LMFDB - Newform orbit 444.2.g.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [444,2,Mod(443,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.443"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 444 = 2^{2} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	444.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$3.54535784974$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 9x^{12} + 43x^{8} - 144x^{4} + 256 $ x^16 - 9x^12 + 43x^8 - 144*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{2} - \beta_{4} q^{3} - \beta_{8} q^{4} - \beta_{12} q^{5} + \beta_{6} q^{6} + ( - \beta_{14} + \beta_{11}) q^{7} + \beta_{5} q^{8} - 3 q^{9} + ( - \beta_{14} + \beta_{4} + \beta_{3}) q^{10}+ \cdots + ( - 4 \beta_{13} - 4 \beta_{12} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) $ q - b1 * q^2 - b4 * q^3 - b8 * q^4 - b12 * q^5 + b6 * q^6 + (-b14 + b11) * q^7 + b5 * q^8 - 3 * q^9 + (-b14 + b4 + b3) * q^10 + (-b14 + b11 - b7) * q^12 + (-b13 + b12 - b5 + b2) * q^14 + (b15 - b9 - b2 + b1) * q^15 + (b4 - b3) * q^16 + (-b15 + b9 - b2 + b1) * q^17 + 3b1 q^18 + (-b13 - b12 + b9 - b6 + b2) * q^20 + (-b14 - b11 + 2b8) q^21 + (b12 + 2b6 - 2b5) * q^23 + (-b9 + b2) * q^24 + (-b14 + b11 - 2b8 - 2b7 + 5) * q^25 + 3b4 q^27 + (-b10 - 3b4 - b3) q^28 + (-b15 - b9 - b2 - 3b1) q^29 + (-b11 + b10 + b8 - b7 + 2) * q^30 + (b13 + b12 - b6) * q^32 + (b11 + b10 + b8 + b7 - 2) * q^34 + (b15 + 2b13 - b12 + b9 - 2b6 + 2b5 - b2 - 3b1) * q^35 + 3b8 q^36 + (-b4 - 2b3) q^37 + (-b14 - b11 - b10 + b7 - 4) * q^40 + (b13 - b12 + 2b9 - b5 + b2) q^42 + 3b12 q^45 + (-b14 + 2b11 - 2b7 - 3b4 + b3) q^46 + (-b10 + 4) * q^48 + (2b4 + 4b3 - 7) * q^49 + (b13 - b12 - 2b9 + 3b5 + b2 - 5b1) q^50 + (-2b13 - b12 - 2b6 + 2b5) q^51 - 3b6 q^54 + (-2b15 + b13 + b12 + b9 + 3b6 - b1) * q^56 + (b11 + b10 - 3b8 + b7 + 6) q^58 + (b15 - 3b9 - b2 + 5b1) * q^59 + (2b15 + 2b13 - 2b12 - b9 + b5 - b1) q^60 + (3b14 - 3b11) * q^63 + (3b14 - b11 + b7) q^64 + (-b14 - b11 - 4b8 + 2b7) * q^67 + (2b15 - 2b13 + 2b12 - b9 - 3b5 + 3b1) q^68 + (-b15 + 3b9 - b2 + 5b1) * q^69 + (3b14 - 3b11 + b10 - 3b8 + b7 + 5b4 + b3 - 6) * q^70 - 3b5 q^72 + (3b14 + b11 - 2b8 + 2b7) q^73 + (2b13 + 2b12 + b6) * q^74 + (-b14 + 3b11 + 4b8 - 2b7 - 5b4) * q^75 + (-2b15 + 4b9 + b2 + 3b1) q^80 + 9 * q^81 + (-b10 + b4 + 3b3 - 8) q^84 + (3b14 - b11 + 2b8 + 4b7 - 2b4 - 4b3) q^85 + (-4b13 + b12 + 2b6 - 2b5) q^87 + (-2b13 + b12 - 4b6 - 4b5) q^89 + (3b14 - 3b4 - 3b3) q^90 + (-b13 - b12 - 3b9 + 3b6 + b2) * q^92 + (-2b15 + b9 - 5b1) * q^96 + (-4b13 - 4b12 - 2b6 + 7b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} + 80 q^{25} + 8 q^{28} + 24 q^{30} - 40 q^{34} - 56 q^{40} + 72 q^{48} - 112 q^{49} + 88 q^{58} - 104 q^{70} + 144 q^{81} - 120 q^{84}+O(q^{100}) $ 16 * q - 48 * q^9 + 80 * q^25 + 8 * q^28 + 24 * q^30 - 40 * q^34 - 56 * q^40 + 72 * q^48 - 112 * q^49 + 88 * q^58 - 104 * q^70 + 144 * q^81 - 120 * q^84

