LMFDB - Newform orbit 3840.2.f.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(769,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3840.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.f (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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 120)
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_{3} q^{3} + \beta_{6} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10}) $ q + b3 * q^3 + b6 * q^5 - b1 * q^7 - q^9	$ q + \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_1 q^{7} - q^{9} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{11} - \beta_{3}) q^{13} + \beta_{4} q^{15} + (\beta_{10} + \beta_{7} + \beta_{4}) q^{17} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{19} + \beta_{8} q^{21} + (2 \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{23}+ \cdots + ( - \beta_{9} + \beta_{8}) q^{99}+O(q^{100}) $ q + b3 * q^3 + b6 * q^5 - b1 * q^7 - q^9 + (b9 - b8) * q^11 + (b11 - b3) * q^13 + b4 * q^15 + (b10 + b7 + b4) * q^17 + (b8 + b6 + b2) * q^19 + b8 * q^21 + (2b10 + b7 + b4 + b1) q^23 + (-b10 - b5) * q^25 - b3 * q^27 + (2b9 - b6 - b2) q^29 + (b7 + b5 - b4 - 3) * q^31 + (-b10 - b1) * q^33 + (-b11 - b9 - b8 + b6 - b3 + b2) * q^35 + (b11 + 3b3) q^37 + (b5 + 1) * q^39 + (2b7 + 2b5 - 2b4) q^41 + (-2b6 + 2b2) * q^43 - b6 * q^45 + (-b7 - b4 - b1) * q^47 + (2b7 - 2b4 - 1) * q^49 + (b9 - b6 - b2) * q^51 + (-b6 - 4b3 + b2) q^53 + (-2b10 + b7 + b5 - b4 - b1 + 1) q^55 + (b7 + b4 + b1) * q^57 + (-b9 + 3b8) q^59 + (-2b8 + 2b6 + 2b2) q^61 + b1 * q^63 + (b10 + 3b7 - b4 + 2b1 - 2) * q^65 + (-2b6 + 2b2) * q^67 + (2b9 - b8 - b6 - b2) q^69 + (-2b7 - 2b5 + 2b4 - 2) q^71 + (2b10 + 4b7 + 4b4) q^73 + (b11 - b9) * q^75 + 4b3 q^77 + (-b7 - b5 + b4 + 3) * q^79 + q^81 + (2b11 - 2b3) * q^83 + (b11 - 2b8 + 3b3 + 2b2) q^85 + (-2b10 - b7 - b4) q^87 + (2b7 - 2b5 - 2b4 + 4) q^89 + (-2b9 - 2b8 + 4b6 + 4b2) * q^91 + (-b11 + b6 - 3b3 - b2) q^93 + (b7 - 2b5 + b4 - b1 + 6) q^95 + (-2b10 + 2b7 + 2b4) q^97 + (-b9 + b8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} - 4 q^{25} - 32 q^{31} + 16 q^{39} + 8 q^{41} - 12 q^{49} + 16 q^{55} - 24 q^{65} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89} + 64 q^{95}+O(q^{100}) $ 12 * q - 12 * q^9 - 4 * q^25 - 32 * q^31 + 16 * q^39 + 8 * q^41 - 12 * q^49 + 16 * q^55 - 24 * q^65 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89 + 64 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8*x^6 + 16*x^2 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 $ (v^8 + v^6 + 4v^4 - 4v^2) / 8
$\beta_{2}$	$=$	$ ( \nu^{11} + 3\nu^{9} + 6\nu^{7} - 12\nu^{5} + 24\nu^{3} ) / 64 $ (v^11 + 3v^9 + 6v^7 - 12v^5 + 24v^3) / 64
$\beta_{3}$	$=$	$ ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 $ (v^11 - v^9 + 2v^7 + 4v^5 + 8*v^3) / 64
$\beta_{4}$	$=$	$ ( -\nu^{6} + \nu^{4} + 2\nu^{2} + 4 ) / 4 $ (-v^6 + v^4 + 2*v^2 + 4) / 4
$\beta_{5}$	$=$	$ ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 $ (-v^10 + v^6 + 4v^4 + 4v^2 - 8) / 8
$\beta_{6}$	$=$	$ ( \nu^{11} - 5\nu^{9} - 2\nu^{7} - 12\nu^{5} + 24\nu^{3} - 64\nu ) / 64 $ (v^11 - 5v^9 - 2v^7 - 12v^5 + 24v^3 - 64*v) / 64
$\beta_{7}$	$=$	$ ( \nu^{8} - \nu^{6} - 2\nu^{4} - 8\nu^{2} + 8 ) / 8 $ (v^8 - v^6 - 2v^4 - 8v^2 + 8) / 8
$\beta_{8}$	$=$	$ ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 $ (-v^11 - 3v^9 + 2v^7 + 4v^5 + 24v^3) / 32
$\beta_{9}$	$=$	$ ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 $ (v^11 - v^9 - 6v^7 - 4v^5 + 40v^3 + 32v) / 32
$\beta_{10}$	$=$	$ ( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} - 32 ) / 16 $ (-v^10 - 3v^8 + 2v^6 + 4v^4 - 8v^2 - 32) / 16
$\beta_{11}$	$=$	$ ( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64 $ (3v^11 - 3v^9 - 10v^7 - 36v^5 - 8v^3 + 128v) / 64

