LMFDB - Newform orbit 3375.1.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3375,1,Mod(1376,3375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("3375.1376");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3375 = 3^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3375.c (of order $2$, degree $1$, 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:	$1.68434441764$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.2463153133392333984375.1

$q$-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_{5} + \beta_{3} + \beta_1) q^{2} + (\beta_{6} - \beta_{2} - 2) q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{8}+O(q^{10}) $ q + (b5 + b3 + b1) * q^2 + (b6 - b2 - 2) * q^4 + (-b5 - 2b3 - b1) q^8	$ q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + (\beta_{6} - \beta_{2} - 2) q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{8} + ( - \beta_{4} + \beta_{2} + 2) q^{16} + ( - \beta_{7} - \beta_{3}) q^{17} - \beta_{4} q^{19} - \beta_{7} q^{23} - \beta_{2} q^{31} + (2 \beta_{3} + \beta_1) q^{32} + (\beta_{6} + \beta_{2}) q^{34} + ( - \beta_{7} - 2 \beta_{5}) q^{38} + (\beta_{6} + \beta_{4} - 1) q^{46} + ( - \beta_{5} - \beta_{3}) q^{47} - q^{49} + \beta_1 q^{53} + (\beta_{6} - \beta_{4}) q^{61} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3}) q^{62} + (\beta_{6} + \beta_{4} - \beta_{2} - 2) q^{64} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{68} + (\beta_{6} + 2 \beta_{4} - 1) q^{76} + ( - \beta_{6} + \beta_{4}) q^{79} + \beta_1 q^{83} + (\beta_{7} + 2 \beta_{5} - \beta_1) q^{92} + (\beta_{2} + 1) q^{94} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{98}+O(q^{100}) $ q + (b5 + b3 + b1) * q^2 + (b6 - b2 - 2) * q^4 + (-b5 - 2b3 - b1) q^8 + (-b4 + b2 + 2) * q^16 + (-b7 - b3) * q^17 - b4 * q^19 - b7 * q^23 - b2 * q^31 + (2b3 + b1) q^32 + (b6 + b2) * q^34 + (-b7 - 2b5) q^38 + (b6 + b4 - 1) * q^46 + (-b5 - b3) * q^47 - q^49 + b1 * q^53 + (b6 - b4) * q^61 + (-b7 - b5 - 2b3) q^62 + (b6 + b4 - b2 - 2) * q^64 + (b7 + 2b5 + 2b3) * q^68 + (b6 + 2b4 - 1) q^76 + (-b6 + b4) * q^79 + b1 * q^83 + (b7 + 2b5 - b1) q^92 + (b2 + 1) * q^94 + (-b5 - b3 - b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{4}+O(q^{10}) $ 8 * q - 10 * q^4	$ 8 q - 10 q^{4} + 12 q^{16} - 2 q^{19} + 2 q^{31} + 2 q^{34} - 2 q^{46} - 8 q^{49} + 2 q^{61} - 8 q^{64} - 2 q^{79} + 6 q^{94}+O(q^{100}) $ 8 * q - 10 * q^4 + 12 * q^16 - 2 * q^19 + 2 * q^31 + 2 * q^34 - 2 * q^46 - 8 * q^49 + 2 * q^61 - 8 * q^64 - 2 * q^79 + 6 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9*x^6 + 26*x^4 + 24*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 2 $ v^2 + 2
$\beta_{3}$	$=$	$ \nu^{3} + 3\nu $ v^3 + 3*v
$\beta_{4}$	$=$	$ \nu^{4} + 4\nu^{2} + 2 $ v^4 + 4*v^2 + 2
$\beta_{5}$	$=$	$ \nu^{5} + 5\nu^{3} + 5\nu $ v^5 + 5v^3 + 5v
$\beta_{6}$	$=$	$ \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 $ v^6 + 6v^4 + 9v^2 + 2
$\beta_{7}$	$=$	$ \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu $ v^7 + 7v^5 + 14v^3 + 7*v

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 2 $ b2 - 2
$\nu^{3}$	$=$	$ \beta_{3} - 3\beta_1 $ b3 - 3*b1
$\nu^{4}$	$=$	$ \beta_{4} - 4\beta_{2} + 6 $ b4 - 4*b2 + 6
$\nu^{5}$	$=$	$ \beta_{5} - 5\beta_{3} + 10\beta_1 $ b5 - 5b3 + 10b1
$\nu^{6}$	$=$	$ \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 $ b6 - 6b4 + 15b2 - 20
$\nu^{7}$	$=$	$ \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 $ b7 - 7b5 + 21b3 - 35*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3375\mathbb{Z}\right)^\times$.

