LMFDB - Newform orbit 880.2.bo.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,2,Mod(81,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.81"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.bo (of order $5$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity $\zeta_{10}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{3} + \zeta_{10} q^{5} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots + 1) q^{9} + ( - \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{11} + (4 \zeta_{10}^{3} - 4) q^{13}+ \cdots + ( - 8 \zeta_{10}^{3} - \zeta_{10}^{2} + \cdots + 1) q^{99}+O(q^{100}) $ q + (-z^3 + 2z^2 - z) q^3 + z * q^5 + (-z^3 - 2z^2 + 2z + 1) * q^9 + (-z^2 + 2z + 2) q^11 + (4z^3 - 4) q^13 + (z^3 - z + 1) * q^15 + (5z^2 - 3z + 5) * q^17 + (-3z^3 + z^2 - 3z) * q^19 + 6 * q^23 + z^2 * q^25 + (2z^2 - z + 2) q^27 + (-6z^3 + 2z - 2) * q^29 + (6z^2 - 6z) * q^31 + (-z^3 + 6z^2 - 6z + 3) * q^33 + 2z^3 q^37 + (-4z^2 + 4z - 4) * q^39 + (-3z^3 - 3z) * q^41 + (-3z^3 + 3z^2 + 1) * q^43 + (-3z^3 + 3z^2 + 1) * q^45 + (2z^3 + 8z^2 + 2z) q^47 + 7z q^49 + (-3z^3 + 8z - 8) * q^51 + (-6z^3 + 10z^2 - 10z + 6) q^53 + (-z^3 + 2z^2 + 2z) * q^55 + (z^3 - 5z^2 + 5z - 1) * q^57 + (6z^3 + 9z - 9) * q^59 + (6z^2 - 2z + 6) * q^61 + (4z^3 - 4z^2 - 4) * q^65 + (-z^3 + z^2) * q^67 + (-6z^3 + 12z^2 - 6z) q^69 + (2z^2 - 8z + 2) * q^71 + (-3z^3 + 3z - 3) * q^73 + (z^3 - 2z^2 + 2z - 1) * q^75 + (-6z^3 + 4z^2 - 4z + 6) q^79 + (-4z^3 - 6z + 6) * q^81 + (z^2 - 14z + 1) q^83 + (5z^3 - 3z^2 + 5z) q^85 + (10z^3 - 10z^2 + 8) * q^87 + (5z^3 - 5z^2 - 1) * q^89 + (-12z^2 + 18z - 12) * q^93 + (-2z^3 - 3z + 3) * q^95 + (-7z^3 + 6z^2 - 6z + 7) q^97 + (-8z^3 - z^2 + 6z + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + q^{5} + 7 q^{9} + 11 q^{11} - 12 q^{13} + 4 q^{15} + 12 q^{17} - 7 q^{19} + 24 q^{23} - q^{25} + 5 q^{27} - 12 q^{29} - 12 q^{31} - q^{33} + 2 q^{37} - 8 q^{39} - 6 q^{41} - 2 q^{43}+ \cdots + 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + q^5 + 7 * q^9 + 11 * q^11 - 12 * q^13 + 4 * q^15 + 12 * q^17 - 7 * q^19 + 24 * q^23 - q^25 + 5 * q^27 - 12 * q^29 - 12 * q^31 - q^33 + 2 * q^37 - 8 * q^39 - 6 * q^41 - 2 * q^43 - 2 * q^45 - 4 * q^47 + 7 * q^49 - 27 * q^51 - 2 * q^53 - q^55 + 7 * q^57 - 21 * q^59 + 16 * q^61 - 8 * q^65 - 2 * q^67 - 24 * q^69 - 2 * q^71 - 12 * q^73 + q^75 + 10 * q^79 + 14 * q^81 - 11 * q^83 + 13 * q^85 + 52 * q^87 + 6 * q^89 - 18 * q^93 + 7 * q^95 + 9 * q^97 + 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$1$	$-\zeta_{10}^{3}$	$1$

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} $

81.1

0.809017	−	0.587785i
−0.309017	+	0.951057i
0.809017	+	0.587785i
−0.309017	−	0.951057i