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

$\nu$	$=$	$ ( \beta_{11} + \beta_{8} - 2\beta_{6} + \beta_{3} ) / 4 $ (b11 + b8 - 2*b6 + b3) / 4
$\nu^{2}$	$=$	$ ( -2\beta_{10} - 2\beta_{7} + \beta_{5} - \beta _1 - 1 ) / 4 $ (-2b10 - 2b7 + b5 - b1 - 1) / 4
$\nu^{3}$	$=$	$ ( -\beta_{11} + 2\beta_{9} + \beta_{8} - \beta_{3} + 2\beta_{2} ) / 4 $ (-b11 + 2b9 + b8 - b3 + 2b2) / 4
$\nu^{4}$	$=$	$ ( 2\beta_{10} - 2\beta_{7} - \beta_{5} + 4\beta_{4} + 5\beta _1 + 1 ) / 4 $ (2b10 - 2b7 - b5 + 4b4 + 5b1 + 1) / 4
$\nu^{5}$	$=$	$ ( -3\beta_{11} + 2\beta_{9} - \beta_{8} - 4\beta_{6} + 13\beta_{3} - 6\beta_{2} ) / 4 $ (-3b11 + 2b9 - b8 - 4b6 + 13b3 - 6*b2) / 4
$\nu^{6}$	$=$	$ ( -2\beta_{10} - 6\beta_{7} + \beta_{5} - 12\beta_{4} + 3\beta _1 + 15 ) / 4 $ (-2b10 - 6b7 + b5 - 12b4 + 3b1 + 15) / 4
$\nu^{7}$	$=$	$ ( 3\beta_{11} - 10\beta_{9} + 9\beta_{8} - 4\beta_{6} + 19\beta_{3} + 14\beta_{2} ) / 4 $ (3b11 - 10b9 + 9b8 - 4b6 + 19b3 + 14b2) / 4
$\nu^{8}$	$=$	$ ( -14\beta_{10} + 6\beta_{7} + 7\beta_{5} - 4\beta_{4} + 5\beta _1 - 23 ) / 4 $ (-14b10 + 6b7 + 7b5 - 4b4 + 5*b1 - 23) / 4
$\nu^{9}$	$=$	$ ( -11\beta_{11} + 10\beta_{9} - 17\beta_{8} - 12\beta_{6} - 27\beta_{3} + 18\beta_{2} ) / 4 $ (-11b11 + 10b9 - 17b8 - 12b6 - 27b3 + 18b2) / 4
$\nu^{10}$	$=$	$ ( -2\beta_{10} - 22\beta_{7} - 31\beta_{5} + 4\beta_{4} + 19\beta _1 - 17 ) / 4 $ (-2b10 - 22b7 - 31b5 + 4b4 + 19*b1 - 17) / 4
$\nu^{11}$	$=$	$ ( 3\beta_{11} + 6\beta_{9} - 39\beta_{8} + 12\beta_{6} + 147\beta_{3} - 2\beta_{2} ) / 4 $ (3b11 + 6b9 - 39b8 + 12b6 + 147b3 - 2b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$1$

