LMFDB - Newform orbit 147.2.o.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,2,Mod(5,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 29]))

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

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

chi := DirichletCharacter("147.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	147.o (of order $42$, degree $12$, 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:	$1.17380090971$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{42}]$

$q$-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 $\zeta_{21}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + ( - 2 \zeta_{21}^{10} - 2 \zeta_{21}^{3}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{9} + \cdots + 1) q^{7}+ \cdots + (3 \zeta_{21}^{9} + 3 \zeta_{21}^{2}) q^{9}+O(q^{10}) $ q + (z^8 + 2z) q^3 + (-2z^10 - 2z^3) * q^4 + (-z^11 + z^9 - z^8 + z^6 - 3z^5 - z^4 + z^3 - z + 1) q^7 + (3z^9 + 3z^2) * q^9	$ q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + ( - 2 \zeta_{21}^{10} - 2 \zeta_{21}^{3}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{9} + \cdots + 1) q^{7}+ \cdots + ( - 3 \zeta_{21}^{11} + 14 \zeta_{21}^{9} + \cdots + 3) q^{97}+O(q^{100}) $ q + (z^8 + 2z) q^3 + (-2z^10 - 2z^3) * q^4 + (-z^11 + z^9 - z^8 + z^6 - 3z^5 - z^4 + z^3 - z + 1) q^7 + (3z^9 + 3z^2) * q^9 + (-4z^11 - 2z^4) * q^12 + (3z^11 + z^10 - z^8 + z^7 + 4z^6 - z^5 + 3z^4 + z^3 - z^2 + 1) q^13 + (4z^11 - 4z^10 + 4z^8 - 4z^7 - 4z^6 + 4z^5 - 4z^3 + 4z^2 - 4) * q^16 + (2z^11 - 7z^9 + 2z^8 - 2z^6 + 5z^5 + 2z^4 - 2z^3 - 2z^2 + 2z - 2) q^19 + (-4z^11 + 4z^10 - 4z^8 + 4z^7 - z^6 - 4z^5 + 4z^3 - 4z^2 + 4) q^21 - 5z^2 q^25 + (6z^10 + 3z^3) * q^27 + (-2z^8 - 6z) * q^28 + (z^11 + z^10 - 5z^4 + 6z^3) * q^31 + (-6z^11 + 6z^9 - 6z^8 + 6z^6 - 6z^4 + 6z^3 - 6z + 6) q^36 + (-7z^8 - 3z^7 - 3z - 7) q^37 + (7z^11 - 7z^9 + 7z^8 + 5z^7 - 7z^6 + 2z^5 + 7z^4 - 7z^3 + 7z - 9) q^39 + (-z^11 + 6z^8 + 6z^4 - z) * q^43 + (-8z^7 - 4) q^48 + (3z^10 - 5z^3) * q^49 + (-2z^9 + 8z^7 + 6z^2 + 6) q^52 + (7z^11 - 14z^10 + 7z^8 - 7z^7 + z^6 + 7z^5 - 6z^3 + 7z^2 - 7) q^57 + (-4z^8 - 5z^7 + 5z + 4) q^61 + (3z^7 + 9) q^63 + 8z^9 q^64 + (-9z^11 + 9z^10 + 9z^7 + 7z^6 - 9z^5 + 9z^3 - 9z^2 + 7z + 9) * q^67 + (-8z^11 - z^10 + 9z^9 - 8z^8 - z^7 - 7z^5 - 9z^4 + 8z^3 + z^2 - 9z + 8) q^73 + (-5z^10 - 10z^3) * q^75 + (4z^11 - 4z^9 + 8z^8 - 4z^6 - 6z^5 + 4z^4 - 4z^3 + 14z - 4) * q^76 + (-7z^11 - 10z^10 - 10z^4 - 7z^3) * q^79 + 9z^11 q^81 + (-8z^9 - 10z^2) * q^84 + (-9z^11 + 6z^9 + z^4 + 11z^2) q^91 + (3z^11 + 4z^9 - 4z^8 + 4z^6 - 11z^5 + 7z^4 + 4z^3 - 4z + 4) * q^93 + (-3z^11 + 14z^9 - 3z^8 + 3z^6 + 8z^5 - 3z^4 + 3z^3 + 8z^2 - 3z + 3) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9	$ 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9} - 6 q^{12} + 4 q^{16} + 9 q^{19} + 6 q^{21} - 5 q^{25} - 8 q^{28} - 15 q^{31} + 12 q^{36} - 76 q^{37} - 66 q^{39} + 10 q^{43} + 13 q^{49} + 34 q^{52} - 18 q^{57} + 79 q^{61} + 90 q^{63} - 16 q^{64} + 11 q^{67} + 27 q^{73} + 15 q^{75} - 13 q^{79} + 9 q^{81} + 6 q^{84} - 9 q^{91} + 15 q^{93}+O(q^{100}) $ 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9 - 6 * q^12 + 4 * q^16 + 9 * q^19 + 6 * q^21 - 5 * q^25 - 8 * q^28 - 15 * q^31 + 12 * q^36 - 76 * q^37 - 66 * q^39 + 10 * q^43 + 13 * q^49 + 34 * q^52 - 18 * q^57 + 79 * q^61 + 90 * q^63 - 16 * q^64 + 11 * q^67 + 27 * q^73 + 15 * q^75 - 13 * q^79 + 9 * q^81 + 6 * q^84 - 9 * q^91 + 15 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$\zeta_{21}^{2} + \zeta_{21}^{9}$

