LMFDB - Newform orbit 1680.2.bl.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(127,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.127"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.bl (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.426337261060096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{6} q^{3} + \beta_{5} q^{5} + \beta_1 q^{7} - \beta_{7} q^{9} + ( - 2 \beta_{6} - 2 \beta_1) q^{11} + (\beta_{5} + \beta_{4}) q^{13} - \beta_{2} q^{15} + ( - 2 \beta_{10} + 2 \beta_{8} + \cdots + \beta_{4}) q^{17}+ \cdots + (2 \beta_{6} - 2 \beta_1) q^{99}+O(q^{100}) $ q + b6 * q^3 + b5 * q^5 + b1 * q^7 - b7 * q^9 + (-2b6 - 2b1) * q^11 + (b5 + b4) * q^13 - b2 * q^15 + (-2b10 + 2b8 - b5 + b4) * q^17 + (-b6 + b3 + b2 + b1) * q^19 - q^21 + (-b11 + 2b9 + 2b6 + b3 + 2b2) q^23 + (-b10 + b8 - b7 + 2b4 - 2) q^25 + b1 * q^27 + (-b10 - b8 - 2b7 + b5 - b4) q^29 + (b11 - b9 - b6 + 2b3 - 2b2 - b1) * q^31 + (2b7 + 2) q^33 + b11 * q^35 + (-b7 + 1) * q^37 + (-b3 - b2) * q^39 + (-b10 - 2b8 - b5 + 2b4) * q^41 + (-2b9 + 4b6 - 2b2) q^43 - b8 * q^45 + (2b11 + 2b3) * q^47 + b7 * q^49 + (-2b11 + 2b9 - b3 + b2) * q^51 + (-2b10 - 2b8 + 2b7 - b5 - b4 + 2) q^53 + (-2b11 + 2b2) * q^55 + (-b10 + b8 + b7 - 1) * q^57 + (2b11 + 2b9 - 2b6 + 2b1) * q^59 + (-3b10 - b8 - 3b5 + b4 - 2) * q^61 - b6 * q^63 + (-b10 + b8 + 4b7 + 2b4 - 2) * q^65 + (-2b11 - 2b3) * q^67 + (-b10 + 2b8 - 2b7 + b5 + 2b4) q^69 + (-b11 + b9 - 4b6 - 3b3 + 3b2 - 4b1) * q^71 + (-2b10 - 2b8 + 4b7 + b5 + b4 + 4) q^73 + (-b11 + b9 - 2b6 - 2b3 + b1) * q^75 + (-2b7 + 2) q^77 + (-2b11 - 2b9 + 2b6 - 2b1) * q^79 - q^81 + (2b11 + 4b6 - 2b3) q^83 + (-3b10 - b8 + 2b7 - 2b5 - 4b4 - 6) * q^85 + (b11 + b9 + b3 - b2 + 2b1) q^87 + (3b8 - 4b7 + 3b4) q^89 + (b11 - b9) * q^91 + (-2b10 - 2b8 + b7 - b5 - b4 + 1) * q^93 + (2b11 - b9 + 2b6 + 2b3 + b2 + 4b1) * q^95 + (-b10 + b8 - 2b7 - 2b5 + 2b4 + 2) q^97 + (2b6 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{13} - 4 q^{17} - 12 q^{21} - 20 q^{25} + 24 q^{33} + 12 q^{37} + 16 q^{41} + 4 q^{45} + 28 q^{53} - 16 q^{57} - 16 q^{61} - 20 q^{65} + 60 q^{73} + 24 q^{77} - 12 q^{81} - 84 q^{85} + 16 q^{93}+ \cdots + 28 q^{97}+O(q^{100}) $ 12 * q + 4 * q^13 - 4 * q^17 - 12 * q^21 - 20 * q^25 + 24 * q^33 + 12 * q^37 + 16 * q^41 + 4 * q^45 + 28 * q^53 - 16 * q^57 - 16 * q^61 - 20 * q^65 + 60 * q^73 + 24 * q^77 - 12 * q^81 - 84 * q^85 + 16 * q^93 + 28 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4*x^9 - 3*x^8 + 4*x^7 + 8*x^6 + 8*x^5 - 12*x^4 - 32*x^3 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{11} + \nu^{10} + 4\nu^{9} - 2\nu^{8} - 3\nu^{7} + 5\nu^{6} + 24\nu^{5} + 2\nu^{4} - 8\nu^{3} - 24\nu^{2} - 48\nu ) / 160 $ (v^11 + v^10 + 4v^9 - 2v^8 - 3v^7 + 5v^6 + 24v^5 + 2v^4 - 8v^3 - 24v^2 - 48*v) / 160
