LMFDB - Newform orbit 148.1.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [148,1,Mod(7,148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(148, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 16]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("148.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 148 = 2^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	148.p (of order $18$, degree $6$, 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:	$0.0738616218697$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.899194740203776.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{4}) q^{5} - \zeta_{18}^{3} q^{8} - \zeta_{18}^{7} q^{9} +O(q^{10}) $ q - z * q^2 + z^2 * q^4 + (z^6 + z^4) * q^5 - z^3 * q^8 - z^7 * q^9	\( q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{4}) q^{5} - \zeta_{18}^{3} q^{8} - \zeta_{18}^{7} q^{9} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{10} - \zeta_{18}^{2} q^{13} + \zeta_{18}^{4} q^{16} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{17} + \zeta_{18}^{8} q^{18} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{20} + (\zeta_{18}^{8} - \zeta_{18}^{3} - \zeta_{18}) q^{25} + \zeta_{18}^{3} q^{26} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{29} - \zeta_{18}^{5} q^{32} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{34} + q^{36} + \zeta_{18}^{2} q^{37} + ( - \zeta_{18}^{7} + 1) q^{40} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{41} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{45} - \zeta_{18} q^{49} + (\zeta_{18}^{4} + \zeta_{18}^{2} + 1) q^{50} - \zeta_{18}^{4} q^{52} - \zeta_{18}^{4} q^{53} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{58} + (\zeta_{18}^{4} + 1) q^{61} + \zeta_{18}^{6} q^{64} + ( - \zeta_{18}^{8} - \zeta_{18}^{6}) q^{65} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{68} - \zeta_{18} q^{72} - q^{73} - \zeta_{18}^{3} q^{74} + (\zeta_{18}^{8} - \zeta_{18}) q^{80} - \zeta_{18}^{5} q^{81} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{82} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \cdots + 1) q^{85} + \cdots + \zeta_{18}^{2} q^{98} +O(q^{100}) \) q - z * q^2 + z^2 * q^4 + (z^6 + z^4) * q^5 - z^3 * q^8 - z^7 * q^9 + (-z^7 - z^5) * q^10 - z^2 * q^13 + z^4 * q^16 + (-z^3 + z^2) * q^17 + z^8 * q^18 + (z^8 + z^6) * q^20 + (z^8 - z^3 - z) * q^25 + z^3 * q^26 + (-z^5 - z) * q^29 - z^5 * q^32 + (z^4 - z^3) * q^34 + q^36 + z^2 * q^37 + (-z^7 + 1) * q^40 + (-z^7 + z^6) * q^41 + (z^4 + z^2) * q^45 - z * q^49 + (z^4 + z^2 + 1) * q^50 - z^4 * q^52 - z^4 * q^53 + (z^6 + z^2) * q^58 + (z^4 + 1) * q^61 + z^6 * q^64 + (-z^8 - z^6) * q^65 + (-z^5 + z^4) * q^68 - z * q^72 - q^73 - z^3 * q^74 + (z^8 - z) * q^80 - z^5 * q^81 + (z^8 - z^7) * q^82 + (z^8 - z^7 + z^6 + 1) * q^85 + (z^8 + 1) * q^89 + (-z^5 - z^3) * q^90 + (z^8 + z^4) * q^97 + z^2 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 6 * q - 3 * q^5 - 3 * q^8	$ 6 q - 3 q^{5} - 3 q^{8} - 3 q^{17} - 3 q^{20} - 3 q^{25} + 3 q^{26} - 3 q^{34} + 6 q^{36} + 6 q^{40} - 3 q^{41} + 6 q^{50} - 3 q^{58} + 6 q^{61} - 3 q^{64} + 3 q^{65} - 6 q^{73} - 3 q^{74} + 3 q^{85} + 6 q^{89} - 3 q^{90}+O(q^{100}) $ 6 * q - 3 * q^5 - 3 * q^8 - 3 * q^17 - 3 * q^20 - 3 * q^25 + 3 * q^26 - 3 * q^34 + 6 * q^36 + 6 * q^40 - 3 * q^41 + 6 * q^50 - 3 * q^58 + 6 * q^61 - 3 * q^64 + 3 * q^65 - 6 * q^73 - 3 * q^74 + 3 * q^85 + 6 * q^89 - 3 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/148\mathbb{Z}\right)^\times$.