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 9x^{12} + 43x^{8} - 144x^{4} + 256 $ x^16 - 9*x^12 + 43*x^8 - 144*x^4 + 256 :

$\beta_{1}$	$=$	$ ( \nu^{15} + 15\nu^{11} - 45\nu^{7} + 120\nu^{3} ) / 448 $ (v^15 + 15v^11 - 45v^7 + 120*v^3) / 448
$\beta_{2}$	$=$	$ ( -\nu^{13} + 41\nu^{9} - 11\nu^{5} + 48\nu ) / 224 $ (-v^13 + 41v^9 - 11v^5 + 48*v) / 224
$\beta_{3}$	$=$	$ ( -\nu^{12} + 13\nu^{8} - 39\nu^{4} + 104 ) / 28 $ (-v^12 + 13v^8 - 39v^4 + 104) / 28
$\beta_{4}$	$=$	$ ( -3\nu^{12} + 11\nu^{8} - 33\nu^{4} + 88 ) / 56 $ (-3v^12 + 11v^8 - 33*v^4 + 88) / 56
$\beta_{5}$	$=$	$ ( -\nu^{13} + 9\nu^{9} - 43\nu^{5} + 144\nu ) / 32 $ (-v^13 + 9v^9 - 43v^5 + 144*v) / 32
$\beta_{6}$	$=$	$ ( -\nu^{15} + 6\nu^{11} - 32\nu^{7} + 111\nu^{3} ) / 56 $ (-v^15 + 6v^11 - 32v^7 + 111*v^3) / 56
$\beta_{7}$	$=$	$ ( 9\nu^{14} - 33\nu^{10} + 211\nu^{6} + 128\nu^{2} ) / 448 $ (9v^14 - 33v^10 + 211v^6 + 128v^2) / 448
$\beta_{8}$	$=$	$ ( -9\nu^{14} + 33\nu^{10} - 211\nu^{6} + 768\nu^{2} ) / 448 $ (-9v^14 + 33v^10 - 211v^6 + 768v^2) / 448
$\beta_{9}$	$=$	$ ( -3\nu^{13} + 11\nu^{9} - 33\nu^{5} + 144\nu ) / 56 $ (-3v^13 + 11v^9 - 33v^5 + 144v) / 56
$\beta_{10}$	$=$	$ ( \nu^{12} - 6\nu^{8} + 32\nu^{4} - 83 ) / 7 $ (v^12 - 6v^8 + 32v^4 - 83) / 7
$\beta_{11}$	$=$	$ ( -15\nu^{14} + 55\nu^{10} - 53\nu^{6} + 384\nu^{2} ) / 448 $ (-15v^14 + 55v^10 - 53v^6 + 384v^2) / 448
$\beta_{12}$	$=$	$ ( 4\nu^{15} - 13\nu^{13} - 24\nu^{11} + 85\nu^{9} + 128\nu^{7} - 367\nu^{5} - 220\nu^{3} + 848\nu ) / 224 $ (4v^15 - 13v^13 - 24v^11 + 85v^9 + 128v^7 - 367v^5 - 220v^3 + 848v) / 224
$\beta_{13}$	$=$	$ ( 4\nu^{15} + 13\nu^{13} - 24\nu^{11} - 85\nu^{9} + 128\nu^{7} + 367\nu^{5} - 220\nu^{3} - 848\nu ) / 224 $ (4v^15 + 13v^13 - 24v^11 - 85v^9 + 128v^7 + 367v^5 - 220v^3 - 848v) / 224
$\beta_{14}$	$=$	$ ( -29\nu^{14} + 181\nu^{10} - 655\nu^{6} + 1504\nu^{2} ) / 448 $ (-29v^14 + 181v^10 - 655v^6 + 1504v^2) / 448
$\beta_{15}$	$=$	$ ( -11\nu^{15} - 6\nu^{13} + 59\nu^{11} + 22\nu^{9} - 177\nu^{7} - 66\nu^{5} + 472\nu^{3} + 288\nu ) / 224 $ (-11v^15 - 6v^13 + 59v^11 + 22v^9 - 177v^7 - 66v^5 + 472v^3 + 288v) / 224

