LMFDB - Newform orbit 1344.2.i.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(545,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.545");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.i (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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.6040479020157644046336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 $ x^16 - 7x^12 - 32x^8 - 567*x^4 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{6} - 1) q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} + ( - 2 \beta_{6} - 1) q^{9}+O(q^{10}) $ q + (b6 - 1) * q^3 + b5 * q^5 - b2 * q^7 + (-2b6 - 1) q^9	$ q + (\beta_{6} - 1) q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} + ( - 2 \beta_{6} - 1) q^{9} - \beta_{10} q^{11} - \beta_1 q^{13} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{15}+ \cdots + (\beta_{10} - 2 \beta_{9}) q^{99}+O(q^{100}) $ q + (b6 - 1) * q^3 + b5 * q^5 - b2 * q^7 + (-2b6 - 1) q^9 - b10 * q^11 - b1 * q^13 + (-b5 - b3 + b2 - b1) * q^15 + b10 * q^17 + (b4 - 1) * q^19 + (b12 + b5 + b2) * q^21 + (-b12 - b8 - b5) * q^23 + (b4 - 2) * q^25 + (b6 + 5) * q^27 + (-b12 + b8) * q^29 + (b11 + b3 + b2) * q^31 + (b10 + b9) * q^33 + (b14 - b10 + 2b6) q^35 + (b3 + b2) * q^37 + (b12 + b8 + b1) * q^39 + b10 * q^41 + (-b9 + 2b7) q^43 + (-b5 + 2b3 - 2b2 + 2b1) q^45 + b15 * q^47 + (b7 + b4) * q^49 + (-b10 - b9) * q^51 + (-b15 - b12 + b8) * q^53 + (-b11 + 2b3 + 2b2) * q^55 + (b13 - b6 - b4 + 1) * q^57 + (b13 - b6) * q^59 + b1 * q^61 + (-2b12 - 2b5 + b2) * q^63 + (b13 - 3b6) q^65 + b9 * q^67 + (b12 + b8 + b5 + b3 - b2 - b1) * q^69 + (b12 + b8 + 3b5) q^71 + 2b7 q^73 + (b13 - 2b6 - b4 + 2) q^75 + (b15 + b12 + b8 - b5) * q^77 + (-b3 + b2 - 2b1) q^79 + (4b6 - 7) q^81 + (b13 + 3b6) q^83 + (b11 - 2b3 - 2b2) * q^85 + (b12 - b8 - 2b3 - 2b2) * q^87 + (2b14 - b10) q^89 + (b9 - b7 + b4 + 3) * q^91 + (b15 - b12 - b11 + b8 - b3 - b2) * q^93 + (2b12 + 2b8 - 4b5) q^95 - 2b7 q^97 + (b10 - 2b9) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 - 16 * q^9	$ 16 q - 16 q^{3} - 16 q^{9} - 16 q^{19} - 32 q^{25} + 80 q^{27} + 16 q^{57} + 32 q^{75} - 112 q^{81} + 48 q^{91}+O(q^{100}) $ 16 * q - 16 * q^3 - 16 * q^9 - 16 * q^19 - 32 * q^25 + 80 * q^27 + 16 * q^57 + 32 * q^75 - 112 * q^81 + 48 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 $ x^16 - 7*x^12 - 32*x^8 - 567*x^4 + 6561 :