$\beta_{2}$	$=$	$ ( - 4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + \cdots - 32 ) / 160 $ (-4v^11 + 5v^10 + 8v^9 - 6v^8 + 4v^7 - 27v^6 + 4v^5 + 18v^4 + 60v^3 + 104v^2 + 16*v - 32) / 160
$\beta_{3}$	$=$	$ ( 3 \nu^{11} - \nu^{10} - 22 \nu^{9} - 2 \nu^{8} + 19 \nu^{7} + 67 \nu^{6} + 22 \nu^{5} - 110 \nu^{4} + \cdots + 352 ) / 160 $ (3v^11 - v^10 - 22v^9 - 2v^8 + 19v^7 + 67v^6 + 22v^5 - 110v^4 - 172v^3 + 40v^2 + 272v + 352) / 160
$\beta_{4}$	$=$	$ ( - 5 \nu^{11} + 8 \nu^{10} + 8 \nu^{9} + 12 \nu^{8} - \nu^{7} - 44 \nu^{6} + 112 \nu^{4} + 76 \nu^{3} + \cdots - 64 ) / 160 $ (-5v^11 + 8v^10 + 8v^9 + 12v^8 - v^7 - 44v^6 + 112v^4 + 76v^3 + 96v^2 - 352*v - 64) / 160
$\beta_{5}$	$=$	$ ( 3 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} + 20 \nu^{8} + 23 \nu^{7} - 12 \nu^{6} - 88 \nu^{5} - 8 \nu^{4} + \cdots - 512 ) / 160 $ (3v^11 - 8v^10 + 16v^9 + 20v^8 + 23v^7 - 12v^6 - 88v^5 - 8v^4 + 124v^3 + 16v^2 + 160*v - 512) / 160
$\beta_{6}$	$=$	$ ( - \nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + \cdots - 192 \nu ) / 160 $ (-v^11 + 9v^10 + 6v^9 + 2v^8 - 17v^7 - 35v^6 + 26v^5 + 38v^4 + 28v^3 - 56v^2 - 192v) / 160
$\beta_{7}$	$=$	$ ( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80 $ (3v^11 + v^10 - 4v^8 - 5v^7 + v^6 + 12v^5 + 8v^4 + 12v^3 - 36v^2 - 16v - 64) / 80
$\beta_{8}$	$=$	$ ( 3 \nu^{11} + 2 \nu^{10} + 6 \nu^{9} - 17 \nu^{7} - 2 \nu^{6} - 18 \nu^{5} + 12 \nu^{4} + 44 \nu^{3} + \cdots - 192 ) / 80 $ (3v^11 + 2v^10 + 6v^9 - 17v^7 - 2v^6 - 18v^5 + 12v^4 + 44v^3 - 24v^2 - 40v - 192) / 80
$\beta_{9}$	$=$	$ ( - 4 \nu^{11} - 7 \nu^{10} - 24 \nu^{9} + 6 \nu^{8} + 28 \nu^{7} + 49 \nu^{6} + 4 \nu^{5} - 130 \nu^{4} + \cdots + 384 ) / 160 $ (-4v^11 - 7v^10 - 24v^9 + 6v^8 + 28v^7 + 49v^6 + 4v^5 - 130v^4 - 44v^3 + 144v + 384) / 160
$\beta_{10}$	$=$	$ ( - 5 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 12 \nu^{8} - \nu^{7} - 14 \nu^{6} - 10 \nu^{5} - 8 \nu^{4} + \cdots + 96 ) / 80 $ (-5v^11 - 2v^10 - 2v^9 + 12v^8 - v^7 - 14v^6 - 10v^5 - 8v^4 + 36v^3 + 96v^2 + 8v + 96) / 80
$\beta_{11}$	$=$	$ ( 7 \nu^{11} + 15 \nu^{10} + 6 \nu^{9} - 22 \nu^{8} - 57 \nu^{7} + 11 \nu^{6} + 98 \nu^{5} + 126 \nu^{4} + \cdots - 64 ) / 160 $ (7v^11 + 15v^10 + 6v^9 - 22v^8 - 57v^7 + 11v^6 + 98v^5 + 126v^4 + 20v^3 - 352v^2 - 208*v - 64) / 160