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

$\nu$	$=$	$ ( \beta_{13} - \beta_{12} + \beta_{9} + 2\beta_{5} ) / 4 $ (b13 - b12 + b9 + 2*b5) / 4
$\nu^{2}$	$=$	$ ( \beta_{8} + \beta_{7} ) / 2 $ (b8 + b7) / 2
$\nu^{3}$	$=$	$ ( \beta_{13} + \beta_{12} + 2\beta_{6} ) / 2 $ (b13 + b12 + 2*b6) / 2
$\nu^{4}$	$=$	$ ( \beta_{10} + 2\beta_{4} + \beta_{3} + 5 ) / 2 $ (b10 + 2*b4 + b3 + 5) / 2
$\nu^{5}$	$=$	$ ( 3\beta_{13} - 3\beta_{12} + 5\beta_{9} + 2\beta_{5} + 4\beta_{2} ) / 4 $ (3b13 - 3b12 + 5b9 + 2b5 + 4*b2) / 4
$\nu^{6}$	$=$	$ ( 3\beta_{11} - 2\beta_{8} + 3\beta_{7} ) / 2 $ (3b11 - 2b8 + 3*b7) / 2
$\nu^{7}$	$=$	$ ( 6\beta_{15} + 9\beta_{13} + 9\beta_{12} - 3\beta_{9} + 2\beta_{6} + 4\beta_1 ) / 4 $ (6b15 + 9b13 + 9b12 - 3b9 + 2b6 + 4b1) / 4
$\nu^{8}$	$=$	$ ( 3\beta_{10} + 2\beta_{4} + 9\beta_{3} - 1 ) / 2 $ (3b10 + 2b4 + 9*b3 - 1) / 2
$\nu^{9}$	$=$	$ ( -\beta_{9} + 12\beta_{2} ) / 2 $ (-b9 + 12*b2) / 2
$\nu^{10}$	$=$	$ ( 12\beta_{14} - \beta_{11} - 25\beta_{8} + 12\beta_{7} ) / 2 $ (12b14 - b11 - 25b8 + 12*b7) / 2
$\nu^{11}$	$=$	$ ( 22\beta_{15} + 11\beta_{13} + 11\beta_{12} - 11\beta_{9} - 26\beta_{6} + 100\beta_1 ) / 4 $ (22b15 + 11b13 + 11b12 - 11b9 - 26b6 + 100b1) / 4
$\nu^{12}$	$=$	$ -26\beta_{4} + 11\beta_{3} $ -26b4 + 11b3
$\nu^{13}$	$=$	$ ( 15\beta_{13} - 15\beta_{12} - 89\beta_{9} + 74\beta_{5} + 44\beta_{2} ) / 4 $ (15b13 - 15b12 - 89b9 + 74b5 + 44*b2) / 4
$\nu^{14}$	$=$	$ ( 44\beta_{14} - 74\beta_{11} - 59\beta_{8} + 59\beta_{7} ) / 2 $ (44b14 - 74b11 - 59b8 + 59b7) / 2
$\nu^{15}$	$=$	$ ( -30\beta_{15} + 15\beta_{9} + 236\beta_1 ) / 2 $ (-30b15 + 15b9 + 236*b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/444\mathbb{Z}\right)^\times$.