$\beta_{1}$	$=$	$ ( 25\nu^{14} - 256\nu^{10} + 496\nu^{6} - 34911\nu^{2} ) / 29160 $ (25v^14 - 256v^10 + 496v^6 - 34911v^2) / 29160
$\beta_{2}$	$=$	$ ( 10\nu^{14} - 69\nu^{12} - 16\nu^{10} + 240\nu^{8} - 1184\nu^{6} - 1680\nu^{4} - 11286\nu^{2} + 43011 ) / 19440 $ (10v^14 - 69v^12 - 16v^10 + 240v^8 - 1184v^6 - 1680v^4 - 11286*v^2 + 43011) / 19440
$\beta_{3}$	$=$	$ ( -10\nu^{14} - 69\nu^{12} + 16\nu^{10} + 240\nu^{8} + 1184\nu^{6} - 1680\nu^{4} + 11286\nu^{2} + 43011 ) / 19440 $ (-10v^14 - 69v^12 + 16v^10 + 240v^8 + 1184v^6 - 1680v^4 + 11286*v^2 + 43011) / 19440
$\beta_{4}$	$=$	$ ( -2\nu^{12} + 14\nu^{8} + 226\nu^{4} + 567 ) / 405 $ (-2v^12 + 14v^8 + 226*v^4 + 567) / 405
$\beta_{5}$	$=$	$ ( 19\nu^{15} + 27\nu^{13} - 52\nu^{11} + 540\nu^{9} + 1012\nu^{7} - 3780\nu^{5} - 22113\nu^{3} - 12393\nu ) / 43740 $ (19v^15 + 27v^13 - 52v^11 + 540v^9 + 1012v^7 - 3780v^5 - 22113v^3 - 12393v) / 43740
$\beta_{6}$	$=$	$ ( 9\nu^{15} + 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} - 3231\nu^{3} - 20979\nu ) / 19440 $ (9v^15 + 37v^13 - 16v^9 - 1184v^5 - 3231v^3 - 20979v) / 19440
$\beta_{7}$	$=$	$ ( 14\nu^{14} + 64\nu^{10} - 124\nu^{6} - 13122\nu^{2} ) / 3645 $ (14v^14 + 64v^10 - 124v^6 - 13122v^2) / 3645
$\beta_{8}$	$=$	$ ( 4\nu^{15} + 21\nu^{13} + 26\nu^{11} + 96\nu^{9} - 506\nu^{7} - 186\nu^{5} - 1080\nu^{3} - 12393\nu ) / 7290 $ (4v^15 + 21v^13 + 26v^11 + 96v^9 - 506v^7 - 186v^5 - 1080v^3 - 12393v) / 7290
$\beta_{9}$	$=$	$ ( 31\nu^{14} + 26\nu^{10} - 506\nu^{6} - 18063\nu^{2} ) / 3645 $ (31v^14 + 26v^10 - 506v^6 - 18063v^2) / 3645
$\beta_{10}$	$=$	$ ( - 169 \nu^{15} + 837 \nu^{13} - 1328 \nu^{11} + 2160 \nu^{9} - 1072 \nu^{7} - 15120 \nu^{5} + \cdots - 836163 \nu ) / 174960 $ (-169v^15 + 837v^13 - 1328v^11 + 2160v^9 - 1072v^7 - 15120v^5 + 141183v^3 - 836163v) / 174960
$\beta_{11}$	$=$	$ ( -31\nu^{12} - 80\nu^{8} + 560\nu^{4} + 22329 ) / 1080 $ (-31v^12 - 80v^8 + 560*v^4 + 22329) / 1080
$\beta_{12}$	$=$	$ ( 31\nu^{15} + 36\nu^{13} + 26\nu^{11} - 252\nu^{9} - 506\nu^{7} + 3222\nu^{5} - 25353\nu^{3} - 24786\nu ) / 21870 $ (31v^15 + 36v^13 + 26v^11 - 252v^9 - 506v^7 + 3222v^5 - 25353v^3 - 24786v) / 21870
$\beta_{13}$	$=$	$ ( 257 \nu^{15} + 1341 \nu^{13} + 2656 \nu^{11} + 4464 \nu^{9} + 2144 \nu^{7} - 19584 \nu^{5} + \cdots - 1483515 \nu ) / 174960 $ (257v^15 + 1341v^13 + 2656v^11 + 4464v^9 + 2144v^7 - 19584v^5 - 253287v^3 - 1483515v) / 174960
$\beta_{14}$	$=$	$ ( -9\nu^{15} + 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} + 3231\nu^{3} - 20979\nu ) / 4860 $ (-9v^15 + 37v^13 - 16v^9 - 1184v^5 + 3231v^3 - 20979v) / 4860
$\beta_{15}$	$=$	$ ( -35\nu^{15} + 75\nu^{13} - 52\nu^{11} + 204\nu^{9} + 1012\nu^{7} + 516\nu^{5} + 26433\nu^{3} - 45441\nu ) / 7290 $ (-35v^15 + 75v^13 - 52v^11 + 204v^9 + 1012v^7 + 516v^5 + 26433v^3 - 45441v) / 7290