0.118034

−

0.363271i

0.809017

−

0.587785i

2.30902

1.67760i

401.1

−2.11803

−

1.53884i

−0.309017

0.951057i

1.19098

3.66547i

641.1

0.118034

0.363271i

0.809017

0.587785i

2.30902

−

1.67760i

801.1

−2.11803

1.53884i

−0.309017

−

0.951057i

1.19098

−

3.66547i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	880.2.bo.b		4
4.b	odd	2	1		110.2.g.b	✓	4
11.c	even	5	1	inner	880.2.bo.b		4
11.c	even	5	1		9680.2.a.bx		2
11.d	odd	10	1		9680.2.a.bw		2
12.b	even	2	1		990.2.n.c		4
20.d	odd	2	1		550.2.h.b		4
20.e	even	4	2		550.2.ba.b		8
44.g	even	10	1		1210.2.a.q		2
44.h	odd	10	1		110.2.g.b	✓	4
44.h	odd	10	1		1210.2.a.n		2
132.o	even	10	1		990.2.n.c		4
220.n	odd	10	1		550.2.h.b		4
220.n	odd	10	1		6050.2.a.cw		2
220.o	even	10	1		6050.2.a.ch		2
220.v	even	20	2		550.2.ba.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
110.2.g.b	✓	4	4.b	odd	2	1
110.2.g.b	✓	4	44.h	odd	10	1
550.2.h.b		4	20.d	odd	2	1
550.2.h.b		4	220.n	odd	10	1
550.2.ba.b		8	20.e	even	4	2
550.2.ba.b		8	220.v	even	20	2
880.2.bo.b		4	1.a	even	1	1	trivial
880.2.bo.b		4	11.c	even	5	1	inner
990.2.n.c		4	12.b	even	2	1
990.2.n.c		4	132.o	even	10	1
1210.2.a.n		2	44.h	odd	10	1
1210.2.a.q		2	44.g	even	10	1
6050.2.a.ch		2	220.o	even	10	1
6050.2.a.cw		2	220.n	odd	10	1
9680.2.a.bw		2	11.d	odd	10	1
9680.2.a.bx		2	11.c	even	5	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} - T_{3} + 1 $ T3^4 + 4*T3^3 + 6*T3^2 - T3 + 1 acting on $S_{2}^{\mathrm{new}}(880, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 4 T^{3} + \cdots + 1 $ T^4 + 4T^3 + 6T^2 - T + 1
$5$	$ T^{4} - T^{3} + T^{2} + \cdots + 1 $ T^4 - T^3 + T^2 - T + 1
$7$	$ T^{4} $ T^4
$11$	$ T^{4} - 11 T^{3} + \cdots + 121 $ T^4 - 11T^3 + 51T^2 - 121*T + 121
$13$	$ T^{4} + 12 T^{3} + \cdots + 256 $ T^4 + 12T^3 + 64T^2 + 128*T + 256
$17$	$ T^{4} - 12 T^{3} + \cdots + 961 $ T^4 - 12T^3 + 94T^2 - 403*T + 961
$19$	$ T^{4} + 7 T^{3} + \cdots + 121 $ T^4 + 7T^3 + 34T^2 + 88*T + 121
$23$	$ (T - 6)^{4} $ (T - 6)^4
$29$	$ T^{4} + 12 T^{3} + \cdots + 1936 $ T^4 + 12T^3 + 64T^2 + 88*T + 1936
$31$	$ T^{4} + 12 T^{3} + \cdots + 1296 $ T^4 + 12T^3 + 144T^2 + 648*T + 1296
$37$	$ T^{4} - 2 T^{3} + \cdots + 16 $ T^4 - 2T^3 + 4T^2 - 8*T + 16
$41$	$ T^{4} + 6 T^{3} + \cdots + 81 $ T^4 + 6T^3 + 36T^2 + 81*T + 81
$43$	$ (T^{2} + T - 11)^{2} $ (T^2 + T - 11)^2
$47$	$ T^{4} + 4 T^{3} + \cdots + 5776 $ T^4 + 4T^3 + 96T^2 + 1064*T + 5776
$53$	$ T^{4} + 2 T^{3} + \cdots + 1936 $ T^4 + 2T^3 + 124T^2 - 792*T + 1936
$59$	$ T^{4} + 21 T^{3} + \cdots + 9801 $ T^4 + 21T^3 + 306T^2 + 2376*T + 9801
$61$	$ T^{4} - 16 T^{3} + \cdots + 1936 $ T^4 - 16T^3 + 136T^2 - 616*T + 1936
$67$	$ (T^{2} + T - 1)^{2} $ (T^2 + T - 1)^2
$71$	$ T^{4} + 2 T^{3} + \cdots + 1936 $ T^4 + 2T^3 + 64T^2 + 528*T + 1936