$n$	$75$	$113$
$\chi(n)$	$-1$	$\zeta_{18}^{2}$

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} $

7.1

0.939693	−	0.342020i
−0.766044	−	0.642788i
−0.173648	−	0.984808i
−0.173648	+	0.984808i
−0.766044	+	0.642788i
0.939693	+	0.342020i

−0.939693

0.342020i

0.766044

−

0.642788i

−0.326352

−

1.85083i

−0.500000

0.866025i

0.766044

0.642788i

0.939693

1.62760i

71.1

0.766044

0.642788i

0.173648

0.984808i

−1.43969

−

0.524005i

−0.500000

0.866025i

0.173648

−

0.984808i

−0.766044

−

1.32683i

83.1

0.173648

0.984808i

−0.939693

0.342020i

0.266044

0.223238i

−0.500000

−

0.866025i

−0.939693

−

0.342020i

−0.173648

0.300767i

107.1

0.173648

−

0.984808i

−0.939693

−

0.342020i

0.266044

−

0.223238i

−0.500000

0.866025i

−0.939693

0.342020i

−0.173648

−

0.300767i

123.1

0.766044

−

0.642788i

0.173648

−

0.984808i

−1.43969

0.524005i

−0.500000

−

0.866025i

0.173648

0.984808i

−0.766044

1.32683i

127.1

−0.939693

−

0.342020i

0.766044

0.642788i

−0.326352

1.85083i

−0.500000

−

0.866025i

0.766044

−

0.642788i

0.939693

−

1.62760i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
37.f	even	9	1	inner
148.p	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	148.1.p.a	✓	6
3.b	odd	2	1		1332.1.cz.a		6
4.b	odd	2	1	CM	148.1.p.a	✓	6
5.b	even	2	1		3700.1.ce.a		6
5.c	odd	4	2		3700.1.cb.a		12
8.b	even	2	1		2368.1.ck.a		6
8.d	odd	2	1		2368.1.ck.a		6
12.b	even	2	1		1332.1.cz.a		6
20.d	odd	2	1		3700.1.ce.a		6
20.e	even	4	2		3700.1.cb.a		12
37.f	even	9	1	inner	148.1.p.a	✓	6
111.p	odd	18	1		1332.1.cz.a		6
148.p	odd	18	1	inner	148.1.p.a	✓	6
185.x	even	18	1		3700.1.ce.a		6
185.bd	odd	36	2		3700.1.cb.a		12
296.bb	odd	18	1		2368.1.ck.a		6
296.bc	even	18	1		2368.1.ck.a		6
444.z	even	18	1		1332.1.cz.a		6
740.bs	odd	18	1		3700.1.ce.a		6
740.cf	even	36	2		3700.1.cb.a		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
148.1.p.a	✓	6	1.a	even	1	1	trivial
148.1.p.a	✓	6	4.b	odd	2	1	CM
148.1.p.a	✓	6	37.f	even	9	1	inner
148.1.p.a	✓	6	148.p	odd	18	1	inner
1332.1.cz.a		6	3.b	odd	2	1
1332.1.cz.a		6	12.b	even	2	1
1332.1.cz.a		6	111.p	odd	18	1
1332.1.cz.a		6	444.z	even	18	1
2368.1.ck.a		6	8.b	even	2	1
2368.1.ck.a		6	8.d	odd	2	1
2368.1.ck.a		6	296.bb	odd	18	1
2368.1.ck.a		6	296.bc	even	18	1
3700.1.cb.a		12	5.c	odd	4	2
3700.1.cb.a		12	20.e	even	4	2
3700.1.cb.a		12	185.bd	odd	36	2
3700.1.cb.a		12	740.cf	even	36	2
3700.1.ce.a		6	5.b	even	2	1
3700.1.ce.a		6	20.d	odd	2	1
3700.1.ce.a		6	185.x	even	18	1
3700.1.ce.a		6	740.bs	odd	18	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(148, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$17$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$19$	$ T^{6} $ T^6
$23$	$ T^{6} $ T^6
$29$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$31$	$ T^{6} $ T^6
$37$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$41$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$43$	$ T^{6} $ T^6
$47$	$ T^{6} $ T^6
$53$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$59$	$ T^{6} $ T^6
$61$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$67$	$ T^{6} $ T^6
$71$	$ T^{6} $ T^6
$73$	$ (T + 1)^{6} $ (T + 1)^6
$79$	$ T^{6} $ T^6
$83$	$ T^{6} $ T^6
$89$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$97$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 148 = 2^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	148.p (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{4}) q^{5} - \zeta_{18}^{3} q^{8} - \zeta_{18}^{7} q^{9} +O(q^{10}) \) q - z * q^2 + z^2 * q^4 + (z^6 + z^4) * q^5 - z^3 * q^8 - z^7 * q^9	