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

$\nu$	$=$	$ ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 $ (b11 + b10 - b9 - b8 + b5 - b4 + b3 + b2 - 2*b1) / 4
$\nu^{2}$	$=$	$ ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 2 \beta_1 ) / 4 $ (-b11 + b10 - b9 + b8 + 2b7 - 2b6 - b5 + b4 + b3 + b2 + 2*b1) / 4
$\nu^{3}$	$=$	$ ( \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 4 ) / 4 $ (b11 + b10 + 3b9 + b8 + 4b7 - 2b6 + b5 + b4 - b3 + 3b2 + 4) / 4
$\nu^{4}$	$=$	$ ( 3 \beta_{11} + \beta_{10} - 3 \beta_{9} - 3 \beta_{8} - 6 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \cdots + 2 ) / 4 $ (3b11 + b10 - 3b9 - 3b8 - 6b6 + b5 + 3b4 + b3 - b2 - 6b1 + 2) / 4
$\nu^{5}$	$=$	$ ( \beta_{11} + 5 \beta_{10} - \beta_{9} - 5 \beta_{8} + 8 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 8 ) / 4 $ (b11 + 5b10 - b9 - 5b8 + 8b7 + b5 - b4 + b3 + b2 + 14b1 - 8) / 4
$\nu^{6}$	$=$	$ ( 5 \beta_{11} - \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - 10 \beta_{7} - 14 \beta_{6} + \beta_{5} + \cdots + 14 \beta_1 ) / 4 $ (5b11 - b10 + 5b9 + 7b8 - 10b7 - 14b6 + b5 + 7b4 + 3b3 + 3b2 + 14*b1) / 4
$\nu^{7}$	$=$	$ ( - \beta_{11} - 9 \beta_{10} + 5 \beta_{9} - 9 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + \cdots + 4 ) / 4 $ (-b11 - 9b10 + 5b9 - 9b8 + 4b7 + 2b6 + 7b5 + 7b4 + b3 + 5b2 + 4) / 4
$\nu^{8}$	$=$	$ ( 5 \beta_{11} + 15 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} + 14 \beta_{6} + 15 \beta_{5} + 5 \beta_{4} + \cdots - 2 ) / 4 $ (5b11 + 15b10 - 5b9 - 5b8 + 14b6 + 15b5 + 5b4 + 15b3 - 15b2 + 14b1 - 2) / 4
$\nu^{9}$	$=$	$ ( - \beta_{11} - 5 \beta_{10} - 7 \beta_{9} + 5 \beta_{8} - 32 \beta_{7} + 7 \beta_{5} - 7 \beta_{4} + \cdots + 32 ) / 4 $ (-b11 - 5b10 - 7b9 + 5b8 - 32b7 + 7b5 - 7b4 - b3 + 7b2 + 66b1 + 32) / 4
$\nu^{10}$	$=$	$ ( 11 \beta_{11} - 15 \beta_{10} + 11 \beta_{9} + 17 \beta_{8} - 22 \beta_{7} + 38 \beta_{6} + \cdots - 38 \beta_1 ) / 4 $ (11b11 - 15b10 + 11b9 + 17b8 - 22b7 + 38b6 + 15b5 + 17b4 + 37b3 + 37b2 - 38*b1) / 4
$\nu^{11}$	$=$	$ ( - 23 \beta_{11} + \beta_{10} - 21 \beta_{9} + \beta_{8} + 100 \beta_{7} + 14 \beta_{6} + 9 \beta_{5} + \cdots + 100 ) / 4 $ (-23b11 + b10 - 21b9 + b8 + 100b7 + 14b6 + 9b5 + 9b4 + 23b3 - 21b2 + 100) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$-\beta_{7}$	$1$	$1$	$-1$

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} $

127.1

−1.35818	−	0.394157i
−0.760198	+	1.19252i
1.41127	−	0.0912546i
−0.394157	−	1.35818i
1.19252	−	0.760198i
−0.0912546	+	1.41127i
−1.35818	+	0.394157i
−0.760198	−	1.19252i
1.41127	+	0.0912546i
−0.394157	+	1.35818i
1.19252	+	0.760198i
−0.0912546	−	1.41127i