$n$	$149$	$223$	$409$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

443.1

0.480758	−	1.32999i
0.480758	+	1.32999i
−0.120150	−	1.40910i
−0.120150	+	1.40910i
1.40910	+	0.120150i
1.40910	−	0.120150i
1.32999	−	0.480758i
1.32999	+	0.480758i
−1.32999	−	0.480758i
−1.32999	+	0.480758i
−1.40910	+	0.120150i
−1.40910	−	0.120150i
0.120150	−	1.40910i
0.120150	+	1.40910i
−0.480758	−	1.32999i
−0.480758	+	1.32999i

−1.39218

−

0.248646i

−

1.73205i

1.87635

0.692322i

4.01883

−0.430667

2.41133i

−

5.11522i

−2.44008

−

1.43039i

−3.00000

−5.59495

−

0.999265i

443.2

−1.39218

0.248646i

1.73205i

1.87635

−

0.692322i

4.01883

−0.430667

−

2.41133i

5.11522i

−2.44008

1.43039i

−3.00000

−5.59495

0.999265i

443.3

−1.16024

−

0.808603i

−

1.73205i

0.692322

1.87635i

−4.22901

−1.40054

2.00960i

1.35443i

0.713962

−

2.73683i

−3.00000

4.90667

3.41959i

443.4

−1.16024

0.808603i

1.73205i

0.692322

−

1.87635i

−4.22901

−1.40054

−

2.00960i

−

1.35443i

0.713962

2.73683i

−3.00000

4.90667

−

3.41959i

443.5

−0.808603

−

1.16024i

1.73205i

−0.692322

1.87635i

−1.45447

2.00960

−

1.40054i

1.35443i

2.73683

−

0.713962i

−3.00000

1.17609

1.68754i

443.6

−0.808603

1.16024i

−

1.73205i

−0.692322

−

1.87635i

−1.45447

2.00960

1.40054i

−

1.35443i

2.73683

0.713962i

−3.00000

1.17609

−

1.68754i

443.7

−0.248646

−

1.39218i

1.73205i

−1.87635

0.692322i

1.96189

2.41133

−

0.430667i

−

5.11522i

1.43039

2.44008i

−3.00000

−0.487817

−

2.73132i

443.8

−0.248646

1.39218i

−

1.73205i

−1.87635

−

0.692322i

1.96189

2.41133

0.430667i

5.11522i

1.43039

−

2.44008i

−3.00000

−0.487817

2.73132i

443.9

0.248646

−

1.39218i

−

1.73205i

−1.87635

−

0.692322i

−1.96189

−2.41133

−

0.430667i

5.11522i

−1.43039

2.44008i

−3.00000

−0.487817

2.73132i

443.10

0.248646

1.39218i

1.73205i

−1.87635

0.692322i

−1.96189

−2.41133

0.430667i

−

5.11522i

−1.43039

−

2.44008i

−3.00000

−0.487817

−

2.73132i

443.11

0.808603

−

1.16024i

−

1.73205i

−0.692322

−

1.87635i

1.45447

−2.00960

−

1.40054i

−

1.35443i

−2.73683

−

0.713962i

−3.00000

1.17609

−

1.68754i

443.12

0.808603

1.16024i

1.73205i

−0.692322

1.87635i

1.45447

−2.00960

1.40054i

1.35443i

−2.73683

0.713962i

−3.00000

1.17609

1.68754i

443.13

1.16024

−

0.808603i

1.73205i

0.692322

−

1.87635i

4.22901

1.40054

2.00960i

−

1.35443i

−0.713962

−

2.73683i

−3.00000

4.90667

−

3.41959i

443.14

1.16024

0.808603i

−

1.73205i

0.692322

1.87635i

