + b3 - 2b1 - 5) / 4
$\nu^{4}$	$=$	$ ( \beta_{11} - 17 \beta_{10} - 6 \beta_{9} + 7 \beta_{8} - 34 \beta_{7} + 7 \beta_{6} - 7 \beta_{5} + \cdots - 28 ) / 8 $ (b11 - 17b10 - 6b9 + 7b8 - 34b7 + 7b6 - 7b5 - 7b4 - 2b3 + 5b2 - 5b1 - 28) / 8
$\nu^{5}$	$=$	$ ( - 9 \beta_{11} - 11 \beta_{10} - 8 \beta_{9} + 15 \beta_{8} - 80 \beta_{7} + 17 \beta_{6} - 3 \beta_{5} + \cdots - 62 ) / 8 $ (-9b11 - 11b10 - 8b9 + 15b8 - 80b7 + 17b6 - 3b5 - 13b4 - 12b3 + 17b2 + 15*b1 - 62) / 8
$\nu^{6}$	$=$	$ ( - 12 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 11 \beta_{8} - 54 \beta_{7} + 11 \beta_{6} + 11 \beta_{5} + \cdots - 58 ) / 4 $ (-12b11 + 7b10 - 3b9 + 11b8 - 54b7 + 11b6 + 11b5 - 10b4 - 18b3 + 19b2 + 46*b1 - 58) / 4
$\nu^{7}$	$=$	$ ( - 31 \beta_{11} + 71 \beta_{10} + 18 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 81 \beta_{5} + \cdots - 192 ) / 8 $ (-31b11 + 71b10 + 18b9 + 3b8 + 2b7 - 9b6 + 81b5 - 31b4 - 86b3 + 73b2 + 255*b1 - 192) / 8
$\nu^{8}$	$=$	$ ( - 9 \beta_{11} + 219 \beta_{10} + 106 \beta_{9} - 101 \beta_{8} + 526 \beta_{7} - 139 \beta_{6} + \cdots - 308 ) / 8 $ (-9b11 + 219b10 + 106b9 - 101b8 + 526b7 - 139b6 + 155b5 - 51b4 - 158b3 + 127b2 + 463*b1 - 308) / 8
$\nu^{9}$	$=$	$ ( 17 \beta_{11} + 323 \beta_{10} + 168 \beta_{9} - 186 \beta_{8} + 943 \beta_{7} - 234 \beta_{6} + \cdots - 225 ) / 4 $ (17b11 + 323b10 + 168b9 - 186b8 + 943b7 - 234b6 + 71b5 - 40b4 - 91b3 + 83b2 + 241*b1 - 225) / 4
$\nu^{10}$	$=$	$ ( - 25 \beta_{11} + 1737 \beta_{10} + 830 \beta_{9} - 851 \beta_{8} + 4358 \beta_{7} - 1099 \beta_{6} + \cdots - 216 ) / 8 $ (-25b11 + 1737b10 + 830b9 - 851b8 + 4358b7 - 1099b6 - 177b5 - 53b4 + 70b3 + 3b2 - 315*b1 - 216) / 8
$\nu^{11}$	$=$	$ ( - 489 \beta_{11} + 3897 \beta_{10} + 1672 \beta_{9} - 1389 \beta_{8} + 7288 \beta_{7} - 1983 \beta_{6} + \cdots + 2194 ) / 8 $ (-489b11 + 3897b10 + 1672b9 - 1389b8 + 7288b7 - 1983b6 - 1075b5 + 359b4 + 1124b3 - 927b2 - 3205*b1 + 2194) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$1$	$-1$	$1$	$1$

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} $

2353.1

0.410511	+	1.28550i
1.97872	−	1.02084i
−0.349377	+	0.423297i
0.104733	−	1.09476i
1.26951	+	1.38830i
−0.414092	+	0.0185074i
−0.414092	−	0.0185074i
1.26951	−	1.38830i
0.104733	+	1.09476i
−0.349377	−	0.423297i
1.97872	+	1.02084i
0.410511	−	1.28550i