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

$\nu$	$=$	$ ( \beta_{15} + \beta_{14} - 2\beta_{13} + \beta_{12} - 4\beta_{10} + 3\beta_{8} + 2\beta_{6} + 4\beta_{5} ) / 16 $ (b15 + b14 - 2b13 + b12 - 4b10 + 3b8 + 2b6 + 4*b5) / 16
$\nu^{2}$	$=$	$ ( 2\beta_{9} - 3\beta_{7} + 2\beta_{3} - 2\beta_{2} - 4\beta_1 ) / 8 $ (2b9 - 3b7 + 2b3 - 2b2 - 4*b1) / 8
$\nu^{3}$	$=$	$ ( \beta_{15} - 4\beta_{14} - 5\beta_{12} + \beta_{8} + 16\beta_{6} - 8\beta_{5} ) / 8 $ (b15 - 4b14 - 5b12 + b8 + 16b6 - 8b5) / 8
$\nu^{4}$	$=$	$ ( \beta_{11} + 10\beta_{4} - 11\beta_{3} - 11\beta_{2} + 14 ) / 8 $ (b11 + 10b4 - 11b3 - 11*b2 + 14) / 8
$\nu^{5}$	$=$	$ ( 15\beta_{15} - 15\beta_{14} - 2\beta_{13} + 47\beta_{12} - 4\beta_{10} + 13\beta_{8} - 62\beta_{6} - 4\beta_{5} ) / 16 $ (15b15 - 15b14 - 2b13 + 47b12 - 4b10 + 13b8 - 62b6 - 4b5) / 16
$\nu^{6}$	$=$	$ ( 5\beta_{7} + 32\beta_{3} - 32\beta_{2} + 16\beta_1 ) / 4 $ (5b7 + 32b3 - 32b2 + 16b1) / 4
$\nu^{7}$	$=$	$ ( 35 \beta_{15} - 35 \beta_{14} + 26 \beta_{13} - 9 \beta_{12} - 52 \beta_{10} - 131 \beta_{8} + \cdots + 52 \beta_{5} ) / 16 $ (35b15 - 35b14 + 26b13 - 9b12 - 52b10 - 131b8 + 166b6 + 52b5) / 16
$\nu^{8}$	$=$	$ ( -39\beta_{11} + 70\beta_{4} + 109\beta_{3} + 109\beta_{2} + 226 ) / 8 $ (-39b11 + 70b4 + 109b3 + 109b2 + 226) / 8
$\nu^{9}$	$=$	$ ( 37\beta_{15} - 68\beta_{14} - 37\beta_{12} + 185\beta_{8} - 272\beta_{6} + 296\beta_{5} ) / 8 $ (37b15 - 68b14 - 37b12 + 185b8 - 272b6 + 296b5) / 8
$\nu^{10}$	$=$	$ ( -62\beta_{9} + 253\beta_{7} + 62\beta_{3} - 62\beta_{2} - 444\beta_1 ) / 8 $ (-62b9 + 253b7 + 62b3 - 62b2 - 444*b1) / 8
$\nu^{11}$	$=$	$ ( 93 \beta_{15} + 93 \beta_{14} + 506 \beta_{13} - 599 \beta_{12} - 1012 \beta_{10} + \cdots - 1012 \beta_{5} ) / 16 $ (93b15 + 93b14 + 506b13 - 599b12 - 1012b10 + 227b8 + 134b6 - 1012b5) / 16
$\nu^{12}$	$=$	$ -20\beta_{11} - 60\beta_{3} - 60\beta_{2} + 679 $ -20b11 - 60b3 - 60*b2 + 679
$\nu^{13}$	$=$	$ ( 1079 \beta_{15} + 1079 \beta_{14} - 1198 \beta_{13} + 2039 \beta_{12} - 2396 \beta_{10} + \cdots + 2396 \beta_{5} ) / 16 $ (1079b15 + 1079b14 - 1198b13 + 2039b12 - 2396b10 + 2277b8 + 3118b6 + 2396b5) / 16
$\nu^{14}$	$=$	$ ( 2158\beta_{9} - 1797\beta_{7} + 2158\beta_{3} - 2158\beta_{2} - 1436\beta_1 ) / 8 $ (2158b9 - 1797b7 + 2158b3 - 2158b2 - 1436*b1) / 8
$\nu^{15}$	$=$	$ ( 359\beta_{15} - 3596\beta_{14} - 1795\beta_{12} + 359\beta_{8} + 14384\beta_{6} - 2872\beta_{5} ) / 8 $ (359b15 - 3596b14 - 1795b12 + 359b8 + 14384b6 - 2872b5) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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} $

545.1

−0.912166	−	1.47240i
0.912166	−	1.47240i
1.73122	−	0.0537601i
−1.73122	−	0.0537601i
0.0537601	+	1.73122i
−0.0537601	+	1.73122i
1.47240	−	0.912166i
−1.47240	−	0.912166i
−1.47240	+	0.912166i
1.47240	+	0.912166i
−0.0537601	−	1.73122i
0.0537601	−	1.73122i
−1.73122	+	0.0537601i
1.73122	+	0.0537601i
0.912166	+	1.47240i
−0.912166	+	1.47240i