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} $

5.1

0.0747301	−	0.997204i
−0.733052	+	0.680173i
−0.733052	−	0.680173i
0.365341	−	0.930874i
−0.988831	+	0.149042i
0.0747301	+	0.997204i
0.365341	+	0.930874i
0.826239	+	0.563320i
0.955573	+	0.294755i
−0.988831	−	0.149042i
0.826239	−	0.563320i
0.955573	−	0.294755i

0.975699

−

1.43109i

1.91115

−

0.589510i

−1.71951

2.01079i

−1.09602

−

2.79262i

17.1

−0.510531

1.65510i

−1.97766

0.298085i

−1.57775

2.12384i

−2.47872

−

1.68996i

26.1

−0.510531

−

1.65510i

−1.97766

−

0.298085i

−1.57775

−

2.12384i

−2.47872

1.68996i

38.1

−0.258149

−

1.71271i

0.149460

−

1.99441i

−2.64420

0.0906624i

−2.86672

0.884266i

47.1

−1.61232

−

0.632789i

1.65248

1.12664i

2.42168

−

1.06559i

2.19916

2.04052i

59.1

0.975699

1.43109i

1.91115

0.589510i

−1.71951

−

2.01079i

−1.09602

2.79262i

89.1

−0.258149

1.71271i

0.149460

1.99441i

−2.64420

−

0.0906624i

−2.86672

−

0.884266i

101.1

1.72721

0.129436i

−1.46610

−

1.36035i

2.34300

−

1.22896i

2.96649

0.447127i

110.1

1.17809

1.26968i

0.730682

−

1.86175i

0.676779

−

2.55773i

−0.224190

2.99161i

122.1

−1.61232

0.632789i

1.65248

−

1.12664i

2.42168

1.06559i

2.19916

−

2.04052i

131.1

1.72721

−

0.129436i

−1.46610

1.36035i

2.34300

1.22896i

2.96649

−

0.447127i

143.1

1.17809

−

1.26968i

0.730682

1.86175i

0.676779

2.55773i

−0.224190

−

2.99161i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
49.h	odd	42	1	inner
147.o	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.2.o.a	✓	12
3.b	odd	2	1	CM	147.2.o.a	✓	12
49.h	odd	42	1	inner	147.2.o.a	✓	12
147.o	even	42	1	inner	147.2.o.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
147.2.o.a	✓	12	1.a	even	1	1	trivial
147.2.o.a	✓	12	3.b	odd	2	1	CM
147.2.o.a	✓	12	49.h	odd	42	1	inner
147.2.o.a	✓	12	147.o	even	42	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{2}^{\mathrm{new}}(147, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - 3 T^{11} + \cdots + 729 $ T^12 - 3T^11 + 6T^10 - 9T^9 + 9T^8 - 27T^6 + 81T^4 - 243T^3 + 486T^2 - 729*T + 729
$5$	$ T^{12} $ T^12
$7$	$ T^{12} + T^{11} + \cdots + 117649 $ T^12 + T^11 - 6T^10 - 13T^9 + 29T^8 + 120T^7 - 83T^6 + 840T^5 + 1421T^4 - 4459T^3 - 14406T^2 + 16807T + 117649
$11$	$ T^{12} $ T^12
$13$	$ T^{12} - 3 T^{10} + \cdots + 82319329 $ T^12 - 3T^10 - 637T^9 + 9T^8 + 8281T^7 + 134835T^6 - 24843T^5 - 1986176T^4 - 9721894T^3 + 21674956T^2 + 75133513T + 82319329