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} $

769.1

0.450129	−	1.34067i
0.806504	+	1.16170i
1.37729	−	0.321037i
−1.37729	−	0.321037i
−0.806504	+	1.16170i
−0.450129	−	1.34067i
0.450129	+	1.34067i
0.806504	−	1.16170i
1.37729	+	0.321037i
−1.37729	+	0.321037i
−0.806504	−	1.16170i
−0.450129	+	1.34067i

−

1.00000i

−2.22158

−

0.254102i

−

2.64265i

−1.00000

769.2

−

1.00000i

−1.23992

−

1.86081i

0.746175i

−1.00000

769.3

−

1.00000i

−0.726062

2.11491i

4.05705i

−1.00000

769.4

−

1.00000i

0.726062

2.11491i

−

4.05705i

−1.00000

769.5

−

1.00000i

1.23992

−

1.86081i

−

0.746175i

−1.00000

769.6

−

1.00000i

2.22158

−

0.254102i

2.64265i

−1.00000

769.7

1.00000i

−2.22158

0.254102i

2.64265i

−1.00000

769.8

1.00000i

−1.23992

1.86081i

−

0.746175i

−1.00000

769.9

1.00000i

−0.726062

−

2.11491i

−

4.05705i

−1.00000

769.10

1.00000i

0.726062

−

2.11491i

4.05705i

−1.00000

769.11

1.00000i

1.23992

1.86081i

0.746175i

−1.00000

769.12

1.00000i

2.22158

0.254102i

−

2.64265i

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3840.2.f.l		12
4.b	odd	2	1		3840.2.f.m		12
5.b	even	2	1	inner	3840.2.f.l		12
8.b	even	2	1	inner	3840.2.f.l		12
8.d	odd	2	1		3840.2.f.m		12
16.e	even	4	1		120.2.d.a	✓	6
16.e	even	4	1		120.2.d.b	yes	6
16.f	odd	4	1		480.2.d.a		6
16.f	odd	4	1		480.2.d.b		6
20.d	odd	2	1		3840.2.f.m		12
40.e	odd	2	1		3840.2.f.m		12
40.f	even	2	1	inner	3840.2.f.l		12
48.i	odd	4	1		360.2.d.e		6
48.i	odd	4	1		360.2.d.f		6
48.k	even	4	1		1440.2.d.e		6
48.k	even	4	1		1440.2.d.f		6
80.i	odd	4	2		600.2.k.f		12
80.j	even	4	2		2400.2.k.f		12
80.k	odd	4	1		480.2.d.a		6
80.k	odd	4	1		480.2.d.b		6
80.q	even	4	1		120.2.d.a	✓	6
80.q	even	4	1		120.2.d.b	yes	6
80.s	even	4	2		2400.2.k.f		12
80.t	odd	4	2		600.2.k.f		12
240.t	even	4	1		1440.2.d.e		6
240.t	even	4	1		1440.2.d.f		6
240.z	odd	4	2		7200.2.k.u		12
240.bb	even	4	2		1800.2.k.u		12
240.bd	odd	4	2		7200.2.k.u		12
240.bf	even	4	2		1800.2.k.u		12
240.bm	odd	4	1		360.2.d.e		6
240.bm	odd	4	1		360.2.d.f		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.2.d.a	✓	6	16.e	even	4	1
120.2.d.a	✓	6	80.q	even	4	1
120.2.d.b	yes	6	16.e	even	4	1
120.2.d.b	yes	6	80.q	even	4	1
360.2.d.e		6	48.i	odd	4	1
360.2.d.e		6	240.bm	odd	4	1
360.2.d.f		6	48.i	odd	4	1
360.2.d.f		6	240.bm	odd	4	1
480.2.d.a		6	16.f	odd	4	1
480.2.d.a		6	80.k	odd	4	1
480.2.d.b		6	16.f	odd	4	1
480.2.d.b		6	80.k	odd	4	1
600.2.k.f		12	80.i	odd	4	2
600.2.k.f		12	80.t	odd	4	2
1440.2.d.e		6	48.k	even	4	1
1440.2.d.e		6	240.t	even	4	1
1440.2.d.f		6	48.k	even	4	1
1440.2.d.f		6	240.t	even	4	1
1800.2.k.u		12	240.bb	even	4	2
1800.2.k.u		12	240.bf	even	4	2
2400.2.k.f		12	80.j	even	4	2
2400.2.k.f		12	80.s	even	4	2
3840.2.f.l		12	1.a	even	1	1	trivial
3840.2.f.l		12	5.b	even	2	1	inner
3840.2.f.l		12	8.b	even	2	1	inner
3840.2.f.l		12	40.f	even	2	1	inner
3840.2.f.m		12	4.b	odd	2	1
3840.2.f.m		12	8.d	odd	2	1
3840.2.f.m		12	20.d	odd	2	1
3840.2.f.m		12	40.e	odd	2	1
7200.2.k.u		12	240.z	odd	4	2
7200.2.k.u		12	240.bd	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3840, [\chi])$:

$ T_{7}^{6} + 24T_{7}^{4} + 128T_{7}^{2} + 64 $

$ T_{11}^{6} - 32T_{11}^{4} + 96T_{11}^{2} - 64 $

$ T_{29}^{6} - 108T_{29}^{4} + 3120T_{29}^{2} - 12544 $

$ T_{31}^{3} + 8T_{31}^{2} - 4T_{31} - 64 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$5$	$ T^{12} + 2 T^{10} + \cdots + 15625 $ T^12 + 2T^10 - 9T^8 - 196T^6 - 225T^4 + 1250*T^2 + 15625
$7$	$ (T^{6} + 24 T^{4} + \cdots + 64)^{2} $ (T^6 + 24T^4 + 128T^2 + 64)^2
$11$	$ (T^{6} - 32 T^{4} + \cdots - 64)^{2} $ (T^6 - 32T^4 + 96T^2 - 64)^2
$13$	$ (T^{6} + 48 T^{4} + \cdots + 3136)^{2} $ (T^6 + 48T^4 + 704T^2 + 3136)^2
$17$	$ (T^{6} + 36 T^{4} + \cdots + 1024)^{2} $ (T^6 + 36T^4 + 368T^2 + 1024)^2
$19$	$ (T^{6} - 60 T^{4} + \cdots - 1024)^{2} $ (T^6 - 60T^4 + 512T^2 - 1024)^2
$23$	$ (T^{6} + 92 T^{4} + \cdots + 16384)^{2} $ (T^6 + 92T^4 + 2304T^2 + 16384)^2
$29$	$ (T^{6} - 108 T^{4} + \cdots - 12544)^{2} $ (T^6 - 108T^4 + 3120T^2 - 12544)^2
$31$	$ (T^{3} + 8 T^{2} - 4 T - 64)^{4} $ (T^3 + 8T^2 - 4T - 64)^4
$37$	$ (T^{6} + 64 T^{4} + \cdots + 64)^{2} $ (T^6 + 64T^4 + 128T^2 + 64)^2
$41$	$ (T^{3} - 2 T^{2} - 100 T - 56)^{4} $ (T^3 - 2T^2 - 100T - 56)^4
$43$	$ (T^{6} + 128 T^{4} + \cdots + 4096)^{2} $ (T^6 + 128T^4 + 4096T^2 + 4096)^2
$47$	$ (T^{6} + 60 T^{4} + \cdots + 1024)^{2} $ (T^6 + 60T^4 + 512T^2 + 1024)^2
$53$	$ (T^{6} + 80 T^{4} + \cdots + 64)^{2} $ (T^6 + 80T^4 + 1216T^2 + 64)^2
$59$	$ (T^{6} - 176 T^{4} + \cdots - 179776)^{2} $ (T^6 - 176T^4 + 9888T^2 - 179776)^2
$61$	$ (T^{6} - 176 T^{4} + \cdots - 65536)^{2} $ (T^6 - 176T^4 + 7168T^2 - 65536)^2
$67$	$ (T^{6} + 128 T^{4} + \cdots + 4096)^{2} $ (T^6 + 128T^4 + 4096T^2 + 4096)^2
$71$	$ (T^{3} + 8 T^{2} + \cdots - 128)^{4} $ (T^3 + 8T^2 - 80T - 128)^4
$73$	$ (T^{6} + 384 T^{4} + \cdots + 16384)^{2} $ (T^6 + 384T^4 + 34560T^2 + 16384)^2
$79$	$ (T^{3} - 8 T^{2} - 4 T + 64)^{4} $ (T^3 - 8T^2 - 4T + 64)^4
$83$	$ (T^{6} + 192 T^{4} + \cdots + 200704)^{2} $ (T^6 + 192T^4 + 11264T^2 + 200704)^2
$89$	$ (T^{3} - 10 T^{2} + \cdots + 1384)^{4} $ (T^3 - 10T^2 - 164T + 1384)^4
$97$	$ (T^{6} + 336 T^{4} + \cdots + 262144)^{2} $ (T^6 + 336T^4 + 28416T^2 + 262144)^2