−

1.00000i

−

2.24914i

−1.00000

2353.2

−

1.00000i

−

2.24914i

−1.00000

2353.3

−

1.00000i

1.14637i

−1.00000

2353.4

−

1.00000i

1.14637i

−1.00000

2353.5

−

1.00000i

3.10278i

−1.00000

2353.6

−

1.00000i

3.10278i

−1.00000

2353.7

1.00000i

−

3.10278i

−1.00000

2353.8

1.00000i

−

3.10278i

−1.00000

2353.9

1.00000i

−

1.14637i

−1.00000

2353.10

1.00000i

−

1.14637i

−1.00000

2353.11

1.00000i

2.24914i

−1.00000

2353.12

1.00000i

2.24914i

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.b	even	2	1	inner
56.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4704.2.c.d		12
4.b	odd	2	1		1176.2.c.d	✓	12
7.b	odd	2	1	inner	4704.2.c.d		12
8.b	even	2	1	inner	4704.2.c.d		12
8.d	odd	2	1		1176.2.c.d	✓	12
28.d	even	2	1		1176.2.c.d	✓	12
56.e	even	2	1		1176.2.c.d	✓	12
56.h	odd	2	1	inner	4704.2.c.d		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.2.c.d	✓	12	4.b	odd	2	1
1176.2.c.d	✓	12	8.d	odd	2	1
1176.2.c.d	✓	12	28.d	even	2	1
1176.2.c.d	✓	12	56.e	even	2	1
4704.2.c.d		12	1.a	even	1	1	trivial
4704.2.c.d		12	7.b	odd	2	1	inner
4704.2.c.d		12	8.b	even	2	1	inner
4704.2.c.d		12	56.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4704, [\chi])$:

$ T_{5}^{6} + 16T_{5}^{4} + 68T_{5}^{2} + 64 $

$ T_{17}^{6} - 80T_{17}^{4} + 1604T_{17}^{2} - 5888 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$5$	$ (T^{6} + 16 T^{4} + \cdots + 64)^{2} $ (T^6 + 16T^4 + 68T^2 + 64)^2
$7$	$ T^{12} $ T^12
$11$	$ (T^{6} + 44 T^{4} + \cdots + 368)^{2} $ (T^6 + 44T^4 + 260T^2 + 368)^2
$13$	$ (T^{6} + 40 T^{4} + \cdots + 256)^{2} $ (T^6 + 40T^4 + 272T^2 + 256)^2
$17$	$ (T^{6} - 80 T^{4} + \cdots - 5888)^{2} $ (T^6 - 80T^4 + 1604T^2 - 5888)^2
$19$	$ (T^{6} + 56 T^{4} + \cdots + 256)^{2} $ (T^6 + 56T^4 + 784T^2 + 256)^2
$23$	$ (T^{3} - 2 T^{2} - 14 T + 32)^{4} $ (T^3 - 2T^2 - 14T + 32)^4
$29$	$ (T^{6} + 88 T^{4} + \cdots + 5888)^{2} $ (T^6 + 88T^4 + 2064T^2 + 5888)^2
$31$	$ (T^{6} - 152 T^{4} + \cdots - 94208)^{2} $ (T^6 - 152T^4 + 6800T^2 - 94208)^2
$37$	$ (T^{6} + 88 T^{4} + \cdots + 5888)^{2} $ (T^6 + 88T^4 + 2064T^2 + 5888)^2
$41$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9956T^2 - 94208)^2
$43$	$ (T^{6} + 148 T^{4} + \cdots + 1472)^{2} $ (T^6 + 148T^4 + 1072T^2 + 1472)^2
$47$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9792T^2 - 94208)^2
$53$	$ (T^{6} + 168 T^{4} + \cdots + 5888)^{2} $ (T^6 + 168T^4 + 2192T^2 + 5888)^2
$59$	$ (T^{6} + 272 T^{4} + \cdots + 369664)^{2} $ (T^6 + 272T^4 + 18496T^2 + 369664)^2
$61$	$ (T^{6} + 216 T^{4} + \cdots + 135424)^{2} $ (T^6 + 216T^4 + 12176T^2 + 135424)^2
$67$	$ (T^{6} + 100 T^{4} + \cdots + 1472)^{2} $ (T^6 + 100T^4 + 1264T^2 + 1472)^2
$71$	$ (T^{3} + 2 T^{2} + \cdots - 176)^{4} $ (T^3 + 2T^2 - 54T - 176)^4
$73$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9792T^2 - 94208)^2
$79$	$ (T^{3} - 4 T^{2} - 24 T + 64)^{4} $ (T^3 - 4T^2 - 24T + 64)^4
$83$	$ (T^{6} + 208 T^{4} + \cdots + 123904)^{2} $ (T^6 + 208T^4 + 11840T^2 + 123904)^2
$89$	$ (T^{6} - 544 T^{4} + \cdots - 1701632)^{2} $ (T^6 - 544T^4 + 80356T^2 - 1701632)^2
$97$	$ (T^{6} - 192 T^{4} + \cdots - 94208)^{2} $ (T^6 - 192T^4 + 9792T^2 - 94208)^2
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{7} q^{3} + ( - \beta_{10} + \beta_{7}) q^{5} - q^{9}+O(q^{10}) \) q - b7 * q^3 + (-b10 + b7) * q^5 - q^9	\( q - \beta_{7} q^{3} + ( - \beta_{10} + \beta_{7}) q^{5} - q^{9} - \beta_{11} q^{11} + (\beta_{10} + \beta_{8} - 2 \beta_{7}) q^{13} + (\beta_1 + 1) q^{15} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{17} + (\beta_{10} - \beta_{8}) q^{19} + (\beta_{4} + 