LMFDB - Newform orbit 121.5.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,5,Mod(40,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([7]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("121.40");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	121.d (of order $10$, degree $4$, 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:	$12.5077655331$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

$q$-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 + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + \cdots - 7) q^{3}+ \cdots + 32 \zeta_{10}^{3} q^{9}+O(q^{10}) $ q + (7z^3 - 7z^2 + 7z - 7) q^3 - 16z q^4 - 49z^2 q^5 + 32z^3 q^9	$ q + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + \cdots - 7) q^{3}+ \cdots + 9793 \zeta_{10}^{3} q^{97}+O(q^{100}) $ q + (7z^3 - 7z^2 + 7z - 7) q^3 - 16z q^4 - 49z^2 q^5 + 32z^3 q^9 + 112 * q^12 + 343z q^15 + 256z^2 q^16 + 784z^3 q^20 + 167 * q^23 + (1776z^3 - 1776z^2 + 1776z - 1776) q^25 - 791z^2 q^27 + 553z^3 q^31 + (-512z^3 + 512z^2 - 512z + 512) q^36 + 2113z q^37 + 1568 * q^45 + (-1918z^3 + 1918z^2 - 1918z + 1918) q^47 - 1792z q^48 + 2401z^2 q^49 + 718z^3 q^53 - 4487z q^59 - 5488z^2 q^60 - 4096z^3 q^64 - 7753 * q^67 + (1169z^3 - 1169z^2 + 1169z - 1169) q^69 + 7607z^2 q^71 - 12432z^3 q^75 + (-12544z^3 + 12544z^2 - 12544z + 12544) q^80 + 2945z q^81 - 6433 * q^89 - 2672z q^92 - 3871z^2 q^93 + 9793z^3 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 7 q^{3} - 16 q^{4} + 49 q^{5} + 32 q^{9}+O(q^{10}) $ 4 * q - 7 * q^3 - 16 * q^4 + 49 * q^5 + 32 * q^9	$ 4 q - 7 q^{3} - 16 q^{4} + 49 q^{5} + 32 q^{9} + 448 q^{12} + 343 q^{15} - 256 q^{16} + 784 q^{20} + 668 q^{23} - 1776 q^{25} + 791 q^{27} + 553 q^{31} + 512 q^{36} + 2113 q^{37} + 6272 q^{45} + 1918 q^{47} - 1792 q^{48} - 2401 q^{49} + 718 q^{53} - 4487 q^{59} + 5488 q^{60} - 4096 q^{64} - 31012 q^{67} - 1169 q^{69} - 7607 q^{71} - 12432 q^{75} + 12544 q^{80} + 2945 q^{81} - 25732 q^{89} - 2672 q^{92} + 3871 q^{93} + 9793 q^{97}+O(q^{100}) $ 4 * q - 7 * q^3 - 16 * q^4 + 49 * q^5 + 32 * q^9 + 448 * q^12 + 343 * q^15 - 256 * q^16 + 784 * q^20 + 668 * q^23 - 1776 * q^25 + 791 * q^27 + 553 * q^31 + 512 * q^36 + 2113 * q^37 + 6272 * q^45 + 1918 * q^47 - 1792 * q^48 - 2401 * q^49 + 718 * q^53 - 4487 * q^59 + 5488 * q^60 - 4096 * q^64 - 31012 * q^67 - 1169 * q^69 - 7607 * q^71 - 12432 * q^75 + 12544 * q^80 + 2945 * q^81 - 25732 * q^89 - 2672 * q^92 + 3871 * q^93 + 9793 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$\zeta_{10}$