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10}) \) q + b3 * q^3 + b6 * q^5 - b1 * q^7 - q^9	\( q + \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_1 q^{7} - q^{9} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{11} - \beta_{3}) q^{13} + \beta_{4} q^{15} + (\beta_{10} + \beta_{7} + \beta_{4}) q^{17} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{19} + \beta_{8} q^{21} + (2 \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{23}+ \cdots + ( - \beta_{9} + \beta_{8}) q^{99}+O(q^{100}) \) q + b3 * q^3 + b6 * q^5 - b1 * q^7 - q^9 + (b9 - b8) * q^11 + (b11 - b3) * q^13 + b4 * q^15 + (b10 + b7 + b4) * q^17 + (b8 + b6 + b2) * q^19 + b8 * q^21 + (2b10 + b7 + b4 + b1) q^23 + (-b10 - b5) * q^25 - b3 * q^27 + (2b9 - b6 - b2) q^29 + (b7 + b5 - b4 - 3) * q^31 + (-b10 - b1) * q^33 + (-b11 - b9 - b8 + b6 - b3 + b2) * q^35 + (b11 + 3b3) q^37 + (b5 + 1) * q^39 + (2b7 + 2b5 - 2b4) q^41 + (-2b6 + 2b2) * q^43 - b6 * q^45 + (-b7 - b4 - b1) * q^47 + (2b7 - 2b4 - 1) * q^49 + (b9 - b6 - b2) * q^51 + (-b6 - 4b3 + b2) q^53 + (-2b10 + b7 + b5 - b4 - b1 + 1) q^55 + (b7 + b4 + b1) * q^57 + (-b9 + 3b8) q^59 + (-2b8 + 2b6 + 2b2) q^61 + b1 * q^63 + (b10 + 3b7 - b4 + 2b1 - 2) * q^65 + (-2b6 + 2b2) * q^67 + (2b9 - b8 - b6 - b2) q^69 + (-2b7 - 2b5 + 2b4 - 2) q^71 + (2b10 + 4b7 + 4b4) q^73 + (b11 - b9) * q^75 + 4b3 q^77 + (-b7 - b5 + b4 + 3) * q^79 + q^81 + (2b11 - 2b3) * q^83 + (b11 - 2b8 + 3b3 + 2b2) q^85 + (-2b10 - b7 - b4) q^87 + (2b7 - 2b5 - 2b4 + 4) q^89 + (-2b9 - 2b8 + 4b6 + 4b2) * q^91 + (-b11 + b6 - 3b3 - b2) q^93 + (b7 - 2b5 + b4 - b1 + 6) q^95 + (-2b10 + 2b7 + 2b4) q^97 + (-b9 + b8) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 4 q^{25} - 32 q^{31} + 16 q^{39} + 8 q^{41} - 12 q^{49} + 16 q^{55} - 24 q^{65} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89} + 64 q^{95}+O(q^{100}) \) 12 * q - 12 * q^9 - 4 * q^25 - 32 * q^31 + 16 * q^39 + 8 * q^41 - 12 * q^49 + 16 * q^55 - 24 * q^65 - 32 * q^71 + 32 * q^79 + 12 * q^81 + 40 * q^89 + 64 * q^95

Introduction

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Newspace parameters

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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	3840.2.f.l