1) q^{23} + ( - \beta_{4} - \beta_1 - 1) q^{25} + \beta_{7} q^{27} + (\beta_{11} - \beta_{6}) q^{29} + ( - \beta_{5} - \beta_{2}) q^{31} - \beta_{5} q^{33} + (\beta_{11} - \beta_{6}) q^{37} + ( - \beta_{4} - \beta_1 - 2) q^{39} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{41} + (\beta_{9} - \beta_{6}) q^{43} + (\beta_{10} - \beta_{7}) q^{45} + 2 \beta_{3} q^{47} + (\beta_{11} - \beta_{9} - \beta_{6}) q^{51} + ( - \beta_{11} + 2 \beta_{9} + \beta_{6}) q^{53} + (3 \beta_{5} - \beta_{2}) q^{55} + (\beta_{4} - \beta_1) q^{57} + ( - 4 \beta_{10} - 2 \beta_{8} + 2 \beta_{7}) q^{59} + ( - 3 \beta_{10} - \beta_{8} - 4 \beta_{7}) q^{61} + (2 \beta_{4} + 4 \beta_1 + 6) q^{65} + (\beta_{9} + \beta_{6}) q^{67} + (\beta_{8} - \beta_{7}) q^{69} + (\beta_{4} - 2 \beta_1 - 1) q^{71} + 2 \beta_{3} q^{73} + ( - \beta_{10} - \beta_{8} + \beta_{7}) q^{75} + (2 \beta_1 + 2) q^{79} + q^{81} + ( - 2 \beta_{8} + 6 \beta_{7}) q^{83} + ( - 3 \beta_{11} + 2 \beta_{9} + \beta_{6}) q^{85} + (\beta_{5} - \beta_{2}) q^{87} + ( - 3 \beta_{5} + \beta_{3} - \beta_{2}) q^{89} + (\beta_{11} + \beta_{6}) q^{93} + ( - 2 \beta_1 + 6) q^{95} - 2 \beta_{3} q^{97} + \beta_{11} q^{99}+O(q^{100}) \) q - b7 * q^3 + (-b10 + b7) * q^5 - q^9 - b11 * q^11 + (b10 + b8 - 2b7) q^13 + (b1 + 1) * q^15 + (-b5 + b3 + b2) * q^17 + (b10 - b8) * q^19 + (b4 + 1) * q^23 + (-b4 - b1 - 1) * q^25 + b7 * q^27 + (b11 - b6) * q^29 + (-b5 - b2) * q^31 - b5 * q^33 + (b11 - b6) * q^37 + (-b4 - b1 - 2) * q^39 + (-b5 - b3 + b2) * q^41 + (b9 - b6) * q^43 + (b10 - b7) * q^45 + 2b3 q^47 + (b11 - b9 - b6) * q^51 + (-b11 + 2b9 + b6) q^53 + (3b5 - b2) q^55 + (b4 - b1) * q^57 + (-4b10 - 2b8 + 2b7) q^59 + (-3b10 - b8 - 4b7) * q^61 + (2b4 + 4b1 + 6) * q^65 + (b9 + b6) * q^67 + (b8 - b7) * q^69 + (b4 - 2b1 - 1) q^71 + 2b3 q^73 + (-b10 - b8 + b7) * q^75 + (2b1 + 2) q^79 + q^81 + (-2b8 + 6b7) * q^83 + (-3b11 + 2b9 + b6) * q^85 + (b5 - b2) * q^87 + (-3b5 + b3 - b2) q^89 + (b11 + b6) * q^93 + (-2b1 + 6) q^95 - 2b3 q^97 + b11 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 8 q^{15} + 8 q^{23} - 4 q^{25} - 16 q^{39} + 48 q^{65} - 8 q^{71} + 16 q^{79} + 12 q^{81} + 80 q^{95}+O(q^{100}) \) 12 * q - 12 * q^9 + 8 * q^15 + 8 * q^23 - 4 * q^25 - 16 * q^39 + 48 * q^65 - 8 * q^71 + 16 * q^79 + 12 * q^81 + 80 * q^95