−0.707107

−

0.707107i

−1.75233

−

1.38900i

0.707107

−

0.707107i

1.00000i

127.2

−0.707107

−

0.707107i

0.432320

2.19388i

0.707107

−

0.707107i

1.00000i

127.3

−0.707107

−

0.707107i

1.32001

−

1.80487i

0.707107

−

0.707107i

1.00000i

127.4

0.707107

0.707107i

−1.75233

−

1.38900i

−0.707107

0.707107i

1.00000i

127.5

0.707107

0.707107i

0.432320

2.19388i

−0.707107

0.707107i

1.00000i

127.6

0.707107

0.707107i

1.32001

−

1.80487i

−0.707107

0.707107i

1.00000i

463.1

−0.707107

0.707107i

−1.75233

1.38900i

0.707107

0.707107i

−

1.00000i

463.2

−0.707107

0.707107i

0.432320

−

2.19388i

0.707107

0.707107i

−

1.00000i

463.3

−0.707107

0.707107i

1.32001

1.80487i

0.707107

0.707107i

−

1.00000i

463.4

0.707107

−

0.707107i

−1.75233

1.38900i

−0.707107

−

0.707107i

−

1.00000i

463.5

0.707107

−

0.707107i

0.432320

−

2.19388i

−0.707107

−

0.707107i

−

1.00000i

463.6

0.707107

−

0.707107i

1.32001

1.80487i

−0.707107

−

0.707107i

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.2.bl.c	✓	12
4.b	odd	2	1	inner	1680.2.bl.c	✓	12
5.c	odd	4	1	inner	1680.2.bl.c	✓	12
20.e	even	4	1	inner	1680.2.bl.c	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1680.2.bl.c	✓	12	1.a	even	1	1	trivial
1680.2.bl.c	✓	12	4.b	odd	2	1	inner
1680.2.bl.c	✓	12	5.c	odd	4	1	inner
1680.2.bl.c	✓	12	20.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1680, [\chi])$:

$ T_{11}^{2} + 8 $

T11^2 + 8

$ T_{13}^{6} - 2T_{13}^{5} + 2T_{13}^{4} + 32T_{13}^{3} + 144T_{13}^{2} + 96T_{13} + 32 $

T13^6 - 2*T13^5 + 2*T13^4 + 32*T13^3 + 144*T13^2 + 96*T13 + 32

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$5$	$ (T^{6} + 5 T^{4} + \cdots + 125)^{2} $ (T^6 + 5T^4 + 8T^3 + 25*T^2 + 125)^2
$7$	$ (T^{4} + 1)^{3} $ (T^4 + 1)^3
$11$	$ (T^{2} + 8)^{6} $ (T^2 + 8)^6
$13$	$ (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} $ (T^6 - 2T^5 + 2T^4 + 32T^3 + 144T^2 + 96*T + 32)^2
$17$	$ (T^{6} + 2 T^{5} + \cdots + 3200)^{2} $ (T^6 + 2T^5 + 2T^4 - 176T^3 + 2304T^2 - 3840*T + 3200)^2
$19$	$ (T^{6} - 36 T^{4} + \cdots - 32)^{2} $ (T^6 - 36T^4 + 68T^2 - 32)^2
$23$	$ T^{12} + \cdots + 1600000000 $ T^12 + 4232T^8 + 5168656T^4 + 1600000000
$29$	$ (T^{6} + 72 T^{4} + \cdots + 1024)^{2} $ (T^6 + 72T^4 + 1040T^2 + 1024)^2
$31$	$ (T^{6} + 132 T^{4} + \cdots + 3872)^{2} $ (T^6 + 132T^4 + 1796T^2 + 3872)^2
$37$	$ (T^{2} - 2 T + 2)^{6} $ (T^2 - 2*T + 2)^6
$41$	$ (T^{3} - 4 T^{2} + \cdots + 400)^{4} $ (T^3 - 4T^2 - 90T + 400)^4
$43$	$ T^{12} + \cdots + 655360000 $ T^12 + 8576T^8 + 8687616T^4 + 655360000
$47$	$ T^{12} + 3200 T^{8} + \cdots + 16777216 $ T^12 + 3200T^8 + 1904640T^4 + 16777216
$53$	$ (T^{6} - 14 T^{5} + \cdots + 32)^{2} $ (T^6 - 14T^5 + 98T^4 + 400T^3 + 784T^2 + 224*T + 32)^2
$59$	$ (T^{6} - 144 T^{4} + \cdots - 8192)^{2} $ (T^6 - 144T^4 + 4160T^2 - 8192)^2
$61$	$ (T^{3} + 4 T^{2} + \cdots - 640)^{4} $ (T^3 + 4T^2 - 116T - 640)^4
$67$	$ T^{12} + 3200 T^{8} + \cdots + 16777216 $ T^12 + 3200T^8 + 1904640T^4 + 16777216
$71$	$ (T^{6} + 336 T^{4} + \cdots + 881792)^{2} $ (T^6 + 336T^4 + 31184T^2 + 881792)^2
$73$	$ (T^{6} - 30 T^{5} + \cdots + 468512)^{2} $ (T^6 - 30T^5 + 450T^4 - 2528T^3 + 2704T^2 + 50336*T + 468512)^2
$79$	$ (T^{6} - 144 T^{4} + \cdots - 8192)^{2} $ (T^6 - 144T^4 + 4160T^2 - 8192)^2
$83$	$ T^{12} + \cdots + 68719476736 $ T^12 + 24704T^8 + 144707584T^4 + 68719476736
$89$	$ (T^{6} + 228 T^{4} + \cdots + 6400)^{2} $ (T^6 + 228T^4 + 3684T^2 + 6400)^2
$97$	$ (T^{6} - 14 T^{5} + \cdots + 32)^{2} $ (T^6 - 14T^5 + 98T^4 + 400T^3 + 784T^2 + 224*T + 32)^2
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Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bl (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + \beta_{5} q^{5} + \beta_1 q^{7} - \beta_{7} q^{9} + ( - 2 \beta_{6} - 2 \beta_1) q^{11} + (\beta_{5} + \beta_{4}) q^{13} - \beta_{2} q^{15} + ( - 2 \beta_{10} + 2 \beta_{8} + \cdots + \beta_{4}) q^{17}+ \cdots + (2 \beta_{6} - 2 \beta_1) q^{99}+O(q^{100}) \) q + b6 * q^3 + b5 * q^5 + b1 * q^7 - b7 * q^9 + (-2b6 - 2b1) * q^11 + (b5 + b4) * q^13 - b2 * q^15 + (-2b10 + 2b8 - b5 + b4) * q^17 + (-b6 + b3 + b2 + b1) * q^19 - q^21 + (-b11 + 2b9 + 2b6 + b3 + 2b2) q^23 + (-b10 + b8 - b7 + 2b4 - 2) q^25 + b1 * q^27 + (-b10 - b8 - 2b7 + b5 - b4) q^29 + (b11 - b9 - b6 + 2b3 - 2b2 - b1) * q^31 + (2b7 + 2) q^33 + b11 * q^35 + (-b7 + 1) * q^37 + (-b3 - b2) * q^39 + (-b10 - 2b8 - b5 + 2b4) * q^41 + (-2b9 + 4b6 - 2b2) q^43 - b8 * q^45 + (2b11 + 2b3) * q^47 + b7 * q^49 + (-2b11 + 2b9 - b3 + b2) * q^51 + (-2b10 - 2b8 + 2b7 - b5 - b4 + 2) q^53 + (-2b11 + 2b2) * q^55 + (-b10 + b8 + b7 - 1) * q^57 + (2b11 + 2b9 - 2b6 + 2b1) * q^59 + (-3b10 - b8 - 3b5 + b4 - 2) * q^61 - b6 * q^63 + (-b10 + b8 + 4b7 + 2b4 - 2) * q^65 + (-2b11 - 2b3) * q^67 + (-b10 + 2b8 - 2b7 + b5 + 2b4) q^69 + (-b11 + b9 - 4b6 - 3b3 + 3b2 - 4b1) * q^71 + (-2b10 - 2b8 + 4b7 + b5 + b4 + 4) q^73 + (-b11 + b9 - 2b6 - 2b3 + b1) * q^75 + (-2b7 + 2) q^77 + (-2b11 - 2b9 + 2b6 - 2b1) * q^79 - q^81 + (2b11 + 4b6 - 2b3) q^83 + (-3b10 - b8 + 2b7 - 2b5 - 4b4 - 6) * q^85 + (b11 + b9 + b3 - b2 + 2b1) q^87 + (3b8 - 4b7 + 3b4) q^89 + (b11 - b9) * q^91 + (-2b10 - 2b8 + b7 - b5 - b4 + 1) * q^93 + (2b11 - b9 + 2b6 + 2b3 + b2 + 4b1) * q^95 + (-b10 + b8 - 2b7 - 2b5 + 2b4 + 2) q^97 + (2b6 - 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{13} - 4 q^{17} - 12 q^{21} - 20 q^{25} + 24 q^{33} + 12 q^{37} + 16 q^{41} + 4 q^{45} + 28 q^{53} - 16 q^{57} - 16 q^{61} - 20 q^{65} + 60 q^{73} + 24 q^{77} - 12 q^{81} - 84 q^{85} + 16 q^{93}+ \cdots + 28 q^{97}+O(q^{100}) \) 12 * q + 4 * q^13 - 4 * q^17 - 12 * q^21 - 20 * q^25 + 24 * q^33 + 12 * q^37 + 16 * q^41 + 4 * q^45 + 28 * q^53 - 16 * q^57 - 16 * q^61 - 20 * q^65 + 60 * q^73 + 24 * q^77 - 12 * q^81 - 84 * q^85 + 16 * q^93 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	1680.2.bl.c