−1.00000

−

1.41421i

−

3.56995i

0.792287

−

2.52434i

−1.00000

2.82843i

545.2

−1.00000

−

1.41421i

−

3.56995i

0.792287

2.52434i

−1.00000

2.82843i

545.3

−1.00000

−

1.41421i

−

1.12046i

2.52434

−

0.792287i

−1.00000

2.82843i

545.4

−1.00000

−

1.41421i

−

1.12046i

2.52434

0.792287i

−1.00000

2.82843i

545.5

−1.00000

−

1.41421i

1.12046i

−2.52434

−

0.792287i

−1.00000

2.82843i

545.6

−1.00000

−

1.41421i

1.12046i

−2.52434

0.792287i

−1.00000

2.82843i

545.7

−1.00000

−

1.41421i

3.56995i

−0.792287

−

2.52434i

−1.00000

2.82843i

545.8

−1.00000

−

1.41421i

3.56995i

−0.792287

2.52434i

−1.00000

2.82843i

545.9

−1.00000

1.41421i

−

3.56995i

−0.792287

−

2.52434i

−1.00000

−

2.82843i

545.10

−1.00000

1.41421i

−

3.56995i

−0.792287

2.52434i

−1.00000

−

2.82843i

545.11

−1.00000

1.41421i

−

1.12046i

−2.52434

−

0.792287i

−1.00000

−

2.82843i

545.12

−1.00000

1.41421i

−

1.12046i

−2.52434

0.792287i

−1.00000

−

2.82843i

545.13

−1.00000

1.41421i

1.12046i

2.52434

−

0.792287i

−1.00000

−

2.82843i

545.14

−1.00000

1.41421i

1.12046i

2.52434

0.792287i

−1.00000

−

2.82843i

545.15

−1.00000

1.41421i

3.56995i

0.792287

−

2.52434i

−1.00000

−

2.82843i

545.16

−1.00000

1.41421i

3.56995i

0.792287

2.52434i

−1.00000

−

2.82843i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
24.f	even	2	1	inner
28.d	even	2	1	inner
56.h	odd	2	1	inner
84.h	odd	2	1	inner
168.i	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1344.2.i.f	✓	16
3.b	odd	2	1	inner	1344.2.i.f	✓	16
4.b	odd	2	1		1344.2.i.g	yes	16
7.b	odd	2	1		1344.2.i.g	yes	16
8.b	even	2	1		1344.2.i.g	yes	16
8.d	odd	2	1	inner	1344.2.i.f	✓	16
12.b	even	2	1		1344.2.i.g	yes	16
21.c	even	2	1		1344.2.i.g	yes	16
24.f	even	2	1	inner	1344.2.i.f	✓	16
24.h	odd	2	1		1344.2.i.g	yes	16
28.d	even	2	1	inner	1344.2.i.f	✓	16
56.e	even	2	1		1344.2.i.g	yes	16
56.h	odd	2	1	inner	1344.2.i.f	✓	16
84.h	odd	2	1	inner	1344.2.i.f	✓	16
168.e	odd	2	1		1344.2.i.g	yes	16
168.i	even	2	1	inner	1344.2.i.f	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1344.2.i.f	✓	16	1.a	even	1	1	trivial
1344.2.i.f	✓	16	3.b	odd	2	1	inner
1344.2.i.f	✓	16	8.d	odd	2	1	inner
1344.2.i.f	✓	16	24.f	even	2	1	inner
1344.2.i.f	✓	16	28.d	even	2	1	inner
1344.2.i.f	✓	16	56.h	odd	2	1	inner
1344.2.i.f	✓	16	84.h	odd	2	1	inner
1344.2.i.f	✓	16	168.i	even	2	1	inner
1344.2.i.g	yes	16	4.b	odd	2	1
1344.2.i.g	yes	16	7.b	odd	2	1
1344.2.i.g	yes	16	8.b	even	2	1
1344.2.i.g	yes	16	12.b	even	2	1
1344.2.i.g	yes	16	21.c	even	2	1
1344.2.i.g	yes	16	24.h	odd	2	1
1344.2.i.g	yes	16	56.e	even	2	1
1344.2.i.g	yes	16	168.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1344, [\chi])$:

$ T_{5}^{4} + 14T_{5}^{2} + 16 $

$ T_{13}^{2} - 12 $

$ T_{19}^{2} + 2T_{19} - 32 $

$ T_{29}^{4} - 56T_{29}^{2} + 256 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 2 T + 3)^{8} $ (T^2 + 2*T + 3)^8
$5$	$ (T^{4} + 14 T^{2} + 16)^{4} $ (T^4 + 14*T^2 + 16)^4
$7$	$ (T^{8} - 34 T^{4} + 2401)^{2} $ (T^8 - 34*T^4 + 2401)^2
$11$	$ (T^{4} - 34 T^{2} + 256)^{4} $ (T^4 - 34*T^2 + 256)^4
$13$	$ (T^{2} - 12)^{8} $ (T^2 - 12)^8
$17$	$ (T^{4} - 34 T^{2} + 256)^{4} $ (T^4 - 34*T^2 + 256)^4
$19$	$ (T^{2} + 2 T - 32)^{8} $ (T^2 + 2*T - 32)^8
$23$	$ (T^{4} + 38 T^{2} + 64)^{4} $ (T^4 + 38*T^2 + 64)^4
$29$	$ (T^{4} - 56 T^{2} + 256)^{4} $ (T^4 - 56*T^2 + 256)^4
$31$	$ (T^{4} + 112 T^{2} + 1024)^{4} $ (T^4 + 112*T^2 + 1024)^4
$37$	$ (T^{4} + 28 T^{2} + 64)^{4} $ (T^4 + 28*T^2 + 64)^4
$41$	$ (T^{4} - 34 T^{2} + 256)^{4} $ (T^4 - 34*T^2 + 256)^4
$43$	$ (T^{4} + 164 T^{2} + 256)^{4} $ (T^4 + 164*T^2 + 256)^4
$47$	$ (T^{4} - 152 T^{2} + 1024)^{4} $ (T^4 - 152*T^2 + 1024)^4
$53$	$ (T^{2} - 96)^{8} $ (T^2 - 96)^8
$59$	$ (T^{4} + 136 T^{2} + 4096)^{4} $ (T^4 + 136*T^2 + 4096)^4
$61$	$ (T^{2} - 12)^{8} $ (T^2 - 12)^8
$67$	$ (T^{4} + 68 T^{2} + 1024)^{4} $ (T^4 + 68*T^2 + 1024)^4
$71$	$ (T^{4} + 102 T^{2} + 2304)^{4} $ (T^4 + 102*T^2 + 2304)^4
$73$	$ (T^{2} + 64)^{8} $ (T^2 + 64)^8
$79$	$ (T^{4} - 76 T^{2} + 256)^{4} $ (T^4 - 76*T^2 + 256)^4
$83$	$ (T^{4} + 168 T^{2} + 2304)^{4} $ (T^4 + 168*T^2 + 2304)^4
$89$	$ (T^{4} - 258 T^{2} + 9216)^{4} $ (T^4 - 258*T^2 + 9216)^4
$97$	$ (T^{2} + 64)^{8} $ (T^2 + 64)^8
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{6} - 1) q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} + ( - 2 \beta_{6} - 1) q^{9}+O(q^{10}) \) q + (b6 - 1) * q^3 + b5 * q^5 - b2 * q^7 + (-2b6 - 1) q^9	\( q + (\beta_{6} - 1) q^{3} + \beta_{5} q^{5} - \beta_{2} q^{7} + ( - 2 \beta_{6} - 1) q^{9} - \beta_{10} q^{11} - \beta_1 q^{13} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{15}+ \cdots + (\beta_{10} - 2 \beta_{9}) q^{99}+O(q^{100}) \) q + (b6 - 1) * q^3 + b5 * q^5 - b2 * q^7 + (-2b6 - 1) q^9 - b10 * q^11 - b1 * q^13 + (-b5 - b3 + b2 - b1) * q^15 + b10 * q^17 + (b4 - 1) * q^19 + (b12 + b5 + b2) * q^21 + (-b12 - b8 - b5) * q^23 + (b4 - 2) * q^25 + (b6 + 5) * q^27 + (-b12 + b8) * q^29 + (b11 + b3 + b2) * q^31 + (b10 + b9) * q^33 + (b14 - b10 + 2b6) q^35 + (b3 + b2) * q^37 + (b12 + b8 + b1) * q^39 + b10 * q^41 + (-b9 + 2b7) q^43 + (-b5 + 2b3 - 2b2 + 2b1) q^45 + b15 * q^47 + (b7 + b4) * q^49 + (-b10 - b9) * q^51 + (-b15 - b12 + b8) * q^53 + (-b11 + 2b3 + 2b2) * q^55 + (b13 - b6 - b4 + 1) * q^57 + (b13 - b6) * q^59 + b1 * q^61 + (-2b12 - 2b5 + b2) * q^63 + (b13 - 3b6) q^65 + b9 * q^67 + (b12 + b8 + b5 + b3 - b2 - b1) * q^69 + (b12 + b8 + 3b5) q^71 + 2b7 q^73 + (b13 - 2b6 - b4 + 2) q^75 + (b15 + b12 + b8 - b5) * q^77 + (-b3 + b2 - 2b1) q^79 + (4b6 - 7) q^81 + (b13 + 3b6) q^83 + (b11 - 2b3 - 2b2) * q^85 + (b12 - b8 - 2b3 - 2b2) * q^87 + (2b14 - b10) q^89 + (b9 - b7 + b4 + 3) * q^91 + (b15 - b12 - b11 + b8 - b3 - b2) * q^93 + (2b12 + 2b8 - 4b5) q^95 - 2b7 q^97 + (b10 - 2b9) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 - 16 * q^9	\( 16 q - 16 q^{3} - 16 q^{9} - 16 q^{19} - 32 q^{25} + 80 q^{27} + 16 q^{57} + 32 q^{75} - 112 q^{81} + 48 q^{91}+O(q^{100}) \) 16 * q - 16 * q^3 - 16 * q^9 - 16 * q^19 - 32 * q^25 + 80 * q^27 + 16 * q^57 + 32 * q^75 - 112 * q^81 + 48 * q^91