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	4704.2.c.d

Level	$4704$
Weight	$2$
Character orbit	4704.c
Analytic conductor	$37.562$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(37.5616291108\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.84396412309504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 39 x^{8} - 28 x^{7} + 13 x^{6} + 10 x^{5} + 6 x^{4} + \cdots + 2 \) x^12 - 6x^11 + 18x^10 - 32x^9 + 39x^8 - 28x^7 + 13x^6 + 10x^5 + 6x^4 + 14x^3 + 17x^2 + 10*x + 2
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	no (minimal twist has level 1176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 5 \nu^{11} - 30 \nu^{10} + 91 \nu^{9} - 166 \nu^{8} + 210 \nu^{7} - 154 \nu^{6} + 59 \nu^{5} + \cdots + 48 ) / 16 \) (5v^11 - 30v^10 + 91v^9 - 166v^8 + 210v^7 - 154v^6 + 59v^5 + 80v^4 - 3v^3 + 102v^2 + 78*v + 48) / 16
\(\beta_{2}\)	\(=\)	\( ( - 5 \nu^{11} + 32 \nu^{10} - 107 \nu^{9} + 224 \nu^{8} - 326 \nu^{7} + 286 \nu^{6} - 119 \nu^{5} + \cdots - 12 ) / 16 \) (-5v^11 + 32v^10 - 107v^9 + 224v^8 - 326v^7 + 286v^6 - 119v^5 - 114v^4 + 71v^3 - 120v^2 - 114*v - 12) / 16
\(\beta_{3}\)	\(=\)	\( ( - 16 \nu^{11} + 103 \nu^{10} - 330 \nu^{9} + 633 \nu^{8} - 818 \nu^{7} + 630 \nu^{6} - 238 \nu^{5} + \cdots - 70 ) / 16 \) (-16v^11 + 103v^10 - 330v^9 + 633v^8 - 818v^7 + 630v^6 - 238v^5 - 279v^4 + 128v^3 - 241v^2 - 222*v - 70) / 16
\(\beta_{4}\)	\(=\)	\( ( 19 \nu^{11} - 118 \nu^{10} + 361 \nu^{9} - 650 \nu^{8} + 782 \nu^{7} - 542 \nu^{6} + 189 \nu^{5} + \cdots + 64 ) / 16 \) (19v^11 - 118v^10 + 361v^9 - 650v^8 + 782v^7 - 542v^6 + 189v^5 + 276v^4 - 41v^3 + 274v^2 + 218*v + 64) / 16
\(\beta_{5}\)	\(=\)	\( ( - 21 \nu^{11} + 130 \nu^{10} - 403 \nu^{9} + 746 \nu^{8} - 938 \nu^{7} + 690 \nu^{6} - 259 \nu^{5} + \cdots - 152 ) / 16 \) (-21v^11 + 130v^10 - 403v^9 + 746v^8 - 938v^7 + 690v^6 - 259v^5 - 340v^4 + 67v^3 - 362v^2 - 294*v - 152) / 16
\(\beta_{6}\)	\(=\)	\( ( 24 \nu^{11} - 151 \nu^{10} + 478 \nu^{9} - 917 \nu^{8} + 1226 \nu^{7} - 1062 \nu^{6} + 670 \nu^{5} + \cdots + 158 ) / 16 \) (24v^11 - 151v^10 + 478v^9 - 917v^8 + 1226v^7 - 1062v^6 + 670v^5 - 9v^4 + 204v^3 + 261v^2 + 414*v + 158) / 16
\(\beta_{7}\)	\(=\)	\( ( 85 \nu^{11} - 545 \nu^{10} + 1753 \nu^{9} - 3431 \nu^{8} + 4692 \nu^{7} - 4248 \nu^{6} + 2793 \nu^{5} + \cdots + 422 ) / 32 \) (85v^11 - 545v^10 + 1753v^9 - 3431v^8 + 4692v^7 - 4248v^6 + 2793v^5 - 285v^4 + 657v^3 + 887v^2 + 1092*v + 422) / 32
\(\beta_{8}\)	\(=\)	\( ( - 129 \nu^{11} + 835 \nu^{10} - 2721 \nu^{9} + 5437 \nu^{8} - 7656 \nu^{7} + 7284 \nu^{6} - 5089 \nu^{5} + \cdots - 570 ) / 32 \) (-129v^11 + 835v^10 - 2721v^9 + 5437v^8 - 7656v^7 + 7284v^6 - 5089v^5 + 931v^4 - 965v^3 - 1549v^2 - 1504*v - 570) / 32
\(\beta_{9}\)	\(=\)	\( ( - 76 \nu^{11} + 497 \nu^{10} - 1630 \nu^{9} + 3267 \nu^{8} - 4574 \nu^{7} + 4290 \nu^{6} - 2910 \nu^{5} + \cdots - 322 ) / 16 \) (-76v^11 + 497v^10 - 1630v^9 + 3267v^8 - 4574v^7 + 4290v^6 - 2910v^5 + 487v^4 - 536v^3 - 819v^2 - 826*v - 322) / 16
\(\beta_{10}\)	\(=\)	\( ( - 181 \nu^{11} + 1159 \nu^{10} - 3725 \nu^{9} + 7297 \nu^{8} - 10024 \nu^{7} + 9172 \nu^{6} + \cdots - 866 ) / 32 \) (-181v^11 + 1159v^10 - 3725v^9 + 7297v^8 - 10024v^7 + 9172v^6 - 6133v^5 + 727v^4 - 1409v^3 - 1953v^2 - 2256*v - 866) / 32
\(\beta_{11}\)	\(=\)	\( ( - 262 \nu^{11} + 1683 \nu^{10} - 5428 \nu^{9} + 10677 \nu^{8} - 14718 \nu^{7} + 13522 \nu^{6} + \cdots - 1238 ) / 16 \) (-262v^11 + 1683v^10 - 5428v^9 + 10677v^8 - 14718v^7 + 13522v^6 - 9064v^5 + 1153v^4 - 2006v^3 - 2829v^2 - 3218*v - 1238) / 16