Level	$1680$
Weight	$2$
Character orbit	1680.bl
Analytic conductor	$13.415$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(13.4148675396\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(i)\)
Coefficient field:	12.0.426337261060096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} + \nu^{10} + 4\nu^{9} - 2\nu^{8} - 3\nu^{7} + 5\nu^{6} + 24\nu^{5} + 2\nu^{4} - 8\nu^{3} - 24\nu^{2} - 48\nu ) / 160 \) (v^11 + v^10 + 4v^9 - 2v^8 - 3v^7 + 5v^6 + 24v^5 + 2v^4 - 8v^3 - 24v^2 - 48*v) / 160
\(\beta_{2}\)	\(=\)	\( ( - 4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + \cdots - 32 ) / 160 \) (-4v^11 + 5v^10 + 8v^9 - 6v^8 + 4v^7 - 27v^6 + 4v^5 + 18v^4 + 60v^3 + 104v^2 + 16*v - 32) / 160
\(\beta_{3}\)	\(=\)	\( ( 3 \nu^{11} - \nu^{10} - 22 \nu^{9} - 2 \nu^{8} + 19 \nu^{7} + 67 \nu^{6} + 22 \nu^{5} - 110 \nu^{4} + \cdots + 352 ) / 160 \) (3v^11 - v^10 - 22v^9 - 2v^8 + 19v^7 + 67v^6 + 22v^5 - 110v^4 - 172v^3 + 40v^2 + 272v + 352) / 160
\(\beta_{4}\)	\(=\)	\( ( - 5 \nu^{11} + 8 \nu^{10} + 8 \nu^{9} + 12 \nu^{8} - \nu^{7} - 44 \nu^{6} + 112 \nu^{4} + 76 \nu^{3} + \cdots - 64 ) / 160 \) (-5v^11 + 8v^10 + 8v^9 + 12v^8 - v^7 - 44v^6 + 112v^4 + 76v^3 + 96v^2 - 352*v - 64) / 160
\(\beta_{5}\)	\(=\)	\( ( 3 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} + 20 \nu^{8} + 23 \nu^{7} - 12 \nu^{6} - 88 \nu^{5} - 8 \nu^{4} + \cdots - 512 ) / 160 \) (3v^11 - 8v^10 + 16v^9 + 20v^8 + 23v^7 - 12v^6 - 88v^5 - 8v^4 + 124v^3 + 16v^2 + 160*v - 512) / 160
\(\beta_{6}\)	\(=\)	\( ( - \nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + \cdots - 192 \nu ) / 160 \) (-v^11 + 9v^10 + 6v^9 + 2v^8 - 17v^7 - 35v^6 + 26v^5 + 38v^4 + 28v^3 - 56v^2 - 192v) / 160
\(\beta_{7}\)	\(=\)	\( ( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80 \) (3v^11 + v^10 - 4v^8 - 5v^7 + v^6 + 12v^5 + 8v^4 + 12v^3 - 36v^2 - 16v - 64) / 80
\(\beta_{8}\)	\(=\)	\( ( 3 \nu^{11} + 2 \nu^{10} + 6 \nu^{9} - 17 \nu^{7} - 2 \nu^{6} - 18 \nu^{5} + 12 \nu^{4} + 44 \nu^{3} + \cdots - 192 ) / 80 \) (3v^11 + 2v^10 + 6v^9 - 17v^7 - 2v^6 - 18v^5 + 12v^4 + 44v^3 - 24v^2 - 40v - 192) / 80
\(\beta_{9}\)	\(=\)	\( ( - 4 \nu^{11} - 7 \nu^{10} - 24 \nu^{9} + 6 \nu^{8} + 28 \nu^{7} + 49 \nu^{6} + 4 \nu^{5} - 130 \nu^{4} + \cdots + 384 ) / 160 \) (-4v^11 - 7v^10 - 24v^9 + 6v^8 + 28v^7 + 49v^6 + 4v^5 - 130v^4 - 44v^3 + 144v + 384) / 160
\(\beta_{10}\)	\(=\)	\( ( - 5 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 12 \nu^{8} - \nu^{7} - 14 \nu^{6} - 10 \nu^{5} - 8 \nu^{4} + \cdots + 96 ) / 80 \) (-5v^11 - 2v^10 - 2v^9 + 12v^8 - v^7 - 14v^6 - 10v^5 - 8v^4 + 36v^3 + 96v^2 + 8v + 96) / 80
\(\beta_{11}\)	\(=\)	\( ( 7 \nu^{11} + 15 \nu^{10} + 6 \nu^{9} - 22 \nu^{8} - 57 \nu^{7} + 11 \nu^{6} + 98 \nu^{5} + 126 \nu^{4} + \cdots - 64 ) / 160 \) (7v^11 + 15v^10 + 6v^9 - 22v^8 - 57v^7 + 11v^6 + 98v^5 + 126v^4 + 20v^3 - 352v^2 - 208*v - 64) / 160