4.22901

1.40054

−

2.00960i

1.35443i

−0.713962

2.73683i

−3.00000

4.90667

3.41959i

443.15

1.39218

−

0.248646i

1.73205i

1.87635

−

0.692322i

−4.01883

0.430667

2.41133i

5.11522i

2.44008

−

1.43039i

−3.00000

−5.59495

0.999265i

443.16

1.39218

0.248646i

−

1.73205i

1.87635

0.692322i

−4.01883

0.430667

−

2.41133i

−

5.11522i

2.44008

1.43039i

−3.00000

−5.59495

−

0.999265i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
111.d	odd	2	1	CM by $\Q(\sqrt{-111}) $
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
37.b	even	2	1	inner
148.b	odd	2	1	inner
444.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	444.2.g.c	✓	16
3.b	odd	2	1	inner	444.2.g.c	✓	16
4.b	odd	2	1	inner	444.2.g.c	✓	16
12.b	even	2	1	inner	444.2.g.c	✓	16
37.b	even	2	1	inner	444.2.g.c	✓	16
111.d	odd	2	1	CM	444.2.g.c	✓	16
148.b	odd	2	1	inner	444.2.g.c	✓	16
444.g	even	2	1	inner	444.2.g.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
444.2.g.c	✓	16	1.a	even	1	1	trivial
444.2.g.c	✓	16	3.b	odd	2	1	inner
444.2.g.c	✓	16	4.b	odd	2	1	inner
444.2.g.c	✓	16	12.b	even	2	1	inner
444.2.g.c	✓	16	37.b	even	2	1	inner
444.2.g.c	✓	16	111.d	odd	2	1	CM
444.2.g.c	✓	16	148.b	odd	2	1	inner
444.2.g.c	✓	16	444.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} - 40T_{5}^{6} + 500T_{5}^{4} - 2000T_{5}^{2} + 2352 $ T5^8 - 40*T5^6 + 500*T5^4 - 2000*T5^2 + 2352 acting on $S_{2}^{\mathrm{new}}(444, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 5T^{8} + 256 $ T^16 - 5*T^8 + 256
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{8} - 40 T^{6} + \cdots + 2352)^{2} $ (T^8 - 40T^6 + 500T^4 - 2000*T^2 + 2352)^2
$7$	$ (T^{4} + 28 T^{2} + 48)^{4} $ (T^4 + 28*T^2 + 48)^4
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} - 136 T^{6} + \cdots + 255792)^{2} $ (T^8 - 136T^6 + 5780T^4 - 78608*T^2 + 255792)^2
$19$	$ T^{16} $ T^16
$23$	$ (T^{8} + 184 T^{6} + \cdots + 330672)^{2} $ (T^8 + 184T^6 + 10580T^4 + 194672*T^2 + 330672)^2
$29$	$ (T^{8} - 232 T^{6} + \cdots + 1436592)^{2} $ (T^8 - 232T^6 + 16820T^4 - 390224*T^2 + 1436592)^2
$31$	$ T^{16} $ T^16
$37$	$ (T^{2} - 37)^{8} $ (T^2 - 37)^8
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ (T^{8} + 472 T^{6} + \cdots + 45442992)^{2} $ (T^8 + 472T^6 + 69620T^4 + 3286064*T^2 + 45442992)^2
$61$	$ T^{16} $ T^16
$67$	$ (T^{4} + 268 T^{2} + 48)^{4} $ (T^4 + 268*T^2 + 48)^4
$71$	$ T^{16} $ T^16
$73$	$ (T^{4} - 292 T^{2} + 21168)^{4} $ (T^4 - 292*T^2 + 21168)^4
$79$	$ T^{16} $ T^16
$83$	$ T^{16} $ T^16
$89$	$ (T^{8} - 712 T^{6} + \cdots + 250180272)^{2} $ (T^8 - 712T^6 + 158420T^4 - 11279504*T^2 + 250180272)^2