\(\nu\)	\(=\)	\( ( \beta_{11} - \beta_{10} - \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 4 ) / 8 \) (b11 - b10 - b8 + 2*b7 + b6 + b5 + b4 - b2 - b1 + 4) / 8
\(\nu^{2}\)	\(=\)	\( ( 3 \beta_{11} - 5 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - \beta_1 ) / 8 \) (3b11 - 5b10 - 2b9 - b8 + 2b7 + b6 - b5 + b4 + 2*b3 - b2 - b1) / 8
\(\nu^{3}\)	\(=\)	\( ( 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots - 5 ) / 4 \) (2b11 - 6b10 - 2b9 + b8 - 3b7 + b6 - 2b5 - b4 + b3 - 2b1 - 5) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{11} - 17 \beta_{10} - 6 \beta_{9} + 7 \beta_{8} - 34 \beta_{7} + 7 \beta_{6} - 7 \beta_{5} + \cdots - 28 ) / 8 \) (b11 - 17b10 - 6b9 + 7b8 - 34b7 + 7b6 - 7b5 - 7b4 - 2b3 + 5b2 - 5b1 - 28) / 8
\(\nu^{5}\)	\(=\)	\( ( - 9 \beta_{11} - 11 \beta_{10} - 8 \beta_{9} + 15 \beta_{8} - 80 \beta_{7} + 17 \beta_{6} - 3 \beta_{5} + \cdots - 62 ) / 8 \) (-9b11 - 11b10 - 8b9 + 15b8 - 80b7 + 17b6 - 3b5 - 13b4 - 12b3 + 17b2 + 15*b1 - 62) / 8
\(\nu^{6}\)	\(=\)	\( ( - 12 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 11 \beta_{8} - 54 \beta_{7} + 11 \beta_{6} + 11 \beta_{5} + \cdots - 58 ) / 4 \) (-12b11 + 7b10 - 3b9 + 11b8 - 54b7 + 11b6 + 11b5 - 10b4 - 18b3 + 19b2 + 46*b1 - 58) / 4
\(\nu^{7}\)	\(=\)	\( ( - 31 \beta_{11} + 71 \beta_{10} + 18 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 81 \beta_{5} + \cdots - 192 ) / 8 \) (-31b11 + 71b10 + 18b9 + 3b8 + 2b7 - 9b6 + 81b5 - 31b4 - 86b3 + 73b2 + 255*b1 - 192) / 8
\(\nu^{8}\)	\(=\)	\( ( - 9 \beta_{11} + 219 \beta_{10} + 106 \beta_{9} - 101 \beta_{8} + 526 \beta_{7} - 139 \beta_{6} + \cdots - 308 ) / 8 \) (-9b11 + 219b10 + 106b9 - 101b8 + 526b7 - 139b6 + 155b5 - 51b4 - 158b3 + 127b2 + 463*b1 - 308) / 8
\(\nu^{9}\)	\(=\)	\( ( 17 \beta_{11} + 323 \beta_{10} + 168 \beta_{9} - 186 \beta_{8} + 943 \beta_{7} - 234 \beta_{6} + \cdots - 225 ) / 4 \) (17b11 + 323b10 + 168b9 - 186b8 + 943b7 - 234b6 + 71b5 - 40b4 - 91b3 + 83b2 + 241*b1 - 225) / 4
\(\nu^{10}\)	\(=\)	\( ( - 25 \beta_{11} + 1737 \beta_{10} + 830 \beta_{9} - 851 \beta_{8} + 4358 \beta_{7} - 1099 \beta_{6} + \cdots - 216 ) / 8 \) (-25b11 + 1737b10 + 830b9 - 851b8 + 4358b7 - 1099b6 - 177b5 - 53b4 + 70b3 + 3b2 - 315*b1 - 216) / 8
\(\nu^{11}\)	\(=\)	\( ( - 489 \beta_{11} + 3897 \beta_{10} + 1672 \beta_{9} - 1389 \beta_{8} + 7288 \beta_{7} - 1983 \beta_{6} + \cdots + 2194 ) / 8 \) (-489b11 + 3897b10 + 1672b9 - 1389b8 + 7288b7 - 1983b6 - 1075b5 + 359b4 + 1124b3 - 927b2 - 3205*b1 + 2194) / 8