\(\nu\)	\(=\)	\( ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) (b11 + b10 - b9 - b8 + b5 - b4 + b3 + b2 - 2*b1) / 4
\(\nu^{2}\)	\(=\)	\( ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 2 \beta_1 ) / 4 \) (-b11 + b10 - b9 + b8 + 2b7 - 2b6 - b5 + b4 + b3 + b2 + 2*b1) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 4 ) / 4 \) (b11 + b10 + 3b9 + b8 + 4b7 - 2b6 + b5 + b4 - b3 + 3b2 + 4) / 4
\(\nu^{4}\)	\(=\)	\( ( 3 \beta_{11} + \beta_{10} - 3 \beta_{9} - 3 \beta_{8} - 6 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \cdots + 2 ) / 4 \) (3b11 + b10 - 3b9 - 3b8 - 6b6 + b5 + 3b4 + b3 - b2 - 6b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( \beta_{11} + 5 \beta_{10} - \beta_{9} - 5 \beta_{8} + 8 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 8 ) / 4 \) (b11 + 5b10 - b9 - 5b8 + 8b7 + b5 - b4 + b3 + b2 + 14b1 - 8) / 4
\(\nu^{6}\)	\(=\)	\( ( 5 \beta_{11} - \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - 10 \beta_{7} - 14 \beta_{6} + \beta_{5} + \cdots + 14 \beta_1 ) / 4 \) (5b11 - b10 + 5b9 + 7b8 - 10b7 - 14b6 + b5 + 7b4 + 3b3 + 3b2 + 14*b1) / 4
\(\nu^{7}\)	\(=\)	\( ( - \beta_{11} - 9 \beta_{10} + 5 \beta_{9} - 9 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + \cdots + 4 ) / 4 \) (-b11 - 9b10 + 5b9 - 9b8 + 4b7 + 2b6 + 7b5 + 7b4 + b3 + 5b2 + 4) / 4
\(\nu^{8}\)	\(=\)	\( ( 5 \beta_{11} + 15 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} + 14 \beta_{6} + 15 \beta_{5} + 5 \beta_{4} + \cdots - 2 ) / 4 \) (5b11 + 15b10 - 5b9 - 5b8 + 14b6 + 15b5 + 5b4 + 15b3 - 15b2 + 14b1 - 2) / 4
\(\nu^{9}\)	\(=\)	\( ( - \beta_{11} - 5 \beta_{10} - 7 \beta_{9} + 5 \beta_{8} - 32 \beta_{7} + 7 \beta_{5} - 7 \beta_{4} + \cdots + 32 ) / 4 \) (-b11 - 5b10 - 7b9 + 5b8 - 32b7 + 7b5 - 7b4 - b3 + 7b2 + 66b1 + 32) / 4
\(\nu^{10}\)	\(=\)	\( ( 11 \beta_{11} - 15 \beta_{10} + 11 \beta_{9} + 17 \beta_{8} - 22 \beta_{7} + 38 \beta_{6} + \cdots - 38 \beta_1 ) / 4 \) (11b11 - 15b10 + 11b9 + 17b8 - 22b7 + 38b6 + 15b5 + 17b4 + 37b3 + 37b2 - 38*b1) / 4
\(\nu^{11}\)	\(=\)	\( ( - 23 \beta_{11} + \beta_{10} - 21 \beta_{9} + \beta_{8} + 100 \beta_{7} + 14 \beta_{6} + 9 \beta_{5} + \cdots + 100 ) / 4 \) (-23b11 + b10 - 21b9 + b8 + 100b7 + 14b6 + 9b5 + 9b4 + 23b3 - 21b2 + 100) / 4