$97$	$ T^{16} $ T^16
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	444.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} - \beta_{4} q^{3} - \beta_{8} q^{4} - \beta_{12} q^{5} + \beta_{6} q^{6} + ( - \beta_{14} + \beta_{11}) q^{7} + \beta_{5} q^{8} - 3 q^{9} + ( - \beta_{14} + \beta_{4} + \beta_{3}) q^{10}+ \cdots + ( - 4 \beta_{13} - 4 \beta_{12} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) \) q - b1 * q^2 - b4 * q^3 - b8 * q^4 - b12 * q^5 + b6 * q^6 + (-b14 + b11) * q^7 + b5 * q^8 - 3 * q^9 + (-b14 + b4 + b3) * q^10 + (-b14 + b11 - b7) * q^12 + (-b13 + b12 - b5 + b2) * q^14 + (b15 - b9 - b2 + b1) * q^15 + (b4 - b3) * q^16 + (-b15 + b9 - b2 + b1) * q^17 + 3b1 q^18 + (-b13 - b12 + b9 - b6 + b2) * q^20 + (-b14 - b11 + 2b8) q^21 + (b12 + 2b6 - 2b5) * q^23 + (-b9 + b2) * q^24 + (-b14 + b11 - 2b8 - 2b7 + 5) * q^25 + 3b4 q^27 + (-b10 - 3b4 - b3) q^28 + (-b15 - b9 - b2 - 3b1) q^29 + (-b11 + b10 + b8 - b7 + 2) * q^30 + (b13 + b12 - b6) * q^32 + (b11 + b10 + b8 + b7 - 2) * q^34 + (b15 + 2b13 - b12 + b9 - 2b6 + 2b5 - b2 - 3b1) * q^35 + 3b8 q^36 + (-b4 - 2b3) q^37 + (-b14 - b11 - b10 + b7 - 4) * q^40 + (b13 - b12 + 2b9 - b5 + b2) q^42 + 3b12 q^45 + (-b14 + 2b11 - 2b7 - 3b4 + b3) q^46 + (-b10 + 4) * q^48 + (2b4 + 4b3 - 7) * q^49 + (b13 - b12 - 2b9 + 3b5 + b2 - 5b1) q^50 + (-2b13 - b12 - 2b6 + 2b5) q^51 - 3b6 q^54 + (-2b15 + b13 + b12 + b9 + 3b6 - b1) * q^56 + (b11 + b10 - 3b8 + b7 + 6) q^58 + (b15 - 3b9 - b2 + 5b1) * q^59 + (2b15 + 2b13 - 2b12 - b9 + b5 - b1) q^60 + (3b14 - 3b11) * q^63 + (3b14 - b11 + b7) q^64 + (-b14 - b11 - 4b8 + 2b7) * q^67 + (2b15 - 2b13 + 2b12 - b9 - 3b5 + 3b1) q^68 + (-b15 + 3b9 - b2 + 5b1) * q^69 + (3b14 - 3b11 + b10 - 3b8 + b7 + 5b4 + b3 - 6) * q^70 - 3b5 q^72 + (3b14 + b11 - 2b8 + 2b7) q^73 + (2b13 + 2b12 + b6) * q^74 + (-b14 + 3b11 + 4b8 - 2b7 - 5b4) * q^75 + (-2b15 + 4b9 + b2 + 3b1) q^80 + 9 * q^81 + (-b10 + b4 + 3b3 - 8) q^84 + (3b14 - b11 + 2b8 + 4b7 - 2b4 - 4b3) q^85 + (-4b13 + b12 + 2b6 - 2b5) q^87 + (-2b13 + b12 - 4b6 - 4b5) q^89 + (3b14 - 3b4 - 3b3) q^90 + (-b13 - b12 - 3b9 + 3b6 + b2) * q^92 + (-2b15 + b9 - 5b1) * q^96 + (-4b13 - 4b12 - 2b6 + 7b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 80 q^{25} + 8 q^{28} + 24 q^{30} - 40 q^{34} - 56 q^{40} + 72 q^{48} - 112 q^{49} + 88 q^{58} - 104 q^{70} + 144 q^{81} - 120 q^{84}+O(q^{100}) \) 16 * q - 48 * q^9 + 80 * q^25 + 8 * q^28 + 24 * q^30 - 40 * q^34 - 56 * q^40 + 72 * q^48 - 112 * q^49 + 88 * q^58 - 104 * q^70 + 144 * q^81 - 120 * q^84