Level	$3840$
Weight	$2$
Character orbit	3840.f
Analytic conductor	$30.663$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(30.6625543762\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 \) (v^8 + v^6 + 4v^4 - 4v^2) / 8
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} + 3\nu^{9} + 6\nu^{7} - 12\nu^{5} + 24\nu^{3} ) / 64 \) (v^11 + 3v^9 + 6v^7 - 12v^5 + 24v^3) / 64
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 \) (v^11 - v^9 + 2v^7 + 4v^5 + 8*v^3) / 64
\(\beta_{4}\)	\(=\)	\( ( -\nu^{6} + \nu^{4} + 2\nu^{2} + 4 ) / 4 \) (-v^6 + v^4 + 2*v^2 + 4) / 4
\(\beta_{5}\)	\(=\)	\( ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 \) (-v^10 + v^6 + 4v^4 + 4v^2 - 8) / 8
\(\beta_{6}\)	\(=\)	\( ( \nu^{11} - 5\nu^{9} - 2\nu^{7} - 12\nu^{5} + 24\nu^{3} - 64\nu ) / 64 \) (v^11 - 5v^9 - 2v^7 - 12v^5 + 24v^3 - 64*v) / 64
\(\beta_{7}\)	\(=\)	\( ( \nu^{8} - \nu^{6} - 2\nu^{4} - 8\nu^{2} + 8 ) / 8 \) (v^8 - v^6 - 2v^4 - 8v^2 + 8) / 8
\(\beta_{8}\)	\(=\)	\( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 \) (-v^11 - 3v^9 + 2v^7 + 4v^5 + 24v^3) / 32
\(\beta_{9}\)	\(=\)	\( ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 \) (v^11 - v^9 - 6v^7 - 4v^5 + 40v^3 + 32v) / 32
\(\beta_{10}\)	\(=\)	\( ( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} - 32 ) / 16 \) (-v^10 - 3v^8 + 2v^6 + 4v^4 - 8v^2 - 32) / 16
\(\beta_{11}\)	\(=\)	\( ( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64 \) (3v^11 - 3v^9 - 10v^7 - 36v^5 - 8v^3 + 128v) / 64

\(\nu\)	\(=\)	\( ( \beta_{11} + \beta_{8} - 2\beta_{6} + \beta_{3} ) / 4 \) (b11 + b8 - 2*b6 + b3) / 4
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{10} - 2\beta_{7} + \beta_{5} - \beta _1 - 1 ) / 4 \) (-2b10 - 2b7 + b5 - b1 - 1) / 4
\(\nu^{3}\)	\(=\)	\( ( -\beta_{11} + 2\beta_{9} + \beta_{8} - \beta_{3} + 2\beta_{2} ) / 4 \) (-b11 + 2b9 + b8 - b3 + 2b2) / 4
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{10} - 2\beta_{7} - \beta_{5} + 4\beta_{4} + 5\beta _1 + 1 ) / 4 \) (2b10 - 2b7 - b5 + 4b4 + 5b1 + 1) / 4
\(\nu^{5}\)	\(=\)	\( ( -3\beta_{11} + 2\beta_{9} - \beta_{8} - 4\beta_{6} + 13\beta_{3} - 6\beta_{2} ) / 4 \) (-3b11 + 2b9 - b8 - 4b6 + 13b3 - 6*b2) / 4
\(\nu^{6}\)	\(=\)	\( ( -2\beta_{10} - 6\beta_{7} + \beta_{5} - 12\beta_{4} + 3\beta _1 + 15 ) / 4 \) (-2b10 - 6b7 + b5 - 12b4 + 3b1 + 15) / 4
\(\nu^{7}\)	\(=\)	\( ( 3\beta_{11} - 10\beta_{9} + 9\beta_{8} - 4\beta_{6} + 19\beta_{3} + 14\beta_{2} ) / 4 \) (3b11 - 10b9 + 9b8 - 4b6 + 19b3 + 14b2) / 4
\(\nu^{8}\)	\(=\)	\( ( -14\beta_{10} + 6\beta_{7} + 7\beta_{5} - 4\beta_{4} + 5\beta _1 - 23 ) / 4 \) (-14b10 + 6b7 + 7b5 - 4b4 + 5*b1 - 23) / 4
\(\nu^{9}\)	\(=\)	\( ( -11\beta_{11} + 10\beta_{9} - 17\beta_{8} - 12\beta_{6} - 27\beta_{3} + 18\beta_{2} ) / 4 \) (-11b11 + 10b9 - 17b8 - 12b6 - 27b3 + 18b2) / 4
\(\nu^{10}\)	\(=\)	\( ( -2\beta_{10} - 22\beta_{7} - 31\beta_{5} + 4\beta_{4} + 19\beta _1 - 17 ) / 4 \) (-2b10 - 22b7 - 31b5 + 4b4 + 19*b1 - 17) / 4
\(\nu^{11}\)	\(=\)	\( ( 3\beta_{11} + 6\beta_{9} - 39\beta_{8} + 12\beta_{6} + 147\beta_{3} - 2\beta_{2} ) / 4 \) (3b11 + 6b9 - 39b8 + 12b6 + 147b3 - 2b2) / 4