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(1\)	\(-\beta_{7}\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 127.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$5$	\( (T^{6} + 5 T^{4} + \cdots + 125)^{2} \) (T^6 + 5T^4 + 8T^3 + 25*T^2 + 125)^2
$7$	\( (T^{4} + 1)^{3} \) (T^4 + 1)^3
$11$	\( (T^{2} + 8)^{6} \) (T^2 + 8)^6
$13$	\( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \) (T^6 - 2T^5 + 2T^4 + 32T^3 + 144T^2 + 96*T + 32)^2
$17$	\( (T^{6} + 2 T^{5} + \cdots + 3200)^{2} \) (T^6 + 2T^5 + 2T^4 - 176T^3 + 2304T^2 - 3840*T + 3200)^2
$19$	\( (T^{6} - 36 T^{4} + \cdots - 32)^{2} \) (T^6 - 36T^4 + 68T^2 - 32)^2
$23$	\( T^{12} + \cdots + 1600000000 \) T^12 + 4232T^8 + 5168656T^4 + 1600000000
$29$	\( (T^{6} + 72 T^{4} + \cdots + 1024)^{2} \) (T^6 + 72T^4 + 1040T^2 + 1024)^2
$31$	\( (T^{6} + 132 T^{4} + \cdots + 3872)^{2} \) (T^6 + 132T^4 + 1796T^2 + 3872)^2
$37$	\( (T^{2} - 2 T + 2)^{6} \) (T^2 - 2*T + 2)^6
$41$	\( (T^{3} - 4 T^{2} + \cdots + 400)^{4} \) (T^3 - 4T^2 - 90T + 400)^4
$43$	\( T^{12} + \cdots + 655360000 \) T^12 + 8576T^8 + 8687616T^4 + 655360000
$47$	\( T^{12} + 3200 T^{8} + \cdots + 16777216 \) T^12 + 3200T^8 + 1904640T^4 + 16777216
$53$	\( (T^{6} - 14 T^{5} + \cdots + 32)^{2} \) (T^6 - 14T^5 + 98T^4 + 400T^3 + 784T^2 + 224*T + 32)^2
$59$	\( (T^{6} - 144 T^{4} + \cdots - 8192)^{2} \) (T^6 - 144T^4 + 4160T^2 - 8192)^2
$61$	\( (T^{3} + 4 T^{2} + \cdots - 640)^{4} \) (T^3 + 4T^2 - 116T - 640)^4
$67$	\( T^{12} + 3200 T^{8} + \cdots + 16777216 \) T^12 + 3200T^8 + 1904640T^4 + 16777216
$71$	\( (T^{6} + 336 T^{4} + \cdots + 881792)^{2} \) (T^6 + 336T^4 + 31184T^2 + 881792)^2
$73$	\( (T^{6} - 30 T^{5} + \cdots + 468512)^{2} \) (T^6 - 30T^5 + 450T^4 - 2528T^3 + 2704T^2 + 50336*T + 468512)^2
$79$	\( (T^{6} - 144 T^{4} + \cdots - 8192)^{2} \) (T^6 - 144T^4 + 4160T^2 - 8192)^2
$83$	\( T^{12} + \cdots + 68719476736 \) T^12 + 24704T^8 + 144707584T^4 + 68719476736
$89$	\( (T^{6} + 228 T^{4} + \cdots + 6400)^{2} \) (T^6 + 228T^4 + 3684T^2 + 6400)^2
$97$	\( (T^{6} - 14 T^{5} + \cdots + 32)^{2} \) (T^6 - 14T^5 + 98T^4 + 400T^3 + 784T^2 + 224*T + 32)^2
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