Introduction

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Newspace parameters

Newform invariants

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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	444.2.g.c

Level	$444$
Weight	$2$
Character orbit	444.g
Analytic conductor	$3.545$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-111
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.54535784974\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 9x^{12} + 43x^{8} - 144x^{4} + 256 \) x^16 - 9x^12 + 43x^8 - 144*x^4 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{11} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{15} + 15\nu^{11} - 45\nu^{7} + 120\nu^{3} ) / 448 \) (v^15 + 15v^11 - 45v^7 + 120*v^3) / 448
\(\beta_{2}\)	\(=\)	\( ( -\nu^{13} + 41\nu^{9} - 11\nu^{5} + 48\nu ) / 224 \) (-v^13 + 41v^9 - 11v^5 + 48*v) / 224
\(\beta_{3}\)	\(=\)	\( ( -\nu^{12} + 13\nu^{8} - 39\nu^{4} + 104 ) / 28 \) (-v^12 + 13v^8 - 39v^4 + 104) / 28
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{12} + 11\nu^{8} - 33\nu^{4} + 88 ) / 56 \) (-3v^12 + 11v^8 - 33*v^4 + 88) / 56
\(\beta_{5}\)	\(=\)	\( ( -\nu^{13} + 9\nu^{9} - 43\nu^{5} + 144\nu ) / 32 \) (-v^13 + 9v^9 - 43v^5 + 144*v) / 32
\(\beta_{6}\)	\(=\)	\( ( -\nu^{15} + 6\nu^{11} - 32\nu^{7} + 111\nu^{3} ) / 56 \) (-v^15 + 6v^11 - 32v^7 + 111*v^3) / 56
\(\beta_{7}\)	\(=\)	\( ( 9\nu^{14} - 33\nu^{10} + 211\nu^{6} + 128\nu^{2} ) / 448 \) (9v^14 - 33v^10 + 211v^6 + 128v^2) / 448
\(\beta_{8}\)	\(=\)	\( ( -9\nu^{14} + 33\nu^{10} - 211\nu^{6} + 768\nu^{2} ) / 448 \) (-9v^14 + 33v^10 - 211v^6 + 768v^2) / 448
\(\beta_{9}\)	\(=\)	\( ( -3\nu^{13} + 11\nu^{9} - 33\nu^{5} + 144\nu ) / 56 \) (-3v^13 + 11v^9 - 33v^5 + 144v) / 56
\(\beta_{10}\)	\(=\)	\( ( \nu^{12} - 6\nu^{8} + 32\nu^{4} - 83 ) / 7 \) (v^12 - 6v^8 + 32v^4 - 83) / 7
\(\beta_{11}\)	\(=\)	\( ( -15\nu^{14} + 55\nu^{10} - 53\nu^{6} + 384\nu^{2} ) / 448 \) (-15v^14 + 55v^10 - 53v^6 + 384v^2) / 448
\(\beta_{12}\)	\(=\)	\( ( 4\nu^{15} - 13\nu^{13} - 24\nu^{11} + 85\nu^{9} + 128\nu^{7} - 367\nu^{5} - 220\nu^{3} + 848\nu ) / 224 \) (4v^15 - 13v^13 - 24v^11 + 85v^9 + 128v^7 - 367v^5 - 220v^3 + 848v) / 224
\(\beta_{13}\)	\(=\)	\( ( 4\nu^{15} + 13\nu^{13} - 24\nu^{11} - 85\nu^{9} + 128\nu^{7} + 367\nu^{5} - 220\nu^{3} - 848\nu ) / 224 \) (4v^15 + 13v^13 - 24v^11 - 85v^9 + 128v^7 + 367v^5 - 220v^3 - 848v) / 224
\(\beta_{14}\)	\(=\)	\( ( -29\nu^{14} + 181\nu^{10} - 655\nu^{6} + 1504\nu^{2} ) / 448 \) (-29v^14 + 181v^10 - 655v^6 + 1504v^2) / 448
\(\beta_{15}\)	\(=\)	\( ( -11\nu^{15} - 6\nu^{13} + 59\nu^{11} + 22\nu^{9} - 177\nu^{7} - 66\nu^{5} + 472\nu^{3} + 288\nu ) / 224 \) (-11v^15 - 6v^13 + 59v^11 + 22v^9 - 177v^7 - 66v^5 + 472v^3 + 288v) / 224

