LMFDB - Newform orbit 1100.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,2,Mod(593,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1100.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1100.k (of order $4$, degree $2$, 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:	$8.78354422234$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) $ q - b1 * q^3 + (-b5 + b4 - 4b2 - b1) q^9	$ q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+ \cdots + (4 \beta_{5} + \beta_{4} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) $ q - b1 * q^3 + (-b5 + b4 - 4b2 - b1) q^9 - b7 * q^11 + (b6 + b4 - 3b2 - b1 + 2) q^23 + (-b7 - 3b5 + b3 - 4b2 - 4) * q^27 + (-2b7 + b6 + b5 - b1) q^31 + (b7 - b6 - b4 + b3 + 4b2 + 3b1 - 3) * q^33 + (-b7 - b6 + b5 + b4 + b3 - 2b2 - 1) q^37 + (-b7 + b3 + 6b2 + 6) q^47 + 7b2 q^49 + (2b7 + 2b3 - 3b2 + 3) q^53 + (3b5 - 3b4 + 2b3 + 3b2 + 3b1) q^59 + (-2b7 + b6 + 5b5 - b4 + 2b3 + 5b2 + 4) * q^67 + (-4b5 - b4 + 2b3 - b2 - 4b1) q^69 + (-2b7 - b6 + 4b5 - 4b1 + 4) q^71 + (2b7 - 2b6 - 7b5 + 7b1 - 15) * q^81 + (4b5 + b4 + 2b3 - 3b2 + 4b1) * q^89 + (3b7 - 2b6 - 2b4 + 3b3 + 4b2 + 3b1 - 2) * q^93 + (-b7 + 2b6 + b5 - 2b4 + b3 - 3b2 - 5) q^97 + (4b5 + b4 - b3 + 7b2 + 4b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3}+O(q^{10}) $ 8 * q - 2 * q^3	$ 8 q - 2 q^{3} + 18 q^{23} - 26 q^{27} - 22 q^{33} - 14 q^{37} + 48 q^{47} + 24 q^{53} + 26 q^{67} + 12 q^{71} - 100 q^{81} - 18 q^{93} - 34 q^{97}+O(q^{100}) $ 8 * q - 2 * q^3 + 18 * q^23 - 26 * q^27 - 22 * q^33 - 14 * q^37 + 48 * q^47 + 24 * q^53 + 26 * q^67 + 12 * q^71 - 100 * q^81 - 18 * q^93 - 34 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5*x^6 + 16*x^4 + 45*x^2 + 81 :

$\beta_{1}$	$=$	$ ( 5\nu^{6} + 16\nu^{4} + 8\nu^{2} + 72\nu + 81 ) / 72 $ (5v^6 + 16v^4 + 8v^2 + 72v + 81) / 72
$\beta_{2}$	$=$	$ ( -5\nu^{7} - 16\nu^{5} - 8\nu^{3} - 81\nu ) / 216 $ (-5v^7 - 16v^5 - 8v^3 - 81v) / 216
$\beta_{3}$	$=$	$ ( -\nu^{6} - 5 ) / 8 $ (-v^6 - 5) / 8
$\beta_{4}$	$=$	$ ( \nu^{7} + 10\nu^{6} + 8\nu^{5} + 32\nu^{4} + 40\nu^{3} + 160\nu^{2} + 165\nu + 306 ) / 72 $ (v^7 + 10v^6 + 8v^5 + 32v^4 + 40v^3 + 160v^2 + 165v + 306) / 72
$\beta_{5}$	$=$	$ ( 8\nu^{7} - 9\nu^{6} + 40\nu^{5} - 72\nu^{4} + 128\nu^{3} - 144\nu^{2} + 360\nu - 405 ) / 216 $ (8v^7 - 9v^6 + 40v^5 - 72v^4 + 128v^3 - 144v^2 + 360*v - 405) / 216
$\beta_{6}$	$=$	$ ( -\nu^{7} + 3\nu^{6} + 4\nu^{5} + 15\nu^{4} + 2\nu^{3} + 21\nu^{2} - 9\nu + 81 ) / 27 $ (-v^7 + 3v^6 + 4v^5 + 15v^4 + 2v^3 + 21v^2 - 9v + 81) / 27
$\beta_{7}$	$=$	$ ( 5\nu^{7} + 16\nu^{5} + 62\nu^{3} + 81\nu ) / 54 $ (5v^7 + 16v^5 + 62v^3 + 81v) / 54