$17$	$ T^{12} $ T^12
$19$	$ T^{12} + \cdots + 124791241 $ T^12 - 9T^11 - 79T^10 + 954T^9 + 6182T^8 - 81396T^7 - 10506T^6 + 2191707T^5 - 284514T^4 - 43243776T^3 + 154218160T^2 - 220381488*T + 124791241
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} + 15 T^{11} + \cdots + 3108169 $ T^12 + 15T^11 - 67T^10 - 2130T^9 + 6710T^8 + 260400T^7 + 900306T^6 - 5998965T^5 - 21749838T^4 + 117936240T^3 + 594624652T^2 + 74151780*T + 3108169
$37$	$ T^{12} + \cdots + 112669649569 $ T^12 + 76T^11 + 2723T^10 + 60880T^9 + 947666T^8 + 10849300T^7 + 94043391T^6 + 625802254T^5 + 3217002277T^4 + 12858157502T^3 + 39881159388T^2 + 89829460734*T + 112669649569
$41$	$ T^{12} $ T^12
$43$	$ T^{12} + \cdots + 6707446201 $ T^12 - 10T^11 + 75T^10 - 500T^9 + 21486T^8 - 72930T^7 + 185227T^6 - 848010T^5 + 171370674T^4 + 2359170230T^3 + 11029368664T^2 - 2569581125*T + 6707446201
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ T^{12} + \cdots + 1476788041 $ T^12 - 79T^11 + 2980T^10 - 70649T^9 + 1169393T^8 - 14214830T^7 + 130014669T^6 - 902913704T^5 + 4851250195T^4 - 20708224805T^3 + 56369725816T^2 - 10476936699*T + 1476788041
$67$	$ T^{12} + \cdots + 163181257849 $ T^12 - 11T^11 + 469T^10 - 3828T^9 + 142474T^8 - 1104026T^7 + 20169024T^6 - 83828591T^5 + 1443877384T^4 - 7823405876T^3 + 43660561658T^2 - 92149862926*T + 163181257849
$71$	$ T^{12} $ T^12
$73$	$ T^{12} + \cdots + 175555134049 $ T^12 - 27T^11 + 486T^10 - 6561T^9 + 132122T^8 - 2719031T^7 + 40669952T^6 - 426051360T^5 + 3012102193T^4 - 12831927785T^3 + 38214846291T^2 - 107051454521*T + 175555134049
$79$	$ T^{12} + \cdots + 120762505081 $ T^12 + 13T^11 + 553T^10 + 4992T^9 + 189874T^8 + 1624714T^7 + 30447852T^6 + 116727793T^5 + 2097436228T^4 + 10037916148T^3 + 77577714494T^2 + 101546994926*T + 120762505081
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ T^{12} + \cdots + 902334707569 $ T^12 + 1166T^10 + 500621T^8 + 95542098T^6 + 7683525798T^4 + 207881737156*T^2 + 902334707569
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + ( - 2 \zeta_{21}^{10} - 2 \zeta_{21}^{3}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{9} + \cdots + 1) q^{7}+ \cdots + (3 \zeta_{21}^{9} + 3 \zeta_{21}^{2}) q^{9}+O(q^{10}) \) q + (z^8 + 2z) q^3 + (-2z^10 - 2z^3) * q^4 + (-z^11 + z^9 - z^8 + z^6 - 3z^5 - z^4 + z^3 - z + 1) q^7 + (3z^9 + 3z^2) * q^9	\( q + (\zeta_{21}^{8} + 2 \zeta_{21}) q^{3} + ( - 2 \zeta_{21}^{10} - 2 \zeta_{21}^{3}) q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{9} + \cdots + 1) q^{7}+ \cdots + ( - 3 \zeta_{21}^{11} + 14 \zeta_{21}^{9} + \cdots + 3) q^{97}+O(q^{100}) \) q + (z^8 + 2z) q^3 + (-2z^10 - 2z^3) * q^4 + (-z^11 + z^9 - z^8 + z^6 - 3z^5 - z^4 + z^3 - z + 1) q^7 + (3z^9 + 3z^2) * q^9 + (-4z^11 - 2z^4) * q^12 + (3z^11 + z^10 - z^8 + z^7 + 4z^6 - z^5 + 3z^4 + z^3 - z^2 + 1) q^13 + (4z^11 - 4z^10 + 4z^8 - 4z^7 - 4z^6 + 4z^5 - 4z^3 + 4z^2 - 4) * q^16 + (2z^11 - 7z^9 + 2z^8 - 2z^6 + 5z^5 + 2z^4 - 2z^3 - 2z^2 + 2z - 2) q^19 + (-4z^11 + 4z^10 - 4z^8 + 4z^7 - z^6 - 4z^5 + 4z^3 - 4z^2 + 4) q^21 - 5z^2 q^25 + (6z^10 + 3z^3) * q^27 + (-2z^8 - 6z) * q^28 + (z^11 + z^10 - 5z^4 + 6z^3) * q^31 + (-6z^11 + 6z^9 - 6z^8 + 6z^6 - 6z^4 + 6z^3 - 6z + 6) q^36 + (-7z^8 - 3z^7 - 3z - 7) q^37 + (7z^11 - 7z^9 + 7z^8 + 5z^7 - 7z^6 + 2z^5 + 7z^4 - 7z^3 + 7z - 9) q^39 + (-z^11 + 6z^8 + 6z^4 - z) * q^43 + (-8z^7 - 4) q^48 + (3z^10 - 5z^3) * q^49 + (-2z^9 + 8z^7 + 6z^2 + 6) q^52 + (7z^11 - 14z^10 + 7z^8 - 7z^7 + z^6 + 7z^5 - 6z^3 + 7z^2 - 7) q^57 + (-4z^8 - 5z^7 + 5z + 4) q^61 + (3z^7 + 9) q^63 + 8z^9 q^64 + (-9z^11 + 9z^10 + 9z^7 + 7z^6 - 9z^5 + 9z^3 - 9z^2 + 7z + 9) * q^67 + (-8z^11 - z^10 + 9z^9 - 8z^8 - z^7 - 7z^5 - 9z^4 + 8z^3 + z^2 - 9z + 8) q^73 + (-5z^10 - 10z^3) * q^75 + (4z^11 - 4z^9 + 8z^8 - 4z^6 - 6z^5 + 4z^4 - 4z^3 + 14z - 4) * q^76 + (-7z^11 - 10z^10 - 10z^4 - 7z^3) * q^79 + 9z^11 q^81 + (-8z^9 - 10z^2) * q^84 + (-9z^11 + 6z^9 + z^4 + 11z^2) q^91 + (3z^11 + 4z^9 - 4z^8 + 4z^6 - 11z^5 + 7z^4 + 4z^3 - 4z + 4) * q^93 + (-3z^11 + 14z^9 - 3z^8 + 3z^6 + 8z^5 - 3z^4 + 3z^3 + 8z^2 - 3z + 3) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9	\( 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9} - 6 q^{12} + 4 q^{16} + 9 q^{19} + 6 q^{21} - 5 q^{25} - 8 q^{28} - 15 q^{31} + 12 q^{36} - 76 q^{37} - 66 q^{39} + 10 q^{43} + 13 q^{49} + 34 q^{52} - 18 q^{57} + 79 q^{61} + 90 q^{63} - 16 q^{64} + 11 q^{67} + 27 q^{73} + 15 q^{75} - 13 q^{79} + 9 q^{81} + 6 q^{84} - 9 q^{91} + 15 q^{93}+O(q^{100}) \) 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9 - 6 * q^12 + 4 * q^16 + 9 * q^19 + 6 * q^21 - 5 * q^25 - 8 * q^28 - 15 * q^31 + 12 * q^36 - 76 * q^37 - 66 * q^39 + 10 * q^43 + 13 * q^49 + 34 * q^52 - 18 * q^57 + 79 * q^61 + 90 * q^63 - 16 * q^64 + 11 * q^67 + 27 * q^73 + 15 * q^75 - 13 * q^79 + 9 * q^81 + 6 * q^84 - 9 * q^91 + 15 * q^93

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