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1344.2.i.f

Level	$1344$
Weight	$2$
Character orbit	1344.i
Analytic conductor	$10.732$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(10.7318940317\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.6040479020157644046336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} - 32x^{8} - 567x^{4} + 6561 \) x^16 - 7x^12 - 32x^8 - 567*x^4 + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{26} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 25\nu^{14} - 256\nu^{10} + 496\nu^{6} - 34911\nu^{2} ) / 29160 \) (25v^14 - 256v^10 + 496v^6 - 34911v^2) / 29160
\(\beta_{2}\)	\(=\)	\( ( 10\nu^{14} - 69\nu^{12} - 16\nu^{10} + 240\nu^{8} - 1184\nu^{6} - 1680\nu^{4} - 11286\nu^{2} + 43011 ) / 19440 \) (10v^14 - 69v^12 - 16v^10 + 240v^8 - 1184v^6 - 1680v^4 - 11286*v^2 + 43011) / 19440
\(\beta_{3}\)	\(=\)	\( ( -10\nu^{14} - 69\nu^{12} + 16\nu^{10} + 240\nu^{8} + 1184\nu^{6} - 1680\nu^{4} + 11286\nu^{2} + 43011 ) / 19440 \) (-10v^14 - 69v^12 + 16v^10 + 240v^8 + 1184v^6 - 1680v^4 + 11286*v^2 + 43011) / 19440
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{12} + 14\nu^{8} + 226\nu^{4} + 567 ) / 405 \) (-2v^12 + 14v^8 + 226*v^4 + 567) / 405
\(\beta_{5}\)	\(=\)	\( ( 19\nu^{15} + 27\nu^{13} - 52\nu^{11} + 540\nu^{9} + 1012\nu^{7} - 3780\nu^{5} - 22113\nu^{3} - 12393\nu ) / 43740 \) (19v^15 + 27v^13 - 52v^11 + 540v^9 + 1012v^7 - 3780v^5 - 22113v^3 - 12393v) / 43740
\(\beta_{6}\)	\(=\)	\( ( 9\nu^{15} + 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} - 3231\nu^{3} - 20979\nu ) / 19440 \) (9v^15 + 37v^13 - 16v^9 - 1184v^5 - 3231v^3 - 20979v) / 19440
\(\beta_{7}\)	\(=\)	\( ( 14\nu^{14} + 64\nu^{10} - 124\nu^{6} - 13122\nu^{2} ) / 3645 \) (14v^14 + 64v^10 - 124v^6 - 13122v^2) / 3645
\(\beta_{8}\)	\(=\)	\( ( 4\nu^{15} + 21\nu^{13} + 26\nu^{11} + 96\nu^{9} - 506\nu^{7} - 186\nu^{5} - 1080\nu^{3} - 12393\nu ) / 7290 \) (4v^15 + 21v^13 + 26v^11 + 96v^9 - 506v^7 - 186v^5 - 1080v^3 - 12393v) / 7290
\(\beta_{9}\)	\(=\)	\( ( 31\nu^{14} + 26\nu^{10} - 506\nu^{6} - 18063\nu^{2} ) / 3645 \) (31v^14 + 26v^10 - 506v^6 - 18063v^2) / 3645
\(\beta_{10}\)	\(=\)	\( ( - 169 \nu^{15} + 837 \nu^{13} - 1328 \nu^{11} + 2160 \nu^{9} - 1072 \nu^{7} - 15120 \nu^{5} + \cdots - 836163 \nu ) / 174960 \) (-169v^15 + 837v^13 - 1328v^11 + 2160v^9 - 1072v^7 - 15120v^5 + 141183v^3 - 836163v) / 174960
\(\beta_{11}\)	\(=\)	\( ( -31\nu^{12} - 80\nu^{8} + 560\nu^{4} + 22329 ) / 1080 \) (-31v^12 - 80v^8 + 560*v^4 + 22329) / 1080
\(\beta_{12}\)	\(=\)	\( ( 31\nu^{15} + 36\nu^{13} + 26\nu^{11} - 252\nu^{9} - 506\nu^{7} + 3222\nu^{5} - 25353\nu^{3} - 24786\nu ) / 21870 \) (31v^15 + 36v^13 + 26v^11 - 252v^9 - 506v^7 + 3222v^5 - 25353v^3 - 24786v) / 21870
\(\beta_{13}\)	\(=\)	\( ( 257 \nu^{15} + 1341 \nu^{13} + 2656 \nu^{11} + 4464 \nu^{9} + 2144 \nu^{7} - 19584 \nu^{5} + \cdots - 1483515 \nu ) / 174960 \) (257v^15 + 1341v^13 + 2656v^11 + 4464v^9 + 2144v^7 - 19584v^5 - 253287v^3 - 1483515v) / 174960
\(\beta_{14}\)	\(=\)	\( ( -9\nu^{15} + 37\nu^{13} - 16\nu^{9} - 1184\nu^{5} + 3231\nu^{3} - 20979\nu ) / 4860 \) (-9v^15 + 37v^13 - 16v^9 - 1184v^5 + 3231v^3 - 20979v) / 4860
\(\beta_{15}\)	\(=\)	\( ( -35\nu^{15} + 75\nu^{13} - 52\nu^{11} + 204\nu^{9} + 1012\nu^{7} + 516\nu^{5} + 26433\nu^{3} - 45441\nu ) / 7290 \) (-35v^15 + 75v^13 - 52v^11 + 204v^9 + 1012v^7 + 516v^5 + 26433v^3 - 45441v) / 7290