\( q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{4}) q^{5} - \zeta_{18}^{3} q^{8} - \zeta_{18}^{7} q^{9} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{10} - \zeta_{18}^{2} q^{13} + \zeta_{18}^{4} q^{16} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{17} + \zeta_{18}^{8} q^{18} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{20} + (\zeta_{18}^{8} - \zeta_{18}^{3} - \zeta_{18}) q^{25} + \zeta_{18}^{3} q^{26} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{29} - \zeta_{18}^{5} q^{32} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{34} + q^{36} + \zeta_{18}^{2} q^{37} + ( - \zeta_{18}^{7} + 1) q^{40} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{41} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{45} - \zeta_{18} q^{49} + (\zeta_{18}^{4} + \zeta_{18}^{2} + 1) q^{50} - \zeta_{18}^{4} q^{52} - \zeta_{18}^{4} q^{53} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{58} + (\zeta_{18}^{4} + 1) q^{61} + \zeta_{18}^{6} q^{64} + ( - \zeta_{18}^{8} - \zeta_{18}^{6}) q^{65} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{68} - \zeta_{18} q^{72} - q^{73} - \zeta_{18}^{3} q^{74} + (\zeta_{18}^{8} - \zeta_{18}) q^{80} - \zeta_{18}^{5} q^{81} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{82} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \cdots + 1) q^{85} + \cdots + \zeta_{18}^{2} q^{98} +O(q^{100}) \) q - z * q^2 + z^2 * q^4 + (z^6 + z^4) * q^5 - z^3 * q^8 - z^7 * q^9 + (-z^7 - z^5) * q^10 - z^2 * q^13 + z^4 * q^16 + (-z^3 + z^2) * q^17 + z^8 * q^18 + (z^8 + z^6) * q^20 + (z^8 - z^3 - z) * q^25 + z^3 * q^26 + (-z^5 - z) * q^29 - z^5 * q^32 + (z^4 - z^3) * q^34 + q^36 + z^2 * q^37 + (-z^7 + 1) * q^40 + (-z^7 + z^6) * q^41 + (z^4 + z^2) * q^45 - z * q^49 + (z^4 + z^2 + 1) * q^50 - z^4 * q^52 - z^4 * q^53 + (z^6 + z^2) * q^58 + (z^4 + 1) * q^61 + z^6 * q^64 + (-z^8 - z^6) * q^65 + (-z^5 + z^4) * q^68 - z * q^72 - q^73 - z^3 * q^74 + (z^8 - z) * q^80 - z^5 * q^81 + (z^8 - z^7) * q^82 + (z^8 - z^7 + z^6 + 1) * q^85 + (z^8 + 1) * q^89 + (-z^5 - z^3) * q^90 + (z^8 + z^4) * q^97 + z^2 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 6 * q - 3 * q^5 - 3 * q^8	\( 6 q - 3 q^{5} - 3 q^{8} - 3 q^{17} - 3 q^{20} - 3 q^{25} + 3 q^{26} - 3 q^{34} + 6 q^{36} + 6 q^{40} - 3 q^{41} + 6 q^{50} - 3 q^{58} + 6 q^{61} - 3 q^{64} + 3 q^{65} - 6 q^{73} - 3 q^{74} + 3 q^{85} + 6 q^{89} - 3 q^{90}+O(q^{100}) \) 6 * q - 3 * q^5 - 3 * q^8 - 3 * q^17 - 3 * q^20 - 3 * q^25 + 3 * q^26 - 3 * q^34 + 6 * q^36 + 6 * q^40 - 3 * q^41 + 6 * q^50 - 3 * q^58 + 6 * q^61 - 3 * q^64 + 3 * q^65 - 6 * q^73 - 3 * q^74 + 3 * q^85 + 6 * q^89 - 3 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

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	148.1.p.a

Level	$148$
Weight	$1$
Character orbit	148.p
Analytic conductor	$0.074$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.0738616218697\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.899194740203776.1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
37.f	even	9	1	inner
148.p	odd	18	1	inner

$p$	$F_p(T)$
$2$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$3$	\( T^{6} \) T^6
$5$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$7$	\( T^{6} \) T^6
$11$	\( T^{6} \) T^6
$13$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$17$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$19$	\( T^{6} \) T^6
$23$	\( T^{6} \) T^6
$29$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$31$	\( T^{6} \) T^6
$37$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$41$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 3T^2 - 3*T + 1
$43$	\( T^{6} \) T^6
$47$	\( T^{6} \) T^6
$53$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$59$	\( T^{6} \) T^6
$61$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$67$	\( T^{6} \) T^6
$71$	\( T^{6} \) T^6
$73$	\( (T + 1)^{6} \) (T + 1)^6
$79$	\( T^{6} \) T^6
$83$	\( T^{6} \) T^6
$89$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$97$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
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\(n\)	\(75\)	\(113\)
\(\chi(n)\)	\(-1\)	\(\zeta_{18}^{2}\)