\(\nu\)	\(=\)	\( ( \beta_{13} - \beta_{12} + \beta_{9} + 2\beta_{5} ) / 4 \) (b13 - b12 + b9 + 2*b5) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{8} + \beta_{7} ) / 2 \) (b8 + b7) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{13} + \beta_{12} + 2\beta_{6} ) / 2 \) (b13 + b12 + 2*b6) / 2
\(\nu^{4}\)	\(=\)	\( ( \beta_{10} + 2\beta_{4} + \beta_{3} + 5 ) / 2 \) (b10 + 2*b4 + b3 + 5) / 2
\(\nu^{5}\)	\(=\)	\( ( 3\beta_{13} - 3\beta_{12} + 5\beta_{9} + 2\beta_{5} + 4\beta_{2} ) / 4 \) (3b13 - 3b12 + 5b9 + 2b5 + 4*b2) / 4
\(\nu^{6}\)	\(=\)	\( ( 3\beta_{11} - 2\beta_{8} + 3\beta_{7} ) / 2 \) (3b11 - 2b8 + 3*b7) / 2
\(\nu^{7}\)	\(=\)	\( ( 6\beta_{15} + 9\beta_{13} + 9\beta_{12} - 3\beta_{9} + 2\beta_{6} + 4\beta_1 ) / 4 \) (6b15 + 9b13 + 9b12 - 3b9 + 2b6 + 4b1) / 4
\(\nu^{8}\)	\(=\)	\( ( 3\beta_{10} + 2\beta_{4} + 9\beta_{3} - 1 ) / 2 \) (3b10 + 2b4 + 9*b3 - 1) / 2
\(\nu^{9}\)	\(=\)	\( ( -\beta_{9} + 12\beta_{2} ) / 2 \) (-b9 + 12*b2) / 2
\(\nu^{10}\)	\(=\)	\( ( 12\beta_{14} - \beta_{11} - 25\beta_{8} + 12\beta_{7} ) / 2 \) (12b14 - b11 - 25b8 + 12*b7) / 2
\(\nu^{11}\)	\(=\)	\( ( 22\beta_{15} + 11\beta_{13} + 11\beta_{12} - 11\beta_{9} - 26\beta_{6} + 100\beta_1 ) / 4 \) (22b15 + 11b13 + 11b12 - 11b9 - 26b6 + 100b1) / 4
\(\nu^{12}\)	\(=\)	\( -26\beta_{4} + 11\beta_{3} \) -26b4 + 11b3
\(\nu^{13}\)	\(=\)	\( ( 15\beta_{13} - 15\beta_{12} - 89\beta_{9} + 74\beta_{5} + 44\beta_{2} ) / 4 \) (15b13 - 15b12 - 89b9 + 74b5 + 44*b2) / 4
\(\nu^{14}\)	\(=\)	\( ( 44\beta_{14} - 74\beta_{11} - 59\beta_{8} + 59\beta_{7} ) / 2 \) (44b14 - 74b11 - 59b8 + 59b7) / 2
\(\nu^{15}\)	\(=\)	\( ( -30\beta_{15} + 15\beta_{9} + 236\beta_1 ) / 2 \) (-30b15 + 15b9 + 236*b1) / 2

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 443.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 5T^{8} + 256 \) T^16 - 5*T^8 + 256
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{8} - 40 T^{6} + \cdots + 2352)^{2} \) (T^8 - 40T^6 + 500T^4 - 2000*T^2 + 2352)^2
$7$	\( (T^{4} + 28 T^{2} + 48)^{4} \) (T^4 + 28*T^2 + 48)^4
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} - 136 T^{6} + \cdots + 255792)^{2} \) (T^8 - 136T^6 + 5780T^4 - 78608*T^2 + 255792)^2
$19$	\( T^{16} \) T^16
$23$	\( (T^{8} + 184 T^{6} + \cdots + 330672)^{2} \) (T^8 + 184T^6 + 10580T^4 + 194672*T^2 + 330672)^2
$29$	\( (T^{8} - 232 T^{6} + \cdots + 1436592)^{2} \) (T^8 - 232T^6 + 16820T^4 - 390224*T^2 + 1436592)^2
$31$	\( T^{16} \) T^16
$37$	\( (T^{2} - 37)^{8} \) (T^2 - 37)^8
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( (T^{8} + 472 T^{6} + \cdots + 45442992)^{2} \) (T^8 + 472T^6 + 69620T^4 + 3286064*T^2 + 45442992)^2
$61$	\( T^{16} \) T^16
$67$	\( (T^{4} + 268 T^{2} + 48)^{4} \) (T^4 + 268*T^2 + 48)^4
$71$	\( T^{16} \) T^16
$73$	\( (T^{4} - 292 T^{2} + 21168)^{4} \) (T^4 - 292*T^2 + 21168)^4
$79$	\( T^{16} \) T^16
$83$	\( T^{16} \) T^16
$89$	\( (T^{8} - 712 T^{6} + \cdots + 250180272)^{2} \) (T^8 - 712T^6 + 158420T^4 - 11279504*T^2 + 250180272)^2
$97$	\( T^{16} \) T^16
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\(n\)	\(149\)	\(223\)	\(409\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)