\(\nu\)	\(=\)	\( ( \beta_{15} + \beta_{14} - 2\beta_{13} + \beta_{12} - 4\beta_{10} + 3\beta_{8} + 2\beta_{6} + 4\beta_{5} ) / 16 \) (b15 + b14 - 2b13 + b12 - 4b10 + 3b8 + 2b6 + 4*b5) / 16
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{9} - 3\beta_{7} + 2\beta_{3} - 2\beta_{2} - 4\beta_1 ) / 8 \) (2b9 - 3b7 + 2b3 - 2b2 - 4*b1) / 8
\(\nu^{3}\)	\(=\)	\( ( \beta_{15} - 4\beta_{14} - 5\beta_{12} + \beta_{8} + 16\beta_{6} - 8\beta_{5} ) / 8 \) (b15 - 4b14 - 5b12 + b8 + 16b6 - 8b5) / 8
\(\nu^{4}\)	\(=\)	\( ( \beta_{11} + 10\beta_{4} - 11\beta_{3} - 11\beta_{2} + 14 ) / 8 \) (b11 + 10b4 - 11b3 - 11*b2 + 14) / 8
\(\nu^{5}\)	\(=\)	\( ( 15\beta_{15} - 15\beta_{14} - 2\beta_{13} + 47\beta_{12} - 4\beta_{10} + 13\beta_{8} - 62\beta_{6} - 4\beta_{5} ) / 16 \) (15b15 - 15b14 - 2b13 + 47b12 - 4b10 + 13b8 - 62b6 - 4b5) / 16
\(\nu^{6}\)	\(=\)	\( ( 5\beta_{7} + 32\beta_{3} - 32\beta_{2} + 16\beta_1 ) / 4 \) (5b7 + 32b3 - 32b2 + 16b1) / 4
\(\nu^{7}\)	\(=\)	\( ( 35 \beta_{15} - 35 \beta_{14} + 26 \beta_{13} - 9 \beta_{12} - 52 \beta_{10} - 131 \beta_{8} + \cdots + 52 \beta_{5} ) / 16 \) (35b15 - 35b14 + 26b13 - 9b12 - 52b10 - 131b8 + 166b6 + 52b5) / 16
\(\nu^{8}\)	\(=\)	\( ( -39\beta_{11} + 70\beta_{4} + 109\beta_{3} + 109\beta_{2} + 226 ) / 8 \) (-39b11 + 70b4 + 109b3 + 109b2 + 226) / 8
\(\nu^{9}\)	\(=\)	\( ( 37\beta_{15} - 68\beta_{14} - 37\beta_{12} + 185\beta_{8} - 272\beta_{6} + 296\beta_{5} ) / 8 \) (37b15 - 68b14 - 37b12 + 185b8 - 272b6 + 296b5) / 8
\(\nu^{10}\)	\(=\)	\( ( -62\beta_{9} + 253\beta_{7} + 62\beta_{3} - 62\beta_{2} - 444\beta_1 ) / 8 \) (-62b9 + 253b7 + 62b3 - 62b2 - 444*b1) / 8
\(\nu^{11}\)	\(=\)	\( ( 93 \beta_{15} + 93 \beta_{14} + 506 \beta_{13} - 599 \beta_{12} - 1012 \beta_{10} + \cdots - 1012 \beta_{5} ) / 16 \) (93b15 + 93b14 + 506b13 - 599b12 - 1012b10 + 227b8 + 134b6 - 1012b5) / 16
\(\nu^{12}\)	\(=\)	\( -20\beta_{11} - 60\beta_{3} - 60\beta_{2} + 679 \) -20b11 - 60b3 - 60*b2 + 679
\(\nu^{13}\)	\(=\)	\( ( 1079 \beta_{15} + 1079 \beta_{14} - 1198 \beta_{13} + 2039 \beta_{12} - 2396 \beta_{10} + \cdots + 2396 \beta_{5} ) / 16 \) (1079b15 + 1079b14 - 1198b13 + 2039b12 - 2396b10 + 2277b8 + 3118b6 + 2396b5) / 16
\(\nu^{14}\)	\(=\)	\( ( 2158\beta_{9} - 1797\beta_{7} + 2158\beta_{3} - 2158\beta_{2} - 1436\beta_1 ) / 8 \) (2158b9 - 1797b7 + 2158b3 - 2158b2 - 1436*b1) / 8
\(\nu^{15}\)	\(=\)	\( ( 359\beta_{15} - 3596\beta_{14} - 1795\beta_{12} + 359\beta_{8} + 14384\beta_{6} - 2872\beta_{5} ) / 8 \) (359b15 - 3596b14 - 1795b12 + 359b8 + 14384b6 - 2872b5) / 8