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

$\nu$	$=$	$ ( -2\beta_{7} - \beta_{6} + 3\beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 5\beta_1 ) / 10 $ (-2b7 - b6 + 3b5 + b4 + 2b3 - b2 + 5b1) / 10
$\nu^{2}$	$=$	$ ( -2\beta_{7} - \beta_{6} - \beta_{5} + 5\beta_{4} - 5\beta_{2} - 9\beta _1 - 10 ) / 10 $ (-2b7 - b6 - b5 + 5b4 - 5b2 - 9b1 - 10) / 10
$\nu^{3}$	$=$	$ \beta_{7} + 4\beta_{2} $ b7 + 4*b2
$\nu^{4}$	$=$	$ ( 10\beta_{7} + 5\beta_{6} - 13\beta_{5} - 7\beta_{4} + 16\beta_{3} + 7\beta_{2} + 27\beta _1 - 30 ) / 10 $ (10b7 + 5b6 - 13b5 - 7b4 + 16b3 + 7b2 + 27*b1 - 30) / 10
$\nu^{5}$	$=$	$ ( -18\beta_{7} + 31\beta_{6} + 27\beta_{5} - \beta_{4} - 2\beta_{3} - 79\beta_{2} - 35\beta_1 ) / 10 $ (-18b7 + 31b6 + 27b5 - b4 - 2b3 - 79b2 - 35b1) / 10
$\nu^{6}$	$=$	$ -8\beta_{3} - 5 $ -8*b3 - 5
$\nu^{7}$	$=$	$ ( 74\beta_{7} - 83\beta_{6} - 135\beta_{5} - 13\beta_{4} - 26\beta_{3} - 227\beta_{2} + 31\beta_1 ) / 10 $ (74b7 - 83b6 - 135b5 - 13b4 - 26b3 - 227b2 + 31*b1) / 10

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$\beta_{2}$	$1$

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} $

593.1

0.396143	+	1.68614i
−0.396143	+	1.68614i
1.26217	−	1.18614i
−1.26217	−	1.18614i
0.396143	−	1.68614i
−0.396143	−	1.68614i
1.26217	+	1.18614i
−1.26217	+	1.18614i

−2.08228

−

2.08228i

5.67181i

593.2

−1.29000

−

1.29000i

0.328185i

593.3

−0.0760282

−

0.0760282i

−

2.98844i

593.4

2.44831

2.44831i

8.98844i

857.1

−2.08228

2.08228i

−

5.67181i

857.2

−1.29000

1.29000i

−

0.328185i

857.3

−0.0760282

0.0760282i

2.98844i

857.4

2.44831

−

2.44831i

−

8.98844i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
5.c	odd	4	1	inner
55.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1100.2.k.b		8
5.b	even	2	1		220.2.k.b	✓	8
5.c	odd	4	1		220.2.k.b	✓	8
5.c	odd	4	1	inner	1100.2.k.b		8
11.b	odd	2	1	CM	1100.2.k.b		8
15.d	odd	2	1		1980.2.y.b		8
15.e	even	4	1		1980.2.y.b		8
20.d	odd	2	1		880.2.bd.h		8
20.e	even	4	1		880.2.bd.h		8
55.d	odd	2	1		220.2.k.b	✓	8
55.e	even	4	1		220.2.k.b	✓	8
55.e	even	4	1	inner	1100.2.k.b		8
165.d	even	2	1		1980.2.y.b		8
165.l	odd	4	1		1980.2.y.b		8
220.g	even	2	1		880.2.bd.h		8
220.i	odd	4	1		880.2.bd.h		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.2.k.b	✓	8	5.b	even	2	1
220.2.k.b	✓	8	5.c	odd	4	1
220.2.k.b	✓	8	55.d	odd	2	1
220.2.k.b	✓	8	55.e	even	4	1
880.2.bd.h		8	20.d	odd	2	1
880.2.bd.h		8	20.e	even	4	1
880.2.bd.h		8	220.g	even	2	1
880.2.bd.h		8	220.i	odd	4	1
1100.2.k.b		8	1.a	even	1	1	trivial
1100.2.k.b		8	5.c	odd	4	1	inner
1100.2.k.b		8	11.b	odd	2	1	CM
1100.2.k.b		8	55.e	even	4	1	inner
1980.2.y.b		8	15.d	odd	2	1
1980.2.y.b		8	15.e	even	4	1
1980.2.y.b		8	165.d	even	2	1
1980.2.y.b		8	165.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1100, [\chi])$:

$ T_{3}^{8} + 2T_{3}^{7} + 2T_{3}^{6} + 6T_{3}^{5} + 125T_{3}^{4} + 312T_{3}^{3} + 392T_{3}^{2} + 56T_{3} + 4 $

$ T_{19} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 2 T^{7} + \cdots + 4 $ T^8 + 2T^7 + 2T^6 + 6T^5 + 125T^4 + 312T^3 + 392T^2 + 56*T + 4
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ (T^{2} - 11)^{4} $ (T^2 - 11)^4
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ T^{8} - 18 T^{7} + \cdots + 131044 $ T^8 - 18T^7 + 162T^6 - 414T^5 + 845T^4 - 13248T^3 + 187272T^2 - 221544*T + 131044
$29$	$ T^{8} $ T^8
$31$	$ (T^{4} - 87 T^{2} + 36)^{2} $ (T^4 - 87*T^2 + 36)^2
$37$	$ T^{8} + 14 T^{7} + \cdots + 12124324 $ T^8 + 14T^7 + 98T^6 + 518T^5 + 16765T^4 + 237244T^3 + 1812608T^2 + 6629728*T + 12124324
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ (T^{4} - 24 T^{3} + \cdots + 2500)^{2} $ (T^4 - 24T^3 + 288T^2 - 1200*T + 2500)^2
$53$	$ (T^{4} - 12 T^{3} + \cdots + 4900)^{2} $ (T^4 - 12T^3 + 72T^2 + 840*T + 4900)^2
$59$	$ (T^{4} + 343 T^{2} + 27556)^{2} $ (T^4 + 343*T^2 + 27556)^2
$61$	$ T^{8} $ T^8
$67$	$ T^{8} - 26 T^{7} + \cdots + 149866564 $ T^8 - 26T^7 + 338T^6 - 1742T^5 + 34285T^4 - 745576T^3 + 9313928T^2 - 52836472*T + 149866564
$71$	$ (T^{2} - 3 T - 204)^{4} $ (T^2 - 3*T - 204)^4
$73$	$ T^{8} $ T^8
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ (T^{4} + 453 T^{2} + 34596)^{2} $ (T^4 + 453*T^2 + 34596)^2
$97$	$ T^{8} + 34 T^{7} + \cdots + 368716804 $ T^8 + 34T^7 + 578T^6 + 3298T^5 + 48205T^4 + 1312604T^3 + 22204448T^2 + 127962128*T + 368716804
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1100.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) q - b1 * q^3 + (-b5 + b4 - 4b2 - b1) q^9	\( q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+ \cdots + (4 \beta_{5} + \beta_{4} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) q - b1 * q^3 + (-b5 + b4 - 4b2 - b1) q^9 - b7 * q^11 + (b6 + b4 - 3b2 - b1 + 2) q^23 + (-b7 - 3b5 + b3 - 4b2 - 4) * q^27 + (-2b7 + b6 + b5 - b1) q^31 + (b7 - b6 - b4 + b3 + 4b2 + 3b1 - 3) * q^33 + (-b7 - b6 + b5 + b4 + b3 - 2b2 - 1) q^37 + (-b7 + b3 + 6b2 + 6) q^47 + 7b2 q^49 + (2b7 + 2b3 - 3b2 + 3) q^53 + (3b5 - 3b4 + 2b3 + 3b2 + 3b1) q^59 + (-2b7 + b6 + 5b5 - b4 + 2b3 + 5b2 + 4) * q^67 + (-4b5 - b4 + 2b3 - b2 - 4b1) q^69 + (-2b7 - b6 + 4b5 - 4b1 + 4) q^71 + (2b7 - 2b6 - 7b5 + 7b1 - 15) * q^81 + (4b5 + b4 + 2b3 - 3b2 + 4b1) * q^89 + (3b7 - 2b6 - 2b4 + 3b3 + 4b2 + 3b1 - 2) * q^93 + (-b7 + 2b6 + b5 - 2b4 + b3 - 3b2 - 5) q^97 + (4b5 + b4 - b3 + 7b2 + 4b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3}+O(q^{10}) \) 8 * q - 2 * q^3	\( 8 q - 2 q^{3} + 18 q^{23} - 26 q^{27} - 22 q^{33} - 14 q^{37} + 48 q^{47} + 24 q^{53} + 26 q^{67} + 12 q^{71} - 100 q^{81} - 18 q^{93} - 34 q^{97}+O(q^{100}) \) 8 * q - 2 * q^3 + 18 * q^23 - 26 * q^27 - 22 * q^33 - 14 * q^37 + 48 * q^47 + 24 * q^53 + 26 * q^67 + 12 * q^71 - 100 * q^81 - 18 * q^93 - 34 * q^97

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