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} $

40.1

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

2.16312

−

6.65740i

4.94427

15.2169i

39.6418

−

28.8015i

25.8885

18.8091i

94.1

−5.66312

−

4.11450i

−12.9443

9.40456i

−15.1418

46.6018i

−9.88854

−

30.4338i

112.1

−5.66312

4.11450i

−12.9443

−

9.40456i

−15.1418

−

46.6018i

−9.88854

30.4338i

118.1

2.16312

6.65740i

4.94427

−

15.2169i

39.6418

28.8015i

25.8885

−

18.8091i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
11.c	even	5	3	inner
11.d	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	121.5.d.a		4
11.b	odd	2	1	CM	121.5.d.a		4
11.c	even	5	1		11.5.b.a	✓	1
11.c	even	5	3	inner	121.5.d.a		4
11.d	odd	10	1		11.5.b.a	✓	1
11.d	odd	10	3	inner	121.5.d.a		4
33.f	even	10	1		99.5.c.a		1
33.h	odd	10	1		99.5.c.a		1
44.g	even	10	1		176.5.h.a		1
44.h	odd	10	1		176.5.h.a		1
55.h	odd	10	1		275.5.c.a		1
55.j	even	10	1		275.5.c.a		1
55.k	odd	20	2		275.5.d.a		2
55.l	even	20	2		275.5.d.a		2
88.k	even	10	1		704.5.h.b		1
88.l	odd	10	1		704.5.h.b		1
88.o	even	10	1		704.5.h.a		1
88.p	odd	10	1		704.5.h.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.5.b.a	✓	1	11.c	even	5	1
11.5.b.a	✓	1	11.d	odd	10	1
99.5.c.a		1	33.f	even	10	1
99.5.c.a		1	33.h	odd	10	1
121.5.d.a		4	1.a	even	1	1	trivial
121.5.d.a		4	11.b	odd	2	1	CM
121.5.d.a		4	11.c	even	5	3	inner
121.5.d.a		4	11.d	odd	10	3	inner
176.5.h.a		1	44.g	even	10	1
176.5.h.a		1	44.h	odd	10	1
275.5.c.a		1	55.h	odd	10	1
275.5.c.a		1	55.j	even	10	1
275.5.d.a		2	55.k	odd	20	2
275.5.d.a		2	55.l	even	20	2
704.5.h.a		1	88.o	even	10	1
704.5.h.a		1	88.p	odd	10	1
704.5.h.b		1	88.k	even	10	1
704.5.h.b		1	88.l	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{5}^{\mathrm{new}}(121, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 7 T^{3} + \cdots + 2401 $ T^4 + 7T^3 + 49T^2 + 343*T + 2401
$5$	$ T^{4} - 49 T^{3} + \cdots + 5764801 $ T^4 - 49T^3 + 2401T^2 - 117649*T + 5764801
$7$	$ T^{4} $ T^4
$11$	$ T^{4} $ T^4
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ (T - 167)^{4} $ (T - 167)^4
$29$	$ T^{4} $ T^4
$31$	$ T^{4} + \cdots + 93519144481 $ T^4 - 553T^3 + 305809T^2 - 169112377*T + 93519144481
$37$	$ T^{4} + \cdots + 19934162223361 $ T^4 - 2113T^3 + 4464769T^2 - 9434056897*T + 19934162223361
$41$	$ T^{4} $ T^4
$43$	$ T^{4} $ T^4
$47$	$ T^{4} + \cdots + 13533010268176 $ T^4 - 1918T^3 + 3678724T^2 - 7055792632*T + 13533010268176
$53$	$ T^{4} + \cdots + 265764994576 $ T^4 - 718T^3 + 515524T^2 - 370146232*T + 265764994576
$59$	$ T^{4} + \cdots + 405344493982561 $ T^4 + 4487T^3 + 20133169T^2 + 90337529303*T + 405344493982561
$61$	$ T^{4} $ T^4
$67$	$ (T + 7753)^{4} $ (T + 7753)^4
$71$	$ T^{4} + \cdots + 33\!\cdots\!01 $ T^4 + 7607T^3 + 57866449T^2 + 440190077543*T + 3348525919869601
$73$	$ T^{4} $ T^4
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ (T + 6433)^{4} $ (T + 6433)^4
$97$	$ T^{4} + \cdots + 91\!\cdots\!01 $ T^4 - 9793T^3 + 95902849T^2 - 939176600257*T + 9197356446316801
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Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	121.d (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + \cdots - 7) q^{3}+ \cdots + 32 \zeta_{10}^{3} q^{9}+O(q^{10}) \) q + (7z^3 - 7z^2 + 7z - 7) q^3 - 16z q^4 - 49z^2 q^5 + 32z^3 q^9	\( q + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + \cdots - 7) q^{3}+ \cdots + 9793 \zeta_{10}^{3} q^{97}+O(q^{100}) \) q + (7z^3 - 7z^2 + 7z - 7) q^3 - 16z q^4 - 49z^2 q^5 + 32z^3 q^9 + 112 * q^12 + 343z q^15 + 256z^2 q^16 + 784z^3 q^20 + 167 * q^23 + (1776z^3 - 1776z^2 + 1776z - 1776) q^25 - 791z^2 q^27 + 553z^3 q^31 + (-512z^3 + 512z^2 - 512z + 512) q^36 + 2113z q^37 + 1568 * q^45 + (-1918z^3 + 1918z^2 - 1918z + 1918) q^47 - 1792z q^48 + 2401z^2 q^49 + 718z^3 q^53 - 4487z q^59 - 5488z^2 q^60 - 4096z^3 q^64 - 7753 * q^67 + (1169z^3 - 1169z^2 + 1169z - 1169) q^69 + 7607z^2 q^71 - 12432z^3 q^75 + (-12544z^3 + 12544z^2 - 12544z + 12544) q^80 + 2945z q^81 - 6433 * q^89 - 2672z q^92 - 3871z^2 q^93 + 9793z^3 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 7 q^{3} - 16 q^{4} + 49 q^{5} + 32 q^{9}+O(q^{10}) \) 4 * q - 7 * q^3 - 16 * q^4 + 49 * q^5 + 32 * q^9	\( 4 q - 7 q^{3} - 16 q^{4} + 49 q^{5} + 32 q^{9} + 448 q^{12} + 343 q^{15} - 256 q^{16} + 784 q^{20} + 668 q^{23} - 1776 q^{25} + 791 q^{27} + 553 q^{31} + 512 q^{36} + 2113 q^{37} + 6272 q^{45} + 1918 q^{47} - 1792 q^{48} - 2401 q^{49} + 718 q^{53} - 4487 q^{59} + 5488 q^{60} - 4096 q^{64} - 31012 q^{67} - 1169 q^{69} - 7607 q^{71} - 12432 q^{75} + 12544 q^{80} + 2945 q^{81} - 25732 q^{89} - 2672 q^{92} + 3871 q^{93} + 9793 q^{97}+O(q^{100}) \) 4 * q - 7 * q^3 - 16 * q^4 + 49 * q^5 + 32 * q^9 + 448 * q^12 + 343 * q^15 - 256 * q^16 + 784 * q^20 + 668 * q^23 - 1776 * q^25 + 791 * q^27 + 553 * q^31 + 512 * q^36 + 2113 * q^37 + 6272 * q^45 + 1918 * q^47 - 1792 * q^48 - 2401 * q^49 + 718 * q^53 - 4487 * q^59 + 5488 * q^60 - 4096 * q^64 - 31012 * q^67 - 1169 * q^69 - 7607 * q^71 - 12432 * q^75 + 12544 * q^80 + 2945 * q^81 - 25732 * q^89 - 2672 * q^92 + 3871 * q^93 + 9793 * q^97

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