\(n\)	\(1471\)	\(1765\)	\(3137\)	\(4609\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2353.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$5$	\( (T^{6} + 16 T^{4} + \cdots + 64)^{2} \) (T^6 + 16T^4 + 68T^2 + 64)^2
$7$	\( T^{12} \) T^12
$11$	\( (T^{6} + 44 T^{4} + \cdots + 368)^{2} \) (T^6 + 44T^4 + 260T^2 + 368)^2
$13$	\( (T^{6} + 40 T^{4} + \cdots + 256)^{2} \) (T^6 + 40T^4 + 272T^2 + 256)^2
$17$	\( (T^{6} - 80 T^{4} + \cdots - 5888)^{2} \) (T^6 - 80T^4 + 1604T^2 - 5888)^2
$19$	\( (T^{6} + 56 T^{4} + \cdots + 256)^{2} \) (T^6 + 56T^4 + 784T^2 + 256)^2
$23$	\( (T^{3} - 2 T^{2} - 14 T + 32)^{4} \) (T^3 - 2T^2 - 14T + 32)^4
$29$	\( (T^{6} + 88 T^{4} + \cdots + 5888)^{2} \) (T^6 + 88T^4 + 2064T^2 + 5888)^2
$31$	\( (T^{6} - 152 T^{4} + \cdots - 94208)^{2} \) (T^6 - 152T^4 + 6800T^2 - 94208)^2
$37$	\( (T^{6} + 88 T^{4} + \cdots + 5888)^{2} \) (T^6 + 88T^4 + 2064T^2 + 5888)^2
$41$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9956T^2 - 94208)^2
$43$	\( (T^{6} + 148 T^{4} + \cdots + 1472)^{2} \) (T^6 + 148T^4 + 1072T^2 + 1472)^2
$47$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9792T^2 - 94208)^2
$53$	\( (T^{6} + 168 T^{4} + \cdots + 5888)^{2} \) (T^6 + 168T^4 + 2192T^2 + 5888)^2
$59$	\( (T^{6} + 272 T^{4} + \cdots + 369664)^{2} \) (T^6 + 272T^4 + 18496T^2 + 369664)^2
$61$	\( (T^{6} + 216 T^{4} + \cdots + 135424)^{2} \) (T^6 + 216T^4 + 12176T^2 + 135424)^2
$67$	\( (T^{6} + 100 T^{4} + \cdots + 1472)^{2} \) (T^6 + 100T^4 + 1264T^2 + 1472)^2
$71$	\( (T^{3} + 2 T^{2} + \cdots - 176)^{4} \) (T^3 + 2T^2 - 54T - 176)^4
$73$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9792T^2 - 94208)^2
$79$	\( (T^{3} - 4 T^{2} - 24 T + 64)^{4} \) (T^3 - 4T^2 - 24T + 64)^4
$83$	\( (T^{6} + 208 T^{4} + \cdots + 123904)^{2} \) (T^6 + 208T^4 + 11840T^2 + 123904)^2
$89$	\( (T^{6} - 544 T^{4} + \cdots - 1701632)^{2} \) (T^6 - 544T^4 + 80356T^2 - 1701632)^2
$97$	\( (T^{6} - 192 T^{4} + \cdots - 94208)^{2} \) (T^6 - 192T^4 + 9792T^2 - 94208)^2
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