\(n\)	\(127\)	\(449\)	\(577\)	\(1093\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 545.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 2 T + 3)^{8} \) (T^2 + 2*T + 3)^8
$5$	\( (T^{4} + 14 T^{2} + 16)^{4} \) (T^4 + 14*T^2 + 16)^4
$7$	\( (T^{8} - 34 T^{4} + 2401)^{2} \) (T^8 - 34*T^4 + 2401)^2
$11$	\( (T^{4} - 34 T^{2} + 256)^{4} \) (T^4 - 34*T^2 + 256)^4
$13$	\( (T^{2} - 12)^{8} \) (T^2 - 12)^8
$17$	\( (T^{4} - 34 T^{2} + 256)^{4} \) (T^4 - 34*T^2 + 256)^4
$19$	\( (T^{2} + 2 T - 32)^{8} \) (T^2 + 2*T - 32)^8
$23$	\( (T^{4} + 38 T^{2} + 64)^{4} \) (T^4 + 38*T^2 + 64)^4
$29$	\( (T^{4} - 56 T^{2} + 256)^{4} \) (T^4 - 56*T^2 + 256)^4
$31$	\( (T^{4} + 112 T^{2} + 1024)^{4} \) (T^4 + 112*T^2 + 1024)^4
$37$	\( (T^{4} + 28 T^{2} + 64)^{4} \) (T^4 + 28*T^2 + 64)^4
$41$	\( (T^{4} - 34 T^{2} + 256)^{4} \) (T^4 - 34*T^2 + 256)^4
$43$	\( (T^{4} + 164 T^{2} + 256)^{4} \) (T^4 + 164*T^2 + 256)^4
$47$	\( (T^{4} - 152 T^{2} + 1024)^{4} \) (T^4 - 152*T^2 + 1024)^4
$53$	\( (T^{2} - 96)^{8} \) (T^2 - 96)^8
$59$	\( (T^{4} + 136 T^{2} + 4096)^{4} \) (T^4 + 136*T^2 + 4096)^4
$61$	\( (T^{2} - 12)^{8} \) (T^2 - 12)^8
$67$	\( (T^{4} + 68 T^{2} + 1024)^{4} \) (T^4 + 68*T^2 + 1024)^4
$71$	\( (T^{4} + 102 T^{2} + 2304)^{4} \) (T^4 + 102*T^2 + 2304)^4
$73$	\( (T^{2} + 64)^{8} \) (T^2 + 64)^8
$79$	\( (T^{4} - 76 T^{2} + 256)^{4} \) (T^4 - 76*T^2 + 256)^4
$83$	\( (T^{4} + 168 T^{2} + 2304)^{4} \) (T^4 + 168*T^2 + 2304)^4
$89$	\( (T^{4} - 258 T^{2} + 9216)^{4} \) (T^4 - 258*T^2 + 9216)^4
$97$	\( (T^{2} + 64)^{8} \) (T^2 + 64)^8
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