$n$	$1001$	$2377$
$\chi(n)$	$-1$	$1$

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} $

1376.1

	−	1.33826i
	−	0.209057i
	−	1.95630i
	−	1.82709i
		1.82709i
		1.95630i
		0.209057i
		1.33826i

−

1.95630i

−2.82709

3.57433i

1376.2

−

1.82709i

−2.33826

2.44512i

1376.3

−

1.33826i

−0.790943

−

0.279773i

1376.4

−

0.209057i

0.956295

−

0.408977i

1376.5

0.209057i

0.956295

0.408977i

1376.6

1.33826i

−0.790943

0.279773i

1376.7

1.82709i

−2.33826

−

2.44512i

1376.8

1.95630i

−2.82709

−

3.57433i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3375.1.c.a	✓	8
3.b	odd	2	1	inner	3375.1.c.a	✓	8
5.b	even	2	1	inner	3375.1.c.a	✓	8
5.c	odd	4	1		3375.1.d.a		4
5.c	odd	4	1		3375.1.d.b		4
15.d	odd	2	1	CM	3375.1.c.a	✓	8
15.e	even	4	1		3375.1.d.a		4
15.e	even	4	1		3375.1.d.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3375.1.c.a	✓	8	1.a	even	1	1	trivial
3375.1.c.a	✓	8	3.b	odd	2	1	inner
3375.1.c.a	✓	8	5.b	even	2	1	inner
3375.1.c.a	✓	8	15.d	odd	2	1	CM
3375.1.d.a		4	5.c	odd	4	1
3375.1.d.a		4	15.e	even	4	1
3375.1.d.b		4	5.c	odd	4	1
3375.1.d.b		4	15.e	even	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3375, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 9 T^{6} + \cdots + 1 $ T^8 + 9T^6 + 26T^4 + 24*T^2 + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} $ T^8
$17$	$ T^{8} + 9 T^{6} + \cdots + 1 $ T^8 + 9T^6 + 26T^4 + 24*T^2 + 1
$19$	$ (T^{4} + T^{3} - 4 T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 - 4T^2 - 4T + 1)^2
$23$	$ T^{8} + 9 T^{6} + \cdots + 1 $ T^8 + 9T^6 + 26T^4 + 24*T^2 + 1
$29$	$ T^{8} $ T^8
$31$	$ (T^{4} - T^{3} - 4 T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 - 4T^2 + 4T + 1)^2
$37$	$ T^{8} $ T^8
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ (T^{4} + 3 T^{2} + 1)^{2} $ (T^4 + 3*T^2 + 1)^2
$53$	$ T^{8} + 9 T^{6} + \cdots + 1 $ T^8 + 9T^6 + 26T^4 + 24*T^2 + 1
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} - T^{3} - 4 T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 - 4T^2 + 4T + 1)^2
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ T^{8} $ T^8
$79$	$ (T^{4} + T^{3} - 4 T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 - 4T^2 - 4T + 1)^2
$83$	$ T^{8} + 9 T^{6} + \cdots + 1 $ T^8 + 9T^6 + 26T^4 + 24*T^2 + 1
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3375 = 3^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3375.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + (\beta_{6} - \beta_{2} - 2) q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) q + (b5 + b3 + b1) * q^2 + (b6 - b2 - 2) * q^4 + (-b5 - 2b3 - b1) q^8	\( q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + (\beta_{6} - \beta_{2} - 2) q^{4} + ( - \beta_{5} - 2 \beta_{3} - \beta_1) q^{8} + ( - \beta_{4} + \beta_{2} + 2) q^{16} + ( - \beta_{7} - \beta_{3}) q^{17} - \beta_{4} q^{19} - \beta_{7} q^{23} - \beta_{2} q^{31} + (2 \beta_{3} + \beta_1) q^{32} + (\beta_{6} + \beta_{2}) q^{34} + ( - \beta_{7} - 2 \beta_{5}) q^{38} + (\beta_{6} + \beta_{4} - 1) q^{46} + ( - \beta_{5} - \beta_{3}) q^{47} - q^{49} + \beta_1 q^{53} + (\beta_{6} - \beta_{4}) q^{61} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3}) q^{62} + (\beta_{6} + \beta_{4} - \beta_{2} - 2) q^{64} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{68} + (\beta_{6} + 2 \beta_{4} - 1) q^{76} + ( - \beta_{6} + \beta_{4}) q^{79} + \beta_1 q^{83} + (\beta_{7} + 2 \beta_{5} - \beta_1) q^{92} + (\beta_{2} + 1) q^{94} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) q + (b5 + b3 + b1) * q^2 + (b6 - b2 - 2) * q^4 + (-b5 - 2b3 - b1) q^8 + (-b4 + b2 + 2) * q^16 + (-b7 - b3) * q^17 - b4 * q^19 - b7 * q^23 - b2 * q^31 + (2b3 + b1) q^32 + (b6 + b2) * q^34 + (-b7 - 2b5) q^38 + (b6 + b4 - 1) * q^46 + (-b5 - b3) * q^47 - q^49 + b1 * q^53 + (b6 - b4) * q^61 + (-b7 - b5 - 2b3) q^62 + (b6 + b4 - b2 - 2) * q^64 + (b7 + 2b5 + 2b3) * q^68 + (b6 + 2b4 - 1) q^76 + (-b6 + b4) * q^79 + b1 * q^83 + (b7 + 2b5 - b1) q^92 + (b2 + 1) * q^94 + (-b5 - b3 - b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{4}+O(q^{10}) \) 8 * q - 10 * q^4	\( 8 q - 10 q^{4} + 12 q^{16} - 2 q^{19} + 2 q^{31} + 2 q^{34} - 2 q^{46} - 8 q^{49} + 2 q^{61} - 8 q^{64} - 2 q^{79} + 6 q^{94}+O(q^{100}) \) 8 * q - 10 * q^4 + 12 * q^16 - 2 * q^19 + 2 * q^31 + 2 * q^34 - 2 * q^46 - 8 * q^49 + 2 * q^61 - 8 * q^64 - 2 * q^79 + 6 * q^94

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