$73$	$ T^{4} + 12 T^{3} + \cdots + 81 $ T^4 + 12T^3 + 54T^2 - 27*T + 81
$79$	$ T^{4} - 10 T^{3} + \cdots + 400 $ T^4 - 10T^3 + 40T^2 + 400
$83$	$ T^{4} + 11 T^{3} + \cdots + 32761 $ T^4 + 11T^3 + 186T^2 + 2896*T + 32761
$89$	$ (T^{2} - 3 T - 29)^{2} $ (T^2 - 3*T - 29)^2
$97$	$ T^{4} - 9 T^{3} + \cdots + 1681 $ T^4 - 9T^3 + 46T^2 - 164*T + 1681
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Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.bo (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10}) q^{3} + \zeta_{10} q^{5} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots + 1) q^{9} + ( - \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{11} + (4 \zeta_{10}^{3} - 4) q^{13}+ \cdots + ( - 8 \zeta_{10}^{3} - \zeta_{10}^{2} + \cdots + 1) q^{99}+O(q^{100}) \) q + (-z^3 + 2z^2 - z) q^3 + z * q^5 + (-z^3 - 2z^2 + 2z + 1) * q^9 + (-z^2 + 2z + 2) q^11 + (4z^3 - 4) q^13 + (z^3 - z + 1) * q^15 + (5z^2 - 3z + 5) * q^17 + (-3z^3 + z^2 - 3z) * q^19 + 6 * q^23 + z^2 * q^25 + (2z^2 - z + 2) q^27 + (-6z^3 + 2z - 2) * q^29 + (6z^2 - 6z) * q^31 + (-z^3 + 6z^2 - 6z + 3) * q^33 + 2z^3 q^37 + (-4z^2 + 4z - 4) * q^39 + (-3z^3 - 3z) * q^41 + (-3z^3 + 3z^2 + 1) * q^43 + (-3z^3 + 3z^2 + 1) * q^45 + (2z^3 + 8z^2 + 2z) q^47 + 7z q^49 + (-3z^3 + 8z - 8) * q^51 + (-6z^3 + 10z^2 - 10z + 6) q^53 + (-z^3 + 2z^2 + 2z) * q^55 + (z^3 - 5z^2 + 5z - 1) * q^57 + (6z^3 + 9z - 9) * q^59 + (6z^2 - 2z + 6) * q^61 + (4z^3 - 4z^2 - 4) * q^65 + (-z^3 + z^2) * q^67 + (-6z^3 + 12z^2 - 6z) q^69 + (2z^2 - 8z + 2) * q^71 + (-3z^3 + 3z - 3) * q^73 + (z^3 - 2z^2 + 2z - 1) * q^75 + (-6z^3 + 4z^2 - 4z + 6) q^79 + (-4z^3 - 6z + 6) * q^81 + (z^2 - 14z + 1) q^83 + (5z^3 - 3z^2 + 5z) q^85 + (10z^3 - 10z^2 + 8) * q^87 + (5z^3 - 5z^2 - 1) * q^89 + (-12z^2 + 18z - 12) * q^93 + (-2z^3 - 3z + 3) * q^95 + (-7z^3 + 6z^2 - 6z + 7) q^97 + (-8z^3 - z^2 + 6z + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + q^{5} + 7 q^{9} + 11 q^{11} - 12 q^{13} + 4 q^{15} + 12 q^{17} - 7 q^{19} + 24 q^{23} - q^{25} + 5 q^{27} - 12 q^{29} - 12 q^{31} - q^{33} + 2 q^{37} - 8 q^{39} - 6 q^{41} - 2 q^{43}+ \cdots + 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + q^5 + 7 * q^9 + 11 * q^11 - 12 * q^13 + 4 * q^15 + 12 * q^17 - 7 * q^19 + 24 * q^23 - q^25 + 5 * q^27 - 12 * q^29 - 12 * q^31 - q^33 + 2 * q^37 - 8 * q^39 - 6 * q^41 - 2 * q^43 - 2 * q^45 - 4 * q^47 + 7 * q^49 - 27 * q^51 - 2 * q^53 - q^55 + 7 * q^57 - 21 * q^59 + 16 * q^61 - 8 * q^65 - 2 * q^67 - 24 * q^69 - 2 * q^71 - 12 * q^73 + q^75 + 10 * q^79 + 14 * q^81 - 11 * q^83 + 13 * q^85 + 52 * q^87 + 6 * q^89 - 18 * q^93 + 7 * q^95 + 9 * q^97 + 3 * q^99

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