\(n\)	\(511\)	\(1537\)	\(2561\)	\(2821\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 769.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$5$	\( T^{12} + 2 T^{10} + \cdots + 15625 \) T^12 + 2T^10 - 9T^8 - 196T^6 - 225T^4 + 1250*T^2 + 15625
$7$	\( (T^{6} + 24 T^{4} + \cdots + 64)^{2} \) (T^6 + 24T^4 + 128T^2 + 64)^2
$11$	\( (T^{6} - 32 T^{4} + \cdots - 64)^{2} \) (T^6 - 32T^4 + 96T^2 - 64)^2
$13$	\( (T^{6} + 48 T^{4} + \cdots + 3136)^{2} \) (T^6 + 48T^4 + 704T^2 + 3136)^2
$17$	\( (T^{6} + 36 T^{4} + \cdots + 1024)^{2} \) (T^6 + 36T^4 + 368T^2 + 1024)^2
$19$	\( (T^{6} - 60 T^{4} + \cdots - 1024)^{2} \) (T^6 - 60T^4 + 512T^2 - 1024)^2
$23$	\( (T^{6} + 92 T^{4} + \cdots + 16384)^{2} \) (T^6 + 92T^4 + 2304T^2 + 16384)^2
$29$	\( (T^{6} - 108 T^{4} + \cdots - 12544)^{2} \) (T^6 - 108T^4 + 3120T^2 - 12544)^2
$31$	\( (T^{3} + 8 T^{2} - 4 T - 64)^{4} \) (T^3 + 8T^2 - 4T - 64)^4
$37$	\( (T^{6} + 64 T^{4} + \cdots + 64)^{2} \) (T^6 + 64T^4 + 128T^2 + 64)^2
$41$	\( (T^{3} - 2 T^{2} - 100 T - 56)^{4} \) (T^3 - 2T^2 - 100T - 56)^4
$43$	\( (T^{6} + 128 T^{4} + \cdots + 4096)^{2} \) (T^6 + 128T^4 + 4096T^2 + 4096)^2
$47$	\( (T^{6} + 60 T^{4} + \cdots + 1024)^{2} \) (T^6 + 60T^4 + 512T^2 + 1024)^2
$53$	\( (T^{6} + 80 T^{4} + \cdots + 64)^{2} \) (T^6 + 80T^4 + 1216T^2 + 64)^2
$59$	\( (T^{6} - 176 T^{4} + \cdots - 179776)^{2} \) (T^6 - 176T^4 + 9888T^2 - 179776)^2
$61$	\( (T^{6} - 176 T^{4} + \cdots - 65536)^{2} \) (T^6 - 176T^4 + 7168T^2 - 65536)^2
$67$	\( (T^{6} + 128 T^{4} + \cdots + 4096)^{2} \) (T^6 + 128T^4 + 4096T^2 + 4096)^2
$71$	\( (T^{3} + 8 T^{2} + \cdots - 128)^{4} \) (T^3 + 8T^2 - 80T - 128)^4
$73$	\( (T^{6} + 384 T^{4} + \cdots + 16384)^{2} \) (T^6 + 384T^4 + 34560T^2 + 16384)^2
$79$	\( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \) (T^3 - 8T^2 - 4T + 64)^4
$83$	\( (T^{6} + 192 T^{4} + \cdots + 200704)^{2} \) (T^6 + 192T^4 + 11264T^2 + 200704)^2
$89$	\( (T^{3} - 10 T^{2} + \cdots + 1384)^{4} \) (T^3 - 10T^2 - 164T + 1384)^4
$97$	\( (T^{6} + 336 T^{4} + \cdots + 262144)^{2} \) (T^6 + 336T^4 + 28416T^2 + 262144)^2
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